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+The Project Gutenberg EBook of Encyclopaedia Britannica, 11th Edition,
+Volume 8, Slice 4, by Various
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+Title: Encyclopaedia Britannica, 11th Edition, Volume 8, Slice 4
+       "Diameter" to "Dinarchus"
+
+Author: Various
+
+Release Date: May 30, 2010 [EBook #32607]
+
+Language: English
+
+Character set encoding: ISO-8859-1
+
+*** START OF THIS PROJECT GUTENBERG EBOOK ENCYC. BRITANNICA, VOL 8 SL 4 ***
+
+
+
+
+Produced by Marius Masi, Don Kretz and the Online
+Distributed Proofreading Team at https://www.pgdp.net
+
+
+
+
+
+
+
+
+
+Transcriber's notes:
+
+(1) Numbers following letters (without space) like C2 were originally
+      printed in subscript. Letter subscripts are preceded by an
+      underscore, like C_n.
+
+(2) Characters following a carat (^) were printed in superscript.
+
+(3) Side-notes were relocated to function as titles of their respective
+      paragraphs.
+
+(4) Letters topped by Macron are represented as [=x].
+
+(5) dP stands for the partial-derivative symbol, or curled 'd'.
+
+(6) [oo] stands for the infinity symbol, and [int] for the integral
+       symbol.
+
+(7) The following typographical errors have been corrected:
+
+    Article DIAMOND: "If this be so the form of the diamond is really
+      the tetrahedron (and the various figures derived symmetrically from
+      it) and not the octahedron". 'octahedron' amended from
+      'octadehron'.
+
+    Article DIARY: "diaries began to be largely written in England,
+      although in most cases without any idea of even eventual
+      publication". 'largely' amended from 'largly'.
+
+    Article DICOTYLEDONS: "The arrangement of the conducting tissue in
+      the stem is characteristic; a transverse section of the very young
+      stem shows a number of distinct conducting strands". 'number'
+      amended from 'nunber'.
+
+    Article DIEKIRCH: "It remained more or less fortified until the
+      beginning of the 19th century when the French during their
+      occupation levelled the old walls, and substituted the avenues of
+      trees that now encircle the town". 'or' amended from 'for'.
+
+    Article DIFFERENCES, CALCULUS OF: "as the second difference of u_n,
+      and therefore as corresponding to the value x_n"; 'difference'
+      amended from 'dfference'.
+
+    Article DINAJPUR: "a town (with a population in 1901 of 13,430) and
+     district of British India, in the Rajshahi division of Eastern
+     Bengal and Assam". 'British' amended from 'Britsh'.
+
+
+
+
+          ENCYCLOPAEDIA BRITANNICA
+
+  A DICTIONARY OF ARTS, SCIENCES, LITERATURE
+           AND GENERAL INFORMATION
+
+              ELEVENTH EDITION
+
+
+           VOLUME VIII, SLICE IV
+
+           Diameter to Dinarchus
+
+
+
+
+ARTICLES IN THIS SLICE:
+
+
+  DIAMETER                             DIEDENHOFEN
+  DIAMOND                              DIEKIRCH
+  DIAMOND NECKLACE, THE AFFAIR OF THE  DIELECTRIC
+  DIANA                                DIELMANN, FREDERICK
+  DIANA MONKEY                         DIEMEN, ANTHONY VAN
+  DIANE DE FRANCE                      DIEPENBECK, ABRAHAM VAN
+  DIANE DE POITIERS                    DIEPPE
+  DIAPASON                             DIERX, LÉON
+  DIAPER                               DIES, CHRISTOPH ALBERT
+  DIAPHORETICS                         DIEST
+  DIAPHRAGM                            DIESTERWEG, FRIEDRICH ADOLF WILHELM
+  DIARBEKR                             DIET
+  DIARRHOEA                            DIETARY
+  DIARY                                DIETETICS
+  DIASPORE                             DIETRICH, CHRISTIAN WILHELM ERNST
+  DIASTYLE                             DIETRICH OF BERN
+  DIATOMACEAE                          DIEZ, FRIEDRICH CHRISTIAN
+  DIAULOS                              DIEZ
+  DIAVOLO, FRA                         DIFFERENCES, CALCULUS OF
+  DIAZ, NARCISSE VIRGILIO              DIFFERENTIAL EQUATION
+  DIAZ, PORFIRIO                       DIFFLUGIA
+  DIAZ DE NOVAES, BARTHOLOMEU          DIFFRACTION OF LIGHT
+  DIAZO COMPOUNDS                      DIFFUSION
+  DIAZOMATA                            DIGBY, SIR EVERARD
+  DIBDIN, CHARLES                      DIGBY, SIR KENELM
+  DIBDIN, THOMAS FROGNALL              DIGBY, KENELM HENRY
+  DIBDIN, THOMAS JOHN                  DIGENES ACRITAS, BASILIUS
+  DIBRA                                DIGEST
+  DIBRUGARH                            DIGESTIVE ORGANS
+  DICAEARCHUS                          DIGGES, WEST
+  DICE                                 DIGIT
+  DICETO, RALPH DE                     DIGITALIS
+  DICEY, EDWARD                        DIGNE
+  DICHOTOMY                            DIGOIN
+  DICK, ROBERT                         DIJON
+  DICK, THOMAS                         DIKE
+  DICKENS, CHARLES JOHN HUFFAM         DIKKA
+  DICKINSON, ANNA ELIZABETH            DILAPIDATION
+  DICKINSON, JOHN                      DILATATION
+  DICKSON, SIR ALEXANDER               DILATORY
+  DICKSON, SIR JAMES ROBERT            DILEMMA
+  DICOTYLEDONS                         DILETTANTE
+  DICTATOR                             DILIGENCE
+  DICTIONARY                           DILKE, SIR CHARLES WENTWORTH
+  DICTYOGENS                           DILL
+  DICTYS CRETENSIS                     DILLEN, JOHANN JAKOB
+  DICUIL                               DILLENBURG
+  DIDACHE, THE                         DILLENS, JULIEN
+  DIDACTIC POETRY                      DILLINGEN
+  DIDEROT, DENIS                       DILLMANN, CHRISTIAN AUGUST
+  DIDIUS SALVIUS JULIANUS, MARCUS      DILLON, ARTHUR RICHARD
+  DIDO                                 DILLON, JOHN
+  DIDON, HENRI                         DILUVIUM
+  DIDOT                                DIME
+  DIDRON, ADOLPHE NAPOLÉON             DIMENSION
+  DIDYMI                               DIMITY
+  DIDYMIUM                             DINAJPUR
+  DIDYMUS                              DINAN
+  DIDYMUS CHALCENTERUS                 DINANT
+  DIE (town of France)                 DINAPUR
+  DIE (datum)                          DINARCHUS
+  DIEBITSCH, HANS KARL FRIEDRICH ANTON
+
+
+
+
+DIAMETER (from the Gr. [Greek: dia], through, [Greek: metron],
+measure), in geometry, a line passing through the centre of a circle or
+conic section and terminated by the curve; the "principal diameters" of
+the ellipse and hyperbola coincide with the "axes" and are at right
+angles; "conjugate diameters" are such that each bisects chords parallel
+to the other. The diameter of a quadric surface is a line at the
+extremities of which the tangent planes are parallel. Newton defined the
+diameter of a curve of any order as the locus of the centres of the mean
+distances of the points of intersection of a system of parallel chords
+with the curve; this locus may be shown to be a straight line. The word
+is also used as a unit of linear measurement of the magnifying power of
+a lens or microscope.
+
+In architecture, the term is used to express the measure of the lower
+part of the shaft of a column. It is employed by Vitruvius (iii. 2) to
+determine the height of a column, which should vary from eight to ten
+diameters according to the intercolumniation: and it is generally the
+custom to fix the lower diameter of the shaft by the height required and
+the Order employed. Thus the diameter of the Roman Doric should be about
+one-eighth of the height, that of the Ionic one-ninth, and of the
+Corinthian one-tenth (see ORDER).
+
+
+
+
+DIAMOND, a mineral universally recognized as chief among precious
+stones; it is the hardest, the most imperishable, and also the most
+brilliant of minerals.[1] These qualities alone have made it supreme as
+a jewel since early times, and yet the real brilliancy of the stone is
+not displayed until it has been faceted by the art of the lapidary
+(q.v.); and this was scarcely developed before the year 1746. The
+consummate hardness of the diamond, in spite of its high price, has made
+it most useful for purposes of grinding, polishing and drilling.
+Numerous attempts have been made to manufacture the diamond by
+artificial means, and these attempts have a high scientific interest on
+account of the mystery which surrounds the natural origin of this
+remarkable mineral. Its physical and chemical properties have been the
+subject of much study, and have a special interest in view of the
+extraordinary difference between the physical characters of the diamond
+and those of graphite (blacklead) or charcoal, with which it is
+chemically identical, and into which it can be converted by the action
+of heat or electricity. Again, on account of the great value of the
+diamond, much of the romance of precious stones has centred round this
+mineral; and the history of some of the great diamonds of historic times
+has been traced through many extraordinary vicissitudes.
+
+The name [Greek: Adamas], "the invincible," was probably applied by the
+Greeks to hard metals, and thence to corundum (emery) and other hard
+stones. According to Charles William King, the first undoubted
+application of the name to the diamond is found in Manilius (A.D.
+16),--_Sic Adamas_, _punctum lapidis_, _pretiosior auro_,--and Pliny
+(A.D. 100) speaks of the rarity of the stone, "the most valuable of
+gems, known only to kings." Pliny described six varieties, among which
+the Indian, having six pointed angles, and also resembling two pyramids
+(_turbines_, whip-tops) placed base to base, may probably be identified
+as the ordinary octahedral crystal (fig. 1). The "diamond" (_Yahalom_)
+in the breastplate of the high priest (Ex. xxxix. 11) was certainly some
+other stone, for it bore the name of a tribe, and methods of engraving
+the true diamond cannot have been known so early. The stone can hardly
+have become familiar to the Romans until introduced from India, where it
+was probably mined at a very early period. But one or other of the
+remaining varieties mentioned by Pliny (the Macedonian, the Arabian, the
+Cyprian, &c.) may be the true diamond, which was in great request for
+the tool of the gem-engraver. Later Roman authors mentioned various
+rivers in India as yielding the _Adamas_ among their sands. The name
+_Adamas_ became corrupted into the forms _adamant_, _diamaunt_,
+_diamant_, _diamond_; but the same word, owing to a medieval
+misinterpretation which derived it from _adamare_ (compare the French
+word _aimant_), was also applied to the lodestone.
+
+Like all the precious stones, the diamond was credited with many
+marvellous virtues; among others the power of averting insanity, and of
+rendering poison harmless; and in the middle ages it was known as the
+"pietra della reconciliazione," as the peacemaker between husband and
+wife.
+
+_Scientific Characters._--The majority of minerals are found most
+commonly in masses which can with difficulty be recognized as aggregates
+of crystalline grains, and occur comparatively seldom as distinct
+crystals; but the diamond is almost always found in single crystals,
+which show no signs of previous attachment to any matrix; the stones
+were, until the discovery of the South African mines, almost entirely
+derived from sands or gravels, but owing to the hardness of the mineral
+it is rarely, if ever, water-worn, and the crystals are often very
+perfect. The crystals belong to the cubic system, generally assuming the
+form of the octahedron (fig. 1), but they may, in accordance with the
+principles of crystallography, also occur in other forms symmetrically
+derived from the octahedron,--for example, the cube, the 12-faced figure
+known as the rhombic dodecahedron (fig. 2), or the 48-faced figure known
+as the hexakis-octahedron (fig. 3), or in combinations of these. The
+octahedron faces are usually smooth; most of the other faces are rounded
+(fig. 4). The cube faces are rough with protruding points. The cube is
+sometimes found in Brazil, but is very rare among the S. African stones;
+and the dodecahedron is perhaps more common in Brazil than elsewhere.
+There is often a furrow running along the edges of the octahedron, or
+across the edges of the cube, and this indicates that the apparently
+simple crystal may really consist of eight individuals meeting at the
+centre; or, what comes to the same thing, of two individuals
+interpenetrating and projecting through each other. If this be so the
+form of the diamond is really the tetrahedron (and the various figures
+derived symmetrically from it) and not the octahedron. Fig. 5 shows how
+the octahedron with furrowed edge may be constructed from two
+interpenetrating tetrahedra (shown in dotted lines). If the grooves be
+left out of account, the large faces which have replaced each
+tetrahedron corner then make up a figure which has the aspect of a
+simple octahedron. Such regular interpenetrations are known in
+crystallography as "twins." There are also twins of diamond in which two
+octahedra (fig. 6) are united by contact along a surface parallel to an
+octahedron face without interpenetration. On account of their
+resemblance to the twins of the mineral spinel (which crystallizes in
+octahedra) these are known as "spinel twins." They are generally
+flattened along the plane of union. The crystals often display
+triangular markings, either elevations or pits, upon the octahedron
+faces; the latter are particularly well defined and have the form of
+equilateral triangles (fig. 7). They are similar to the "etched figures"
+produced by moistening an octahedron of alum, and have probably been
+produced, like them, by the action of some solvent. Similar, but
+somewhat different markings are produced by the combustion of diamond in
+oxygen, unaccompanied by any rounding of the edges.
+
+[Illustration: FIG. 1.]
+
+[Illustration: FIG. 2.]
+
+[Illustration: FIG. 3.]
+
+[Illustration: FIG. 4.]
+
+[Illustration: FIG. 5.]
+
+[Illustration: FIG. 6.]
+
+[Illustration: FIG. 7.]
+
+Diamond possesses a brilliant "adamantine" lustre, but this tends to be
+greasy on the surface of the natural stones and gives the rounded
+crystals somewhat the appearance of drops of gum. Absolutely colourless
+stones are not so common as cloudy and faintly coloured specimens; the
+usual tints are grey, brown, yellow or white; and as rarities, red,
+green, blue and black stones have been found. The colour can sometimes
+be removed or changed at a high temperature, but generally returns on
+cooling. It is therefore more probably due to metallic oxides than to
+hydrocarbons. Sir William Crookes has, however, changed a pale yellow
+diamond to a bluish-green colour by keeping it embedded in radium
+bromide for eleven weeks. The black coloration upon the surface produced
+by this process, as also by the electric bombardment in a vacuum tube,
+appears to be due to a conversion of the surface film into graphite.
+Diamond may break with a conchoidal fracture, but the crystals always
+cleave readily along planes parallel to the octahedron faces: of this
+property the diamond cutters avail themselves when reducing the stone to
+the most convenient form for cutting; a sawing process, has, however,
+now been introduced, which is preferable to that of cleavage. It is the
+hardest known substance (though tantalum, or an alloy of tantalum now
+competes with it) and is chosen as 10 in the mineralogist's scale of
+hardness; but the difference in hardness between diamond (10) and
+corundum (9) is really greater than that between corundum (9) and talc
+(1); there is a difference in the hardness of the different faces; the
+Borneo stones are also said to be harder than those of Australia, and
+the Australian harder than the African, but this is by no means certain.
+The specific gravity ranges from 3.56 to 3.50, generally about 3.52. The
+coefficient of expansion increases very rapidly above 750°, and
+diminishes very rapidly at low temperatures; the maximum density is
+attained about -42° C.
+
+The very high refractive power (index = 2.417 for sodium light) gives
+the stone its extraordinary brilliancy; for light incident within a
+diamond at a greater angle than 24½° is reflected back into the stone
+instead of passing through it; the corresponding angle for glass is
+40½°. The very high dispersion (index for red light = 2.402, for blue
+light = 2.460) gives it the wonderful "fire" or display of spectral
+colours. Certain absorption bands at the blue end of the spectrum are
+supposed to be due to rare elements such as samarium. Unlike other cubic
+crystals, diamond experiences a diminution of refractive index with
+increase of temperature. It is very transparent for Röntgen rays,
+whereas paste imitations are opaque. It is a good conductor of heat, and
+therefore feels colder to the touch than glass and imitation stones. The
+diamond has also a somewhat greasy feel. The specific heat increases
+rapidly with rising temperature up to 60° C., and then more slowly.
+Crystals belonging to the cubic system should not be birefringent unless
+strained; diamond often displays double refraction particularly in the
+neighbourhood of inclusions, both liquid and solid; this is probably due
+to strain, and the spontaneous explosion of diamonds has often been
+observed. Diamond differs from graphite in being a bad conductor of
+electricity: it becomes positively electrified by friction. The
+electrical resistance is about that of ordinary glass, and is diminished
+by one-half during exposure by Röntgen rays; the dielectric constant
+(16) is greater than that which should correspond to the specific
+gravity.
+
+The phosphorescence produced by friction has been known since the time
+of Robert Boyle (1663); the diamond becomes luminous in a dark room
+after exposure to sunlight or in the presence of radium; and many stones
+phosphoresce beautifully (generally with a pale green light) when
+subjected to the electric discharge in a vacuum tube. Some diamonds are
+more phosphorescent than others, and different faces of a crystal may
+display different tints. The combustibility of the diamond was predicted
+by Sir Isaac Newton on account of its high refractive power; it was
+first established experimentally by the Florentine Academicians in 1694.
+In oxygen or air diamond burns at about 850°, and only continues to do
+so if maintained at a high temperature; but in the absence of oxidising
+agents it may be raised to a much higher temperature. It is, however,
+infusible at the temperature of the electric arc, but becomes converted
+superficially into graphite. Experiments on the combustion of diamond
+were made by Smithson Tennant (1797) and Sir Humphry Davy (1816), with
+the object of proving that it is pure carbon; they showed that burnt in
+oxygen it yields exactly the same amount of carbon dioxide as that
+produced by burning the same weight of carbon. Still more convincing
+experiments were made by A. Krause in 1890. Similarly Guyton de Morveau
+showed that, like charcoal, diamond converts soft iron into steel.
+Diamond is insoluble in acid and alkalis, but is oxidised on heating
+with potassium bichromate and sulphuric acid.
+
+Bort (or Boart) is the name given to impure crystals or fragments
+useless for jewels; it is also applied to the rounded crystalline
+aggregates, which generally have a grey colour, a rough surface, often a
+radial structure, and are devoid of good cleavage. They are sometimes
+spherical ("shot bort"). Carbonado or "black diamond," found in Bahia
+(also recently in Minas Geraes), is a black material with a minutely
+crystalline structure somewhat porous, opaque, resembling charcoal in
+appearance, devoid of cleavage, rather harder than diamond, but of less
+specific gravity; it sometimes displays a rude cubic crystalline form.
+The largest specimen found (1895) weighed 3078 carats. Both bort and
+carbonado seem to be really aggregates of crystallized diamond, but the
+carbonado is so nearly structureless that it was till recently regarded
+as an amorphous modification of carbon.
+
+_Uses of the Diamond._--The use of the diamond for other purposes than
+jewelry depends upon its extreme hardness: it has always been the only
+material used for cutting or engraving the diamond itself. The
+employment of powdered bort and the lapidary's wheel for faceting
+diamonds was introduced by L. von Berquen of Bruges in 1476. Diamonds
+are now employed not only for faceting precious stones, but also for
+cutting and drilling glass, porcelain, &c,; for fine engraving such as
+scales; in dentistry for drilling; as a turning tool for electric-light
+carbons, hard rubber, &c.; and occasionally for finishing accurate
+turning work such as the axle of a transit instrument. For these tools
+the stone is actually shaped to the best form: it is now electroplated
+before being set in its metal mount in order to secure a firm fastening.
+It is also used for bearings in watches and electric meters. The best
+glaziers' diamonds are chosen from crystals such that a natural curved
+edge can be used. For rock drills, and revolving saws for stone cutting,
+either diamond, bort or carbonado is employed, set in steel tubes, disks
+or bands. Rock drilling is the most important industrial application;
+and for this, owing to its freedom from cleavage, the carbonado is more
+highly prized than diamond; it is broken into fragments about 3 carats
+in weight; and in 1905 the value of carbonado was no less than from £10
+to £14 a carat. It has been found that the "carbons" in drills can
+safely be subjected to a pressure of over 60 kilograms per square
+millimetre, and a speed of 25 metres per second. A recent application of
+the diamond is for wire drawing; a hole tapering towards the centre is
+drilled through a diamond, and the metal is drawn through this. No other
+tool is so endurable, or gives such uniform thickness of wire.
+
+_Distribution and Mining._--The most important localities for diamonds
+have been: (1) India, where they were mined from the earliest times till
+the close of the 19th century; (2) South America, where they have been
+mined since the middle of the 18th century; and (3) South Africa, to
+which almost the whole of the diamond-mining industry has been
+transferred since 1870.
+
+  _India._--The diamond is here found in ancient sandstones and
+  conglomerates, and in the river gravels and sands derived from them.
+  The sandstones and conglomerates belong to the Vindhyan formation and
+  overlie the old crystalline rocks: the diamantiferous beds are well
+  defined, often not more than 1 ft. in thickness, and contain pebbles
+  of quartzite, jasper, sandstone, slate, &c. The mines fall into five
+  groups situated on the eastern side of the Deccan plateau about the
+  following places (beginning from the south), the first three being in
+  Madras. (1) Chennur near Cuddapah on the river Pennar. (2) Kurnool
+  near Baneganapalle between the rivers Pennar and Kistna. (3) Kollar
+  near Bezwada on the river Kistna. (4) Sambalpur on the river Mahanadi
+  in the Central Provinces. (5) Panna near Allahabad, in Bundelkhand.
+  The mining has always been carried on by natives of low caste, and by
+  primitive methods which do not differ much from those described by the
+  French merchant Jean Baptiste Tavernier (1605-1689), who paid a
+  prolonged visit to most of the mines between 1638 and 1665 as a
+  dealer in precious stones. According to his description shallow pits
+  were sunk, and the gravel excavated was gathered into a walled
+  enclosure where it was crushed and water was poured over it, and it
+  was finally sifted in baskets and sorted by hand. The buying and
+  selling was at that period conducted by young children. In more modern
+  times there has been the same excavation of shallow pits, and
+  sluicing, sifting and sorting, by hand labour, the only machinery used
+  being chain pumps made of earthen bowls to remove the water from the
+  deeper pits.
+
+  At some of the Indian localities spasmodic mining has been carried on
+  at different periods for centuries, at some the work which had been
+  long abandoned was revived in recent times, at others it has long been
+  abandoned altogether. Many of the large stones of antiquity were
+  probably found in the Kollar group, where Tavernier found 60,000
+  workers in 1645 (?), the mines having, according to native accounts,
+  been discovered about 100 years previously. Golconda was the fortress
+  and the market for the diamond industry at this group of mines, and so
+  gave its name to them. The old mines have now been completely
+  abandoned, but in 1891 about 1000 carats were being raised annually in
+  the neighbourhood of Hyderabad. The Sambalpur group appear to have
+  been the most ancient mines of all, but they were not worked later
+  than 1850. The Panna group were the most productive during the 19th
+  century. India was no doubt the source of all the large stones of
+  antiquity; a stone of 67-3/8 carats was found at Wajra Karur in the
+  Chennur group in 1881, and one of 210½ carats at Hira Khund in 1809.
+  Other Indian localities besides those mentioned above are Simla, in
+  the N.W. Provinces, where a few stones have been found, and a district
+  on the Gouel and the Sunk rivers in Bengal, which V. Ball has
+  identified with the Soumelpour mentioned by Tavernier. The mines of
+  Golconda and Kurnool were described as early as 1677 in the twelfth
+  volume of the _Philosophical Transactions_ of the Royal Society. At
+  the present time very few Indian diamonds find their way out of the
+  country, and, so far as the world's supply is concerned, Indian mining
+  of diamonds may be considered extinct. The first blow to this industry
+  was the discovery of the Brazilian mines in Minas Geraes and Bahia.
+
+  _Brazil._---Diamonds were found about 1725 at Tejuco (now Diamantina)
+  in Minas Geraes, and the mining became important about 1740. The chief
+  districts in Minas Geraes are (1) Bagagem on the W. side of the Serra
+  da Mata da Corda; (2) Rio Abaete on the E. side of the same range;
+  these two districts being among the head waters of the Rio de San
+  Francisco and its tributaries; (3) Diamantina, on and about the
+  watershed separating the Rio de San Francisco from the Rio
+  Jequitinhonha; and (4) Grao Mogul, nearly 200 m. to the N.E. of
+  Diamantina on the latter river.
+
+  The Rio Abaete district was worked on a considerable scale between
+  1785 and 1807, but is now abandoned. Diamantina is at present the most
+  important district; it occupies a mountainous plateau, and the
+  diamonds are found both on the plateau and in the river valleys below
+  it. The mountains consist here of an ancient laminated micaceous
+  quartzite, which is in parts a flexible sandstone known as
+  itacolumite, and in parts a conglomerate; it is interbedded with
+  clay-slate, mica-schist, hornblende-schist and haematite-schist, and
+  intersected by veins of quartz. This series is overlain unconformably
+  by a younger quartzite of similar character, and itself rests upon the
+  crystalline schists. The diamond is found under three conditions: (1)
+  in the gravels of the present rivers, embedded in a ferruginous
+  clay-cemented conglomerate known as _cascalho_; (2) in terraces
+  (gupiarras) in a similar conglomerate occupying higher levels in the
+  present valleys; (3) in plateau deposits in a coarse surface
+  conglomerate known as _gurgulho_, the diamond and other heavy minerals
+  being embedded in the red clay which cements the larger blocks. Under
+  all these three conditions the diamond is associated with fragments of
+  the rocks of the country and the minerals derived from them,
+  especially quartz, hornstone, jasper, the polymorphous oxide of
+  titanium (rutile, anatase and brookite), oxides and hydrates of iron
+  (magnetite, ilmenite, haematite, limonite), oxide of tin, iron
+  pyrites, tourmaline, garnet, xenotime, monazite, kyanite, diaspore,
+  sphene, topaz, and several phosphates, and also gold. Since the heavy
+  minerals of the _cascalho_ in the river beds are more worn than those
+  of the terraces, it is highly probable that they have been derived by
+  the cutting down of the older river gravels represented by the
+  terraces; and since in both deposits the heavy minerals are more
+  abundant near the heads of the valleys in the plateau, it is also
+  highly probable that both have really been derived from the plateau
+  deposit. In the latter, especially at São João da Chapada, the
+  minerals accompanying the diamond are scarcely worn at all; in the
+  terraces and the river beds they are more worn and more abundant; the
+  terraces, therefore, are to be regarded as a first concentration of
+  the plateau material by the old rivers; and the _cascalho_ as a second
+  concentration by the modern rivers. The mining is carried on by
+  negroes under the supervision of overseers; the _cascalho_ is dug out
+  in the dry season and removed to a higher level, and is afterwards
+  washed out by hand in running water in shallow wooden basins
+  (_bateas_). The terraces can be worked at all seasons, and the
+  material is partly washed out by leading streams on to it. The washing
+  of the plateau material is effected in reservoirs of rain water.
+
+  It is difficult to obtain an estimate of the actual production of the
+  Minas Geraes mines, for no official returns have been published, but
+  in recent years it has certainly been rivalled by the yield in Bahia.
+  The diamond here occurs in river gravels and sands associated with
+  the same minerals as in Minas Geraes; since 1844 the richest mines
+  have been worked in the Serra de Cincora, where the mountains are
+  intersected by the river Paraguassu and its tributaries; it is said
+  that there were as many as 20,000 miners working here in 1845, and it
+  was estimated that 54,000 carats were produced in Bahia in 1858. The
+  earlier workings were in the Serra de Chapada to the N.W. of the mines
+  just mentioned. In 1901 there were about 5000 negroes employed in the
+  Bahia mines; methods were still primitive; the _cascalho_ was dug out
+  from the river beds or tunnelled out from the valley side, and washed
+  once a week in sluices of running water, where it was turned over with
+  the hoe, and finally washed in wooden basins and picked over by hand;
+  sometimes also the diamantiferous material is scooped out of the bed
+  of the shallow rivers by divers, and by men working under water in
+  caissons. It is almost exclusively in the mines of Bahia, and in
+  particular in the Cincora district, that the valuable carbonado is
+  found. The carbonado and the diamond have been traced to an extensive
+  hard conglomerate which occurs in the middle of the sandstone
+  formation. Diamonds are also mined at Salobro on the river Pardo not
+  far inland from the port of Canavieras in the S.E. corner of Bahia.
+  The enormous development of the South African mines, which supplied in
+  1906, about 90% of the world's produce, has thrown into the shade the
+  Brazilian production; but the _Bulletin_ for Feb. 1909 of the
+  International Bureau of American Republics gave a very confident
+  account of its future, under improved methods.
+
+  _South Africa._---The first discovery was made in 1867 by Dr W. G.
+  Atherstone, who identified as diamond a pebble obtained from a child
+  in a farm on the banks of the Orange river and brought by a trader to
+  Grahamstown; it was bought for £500 and displayed in the Paris
+  Exhibition of that year. In 1869 a stone weighing 83½ carats was found
+  near the Orange river; this was purchased by the earl of Dudley for
+  £25,000 and became famous as the "Star of South Africa." A rush of
+  prospectors at once took place to the banks of the Orange and Vaal
+  rivers, and resulted in considerable discoveries, so that in 1870
+  there was a mining camp of no less than 10,000 persons on the "River
+  Diggings." In the River Diggings the mining was carried on in the
+  coarse river gravels, and by the methods of the Brazilian negroes and
+  of gold placer-miners. A diggers' committee limited the size of claims
+  to 30 ft. square, with free access to the river bank; the gravel and
+  sand were washed in cradles provided with screens of perforated metal,
+  and the concentrates were sorted by hand on tables by means of an iron
+  scraper.
+
+  But towards the close of 1870 stones were found at Jagersfontein and
+  at Dutoitspan, far from the Vaal river, and led to a second great rush
+  of prospectors, especially to Dutoitspan, and in 1871 to what is now
+  the Kimberley mine in the neighbourhood of the latter. At each of
+  these spots the diamantiferous area was a roughly circular patch of
+  considerable size, and in some occupied the position of one of those
+  depressions or "pans" so frequent in S. Africa. These "dry diggings"
+  were therefore at first supposed to be alluvial in origin like the
+  river gravels; but it was soon discovered that, below the red surface
+  soil and the underlying calcareous deposit, diamonds were also found
+  in a layer of yellowish clay about 50 ft. thick known as "yellow
+  ground." Below this again was a hard bluish-green serpentinous rock
+  which was at first supposed to be barren bed-rock; but this also
+  contained the precious stone, and has become famous, under the name of
+  "blue ground," as the matrix of the S. African diamonds. The yellow
+  ground is merely decomposed blue ground. In the Kimberley district
+  five of these round patches of blue ground were found within an area
+  little more than 3 m. in diameter; that at Kimberley occupying 10
+  acres, that at Dutoitspan 23 acres. There were soon 50,000 workers on
+  this field, the canvas camp was replaced by a town of brick and iron
+  surrounded by the wooden huts of the natives, and Kimberley became an
+  important centre.
+
+  It was soon found that each mine was in reality a huge vertical funnel
+  or crater descending to an unknown depth, and filled with
+  diamantiferous blue ground. At first each claim was an independent pit
+  31 ft. square sunk into the blue ground; the diamantiferous rock was
+  hoisted by bucket and windlass, and roadways were left across the pit
+  to provide access to the claims. But the roadways soon fell in, and
+  ultimately haulage from the claims could only be provided by means of
+  a vast system of wire ropes extending from a triple staging of
+  windlasses erected round the entire edge of the mine, which had by
+  this time become a huge open pit; the ropes from the upper windlasses
+  extended to the centre, and those from the lower tier to the sides of
+  the pit; covering the whole mass like a gigantic cobweb. (See Plate
+  II. fig. 12.) The buckets of blue ground were hauled up these ropes by
+  means of horse whims, and in 1875 steam winding engines began to be
+  employed. By this time also improved methods in the treatment of the
+  blue ground were introduced. It was carried off in carts to open
+  spaces, where an exposure of some weeks to the air was found to
+  pulverize the hard rock far more efficiently than the old method of
+  crushing with mallets. The placer-miner's cradle and rocking-trough
+  were replaced by puddling troughs stirred by a revolving comb worked
+  by horse power; reservoirs were constructed for the scanty
+  water-supply, bucket elevators were introduced to carry away the
+  tailings; and the natives were confined in compounds. For these
+  improvements co-operation was necessary; the better claims, which in
+  1872 had risen from £100 to more than £4000 in value, began to be
+  consolidated, and a Mining Board was introduced.
+
+
+  PLATE I.
+
+  [Illustration: FIG. 9.--DE BEERS MINE, 1874.]
+
+  [Illustration: FIG. 10.--KIMBERLEY MINE, 1874.]
+
+  [Illustration: FIG. 11.--DE BEERS MINE, 1873. (From photographs by C.
+  Evans.)]
+
+
+  PLATE II.
+
+  [Illustration: _Fig. 12._--KIMBERLEY MINE, 1874.]
+
+  [Illustration: _Fig. 13._--KIMBERLEY MINE, 1902. (From Photographs by
+  C. Evans.)]
+
+
+  In a very few years, however, the open pit mining was rendered
+  impossible by the mud rushes, by the falls of the masses of barren
+  rock known as "reef," which were left standing in the mine, and by
+  landslips from the sides, so that in 1883, when the pit had reached a
+  depth of about 400 ft., mining in the Kimberley crater had become
+  almost impossible. By 1889, in the whole group of mines, Kimberley,
+  Dutoitspan, De Beers and Bultfontein, open pit working was practically
+  abandoned. Meanwhile mining below the bottom of the pits by means of
+  shafts and underground tunnels had been commenced; but the full
+  development of modern methods dates from the year 1889 when Cecil
+  Rhodes and Alfred Beit, who had already secured control of the De
+  Beers mine, acquired also the control of the Kimberley mine, and
+  shortly afterwards consolidated the entire group in the hands of the
+  De Beers Company. (See KIMBERLEY.)
+
+  The scene of native mining was now transferred from the open pit to
+  underground tunnels; the vast network of wire ropes (Plate II. fig.
+  12) with their ascending and descending buckets disappeared, and with
+  it the cosmopolitan crowd of busy miners working like ants at the
+  bottom of the pit. In place of all this, the visitor to Kimberley
+  encounters at the edge of the town only a huge crater, silent and
+  apparently deserted, with no visible sign of the great mining
+  operations which are conducted nearly half a mile below the surface.
+  The aspect of the Kimberley pit in 1906 is shown in fig. 13 of Plate
+  II., which may be compared with the section of fig. 8.
+
+  In fig. 13, Plate II., the sequence of the basalt, shale and melaphyre
+  is clearly visible on the sides of the pit; and fig. 8 shows how the
+  crater or "pipe" of blue ground has penetrated these rocks and also
+  the underlying quartzite. The workings at De Beers had extended into
+  the still more deeply seated granite in 1906. Figure 9, Plate I.,
+  shows the top of the De Beers' crater with basalt overlying the shale.
+  Figure 8 also explains the modern system of mining introduced by
+  Gardner Williams. A vertical shaft is sunk in the vicinity of the
+  mine, and from this horizontal tunnels are driven into the pipe at
+  different levels separated by intervals of 40 ft. Through the blue
+  ground itself on each level a series of parallel tunnels about 120 ft.
+  apart are driven to the opposite side of the pipe, and at right angles
+  to these, and 36 ft. apart, another series of tunnels. When the
+  tunnels reach the side of the mine they are opened upwards and
+  sideways so as to form a large chamber, and the overlying mass of blue
+  ground and débris is allowed to settle down and fill up the gallery.
+  On each level this process is carried somewhat farther back than on
+  the level below (fig. 8); material is thus continually withdrawn from
+  one side of the mine and extracted by means of the rock shaft on the
+  opposite side, while the superincumbent débris is continually sinking,
+  and is allowed to fall deeper on the side farthest from the shaft as
+  the blue ground is withdrawn from beneath it. In 1905 the main shaft
+  had been sunk to a depth of 2600 ft. at the Kimberley mine.
+
+  For the extraction and treatment of the blue ground the De Beers
+  Company in its great winding and washing plant employs labour-saving
+  machinery on a gigantic scale. The ground is transferred in trucks to
+  the shaft where it is automatically tipped into skips holding 96 cubic
+  ft. (six truck loads); these are rapidly hoisted to the surface, where
+  their contents are automatically dumped into side-tipping trucks, and
+  these in turn are drawn away in a continual procession by an endless
+  wire rope along the tram lines leading to the vast "distributing
+  floors." These are open tracts upon which the blue ground is spread
+  out and left exposed to sun and rain until it crumbles and
+  disintegrates, the process being hastened by harrowing with steam
+  ploughs; this may require a period of three or six months, or even a
+  year. The stock of blue ground on the floors at one time in 1905 was
+  nearly 4,500,000 loads. The disintegrated ground is then brought back
+  in the trucks and fed through perforated cylinders into the washing
+  pans; the hard blue which has resisted disintegration on the floors,
+  and the lumps which are too big to pass the cylindrical sieves, are
+  crushed before going to the pans. These are shallow cylindrical
+  troughs containing muddy water in which the diamonds and other heavy
+  minerals (concentrates) are swept to the rim by revolving toothed
+  arms, while the lighter stuff escapes near the centre of the pan. The
+  concentrates are then passed over sloping tables (pulsator) and shaken
+  to and fro under a stream of water which effects a second
+  concentration of the heaviest material.
+
+  Until recently the final separation of the diamond from the
+  concentrates was made by hand picking, but even this has now been
+  replaced by machinery, owing to the remarkable discovery that a
+  greased surface will hold a diamond while allowing the other heavy
+  minerals to pass over it. The concentrates are washed down a sloping
+  table of corrugated iron which is smeared with grease, and it is found
+  that practically all the diamonds adhere to the table, and the other
+  minerals are washed away. At the large and important Premier mine in
+  the Transvaal the Elmore process, used in British Columbia and in
+  Wales for the separation of metallic ores, has been also introduced.
+  In the Elmore process oil is employed to float off the materials which
+  adhere to it, while the other materials remain in the water, the oil
+  being separated from the water by centrifugal action. The other
+  minerals found in the concentrates are pebbles and fragments of
+  pyrope, zircon, cyanite, chrome-diopside, enstatite, a green pyroxene,
+  mica, ilmenite, magnetite, chromite, hornblende, olivine, barytes,
+  calcite and pyrites.
+
+  In all the S. African mines the diamonds are not only crystals of
+  various weights from fractions of a carat to 150 carats, but also
+  occur as microscopic crystals disseminated through the blue ground. In
+  spite of this, however, the average yield in the profitable mines is
+  only from 0.2 carat to 0.6 carat per load of 1600 lb., or on an average
+  about 1½ grs. per ton. The annual output of diamonds from the De Beers
+  mines was valued in 1906 at nearly £5,000,000; the value per carat
+  ranging from about 35s. to 70s.
+
+  [Illustration: SECTION OF KIMBERLEY MINE LOOKING EAST
+
+  From Gardner Williams's _Diamond Mines of South Africa_. FIG. 8.]
+
+  Pipes similar to those which surround Kimberley have been found in
+  other parts of S. Africa. One of the best known is that of
+  Jagersfontein, which was really the first of the dry diggings
+  (discovered in 1870). This large mine is near Fauresmith and 80 m. to
+  the south of Kimberley. In 1905 the year's production from the Orange
+  River Colony mines was more than 320,000 carats, valued at £938,000.
+  But by far the largest of all the pipes hitherto discovered is the
+  Premier mine in the Transvaal, about 300 m. to the east of Kimberley.
+  This was discovered in 1902 and occupies an area of about 75 acres. In
+  1906 it was being worked as a shallow open mine; but the description
+  of the Kimberley methods given above is applicable to the washing
+  plant at that time being introduced into the Premier mine upon a very
+  large scale. Comparatively few of the pipes which have been discovered
+  are at all rich in diamonds, and many are quite barren; some are
+  filled with "hard blue" which even if diamantiferous may be too
+  expensive to work.
+
+  The most competent S. African geologists believe all these remarkable
+  pipes to be connected with volcanic outbursts which occurred over the
+  whole of S. Africa during the Cretaceous period (after the deposition
+  of the Stormberg beds), and drilled these enormous craters through all
+  the later formations. With the true pipes are associated dykes and
+  fissures also filled with diamantiferous blue ground. It is only in
+  the more northerly part of the country that the pipes are filled with
+  blue ground (or "kimberlite"), and that they are diamantiferous; but
+  over a great part of Cape Colony have been discovered what are
+  probably similar pipes filled with agglomerates, breccias and tuffs,
+  and some with basic lavas; one, in particular, in the Riversdale
+  Division near the southern coast, being occupied by a melilite-basalt.
+  It is quite clear that the occurrence of the diamond in the S. African
+  pipes is quite different from the occurrences in alluvial deposits
+  which have been described above. The question of the origin of the
+  diamond in S. Africa and elsewhere is discussed below.
+
+  The River Diggings on the Vaal river are still worked upon a small
+  scale, but the production from this source is so limited that they are
+  of little account in comparison with the mines in the blue ground. The
+  stones, however, are good; since they differ somewhat from the
+  Kimberley crystals it is probable that they were not derived from the
+  present pipes. Another S. African locality must be mentioned;
+  considerable finds were reported in 1905 and 1906 from gravels at
+  Somabula near Gwelo in Rhodesia where the diamond is associated with
+  chrysoberyl, corundum (both sapphire and ruby), topaz, garnet,
+  ilmenite, staurolite, rutile, with pebbles of quartz, granite,
+  chlorite-schist, &c. Diamond has also been reported from kimberlite
+  "pipes" in Rhodesia.
+
+  _Other Localities._--In addition to the South American localities
+  mentioned above, small diamonds have also been mined since their
+  discovery in 1890 on the river Mazaruni in British Guiana, and finds
+  have been reported in the gold washings of Dutch Guiana. Borneo has
+  possessed a diamond industry since the island was first settled by the
+  Malays; the references in the works of Garcia de Orta, Linschoten, De
+  Boot, De Laet and others, to Malacca as a locality relate to Borneo.
+  The large Borneo stone, over 360 carats in weight, known as the Matan,
+  is in all probability not a diamond. The chief mines are situated on
+  the river Kapuas in the west and near Bandjarmassin in the south-east
+  of the island, and the alluvial deposits in which they occur are
+  worked by a small number of Chinese and Malays. Australia has yielded
+  diamonds in alluvial deposits near Bathurst (where the first discovery
+  was made in 1851) and Mudgee in New South Wales, and also near Bingara
+  and Inverell in the north of the colony. At Mount Werong a stone
+  weighing 29 carats was found in 1905. At Ruby Hill near Bingara they
+  were found in a breccia filling a volcanic pipe. At Ballina, in New
+  England, diamonds have been found in the sea sand. Other Australian
+  localities are Echunga in South Australia; Beechworth, Arena and
+  Melbourne in Victoria; Freemantle and Nullagine in Western Australia;
+  the Palmer and Gilbert rivers in Queensland. These have been for the
+  most part discoveries in alluvial deposits of the goldfields, and the
+  stones were small. In Tasmania also diamonds have been found in the
+  Corinna goldfields. Europe has produced few diamonds. Humboldt
+  searched for them in the Urals on account of the similarity of the
+  gold and platinum deposits to those of Brazil, and small diamonds were
+  ultimately found (1829) in the gold washings of Bissersk, and later at
+  Ekaterinburg and other spots in the Urals. In Lapland they have been
+  found in the sands of the Pasevig river. Siberia has yielded isolated
+  diamonds from the gold washings of Yenisei. In North America a few
+  small stones have been found in alluvial deposits, mostly auriferous,
+  in Georgia, N. and S. Carolina, Kentucky, Virginia, Tennessee,
+  Wisconsin, California, Oregon and Indiana. A crystal weighing 23¾
+  carats was found in Virginia in 1855, and one of 21¼ carats in
+  Wisconsin in 1886. In 1906 a number of small diamonds were discovered
+  in an altered peridotite somewhat resembling the S. African blue
+  ground, at Murfreesboro, Pike county, Arkansas. Considerable interest
+  attaches to the diamonds found in Wisconsin, Michigan and Ohio near
+  the Great Lakes, for they are here found in the terminal moraines of
+  the great glacial sheet which is supposed to have spread southwards
+  from the region of Hudson Bay; several of the drift minerals of the
+  diamantiferous region of Indiana have been identified as probably of
+  Canadian origin; no diamonds have however yet been found in the
+  intervening country of Ontario. A rock similar to the blue ground of
+  Kimberley has been found in the states of Kentucky and New York. The
+  occurrence of diamond in meteorites is described below.
+
+  _Origin of the Diamond in Nature._--It appears from the foregoing
+  account that at most localities the diamond is found in alluvial
+  deposits probably far from the place where it originated. The minerals
+  associated with it do not afford much clue to the original conditions;
+  they are mostly heavy minerals derived from the neighbouring rocks, in
+  which the diamond itself has not been observed. Among the commonest
+  associates of the diamond are quartz, topaz, tourmaline, rutile,
+  zircon, magnetite, garnet, spinel and other minerals which are common
+  accessory constituents of granite, gneiss and the crystalline schists.
+  Gold (also platinum) is a not infrequent associate, but this may only
+  mean that the sands in which the diamond is found have been searched
+  because they were known to be auriferous; also that both gold and
+  diamond are among the most durable of minerals and may have survived
+  from ancient rocks of which other traces have been lost.
+
+  The localities at which the diamond has been supposed to occur in its
+  original matrix are the following:--at Wajra Karur, in the Cuddapah
+  district, India, M. Chaper found diamond with corundum in a decomposed
+  red pegmatite vein in gneiss. At S[=a]o João da Chapada, in Minas
+  Geraes, diamonds occur in a clay interstratified with the itacolumite,
+  and are accompanied by sharp crystals of rutile and haematite in the
+  neighbourhood of decomposed quartz veins which intersect the
+  itacolumite. It has been suggested that these three minerals were
+  originally formed in the quartz veins. In both these occurrences the
+  evidence is certainly not sufficient to establish the presence of an
+  original matrix. At Inverell in New South Wales a diamond (1906) has
+  been found embedded in a hornblende diabase which is described as a
+  dyke intersecting the granite. Finally there is the remarkable
+  occurrence in the blue ground of the African pipes.
+
+  There has been much controversy concerning the nature and origin of
+  the blue ground itself; and even granted that (as is generally
+  believed) the blue ground is a much serpentinized volcanic breccia
+  consisting originally of an olivine-bronzite-biotite rock (the
+  so-called kimberlite), it contains so many rounded and angular
+  fragments of various rocks and minerals that it is difficult to say
+  which of them may have belonged to the original rock, and whether any
+  were formed _in situ_, or were brought up from below as inclusions.
+  Carvill Lewis believed the blue ground to be true eruptive rock, and
+  the carbon to have been derived from the bituminous shales of which it
+  contains fragments. The Kimberley shales, which are penetrated by the
+  De Beers group of pipes, were, however, certainly not the source of
+  the carbon at the Premier (Transvaal) mine, for at this locality the
+  shales do not exist. The view that the diamond may have crystallized
+  out from solution in its present matrix receives some support from the
+  experiments of W. Luzi, who found that it can be corroded by the
+  solvent action of fused blue ground; from the experiments of J.
+  Friedländer, who obtained diamond by dissolving graphite in fused
+  olivine; and still more from the experiments of R. von Hasslinger and
+  J. Wolff, who have obtained it by dissolving graphite in a fused
+  mixture of silicates having approximately the composition of the blue
+  ground. E. Cohen, who regarded the pipes as of the nature of a mud
+  volcano, and the blue ground as a kimberlite breccia altered by
+  hydrothermal action, thought that the diamond and accompanying
+  minerals had been brought up from deep-seated crystalline schists.
+  Other authors have sought the origin of the diamond in the action of
+  the hydrated magnesian silicates on hydrocarbons derived from
+  bituminous schists, or in the decomposition of metallic carbides.
+
+  Of great scientific interest in this connexion is the discovery of
+  small diamonds in certain meteorites, both stones and irons; for
+  example, in the stone which fell at Novo-Urei in Penza, Russia, in
+  1886, in a stone found at Carcote in Chile, and in the iron found at
+  Cañon Diablo in Arizona. Graphitic carbon in cubic form (cliftonite)
+  has also been found in certain meteoric "irons," for example in those
+  from Magura in Szepes county, Hungary, and Youndegin near York in
+  Western Australia. The latter is now generally believed to be altered
+  diamond. The fact that H. Moissan has produced the diamond
+  artificially, by allowing dissolved carbon to crystallize out at a
+  high temperature and pressure from molten iron, coupled with the
+  occurrence in meteoric iron, has led Sir William Crookes and others to
+  conclude that the mineral may have been derived from deep-seated iron
+  containing carbon in solution (see the article GEM, ARTIFICIAL). Adolf
+  Knop suggested that this may have first yielded hydrocarbons by
+  contact with water, and that from these the crystalline diamond has
+  been formed. The meteoric occurrence has even suggested the fanciful
+  notion that all diamonds were originally derived from meteorites. The
+  meteoric iron of Arizona, some of which contains diamond, is actually
+  found in and about a huge crater which is supposed by some to have
+  been formed by an immense meteorite penetrating the earth's crust.
+
+  It is, at any rate, established that carbon can crystallize as diamond
+  from solution in iron, and other metals; and it seems that high
+  temperature and pressure and the absence of oxidizing agents are
+  necessary conditions. The presence of sulphur, nickel, &c., in the
+  iron appears to favour the production of the diamond. On the other
+  hand, the occurrence in meteoric stones, and the experiments mentioned
+  above, show that the diamond may also crystallize from a basic magma,
+  capable of yielding some of the metallic oxides and ferro-magnesian
+  silicates; a magma, therefore, which is not devoid of oxygen. This is
+  still more forcibly suggested by the remarkable eclogite boulder found
+  in the blue ground of the Newlands mine, not far from the Vaal river,
+  and described by T. G. Bonney. The boulder is a crystalline rock
+  consisting of pyroxene (chrome-diopside), garnet, and a little
+  olivine, and is studded with diamond crystals; a portion of it is
+  preserved in the British Museum (Natural History). In another eclogite
+  boulder, diamond was found partly embedded in pyrope. Similar boulders
+  have also been found in the blue ground elsewhere. Specimens of pyrope
+  with attached or embedded diamond had previously been found in the
+  blue ground of the De Beers mines. In the Newlands boulder the
+  diamonds have the appearance of being an original constituent of the
+  eclogite. It seems therefore that a holocrystalline pyroxene-garnet
+  rock may be one source of the diamond found in blue ground. On the
+  other hand many tons of the somewhat similar eclogite in the De Beers
+  mine have been crushed and have not yielded diamond. Further, the
+  ilmenite, which is the most characteristic associate of the diamond in
+  blue ground, and other of the accompanying minerals, may have come
+  from basic rocks of a different nature.
+
+  The Inverell occurrence may prove to be another example of diamond
+  crystallized from a basic rock.
+
+  In both occurrences, however, there is still the possibility that the
+  eclogite or the basalt is not the original matrix, but may have caught
+  up the already formed diamond from some other matrix. Some regard the
+  eclogite boulders as derived from deep-seated crystalline rocks,
+  others as concretions in the blue ground.
+
+  None of the inclusions in the diamond gives any clue to its origin;
+  diamond itself has been found as an inclusion, as have also black
+  specks of some carbonaceous materials. Other black specks have been
+  identified as haematite and ilmenite; gold has also been found; other
+  included minerals recorded are rutile, topaz, quartz, pyrites,
+  apophyllite, and green scales of chlorite (?). Some of these are of
+  very doubtful identification; others (e.g. apophyllite and chlorite)
+  may have been introduced along cracks. Some of the fibrous inclusions
+  were identified by H. R. Göppert as vegetable structures and were
+  supposed to point to an organic origin, but this view is no longer
+  held. Liquid inclusions, some of which are certainly carbon dioxide,
+  have also been observed.
+
+  Finally, then, both experiment and the natural occurrence in rocks and
+  meteorites suggest that diamond may crystallize not only from iron but
+  also from a basic silicate magma, possibly from various rocks
+  consisting of basic silicates. The blue ground of S. Africa may be
+  the result of the serpentinization of several such rocks, and
+  although now both brecciated and serpentinized some of these may have
+  been the original matrix. A circumstance often mentioned in support of
+  this view is the fact that the diamonds in one pipe generally differ
+  somewhat in character from those of another, even though they be near
+  neighbours.
+
+_History._--All the famous diamonds of antiquity must have been Indian
+stones. The first author who described the Indian mines at all fully was
+the Portuguese, Garcia de Orta (1565), who was physician to the viceroy
+of Goa. Before that time there were only legendary accounts like that of
+Sindbad's "Valley of the Diamonds," or the tale of the stones found in
+the brains of serpents. V. Ball thinks that the former legend originated
+in the Indian practice of sacrificing cattle to the evil spirits when a
+new mine is opened; birds of prey would naturally carry off the flesh,
+and might give rise to the tale of the eagles carrying diamonds adhering
+to the meat.
+
+The following are some of the most famous diamonds of the world:--
+
+A large stone found in the Golconda mines and said to have weighed 787
+carats in the rough, before being cut by a Venetian lapidary, was seen
+in the treasury of Aurangzeb in 1665 by Tavernier, who estimated its
+weight after cutting as 280 (?) carats, and described it as a rounded
+rose-cut-stone, tall on one side. The name _Great Mogul_ has been
+frequently applied to this stone. Tavernier states that it was the
+famous stone given to Shah Jahan by the emir Jumla. The _Orloff_, stolen
+by a French soldier from the eye of an idol in a Brahmin temple, stolen
+again from him by a ship's captain, was bought by Prince Orloff for
+£90,000, and given to the empress Catharine II. It weighs 194¾ carats,
+is of a somewhat yellow tinge, and is among the Russian crown jewels.
+The _Koh-i-nor_, which was in 1739 in the possession of Nadir Shah, the
+Persian conqueror, and in 1813 in that of the raja of Lahore, passed
+into the hands of the East India Company and was by them presented to
+Queen Victoria in 1850. It then weighed 186-1/16 carats, but was recut in
+London by Amsterdam workmen, and now weighs 106-1/16 carats. There has
+been much discussion concerning the possibility of this stone and the
+Orloff being both fragments of the Great Mogul. The Mogul Baber in his
+memoirs (1526) relates how in his conquest of India he captured at Agra
+the great stone weighing 8 mishkals, or 320 ratis, which may be
+equivalent to about 187 carats. The Koh-i-nor has been identified by
+some authors with this stone and by others with the stone seen by
+Tavernier. Tavernier, however, subsequently described and sketched the
+diamond which he saw as shaped like a bisected egg, quite different
+therefore from the Koh-i-nor. Nevil Story Maskelyne has shown reason for
+believing that the stone which Tavernier saw was really the Koh-i-nor
+and that it is identical with the great diamond of Baber; and that the
+280 carats of Tavernier is a misinterpretation on his part of the Indian
+weights. He suggests that the other and larger diamond of antiquity
+which was given to Shah Jahan may be one which is now in the treasury of
+Teheran, and that this is the true Great Mogul which was confused by
+Tavernier with the one he saw. (See Ball, Appendix I. to Tavernier's
+_Travels_ (1889); and Maskelyne, _Nature_, 1891, 44, p. 555.).
+
+The _Regent_ or _Pitt_ diamond is a magnificent stone found in either
+India or Borneo; it weighed 410 carats and was bought for £20,400 by
+Pitt, the governor of Madras; it was subsequently, in 1717, bought for
+£80,000 (or, according to some authorities, £135,000) by the duke of
+Orleans, regent of France; it was reduced by cutting to 1361-4/16 carats;
+was stolen with the other crown jewels during the Revolution, but was
+recovered and is still in France. The _Akbar Shah_ was originally a
+stone of 116 carats with Arabic inscriptions engraved upon it; after
+being cut down to 71 carats it was bought by the gaikwar of Baroda for
+£35,000. The _Nizam_, now in the possession of the nizam of Hyderabad,
+is supposed to weigh 277 carats; but it is only a portion of a stone
+which is said to have weighed 440 carats before it was broken. The
+_Great Table_, a rectangular stone seen by Tavernier in 1642 at
+Golconda, was found by him to weigh 242-3/16 carats; Maskelyne regards it
+as identical with the _Darya-i-nur_, which is also a rectangular stone
+weighing about 186 carats in the possession of the shah of Persia.
+Another stone, the _Taj-e-mah_, belonging to the shah, is a pale rose
+pear-shaped stone and is said to weigh 146 carats.
+
+Other famous Indian diamonds are the following:--The _Sancy_, weighing
+531-2/16 carats, which is said to have been successively the property of
+Charles the Bold, de Sancy, Queen Elizabeth, Henrietta Maria, Cardinal
+Mazarin, Louis XIV.; to have been stolen with the Pitt during the French
+Revolution; and subsequently to have been the property of the king of
+Spain, Prince Demidoff and an Indian prince. The _Nassak_, 78-5/8 carats,
+the property of the duke of Westminster. The _Empress Eugénie_, 51
+carats, the property of the gaikwar of Baroda. The _Pigott_, 49
+carats(?), which cannot now be traced. The _Pasha_, 40 carats. The
+_White Saxon_, 48¾ carats. The _Star of Este_, 251-3/32 carats.
+
+Coloured Indian diamonds of large size are rare; the most famous are:--a
+beautiful blue brilliant, 67-2/16 carats, cut from a stone weighing
+112-3/16 carats brought to Europe by Tavernier. It was stolen from the
+French crown jewels with the Regent and was never recovered. The _Hope_,
+44¼ carats, has the same colour and is probably a portion of the missing
+stone: it was so-called as forming part of the collection of H. T. Hope
+(bought for £18,000), and was sold again in 1906 (resold 1909). Two
+other blue diamonds are known, weighing 13¾ and 1¾ carats, which may
+also be portions of the French diamond. The _Dresden Green_, one of the
+Saxon crown jewels, 40 carats, has a fine apple-green colour. The
+_Florentine_, 133-1/5 carats, one of the Austrian crown jewels, is a very
+pale yellow.
+
+The most famous Brazilian stones are:--The _Star of the South_, found in
+1853, when it weighed 254½ carats and was sold for £40,000; when cut it
+weighed 125 carats and was bought by the gaikwar of Baroda for £80,000.
+Also a diamond belonging to Mr Dresden, 119 carats before, and 76½
+carats after cutting.
+
+Many large stones have been found in South Africa; some are yellow but
+some are as colourless as the best Indian or Brazilian stones. The most
+famous are the following:--the _Star of South Africa_, or _Dudley_,
+mentioned above, 83½ carats rough, 46½ carats cut. The _Stewart_, 288-3/8
+carats rough, 120 carats cut. Both these were found in the river
+diggings. The _Porter Rhodes_ from Kimberley, of the finest water,
+weighed about 150 carats. The _Victoria_, 180 carats, was cut from an
+octahedron weighing 457½ carats, and was sold to the nizam of Hyderabad
+for £400,000. The _Tiffany_, a magnificent orange-yellow stone, weighs
+125½ carats cut. A yellowish octahedron found at De Beers weighed 428½
+carats, and yielded a brilliant of 288½ carats. Some of the finest and
+largest stones have come from the Jagersfontein mine; one, the
+_Jubilee_, found in 1895, weighed 640 carats in the rough and 239 carats
+when cut. Until 1905 the largest known diamond in the world was the
+_Excelsior_, found in 1893 at Jagersfontein by a native while loading a
+truck. It weighed 971 carats, and was ultimately cut into ten stones
+weighing from 68 to 13 carats. But all previous records were surpassed
+in 1905 by a magnificent stone more than three times the size of any
+known diamond, which was found in the yellow ground at the newly
+discovered Premier mine in the Transvaal. This extraordinary diamond
+weighed 3025¾ carats (11/3 lb.) and was clear and water white; the
+largest of its surfaces appeared to be a cleavage plane, so that it
+might be only a portion of a much larger stone. It was known as the
+_Cullinan Diamond_. This stone was purchased by the Transvaal government
+in 1907 and presented to King Edward VII. It was sent to Amsterdam to be
+cut, and in 1908 was divided into nine large stones and a number of
+small brilliants. The four largest stones weigh 516½ carats, 309-3/16
+carats, 92 carats and 62 carats respectively. Of these the first and
+second are the largest brilliants in existence. All the stones are
+flawless and of the finest quality.
+
+  BIBLIOGRAPHY.--Boetius de Boot, _Gemmarum et lapidum historia_ (1609);
+  D. Jeffries, _A Treatise on Diamonds and Pearls_ (1757); J. Mawe,
+  _Travels in the Interior of Brazil_ (1812); _Treatise on Diamonds and
+  Precious Stones_ (1813): Pinder, _De adamante_ (1829); Murray, _Memoir
+  on the Nature of the Diamond_ (1831); C. Zerenner, _De adamante
+  dissertatio_ (1850); H. Emanuel, _Diamonds and Precious Stones_
+  (1865); A. Schrauf, _Edelsteinkunde_ (1869); N. Jacobs and N.
+  Chatrian, _Monographie du diamant_ (1880); V. Ball, _Geology of India_
+  (1881); C. W. King, _The Natural History of Precious Stones_ _and
+  Precious Metals_ (1883); M. E. Boutan, _Le Diamant_ (1886); S. M.
+  Burnham, _Precious Stones in Nature, Art and Literature_ (1887); P.
+  Groth, _Grundriss der Edelsteinkunde_ (1887); A. Liversidge, _The
+  Minerals of New South Wales_ (1888); _Tavernier's Travels in India_,
+  translated by V. Ball (1889); E. W. Streeter, _The Great Diamonds of
+  the World_ (1896); H. C. Lewis, _The Genesis and Matrix of the
+  Diamond_ (1897); L. de Launay, _Les Diamants du Cap_ (1897); C.
+  Hintze, _Handbuch der Mineralogie_ (1898); E. W. Streeter, _Precious
+  Stones and Gems_ (6th ed., 1898); Dana, _System of Mineralogy_ (1899);
+  Kunz and others, _The Production of Precious Stones_ (in annual,
+  _Mineral Resources of the United States_); M. Bauer, _Precious Stones_
+  (trans. L. J. Spencer, 1904); A. W. Rogers, _An Introduction to the
+  Geology of Cape Colony_ (1905); Gardner F. Williams, _The Diamond
+  Mines of South Africa_ (revised edition, 1906); George F. Kunz,
+  "Diamonds, a study of their occurrence in the United States, with
+  descriptions and comparisons of those from all known localities" (U.S.
+  Geol. Survey, 1909); P. A. Wagner, _Die Diamantführenden Gesteine
+  Südafrikas_ (1909).
+
+  Among papers in scientific periodicals may be mentioned articles by
+  Adler, Ball, Baumhauer, Beck, Bonney, Brewster, Chaper, Cohen,
+  Crookes, Daubrée, Derby, Des Cloizeaux, Doelter, Dunn, Flight,
+  Friedel, Gorceix, Gürich, Goeppert, Harger, Hudleston, Hussak,
+  Jannettaz, Jeremejew, de Launay, Lewis, Maskelyne, Meunier, Moissan,
+  Molengraaff, Moulle, Rose, Sadebeck, Scheibe, Stelzner, Stow. See
+  generally Hintze's _Handbuch der Mineralogie_. (H. A. MI.)
+
+
+FOOTNOTE:
+
+  [1] Diamonds are invariably weighed in carats and in ½, ¼, 1/8, 1/16,
+    1/32, 1/64 of a carat. One (English) carat = 3.17 grains = .2054
+    gram. One ounce = 151½ carats. (See CARAT.)
+
+
+
+
+DIAMOND NECKLACE, THE AFFAIR OF THE, a mysterious incident at the court
+of Louis XVI. of France, which involved the queen Marie Antoinette. The
+Parisian jewellers Boehmer and Bassenge had spent some years collecting
+stones for a necklace which they hoped to sell to Madame Du Barry, the
+favourite of Louis XV., and after his death to Marie Antoinette. In 1778
+Louis XVI. proposed to the queen to make her a present of the necklace,
+which cost 1,600,000 livres. But the queen is said to have refused it,
+saying that the money would be better spent equipping a man-of-war.
+According to others, Louis XVI. himself changed his mind. After having
+vainly tried to place the necklace outside of France, the jewellers
+attempted again in 1781 to sell it to Marie Antoinette after the birth
+of the dauphin. It was again refused, but it was evident that the queen
+regretted not being able to acquire it.
+
+At that time there was a personage at the court whom Marie Antoinette
+particularly detested. It was the cardinal Louis de Rohan, formerly
+ambassador at Vienna, whence he had been recalled in 1774, having
+incurred the queen's displeasure by revealing to the empress Maria
+Theresa the frivolous actions of her daughter, a disclosure which
+brought a maternal reprimand, and for having spoken lightly of Maria
+Theresa in a letter of which Marie Antoinette learned the contents.
+After his return to France the cardinal was anxious to regain the favour
+of the queen in order to obtain the position of prime minister. In March
+1784 he entered into relations with a certain Jeanne de St Remy de
+Valois, a descendant of a bastard of Henry II., who after many
+adventures had married a _soi-disant_ comte de Lamotte, and lived on a
+small pension which the king granted her. This adventuress soon gained
+the greatest ascendancy over the cardinal, with whom she had intimate
+relations. She persuaded him that she had been received by the queen and
+enjoyed her favour; and Rohan resolved to use her to regain the queen's
+good will. The comtesse de Lamotte assured the cardinal that she was
+making efforts on his behalf, and soon announced to him that he might
+send his justification to Marie Antoinette. This was the beginning of a
+pretended correspondence between Rohan and the queen, the adventuress
+duly returning replies to Rohan's notes, which she affirmed to come from
+the queen. The tone of the letters became very warm, and the cardinal,
+convinced that Marie Antoinette was in love with him, became ardently
+enamoured of her. He begged the countess to obtain a secret interview
+for him with the queen, and a meeting took place in August 1784 in a
+grove in the garden at Versailles between him and a lady whom the
+cardinal believed to be the queen herself. Rohan offered her a rose, and
+she promised him that she would forget the past. Later a certain Marie
+Lejay (renamed by the comtesse "Baronne Gay d'Oliva," the last word
+being apparently an anagram of Valoi), who resembled Marie Antoinette,
+stated that she had been engaged to play the role of queen in this
+comedy. In any case the countess profited by the cardinal's conviction
+to borrow from him sums of money destined ostensibly for the queen's
+works of charity. Enriched by these, the countess was able to take an
+honourable place in society, and many persons believed her relations
+with Marie Antoinette, of which she boasted openly and unreservedly, to
+be genuine. It is still an unsettled question whether she simply
+mystified people, or whether she was really employed by the queen for
+some unknown purpose, perhaps to ruin the cardinal. In any case the
+jewellers believed in the relations of the countess with the queen, and
+they resolved to use her to sell their necklace. She at first refused
+their commission, then accepted it. On the 21st of January 1785 she
+announced that the queen would buy the necklace, but that not wishing to
+treat directly, she left the affair to a high personage. A little while
+later Rohan came to negotiate the purchase of the famous necklace for
+the 1,600,000 livres, payable in instalments. He said that he was
+authorized by the queen, and showed the jewellers the conditions of the
+bargain approved in the handwriting of Marie Antoinette. The necklace
+was given up. Rohan took it to the countess's house, where a man, in
+whom Rohan believed he recognized a valet of the queen, came to fetch
+it. Madame de Lamotte had told the cardinal that Marie Antoinette would
+make him a sign to indicate her thanks, and Rohan believed that she did
+make him a sign. Whether it was so, or merely chance or illusion, no one
+knows. But it is certain that the cardinal, convinced that he was acting
+for the queen, had engaged the jewellers to thank her; that Boehmer and
+Bassenge, before the sale, in order to be doubly sure, had sent word to
+the queen of the negotiations in her name; that Marie Antoinette had
+allowed the bargain to be concluded, and that after she had received a
+letter of thanks from Boehmer, she had burned it. Meanwhile the "comte
+de Lamotte" appears to have started at once for London, it is said with
+the necklace, which he broke up in order to sell the stones.
+
+When the time came to pay, the comtesse de Lamotte presented the
+cardinal's notes; but these were insufficient, and Boehmer complained to
+the queen, who told him that she had received no necklace and had never
+ordered it. She had the story of the negotiations repeated for her. Then
+followed a _coup de théâtre_. On the 15th of August 1785, Assumption
+day, when the whole court was awaiting the king and queen in order to go
+to the chapel, the cardinal de Rohan, who was preparing to officiate,
+was arrested and taken to the Bastille. He was able, however, to destroy
+the correspondence exchanged, as he thought, with the queen, and it is
+not known whether there was any connivance of the officials, who did not
+prevent this, or not. The comtesse de Lamotte was not arrested until the
+18th of August, after having destroyed her papers. The police set to
+work to find all her accomplices, and arrested the girl Oliva and a
+certain Reteaux de Villette, a friend of the countess, who confessed
+that he had written the letters given to Rohan in the queen's name, and
+had imitated her signature on the conditions of the bargain. The famous
+charlatan Cagliostro was also arrested, but it was recognized that he
+had taken no part in the affair. The cardinal de Rohan accepted the
+parlement of Paris as judges. A sensational trial resulted (May 31,
+1786) in the acquittal of the cardinal, of the girl Oliva and of
+Cagliostro. The comtesse de Lamotte was condemned to be whipped, branded
+and shut up in the Salpetrière. Her husband was condemned, in his
+absence, to the galleys for life. Villette was banished.
+
+Public opinion was much excited by this trial. It is generally believed
+that Marie Antoinette was stainless in the matter, that Rohan was an
+innocent dupe, and that the Lamottes deceived both for their own ends.
+People, however, persisted in the belief that the queen had used the
+countess as an instrument to satisfy her hatred of the cardinal de
+Rohan. Various circumstances fortified this belief, which contributed to
+render Marie Antoinette very unpopular--her disappointment at Rohan's
+acquittal, the fact that he was deprived of his charges and exiled to
+the abbey of la Chaise-Dieu, and finally the escape of the comtesse de
+Lamotte from the Salpetrière, with the connivance, as people believed,
+of the court. The adventuress, having taken refuge abroad, published
+_Mémoires_ in which she accused the queen. Her husband also wrote
+_Mémoires_, and lived until 1831, after having, it is said, received
+subsidies from Louis XVIII.
+
+  See M. Tourneux, _Marie Antoinette devant l'histoire: Essai
+  bibliographique_ (2nd ed., Paris, 1901); Émile Campardon, _Marie
+  Antoinette et le procès du collier_ (Paris, 1863); P. Audebert,
+  _L'Affaire du collier de la reine, d'après la correspondance inédite
+  du chevalier de Pujol_ (Rouen, 1901); F. d'Albini, _Marie Antoinette
+  and the Diamond Necklace from another Point of View_ (London, 1900);
+  Funck-Brentano, _L'Affaire du collier_ (1903); A. Lang, _Historical
+  Mysteries_ (1904). Carlyle's essay on _The Diamond Necklace_ (first
+  published in 1837 in _Fraser's Magazine_) is of historical literary
+  interest.
+
+
+
+
+DIANA, in Roman mythology, an old Italian goddess, in later times
+identified with the Greek Artemis (q.v.). That she was originally an
+independent Italian deity is shown by her name, which is the feminine
+form of Janus (= Dianus). She is essentially the goddess of the moon and
+light generally, and presides over wood, plain and water, the chase and
+war. As the goddess of childbirth, she was known, like Juno, by the name
+of Lucina, the "bringer to light." As the moon-goddess she was also
+identified with Hecate, and invoked as "three-formed" in reference to
+the phases of the moon. Her most celebrated shrine was in a grove at
+Aricia (whence her title of Nemorensis) near the modern lake of Nemi.
+Here she was worshipped side by side with a male deity Virbius, a god of
+the forest and the chase. This Virbius was subsequently identified with
+Hippolytus, the favourite of Artemis, who was said to have been brought
+to life by Aesculapius and conducted by Diana to Aricia (Ovid, _Fasti_,
+iii. 263, vi. 731, _Metam._ xv. 497; Virgil, _Aeneid_, vii. 761). A
+barbarous custom, perhaps reminiscent of human sacrifice once offered to
+her, prevailed in connexion with her ritual here; her priest, called
+_Rex Nemorensis_, who was a runaway slave, was obliged to qualify for
+office by slaying his predecessor in single combat (Strabo v. p. 239;
+Suetonius, _Caligula_, 35). This led to the identification of Diana with
+the Tauric Artemis, whose image was said to have been removed by Orestes
+to the grove of Aricia (see ARICINI).
+
+After the destruction of Alba Longa this grove was for a long time the
+united sanctuary of the neighbouring Latin and Rutulian cities, until at
+last it was extinguished beneath the supremacy of Rome. The festival of
+the goddess was on the ides (13th) of August, the full moon of the hot
+season. She was worshipped with torches, her aid was sought by women
+seeking a happy deliverance in childbirth, and many votive offerings
+have been found on the site. The worship of Diana was brought to Rome by
+Latin plebeians, and hence she was regarded as the protectress of the
+lower classes, and especially of slaves. In accordance with this, her
+most important temple was that on the Aventine, the chief seat of the
+plebeians, founded by Servius Tullius, originally as a sanctuary of the
+Latin league (Dion. Halic. iv. 26). No man was allowed to enter the
+temple, and on the day of its dedication (August 13) the slaves kept
+holiday (Plutarch, _Quaest. Rom._ 100). This Diana was identified with
+the sister of Apollo, and at the secular games she was worshipped simply
+as Artemis. Another celebrated sanctuary of Diana was that on the slopes
+of Mount Tifata near Capua (where she was worshipped under the name of
+Tifatina), a sanctuary specially favoured by Sulla and Vespasian. As
+Noctiluca ("giving light by night") she had a sanctuary on the Palatine
+which was kept illuminated throughout the night (Varro, _L.L._ v. 68).
+On the Nemi priesthood see J. G. Frazer, _Golden Bough_.
+
+
+
+
+DIANA MONKEY, a West African representative of the guenon monkeys taking
+its name, _Cercopithecus diana_, from the presence of a white crescent
+on the forehead; another characteristic feature being the pointed white
+beard. The general colour of the fur is greyish, with a deep tinge of
+chestnut from the middle of the back to the root of the tail. Together
+with _C. neglectus_ of East and Central Africa, _C. ignitus_ of Liberia,
+and _C. roloway_ of the Gold Coast, the diana represents the special
+subgenus of guenons known as _Pogonocebus_. Although the diana monkey is
+commonly seen in menageries, little is known of its habits in the wild
+state.
+
+
+
+
+DIANE DE FRANCE (1538-1619), duchess of Montmorency and Angoulême, was
+the natural daughter of Henry II. of France and a young Piedmontese,
+Filippe Duc. The constable de Montmorency went so far as to assert that
+of all the children of Henry II. Diane was the only one who resembled
+him. Catherine de' Medici was greatly incensed at this affront, and took
+her revenge by having the constable disgraced on the death of Henry II.
+Brantôme is loud in praise of Diane. She was a perfect horsewoman and
+dancer, played several musical instruments, knew Spanish and Italian,
+and "estoit très belle de visage et de taille." Legitimated in 1547, she
+was married in 1553 to Horace Farnese, second son of the duke of Parma,
+but her husband was killed soon afterwards at the siege of Hesdin. In
+order to assure his position, the constable de Montmorency wished to
+marry her to his eldest son, Francis. This was a romantic adventure, for
+Francis had clandestinely married Mademoiselle de Piennes. The constable
+dissolved this union, and after lengthy negotiations obtained the
+dispensation of the pope. On the 3rd of May 1559 Francis married Diane.
+A wise and moderate woman, Diane undoubtedly helped to make Francis de
+Montmorency one of the leaders of the party of the _politiques_. Again a
+widow in 1579, she had some influence at the court of Henry III., and
+negotiated his reconciliation with Henry of Navarre (1588). She retained
+her influence in the reign of Henry IV., conveyed the bodies of
+Catherine de' Medici and Henry III. to St Denis, and died in 1619 at her
+hôtel of Angoulême.
+
+  See Brantôme, ed. by Lalanne, in the _Coll de la société d'histoire de
+  France_, vol. viii. (1875); J. de Thou, _Historia sui temporis..._
+  (1733); Matthieu de Morgues, _Oraison funèbre de Diane de France_
+  (Paris, 1619).
+
+
+
+
+DIANE DE POITIERS (1499-1566), duchess of Valentinois, and mistress of
+Henry II. of France, was the daughter of Jean de Poitiers, seigneur de
+St Vallier, who came of an old family of Dauphiné. In 1515 she married
+Louis de Brézé, grand seneschal of Normandy, by whom she had two
+daughters. She became a widow in 1533, but soon replaced her husband by
+a more illustrious lover, the king's second son, Henry, who became
+dauphin in 1536. Although he was ten years younger than Diane, she
+inspired the young prince with a profound passion, which lasted until
+his death. The accession of Henry II. in 1547 was also the accession of
+Diane: she was virtual queen, while Henry's lawful wife, Catherine de'
+Medici, lived in comparative obscurity. The part Diane played, however,
+must not be exaggerated. More rapacious than ambitious, she concerned
+herself little with government, but devoted her energies chiefly to
+augmenting her income, and providing for her family and friends. Henry
+was the most prodigal of lovers, and gave her all rights over the duchy
+of Valentinois. Although she showed great tact in her dealings with the
+queen, Catherine drove her from the court after Henry's death, and
+forced her to restore the crown jewels and to accept Chaumont in
+exchange for Chenonceaux. Diane retired to her château at Anet, where
+she died in 1566.
+
+Several historians relate that she had been the mistress of Francis I.
+before she became the dauphin's mistress, and that she gave herself to
+the king in order to obtain the pardon of her father, who had been
+condemned to death as an accomplice of the constable de Bourbon. This
+rumour, however, has no serious foundation. Men vied with each other in
+celebrating Diane's beauty, which, if we may judge from her portraits,
+has been slightly exaggerated. She was a healthy, vigorous woman, and,
+by dint of great pains, succeeded in retaining her beauty late into
+life. It is said that even on the coldest mornings she would wash her
+face with well water. Diane was a patroness of the arts. She entrusted
+to Philibert de l'Orme the building of her château at Anet, and it was
+for her that Jean Goujon executed his masterpiece, the statue of Diana,
+now in the Louvre.
+
+  See G. Guiffrey, _Lettres inédites de Diane de Poytiers_ (Paris, 1866)
+  and _Procès criminel de Jehan de Poytiers_ (Paris, 1867); Capefigue,
+  _Diane de Poitiers_ (Paris, 1860); Hay, _Madame Dianne de Poytiers_
+  (London, 1900).
+
+
+
+
+DIAPASON (Gr. [Greek: dia pasôn], through all), a term in music,
+originally for an interval of an octave. The Greek is an abbreviation of
+[Greek: hê dia pasôn chordôn symphônia], a consonance through all the
+tones of the scale. In this sense it is only used now, loosely, for the
+compass of an instrument or voice, or for a harmonious melody. The name
+is given to the two foundation stops of an organ, the open and the
+stopped diapason (see ORGAN), and to a standard of musical pitch, as in
+the French _diapason normal_ (see PITCH, MUSICAL).
+
+
+
+
+DIAPER (derived through the Fr, from the Gr. [Greek: dia], through, and
+[Greek: aspros], white; the derivation from the town of Ypres,
+"d'Ypres," in Belgium is unhistorical, as diapers were known for
+centuries before its existence), the name given to a textile fabric,
+formerly of a rich and costly nature with embroidered ornament, but now
+of linen or cotton, with a simple woven pattern; and particularly
+restricted to small napkins. In architecture, the term "diaper" is given
+to any small pattern of a conventional nature repeated continuously and
+uniformly over a surface; the designs may be purely geometrical, or
+based on floral forms, and in early examples were regulated by the
+process of their textile origin. Subsequently, similar patterns were
+employed in the middle ages for the surface decoration of stone, as in
+Westminster Abbey and Bayeux cathedral in the spandrils of the arcades
+of the choir and nave; also in mural painting, stained glass, incised
+brasses, encaustic tiles, &c. Probably in most cases the pattern was
+copied, so far as the general design is concerned, from the tissues and
+stuffs of Byzantine manufacture, which came over to Europe and were
+highly prized as ecclesiastical vestments.
+
+[Illustration: A B C]
+
+  In its textile use, the term diaper was originally applied to silk
+  patterns of a geometrical pattern; it is now almost exclusively used
+  for diamond patterns made from linen or cotton yarns. An illustration
+  of two patterns of this nature is shown in the figure. The floats of
+  the warp and the weft are mostly in three; indeed the patterns are
+  made from a base weave which is composed entirely of floats of this
+  number. It will be seen that both designs are formed of what may be
+  termed concentric figures--alternately black and white. Pattern B
+  differs from pattern A only in that more of these concentric figures
+  are used for the complete figure. If pattern B, which shows only one
+  unit, were extended, the effect would be similar to A, except for the
+  size of the unit. In A there are four complete units, and hence the
+  pattern appears more striking. Again, the repeating of B would cause
+  the four corner pieces to join and to form a diamond similar to the
+  one in the centre. The two diamonds in B would then alternate
+  diagonally to left and right. Special names are given to certain kinds
+  of diapers, e.g. "bird's-eye," "pheasant's-eye"; these terms indicate,
+  to a certain extent, the size of the complete diamond in the
+  cloth--the smaller kind taking the name "bird's-eye." The size of the
+  pattern on paper has little connexion with the size of the pattern in
+  the cloth, for it is clearly the number of threads and picks per inch
+  which determine the size of the pattern in the cloth from any given
+  design. Although A is larger than what is usually termed the
+  "bird's-eye" pattern, it is evident that it may be made to appear as
+  such, provided that the cloth is fine enough. These designs, although
+  adapted mostly for cloths such as nursery-diapers, for pinafores, &c.,
+  are sometimes used in the production of towels and table-cloths. In
+  the figure, the first pick in A is identical with the first pick in B,
+  and the part C shows how each interweaves with the twenty-four
+  threads.
+
+
+
+
+DIAPHORETICS (from Gr. [Greek: diaphorein], to carry through), the name
+given to those remedies which promote perspiration. In health there is
+constantly taking place an exhalation of watery vapour from the skin, by
+which not only are many of the effete products of nutrition eliminated,
+but the body is kept cool. Under exertion or in a heated atmosphere this
+natural function of the skin is increased, sweating more or less profuse
+follows, and, evaporation going on rapidly over the whole surface,
+little or no rise in the temperature of the body takes place. In many
+forms of disease, such as fevers and inflammatory affections, the
+action of the skin is arrested, and the surface of the body feels harsh
+and dry, while the temperature is greatly elevated. The occurrence of
+perspiration not unfrequently marks a crisis in such diseases, and is in
+general regarded as a favourable event. In some chronic diseases, such
+as diabetes and some cases of Bright's disease, the absence of
+perspiration is a marked feature; while, on the other hand, in many
+wasting diseases, such as phthisis, the action of the skin is increased,
+and copious exhausting sweating occurs. Many means can be used to induce
+perspiration, among the best known being baths, either in the form of
+hot vapour or hot water baths, or in that part of the process of the
+Turkish bath which consists in exposing the body to a dry and hot
+atmosphere. Such measures, particularly if followed by the drinking of
+hot liquids and the wrapping of the body in warm clothing, seldom fail
+to excite copious perspiration. Numerous medicinal substances have the
+same effect.
+
+
+
+
+DIAPHRAGM (Gr. [Greek: diaphragma], a partition). The diaphragm or
+midriff (Anglo-Saxon, _mid_, middle, _hrif_, belly) in human anatomy is
+a large fibro-muscular partition between the cavities of the thorax and
+abdomen; it is convex toward the thorax, concave toward the abdomen, and
+consists of a central tendon and a muscular margin. The _central tendon_
+(q, fig. 1) is trefoil in shape, its leaflets being right, left and
+anterior; of these the right is the largest and the left the smallest.
+The fleshy fibres rise, in front from the back of the xiphoid cartilage
+of the sternum (d), laterally by six serrations, from the inner surfaces
+of the lower six ribs, interdigitating with the transversalis,
+posteriorly from the arcuate ligaments, of which there are five, a pair
+of external, a pair of internal, and a single median one. The _external
+arcuate ligament_ (h) stretches from the tip of the twelfth rib (b) to
+the costal process of the first lumbar vertebra in front of the
+quadratus lumborum muscle (o), the _internal_ and _middle_ are
+continuations of the _crura_ which rise from the ventro-lateral aspects
+of the bodies of the lumbar vertebrae, the right (e) coming from three,
+the left (f) from two. On reaching the level of the twelfth thoracic
+vertebra each crus spreads out into a fan-shaped mass of fibres, of
+which the innermost join their fellows from the opposite crus, in front
+of the aortic opening (k), to form the _middle arcuate ligament_; the
+outer ones (g) arch in front of the psoas muscle (n) to the tip of the
+costal process of the first lumbar vertebra to form the _internal
+arcuate ligament_, while the intermediate ones pass to the central
+tendon. There are three large openings in the diaphragm; the _aortic_
+(k) is behind the middle arcuate ligament and transmits the aorta, the
+vena azygos major, and the thoracic duct. In the right leaflet is an
+opening (sometimes called the _hiatus quadratus_) for the inferior vena
+cava and a branch of the right phrenic nerve (m), while in front and a
+little to the left of the aortic opening is one for the oesophagus and
+the two pneumogastric nerves (l), the left being in front and the right
+behind. The fleshy fibres on each side of this opening act as a
+sphincter. Passing between the xiphoid and costal origins in front are
+the superior epigastric arteries, while the other terminal branches of
+the internal mammaries, the musculo-phrenics, pass through between two
+costal origins.
+
+[Illustration: FIG. 1.--Abdominal Surface of the Diaphragm.]
+
+Through the crura pass the splanchnic nerves, and in addition to these
+the left crus is pierced by the vena azygos minor. The sympathetic
+nerves usually enter the abdomen behind the internal arcuate ligaments.
+The phrenic nerves, which are the main supply of the diaphragm, divide
+before reaching the muscle and pierce it in a number of places to enter
+its abdominal surface, but some of the lower intercostal nerves assist
+in the supply. The last thoracic or subcostal nerves pass behind the
+external arcuate ligament.
+
+For the action of the diaphragm see RESPIRATORY SYSTEM.
+
+  _Embryology._--The diaphragm is at first developed in the neck region
+  of the embryo, and this accounts for the phrenic nerves, which supply
+  it, rising from the fourth and fifth cervical. From the mesoderm on
+  the caudal side of the pericardium is developed the _septum
+  transversum_, and in this the central tendon is formed. The fleshy
+  portion is developed on each side in two parts, an anterior or
+  sterno-costal which is derived from the longitudinal neck musculature,
+  probably the same layer from which the sternothyroid comes, and a
+  spinal part which is a derivative of the transversalis sheet of the
+  trunk. Between these two parts is at one time a gap, the _spino-costal
+  hiatus_, and this is obliterated by the growth of the
+  pleuro-peritoneal membrane, which may occasionally fail to close and
+  so may form the site of a phrenic hernia. With the growth of the body
+  and the development of the lungs the diaphragm shifts its position
+  until it becomes the septum between the thoracic and abdominal
+  cavities. (See A. Keith, "On the Development of the Diaphragm," _Jour.
+  of Anat. and Phys._ vol. 39.) A. Paterson has recorded cases in which
+  the left half of the diaphragm is wanting (_Proceedings_ of the
+  Anatomical Society of Gt. Britain, June 1900; _Jour. of Anat. and
+  Phys._ vol. 34), and occasionally deficiencies are found elsewhere,
+  especially in the sternal portion. For further details see Quain's
+  _Anatomy_, vol. i. (London, 1908).
+
+  _Comparative Anatomy._--A complete diaphragm, separating the thoracic
+  from the abdominal parts of the coelom, is characteristic of the
+  Mammalia; it usually has the human structure and relations except that
+  below the Anthropoids it is separated from the pericardium by the
+  azygous lobe of the lung. In some Mammals, e.g. Echidna and Phocoena,
+  it is entirely muscular. In the Cetacea it is remarkable for its
+  obliquity; its vertebral attachment is much nearer the tail than its
+  sternal or ventral one; this allows a much larger lung space in the
+  dorsal than in the ventral part of the thorax, and may be concerned
+  with the equipoise of the animal. (Otto Müller, "Untersuchungen über
+  die Veränderung, welche die Respirationsorgane der Säugetiere durch
+  die Anpassung an das Leben im Wasser erlitten haben," _Jen. Zeitschr.
+  f. Naturwiss._, 1898, p. 93.) In the Ungulata only one crus is found
+  (Windle and Parsons, "Muscles of the Ungulata," _Proc. Zool. Soc._,
+  1903, p. 287). Below the Mammals incomplete partitions between the
+  pleural and peritoneal cavities are found in Chelonians, Crocodiles
+  and Birds, and also in Amphibians (Xenopus and Pipa).     (F. G. P.)
+
+
+
+
+DIARBEKR[1] (_Kara Amid_ or Black Amid; the Roman _Amida_), the chief
+town of a vilayet of Asiatic Turkey, situated on a basaltic plateau on
+the right bank of the Tigris, which here flows in a deep open valley.
+The town is still surrounded by the masonry walls of black basalt which
+give it the name of _Kara_ or Black Amid; they are well built and
+imposing on the west facing the open country, but almost in ruins where
+they overlook the river. A mass of gardens and orchards cover the slope
+down to the river on the S.W., but there are no suburbs outside the
+walls. The houses are rather crowded but only partially fill the walled
+area. The population numbers 38,000, nearly half being Christian,
+comprising Turks, Kurds, Arabs, Turkomans, Armenians, Chaldeans,
+Jacobites and a few Greeks. The streets are 10 ft. to 15 ft. wide, badly
+paved and dirty; the houses and shops are low, mostly of stone, and some
+of stone and mud. The bazaar is a good one, and gold and silver filigree
+work is made, peculiar in character and design. The cotton industry is
+declining, but manufacture of silk is increasing. Fruit is good and
+abundant as the rich volcanic soil is well watered from the town
+springs. The size of the melons is specially famous. To the south, the
+walls are some 40 ft. high, faced with large cut stone blocks of very
+solid construction, with towers and square bastions rising to 500 ft.
+There are four gates: on the north the Kharput gate, on the west the
+Rum, on the south the Mardin, and on the east the Yeni Kapu or new
+gate. A citadel enclosure stands at the N. E. corner and is now partly
+in ruins, but the interior space is occupied by the government konak.
+The summer climate in the confined space within the town is excessively
+hot and unhealthy. Epidemics of typhus are not unknown, as well as
+ophthalmia. The Diarbekr boil is like the "Aleppo button," lasting a
+long time and leaving a deep scar. Winters are frequently severe but do
+not last long. Snow sometimes lies, and ice is stored for summer use.
+Scorpions noted for the virulence of their poison abound as well as
+horse leeches in the tanks. The town is supplied with water both by
+springs inside the town and by aqueducts from fountains at Ali Punar and
+Hamervat. The principal exports are wool, mohair and copper ore, and
+imports are cotton and woollen goods, indigo, coffee, sugar, petroleum,
+&c.
+
+The Great Mosque, Ulu Jami, formerly a Christian church, occupies the
+site of a Sassanian palace and was built with materials from an older
+palace, probably that of Tigranes II. The remains consist of the façades
+of two palaces 400 ft. apart, each formed by a row of Corinthian columns
+surmounted by an equal number of a Byzantine type. Kufic inscriptions
+run across the fronts under the entablature. The court of the mosque is
+entered by a gateway on which lions and other animals are sculptured.
+The churches of greatest interest are those of SS. Cosmas and Damian
+(Jacobite) and the church of St James (Greek). In the 19th century
+Diarbekr was one of the largest and most flourishing cities of Asia, and
+as a commercial centre it now stands at the meeting-point of several
+important routes. It is at the head of the navigation of the Tigris,
+which is traversed down stream by _keleks_ or rafts supported by
+inflated skins. There is a good road to Aleppo and Alexandretta on the
+Mediterranean, and to Samsun on the Black Sea by Kharput, Malatia and
+Sivas. There are also routes to Mosul and Bitlis.
+
+Diarbekr became a Roman colony in A.D. 230 under the name of Amida, and
+received a Christian bishop in A.D. 325. It was enlarged and
+strengthened by Constantius II., in whose reign it was taken after a
+long siege by Shapur (Sapor) II., king of Persia. The historian Ammianus
+Marcellinus, who took part in the defence, gives a detailed account of
+it. In the later wars between the Persians and Romans it more than once
+changed hands. Though ceded by Jovian to the Persians it again became
+annexed to the Roman empire, and in the reign of Anastasius (A.D. 502)
+was once more taken by the Persians, when 80,000 of its inhabitants were
+slain. It was taken c. 638 by the Arabs, and afterwards passed into the
+hands of the Seljuks and Persians, from whom it was finally captured by
+Selim I. in 1515; and since that date it has remained under Ottoman
+rule. About 2 m. below the town is a masonry bridge over the Tigris; the
+older portion being probably Roman, and the western part, which bears a
+Kufic inscription, being Arab.
+
+The vilayet of Diarbekr extends south from Palu on the Euphrates to
+Mardin and Nisibin on the edge of the Mesopotamian plain, and is divided
+into three sanjaks--Arghana, Diarbekr and Mardin. The headwaters of the
+main arm of the Tigris have their source in the vilayet.
+
+Cereals, cotton, tobacco, rice and silk are produced, but most of the
+fertile lands have been abandoned to semi-nomads, who raise large
+quantities of live stock. The richest portion of the vilayet lies east
+of the capital in the rolling plains watered by tributaries of the
+Tigris. An exceptionally rich copper mine exists at Arghana Maden, but
+it is very imperfectly worked; galena mineral oil and silicious sand are
+also found.     (C. W. W.; F. R. M.)
+
+
+FOOTNOTE:
+
+  [1] From _Diar_, land, and Bekr (i.e. Abu Bekr, the caliph).
+
+
+
+
+DIARRHOEA (from Gr. [Greek: dia], through, [Greek: rheô], flow), an
+excessive looseness of the bowels, a symptom of irritation which may be
+due to various causes, or may be associated with some specific disease.
+The treatment in such latter cases necessarily varies, since the symptom
+itself may be remedial, but in ordinary cases depends on the removal of
+the cause of irritation by the use of aperients, various sedatives being
+also prescribed. In chronic diarrhoea careful attention to the diet is
+necessary.
+
+
+
+
+DIARY, the Lat. _diarium_ (from _dies_, a day), the book in which are
+preserved the daily memoranda regarding events and actions which come
+under the writer's personal observation, or are related to him by
+others. The person who keeps this record is called a diarist. It is not
+necessary that the entries in a diary should be made each day, since
+every life, however full, must contain absolutely empty intervals. But
+it is essential that the entry should be made during the course of the
+day to which it refers. When this has evidently not been done, as in the
+case of Evelyn's diary, there is nevertheless an effort made to give the
+memoranda the effect of being so recorded, and in point of fact, even in
+a case like that of Evelyn, it is probable that what we now read is an
+enlargement of brief notes jotted down on the day cited. When this is
+not approximately the case, the diary is a fraud, for its whole value
+depends on its instantaneous transcript of impressions.
+
+In its primitive form, the diary must always have existed; as soon as
+writing was invented, men and women must have wished to note down, in
+some almanac or journal, memoranda respecting their business, their
+engagements or their adventures. But the literary value of these would
+be extremely insignificant until the spirit of individualism had crept
+in, and human beings began to be interesting to other human beings for
+their own sake. It is not, therefore, until the close of the Renaissance
+that we find diaries beginning to have literary value, although, as the
+study of sociology extends, every scrap of genuine and unaffected record
+of early history possesses an ethical interest. In the 17th century,
+diaries began to be largely written in England, although in most cases
+without any idea of even eventual publication. Sir William Dugdale
+(1605-1686) had certainly no expectation that his slight diary would
+ever see the light. There is no surviving record of a journal kept by
+Clarendon, Richard Baxter, Lucy Hutchinson and other autobiographical
+writers of the middle of the century, but we may take it for granted
+that they possessed some such record, kept from day to day. Bulstrode
+Whitelocke (1605-1675), whose _Memorials of the English Affairs_ covers
+the ground from 1625 to 1660, was a genuine diarist. So was the elder
+George Fox (1624-1690), who kept not merely "a great journal," but "the
+little journal books," and whose work was published in 1694. The famous
+diary of John Evelyn (1620-1706) professes to be the record of seventy
+years, and, although large tracts of it are covered in a very
+perfunctory manner, while in others many of the entries have the air of
+having been written in long after the event, this is a very interesting
+and amusing work; it was not published until 1818. In spite of all its
+imperfections there is a great charm about the diary of Evelyn, and it
+would hold a still higher position in the history of literature than it
+does if it were not overshadowed by what is unquestionably the most
+illustrious of the diaries of the world, that of Samuel Pepys
+(1633-1703). This was begun on the 1st of January 1660 and was carried
+on until the 29th of May 1669. The extraordinary value of Pepys' diary
+consists in its fidelity to the portraiture of its author's character.
+He feigns nothing, conceals nothing, sets nothing down in malice or
+insincerity. He wrote in a form of shorthand intelligible to no one but
+himself, and not a phrase betrays the smallest expectation that any eye
+but his own would ever investigate the pages of his confession. The
+importance of this wonderful document, in fact, lay unsuspected until
+1819, when the Rev. John Smith of Baldock began to decipher the MS. in
+Magdalene College, Cambridge. It was not until 1825 that Lord Braybrooke
+published part of what was only fully edited, under the care of Mr
+Wheatley, in 1893-1896. In the age which succeeded that of Pepys, a
+diary of extraordinary emotional interest was kept by Swift from 1710 to
+1713, and was sent to Ireland in the form of a "Journal to Stella"; it
+is a surprising amalgam of ambition, affection, wit and freakishness.
+John Byrom (1692-1763), the Manchester poet, kept a journal, which was
+published in 1854. The diary of the celebrated dissenting divine, Philip
+Doddridge (1702-1751), was printed in 1829. Of far greater interest are
+the admirably composed and vigorously written journals of John Wesley
+(1703-1791). But the most celebrated work of this kind produced in the
+latter half of the 18th century was the diary of Fanny Burney (Madame
+D'Arblay), published in 1842-1846. It will be perceived that, without
+exception, these works were posthumously published, and the whole
+conception of the diary has been that it should be written for the
+writer alone, or, if for the public, for the public when all prejudice
+shall have passed away and all passion cooled down. Thus, and thus only,
+can the diary be written so as to impress upon its eventual readers a
+sense of its author's perfect sincerity and courage.
+
+Many of the diaries described above were first published in the opening
+years of the 19th century, and it is unquestionable that the interest
+which they awakened in the public led to their imitation. Diaries ceased
+to be rare, but as a rule the specimens which have hitherto appeared
+have not presented much literary interest. Exception must be made in
+favour of the journals of two minor politicians, Charles Greville
+(1794-1865) and Thomas Creevey (1768-1838), whose indiscretions have
+added much to the gaiety of nations; the papers of the former appeared
+in 1874-1887, those of the latter in 1903. The diary of Henry Crabb
+Robinson (1775-1867), printed in 1869, contains excellent biographical
+material. Tom Moore's journal, published in 1856 by Lord John Russell,
+disappointed its readers. But it is probable, if we reason by the
+analogy of the past, that the most curious and original diaries of the
+19th century are still unknown to us, and lie jealously guarded under
+lock and key by the descendants of those who compiled them.
+
+It was natural that the form of the diary should appeal to a people so
+sensitive to social peculiarities and so keen in the observation of them
+as the French. A medieval document of immense value is the diary kept by
+an anonymous _curé_ during the reigns of Charles VI. and Charles VII.
+This _Journal d'un bourgeois de Paris_ was kept from 1409 to 1431, and
+was continued by another hand down to 1449. The marquis de Dangeau
+(1638-1720) kept a diary from 1684 till the year of his death; this
+although dull, and as Saint-Simon said "of an insipidity to make you
+sick," is an inexhaustible storehouse of facts about the reign of Louis
+XIV. Saint-Simon's own brilliant memoirs, written from 1691 to 1723, may
+be considered as a sort of diary. The lawyer, Edmond Barbier
+(1689-1771), wrote a journal of the anecdotes and little facts which
+came to his knowledge from 1718 to 1762. The studious care which he took
+to be correct, and his manifest candour, give a singular value to
+Barbier's record; his diary was not printed at all until 1847, nor, in
+its entirety, until 1857. The song-writer, Charles Collé (1709-1783),
+kept a _journal historique_ from 1758 to 1782; it is full of vivacity,
+but very scandalous and spiteful. It saw the light in 1805, and
+surprised those to whom Collé, in his lifetime, had seemed the most
+placid and good-natured of men. Petit de Bachaumont (1690-1770) had
+access to remarkable sources of information, and his _Mémoires secrets_
+(a diary the publication of which began in 1762 and was continued after
+Bachaumont's death, until 1787, by other persons) contains a valuable
+mass of documents. The marquis d'Argenson (1694-1757) kept a diary, of
+which a comparatively full text was first published in 1859. In recent
+times the posthumous publication of the diaries of the Russian artist,
+Marie Bashkirtseff (1860-1884), produced a great sensation in 1887, and
+revealed a most remarkable temperament. The brothers Jules and Edmond de
+Goncourt kept a very minute diary of all that occurred around them in
+artistic and literary Paris; after the death of Jules, in 1870, this was
+continued by Edmond, who published the three first volumes in 1888. The
+publication of this work was continued, and it produced no little
+scandal. It is excessively ill-natured in parts, but of its vivid
+picturesqueness, and of its general accuracy as a transcript of
+conversation, there can be no two opinions.     (E. G.)
+
+
+
+
+DIASPORE, a native aluminium hydroxide, AlO(OH), crystallizing in the
+orthorhombic system and isomorphous with göthite and manganite. It
+occurs sometimes as flattened crystals, but usually as lamellar or scaly
+masses, the flattened surface being a direction of perfect cleavage on
+which the lustre is markedly pearly in character. It is colourless or
+greyish-white, yellowish, sometimes violet in colour, and varies from
+translucent to transparent. It may be readily distinguished from other
+colourless transparent minerals, with a perfect cleavage and pearly
+lustre--mica, talc, brucite, gypsum--by its greater hardness of 6½-7.
+The specific gravity is 3.4. When heated before the blowpipe it
+decrepitates violently, breaking up into white pearly scales; it was
+because of this property that the mineral was named diaspore by R. J.
+Hauy in 1801, from [Greek: diaspeirein], "to scatter." The mineral
+occurs as an alteration product of corundum or emery, and is found in
+granular limestone and other crystalline rocks. Well-developed crystals
+are found in the emery deposits of the Urals and at Chester,
+Massachusetts, and in kaolin at Schemnitz in Hungary. If obtainable in
+large quantity it would be of economic importance as a source of
+alumina.     (L. J. S)
+
+
+
+
+DIASTYLE (from Gr. [Greek: dia], through, and [Greek: stylos], column),
+in architecture, a term used to designate an intercolumniation of three
+or four diameters.
+
+
+
+
+DIATOMACEAE. For the knowledge we possess of these beautiful plants, so
+minute as to be undiscernible by our unaided vision, we are indebted to
+the assistance of the microscope. It was not till towards the close of
+the 18th century that the first known forms of this group were
+discovered by O. F. Muller. And so slow was the process of discovery in
+this field of scientific research that in the course of half a century,
+when Agardh published his _Systema algarum_ in 1824, only forty-nine
+species included under eight genera had been described. Since that time,
+however, with modern microscopes and microscopic methods, eminent
+botanists in all parts of the civilized world have studied these minute
+plants, with the result that the number of known genera and species has
+been greatly increased. Over 10,000 species of diatoms have been
+described, and about 1200 species and numerous varieties occur in the
+fresh waters and on the coasts of Great Britain and Ireland. Rabenhorst,
+in the index to his _Flora Europaea algarum_ (1864) enumerated about
+4000 forms which had up to that time been discovered throughout the
+continent of Europe.
+
+[Illustration: FIG. 1. A and B, _Melosira arenaria._ C-E, _Melosira
+varians._ E, showing formation of auxospore.]
+
+[Illustration: FIG. 2.--_Synedra Ulna._]
+
+The diatoms are more commonly known among systematic botanists as the
+Bacillarieae, particularly on the continent of Europe, and although such
+an immense number of very diverse forms are included in it, the group as
+a whole exhibits a remarkable uniformity of structure. The Bacillarieae
+is one of the large groups of Algae, placed by some in close proximity
+to the Conjugatae and by others as an order of the Brown Algae (or
+Phaeophyceae), but their characters are so distinctive and their
+structure is so uniform as to warrant the separation of the diatoms as a
+distinct class. The affinities of the group are doubtful.
+
+The diatoms exhibit great variety of form. While some species are
+circular and more or less disk-shaped, others are oval in outline. Some
+are linear, as _Synedra Ulna_ (fig. 2), others more or less crescentic;
+others again are cuneate, as _Podosphenia Lyngbyii_ (fig. 3); some few
+have a sigmoid outline, as _Pleurosigma balticum_ (fig. 4); but the
+prevailing forms are naviculoid, as in the large family Naviculaceae, of
+which the genus _Navicula_ embraces upwards of 1000 species. They vary
+also in their modes of growth,--some being free-floating, others
+attached to foreign bodies by simple or branched gelatinous stalks,
+which in some species are short and thick, while in others they are long
+and slender. In some genera the forms are simple, while in others the
+frustules are connected together in ribbon-like filaments, or form, as
+in other cases, zigzag chains. In some genera the individuals are naked,
+while in many others they are enclosed in a more or less definite
+gelatinous investment. The conditions necessary to their growth are
+moisture and light. Wherever these circumstances coexist, diatomaceous
+forms will almost invariably be found. They occur mixed with other
+organisms on the surface of moist rocks; in streamlets and pools, they
+form a brownish stratum on the surface of the mud, or cover the stems
+and leaves of water plants or floating twigs with a furry investment.
+Marine forms are usually attached to various sea-weeds, and many are
+found in the stomachs of molluscs, holothurians, ascidians and other
+denizens of the ocean. The fresh-water forms are specifically distinct
+from those incidental to salt or brackish water,--fresh-water species,
+however, are sometimes carried some distance into the sea by the force
+of the current, and in tidal rivers marine forms are carried up by the
+force of the tide. Some notion may be formed of the extreme minuteness
+of these forms from the fact that one the length of which is 1/200th of
+an inch may be considered as beyond the medium size. Some few, indeed,
+are much larger, but by far the greater proportion are of very much
+smaller dimensions.
+
+[Illustration: FIG. 3.--_Podosphenia Lyngbyii._]
+
+[Illustration: FIG. 4.--_Pleurosigma balticum._]
+
+[Illustration: FIG. 5. A-C, _Tetracyclus lacustris._ D and E,
+_Tabellaria fenestrata._ F and G, _Tabellaria flocculosa._]
+
+Diatoms are unicellular plants distinguished from kindred forms by the
+fact of having their soft vegetative part covered by a siliceous case.
+Each individual is known as a frustule, and the cell-wall consists of
+two similar valves nearly parallel to each other, each valve being
+furnished with a rim (or connecting-band) projecting from it at a right
+angle.
+
+One of these valves with its rim is slightly smaller than the other,
+the smaller fitting into the larger pretty much as a pill-box fits into
+its cover. This peculiarity of structure affords ample scope for the
+growth of the protoplasmic cell-contents, for as the latter increase in
+volume the siliceous valves are pushed out, and their corresponding
+siliceous rims become broader. The connecting-bands although closely
+fitting their respective valves are distinct from them, and together the
+two bands form the girdle.
+
+An individual diatom is usually described from two aspects, one in which
+the surface of the valve is exposed to view--the valve view, and one in
+which the girdle side is exposed--the girdle view. The valves are thin
+and transparent, convex on the outside, and generally ornamented with a
+variety of sculptured markings. These sculptures often present the
+aspect of striae across the face of the valve, and the best lenses have
+shown them to consist of a series of small cavities within the siliceous
+wall of the cell. The valves of some of the marine genera exhibit a
+beautiful areolated structure due to the presence of larger chambers
+within the siliceous cell-wall. Many diatoms possess thickenings of the
+cell-wall, visible in the valve view, in the centre of the valve and at
+each extremity. These thickenings are known as the nodules, and they are
+generally connected by a long median line, the raphe, which is a cleft
+in the siliceous valve, extending at least some part of its length.
+
+The protoplasmic contents of this siliceous box-like unicell are very
+similar to the contents of many other algal cells. There is a living
+protoplasmic layer or primordial utricle, connected either by two broad
+bands or by a number of anastomosing threads with a central mass of
+protoplasm in which the nucleus is embedded. The greater part of the
+cavity of the cell is occupied by one or several fluid vacuoles. The
+characteristic brown colour of diatoms is due to the presence of
+chromatophores embedded in the lining layer of protoplasm. In number and
+form these chromatophores are variable. They contain chlorophyll, but
+the green colour is masked by the presence of diatomin, a brown pigment
+which resembles that which occurs in the Brown Algae or Phaeophyceae.
+The chromatophores contain a variable number of pyrenoids, colourless
+proteid bodies of a crystalloidal character.
+
+One of the first phenomena which comes under the notice of the observer
+is the extraordinary power of motion with which the frustules are
+endowed. Some species move slowly backwards and forwards in pretty much
+the same line, but in the case of _Bacillaria paradoxa_ the motion is
+very rapid, the frustules darting through the water in a zigzag course.
+To account for this motion various theories have been suggested, none of
+which appear to be altogether satisfactory. There is little doubt that
+the movements are connected with the raphe, and in some diatoms there is
+much evidence to prove that they are due to an exudation of mucilage.
+
+_Classification._--The most natural system of classification of the
+Bacillarieae is the one put forward by Schütt (1896), and since
+generally followed by systematists. He separates them into two primary
+divisions, the 'Centricae' and the 'Pennatae.' The former includes all
+those diatoms which in the valve view possess a radial symmetry around a
+central point, and which are destitute of a raphe (or a pseudoraphe).
+The latter includes those which are zygomorphic or otherwise irregular,
+and in which the valve view is generally boat-shaped or needle-shaped,
+with the markings arranged in a sagittal manner on each side of a raphe
+or pseudoraphe.
+
+_Reproduction._--In the Diatomaceae, as well as in the Desmidieae, the
+ordinary mode of increase is by simple cell-division. The cell-contents
+within the enclosure of the siliceous case separate into two distinct
+masses. As these two daughter-masses become more and more developed, the
+valves of the mother-cell are pushed more and more widely apart. A new
+siliceous valve is secreted by each of the two masses on the side
+opposite to the original valve, the new valves being situated within the
+girdle of the original frustule. When this process has been completed
+the girdle of the mother frustule gives way, and two distinct frustules
+are formed, the siliceous valves in each of these new frustules being
+one of the valves of the mother-cell, and a newly formed valve similar
+and more or less parallel to it.
+
+During the life of the plant this process of self-division is continued
+with an almost incredible rapidity. On this subject the observation of
+Professor William Smith, writing in 1853, is worthy of special
+notice:--"I have been unable to ascertain the time occupied in a single
+act of self-division, but supposing it to be completed in twenty-four
+hours we should have, as the progeny of a single frustule, the amazing
+number of 1,000,000,000 in a single month, a circumstance which will in
+some degree explain the sudden, or at least rapid, appearance of these
+organisms in localities where they were a short time previously either
+unrecognized or sparingly diffused" (_British Diatomaceae_, vol. i. p.
+25).
+
+[Illustration: FIG. 6.--Formation of Auxospores.
+
+  A. _Navicula limosa._
+  B. _Achnanthes flexella._
+  C. _Navicula Amphisbaena._
+  D. _Navicula viridis._]
+
+Individual diatoms when once produced by cell-division are incapable of
+any increase in size owing to the rigidity of their siliceous
+cell-walls, and since the new valves are always formed _within_ the
+girdle of the old ones, it would follow that every succeeding generation
+is reduced in size by the thickness of the girdle. In some diatoms,
+however, this is not strictly true as daughter-cells are sometimes
+produced of larger size than the parent-cells. Thus, the reduction in
+size of the individuals is not always proportionate to the number of
+cell-divisions.
+
+On the diminution in size having reached a limit in any species, the
+maximum size is regained by the formation of an auxospore. There are
+five known methods of reproduction by auxospores, but it is unnecessary
+here to enter into details of these methods. Suffice it to say that a
+normal auxospore is produced by the conjugation of two parent-cells, its
+distinguishing feature being a rejuvenescence accompanied by a marked
+increase in size. These auxospores formed without conjugation are
+parthenogenetic.
+
+_Mode of Preparation._--The Diatomaceae are usually gathered in small
+bottles, and special care should be taken to collect them as free as
+possible from extraneous matter. A small portion having been examined
+under the microscope, should the gathering be thought worthy of
+preservation, some of the material is boiled in acid for the purpose of
+cleaning it. The acids usually employed are hydrochloric, nitric or
+sulphuric, according as circumstances require. When the operator
+considers that by this process all foreign matter has been eliminated,
+the residuum is put into a precipitating jar of a conical shape, broader
+at the bottom than at the top, and covered to the brim with filtered or
+distilled water. When the diatoms have settled in the bottom of the jar,
+the supernatant fluid is carefully removed by a syringe or some similar
+instrument, so that the sediment be not disturbed. The jar is again
+filled with water, and the process repeated till the acid has been
+completely removed. It is desirable afterwards to boil the sediment for
+a short time with supercarbonate of soda, the alkali being removed in
+the same manner as the acid. A small portion may then be placed with a
+pipette upon a slip of glass, and, when the moisture has been thoroughly
+evaporated, the film that remains should be covered with dilute Canada
+balsam, and, a thin glass cover having been gently laid over the balsam,
+the preparation should be laid aside for a short time to harden, and
+then is ready for observation.
+
+_General Remarks._--Diatoms are most abundant in cold latitudes, having
+a general preference for cold water. In the pelagic waters of lakes and
+of the oceans they are often very abundant, and in the cold waters of
+the Arctic and Antarctic Oceans they exist in prodigious numbers. They
+thus form a large proportion of both the marine and the fresh-water
+plankton.
+
+Large numbers of fossil diatoms are known. Not only are these minute
+plants assisting at the present time in the accumulation of oceanic and
+lake deposits, but in former ages they have been sufficiently active to
+give rise to considerable deposits of diatomaceous earths. When the
+plant has fulfilled its natural course the siliceous covering sinks to
+the bottom of the water in which it had lived, and there forms part of
+the sediment. When in the process of ages, as it has often happened, the
+accumulated sediment has been hardened into solid rock, the siliceous
+frustules of the diatoms remain unaltered, and, if the rock be
+disintegrated by natural or artificial means, may be removed from the
+enveloping matrix and subjected to examination under the microscope. The
+forms found may from their character help in some degree to illustrate
+the conditions under which the stratum of rock had been originally
+deposited. These earths are generally of a white or grey colour. Some of
+them are hard, but most are soft and friable. Many of them are of
+economic importance, being used as polishing powders ("Tripoli"), as
+absorbents for nitroglycerin in the manufacture of dynamite
+("Kieselguhr"), as a dentifrice, and more recently they have been used
+to a large extent in the manufacture of non-conducting and sound-proof
+materials. Most of these diatomaceous earths are associated with rocks
+of Tertiary formations, although it is generally regarded that the
+earliest appearance of diatoms is in the Upper Cretaceous (chalk).
+
+Vast deposits of Diatomaceous earths have been discovered in various
+parts of the world,--some the deposit of fresh, others of salt water. Of
+these deposits the most remarkable for extent, as well as for the number
+and beauty of the species contained in it, is that of Richmond, in
+Virginia, one of the United States of America. It extends for many
+miles, and is in some places at least 40 ft. deep. It is a remarkable
+fact that though the generations of a diatom in the space of a few
+months far exceed in number the generation of man during the period
+usually assigned to the existence of the race, the fossil genera and
+species are in most respects to the most minute details identical with
+the numerous living representatives of their class.
+     (E. O'M.; G. S. W.*)
+
+
+
+
+DIAULOS (from Gr. [Greek: di-], double, and [Greek: aulos], pipe), in
+architecture, the peristyle round the great court of the palaestra,
+described by Vitruvius (v. II), which measured two stadia (1200 ft.) in
+length; on the south side this peristyle had two rows of columns, so
+that in stormy weather the rain might not be driven into the inner part.
+The word was also used in ancient Greece for a foot-race of twice the
+usual length.
+
+
+
+
+DIAVOLO, FRA (1771-1806), the popular name given to a famous Italian
+brigand associated with the political revolutions of southern Italy at
+the time of the French invasion. His real name was Michele Pezza, and he
+was born of low parentage at Itri; he had committed many murders and
+robberies in the Terra di Lavoro, but by good luck combined with
+audacity he always escaped capture, whence his name of Fra Diavolo,
+popular superstition having invested him with the characters of a monk
+and a demon, and it seems that at one time he actually was a monk. When
+the kingdom of Naples was overrun by the French and the Parthenopaean
+Republic established (1799), Cardinal Ruffo, acting on behalf of the
+Bourbon king Ferdinand IV., who had fled to Sicily, undertook the
+reconquest of the country, and for this purpose he raised bands of
+peasants, gaol-birds, brigands, &c., under the name of Sanfedisti or
+_bande della Santa Fede_ ("bands of the Holy Faith"). Fra Diavolo was
+made leader of one of them, and waged untiring war against the French
+troops, cutting off isolated detachments and murdering stragglers and
+couriers. Owing to his unrivalled knowledge of the country, he succeeded
+in interrupting the enemy's communications between Rome and Naples. But
+although, like his fellow-brigands under Ruffo, he styled himself "the
+faithful servant and subject of His Sicilian Majesty," wore a military
+uniform and held military rank, and was even created duke of Cassano,
+his atrocities were worthy of a bandit chief. On one occasion he threw
+some of his prisoners, men, women and children, over a precipice, and
+on another he had a party of seventy shot. His excesses while at Albano
+were such that the Neapolitan general Naselli had him arrested and
+imprisoned in the castle of St Angelo, but he was liberated soon after.
+When Joseph Bonaparte was made king of Naples, extraordinary tribunals
+were established to suppress brigandage, and a price was put on Fra
+Diavolo's head. After spreading terror through Calabria, he crossed over
+to Sicily, where he concerted further attacks on the French. He returned
+to the mainland at the head of 200 convicts, and committed further
+excesses in the Terra di Lavoro; but the French troops were everywhere
+on the alert to capture him and he had to take refuge in the woods of
+Lenola. For two months he evaded his pursuers, but at length, hungry and
+ill, he went in disguise to the village of Baronissi, where he was
+recognized and arrested, tried by an extraordinary tribunal, condemned
+to death and shot. In his last moments he cursed both the Bourbons and
+Admiral Sir Sidney Smith for having induced him to engage in this
+reckless adventure (1806). Although his cruelty was abominable, he was
+not altogether without generosity, and by his courage and audacity he
+acquired a certain romantic popularity. His name has gained a world-wide
+celebrity as the title of a famous opera by Auber.
+
+  The best known account of Fra Diavolo is in Pietro Colletta's _Storia
+  del reame di Napoli_ (2nd ed., Florence, 1848); B. Amante's _Fra
+  Diavolo e il suo tempo_ (Florence, 1904) is an attempted
+  rehabilitation; but A. Luzio, whose account in _Profili e bozzetti
+  storici_ (Milan, 1906) gives the latest information on the subject,
+  has demolished Amante's arguments.     (L. V.*)
+
+
+
+
+DIAZ, NARCISSE VIRGILIO (1808-1876), French painter, was born in
+Bordeaux of Spanish parents, on the 25th of August 1808. At first a
+figure-painter who indulged in strong colour, in his later life Diaz
+became a painter of the forest and a "tone artist" of the first order.
+He spent much time at Barbizon; and although he is the least exalted of
+the half-dozen great artists who are usually grouped round that name, he
+sometimes produced works of the highest quality. At the age of ten Diaz
+became an orphan, and misfortune dogged his earlier years. His foot was
+bitten by a reptile in Meudon wood, near Sèvres, where he had been taken
+to live with some friends of his mother. The bite was badly dressed, and
+ultimately it cost him his leg. Afterwards his wooden stump became
+famous. At fifteen he entered the studios at Sèvres, where the
+decoration of porcelain occupied him; but tiring of the restraint of
+fixed hours, he took to painting Eastern figures dressed in richly
+coloured garments. Turks and Oriental scenes attracted him, and many
+brilliant gems remain of this period. About 1831 Diaz encountered
+Théodore Rousseau, for whom he entertained a great veneration, although
+Rousseau was four years his junior; but it was not until ten years later
+that the remarkable incident took place of Rousseau teaching Diaz to
+paint trees. At Fontainebleau Diaz found Rousseau painting his wonderful
+forest pictures, and determined to paint in the same way if possible.
+Rousseau, then in poor health, worried at home, and embittered against
+the world, was difficult to approach. Diaz followed him surreptitiously
+to the forest,--wooden leg not hindering,--and he dodged round after the
+painter, trying to observe his method of work. After a time Diaz found a
+way to become friendly with Rousseau, and revealed his anxiety to
+understand his painting. Rousseau was touched with the passionate words
+of admiration, and finally taught Diaz all he knew. Diaz exhibited many
+pictures at the Paris Salon, and was decorated in 1851. During the
+Franco-German War he went to Brussels. After 1871 he became fashionable,
+his works gradually rose in the estimation of collectors, and he worked
+constantly and successfully. In 1876 he caught cold at his son's grave,
+and on the 18th of November of that year he died at Mentone, whither he
+had gone to recruit his health. Diaz's finest pictures are his forest
+scenes and storms, and it is on these, and not on his pretty figures,
+that his fame is likely to rest. There are several fairly good examples
+of the master in the Louvre, and three small figure pictures in the
+Wallace collection, Hertford House. Perhaps the most notable of Diaz's
+works are "La Fée aux Perles" (1857), in the Louvre; "Sunset in the
+Forest" (1868); "The Storm," and "The Forest of Fontainebleau" (1870)
+at Leeds. Diaz had no well-known pupils, but Léon Richet followed
+markedly his methods of tree-painting, and J. F. Millet at one period
+painted small figures in avowed imitation of Diaz's then popular
+subjects.
+
+  See A. Hustin, _Les Artistes célèbres: Diaz_ (Paris); D. Croal
+  Thomson, _The Barbizon School of Painters_ (London, 1890); J. W.
+  Mollett, _Diaz_ (London, 1890); J. Claretie, _Peintres et sculpteurs
+  contemporains: Diaz_ (Paris, 1882); Albert Wolff, _La Capitale de
+  l'art: Narcisse Diaz_ (Paris, 1886); Ph. Burty, _Maîtres et
+  petit-maîtres: N. Diaz_ (Paris, 1877). (D.C.T.)
+
+
+
+
+DIAZ, PORFIRIO (1830-   ), president of the republic of Mexico (q.v.),
+was born in the southern state of Oaxaca, on the 15th of September 1830.
+His father was an innkeeper in the little capital of that province, and
+died three years after the birth of Porfirio, leaving a family of seven
+children. The boy, who had Indian blood in his veins, was educated for
+the Catholic Church, a body having immense influence in the country at
+that time and ordering and controlling revolutions by the strength of
+their filled coffers. Arrived at the age of sixteen Porfirio Diaz threw
+off the authority of the priests. Fired with enthusiasm by stories told
+by the revolutionary soldiers continually passing through Oaxaca, and
+hearing about the war with the United States, a year later he determined
+to set out for Mexico city and join the National Guard. There being no
+trains, and he being too poor to ride, he walked the greater part of the
+250 m., but arrived there too late, as the treaty of Guadalupe-Hidalgo
+(1848) had been already signed, and Texas finally ceded to the United
+States. Thus his entering the army was for the time defeated. Thereupon
+he returned to his native town and began studying law. He took pupils in
+order to pay his own fees at the Law Institute, and help his mother. At
+this time he came under the notice and influence of Don Marcos Pérez and
+Benito Juárez, the first a judge, the second a governor of the state of
+Oaxaca, and soon to become famous as the deliverer of Mexico from the
+priesthood (War of Reform). Diaz continued in his native town until
+1854, when, refusing to vote for the dictator, Santa Anna, he was stung
+by a taunt of cowardice, and hastily pushing his way to the voting
+place, he recorded his vote in favour of Alvarez and the revolutionists.
+Orders were given for his arrest, but seizing a rifle and mounting a
+horse he placed himself at the head of a few revolting peasants, and
+from that moment became one of the leading spirits in that long struggle
+for reform, known as the War of Reform, which, under the leadership of
+Juárez, followed the overthrow of Santa Anna. Promotion succeeded
+promotion, as Diaz led his troops from victory to victory, amid great
+privations and difficulties. He was made captain (1856),
+lieutenant-colonel and colonel (1859), brigadier-general (1861), and
+general of division for the army (1863). Closely following on civil war,
+political strife, open rebellion and the great War of Reform, came the
+French invasion of 1862, and the landing of the emperor Maximilian in
+1864. From the moment the French disclosed their intentions of settling
+in Mexico in 1862, Diaz took a prominent part against the foreign
+invasion. He was twice seriously wounded, imprisoned on three different
+occasions, had two hairbreadth escapes, and took part in many daring
+engagements. So important a personage did he become that both Marshal
+Bazaine and the emperor Maximilian made overtures to him. At the time of
+Maximilian's death (with which Diaz personally had nothing to do) he was
+carrying on the siege of Mexico city, which ended in the surrender of
+the town two days after the emperor was shot at Quérétaro between his
+two leading generals. Diaz at once set to work to pay up arrears due to
+his soldiers, proclaimed death as the penalty of plunder and theft, and
+in the few weeks that followed showed his great administrative powers,
+the officers as well as the rank and file receiving arrears of pay. On
+the very day that he occupied Mexico city, the great commander of the
+army of the east, to everyone's surprise, sent in his resignation. He
+was, indeed, appointed to the command of the second division of the army
+by President Juárez in his military reorganization, but Diaz, seeing men
+who had given great and loyal service to the state dismissed from their
+positions in the government, and disgusted at this course, retired to
+the little city of Oaxaca; there he lived, helping in the
+reorganization of the army but taking no active part in the government
+until 1871.
+
+On Juárez' death Lerdo succeeded as president, in 1872. His term of
+office again brought discord, and when it was known that he was
+attempting to be re-elected in 1876, the storm broke. Diaz came from
+retirement, took up the leadership against Lerdo, and after desperate
+struggles and a daring escape finally made a triumphal entry into Mexico
+city on the 24th of November 1876, as provisional president, quickly
+followed by the full presidentship. His term of office marks a prominent
+change in the history of Mexico; from that date he at once forged ahead
+with financial and political reform, the scrupulous settlement of all
+national debts, the welding together of the peoples and tribes (there
+are 150 different Indian tribes) of his country, the establishment of
+railroads and telegraphs, and all this in a land which had been upheaved
+for a century with revolutions and bloodshed, and which had had
+fifty-two dictators, presidents and rulers in fifty-nine years. In 1880
+Diaz was succeeded by Gonzalez, the former minister of war, for four
+years (owing to the limit of the presidential office), but in 1884 he
+was unanimously re-elected. The government having set aside the
+above-mentioned limitation, Diaz was continually re-elected to the
+presidency. He married twice and had a son and two daughters. His gifted
+second wife (Carmelita), very popular in Mexico, was many years younger
+than himself. King Edward VII. made him an honorary grand commander of
+the Bath in June 1906, in recognition of his wonderful administration as
+perpetual president for over a quarter of a century.
+
+  See also Mrs Alec Tweedie, _Porfirio Diaz, Seven Times President of
+  Mexico_ (1906), and _Mexico as I saw it_ (1901); Dr Noll, _From Empire
+  to Republic_ (1890); Lieut. Seaton Schroeder, _Fall of Maximilian's
+  Empire_ (New York, 1887); R. de Z. Enriquez, _P. Diaz_ (1908); and an
+  article by Percy Martin in _Quarterly Review_ for October 1909.
+       (E. A. T.)
+
+
+
+
+DIAZ DE NOVAES, BARTHOLOMEU (fl. 1481-1500), Portuguese explorer,
+discoverer of the Cape of Good Hope, was probably a kinsman of João
+Diaz, one of the first Portuguese to round Cape Bojador (1434), and of
+Diniz Diaz, the discoverer of Cape Verde (1445). In 1478 a Bartholomeu
+Diaz, probably identical with the discoverer, was exempted from certain
+customary payments on ivory brought from the Guinea coast. In 1481 he
+commanded one of the vessels sent by King John II. under Diogo
+d'Azambuja to the Gold Coast. In 1486 he seems to have been a cavalier
+of the king's household, and superintendent of the royal warehouses; on
+the 10th of October in this year he received an annuity of 6000 reis
+from King John for "services to come"; and some time after this
+(probably about July or August 1487, rather than July 1486, the
+traditional date) he left Lisbon with three ships to carry on the work
+of African exploration so greatly advanced by Diogo Cão (1482-1486).
+Passing Cão's farthest point near Cape Cross (in the modern German
+South-west Africa and) in 21° 50´ S., he erected a pillar on what is now
+known as Diaz Point, south of Angra Pequena or Lüderitz Bay, in 26° 38´
+S.; of this fragments still exist. From this point (according to De
+Barros) Diaz ran thirteen days southwards before strong winds, which
+freshened to dangerous stormy weather, in a comparatively high southern
+latitude, considerably south of the Cape. When the storm subsided the
+Portuguese stood east; and failing, after several days' search, to find
+land, turned north, and so struck the south coast of Cape Colony at
+Mossel Bay (Diaz' Bahia dos Vaqueiros), half way between the Cape of
+Good Hope and Port Elizabeth (February 3, 1488). Thence they coasted
+eastward, passing Algoa Bay (Diaz' Bahia da Roca), erecting pillars (or
+perhaps wooden crosses), it is said, on one of the islands in this bay
+and at or near Cape Padrone farther east; of these no traces remain. The
+officers and men now began to insist on return, and Diaz could only
+persuade them to go as far as the estuary of the Great Fish River (Diaz'
+Rio do Iffante, so named from his colleague, Captain João Iffante).
+Here, however, half way between Port Elizabeth and East London (and
+indeed from Cape Padrone), the north-easterly trend of the coast became
+unmistakable; the way round Africa had been laid open. On his return
+Diaz perhaps named Cape Agulhas after St Brandan; while on the
+southernmost projection of the modern Cape peninsula, whose remarkable
+highlands (Table Mountain, &c.) doubtless impressed him as the practical
+termination of the continent, he bestowed, says De Barros, the name of
+Cape of Storms (_Cabo Tormentoso_) in memory of the storms he had
+experienced in these far southern waters; this name (in the ordinary
+tradition) was changed by King John to that of Good Hope (_Cabo da Boa
+Esperança_). Some excellent authorities, however, make Diaz himself give
+the Cape its present name. Hard by this "so many ages unknown
+promontory" the explorer probably erected his last pillar. After
+touching at the Ilha do Principe (Prince's Island, south-west of the
+Cameroons) as well as at the Gold Coast, he appeared at Lisbon in
+December 1488. He had discovered 1260 m. of hitherto unknown coast; and
+his voyage, taken with the letters soon afterwards received from Pero de
+Covilhão (who by way of Cairo and Aden had reached Malabar on one side
+and the "Zanzibar coast" on the other as far south as Sofala, in
+1487-1488) was rightly considered to have solved the question of an
+ocean route round Africa to the Indies and other lands of South and East
+Asia.
+
+No record has yet been found of any adequate reward for Diaz: on the
+contrary, when the great Indian expedition was being prepared (for Vasco
+da Gama's future leadership) Bartolomeu only superintended the building
+and outfit of the ships; when the fleet sailed in 1497, he only
+accompanied da Gama to the Cape Verde Islands, and after this was
+ordered to El Mina on the Gold Coast. On Cabral's voyage of 1500 he was
+indeed permitted to take part in the discovery of Brazil (April 22), and
+thence should have helped to guide the fleet to India; but he perished
+in a great storm off his own Cabo Tormentoso. Like Moses, as Galvano
+says, he was allowed to see the Promised Land, but not to enter in.
+
+  See João de Barros, _Asia_, Dec. I. bk. iii. ch. 4; Duarte Pacheco
+  Pereira, _Esmeraldo de situ orbis_, esp. pp. 15, 90, 92, 94 and
+  Raphael Bastos's introduction to the edition of 1892 (Pacheco met
+  Diaz, returning from his great voyage, at the Ilha do Principe); a
+  marginal note, probably by Christopher Columbus himself, on fol. 13 of
+  a copy of Pierre d'Ailly's _Imago mundi_, now in the Colombina at
+  Seville (the writer of this note fixes Diaz's return to Lisbon,
+  December 1488, and says he was present at Diaz's interview with the
+  king of Portugal, when the explorer described his voyage and showed
+  his route upon the chart he had kept); a similar but briefer note in a
+  copy of Pope Pius II.'s _Historia rerum ubique gestarum_, from the
+  same hand; the _Roteiro_ of Vasco da Gama's First Voyage (_Journal of
+  the First Voyage of ... Da Gama_, Hakluyt Soc., ed. E. G. Ravenstein
+  (1898), pp. 9, 14); Ramusio, _Navigationi_ (3rd ed.), vol. i. fol.
+  144; Castanheda, _Historia_, bk. i. ch. 1; Galvano, _Descobrimentos
+  (Discoveries of the World)_, Hakluyt Soc. (1862), p. 77; E. G.
+  Ravenstein, "Voyages of ... Cão and ... Dias," in _Geog. Journ._
+  (London, December 1900), vol. xvi. pp. 638-655), an excellent critical
+  summary in the light of the most recent investigations of all the
+  material. The fragments of Diaz's only remaining pillar (from Diaz
+  Point) are now partly at the Cape Museum, partly at Lisbon: the latter
+  are photographed in Ravenstein's paper in _Geog. Journ._ (December
+  1900, p. 642).     (C. R. B.)
+
+
+
+
+DIAZO COMPOUNDS, in organic chemistry, compounds of the type R·N·2·X
+(where R = a hydrocarbon radical, and X = an acid radical or a hydroxyl
+group). These compounds may be divided into two classes, namely, the
+true diazo compounds, characterized by the grouping - N = N -, and the
+diazonium compounds, characterized by the grouping N:·N<.
+
+The diazonium compounds were first discovered by P. Griess (_Ann._,
+1858, 106, pp. 123 et seq.), and may be prepared by the action of
+nitrous fumes on a well-cooled solution of a salt of a primary amine,
+
+  C6H5NH2·HNO3 + HNO2 = C6H5N2·NO3 + 2H2O,
+
+or, as is more usually the case (since the diazonium salts themselves
+are generally used only in aqueous solution) by the addition of a
+well-cooled solution of potassium or sodium nitrite to a well-cooled
+dilute acid solution of the primary amine. In order to isolate the
+anhydrous diazonium salts, the method of E. Knoevenagel (_Ber._, 1890,
+23, p. 2094) may be employed. In this process the amine salt is
+dissolved in absolute alcohol and diazotized by the addition of amyl
+nitrite; a crystalline precipitate of the diazonium salt is formed on
+standing, or on the addition of a small quantity of ether. The diazonium
+salts are also formed by the action of zinc-dust and acids on the
+nitrates of primary amines (R. Mohlau, _Ber._, 1883, 16, p. 3080), and
+by the action of hydroxylamine on nitrosobenzenes. They are colourless
+crystalline solids which turn brown on exposure. They dissolve easily in
+water, but only to a slight extent in alcohol and ether. They are very
+unstable, exploding violently when heated or rubbed. _Benzene diazonium
+nitrate_, C6H5N(NO3):·N, crystallizes in long silky needles. The
+sulphate and chloride are similar, but they are not quite so unstable as
+the nitrate. The bromide may be prepared by the addition of bromine to
+an ethereal solution of diazo-amino-benzene (tribromaniline remaining in
+solution). By the addition of potassium bromide and bromine water to
+diazonium salts they are converted into a _perbromide_, e.g. C6H5N2Br3,
+which crystallizes in yellow plates.
+
+  The diazonium salts are characterized by their great reactivity and
+  consequently are important reagents in synthetical processes, since by
+  their agency the amino group in a primary amine may be exchanged for
+  other elements or radicals. The chief reactions are as follows:--
+
+  1. _Replacement of -NH2 by -OH_:--The amine is diazotized and the
+  aqueous solution of the diazonium salt is heated, nitrogen being
+  eliminated and a phenol formed.
+
+  2. _Replacement of -NH2 by halogens and by the -CN and -CNO
+  groups_:--The diazonium salt is warmed with an acid solution of the
+  corresponding cuprous salt (T. Sandmeyer, _Ber._, 1884, 17, p. 2650),
+  or with copper powder (L. Gattermann, Ber., 1890, 23, p. 1218; 1892,
+  25, p. 1074). In the case of iodine, the substitution is effected by
+  adding a warm solution of potassium iodide to the diazonium solution,
+  no copper or cuprous salt being necessary; whilst for the production
+  of nitriles a solution of potassium cuprous cyanide is used. This
+  reaction (the so-called "Sandmeyer" reaction) has been investigated by
+  A. Hantzsch and J. W. Blagden (_Ber._, 1900, 33, p. 2544), who
+  consider that three simultaneous reactions occur, namely, the
+  formation of labile double salts which decompose in such a fashion
+  that the radical attached to the copper atom wanders to the aromatic
+  nucleus; a catalytic action, in which nitrogen is eliminated and the
+  acid radical attaches itself to the aromatic nucleus; and finally, the
+  formation of azo compounds.
+
+  3. _Replacement of -NH2 by -NO2_:--A well-cooled concentrated solution
+  of potassium mercuric nitrate is added to a cooled solution of benzene
+  diazonium nitrate, when the crystalline salt 2C6H5N2·NO3, Hg(NO2)2 is
+  precipitated. On warming this with copper powder, it gives a
+  quantitative yield of nitrobenzene (A. Hantzsch, _Ber._, 1900, 33, p.
+  2551).
+
+  4. _Replacement of -NH2 by hydrogen_:--This exchange is brought about,
+  in some cases, by boiling the diazonium salt with alcohol; but I.
+  Remsen and his pupils (_Amer. Chem. Journ._, 1888, 9, pp. 389 et seq.)
+  have shown that the main product of this reaction is usually a
+  phenolic ether. This reaction has also been investigated by A.
+  Hantzsch and E. Jochem (_Ber._, 1901, 34, p. 3337), who arrived at the
+  conclusion that the normal decomposition of diazonium salts by
+  alcohols results in the formation of phenolic ethers, but that an
+  increase in the molecular weight of the alcohol, or the accumulation
+  of negative groups in the aromatic nucleus, diminishes the yield of
+  the ether and increases the amount of the hydrocarbon formed. The
+  replacement is more readily brought about by the use of sodium
+  stannite (P. Friedlander, _Ber._, 1889, 22, p. 587), or by the use of
+  a concentrated solution of hypophosphorous acid (J. Mai, _Ber._, 1902,
+  35, p. 162). A. Hantzsch (_Ber._, 1896, 29, p. 947; 1898, 31, p. 1253)
+  has shown that the chlor- and brom- diazoniumthiocyanates, when
+  dissolved in alcohol containing a trace of hydrochloric acid, become
+  converted into the isomeric thiocyanbenzene diazonium chlorides and
+  bromides. This change only occurs when the halogen atom is in the
+  ortho- or para- position to the -N2- group.
+
+  _Metallic Diazo Derivatives._--Benzene diazonium chloride is
+  decomposed by silver oxide in aqueous solution, with the formation of
+  _benzene diazonium hydroxide_, C6H5·N(OH):·N. This hydroxide, although
+  possessing powerful basic properties, is unstable in the presence of
+  alkalis and neutralizes them, being converted first into the isomeric
+  benzene-diazotic acid, the potassium salt of which is obtained when
+  the diazonium chloride is added to an excess of cold concentrated
+  potash (A. Hantzsch and W. B. Davidson, _Ber._, 1898, 31, p. 1612).
+  _Potassium benzene diazotate_, C6H5N2·OK, crystallizes in colourless
+  silky needles. The free acid is not known; by the addition of the
+  potassium salt to 50% acetic acid at -20° C., the acid anhydride,
+  _benzene diazo oxide_, (C6H5N2)2O, is obtained as a very unstable,
+  yellow, insoluble compound, exploding spontaneously at 0° C. Strong
+  acids convert it into a diazonium salt, and potash converts it into
+  the diazotate. On the constitution, of these anhydrides see E.
+  Bamberger, _Ber._, 1896, 29, p. 446, and A. Hantzsch, _Ber._, 1896,
+  29, p. 1067; 1898, 31, p. 636. By the addition of the diazonium salts
+  to a hot concentrated solution of a caustic alkali, C. Schraube and C.
+  Schmidt (_Ber._, 1894, 27, p. 520) obtained an isomer of potassium
+  benzene diazotate. These _iso-_diazotates are formed much more readily
+  when the aromatic nucleus in the diazonium salt contains negative
+  radicals. _Potassium benzene iso-diazotate_ resembles the normal salt,
+  but is more stable, and is more highly ionized. Carbon dioxide
+  converts it into _phenyl nitrosamine_, C6H5NH·NO (A. Hantzsch). The
+  potassium salt of the iso-diazo hydroxide yields on methylation a
+  nitrogen ether, R·N(CH3)·NO, whilst the silver salt yields an oxygen
+  ether, R·N:N·OCH3. These results point to the conclusion that the
+  iso-diazo hydroxide is a tautomeric substance. The same oxygen ether
+  is formed by the methylation of the silver salt of the normal diazo
+  hydroxide; this points to the conclusion that the isomeric hydroxides,
+  corresponding with the silver derivatives, have the same structural
+  formulae, namely, R·N:N·OH. These oxygen ethers contain the grouping
+  -N:N-, since they couple very readily with the phenols in alkaline
+  solution to form azo compounds (q.v.) (E. Bamberger, _Ber._, 1895, 28,
+  p. 225); they are also explosive.
+
+  By oxidizing potassium benzene iso-diazotate with alkaline potassium
+  ferricyanide, E. Bamberger (_Ber._, 1894, 27, p. 914) obtained the
+  _diazoic acids_, R·NH·NO2, substances which he had previously prepared
+  by similarly oxidizing the diazonium salts, by dehydrating the
+  nitrates of primary amines with acetic anhydride, and by the action of
+  nitric anhydride on the primary amines. Concentrated acids convert
+  them into the isomeric nitro-amines, the -NO2 group going into the
+  nucleus in the ortho- or para- position to the amine nitrogen; this
+  appears to indicate that the compounds are nitramines. They behave,
+  however, as tautomeric substances, since their alkali salts on
+  methylation give nitrogen ethers, whilst their silver salts yield
+  oxygen ethers:
+
+            /-->  potassium salt --> R·N(CH3)·NO2 nitramine.
+  R·NH·NO2<
+            \-->  silver salt    --> R·N:N·O·OCH3 diazoate.
+
+  _Phenyl nitramine_, C6H5NH·NO2, is a colourless crystalline solid,
+  which melts at 46° C. Sodium amalgam in alkaline solution reduces it
+  to phenylhydrazine.
+
+  _Constitution of the Diazo Compounds._--P. Griess (_Ann._, 1866, 137,
+  p. 39) considered that the diazo compounds were formed by the addition
+  of complex groupings of the type C6H4N2- to the inorganic acids;
+  whilst A. Kekulé (_Zeit. f. Chemie_, 1866, 2, p. 308), on account of
+  their ready condensation to form azo compounds and their easy
+  reduction to hydrazines, assumed that they were substances of the type
+  R·N:N·Cl. The constitution of the diazonium group -N2·X, may be
+  inferred from the following facts:--The group C6H5N2- behaves in many
+  respects similarly to an alkali metal, and even more so to the
+  ammonium group, since it is capable of forming colourless neutral
+  salts with mineral acids, which in dilute aqueous solution are
+  strongly ionized, but do not show any trace of hydrolytic dissociation
+  (A. Hantzsch, _Ber._, 1895, 28, p. 1734). Again, the diazonium
+  chlorides combine with platinic chloride to form difficultly soluble
+  double platinum salts, such as (C6H5N2Cl)2·PtCl4; similar gold salts,
+  C6H5N2Cl·AuCl3, are known. Determinations of the electrical
+  conductivity of the diazonium chloride and nitrate also show that the
+  diazonium radical is strictly comparable with other quaternary
+  ammonium ions. For these reasons, one must assume the existence of
+  pentavalent nitrogen in the diazonium salts, in order to account for
+  their basic properties.
+
+  The constitution of the isomeric diazo hydroxides has given rise to
+  much discussion. E. Bamberger (_Ber._, 1895, 28, pp. 444 et seq.) and
+  C. W. Blomstrand (_Journ. prakt. Chem._, 1896, 53, pp. 169 et seq.)
+  hold that the compounds are structurally different, the normal
+  diazo-hydroxide being a diazonium derivative of the type
+  R·N([3:]N)·OH. The recent work of A. Hantzsch and his pupils seems to
+  invalidate this view (_Ber._, 1894, 27, pp. 1702 et seq.; see also A.
+  Hantzsch, _Die Diazoverbindungen_). According to Hantzsch the isomeric
+  diazo hydroxides are structurally identical, and the differences in
+  behaviour are due to stereo-chemical relations, the isomerism being
+  comparable with that of the oximes (q.v.). On such a hypothesis, the
+  relatively unstable normal diazo hydroxides would be the
+  _syn-_compounds, since here the nitrogen atoms would be more easily
+  eliminated, whilst the stable iso-diazo derivatives would be the
+  _anti-_compounds, thus:
+
+         R · N             R · N
+             ..                ..
+        HO · N                 N · OH
+    Normal hydroxide      Iso hydroxide
+    (Syn-compound)       (Anti-compound)
+
+  In support of this theory, Hantzsch has succeeded in isolating a
+  series of syn- and anti-diazo-cyanides and -sulphonates (_Ber._, 1895,
+  28, p. 666; 1900, 33, p. 2161; 1901, 34, p. 4166). By diazotizing
+  para-chloraniline and adding a cold solution of potassium cyanide, a
+  salt (melting at 29° C.) is obtained, which readily loses nitrogen,
+  and forms para-chlorbenzonitrile on the addition of copper powder. By
+  dissolving this diazocyanide in alcohol and reprecipitating it by
+  water, it is converted into the isomeric diazocyanide (melting at
+  105-106° C.), which does not yield para-chlorbenzonitrile when treated
+  with copper powder. Similar results have been obtained by using
+  diazotized para-anisidine, a syn- and an anti- compound being formed,
+  as well as a third isomeric cyanide, obtained by evaporating
+  para-methoxy-benzenediazonium hydroxide in the presence of an excess
+  of hydrocyanic acid at ordinary temperatures. This salt is a
+  colourless crystalline substance of composition
+  CH3O·C6H4·N2·CN·HCN·2H2O, and has the properties of a metallic salt;
+  it is very soluble in water and its solution is an electrolyte,
+  whereas the solutions of the syn-and anti- compounds are not
+  electrolytes. The isolation of these compounds is a powerful argument
+  in favour of the Hantzsch hypothesis which requires the existence of
+  these three different types, whilst the Bamberger-Blomstrand view only
+  accounts for the formation of two isomeric cyanides, namely, one of
+  the normal diazonium type and one of the iso-diazocyanide type.
+
+  Benzene diazonium hydroxide, although a strong base, reacts with the
+  alkaline hydroxides to form salts with the evolution of heat, and
+  generally behaves as a weak acid. On mixing dilute solutions of the
+  diazonium hydroxide and the alkali together, it is found that the
+  molecular conductivity of the mixture is much less than the sum of the
+  two electrical conductivities of the solutions separately, from which
+  it follows that a portion of the ions present have changed to the
+  non-ionized condition. This behaviour is explained by considering the
+  non-ionized part of the diazonium hydroxide to exist in solution in a
+  hydrated form, the equation of equilibrium being:
+
+          C6H5·N·       --> C6H5·N·OH
+    H2O +     ... + OH'          |
+               N        <--   HO·N·H
+
+  On adding the alkaline hydroxide to the solution, this hydrate is
+  supposed to lose water, yielding the syn-diazo hydroxide, which then
+  gives rise to a certain amount of the sodium salt (A. Hantzsch,
+  _Ber._, 1898, 31, p. 1612),
+
+           ....
+    C6H5·N·:OH:  -->  C6H5·N  -->  C6H5·N
+         | :  :            ||           ||
+      HO·N·:H :  <--    HO·N  <--   NaO·N
+           ....
+
+  This assumption also shows the relationship of the diazonium
+  hydroxides to other quaternary ammonium compounds, for most of the
+  quaternary ammonium hydroxides (except such as have the nitrogen atom
+  attached to four saturated hydrocarbon radicals) are unstable, and
+  readily pass over into compounds in which the hydroxyl group is no
+  longer attached to the amine nitrogen; thus the syn-diazo hydroxides
+  are to be regarded as pseudo-diazonium derivatives. (A. Hantzsch,
+  _Ber._, 1899, 32, p. 3109; 1900, 33, p. 278.) It is generally accepted
+  that the iso-diazo hydroxides possess the oxime structure R·N:N·OH.
+
+  Hantzsch explains the characteristic reactions of the diazonium
+  compounds by the assumption that an addition compound is first formed,
+  which breaks down with the elimination of the hydride of the acid
+  radical, and the formation of an unstable syn-diazo compound, which,
+  in its turn, decomposes with evolution of nitrogen (_Ber._, 1897, 30,
+  p. 2548; 1898, 31, p. 2053).
+
+     R         X     R      X     R   X
+      \        |      \    /      |   |
+        N·:N + | -->   N·:N   --> |   | + HCl --> R·X + N2.
+      /        |      /    \      |   |
+    Cl         H    Cl      H     N = N
+
+  J. Cain (_Jour. Chem. Soc._, 1907, 91, p. 1049) suggested a quinonoid
+  formula for diazonium salts, which has been combated by Hantzsch
+  (_Ber._, 1908, 41, pp. 3532 et seq.). G. T. Morgan and F. M. G.
+  Micklethwaite (_Jour. Chem. Soc._, 1908, 93, p. 617; 1909, 95, p.
+  1319) have pointed out that the salts may possess a dynamic formula,
+  Cain's representing the middle stage, thus:
+
+        /||\                /||\                /||\
+       / || \              / || \              / || \
+    H / N·Cl \ H        H / N·Cl \ H        H / N·Cl \ H
+     |\  ||  ||  --->    ||  ||   |  --->    ||  ||  /|
+     | \ ||  ||  <---    ||  ||   |  <---    ||  || / |
+     |  \N   ||          ||  N    |          ||   N   |
+    H\\      / H        H\\  |   / H        H \      //H
+      \\    /             \\ |  /              \    //
+       \\  /               \\| /                \  //
+         H                   H                   H
+
+  _Diazoamines._--The diazoamines, R·N2·NHR, may be prepared by the
+  action of the primary and secondary amines on the diazonium salts, or
+  by the action of nitrous acid on the free primary amine. In the latter
+  reaction it is assumed that the isodiazohydroxide first formed is
+  immediately attacked by a second molecule of the amine. They are
+  yellow crystalline solids, which do not unite with acids. Nitrous acid
+  converts them, in acid solution, into diazonium salts.
+
+    C6H5N2·NHC6H5 + 2HCl + HNO2 = 2C6H5N2Cl + 2H2O.
+
+  They are readily converted into the isomeric aminoazo compounds,
+  either by standing in alcoholic solution, or by warming with a mixture
+  of the parent base and its hydrochloride; the diazo group preferably
+  going into the para-position to the amino group. When the
+  para-position is occupied, the diazo group takes the ortho-position.
+  H. Goldschmidt and R. U. Reinders (_Ber._, 1896, 29, p. 1369, 1899)
+  have shown that the transformation is a monomolecular reaction, the
+  velocity of transformation in moderately dilute solution being
+  independent of the concentration, but proportional to the amount of
+  the catalyst present (amine hydrochloride) and to the temperature. It
+  has also been shown that when different salts of the amine are used,
+  their catalytic influence varies in amount and is almost proportional
+  to their degree of ionization in aqueous solution. Diazoaminobenzene,
+  C6H5N2·NHC6H5, crystallizes in golden yellow laminae, which melt at
+  96° C. and explode at a slightly higher temperature. It is readily
+  soluble in alcohol, ether and benzene. Concentrated hydrochloric acid
+  converts it into chlorbenzene, aniline and nitrogen. Zinc dust and
+  alcoholic acetic acid reduce it to aniline and phenylhydrazine.
+
+  _Diazoimino compounds_, R·N3, may be regarded as derivatives of
+  azoimide (q.v.); they are formed by the action of ammonia on the
+  diazoperbromides, or by the action of hydroxylamine on the diazonium
+  sulphates (J. Mai, _Ber._, 1892, 25, p. 372; T. Curtius, _Ber._, 1893,
+  26, p. 1271). Diazobenzeneimide, C6H5N3, is a yellowish oil of
+  stupefying odour. It boils at 59° C. (12 mm.), and explodes when
+  heated. Concentrated hydrochloric acid decomposes it with formation of
+  chloranilines and elimination of nitrogen, whilst on boiling with
+  sulphuric acid it is converted into aminophenols.
+
+  _Aliphatic Diazo Compounds._--The esters of the aliphatic amino acids
+  may be diazotized in a manner similar to the primary aromatic amines,
+  a fact discovered by T. Curtius (_Ber._, 1833, 16, p. 2230). The first
+  aliphatic diazo compound to be isolated was _diazoacetic ester_,
+  CH·N2·CO2C2H5, which is prepared by the action of potassium nitrite on
+  the ethyl ester of glycocoll hydrochloride, HCl·NH2·CH2·CO2C2H5 + KNO2
+  = CHN2·CO2C2H5 + KCl + 2H2O. It is a yellowish oil which melts at -24°
+  C.; it boils at 143-144° C., but cannot be distilled safely as it
+  decomposes violently, giving nitrogen and ethyl fumarate. It explodes
+  in contact with concentrated sulphuric acid. On reduction it yields
+  ammonia and glycocoll (aminoacetic acid). When heated with water it
+  forms ethyl hydroxy-acetate; with alcohol it yields ethyl
+  ethoxyacetate. Halogen acids convert it into monohalogen fatty acids,
+  and the halogens themselves convert it into dihalogen fatty acids. It
+  unites with aldehydes to form esters of ketonic acids, and with
+  aniline yields anilido-acetic acid. It forms an addition product with
+  acrylic ester, which on heating loses nitrogen and leaves trimethylene
+  dicarboxylic ester. Concentrated ammonia converts it into
+  _diazoacetamide_, CHN2·CONH2, which crystallizes in golden yellow
+  plates which melt at 114° C. For other reactions see HYDRAZINE. The
+  constitution of the diazo fatty esters is inferred from the fact that
+  the two nitrogen atoms, when split off, are replaced by two monovalent
+  elements or groups, thus leading to the formula
+
+     N \
+    ..  > CH·CO2C2H5, for diazoacetic ester.
+     N /
+
+  _Diazosuccinic ester_, N2·C(CO2C2H5)2, is similarly prepared by the
+  action of nitrous acid on the hydrochloride of aspartic ester. It is
+  decomposed by boiling water and yields fumaric ester.
+
+  _Diazomethane_, CH2N2, was first obtained in 1894 by H. v. Pechmann
+  (_Ber._, 1894, 27, p. 1888; 1895, 28, p. 855). It is prepared by the
+  action of aqueous or alcoholic solutions of the caustic alkalis on the
+  nitroso-acidyl derivatives of methylamine (such, for example, as
+  _nitrosomethyl urethane_, NO·N(CH3)·CO2C2H5, which is formed on
+  passing nitrous fumes into an ethereal solution of methyl urethane).
+  E. Bamberger (_Ber._, 1895, 28, p. 1682) regards it as the anhydride
+  of iso-diazomethane, CH3·N:N·OH, and has prepared it by a method
+  similar to that used for the preparation of iso-diazobenzene. By the
+  action of bleaching powder on methylamine hydrochloride, there is
+  obtained a volatile liquid (_methyldichloramine_, CH3·N·Cl2), boiling
+  at 58-60° C., which explodes violently when heated with water,
+  yielding hydrocyanic acid (CH3NCl2 = HCN + 2HCl). Well-dried
+  hydroxylamine hydrochloride is dissolved in methyl alcohol and mixed
+  with sodium methylate; a solution of methyldichloramine in absolute
+  ether is then added and an ethereal solution of diazomethane distils
+  over. Diazomethane is a yellow inodorous gas, very poisonous and
+  corrosive. It may be condensed to a liquid, which boils at about 0° C.
+  It is a powerful methylating agent, reacting with water to form methyl
+  alcohol, and converting acetic acid into methylacetate, hydrochloric
+  acid into methyl chloride, hydrocyanic acid into acetonitrile, and
+  phenol into anisol, nitrogen being eliminated in each case. It is
+  reduced by sodium amalgam (in alcoholic solution) to
+  _methylhydrazine_, CH3·NH·NH2. It unites directly with acetylene to
+  form pyrazole (H. v. Pechmann, _Ber._, 1898, 31, p. 2950) and with
+  fumaric methyl ester it forms pyrazolin dicarboxylic ester.
+       (F. G. P.*)
+
+    See G. T. Morgan, _B.A. Rep._, 1902; J. Cain, _Diazo Compounds_,
+    1908.
+
+
+
+
+DIAZOMATA (Gr. [Greek: diazôma], a girdle), in architecture, the landing
+places and passages which were carried round the semicircle and
+separated the upper and lower tiers in a Greek theatre.
+
+
+
+
+DIBDIN, CHARLES (1745-1814), British musician, dramatist, novelist,
+actor and song-writer, the son of a parish clerk, was born at
+Southampton on or before the 4th of March 1745, and was the youngest of
+a family of eighteen. His parents designing him for the church, he was
+sent to Winchester; but his love of music early diverted his thoughts
+from the clerical profession. After receiving some instruction from the
+organist of Winchester cathedral, where he was a chorister from 1756 to
+1759, he went to London at the age of fifteen. Here he was placed in a
+music warehouse in Cheapside, but he soon abandoned this employment to
+become a singing actor at Covent Garden. On the 21st of May 1762 his
+first work, an operetta entitled _The Shepherd's Artifice_, with words
+and music by himself, was produced at this theatre. Other works
+followed, his reputation being firmly established by the music to the
+play of _The Padlock_, produced at Drury Lane under Garrick's management
+in 1768, the composer himself taking the part of Mungo with conspicuous
+success. He continued for some years to be connected with Drury Lane,
+both as composer and as actor, and produced during this period two of
+his best known works, _The Waterman_ (1774) and _The Quaker_ (1775). A
+quarrel with Garrick led to the termination of his engagement. In _The
+Comic Mirror_ he ridiculed prominent contemporary figures through the
+medium of a puppet show. In 1782 he became joint manager of the Royal
+circus, afterwards known as the Surrey theatre. In three years he lost
+this position owing to a quarrel with his partner. His opera _Liberty
+Hall_, containing the successful songs "Jock Ratlin," "The Highmettled
+Racer," and "The Bells of Aberdovey," was produced at Drury Lane theatre
+on the 8th of February 1785. In 1788 he sailed for the East Indies, but
+the vessel having put in to Torbay in stress of weather, he changed his
+mind and returned to London. In a musical variety entertainment called
+_The Oddities_, he succeeded in winning marked popularity with a number
+of songs that included "'Twas in the good ship 'Rover'," "Saturday Night
+at Sea," "I sailed from the Downs in the 'Nancy,'" and the immortal "Tom
+Bowling," written on the death of his eldest brother, Captain Thomas
+Dibdin, at whose invitation he had planned his visit to India. A series
+of monodramatic entertainments which he gave at his theatre, Sans Souci,
+in Leicester Square, brought his songs, music and recitations more
+prominently into notice, and permanently established his fame as a lyric
+poet. It was at these entertainments that he first introduced many of
+those sea-songs which so powerfully influenced the national spirit. The
+words breathe the simple loyalty and dauntless courage that are the
+cardinal virtues of the British sailor, and the music was appropriate
+and naturally melodious. Their effect in stimulating and ennobling the
+spirit of the navy during the war with France was so marked as to call
+for special acknowledgment. In 1803 Dibdin was rewarded by government
+with a pension of £200 a year, of which he was only for a time deprived
+under the administration of Lord Grenville. During this period he opened
+a music shop in the Strand, but the venture was a failure. Dibdin died
+of paralysis in London on the 25th of July 1814. Besides his _Musical
+Tour through England_ (1788), his _Professional Life_, an autobiography
+published in 1803, a _History of the Stage_ (1795), and several smaller
+works, he wrote upwards of 1400 songs and about thirty dramatic pieces.
+He also wrote the following novels:--_The Devil_ (1785); _Hannah Hewitt_
+(1792); _The Younger Brother_ (1793). An edition of his songs by G.
+Hogarth (1843) contains a memoir of his life. His two sons, Charles and
+Thomas John Dibdin (q.v.), whose works are often confused with those of
+their father, were also popular dramatists in their day.
+
+
+
+
+DIBDIN, THOMAS FROGNALL (1776-1847), English bibliographer, born at
+Calcutta in 1776, was the son of Thomas Dibdin, the sailor brother of
+Charles Dibdin. His father and mother both died on the way home to
+England in 1780, and Thomas was brought up by a maternal uncle. He was
+educated at St John's College, Oxford, and studied for a time at
+Lincoln's Inn. After an unsuccessful attempt to obtain practice as a
+provincial counsel at Worcester, he was ordained a clergyman at the
+close of 1804, being appointed to a curacy at Kensington. It was not
+until 1823 that he received the living of Exning in Sussex. Soon
+afterwards he was appointed by Lord Liverpool to the rectory of St
+Mary's, Bryanston Square, which he held until his death on the 18th of
+November 1847. The first of his numerous bibliographical works was his
+_Introduction to the Knowledge of Editions of the Classics_ (1802),
+which brought him under the notice of the third Earl Spencer, to whom he
+owed much important aid in his bibliographical pursuits. The rich
+library at Althorp was thrown open to him; he spent much of his time in
+it, and in 1814-1815 published his _Bibliotheca Spenceriana_. As the
+library was not open to the general public, the information given in the
+_Bibliotheca_ was found very useful, but since its author was unable
+even to read the characters in which the books he described were
+written, the work was marred by the errors which more or less
+characterize all his productions. This fault of inaccuracy however was
+less obtrusive in his series of playful, discursive works in the form of
+dialogues on his favourite subject, the first of which, _Bibliomania_
+(1809), was republished with large additions in 1811, and was very
+popular, passing through numerous editions. To the same class belonged
+the _Bibliographical Decameron_, a larger work, which appeared in 1817.
+In 1810 he began the publication of a new and much extended edition of
+Ames's _Typographical Antiquities_. The first volume was a great
+success, but the publication was checked by the failure of the fourth
+volume, and was never completed. In 1818 Dibdin was commissioned by
+Earl Spencer to purchase books for him on the continent, an expedition
+described in his sumptuous _Bibliographical, Antiquarian and Picturesque
+Tour in France and Germany_ (1821). In 1824 he made an ambitious venture
+in his _Library Companion, or the Young Man's Guide and Old Man's
+Comfort in the Choice of a Library_, intended to point out the best
+works in all departments of literature. His culture was not broad
+enough, however, to render him competent for the task, and the work was
+severely criticized. For some years Dibdin gave himself up chiefly to
+religious literature. He returned to bibliography in his _Bibliophobia,
+or Remarks on the Present Depression in the State of Literature and the
+Book Trade_ (1832), and the same subject furnishes the main interest of
+his _Reminiscences of a Literary Life_ (1836), and his _Bibliographical,
+Antiquarian and Picturesque Tour in the Northern Counties of England and
+Scotland_ (1838). Dibdin was the originator and vice-president, Lord
+Spencer being the president, of the Roxburghe Club, founded in
+1812,--the first of the numerous book clubs which have done such service
+to literature.
+
+
+
+
+DIBDIN, THOMAS JOHN (1771-1841), English dramatist and song-writer, son
+of Charles Dibdin, the song-writer, and of Mrs Davenet, an actress whose
+real name was Harriet Pitt, was born on the 21st of March 1771. He was
+apprenticed to his maternal uncle, a London upholsterer, and later to
+William Rawlins, afterwards sheriff of London. He summoned his second
+master unsuccessfully for rough treatment; and after a few years of
+service he ran away to join a company of country players. From 1789 to
+1795 he played in all sorts of parts; he acted as scene painter at
+Liverpool in 1791; and during this period he composed more than 1000
+songs. He made his first attempt as a dramatic writer in _Something
+New_, followed by _The Mad Guardian_ in 1795. He returned to London in
+1795, having married two years before; and in the winter of 1798-1799
+his _Jew and the Doctor_ was produced at Covent Garden. From this time
+he contributed a very large number of comedies, operas, farces, &c., to
+the public entertainment. Some of these brought immense popularity to
+the writer and immense profits to the theatres. It is stated that the
+pantomime of _Mother Goose_ (1807) produced more than £20,000 for the
+management at Covent Garden theatre, and the _High-mettled Racer_,
+adapted as a pantomime from his father's play, £18,000 at Astley's.
+Dibdin was prompter and pantomime writer at Drury Lane until 1816, when
+he took the Surrey theatre. This venture proved disastrous and he became
+bankrupt. After this he was manager of the Haymarket, but without his
+old success, and his last years were passed in comparative poverty. In
+1827 he published two volumes of _Reminiscences_; and at the time of his
+death he was preparing an edition of his father's sea songs, for which a
+small sum was allowed him weekly by the lords of the admiralty. Of his
+own songs "The Oak Table" and "The Snug Little Island" are well-known
+examples. He died in London on the 16th of September 1841.
+
+
+
+
+DIBRA (Slav. _Debra_), the capital of a sanjak bearing the same name, in
+the vilayet of Monastir, eastern Albania, Turkey. Pop. (1900) about
+15,000. Dibra occupies a valley enclosed by mountains, and watered by
+the Tsrni Drin and Radika rivers, which meet 3 m. S. It is a fortified
+city, and the only episcopal see of the Bulgarian exarchate in Albania;
+most of the inhabitants are Albanians, but there is a strong Bulgarian
+colony. The local trade is almost entirely agricultural.
+
+
+
+
+DIBRUGARH, a town of British India, in the Lakhimpur district of eastern
+Bengal and Assam, of which it is the headquarters, situated on the Dibru
+river about 4 m. above its confluence with the Brahmaputra. Pop. (1901)
+11,227. It is the terminus of steamer navigation on the Brahmaputra, and
+also of a railway running to important coal-mines and petroleum wells,
+which connects with the Assam-Bengal system. Large quantities of coal
+and tea are exported. There are a military cantonment, the headquarters
+of the volunteer corps known as the Assam Valley Light Horse; a
+government high school, a training school for masters; and an aided
+school for girls. In 1900 a medical school for the province was
+established, out of a bequest left by Brigade-Surgeon J. Berry-White,
+which is maintained by the government, to train hospital assistants for
+the tea gardens. The Williamson artisan school is entirely supported by
+an endowment.
+
+
+
+
+DICAEARCHUS, of Messene in Sicily, Peripatetic philosopher and pupil of
+Aristotle, historian, and geographer, flourished about 320 B.C. He was a
+friend of Theophrastus, to whom he dedicated the majority of his works.
+Of his writings, which comprised treatises on a great variety of
+subjects, only the titles and a few fragments survive. The most
+important of them was his [Greek: bios tês Hellados] (_Life in Greece_),
+in which the moral, political and social condition of the people was
+very fully discussed. In his _Tripoliticos_ he described the best form
+of government as a mixture of monarchy, aristocracy and democracy, and
+illustrated it by the example of Sparta. Among the philosophical works
+of Dicaearchus may be mentioned the _Lesbiaci_, a dialogue in three
+books, in which the author endeavours to prove that the soul is mortal,
+to which he added a supplement called _Corinthiaci_. He also wrote a
+_Description of the World_ illustrated by maps, in which was probably
+included his _Measurements of Mountains_. A description of Greece (150
+iambics, in C. Müller, _Frag. hist. Graec_. i. 238-243) was formerly
+attributed to him, but, as the initial letters of the first twenty-three
+lines show, was really the work of Dionysius, son of Calliphon. Three
+considerable fragments of a prose description of Greece (Müller, i.
+97-110) are now assigned to an unknown author named Heracleides. The _De
+re publica_ of Cicero is supposed to be founded on one of Dicaearchus's
+works.
+
+  The best edition of the fragments is by M. Fuhr (1841), a work of
+  great learning; see also a dissertation by F. G. Osann, _Beiträge zur
+  röm. und griech. Litteratur_, ii. pp. 1-117 (1839); Pauly-Wissowa,
+  _Realencyclopädie der klass. Altertumswiss_. v. pt. 1 (1905).
+
+
+
+
+DICE (plural of die, O. Fr. _de_, derived from Lat. _dare_, to give),
+small cubes of ivory, bone, wood or metal, used in gaming. The six sides
+of a die are each marked with a different number of incised dots in such
+a manner that the sum of the dots on any two opposite sides shall be 7.
+Dice seem always to have been employed, as is the case to-day, for
+gambling purposes, and they are also used in such games as backgammon.
+There are many methods of playing, from one to five dice being used,
+although two or three are the ordinary numbers employed in Great Britain
+and America. The dice are thrown upon a table or other smooth surface
+either from the hand or from a receptacle called a dice-box, the latter
+method having been in common use in Greece, Rome and the Orient in
+ancient times. Dice-boxes have been made in many shapes and of various
+materials, such as wood, leather, agate, crystal, metal or paper. Many
+contain bars within to ensure a proper agitation of the dice, and thus
+defeat trickery. Some, formerly used in England, were employed with
+unmarked dice, and allowed the cubes to fall through a kind of funnel
+upon a board marked off into six equal parts numbered from 1 to 6. It is
+a remarkable fact, that, wherever dice have been found, whether in the
+tombs of ancient Egypt, of classic Greece, or of the far East, they
+differ in no material respect from those in use to-day, the elongated
+ones with rounded ends found in Roman graves having been, not dice but
+_tali_, or knucklebones. Eight-sided dice have comparatively lately been
+introduced in France as aids to children in learning the multiplication
+table. The teetotum, or spinning die, used in many modern games, was
+known in ancient times in China and Japan. The increased popularity of
+the more elaborate forms of gaming has resulted in the decline of
+dicing. The usual method is to throw three times with three dice. If one
+or more sixes or fives are thrown the first time they may be reserved,
+the other throws being made with the dice that are left. The object is
+to throw three sixes = 18 or as near that number as possible, the
+highest throw winning, or, when drinks are to be paid for, the lowest
+throw losing. (For other methods of throwing consult the _Encyclopaedia
+of Indoor Games_, by R. F. Foster, 1903.) The most popular form of pure
+gambling with dice at the present day, particularly with the lower
+classes in America, is _Craps_, or _Crap-Shooting_, a simple form of
+_Hazard_, of French origin. Two dice are used. Each player puts up a
+stake and the first caster may cover any or all of the bets. He then
+_shoots_, i.e. throws the dice from his open hand upon the table. If the
+sum of the dice is 7 or 11 the throw is a _nick_, or _natural_, and the
+caster wins all stakes. If the throw is either 2, 3 or 12 it is a
+_crap_, and the caster loses all. If any other number is thrown it is a
+_point_, and the caster continues until he throws the same number again,
+in which case he wins, or a 7, in which case he loses. The now
+practically obsolete game of Hazard was much more complicated than
+_Craps_. (Consult _The Game of Hazard Investigated_, by George Lowbut.)
+_Poker dice_ are marked with ace, king, queen, jack and ten-spot. Five
+are used and the object is, in three throws, to make pairs, triplets,
+full hands or fours and fives of a kind, five aces being the highest
+hand. Straights do not count. In throwing to decide the payment of
+drinks the usual method is called _horse and horse_, in which the
+highest throws retire, leaving the two lowest to decide the loser by the
+best two in three throws. Should each player win one throw both are said
+to be _horse and horse_, and the next throw determines the loser. The
+two last casters may also agree to _sudden death_, i.e. a single throw.
+_Loaded dice_, i.e. dice weighted slightly on the side of the lowest
+number, have been used by swindlers from the very earliest times to the
+present day, a fact proved by countless literary allusions. Modern dice
+are often rounded at the corners, which are otherwise apt to wear off
+irregularly.
+
+_History._--Dice were probably evolved from knucklebones. The antiquary
+Thomas Hyde, in his _Syntagma_, records his opinion that the game of
+"odd or even," played with pebbles, is nearly coeval with the creation
+of man. It is almost impossible to trace clearly the development of dice
+as distinguished from knucklebones, on account of the confusing of the
+two games by the ancient writers. It is certain, however, that both were
+played in times antecedent to those of which we possess any written
+records. Sophocles, in a fragment, ascribed their invention to
+Palamedes, a Greek, who taught them to his countrymen during the siege
+of Troy, and who, according to Pausanias (on Corinth, xx.), made an
+offering of them on the altar of the temple of Fortune. Herodotus
+(_Clio_) relates that the Lydians, during a period of famine in the days
+of King Atys, invented dice, knucklebones and indeed all other games
+except chess. The fact that dice have been used throughout the Orient
+from time immemorial, as has been proved by excavations from ancient
+tombs, seems to point clearly to an Asiatic origin. Dicing is mentioned
+as an Indian game in the _Rig-veda_. In its primitive form knucklebones
+was essentially a game of skill, played by women and children, while
+dice were used for gambling, and it was doubtless the gambling spirit of
+the age which was responsible for the derivative form of knucklebones,
+in which four sides of the bones received different values, which were
+then counted, like dice. Gambling with three, sometimes two, dice
+([Greek: kuboi]) was a very popular form of amusement in Greece,
+especially with the upper classes, and was an almost invariable
+accompaniment to the symposium, or drinking banquet. The dice were cast
+from conical beakers, and the highest throw was three sixes, called
+_Aphrodite_, while the lowest, three aces, was called the _dog_. Both in
+Greece and Rome different modes of counting were in vogue. Roman dice
+were called _tesserae_ from the Greek word for four, indicative of the
+four sides. The Romans were passionate gamblers, especially in the
+luxurious days of the Empire, and dicing was a favourite form, though it
+was forbidden except during the Saturnalia. The emperor Augustus wrote
+in a letter to Suetonius concerning a game that he had played with his
+friends: "Whoever threw a _dog_ or a six paid a _denarius_ to the bank
+for every die, and whoever threw a _Venus_ (the highest) won
+everything." In the houses of the rich the dice-beakers were of carved
+ivory and the dice of crystal inlaid with gold. Mark Antony wasted his
+time at Alexandria with dicing, while, according to Suetonius, the
+emperors Augustus, Nero and Claudius were passionately fond of it, the
+last named having written a book on the game. Caligula notoriously
+cheated at the game; Domitian played it, and Commodus set apart special
+rooms in his palace for it. The emperor Verus, adopted son of Antonine,
+is known to have thrown dice whole nights together. Fashionable society
+followed the lead of its emperors, and, in spite of the severity of the
+laws, fortunes were squandered at the dicing table. Horace derided the
+youth of the period, who wasted his time amid the dangers of dicing
+instead of taming his charger and giving himself up to the hardships of
+the chase. Throwing dice for money was the cause of many special laws in
+Rome, according to one of which no suit could be brought by a person who
+allowed gambling in his house, even if he had been cheated or assaulted.
+Professional gamblers were common, and some of their loaded dice are
+preserved in museums. The common public-houses were the resorts of
+gamblers, and a fresco is extant showing two quarrelling dicers being
+ejected by the indignant host. Virgil, in the _Copa_ generally ascribed
+to him, characterizes the spirit of that age in verse, which has been
+Englished as follows:--
+
+  "What ho! Bring dice and good wine!
+     Who cares for the morrow?
+   Live--so calls grinning Death--
+     Live, for I come to you soon!"
+
+That the barbarians were also given to gaming, whether or not they
+learned it from their Roman conquerors, is proved by Tacitus, who states
+that the Germans were passionately fond of dicing, so much so, indeed,
+that, having lost everything, they would even stake their personal
+liberty. Centuries later, during the middle ages, dicing became the
+favourite pastime of the knights, and both dicing schools (_scholae
+deciorum_) and gilds of dicers existed. After the downfall of feudalism
+the famous German mercenaries called _landsknechts_ established a
+reputation as the most notorious dicing gamblers of their time. Many of
+the dice of the period were curiously carved in the images of men and
+beasts. In France both knights and ladies were given to dicing, which
+repeated legislation, including interdictions on the part of St Louis in
+1254 and 1256, did not abolish. In Japan, China, Korea, India and other
+Asiatic countries dice have always been popular and are so still.
+
+  See Foster's _Encyclopaedia of Indoor Games_ (1903); Raymond's
+  _Illustriertes Knobelbrevier_ (Oranienburg, 1888); _Les Jeux des
+  Anciens_, by L. Becq de Fouquières (Paris, 1869); _Das Knöchelspiel
+  der Alten_, by Bolle (Wismar, 1886); _Die Spiele der Griechen und
+  Römer_, by W. Richter (Leipzig, 1887); Raymond's _Alte und neue
+  Würfelspiele_; _Chinese Games with Dice_, by Stewart Culin
+  (Philadelphia, 1889); _Korean Games_, by Stewart Culin (Philadelphia,
+  1895).
+
+
+
+
+DICETO, RALPH DE (d. c. 1202), dean of St Paul's, London, and
+chronicler, is first mentioned in 1152, when he received the
+archdeaconry of Middlesex. He was probably born between 1120 and 1130;
+of his parentage and nationality we know nothing. The common statement
+that he derived his surname from Diss in Norfolk is a mere conjecture;
+Dicetum may equally well be a Latinized form of Dissai, or Dicy, or
+Dizy, place names which are found in Maine, Picardy, Burgundy and
+Champagne. In 1152 Diceto was already a master of arts; presumably he
+had studied at Paris. His reputation for learning and integrity stood
+high; he was regarded with respect and favour by Arnulf of Lisieux and
+Gilbert Foliot of Hereford (afterwards of London), two of the most
+eminent bishops of their time. Quite naturally, the archdeacon took in
+the Becket question the same side as his friends. Although his narrative
+is colourless, and although he was one of those who showed some sympathy
+for Becket at the council of Northampton (1164), the correspondence of
+Diceto shows that he regarded the archbishop's conduct as
+ill-considered, and that he gave advice to those whom Becket regarded as
+his chief enemies. Diceto was selected, in 1166, as the envoy of the
+English bishops when they protested against the excommunications
+launched by Becket. But, apart from this episode, which he
+characteristically omits to record, he remained in the background. The
+natural impartiality of his intellect was accentuated by a certain
+timidity, which is apparent in his writings no less than in his life.
+About 1180 he became dean of St Paul's. In this office he distinguished
+himself by careful management of the estates, by restoring the
+discipline of the chapter, and by building at his own expense a
+deanery-house. A scholar and a man of considerable erudition, he showed
+a strong preference for historical studies; and about the time when he
+was preferred to the deanery he began to collect materials for the
+history of his own times. His friendships with Richard Fitz Nigel, who
+succeeded Foliot in the see of London, with William Longchamp, the
+chancellor of Richard I., and with Walter of Coutances, the archbishop
+of Rouen, gave him excellent opportunities of collecting information.
+His two chief works, the _Abbreviationes Chronicorum_ and the _Ymagines
+Historiarum_, cover the history of the world from the birth of Christ to
+the year 1202. The former, which ends in 1147, is a work of learning and
+industry, but almost entirely based upon extant sources. The latter,
+beginning as a compilation from Robert de Monte and the letters of
+Foliot, becomes an original authority about 1172, and a contemporary
+record about 1181. In precision and fulness of detail the _Ymagines_ are
+inferior to the chronicles of the so-called Benedict and of Hoveden.
+Though an annalist, Diceto is careless in his chronology; and the
+documents which he incorporates, while often important, are selected on
+no principle. He has little sense of style; but displays considerable
+insight when he ventures to discuss a political situation. For this
+reason, and on account of the details with which they supplement the
+more important chronicles of the period, the _Ymagines_ are a valuable
+though a secondary source.
+
+  See W. Stubbs' edition of the _Historical Works_ of Diceto (Rolls ed.
+  1876, 2 vols.), and especially the introduction. The second volume
+  contains minor works which are the barest compendia of facts taken
+  from well-known sources. Diceto's fragmentary Domesday of the
+  capitular estates has been edited by Archdeacon Hale in _The Domesday
+  of St Paul's_, pp. 109 ff. (Camden Society, 1858).
+
+
+
+
+DICEY, EDWARD (1832- ), English writer, son of T. E. Dicey of Claybrook
+Hall, Leicestershire, was born in 1832. Educated at Trinity College,
+Cambridge, where he took mathematical and classical honours, he became
+an active journalist, contributing largely to the principal reviews. He
+was called to the bar in 1875, became a bencher of Gray's Inn in 1896,
+and was treasurer in 1903-1904. He was connected with the _Daily
+Telegraph_ as leader writer and then as special correspondent, and after
+a short spell in 1870 as editor of the _Daily News_ he became editor of
+the _Observer_, a position which he held until 1889. Of his many books
+on foreign affairs perhaps the most important are his _England and
+Egypt_ (1884), _Bulgaria, the Peasant State_ (1895), _The Story of the
+Khedivate_ (1902), and _The Egypt of the Future_ (1907). He was created
+C.B. in 1886.
+
+His brother ALBERT VENN DICEY (b. 1835), English jurist, was educated at
+Balliol College, Oxford, where he took a first class in the classical
+schools in 1858. He was called to the bar at the Inner Temple in 1863.
+He held fellowships successively at Balliol, Trinity and All Souls', and
+from 1882 to 1909 was Vinerian professor of law. He became Q.C. in 1890.
+His chief works are the _Introduction to the Study of the Law of the
+Constitution_ (1885, 6th ed. 1902), which ranks as a standard work on
+the subject; _England's Case against Home Rule_ (1886); _A Digest of the
+Law of England with Reference to the Conflict of Laws_ (1896), and
+_Lectures on the Relation between Law and Public Opinion in England
+during the 19th century_ (1905).
+
+
+
+
+DICHOTOMY (Gr. [Greek: dicha], apart, [Greek: temnein], to cut),
+literally a cutting asunder, the technical term for a form of logical
+division, consisting in the separation of a genus into two species, one
+of which has and the other has not, a certain quality or attribute. Thus
+men may be thus divided into white men, and men who are not white; each
+of these may be subdivided similarly. On the principle of contradiction
+this division is both exhaustive and exclusive; there can be no
+overlapping, and no members of the original genus or the lower groups
+are omitted. This method of classification, though formally accurate,
+has slight value in the exact sciences, partly because at every step one
+of the two groups is merely negatively characterized and therefore
+incapable of real subdivision; it is useful, however, in setting forth
+clearly the gradual descent from the most inclusive genus (_summum
+genus_) through species to the lowest class (_infima species_), which is
+divisible only into individual persons or things. (See further
+DIVISION.) In astronomy the term is used for the aspect of the moon or
+of a planet when apparently half illuminated, so that its disk has the
+form of a semicircle.
+
+
+
+
+DICK, ROBERT (1811-1866), Scottish geologist and botanist, was born at
+Tullibody, in Clackmannanshire, in January 1811. His father was an
+officer of excise. At the age of thirteen, after receiving a good
+elementary education at the parish school, Robert Dick was apprenticed
+to a baker, and served for three years. In these early days he became
+interested in wild flowers--he made a collection of plants and gradually
+acquired some knowledge of their names from an old encyclopaedia. When
+his time was out he left Tullibody and gained employment as a journeyman
+baker at Leith, Glasgow and Greenock. Meanwhile his father, who in 1826
+had been removed to Thurso, as supervisor of excise, advised his son to
+set up a baker's shop in that town. Thither Robert Dick went in 1830, he
+started in business as a baker and worked laboriously until he died on
+the 24th of December 1866. Throughout this period he zealously devoted
+himself to studying and collecting the plants, mollusca and insects of a
+wide area of Caithness, and his attention was directed soon after he
+settled in Thurso to the rocks and fossils. In 1835 he first found
+remains of fossil fishes; but it was not till some years later that his
+interest became greatly stirred. Then he obtained a copy of Hugh
+Miller's _Old Red Sandstone_ (published in 1841), and he began
+systematically to collect with hammer and chisel the fossils from the
+Caithness flags. In 1845 he found remains of _Holoptychius_ and
+forwarded specimens to Hugh Miller, and he continued to send the best of
+his fossil fishes to that geologist, and to others after the death of
+Miller. In this way he largely contributed to the progress of geological
+knowledge, although he himself published nothing and was ever averse
+from publicity. His herbarium, which consisted of about 200 folios of
+mosses, ferns and flowering plants "almost unique in its completeness,"
+is now stored, with many of his fossils, in the museum at Thurso. Dick
+had a hard struggle for existence, especially through competition during
+his late years, when he was reduced almost to beggary: but of this few,
+if any, of his friends were aware until it was too late. A monument
+erected in the new cemetery at Thurso testifies to the respect which his
+life-work created, when the merits of this enthusiastic naturalist came
+to be appreciated.
+
+  See _Robert Dick, Baker of Thurso, Geologist and Botanist_, by Samuel
+  Smiles (1878).
+
+
+
+
+DICK, THOMAS (1774-1857), Scottish writer on astronomy, was born at
+Dundee on the 24th of November 1774. The appearance of a brilliant
+meteor inspired him, when in his ninth year, with a passion for
+astronomy; and at the age of sixteen he forsook the loom, and supported
+himself by teaching. In 1794 he entered the university of Edinburgh, and
+set up a school on the termination of his course; then, in 1801, took
+out a licence to preach, and officiated for some years as probationer in
+the United Presbyterian church. From about 1807 to 1817 he taught in the
+secession school at Methven in Perthshire, and during the ensuing decade
+in that of Perth, where he composed his first substantive book, _The
+Christian Philosopher_ (1823, 8th ed. 1842). Its success determined his
+vocation as an author; he built himself, in 1827, a cottage at Broughty
+Ferry, near Dundee, and devoted himself wholly to literary and
+scientific pursuits. They proved, however, owing to his unpractical turn
+of mind, but slightly remunerative, and he was in 1847 relieved from
+actual poverty by a crown pension of £50 a year, eked out by a local
+subscription. He died on the 29th of July 1857. His best-known works
+are: _Celestial Scenery_ (1837), _The Sidereal Heavens_ (1840), and _The
+Practical Astronomer_ (1845), in which is contained (p. 204) a
+remarkable forecast of the powers and uses of celestial photography.
+Written with competent knowledge, and in an agreeable style, they
+obtained deserved and widespread popularity.
+
+  See R. Chambers's _Eminent Scotsmen_ (ed. 1868); _Monthly Notices Roy.
+  Astr. Society_, xviii. 98; _Athenaeum_ (1857), p. 1008. (A. M. C.)
+
+
+
+
+DICKENS, CHARLES JOHN HUFFAM (1812-1870), English novelist, was born on
+the 7th of February 1812 at a house in the Mile End Terrace, Commercial
+Road, Landport (Portsea)--a house which was opened as a Dickens Museum
+on 22nd July 1904. His father John Dickens (d. 1851), a clerk in the
+navy-pay office on a salary of £80 a year, and stationed for the time
+being at Portsmouth, had married in 1809 Elizabeth, daughter of Thomas
+Barrow, and she bore him a family of eight children, Charles being the
+second. In the winter of 1814 the family moved from Portsea in the snow,
+as he remembered, to London, and lodged for a time near the Middlesex
+hospital. The country of the novelist's childhood, however, was the
+kingdom of Kent, where the family was established in proximity to the
+dockyard at Chatham from 1816 to 1821. He looked upon himself in later
+years as a man of Kent, and his capital abode as that in Ordnance
+Terrace, or 18 St Mary's Place, Chatham, amid surroundings classified in
+Mr Pickwick's notes as "appearing" to be soldiers, sailors, Jews, chalk,
+shrimps, officers and dockyard men. He fell into a family the general
+tendency of which was to go down in the world, during one of its easier
+periods (John Dickens was now fifth clerk on £250 a year), and he always
+regarded himself as belonging by right to a comfortable, genteel, lower
+middle-class stratum of society. His mother taught him to read; to his
+father he appeared very early in the light of a young prodigy, and by
+him Charles was made to sit on a tall chair and warble popular ballads,
+or even to tell stories and anecdotes for the benefit of fellow-clerks
+in the office. John Dickens, however, had a small collection of books
+which were kept in a little room upstairs that led out of Charles's own,
+and in this attic the boy found his true literary instructors in
+_Roderick Random_, _Peregrine Pickle_, _Humphry Clinker_, _Tom Jones_,
+_The Vicar of Wakefield_, _Don Quixote_, _Gil Blas_ and _Robinson
+Crusoe_. The story of how he played at the characters in these books and
+sustained his idea of Roderick Random for a month at a stretch is
+picturesquely told in _David Copperfield_. Here as well as in his first
+and last books and in what many regard as his best, _Great
+Expectations_, Dickens returns with unabated fondness and mastery to the
+surroundings of his childhood. From seven to nine years he was at a
+school kept in Clover Lane, Chatham, by a Baptist minister named William
+Giles, who gave him Goldsmith's _Bee_ as a keepsake when the call to
+Somerset House necessitated the removal of the family from Rochester to
+a shabby house in Bayham Street, Camden Town. At the very moment when a
+consciousness of capacity was beginning to plump his youthful ambitions,
+the whole flattering dream vanished and left not a rack behind.
+Happiness and Chatham had been left behind together, and Charles was
+about to enter a school far sterner and also far more instructive than
+that in Clover Lane. The family income had been first decreased and then
+mortgaged; the creditors of the "prodigal father" would not give him
+time; John Dickens was consigned to the Marshalsea; Mrs Dickens started
+an "Educational Establishment" as a forlorn hope in Upper Gower Street;
+and Charles, who had helped his mother with the children, blacked the
+boots, carried things to the pawnshop and done other menial work, was
+now sent out to earn his own living as a young hand in a blacking
+warehouse, at Old Hungerford Stairs, on a salary of six shillings a
+week. He tied, trimmed and labelled blacking pots for over a year,
+dining off a saveloy and a slice of pudding, consorting with two very
+rough boys, Bob Fagin and Pol Green, and sleeping in an attic in Little
+College Street, Camden Town, in the house of Mrs Roylance (Pipchin),
+while on Sunday he spent the day with his parents in their comfortable
+prison, where they had the services of a "marchioness" imported from the
+Chatham workhouse.
+
+Already consumed by ambition, proud, sensitive and on his dignity to an
+extent not uncommon among boys of talent, he felt his position keenly,
+and in later years worked himself up into a passion of self-pity in
+connexion with the "degradation" and "humiliation" of this episode. The
+two years of childish hardship which ate like iron into his soul were
+obviously of supreme importance in the growth of the novelist.
+Recollections of the streets and the prison and its purlieus supplied
+him with a store of literary material upon which he drew through all the
+years of his best activity. And the bitterness of such an experience was
+not prolonged sufficiently to become sour. From 1824 to 1826, having
+been rescued by a family quarrel and by a windfall in the shape of a
+legacy to his father, from the warehouse, he spent two years at an
+academy known as Wellington House, at the corner of Granby Street and
+the Hampstead Road (the lighter traits of which are reproduced in Salem
+House), and was there known as a merry and rather mischievous boy.
+Fortunately he learned nothing there to compromise the results of
+previous instruction. His father had now emerged from the Marshalsea and
+was seeking employment as a parliamentary reporter. A Gray's Inn
+solicitor with whom he had had dealings was attracted by the bright,
+clever look of Charles, and took him into his office as a boy at a
+salary of thirteen and sixpence (rising to fifteen shillings) a week. He
+remained in Mr Blackmore's office from May 1827 to November 1828, but he
+had lost none of his eager thirst for distinction, and spent all his
+spare time mastering Gurney's shorthand and reading early and late at
+the British Museum. A more industrious apprentice in the lower grades of
+the literary profession has never been known, and the consciousness of
+opportunities used to the most splendid advantage can hardly have been
+absent from the man who was shortly to take his place at the head of it
+as if to the manner born. Lowten and Guppy, and Swiveller had been
+observed from this office lad's stool; he was now greatly to widen his
+area of study as a reporter in Doctors' Commons and various police
+courts, including Bow Street, working all day at law and much of the
+night at shorthand. Some one asked John Dickens, during the first eager
+period of curiosity as to the man behind "Pickwick," where his son
+Charles was educated. "Well really," said the prodigal father, "he may
+be said--haw--haw--to have educated himself." He was one of the most
+rapid and accurate reporters in London when, at nineteen years of age,
+in 1831, he realized his immediate ambition and "entered the gallery" as
+parliamentary reporter to the _True Sun_. Later he was reporter to the
+_Mirror of Parliament_ and then to the _Morning Chronicle_. Several of
+his earliest letters are concerned with his exploits as a reporter, and
+allude to the experiences he had, travelling fifteen miles an hour and
+being upset in almost every description of known vehicle in various
+parts of Britain between 1831 and 1836. The family was now living in
+Bentwick Street, Manchester Square, but John Dickens was still no
+infrequent inmate of the sponging-houses. With all the accessories of
+these places of entertainment his son had grown to be excessively
+familiar. Writing about 1832 to his school friend Tom Mitton, Dickens
+tells him that his father has been arrested at the suit of a wine firm,
+and begs him go over to Cursitor Street and see what can be done. On
+another occasion of a paternal disappearance he observes: "I own that
+his absence does not give me any great uneasiness, knowing how apt he is
+to get out of the way when anything goes wrong." In yet another letter
+he asks for a loan of four shillings.
+
+In the meanwhile, however, he had commenced author in a more creative
+sense by penning some sketches of contemporary London life, such as he
+had attempted in his school days in imitation of the sketches published
+in the _London_ and other magazines of that day. The first of these
+appeared in the December number of the _Old Monthly Magazine_ for 1833.
+By the following August, when the signature "Boz" was first given, five
+of these sketches had appeared. By the end of 1834 we find him settled
+in rooms in Furnival's Inn, and a little later his salary on the
+_Morning Chronicle_ was raised, owing to the intervention of one of its
+chiefs, George Hogarth, the father of (in addition to six sons) eight
+charming daughters, to one of whom, Catherine, Charles was engaged to be
+married before the year was out. Clearly as his career now seemed
+designated, he was at this time or a little before it coquetting very
+seriously with the stage: but circumstances were rapidly to determine
+another stage in his career. A year before Queen Victoria's accession
+appeared in two volumes _Sketches by Boz_, _Illustrative of Everyday
+Life and Everyday People_. The book came from a prentice hand, but like
+the little tract on the Puritan abuse of the Sabbath entitled "Sunday
+under three Heads" which appeared a few months later, it contains in
+germ all, or almost all, the future Dickens. Glance at the headings of
+the pages. Here we have the Beadle and all connected with him, London
+streets, theatres, shows, the pawnshop, Doctors' Commons, Christmas,
+Newgate, coaching, the river. Here comes a satirical picture of
+parliament, fun made of cheap snobbery, a rap on the knuckles of
+sectarianism. And what could be more prophetic than the title of the
+opening chapter--Our Parish? With the Parish--a large one
+indeed--Dickens to the end concerned himself; he began with a rapid
+survey of his whole field, hinting at all he might accomplish,
+indicating the limits he was not to pass. This year was to be still more
+momentous to Dickens, for, on the 2nd of April 1836, he was married to
+George Hogarth's eldest daughter Catherine. He seems to have fallen in
+love with the daughters collectively, and, judging by subsequent events,
+it has been suggested that perhaps he married the wrong one. His wife's
+sister Mary was the romance of his early married life, and another
+sister, Georgina, was the dearest friend of his last ten years.
+
+A few days before the marriage, just two months after the appearance of
+the _Sketches_, the first part of _The Posthumous Papers of the Pickwick
+Club_ was announced. One of the chief vogues of the day was the issue of
+humorous, sporting or anecdotal novels in parts, with plates, and some
+of the best talent of the day, represented by Ainsworth, Bulwer,
+Marryat, Maxwell, Egan, Hook and Surtees, had been pressed into this
+kind of enterprise. The publishers of the day had not been slow to
+perceive Dickens's aptitude for this species of "letterpress." A member
+of the firm of Chapman & Hall called upon him at Furnival's Inn in
+December 1835 with a proposal that he should write about a Nimrod Club
+of amateur sportsmen, foredoomed to perpetual ignominies, while the
+comic illustrations were to be etched by Seymour, a well-known rival of
+Cruikshank (the illustrator of _Boz_). The offer was too tempting for
+Dickens to refuse, but he changed the idea from a club of Cockney
+sportsmen to that of a club of eccentric peripatetics, on the sensible
+grounds, first that sporting sketches were stale, and, secondly, that he
+knew nothing worth speaking of about sport. The first seven pictures
+appeared with the signature of Seymour and the letterpress of Dickens.
+Before the eighth picture appeared Seymour had blown his brains out.
+After a brief interval of Buss, Dickens obtained the services of Hablot
+K. Browne, known to all as "Phiz." Author and illustrator were as well
+suited to one another and to the common creation of a unique thing as
+Gilbert and Sullivan. Having early got rid of the sporting element,
+Dickens found himself at once. The subject exactly suited his knowledge,
+his skill in arranging incidents--nay, his very limitations too. No
+modern book is so incalculable. We commence laughing heartily at
+Pickwick and his troupe. The laugh becomes kindlier. We are led on
+through a tangle of adventure, never dreaming what is before us. The
+landscape changes: Pickwick becomes the symbol of kindheartedness,
+simplicity and innocent levity. Suddenly in the Fleet Prison a deeper
+note is struck. The medley of human relationships, the loneliness, the
+mystery and sadness of human destinies are fathomed. The tragedy of
+human life is revealed to us amid its most farcical elements. The droll
+and laughable figure of the hero is transfigured by the kindliness of
+human sympathy into a beneficent and bespectacled angel in shorts and
+gaiters. By defying accepted rules, Dickens had transcended the limited
+sphere hitherto allotted to his art: he had produced a book to be
+enshrined henceforth in the inmost hearts of all sorts and conditions of
+his countrymen, and had definitely enlarged the boundaries of English
+humour and English fiction. As for Mr Pickwick, he is a fairy like Puck
+or Santa Claus, while his creator is "the last of the mythologists and
+perhaps the greatest."
+
+When _The Pickwick Papers_ appeared in book form at the close of 1837
+Dickens's popular reputation was made. From the appearance of Sam Weller
+in part v. the universal hunger for the monthly parts had risen to a
+furore. The book was promptly translated into French and German. The
+author had received little assistance from press or critics, he had no
+influential connexions, his class of subjects was such as to "expose him
+at the outset to the fatal objections of vulgarity," yet in less than
+six months from the appearance of the first number, as the _Quarterly
+Review_ almost ruefully admits, the whole reading world was talking
+about the Pickwickians. The names of Winkle, Wardle, Weller, Jingle,
+Snodgrass, Dodson & Fogg, were as familiar as household words. Pickwick
+chintzes figured in the linendrapers' windows, and Pickwick cigars in
+every tobacconist's; Weller corduroys became the stock-in-trade of every
+breeches-maker; Boz cabs might be seen rattling through the streets, and
+the portrait of the author of _Pelham_ and _Crichton_ was scraped down
+to make way for that of the new popular favourite on the omnibuses. A
+new and original genius had suddenly sprung up, there was no denying it,
+even though, as the _Quarterly_ concluded, "it required no gift of
+prophecy to foretell his fate--he has risen like a rocket and he will
+come down like the stick." It would have needed a very emphatic gift of
+prophecy indeed to foretell that Dickens's reputation would have gone on
+rising until at the present day (after one sharp fall, which reached an
+extreme about 1887) it stands higher than it has ever stood before.
+
+Dickens's assumption of the literary purple was as amazing as anything
+else about him. Accepting the homage of the luminaries of the literary,
+artistic and polite worlds as if it had been his natural due, he
+arranges for the settlement of his family, decrees, like another Edmund
+Kean, that his son is to go to Eton, carries on the most complicated
+negotiations with his publishers and editors, presides and orates with
+incomparable force at innumerable banquets, public and private, arranges
+elaborate villegiatures in the country, at the seaside, in France or in
+Italy, arbitrates in public on every topic, political, ethical,
+artistic, social or literary, entertains and legislates for an
+increasingly large domestic circle, both juvenile and adult, rules
+himself and his time-table with a rod of iron. In his letter-writing
+alone, Dickens did a life's literary work. Nowadays no one thinks of
+writing such letters; that is to say, letters of such length and detail,
+for the quality is Dickens's own. He evidently enjoyed this use of the
+pen. Page after page of Forster's _Life_ (750 pages in the _Letters_
+edited by his daughter and sister-in-law) is occupied with transcription
+from private correspondence, and never a line of this but is thoroughly
+worthy of print and preservation. If he makes a tour in any part of the
+British Isles, he writes a full description of all he sees, of
+everything that happens, and writes it with such gusto, such mirth, such
+strokes of fine picturing, as appear in no other private letters ever
+given to the public. Naturally buoyant in all circumstances, a holiday
+gave him the exhilaration of a schoolboy. See how he writes from
+Cornwall, when on a trip with two or three friends, in 1843. "Heavens!
+if you could have seen the necks of bottles, distracting in their
+immense variety of shape, peering out of the carriage pockets! If you
+could have witnessed the deep devotion of the post-boys, the maniac glee
+of the waiters! If you could have followed us into the earthy old
+churches we visited, and into the strange caverns on the gloomy
+seashore, and down into the depths of mines, and up to the tops of giddy
+heights, where the unspeakably green water was roaring, I don't know how
+many hundred feet below.... I never laughed in my life as I did on this
+journey. It would have done you good to hear me. I was choking and
+gasping and bursting the buckles off the back of my stock, all the way.
+And Stanfield"--the painter--"got into such apoplectic entanglements
+that we were obliged to beat him on the back with portmanteaus before we
+could recover him."
+
+The animation of Dickens's look would attract the attention of any one,
+anywhere. His figure was not that of an Adonis, but his brightness made
+him the centre and pivot of every society he was in. The keenness and
+vivacity of his eye combined with his inordinate appetite for life to
+give the unique quality to all that he wrote. His instrument is that of
+the direct, sinewy English of Smollett, combined with much of the
+humorous grace of Goldsmith (his two favourite authors), but modernized
+to a certain extent under the influence of Washington Irving, Sydney
+Smith, Jeffrey, Lamb, and other writers of the _London Magazine_. He
+taught himself to speak French and Italian, but he could have read
+little in any language. His ideas were those of the inchoate and insular
+liberalism of the 'thirties. His unique force in literature he was to
+owe to no supreme artistic or intellectual quality, but almost entirely
+to his inordinate gift of observation, his sympathy with the humble, his
+power over the emotions and his incomparable endowment of unalloyed
+human fun. To contemporaries he was not so much a man as an
+institution, at the very mention of whose name faces were puckered with
+grins or wreathed in smiles. To many his work was a revelation, the
+revelation of a new world and one far better than their own. And his
+influence went further than this in the direction of revolution or
+revival. It gave what were then universally referred to as "the lower
+orders" a new sense of self-respect, a new feeling of citizenship. Like
+the defiance of another Luther, or the Declaration of a new
+Independence, it emitted a fresh ray of hope across the firmament. He
+did for the whole English-speaking race what Burns had done for
+Scotland--he gave it a new conceit of itself. He knew what a people
+wanted and he told what he knew. He could do this better than anybody
+else because his mind was theirs. He shared many of their "great useless
+virtues," among which generosity ranks before justice, and sympathy
+before truth, even though, true to his middle-class vein, he exalts
+piety, chastity and honesty in a manner somewhat alien to the mind of
+the low-bred man. This is what makes Dickens such a demigod and his
+public success such a marvel, and this also is why any exclusively
+literary criticism of his work is bound to be so inadequate. It should
+also help us to make the necessary allowances for the man. Dickens, even
+the Dickens of legend that we know, is far from perfect. The Dickens of
+reality to which Time may furnish a nearer approximation is far less
+perfect. But when we consider the corroding influence of adulation, and
+the intoxication of unbridled success, we cannot but wonder at the
+relatively high level of moderation and self-control that Dickens almost
+invariably observed. Mr G. K. Chesterton remarks suggestively that
+Dickens had all his life the faults of the little boy who is kept up too
+late at night. He is overwrought by happiness to the verge of
+exasperation, and yet as a matter of fact he does keep on the right side
+of the breaking point. The specific and curative in his case was the
+work in which he took such anxious pride, and such unmitigated delight.
+He revelled in punctual and regular work; at his desk he was often in
+the highest spirits. Behold how he pictured himself, one day at
+Broadstairs, where he was writing _Chuzzlewit_. "In a bay-window in a
+one-pair sits, from nine o'clock to one, a gentleman with rather long
+hair and no neckcloth, who writes and grins, as if he thought he was
+very funny indeed. At one he disappears, presently emerges from a
+bathing-machine, and may be seen, a kind of salmon-colour porpoise,
+splashing about in the ocean. After that, he may be viewed in another
+bay-window on the ground-floor eating a strong lunch; and after that,
+walking a dozen miles or so, or lying on his back on the sand reading a
+book. Nobody bothers him, unless they know he is disposed to be talked
+to, and I am told he is very comfortable indeed. He's as brown as a
+berry, and they do say he is as good as a small fortune to the
+innkeeper, who sells beer and cold punch." Here is the secret of such
+work as that of Dickens; it is done with delight--done (in a sense)
+easily, done with the mechanism of mind and body in splendid order. Even
+so did Scott write; though more rapidly and with less conscious care:
+his chapter finished before the world had got up to breakfast. Later,
+Dickens produced novels less excellent with much more of mental strain.
+The effects of age could not have shown themselves so soon, but for the
+unfortunate loss of energy involved in his non-literary labours.
+
+While the public were still rejoicing in the first sprightly runnings of
+the "new humour," the humorist set to work desperately on the grim
+scenes of _Oliver Twist_, the story of a parish orphan, the nucleus of
+which had already seen the light in his _Sketches_. The early scenes are
+of a harrowing reality, despite the germ of forced pathos which the
+observant reader may detect in the pitiful parting between Oliver and
+little Dick; but what will strike every reader at once in this book is
+the directness and power of the English style, so nervous and unadorned:
+from its unmistakable clearness and vigour Dickens was to travel far as
+time went on. But the full effect of the old simplicity is felt in such
+masterpieces of description as the drive of Oliver and Sikes to
+Chertsey, the condemned-cell ecstasy of Fagin, or the unforgettable
+first encounter between Oliver and the Artful Dodger. Before November
+1837 had ended, Charles Dickens entered on an engagement to write a
+successor to _Pickwick_ on similar lines of publication. _Oliver Twist_
+was then in mid-career; a _Life of Grimaldi_ and _Barnaby Rudge_ were
+already covenanted for. Dickens forged ahead with the new tale of
+_Nicholas Nickleby_ and was justified by the results, for its sale far
+surpassed even that of _Pickwick_. As a conception it is one of his
+weakest. An unmistakably 18th-century character pervades it. Some of the
+vignettes are among the most piquant and besetting ever written. Large
+parts of it are totally unobserved conventional melodrama; but the
+Portsmouth Theatre and Dotheboys Hall and Mrs Nickleby (based to some
+extent, it is thought, upon Miss Bates in _Emma_, but also upon the
+author's Mamma) live for ever as Dickens conceived them in the pages of
+_Nicholas Nickleby_.
+
+Having got rid of _Nicholas Nickleby_ and resigned his editorship of
+_Bentley's Miscellany_, in which _Oliver Twist_ originally appeared,
+Dickens conceived the idea of a weekly periodical to be issued as
+_Master Humphrey's Clock_, to comprise short stories, essays and
+miscellaneous papers, after the model of Addison's _Spectator_. To make
+the weekly numbers "go," he introduced Mr Pickwick, Sam Weller and his
+father in friendly intercourse. But the public requisitioned "a story,"
+and in No. 4 he had to brace himself up to give them one. Thus was
+commenced _The Old Curiosity Shop_, which was continued with slight
+interruptions, and followed by _Barnaby Rudge_. For the first time we
+find Dickens obsessed by a highly complicated plot. The tonality
+achieved in _The Old Curiosity Shop_ surpassed anything he had attempted
+in this difficult vein, while the rich humour of Dick Swiveller and the
+Marchioness, and the vivid portraiture of the wandering Bohemians,
+attain the very highest level of Dickensian drollery; but in the
+lamentable tale of Little Nell (though Landor and Jeffrey thought the
+character-drawing of this infant comparable with that of Cordelia), it
+is generally admitted that he committed an indecent assault upon the
+emotions by exhibiting a veritable monster of piety and long-suffering
+in a child of tender years. In _Barnaby Rudge_ he was manifestly
+affected by the influence of Scott, whose achievements he always
+regarded with a touching veneration. The plot, again, is of the utmost
+complexity, and Edgar Allan Poe (who predicted the conclusion) must be
+one of the few persons who ever really mastered it. But few of Dickens's
+books are written in a more admirable style.
+
+_Master Humphrey's Clock_ concluded, Dickens started in 1842 on his
+first visit to America--an episode hitherto without parallel in English
+literary history, for he was received everywhere with popular
+acclamation as the representative of a grand triumph of the English
+language and imagination, without regard to distinctions of nationality.
+He offended the American public grievously by a few words of frank
+description and a few quotations of the advertisement columns of
+American papers illustrating the essential barbarity of the old slave
+system (_American Notes_). Dickens was soon pining for home--no English
+writer is more essentially and insularly English in inspiration and
+aspiration than he is. He still brooded over the perverseness of America
+on the copyright question, and in his next book he took the opportunity
+of uttering a few of his impressions about the objectionable sides of
+American democracy, the result being that "all Yankee-doodle-dom blazed
+up like one universal soda bottle," as Carlyle said. _Martin Chuzzlewit_
+(1843-1844) is important as closing his great character period. His
+_sève originale_, as the French would say, was by this time to a
+considerable extent exhausted, and he had to depend more upon artistic
+elaboration, upon satires, upon _tours de force_ of description, upon
+romantic and ingenious contrivances. But all these resources combined
+proved unequal to his powers as an original observer of popular types,
+until he reinforced himself by autobiographic reminiscence, as in _David
+Copperfield_ and _Great Expectations_, the two great books remaining to
+his later career.
+
+After these two masterpieces and the three wonderful books with which he
+made his début, we are inclined to rank _Chuzzlewit_. Nothing in Dickens
+is more admirably seen and presented than Todgers's, a bit of London
+particular cut out with a knife. Mr Pecksniff and Mrs Gamp, Betsy Prig
+and "Mrs Harris" have passed into the national language and life. The
+coach journey, the windy autumn night, the stealthy trail of Jonas, the
+undertone of tragedy in the Charity and Mercy and Chuffey episodes
+suggest a blending of imaginative vision and physical penetration hardly
+seen elsewhere. Two things are specially notable about this novel--the
+exceptional care taken over it (as shown by the interlineations in the
+MS.) and the caprice or nonchalance of the purchasing public, its sales
+being far lower than those of any of its monthly predecessors.
+
+At the close of 1843, to pay outstanding debts of his now lavish
+housekeeping, he wrote that pioneer of Christmas numbers, that national
+benefit as Thackeray called it, _A Christmas Carol_. It failed to
+realize his pecuniary anticipations, and Dickens resolved upon a drastic
+policy of retrenchment and reform. He would save expense by living
+abroad and would punish his publishers by withdrawing his custom from
+them, at least for a time. Like everything else upon which he ever
+determined, this resolution was carried out with the greatest possible
+precision and despatch. In June 1844 he set out for Marseilles with his
+now rapidly increasing family (the journey cost him £200). In a villa on
+the outskirts of Genoa he wrote _The Chimes_, which, during a brief
+excursion to London before Christmas, he read to a select circle of
+friends (the germ of his subsequent lecture-audiences), including
+Forster, Carlyle, Stanfield, Dyce, Maclise and Jerrold. He was again in
+London in 1845, enjoying his favourite diversion of private theatricals;
+and in January 1846 he experimented briefly as the editor of a London
+morning paper--the _Daily News_. By early spring he was back at
+Lausanne, writing his customary vivid letters to his friends, craving as
+usual for London streets, commencing _Dombey and Son_, and walking his
+fourteen miles daily. The success of _Dombey and Son_ completely
+rehabilitated the master's finances, enabled him to return to England,
+send his son to Eton and to begin to save money. Artistically it is less
+satisfactory; it contains some of Dickens's prime curios, such as
+Cuttle, Bunsby, Toots, Blimber, Pipchin, Mrs MacStinger and young Biler;
+it contains also that masterpiece of sentimentality which trembles upon
+the borderland of the sublime and the ridiculous, the death of Paul
+Dombey ("that sweet Paul," as Jeffrey, the "critic laureate," called
+him), and some grievous and unquestionable blemishes. As a narrative,
+moreover, it tails off into a highly complicated and exacting plot. It
+was followed by a long rest at Broadstairs before Dickens returned to
+the native home of his genius, and early in 1849 "began to prepare for
+_David Copperfield_."
+
+"Of all my books," Dickens wrote, "I like this the best; like many fond
+parents I have my favourite child, and his name is David Copperfield."
+In some respects it stands to Dickens in something of the same relation
+in which the contemporary _Pendennis_ stands to Thackeray. As in that
+book, too, the earlier portions are the best. They gained in intensity
+by the autobiographical form into which they are thrown; as Thackeray
+observed, there was no writing against such power. The tragedy of Emily
+and the character of Rosa Dartle are stagey and unreal; Uriah Heep is
+bad art; Agnes, again, is far less convincing as a consolation than
+Dickens would have us believe; but these are more than compensated by
+the wonderful realization of early boyhood in the book, by the picture
+of Mr Creakle's school, the Peggottys, the inimitable Mr Micawber, Betsy
+Trotwood and that monument of selfish misery, Mrs Gummidge.
+
+At the end of March 1850 commenced the new twopenny weekly called
+_Household Words_, which Dickens planned to form a direct means of
+communication between himself and his readers, and as a means of
+collecting around him and encouraging the talents of the younger
+generation. No one was better qualified than he for this work, whether
+we consider his complete freedom from literary jealousy or his magical
+gift of inspiring young authors. Following the somewhat dreary and
+incoherent _Bleak House_ of 1852, _Hard Times_ (1854)--an
+anti-Manchester School tract, which Ruskin regarded as Dickens's best
+work--was the first long story written for _Household Words_. About this
+time Dickens made his final home at Gad's Hill, near Rochester, and put
+the finishing touch to another long novel published upon the old plan,
+_Little Dorrit_ (1855-1857). In spite of the exquisite comedy of the
+master of the Marshalsea and the final tragedy of the central figure,
+_Little Dorrit_ is sadly deficient in the old vitality, the humour is
+often a mock reality, and the repetition of comic catch-words and
+overstrung similes and metaphors is such as to affect the reader with
+nervous irritation. The plot and characters ruin each other in this
+amorphous production. The _Tale of Two Cities_, commenced in _All the
+Year Round_ (the successor of _Household Words_) in 1859, is much
+better: the main characters are powerful, the story genuinely tragic,
+and the atmosphere lurid; but enormous labour was everywhere expended
+upon the construction of stylistic ornament.
+
+The _Tale of Two Cities_ was followed by two finer efforts at
+atmospheric delineation, the best things he ever did of this kind:
+_Great Expectations_ (1861), over which there broods the mournful
+impression of the foggy marshes of the Lower Thames; and _Our Mutual
+Friend_ (1864-1865), in which the ooze and mud and slime of Rotherhithe,
+its boatmen and loafers, are made to pervade the whole book with
+cumulative effect. The general effect produced by the stories is,
+however, very different. In the first case, the foreground was supplied
+by autobiographical material of the most vivid interest, and the
+lucidity of the creative impulse impelled him to write upon this
+occasion with the old simplicity, though with an added power. Nothing
+therefore, in the whole range of Dickens surpassed the early chapters of
+_Great Expectations_ in perfection of technique or in mastery of all the
+resources of the novelist's art. To have created Abel Magwitch alone is
+to be a god indeed, says Mr Swinburne, among the creators of deathless
+men. Pumblechook is actually better and droller and truer to imaginative
+life than Pecksniff; Joe Gargery is worthy to have been praised and
+loved at once by Fielding and by Sterne: Mr Jaggers and his clients, Mr
+Wemmick and his parent and his bride, are such figures as Shakespeare,
+when dropping out of poetry, might have created, if his lot had been
+cast in a later century. "Can as much be said," Mr Swinburne boldly
+asks, "for the creatures of any other man or god?"
+
+In November 1867 Dickens made a second expedition to America, leaving
+all the writing that he was ever to complete behind him. He was to make
+a round sum of money, enough to free him from all embarrassments, by a
+long series of exhausting readings, commencing at the Tremont Temple,
+Boston, on the 2nd of December. The strain of Dickens's ordinary life
+was so tense and so continuous that it is, perhaps, rash to assume that
+he broke down eventually under this particular stress; for other
+reasons, however, his persistence in these readings, subsequent to his
+return, was strongly deprecated by his literary friends, led by the
+arbitrary and relentless Forster. It is a long testimony to Dickens's
+self-restraint, even in his most capricious and despotic moments, that
+he never broke the cord of obligation which bound him to his literary
+mentor, though sparring matches between them were latterly of frequent
+occurrence. His farewell reading was given on the 15th of March 1870, at
+St James's Hall. He then vanished from "those garish lights," as he
+called them, "for evermore." Of the three brief months that remained to
+him, his last book, _The Mystery of Edwin Drood_, was the chief
+occupation. It hardly promised to become a masterpiece (Longfellow's
+opinion) as did Thackeray's _Denis Duval_, but contained much fine
+descriptive technique, grouped round a scene of which Dickens had an
+unrivalled sympathetic knowledge.
+
+In March and April 1870 Dickens, as was his wont, was mixing in the best
+society; he dined with the prince at Lord Houghton's and was twice at
+court, once at a long deferred private interview with the queen, who had
+given him a presentation copy of her _Leaves from a Journal of our Life
+in the Highlands_ with the inscription "From one of the humblest of
+authors to one of the greatest"; and who now begged him on his
+persistent refusal of any other title to accept the nominal distinction
+of a privy councillor. He took for four months the Milner Gibsons' house
+at 5 Hyde Park Place, opposite the Marble Arch, where he gave a
+brilliant reception on the 7th of April. His last public appearance was
+made at the Royal Academy banquet early in May. He returned to his
+regular methodical routine of work at Gad's Hill on the 30th of May, and
+one of the last instalments he wrote of _Edwin Drood_ contained an
+ominous speculation as to the next two people to die at Cloisterham:
+"Curious to make a guess at the two, or say at one of the two." Two
+letters bearing the well-known superscription "Gad's Hill Place, Higham
+by Rochester, Kent" are dated the 8th of June, and, on the same
+Thursday, after a long spell of writing in the Châlet where he
+habitually wrote, he collapsed suddenly at dinner. Startled by the
+sudden change in the colour and expression of his face, his
+sister-in-law (Miss Hogarth) asked him if he was ill; he said "Yes, very
+ill," but added that he would finish dinner and go on afterwards to
+London. "Come and lie down," she entreated; "Yes, on the ground," he
+said, very distinctly; these were the last words he spoke, and he slid
+from her arms and fell upon the floor. He died at 6-10 P.M. on Friday,
+the 9th of June, and was buried privately in Poets' Corner, Westminster
+Abbey, in the early morning of the 14th of June. One of the most
+appealing memorials was the drawing by his "new illustrator" Luke Fildes
+in the _Graphic_ of "The Empty Chair; Gad's Hill: ninth of June, 1870."
+"Statesmen, men of science, philanthropists, the acknowledged
+benefactors of their race, might pass away, and yet not leave the void
+which will be caused by the death of Charles Dickens" (_The Times_). In
+his will he enjoined his friends to erect no monument in his honour, and
+directed his name and dates only to be inscribed on his tomb, adding
+this proud provision, "I rest my claim to the remembrance of my country
+on my published works."
+
+Dickens had no artistic ideals worth speaking about. The sympathy of his
+readers was the one thing he cared about and, like Cobbett, he went
+straight for it through the avenue of the emotions. In personality,
+intensity and range of creative genius he can hardly be said to have any
+modern rival. His creations live, move and have their being about us
+constantly, like those of Homer, Virgil, Chaucer, Rabelais, Cervantes,
+Shakespeare, Bunyan, Molière and Sir Walter Scott. As to the books
+themselves, the backgrounds on which these mighty figures are projected,
+they are manifestly too vast, too chaotic and too unequal ever to become
+classics. Like most of the novels constructed upon the unreformed model
+of Smollett and Fielding, those of Dickens are enormous stock-pots into
+which the author casts every kind of autobiographical experience,
+emotion, pleasantry, anecdote, adage or apophthegm. The fusion is
+necessarily very incomplete and the hotch-potch is bound to fall to
+pieces with time. Dickens's plots, it must be admitted, are strangely
+unintelligible, the repetitions and stylistic decorations of his work
+exceed all bounds, the form is unmanageable and insignificant. The
+diffuseness of the English novel, in short, and its extravagant
+didacticism cannot fail to be most prejudicial to its perpetuation. In
+these circumstances there is very little fiction that will stand
+concentration and condensation so well as that of Dickens.
+
+For these reasons among others our interest in Dickens's novels as
+integers has diminished and is diminishing. But, on the other hand, our
+interest and pride in him as a man and as a representative author of his
+age and nation has been steadily augmented and is still mounting. Much
+of the old criticism of his work, that it was not up to a sufficiently
+high level of art, scholarship or gentility, that as an author he is
+given to caricature, redundancy and a shameless subservience to popular
+caprice, must now be discarded as irrelevant.
+
+As regards formal excellence it is plain that Dickens labours under the
+double disadvantage of writing in the least disciplined of all literary
+genres in the most lawless literary milieu of the modern world, that of
+Victorian England. In spite of these defects, which are those of masters
+such as Rabelais, Hugo and Tolstoy, the work of Dickens is more and more
+instinctively felt to be true, original and ennobling. It is already
+beginning to undergo a process of automatic sifting, segregation and
+crystallization, at the conclusion of which it will probably occupy a
+larger segment in the literary consciousness of the English-spoken race
+than ever before.
+
+Portraits of Dickens, from the gay and alert "Boz" of Samuel Lawrence,
+and the self-conscious, rather foppish portrait by Maclise which served
+as frontispiece to _Nicholas Nickleby_, to the sketch of him as Bobadil
+by C. R. Leslie, the Drummond and Ary Scheffer portraits of middle age
+and the haggard and drawn representations of him from photographs after
+his shattering experiences as a public entertainer from 1856 (the year
+of his separation from his wife) onwards, are reproduced in Kitton, in
+Forster and Gissing and in the other biographies. Sketches are also
+given in most of the books of his successive dwelling places at Ordnance
+Terrace and 18 St Mary's Place, Chatham; Bayham Street, Camden Town; 15
+Furnival's Inn; 48 Doughty Street; 1 Devonshire Terrace, Regent's Park;
+Tavistock House, Tavistock Square; and Gad's Hill Place. The manuscripts
+of all the novels, with the exception of the _Tale of Two Cities_ and
+_Edwin Drood_, were given to Forster, and are now preserved in the Dyce
+and Forster Museum at South Kensington. The work of Dickens was a prize
+for which publishers naturally contended both before and after his
+death. The first collective edition of his works was begun in April
+1847, and their number is now very great. The most complete is still
+that of Messrs Chapman & Hall, the original publishers of _Pickwick_;
+others of special interest are the Harrap edition, originally edited by
+F. G. Kitton; Macmillan's edition with original illustrations and
+introduction by Charles Dickens the younger; and the edition in the
+World's Classics with introductions by G.K. Chesterton. Of the
+translations the best known is that done into French by Lorain, Pichot
+and others, with B.H. Gausseron's excellent _Pages Choisies_ (1903).
+
+  BIBLIOGRAPHY.--During his lifetime Dickens's biographer was clearly
+  indicated in his guide, philosopher and friend, John Forster, who had
+  known the novelist intimately since the days of his first triumph with
+  _Pickwick_, who had constituted himself a veritable encyclopaedia of
+  information about Dickens, and had clung to his subject (in spite of
+  many rebuffs which his peremptory temper found it hard to digest) as
+  tightly as ever Boswell had enveloped Johnson. Two volumes of
+  Forster's _Life of Charles Dickens_ appeared in 1872 and a third in
+  1874. He relied much on Dickens's letters to himself and produced what
+  must always remain the authoritative work. The first two volumes are
+  put together with much art, the portrait as a whole has been regarded
+  as truthful, and the immediate success was extraordinary. In the
+  opinion of Carlyle, Forster's book was not unworthy to be named after
+  that of Boswell. A useful abridgment was carried out in 1903 by the
+  novelist George Gissing. Gissing also wrote _Charles Dickens: A
+  Critical Study_ (1898), which ranks with G.K. Chesterton's _Charles
+  Dickens_(1906) as a commentary inspired by deep insight and adorned by
+  great literary talent upon the genius of the master-novelist. The
+  names of other lives, sketches, articles and estimates of Dickens and
+  his works would occupy a large volume in the mere enumeration. See
+  R.H. Shepherd, _The Bibliography of Dickens_ (1880); _James Cooke's
+  Bibliography of the Writings of Charles Dickens_ (1879);
+  _Dickensiana_, by F. G. Kitton (1886); and _Bibliography_ by J.P.
+  Anderson, appended to Sir F.T. Marzials's _Life of Charles Dickens_
+  (1887). Among the earlier sketches may be specially cited the lives by
+  J. C. Hotten and G. A. Sala (1870), the Anecdote-Biography edited by
+  the American R. H. Stoddard (1874), Dr A. W. Ward in the English Men
+  of Letters Series (1878), that by Sir Leslie Stephen in the
+  _Dictionary of National Biography_, and that by Professor Minto in the
+  eighth edition of the _Encyclopaedia Britannica_. The _Letters_ were
+  first issued in two volumes edited by his daughter and sister-in-law
+  in 1880. For Dickens's connexion with Kent the following books are
+  specially valuable:--Robert Langton's _Childhood and Youth of Charles
+  Dickens_ (1883); Langton's _Dickens and Rochester_ (1880); Thomas
+  Frost's _In Kent with Charles Dickens_ (1880); F. G. Kitton's _The
+  Dickens Country_ (1905); H. S. Ward's _The Real Dickens Land_ (1904);
+  R. Allbut's _Rambles in Dickens Land_ (1899 and 1903). For Dickens's
+  reading tours see G. Dolby's _Charles Dickens as I knew him_ (1884);
+  J. T. Fields's _In and Out of Doors with Charles Dickens_ (1876);
+  Charles Kent's _Dickens as a Reader_ (1872). And for other aspects of
+  his life see M. Dickens's _My Father as I recall him_ (1897); P. H.
+  Fitzgerald's _Life of C. Dickens as revealed in his Writings_ (1905),
+  and _Bozland_ (1895); F. G. Kitton's _Charles Dickens, his Life,
+  Writings and Personality_, a useful compendium (1902); T. E.
+  Pemberton's _Charles Dickens and the Stage_, and _Dickens's London_
+  (1876); F. Miltoun's _Dickens's London_ (1904); Kitton's _Dickens and
+  his Illustrators_; W. Teignmouth Shore's _Charles Dickens and his
+  Friends_ (1904 and 1909); B. W. Matz, _Story of Dickens's Life and
+  Work_ (1904), and review of solutions to _Edwin Drood_ in _The
+  Bookman_ for March 1908; the recollections of Edmund Yates, Trollope,
+  James Payn, Lehmann, R. H. Horne, Lockwood and many others. _The
+  Dickensian_, a magazine devoted to Dickensian subjects, was started in
+  1905; it is the organ of the Dickens Fellowship, and in a sense of the
+  Boz Club. _A Dickens Dictionary_ (by G. A. Pierce) appeared in 1872
+  and 1878; another (by A. J. Philip) in 1909; and a _Dickens
+  Concordance_ by Mary Williams in 1907.     (T. SE.)
+
+
+
+
+DICKINSON, ANNA ELIZABETH (1842-   ), American author and lecturer, was
+born, of Quaker parentage, at Philadelphia, Pennsylvania, on the 28th of
+October 1842. She was educated at the Friends' Free School in
+Philadelphia, and was for a time a teacher. In 1861 she obtained a
+clerkship in the United States mint, but was removed for criticizing
+General McClellan at a public meeting. She had gradually become widely
+known as an eloquent and persuasive public speaker, one of the first of
+her sex to mount the platform to discuss the burning questions of the
+hour. Before the Civil War she lectured on anti-slavery topics, during
+the war she toured the country on behalf of the Sanitary Commission, and
+also lectured on reconstruction, temperance and woman's rights. She
+wrote several plays, including _The Crown of Thorns_ (1876); _Mary
+Tudor_ (1878), in which she appeared in the title rôle; _Aurelian_
+(1878); and _An American Girl_ (1880), successfully acted by Fanny
+Davenport. She also published a novel, _Which Answer?_ (1868); _A Paying
+Investment, a Plea for Education_ (1876); and _A Ragged Register of
+People, Places and Opinions_ (1879).
+
+
+
+
+DICKINSON, JOHN (1732-1808), American statesman and pamphleteer, was
+born in Talbot county, Maryland, on the 8th of November 1732. He removed
+with his father to Kent county, Delaware, in 1740, studied under private
+tutors, read law, and in 1753 entered the Middle Temple, London.
+Returning to America in 1757, he began the practice of law in
+Philadelphia, was speaker of the Delaware assembly in 1760, and was a
+member of the Pennsylvania assembly in 1762-1765 and again in
+1770-1776.[1] He represented Pennsylvania in the Stamp Act Congress
+(1765) and in the Continental Congress from 1774 to 1776, when he was
+defeated owing to his opposition to the Declaration of Independence. He
+then retired to Delaware, served for a time as private and later as
+brigader-general in the state militia, and was again a member of the
+Continental Congress (from Delaware) in 1779-1780. He was president of
+the executive council, or chief executive officer, of Delaware in
+1781-1782, and of Pennsylvania in 1782-1785, and was a delegate from
+Delaware to the Annapolis convention of 1786 and the Federal
+Constitutional convention of 1787. Dickinson has aptly been called the
+"Penman of the Revolution." No other writer of the day presented
+arguments so numerous, so timely and so popular. He drafted the
+"Declaration of Rights" of the Stamp Act Congress, the "Petition to the
+King" and the "Address to the Inhabitants of Quebec" of the Congress of
+1774, and the second "Petition to the King"[2] and the "Articles of
+Confederation" of the second Congress. Most influential of all, however,
+were _The Letters of a Farmer in Pennsylvania_, written in 1767-1768 in
+condemnation of the Townshend Acts of 1767, in which he rejected
+speculative natural rights theories and appealed to the common sense of
+the people through simple legal arguments. By opposing the Declaration
+of Independence, he lost his popularity and was never able entirely to
+regain it. As the representative of a small state, he championed the
+principle of state equality in the constitutional convention, but was
+one of the first to advocate the compromise, which was finally adopted,
+providing for equal representation, in one house and proportional
+representation in the other. He was probably influenced by Delaware
+prejudice against Pennsylvania when he drafted the clause which forbids
+the creation of a new state by the junction of two or more states or
+parts of states without the consent of the states concerned as well as
+of congress. After the adjournment of the convention he defended its
+work in a series of letters signed "Fabius," which will bear comparison
+with the best of the Federalist productions. It was largely through his
+influence that Delaware and Pennsylvania were the first two states to
+ratify the Constitution. Dickinson's interests were not exclusively
+political. He helped to found Dickinson College (named in his honour) at
+Carlisle, Pennsylvania, in 1783, was the first president of its board of
+trustees, and was for many years its chief benefactor. He died on the
+14th of February 1808 and was buried in the Friends' burial ground in
+Wilmington, Del.
+
+  See C. J. Stillé, _Life and Times of John Dickinson_, and P. L. Ford
+  (editor), _The Writings of John Dickinson_, in vols. xiii. and xiv.
+  respectively of the _Memoirs of the Historical Society of
+  Pennsylvania_ (Philadelphia, 1891 and 1895).
+
+
+FOOTNOTES:
+
+  [1] Being under the same proprietor and the same governor,
+    Pennsylvania and Delaware were so closely connected before the
+    Revolution that there was an interchange of public men.
+
+  [2] The "Declaration of the United Colonies of North America ...
+    setting forth the Causes and the Necessity of their Taking up Arms"
+    (often erroneously attributed to Thomas Jefferson).
+
+
+
+
+DICKSON, SIR ALEXANDER (1777-1840), British artillerist, entered the
+Royal Military Academy in 1793, passing out as second lieutenant in the
+Royal Artillery in the following year. As a subaltern he saw service in
+Minorca in 1798 and at Malta in 1800. As a captain he took part in the
+unfortunate Montevideo Expedition of 1806-07, and in 1809 he accompanied
+Howorth to the Peninsular War as brigade-major of the artillery. He soon
+obtained a command in the Portuguese artillery, and as a
+lieutenant-colonel of the Portuguese service took part in the various
+battles of 1810-11. At the two sieges of Budazoz, Ciudad Rodrigo, the
+Salamanca forts and Burgos, he was entrusted by Wellington (who had the
+highest opinion of him) with most of the detailed artillery work, and at
+Salamanca battle he commanded the reserve artillery. In the end he
+became commander of the whole of the artillery of the allied army, and
+though still only a substantive captain in the British service he had
+under his orders some 8000 men. At Vitoria, the Pyrenees battles and
+Toulouse he directed the movements of the artillery engaged, and at the
+end of the war received handsome presents from the officers who had
+served under him, many of whom were his seniors in the army list. He was
+at the disastrous affair of New Orleans, but returned to Europe in time
+for the Waterloo campaign. He was present at Quatre Bras and Waterloo on
+the artillery staff of Wellington's army, and subsequently commanded the
+British battering train at the sieges of the French fortresses left
+behind the advancing allies. For the rest of his life he was on home
+service, principally as a staff officer of artillery. He died, a
+major-general and G.C.B., in 1840. A memorial was erected at Woolwich in
+1847. Dickson was one of the earliest fellows of the Royal Geographical
+Society.
+
+  His diaries kept in the Peninsula were the main source of information
+  used in Duncan's _History of the Royal Artillery_.
+
+
+
+
+DICKSON, SIR JAMES ROBERT (1832-1901), Australian statesman, was born in
+Plymouth on the 30th of November 1832. He was brought up in Glasgow,
+receiving his education at the high school, and became a clerk in the
+City of Glasgow Bank. In 1854 he emigrated to Victoria, but after some
+years spent in that colony and in New South Wales, he settled in 1862 in
+Queensland, where he was connected with many important business
+enterprises, among them the Royal Bank of Queensland. He entered the
+Queensland House of Assembly in 1872, and became minister of works
+(1876), treasurer (1876-1879, and 1883-1887), acting premier (1884), but
+resigned in 1887 on the question of taxing land. In 1889 he retired from
+business, and spent three years in Europe before resuming political
+life. He fought for the introduction of Polynesian labour on the
+Queensland sugar plantations at the general election of 1892, and was
+elected to the House of Assembly in that year and again at the elections
+of 1893 and 1896. He became secretary for railways in 1897, minister for
+home affairs in 1898, represented Queensland in the federal council of
+Australia in 1896 and at the postal conference at Hobart in 1898, and in
+1898 became premier. His energies were now devoted to the formation of
+an Australian commonwealth. He secured the reference of the question to
+a plebiscite, the result of which justified his anticipations. He
+resigned the premiership in November 1899, but in the ministry of Robert
+Philp, formed in the next month, he was reappointed to the offices of
+chief secretary and vice-president of the executive council which he had
+combined with the office of premier. He represented Queensland in 1900
+at the conference held in London to consider the question of Australian
+unity, and on his return was appointed minister of defence in the first
+government of the Australian Commonwealth. He did not long survive the
+accomplishment of his political aims, dying at Sydney on the 10th of
+January 1901, in the midst of the festivities attending the inauguration
+of the new state.
+
+
+
+
+DICOTYLEDONS, in botany, the larger of the two great classes of
+angiosperms, embracing most of the common flower-bearing plants. The
+name expresses the most universal character of the class, the importance
+of which was first noticed by John Ray, namely, the presence of a pair
+of seed-leaves or cotyledons, in the plantlet or embryo contained in the
+seed. The embryo is generally surrounded by a larger or smaller amount
+of foodstuff (endosperm) which serves to nourish it in its development
+to form a seedling when the seed germinates; frequently, however, as in
+pea or bean and their allies, the whole of the nourishment for future
+use is stored up in the cotyledons themselves, which then become thick
+and fleshy. In germination of the seed the root of the embryo (radicle)
+grows out to get a holdfast for the plant; this is generally followed by
+the growth of the short stem immediately above the root, the so-called
+"hypocotyl," which carries up the cotyledons above the ground, where
+they spread to the light and become the first green leaves of the plant.
+Protected between the cotyledons and terminating the axis of the plant
+is the first stem-bud (the plumule of the embryo), by the further growth
+and development of which the aerial portion of the plant, consisting of
+stem, leaves and branches, is formed, while the development of the
+radicle forms the root-system. The size and manner of growth of the
+adult plant show a great variety, from the small herb lasting for one
+season only, to the forest tree living for centuries. The arrangement of
+the conducting tissue in the stem is characteristic; a transverse
+section of the very young stem shows a number of distinct conducting
+strands--vascular bundles--arranged in a ring round the pith; these soon
+become united to form a closed ring of bast and wood, separated by a
+layer of formative tissue (cambium). In perennials the stem shows a
+regular increase in thickness each year by the addition of a new ring of
+wood outside the old one--for details of structure see PLANTS: Anatomy.
+A similar growth occurs in the root. This increase in the diameter of
+stem and root is correlated with the increase in leaf-area each season,
+due to the continued production of new leaf-bearing branches. A
+characteristic of the class is afforded by the complicated network
+formed by the leaf-veins,--well seen in a skeleton leaf, from which the
+soft parts have been removed by maceration. The parts of the flower are
+most frequently arranged in fives, or multiples of fives; for instance,
+a common arrangement is as follows,--five sepals, succeeded by five
+petals, ten stamens in two sets of five, and five or fewer carpels; an
+arrangement in fours is less frequent, while the arrangement in threes,
+so common in monocotyledons, is rare in dicotyledons. In some orders the
+parts are numerous, chiefly in the case of the stamens and the carpels,
+as in the buttercup and other members of the order Ranunculaceae. There
+is a very wide range in the general structure and arrangement of the
+parts of the flower, associated with the means for ensuring the
+transference of pollen; in the simplest cases the flower consists only
+of a few stamens or carpels, with no enveloping sepals or petals, as in
+the willow, while in the more elaborate type each series is represented,
+the whole forming a complicated structure closely correlated with the
+size, form and habits of the pollinating agent (see FLOWER). The
+characters of the fruit and seed and the means for ensuring the
+dispersal of the seeds are also very varied (see FRUIT).
+
+
+
+
+DICTATOR (from the Lat. _dictare_, frequentative of _dicere_, to speak).
+In modern usage this term is loosely used for a personal ruler enjoying
+extraordinary and extra-constitutional power. The etymological sense of
+one who "dictates"--i.e. one whose word (_dictum_) is law (from which
+that of one who "dictates," i.e. speaks for some writer to record, is to
+be distinguished)--has been assisted by the historical use of the term,
+in ancient times, for an extraordinary magistrate in the Roman
+commonwealth. It is unknown precisely how the Roman word came into use,
+though an explanation of the earlier official title, magister populi,
+throws some light on the subject. That designation may mean "head of the
+(infantry) host" as opposed to his subordinate, the magister equitum,
+who was "head of the cavalry." If this explanation be accepted, emphasis
+was thus laid in early times on the military aspect of the dictatorship,
+and in fact the office seems to have been instituted for the purpose of
+meeting a military crisis such as might have proved too serious for the
+annual consuls with their divided command. Later constitutional theory
+held that the repression of civil discord was also one of the motives
+for the institution of a dictatorship. Such is the view expressed by
+Cicero in the _De legibus_ (iii. 3, 9) and by the emperor Claudius in
+his extant _Oratio_ (i. 28). This function of the office, although it
+may not have been contemplated at first, is attested by the internal
+history of Rome. In the crisis of the agitation that gathered round the
+Licinian laws (367 B.C.) a dictator was appointed, and in 314 B.C. we
+have the notice of a dictator created for purposes of criminal
+jurisdiction (_quaestionibus exercendis_). The dictator appointed to
+meet the dangers of war, sedition or crime was technically described as
+"the administrative dictator" (_rei gerundae causa_). Minor, or merely
+formal, needs of the state might lead to the creation of other types of
+this office. Thus we find dictators destined to hold the elections, to
+make out the list of the senate, to celebrate games, to establish
+festivals, and to drive the nail into the temple of Jupiter--an act of
+natural magic which was believed to avert pestilence. These dictators
+appointed for minor purposes were expected to retire from office as soon
+as their function was completed. The "administrative dictator" held
+office for at least six months.
+
+The powers of a dictator were a temporary revival of those of the kings;
+but there were some limitations to his authority. He was never concerned
+with civil jurisdiction, and was dependent on the senate for supplies of
+money. His military authority was confined to Italy; and his power of
+life and death over the citizens was at an early period limited by law.
+It was probably the _lex Valeria_ of 300 B.C. that made him subject to
+the right of criminal appeal (_provocatio_) within the limits of the
+city. But during his tenure of power all the magistrates of the people
+were regarded as his subordinates; and it was even held that the right
+of assistance (_auxilium_), furnished by the tribunes of the plebs to
+members of the citizen body, should not be effectively exercised when
+the state was under this type of martial law. The dictator was nominated
+by one of the consuls. But here as elsewhere the senate asserted its
+authority over the magistrates, and the view was finally held that the
+senate should not only suggest the need of nomination but also the name
+of the nominee. After the nomination, the imperium of the dictator was
+confirmed by a _lex curiata_ (see COMITIA). To emphasize the superiority
+of this imperium over that of the consuls, the dictator might be
+preceded by twenty-four lictors, not by the usual twelve; and, at least
+in the earlier period of the office, these lictors bore the axes, the
+symbols of life and death, within the city walls.
+
+Tradition represents the dictatorship as having a life of three
+centuries in the history of the Roman state. The first dictator is said
+to have been created in 501 B.C.; the last of the "administrative"
+dictators belongs to the year 216 B.C. It was an office that was
+incompatible both with the growing spirit of constitutionalism and with
+the greater security of the city; and the epoch of the Second Punic War
+was marked by experiments with the office, such as the election of Q.
+Fabius Maximus by the people, and the co-dictatorship of M. Minucius
+with Fabius, which heralded its disuse (see PUNIC WARS). The emergency
+office of the early and middle Republic has few points of contact,
+except those of the extraordinary position and almost unfettered
+authority of its holder, with the dictatorship as revised by Sulla and
+by Caesar. Sulla's dictatorship was the form taken by a provisional
+government. He was created "for the establishment of the Republic." It
+is less certain whether the dictatorships held by Caesar were of a
+consciously provisional character. Since the office represented the only
+supreme _Imperium_ in Rome, it was the natural resort of the founder of
+a monarchy (see SULLA and CAESAR). Ostensibly to prevent its further use
+for such a purpose, M. Antonius in 44 B.C. carried a law abolishing the
+dictatorship as a part of the constitution.
+
+  BIBLIOGRAPHY.--Mommsen, _Römisches Staatsrecht_, ii. 141 foll. (3rd
+  ed., Leipzig, 1887); Herzog, _Geschichte und System der römischen
+  Staatsverfassung_, i. 718 foll. (Leipzig, 1884); Pauly-Wissowa,
+  _Realencyclopädie_, v. 370 foll. (new edition, Stuttgart. 1893, &c.);
+  Lange, _Römische Alterthümer_, i. 542 foll. (Berlin, 1856, &c.);
+  Daremberg-Saglio, _Dictionnaire des antiquités grecques et romaines_,
+  ii. 161 foll. (1875, &c.); Haverfield, "The Abolition of the
+  Dictatorship," in _Classical Review_, iii. 77.     (A. H. J. G.)
+
+
+
+
+DICTIONARY.
+
+  Definition and history.
+
+In its proper and most usual meaning a dictionary is a book containing a
+collection of the words of a language, dialect or subject, arranged
+alphabetically or in some other definite order, and with explanations in
+the same or some other language. When the words are few in number, being
+only a small part of those belonging to the subject, or when they are
+given without explanation, or some only are explained, or the
+explanations are partial, the work is called a _vocabulary_; and when
+there is merely a list of explanations of the technical words and
+expressions in some particular subject, a _glossary_. An alphabetical
+arrangement of the words of some book or author with references to the
+places where they occur is called an index (q.v.). When under each word
+the phrases containing it are added to the references, the work is
+called a _concordance_. Sometimes, however, these names are given to
+true dictionaries; thus the great Italian dictionary of the _Accademia
+della Crusca_, in six volumes folio, is called _Vocabolario_, and
+Ernesti's dictionary to Cicero is called _Index_. When the words are
+arranged according to a definite system of classification under heads
+and subdivisions, according to their nature or their meaning, the book
+is usually called a classed vocabulary; but when sufficient explanations
+are given it is often accepted as a dictionary, like the _Onomasticon_
+of Julius Pollux, or the native dictionaries of Sanskrit, Manchu and
+many other languages.
+
+Dictionaries were originally books of reference explaining the words of
+a language or of some part of it. As the names of things, as well as
+those of persons and places, are words, and often require explanation
+even more than other classes of words, they were necessarily included in
+dictionaries, and often to a very great extent. In time, books were
+devoted to them alone, and were limited to special subjects, and these
+have so multiplied, that dictionaries of things now rival in number and
+variety those of words or of languages, while they often far surpass
+them in bulk. There are dictionaries of biography and history, real and
+fictitious, general and special, relating to men of all countries,
+characters and professions; the English _Dictionary of National
+Biography_ (see BIOGRAPHY) is a great instance of one form of these;
+dictionaries of bibliography, relating to all books, or to those of some
+particular kind or country; dictionaries of geography (sometimes called
+_gazetteers_) of the whole world, of particular countries, or of small
+districts, of towns and of villages, of castles, monasteries and other
+buildings. There are dictionaries of philosophy; of the Bible; of
+mathematics; of natural history, zoology, botany; of birds, trees,
+plants and flowers; of chemistry, geology and mineralogy; of
+architecture, painting and music; of medicine, surgery, anatomy,
+pathology and physiology; of diplomacy; of law, canon, civil, statutory
+and criminal; of political and social sciences; of agriculture, rural
+economy and gardening; of commerce, navigation, horsemanship and the
+military arts; of mechanics, machines and the manual arts. There are
+dictionaries of antiquities, of chronology, of dates, of genealogy, of
+heraldry, of diplomatics, of abbreviations, of useful receipts, of
+monograms, of adulterations and of very many other subjects. These works
+are separately referred to in the bibliographies attached to the
+articles on the separate subjects. And lastly, there are dictionaries of
+the arts and sciences, and their comprehensive offspring, encyclopaedias
+(q.v.), which include in themselves every branch of knowledge. Neither
+under the heading of _dictionary_ nor under that of _encyclopaedia_ do
+we propose to include a mention of every work of its class, but many of
+these will be referred to in the separate articles on the subjects to
+which they pertain. And in this article we confine ourselves to an
+account of those dictionaries which are primarily word-books. This is
+practically the most convenient distinction from the subject-book or
+encyclopaedia; though the two characters are often combined in one work.
+Thus the _Century Dictionary_ has encyclopaedic features, while the
+present edition of the _Encyclopaedia Britannica_, restoring its
+earlier tradition but carrying out the idea more systematically, also
+embodies dictionary features.
+
+_Dictionarium_ is a word of low or modern Latinity;[1] _dictio_, from
+which it was formed, was used in medieval Latin to mean a word.
+_Lexicon_ is a corresponding word of Greek origin, meaning a book of or
+for words--a dictionary. A _glossary_ is properly a collection of
+unusual or foreign words requiring explanation. It is the name
+frequently given to English dictionaries of dialects, which the Germans
+usually call _idioticon_, and the Italians _vocabolario_. _Wörterbuch_,
+a book of words, was first used among the Germans, according to Grimm,
+by Kramer (1719), imitated from the Dutch _woordenboek_. From the
+Germans the Swedes and Danes adopted _ordbok_, _ordbog_. The Icelandic
+_ordabôk_, like the German, contains the genitive plural. The Slavonic
+nations use _slovar_, _slovnik_, and the southern Slavs _ryetshnik_,
+from _slovo_, _ryetsh_, a word, formed, like dictionary and lexicon,
+without composition. Many other names have been given to dictionaries,
+as _thesaurus_, _Sprachschatz_, _cornucopia_, _gazophylacium_,
+_comprehensorium_, _catholicon_, to indicate their completeness;
+_manipulus predicantium_, _promptorium puerorum_, _liber memorialis_,
+_hortus vocabulorum_, _ionia_ (a violet bed), _alveary_ (a beehive),
+_kamoos_ (the sea), _haft kulzum_ (the seven seas), _tsze tien_ (a
+standard of character), _onomasticon_, _nomenclator_, _bibliotheca_,
+_elucidario_, _Mundart-sammlung_, _clavis_, _scala_, _pharetra_,[2] _La
+Crusca_ from the great Italian dictionary, and _Calepino_ (in Spanish
+and Italian) from the Latin dictionary of Calepinus.
+
+The tendency of great dictionaries is to unite in themselves all the
+peculiar features of special dictionaries. A large dictionary is most
+useful when a word is to be thoroughly studied, or when there is
+difficulty in making out the meaning of a word or phrase. Special
+dictionaries are more useful for special purposes; for instance,
+synonyms are best studied in a dictionary of synonyms. And small
+dictionaries are more convenient for frequent use, as in translating
+from an unfamiliar language, for words may be found more quickly, and
+they present the words and their meanings in a concentrated and compact
+form, instead of being scattered over a large space, and separated by
+other matter. Dictionaries of several languages, called _polyglots_, are
+of different kinds. Some are polyglot in the vocabulary, but not in the
+explanation, like Johnson's dictionary of Persian and Arabic explained
+in English; some in the interpretation, but not in the vocabulary or
+explanation, like _Calepini octoglotton_, a Latin dictionary of Latin,
+with the meanings in seven languages. Many great dictionaries are now
+polyglot in this sense. Some are polyglot in the vocabulary and
+interpretation, but are explained in one language, like Jal's _Glossaire
+nautique_, a glossary of sea terms in many languages, giving the
+equivalents of each word in the other languages, but the explanation in
+French. Pauthier's _Annamese Dictionary_ is polyglot in a peculiar way.
+It gives the Chinese characters with their pronunciation in Chinese and
+Annamese. Special dictionaries are of many kinds. There are technical
+dictionaries of etymology, foreign words, dialects, secret languages,
+slang, neology, barbarous words, faults of expression, choice words,
+prosody, pronunciation, spelling, orators, poets, law, music, proper
+names, particular authors, nouns, verbs, participles, particles, double
+forms, difficulties and many others. Fick's dictionary (Göttingen, 1868,
+8vo; 1874-1876, 8vo, 4 vols.) is a remarkable attempt to ascertain the
+common language of the Indo-European nations before each of their great
+separations. In the second edition of his _Etymologische Forschungen_
+(Lemgo and Detmoldt, 1859-1873, 8vo, 7217 pages) Pott gives a
+comparative lexicon of Indo-European roots, 2226 in number, occupying
+5140 pages.
+
+
+  Methods.
+
+At no time was progress in the making of general dictionaries so rapid
+as during the second half of the 19th century. It is to be seen in three
+things: in the perfecting of the theory of what a general dictionary
+should be; in the elaboration of methods of collecting and editing
+lexicographic materials; and in the magnitude and improved quality of
+the work which has been accomplished or planned. Each of these can best
+be illustrated from English lexicography, in which the process of
+development has in all directions been carried farthest. The advance
+that has been made in theory began with a radical change of opinion with
+regard to the chief end of the general dictionary of a language. The
+older view of the matter was that the lexicographer should furnish a
+standard of usage--should register only those words which are, or at
+some period of the language have been, "good" from a literary point of
+view, with their "proper" senses and uses, or should at least furnish
+the means of determining what these are. In other words, his chief duty
+was conceived to be to sift and refine, to decide authoritatively
+questions with regard to good usage, and thus to fix the language as
+completely as might be possible within the limits determined by the
+literary taste of his time. Thus the Accademia della Crusca, founded
+near the close of the 16th century, was established for the purpose of
+purifying in this way the Italian tongue, and in 1612 the _Vocabolario
+degli Accademici della Crusca_, long the standard of that language, was
+published. The Académie Française, the first edition of whose dictionary
+appeared in 1694, had a similar origin. In England the idea of
+constructing a dictionary upon this principle arose during the second
+quarter of the 18th century. It was imagined by men of letters--among
+them Alexander Pope--that the English language had then attained such
+perfection that further improvement was hardly possible, and it was
+feared that if it were not fixed by lexicographic authority
+deterioration would soon begin. Since there was no English "Academy," it
+was necessary that the task should fall to some one whose judgment would
+command respect, and the man who undertook it was Samuel Johnson. His
+dictionary, the first edition of which, in two folio volumes, appeared
+in 1755, was in many respects admirable, but it was inadequate even as a
+standard of the then existing literary usage. Johnson himself did not
+long entertain the belief that the natural development of a language can
+be arrested in that or in any other way. His work was, however,
+generally accepted as a final authority, and the ideas upon which it was
+founded dominated English lexicography for more than a century. The
+first effective protest in England against the supremacy of this
+literary view was made by Dean (later Archbishop) Trench, in a paper on
+"Some Deficiencies in Existing English Dictionaries" read before the
+Philological Society in 1857. "A dictionary," he said, "according to
+that idea of it which seems to me alone capable of being logically
+maintained, is an _inventory of the language_; much more, but this
+primarily.... It is no task of the maker of it to select the _good_
+words of the language.... The business which he has undertaken is to
+collect and arrange _all_ words, whether good or bad, whether they
+commend themselves to his judgment or otherwise.... _He is an historian
+of_ [the language], _not a critic._" That is, for the literary view of
+the chief end of the general dictionary should be substituted the
+philological or scientific. In Germany this substitution had already
+been effected by Jacob and Wilhelm Grimm in their dictionary of the
+German language, the first volume of which appeared in 1854. In brief,
+then, the modern view is that the general dictionary of a language
+should be a record of all the words--current or obsolete--of that
+language, with all their meanings and uses, but should not attempt to
+be, except secondarily or indirectly, a guide to "good" usage. A
+"standard" dictionary has, in fact, been recognized to be an
+impossibility, if not an absurdity.
+
+This theoretical requirement must, of course, be modified considerably
+in practice. The date at which a modern language is to be regarded by
+the lexicographer as "beginning" must, as a rule, be somewhat
+arbitrarily chosen; while considerable portions of its earlier
+vocabulary cannot be recovered because of the incompleteness of the
+literary record. Moreover, not even the most complete dictionary can
+include all the words which the records--earlier and later--actually
+contain. Many words, that is to say, which are found in the literature
+of a language cannot be regarded as, for lexicographic purposes,
+belonging to that language; while many more may or may not be held to
+belong to it, according to the judgment--almost the whim--of the
+individual lexicographer. This is especially true of the English tongue.
+"That vast aggregate of words and phrases which constitutes the
+vocabulary of English-speaking men presents, to the mind that endeavours
+to grasp it as a definite whole, the aspect of one of those nebulous
+masses familiar to the astronomer, in which a clear and unmistakable
+nucleus shades off on all sides, through zones of decreasing brightness,
+to a dim marginal film that seems to end nowhere, but to lose itself
+imperceptibly in the surrounding darkness" (Dr J. A. H. Murray, _Oxford
+Dict._ General Explanations, p. xvii). This "marginal film" of words
+with more or less doubtful claims to recognition includes thousands of
+the terms of the natural sciences (the New-Latin classificatory names of
+zoology and botany, names of chemical compounds and of minerals, and the
+like); half-naturalized foreign words; dialectal words; slang terms;
+trade names (many of which have passed or are passing into common use);
+proper names and many more. Many of these even the most complete
+dictionary should exclude; others it should include; but where the line
+shall be drawn will always remain a vexed question.
+
+Another important principle upon which Trench insisted, and which also
+expresses a requirement of modern scientific philology, is that the
+dictionary shall be not merely a record, but also an _historical_ record
+of words and their uses. From the literary point of view the most
+important thing is present usage. To that alone the idea of a "standard"
+has any application. Dictionaries of the older type, therefore, usually
+make the common, or "proper" or "root" meaning of a word the starting
+point of its definition, and arrange its other senses in a logical or
+accidental order commonly ignoring the historical order in which the
+various meanings arose. Still less do they attempt to give data from
+which the vocabulary of the language at any previous period may be
+determined. The philologist, however, for whom the growth, or
+progressive alteration, of a language is a fact of central importance,
+regards no record of a language as complete which does not exhibit this
+growth in its successive stages. He desires to know when and where each
+word, and each form and sense of it, are first found in the language; if
+the word or sense is obsolete, when it died; and any other fact that
+throws light upon its history. He requires, accordingly, of the
+lexicographer that, having ascertained these data, he shall make them
+the foundation of his exposition--in particular, of the division and
+arrangement of his definitions, that sense being placed first which
+appeared first in order of time. In other words, each article in the
+dictionary should furnish an orderly biography of the word of which it
+treats, each word and sense being so dated that the exact time of its
+appearance and the duration of its use may as nearly as possible be
+determined. This, in principle, is the method of the new lexicography.
+In practice it is subject to limitations similar to those of the
+vocabulary mentioned above. Incompleteness of the early record is here
+an even greater obstacle; and there are many words whose history is, for
+one reason or another, so unimportant that to treat it elaborately would
+be a waste of labour and space.
+
+The adoption of the historical principle involves a further noteworthy
+modification of older methods, namely, an important extension of the use
+of quotations. To Dr Johnson belongs the credit of showing how useful,
+when properly chosen, they may be, not only in corroborating the
+lexicographer's statements, but also in revealing special shades of
+meaning or variations of use which his definitions cannot well express.
+No part of Johnson's work is more valuable than this. This idea was more
+fully developed and applied by Dr Charles Richardson, whose _New
+Dictionary of the English Language ... Illustrated by Quotations from
+the Best Authors_ (1835-1836) still remains a most valuable collection
+of literary illustrations. Lexicographers, however, have, with few
+exceptions, until a recent date, employed quotations chiefly for the
+ends just mentioned--as instances of use or as illustrations of correct
+usage--with scarcely any recognition of their value as historical
+evidence; and they have taken them almost exclusively from the works of
+the "best" authors. But since all the data upon which conclusions with
+regard to the history of a word can be based must be collected from the
+literature of the language, it is evident that, in so far as the
+lexicographer is required to furnish evidence for an historical
+inference, a quotation is the best form in which he can give it. In
+fact, extracts, properly selected and grouped, are generally sufficient
+to show the entire meaning and biography of a word without the aid of
+elaborate definitions. The latter simply save the reader the trouble of
+drawing the proper conclusions for himself. A further rule of the new
+lexicography, accordingly, is that quotations should be used, primarily,
+as historical evidence, and that the history of words and meanings
+should be exhibited by means of them. The earliest instance of use that
+can be found, and (if the word or sense is obsolete) the latest, are as
+a rule to be given; while in the case of an important word or sense,
+instances taken from successive periods of its currency also should be
+cited. Moreover, a quotation which contains an important bit of
+historical evidence must be used, whether its source is "good," from the
+literary point of view, or not--whether it is a classic of the language
+or from a daily newspaper; though where choice is possible, preference
+should, of course, be given to quotations extracted from the works of
+the best writers. This rule does not do away with the illustrative use
+of quotations, which is still recognized as highly important, but it
+subordinates it to their historical use. It is necessary to add that it
+implies that the extracts must be given exactly, and in the original
+spelling and capitalization, accurately dated, and furnished with a
+precise reference to author, book, volume, page and edition; for
+insistence upon these requirements--which are obviously important,
+whatever the use of the quotation may be--is one of the most noteworthy
+of modern innovations. Johnson usually gave simply the author's name,
+and often quoted from memory and inaccurately; and many of his
+successors to this day have followed--altogether or to some extent--his
+example.
+
+The chief difficulty in the way of this use of quotations--after the
+difficulty of collection--is that of finding space for them in a
+dictionary of reasonable size. Preference must be given to those which
+are essential, the number of those which are cited merely on methodical
+grounds being made as small as possible. It is hardly necessary to add
+that the negative evidence furnished by quotations is generally of
+little value; one can seldom, that is, be certain that the lexicographer
+has actually found the earliest or the latest use, or that the word or
+sense has not been current during some intermediate period from which he
+has no quotations.
+
+Lastly, a much more important place in the scheme of the ideal
+dictionary is now assigned to the _etymology_ of words. This may be
+attributed, in part, to the recent rapid development of etymology as a
+science, and to the greater abundance of trustworthy data; but it is
+chiefly due to the fact that from the historical point of view the
+connexion between that section of the biography of a word which lies
+within the language--subsequent, that is, to the time when the language
+may, for lexicographical purposes, be assumed to have begun, or to the
+time when the word was adopted or invented--and its antecedent history
+has become more vital and interesting. Etymology, in other words, is
+essentially the history of the _form_ of a word up to the time when it
+became a part of the language, and is, in a measure, an extension of the
+history of the development of the word in the language. Moreover, it is
+the only means by which the exact relations of allied words can be
+ascertained, and the separation of words of the same form but of diverse
+origin (homonyms) can be effected, and is thus, for the dictionary, the
+foundation of all _family history_ and correct _genealogy_. In fact, the
+attention that has been paid to these two points in the best recent
+lexicography is one of its distinguishing and most important
+characteristics. Related to the etymology of words are the changes in
+their form which may have occurred while they have been in use as parts
+of the language--modifications of their pronunciation, corruptions by
+popular etymology or false associations, and the like. The facts with
+regard to these things which the wide research necessitated by the
+historical method furnishes abundantly to the modern lexicographer are
+often among the most novel and interesting of his acquisitions.
+
+It should be added that even approximate conformity to the theoretical
+requirements of modern lexicography as above outlined is possible only
+under conditions similar to those under which the Oxford _New English
+Dictionary_ was undertaken (see below). The labour demanded is too vast,
+and the necessary bulk of the dictionary too great. When, however, a
+language is recorded in one such dictionary, those of smaller size and
+more modest pretensions can rest upon it as an authority and conform to
+it as a model so far as their special limitations permit.
+
+The ideal thus developed is primarily that of the general dictionary of
+the purely philological type, but it applies also to the encyclopaedic
+dictionary. In so far as the latter is strictly lexicographic--deals
+with words as words, and not with the things they denote--it should be
+made after the model of the former, and is defective to the extent in
+which it deviates from it. The addition of encyclopaedic matter to the
+philological in no way affects the general principles involved. It may,
+however, for practical reasons, modify their application in various
+ways. For example, the number of obsolete and dialectal words included
+may be much diminished and the number of scientific terms (for instance,
+new Latin botanical and zoological names) be increased; and the relative
+amount of space devoted to etymologies and quotations may be lessened.
+In general, since books of this kind are designed to serve more or less
+as works of general reference, the making of them must be governed by
+considerations of practical utility which the compilers of a purely
+philological dictionary are not obliged to regard. The encyclopaedic
+type itself, although it has often been criticized as hybrid--as a
+mixture of two things which should be kept distinct--is entirely
+defensible. Between the dictionary and the encyclopaedia the dividing
+line cannot sharply be drawn. There are words the meaning of which
+cannot be explained fully without some description of things, and, on
+the other hand, the description of things and processes often involves
+the definition of names. To the combination of the two objection cannot
+justly be made, so long as it is effected in a way--with a selection of
+material--that leaves the dictionary essentially a dictionary and not an
+encyclopaedia. Moreover, the large vocabulary of the general dictionary
+makes it possible to present certain kinds of encyclopaedic matter with
+a degree of fulness and a convenience of arrangement which are possible
+in no single work of any other class. In fact, it may be said that if
+the encyclopaedic dictionary did not exist it would have to be invented;
+that its justification is its indispensableness. Not the least of its
+advantages is that it makes legitimate the use of diagrams and pictorial
+illustrations, which, if properly selected and executed, are often
+valuable aids to definition.
+
+On its practical side the advance in lexicography has consisted in the
+elaboration of methods long in use rather than in the invention of new
+ones. The only way to collect the data upon which the vocabulary, the
+definitions and the history are to be based is, of course, to search for
+them in the written monuments of the language, as all lexicographers who
+have not merely borrowed from their predecessors have done. But the
+wider scope and special aims of the new lexicography demand that the
+investigation shall be vastly more comprehensive, systematic and
+precise. It is necessary, in brief, that, as far as may be possible, the
+literature (of all kinds) of every period of the language shall be
+examined systematically, in order that all the words, and senses and
+forms of words, which have existed during any period may be found, and
+that enough excerpts (carefully verified, credited and dated) to cover
+all the essential facts shall be made. The books, pamphlets, journals,
+newspapers, and so on which must thus be searched will be numbered by
+thousands, and the quotations selected may (as in the case of the Oxford
+_New English Dictionary_) be counted by millions. This task is beyond
+the powers of any one man, even though he be a Johnson, or a Littré or a
+Grimm, and it is now assigned to a corps of readers whose number is
+limited only by the ability of the editor to obtain such assistance. The
+modern method of editing the material thus accumulated--the actual work
+of compilation--also is characterized by the application of the
+principle of the division of labour. Johnson boasted that his dictionary
+was written with but little assistance from the learned, and the same
+was in large measure true of that of Littré. Such attempts on the part
+of one man to write practically the whole of a general dictionary are no
+longer possible, not merely because of the vast labour and philological
+research necessitated by modern aims, but more especially because the
+immense development of the vocabulary of the special sciences renders
+indispensable the assistance, in the work of definition, of persons who
+are expert in those sciences. The tendency, accordingly, has been to
+enlarge greatly the editorial staff of the dictionary, scores of
+sub-editors and contributors being now employed where a dozen or fewer
+were formerly deemed sufficient. In other words, the making of a
+"complete" dictionary has become a co-operative enterprise, to the
+success of which workers in all the fields of literature and science
+contribute.
+
+The most complete exemplification of these principles and methods is the
+_Oxford New English Dictionary, on historical principles, founded mainly
+on materials collected by the Philological Society_. This monumental
+work originated in the suggestion of Trench that an attempt should be
+made, under the direction of the Philological Society, to complete the
+vocabulary of existing dictionaries and to supply the historical
+information which they lacked. The suggestion was adopted, considerable
+material was collected, and Mr Herbert Coleridge was appointed general
+editor. He died in 1861, and was succeeded by Dr F. J. Furnivall.
+Little, however, was done, beyond the collection of quotations--about
+2,000,000 of which were gathered--until in 1878 the expense of printing
+and publishing the proposed dictionary was assumed by the Delegates of
+the University Press, and the editorship was entrusted to Dr (afterwards
+Sir) J. A. H. Murray. As the historical point of beginning, the middle
+of the 12th century was selected, all words that were obsolete at that
+date being excluded, though the history of words that were current both
+before and after that date is given in its entirety; and it was decided
+that the search for quotations--which, according to the original design,
+was to cover the entire literature down to the beginning of the 16th
+century and as much of the subsequent literature (especially the works
+of the more important writers and works on special subjects) as might be
+possible--should be made more thorough. More than 800 readers, in all
+parts of the world, offered their aid; and when the preface to the first
+volume appeared in 1888, the editor was able to announce that the
+readers had increased to 1300, and that 3,500,000 of quotations, taken
+from the writings of more than 5000 authors, had already been amassed.
+The whole work was planned to be completed in ten large volumes, each
+issued first in smaller parts. The first part was issued in 1884, and by
+the beginning of 1910 the first part of the letter S had been reached.
+
+The historical method of exposition, particularly by quotations, is
+applied in the _New English Dictionary_, if not in all cases with entire
+success, yet, on the whole, with a regularity and a precision which
+leave little to be desired. A minor fault is that excerpts from second
+or third rate authors have occasionally been used where better ones from
+writers of the first class either must have been at hand or could have
+been found. As was said above, the literary quality of the question is
+highly important even in historical lexicography, and should not be
+neglected unnecessarily. Other special features of the book are the
+completeness with which variations of pronunciation and orthography
+(with dates) are given; the fulness and scientific excellence of the
+etymologies, which abound in new information and corrections of old
+errors; the phonetic precision with which the present (British)
+pronunciation is indicated; and the elaborate subdivision of meanings.
+The definitions as a whole are marked by a high degree of accuracy,
+though in a certain number of cases (not explicable by the date of the
+volumes) the lists of meanings are not so good as one would expect, as
+compared (say) with the _Century Dictionary_. Work of such magnitude
+and quality is possible, practically, only when the editor of the
+dictionary can command not merely the aid of a very large number of
+scholars and men of science, but their gratuitous aid. In this the _New
+English Dictionary_ has been singularly fortunate. The conditions under
+which it originated, and its aim, have interested scholars everywhere,
+and led them to contribute to the perfecting of it their knowledge and
+time. The long list of names of such helpers in Sir J. A. H. Murray's
+preface is in curious contrast with their absence from Dr Johnson's and
+the few which are given in that of Littré. The editor's principal
+assistants were Dr Henry Bradley and Dr W. A. Craigie. Of the dictionary
+as a whole it may be said that it is one of the greatest achievements,
+whether in literature or science, of modern English scholarship and
+research.
+
+  The _New English Dictionary_ furnishes for the first time data from
+  which the extent of the English word-store at any given period, and
+  the direction and rapidity of its growth, can fairly be estimated. For
+  this purpose the materials furnished by the older dictionaries are
+  quite insufficient, on account of their incompleteness and
+  unhistorical character. For example 100 pages of the _New English
+  Dictionary_ (from the letter H) contain 1002 words, of which, as the
+  dated quotations show, 585 were current in 1750 (though some, of
+  course, were very rare, some dialectal, and so on), 191 were obsolete
+  at that date, and 226 have since come into use. But of the more than
+  700 words--current or obsolete--which Johnson might thus have
+  recorded, he actually did record only about 300. Later dictionaries
+  give more of them, but they in no way show their status at the date in
+  question. It is worth noting that the figures given seem to indicate
+  that not very many more words have been added to the vocabulary of the
+  language during the past 150 years than had been lost by 1750. The
+  pages selected, however, contain comparatively few recent scientific
+  terms. A broader comparison would probably show that the gain has been
+  more than twice as great as the loss.
+
+In the _Deutsches Wörterbuch_ of Jacob and Wilhelm Grimm the scientific
+spirit, as was said above, first found expression in general
+lexicography. The desirability of a complete inventory and investigation
+of German words was recognized by Leibnitz and by various 18th-century
+scholars, but the plan and methods of the Grimms were the direct product
+of the then new scientific philology. Their design, in brief, was to
+give an exhaustive account of the words of the literary language (New
+High German) from about the end of the 15th century, including their
+earlier etymological and later history, with references to important
+dialectal words and forms; and to illustrate their use and history
+abundantly by quotations. The first volume appeared in 1854. Jacob Grimm
+(died 1863) edited the first, second (with his brother, who died in
+1859), third and a part of the fourth volumes; the others have been
+edited by various distinguished scholars. The scope and methods of this
+dictionary have been broadened somewhat as the work has advanced. In
+general it may be said that it differs from the _New English Dictionary_
+chiefly in its omission of pronunciations and other pedagogic matter;
+its irregular treatment of dates; its much less systematic and less
+lucid statement of etymologies; its less systematic and less fruitful
+use of quotations; and its less convenient and less intelligible
+arrangement of material and typography.
+
+These general principles lie also at the foundation of the scholarly
+_Dictionnaire de la langue française_ of E. Littré, though they are
+there carried out less systematically and less completely. In the
+arrangement of the definitions the first place is given to the most
+primitive meaning of the word instead of to the most common one, as in
+the dictionary of the Academy; but the other meanings follow in an order
+that is often logical rather than historical. Quotations also are
+frequently used merely as literary illustrations, or are entirely
+omitted; in the special paragraphs on the history of words before the
+16th century, however, they are put to a strictly historical use. This
+dictionary--perhaps the greatest ever compiled by one man--was published
+1863-1872. (Supplement, 1878.)
+
+The _Thesaurus Linguae Latinae_, prepared under the auspices of the
+German Academies of Berlin, Göttingen, Leipzig, Munich and Vienna, is a
+notable application of the principles and practical co-operative method
+of modern lexicography to the classical tongues. The plan of the work is
+to collect quotations which shall register, with its full context, every
+word (except the most familiar particles) in the text of each Latin
+author down to the middle of the 2nd century A.D., and to extract all
+important passages from all writers of the following centuries down to
+the 7th; and upon these materials to found a complete historical
+dictionary of the Latin language. The work of collecting quotations was
+begun in 1894, and the first part of the first volume has been
+published.
+
+In the making of all these great dictionaries (except, of course, the
+last) the needs of the general public as well as those of scholars have
+been kept in view. But the type to which the general dictionary designed
+for popular use has tended more and more to conform is the
+_encyclopaedic_. This combination of lexicon and encyclopaedia is
+exhibited in an extreme--and theoretically objectionable--form in the
+_Grand dictionnaire universel du XIX^e siècle_ of Pierre Larousse.
+Besides common words and their definitions, it contains a great many
+proper names, with a correspondingly large number of biographical,
+geographical, historical and other articles, the connexion of which with
+the strictly lexicographical part is purely mechanical. Its utility,
+which--notwithstanding its many defects--is very great, makes it,
+however, a model in many respects. Fifteen volumes were published
+(1866-1876), and supplements were brought out later (1878-1890). The
+_Nouveau Larousse illustré_ started publication in 1901, and was
+completed in 1904 (7 vols.). This is not an abridgment or a fresh
+edition of the _Grand Dictionnaire_ of Pierre Larousse, but a new and
+distinct publication.
+
+The most notable work of this class, in English, is the _Century
+Dictionary_, an American product, edited by Professor W. D. Whitney, and
+published 1889-1891 in six volumes, containing 7046 pages (large
+quarto). It conforms to the philological mode in giving with great
+fulness the older as well as the present vocabulary of the language, and
+in the completeness of its etymologies; but it does not attempt to give
+the full history of every word within the language. Among its other more
+noteworthy characteristics are the inclusion of a great number of modern
+scientific and technical words, and the abundance of its quotations. The
+quotations are for the most part provided with references, but they are
+not dated. Even when compared with the much larger _New English
+Dictionary_, the _Century's_ great merit is the excellent enumeration of
+meanings, and the accuracy of its explanations; in this respect it is
+often better and fuller than the _New English_. In the application of
+the encyclopaedic method this dictionary is conservative, excluding,
+with a few exceptions, proper names, and restricting, for the most part,
+the encyclopaedic matter to descriptive and other details which may
+legitimately be added to the definitions. Its pictorial illustrations
+are very numerous and well executed. In the manner of its compilation it
+is a good example of modern cooperative dictionary-making, being the
+joint product of a large number of specialists. Next to the _New English
+Dictionary_ it is the most complete and scholarly of English lexicons.
+
+_Bibliography._--The following list of dictionaries (from the 9th
+edition of this work, with occasional corrections) is given for its
+historical interest, but in recent years dictionary-making has been so
+abundant that no attempt is made to be completely inclusive of later
+works; the various articles on languages may be consulted for these. The
+list is arranged geographically by families of languages, or by regions.
+In each group the order, when not alphabetical, is usually from north to
+south, extinct languages generally coming first, and dialects being
+placed under their language. Dictionaries forming parts of other works,
+such as travels, histories, transactions, periodicals, reading-books,
+&c., are generally excluded. The system here adopted was chosen as on
+the whole the one best calculated to keep together dictionaries
+naturally associated. The languages to be considered are too many for an
+alphabetical arrangement, which ignores all relations both natural and
+geographical, and too few to require a strict classification by
+affinities, by which the European languages, which for many reasons
+should be kept together, would be dispersed. Under either system,
+Arabic, Persian and Turkish, whose dictionaries are so closely
+connected, would be widely separated. A wholly geographical arrangement
+would be inconvenient, especially in Europe. Any system, however, which
+attempts to arrange in a consecutive series the great network of
+languages by which the whole world is enclosed, must be open to some
+objections; and the arrangement adopted in this list has produced some
+anomalies and dispersions which might cause inconvenience if not pointed
+out. The old Italic languages are placed under Latin, all dialects of
+France under French (but Provençal as a distinct language), and
+Wallachian among Romanic languages. Low German and its dialects are not
+separated from High German. Basque is placed after Celtic; Albanian,
+Gipsy and Turkish at the end of Europe, the last being thus separated
+from its dialects and congeners in Northern and Central Asia, among
+which are placed the Kazan dialect of Tatar, Samoyed and Ostiak.
+Accadian is placed after Assyrian among the Semitic languages, and
+Maltese as a dialect of Arabic; while the Ethiopic is among African
+languages as it seemed undesirable to separate it from the other
+Abyssinian languages, or these from their neighbours to the north and
+south. Circassian and Ossetic are joined to the first group of Aryan
+languages lying to the north-west of Persia, and containing Armenian,
+Georgian and Kurd. The following is the order of the groups, some of the
+more important languages, that is, of those best provided with
+dictionaries, standing alone:--
+
+EUROPE: Greek, Latin, French, Romance, Teutonic (Scandinavian and
+German), Celtic, Basque, Baltic, Slavonic, Ugrian, Gipsy, Albanian.
+
+ASIA: Semitic, Armenian, Persian, Sanskrit, Indian, Indo-Chinese, Malay
+Archipelago, Philippines, Chinese, Japanese, Northern and Central Asia.
+
+AFRICA: Egypt and Abyssinia, Eastern Africa, Southern, Western, Central,
+Berber.
+
+AUSTRALIA AND POLYNESIA.
+
+AMERICA: North, Central (with Mexico), South.
+
+
+EUROPE
+
+ Greek.---Athenaeus quotes 35 writers of works, known or supposed to be
+  dictionaries, for, as they are all lost, it is often difficult to
+  decide on their nature. Of these, Anticlides, who lived after the
+  reign of Alexander the Great, wrote [Greek: Exêgêtikos], which seems
+  to have been a sort of dictionary, perhaps explaining the words and
+  phrases occurring in ancient stories. Zenodotus, the first
+  superintendent of the great library of Alexandria, who lived in the
+  reigns of Ptolemy I. and Ptolemy II., wrote [Greek: Glôssai], and also
+  [Greek: Lexeis ethnikai], a dictionary of barbarous or foreign
+  phrases. Aristophanes of Byzantium, son of Apelles the painter, who
+  lived in the reigns of Ptolemy II. and Ptolemy III., and had the
+  supreme management of the Alexandrian library, wrote a number of
+  works, as [Greek: Attikai Lexeis, Lakônikai Glôssai] which, from the
+  titles, should be dictionaries, but a fragment of his [Greek: Lexeis]
+  printed by Boissonade, in his edition of Herodian (London, 1869, 8vo,
+  pp. 181-189), is not alphabetical. Artemidorus, a pupil of
+  Aristophanes, wrote a dictionary of technical terms used in cookery.
+  Nicander Colophonius, hereditary priest of Apollo Clarius, born at
+  Claros, near Colophon in Ionia, in reputation for 50 years, from 181
+  to 135, wrote [Greek: Glôssai] in at least three books. Parthenius, a
+  pupil of the Alexandrian grammarian Dionysius (who lived in the 1st
+  century before Christ), wrote on choice words used by historians.
+  Didymus, called [Greek: chalkenteros], who, according to Athenaeus,
+  wrote 3500 books, and, according to Seneca, 4000, wrote lexicons of
+  the tragic poets (of which book 28 is quoted), of the comic poets, of
+  ambiguous words and of corrupt expressions. Glossaries of Attic words
+  were written by Crates, Philemon, Philetas and Theodorus; of Cretan,
+  by Hermon or Hermonax; of Phrygian, by Neoptolemus; of Rhodian, by
+  Moschus; of Italian, by Diodorus of Tarsus; of foreign words, by
+  Silenus; of synonyms, by Simaristus; of cookery, by Heracleon; and of
+  drinking vessels, by Apollodorus of Cyrene. According to Suidas, the
+  most ancient Greek lexicographer was Apollonius the sophist, son of
+  Archibius. According to the common opinion, he lived in the time of
+  Augustus at Alexandria. He composed a lexicon of words used by Homer,
+  [Greek: Lexeis Homêrikai], a very valuable and useful work, though
+  much interpolated, edited by Villoison, from a MS. of the 10th
+  century, Paris, 1773, 4to, 2 vols.; and by Tollius, Leiden, 1788, 8vo;
+  ed. Bekker, Berlin, 1833, 8vo. Erotian or Herodian, physician to Nero,
+  wrote a lexicon on Hippocrates, arranged in alphabetical order,
+  probably by some copyist, whom Klein calls "homo sciolus." It was
+  first published in Greek in H. Stephani _Dictionarium Medicum_, Paris,
+  1564, 8vo; ed. Klein, Lipsiae, 1865, 8vo, with additional fragments.
+  Timaeus the sophist, who, according to Ruhnken, lived in the 3rd
+  century, wrote a very short lexicon to Plato, which, though much
+  interpolated, is of great value, 1st ed. Ruhnken, Leiden, 1754; ed.
+  locupletior, Lugd. Bat. 1789, 8vo. Aelius Moeris, called the Atticist,
+  lived about 190 A.D., and wrote an Attic lexicon, 1st ed. Hudson,
+  Oxf. 1712, Bekker, 1833. Julius Pollux ([Greek: Ioulios Polydeukês])
+  of Naucratis, in Egypt, died, aged fifty-eight, in the reign of
+  Commodus (180-192), who made him professor of rhetoric at Athens. He
+  wrote, besides other lost works, an Onomasticon in ten books, being a
+  classed vocabulary, intended to supply all the words required by each
+  subject with the usage of the best authors. It is of the greatest
+  value for the knowledge both of language and of antiquities. First
+  printed by Aldus, Venice, 1500, fol.; often afterwards; ed. Lederlinus
+  and Hemsterhuis, Amst. 1706, 2 vols.; Dindorf, 1824, 5 vols., Bethe
+  (1900 f.). Harpocration of Alexandria, probably of the 2nd century,
+  wrote a lexicon on the ten Attic orators, first printed by Aldus, Ven.
+  1503, fol.; ed. Dindorf, Oxford, 1853, 8vo, 2 vols. from 14 MSS.
+  Orion, a grammarian of Thebes, in Egypt, who lived between 390 and
+  460, wrote an etymological dictionary, printed by Sturz, Leipzig,
+  1820, 4to. Helladius a priest of Jupiter at Alexandria, when the
+  heathen temples there were destroyed by Theophilus in 389 or 391
+  escaped to Constantinople, where he was living in 408. He wrote an
+  alphabetical lexicon, now lost, chiefly of prose, called by Photius
+  the largest ([Greek: polystichôtaton]) which he knew. Ammonius,
+  professor of grammar at Alexandria, and priest of the Egyptian ape,
+  fled to Constantinople with Helladius, and wrote a dictionary of words
+  similar in sound but different in meaning, which has been often
+  printed in Greek lexicons, as Aldus, 1497, Stephanus, and separately
+  by Valckenaer, Lugd. Bat. 1739, 4to, 2 vols., and by others. Zenodotus
+  wrote on the cries of animals, printed in Valckenaer's _Ammonius_;
+  with this may be compared the work of Vincentio Caralucci, _Lexicon
+  vocum quae a brutis animalibus emittuntur_, Perusia, 1779, 12mo.
+  Hesychius of Alexandria wrote a lexicon, important for the knowledge
+  of the language and literature, containing many dialectic and local
+  expressions and quotations from other authors, 1st ed. Aldus, Ven.
+  1514, fol.; the best is Alberti and Ruhnken, Lugd. Bat. 1746-1766,
+  fol. 2 vols.; collated with the MS. in St Mark's library, Venice, the
+  only MS. existing, by Niels Iversen Schow, Leipzig, 1792, 8vo; ed.
+  Schmidt, Jena, 1867, 8vo. The foundation of this lexicon is supposed
+  to have been that of Pamphilus, an Alexandrian grammarian, quoted by
+  Athenaeus, which, according to Suidas, was in 95 books from [Epsilon]
+  to [Omega]; [Alpha] to [Delta] had been compiled by Zopirion. Photius,
+  consecrated patriarch of Constantinople, 25th December 857, living in
+  886, left a lexicon, partly extant, and printed with Zonaras, Lips.
+  1808, 4to, 3 vols., being vol. iii.; ed. Naber, Leidae, 1864-1865,
+  8vo, 2 vols. The most celebrated of the Greek glossaries is that of
+  Suidas, of whom nothing is known. He probably lived in the 10th
+  century. His lexicon is an alphabetical dictionary of words including
+  the names of persons and places--a compilation of extracts from Greek
+  writers, grammarians, scholiasts and lexicographers, very carelessly
+  and unequally executed. It was first printed by Demetrius
+  Chalcondylas, Milan, 1499, fol.; the best edition, Bernhardy, Halle,
+  1853, 4to, 2 vols. John Zonaras, a celebrated Byzantine historian and
+  theologian, who lived in the 12th century, compiled a lexicon, first
+  printed by Tittmann, Lips. 1808. 4to, 2 vols. An anonymous Greek
+  glossary, entitled [Greek: Etymologikon mega], _Etymologicum magnum_,
+  has been frequently printed. The first edition is by Musurus, Venitia,
+  1499, fol.; the best by Gaisford, Oxonii, 1848, fol. It contains many
+  grammatical remarks by famous authorities, many passages of authors,
+  and mythological and historical notices. The MSS. vary so much that
+  they look like the works of different authors. To Eudocia Augusta of
+  Makrembolis, wife of the emperors Constantine XI. and Romanus IV.
+  (1059 to 1071), was ascribed a dictionary of history and mythology,
+  [Greek: Iônia] (bed of violets), first printed by D'Ansse de
+  Villoison, _Anecdota Graeca_, Venetiis, 1781, 4to, vol. i. pp. 1-442.
+  It was supposed to have been of much value before it was published.
+  Thomas, Magister Officiorum under Andronicus Palaeologus, afterward
+  called as a monk Theodulus, wrote [Greek: Eklogai onomatôn Attikôn],
+  printed by Callierges, Romae, 1517, 8vo: Papias, _Vocabularium_,
+  Mediolani, 1476, fol.: Craston, an Italian Carmelite monk of Piacenza,
+  compiled a Greek and Latin lexicon, edited by Bonus Accursius, printed
+  at Milan, 1478, fol.: Aldus, Venetiis, 1497, fol.: Guarino, born about
+  1450 at Favora, near Camarino, who called himself both Phavorinus and
+  Camers, published his _Thesaurus_ in 1504. These three lexicons were
+  frequently reprinted. Estienne, _Thesaurus_, Genevae, 1572, fol., 4
+  vols.; ed. Valpy, Lond. 1816-1826, 6 vols. fol.; Paris, 1831-1865, 9
+  vols. fol., 9902 pages: [Greek: Kibôtos], the ark, was intended to
+  give the whole language, ancient and modern, but vol. i.,
+  Constantinople, 1819, fol., 763 pages, [Alpha] to [Delta], only
+  appeared, as the publication was put an end to by the events of 1821.
+  ENGLISH.--Jones, London, 1823, 8vo: Dunbar, Edin. 3rd ed. 1850, 4to:
+  Liddell and Scott, 8th ed. Oxford, 1897, 4to. FRENCH.--Alexandre, 12th
+  ed. Paris, 1863, 8vo; 1869-1871, 2 vols: Chassang, ib. 1872, 8vo.
+  ITALIAN.--Camini, Torino, 1865, 8vo, 972 pages: Müller, ib. 1871, 8vo.
+  SPANISH.--_Diccionario manual, por les padres Esculapios_, Madrid,
+  1859, 8vo. GERMAN.--Passow, 5th ed. Leipzig, 1841-1857, 4to: Jacobitz
+  and Seiler, 4th ed. ib. 1856, 8vo: Benseler, ib. 1859, 8vo: Pape,
+  Braunschweig, 1870-1874, 8vo, 4 vols. Prellwitz, _Etymologisches
+  Wörterbuch der griechischen Sprache_, new edition, 1906: Herwerden,
+  _Lexicon Graecum suppletorium et dialecticum_, 1902.
+  DIALECTS.--_Attic_: Moeris, ed. Pierson, Lugd. Bat. 1759. 8vo. _Attic
+  Orators_: Reiske, Oxon. 1828, 8vo, 2 vols. _Doric_: Portus, Franckof.
+  1605, 8vo. _Ionic_: Id. ib. 1603, 8vo; 1817; 1825. PROSODY.--Morell,
+  Etonae, 1762, 4to; ed. Maltby, Lond. 1830, 4to: Brasse, Lond. 1850,
+  8vo. RHETORIC.--Ernesti, Lips. 1795, 8vo. MUSIC.--Drieberg, Berlin,
+  1855. ETYMOLOGY.--Curtius, Leipzig, 1858-1862: Lancelot, Paris, 1863,
+  8vo. SYNONYMS.--Peucer, Dresden, 1766, 8vo: Pillon, Paris, 1847, 8vo.
+  PROPER NAMES.--Pape, ed. Sengebusch, 1866, 8vo, 969 pages.
+  VERBS.--Veitch, 2nd ed. Oxf. 1866. TERMINATIONS.--Hoogeveen, Cantab.
+  1810, 4to: Pape, Berlin, 1836, 8vo. PARTICULAR AUTHORS.--_Aeschylus_:
+  Wellauer, 2 vols. Lips. 1830-1831, 8vo. _Aristophanes_: Caravella,
+  Oxonii, 1822, 8vo. _Demosthenes_: Reiske, Lips. 1775, 8vo.
+  _Euripides_: Beck, Cantab. 1829, 8vo. _Herodotus_: Schweighäuser,
+  Strassburg, 1824, 8vo, 2 vols. _Hesiod_: Osoruis, Neapol. 1791, 8vo.
+  _Homer_: Apollonius Sophista, ed. Tollius, Lugd. Bat., 1788, 8vo:
+  Schaufelberger, Zürich, 1761-1768, 8vo, 8 vols.: Crusius, Hanover,
+  1836, 8vo: Wittich, London, 1843, 8vo: Döderlein, Erlangen, 8vo, 3
+  vols.: Eberling, Lipsiae, 1875, 8vo: Autenrieth, Leipzig, 1873, 8vo;
+  London, 1877, 8vo. _Isocrates_: Mitchell, Oxon. 1828, 8vo. _Pindar_:
+  Portus, Hanov. 1606, 8vo. _Plato_: Timaeus, ed. Koch, Lips. 1828, 8vo:
+  Mitchell, Oxon. 1832, 8vo: Ast, Lips. 1835-1838, 8vo, 3 vols.
+  _Plutarch_: Wyttenbach, Lips. 1835, 8vo, 2 vols. _Sophocles_: Ellendt,
+  Regiomonti, 1834-1835, 8vo ed.; Genthe, Berlin, 1872, 8vo.
+  _Thucydides_: Bétant, Geneva, 1843-1847, 8vo, 2 vols. _Xenophon_:
+  Sturtz, Lips. 1801-1804, 8vo, 4 vols.: Cannesin (Anabasis,
+  Gr.-Finnish), Helsirgissä, 1868, 8vo: Sauppe, Lipsiae, 1869, 8vo.
+  _Septuagint_: Hutter, Noribergae, 1598, 4to: Biel, Hagae, 1779-1780,
+  8vo. _New Testament_: Lithocomus, Colon, 1552, 8vo: Parkhurst, ed.
+  Major, London, 1845, 8vo: Schleusner (juxta ed. Lips. quartam),
+  Glasguae, 1824, 4to.
+
+
+ Medieval and Modern Greek.--Meursius, Lugd. Bat. 1614, 4to:
+  Critopulos, Stendaliae, 1787, 8vo: Portius, Par. 1635, 4to: Du Cange,
+  Paris, 1682, fol., 2 vols.; Ludg. 1688, fol. ENGLISH.--Polymera,
+  Hermopolis, 1854, 8vo: Sophocles, Cambr. Mass. 1860-1887: Contopoulos,
+  Athens, 1867, 8vo; Smyrna, 1868-1870, 8vo, 2 parts, 1042 pages.
+  FRENCH.--Skarlatos, Athens, 1852, 4to: Byzantius, ib. 1856, 8vo, 2
+  vols.: Varvati, 4th ed. ib., 1860, 8vo. ITALIAN.--Germano, Romae,
+  1622, 8vo: Somavera, Parigi, 1709, fol., 2 vols.: Pericles,
+  Hermopolis, 1857, 8vo. GERMAN.--Schmidt, Lips. 1825-1827, 12mo, 2
+  vols.: Petraris, Leipz. 1897. POLYGLOTS.--Koniaz (Russian and Fr.),
+  Moscow, 1811, 4to; Schmidt (Fr.-Germ.), Leipzig, 1837-1840, 12mo, 3
+  vols.: Theocharopulas de Patras (Fr.-Eng.), Munich, 1840, 12mo.
+
+
+ Latin.--Johannes de Janua, _Catholicon_ or _Summa_, finished in 1286,
+  printed Moguntiæ 1460, fol.; Venice, 1487; and about 20 editions
+  before 1500: Johannes, _Comprehensorium_, Valentia, 1475, fol.: Nestor
+  Dionysius, _Onomasticon_, Milan, 1477, fol.: Stephanus, Paris, 1531,
+  fol., 2 vols.: Gesner, Lips. 1749, fol., 4 vols.: Forcellini, Patavii,
+  1771, fol., 4 vols. POLYGLOT.--Calepinus, Reggio, 1502, fol. (Aldus
+  printed 16 editions, with the Greek equivalents of the Latin words;
+  Venetiis, 1575, fol., added Italian, French and Spanish; Basileae,
+  1590, fol., is in 11 languages; several editions, from 1609, are
+  called Octolingue; many of the latter 2 vol. editions were edited by
+  John Facciolati): Verantius (Ital., Germ., Dalmatian, Hungarian),
+  Venetiis, 1595, 4to: Lodereckerus (Ital., Germ., Dalm., Hungar.,
+  Bohem., Polish), Pragae, 1605, 4to. ENGLISH.--_Promptorium
+  parvulorum_, compiled in 1440 by Galfridus Grammaticus, a Dominican
+  monk of Lynn Episcopi, in Norfolk, was printed by Pynson, 1499; 8
+  editions, 1508-1528, ed. Way, Camden Society, 1843-1865, 3 vols. 4to;
+  _Medulla grammaticis_, probably by the same author, MS. written 1483;
+  printed as _Ortus vocabulorum_, by Wynkyn de Worde, 1500; 13 editions
+  1509-1523; Sir Thomas Elyot, London, 1538, fol.; 2nd ed. 1543;
+  _Bibliotheca Eliotae_, ed. Cooper, ib. 1545, fol.: Huloet,
+  _Abecedarium_, London, 1552, fol.; _Dictionarie_, 1572, fol.: Cooper,
+  London, 1565, fol.; 4th edition, 1584, fol.: Baret, _Alvearie_, ib.
+  1575, fol.; 1580, fol.: Fleming, ib. 1583, fol.: Ainsworth, London,
+  1736, 4to; ed. Morell, London, 1796, 4to, 2 vols.; ed. Beatson and
+  Ellis, ib. 1860, 8vo: Scheller, translated by Riddle, Oxford, 1835,
+  fol.: Smith, London, 1855, 8vo; 1870: Lewis and Short, Oxford, 1879.
+  ENG.-LATIN.--Levins, _Manipulus puerorum_, Lond. 1570, 4to: Riddle,
+  ib. 1838, 8vo: Smith, ib. 1855, 8vo. FRENCH.--_Catholicon parvum_,
+  Geneva, 1487: Estienne, _Dictionnaire_, Paris, 1539, fol. 675 pages;
+  enlarged 1549; ed. Huggins, Lond. 1572: Id. _Dictionarium
+  Latino-Gallicum_, Lutetiae, 1546, fol.; Paris, 1552; 1560: Id.,
+  _Dictionariolum puerorum_, Paris, 1542, 4to: _Les Mots français_,
+  Paris, 1544, 4to; the copy in the British Museum has the autograph of
+  Queen Catherine Parr: Thierry (Fr.-Lat.), Paris, 1564, fol.: Danet, Ad
+  usum Delphini, Paris, 1700, 4to, 2 vols.; and frequently: Quicherat,
+  9th ed. Paris, 1857, 8vo: Theil, 3rd ed. Paris, 1863, 8vo: Freund, ib.
+  1835-1865, 4to, 3 vols. GERMAN.--Joh. Melber, of Gerolzhofen,
+  _Vocabularius praedicantium_, of which 26 editions are described by
+  Hain (_Repertorium_, No. 11,022, &c.), 15 undated, 7 dated 1480-1495,
+  4to, and 3 after 1504: _Vocabularius gemma gemmarum_, Antwerp, 1484,
+  4to; 1487; 12 editions, 1505-1518: Herman Torentinus, _Elucidarius
+  carminum_, Daventri, 1501, 4to; 22 editions, 1504-1536: Binnart, Ant.
+  1649, 8vo: Id., _Biglotton_, ib. 1661; 4th ed. 1688: Faber, ed.
+  Gesner, Hagae Com. 1735, fol., 2 vols.: Hederick, Lips. 1766, 8vo, 2
+  vols.: Ingerslev, Braunschweig, 1835-1855, 8vo, 2 vols.: _Thesaurus
+  linguae Latinae_, Leipzig, 1900: Walde, _Lateinisches etymologisches
+  Wörterbuch_, 1906. ITALIAN.--Seebar (Sicilian translation of Lebrixa),
+  Venet. 1525, 8vo: Venuti, 1589, 8vo: Galesini, Venez. 1605, 8vo:
+  Bazzarini and Bellini, Torino, 1864, 4to, 2 vols. 3100 pages.
+  SPANISH.--Salmanticae, 1494, fol.; Antonio de Lebrixa, Nebrissenis,
+  Compluti, 1520, fol., 2 vols.: Sanchez de la Ballesta, Salamanca,
+  1587, 4to: Valbuena, Madrid, 1826, fol. PORTUGUESE.--Bluteau, Lisbon,
+  1712-1728, fol., 10 vols: Fonseca, ib. 1771, fol.: Ferreira, Paris,
+  1834, 4to; 1852. ROMANSCH.--_Promptuario di voci volgari_, Valgrisii,
+  1565, 4to. VLACH.--Divalitu, Bucuresci, 1852, 8vo.
+  SWEDISH.--_Vocabula_, Rostock, 1574, 8vo; Stockholm, 1579: Lindblom,
+  Upsala, 1790, 4to. DUTCH.--Binnart, Antw. 1649, 8vo: Scheller, Lugd.
+  Bat. 1799, 4to, 2 vols. FLEMISH.--Paludanus, Gandavi, 1544, 4to.
+  POLISH.--Macinius, Königsberg, 1564, fol.: Garszynski, Breslau, 1823,
+  8vo, 2 vols. BOHEMIAN.--Johannes Aquensis, Pilsnae, 1511, 4to:
+  Reschel, Olmucii, 1560-1562, 4to, 2 vols.: Cnapius, Cracovia, 1661,
+  fol., 3 vols. ILLYRIAN.--Bellosztenecz, Zagrab, 1740, 4to: Jambresich
+  (also Germ. and Hungar.), Zagrab, 1742, 4to. SERVIAN.--Swotlik, Budae,
+  1721, 8vo. HUNGARIAN.--Molnar, Frankf. a. M. 1645, 8vo: Pariz-Papai,
+  Leutschen, 1708, 8vo; 1767. FINNISH.--Rothsen, Helsingissä, 1864, 8vo.
+  POETIC.--_Epithetorum et synonymorum thesaurus_, Paris, 1662, 8vo,
+  attributed to Chatillon; reprinted by Paul Aler, a German Jesuit, as
+  _Gradus ad Parnassum_, Paris, 1687, 8vo; many subsequent editions:
+  _Schirach_, Hal. 1768, 8vo: Noel, Paris, 1810, 8vo; 1826: Quicherat,
+  Paris, 1852, 8vo: Young, London, 1856, 8vo. EROTIC.--Rambach,
+  Stuttgart, 1836, 8vo. RHETORICAL.--Ernesti, Lips. 1797, 8vo. CIVIL
+  LAW.--Dirksen, Berolini, 1837, 4to. SYNONYMS.--Hill, Edinb. 1804, 4to:
+  Döderlein, Lips. 1826-1828, 8vo, 6 vols. ETYMOLOGY.--Danet, Paris,
+  1677, 8vo: Vossius, Neap. 1762, fol., 2 vols.: Salmon, London, 1796,
+  8vo, 2 vols.: Nagel, Berlin, 1869, 8vo; Latin roots, with their French
+  and English derivatives, explained in German: Zehetmayr, Vindobonae,
+  1873, 8vo: Vani[vc]ek, Leipz. 1874, 8vo. BARBAROUS.--Marchellus,
+  Mediol. 1753, 4to; Krebs, Frankf. a. M. 1834, 8vo; 1837. PARTICULAR
+  AUTHORS.--_Caesar_: Crusius, Hanov. 1838, 8vo. _Cicero_: Nizzoli,
+  Brescia, 1535, fol.; ed. Facciolati, Patavii, 1734, fol.; London,
+  1820, 8vo, 3 vols.: Ernesti, Lips. 1739, 8vo; Halle, 1831. _Cornelius
+  Nepos_: Schmieder, Halle, 1798, 8vo; 1816: Billerbeck, Hanover, 1825,
+  8vo. _Curtius Rufus_: Crusius, Hanov. 1844, 8vo. _Horace_: Ernesti,
+  Berlin, 1802-1804, 8vo, 3 vols.: Döring, Leipz. 1829, 8vo. _Justin_:
+  Meinecke, Lemgo, 1793, 8vo; 2nd ed. 1818. _Livy_: Ernesti, Lips. 1784,
+  8vo; ed Schäfer, 1804. _Ovid_: Gierig, Leipz. 1814: (Metamorphoses)
+  Meinecke, 2nd ed., Lemgo, 1825, 8vo: Billerbeck (Do.), Hanover, 1831,
+  8vo. _Phaedrus_: Oertel, Nürnberg, 1798, 8vo: Hörstel, Leipz. 1803,
+  8vo: Billerbeck Hanover, 1828, 8vo. _Plautus_: Paraeus, Frankf. 1614,
+  8vo. _Pliny_: Denso, Rostock, 1766, 8vo_. Pliny, jun._: Wensch,
+  Wittenberg, 1837-1839, 4to. _Quintilian_: Bonnellus, Leipz. 1834, 8vo.
+  _Sallust_: Schneider, Leipz. 1834, 8vo: Crusius, Hanover, 1840, 8vo.
+  _Tacitus_: Bötticher, Berlin, 1830, 8vo. _Velleius Paterculus_: Koch,
+  Leipz. 1857, 8vo. _Virgil_: _Clavis_, London, 1742, 8vo: Braunhard,
+  Coburg, 1834, 8vo. _Vitruvius_: Rode, Leipz. 1679, 4to, 2 vols.:
+  Orsini, Perugia, 1801, 8vo.
+
+  OLD ITALIAN LANGUAGES.--Fabretti, Torini, 1858, 4to. _Umbrian_:
+  Huschke, Leipz. 1860, 8vo. _Oscan and Sabellian_: Id. Elberfeld, 1856,
+  8vo.
+
+  MEDIEVAL LATIN.--Du Cange, _Glossarium_, Paris, 1733-1736, fol., 6
+  vols.; Carpentier, Suppl., Paris, 1766, fol., 4 vols.; ed. Adelung,
+  Halae, 1772-1784, 8vo, 6 vols.; ed. Henschel, Paris, 1840-1850, 4to, 7
+  vols. (vol. vii. contains a glossary of Old French): Brinckmeier,
+  Gotha, 1850-1863, 8vo, 2 vols.: Hildebrand (_Glossarium saec. ix._),
+  Götting. 1854, 4to: Diefenbach, _Glossarium_, Frankf. 1857, 4to: Id.
+  _Gloss. novum_, ib. 1867, 4to. ECCLESIASTICAL.--Magri, Messina, 1644,
+  4to; 8th ed. Venezia, 1732; Latin translation, _Magri Hierolexicon_,
+  Romae, 1677, fol.; 6th ed. Bologna, 1765, 4to, 2 vols.
+
+
+_Romance Languages. _
+
+ Romance Languages generally.--Diez, Bonn, 1853, 8vo; 2nd ed. ib.
+  1861-1862, 8vo, 2 vols.; 3rd ed. ib. 1869-1870, 8vo, 2 vols.; transl.
+  by Donkin, 1864, 8vo.
+
+
+ French.--Ranconet, _Thresor_, ed. Nicot, Paris, 1606, fol.; ib. 1618,
+  4to: Richelet, Genève, 1680, fol., 2 vols.; ed. Gattel, Paris, 1840,
+  8vo, 2 vols.
+
+  The French Academy, after five years' consideration, began their
+  dictionary, on the 7th of February 1639, by examining the letter A,
+  which took them nine months to go through. The word Académie was for
+  some time omitted by oversight. They decided, on the 8th of March
+  1638, not to cite authorities, and they have since always claimed the
+  right of making their own examples. Olivier justifies them by saying
+  that for eighty years all the best writers belonged to their body, and
+  they could not be expected to cite each other. Their design was to
+  raise the language to its last perfection, and to open a road to reach
+  the highest eloquence. Antoine Furetière, one of their members,
+  compiled a dictionary which he says cost him forty years' labour for
+  ten hours a day, and the manuscript filled fifteen chests. He gave
+  words of all kinds, especially technical, names of persons and places,
+  and phrases. As a specimen, he published his _Essai_, Paris, 1684,
+  4to; Amst. 1685, 12mo. The Academy charged him with using the
+  materials they had prepared for their dictionary, and expelled him, on
+  the 22nd of January 1685, for plagiarism. He died on the 14th of May
+  1688, in the midst of the consequent controversy and law suit. His
+  complete work was published, with a preface by Bayle, La Haye and
+  Rotterdam, 1690, fol., 3 vols.; again edited by Basnage de Beauval,
+  1701; La Haye, 1707, fol., 4 vols. From the edition of 1701 the very
+  popular so-called _Dictionnaire de Trevoux_, Trevoux, 1704, fol., 2
+  vols., was made by the Jesuits, who excluded everything that seemed to
+  favour the Calvinism of Basnage. The last of its many editions is
+  Paris, 1771, fol., 8 vols. The Academy's dictionary was first printed
+  Paris, 1694, fol., 2 vols. They began the revision in 1700; second
+  edition 1718, fol., 2 vols.; 3rd, 1740, fol., 2 vols.; 6th, 1835, 2
+  vols. 4to, reprinted 1855; Supplément, by F. Raymond, 1836, 4to;
+  Complément, 1842, 4to, reprinted 1856; _Dictionnaire historique_,
+  Paris, 1858-1865, 4to, 2 parts (A to Actu), 795 pages, published by
+  the Institut: Dochez, Paris, 1859, 4to: Bescherelle, ib. 1844, 4to, 2
+  vols.; 5th ed. Paris, 1857, 4to, 2 vols.; 1865; 1887: Landais, Paris,
+  1835; 12th ed. ib. 1854, 4to, 2 vols.: Littré, Paris, 1863-1873, 4to,
+  4 vols. 7118 pages: Supplément, Paris, 1877, 4to: Godefroy (with
+  dialects from 9th to 15th cent.), Paris, 1881-1895, and _Complément_:
+  Hatzfield, Darmesteter, and Thomas, Paris, 1890-1900: Larive and
+  Fleury, (_mots et choses, illustré_), Paris, 1884-1891.
+  ENGLISH.--Palsgrave, _Lesclaircissement de la langue francoyse_,
+  London, 1530, 4to, 2 parts; 1852: Hollyband, London, 1533, 4to:
+  Cotgrave, ib. 1611, fol.: Boyer, La Haye, 1702, 4to, 2 vols.; 37th ed.
+  Paris, 1851, 8vo, 2 vols.: Fleming and Tibbins, Paris, 1846-1849, 4to,
+  2 vols.; ib. 1854, 4to, 2 vols.; ib. 1870-1872, 4to, 2 vols.: Tarver,
+  London, 1853-1854, 8vo, 2 vols.; 1867-1872: Bellows, Gloucester, 1873,
+  16mo; ib. 1876. IDEOLOGICAL, OR ANALOGICAL.--Robertson, Paris, 1859,
+  8vo: Boissière, Paris, 1862, 8vo. ETYMOLOGY.--Lebon, Paris, 1571, 8vo:
+  Ménage, ib. 1650, 4to. Pougens projected a _Trésor des origines_, his
+  extracts for which, filling nearly 100 volumes folio, are in the
+  library of the Institut. He published a specimen, Paris, 1819, 4to.
+  After his death, _Archéologie française_, Paris, 1821, 8vo, 2 vols.,
+  was compiled from his MSS., which were much used by Littré: Scheler,
+  Bruxelles, 1862, 8vo; 1873: Brachet, 2nd ed. Paris, 1870, 12mo;
+  English trans. Kitchin, Oxf. 1866, 8vo. GREEK WORDS.--Trippault,
+  Orleans, 1580, 8vo: Morin, Paris, 1809, 8vo. GERMAN WORDS.--Atzler,
+  Cöthen, 1867, 8vo. ORIENTAL WORDS.--Pihan, Paris, 1847, 8vo; 1866:
+  Devic, ib. 1876, 8vo. NEOLOGY.--Desfontaines, 3rd ed. Amst. 1728,
+  12mo: Mercier, Paris, 1801, 8vo, 2 vols.: Richard, ib. 1842, 8vo; 2nd
+  ed. 1845. POETIC.--_Dict. des rimes_ (by La Noue), Geneve, 1596, 8vo;
+  Cologny, 1624, 8vo: Carpentier, _Le Gradus français_, Paris, 1825,
+  8vo, 2 vols. EROTIC.--De Landes, Bruxelles, 1861, 12mo.
+  ORATORY.--Demandre and Fontenai, Paris, 1802, 8vo: Planche, ib.
+  1819-1820, 8vo, 3 vols. PRONUNCIATION.--Féline, ib. 1857, 8vo. DOUBLE
+  FORMS.--Brachet, ib. 1871, 8vo. EPITHETS.--Daire, ib. 1817, 8vo.
+  VERBS.--Bescherelle, ib. 1855, 8vo, 2 vols.: 3rd ed. 1858.
+  PARTICIPLES.--Id., ib. 1861, 12mo. DIFFICULTIES.--Boiste, London,
+  1828, 12mo: Laveaux, Paris, 1872, 8vo, 843 pages.
+  SYNONYMS.--Boinvilliers, Paris, 1826, 8vo: Lafaye, ib. 1858, 8vo;
+  1861; 1869: Guizot, ib. 1809, 8vo; 6th ed. 1863; 1873.
+  HOMONYMS.--Zlatagorski (Germ., Russian, Eng.), Leipzig, 1862, 8vo, 664
+  pages. IMITATIVE WORDS.--Nodier, _Onomatopées_, ib. 1828, 8vo.
+  TECHNOLOGY.--D'Hautel, ib. 1808, 8vo, 2 vols.: Desgranges, ib. 1821,
+  8vo: Tolhausen (Fr., Eng., Germ.), Leipz. 1873, 8vo, 3 vols. FAULTS OF
+  EXPRESSION.--Roland, Gap, 1823, 8vo: Blondin, Paris, 1823, 8vo.
+  PARTICULAR AUTHORS.--_Corneille_: Godefroy, ib. 1862, 8vo, 2 vols.:
+  Marty-Laveaux, ib. 1868, 8vo, 2 vols. _La Fontaine_: Lorin, ib. 1852,
+  8vo. _Malherbe_: Regnier, ib. 1869, 8vo. _Molière_: Genin, ib. 1846,
+  8vo: Marty-Laveaux, ib. 8vo. _Racine_: Marty-Laveaux, ib. 1873, 8vo, 2
+  vols. _M^me de Sévigné_: Sommer, ib. 1867, 8vo, 2 vols. OLD
+  FRENCH.--La Curne de St Palaye prepared a dictionary, of which he only
+  published _Projet d'un glossaire_, Paris, 1756, 4to. His MSS. in many
+  volumes are in the National Library, and were much used by Littré.
+  They were printed by L. Favre, and fasciculi 21-30 (tom. iii.), Niort,
+  4to, 484 pages, were published in February 1877. Lacombe (vieux
+  langage), Paris, 1766, 2 vols. 4to: Kelham (Norman and Old French),
+  London, 1779, 8vo: Roquefort (langue romane), Paris, 1808, 8vo;
+  Supplément, ib. 1820, 8vo: Pougens, _Archéologie_, ib. 1821, 8vo, 2
+  vols.: Burguy, Berlin, 1851-1856, 8vo, 3 vols.: Laborde (_Notice des
+  émaux ... du Louvre_, part ii.), Paris, 1853, 8vo, 564 pages:[3]
+  Gachet (rhymed chronicles), Bruxelles, 1859, 4to: Le Héricher (Norman,
+  English and French), Paris, 1862, 3 vols. 8vo: Hippeau (12th and 13th
+  centuries), Paris, 1875, 8vo. DIALECTS.--Jaubert (central), Paris,
+  1856-1857, 8vo, 2 vols.: Baumgarten (north and centre), Coblentz,
+  1870, 8vo: Azais, _Idiomes romans du midi_, Montpellier, 1877.
+  _Austrasian_: François. Metz, 1773, 8vo. _Auvergne_: Mège, Riom, 1861,
+  12mo. _Bearn_: Lespi, Pau, 1858, 8vo. _Beaucaire_: Bonnet (Bouguirén),
+  Nismes, 1840, 8vo. _Pays de Bray_: Decorde, Neufchâtel, 1852, 8vo.
+  _Burgundy_: Mignard, Dijon, 1870, 8vo. _Pays de Castres_: Couzinié,
+  Castres, 1850, 4to. _Dauphiné_: Champollion-Figeac, Paris, 1809, 8vo:
+  Jules, Valence, 1835, 8vo; Paris, 1840, 4to. _Dep. of Doubs_: Tissot
+  (Patois des Fourg, arr. de Pontarlier) Besançon, 1865, 8vo. _Forez_:
+  Gras, Paris, 1864, 8vo; Neolas, Lyon, 1865, 8vo. _Franche Comté_:
+  Maisonforte, 2nd ed. Besançon, 1753, 8vo. _Gascony_: Desgrouais
+  (Gasconismes corrigés), Toulouse, 1766, 8vo; 1769; 1812, 12mo, 2
+  vols.; 1825, 8vo, 2 vols. _Dep. of Gers_: Cenac-Montaut, Paris, 1863,
+  8vo. _Geneva_: Humbert, Geneve, 1820, 8vo. _Languedoc_: Odde, Tolose,
+  1578, 8vo: Doujat, Toulouse, 1638, 8vo: De S.[auvages], Nismes, 1756,
+  2 vols.; 1785; Alais, 1820: Azais, Beziers, 1876, &c., 8vo: Hombres,
+  Alais, 1872, 4to: Thomas (_Greek words_) Montpellier, 1843, 4to.
+  _Liége_: Forir, Liége, 1866, 8vo, vol i. 455 pages. _Lille_: Vermesse,
+  Lille, 1861, 12mo: Debuire du Buc ib., 1867, 8vo. _Limousin_: Beronie,
+  ed. Vialle (Corrèze), Tulle, 1823, 4to. _Lyonnais, Forez,
+  Beaujolais_: Onofrio, Lyon, 1864, 8vo. _Haut Maine_: R[aoul] de
+  M.[ontesson], Paris, 1857; 1859, 503 pages. _Mentone_: Andrews, Nice,
+  1877, 12mo. _Dep. de la Meuse_: Cordier, Paris, 1853, 8vo. _Norman_:
+  Edélestand and Alfred Duméril, Caen, 1849, 8vo: Dubois, ib. 1857, 8vo:
+  Le Héricher (_Philologie topographique_), Caen, 1863, 4to: Id.
+  (éléments scandinaves), Avranches, 1861, 12mo: Metivier (Guernsey),
+  London, 1870, 8vo: Vasnier (arrond de Pont Audemer), Rouen, 1861, 8vo:
+  Delboulle (Vallée d'Yères), Le Havre, 1876. _Picardy_: Corblet,
+  Amiens, 1851, 8vo. _Poitou, Saintonge, Aunis_: Favre, Niort, 1867,
+  8vo. _Poitou_: Beauchet-Filleau, Paris, 1864, 8vo: Levrier, Niort,
+  1867, 8vo: Lalanne, Poitiers, 1868, 8vo. _Saintonge_: Boucherie,
+  Angoulême, 1865, 8vo: Jonain, Royan, 1867, 8vo. _Savoy_: Pont
+  (Terratzu de la Tarantaise), Chambery, 1869, 8vo. _La Suisse Romande_:
+  Bridel, Lausanne, 1866, 8vo. _Dep. of Tarn_: Gary, Castre, 1845, 8vo.
+  _Dep. of Vaucluse_: Barjavel, Carpentras, 1849, 8vo. _Walloon
+  (Rouchi)_: Cambresier, Liége, 1787, 8vo: Grandgagnage, ib. 1845-1850,
+  8vo. 2 vols.: Chavée, Paris, 1857, 18mo: Vermesse, Doudi, 1867, 8vo.
+  Sigart (_Montois_), Bruxelles, 1870, 8vo. SLANG.--Oudin, _Curiositez
+  Françaises_, Paris, 1640, 8vo: Baudeau de Saumaise (Précieuses, Langue
+  de Ruelles), Paris, 1660, 12mo; ed. Livet, ib. 1856: Le Roux, _Dict.
+  Comique_, Amst. 1788, and 6 other editions: Carême Prenant [i.e.
+  Taumaise], (argot réforme), Paris, 1829, 8vo: Larchey (excentricitées
+  du langage), Paris, 1860, 12mo; 5th ed. 1865: Delvau (langue verte,
+  Parisian), Paris, 1867, 8vo: Larchey, Paris, 1873, 4to, 236 pages.
+
+
+ Provençal.--Pallas, Avignon, 1723, 4to: Bastero, _La Crusca
+  Provenzale_, Roma, 1724, fol. vol. i. only: Raynouard, Paris,
+  1836-1844, 8vo, 6 vols.: Garcin, Draguignand, 1841, 8vo, 2 vols.:
+  Honnorat, Digne, 1846-1849, 4to, 4 vols. 107,201 words: Id., _Vocab.
+  fr. prov._, ib. 1848, 12mo, 1174 pages.
+
+
+ Spanish.--Covarruvias Orosco, Madrid, 1611, fol.: ib. 1673-1674, fol.
+  2 vols.; Academia Española, Madrid, 1726-1739, fol. 6 vols.; 8th ed.
+  1837: Caballero, Madrid, 1849, fol.; 8th ed. ib. 1860, 4to, 2 vols.:
+  Cuesta, ib. 1872, fol. 2 vols.: Campano, Paris, 1876, 18mo, 1015
+  pages. Cuervo, 1886-1894; Monlau, 1881; Zerola, Toro y Gomes, and
+  Isaza, 1895; Serrano (encyclopaedic) 1876-1881. ENGLISH.--Percivall,
+  London, 1591, 4to: Pineda, London, 1740, fol.: Connelly and Higgins,
+  Madrid, 1797-1798, 4to, 4 vols.: Neuman and Baretti, 9th ed. London,
+  1831, 8vo, 2 vols.; 1874. FRENCH.--Oudin, Paris, 1607, 4to, 1660;
+  Gattel, Lyon, 1803, 4to, 2 vols.: Dominguez, Madrid, 1846, 8vo, 6
+  vols.: Blanc, Paris, 1862, 8vo, 2 vols. GERMAN.--Wagener, Hamb.
+  1801-1805, 8vo, 4 vols.: Seckendorp, ib. 1823, 8vo, 3 vols.:
+  Franceson, 3rd ed. Leipzig, 1862, 8vo, 2 vols. ITALIAN.--Franciosini,
+  Venezia, 1735, 8vo, 2 vols.; Cormon y Manni, Leon, 1843, 16mo, 2
+  vols.: Romero, Madrid, 1844, 4to. SYNONYMS.--_Diccionario de
+  Sinonimos_, Paris, 1853, 4to. ETYMOLOGY.--Aldrete, Madrid, 1682, fol.:
+  Monlau y Roca, ib. 1856, 12mo; Barcia, 1881-1883. ARABIC
+  WORDS.--Hammer Purgstall, Wien, 1855, 8vo: Dozy and Engelmann, 2d ed.
+  Leiden, 1869, 8vo. ANCIENT.--Sanchez, Paris, 1842, 8vo.
+  RHYMING.--Garcia de Rengifo (consonancias) Salmantica, 1592, 4to;
+  1876. DON QUIXOTE.--Beneke (German), Leipzig, 1800, 16mo; 4th ed.
+  Berlin, 1841, 16mo. DIALECTS.--_Aragonese_: Peralta, Zaragoza, 1836,
+  8vo: Borao, ib. 1859, 4to. _Catalan_: Rocha de Girona (Latin),
+  Barcinone, 1561, fol.: _Dictionari Catala_ (Lat. Fr. Span.),
+  Barcelona, 1642, 8vo: Lacavalleria (Cat.-Lat.), ib. 1696, fol.:
+  Esteve, ed. Belvitges, &c. (Catal.-Sp. Lat.), Barcelona, 1805-1835,
+  fol. 2 vols.: Saura (Cat.-Span.), ib. 1851, 16mo; 2nd ed.(Span.-Cat.),
+  ib. 1854; 3rd ed. (id.) ib. 1862, 8vo: Labernia, ib. 1844-1848, 8vo, 2
+  vols. 1864. _Gallegan_: Rodriguez, Coruña, 1863, 4to: Cuveira y Piñol,
+  Madrid, 1877, 8vo. Majorca: Figuera, Palma, 1840, 4to: Amengual, ib.
+  1845, 4to. _Minorca_: _Diccionario_, Madrid, 1848, 8vo. _Valencian_:
+  Palmyreno, Valentiae, 1569: Ros, Valencia, 1764, 8vo: Fuster, ib.
+  1827, 8vo: Lamarca, 2nd ed. ib. 1842, 16mo. _Cuba_: _Glossary of
+  Creole Words_, London, 1840, 8vo: Pichardo, 1836; 2nd ed. Havana,
+  1849, 8vo; 3rd ed. ib. 1862, 8vo; Madrid, 1860, 4to.
+
+
+ Portuguese.--Lima, Lisbon, 1783, 4to: Moraes da Silva, ib. 1789, 4to,
+  2 vols.; 6th ed. 1858: Academia real das Sciencas, ib. 1793, tom. i.,
+  ccvi. and 544 pages (A to Azurrar); Faria, ib. 1849, fol. 2 vols.; 3rd
+  ed. ib. 1850-1857, fol. 2 vols. 2220 pages. ENGLISH.--Vieyra, London,
+  1773, 2 vols. 4to: Lacerda, Lisboa, 1866-1871, 4to, 2 vols.
+  FRENCH.--Marquez, Lisboa, 1756-1761, fol. 2 vols.: Roquette, Paris,
+  1841, 8vo, 2 vols.; 4th ed. 1860: Marques, Lisbonne, 1875, fol. 2
+  vols.: Souza Pinto, Paris, 1877, 32mo, 1024 pages. GERMAN.--Wagener,
+  Leipzig, 1811-1812, 8vo, 2 vols.: Wollheim, ib. 1844, 12mo, 2 vols.:
+  Bösche, Hamburg, 1858, 8vo, 2 vols. 1660 pages. ITALIAN.--Costa e Sá,
+  Lisboa, 1773-1774, fol. 2 vols. 1652 pages: Prefumo, Lisboa, 1853,
+  8vo, 1162 pages. ANCIENT.--Joaquim de Sancta Rosa de Viterbo, ib.
+  1798, fol. 2 vols.; 1824, 8vo. ARABIC WORDS.--Souza, ib. 1789, 4to;
+  2nd ed. by S. Antonio Moura, ib. 1830, 224 pages. ORIENTAL AND AFRICAN
+  WORDS, NOT ARABIC.--Saõ Luiz, ib. 1837, 4to, 123 pages. FRENCH
+  WORDS.--Id., ib. 1827, 4to; 2nd ed. Rio de Janeiro, 1835, 8vo.
+  SYNONYMS.--Id., ib. 1821, 4to; 2nd ed. ib. 1824-1828, 8vo. Fonseca,
+  Paris, 1833, 8vo; 1859, 18mo, 863 pages. HOMONYMS.--De Couto, Lisboa,
+  1842, fol. POETIC.--Luzitano (i.e. Freire), ib. 1765, 8vo, 2 vols.;
+  3rd ed. ib. 1820, 4to, 2 vols. RHYMING.--Couto Guerreiro, Lisboa,
+  1763, 4to. NAVAL.--Tiberghien, Rio de Janeiro, 1870, 8vo.
+  CEYLON-PORTUGUESE.--Fox, Colombo, 1819, 8vo: Callaway, ib. 1823, 8vo.
+
+
+ Italian.--Accarigi, _Vocabulario_, Cento, 1543, 4to: Alunno, _La_
+  _fabrica del mundo_, Vinezia, 1548, fol.: Porccachi, Venetia, 1588,
+  fol.: Accademici della Crusca, _Vocabulario_, Venez. 1612, fol.; 4th
+  ed. Firenze, 1729-1738, fol. 6 vols.: Costa and Cardinali, Bologna,
+  1819-1826, 4to, 7 vols.: Tommaseo and Bellini, Torino, 1861, &c., 4to,
+  4 vols.: Petrocchi, 1884-1891. ENGLISH.--Thomas, London, 1598, 4to:
+  Florio, London, 1598, 4to, 1611: Baretti, London, 1794, 2 vols.: 1854,
+  8vo, 2 vols.: Petronj and Davenport, Londra, 1828, 8vo, 3 vols.:
+  Grassi, Leipz. 1854, 12mo: Millhouse, Lond., 1868, 8vo, 2 vols. 1348
+  pages. FRENCH.--Alberti, Paris, 1771, 4to, 2 vols.; Milan, 1862:
+  Barberi, Paris, 1838, 4to, 2 vols.: Renzi, Paris, 1850, 8vo.
+  GERMAN.--_Libro utilissimo_, Venetiis, 1499, 4to: Valentini, Leipzig,
+  1834-1836, 4to, 4 vols. ETYMOLOGY.--Menage, Geneva, 1685, fol.: Bolza,
+  Vienna, 1852, 4to. PROVENÇAL WORDS.--Nannucci, Firenze, 1840, 8vo.
+  SYNONYMS.--Rabbi, Venezia, 1774, 4to; 10th ed. 1817; Tommaseo,
+  Firenze, 1839-1840, 4to, 2 vols.: Milano, 1856, 8vo; 1867.
+  VERBS.--Mastrofini, Roma, 1814, 4to, 2 vols. SELECT WORDS AND
+  PHRASES.--Redi, Brescia, 1769, 8vo. INCORRECT WORDS AND
+  PHRASES.--Molassi, Parma, 1830-1841, 8vo, 854 pages. SUPPOSED
+  GALLICISMS.--Viani, Firenze, 1858-1860, 8vo, 2 vols. ADDITIONS TO THE
+  DICTIONARIES.--Gherardini, Milano, 1819-1821, 8vo, 2 vols.; ib.
+  1852-1857, 8vo, 6 vols. RHYMING.--Falco, Napoli, 1535, 4to: Ruscelli,
+  Venetia, 1563, 8vo; 1827: Stigliani, Roma, 1658, 8vo: Rosasco, Padova,
+  1763, 4to; Palermo, 1840, 8vo. TECHNICAL.--Bonavilla-Aquilino, Mil.
+  1819-1821, 8vo, 5 vols.; 2nd ed. 1829-1831, 4to, 2 vols.: Vogtberg
+  (Germ.), Wein, 1831, 8vo. PARTICULAR AUTHORS.--_Boccaccio_: Aluno, _Le
+  ricchezze della lingua volgare_, Vinegia, 1543. fol. _Dante_: Blanc,
+  Leipzig, 1852, 8vo; Firenze, 1859, 8vo. DIALECTS.--_Bergamo_:
+  Gasparini, Mediol. 1565: Zappetini, Bergamo, 1859, 8vo: Tiraboschi
+  (anc. and mod.), Turin, 1873, 8vo. _Bologna_: Bumaldi, Bologna, 1660,
+  12mo: Ferrari, ib. 1820, 8vo; 1838, 4to. _Brescia_: Gagliardi,
+  Brescia, 1759, 8vo: Melchiori, ib. 1817-1820, 8vo: _Vocabularietto_,
+  ib. 1872, 4to. _Como_: Monti, Milano, 1845, 8vo. _Ferrara_: Manini,
+  Ferrara, 1805, 8vo: Azzi, ib. 1857, 8vo. _Friuli_: Scala, Pordenone,
+  1870, 8vo. _Genoa_: Casaccia, Gen. 1842-1851, 8vo; 1873, &c.:
+  Paganini, ib. 1857, 8vo. _Lombardy_: Margharini, Tuderti, 1870, 8vo.
+  _Mantua_: Cherubini, Milano, 1827, 4to. _Milan_: Varon, ib. 1606, 8vo:
+  Cherubini, ib. 1814, 8vo, 2 vols.; 1841-1844, 8vo, 4 vols.; 1851-1861,
+  8vo, 5 vols.: Banfi, ib. 1857, 8vo: 1870, 8vo. _Modena_: Galvani,
+  Modena, 1868, 8vo. _Naples_: Galiani, Napoli, 1789, 12mo, 2 vols.
+  _Parma_: Peschieri, Parma, 1828-1831, 8vo, 3 vols. 1840; Malespina,
+  ib. 1856, 8vo, 2 vols. _Pavia_: _Dizionario domestico pavese_, Pavia,
+  1829, 8vo: Gambini, ib. 1850, 4to, 346 pages. _Piacenza_: Nicolli,
+  Piacenza, 1832: Foresti, ib. 1837-1838, 8vo, 2 pts. _Piedmont_: Pino,
+  Torino, 1784, 4to: Capello (Fr.), Turin, 1814, 8vo, 2 pts.: Zalli
+  (Ital. Lat. Fr.), Carmagnola, 1815, 8vo, 2 vols: Sant' Albino, Torino,
+  1860, 4to. _Reggio_: _Vocabulario Reggiano_, 1832. _Romagna_: Morri,
+  Fienza, 1840. _Rome_: _Raccolto di voci Romani e Marchiani_, Osimo,
+  1769, 8vo. _Roveretano and Trentino_: Azzolini, Venezia, 1856, 8vo.
+  _Sardinia_: Porru, Casteddu, 1832, fol.: Spano, Cagliari, 1851-1852,
+  fol. 3 vols. _Sicily_: Bono (It. Lat.), Palermo, 1751-1754, 4to, 3
+  vols.; 1783-1785, 4to, 5 vols.: Pasqualino, ib. 1785-1795, 4to, 5
+  vols.: Mortillaro, ib. 1853, 4to, 956 pages: Biundi, ib. 1857, 12mo,
+  578 pages: Traina, ib. 1870, 8vo. _Siena_: Barbagli, Siena, 1602, 4to.
+  _Taranto_: Vincentiis, Taranto, 1872, 8vo. _Turin_: Somis di Chavrie,
+  Torino, 1843, 8vo. _Tuscany_: Luna, Napoli, 1536, 4to: Politi, Roma,
+  1604, 8vo; Venezia, 1615; 1628; 1665; Paulo, ib. 1740, 4to. _Vaudois_:
+  Callet, Lausanne, 1862, 12mo. _Venetian_: Patriarchi (_Veneziano e
+  padevano_), Padova, 1755, 4to; 1796, 1821: Boerio, Venezia, 1829, 4to;
+  1858-1859; 1861. _Verona_: Angeli, Verona, 1821, 8vo. _Vicenza_:
+  Conti, Vicenza, 1871, 8vo. LINGUA FRANCA.--_Dictionnaire de la langue
+  Franque, ou Petit Mauresque_, Marseille, 1830, 16mo, 107 pages.
+  SLANG.--Sabio (lingua Zerga), Venetia, 1556, 8vo; 1575: _Trattato
+  degli bianti_, Pisa, 1828, 8vo.
+
+
+ Romansh.--_Promptuario de voci volgari e Latine_, Valgrisii, 1565,
+  4to: _Der, die, das, oder Nomenclatura_ (German nouns explained in
+  Rom.), Scoul, 1744, 8vo: Conradi, Zurich, 1820, 8vo; 1826, 12mo, 2
+  vols.: Carisch, Chur, 1821, 8vo; 1852, 16mo.
+
+
+ Vlach.--_Lesicon Rumanese_ (Lat. Hung. Germ.), Budae, 1825, 4to: Bobb
+  (Lat. Hung.), Clus, 1822-1823, 4to, 2 vols. FRENCH.--Vaillant,
+  Boucoureshti, 1840, 8vo: Poyenar, Aaron and Hill, Boucourest,
+  1840-1841, 4to, 2 vols.; Jassi, 1852, 16mo, 2 vols.: De Pontbriant,
+  Bucuresci, 1862, 8vo: Cihac, Frankf. 1870, 8vo: Costinescu, Bucuresci,
+  1870, 8vo, 724 pages: Antonescu, Bucharest, 1874, 16mo, 2 vols. 919
+  pages. GERMAN.--Clemens, Hermanstadt, 1823, 8vo: Isser, Kronstadt,
+  1850: Polyzu, ib. 1857, 8vo.
+
+
+TEUTONIC: (1) _Scandinavian._
+
+ Icelandic.--LATIN.--Andreae, Havniae, 1683, 8vo: Halderson (Lat.
+  Danish), ib. 1814, 4to, 2 vols. ENGLISH.--Cleasby-Vigfusson, Oxford,
+  1874, 4to. GERMAN.--Dieterich, Stockholm, 1844, 8vo: Möbius, Leipzig,
+  1866, 8vo. DANISH.--Jonssen, Kjöbenhavn, 1863, 8vo. NORWEGIAN.--Kraft,
+  Christiania, 1863, 8vo: Fritzner, Kristiania, 1867, 8vo.
+  POETIC.--Egilsson (Latin), Hafniae, 1860, 8vo; 1864.
+
+
+ Swedish.--Kindblad, Stockholm, 1840, 4to: Almqvist, Örebro, 1842-1844,
+  8vo: Dalin, _Ordbog._ Stockholm, 1850-1853, 8vo, 2 vols. 1668 pages;
+  1867, &c. 4to (vol. i. ii., A to Fjermare, 928 pages): Id.,
+  _Handordbog_, ib. 1868, 12mo, 804 pages; Svenska Academien. Stockholm,
+  1870, 4to (A) pp. 187. LATIN.--Stjernhjelm, Holm, 1643, 4to: Verelius,
+  Upsala, 1691, 8vo: Ihre (Sueo-Gothicum), Upsala, 1769, fol. 2 vols.
+  ENGLISH.--Serenius, Nyköping, 1757, 4to: Brisnon, Upsala, 1784, 4to:
+  Widegren, Stockholm, 1788, 4to; Brisman, Upsala, 1801, 4to; 3rd ed.
+  1815, 2 vols.: Deleen Örebro, 1829, 8vo: Granberg, ib. 1832, 12mo:
+  Nilssen, Widmark, &c., Stockholm, 1875, 8vo. FRENCH.--Möller,
+  Stockholm, 1745, 4to: Björkengren, ib. 1795, 2 vols.: Nordforss, ib.
+  1805, 8vo, 2 vols.: 2nd ed. Örebro, 1827, 12mo: West, Stockh. 1807,
+  8vo: Dalin, ib. 1842-1843, 4to, 2 vols.; 1872. GERMAN.--Dähnert,
+  Holmiae, 1746, 4to: Heinrich, Christiansund, 1814, 4to, 2 vols.; 4th
+  ed. Örebro, 1841, 12mo: Helms, Leipzig, 1858, 8vo; 1872.
+  DANISH.--Höst, Kjöbenhavn, 1799, 4to: Welander, Stockholm, 1844, 8vo:
+  Dalin, ib. 1869, 16mo: Kaper, Kjöbenhavn, 1876, 16mo.
+  ETYMOLOGY.--Tamm, Upsala, 1874, &c., 8vo (A and B), 200 pages. FOREIGN
+  WORDS.--Sahlstedt, Wästerås, 1769, 8vo: Andersson (20,000), Stockholm,
+  1857, 16mo: Tullberg, ib. 1868, 8vo: Ekbohrn, ib. 1870, 12mo: Dalin,
+  ib. 1870, &c., 8vo. SYNONYMS.--Id., ib. 1870, 12mo. NAVAL.--Ramsten,
+  ib. 1866, 8vo. TECHNICAL.--Jungberg, ib. 1873, 8vo. DIALECTS.--Ihre,
+  Upsala, 1766, 4to: Rietz, Lund, 1862-1867, 4to, 859 pages. _Bohuslän_:
+  _Idioticon Bohusiense_, Götaborg, 1776, 4to. _Dalecarlia_: Arborelius,
+  Upsala, 1813, 4to. _Gothland_: Hof (Sven), Stockholmiae, 1772, 8vo:
+  Rääf (Ydre), Örebro, 1859, 8vo. _Halland_: Möller, Lund, 158, 8vo.
+  _Helsingland_: Lenström, ib. 1841, 8vo: Fornminnessällskap,
+  Hudikswall, 1870, 8vo.
+
+
+ Norwegian.--Jenssen, Kjöbenhavn, 1646, 8vo: Pontoppidan, Bergen, 1749,
+  8vo: Hanson (German), Christiania, 1840, 8vo: Aasen, ib. 1873, 8vo,
+  992 pages.
+
+
+ Danish.--Aphelen, Kopenh, 1764, 4to, 2 vols.; 1775, 4to, 3 vols.:
+  Molbech, Kjöbenhavn, 1833, 8vo, 2 vols.: ib. 1859, 2 vols.:
+  Videnskabernes Selskab, ib. 1793-1865, Kalkar. ENGLISH.--Berthelson
+  (Eng. Dan.), 1754, 4to: Wolff, London, 1779, 4to. Bay, ib. 1807, 8vo,
+  2 vols.; 1824, 8vo: Hornbeck, ib. 1863, 8vo: Ferrall and Repp, ib.
+  1814, 16mo; 1873, 8vo: Rosing, Copenhagen, 1869, 8vo: Ancker, ib.
+  1874, 8vo. FRENCH.--Aphelen, 1754, 8vo: Id., ib. 1759, 4to, 2 vols.;
+  2nd ed. 1772-1777, vol. i. ii. GERMAN.--Id., ib. 1764, 4to, 2 vols.:
+  Grönberg, 2nd ed. Kopenh. 1836-1839, 12mo, 2 vols.; 1851, Helms,
+  Leipzig, 1858, 8vo. SYNONYMS.--Müller, Kjöbenhavn, 1853, 8vo. FOREIGN
+  WORDS.--Hansen, Christiania, 1842, 12mo. NAVAL.--Wilsoet, Copenhagen,
+  1830, 8vo: Fisker (French), Kjöbenhavn, 1839, 8vo. OLD
+  DANISH.--Molbech, ib. 1857-1868, 8vo, 2 vols. DIALECTS.--Id., ib.
+  1841, 8vo. _Bornholm_: Adler, _ib._ 1856, 8vo. _South Jutland_: Kok,
+  1867, 8vo. SLANG.--Kristiansen (Gadesproget), ib. 1866, 8vo. p. 452.
+
+
+(2) _Germanic._
+
+ Teutonic.--COMPARATIVE.--Meidinger, Frankf. a. M. 1833, 8vo, 2nd ed.
+  1836, 8vo.
+
+
+ Gothic.--Junius, Dortrecht, 1665, 4to: 1671; 1684, Diefenbach
+  (comparative), Franckf. a. M. 1846-1851, 2 vols. 8vo: Schulze,
+  Magdeburg, 1848, 4to: 1867, 8vo: Skeat, London, 1868, 4to: Balg
+  (_Comparative Glossary_), Magvike, Wisconsin, 1887-1889. ULPHILAS
+  (editions with dictionaries).--Castilionaeus, Mediol, 1829, 4to:
+  Gabelentz and Löbe, Altenburg, 1836-1843, 4to, 2 vols.: Gaugengigl,
+  Passau, 1848, 8vo: Stamm, Paderborn, 1857: Stamm and Heyne, ib. 1866,
+  8vo.
+
+
+ Anglo-Saxon.--LATIN.--Somner (Lat. Eng.), Oxonii, 1659, fol.: Benson,
+  ib. 1701, 8vo: Lye (A.-S. and Gothic), London, 1772, fol. 2 vols.:
+  Ettmüller, Quedlinburg, 1851, 8vo. 838 pages. ENGLISH.--Bosworth,
+  London, 1838, 8vo, 721 pages: Id. (_Compendious_), 1848, 278 pages.
+  Corson (A.-S. and Early English), New York, 1871, 8vo, 587 pages;
+  Toller (based on Bosworth), Oxford, 1882-1898. GERMAN.--Bouterwek,
+  Gütersloh, 1850, 8vo, 418 pages: Grein (Poets), Göttingen, 1861-1863,
+  8vo, 2 vols.: Leo, Halle, 1872, 8vo.
+
+
+ English.--Cockeram, London, 1623, 8vo: 9th ed. 1650: Blount, ib. 1656,
+  8vo: Philips, The new World of Words, London, 1658, fol.: Bailey,
+  London, 1721, 8vo; 2nd ed. ib. 1736, fol.; 24th ed. ib. 1782, 8vo:
+  Johnson, ib. 1755, fol. 2 vols.; ed. Todd, London, 1818, 4to, 4 vols.;
+  ib. 1827. 4to, 3 vols.; ed. Latham, ib. 1866-1874, 4to, 4 vols. (2 in
+  4 parts): Barclay, London, 1774, 4to; ed. Woodward, ib. 1848:
+  Sheridan, ib. 1780, 4to, 2 vols.: Webster, New York, 1828, 4to, 2
+  vols.; London, 1832, 4to, 2 vols.; ed. Goodrich and Porter, 1865, 4to:
+  Richardson, ib. 1836, 4to, 2 vols.; Supplement, 1856: Ogilvie,
+  _Imperial Dictionary_, Glasgow, 1850-1855, 8vo, 3 vols. (the new
+  edition of Ogilvie by Charles Annandale, 4 vols., 1882, was an
+  encyclopaedic dictionary, which served to some extent as the
+  foundation of the _Century Dictionary_); Boag, _Do._, Edinburgh,
+  1852-1853, 8vo, 2 vols.: Craik, ib. 1856, 8vo: Worcester, Boston,
+  1863, 4to. Stormouth and Bayne, 1885; Murray and Bradley, _The Oxford
+  English Dictionary_, 1884- ; Whitney, _The Century Dict._, New York,
+  1889-1891; Porter, _Webster's Internat. Dict._, Springfield,
+  Massachusetts, 1890; Funk, _Standard Dict._, New York, 1894; Hunter,
+  _The Encyclopaedic Dict._, 1879-1888. ETYMOLOGY.--Skinner, Londini,
+  1671, fol.: Junius, Oxonii, 1743, fol.: Wedgewood, London, 1859-1865,
+  3 vols.; ib. 1872, 8vo. Skeat, Oxford, 1881; Fennell (Anglicized
+  words), Camb. 1892. PRONOUNCING.--Walker, London, 1774, 4to: by Smart,
+  2nd ed. ib. 1846, 8vo. PRONOUNCING IN GERMAN.--Hausner, Frankf. 1793,
+  8vo; 3rd ed. 1807; Winkelmann, Berlin, 1818, 8vo: Voigtmann, Coburg,
+  1835, 8vo: Albert, Leipz. 1839, 8vo: Bassler, ib. 1840, 16mo.
+  ANALYTICAL.--Booth, Bath, 1836, 4to: Roget, _Thesaurus_, London, 1852,
+  8vo; 6th ed. 1857; Boston, 1874. SYNONYMS.--Piozzi, London, 1794, 8vo,
+  2 vols.: L. [abarthe], Paris, 1803, 8vo, 2 vols.: Crabb, London,
+  1823, 8vo; 11th ed. 1859: C. J. Smith, ib. 1871, 8vo, 610 pages.
+  REDUPLICATED WORDS.--Wheatley, ib. 1866, 8vo. SURNAMES.--Arthur, New
+  York, 1857, 12mo, about 2600 names: Lower, ib. 1860, 4to.
+  PARTICLES.--Le Febure de Villebrune, Paris, 1774, 8vo.
+  RHYMING.--Levins, _Manipulus Puerorum_, London, 1570, 4to; ed.
+  Wheatley, ib. 1867, 8vo: Walker, London, 1775, 8vo; 1865, 8vo.
+  SHAKESPEARE.--Nares, Berlin, 1822, 4to; ed. Halliwell and Wright,
+  London, 1859, 8vo: Schmidt, Berlin, 1874. OLD ENGLISH.--Spelman,
+  London [1626], fol. (A to I only); 1664 (completed); 1687 (best ed.):
+  Coleridge (1250-1300), ib. 1859, 8vo: Stratmann (Early Eng.), Krefeld,
+  1867, 8vo; 2nd ed. 1873, 4to: Bradley (new edition of Stratman),
+  Oxford, 1891; Matzner and Bieling, Berlin, 1878- . OLD AND
+  PROVINCIAL.--Halliwell, London, 1844-1846, 8vo; 2nd ed. ib. 1850, 2
+  vols.: 6th ed. 1904: Wright, ib. 1857, 8vo, 2 vols.; 1862.
+  DIALECTS.--Ray, ib. 1674, 12mo: Grose, ib. 1787, 8vo; 1790: Holloway,
+  Lewes, 1840, 8vo; Wright, _Eng. Dialect Dict._, London, 1898-1905, 28
+  vols. _Scotch_: Jamieson, Edin. 1806, 4to, 2 vols.; Supplement, 1826,
+  2 vols.; abridged by Johnstone, ib. 1846, 8vo: Brown, Edin, 1845, 8vo:
+  Motherby (German), Königsberg, 1826-1828, 8vo: (_Shetland and
+  Orkney_), Edmonston, London, 1866, 8vo: (_Banffshire_), Gregor, ib.
+  1866, 8vo. _North Country_: Brockett, London, 1839, 8vo, 2 vols.
+  _Berkshire_: [Lousley] ib. 1852, 8vo, _Cheshire_: Wilbraham, ib. 1817,
+  4to; 1826, 12mo: Leigh, Chester, 1877, 8vo. _Cumberland_: _Glossary_,
+  ib. 1851, 12mo: Dickenson, Whitehaven, 1854, 12mo; Supplement, 1867:
+  Ferguson (Scandinavian Words), London, 1856, 8vo. _Derbyshire_: Hooson
+  (mining), Wrexham, 1747, 8vo: Sleigh, London, 1865, 8vo. _Dorset_:
+  Barnes, Berlin, 1863, 8vo. _Durham_: [Dinsdale] (Teesdale), London,
+  1849, 12mo. _Gloucestershire_: Huntley (Cotswold), ib. 1868, 8vo.
+  _Herefordshire_: [Sir George Cornewall Lewis,] London, 1839, 12mo.
+  _Lancashire_: Nodal and Milner, Manchester Literary Club, 1875, 8vo,
+  Morris (Furness), London, 1869, 8vo: R. B. Peacock (Lonsdale, North
+  and South of the Sands), ib. 1869, 8vo. _Leicestershire_: A. B. Evans,
+  ib. 1848, 8vo. _Lincolnshire_: Brogden, ib. 1866, 12mo: Peacock
+  (Manley & Corringham), ib. 1877, 8vo. _Norfolk and Suffolk_; Forby,
+  London, 1830, 8vo, 2 vols. _Northamptonshire_: Sternberg, ib. 1851,
+  8vo: Miss Anne E. Baker, ib. 1866, 8vo, 2 vols. 868 pages.
+  _Somersetshire_: Jennings, ib. 1869, 8vo: W. P. Williams and W. A.
+  Jones, Taunton, 1873, 8vo. _Suffolk_: Moor, Woodbridge, 1823, 12mo:
+  Bowditch (Surnames), Boston, U.S., 1851, 8vo; 1858; 3rd ed. London,
+  1861, 8vo, 784 pages. _Sussex_: Cooper, Brighton, 1836, 8vo: Parish,
+  Farncombe, 1875, 8vo. _Wiltshire_: Akerman, London, 1842, 12mo.
+  _Yorkshire (North and East)_, Toone, ib. 1832, 8vo: (_Craven_), Carr,
+  2nd ed. London, 1828, 8vo, 2 vols.: (_Swaledale_), Harland, ib. 1873,
+  8vo: (_Cleveland_), Atkinson, ib. 1868, 4to, 653 pages: (_Whitby_) [F.
+  K. Robinson], ib. 1876, 8vo: (_Mid-Yorkshire and Lower Niddersdale_),
+  C. Clough Robinson, ib. 1876, 8vo: (_Leeds_), Id., ib. 1861, 12mo:
+  (_Wakefield_), Banks, ib. 1865, 16mo: (_Hallamshire_), Hunter, London,
+  1829, 8vo. _Ireland: (Forth and Bargy, Co. Wexford)_, Poole, London,
+  1867, 8vo. _America_: Pickering, Boston, 1816, 8vo: Bartlett, New
+  York, 1848, 8vo; 3rd ed. Boston, 1860. 8vo; Dutch transl. by Keijzer,
+  Gorinchen, 1854, 12mo; Germ. transl. by Köhler, Leipz. 1868, 8vo.
+  Elwyn, Philadelphia, 1859. 8vo. _Negro English_: Kingos, St Croix,
+  1770, 8vo: Focke (Dutch), Leiden, 1855, 8vo: Wullschlaegel, Löbau,
+  1856, 8vo. 350 pages. SLANG.--Grose, London, 1785, 8vo; 1796: Hotten,
+  ib. 1864, 8vo; 1866; Farmer & Henley (7 vols., 1890-1904).
+
+
+ Frisic.--Wassenbergh, Leeuwarden, 1802, 8vo: Franeker, 1806, 8vo:
+  Outzen, Kopenh. 1837, 4to: Hettema (Dutch), Leuwarden, 1832, 8vo;
+  1874, 8vo, 607 pages: Winkler (Nederdeutsch en Friesch Dialectikon),
+  's Gravenhage, 1874, 8vo, 2 vols. 1025 pages. OLD FRISIC.--Wiarda
+  (Germ.), Aurich, 1786, 8vo: Richthofen, Göttingen, 1840, 4to. NORTH
+  FRISIC.--Bendson (Germ.), Leiden, 1860, 8vo: Johansen (Föhringer und
+  Amrumer Mundart), Kiel, 1862, 8vo. EAST FRISIC.--Stürenburg, Aurich,
+  1857, 8vo. HELIGOLAND.--Oelrichs, s. l., 1836, 16mo.
+
+
+ Dutch.--Kok, 2nd ed. Amst. 1785-1798, 8vo, 38 vols.: Weiland, Amst.
+  1790-1811, 8vo, 11 vols.: Harrebomée, Utrecht, 1857, 4to; 1862-1870,
+  8vo, 3 vols.: De Vries and Te Winkel, Gravenh. 1864, &c., 4to (new ed.
+  1882- ); Dale, ib. 4th ed. 1898; ENGLISH.--Hexham, ed. Manley,
+  Rotterdam, 1675-1678, 4to: Holtrop, Dortrecht, 1823-1824, 8vo, 2
+  vols.: Bomhoff, Nimeguen, 1859, 8vo, 2 vols. 2323 pages: Jaeger,
+  Gouda, 1862, 16mo: Calisch, Tiel, 1871, &c., 8vo. FRENCH.--Halma,
+  Amst. 1710, 4to; 4th ed. 1761: Marin, ib. 1793, 4to, 2 vols.:
+  Winkelman, ib. 1793, 4to, 2 vols.: Mook, Zutphen, 1824-1825, 8vo, 4
+  vols.; Gouda, 1857, 8vo, 2 vols. 2818 pages: Kramers, ib. 1859-1862, 2
+  vols. 16mo. GERMAN.--Kramer, Nürnb. 1719, fol.; 1759, 4to, 2 vols.;
+  ed. Titius, 1784, Weiland, Haag, 1812, 8vo: Terwen, Amst. 1844, 8vo.
+  ETYMOLOGY.--Franck, 1884-1892. ORIENTAL WORDS.--Dozy, 's Gravenhage,
+  1867, 8vo. GENDERS OF NOUNS.--Bilderdijk, Amst. 1822, 8vo, 2 vols.
+  SPELLING.--Id., 's Gravenhage, 1829, 8vo. FREQUENTATIVES.--De Jager,
+  Gouda, 1875, 8vo, vol. i. OLD DUTCH.--Suringer, Leyden, 1865, 8vo.
+  MIDDLE DUTCH.--De Vries, 's Gravenhage, 1864, &c., 4to. Verwijs and
+  Verdam, ib. 1885- .
+
+
+ Flemish.--Kilian, Antw. 1511, 8vo; ed. Hasselt, Utrecht, 1777, 4to, 2
+  vols. FRENCH.--Berlemont, Anvers, 1511, 4to: Meurier, ib. 1557, 8vo:
+  Rouxell and Halma, Amst. 1708, 4to; 6th ed. 1821: Van de Velde and
+  Sleeckx, Brux. 1848-1851, 8vo, 2440 pages; ib. 1860, 8vo, 2 vols.
+  ANCIENT NAMES OF PLACES.--Grandgagnage (East Belgium), Bruxelles,
+  1859, 8vo.
+
+
+ German.--Josua Pictorius (Maaler), _Die teütsch Spraach_, Tiguri,
+  1561, 8vo; Stieler, Nürnb. 1691, 4to: Adelung, Leipz. 1774-1786, 4to,
+  5 vols.; 1793-1818, 5 vols.: Campe, Braunschweig, 1807-1811, 4to, 5
+  vols.: Grimm, Leipzig, 1854, &c., 4to: Sanders, ib. 1860-1865, 4to, 3
+  vols. 1885: Diefenbach and Wülcker (High and Low German, to supplement
+  Grimm), Frankf. a. M. 1874, 1885, 8vo.; Kluge, Strassburg, 1883;
+  Heine, Leipzig, 1890-1895; Weigand, Giessen, 1873. ENGLISH.--Adelung,
+  1783-1796, 8vo, 3 vols.: Hilpert, Karlsruhe, 1828-1829, 8vo, 2 vols.;
+  1845-1846, 4to, 2 vols.: Flügel, Leipz. 1830, 8vo, 2 vols.; London,
+  1857, 8vo; Leipzig, 1870: Müller, Cöthen, 1867, 8vo, 2 vols.
+  FRENCH.--Laveaux, Strassburg, 1812, 4to: Mozin, Stuttgard, 1811-1812,
+  4to, 4 vols.; 1842-1846, 8vo, 4 vols., 3rd ed. 1850-1851, 8vo:
+  Schuster, Strasb. 1859, 8vo: Daniel, Paris, 1877, 16mo. OLD HIGH
+  GERMAN.--Haltaeus, Lipsiae, 1758, fol. 2 vols.: Graff, Berlin,
+  1834-1846, 4to, 7 vols.: Brinckmeier, Gotha, 1850-1863, 4to, 2 vols.:
+  Kehrein (from Latin records), Nordhausen, 1863, 8vo. Schade, Halle,
+  1872-1882. MIDDLE HIGH GERMAN.--Ziemann, Quedlinburg, 1838, 8vo:
+  Benecke, Müller and Zarnche, Leipz. 1854-1866, 8vo, 3 vols.: Lexer,
+  Leipzig, 1870, 8vo. MIDDLE LOW GERMAN.--Schiller and Lübben, Bremen,
+  1872, &c., 8vo, in progress. LOW GERMAN.--Vollbeding, Zerbst, 1806,
+  8vo: Kosegarten, Griefswald, 1839, 4to; 1856, &c., 4to.
+  ETYMOLOGY.--Helvigius, Hanov. 1620, 8vo: Wachter, Lipsiae, 1737, fol.
+  2 vols.: Kaindl, Salzbach, 1815-1830, 8vo, 7 vols.: Heyse, Magdeburg,
+  1843-1849, 8vo, 3 vols.: Kehrein, Wiesbaden, 1847-1852, 2 vols.
+  SYNONYMS.--Eberhard, Maas, and Grüber, 4th ed. Leipzig, 1852-1863,
+  8vo, 4 vols.: Aue (Engl.), Edinb. 1836, 8vo: Eberhard, 11th ed.
+  Berlin, 1854, 12mo: Sanders, Hamburg, 1872, 8vo, 743 pages. FOREIGN
+  WORDS.--Campe, Braunschweig, 1813, 4to: Heyse, _Fremdwörterbuch_,
+  Hannover, 1848, 8vo. NAMES.--Pott. Leipz. 1853, 8vo: Michaelis
+  (Taufnamen), Berlin, 1856, 8vo: Förstemann (Old Germ.) Nordhausen,
+  1856-1859, 4to, 2 vols. 1573 pages, 12,000 names: Steub
+  (Oberdeutschen), München, 1871, 8vo. LUTHER.--Dietz, Leipzig,
+  1869-1872, 8vo, 2 vols. DIALECTS.--Popowitsch, Wien, 1780, 8vo: Fulda,
+  Berlin, 1788, 8vo: Klein, Frankf. 1792, 8vo, 2 vols.: Kaltschmidt,
+  Nordlingen, 1851, 4to; 1854, 5th ed. 1865. _Aix-la-Chapelle_, Müller
+  and Weitz, Aachen, 1836, 12mo. _Appenzell_: Tobler, Zürich, 1837, 8vo.
+  _Austria_: Höfer, Linz, 1815, 8vo; Castelli, Wien, 1847, 12mo:
+  Scheuchenstül (mining), ib. 1856, 8vo. _Bavaria_: Zaupser, München,
+  1789, 8vo: Deling, ib. 1820, 2 vols.: Schmeller, Stuttg. 1827-1837,
+  8vo, 4 vols.; 2nd ed. München, 1872, 4to, vol. i. 1799 pages.
+  _Berlin_: Trachsel. Berlin, 1873, 8vo. _Bremen_: Bremisch Deutsch
+  Gesellschaft, Bremen, 1767-1771, 1869, 8vo, 6 vols. Oelrich (anc.
+  statutes), Frankf. a. M. 1767, 8vo. _Carinthia_: Ueberfelder,
+  Klagenfurt, 1862, 8vo: Lexe, Leipzig, 1862, 8vo. _Cleves_: De
+  Schueren, _Teuthonista_, Colon, 1477, fol.; Leiden, 1804, 4to.
+  _Göttingen_: Schambach, Hannover, 1838, 8vo. _Hamburg_: Richey, Hamb.
+  1873, 4to; 1755, 8vo. _Henneberg_: Reinwold, Berlin and Stettin, 1793,
+  1801, 8vo, 2 vols.: Brückner, Meiningen, 1843, 4to. _Hesse_: Vilmar,
+  Marburg, 1868, 8vo, 488 pages. _Holstein_: Schütz Hamb. 1800-1806,
+  8vo, 4 vols. _Hungary_: Schoer, Wien, 1858. _Livonia_: Bergmann,
+  Salisburg, 1785, 8vo: Gutzeit, Riga, 1859-1864, 8vo, 2 parts. _Upper
+  Lusatia_: Anton, Görlitz, 1825-1839, 13 parts. _Luxembourg_: Gangler,
+  Lux. 1847, 8vo, 406 pages. _Mecklenburg and Western Pomerania_: M.,
+  Leipzig, 1876, 8vo, 114 pages. _Nassau_: Kehrein, Weilburg, 1860, 8vo.
+  _Osnaburg_: Strodtmann, Leipz. 1756, 8vo. _Pomerania and Rügen_:
+  Dähnert, Stralsund, 1781, 4to. _Posen_: Bernd, Bonn, 1820, 8vo.
+  _Prussia_: Bock, Königsb. 1759, 8vo: Hennig, ib. 1785, 8vo. _Saxony_:
+  Schmeller (from Heliand, &c.), Stuttg. 1840, 4to. _Silesia_: Berndt,
+  Stendal, 1787, 8vo. _Swabia_: Schmid, Berlin, 1795, 8vo; Stuttg. 1831,
+  8vo. _Switzerland_: Stalder, Aarau, 1807-1813, 8vo, 2 vols.
+  _Thuringia_: Keller, Jena, 1819, 8vo. _Transylvania_: Schuller, Prag,
+  1865, 8vo. _Tirol_: Schöpf, Innspruck, 1866, 8vo. _Venetian Alps_:
+  Schmeller, Wien, 1854, 8vo. _Vienna_: Hugel, ib. 1873, 8vo.
+  HUNTING.--_Westerwald_: Schmidt, Hadamar, 1800, 8vo; Kehrein,
+  Wiesbaden, 1871, 12mo. SLANG.--_Gauner Sprache_: Schott, Erlangen,
+  1821, 8vo: Grolmann, Giessen, 1822, 8vo: Train, Meissen, 1833, 8vo:
+  Anton, 2nd ed. Magdeburg, 1843, 8vo; 1859: Avé-Lallemant, _Das
+  Deutsche Gaunerthun_, Leipzig, 1858-1862, 8vo, vol. iv. pp. 515-628.
+  _Student Slang_: Vollmann (Burschicoses), Ragaz, 1846, 16mo, 562
+  pages.
+
+
+_Celtic._
+
+ Celtic generally.--Lluyd, Archaeologia Britannica, Oxford, 1707,
+  folio: Bullet, Besançon, 1754-1860, fol. 2 vols.
+
+
+ Irish.--Cormac, bishop of Cashel, born 831, slain in battle 903, wrote
+  a Glossary, _Sanas Cormaic_, printed by Dr Whitley Stokes, London,
+  1862, 8vo, with another, finished in 1569, by O'Davoren, a
+  schoolmaster at Burren Castle, Co. Clare: O'Clery, Lovanii, 1643, 8vo:
+  MacCuirtin (Eng.-Irish), Paris, 1732, 4to: O'Brien, ib. 1768, 4to;
+  Dublin, 1832, 8vo: O'Reilly, 1817, 4to: 1821; ed. O'Donovan, ib. 1864,
+  4to, 725 pages: Foley (Eng.-Irish), ib. 1855, 8vo: Connellan (do.),
+  1863, 8vo.
+
+
+ Gaelic.--Macdonald, Edin. 1741, 8vo: Shaw, London, 1780, 4to, 2 vols.:
+  Allan, Edin. 1804, 4to: Armstrong, London, 1825, 4to: Highland
+  Society, ib. 1828, 4to, 2 vols.: Macleod and Dewar, Glasgow, 1853,
+  8vo.
+
+
+ Manx.--Cregeen, Douglas, 1835, 8vo: Kelly, ib. 1866, 8vo, 2 vols.
+
+
+ Welsh.--LATIN.--Davies, London, 1632, fol.: Boxhornius, Amstelodami,
+  1654, 4to. ENGLISH.--Salesbury, London, 1547, 4to: 1551: Richards,
+  Bristol, 1759, 8vo: Owen (W.), London, 1793-1794, 8vo, 2 vols.; 1803,
+  4to, 3 vols.: Walters, ib. 1794, 4to: Owen-Pughe, Denbigh, 1832, 8vo;
+  3rd ed. Pryse, ib. 1866, 8vo: D. S. Evans (Eng.-Welsh), ib. 1852-1853,
+  8vo; 1887.
+
+
+ Cornish.--Pryce, _Archaeologia_, Sherborne, 1770, 4to: Williams,
+  Llandovery, 1862-1865, 4to. NAMES.--Bannister (20,000), Truro,
+  1869-1871, 8vo.
+
+
+ Breton.--Legadeuc, _Le Catholicon breton_, finished 1464, printed at
+  Lantrequier, 1499, fol. 210 pages; 1501, 4to; L'Orient, 1868, 8vo:
+  Quicquer de Roskoff, Morlaix, 1633, 8vo: Rostrenen, Rennes, 1732, 4to,
+  978 pages; ed. Jolivet, Guingamps, 1834, 8vo, 2 vols.: l'A.[rmerie],
+  Leyde, 1744, 8vo; La Haye, 1756: Lepelletier, Paris, 1752, fol.:
+  Legonidec, Angouleme, 1821, 8vo; St Brieuc, 1847-1850, 4to, 924 pages.
+  DIALECT OF LÉON.--Troude (Fr.-Bret.), Brest, 1870, 8vo; Id.
+  (Bret.-Fr.), ib. 1876, 8vo, 845 pages. DIOCESE OF VANNES.--Armerie,
+  Leyde, 1774, 8vo.
+
+
+_Basque._
+
+ Basque.--Larramendi, St Sebastian, 1745, fol. 2 vols.; ed. Zuazua, ib.
+  1854, fol.; Chaho, Bayonne, 1856, 4to, 1867: Fabre, ib. 1870, 8vo: Van
+  Eys, Paris, 1873, 8vo: Egúren, Madrid, 1877.
+
+
+_Baltic._
+
+ Lithuanian.--Szyrwid, 3rd ed., Vilnae, 1642, 8vo; 5th ed. 1713:
+  Schleicher, Prag, 1856-1857, 8vo, 2 vols.: Kurmin, Wilno, 1858, 8vo:
+  Kurschat, Halle, 1870, &c., 8vo.
+
+
+ Lettic.--Mancelius, Riga, 1638, 4to: Elvers, ib. 1748, 8vo: Lange,
+  Mitau, 1777, 4to: Sjögren, Petersburg, 1861, 4to: Ulmann, ed.
+  Bielenstein, Riga, 1872, &c., 8vo.
+
+
+ Prussian.--Bock, Königsberg, 1759, 8vo: Hennig, ib. 1785, 8vo:
+  Nesselmann, Berlin, 1873, 8vo: Pierson, ib. 1875, 8vo.
+
+
+_Slavonic_.
+
+ Slavonic generally.--Franta-Sumavski (Russ. Bulg. Old Slav. Boh.
+  Polish), Praga, 1857, 8vo, Miklosich, Wien, 1886.
+
+
+ Old Slavonic.--Beruinda, Kiev, 1627, 8vo; Kuteinsk, 1653, 4to:
+  Polycarpi (Slav. Greek, Latin), Mosque, 1704, 4to: Alexyeev, St
+  Petersb. 1773, 8vo; 4th ed. ib. 1817-1819, 8vo, 5 vols.: Russian Imp.
+  Academy, ib. 1847, 4to, 4 vols.: Miklosich, Vindobonae, 1850: 4to;
+  1862-1865, 8vo, Mikhailovski, St Petersb. 1875, 8vo: Charkovski,
+  Warschaw, 1873, 8vo.
+
+
+ Russian.--Russian Academy, St Petersburg, 1789-1794, 4to, 6 vols.;
+  1806-1822, ib. 1869, 8vo, 3 vols.: Dahl, Moskva, 1862-1866, fol. 4
+  vols.; d., ib. 1873, &c., 4to; a 3rd edition, 1903, &c.
+  FRENCH-GERM.-ENG.--Reiff, ib. 1852-1854, 4to. GERMAN,
+  LATIN.--Holterhof, Moskva, 1778, 8vo, 2 vols.; 3rd ed. 1853-1855, 8vo,
+  2 vols.: Weismann, ib. 1731, 4to; 1782, and frequently. FRENCH,
+  GERMAN.--Nordstet, ib. 1780-1782, 4to, 2 vols.: Heym, Moskau,
+  1796-1805, 4to, 4 vols.: Booch-Arkossi and Frey, Leipzig, 1871, &c.,
+  8vo. ENGLISH.--Nordstet, London, 1780, 4to: Grammatin and Parenogo,
+  Moskva, 1808-1817, 4to, 4 vols. FRENCH.--Tatischeff, 2nd ed. St
+  Petersb. 1798, 8vo, 2 vols.; Moskau, 1816, 4to, 2 vols.: Reiff, St
+  Petersb. 1835-1836, 8vo, 2 vols.: Makaroff, ib. 1872, 8vo, 2 vols,
+  1110 pages; 1873-1874, 12mo, 2 vols. GERMAN.--Pawlowski, Riga, 1859,
+  8vo: Lenström, Mitau, 1871, 8vo. SWEDISH.--Geitlin, Helsingfors, 1833,
+  12mo: Meurmann, ib. 1846, 8vo. POLISH.--Jakubowicz, Warszawa,
+  1825-1828, 8vo, 2 vols.: Amszejewicz, ib. 1866, 8vo: Szlezigier, ib.
+  1867, 8vo. TECHNICAL.--Grakov (Germ.), St Petersb. 1872, 8vo.
+  NAVAL.--Butakov, ib. 1837. DIALECTS.--_North-west Russia_:
+  Gorbachevski (old language, in Russian), Vilna, 1874, 8vo, 418 pages.
+  _White Russia_: Nosovich (Russian), St Petersburg, 1870, 4to, 760
+  pages. _Red Russia_: Patritzkii (German), Lemberg, 1867, 8vo, 2 vols.
+  842 pages. _Ukraine_: Piskanov (Russian), Odessa, 1873, 4to, 156
+  pages.
+
+
+ Polish.--Linde (explained in Lat. Germ. and 13 Slav dialects),
+  Warszawie, 1807-1814, 4to, 6 vols. 4574 pages.
+  ENGLISH.--[Rykaczewski], _Complete Dictionary_, Berlin, 1849-1851,
+  8vo, 2 vols.: Rykaczewski, Berlin, 1866, 16mo, 1161 pages. FRENCH AND
+  GERMAN.--Troc, Leipz. 1742-1764, 8vo, 4 vols.; 4th ed. ib. 1806-1822,
+  4to, 4 vols.: Bandtke, Breslau, 1806, 8vo, 2 vols.; 1833-1839, 8vo.
+  FRENCH.--Schmidt, Leipzig, 1870, 16mo. RUSSIAN AND GERMAN.--Schmidt
+  (J. A. E.), Breslau, 1834, 8vo. GERMAN.--Mrongovius, Königsberg, 1765;
+  1835, 4to; 1837: Troianski, Berlin, 1835-1838, 8vo, 2 vols.:
+  Booch-Arkossi, Leipzig, 1864-1868, 8vo, 2 vols.: Jordan, ib. 1866,
+  8vo. ITALIAN.--Plazowski, Warszawa, 1860, 8vo. 2 vols. 730 pages.
+  RUSSIAN.--Potocki, Lipsk, 1873, &c., 12mo.
+
+
+ Wendish.--Matthäi, Budissen, 1721, 8vo: Bose, Grimma, 1840, 8vo:
+  Pfuhl, w Budzsinje, 1866, 8vo, 1210 pages. UPPER LUSATIAN.--Pfuhl and
+  Jordan, Leipz. 1844, 8vo. LOWER LUSATIAN.--Zwahr, Spremberg, 1847,
+  8vo.
+
+
+ Czech.--Rohn (Germ. Lat.), Prag, 1780, 4to, 4 vols.: Dobrowski and
+  Hanka, ib. 1802-1821, 4to, 2 vols. LAT. GERM. HUNGAR.--Jungmann,
+  Praze, 1835-1839, 6 vols. 4to, 5316 pages. GERMAN.--Thàm, Prag.
+  1805-1807, 8vo, 2 vols.: Sumavski, ib. 1844-1846, 8vo, 2 vols.:
+  Koneney, ib. 1855, 18mo, 2 vols.: Rank (Germ. Boh.), ib. 1860, 16mo,
+  775 pages. TECHNICAL.--Spatny, ib. 1864, 8vo: Kheil (names of goods,
+  Germ. Boh.), ib. 1864, 8vo, 432 pages. HUNTING.--Spatny, ib. 1870,
+  8vo, 137 pages.
+
+
+ South Slavic.--Richter and Ballman, Wien, 1839-1840, 8vo, 2 vols.
+  SERVIAN.--Karajic (Germ. Lat.), ib. 1818, 8vo; 1852: Lavrovski
+  (Russian), St Petersb. 1870, 8vo, 814 pages. BOSNIAN.--Micalia,
+  Laureti, 1649, 8vo. SLOVAK.--Bernolak (Lat. Germ. Hung.), Budae,
+  1825-1827, 8vo, 6 vols.: Loos (Hung. and Germ.), Pest, 1869, &c., 3
+  vols. SLOVENE.--Gutsmann, Klagenfurt, 1789, 4to: Relkovich, Wien,
+  1796, 4to, 2 vols.: Murko, Grätz, 1838, 8vo, 2 vols.: Janezic,
+  Klagenfurt, 1851, 12mo. DALMATIAN.--Ardelio della Bella, Venezia,
+  1728, 8vo; 2nd ed. Ragusae, 1785, 4to: Stulli, ib. 1801-1810, 4to, 2
+  vols. CROATIAN.--Habdelich, Grätz, 1670, 8vo: Sulek, Agram, 1854-1860,
+  8vo, 2 vols. 1716 pages. CARINTHIAN.--Lexer, Leipzig, 1862, 8vo. OLD
+  SERVIAN.--Danitziye (Servian), Belgrad, 1864, 8vo, 3 vols.
+
+
+ Bulgarian.--Daniel (Romaic, Albanian, Rumanian, and Bulgarian),
+  Moschopolis, 1770; Venice, 1802, 4to. ENGLISH.--Morse and Vassiliev,
+  Constantinople, 1860, 8vo. RUSSIAN.--Borogoff, Vienna, 1872, &c., 8vo.
+
+
+_Ugrian._
+
+ Ugrian, Comparative.--Donner, Helsingfors, 1874, 8vo, in progress:
+  Budenz (Ugrian-Magyar), Budapest, 1872-1875, 8vo.
+
+
+ Lappish.--_Manuale_, Holmiae, 1648, 8vo: Fjellström, ib. 1738, 8vo:
+  Leem and Sandberg, Havn. 1768-1781, 4to, 2 parts: Lindahl and
+  Oehrling, Holm. 1780, 8vo. NORTH LAPPISH.--Stockfleht, Christiania,
+  1852, 8vo.
+
+
+ Finnish.--Juslenius, Holmiae, 1745, 4to, 567 pages: Renvall, Aboae,
+  1826, 4to, 2 vols.: Europaeus, Helsingissä, 1852-1853, 16mo, 2 vols.
+  742 pages: Lunin, Derpt, 1853, 8vo: Eurén, Tavashuus, 1860, 8vo:
+  Ahlman, ib. 1864, 8vo: Wiedemann, St Petersb. 1869, 4to: Godenhjelm
+  (Germ.), Helsingfors, 1871: Lönnrot, Helsingissä, 1874.
+  NAVAL.--Stjerncreutz, ib. 1863, 8vo.
+
+
+ Esthonian.--Hupel, Mitau, 1818, 8vo, 832 pages: Körber, Dorpat, 1860,
+  8vo: Wiedemann, St Petersb. 1869, 4to, 1002 pages: Aminoff
+  (Esth.-Finnish), Helsingissä, 1869, 8vo: Meves (Russian), Riga, 1876,
+  12mo.
+
+
+ Permian.--Rogord (Russian), St Petersb. 1869, 8vo, 420 pages.
+
+
+ Votiak.--Wiedemann, Reval, 1847, 8vo: Ahlquist, Helsingfors, 1856,
+  4to.
+
+
+ Cheremiss.--Budenz, Pest, 1866, 8vo.
+
+
+ Ersa-Mordvine.--Wiedemann, St Petersb. 1865, 4to.
+  MOKSHA-MORDVINE.--Ahlquist, ib. 1862, 8vo.
+
+
+ Magyar.--Szabo, Kassan, 1792, 8vo: Guczor and Fogarazi (Hung.
+  Academy), Pesth, 1862, 8vo, in progress. ENGLISH.--Dallos, Pesth,
+  1860, 8vo. FRENCH.--Kiss, ib. 1844, 12mo, 2 vols.: Karady, Leipz.
+  1848, 12mo: Mole, Pesth, 1865, 8vo, 2 vols. GERMAN.--Schuster, Wien,
+  1838, 8vo: Bloch, Pesth, 1857, 4to, 2 vols.: Ballagi, ib. 1857, 8vo;
+  6th ed. 1905, 8vo, 2 vols.: Loos, ib. 1870, 8vo, 914 pages.
+  ETYMOLOGICAL.--Dankovsky (Lat.-Germ.), Pressburg, 1853, 8vo:
+  Kresznerics (under roots, in Hung.), Budân, 1831-1832, 4to, 2 vols.:
+  Podhorsky (from Chinese roots, in Germ.), Budapest, 1877, 8vo. NEW
+  WORDS.--Kunoss, Pesth, 1836, 8vo; 1844.
+
+
+ Turkish.--ARAB. PERS.--Esaad Effendi, Constantinople, 1802, fol.
+  ROMAIC.--Alexandrides, Vienna, 1812, 4to. POLYGLOTTS.--Pianzola (Ital.
+  Grec. volgare, e Turca), Padova, 1789, 4to: Ciakciak (Ital. Armeno,
+  Turco), Venice, 1804, 4to; 2nd ed. 1829: Azarian (Ellenico, Ital. Arm.
+  Turco), Vienna, 1848, 8vo: Mechitarist Congregation (Ital. Francese,
+  Arm. Turco), ib. 1846, 8vo. LATIN.--Mesgnien-Meninski, Viennae, 1680,
+  fol. 3 vols.; ed. Jenisch and Klezl, ib. 1780-1802, fol. 4 vols.
+  ENGLISH.--Sauerwein, London, 1855, 12mo: Redhouse, ib. 1856, 8vo, 1176
+  pages: Id., Eng. Turkish, ib. 860, 8vo. FRENCH.--Kieffer and Bianchi
+  (Turk.-Fr.), Paris, 1835-1837, 2 vols. 2118 pages: Bianchi (Fr.-Turk.)
+  Paris, 1843-1846, 8vo, 2 vols. 2287 pages; 1850, 8vo, 2 vols.:
+  Mallouf, ib. 1863-1867, 8vo, 2 vols. FRENCH AND GERMAN.--Zenker (Arab.
+  Pers.), Leipz, 1862-1876, 4to, 2 vols, 982 pages. GERMAN.--Korabinsky,
+  Pressburg, 1788, 8vo: Vambéry, Constantinople, 1858, 8vo.
+  ITALIAN.--Molina, Roma, 1641, 8vo: Masais, Firenze, 1677, 8vo:
+  Ciadyrgy, Milano, 1832-1834, 4to, 2 vols. RUSSIAN.--Budagov
+  (Comparative lexicon of the Turkish-Tartar dialects), St Petersburg,
+  1869, 8vo, 2 vols.
+
+
+ Gipsy.--Bischoff, Ilmenau, 1827, 8vo: Truxillo, Madrid, 1844, 8vo:
+  Jimenes, Sevilla, 1846, 16mo: Baudrimont, Bordeaux, 1862, 8vo:
+  Vaillant, Paris, 1868, 8vo: Paspati; Constantinople, 1870, 4to:
+  Borrow, _Romany Lavo Lil_, London, 1874, 8vo: Smart and Crofton,
+  London, 1875, 8vo.
+
+
+ Albanian.--Blanchus, Romae, 1635, 8vo: Kaballioti (Romaic, Wallach.
+  Alb.), Venice, 1770, 8vo: Xylander, Frankfurt a. M. 1835, 8vo: Hahn,
+  Jena, 1854, 4to: Rossi da Montalto, Roma, 1866, 8vo.
+
+
+ASIA
+
+ Semitic.--POLYGLOTTS.--Thurneissius, Berolini, 1585, fol.: Thorndike,
+  London, 1635, fol.: Schindler, Pentaglotton, Frankf, ad M. 1653, fol.:
+  Hottinger, Heptaglotton, ib. 1661, fol.: Castellus, London, 1669, fol.
+  2 vols. (Hebrew, Chaldaic, Syriac, Samaritan, Aethiopic and Arabic in
+  one alphabet; Persian separately. It occupied him for seventeen years,
+  during which he worked sixteen to eighteen hours a day): Otho, Frankf.
+  a. M. 1702, 4to (the same languages with Rabbinical).
+
+
+ Hebrew.--About 875, Zemah, head of the school of Pumbeditha, wrote a
+  Talmudical dictionary of words and things, arranged in alphabetical
+  order, which is lost. About 880, Jehudah ben 'Alan, of Tiberias, and
+  Jehudah ibn Koreish, of Tahurt, in Morocco wrote Hebrew dictionaries.
+  Saadia ben Joseph (born 892, died 942), of Fayum, in Upper Egypt,
+  wrote [Hebrew: Kefer Igaron], probably a Hebrew-Arabic dictionary.
+  Menahem ben Jacob Ibn Sar[=u]q (born 910, died about 970), of Tortosa
+  and Cordova, wrote a copious Hebrew dictionary, first printed by
+  Herschell F. Filipowski, Edinburgh, 1855, 8vo, from five MSS. David
+  ben Abraham, of F[=a]s, wrote, in Arabic, a large Hebrew dictionary,
+  the MS. of which, a quarto of 313 leaves on cotton paper, was found
+  about 1830 by A. Firkowitz, of Eupatoria, in the cellar of a Qaraite
+  synagogue in Jerusalem. The age of this work cannot be ascertained.
+  About 1050, Ali ben Suleiman wrote a dictionary in Arabic, on the plan
+  of that of David ben Abraham. The MS. of 429 leaves belongs to
+  Firkowitz. Haja ben Sherira, the famous teacher of the Academy of
+  Pumbeditha, wrote a Hebrew dictionary in Arabic, called _al H[=a]vi_
+  (The Gathering), arranged alphabetically in the order of the last
+  radical letter. This dictionary is lost, as well as that of the
+  Spaniard Isaac ben Saul, of Lucena. Iona ibn Ganah, of Cordova, born
+  about 985, wrote a Hebrew dictionary in Arabic called _Kit[=a]b al
+  Azul_ (Book of Roots). This, as well as a Hebrew translation by Samuel
+  ibn Tab[=o]n, is extant in MS., and was used by Gesenius in his
+  _Thesaurus_. Rabbi David ben Joseph Kimhi died soon after 1232. His
+  lexicon of roots, called [Hebrew: Shorashim], was printed at Naples
+  1490, fol.; Constantinople, 1513, fol.; Naples, 1491, 8vo; Venice,
+  1552; Berolini, 1838, 4to. _Tishbi_ (The Tishbite), by Elijah ben
+  Asher, the Levite, so called because it contained 712 roots, was
+  printed at Isny 1541, 8vo and 4to, and often afterwards.
+  LATIN.--Münster, Basileae, 1523, 8vo; 5 editions to 1564: Zamora,
+  Compluti, 1526, fol.: Pellicanus, Argentorati, 1540, fol.: Reuchlin,
+  Basil, 1556, fol.: Avenarius, Wittebergae, 1568, fol.; auctus, 1589:
+  Pagnini, Lugd. Bat. 1575, fol.; 1577; Genevae, 1614; Buxtorf, Basil.
+  1607, 8vo; 1615, and many other editions: Frey (Lat.-Eng.), 2nd ed.
+  London, 1815, 8vo: Gesenius, _Thesaurus_, Leipz. 1829-1858, 4to, 3
+  vols. ENGLISH.--Bale, London, 1767, 4to: Parkhurst, ib. 1792, 4to:
+  Lee, ib. 1840, 8vo: Gesenius, translated by Robinson, ib. 1844, 8vo;
+  by Tregelles, ib. 1846, 4to: Fuerst, 4th ed. transl. by Davidson, ib.
+  1866, 8vo: 1871, 8vo, 1547 pages. FRENCH.--Leigh, Amst. 1703, 4to:
+  Glaire, Paris, 1830, 8vo; 1843. GERMAN.--Gesenius, Leipzig, 1810-1812,
+  8vo, 2 vols.: Fuerst, ib. 1842, 16mo: ib. 1876, 8vo, 2 vols.
+  ITALIAN.--Modena, Venetia, 1612, 4to; 1640; Coen, Reggio, 1811, 8vo:
+  Fontanella, Venezia, 1824, 8vo. DUTCH.--Waterman, Rotterdam, 1859,
+  &c., 8vo. HUNGARIAN.--Ehrentheil (Pentateuch), Pest, 1868, 8vo.
+  ROMAIC.--Loundes, Melité. 1845, 8vo, 987 pages.
+
+
+ Rabbinical and Chaldee.--Nathan ben Yehiel of Rome wrote in the
+  beginning of the 12th century a Talmudic dictionary, _Aruch_, printed
+  1480 (?), s. l., fol.; Pesaro, 1517, fol.; Venice, 1531; and often:
+  Isaiah ben Loeb, Berlin, wrote a supplement to _Aruch_, vol. i.
+  Breslau, 1830, 8vo; vol. ii. ([Hebrew: L] to [Hebrew: T]), Wien, 1859,
+  8vo: Münster, Basil. 1527, 4to, 1530, fol.: Elijah ben Asher, the
+  Levite, transl. by Fagius, Isnae, 1541, fol.; Venet. 1560: David ben
+  Isaac de Pomis, _Zamah David_, Venet. 1587, fol.: Buxtorf,
+  Basileae, 1639, fol.: ed. Fischer, Leipz. 1866-1875, 4to: Otho,
+  Geneva, 1675, 8vo; Altona, 1757, 8vo: Zanolini, Patavii, 1747, 8vo:
+  Hornheim, Halle, 1807, 8vo: Landau, Prag, 1819-1824, 8vo, 5 vols.:
+  Dessauer, Erlangen, 1838, 8vo: Nork (i.e. Korn), Grimma, 1842, 4to:
+  Schönhak, Warschau, 1858, 8vo, 2 vols. TARGUMS.--Levy, Leipzig,
+  1866-68 4to, 2 vols.; 1875: Id. (Eng.), London, 1869, 8vo, 2 vols.
+  TALMUD.--Löwy (in Heb.), Wien, 1863, 8vo: Levy, Leipzig, 1876, &c.,
+  4to. PRAYER-BOOK.--Hecht, Kreuznach, 1860, 8vo: Nathan, Berlin, 1854,
+  12mo. SYNONYMS.--Pantavitius, Lodevae, 1640, fol. FOREIGN
+  WORDS.--Rabeini, Lemberg, 1857, 8vo, &c. JEWISH-GERMAN.--Callenberg,
+  Halle, 1736, 8vo: Vollbeding, Hamburg, 1808, 8vo: Stern, München,
+  1833, 8vo, 2 vols.: Theile, Berlin, 1842-1843, 8vo, 2 vols.:
+  Avé-Lallemant, _Das deutsche Gaunerthum_, Leipzig, 1858, 8vo, 4 vols.;
+  vol. iv. pp. 321-512.
+
+
+ Ph[oe]nician.--M. A. Levy, Breslau, 1864, 8vo.
+
+
+ Samaritan.--Crinesius, Altdorphi, 1613, 4to: Morini, Parisiis, 1657,
+  12mo: Hilligerus, Wittebergae, 1679, 4to: Cellarius, Cizae, 1682, 4to;
+  Frankof. 1705: Uhlemann, Leipsiae, 1837, 8vo: Nicholls, London, 1859,
+  8vo.
+
+
+ Assyrian.--Norris, London, 1868, 8vo, 3 vols. PROPER NAMES.--Menant,
+  Paris, 1861, 8vo.
+
+
+ Accadian.--Lenormant, Paris, 1875, 8vo.
+
+
+ Syriac.--Joshua ben Ali, a physician, who lived about 885, made a
+  Syro-Arabic lexicon, of which there is a MS. in the Vatican. Hoffmann
+  printed this lexicon from Alif to Mim, from a Gotha MS., Kiel, 1874,
+  4to. Joshua bar Bahlul, living 963, wrote another, great part of which
+  Castelli put into his lexicon. His MS. is now at Cambridge, and, with
+  those at Florence and Oxford, was used by Bernstein. Elias bar
+  Shinaya, born 975, metropolitan of Nisibis, 1009, wrote a Syriac and
+  Arabic lexicon, entitled _Kit[=a]b [=u]t Tarjuman fi Taalem Loghat es
+  S[=u]ri[=a]n_ (Book called the Interpreter for teaching the Language
+  of the Syrians), of which there is a MS. in the British Museum. It was
+  translated into Latin by Thomas à Novaria, a Minorite friar, edited by
+  Germanus, and published at Rome by Obicinus, 1636, 8vo. It is a
+  classified vocabulary, divided in 30 chapters, each containing several
+  sections. Crinesius, Wittebergae, 1612, 4to: Buxforf, Basileae, 1622,
+  4to: Ferrarius, Romae, 1622, 4to: Trost, Cothenis Anhaltor, 1643, 4to:
+  Gutbir, Hamburgi, 1667, 8vo: Schaaf, Lugd. Bat, 1708, 4to: Zanolini,
+  Patavii, 1742, 4to: Castellus, ed. Michaelis, Göttingen, 1788, 4to, 2
+  vols.: Bernstein, Berlin, 1857, &c. fol.: Smith (Robt. Paine), Dean of
+  Canterbury, Oxonii, 1868, &c. fol.: fasc. 1-3 contain 538 pages:
+  Zingerle, Romae, 1873, 8vo, 148 pages.
+
+
+ Arabic.--The native lexicons are very many, voluminous and copious. In
+  the preface to his great Arabic-English lexicon, Lane describes 33,
+  the most remarkable of which are-the _'Ain_, so called from the letter
+  which begins its alphabet, commonly ascribed to al Khalil (who died
+  before A.H. 175 [A.D. 791], aged seventy-four): the _Sihah_ of Jauhari
+  (died 398 [1003]): the _Mohkam_ of Ibn Sidah the Andalusian, who was
+  blind, and died A.H. 458 [A.D. 1066], aged about sixty: the _Asas_ of
+  Zamakhshari (born 467 [1075], died 538 [1144]), "a most excellent
+  repertory of choice words and phrases": the _Lis[=a]n el 'Arab_ of Ibn
+  Mukarram (born 630 [1232], died 711 [1311]); Lane's copy is in 28
+  vols. 4to: the _Kamus_ (The Sea) of Fairuzabadi (born 729 [1328], died
+  816 [1413]),: the _Taj el Arus_, by Murtada Ez Zebadi (born A.D. 1732,
+  died 1791)--the copy made for Lane is in 24 vols. thick 4to. The
+  _Sihah_ was printed Hardervici Getorum, 1774, 4to; Bulak, 1865, fol. 2
+  vols.: _Kamus_, Calcutta, 1817, fol. 2 vols.; Bombay, 1855, fol. 920
+  pages: _Sirr el Lagal_, by Farish esh Shidiac, Tunis, fol. 609 pages:
+  _Muh[=i]t al Muh[=i]t_, by Beitrus Al Bustani Beirut, 1867-1870, 2
+  vols. 4to, 2358 pages (abridged as _Katr Al Muhit_, ib. 1867-1869, 2
+  vols. 8vo, 2352 pages), is excellent for spoken Arabic. PERSIAN.--The
+  _Surah_, by Jumal, Calcutta, 1812-1815, 2 vols. 4to: _Samachsharii
+  Lexicon_, ed. Wetzstein, Leipz. 1845, 4to; 1850: _Muntakhal al
+  Loghat_, Calcutta, 1808; ib. 1836; Lucknow, 1845; Bombay, 1862, 8vo, 2
+  vols.: _Muntaha l'Arabi_, 4 vols. fol. 1840: _Shams al Loghat_,
+  Bombay, 1860, fol. 2 vols. 509 pages. TURKISH.--_Achteri Kabir_,
+  Constantinople. 1827, fol.: _El Kamus_, ib. 1816, fol. 3 vols.;
+  translated by Açan Effendi, Bulak, fol. 3 vols.; _El Sihah_,
+  translated by Al Vani, Constantinople, 1728, fol. 2 vols.: 1755-1756;
+  Scutari, 1802, fol. 2 vols. LATIN.--Raphelengius, Leiden, 1613, fol.:
+  Giggeius, Mediolani, 1632, fol. 4 vols.: Golius Lugd. Bat. 1653, fol.
+  (the best before Lane's): Jahn, Vindobonae, 1802, 8vo: Freytag, Halle,
+  1830-1838, 4 vols. 4to; abridged, ib. 1837, 4to. ENGLISH.--Catafago
+  (Arab.-Eng. and Eng.-Arab.), London, 1858, 8vo, 2 vols.; 2nd ed. 1873,
+  8vo: Lane, London, 1863-1893 (edited after Lane's death, from 1876, by
+  his grandnephew, Stanley Lane-Poole. The Arabic title is _Medd el
+  Kamoos_, meaning either the Flow of the Sea, or The Extension of the
+  Kamus. It was undertaken in 1842, at the suggestion and at the cost of
+  the 6th duke of Northumberland, then Lord Prudhoe, by Mr Lane, who
+  returned to Egypt for the purpose, and lived in Cairo for seven years
+  to study, and obtain copies of, the great MS. lexicons in the
+  libraries of the mosques, few of which had ever been seen by a
+  European, and which were so quickly disappearing through decay,
+  carelessness and theft, that the means of composing such a work would
+  not long have existed). Newman (modern), ib. 1872, 8vo, 2 vols. 856
+  pages. FRENCH.--Ruphy (Fr.-Ar.), Paris, 1802, 4to: Bochtor (do.),
+  Paris, 1828, 4to, 2 vols.; 2nd ed. ib. 1850: Roland de Bussy (Algiers,
+  Fr.-Ar.), Alger, 1835, 16mo: Id., 1836, 8vo; 1839: Berggren (Fr.-vulg.
+  Ar., Syria and Egypt.), Upsala, 1844, 4to: Farhat (Germanos), revu par
+  Rochaid ed Dahdah, Marseille, 1849, 4to: Biberstein Kasimirski, Paris,
+  1846, 8vo, 2 vols.; 1853-1856; 1860, 2 vols. 3032 pages: Marcel
+  (vulgar dialects of Africa), Paris, 1830; 1835, 8vo; 1837; enlarged,
+  1869, 8vo; Paulmier (Algeria), 2nd ed. Paris, 1860, 8vo, 931 pages;
+  1872: Bernard (Egypt), Lyon, 1864, 18mo: Cuche, Beirut, 1862, 8vo;
+  1867: Nar Bey (A. Calfa), 2nd ed. Paris, 1872, 12mo, 1042 pages:
+  Cherbonneau (written language), Paris, 1876, 2 vols. 8vo: Id.
+  (Fr.-Ar.), Paris, 1872, 8vo: Beausier (Algiers, Tunis, legal,
+  epistolary), Alger, 1871, 4to, 764 pages; 1873. GERMAN.--Seyfarth
+  (Algeria), Grimma, 1849, 16mo: Wolff (Mod. Ar.), Leipzig, 1867, 8vo:
+  Wahrmund (do.), Giessen, 1870-1875, 8vo, 4 vols. ITALIAN.--Germano,
+  Roma, 1636, 8vo; (Ar. Lat. It.), Romae, 1639, fol.: _Dizionario_,
+  Bulak. 1824, 4to: Schiaparelli, Firenze, 1871, 4to, 641 pages.
+  SPANISH.--Alcala, Grenada, 1505, 4to: Cañes, Madrid, 1787, fol. 3
+  vols. SUFI TECHNICAL TERMS.--Abd Errahin, ed. Sprenger, Calcutta,
+  1845, 8vo. TECHNICAL TERMS OF THE MUSSULMAN SCIENCES.--Abd al Hagg and
+  Gholam Kadir, Calcutta, 1853-1862, 4to, 1593 pages. MEDICAL
+  TERMS.--Pharaon and Bertherand, Paris, 1860, 12mo. MATERIA
+  MEDICA.--Muhammed Abd Allah Shirazi, _Ulfaz Udwiyeh_, translated by
+  Gladwin (Eng. Pers. Hindi), Calcutta, 1793, 4to, 1441 words. NOMS DES
+  VÊTEMENTS.--Dozy, Amst. 1845, 8vo. WÖRTER IN ENTGEGENGESETZTEN
+  BEDEUTUNGEN.--Redslob, Göttingen, 1873, 8vo. KORAN.--Willmet (also in
+  Haririum et vitam Timuri), Lugd. Bat. 1784, 4to; Amst. 1790: Fluegel,
+  _Concordantia_, Leipz. 1842, 4to: Penrice, _Dictionary and Glossary_,
+  London, 1873, 4to. EL TABRIZI'S LOGIC.--Mir Abufeth (French), Bulak,
+  1842, 8vo. MALTESE.--Vassali, Romae, 1796, 4to: Falzon (Malt. Ital.
+  Eng.), Malta, _s.a._ 8vo: Vella, Livorno, 1843, 8vo.
+
+
+ Armenian.--Mechitar, Venice, 1749-1769, 4to, 2 vols.: Avedichiam,
+  Sürmelian and Aucher (Aukerian), ib. 1836-1837, 4to, 2 vols.: Aucher,
+  ib. 1846, 4to. POLYGLOT.--Villa (Arm.-vulg., litteralis, Lat. Indicae
+  et Gallicae), Romae, 1780. GREEK AND LATIN.--Lazarists, Venice,
+  1836-1837, 4to, 2 vols. 2217 pages. LATIN.--Rivola, Mediolani, 1621,
+  fol.: Nierszesovicz, Romae, 1695, 4to; Villotte, ib. 1714, fol.:
+  Mechitar, Venetiae, 1747-1763, 4to, 2 vols. ENGLISH.--Aucher, Venice,
+  1821-1825, 4to, 2 vols. FRENCH.--Aucher, Venise, 1812-1817, 8vo, 2
+  vols.; (Fr.-Arm. Turc.), ib. 1840, 4to: Eminian, Vienna, 1853, 4to:
+  Calfa, Paris, 1861, 8vo, 1016 pages; 1872. ITALIAN.--Ciakciak,
+  Venezia, 1837, 4to. RUSSIAN.--Khudobashev [Khutapashian], Moskva,
+  1838, 8vo, 2 vols. RUSS. ARM.--Adamdarov, ib. 1821, 8vo: Popov, ib.
+  1841, 8vo, 2 vols. MODERN WORDS.--Riggs, Smyrna, 1847, 8vo.
+
+
+ Georgian.--Paolini (Ital.), Roma, 1629, 4to: Klaproth (Fr.), Paris,
+  1827, 8vo: Tshubinov (Russian, French), St Petersburg, 1840, 4to;
+  1846, 8vo, 2 vols. 1187 pages.
+
+
+ Circassian.--Loewe, London, 1854, 8vo.
+
+
+ Ossetic.--Sjörgen, St Petersb. 1844, 4to.
+
+
+ Kurd.--Garzoni, Roma, 1787, 8vo: Lerch (German), St Petersburg, 1857,
+  8vo: Id. (Russian), ib. 1856-1858, 8vo.
+
+
+ Persian.--_Burhani Qatiu_, arranged by J. Roebuck, Calcutta, 1818,
+  4to: _Burhan i Kati_, Bulak, 1836, fol.: Muhammed Kazim, Tabriz, 1844,
+  fol.: _Haft Kulzum_ (The Seven Seas), by Ghazi ed din Haidar, King of
+  Oude, Lucknow, 1822, fol. 7 vols. ARABIC.--_Shums ul Loghat_,
+  Calcutta, 1806, 4to, 2 vols. TURKISH.--Ibrahim Effendi, _Farhangi
+  Shu'uri_, ib. 1742, fol. 2 vols. 22,530 words, and 22,450 poetical
+  quotations: _Burhan Kati_, by Ibn Kalif, translated by Ahmed Asin
+  Aintabi, ib. 1799, fol.; Bulak, 1836, fol.: Hayret Effendi, ib. 1826,
+  8vo. ARMENIAN.--Douzean, Constantinople, 1826, fol. BENGALI.--Jay
+  Gopal, Serampore, 1818, 8vo. LATIN.--Vullers (Zend appendix), Bonnae
+  ad Rhen, 1855-1868, 4to, 2 vols. 2544 pages; Supplement of Roots,
+  1867, 142 pages. ENGLISH.--Gladwin, Malda in Bengal, 1780, 4to;
+  Calcutta, 1797: Kirkpatrick, London, 1785, 4to: Moises, Newcastle,
+  1794, 4to: Rousseau, London, 1802, 8vo; 1810: Richardson (Arab, and
+  Pers.), ib. 1780-1800, fol. 2 vols.; ed. Wilkins, ib. 1806-1810, 4to,
+  2 vols.; ed Johnson, ib. 1829, 4to: Ramdhen Sen, Calcutta, 1829, 8vo;
+  1831: Tucker (Eng.-Pers.), London, 1850, 4to: Johnson (Pers. and
+  Arab.), ib. 1852, 4to: Palmer, ib. 1876, 8vo, 726 pages.
+  FRENCH.--Handjeri (Pers. Arab. and Turkish), Moscou, 1841, 4to, 3
+  vols. 2764 pages: Bergé, Leipzig, 1869, 12mo. GERMAN.--Richardson,
+  translated by Wahl as _Orientalische Bibliotheque_, Lemg, 1788-1792,
+  8vo, 3 vols. ITALIAN.--Angelus a S. Josepho [i.e. Labrosse] (Ital.
+  Lat. Fr.), Amst. 1684, fol.
+
+
+ Old Persian.--(Cuneiform), Benfey (German), Leipzig, 1847, 8vo:
+  Spiegel (id.), ib. 1862, 8vo: Kossovich (Latin), Petropoli, 1872, 8vo.
+
+
+ Zend.--Justi, Leipzig, 1864, 4to: Vullers, Persian Lexicon, Appendix:
+  Lagarde, Leipzig, 1868, 8vo.
+
+
+ Pahlavi.--_An old Pahlavi and Pazend Glossary_, translated by Destur
+  Hoshengi Jamaspji, ed. Haug, London, 1867, 8vo; 1870, 8vo: West,
+  Bombay, 1874, 8vo.
+
+
+  INDIAN TERMS.--_The Indian Vocabulary_, London, 1788, 16mo: Gladwin,
+  Calcutta, 1797, 4to: Roberts, London, 1800, 8vo: Rousseau, ib. 1802,
+  8vo: Roebuck (naval), ib. 1813, 12mo: C. P. Brown, _Zillah Dict._,
+  Madras, 1852, 8vo: Robinson (Bengal Courts), Calcutta, 1854, 8vo;
+  1860: Wilson, London, 1855, 4to: Fallon, Calcutta, 1858, 8vo.
+
+
+ Sanskrit.--Amarasimha (lived before A.D. 1000), _Amarakosha_ Calcutta,
+  1807, 8vo; ib. 1834, 4to; Bombay, 1860, 4to; Lucknow, 1863, 4to;
+  Madras, 1870, 8vo, in Grantha characters; Cottayam, 1873, 8vo, in
+  Malaylim characters; Benares, 1867, fol. with _Amaraviveka_, a
+  commentary by Mahesvara: Rajah Radhakanta Deva, _Sabdakalpadruma_,
+  Calcutta, 1821-1857, 4to, 8 vols. 8730 pages: 2nd ed. 1874, &c.:
+  Bhattachdrya, _Sabdastoma Mahanidhi_, Calcutta, 1869-1870, 8vo, parts
+  i.-vii. 528 pages: _Abhidhanaratnamala_, by Halayudha, ed. Aufrecht,
+  London, 1861, 8vo: VACHASPATYA, by Taranatha Tarkavachaspati,
+  Calcutta, 1873, &c., 4to (parts i.-vii., 1680 pages).
+  BENGALI.--_Sabdasindhu_, Calcutta, 1808: _Amarakosa_, translated by
+  Ramodoyu Bidjalunker, Calcutta, 1831, 4to: Mathurana Tarkaratna,
+  _Sabdasandarbhasindhu_, Calcutta, 1863, 4to. MARATHI.--Ananta Sastri
+  Talekar, Poona, 1853, 8vo, 495 pages: Madhava Chandora, Bombay, 1870,
+  4to, 695 pages. TELUGU.--_Amarakosha_, Madras, 1861, ed. Kala, with
+  _Gurubalala prabodhika_, a commentary, ib. 1861, 4to; with the same,
+  ib. 1875, 4to, 516 pages; with _Amarapadaparijata_ (Sans. and Tel.),
+  by Vavilla Ramasvani Sastri, ib. 1862, 4to; ib. 1863, 8vo; 3rd ed. by
+  Jaganmohana Tarkalankara and Khetramohana, 1872, &c., parts i.-iv. 600
+  pages: Suria Pracasa Row, _Sarva-Sabda-Sambodhini_, ib. 1875, 4to,
+  1064 pages. TIBETAN AND MONGOL.--Schiefner, _Buddhistische Triglotte_,
+  St Petersburg, 1859, fol., the _Vyupatti_ or _Mahavyupatti_ from the
+  _Tanguir_, vol. 123 of the Sutra. LATIN.--Paulinus a Sancto
+  Bartholomeo, Amarasinha, sectio i. de coelo, Romae, 1798, 4to: Bopp.
+  Berlin, 1828-1830, 4to; 2nd ed. 1840-1844; 3rd, 1866, 4to.
+  ENGLISH.--_Amarakosha_, trans. by Colebrooke, Serampore, 1808, 4to;
+  1845, 8vo: Rousseau, London, 1812, 4to: Wilson, Calcutta, 1819, 4to;
+  2nd ed. 1832: ed. Goldstücker, Berlin, 1862, &c., folio, to be in 20
+  parts: Yates, Calcutta, 1846, 4to: Benfey, London, 1865, 8vo: Ram
+  Jasen, Benares, 1871, 8vo, 713 pages: Williams, Oxford, 1872, 4to.
+  ENGLISH-SANSKRIT.--Williams, London, 1851, 4to. FRENCH.--Amarakosha,
+  transl. by Loiseleur Deslongchamps, Paris, 1839-1845, 8vo, 2 vols. 796
+  pages: Burnouf and Leupol, Nancy, 1863-1864, 8vo. GERMAN.--Böhtlingk
+  and Roth, St Petersb. 1853, &c., 4to, 7 vols. to 1875.
+  ITALIAN.--Gubernatis, Torino, 1856, &c. 8vo, unfinished, 2 parts.
+  RUSSIAN.--Kossovich, St Petersburg, 1859, 8vo. ROOTS.--Wilkins,
+  London, 1815, 4to: Rosen, Berolini, 1827, 8vo: Westergaard, Bonnae,
+  1840-1841, 8vo: Vishnu Parasurama Sastri Pandita (Sans. and Marathi),
+  Bombay, 1865, 8vo: Taranatha Tarkavachaspati, _Dhatupadarsa_,
+  Calcutta, 1869, 8vo: Leupol, Paris, 1870, 8vo.
+  SYNONYMS.--_Abhidhanacintamani_, by Hemachadra, ed. Colebrooke,
+  Calcutta, 1807, 8vo; translated by Böhtlingk and Rieu (German), St
+  Petersburg, 1847, 8vo. HOMONYMS.--Medinikara, _Medinikosha_, Benares,
+  1865, 4to; Calcutta, 1869, 8vo; ib. 1872, 8vo. DERIVATIVES.--Hirochand
+  and Rooji Rangit, _Dhatumanjari_, Bombay, 1865, 8vo. TECHNICAL TERMS
+  OF THE NYÂYA PHILOSOPHY.--_Nyâyakosa_, by Bhimachârya Jhalakîkar
+  (Sanskrit), Bombay, 1875, 8vo, 183 pages. RIG VEDA.--Grassmann,
+  Leipzig, 1873-1875, 8vo.
+
+
+ Bengali.--Manoel, Lisboa, 1743, 8vo: Forster, Calcutta, 1799-1802,
+  4to, 2 vols. 893 pages: Carey, Serampore, 1815-1825, 4to, 2 vols.; ed.
+  Marshman, ib. 1827-1828, 8vo, 2 vols.; 3rd ed. ib. 1864-1867, 8vo;
+  abridged by Marshman, ib. 1865, 8vo; ib. 1871, 8vo, 2 vols. 936 pages:
+  Morton, Calcutta, 1828, 8vo: Houghton, London, 1833, 4to: Adea,
+  _Shabdabudhi_, Calcutta, 1854, 604 pages. ENGLISH.--Ram Comul Sen, ib.
+  1834, 4to, 2 vols.; London, 1835, 4to: D'Rozario, Calcutta, 1837, 8vo:
+  Adea, _Abhidan_, Calcutta, 1854, 761 pages. ENGLISH LAT.--Ramkissen
+  Sen, ib. 1821, 4to. ENG.-BENG. AND MANIPURI.--[Gordon], Calcutta,
+  1837, 8vo.
+
+
+ Canarese.--Reeve, Madras, 1824-1832, 4to, 2 vols.; ed. Sanderson,
+  Bangalore, 1858, 8vo, 1040 pages; abridged by the same, 1858, 8vo, 276
+  pages: _Dictionarium Canarense_, Bengalori, 1855, 8vo: _School
+  Dictionary_, Mangalore, 1876, 8vo, 575 pages.
+
+
+ Dardic Languages.--Leitner (Astori, Ghilghiti, Chilasi, and dialects
+  of Shina, viz. Arnyia, Khajuna and Kalasha), Lahore, 1868, 4to.
+
+
+ Guzarati.--(English) Mirza Mohammed Cauzim, Bombay, 1846, 4to;
+  Shapurji Edalji, ib. 1868, 8vo, 896 pages: Karsandas Mulji, ib. 1868,
+  8vo, 643 pages.
+
+
+ Hindi.--Rousseau, London, 1812, 4to: Adam, Calcutta, 1829, 8vo:
+  Thompson, ib. 1846, 8vo: J. D. Bate, London, 1876, 8vo, 809 pages.
+  ENGLISH.--Adam, Calcutta, 1833, 8vo. ENGLISH, URDU AND
+  HINDI.--Mathuraprasada Mirsa, Benares, 1865, 8vo, 1345 pages.
+
+
+ Hindustani.--Ferguson, London, 1773, 4to: Gilchrist, Calcutta, 1800,
+  8vo; ed. Hunter, Edinb. 1810; Lond. 1825: Taylor, Calcutta, 1808, 4to,
+  2 vols.: Gladwin (Persian and Hind.), Calcutta, 1809, 8vo, 2 vols.:
+  Shakespeare, London, 1817, 4to; 1820; 1834; 1849: Forbes, London,
+  1847, 8vo; 1857: Bertrand (French), Paris, 1858, 8vo: Brice, London,
+  1864, 12mo: Fallon, Banaras, 1876, &c., to be in about 25 parts and
+  1200 pages. ENGLISH.--Gilchrist, 1787-1780, 4to, 2 parts: Thompson,
+  Serampore, 1838, 8vo.
+
+
+ Kashmiri.--Elmslie, London, 1872, 12mo.
+
+
+ Khassia.--Roberts, Calcutta, 1875, 12mo.
+
+
+ Malayalim.--Fabricius and Breithaupt, Weperg, 1779, 4to: Bailey,
+  Cottayam, 1846, 8vo: Gundert, Mangalore, 1871, 8vo, 1171 pages.
+
+
+ Marathi.--Carey, Serampore, 1810, 8vo: Kennedy, Bombay, 1824, fol.:
+  Jugunnauth Shastri Kramavant, Bombay, 1829-1831, 4to, 3 vols.:
+  Molesworth, ib. 1831, 4to; 2nd ed. 1847, 4to; ed. Candy, Bombay, 1857,
+  4to, 957 pages; abridged by Baba Padmanji, ib. 1863, 8vo; 2nd ed.
+  (abridged), London, 1876, 8vo, 644 pages. ENGLISH.--Molesworth,
+  Bombay, 1847, 4to.
+
+
+ Oriya.--Mohunpersaud Takoor, Serampore, 1811, 8vo: Sutton, Cuttack,
+  1841-1848, 8vo, 3 vols. 856 pages.
+
+
+ Pali.--Clough, Colombo, 1824, 8vo: Moggallana Thero (a Sinhalese
+  priest of the 12th century), _Abhidhanappika_ (Pali, Eng. Sinhalese),
+  ed. Waskeduwe Subheti, Colombo, 1865, 8vo: Childers, London,
+  1872-1875, 8vo, 658 pages. ROOTS.--Silavansa, _Dhatumanjusa_ (Pali
+  Sing. and Eng.), Colombo, 1872, 8vo.
+
+
+ Prakrit.--Delius, _Radices_, Bonnae ad Rh., 1839, 8vo.
+
+
+ Punjabi.--Starkey, 1850, 8vo; Lodiana Mission, Lodiana, 1854-1860, 444
+  pages.
+
+
+ Pushtu or Afghan.--Dorn, St Petersb. 1845, 4to: Raverty, London, 1860,
+  4to; 2nd ed. ib. 1867, 4to: Bellew, 1867, 8vo.
+
+
+ Sindhi.--Eastwick, Bombay, 1843, fol. 73 pages: Stack, ib. 1855, 8vo,
+  2 vols.
+
+
+ Sinhalese.--Clough, Colombo, 1821-1830, 8vo, 2 vols.: Callaway (Eng.,
+  Portuguese and Sinhalese), ib. 1818, 8vo: Id., _School Dictionary_,
+  ib. 1821, 8vo: Bridgenell (Sinh.-Eng.), ib. 1847, 18mo: Nicholson
+  (Eng.-Sinh.), 1864, 32mo, 646 pages.
+
+
+ Tamil.--Provenza (Portug.), Ambalacotae, 1679, 8vo: _Sadur Agurardi_,
+  written by Beschi in 1732, Madras, 1827, fol.; Pondicherry, 1875, 8vo:
+  Blin (French), Paris, 1834, 8vo: Rottler, Madras, 1834-1841, 4to, 4
+  vols.: Jaffna Book Society (Tamil), Jaffna, 1842, 8vo, about 58,500
+  words: Knight and Spaulding (Eng. Tam.), ib. 1844, 8vo; _Dictionary_,
+  ib. 1852, 4to: Pope, 2nd ed. ib. 1859, 8vo: Winslow, Madras, 1862,
+  4to, 992 pages, 67,452 words.
+
+
+ Telugu.--Campbell, Madras, 1821, 4to: C. P. Brown, Madras (Eng.-Tel.),
+  1852, 8vo, 1429 pages: Id. (Tel.-Eng.), ib. 1852, 8vo, 1319 pages.
+  MIXED TELUGU.--Id., ib. 1854, 8vo.
+
+
+ Thuggee.--Sleeman, Calcutta, 1830, 8vo, 680 Ramasi words.
+
+
+ Indo-Chinese Languages.--Leyden, _Comparative Vocabulary of Barma,
+  Malaya and Thai_, Serampore, 1810, 8vo. _Annamese_: Rhodes (Portug.
+  and Lat.), Romae, 1651, 4to: Pigneaux and Taberd, Fredericinagori,
+  1838, 4to; Legrand de la Liraye, Paris, 1874, 8vo: Pauthier (Chin.
+  Ann.-Fr. Lat.), Paris, 1867, &c., 8vo. _Assamese_: Mrs Cutter, Saipur,
+  1840, 12 mo; Bronson, London, 1876, 8vo, 617 pages. _Burmese_: Hough
+  (Eng.-Burm.), Serampore, 1825, Moulmain, 1845, 8vo, 2 vols. 955 pages:
+  Judson, Calcutta, 1826, 8vo; (Eng. Burm.), Moulmain, 1849, 4to; (Burm.
+  Eng.), ib. 1852, 8vo; 2nd ed., Rangoon, 1866, 8vo, 2 vols. 968 pages:
+  Lane, Calcutta, 1841, 4to. _Cambodian_: Aymonier (Fr.-Camb.), Saigon,
+  1874, 4to; Id. (Camb.-Fr.), ib. 1875, fol. _Karen_: Sau-kau Too
+  (Karen), Tavoy, 1847, 12mo, 4 vols.: Mason, Tavoy, 1840, 4to.
+  _Sgau-Karen_: Wade, ib. 1849, 8vo. _Siamese (Thai)_: Pallegoix (Lat.
+  French, Eng.), Paris, 1854, 4to: _Dictionarium Latinum Thai_, Bangkok,
+  1850, 4to, 498 pages.
+
+
+ Malay.--LATIN.--Haex, Romae, 1631, 4to; Batavia, 1707.
+  DUTCH.--Houtmann (Malay and Malagasy), Amst. 1603, 4to; 1673; 1680;
+  1687; 1703; Batavia, 1707: Wiltens and Dankaarts, Gravenhage, 1623,
+  4to; Amst. 1650; 1677; Batavia, 1708, 4to: Heurnius, Amst. 1640, 4to:
+  Gueynier, Batavia, 1677, 4to; 1708: Loder, ib. 1707-1708, 4to: Van der
+  Worm, ib. 1708, 4to: Roorda van Eysinga (Low), ib. 1824-1825, 8vo, 2
+  vols.; 12th ed. 's Gravenhage, 1863, 8vo; Id. (Hof, Volks en Lagen
+  Taal), ib. 1855, 8vo: Dissel and Lucardie (High Malay), Leiden, 1860,
+  12mo: Pijnappel, Amst. 1863, 8vo: Badings, Schoonhoven, 1873, 8vo.
+  ENGLISH.--Houtmann (Malay and Malagasy), translated by A. Spaulding,
+  London, 1614, 4to: Bowrey, ib. 1701, 4to: Howison, ib. 1801, 4to:
+  Marsden, ib. 1812, 4to: Thomsen, Malacca, 1820, 8vo; 1827: Crawford,
+  London, 1851, 8vo, 2 vols. FRENCH.--Boze, Paris, 1825, 16mo: Elout
+  (Dutch-Malay and French-Malay), Harlem, 1826, 4to: Bougourd, Le Havre,
+  1856, 8vo: Richard, Paris, 1873, 8vo, 2 vols.: Favre, Vienna, 1875,
+  8vo, 2 vols.
+
+
+ Malay Archipelago.--_Batak_: Van der Tuuk, Amsterdam, 1861, 8vo, 564
+  pages. _Bugis_: Mathes, Gravenh. 1874, 8vo, 1188 pages: Thomsen
+  (Eng.-Bugis and Malay), Singapore, 1833, 8vo. _Dyak_: Hardeland
+  (German), Amst. 1859, 8vo, 646 pages. _Javanese_: Senerpont Domis,
+  Samarang, 1827, 4to, 2 vols.: Roorda van Eysinga, Kampen, 1834-1835,
+  8vo, 2 vols.: Gericke, Amst. 1847, 8vo; ed. Taco Roorda, ib. 1871, &c.
+  parts i.-v., 880 pages: Jansz and Klinkert, Samarang, 1851, 8vo; 1865:
+  Favre (French), Vienne, 1870, 8vo. _Macassar_: Matthes, Amst. 1859,
+  8vo, 951 pages. Sunda: De Wilde (Dutch, Malay and Sunda), Amsterdam,
+  1841, 8vo: Rigg (Eng.), Batavia, 1862, 4to, 573 pages. _Formosa_:
+  Happart (Favorlang dialect, written about 1650), Parrapattan, 1840,
+  12mo.
+
+
+ Philippines.--_Bicol_: Marcos, Sampaloc, 1754, fol. _Bisaya_: Sanchez,
+  Manila, 1711, fol.: Bergaño, ib. 1735, fol.: Noceda, ib. 1841:
+  Mentrida (also Hiliguena and Haraya) ib. 1637, 4to; 1841, fol. 827
+  pages: Felis de la Encarnacion, ib. 1851, 4to, 2 vols. 1217 pages.
+  _Ibanac_: Bugarin, ib. 1854, 4to. _Ilocana_, Carro, ib. 1849, fol.
+  _Pampanga_: Bergaño, ib. 1732, fol. _Tagala_: Santos, Toyabas, 1703,
+  fol.; ib. 1835, 4to, 857 pages: Noceda and San Lucar, Manila, 1754,
+  fol.; 1832.
+
+
+ Chinese.--Native Dictionaries are very numerous. Many are very copious
+  and voluminous, and have passed through many editions. _Shwo wan_, by
+  Hü Shin, is a collection of the ancient characters, about 10,000 in
+  number, arranged under 540 radicals, published 150 B.C., usually in 12
+  vols.: _Yu pien_, by Ku Ye Wang, published A.D. 530, arranged under
+  542 radicals, is the basis of the Chinese Japanese Dictionaries used
+  in Japan: _Ping tseu loui pien_, Peking, 1726, 8vo, 130 vols.: _Pei
+  wan yün fu_ (Thesaurus of Literary Phrases), 1711, 131 vols. 8vo,
+  prepared by 66 doctors of the Han lin Academy in seven years. It
+  contains 10,362 characters, and countless combinations of two, three
+  or four characters, forming compound words and idioms, with numerous
+  and copious quotations. According to Williams (_On the word Shin_, p.
+  79), an English translation would fill 140 volumes octavo of 1000
+  pages each. _Kanghi tsze tien_ (Kanghi's Standard or Canon of the
+  Character), the dictionary of Kanghi, the first emperor of the present
+  dynasty, was composed by 30 members of the Han lin, and published in
+  1716, 40 vols. 4to, with a preface by the emperor. It contains 49,030
+  characters, arranged under the 214 radicals. It is generally in 12
+  vols., and is universally used in China, being the standard authority
+  among native scholars for the readings as well as the meanings of
+  characters. LATIN.--De Guignes (French, Lat.), Paris, 1813, fol.;
+  Klaproth, Supplément, 1819; ed. Bazil (Latin), Hong-Kong, 1853, 4to:
+  Gonçalves (Lat.-Chin.), Macao, 1841, fol.: Callery, _Systema
+  phoneticum_, Macao, 1841, 8vo: Schott, _Vocabularium_, Berlin, 1844,
+  4to. ENGLISH.--Raper, London, 1807, fol. 4 vols.: Morrison, Macao,
+  1815-1823, 4to, 3 parts in 6 vols.: Medhurst, Batavia, 1842-1843, 8vo,
+  2 vols.: Thom, Canton, 1843, 8vo: Lobscheid, Hong-Kong, 1871, 4to:
+  Williams, Shanghai, 1874, 4to. ENG. CHINESE.--Morrison, part iii.:
+  Williams, Macao, 1844, 8vo: Medhurst, Shanghai, 1847-1848, 8vo, 2
+  vols.: Hung Maou, _Tung yung fan hwa_ (Common words of the Red-haired
+  Foreigners), 1850, 8vo. Doolittle, Foochow, 1872, 4to, vol. i. 550
+  pages. FRENCH,--Callery, _Dict. encyclopédique_, Macao and Paris, 1845
+  (radicals 1-20 only): M. A. H., 1876, 8vo, autographié, 1730 pages.
+  FRENCH-CHIN.--Perny (Fr.-Latin, Spoken Mandarin), Paris, 1869, 4to;
+  Appendice, 1770; Lemaire and Giguel, Shanghai, 1874, 16mo.
+  PORTUGUESE.--Gonçalves (Port.-Chin.), Macao, 1830, 8vo, 2 vols.: Id.
+  (Chin.-Port.), ib. 1833, 8vo. IDIOMS.--Giles, Shanghai, 1873, 4to.
+  PHRASES.--Yaou Pei-keen, _Luy yih_, 1742-1765, 8vo, 55 vols.: Tseen
+  Ta-hin, _Shing luy_, 1853, 8vo, 4 vols. CLASSICAL EXPRESSIONS.--Keang
+  Yang and 30 others, _Sze Shoo teen Lin_, 1795, 8vo, 30 vols. ELEGANT
+  EXPRESSIONS.--Chang ting yuh, _Fun luy tsze kin_, 1722, 8vo, 64 vols.
+  PHRASES OF THREE WORDS.--Julien (Latin), Paris, 1864, 8vo.
+  POETICAL.--_Pei wan she yun_, 1800, 8vo, 5 vols. PROPER NAMES.--F.
+  Porter Smith (China, Japan, Corea, Annam, &c., Chinese-Eng.),
+  Shanghai, 1870, 8vo. TOPOGRAPHY.--Williams, Canton, 1841, 8vo. NAMES
+  OF TOWNS.--Biot, Paris, 1842, 8vo. ANCIENT CHARACTERS.--Foo
+  Lwantseang, _Luh shoo fun luy_, 1800, 8vo, 12 vols. SEAL
+  CHARACTER.--Heu Shin, _Shwo wan_, ed. Seu Heuen, 1527, 8vo, 12 vols.
+  RUNNING HAND.--St Aulaire and Groeneveld (Square Characters, Running
+  Hand; Running, Square), Amst. 1861, 4to, 117 pages. TECHNICAL TERMS
+  (in Buddhist translations from Sanskrit)--Yuen Ying, _Yih 'see king
+  pin e_, 1848, 8vo. DIALECTS.--_Amoy_: Douglas, London, 1873, 4to, 632
+  pages: Macgowan, Hong-Kong, 1869, 8vo. _Canton_: Yu Heo-poo and Wan
+  ke-shih, _Keang hoo chih tuh fun yun tso yaou ho tseih_, Canton, 1772,
+  8vo, 4 vols.; 1803, 8vo, 4 vols.; Fuh-shan, 1833, 8vo, 4 vols.:
+  Morrison, Macao, 1828, 8vo: Wan ke shih, Canton, 1856, 8vo: Williams
+  (tonic, Eng.-Chinese), Canton, 1856, 8vo: Chalmers, Hong-Kong, 1859,
+  12mo; 3rd ed. 1873, 8vo. _Changchow in Fuhkeen_: Seay Sew-lin, _Ya suh
+  tung shih woo yin_, 1818, 8vo, 8 vols.; 1820. _Foo-chow_: Tseih (a
+  Japanese general) and Lin Peih shan, _Pa yin ho ting_, ed. Tsin Gan,
+  1841, 8vo: Maclay and Baldwin, Foochow, 1870, 8vo, 1123 pages.
+  _Hok-keen_: Medhurst, Macao, 1832, 4to: _Peking_, Stent, Shanghai,
+  1871, 8vo.
+
+
+ Corean.--CHINESE, COREAN AND JAPANESE.--_Cham Seen Wo Kwo tsze mei_,
+  translated by Medhurst, Batavia, 1835, 8vo. RUSSIAN.--Putzillo, St
+  Petersburg, 1874, 12mo, 746 pages.
+
+
+ Japanese.--_Sio Ken Zi Ko_ (Examination of Words and Characters),
+  1608, 8vo, 10 vols.: _Wa Kan Won Se Ki Sio Gen Zi Ko_, lithographed by
+  Siebold, Lugd. Bat., 1835, fol. JAP.-CHINESE.--_Faga biki set yo siu_.
+  CHINESE-JAP.--_Kanghi Tse Tein_, 30 vols. 12mo: _Zi rin gioku ben_.
+  DUTCH DICTIONARIES PRINTED BY JAPANESE.--_Nieeu verzameld Japansch en
+  Hollandsch Woordenbock_, by the interpreter, B. Sadayok, 1810:
+  Minamoto Masataka, Prince of Nakats (Jap. Chinese-Dutch), 5 vols. 4to,
+  printed at Kakats by his servants: _Jedo-Halma_ (Dutch-Jap.), Jedo,
+  4to, 20 vols.: _Nederduitsche taal_, Dutch Chinese, for the use of
+  interpreters. LATIN AND PORTUGUESE.--Calepinus, _Dictionarium_,
+  Amacusa, 1595, 4to. LATIN.--Collado, _Compendium_, Romae, 1632, 4to:
+  _Lexicon_, Romae, 1870, 4to, from Calepinus. ENGLISH.--Medhurst,
+  Batavia, 1830, 8vo: Hepburn, Shanghai, 1867, 8vo; 1872.
+  ENG.-JAP.--Hori Tatnoskoy, Yedo, 1862, 8vo; 2nd ed. Yeddo, 1866, 8vo:
+  Satow and Ishibashi Masakata (spoken language), London, 1876, 8vo.
+  FRENCH.--Rosny (Jap. Fr. Eng.), Paris, 1857, 4to, vol. i.: Pagés,
+  Paris, 1869, 4to, translated from Calepinus. FR.-JAP.--Soutcovey,
+  Paris, 1864, 8vo. FR. ENG. JAP.--Mermet de Cachon, Paris, 1866, 8vo,
+  unfinished. GERMAN.--Pfizmaier (Jap.-Ger., Eng.), Wien, 1851, 4to,
+  unfinished. SPANISH.--_Vocabulario del Japon_, Manila, 1630, 4to,
+  translated from the next. PORTUGUESE.--_Vocabulario da Lingua de
+  Japam_, Nagasaki, 1603, 4to. RUSSIAN.--Goshkevich, St Petersburg,
+  1857, 8vo, 487 pages. CHINESE CHARACTERS WITH JAPANESE
+  PRONUNCIATION.--Rosny, Paris, 1867, 8vo. CHINESE AND JAPANESE NAMES OF
+  PLANTS.--Hoffmann, Leyde, 1864, 8vo.
+
+
+ Aino.--Pfizmaier, Wien, 1854, 4to.
+
+
+ Northern and Central Asia.--_Buriat_: Castrén, St Petersburg, 1857,
+  8vo. _Calmuck_: Zwick, Villingen, 1853, 4to: Smirnov, Kazan, 1857,
+  12mo: Jügl, _Siddhi Kur_, Leipzig, 1866, 8vo. _Chuvash_: Clergy of the
+  school of the Kazan Eparchia, Kazan, 1836, 8vo, 2481 words: Lyulé
+  (Russ.-Chuv. French), Odessa, 1846, 8vo, 244 pages: Zolotnitski,
+  Kazan, 1875, 8vo, 287 pages. _Jagatai_: Mir Ali Shir, _Abuska_, ed.
+  Vámbéry, with Hungarian translation, Pesth, 1862, 8vo: Vámbéry,
+  Leipzig, 1867, 8vo: Pavet de Courteille, Paris, 1870, 8vo. _Koibal and
+  Karagas_: Castrén, St Petersburg, 1857, 8vo. _Manchu_: _Yutchi tseng
+  ting tsing wen kian_ (Manchu Chinese), 1771, 4to, 6 vols.: _Sze li hoh
+  pik wen kian_ (Manchu-Mongol, Tibetan, Chinese) 10 vols. 4to, the
+  Chinese pronunciation represented in Manchu: _San hoh pien lan_
+  (Manchu-Chinese, Mongol), 1792, 8vo, 12 vols.;--all three classed
+  vocabularies: Langlès (French), Paris, 1789-1790, 4to, 3 vols.:
+  Gabelentz (German), Leipzig, 1864, 8vo: Zakharov (Russian), St
+  Petersburg, 1875, 8vo, 1235 pages: _Mongol_: I. J. Schmidt (German,
+  Russian), St Petersburg, 1835, 4to: Schergin, Kazan, 1841, 8vo:
+  Kovalevski, Kasan, 1844-1849, 4to, 3 vols. 2703 pages. _Ostiak_:
+  Castrén, St Petersb. 1858, 8vo. _Samoyed_: Castrén, St Petersb. 1855,
+  8vo, 308 pages. Tartar: Giganov (Tobolsk), St Petersburg, 1804, 4to;
+  (Russ.-Tartar), ib. 1840, 4to: Troyanski (Karan), Kasan, 1835-1855,
+  4to. _Tibetan_: _Minggi djamtoo_ (Tibet-Mongol): _Bodschi dajig togpar
+  lama_: _Kad shi schand scharwi melonggi jige_
+  (Manchu-Mongol-Tibetan-Chinese), Kanghi's Dictionary with the Tibetan
+  added in the reign of Khian lung (1736-1795); Csoma de Körös (Eng.),
+  Calcutta, 1834, 4to: I. J. Schmidt (German), St Petersburg, 1841, 4to:
+  Id. (Russian), ib. 1843, 4to: Jaeschke (Eng.), London, 1870, 8vo, 160
+  pages: Id. (Germ.), Gnadau, 1871, 658 pages: (Bhotanta), Schroeter,
+  Serampore, 1826, 4to. _Tungusian_: Castrén, St Petersburg, 1856, 8vo,
+  632 pages. _Uigur_: Vámbéry, Innspruck, 1870, 4to. _Yakut_: Böhtlingk,
+  ib. 1854, 4to, 2 vols. _Yenissei Ostiak_: Castrén, ib. 1849, 8vo.
+
+
+AFRICA
+
+ Egyptian.--Young (enchorial), London, 1830-1831, 8vo: Sharpe, London,
+  1837, 4to: Birch, London. 1838, 4to: Champollion (died March 4, 1832),
+  _Dictionnaire égyptien_, Paris, 1841, 4to: Brugsch,
+  _Hieroglyphisch-Demotisches Wörterbuch_, Leipzig, 1867-1868, 4to, 4
+  vols. 1775 pages, nearly 4700 words, arranged according to the
+  hieroglyphic alphabet of 28 letters: Pierret, _Vocabulaire hiérog._,
+  Paris, 1875, 8vo, containing also names of persons and places: Birch,
+  in vol. v. pp. 337-580 of Bunsen's _Egypt's Place_, 2nd ed. London,
+  1867, &c. 8vo, 5010 words. PROPER NAMES.--Brugsch, Berlin, 1851, 8vo,
+  726 names: Parthey, ib. 1864, 8vo, about 1500 names: Lieblein,
+  Christiania, 1871, 8vo, about 3200 from hieroglyphic texts. BOOK OF
+  THE DEAD.--Id., Paris, 1875, 12mo.
+
+
+ Coptic.--Veyssière de la Croze, Oxon. 1775, 8vo: Rossi, Romae, 1807,
+  4to: Tattam, Oxon. 1855, 8vo: Peyron, 1835, 4to (the standard):
+  Parthey, Berolini, 1844, 8vo.
+
+
+ Ethiopic.--Wemmer, Romae, 1638, 4to: Ludolf, London, 1661, 4to:
+  Francof. ad M., 1699, fol.: Dillmann (Tigré appendix), Leipzig,
+  1863-1865, 4to, 828 pages.
+
+
+ Amharic.--Ludolphus, Franc. ad Maenum, 1698, fol.: Isenberg, London,
+  1841, 4to, 442 pages. _Tigré_: Munzinger, Leipzig, 1865, 8vo:
+  Beurmann, ib. 1868, 8vo.
+
+
+ East Coast.--_Dankali_: Isenberg, London, 1840, 12mo. _Galla_: Krapf,
+  London, 1842, 8vo: Tutschek, München, 1844, 8vo. _Engutuk Iloigob_:
+  Erhardt, Ludwigsberg, 1857, 8vo. _Kisuaheli_: _Vocabulary of the
+  Soahili_, Cambridge, U.S. 1845, 8vo: Steere, London, 1870, 8vo, about
+  5800 words. _Kisuaheli, Kinika, Kikamba, Kipokono, Kikian, Kigalla_:
+  Krapf, Tübingen, 1850, 8vo.
+
+
+ Malagasy.--Houtmann (Malaysche en Madagask Talen), Amst. 1603, 2nd ed.
+  Matthysz, ib. 1680, 8vo: Huet de Froberville, Isle de France, fol. 2
+  vols.: Flacourt, Paris, 1658, 8vo: Challand (Southern), Isle de
+  France, 1773, 4to: Freeman and Johns, London, 1835, 8vo, 2 vols.:
+  Dalmont (Malgache, Salalave, et Betsimara), 1842, 8vo: Kessler,
+  London, 1870, 8vo.
+
+
+ Southern Africa.--Bleek, _The Languages of Mozambique_, London, 1856,
+  8vo. _Kaffre_: Bennie, Lovedale, 1826, 16mo: Ayliffe, Graham's Town,
+  1846, 12mo: Appleyard, 1850, 8vo: Bleek, Bonn, 1853, 4to, 646 pages.
+  _Zulu-Kaffre_: Perrin (Kaffre-Eng.), London, 1855, 24mo, 172 pages:
+  Id. (Eng.-Kaffre), Pietermaritzburg, 1855, 24mo, 227 pages: Id.
+  (Eng.-Zulu), ib. 1865, 12mo, 226 pages: Dohne, Cape Town, 1857, 8vo,
+  428 pages: Colenso, Pietermaritzburg, 1861, 8vo, 560 pages, about 8000
+  words. _Hottentot_: Bleek, Cape Town, 1857, 4to, 261 pages. _Namaqua_:
+  Tindall, ib. 1852, 8vo: _Vocabulary_, Barmen, 1854, 8vo: Hahn,
+  Leipzig, 1870, 12mo. Sechuana: Casalis, Paris, 1841, 8vo. _Herero_:
+  Hahn, Berlin, 1857, 8vo, 207 pages, 4300 words.
+
+
+ Western Africa.--_Akra_ or _Ga_: Zimmermann, Stuttgart, 1858, 8vo, 690
+  pages. _Ashantee_: Christaller (also Akra), Basel, 1874, 8vo, 299
+  pages. _Bullom_: Nylander, London, 1814, 12mo. _Bunda or Angola_:
+  Cannecatim, Lisboa, 1804, 4to, 722 pages. _Dualla Grammatical
+  Elements_, &c., Cameroons, 1855, 8vo. _Efik_ or _Old Calabar_:
+  Waddell, Old Calabar, 1846, 16mo, 126 pages; Edinb, 1849, 8vo, 95
+  pages. _Eyo_: Raban, London, 1830-1831, 12mo, 2 parts. _Grebo_:
+  _Vocabulary_, Cape Palmas, 1837, 8vo; _Dictionary_, ib. 1839, 8vo, 119
+  pages. _Ifa_: Schlegel, Stuttgart, 1857, 8vo. _Mpongwe_: De Lorme
+  (Franç.-Pongoué), Paris, 1876, 12mo, 354 pages. _Oji_: Riis, Basel,
+  1854, 8vo, 284 pages. _Sherbro'_: Schön, _s. a. et l._ 8vo, written in
+  1839, 42 pages. _Susu_: Brunton, Edinburgh, 1802, 8vo, 145 pages.
+  _Vei_: Koelle, London, 1854, 8vo, 266 pages. _Wolof and Bambarra_:
+  Dard, Paris, 1825, 8vo. _Wolof_: Roger, ib. 1829, 8vo: Missionnaires
+  de S. Esprit, Dakar, 1855, &c. 16mo. Faidherbe (French-Wolof, Poula
+  and Soninke), St Louis, Senegambia, 1860, 12mo. _Yoruba_: Crowther,
+  London, 1843, 8vo; 1852, 298 pages: Vidal, ib. 1852, 8vo: Bowen,
+  Washington, 1858, 4to.
+
+
+ Central Africa.--Barth, _Vocabularies_. Gotha, 1862-1866, 4to. _Bari_:
+  Mitterreutzner, Brixen, 1867, 8vo: Reinisch, Vienna, 1874, 8vo.
+  _Dinka_: Mitterreutzner, Brixen, 1866, 8vo. _Haussa_: Schön (Eng.),
+  London, 1843, 8vo.
+
+
+ Berber.--Venture de Paradis, Paris, 1844, 8vo: Brosselard, ib. 1844,
+  8vo: Delaporte, ib. 1844, 4to, by order of the Minister of War:
+  Creusat, Franç.-Kabyle (Zouaoua), Alger, 1873, 8vo. _Siwah_: Minutoli,
+  Berlin, 1827, 4to.
+
+
+AUSTRALIA AND POLYNESIA
+
+ Australia.--_New South Wales_: Threlkeld (Lake Macquarie Language),
+  Sydney, 1834, 8vo. _Victoria_: Bunce, Melbourne, 1856, 12mo, about
+  2200 words. _South Australia_: Williams, South Australia, 1839, 8vo:
+  Teichelmann and Schürmann, Adelaide, 1840, 8vo: Meyer, ib. 1843, 8vo.
+  _Murray River_: Moorhouse, ib. 1846, 8vo. _Parnkalla_: Schürmann,
+  Adelaide, 1844, 8vo. _Woolner District_: _Vocabulary_, ib. 1869, 12mo.
+  _Western Australia_: Sir George Grey, Perth, 1839, 4to; London, 1840,
+  8vo: Moore, ib. 1843: Brady, Roma, 1845, 24mo, 8vo, 187 pages.
+  _Tasmania_: Millegan, Tasmania, 1857.
+
+
+ Polynesia.--Hale, _Grammars and Vocabularies of all the Polynesian
+  Languages_, Philadelphia, 1846, 4to. _Marquesas, Sandwich Gambier_:
+  Mosblech, Paris, 1843, 8vo. _Hawaiian_: Andrews, _Vocabulary_,
+  Lahainaluna, 1636, 8vo: Id., _Dictionary_, Honolulu, 1865, 8vo, 575
+  pages, about 15,500 words. _Marquesas_: Pierquin, de Gembloux,
+  Bourges, 1843, 8vo: Buschmann, Berlin, 1843, 8vo. _Samoan_:
+  _Dictionary_, Samoa, 1862, 8vo. _Tahitian_: _A Tahitian and English
+  Dictionary_, Tahiti, 1851, 8vo, 314 pages. _Tonga_: Rabone, Vavau,
+  1845, 8vo. _Fijian_: Hazlewood (Fiji-Eng.), Vewa. 1850, 12mo: Id.
+  (Eng.-Fiji), ib. 1852, 12mo: Id., London, 1872, 8vo. _Maori_: Kendall,
+  1820, 12mo: Williams, Paihia, 1844, 8vo; 3rd ed. London, 1871, 8vo:
+  Taylor, Auckland, 1870, 12mo.
+
+
+AMERICA
+
+ North America.--_Eskimo_: Washington, London, 1850, 8vo: Petitot
+  (Mackenzie and Anderson Rivers), Paris, 1876, 4to. _Kinai_: Radloff,
+  St Petersburg, 1874, 4to. _Greenland_: Egede (Gr. Dan. Lat., 3 parts),
+  Hafn, 1750, 8vo; 1760, Fabricius, Kjöbenhavn, 1804, 4to. _Hudson's Bay
+  Indians_: Bowrey, London, 1701, fol. _Abnaki_: Rasles, Cambridge,
+  U.S., 1833, 4to. _Chippewa_: Baraga, Cincinnati, 1853, 12mo, 622
+  pages: Petitot, Paris, 1876, 4to, 455 pages. _Massachusetts_ or
+  _Natick_: Cotton, Cambridge, U.S. 1829, 8vo. _Onondaga_: Shea
+  (French-Onon.), from a MS. (of 17th century), London, 1860, 4to, 109
+  pages. _Dacota_: Riggs, New York, 1851, 4to, 424 pages: Williamson
+  (Eng. Dac.), Santos Agency, Nebraska, 12mo, 139 pages. _Mohawk_:
+  Bruyas, New York, 1863, 8vo. _Hidatsa (Minnetarees, Gros Ventres of
+  the Missouri)_: Matthews, ib. 1874, 8vo. _Choctaw_: Byington, ib.
+  1852, 16mo. _Clallam and Lummi_: Gibbs, ib. 1863, 8vo. _Yakama_:
+  Pandosy, translated by Gibbs and Shea, ib. 1862, 8vo. _Chinook_:
+  Gibbs, New York, 1863, 4to. _Chinook Jargon, the trade language of
+  Oregon_: Id., ib. 1863, 8vo. _Tatche_ or _Telamé_: Sitjar, ib. 1841,
+  8vo.
+
+
+ Mexico and Central America.--_Tepehuan_: Rinaldini, Mexico, 1743, 4to.
+  _Cora_: Ortega, Mexico, 1732, 4to. _Tarahumara_: Steffel, Brünn, 1791,
+  8vo. _Otomi_: Carochi, Mexico, 1645, 4to: Neve y Molina, ib. 1767,
+  8vo: Yepes, ib. 1826, 4to: Piccolomini, Roma, 1841, 8vo. _Mexican_ or
+  _Aztec_: Molina, Mexico, 1555, 4to; 1571, fol. 2 vols.: Arenas, ib.
+  1583; 1611, 8vo; 1683; 1725; 1793, 12mo: Biondelli, Milan, 1869, fol.
+  _Mexican, Tontonacan, and Huastecan_: Olmos, Mexico, 1555-1560, 4to, 2
+  vols. _Huastecan_: Tapia Zenteno, ib. 1767, 4to, 128 pages. _Opata_ or
+  _Tequima_: Lombardo, ib. 1702, 4to. _Tarasca_: Gilberti, ib. 1559,
+  4to: Lagunas, ib. 1574, 8vo. _Mixtecan_: Alvarado, Mexico, 1593, 4to.
+  _Zapoteca_: Cordova, ib. 1578, 4to. _Maya_: Beltran de Santa Rosa
+  Maria, ib. 1746, 4to; Merida de Yucatan, 1859, 4to, 250 pages:
+  Brasseur de Bourbourg, Paris, 1874, 8vo, 745 pages. _Quiché_: Id.
+  (also Cakchiquel and Trutuhil dialects), ib. 1862, 8vo.
+
+
+ South America.--_Chibcha_: Uricoechea, Paris, 1871, 8vo. _Chayma_:
+  Tauste, Madrid, 1680, 4to: Yanguas, Burgos, 1683, 4to. _Carib_:
+  Raymond, Auxerre, 1665-1666, 8vo. _Galibi_: D.[e]. L.[a] S.[auvage],
+  Paris, 1763, 8vo. _Tupi_: Costa Rubim, Rio de Janeiro, 1853, 8vo:
+  Silva Guimaräes, Bahia, 1854, 8vo: Diaz, Lipsia, 1858, 16mo.
+  _Guarani_: Ruiz de Montoyo, Madrid, 1639, 4to; 1640; 1722, 4to; ed.
+  Platzmann, Leipzig, 1876, &c., 8vo, to be in 4 vols. 1850 pages.
+  _Moxa_: Marban, Lima, 1701, 8vo. _Lule_: Machoni de Corderia, Madrid,
+  1732, 12mo. _Quichua_: Santo Thomas, Ciudad de los Reyes, 1586, 8vo:
+  Torres Rubio, Sevilla, 1603, 8vo; Lima, 1609, 8vo; ed. Figueredo,
+  Lima, 1754, 8vo; Holguin, Ciudad de los Reyes, 1608, 8vo: Tschudi,
+  Wien, 1853, 8vo, 2 vols.: Markham, London, 1864, 8vo: Lopez, _Les
+  Races Aryennes de Perou_, Paris, 1871, 8vo, comparative vocabulary,
+  pp. 345-421. _Aymara_: Bertonio, Chicuyto, 1612, 4to, 2 vols.
+  _Chileno_: Valdivia (also Allentiac and Milcocayac), Lima, 1607, 8vo:
+  Febres, ib. 1765, 12mo; ed. Hernandez y Caluza, Santiago, 1846, 8vo, 2
+  vols. _Tsonecan_ (Patagonian): Schmid, Bristol, 1860, 12mo.
+
+The above article incorporates the salient features of the 9th-edition
+article by the Rev. Ponsonby A. Lyons, and the 10th-edition article by
+Benjamin E. Smith.
+
+
+FOOTNOTES:
+
+  [1] Joannes de Garlandia (John Garland; fl. 1202-1252) gives the
+    following explanation in his _Dictionarius_, which is a classed
+    vocabulary:--"Dictionarius dicitur libellus iste a dictionibus magis
+    necessariis, quas tenetur quilibet scolaris, non tantum in scrinio de
+    lignis facto, sed in cordis armariolo firmiter retinere." This has
+    been supposed to be the first use of the word.
+
+  [2] An excellent dictionary of quotations, perhaps the first of the
+    kind; a large folio volume printed in Strassburg about 1475 is
+    entitled "Pharetra auctoritates et dicta doctorum, philosophorum, et
+    poetarum continens."
+
+  [3] This volume was issued with a new title-page as _Glossaire du
+    moyen âge_, Paris, 1872.
+
+
+
+
+DICTYOGENS (Gr. [Greek: diktyon], a net, and the termination [Greek:
+-genês], produced), a botanical name proposed by John Lindley for a
+class including certain families of Monocotyledons which have net-veined
+leaves. The class was not generally recognized.
+
+
+
+
+DICTYS CRETENSIS, of Cnossus in Crete, the supposed companion of
+Idomeneus during the Trojan War, and author of a diary of its events.
+The MS. of this work, written in Phoenician characters, was said to have
+been found in his tomb (enclosed in a leaden box) at the time of an
+earthquake during the reign of Nero, by whose order it was translated
+into Greek. In the 4th century A.D. a certain Lucius Septimius brought
+out _Dictys Cretensis Ephemeris belli Trojani_, which professed to be a
+Latin translation of the Greek version. Scholars were not agreed whether
+any Greek original really existed; but all doubt on the point was
+removed by the discovery of a fragment in Greek amongst the papyri found
+by B. P. Grenfell and A. S. Hunt in 1905-1906. Possibly the Latin
+Ephemeris was the work of Septimius himself. Its chief interest lies in
+the fact that (together with Dares Phrygius's _De excidio Trojae_) it
+was the source from which the Homeric legends were introduced into the
+romantic literature of the middle ages.
+
+  Best edition by F. Meister (1873), with short but useful introduction
+  and index of Latinity; see also G. Körting, _Diktys und Dares_
+  (1874), with concise bibliography; H. Dunger, _Die Sage vom
+  trojanischen Kriege in den Bearbeitungen des Mittelalters und ihren
+  antiken Quellen_ (1869, with a literary genealogical table); E.
+  Collilieux, _Étude sur Dictys de Crète et Darès de Phrygie_ (1887),
+  with bibliography; W. Greif, "Die mittelalterlichen Bearbeitungen der
+  Trojanersage," in E. M. Stengel's _Ausgaben und Abhandlungen aus dem
+  Gebiete der romanischen Philologie_, No. 61 (1886, esp. sections 82,
+  83, 168-172); F. Colagrosso, "Ditte Cretese" in _Atti della r.
+  Accademia di Archeologia_ (Naples, 1897, vol. 18, pt. ii. 2); F.
+  Noack, "Der griechische Dictys," in _Philologus_, supp. vi. 403 ff.;
+  N. E. Griffin, _Dares and Dictys, Introduction to the Study of the
+  Medieval Versions of the Story of Troy_ (1907).
+
+
+
+
+DICUIL (fl. 825), Irish monastic scholar, grammarian and geographer. He
+was the author of the _De mensura orbis terrae_, finished in 825, which
+contains the earliest clear notice of a European discovery of and
+settlement in Iceland and the most definite Western reference to the old
+freshwater canal between the Nile and the Red Sea, finally blocked up in
+767. In 795 (February 1-August 1) Irish hermits had visited Iceland; on
+their return they reported the marvel of the perpetual day at midsummer
+in "Thule," where there was then "no darkness to hinder one from doing
+what one would." These eremites also navigated the sea north of Iceland
+on their first arrival, and found it ice-free for one day's sail, after
+which they came to the ice-wall. Relics of this, and perhaps of other
+Irish religious settlements, were found by the permanent Scandinavian
+colonists of Iceland in the 9th century. Of the old Egyptian freshwater
+canal Dicuil learnt from one "brother Fidelis," probably another Irish
+monk, who, on his way to Jerusalem, sailed along the "Nile" into the Red
+Sea--passing on his way the "Barns of Joseph" or Pyramids of Giza, which
+are well described. Dicuil's knowledge of the islands north and west of
+Britain is evidently intimate; his references to Irish exploration and
+colonization, and to (more recent) Scandinavian devastation of the same,
+as far as the Faeroes, are noteworthy, like his notice of the elephant
+sent by Harun al-Rashid (in 801) to Charles the Great, the most curious
+item in a political and diplomatic intercourse of high importance.
+Dicuil's reading was wide; he quotes from, or refers to, thirty Greek
+and Latin writers, including the classical Homer, Hecataeus, Herodotus,
+Thucydides, Virgil, Pliny and King Juba, the sub-classical Solinus, the
+patristic St Isidore and Orosius, and his contemporary the Irish poet
+Sedulius;--in particular, he professes to utilize the alleged surveys of
+the Roman world executed by order of Julius Caesar, Augustus and
+Theodosius (whether Theodosius the Great or Theodosius II. is
+uncertain). He probably did not know Greek; his references to Greek
+authors do not imply this. Though certainly Irish by birth, it has been
+conjectured (from his references to Sedulius and the caliph's elephant)
+that he was in later life in an Irish monastery in the Frankish empire.
+Letronne inclines to identify him with Dicuil or Dichull, abbot of
+Pahlacht, born about 760.
+
+  There are seven chief MSS. of the _De mensura_ (Dicuil's tract on
+  grammar is lost); of these the earliest and best are (1) Paris,
+  National Library, Lat. 4806; (2) Dresden, Regius D. 182; both are of
+  the 10th century. Three editions exist: (1) C. A. Walckenaer's, Paris,
+  1807; (2) A. Letronne's, Paris, 1814, best as to commentary; (3) G.
+  Parthey's, Berlin, 1870, best as to text. See also C. R. Beazley,
+  _Dawn of Modern Geography_ (London, 1897), i. 317-327, 522-523, 529;
+  T. Wright, _Biographia Britannica literaria, Anglo-Saxon Period_
+  (London, 1842), pp. 372-376.     (C. R. B.)
+
+
+
+
+DIDACHE, THE, or _Teaching of the (twelve) Apostles_,--the most
+important of the recent recoveries in the region of early Christian
+literature (see APOCRYPHAL LITERATURE). It was previously known by name
+from lists of canonical and extra-canonical books compiled by Eusebius
+and other writers. Moreover, it had come to be suspected by several
+scholars that a lost book, variously entitled _The Two Ways_ or _The
+Judgment of Peter_, had been freely used in a number of works, of which
+mention must presently be made. In 1882 a critical reconstruction of
+this book was made by Adam Krawutzcky with marvellous accuracy, as was
+shown when in the very next year the Greek bishop and metropolitan,
+Philotheus Bryennius, published _The Teaching of the Twelve Apostles_
+from the same manuscript from which he had previously published the
+complete form of the Epistle of Clement.[1]
+
+_The Didach[=e]_, as we now have it in the Greek, falls into two marked
+divisions: (a) a book of moral precepts, opening with the words, "There
+are two ways"; (b) a manual of church ordinances, linked on to the
+foregoing by the words, "Having first said all these things, baptize,
+&c." Each of these must be considered separately before we approach the
+question of the locality and date of the whole book in its present form.
+
+1. _The Two Ways._--The author of the complete work, as we now have it,
+has modified the original _Two Ways_ by inserting near the beginning a
+considerable section containing, among other matter, passages from the
+Sermon on the Mount, in which the language of St Matthew's Gospel is
+blended with that of St Luke's. He has also added at the close a few
+sentences, beginning, "If thou canst not bear (the whole yoke of the
+Lord), bear what thou canst" (vi. 2); and among minor changes he has
+introduced, in dealing with confession, reference to "the church" (iv.
+14). No part of this matter is to be found in the following documents,
+which present us in varying degrees of accuracy with _The Two Ways_:
+(i.) the Epistle of Barnabas, chaps. xix., xx. (in which the order of
+the book has been much broken up, and a good deal has been omitted);
+(ii.) the _Ecclesiastical Canons of the Holy Apostles_, usually called
+the _Apostolic Church Order_, a book which presents a parallel to the
+_Teaching_, in so far as it consists first of a form of _The Two Ways_,
+and secondly of a number of church ordinances (here, however, as in the
+Syriac _Didascalia_, which gives about the same amount of _The Two
+Ways_, various sections are ascribed to individual apostles, e.g. "John
+said, There are two ways," &c.); (iii.) a discourse of the Egyptian monk
+Schnudi (d. 451), preserved in Arabic (see Iselin, _Texte u. Unters._,
+1895); (iv.) a Latin version, of which a fragment was published by O.
+von Gebhardt in 1884, and the whole by J. Schlecht in 1900. When by the
+aid of this evidence _The Two Ways_ is restored to us free of glosses,
+it has the appearance of being a Jewish manual which has been carried
+over into the use of the Christian church. This is of course only a
+probable inference; there is no prototype extant in Jewish literature,
+and, comparing the moral (non-doctrinal) instruction for Christian
+catechumens in Hermas, _Shepherd_ (_Mand._ i.-ix.), no real need to
+assume one. There was a danger of admitting Gentile converts to the
+church on too easy moral terms; hence the need of such insistence on the
+ideal as in The Two Ways and the _Mandates_. The recent recovery of the
+Latin version is of singular interest, as showing that, even without the
+distinctively Christian additions and interpolations which our full form
+of the _Teaching_ presents, it was circulating under the title _Doctrina
+apostolorum_.[2]
+
+2. The second part of our _Teaching_ might be called a church directory.
+It consists of precepts relating to church life, which are couched in
+the second person plural; whereas _The Two Ways_ uses throughout the
+second person singular. It appears to be a composite work. First (vii.
+1-xi. 2) is a short sacramental manual intended for the use of local
+elders or presbyters, though such are not named, for they were not yet a
+distinctive order or clergy. This section was probably added to _The Two
+Ways_ before the addition of the remainder. It orders baptism in the
+threefold name, making a distinction as to waters which has Jewish
+parallels, and permitting a threefold pouring on the head, if sufficient
+water for immersion cannot be had. It prescribes a fast before baptism
+for the baptizer as well as the candidate. Fasts are to be kept on
+Wednesday and Friday, not Monday and Thursday, which are the fast days
+of "the hypocrites," i.e. by a perversion of the Lord's words, the Jews.
+"Neither pray ye as the hypocrites; but as the Lord commanded in His
+Gospel." Then follows the Lord's Prayer, almost exactly as in St
+Matthew, with a brief doxology--"for Thine is the power and the glory
+forever." This is to be said three times a day. Next come three
+eucharistic prayers, the language of which is clearly marked off from
+that of the rest of the book, and shows parallels with the diction of St
+John's Gospel. They are probably founded on Jewish thanksgivings, and it
+is of interest to note that a portion of them is prescribed as a grace
+before meat in (pseudo-) Athanasius' _De virginitate_. A trace of them
+is found in one of the liturgical prayers of Serapion, bishop of Thmui,
+in Egypt, but they have left little mark on the liturgies of the church.
+As in Ignatius and other early writers, the eucharist, a real meal (x.
+1) of a family character, is regarded as producing immortality (cf.
+"spiritual food and drink and eternal life"). None are to partake of it
+save those who have been "baptized in the name of the Lord" (an
+expression which is of interest in a document which prescribes the
+threefold formula). The prophets are not to be confined to these forms,
+but may "give thanks as much as they will." This appears to show that a
+prophet, if present, would naturally preside over the eucharist. The
+next section (xi. 3-xiii.) deals with the ministry of spiritual gifts as
+exercised by apostles, prophets and teachers. An apostle is to be
+"received as the Lord"; but he must follow the Gospel precepts, stay but
+one or two days, and take no money, but only bread enough for a day's
+journey. Here we have that wider use of the term "apostle" to which
+Lightfoot had already drawn attention. A prophet, on the contrary, may
+settle if he chooses, and in that case he is to receive tithes and
+first-fruits; "for they are your high priests." If he be once approved
+as a true prophet, his words and acts are not to be criticized; for this
+is the sin that shall not be forgiven. Next comes a section (xiv., xv.)
+reflecting a somewhat later development concerning fixed services and
+ministry; the desire for a stated service, and the need of regular
+provision for it, is leading to a new order of things. The eucharist is
+to be celebrated every Lord's Day, and preceded by confession of sins,
+"that your sacrifice may be pure ... for this is that sacrifice which
+was spoken of by the Lord, In every place and time to offer unto Me a
+pure sacrifice. Appoint therefore unto yourselves bishops and deacons,
+worthy of the Lord, men meek and uncovetous, and true and approved; for
+they also minister unto you the ministration of the prophets and
+teachers. Therefore despise them not; for they are your honoured ones,
+together with the prophets and teachers." This is an arrangement
+recommended by one who has tried it, and he reassures the old-fashioned
+believer who clings to the less formal régime (and whose protest was
+voiced in the Montanist movement), that there will be no spiritual loss
+under the new system. The book closes (chap. xvi.) with exhortations to
+steadfastness in the last days, and to the coming of the
+"world-deceiver" or Antichrist, which will precede the coming of the
+Lord. This section is perhaps the actual utterance of a Christian
+prophet, and may be of earlier origin than the two preceding sections.
+
+3. It will now be clear that indications of the locality and date of our
+present _Teaching_ must be sought for only in the second part, and in
+the Christian interpolations in the first part. We have no ground for
+thinking that the second part ever existed independently as a separate
+book. The whole work was in the hands of the writer of the seventh book
+of the _Apostolic Constitutions_, who embodies almost every sentence of
+it, interspersing it with passages of Scripture, and modifying the
+precepts of the second part to suit a later (4th-century) stage of
+church development; this writer was also the interpolator of the
+Epistles of Ignatius, and belonged to the Syrian Church. Whether the
+second part was known to the writer of the _Apostolic Church Order_ is
+not clear, as his only quotation of it comes from one of the eucharistic
+prayers. The allusions of early writers seem to point to Egypt, but
+their references are mostly to the first part, so that we must be
+careful how we argue from them as to the provenance of the book as a
+whole. Against Egypt has been urged the allusion in one of the
+eucharistic prayers to "corn upon the mountains." This is found in the
+Prayer-book of Serapion (c. 350) but omitted in a later Egyptian
+prayer; the form as we have it in _The Didach[=e]_ may have passed into
+Egypt with the authority of tradition which was afterwards weakened. The
+anti-Jewish tone of the second part suggests the neighbourhood of Jews,
+from whom the Christians were to be sharply distinguished. Either Egypt
+or Syria would satisfy this condition, and in favour of Syria is the
+fact that the presbyterate there was to a late date regarded as a rank
+rather than an office. If we can connect the injunctions (vi. 3)
+concerning (abstinence from certain) food and that which is offered to
+idols with the old trouble that arose at Antioch (Acts xv. 1) and was
+legislated for by the Jerusalem council, we have additional support for
+the Syrian claim. But all that we can safely say as to locality is that
+the community here represented seems to have been isolated, and out of
+touch with the larger centres of Christian life.
+
+This last consideration helps us in discussing the question of date. For
+such an isolated community may have preserved primitive customs for some
+time after they had generally disappeared. Certainly the stage of
+development is an early one, as is shown, e.g., by the prominence of
+prophets, and the need that was felt for the vindication of the position
+of the bishops and deacons (there is no mention at all of presbyters);
+moreover, there is no reference to a canon of Scripture (though the
+written Gospel is expressly mentioned) or to a creed. On the other hand
+the "apostles" of the second part are obviously not "the twelve
+apostles" of the title; and the prophets seem in some instances to have
+proved unworthy of their high position. The ministry of enthusiasm which
+they represent is about to give way to the ministry of office, a
+transition which is reflected in the New Testament in the 3rd Epistle of
+John. Three of the Gospels have clearly been for some time in
+circulation; St Matthew's is used several times, and there are phrases
+which occur only in St Luke's, while St John's Gospel lies behind the
+eucharistic prayers which the writer has embodied in his work. There are
+no indications of any form of doctrinal heresy as needing rebuke; the
+warnings against false teaching are quite general. While the first part
+must be dated before the Epistle of Barnabas, i.e. before A.D. 90, it
+seems wisest not to place the complete work much earlier than A.D. 120,
+and there are passages which may well be later.
+
+  A large literature has sprung up round The _Didach[=e]_ since 1884.
+  Harnack's edition in _Texte u. Unters._ vol. ii. (1884) is
+  indispensable to the student; and his discussions in _Altchristl.
+  Litteratur_ and _Chronologie_ give clear summaries of his work. Other
+  editions of the text are those of F. X. Funk, _Patres Apostolici_,
+  vol. i. (Tübingen, 1901); H. Lietzmann (Bonn, 1903; with Latin
+  version). Dr J. E. Odgers has published an English translation with
+  introduction and notes (London, 1906). Dr C. Taylor in 1886 drew
+  attention to some important parallels in Jewish literature; his
+  edition contains an English translation. Dr Rendel Harris published in
+  1887 a complete facsimile, and gathered a great store of patristic
+  illustration. Text and translation will also be found in Lightfoot's
+  _Apostolic Fathers_ (ed. min.) The fullest critical treatment in
+  English is by Dr Vernon Bartlet in the extra volume of Hastings's
+  _Dictionary of the Bible_; the most complete commentary on the text is
+  by P. Drews in Hennecke's _Handbuch zu den N.T. Apocryphen_ (1904).
+  Other references to the literature may be found by consulting
+  Harnack's _Altchristl. Litteratur_.
+
+
+FOOTNOTES:
+
+  [1] The MS. was found in the Library of the Jerusalem Monastery of
+    the Most Holy Sepulchre, in Phanar, the Greek quarter of
+    Constantinople. It is a small octavo volume of 120 parchment leaves,
+    written throughout by Leo, "notary and sinner," who finished his task
+    on the 11th of June 1156. Besides The _Didach[=e]_ and the Epistles
+    of Clement it contains several spurious Ignatian epistles.
+
+  [2] The word _twelve_ had no place in the original title and was
+    inserted when the original _Didach[=e]_ or _Teaching_ (e.g. _The Two
+    Ways_) was combined with the church manual which mentions apostles
+    outside of the twelve. It may be noted that the division of the
+    _Didach[=e]_ into chapters is due to Bryennius, that into verses to
+    A. Harnack.
+
+
+
+
+DIDACTIC POETRY, that form of verse the aim of which is, less to excite
+the hearer by passion or move him by pathos, than to instruct his mind
+and improve his morals. The Greek word [Greek: didaktikos] signifies a
+teacher, from the verb [Greek: didaskein], and poetry of the class under
+discussion approaches us with the arts and graces of a schoolmaster. At
+no time was it found convenient to combine lyrical verse with
+instruction, and therefore from the beginning of literature the didactic
+poets have chosen a form approaching the epical. Modern criticism, which
+discourages the epic, and is increasingly anxious to limit the word
+"poetry" to lyric, is inclined to exclude the term "didactic poetry"
+from our nomenclature, as a phrase absurd in itself. It is indeed more
+than probable that didactic verse is hopelessly obsolete. Definite
+information is now to be found in a thousand shapes, directly and boldly
+presented in clear and technical prose. No farmer, however elegant,
+will, any longer choose to study agriculture in hexameters, or even in
+Tusser's shambling metre. The sciences and the professions will not
+waste their time on methods of instruction which must, from their very
+nature, be artless, inexact and vague. But in the morning of the world,
+those who taught with authority might well believe that verse was the
+proper, nay, the only serious vehicle of their instruction. What they
+knew was extremely limited, and in its nature it was simple and
+straightforward; it had little technical subtlety; it constantly lapsed
+into the fabulous and the conjectural. Not only could what early sages
+knew, or guessed, about astronomy and medicine and geography be
+conveniently put into rolling verse, but, in the absence of all written
+books, this was the easiest way in which information could be made
+attractive to the ear and be retained by the memory.
+
+In the prehistoric dawn of Greek civilization there appear to have been
+three classes of poetry, to which the literature of Europe looks back as
+to its triple fountain-head. There were romantic epics, dealing with the
+adventures of gods and heroes; these Homer represents. There were mystic
+chants and religious odes, purely lyrical in character, of which the
+best Orphic Hymns must have been the type. And lastly there was a great
+body of verse occupied entirely with increasing the knowledge of
+citizens in useful branches of art and observation; these were the
+beginnings of didactic poetry, and we class them together under the dim
+name of Hesiod. It is impossible to date these earliest didactic poems,
+which nevertheless set the fashion of form which has been preserved ever
+since. The _Works and Days_, which passes as the direct masterpiece of
+Hesiod (q.v.), is the type of all the poetry which has had education as
+its aim. Hesiod is supposed to have been a tiller of the ground in a
+Boeotian village, who determined to enrich his neighbours' minds by
+putting his own ripe stores of useful information into sonorous metre.
+Historically examined, the legend of Hesiod becomes a shadow, but the
+substance of the poems attributed to him remains. The genuine parts of
+the _Works and Days_, which Professor Gilbert Murray has called "a slow,
+lowly, simple poem," deal with rules for agriculture. The _Theogony_ is
+an annotated catalogue of the gods. Other poems attributed to Hesiod,
+but now lost, were on astronomy, on auguries by birds, on the character
+of the physical world; still others seem to have been genealogies of
+famous women. All this mass of Boeotian verse was composed for
+educational purposes, in an age when even preposterous information was
+better than no knowledge at all. In slightly later times, as the Greek
+nation became better supplied with intellectual appliances, the stream
+of didactic poetry flowed more and more closely in one, and that a
+theological, channel. The great poem of Parmenides _On Nature_ and those
+of Empedocles exist only in fragments, but enough remains to show that
+these poets carried on the didactic method in mythology. Cleostratus of
+Tenedos wrote an astronomical poem in the 6th century, and Periander a
+medical one in the 4th, but didactic poetry did not flourish again in
+Greece until the 3rd century, when Aratus, in the Alexandrian age, wrote
+his famous _Phenomena_, a poem about things seen in the heavens. Other
+later Greek didactic poets were Nicander, and perhaps Euphorion.
+
+It was from the hands of these Alexandrian writers that the genius of
+didactic poetry passed over to Rome, since, although it is possible that
+some of the lost works of the early republic, and in particular those of
+Ennius, may have possessed an educational character, the first and by
+far the greatest didactic Latin poet known to us is Lucretius. A highly
+finished translation by Cicero into Latin hexameters of the principal
+works of Aratus is believed to have drawn the attention of Lucretius to
+this school of Greek poetry, and it was not without reference to the
+Greeks, although in a more archaic and far purer taste, that he
+composed, in the 1st century before Christ, his magnificent _De rerum
+natura_. By universal consent, this is the noblest didactic poem in the
+literature of the world. It was intended to instruct mankind in the
+interpretation and in the working of the system of philosophy revealed
+by Epicurus, which at that time was exciting the sympathetic attention
+of all classes of Roman society. What gave the poem of Lucretius its
+extraordinary interest, and what has prolonged and even increased its
+vitality, was the imaginative and illustrative insight of the author,
+piercing and lighting up the recesses of human experience. On a lower
+intellectual level, but of a still greater technical excellence, was the
+_Georgics_ of Virgil, a poem on the processes of agriculture, published
+about 30 B.C. The brilliant execution of this famous work has justly
+made it the type and unapproachable standard of all poetry which desires
+to impart useful information in the guise of exquisite literature.
+Himself once a farmer on the banks of the Mincio, Virgil, at the apex of
+his genius, set himself in his Campanian villa to recall whatever had
+been essential in the agricultural life of his boyish home, and the
+result, in spite of the ardours of the subject, was what J. W. Mackail
+has called "the most splendid literary production of the Empire." In the
+rest of surviving Latin didactic poetry, the influence and the imitation
+of Virgil and Lucretius are manifest. Manilius, turning again to
+Alexandria, produced a fine _Astronomica_ towards the close of the reign
+of Augustus. Columella, regretting that Virgil had omitted to sing of
+gardens, composed a smooth poem on horticulture. Natural philosophy
+inspired Lucilius junior, of whom a didactic poem on Etna survives. Long
+afterwards, under Diocletian, a poet of Carthage, Nemesianus, wrote in
+the manner of Virgil the _Cynegetica_, a poem on hunting with dogs,
+which has had numerous imitations in later European literatures. These
+are the most important specimens of didactic poetry which ancient Rome
+has handed down to us.
+
+In Anglo-Saxon and early English poetic literature, and especially in
+the religious part of it, an element of didacticism is not to be
+overlooked. But it would be difficult to say that anything of importance
+was written in verse with the sole purpose of imparting information,
+until we reach the 16th century. Some of the later medieval allegories
+are didactic or nothing. The first poem, however, which we can in any
+reasonable way compare with the classic works of which we have been
+speaking is the _Hundreth Pointes of Good Husbandrie_, published in 1557
+by Thomas Tusser; these humble Georgics aimed at a practical description
+of the whole art of English farming. Throughout the early part of the
+17th century, when our national poetry was in its most vivid and
+brilliant condition, the last thing a poet thought of doing was the
+setting down of scientific facts in rhyme. We come across, however, one
+or two writers who were as didactic as the age would permit them to be,
+Samuel Daniel with his philosophy, Fulke Greville, Lord Brooke with his
+"treatises" of war and monarchy. After the Restoration, as the lyrical
+element rapidly died out of English poetry, there was more and more room
+left for educational rhetoric in verse. The poems about prosody, founded
+upon Horace, and signed by John Sheffield, 3rd earl of Mulgrave
+(1648-1721), and Lord Roscommon, were among the earliest purely didactic
+verse-studies in English. John Philips deserves a certain pre-eminence,
+as his poem called Cyder, in 1706, set the fashion which lasted all down
+the 18th century, of writing precisely in verse about definite branches
+of industry or employment. None of the greater poets of the age of Anne
+quite succumbed to the practice, but there is a very distinct flavour of
+the purely didactic about a great deal of the verse of Pope and Gay. In
+such productions as Gilbert West's (1703-1756) _Education_, Dyer's
+_Fleece_, and Somerville's _Chase_, we see technical information put
+forward as the central aim of the poet. Instead of a passionate
+pleasure, or at least an uplifted enthusiasm, being the poet's object,
+he frankly admits that, first and foremost, he has some facts about wool
+or dogs or schoolmasters which he wishes to bring home to his readers,
+and that, secondly, he consents to use verse, as brilliantly as he can,
+for the purpose of gilding the pill and attracting an unwilling
+attention. As we descend the 18th century, these works become more and
+more numerous, and more dry, especially when opposed by the descriptive
+and rural poets of the school of Thomson, the poet of _The Seasons_. But
+Thomson himself wrote a huge poem of _Liberty_ (1732), for which we have
+no name if we must not call it didactic. Even Gray began, though he
+failed to finish, a work of this class, on _The Alliance of Education
+and Government_. These poems were discredited by the publication of _The
+Sugar-Cane_ (1764), a long verse-treatise about the cultivation of sugar
+by negroes in the West Indies, by James Grainger (1721-1766), but,
+though liable to ridicule, such versified treatises continued to
+appear. Whether so great a writer as Cowper is to be counted among the
+didactic poets is a question on which readers of _The Task_ may be
+divided; this poem belongs rather to the class of descriptive poetry,
+but a strong didactic tendency is visible in parts of it. Perhaps the
+latest frankly educational poem which enjoyed a great popularity was
+_The Course of Time_ by Robert Pollok (1798-1827), in which a system of
+Calvinistic divinity is laid down with severity and in the pomp of blank
+verse. This kind of literature had already been exposed, and
+discouraged, by the teaching of Wordsworth, who had insisted on the
+imperative necessity of charging all poetry with imagination and
+passion. Oddly enough, _The Excursion_ of Wordsworth himself is perhaps
+the most didactic poem of the 19th century, but it must be acknowledged
+that his influence, in this direction, was saner than his practice.
+Since the days of Coleridge and Shelley it has been almost impossible to
+conceive a poet of any value composing in verse a work written with the
+purpose of inculcating useful information.
+
+The history of didactic poetry in France repeats, in great measure, but
+in drearier language, that of England. Boileau, like Pope, but with a
+more definite purpose as a teacher, offered instruction in his _Art
+poétique_ and in his _Epistles_. But his doctrine was always literary,
+not purely educational. At the beginning of the 18th century, the
+younger Racine (1692-1763) wrote sermons in verse, and at the close of
+it the Abbé Delille (1738-1813) tried to imitate Virgil in poems about
+horticulture. Between these two there lies a vast mass of verse written
+for the indulgence of intellect rather than at the dictates of the
+heart; wherever this aims at increasing knowledge, it at once becomes
+basely and flatly didactic. There is nothing in French literature of the
+transitional class that deserves mention beside _The Task_ or _The
+Excursion_.
+
+During the century which preceded the Romantic revival of poetry in
+Germany, didactic verse was cultivated in that country on the lines of
+imitation of the French, but with a greater dryness and on a lower level
+of utility. Modern German literature began with Martin Opitz (1597-1639)
+and the Silesian School, who were in their essence rhetorical and
+educational, and who gave their tone to German verse. Albrecht von
+Haller (1708-1777) brought a very considerable intellectual force to
+bear on his huge poems, _The Origin of Evil_, which was theological, and
+_The Alps_ (1729), botanical and topographical. Johann Peter Uz
+(1720-1796) wrote a _Theodicée_, which was very popular, and not without
+dignity. Johann Jacob Dusch (1725-1787) undertook to put _The Sciences_
+into the eight books of a great didactic poem. Tiedge (1752-1840) was
+the last of the school; in a once-famous _Urania_, he sang of God and
+Immortality and Liberty. These German pieces were the most unswervingly
+didactic that any modern European literature has produced. There was
+hardly the pretence of introducing into them descriptions of natural
+beauty, as the English poets did, or of grace and wit like the French.
+The German poets simply poured into a lumbering mould of verse as much
+solid information and direct instruction as the form would hold.
+
+Didactic poetry has, in modern times, been antipathetic to the spirit of
+the Latin peoples, and neither Italian nor Spanish literature has
+produced a really notable work in this class. An examination of the
+poems, ancient and modern, which have been mentioned above, will show
+that from primitive times there have been two classes of poetic work to
+which the epithet didactic has been given. It is desirable to
+distinguish these a little more exactly. One is the pure instrument of
+teaching, the poetry which desires to impart all that it knows about the
+growing of cabbages or the prevention of disasters at sea, the
+revolution of the planets or the blessings of inoculation. This is
+didactic poetry proper, and this, it is almost certain, became
+irrevocably obsolete at the close of the 18th century. No future Virgil
+will give the world a second _Georgics_. But there is another species
+which it is very improbable that criticism has entirely dislodged; that
+is the poetry which combines, with philosophical instruction, an impetus
+of imaginative movement, and a certain definite cultivation of fire and
+beauty. In hands so noble as those of Lucretius and Goethe this species
+of didactic poetry has enriched the world with durable masterpieces,
+and, although the circle of readers which will endure scientific
+disquisition in the bonds of verse grows narrower and narrower, it is
+probable that the great poet who is also a great thinker will now and
+again insist on being heard. In Sully-Prudhomme France has possessed an
+eminent writer whose methods are directly instructive, and both _La
+Justice_ (1878) and _Le Bonheur_ (1888) are typically didactic poems.
+Perhaps future historians may name these as the latest of their class.
+     (E. G.)
+
+
+
+
+DIDEROT, DENIS (1713-1784), French man of letters and encyclopaedist,
+was born at Langres on the 5th of October 1713. He was educated by the
+Jesuits, like most of those who afterwards became the bitterest enemies
+of Catholicism; and, when his education was at an end, he vexed his
+brave and worthy father's heart by turning away from respectable
+callings, like law or medicine, and throwing himself into the vagabond
+life of a bookseller's hack in Paris. An imprudent marriage (1743) did
+not better his position. His wife, Anne Toinette Champion, was a devout
+Catholic, but her piety did not restrain a narrow and fretful temper,
+and Diderot's domestic life was irregular and unhappy. He sought
+consolation for chagrins at home in attachments abroad, first with a
+Madame Puisieux, a fifth-rate female scribbler, and then with Sophie
+Voland, to whom he was constant for the rest of her life. His letters to
+her are among the most graphic of all the pictures that we have of the
+daily life of the philosophic circle in Paris. An interesting contrast
+may be made between the Bohemianism of the famous English literary set
+who supped at the Turk's Head with the Tory Johnson and the Conservative
+Burke for their oracles, and the Bohemianism of the French set who about
+the same time dined once a week at the baron D'Holbach's, to listen to
+the wild sallies and the inspiring declamations of Diderot. For Diderot
+was not a great writer; he stands out as a fertile, suggestive and
+daring thinker, and a prodigious and most eloquent talker.
+
+Diderot's earliest writings were of as little importance as Goldsmith's
+_Enquiry into the State of Polite Learning_ or Burke's _Abridgement of
+English History_. He earned 100 crowns by translating Stanyan's _History
+of Greece_ (1743); with two colleagues he produced a translation of
+James's _Dictionary of Medicine_ (1746-1748) and about the same date he
+published a free rendering of Shaftesbury's _Inquiry Concerning Virtue
+and Merit_ (1745), with some original notes of his own. With strange and
+characteristic versatility, he turned from ethical speculation to the
+composition of a volume of stories, the _Bijoux indiscrets_ (1748),
+gross without liveliness, and impure without wit. In later years he
+repented of this shameless work, just as Boccaccio is said in the day of
+his grey hairs to have thought of the sprightliness of the _Decameron_
+with strong remorse. From tales Diderot went back to the more congenial
+region of philosophy. Between the morning of Good Friday and the evening
+of Easter Monday he wrote the _Pensées philosophiques_ (1746), and he
+presently added to this a short complementary essay on the sufficiency
+of natural religion. The gist of these performances is to press the
+ordinary rationalistic objections to a supernatural revelation; but
+though Diderot did not at this time pass out into the wilderness beyond
+natural religion, yet there are signs that he accepted that less as a
+positive doctrine, resting on grounds of its own, than as a convenient
+point of attack against Christianity. In 1747 he wrote the _Promenade du
+sceptique_, a rather poor allegory--pointing first to the extravagances
+of Catholicism; second, to the vanity of the pleasures of that world
+which is the rival of the church; and third, to the desperate and
+unfathomable uncertainty of the philosophy which professes to be so high
+above both church and world.
+
+Diderot's next piece was what first introduced him to the world as an
+original thinker, his famous _Lettre sur les aveugles_ (1749). The
+immediate object of this short but pithy writing was to show the
+dependence of men's ideas on their five senses. It considers the case of
+the intellect deprived of the aid of one of the senses; and in a second
+piece, published afterwards, Diderot considered the case of a similar
+deprivation in the deaf and dumb. The _Lettre sur les sourds et muets_,
+however, is substantially a digressive examination of some points in
+aesthetics. The philosophic significance of the two essays is in the
+advance they make towards the principle of Relativity. But what
+interested the militant philosophers of that day was an episodic
+application of the principle of relativity to the master-conception of
+God. What makes the _Lettre sur les aveugles_ interesting is its
+presentation, in a distinct though undigested form, of the modern theory
+of variability, and of survival by superior adaptation. It is worth
+noticing, too, as an illustration of the comprehensive freedom with
+which Diderot felt his way round any subject that he approached, that in
+this theoretic essay he suggests the possibility of teaching the blind
+to read through the sense of touch. If the _Lettre sur les aveugles_
+introduced Diderot into the worshipful company of the philosophers, it
+also introduced him to the penalties of philosophy. His speculation was
+too hardy for the authorities, and he was thrown into the prison of
+Vincennes. Here he remained for three months; then he was released, to
+enter upon the gigantic undertaking of his life.
+
+The bookseller Lebreton had applied to him with a project for the
+publication of a translation into French of Ephraim Chambers's
+_Cyclopaedia_, undertaken in the first instance by an Englishman, John
+Mills, and a German, Gottfried Sellius (for particulars see
+ENCYCLOPAEDIA). Diderot accepted the proposal, but in his busy and
+pregnant intelligence the scheme became transformed. Instead of a mere
+reproduction of Chambers, he persuaded the bookseller to enter upon a
+new work, which should collect under one roof all the active writers,
+all the new ideas, all the new knowledge, that were then moving the
+cultivated class to its depths, but still were comparatively ineffectual
+by reason of their dispersion. His enthusiasm infected the publishers;
+they collected a sufficient capital for a vaster enterprise than they
+had at first planned; D'Alembert was persuaded to become Diderot's
+colleague; the requisite permission was procured from the government; in
+1750 an elaborate prospectus announced the project to a delighted
+public; and in 1751 the first volume was given to the world. The last of
+the letterpress was issued in 1765, but it was 1772 before the
+subscribers received the final volumes of the plates. These twenty years
+were to Diderot years not merely of incessant drudgery, but of harassing
+persecution, of sufferings from the cabals of enemies, and of injury
+from the desertion of friends. The ecclesiastical party detested the
+_Encyclopaedia_, in which they saw a rising stronghold for their
+philosophic enemies. By 1757 they could endure the sight no longer. The
+subscribers had grown from 2000 to 4000, and this was a right measure of
+the growth of the work in popular influence and power. To any one who
+turns over the pages of these redoubtable volumes now, it seems
+surprising that their doctrines should have stirred such portentous
+alarm. There is no atheism, no overt attack on any of the cardinal
+mysteries of the faith, no direct denunciation even of the notorious
+abuses of the church. Yet we feel that the atmosphere of the book may
+well have been displeasing to authorities who had not yet learnt to
+encounter the modern spirit on equal terms. The _Encyclopaedia_ takes
+for granted the justice of religious tolerance and speculative freedom.
+It asserts in distinct tones the democratic doctrine that it is the
+common people in a nation whose lot ought to be the main concern of the
+nation's government. From beginning to end it is one unbroken process of
+exaltation of scientific knowledge on the one hand, and pacific industry
+on the other. All these things were odious to the old governing classes
+of France; their spirit was absolutist, ecclesiastical and military.
+Perhaps the most alarming thought of all was the current belief that the
+_Encyclopaedia_ was the work of an organized band of conspirators
+against society, and that a pestilent doctrine was now made truly
+formidable by the confederation of its preachers into an open league.
+When the seventh volume appeared, it contained an article on "Geneva,"
+written by D'Alembert. The writer contrived a panegyric on the pastors
+of Geneva, of which every word was a stinging reproach to the abbés and
+prelates of Versailles. At the same moment Helvétius's book, _L'Esprit_,
+appeared, and gave a still more profound and, let us add, a more
+reasonable shock to the ecclesiastical party. Authority could brook no
+more, and in 1759 the _Encyclopaedia_ was formally suppressed.
+
+The decree, however, did not arrest the continuance of the work. The
+connivance of the authorities at the breach of their own official orders
+was common in those times of distracted government. The work went on,
+but with its difficulties increased by the necessity of being
+clandestine. And a worse thing than troublesome interference by the
+police now befell Diderot. D'Alembert, wearied of shifts and
+indignities, withdrew from the enterprise. Other powerful colleagues,
+Turgot among them, declined to contribute further to a book which had
+acquired an evil fame. Diderot was left to bring the task to an end as
+he best could. For seven years he laboured like a slave at the oar. He
+wrote several hundred articles, some of them very slight, but many of
+them most laborious, comprehensive and ample. He wore out his eyesight
+in correcting proofs, and he wearied his soul in bringing the manuscript
+of less competent contributors into decent shape. He spent his days in
+the workshops, mastering the processes of manufactures, and his nights
+in reproducing on paper what he had learnt during the day. And he was
+incessantly harassed all the time by alarms of a descent from the
+police. At the last moment, when his immense work was just drawing to an
+end, he encountered one last and crowning mortification: he discovered
+that the bookseller, fearing the displeasure of the government, had
+struck out from the proof sheets, after they had left Diderot's hands,
+all passages that he chose to think too hardy. The monument to which
+Diderot had given the labour of twenty long and oppressive years was
+irreparably mutilated and defaced. It is calculated that the average
+annual salary received by Diderot for his share in the _Encyclopaedia_
+was about £120 sterling. "And then to think," said Voltaire, "that an
+army contractor makes £800 in a day!"
+
+Although the _Encyclopaedia_ was Diderot's monumental work, he is the
+author of a shower of dispersed pieces that sowed nearly every field of
+intellectual interest with new and fruitful ideas. We find no
+masterpiece, but only thoughts for masterpieces; no creation, but a
+criticism with the quality to inspire and direct creation. He wrote
+plays--_Le Fils naturel_ (1757) and _Le Père de famille_ (1758)--and
+they are very insipid performances in the sentimental vein. But he
+accompanied them by essays on dramatic poetry, including especially the
+_Paradoxe sur le comédien_, in which he announced the principles of a
+new drama,--the serious, domestic, bourgeois drama of real life, in
+opposition to the stilted conventions of the classic French stage. It
+was Diderot's lessons and example that gave a decisive bias to the
+dramatic taste of Lessing, whose plays, and his _Hamburgische
+Dramaturgie_ (1768), mark so important an epoch in the history of the
+modern theatre. In the pictorial art, Diderot's criticisms are no less
+rich, fertile and wide in their ideas. His article on "Beauty" in the
+_Encyclopaedia_ shows that he had mastered and passed beyond the
+metaphysical theories on the subject, and the _Essai sur la peinture_
+was justly described by Goethe, who thought it worth translating, as "a
+magnificent work, which speaks even more helpfully to the poet than to
+the painter, though to the painter too it is as a blazing torch."
+Diderot's most intimate friend was Grimm, one of the conspicuous figures
+of the philosophic body. Grimm wrote news-letters to various high
+personages in Germany, reporting what was going on in the world of art
+and literature in Paris, then without a rival as the capital of the
+intellectual activity of Europe. Diderot helped his friend at one time
+and another between 1759 and 1779, by writing for him an account of the
+annual exhibitions of paintings. These _Salons_ are among the most
+readable of all pieces of art criticism. They have a freshness, a
+reality, a life, which take their readers into a different world from
+the dry and conceited pedantries of the ordinary virtuoso. As has been
+said by Sainte-Beuve, they initiated the French into a new sentiment,
+and introduced people to the mystery and purport of colour by ideas.
+"Before Diderot," Madame Necker said, "I had never seen anything in
+pictures except dull and lifeless colours; it was his imagination that
+gave them relief and life, and it is almost a new sense for which I am
+indebted to his genius."
+
+Greuze was Diderot's favourite among contemporary artists, and it is
+easy to see why. Greuze's most characteristic pictures were the
+rendering in colour of the same sentiment of domestic virtue and the
+pathos of common life, which Diderot attempted with inferior success to
+represent upon the stage. For Diderot was above all things interested in
+the life of men,--not the abstract life of the race, but the incidents
+of individual character, the fortunes of a particular family, the
+relations of real and concrete motives in this or that special case. He
+delighted with the enthusiasm of a born casuist in curious puzzles of
+right and wrong, and in devising a conflict between the generalities of
+ethics and the conditions of an ingeniously contrived practical dilemma.
+Mostly his interest expressed itself in didactic and sympathetic form;
+in two, however, of the most remarkable of all his pieces, it is not
+sympathetic, but ironical. _Jacques le fataliste_ (written in 1773, but
+not published until 1796) is in manner an imitation of _Tristram Shandy_
+and _The Sentimental Journey_. Few modern readers will find in it any
+true diversion. In spite of some excellent criticisms dispersed here and
+there, and in spite of one or two stories that are not without a certain
+effective realism, it must as a whole be pronounced savourless, forced,
+and as leaving unmoved those springs of laughter and of tears which are
+the common fountain of humour. _Le Neveu de Rameau_ is a far superior
+performance. If there were any inevitable compulsion to name a
+masterpiece for Diderot, one must select this singular "farce-tragedy."
+Its intention has been matter of dispute; whether it was designed to be
+merely a satire on contemporary manners, or a reduction of the theory of
+self-interest to an absurdity, or the application of an ironical
+clincher to the ethics of ordinary convention, or a mere setting for a
+discussion about music, or a vigorous dramatic sketch of a parasite and
+a human original. There is no dispute as to its curious literary
+flavour, its mixed qualities of pungency, bitterness, pity and, in
+places, unflinching shamelessness. Goethe's translation (1805) was the
+first introduction of _Le Neveu de Rameau_ to the European public. After
+executing it, he gave back the original French manuscript to Schiller,
+from whom he had it. No authentic French copy of it appeared until the
+writer had been nearly forty years in his grave (1823).
+
+It would take several pages merely to contain the list of Diderot's
+miscellaneous pieces, from an infinitely graceful trifle like the
+_Regrets sur ma vieille robe de chambre_ up to _Le Rêve de D'Alembert_,
+where he plunges into the depths of the controversy as to the ultimate
+constitution of matter and the meaning of life. It is a mistake to set
+down Diderot for a coherent and systematic materialist. We ought to look
+upon him "as a philosopher in whom all the contradictions of the time
+struggle with one another" (Rosenkranz). That is to say, he is critical
+and not dogmatic. There is no unity in Diderot, as there was in Voltaire
+or in Rousseau. Just as in cases of conduct he loves to make new ethical
+assumptions and argue them out as a professional sophist might have
+done, so in the speculative problems as to the organization of matter,
+the origin of life, the compatibility between physiological machinery
+and free will, he takes a certain standpoint, and follows it out more or
+less digressively to its consequences. He seizes a hypothesis and works
+it to its end, and this made him the inspirer in others of materialist
+doctrines which they held more definitely than he did. Just as Diderot
+could not attain to the concentration, the positiveness, the finality of
+aim needed for a masterpiece of literature, so he could not attain to
+those qualities in the way of dogma and system. Yet he drew at last to
+the conclusions of materialism, and contributed many of its most
+declamatory pages to the _Système de la nature_ of his friend
+D'Holbach,--the very Bible of atheism, as some one styled it. All that
+he saw, if we reduce his opinions to formulae, was motion in space:
+"attraction and repulsion, the only truth." If matter produces life by
+spontaneous generation, and if man has no alternative but to obey the
+compulsion of nature, what remains for God to do?
+
+In proportion as these conclusions deepened in him, the more did
+Diderot turn for the hope of the race to virtue; in other words, to such
+a regulation of conduct and motive as shall make us tender, pitiful,
+simple, contented. Hence his one great literary passion, his enthusiasm
+for Richardson, the English novelist. Hence, also, his deepening
+aversion for the political system of France, which makes the realization
+of a natural and contented domestic life so hard. Diderot had almost as
+much to say against society as even Rousseau himself. The difference
+between them was that Rousseau was a fervent theist. The atheism of the
+Holbachians, as he called Diderot's group, was intolerable to him; and
+this feeling, aided by certain private perversities of humour, led to a
+breach of what had once been an intimate friendship between Rousseau and
+Diderot (1757). Diderot was still alive when Rousseau's _Confessions_
+appeared, and he was so exasperated by Rousseau's stories about Grimm,
+then and always Diderot's intimate, that in 1782 he transformed a life
+of Seneca, that he had written four years earlier, into an _Essai sur
+les règnes de Claude et de Néron_ (1778-1782), which is much less an
+account of Seneca than a vindication of Diderot and Grimm, and is one of
+the most rambling and inept productions in literature. As for the merits
+of the old quarrel between Rousseau and Diderot, we may agree with the
+latter, that too many sensible people would be in the wrong if Jean
+Jacques was in the right.
+
+Varied and incessant as was Diderot's mental activity, it was not of a
+kind to bring him riches. He secured none of the posts that were
+occasionally given to needy men of letters; he could not even obtain
+that bare official recognition of merit which was implied by being
+chosen a member of the Academy. The time came for him to provide a dower
+for his daughter, and he saw no other alternative than to sell his
+library. When the empress Catherine of Russia heard of his straits, she
+commissioned an agent in Paris to buy the library at a price equal to
+about £1000 of English money, and then handsomely requested the
+philosopher to retain the books in Paris until she required them, and to
+constitute himself her librarian, with a yearly salary. In 1773 Diderot
+started on an expedition to thank his imperial benefactress in person,
+and he passed some months at St Petersburg. The empress received him
+cordially. The strange pair passed their afternoons in disputes on a
+thousand points of high philosophy, and they debated with a vivacity and
+freedom not usual in courts. "_Fi, donc,_" said Catherine one day, when
+Diderot hinted that he argued with her at a disadvantage, "_is there any
+difference among men?_" Diderot returned home in 1774. Ten years
+remained to him, and he spent them in the industrious acquisition of new
+knowledge, in the composition of a host of fragmentary pieces, some of
+them mentioned above, and in luminous declamations with his friends. All
+accounts agree that Diderot was seen at his best in conversation. "He
+who only knows Diderot in his writings," says Marmontel, "does not know
+him at all. When he grew animated in talk, and allowed his thoughts to
+flow in all their abundance, then he became truly ravishing. In his
+writings he had not the art of ensemble; the first operation which
+orders and places everything was too slow and too painful to him."
+Diderot himself was conscious of the want of literary merit in his
+pieces. In truth he set no high value on what he had done. It is
+doubtful whether he was ever alive to the waste that circumstance and
+temperament together made of an intelligence from which, if it had been
+free to work systematically, the world of thought had so much to hope.
+He was one of those simple, disinterested and intellectually sterling
+workers to whom their own personality is as nothing in presence of the
+vast subjects that engage the thoughts of their lives. He wrote what he
+found to write, and left the piece, as Carlyle has said, "on the waste
+of accident, with an ostrich-like indifference." When he heard one day
+that a collected edition of his works was in the press at Amsterdam, he
+greeted the news with "peals of laughter," so well did he know the haste
+and the little heed with which those works had been dashed off.
+
+Diderot died on the 30th of July 1784, six years after Voltaire and
+Rousseau, one year after his old colleague D'Alembert, and five years
+before D'Holbach, his host and intimate for a lifetime. Notwithstanding
+Diderot's peals of laughter at the thought, an elaborate and exhaustive
+collection of his writings in twenty stout volumes, edited by MM.
+Assézat and Tourneux, was completed in 1875-1877.
+
+  AUTHORITIES.--Studies on Diderot by Scherer (1880); by E. Faguet
+  (1890); by Sainte-Beuve in the _Causeries du lundi_; by F. Brunetière
+  in the _Études critiques_, 2nd series, may be consulted. In English,
+  Diderot has been the subject of a biography by John Morley [Viscount
+  Morley of Blackburn] (1878). See also Karl Rosenkranz, _Diderots Leben
+  und Werke_ (1866). For a discussion of the authenticity of the
+  posthumous works of Diderot see R. Dominic in the _Revue des deux
+  mondes_ (October 15, 1902).     (J. Mo.)
+
+
+
+
+DIDIUS SALVIUS JULIANUS, MARCUS, Roman emperor for two months (March
+28-June 2) during the year A.D. 193. He was the grandson of the famous
+jurist Salvius Julianus (under Hadrian and the Antonines), and the son
+of a distinguished general, who might have ascended the throne after the
+death of Antoninus Pius, had not his loyalty to the ruling house
+prevented him. Didius filled several civil and military offices with
+distinguished success, but subsequently abandoned himself to
+dissipation. On the death of Pertinax, the praetorian guards offered the
+throne to the highest bidder. Flavius Sulpicianus, the father-in-law of
+Pertinax and praefect of the city, had already made an offer; Didius,
+urged on by the members of his family, his freedmen and parasites,
+hurried to the praetorian camp to contend for the prize. He and
+Sulpicianus bid against each other, and finally the throne was knocked
+down to Didius. The senate and nobles professed their loyalty; but the
+people made no attempt to conceal their indignation at this insult to
+the state, and the armies of Britain, Syria and Illyricum broke out into
+open revolt. Septimius Severus, the commander of the Pannonian legions,
+was declared emperor and hastened by forced marches to Italy. Didius,
+abandoned by the praetorians, was condemned and executed by order of the
+senate, which at once acknowledged Severus.
+
+  AUTHORITIES.--Dio Cassius lxxiii. 11-17, who was actually in Rome at
+  the time; Aelius Spartianus, _Didius Julianus_; Julius Capitolinus,
+  _Pertinax_; Herodian ii.; Aurelius Victor, _De Caesaribus_, 19;
+  Zosimus i. 7; Gibbon, _Decline and Fall_, chap. 5.
+
+
+
+
+DIDO, or ELISSA, the reputed founder of Carthage (q.v.), in Africa,
+daughter of the Tyrian king Metten (Mutto, Methres, Belus), wife of
+Acerbas (more correctly Sicharbas; Sychaeus in Virgil), a priest of
+Hercules. Her husband having been slain by her brother Pygmalion, Dido
+fled to Cyprus, and thence to the coast of Africa, where she purchased
+from a local chieftain Iarbas a piece of land on which she built
+Carthage. The city soon began to prosper and Iarbas sought Dido's hand
+in marriage, threatening her with war in case of refusal. To escape from
+him, Dido constructed a funeral pile, on which she stabbed herself
+before the people (Justin xviii. 4-7). Virgil, in defiance of the
+usually accepted chronology, makes Dido a contemporary of Aeneas, with
+whom she fell in love after his landing in Africa, and attributes her
+suicide to her abandonment by him at the command of Jupiter (_Aeneid_,
+iv.). Dido was worshipped at Carthage as a divinity under the name of
+Caelestis, the Roman counterpart of Tanit, the tutelary goddess of
+Carthage. According to Timaeus, the oldest authority for the story, her
+name was Theiosso, in Phoenician Helissa, and she was called Dido from
+her wanderings, Dido being the Phoenician equivalent of [Greek:
+planêtis] (_Etymologicum Magnum_, s.v.); some modern scholars, however,
+translate the name by "beloved." Timaeus makes no mention of Aeneas, who
+seems to have been introduced by Naevius in his _Bellum Poenicum_,
+followed by Ennius in his _Annales_.
+
+  For the variations of the legend in earlier and later Latin authors,
+  see O. Rossbach in Pauly-Wissowa's _Realencyclopädie_, v. pt. 1
+  (1905); O. Meltzer's _Geschichte der Karthager_, i. (1879), and his
+  article in Roscher's _Lexikon der Mythologie_.
+
+
+
+
+DIDON, HENRI (1840-1900), French Dominican, was born at Trouvet, Isère,
+on the 17th of March 1840. He joined the Dominicans, under the influence
+of Lacordaire, in 1858, and completed his theological studies at the
+Minerva convent at Rome. The influence of Lacordaire was shown in the
+zeal displayed by Didon in favour of a reconciliation between philosophy
+and science. In 1871 his fame had so much grown that he was chosen to
+deliver the funeral oration over the murdered archbishop of Paris,
+Monseigneur G. Darboy. He also delivered some discourses at the church
+of St Jean de Beauvais in Paris on the relations between science and
+religion; but his utterances, especially on the question of divorce,
+were deemed suspicious by his superiors, and his intimacy with Claude
+Bernard the physiologist was disapproved. He was interdicted from
+preaching and sent into retirement at the convent of Corbara in Corsica.
+After eighteen months he emerged, and travelled in Germany, publishing
+an interesting work upon that country, entitled _Les Allemands_ (English
+translation by R. Ledos de Beaufort, London, 1884). On his return to
+France in 1890 he produced his best known work, _Jésus-Christ_ (2 vols.,
+Paris), for which he had qualified himself by travel in the Holy Land.
+In the same year he became director of the Collège Albert-le-Grand at
+Arcueil, and founded three auxiliary institutions, École Lacordaire,
+École Laplace and École St Dominique. He wrote, in addition, several
+works on educational questions, and augmented his fame as an eloquent
+preacher by discourses preached during Lent and Advent. He died at
+Toulouse on the 13th of March 1900.
+
+  See the biographies by J. de Romano (1891), and A. de Coulanges
+  (Paris, 1900); and especially the work of Stanislas Reynaud, entitled
+  _Le Père Didon, sa vie et son oeuvre_ (Paris, 1904).
+
+
+
+
+DIDOT, the name of a family of learned French printers and publishers.
+FRANÇOIS DIDOT (1689-1757), founder of the family, was born at Paris. He
+began business as a bookseller and printer in 1713, and among his
+undertakings was a collection of the travels of his friend the Abbé
+Prévost, in twenty volumes (1747). It was remarkable for its
+typographical perfection, and was adorned with many engravings and maps.
+FRANÇOIS AMBROISE DIDOT (1730-1804), son of François, made important
+improvements in type-founding, and was the first to attempt printing on
+vellum paper. Among the works which he published was the famous
+collection of French classics prepared by order of Louis XVI. for the
+education of the Dauphin, and the folio edition of _L'Art de vérifier
+les dates_. PIERRE FRANÇOIS DIDOT (1732-1795), his brother, devoted much
+attention to the art of type-founding and to paper-making. Among the
+works which issued from his press was an edition in folio of the
+_Imitatio Christi_ (1788). HENRI DIDOT (1765-1852), son of Pierre
+François, is celebrated for his "microscopic" editions of various
+standard works, for which he engraved the type when nearly seventy years
+of age. He was also the engraver of the _assignats_ issued by the
+Constituent and Legislative Assemblies and the Convention. DIDOT
+SAINT-LÉGER, second son of Pierre François, was the inventor of the
+paper-making machine known in England as the Didot machine. PIERRE DIDOT
+(1760-1853), eldest son of François Ambroise, is celebrated as the
+publisher of the beautiful "Louvre" editions of Virgil, Horace and
+Racine. The Racine, in three volumes folio, was pronounced in 1801 to be
+"the most perfect typographical production of all ages." FIRMIN DIDOT
+(1764-1836), his brother, second son of François Ambroise, sustained the
+reputation of the family both as printer and type-founder. He revived
+(if he did not invent--a distinction which in order of time belongs to
+William Ged) the process of stereotyping, and coined its name, and he
+first used the process in his edition of Callet's _Tables of Logarithms_
+(1795), in which he secured an accuracy till then unattainable. He
+published stereotyped editions of French, English and Italian classics
+at a very low price. He was the author of two tragedies--_La Reine de
+Portugal_ and _La Mort d'Annibal_; and he wrote metrical translations
+from Virgil, Tyrtaeus and Theocritus. AMBROISE FIRMIN DIDOT (1790-1876)
+was his eldest son. After receiving a classical education, he spent
+three years in Greece and in the East; and on the retirement of his
+father in 1827 he undertook, in conjunction with his brother Hyacinthe,
+the direction of the publishing business. Their greatest undertaking was
+a new edition of the _Thesaurus Graecae linguae_ of Henri Estienne,
+under the editorial care of the brothers Dindorf and M. Hase (9 vols.,
+1855-1859). Among the numerous important works published by the
+brothers, the 200 volumes forming the _Bibliothèque des auteurs grecs_,
+_Bibliothèque latine_, and _Bibliothèque française_ deserve special
+mention. Ambroise Firmin Didot was the first to propose (1823) a
+subscription in favour of the Greeks, then in insurrection against
+Turkish tyranny. Besides a translation of Thucydides (1833), he wrote
+the articles "Estienne" in the _Nouvelle Biographie générale_, and
+"Typographie" in the _Ency. mod._, as well as _Observations sur
+l'orthographie française_ (1867), &c. In 1875 he published a very
+learned and elaborate monograph on Aldus Manutius. His collection of
+MSS., the richest in France, was said to have been worth, at the time of
+his death, not less than 2,000,000 francs.
+
+
+
+
+DIDRON, ADOLPHE NAPOLÉON (1806-1867), French archaeologist, was born at
+Hautvillers, in the department of Marne, on the 13th of March 1806. At
+first a student of law, he began in 1830, by the advice of Victor Hugo,
+a study of the Christian archaeology of the middle ages. After visiting
+and examining the principal churches, first of Normandy, then of central
+and southern France, he was on his return appointed by Guizot secretary
+to the Historical Committee of Arts and Monuments (1835); and in the
+following years he delivered several courses of lectures on Christian
+iconography at the Bibliothèque Royale. In 1839 he visited Greece for
+the purpose of examining the art of the Eastern Church, both in its
+buildings and its manuscripts. In 1844 he originated the _Annales
+archéologiques_, a periodical devoted to his favourite subject, which he
+edited until his death. In 1845 he established at Paris a special
+archaeological library, and at the same time a manufactory of painted
+glass. In the same year he was admitted to the Legion of Honour. His
+most important work is the _Iconographie chrétienne_, of which, however,
+the first portion only, _Histoire de Dieu_ (1843), was published. It was
+translated into English by E. J. Millington. Among his other works may
+be mentioned the _Manuel d'iconographie chrétienne grecque et latine_
+(1845), the _Iconographie des chapiteaux du palais ducal de Venise_
+(1857), and the _Manuel des objets de bronze et d'orfèvrerie_ (1859). He
+died on the 13th of November 1867.
+
+
+
+
+DIDYMI, or DIDYMA (mod. _Hieronta_), an ancient sanctuary of Apollo in
+Asia Minor situated in the territory of Miletus, from which it was
+distant about 10 m. S. and on the promontory Poseideion. It was
+sometimes called _Branchidae_ from the name of its priestly caste which
+claimed descent from Branchus, a youth beloved by Apollo. As the seat of
+a famous oracle, the original temple attracted offerings from Pharaoh
+Necho (in whose army there was a contingent of Milesian mercenaries),
+and the Lydian Croesus, and was plundered by Darius of Persia. Xerxes
+finally sacked and burnt it (481 B.C.) and exiled the Branchidae to the
+far north-east of his empire. This exile was believed to be voluntary,
+the priests having betrayed their treasures to the Persian; and on this
+belief Alexander the Great acted 150 years later, when, finding the
+descendants of the Branchidae established in a city beyond the Oxus, he
+ordered them to be exterminated for the sin of their fathers (328). The
+celebrated cult-statue of Apollo by Canachus, familiar to us from
+reproductions on Milesian coins, was also carried to Persia, there to
+remain till restored by Seleucus I. in 295, and the oracle ceased to
+speak for a century and a half. The Milesians were not able to undertake
+the rebuilding till about 332 B.C., when the oracle revived at the
+bidding of Alexander. The work proved too costly, and despite a special
+effort made by the Asian province nearly 400 years later, at the bidding
+of the emperor Caligula, the structure was never quite finished: but
+even as it was, Strabo ranked the Didymeum the greatest of Greek temples
+and Pliny placed it among the four most splendid and second only to the
+Artemisium at Ephesus. In point of fact it was a little smaller than the
+Samian Heraeum and the temple of Cybele at Sardis, and almost exactly
+the same size as the Artemisium. The area covered by the platform
+measures roughly 360 × 160 ft.
+
+When Cyriac of Ancona visited the spot in 1446, it seems that the temple
+was still standing in great part, although the _cella_ had been
+converted into a fortress by the Byzantines: but when the next European
+visitor, the Englishman Dr Pickering, arrived in 1673, it had collapsed.
+It is conjectured that the cause was the great earthquake of 1493. The
+Society of Dilettanti sent two expeditions to explore the ruins, the
+first in 1764 under Richard Chandler, the second in 1812 under Sir Wm.
+Gell; and the French "Rothschild Expedition" of 1873 under MM. O. Rayet
+and A. Thomas sent a certain amount of architectural sculpture to the
+Louvre. But no excavation was attempted till MM. E. Pontremoli and B.
+Haussoullier were sent out by the French Schools of Rome and Athens in
+1895. They cleared the western façade and the _prodomos_, and discovered
+inscriptions giving information about other parts which they left still
+buried. Finally the site was purchased by, and the French rights were
+ceded to, Dr Th. Wiegand, the German explorer of Miletus, who in 1905
+began a thorough clearance of what is incomparably the finest temple
+ruin in Asia Minor.
+
+The temple was a decastyle peripteral structure of the Ionic order,
+standing on seven steps and possessing double rows of outer columns 60
+ft. high, twenty-one in each row on the flanks. It is remarkable not
+only for its great size, but (_inter alia_) for (1) the rich ornament of
+its column bases, which show great variety of design; (2) its various
+developments of the Ionic capital, e.g. heads of gods, probably of
+Pergamene art, spring from the "eyes" of the volutes with bulls' heads
+between them; (3) the massive building two storeys high at least, which
+served below for _prodomos_, and above for a dispensary of oracles
+([Greek: chrêsmographia] mentioned in the inscriptions) and a treasury;
+two flights of stairs called "labyrinths" in the inscriptions, led up to
+these chambers; (4) the pylon and staircase at the west; (5) the frieze
+of Medusa heads and foliage. Two outer columns are still erect on the
+north-east flank, carrying their entablature, and one of the inner order
+stands on the south-west. The fact that the temple was never finished is
+evident from the state in which some bases still remain at the west.
+There were probably no pedimental sculptures. A sacred way led from the
+temple to the sea at Panormus, which was flanked with rows of archaic
+statues, ten of which were excavated and sent to the British Museum in
+1858 by C. T. Newton. Fragments of architectural monuments, which once
+adorned this road, have also been found. Modern Hieronta is a large and
+growing Greek village, the only settlement within a radius of several
+miles. Its harbour is Kovella, distant about 2½ m., and on the N. of the
+promontory.
+
+  See Dilettanti Society, _Ionian Antiquities_, ii. (1821); C. T.
+  Newton, _Hist. of Discoveries_, &c. (1862) and _Travels in the
+  Levant_, ii. (1865); O. Rayet and A. Thomas, _Milet et le Golfe
+  Latmique_ (1877); E. Pontremoli and B. Haussoullier, _Didymes_ (1904).
+       (D. G. H.)
+
+
+
+
+DIDYMIUM (from the Gr. [Greek: didymos], twin), the name given to the
+supposed element isolated by C. G. Mosander from cerite (1839-1841). In
+1879, however, Lecoq de Boisbaudran showed that Mosander's "didymium"
+contained samarium; while the residual "didymium," after removal of
+samarium, was split by Auer v. Welsbach (_Monats. f. Chemie_, 1885, 6,
+477) into two components (known respectively as neodymium and
+praseodymium) by repeated fractional crystallization of the double
+nitrate of ammonium and didymium in nitric acid. _Neodymium_ (Nd) forms
+the chief portion of the old "didymium." Its salts are reddish violet in
+colour, and give a characteristic absorption spectrum. It forms oxides
+of composition Nd2O3 and Nd2O5, the latter being obtained by ignition of
+the nitrate (B. Brauner). The atomic weight of neodymium is 143.6 (B.
+Brauner, _Proc. Chem. Soc._, 1897-1898, p. 70). _Praseodymium_ (Pr)
+forms oxides of composition Pr2O3, Pr2O5, xH2O (B. Brauner), and Pr4O7.
+The peroxide, Pr4O7, forms a dark brown powder, and is obtained by
+ignition of the oxalate or nitrate. The sesquioxide, Pr2O3, is obtained
+as a greenish white mass by the reduction of the peroxide. The salts of
+praseodymium are green in colour, and give a characteristic spark
+spectrum. The atomic weight of praseodymium is 140.5.
+
+
+
+
+DIDYMUS (?309-?394), surnamed "the Blind," ecclesiastical writer of
+Alexandria, was born about the year 309. Although he became blind at the
+age of four, before he had learned to read; he succeeded in mastering
+the whole circle of the sciences then known; and on entering the service
+of the Church he was placed at the head of the Catechetical school in
+Alexandria, where he lived and worked till almost the close of the
+century. Among his pupils were Jerome and Rufinus. He was a loyal
+follower of Origen, though stoutly opposed to Arian and Macedonian
+teaching. Such of his writings as survive show a remarkable knowledge of
+scripture, and have distinct value as theological literature. Among them
+are the _De Trinitate_, _De Spiritu Sancto_ (Jerome's Latin
+translation), _Adversus Manichaeos_, and notes and expositions of
+various books, especially the Psalms and the Catholic Epistles.
+
+  See Migne, _Patrol. Graec._ xxxix.; O. Bardenhewer, _Patrologie_, pp.
+  290-293 (Freiburg, 1894).
+
+
+
+
+DIDYMUS CHALCENTERUS (c. 63 B.C.-A.D. 10), Greek scholar and grammarian,
+flourished in the time of Cicero and Augustus. His surname (Gr. [Greek:
+Chalkenteros], brazen-bowelled) came from his indefatigable industry; he
+was said to have written so many books (more than 3500) that he was
+unable to recollect their names ([Greek: bibliolathas]). He lived and
+taught in Alexandria and Rome, where he became the friend of Varro. He
+is chiefly important as having introduced Alexandrian learning to the
+Romans. He was a follower of the school of Aristarchus, upon whose
+recension of Homer he wrote a treatise, fragments of which have been
+preserved in the Venetian Scholia. He also wrote commentaries on many
+other Greek poets and prose authors. In his work on the lyric poets he
+treated of the various classes of poetry and their chief
+representatives, and his lists of words and phrases (used in tragedy and
+comedy and by orators and historians), of words of doubtful meaning, and
+of corrupt expressions, furnished the later grammarians with valuable
+material. His activity extended to all kinds of subjects: grammar
+(orthography, inflexions), proverbs, wonderful stories, the law-tablets
+([Greek: axones]) of Solon, stones, and different kinds of wood. His
+polemic against Cicero's _De republica_ (Ammianus Marcellinus xxii. 16)
+provoked a reply from Suetonius. In spite of his stupendous industry,
+Didymus was little more than a compiler, of little critical judgment and
+doubtful accuracy, but he deserves recognition for having incorporated
+in his numerous writings the works of earlier critics and commentators.
+
+  See M. W. Schmidt, _De Didymo Chalcentero_ (1853) and _Didymi
+  Chalcenteri fragmenta_ (1854); also F. Susemihl, _Geschichte der
+  griech. Literatur in der Alexandrinerzeit_, ii. (1891); J. E. Sandys,
+  _History of Classical Scholarship_, i. (1906).
+
+
+
+
+DIE, a town of south-eastern France, capital of an arrondissement in the
+department of Drôme, 43 m. E.S.E. of Valence on the Paris-Lyon railway.
+Pop. (1906) 3090. The town is situated in a plain enclosed by mountains
+on the right bank of the Drôme below its confluence with the Meyrosse,
+which supplies power to some of the industries. The most interesting
+structures of Die are the old cathedral, with a porch of the 11th
+century supported on granite columns from an ancient temple of Cybele;
+and the Porte St Marcel, a Roman gateway flanked by massive towers. The
+Roman remains also include the ruins of aqueducts and altars. Die is the
+seat of a sub-prefect, and of a tribunal of first instance. The
+manufactures are silk, furniture, cloth, lime and cement, and there are
+flour and saw mills. Trade is in timber, especially walnut, and in white
+wine known as _clairette de Die_. The mulberry is largely grown for the
+rearing of silkworms. Under the Romans, Die (_Dea Augusta Vocontiorum_)
+was an important colony. It was formerly the seat of a bishopric, united
+to that of Valence from 1276 to 1687 and suppressed in 1790. Previous to
+the revocation of the edict of Nantes in 1685 it had a Calvinistic
+university.
+
+
+
+
+DIE (Fr. _dé_, from Lat. _datum_, given), a word used in various senses,
+for a small cube of ivory, &c. (see DICE), for the engraved stamps used
+in coining money, &c., and various mechanical appliances in engineering.
+In architecture a "die" is the term used for the square base of a
+column, and it is applied also to the vertical face of a pedestal or
+podium.
+
+The fabrics known as "dice" take their name from the rectangular form of
+the figure. The original figures would probably be perfectly square, but
+to-day the same principle of weaving is applied, and the name dice is
+given to all figures of rectangular form. The different effects in the
+adjacent squares or rectangles are due to precisely the same reasons as
+those explained in connexion with the ground and the figure of damasks.
+The same weaves are used in both damasks and dices, but simpler weaves
+are generally employed for the commoner classes of the latter. The
+effect is, in every case, obtained by what are technically called warp
+and weft float weaves. The illustration B shows the two double damask
+weaves arranged to form a dice pattern, while A shows a similar pattern
+made from two four-thread twill weaves. C and D represent respectively
+the disposition of the threads in A and B with the first pick, and the
+solid marks represent the floats of warp. The four squares, which are
+almost as pronounced in the cloth as those of a chess-board, may be made
+of any size by repeating each weave for the amount of surface required.
+It is only in the finest cloths that the double damask weaves B are used
+for dice patterns, the single damask weaves and the twill weaves being
+employed to a greater extent. This class of pattern is largely employed
+for the production of table-cloths of lower and medium qualities. The
+term damask is also often applied to cloths of this character, and
+especially so when the figure is formed by rectangles of different
+sizes.
+
+[Illustration: A B C D]
+
+
+
+
+DIEBITSCH, HANS KARL FRIEDRICH ANTON, count von Diebitsch and Narden,
+called by the Russians Ivan Ivanovich, Count Diebich-Zabalkansky
+(1785-1831), Russian field-marshal, was born in Silesia on the 13th of
+May 1785. He was educated at the Berlin cadet school, but by the desire
+of his father, a Prussian officer who had passed into the service of
+Russia, he also did the same in 1801. He served in the campaign of 1805,
+and was wounded at Austerlitz, fought at Eylau and Friedland, and after
+Friedland was promoted captain. During the next five years of peace he
+devoted himself to the study of military science, engaging once more in
+active service in the War of 1812. He distinguished himself very greatly
+in Wittgenstein's campaign, and in particular at Polotzk (October 18 and
+19), after which combat he was raised to the rank of major-general. In
+the latter part of the campaign he served against the Prussian
+contingent of General Yorck (von Wartenburg), with whom, through
+Clausewitz, he negotiated the celebrated convention of Tauroggen,
+serving thereafter with Yorck in the early part of the War of
+Liberation. After the battle of Lützen he served in Silesia and took
+part in negotiating the secret treaty of Reichenbach. Having
+distinguished himself at the battles of Dresden and Leipzig he was
+promoted lieutenant-general. At the crisis of the campaign of 1814 he
+strongly urged the march of the allies on Paris; and after their entry
+the emperor Alexander conferred on him the order of St Alexander Nevsky.
+In 1815 he attended the congress of Vienna, and was afterwards made
+adjutant-general to the emperor, with whom, as also with his successor
+Nicholas, he had great influence. By Nicholas he was created baron, and
+later count. In 1820 he had become chief of the general staff, and in
+1825 he assisted in suppressing the St Petersburg _émeute_. His greatest
+exploits were in the Russo-Turkish War of 1828-1829, which, after a
+period of doubtful contest, was decided by Diebitsch's brilliant
+campaign of Adrianople; this won him the rank of field-marshal and the
+honorary title of Zabalkanski to commemorate his crossing of the
+Balkans. In 1830 he was appointed to command the great army destined to
+suppress the insurrection in Poland. He won the terrible battle of
+Gróchow on the 25th of February, and was again victorious at Ostrolenka
+on the 26th of May, but soon afterwards he died of cholera (or by his
+own hand) at Klecksewo near Pultusk, on the 10th of June 1831.
+
+  See Belmont (Schümberg), _Graf Diebitsch_ (Dresden, 1830); Stürmer,
+  _Der Tod des Grafen Diebitsch_ (Berlin, 1832); Bantych-Kamenski,
+  _Biographies of Russian Field-Marshals_ (in Russian, St Petersburg,
+  1841).
+
+
+
+
+DIEDENHOFEN (Fr. _Thionville_), a fortified town of Germany, in
+Alsace-Lorraine, dist. Lorraine, on the Mosel, 22 m. N. from Metz by
+rail. Pop. (1905) 6047. It is a railway junction of some consequence,
+with cultivation of vines, fruit and vegetables, brewing, tanning, &c.
+Diedenhofen is an ancient Frank town (Theudonevilla, Totonisvilla), in
+which imperial diets were held in the 8th century; was captured by Condé
+in 1643 and fortified by Vauban; capitulated to the Prussians, after a
+severe bombardment, on the 25th of November 1870.
+
+
+
+
+DIEKIRCH, a small town in the grand duchy of Luxemburg, charmingly
+situated on the banks of the Sûre. Pop. (1905) 3705. Its name is said to
+be derived from Dide or Dido, granddaughter of Odin and niece of Thor.
+The mountain at the foot of which the town lies, now called Herrenberg,
+was formerly known as Thorenberg, or Thor's mountain. On the summit of
+this rock rises a perennial stream which flows down into the town under
+the name of Bellenflesschen. Diekirch was an important Roman station,
+and in the 14th century John of Luxemburg, the blind king of Bohemia,
+fortified it, surrounding the place with a castellated wall and a ditch
+supplied by the stream mentioned. It remained more or less fortified
+until the beginning of the 19th century when the French during their
+occupation levelled the old walls, and substituted the avenues of trees
+that now encircle the town. Diekirch is the administrative centre of one
+of the three provincial divisions of the grand duchy. It is visited
+during the summer by many thousand tourists and travellers from Holland,
+Belgium and Germany.
+
+
+
+
+DIELECTRIC, in electricity, a non-conductor of electricity; it is the
+same as insulator. The "dielectric constant" of a medium is its specific
+inductive capacity, and on the electromagnetic theory of light it equals
+the square of its refractive index for light of infinite wave length
+(see ELECTROSTATICS; MAGNETO-OPTICS).
+
+
+
+
+DIELMANN, FREDERICK (1847- ), American portrait and figure painter, was
+born at Hanover, Germany, on the 25th of December 1847. He was taken to
+the United States in early childhood; studied under Diez at the Royal
+Academy at Munich; was first an illustrator, and became a distinguished
+draughtsman and painter of genre pictures. His mural decorations and
+mosaic panels for the Congressional library, Washington, are notable. He
+was elected in 1899 president of the National Academy of Design.
+
+
+
+
+DIEMEN, ANTHONY VAN (1593-1645), Dutch admiral and governor-general of
+the East Indian settlements, was born at Kuilenburg in 1593. He was
+educated in commerce, and on entering the service of the East India
+Company speedily attained high rank. In 1631 he led a Dutch fleet from
+the Indies to Holland, and in 1636 he was raised to the
+governor-generalship. He came into conflict with the Portuguese, and
+took their possessions in Ceylon and Malacca from them. He greatly
+extended the commercial relationships of the Dutch, opening up trade
+with Tong-king, China and Japan. As an administrator also he showed
+ability, and the foundation of a Latin school and several churches in
+Batavia is to be ascribed to him. Exploring expeditions were sent to
+Australia under his auspices in 1636 and 1642, and Abel Tasman named
+after him (Van Diemen's Land) the island now called Tasmania. Van Diemen
+died at Batavia on the 19th of April 1645.
+
+
+
+
+DIEPENBECK, ABRAHAM VAN (1599-1675), Flemish painter, was born at
+Herzogenbusch, and studied painting at Antwerp, where he became one of
+Rubens's "hundred pupils." But he was not one of the cleverest of
+Rubens's followers, and he succeeded, at the best, in imitating the
+style and aping the peculiarities of his master. We see this in his
+earliest pictures--a portrait dated 1629 in the Munich Pinakothek, and a
+"Distribution of Alms" of the same period in the same collection. Yet
+even at this time there were moments when Diepenbeck probably fancied
+that he might take another path. A solitary copperplate executed with
+his own hand in 1630 represents a peasant sitting under a tree holding
+the bridle of an ass, and this is a minute and finished specimen of the
+engraver's art which shows that the master might at one time have hoped
+to rival the animal draughtsmen who flourished in the schools of
+Holland. However, large commissions now poured in upon him; he was asked
+for altarpieces, subject-pieces and pagan allegories. He was tempted to
+try the profession of a glass-painter, and at last he gave up every
+other occupation for the lucrative business of a draughtsman and
+designer for engravings. Most of Diepenbeck's important canvases are in
+continental galleries. The best are the "Marriage of St Catherine" at
+Berlin and "Mary with Angels Wailing over the Dead Body of Christ" in
+the Belvedere at Vienna, the first a very fair specimen of the artist's
+skill, the second a picture of more energy and feeling than might be
+expected from one who knew more of the outer form than of the spirit of
+Rubens. Then we have the fine "Entombment" at Brunswick, and "St Francis
+Adoring the Sacrament" at the museum at Brussels, "Clelia and her Nymphs
+Flying from the Presence and Pursuit of Porsenna" in two examples at
+Berlin and Paris, and "Neptune and Amphitrite" at Dresden. In all these
+compositions the drawing and execution are after the fashion of Rubens,
+though inferior to Rubens in harmony of tone and force of contrasted
+light and shade. Occasionally a tendency may be observed to imitate the
+style of Vandyck, for whom, in respect of pictures, Diepenbeck in his
+lifetime was frequently taken. But Diepenbeck spent much less of his
+leisure on canvases than on glass-painting. Though he failed to master
+the secrets of gorgeous tinting, which were lost, apparently for ever in
+the 16th century, he was constantly employed during the best years of
+his life in that branch of his profession. In 1635 he finished forty
+scenes from the life of St Francis of Paula in the church of the Minimes
+at Antwerp. In 1644 he received payment for four windows in St Jacques
+of Antwerp, two of which are still preserved, and represent Virgins to
+whom Christ appears after the Resurrection. The windows ascribed to him
+at St Gudule of Brussels were executed from the cartoons of Theodore van
+Thulden. On the occasion of his matriculation at Antwerp in 1638-1639,
+Diepenbeck was registered in the guild of St Luke as a glass-painter. He
+resigned his membership in the Artist Club of the Violette in 1542,
+apparently because he felt hurt by a valuation then made of drawings
+furnished for copperplates to the engraver Pieter de Jode. The earliest
+record of his residence at Antwerp is that of his election to the
+brotherhood (Sodalität) "of the Bachelors" in 1634. It is probable that
+before this time he had visited Rome and London, as noted in the work of
+Houbraken. In 1636 he was made a burgess of Antwerp. He married twice,
+in 1637 and 1652. He died in December 1675, and was buried at St Jacques
+of Antwerp.
+
+
+
+
+DIEPPE, a seaport of northern France, capital of an arrondissement in
+the department of Seine-Inférieure, on the English Channel, 38 m. N. of
+Rouen, and 105 m. N.W. of Paris by the Western railway. Pop. (1906)
+22,120. It is situated at the mouth of the river Arques in a valley
+bordered on each side by steep white cliffs. The main part of the town
+lies to the west, and the fishing suburb of Le Pollet to the east of the
+river and harbour. The sea-front of Dieppe, which in summer attracts
+large numbers of visitors, consists of a pebbly beach backed by a
+handsome marine promenade. Dieppe has a modern aspect; its streets are
+wide and its houses, in most cases, are built of brick. Two squares side
+by side and immediately to the west of the outer harbour form the
+nucleus of the town, the Place Nationale, overlooked by the statue of
+Admiral A. Duquesne, and the Place St Jacques, named after the beautiful
+Gothic church which stands in its centre. The Grande Rue, the busiest
+and handsomest street, leads westward from the Place Nationale. The
+church of St Jacques was founded in the 13th century, but consists in
+large measure of later workmanship and was in some portions restored in
+the 19th century. The castle, overlooking the beach from the summit of
+the western cliff, was erected in 1435. The church of Notre-Dame de Bon
+Secours on the opposite cliff, and the church of St Remy, of the 16th
+and 17th centuries, are other noteworthy buildings. A well-equipped
+casino stands at the west end of the sea-front. The public institutions
+include the subprefecture, tribunals of first instance and commerce, a
+chamber of commerce, a communal college and a school of navigation.
+
+Dieppe has one of the safest and deepest harbours on the English
+Channel. A curved passage cut in the bed of the Arques and protected by
+an eastern and a western jetty gives access to the outer harbour, which
+communicates at the east end by a lockgate with the Bassin Duquesne and
+the Bassin Bérigny, and at the west end by the New Channel, with an
+inner tidal harbour and two other basins. Vessels drawing 20 ft. can
+enter the new docks at neap tide. A dry-dock and a gridiron are included
+among the repairing facilities of the port. The harbour railway station
+is on the north-west quay of the outer harbour alongside which the
+steamers from Newhaven lie. The distance of Dieppe from Newhaven, with
+which there has long been daily communication, is 64 m. The imports
+include silk and cotton goods, thread, oil-seeds, timber, coal and
+mineral oil; leading exports are wine, silk, woollen and cotton fabrics,
+vegetables and fruit and flint-pebbles. The average annual value of
+imports for the five years 1901-1905 was £4,916,000 (£4,301,000 for the
+years 1896-1900); the exports were valued at £9,206,000 (£7,023,000 for
+years 1896-1900). The industries comprise shipbuilding, cotton-spinning,
+steam-sawing, the manufacture of machinery, porcelain, briquettes, lace,
+and articles in ivory and bone, the production of which dates from the
+15th century. There is also a tobacco factory of some importance. The
+fishermen of Le Pollet, to whom tradition ascribes a Venetian origin,
+are among the main providers of the Parisian market. The sea-bathing
+attracts many visitors in the summer. Two miles to the north-east of the
+town is the ancient camp known as the Cité de Limes, which perhaps
+furnished the nucleus of the population of Dieppe.
+
+It is suggested on the authority of its name, that Dieppe owed its
+origin to a band of Norman adventurers, who found its "diep" or inlet
+suitable for their ships, but it was unimportant till the latter half of
+the 12th century. Its first castle was probably built in 1188 by Henry
+II. of England, and it was counted a place of some consideration when
+Philip Augustus attacked it in 1195. By Richard I. of England it was
+bestowed in 1197 on the archbishop of Rouen in return for certain
+territory in the neighbourhood of the episcopal city. In 1339 it was
+plundered by the English, but it soon recovered from the blow, and in
+spite of the opposition of the lords of Hantot managed to surround
+itself with fortifications. Its commercial activity was already great,
+and it is believed that its seamen visited the coast of Guinea in 1339,
+and founded there a Petit Dieppe in 1365. The town was occupied by the
+English from 1420 to 1435. A siege undertaken in 1442 by John Talbot,
+first earl of Shrewsbury, was raised by the dauphin, afterwards Louis
+XI., and the day of the deliverance continued for centuries to be
+celebrated by a great procession and miracle plays. In the beginning of
+the 16th century Jean Parmentier, a native of the town, made voyages to
+Brazil and Sumatra; and a little later its merchant prince, Jacques
+Ango, was able to blockade the Portuguese fleet in the Tagus. Francis I.
+began improvements which were continued under his successor. Its
+inhabitants in great number embraced the reformed religion; and they
+were among the first to acknowledge Henry IV., who fought one of his
+great battles at the neighbouring village of Arques. Few of the cities
+of France suffered more from the revocation of the edict of Nantes in
+1685; and this blow was followed in 1694 by a terrible bombardment on
+the part of the English and Dutch. The town was rebuilt after the peace
+of Ryswick, but the decrease of its population and the deterioration of
+its port prevented the restoration of its commercial prosperity. During
+the 19th century it made rapid advances, partly owing to Marie Caroline,
+duchess of Berry, who brought it into fashion as a watering-place; and
+also because the establishment of railway communication with Paris gave
+an impetus to its trade. During the Franco-German War the town was
+occupied by the Germans from December 1870 till July 1871.
+
+  See L. Vitet, _Histoire de Dieppe_ (Paris, 1844); D. Asseline, _Les
+  Antiquités et chroniques de la ville de Dieppe_, a 17th-century
+  account published at Paris in 1874.
+
+
+
+
+DIERX, LÉON (1838-   ), French poet, was born in the island of Réunion
+in 1838. He came to Paris to study at the Central School of Arts and
+Manufactures, and subsequently settled there, taking up a post in the
+education office. He became a disciple of Leconte de Lisle and one of
+the most distinguished of the Parnassians. In the death of Stéphane
+Mallarmé in 1898 he was acclaimed "prince of poets" by "les jeunes." His
+works include: _Poèmes et poésies_ (1864); _Lèvres closes_ (1867);
+_Paroles d'un vaincu_ (1871); _La Rencontre_, a dramatic scene (1875)
+and _Les Amants_ (1879). His _Poésies complètes_ (1872) were crowned by
+the French Academy. A complete edition of his works was published in 2
+vols., 1894-1896.
+
+
+
+
+DIES, CHRISTOPH ALBERT (1755-1822), German painter, was born at Hanover,
+and learned the rudiments of art in his native place. For one year he
+studied in the academy of Dusseldorf, and then he started at the age of
+twenty with thirty ducats in his pocket for Rome. There he lived a
+frugal life till 1796. Copying pictures, chiefly by Salvator Rosa, for a
+livelihood, his taste led him to draw and paint from nature in Tivoli,
+Albano and other picturesque places in the vicinity of Rome. Naples, the
+birthplace of his favourite master, he visited more than once for the
+same reasons. In this way he became a bold executant in water-colours
+and in oil, though he failed to acquire any originality of his own. Lord
+Bristol, who encouraged him as a copyist, predicted that he would be a
+second Salvator Rosa. But Dies was not of the wood which makes original
+artists. Besides other disqualifications, he had necessities which
+forced him to give up the great career of an independent painter. David,
+then composing his Horatii at Rome, wished to take him to Paris. But
+Dies had reasons for not accepting the offer. He was courting a young
+Roman whom he subsequently married. Meanwhile he had made the
+acquaintance of Volpato, for whom he executed numerous drawings, and
+this no doubt suggested the plan, which he afterwards carried out, of
+publishing, in partnership with Méchan, Reinhardt and Frauenholz, the
+series of plates known as the _Collection de vues pittoresques de
+l'Italie_, published in seventy-two sheets at Nuremberg in 1799. With so
+many irons in the fire Dies naturally lost the power of concentration.
+Other causes combined to affect his talent. In 1787 he swallowed by
+mistake three-quarters of an ounce of sugar of lead. His recovery from
+this poison was slow and incomplete. He settled at Vienna, and lived
+there on the produce of his brush as a landscape painter, and on that of
+his pencil or graver as a draughtsman and etcher. But instead of getting
+better, his condition became worse, and he even lost the use of one of
+his hands. In this condition he turned from painting to music, and spent
+his leisure hours in the pleasures of authorship. He did not long
+survive, dying at Vienna in 1822, after long years of chronic suffering.
+From two pictures now in the Belvedere gallery, and from numerous
+engraved drawings from the neighbourhood of Tivoli, we gather that Dies
+was never destined to rise above a respectable mediocrity. He followed
+Salvator Rosa's example in imitating the manner of Claude Lorraine. But
+Salvator adapted the style of Claude, whilst Dies did no more than copy
+it.
+
+
+
+
+DIEST, a small town in the province of Brabant, Belgium, situated on the
+Demer at its junction with the Bever. Pop. (1904) 8383. It lies about
+half-way between Hasselt and Louvain, and is still one of the five
+fortified places in Belgium. It contains many breweries, and is famous
+for the excellence of its beer.
+
+
+
+
+DIESTERWEG, FRIEDRICH ADOLF WILHELM (1790-1866), German educationist,
+was born at Siegen on the 29th of October 1790. Educated at Herborn and
+Tübingen universities, he took to the profession of teaching in 1811. In
+1820 he was appointed director of the new school at Mörs, where he put
+in practice the methods of Pestalozzi. In 1832 he was summoned to Berlin
+to direct the new state-schools seminary in that city. Here he proved
+himself a strong supporter of unsectarian religious teaching. In 1846 he
+established the Pestalozzi institution at Pankow, and the Pestalozzi
+societies for the support of teachers' widows and orphans. In 1850 he
+retired on a pension, but continued vigorously to advocate his
+educational views. In 1858 he was elected to the chamber of deputies as
+member for the city of Berlin, and voted with the Liberal opposition. He
+died in Berlin on the 7th of July 1866. Diesterweg was a voluminous
+writer on educational subjects, and was the author of various school
+text-books.
+
+
+
+
+DIET, a term used in two senses, (1) food or the regulation of feeding
+(see DIETARY and DIETETICS), (2) an assembly or council (Fr. _diète_;
+It. _dieta_; Low Lat. _diaeta_; Ger. _Tag_). We are here concerned only
+with this second sense. In modern usage, though in Scotland the term is
+still sometimes applied to any assembly or session, it is practically
+confined to the sense of an assembly of estates or of national or
+federal representatives. The origin of the word in this connotation is
+somewhat complicated. It is undoubtedly ultimately derived from the
+Greek [Greek: diaita] (Lat. _diaeta_), which meant "mode of life" and
+thence "prescribed mode of life," the English "diet" or "regimen." This
+was connected with the verb [Greek: diaitan], in the sense of "to rule,"
+"to regulate"; compare the office of [Greek: diaitêtês] at Athens, and
+_dieteta_, "umpire," in Late Latin. In both Greek and Latin, too, the
+word meant "a room," from which the transition to "a place of assembly"
+and so to "an assembly" would be easy. In the latter sense the word,
+however, actually occurs only in Low Latin, Du Cange (_Glossarium_,
+s.v.) deriving it from the late sense of "meal" or "feast," the Germans
+being accustomed to combine their political assemblies with feasting. It
+is clear, too, that the word _diaeta_ early became confused with Lat.
+_dies_, "day" (Ger. _Tag_), "especially a set day, a day appointed for
+public business; whence, by extension, meeting for business, an
+assembly" (Skeat). Instances of this confusion are given by Du Cange,
+e.g. _diaeta_ for _dieta_, "a day's journey" (also an obsolete sense of
+"diet" in English), and _dieta_ for "the ordinary course of the church,"
+i.e. "the daily office," which suggests the original sense of _diaeta_
+as "a prescribed mode of life."
+
+The word "diet" is now used in English for the _Reichstag_, "imperial
+diet" of the old Holy Roman Empire; for the _Bundestag_, "federal diet,"
+of the former Germanic confederation; sometimes for the _Reichstag_ of
+the modern German empire; for the _Landtage_, "territorial diets" of the
+constituent states of the German and Austrian empires; as well as for
+the former or existing federal or national assemblies of Switzerland,
+Hungary, Poland, &c. Although, however, the word is still sometimes used
+of all the above, the tendency is to confine it, so far as contemporary
+assemblies are concerned, to those of subordinate importance. Thus
+"parliament" is often used of the German _Reichstag_ or of the Russian
+Landtag, while the _Landtag_, e.g. of Styria, would always be rendered
+"diet." In what follows we confine ourselves to the diet of the Holy
+Roman Empire and its relation to its successors in modern Germany.
+
+The origin of the diet, or deliberative assembly, of the Holy Roman
+Empire must be sought in the _placitum_ of the Frankish empire. This
+represented the tribal assembly of the Franks, meeting (originally in
+March, but after 755 in May, whence it is called the Campus Maii) partly
+for a military review on the eve of the summer campaign, partly for
+deliberation on important matters of politics and justice. By the side
+of this larger assembly, however, which contained in theory, if not in
+practice, the whole body of Franks available for war, there had
+developed, even before Carolingian times, a smaller body composed of the
+magnates of the Empire, both lay and ecclesiastical. The germ of this
+smaller body is to be found in the episcopal synods, which, afforced by
+the attendance of lay magnates, came to be used by the king for the
+settlement of national affairs. Under the Carolingians it was usual to
+combine the assembly of magnates with the _generalis conventus_ of the
+"field of May," and it was in this inner assembly, rather than in the
+general body (whose approval was merely formal, and confined to matters
+momentous enough to be referred to a general vote), that the centre of
+power really lay. It is from the assembly of magnates that the diet of
+medieval Germany springs. The general assembly became meaningless and
+unnecessary, as the feudal array gradually superseded the old levy _en
+masse_, in which each freeman had been liable to service; and after the
+close of the 10th century it no longer existed.
+
+The imperial diet (_Reichstag_) of the middle ages might sometimes
+contain representatives of Italy, the _regnum Italicum_; but it was
+practically always confined to the magnates of Germany, the _regnum
+Teutonicum_. Upon occasion a summons to the diet might be sent even to
+the knights, but the regular members were the princes (_Fürsten_), both
+lay and ecclesiastical. In the 13th century the seven electors began to
+disengage themselves from the prince as a separate element, and the
+Golden Bull (1356) made their separation complete; from the 14th century
+onwards the nobles (both counts and other lords) are regarded as regular
+members; while after 1250 the imperial and episcopal towns often appear
+through their representatives. By the 14th century, therefore, the
+originally homogeneous diet of princes is already, at any rate
+practically if not yet in legal form, divided into three colleges--the
+electors, the princes and nobles, and the representatives of the towns
+(though, as we shall see, the latter can hardly be reckoned as regular
+members until the century of the Reformation). Under the Hohenstaufen it
+is still the rule that every member of the diet must attend personally,
+or lose his vote; at a later date the principle of representation by
+proxy, which eventually made the diet into a mere congress of envoys,
+was introduced. By the end of the 13th century the vote of the majority
+had come to be regarded as decisive; but in accordance with the strong
+sense of social distinctions which marks German history, the quality as
+well as the quantity of votes was weighed, and if the most powerful of
+the princes were agreed, the opinion of the lesser magnates was not
+consulted. The powers of the medieval diet extended to matters like
+legislation, the decision upon expeditions (especially the _expeditio
+Romana_), taxation and changes in the constitution of the principalities
+or the Empire. The election of the king, which was originally regarded
+as one of the powers of the diet, had passed to the electors by the
+middle of the 13th century.
+
+A new era in the history of the diet begins with the Reformation. The
+division of the diet into three colleges becomes definite and precise;
+the right of the electors, for instance, to constitute a separate
+college is explicitly recognized as a matter of established custom in
+1544. The representatives of the towns now become regular members. In
+the 15th century they had only attended when special business, such as
+imperial reform or taxation, fell under discussion; in 1500, however,
+they were recognized as a separate and regular estate, though it was not
+until 1648 that they were recognized as equal to the other estates of
+the diet. The estate of the towns, or college of municipal
+representatives, was divided into two benches, the Rhenish and the
+Swabian. The estate of the princes and counts, which stood midway
+between the electors and the towns, also attained, in the years that
+followed the Reformation, its final organization. The vote of the great
+princes ceased to be personal, and began to be territorial. This had two
+results. The division of a single territory among the different sons of
+a family no longer, as of old, multiplied the voting power of the
+family; while in the opposite case, the union of various territories in
+the hands of a single person no longer meant the extinction of several
+votes, since the new owner was now allowed to give a vote for each of
+his territories. The position of the counts and other lords, who joined
+with the princes in forming the middle estate, was finally fixed by the
+middle of the 17th century. While each of the princes enjoyed an
+individual vote, the counts and other lords were arranged in groups,
+each of which voted as a whole, though the whole of its vote
+(_Kuriatstimme_) only counted as equal to the vote of a single prince
+(_Virilstimme_). There were six of these groups; but as the votes of the
+whole college of princes and counts (at any rate in the 18th century)
+numbered 100, they could exercise but little weight.
+
+The last era in the history of the diet may be said to open with the
+treaty of Westphalia (1648). The treaty acknowledged that Germany was no
+longer a unitary state, but a loose confederation of sovereign princes;
+and the diet accordingly ceased to bear the character of a national
+assembly, and became a mere congress of envoys. The "last diet" which
+issued a regular recess (_Reichsabschied_--the term applied to the
+_acta_ of the diet, as formally compiled and enunciated at its
+dissolution) was that of Regensburg in 1654. The next diet, which met at
+Regensburg in 1663, never issued a recess, and was never dissolved; it
+continued in permanent session, as it were, till the dissolution of the
+Empire in 1806. This result was achieved by the process of turning the
+diet from an assembly of principals into a congress of envoys. The
+emperor was represented by two _commissarii_; the electors, princes and
+towns were similarly represented by their accredited agents. Some
+legislation was occasionally done by this body; a _conclusum imperii_
+(so called in distinction from the old _recessus imperii_ of the period
+before 1663) might slowly (very slowly--for the agents, imperfectly
+instructed, had constantly to refer matters back to their principals) be
+achieved; but it rested with the various princes to promulgate and
+enforce the _conclusum_ in their territories, and they were sufficiently
+occupied in issuing and enforcing their own decrees. In practice the
+diet had nothing to do; and its members occupied themselves in
+"wrangling about chairs"--that is to say, in unending disputes about
+degrees and precedences.
+
+In the Germanic Confederation, which occupies the interval between the
+death of the Holy Roman Empire and the formation of the North German
+Confederation (1815-1866), a diet (_Bundestag_) existed, which was
+modelled on the old diet of the 18th century. It was a standing congress
+of envoys at Frankfort-on-Main. Austria presided in the diet, which, in
+the earlier years of its history, served, under the influence of
+Metternich, as an organ for the suppression of Liberal opinion. In the
+North German Confederation (1867-1870) a new departure was made, which
+has been followed in the constitution of the present German empire. Two
+bodies were instituted--a _Bundesrat_, which resembles the old diet in
+being a congress of envoys sent by the sovereigns of the different
+states of the confederation, and a _Reichstag_, which bears the name of
+the old diet, but differs entirely in composition. The new Reichstag is
+a popular representative assembly, based on wide suffrage and elected by
+ballot; and, above all, it is an assembly representing, not the several
+states, but the whole Empire, which is divided for this purpose into
+electoral districts. Both as a popular assembly, and as an assembly
+which represents the whole of a united Germany, the new Reichstag goes
+back, one may almost say, beyond the diet even of the middle ages, to
+the days of the old Teutonic folk-moot.
+
+  See R. Schröder, _Lehrbuch der deutschen Rechtsgeschichte_ (1902), pp.
+  149, 508, 820, 880. Schröder gives a bibliography of monographs
+  bearing on the history of the medieval diet.     (E. BR.)
+
+
+
+
+DIETARY, in a general sense, a system or course of diet, in the sense of
+food; more particularly, such an allowance and regulation of food as
+that supplied to workhouses, the army and navy, prisons, &c. Lowest in
+the scale of such dietaries comes what is termed "bare existence" diet,
+administered to certain classes of the community who have a claim on
+their fellow-countrymen that their lives and health shall be preserved
+_in statu quo_, but nothing further. This applies particularly to the
+members of a temporarily famine-stricken community. Before the days of
+prison reform, too, the dietary scale of many prisons was to a certain
+extent penal, in that the food supplied to prisoners was barely
+sufficient for existence. Nowadays more humane principles apply; there
+is no longer the obvious injustice of applying the same scale of
+quantity and quality to all prisoners under varying circumstances of
+constitution and surroundings, and whether serving long or short periods
+of imprisonment.
+
+  The system of dietary in force in the local and convict prisons of
+  England and Wales is that recommended by the Home Office on the advice
+  of a departmental committee. As to the local prison dietary, its
+  application is based on (1) the principle of variation of diet with
+  length of sentence; (2) the system of progressive dietary; (3) the
+  distinction between hard labour diets and non-hard labour diets; (4)
+  the differentiation of diet according to age and sex. There are three
+  classes of diet, classes A, B and C. Class A diet is given to
+  prisoners undergoing not more than seven days' imprisonment. The food
+  is good and wholesome, but sufficiently plain and unattractive, so as
+  not to offer temptation to the loafer or mendicant. It is given in
+  quantity sufficient to maintain health and strength during the single
+  week. Prisoners sentenced to more than seven days and not more than
+  fourteen days are given class A diet for the first seven days and
+  class B for the remainder of the sentence. In most of the local
+  prisons in England and Wales prisoners sentenced to hard labour
+  received hard labour diet, although quite 60% were unable to perform
+  the hardest forms of prison labour either through physical defect, age
+  or infirmity. The departmental committee of 1899 in their report
+  recommended that no distinction should be made between hard labour and
+  non-hard labour diets. Class A diet is as follows:--_Breakfast_,
+  Bread, 8 oz. daily (6 oz. for women and juveniles) with 1 pint of
+  gruel. Juveniles (males and females under sixteen years of age) get,
+  in addition, ½ pint of milk. _Dinner_, 8 oz. of bread daily, with 1
+  pint of porridge on three days of the week, 8 oz. of potatoes
+  (representing the vegetable element) on two other days, and 8 oz. of
+  suet pudding (representing the fatty element) on the other two days.
+  _Supper_, the breakfast fare repeated.
+
+  Class B diet, which is also given to (1) prisoners on remand or
+  awaiting trial, (2) offenders of the 1st division who do not maintain
+  themselves, (3) offenders of the 2nd division and (4) debtors, is as
+  shown in Table I.
+
+  Class C diet is class B amplified, and is given to those prisoners
+  serving sentences of three months and over.
+
+
+      TABLE I.
+
+    +------------+------------------+--------+--------+-----------+
+    |            |                  |  Men.  | Women. | Juveniles.|
+    +------------+------------------+--------+--------+-----------+
+    |            |                  |        |        |           |
+    | Breakfast. | Daily:--         |        |        |           |
+    |            |    Bread         |  8 oz. |  6 oz. |   6 oz.   |
+    |            |    Gruel         |  1 pt. |  1 pt. |   1 pt.   |
+    |            |    Milk          |    ..  |    ..  |   ½ pt.   |
+    |            |                  |        |        |           |
+    | Dinner.    | Sunday:--        +--------+--------+-----------+
+    |            |    Bread         |  6 oz. |       6 oz.        |
+    |            |    Potatoes      |  8  "  |       8  "         |
+    |            |    Cooked meat,  |        |                    |
+    |            |      preserved   |  4  "  |       3  "         |
+    |            |      by heat     |        |                    |
+    |            | Monday:--        |        |                    |
+    |            |    Bread         |  6 oz. |       6 oz.        |
+    |            |    Potatoes      |  8  "  |       8  "         |
+    |            |    Beans         | 10  "  |       8  "         |
+    |            |    Fat bacon     |  2  "  |       1  "         |
+    |            |                  |        |                    |
+    |            | Tuesday:--       |        |                    |
+    |            |    Bread         |  6 oz  |       6 oz.        |
+    |            |    Potatoes      |  8  "  |       8  "         |
+    |            |    Soup          |  1 pt. |       1 pt.        |
+    |            |                  |        |                    |
+    |            | Wednesday:--     |        |                    |
+    |            |    Bread         |  6 oz. |       6 oz.        |
+    |            |    Potatoes      |  8  "  |       8  "         |
+    |            |    Suet pudding  | 10  "  |       8  "         |
+    |            |                  |        |                    |
+    |            | Thursday:--      |        |                    |
+    |            |    Bread         |  6 oz. |       6 oz.        |
+    |            |    Potatoes      |  8  "  |       8  "         |
+    |            |    Cooked beef,  |        |                    |
+    |            |      without bone|  4  "  |       3  "         |
+    |            |                  |        |                    |
+    |            | Friday:--        |        |                    |
+    |            |    Bread         |  6 oz. |       6 oz.        |
+    |            |    Potatoes      |  8  "  |       8  "         |
+    |            |    Soup          |  1 pt. |       1 pt.        |
+    |            |                  |        |                    |
+    |            | Saturday:--      |        |                    |
+    |            |    Bread         |  6 oz. |       6 oz.        |
+    |            |    Potatoes      |  8  "  |       8  "         |
+    |            |    Suet pudding  | 10  "  |       8  "         |
+    |            |                  |        |                    |
+    | Supper.    | Daily:--         |        +---------+----------+
+    |            |    Bread         |  8 oz. |   6 oz. |   6 oz.  |
+    |            |    Porridge      |  1 pt. |         |          |
+    |            |    Gruel         |        |   1 pt. |          |
+    |            |    Cocoa         |        |         |   1 pt.  |
+    +------------+------------------+--------+---------+----------+
+
+
+      TABLE II.
+    +------------------+-----+------+------+------+------+------+------+
+    |                  |Sun. | Mon. | Tue. | Wed. | Thu. | Fri. | Sat. |
+    +------------------+-----+------+------+------+------+------+------+
+    | Breakfast.       |     |      |      |      |      |      |      |
+    |  Bread.       oz.|  8  |   4  |   4  |   4  |   4  |   4  |   4  |
+    |  Porridge.    pt.|  *  |   1½ |   1½ |   1½ |   1½ |   1½ |   1½ |
+    +------------------+-----+------+------+------+------+------+------+
+    | Dinner.          |     |      |      |      |      |      |      |
+    |  Bread.       oz.|  4  |   6  |  ..  |   4  |   4  |   8  |   6  |
+    |  Beef.        oz.|  4½ |  ..  |  ..  |  ..  |   4½ |  ..  |  ..  |
+    |  Vegetables.  oz.| 12  |  ..  |  ..  |  12  |  12  |  ..  |  ..  |
+    |  Barley Soup. pt.| ..  |   1½ |  ..  |  ..  |  ..  |  ..  |  ..  |
+    |  Pork.        oz.| ..  |  ..  |   4½ |  ..  |  ..  |  ..  |  ..  |
+    |  Beans.       oz.| ..  |  ..  |  12  |  ..  |  ..  |  ..  |  ..  |
+    |  Fish.        oz.| ..  |  ..  |  ..  |  10  |  ..  |  ..  |  ..  |
+    |  Cheese.      oz.| ..  |  ..  |  ..  |  ..  |  ..  |   3  |  ..  |
+    |  Broth.       pt.| ..  |  ..  |  ..  |  ..  |  ..  |   1  |  ..  |
+    |  Irish Stew.  pt.| ..  |  ..  |  ..  |  ..  |  ..  |  ..  |   1  |
+    +------------------+-----+------+------+------+------+------+------+
+    | Supper.          |     |      |      |      |      |      |      |
+    |  Bread.       oz.|  8  |   6  |   6  |   6  |   8  |   6  |   6  |
+    |  Butter.      oz.|   ½ |  ..  |  ..  |  ..  |  ..  |  ..  |  ..  |
+    |  Tea.         pt.|  1  |  ..  |  ..  |  ..  |  ..  |  ..  |  ..  |
+    |  Gruel.       pt.| ..  |   1½ |   1½ |   1½ |  ..  |   1½ |   1½ |
+    |  Broth.       pt.| ..  |  ..  |  ..  |  ..  |   1  |  ..  |  ..  |
+    |  Cheese.      oz.| ..  |  ..  |  ..  |  ..  |   2  |  ..  |  ..  |
+    +------------------+-----+------+------+------+------+------+------+
+        * On Sundays 1 pint of tea and 2½ oz. of butter are given instead
+            of porridge.
+
+  The dietary of convict prisons, in which prisoners are all under long
+  sentence, is divided into a diet for convicts employed at hard labour
+  and a diet for convicts employed at sedentary, indoor and light
+  labour. It will be found set forth in the Blue-book mentioned above.
+  The sparest of all prison diets is called "punishment diet," and is
+  administered for offences against the internal discipline of the
+  prison. It is limited to a period of three days. It consists of 1 lb.
+  of bread and as much water as the prisoner chooses to drink.
+
+  In French prisons the dietary is nearly two pounds weight of bread,
+  with two meals of thin soup (breakfast and dinner) made from potatoes,
+  beans or other vegetables, and on two days a week made from meat. In
+  France the canteen system is in vogue, additional food, such as
+  sausages, cheese, fruit, &c., may be obtained by the prisoner,
+  according to the wages he receives for his labours. The dietary of
+  Austrian prisons is 1½ lb. of bread daily, a dinner of soup on four
+  days of the week, and of meat on the other three days, with a supper
+  of soup or vegetable stew. Additional food can be purchased by the
+  prisoner out of his earnings.
+
+  These dietaries may be taken as more or less typical of the ordinary
+  prison fare in most civilized countries, though in some countries it
+  may err on the side of severity, as in Sweden, prisoners being given
+  only two meals a day, one at mid-day and one at seven p.m., porridge
+  or gruel being the principal element in both meals. On the other hand,
+  the prison dietaries of many of the United States prisons go to the
+  other extreme, fresh fish, green vegetables, even coffee and fruit,
+  figuring in the dietary.
+
+  Another class of dietary is that given to paupers. In England, until
+  1900, almost every individual workhouse had its own special dietary,
+  with the consequence that many erred on the side of scantiness and
+  unsuitability, while others were too lavish. By an order of the Local
+  Government Board of that year, acting on a report of a committee, all
+  inmates of workhouses, with the exception of the sick, children under
+  three years of age, and certain other special cases, are dieted in
+  accordance with certain dietary tables as framed and settled by the
+  board. The order contained a great number of different rations, it
+  being left to the discretion of the guardians as to the final
+  settlement of the tables. For adult inmates the dietary tables are for
+  each sex respectively, two in number, one termed "plain diet" and the
+  other "infirm diet." All male inmates certified as healthy able-bodied
+  persons receive plain diet only. All inmates, however, in workhouses
+  are kept employed according to their capacity and ability, and this is
+  taken into consideration in giving allowances of food. For instance,
+  for work with sustained exertion, such as stone-breaking, digging,
+  &c., more food is given than for work without sustained exertion, such
+  as wood-chopping, weeding or sewing. Table II. shows an example of a
+  workhouse dietary.
+
+  In the casual wards of workhouses the dietary is plainer, consisting
+  of 8 oz. of bread, or 6 oz. of bread and one pint of gruel or broth
+  for breakfast; the same for supper; for dinner 8 oz. of bread and 1½
+  oz. of cheese or 6 oz. of bread and one pint of soup. The American
+  poor law system is based broadly on that of England, and the methods
+  of relief are much the same. Each state, however, makes its own
+  regulations, and there is considerable diversity in workhouse
+  dietaries in consequence. The German system of poor relief is more
+  methodical than those of England and America. The really deserving are
+  treated with more commiseration, and a larger amount of outdoor
+  relief is given than in England. There is no casual ward, tramps and
+  beggars being liable to penal treatment, but there are "relief
+  stations," somewhat corresponding to casual wards, where destitute
+  persons tramping from one place to another can obtain food and lodging
+  in return for work done.
+
+  In the British navy certain staple articles of diet are supplied to
+  the men to the value approximately of 6d. per diem--the standard
+  government ration--and, in addition, a messing allowance of 4d. per
+  diem, which may either be expended on luxuries in the canteen, or in
+  taking up government provisions on board ship, in addition to the
+  standard ration. The standard ration as recommended in 1907 by a
+  committee appointed to inquire into the question of victualling in the
+  navy is as follows:--
+
+  _Service Afloat._
+
+    1 lb. bread (or ¾ lb. bread and ¼ lb. trade flour).
+    ½ lb. fresh meat.
+    1 lb. fresh vegetables.
+    1/8 pint spirit.
+    4 oz. sugar.
+    ½ oz. tea (or 1 oz. coffee for every ¼ oz. tea).
+    ½ oz. ordinary or soluble chocolate (or 1 oz. coffee).
+    ¾ oz. condensed milk.
+    1 oz. jam or marmalade.
+    4 oz. preserved meat on _one_ day of the week in harbour, or on
+       _two_ days at sea.
+
+  Mustard, pepper, vinegar, and salt as required.
+
+  Substitute for soft bread when the latter is not available--
+      ½ lb. biscuit (new type) or 1 lb. flour.
+
+  Substitutes for fresh meat when the latter is not available:--
+
+              /(1) Salt pork day:--
+             |     ½ lb. salt pork.
+             |     ¼ lb. split peas.
+             |     Celery seed, ½ oz. to every 8 lb. of split peas put
+             |       into the coppers.
+      On     |     ½ lb. potatoes (or 1 oz. compressed vegetables).
+   alternate<
+     days    | (2) Preserved meat day:--
+             |     6 oz. preserved meat.
+             |     8 oz. trade flour   \
+             |     ¾ oz. refined suet   > or 4 oz. rice.
+             |     2 oz. raisins       /
+              \    ½ lb. potatoes (or 1 oz. compressed vegetables).
+
+  On shore establishments and depot ships ¼ pt. fresh milk is issued in
+  lieu of the ¾ oz. of condensed milk.
+
+  In the United States navy there is more liberality and variety of
+  diet, the approximate daily cost of the rations supplied being 1s. 3d.
+  per head. In the American mercantile marine, too, according to the
+  scale sanctioned by act of Congress (December 21, 1898) for American
+  ships, the seaman is better off than in the British merchant service.
+  The scale is shown in Table III.
+
+
+      TABLE III.
+
+    +---------+-----------------------++-----------+----------------+
+    | Weekly  |       Articles.       ||  Weekly   |    Articles.   |
+    | Scale.  |                       ||  Scale.   |                |
+    +---------+-----------------------++-----------+----------------+
+    | 3½  lb. | Biscuits.             || 7/8  oz.  | Tea.           |
+    | 3¾   "  | Salt beef.            || 21    "   | Sugar.         |
+    | 3    "  |  "   pork.            ||  1½  lb.  | Molasses.      |
+    | 1½   "  | Flour.                ||  9   oz.  | Fruits, dried. |
+    | 2    "  | Meats, preserved.     ||   ¾  pt.  | Pickles.       |
+    | 10½  "  | Bread, fresh (8 lb.   ||   1   "   | Vinegar.       |
+    |         |   flour in lieu).     ||  8   oz.  | Corn Meal.     |
+    | 1    "  | Fish, dried.          || 12    "   | Onions.        |
+    | 7    "  | Potatoes or yams.     ||  7    "   | Lard.          |
+    | 1    "  | Tomatoes, preserved.  ||  7    "   | Butter.        |
+    |2/3   "  | Peas.                 ||   ¼   "   | Mustard.       |
+    |2/3   "  | Calavances.           ||   ¼   "   | Pepper.        |
+    |2/3   "  | Rice.                 ||   ¼   "   | Salt.          |
+    | 5¼  oz. | Coffee, green.        ||           |                |
+    +---------+-----------------------++-----------+----------------+
+
+  In the British mercantile marine there is no scale of provisions
+  prescribed by the Board of Trade; there is, however, a traditional
+  scale very generally adopted, having the sanction of custom only and
+  seldom adhered to. The following dietary scale for steerage
+  passengers, laid down in the 12th schedule of the Merchant Shipping
+  Act 1894, is of interest. See Table IV.
+
+  Certain substitutions may be made in this scale at the option of the
+  master of any emigrant ship, provided that the substituted articles
+  are set forth in the contract tickets of the steerage passengers.
+
+  In the British army the soldier is fed partly by a system of
+  co-operation. He gets a free ration from government of 1 lb. of bread
+  and ¾ lb. of meat; in addition there is a messing allowance of 3½d.
+  per man per day. He is able to supplement his food by purchases from
+  the canteen. Much depends on the individual management in each
+  regiment as to the satisfactory expenditure of the messing allowance.
+  In some regiments an allowance is made from the canteen funds towards
+  messing in addition to that granted by the government. The ordinary
+  _field_ ration of the British soldier is 1½ lb. of bread or 1 lb. of
+  biscuit; 1 lb. of fresh, salt or preserved meat; ½ oz. of coffee; 1/6
+  oz. of tea; 2 oz. of sugar; ½ oz. of salt, 1/36 oz. of pepper, the
+  whole weighing something over 2 lb. 3 oz. This cannot be looked on as
+  a fixed ration, as it varies in different campaigns, according to the
+  country into which the troops may be sent. The Prussian soldier during
+  peace gets weekly from his canteen 11 lb. 1 oz. of rye bread and not
+  quite 2½ lb. of meat. This is obviously insufficient, but under
+
+
+      TABLE IV.--_Weekly, per Statute Adult._
+
+    +---------------------------+-------------------+-------------------+
+    |                           |      Scale A.     |      Scale B.     |
+    |                           |For voyages not    |For voyages        |
+    |                           | exceeding 84 days | exceeding 84 days |
+    |                           | for sailing ships | for sailing ships |
+    |                           | or 50 days        | or 50 days        |
+    |                           | for steamships.   | for steamships.   |
+    +---------------------------+-------------------+-------------------+
+    |                           |     lb.   oz.     |     lb.   oz.     |
+    | Bread or biscuit, not     |                   |                   |
+    |  inferior to navy biscuit |      3    8       |      3    8       |
+    | Wheaten flour             |      1    0       |      2    0       |
+    | Oatmeal                   |      1    8       |      1    0       |
+    | Rice                      |      1    8       |      0    8       |
+    | Peas                      |      1    8       |      1    8       |
+    | Beef                      |      1    4       |      1    4       |
+    | Pork                      |      1    0       |      1    0       |
+    | Butter                    |        ..         |      0    4       |
+    | Potatoes                  |      2    0       |      2    0       |
+    | Sugar                     |      1    0       |      1    0       |
+    | Tea                       |      0    2       |      0    2       |
+    | Salt                      |      0    2       |      0    2       |
+    | Pepper (white or          |                   |                   |
+    |  black), ground           |      0    0½      |      0    0½      |
+    | Vinegar                   |      1 gill       |      1 gill       |
+    | Preserved meat            |        ..         |      1    0       |
+    | Suet                      |                   |      0    6       |
+    | Raisins                   |                   |      0    8       |
+    | Lime juice                |                   |      0    6       |
+    +---------------------------+-------------------+-------------------+
+
+  the conscription system it is reckoned that he will be able to make up
+  the deficiency out of his own private means, or obtain charitable
+  contributions from his friends. In the French infantry of the line
+  each man during peace gets weekly 15 lb. of bread, 3-3/10 lb. of meat,
+  2½ lb. of haricot beans or other vegetables, with salt and pepper, and
+  1¾ oz. of brandy.
+
+  An Austrian under the same circumstances receives 13.9 lb. of bread, ½
+  lb. of flour and 3.3 lb. of meat.
+
+  The Russian conscript is allowed weekly:--
+
+    Black bread              7 lb.
+    Meat                     7 lb.
+    Kvass (beer)             7.7 quarts.
+    Sour cabbage             24½ gills = 122½ oz.
+    Barley                   24½ gills = 122½ oz.
+    Salts                    10½ oz.
+    Horse-radish             28 grains.
+    Pepper                   28 grains.
+    Vinegar                  5½ gills = 26½ oz.
+
+
+
+
+DIETETICS, the science of diet, i.e. the food and nutrition of man in
+health and disease (see NUTRITION). This article deals mainly with that
+part of the subject which has to do with the composition and nutritive
+values of foods and their adaptation to the use of people in health. The
+principal topics considered are: (1) Food and its functions; (2)
+Metabolism of matter and energy; (3) Composition of food materials; (4)
+Digestibility of food; (5) Fuel value of food; (6) Food consumption; (7)
+Quantities of nutrients needed; (8) Hygienic economy of food; (9)
+Pecuniary economy of food.
+
+1. _Food and its Functions._--For practical purposes, food may be
+defined as that which, when taken into the body, may be utilized for the
+formation and repair of body tissue, and the production of energy. More
+specifically, food meets the requirements of the body in several ways.
+It is used for the formation of the tissues and fluids of the body, and
+for the restoration of losses of substance due to bodily activity. The
+potential energy of the food is converted into heat or muscular work or
+other forms of energy. In being thus utilized, food protects body
+substance or previously acquired nutritive material from consumption.
+When the amount of food taken into the body is in excess of immediate
+needs, the surplus may be stored for future consumption.
+
+Ordinary food materials, such as meat, fish, eggs, vegetables, &c.,
+consist of inedible materials, or _refuse_, e.g. bone of meat and fish,
+shell of eggs, rind and seed of vegetables; and _edible material_, as
+flesh of meat and fish, white and yolk of eggs, wheat flour, &c. The
+edible material is by no means a simple substance, but consists of
+_water_, and some or all of the compounds variously designated as food
+stuffs, proximate principles, nutritive ingredients or nutrients, which
+are classified as _protein_, _fats_, _carbohydrates_ and _mineral
+matters_. These have various functions in the nourishment of the body.
+
+The _refuse_ commonly contains compounds similar to those in the food
+from which it is derived, but since it cannot be eaten, it is usually
+considered as a non-nutrient. It is of importance chiefly in a
+consideration of the pecuniary economy of food. _Water_ is also
+considered as a non-nutrient, because although it is a constituent of
+all the tissues and fluids of the body, the body may obtain the water it
+needs from that drunk; hence, that contained in the food materials is of
+no special significance as a nutrient.
+
+_Mineral matters_, such as sulphates, chlorides, phosphates and
+carbonates of sodium, potassium, calcium, &c., are found in different
+combinations and quantities in most food materials. These are used by
+the body in the formation of the various tissues, especially the
+skeletal and protective tissues, in digestion, and in metabolic
+processes within the body. They yield little or no energy, unless
+perhaps the very small amount involved in their chemical transformation.
+
+Protein[1] is a term used to designate the whole group of nitrogenous
+compounds of food except the nitrogenous fats. It includes the
+albuminoids, as albumin of egg-white, and of blood serum, myosin of meat
+(muscle), casein of milk, globulin of blood and of egg yolk, fibrin of
+blood, gluten of flour; the gelatinoids, as gelatin and allied
+substances of connective tissue, collagen of tendon, ossein of bone and
+the so-called extractives (e.g. creatin) of meats; and the amids (e.g.
+asparagin) and allied compounds of vegetables and fruits.
+
+The albuminoids and gelatinoids, classed together as proteids, are the
+most important constituents of food, because they alone can supply the
+nitrogenous material necessary for the formation of the body tissues.
+For this purpose, the albuminoids are most valuable. Both groups of
+compounds, however, supply the body with energy, and the gelatinoids in
+being thus utilized protect the albuminoids from consumption for this
+purpose. When their supply in the food is in excess of the needs of the
+body, the surplus proteids may be converted into body fat and stored.
+
+The so-called extractives, which are the principal constituents of meat
+extract, beef tea and the like, act principally as stimulants and
+appetizers. It has been believed that they serve neither to build tissue
+nor to yield energy, but recent investigations[2] indicate that creatin
+may be metabolized in the body.
+
+The _fats_ of food include both the animal fats and the vegetable oils.
+The _carbohydrates_ include such compounds as starches, sugars and the
+fibre of plants or cellulose, though the latter has but little value as
+food for man. The more important function of both these classes of
+nutrients is to supply energy to the body to meet its requirements above
+that which it may obtain from the proteids. It is not improbable that
+the atoms of their molecules as well as those from the proteids are
+built up into the protoplasmic substance of the tissues. In this sense,
+these nutrients may be considered as being utilized also for the
+formation of tissue; but they are rather the accessory ingredients,
+whereas the proteids are the essential ingredients for this purpose. The
+fats in the food in excess of the body requirements may be stored as
+body fat, and the surplus carbohydrates may also be converted into fat
+and stored.
+
+To a certain extent, then, the nutrients of the food may substitute each
+other. All may be incorporated into the protoplasmic structure of body
+tissue, though only the proteids can supply the essential nitrogenous
+ingredients; and apart from the portion of the proteid material that is
+indispensable for this purpose, all the nutrients are used as a source
+of energy. If the supply of energy in the food is not sufficient, the
+body will use its own proteid and fat for this purpose. The gelatinoids,
+fats and carbohydrates in being utilized for energy protect the body
+proteids from consumption. The fat stored in the body from the excess of
+food is a reserve of energy material, on which the body may draw when
+the quantity of energy in the food is insufficient for its immediate
+needs.
+
+What compounds are especially concerned in intellectual activity is not
+known. The belief that fish is especially rich in phosphorus and
+valuable as a brain food has no foundation in observed fact.
+
+2. _Metabolism of Matter and Energy._--The processes of nutrition thus
+consist largely of the transformation of food into body material and the
+conversion of the potential energy of both food and body material into
+the kinetic energy of heat and muscular work and other forms of energy.
+These various processes are generally designated by the term metabolism.
+The metabolism of matter in the body is governed largely by the needs of
+the body for energy. The science of nutrition, of which the present
+subject forms a part, is based on the principle that the transformations
+of matter and energy in the body occur in accordance with the laws of
+the conservation of matter and of energy. That the body can neither
+create nor destroy matter has long been universally accepted. It would
+seem that the transformation of energy must likewise be governed by the
+law of the conservation of energy; indeed there is every reason a priori
+to believe that it must; but the experimental difficulties in the way of
+absolute demonstration of the principle are considerable. For such
+demonstration it is necessary to prove that the income and expenditure
+of energy are equal. Apparatus and methods of inquiry devised in recent
+years, however, afford means for a comparison of the amounts of both
+matter and energy received and expended by the body, and from the
+results obtained in a large amount of such research, it seems probable
+that the law obtains in the living organism in general.
+
+The first attempt at such demonstration was made by M. Rubner[3] in
+1894, experimenting with dogs doing no external muscular work. The
+income of energy (as heat) was computed, but the heat eliminated was
+measured. In the average of eight experiments continuing forty-five
+days, the two quantities agreed within 0.47%, thus demonstrating what it
+was desired to prove--that the heat given off by the body came solely
+from the oxidation of food within it. Results in accordance with these
+were reported by Studenski[4] in 1897, and by Laulanie[5] in 1898.
+
+The most extensive and complete data yet available on the subject have
+been obtained by W. O. Atwater, F. G. Benedict and associates[6] in
+experiments with men in the respiration calorimeter, in which a subject
+may remain for several consecutive days and nights. These experiments
+involve actual weighing and analyses of the food and drink, and of the
+gaseous, liquid and solid excretory products; determinations of
+potential energy (heat of oxidation) of the oxidizable material received
+and given off by the body (including estimation of the energy of the
+material gained or lost by the body); and measurements of the amounts of
+energy expended as heat and as external muscular work. By October 1906
+eighty-eight experiments with fifteen different subjects had been
+completed. The separate experiments continued from two to thirteen days,
+making a total of over 270 days.
+
+
+    TABLE I.--_Percentage Composition of some Common Food Materials._
+
+  +--------------------------+--------+--------+---------+------+---------+--------+-----------+
+  |     Food Material.       | Refuse.|  Water.| Protein.| Fat. | Carbo-  | Mineral| Fuel Value|
+  |                          |        |        |         |      |hydrates.| Matter.| per lb.   |
+  +--------------------------+--------+--------+---------+------+---------+--------+-----------+
+  |                          |        |        |         |      |         |        |           |
+  |                          |    %   |    %   |    %    |   %  |    %    |    %   | Calories. |
+  |Beef, fresh (medium fat)--|        |        |         |      |         |        |           |
+  |  Chuck                   |  16.3  |  52.6  |  15.5   | 15.0 |   . .   |   0.8  |    910    |
+  |  Loin                    |  13.3  |  52.5  |  16.1   | 17.5 |   . .   |   0.9  |   1025    |
+  |  Ribs                    |  20.8  |  43.8  |  13.9   | 21.2 |   . .   |   0.7  |   1135    |
+  |  Round                   |   7.2  |  60.7  |  19.0   | 12.8 |   . .   |   1.0  |    890    |
+  |  Shoulder                |  16.4  |  56.8  |  16.4   |  9.8 |   . .   |   0.9  |    715    |
+  |Beef, dried and smoked    |   4.7  |  53.7  |  26.4   |  6.9 |   . .   |   8.9  |    790    |
+  |Veal--                    |        |        |         |      |         |        |           |
+  |  Leg                     |  14.2  |  60.1  |  15.5   |  7.9 |   . .   |   0.9  |    625    |
+  |  Loin                    |  16.5  |  57.6  |  16.6   |  9.0 |   . .   |   0.9  |    685    |
+  |  Breast                  |  21.3  |  52.0  |  15.4   | 11.0 |   . .   |   0.8  |    745    |
+  |Mutton--                  |        |        |         |      |         |        |           |
+  |  Leg                     |  18.4  |  51.2  |  15.1   | 14.7 |   . .   |   0.8  |    890    |
+  |  Loin                    |  16.0  |  42.0  |  13.5   | 28.3 |   . .   |   0.7  |   1415    |
+  |  Flank                   |   9.9  |  39.0  |  13.8   | 36.9 |   . .   |   0.6  |   1770    |
+  |Pork--                    |        |        |         |      |         |        |           |
+  |  Loin                    |  19.7  |  41.8  |  13.4   | 24.2 |   . .   |   0.8  |   1245    |
+  |  Ham, fresh              |  10.7  |  48.0  |  13.5   | 25.9 |   . .   |   0.8  |   1320    |
+  |  Ham, smoked and salted  |  13.6  |  34.8  |  14.2   | 33.4 |   . .   |   4.2  |   1635    |
+  |  Fat, salt               |  . .   |   7.9  |   1.9   | 86.2 |   . .   |   3.9  |   3555    |
+  |  Bacon                   |   7.7  |  17.4  |   9.1   | 62.2 |   . .   |   4.1  |   2715    |
+  |  Lard, refined           |  . .   |  . .   |  . .    |100.0 |   . .   |  . .   |   4100    |
+  |Chicken                   |  25.9  |  47.1  |  13.7   | 12.3 |   . .   |   0.7  |    765    |
+  |Turkey                    |  22.7  |  42.4  |  16.1   | 18.4 |   . .   |   0.8  |   1060    |
+  |Goose                     |  17.6  |  38.5  |  13.4   | 29.8 |   . .   |   0.7  |   1475    |
+  |Eggs                      |  11.2  |  65.5  |  13.1   |  9.3 |   . .   |   0.9  |    635    |
+  |Cod, fresh                |  29.9  |  58.5  |  11.1   |  0.2 |   . .   |   0.8  |    220    |
+  |Cod, salted               |  24.9  |  40.2  |  16.0   |  0.4 |   . .   |  18.5  |    325    |
+  |Mackerel, fresh           |  44.7  |  40.4  |  10.2   |  4.2 |   . .   |   0.7  |    370    |
+  |Herring, smoked           |  44.4  |  19.2  |  20.5   |  8.8 |   . .   |   7.4  |    755    |
+  |Salmon, tinned            |  . .   |  63.5  |  21.8   | 12.1 |   . .   |   2.6  |    915    |
+  |Oysters, shelled          |  . .   |  88.3  |   6.0   |  1.3 |    3.3  |   1.1  |    225    |
+  |Butter                    |  . .   |  11.0  |   1.0   | 85.0 |   . .   |   3.0  |   3410    |
+  |Cheese                    |  . .   |  34.2  |  25.9   | 33.7 |    2.4  |   3.8  |   1885    |
+  |Milk, whole               |  . .   |  87.0  |   3.3   |  4.0 |    5.0  |   0.7  |    310    |
+  |Milk, skimmed             |  . .   |  90.5  |   3.4   |  0.3 |    5.1  |   0.7  |    165    |
+  |Oatmeal                   |  . .   |   7.7  |  16.7   |  7.3 |   66.2  |   2.1  |   1800    |
+  |Corn (maize) meal         |  . .   |  12.5  |   9.2   |  1.9 |   75.4  |   1.0  |   1635    |
+  |Rye flour                 |  . .   |  12.9  |   6.8   |  0.9 |   78.7  |   0.7  |   1620    |
+  |Buckwheat flour           |  . .   |  13.6  |   6.4   |  1.2 |   77.9  |   0.9  |   1605    |
+  |Rice                      |  . .   |  12.3  |   8.0   |  0.3 |   79.0  |   0.4  |   1620    |
+  |Wheat flour, white        |  . .   |  12.0  |  11.4   |  1.0 |   75.1  |   0.5  |   1635    |
+  |Wheat flour, graham       |  . .   |  11.3  |  13.3   |  2.2 |   71.4  |   1.8  |   1645    |
+  |Wheat, breakfast food     |  . .   |   9.6  |  12.1   |  1.8 |   75.2  |   1.3  |   1680    |
+  |Wheat bread, white        |  . .   |  35.3  |   9.2   |  1.3 |   53.1  |   1.1  |   1200    |
+  |Wheat bread, graham       |  . .   |  35.7  |   8.9   |  1.8 |   52.1  |   1.5  |   1195    |
+  |Rye bread                 |  . .   |  35.7  |   9.0   |  0.6 |   53.2  |   1.5  |   1170    |
+  |Biscuit (crackers)        |  . .   |   6.8  |   9.7   | 12.1 |   69.7  |   1.7  |   1925    |
+  |Macaroni                  |  . .   |  10.3  |  13.4   |  0.9 |   74.1  |   1.3  |   1645    |
+  |Sugar                     |  . .   |  . .   |  . .    | . .  |  100.0  |  . .   |   1750    |
+  |Starch (corn starch)      |  . .   |  . .   |  . .    | . .  |   90.0  |  . .   |   1680    |
+  |Beans, dried              |  . .   |  12.6  |  22.5   |  1.8 |   59.6  |   3.5  |   1520    |
+  |Peas, dried               |  . .   |  9.5   |  24.6   |  1.0 |   62.0  |   2.9  |   1565    |
+  |Beets                     |  20.0  |  70.0  |   1.3   |  0.1 |    7.7  |   0.9  |    160    |
+  |Cabbage                   |  50.0  |  44.2  |   0.7   |  0.2 |    4.5  |   0.4  |    100    |
+  |Potatoes                  |  20.0  |  62.6  |   1.8   |  0.1 |   14.7  |   0.8  |    295    |
+  |Sweet potatoes            |  20.0  |  55.2  |   1.4   |  0.6 |   21.9  |   0.9  |    440    |
+  |Tomatoes                  |  . .   |  94.3  |   0.9   |  0.4 |    3.9  |   0.5  |    100    |
+  |Apples                    |  25.0  |  63.3  |   0.3   |  0.3 |   10.8  |   0.3  |    190    |
+  |Bananas                   |  35.0  |  48.9  |   0.8   |  0.4 |   14.3  |   0.6  |    260    |
+  |Grapes                    |  25.0  |  58.0  |   1.0   |  1.2 |   14.4  |   0.4  |    295    |
+  |Strawberries              |   5.0  |  85.9  |   0.9   |  0.6 |    7.0  |   0.6  |    150    |
+  |Almonds                   |  45.0  |   2.7  |  11.5   | 30.2 |    9.5  |   1.1  |   1515    |
+  |Brazil nuts               |  49.6  |   2.6  |   8.6   | 33.7 |    3.5  |   2.0  |   1485    |
+  |Chestnuts                 |  16.0  |  37.8  |   5.2   |  4.5 |   35.4  |   1.1  |    915    |
+  |Walnuts                   |  58.1  |   1.0  |   6.9   | 26.6 |    6.8  |   0.6  |   1250    |
+  +--------------------------+--------+--------+---------+------+---------+--------+-----------+
+
+In some cases the subjects were at rest; in others they performed
+varying amounts of external muscular work on an apparatus by means of
+which the amount of work done was measured. In some cases they fasted,
+and in others they received diets generally not far from sufficient to
+maintain nitrogen, and usually carbon, equilibrium in the body. In these
+experiments the amount of energy expended by the body as heat and as
+external muscular work measured in terms of heat agreed on the average
+very closely with the amount of heat that would be produced by the
+oxidation of all the matter metabolized in the body. The variations for
+individual days, and in the average for individual experiments as well,
+were in some cases appreciable, amounting to as much as 6%, which is not
+strange in view of the uncertainties in physiological experimenting; but
+in the average of all the experiments the energy of the expenditure was
+above 99.9% of the energy of the income,--an agreement within one part
+in 1000. While these results do not absolutely prove the application of
+the law of the conservation of energy in the human body, they certainly
+approximate very closely to such demonstration. It is of course possible
+that energy may have given off from the body in other forms than heat
+and external muscular work. It is conceivable, for example, that
+intellectual activity may involve the transformation of physical energy,
+and that the energy involved may be eliminated in some form now unknown.
+But if the body did give off energy which was not measured in these
+experiments, the quantity must have been extremely small. It seems fair
+to infer from the results obtained that the metabolism of energy in the
+body occurred in conformity with the law of the conservation of energy.
+
+3. _Composition of Food Materials._--The composition of food is
+determined by chemical analyses, the results of which are conventionally
+expressed in terms of the nutritive ingredients previously described. As
+a result of an enormous amount of such investigation in recent years,
+the kinds and proportions of nutrients in our common sorts of food are
+well known. Average values for percentage composition of some ordinary
+food materials are shown in Table I. (Table I. also includes figures for
+fuel value.)
+
+It will be observed that different kinds of food materials vary widely
+in their proportions of nutrients. In general the animal foods contain
+the most protein and fats, and vegetable foods are rich in
+carbohydrates. The chief nutrient of lean meat and fish is protein; but
+in medium fat meats the proportion of fat is as large as that of
+protein, and in the fatter meats it is larger. Cheese is rich in both
+protein and fat. Among the vegetable foods, dried beans and peas are
+especially rich in protein. The proportion in oatmeal is also fairly
+large, in wheat it is moderate, and in maize meal and rice it is rather
+small. Oats contain more oil than any of the common cereals, but in none
+of them is the proportion especially large. The most abundant nutrient
+in all the cereals is starch, which comprises from two-thirds to
+three-fourths or more of their total nutritive substance. Cotton-seed is
+rich in edible oil, and so are olives. Some of the nuts contain fairly
+large proportions of both protein and fat. The nutrient of potatoes is
+starch, present in fair proportion. Fruits contain considerable
+carbohydrates, chiefly sugar. Green vegetables are not of much account
+as sources of any of the nutrients or energy.
+
+Similar food materials from different sources may also differ
+considerably in composition. This is especially true of meats. Thus, the
+leaner portions from a fat animal may contain nearly as much fat as the
+fatter portions from a lean animal. The data here presented are largely
+those for American food products, but the available analyses of English
+food materials indicate that the latter differ but little from the
+former in composition. The analyses of meats produced in Europe imply
+that they commonly contain somewhat less fat and more water, and often
+more protein, than American meats. The meats of English production
+compare with the American more than with the European meats. Similar
+vegetable foods from the different countries do not differ so much in
+composition.
+
+4. _Digestibility or Availability of Food Materials._--The value of any
+food material for nutriment depends not merely upon the kinds and
+amounts of nutrients it contains, but also upon the ease and convenience
+with which the nutrients may be digested, and especially upon the
+proportion of the nutrients that will be actually digested and
+absorbed. Thus, two foods may contain equal amounts of the same
+nutrient, but the one most easily digested will really be of most value
+to the body, because less effort is necessary to utilize it.
+Considerable study of this factor is being made, and much valuable
+information is accumulating, but it is of more especial importance in
+cases of disordered digestion.
+
+    TABLE II.--_Coefficients of Digestibility (or Availability) of
+    Nutrients in Different Classes of Food Materials._
+
+  +--------------------------+----------+----------+----------------+
+  |       Kind of Food.      | Protein. |   Fat.   | Carbohydrates. |
+  +--------------------------+----------+----------+----------------+
+  |                          |          |          |                |
+  |                          |     %    |     %    |        %       |
+  | Meats                    |    98    |    98    |       ..       |
+  | Fish                     |    96    |    97    |       ..       |
+  | Poultry                  |    96    |    97    |       ..       |
+  | Eggs                     |    97    |    98    |       ..       |
+  | Dairy products           |    97    |    96    |       98       |
+  | Total animal food of     |          |          |                |
+  |   mixed diet             |    97    |    97    |       98       |
+  | Potatoes                 |    73    |    ..    |       98       |
+  | Beets, carrots, &c.      |    72    |    ..    |       97       |
+  | Cabbage, lettuce, &c.    |    ..    |    ..    |       83       |
+  | Legumes                  |    78    |    90    |       95       |
+  | Oatmeal                  |    78    |    90    |       97       |
+  | Corn meal                |    80    |    ..    |       99       |
+  | Wheat meals without bran |    83    |    ..    |       93       |
+  | Wheat meals with bran    |    75    |    ..    |       92       |
+  | White bread              |    88    |    ..    |       98       |
+  | Entire wheat bread       |    82    |    ..    |       94       |
+  | Graham bread             |    76    |    ..    |       90       |
+  | Rice                     |    76    |    ..    |       91       |
+  | Fruits and nuts          |    80    |    86    |       96       |
+  | Sugars and starches      |    ..    |    ..    |       98       |
+  | Total vegetable food of  |          |          |                |
+  |   mixed diet             |    85    |    90    |       97       |
+  | Total food of mixed diet |    92    |    95    |       97       |
+  +--------------------------+----------+----------+----------------+
+
+The digestibility of food in the sense of thoroughness of digestion,
+however, is of particular importance in the present discussion. Only
+that portion of the food that is digested and absorbed is available to
+the body for the building of tissue and the production of energy. Not
+all the food eaten is thus actually digested; undigested material is
+excreted in the faeces. The thoroughness of digestion is determined
+experimentally by weighing and analysing the food eaten and the faeces
+pertaining to it. The difference between the corresponding ingredients
+of the two is commonly considered to represent the amounts of the
+ingredients digested. Expressed in percentages, these are called
+coefficients of digestibility. See Table II.
+
+Such a method is not strictly accurate, because the faeces do not
+consist entirely of undigested food but contain in addition to this the
+so-called metabolic products, which include the residuum of digestive
+juices not resorbed, fragments of intestinal epithelium, &c. Since there
+is as yet no satisfactory method of separating these constituents of the
+excreta, the actual digestibility of the food is not determined. It has
+been suggested that since these materials must originally come from
+food, they represent, when expressed in terms of food ingredients, the
+cost of digestion; hence that the values determined as above explained
+represent the portion of food available to the body for the building of
+tissue and the yielding of energy, and what is commonly designated as
+digestibility should be called availability. Other writers retain the
+term "digestibility," but express the results as "apparent
+digestibility," until more knowledge regarding the metabolic products of
+the excreta is available and the actual digestibility may be
+ascertained.
+
+Experimental inquiry of this nature has been very active in recent
+years, especially in Europe, the United States and Japan; and the
+results of considerably over 1000 digestion experiments with single
+foods or combinations of food materials are available. These were mostly
+with men, but some were with women and with children. The larger part of
+these have been taken into account in the following estimations of the
+digestibility of the nutrients in different classes of food materials.
+The figures here shown are subject to revision as experimental data
+accumulate. They are not to be taken as exact measures of the
+digestibility (or availability) of every kind of food in each given
+class, but they probably represent fairly well the average digestibility
+of the classes of food materials as ordinarily utilized in the mixed
+diet.
+
+5. _Fuel Value of Food._--The potential energy of food is commonly
+measured as the amount of heat evolved when the food is completely
+oxidized. In the laboratory this is determined by burning the food in
+oxygen in a calorimeter. The results, which are known as the heat of
+combustion of the food, are expressed in calories, one calory being the
+amount of heat necessary to raise the temperature of one kilogram of
+water one degree centigrade. But it is to be observed that this unit is
+employed simply from convenience, and without implication as to what
+extent the energy of food is converted into heat in the body. The unit
+employed in the measurement of some other form of energy might be used
+instead, as, for example, the foot-ton, which represents the amount of
+energy necessary to raise one ton through one foot.
+
+  TABLE III.--_Estimates of Heats of Combustion and of Fuel Value of
+  Nutrients in Ordinary Mixed Diet._
+
+  +---------------------------+-------------+-------------+
+  |         Nutrients.        |   Heat of   | Fuel Value. |
+  |                           | Combustion. |             |
+  +---------------------------+-------------+-------------+
+  |                           |             |             |
+  |                           |  Calories.  |  Calories.  |
+  |                           |             |             |
+  | One gram of protein       |    5.65     |    4.05     |
+  | One gram of fats          |    9.40     |    8.93     |
+  | One gram of carbohydrates |    4.15     |    4.03     |
+  |                           |             |             |
+  +---------------------------+-------------+-------------+
+
+  The amount of energy which a given quantity of food will produce on
+  complete oxidation outside the body, however, is greater than that
+  which the body will actually derive from it. In the first place, as
+  previously shown, part of the food will not be digested and absorbed.
+  In the second place, the nitrogenous compounds absorbed are not
+  completely oxidized in the body, the residuum being excreted in the
+  urine as urea and other bodies that are capable of further oxidation
+  in the calorimeter. The total heat of combustion of the food eaten
+  must therefore be diminished by the heat of combustion of the
+  oxidizable material rejected by the body, to find what amount of
+  energy is actually available to the organism for the production of
+  work and heat. The amount thus determined is commonly known as the
+  fuel value of food.
+
+  Rubner's[7] commonly quoted estimates for the fuel value of the
+  nutrients of mixed diet are,--for protein and carbohydrates 4.1, and
+  for fats 9.3 calories per gram. According to the method of deduction,
+  however, these factors were more applicable to digested than to total
+  nutrients. Atwater[8] and associates have deduced, from data much more
+  extensive than those available to Rubner, factors for total nutrients
+  somewhat lower than these, as shown in Table III. These estimates seem
+  to represent the best average factors at present available, but are
+  subject to revision as knowledge is extended.
+
+
+      TABLE IV.--_Quantities of Available Nutrients and Energy in Daily
+      Food Consumption of Persons in Different Circumstances._
+
+    +------------------------------------------+--------+--------------------------------+
+    |                                          |        |     Nutrients and Energy       |
+    |                                          | Number |        per Man per Day.        |                                |
+    |                                          |   of   +------+------+--------+---------+
+    |                                          |Studies.| Pro- | Fat. |Carbohy-|  Fuel   |
+    |                                          |        | tein.|      | drates.| Value.  |
+    +------------------------------------------+--------+------+------+--------+---------+
+    |                                          |        |      |      |        |         |
+    |     _Persons with Active Work._          |        |Grams.|Grams.|  Grams.|Calories.|
+    | English royal engineers                  |    1   |  132 |   79 |   612  |   3835  |
+    | Prussian machinists                      |    1   |  129 |  107 |   657  |   4265  |
+    | Swedish mechanics                        |    5   |  174 |  105 |   693  |   4590  |
+    | Bavarian lumbermen                       |    3   |  120 |  277 |   702  |   6015  |
+    | American lumbermen                       |    5   |  155 |  327 |   804  |   6745  |
+    | Japanese rice cleaner                    |    1   |  103 |   11 |   917  |   4415  |
+    | Japanese jinrikshaw runner               |    1   |  137 |   22 |  1010  |   5050  |
+    | Chinese farm labourers in California     |    1   |  132 |   90 |   621  |   3980  |
+    | American athletes                        |   19   |  178 |  192 |   525  |   4740  |
+    | American working-men's families          |   13   |  156 |  226 |   694  |   5650  |
+    |                                          |        |      |      |        |         |
+    |                                          |        |      |      |        |         |
+    |     _Persons with Ordinary Work._        |        |      |      |        |         |
+    | Bavarian mechanics.                      |   11   |  112 |   32 |   553  |   3060  |
+    | Bavarian farm labourers                  |    5   |  126 |   52 |   526  |   3200  |
+    | Russian peasants                         |   ..   |  119 |   31 |   571  |   3155  |
+    | Prussian prisoners                       |    1   |  117 |   28 |   620  |   3320  |
+    | Swedish mechanics.                       |    6   |  123 |   75 |   507  |   3325  |
+    | American working-men's families          |   69   |  105 |  135 |   426  |   3480  |
+    |                                          |        |      |      |        |         |
+    |     _Persons with Light Work._           |        |      |      |        |         |
+    | American artisans' families              |   21   |   93 |  107 |   358  |   2880  |
+    | English tailors (prisoners)              |    1   |  121 |   37 |   509  |   2970  |
+    | German shoemakers                        |    1   |   99 |   73 |   367  |   2629  |
+    | Japanese prisoners                       |    1   |   43 |    6 |   444  |   2110  |
+    |                                          |        |      |      |        |         |
+    |     _Professional and Business Men._     |        |      |      |        |         |
+    | Japanese professional men.               |   13   |   75 |   15 |   408  |   2190  |
+    | Japanese students                        |    8   |   85 |   18 |   537  |   2800  |
+    | Japanese military cadets                 |   11   |   98 |   20 |   611  |   3185  |
+    | German physicians                        |    2   |  121 |   90 |   317  |   2685  |
+    | Swedish medical students                 |    5   |  117 |  108 |   291  |   2725  |
+    | Danish physicians                        |    1   |  124 |  133 |   242  |   2790  |
+    | American professional and business       |        |      |      |        |         |
+    |   men and students                       |   51   |   98 |  125 |   411  |   3285  |
+    |                                          |        |      |      |        |         |
+    |     _Persons with Little or no Exercise._|        |      |      |        |         |
+    | Prussian prisoners                       |    2   |   90 |   27 |   427  |   2400  |
+    | Japanese prisoners                       |    1   |   36 |    6 |   360  |   1725  |
+    | Inmates of home for aged--Germany        |    1   |   85 |   43 |   322  |   2097  |
+    | Inmates of hospitals for insane--America |   49   |   80 |   86 |   353  |   2590  |
+    |                                          |        |      |      |        |         |
+    |     _Persons in Destitute Circumstances._|        |      |      |        |         |
+    | Prussian working people                  |   13   |   63 |   43 |   372  |   2215  |
+    | Italian mechanics                        |    5   |   70 |   36 |   384  |   2225  |
+    | American working-men's families          |   11   |   69 |   75 |   263  |   2085  |
+    +------------------------------------------+--------+------+------+--------+---------+
+
+The heats of combustion of all the fats in an ordinary mixed diet would
+average about 9.40 calories per gram, but as only 95% of the fat would
+be available to the body, the fuel value per gram would be (9.40 × 0.95
+=) 8.93 calories. Similarly, the average heat of combustion of
+carbohydrates of the diet would be about 4.15 calories per gram, and as
+97% of the total quantity is available to the body, the fuel value per
+gram would be 4.03. (It is commonly assumed that the resorbed fats and
+carbohydrates are completely oxidized in the body.) The heats of
+combustion of all the kinds of protein in the diet would average about
+5.65 calories per gram. Since about 92% of the total protein would be
+available to the body, the potential energy of the available protein
+would be equivalent to (5.65 × 0.92 =) 5.20 calories; but as the
+available protein is not completely oxidized allowance must be made for
+the potential energy of the incompletely oxidized residue. This is
+estimated as equivalent to 1.15 calories for the 0.92 gram of available
+protein; hence, the fuel value of the total protein is (5.20 - 1.15 =)
+4.05 calories per gram. Nutrients of the same class, but from different
+food materials, vary both in digestibility and in heat of combustion,
+and hence in fuel value. These factors are therefore not so applicable
+to the nutrients of the separate articles in a diet as to those of the
+diet as a whole.
+
+6. _Food Consumption._--Much information regarding the food consumption
+of people in various circumstances in different parts of the world has
+accumulated during the past twenty years, as a result of studies of
+actual dietaries in England, Germany, Italy, Russia, Sweden and
+elsewhere in Europe, in Japan and other oriental countries, and
+especially in the United States. These studies commonly consist in
+ascertaining the kinds, amounts and composition of the different food
+materials consumed by a group of persons during a given period and the
+number of meals taken by each member of the group, and computing the
+quantities of the different nutrients in the food on the basis of one
+man for one day. When the members of the group are of different age,
+sex, occupation, &c., account must be taken of the effect of these
+factors on consumption in estimating the value "per man." Men as a rule
+eat more than women under similar conditions, women more than children,
+and persons at active work more than those at sedentary occupation. The
+navvy, for example, who is constantly using up more nutritive material
+or body tissue to supply the energy required for his muscular work needs
+more protein and energy in his food than a bookkeeper who sits at his
+desk all day.
+
+In making allowance for these differences, the various individuals are
+commonly compared with a man at moderately active muscular work, who is
+taken as unity. A man at hard muscular work is reckoned at 1.2 times
+such an individual; a man with light muscular work or a boy 15-16 years
+old, .9; a man at sedentary occupation, woman at moderately active
+muscular work, boy 13-14 or girl 15-16 years old, .8; woman at light
+work, boy 12 or girl 13-14 years old, .7; boy 10-11 or girl 10-12 years
+old, .6; child 6-9 years old, .5; child 2-5 years old, .4; child under 2
+years, .3. These factors are by no means absolute or final, but are
+based in part upon experimental data and in part upon arbitrary
+assumption.
+
+The total number of dietary studies on record is very large, but not all
+of them are complete enough to furnish reliable data. Upwards of 1000
+are sufficiently accurate to be included in statistical averages of food
+consumed by people in different circumstances, nearly half of which have
+been made in the United States in the past decade. The number of persons
+in the individual studies has ranged from one to several hundred. Some
+typical results are shown in Table IV.
+
+7. _Quantities of Nutrients needed._--For the proper nourishment of the
+body, the important problem is how much protein, fats and carbohydrates,
+or more simply, what amounts of protein and potential energy are needed
+under varying circumstances, to build and repair muscular and other
+tissues and to supply energy for muscular work, heat and other forms of
+energy. The answer to the problem is sought in the data obtained in
+dietary studies with considerable numbers of people, and in metabolism
+experiments with individuals in which the income and expenditure of the
+body are measured. From the information thus derived, different
+investigators have proposed so-called dietary standards, such as are
+shown in the table below, but unfortunately the experimental data are
+still insufficient for entirely trustworthy figures of this sort; hence
+the term "standard" as here used is misleading. The figures given are
+not to be considered as exact and final as that would suggest; they are
+merely tentative estimates of the average daily amounts of nutrients and
+energy required. (It is to be especially noted that these are available
+nutrients and fuel value rather than total nutrients and energy.) Some
+of the values proposed by other investigators are slightly larger than
+these, and others are decidedly smaller, but these are the ones that
+have hitherto been most commonly accepted in Europe and America.
+
+
+    TABLE V.--_Standards for Dietaries. Available Nutrients and Energy
+    per Man per Day._
+
+  +---------------------------+---------+--------+---------+---------+
+  |                           | Protein.|  Fat.  | Carbo-  |  Fuel   |
+  |                           |         |        |hydrates.|  Value. |
+  +---------------------------+---------+--------+---------+---------+
+  |                           |         |        |         |         |
+  |   _Voit's Standards._     |Grams.[9]| Grams. |  Grams. |Calories.|
+  | Man at hard work          |   133   |   95   |   437   |  3270   |
+  | Man at moderate work      |   109   |   53   |   485   |  2965   |
+  |    _Atwater's Standards._ |         |        |         |         |
+  | Man at very hard          |         |        |         |         |
+  |   muscular work           |   161   |  ..[10]|   ..[10]|  5500   |
+  | Man at hard muscular work |   138   |   ..   |   ..    |  4150   |
+  | Man at moderately         |         |        |         |         |
+  |   active muscular work    |   115   |   ..   |   ..    |  3400   |
+  | Man at light to moderate  |         |        |         |         |
+  |    muscular work          |   103   |   ..   |   ..    |  3050   |
+  | Man at "sedentary"        |         |        |         |         |
+  |    or woman at moderately |         |        |         |         |
+  |    active work            |    92   |   ..   |   ..    |  2700   |
+  | Woman at light muscular   |         |        |         |         |
+  |    work, or man without   |         |        |         |         |
+  |    muscular exercise      |    83   |   ..   |   ..    |  2450   |
+  |                           |         |        |         |         |
+  +---------------------------+---------+--------+---------+---------+
+
+8. _Hygienic Economy of Food._--For people in good health, there are two
+important rules to be observed in the regulation of the diet. One is to
+choose the foods that "agree" with them, and to avoid those which they
+cannot digest and assimilate without harm; and the other is to use such
+sorts and quantities of foods as will supply the kinds and amounts of
+nutrients needed by the body and yet to avoid burdening it with
+superfluous material to be disposed of at the cost of health and
+strength.
+
+As for the first-mentioned rule, it is practically impossible to give
+information that may be of more than general application. There are
+people who, because of some individual peculiarity, cannot use foods
+which for people in general are wholesome and nutritious. Some persons
+cannot endure milk, others suffer if they eat eggs, others have to
+eschew certain kinds of meat, or are made uncomfortable by fruit; but
+such cases are exceptions. Very little is known regarding the cause of
+these conditions. It is possible that in the metabolic processes to
+which the ingredients of the food are subjected in the body, or even
+during digestion before the substances are actually taken into the body,
+compounds may be formed that are in one way or another injurious.
+Whatever the cause may be, it is literally true in this sense that "what
+is one man's meat is another man's poison," and each must learn for
+himself what foods "agree" with him and what ones do not. But for the
+great majority of people in health, suitable combinations of the
+ordinary sorts of wholesome food materials make a healthful diet. On the
+other hand, some foods are of particular value at times, aside from
+their use for nourishment. Fruits and green vegetables often benefit
+people greatly, not as nutriment merely, for they may have very little
+actual nutritive material, but because of fruit or vegetable acids or
+other substances which they contain, and which sometimes serve a most
+useful purpose.
+
+
+    TABLE VI.--_Amounts of Nutrients and Energy Furnished for One
+    Shilling in Food Materials at Ordinary Prices._
+
+  +----------------------+-------+----------------------------------------------+
+  |                      |       |            One Shilling will buy             |
+  |                      |       +----------+-------------------------+---------+
+  |   Food Materials     |Prices |          |   Available Nutrients.  |         |
+  |    as Purchased.     |  per  |Total Food+-------------------------+   Fuel  |
+  |                      |  lb.  |materials.|        |      | Carbo-  |  Value. |
+  |                      |       |          |Protein.| Fat. |hydrates.|         |
+  +----------------------+-------+----------+--------+------+---------+---------+
+  |                      | s. d. |    lb.   |    lb. |  lb. |    lb.  |Calories.|
+  | Beef, round          | 0 10  |   1.20   |   .22  |  .14 |    ..   |  1,155  |
+  |                      | 0  8½ |   1.41   |   .26  |  .17 |    ..   |  1,235  |
+  |                      | 0  5  |   2.40   |   .44  |  .29 |    ..   |  2,105  |
+  |                      |       |          |        |      |         |         |
+  | Beef, sirloin        | 0 10  |   1.20   |   .19  |  .20 |    ..   |  1,225  |
+  |                      | 0  9  |   1.33   |   .21  |  .22 |    ..   |  1,360  |
+  |                      | 0  8  |   1.50   |    ..  |   .. |    ..   |    ..   |
+  |                      | 0  5  |   2.40   |    ..  |   .. |    ..   |    ..   |
+  |                      |       |          |        |      |         |         |
+  | Beef, rib            | 0  9  |   1.33   |   .19  |  .19 |    ..   |  1,200  |
+  |                      | 0  7½ |   1.60   |    ..  |   .. |    ..   |    ..   |
+  |                      | 0  4½ |   2.67   |    ..  |   .. |    ..   |    ..   |
+  |                      |       |          |        |      |         |         |
+  | Mutton, leg          | 0  9  |   1.33   |   .20  |  .20 |    ..   |  1,245  |
+  |                      | 0  5  |   2.40   |   .37  |  .35 |    ..   |  2,245  |
+  |                      |       |          |        |      |         |         |
+  | Pork, spare-rib      | 0  9  |   1.33   |   .17  |  .31 |    ..   |  1,645  |
+  |                      | 0  7  |   1.71   |   .22  |  .39 |    ..   |  2,110  |
+  |                      |       |          |        |      |         |         |
+  | Pork, salt, fat      | 0  7  |   1.71   |   .03  | 1.40 |    ..   |  6,025  |
+  |                      | 0  5  |   2.40   |   .04  | 1.97 |    ..   |  8,460  |
+  |                      |       |          |        |      |         |         |
+  | Pork, smoked ham     | 0  8  |   1.50   |   .20  |  .48 |    ..   |  2,435  |
+  |                      | 0  4½ |   2.67   |   .36  |  .85 |    ..   |  4,330  |
+  |                      |       |          |        |      |         |         |
+  | Fresh cod            | 0  4  |   3.00   |   .34  |  .01 |    ..   |    710  |
+  |                      | 0  3  |   4.00   |   .45  |  .01 |    ..   |    945  |
+  |                      |       |          |        |      |         |         |
+  | Salt cod             | 0  3½ |   3.43   |   .54  |  .07 |    ..   |  1,370  |
+  |                      | 0 10  |   1.20   |   .07  |  .01 |   .04   |    275  |
+  |                      |       |          |        |      |         |         |
+  |Milk, whole, 4d. a qt.| 0  2  |   6.00   |   .19  |  .23 |   .30   |  1,915  |
+  |    "        3d. a qt.| 0  1½ |   8.00   |   .26  |  .30 |   .40   |  2,550  |
+  |    "        2d. a qt.| 0  1  |  12.00   |   .38  |  .46 |   .60   |  3,825  |
+  |                      |       |          |        |      |         |         |
+  | Milk, skimmed, 2d. a | 0  1  |  12.00   |   .40  |  .03 |   .61   |  2,085  |
+  |  qt.                 |       |          |        |      |         |         |
+  | Butter               | 1  6  |    .67   |   .01  |  .54 |    ..   |  2,320  |
+  |                      | 1  3  |    .80   |   .01  |  .64 |    ..   |  2,770  |
+  |                      | 1  0  |   1.00   |   .01  |  .81 |    ..   |  3,460  |
+  |                      |       |          |        |      |         |         |
+  | Margarine            | 0  4  |   3.00   |    ..  | 2.37 |    ..   | 10,080  |
+  |                      |       |          |        |      |         |         |
+  | Eggs, 2s. a dozen    | 1  4  |    .75   |   .10  |  .07 |    ..   |    475  |
+  |   "   1½s. a dozen   | 1  0  |   1.00   |   .13  |  .09 |    ..   |    635  |
+  |   "    1s. a dozen   | 0  8  |   1.50   |   .19  |  .13 |    ..   |    950  |
+  |                      |       |          |        |      |         |         |
+  | Cheese               | 0  8  |   1.50   |   .38  |  .48 |   .04   |  2,865  |
+  |                      | 0  7  |   1.71   |   .43  |  .55 |   .04   |  3,265  |
+  |                      | 0  5  |   2.40   |   .60  |  .77 |   .06   |  4,585  |
+  |                      |       |          |        |      |         |         |
+  | Wheat bread          |0 1-1/8|  10.67   |   .76  |  .13 |  5.57   | 12,421  |
+  |                      |       |          |        |      |         |         |
+  | Wheat flour          |0 1-3/5|   7.64   |   .67  |  .07 |  5.63   | 12,110  |
+  |                      | 0 1½  |   8.16   |   .72  |  .07 |  6.01   | 12,935  |
+  |                      |       |          |        |      |         |         |
+  | Oatmeal              |0 1-2/5|   8.39   |  1.11  |  .54 |  5.54   | 14,835  |
+  |                      | 0 1½  |   8.16   |  1.08  |  .53 |  5.39   | 14,430  |
+  |                      |       |          |        |      |         |         |
+  | Rice                 | 0 1¾  |   6.86   |   .45  |  .02 |  5.27   | 10,795  |
+  |                      |       |          |        |      |         |         |
+  | Potatoes             |0 0-2/3|  18.00   |   .25  |  .02 |  2.70   |  5,605  |
+  |                      | 0 0½  |  24.00   |   .34  |  .02 |  3.60   |  7,470  |
+  |                      |       |          |        |      |         |         |
+  | Beans                | 0 2   |   6.00   |  1.05  |  .10 |  3.47   |  8,960  |
+  |                      |       |          |        |      |         |         |
+  | Sugar                | 1 ¾   |   6.86   |   ..   |  ..  |  6.86   | 12,760  |
+  +----------------------+-------+----------+--------+------+---------+---------+
+
+The proper observance of the second rule mentioned requires information
+regarding the demands of the body for food under different
+circumstances. To supply this information is one purpose of the effort
+to determine the so-called dietary standards mentioned above. It should
+be observed, however, that these are generally more applicable to the
+proper feeding of a group or class of people as a whole than for
+particular individuals in this class. The needs of individuals will vary
+largely from the average in accordance with the activity and
+individuality. Moreover, it is neither necessary nor desirable for the
+individual to follow any standard exactly from day to day. It is
+requisite only that the average supply shall be sufficient to meet the
+demands of the body during a given period.
+
+The cooking of food and other modes of preparing it for consumption have
+much to do with its nutritive value. Many materials which, owing to
+their mechanical condition or to some other cause, are not particularly
+desirable food materials in their natural state, are quite nutritious
+when cooked or otherwise prepared for consumption. It is also a matter
+of common experience that well-cooked food is wholesome and appetizing,
+whereas the same material poorly prepared is unpalatable. There are
+three chief purposes of cooking; the first is to change the mechanical
+condition of the food. Heating changes the structure of many food
+materials very materially, so that they may be more easily chewed and
+brought into a condition in which the digestive juices can act upon them
+more freely, and in this way probably influencing the ease and
+thoroughness of digestion. The second is to make the food more
+appetizing by improving the appearance or flavour or both. Food which is
+attractive to the eye and pleasing to the palate quickens the flow of
+saliva and other digestive juices and thus aids digestion. The third is
+to kill, by heat, disease germs, parasites or other dangerous organisms
+that may be contained in food. This is often a very important matter and
+applies to both animal and vegetable foods. Scrupulous neatness should
+always be observed in storing, handling and serving food. If ever
+cleanliness is desirable it must be in the things we eat, and every care
+should be taken to ensure it for the sake of health as well as of
+decency. Cleanliness in this connexion means not only absence of visible
+dirt, but freedom from undesirable bacteria and other minute organisms
+and from worms and other parasites. If food, raw or cooked, is kept in
+dirty places, peddled from dirty carts, prepared in dirty rooms and in
+dirty dishes, or exposed to foul air, disease germs and other offensive
+and dangerous substances may easily enter it.
+
+9. _Pecuniary Economy of Food._--Statistics of economy and of cost of
+living in Great Britain, Germany and the United States show that at
+least half, and commonly more, of the income of wage-earners and other
+people in moderate circumstances is expended for subsistence. The
+relatively large cost of food, and the important influence of diet upon
+health and strength, make a more widespread understanding of the subject
+of dietetics very desirable. The maxim that "the best is the cheapest"
+does not apply to food. The "best" food, in the sense of that which is
+the finest in appearance and flavour and which is sold at the highest
+price, is not generally the most economical.
+
+The price of food is not regulated largely by its value for nutriment.
+Its agreeableness to the palate or to the buyer's fancy is a large
+factor in determining the current demand and market price. There is no
+more nutriment in an ounce of protein or fat from the tender-loin of
+beef than from the round or shoulder. The protein of animal food has,
+however, some advantage over that of vegetable foods in that it is more
+thoroughly, and perhaps more easily, digested, for which reason it would
+be economical to pay somewhat more for the same quantity of nutritive
+material in the animal food. Furthermore, animal foods such as meats,
+fish and the like, gratify the palate as most vegetable foods do not.
+For persons in good health, foods in which the nutrients are the most
+expensive are like costly articles of adornment. People who can well
+afford them may be justified in buying them, but they are not
+economical. The most economical food is that which is at the same time
+most healthful and cheapest.
+
+The variations in the cost of the actual nutriment in different food
+materials may be illustrated by comparison of the amounts of nutrients
+obtained for a given sum in the materials as bought at ordinary market
+prices. This is done in Table VI., which shows the amounts of available
+nutrients contained in the quantities of different food materials that
+may be purchased for one shilling at prices common in England.
+
+When proper attention is given to the needs of the body for food and the
+relation between cost and nutritive value of food materials, it will be
+found that with care in the purchase and skill in the preparation of
+food, considerable control may be had over the expensiveness of a
+palatable, nutritious and healthful diet.
+
+  AUTHORITIES.--COMPOSITION OF FOODS:--König, _Chemie der menschlichen
+  Nahrungs- und Genussmittel_; Atwater and Bryant, "Composition of
+  American Food Materials," Bul. 28, Office of Experiment Stations, U.S.
+  Department of Agriculture. NUTRITION AND DIETETICS:--Armsby,
+  _Principles of Animal Nutrition_; Lusk, _The Science of Nutrition_;
+  Burney Yeo, _Food in Health and Disease_; Munk and Uffelmann, _Die
+  Ernährung des gesunden und kranken Menschen_; Von Leyden,
+  _Ernährungstherapie und Diätetik_; Dujardin-Beaumetz, Hygiène
+  alimentaire; Hutchison, _Food and Dietetics_; R. H. Chittenden,
+  _Physiological Economy in Nutrition_ (1904), _Nutrition of Man_
+  (1907); Atwater, "Chemistry and Economy of Food," Bul. 21, Office of
+  Experiment Stations, U.S. Department of Agriculture. See also other
+  Bulletins of the same office on composition of food, results of
+  dietary studies, metabolism experiments, &c., in the United States.
+  GENERAL METABOLISM:--Voit, _Physiologie des allgemeinen Stoffwechsels
+  und der Ernährung_; Hermann, _Handbuch der Physiologie_, Bd. vi.; Von
+  Noorden, _Pathologie des Stoffwechsels_; Schäfer, _Text-Book of
+  Physiology_, vol. i.; Atwater and Langworthy, "Digest of Metabolism
+  Experiments," Bull. 45, Office of Experiment Stations, U.S. Department
+  of Agriculture.     (W. O. A.; R. D. M.)
+
+
+FOOTNOTES:
+
+  [1] The terms applied by different writers to these nitrogenous
+    compounds are conflicting. For instance, the term "proteid" is
+    sometimes used as protein is here used, and sometimes to designate
+    the group here called albuminoids. The classification and terminology
+    here followed are those tentatively recommended by the Association of
+    American Agricultural Colleges and Experiment Stations.
+
+  [2] Folin, _Festschrift für Olaf Hammarsten_, iii. (Upsala, 1906).
+
+  [3] _Ztschr. Biol._ 30, 73.
+
+  [4] In Russian. Cited in United States Department of Agriculture,
+    Office of Experiment Stations, Bul. No. 45, _A Digest of Metabolism
+    Experiments_, by W. O. Atwater and C. F. Langworthy.
+
+  [5] _Arch. physiol. norm. et path._ (1894) 4.
+
+  [6] U.S. Department of Agriculture, Office of Experiment Stations,
+    Bulletins Nos. 63, 69, 109, 136, 175. For a description of the
+    respiration calorimeter here mentioned see also publication No. 42 of
+    the Carnegie Institution of Washington.
+
+  [7] _Ztschr. Biol._ 21 (1885), p. 377.
+
+  [8] _Connecticut_ (Storrs) _Agricultural Experiment Station Report_
+    (1899), 73.
+
+  [9] One ounce equals 28.35 grams.
+
+  [10] As the chief function of both fats and carbohydrates is to
+    furnish energy, their exact proportion in the diet is of small
+    account. The amount of either may vary largely according to taste,
+    available supply, or other condition, as long as the total amount of
+    both is sufficient, together with the protein to furnish the required
+    energy.
+
+
+
+
+DIETRICH, CHRISTIAN WILHELM ERNST (1712-1774), German painter, was born
+at Weimar, where he was brought up early to the profession of art by his
+father Johann George, then painter of miniatures to the court of the
+duke. Having been sent to Dresden to perfect himself under the care of
+Alexander Thiele, he had the good fortune to finish in two hours, at the
+age of eighteen, a picture which attracted the attention of the king of
+Saxony. Augustus II. was so pleased with Dietrich's readiness of hand
+that he gave him means to study abroad, and visit in succession the
+chief cities of Italy and the Netherlands. There he learnt to copy and
+to imitate masters of the previous century with a versatility truly
+surprising. Winckelmann, to whom he had been recommended, did not
+hesitate to call him the Raphael of landscape. Yet in this branch of his
+practice he merely imitated Salvator Rosa and Everdingen. He was more
+successful in aping the style of Rembrandt, and numerous examples of
+this habit may be found in the galleries of St Petersburg, Vienna and
+Dresden. At Dresden, indeed, there are pictures acknowledged to be his,
+bearing the fictitious dates of 1636 and 1638, and the name of
+Rembrandt. Among Dietrich's cleverest reproductions we may account that
+of Ostade's manner in the "Itinerant Singers" at the National Gallery.
+His skill in catching the character of the later masters of Holland is
+shown in candlelight scenes, such as the "Squirrel and the Peep-Show" at
+St Petersburg, where we are easily reminded of Godfried Schalcken.
+Dietrich tried every branch of art except portraits, painting Italian
+and Dutch views alternately with Scripture scenes and still life. In
+1741 he was appointed court painter to Augustus III. at Dresden, with an
+annual salary of 400 thalers (£60), conditional on the production of
+four cabinet pictures a year. This condition, no doubt, accounts for the
+presence of fifty-two of the master's panels and canvases in one of the
+rooms at the Dresden museum. Dietrich, though popular and probably the
+busiest artist of his time, never produced anything of his own; and his
+imitations are necessarily inferior to the originals which he affected
+to copy. His best work is certainly that which he gave to engravings. A
+collection of these at the British Museum, produced on the general lines
+of earlier men, such as Ostade and Rembrandt, reveal both spirit and
+skill. Dietrich, after his return from the Peninsula, generally signed
+himself "Dietericij," and with this signature most of his extant
+pictures are inscribed. He died at Dresden, after he had successively
+filled the important appointments of director of the school of painting
+at the Meissen porcelain factory and professor of the Dresden academy of
+arts.
+
+
+
+
+DIETRICH OF BERN, the name given in German popular poetry to Theodoric
+the Great. The legendary history of Dietrich differs so widely from the
+life of Theodoric that it has been suggested that the two were
+originally unconnected. Medieval chroniclers, however, repeatedly
+asserted the identity of Dietrich and Theodoric, although the more
+critical noted the anachronisms involved in making Ermanaric (d. 376)
+and Attila (d. 453) contemporary with Theodoric (b. 455). That the
+legend is based on vague historical reminiscences is proved by the
+retention of the names of Theodoric (Thiuda-reiks, Dietrich) and his
+father Theudemir (Dietmar), by Dietrich's connexion with Bern (Verona)
+and Raben (Ravenna). Something of the Gothic king's character descended
+to Dietrich, familiarly called the Berner, the favourite of German
+medieval saga heroes, although his story did not leave the same mark on
+later German literature as did that of the Nibelungs. The cycle of songs
+connected with his name in South Germany is partially preserved in the
+Heldenbuch (q.v.) in _Dietrich's Flucht_, the _Rabenschlacht_ and
+_Alpharts Tod_; but it was reserved for an Icelandic author, writing in
+Norway in the 13th century, to compile, with many romantic additions, a
+consecutive account of Dietrich. In this Norse prose redaction, known as
+the _Vilkina Saga_, or more correctly the _Thidrekssaga_, is
+incorporated much extraneous matter from the Nibelungen and Wayland
+legends, in fact practically the whole of south German heroic tradition.
+
+There are traces of a form of the Dietrich legend in which he was
+represented as starting out from Byzantium, in accordance with
+historical tradition, for his conquest of Italy. But this early
+disappeared, and was superseded by the existing legend, in which,
+perhaps by an "epic fusion" with his father Theudemir, he was associated
+with Attila, and then by an easy transition with Ermanaric. Dietrich was
+driven from his kingdom of Bern by his uncle Ermanaric. After years of
+exile at the court of Attila he returned with a Hunnish army to Italy,
+and defeated Ermanaric in the Rabenschlacht, or battle of Ravenna.
+Attila's two sons, with Dietrich's brother, fell in the fight, and
+Dietrich returned to Attila's court to answer for the death of the young
+princes. This very improbable renunciation of the advantages of his
+victory suggests that in the original version of the story the
+Rabenschlacht was a defeat. In the poem of _Ermenrichs Tod_ he is
+represented as slaying Ermanaric, as in fact Theodoric slew Odoacer.
+"Otacher" replaces Ermanaric as his adversary in the _Hildebrandslied_,
+which relates how thirty years after the earlier attempt he reconquered
+his Lombard kingdom. Dietrich's long residence at Attila's court
+represents the youth and early manhood of Theodoric spent at the
+imperial court and fighting in the Balkan peninsula, and, in accordance
+with epic custom, the period of exile was adorned with war-like
+exploits, with fights with dragons and giants, most of which had no
+essential connexion with the cycle. The romantic poems of _König
+Laurin_, _Sigenot_, _Eckenlied_ and _Virginal_ are based largely on
+local traditions originally independent of Dietrich. The court of Attila
+(Etzel) was a ready bridge to the Nibelungen legend. In the final
+catastrophe he was at length compelled, after steadily holding aloof
+from the combat, to avenge the slaughter of his Amelungs by the
+Burgundians, and delivered Hagen bound into the hands of Kriemhild. The
+flame breath which anger induced from him shows the influence of pure
+myth, but the tales of his demonic origin and of his being carried off
+by the devil in the shape of a black horse may safely be put down to the
+clerical hostility to Theodoric's Arianism.
+
+Generally speaking, Dietrich of Bern was the wise and just monarch as
+opposed to Ermanaric, the typical tyrant of Germanic legend. He was
+invariably represented as slow of provocation and a friend of peace, but
+once roused to battle not even Siegfried could withstand his onslaught.
+But probably Dietrich's fight with Siegfried in Kriemhild's rose garden
+at Worms is a late addition to the Rosengarten myth. The chief heroes of
+the Dietrich cycle are his tutor and companion in arms, Hildebrand (see
+HILDEBRAND, lay of), with his nephews the Wolfings Alphart and Wolfhart;
+Wittich, who renounced his allegiance to Dietrich and slew the sons of
+Attila; Heime and Biterolf.
+
+  The contents of the poems dealing with the Dietrich cycle are
+  summarized by Uhland in _Schriften zur Geschichte der Dichtung und
+  Sage_ (Stuttgart, 1873). The _Thidrekssaga_ (ed. C. Unger,
+  Christiania, 1853) is translated into German by F. H. v. der Hagen in
+  _Altdeutsche und altnordische Heldensagen_ (vols. i. and ii. 3rd ed.,
+  Breslau, 1872). A summary of it forms the concluding chapter of T.
+  Hodgkin's _Theodoric the Goth_ (1891). The variations in the Dietrich
+  legend in the Latin historians, in Old and Middle High German
+  literature, and in the northern saga, can be studied in W. Grimm's
+  _Deutsche Heldensage_ (2nd ed., Berlin, 1867). There is a good account
+  in English in F. E. Sandbach's _Heroic Saga-cycle of Dietrich of Bern_
+  (1906), forming No. 15 of Alfred Nutt's _Popular Studies in
+  Mythology_, and another in M. Bentinck Smith's translation of Dr O. L.
+  Jiriczek's _Deutsche Heldensage_ (_Northern Legends_, London, 1902).
+  For modern German authorities and commentators see B. Symons,
+  "Deutsche Heldensage" in H. Paul's _Grd. d. german. Phil._
+  (Strassburg, new ed., 1905); also Goedeke, _Geschichte der deutschen
+  Dichtung_ (i. 241-246).
+
+
+
+
+DIEZ, FRIEDRICH CHRISTIAN (1794-1876), German philologist, was born at
+Giessen, in Hesse-Darmstadt, on the 15th of March 1794. He was educated
+first at the gymnasium and then at the university of his native town.
+There he studied classics under Friedrich Gottlieb Welcker (1784-1868)
+who had just returned from a two years' residence in Italy to fill the
+chair of archaeology and Greek literature. It was Welcker who kindled in
+him a love of Italian poetry, and thus gave the first bent to his
+genius. In 1813 he joined the Hesse corps as a volunteer and served in
+the French campaign. Next year he returned to his books, and this short
+taste of military service was the only break in a long and uneventful
+life of literary labours. By his parents' desire he applied himself for
+a short time to law, but a visit to Goethe in 1818 gave a new direction
+to his studies, and determined his future career. Goethe had been
+reading Raynouard's _Selections from the Romance Poets_, and advised the
+young scholar to explore the rich mine of Provençal literature which the
+French savant had opened up. This advice was eagerly followed, and
+henceforth Diez devoted himself to Romance literature. He thus became
+the founder of Romance philology. After supporting himself for some
+years by private teaching, he removed in 1822 to Bonn, where he held the
+position of privatdocent. In 1823 he published his first work, _An
+Introduction to Romance Poetry_; in the following year appeared _The
+Poetry of the Troubadours_, and in 1829 _The Lives and Works of the
+Troubadours_. In 1830 he was called to the chair of modern literature.
+The rest of his life was mainly occupied with the composition of the two
+great works on which his fame rests, the _Grammar of the Romance
+Languages_ (1836-1844), and the _Lexicon of the Romance
+Languages--Italian, Spanish and French_ (1853); in these two works Diez
+did for the Romance group of languages what Jacob Grimm did for the
+Teutonic family. He died at Bonn on the 29th of May 1876.
+
+  The earliest French philologists, such as Perion and Henri Estienne,
+  had sought to discover the origin of French in Greek and even in
+  Hebrew. For more than a century Ménage's _Etymological Dictionary_
+  held the field without a rival. Considering the time at which it was
+  written (1650), it was a meritorious work, but philology was then in
+  the empirical stage, and many of Ménage's derivations (such as that of
+  "rat" from the Latin "mus," or of "haricot" from "faba") have since
+  become bywords among philologists. A great advance was made by
+  Raynouard, who by his critical editions of the works of the
+  Troubadours, published in the first years of the 19th century, laid
+  the foundations on which Diez afterwards built. The difference between
+  Diez's method and that of his predecessors is well stated by him in
+  the preface to his dictionary. In sum it is the difference between
+  science and guess-work. The scientific method is to follow implicitly
+  the discovered principles and rules of phonology, and not to swerve a
+  foot's breadth from them unless plain, actual exceptions shall justify
+  it; to follow the genius of the language, and by cross-questioning to
+  elicit its secrets; to gauge each letter and estimate the value which
+  attaches to it in each position; and lastly to possess the true
+  philosophic spirit which is prepared to welcome any new fact, though
+  it may modify or upset the most cherished theory. Such is the
+  historical method which Diez pursues in his grammar and dictionary. To
+  collect and arrange facts is, as he tells us, the sole secret of his
+  success, and he adds in other words the famous apophthegm of Newton,
+  "hypotheses non fingo." The introduction to the grammar consists of
+  two parts:--the first discusses the Latin, Greek and Teutonic elements
+  common to the Romance languages; the second treats of the six dialects
+  separately, their origin and the elements peculiar to each. The
+  grammar itself is divided into four books, on phonology, on flexion,
+  on the formation of words by composition and derivation, and on
+  syntax.
+
+  His dictionary is divided into two parts. The first contains words
+  common to two at least of the three principal groups of
+  Romance:--Italian, Spanish and Portuguese, and Provençal and French.
+  The Italian, as nearest the original, is placed at the head of each
+  article. The second part treats of words peculiar to one group. There
+  is no separate glossary of Wallachian.
+
+  Of the introduction to the grammar there is an English translation by
+  C. B. Cayley. The dictionary has been published in a remodelled form
+  for English readers by T. C. Donkin.
+
+
+
+
+DIEZ, a town of Germany, in the Prussian province of Hesse-Nassau,
+romantically situated in the deep valley of the Lahn, here crossed by an
+old bridge, 30 m. E. from Coblenz on the railway to Wetzlar. Pop. 4500.
+It is overlooked by a former castle of the counts of Nassau-Dillenburg,
+now a prison. Close by, on an eminence above the river, lies the castle
+of Oranienstein, formerly a Benedictine nunnery and now a cadet school,
+with beautiful gardens. There are a Roman Catholic and two Evangelical
+churches. The new part of the town is well built and contains numerous
+pretty villa residences. In addition to extensive iron-works there are
+sawmills and tanneries. In the vicinity are Fachingen, celebrated for
+its mineral waters, and the majestic castle of Schaumburg belonging to
+the prince of Waldeck-Pyrmont.
+
+
+
+
+DIFFERENCES, CALCULUS OF (_Theory of Finite Differences_), that branch
+of mathematics which deals with the successive differences of the terms
+of a series.
+
+1. The most important of the cases to which mathematical methods can be
+applied are those in which the terms of the series are the values, taken
+at stated intervals (regular or irregular), of a continuously varying
+quantity. In these cases the formulae of finite differences enable
+certain quantities, whose exact value depends on the law of variation
+(i.e. the law which governs the relative magnitude of these terms) to be
+calculated, often with great accuracy, from the given terms of the
+series, without explicit reference to the law of variation itself. The
+methods used may be extended to cases where the series is a double
+series (series of double entry), i.e. where the value of each term
+depends on the values of a pair of other quantities.
+
+2. The _first differences_ of a series are obtained by subtracting from
+each term the term immediately preceding it. If these are treated as
+terms of a new series, the first differences of this series are the
+_second differences_ of the original series; and so on. The successive
+differences are also called _differences of the first, second, ...
+order_. The differences of successive orders are most conveniently
+arranged in successive columns of a table thus:--
+
+  +-----+----------+-----------+-----------------+----------------------+
+  |Term.| 1st Diff.| 2nd Diff. |    3rd Diff.    |      4th Diff.       |
+  +-----+----------+-----------+-----------------+----------------------+
+  |     |          |           |                 |                      |
+  |  a  |          |           |                 |                      |
+  |     |   b - a  |           |                 |                      |
+  |  b  |          | c - 2b +a |                 |                      |
+  |     |   c - b  |           | d - 3c + 3b - a |                      |
+  |  c  |          | d - 2c +b |                 | e - 4d + 6c - 4b + a |
+  |     |   d - c  |           | e - 3d + 3c - b |                      |
+  |  d  |          | e - 2d +c |                 |                      |
+  |     |   e - d  |           |                 |                      |
+  |  e  |          |           |                 |                      |
+  +-----+----------+-----------+-----------------+----------------------+
+
+    _Algebra of Differences and Sums._
+
+  [Illustration: FIG. 1.]
+
+  3. The formal relations between the terms of the series and the
+  differences may be seen by comparing the arrangements (A) and (B) in
+  fig. 1. In (A) the various terms and differences are the same as in §
+  2, but placed differently. In (B) we take a new series of terms
+  [alpha], [beta], [gamma], [delta], commencing with the same term
+  [alpha], and take the successive sums of pairs of terms, instead of
+  the successive differences, but place them to the left instead of to
+  the right. It will be seen, in the first place, that the successive
+  terms in (A), reading downwards to the right, and the successive terms
+  in (B), reading downwards to the left, consist each of a series of
+  terms whose coefficients follow the binomial law; i.e. the
+  coefficients in b - a, c - 2b + a, d - 3c + 3b - a, ... and in [alpha]
+  + [beta], [alpha] + 2[beta] + [gamma], [alpha] + 3[beta] + 3[gamma] +
+  [delta], ... are respectively the same as in y - x, (y - x)², (y -
+  x)³, ... and in x + y, (x + y)², (x + y)³,.... In the second place, it
+  will be seen that the relations between the various terms in (A) are
+  identical with the relations between the similarly placed terms in
+  (B); e.g. [beta] + [gamma] is the difference of [alpha] + 2[beta] +
+  [gamma] and [alpha] + [beta], just as c - b is the difference of c and
+  b: and d - c is the sum of c - b and d - 2c + b, just as [beta] +
+  2[gamma] + [delta] is the sum of [beta] + [gamma] and [gamma] +
+  [delta]. Hence if we take [beta], [gamma], [delta], ... of (B) as
+  being the same as b - a, c - 2b + a, d -3c + 3b - a, ... of (A), all
+  corresponding terms in the two diagrams will be the same.
+
+  Thus we obtain the two principal formulae connecting terms and
+  differences. If we provisionally describe b - a, c - 2b + a, ... as
+  the first, second, ... differences of the particular term a (§ 7),
+  then (i.) the nth difference of a is
+
+                              n·n - 1
+    l - nk + ... + (-1)^(n-2) ------- c + (-1)^(n-1) nb + (-1)^n a,
+                                1·2
+
+  where l, k ... are the (n + 1)th, nth, ... terms of the series a, b,
+  c, ...; the coefficients being those of the terms in the expansion of
+  (y -x)^n: and (ii.) the (n + 1)th term of the series, i.e. the nth
+  term after a, is
+
+                  n·n - 1
+    a + n[beta] + ------- [gamma] + ...
+                    1·2
+
+  where [beta], [gamma], ... are the first, second, ... differences of
+  a; the coefficients being those of the terms in the expansion of (x +
+  y)^n.
+
+  4. Now suppose we treat the terms a, b, c, ... as being themselves the
+  first differences of another series. Then, if the first term of this
+  series is N, the subsequent terms are N + a, N + a + b, N + a + b + c,
+  ...; i.e. the difference between the (n + 1)th term and the first term
+  is the sum of the first n terms of the original series. The term N, in
+  the diagram (A), will come above and to the left of a; and we see, by
+  (ii.) of § 3, that the sum of the first n terms of the original series
+  is
+
+     /         n·n - 1            \             n·n - 1          n·n - 1·n - 2
+    ( N + na + ------- [beta] + ...) - N = na + ------- [beta] + ------------- [gamma] + ...
+     \           1·2              /               1·2              1 · 2 · 3
+
+  5. As an example, take the arithmetical series
+
+    a, a + p, a + 2p, ...
+
+  The first differences are p, p, p, ... and the differences of any
+  higher order are zero. Hence, by (ii.) of § 3, the (n + 1)th term is a
+  + np, and, by § 4, the sum of the first n terms is na + ½n(n - 1)p =
+  ½n{2a + (n - 1)p}.
+
+  6 As another example, take the series 1, 8, 27, ... the terms of which
+  are the cubes of 1, 2, 3, ... The first, second and third differences
+  of the first term are 7, 12 and 6, and it may be shown (§ 14 (i.))
+  that all differences of a higher order are zero. Hence the sum of the
+  first n terms is
+
+          n·n - 1      n·n - 1·n - 2     n·n - 1·n - 2·n - 3
+    n + 7 ------- + 12 ------------- + 6 ------------------- =
+            1·2            1·2·3              1·2·3·4
+
+      ¼n^4 + ½n³ + ¼n² = {½n(n + 1)}².
+
+  7. In § 3 we have described b - a, c - 2b + a, ... as the first,
+  second, ... differences of a. This ascription of the differences to
+  particular terms of the series is quite arbitrary. If we read the
+  differences in the table of § 2 upwards to the right instead of
+  downwards to the right, we might describe e - d, e - 2d + c, ... as
+  the first, second, ... differences of e. On the other hand, the term
+  of greatest weight in c -2b + a, i.e. the term which has the
+  numerically greatest coefficient, is b, and therefore c - 2b + a might
+  properly be regarded as the second difference of b, and similarly e -
+  4d + 6c - 4b + a might be regarded as the fourth difference of c.
+  These three methods of regarding the differences lead to three
+  different systems of notation, which are described in §§ 9, 10 and 11.
+
+
+  _Notation of Differences and Sums._
+
+  8. It is convenient to denote the terms a, b, c, ... of the series by
+  u0, u1, u2, u3, ... If we merely have the terms of the series, un may
+  be regarded as meaning the (n + 1)th term. Usually, however, the terms
+  are the values of a quantity u, which is a function of another
+  quantity x, and the values of x, to which a, b, c, ... correspond,
+  proceed by a constant difference h. If x0 and u0 are a pair of
+  corresponding values of x and u, and if any other value x0 + mh of x
+  and the corresponding value of u are denoted by xm and um, then the
+  terms of the series will be ... u_(n-2), u_(n-1), u_n, u_(n+1),
+  u_(n+2) ..., corresponding to values Of x denoted by ... x_(n-2),
+  x_(n-1), x_n, x_(n+1), x_(n+2)....
+
+  9. In the _advancing-difference notation_ u_(n+1) - u_n is denoted by
+  [Delta]un. The differences [Delta]u0, [Delta]u1, [Delta]u2 ... may
+  then be regarded as values of a function [Delta]u corresponding to
+  values of x proceeding by constant difference h; and therefore
+  [Delta]u_(n+1) -[Delta]u_n denoted by [Delta][Delta]u_n, or, more
+  briefly, [Delta]²u_n; and so on. Hence the table of differences in §
+  2, with the corresponding values of x and of u placed opposite each
+  other in the ordinary manner of mathematical tables, becomes
+
+    +---------+---------+----------------+-----------------+-----------------+----------------------+
+    |    x    |    u    |    1st Diff.   |    2nd Diff.    |     3rd Diff.   |       4th Diff.      |
+    +---------+---------+----------------+-----------------+-----------------+----------------------+
+    |    .    |    .    |        .       |        .        |        .        |          .           |
+    |    .    |    .    |        .       |        .        |        .        |          .           |
+    |    .    |    .    |        .       |        .        |        .        |          .           |
+    |         |         |                |                 |                 |                      |
+    | x_(n-2) | u_(n-2) |                | [Delta]²u_(n-3) |                 | [Delta]^4u_(n-4) ... |
+    |         |         | [Delta]u_(n-2) |                 | [Delta]³u_(n-3) |                      |
+    | x_(n-1) | u_(n-1) |                | [Delta]²u_(n-2) |                 | [Delta]^4u_(n-3) ... |
+    |         |         | [Delta]u_(n-1) |                 | [Delta]³u_(n-2) |                      |
+    | xn      | u_n     |                | [Delta]²u_(n-1) |                 | [Delta]^4u_(n-2) ... |
+    |         |         | [Delta]u_n     |                 | [Delta]³u_(n-1) |                      |
+    | x_(n+1) | u_(n+1) |                | [Delta]²u_n     |                 | [Delta]^4u_(n-1) ... |
+    |         |         | [Delta]u_(n+1) |                 | [Delta]³u_n     |                      |
+    | x_(n+2) | u_(n+2) |                | [Delta]²u_(n+1) |                 | [Delta]^4u_n     ... |
+    |    .    |    .    |        .       |        .        |        .        |          .           |
+    |    .    |    .    |        .       |        .        |        .        |          .           |
+    |    .    |    .    |        .       |        .        |        .        |          .           |
+    +---------+---------+----------------+-----------------+-----------------+----------------------+
+
+  The terms of the series of which ... u_(n-1), u_n, u_(n+1), ... are
+  the first differences are denoted by [Sigma]u, with proper suffixes,
+  so that this series is ... [Sigma]u_(n-1), [Sigma]u_n,
+  [Sigma]u_(n+1).... The suffixes are chosen so that we may have
+  [Delta][Sigma]un = un, whatever n may be; and therefore (§ 4)
+  [Sigma]un may be regarded as being the sum of the terms of the series
+  up to and including un-1. Thus if we write [Sigma]u_(n-1) = C + un-2,
+  where C is any constant, we shall have
+
+    [Sigma]u_n = [Sigma]u_(n-1) + [Delta][Sigma]u_(n-1) = C + u_(n-2) + u_(n-1),
+    [Sigma]u_(n+1) = C + u_(n-2) + u_(n-1) + u_n,
+
+  and so on. This is true whatever C may be, so that the knowledge of
+  ... u_n-1, u_n, ... gives us no knowledge of the exact value of
+  [Sigma]u_n; in other words, C is an arbitrary constant, the value of
+  which must be supposed to be the same throughout any operations in
+  which we are concerned with values of [Sigma]_u corresponding to
+  different suffixes.
+
+  There is another symbol E, used in conjunction with u to denote the
+  next term in the series. Thus Eun means u_(n+1), so that Eun = u_n +
+  [Delta]u_n.
+
+  10. Corresponding to the advancing-difference notation there is a
+  _receding-difference_ notation, in which u_(n+1) - u_n is regarded as
+  a difference of u_(n+1), and may be denoted by [Delta]'u_(n+1), and
+  similarly u_(n+1) - 2u_n + u_(n-1) may be denoted by [Delta]'²u_(n+1).
+  This notation is only required for certain special purposes, and the
+  usage is not settled (§ 19 (ii.)).
+
+  11. The _central-difference_ notation depends on treating u_(n+1) -
+  2u_n -u_(n-1) as the second difference of un, and therefore as
+  corresponding to the value x_n; but there is no settled system of
+  notation. The following seems to be the most convenient. Since un is a
+  function of x_n, and the second difference u_(n+2) - 2u_(n+1) + u_n is
+  a function of x_(n+1), the first difference u_(n+1) - u_n must be
+  regarded as a function of x_(n+½), i.e. of ½{x_n + x_(n+1)}. We
+  therefore write u_(n+1) - u_n = [delta]u_(n+½), and each difference in
+  the table in § 9 will have the same suffix as the value of x in the
+  same horizontal line; or, if the difference is of an odd order, its
+  suffix will be the means of those of the two nearest values of x. This
+  is shown in the table below.
+
+  In this notation, instead of using the symbol E, we use a symbol [mu]
+  to denote the mean of two consecutive values of u, or of two
+  consecutive differences of the same order, the suffixes being assigned
+  on the same principle as in the case of the differences. Thus
+
+    [mu]u_(n+½) = ½{u_n + u_(n+1)}, [mu][delta]u_n = ½{[delta]u_(n-½)} + [delta]u_(n+½), &c.
+
+  If we take the means of the differences of odd order immediately above
+  and below the horizontal line through any value of x, these means,
+  with the differences of even order in that line, constitute the
+  _central differences_ of the corresponding value of u. Thus the table
+  of central differences is as follows, the values obtained as means
+  being placed in brackets to distinguish them from the actual
+  differences:--
+
+    +-------+-------+---------------------+----------------+----------------------+----------------------+
+    |   x   |   u   |      1st Diff.      |    2nd Diff.   |       3rd Diff.      |      4th Diff.       |
+    +-------+-------+---------------------+----------------+----------------------+----------------------+
+    |   .   |   .   |          .          |        .       |           .          |           .          |
+    |   .   |   .   |          .          |        .       |           .          |           .          |
+    |   .   |   .   |          .          |        .       |           .          |           .          |
+    |x_(n-2)|u_(n-2)| {[mu][delta]u_(n-2)}| [delta]²u_(n-2)| {[mu][delta]³u_(n-2)}| [delta]^4u_(n-2) ... |
+    |       |       |   [delta]u_(n-3/2)  |                |  [delta]³u_(n-3/2)   |                      |
+    |x_(n-1)|u_(n-1)| {[mu][delta]u_(n-1)}| [delta]²u_(n-1)| {[mu][delta]³u_(n-1)}| [delta]^4u_(n-1) ... |
+    |       |       |   [delta]u_(n-½)    |                |    [delta]³u_(n-2    |                      |
+    |x_n    |u_n    |  ([mu][delta]u_n)   | [delta]²u_n    |  ([mu][delta]³u_n)   | [delta]^4u_n     ... |
+    |       |       |   [delta]u_(n+½)    |                |     [delta]³u_(n+½)  |                      |
+    |x_(n+1)|u_(n+1)| {[mu][delta]u_(n+1)}| [delta]²u_(n+1)| {[mu][delta]³u_(n+1)}| [delta]^4u_(n+1) ... |
+    |       |       |   [delta]u_(n+3/2)  |                |  [delta]³u_(n+3/2)   |                      |
+    |x_(n+2)|u_(n+2)| {[mu][delta]u_(n+2)}| [delta]²u_(n+2)| {[mu][delta]³u_(n+2)}| [delta]^4u_(n+2) ... |
+    |   .   |   .   |          .          |        .       |           .          |           .          |
+    |   .   |   .   |          .          |        .       |           .          |           .          |
+    |   .   |   .   |          .          |        .       |           .          |           .          |
+    +-------+-------+---------------------+----------------+----------------------+----------------------+
+
+  Similarly, by taking the means of consecutive values of u and also of
+  consecutive differences of even order, we should get a series of terms
+  and differences central to the intervals x_(n-2) to x_(n-1), x_(n-1)
+  to x_n, ....
+
+  The terms of the series of which the values of u are the first
+  differences are denoted by [sigma]u, with suffixes on the same
+  principle; the suffixes being chosen so that [delta][sigma]un shall be
+  equal to un. Thus, if
+
+    [sigma]u_(n-3/2) = C + u_(n-2),
+
+  then
+
+    [sigma]u_(n-½) = C + u_(n-2) + u_(n-1), [sigma]_(n+½)
+       = C + u_(n-2) + u_(n-1) + u_n, &c.,
+
+  and also
+
+    [mu][sigma]u_(n-1) = C + u_(n-2) + ½u_(n-1), [mu][sigma]u_n
+       = C + u_(n-2) + u_(n-1) + ½u_n, &c.,
+
+  C being an arbitrary constant which must remain the same throughout
+  any series of operations.
+
+
+  _Operators and Symbolic Methods._
+
+  12. There are two further stages in the use of the symbols [Delta],
+  [Sigma], [delta], [sigma], &c., which are not essential for elementary
+  treatment but lead to powerful methods of deduction.
+
+  (i.) Instead of treating [Delta]u as a function of x, so that
+  [Delta]u_n means ([Delta]u)_n, we may regard [Delta] as denoting an
+  _operation_ performed on u, and take [Delta]un as meaning [Delta].u_n.
+  This applies to the other symbols E, [delta], &c., whether taken
+  simply or in combination. Thus [Delta]Eu_n means that we first replace
+  un by un+1, and then replace this by u_(n+2) - u_(n+1).
+
+  (ii.) The operations [Delta], E, [delta], and [mu], whether performed
+  separately or in combination, or in combination also with numerical
+  multipliers and with the operation of differentiation denoted by D (:=
+  d/dx), follow the ordinary rules of algebra: e.g. [Delta](u_n + v_n) =
+  [Delta]u_n + [Delta]v_n, [Delta]Du_n = D[Delta]u_n, &c. Hence the
+  symbols can be separated from the functions on which the operations
+  are performed, and treated as if they were algebraical quantities. For
+  instance, we have
+
+    E·u_n = u_(n+1) = u_n + [Delta]u_n = 1·u_n + [Delta]·u_n,
+
+  so that we may write E = 1 + [Delta], or [Delta] = E - 1. The first of
+  these is nothing more than a statement, in concise form, that if we
+  take two quantities, subtract the first from the second, and add the
+  result to the first, we get the second. This seems almost a truism.
+  But, if we deduce E^n = (1 + [Delta])^n, [Delta]^n = (E-1)^n, and
+  expand by the binomial theorem and then operate on u0, we get the
+  general formulae
+
+                           n·n - 1
+    un = u0 + n[Delta]u0 + ------- [Delta]^2u0 + ... + [Delta]^nu0,
+                             1·2
+                                   n·n - 1
+    [Delta]^nu0 = u_n - nu_(n-1) + ------- u_(n-2) + ... + (-1)^nu0,
+                                     1·2
+
+  which are identical with the formulae in (ii.) and (i.) of § 3.
+
+  (iii.) What has been said under (ii.) applies, with certain
+  reservations, to the operations [Sigma] and [sigma], and to the
+  operation which represents integration. The latter is sometimes
+  denoted by D^-1; and, since [Delta][Sigma]un = un, and
+  [delta][sigma]u_n = u_n, we might similarly replace [Sigma] and
+  [sigma] by [Delta]^-1 and [delta]^-1. These symbols can be combined
+  with [Delta], E, &c. according to the ordinary laws of algebra,
+  provided that proper account is taken of the arbitrary constants
+  introduced by the operations D^-1, [Delta]^-1, [delta]^-1.
+
+
+  _Applications to Algebraical Series._
+
+  13. _Summation of Series._--If ur, denotes the (r+1)th term of a
+  series, and if vr is a function of r such that [Delta]v_r = u_r for
+  all integral values of r, then the sum of the terms u_m, u_(m+1), ...
+  un is v_(n+1) -v_m. Thus the sum of a number of terms of a series may
+  often be found by inspection, in the same kind of way that an integral
+  is found.
+
+  14. _Rational Integral Functions._--(i.) If u_r is a rational integral
+  function of r of degree p, then [Delta]ur, is a rational integral
+  function of r of degree p-1.
+
+  (ii.) A particular case is that of a _factorial_, i.e. a product of
+  the form (r+a+1) (r+a+2) ... (r+b), each factor exceeding the
+  preceding factor by 1. We have
+
+    [Delta]·(r+a+1) (r+a+2) ... (r+b) = (b-a)·(r+a+2) ... (r+b),
+
+  whence, changing a into a-1,
+
+    [Sigma](r+a+1)(r+a+2) ... (r+b) = _const._ + (r+a)(r+a+1) ...
+      (r+b)/(b-a+1).
+
+  A similar method can be applied to the series whose (r+1)th term is of
+  the form 1/(r+a+1) (r+a+2) ... (r+b).
+
+  (iii.) Any rational integral function can be converted into the sum of
+  a number of factorials; and thus the sum of a series of which such a
+  function is the general term can be found. For example, it may be
+  shown in this way that the sum of the pth powers of the first n
+  natural numbers is a rational integral function of n of degree p+1,
+  the coefficient of n^p+1 being 1/(p+1).
+
+  15. _Difference-equations._--The summation of the series ... + u_(n+2)
+  + u_(n-1) + u_n is a solution of the _difference-equation_ [Delta]v_n
+  = u_(n+1), which may also be written (E-1)v_n = u_(n+1). This is a
+  simple form of difference-equation. There are several forms which have
+  been investigated; a simple form, more general than the above, is the
+  _linear equation_ with _constant coefficients_--
+
+    v_(n+m) + a1v_(n+m-1) + a2v_(n+m-2) + ... + a_mv_n = N,
+
+  where a1, a2, ... am are constants, and N is a given function of n.
+  This may be written
+
+    (E^m + a1E^(m-1) + ... + a_m)v_n = N
+
+  or
+
+    (E-p1)(E-p2) ... (E-p_m)v_n = N.
+
+  The solution, if p1, p2, ... pm are all different, is vn = C1p1^n +
+  C2p2^n + ... + C_mp_m^n + V_n, where C1, C2 ... are constants, and v_n
+  = V_n is any one solution of the equation. The method of finding a
+  value for Vn depends on the form of N. Certain modifications are
+  required when two or more of the p's are equal.
+
+  It should be observed, in all cases of this kind, that, in describing
+  C1, C2 as "constants," it is meant that the value of any one, as C1,
+  is the same for all values of n occurring in the series. A "constant"
+  may, however, be a periodic function of n.
+
+
+  _Applications to Continuous Functions._
+
+  16. The cases of greatest practical importance are those in which u is
+  a continuous function of x. The terms u1, u2 ... of the series then
+  represent the successive values of u corresponding to x = x1, x2....
+  The important applications of the theory in these cases are to (i.)
+  relations between differences and differential coefficients, (ii.)
+  interpolation, or the determination of intermediate values of u, and
+  (iii.) relations between sums and integrals.
+
+  17. Starting from any pair of values x0 and u0, we may suppose the
+  interval h from x0 to x1 to be divided into q equal portions. If we
+  suppose the corresponding values of u to be obtained, and their
+  differences taken, the successive advancing differences of u0 being
+  denoted by dPu0, dP²u0 ..., we have (§ 3 (ii.))
+
+                         q·q - 1
+    u1  = u0  + qdPu0  + ------- dP²u0  + ....
+                           1·2
+
+  When q is made indefinitely great, this (writing f(x) for u) becomes
+  Taylor's Theorem (INFINITESIMAL CALCULUS)
+
+                                h²
+    f(x + h) = f(x) + hf'(x) + --- f"(x) + ...,
+                               1·2
+
+  which, expressed in terms of operators, is
+
+                  h²       h³
+    E = 1 + hD + ---D² + ----- D³ + ... = e^(hD).
+                 1·2     1·2·3
+
+  This gives the relation between [Delta] and D. Also we have
+
+                         2q·2q - 1
+    u2 = u0  + 2qdPu0  + --------- dP²u0  + ...
+                            1·2
+
+                         3q·3q - 1
+    u3 = u0  + 3qdPu0  + --------- dP²u0  + ...
+                            1·2
+    .            .
+    .            .
+    .            .
+
+  and, if p is any integer,
+
+                           p·p - 1
+    u_(p/q) = u0 + pdPu0 + ------- dP²u0 + ....
+                             1·2
+
+  From these equations up/q could be expressed in terms of u0, u1, u2,
+  ...; this is a particular case of interpolation (q.v.).
+
+  18. _Differences and Differential Coefficients._--The various formulae
+  are most quickly obtained by symbolical methods; i.e. by dealing with
+  the operators [Delta], E, D, ... as if they were algebraical
+  quantities. Thus the relation E = e^(hD) (§ 17) gives
+
+    hD = log_e  (1 + [Delta]) = [Delta] - ½[Delta]² + 1/3 [Delta]³ ...
+
+         /du\
+    or h( -- ) = [Delta]u0 - ½[Delta]²u0 + 1/3 [Delta]³u0  ....
+         \dx/0
+
+  The formulae connecting central differences with differential
+  coefficients are based on the relations [mu] = cosh ½hD = ½(e^ ½hD +
+  e^ -½hD), [delta] = 2 sinh ½hD - e^ ½hD - e^ -½hD, and may be grouped
+  as follows:--
+
+              u0  = u0                                              \
+                                                                    |
+    [mu][delta]u0 = (hD + 1/6 h³D³ + 1/120 h^5 D^5 + ...)u0         |
+                                                                    |
+       [delta]²u0 = (h²D² + 1/12 h^4 D^4 + 1/360 h^6 D^6 + ...)u0    >
+                                                                    |
+   [mu][delta]³u0 = (h³D³ + 1/4 h^5 D^5 + ...)u0                    |
+                                                                    |
+     [delta]^4 u0 = (h^4 D^4 + 1/6 h^6 D^6 + ...)u0                 /
+
+                  .                .             .
+                  .                .             .
+                  .                .             .
+
+          [mu]u_½ = (1 + 1/8 h²D² + 1/384 h^4 D^4 + 1/46080 h^6 D^6 + ...)u_½ \
+                                                                              |
+       [delta]u_½ = (hD + 1/24 h³D³ + 1/1920 h^5 D^5 + ...)u_½                |
+                                                                              |
+   [mu][delta]²u_½ = (h²D² + 5/24 h^4 D^4 + 91/5760 h^6 D^6 + ...)u_½          >
+                                                                              |
+      [delta]³u_½ = (h³D³ + 1/8 h^5 D^5 + ...)u_½                             |
+                                                                              |
+   [mu][delta]^4 u_½ = (h^4 D^4 + 7/24 h^6 D^6 + ...)u_½                      /
+
+                  .                .             .
+                  .                .             .
+                  .                .             .
+
+               u0 = u0                                                             \
+                                                                                   |
+            hDu0  = ([mu][delta] - 1/6 [mu][delta]³ + 1/30 [mu][delta]^5  - ...)u0 |
+                                                                                   |
+           h²D²u0 = ([delta]² - 1/12 [delta]^4 + 1/90 [delta]^6 - ...)u0            >
+                                                                                   |
+          h³D³u0  = ([mu][delta]³ 1/4 [mu][delta]^5  + ...)u0                      |
+                                                                                   |
+      h^4 D^4 u_0 = ([delta]^4 - 1/6 [delta]^6 + ...)u0                            /
+
+                  .                .             .
+                  .                .             .
+                  .                .             .
+
+              u_½ = ([mu] - 1/8 [mu][delta]² + 3/128 [mu][delta]^4 - 5/1024 [mu][delta]^6 + ...)u_½ \
+                                                                                                    |
+           hDu_½  = ([delta] - 1/24 [delta]³ + 3/640 [delta]^5 - ...)u_½                            |
+                                                                                                    |
+          h²D²u_½ = ([mu][delta]² - 5/24 [mu][delta]^ + 259/5760 [mu][delta]^6 - ...)u_½             >
+                                                                                                    |
+         h³D³u_½  = ([delta]³ - 1/8 [delta]^5 + ...)u_½                                             |
+                                                                                                    |
+      h^4 D^4 u_½ = ([mu][delta]^4 - 7/24 [mu][delta]^6 + ...)u_½                                   /
+
+                  .                .             .
+                  .                .             .
+                  .                .             .
+
+  When u is a rational integral function of x, each of the above series
+  is a terminating series. In other cases the series will be an infinite
+  one, and may be divergent; but it may be used for purposes of
+  approximation up to a certain point, and there will be a "remainder,"
+  the limits of whose magnitude will be determinate.
+
+  19. _Sums and Integrals._--The relation between a sum and an integral
+  is usually expressed by the _Euler-Maclaurin formula_. The principle
+  of this formula is that, if um and um+1, are ordinates of a curve,
+  distant h from one another, then for a first approximation to the area
+  of the curve between um and um+1 we have ½h(u_m + u_m+1), and the
+  difference between this and the true value of the area can be
+  expressed as the difference of two expressions, one of which is a
+  function of x_m, and the other is the same function of x_m+1. Denoting
+  these by [phi](x_m) and [phi](xm+1), we have
+
+      _ x_m+1
+     /
+     |   udx = ½h(u_m  + u_m+1) + [phi](x_m+1 ) - [phi](x_m).
+    _/x_m
+
+  Adding a series of similar expressions, we find
+
+      _ x_n
+     /
+     |  udx = h{½u_m + u_m+1 + u_m+2 + ... + u_n-1 + ½u_n} + [phi](x_n) - [phi](x_m).
+    _/x_m
+
+  The function [phi](x) can be expressed in terms either of differential
+  coefficients of u or of advancing or central differences; thus there
+  are three formulae.
+
+  (i.) The Euler-Maclaurin formula, properly so called, (due
+  independently to Euler and Maclaurin) is
+
+      _ x_n
+     /                             1    du_n    1       d³u_n     1       d^5 u_n
+     |   udx = h·[mu][sigma]u_n - -- h² ---- + --- h^4  ----- - ----- h^6 ------- + ...
+    _/x_m                         12     dx    720       dx³    30240       dx^5
+
+                                  B1    du_n   B2     d³u_n   B3     d^5u_n
+             = h·[mu][sigma]u_n - -- h2 ---- + -- h^4 ----- - -- h^6 ------ + ...,
+                                  2!     dx    4!      dx³    6!      dx^5
+
+  where B1, B2, B3 ... are _Bernoulli's numbers_.
+
+  (ii.) If we express differential coefficients in terms of advancing
+  differences, we get a theorem which is due to Laplace:--
+
+        _ x_n
+    1  /
+    -  |  udx = [mu][sigma](u_n - u0) - 1/12 ([Delta]u_n - [Delta]u0) + 1/24 ( [Delta]²u_n - [Delta]²u0)
+    h _/x0
+
+           - 19/720 ([Delta]³u_n - [Delta]³u_0) + 3/160 ([Delta]^4 u_n - [Delta]^4 u0) - ...
+
+
+  For practical calculations this may more conveniently be written
+
+        _ x_n
+    1  /
+    -  |  udx = [mu][sigma](u_n - u0) + 1/12 ([Delta]u0 - ½[Delta]²u0 + 19/60 [Delta]³u0 - ...)
+    h _/x0
+
+                 + 1/12 ([Delta]'u_n - ½[Delta]'²u_n + 19/60 [Delta]'³u_n - ...),
+
+  where accented differences denote that the values of u are read
+  backwards from un; i.e. [Delta]'un denotes u_n-1 - u_n, not (as in §
+  10) u_n - u_n-1.
+
+  (iii.) Expressed in terms of central differences this becomes
+
+        _ x_n
+    1  /
+    -  |   udx = [mu][sigma](u_n - u0) - 1/12 [mu][delta]u_n + 11/720 [mu][delta]³u_n - ...
+    h _/x0
+                            + 1/12 [mu][delta]u0  - 11/720 [mu][delta]³u0 + ...
+
+           /          1            11             191               2497                 \  /      \
+    = [mu]([sigma] - -- [delta] + --- [delta]³ - ----- [delta]^5 + ------- [delta]^7 - ...)(u_n - u0).
+           \         12           720            60480             3628800               /  \      /
+
+  (iv.) There are variants of these formulae, due to taking hum+½ as the
+  first approximation to the area of the curve between um and um+1; the
+  formulae involve the sum u_½ + u_3/2 + ... + u_n-½ := [sigma](u_n -
+  u0) (see MENSURATION).
+
+  20. The formulae in the last section can be obtained by symbolical
+  methods from the relation
+
+        _
+    1  /       1         1
+    -  | udx = - D^1 u = --·u.
+    h _/       h         hD
+
+  Thus for central differences, if we write [theta] := ½hD, we have [mu]
+  = cosh [theta], [delta] = 2 sinh [theta], [sigma] = [delta]^-1, and
+  the result in (iii.) corresponds to the formula
+
+                                        / /   1                 2                   2·4                    \
+    sinh [theta] = [theta] cosh [theta]/ (1 + - sinh²[theta] - --- sinh^4[theta] + ----- sinh^6[theta] - ...).
+                                      /   \   3                3·5                 3·5·7                   /
+
+  REFERENCES.--There is no recent English work on the theory of finite
+  differences as a whole. G. Boole's _Finite Differences_ (1st ed.,
+  1860, 2nd ed., edited by J. F. Moulton, 1872) is a comprehensive
+  treatise, in which symbolical methods are employed very early. A. A.
+  Markoff's _Differenzenrechnung_ (German trans., 1896) contains general
+  formulae. (Both these works ignore central differences.) _Encycl. der
+  math. Wiss._ vol. i. pt. 2, pp. 919-935, may also be consulted. An
+  elementary treatment of the subject will be found in many text-books,
+  e.g. G. Chrystal's _Algebra_ (pt. 2, ch. xxxi.). A. W. Sunderland,
+  _Notes on Finite Differences_ (1885), is intended for actuarial
+  students. Various central-difference formulae with references are
+  given in _Proc. Lond. Math. Soc._ xxxi. pp. 449-488. For other
+  references see INTERPOLATION.     (W. F. SH.)
+
+
+
+
+DIFFERENTIAL EQUATION, in mathematics, a relation between one or more
+functions and their differential coefficients. The subject is treated
+here in two parts: (1) an elementary introduction dealing with the more
+commonly recognized types of differential equations which can be solved
+by rule; and (2) the general theory.
+
+
+  _Part I.--Elementary Introduction._
+
+  Of equations involving only one independent variable, x (known as
+  _ordinary_ differential equations), and one dependent variable, y, and
+  containing only the first differential coefficient dy/dx (and
+  therefore said to be of the first _order_), the simplest form is that
+  reducible to the type
+
+    dy/dx = f(x)/F(y),
+
+  leading to the result fF(y)dy - ff(x)dx = A, where A is an arbitrary
+  constant; this result is said to solve the differential equation, the
+  problem of evaluating the integrals belonging to the integral
+  calculus.
+
+
+  Another simple form is
+
+    dy/dx + yP = Q,
+
+  where P, Q are functions of x only; this is known as the linear
+  equation, since it contains y and dy/dx only to the first degree. If
+
+    fPdx = u, we clearly have
+
+     d             /dy    \
+    --(ye^u) =e^u ( -- + Py) = e^u Q,
+    dx             \dx    /
+
+  so that y = e^-u(fe^u Qdx + A) solves the equation, and is the only
+  possible solution, A being an arbitrary constant. The rule for the
+  solution of the linear equation is thus to multiply the equation by
+  e^u, where u = fPdx.
+
+  A third simple and important form is that denoted by
+
+    y = px + f(p),
+
+  where p is an abbreviation for dy/dx; this is known as Clairaut's
+  form. By differentiation in regard to x it gives
+
+             dp        dp
+    p = p + x-- + f'(p)--,
+             dx        dx
+
+  where
+
+             d
+    f'(p) = -- f(p);
+            dp
+
+  thus, either (i.) dp/dx = 0, that is, p is constant on the curve
+  satisfying the differential equation, which curve is thus any one of
+  the straight lines y = cx = f(c), where c is an arbitrary constant, or
+  else, (ii.) x + [f]'(p) = 0; if this latter hypothesis be taken, and p
+  be eliminated between x + f'(p) = 0 and y = px + f(p), a relation
+  connecting x and y, not containing an arbitrary constant, will be
+  found, which obviously represents the envelope of the straight lines y
+  = cx + f(c).
+
+  In general if a differential equation [phi](x, y, dy/dx) = 0 be
+  satisfied by any one of the curves F(x, y, c) = 0, where c is an
+  arbitrary constant, it is clear that the envelope of these curves,
+  when existent, must also satisfy the differential equation; for this
+  equation prescribes a relation connecting only the co-ordinates x, y
+  and the differential coefficient dy/dx, and these three quantities are
+  the same at any point of the envelope for the envelope and for the
+  particular curve of the family which there touches the envelope. The
+  relation expressing the equation of the envelope is called a
+  _singular_ solution of the differential equation, meaning an
+  _isolated_ solution, as not being one of a family of curves depending
+  upon an arbitrary parameter.
+
+  An extended form of Clairaut's equation expressed by
+
+    y = xF(p) + f(p)
+
+  may be similarly solved by first differentiating in regard to p, when
+  it reduces to a linear equation of which x is the dependent and p the
+  independent variable; from the integral of this linear equation, and
+  the original differential equation, the quantity p is then to be
+  eliminated.
+
+  Other types of solvable differential equations of the first order are
+  (1)
+
+    M dy/dx = N,
+
+  where M, N are homogeneous polynomials in x and y, of the same order;
+  by putting v = y/x and eliminating y, the equation becomes of the
+  first type considered above, in v and x. An equation (aB <> bA)
+
+    (ax + by + c)dy/dx = Ax + By + C
+
+  may be reduced to this rule by first putting x + h, y + k for x and y,
+  and determining h, k so that ah + bk + c = 0, Ah + Bk + C = 0.
+
+  (2) An equation in which y does not explicitly occur,
+
+    f(x, dy/dx) = 0,
+
+  may, theoretically, be reduced to the type dy/dx = F(x); similarly an
+  equation F(y, dy/dx) = 0.
+
+  (3) An equation
+
+    f(dy/dx, x, y) = 0,
+
+  which is an integral polynomial in dy/dx, may, theoretically, be
+  solved for dy/dx, as an algebraic equation; to any root dy/dx = F1(x,
+  y) corresponds, suppose, a solution [phi]1(x, y, c) = 0, where c is an
+  arbitrary constant; the product equation [phi]1(x, y, c)[phi]2(x, y,
+  c) ... = 0, consisting of as many factors as there were values of
+  dy/dx, is effectively as general as if we wrote [phi]1(x, y, c1)
+  [phi]2(x, y, c2) ... = 0; for, to evaluate the first form, we must
+  necessarily consider the factors separately, and nothing is then
+  gained by the multiple notation for the various arbitrary constants.
+  The equation [phi]1(x, y, c)[phi]2(x, y, c) ... = 0 is thus the
+  solution of the given differential equation.
+
+  In all these cases there is, except for cases of singular solutions,
+  one and only one arbitrary constant in the most general solution of
+  the differential equation; that this must necessarily be so we may
+  take as obvious, the differential equation being supposed to arise by
+  elimination of this constant from the equation expressing its solution
+  and the equation obtainable from this by differentiation in regard to
+  x.
+
+  A further type of differential equation of the first order, of the
+  form
+
+    dy/dx = A + By + Cy²
+
+  in which A, B, C are functions of x, will be briefly considered below
+  under differential equations of the second order.
+
+  When we pass to ordinary differential equations of the second order,
+  that is, those expressing a relation between x, y, dy/dx and d²y/dx²,
+  the number of types for which the solution can be found by a known
+  procedure is very considerably reduced. Consider the general linear
+  equation
+
+    d²y    dy
+    --- + P-- + Qy = R,
+    dx²    dx
+
+  where P, Q, R are functions of x only. There is no method always
+  effective; the main general result for such a linear equation is that
+  if any particular function of x, say y1, can be discovered, for which
+
+    d²y1    dy1
+    ---- + P--- + Qy1 = 0,
+    dx²     dx
+
+  then the substitution y = y1[eta] in the original equation, with R on
+  the right side, reduces this to a linear equation of the first order
+  with the dependent variable d[eta]/dx. In fact, if y = y1[eta] we have
+
+    dy     d[eta]        dy1     d²y     d²[eta]    dy1 d[eta]        d²y1
+    -- = y1------ + [eta]--- and --- = y1------- + 2--- ------ + [eta]----,
+    dx       dx          dx      dx²       dx²      dx    dx           dx²
+
+  and thus
+
+    d²y     dy          d²[eta]    / dy1     \  d[eta]    /d²y1    dy1     \
+    --- + P -- + Qy = y1------- + ( 2--- + Py1) ------ + ( ---- + P--- + Qy1)[eta];
+    dx²     dx            dx²      \ dx      /    dx      \ dx²    dx      /
+
+  if then
+
+    d²y1     dy1
+    ---- + P --- + Qy1 = 0,
+    dx²      dx
+
+  and z denote d[eta]/dx, the original differential equation becomes
+
+      dz    / dy1     \
+    y1-- + ( 2--- + Py1)z = R.
+      dx    \ dx      /
+
+  From this equation z can be found by the rule given above for the
+  linear equation of the first order, and will involve one arbitrary
+  constant; thence y = y1 [eta] = y1 [int] zdx + Ay1, where A is another
+  arbitrary constant, will be the general solution of the original
+  equation, and, as was to be expected, involves two arbitrary
+  constants.
+
+  The case of most frequent occurrence is that in which the coefficients
+  P, Q are constants; we consider this case in some detail. If [t]*
+  be a root of the quadratic equation [t]² + [t]P + Q = 0, it
+  can be at once seen that a particular integral of the differential
+  equation with zero on the right side is y1 = e^[theta]x. Supposing
+  first the roots of the quadratic equation to be different, and [phi]
+  to be the other root, so that [p] + [t] = -P, the auxiliary
+  differential equation for z, referred to above, becomes dz/dx +
+  ([t] - [p])z = Re^(-[t]^x), which leads to
+  ze^{([t]-[p])^x} = B + [int] Re^(-[p]^x)dx, where B is an
+  arbitrary constant, and hence to
+
+(*) [t] = [theta]; [p] = [phi].
+                               _                             _               _
+                              /                             /               /
+    y = Ae^([t]^x) + e^([t]^x)| Be^([p]-[t])^x dx + e^[t]^x | e^([p]-[t])^x | Re^-[p]^x dxdx,
+                             _/                            _/              _/
+
+  or say to y = Ae^[t]^x + Ce^[p]^x + U, where A, C are arbitrary
+  constants and U is a function of x, not present at all when R = 0. If
+  the quadratic equation [t]² + P[t] + Q = 0 has equal roots, so that
+  2[t] = -P, the auxiliary equation in z becomes dz/dx = Re^-[t]^x,
+  giving z = B + [int] Re^-[t]^x dx, where B is an arbitrary constant,
+  and hence
+                                   _  _
+                                  /  /
+    y = (A + Bx)e^[t]^x + e^[t]^x |  | Re^-[t]^x dxdx,
+                                 _/ _/
+
+  or, say, y = (A + Bx)e^[t]^x + U, where A, B are arbitrary constants,
+  and U is a function of x not present at all when R = 0. The portion
+  Ae^[t]^x + Be^[p]^x or (A + Bx)e^[t]^x of the solution, which is known
+  as the _complementary function_, can clearly be written down at once
+  by inspection of the given differential equation. The remaining
+  portion U may, by taking the constants in the complementary function
+  properly, be replaced by any particular solution whatever of the
+  differential equation
+
+    d²v     dy
+    --- + P -- + Qy = R;
+    dx²     dx
+
+  for if u be any particular solution, this has a form
+
+    u = A0  e^[t]^x + B0 e^[p]^x + U,
+
+  or a form
+
+    u = (A0 + B0x)e^[t]^x + U;
+
+  thus the general solution can be written
+
+    (A - A0)e^[t]^x + (B - B0)e^[p]^x + u,
+
+  or
+
+    {A - A0  + (B - B0)x}e^[t]^x + u,
+
+  where A - A0, B - B0, like A, B, are arbitrary constants.
+
+  A similar result holds for a linear differential equation of any
+  order, say
+
+    d^n y      d^n-1 y
+    ----- + P1 ------- + ... + P_n y = R,
+    dx_n       dx^n-1
+
+  where P1, P2, ... Pn are constants, and R is a function of x. If we
+  form the algebraic equation [t]^n + P1[t]^n-1 + ... + P_n = 0, and all
+  the roots of this equation be different, say they are [t]1, [t]2, ...
+  [t]n, the general solution of the differential equation is
+
+    y = A1 e^[t]1^x + A2 e^[t]2^x + ... + A_n e^[t]_n^x + u,
+
+  where A1, A2, ... An are arbitrary constants, and u is any particular
+  solution whatever; but if there be one root [t]1 repeated r times, the
+  terms A1 e^[t]1^x + ... + A_r e^[t]_r^x must be replaced by (A1 + A2x
+  + ... + A_r x^r-1)e^[t]1x where A1, ... An are arbitrary constants;
+  the remaining terms in the complementary function will similarly need
+  alteration of form if there be other repeated roots.
+
+  To complete the solution of the differential equation we need some
+  method of determining a particular integral u; we explain a procedure
+  which is effective for this purpose in the cases in which R is a sum
+  of terms of the form e^ax[p](x), where [p](x) is an integral
+  polynomial in x; this includes cases in which R contains terms of the
+  form cos bx·[p](x) or sin bx·[p](x). Denote d/dx by D; it is clear
+  that if u be any function of x, D(e^ax u) = e^ax Du + ae^ax u, or say,
+  D(e^ax u) = e^ax (D + a)u; hence D²(e^ax u), i.e. d²/dx² (e^ax u),
+  being equal to D(e^ax v), where v=(D + a)u, is equal to e^ax(D + a)v,
+  that is to e^ax(D + a)²u. In this way we find D^n(e^ax u) = e^ax(D +
+  a)^n u, where n is any positive integer. Hence if [psi](D) be any
+  polynomial in D with constant coefficients, [psi](D)(e^ax u) = e^ax
+  [psi](D + a)u. Next, denoting [int] udx by D^-1 u, and any solution of
+  the differential equation dz/dx + az = u by z = (d + a)^-1 u, we have
+  D[e^ax(D + a)^-1 u] = D(e^ax z) = e^ax(D + a)z = e^ax u, so that we
+  may write D^-1(e^ax u) = e^ax(D+a)^-1 u, where the meaning is that one
+  value of the left side is equal to one value of the right side; from
+  this, the expression D^-2(e^axu), which means D^-1[D^-1(e^ax u)], is
+  equal to D^-1(e^ax z) and hence to e^ax(D + a)^-1 z, which we write
+  e^ax(D + a)^-2 u; proceeding thus we obtain
+
+    D^-n(e^ax u) = e^ax(D + a)^-n u,
+
+  where n is any positive integer, and the meaning, as before, is that
+  one value of the first expression is equal to one value of the second.
+  More generally, if [psi](D) be any polynomial in D with constant
+  coefficients, and we agree to denote by 1/[psi](D) u any solution z of
+  the differential equation [psi](D)z = u, we have, if v = 1/[psi](D +
+  a) u, the identity [psi](D)(e^ax v) = e^ax [psi](D + a)v = e^ax u,
+  which we write in the form
+
+        1                        1
+    --------(e^ax u) = e^ax ------------ u.
+    [psi](D)                [psi](D + a)
+
+  This gives us the first step in the method we are explaining, namely
+  that a solution of the differential equation [psi](D)y = e^ax u + e^bx
+  v + ... where u, v, ... are any functions of x, is any function
+  denoted by the expression
+
+              1                     1
+    e^ax ------------ u + e^ax ------------ v + ....
+         [psi](D + a)          [psi](D + b)
+
+  It is now to be shown how to obtain one value of 1/[psi](D + a) u,
+  when u is a polynomial in x, namely one solution of the differential
+  equation [psi](D + a)z = u. Let the highest power of x entering in u
+  be x^m; if t were a variable quantity, the rational fraction in t,
+  1/[psi](t + a), by first writing it as a sum of partial fractions, or
+  otherwise, could be identically written in the form
+
+    K_r t^-r + K_r-1 t^-r+1 + ... + K1 t^-1 + H + H1t + ... + H_m t^m + t^m+1 [p](t)/[psi](t + a),
+
+  where [p](t) is a polynomial in t; this shows that there exists an
+  identity of the form
+
+    1 = [psi](t + a)(K_r t^-r + ... + K1t^-1 + H + H1t + ... + H_m t^m) + [p](t)t^m+1,
+
+  and hence an identity
+
+    u = [psi](D + a)[K_r D^-r + ... + K1D^-1 + H + H1D + ... + H_m D^m]u + [p](D)D^m+1 u;
+
+  in this, since u contains no power of x higher than x^m, the second
+  term on the right may be omitted. We thus reach the conclusion that a
+  solution of the differential equation [psi](D + a)z = u is given by
+
+    z = (K_r D^-r + ... + K1D^-1 + H + H1D + ... + H_m D^m)u,
+
+  of which the operator on the right is obtained simply by expanding
+  1/[psi](D + a) in ascending powers of D, as if D were a numerical
+  quantity, the expansion being carried as far as the highest power of D
+  which, operating upon u, does not give zero. In this form every term
+  in z is capable of immediate calculation.
+
+  _Example._--For the equation
+
+    d^4v    d²y
+    ---- + 2--- + y = x³ cos x or (D² + 1)²y = x³ cos x,
+    dx^4    dx³
+
+  the roots of the associated algebraic equation ([t]²+1)² = 0 are [t] =
+  ±i, each repeated; the complementary function is thus
+
+    (A + Bx)e^ix + (C + Dx)e^ix,
+
+  where A, B, C, D are arbitrary constants; this is the same as
+
+    (H + Kx) cos x + (M + Nx) sin x,
+
+  where H, K, M, N are arbitrary constants. To obtain a particular
+  integral we must find a value of (1 + D²)^-2 x³ cos x; this is the
+  real part of (1+D²)^-2 e^ix x³ and hence of e^ix [1 + (D + i)²]^-2 x³
+
+  or    e^ix [2iD(1 + ½iD)]^-2 x³,
+
+  or    -¼e^ix D^-2 (1 + iD - ¾D² - ½iD³ + 5/16 D^4 + 3/16 iD^5 ...)x³,
+
+  or    -¼e^ix(1/20 x^5 + ¼ix^4 - ¾x³ - 3/2 ix² + 15/8 x + 9/8 i);
+
+  the real part of this is
+
+    -¼(1/20 x^5 - ¾x² + 15/8 x) cos x + ¼(¼x^4 - 3/2 x² + 9/8) sin x.
+
+  This expression added to the complementary function found above gives
+  the complete integral; and no generality is lost by omitting from the
+  particular integral the terms -15/32 x cos x + 9/32 sin x, which are
+  of the types of terms already occurring in the complementary function.
+
+  The symbolical method which has been explained has wider applications
+  than that to which we have, for simplicity of explanation, restricted
+  it. For example, if [psi](x) be any function of x, and a1, a2, ... an
+  be different constants, and [(t + a1) (t + a2) ... (t + an)]^-1 when
+  expressed in partial fractions be written [Sigma]c_m(t + a_m)^-1, a
+  particular integral of the differential equation (D + a1)(D + a2) ...
+  (D + a_n)y = [psi](x) is given by
+
+    y = [Sigma]c_m(D + a_m)^-1 [psi](x) = [Sigma]c_m(D + a_m)^-1 e^-a m^x e^a m^x [psi](x) =
+
+    [Sigma]c_m e^-a m^x D^-1 (e^a m^x [psi](x)) = [Sigma]c_m e^-a m^x [int] e^a m^x [psi](x)dx.
+
+  The particular integral is thus expressed as a sum of n integrals.
+
+  A linear differential equation of which the left side has the form
+
+        d^ny           d^n-1 y                dy
+    x^n ---- + P1x^n-1 ------- + ... + P_n-1 x-- + P_n y,
+        dx^n           dx^n-1                 dx
+
+  where P1, ... Pn are constants, can be reduced to the case considered
+  above. Writing x = e^t we have the identity
+
+        d^mu
+    x^m ----  = [t]([t] - 1)([t] - 2) ... ([t] - m + 1)u, where [t] = d/dt.
+        dx^m
+
+  When the linear differential equation, which we take to be of the
+  second order, has variable coefficients, though there is no general
+  rule for obtaining a solution in finite terms, there are some results
+  which it is of advantage to have in mind. We have seen that if one
+  solution of the equation obtained by putting the right side zero, say
+  y1, be known, the equation can be solved. If y2 be another solution of
+
+    d²y    dy
+    --- + P-- + Qy = 0,
+    dx²    dx
+
+  there being no relation of the form my1 + ny2 = k, where m, n, k are
+  constants, it is easy to see that
+
+    d/dx(y1'y2 - y1y2') = P(y1'y2 - y1y2'),
+
+  so that we have
+
+    y1'y2 - y1y2' = A exp.([int] Pdx),
+
+  where A is a suitably chosen constant, and exp. z denotes e^z. In
+  terms of the two solutions y1, y2 of the differential equation having
+  zero on the right side, the general solution of the equation with R =
+  [phi](x) on the right side can at once be verified to be Ay1 + By2 +
+  y1u - y2v, where u, v respectively denote the integrals
+         _                                     _
+        /                                     /
+    u = |y2[phi](x)(y1'y2 - y2'y1)^-1 dx, v = |y1[phi](x)(y1'y2 - y2'y1)^-1 dx.
+       _/                                    _/
+
+  The equation
+
+    d²y    dy
+    --- + P-- + Qy = 0,
+    dx²    dx
+
+  by writing y = v exp. (-½ [int] Pdx), is at once seen to be reduced to
+  d²v/dx² + 1v = 0, where 1 = Q - ½dP/dx - ¼P². If [eta] = - 1/v dv/dx,
+  the equation d²v/dx² + 1v = 0 becomes d[eta]/dx = 1 + [eta]², a
+  non-linear equation of the first order.
+
+  More generally the equation
+
+    d[eta]
+    ------ = A + B[eta] + C[eta]²,
+      dx
+
+  where A, B, C are functions of x, is, by the substitution
+
+               1 dy
+    [eta] = - -- --,
+              Cy dx
+
+  reduced to the linear equation
+
+  d²y    /    1 dC\ dy
+  --- - ( B + - -- )-- + ACy = 0.
+  dx²    \    C dx/ dx
+
+  The equation
+
+    d[eta]
+    ------ = A + B[eta] + C[eta]²,
+      dx
+
+  known as Riccati's equation, is transformed into an equation of the
+  same form by a substitution of the form [eta] = (aY + b)/(cY + d),
+  where a, b, c, d are any functions of x, and this fact may be utilized
+  to obtain a solution when A, B, C have special forms; in particular if
+  any particular solution of the equation be known, say [eta]0, the
+  substitution [eta] = [eta]0 - 1/Y enables us at once to obtain the
+  general solution; for instance, when
+
+         d      /A\
+    2B = -- log( - ),
+         dx     \C/
+
+  a particular solution is [eta]0 = [root](-A/C). This is a case of the
+  remark, often useful in practice, that the linear equation
+
+            d²y    d[phi] dy
+    [phi](x)--- + ½------ -- + [mu]y = 0,
+            dx²      dx   dx
+
+  where [mu] is a constant, is reducible to a standard form by taking a
+  new independent variable
+         _
+        /
+    z = | dx[[p](x)]^-½.
+       _/
+
+  We pass to other types of equations of which the solution can be
+  obtained by rule. We may have cases in which there are two dependent
+  variables, x and y, and one independent variable t, the differential
+  coefficients dx/dt, dy/dt being given as functions of x, y and t. Of
+  such equations a simple case is expressed by the pair
+
+    dx                dy
+    -- = ax + by + c, -- = a'x + b'y + c',
+    dt                dt
+
+  wherein the coefficients a, b, c, a', b', c', are constants. To
+  integrate these, form with the constant [lambda] the differential
+  coefficient of z = x + [lambda]y, that is dz/dt = (a + [lambda]a')x +
+  (b + [lambda]b')y + c + [lambda]c', the quantity [lambda] being so
+  chosen that b + [lambda]b' = [lambda](a + [lambda]a'), so that we have
+  dz/dt = (a + [lambda]a')z + c + [lambda]c'; this last equation is at
+  once integrable in the form z(a + [lambda]a') + c + [lambda]c' = Ae^(a
+  + [lambda]a')t, where A is an arbitrary constant. In general, the
+  condition b + [lambda]b' = [lambda](a + [lambda]a') is satisfied by
+  two different values of [lambda], say [lambda]1, [lambda]2; the
+  solutions corresponding to these give the values of x +[lambda]1y and
+  x + [lambda]2y, from which x and y can be found as functions of t,
+  involving two arbitrary constants. If, however, the two roots of the
+  quadratic equation for [lambda] are equal, that is, if (a - b')² +
+  4a'b = 0, the method described gives only one equation, expressing x +
+  [lambda]y in terms of t; by means of this equation y can be eliminated
+  from dx/dt = ax + by + c, leading to an equation of the form dx/dt =
+  Px + Q + Re^(a + [lambda]a')t, where P, Q, R are constants. The
+  integration of this gives x, and thence y can be found.
+
+  A similar process is applicable when we have three or more dependent
+  variables whose differential coefficients in regard to the single
+  independent variables are given as linear functions of the dependent
+  variables with constant coefficients.
+
+  Another method of solution of the equations
+
+    dx/dt = ax + by + c, dy/dt = a'x + b'y + c',
+
+  consists in differentiating the first equation, thereby obtaining
+
+    d²x    dx    dy
+    --- = a-- + b--;
+    dt²    dt    dx
+
+  from the two given equations, by elimination of y, we can express
+  dy/dt as a linear function of x and dx/dt; we can thus form an
+  equation of the shape d²x/dt² = P + Qx + Rdx/dt, where P, Q, R are
+  constants; this can be integrated by methods previously explained, and
+  the integral, involving two arbitrary constants, gives, by the
+  equation dx/dt = ax + by + c, the corresponding value of y. Conversely
+  it should be noticed that any single linear differential equation
+
+    d²x             dx
+    --- = u + vx + w--,
+    dt²             dt
+
+  where u, v, w are functions of t, by writing y for dx/dt, is
+  equivalent with the two equations dx/dt = y, dy/dt = u + vx + wy. In
+  fact a similar reduction is possible for any system of differential
+  equations with one independent variable.
+
+  Equations occur to be integrated of the form
+
+    Xdx + Ydy + Zdz = 0,
+
+  where X, Y, Z are functions of x, y, z. We consider only the case in
+  which there exists an equation [phi](x, y, z) = C whose differential
+
+    dP[phi]     dP[phi]     dP[phi]
+    -------dx + -------dy + -------dz = 0
+      dPx         dPy         dPz
+
+  is equivalent with the given differential equation; that is, [mu]
+  being a proper function of x, y, z, we assume that there exist
+  equations
+
+    dP[phi]          dP[phi]          v[phi]
+    ------- = [mu]X, ------- = [mu]Y, ------ = [mu]Z;
+      dPx              vy               vz
+
+  these equations require
+
+    dP           dP
+    ---([mu]Y) = ---([mu]Z), &c.,
+    dPz          dPy
+
+  and hence
+
+      /dPZ   dPY\      /dPX   dPZ\      /dPY   dPX\
+    X( --- - --- ) + Y( --- - --- ) + Z( --- - --- ) = 0;
+      \dPy   dPz/      \dPz   dPx/      \dPx   dPy/
+
+  conversely it can be proved that this is sufficient in order that [mu]
+  may exist to render [mu](Xdx + Ydy + Zdz) a perfect differential; in
+  particular it may be satisfied in virtue of the three equations such
+  as
+
+    dPZ   dPY
+    --- - --- = 0;
+    dPy   dPz
+
+  in which case we may take [mu] = 1. Assuming the condition in its
+  general form, take in the given differential equation a plane section
+  of the surface [phi] = C parallel to the plane z, viz. put z constant,
+  and consider the resulting differential equation in the two variables
+  x, y, namely Xdx + Ydy = 0; let [psi](x, y, z) = constant, be its
+  integral, the constant z entering, as a rule, in [psi] because it
+  enters in X and Y. Now differentiate the relation [psi](x, y, z) =
+  [f](z), where [f] is a function to be determined, so obtaining
+
+    dP[psi]     dP[psi]      /dP[psi]   df\
+    -------dx + -------dy + ( ------- - -- )dz = 0;
+      dPx         dPy        \  dPz     dz/
+
+  there exists a function [sigma] of x, y, z such that
+
+    dP[psi]              dP[psi]
+    -------- = [sigma]X, ------- = [sigma]Y,
+      dPx                  dPy
+
+  because [psi] = constant, is the integral of Xdx + Ydy = 0; we desire
+  to prove that [f] can be chosen so that also, in virtue of [psi](x, y,
+  z) = f(z), we have
+
+    dP[psi]   df                    df   dP[psi]
+    ------- - -- = [sigma]Z, namely -- = ------- - [sigma]Z;
+      dPz     dz                    dz     dPz
+
+  if this can be proved the relation [psi](x, y, z) - f(z) = constant,
+  will be the integral of the given differential equation. To prove this
+  it is enough to show that, in virtue of [psi](x, y, z) = [f](z), the
+  function dP[psi]/dPx - [sigma]Z can be expressed in terms of z only.
+  Now in consequence of the originally assumed relations,
+
+    dP[psi]          dP[phi]          dP[phi]
+    ------- = [mu]X, ------- = [mu]Y, ------- = [mu]Z,
+      dPx              dPy              dPz
+
+  we have
+
+    dP[psi]  /dP[phi]   [sigma]   dP[psi]  /dP[phi]
+    ------- / ------- = ------- = ------- / -------,
+      dPx  /    dPx      [mu]       dPy  /    dPy
+
+  and hence
+
+    dP[psi] dP[phi]   dP[psi] dP[phi]
+    ------- ------- - ------- ------- = 0;
+      dPx     dPy       dPy     dPx
+
+  this shows that, as functions of x and y, [psi] is a function of [phi]
+  (see the note at the end of part i. of this article, on Jacobian
+  determinants), so that we may write [psi] = F(z, [phi]), from which
+
+    [sigma]     dPF         dP[psi]   dPF     dPF   dP[phi]   dPF   [sigma]           dPF
+    ------- = -------; then ------- = --- + ------- ------- = --- + ------- · [mu]Z = --- + [sigma]Z
+     [mu]     dP[phi]         dPz     dPz   dP[phi]   dPz     dPz    [mu]             dPz
+
+         dP[psi]              dPF
+      or ------- - [sigma]Z = ---;
+           dPz                dPz
+
+  in virtue of [psi](x, y, z) = f(z), and [psi] = F(z, [phi]), the
+  function [phi] can be written in terms of z only, thus dPF/dPz can be
+  written in terms of z only, and what we required to prove is proved.
+
+  Consider lastly a simple type of differential equation containing
+  _two_ independent variables, say x and y, and one dependent variable
+  z, namely the equation
+
+     dPz    dPz
+    P--- + Q--- = R,
+     dPx    dPy
+
+  where P, Q, R are functions of x, y, z. This is known as Lagrange's
+  linear partial differential equation of the first order. To integrate
+  this, consider first the ordinary differential equations dx/dz = P/R,
+  dy/dz = Q/R, and suppose that two functions u, v, of x, y, z can be
+  determined, independent of one another, such that the equations u = a,
+  v = b, where a, b are arbitrary constants, lead to these ordinary
+  differential equations, namely such that
+
+     dPu    dPu    dPu          dPv    dPv    dPv
+    P--- + Q--- = R--- = 0 and P--- + Q--- = R--- = 0.
+     dPx    dPy    dPz          dPx    dPy    dPz
+
+  Then if F(x, y, z) = 0 be a relation satisfying the original
+  differential equations, this relation giving rise to
+
+    dPF   dPF dPz         dPF   dPF dPz               dPF    dPF    dPF
+    --- + --- --- = 0 and --- + --- --- = 0, we have P--- + Q--- = R--- = 0.
+    dPx   dPz dPx         dPy   dPz dPy               dPx    dPy    dPz
+
+  It follows that the determinant of three rows and columns vanishes
+  whose first row consists of the three quantities dPF/dPx, dPF/dPy,
+  dPF/dPz, whose second row consists of the three quantities dPu/dPx,
+  dPu/dPy, dPu/dPz, whose third row consists similarly of the partial
+  derivatives of v. The vanishing of this so-called Jacobian determinant
+  is known to imply that F is expressible as a function of u and v,
+  unless these are themselves functionally related, which is contrary to
+  hypothesis (see the note below on Jacobian determinants). Conversely,
+  any relation [phi](u, v) = 0 can easily be proved, in virtue of the
+  equations satisfied by u and v, to lead to
+
+     dz    dz
+    P-- + Q-- = R.
+     dx    dx
+
+  The solution of this partial equation is thus reduced to the solution
+  of the two ordinary differential equations expressed by dx/P = dy/Q =
+  dz/R. In regard to this problem one remark may be made which is often
+  of use in practice: when one equation u = a has been found to satisfy
+  the differential equations, we may utilize this to obtain the second
+  equation v = b; for instance, we may, by means of u = a, eliminate
+  z--when then from the resulting equations in x and y a relation v = b
+  has been found containing x and y and a, the substitution a = u will
+  give a relation involving x, y, z.
+
+  _Note on Jacobian Determinants._--The fact assumed above that the
+  vanishing of the Jacobian determinant whose elements are the partial
+  derivatives of three functions F, u, v, of three variables x, y, z,
+  involves that there exists a functional relation connecting the three
+  functions F, u, v, may be proved somewhat roughly as follows:--
+
+  The corresponding theorem is true for any number of variables.
+  Consider first the case of two functions p, q, of two variables x, y.
+  The function p, not being constant, must contain one of the variables,
+  say x; we can then suppose x expressed in terms of y and the function
+  p; thus the function q can be expressed in terms of y and the function
+  p, say q = Q(p, y). This is clear enough in the simplest cases which
+  arise, when the functions are rational. Hence we have
+
+    dPq   dPQ dPp     dPq   dPQ dPp   dPQ
+    --- = --- --- and --- = --- --- + ---;
+    dPx   dPp dPx     dPy   dPp dPy   dPy
+
+  these give
+
+    dPp dPq   dPp dPq   dPp dPQ
+    --- --- - --- --- = --- ---;
+    dPx dPy   dPy dPx   dPx dPy
+
+  by hypothesis dPp/dPx is not identically zero; therefore if the
+  Jacobian determinant of p and q in regard to x and y is zero
+  identically, so is dPQ/dPy, or Q does not contain y, so that q is
+  expressible as a function of p only. Conversely, such an expression
+  can be seen at once to make the Jacobian of p and q vanish
+  identically.
+
+  Passing now to the case of three variables, suppose that the Jacobian
+  determinant of the three functions F, u, v in regard to x, y, z is
+  identically zero. We prove that if u, v are not themselves
+  functionally connected, F is expressible as a function of u and v.
+  Suppose first that the minors of the elements of dPF/dPx, dPF/dPy,
+  dPF/dPz in the determinant are all identically zero, namely the three
+  determinants such as
+
+    dPu dPv   dPu dPv
+    --- --- - --- ---;
+    dPy dPz   dPz dPy
+
+  then by the case of two variables considered above there exist three
+  functional relations. [psi]1(u, v, x) = 0, [psi]2(u, v, y) = 0,
+  [psi]3(u, v, z) = 0, of which the first, for example, follows from the
+  vanishing of
+
+    dPu dPv   dPu dPv
+    --- --- - --- ---.
+    dPy dPz   dPz dPy
+
+  We cannot assume that x is absent from [psi]1, or y from [psi]2, or z
+  from [psi]3; but conversely we cannot simultaneously have x entering
+  in [psi]1, and y in [psi]2, and z in [psi]3, or else by elimination of
+  u and v from the three equations [psi]1 = 0, [psi]2 = 0, [psi]3 = 0,
+  we should find a necessary relation connecting the three independent
+  quantities x, y, z; which is absurd. Thus when the three minors of
+  dPF/dPx, dPF/dPy, dPF/dPz in the Jacobian determinant are all zero,
+  there exists a functional relation connecting u and v only. Suppose no
+  such relation to exist; we can then suppose, for example, that
+
+    dPu dPv   dPu dPv
+    --- --- - --- ---
+    dPy dPz   dPz dPy
+
+  is not zero. Then from the equations u(x, y, z) = u, v(x, y, z) = v we
+  can express y and z in terms of u, v, and x (the attempt to do this
+  could only fail by leading to a relation connecting u, v and x, and
+  the existence of such a relation would involve that the determinant
+
+    dPu dPv   dPu dPv
+    --- --- - --- ---
+    dPy dPz   dPz dPy
+
+  was zero), and so write F in the form F(x, y, z) = [Phi](u, v, x). We
+  then have
+
+    dPF   dP[Phi] dPu   dP[Phi] dPv   dP[Phi]  dPF   dP[Phi] dPu   dP[Phi] dPv  dPF   dP[Phi] dPu   dP[Phi] dPv
+    --- = ------- --- + ------- --- + -------, --- = ------- --- + ------- ---, --- = ------- --- + ------- ---;
+    dPx     dPu   dPx     dPv   dPx     dPx    dPy     dPu   dPy     dPv   dPy  dPz     dPu   dPz     dPv   dPz
+
+  thereby the Jacobian determinant of F, u, v is reduced to
+
+    dP[Phi] /dPu dPv   dPu dPv\
+    -------( --- --- - --- --- );
+      dPx   \dPy dPz   dPz dPy/
+
+  by hypothesis the second factor of this does not vanish identically;
+  hence dP[Phi]/dPx = 0 identically, and [Phi] does not contain x; so
+  that F is expressible in terms of u, v only; as was to be proved.
+
+
+_Part II.--General Theory._
+
+Differential equations arise in the expression of the relations between
+quantities by the elimination of details, either unknown or regarded as
+unessential to the formulation of the relations in question. They give
+rise, therefore, to the two closely connected problems of determining
+what arrangement of details is consistent with them, and of developing,
+apart from these details, the general properties expressed by them. Very
+roughly, two methods of study can be distinguished, with the names
+Transformation-theories, Function-theories; the former is concerned with
+the reduction of the algebraical relations to the fewest and simplest
+forms, eventually with the hope of obtaining explicit expressions of the
+dependent variables in terms of the independent variables; the latter is
+concerned with the determination of the general descriptive relations
+among the quantities which are involved by the differential equations,
+with as little use of algebraical calculations as may be possible. Under
+the former heading we may, with the assumption of a few theorems
+belonging to the latter, arrange the theory of partial differential
+equations and Pfaff's problem, with their geometrical interpretations,
+as at present developed, and the applications of Lie's theory of
+transformation-groups to partial and to ordinary equations; under the
+latter, the study of linear differential equations in the manner
+initiated by Riemann, the applications of discontinuous groups, the
+theory of the singularities of integrals, and the study of potential
+equations with existence-theorems arising therefrom. In order to be
+clear we shall enter into some detail in regard to partial differential
+equations of the first order, both those which are linear in any number
+of variables and those not linear in two independent variables, and also
+in regard to the function-theory of linear differential equations of the
+second order. Space renders impossible anything further than the
+briefest account of many other matters; in particular, the theories of
+partial equations of higher than the first order, the function-theory of
+the singularities of ordinary equations not linear and the applications
+to differential geometry, are taken account of only in the bibliography.
+It is believed that on the whole the article will be more useful to the
+reader than if explanations of method had been further curtailed to
+include more facts.
+
+When we speak of a function without qualification, it is to be
+understood that in the immediate neighbourhood of a particular set x0,
+y0, ... of values of the independent variables x, y, ... of the
+function, at whatever point of the range of values for x, y, ... under
+consideration x0, y0, ... may be chosen, the function can be expressed
+as a series of positive integral powers of the differences x - x0, y
+-y0, ..., convergent when these are sufficiently small (see FUNCTION:
+Functions of Complex Variables). Without this condition, which we
+express by saying that the function is developable about x0, y0, ...,
+many results provisionally stated in the transformation theories would
+be unmeaning or incorrect. If, then, we have a set of k functions, f1
+... fk of n independent variables x1 ... xn, we say that they are
+independent when n >= k and not every determinant of k rows and columns
+vanishes of the matrix of k rows and n columns whose r-th row has the
+constituents dfr/dx1, ... dfr/dxn; the justification being in the
+theorem, which we assume, that if the determinant involving, for
+instance, the first k columns be not zero for x1 = x1^0 ... xn = xn^0,
+and the functions be developable about this point, then from the
+equations f1 = c1, ... fk = ck we can express x1, ... xk by convergent
+power series in the differences x_k+1 - x_k+1^0, ... x_n - x_n^0, and so
+regard x1, ... xk as functions of the remaining variables. This we often
+express by saying that the equations f1 = c1, ... fk = ck can be solved
+for x1, ... xk. The explanation is given as a type of explanation often
+understood in what follows.
+
+
+    Ordinary equations of the first order.
+
+    Single homogeneous partial equation of the first order.
+
+    Proof of the existence of integrals.
+
+  We may conveniently begin by stating the theorem: If each of the n
+  functions [phi]1, ... [phi]n of the (n + 1) variables x1, ... x_nt be
+  developable about the values x1^0, ... x_n^0t^0, the n differential
+  equations of the form dx1/dt = [phi]1(tx1, ... xn) are satisfied by
+  convergent power series
+
+    x_r = x_r^0 + (t - t^0 ) A_r1 + (t - t0 )²A_r2 + ...
+
+  reducing respectively to x1^0, ... xn^0 when t = t^0; and the only
+  functions satisfying the equations and reducing respectively to x1^0,
+  ... xn^0 when t = t^0, are those determined by continuation of these
+  series. If the result of solving these n equations for x1^0, ... xn^0
+  be written in the form [omega]1(x1, ... xnt) = x1^0, ... [omega]n(x1,
+  ... xnt) = xn^0, it is at once evident that the differential equation
+
+    df/dt + [phi]1 df/dx1 + ... + [phi]n df/dxn = 0
+
+  possesses n integrals, namely, the functions [omega]1, ... [omega]n,
+  which are developable about the values (x1^0 ... xn^0t^0) and reduce
+  respectively to x1, ... xn when t = t^0. And in fact it has no other
+  integrals so reducing. Thus this equation also possesses a unique
+  integral reducing when t = t^0 to an arbitrary function [psi](x1, ...
+  xn), this integral being. [psi]([omega]1, ... [omega]n). Conversely
+  the existence of these _principal_ integrals [omega]1, ... [omega]n of
+  the partial equation establishes the existence of the specified
+  solutions of the ordinary equations dxi/dt = [phi]i. The following
+  sketch of the proof of the existence of these principal integrals for
+  the case n = 2 will show the character of more general investigations.
+  Put x for x - x^0, &c., and consider the equation a(xyt) df/dx +
+  b(xyt) df/dy = df/dt, wherein the functions a, b are developable about
+  x = 0, y = 0, t = 0; say
+
+    a(xyt) = a0 + ta1 + t²a2/2! + ..., b(xyt) = b0 + tb1 + t²b2/2! + ...,
+
+  so that
+
+    ad/dx + bd/dy = [delta]0 + t[delta]1 + ½t²[delta]2 + ...,
+
+  where [delta] = a_r d/dx + b_r d/dy. In order that
+
+    f = p0 + tp1 + t²p2/2! + ...
+
+  wherein p0, p1 ... are power series in x, y, should satisfy the
+  equation, it is necessary, as we find by equating like terms, that
+
+    p1 = [delta]0 p0, p2 = [delta]0 p1 + [delta]1 p0, &c.
+
+  and in general
+
+    p_s+1 = [delta]0 p_s + s1 [delta]1 p_s-1 + ... + [delta]_s p0,
+
+  where s_r = (s!)/(r!) (s - r)!
+
+  Now compare with the given equation another equation
+
+    A(xyt)dF/dx + B(xyt)dF/dy = dF/dt,
+
+  wherein each coefficient in the expansion of either A or B is real and
+  positive, and not less than the absolute value of the corresponding
+  coefficient in the expansion of a or b. In the second equation let us
+  substitute a series
+
+    F = P0 + tP1 + t²P2/2! + ...,
+
+  wherein the coefficients in P0 are real and positive, and each not
+  less than the absolute value of the corresponding coefficient in p0;
+  then putting [Delta]r = A_r d/dx + B_r d/dy we obtain necessary
+  equations of the same form as before, namely,
+
+    P1 = [Delta]0 P0, P2= [Delta]0 P1 + [Delta]1 P0, ...
+
+  and in general P_s+1 = [Delta]0 P_s, + s1[Delta]1 P_s-1 + ... +
+  [Delta]_s P0. These give for every coefficient in Ps+1 an integral
+  aggregate with real positive coefficients of the coefficients in P_s,
+  P_s-1, ..., P0 and the coefficients in A and B; and they are the same
+  aggregates as would be given by the previously obtained equations for
+  the corresponding coefficients in p_s+1 in terms of the coefficients
+  in ps, p_s-1, ..., p0 and the coefficients in a and b. Hence as the
+  coefficients in P0 and also in A, B are real and positive, it follows
+  that the values obtained in succession for the coefficients in P1, P2,
+  ... are real and positive; and further, taking account of the fact
+  that the absolute value of a sum of terms is not greater than the sum
+  of the absolute values of the terms, it follows, for each value of s,
+  that every coefficient in p_s+1 is, in absolute value, not greater
+  than the corresponding coefficient in P_s+1. Thus if the series for F
+  be convergent, the series for f will also be; and we are thus reduced
+  to (1), specifying functions A, B with real positive coefficients,
+  each in absolute value not less than the corresponding coefficient in
+  a, b; (2) proving that the equation
+
+    AdF/dx + BdF/dy = dF/dt
+
+  possesses an integral P0 + tP1 + t²P2/2! + ... in which the
+  coefficients in P0 are real and positive, and each not less than the
+  absolute value of the corresponding coefficient in p0. If a, b be
+  developable for x, y both in absolute value less than r and for t less
+  in absolute value than R, and for such values a, b be both less in
+  absolute value than the real positive constant M, it is not difficult
+  to verify that we may take
+
+              /    x + y\-1 /    t\-1
+    A = B = M( 1 - ----- ) ( 1 - - ),
+              \      r  /   \    R/
+
+  and obtain
+                         _                                 _
+                        |     4MR /   x + y\-2     /   t\-1 |½
+    F = r - (r - x - y) | 1 - ---(1 - ------) log (1 - - )  |,
+                        |_     r  \     r  /       \   R/  _|
+
+  and that this solves the problem when x, y, t are sufficiently small
+  for the two cases p0 = x, p0 = y. One obvious application of the
+  general theorem is to the proof of the existence of an integral of an
+  ordinary linear differential equation given by the n equations dy/dx =
+  y1, dy1/dx = y2, ...,
+
+    dy_n-1/dx = p - p1 y_n-1 - ... - p_n y;
+
+  but in fact any simultaneous system of ordinary equations is reducible
+  to a system of the form
+
+    dx1/dt = [phi](tx1, ... x_n).
+
+
+    Simultaneous linear partial equations.
+
+    Complete systems of linear partial equations.
+
+    Jacobian systems.
+
+  Suppose we have k homogeneous linear partial equations of the first
+  order in n independent variables, the general equation being
+  a_[sigma]1 df/dx1 + ... + a_[sigma]n df/dx_n = 0, where [sigma] = 1,
+  ... k, and that we desire to know whether the equations have common
+  solutions, and if so, how many. It is to be understood that the
+  equations are linearly independent, which implies that k <= n and not
+  every determinant of k rows and columns is identically zero in the
+  matrix in which the i-th element of the [sigma]-th row is a[sigma]_i(i
+  = 1, ... n, [sigma] = 1, ... k). Denoting the left side of the
+  [sigma]-th equation by P[sigma]f, it is clear that every common
+  solution of the two equations P_[sigma]f = 0, P_[rho]f = 0, is also a
+  solution of the equation P_[rho](P_[sigma]f), P_[sigma](P_[rho]f), We
+  immediately find, however, that this is also a linear equation,
+  namely, [Sigma]H_i df/dx_i = 0 where H_i = P[rho]a[sigma]_i -
+  P[sigma]a[rho]_i, and if it be not already contained among the given
+  equations, or be linearly deducible from them, it may be added to
+  them, as not introducing any additional limitation of the possibility
+  of their having common solutions. Proceeding thus with every pair of
+  the original equations, and then with every pair of the possibly
+  augmented system so obtained, and so on continually, we shall arrive
+  at a system of equations, linearly independent of each other and
+  therefore not more than n in number, such that the combination, in the
+  way described, of every pair of them, leads to an equation which is
+  linearly deducible from them. If the number of this so-called
+  _complete system_ is n, the equations give df/dx1 = 0 ... df/dxn = 0,
+  leading to the nugatory result f = a constant. Suppose, then, the
+  number of this system to be r < n; suppose, further, that from the
+  matrix of the coefficients a determinant of r rows and columns not
+  vanishing identically is that formed by the coefficients of the
+  differential coefficients of f in regard to x1 ... x_r; also that the
+  coefficients are all developable about the values x1 = x1^0, ... xn=
+  xn^0, and that for these values the determinant just spoken of is not
+  zero. Then the main theorem is that the complete system of r
+  equations, and therefore the originally given set of k equations,
+  have in common n - r solutions, say [omega]r+1, ... [omega]n, which
+  reduce respectively to x_r+1, ... x_n when in them for x1, ... x_r are
+  respectively put x1^0, ... x_r^0; so that also the equations have in
+  common a solution reducing when x1 = x1^0, ... x_r = x_r^0 to an
+  arbitrary function [psi](x_r+1, ... x_n) which is developable about
+  x_r+1^0, ... x_n^0, namely, this common solution is [psi]([omega]_r+1,
+  ... [omega]_n). It is seen at once that this result is a
+  generalization of the theorem for r = 1, and its proof is conveniently
+  given by induction from that case. It can be verified without
+  difficulty (1) that if from the r equations of the complete system we
+  form r independent linear aggregates, with coefficients not
+  necessarily constants, the new system is also a complete system; (2)
+  that if in place of the independent variables x1, ... xn we introduce
+  any other variables which are independent functions of the former, the
+  new equations also form a complete system. It is convenient, then,
+  from the complete system of r equations to form r new equations by
+  solving separately for df/dx1, ..., df/dx_r; suppose the general
+  equation of the new system to be
+
+    Q_[sigma]f = df/dx_[sigma] + c_[sigma],r+1 df/dx_r+1 + ... + c_[sigma]n df/dx_n = 0 ([sigma] = 1, ... r).
+
+  Then it is easily obvious that the equation Q_[rho]Q_[sigma]f -
+  Q_[sigma]Q_[rho]f = 0 contains only the differential coefficients of f
+  in regard to x_r+1 ... xn; as it is at most a linear function of Q1f,
+  ... Qrf, it must be identically zero. So reduced the system is called
+  a Jacobian system. Of this system Q1f=0 has n - 1 principal solutions
+  reducing respectively to x2, ... xn when
+
+    x1 = x1^0,
+
+  and its form shows that of these the first r - 1 are exactly x2 ...
+  xr. Let these n - 1 functions together with x1 be introduced as n new
+  independent variables in all the r equations. Since the first equation
+  is satisfied by n - 1 of the new independent variables, it will
+  contain no differential coefficients in regard to them, and will
+  reduce therefore simply to df/dx1 = 0, expressing that any common
+  solution of the r equations is a function only of the n - 1 remaining
+  variables. Thereby the investigation of the common solutions is
+  reduced to the same problem for r - 1 equations in n - 1 variables.
+  Proceeding thus, we reach at length one equation in n - r + 1
+  variables, from which, by retracing the analysis, the proposition
+  stated is seen to follow.
+
+
+    System of total differential equations.
+
+  The analogy with the case of one equation is, however, still closer.
+  With the coefficients c_[sigma]j, of the equations Q_[sigma]f = 0 in
+  transposed array ([sigma] = 1, ... r, j = r + 1, ... n) we can put
+  down the (n - r) equations, dx_j = c1_j dx1 + ... + c_rj dx_r,
+  equivalent to the r(n - r) equations dx_j/dx_[sigma] = c_[sigma]r.
+  That consistent with them we may be able to regard x_r+1, ... x_n as
+  functions of x1, ... x_r, these being regarded as independent
+  variables, it is clearly necessary that when we differentiate
+  c_[sigma]j in regard to x_[rho] on this hypothesis the result should
+  be the same as when we differentiate c[rho]j, in regard to x[sigma] on
+  this hypothesis. The differential coefficient of a function f of x1,
+  ... xn on this hypothesis, in regard to x_[rho]j is, however,
+
+    df/dx_[rho] + c_[rho],r+1 df/dx_r+1 + ... + c_[rho]n df/dx_n,
+
+  namely, is Q_[rho]f. Thus the consistence of the n - r total equations
+  requires the conditions Q_[rho]c_[sigma]j - Q_[sigma]c_[rho]j = 0,
+  which are, however, verified in virtue of Q[rho](Q[sigma][f]) -
+  Q_[sigma](Q_[rho]f) = 0. And it can in fact be easily verified that if
+  [omega]_r+1, ... [omega]_n be the principal solutions of the Jacobian
+  system, Q_[sigma]f = 0, reducing respectively to x_r+1, ... xn when x1
+  = x1^0, ... x_r = x_r^0, and the equations [omega]_r+1 = x_r+1^0, ...
+  [omega]_n = x_n^0 be solved for x_r+1, ... x_n to give x_j =
+  [psi]_j(x1, ... x_r, x_r+1^0, ... x_n^0), these values solve the total
+  equations and reduce respectively to x_r+1^0, ... x_n^0 when x1 = x1^0
+  ... x_r = x_r^0. And the total equations have no other solutions with
+  these initial values. Conversely, the existence of these solutions of
+  the total equations can be deduced a priori and the theory of the
+  Jacobian system based upon them. The theory of such total equations,
+  in general, finds its natural place under the heading _Pfaffian
+  Expressions_, below.
+
+
+    Geometrical interpretation and solution.
+
+    Mayer's method of integration.
+
+  A practical method of reducing the solution of the r equations of a
+  Jacobian system to that of a single equation in n - r + 1 variables
+  may be explained in connexion with a geometrical interpretation which
+  will perhaps be clearer in a particular case, say n = 3, r = 2. There
+  is then only one total equation, say dz = adz + bdy; if we do not take
+  account of the condition of integrability, which is in this case da/dy
+  + bda/dz = db/dx + adb/dz, this equation may be regarded as defining
+  through an arbitrary point (x0, y0, z0) of three-dimensioned space
+  (about which a, b are developable) a plane, namely, z - z0 = a0(x -
+  x0) + b0(y - y0), and therefore, through this arbitrary point [oo]²
+  directions, namely, all those in the plane. If now there be a surface
+  z = [psi](x, y), satisfying dz = adz + bdy and passing through (x0,
+  y0, z0), this plane will touch the surface, and the operations of
+  passing along the surface from (x0, y0, z0) to
+
+    (x0 + dx0, y0, z0 + dz0)
+
+  and then to (x0 + dx0, y0 + dy0, Z0 + d¹z0), ought to lead to the same
+  value of d^1z0 as do the operations of passing along the surface from
+  (x0, y0, z0) to (x0, y0 + dy0, z0 + [delta]z0), and then to
+
+  (x_  + dx_  , y_  + dy_  , Z_  + [delta]¹z_  ),
+    0    0   0    0   0           0
+
+  namely, [delta]¹z0 ought to be equal to d¹z0. But we find
+
+    d¹z0 = a0dx0 + b(x0 + dx0 , y0, z0 + a0dx0)dy0 =
+
+                           /db      db \
+    a0dx0 + b0dy0 + dx0dy0( --- + a0--- ),
+                           \dx0     dz0/
+
+  and so at once reach the condition of integrability. If now we put x
+  = x0 + t, y = y0 + mt, and regard m as constant, we shall in fact be
+  considering the section of the surface by a fixed plane y - y0 = m(x -
+  x0); along this section dz = dt(a + bm); if we then integrate the
+  equation dx/dt = a + bm, where a, b are expressed as functions of m
+  and t, with m kept constant, finding the solution which reduces to z0
+  for t = 0, and in the result again replace m by (y - y0)/(x - x0), we
+  shall have the surface in question. In the general case the equations
+
+    dx_j - c_1j dx1 + ... c_rj dx_r
+
+  similarly determine through an arbitrary point x1^0, ... xn^0 a planar
+  manifold of r dimensions in space of n dimensions, and when the
+  conditions of integrability are satisfied, every direction in this
+  manifold through this point is tangent to the manifold of r
+  dimensions, expressed by [omega]_r+1 = x_r+1^0, ... [omega]_n = x_n^0,
+  which satisfies the equations and passes through this point. If we put
+  x1 = x1^0 = t, x2 = x2^0 = m2t, ... xr = xr^0 = mrt, and regard m2,
+  ... mr as fixed, the (n-r) total equations take the form dx_j/dt =
+  c_1j + m2c_2j + ... + m_rc_rj, and their integration is equivalent to
+  that of the single partial equation
+
+                n
+    df/dt + [Sigma](c_1j + m2c_2j + ... + m_rc_rj)df/dx_j = 0
+              j=r+1
+
+  in the n - r + 1 variables t, xr+1, ... xn. Determining the solutions
+  [Omega]_r+1, ... [Omega]_n which reduce to respectively x_r+1, ... x_n
+  when t = 0, and substituting t = x1 - x1^0, m2 = (x2 - x2^0)/(x1 -
+  x1^0), ... mr = (xr - xr^0)/(x1 - x1^0), we obtain the solutions of
+  the original system of partial equations previously denoted by
+  [omega]_r+1, ... [omega]_n. It is to be remarked, however, that the
+  presence of the fixed parameters m2, ... mr in the single integration
+  may frequently render it more difficult than if they were assigned
+  numerical quantities.
+
+
+    Pfaffian Expressions.
+
+  We have above considered the integration of an equation
+
+    dz = adz + bdy
+
+  on the hypothesis that the condition
+
+    da/dy + bda/dz = db/dz + adb/dz.
+
+  It is natural to inquire what relations among x, y, z, if any, are
+  implied by, or are consistent with, a differential relation adx + bdy
+  + cdx = 0, when a, b, c are unrestricted functions of x, y, z. This
+  problem leads to the consideration of the so-called _Pfaffian
+  Expression_ adx + bdy + cdz. It can be shown (1) if each of the
+  quantities db/dz - dc/dy, dc/dx - da/dz, da/dy - db/dz, which we shall
+  denote respectively by u23, u31, u12, be identically zero, the
+  expression is the differential of a function of x, y, z, equal to dt
+  say; (2) that if the quantity au23 + bu31 + cu12 is identically zero,
+  the expression is of the form udt, i.e. it can be made a perfect
+  differential by multiplication by the factor 1/u; (3) that in general
+  the expression is of the form dt + u1dt1. Consider the matrix of four
+  rows and three columns, in which the elements of the first row are a,
+  b, c, and the elements of the (r+1)-th row, for r = 1, 2, 3, are the
+  quantities u_r1, u_r2, u_r3, where u11 = u22 = u33 = 0. Then it is
+  easily seen that the cases (1), (2), (3) above correspond respectively
+  to the cases when (1) every determinant of this matrix of two rows and
+  columns is zero, (2) every determinant of three rows and columns is
+  zero, (3) when no condition is assumed. This result can be generalized
+  as follows: if a1, ... an be any functions of x1, ... xn, the
+  so-called Pfaffian expression a1dx1 + ... + a_ndx_n can be reduced to
+  one or other of the two forms
+
+    u1dt1 + ... + u_kdt_k, dt + u1dt1 +  ... + u_k-1 dt_k-1,
+
+  wherein t, u1 ..., t1, ... are independent functions of x1, ... xn,
+  and k is such that in these two cases respectively 2k or 2k - 1 is the
+  rank of a certain matrix of n + 1 rows and n columns, that is, the
+  greatest number of rows and columns in a non-vanishing determinant of
+  the matrix; the matrix is that whose first row is constituted by the
+  quantities a1, ... an, whose s-th element in the (r+1)-th row is the
+  quantity da_r/dx_s - da_s/dx_r. The proof of such a reduced form can
+  be obtained from the two results: (1) If t be any given function of
+  the 2m independent variables u1, ... um, t1, ... tm, the expression dt
+  + u1 dt1 + ... + u_m dt_m can be put into the form u'1 dt'1 + ... +
+  u'_mdt'_m. (2) If the quantities u1, ..., u1, t1, ... tm be connected
+  by a relation, the expression n1dt1 + ... + umdtm can be put into the
+  format dt' + u'1 dt'1 + ... + u'_m-1 dt'_m-1; and if the relation
+  connecting u1, um, t1, ... tm be homogeneous in u1, ... um, then t'
+  can be taken to be zero. These two results are deductions from the
+  theory of _contact transformations_ (see below), and their
+  demonstration requires, beside elementary algebraical considerations,
+  only the theory of complete systems of linear homogeneous partial
+  differential equations of the first order. When the existence of the
+  reduced form of the Pfaffian expression containing only independent
+  quantities is thus once assured, the identification of the number k
+  with that defined by the specified matrix may, with some difficulty,
+  be made _a posteriori_.
+
+
+    Single linear Pfaffian equation.
+
+  In all cases of a single Pfaffian equation we are thus led to consider
+  what is implied by a relation dt - u1dt1 - ... - umdtm = 0, in which
+  t, u1, ... um, t1 ..., tm are, except for this equation, independent
+  variables. This is to be satisfied in virtue of one or several
+  relations connecting the variables; these must involve relations
+  connecting t, t1, ... tm only, and in one of these at least t must
+  actually enter. We can then suppose that in one actual system of
+  relations in virtue of which the Pfaffian equation is satisfied, all
+  the relations connecting t, t1 ... tm only are given by
+
+    t = [psi](t_s+1 ... t_m), t1 = [psi]1(t_s+1 ... t_m), ... t_s = [psi]_s(t_s+1 ... t_m);
+
+  so that the equation
+
+    d[psi] - u1d[psi]1 - ... - u_s d[psi]_s - u_s+1 dt_s+1 - ... - u_m dt_m = 0
+
+  is identically true in regard to u1, ... um, t_s+1 ..., t_m; equating
+  to zero the coefficients of the differentials of these variables, we
+  thus obtain m - s relations of the form
+
+    d[psi]/dt_j - u1 d[psi]1/dt_j - ... - u_s d[psi]_s/dt_j - u_j = 0;
+
+  these m - s relations, with the previous s + 1 relations, constitute a
+  set of m + 1 relations connecting the 2m + 1 variables in virtue of
+  which the Pfaffian equation is satisfied independently of the form of
+  the functions [psi],[psi]1, ... [psi]s. There is clearly such a set
+  for each of the values s = 0, s = 1, ..., s = m - 1, s = m. And for
+  any value of s there may exist relations additional to the specified m
+  + 1 relations, provided they do not involve any relation connecting t,
+  t1, ... tm only, and are consistent with the m - s relations
+  connecting u1, ... um. It is now evident that, essentially, the
+  integration of a Pfaffian equation
+
+    a1dx1 + ... + a_n dx_n = 0,
+
+  wherein a1, ... an are functions of x1, ... xn, is effected by the
+  processes necessary to bring it to its reduced form, involving only
+  independent variables. And it is easy to see that if we suppose this
+  reduction to be carried out in all possible ways, there is no need to
+  distinguish the classes of integrals corresponding to the various
+  values of s; for it can be verified without difficulty that by putting
+  t' = t - u1t1 - ... - u_s t_s, t'1 = u1, ... t'_s = u_s, u'1 = -t1,
+  ..., u'_s = -t_s, t'_s+1 = t_s+1, ... t'_m = t_m, u'_s+1 = u_s+1, ...
+  u'_m = u_m, the reduced equation becomes changed to dt' - u'1 dt'1 -
+  ... - u'_m dt'_m = 0, and the general relations changed to
+
+    t' = [psi](t'_s+l, ... t'_m) - t'1[psi]1(t'_s+1, ... t'_m) -  ... -t'_s[psi]_s(t'_s+1, ... t'_m), = [phi],
+
+  say, together with u'1 = d[phi]/dt'1, ..., u'm = d[phi]/dt'm, which
+  contain only one relation connecting the variables t', t'1, ... t'm
+  only.
+
+
+    Simultaneous Pfaffian equations.
+
+  This method for a single Pfaffian equation can, strictly speaking, be
+  generalized to a simultaneous system of (n - r) Pfaffian equations dxj
+  = c_1j dx1 + ... + c_rj dxr only in the case already treated, when
+  this system is satisfied by regarding x_r+1, ... x_n as suitable
+  functions of the independent variables x1, ... xr; in that case the
+  integral manifolds are of r dimensions. When these are non-existent,
+  there may be integral manifolds of higher dimensions; for if
+
+  d[phi] = [phi]1 dx_r +  ... + [phi]_r dx_r + [phi]_r+1(c_1,r+1 dx1 +  ...  + c_r,r+1 dx_r) + [phi]_r+2 ( ) +  ...
+
+  be identically zero, then [phi][sigma] + c[sigma]_,r+1 [phi]_r+1 + ...
+  + c[sigma]_,n [phi]_n = 0, or [phi] satisfies the r partial
+  differential equations previously associated with the total equations;
+  when these are not a complete system, but included in a complete
+  system of r - [mu] equations, having therefore n - r - [mu]
+  independent integrals, the total equations are satisfied over a
+  manifold of r + [mu] dimensions (see E. v. Weber, _Math. Annal._ 1v.
+  (1901), p. 386).
+
+
+    Contact transformations.
+
+  It seems desirable to add here certain results, largely of algebraic
+  character, which naturally arise in connexion with the theory of
+  contact transformations. For any two functions of the 2n independent
+  variables x1, ... xn, p1, ... pn we denote by ([phi][psi]) the sum of
+  the n terms such as d[phi]d[psi]/dp_idx_i - d[psi]d[phi]/dp_idx_i. For
+  two functions of the (2n + 1) independent variables z, x1, ... xn, p1,
+  ... pn we denote by [phi][psi] the sum of the n terms such as
+
+    d[phi] /d[psi]      d[psi]\    d[psi] /d[phi]      d[phi]\
+    ------( ------ + p_i------ ) - ------( ------ + p_i------ ).
+     dpi   \ dxi          dz  /     dpi   \ dxi          dz  /
+
+  It can at once be verified that for any three functions
+  [f[[phi][psi]]] + [[phi][psi]f]] + [[psi][f[phi]]] = df/dz
+  [[phi][psi]] + d[phi]/dz [[psi]f] + d[psi]/dz [f[phi]], which when f,
+  [phi],[psi] do not contain z becomes the identity (f([phi][psi])) +
+  (phi([psi]f)) + ([psi](f[phi])) = 0. Then, if X1, ... Xn, P1, ... Pn
+  be such functions Of x1, ... xn, p1 ... pn that P1 dX1 + ... + Pn dXn
+  is identically equal to p1dx1 + ... + pn dxn, it can be shown by
+  elementary algebra, after equating coefficients of independent
+  differentials, (1) that the functions X1, ... Pn are independent
+  functions of the 2n variables x1, ... pn, so that the equations x'i =
+  Xi, p'i = Pi can be solved for x1, ... xn, p1, ... pn, and represent
+  therefore a transformation, which we call a homogeneous contact
+  transformation; (2) that the X1, ... Xn are homogeneous functions of
+  p1, ... pn of zero dimensions, the P1, ... Pn are homogeneous
+  functions of p1, ... pn of dimension one, and the ½n(n - 1) relations
+  (Xi Xj) = 0 are verified. So also are the n² relations (Pi Xi) = 1,
+  (Pi Xj) = 0, (Pi Pj) = 0. Conversely, if X1, ... Xn be independent
+  functions, each homogeneous of zero dimension in p1, ... pn satisfying
+  the ½n(n - 1) relations (Xi Xj) = 0, then P1, ... Pn can be uniquely
+  determined, by solving linear algebraic equations, such that P1 dX1 +
+  ... + Pn dXn = p1 dx1 + ... + pn dxn. If now we put n + 1 for n, put z
+  for x_n+1, Z for X_n+1, Qi for -Pi/P_n+1, for i = 1, ... n, put qi for
+  -p_i/p_n+1 and [sigma] for q_n+1/Q_n+1, and then finally write P1, ...
+  Pn, p1, ... pn for Q1, ... Qn, q1, ... qn, we obtain the following
+  results: If ZX1 ... Xn, P1, ... Pn be functions of z, x1, ... xn, p1,
+  ... pn, such that the expression dZ - P1 dX1 - ... - Pn dXn is
+  identically equal to [sigma](dz - p1 dx1 - ... - pn dxn), and [sigma]
+  not zero, then (1) the functions Z, X1, ... Xn, P1, ... Pn are
+  independent functions of z, x1, ... xn, p1, ... pn, so that the
+  equations z' = Z, x'i = Xi, p'i = Pi can be solved for z, x1, ... xn,
+  p1, ... pn and determine a transformation which we call a
+  (non-homogeneous) contact transformation; (2) the Z, X1, ... Xn verify
+  the ½n(n + 1) identities [Z Xi] = 0, [Xi Xj] = 0. And the further
+  identities
+
+    [Pi Xi] = [sigma], [Pi Xj] = 0, [Pi Z] = [sigma]Pi, [Pi Pj] = 0,
+
+                        dZ                                   dXi                        dPi
+    [Z[sigma]] = [sigma]-- - [sigma]², [Xi [sigma]] = [sigma]---, [Pi [sigma]] = [sigma]---
+                        dz                                   dz                         dz
+
+  are also verified. Conversely, if Z, x1, ... Xn be independent
+  functions satisfying the identities [Z Xi] = 0, [Xi Xj] = 0, then
+  [sigma], other than zero, and P1, ... Pn can be uniquely determined,
+  by solution of algebraic equations, such that
+
+    dZ - P1 dX1 - ... - Pn dXn = [sigma](dz - p1 dx1 - ... - p_n dx_n).
+
+  Finally, there is a particular case of great importance arising when
+  [sigma] = 1, which gives the results: (1) If U, X1, ... Xn, P1, ... Pn
+  be 2n + 1 functions of the 2n independent variables x1, ... xn, p1,
+  ... pn, satisfying the identity
+
+    dU + P1 dx1 + ... + Pn dXn = p1 dx1 + ... + p_n dx_n,
+
+  then the 2n functions P1, ... Pn, X1, ... Xn are independent, and we
+  have
+
+    (Xi Xj) = 0, (Xi U) = [delta]Xi, (Pi Xi) = 1, (Pi Xj) = 0, (Pi Pj ) = 0, (Pi U) + Pi = [delta]Pi,
+
+  where [delta] denotes the operator p1d/dp1 + ... + pnd/dpn; (2) If X1,
+  ... Xn be independent functions of x1, ... xn, p1, ... pn, such that
+  (Xi Xj) = 0, then U can be found by a quadrature, such that
+
+    (Xi U) = [delta]Xi;
+
+  and when Xi, ... Xn, U satisfy these ½n(n + 1) conditions, then P1,
+  ... Pn can be found, by solution of linear algebraic equations, to
+  render true the identity dU + P1 dX1 + ... + Pn dXn = p1 dx1 + ... +
+  pn dxn; (3) Functions X1, ... Xn, P1, ... Pn can be found to satisfy
+  this differential identity when U is an arbitrary given function of
+  x1, ... xn, p1, ... pn; but this requires integrations. In order to
+  see what integrations, it is only necessary to verify the statement
+  that if U be an arbitrary given function of x1, ... xn, p1, ... pn,
+  and, for r < n, X1, ... Xr be independent functions of these
+  variables, such that (X_[sigma] U) = [delta]X_[sigma], (X_[rho]
+  X_[sigma]) = 0, for [rho], [sigma] = 1 ... r, then the r + 1
+  homogeneous linear partial differential equations of the first order
+  (Uf) + [delta]f = 0, (X[rho]f) = 0, form a complete system. It will be
+  seen that the assumptions above made for the reduction of Pfaffian
+  expressions follow from the results here enunciated for contact
+  transformations.
+
+
+  Partial differential equation of the first order.
+
+  Meaning of a solution of the equation.
+
+We pass on now to consider the solution of any partial differential
+equation of the first order; we attempt to explain certain ideas
+relatively to a single equation with any number of independent variables
+(in particular, an ordinary equation of the first order with one
+independent variable) by speaking of a single equation with two
+independent variables x, y, and one dependent variable z. It will be
+seen that we are naturally led to consider systems of such simultaneous
+equations, which we consider below. The central discovery of the
+transformation theory of the solution of an equation F(x, y, z, dz/dx,
+dz/dy) = 0 is that its solution can always be reduced to the solution of
+partial equations which are _linear_. For this, however, we must regard
+dz/dx, dz/dy, during the process of integration, not as the differential
+coefficients of a function z in regard to x and y, but as variables
+independent of x, y, z, the too great indefiniteness that might thus
+appear to be introduced being provided for in another way. We notice
+that if z = [psi](x, y) be a solution of the differential equation, then
+dz = dxd[psi]/dx + dyd[psi]/dy; thus if we denote the equation by F(x,
+y, z, p, q,) = 0, and prescribe the condition dz = pdx + qdy for every
+solution, any solution such as z = [psi](x, y) will necessarily be
+associated with the equations p = dz/dx, q = dz/dy, and z will satisfy
+the equation in its original form. We have previously seen (under
+_Pfaffian Expressions_) that if five variables x, y, z, p, q, otherwise
+independent, be subject to dz - pdx - qdy = 0, they must in fact be
+subject to at least three mutual relations. If we associate with a point
+(x, y, z) the plane
+
+  Z - z = p(X - x) + q(Y - y)
+
+passing through it, where X, Y, Z are current co-ordinates, and call
+this association a surface-element; and if two consecutive elements of
+which the point(x + dx, y + dy, z + dz) of one lies on the plane of the
+other, for which, that is, the condition dz = pdx + qdy is satisfied, be
+said to be _connected,_ and an infinity of connected elements following
+one another continuously be called a _connectivity_, then our statement
+is that a connectivity consists of not more than [oo]² elements, the
+whole number of elements (x, y, z, p, q) that are possible being called
+[oo]^5. The solution of an equation F(x, y, z, dz/dx, dz/dy) = 0 is then
+to be understood to mean finding in all possible ways, from the [oo]^4
+elements (x, y, z, p, q) which satisfy F(x, y, z, p, q) = 0 a set of
+[oo]² elements forming a connectivity; or, more analytically, finding in
+all possible ways two relations G = 0, H = 0 connecting x, y, z, p, q
+and independent of F = 0, so that the three relations together may
+involve
+
+  dz = pdx + qdy.
+
+Such a set of three relations may, for example, be of the form z =
+[psi](x, y), p = d[psi]/dx, q = d[psi]/dy; but it may also, as another
+case, involve two relations z = [psi](y), x = [psi]1(y) connecting x, y,
+z, the third relation being
+
+  [psi]'(y) = p[psi]'1(y) + q,
+
+the connectivity consisting in that case, geometrically, of a curve in
+space taken with [oo]¹ of its tangent planes; or, finally, a
+connectivity is constituted by a fixed point and all the planes passing
+through that point. This generalized view of the meaning of a solution
+of F = 0 is of advantage, moreover, in view of anomalies otherwise
+arising from special forms of the equation itself. For instance, we may
+include the case, sometimes arising when the equation to be solved is
+obtained by transformation from another equation, in which F does not
+contain either p or q. Then the equation has [oo]² solutions, each
+consisting of an arbitrary point of the surface F = 0 and all the [oo]²
+planes passing through this point; it also has [oo]² solutions, each
+consisting of a curve drawn on the surface F = 0 and all the tangent
+planes of this curve, the whole consisting of [oo]² elements; finally,
+it has also an isolated (or singular) solution consisting of the points
+of the surface, each associated with the tangent plane of the surface
+thereat, also [oo]² elements in all. Or again, a linear equation F = Pp
++ Qq - R = 0, wherein P, Q, R are functions of x, y, z only, has [oo]²
+solutions, each consisting of one of the curves defined by
+
+  dx/P = dy/Q = dz/R
+
+taken with all the tangent planes of this curve; and the same equation
+has [oo]² solutions, each consisting of the points of a surface
+containing [oo]¹ of these curves and the tangent planes of this surface.
+And for the case of n variables there is similarly the possibility of n
++ 1 kinds of solution of an equation F(x1, ... xn, z, p1, ... pn) = 0;
+these can, however, by a simple contact transformation be reduced to one
+kind, in which there is only one relation z' = [psi](x'1, ... x'n)
+connecting the new variables x'1, ... x'n, z' (see under PFAFFIAN
+EXPRESSIONS); just as in the case of the solution
+
+  z = [psi](y), x = [psi]1(y), [psi]'(y) = p[psi]'1(y) + q
+
+of the equation Pp + Qq = R the transformation z' = z - px, x' = p, p' =
+-x, y' = y, q' = q gives the solution
+
+  z' = [psi](y') + x'[psi]1(y'), p' = dz'/dx', q' = dz'/dy'
+
+of the transformed equation. These explanations take no account of the
+possibility of p and q being infinite; this can be dealt with by writing
+p = -u/w, q = -v/w, and considering homogeneous equations in u, v, w,
+with udx + vdy + wdz = 0 as the differential relation necessary for a
+connectivity; in practice we use the ideas associated with such a
+procedure more often without the appropriate notation.
+
+
+  Order of the ideas.
+
+In utilizing these general notions we shall first consider the theory of
+characteristic chains, initiated by Cauchy, which shows well the nature
+of the relations implied by the given differential equation; the
+alternative ways of carrying out the necessary integrations are
+suggested by considering the method of Jacobi and Mayer, while a good
+summary is obtained by the formulation in terms of a Pfaffian
+expression.
+
+
+    Characteristic chains.
+
+  Consider a solution of F = 0 expressed by the three independent
+  equations F = 0, G = 0, H = 0. If it be a solution in which there is
+  more than one relation connecting x, y, z, let new variables x', y',
+  z', p', q' be introduced, as before explained under PFAFFIAN
+  EXPRESSIONS, in which z' is of the form
+
+    z' = z - p1x1 - ... - p_s x_s (s = 1 or 2),
+
+  so that the solution becomes of a form z' = [psi](x'y'), p' =
+  d[psi]/dx', q' = d[psi]/dy', which then will identically satisfy the
+  transformed equations F' = 0, G' = 0, H' = 0. The equation F' = 0, if
+  x', y', z' be regarded as fixed, states that the plane Z - z' = p'(X -
+  x') + q'(Y - y') is tangent to a certain cone whose vertex is (x', y',
+  z'), the consecutive point (x' + dx', y' + dy', z' + dz') of the
+  generator of contact being such that
+
+        /dF'       /dF'       / /  dF'      dF'\
+    dx'/ --  = dy'/ --  = dz'/ ( p'--- + q' --- ).
+      /  dp'     /  dq'     /   \  dp'      dq'/
+
+
+  Passing in this direction on the surface z' = [psi](x', y') the
+  tangent plane of the surface at this consecutive point is (p' + dp',
+  q' + dq'), where, since F'(x', y', [psi], d[psi]/dx', d[psi]/dy') = 0
+  is identical, we have dx' (dF'/dx' + p'dF'/dz') + dp'dF'/dp' = 0. Thus
+  the equations, which we shall call the characteristic equations,
+
+        /dF'       /dF'       //   dF'     dF'\        //  dF'     dF'\
+    dx'/ --- = dy'/ --- = dz'/( p' --- + q'--- ) = dp'/( - --- - p'--- )
+      /  dp'     /  dq'     /  \   dp'     dq'/      /  \  dx'     dz'/
+
+                                                       //  dF'     dF'\
+                                                 = dq'/( - --- - q'--- )
+                                                     /  \  dy'     dz'/
+
+  are satisfied along a connectivity of [oo]¹ elements consisting of a
+  curve on z' = [psi](x', y') and the tangent planes of the surface
+  along this curve. The equation F' = 0, when p', q' are fixed,
+  represents a curve in the plane Z - z' = p'(X - x') + q'(Y - y')
+  passing through (x', y', z'); if (x' + [delta]x', y' + [delta]y', z' +
+  [delta]z') be a consecutive point of this curve, we find at once
+
+              /dF'     dF'\              /dF'     dF'\
+    [delta]x'( --- + p'--- ) + [delta]y'( --- + q'--- ) = 0;
+              \dx'     dz'/              \dy'     dz'/
+
+  thus the equations above give [delta]x'dp' + [delta]y'dq' = 0, or the
+  tangent line of the plane curve, is, on the surface z' = [psi](x',
+  y'), in a direction conjugate to that of the generator of the cone.
+  Putting each of the fractions in the characteristic equations equal to
+  dt, the equations enable us, starting from an arbitrary element x'0,
+  y'0, z'0, p'0, q'0, about which all the quantities F', dF'/dp', &c.,
+  occurring in the denominators, are developable, to define, from the
+  differential equation F' = 0 alone, a connectivity of [oo]¹ elements,
+  which we call a _characteristic chain_; and it is remarkable that when
+  we transform again to the original variables (x, y, z, p, q), the form
+  of the differential equations for the chain is unaltered, so that they
+  can be written down at once from the equation F = 0. Thus we have
+  proved that the characteristic chain starting from any ordinary
+  element of any integral of this equation F = 0 consists only of
+  elements belonging to this integral. For instance, if the equation do
+  not contain p, q, the characteristic chain, starting from an arbitrary
+  plane through an arbitrary point of the surface F = 0, consists of a
+  pencil of planes whose axis is a tangent line of the surface F = 0. Or
+  if F = 0 be of the form Pp + Qq = R, the chain consists of a curve
+  satisfying dx/P = dy/Q = dz/R and a single infinity of tangent planes
+  of this curve, determined by the tangent plane chosen at the initial
+  point. In all cases there are [oo]³ characteristic chains, whose
+  aggregate may therefore be expected to exhaust the [oo]^4 elements
+  satisfying F = 0.
+
+
+    Complete integral constructed with characteristic chains.
+
+  Consider, in fact, a single infinity of connected elements each
+  satisfying F = 0, say a chain connectivity T, consisting of elements
+  specified by x0, y0, z0, p0, q0, which we suppose expressed as
+  functions of a parameter u, so that
+
+    U0 = dz0/du - p0dx0/du - q0dy0/du
+
+  is everywhere zero on this chain; further, suppose that each of F,
+  dF/dp, ... , dF/dx + pdF/dz is developable about each element of this
+  chain T, and that T is _not_ a characteristic chain. Then consider the
+  aggregate of the characteristic chains issuing from all the elements
+  of T. The [oo]² elements, consisting of the aggregate of these
+  characteristic chains, satisfy F = 0, provided the chain connectivity
+  T consists of elements satisfying F = 0; for each characteristic chain
+  satisfies dF = 0. It can be shown that these chains are connected; in
+  other words, that if x, y, z, p, q, be any element of one of these
+  characteristic chains, not only is
+
+    dz/dt - pdx/dt - qdy/dt = 0,
+
+  as we know, but also U = dz/du - pdx/du - qdy/du is also zero. For we
+  have
+
+    dU   d  /dz    dx    dy\    d  /dz    dx    dy\
+    -- = --( -- - p-- - q-- ) - --( -- - p-- - q-- )
+    dt   dt \du    du    du/    du \dt    dt    dt/
+
+         dp dx   dp dx   dq dy   dq dy
+       = -- -- - -- -- + -- -- - -- -- ,
+         du dt   dt du   du dt   dt du
+
+  which is equal to
+
+    dp dF   dx /dF    dF\    dq dF   dy /dF    dF\      dF
+    -- -- + --( -- + p-- ) + -- -- + --( -- + q-- ) = - -- U.
+    du dp   du \dx    dz/    du dq   du \dy    dz/      dz
+
+     dF
+  As -- is a developable function of t, this, giving
+     dz
+                       _
+                  /   / t  dF  \
+    U = U_{0} exp( -  |    --dt ),
+                  \  _/t0  dz  /
+
+  shows that U is everywhere zero. Thus integrals of F = 0 are
+  obtainable by considering the aggregate of characteristic chains
+  issuing from arbitrary chain connectivities T satisfying F = 0; and
+  such connectivities T are, it is seen at once, determinable without
+  integration. Conversely, as such a chain connectivity T can be taken
+  out from the elements of any given integral all possible integrals are
+  obtainable in this way. For instance, an arbitrary curve in space,
+  given by x0 = [theta](u), y0 = [phi](u), z0 = [psi](u), determines by
+  the two equations F(x0, y0, z0, p0, q0) = 0, [psi]'(u) = p0[theta]'(u)
+  + q0[phi]'(u), such a chain connectivity T, through which there passes
+  a perfectly definite integral of the equation F = 0. By taking [oo]²
+  initial chain connectivities T, as for instance by taking the curves
+  x0 = [theta], y0 = [phi], z0 = [psi] to be the [oo]² curves upon an
+  arbitrary surface, we thus obtain [oo]² integrals, and so [oo]^4
+  elements satisfying F = 0. In general, if functions G, H, independent
+  of F, be obtained, such that the equations F = 0, G = b, H = c
+  represent an integral for all values of the constants b, c, these
+  equations are said to constitute a _complete integral_. Then [oo]^4
+  elements satisfying F = 0 are known, and in fact every other form of
+  integral can be obtained without further integrations.
+
+
+    Operations necessary for integration of F = a.
+
+  In the foregoing discussion of the differential equations of a
+  characteristic chain, the denominators dF/dp, ... may be supposed to
+  be modified in form by means of F = 0 in any way conducive to a simple
+  integration. In the immediately following explanation of ideas,
+  however, we consider indifferently all equations F = constant; when a
+  function of x, y, z, p, q is said to be zero, it is meant that this is
+  so identically, not in virtue of F = 0; in other words, we consider
+  the integration of F = a, where a is an arbitrary constant. In the
+  theory of linear partial equations we have seen that the integration
+  of the equations of the characteristic chains, from which, as has just
+  been seen, that of the equation F = a follows at once, would be
+  involved in completely integrating the single linear homogeneous
+  partial differential equation of the first order [Ff] = 0 where the
+  notation is that explained above under CONTACT TRANSFORMATIONS. One
+  obvious integral is f = F. Putting F = a, where a is arbitrary, and
+  eliminating one of the independent variables, we can reduce this
+  equation [Ff] = 0 to one in four variables; and so on. Calling, then,
+  the determination of a single integral of a single homogeneous partial
+  differential equation of the first order in n independent variables,
+  _an operation of order_ n - 1, the characteristic chains, and
+  therefore the most general integral of F = a, can be obtained by
+  successive operations of orders 3, 2, 1. If, however, an integral of F
+  = a be represented by F = a, G = b, H = c, where b and c are arbitrary
+  constants, the expression of the fact that a characteristic chain of F
+  = a satisfies dG = 0, gives [FG] = 0; similarly, [FH] = 0 and [GH] =
+  0, these three relations being identically true. Conversely, suppose
+  that an integral G, independent of F, has been obtained of the
+  equation [Ff] = 0, which is an operation of order three. Then it
+  follows from the identity [f[[phi][psi]]] + [[phi][[psi]f]] +
+  [[psi][f[phi]]] = df/dz [[psi][phi]] + d[phi]/dz [psif] + d[psi]/dz
+  [f[phi]] before remarked, by putting [phi] = F, [psi] = G, and then
+  [Ff] = A(f), [Gf] = B(f), that AB(f) - BA(f) = dF/dz B(f) - dG/dz
+  A(f), so that the two linear equations [Ff] = 0, [Gf] = 0 form a
+  complete system; as two integrals F, G are known, they have a common
+  integral H, independent of F, G, determinable by an operation of order
+  one only. The three functions F, G, H thus identically satisfy the
+  relations [FG] = [GH] = [FH] = 0. The [oo]² elements satisfying F = a,
+  G = b, H = c, wherein a, b, c are assigned constants, can then be seen
+  to constitute an integral of F = a. For the conditions that a
+  characteristic chain of G = b issuing from an element satisfying F =
+  a, G = b, H = c should consist only of elements satisfying these three
+  equations are simply [FG] = 0, [GH] = 0. Thus, starting from an
+  arbitrary element of (F = a, G = b, H = c), we can single out a
+  connectivity of elements of (F = a, G = b, H = c) forming a
+  characteristic chain of G = b; then the aggregate of the
+  characteristic chains of F = a issuing from the elements of this
+  characteristic chain of G = b will be a connectivity consisting only
+  of elements of
+
+    (F = a, G = b, H = c),
+
+  and will therefore constitute an integral of F = a; further, it will
+  include all elements of (F = a, G = b, H = c). This result follows
+  also from a theorem given under CONTACT TRANSFORMATIONS, which shows,
+  moreover, that though the characteristic chains of F = a are not
+  determined by the three equations F = a, G = b, H = c, no further
+  integration is now necessary to find them. By this theorem, since
+  identically [FG] = [GH] = [FH] = 0, we can find, by the solution of
+  linear algebraic equations only, a non-vanishing function [sigma] and
+  two functions A, C, such that
+
+    dG - AdF - CdH = [sigma](dz - pdz - qdy);
+
+  thus all the elements satisfying F = a, G = b, H = c, satisfy dz = pdx
+  + qdy and constitute a connectivity, which is therefore an integral of
+  F = a. While, further, from the associated theorems, F, G, H, A, C are
+  independent functions and [FC] = 0. Thus C may be taken to be the
+  remaining integral independent of G, H, of the equation [Ff] = 0,
+  whereby the characteristic chains are entirely determined.
+
+
+    The single equation F = 0 and Pfaffian formulations.
+
+  When we consider the particular equation F = 0, neglecting the case
+  when neither p nor q enters, and supposing p to enter, we may express
+  p from F = 0 in terms of x, y, z, q, and then eliminate it from all
+  other equations. Then instead of the equation [Ff] = 0, we have, if F
+  = 0 give p = [psi](x, y, z, q), the equation
+
+                  /df         df\    d[psi]  /df     df\     /d[psi]     d[psi]\  df
+    [Sigma]f = - ( -- + [psi] -- ) + ------ ( -- + q -- ) - ( ------ + q ------ ) -- = 0,
+                  \dx         dz/      dq    \dy     dz/     \  dy         dz  /  dq
+
+  moreover obtainable by omitting the term in df/dp in [p-[psi], f] = 0.
+  Let x0, y0, z0, q0, be values about which the coefficients in this
+  equation are developable, and let [zeta], [eta], [omega] be the
+  principal solutions reducing respectively to z, y and q when x = x0.
+  Then the equations p = [psi], [zeta] = z0, [eta] = y0, [omega] = q0
+  represent a characteristic chain issuing from the element x0, y0, z0,
+  [psi]0, q0; we have seen that the aggregate of such chains issuing
+  from the elements of an arbitrary chain satisfying
+
+    dz0 = p0dx0 - q0dy0 = 0
+
+  constitute an integral of the equation p = [psi]. Let this arbitrary
+  chain be taken so that x0 is constant; then the condition for initial
+  values is only
+
+    dz0 - q0dy0 = 0,
+
+  and the elements of the integral constituted by the characteristic
+  chains issuing therefrom satisfy
+
+    d[zeta] - [omega]d[eta] = 0.
+
+  Hence this equation involves dz - [psi]dx - qdy = 0, or we have
+
+    dz - [psi]dx - qdy = [sigma](d[zeta] - [omega]d[eta]),
+
+  where [sigma] is not zero. Conversely, the integration of p = [psi]
+  is, essentially, the problem of writing the expression dz - [psi]dx -
+  qdy in the form [sigma](d[zeta] - [omega]d[eta]), as must be possible
+  (from what was said under _Pfaffian Expressions_).
+
+
+    System of equations of the first order.
+
+  To integrate a system of simultaneous equations of the first order X1
+  = a1, ... Xr = ar in n independent variables x1, ... xn and one
+  dependent variable z, we write p1 for dz/dx1, &c., and attempt to find
+  n + 1 - r further functions Z, X_r+1 ... Xn, such that the equations Z
+  = a, Xi = ai,(i = 1, ... n) involve dz - p1dx1 - ... - pndxn = 0. By
+  an argument already given, the common integral, if existent, must be
+  satisfied by the equations of the characteristic chains of any one
+  equation Xi = ai; thus each of the expressions [Xi Xj] must vanish in
+  virtue of the equations expressing the integral, and we may without
+  loss of generality assume that each of the corresponding ½r(r - 1)
+  expressions formed from the r given differential equations vanishes in
+  virtue of these equations. The determination of the remaining n + 1 -
+  r functions may, as before, be made to depend on characteristic
+  chains, which in this case, however, are manifolds of r dimensions
+  obtained by integrating the equations [X1f] = 0, ... [Xrf] = 0; or
+  having obtained one integral of this system other than X1, ... Xr, say
+  Xr+1, we may consider the system [X1f] = 0, ... [X_r+1 f] = 0, for
+  which, again, we have a choice; and at any stage we may use Mayer's
+  method and reduce the simultaneous linear equations to one equation
+  involving parameters; while if at any stage of the process we find
+  some but not all of the integrals of the simultaneous system, they can
+  be used to simplify the remaining work; this can only be clearly
+  explained in connexion with the theory of so-called function groups
+  for which we have no space. One result arising is that the
+  simultaneous system p1 = [phi]1, ... pr = [phi]r, wherein p1, ... pr
+  are not involved in [phi]1, ... [phi]r, if it satisfies the ½r(r - 1)
+  relations [pi - [phi]i, pj - [phi]j] = 0, has a solution z = [psi](x1,
+  ... xn), p1 = d[psi]/dx1, ... pn = d[psi]/dxn, reducing to an
+  arbitrary function of x_r+1, ... xn only, when x1 = x1^0, ... xr =
+  xr^0 under certain conditions as to developability; a generalization
+  of the theorem for linear equations. The problem of integration of
+  this system is, as before, to put
+
+    dz - [phi]1dx1 - ... - [phi]_r dx_r - p_r+1 dx_r+1 - ... - p_n dx_n
+
+  into the form [sigma](d[zeta] - [omega]_r+1 + d[xi]_r+1 - ... -
+  [omega]_n d[xi]_n); and here [zeta], [xi]_r+1, ... [xi]_n,
+  [omega]_r+1, ... [omega]_n may be taken, as before, to be principal
+  integrals of a certain complete system of linear equations; those,
+  namely, determining the characteristic chains.
+
+
+    Equations of dynamics.
+
+  If L be a function of t and of the 2n quantities x1, ... xn, [.x]1,
+  ... [.x]n, where [.x]i, denotes dxi/dt, &c., and if in the n equations
+
+     d   /  dL  \     dL
+    --- (--------) = ----
+    dt   \ dx_i /    dx_i
+
+  we put p_i = dL/d[.x]_i, and so express [.x]1 , ... [.x]_n in terms of
+  t, x_i, ... x_n, p1, ... p_n, assuming that the determinant of the
+  quantities d²L/dx_i d[.x]_j is not zero; if, further, H denote the
+  function of t, x1, ... xn, p1, ... pn, numerically equal to p1[.x]1 +
+  ... + pn[.x]n - L, it is easy to prove that dpi/dt = -dH/dxi, dxi/dt =
+  dH/dp_i. These so-called _canonical_ equations form part of those for
+  the characteristic chains of the single partial equation dz/dt + H(t,
+  x1, ... xn, dz/dx1, ..., dz/dx_n) = 0, to which then the solution of
+  the original equations for x1 ... xn can be reduced. It may be shown
+  (1) that if z = [psi](t, x1, ... xn, c1, .. cn) + c be a complete
+  integral of this equation, then pi = d[psi]/dx_i, d[psi]/dc_i = e_i are
+  2n equations giving the solution of the canonical equations referred
+  to, where c1 ... cn and e1, ... en are arbitrary constants; (2) that
+  if xi = Xi(t, x^01, ... pn^0), pi=Pi(t, x1^0, ... p^0n) be the
+  principal solutions of the canonical equations for t = t^0, and
+  [omega] denote the result of substituting these values in p1dH/dp1 +
+  ... + pndH/dpn - H, and [Omega] = [int] [t0 to t] [omega]dt, where,
+  after integration, [Omega] is to be expressed as a function of t, x1,
+  ... xn, x1^0, ... xn^0, then z = [Omega] + z^0 is a complete integral
+  of the partial equation.
+
+
+  Application of theory of continuous groups to formal theories.
+
+A system of differential equations is said to allow a certain continuous
+group of transformations (see GROUPS, THEORY OF) when the introduction
+for the variables in the differential equations of the new variables
+given by the equations of the group leads, for all values of the
+parameters of the group, to the same differential equations in the new
+variables. It would be interesting to verify in examples that this is
+the case in at least the majority of the differential equations which
+are known to be integrable in finite terms. We give a theorem of very
+general application for the case of a simultaneous complete system of
+linear partial homogeneous differential equations of the first order, to
+the solution of which the various differential equations discussed have
+been reduced. It will be enough to consider whether the given
+differential equations allow the infinitesimal transformations of the
+group.
+
+  It can be shown easily that sufficient conditions in order that a
+  complete system [Pi]1f = 0 ... [Pi]kf = 0, in n independent variables,
+  should allow the infinitesimal transformation Pf = 0 are expressed by
+  k equations [Pi]_i Pf - P[Pi]_i f = [lambda]_i1 [Pi]1f + ... +
+  [lambda]_ik [Pi]_kf. Suppose now a complete system of n - r equations
+  in n variables to allow a group of r infinitesimal transformations
+  (P1f, ..., Prf) which has an invariant subgroup of r - 1 parameters
+  (P1f, ..., Pr-1f), it being supposed that the n quantities [Pi]1f,
+  ..., [Pi]_n-r f, P1 f, ..., P_r f are not connected by an identical
+  linear equation (with coefficients even depending on the independent
+  variables). Then it can be shown that one solution of the complete
+  system is determinable by a quadrature. For each of [Pi]_i P_[sigma] f
+  - P_[sigma] [Pi]_i f is a linear function of [Pi]1f, ..., [Pi]_n-r f
+  and the simultaneous system of independent equations [Pi]1f = 0, ...
+  [Pi]_n-r f = 0, P1f = 0, ... P_r-1 f = 0 is therefore a complete
+  system, allowing the infinitesimal transformation Prf. This complete
+  system of n - 1 equations has therefore one common solution [omega],
+  and P_r([omega]) is a function of [omega]. By choosing [omega]
+  suitably, we can then make Pr([omega]) = 1. From this equation and the
+  n - 1 equations [Pi]_i[omega] = 0, P_[sigma][omega] = 0, we can
+  determine [omega] by a quadrature only. Hence can be deduced a much
+  more general result, _that if the group of r parameters be integrable,
+  the complete system can be entirety solved by quadratures_; it is only
+  necessary to introduce the solution found by the first quadrature as
+  an independent variable, whereby we obtain a complete system of n - r
+  equations in n - 1 variables, subject to an integrable group of r - 1
+  parameters, and to continue this process. We give some examples of the
+  application of the theorem. (1) If an equation of the first order y' =
+  [psi](x, y) allow the infinitesimal transformation [xi]df/dx +
+  [eta]df/dy, the integral curves [omega](x, y) = y°, wherein [omega](x,
+  y) is the solution of df/dx + [psi](x, y) df/dy = 0 reducing to y for
+  x = x°, are interchanged among themselves by the infinitesimal
+  transformation, or [omega](x, y) can be chosen to make [xi]d[omega]/dx
+  + [eta]d[omega]/dy = 1; this, with d[omega]/dx + [psi]d[omega]/dy = 0,
+  determines [omega] as the integral of the complete differential (dy -
+  [psi]dx)/([eta] - [psi][xi]). This result itself shows that every
+  ordinary differential equation of the first order is subject to an
+  infinite number of infinitesimal transformations. But every
+  infinitesimal transformation [xi]df/dx + [eta]df/dy can by change of
+  variables (after integration) be brought to the form df/dy, and all
+  differential equations of the first order allowing this group can then
+  be reduced to the form F(x, dy/dx) = 0. (2) In an ordinary equation of
+  the second order y" = [psi](x, y, y'), equivalent to dy/dx = y1,
+  dy1/dx = [psi](x, y, y1), if H, H1 be the solutions for y and y1
+  chosen to reduce to y^0 and y1° when x = x°, and the equations H = y,
+  H1= y1 be equivalent to [omega] = y°, [omega]1 = y1°, then [omega],
+  [omega]1 are the principal solutions of [Pi]f = df/dx + y1df/dy +
+  [psi]df/dy1 = 0. If the original equation allow an infinitesimal
+  transformation whose first _extended_ form (see GROUPS) is Pf =
+  [xi]df/dx + [eta]df/dy + [eta]1df/dy1, where [eta]1[delta]t is the
+  increment of dy/dx when [xi][delta]t, [eta][delta]t are the increments
+  of x, y, and is to be expressed in terms of x, y, y1, then each of
+  P[omega] and P[omega]1 must be functions of [omega] and [omega]1, or
+  the partial differential equation [Pi]f must allow the group Pf. Thus
+  by our general theorem, if the differential equation allow a group of
+  two parameters (and such a group is always integrable), it can be
+  solved by quadratures, our explanation sufficing, however, only
+  provided the form [Pi]f and the two infinitesimal transformations are
+  not linearly connected. It can be shown, from the fact that [eta]1 is
+  a quadratic polynomial in y1, that no differential equation of the
+  second order can allow more than 8 really independent infinitesimal
+  transformations, and that every homogeneous linear differential
+  equation of the second order allows just 8, being in fact reducible to
+  d²y/dx² = 0. Since every group of more than two parameters has
+  subgroups of two parameters, a differential equation of the second
+  order allowing a group of more than two parameters can, as a rule, be
+  solved by quadratures. By transforming the group we see that if a
+  differential equation of the second order allows a single
+  infinitesimal transformation, it can be transformed to the form F(x,
+  d[gamma]/dx, d²[gamma]/dx²); this is not the case for every
+  differential equation of the second order. (3) For an ordinary
+  differential equation of the third order, allowing an integrable group
+  of three parameters whose infinitesimal transformations are not
+  linearly connected with the partial equation to which the solution of
+  the given ordinary equation is reducible, the similar result follows
+  that it can be integrated by quadratures. But if the group of three
+  parameters be simple, this result must be replaced by the statement
+  that the integration is reducible to quadratures and that of a
+  so-called Riccati equation of the first order, of the form dy/dx = A +
+  By + Cy², where A, B, C are functions of x. (4) Similarly for the
+  integration by quadratures of an ordinary equation yn = [psi](x, y,
+  y1, ... yn-1) of any order. Moreover, the group allowed by the
+  equation may quite well consist of extended contact transformations.
+  An important application is to the case where the differential
+  equation is the resolvent equation defining the group of
+  transformations or rationality group of another differential equation
+  (see below); in particular, when the rationality group of an ordinary
+  linear differential equation is integrable, the equation can be solved
+  by quadratures.
+
+
+  Consideration of function theories of differential equations.
+
+Following the practical and provisional division of theories of
+differential equations, to which we alluded at starting, into
+transformation theories and function theories, we pass now to give some
+account of the latter. These are both a necessary logical complement of
+the former, and the only remaining resource when the expedients of the
+former have been exhausted. While in the former investigations we have
+dealt only with values of the independent variables about which the
+functions are developable, the leading idea now becomes, as was long ago
+remarked by G. Green, the consideration of the neighbourhood of the
+values of the variables for which this developable character ceases.
+Beginning, as before, with existence theorems applicable for ordinary
+values of the variables, we are to consider the cases of failure of such
+theorems.
+
+
+  A general existence theorem.
+
+When in a given set of differential equations the number of equations is
+greater than the number of dependent variables, the equations cannot be
+expected to have common solutions unless certain conditions of
+compatibility, obtainable by equating different forms of the same
+differential coefficients deducible from the equations, are satisfied.
+We have had examples in systems of linear equations, and in the case of
+a set of equations p1 = [phi]1, ..., pr = [phi]r. For the case when the
+number of equations is the same as that of dependent variables, the
+following is a general theorem which should be referred to: Let there be
+r equations in r dependent variables z1, ... zr and n independent
+variables x1, ... xn; let the differential coefficient of z[sigma] of
+highest order which enters be of order h[sigma], and suppose d^h_[sigma]
+z_[sigma]/dx1^h_[sigma] to enter, so that the equations can be written
+d^h_[sigma] z_[sigma]/dx1^h_[sigma] = [Phi]_[sigma], where in the
+general differential coefficient of z_[rho] which enters in
+[Phi]_[sigma], say
+
+  d^(k1 + ... + kn) z_[rho]/dx1^k1 ... dx_n^k_n,
+
+we have k1 < h_[rho] and k1 + ... + k_n <= h_[rho]. Let a1, ... an, b1,
+... br, and b[rho]_(k1 ... kn) be a set of values of
+
+  x1, ... x_n, z1, ... z_r
+
+and of the differential coefficients entering in [Phi]_[sigma] about
+which all the functions [Phi]1, ... [Phi]_r, are developable.
+Corresponding to each dependent variable z_[sigma], we take now a set of
+h_[sigma] functions of x2, ... xn, say [phi][sigma], [phi][sigma]^(1),
+..., [phi][sigma]^(h-1) arbitrary save that they must be developable
+about a2, a3, ... an, and such that for these values of x2, ... xn, the
+function [phi]_[rho] reduces to b_[rho], and the differential
+coefficient
+
+  d^(k2 + ... + kn) [phi]_[rho]^(k1)/dx2^k2 ... dx_n^kn
+
+reduces to b^kn_(k1 ... kn). Then the theorem is that there exists one,
+and only one, set of functions z1, ... z_r, of x2, ... x_n developable
+about a1, ... an satisfying the given differential equations, and such
+that for x1 = a1 we have
+
+  z_[sigma] = [phi]_[sigma], dz_[sigma]/dx1 = [phi]_[sigma]^(1), ...
+         d^(h_[sigma]-1) z_[sigma]/d^(h_[sigma]-1) x1 = [phi][sigma]^(h_[sigma]-1).
+
+And, moreover, if the arbitrary functions [phi]_[sigma],
+[phi]_[sigma]^(1) ... contain a certain number of arbitrary variables
+t1, ... tm, and be developable about the values t1°, ... tm° of these
+variables, the solutions z1, ... zr will contain t1, ... tm, and be
+developable about t1°, ... tm°.
+
+
+    Singular points of solutions.
+
+  The proof of this theorem may be given by showing that if ordinary
+  power series in x1 - -a1, ... xn - an, t1 - t1°, ... tm - tm° be
+  substituted in the equations wherein in z[sigma] the coefficients of
+  (x1 - a1)°, x1 - a1, ..., (x1 - a1)^(h_[sigma]-1) are the arbitrary
+  functions [phi]_[sigma], [phi]_[sigma]^(1), ..., [phi]_[sigma]^h-1,
+  divided respectively by 1, 1!, 2!, &c., then the differential
+  equations determine uniquely all the other coefficients, and that the
+  resulting series are convergent. We rely, in fact, upon the theory of
+  monogenic analytical functions (see FUNCTION), a function being
+  determined entirely by its development in the neighbourhood of one set
+  of values of the independent variables, from which all its other
+  values arise by _continuation_; it being of course understood that the
+  coefficients in the differential equations are to be continued at the
+  same time. But it is to be remarked that there is no ground for
+  believing, if this method of continuation be utilized, that the
+  function is single-valued; we may quite well return to the same values
+  of the independent variables with a different value of the function;
+  belonging, as we say, to a different branch of the function; and there
+  is even no reason for assuming that the number of branches is finite,
+  or that different branches have the same singular points and regions
+  of existence. Moreover, and this is the most difficult consideration
+  of all, all these circumstances may be dependent upon the values
+  supposed given to the arbitrary constants of the integral; in other
+  words, the singular points may be either _fixed_, being determined by
+  the differential equations themselves, or they may be _movable_ with
+  the variation of the arbitrary constants of integration. Such
+  difficulties arise even in establishing the reversion of an elliptic
+  integral, in solving the equation
+
+     /dx\²
+    ( -- ) = (x-a1)(x - a2)(x - a3)(x - a4);
+     \ds/
+
+  about an ordinary value the right side is developable; if we put x -
+  a1 = t1², the right side becomes developable about t1 = 0; if we put x
+  = 1/t, the right side of the changed equation is developable about t =
+  0; it is quite easy to show that the integral reducing to a definite
+  value x0 for a value s0 is obtainable by a series in integral powers;
+  this, however, must be supplemented by showing that for no value of s
+  does the value of x become entirely undetermined.
+
+
+    Linear differential equations with rational coefficients.
+
+  These remarks will show the place of the theory now to be sketched of
+  a particular class of ordinary linear homogeneous differential
+  equations whose importance arises from the completeness and generality
+  with which they can be discussed. We have seen that if in the
+  equations dy/dx = y1, dy1/dx = y2, ..., dy_n-2/dx = y_n-1,
+
+    dy_n-1/dx = a_n y + a_n-1 y1 + ... + a1 y_n-1,
+
+  where a1, a2, ..., an are now to be taken to be rational functions of
+  x, the value x = xº be one for which no one of these rational
+  functions is infinite, and yº, yº1, ..., yº_n-1 be quite arbitrary
+  finite values, then the equations are satisfied by
+
+    y = yºu + yº1u1 + ... + yº_n-1 u_n-1,
+
+  where u, u1, ..., un-1 are functions of x, independent of yº, ...
+  yº_n-1, developable about x = xº; this value of y is such that for x =
+  xº the functions y, y1 ... y_n-1 reduce respectively to yº, yº1, ...
+  yº_n-1; it can be proved that the region of existence of these series
+  extends within a circle centre xº and radius equal to the distance
+  from xº of the nearest point at which one of a1, ... an becomes
+  infinite. Now consider a region enclosing xº and only one of the
+  places, say [Sigma], at which one of a1, ... an becomes infinite. When
+  x is made to describe a closed curve in this region, including this
+  point [Sigma] in its interior, it may well happen that the
+  continuations of the functions u, u1, ..., u_n-1 give, when we have
+  returned to the point x, values v, v1, ..., v_n-1, so that the
+  integral under consideration becomes changed to yº + yº1v1 + ... +
+  yº_n-1 v_n-1. At xº let this branch and the corresponding values of
+  y1, ... y_n-1 be [eta]º, [eta]º1, ... [eta]º_n-1; then, as there is
+  only one series satisfying the equation and reducing to ([eta]º,
+  [eta]º1, ... [eta]º_n-1) for x = xº and the coefficients in the
+  differential equation are single-valued functions, we must have
+  [eta]ºu + [eta]º1u1 + ... + [eta]º_n-1 u_n-1 = yºv + yº1v1 + ... +
+  yº_n-1 v_n-1; as this holds for arbitrary values of yº ... yº_n-1,
+  upon which u, ... u_n-1 and v, ... v_n-1 do not depend, it follows
+  that each of v, ... v_n-1 is a linear function of u, ... u_n-1 with
+  constant coefficients, say v_i = A_i1 u + ... + A_in u_n-1. Then
+
+    yºv + ... + yº_n-1 v_n-1 = ([Sigma]_i A_i1 y_iº)u + ... + ([Sigma]_i A_in yº_i)u_n-1;
+
+  this is equal to [mu](yºu + ... + yº_n-1 u_n-1) if [Sigma]_i A_ir yº_i
+  = [mu]yº_r-1; eliminating yº ... yº_n-1 from these linear equations,
+  we have a determinantal equation of order n for [mu]; let [mu]1 be one
+  of its roots; determining the ratios of yº, y1º, ... yº_n-1 to satisfy
+  the linear equations, we have thus proved that there exists an
+  integral, H, of the equation, which when continued round the point
+  [Sigma] and back to the starting-point, becomes changed to H1 =
+  [mu]1H. Let now [xi] be the value of x at [Sigma] and r1 one of the
+  values of (1/2[pi]i) log [mu]1; consider the function (x - [xi])^r1 H;
+  when x makes a circuit round x = [xi], this becomes changed to
+
+    exp(-2[pi]ir1) (x - [xi])^-r1 [mu]H,
+
+  that is, is unchanged; thus we may put H = (x - [xi])^r1 [phi]1,
+  [phi]1 being a function single-valued for paths in the region
+  considered described about [Sigma], and therefore, by Laurent's
+  Theorem (see FUNCTION), capable of expression in the annular region
+  about this point by a series of positive and negative integral powers
+  of x - [xi], which in general may contain an infinite number of
+  negative powers; there is, however, no reason to suppose r1 to be an
+  integer, or even real. Thus, if all the roots of the determinantal
+  equation in [mu] are different, we obtain n integrals of the forms (x
+  -[xi])^r1 phi1, ..., (x - [xi])^rn [phi]_n. In general we obtain as
+  many integrals of this form as there are really different roots; and
+  the problem arises to discover, in case a root be k times repeated, k
+  - 1 equations of as simple a form as possible to replace the k - 1
+  equations of the form yº + ... + yº_n-1 v_n-1 = [mu](yº + ... + yº_n-1
+  u_n-1) which would have existed had the roots been different. The most
+  natural method of obtaining a suggestion lies probably in remarking
+  that if r2 = r1 + h, there is an integral [(x - [xi])^(r1 + h) [phi]2
+  - (x -[xi])^r1 [phi]1]/h, where the coefficients in [phi]2 are the
+  same functions of r1 + h as are the coefficients in [phi]1 of r1; when
+  h vanishes, this integral takes the form
+                    _                              _
+                   | d[phi]1                        |
+    (x - [xi])^r1  | ------- + [phi]1 log (x - [xi])|,
+                   |_  dr1                         _|
+
+  or say       (x-[xi])^r1 [[phi]1 + [psi]1 log (x - [xi])];
+
+  denoting this by 2[pi]i[mu]1K, and (x-[xi])^r1 [phi]1 by H, a circuit
+  of the point [xi] changes K into
+
+              1
+    K' = ----------- [e^(2[pi]ir1) (x - [xi])^r1 [psi]1 + e^(2[pi]ir1) (x - [xi])^r1 [phi]1 (2[pi]i + log(x - [xi]))]
+         2[pi]i[mu]1
+
+       = [mu]1K + H.
+
+  A similar artifice suggests itself when three of the roots of the
+  determinantal equation are the same, and so on. We are thus led to the
+  result, which is justified by an examination of the algebraic
+  conditions, that whatever may be the circumstances as to the roots of
+  the determinantal equation, n integrals exist, breaking up into
+  batches, the values of the constituents H1, H2, ... of a batch after
+  circuit about x = [xi] being H1' = [mu]1H1, H2' = [mu]1H2 + H1, H3' =
+  [mu]1H3 + H2, and so on. And this is found to lead to the forms (x -
+  [xi])^r1 [phi]1, (x - [xi])^r1 [[psi]1 + [phi]1 log (x - [xi])], (x -
+  [xi])^r1 [[chi]1 + [chi]2 log (x - [xi]) + [phi]1(log(x - [xi]))²],
+  and so on. Here each of [phi]1, [psi]1, [chi]1, [chi]2, ... is a
+  series of positive and negative integral powers of x - [xi] in which
+  the number of negative powers may be infinite.
+
+
+    Regular equations.
+
+  It appears natural enough now to inquire whether, under proper
+  conditions for the forms of the rational functions a1, ... an, it may
+  be possible to ensure that in each of the series [phi]1, [psi]1,
+  [chi]1, ... the number of negative powers shall be finite. Herein
+  lies, in fact, the limitation which experience has shown to be
+  justified by the completeness of the results obtained. Assuming n
+  integrals in which in each of [phi]1, [psi]1, [chi]1 ... the number of
+  negative powers is finite, there is a definite homogeneous linear
+  differential equation having these integrals; this is found by forming
+  it to have the form
+
+    y'^n = (x - [xi])^-1 b1y'^(n-1) + (x - [xi])^-2 b2y'^(n-2) + ... +(x - [xi])^-n b_n y,
+
+  where b1, ... bn are finite for x = [xi]. Conversely, assume the
+  equation to have this form. Then on substituting a series of the form
+  (x - [xi])^r [1 + A1(x - [xi]) + A2(x - [xi])² + ... ] and equating
+  the coefficients of like powers of x-[xi], it is found that r must be
+  a root of an algebraic equation of order n; this equation, which we
+  shall call the index equation, can be obtained at once by substituting
+  for y only (x - [xi])^r and replacing each of b1, ... bn by their
+  values at x = [xi]; arrange the roots r1, r2, ... of this equation so
+  that the real part of ri is equal to, or greater than, the real part
+  of r_i+1, and take r equal to r1; it is found that the coefficients
+  A1, A2 ... are uniquely determinate, and that the series converges
+  within a circle about x = [xi] which includes no other of the points
+  at which the rational functions a1 ... an become infinite. We have
+  thus a solution H1 = (x -[xi])^r1 [phi]1 of the differential equation.
+  If we now substitute in the equation y = H1 f[eta]dx, it is found to
+  reduce to an equation of order n - 1 for [eta] of the form
+
+    [eta]'^(n-1) = (x - [xi])^-1 c1[eta]'^(n-2) + ... + (x-[xi])^(n-1) c_n-1 [eta],
+
+  where c1, ... c_n-1 are not infinite at x = [xi]. To this equation
+  precisely similar reasoning can then be applied; its index equation
+  has in fact the roots r2 - r1 - 1, ... , rn - r1 - 1; if r2 - r1 be
+  zero, the integral (x - [xi])^-1 [psi]1 of the [eta] equation will
+  give an integral of the original equation containing log (x - [xi]);
+  if r2 - r1 be an integer, and therefore a negative integer, the same
+  will be true, unless in [psi]1 the term in (x - [xi])^(r1 - r2) be
+  absent; if neither of these arise, the original equation will have an
+  integral (x -[xi])^r2 [phi]2. The [eta] equation can now, by means of
+  the one integral of it belonging to the index r2 - r1 - 1, be
+  similarly reduced to one of order n - 2, and so on. The result will be
+  that stated above. We shall say that an equation of the form in
+  question is _regular_ about x = [xi].
+
+
+    Fuchsian equations.
+
+    Equation of the second order.
+
+  We may examine in this way the behaviour of the integrals at all the
+  points at which any one of the rational functions a1 ... an becomes
+  infinite; in general we must expect that beside these the value x =
+  [oo] will be a singular point for the solutions of the differential
+  equation. To test this we put x = 1/t throughout, and examine as
+  before at t = 0. For instance, the ordinary linear equation with
+  constant coefficients has no singular point for finite values of x; at
+  x = [oo] it has a singular point and is not regular; or again,
+  Bessel's equation x² + xy' + (x² - n²)y = 0 is regular about x = 0,
+  but not about x = [oo]. An equation regular at all the finite
+  singularities and also at x = [oo] is called a Fuchsian equation. We
+  proceed to examine particularly the case of an equation of the second
+  order
+
+    y" + ay' + by = 0.
+
+  Putting x = 1/t, it becomes
+
+    d²y/dt² + (2t^-1 - at^-2)dy/dt + bt^-4 y = 0,
+
+  which is not regular about t = 0 unless 2 - at^-1 and bt^-2, that is,
+  unless ax and bx² are finite at x =[oo]; which we thus assume; putting
+  y = t^r(1 + A1t + ... ), we find for the index equation at x =
+  [inifinity] the equation r(r - 1) + r(2 - ax)_0 + (bx²)_0 = 0. If
+  there be finite singular points at [xi]1, ... [xi]m, where we assume
+  m>1, the cases m = 0, m = 1 being easily dealt with, and if [phi](x) =
+  (x - [xi]1) ... (x -[xi]m), we must have a.[phi](x) and b·[[phi](x)]²
+  finite for all finite values of x, equal say to the respective
+  polynomials [psi](x) and [theta](x), of which by the conditions at x =
+  [oo] the highest respective orders possible are m - 1 and 2(m - 1).
+  The index equation at x = [xi]1 is r(r - 1) +
+  r[psi]([xi]1)/[phi]'([xi]1) + [theta]([xi])1/[[phi]'([xi]1)]² = 0, and
+  if [alpha]1, [beta]1 be its roots, we have [alpha]1 + [beta]1 = 1 -
+  [psi]([xi]1)/[phi]'([xi]1) and [alpha]1[beta]1 =
+  [theta]([xi])1/[[phi]'([xi]1)]². Thus by an elementary theorem of
+  algebra, the sum [Sigma](1 - [alpha]i - [beta]i)/(x - [xi]i), extended
+  to the m finite singular points, is equal to [psi](x)/[phi](x), and
+  the sum [Sigma](1 - [alpha]i - [beta]i) is equal to the ratio of the
+  coefficients of the highest powers of x in [psi](x) and [phi](x), and
+  therefore equal to 1 + [alpha] + [beta], where [alpha], [beta] are the
+  indices at x = [oo]. Further, if (x, 1)m-2 denote the integral part of
+  the quotient [theta](x)/[phi](x), we have
+  [Sigma][alpha]_i[beta]_i[phi]'([xi]_i)/(x - [xi]_i) equal to -(x,
+  1)_m-2 + [theta](x)/[phi](x), and the coefficient of x^m-2 in (x,
+  1)_m-2 is [alpha][beta]. Thus the differential equation has the form
+
+    y" + y'[Sigma](1 - [alpha]_i - [beta]_i)/(x - [xi]_i) + y[(x, 1)_m-2 +
+         [Sigma][alpha]_i[beta]_i[phi]'([xi]_i)/(x - [xi]_i)]/[phi](x) = 0.
+
+  If, however, we make a change in the dependent variable, putting y =
+  (x - [xi]1)^[alpha]1 ... (x - [xi]_m)^[alpha] m[eta], it is easy to
+  see that the equation changes into one having the same singular points
+  about each of which it is regular, and that the indices at x = [xi]_i
+  become 0 and [beta]_i - [alpha]_i, which we shall denote by [lambda]i,
+  for (x -[xi]_i)^[alpha]j can be developed in positive integral powers
+  of x -[xi]_i about x = [xi]_i; by this transformation the indices at x
+  = [oo] are changed to
+
+    [alpha] + [alpha]1 + ... + [alpha]m, [beta] + [beta]1 + ... + [beta]m
+
+  which we shall denote by [lambda], [mu]. If we suppose this change to
+  have been introduced, and still denote the independent variable by y,
+  the equation has the form
+
+    y" + y'[Sigma](1 - [lambda]_i)/(x - [xi]_i) + y(x, 1)_m-2/[phi](x) = 0,
+
+  while [lambda] + [mu] + [lambda]1 + ... + [lambda]_m = m - 1.
+  Conversely, it is easy to verify that if [lambda][mu] be the
+  coefficient of x^m-2 in (x, 1)_m-2, this equation has the specified
+  singular points and indices whatever be the other coefficients in (x,
+  1)_m-2.
+
+
+    Hypergeometric equation.
+
+  Thus we see that (beside the cases m = 0, m = 1) the "Fuchsian
+  equation" of the second order with _two_ finite singular points is
+  distinguished by the fact that it has a definite form when the
+  singular points and the indices are assigned. In that case, putting (x
+  - [xi]1)/(x - [xi]2) = t/(t - 1), the singular points are transformed
+  to 0, 1, [oo], and, as is clear, without change of indices. Still
+  denoting the independent variable by x, the equation then has the form
+
+    x(1 - x)y" + y'[1 - [lambda]1 - x(1  + [lambda] + [mu])] - [lambda][mu]y = 0,
+
+  which is the ordinary hypergeometric equation. Provided none of
+  [lambda]1, [lambda]2, [lambda] - [mu] be zero or integral about x = 0,
+  it has the solutions
+
+    F([lambda], [mu], 1 - [lambda]1, x), x^[lambda]1 F([lambda] + [lambda]1, [mu] + [lambda]1, 1 + [lambda]1, x);
+
+  about x = 1 it has the solutions
+
+    F([lambda], [mu], 1 - [lambda]2, 1 - x), (1 - x)^[lambda]1 F([lambda] + [lambda]2, [mu] + [lambda]2, 1 + [lambda]2, 1 - x),
+
+  where [lambda] + [mu] + [lambda]1 + [lambda]2 = 1; about x = [oo] it
+  has the solutions
+
+    x^-[lambda] F([lambda], [lambda] + [lambda]1, [lambda] - [mu] + 1, x^-1),
+      x^-[mu] F([mu], [mu] + [lambda]1, [mu] - [lambda] + 1, x^-1),
+
+  where F([alpha], [beta], [gamma], x) is the series
+
+        [alpha][beta]x   [alpha]([alpha] + 1)[beta]([beta] + 1)x²
+    1 + -------------- + ---------------------------------------- ...,
+           [gamma]               1·2·[gamma]([gamma] + 1)
+
+  which converges when |x| < 1, whatever [alpha], [beta], [gamma] may
+  be, converges for all values of x for which |x| = 1 provided the real
+  part of [gamma] - [alpha] - [beta] < 0 algebraically, and converges
+  for all these values except x = 1 provided the real part of [gamma] -
+  [alpha] -[beta] > -1 algebraically.
+
+  In accordance with our general theory, logarithms are to be expected
+  in the solution when one of [lambda]1, [lambda]2, [lambda] - [mu] is
+  zero or integral. Indeed when [lambda]1 is a negative integer, not
+  zero, the second solution about x = 0 would contain vanishing factors
+  in the denominators of its coefficients; in case [lambda] or [mu] be
+  one of the positive integers 1, 2, ... (-[lambda]1), vanishing factors
+  occur also in the numerators; and then, in fact, the second solution
+  about x = 0 becomes x^[lambda]1 times an integral polynomial of degree
+  (-[lambda]1) - [lambda] or of degree (-[lambda]1) - [mu]. But when
+  [lambda]1 is a negative integer including zero, and neither [lambda]
+  nor [mu] is one of the positive integers 1, 2 ... (-[lambda]1), the
+  second solution about x = 0 involves a term having the factor log x.
+  When [lambda]1 is a positive integer, not zero, the second solution
+  about x = 0 persists as a solution, in accordance with the order of
+  arrangement of the roots of the index equation in our theory; the
+  first solution is then replaced by an integral polynomial of degree
+  -[lambda] or -[mu]1, when [lambda] or [mu] is one of the negative
+  integers 0, -1, -2, ..., 1 - [lambda]1, but otherwise contains a
+  logarithm. Similarly for the solutions about x = 1 or x = [oo]; it
+  will be seen below how the results are deducible from those for x = 0.
+
+
+    March of the Integral.
+
+  Denote now the solutions about x = 0 by u1, u2; those about x = 1 by
+  v1, v2; and those about x = [oo] by w1, w2; in the region (S0S1)
+  common to the circles S0, S1 of radius 1 whose centres are the points
+  x = 0, x = 1, all the first four are valid, and there exist equations
+  u1 =Av1 + Bv2, u2 = Cv1 + Dv2 where A, B, C, D are constants; in the
+  region (S1S) lying inside the circle S1 and outside the circle S0,
+  those that are valid are v1, v2, w1, w2, and there exist equations v1
+  = Pw1 + Qw2, v2 = Rw1 + Tw2, where P, Q, R, T are constants; thus
+  considering any integral whose expression within the circle S0 is au1
+  + bu2, where a, b are constants, the same integral will be represented
+  within the circle S1 by (aA + bC)v1 + (aB + bD)v2, and outside these
+  circles will be represented by
+
+    [(aA + bC)P + (aB + bD)R]w1 + [(aA + bC)Q + (aB + bD)T]w2.
+
+  A single-valued branch of such integral can be obtained by making a
+  barrier in the plane joining [oo] to 0 and 1 to [oo]; for instance, by
+  excluding the consideration of real negative values of x and of real
+  positive values greater than 1, and defining the phase of x and x - 1
+  for real values between 0 and 1 as respectively 0 and [pi].
+
+
+    Transformation of the equation into itself.
+
+  We can form the Fuchsian equation of the second order with three
+  arbitrary singular points [xi]1, [xi]2, [xi]3, and no singular point
+  at x = [oo], and with respective indices [alpha]1, [beta]1, [alpha]2,
+  [beta]2, [alpha]3, [beta]3 such that [alpha]1 + [beta]1 + [alpha]2 +
+  [beta]2 + [alpha]3 + [beta]3 = 1. This equation can then be
+  transformed into the hypergeometric equation in 24 ways; for out of
+  [xi]1, [xi]2, [xi]3 we can in six ways choose two, say [xi]1, [xi]2,
+  which are to be transformed respectively into 0 and 1, by (x -
+  [xi]1)/(x - [xi]2) = t(t - 1); and then there are four possible
+  transformations of the dependent variable which will reduce one of the
+  indices at t = 0 to zero and one of the indices at t = 1 also to zero,
+  namely, we may reduce either [alpha]1 or [beta]1 at t = 0, and
+  simultaneously either [alpha]2 or [beta]2 at t = 1. Thus the
+  hypergeometric equation itself can be transformed into itself in 24
+  ways, and from the expression F([lambda], [mu], 1 - [lambda]1, x)
+  which satisfies it follow 23 other forms of solution; they involve
+  four series in each of the arguments, x, x-1, 1/x, 1/(1-x), (x-1)/x,
+  x/(x-1). Five of the 23 solutions agree with the fundamental solutions
+  already described about x = 0, x = 1, x = [oo]; and from the
+  principles by which these were obtained it is immediately clear that
+  the 24 forms are, in value, equal in fours.
+
+
+    Inversion. Modular functions.
+
+  The quarter periods K, K' of Jacobi's theory of elliptic functions, of
+  which K = [int] [0 to [pi]/2] (1 - h sin²[theta])^-½ d[theta], and K'
+  is the same function of 1-h, can easily be proved to be the solutions
+  of a hypergeometric equation of which h is the independent variable.
+  When K, K' are regarded as defined in terms of h by the differential
+  equation, the ratio K'/K is an infinitely many valued function of h.
+  But it is remarkable that Jacobi's own theory of theta functions leads
+  to an expression for h in terms of K'/K (see FUNCTION) in terms of
+  single-valued functions. We may then attempt to investigate, in
+  general, in what cases the independent variable x of a hypergeometric
+  equation is a single-valued function of the ratio s of two independent
+  integrals of the equation. The same inquiry is suggested by the
+  problem of ascertaining in what cases the hypergeometric series
+  F([alpha], [beta], [gamma], x) is the expansion of an algebraic
+  (irrational) function of x. In order to explain the meaning of the
+  question, suppose that the plane of x is divided along the real axis
+  from -[oo] to 0 and from 1 to +[oo], and, supposing logarithms not to
+  enter about x = 0, choose two quite definite integrals y1, y2 of the
+  equation, say
+
+    y1 = F([lambda], [mu], 1-[lambda]1, x),
+    y2 = x^[lambda]1 F([lambda] + [lambda]1, [mu] + [lambda]1, 1 + [lambda]1, x),
+
+  with the condition that the phase of x is zero when x is real and
+  between 0 and 1. Then the value of [sigma] = y2/y1 is definite for all
+  values of x in the divided plane, [sigma] being a single-valued
+  monogenic branch of an analytical function existing and without
+  singularities all over this region. If, now, the values of [sigma]
+  that so arise be plotted on to another plane, a value p + iq of
+  [sigma] being represented by a point (p, q) of this [stigma]-plane, and
+  the value of x from which it arose being mentally associated with this
+  point of the [sigma]-plane, these points will fill a connected region
+  therein, with a continuous boundary formed of four portions
+  corresponding to the two sides of the two barriers of the x-plane. The
+  question is then, firstly, whether the same value of s can arise for
+  two different values of x, that is, whether the same point (p, q) of
+  the [sigma]-plane can arise twice, or in other words, whether the
+  region of the [sigma]-plane overlaps itself or not. Supposing this is
+  not so, a second part of the question presents itself. If in the
+  x-plane the barrier joining -[oo] to 0 be momentarily removed, and x
+  describe a small circle with centre at x = 0 starting from a point x =
+  -h - ik, where h, k are small, real, and positive and coming back to
+  this point, the original value s at this point will be changed to a
+  value [sigma], which in the original case did not arise for this value
+  of x, and possibly not at all. If, now, after restoring the barrier
+  the values arising by continuation from [sigma] be similarly plotted
+  on the s-plane, we shall again obtain a region which, while not
+  overlapping itself, may quite possibly overlap the former region. In
+  that case two values of x would arise for the same value or values of
+  the quotient y2/y1, arising from two different branches of this
+  quotient. We shall understand then, by the condition that x is to be a
+  single-valued function of x, that the region in the [stimga]-plane
+  corresponding to any branch is not to overlap itself, and that no two
+  of the regions corresponding to the different branches are to overlap.
+  Now in describing the circle about x = 0 from x = -h - ik to -h + ik,
+  where h is small and k evanescent,
+
+    [stigma] = x^[lambda]1 F([lambda] + [lambda]1, [mu] + [lambda]1, 1 + [lambda]1, x)/F([lambda], [mu], 1 - [lambda]1, x)
+
+  is changed to [sigma] = [stigma]e^(2[pi]i[lambda])1. Thus the two
+  portions of boundary of the s-region corresponding to the two sides of
+  the barrier (-[oo], 0) meet (at [sigmaf] = 0 if the real part of
+  [lambda]1 be positive) at an angle 2[pi]L1, where L1 is the absolute
+  value of the real part of [lambda]1; the same is true for the
+  [sigma]-region representing the branch [sigma]. The condition that the
+  s-region shall not overlap itself requires, then, L1 = 1. But,
+  further, we may form an infinite number of branches [sigma] =
+  [stigma]e^(2[pi]i[lambda])1, [sigma]1 = e^(2[pi]i[lambda])1, ... in
+  the same way, and the corresponding regions in the plane upon which
+  y2/y1 is represented will have a common point and each have an angle
+  2[pi]L1; if neither overlaps the preceding, it will happen, if L1 is
+  not zero, that at length one is reached overlapping the first, unless
+  for some positive integer [alpha] we have 2[pi][alpha]L1 = 2[pi], in
+  other words L1 = 1/a. If this be so, the branch [sigma]_a-1 =
+  [stigma]e^(2[pi]ia[lambda])1 will be represented by a region having
+  the angle at the common point common with the region for the branch
+  [stigma]; but not altogether coinciding with this last region unless
+  [lambda]1 be real, and therefore = ±1/a; then there is only a finite
+  number, a, of branches obtainable in this way by crossing the barrier
+  (-[oo], 0). In precisely the same way, if we had begun by taking the
+  quotient
+
+    [stigma]' = (x - 1)^[lambda]2 F([lambda] + [lambda]2, [mu] + [lambda]2, 1 + [lambda]2, 1 - x)/F([lambda], [mu], 1 - [lambda]2, 1 - x)
+
+  of the two solutions about x = 1, we should have found that x is not a
+  single-valued function of [stigma]' unless [lambda]2 is the inverse of
+  an integer, or is zero; as [stigma]' is of the form (A[stigma] +
+  B)/(C[stigma] + D), A, B, C, D constants, the same is true in our
+  case; equally, by considering the integrals about x = [oo] we find, as
+  a third condition necessary in order that x may be a single-valued
+  function of [stigma], that [lambda] - [mu] must be the inverse of an
+  integer or be zero. These three differences of the indices, namely,
+  [lambda]1, [lambda]2, [lambda] - [mu], are the quantities which enter
+  in the differential equation satisfied by x as a function of [stigma],
+  which is easily found to be
+
+       x111   3²x²11
+    -  ---- + ------ = ½(h - h1  - h2)x^-1 (x - 1)^-1 + ½h1 x^-2 + ½h2(x - 1)^-2,
+       x1³    2x1^4
+
+  where x1 = dx/d[stigma], &c.; and h1 = 1 - y1², h2 = 1 - [lambda]2²,
+  h3 = 1 - ([lambda] - [mu])². Into the converse question whether the
+  three conditions are sufficient to ensure (1) that the [stigma] region
+  corresponding to any branch does not overlap itself, (2) that no two
+  such regions overlap, we have no space to enter. The second question
+  clearly requires the inquiry whether the group (that is, the monodromy
+  group) of the differential equation is properly discontinuous. (See
+  GROUPS, THEORY OF.)
+
+The foregoing account will give an idea of the nature of the function
+theories of differential equations; it appears essential not to exclude
+some explanation of a theory intimately related both to such theories
+and to transformation theories, which is a generalization of Galois's
+theory of algebraic equations. We deal only with the application to
+homogeneous linear differential equations.
+
+
+    Rationality group of a linear equation.
+
+    Irreducibility of a rational equation.
+
+  In general a function of variables x1, x2 ... is said to be rational
+  when it can be formed from them and the integers 1, 2, 3, ... by a
+  finite number of additions, subtractions, multiplications and
+  divisions. We generalize this definition. Assume that we have assigned
+  a fundamental series of quantities and functions of x, in which x
+  itself is included, such that all quantities formed by a finite number
+  of additions, subtractions, multiplications, divisions _and
+  differentiations in regard to x_, of the terms of this series, are
+  themselves members of this series. Then the quantities of this series,
+  and only these, are called _rational_. By a rational function of
+  quantities p, q, r, ... is meant a function formed from them and any
+  of the fundamental rational quantities by a finite number of the five
+  fundamental operations. Thus it is a function which would be called,
+  simply, rational if the fundamental series were widened by the
+  addition to it of the quantities p, q, r, ... and those derivable from
+  them by the five fundamental operations. A rational ordinary
+  differential equation, with x as independent and y as dependent
+  variable, is then one which equates to zero a rational function of y,
+  the order k of the differential equation being that of the highest
+  differential coefficient y^(k) which enters; only such equations are
+  here discussed. Such an equation P = 0 is called _irreducible_ when,
+  firstly, being arranged as an integral polynomial in y^(k), this
+  polynomial is not the product of other polynomials in y^(k) also of
+  rational form; and, secondly, the equation has no solution satisfying
+  also a rational equation of lower order. From this it follows that if
+  an irreducible equation P = 0 have one solution satisfying another
+  rational equation Q = 0 of the same or higher order, then all the
+  solutions of P = 0 also satisfy Q = 0. For from the equation P = 0 we
+  can by differentiation express y^(k+1), y^(k+2), ... in terms of x, y,
+  y^(1), ... , y^(k), and so put the function Q rationally in terms of
+  these quantities only. It is sufficient, then, to prove the result
+  when the equation Q = 0 is of the same order as P = 0. Let both the
+  equations be arranged as integral polynomials in y^(k); their
+  algebraic eliminant in regard to y^(k) must then vanish identically,
+  for they are known to have one common solution not satisfying an
+  equation of lower order; thus the equation P = 0 involves Q = 0 for
+  all solutions of P = 0.
+
+
+    The variant function for a linear equation.
+
+  Now let y^(n) = [alpha]1y^(n-1) + ... + [alpha]_n y be a given
+  rational homogeneous linear differential equation; let y1, ... yn be n
+  particular functions of x, unconnected by any equation with constant
+  coefficients of the form c1y1 + ... + cnyn = 0, all satisfying the
+  differential equation; let [eta]1, ... [eta]n be linear functions of
+  y1, ... yn, say [eta]i = A_i1 y1 + ... + A_in yn, where the constant
+  coefficients Aij have a non-vanishing determinant; write ([eta]) =
+  A(y), these being the equations of a general linear homogeneous group
+  whose transformations may be denoted by A, B, .... We desire to form a
+  rational function [phi]([eta]), or say [phi](A(y)), of [eta]1, ...
+  [eta], in which the [eta]² constants Aij shall all be essential, and
+  not reduce effectively to a fewer number, as they would, for instance,
+  if the y1, ... yn were connected by a linear equation with constant
+  coefficients. Such a function is in fact given, if the solutions y1,
+  ... yn be developable in positive integral powers about x = a, by
+  [phi]([eta]) = [eta]1 + (x - a)^n[eta]2 + ... + (x - a)^(n-1)n[eta]n.
+  Such a function, V, we call a _variant_.
+
+
+    The resolvent eqution.
+
+  Then differentiating V in regard to x, and replacing [eta]i^(n) by
+  its value a1[eta]^(n-1) + ... + an[eta], we can arrange dV/dx, and
+  similarly each of d²/dx² ... d^NV/dx^N, where N = n², as a linear
+  function of the N quantities [eta]1, ... [eta]n, ... [eta]1^(n-1), ...
+  [eta]n^(n-1), and thence by elimination obtain a linear differential
+  equation for V of order N with rational coefficients. This we denote
+  by F = 0. Further, each of [eta]1 ... [eta]n is expressible as a
+  linear function of V, dV/dx, ... d^(N-1)V/dx^(N-1), with rational
+  coefficients not involving any of the n² coefficients A_ij, since
+  otherwise V would satisfy a linear equation of order less than N,
+  which is impossible, as it involves (linearly) the n² arbitrary
+  coefficients Aij, which would not enter into the coefficients of the
+  supposed equation. In particular, y1 ,.. yn are expressible rationally
+  as linear functions of [omega], d[omega]/dx, ...
+  d^(N-1)[omega]/dx^(N-1), where [omega] is the particular function
+  [phi](y). Any solution W of the equation F = 0 is derivable from
+  functions [zeta]1, ... [zeta]n, which are linear functions of y1, ...
+  yn, just as V was derived from [eta]1, ... [eta]n; but it does not
+  follow that these functions [zeta]i, ... [zeta]n are obtained from y1,
+  ... yn by a transformation of the linear group A, B, ... ; for it may
+  happen that the determinant d([zeta]1, ... [zeta]n)/(dy1, ... yn) is
+  zero. In that case [zeta]1, ... [zeta]n may be called a singular set,
+  and W a singular solution; it satisfies an equation of lower than the
+  N-th order. But every solution V, W, ordinary or singular, of the
+  equation F = 0, is expressible rationally in terms of [omega],
+  d[omega]/dx, ... d^(N-1)[omega]/dx^(N-1); we shall write, simply, V =
+  r([omega]). Consider now the rational irreducible equation of lowest
+  order, not necessarily a linear equation, which is satisfied by
+  [omega]; as y1, ... yn are particular functions, it may quite well be
+  of order less than N; we call it the _resolvent equation_, suppose it
+  of order p, and denote it by [gamma](v). Upon it the whole theory
+  turns. In the first place, as [gamma](v) = 0 is satisfied by the
+  solution [omega] of F = 0, all the solutions of [gamma](v) are
+  solutions F = 0, and are therefore rationally expressible by [omega];
+  any one may then be denoted by r([omega]). If this solution of F = 0
+  be not singular, it corresponds to a transformation A of the linear
+  group (A, B, ...), effected upon y1, ... yn. The coefficients Aij of
+  this transformation follow from the expressions before mentioned for
+  [eta]1 ... [eta]n in terms of V, dV/dx, d²V/dx², ... by substituting V
+  = r([omega]); thus they depend on the p arbitrary parameters which
+  enter into the general expression for the integral of the equation
+  [gamma](v) = 0. Without going into further details, it is then clear
+  enough that the resolvent equation, being irreducible and such that
+  any solution is expressible rationally, with p parameters, in terms of
+  the solution [omega], enables us to define a linear homogeneous group
+  of transformations of y1 ... yn depending on p parameters; and every
+  operation of this (continuous) group corresponds to a rational
+  transformation of the solution of the resolvent equation. This is the
+  group called the _rationality group_, or the _group of
+  transformations_ of the original homogeneous linear differential
+  equation.
+
+  The group must not be confounded with a subgroup of itself, the
+  _monodromy group_ of the equation, often called simply the group of
+  the equation, which is a set of transformations, not depending on
+  arbitrary variable parameters, arising for one particular fundamental
+  set of solutions of the linear equation (see GROUPS, THEORY OF).
+
+
+    The fundamental theorem in regard to the rationality group.
+
+  The importance of the rationality group consists in three
+  propositions. (1) Any rational function of y1, ... yn which is
+  unaltered in value by the transformations of the group can be written
+  in rational form. (2) If any rational function be changed in form,
+  becoming a rational function of y1, ... yn, a transformation of the
+  group applied to its new form will leave its value unaltered. (3) Any
+  homogeneous linear transformation leaving unaltered the value of every
+  rational function of y1, ... yn which has a rational value, belongs to
+  the group. It follows from these that any group of linear homogeneous
+  transformations having the properties (1) (2) is identical with the
+  group in question. It is clear that with these properties the group
+  must be of the greatest importance in attempting to discover what
+  functions of x must be regarded as rational in order that the values
+  of y1 ... yn may be expressed. And this is the problem of solving the
+  equation from another point of view.
+
+  LITERATURE.--([alpha]) _Formal or Transformation Theories for
+  Equations of the First Order_:--E. Goursat, _Leçons sur l'intégration
+  des équations aux dérivées partielles du premier ordre_ (Paris, 1891);
+  E. v. Weber, _Vorlesungen über das Pfaff'sche Problem und die Theorie
+  der partiellen Differentialgleichungen erster Ordnung_ (Leipzig,
+  1900); S. Lie und G. Scheffers, _Geometrie der
+  Berührungstransformationen_, Bd. i. (Leipzig, 1896); Forsyth, _Theory
+  of Differential Equations, Part i., Exact Equations and Pfaff's
+  Problem_ (Cambridge, 1890); S. Lie, "Allgemeine Untersuchungen über
+  Differentialgleichungen, die eine continuirliche endliche Gruppe
+  gestatten" (Memoir), _Mathem. Annal._xxv. (1885), pp. 71-151; S. Lie
+  und G. Scheffers, _Vorlesungen über Differentialgleichungen mit
+  bekannten infinitesimalen Transformationen_ (Leipzig, 1891). A very
+  full bibliography is given in the book of E. v. Weber referred to;
+  those here named are perhaps sufficiently representative of modern
+  works. Of classical works may be named: Jacobi, _Vorlesungen über
+  Dynamik_ (von A. Clebsch, Berlin, 1866); _Werke, Supplementband_; G
+  Monge, _Application de l'analyse à la géométrie_ (par M. Liouville,
+  Paris, 1850); J. L. Lagrange, _Leçons sur le calcul des fonctions_
+  (Paris, 1806), and _Théorie des fonctions analytiques_ (Paris,
+  Prairial, an V); G. Boole, _A Treatise on Differential Equations_
+  (London, 1859); and _Supplementary Volume_ (London, 1865); Darboux,
+  _Leçons sur la théorie générale des surfaces_, tt. i.-iv. (Paris,
+  1887-1896); S. Lie, _Théorie der transformationsgruppen_ ii. (on
+  Contact Transformations) (Leipzig, 1890).
+
+  ([beta]) _Quantitative or Function Theories for Linear Equations_:--C.
+  Jordan, _Cours d'analyse_, t. iii. (Paris, 1896); E. Picard, _Traité
+  d'analyse_, tt. ii. and iii. (Paris, 1893, 1896); Fuchs, _Various
+  Memoirs, beginning with that in Crelle's Journal_, Bd. lxvi. p. 121;
+  Riemann, _Werke_, 2^r Aufl. (1892); Schlesinger, _Handbuch der Theorie
+  der linearen Differentialgleichungen_, Bde. i.-ii. (Leipzig,
+  1895-1898); Heffter, _Einleitung in die Theorie der linearen
+  Differentialgleichungen mit einer unabhängigen Variablen_ (Leipzig,
+  1894); Klein, _Vorlesungen über lineare Differentialgleichungen der
+  zweiten Ordnung_ (Autographed, Göttingen, 1894); and _Vorlesungen über
+  die hypergeometrische Function_ (Autographed, Göttingen, 1894);
+  Forsyth, _Theory of Differential Equations, Linear Equations_.
+
+  ([gamma]) _Rationality Group (of Linear Differential
+  Equations)_:--Picard, _Traité d'Analyse_, as above, t. iii.; Vessiot,
+  _Annales de l'École Normale_, série III. t. ix. p. 199 (Memoir); S.
+  Lie, _Transformationsgruppen_, as above, iii. A connected account is
+  given in Schlesinger, as above, Bd. ii., erstes Theil.
+
+  ([delta]) _Function Theories of Non-Linear Ordinary
+  Equations_:--Painlevé, _Leçons sur la théorie analytique des équations
+  différentielles_ (Paris, 1897, Autographed); Forsyth, _Theory of
+  Differential Equations, Part ii., Ordinary Equations not Linear_ (two
+  volumes, ii. and iii.) (Cambridge, 1900); Königsberger, _Lehrbuch der
+  Theorie der Differentialgleichungen_ (Leipzig, 1889); Painlevé,
+  _Leçons sur l'intégration des équations differentielles de la
+  mécanique et applications_ (Paris, 1895).
+
+  ([epsilon]) _Formal Theories of Partial Equations of the Second and
+  Higher Orders_:--E. Goursat, _Leçons sur l'intégration des équations
+  aux dérivées partielles du second ordre_, tt. i. and ii. (Paris, 1896,
+  1898); Forsyth, _Treatise on Differential Equations_ (London, 1889);
+  and _Phil. Trans. Roy. Soc._ (A.), vol. cxci. (1898), pp. 1-86.
+
+  ([zeta]) See also the six extensive articles in the second volume of
+  the German _Encyclopaedia of Mathematics_.     (H. F. BA.)
+
+
+
+
+DIFFLUGIA (L. Leclerc), a genus of lobose Rhizopoda, characterized by a
+shell formed of sand granules cemented together; these are swallowed by
+the animal, and during the process of bud-fission they pass to the
+surface of the daughter-bud and are cemented there. _Centropyxis_
+(Steia) and _Lecqueureuxia_ (Schlumberg) differ only in minor points.
+
+
+
+
+DIFFRACTION OF LIGHT.--1. When light proceeding from a small source
+falls upon an opaque object, a shadow is cast upon a screen situated
+behind the obstacle, and this shadow is found to be bordered by
+alternations of brightness and darkness, known as "diffraction bands."
+The phenomena thus presented were described by Grimaldi and by Newton.
+Subsequently T. Young showed that in their formation interference plays
+an important part, but the complete explanation was reserved for A. J.
+Fresnel. Later investigations by Fraunhofer, Airy and others have
+greatly widened the field, and under the head of "diffraction" are now
+usually treated all the effects dependent upon the limitation of a beam
+of light, as well as those which arise from irregularities of any kind
+at surfaces through which it is transmitted, or at which it is
+reflected.
+
+2. _Shadows._--In the infancy of the undulatory theory the objection
+most frequently urged against it was the difficulty of explaining the
+very existence of shadows. Thanks to Fresnel and his followers, this
+department of optics is now precisely the one in which the theory has
+gained its greatest triumphs. The principle employed in these
+investigations is due to C. Huygens, and may be thus formulated. If
+round the origin of waves an ideal closed surface be drawn, the whole
+action of the waves in the region beyond may be regarded as due to the
+motion continually propagated across the various elements of this
+surface. The wave motion due to any element of the surface is called a
+_secondary_ wave, and in estimating the total effect regard must be paid
+to the phases as well as the amplitudes of the components. It is usually
+convenient to choose as the surface of resolution a _wave-front_, i.e. a
+surface at which the primary vibrations are in one phase. Any obscurity
+that may hang over Huygens's principle is due mainly to the
+indefiniteness of thought and expression which we must be content to put
+up with if we wish to avoid pledging ourselves as to the character of
+the vibrations. In the application to sound, where we know what we are
+dealing with, the matter is simple enough in principle, although
+mathematical difficulties would often stand in the way of the
+calculations we might wish to make. The ideal surface of resolution may
+be there regarded as a flexible lamina; and we know that, if by forces
+locally applied every element of the lamina be made to move normally to
+itself exactly as the air at that place does, the external aerial motion
+is fully determined. By the principle of superposition the whole effect
+may be found by integration of the partial effects due to each element
+of the surface, the other elements remaining at rest.
+
+  We will now consider in detail the important case in which uniform
+  plane waves are resolved at a surface coincident with a wave-front
+  (OQ). We imagine a wave-front divided into elementary rings or
+  zones--often named after Huygens, but better after Fresnel--by spheres
+  described round P (the point at which the aggregate effect is to be
+  estimated), the first sphere, touching the plane at O, with a radius
+  equal to PO, and the succeeding spheres with radii increasing at each
+  step by ½[lambda]. There are thus marked out a series of circles,
+  whose radii x are given by x² + r² = (r + ½n[lambda])², or x² =
+  n[lambda]r nearly; so that the rings are at first of nearly equal
+  area. Now the effect upon P of each element of the plane is
+  proportional to its area; but it depends also upon the distance from
+  P, and possibly upon the inclination of the secondary ray to the
+  direction of vibration and to the wave-front.
+
+        O     x          Q
+    ---------------------------
+        |                /
+        |               /
+        |              /
+        |             /
+        |            /
+        |           /
+        |          /
+       r|         /
+        |        /
+        |       /
+        |      /
+        |     /
+        |    /
+        |   /
+        |  /
+        | /
+       P|/
+
+    FIG. 1.
+
+  The latter question can only be treated in connexion with the
+  dynamical theory (see below, § 11); but under all ordinary
+  circumstances the result is independent of the precise answer that may
+  be given. All that it is necessary to assume is that the effects of
+  the successive zones gradually diminish, whether from the increasing
+  obliquity of the secondary ray or because (on account of the
+  limitation of the region of integration) the zones become at last more
+  and more incomplete. The component vibrations at P due to the
+  successive zones are thus nearly equal in amplitude and opposite in
+  phase (the phase of each corresponding to that of the infinitesimal
+  circle midway between the boundaries), and the series which we have to
+  sum is one in which the terms are alternately opposite in sign and,
+  while at first nearly constant in numerical magnitude, gradually
+  diminish to zero. In such a series each term may be regarded as very
+  nearly indeed destroyed by the halves of its immediate neighbours, and
+  thus the sum of the whole series is represented by half the first
+  term, which stands over uncompensated. The question is thus reduced to
+  that of finding the effect of the first zone, or central circle, of
+  which the area is [pi][lambda]r.
+
+  We have seen that the problem before us is independent of the law of
+  the secondary wave as regards obliquity; but the result of the
+  integration necessarily involves the law of the intensity and phase of
+  a secondary wave as a function of r, the distance from the origin. And
+  we may in fact, as was done by A. Smith (_Camb. Math. Journ._, 1843,
+  3, p. 46), determine the law of the secondary wave, by comparing the
+  result of the integration with that obtained by supposing the primary
+  wave to pass on to P without resolution.
+
+  Now as to the phase of the secondary wave, it might appear natural to
+  suppose that it starts from any point Q with the phase of the primary
+  wave, so that on arrival at P, it is retarded by the amount
+  corresponding to QP. But a little consideration will prove that in
+  that case the series of secondary waves could not reconstitute the
+  primary wave. For the aggregate effect of the secondary waves is the
+  half of that of the first Fresnel zone, and it is the central element
+  only of that zone for which the distance to be travelled is equal to
+  r. Let us conceive the zone in question to be divided into
+  infinitesimal rings of equal area. The effects due to each of these
+  rings are equal in amplitude and of phase ranging uniformly over half
+  a complete period. The phase of the resultant is midway between those
+  of the extreme elements, that is to say, a quarter of a period behind
+  that due to the element at the centre of the circle. It is accordingly
+  necessary to suppose that the secondary waves start with a phase
+  one-quarter of a period in advance of that of the primary wave at the
+  surface of resolution.
+
+  Further, it is evident that account must be taken of the variation of
+  phase in estimating the magnitude of the effect at P of the first
+  zone. The middle element alone contributes without deduction; the
+  effect of every other must be found by introduction of a resolving
+  factor, equal to cos [theta], if [theta] represent the difference of
+  phase between this element and the resultant. Accordingly, the
+  amplitude of the resultant will be less than if all its components had
+  the same phase, in the ratio
+
+      _ +½[pi]
+     /
+     |      cos [theta]d[theta] : [pi],
+    _/-½[pi]
+
+  or 2 : [pi]. Now 2 area /[pi] = 2[lambda]r; so that, in order to
+  reconcile the amplitude of the primary wave (taken as unity) with the
+  half effect of the first zone, the amplitude, at distance r, of the
+  secondary wave emitted from the element of area dS must be taken to be
+
+    dS/[lambda]r                   (1).
+
+
+  By this expression, in conjunction with the quarter-period
+  acceleration of phase, the law of the secondary wave is determined.
+
+  That the amplitude of the secondary wave should vary as r^-1 was to be
+  expected from considerations respecting energy; but the occurrence of
+  the factor [lambda]^-1, and the acceleration of phase, have sometimes
+  been regarded as mysterious. It may be well therefore to remember that
+  precisely these laws apply to a secondary wave of sound, which can be
+  investigated upon the strictest mechanical principles.
+
+  The recomposition of the secondary waves may also be treated
+  analytically. If the primary wave at O be cos kat, the effect of the
+  secondary wave proceeding from the element dS at Q is
+
+         dS                                           dS
+    ------------- cos k(at - [rho] + ¼[lambda]) = ------------- sin k(at - [rho]).
+    [lambda][rho]                                 [lambda][rho]
+
+
+  If dS = 2[pi]xdx, we have for the whole effect
+
+                  _[oo]
+       2[pi]     /   sin k(at - [rho])x dx
+    - --------   |   ---------------------,
+      [lambda]  _/ 0         [rho]
+
+  or, since xdx = [rho]d[rho], k = 2[pi]/[lambda],
+
+        _[oo]                       _                 _
+       /                           |                   |[oo]
+    -k | sin k(at - [rho])d[rho] = | -cos k(at - [rho])|    .
+      _/r                          |_                 _|r
+
+  In order to obtain the effect of the primary wave, as retarded by
+  traversing the distance r, viz. cos k(at - r), it is necessary to
+  suppose that the integrated term vanishes at the upper limit. And it
+  is important to notice that without some further understanding the
+  integral is really ambiguous. According to the assumed law of the
+  secondary wave, the result must actually depend upon the precise
+  radius of the outer boundary of the region of integration, supposed to
+  be exactly circular. This case is, however, at most very special and
+  exceptional. We may usually suppose that a large number of the outer
+  rings are incomplete, so that the integrated term at the upper limit
+  may properly be taken to vanish. If a formal proof be desired, it may
+  be obtained by introducing into the integral a factor such as
+  e^-h[rho], in which h is ultimately made to diminish without limit.
+
+  When the primary wave is plane, the area of the first Fresnel zone is
+  [pi][lambda]r, and, since the secondary waves vary as r^-1, the
+  intensity is independent of r, as of course it should be. If, however,
+  the primary wave be spherical, and of radius a at the wave-front of
+  resolution, then we know that at a distance r further on the amplitude
+  of the primary wave will be diminished in the ratio a:(r + a). This
+  may be regarded as a consequence of the altered area of the first
+  Fresnel zone. For, if x be its radius, we have
+
+                                /
+    {(r + ½[lambda])² - x²} + \/ {a² - x²} = r + a,
+
+  so that
+
+    x² = [lambda]ar/(a + r) nearly.
+
+  Since the distance to be travelled by the secondary waves is still r,
+  we see how the effect of the first zone, and therefore of the whole
+  series is proportional to a/(a + r). In like manner may be treated
+  other cases, such as that of a primary wave-front of unequal principal
+  curvatures.
+
+  The general explanation of the formation of shadows may also be
+  conveniently based upon Fresnel's zones. If the point under
+  consideration be so far away from the geometrical shadow that a large
+  number of the earlier zones are complete, then the illumination,
+  determined sensibly by the first zone, is the same as if there were no
+  obstruction at all. If, on the other hand, the point be well immersed
+  in the geometrical shadow, the earlier zones are altogether missing,
+  and, instead of a series of terms beginning with finite numerical
+  magnitude and gradually diminishing to zero, we have now to deal with
+  one of which the terms diminish to zero _at both ends_. The sum of
+  such a series is very approximately zero, each term being neutralized
+  by the halves of its immediate neighbours, which are of the opposite
+  sign. The question of light or darkness then depends upon whether the
+  series begins or ends abruptly. With few exceptions, abruptness can
+  occur only in the presence of the first term, viz. when the secondary
+  wave of least retardation is unobstructed, or when a _ray_ passes
+  through the point under consideration. According to the undulatory
+  theory the light cannot be regarded strictly as travelling along a
+  ray; but the existence of an unobstructed ray implies that the system
+  of Fresnel's zones can be commenced, and, if a large number of these
+  zones are fully developed and do not terminate abruptly, the
+  illumination is unaffected by the neighbourhood of obstacles.
+  Intermediate cases in which a few zones only are formed belong
+  especially to the province of diffraction.
+
+  An interesting exception to the general rule that full brightness
+  requires the existence of the first zone occurs when the obstacle
+  assumes the form of a small circular disk parallel to the plane of the
+  incident waves. In the earlier half of the 18th century R. Delisle
+  found that the centre of the circular shadow was occupied by a bright
+  point of light, but the observation passed into oblivion until S. D.
+  Poisson brought forward as an objection to Fresnel's theory that it
+  required at the centre of a circular shadow a point as bright as if no
+  obstacle were intervening. If we conceive the primary wave to be
+  broken up at the plane of the disk, a system of Fresnel's zones can be
+  constructed which begin from the circumference; and the first zone
+  external to the disk plays the part ordinarily taken by the centre of
+  the entire system. The whole effect is the half of that of the first
+  existing zone, and this is sensibly the same as if there were no
+  obstruction.
+
+  When light passes through a small circular or annular aperture, the
+  illumination at any point along the axis depends upon the precise
+  relation between the aperture and the distance from it at which the
+  point is taken. If, as in the last paragraph, we imagine a system of
+  zones to be drawn commencing from the inner circular boundary of the
+  aperture, the question turns upon the manner in which the series
+  terminates at the outer boundary. If the aperture be such as to fit
+  exactly an integral number of zones, the aggregate effect may be
+  regarded as the half of those due to the first and last zones. If the
+  number of zones be even, the action of the first and last zones are
+  antagonistic, and there is complete darkness at the point. If on the
+  other hand the number of zones be odd, the effects conspire; and the
+  illumination (proportional to the square of the amplitude) is four
+  times as great as if there were no obstruction at all.
+
+  The process of augmenting the resultant illumination at a particular
+  point by stopping some of the secondary rays may be carried much
+  further (Soret, _Pogg. Ann._, 1875, 156, p. 99). By the aid of
+  photography it is easy to prepare a plate, transparent where the zones
+  of odd order fall, and opaque where those of even order fall. Such a
+  plate has the power of a condensing lens, and gives an illumination
+  out of all proportion to what could be obtained without it. An even
+  greater effect (fourfold) can be attained by providing that the
+  stoppage of the light from the alternate zones is replaced by a
+  phase-reversal without loss of amplitude. R. W. Wood (_Phil. Mag._,
+  1898, 45, p 513) has succeeded in constructing zone plates upon this
+  principle.
+
+  In such experiments the narrowness of the zones renders necessary a
+  pretty close approximation to the geometrical conditions. Thus in the
+  case of the circular disk, equidistant (r) from the source of light
+  and from the screen upon which the shadow is observed, the width of
+  the first exterior zone is given by
+
+    dx = [lambda](2r)/4(2x),
+
+  2x being the diameter of the disk. If 2r = 1000 cm., 2x = 1 cm.,
+  [lambda] = 6 × 10^-5 cm., then dx = .0015 cm. Hence, in order that
+  this zone may be perfectly formed, there should be no error in the
+  circumference of the order of .001 cm. (It is easy to see that the
+  radius of the bright spot is of the same order of magnitude.) The
+  experiment succeeds in a dark room of the length above mentioned, with
+  a threepenny bit (supported by three threads) as obstacle, the origin
+  of light being a small needle hole in a plate of tin, through which
+  the sun's rays shine horizontally after reflection from an external
+  mirror. In the absence of a heliostat it is more convenient to obtain
+  a point of light with the aid of a lens of short focus.
+
+  The amplitude of the light at any point in the axis, when plane waves
+  are incident perpendicularly upon an annular aperture, is, as above,
+
+    cos k(at - r1) - cos k(at - r2) = 2 sin kat sin k(r1 - r2),
+
+  r2, r1 being the distances of the outer and inner boundaries from the
+  point in question. It is scarcely necessary to remark that in all such
+  cases the calculation applies in the first instance to homogeneous
+  light, and that, in accordance with Fourier's theorem, each
+  homogeneous component of a mixture may be treated separately. When the
+  original light is white, the presence of some components and the
+  absence of others will usually give rise to coloured effects, variable
+  with the precise circumstances of the case.
+
+  [Illustration: FIG. 2.]
+
+  Although the matter can be fully treated only upon the basis of a
+  dynamical theory, it is proper to point out at once that there is an
+  element of assumption in the application of Huygens's principle to the
+  calculation of the effects produced by opaque screens of limited
+  extent. Properly applied, the principle could not fail; but, as may
+  readily be proved in the case of sonorous waves, it is not in
+  strictness sufficient to assume the expression for a secondary wave
+  suitable when the primary wave is undisturbed, with mere limitation of
+  the integration to the transparent parts of the screen. But, except
+  perhaps in the case of very fine gratings, it is probable that the
+  error thus caused is insignificant; for the incorrect estimation of
+  the secondary waves will be limited to distances of a few wave-lengths
+  only from the boundary of opaque and transparent parts.
+
+3. _Fraunhofer's Diffraction Phenomena._--A very general problem in
+diffraction is the investigation of the distribution of light over a
+screen upon which impinge divergent or convergent spherical waves after
+passage through various diffracting apertures. When the waves are
+convergent and the recipient screen is placed so as to contain the
+centre of convergency--the image of the original radiant point, the
+calculation assumes a less complicated form. This class of phenomena was
+investigated by J. von Fraunhofer (upon principles laid down by
+Fresnel), and are sometimes called after his name. We may conveniently
+commence with them on account of their simplicity and great importance
+in respect to the theory of optical instruments.
+
+  If f be the radius of the spherical wave at the place of resolution,
+  where the vibration is represented by cos kat, then at any point M
+  (fig. 2) in the recipient screen the vibration due to an element dS of
+  the wave-front is (§ 2)
+
+           dS
+    - ------------- sin k(at - [rho]),
+      [lambda][rho]
+
+  [rho] being the distance between M and the element dS.
+
+  Taking co-ordinates in the plane of the screen with the centre of the
+  wave as origin, let us represent M by [xi], [eta], and P (where dS is
+  situated) by x, y, z. Then
+
+    [rho]² = (x - [xi])² + (y - [eta])² + z², f² = x² + y² + z²;
+
+  so that
+
+    [rho]² = f² - 2x[xi] - 2y[eta] + [xi]² + [eta]².
+
+  In the applications with which we are concerned, [xi], [eta] are very
+  small quantities; and we may take
+
+               /    x[xi] + y[eta]\
+    [rho] = f ( 1 - -------------- ).
+               \          f²      /
+
+  At the same time dS may be identified with dxdy, and in the
+  denominator [rho] may be treated as constant and equal to f. Thus the
+  expression for the vibration at M becomes
+
+                      _ _
+            1        / /        /         x[xi] + y[eta]\
+    - -------------  | | sin k ( at - f + -------------- ) dxdy    (1);
+      [lambda]²[f]² _/_/        \               f       /
+
+  and for the intensity, represented by the square of the amplitude,
+
+                       _   _ _                         _
+               1      |   / /       x[xi] + y[eta]      |²
+    I² = ------------ |   | | sin k -------------- dxdy |
+          [lambda]²f² |_ _/_/             f            _|
+                      _   _ _                          _
+               1     |   / /        x[xi] + y[eta]      |²
+       + ----------- |   | |  cos k -------------- dxdy |         (2).
+         [lambda]²f² |_ _/_/              f            _|
+
+  This expression for the intensity becomes rigorously applicable when f
+  is indefinitely great, so that ordinary optical aberration disappears.
+  The incident waves are thus plane, and are limited to a plane aperture
+  coincident with a wave-front. The integrals are then properly
+  functions of the _direction_ in which the light is to be estimated.
+
+  In experiment under ordinary circumstances it makes no difference
+  whether the collecting lens is in front of or behind the diffracting
+  aperture. It is usually most convenient to employ a telescope focused
+  upon the radiant point, and to place the diffracting apertures
+  immediately in front of the object-glass. What is seen through the
+  eye-piece in any case is the same as would be depicted upon a screen
+  in the focal plane.
+
+  Before proceeding to special cases it may be well to call attention to
+  some general properties of the solution expressed by (2) (see Bridge,
+  _Phil. Mag._, 1858).
+
+  If when the aperture is given, the wave-length (proportional to k^-1)
+  varies, the composition of the integrals is unaltered, provided [xi]
+  and [eta] are taken universely proportional to [lambda]. A diminution
+  of [lambda] thus leads to a simple proportional shrinkage of the
+  diffraction pattern, attended by an augmentation of brilliancy in
+  proportion to [lambda]^-2.
+
+  If the wave-length remains unchanged, similar effects are produced by
+  an increase in the scale of the aperture. The linear dimension of the
+  diffraction pattern is inversely as that of the aperture, and the
+  brightness at corresponding points is as the _square_ of the area of
+  aperture.
+
+  If the aperture and wave-length increase in the same proportion, the
+  size and shape of the diffraction pattern undergo no change.
+
+  We will now apply the integrals (2) to the case of a rectangular
+  aperture of width a parallel to x and of width b parallel to y. The
+  limits of integration for x may thus be taken to be -½a and +½a, and
+  for y to be -½b, +½b. We readily find (with substitution for k of
+  2[pi]/[lambda])
+
+                            [pi]a[xi]           [pi]b[eta]
+                       sin² ---------      sin² ----------
+              a²b²          f[lambda]           f[lambda]
+    I² = ----------- · ----------------- · ---------------     (3),
+         f²[lambda]²     [pi]²a²[xi]²       [pi]²b²[eta]²
+                         ------------       -------------
+                         f²[lambda]²         f²[lambda]²
+
+  as representing the distribution of light in the image of a
+  mathematical point when the aperture is rectangular, as is often the
+  case in spectroscopes.
+
+  The second and third factors of (3) being each of the form sin²u/u²,
+  we have to examine the character of this function. It vanishes when u
+  = m[pi], m being any whole number other than zero. When u = 0, it
+  takes the value unity. The maxima occur when
+
+    u = tan u,                                                 (4),
+
+  and then
+
+    sin²u/u² = cos²u                                           (5).
+
+  To calculate the roots of (5) we may assume
+
+    u = (m + ½)[pi] - y = U - y,
+
+
+
+  where y is a positive quantity which is small when u is large.
+  Substituting this, we find cot y = U - y, whence
+
+        1  /    y   y-      \    y³ 2y^5   17y^7
+    y = - ( 1 + - + -- + ... ) - -- ---- - -----.
+        U  \    U   U²      /    3   15     315
+
+  This equation is to be solved by successive approximation. It will
+  readily be found that
+
+                           2         13        146
+    u = U - y = U - U^-1 - -- U^-3 - -- U^-5 - --- U^-7 - ...   (6).
+                           3         15        105
+
+  In the first quadrant there is no root after zero, since tan u > u,
+  and in the second quadrant there is none because the signs of u and
+  tan u are opposite. The first root after zero is thus in the third
+  quadrant, corresponding to m = 1. Even in this case the series
+  converges sufficiently to give the value of the root with considerable
+  accuracy, while for higher values of m it is all that could be
+  desired. The actual values of u/[pi] (calculated in another manner by
+  F. M. Schwerd) are 1.4303, 2.4590, 3.4709, 4.4747, 5.4818, 6.4844, &c.
+
+  Since the maxima occur when u = (m + ½)[pi] nearly, the successive
+  values are not very different from
+
+      4        4       4
+    ------, ------, -------, &c.
+    9[pi]²  25[pi]  49[pi]²
+
+  The application of these results to (3) shows that the field is
+  brightest at the centre [xi] = 0, [eta] = 0, viz. at the geometrical
+  image of the radiant point. It is traversed by dark lines whose
+  equations are
+
+    [xi] = mf[lambda]/a, [eta] = mf[lambda]/b.
+
+  Within the rectangle formed by pairs of consecutive dark lines, and
+  not far from its centre, the brightness rises to a maximum; but these
+  subsequent maxima are in all cases much inferior to the brightness at
+  the centre of the entire pattern ([xi] = 0, [eta] = 0).
+
+  By the principle of energy the illumination over the entire focal
+  plane must be equal to that over the diffracting area; and thus, in
+  accordance with the suppositions by which (3) was obtained, its value
+  when integrated from [xi] = [oo] to [xi] = +[oo], and from [eta] =
+  -[oo] to [eta] = +[oo] should be equal to ab. This integration,
+  employed originally by P. Kelland (_Edin. Trans._ 15, p. 315) to
+  determine the absolute intensity of a secondary wave, may be at once
+  effected by means of the known formula
+
+      _+[oo]        _+[oo]
+     / sin²u       / sin u
+     | ----- du =  | ----- du = [pi].
+    _/   u²       _/   u
+     -[oo]         -[oo]
+
+  It will be observed that, while the total intensity is proportional to
+  ab, the intensity at the focal point is proportional to a²b². If the
+  aperture be increased, not only is the total brightness over the focal
+  plane increased with it, but there is also a concentration of the
+  diffraction pattern. The form of (3) shows immediately that, if a and
+  b be altered, the co-ordinates of any characteristic point in the
+  pattern vary as a^-1 and b^-1.
+
+  The contraction of the diffraction pattern with increase of aperture
+  is of fundamental importance in connexion with the resolving power of
+  optical instruments. According to common optics, where images are
+  absolute, the diffraction pattern is supposed to be infinitely small,
+  and two radiant points, however near together, form separated images.
+  This is tantamount to an assumption that [lambda] is infinitely small.
+  The actual finiteness of [lambda] imposes a limit upon the separating
+  or resolving power of an optical instrument.
+
+  This indefiniteness of images is sometimes said to be due to
+  diffraction by the edge of the aperture, and proposals have even been
+  made for curing it by causing the transition between the interrupted
+  and transmitted parts of the primary wave to be less abrupt. Such a
+  view of the matter is altogether misleading. What requires explanation
+  is not the imperfection of actual images so much as the possibility of
+  their being as good as we find them.
+
+  At the focal point ([xi] = 0, [eta] = 0) all the secondary waves agree
+  in phase, and the intensity is easily expressed, whatever be the form
+  of the aperture. From the general formula (2), if A be the _area_ of
+  aperture,
+
+    I0² = A²/[lambda]²f²           (7).
+
+  The formation of a sharp image of the radiant point requires that the
+  illumination become insignificant when [xi], [eta] attain small
+  values, and this insignificance can only arise as a consequence of
+  discrepancies of phase among the secondary waves from various parts of
+  the aperture. So long as there is no sensible discrepancy of phase
+  there can be no sensible diminution of brightness as compared with
+  that to be found at the focal point itself. We may go further, and lay
+  it down that there can be no considerable loss of brightness until the
+  difference of phase of the waves proceeding from the nearest and
+  farthest parts of the aperture amounts to ¼[lambda].
+
+  When the difference of phase amounts to [lambda], we may expect the
+  resultant illumination to be very much reduced. In the particular case
+  of a rectangular aperture the course of things can be readily
+  followed, especially if we conceive f to be infinite. In the direction
+  (suppose horizontal) for which [eta] = 0, [xi]/f = sin [theta], the
+  phases of the secondary waves range over a complete period when sin
+  [theta] = [lambda]/a, and, since all parts of the horizontal aperture
+  are equally effective, there is in this direction a complete
+  compensation and consequent absence of illumination. When sin [theta]
+  = 3/2[lambda]/a, the phases range one and a half periods, and there
+  is revival of illumination. We may compare the brightness with that in
+  the direction [theta] = 0. The phase of the resultant amplitude is the
+  same as that due to the central secondary wave, and the discrepancies
+  of phase among the components reduce the amplitude in the proportion
+
+            _+3/2[pi]
+      1    /
+    -----  | cos [phi] d[phi]: 1,
+    3[pi] _/-3/2[pi]
+
+  or -2/3[pi]:1; so that the brightness in this direction is 4/9[pi]² of
+  the maximum at [theta] = 0. In like manner we may find the
+  illumination in any other direction, and it is obvious that it
+  vanishes when sin [theta] is any multiple of [lamba]/a.
+
+  The reason of the augmentation of resolving power with aperture will
+  now be evident. The larger the aperture the smaller are the angles
+  through which it is necessary to deviate from the principal direction
+  in order to bring in specified discrepancies of phase--the more
+  concentrated is the image.
+
+  In many cases the subject of examination is a luminous line of uniform
+  intensity, the various points of which are to be treated as
+  independent sources of light. If the image of the line be [xi] = 0,
+  the intensity at any point [xi], [eta] of the diffraction pattern may
+  be represented by
+
+                                 [pi]a[xi]
+      _+[oo]                 sin²---------
+     /               a²b         [lambda]f
+     | I²d[eta] = ---------  -------------     (8),
+    _/            [lambda]f  [pi]²a²[xi]²
+     -[oo]                   ------------
+                              [lambda]²f²
+
+  the same law as obtains for a luminous point when horizontal
+  directions are alone considered. The definition of a fine vertical
+  line, and consequently the resolving power for contiguous vertical
+  lines, is thus _independent of the vertical aperture of the
+  instrument_, a law of great importance in the theory of the
+  spectroscope.
+
+  The distribution of illumination in the image of a luminous line is
+  shown by the curve ABC (fig. 3), representing the value of the
+  function sin²u/u² from u = 0 to u = 2[pi]. The part corresponding to
+  negative values of u is similar, OA being a line of symmetry.
+
+  [Illustration: Fig. 3.]
+
+  Let us now consider the distribution of brightness in the image of a
+  double line whose components are of equal strength, and at such an
+  angular interval that the central line in the image of one coincides
+  with the first zero of brightness in the image of the other. In fig. 3
+  the curve of brightness for one component is ABC, and for the other
+  OA'C'; and the curve representing half the combined brightnesses is
+  E'BE. The brightness (corresponding to B) midway between the two
+  central points AA' is .8106 of the brightness at the central points
+  themselves. We may consider this to be about the limit of closeness at
+  which there could be any decided appearance of resolution, though
+  doubtless an observer accustomed to his instrument would recognize the
+  duplicity with certainty. The obliquity, corresponding to u = [pi], is
+  such that the phases of the secondary waves range over a complete
+  period, i.e. such that the projection of the horizontal aperture upon
+  this direction is one wave-length. We conclude that a _double line
+  cannot be fairly resolved unless its components subtend an angle
+  exceeding that subtended by the wave-length of light at a distance
+  equal to the horizontal aperture_. This rule is convenient on account
+  of its simplicity; and it is sufficiently accurate in view of the
+  necessary uncertainty as to what exactly is meant by resolution.
+
+  If the angular interval between the components of a double line be
+  half as great again as that supposed in the figure, the brightness
+  midway between is .1802 as against 1.0450 at the central lines of each
+  image. Such a falling off in the middle must be more than sufficient
+  for resolution. If the angle subtended by the components of a double
+  line be twice that subtended by the wave-length at a distance equal to
+  the horizontal aperture, the central bands are just clear of one
+  another, and there is a line of absolute blackness in the middle of
+  the combined images.
+
+  The resolving power of a telescope with circular or rectangular
+  aperture is easily investigated experimentally. The best object for
+  examination is a grating of fine wires, about fifty to the inch,
+  backed by a sodium flame. The object-glass is provided with diaphragms
+  pierced with round holes or slits. One of these, of width equal, say,
+  to one-tenth of an inch, is inserted in front of the object-glass, and
+  the telescope, carefully focused all the while, is drawn gradually
+  back from the grating until the lines are no longer seen. From a
+  measurement of the maximum distance the least angle between
+  consecutive lines consistent with resolution may be deduced, and a
+  comparison made with the rule stated above.
+
+  Merely to show the dependence of resolving power on aperture it is not
+  necessary to use a telescope at all. It is sufficient to look at wire
+  gauze backed by the sky or by a flame, through a piece of blackened
+  cardboard, pierced by a needle and held close to the eye. By varying
+  the distance the point is easily found at which resolution ceases; and
+  the observation is as sharp as with a telescope. The function of the
+  telescope is in fact to allow the use of a wider, and therefore more
+  easily measurable, aperture. An interesting modification of the
+  experiment may be made by using light of various wave-lengths.
+
+  Since the limitation of the width of the central band in the image of
+  a luminous line depends upon discrepancies of phase among the
+  secondary waves, and since the discrepancy is greatest for the waves
+  which come from the edges of the aperture, the question arises how far
+  the operation of the central parts of the aperture is advantageous. If
+  we imagine the aperture reduced to two equal narrow slits bordering
+  its edges, compensation will evidently be complete when the projection
+  on an oblique direction is equal to ½[lambda], instead of [lambda] as
+  for the complete aperture. By this procedure the width of the central
+  band in the diffraction pattern is halved, and so far an advantage is
+  attained. But, as will be evident, the bright bands bordering the
+  central band are now not inferior to it in brightness; in fact, a band
+  similar to the central band is reproduced an indefinite number of
+  times, so long as there is no sensible discrepancy of phase in the
+  secondary waves proceeding from the various parts of the _same_ slit.
+  Under these circumstances the narrowing of the band is paid for at a
+  ruinous price, and the arrangement must be condemned altogether.
+
+  A more moderate suppression of the central parts is, however,
+  sometimes advantageous. Theory and experiment alike prove that a
+  double line, of which the components are equally strong, is better
+  resolved when, for example, one-sixth of the horizontal aperture is
+  blocked off by a central screen; or the rays quite at the centre may
+  be allowed to pass, while others a little farther removed are blocked
+  off. Stops, each occupying one-eighth of the width, and with centres
+  situated at the points of trisection, answer well the required
+  purpose.
+
+  It has already been suggested that the principle of energy requires
+  that the general expression for I² in (2) when integrated over the
+  whole of the plane [xi], [eta] should be equal to A, where A is the
+  area of the aperture. A general analytical verification has been given
+  by Sir G. G. Stokes (_Edin. Trans._, 1853, 20, p. 317). Analytically
+  expressed--
+
+      _ _+[oo]             _ _
+     / /                  / /
+     | | I² d[xi]d[eta] = | | dxdy  = A        (9).
+    _/_/-[oo]            _/_/
+
+  We have seen that I0² (the intensity at the focal point) was equal to
+  A²/[lambda]²f². If A' be the area over which the intensity must be I0²
+  in order to give the actual total intensity in accordance with
+
+             _ _+[oo]
+            / /
+    A'I0² = | | I² d[xi]d[eta],
+           _/_/-[oo]
+
+  the relation between A and A' is AA' = [lambda]²f². Since A' is in
+  some sense the area of the diffraction pattern, it may be considered
+  to be a rough criterion of the definition, and we infer that the
+  definition of a point depends principally upon the area of the
+  aperture, and only in a very secondary degree upon the shape when the
+  area is maintained constant.
+
+4. _Theory of Circular Aperture._--We will now consider the important
+case where the form of the aperture is circular.
+
+  Writing for brevity
+
+    k[xi]/f = p, k[eta]/f = q,        (1),
+
+  we have for the general expression (§ 11) of the intensity
+
+    [lambda]²f²I² = S² + C²           (2),
+
+  where
+          _ _
+         / /
+    S =  | | sin(px + qy)dx dy,       (3),
+        _/_/
+          _ _
+         / /
+    C =  | | cos(px + qy)dx dy,       (4).
+        _/_/
+
+  When, as in the application to rectangular or circular apertures, the
+  form is symmetrical with respect to the axes both of x and y, S = 0,
+  and C reduces to
+          _ _
+         / /
+    C =  | | cos px cos qy dx dy,     (5).
+        _/_/
+
+  In the case of the circular aperture the distribution of light is of
+  course symmetrical with respect to the focal point p = 0, q = 0; and C
+  is a function of p and q only through [sqrt](p² + q²). It is thus
+  sufficient to determine the intensity along the axis of p. Putting q =
+  0, we get
+          _ _                  _+R
+         / /                  /          /
+    C =  | | cos px dx dy = 2 | cos px \/(R² - x²) dx,
+        _/_/                 _/-R
+
+  R being the radius of the aperture. This integral is the Bessel's
+  function of order unity, defined by
+
+                   _[pi]
+             z    /
+    J1(z) = ----  | cos(z cos [phi]) sin² [phi] d[phi]    (6).
+            [pi] _/0
+
+  Thus, if x = R cos [phi],
+
+               2J1(pR)
+    C = [pi]²R -------                                    (7);
+                 pR
+
+  and the illumination at distance r from the focal point is
+
+                            / 2[pi]Rr \
+                       4J1²( --------- )
+          [pi]²R^4          \f[lambda]/
+    I² = ----------- · -----------------                  (8).
+         [lambda]²f²     / 2[pi]Rr \²
+                        ( --------- )
+                         \f[lambda]/
+
+  The ascending series for J1(z), used by Sir G. B. Airy (_Camb.
+  Trans._, 1834) in his original investigation of the diffraction of a
+  circular object-glass, and readily obtained from (6), is
+
+            z    z³      z^5         z^7
+    J1(z) = - - ---- + ------- - ---------- + ...         (9).
+            2   2²·4   2²·4²·6   2²·4²·6²·8
+
+  When z is great, we may employ the semi-convergent series
+                                          _
+              / /  2  \                  |    3·5·1   /1\²
+    J1(z) =  / ( ----- ) sin (z - ¼[pi]) |1 + ------ ( - )
+           \/   \[pi]z/                  |_    8·16   \z/
+                                          _
+               3·5·7·9·1·3·5  /1\^4        |
+             - ------------- ( - )   + ... |
+                 8·16·24·32   \z/         _|
+                                       _
+           / /  2  \                  | 3   1   3·5·7·1·3  /1\ ³
+       +  / ( ----- ) cos (z - ¼[pi]) | - · - - --------- ( - )
+        \/   \[pi]z/                  |_8   z    8·16·24   \z/
+                                         _
+         3·5·7·9·11·1·3·5·7  /1\^5        |
+       + ------------------ ( - )   - ... | ...          (10).
+           8·16·24·32·40     \z/         _|
+
+  A table of the values of 2z^-1J1(z) has been given by E. C. J. Lommel
+  (_Schlömilch_, 1870, 15, p. 166), to whom is due the first systematic
+  application of Bessel's functions to the diffraction integrals.
+
+  The illumination vanishes in correspondence with the roots of the
+  equation J1(z) = 0. If these be called z1 z2, z3, ... the radii of the
+  dark rings in the diffraction pattern are
+
+    f[lambda]z1  f[lambda]z2
+    -----------, -----------, ...
+      2[pi]R       2[pi]R
+
+  being thus _inversely_ proportional to R.
+
+  The integrations may also be effected by means of polar co-ordinates,
+  taking first the integration with respect to [phi] so as to obtain the
+  result for an infinitely thin annular aperture. Thus, if
+
+    x = [rho] cos [phi], y = [rho] sin [phi],
+
+         _ _                _R  _2[pi]
+        / /                /   /
+    C = | | cos px dx dy = |   |   cos (p[rho] cos [theta]) [rho]d[rho] d[theta].
+       _/_/               _/0 _/0
+
+  Now by definition
+
+                   _½[pi]
+              2   /                              z²    z^4      z^6
+    J0(z) = ----  |  cos(z cos[theta])d[theta] = -- + ----- - -------- + ...  (11).
+            [pi] _/0                             2²   2²·4²   2²·4²·6²
+
+  The value of C for an annular aperture of radius r and width dr is thus
+
+    dC = 2 [pi]J0 (p[rho]) [rho] d[rho],                                       (12).
+
+  For the complete circle,
+
+               _ pR
+        2[pi] /
+    C = ----- | J0(z) zdz
+         p²  _/0
+
+         2[pi]  /p²R²   p^4 R^4   p^6 R^6       \
+      = ------ ( ---- - ------- + -------- - ... )
+          p²    \ 2      2²·4²    2²·4²·6²      /
+
+                 2J1(pR)
+      = [pi]R² · ------- as before.
+                   pR
+
+  In these expressions we are to replace p by k[xi]/f, or rather, since
+  the diffraction pattern is symmetrical, by kr/f, where r is the
+  distance of any point in the focal plane from the centre of the
+  system.
+
+  The roots of J0(z) after the first may be found from
+
+     z               .050561    .053041     .262051
+    ---- = i - .25 + ------- - --------- + ---------- ...   (13),
+    [pi]             4i - 1    (4i - 1)³   (4i - 1)^5
+
+  and those of J1(z) from
+
+     z               .151982    .015399     .245835
+    ---- = i + .25 - ------- + --------- + ---------- ...   (14),
+    [pi]             4i + 1    (4i + 1)³   (4i + 1)^5
+
+  formulae derived by Stokes (_Camb. Trans._, 1850, vol. ix.) from the
+  descending series.[1] The following table gives the actual values:--
+
+    +---+--------------------+--------------------+
+    |   |   z                |  z                 |
+    | i | ---- for J0(z) = 0 | ---- for J1(z) = 0 |
+    |   | [pi]               | [pi]               |
+    +---+--------------------+--------------------+
+    | 1 |        7655        |       1 2197       |
+    | 2 |      1 7571        |       2 2330       |
+    | 3 |      2 7546        |       3 2383       |
+    | 4 |      3 7534        |       4 2411       |
+    | 5 |      4 7527        |       5 2428       |
+    | 6 |      5 7522        |       6 2439       |
+    | 7 |      6 7519        |       7 2448       |
+    | 8 |      7 7516        |       8 2454       |
+    | 9 |      8 7514        |       9 2459       |
+    |10 |      9 7513        |      10 2463       |
+    +---+--------------------+--------------------+
+
+  In both cases the image of a mathematical point is thus a symmetrical
+  ring system. The greatest brightness is at the centre, where
+
+    dC = 2[pi][rho] d[rho], C = [pi]R².
+
+  For a certain distance outwards this remains sensibly unimpaired and
+  then gradually diminishes to zero, as the secondary waves become
+  discrepant in phase. The subsequent revivals of brightness forming the
+  bright rings are necessarily of inferior brilliancy as compared with
+  the central disk.
+
+  The first dark ring in the diffraction pattern of the complete
+  circular aperture occurs when
+
+    r/f = 1.2197 × [lambda]/2R                  (15).
+
+  We may compare this with the corresponding result for a rectangular
+  aperture of width a,
+
+    [xi]/f =[lambda]/a;
+
+  and it appears that in consequence of the preponderance of the central
+  parts, the compensation in the case of the circle does not set in at
+  so small an obliquity as when the circle is replaced by a rectangular
+  aperture, whose side is equal to the diameter of the circle.
+
+  Again, if we compare the complete circle with a narrow annular
+  aperture of the same radius, we see that in the latter case the first
+  dark ring occurs at a much smaller obliquity, viz.
+
+    r/f = .7655 × [lambda]/2R.
+
+  It has been found by Sir William Herschel and others that the
+  definition of a telescope is often improved by stopping off a part of
+  the central area of the object-glass; but the advantage to be obtained
+  in this way is in no case great, and anything like a reduction of the
+  aperture to a narrow annulus is attended by a development of the
+  external luminous rings sufficient to outweigh any improvement due to
+  the diminished diameter of the central area.[2]
+
+  The maximum brightnesses and the places at which they occur are easily
+  determined with the aid of certain properties of the Bessel's
+  functions. It is known (see SPHERICAL HARMONICS) that
+
+    J0'(z) = -J1(z),                    (16);
+
+            1
+    J2(z) = - J1(z) - J1'(z)            (17);
+            z
+
+                    2
+    J0(z) + J2(z) = - J1(z)             (18).
+                    z
+
+  The maxima of C occur when
+
+     d  /J1(z)\    J1'(z)   J1(z)
+    -- (-------) = ------ - ----- = 0;
+    dz  \  z  /      z        z²
+
+  or by 17 when J2(z) = 0. When z has one of the values thus determined,
+
+    2
+    - J1(z) = J0(z).
+    z
+
+  The accompanying table is given by Lommel, in which the first column
+  gives the roots of J2(z) = 0, and the second and third columns the
+  corresponding values of the functions specified. If appears that the
+  maximum brightness in the first ring is only about 1/57 of the
+  brightness at the centre.
+
+    +-------------------------------------------+
+    |       z        2z^-1 J1(z)  4z^-2 J1²(z)  |
+    +-------------------------------------------+
+    |                                           |
+    |    .000000     +1.000000      1.000000    |
+    |   5.135630     - .132279       .017498    |
+    |   8.417236     + .064482       .004158    |
+    |  11.619857     - .040008       .001601    |
+    |  14.795938     + .027919       .000779    |
+    |  17.959820     - .020905       .000437    |
+    +-------------------------------------------+
+
+  We will now investigate the total illumination distributed over the
+  area of the circle of radius r. We have
+
+           [pi]²R^4     4J1²(z)
+    I^2 = ----------- · -------                (19),
+          [lambda]²f²     z²
+
+  where
+
+    z = 2[pi]Rr/[lambda]f                      (20).
+
+  Thus
+           _                     _                  _
+          /         [lambda]²f² /                  /
+    2[pi] | I²rdr = ----------- | I²zdz = [pi]R²·2 | z^-1 J1²(z)dz.
+         _/           2[pi]R²  _/                 _/
+
+  Now by (17), (18)
+
+    z^-1 J1(z) = J0(z) - J1'(z);
+
+  so that
+
+                   d          d
+    z^-1J1²(z) = ½ -- J0² - ½ -- J1²(z),
+                   dz         dz
+
+  and
+
+       _z
+      /
+    2 | z^-1 J1²(z)dz = 1 - J0²(z) - J1²(z)    (21).
+     _/0
+
+  If r, or z, be infinite, J0(z), J1(z) vanish, and the whole
+  illumination is expressed by [pi]R², in accordance with the general
+  principle. In any case the proportion of the whole illumination to be
+  found outside the circle of radius r is given by
+
+    J0²(z) + J1²(z).
+
+  For the dark rings J1(z) = 0; so that the fraction of illumination
+  outside any dark ring is simply J0²(z). Thus for the first, second,
+  third and fourth dark rings we get respectively .161, .090, .062,
+  .047, showing that more than 9/10ths of the whole light is
+  concentrated within the area of the second dark ring (_Phil. Mag._,
+  1881).
+
+  When z is great, the descending series (10) gives
+
+    2J1(z)   2    / /  2  \
+    ------ = -   / ( ----- ) sin(z - ¼[pi])    (22);
+      z      z \/   \[pi]z/
+
+  so that the places of maxima and minima occur at equal intervals.
+
+  The mean brightness varies as z^-3 (or as r^-3), and the integral
+  found by multiplying it by zdz and integrating between 0 and [oo]
+  converges.
+
+  It may be instructive to contrast this with the case of an infinitely
+  narrow annular aperture, where the brightness is proportional to
+  J0²(z). When z is great,
+
+                /  2
+    J0(z) = \  / ----- cos(z^-¼ [pi]).
+             \/  [pi]z
+
+  The mean brightness varies as z^-1; and the integral
+      _
+     / [oo]
+     |   J0²(z)z dz  is not convergent.
+    _/ 0
+
+5. _Resolving Power of Telescopes._--The efficiency of a telescope is of
+course intimately connected with the size of the disk by which it
+represents a mathematical point. In estimating theoretically the
+resolving power on a double star we have to consider the illumination of
+the field due to the superposition of the two independent images. If the
+angular interval between the components of a double star were equal to
+twice that expressed in equation (15) above, the central disks of the
+diffraction patterns would be just in contact. Under these conditions
+there is no doubt that the star would appear to be fairly resolved,
+since the brightness of its external ring system is too small to produce
+any material confusion, unless indeed the components are of very unequal
+magnitude. The diminution of the star disks with increasing aperture was
+observed by Sir William Herschel, and in 1823 Fraunhofer formulated the
+law of inverse proportionality. In investigations extending over a long
+series of years, the advantage of a large aperture in separating the
+components of close double stars was fully examined by W. R. Dawes.
+
+The resolving power of telescopes was investigated also by J. B. L.
+Foucault, who employed a scale of equal bright and dark alternate parts;
+it was found to be proportional to the aperture and independent of the
+focal length. In telescopes of the best construction and of moderate
+aperture the performance is not sensibly prejudiced by outstanding
+aberration, and the limit imposed by the finiteness of the waves of
+light is practically reached. M. E. Verdet has compared Foucault's
+results with theory, and has drawn the conclusion that the radius of the
+visible part of the image of a luminous point was equal to half the
+radius of the first dark ring.
+
+The application, unaccountably long delayed, of this principle to the
+microscope by H. L. F. Helmholtz in 1871 is the foundation of the
+important doctrine of the _microscopic limit_. It is true that in 1823
+Fraunhofer, inspired by his observations upon gratings, had very nearly
+hit the mark.[3] And a little before Helmholtz, E. Abbe published a
+somewhat more complete investigation, also founded upon the phenomena
+presented by gratings. But although the argument from gratings is
+instructive and convenient in some respects, its use has tended to
+obscure the essential unity of the principle of the limit of resolution
+whether applied to telescopes or microscopes.
+
+  [Illustration: Fig. 4.]
+
+  In fig. 4, AB represents the axis of an optical instrument (telescope
+  or microscope), A being a point of the object and B a point of the
+  image. By the operation of the object-glass LL' all the rays issuing
+  from A arrive in the same phase at B. Thus if A be self-luminous, the
+  illumination is a maximum at B, where all the secondary waves agree in
+  phase. B is in fact the centre of the diffraction disk which
+  constitutes the image of A. At neighbouring points the illumination is
+  less, in consequence of the discrepancies of phase which there enter.
+  In like manner if we take a neighbouring point P, also self-luminous,
+  in the plane of the object, the waves which issue from it will arrive
+  at B with phases no longer absolutely concordant, and the discrepancy
+  of phase will increase as the interval AP increases. When the
+  interval is very small the discrepancy, though mathematically
+  existent, produces no practical effect; and the illumination at B due
+  to P is as important as that due to A, the intensities of the two
+  luminous sources being supposed equal. Under these conditions it is
+  clear that A and P are not separated in the image. The question is to
+  what amount must the distance AP be increased in order that the
+  difference of situation may make itself felt in the image. This is
+  necessarily a question of degree; but it does not require detailed
+  calculations in order to show that the discrepancy first becomes
+  conspicuous when the phases corresponding to the various secondary
+  waves which travel from P to B range over a complete period. The
+  illumination at B due to P then becomes comparatively small, indeed
+  for some forms of aperture evanescent. The extreme discrepancy is that
+  between the waves which travel through the outermost parts of the
+  object-glass at L and L'; so that if we adopt the above standard of
+  resolution, the question is where must P be situated in order that the
+  relative retardation of the rays PL and PL' may on their arrival at B
+  amount to a wave-length ([lambda]). In virtue of the general law that
+  the reduced optical path is stationary in value, this retardation may
+  be calculated without allowance for the different paths pursued on the
+  farther side of L, L', so that the value is simply PL - PL'. Now since
+  AP is very small, AL' - PL' = AP sin [alpha], where [alpha] is the
+  angular semi-aperture L'AB. In like manner PL - AL has the same value,
+  so that
+
+    PL - PL' = 2AP sin [alpha].
+
+  According to the standard adopted, the condition of resolution is
+  therefore that AP, or [epsilon], should exceed ½[lambda]/sin [alpha].
+  If [epsilon] be less than this, the images overlap too much; while if
+  [epsilon] greatly exceed the above value the images become
+  unnecessarily separated.
+
+  In the above argument the whole space between the object and the lens
+  is supposed to be occupied by matter of one refractive index, and
+  [lambda] represents the wave-length _in this medium_ of the kind of
+  light employed. If the restriction as to uniformity be violated, what
+  we have ultimately to deal with is the wave-length in the medium
+  immediately surrounding the object.
+
+  Calling the refractive index [mu], we have as the critical value of
+  [epsilon],
+
+    [epsilon] = ½[lambda]0/[mu] sin[alpha],      (1),
+
+  [lambda]0 being the wave-length _in vacuo_. The denominator [mu] sin
+  [alpha] is the quantity well known (after Abbe) as the "numerical
+  aperture."
+
+  The extreme value possible for [alpha] is a right angle, so that for
+  the microscopic limit we have
+
+    [epsilon] = ½[lambda]0/[mu]                  (2).
+
+  The limit can be depressed only by a diminution in [lambda]0, such as
+  photography makes possible, or by an increase in [mu], the refractive
+  index of the medium in which the object is situated.
+
+  The statement of the law of resolving power has been made in a form
+  appropriate to the microscope, but it admits also of immediate
+  application to the telescope. If 2R be the diameter of the
+  object-glass and D the distance of the object, the angle subtended by
+  AP is [epsilon]/D, and the angular resolving power is given by
+
+    [lambda]/2D sin[alpha] = [lambda]/2R         (3).
+
+  This method of derivation (substantially due to Helmholtz) makes it
+  obvious that there is no essential difference of principle between the
+  two cases, although the results are conveniently stated in different
+  forms. In the case of the telescope we have to deal with a linear
+  measure of aperture and an angular limit of resolution, whereas in the
+  case of the microscope the limit of resolution is linear, and it is
+  expressed in terms of angular aperture.
+
+  It must be understood that the above argument distinctly assumes that
+  the different parts of the object are self-luminous, or at least that
+  the light proceeding from the various points is without phase
+  relations. As has been emphasized by G. J. Stoney, the restriction is
+  often, perhaps usually, violated in the microscope. A different
+  treatment is then necessary, and for some of the problems which arise
+  under this head the method of Abbe is convenient.
+
+  The importance of the general conclusions above formulated, as
+  imposing a limit upon our powers of direct observation, can hardly be
+  overestimated; but there has been in some quarters a tendency to
+  ascribe to it a more precise character than it can bear, or even to
+  mistake its meaning altogether. A few words of further explanation may
+  therefore be desirable. The first point to be emphasized is that
+  nothing whatever is said as to the smallness of a single object that
+  may be made visible. The eye, unaided or armed with a telescope, is
+  able to see, as points of light, stars subtending no sensible angle.
+  The visibility of a star is a question of brightness simply, and has
+  nothing to do with resolving power. The latter element enters only
+  when it is a question of recognizing the duplicity of a double star,
+  or of distinguishing detail upon the surface of a planet. So in the
+  microscope there is nothing except lack of light to hinder the
+  visibility of an object however small. But if its dimensions be much
+  less than the half wave-length, it can only be seen as a whole, and
+  its parts cannot be distinctly separated, although in cases near the
+  border line some inference may be possible, founded upon experience of
+  what appearances are presented in various cases. Interesting
+  observations upon particles, _ultra-microscopic_ in the above sense,
+  have been recorded by H. F. W. Siedentopf and R. A. Zsigmondy
+  (_Drude's Ann._, 1903, 10, p. 1).
+
+  In a somewhat similar way a dark linear interruption in a bright
+  ground may be visible, although its actual width is much inferior to
+  the half wave-length. In illustration of this fact a simple experiment
+  may be mentioned. In front of the naked eye was held a piece of copper
+  foil perforated by a fine needle hole. Observed through this the
+  structure of some wire gauze just disappeared at a distance from the
+  eye equal to 17 in., the gauze containing 46 meshes to the inch. On
+  the other hand, a single wire 0.034 in. in diameter remained fairly
+  visible up to a distance of 20 ft. The ratio between the limiting
+  angles subtended by the periodic structure of the gauze and the
+  diameter of the wire was (.022/.034) × (240/17) = 9.1. For further
+  information upon this subject reference may be made to _Phil. Mag._,
+  1896, 42, p. 167; _Journ. R. Micr. Soc._, 1903, p. 447.
+
+6. _Coronas or Glories._--The results of the theory of the diffraction
+patterns due to circular apertures admit of an interesting application
+to _coronas_, such as are often seen encircling the sun and moon. They
+are due to the interposition of small spherules of water, which act the
+part of diffracting obstacles. In order to the formation of a
+well-defined corona it is essential that the particles be exclusively,
+or preponderatingly, of one size.
+
+  If the origin of light be treated as infinitely small, and be seen in
+  focus, whether with the naked eye or with the aid of a telescope, the
+  whole of the light in the absence of obstacles would be concentrated
+  in the immediate neighbourhood of the focus. At other parts of the
+  field the effect is the same, in accordance with the principle known
+  as Babinet's, whether the imaginary screen in front of the
+  object-glass is generally transparent but studded with a number of
+  opaque circular disks, or is generally opaque but perforated with
+  corresponding apertures. Since at these points the resultant due to
+  the whole aperture is zero, any two portions into which the whole may
+  be divided must give equal and opposite resultants. Consider now the
+  light diffracted in a direction many times more oblique than any with
+  which we should be concerned, were the whole aperture uninterrupted,
+  and take first the effect of a single small aperture. The light in the
+  proposed direction is that determined by the size of the small
+  aperture in accordance with the laws already investigated, and its
+  phase depends upon the position of the aperture. If we take a
+  direction such that the light (of given wave-length) from a single
+  aperture vanishes, the evanescence continues even when the whole
+  series of apertures is brought into contemplation. Hence, whatever
+  else may happen, there must be a system of dark rings formed, the same
+  as from a single small aperture. In directions other than these it is
+  a more delicate question how the partial effects should be compounded.
+  If we make the extreme suppositions of an infinitely small source and
+  absolutely homogeneous light, there is no escape from the conclusion
+  that the light in a definite direction is arbitrary, that is,
+  dependent upon the chance distribution of apertures. If, however, as
+  in practice, the light be heterogeneous, the source of finite area,
+  the obstacles in motion, and the discrimination of different
+  directions imperfect, we are concerned merely with the mean brightness
+  found by varying the arbitrary phase-relations, and this is obtained
+  by simply multiplying the brightness due to a single aperture by the
+  number of apertures (n) (see INTERFERENCE OF LIGHT, § 4). The
+  diffraction pattern is therefore that due to a single aperture, merely
+  brightened n times.
+
+  In his experiments upon this subject Fraunhofer employed plates of
+  glass dusted over with lycopodium, or studded with small metallic
+  disks of uniform size; and he found that the diameters of the rings
+  were proportional to the length of the waves and inversely as the
+  diameter of the disks.
+
+  In another respect the observations of Fraunhofer appear at first
+  sight to be in disaccord with theory; for his measures of the
+  diameters of the red rings, visible when white light was employed,
+  correspond with the law applicable to dark rings, and not to the
+  different law applicable to the luminous maxima. Verdet has, however,
+  pointed out that the observation in this form is essentially different
+  from that in which homogeneous red light is employed, and that the
+  position of the red rings would correspond to the _absence_ of
+  blue-green light rather than to the greatest abundance of red light.
+  Verdet's own observations, conducted with great care, fully confirm
+  this view, and exhibit a complete agreement with theory.
+
+  By measurements of coronas it is possible to infer the size of the
+  particles to which they are due, an application of considerable
+  interest in the case of natural coronas--the general rule being the
+  larger the corona the smaller the water spherules. Young employed this
+  method not only to determine the diameters of cloud particles (e.g.
+  1/1000 in.), but also those of fibrous material, for which the theory
+  is analogous. His instrument was called the _eriometer_ (see
+  "Chromatics," vol. iii. of supp. to _Ency. Brit._, 1817).
+
+7. _Influence of Aberration. Optical Power of Instruments._--Our
+investigations and estimates of resolving power have thus far proceeded
+upon the supposition that there are no optical imperfections, whether of
+the nature of a regular aberration or dependent upon irregularities of
+material and workmanship. In practice there will always be a certain
+aberration or error of phase, which we may also regard as the deviation
+of the actual wave-surface from its intended position. In general, we
+may say that aberration is unimportant when it nowhere (or at any rate
+over a relatively small area only) exceeds a small fraction of the
+wave-length ([lamda]). Thus in estimating the intensity at a focal point,
+where, in the absence of aberration, all the secondary waves would have
+exactly the same phase, we see that an aberration nowhere exceeding
+¼[lambda] can have but little effect.
+
+  The only case in which the influence of small aberration upon the
+  entire image has been calculated (_Phil. Mag._, 1879) is that of a
+  rectangular aperture, traversed by a cylindrical wave with aberration
+  equal to cx³. The aberration is here unsymmetrical, the wave being in
+  advance of its proper place in one half of the aperture, but behind in
+  the other half. No terms in x or x² need be considered. The first
+  would correspond to a general turning of the beam; and the second
+  would imply imperfect focusing of the central parts. The effect of
+  aberration may be considered in two ways. We may suppose the aperture
+  (a) constant, and inquire into the operation of an increasing
+  aberration; or we may take a given value of c (i.e. a given
+  wave-surface) and examine the effect of a varying aperture. The
+  results in the second case show that an increase of aperture up to
+  that corresponding to an extreme aberration of half a period has no
+  ill effect upon the central band (§ 3), but it increases unduly the
+  intensity of one of the neighbouring lateral bands; and the practical
+  conclusion is that the best results will be obtained from an aperture
+  giving an extreme aberration of from a quarter to half a period, and
+  that with an increased aperture aberration is not so much a direct
+  cause of deterioration as an obstacle to the attainment of that
+  improved definition which should accompany the increase of aperture.
+
+  If, on the other hand, we suppose the aperture given, we find that
+  aberration begins to be distinctly mischievous when it amounts to
+  about a quarter period, i.e. when the wave-surface deviates at each
+  end by a quarter wave-length from the true plane.
+
+  As an application of this result, let us investigate what amount of
+  temperature disturbance in the tube of a telescope may be expected to
+  impair definition. According to J. B. Biot and F. J. D. Arago, the
+  index [mu] for air at t° C. and at atmospheric pressure is given by
+
+                .00029
+    [mu] - 1 = -----------.
+               1 + .0037 t
+
+  If we take 0° C. as standard temperature,
+
+    [delta][mu] = -1.1 × 10^-6.
+
+  Thus, on the supposition that the irregularity of temperature t
+  extends through a length l, and produces an acceleration of a quarter
+  of a wave-length,
+
+    ¼[lambda] = 1.1 lt × 10^-6;
+
+  or, if we take [lambda] = 5.3 × 10^-5,
+
+    lt = 12,
+
+  the unit of length being the centimetre.
+
+  We may infer that, in the case of a telescope tube 12 cm. long, a
+  stratum of air heated 1° C. lying along the top of the tube, and
+  occupying a moderate fraction of the whole volume, would produce a not
+  insensible effect. If the change of temperature progressed uniformly
+  from one side to the other, the result would be a lateral displacement
+  of the image without loss of definition; but in general both effects
+  would be observable. In longer tubes a similar disturbance would be
+  caused by a proportionally less difference of temperature. S. P.
+  Langley has proposed to obviate such ill-effects by stirring the air
+  included within a telescope tube. It has long been known that the
+  definition of a carbon bisulphide prism may be much improved by a
+  vigorous shaking.
+
+  We will now consider the application of the principle to the formation
+  of images, unassisted by reflection or refraction (_Phil. Mag._,
+  1881). The function of a lens in forming an image is to compensate by
+  its variable thickness the differences of phase which would otherwise
+  exist between secondary waves arriving at the focal point from various
+  parts of the aperture. If we suppose the diameter of the lens to be
+  given (2R), and its focal length f gradually to increase, the original
+  differences of phase at the image of an infinitely distant luminous
+  point diminish without limit. When f attains a certain value, say f1,
+  the extreme error of phase to be compensated falls to ¼[lambda]. But,
+  as we have seen, such an error of phase causes no sensible
+  deterioration in the definition; so that from this point onwards the
+  lens is useless, as only improving an image already sensibly as
+  perfect as the aperture admits of. Throughout the operation of
+  increasing the focal length, the resolving power of the instrument,
+  which depends only upon the aperture, remains unchanged; and we thus
+  arrive at the rather startling conclusion that a telescope of any
+  degree of resolving power might be constructed without an
+  object-glass, if only there were no limit to the admissible focal
+  length. This last proviso, however, as we shall see, takes away almost
+  all practical importance from the proposition.
+
+  To get an idea of the magnitudes of the quantities involved, let us
+  take the case of an aperture of 1/5 in., about that of the pupil of
+  the eye. The distance f1, which the actual focal length must exceed,
+  is given by
+
+      /
+    \/ (f1² + R²) - f1  = ¼[lambda];
+
+  so that
+
+    f1  = 2R²/[lambda]                        (1).
+
+  Thus, if [lambda] = 1/4000, R = 1/10, we find
+
+    f1 = 800 inches.
+
+  The image of the sun thrown upon a screen at a distance exceeding 66
+  ft., through a hole 1/5 in. in diameter, is therefore at least as well
+  defined as that seen direct.
+
+  As the minimum focal length increases with the square of the aperture,
+  a quite impracticable distance would be required to rival the
+  resolving power of a modern telescope. Even for an aperture of 4 in.,
+  f1 would have to be 5 miles.
+
+  A similar argument may be applied to find at what point an achromatic
+  lens becomes sensibly superior to a single one. The question is
+  whether, when the adjustment of focus is correct for the central rays
+  of the spectrum, the error of phase for the most extreme rays (which
+  it is necessary to consider) amounts to a quarter of a wave-length. If
+  not, the substitution of an achromatic lens will be of no advantage.
+  Calculation shows that, if the aperture be 1/5 in., an achromatic lens
+  has no sensible advantage if the focal length be greater than about 11
+  in. If we suppose the focal length to be 66 ft., a single lens is
+  practically perfect up to an aperture of 1.7 in.
+
+  Another obvious inference from the necessary imperfection of optical
+  images is the uselessness of attempting anything like an absolute
+  destruction of spherical aberration. An admissible error of phase of
+  ¼[lambda] will correspond to an error of 1/8[lambda] in a reflecting
+  and ½[lambda] in a (glass) refracting surface, the incidence in both
+  cases being perpendicular. If we inquire what is the greatest
+  admissible longitudinal aberration ([delta]f) in an object-glass
+  according to the above rule, we find
+
+    [delta]f = [lambda][alpha]^-2               (2),
+
+  [alpha] being the angular semi-aperture.
+
+  In the case of a single lens of glass with the most favourable
+  curvatures, [delta]f is about equal to [alpha]²f, so that [alpha]^4
+  must not exceed [lambda]/f. For a lens of 3 ft. focus this condition
+  is satisfied if the aperture does not exceed 2 in.
+
+  When parallel rays fall directly upon a spherical mirror the
+  longitudinal aberration is only about one-eighth as great as for the
+  most favourably shaped single lens of equal focal length and aperture.
+  Hence a spherical mirror of 3 ft. focus might have an aperture of 2½
+  in., and the image would not suffer materially from aberration.
+
+  On the same principle we may estimate the least visible displacement
+  of the eye-piece of a telescope focused upon a distant object, a
+  question of interest in connexion with range-finders. It appears
+  (_Phil. Mag._, 1885, 20, p. 354) that a displacement [delta]f from the
+  true focus will not sensibly impair definition, provided
+
+    [delta]f < f²[lambda]/R²                    (3),
+
+  2R being the diameter of aperture. The linear accuracy required is
+  thus a function of the _ratio_ of aperture to focal length. The
+  formula agrees well with experiment.
+
+  The principle gives an instantaneous solution of the question of the
+  ultimate optical efficiency in the method of "mirror-reading," as
+  commonly practised in various physical observations. A rotation by
+  which one edge of the mirror advances ¼[lambda] (while the other edge
+  retreats to a like amount) introduces a phase-discrepancy of a whole
+  period where before the rotation there was complete agreement. A
+  rotation of this amount should therefore be easily visible, but the
+  limits of resolving power are being approached; and the conclusion is
+  independent of the focal length of the mirror, and of the employment
+  of a telescope, provided of course that the reflected image is seen in
+  focus, and that the full width of the mirror is utilized.
+
+  A comparison with the method of a material pointer, attached to the
+  parts whose rotation is under observation, and viewed through a
+  microscope, is of interest. The limiting efficiency of the microscope
+  is attained when the angular aperture amounts to 180°; and it is
+  evident that a lateral displacement of the point under observation
+  through ½[lambda] entails (at the old image) a phase-discrepancy of a
+  whole period, one extreme ray being accelerated and the other retarded
+  by half that amount. We may infer that the limits of efficiency in the
+  two methods are the same when the length of the pointer is equal to
+  the width of the mirror.
+
+  [Illustraton: FIG. 5.]
+
+  We have seen that in perpendicular reflection a surface error not
+  exceeding 1/8[lambda] may be admissible. In the case of oblique
+  reflection at an angle [phi], the error of retardation due to an
+  elevation BD (fig. 5) is
+
+    QQ' - QS = BD sec [phi](1 - cos SQQ') = BD sec [phi] (1 + cos 2[phi]) = 2BD cos [phi];
+
+  from which it follows that an error of given magnitude in the figure
+  of a surface is less important in oblique than in perpendicular
+  reflection. It must, however, be borne in mind that errors can
+  sometimes be compensated by altering adjustments. If a surface
+  intended to be flat is affected with a slight general curvature, a
+  remedy may be found in an alteration of focus, and the remedy is the
+  less complete as the reflection is more oblique.
+
+  The formula expressing the optical power of prismatic spectroscopes
+  may readily be investigated upon the principles of the wave theory.
+  Let A0B0 be a plane wave-surface of the light before it falls upon the
+  prisms, AB the corresponding wave-surface for a particular part of the
+  spectrum after the light has passed the prisms, or after it has passed
+  the eye-piece of the observing telescope. The path of a ray from the
+  wave-surface A0B0 to A or B is determined by the condition that the
+  optical distance, [int] [mu]ds, is a minimum; and, as AB is by
+  supposition a wave-surface, this optical distance is the same for both
+  points. Thus
+      _                  _
+     /                  /
+     | [mu]ds (for A) = | [mu]ds (for B)          (4).
+    _/                 _/
+
+  We have now to consider the behaviour of light belonging to a
+  neighbouring part of the spectrum. The path of a ray from the
+  wave-surface A0B0 to the point A is changed; but in virtue of the
+  minimum property the change may be neglected in calculating the
+  optical distance, as it influences the result by quantities of the
+  second order only in the changes of refrangibility. Accordingly, the
+  optical distance from A0B0 to A is represented by [int]([mu] +
+  [delta][mu])ds, the integration being along the original path A0 ...
+  A; and similarly the optical distance between A0B0 and B is
+  represented by [int] ([mu] + [delta][mu])ds, the integration being
+  along B0 ... B. In virtue of (4) the difference of the optical
+  distances to A and B is
+      _                                  _
+     /                                  /
+     | [delta][mu]ds (along B0 ... B) - | [delta][mu]ds (along A0 ... A)  (5).
+    _/                                 _/
+
+  The new wave-surface is formed in such a position that the optical
+  distance is constant; and therefore the _dispersion_, or the angle
+  through which the wave-surface is turned by the change of
+  refrangibility, is found simply by dividing (5) by the distance AB.
+  If, as in common flint-glass spectroscopes, there is only one
+  dispersing substance, [int] [delta][mu] ds = [delta][mu]·s, where s is
+  simply the thickness traversed by the ray. If t2 and t1 be the
+  thicknesses traversed by the extreme rays, and a denote the width of
+  the emergent beam, the dispersion [theta] is given by
+
+    [theta] = [delta][mu](t2 - t1)/a,
+
+  or, if t1 be negligible,
+
+    [theta] = [delta][mu]t/a                 (6).
+
+  The condition of resolution of a double line whose components subtend
+  an angle [theta] is that [theta] must exceed [lambda]/a. Hence, in
+  order that a double line may be resolved whose components have indices
+  [mu] and [mu] + [delta][mu], it is necessary that t should exceed the
+  value given by the following equation:--
+
+    t = [lambda]/[delta][mu]                 (7).
+
+8. _Diffraction Gratings._--Under the heading "Colours of Striated
+Surfaces," Thomas Young (_Phil. Trans._, 1802) in his usual summary
+fashion gave a general explanation of these colours, including the law
+of sines, the striations being supposed to be straight, parallel and
+equidistant. Later, in his article "Chromatics" in the supplement to the
+5th edition of this encyclopaedia, he shows that the colours "lose the
+mixed character of periodical colours, and resemble much more the
+ordinary prismatic spectrum, with intervals completely dark interposed,"
+and explains it by the consideration that any phase-difference which may
+arise at neighbouring striae is multiplied in proportion to the total
+number of striae.
+
+The theory was further developed by A. J. Fresnel (1815), who gave a
+formula equivalent to (5) below. But it is to J. von Fraunhofer that we
+owe most of our knowledge upon this subject. His recent discovery of the
+"fixed lines" allowed a precision of observation previously impossible.
+He constructed gratings up to 340 periods to the inch by straining fine
+wire over screws. Subsequently he ruled gratings on a layer of gold-leaf
+attached to glass, or on a layer of grease similarly supported, and
+again by attacking the glass itself with a diamond point. The best
+gratings were obtained by the last method, but a suitable diamond point
+was hard to find, and to preserve. Observing through a telescope with
+light perpendicularly incident, he showed that the position of any ray
+was dependent only upon the grating interval, viz. the distance from the
+centre of one wire or line to the centre of the next, and not otherwise
+upon the thickness of the wire and the magnitude of the interspace. In
+different gratings the lengths of the spectra and their distances from
+the axis were inversely proportional to the grating interval, while with
+a given grating the distances of the various spectra from the axis were
+as 1, 2, 3, &c. To Fraunhofer we owe the first accurate measurements of
+wave-lengths, and the method of separating the overlapping spectra by a
+prism dispersing in the perpendicular direction. He described also the
+complicated patterns seen when a point of light is viewed through two
+superposed gratings, whose lines cross one another perpendicularly or
+obliquely. The above observations relate to transmitted light, but
+Fraunhofer extended his inquiry to the light _reflected_. To eliminate
+the light returned from the hinder surface of an engraved grating, he
+covered it with a black varnish. It then appeared that under certain
+angles of incidence parts of the resulting spectra were _completely
+polarized_. These remarkable researches of Fraunhofer, carried out in
+the years 1817-1823, are republished in his _Collected Writings_
+(Munich, 1888).
+
+  The principle underlying the action of gratings is identical with that
+  discussed in § 2, and exemplified in J. L. Soret's "zone plates." The
+  alternate Fresnel's zones are blocked out or otherwise modified; in
+  this way the original compensation is upset and a revival of light
+  occurs in unusual directions. If the source be a point or a line, and
+  a collimating lens be used, the incident waves may be regarded as
+  plane. If, further, on leaving the grating the light be received by a
+  focusing lens, e.g. the object-glass of a telescope, the Fresnel's
+  zones are reduced to parallel and equidistant straight strips, which
+  at certain angles coincide with the ruling. The directions of the
+  lateral spectra are such that the passage from one element of the
+  grating to the corresponding point of the next implies a retardation
+  of an integral number of wave-lengths. If the grating be composed of
+  alternate transparent and opaque parts, the question may be treated by
+  means of the general integrals (§ 3) by merely limiting the
+  integration to the transparent parts of the aperture. For an
+  investigation upon these lines the reader is referred to Airy's
+  _Tracts_, to Verdet's _Leçons_, or to R. W. Wood's _Physical Optics_.
+  If, however, we assume the theory of a simple rectangular aperture (§
+  3); the results of the ruling can be inferred by elementary methods,
+  which are perhaps more instructive.
+
+  Apart from the ruling, we know that the image of a mathematical line
+  will be a series of narrow bands, of which the central one is by far
+  the brightest. At the middle of this band there is complete agreement
+  of phase among the secondary waves. The dark lines which separate the
+  bands are the places at which the phases of the secondary wave range
+  over an integral number of periods. If now we suppose the aperture AB
+  to be covered by a great number of opaque strips or bars of width d,
+  separated by transparent intervals of width a, the condition of things
+  in the directions just spoken of is not materially changed. At the
+  central point there is still complete agreement of phase; but the
+  amplitude is diminished in the ratio of a : a + d. In another
+  direction, making a small angle with the last, such that the
+  projection of AB upon it amounts to a few wave-lengths, it is easy to
+  see that the mode of interference is the same as if there were no
+  ruling. For example, when the direction is such that the projection of
+  AB upon it amounts to one wave-length, the elementary components
+  neutralize one another, because their phases are distributed
+  symmetrically, though discontinuously, round the entire period. The
+  only effect of the ruling is to diminish the amplitude in the ratio a
+  : a + d; and, except for the difference in illumination, the
+  appearance of a line of light is the same as if the aperture were
+  perfectly free.
+
+  The lateral (spectral) images occur in such directions that the
+  projection of the element (a + d) of the grating upon them is an exact
+  multiple of [lambda]. The effect of each of the n elements of the
+  grating is then the same; and, unless this vanishes on account of a
+  particular adjustment of the ratio a : d, the resultant amplitude
+  becomes comparatively very great. These directions, in which the
+  retardation between A and B is exactly mn[lambda], may be called the
+  principal directions. On either side of any one of them the
+  illumination is distributed according to the same law as for the
+  central image (m = 0), vanishing, for example, when the retardation
+  amounts to (mn ± 1)[lambda]. In considering the relative brightnesses
+  of the different spectra, it is therefore sufficient to attend merely
+  to the principal directions, provided that the whole deviation be not
+  so great that its cosine differs considerably from unity.
+
+  We have now to consider the amplitude due to a single element, which
+  we may conveniently regard as composed of a transparent part a bounded
+  by two opaque parts of width ½d. The phase of the resultant effect is
+  by symmetry that of the component which comes from the middle of a.
+  The fact that the other components have phases differing from this by
+  amounts ranging between ± am[pi]/(a + d) causes the resultant
+  amplitude to be less than for the central image (where there is
+  complete phase agreement). If Bm denote the brightness of the m^th
+  lateral image, and B0 that of the central image, we have
+
+                _    _+ am[pi]/(a + d)       _
+               |    /                 2am[pi] |²   /a + d \²      am[pi]
+    B_m : B0 = |    |       cosx dx ÷ ------- | = ( ------ ) sin² ------ (1).
+               |_  _/                  a + d _|    \am[pi]/       a + d
+                     -am[pi]/(a + d)
+
+  If B denotes the brightness of the central image when the whole of the
+  space occupied by the grating is transparent, we have
+
+    B0 : B = a² : (a + d)²,
+
+  and thus
+
+                1         am[pi]
+    Bm : B = ------- sin² ------          (2).
+             m²[pi]²      a + d
+
+  The sine of an angle can never be greater than unity; and consequently
+  under the most favourable circumstances only 1/m²[pi]² of the original
+  light can be obtained in the m^th spectrum. We conclude that, with a
+  grating composed of transparent and opaque parts, the utmost light
+  obtainable in any one spectrum is in the first, and there amounts to
+  1/[pi]², or about 1/10, and that for this purpose a and d must be
+  equal. When d = a the general formula becomes
+
+              sin² ½m[pi]
+    Bm : B =  -----------                 (3),
+                m²[pi]²
+
+  showing that, when m is even, Bm vanishes, and that, when m is odd,
+
+    Bm : B = 1/m²[pi]².
+
+  The third spectrum has thus only 1/9 of the brilliancy of the first.
+
+  Another particular case of interest is obtained by supposing a small
+  relatively to (a + d). Unless the spectrum be of very high order, we
+  have simply
+
+    Bm : B = a/(a + d)²                   (4);
+
+  so that the brightnesses of all the spectra are the same.
+
+  The light stopped by the opaque parts of the grating, together with
+  that distributed in the central image and lateral spectra, ought to
+  make up the brightness that would be found in the central image, were
+  all the apertures transparent. Thus, if a = d, we should have
+
+        1   1     2    /    1    1      \
+    1 = - + - + ----- ( 1 + - + -- + ... ),
+        2   4   [pi]²  \    9   25      /
+
+  which is true by a known theorem. In the general case
+
+                              ___m=[oo]
+      a      /  a  \²     2   \   1       /m[pi]a\
+    ----- = ( ----- ) + -----  >  -- sin²( ------ ),
+    a + d    \a + d/    [pi]² /__ m²      \ a + d/
+                               m=1
+
+  a formula which may be verified by Fourier's theorem.
+
+  According to a general principle formulated by J. Babinet, the
+  brightness of a lateral spectrum is not affected by an interchange of
+  the transparent and opaque parts of the grating. The vibrations
+  corresponding to the two parts are precisely antagonistic, since if
+  both were operative the resultant would be zero. So far as the
+  application to gratings is concerned, the same conclusion may be
+  derived from (2).
+
+  [Illustration: FIG. 6.]
+
+  From the value of Bm : B0 we see that no lateral spectrum can surpass
+  the central image in brightness; but this result depends upon the
+  hypothesis that the ruling acts by opacity, which is generally very
+  far from being the case in practice. In an engraved glass grating
+  there is no opaque material present by which light could be absorbed,
+  and the effect depends upon a difference of retardation in passing the
+  alternate parts. It is possible to prepare gratings which give a
+  lateral spectrum brighter than the central image, and the explanation
+  is easy. For if the alternate parts were equal and alike transparent,
+  but so constituted as to give a relative retardation of ½[lambda], it
+  is evident that the central image would be entirely extinguished,
+  while the first spectrum would be four times as bright as if the
+  alternate parts were opaque. If it were possible to introduce at every
+  part of the aperture of the grating an arbitrary retardation, all the
+  light might be concentrated in any desired spectrum. By supposing the
+  retardation to vary uniformly and continuously we fall upon the case
+  of an ordinary prism: but there is then no diffraction spectrum in the
+  usual sense. To obtain such it would be necessary that the retardation
+  should gradually alter by a wave-length in passing over any element of
+  the grating, and then fall back to its previous value, thus springing
+  suddenly over a wave-length (_Phil. Mag._, 1874, 47, p. 193). It is
+  not likely that such a result will ever be fully attained in practice;
+  but the case is worth stating, in order to show that there is no
+  theoretical limit to the concentration of light of assigned
+  wave-length in one spectrum, and as illustrating the frequently
+  observed unsymmetrical character of the spectra on the two sides of
+  the central image.[4]
+
+  We have hitherto supposed that the light is incident perpendicularly
+  upon the grating; but the theory is easily extended. If the incident
+  rays make an angle [theta] with the normal (fig. 6), and the
+  diffracted rays make an angle [phi] (upon the same side), the relative
+  retardation from each element of width (a + d) to the next is (a + d)
+  (sin[theta] + sin[phi]); and this is the quantity which is to be
+  equated to m[lambda]. Thus
+
+    sin[theta] + sin[phi] = 2 sin ½([theta] + [phi]) cos ½([theta] - [phi]) = m[lambda]/(a + d) (5).
+
+  The "deviation" is ([theta] + [phi]), and is therefore a minimum when
+  [theta] = [phi], i.e. when the grating is so situated that the angles
+  of incidence and diffraction are equal.
+
+  In the case of a reflection grating the same method applies. If
+  [theta] and [phi] denote the angles with the normal made by the
+  incident and diffracted rays, the formula (5) still holds, and, if the
+  deviation be reckoned from the direction of the regularly reflected
+  rays, it is expressed as before by ([theta] + [phi]), and is a minimum
+  when [theta] = [phi], that is, when the diffracted rays return upon
+  the course of the incident rays.
+
+  [Illustration: FIG. 7.]
+
+  In either case (as also with a prism) the position of minimum
+  deviation leaves the width of the beam unaltered, i.e. neither
+  magnifies nor diminishes the angular width of the object under view.
+
+  From (5) we see that, when the light falls perpendicularly upon a
+  grating ([theta] = 0), there is no spectrum formed (the image
+  corresponding to m = 0 not being counted as a spectrum), if the
+  grating interval [sigma] or (a + d) is less than [lambda]. Under these
+  circumstances, if the material of the grating be completely
+  transparent, the whole of the light must appear in the direct image,
+  and the ruling is not perceptible. From the absence of spectra
+  Fraunhofer argued that there must be a microscopic limit represented
+  by [lambda]; and the inference is plausible, to say the least (_Phil.
+  Mag._, 1886). Fraunhofer should, however, have fixed the microscopic
+  limit at ½[lambda], as appears from (5), when we suppose [theta] =
+  ½[pi], [phi] = ½[pi].
+
+  [Illustration: FIG. 8.]
+
+  We will now consider the important subject of the resolving power of
+  gratings, as dependent upon the number of lines (n) and the order of
+  the spectrum observed (m). Let BP (fig. 8) be the direction of the
+  principal maximum (middle of central band) for the wave-length
+  [lambda] in the m^th spectrum. Then the relative retardation of the
+  extreme rays (corresponding to the edges A, B of the grating) is
+  mn[lambda]. If BQ be the direction for the first minimum (the darkness
+  between the central and first lateral band), the relative retardation
+  of the extreme rays is (mn + 1)[lambda]. Suppose now that [lambda] +
+  [delta][lambda] is the wave-length for which BQ gives the principal
+  maximum, then
+
+    (mn + 1)[lambda] = mn([lambda] + [delta][lambda]);
+
+  whence
+
+    [delta][lambda]/[lambda] = 1/mn            (6).
+
+  According to our former standard, this gives the smallest difference
+  of wave-lengths in a double line which can be just resolved; and we
+  conclude that the resolving power of a grating depends only upon the
+  total number of lines, and upon the order of the spectrum, without
+  regard to any other considerations. It is here of course assumed that
+  the n lines are really utilized.
+
+  In the case of the D lines the value of [delta][lambda]/[lambda] is
+  about 1/1000; so that to resolve this double line in the first
+  spectrum requires 1000 lines, in the second spectrum 500, and so on.
+
+  It is especially to be noticed that the resolving power does not
+  depend directly upon the closeness of the ruling. Let us take the case
+  of a grating 1 in. broad, and containing 1000 lines, and consider the
+  effect of interpolating an additional 1000 lines, so as to bisect the
+  former intervals. There will be destruction by interference of the
+  first, third and odd spectra generally; while the advantage gained in
+  the spectra of even order is not in dispersion, nor in resolving
+  power, but simply in brilliancy, which is increased four times. If we
+  now suppose half the grating cut away, so as to leave 1000 lines in
+  half an inch, the dispersion will not be altered, while the brightness
+  and resolving power are halved.
+
+  There is clearly no theoretical limit to the resolving power of
+  gratings, even in spectra of given order. But it is possible that, as
+  suggested by Rowland,[5] the structure of natural spectra may be too
+  coarse to give opportunity for resolving powers much higher than those
+  now in use. However this may be, it would always be possible, with the
+  aid of a grating of given resolving power, to construct artificially
+  from white light mixtures of slightly different wave-length whose
+  resolution or otherwise would discriminate between powers inferior and
+  superior to the given one.[6]
+
+  If we define as the "dispersion" in a particular part of the spectrum
+  the ratio of the angular interval d[theta] to the corresponding
+  increment of wave-length d[lambda], we may express it by a very simple
+  formula. For the alteration of wave-length entails, at the two limits
+  of a diffracted wave-front, a relative retardation equal to
+  mnd[lambda]. Hence, if a be the width of the diffracted beam, and
+  d[theta] the angle through which the wave-front is turned,
+
+    ad[theta] = mn d[lambda],
+
+  or  dispersion = mn/a                       (7).
+
+  The resolving power and the width of the emergent beam fix the optical
+  character of the instrument. The latter element must eventually be
+  decreased until less than the diameter of the pupil of the eye. Hence
+  a wide beam demands treatment with further apparatus (usually a
+  telescope) of high magnifying power.
+
+  In the above discussion it has been supposed that the ruling is
+  accurate, and we have seen that by increase of m a high resolving
+  power is attainable with a moderate number of lines. But this
+  procedure (apart from the question of illumination) is open to the
+  objection that it makes excessive demands upon accuracy. According to
+  the principle already laid down it can make but little difference in
+  the principal direction corresponding to the first spectrum, provided
+  each line lie within a quarter of an interval (a + d) from its
+  theoretical position. But, to obtain an equally good result in the
+  m^th spectrum, the error must be less than 1/m of the above amount.[7]
+
+  There are certain errors of a systematic character which demand
+  special consideration. The spacing is usually effected by means of a
+  screw, to each revolution of which corresponds a large number (e.g.
+  one hundred) of lines. In this way it may happen that although there
+  is almost perfect periodicity with each revolution of the screw after
+  (say) 100 lines, yet the 100 lines themselves are not equally spaced.
+  The "ghosts" thus arising were first described by G. H. Quincke
+  (_Pogg. Ann._, 1872, 146, p. 1), and have been elaborately
+  investigated by C. S. Peirce (_Ann. Journ. Math._, 1879, 2, p. 330),
+  both theoretically and experimentally. The general nature of the
+  effects to be expected in such a case may be made clear by means of an
+  illustration already employed for another purpose. Suppose two similar
+  and accurately ruled transparent gratings to be superposed in such a
+  manner that the lines are parallel. If the one set of lines exactly
+  bisect the intervals between the others, the grating interval is
+  practically halved, and the previously existing spectra of odd order
+  vanish. But a very slight relative displacement will cause the
+  apparition of the odd spectra. In this case there is approximate
+  periodicity in the half interval, but complete periodicity only after
+  the whole interval. The advantage of approximate bisection lies in the
+  superior brilliancy of the surviving spectra; but in any case the
+  compound grating may be considered to be perfect in the longer
+  interval, and the definition is as good as if the bisection were
+  accurate.
+
+  [Illustration:
+
+     | | | | |         ( ( (            | |  |   |         ) | (
+
+    FIG. 9.--x².    FIG. 10.--y².     FIG. 11.--x³.    FIG. 12.--xy².
+
+                                        /    /    /
+      \ | | /           |  \  |          |    |    |
+                                        /    /    /
+
+    FIG. 13.--xy.   FIG. 14.--x²y.    FIG. 15.--y³.]
+
+  The effect of a gradual increase in the interval (fig. 9) as we pass
+  across the grating has been investigated by M. A. Cornu (_C.R._, 1875,
+  80, p. 655), who thus explains an anomaly observed by E. E. N.
+  Mascart. The latter found that certain gratings exercised a converging
+  power upon the spectra formed upon one side, and a corresponding
+  diverging power upon the spectra on the other side. Let us suppose
+  that the light is incident perpendicularly, and that the grating
+  interval increases from the centre towards that edge which lies
+  nearest to the spectrum under observation, and decreases towards the
+  hinder edge. It is evident that the waves from _both_ halves of the
+  grating are accelerated in an increasing degree, as we pass from the
+  centre outwards, as compared with the phase they would possess were
+  the central value of the grating interval maintained throughout. The
+  irregularity of spacing has thus the effect of a convex lens, which
+  accelerates the marginal relatively to the central rays. On the other
+  side the effect is reversed. This kind of irregularity may clearly be
+  present in a degree surpassing the usual limits, without loss of
+  definition, when the telescope is focused so as to secure the best
+  effect.
+
+  It may be worth while to examine further the other variations from
+  correct ruling which correspond to the various terms expressing the
+  deviation of the wave-surface from a perfect plane. If x and y be
+  co-ordinates in the plane of the wave-surface, the axis of y being
+  parallel to the lines of the grating, and the origin corresponding to
+  the centre of the beam, we may take as an approximate equation to the
+  wave-surface
+
+          x²              y²
+    z = ------ + Bxy + ------- + [alpha]x³ + [beta]x²y + [gamma]xy² + [delta]y³ + ... (8);
+        2[rho]         2[rho]'
+
+  and, as we have just seen, the term in x² corresponds to a linear
+  error in the spacing. In like manner, the term in y² corresponds to a
+  general _curvature_ of the lines (fig. 10), and does not influence the
+  definition at the (primary) focus, although it may introduce
+  astigmatism.[8] If we suppose that everything is symmetrical on the
+  two sides of the primary plane y = 0, the coefficients B, [beta],
+  [delta] vanish. In spite of any inequality between [rho] and [rho]',
+  the definition will be good to this order of approximation, provided
+  [alpha] and [gamma] vanish. The former measures the _thickness_ of the
+  primary focal line, and the latter measures its _curvature_. The error
+  of ruling giving rise to [alpha] is one in which the intervals
+  increase or decrease in _both_ directions from the centre outwards
+  (fig. 11), and it may often be compensated by a slight rotation in
+  azimuth of the object-glass of the observing telescope. The term in
+  [gamma] corresponds to a _variation_ of curvature in crossing the
+  grating (fig. 12).
+
+  When the plane zx is not a plane of symmetry, we have to consider the
+  terms in xy, x²y, and y³. The first of these corresponds to a
+  deviation from parallelism, causing the interval to alter gradually as
+  we pass _along_ the lines (fig. 13). The error thus arising may be
+  compensated by a rotation of the object-glass about one of the
+  diameters y = ± x. The term in x²y corresponds to a deviation from
+  parallelism in the same direction on both sides of the central line
+  (fig. 14); and that in y³ would be caused by a curvature such that
+  there is a point of inflection at the middle of each line (fig. 15).
+
+  All the errors, except that depending on [alpha], and especially those
+  depending on [gamma] and [delta], can be diminished, without loss of
+  resolving power, by contracting the _vertical_ aperture. A linear
+  error in the spacing, and a general curvature of the lines, are
+  eliminated in the ordinary use of a grating.
+
+  The explanation of the difference of focus upon the two sides as due
+  to unequal spacing was verified by Cornu upon gratings purposely
+  constructed with an increasing interval. He has also shown how to rule
+  a plane surface with lines so disposed that the grating shall of
+  itself give well-focused spectra.
+
+  [Illustration: FIG. 16.]
+
+  A similar idea appears to have guided H. A. Rowland to his brilliant
+  invention of concave gratings, by which spectra can be photographed
+  without any further optical appliance. In these instruments the lines
+  are ruled upon a spherical surface of speculum metal, and mark the
+  intersections of the surface by a system of parallel and equidistant
+  planes, of which the middle member passes through the centre of the
+  sphere. If we consider for the present only the primary plane of
+  symmetry, the figure is reduced to two dimensions. Let AP (fig. 16)
+  represent the surface of the grating, O being the centre of the
+  circle. Then, if Q be any radiant point and Q' its image (primary
+  focus) in the spherical mirror AP, we have
+
+    1    1   2cos[phi]
+    -- + - = ---------,
+    v1   u       a
+
+  where v1 = AQ', u = AQ, a = OA, [phi] = angle of incidence QAO, equal
+  to the angle of reflection Q'AO. If Q be on the circle described upon
+  OA as diameter, so that u = a cos [phi], then Q' lies also upon the
+  same circle; and in this case it follows from the symmetry that the
+  unsymmetrical aberration (depending upon a) vanishes.
+
+  This disposition is adopted in Rowland's instrument; only, in addition
+  to the central image formed at the angle [phi]' = [phi], there are a
+  series of spectra with various values of [phi]', but all disposed upon
+  the same circle. Rowland's investigation is contained in the paper
+  already referred to; but the following account of the theory is in the
+  form adopted by R. T. Glazebrook (_Phil. Mag._, 1883).
+
+  In order to find the difference of optical distances between the
+  courses QAQ', QPQ', we have to express QP - QA, PQ' - AQ'. To find the
+  former, we have, if OAQ = [phi], AOP = [omega],
+
+    QP² = u² + 4a²sin²½[omega] - 4au sin ½[omega] sin (½[omega] - [phi])
+   = (u + a sin[phi] sin[omega])² - a² sin²[phi] sin²[omega] + 4a sin² ½[omega](a - u cos[phi]).
+
+  Now as far as [omega]^4
+
+    4 sin² ½[omega] = sin²[omega] + ¼sin^4[omega],
+
+  and thus to the same order
+
+      QP² = (u + a sin [phi] sin [omega])²
+    -a cos [phi](u - a cos [phi]) sin²[omega] + ¼ a(a - u cos[phi]) sin^4 [omega].
+
+  But if we now suppose that Q lies on the circle u = a cos [phi], the
+  middle term vanishes, and we get, correct as far as [omega]^4,
+
+                                       / /    a² sin²[phi] sin^4[omega]\
+    QP = (u + a sin[phi] sin[omega])  / ( 1 + ------------------------- );
+                                    \/   \              4u             /
+  so that
+
+    QP - u = a sin [phi] sin [omega] + 1/8 a sin[phi] tan[phi] sin^4 [omega] (9),
+
+  in which it is to be noticed that the adjustment necessary to secure
+  the disappearance of sin²[omega] is sufficient also to destroy the
+  term in sin³[omega].
+
+  A similar expression can be found for Q'P - Q'A; and thus, if Q'A = v,
+  Q'AO = [phi]', where v = a cos [phi]', we get
+
+    QP + PQ' - QA -AQ' = a sin[omega] (sin[phi] - sin[phi]')
+       + 1/8 a sin^4 [omega] (sin[phi] tan[phi] + sin[phi]' tan[phi]')         (10).
+
+  If [phi]' = [phi], the term of the first order vanishes, and the
+  reduction of the difference of path _via_ P and _via_ A to a term of
+  the fourth order proves not only that Q and Q' are conjugate foci, but
+  also that the foci are exempt from the most important term in the
+  aberration. In the present application [phi]' is not necessarily equal
+  to [phi]; but if P correspond to a line upon the grating, the
+  difference of retardations for consecutive positions of P, so far as
+  expressed by the term of the first order, will be equal to [-+]
+  m[lambda] (m integral), and therefore without influence, provided
+
+    [sigma] (sin[phi] - sin[phi]') = ± m[lambda]      (11),
+
+  where [sigma] denotes the constant interval between the planes
+  containing the lines. This is the ordinary formula for a reflecting
+  plane grating, and it shows that the spectra are formed in the usual
+  directions. They are here focused (so far as the rays in the primary
+  plane are concerned) upon the circle OQ'A, and the outstanding
+  aberration is of the fourth order.
+
+  In order that a large part of the field of view may be in focus at
+  once, it is desirable that the locus of the focused spectrum should be
+  nearly perpendicular to the line of vision. For this purpose Rowland
+  places the eye-piece at O, so that [phi] = 0, and then by (11) the
+  value of [phi]' in the m^th spectrum is
+
+    [sigma] sin [phi]' = ± m[lambda]                  (12).
+
+  If [omega] now relate to the edge of the grating, on which there are
+  altogether n lines,
+
+  n[sigma] = 2a sin [omega],
+
+  and the value of the last term in (10) becomes
+
+    1/16 n[sigma] sin³[omega] sin[phi]' tan[phi]',
+
+  or
+
+    1/16 mn[lambda] sin³[omega] tan [phi]'            (13).
+
+  This expresses the retardation of the extreme relatively to the
+  central ray, and is to be reckoned positive, whatever may be the signs
+  of [omega], and [phi]'. If the semi-angular aperture ([omega]) be
+  1/100, and tan [phi]' = 1, mn might be as great as four millions
+  before the error of phase would reach ¼[lambda]. If it were desired to
+  use an angular aperture so large that the aberration according to (13)
+  would be injurious, Rowland points out that on his machine there would
+  be no difficulty in applying a remedy by making [sigma] slightly
+  variable towards the edges. Or, retaining [sigma] constant, we might
+  attain compensation by so polishing the surface as to bring the
+  circumference slightly forward in comparison with the position it
+  would occupy upon a true sphere.
+
+  It may be remarked that these calculations apply to the rays in the
+  primary plane only. The image is greatly affected with astigmatism;
+  but this is of little consequence, if [gamma] in (8) be small enough.
+  Curvature of the primary focal line having a very injurious effect
+  upon definition, it may be inferred from the excellent performance of
+  these gratings that [gamma] is in fact small. Its value does not
+  appear to have been calculated. The other coefficients in (8) vanish
+  in virtue of the symmetry.
+
+  The mechanical arrangements for maintaining the focus are of great
+  simplicity. The grating at A and the eye-piece at O are rigidly
+  attached to a bar AO, whose ends rest on carriages, moving on rails
+  OQ, AQ at right angles to each other. A tie between the middle point
+  of the rod OA and Q can be used if thought desirable.
+
+  The absence of chromatic aberration gives a great advantage in the
+  comparison of overlapping spectra, which Rowland has turned to
+  excellent account in his determinations of the relative wave-lengths of
+  lines in the solar spectrum (_Phil. Mag._, 1887).
+
+  For absolute determinations of wave-lengths plane gratings are used.
+  It is found (Bell, _Phil. Mag._, 1887) that the angular measurements
+  present less difficulty than the comparison of the grating interval
+  with the standard metre. There is also some uncertainty as to the
+  actual temperature of the grating when in use. In order to minimize
+  the heating action of the light, it might be submitted to a
+  preliminary prismatic analysis before it reaches the slit of the
+  spectrometer, after the manner of Helmholtz.
+
+In spite of the many improvements introduced by Rowland and of the care
+with which his observations were made, recent workers have come to the
+conclusion that errors of unexpected amount have crept into his
+measurements of wave-lengths, and there is even a disposition to discard
+the grating altogether for fundamental work in favour of the so-called
+"interference methods," as developed by A. A. Michelson, and by C. Fabry
+and J. B. Pérot. The grating would in any case retain its utility for
+the reference of new lines to standards otherwise fixed. For such
+standards a relative accuracy of at least one part in a million seems
+now to be attainable.
+
+Since the time of Fraunhofer many skilled mechanicians have given their
+attention to the ruling of gratings. Those of Nobert were employed by A.
+J. Ångström in his celebrated researches upon wave-lengths. L. M.
+Rutherfurd introduced into common use the reflection grating, finding
+that speculum metal was less trying than glass to the diamond point,
+upon the permanence of which so much depends. In Rowland's dividing
+engine the screws were prepared by a special process devised by him, and
+the resulting gratings, plane and concave, have supplied the means for
+much of the best modern optical work. It would seem, however, that
+further improvements are not excluded.
+
+There are various copying processes by which it is possible to reproduce
+an original ruling in more or less perfection. The earliest is that of
+Quincke, who coated a glass grating with a chemical silver deposit,
+subsequently thickened with copper in an electrolytic bath. The metallic
+plate thus produced formed, when stripped from its support, a reflection
+grating reproducing many of the characteristics of the original. It is
+best to commence the electrolytic thickening in a silver acetate bath.
+At the present time excellent reproductions of Rowland's speculum
+gratings are on the market (Thorp, Ives, Wallace), prepared, after a
+suggestion of Sir David Brewster, by coating the original with a
+varnish, e.g. of celluloid. Much skill is required to secure that the
+film when stripped shall remain undeformed.
+
+A much easier method, applicable to glass originals, is that of
+photographic reproduction by contact printing. In several papers dating
+from 1872, Lord Rayleigh (see _Collected Papers_, i. 157, 160, 199, 504;
+iv. 226) has shown that success may be attained by a variety of
+processes, including bichromated gelatin and the old bitumen process,
+and has investigated the effect of imperfect approximation during the
+exposure between the prepared plate and the original. For many purposes
+the copies, containing lines up to 10,000 to the inch, are not inferior.
+It is to be desired that transparent gratings should be obtained from
+first-class ruling machines. To save the diamond point it might be
+possible to use something softer than ordinary glass as the material of
+the plate.
+
+9. _Talbot's Bands._--These very remarkable bands are seen under certain
+conditions when a tolerably pure spectrum is regarded with the naked
+eye, or with a telescope, _half the aperture being covered by a thin
+plate_, e.g. _of glass or mica_. The view of the matter taken by the
+discoverer (_Phil. Mag._, 1837, 10, p. 364) was that any ray which
+suffered in traversing the plate a retardation of an odd number of half
+wave-lengths would be extinguished, and that thus the spectrum would be
+seen interrupted by a number of dark bars. But this explanation cannot
+be accepted as it stands, being open to the same objection as Arago's
+theory of stellar scintillation.[9] It is as far as possible from being
+true that a body emitting homogeneous light would disappear on merely
+covering half the aperture of vision with a half-wave plate. Such a
+conclusion would be in the face of the principle of energy, which
+teaches plainly that the retardation in question leaves the aggregate
+brightness unaltered. The actual formation of the bands comes about in
+a very curious way, as is shown by a circumstance first observed by
+Brewster. When the retarding plate is held on the side towards the red
+of the spectrum, _the bands are not seen_. Even in the contrary case,
+the thickness of the plate must not exceed a certain limit, dependent
+upon the purity of the spectrum. A satisfactory explanation of these
+bands was first given by Airy (_Phil. Trans._, 1840, 225; 1841, 1), but
+we shall here follow the investigation of Sir G. G. Stokes (_Phil.
+Trans._, 1848, 227), limiting ourselves, however, to the case where the
+retarded and unretarded beams are contiguous and of equal width.
+
+  The aperture of the unretarded beam may thus be taken to be limited by
+  x = -h, x = 0, y = -l, y= +l; and that of the beam retarded by R to be
+  given by x = 0, x = h, y= -l, y = +l. For the former (1) § 3 gives
+                   _     _
+          1       / 0   / +l       /        x[xi] + y[eta]\
+    - ---------   |     |   sin k (at - f + -------------- )dxdy
+      [lambda]f  _/-h  _/-l        \              f       /
+
+           2lh         f        k[eta]l     2f       k[xi]h          /       [xi]h \
+    = - --------- · ------- sin ------- · ------ sin ------ · sin k (at - f - ----- ) (1),
+        [lambda]f   k[eta]l        f      k[xi]h       2f            \         2f  /
+
+  on integration and reduction.
+
+  For the retarded stream the only difference is that we must subtract R
+  from at, and that the limits of x are 0 and +h. We thus get for the
+  disturbance at [xi], [eta], due to this stream
+
+         2lh         f        k[eta]l     2f       k[xi]h          /           [xi]h \
+    - --------- · ------- sin ------- · ------ sin ------ . sin k (at - f - R + ----- ) (2).
+      [lambda]f   k[eta]l        f      k[xi]h       2f            \            2f   /
+
+  If we put for shortness [pi] for the quantity under the last circular
+  function in (1), the expressions (1), (2) may be put under the forms u
+  sin [tau], v sin ([tau] - [alpha]) respectively; and, if I be the
+  intensity, I will be measured by the sum of the squares of the
+  coefficients of sin [tau] and cos [tau] in the expression
+
+    u sin[tau] + v sin([tau] - [alpha]),
+
+  so that
+
+    I = u² + v² + 2uv cos[alpha],
+
+  which becomes on putting for u, v, and [alpha] their values, and
+  putting
+
+     /   f        k[eta]l \²
+    ( ------- sin -------  )  = Q           (3),
+     \k[eta]l        f    /
+                                       _                                  _
+                4l²         [pi][xi]h |           / 2[pi]R    2[pi][xi]h\  |
+    I = Q · ---------- sin² --------- |2 + 2 cos ( -------- - ---------- ) | (4).
+            [pi]²[xi]²      [lambda]f |_          \[lambda]   [lambda]f / _|
+
+  If the subject of examination be a luminous line parallel to [eta], we
+  shall obtain what we require by integrating (4) with respect to [eta]
+  from -[oo] to +[oo]. The constant multiplier is of no especial
+  interest so that we may take as applicable to the image of a line
+                              _                                 _
+          2        [pi][xi]h |         / 2[pi]R    2[pi][xi]h \  |
+    I = ----- sin² --------- |1 + cos ( -------- - ----------  ) | (5).
+        [xi]²      [lambda]f |_        \[lambda]   [lambda]f  / _|
+
+  If R = ½[lambda], I vanishes at [xi]= 0; but the whole illumination,
+  represented by
+      _
+     / +[oo]
+     |    I d[xi], is independent of the value of R. If R = 0,
+    _/-[oo]
+
+          1        2[pi][xi]h
+    I = ----- sin² ----------,
+        [xi]²      [lambda]f
+
+  in agreement with § 3, where a has the meaning here attached to 2h.
+
+  The expression (5) gives the illumination at [xi] due to that part of
+  the complete image whose geometrical focus is at [xi] = 0, the
+  retardation for this component being R. Since we have now to integrate
+  for the whole illumination at a particular point O due to all the
+  components which have their foci in its neighbourhood, we may
+  conveniently regard O as origin. [xi] is then the co-ordinate
+  relatively to O of any focal point O' for which the retardation is R;
+  and the required result is obtained by simply integrating (5) with
+  respect to [xi] from -[oo] to +[oo]. To each value of [xi] corresponds
+  a different value of [lambda], and (in consequence of the dispersing
+  power of the plate) of R. The variation of [lambda] may, however, be
+  neglected in the integration, except in 2[pi]R/[lambda], where a small
+  variation of [lambda] entails a comparatively large alteration of
+  phase. If we write
+
+    [rho] = 2[pi]R/[lambda]                                                     (6),
+
+  we must regard [rho] as a function of [xi], and we may take with
+  sufficient approximation under any ordinary circumstances
+
+    [rho] = [rho]' + [=omega][xi]                                               (7),
+
+  where [rho]' denotes the value of [rho] at O, and [=omega] is a
+  constant, which is positive when the retarding plate is held at the
+  side on which the lue of the spectrum _is seen_. The possibility of
+  dark bands depends upon [=omega] being positive. Only in this case can
+
+    cos {[rho]' + ([=omega] - 2[pi]h/[lambda]f)[xi]}
+
+  retain the constant value -1 throughout the integration, and then only
+  when
+
+    [=omega] = 2[pi]h / [lambda]f                                               (8)
+
+  and
+
+    cos [rho]' = -1                                                             (9).
+
+  The first of these equations is the condition for the formation of
+  dark bands, and the second marks their situation, which is the same
+  as that determined by the imperfect theory.
+
+  The integration can be effected without much difficulty. For the first
+  term in (5) the evaluation is effected at once by a known formula. In
+  the second term if we observe that
+
+    cos {[rho]' +([=omega] - 2[pi]h/[lambda]f)[xi]} = cos {[rho]'- g1[xi]}
+      = cos [rho]' cos g1[xi] + sin [rho]' sin g1[xi],
+
+  we see that the second part vanishes when integrated, and that the
+  remaining integral is of the form
+
+          _+[oo]
+         /                        d[xi]
+    w =  | sin² h1[xi] cos g1[xi] -----,
+        _/-[oo]                   [xi]²
+
+  where
+
+    h1 = [pi]h/[lambda]f,  g1 = [omega] - 2[pi]h/[lambda]f    (10).
+
+  By differentiation with respect to g1 it may be proved that
+
+    w = 0                  from g1 = -[oo] to g1 = -2h1,
+    w = ½[pi](2h1 + g1)    from g1 = -2h1  to g1 = 0,
+    w = ½[pi](2h1 - g1)    from g1 =  0    to g1 = 2h1,
+    w = 0                  from g1 = 2h1   to g1 = [oo].
+
+  The integrated intensity, I', or
+
+    2[pi]h1 + 2 cos[rho]w,
+
+  is thus
+
+    I' = 2[pi]h1              (11),
+
+  when g1 numerically exceeds 2h1; and, when g1 lies between ±2h1,
+
+    I = [pi]2h1 + (2h1 - [sqrt] g1²) cos[rho]'     (12).
+
+  It appears therefore that there are no bands at all unless [omega]
+  lies between 0 and +4h1, and that within these limits the best bands
+  are formed at the middle of the range when [omega] = 2h1. The
+  formation of bands thus requires that the retarding plate be held upon
+  the side already specified, so that [omega] be positive; and that the
+  thickness of the plate (to which [omega] is proportional) do not
+  exceed a certain limit, which we may call 2T0. At the best thickness
+  T0 the bands are black, and not otherwise.
+
+  The linear width of the band (e) is the increment of [xi] which alters
+  [rho] by 2[pi], so that
+
+    e = 2[pi]/[=omega]                             (13).
+
+  With the best thickness
+
+    [=omega] = 2[pi]h/[lambda]f                    (14),
+
+  so that in this case
+
+    e = [lambda]f/h                                (15).
+
+  The bands are thus of the same width as those due to two infinitely
+  narrow apertures coincident with the central lines of the retarded and
+  unretarded streams, the subject of examination being itself a fine
+  luminous line.
+
+  If it be desired to see a given number of bands in the whole or in any
+  part of the spectrum, the thickness of the retarding plate is thereby
+  determined, independently of all other considerations. But in order
+  that the bands may be really visible, and still more in order that
+  they may be black, another condition must be satisfied. It is
+  necessary that the aperture of the pupil be accommodated to the
+  angular extent of the spectrum, or reciprocally. Black bands will be
+  too fine to be well seen unless the aperture (2h) of the pupil be
+  somewhat contracted. One-twentieth to one-fiftieth of an inch is
+  suitable. The aperture and the number of bands being both fixed, the
+  condition of blackness determines the angular magnitude of a band and
+  of the spectrum. The use of a grating is very convenient, for not only
+  are there several spectra in view at the same time, but the dispersion
+  can be varied continuously by sloping the grating. The slits may be
+  cut out of tin-plate, and half covered by mica or "microscopic glass,"
+  held in position by a little cement.
+
+  If a telescope be employed there is a distinction to be observed,
+  according as the half-covered aperture is between the eye and the
+  ocular, or in front of the object-glass. In the former case the
+  function of the telescope is simply to increase the dispersion, and
+  the formation of the bands is of course independent of the particular
+  manner in which the dispersion arises. If, however, the half-covered
+  aperture be in front of the object-glass, the phenomenon is magnified
+  as a whole, and the desirable relation between the (unmagnified)
+  dispersion and the aperture is the same as without the telescope.
+  There appears to be no further advantage in the use of a telescope
+  than the increased facility of accommodation, and for this of course a
+  very low power suffices.
+
+  The original investigation of Stokes, here briefly sketched, extends
+  also to the case where the streams are of unequal width h, k, and are
+  separated by an interval 2g. In the case of unequal width the bands
+  cannot be black; but if h = k, the finiteness of 2g does not preclude
+  the formation of black bands.
+
+  The theory of Talbot's bands with a half-covered _circular_ aperture
+  has been considered by H. Struve (_St Peters. Trans._, 1883, 31, No.
+  1).
+
+  The subject of "Talbot's bands" has been treated in a very instructive
+  manner by A. Schuster (_Phil. Mag._, 1904), whose point of view offers
+  the great advantage of affording an instantaneous explanation of the
+  peculiarity noticed by Brewster. A plane _pulse_, i.e. a disturbance
+  limited to an infinitely thin slice of the medium, is supposed to fall
+  upon a parallel grating, which again may be regarded as formed of
+  infinitely thin wires, or infinitely narrow lines traced upon glass.
+  The secondary pulses diverted by the ruling fall upon an object-glass
+  as usual, and on arrival at the focus constitute a procession equally
+  spaced in time, the interval between consecutive members depending
+  upon the obliquity. If a retarding plate be now inserted so as to
+  operate upon the pulses which come from one side of the grating, while
+  leaving the remainder unaffected, we have to consider what happens at
+  the focal point chosen. A full discussion would call for the formal
+  application of Fourier's theorem, but some conclusions of importance
+  are almost obvious.
+
+  Previously to the introduction of the plate we have an effect
+  corresponding to wave-lengths closely grouped around the principal
+  wave-length, viz. [sigma] sin [phi], where [sigma] is the
+  grating-interval and [phi] the obliquity, the closeness of the
+  grouping increasing with the number of intervals. In addition to these
+  wave-lengths there are other groups centred round the wave-lengths
+  which are submultiples of the principal one--the overlapping spectra
+  of the second and higher orders. Suppose now that the plate is
+  introduced so as to cover naif the aperture and that it retards those
+  pulses which would otherwise arrive first. The consequences must
+  depend upon the amount of the retardation. As this increases from
+  zero, the two processions which correspond to the two halves of the
+  aperture begin to overlap, and the overlapping gradually increases
+  until there is almost complete superposition. The stage upon which we
+  will fix our attention is that where the one procession bisects the
+  intervals between the other, so that a new simple procession is
+  constituted, containing the same number of members as before the
+  insertion of the plate, but now spaced at intervals only half as
+  great. It is evident that the effect at the focal point is the
+  obliteration of the first and other spectra of odd order, so that as
+  regards the spectrum of the first order we may consider that the two
+  beams _interfere_. The formation of black bands is thus explained, and
+  it requires that the plate be introduced upon one particular side, and
+  that the amount of the retardation be adjusted to a particular value.
+  If the retardation be too little, the overlapping of the processions
+  is incomplete, so that besides the procession of half period there are
+  residues of the original processions of full period. The same thing
+  occurs if the retardation be too great. If it exceed the double of the
+  value necessary for black bands, there is again no overlapping and
+  consequently no interference. If the plate be introduced upon the
+  other side, so as to retard the procession originally in arrear, there
+  is no overlapping, whatever may be the amount of retardation. In this
+  way the principal features of the phenomenon are accounted for, and
+  Schuster has shown further how to extend the results to spectra having
+  their origin in prisms instead of gratings.
+
+10. _Diffraction when the Source of Light is not seen in Focus._--The
+phenomena to be considered under this head are of less importance than
+those investigated by Fraunhofer, and will be treated in less detail;
+but in view of their historical interest and of the ease with which many
+of the experiments may be tried, some account of their theory cannot be
+omitted. One or two examples have already attracted our attention when
+considering Fresnel's zones, viz. the shadow of a circular disk and of a
+screen circularly perforated.
+
+Fresnel commenced his researches with an examination of the fringes,
+external and internal, which accompany the shadow of a narrow opaque
+strip, such as a wire. As a source of light he used sunshine passing
+through a very small hole perforated in a metal plate, or condensed by a
+lens of short focus. In the absence of a heliostat the latter was the
+more convenient. Following, unknown to himself, in the footsteps of
+Young, he deduced the principle of interference from the circumstance
+that the darkness of the interior bands requires the co-operation of
+light from both sides of the obstacle. At first, too, he followed Young
+in the view that the exterior bands are the result of interference
+between the direct light and that reflected from the edge of the
+obstacle, but he soon discovered that the character of the edge--e.g.
+whether it was the cutting edge or the back of a razor--made no material
+difference, and was thus led to the conclusion that the explanation of
+these phenomena requires nothing more than the application of Huygens's
+principle to the unobstructed parts of the wave. In observing the bands
+he received them at first upon a screen of finely ground glass, upon
+which a magnifying lens was focused; but it soon appeared that the
+ground glass could be dispensed with, the diffraction pattern being
+viewed in the same way as the image formed by the object-glass of a
+telescope is viewed through the eye-piece. This simplification was
+attended by a great saving of light, allowing measures to be taken such
+as would otherwise have presented great difficulties.
+
+  In theoretical investigations these problems are usually treated as of
+  two dimensions only, everything being referred to the plane passing
+  through the luminous point and perpendicular to the diffracting edges,
+  supposed to be straight and parallel. In strictness this idea is
+  appropriate only when the source is a luminous line, emitting
+  cylindrical waves, such as might be obtained from a luminous point
+  with the aid of a cylindrical lens. When, in order to apply Huygens's
+  principle, the wave is supposed to be broken up, the phase is the same
+  at every element of the surface of resolution which lies upon a line
+  perpendicular to the plane of reference, and thus the effect of the
+  whole line, or rather infinitesimal strip, is related in a constant
+  manner to that of the element which lies in the plane of reference,
+  and may be considered to be represented thereby. The same method of
+  representation is applicable to spherical waves, issuing from a
+  _point_, if the radius of curvature be large; for, although there is
+  variation of phase along the length of the infinitesimal strip, the
+  whole effect depends practically upon that of the central parts where
+  the phase is sensibly constant.[10]
+
+  [Illustration: FIG. 17.]
+
+  In fig. 17 APQ is the arc of the circle representative of the
+  wave-front of resolution, the centre being at O, and the radius QA
+  being equal to a. B is the point at which the effect is required,
+  distant a + b from O, so that AB = b, AP = s, PQ = ds.
+
+  Taking as the standard phase that of the secondary wave from A, we may
+  represent the effect of PQ by
+
+               /t   [delta] \
+    cos 2[pi] ( - - -------- )·ds,
+               \r   [lambda]/
+
+  where [delta] = BP - AP is the retardation at B of the wave from P
+  relatively to that from A.
+
+  Now
+
+    [delta] = (a + b) s²/2ab                    (1),
+
+  so that, if we write
+
+    2[pi][delta] =  [pi](a + b)s²     [pi]v²
+    ------------   ---------------  = ------    (2),
+      [lambda]       ab[lambda]         2
+
+  the effect at B is
+                                 _                             _
+     /ab[lambda]\½  /    2[pi]t /                      2[pi]t /               \
+    ( ---------- ) ( cos ------ | cos ½[pi]v²·dv + sin ------ | sin ½[pi]v²·dv ) (3),
+     \2(a + b)  /   \    [tau] _/                      [tau] _/               /
+
+  the limits of integration depending upon the disposition of the
+  diffracting edges. When a, b, [lambda] are regarded as constant, the
+  first factor may be omitted,--as indeed should be done for
+  consistency's sake, inasmuch as other factors of the same nature have
+  been omitted already.
+
+  The intensity I², the quantity with which we are principally
+  concerned, may thus be expressed
+
+             _                        _
+         /  /               \²    /  /               \²
+    I²= (   | cos ½[pi]v²·dv ) + (   | sin ½[pi]v²·dv ) (4).
+         \ _/               /     \ _/               /
+
+  These integrals, taken from v = 0, are known as Fresnel's integrals;
+  we will denote them by C and S, so that
+         _                       _
+        / v                     / v
+    C = | cos ½[pi]v²·dv,   S = | cos ½[pi]v²·dv               (5).
+       _/0                     _/0
+
+  When the upper limit is infinity, so that the limits correspond to the
+  inclusion of half the primary wave, C and S are both equal to ½, by a
+  known formula; and on account of the rapid fluctuation of sign the
+  parts of the range beyond very moderate values of v contribute but
+  little to the result.
+
+  Ascending series for C and S were given by K. W. Knockenhauer, and are
+  readily investigated. Integrating by parts, we find
+
+              _v                                        _v
+             /   i·½[pi]v²      i·½[pi]v²      1       /   i·½[pi]v²
+    C + iS = |  e         dv = e         · v - - i[pi] |  e         dv³;
+            _/0                                3      _/0
+
+  and, by continuing this process,
+
+              i.½[pi]v² /    i[pi]      i[pi] i[pi]       i[pi] i[pi] i[pi]          \
+    C + iS = e         ( v - ----- v³ + ----- ----- v^5 - ----- ----- ----- v^7 + ... ).
+                        \      3          3     5           3     5     7            /
+
+  By separation of real and imaginary parts,
+
+    C = M cos ½[pi]v² - N sin ½[pi]v²  \
+    S = M sin ½[pi]v² - N cos ½[pi]v²  /          (6)
+
+  where
+
+        v   [pi]²v^5    [pi]^4v^9
+    M = - - --------- + --------- - ...           (7)
+        1     3·5        3·5·7·9
+
+        [pi]v³   [pi]^3v^7    [pi]^5v^11
+    N = ------ - --------- + ------------ ...     (8)
+         1·3      1·3·5·7    1·3·5·7·9·11
+
+  These series are convergent for all values of v, but are practically
+  useful only when v is small.
+
+  Expressions suitable for discussion when v is large were obtained by
+  L. P. Gilbert (_Mem. cour. de l'Acad. de Bruxelles_, 31, p. 1). Taking
+
+    ½[pi]v² = u                                   (9),
+
+  we may write
+                                _
+                    1          /u  e^iu du
+    C + iS =  -------------    |   --------      (10).
+              [sqrt](2[pi])   _/0  [sqrt] u
+
+Again, by a known formula,
+
+                             _[oo]
+        1           1       /  e^-ux dx
+    --------  = ----------  |  --------         (11).
+    [sqrt] u    [sqrt][pi] _/0  [sqrt]x
+
+  Substituting this in (10), and inverting the order of integration, we
+  get
+
+                       _[oo]         _u
+                1     /     dx      /  e^u(i - x)
+    C + iS = -------  |  --------   |  ----------- dx
+             [sqrt]2 _/0 [sqrt] x  _/0   [sqrt]x
+
+                   _[oo]
+           1     /     dx    e^u(i - x) - 1
+      = -------  |  -------- -------------- dx     (12).
+        [sqrt]2 _/0 [sqrt] x     i - x
+
+  Thus, if we take
+
+                      _[oo]
+             1       /  e^-ux [sqrt](x)·dx
+    G = -----------  |  ------------------,
+        [pi][sqrt]2 _/0       1 + x²
+
+                      _[oo]
+             1       /      e^-ux dx
+    H = -----------  | ------------------         (13).
+        [pi][sqrt]2 _/ [sqrt]x · (1 + x²)
+                     0
+
+    C = ½ - G cos u + H sin u,  S = ½ - G sin u - H cos u   (14).
+
+  The constant parts in (14), viz. ½, may be determined by direct
+  integration of (12), or from the observation that by their
+  constitution G and H vanish when u = [oo], coupled with the fact that
+  C and S then assume the value ½.
+
+  Comparing the expressions for C, S in terms of M, N, and in terms of
+  G, H, we find that
+
+    G = ½ (cos u + sin u) - M,  H = ½ (cos u - sin u) + N  (15),
+
+  formulae which may be utilized for the calculation of G, H when u (or
+  v) is small. For example, when u = 0, M = 0, N = 0, and consequently G
+  = H = ½.
+
+  Descending series of the semi-convergent class, available for
+  numerical calculation when u is moderately large, can be obtained from
+  (12) by writing x = uy, and expanding the denominator in powers of y.
+  The integration of the several terms may then be effected by the
+  formula
+
+      _ [oo]
+     /  -y   q-½
+     | e    y    dy = [Gamma](q + ½) = (q - ½)(q - 3/2) ... ½[sqrt][pi];
+    _/0
+
+  and we get in terms of v
+
+           1         1·3·5       1·3·5·9
+    G = ------- - ---------- + ----------- -          (16),
+        [pi]²v³   [pi]^4 v^7   [pi]^6 v^11
+
+          1        1·3        1·3·5·7
+    H = ----- - --------- + ---------- -              (17).
+        [pi]v   [pi]³ v^5   [pi]^5 v^9
+
+  The corresponding values of C and S were originally derived by A. L.
+  Cauchy, without the use of Gilbert's integrals, by direct integration
+  by parts.
+
+  From the series for G and H just obtained it is easy to verify that
+
+    dH             dG
+    -- = - [pi]vG, -- = [pi]vH - 1                    (18).
+    dv             dv
+
+  We now proceed to consider more particularly the distribution of light
+  upon a screen PBQ near the shadow of a straight edge A. At a point P
+  within the geometrical shadow of the obstacle, the half of the wave to
+  the right of C (fig. 18), the nearest point on the wave-front, is
+  wholly intercepted, and on the left the integration is to be taken
+  from s = CA to s = [oo]. If V be the value of v corresponding to CA,
+  viz.
+
+         / /  2(a + b)  \
+    V=  / (  ----------  )·CA,                       (19),
+      \/   \ ab[lambda] /
+
+  we may write
+
+              _[oo]                    _[oo]
+          /  /               \²    /  /               \²
+    I² = (   | cos ½[pi]v²·dv ) + (   | sin ½[pi]v²·dv ) (20),
+          \ _/v              /     \ _/v              /
+
+  or, according to our previous notation,
+
+    I² = (½ - Cv)² + (½ - Sv)² = G² + H²             (21).
+
+  Now in the integrals represented by G and H every element diminishes
+  as V increases from zero. Hence, as CA increases, viz. as the point P
+  is more and more deeply immersed in the shadow, the illumination
+  _continuously_ decreases, and that without limit. It has long been
+  known from observation that there are no bands on the interior side of
+  the shadow of the edge.
+
+  [Illustration: FIG. 18.]
+
+  The law of diminution when V is moderately large is easily expressed
+  with the aid of the series (16), (17) for G, H. We have ultimately G =
+  0, H = ([pi]V)^-1, so that
+
+    I² = 1/[pi]²V²,
+
+  or the illumination is inversely as the square of the distance from
+  the shadow of the edge.
+
+  For a point Q outside the shadow the integration extends over _more_
+  than half the primary wave. The intensity may be expressed by
+
+    I² = (½ + Cv)² + (½ + Sv)²                   (22);
+
+  and the maxima and minima occur when
+
+              dC             dS
+    (½ + C_v) -- + (½ + S_v) -- = 0,
+              dV             dV
+
+  whence
+
+    sin ½[pi]V² + cos ½[pi]V² = G               (23).
+
+  When V = 0, viz. at the edge of the shadow, I² = ½; when V = [oo], I²
+  = 2, on the scale adopted. The latter is the intensity due to the
+  uninterrupted wave. The quadrupling of the intensity in passing
+  outwards from the edge of the shadow is, however, accompanied by
+  fluctuations giving rise to bright and dark bands. The position of
+  these bands determined by (23) may be very simply expressed when V is
+  large, for then sensibly G = 0, and
+
+    ½[pi]V² = ¾[pi] + n[pi]                     (24),
+
+  n being an integer. In terms of [delta], we have from (2)
+
+    [delta] = (3/8 + ½n)[lambda]                (25).
+
+  The first maximum in fact occurs when [delta] = 3/8[lambda]
+  -.0046[lambda], and the first minimum when [delta] = 7/8[lambda]
+  -.0016[lambda], the corrections being readily obtainable from a table
+  of G by substitution of the approximate value of V.
+
+  The position of Q corresponding to a given value of V, that is, to a
+  band of given order, is by (19)
+
+         a + b          / / b[lambda](a + b) \
+    BQ = ----- AD = V  / (  ----------------- )                               (26).
+           a         \/   \        2a        /
+
+  By means of this expression we may trace the locus of a band of given
+  order as b varies. With sufficient approximation we may regard BQ and
+  b as rectangular co-ordinates of Q. Denoting them by x, y, so that AB
+  is axis of y and a perpendicular through A the axis of x, and
+  rationalizing (26), we have
+
+    2ax² - V²[lambda]y² - V²a[lambda]y = 0,
+
+  which represents a hyperbola with vertices at O and A.
+
+  From (24), (26) we see that the width of the bands is of the order
+  [sqrt] {b[lambda](a + b)/a}. From this we may infer the limitation
+  upon the width of the source of light, in order that the bands may be
+  properly formed. If [omega] be the apparent magnitude of the source
+  seen from A, [omega]b should be much smaller than the above quantity,
+  or
+
+    [omega] < [sqrt] {[lambda](a + b)/ab}        (27).
+
+  If a be very great in relation to b, the condition becomes
+
+    [omega] < [sqrt] ([lambda]/b)                (28).
+
+  so that if b is to be moderately great (1 metre), the apparent
+  magnitude of the sun must be greatly reduced before it can be used as
+  a source. The values of V for the maxima and minima of intensity, and
+  the magnitudes of the latter, were calculated by Fresnel. An extract
+  from his results is given in the accompanying table.
+
+    +--------------------+----------+------------+
+    |                    |     V    |     I²     |
+    +--------------------+----------+------------+
+    | First maximum      |  1.2172  |   2.7413   |
+    | First minimum      |  1.8726  |   1.5570   |
+    | Second maximum     |  2.3449  |   2.3990   |
+    | Second minimum     |  2.7392  |   1.6867   |
+    | Third maximum.     |  3.0820  |   2.3022   |
+    | Third minimum      |  3.3913  |   1.7440   |
+    +--------------------+----------+------------+
+
+  A very thorough investigation of this and other related questions,
+  accompanied by fully worked-out tables of the functions concerned,
+  will be found in a paper by E. Lommel (_Abh. bayer. Akad. d. Wiss._
+  II. CI., 15, Bd., iii. Abth., 1886).
+
+  When the functions C and S have once been calculated, the discussion
+  of various diffraction problems is much facilitated by the idea, due
+  to M. A. Cornu (_Journ. de Phys._, 1874, 3, p. 1; a similar suggestion
+  was made independently by G. F. Fitzgerald), of exhibiting as a curve
+  the relationship between C and S, considered as the rectangular
+  co-ordinates (x, y) of a point. Such a curve is shown in fig. 19,
+  where, according to the definition (5) of C, S,
+
+          _ v                        _ v
+         /                          /
+    x =  | cos ½[pi]v²·dv,      y = | sin ½[pi]v²·dv   (29).
+        _/0                        _/0
+
+  The origin of co-ordinates O corresponds to v = 0; and the asymptotic
+  points J, J', round which the curve revolves in an ever-closing
+  spiral, correspond to v = ±[oo].
+
+  The intrinsic equation, expressing the relation between the arc
+  [sigma] (measured from O) and the inclination [phi] of the tangent at
+  any points to the axis of x, assumes a very simple form. For
+
+    dx = cos ½[pi]v²·dv,   dy = sin ½[pi]v²·dv;
+
+  so that
+               _
+              /
+    [sigma] = | [sqrt] (dx² + dy²) = v,                (30),
+             _/
+
+    [phi] = tan^-1 (dy/dx) = ½[pi]v²                   (31).
+
+
+
+  Accordingly,
+
+    [phi] = ½[pi][sigma]²                              (32);
+
+  and for the curvature,
+
+    d[phi]/d[sigma] = [pi][sigma]                      (33).
+
+  Cornu remarks that this equation suffices to determine the general
+  character of the curve. For the osculating circle at any point
+  includes the whole of the curve which lies beyond; and the successive
+  convolutions envelop one another without intersection.
+
+  [Illustration: Fig. 19.]
+
+  The utility of the curve depends upon the fact that the elements of
+  arc represent, in amplitude and phase, the component vibrations due to
+  the corresponding portions of the primary wave-front. For by (30)
+  d[sigma] = dv, and by (2) dv is proportional to ds. Moreover by (2)
+  and (31) the retardation of phase of the elementary vibration from PQ
+  (fig. 17) is 2[pi][delta]/[lambda], or [phi]. Hence, in accordance
+  with the rule for compounding vector quantities, the resultant
+  vibration at B, due to any finite part of the primary wave, is
+  represented in amplitude and phase by the chord joining the
+  extremities of the corresponding arc ([sigma]2 - [sigma]1).
+
+  In applying the curve in special cases of diffraction to exhibit the
+  effect at any point P (fig. 18) the centre of the curve O is to be
+  considered to correspond to that point C of the primary wave-front
+  which lies nearest to P. The operative part, or parts, of the curve
+  are of course those which represent the unobstructed portions of the
+  primary wave.
+
+  Let us reconsider, following Cornu, the diffraction of a screen
+  unlimited on one side, and on the other terminated by a straight edge.
+  On the illuminated side, at a distance from the shadow, the vibration
+  is represented by JJ'. The co-ordinates oí J, J' being (½, ½), (-½,
+  -½), I² is 2; and the phase is 1/8 period in arrear of that of the
+  element at O. As the point under contemplation is supposed to approach
+  the shadow, the vibration is represented by the chord drawn from J to
+  a point on the other half of the curve, which travels inwards from J'
+  towards O. The amplitude is thus subject to fluctuations, which
+  increase as the shadow is approached. At the point O the intensity is
+  one-quarter of that of the entire wave, and after this point is
+  passed, that is, when we have entered the geometrical shadow, the
+  intensity falls off gradually to zero, _without fluctuations_. The
+  whole progress of the phenomenon is thus exhibited to the eye in a
+  very instructive manner.
+
+  We will next suppose that the light is transmitted by a slit, and
+  inquire what is the effect of varying the width of the slit upon the
+  illumination at the projection of its centre. Under these
+  circumstances the arc to be considered is bisected at O, and its
+  length is proportional to the width of the slit. It is easy to see
+  that the length of the chord (which passes in all cases through O)
+  increases to a maximum near the place where the phase-retardation is
+  3/8 of a period, then diminishes to a minimum when the retardation is
+  about 7/8 of a period, and so on.
+
+  If the slit is of constant width and we require the illumination at
+  various points on the screen behind it, we must regard the arc of the
+  curve as of _constant length_. The intensity is then, as always,
+  represented by the square of the length of the chord. If the slit be
+  narrow, so that the arc is short, the intensity is constant over a
+  wide range, and does not fall off to an important extent until the
+  discrepancy of the extreme phases reaches about a quarter of a period.
+
+  We have hitherto supposed that the shadow of a diffracting obstacle is
+  received upon a diffusing screen, or, which comes to nearly the same
+  thing, is observed with an eye-piece. If the eye, provided if
+  necessary with a perforated plate in order to reduce the aperture, be
+  situated inside the shadow at a place where the illumination is still
+  sensible, and be focused upon the diffracting edge, the light which it
+  receives will appear to come from the neighbourhood of the edge, and
+  will present the effect of a silver lining. This is doubtless the
+  explanation of a "pretty optical phenomenon, seen in Switzerland, when
+  the sun rises from behind distant trees standing on the summit of a
+  mountain."[11]
+
+II. _Dynamical Theory of Diffraction._--The explanation of diffraction
+phenomena given by Fresnel and his followers is independent of special
+views as to the nature of the aether, at least in its main features; for
+in the absence of a more complete foundation it is impossible to treat
+rigorously the mode of action of a solid obstacle such as a screen. But,
+without entering upon matters of this kind, we may inquire in what
+manner a primary wave may be resolved into elementary secondary waves,
+and in particular as to the law of intensity and polarization in a
+secondary wave as dependent upon its direction of propagation, and upon
+the character as regards polarization of the primary wave. This question
+was treated by Stokes in his "Dynamical Theory of Diffraction" (_Camb.
+Phil. Trans._, 1849) on the basis of the elastic solid theory.
+
+  Let x, y, z be the co-ordinates of any particle of the medium in its
+  natural state, and [chi], [eta], [zeta] the displacements of the same
+  particle at the end of time t, measured in the directions of the three
+  axes respectively. Then the first of the equations of motion may be
+  put under the form
+
+    d²[xi]      /d²[xi]   d²[xi]   d²[xi]\             d² /d²[xi]   d²[eta]   d²[zeta]\
+    ------ = b²( ------ + ------ + ------ ) + (a² - b²)--( ------ + ------- + -------- ),
+     dt²        \ dx²      dy²      dz²  /             dx \ dx²       dy²        dz²  /
+
+  where a2 and b2 denote the two arbitrary constants. Put for shortness
+
+    d²[xi]   d²[eta]   d²[zeta]
+    ------ + ------- + -------- = [delta]    (1),
+     dx²       dy²        dz²
+
+  and represent by [Delta]²[chi] the quantity multiplied by b².
+  According to this notation, the three equations of motion are
+
+    d²[xi]                                  d[delta] \
+    ------   = b²[Delta]²[xi]   + (a² - b²) -------- |
+      dt²                                      dx    |
+                                                     |
+    d²[eta]                                 d[delta] |
+    -------  = b²[Delta]²[eta]  + (a² - b²) --------  >   (2).
+      dt²                                      dy    |
+                                                     |
+    d²[zeta]                                d[delta] |
+    -------- = b²[Delta]²[zeta] + (a² - b²) -------- |
+      dt²                                      dz    /
+
+  It is to be observed that S denotes the dilatation of volume of the
+  element situated at (x, y, z). In the limiting case in which the
+  medium is regarded as absolutely incompressible [delta] vanishes; but,
+  in order that equations (2) may preserve their generality, we must
+  suppose a at the same time to become infinite, and replace a²[delta]
+  by a new function of the co-ordinates.
+
+  These equations simplify very much in their application to plane
+  waves. If the ray be parallel to OX, and the direction of vibration
+  parallel to OZ, we have [xi] = 0, [eta] = 0, while [zeta] is a
+  function of x and t only. Equation (1) and the first pair of equations
+  (2) are thus satisfied identically. The third equation gives
+
+    d²[zeta]   d²[zeta]
+    -------- = --------                   (3),
+       dt²        dx²
+
+  of which the solution is
+
+    [zeta] = f(bt - x)                    (4),
+
+  where f is an arbitrary function.
+
+  The question as to the law of the secondary waves is thus answered by
+  Stokes. "Let [xi] = 0, [eta] = 0, [zeta] = f(bt-x) be the
+  displacements corresponding to the incident light; let O1 be any point
+  in the plane P (of the wave-front), dS an element of that plane
+  adjacent to O1, and consider the disturbance due to that portion only
+  of the incident disturbance which passes continually across dS. Let O
+  be any point in the medium situated at a distance from the point O1
+  which is large in comparison with the length of a wave; let O1O = r,
+  and let this line make an angle [theta] with the direction of
+  propagation of the incident light, or the axis of x, and [phi] with
+  the direction of vibration, or axis of z. Then the displacement at O
+  will take place in a direction perpendicular to O1O, and lying in the
+  plane ZO1O; and, if [zeta]' be the displacement at O, reckoned
+  positive in the direction nearest to that in which the incident
+  vibrations are reckoned positive,
+
+                dS
+    [zeta]' = ------ ( 1 + cos[theta]) sin[phi] f'(bt - r).
+              4[pi]r
+
+  In particular, if
+
+                          2[pi]
+    f(bt - x) =   c sin -------- (bt - x)        (5),
+                        [lambda]
+
+  we shall have
+
+                   cdS                                  2[pi]
+    [zeta]' =  ---------- (1 + cos[theta]) sin[phi]cos -------- (bt - r)  (6)."
+               2[lambda]r                              [lambda]
+
+  It is then verified that, after integration with respect to dS, (6)
+  gives the same disturbance as if the primary wave had been supposed to
+  pass on unbroken.
+
+  The occurrence of sin [phi] as a factor in (6) shows that the relative
+  intensities of the primary light and of that diffracted in the
+  direction [theta] depend upon the condition of the former as regards
+  polarization. If the direction of primary vibration be perpendicular
+  to the plane of diffraction (containing both primary and secondary
+  rays), sin [phi] = 1; but, if the primary vibration be in the plane of
+  diffraction, sin [phi] = cos [theta]. This result was employed by
+  Stokes as a criterion of the direction of vibration; and his
+  experiments, conducted with gratings, led him to the conclusion that
+  the vibrations of polarized light are executed in a direction
+  _perpendicular_ to the plane of polarization.
+
+  The factor (1 + cos [theta]) shows in what manner the secondary
+  disturbance depends upon the direction in which it is propagated with
+  respect to the front of the primary wave.
+
+  If, as suffices for all practical purposes, we limit the application
+  of the formulae to points in advance of the plane at which the wave is
+  supposed to be broken up, we may use simpler methods of resolution
+  than that above considered. It appears indeed that the purely
+  mathematical question has no definite answer. In illustration of this
+  the analogous problem for sound may be referred to. Imagine a flexible
+  lamina to be introduced so as to coincide with the plane at which
+  resolution is to be effected. The introduction of the lamina (supposed
+  to be devoid of inertia) will make no difference to the propagation of
+  plane parallel sonorous waves through the position which it occupies.
+  At every point the motion of the lamina will be the same as would have
+  occurred in its absence, the pressure of the waves impinging from
+  behind being just what is required to generate the waves in front. Now
+  it is evident that the aerial motion in front of the lamina is
+  determined by what happens at the lamina without regard to the cause
+  of the motion there existing. Whether the necessary forces are due to
+  aerial pressures acting on the rear, or to forces directly impressed
+  from without, is a matter of indifference. The conception of the
+  lamina leads immediately to two schemes, according to which a primary
+  wave may be supposed to be broken up. In the first of these the
+  element dS, the effect of which is to be estimated, is supposed to
+  execute its actual motion, while every other element of the plane
+  lamina is maintained at rest. The resulting aerial motion in front is
+  readily calculated (see Rayleigh, _Theory of Sound_, § 278); it is
+  symmetrical with respect to the origin, i.e. independent of [theta].
+  When the secondary disturbance thus obtained is integrated with
+  respect to dS over the entire plane of the lamina, the result is
+  necessarily the same as would have been obtained had the primary wave
+  been supposed to pass on without resolution, for this is precisely the
+  motion generated when every element of the lamina vibrates with a
+  common motion, equal to that attributed to dS. The only assumption
+  here involved is the evidently legitimate one that, when two systems
+  of variously distributed motion at the lamina are superposed, the
+  corresponding motions in front are superposed also.
+
+  The method of resolution just described is the simplest, but it is
+  only one of an indefinite number that might be proposed, and which are
+  all equally legitimate, so long as the question is regarded as a
+  merely mathematical one, without reference to the physical properties
+  of actual screens. If, instead of supposing the _motion_ at dS to be
+  that of the primary wave, and to be zero elsewhere, we suppose the
+  _force_ operative over the element dS of the lamina to be that
+  corresponding to the primary wave, and to vanish elsewhere, we obtain
+  a secondary wave following quite a different law. In this case the
+  motion in different directions varies as cos[theta], vanishing at
+  right angles to the direction of propagation of the primary wave. Here
+  again, on integration over the entire lamina, the aggregate effect of
+  the secondary waves is necessarily the same as that of the primary.
+
+  In order to apply these ideas to the investigation of the secondary
+  wave of light, we require the solution of a problem, first treated by
+  Stokes, viz. the determination of the motion in an infinitely extended
+  elastic solid due to a locally applied periodic force. If we suppose
+  that the force impressed upon the element of mass D dx dy dz is
+
+    DZ dx dy dz,
+
+  being everywhere parallel to the axis of Z, the only change required
+  in our equations (1), (2) is the addition of the term Z to the second
+  member of the third equation (2). In the forced vibration, now under
+  consideration, Z, and the quantities [xi], [eta], [zeta], [delta]
+  expressing the resulting motion, are to be supposed proportional to
+  e^int, where i = [sqrt](-1), and n = 2[pi]/[tau], [tau] being the
+  periodic time. Under these circumstances the double differentiation
+  with respect to t of any quantity is equivalent to multiplication by
+  the factor -n², and thus our equations take the form
+
+                                        d[delta]      \
+    (b²[Delta]² + n²)[xi]   + (a² - b²) -------- = 0  |
+                                           dx         |
+                                                      |
+                                        d[delta]      |
+    (b²[Delta]² + n²)[eta]  + (a² - b²) -------- = 0   >   (7).
+                                           dx         |
+                                                      |
+                                        d[delta]      |
+    (b²[Delta]² + n²)[zeta] + (a² - b²) -------- = -Z |
+                                           dx         /
+
+  It will now be convenient to introduce the quantities.[=omega]1,
+  [=omega]2, [=omega]3 which express the _rotations_ of the elements of
+  the medium round axes parallel to those of co-ordinates, in accordance
+  with the equations
+
+                d[xi]   d[eta]              d[eta]   d[zeta]
+    [=omega]3 = ----- - ------, [=omega]1 = ------ - -------,
+                 dy       dx'                 dz       dy
+
+                d[zeta]   d[xi]
+    [=omega]2 = ------- - -----                (8).
+                  dx       dz
+
+  In terms of these we obtain from (7), by differentiation and
+  subtraction,
+
+    (b²[Delta]² + n²) [=omega]3 = 0       \
+    (b²[Delta]² + n²) [=omega]1 = dZ/dy    >   (9).
+    (b²[Delta]² + n²) [=omega]2 = -dZ/dx  /
+
+
+  The first of equations (9) gives
+
+    [=omega]3 = 0                             (10).
+
+  For =[omega]1, we have
+                          _ _ _    -ikr
+                   1     / / / dZ e
+    [=omega]1 = -------  | | | -- ----- dx dy dz   (11),
+                4[pi]b² _/_/_/ dy   r
+
+  where r is the distance between the element dx dy dz and the point
+  where [=omega]1 is estimated, and
+
+    k = n/b = 2[pi]/[lambda]                  (12),
+
+  [lambda] being the wave-length.
+
+  (This solution may be verified in the same manner as Poisson's
+  theorem, in which k = 0.)
+
+  We will now introduce the supposition that the force Z acts only
+  within a small space of volume T, situated at (x, y, z), and for
+  simplicity suppose that it is at the origin of co-ordinates that the
+  rotations are to be estimated. Integrating by parts in (11), we get
+
+      _ -ikr           _       _     _
+     / e      dZ      | Ze^-ikr |   /   d   / e^-ikr\
+     | ------ -- dy = | ------- | - | Z -- ( ------- ) dy,
+    _/    r   dy      |_   r   _|  _/   dy  \    r  /
+
+  in which the integrated terms at the limits vanish, Z being finite
+  only within the region T. Thus
+
+                          _ _ _         -ikr
+                   1     / / /   d   /e^      \
+    [=omega]1 = -------  | | | Z -- ( -------- ) dx dy dz.
+                4[pi]b² _/_/_/   dy  \    r   /
+
+
+  Since the dimensions of T are supposed to be very small in comparison
+  with [lambda], the factor d/dy (e^-ikr / r) is sensibly constant; so
+  that, if Z stand for the mean value of Z over the volume T, we may
+  write
+
+                  TZ      y   d   / e^-ikr \
+    [=omega]1 = ------- · - · -- (  ------  )        (13).
+                4[pi]b²   r   dr  \   r    /
+
+  In like manner we find
+
+                  TZ      x   d   / e^-ikr \
+    [=omega]2 = ------  · - · -- ( -------  )        (14).
+                4[pi]b²   r   dr  \     r  /
+
+
+  From (10), (13), (14) we see that, as might have been expected, the
+  rotation at any point is about an axis perpendicular both to the
+  direction of the force and to the line joining the point to the source
+  of disturbance. If the resultant rotation be [omega], we have
+
+                 TZ      [sqrt](x² + y²)   d   /e^-ikr\
+    [=omega] = ------- · --------------- · -- ( ------ ) =
+               4[pi]b²          r          dr  \   r  /
+
+                TZ sin[phi] d   /e^-ikr\
+             =  ----------- -- ( ------ ),
+                  4[pi]b²   dr  \   r  /
+
+  [phi] denoting the angle between r and z. In differentiating
+  e^(-ikr)/r with respect to r, we may neglect the term divided by r² as
+  altogether insensible, kr being an exceedingly great quantity at any
+  moderate distance from the origin of disturbance. Thus
+
+               -ik·TZ sin[phi]    /e^-ikr\
+    [=omega] = --------------- · ( ------ )         (15),
+                   4[pi]b²        \  r   /
+
+  which completely determines the rotation at any point. For a
+  disturbing force of given integral magnitude it is seen to be
+  everywhere about an axis perpendicular to r and the direction of the
+  force, and in magnitude dependent only upon the angle ([phi]) between
+  these two directions and upon the distance (r).
+
+  The intensity of light is, however, more usually expressed in terms of
+  the actual displacement in the plane of the wave. This displacement,
+  which we may denote by [zeta]', is in the plane containing z and r,
+  and perpendicular to the latter. Its connexion with [=omega]is
+  expressed by [=omega] = d[zeta]'/dr; so that
+
+              TZ sin [phi]   /e^-ikr\
+    [zeta]' = ----------- · ( ------ )              (16),
+                4[pi]b²      \  r   /
+
+  where the factor e^int is restored.
+
+  Retaining only the real part of (16), we find, as the result of a
+  local application of force equal to
+
+    DTZ cos nt                                      (17),
+
+  the disturbance expressed by
+
+              TZ sin [phi]    /cos(nt - kr)\
+    [zeta]' = ------------ · ( ------------ )       (18).
+                4[pi]b²       \     r      /
+
+  The occurrence of sin [phi] shows that there is no disturbance
+  radiated in the direction of the force, a feature which might have
+  been anticipated from considerations of symmetry.
+
+  We will now apply (18) to the investigation of a law of secondary
+  disturbance, when a primary wave
+
+    [zeta] = sin(nt - kx)                           (19)
+
+  is supposed to be broken up in passing the plane x = 0. The first step
+  is to calculate the force which represents the reaction between the
+  parts of the medium separated by x = 0. The force operative upon the
+  positive half is parallel to OZ, and of amount per unit of area equal
+  to
+
+    -b²D d[zeta]/dx = b²kD cos nt;
+
+  and to this force acting over the whole of the plane the actual motion
+  on the positive side may be conceived to be due. The secondary
+  disturbance corresponding to the element dS of the plane may be
+  supposed to be that caused by a force of the above magnitude acting
+  over dS and vanishing elsewhere; and it only remains to examine what
+  the result of such a force would be.
+
+  Now it is evident that the force in question, supposed to act upon the
+  positive half only of the medium, produces just double of the effect
+  that would be caused by the same force if the medium were undivided,
+  and on the latter supposition (being also localized at a point) it
+  comes under the head already considered. According to (18), the effect
+  of the force acting at dS parallel to OZ, and of amount equal to
+
+    2b²kD dS cos nt,
+
+  will be a disturbance
+
+              dS sin [phi]
+    [zeta]' = ------------ cos(nt - kr)            (20),
+               [lambda]r
+
+  regard being had to (12). This therefore expresses the secondary
+  disturbance at a distance r and in a direction making an angle [phi]
+  with OZ (the direction of primary vibration) due to the element dS of
+  the wave-front.
+
+  The proportionality of the secondary disturbance to sin [phi] is
+  common to the present law and to that given by Stokes, but here there
+  is no dependence upon the angle [theta] between the primary and
+  secondary rays. The occurrence of the factor [lambda]r^-1, and the
+  necessity of supposing the phase of the secondary wave accelerated by
+  a quarter of an undulation, were first established by Archibald Smith,
+  as the result of a comparison between the primary wave, supposed to
+  pass on without resolution, and the integrated effect of all the
+  secondary waves (§ 2). The occurrence of factors such as sin [phi], or
+  ½(1 + cos [theta]), in the expression of the secondary wave has no
+  influence upon the result of the integration, the effects of all the
+  elements for which the factors differ appreciably from unity being
+  destroyed by mutual interference.
+
+  The choice between various methods of resolution, all mathematically
+  admissible, would be guided by physical considerations respecting the
+  mode of action of obstacles. Thus, to refer again to the acoustical
+  analogue in which plane waves are incident upon a perforated rigid
+  screen, the circumstances of the case are best represented by the
+  first method of resolution, leading to symmetrical secondary waves, in
+  which the normal motion is supposed to be zero over the unperforated
+  parts. Indeed, if the aperture is very small, this method gives the
+  correct result, save as to a constant factor. In like manner our
+  present law (20) would apply to the kind of obstruction that would be
+  caused by an actual physical division of the elastic medium, extending
+  over the whole of the area supposed to be occupied by the intercepting
+  screen, but of course not extending to the parts supposed to be
+  perforated.
+
+  On the electromagnetic theory, the problem of diffraction becomes
+  definite when the properties of the obstacle are laid down. The
+  simplest supposition is that the material composing the obstacle is
+  perfectly conducting, i.e. perfectly reflecting. On this basis A. J.
+  W. Sommerfeld (_Math. Ann._, 1895, 47, p. 317), with great
+  mathematical skill, has solved the problem of the shadow thrown by a
+  semi-infinite plane screen. A simplified exposition has been given by
+  Horace Lamb (_Proc. Lond. Math. Soc._, 1906, 4, p. 190). It appears
+  that Fresnel's results, although based on an imperfect theory, require
+  only insignificant corrections. Problems not limited to two
+  dimensions, such for example as the shadow of a circular disk, present
+  great difficulties, and have not hitherto been treated by a rigorous
+  method; but there is no reason to suppose that Fresnel's results would
+  be departed from materially.     (R.)
+
+
+FOOTNOTES:
+
+  [1] The descending series for J0(z) appears to have been first given
+    by Sir W. Hamilton in a memoir on "Fluctuating Functions," _Roy.
+    Irish Trans._, 1840.
+
+  [2] Airy, loc. cit. "Thus the magnitude of the central spot is
+    diminished, and the brightness of the rings increased, by covering
+    the central parts of the object-glass."
+
+  [3] _"Man kann daraus schliessen, was moglicher Weise durch
+    Mikroskope noch zu sehen ist. Ein mikroskopischer Gegenstand z. B,
+    dessen Durchmesser = ([lambda]) ist, und der aus zwei Theilen
+    besteht, kann nicht mehr als aus zwei Theilen bestehend erkannt
+    werden. Dieses zeigt uns eine Grenze des Sehvermogens durch
+    Mikroskope"_ (_Gilbert's Ann._ 74, 337). Lord Rayleigh has recorded
+    that he was himself convinced by Fraunhofer's reasoning at a date
+    antecedent to the writings of Helmholtz and Abbe.
+
+  [4] The last sentence is repeated from the writer's article "Wave
+    Theory" in the 9th edition of this work, but A. A. Michelson's
+    ingenious échelon grating constitutes a realization in an unexpected
+    manner of what was thought to be impracticable.--[R.]
+
+  [5] Compare also F. F. Lippich, _Pogg. Ann._ cxxxix. p. 465, 1870;
+    Rayleigh, _Nature_ (October 2, 1873).
+
+  [6] The power of a grating to construct light of nearly definite
+   wave-length is well illustrated by Young's comparison with the
+   production of a musical note by reflection of a sudden sound from a
+   row of palings. The objection raised by Herschel (_Light_, § 703) to
+   this comparison depends on a misconception.
+
+  [7] It must not be supposed that errors of this order of magnitude
+    are unobjectionable in all cases. The position of the middle of the
+    bright band representative of a mathematical line can be fixed with a
+    spider-line micrometer within a small fraction of the width of the
+    band, just as the accuracy of astronomical observations far
+    transcends the separating power of the instrument.
+
+  [8] "In the same way we may conclude that in flat gratings any
+    departure from a straight line has the effect of causing the dust in
+    the slit and the spectrum to have different foci--a fact sometimes
+    observed." (Rowland, "On Concave Gratings for Optical Purposes,"
+    _Phil. Mag._, September 1883).
+
+  [9] On account of inequalities in the atmosphere giving a variable
+    refraction, the light from a star would be irregularly distributed
+    over a screen. The experiment is easily made on a laboratory scale,
+    with a small source of light, the rays from which, in their course
+    towards a rather distant screen, are disturbed by the neighbourhood
+    of a heated body. At a moment when the eye, or object-glass of a
+    telescope, occupies a dark position, the star vanishes. A fraction of
+    a second later the aperture occupies a bright place, and the star
+    reappears. According to this view the chromatic effects depend
+    entirely upon atmospheric dispersion.
+
+  [10] In experiment a line of light is sometimes substituted for a
+    point in order to increase the illumination. The various parts of the
+    line are here _independent_ sources, and should be treated
+    accordingly. To assume a cylindrical form of primary wave would be
+    justifiable only when there is synchronism among the secondary waves
+    issuing from the various centres.
+
+  [11] H. Necker (_Phil. Mag._, November 1832); Fox Talbot (_Phil.
+    Mag._, June 1833). "When the sun is about to emerge ... every branch
+    and leaf is lighted up with a silvery lustre of indescribable
+    beauty.... The birds, as Mr Necker very truly describes, appear like
+    flying brilliant sparks." Talbot ascribes the appearance to
+    diffraction; and he recommends the use of a telescope.
+
+
+
+
+DIFFUSION (from the Lat. _diffundere; dis-_, asunder, and _fundere_, to
+pour out), in general, a spreading out, scattering or circulation; in
+physics the term is applied to a special phenomenon, treated below.
+
+1. _General Description._--When two different substances are placed in
+contact with each other they sometimes remain separate, but in many
+cases a gradual mixing takes place. In the case where both the
+substances are gases the process of mixing continues until the result is
+a uniform mixture. In other cases the proportions in which two different
+substances can mix lie between certain fixed limits, but the mixture is
+distinguished from a chemical compound by the fact that between these
+limits the composition of the mixture is capable of continuous
+variation, while in chemical compounds, the proportions of the different
+constituents can only have a discrete series of numerical values, each
+different ratio representing a different compound. If we take, for
+example, air and water in the presence of each other, air will become
+dissolved in the water, and water will evaporate into the air, and the
+proportions of either constituent absorbed by the other will vary
+continuously. But a limit will come when the air will absorb no more
+water, and the water will absorb no more air, and throughout the change
+a definite surface of separation will exist between the liquid and the
+gaseous parts. When no surface of separation ever exists between two
+substances they must necessarily be capable of mixing in all
+proportions. If they are not capable of mixing in all proportions a
+discontinuous change must occur somewhere between the regions where the
+substances are still unmixed, thus giving rise to a surface of
+separation.
+
+The phenomena of mixing thus involves the following processes:--(1) A
+motion of the substances relative to one another throughout a definite
+_region_ of space in which mixing is taking place. This relative motion
+is called "diffusion." (2) The passage of portions of the mixing
+substances across the _surface_ of separation when such a surface
+exists. These surface actions are described under various terms such as
+solution, evaporation, condensation and so forth. For example, when a
+soluble salt is placed in a liquid, the process which occurs at the
+surface of the salt is called "solution," but the salt which enters the
+liquid by solution is transported from the surface into the interior of
+the liquid by "diffusion."
+
+Diffusion may take place in solids, that is, in regions occupied by
+matter which continues to exhibit the properties of the solid state.
+Thus if two liquids which can mix are separated by a membrane or
+partition, the mixing may take place through the membrane. If a solution
+of salt is separated from pure water by a sheet of parchment, part of
+the salt will pass through the parchment into the water. If water and
+glycerin are separated in this way most of the water will pass into the
+glycerin and a little glycerin will pass through in the opposite
+direction, a property frequently used by microscopists for the purpose
+of gradually transferring minute algae from water into glycerin. A still
+more interesting series of examples is afforded by the passage of gases
+through partitions of metal, notably the passage of hydrogen through
+platinum and palladium at high temperatures. When the process is
+considered with reference to a membrane or partition taken as a whole,
+the passage of a substance from one side to the other is commonly known
+as "osmosis" or "transpiration" (see SOLUTION), but what occurs in the
+material of the membrane itself is correctly described as diffusion.
+
+Simple cases of diffusion are easily observed qualitatively. If a
+solution of a coloured salt is carefully introduced by a funnel into the
+bottom of a jar containing water, the two portions will at first be
+fairly well defined, but if the mixture can exist in all proportions,
+the surface of separation will gradually disappear; and the rise of the
+colour into the upper part and its gradual weakening in the lower part,
+may be watched for days, weeks or even longer intervals. The diffusion
+of a strong aniline colouring matter into the interior of gelatine is
+easily observed, and is commonly seen in copying apparatus. Diffusion of
+gases may be shown to exist by taking glass jars containing vapours of
+hydrochloric acid and ammonia, and placing them in communication with
+the heavier gas downmost. The precipitation of ammonium chloride shows
+that diffusion exists, though the chemical action prevents this example
+from forming a typical case of diffusion. Again, when a film of Canada
+balsam is enclosed between glass plates, the disappearance during a few
+weeks of small air bubbles enclosed in the balsam can be watched under
+the microscope.
+
+In fluid media, whether liquids or gases, the process of mixing is
+greatly accelerated by stirring or agitating the fluids, and liquids
+which might take years to mix if left to themselves can thus be mixed in
+a few seconds. It is necessary to carefully distinguish the effects of
+agitation from those of diffusion proper. By shaking up two liquids
+which do not mix we split them up into a large number of different
+portions, and so greatly increase the area of the surface of separation,
+besides decreasing the thicknesses of the various portions. But even
+when we produce the appearance of a uniform turbid mixture, the small
+portions remain quite distinct. If however the fluids can really mix,
+the final process must in every case depend on diffusion, and all we do
+by shaking is to increase the sectional area, and decrease the thickness
+of the diffusing portions, thus rendering the completion of the
+operation more rapid. If a gas is shaken up in a liquid the process of
+absorption of the bubbles is also accelerated by capillary action, as
+occurs in an ordinary sparklet bottle. To state the matter precisely,
+however finely two fluids have been subdivided by agitation, the
+molecular constitution of the different portions remains unchanged. The
+ultimate process by which the individual molecules of two different
+substances become mixed, producing finally a homogeneous mixture, is in
+every case diffusion. In other words, diffusion is that relative motion
+of the molecules of two different substances by which the proportions of
+the molecules in any region containing a finite number of molecules are
+changed.
+
+  In order, therefore, to make accurate observations of diffusion in
+  fluids it is necessary to guard against any cause which may set up
+  currents; and in some cases this is exceedingly difficult. Thus, if
+  gas is absorbed at the upper surface of a liquid, and if the gaseous
+  solution is heavier than the pure liquid, currents may be set up, and
+  a steady state of diffusion may cease to exist. This has been tested
+  experimentally by C. G. von Hüfner and W. E. Adney. The same thing may
+  happen when a gas is evolved into a liquid at the surface of a solid
+  even if no bubbles are formed; thus if pieces of aluminium are placed
+  in caustic soda, the currents set up by the evolution of hydrogen are
+  sufficient to set the aluminium pieces in motion, and it is probable
+  that the motions of the Diatomaceae are similarly caused by the
+  evolution of oxygen. In some pairs of substances diffusion may take
+  place more rapidly than in others. Of course the progress of events in
+  any experiment necessarily depends on various causes, such as the size
+  of the containing vessels, but it is easy to see that when experiments
+  with different substances are carried out under similar conditions,
+  however these "similar conditions" be defined, the rates of diffusion
+  must be capable of numerical comparison, and the results must be
+  expressible in terms of at least one physical quantity, which for any
+  two substances can be called their coefficient of diffusion. How to
+  select this quantity we shall see later.
+
+2 _Quantitative Methods of observing Diffusion._--The simplest plan of
+determining the progress of diffusion between two liquids would be to
+draw off and examine portions from different strata at some stage in the
+process; the disturbance produced would, however, interfere with the
+subsequent process of diffusion, and the observations could not be
+continued. By placing in the liquid column hollow glass beads of
+different average densities, and observing at what height they remain
+suspended, it is possible to trace the variations of density of the
+liquid column at different depths, and different times. In this method,
+which was originally introduced by Lord Kelvin, difficulties were caused
+by the adherence of small air bubbles to the beads.
+
+In general, optical methods are the most capable of giving exact
+results, and the following may be distinguished, (a) _By refraction in a
+horizontal plane._ If the containing vessel is in the form of a prism,
+the deviation of a horizontal ray of light in passing through the prism
+determines the index of refraction, and consequently the density of the
+stratum through which the ray passes, (b) _By refraction in a vertical
+plane._ Owing to the density varying with the depth, a horizontal ray
+entering the liquid also undergoes a small vertical deviation, being
+bent downwards towards the layers of greater density. The observation of
+this vertical deviation determines not the actual density, but its rate
+of variation with the depth, i.e. the "density gradient" at any point,
+(c) _By the saccharimeter._ In the cases of solutions of sugar, which
+cause rotation of the plane of polarized light, the density of the sugar
+at any depth may be determined by observing the corresponding angle of
+rotation, this was done originally by W. Voigt.
+
+3. _Elementary Definitions of Coefficient of Diffusion._--The simplest
+case of diffusion is that of a substance, say a gas, diffusing in the
+interior of a homogeneous solid medium, which remains at rest, when no
+external forces act on the system. We may regard it as the result of
+experience that: (1) if the density of the diffusing substance is
+everywhere the same no diffusion takes place, and (2) if the density of
+the diffusing substance is different at different points, diffusion will
+take place from places of greater to those of lesser density, and will
+not cease until the density is everywhere the same. It follows that the
+rate of flow of the diffusing substance at any point in any direction
+must depend on the density gradient at that point in that direction,
+i.e. on the rate at which the density of the diffusing substance
+decreases as we move in that direction. We may define the _coefficient
+of diffusion_ as the ratio of the total mass per unit area which flows
+across any small section, to the rate of decrease of the density per
+unit distance in a direction perpendicular to that section.
+
+  In the case of steady diffusion parallel to the axis of x, if [rho] be
+  the density of the diffusing substance, and q the mass which flows
+  across a unit of area in a plane perpendicular to the axis of x, then
+  the density gradient is -d[rho]/dx and the ratio of q to this is
+  called the "coefficient of diffusion." By what has been said this
+  ratio remains finite, however small the actual gradient and flow may
+  be., and it is natural to assume, at any rate as a first
+  approximation, that it is constant as far as the quantities in
+  question are concerned. Thus if the coefficient of diffusion be
+  denoted by K we have q= -K(d[rho]/dx).
+
+  Further, the rate at which the quantity of substance is increasing in
+  an element between the distances x and x+dx is equal to the difference
+  of the rates of flow in and out of the two faces, whence as in
+  hydrodynamics, we have d[rho]/dt =-dq/dx.
+
+  It follows that the equation of diffusion in this case assumes the
+  form
+
+    d[rho]   d   /  d[rho] \
+    ------ = -- ( K ------  ),
+      dt     dx  \    dx   /
+
+  which is identical with the equations representing conduction of heat,
+  flow of electricity and other physical phenomena. For motion in three
+  dimensions we have in like manner
+
+    d[rho]   d   /  d[rho]\    d   /  d[rho]\    d   /  d[rho]\
+    ------ = -- ( K ------ ) + -- ( K ------ ) + -- ( K ------ );
+      dt     dx  \   dx   /    dy  \   dy   /    dz  \   dz   /
+
+  and the corresponding equations in electricity and heat for
+  anisotropic substances would be available to account for any parallel
+  phenomena, which may arise, or might be conceived, to exist in
+  connexion with diffusion through a crystalline solid.
+
+In the case of a very dilute solution, the coefficient of diffusion of
+the dissolved substance can be defined in the same way as when the
+diffusion takes place in a solid, because the effects of diffusion will
+not have any perceptible influence on the solvent, and the latter may
+therefore be regarded as remaining practically at rest. But in most
+cases of diffusion between two fluids, both of the fluids are in motion,
+and hence there is far greater difficulty in determining the motion, and
+even in defining the coefficient of diffusion. It is important to notice
+in the first instance, that it is only the relative motion of the two
+substances which constitutes diffusion. Thus when a current of air is
+blowing, under ordinary circumstances the changes which take place are
+purely mechanical, and do not depend on the separate diffusions of the
+oxygen and nitrogen of which the air is mainly composed. It is only when
+two gases are flowing with unequal velocity, that is, when they have a
+relative motion, that these changes of relative distribution, which are
+called diffusion, take place. The best way out of the difficulty is to
+investigate the separate motions of the two fluids, taking account of
+the mechanical actions exerted on them, and supposing that the mutual
+action of the fluids causes either fluid to resist the relative motion
+of the other.
+
+4. _The Coefficient of Resistance._--Let us call the two diffusing
+fluids A and B. If B were absent, the motion of the fluid A would be
+determined entirely by the variations of pressure of the fluid A, and by
+the external forces, such as that due to gravity acting on A. Similarly
+if A were absent, the motion of B would be determined entirely by the
+variations of pressure due to the fluid B, and by the external forces
+acting on B. When both fluids are mixed together, each fluid tends to
+resist the relative motion of the other, and by the law of equality of
+action and reaction, the resistance which A experiences from B is
+everywhere equal and opposite to the resistance which B experiences from
+A. If the amount of this resistance per unit volume be divided by the
+relative velocity of the two fluids, and also by the product of their
+densities, the quotient is called the "coefficient of resistance." If
+then [rho]1, [rho]2 are the densities cf the two fluids, u1, u2 their
+velocities, C the coefficient of resistance, then the portion of the
+fluid A contained in a small element of volume v will experience from
+the fluid B a resistance C[rho]1[rho]2v(u1- u2), and the fluid B
+contained in the same volume element will experience from the fluid A an
+equal and opposite resistance, C[rho]1[rho]2v(u2 - u1).
+
+This definition implies the following laws of resistance to diffusion,
+which must be regarded as based on experience, and not as self-evident
+truths: (1) each fluid tends to assume, so far as diffusion is
+concerned, the same equüibrium distribution that it would assume if its
+motion were unresisted by the presence of the other fluid. (Of course,
+the mutual attraction of gravitation of the two fluids might affect the
+final distribution, but this is practically negligible. Leaving such
+actions as this out of account the following statement is correct.) In
+a state of equilibrium, the density of each fluid at any point thus
+depends only on the partial pressure of that fluid alone, and is the
+same as if the other fluids were absent. It does not depend on the
+partial pressures of the other fluids. If this were not the case, the
+resistance to diffusion would be analogous to friction, and would
+contain terms which were independent of the relative velocity u2 - u1.
+(2) For slow motions the resistance to diffusion is (approximately at
+any rate) proportional to the relative velocity. (3) The coefficient of
+resistance C is not necessarily always constant; it may, for example,
+and, in general, does, depend on the temperature.
+
+  If we form the equations of hydrodynamics for the different fluids
+  occurring in any mixture, taking account of diffusion, but neglecting
+  viscosity, and using suffixes 1, 2 to denote the separate fluids,
+  these assume the form given by James Clerk Maxwell ("Diffusion," in
+  _Ency. Brit._, 9th ed.):--
+
+          Du1   dp1
+    [rho] --- + --- - X1[rho]1 + C12[rho]1[rho]2(u1 - u2) + &c. = 0,
+          Dt    dx
+
+  where
+
+    Du1   du1      du1      du1      du1
+    --- = --- + u1 --- + v1 --- + w1 ---,
+    Dt    dt       dx       dy       dz
+
+  and these equations imply that when diffusion and other motions cease,
+  the fluids satisfy the separate conditions of equilibrium dp1/dx -
+  X1[rho]1 = 0. The assumption made in the following account is that
+  terms such as Du1/Dt may be neglected in the cases considered.
+
+A further property based on experience is that the motions set up in a
+mixture by diffusion are very slow compared with those set up by
+mechanical actions, such as differences of pressure. Thus, if two gases
+at equal temperature and pressure be allowed to mix by diffusion, the
+heavier gas being below the lighter, the process will take a long time;
+on the other hand, if two gases, or parts of the same gas, at different
+pressures be connected, equalization of pressure will take place almost
+immediately. It follows from this property that the forces required to
+overcome the "inertia" of the fluids in the motions due to diffusion are
+quite imperceptible. At any stage of the process, therefore, any one of
+the diffusing fluids may be regarded as in equilibrium under the action
+of its own partial pressure, the external forces to which it is
+subjected and the resistance to diffusion of the other fluids.
+
+5. _Slow Diffusion of two Gases. Relation between the Coefficients of
+Resistance and of Diffusion._--We now suppose the diffusing substances
+to be two gases which obey Boyle's law, and that diffusion takes place
+in a closed cylinder or tube of unit sectional area at constant
+temperature, the surfaces of equal density being perpendicular to the
+axis of the cylinder, so that the direction of diffusion is along the
+length of the cylinder, and we suppose no external forces, such as
+gravity, to act on the system.
+
+  The densities of the gases are denoted by [rho]1, [rho]2, their
+  velocities of diffusion by u1, u2, and if their partial pressures are
+  p1, p2, we have by Boyle's law p1 = k1[rho]1, p2 = k2[rho]2, where
+  k1, k2 are constants for the two gases, the temperature being constant.
+  The axis of the cylinder is taken as the axis of x.
+
+  From the considerations of the preceding section, the effects of
+  inertia of the diffusing gases may be neglected, and at any instant of
+  the process either of the gases is to be treated as kept in
+  equilibrium by its partial pressure and the resistance to diffusion
+  produced by the other gas. Calling this resistance per unit volume R,
+  and putting R = C[rho]1[rho]2(u1 - u2), where C is the coefficient of
+  resistance, the equations of equilibrium give
+
+    dp1                                  dp2
+    --- + C[rho]1[rho]2(u1 - u2)= 0, and --- + C[rho]1[rho]2(u2 - u1)= 0 (1).
+    dx                                   dx
+
+  These involve
+
+    dp1   dp2
+    --- + --- = 0 or p1 + p2 = P                      (2)
+    dx    dx
+
+  where P is the total pressure of the mixture, and is everywhere
+  constant, consistently with the conditions of mechanical equilibrium.
+
+  Now dp1/dx is the pressure-gradient of the first gas, and is, by
+  Boyle's law, equal to k1 times the corresponding density-gradient.
+  Again [rho]1u1 is the mass of gas flowing across any section per unit
+  time, and k1[rho]1u1 or p1u1 can be regarded as representing the flux
+  of partial pressure produced by the motion of the gas. Since the total
+  pressure is everywhere constant, and the ends of the cylinder are
+  supposed fixed, the fluxes of partial pressure due to the two gases
+  are equal and opposite, so that
+
+    p1u1 + p2u2 = 0 or k1[rho]1u1 + k2[rho]2u2 = 0    (3).
+
+  From (2) (3) we find by elementary algebra
+
+    u1/p2 = - u2/p1 = (u1 - u2)/(p1 + p2) = (u1 - u2)/P,
+
+  and therefore
+
+    p2u1 = - p2u2 = p1p2(u1 - u2)/P = k1k2[rho]1[rho]2(u1 - u2)/P
+
+  Hence equations (1) (2) gives
+
+    dp1    CP                   dp2    CP
+    --- + ---- (p1u1) = 0, and  --- + ---- (p2u2) = 0;
+    dx    k1k2                  dx    k1k2
+
+  whence also substituting p1 = k1[rho]1, p2 = k2[rho]2, and by
+  transposing
+
+                 k1k2 d[rho]1                   k1k2 d[rho]2
+    [rho]1u1 = - ---- -------, and [rho]2u2 = - ---- -------.
+                  CP    dx                       CP     dx
+
+  We may now define the "coefficient of diffusion" of either gas as the
+  ratio of the rate of flow of that gas to its density-gradient. With
+  this definition, the coefficients of diffusion of both the gases in a
+  mixture are equal, each being equal to k1k2/CP. The ratios of the
+  fluxes of partial pressure to the corresponding pressure-gradients are
+  also equal to the same coefficient. Calling this coefficient K, we
+  also observe that the equations of continuity for the two gases are
+
+    d[rho]1   d([rho]1u1)          d[rho]2   d([rho]2u2)
+    ------- + ----------- = 0, and ------- + ----------- = 0,
+       dt         dx                  dt         dx
+
+  leading to the equations of diffusion
+
+    d[rho]1   d   /  d[rho]1\        d[rho]2   d   /  d[rho]2\
+    ------- = -- ( K ------- ) , and ------- = -- ( K ------- ),
+      dt      dx  \    dx   /          dt      dx  \    dx   /
+
+  exactly as in the case of diffusion through a solid.
+
+If we attempt to treat diffusion in liquids by a similar method, it is,
+in the first place, necessary to define the "partial pressure" of the
+components occurring in a liquid mixture. This leads to the conception
+of "osmotic pressure," which is dealt with in the article SOLUTION. For
+dilute solutions at constant temperature, the assumption that the
+osmotic pressure is proportional to the density, leads to results
+agreeing fairly closely with experience, and this fact may be
+represented by the statement that a substance occurring in a dilute
+solution behaves like a perfect gas.
+
+6. _Relation of the Coefficient of Diffusion to the Units of Length and
+Time._--We may write the equation defining K in the form
+
+             I   d[rho]
+  u = -K × ----- ------.
+           [rho]   dx
+
+Here -d[rho]/[rho]dx represents the "percentage rate" at which the
+density decreases with the distance x; and we thus see that the
+coefficient of diffusion represents the ratio of the velocity of flow to
+the percentage rate at which the density decreases with the distance
+measured in the direction of flow. This percentage rate being of the
+nature of a number divided by a length, and the velocity being of the
+nature of a length divided by a time, we may state that K is of two
+dimensions in length and - 1 in time, i.e. dimensions L²/T.
+
+  _Example 1._ Taking K = 0.1423 for carbon dioxide and air (at
+  temperature 0° C. and pressure 76 cm. of mercury) referred to a
+  centimetre and a second as units, we may interpret the result as
+  follows:--Supposing in a mixture of carbon dioxide and air, the
+  density of the carbon dioxide decreases by, say, 1, 2 or 3% of itself
+  in a distance of 1 cm., then the corresponding velocities of the
+  diffusing carbon dioxide will be respectively 0.01, 0.02 and 0.03
+  times 0.1423, that is, 0.001423, 0.002846 and 0.004269 cm. per second
+  in the three cases.
+
+  _Example 2._ If we wished to take a foot and a second as our units, we
+  should have to divide the value of the coefficient of diffusion in
+  Example 1 by the square of the number of centimetres in 1 ft., that
+  is, roughly speaking, by 900, giving the new value of K = 0.00016
+  roughly.
+
+7. _Numerical Values of the Coefficient of Diffusion._--The table on p.
+258 gives the values of the coefficient of diffusion of several of the
+principal pairs of gases at a pressure of 76 cm. of mercury, and also of
+a number of other substances. In the gases the centimetre and second are
+taken as fundamental units, in other cases the centimetre and day.
+
+8. _Irreversible Changes accompanying Diffusion._--The diffusion of two
+gases at constant pressure and temperature is a good example of an
+"irreversible process." The gases always tend to mix, never to separate.
+In order to separate the gases a change must be effected in the external
+conditions to which the mixture is subjected, either by liquefying one
+of the gases, or by separating them by diffusion through a membrane, or
+by bringing other outside influences to bear on them. In the case of
+liquids, electrolysis affords a means of separating the constituents of
+a mixture. Every such method involves some change taking place outside
+the mixture, and this change may be regarded as a "compensating
+transformation." We thus have an instance of the property that every
+irreversible change leaves an indelible imprint somewhere or other on
+the progress of events in the universe. That the process of diffusion
+obeys the laws of irreversible thermodynamics (if these laws are
+properly stated) is proved by the fact that the compensating
+transformations required to separate mixed gases do not essentially
+involve anything but transformation of energy. The process of allowing
+gases to mix by diffusion, and then separating them by a compensating
+transformation, thus constitutes an irreversible cycle, the outside
+effects of which are that energy somewhere or other must be less capable
+of transformation than it was before the change. We express this fact by
+stating that an irreversible process essentially implies a loss of
+availability. To measure this loss we make use of the laws of
+thermodynamics, and in particular of Lord Kelvin's statement that "It is
+impossible by means of inanimate material agency to derive mechanical
+effect from any portion of matter by cooling it below the temperature of
+the coldest of the surrounding objects."
+
+  +-------------------------------------------+---------+---------------------+--------------+
+  |                Substances.                |  Temp.  |         K.          |   Author.    |
+  +-------------------------------------------+---------+---------------------+--------------+
+  | Carbon dioxide and air                    |    0°C. | 0.1423 cm²/sec.     | J. Loschmidt.|
+  |    "      "        hydrogen               |    0°C. | 0.5558    "         |      "       |
+  |    "      "        oxygen                 |    0°C. | 0.1409    "         |      "       |
+  |    "      "        carbon monoxide        |    0°C. | 0.1406    "         |      "       |
+  |    "      "        marsh gas (methane)    |    0°C. | 0.1586    "         |      "       |
+  |    "      "        nitrous oxide          |    0°C. | 0.0983    "         |      "       |
+  | Hydrogen and oxygen                       |    0°C. | 0.7214    "         |      "       |
+  |    "      "  carbon monoxide              |    0°C. | 0.6422    "         |      "       |
+  |    "      "  sulphur dioxide              |    0°C. | 0.4800    "         |      "       |
+  | Oxygen and carbon monoxide                |    0°C. | 0.1802    "         |      "       |
+  | Water and ammonia                         |   20°C. | 1.250     "         | G. Hüfner.   |
+  |    "        "                             |    5°C. | 0.822     "         |      "       |
+  |    "      common salt (density 1.0269)    |         | 0.355 cm²/hour.     | J. Graham.   |
+  |    "         "          "        "        |14.33°C. | 1.020, 0.996, 0.972,|      "       |
+  |                                           |         | 0.932 cm²/day.      | F. Heimbrodt.|
+  |    "      zinc sulphate (0.312 gm/cm³)    |         | 0.1162    "         | W. Seitz.    |
+  |    "      zinc sulphate (normal)          |         | 0.2355    "         |      "       |
+  |    "      zinc acetate (double normal)    |         | 0.1195    "         |      "       |
+  |    "      zinc formate (half normal)      |         | 0.4654    "         |      "       |
+  |    "      cadmium sulphate (double normal)|         | 0.2456    "         |      "       |
+  |    "      glycerin (1/8n, ½n, 7/8n, 7/8n) |10.14°C. | 0.356, 0.350, 0.342,| F. Heimbrodt.|
+  |                                           |         | 0.315 cm²/day.      |      "       |
+  |    "      urea          "         "       |14.83°C. | 0.973, 0.946, 0.926,|      "       |
+  |                                           |         | 0.883 cm²/day.      |      "       |
+  |    "      hydrochloric acid               |14.30°C. | 2.208, 2.331,       |      "       |
+  |                                           |         | 2.480 cm²/day       |      "       |
+  | Gelatin 20% and ammonia                   |   17°C. | 127.1     "         | A. Hagenbach.|
+  |    "     "      carbon dioxide            |         | 0.845     "         |      "       |
+  |    "     "      nitrous oxide             |         | 0.509     "         |      "       |
+  |    "     "      oxygen                    |         | 0.230     "         |      "       |
+  |    "     "      hydrogen                  |         | 0.0565    "         |      "       |
+  +-------------------------------------------+---------+---------------------+--------------+
+
+  Let us now assume that we have any syste m such as the gases above
+  considered, and that it is in the presence of an indefinitely extended
+  medium which we shall call the "auxiliary medium." If heat be taken
+  from any part of the system, only part of this heat can be converted
+  into work by means of thermodynamic engines; and the rest will be
+  given to the auxiliary medium, and will constitute unavailable energy
+  or waste. To understand what this means, we may consider the case of a
+  condensing steam engine. Only part of the energy liberated by the
+  combustion of the coal is available for driving the engine, the rest
+  takes the form of heat imparted to the condenser. The colder the
+  condenser the more efficient is the engine, and the smaller is the
+  quantity of waste.
+
+  The amount of unavailable energy associated with any given
+  transformation is proportional to the absolute temperature of the
+  auxiliary medium. When divided by that temperature the quotient is
+  called the change of "entropy" associated with the given change (see
+  THERMODYNAMICS). Thus if a body at temperature T receives a quantity
+  of heat Q, and if T0 is the temperature of the auxiliary medium, the
+  quantity of work which could be obtained from Q by means of ideal
+  thermodynamic engines would be Q(1 - T0/T), and the balance, which is
+  QT0/T, would take the form of unavailable or waste energy given to the
+  medium. The quotient of this, when divided by T0, is Q/T, and this
+  represents the quantity of entropy associated with Q units of heat at
+  temperature T.
+
+  Any irreversible change for which a compensating transformation of
+  energy exists represents, therefore, an increase of unavailable
+  energy, which is measurable in terms of entropy. The increase of
+  entropy is independent of the temperature of the auxiliary medium. It
+  thus affords a measure of the extent to which energy has run to waste
+  during the change. Moreover, when a body is heated, the increase of
+  entropy is the factor which determines how much of the energy imparted
+  to the body is unavailable for conversion into work under given
+  conditions. In all cases we have
+
+    increase of unavailable energy
+    ------------------------------- = increase of entropy.
+    temperature of auxiliary medium
+
+  When diffusion takes place between two gases inside a closed vessel at
+  uniform pressure and temperature no energy in the form of heat or work
+  is received from without, and hence the entropy gained by the gases
+  from without is zero. But the irreversible processes inside the vessel
+  may involve a gain of entropy, and this can only be estimated by
+  examining by what means mixed gases can be separated, and, in
+  particular, under what conditions the process of mixing and separating
+  the gases could (theoretically) be made reversible.
+
+9. _Evidence derived from Liquefaction of one or both of the
+Gases._--The gases in a mixture can often be separated by liquefying, or
+even solidifying, one or both of the components. In connexion with this
+property we have the important law according to which "The pressure of a
+vapour in equilibrium with its liquid depends only on the temperature
+and is independent of the pressures of any other gases or vapours which
+may be mixed with it." Thus if two closed vessels be taken containing
+some water and one be exhausted, the other containing air, and if the
+temperatures be equal, evaporation will go on until the pressure of the
+vapour in the exhausted vessel is equal to its _partial_ pressure in the
+other vessel, notwithstanding the fact that the _total_ pressure in the
+latter vessel is greater by the pressure of the air.
+
+  To separate mixed gases by liquefaction, they must be compressed and
+  cooled till one separates in the form of a liquid. If no changes are
+  to take place outside the system, the separate components must be
+  allowed to expand until the work of expansion is equal to the work of
+  compression, and the heat given out in compression is reabsorbed in
+  expansion. The process may be made as nearly reversible as we like by
+  performing the operations so slowly that the substances are
+  practically in a state of equilibrium at every stage. This is a
+  consequence of an important axiom in thermodynamics according to which
+  "any small change in the neighbourhood of a state of equilibrium is to
+  a first approximation reversible."
+
+  Suppose now that at any stage of the compression the partial pressures
+  of the two gases are p1 and p2, and that the volume is changed from V
+  to V - dV. The work of compression is (p1 + p2)dV, and this work will
+  be restored at the corresponding stage if each of the separated gases
+  increases in volume from V - dV to V. The ultimate state of the
+  separated gases will thus be one in which each gas occupies the volume
+  V originally occupied by the mixture.
+
+  We may now obtain an estimate of the amount of energy rendered
+  unavailable by diffusion. We suppose two gases occupying volumes V1
+  and V2 at equal pressure p to mix by diffusion, so that the final
+  volume is V1 + V2. Then if before mixing each gas had been allowed to
+  expand till its volume was V1 + V2, work would have been done in the
+  expansion, and the gases could still have been mixed by a reversal of
+  the process above described. In the actual diffusion this work of
+  expansion is lost, and represents energy rendered unavailable at the
+  temperature at which diffusion takes place. When divided by that
+  temperature the quotient gives the increase of entropy. Thus the
+  irreversible processes, and, in particular, the entropy changes
+  associated with diffusion of two gases at uniform pressure, are the
+  same as would take place if each of the gases in turn were to expand
+  by rushing into a vacuum, till it occupied the whole volume of the
+  mixture. A more rigorous proof involves considerations of the
+  thermodynamic potentials, following the methods of J. Willard Gibbs
+  (see ENERGETICS).
+
+  Another way in which two or more mixed gases can be separated is by
+  placing them in the presence of a liquid which can freely absorb one
+  of the gases, but in which the other gas or gases are insoluble. Here
+  again it is found by experience that when equilibrium exists at a
+  given temperature between the dissolved and undissolved portions of
+  the first gas, the partial pressure of that gas in the mixture depends
+  on the temperature alone, and is independent of the partial pressures
+  of the insoluble gases with which it is mixed, so that the conclusions
+  are the same as before.
+
+10. _Diffusion through a Membrane or Partition. Theory of the
+semi-permeable Membrane._--It has been pointed out that diffusion of
+gases frequently takes place in the interior of solids; moreover,
+different gases behave differently with respect to the same solid at the
+same temperature. A membrane or partition formed of such a solid can
+therefore be used to effect a more or less complete separation of gases
+from a mixture. This method is employed commercially for extracting
+oxygen from the atmosphere, in particular for use in projection lanterns
+where a high degree of purity is not required. A similar method is often
+applied to liquids and solutions and is known as "dialysis."
+
+In such cases as can be tested experimentally it has been found that a
+gas always tends to pass through a membrane from the side where its
+density, and therefore its partial pressure, is greater to the side
+where it is less; so that for equilibrium the partial pressures on the
+two sides must be equal. This result is unaffected by the presence of
+other gases on one or both sides of the membrane. For example, if
+different gases at the same pressure are separated by a partition
+through which one gas can pass more rapidly than the other, the
+diffusion will give rise to a difference of pressure on the two sides,
+which is capable of doing mechanical work in moving the partition. In
+evidence of this conclusion Max Planck quotes a test experiment made by
+him in the Physical Institute of the university of Munich in 1883,
+depending on the fact that platinum foil at white heat is permeable to
+hydrogen but impermeable to air, so that if a platinum tube filled with
+hydrogen be heated the hydrogen will diffuse out, leaving a vacuum.
+
+  The details of the experiment may be quoted here:--"A glass tube of
+  about 5 mm. internal diameter, blown out to a bulb at the middle, was
+  provided with a stop-cock at one end. To the other a platinum tube 10
+  cm. long was fastened, and closed at the end. The whole tube was
+  exhausted by a mercury pump, filled with hydrogen at ordinary
+  atmospheric pressure, and then closed. The closed end of the platinum
+  portion was then heated in a horizontal position by a Bunsen burner.
+  The connexion between the glass and platinum tubes, having been made
+  by means of sealing-wax, had to be kept cool by a continuous current
+  of water to prevent the softening of the wax. After four hours the
+  tube was taken from the flame, cooled to the temperature of the room,
+  and the stop-cock opened under mercury. The mercury rose rapidly,
+  almost completely filling the tube, proving that the tube had been
+  very nearly exhausted."
+
+[Illustration]
+
+In order that diffusion through a membrane may be reversible so far as a
+particular gas is concerned, the process must take place so slowly that
+equilibrium is set up at every stage (see § 9 above). In order to
+separate one gas from another consistently with this condition it is
+necessary that no diffusion of the latter gas should accompany the
+process. The name "semi-permeable" is applied to an ideal membrane or
+partition through which one gas can pass, and which offers an
+insuperable barrier to any diffusion whatever of a second gas. By means
+of two semi-permeable partitions acting oppositely with respect to two
+different gases A and B these gases could be mixed or separated by
+reversible methods. The annexed figure shows a diagrammatic
+representation of the process.
+
+  We suppose the gases contained in a cylindrical tube; P, Q, R, S are
+  four pistons, of which P and R are joined to one connecting rod, Q and
+  S to another. P, S are impermeable to both gases; Q is semi-permeable,
+  allowing the gas A to pass through but not B, similarly R allows the
+  gas B to pass through but not A. The distance PR is equal to the
+  distance QS, so that if the rods are pushed towards each other as far
+  as they will go, P and Q will be in contact, as also R and S. Imagine
+  the space RQ filled with a mixture of the two gases under these
+  conditions. Then by slowly drawing the connecting rods apart until R,
+  Q touch, the gas A will pass into the space PQ, and B will pass into
+  the space RS, and the gases will finally be completely separated;
+  similarly, by pushing the connecting rods together, the two gases will
+  be remixed in the space RQ. By performing the operations slowly enough
+  we may make the processes as nearly reversible as we please, so that
+  no available energy is lost in either change. The gas A being at every
+  instant in equilibrium on the two sides of the piston Q, its density,
+  and therefore its partial pressure, is the same on both sides, and the
+  same is true regarding the gas B on the two sides of R. Also _no work
+  is done in moving the pistons_, for the partial pressures of B on the
+  two sides of R balance each other, consequently, the resultant thrust
+  on R is due to the gas A alone, and is equal and opposite to its
+  resultant thrust on P, so that the connecting rods are at every
+  instant in a state of mechanical equilibrium so far as the pressures
+  of the gases A and B are concerned. We conclude that in the reversible
+  separation of the gases by this method at constant temperature without
+  the production or absorption of mechanical work, the densities and the
+  partial pressures of the two separated gases are the same as they were
+  in the mixture. These conclusions are in entire agreement with those
+  of the preceding section. If this agreement did not exist it would be
+  possible, theoretically, to obtain perpetual motion from the gases in
+  a way that would be inconsistent with the second law of
+  thermodynamics.
+
+Most physicists admit, as Planck does, that it is impossible to obtain
+an ideal semi-permeable substance; indeed such a substance would
+necessarily have to possess an infinitely great resistance to diffusion
+for such gases as could not penetrate it. But in an experiment performed
+under actual conditions the losses of available energy arising from this
+cause would be attributable to the imperfect efficiency of the
+partitions and not to the gases themselves; moreover, these losses are,
+in every case, found to be completely in accordance with the laws of
+irreversible thermodynamics. The reasoning in this article being
+somewhat condensed the reader must necessarily be referred to treatises
+on thermodynamics for further information on points of detail connected
+with the argument. Even when he consults these treatises he may find
+some points omitted which have been examined in full detail at some time
+or other, but are not sufficiently often raised to require mention in
+print.
+
+II. _Kinetic Models of Diffusion._--Imagine in the first instance that a
+very large number of red balls are distributed over one half of a
+billiard table, and an equal number of white balls over the other half.
+If the balls are set in motion with different velocities in various
+directions, diffusion will take place, the red balls finding their way
+among the white ones, and vice versa; and the process will be retarded
+by collisions between the balls. The simplest model of a perfect gas
+studied in the kinetic theory of gases (see MOLECULE) differs from the
+above illustration in that the bodies representing the molecules move in
+space instead of in a plane, and, unlike billiard balls, their motion is
+unresisted, and they are perfectly elastic, so that no kinetic energy is
+lost either during their free motions, or at a collision.
+
+  The mathematical analysis connected with the application of the
+  kinetic theory to diffusion is very long and cumbersome. We shall
+  therefore confine our attention to regarding a medium formed of
+  elastic spheres as a mechanical model, by which the most important
+  features of diffusion can be illustrated. We shall assume the results
+  of the kinetic theory, according to which:--(1) In a dynamical model
+  of a perfect gas the mean kinetic energy of translation of the
+  molecules represents the absolute temperature of the gas. (2) The
+  pressure at any point is proportional to the product of the number of
+  molecules in unit volume about that point into the mean square of the
+  velocity. (The mean square of the velocity is different from but
+  proportional to the square of the mean velocity, and in the subsequent
+  arguments either of these two quantities can generally be taken.) (3)
+  In a gas mixture represented by a mixture of molecules of unequal
+  masses, the mean kinetic energies of the different kinds are equal.
+
+  Consider now the problem of diffusion in a region containing two kinds
+  of molecules A and B of unequal mass. The molecules of A in the
+  neighbourhood of any point will, by their motion, spread out in every
+  direction until they come into collision with other molecules of
+  either kind, and this spreading out from every point of the medium
+  will give rise to diffusion. If we imagine the velocities of the A
+  molecules to be equally distributed in all directions, as they would
+  be in a homogeneous mixture, it is obvious that the process of
+  diffusion will be greater, _ceteris paribus_, the greater the velocity
+  of the molecules, and the greater the length of the free path before a
+  collision takes place. If we assume consistently with this, that the
+  coefficient of diffusion of the gas A is proportional to the mean
+  value of Wala, where wa is the velocity and la is the length of the
+  path of a molecule of A, this expression for the coefficient of
+  diffusion is of the right dimensions in length and time. If, moreover,
+  we observe that when diffusion takes place in a fixed direction, say
+  that of the axis of x, it depends only on the resolved part of the
+  velocity and length of path in that direction: this hypothesis readily
+  leads to our taking the mean value of 1/3w_a l_a as the coefficient of
+  diffusion for the gas A. This value was obtained by O. E. Meyer and
+  others.
+
+  Unfortunately, however, it makes the coefficients of diffusion unequal
+  for the two gases, a result inconsistent with that obtained above from
+  considerations of the coefficient of resistance, and leading to the
+  consequence that differences of pressure would be set up in different
+  parts of the gas. To equalize these differences of pressure, Meyer
+  assumed that a counter current is set up, this current being, of
+  course, very slow in practice; and J. Stefan assumed that the
+  diffusion of one gas was not affected by collisions between molecules
+  of the _same gas_. When the molecules are mixed in equal proportions
+  both hypotheses lead to the value 1/6([w_a l_a] + [w_b l_b]), (square
+  brackets denoting mean values). When one gas preponderates largely
+  over the other, the phenomena of diffusion are too difficult of
+  observation to allow of accurate experimental tests being made.
+  Moreover, in this case no difference exists unless the molecules are
+  different in size or mass.
+
+  Instead of supposing a velocity of translation added after the
+  mathematical calculations have been performed, a better plan is to
+  assume from the outset that the molecules of the two gases have small
+  velocities of translation in opposite directions, superposed on the
+  distribution of velocity, which would occur in a medium representing a
+  gas at rest. When a collision occurs between molecules of different
+  gases a transference of momentum takes place between them, and the
+  quantity of momentum so transferred in one second in a unit of volume
+  gives a dynamical measure of the resistance to diffusion. It is to be
+  observed that, however small the relative velocity of the gases A and
+  B, it plays an all-important part in determining the coefficient of
+  resistance; for without such relative motion, and with the velocities
+  evenly distributed in all directions, no transference of momentum
+  could take place. The coefficient of resistance being found, the
+  motion of each of the two gases may be discussed separately.
+
+One of the most important consequences of the kinetic theory is that if
+the volume be kept constant the coefficient of diffusion varies as the
+square root of the absolute temperature. To prove this, we merely have
+to imagine the velocity of each molecule to be suddenly increased n
+fold; the subsequent processes, including diffusion, will then go on n
+times as fast; and the temperature T, being proportional to the kinetic
+energy, and therefore to the square of the velocity, will be increased
+n² fold. Thus K, the coefficient of diffusion, varies as [sqrt]T.
+
+The relation of K to the density when the temperature remains constant
+is more difficult to discuss, but it may be sufficient to notice that if
+the number of molecules is increased n fold, the chances of a collision
+are n times as great, and the distance traversed between collisions is
+(not _therefore_ but as the result of more detailed reasoning) on the
+average 1/n of what it was before. Thus the free path, and therefore the
+coefficient of diffusion, varies inversely as the density, or directly
+as the volume. If the pressure p and temperature T be taken as
+variables, K varies inversely as p and directly as [sqrt]T³.
+
+Now according to the experiments first made by J. C. Maxwell and J.
+Loschmidt, it appeared that with constant density K was proportional to
+T more nearly than to [sqrt]T. The inference is that in this respect a
+medium formed of colliding spheres fails to give a correct mechanical
+model of gases. It has been found by L. Boltzmann, Maxwell and others
+that a system of particles whose mutual actions vary according to the
+inverse fifth power of the distance between them represents more
+correctly the relation between the coefficient of diffusion and
+temperature in actual gases. Other recent theories of diffusion have
+been advanced by M. Thiesen, P. Langevin and W. Sutherland. On the other
+hand, J. Thovert finds experimental evidence that the coefficient of
+diffusion is proportional to molecular velocity in the cases examined of
+non-electrolytes dissolved in water at 18° at 2.5 grams per litre.
+
+  BIBLIOGRAPHY.--The best introduction to the study of theories of
+  diffusion is afforded by O. E. Meyer's Kinetic _Theory of Gases_,
+  translated by Robert E. Baynes (London, 1899). The mathematical
+  portion, though sufficient for ordinary purposes, is mostly of the
+  simplest possible character. Another useful treatise is R. Ruhlmann's
+  _Handbuch der mechanischen Wärmetheorie_ (Brunswick, 1885). For a
+  shorter sketch the reader may refer to J. C. Maxwell's _Theory of
+  Heat_, chaps, xix. and xxii., or numerous other treatises on physics.
+  The theory of the semi-permeable membrane is discussed by M. Planck
+  in his _Treatise on Thermodynamics_, English translation by A. Ogg
+  (1903), also in treatises on thermodynamics by W. Voigt and other
+  writers. For a more detailed study of diffusion in general the
+  following papers may be consulted:--L. Boltzmann, "Zur Integration der
+  Diffusionsgleichung," _Sitzung. der k. bayer. Akad math.-phys. Klasse_
+  (May 1894); T. des Coudres, "Diffusionsvorgänge in einem Zylinder,"
+  _Wied. Ann._ lv. (1895), p. 213; J. Loschmidt,
+  "Experimentaluntersuchungen über Diffusion," _Wien. Sitz._ lxi., lxii.
+  (1870); J. Stefan, "Gleichgewicht und ... Diffusion von Gasmengen,"
+  _Wien. Sitz._ lxiii., "Dynamische Theorie der Diffusion," _Wien.
+  Sitz._ lxv. (April 1872); M. Toepler, "Gas-diffusion," _Wied. Ann._
+  lviii. (1896), p. 599; A. Wretschko, "Experimentaluntersuchungen über
+  die Diffusion von Gasmengen," _Wien. Sitz._ lxii. The mathematical
+  theory of diffusion, according to the kinetic theory of gases, has
+  been treated by a number of different methods, and for the study of
+  these the reader may consult L. Boltzmann, _Vorlesungen über
+  Gastheorie_ (Leipzig, 1896-1898); S. H. Burbury, _Kinetic Theory of
+  Gases_ (Cambridge, 1899), and papers by L. Boltzmann in _Wien. Sitz._
+  lxxxvi. (1882), lxxxvii. (1883); P. G. Tait, "Foundations of the
+  Kinetic Theory of Gases," _Trans. R.S.E._ xxxiii., xxxv., xxvi., or
+  _Scientific Papers_, ii. (Cambridge, 1900). For recent work reference
+  should be made to the current issues of _Science Abstracts_ (London),
+  and entries under the heading "Diffusion" will be found in the general
+  index at the end of each volume.     (G. H. BR.)
+
+
+
+
+DIGBY, SIR EVERARD (1578-1606), English conspirator, son of Everard
+Digby of Stoke Dry, Rutland, was born on the 16th of May 1578. He
+inherited a large estate at his father's death in 1592, and acquired a
+considerable increase by his marriage in 1596 to Mary, daughter and heir
+of William Mulsho of Gothurst (now Gayhurst), in Buckinghamshire. He
+obtained a place in Queen Elizabeth's household and as a ward of the
+crown was brought up a Protestant; but about 1599 he came under the
+influence of the Jesuit, John Gerard, and soon afterwards joined the
+Roman Catholics. He supported James's accession and was knighted by the
+latter on the 23rd of April 1603. In a letter to Salisbury, the date of
+which has been ascribed to May 1605, Digby offered to go on a mission to
+the pope to obtain from the latter a promise to prevent Romanist
+attempts against the government in return for concessions to the Roman
+Catholics; adding that if severe measures were again taken against them
+"within brief there will be massacres, rebellions and desperate attempts
+against the king and state." Digby had suffered no personal injury or
+persecution on account of his religion, but he sympathized with his
+co-religionists; and when at Michaelmas, 1605, the government had fully
+decided to return to the policy of repression, the authors of the
+Gunpowder Plot (q.v.) sought his financial support, and he joined
+eagerly in the conspiracy. His particular share in the plan was the
+organization of a rising in the Midlands; and on the pretence of a
+hunting party he assembled a body of gentlemen together at Danchurch in
+Warwickshire on the 5th of November, who were to take action immediately
+the news arrived from London of the successful destruction of the king
+and the House of Lords, and to seize the person of the princess
+Elizabeth, who was residing in the neighbourhood. The conspirators
+arrived late on the evening of the 6th to tell their story of failure
+and disaster, and Digby, who possibly might have escaped the more
+serious charge of high treason, was persuaded by Catesby, with a false
+tale that the king and Salisbury were dead, to further implicate himself
+in the plot and join the small band of conspirators in their hopeless
+endeavour to raise the country. He accompanied them, the same day, to
+Huddington in Worcestershire and on the 7th to Holbeche in
+Staffordshire. The following morning, however, he abandoned his
+companions, dismissed his servants except two, who declared "they would
+never leave him but against their will," and attempted with these to
+conceal himself in a pit. He was, however, soon discovered and
+surrounded. He made a last effort to break through his captors on
+horseback, but was taken and conveyed a prisoner to the Tower. His trial
+took place in Westminster Hall, on the 27th of January 1606, and alone
+among the conspirators he pleaded guilty, declaring that the motives of
+his crime had been his friendship for Catesby and his devotion to his
+religion. He was condemned to death, and his execution, which took place
+on the 31st, in St Paul's Churchyard, was accompanied by all the
+brutalities exacted by the law.
+
+Digby was a handsome man, of fine presence. Father Gerard extols his
+skill in sport, his "riding of great horses," as well as his skill in
+music, his gifts of mind and his religious devotion, and concludes "he
+was as complete a man in all things, that deserved estimation or might
+win affection as one should see in a kingdom." Some of Digby's letters
+and papers, which include a poem before his execution, a last letter to
+his infant sons and correspondence with his wife from the Tower, were
+published in _The Gunpowder Treason_ by Thomas Barlow, bishop of
+Lincoln, in 1679. He left two sons, of whom the elder, Sir Kenelm Digby,
+was the well-known author and diplomatist.
+
+  See works on the Gunpowder Plot; Narrative of Father Gerard, in
+  _Condition of the Catholics under James I._ by J. Morris (1872), &c. A
+  life of Digby under the title of _A Life of a Conspirator_, by a
+  Romish Recusant (Thomas Longueville), was published in 1895.
+       (P. C. Y.)
+
+
+
+
+DIGBY, SIR KENELM (1603-1665), English author, diplomatist and naval
+commander, son of Sir Everard Digby (q.v.), was born on the 11th of July
+1603, and after his father's execution in 1606 resided with his mother
+at Gayhurst, being brought up apparently as a Roman Catholic. In 1617 he
+accompanied his cousin, Sir John Digby, afterwards 1st earl of Bristol,
+and then ambassador in Spain, to Madrid. On his return in April 1618 he
+entered Gloucester Hall (now Worcester College), Oxford, and studied
+under Thomas Allen (1542-1632), the celebrated mathematician, who was
+much impressed with his abilities and called him the _Mirandula_, i.e.
+the infant prodigy, of his age.[1] He left the university without taking
+a degree in 1620, and travelled in France, where, according to his own
+account, he inspired an uncontrollable passion in the queen-mother,
+Marie de' Medici, now a lady of more than mature age and charms; he
+visited Florence, and in March 1623 joined Sir John Digby again at
+Madrid, at the time when Prince Charles and Buckingham arrived on their
+adventurous expedition. He joined the prince's household and returned
+with him to England on the 5th of October 1623, being knighted by James
+I. on the 23rd of October and receiving the appointment of gentleman of
+the privy chamber to Prince Charles. In 1625 he married secretly
+Venetia, daughter of Sir Edward Hanley of Tonge Castle, Shropshire, a
+lady of extraordinary beauty and intellectual attainments, but of
+doubtful virtue. Digby was a man of great stature and bodily strength.
+Edward Hyde, afterwards earl of Clarendon, who with Ben Jonson was
+included among his most intimate friends, describes him as "a man of
+very extraordinary person and presence which drew the eyes of all men
+upon him, a wonderful graceful behaviour, a flowing courtesy and
+civility, and such a volubility of language as surprised and
+delighted."[2] Digby for some time was excluded from public employment
+by Buckingham's jealousy of his cousin, Lord Bristol. At length in 1627,
+on the latter's advice, Digby determined to attempt "some generous
+action," and on the 22nd of December, with the approval of the king,
+embarked as a privateer with two ships, with the object of attacking the
+French ships in the Venetian harbour of Scanderoon. On the 18th of
+January he arrived off Gibraltar and captured several Spanish and
+Flemish vessels. From the 15th of February to the 27th of March he
+remained at anchor off Algiers on account of the sickness of his men,
+and extracted a promise from the authorities of better treatment of the
+English ships. He seized a rich Dutch vessel near Majorca, and after
+other adventures gained a complete victory over the French and Venetian
+ships in the harbour of Scanderoon on the 11th of June. His successes,
+however, brought upon the English merchants the risk of reprisals, and
+he was urged to depart. He returned home in triumph in February 1629,
+and was well received by the king, and was made a commissioner of the
+navy in October 1630, but his proceedings were disavowed on account of
+the complaints of the Venetian ambassador. In 1633 Lady Digby died, and
+her memory was celebrated by Ben Jonson in a series of poems entitled
+_Eupheme_, and by other poets of the day. Digby retired to Gresham
+College, and exhibited extravagant grief, maintaining a seclusion for
+two years. About this time Digby professed himself a Protestant, but by
+October 1635, while in France, he had already returned to the Roman
+Catholic faith.[3] In a letter dated the 27th of March 1636 Laud
+remonstrates with him, but assures him of the continuance of his
+friendship.[4] In 1638 he published _A Conference with a Lady about
+choice of a Religion_, in which he argues that the Roman Church,
+possessing alone the qualifications of universality, unity of doctrine
+and uninterrupted apostolic succession, is the only true church, and
+that the intrusion of error into it is impossible. The same subject is
+treated in letters to George Digby, afterwards 2nd earl of Bristol,
+dated the 2nd of November 1638 and the 29th of November 1639, which were
+published in 1651, as well as in a further _Discourse concerning
+Infallibility in Religion_ in 1652. Returning to England he associated
+himself with the queen and her Roman Catholic friends, and joined in the
+appeal to the English Romanists for money to support the king's Scottish
+expedition.[5] In consequence he was summoned to the bar of the House of
+Commons on the 27th of January 1641, and the king was petitioned to
+remove him with other recusants from his councils. He left England, and
+while at Paris killed in a duel a French lord who had insulted Charles
+I. in his presence. Louis XIII. took his part, and furnished him with a
+military escort into Flanders. Returning home he was imprisoned, by
+order of the House of Commons, early in 1642, successively in the "Three
+Tobacco Pipes nigh Charing Cross," where his delightful conversation is
+said to have transformed the prison into "a place of delight,"[6] and at
+Winchester House. He was finally released and allowed to go to France on
+the 30th of July 1643, through the intervention of the queen of France,
+Anne of Austria, on condition that he would neither promote nor conceal
+any plots abroad against the English government.
+
+Before leaving England an attempt was made to draw from him an admission
+that Laud, with whom he had been intimate, had desired to be made a
+cardinal, but Digby denied that the archbishop had any leanings towards
+Rome. On the 1st of November 1643 it was resolved by the Commons to
+confiscate his property. He published in London the same year
+_Observations on the 22nd stanza in the 9th canto of the 2nd book of
+Spenser's "Faërie Queene,"_ the MS. of which is in the Egerton
+collection (British Museum, No. 2725 f. 117 b), and _Observations_ on a
+surreptitious and unauthorized edition of the _Religio Medici_, by Sir
+Thomas Browne, from the Roman Catholic point of view, which drew a
+severe rebuke from the author. After his arrival in Paris he published
+his chief philosophical works, _Of Bodies_ and _Of the Immortality of
+Man's Soul_ (1644), autograph MSS. of which are in the Bibliothèque Ste
+Geneviève at Paris, and made the acquaintance of Descartes. He was
+appointed by Queen Henrietta Maria her chancellor, and in the summer of
+1645 he was despatched by her to Rome to obtain assistance. Digby
+promised the conversion of Charles and of his chief supporters. At first
+his eloquence made a great impression. Pope Innocent X. declared that he
+spoke not merely as a Catholic but as an ecclesiastic. But the absence
+of any warrant from Charles himself roused suspicions as to the solidity
+of his assurances, and he obtained nothing but a grant of 20,000 crowns.
+A violent quarrel with the pope followed, and he returned in 1646,
+having consented in the queen's name to complete religious freedom for
+the Roman Catholics, both in England and Ireland, to an independent
+parliament in Ireland, and to the surrender of Dublin and all the Irish
+fortresses into the hands of the Roman Catholics, the king's troops to
+be employed in enforcing the articles and the pope granting about
+£36,000 with a promise of further payments in obtaining direct
+assistance. In February 1649 Digby was invited to come to England to
+arrange a proposed toleration of the Roman Catholics, but on his arrival
+in May the scheme had already been abandoned. He was again banished on
+the 31st of August, and it was not till 1654 that he was allowed by the
+council of state to return. He now entered into close relations with
+Cromwell, from whom he hoped to obtain toleration for the Roman
+Catholics, and whose alliance he desired to secure for France rather
+than for Spain, and was engaged by Cromwell, much to the scandal of
+both Royalists and Roundheads, in negotiations abroad, of which the aim
+was probably to prevent a union between those two foreign powers. He
+visited Germany, in 1660 was in Paris, and at the Restoration returned
+to England. He was well received in spite of his former relations with
+Cromwell, and was confirmed in his post as Queen Henrietta Maria's
+chancellor. In January 1661 he delivered a lecture, which was published
+the same month, at Gresham College, on the vegetation of plants, and
+became an original member of the Royal Society in 1663. In January 1664
+he was forbidden to appear at court, the cause assigned being that he
+had interposed too far in favour of the 2nd earl of Bristol, disgraced
+by the king on account of the charge of high treason brought by him
+against Clarendon into the House of Lords. The rest of his life was
+spent in the enjoyment of literary and scientific society at his house
+in Covent Garden. He died on the 11th of June 1665. He had five
+children, of whom two, a son and one daughter, survived him.
+
+Digby, though he possessed for the time a considerable knowledge of
+natural science, and is said to have been the first to explain the
+necessity of oxygen to the existence of plants, bears no high place in
+the history of science. He was a firm believer in astrology and alchemy,
+and the extraordinary fables which he circulated on the subject of his
+discoveries are evidence of anything rather than of the scientific
+spirit. In 1656 he made public a marvellous account of a city in
+Tripoli, petrified in a few hours, which he printed in the _Mercurius
+Politicus_. Malicious reports had been current that his wife had been
+poisoned by one of his prescriptions, viper wine, taken to preserve her
+beauty. Evelyn, who visited him in Paris in 1651, describes him as an
+"errant mountebank." Henry Stubbes characterizes him as "the very Pliny
+of our age for lying," and Lady Fanshawe refers to the same
+"infirmity."[7] His famous "powder of sympathy," which seems to have
+been only powder of "vitriol," healed without any contact, by being
+merely applied to a rag or bandage taken from the wound, and Digby
+records a miraculous cure by this means in a lecture given by him at
+Montpellier on this subject in 1658, published in French and English the
+same year, in German in 1660 and in Dutch in 1663; but Digby's claim to
+its original discovery is doubtful, Nathaniel Highmore in his _History
+of Generation_ (1651, p. 113) calling the powder "Talbot's powder," and
+ascribing its invention to Sir Gilbert Talbot. Some of Digby's pills and
+preparations, however, described in _The Closet of the Eminently Learned
+Sir Kenelm Digby Knt. Opened_ (publ. 1677), are said to make less demand
+upon the faith of patients, and his injunction on the subject of the
+making of tea, to let the water "remain upon it no longer than you can
+say the Miserere Psalm very leisurely," is one by no means to be
+ridiculed. As a philosopher and an Aristotelian Digby shows little
+originality and followed the methods of the schoolmen. His Roman
+Catholic orthodoxy mixed with rationalism, and his political opinions,
+according to which any existing authority should receive support, were
+evidently derived from Thomas White (1582-1676), the Roman Catholic
+philosopher, who lived with him in France. White published in 1651
+_Institutionum Peripateticorum libri quinque_, purporting to expound
+Digby's "peripatetic philosophy," but going far beyond Digby's published
+treatises. Digby's _Memoirs_ are composed in the high-flown fantastic
+manner then usual when recounting incidents of love and adventure, but
+the style of his more sober works is excellent. In 1632 he presented to
+the Bodleian library a collection of 236 MSS., bequeathed to him by his
+former tutor Thomas Allen, and described in _Catalogi codicum
+manuscriptorum bibliothecae Bodleianae_, by W. D. Macray, part ix.
+Besides the works already mentioned Digby translated _A Treatise of
+adhering to God written by Albert the Great, Bishop of Ratisbon_ (1653);
+and he was the author of _Private Memoirs_, published by Sir N. H.
+Nicholas from _Harleian MS. 6758_ with introduction (1827); _Journal of
+the Scanderoon Voyage in 1628_, printed by J. Bruce with preface (Camden
+Society, 1868); _Poems from Sir Kenelm Digby's Papers_... with preface
+and notes (Roxburghe Club, 1877); in the _Add. MSS._ 34,362 f. 66 is a
+poem _Of the Miserys of Man_, probably by Digby; _Choice of Experimental
+Receipts in Physick and Chirurgery_ ... _collected by Sir K. Digby_
+(1668), and _Chymical Secrets and Rare Experiments_ (1683), were
+published by G. Hartman, who describes himself as Digby's steward and
+laboratory assistant.
+
+  See the _Life of Sir Kenelm Digby by one of his Descendants_ (T.
+  Longueville), 1896.     (P. C. Y.)
+
+
+FOOTNOTES:
+
+  [1] _Letters by Eminent Persons_ (Aubrey's Lives), ii. 324.
+
+  [2] _Life and Continuation._
+
+  [3] Strafford's _Letters_, i. 474.
+
+  [4] Laud's _Works_, vi. 447.
+
+  [5] _Thomason Tracts_, Brit. Mus. E 164 (15).
+
+  [6] _Archaeologia Cantiana_, ii. 190.
+
+  [7] _Dict. of Nat. Biog._ sub "Digby." See also Robert Boyle's _Works_
+    (1744), v. 302.
+
+
+
+
+DIGBY, KENELM HENRY (1800-1880), English writer, youngest son of William
+Digby, dean of Clonfert, was born at Clonfert, Ireland, in 1800. He was
+educated at Trinity College, Cambridge, and soon after taking his B.A.
+degree there in 1819 became a Roman Catholic. He spent most of his life,
+which was mainly devoted to literary pursuits, in London, where he died
+on the 22nd of March 1880. Digby's reputation rests chiefly on his
+earliest publication, _The Broadstone of Honour, or Rules for the
+Gentlemen of England_ (1822), which contains an exhaustive survey of
+medieval customs, full of quotations from varied sources. The work was
+subsequently enlarged and issued (1826-1827) in four volumes entitled:
+_Godefridus_, _Tancredus_, _Morus_ and _Orlandus_ (numerous
+re-impressions, the best of which is the edition brought out by B.
+Quaritch in five volumes, 1876-1877).
+
+  Among Digby's other works are: _Mores Catholici, or Ages of Faith_ (11
+  vols., London, 1831-1840); _Compitum; or the Meeting of the Ways at
+  the Catholic Church_ (7 vols., London, 1848-1854); _The Lovers' Seat,
+  Kathemérina; or Common Things in relation to Beauty, Virtue and Faith_
+  (2 vols., London, 1856). A complete list is given in J. Gillow's
+  _Bibliographical Dictionary of English Catholics_, ii. 81-83.
+
+
+
+
+DIGENES ACRITAS, BASILIUS, Byzantine national hero, probably lived in
+the 10th century. He is named Digenes (of double birth) as the son of a
+Moslem father and a Christian mother; Acritas ([Greek: akra], frontier,
+boundary), as one of the frontier guards of the empire, corresponding to
+the Roman _milites limitanei_. The chief duty of these _acritae_
+consisted in repelling Moslem inroads and the raids of the _apelatae_
+(cattle-lifters), brigands who may be compared with the more modern
+Klephts. The original Digenes epic is lost, but four poems are extant,
+in which the different incidents of the legend have been worked up by
+different hands. The first of these consists of about 4000 lines,
+written in the so-called "political" metre, and was discovered in the
+latter part of the 19th century, in a 16th-century MS., at Trebizond;
+the other three MSS. were found at Grotta Ferrata, Andros and Oxford.
+The poem, which has been compared with the _Chanson de Roland_ and the
+_Romance of the Cid_, undoubtedly contains a kernel of fact, although it
+cannot be regarded as in any sense an historical record. The scene of
+action is laid in Cappadocia and the district of the Euphrates.
+
+  Editions of the Trebizond MS. by C. Sathas and E. Legrand in the
+  _Collection des monuments pour servir à l'étude de la langue
+  néohellénique_, new series, vi. (1875), and by S. Joannides
+  (Constantinople, 1887). See monographs by A. Luber (Salzburg, 1885)
+  and G. Wartenberg (Berlin, 1897). Full information will be found in C.
+  Krumbacher's _Geschichte der byzantinischen Litteratur_, p. 827 (2nd
+  ed., 1897); see also G. Schlumberger, _L'Épopée Byzantine à la fin du
+  dixième siècle_ (1897).
+
+
+
+
+DIGEST, a term used generally of any digested or carefully arranged
+collection or compendium of written matter, but more particularly in law
+of a compilation in condensed form of a body of law digested in a
+systematical method; e.g. the Digest (_Digesta_) or Pandects ([Greek:
+Pandektai]) of Justinian, a collection of extracts from the earlier
+jurists compiled by order of the emperor Justinian. The word is also
+given to the compilations of the main points (marginal or hand-notes) of
+decided cases, usually arranged in alphabetical and subject order, and
+published under such titles as "Common Law Digest," "Annual Digest," &c.
+
+
+
+
+DIGESTIVE ORGANS (PATHOLOGY). Several facts of importance have to be
+borne in mind for a proper appreciation of the pathology of the organs
+concerned in digestive processes (for the anatomy see ALIMENTARY CANAL
+and allied articles). In the first place, more than all other systems,
+the digestive comprises greater range of structure and exhibits wider
+diversity of function within its domain. Each separate structure and
+each different function presents special pathological signs and
+symptoms. Again, the duties imposed upon the system have to be performed
+notwithstanding constant variations in the work set them. The crude
+articles of diet offered them vary immensely in nature, bulk and
+utility, from which they must elaborate simple food-elements for
+absorption, incorporate them after absorption into complex organic
+substances properly designed to supply the constant needs of cellular
+activity, of growth and repair, and fitly harmonized to fulfil the many
+requirements of very divergent processes and functions. Any form of
+unphysiological diet, each failure to cater for the wants of any special
+tissue engaged in, or of any processes of, metabolism, carry with them
+pathological signs. Perhaps in greater degree than elsewhere are the
+individual sections of the digestive system dependent upon, and closely
+correlated with, one another. The lungs can only yield oxygen to the
+blood when the oxygen is uncombined; no compounds are of use. The
+digestive organs have to deal with an enormous variety of compound
+bodies, from which to obtain the elements necessary for protoplasmic
+upkeep and activity. Morbid lesions of the respiratory and circulatory
+systems are frequently capable of compensation through increased
+activity elsewhere, and the symptoms they give rise to follow chiefly
+along one line; diseases of the digestive organs are more liable to
+occasion disorders elsewhere than to excite compensatory actions. The
+digestive system includes every organ, function and process concerned
+with the utilization of food-stuffs, from the moment of their entrance
+into the mouth, their preparation in the canal, assimilation with the
+tissues, their employment therein, up to their excretion or expulsion in
+the form of waste. Each portion resembles a link of a continuous chain;
+each link depends upon the integrity of the others, the weakening or
+breaking of one straining or making impotent the chain as a whole.
+
+The mucous membrane lining the alimentary tract is the part most subject
+to pathological alterations, and in this connexion it should be
+remembered that this membrane differs both in structure and functions
+throughout the tract. Chiefly protective from the mouth to the cardia,
+it is secretory and absorbent in the stomach and bowel; while the
+glandular cells forming part of it secrete both acid and alkaline
+fluids, several ferments or mucus. Over the dorsum of the tongue its
+modified cells subserve the sense of taste. Without, connected with it
+by the submucous connective tissue, is placed the muscular coat, and
+externally over the greater portion of its length the peritoneal serous
+membrane. All parts are supplied with blood-vessels, lymph-ducts and
+nerves, the last belonging either to local or to central circuits.
+Associated with the tract are the salivary glands, the liver and the
+pancreas; while, in addition, lymphoid tissue is met with diffusely
+scattered throughout the lining membranes in the tonsils, appendix,
+solitary glands and Peyer's patches, and the mesenteric glands. The
+functions of the various parts of the system in whose lesions we are
+here interested are many in number, and can only be summarized here.
+(For the physiology of digestion see NUTRITION.) Broadly, they maybe
+given as: (1) Ingestion and swallowing of food, transmission of it
+through the tract, and expulsion of the waste material; (2) secretion of
+acids and alkalis for the performance of digestive processes, aided by
+(3) elaboration and addition of complex bodies, termed enzymes or
+ferments; (4) secretion of mucus; (5) protection of the body against
+organismal infection, and against toxic products; (6) absorption of food
+elements and reconstitution of them into complex substances fitted for
+metabolic application; and (7) excretion of the waste products of
+protoplasmic action. These functions may be altered by disease, singly
+or in conjunction; it is rare, however, to find but one affected, while
+an apparently identical disturbance of function may often arise from
+totally different organic lesions. Another point of importance is seen
+in the close interdependence which exists between the secretions of acid
+and those of alkaline reaction. The difference in reaction seems to act
+_mutatis mutandis_ as a stimulant in each instance.
+
+
+_General Diseases._
+
+  Vascular lesions.
+
+In all sections of the alimentary canal actively engaged in the
+digestion of food, a well-marked local engorgement of the blood-vessels
+supplying the walls occurs. The hyperaemia abates soon after completion
+of the special duties of the individual sections. This normal condition
+may be abnormally exaggerated by overstimulation from irritant poisons
+introduced into the canal; from too rich, too copious or indigestible
+articles of diet; or from too prolonged an experience of some unvaried
+kind of food-stuff, especially if large quantities of it are necessary
+for metabolic needs; entering into the first stage of inflammation,
+acute hyperaemia. More important, because productive of less tractable
+lesions, is passive congestion of the digestive organs. Whenever the
+flow of blood into the right side of the heart is hindered, whether it
+arise from disease of the heart itself, or of the lungs, or proceed from
+obstruction in some part of the portal system, the damming-back of the
+venous circulation speedily produces a more or less pronounced stasis of
+the blood in the walls of the alimentary canal and in the associated
+abdominal glands. The lack of a sufficiently vigorous flow of blood is
+followed by deficient secretion of digestive agents from the glandular
+elements involved, by decreased motility of the muscular coats of the
+stomach and bowel, and lessened adaptability throughout for dealing with
+even slight irregular demands on their powers. The mucous membrane of
+the stomach and bowel, less able to withstand the effects of irritation,
+even of a minor character, readily passes into a condition of chronic
+catarrh, while it frequently is the seat of small abrasions,
+haemorrhagic erosions, which may cause vomiting of blood and the
+appearance of blood in the stools. Obstruction to the flow of blood from
+the liver leads to dilatation of its blood-vessels, consequent pressure
+upon the hepatic cells adjoining them, and their gradual loss of
+function, or even atrophy and degeneration. In addition to the results
+of such passive congestion exhibited by the stomach and bowel as noted
+above, passive congestion of the liver is often accompanied by varicose
+enlargement of the abdominal veins, in particular of those which
+surround the lower end of the oesophagus, the lowest part of the rectum
+and anus. In the latter position these dilated veins constitute what are
+known as haemorrhoids or piles, internal or external as their site lies
+within or outside the anal aperture.
+
+
+  Inflammatory lesions.
+
+The mucous and serous membranes of the canal and the glandular elements
+of the associated organs are the parts most subject to inflammatory
+affections. Among the several sections of the digestive tract itself,
+the oesophagus and jejunum are singularly exempt from inflammatory
+processes; the fauces, stomach, caecum and appendix, ileum, mouth and
+duodenum (including the opening of the common bile-duct), are more
+commonly involved. _Stomatitis_, or inflammation of the mouth, has many
+predisposing factors, but it has now been definitely determined that its
+exciting cause is always some form of micro-organism. Any condition
+favouring oral sepsis, as carious teeth, pyorrhoea alveolaris (a
+discharge of pus due to inflamed granulations round carious teeth),
+granulations beneath thick crusts of tartar, or an irritating tooth
+plate, favours the growth of pyogenic organisms and hence of stomatitis.
+Many varieties of this disease have been described, but all are forms of
+"pyogenic" or "septic stomatitis." This in its mildest form is catarrhal
+or erythematous, and is attended only by slight swelling tenderness and
+salivation. In its next stage of acuteness it is known as "membranous,"
+as a false membrane is produced somewhat resembling that due to
+diphtheria, though caused by a staphylococcus only. A still more acute
+form is "ulcerative," which may go on to the formation of an abscess
+beneath the tongue. Scarlet fever usually gives rise to a slight
+inflammation of the mouth followed by desquamation, but more rarely it
+is accompanied by a most severe oedematous stomatitis with glossitis and
+tonsillitis. Erysipelas on the face may infect the mouth, and an acute
+stomatitis due to the diphtheria bacillus, Klebs-Loeffler bacillus, has
+been described. A distinct and very dangerous form of stomatitis in
+infants and young children is known as "aphthous stomatitis" or
+"thrush." This is caused by the growth of _Oidium albicans_. It is
+always preceded by a gastro-enteritis and dry mouth, and if this is not
+attended to, soon attracts attention by the little white raised patches
+surrounded by a dusky red zone scattered on tongue and cheeks.
+Epidemics have occurred in hospitals and orphanages. Mouth breathing is
+the cause of many ills. As a result of this, the mucous membrane of the
+tongue, &c., becomes dry, micro-organisms multiply and the mouth becomes
+foul. Also from disease of the nose, the upper jaw, palate and teeth do
+not make proper progress in development. There is overgrowth of tonsils,
+and adenoids, with resulting deafness, and the child's mental
+development suffers. An ordinary "sore throat" usually signifies acute
+catarrh of the fauces, and is of purely organismal origin, "catching
+cold" being only a secondary and minor cause. In "relaxed throats" there
+is a chronic catarrhal state of the lining membrane, with some passive
+congestion. The tonsils are peculiarly liable to catarrhal attacks, as
+might a priori be expected by reason of their Cerberus-like function
+with regard to bacterial intruders. Still, acute attacks of tonsillitis
+appear on good evidence to be more common among individuals predisposed
+constitutionally to rheumatic manifestations. Cases of acute tonsillitis
+may or may not go on to suppuration or quinsy; in all there is great
+congestion of the glands, increased mucus secretion, and often secondary
+involvement of the lymphatic glands of the neck. Repeated acute attacks
+often lead to chronic inflammation, in which the glands are enlarged,
+and often hypertrophied in the true sense of the term. The oesophagus is
+the seat of inflammation but seldom. In infants and young children
+thrush due to _Oidium albicans_ may spread from the mouth, and also a
+diphtheritic inflammation spreads from the fauces into the oesophagus. A
+catarrhal oesophagitis is rarely seen, but the commonest form is
+traumatic, due to the swallowing of boiling water, corrosive or irritant
+substances, &c. A non-malignant ulceration may result which later leads
+on to an oesophageal stricture. The physical changes presented by the
+coats of the stomach and the intestine, the subjects of catarrhal
+attacks, closely resemble one another, but differ symptomatically. Acute
+catarrh of the stomach is associated with intense hyperaemia of its
+lining coats, with visible engorgement and swelling of the mucous
+membrane, and an excessive secretion of mucus. The formation of active
+gastric juice is arrested, digestion ceases, peristaltic movements are
+sluggish or absent, unless so over-stimulated that they act in a
+direction the reverse of the normal, and induce expulsion of the gastric
+contents by vomiting. The gastric contents, in whatever degree of
+dilution or concentration they may have been ingested, when ejected are
+of porridge-thick consistency, and often but slightly digested. Such
+conditions may succeed a severe alcoholic bout, be caused by irritant
+substances taken in by the mouth or arise from fermentative processes in
+the stomach contents themselves. Should the irritating material succeed
+in passing from the stomach into the bowel, similar physical signs are
+present; but as the quickest path offered for the expulsion of the
+offending substances from the body is downwards, peristalsis is
+increased, the flow of fluid from the intestinal glands is larger in
+bulk, though of less potency as regards its normal actions, than in
+health, and diarrhoea, with removal of the irritant, follows. As a
+general rule, the more marked the involvement of the large bowel, the
+severer and more fluid is the resultant diarrhoea. Inflammation of the
+stomach may be due to mechanical injury, thermal or chemical irritants
+or invasion by micro-organisms. Also all the symptoms of gastric catarrh
+may be brought on by any acute emotion. The commonest mechanical injury
+is that due to an excess of food, especially when following on a fast;
+poisons act as irritants, and also the weevils of cheese and the larvae
+of insects.
+
+Inflammatory affections of the caecum and its attached appendix
+vermiformis are very common, and give rise to several special symptoms
+and signs. Acute inflammatory appendicitis appears to be increasing in
+frequency, and is associated by many with the modern deterioration in
+the teeth. Constipation certainly predisposes to it, and it appears to
+be more prevalent among medical men, commercial travellers, or any
+engaged in arduous callings, subjected to irregular meals, fatigue and
+exposure. A foreign body is the exciting cause in many cases, though
+less commonly so than was formerly imagined. The inflammation in the
+appendix varies in intensity from a very slight catarrhal or simple
+form to an ulcerative variety, and much more rarely to the acute
+fulminating appendicitis in which necrosis of the appendix with abscess
+formation occurs. It is always accompanied by more or less peritonitis,
+which is protective in nature, shutting in the inflammatory process.
+Very similar symptomatically is the condition termed perityphlitis,
+doubtless in former days frequently due to the appendix, an acute or
+chronic inflammation of the walls of the caecum often leading to abscess
+formation outside the gut, with or without direct communication with the
+canal. The colon is subject to three main forms of inflammation. In
+simple _colitis_ the mucous membrane of the colon is intensely injected,
+bright red in colour, and secreting a thick mucus, but there is no
+accompanying ulceration. It is often found in association with some
+constitutional disease, as Bright's disease, and also with cancer of the
+bowel. But when it has no association with other trouble it is probably
+bacterial in origin, the _Bacillus enteritidis spirogenes_ having been
+isolated in many cases. The motions always contain large quantities of
+mucus and more or less blood. A second very severe form of inflammation
+of the colon is known as "membranous colitis," and this may be either
+dyspeptic, or secondary to other diseases. In this trouble membranes are
+passed _per anum_, accompanied by a pain so intense as often to cause
+fainting. In severe cases complete tubular casts of the intestine have
+been found. Often the motions contain very little faecal matter, but
+consist only of membranes, mucus and a little blood. A third form is
+that known as "ulcerative colitis." Any part of the large intestine may
+be affected, and the ulceration shows no special distribution. In severe
+cases the muscular coat is exposed, and perforation may ensue. The
+number of ulcers varies from a few to many dozen, and in size from a pea
+to a five-shilling piece. Like all chronic intestinal ulcers they show a
+tendency to become transverse.
+
+Chronic catarrhal affections of the stomach are very common, and often
+follow upon repeated acute attacks. In them the connective tissue
+increases at the expense of the glandular elements; the mucous membrane
+becomes thickened and less active in function. Should the muscular coat
+be involved, the elasticity and contractility of the organ suffer;
+peristaltic movement is weakened; expulsion of the contents through the
+pylorus hindered; and, aggravated by these effects, the condition
+becomes worse, atonic dyspepsia in its most pronounced form results,
+with or without dilatation. Chronic vascular congestion may occasion in
+process of time similar signs and symptoms.
+
+Duodenal catarrh is constantly associated with jaundice, indeed is most
+probably the commonest cause of catarrhal jaundice; often it is
+accompanied by catarrh of the common bile-duct. Chronic inflammation of
+the small intestine gives rise to less prominent symptoms than in the
+stomach. It generally arises from more than one cause; or rather
+secondary causes rapidly become as important as the primary in its
+incidence. Chronic congestion and prolonged irritation lead to deficient
+secretion and sluggish peristalsis; these effects encourage intestinal
+putrefaction and auto-intoxication; and these latter, in turn, increase
+the local unrest.
+
+
+  Infective lesions.
+
+The intestinal mucous membrane, the peritoneum and the mesenteric glands
+are the chief sites of tubercular infection in the digestive organs.
+Rarely met with in the gullet and stomach, and comparatively seldom in
+the mouth and lips, tubercular inflammation of the small intestine and
+peritoneum is common. Tubercular enteritis is a frequent accompaniment
+of phthisis, but may occur apart from tubercle of other organs. Children
+are especially subject to the primary form. Tubercular peritonitis often
+is present also. The inflammatory process readily tends towards ulcer
+formation, with haemorrhage and sometimes perforation. If in the large
+bowel, the symptoms are usually less acute than those characterizing
+tubercular inflammation of the small intestine. The appendix has been
+found to be the seat of tubercular processes; in the rectum they form
+the general cause of the fistulae and abscesses so commonly met with
+here. Tubercular peritonitis may be primary or secondary, acute or
+chronic; occasionally very acute cases are seen running a rapid course;
+the majority are chronic in type. The tubercles spread over the surface
+of the serous membrane, and if small and not very numerous may give rise
+in chronic cases to few symptoms; if larger, and especially when they
+involve and obstruct the lymph- and blood-vessels, ascites follows. It
+is hardly possible that tubercular invasion of the mesenteric glands can
+ever occur unaccompanied by peritoneal infection; but when the infection
+of the glands constitutes the most prominent sign, the term _tabes
+mesenterica_ is sometimes employed. Here the glands, enlarged, form a
+doughy mass in the abdomen, leading to marked protrusion of the
+abdominal walls, with wasting elsewhere and diarrhoea.
+
+The liver is seldom attacked by tubercle, unless in cases of general
+miliary tuberculosis. Now and then it contains large caseous tubercular
+masses in its substance.
+
+An important fact with regard to the tubercular processes in the
+digestive organs lies in the ready response to treatment shown by many
+cases of peritoneal or mesenteric invasion, particularly in the young.
+
+The later sequelae of syphilis display a predilection for the rectum and
+the liver, usually leading to the development of a stricture in the
+former, to a diffuse hepatitis or the formation of gummata in the
+second. In inherited syphilis the temporary teeth usually appear early,
+are discoloured and soon crumble away. The permanent teeth may be sound
+and healthy, but are often--especially the upper incisors--notched and
+stunted, when they are known as "Hutchinson's teeth." As the result both
+of syphilis and of tubercle, the tissues of the liver and bowel may
+present a peculiar alteration; they become amyloid, or lardaceous, a
+condition in which they appear "waxy," are coloured dark mahogany brown
+with dilute iodine solutions, and show degenerative changes in the
+connective tissue.
+
+The _Bacillus typhosus_ discovered by Eberth is the causal agent of
+typhoid fever, and has its chief seat of activity in the small
+intestine, more especially in the lower half of the ileum. Attacking the
+lymphoid follicles in the mucous membrane, it causes first inflammatory
+enlargement, then necrosis and ulceration. The adjacent portions of the
+mucous membrane show acute catarrhal changes. Diarrhoea, of a special
+"pea-soup" type, may or may not be present; while haemorrhage from the
+bowel, if ulcers have formed, is common. As the ulcers frequently extend
+down to the peritoneal coat of the bowel, perforation of this membrane
+and extravasation into the peritoneal cavity is easily induced by
+irritants introduced into or elaborated in the bowel, acting physically
+or by the excitation of hyper-peristalsis.
+
+True Asiatic cholera is due to the comma-bacillus or spirillum of
+cholera, which is found in the rice-water evacuations, in the contents
+of the intestine after death, and in the mucous membrane of the
+intestine just beneath the epithelium. It has not been found in the
+blood. It produces an intense irritation of the bowel, seldom of the
+stomach, without giving rise locally to any marked physical change; it
+causes violent diarrhoea and copious discharges of "rice-water" stools,
+consisting largely of serum swarming with the organism.
+
+Dysentery gives rise to an inflammation of the large intestine and
+sometimes of the lower part of the ileum, resulting in extensive
+ulceration and accompanied by faecal discharges of mucus, muco-pus or
+blood. In some forms a protozoan, the _Amoeba dysenteriae_, is found in
+the stools--this is the amoebic dysentery; in other cases a bacillus,
+_Bacillus dysenteriae_, is found--the bacillary dysentery.
+
+Acute parotitis, or mumps, is an infectious disease of the parotid
+glands, chiefly interesting because of the association between it and
+the testes in males, inflammation of these glands occasionally following
+or replacing the affection of the parotids. The causal agent is probably
+organismal, but has as yet escaped detection.
+
+
+  New growths.
+
+The relative frequency with which malignant growths occur in the
+different organs of the digestive system may be gathered from the
+tabular analysis, on p. 266, of 1768 cases recorded in the books of the
+Edinburgh Royal Infirmary as having been treated in the medical and
+surgical wards between the years 1892 and 1899 inclusive. Of these,
+1263, or 71.44%, were males; 505, or 28.56%, females. (See Table I. p.
+266.)
+
+If the figures there given be classified upon broader lines, the results
+are as given in Table II. p. 266, and speak for themselves.
+
+The digestive organs are peculiarly subject to malignant disease, a
+result of the incessant changes from passive to active conditions, and
+vice versa, called for by repeated introduction of food; while the
+comparative frequency with which different parts are attacked depends,
+in part, upon the degree of irritation or changes of function imposed
+upon them. Scirrhous, encephaloid and colloid forms of carcinoma occur.
+In the stomach and oesophagus the scirrhous form is most common, the
+soft encephaloid form coming next. The most common situation for
+cancerous growth in the stomach is the pyloric region. Walsh out of 1300
+cases found 60.8% near the pylorus, 11.4% over the lesser curvature, and
+4.7% more or less over the whole organ. The small intestine is rarely
+attacked by cancer; the large intestine frequently. The rectum, sigmoid
+flexure, caecum and colon are affected, and in this order, the
+cylindrical-celled form being the most common. Carcinoma of the
+peritoneum is generally colloid in character, and is often secondary to
+growths in other organs. Cancer of the liver follows cancer of the
+stomach and rectum in frequency of occurrence, and is relatively more
+common in females than males. Secondary invasion of the liver is a
+frequent sequel to gastric cancer. The pancreas occasionally is the seat
+of cancerous growth.
+
+Sarcomata are not so often met with in the digestive organs. When
+present, they generally involve the peritoneum or the mesenteric glands.
+The liver is sometimes attacked, the stomach rarely.
+
+Benign tumours are not of common occurrence in the digestive organs.
+Simple growths of the salivary glands, cysts of the pancreas and
+polypoid tumours of the rectum are the most frequent.
+
+
+  Animal parasites.
+
+The intestinal canal is the habitat of the majority of animal parasites
+found in man. Frequently their presence leads to no morbid symptoms,
+local or general; nor are the symptoms, when they do arise, always
+characteristic of the presence of parasites alone. Discovery of their
+bodies, or of their eggs, in the stools is in most instances the only
+satisfactory proof of their presence. The parasites found in the bowel
+belong principally to two natural groups, Protozoa and Metazoa. The
+great class of the Protozoa furnish amoebae, members of Sporozoa and
+Infusoria. The amoebae are almost invariably found in the large
+intestine; one species, indeed, is termed _Amoeba coli_. The frequently
+observed relation between attacks of dysentery and the presence of
+amoebae in the stools has led to the proposition that an _Amoeba
+dysenterica_ exists, causing the disease--a theory supported by the
+detection of amoebae in the contents of dysenteric abscesses of the
+liver. No symptoms of injury to health appear to accompany the presence
+of Sporozoa in the bowel, while the species of Infusoria found in it,
+the _Cercomonas_, and _Trichomonas intestinalis_, and the _Balantidium
+coli_, may or may not be guilty of prolonging conditions within the
+bowel as have previously set up diarrhoea.
+
+The Metazoa supply examples of intestinal parasites from the classes
+Annuloida and Nematoidea. To the former class belong the various
+tapeworms found in the small intestine of man. They, like other
+intestinal parasites, are destitute of any power of active digestion,
+simply absorbing the nutritious proceeds of the digestive processes of
+their hosts. Nematode worms infest both the small and large intestine;
+_Ascaris lumbricoides_, the common round worm, and the male _Oxyuris
+vermicularis_ are found in the small bowel, the adult female _Oxyuris
+vermicularis_ and the _Tricocephalus dispar_ in the large.
+
+The eggs of the _Trichina spiralis_, when introduced with the food,
+develop in the bowel into larval forms which invade the tissues of the
+body, to find in the muscles congenial spots wherein to reach maturity.
+Similarly, the eggs of the Echinococcus are hatched in the bowel, and
+the embryos proceed to take up their abode in the tissues of the body,
+developing into cysts capable of growth into mature worms after their
+ingestion by dogs.
+
+
+  Vegetable parasites.
+
+Numbers of bacterial forms habitually infest the alimentary canal. Many
+of them are non-pathogenic; some develop pathogenic characters only
+under provocation or when a suitable environment induces them to act in
+such a manner; others may form the _materies morbi_ of special lesions,
+or be casual visitors capable of originating disease if opportunity
+occurs. Apart from those organisms associated with acute infective
+diseases, disturbances of function and physical lesions may be the
+result of abnormal bacterial activity in the canal; and these
+disturbances may be both local and general. Many of the bacteria
+commonly present produce putrefactive changes in the contents of the
+tract by their metabolic processes. They render the medium they grow in
+alkaline, produce different gases and elaborate more or less virulent
+toxins. Other species set up an acid fermentation, seldom accompanied by
+gas or toxin formation. The products of either class are inimical to the
+free growth of members of the other. The species which produce acids are
+more resistant to the action of acids. Thus, when the contents of the
+stomach possess a normal or excessive proportion of free hydrochloric
+acid, a much larger number of putrefactive and pathogenic organisms in
+the food are destroyed or inhibited than of the bacteria of acid
+fermentation. Diminished gastric acidity allows of the entry of a
+greater number of putrefactive (and pathogenic) types, with, as a
+consequence, increased facilities for their growth and activity, and the
+appearance of intestinal derangements.
+
+    TABLE I.
+
+  +-----------------------------+--------------------------+--------------------------+
+  |        Males.               |        Females.          |        Both Sexes.       |
+  +---------------------+-------+------------------+-------+------------------+-------+
+  |Organ or Tissue in   | Per-  |Organ or Tissue in| Per-  |Organ or Tissue in| Per-  |
+  |Order of Frequency.  |centage|Order of Frequency|centage|Order of Frequency|centage|
+  +---------------------+-------+------------------+-------+------------------+-------+
+  | 1 Stomach           | 22.56 | 1 Stomach        | 22.37 | 1 Stomach        | 22.49 |
+  | 2 Lip               | 12.94 | 2 Rectum         | 17.24 | 2 Rectum         | 13.12 |
+  | 3 Rectum            | 11.57 | 3 Liver          | 15.50 | 3 Liver          | 10.02 |
+  | 4 Tongue            | 11.36 | 4 Peritoneum     |  7.86 | 4 Lip            |  9.89 |
+  | 5 Oesophagus        | 10.90 | 5 Oesophagus     |  5.33 | 5 Oesophagus     |  9.29 |
+  | 6 Liver             |  7.80 | 6 Sigmoid        |  4.53 | 6 Tongue         |  8.96 |
+  | 7 Jaw               |  6.38 | 7 Pancreas       |  3.52 | 7 Jaw            |  5.65 |
+  | 8 Mouth             |  2.88 | 8 Tongue         |  3.12 | 8 Peritoneum     |  2.94 |
+  | 9 Tonsils           |  2.09 | 9 Omentum        |  2.98 | 9 Sigmoid        |  2.56 |
+  |10 Sigmoid flexure   |  1.77 |10 Lip            |  2.57 |10 Mouth          |  2.40 |
+  |11 Parotid           |  1.10 |11 Jaw            |  1.97 |11 Pancreas       |  1.80 |
+  |12 Pancreas          |   "   |12 Colon          |  1.84 |12 Tonsils        |  1.35 |
+  |13 Caecum            |  0.94 |13 Abdomen        |   "   |13 Omentum        |  1.25 |
+  |14 Peritoneum        |   "   |14 Intestine      |  1.56 |14 Parotid        |  1.12 |
+  |15 Colon             |  0.89 |15 Caecum         |  1.37 |15 Colon          |   "   |
+  |16 Pharynx           |       |16 Mouth          |  1.18 |16 Caecum         |  1.08 |
+  |17 Intestine (site   |       |17 Parotid        |   "   |17 Intestine      |  1.00 |
+  |     unknown)        |  0.79 |18 Splenic flexure|  0.98 |18 Abdomen        |   "   |
+  |18 Abdomen           |  0.71 |19 Jejunum and    |       |19 Pharynx        |  0.62 |
+  |19 Mesentery         |  0.55 |     ileum        |  0.68 |20 Mesentery      |  0.52 |
+  |20 Omentum           |   "   |20 Tonsils        |  0.68 |21 Jejunum and    |       |
+  |21 Hepatic flexure   |  0.39 |21 Pharynx        |  0.40 |     ileum        |  0.44 |
+  |22 Submaxillary gland|  0.31 |22 Hepatic flexure|   "   |22 Hepatic flexure|   "   |
+  |23 Jejunum and ileum |   "   |23 Mesentery      |   "   |23 Splenic flexure|  0.28 |
+  |24 Duodenum          |  0.23 |24 Submaxillary   |  0.20 |24 Submaxillary   |  0.22 |
+  |25 Splenic flexure   |  0.15 |25 Duodenum       |   "   |25 Duodenum       |       |
+  +---------------------+-------+------------------+-------+------------------+-------+
+
+  _Note._--The figures where several organs are bracketed apply to each organ separately.
+
+In a healthy new-born infant the mouth is free from micro-organisms, and
+very few are found in a breast-fed baby, but _Bacillus lactis_ may be
+found where the child is bottle fed. If there is trouble with the first
+dentition and food is allowed to collect, staphylococci, streptococci,
+pneumococci and colon bacilli may be present. Even in healthy babies
+_Oidium albicans_ may be present, and in older children the
+pseudo-diphtheria bacillus. From carious teeth may be isolated
+streptothrix, leptothrix, spirilla and fusiform bacilli. Under
+conditions of health these micro-organisms live in the mouth as
+saprophytes, and show no virulence when cultivated and injected into
+animals. The two common pyogenetic organisms, _Staphylococcus albus_ and
+_brevis_, show no virulence. Also the pneumococcus, though often
+present, must be raised in virulence before it can produce untoward
+results. The foulness of the mouth is supposed to be due to the colon
+bacillus and its allies, but those obtained from the mouth are
+innocuous. Also to enable the _Oidium albicans_ to attack the mucous
+membrane there must be some slight inflammation or injury. The
+micro-organisms found in the stomach gain access to that organ in the
+food or by regurgitation from the small intestine. Most are relatively
+inert, but some have a special fermentative action on the food (see
+NUTRITION). Abelous isolated sixteen distinct species of organism from a
+healthy stomach, including Sarcinae, _B. lactis_, _pyocyaneus_,
+_subtilis_, _lactis erythrogenes_, _amylobacter_, _megatherium_, and
+_Vibrio rugula_.
+
+
+  Physical abnormalities
+
+Hare-lip, cleft palate, hernia and imperforate anus are physical
+abnormalities which are interesting to the surgeon rather than to the
+pathologist. The oesophagus may be the seat of a diverticulum, or blind
+pouch, usually situated in its lower half, which in most instances is
+probably partly acquired and partly congenital; a local weakness
+succumbing to pressure. Hypertrophy of the muscular coat of the pyloric
+region is an infrequent congenital gastric anomaly in infants,
+preventing the passage of food into the bowel, and causing death in a
+short time. Incomplete closure of the vitelline duct results in the
+presence of a diverticulum--Meckel's--generally connected with the
+ileum, mainly important by reason of the readiness with which it
+occasions intestinal obstruction. Idiopathic congenital dilatation of
+the colon has been described.
+
+    TABLE II.
+
+  +------------------+--------+------------------+--------+------------------+--------+
+  |      Males.      |  Per-  |     Females.     |  Per-  |      Total.      |  Per-  |
+  |                  |centage.|                  |centage.|                  |centage.|
+  +------------------+--------+------------------+--------+------------------+--------+
+  | 1 Mouth and      |        | 1 Intestines     |  28.9  | 1 Oesophagus and |        |
+  |     pharynx      | 37.85  | 2 Oesophagus and |        |     stomach      | 31.78  |
+  | 2 Oesophagus and |        |     stomach      |  27.7  | 2 Mouth and      |        |
+  |     stomach      | 33.46  | 3 Liver          |  15.5  |     pharynx      | 30.27  |
+  | 3 Intestines     | 17.04  | 4 Peritoneum     |  13.1  | 3 Intestines     | 20.42  |
+  | 4 Liver          |   7.8  | 5 Mouth and      |        | 4 Liver          | 10.02  |
+  | 5 Peritoneum     |   2.75 |     pharynx      |  11.3  | 5 Peritoneum     |  5.71  |
+  | 6 Pancreas       |   1.1  | 6 Pancreas       |   3.5  | 6 Pancreas       |  1.80  |
+  +------------------+--------+------------------+--------+------------------+--------+
+
+Traction diverticula of the oesophagus not uncommonly occur as sequels
+to suppurative inflammation of cervical lymphatic glands. More
+frequently dilatation of a section is met with, due as a rule to the
+presence of a stricture. The stomach often diverges from the normal in
+size, shape and position. Normally capable in the adult of containing
+from fifty to sixty ounces, either by reason of organic disease, or as
+the result of functional disturbance, its capacity may vary enormously.
+The writer has seen post mortem a stomach which held a gallon (160
+ounces), and again one holding only two ounces. Cancer spread over a
+large area and cirrhosis of the stomach wall cause diminution in
+capacity; pyloric obstruction, weakness of the muscular coat, and
+nervous influences are associated with dilatation. A peculiar distortion
+of the shape of the stomach follows cicatrization of ulcers of greater
+or lesser curvature; the gastric cavity becomes "hour-glass" in shape.
+In addition, the stomach may be displaced downwards as a whole, a
+condition known as gastroptosis: if the pyloric portion only be
+displaced, the lesion is termed pyloroptosis. Ptoses of other abdominal
+organs are described; the liver, transverse colon, spleen and kidneys
+may be involved. Displacements downwards of the stomach and transverse
+colon, along with a movable right kidney and associated with dyspepsia
+and neurasthenia, form the malady termed by Glénard enteroptosis. A
+general visceroptosis often occurs in those patients who have some
+tuberculous lesion of the lungs or elsewhere, this disease causing a
+general weakening and subsequent stretching of all ligaments.
+Displacements of the abdominal viscera are almost invariably accompanied
+by symptoms of dyspepsia of a neurotic type. The rectum is liable to
+prolapse, consequent upon constipation and straining at stool, or
+following local injuries of the perineal floor.
+
+
+  Influence of the nervous system.
+
+Every pathological lesion shown by digestive organs is closely
+associated with the state of the nervous system, general or local; so
+stoppage of active gastric digestive processes after profound nervous
+shock, and occurrence of nervous diarrhoea from the same cause. Gastric
+dyspepsia of nervous origin presents most varied and contradictory
+symptoms: diminished acidity of the gastric juice, hyper-acidity,
+over-production, arrest of secretion, lessened or increased movements,
+greater sensitiveness to the presence of contents, dilatation or spasm.
+Often the nervous cause can be traced back farther,--in females,
+frequently to the pelvic organs; in both sexes, to the condition of the
+blood, the brain or the bowel. Unhealthy conditions related to
+evacuation of the bowel-contents commonly induce reflex nervous
+manifestations of abnormal character referred to the stomach and liver.
+Gastric disturbances similarly react upon the proper conduct of
+intestinal functions.
+
+
+_Local Diseases._
+
+_The Mouth._--The lining membrane of the cheeks inside the mouth, of the
+gums and the under-surface and edges of the tongue, is often the seat of
+small irritable ulcers, usually associated with some digestive
+derangement. A crop of minute vesicles known as Koplik's spots over
+these parts has been lately stated by Koplik to be an early symptom of
+measles. Xerostomia, or dry mouth, is a rare condition, connected with
+lack of salivary secretion. Gangrenous stomatitis, cancrum oris, or
+noma, occasionally attacks debilitated children, or patients
+convalescing from acute fevers, more especially after measles. It
+commences in the gums or cheeks, and causes widespread sloughing of the
+adjacent soft parts--it may be of the bones.
+
+_The Stomach._--It were futile to attempt to enumerate all the protean
+manifestations of disturbance which proceed from a disordered stomach.
+The possible permutations and combinations of the causes of gastric
+vagaries almost reach infinity. Idiosyncrasy, past and present gastric
+education, penury or plethora, actual digestive power, motility, bodily
+requirements and conditions, environment, mental influences, local or
+adjacent organic lesions, and, not least, reflex impressions from other
+organs, all contribute to the variance.
+
+Ulcer of the stomach, however--the perforating gastric ulcer--occupies a
+unique position among diseases of this organ. Gastric ulcers are
+circumscribed, punched out, rarely larger than a sixpenny-bit,
+funnel-shaped, the narrower end towards the peritoneal coat, and
+distributed in those regions of the stomach wall which are most exposed
+to the action of the gastric contents. They occur most frequently in
+females, especially if anaemic, and are usually accompanied by excess of
+acid, actual or relative to the state of the blood, in the stomach
+contents. Local pain, dorsal pain, generally to the left of the eighth
+or ninth dorsal spinous process, and haematernesis and melaena, are
+symptomatic of it. The amount of blood lost varies with the rapidity of
+ulcer formation and the size of vessel opened into. Fatal results arise
+from ulceration into large blood-vessels, followed by copious
+haemorrhage, or by perforation of the ulcer into the peritoneal cavity.
+Scars of such ulcers may be found post mortem, although no symptoms of
+gastric disease have been exhibited during life; gastric ulcers,
+therefore, may be latent.
+
+Irritation of the sensory nerve-endings in the stomach wall from the
+presence of an increased proportion of acid, organic or mineral, in the
+stomach contents is accountable for the well known symptom heartburn.
+Water-brash is a term applied to eructation of a colourless, almost
+tasteless fluid, probably saliva, which has collected in the lower part
+of the oesophagus from failure of the cardiac sphincter of the stomach
+to relax; reversed oesophageal peristalsis causing regurgitation. A
+similar reversed action serves in merycism, or rumination, occasionally
+found in man, to raise part of the food, lately ingested, from the
+stomach to the mouth. Vomiting also is aided by reversed peristaltic
+action, both of the stomach and the oesophagus, with the help of the
+diaphragm and the muscles of the anterior abdominal wall. Emesis may be
+caused both by local nervous influence, and through the central nervous
+mechanism either reflexly or from the direct action of substances
+circulating in the blood. Further, the causal agent acting on the
+central nervous apparatus may be organic or functional, as well as
+medicinal. Vomiting without any apparent cause suggests nervous lesions,
+organic or reflex. The obstinate vomiting of pregnancy is a case in
+point. Here the primary cause proceeds reflexly from the pelvis. In
+females the pelvic organs are often the true source of emesis.
+Haematemesis accompanies gastric ulcer, cancer, chronic congestion with
+haemorrhagic erosion, congestion of the liver, or may follow violent
+acts of vomiting. In cases of ulcer the blood is usually bright and in
+considerable amount; in cancer, darker, like coffee-grounds; and in
+cases of erosion, in smaller quantity and of bright colour. The reaction
+of the stomach contents, if the cause be doubtful, yields valuable aid
+towards a diagnosis. Of increased acidity in gastric ulcer, normal in
+hepatic congestion, it is diminished in cancer; but as the acid present
+in cancer is largely lactic, analysis of the gastric contents must often
+be a _sine qua non_, because hyperacidity from lactic may obscure
+hypoacidity of hydrochloric acid.
+
+Flatulence usually results from fermentative processes in the stomach
+and bowel, as the outcome of bacterial activity. A different form of
+flatulence is common in neurotic individuals: in such the gas evolved
+consists simply in carbonic acid liberated from the blood, and its
+evolution is generally characterized by rapid development and by lack of
+all fermentative signs.
+
+_The Liver._--The liver is an organ frequently libelled for the
+delinquencies of other organs, and regarded as a common source of ill.
+In catarrhal jaundice it is in most cases the bowel that is at fault,
+the liver acting properly, but unable to get rid of all the bile
+produced. The liver suffers, however, from several diseases of its own.
+Its fibrous or connective tissue is very apt to increase at the expense
+of the cellular elements, destroying their functions. This cirrhotic
+process usually follows long-continued irritation, such as is produced
+by too much alcohol absorbed from the bowel habitually, the organ
+gradually becoming harder in texture and smaller in bulk. Hypertrophic
+cirrhosis of the liver is not uncommonly met with, in which the liver is
+much increased in size, the "unilobular" form, also of alcoholic origin.
+In still-born children and in some infants a form of hypertrophic
+cirrhosis is occasionally seen, probably of hereditary syphilitic
+origin. Acute congestion of the liver forms an important symptom of
+malarial fever, and often leads in time to establishment of cirrhotic
+changes; here the liver is generally enlarged, but not invariably so,
+and the part played by alcohol in its causation has still to be
+investigated. Acute yellow atrophy of the liver is a disease _sui
+generis_. Of rare occurrence, possibly of toxic origin, it is marked by
+jaundice, at first of usual type, later becoming most intense; by
+vomiting; haemorrhages widely distributed; rapid diminution in the size
+of the liver; the appearance of leucin and tyrosin in the urine, with
+lessened urea; and in two or three days, death. The liver after death is
+soft, of a reddish colour dotted with yellow patches, and weighs only
+about a third part of the normal--about 1½ lb in place of 3¾ lb. A
+closely analogous affection of the liver, known as Weil's disease, is of
+infectious type, and has been noted in epidemic form. In this the
+spleen and liver are commonly but not always swollen, and the liver is
+often tender on pressure. As a large proportion of the sufferers from
+this disease have been butchers, and the epidemics have occurred in the
+hot season of the year, it probably arises from contact with decomposing
+animal matter. Hepatic abscess may follow on an attack of amoebic
+dysentery, and is produced either by infection through the portal vein,
+or by direct infection from the adjacent colon. In general pyaemia
+multiple small abscesses may occur in the liver.
+
+_The Gall-Bladder._--The formation of biliary calculi in the
+gall-bladder is the chief point of interest here. At least 75% of such
+cases occur in women, especially in those who have borne children.
+Tight-lacing has been stated to act as an exciting cause, owing to the
+consequent retardation of the flow of bile. Gall-stones may number from
+one to many thousands. They are largely composed of cholesterin,
+combined with small amounts of bile-pigments and acids, lime and
+magnesium salts. Their presence may give rise to no symptoms, or may
+cause violent biliary colic, and, if the bile-stream be obstructed, to
+jaundice. Inflammatory processes may be initiated in the gall-bladder or
+the bile-ducts, catarrhal or suppurative in character.
+
+_The Pancreas._--Haemorrhages into the body of the pancreas, acute and
+chronic inflammation, calculi, cysts and tumours, among which cancer is
+by far the most common, are recognized as occurring in this organ; the
+point of greatest interest regarding them lies in the relations
+established between pancreatic disease and diabetes mellitus, affections
+of the gland frequently being complicated by, and probably causing, the
+appearance of sugar in the urine.
+
+_The Small Intestine._--Little remains to be added to the account of
+inflammatory lesions in connexion with the small intestine. It offers
+but few conditions peculiar to itself, save in typhoid fever, and the
+ease with which it contrives to become kinked, or intussuscepted,
+producing obstruction, or to take part in hernial protrusions. The first
+section, the duodenum, is subject to development of ulcers very similar
+to those of the gastric mucous membrane. For long duodenal ulceration
+has been regarded as a complication of extensive burns of the skin, but
+the relationship between them has not yet been quite satisfactorily
+explained. The condition of colic in the bowel usually arises from
+overdistension of some part of the small gut with gas, the frequent
+sharp turns of the gut facilitating temporary closure of its lumen by
+pressure of the dilated gut near a curve against the part beyond. In the
+large bowel accumulations of gas seldom cause such acute symptoms,
+having a readier exit.
+
+_The Large Intestine._--The colon, especially the ascending portion, may
+become immensely dilated, usually after prolonged constipation and
+paralysis of the gut; occasionally the condition is congenital.
+Straining efforts made in defaecation may often account for prolapse of
+the lower end of the rectum through the anus. Haemorrhage from the bowel
+is usually a sign of disease situated in the large intestine: if bright
+in colour, the source is probably low down; if dark, from the caecum or
+from above the ileo-caecal valve. Blood after a short stay in any
+section of the alimentary canal darkens, and eventually becomes almost
+black in colour.     (A. L. G.; M. F.*)
+
+
+
+
+DIGGES, WEST (1720-1786), English actor, made his first stage appearance
+in Dublin in 1749 as Jaffier in _Venice Preserved_; and both there and
+in Edinburgh until 1764 he acted in many tragic rôles with success. He
+was the original "young Norval" in Home's _Douglas_ (1756). His first
+London appearance was as Cato in the Haymarket in 1777, and he
+afterwards played Lear, Macbeth, Shylock and Wolsey. In 1881 he returned
+to Dublin and retired in 1784.
+
+
+
+
+DIGIT (Lat. _digitus_, finger), literally a finger or toe, and so used
+to mean, from counting on the fingers, a single numeral, or, from
+measuring, a finger's breadth. In astronomy a digit is the twelfth part
+of the diameter of the sun or moon; it is used to express the magnitude
+of an eclipse.
+
+
+
+
+DIGITALIS. The leaves of the foxglove (q.v.), gathered from wild plants
+when about two-thirds of their flowers are expanded, deprived usually
+of the petiole and the thicker part of the midrib, bitter taste; and to
+preserve their properties they must be kept excluded from light in
+stoppered bottles. They are occasionally adulterated with the leaves of
+_Inula Conyza_, ploughman's spikenard, which may be distinguished by
+their greater roughness, their less divided margins, and their odour
+when rubbed; also with the leaves of _Symphytum officinale_, comfrey,
+and of _Verbascum Thapsus_, great mullein, which unlike those of the
+foxglove have woolly upper and under surfaces. The earliest known
+descriptions of the foxglove are those given by Leonhard Fuchs and
+Tragus about the middle of the 16th century, but its virtues were
+doubtless known to herbalists at a much remoter period. J. Gerarde, in
+his _Herbal_ (1597), advocates the use of foxglove for a variety of
+complaints; and John Parkinson, in the _Theatrum Botanicum_, or _Theater
+of Plants_ (1640), and later W. Salmon, in _The New London
+Dispensatory_, similarly praised the remedy. Digitalis was first brought
+prominently under the notice of the medical profession by Dr W.
+Withering, who, in his _Account of the Foxglove_ (1785), gave details of
+upwards of 200 cases chiefly dropsical, in which it was used.
+
+Digitalis contains four important glucosides, of which three are cardiac
+stimulants. The most powerful is _digitoxin_ C34H54O11, an extremely
+poisonous and cumulative drug, insoluble in water. _Digitalin_,
+C35H56O14, is crystalline and is also insoluble in water. _Digitalein_
+is amorphous but readily soluble in water. It can therefore be
+administered subcutaneously, in doses of about one-hundredth of a grain.
+_Digitonin_, on the other hand, is a cardiac depressant, and has been
+found to be identical with saponin, the chief constituent of senega
+root. There are numerous preparations, patent and pharmacopeial, their
+composition being extremely varied, so that, unless one has reason to be
+certain of any particular preparation, it is almost better to use only
+the dried leaves themselves in the form of a powder (dose ½-2 grains).
+The pharmacopeial tincture may be given in doses of five to fifteen
+minims, and the infusion has the unusually small dose of two to four
+drachms--the dose of other infusions being an ounce or more. The
+tincture contains a fair proportion of both digitalin and digitoxin.
+
+Digitalis leaves have no definite external action. Taken by the mouth,
+the drug is apt to cause considerable digestive disturbance, varying in
+different cases and sometimes so severe as to cause serious difficulty.
+This action is probably due to the digitonin, which is thus a
+constituent in every way undesirable. The all-important property of the
+drug is its action on the circulation. Its first action on any of the
+body-tissues is upon unstriped muscle, so that the first consequence of
+its absorption is a contraction of the arteries and arterioles. No other
+known drug has an equally marked action in contracting the arterioles.
+As the vaso-motor centre in the medulla oblongata is also stimulated, as
+well as the contractions of the heart, there is thus trebly caused a
+very great rise in the blood-pressure.
+
+The clinical influence of digitalis upon the heart is very well defined.
+After the taking of a moderate dose the pulse is markedly slowed. This
+is due to a very definite influence upon the different portions of the
+cardiac cycle. The systole is not altered in length, but the diastole is
+very much prolonged, and since this is the period not only of cardiac
+rest but also of cardiac "feeding"--the coronary vessels being
+compressed and occluded during systole--the result is greatly to benefit
+the nutrition of the cardiac muscle. So definite is this that, despite a
+great increase in the force of the contractions and despite experimental
+proof that the heart does more work in a given time under the influence
+of digitalis, the organ subsequently displays all the signs of having
+rested, its improved vigour being really due to its obtaining a larger
+supply of the nutrient blood. Almost equally striking is the fact that
+digitalis causes an irregular pulse to become regular. Added to the
+greater force of cardiac contraction is a permanent tonic contraction of
+the organ, so that its internal capacity is reduced. The bearing of this
+fact on cases of cardiac dilatation is evident. In larger doses a
+remarkable sequel to these actions may be observed. The cardiac
+contractions become irregular, the ventricle assumes curious
+shapes--"hour-glass," &c.--becomes very pale and bloodless, and finally
+the heart stops in a state of spasm, which shortly afterwards becomes
+rigor-mortis. Before this final change the heart may be started again by
+the application of a soluble potassium salt, or by raising the fluid
+pressure within it. Clinically it is to be observed that the drug is
+cumulative, being very slowly excreted, and that after it has been taken
+for some time the pulse may become irregular, the blood-pressure low,
+and the cardiac pulsations rapid and feeble. These symptoms with more or
+less gastro-intestinal irritation and decrease in the quantity of urine
+passed indicate digitalis poisoning. The initial action of digitalis is
+a stimulation of the cardiac terminals of the vagus nerves, so that the
+heart's action is slowed. Thereafter follows the most important effect
+of the drug, which is a direct stimulation of the cardiac muscle. This
+can be proved to occur in a heart so embryonic that no nerves can be
+recognized in it, and in portions of cardiac muscle that contain neither
+nervecells nor nerve-fibres.
+
+The action of this drug on the kidney is of importance only second to
+its action on the circulation. In small or moderate doses it is a
+powerful diuretic. Though Heidenhain asserts that rise in the renal
+blood-pressure has not a diuretic action per se, it seems probable that
+this influence of the drug is due to a rise in the general
+blood-pressure associated with a relatively dilated condition of the
+renal vessels. In large doses, on the other hand, the renal vessels also
+are constricted and the amount of urine falls. It is probable that
+digitalis increases the amount of water rather than that of the urinary
+solids. In large doses the action of digitalis on the circulation causes
+various cerebral symptoms, such as seeing all objects blue, and various
+other disturbances of the special senses. There appears also to be a
+specific action of lowering the reflex excitability of the spinal cord.
+
+Digitalis is used in therapeutics exclusively for its action on the
+circulation. In prescribing this drug it must be remembered that fully
+three days elapse before it gets into the system, and thus it must
+always be combined with other remedies to tide the patient over this
+period. It must never be prescribed in large doses to begin with, as
+some patients are quite unable to take it, intractable vomiting being
+caused. The three days that must pass before any clinical effect is
+obtained renders it useless in an emergency. A certain consequence of
+its use is to cause or increase cardiac hypertrophy--a condition which
+has its own dangers and ultimately disastrous consequences, and must
+never be provoked beyond the positive needs of the case. But digitalis
+is indicated whenever the heart shows itself unequal to the work it has
+to perform. This formula includes the vast majority of cardiac cases.
+The drug is contra-indicated in all cases where the heart is already
+beating too slowly; in aortic incompetence--where the prolongation of
+diastole increases the amount of the blood that regurgitates through the
+incompetent valve; in chronic Bright's disease and in fatty degeneration
+of the heart--since nothing can cause fat to become contractile.
+
+
+
+
+DIGNE, the chief town of the department of the Basses Alpes, in S.E.
+France, 14 m. by a branch line from the main railway line between
+Grenoble and Avignon. Pop. (1906), town, 4628; commune, 7456. The Ville
+Haute is built on a mountain spur running down to the left bank of the
+Bléone river, and is composed of a labyrinth of narrow winding streets,
+above which towers the present cathedral church, dating from the end of
+the 15th century, but largely reconstructed in modern times, and the
+former bishop's palace (now the prison). The fine Boulevard Gassendi
+separates the Ville Haute from the Ville Basse, which is of modern date.
+The old cathedral (Notre Dame du Bourg) is a building of the 13th
+century, but is now disused except for funerals: it stands at the east
+end of the Ville Basse. The neighbourhood of Digne is rich in orchards,
+which have long made the town famous in France for its preserved fruits
+and confections. It is the _Dinia_ of the Romans, and was the capital of
+the Bodiontii. From the early 6th century at least it has been an
+episcopal see, which till 1790 was in the ecclesiastical province of
+Embrun, but since 1802 in that of Aix en Provence. The history of Digne
+in the middle ages is bound up with that of its bishops, under whom it
+prospered greatly. But it suffered much during the religious wars of the
+16th and 17th centuries, when it was sacked several times. A little way
+off, above the right bank of the Bléone, is Champtercier, the birthplace
+of the astronomer Gassendi (1592-1655), whose name has been given to the
+principal thoroughfare of the little town.
+
+  See F. Guichard, _Souvenirs historiques sur la ville de Digne et ses
+  environs_ (Digne, 1847).     (W. A. B. C.)
+
+
+
+
+DIGOIN, a town of east-central France, in the department of
+Saône-et-Loire, on the right bank of the Loire, 55 m. W.N.W. of Mâcon on
+the Paris-Lyon railway. Pop. (1906) 5321. It is situated at the meeting
+places of the Loire, the Lateral canal of the Loire and the Canal du
+Centre, which here crosses the Loire by a fine aqueduct. The town
+carries on considerable manufactures of faience, pottery and porcelain.
+The port on the Canal du Centre has considerable traffic in timber,
+sand, iron, coal and stone.
+
+
+
+
+DIJON, a town of eastern France, capital of the department of Côte d'Or
+and formerly capital of the province of Burgundy, 195 m. S.E. of Paris
+on the Paris-Lyon railway. Pop. (1906) 65,516. It is situated on the
+western border of the fertile plain of Burgundy, at the foot of Mont
+Afrique, the north-eastern summit of the Côte d'Or range, and at the
+confluence of the Ouche and the Suzon; it also has a port on the canal
+of Burgundy. The great strategic importance of Dijon as a centre of
+railways and roads, and its position with reference to an invasion of
+France from the Rhine, have led to the creation of a fortress forming
+part of the Langres group. There is no _enceinte_, but on the east side
+detached forts, 3 to 4 m. distant from the centre, command all the great
+roads, while the hilly ground to the west is protected by Fort
+Hauteville to the N.W. and the "groups" of Motte Giron and Mont Afrique
+to the S.W., these latter being very formidable works. Including a fort
+near Saussy (about 8 m. to the N.W.) protecting the water-supply of
+Dijon, there are eight forts, besides the groups above mentioned. The
+fortifications which partly surrounded the old and central portion of
+the city have disappeared to make way for tree-lined boulevards with
+fine squares at intervals. The old churches and historic buildings of
+Dijon are to be found in the irregular streets of the old town, but
+industrial and commercial activity has been transferred to the new
+quarters beyond its limits. A fine park more than 80 acres in extent
+lies to the south of the city, which is rich in open spaces and
+promenades, the latter including the botanical garden and the Promenade
+de l'Arquebuse, in which there is a black poplar famous for its size and
+age.
+
+The cathedral of St Bénigne, originally an abbey church, was built in
+the latter half of the 13th century on the site of a Romanesque
+basilica, of which the crypt remains. The west front is flanked by two
+towers and the crossing is surmounted by a slender timber spire. The
+plan consists of three naves, short transepts and a small choir, without
+ambulatory, terminating in three apses. In the interior there is a fine
+organ and a quantity of statuary, and the vaults contain the remains of
+Philip the Bold, duke of Burgundy, and Anne of Burgundy, daughter of
+John the Fearless. The site of the abbey buildings is occupied by the
+bishop's palace and an ecclesiastical seminary. The church of
+Notre-Dame, typical of the Gothic style of Burgundy, was erected from
+1252 to 1334, and is distinguished for the grace of its interior and the
+beauty of the western façade. The portal consists of three arched
+openings, above which are two stages of arcades, open to the light and
+supported on slender columns. A row of gargoyles surmounts each storey
+of the façade, which is also ornamented by sculptured friezes. A turret
+to the right of the portal carries a clock called the Jaquemart, on
+which the hours are struck by two figures. The church of St Michel
+belongs to the 15th century. The west façade, the most remarkable
+feature of the church, is, however, of the Renaissance period. The
+vaulting of the three portals is of exceptional depth owing to the
+projection of the lower storey of the façade. Above this storey rise two
+towers of five stages, the fifth stage being formed by an octagonal
+cupola. The columns decorating the façade represent all the four orders.
+The design of this façade is wrongly attributed to Hugues Sambin (fl.
+c. 1540), a native of Dijon, and pupil of Leonardo da Vinci, but the
+sculpture of the portals, including "The Last Judgment" on the tympanum
+of the main portal, is probably from his hand. St Jean (15th century)
+and St Étienne (15th, 16th and 17th centuries), now used as the
+exchange, are the other chief churches. Of the ancient palace of the
+dukes of Burgundy there remain two towers, the Tour de la Terrasse and
+the Tour de Bar, the guard-room and the kitchens; these now form part of
+the hôtel de ville, the rest of which belongs to the 17th and 18th
+centuries. This building contains an archaeological museum with a
+collection of Roman stone monuments; the archives of the town; and the
+principal museum, which, besides valuable paintings and other works of
+art, contains the magnificent tombs of Philip the Bold and John the
+Fearless, dukes of Burgundy. These were transferred from the Chartreuse
+of Dijon (or of Champmol), built by Philip the Bold as a mausoleum, now
+replaced by a lunatic asylum. Relics of it survive in the old Gothic
+entrance, the portal of the church, a tower and the well of Moses, which
+is adorned with statues of Moses and the prophets by Claux Sluter (fl.
+end of 14th century), the Dutch sculptor, who also designed the tomb of
+Philip the Bold. The Palais de Justice, which belongs to the reign of
+Louis XII., is of interest as the former seat of the _parlement_ of
+Burgundy. Dijon possesses several houses of the 15th, 16th and 17th
+centuries, notably the Maison Richard in the Gothic, and the Hôtel Vogüé
+in the Renaissance style. St Bernard, the composer J. P. Rameau and the
+sculptor François Rude have statues in the town, of which they were
+natives. There are also monuments to those inhabitants of Dijon who fell
+in the engagement before the town in 1870, and to President Carnot and
+Garibaldi.
+
+The town is important as the seat of a prefecture, a bishopric, a court
+of appeal and a court of assizes, and as centre of an académie
+(educational district). There are tribunals of first instance and of
+commerce, a board of trade-arbitrators, a chamber of commerce, an
+exchange (occupying the former cathedral of St Étienne), and an
+important branch of the Bank of France. Its educational establishments
+include faculties of law, of science and of letters, a preparatory
+school of medicine and pharmacy, a higher school of commerce, a school
+of fine art, a conservatoire of music, _lycées_ and training colleges,
+and there is a public library with about 100,000 volumes.
+
+Dijon is well known for its mustard, and for the black currant liqueur
+called _cassis de Dijon_; its industries include the manufacture of
+machinery, automobiles, bicycles, soap, biscuits, brandy, leather, boots
+and shoes, candles and hosiery. There are also flour mills, breweries,
+important printing works, vinegar works and, in the vicinity, nursery
+gardens. The state has a large tobacco manufactory in the town. Dijon
+has considerable trade in cereals and wool, and is the second market for
+the wines of Burgundy.
+
+Under the Romans Dijon (_Divonense castrum_) was a _vicus_ in the
+_civitas_ of Langres. In the 2nd century it was the scene of the
+martyrdom of St Benignus (Bénigne, vulg. Berin, Berain), the apostle of
+Burgundy. About 274 the emperor Aurelian surrounded it with ramparts.
+Gregory of Tours, in the 6th century, comments on the strength and
+pleasant situation of the place, expressing surprise that it does not
+rank as a _civitas_. During the middle ages the fortunes of Dijon
+followed those of Burgundy, the dukes of which acquired it early in the
+11th century. The communal privileges, conferred on the town in 1182 by
+Hugh III., duke of Burgundy, were confirmed by Philip Augustus in 1183,
+and in the 13th century the dukes took up their residence there. For the
+decoration of the palace and other monuments built by them, eminent
+artists were gathered from northern France and Flanders, and during this
+period the town became one of the great intellectual centres of France.
+The union of the duchy with the crown in 1477 deprived Dijon of the
+splendour of the ducal court; but to counterbalance this loss it was
+made the capital of the province and seat of a _parlement_. Its fidelity
+to the monarchy was tested in 1513, when the citizens were besieged by
+50,000 Swiss and Germans, and forced to agree to a treaty so
+disadvantageous that Louis XII. refused to ratify it. In the wars of
+religion Dijon sided with the League, and only opened its gates to Henry
+IV. in 1595. The 18th century was a brilliant period for the city; it
+became the seat of a bishopric, its streets were improved, its commerce
+developed, and an academy of science and letters founded; while its
+literary salons were hardly less celebrated than those of Paris. The
+neighbourhood was the scene of considerable fighting during the
+Franco-German War, which was, however, indirectly of some advantage to
+the city owing to the impetus given to its industries by the immigrants
+from Alsace.
+
+  See H. Chabeuf, _Dijon à travers les âges_ (Dijon, 1897), and _Dijon,
+  monuments et souvenirs_ (Dijon, 1894).
+
+
+
+
+DIKE, or DYKE (Old Eng. _dic_, a word which appears in various forms in
+many Teutonic languages, cf. Dutch _dijk_, German _Teich_, Danish
+_dige_, and in French, derived from Teutonic, _digue_; it is the same
+word as "ditch" and is ultimately connected with the root of "dig"),
+properly a trench dug out of the earth for defensive and other purposes.
+Water naturally collects in such trenches, and hence the word is applied
+to natural and artificial channels filled with water, as appears in the
+proverbial expression "February fill-dyke," and in the names of many
+narrow waterways in East Anglia. "Dike" also is naturally used of the
+bank of earth thrown up out of the ditch, and so of any embankment, dam
+or causeway, particularly the defensive works in Holland, the Fen
+district of England, and other low-lying districts which are liable to
+flooding by the sea or rivers (see HOLLAND and FENS). In Scotland any
+wall, fence or even hedge, used as a boundary is called a dyke. In
+geology the term is applied to wall-like masses of rock (sometimes
+projecting beyond the surrounding surface) which fill up vertical or
+highly inclined fissures in the strata.
+
+
+
+
+DIKKA, a term in Mahommedan architecture for the tribune raised upon
+columns, from which the Koran is recited and the prayers intoned by the
+Imam of the mosque.
+
+
+
+
+DILAPIDATION (Lat. for "scattering the stones," _lapides_, of a
+building), a term meaning in general a falling into decay, but more
+particularly used in the plural in English law for (1) the waste
+committed by the incumbent of an ecclesiastical living; (2) the
+disrepair for which a tenant is usually liable when he has agreed to
+give up his premises in good repair (see EASEMENT; FLAT; LANDLORD AND
+TENANT). By the general law a tenant for life has no power to cut down
+timber, destroy buildings, &c., (voluntary waste), or to let buildings
+fall into disrepair (permissive waste). In the eye of the law an
+incumbent of a living is a tenant for life of his benefice, and any
+waste, voluntary or permissive, on his part must be made good by his
+administrators to his successor in office. The principles on which such
+dilapidations are to be ascertained, and the application of the money
+payable in respect thereof, depend partly on old ecclesiastical law and
+partly on acts of parliament. Questions as to ecclesiastical
+dilapidations usually arise in respect of the residence house and other
+buildings belonging to the living. Inclosures, hedges, ditches and the
+like are included in things "of which the beneficed person hath the
+burden and charge of reparation." In a leading case (_Ross_ v. _Adcock_,
+1868, L.R. 3 C.P. 657) it was said that the court was acquainted with no
+precedent or decision extending the liability of the executors of a
+deceased incumbent to any species of waste beyond dilapidation of the
+house, chancel or other buildings or fences of the benefice. And it has
+been held that the mere mismanagement or miscultivation of the
+ecclesiastical lands will not give rise to an action for dilapidations.
+To place the law relating to dilapidations on a more satisfactory
+footing, the Ecclesiastical Dilapidations Act 1871 was passed. The
+buildings to which the act applies are defined to be such houses of
+residence, chancels, walls, fences and other buildings and things as the
+incumbent of the benefice is by law and custom bound to maintain in
+repair. In each diocese a surveyor is appointed by the archdeacons and
+rural deans subject to the approval of the bishop; and such surveyor
+shall by the direction of the bishop examine the buildings on the
+following occasions--viz. (1) when the benefice is sequestrated; (2)
+when it is vacant; (3) at the request of the incumbent or on complaint
+by the archdeacon, rural dean or patron. The surveyor specifies the
+works required, and gives an estimate of their probable cost. In the
+case of a vacant benefice, the new incumbent and the old incumbent or
+his representatives may lodge objections to the surveyor's report on any
+grounds of fact or law, and the bishop, after consideration, may make an
+order for the repairs and their cost, for which the late incumbent or
+his representatives are liable. The sum so stated becomes a debt due
+from the late incumbent or his representatives to the new incumbent, who
+shall pay over the money when recovered to the governors of Queen Anne's
+Bounty. The governors pay for the works on execution on receipt of a
+certificate from the surveyor; and the surveyor, when the works have
+been completed to his satisfaction, gives a certificate to that effect,
+the effect of which, so far as regards the incumbent, is to protect him
+from liability for dilapidations for the next five years. Unnecessary
+buildings belonging to a residence house may, by the authority of the
+bishop and with the consent of the patron, be removed. An amending
+statute of 1872 (Ecclesiastical Dilapidations Act (1871) Amendment)
+relates chiefly to advances by the governors of Queen Anne's Bounty for
+the purposes of the act.
+
+
+
+
+DILATATION (from Lat. _dis-_, distributive, and _latus_, wide), a
+widening or enlarging; a term used in physiology, &c.
+
+
+
+
+DILATORY (from Lat. _dilatus_, from _differre_, to put off or delay),
+delaying, or slow; in law a "dilatory plea" is one made merely for
+delaying the suit.
+
+
+
+
+DILEMMA (Gr. [Greek: dilêmma], a double proposition, from [Greek: di-]
+and [Greek: lambanein]), a term used technically in logic, and popularly
+in common parlance and rhetoric. (1) The latter use has no exact
+definition, but in general it describes a situation wherein from either
+of two (or more) possible alternatives an unsatisfactory conclusion
+results. The alternatives are called the "horns" of the dilemma. Thus a
+nation which has to choose between bankruptcy and the repudiation of its
+debts is on the horns of a dilemma. (2) In logic there is considerable
+divergence of opinion as to the best definition. Whately defined it as
+"a conditional syllogism with two or more antecedents in the major and a
+disjunctive minor." Aulus Gellius gives an example as follows:--"Women
+are either fair or ugly; if you marry a fair woman, she will attract
+other men; if an ugly woman she will not please you; therefore marriage
+is absurd." From either alternative, an unpleasant result follows. Four
+kinds of dilemma are admitted:--(a) _Simple Constructive_: If A, then C;
+if B, then C, but either B or A; therefore C. (b) _Simple Destructive_:
+If A is true, B is true; if A is true, C is true; B and C are not both
+true; therefore A is not true. (c) _Complex Constructive_: If A, then B;
+if C, then D; but either A or C; therefore either B or D. (d) _Complex
+Destructive_: If A is true, B is true; if C is true, D is true; but B
+and D are not both true; hence A and C are not both true. The soundness
+of the dilemmatic argument in general depends on the alternative
+possibilities. Unless the alternatives produced exhaust the
+possibilities of the case, the conclusion is invalid. The logical form
+of the argument makes it especially valuable in public speaking, before
+uncritical audiences. It is, in fact, important rather as a rhetorical
+subtlety than as a serious argument.
+
+_Dilemmist_ is also a term used to translate _Vaibhashikas_, the name of
+a Buddhist school of philosophy.
+
+
+
+
+DILETTANTE, an Italian word for one who delights in the fine arts,
+especially in music and painting, so a lover of the fine arts in
+general. The Ital. _dilettare_ is from Lat. _delectare_, to delight.
+Properly the word refers to an "amateur" as opposed to a "professional"
+cultivation of the arts, but like "amateur" it is often used in a
+depreciatory sense for one who is only a dabbler, or who only has a
+superficial knowledge or interest in art. The Dilettanti Society founded
+in 1733-1734 still exists in England. A history of the society, by
+Lionel Cust, was published in 1898.
+
+
+
+
+DILIGENCE, in law, the care which a person is bound to exercise in his
+relations with others. The possible degrees of diligence are of course
+numerous, and the same degree is not required in all cases. Thus a mere
+depositary would not be held bound to the same degree of diligence as a
+person borrowing an article for his own use and benefit. Jurists,
+following the divisions of the civil law, have concurred in fixing three
+approximate standards of diligence--viz. ordinary (_diligentia_), less
+than ordinary (_levissima diligentia_) and more than ordinary
+(_exactissima diligentia_). Ordinary or common diligence is defined by
+Story (_On Bailments_) as "that degree of diligence which men in general
+exert in respect of their own concerns." So Sir William Jones:--"This
+care, which every person of common prudence and capable of governing a
+family takes of his own concerns, is a proper measure of that which
+would uniformly be required in performing every contract, if there were
+not strong reasons for exacting in some of them a greater and permitting
+in others a less degree of attention" (_Essay on Bailments_). The
+highest degree of diligence would be that which only very prudent
+persons bestow on their own concerns; the lowest, that which even
+careless persons bestow on their own concerns. The want of these various
+degrees of diligence is negligence in corresponding degrees. These
+approximations indicate roughly the greater or less severity with which
+the law will judge the performance of different classes of contracts;
+but English judges have been inclined to repudiate the distinction as a
+useless refinement of the jurists. Thus Baron Rolfe could see no
+difference between negligence and gross negligence; it was the same
+thing with the addition of a vituperative epithet. See NEGLIGENCE.
+
+_Diligence_, in Scots law, is a general term for the process by which
+persons, lands or effects are attached on execution, or in security for
+debt.
+
+
+
+
+DILKE, SIR CHARLES WENTWORTH, Bart. (1810-1869), English politician, son
+of Charles Wentworth Dilke, proprietor and editor of _The Athenaeum_,
+was born in London on the 18th of February 1810, and was educated at
+Westminster school and Trinity Hall, Cambridge. He studied law, and in
+1834 took his degree of LL.B., but did not practise. He assisted his
+father in his literary work, and was for some years chairman of the
+council of the Society of Arts, besides taking a prominent part in the
+affairs of the Royal Horticultural Society and other bodies. He was one
+of the most zealous promoters of the Great Exhibition (1851), and a
+member of the executive committee. At the close of the exhibition he was
+honoured by foreign sovereigns, and the queen offered him knighthood,
+which, however, he did not accept; he also declined a large remuneration
+offered by the royal commission. In 1853 Dilke was one of the English
+commissioners at the New York Industrial Exhibition, and prepared a
+report on it. He again declined to receive any money reward for his
+services. He was appointed one of the five royal commissioners for the
+Great Exhibition of 1862; and soon after the death of the prince consort
+he was created a baronet. In 1865 he entered parliament as member for
+Wallingford. In 1869 he was sent to Russia as representative of England
+at the horticultural exhibition held at St Petersburg. His health,
+however, had been for some time failing, and he died suddenly in that
+city, on the 10th of May 1869. A selection from his writings, _Papers of
+a Critic_ (2 vols., 1875), contains a biographical sketch by his son.
+
+His son, SIR CHARLES WENTWORTH DILKE, BART. (1843-   ), became a
+prominent Liberal politician, as M.P. for Chelsea (1868-1886),
+under-secretary for foreign affairs (1880-1882), and president of the
+local government board (1882-1885); and he was then marked out as one of
+the best-informed and ablest of the advanced Radicals. He was chairman
+of the royal commission on the housing of the working classes in
+1884-1885. But his sensational appearance as co-respondent in a divorce
+case of a peculiarly unpleasant character in 1885 cast a cloud over his
+career. He was defeated in Chelsea in 1886, and did not return to
+parliament till 1892, when he was elected for the Forest of Dean; and
+though his knowledge of foreign affairs and his powers as a critic and
+writer on military and naval questions were admittedly of the highest
+order, his official position in public life could not again be
+recovered. His military writings are _The British Army_ (1888); _Army
+Reform_ (1898) and, with Mr Spenser Wilkinson, _Imperial Defence_
+(1892). On colonial questions he wrote with equal authority. His
+_Greater Britain_ (2 vols., 1866-1867) reached a fourth edition in 1868,
+and was followed by _Problems of Greater Britain_ (2 vols., 1890) and
+_The British Empire_ (1899). He was twice married, his second wife (née
+Emilia Frances Strong), the widow of Mark Pattison, being an
+accomplished art critic and collector. She died in 1904. The most
+important of her books were the studies on _French Painters of the
+Eighteenth Century_ (1899) and three subsequent volumes on the
+architects and sculptors, furniture and decoration, engravers and
+draughtsmen of the same period, the last of which appeared in 1902. A
+posthumous volume, _The Book of the Spiritual Life_ (1905), contains a
+memoir of her by Sir Charles Dilke.
+
+[Illustration: Dill (_Anethum_ or _Peucedanum graveolens_), leaf and
+inflorescence.]
+
+
+
+
+DILL (_Anethum_ or _Peucedanum graveolens_), a member of the natural
+botanical order Umbelliferae, indigenous to the south of Europe, Egypt
+and the Cape of Good Hope. It resembles fennel in appearance. Its root
+is long and fusiform; the stem is round, jointed and about a yard high;
+the leaves have fragrant leaflets; and the fruits are brown, oval and
+concavo-convex. The plant flowers from June till August in England. The
+seeds are sown, preferably as soon as ripe, either broadcast or in
+drills between 6 and 12 in. asunder. The young plants should be thinned
+when 3 or 4 weeks old, so as to be at distances of about 10 in. A
+sheltered spot and dry soil are needed for the production of the seed in
+the climate of England. The leaves of the dill are used in soups and
+sauces, and, as well as the umbels, for flavouring pickles. The seeds
+are employed for the preparation of dill-water and oil of dill; they are
+largely consumed in the manufacture of gin, and, when ground, are eaten
+in the East as a condiment. The British Pharmacopoeia contains the Aqua
+Anethi or dill-water (dose 1-2 oz.), and the Oleum Anethi, almost
+identical in composition with caraway oil, and given in doses of ½-3
+minims. Dill-water is largely used as a carminative for children, and as
+a vehicle for the exhibition of nauseous drugs.
+
+
+
+
+DILLEN [DILLENIUS], JOHANN JAKOB (1684-1747), English botanist, was born
+at Darmstadt in 1684, and was educated at the university of Giessen,
+where he wrote several botanical papers for the _Ephemerides naturae
+curiosorum_, and printed, in 1719, his _Catalogus plantarum sponte circa
+Gissam nascentium_, illustrated with figures drawn and engraved by his
+own hand, and containing descriptions of many new species. In 1721, at
+the instance of the botanist William Sherard (1659-1728), he came to
+England, and in 1724 he published a new edition of Ray's _Synopsis
+stirpium Britannicarum_. In 1732 he published _Hortus Elthamensis_, a
+catalogue of the rare plants growing at Eltham, Kent, in the collection
+of Sherard's younger brother, James (1666-1738), who, after making a
+fortune as an apothecary, devoted himself to gardening and music. For
+this work Dillen himself executed 324 plates, and it was described by
+Linnaeus, who spent a month with him at Oxford in 1736, and afterwards
+dedicated his _Critica botanica_ to him, as "opus botanicum quo
+absolutius mundus non vidit." In 1734 he was appointed Sherardian
+professor of botany at Oxford, in accordance with the will of W.
+Sherard, who at his death in 1728 left the university £3000 for the
+endowment of the chair, as well as his library and herbarium. Dillen,
+who was also the author of an _Historia muscorum_ (1741), died at
+Oxford, of apoplexy, on the 2nd of April 1747. His manuscripts, books
+and collections of dried plants, with many drawings, were bought by his
+successor at Oxford, Dr Humphry Sibthorp (1713-1797), and ultimately
+passed into the possession of the university.
+
+  For an account of his collections preserved at Oxford, see _The
+  Dillenian Herbaria_, by G. Claridge Druce (Oxford, 1907).
+
+
+
+
+DILLENBURG, a town of Germany, in the Prussian province of Hesse-Nassau,
+delightfully situated in the midst of a well-wooded country, on the
+Dill, 25 m. N.W. from. Giessen on the railway to Troisdorf. Pop. 4500.
+On an eminence above it lie the ruins of the castle of Dillenburg,
+founded by Count Henry the Rich of Nassau, about the year 1255, and the
+birthplace of Prince William of Orange (1533). It has an Evangelical
+church, with the vault of the princes of Nassau-Dillenburg, a Roman
+Catholic church, a classical school, a teachers' seminary and a chamber
+of commerce. Its industries embrace iron-works, tanneries and the
+manufacture of cigars. Owing to its beautiful surroundings Dillenburg
+has become a favourite summer resort.
+
+
+
+
+DILLENS, JULIEN (1849-1904), Belgian sculptor, was born at Antwerp on
+the 8th of June 1849, son of a painter. He studied under Eugène Simonis
+at the Brussels Academy of Fine Arts. In 1877 he received the _prix de
+Rome_ for "A Gaulish Chief taken Prisoner by the Romans." At Brussels,
+in 1881, he executed the groups entitled "Justice" and "Herkenbald, the
+Brussels Brutus." For the pediment of the orphanage at Uccle, "Figure
+Kneeling" (Brussels Gallery), and the statue of the lawyer
+Metdepenningen in front of the Palais de Justice at Ghent, he was
+awarded the medal of honour in 1889 at the Paris Universal Exhibition,
+where, in 1900, his "Two Statues of the Anspach Monument" gained him a
+similar distinction. For the town of Brussels he executed "The Four
+Continents" (Maison du Renard, Grand' Place), "The Lansquenets" crowning
+the lucarnes of the Maison de Roi, and the "Monument t' Serclaes" under
+the arcades of the Maison de l'Etoile, and, for the Belgian government,
+"Flemish Art," "German Art," "Classic Art" and "Art applied to Industry"
+(all in the Palais des Beaux Arts, Brussels), "The Laurel" (Botanic
+Garden, Brussels), and the statue of "Bernard van Orley" (Place du petit
+Sablon, Brussels). Mention must also be made of "An Enigma" (1876), the
+bronze busts of "Rogier de la Pasture" and "P. P. Rubens" (1879),
+"Etruria" (1880), "The Painter Leon Frederic" (1888), "Madame Leon
+Herbo," "Hermes," a scheme of decoration for the ogival façade of the
+hôtel de ville at Ghent (1893), "The Genius of the Funeral Monument of
+the Moselli Family," "The Silence of Death" (for the entrance of the
+cemetery of St Gilles), two caryatides for the town hall of St Gilles,
+presentation plaquette to Dr Heger, medals of MM. Godefroid and
+Vanderkindere and of "The Three Burgomasters of Brussels," and the
+ivories "Allegretto," "Minerva" and the "Jamaer Memorial." Dillens died
+at Brussels in November 1904.
+
+
+
+
+DILLINGEN, a town of Germany, in the kingdom of Bavaria, on the left
+bank of the Danube, 25 m. N.E. from Ulm, on the railway to Ingolstadt.
+Pop. (1905) 6078. Its principal buildings are an old palace, formerly
+the residence of the bishops of Augsburg and now government offices, a
+royal gymnasium, a Latin school with a library of 75,000 volumes, seven
+churches (six Roman Catholic), two episcopal seminaries, a Capuchin
+monastery, a Franciscan convent and a deaf and dumb asylum. The
+university, founded in 1549, was abolished in 1804, being converted into
+a lyceum. The inhabitants are engaged in cattle-rearing, the cultivation
+of corn, hops and fruit, shipbuilding and the shipping trade, and the
+manufacture of cloth, paper and cutlery. In the vicinity is the
+Karolinen canal, which cuts off a bend in the Danube between Lauingen
+and Dillingen. In 1488 Dillingen became the residence of the bishops of
+Augsburg; was taken by the Swedes in 1632 and 1648, by the Austrians in
+1702, and on the 17th of June 1800 by the French. In 1803 it passed to
+Bavaria.
+
+
+
+
+DILLMANN, CHRISTIAN FRIEDRICH AUGUST (1823-1894), German orientalist and
+biblical scholar, the son of a Württemberg schoolmaster, was born at
+Illingen on the 25th of April 1823. He was educated at Tübingen, where
+he became a pupil and friend of Heinrich Ewald, and studied under F. C.
+Baur, though he did not join the new Tübingen school. For a short time
+he worked as pastor at Gersheim, near his native place, but he soon came
+to feel that his studies demanded his whole time. He devoted himself to
+the study of Ethiopic MSS. in the libraries of Paris, London and Oxford,
+and this work caused a revival of Ethiopic study in the 19th century. In
+1847 and 1848 he prepared catalogues of the Ethiopic MSS. in the British
+Museum and the Bodleian library at Oxford. He then set to work upon an
+edition of the Ethiopic bible. Returning to Tübingen in 1848, in 1853 he
+was appointed professor extraordinarius. Subsequently he became
+professor of philosophy at Kiel (1854), and of theology at Giessen
+(1864) and Berlin (1869). He died on the 4th of July 1894.
+
+In 1851 he had published the _Book of Enoch_ in Ethiopian (German,
+1853), and at Kiel he completed the first part of the Ethiopic bible,
+_Octateuchus Aethiopicus_ (1853-1855). In 1857 appeared his _Grammatik
+der äthiopischen Sprache_ (2nd ed. by C. Bezold, 1899); in 1859 the
+_Book of Jubilees_; in 1861 and 1871 another part of the Ethiopic bible,
+_Libri Regum_; in 1865 his great _Lexicon linguae aethiopicae_; in 1866
+his _Chrestomathia aethiopica_. Always a theologian at heart, however,
+he returned to theology in 1864. His Giessen lectures were published
+under the titles, _Ursprung der alttestamentlichen Religion_ (1865) and
+_Die Propheten des alten Bundes nach ihrer politischen Wirksamkeit_
+(1868). In 1869 appeared his _Commentar zum Hiob_ (4th ed. 1891) which
+stamped him as one of the foremost Old Testament exegetes. His renown as
+a theologian, however, was mainly founded by the series of commentaries,
+based on those of August Wilhelm Knobels' _Genesis_ (Leipzig, 1875; 6th
+ed. 1892; Eng. trans, by W. B. Stevenson, Edinburgh, 1897); _Exodus und
+Leviticus_, 1880, revised edition by V. Ryssel, 1897; _Numeri,
+Deuteronomium und Josua_, with a dissertation on the origin of the
+Hexateuch, 1886; _Jesaja_, 1890 (revised edition by Rudolf Kittel in
+1898). In 1877 he published the _Ascension of Isaiah_ in Ethiopian and
+Latin. He was also a contributor to D. Schenkel's _Bibellexikon_,
+Brockhaus's _Conversationslexikon_, and Herzog's _Realencyklopädie_. His
+lectures on Old Testament theology, _Vorlesungen über Theologie des
+Allen Testamentes_, were published by Kittel in 1895.
+
+  See the articles in Herzog-Hauck, _Realencyklopädie_, and the
+  _Allgemeine deutsche Biographie_; F. Lichtenberger, _History of German
+  Theology in the Nineteenth Century_ (1889); Wolf Baudissin, _A.
+  Dillmann_ (Leipzig, 1895).
+
+
+
+
+DILLON, ARTHUR RICHARD (1721-1807), French archbishop, was the son of
+Arthur Dillon (1670-1733), an Irish gentleman who became general in the
+French service. He was born at St Germain, entered the priesthood and
+was successively curé of Elan near Mezières, vicar-general of Pontoise
+(1747), bishop of Evreux (1753) and archbishop of Toulouse (1758),
+archbishop of Narbonne in 1763, and in that capacity, president of the
+estates of Languedoc. He devoted himself much less to the spiritual
+direction of his diocese than to its temporal welfare, carrying out many
+works of public utility, bridges, canals, roads, harbours, &c.; had
+chairs of chemistry and of physics created at Montpellier and at
+Toulouse, and tried to reduce the poverty, especially in Narbonne. In
+1787 and in 1788 he was a member of the Assembly of Notables called
+together by Louis XVI., and in 1788 presided over the assembly of the
+clergy. Having refused to accept the civil constitution of the clergy,
+Dillon had to leave Narbonne in 1790, then to emigrate to Coblenz in
+1791. Soon afterwards he went to London, where he lived until his death
+in 1807, never accepting the Concordat, which had suppressed his
+archiepiscopal see.
+
+  See L. Audibret, _Le Dernier Président des États du Languedoc, Mgr.
+  Arthur Richard Dillon, archevêque de Narbonne_ (Bordeaux, 1868); L. de
+  Lavergne, _Les Assemblées provinciales sous Louis XVI_ (Paris, 1864).
+
+
+
+
+DILLON, JOHN (1851-   ), Irish nationalist politician, was the son of
+John Blake Dillon (1816-1866), who sat in parliament for Tipperary, and
+was one of the leaders of "Young Ireland." John Dillon was educated at
+the Roman Catholic university of Dublin, and afterwards studied
+medicine. He entered parliament in 1880 as member for Tipperary, and was
+at first an ardent supporter of C. S. Parnell. In August he delivered a
+speech on the Land League at Kildare which was characterized as "wicked
+and cowardly" by W. E. Forster; he advocated boycotting, and was
+arrested in May 1881 under the Coercion Act, and again after two months
+of freedom in October. In 1883 he resigned his seat for reasons of
+health, but was returned unopposed in 1885 for East Mayo, which he
+continued to represent. He was one of the prime movers in the famous
+"plan of campaign," which provided that the tenant should pay his rent
+to the National League instead of the landlord, and in case of eviction
+be supported by the general fund. Mr Dillon was compelled by the court
+of queen's bench on the 14th of December 1886 to find securities for
+good behaviour, but two days later he was arrested while receiving rents
+on Lord Clanricarde's estates. In this instance the jury disagreed, but
+in June 1888 under the provisions of the new Criminal Law Procedure Bill
+he was condemned to six months' imprisonment. He was, however, released
+in September, and in the spring of 1889 sailed for Australia and New
+Zealand, where he collected funds for the Nationalist party. On his
+return to Ireland he was again arrested, but, being allowed bail, sailed
+to America, and failed to appear at the trial. He returned to Ireland by
+way of Boulogne, where he and Mr W. O'Brien held long and indecisive
+conferences with Parnell. They surrendered to the police in February,
+and on their release from Galway gaol in July declared their opposition
+to Parnell. After the expulsion of Mr T. M. Healy and others from the
+Irish National Federation, Mr Dillon became the chairman (February
+1896). His early friendship with Mr O'Brien gave place to considerable
+hostility, but the various sections of the party were ostensibly
+reconciled in 1900 under the leadership of Mr Redmond. In the autumn of
+1896 he arranged a convention of the Irish race, which included 2000
+delegates from various parts of the world. In 1897 Mr Dillon opposed in
+the House the Address to Queen Victoria on the occasion of the Diamond
+Jubilee, on the ground that her reign had not been a blessing to
+Ireland, and he showed the same uncompromising attitude in 1901 when a
+grant to Lord Roberts was under discussion, accusing him of
+"systematized inhumanity." He was suspended on the 20th of March for
+violent language addressed to Mr Chamberlain. He married in 1895
+Elizabeth (d. 1907), daughter of Lord justice J. C. Mathew.
+
+
+
+
+DILUVIUM (Lat. for "deluge," from _diluere_, to wash away), a term in
+geology for superficial deposits formed by flood-like operations of
+water, and so contrasted with alluvium (q.v.) or alluvial deposits
+formed by slow and steady aqueous agencies. The term was formerly given
+to the "boulder clay" deposits, supposed to have been caused by the
+Noachian deluge.
+
+
+
+
+DIME (from the Lat. _decima_, a tenth, through the O. Fr. _disme_), the
+tenth part, the tithe paid as church dues, or as tribute to a temporal
+power. In this sense it is obsolete, but is found in Wycliffe's
+translation of the Bible--"He gave him dymes of alle thingis" (Gen. xiv.
+20). A dime is a silver coin of the United States, in value 10 cents
+(English equivalent about 5d.) or one-tenth of a dollar; hence
+"dime-novel," a cheap sensational novel, a "penny dreadful"; also
+"dime-museum."
+
+
+
+
+DIMENSION (from Lat. _dimensio_, a measuring), in geometry, a magnitude
+measured in a specified direction, i.e. length, breadth and thickness;
+thus a line has only length and is said to be of one dimension, a
+surface has length and breadth, and has two dimensions, a solid has
+length, breadth and thickness, and has three dimensions. This concept is
+extended to algebra: since a line, surface and solid are represented by
+linear, quadratic and cubic equations, and are of one, two and three
+dimensions; a biquadratic equation has its highest terms of four
+dimensions, and, in general, an equation in any number of variables
+which has the greatest sum of the indices of any term equal to n is said
+to have n dimensions. The "fourth dimension" is a type of non-Euclidean
+geometry, in which it is conceived that a "solid" has one dimension more
+than the solids of experience. For the dimensions of units see UNITS,
+DIMENSIONS OF.
+
+
+
+
+DIMITY, derived from the Gr. [Greek: dimitos] "double thread," through
+the Ital. _dimito_, "a kind of course linzie-wolzie" (Florio, 1611); a
+cloth commonly employed for bed upholstery and curtains, and usually
+white, though sometimes a pattern is printed on it in colours. It is
+stout in texture, and woven in raised patterns.
+
+
+
+
+DINAJPUR, a town (with a population in 1901 of 13,430) and district of
+British India, in the Rajshahi division of Eastern Bengal and Assam. The
+earthquake of the 12th of June 1897 caused serious damage to most of the
+public buildings of the town. There is a railway station and a
+government high school. The district comprises an area of 3946 sq. m. It
+is traversed in every direction by a network of channels and water
+courses. Along the banks of the Kulik river, the undulating ridges and
+long lines of mango-trees give the landscape a beauty which is not
+found elsewhere. Dinajpur forms part of the rich arable tract lying
+between the Ganges and the southern slopes of the Himalayas. Although
+essentially a fluvial district, it does not possess any river navigable
+throughout the year by boats of 4 tons burden. Rice forms the staple
+agricultural product. The climate of the district, although cooler than
+that of Calcutta, is very unhealthy, and the people have a sickly
+appearance. The worst part of the year is at the close of the rains in
+September and October, during which months few of the natives escape
+fever. The average maximum temperature is 92.3° F., and the minimum
+74.8°. The average rainfall is 85.54 in. In 1901 the population was
+1,567,080, showing an increase of 6% in the decade. The district is
+partly traversed by the main line of the Eastern Bengal railway and by
+two branch lines. Save between 1404 and 1442, when it was the seat of an
+independent _raj_, founded by Raja Ganesh, a Hindu turned Mussulman,
+Dinajpur has no separate history. Pillars and copper-plate inscriptions
+have yielded numerous records of the Pal kings who ruled the country
+from the 9th century onwards, and the district is famous for many other
+antiquities, some of which are connected by legend with an immemorial
+past (see _Reports, Arch. Survey of India_, xv.; _Epigraphia Indica_,
+ii.).
+
+
+
+
+DINAN, a town of north-western France, capital of an arrondissement in
+the department of Côtes-du-Nord, 37 m. E. of St Brieuc on the Western
+railway. Pop. (1906) 8588. Dinan is situated on a height on the left
+bank of the Ranee (here canalized), some 17 m. above its mouth at St
+Malo, with which it communicates by means of small steamers. It is
+united to the village of Lanvallay on the right bank of the river by a
+granite viaduct 130 ft. in height. The town is almost entirely encircled
+by the ramparts of the middle ages, strengthened at intervals by towers
+and defended on the south by a castle of the late 14th century, which
+now serves as prison. Three old gateways are also preserved. Dinan has
+two interesting churches; that of St Malo, of late Gothic architecture,
+and St Sauveur, in which the Romanesque and Gothic styles are
+intermingled. In the latter church a granite monument contains the heart
+of Bertrand Du Guesclin, whose connexion with the town is also
+commemorated by an equestrian statue. The quaint winding streets of
+Dinan are often bordered by medieval houses. Its picturesqueness
+attracts large numbers of visitors and there are many English residents
+in the town and its vicinity. About three-quarters of a mile from the
+town are the ruins of the château and the Benedictine abbey at Léhon;
+near the neighbouring village of St Esprit stands the large lunatic
+asylum of Les Bas Foins, founded in 1836; and at no great distance is
+the now dismantled château of La Garaye, which was rendered famous in
+the 18th century by the philanthropic devotion of the count and countess
+whose story is told in Mrs Norton's _Lady of La Garaye_. Dinan is the
+seat of a subprefect and has a tribunal of first instance, and a
+communal college. There is trade in grain, cider, wax, butter and other
+agricultural products. The industries include the manufacture of
+leather, farm-implements and canvas.
+
+The principal event in the history of Dinan, which was a stronghold of
+the dukes of Brittany, is the siege by the English under the duke of
+Lancaster in 1359, during which Du Guesclin and an English knight called
+Thomas of Canterbury engaged in single combat.
+
+
+
+
+DINANT, an ancient town on the right bank of the Meuse in the province
+of Namur, Belgium, connected by a bridge with the left bank, on which
+are the station and the suburb of St Medard. Pop. (1904) 7674. The name
+is supposed to be derived from Diana, and as early as the 7th century it
+was named as one of the dependencies of the bishopric of Tongres. In the
+10th century it passed under the titular sway of Liége, and remained the
+fief of the prince-bishopric till the French revolution put an end to
+that survival of feudalism. In the middle of the 15th century Dinant
+reached the height of its prosperity. With a population of 60,000, and
+8000 workers in copper, it was one of the most flourishing cities in
+Walloon Belgium, until it incurred the wrath of Charles the Bold. Belief
+in the strength of its walls and of the castle that occupied the centre
+bridge, thus effectually commanding navigation by the river, engendered
+arrogance and overconfidence, and the people of Dinant thought they
+could defy the full power of Burgundy. Perhaps they also expected aid
+from France or Liége. In 1466 Charles, in his father's name, laid siege
+to Dinant, and on the 27th of August carried the place by storm. He
+razed the walls and allowed the women, children and priests to retire in
+safety to Liége, but the male prisoners he either hanged or drowned in
+the river by causing them to be cast from the projecting cliff of
+Bouvignes. In 1675 the capture of Dinant formed one of the early
+military achievements of Louis XIV., and it remained in the hands of the
+French for nearly thirty years after that date. The citadel on the
+cliff, 300 ft. or 408 steps above the town, was fortified by the Dutch
+in 1818. It is now dismantled, but forms the chief curiosity of the
+place. The views of the river valley from this eminence are exceedingly
+fine. Half way up the cliff, but some distance south of the citadel, is
+the grotto of Montfat, alleged to be the site of Diana's shrine. The
+church of Notre Dame, dating from the 13th century, stands immediately
+under the citadel and flanking the bridge. It has been restored, and is
+considered by some authorities, although others make the same claim on
+behalf of Huy, the most complete specimen in Belgium of pointed Gothic
+architecture. The baptismal fonts date from the 12th century, and the
+curious spire in the form of an elongated pumpkin and covered with
+slates gives a fantastic and original appearance to the whole edifice.
+The present prosperity of Dinant is chiefly derived from its being a
+favourite summer resort for Belgians as well as foreigners. It has
+facilities for beating and bathing as well as for trips by steamer up
+and down the river Meuse. It is also a convenient central point for
+excursions into the Ardennes. Although there are some indications of
+increased industrial activity in recent years, the population of Dinant
+is not one-eighth of what it was at the time of the Burgundians.
+
+
+
+
+DINAPUR, a town and military station of British India, in the Patna
+district of Bengal, on the right bank of the Ganges, 12 m. W. of Patna
+city by rail. Pop. (1901) 33,699. It is the largest military cantonment
+in Bengal, with accommodation for two batteries of artillery, a European
+and a native infantry regiment. In 1857 the sepoy garrison of the place
+initiated the mutiny of that year in Patna district, but after a
+conflict with the European troops were forced to retire from the town,
+and subsequently laid siege to Arrah.
+
+
+
+
+DINARCHUS, last of the "ten" Attic orators, son of Sostratus (or,
+according to Suidas, Socrates), born at Corinth about 361 B.C. He
+settled at Athens early in life, and when not more than twenty-five was
+already active as a writer of speeches for the law courts. As an alien,
+he was unable to take part in the debates. He had been the pupil both of
+Theophrastus and of Demetrius Phalereus, and had early acquired a
+certain fluency and versatility of style. In 324 the Areopagus, after
+inquiry, reported that nine men had taken bribes from Harpalus, the
+fugitive treasurer of Alexander. Ten public prosecutors were appointed.
+Dinarchus wrote, for one or more of these prosecutors, the three
+speeches which are still extant--_Against Demosthenes_, _Against
+Aristogeiton_, _Against Philocles_. The sympathies of Dinarchus were in
+favour of an Athenian oligarchy under Macedonian control; but it should
+be remembered that he was not an Athenian citizen. Aeschines and Demades
+had no such excuse. In the Harpalus affair, Demosthenes was doubtless
+innocent, and so, probably, were others of the accused. Yet Hypereides,
+the most fiery of the patriots, was on the same side as Dinarchus.
+
+Under the regency of his old master, Demetrius Phalereus, Dinarchus
+exercised much political influence. The years 317-307 were the most
+prosperous of his life. On the fall of Demetrius Phalereus and the
+restoration of the democracy by Demetrius Poliorcetes, Dinarchus was
+condemned to death and withdrew into exile at Chalcis in Euboea. About
+292, thanks to his friend Theophrastus, he was able to return to Attica,
+and took up his abode in the country with a former associate, Proxenus.
+He afterwards brought an action against Proxenus on the ground that he
+had robbed him of some money and plate. Dinarchus died at Athens about
+291.
+
+According to Suidas, Dinarchus wrote 160 speeches; and Dionysius held
+that, out of 85 extant speeches bearing his name, 58 were genuine,--28
+relating to public, 30 to private causes. Although the authenticity of
+the three speeches mentioned above is generally admitted, Demetrius of
+Magnesia doubted that of the speech Against Demosthenes, while A.
+Westermann rejected all three. Dinarchus had little individual style and
+imitated by turns Lysias, Hypereides and Demosthenes. He is called by
+Hermogenes [Greek: o critinos Demosthenes], a metaphor taken from barley compared
+with wheat, or beer compared with wine,--a Demosthenes whose strength
+is rougher, without flavour or sparkle.
+
+  Editions: (text and exhaustive commentary) E. Metzner (1842); (text)
+  T. Thalheim (1887), F. Blass (1888); see L. L. Forman, _Index
+  Andocideus, Lycurgeus, Dinarcheus_ (1897); and, in general, F. Blass,
+  _Attische Beredsamkeit_, iii. There is a valuable treatise on the life
+  and speeches of Dinarchus by Dionysius of Halicarnassus.
+
+
+
+
+
+End of the Project Gutenberg EBook of Encyclopaedia Britannica, 11th
+Edition, Volume 8, Slice 4, by Various
+
+*** END OF THIS PROJECT GUTENBERG EBOOK ENCYC. BRITANNICA, VOL 8 SL 4 ***
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+***** This file should be named 32607-8.txt or 32607-8.zip *****
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+<pre>
+
+The Project Gutenberg EBook of Encyclopaedia Britannica, 11th Edition,
+Volume 8, Slice 4, by Various
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+Title: Encyclopaedia Britannica, 11th Edition, Volume 8, Slice 4
+       "Diameter" to "Dinarchus"
+
+Author: Various
+
+Release Date: May 30, 2010 [EBook #32607]
+
+Language: English
+
+Character set encoding: ISO-8859-1
+
+*** START OF THIS PROJECT GUTENBERG EBOOK ENCYC. BRITANNICA, VOL 8 SL 4 ***
+
+
+
+
+Produced by Marius Masi, Don Kretz and the Online
+Distributed Proofreading Team at https://www.pgdp.net
+
+
+
+
+
+
+</pre>
+
+
+<table border="0" cellpadding="10" style="background-color: #dcdcdc; color: #696969; " summary="Transcriber's note">
+<tr>
+<td style="width:25%; vertical-align:top">
+Transcriber's note:
+</td>
+<td class="norm">
+A few typographical errors have been corrected. They
+appear in the text <span class="correction" title="explanation will pop up">like this</span>, and the
+explanation will appear when the mouse pointer is moved over the marked
+passage. Sections in Greek will yield a transliteration
+when the pointer is moved over them, and words using diacritic characters in the
+Latin Extended Additional block, which may not display in some fonts or browsers, will
+display an unaccented version. <br /><br />
+<a name="artlinks">Links to other EB articles:</a> Links to articles residing in other EB volumes will
+be made available when the respective volumes are introduced online.
+</td>
+</tr>
+</table>
+<div style="padding-top: 3em; ">&nbsp;</div>
+
+
+<h2>THE ENCYCLOP&AElig;DIA BRITANNICA</h2>
+
+<h2>A DICTIONARY OF ARTS, SCIENCES, LITERATURE AND GENERAL INFORMATION</h2>
+
+<h3>ELEVENTH EDITION</h3>
+<div style="padding-top: 3em; ">&nbsp;</div>
+
+<hr class="full" />
+<h3>VOLUME VIII SLICE IV<br /><br />
+Diameter to Dinarchus</h3>
+<hr class="full" />
+<div style="padding-top: 3em; ">&nbsp;</div>
+
+<p class="center1" style="font-size: 150%; font-family: 'verdana';">Articles in This Slice</p>
+<table class="reg" style="width: 90%; font-size: 90%; border: gray 2px solid;" summary="Contents">
+
+<tr><td class="tcl"><a href="#ar1">DIAMETER</a></td>  <td class="tcl"><a href="#ar63">DIEDENHOFEN</a></td></tr>
+<tr><td class="tcl"><a href="#ar2">DIAMOND</a></td>  <td class="tcl"><a href="#ar64">DIEKIRCH</a></td></tr>
+<tr><td class="tcl"><a href="#ar3">DIAMOND NECKLACE, THE AFFAIR OF THE</a></td>  <td class="tcl"><a href="#ar65">DIELECTRIC</a></td></tr>
+<tr><td class="tcl"><a href="#ar4">DIANA</a></td>  <td class="tcl"><a href="#ar66">DIELMANN, FREDERICK</a></td></tr>
+<tr><td class="tcl"><a href="#ar5">DIANA MONKEY</a></td>  <td class="tcl"><a href="#ar67">DIEMEN, ANTHONY VAN</a></td></tr>
+<tr><td class="tcl"><a href="#ar6">DIANE DE FRANCE</a></td>  <td class="tcl"><a href="#ar68">DIEPENBECK, ABRAHAM VAN</a></td></tr>
+<tr><td class="tcl"><a href="#ar7">DIANE DE POITIERS</a></td>  <td class="tcl"><a href="#ar69">DIEPPE</a></td></tr>
+<tr><td class="tcl"><a href="#ar8">DIAPASON</a></td>  <td class="tcl"><a href="#ar70">DIERX, LÉON</a></td></tr>
+<tr><td class="tcl"><a href="#ar9">DIAPER</a></td>  <td class="tcl"><a href="#ar71">DIES, CHRISTOPH ALBERT</a></td></tr>
+<tr><td class="tcl"><a href="#ar10">DIAPHORETICS</a></td>  <td class="tcl"><a href="#ar72">DIEST</a></td></tr>
+<tr><td class="tcl"><a href="#ar11">DIAPHRAGM </a></td>  <td class="tcl"><a href="#ar73">DIESTERWEG, FRIEDRICH ADOLF WILHELM</a></td></tr>
+<tr><td class="tcl"><a href="#ar12">DIARBEKR</a></td>  <td class="tcl"><a href="#ar74">DIET</a></td></tr>
+<tr><td class="tcl"><a href="#ar13">DIARRHOEA</a></td>  <td class="tcl"><a href="#ar75">DIETARY</a></td></tr>
+<tr><td class="tcl"><a href="#ar14">DIARY</a></td>  <td class="tcl"><a href="#ar76">DIETETICS</a></td></tr>
+<tr><td class="tcl"><a href="#ar15">DIASPORE</a></td>  <td class="tcl"><a href="#ar77">DIETRICH, CHRISTIAN WILHELM ERNST</a></td></tr>
+<tr><td class="tcl"><a href="#ar16">DIASTYLE</a></td>  <td class="tcl"><a href="#ar78">DIETRICH OF BERN</a></td></tr>
+<tr><td class="tcl"><a href="#ar17">DIATOMACEAE</a></td>  <td class="tcl"><a href="#ar79">DIEZ, FRIEDRICH CHRISTIAN</a></td></tr>
+<tr><td class="tcl"><a href="#ar18">DIAULOS</a></td>  <td class="tcl"><a href="#ar80">DIEZ</a></td></tr>
+<tr><td class="tcl"><a href="#ar19">DIAVOLO, FRA</a></td>  <td class="tcl"><a href="#ar81">DIFFERENCES, CALCULUS OF</a></td></tr>
+<tr><td class="tcl"><a href="#ar20">DIAZ, NARCISSE VIRGILIO</a></td>  <td class="tcl"><a href="#ar82">DIFFERENTIAL EQUATION</a></td></tr>
+<tr><td class="tcl"><a href="#ar21">DIAZ, PORFIRIO</a></td>  <td class="tcl"><a href="#ar83">DIFFLUGIA</a></td></tr>
+<tr><td class="tcl"><a href="#ar22">DIAZ DE NOVAES, BARTHOLOMEU</a></td>  <td class="tcl"><a href="#ar84">DIFFRACTION OF LIGHT</a></td></tr>
+<tr><td class="tcl"><a href="#ar23">DIAZO COMPOUNDS</a></td>  <td class="tcl"><a href="#ar85">DIFFUSION</a></td></tr>
+<tr><td class="tcl"><a href="#ar24">DIAZOMATA</a></td>  <td class="tcl"><a href="#ar86">DIGBY, SIR EVERARD</a></td></tr>
+<tr><td class="tcl"><a href="#ar25">DIBDIN, CHARLES</a></td>  <td class="tcl"><a href="#ar87">DIGBY, SIR KENELM</a></td></tr>
+<tr><td class="tcl"><a href="#ar26">DIBDIN, THOMAS FROGNALL</a></td>  <td class="tcl"><a href="#ar88">DIGBY, KENELM HENRY</a></td></tr>
+<tr><td class="tcl"><a href="#ar27">DIBDIN, THOMAS JOHN</a></td>  <td class="tcl"><a href="#ar89">DIGENES ACRITAS, BASILIUS</a></td></tr>
+<tr><td class="tcl"><a href="#ar28">DIBRA</a></td>  <td class="tcl"><a href="#ar90">DIGEST</a></td></tr>
+<tr><td class="tcl"><a href="#ar29">DIBRUGARH</a></td>  <td class="tcl"><a href="#ar91">DIGESTIVE ORGANS</a></td></tr>
+<tr><td class="tcl"><a href="#ar30">DICAEARCHUS</a></td>  <td class="tcl"><a href="#ar92">DIGGES, WEST</a></td></tr>
+<tr><td class="tcl"><a href="#ar31">DICE </a></td>  <td class="tcl"><a href="#ar93">DIGIT</a></td></tr>
+<tr><td class="tcl"><a href="#ar32">DICETO, RALPH DE</a></td>  <td class="tcl"><a href="#ar94">DIGITALIS</a></td></tr>
+<tr><td class="tcl"><a href="#ar33">DICEY, EDWARD</a></td>  <td class="tcl"><a href="#ar95">DIGNE</a></td></tr>
+<tr><td class="tcl"><a href="#ar34">DICHOTOMY</a></td>  <td class="tcl"><a href="#ar96">DIGOIN</a></td></tr>
+<tr><td class="tcl"><a href="#ar35">DICK, ROBERT</a></td>  <td class="tcl"><a href="#ar97">DIJON</a></td></tr>
+<tr><td class="tcl"><a href="#ar36">DICK, THOMAS</a></td>  <td class="tcl"><a href="#ar98">DIKE</a></td></tr>
+<tr><td class="tcl"><a href="#ar37">DICKENS, CHARLES JOHN HUFFAM</a></td>  <td class="tcl"><a href="#ar99">DIKKA</a></td></tr>
+<tr><td class="tcl"><a href="#ar38">DICKINSON, ANNA ELIZABETH</a></td>  <td class="tcl"><a href="#ar100">DILAPIDATION</a></td></tr>
+<tr><td class="tcl"><a href="#ar39">DICKINSON, JOHN</a></td>  <td class="tcl"><a href="#ar101">DILATATION</a></td></tr>
+<tr><td class="tcl"><a href="#ar40">DICKSON, SIR ALEXANDER</a></td>  <td class="tcl"><a href="#ar102">DILATORY</a></td></tr>
+<tr><td class="tcl"><a href="#ar41">DICKSON, SIR JAMES ROBERT</a></td>  <td class="tcl"><a href="#ar103">DILEMMA</a></td></tr>
+<tr><td class="tcl"><a href="#ar42">DICOTYLEDONS</a></td>  <td class="tcl"><a href="#ar104">DILETTANTE</a></td></tr>
+<tr><td class="tcl"><a href="#ar43">DICTATOR</a></td>  <td class="tcl"><a href="#ar105">DILIGENCE</a></td></tr>
+<tr><td class="tcl"><a href="#ar44">DICTIONARY</a></td>  <td class="tcl"><a href="#ar106">DILKE, SIR CHARLES WENTWORTH</a></td></tr>
+<tr><td class="tcl"><a href="#ar45">DICTYOGENS</a></td>  <td class="tcl"><a href="#ar107">DILL</a></td></tr>
+<tr><td class="tcl"><a href="#ar46">DICTYS CRETENSIS</a></td>  <td class="tcl"><a href="#ar108">DILLEN, JOHANN JAKOB</a></td></tr>
+<tr><td class="tcl"><a href="#ar47">DICUIL</a></td>  <td class="tcl"><a href="#ar109">DILLENBURG</a></td></tr>
+<tr><td class="tcl"><a href="#ar48">DIDACH&#274;, THE</a></td>  <td class="tcl"><a href="#ar110">DILLENS, JULIEN</a></td></tr>
+<tr><td class="tcl"><a href="#ar49">DIDACTIC POETRY</a></td>  <td class="tcl"><a href="#ar111">DILLINGEN</a></td></tr>
+<tr><td class="tcl"><a href="#ar50">DIDEROT, DENIS</a></td>  <td class="tcl"><a href="#ar112">DILLMANN, CHRISTIAN FRIEDRICH AUGUST</a></td></tr>
+<tr><td class="tcl"><a href="#ar51">DIDIUS SALVIUS JULIANUS, MARCUS</a></td>  <td class="tcl"><a href="#ar113">DILLON, ARTHUR RICHARD</a></td></tr>
+<tr><td class="tcl"><a href="#ar52">DIDO</a></td>  <td class="tcl"><a href="#ar114">DILLON, JOHN</a></td></tr>
+<tr><td class="tcl"><a href="#ar53">DIDON, HENRI</a></td>  <td class="tcl"><a href="#ar115">DILUVIUM</a></td></tr>
+<tr><td class="tcl"><a href="#ar54">DIDOT</a></td>  <td class="tcl"><a href="#ar116">DIME</a></td></tr>
+<tr><td class="tcl"><a href="#ar55">DIDRON, ADOLPHE NAPOLÉON</a></td>  <td class="tcl"><a href="#ar117">DIMENSION</a></td></tr>
+<tr><td class="tcl"><a href="#ar56">DIDYMI</a></td>  <td class="tcl"><a href="#ar118">DIMITY</a></td></tr>
+<tr><td class="tcl"><a href="#ar57">DIDYMIUM</a></td>  <td class="tcl"><a href="#ar119">DINAJPUR</a></td></tr>
+<tr><td class="tcl"><a href="#ar58">DIDYMUS</a></td>  <td class="tcl"><a href="#ar120">DINAN</a></td></tr>
+<tr><td class="tcl"><a href="#ar59">DIDYMUS CHALCENTERUS</a></td>  <td class="tcl"><a href="#ar121">DINANT</a></td></tr>
+<tr><td class="tcl"><a href="#ar60">DIE</a> (town of France)</td>  <td class="tcl"><a href="#ar122">DINAPUR</a></td></tr>
+<tr><td class="tcl"><a href="#ar61">DIE</a> (datum)</td>  <td class="tcl"><a href="#ar123">DINARCHUS</a></td></tr>
+<tr><td class="tcl"><a href="#ar62">DIEBITSCH, HANS KARL FRIEDRICH ANTON</a></td> <td>&nbsp;</td></tr>
+</table>
+
+
+<hr class="art" />
+<p><span class="pagenum"><a name="page158" id="page158"></a>158</span></p>
+<p><span class="bold">DIAMETER<a name="ar1" id="ar1"></a></span> (from the Gr. <span class="grk" title="dia">&#948;&#953;&#940;</span>, through, <span class="grk" title="metron">&#956;&#941;&#964;&#961;&#959;&#957;</span>, measure),
+in geometry, a line passing through the centre of a circle or conic
+section and terminated by the curve; the &ldquo;principal diameters&rdquo; of the
+ellipse and hyperbola coincide with the &ldquo;axes&rdquo; and are at
+right angles; &ldquo;conjugate diameters&rdquo; are such that each bisects
+chords parallel to the other. The diameter of a quadric surface
+is a line at the extremities of which the tangent planes are parallel.
+Newton defined the diameter of a curve of any order as the locus
+of the centres of the mean distances of the points of intersection
+of a system of parallel chords with the curve; this locus may
+be shown to be a straight line. The word is also used as a unit
+of linear measurement of the magnifying power of a lens or
+microscope.</p>
+
+<p>In architecture, the term is used to express the measure of the
+lower part of the shaft of a column. It is employed by Vitruvius
+(iii. 2) to determine the height of a column, which should vary
+from eight to ten diameters according to the intercolumniation:
+and it is generally the custom to fix the lower diameter of the
+shaft by the height required and the Order employed. Thus
+the diameter of the Roman Doric should be about one-eighth of
+the height, that of the Ionic one-ninth, and of the Corinthian
+one-tenth (see <span class="sc"><a href="#artlinks">Order</a></span>).</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIAMOND,<a name="ar2" id="ar2"></a></span> a mineral universally recognized as chief among
+precious stones; it is the hardest, the most imperishable, and
+also the most brilliant of minerals.<a name="fa1a" id="fa1a" href="#ft1a"><span class="sp">1</span></a> These qualities alone
+have made it supreme as a jewel since early times, and yet the
+real brilliancy of the stone is not displayed until it has been
+faceted by the art of the lapidary (<i>q.v.</i>); and this was scarcely
+developed before the year 1746. The consummate hardness of
+the diamond, in spite of its high price, has made it most useful
+for purposes of grinding, polishing and drilling. Numerous
+attempts have been made to manufacture the diamond by artificial
+means, and these attempts have a high scientific interest on
+account of the mystery which surrounds the natural origin of this
+remarkable mineral. Its physical and chemical properties have
+been the subject of much study, and have a special interest
+in view of the extraordinary difference between the physical
+characters of the diamond and those of graphite (blacklead) or
+charcoal, with which it is chemically identical, and into which it
+can be converted by the action of heat or electricity. Again, on
+account of the great value of the diamond, much of the romance
+of precious stones has centred round this mineral; and the
+history of some of the great diamonds of historic times has been
+traced through many extraordinary vicissitudes.</p>
+
+<p>The name <span class="grk" title="Adamas">&#902;&#948;&#940;&#956;&#945;&#962;</span>, &ldquo;the invincible,&rdquo; was probably applied by
+the Greeks to hard metals, and thence to corundum (emery) and
+other hard stones. According to Charles William King, the first
+undoubted application of the name to the diamond is found
+in Manilius (<span class="sc">a.d.</span> 16),&mdash;<i>Sic Adamas</i>, <i>punctum lapidis</i>, <i>pretiosior
+auro</i>,&mdash;and Pliny (<span class="sc">a.d.</span> 100) speaks of the rarity of the stone,
+&ldquo;the most valuable of gems, known only to kings.&rdquo; Pliny described
+six varieties, among which the Indian, having six pointed
+angles, and also resembling two pyramids (<i>turbines</i>, whip-tops)
+placed base to base, may probably be identified as the ordinary
+octahedral crystal (fig. 1). The &ldquo;diamond&rdquo; (<i>Yahalom</i>) in the
+breastplate of the high priest (Ex. xxxix. 11) was certainly some
+other stone, for it bore the name of a tribe, and methods of
+engraving the true diamond cannot have been known so early.
+The stone can hardly have become familiar to the Romans until
+introduced from India, where it was probably mined at a very
+early period. But one or other of the remaining varieties
+mentioned by Pliny (the Macedonian, the Arabian, the Cyprian,
+&amp;c.) may be the true diamond, which was in great request for
+the tool of the gem-engraver. Later Roman authors mentioned
+various rivers in India as yielding the <i>Adamas</i> among their sands.
+The name <i>Adamas</i> became corrupted into the forms <i>adamant</i>,
+<i>diamaunt</i>, <i>diamant</i>, <i>diamond</i>; but the same word, owing to
+a medieval misinterpretation which derived it from <i>adamare</i>
+(compare the French word <i>aimant</i>), was also applied to the
+lodestone.</p>
+
+<p>Like all the precious stones, the diamond was credited with
+many marvellous virtues; among others the power of averting
+insanity, and of rendering poison harmless; and in the middle
+ages it was known as the &ldquo;pietra della reconciliazione,&rdquo; as the
+peacemaker between husband and wife.</p>
+
+<table class="nobctr" summary="Illustration">
+<tr><td class="figcenter" colspan="4"><img style="width:519px; height:132px" src="images/img158a.jpg" alt="" /></td></tr>
+<tr><td class="caption sc">Fig. 1.</td> <td class="caption sc">Fig. 2.</td>
+<td class="caption sc">Fig. 3.</td> <td class="caption sc">Fig. 4.</td></tr></table>
+
+<table class="nobctr" style="float: right; width: 240px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:192px; height:162px" src="images/img158b.jpg" alt="" /></td></tr>
+<tr><td class="caption sc">Fig. 5.</td></tr>
+<tr><td class="figright1"><img style="width:163px; height:148px" src="images/img158c.jpg" alt="" /></td></tr>
+<tr><td class="caption sc">Fig. 6.</td></tr>
+<tr><td class="figright1"><img style="width:177px; height:172px" src="images/img158d.jpg" alt="" /></td></tr>
+<tr><td class="caption sc">Fig. 7.</td></tr></table>
+
+<p><i>Scientific Characters.</i>&mdash;The majority of minerals are found most
+commonly in masses which can with difficulty be recognized as
+aggregates of crystalline grains, and occur comparatively seldom
+as distinct crystals; but the diamond is almost always found
+in single crystals, which show no signs of previous attachment to
+any matrix; the stones were, until the discovery of the South
+African mines, almost entirely derived from sands or gravels,
+but owing to the hardness of the mineral it is rarely, if ever,
+water-worn, and the crystals are often very perfect. The crystals
+belong to the cubic system, generally assuming the form of the
+octahedron (fig. 1), but they may, in accordance with the principles
+of crystallography, also occur in other forms symmetrically
+derived from the octahedron,&mdash;for example, the cube, the
+12-faced figure known as the rhombic dodecahedron (fig. 2), or
+the 48-faced figure known as the hexakis-octahedron (fig. 3), or
+in combinations of these. The octahedron faces are usually
+smooth; most of the other faces are rounded (fig. 4). The cube
+faces are rough with protruding points. The cube is sometimes
+found in Brazil, but is very rare among the S. African stones;
+and the dodecahedron is perhaps more
+common in Brazil than elsewhere.
+There is often a furrow running along
+the edges of the octahedron, or across
+the edges of the cube, and this indicates
+that the apparently simple crystal may
+really consist of eight individuals meeting
+at the centre; or, what comes to the
+same thing, of two individuals interpenetrating
+and projecting through
+each other. If this be so the form of the diamond is really the
+tetrahedron (and the various figures derived symmetrically from
+it) and not the <span class="correction" title="amended from octadehron">octahedron</span> Fig. 5 shows
+how the octahedron with furrowed edge
+may be constructed from two interpenetrating
+tetrahedra (shown in dotted lines).
+If the grooves be left out of account, the
+large faces which have replaced each tetrahedron
+corner then make up a figure which
+has the aspect of a simple octahedron.
+Such regular interpenetrations are known
+in crystallography as &ldquo;twins.&rdquo; There are also twins of diamond
+in which two octahedra (fig. 6) are united by contact along
+a surface parallel to an octahedron face without interpenetration.
+On account of their resemblance to
+the twins of the mineral spinel (which
+crystallizes in octahedra) these are
+known as &ldquo;spinel twins.&rdquo; They are generally
+flattened along the plane of union.
+The crystals often display triangular
+markings, either elevations or pits, upon
+the octahedron faces; the latter are
+particularly well defined and have the form
+of equilateral triangles (fig. 7). They are
+similar to the &ldquo;etched figures&rdquo; produced
+by moistening an octahedron of alum, and have probably been
+produced, like them, by the action of some solvent. Similar, but
+somewhat different markings are produced by the combustion
+of diamond in oxygen, unaccompanied by any rounding of the
+edges.</p>
+
+<p>Diamond possesses a brilliant &ldquo;adamantine&rdquo; lustre, but this
+tends to be greasy on the surface of the natural stones and gives
+<span class="pagenum"><a name="page159" id="page159"></a>159</span>
+the rounded crystals somewhat the appearance of drops of gum.
+Absolutely colourless stones are not so common as cloudy and
+faintly coloured specimens; the usual tints are grey, brown,
+yellow or white; and as rarities, red, green, blue and black
+stones have been found. The colour can sometimes be removed
+or changed at a high temperature, but generally returns on
+cooling. It is therefore more probably due to metallic oxides than
+to hydrocarbons. Sir William Crookes has, however, changed
+a pale yellow diamond to a bluish-green colour by keeping it
+embedded in radium bromide for eleven weeks. The black
+coloration upon the surface produced by this process, as also by
+the electric bombardment in a vacuum tube, appears to be due
+to a conversion of the surface film into graphite. Diamond may
+break with a conchoidal fracture, but the crystals always cleave
+readily along planes parallel to the octahedron faces: of this
+property the diamond cutters avail themselves when reducing
+the stone to the most convenient form for cutting; a sawing
+process, has, however, now been introduced, which is preferable
+to that of cleavage. It is the hardest known substance (though
+tantalum, or an alloy of tantalum now competes with it) and is
+chosen as 10 in the mineralogist&rsquo;s scale of hardness; but the
+difference in hardness between diamond (10) and corundum (9)
+is really greater than that between corundum (9) and talc (1);
+there is a difference in the hardness of the different faces; the
+Borneo stones are also said to be harder than those of Australia,
+and the Australian harder than the African, but this is by no
+means certain. The specific gravity ranges from 3.56 to 3.50,
+generally about 3.52. The coefficient of expansion increases very
+rapidly above 750°, and diminishes very rapidly at low temperatures;
+the maximum density is attained about &minus;42° C.</p>
+
+<p>The very high refractive power (index = 2.417 for sodium light)
+gives the stone its extraordinary brilliancy; for light incident
+within a diamond at a greater angle than 24½° is reflected back
+into the stone instead of passing through it; the corresponding
+angle for glass is 40½°. The very high dispersion (index for red
+light = 2.402, for blue light = 2.460) gives it the wonderful &ldquo;fire&rdquo;
+or display of spectral colours. Certain absorption bands at the
+blue end of the spectrum are supposed to be due to rare elements
+such as samarium. Unlike other cubic crystals, diamond
+experiences a diminution of refractive index with increase of
+temperature. It is very transparent for Röntgen rays, whereas
+paste imitations are opaque. It is a good conductor of heat, and
+therefore feels colder to the touch than glass and imitation stones.
+The diamond has also a somewhat greasy feel. The specific heat
+increases rapidly with rising temperature up to 60° C., and then
+more slowly. Crystals belonging to the cubic system should not
+be birefringent unless strained; diamond often displays double
+refraction particularly in the neighbourhood of inclusions, both
+liquid and solid; this is probably due to strain, and the
+spontaneous explosion of diamonds has often been observed.
+Diamond differs from graphite in being a bad conductor of
+electricity: it becomes positively electrified by friction. The
+electrical resistance is about that of ordinary glass, and is
+diminished by one-half during exposure by Röntgen rays; the
+dielectric constant (16) is greater than that which should
+correspond to the specific gravity.</p>
+
+<p>The phosphorescence produced by friction has been known
+since the time of Robert Boyle (1663); the diamond becomes
+luminous in a dark room after exposure to sunlight or in the
+presence of radium; and many stones phosphoresce beautifully
+(generally with a pale green light) when subjected to the electric
+discharge in a vacuum tube. Some diamonds are more phosphorescent
+than others, and different faces of a crystal may display
+different tints. The combustibility of the diamond was predicted
+by Sir Isaac Newton on account of its high refractive
+power; it was first established experimentally by the Florentine
+Academicians in 1694. In oxygen or air diamond burns at about
+850°, and only continues to do so if maintained at a high temperature;
+but in the absence of oxidising agents it may be raised
+to a much higher temperature. It is, however, infusible at
+the temperature of the electric arc, but becomes converted
+superficially into graphite. Experiments on the combustion of
+diamond were made by Smithson Tennant (1797) and Sir
+Humphry Davy (1816), with the object of proving that it is pure
+carbon; they showed that burnt in oxygen it yields exactly the
+same amount of carbon dioxide as that produced by burning the
+same weight of carbon. Still more convincing experiments were
+made by A. Krause in 1890. Similarly Guyton de Morveau
+showed that, like charcoal, diamond converts soft iron into steel.
+Diamond is insoluble in acid and alkalis, but is oxidised on
+heating with potassium bichromate and sulphuric acid.</p>
+
+<p>Bort (or Boart) is the name given to impure crystals or fragments
+useless for jewels; it is also applied to the rounded
+crystalline aggregates, which generally have a grey colour,
+a rough surface, often a radial structure, and are devoid of
+good cleavage. They are sometimes spherical (&ldquo;shot bort&rdquo;).
+Carbonado or &ldquo;black diamond,&rdquo; found in Bahia (also recently
+in Minas Geraes), is a black material with a minutely crystalline
+structure somewhat porous, opaque, resembling charcoal in
+appearance, devoid of cleavage, rather harder than diamond,
+but of less specific gravity; it sometimes displays a rude cubic
+crystalline form. The largest specimen found (1895) weighed
+3078 carats. Both bort and carbonado seem to be really aggregates
+of crystallized diamond, but the carbonado is so nearly
+structureless that it was till recently regarded as an amorphous
+modification of carbon.</p>
+
+<p><i>Uses of the Diamond.</i>&mdash;The use of the diamond for other
+purposes than jewelry depends upon its extreme hardness: it
+has always been the only material used for cutting or engraving
+the diamond itself. The employment of powdered bort and
+the lapidary&rsquo;s wheel for faceting diamonds was introduced by
+L. von Berquen of Bruges in 1476. Diamonds are now employed
+not only for faceting precious stones, but also for cutting and
+drilling glass, porcelain, &amp;c,; for fine engraving such as scales;
+in dentistry for drilling; as a turning tool for electric-light
+carbons, hard rubber, &amp;c.; and occasionally for finishing accurate
+turning work such as the axle of a transit instrument. For these
+tools the stone is actually shaped to the best form: it is now
+electroplated before being set in its metal mount in order to
+secure a firm fastening. It is also used for bearings in watches
+and electric meters. The best glaziers&rsquo; diamonds are chosen from
+crystals such that a natural curved edge can be used. For rock
+drills, and revolving saws for stone cutting, either diamond, bort
+or carbonado is employed, set in steel tubes, disks or bands. Rock
+drilling is the most important industrial application; and for
+this, owing to its freedom from cleavage, the carbonado is more
+highly prized than diamond; it is broken into fragments about
+3 carats in weight; and in 1905 the value of carbonado was no
+less than from £10 to £14 a carat. It has been found that the
+&ldquo;carbons&rdquo; in drills can safely be subjected to a pressure of over
+60 kilograms per square millimetre, and a speed of 25 metres
+per second. A recent application of the diamond is for wire
+drawing; a hole tapering towards the centre is drilled through
+a diamond, and the metal is drawn through this. No other tool
+is so endurable, or gives such uniform thickness of wire.</p>
+
+<p><i>Distribution and Mining.</i>&mdash;The most important localities for
+diamonds have been: (1) India, where they were mined from
+the earliest times till the close of the 19th century; (2) South
+America, where they have been mined since the middle of the
+18th century; and (3) South Africa, to which almost the whole
+of the diamond-mining industry has been transferred since 1870.</p>
+
+<div class="condensed">
+<p><i>India.</i>&mdash;The diamond is here found in ancient sandstones and conglomerates,
+and in the river gravels and sands derived from them.
+The sandstones and conglomerates belong to the Vindhyan formation
+and overlie the old crystalline rocks: the diamantiferous beds are
+well defined, often not more than 1 ft. in thickness, and contain
+pebbles of quartzite, jasper, sandstone, slate, &amp;c. The mines fall
+into five groups situated on the eastern side of the Deccan plateau
+about the following places (beginning from the south), the first three
+being in Madras. (1) Chennur near Cuddapah on the river Pennar.
+(2) Kurnool near Baneganapalle between the rivers Pennar and
+Kistna. (3) Kollar near Bezwada on the river Kistna. (4) Sambalpur
+on the river Mahanadi in the Central Provinces. (5) Panna near
+Allahabad, in Bundelkhand. The mining has always been carried
+on by natives of low caste, and by primitive methods which do not
+differ much from those described by the French merchant Jean
+Baptiste Tavernier (1605-1689), who paid a prolonged visit to most
+<span class="pagenum"><a name="page160" id="page160"></a>160</span>
+of the mines between 1638 and 1665 as a dealer in precious stones.
+According to his description shallow pits were sunk, and the gravel
+excavated was gathered into a walled enclosure where it was crushed
+and water was poured over it, and it was finally sifted in baskets and
+sorted by hand. The buying and selling was at that period conducted
+by young children. In more modern times there has been the same
+excavation of shallow pits, and sluicing, sifting and sorting, by hand
+labour, the only machinery used being chain pumps made of earthen
+bowls to remove the water from the deeper pits.</p>
+
+<p>At some of the Indian localities spasmodic mining has been carried
+on at different periods for centuries, at some the work which had been
+long abandoned was revived in recent times, at others it has long been
+abandoned altogether. Many of the large stones of antiquity were
+probably found in the Kollar group, where Tavernier found 60,000
+workers in 1645 (?), the mines having, according to native accounts,
+been discovered about 100 years previously. Golconda was the
+fortress and the market for the diamond industry at this group of
+mines, and so gave its name to them. The old mines have now been
+completely abandoned, but in 1891 about 1000 carats were being
+raised annually in the neighbourhood of Hyderabad. The Sambalpur
+group appear to have been the most ancient mines of all, but they
+were not worked later than 1850. The Panna group were the most
+productive during the 19th century. India was no doubt the source
+of all the large stones of antiquity; a stone of 67<span class="spp">3</span>&frasl;<span class="suu">8</span> carats was found
+at Wajra Karur in the Chennur group in 1881, and one of 210½
+carats at Hira Khund in 1809. Other Indian localities besides those
+mentioned above are Simla, in the N.W. Provinces, where a few
+stones have been found, and a district on the Gouel and the Sunk
+rivers in Bengal, which V. Ball has identified with the Soumelpour
+mentioned by Tavernier. The mines of Golconda and Kurnool were
+described as early as 1677 in the twelfth volume of the <i>Philosophical
+Transactions</i> of the Royal Society. At the present time very few
+Indian diamonds find their way out of the country, and, so far as
+the world&rsquo;s supply is concerned, Indian mining of diamonds may be
+considered extinct. The first blow to this industry was the discovery
+of the Brazilian mines in Minas Geraes and Bahia.</p>
+
+<p><i>Brazil.</i>&mdash;-Diamonds were found about 1725 at Tejuco (now Diamantina)
+in Minas Geraes, and the mining became important about
+1740. The chief districts in Minas Geraes are (1) Bagagem on the W.
+side of the Serra da Mata da Corda; (2) Rio Abaete on the E. side of
+the same range; these two districts being among the head waters of
+the Rio de San Francisco and its tributaries; (3) Diamantina, on and
+about the watershed separating the Rio de San Francisco from the
+Rio Jequitinhonha; and (4) Grao Mogul, nearly 200 m. to the N.E.
+of Diamantina on the latter river.</p>
+
+<p>The Rio Abaete district was worked on a considerable scale between
+1785 and 1807, but is now abandoned. Diamantina is at present the
+most important district; it occupies a mountainous plateau, and
+the diamonds are found both on the plateau and in the river valleys
+below it. The mountains consist here of an ancient laminated
+micaceous quartzite, which is in parts a flexible sandstone known as
+itacolumite, and in parts a conglomerate; it is interbedded with
+clay-slate, mica-schist, hornblende-schist and haematite-schist, and
+intersected by veins of quartz. This series is overlain unconformably
+by a younger quartzite of similar character, and itself rests upon the
+crystalline schists. The diamond is found under three conditions:
+(1) in the gravels of the present rivers, embedded in a ferruginous clay-cemented
+conglomerate known as <i>cascalho</i>; (2) in terraces (gupiarras)
+in a similar conglomerate occupying higher levels in the present
+valleys; (3) in plateau deposits in a coarse surface conglomerate
+known as <i>gurgulho</i>, the diamond and other heavy minerals being
+embedded in the red clay which cements the larger blocks. Under
+all these three conditions the diamond is associated with fragments
+of the rocks of the country and the minerals derived from them,
+especially quartz, hornstone, jasper, the polymorphous oxide of
+titanium (rutile, anatase and brookite), oxides and hydrates of iron
+(magnetite, ilmenite, haematite, limonite), oxide of tin, iron pyrites,
+tourmaline, garnet, xenotime, monazite, kyanite, diaspore, sphene,
+topaz, and several phosphates, and also gold. Since the heavy
+minerals of the <i>cascalho</i> in the river beds are more worn than those of
+the terraces, it is highly probable that they have been derived by the
+cutting down of the older river gravels represented by the terraces;
+and since in both deposits the heavy minerals are more abundant
+near the heads of the valleys in the plateau, it is also highly probable
+that both have really been derived from the plateau deposit. In the
+latter, especially at São João da Chapada, the minerals accompanying
+the diamond are scarcely worn at all; in the terraces and the river
+beds they are more worn and more abundant; the terraces, therefore,
+are to be regarded as a first concentration of the plateau material by
+the old rivers; and the <i>cascalho</i> as a second concentration by the
+modern rivers. The mining is carried on by negroes under the supervision
+of overseers; the <i>cascalho</i> is dug out in the dry season and
+removed to a higher level, and is afterwards washed out by hand in
+running water in shallow wooden basins (<i>bateas</i>). The terraces can
+be worked at all seasons, and the material is partly washed out
+by leading streams on to it. The washing of the plateau material is
+effected in reservoirs of rain water.</p>
+
+<p>It is difficult to obtain an estimate of the actual production of the
+Minas Geraes mines, for no official returns have been published, but
+in recent years it has certainly been rivalled by the yield in Bahia.
+The diamond here occurs in river gravels and sands associated with
+the same minerals as in Minas Geraes; since 1844 the richest mines
+have been worked in the Serra de Cincora, where the mountains are
+intersected by the river Paraguassu and its tributaries; it is said
+that there were as many as 20,000 miners working here in 1845, and
+it was estimated that 54,000 carats were produced in Bahia in 1858.
+The earlier workings were in the Serra de Chapada to the N.W. of
+the mines just mentioned. In 1901 there were about 5000 negroes
+employed in the Bahia mines; methods were still primitive; the
+<i>cascalho</i> was dug out from the river beds or tunnelled out from the
+valley side, and washed once a week in sluices of running water,
+where it was turned over with the hoe, and finally washed in wooden
+basins and picked over by hand; sometimes also the diamantiferous
+material is scooped out of the bed of the shallow rivers by divers, and
+by men working under water in caissons. It is almost exclusively in
+the mines of Bahia, and in particular in the Cincora district, that the
+valuable carbonado is found. The carbonado and the diamond have
+been traced to an extensive hard conglomerate which occurs in the
+middle of the sandstone formation. Diamonds are also mined at
+Salobro on the river Pardo not far inland from the port of Canavieras
+in the S.E. corner of Bahia. The enormous development of the South
+African mines, which supplied in 1906, about 90% of the world&rsquo;s produce,
+has thrown into the shade the Brazilian production; but the <i>Bulletin</i>
+for Feb. 1909 of the International Bureau of American Republics gave
+a very confident account of its future, under improved methods.</p>
+
+<p><i>South Africa.</i>&mdash;-The first discovery was made in 1867 by Dr W. G.
+Atherstone, who identified as diamond a pebble obtained from a
+child in a farm on the banks of the Orange river and brought by a
+trader to Grahamstown; it was bought for £500 and displayed in the
+Paris Exhibition of that year. In 1869 a stone weighing 83½ carats
+was found near the Orange river; this was purchased by the earl
+of Dudley for £25,000 and became famous as the &ldquo;Star of South
+Africa.&rdquo; A rush of prospectors at once took place to the banks of
+the Orange and Vaal rivers, and resulted in considerable discoveries, so
+that in 1870 there was a mining camp of no less than 10,000 persons
+on the &ldquo;River Diggings.&rdquo; In the River Diggings the mining was
+carried on in the coarse river gravels, and by the methods of the
+Brazilian negroes and of gold placer-miners. A diggers&rsquo; committee
+limited the size of claims to 30 ft. square, with free access to the river
+bank; the gravel and sand were washed in cradles provided with
+screens of perforated metal, and the concentrates were sorted by
+hand on tables by means of an iron scraper.</p>
+
+<p>But towards the close of 1870 stones were found at Jagersfontein
+and at Dutoitspan, far from the Vaal river, and led to a second great
+rush of prospectors, especially to Dutoitspan, and in 1871 to what
+is now the Kimberley mine in the neighbourhood of the latter. At
+each of these spots the diamantiferous area was a roughly circular
+patch of considerable size, and in some occupied the position of
+one of those depressions or &ldquo;pans&rdquo; so frequent in S. Africa. These
+&ldquo;dry diggings&rdquo; were therefore at first supposed to be alluvial in origin
+like the river gravels; but it was soon discovered that, below the red
+surface soil and the underlying calcareous deposit, diamonds were also
+found in a layer of yellowish clay about 50 ft. thick known as &ldquo;yellow
+ground.&rdquo; Below this again was a hard bluish-green serpentinous rock
+which was at first supposed to be barren bed-rock; but this also
+contained the precious stone, and has become famous, under the
+name of &ldquo;blue ground,&rdquo; as the matrix of the S. African diamonds.
+The yellow ground is merely decomposed blue ground. In the
+Kimberley district five of these round patches of blue ground were
+found within an area little more than 3 m. in diameter; that at
+Kimberley occupying 10 acres, that at Dutoitspan 23 acres. There
+were soon 50,000 workers on this field, the canvas camp was replaced
+by a town of brick and iron surrounded by the wooden huts of the
+natives, and Kimberley became an important centre.</p>
+
+<p>It was soon found that each mine was in reality a huge vertical
+funnel or crater descending to an unknown depth, and filled with
+diamantiferous blue ground. At first each claim was an independent
+pit 31 ft. square sunk into the blue ground; the diamantiferous rock
+was hoisted by bucket and windlass, and roadways were left across
+the pit to provide access to the claims. But the roadways soon fell
+in, and ultimately haulage from the claims could only be provided by
+means of a vast system of wire ropes extending from a triple staging
+of windlasses erected round the entire edge of the mine, which had by
+this time become a huge open pit; the ropes from the upper windlasses
+extended to the centre, and those from the lower tier to the
+sides of the pit; covering the whole mass like a gigantic cobweb.
+(See Plate II. fig. 12.) The buckets of blue ground were hauled up
+these ropes by means of horse whims, and in 1875 steam winding
+engines began to be employed. By this time also improved methods
+in the treatment of the blue ground were introduced. It was carried
+off in carts to open spaces, where an exposure of some weeks to the air
+was found to pulverize the hard rock far more efficiently than the
+old method of crushing with mallets. The placer-miner&rsquo;s cradle and
+rocking-trough were replaced by puddling troughs stirred by a
+revolving comb worked by horse power; reservoirs were constructed
+for the scanty water-supply, bucket elevators were introduced to
+carry away the tailings; and the natives were confined in compounds.
+For these improvements co-operation was necessary; the better
+claims, which in 1872 had risen from £100 to more than £4000 in
+value, began to be consolidated, and a Mining Board was introduced.</p>
+</div>
+
+<p class="noind pt2 sc f80">Plate I.</p>
+
+<table class="nobctr" summary="Illustration">
+<tr><td class="figcenter"><img style="width:320px; height:408px" src="images/img160a1.jpg" alt="" /></td>
+<td class="figcenter"><img style="width:320px; height:407px" src="images/img160a2.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 9.</span>&mdash;DE BEERS MINE, 1874.</td>
+<td class="caption"><span class="sc">Fig. 10.</span>&mdash;KIMBERLEY MINE, 1874.</td></tr>
+
+<tr><td class="figcenter" colspan="2"><img style="width:700px; height:432px" src="images/img160a3.jpg" alt="" /></td></tr>
+<tr><td class="caption" colspan="2"><span class="sc">Fig. 11.</span>&mdash;DE BEERS MINE, 1873.<br />
+<span class="f80">(From photographs by C. Evans.)</span></td></tr></table>
+
+
+<p class="noind pt2 sc f80">Plate II.</p>
+
+<table class="nobctr" summary="Illustration">
+<tr><td class="figcenter"><img style="width:700px; height:439px" src="images/img160b1.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 12.</span>&mdash;KIMBERLEY MINE, 1874.</td></tr>
+
+<tr><td class="figcenter"><img style="width:700px; height:443px" src="images/img160b2.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 13.</span>&mdash;KIMBERLEY MINE, 1902.<br />
+<span class="f80">(From photographs by C. Evans.)</span></td></tr></table>
+
+<p><span class="pagenum"><a name="page161" id="page161"></a>161</span></p>
+
+<div class="condensed">
+<p>In a very few years, however, the open pit mining was rendered
+impossible by the mud rushes, by the falls of the masses of barren
+rock known as &ldquo;reef,&rdquo; which were left standing in the mine, and by
+landslips from the sides, so that in 1883, when the pit had reached a
+depth of about 400 ft., mining in the Kimberley crater had become
+almost impossible. By 1889, in the whole group of mines, Kimberley,
+Dutoitspan, De Beers and Bultfontein, open pit working was practically
+abandoned. Meanwhile mining below the bottom of the pits by
+means of shafts and underground tunnels had been commenced; but
+the full development of modern methods dates from the year 1889
+when Cecil Rhodes and Alfred Beit, who had already secured control
+of the De Beers mine, acquired also the control of the Kimberley mine,
+and shortly afterwards consolidated the entire group in the hands of
+the De Beers Company. (See <span class="sc"><a href="#artlinks">Kimberley</a></span>.)</p>
+
+<p>The scene of native mining was now transferred from the open pit
+to underground tunnels; the vast network of wire ropes (Plate II.
+fig. 12) with their ascending and descending buckets disappeared, and
+with it the cosmopolitan crowd of busy miners working like ants at
+the bottom of the pit. In place of all this, the visitor to Kimberley
+encounters at the edge of the town only a huge crater,
+silent and apparently deserted, with no visible sign of the
+great mining operations which are conducted nearly half
+a mile below the surface. The aspect of the Kimberley
+pit in 1906 is shown in fig. 13 of Plate II., which may
+be compared with the section of fig. 8.</p>
+
+<p>In fig. 13, Plate II., the sequence of the basalt, shale and
+melaphyre is clearly visible on the sides of the pit; and
+fig. 8 shows how the crater or &ldquo;pipe&rdquo; of blue ground has
+penetrated these rocks and also the underlying quartzite.
+The workings at De Beers had extended into the still
+more deeply seated granite in 1906. Figure 9, Plate I.,
+shows the top of the De Beers&rsquo; crater with basalt overlying
+the shale. Figure 8 also explains the modern
+system of mining introduced by Gardner Williams. A
+vertical shaft is sunk in the vicinity of the mine, and from
+this horizontal tunnels are driven into the pipe at different
+levels separated by intervals of 40 ft. Through the
+blue ground itself on each level a series of parallel tunnels
+about 120 ft. apart are driven to the opposite side of the
+pipe, and at right angles to these, and 36 ft. apart,
+another series of tunnels. When the tunnels reach the
+side of the mine they are opened upwards and sideways
+so as to form a large chamber, and the overlying mass of
+blue ground and débris is allowed to settle down and fill
+up the gallery. On each level this process is carried
+somewhat farther back than on the level below (fig. 8);
+material is thus continually withdrawn from one side of
+the mine and extracted by means of the rock shaft on the
+opposite side, while the superincumbent débris is continually
+sinking, and is allowed to fall deeper on the side
+farthest from the shaft as the blue ground is withdrawn
+from beneath it. In 1905 the main shaft had been sunk
+to a depth of 2600 ft. at the Kimberley mine.</p>
+
+<p>For the extraction and treatment of the blue ground
+the De Beers Company in its great winding and washing plant employs
+labour-saving machinery on a gigantic scale. The ground is
+transferred in trucks to the shaft where it is automatically tipped into
+skips holding 96 cubic ft. (six truck loads); these are rapidly hoisted
+to the surface, where their contents are automatically dumped into
+side-tipping trucks, and these in turn are drawn away in a continual
+procession by an endless wire rope along the tram lines leading to the
+vast &ldquo;distributing floors.&rdquo; These are open tracts upon which the blue
+ground is spread out and left exposed to sun and rain until it crumbles
+and disintegrates, the process being hastened by harrowing with
+steam ploughs; this may require a period of three or six months, or
+even a year. The stock of blue ground on the floors at one time in
+1905 was nearly 4,500,000 loads. The disintegrated ground is then
+brought back in the trucks and fed through perforated cylinders into
+the washing pans; the hard blue which has resisted disintegration
+on the floors, and the lumps which are too big to pass the cylindrical
+sieves, are crushed before going to the pans. These are shallow
+cylindrical troughs containing muddy water in which the diamonds
+and other heavy minerals (concentrates) are swept to the rim by
+revolving toothed arms, while the lighter stuff escapes near the centre
+of the pan. The concentrates are then passed over sloping tables
+(pulsator) and shaken to and fro under a stream of water which effects
+a second concentration of the heaviest material.</p>
+
+<p>Until recently the final separation of the diamond from the concentrates
+was made by hand picking, but even this has now been
+replaced by machinery, owing to the remarkable discovery that a
+greased surface will hold a diamond while allowing the other heavy
+minerals to pass over it. The concentrates are washed down a sloping
+table of corrugated iron which is smeared with grease, and it is found
+that practically all the diamonds adhere to the table, and the other
+minerals are washed away. At the large and important Premier mine
+in the Transvaal the Elmore process, used in British Columbia and
+in Wales for the separation of metallic ores, has been also introduced.
+In the Elmore process oil is employed to float off the materials which
+adhere to it, while the other materials remain in the water, the oil
+being separated from the water by centrifugal action. The other
+minerals found in the concentrates are pebbles and fragments of
+pyrope, zircon, cyanite, chrome-diopside, enstatite, a green pyroxene,
+mica, ilmenite, magnetite, chromite, hornblende, olivine, barytes,
+calcite and pyrites.</p>
+
+<p>In all the S. African mines the diamonds are not only crystals of
+various weights from fractions of a carat to 150 carats, but also occur
+as microscopic crystals disseminated through the blue ground. In
+spite of this, however, the average yield in the profitable mines is
+only from 0.2 carat to 0.6 carat per load of 1600 lb, or on an average
+about 1½ grs. per ton. The annual output of diamonds from the De
+Beers mines was valued in 1906 at nearly £5,000,000; the value per
+carat ranging from about 35s. to 70s.</p>
+
+<table class="nobctr" summary="Illustration">
+<tr><td class="figcenter"><img style="width:603px; height:489px" src="images/img161.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="f90">From Gardner Williams&rsquo;s <i>Diamond Mines of South Africa</i>.</span><br />
+<span class="sc">Fig. 8.</span></td></tr></table>
+
+<p>Pipes similar to those which surround Kimberley have been found
+in other parts of S. Africa. One of the best known is that of Jagersfontein,
+which was really the first of the dry diggings (discovered in
+1870). This large mine is near Fauresmith and 80 m. to the south
+of Kimberley. In 1905 the year&rsquo;s production from the Orange River
+Colony mines was more than 320,000 carats, valued at £938,000. But
+by far the largest of all the pipes hitherto discovered is the Premier
+mine in the Transvaal, about 300 m. to the east of Kimberley. This
+was discovered in 1902 and occupies an area of about 75 acres. In
+1906 it was being worked as a shallow open mine; but the description
+of the Kimberley methods given above is applicable to the washing
+plant at that time being introduced into the Premier mine upon a very
+large scale. Comparatively few of the pipes which have been discovered
+are at all rich in diamonds, and many are quite barren; some
+are filled with &ldquo;hard blue&rdquo; which even if diamantiferous may be
+too expensive to work.</p>
+
+<p>The most competent S. African geologists believe all these remarkable
+pipes to be connected with volcanic outbursts which occurred
+over the whole of S. Africa during the Cretaceous period (after the
+deposition of the Stormberg beds), and drilled these enormous craters
+through all the later formations. With the true pipes are associated
+dykes and fissures also filled with diamantiferous blue ground. It
+is only in the more northerly part of the country that the pipes
+are filled with blue ground (or &ldquo;kimberlite&rdquo;), and that they are
+diamantiferous; but over a great part of Cape Colony have been
+discovered what are probably similar pipes filled with agglomerates,
+breccias and tuffs, and some with basic lavas; one, in particular, in
+the Riversdale Division near the southern coast, being occupied by a
+melilite-basalt. It is quite clear that the occurrence of the diamond
+in the S. African pipes is quite different from the occurrences in
+alluvial deposits which have been described above. The question of
+the origin of the diamond in S. Africa and elsewhere is discussed
+below.</p>
+
+<p>The River Diggings on the Vaal river are still worked upon a small
+scale, but the production from this source is so limited that they are
+of little account in comparison with the mines in the blue ground.
+The stones, however, are good; since they differ somewhat from the
+Kimberley crystals it is probable that they were not derived from
+the present pipes. Another S. African locality must be mentioned;
+considerable finds were reported in 1905 and 1906 from gravels
+at Somabula near Gwelo in Rhodesia where the diamond is associated
+with chrysoberyl, corundum (both sapphire and ruby), topaz,
+garnet, ilmenite, staurolite, rutile, with pebbles of quartz, granite,
+<span class="pagenum"><a name="page162" id="page162"></a>162</span>
+chlorite-schist, &amp;c. Diamond has also been reported from kimberlite
+&ldquo;pipes&rdquo; in Rhodesia.</p>
+
+<p><i>Other Localities.</i>&mdash;In addition to the South American localities
+mentioned above, small diamonds have also been mined since their
+discovery in 1890 on the river Mazaruni in British Guiana, and
+finds have been reported in the gold washings of Dutch Guiana.
+Borneo has possessed a diamond industry since the island was first
+settled by the Malays; the references in the works of Garcia de Orta,
+Linschoten, De Boot, De Laet and others, to Malacca as a locality
+relate to Borneo. The large Borneo stone, over 360 carats in weight,
+known as the Matan, is in all probability not a diamond. The chief
+mines are situated on the river Kapuas in the west and near
+Bandjarmassin in the south-east of the island, and the alluvial
+deposits in which they occur are worked by a small number of Chinese
+and Malays. Australia has yielded diamonds in alluvial deposits
+near Bathurst (where the first discovery was made in 1851) and
+Mudgee in New South Wales, and also near Bingara and Inverell
+in the north of the colony. At Mount Werong a stone weighing
+29 carats was found in 1905. At Ruby Hill near Bingara they were
+found in a breccia filling a volcanic pipe. At Ballina, in New England,
+diamonds have been found in the sea sand. Other Australian
+localities are Echunga in South Australia; Beechworth, Arena and
+Melbourne in Victoria; Freemantle and Nullagine in Western
+Australia; the Palmer and Gilbert rivers in Queensland. These have
+been for the most part discoveries in alluvial deposits of the goldfields,
+and the stones were small. In Tasmania also diamonds have
+been found in the Corinna goldfields. Europe has produced few
+diamonds. Humboldt searched for them in the Urals on account of
+the similarity of the gold and platinum deposits to those of Brazil,
+and small diamonds were ultimately found (1829) in the gold washings
+of Bissersk, and later at Ekaterinburg and other spots in the Urals.
+In Lapland they have been found in the sands of the Pasevig river.
+Siberia has yielded isolated diamonds from the gold washings of
+Yenisei. In North America a few small stones have been found in
+alluvial deposits, mostly auriferous, in Georgia, N. and S. Carolina,
+Kentucky, Virginia, Tennessee, Wisconsin, California, Oregon and
+Indiana. A crystal weighing 23¾ carats was found in Virginia in
+1855, and one of 21¼ carats in Wisconsin in 1886. In 1906 a number
+of small diamonds were discovered in an altered peridotite somewhat
+resembling the S. African blue ground, at Murfreesboro, Pike
+county, Arkansas. Considerable interest attaches to the diamonds
+found in Wisconsin, Michigan and Ohio near the Great Lakes, for they
+are here found in the terminal moraines of the great glacial sheet which
+is supposed to have spread southwards from the region of Hudson
+Bay; several of the drift minerals of the diamantiferous region of
+Indiana have been identified as probably of Canadian origin; no
+diamonds have however yet been found in the intervening country of
+Ontario. A rock similar to the blue ground of Kimberley has been
+found in the states of Kentucky and New York. The occurrence of
+diamond in meteorites is described below.</p>
+
+<p><i>Origin of the Diamond in Nature.</i>&mdash;It appears from the foregoing
+account that at most localities the diamond is found in alluvial deposits
+probably far from the place where it originated. The minerals
+associated with it do not afford much clue to the original conditions;
+they are mostly heavy minerals derived from the neighbouring rocks,
+in which the diamond itself has not been observed. Among the
+commonest associates of the diamond are quartz, topaz, tourmaline,
+rutile, zircon, magnetite, garnet, spinel and other minerals which are
+common accessory constituents of granite, gneiss and the crystalline
+schists. Gold (also platinum) is a not infrequent associate, but this
+may only mean that the sands in which the diamond is found have
+been searched because they were known to be auriferous; also that
+both gold and diamond are among the most durable of minerals and
+may have survived from ancient rocks of which other traces have been
+lost.</p>
+
+<p>The localities at which the diamond has been supposed to occur
+in its original matrix are the following:&mdash;at Wajra Karur, in the
+Cuddapah district, India, M. Chaper found diamond with corundum
+in a decomposed red pegmatite vein in gneiss. At S&#257;o João da
+Chapada, in Minas Geraes, diamonds occur in a clay interstratified
+with the itacolumite, and are accompanied by sharp crystals of rutile
+and haematite in the neighbourhood of decomposed quartz veins
+which intersect the itacolumite. It has been suggested that these
+three minerals were originally formed in the quartz veins. In both
+these occurrences the evidence is certainly not sufficient to establish
+the presence of an original matrix. At Inverell in New South Wales
+a diamond (1906) has been found embedded in a hornblende diabase
+which is described as a dyke intersecting the granite. Finally there is
+the remarkable occurrence in the blue ground of the African pipes.</p>
+
+<p>There has been much controversy concerning the nature and origin
+of the blue ground itself; and even granted that (as is generally
+believed) the blue ground is a much serpentinized volcanic breccia
+consisting originally of an olivine-bronzite-biotite rock (the so-called
+kimberlite), it contains so many rounded and angular fragments of
+various rocks and minerals that it is difficult to say which of them
+may have belonged to the original rock, and whether any were formed
+<i>in situ</i>, or were brought up from below as inclusions. Carvill Lewis
+believed the blue ground to be true eruptive rock, and the carbon to
+have been derived from the bituminous shales of which it contains
+fragments. The Kimberley shales, which are penetrated by the De
+Beers group of pipes, were, however, certainly not the source of the
+carbon at the Premier (Transvaal) mine, for at this locality the shales
+do not exist. The view that the diamond may have crystallized out
+from solution in its present matrix receives some support from the
+experiments of W. Luzi, who found that it can be corroded by the
+solvent action of fused blue ground; from the experiments of
+J. Friedländer, who obtained diamond by dissolving graphite in fused
+olivine; and still more from the experiments of R. von Hasslinger
+and J. Wolff, who have obtained it by dissolving graphite in a fused
+mixture of silicates having approximately the composition of the
+blue ground. E. Cohen, who regarded the pipes as of the nature of a
+mud volcano, and the blue ground as a kimberlite breccia altered by
+hydrothermal action, thought that the diamond and accompanying
+minerals had been brought up from deep-seated crystalline schists.
+Other authors have sought the origin of the diamond in the action
+of the hydrated magnesian silicates on hydrocarbons derived from
+bituminous schists, or in the decomposition of metallic carbides.</p>
+
+<p>Of great scientific interest in this connexion is the discovery of
+small diamonds in certain meteorites, both stones and irons; for
+example, in the stone which fell at Novo-Urei in Penza, Russia, in
+1886, in a stone found at Carcote in Chile, and in the iron found at
+Cañon Diablo in Arizona. Graphitic carbon in cubic form (cliftonite)
+has also been found in certain meteoric &ldquo;irons,&rdquo; for example in those
+from Magura in Szepes county, Hungary, and Youndegin near York
+in Western Australia. The latter is now generally believed to be
+altered diamond. The fact that H. Moissan has produced the
+diamond artificially, by allowing dissolved carbon to crystallize out
+at a high temperature and pressure from molten iron, coupled with
+the occurrence in meteoric iron, has led Sir William Crookes and others
+to conclude that the mineral may have been derived from deep-seated
+iron containing carbon in solution (see the article <span class="sc"><a href="#artlinks">Gem, Artificial</a></span>).
+Adolf Knop suggested that this may have first yielded hydrocarbons
+by contact with water, and that from these the crystalline diamond
+has been formed. The meteoric occurrence has even suggested the
+fanciful notion that all diamonds were originally derived from
+meteorites. The meteoric iron of Arizona, some of which contains
+diamond, is actually found in and about a huge crater which is
+supposed by some to have been formed by an immense meteorite
+penetrating the earth&rsquo;s crust.</p>
+
+<p>It is, at any rate, established that carbon can crystallize as diamond
+from solution in iron, and other metals; and it seems that high
+temperature and pressure and the absence of oxidizing agents are
+necessary conditions. The presence of sulphur, nickel, &amp;c., in the
+iron appears to favour the production of the diamond. On the other
+hand, the occurrence in meteoric stones, and the experiments
+mentioned above, show that the diamond may also crystallize from
+a basic magma, capable of yielding some of the metallic oxides and
+ferro-magnesian silicates; a magma, therefore, which is not devoid
+of oxygen. This is still more forcibly suggested by the remarkable
+eclogite boulder found in the blue ground of the Newlands mine, not
+far from the Vaal river, and described by T. G. Bonney. The boulder
+is a crystalline rock consisting of pyroxene (chrome-diopside), garnet,
+and a little olivine, and is studded with diamond crystals; a portion
+of it is preserved in the British Museum (Natural History). In
+another eclogite boulder, diamond was found partly embedded in
+pyrope. Similar boulders have also been found in the blue ground
+elsewhere. Specimens of pyrope with attached or embedded diamond
+had previously been found in the blue ground of the De Beers mines.
+In the Newlands boulder the diamonds have the appearance of being
+an original constituent of the eclogite. It seems therefore that a holocrystalline
+pyroxene-garnet rock may be one source of the diamond
+found in blue ground. On the other hand many tons of the somewhat
+similar eclogite in the De Beers mine have been crushed and have not
+yielded diamond. Further, the ilmenite, which is the most characteristic
+associate of the diamond in blue ground, and other of the
+accompanying minerals, may have come from basic rocks of a
+different nature.</p>
+
+<p>The Inverell occurrence may prove to be another example of
+diamond crystallized from a basic rock.</p>
+
+<p>In both occurrences, however, there is still the possibility that the
+eclogite or the basalt is not the original matrix, but may have caught
+up the already formed diamond from some other matrix. Some
+regard the eclogite boulders as derived from deep-seated crystalline
+rocks, others as concretions in the blue ground.</p>
+
+<p>None of the inclusions in the diamond gives any clue to its origin;
+diamond itself has been found as an inclusion, as have also black
+specks of some carbonaceous materials. Other black specks have been
+identified as haematite and ilmenite; gold has also been found;
+other included minerals recorded are rutile, topaz, quartz, pyrites,
+apophyllite, and green scales of chlorite (?). Some of these are of very
+doubtful identification; others (<i>e.g.</i> apophyllite and chlorite) may
+have been introduced along cracks. Some of the fibrous inclusions
+were identified by H. R. Göppert as vegetable structures and were
+supposed to point to an organic origin, but this view is no longer held.
+Liquid inclusions, some of which are certainly carbon dioxide, have
+also been observed.</p>
+
+<p>Finally, then, both experiment and the natural occurrence in rocks
+and meteorites suggest that diamond may crystallize not only from
+iron but also from a basic silicate magma, possibly from various rocks
+consisting of basic silicates. The blue ground of S. Africa may be
+<span class="pagenum"><a name="page163" id="page163"></a>163</span>
+the result of the serpentinization of several such rocks, and although
+now both brecciated and serpentinized some of these may have been
+the original matrix. A circumstance often mentioned in support of
+this view is the fact that the diamonds in one pipe generally differ
+somewhat in character from those of another, even though they be
+near neighbours.</p>
+</div>
+
+<p><i>History.</i>&mdash;All the famous diamonds of antiquity must have been
+Indian stones. The first author who described the Indian mines
+at all fully was the Portuguese, Garcia de Orta (1565), who was
+physician to the viceroy of Goa. Before that time there were
+only legendary accounts like that of Sindbad&rsquo;s &ldquo;Valley of
+the Diamonds,&rdquo; or the tale of the stones found in the brains of
+serpents. V. Ball thinks that the former legend originated in the
+Indian practice of sacrificing cattle to the evil spirits when a new
+mine is opened; birds of prey would naturally carry off the flesh,
+and might give rise to the tale of the eagles carrying diamonds
+adhering to the meat.</p>
+
+<p>The following are some of the most famous diamonds of the
+world:&mdash;</p>
+
+<p>A large stone found in the Golconda mines and said to have
+weighed 787 carats in the rough, before being cut by a Venetian
+lapidary, was seen in the treasury of Aurangzeb in 1665 by
+Tavernier, who estimated its weight after cutting as 280 (?)
+carats, and described it as a rounded rose-cut-stone, tall on one
+side. The name <i>Great Mogul</i> has been frequently applied to this
+stone. Tavernier states that it was the famous stone given to
+Shah Jahan by the emir Jumla. The <i>Orloff</i>, stolen by a French
+soldier from the eye of an idol in a Brahmin temple, stolen again
+from him by a ship&rsquo;s captain, was bought by Prince Orloff for
+£90,000, and given to the empress Catharine II. It weighs
+194¾ carats, is of a somewhat yellow tinge, and is among the
+Russian crown jewels. The <i>Koh-i-nor</i>, which was in 1739 in the
+possession of Nadir Shah, the Persian conqueror, and in 1813 in
+that of the raja of Lahore, passed into the hands of the East
+India Company and was by them presented to Queen Victoria
+in 1850. It then weighed 186<span class="spp">1</span>&frasl;<span class="suu">16</span> carats, but was recut in London
+by Amsterdam workmen, and now weighs 106<span class="spp">1</span>&frasl;<span class="suu">16</span> carats. There
+has been much discussion concerning the possibility of this stone
+and the Orloff being both fragments of the Great Mogul. The
+Mogul Baber in his memoirs (1526) relates how in his conquest of
+India he captured at Agra the great stone weighing 8 mishkals,
+or 320 ratis, which may be equivalent to about 187 carats. The
+Koh-i-nor has been identified by some authors with this stone and
+by others with the stone seen by Tavernier. Tavernier, however,
+subsequently described and sketched the diamond which he saw
+as shaped like a bisected egg, quite different therefore from the
+Koh-i-nor. Nevil Story Maskelyne has shown reason for believing
+that the stone which Tavernier saw was really the Koh-i-nor
+and that it is identical with the great diamond of Baber; and
+that the 280 carats of Tavernier is a misinterpretation on his part
+of the Indian weights. He suggests that the other and larger
+diamond of antiquity which was given to Shah Jahan may
+be one which is now in the treasury of Teheran, and that this is
+the true Great Mogul which was confused by Tavernier with the
+one he saw. (See Ball, Appendix I. to Tavernier&rsquo;s <i>Travels</i> (1889);
+and Maskelyne, <i>Nature</i>, 1891, 44, p. 555.).</p>
+
+<p>The <i>Regent</i> or <i>Pitt</i> diamond is a magnificent stone found in
+either India or Borneo; it weighed 410 carats and was bought for
+£20,400 by Pitt, the governor of Madras; it was subsequently,
+in 1717, bought for £80,000 (or, according to some authorities,
+£135,000) by the duke of Orleans, regent of France; it was reduced
+by cutting to 136<span class="spp">14</span>&frasl;<span class="suu">16</span> carats; was stolen with the other crown
+jewels during the Revolution, but was recovered and is still in
+France. The <i>Akbar Shah</i> was originally a stone of 116 carats with
+Arabic inscriptions engraved upon it; after being cut down to
+71 carats it was bought by the gaikwar of Baroda for £35,000.
+The <i>Nizam</i>, now in the possession of the nizam of Hyderabad, is
+supposed to weigh 277 carats; but it is only a portion of a stone
+which is said to have weighed 440 carats before it was broken.
+The <i>Great Table</i>, a rectangular stone seen by Tavernier in 1642
+at Golconda, was found by him to weigh 242<span class="spp">3</span>&frasl;<span class="suu">16</span> carats; Maskelyne
+regards it as identical with the <i>Darya-i-nur</i>, which is also a
+rectangular stone weighing about 186 carats in the possession of
+the shah of Persia. Another stone, the <i>Taj-e-mah</i>, belonging to
+the shah, is a pale rose pear-shaped stone and is said to weigh
+146 carats.</p>
+
+<p>Other famous Indian diamonds are the following:&mdash;The <i>Sancy</i>,
+weighing 53<span class="spp">12</span>&frasl;<span class="suu">16</span> carats, which is said to have been successively the
+property of Charles the Bold, de Sancy, Queen Elizabeth,
+Henrietta Maria, Cardinal Mazarin, Louis XIV.; to have been
+stolen with the Pitt during the French Revolution; and subsequently
+to have been the property of the king of Spain, Prince
+Demidoff and an Indian prince. The <i>Nassak</i>, 78<span class="spp">5</span>&frasl;<span class="suu">8</span> carats, the
+property of the duke of Westminster. The <i>Empress Eugénie</i>,
+51 carats, the property of the gaikwar of Baroda. The <i>Pigott</i>,
+49 carats(?), which cannot now be traced. The <i>Pasha</i>, 40 carats.
+The <i>White Saxon</i>, 48¾ carats. The <i>Star of Este</i>, 25<span class="spp">13</span>&frasl;<span class="suu">32</span> carats.</p>
+
+<p>Coloured Indian diamonds of large size are rare; the most
+famous are:&mdash;a beautiful blue brilliant, 67<span class="spp">2</span>&frasl;<span class="suu">16</span> carats, cut from a
+stone weighing 112<span class="spp">3</span>&frasl;<span class="suu">16</span> carats brought to Europe by Tavernier.
+It was stolen from the French crown jewels with the Regent and
+was never recovered. The <i>Hope</i>, 44¼ carats, has the same colour
+and is probably a portion of the missing stone: it was so-called
+as forming part of the collection of H. T. Hope (bought for
+£18,000), and was sold again in 1906 (resold 1909). Two other
+blue diamonds are known, weighing 13¾ and 1¾ carats, which may
+also be portions of the French diamond. The <i>Dresden Green</i>, one
+of the Saxon crown jewels, 40 carats, has a fine apple-green
+colour. The <i>Florentine</i>, 133<span class="spp">1</span>&frasl;<span class="suu">5</span> carats, one of the Austrian crown
+jewels, is a very pale yellow.</p>
+
+<p>The most famous Brazilian stones are:&mdash;The <i>Star of the South</i>,
+found in 1853, when it weighed 254½ carats and was sold for
+£40,000; when cut it weighed 125 carats and was bought by the
+gaikwar of Baroda for £80,000. Also a diamond belonging to
+Mr Dresden, 119 carats before, and 76½ carats after cutting.</p>
+
+<p>Many large stones have been found in South Africa; some are
+yellow but some are as colourless as the best Indian or Brazilian
+stones. The most famous are the following:&mdash;the <i>Star of South
+Africa</i>, or <i>Dudley</i>, mentioned above, 83½ carats rough, 46½ carats
+cut. The <i>Stewart</i>, 288<span class="spp">3</span>&frasl;<span class="suu">8</span> carats rough, 120 carats cut. Both these
+were found in the river diggings. The <i>Porter Rhodes</i> from
+Kimberley, of the finest water, weighed about 150 carats. The
+<i>Victoria</i>, 180 carats, was cut from an octahedron weighing 457½
+carats, and was sold to the nizam of Hyderabad for £400,000.
+The <i>Tiffany</i>, a magnificent orange-yellow stone, weighs 125½
+carats cut. A yellowish octahedron found at De Beers weighed
+428½ carats, and yielded a brilliant of 288½ carats. Some of the
+finest and largest stones have come from the Jagersfontein mine;
+one, the <i>Jubilee</i>, found in 1895, weighed 640 carats in the rough
+and 239 carats when cut. Until 1905 the largest known diamond
+in the world was the <i>Excelsior</i>, found in 1893 at Jagersfontein by
+a native while loading a truck. It weighed 971 carats, and was
+ultimately cut into ten stones weighing from 68 to 13 carats.
+But all previous records were surpassed in 1905 by a magnificent
+stone more than three times the size of any known diamond,
+which was found in the yellow ground at the newly discovered
+Premier mine in the Transvaal. This extraordinary diamond
+weighed 3025¾ carats (1<span class="spp">1</span>&frasl;<span class="suu">3</span> &#8468;) and was clear and water white; the
+largest of its surfaces appeared to be a cleavage plane, so that it
+might be only a portion of a much larger stone. It was known
+as the <i>Cullinan Diamond</i>. This stone was purchased by the
+Transvaal government in 1907 and presented to King Edward VII.
+It was sent to Amsterdam to be cut, and in 1908 was divided into
+nine large stones and a number of small brilliants. The four
+largest stones weigh 516½ carats, 309<span class="spp">3</span>&frasl;<span class="suu">16</span> carats, 92 carats and 62
+carats respectively. Of these the first and second are the largest
+brilliants in existence. All the stones are flawless and of the
+finest quality.</p>
+
+<div class="condensed">
+<p><span class="sc">Bibliography.</span>&mdash;Boetius de Boot, <i>Gemmarum et lapidum
+historia</i> (1609); D. Jeffries, <i>A Treatise on Diamonds and Pearls</i>
+(1757); J. Mawe, <i>Travels in the Interior of Brazil</i> (1812); <i>Treatise on
+Diamonds and Precious Stones</i> (1813): Pinder, <i>De adamante</i> (1829);
+Murray, <i>Memoir on the Nature of the Diamond</i> (1831); C. Zerenner,
+<i>De adamante dissertatio</i> (1850); H. Emanuel, <i>Diamonds and
+Precious Stones</i> (1865); A. Schrauf, <i>Edelsteinkunde</i> (1869); N. Jacobs
+and N. Chatrian, <i>Monographie du diamant</i> (1880); V. Ball, <i>Geology
+of India</i> (1881); C. W. King, <i>The Natural History of Precious Stones</i>
+<span class="pagenum"><a name="page164" id="page164"></a>164</span>
+<i>and Precious Metals</i> (1883); M. E. Boutan, <i>Le Diamant</i> (1886);
+S. M. Burnham, <i>Precious Stones in Nature, Art and Literature</i> (1887);
+P. Groth, <i>Grundriss der Edelsteinkunde</i> (1887); A. Liversidge, <i>The
+Minerals of New South Wales</i> (1888); <i>Tavernier&rsquo;s Travels in India</i>,
+translated by V. Ball (1889); E. W. Streeter, <i>The Great Diamonds
+of the World</i> (1896); H. C. Lewis, <i>The Genesis and Matrix of the
+Diamond</i> (1897); L. de Launay, <i>Les Diamants du Cap</i> (1897);
+C. Hintze, <i>Handbuch der Mineralogie</i> (1898); E. W. Streeter,
+<i>Precious Stones and Gems</i> (6th ed., 1898); Dana, <i>System of Mineralogy</i>
+(1899); Kunz and others, <i>The Production of Precious Stones</i> (in
+annual, <i>Mineral Resources of the United States</i>); M. Bauer, <i>Precious
+Stones</i> (trans. L. J. Spencer, 1904); A. W. Rogers, <i>An Introduction
+to the Geology of Cape Colony</i> (1905); Gardner F. Williams, <i>The
+Diamond Mines of South Africa</i> (revised edition, 1906); George F.
+Kunz, &ldquo;Diamonds, a study of their occurrence in the United States,
+with descriptions and comparisons of those from all known localities&rdquo;
+(U.S. Geol. Survey, 1909); P. A. Wagner, <i>Die Diamantführenden
+Gesteine Südafrikas</i> (1909).</p>
+
+<p>Among papers in scientific periodicals may be mentioned articles
+by Adler, Ball, Baumhauer, Beck, Bonney, Brewster, Chaper, Cohen,
+Crookes, Daubrée, Derby, Des Cloizeaux, Doelter, Dunn, Flight,
+Friedel, Gorceix, Gürich, Goeppert, Harger, Hudleston, Hussak,
+Jannettaz, Jeremejew, de Launay, Lewis, Maskelyne, Meunier,
+Moissan, Molengraaff, Moulle, Rose, Sadebeck, Scheibe, Stelzner,
+Stow. See generally Hintze&rsquo;s <i>Handbuch der Mineralogie</i>.</p>
+</div>
+<div class="author">(H. A. Mi.)</div>
+
+<hr class="foot" /> <div class="note">
+
+<p><a name="ft1a" id="ft1a" href="#fa1a"><span class="fn">1</span></a> Diamonds are invariably weighed in carats and in ½, ¼, <span class="spp">1</span>&frasl;<span class="suu">8</span>, <span class="spp">1</span>&frasl;<span class="suu">16</span>, <span class="spp">1</span>&frasl;<span class="suu">32</span>, <span class="spp">1</span>&frasl;<span class="suu">64</span>
+of a carat. One (English) carat = 3.17 grains = .2054 gram. One
+ounce = 151½ carats. (See <span class="sc"><a href="#artlinks">Carat</a></span>.)</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIAMOND NECKLACE, THE AFFAIR OF THE,<a name="ar3" id="ar3"></a></span> a mysterious
+incident at the court of Louis XVI. of France, which involved
+the queen Marie Antoinette. The Parisian jewellers Boehmer and
+Bassenge had spent some years collecting stones for a necklace
+which they hoped to sell to Madame Du Barry, the favourite of
+Louis XV., and after his death to Marie Antoinette. In 1778
+Louis XVI. proposed to the queen to make her a present of
+the necklace, which cost 1,600,000 livres. But the queen is
+said to have refused it, saying that the money would be better
+spent equipping a man-of-war. According to others, Louis XVI.
+himself changed his mind. After having vainly tried to place the
+necklace outside of France, the jewellers attempted again in 1781
+to sell it to Marie Antoinette after the birth of the dauphin. It
+was again refused, but it was evident that the queen regretted
+not being able to acquire it.</p>
+
+<p>At that time there was a personage at the court whom Marie
+Antoinette particularly detested. It was the cardinal Louis de
+Rohan, formerly ambassador at Vienna, whence he had been
+recalled in 1774, having incurred the queen&rsquo;s displeasure by
+revealing to the empress Maria Theresa the frivolous actions of
+her daughter, a disclosure which brought a maternal reprimand,
+and for having spoken lightly of Maria Theresa in a letter of
+which Marie Antoinette learned the contents. After his return
+to France the cardinal was anxious to regain the favour of the
+queen in order to obtain the position of prime minister. In March
+1784 he entered into relations with a certain Jeanne de St Remy
+de Valois, a descendant of a bastard of Henry II., who after many
+adventures had married a <i>soi-disant</i> comte de Lamotte, and lived
+on a small pension which the king granted her. This adventuress
+soon gained the greatest ascendancy over the cardinal, with whom
+she had intimate relations. She persuaded him that she had been
+received by the queen and enjoyed her favour; and Rohan
+resolved to use her to regain the queen&rsquo;s good will. The comtesse
+de Lamotte assured the cardinal that she was making efforts on
+his behalf, and soon announced to him that he might send his
+justification to Marie Antoinette. This was the beginning of a
+pretended correspondence between Rohan and the queen, the
+adventuress duly returning replies to Rohan&rsquo;s notes, which she
+affirmed to come from the queen. The tone of the letters became
+very warm, and the cardinal, convinced that Marie Antoinette
+was in love with him, became ardently enamoured of her. He
+begged the countess to obtain a secret interview for him with the
+queen, and a meeting took place in August 1784 in a grove in
+the garden at Versailles between him and a lady whom the
+cardinal believed to be the queen herself. Rohan offered her
+a rose, and she promised him that she would forget the past.
+Later a certain Marie Lejay (renamed by the comtesse &ldquo;Baronne
+Gay d&rsquo;Oliva,&rdquo; the last word being apparently an anagram of
+Valoi), who resembled Marie Antoinette, stated that she had
+been engaged to play the role of queen in this comedy. In any
+case the countess profited by the cardinal&rsquo;s conviction to borrow
+from him sums of money destined ostensibly for the queen&rsquo;s
+works of charity. Enriched by these, the countess was able to
+take an honourable place in society, and many persons believed
+her relations with Marie Antoinette, of which she boasted openly
+and unreservedly, to be genuine. It is still an unsettled question
+whether she simply mystified people, or whether she was really
+employed by the queen for some unknown purpose, perhaps
+to ruin the cardinal. In any case the jewellers believed in
+the relations of the countess with the queen, and they resolved
+to use her to sell their necklace. She at first refused their
+commission, then accepted it. On the 21st of January 1785
+she announced that the queen would buy the necklace, but
+that not wishing to treat directly, she left the affair to a high
+personage. A little while later Rohan came to negotiate the
+purchase of the famous necklace for the 1,600,000 livres, payable
+in instalments. He said that he was authorized by the queen,
+and showed the jewellers the conditions of the bargain approved
+in the handwriting of Marie Antoinette. The necklace was
+given up. Rohan took it to the countess&rsquo;s house, where a man,
+in whom Rohan believed he recognized a valet of the queen,
+came to fetch it. Madame de Lamotte had told the cardinal
+that Marie Antoinette would make him a sign to indicate her
+thanks, and Rohan believed that she did make him a sign.
+Whether it was so, or merely chance or illusion, no one knows.
+But it is certain that the cardinal, convinced that he was acting
+for the queen, had engaged the jewellers to thank her; that
+Boehmer and Bassenge, before the sale, in order to be doubly sure,
+had sent word to the queen of the negotiations in her name; that
+Marie Antoinette had allowed the bargain to be concluded, and
+that after she had received a letter of thanks from Boehmer, she
+had burned it. Meanwhile the &ldquo;comte de Lamotte&rdquo; appears to
+have started at once for London, it is said with the necklace,
+which he broke up in order to sell the stones.</p>
+
+<p>When the time came to pay, the comtesse de Lamotte presented
+the cardinal&rsquo;s notes; but these were insufficient, and
+Boehmer complained to the queen, who told him that she had
+received no necklace and had never ordered it. She had the
+story of the negotiations repeated for her. Then followed a <i>coup
+de théâtre</i>. On the 15th of August 1785, Assumption day, when
+the whole court was awaiting the king and queen in order to go to
+the chapel, the cardinal de Rohan, who was preparing to officiate,
+was arrested and taken to the Bastille. He was able, however, to
+destroy the correspondence exchanged, as he thought, with the
+queen, and it is not known whether there was any connivance of
+the officials, who did not prevent this, or not. The comtesse de
+Lamotte was not arrested until the 18th of August, after having
+destroyed her papers. The police set to work to find all her
+accomplices, and arrested the girl Oliva and a certain Reteaux
+de Villette, a friend of the countess, who confessed that he had
+written the letters given to Rohan in the queen&rsquo;s name, and
+had imitated her signature on the conditions of the bargain. The
+famous charlatan Cagliostro was also arrested, but it was recognized
+that he had taken no part in the affair. The cardinal de
+Rohan accepted the parlement of Paris as judges. A sensational
+trial resulted (May 31, 1786) in the acquittal of the cardinal, of
+the girl Oliva and of Cagliostro. The comtesse de Lamotte was
+condemned to be whipped, branded and shut up in the
+Salpetrière. Her husband was condemned, in his absence, to the
+galleys for life. Villette was banished.</p>
+
+<p>Public opinion was much excited by this trial. It is generally
+believed that Marie Antoinette was stainless in the matter, that
+Rohan was an innocent dupe, and that the Lamottes deceived
+both for their own ends. People, however, persisted in the belief
+that the queen had used the countess as an instrument to satisfy
+her hatred of the cardinal de Rohan. Various circumstances
+fortified this belief, which contributed to render Marie Antoinette
+very unpopular&mdash;her disappointment at Rohan&rsquo;s acquittal, the
+fact that he was deprived of his charges and exiled to the abbey of
+la Chaise-Dieu, and finally the escape of the comtesse de Lamotte
+from the Salpetrière, with the connivance, as people believed,
+of the court. The adventuress, having taken refuge abroad,
+published <i>Mémoires</i> in which she accused the queen. Her
+<span class="pagenum"><a name="page165" id="page165"></a>165</span>
+husband also wrote <i>Mémoires</i>, and lived until 1831, after having,
+it is said, received subsidies from Louis XVIII.</p>
+
+<div class="condensed">
+<p>See M. Tourneux, <i>Marie Antoinette devant l&rsquo;histoire: Essai bibliographique</i>
+(2nd ed., Paris, 1901); Émile Campardon, <i>Marie Antoinette
+et le procès du collier</i> (Paris, 1863); P. Audebert, <i>L&rsquo;Affaire du collier
+de la reine, d&rsquo;après la correspondance inédite du chevalier de Pujol</i>
+(Rouen, 1901); F. d&rsquo;Albini, <i>Marie Antoinette and the Diamond Necklace
+from another Point of View</i> (London, 1900); Funck-Brentano,
+<i>L&rsquo;Affaire du collier</i> (1903); A. Lang, <i>Historical Mysteries</i> (1904).
+Carlyle&rsquo;s essay on <i>The Diamond Necklace</i> (first published in 1837 in
+<i>Fraser&rsquo;s Magazine</i>) is of historical literary interest.</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIANA,<a name="ar4" id="ar4"></a></span> in Roman mythology, an old Italian goddess, in later
+times identified with the Greek Artemis (<i>q.v.</i>). That she was
+originally an independent Italian deity is shown by her name,
+which is the feminine form of Janus (= Dianus). She is essentially
+the goddess of the moon and light generally, and presides over
+wood, plain and water, the chase and war. As the goddess of
+childbirth, she was known, like Juno, by the name of Lucina, the
+&ldquo;bringer to light.&rdquo; As the moon-goddess she was also identified
+with Hecate, and invoked as &ldquo;three-formed&rdquo; in reference to the
+phases of the moon. Her most celebrated shrine was in a grove
+at Aricia (whence her title of Nemorensis) near the modern lake of
+Nemi. Here she was worshipped side by side with a male deity
+Virbius, a god of the forest and the chase. This Virbius was
+subsequently identified with Hippolytus, the favourite of Artemis,
+who was said to have been brought to life by Aesculapius and
+conducted by Diana to Aricia (Ovid, <i>Fasti</i>, iii. 263, vi. 731,
+<i>Metam.</i> xv. 497; Virgil, <i>Aeneid</i>, vii. 761). A barbarous custom,
+perhaps reminiscent of human sacrifice once offered to her,
+prevailed in connexion with her ritual here; her priest, called
+<i>Rex Nemorensis</i>, who was a runaway slave, was obliged to qualify
+for office by slaying his predecessor in single combat (Strabo v.
+p. 239; Suetonius, <i>Caligula</i>, 35). This led to the identification of
+Diana with the Tauric Artemis, whose image was said to have been
+removed by Orestes to the grove of Aricia (see <span class="sc"><a href="#artlinks">Aricini</a></span>).</p>
+
+<p>After the destruction of Alba Longa this grove was for a long time
+the united sanctuary of the neighbouring Latin and Rutulian cities,
+until at last it was extinguished beneath the supremacy of Rome.
+The festival of the goddess was on the ides (13th) of August, the
+full moon of the hot season. She was worshipped with torches,
+her aid was sought by women seeking a happy deliverance in
+childbirth, and many votive offerings have been found on the site.
+The worship of Diana was brought to Rome by Latin plebeians,
+and hence she was regarded as the protectress of the lower
+classes, and especially of slaves. In accordance with this, her
+most important temple was that on the Aventine, the chief seat
+of the plebeians, founded by Servius Tullius, originally as a
+sanctuary of the Latin league (Dion. Halic. iv. 26). No man was
+allowed to enter the temple, and on the day of its dedication
+(August 13) the slaves kept holiday (Plutarch, <i>Quaest. Rom.</i> 100).
+This Diana was identified with the sister of Apollo, and at the
+secular games she was worshipped simply as Artemis. Another
+celebrated sanctuary of Diana was that on the slopes of Mount
+Tifata near Capua (where she was worshipped under the name of
+Tifatina), a sanctuary specially favoured by Sulla and Vespasian.
+As Noctiluca (&ldquo;giving light by night&rdquo;) she had a sanctuary on
+the Palatine which was kept illuminated throughout the night
+(Varro, <i>L.L.</i> v. 68). On the Nemi priesthood see J. G. Frazer,
+<i>Golden Bough</i>.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIANA MONKEY,<a name="ar5" id="ar5"></a></span> a West African representative of the
+guenon monkeys taking its name, <i>Cercopithecus diana</i>, from the
+presence of a white crescent on the forehead; another characteristic
+feature being the pointed white beard. The general colour
+of the fur is greyish, with a deep tinge of chestnut from the
+middle of the back to the root of the tail. Together with
+<i>C. neglectus</i> of East and Central Africa, <i>C. ignitus</i> of Liberia, and
+<i>C. roloway</i> of the Gold Coast, the diana represents the special
+subgenus of guenons known as <i>Pogonocebus</i>. Although the diana
+monkey is commonly seen in menageries, little is known of its
+habits in the wild state.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIANE DE FRANCE<a name="ar6" id="ar6"></a></span> (1538-1619), duchess of Montmorency
+and Angoulême, was the natural daughter of Henry II. of France
+and a young Piedmontese, Filippe Duc. The constable de
+Montmorency went so far as to assert that of all the children of
+Henry II. Diane was the only one who resembled him. Catherine
+de&rsquo; Medici was greatly incensed at this affront, and took her
+revenge by having the constable disgraced on the death of Henry
+II. Brantôme is loud in praise of Diane. She was a perfect horsewoman
+and dancer, played several musical instruments, knew
+Spanish and Italian, and &ldquo;estoit très belle de visage et de taille.&rdquo;
+Legitimated in 1547, she was married in 1553 to Horace Farnese,
+second son of the duke of Parma, but her husband was killed soon
+afterwards at the siege of Hesdin. In order to assure his position,
+the constable de Montmorency wished to marry her to his eldest
+son, Francis. This was a romantic adventure, for Francis had
+clandestinely married Mademoiselle de Piennes. The constable
+dissolved this union, and after lengthy negotiations obtained the
+dispensation of the pope. On the 3rd of May 1559 Francis
+married Diane. A wise and moderate woman, Diane undoubtedly
+helped to make Francis de Montmorency one of the leaders of the
+party of the <i>politiques</i>. Again a widow in 1579, she had some
+influence at the court of Henry III., and negotiated his reconciliation
+with Henry of Navarre (1588). She retained her influence
+in the reign of Henry IV., conveyed the bodies of Catherine
+de&rsquo; Medici and Henry III. to St Denis, and died in 1619 at her
+hôtel of Angoulême.</p>
+
+<div class="condensed">
+<p>See Brantôme, ed. by Lalanne, in the <i>Coll de la société d&rsquo;histoire
+de France</i>, vol. viii. (1875); J. de Thou, <i>Historia sui temporis...</i>
+(1733); Matthieu de Morgues, <i>Oraison funèbre de Diane de France</i>
+(Paris, 1619).</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIANE DE POITIERS<a name="ar7" id="ar7"></a></span> (1499-1566), duchess of Valentinois,
+and mistress of Henry II. of France, was the daughter of Jean
+de Poitiers, seigneur de St Vallier, who came of an old family of
+Dauphiné. In 1515 she married Louis de Brézé, grand seneschal
+of Normandy, by whom she had two daughters. She became a
+widow in 1533, but soon replaced her husband by a more illustrious
+lover, the king&rsquo;s second son, Henry, who became dauphin
+in 1536. Although he was ten years younger than Diane, she
+inspired the young prince with a profound passion, which lasted
+until his death. The accession of Henry II. in 1547 was also the
+accession of Diane: she was virtual queen, while Henry&rsquo;s lawful
+wife, Catherine de&rsquo; Medici, lived in comparative obscurity. The
+part Diane played, however, must not be exaggerated. More
+rapacious than ambitious, she concerned herself little with
+government, but devoted her energies chiefly to augmenting her
+income, and providing for her family and friends. Henry was
+the most prodigal of lovers, and gave her all rights over the
+duchy of Valentinois. Although she showed great tact in her
+dealings with the queen, Catherine drove her from the court
+after Henry&rsquo;s death, and forced her to restore the crown jewels
+and to accept Chaumont in exchange for Chenonceaux. Diane
+retired to her château at Anet, where she died in 1566.</p>
+
+<p>Several historians relate that she had been the mistress of
+Francis I. before she became the dauphin&rsquo;s mistress, and that she
+gave herself to the king in order to obtain the pardon of her
+father, who had been condemned to death as an accomplice of the
+constable de Bourbon. This rumour, however, has no serious
+foundation. Men vied with each other in celebrating Diane&rsquo;s
+beauty, which, if we may judge from her portraits, has been
+slightly exaggerated. She was a healthy, vigorous woman, and,
+by dint of great pains, succeeded in retaining her beauty late into
+life. It is said that even on the coldest mornings she would wash
+her face with well water. Diane was a patroness of the arts.
+She entrusted to Philibert de l&rsquo;Orme the building of her château
+at Anet, and it was for her that Jean Goujon executed his masterpiece,
+the statue of Diana, now in the Louvre.</p>
+
+<div class="condensed">
+<p>See G. Guiffrey, <i>Lettres inédites de Diane de Poytiers</i> (Paris, 1866)
+and <i>Procès criminel de Jehan de Poytiers</i> (Paris, 1867); Capefigue,
+<i>Diane de Poitiers</i> (Paris, 1860); Hay, <i>Madame Dianne de Poytiers</i>
+(London, 1900).</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIAPASON<a name="ar8" id="ar8"></a></span> (Gr. <span class="grk" title="dia pasôn">&#948;&#953;&#8048; &#960;&#945;&#963;&#8182;&#957;</span>, through all), a term in music,
+originally for an interval of an octave. The Greek is an abbreviation
+of <span class="grk" title="hê dia pasôn chordôn symphônia">&#7969; &#948;&#953;&#8048; &#960;&#945;&#963;&#8182;&#957; &#967;&#959;&#961;&#948;&#8182;&#957; &#963;&#965;&#956;&#966;&#969;&#957;&#943;&#945;</span>, a consonance
+through all the tones of the scale. In this sense it is only
+used now, loosely, for the compass of an instrument or voice,
+or for a harmonious melody. The name is given to the two
+<span class="pagenum"><a name="page166" id="page166"></a>166</span>
+foundation stops of an organ, the open and the stopped diapason
+(see <span class="sc"><a href="#artlinks">Organ</a></span>), and to a standard of musical pitch, as in the French
+<i>diapason normal</i> (see <span class="sc"><a href="#artlinks">Pitch, Musical</a></span>).</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIAPER<a name="ar9" id="ar9"></a></span> (derived through the Fr, from the Gr. <span class="grk" title="dia">&#948;&#953;&#940;</span>, through,
+and <span class="grk" title="aspros">&#7940;&#963;&#960;&#961;&#959;&#962;</span>, white; the derivation from the town of Ypres,
+&ldquo;d&rsquo;Ypres,&rdquo; in Belgium is unhistorical, as diapers were known
+for centuries before its existence), the name given to a textile
+fabric, formerly of a rich and costly nature with embroidered
+ornament, but now of linen or cotton, with a simple woven
+pattern; and particularly restricted to small napkins. In
+architecture, the term &ldquo;diaper&rdquo; is given to any small pattern of
+a conventional nature repeated continuously and uniformly
+over a surface; the designs may be purely geometrical, or based
+on floral forms, and in early examples were regulated by the process
+of their textile origin. Subsequently, similar patterns were
+employed in the middle ages for the surface decoration of stone,
+as in Westminster Abbey and Bayeux cathedral in the spandrils
+of the arcades of the choir and nave; also in mural painting,
+stained glass, incised brasses, encaustic tiles, &amp;c. Probably in
+most cases the pattern was copied, so far as the general design
+is concerned, from the tissues and stuffs of Byzantine manufacture,
+which came over to Europe and were highly prized as
+ecclesiastical vestments.</p>
+
+<table class="nobctr" summary="Illustration">
+<tr><td class="figcenter1"><img style="width:367px; height:261px" src="images/img166a.jpg" alt="" /></td></tr></table>
+
+<div class="condensed">
+<p>In its textile use, the term diaper was originally applied to silk
+patterns of a geometrical pattern; it is now almost exclusively used
+for diamond patterns made from linen or cotton yarns. An illustration
+of two patterns of this nature is shown in the figure. The floats
+of the warp and the weft are mostly in three; indeed the patterns
+are made from a base weave which is composed entirely of
+floats of this number. It will be seen that both designs are formed
+of what may be termed concentric figures&mdash;alternately black and
+white. Pattern B differs from pattern A only in that more of these
+concentric figures are used for the complete figure. If pattern B,
+which shows only one unit, were extended, the effect would be similar
+to A, except for the size of the unit. In A there are four complete
+units, and hence the pattern appears more striking. Again, the
+repeating of B would cause the four corner pieces to join and to form
+a diamond similar to the one in the centre. The two diamonds in B
+would then alternate diagonally to left and right. Special names are
+given to certain kinds of diapers, <i>e.g.</i> &ldquo;bird&rsquo;s-eye,&rdquo; &ldquo;pheasant&rsquo;s-eye&rdquo;;
+these terms indicate, to a certain extent, the size of the
+complete diamond in the cloth&mdash;the smaller kind taking the name
+&ldquo;bird&rsquo;s-eye.&rdquo; The size of the pattern on paper has little connexion
+with the size of the pattern in the cloth, for it is clearly the number
+of threads and picks per inch which determine the size of the pattern
+in the cloth from any given design. Although A is larger than what
+is usually termed the &ldquo;bird&rsquo;s-eye&rdquo; pattern, it is evident that it may
+be made to appear as such, provided that the cloth is fine enough.
+These designs, although adapted mostly for cloths such as nursery-diapers,
+for pinafores, &amp;c., are sometimes used in the production of
+towels and table-cloths. In the figure, the first pick in A is identical
+with the first pick in B, and the part C shows how each interweaves
+with the twenty-four threads.</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIAPHORETICS<a name="ar10" id="ar10"></a></span> (from Gr. <span class="grk" title="diaphorein">&#948;&#953;&#945;&#966;&#959;&#961;&#949;&#8150;&#957;</span>, to carry through),
+the name given to those remedies which promote perspiration.
+In health there is constantly taking place an exhalation of
+watery vapour from the skin, by which not only are many of the
+effete products of nutrition eliminated, but the body is kept cool.
+Under exertion or in a heated atmosphere this natural function
+of the skin is increased, sweating more or less profuse follows,
+and, evaporation going on rapidly over the whole surface, little
+or no rise in the temperature of the body takes place. In many
+forms of disease, such as fevers and inflammatory affections, the
+action of the skin is arrested, and the surface of the body feels
+harsh and dry, while the temperature is greatly elevated. The
+occurrence of perspiration not unfrequently marks a crisis in such
+diseases, and is in general regarded as a favourable event. In
+some chronic diseases, such as diabetes and some cases of
+Bright&rsquo;s disease, the absence of perspiration is a marked feature;
+while, on the other hand, in many wasting diseases, such as
+phthisis, the action of the skin is increased, and copious exhausting
+sweating occurs. Many means can be used to induce perspiration,
+among the best known being baths, either in the form of hot
+vapour or hot water baths, or in that part of the process of
+the Turkish bath which consists in exposing the body to a dry and
+hot atmosphere. Such measures, particularly if followed by the
+drinking of hot liquids and the wrapping of the body in warm
+clothing, seldom fail to excite copious perspiration. Numerous
+medicinal substances have the same effect.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIAPHRAGM<a name="ar11" id="ar11"></a></span> (Gr. <span class="grk" title="diaphragma">&#948;&#953;&#940;&#966;&#961;&#945;&#947;&#956;&#945;</span>, a partition). The diaphragm
+or midriff (Anglo-Saxon, <i>mid</i>, middle, <i>hrif</i>, belly) in
+human anatomy is a large fibro-muscular partition between the
+cavities of the thorax and abdomen; it is convex toward the
+thorax, concave toward the abdomen, and consists of a central
+tendon and a muscular margin. The <i>central tendon</i> (q, fig. 1) is trefoil
+in shape, its leaflets being right, left and anterior; of these the right
+is the largest and the left the smallest. The fleshy fibres rise, in
+front from the back of the xiphoid cartilage of the sternum (d),
+laterally by six serrations, from the inner surfaces of the lower six
+ribs, interdigitating with the transversalis, posteriorly from the
+arcuate ligaments, of which there are five, a pair of external, a
+pair of internal, and a single median one. The <i>external arcuate
+ligament</i> (h) stretches from the tip of the twelfth rib (b) to the
+costal process of the first lumbar vertebra in front of the quadratus
+lumborum muscle (o), the <i>internal</i> and <i>middle</i> are continuations
+of the <i>crura</i> which rise from the ventro-lateral aspects of
+the bodies of the lumbar vertebrae, the right (e) coming from
+three, the left (f) from two. On reaching the level of the twelfth
+thoracic vertebra each crus spreads out into a fan-shaped mass of
+fibres, of which the innermost join their fellows from the opposite
+crus, in front of the aortic opening (k), to form the <i>middle arcuate
+ligament</i>; the outer ones (g) arch in front of the psoas muscle (n)
+to the tip of the costal process of the first lumbar vertebra to
+form the <i>internal arcuate ligament</i>, while the intermediate ones
+pass to the central tendon. There are three large openings in the
+diaphragm; the <i>aortic</i> (k) is behind the middle arcuate ligament
+and transmits the aorta, the vena azygos major, and the thoracic
+duct. In the right leaflet is an opening (sometimes called the
+<i>hiatus quadratus</i>) for the inferior vena cava and a branch of the
+right phrenic nerve (m), while in front and a little to the left of
+the aortic opening is one for the oesophagus and the two pneumogastric
+nerves (l), the left being in front and the right behind.
+<span class="pagenum"><a name="page167" id="page167"></a>167</span>
+The fleshy fibres on each side of this opening act as a sphincter.
+Passing between the xiphoid and costal origins in front are the
+superior epigastric arteries, while the other terminal branches of
+the internal mammaries, the musculo-phrenics, pass through
+between two costal origins.</p>
+
+<table class="nobctr" summary="Illustration">
+<tr><td class="figcenter"><img style="width:413px; height:430px" src="images/img166b.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 1.</span>&mdash;Abdominal Surface of the Diaphragm.</td></tr></table>
+
+<p>Through the crura pass the splanchnic nerves, and in addition
+to these the left crus is pierced by the vena azygos minor. The
+sympathetic nerves usually enter the abdomen behind the internal
+arcuate ligaments. The phrenic nerves, which are the main
+supply of the diaphragm, divide before reaching the muscle and
+pierce it in a number of places to enter its abdominal surface, but
+some of the lower intercostal nerves assist in the supply. The last
+thoracic or subcostal nerves pass behind the external arcuate
+ligament.</p>
+
+<p>For the action of the diaphragm see <span class="sc"><a href="#artlinks">Respiratory System</a></span>.</p>
+
+<div class="condensed">
+<p><i>Embryology.</i>&mdash;The diaphragm is at first developed in the neck region
+of the embryo, and this accounts for the phrenic nerves, which supply
+it, rising from the fourth and fifth cervical. From the mesoderm on
+the caudal side of the pericardium is developed the <i>septum transversum</i>,
+and in this the central tendon is formed. The fleshy portion is
+developed on each side in two parts, an anterior or sterno-costal
+which is derived from the longitudinal neck musculature, probably
+the same layer from which the sternothyroid comes, and a spinal part
+which is a derivative of the transversalis sheet of the trunk. Between
+these two parts is at one time a gap, the <i>spino-costal hiatus</i>, and this
+is obliterated by the growth of the pleuro-peritoneal membrane, which
+may occasionally fail to close and so may form the site of a phrenic
+hernia. With the growth of the body and the development of the
+lungs the diaphragm shifts its position until it becomes the septum
+between the thoracic and abdominal cavities. (See A. Keith, &ldquo;On the
+Development of the Diaphragm,&rdquo; <i>Jour. of Anat. and Phys.</i> vol. 39.)
+A. Paterson has recorded cases in which the left half of the diaphragm
+is wanting (<i>Proceedings</i> of the Anatomical Society of Gt. Britain,
+June 1900; <i>Jour. of Anat. and Phys.</i> vol. 34), and occasionally
+deficiencies are found elsewhere, especially in the sternal portion.
+For further details see Quain&rsquo;s <i>Anatomy</i>, vol. i. (London, 1908).</p>
+
+<p><i>Comparative Anatomy.</i>&mdash;A complete diaphragm, separating the
+thoracic from the abdominal parts of the coelom, is characteristic of
+the Mammalia; it usually has the human structure and relations
+except that below the Anthropoids it is separated from the pericardium
+by the azygous lobe of the lung. In some Mammals, <i>e.g.</i> Echidna
+and Phocoena, it is entirely muscular. In the Cetacea it is remarkable
+for its obliquity; its vertebral attachment is much nearer the tail
+than its sternal or ventral one; this allows a much larger lung space
+in the dorsal than in the ventral part of the thorax, and may be
+concerned with the equipoise of the animal. (Otto Müller, &ldquo;Untersuchungen
+über die Veränderung, welche die Respirationsorgane der
+Säugetiere durch die Anpassung an das Leben im Wasser erlitten
+haben,&rdquo; <i>Jen. Zeitschr. f. Naturwiss.</i>, 1898, p. 93.) In the Ungulata
+only one crus is found (Windle and Parsons, &ldquo;Muscles of the
+Ungulata,&rdquo; <i>Proc. Zool. Soc.</i>, 1903, p. 287). Below the Mammals
+incomplete partitions between the pleural and peritoneal cavities
+are found in Chelonians, Crocodiles and Birds, and also in Amphibians
+(Xenopus and Pipa).</p>
+</div>
+<div class="author">(F. G. P.)</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIARBEKR<a name="fa1b" id="fa1b" href="#ft1b"><span class="sp">1</span></a><a name="ar12" id="ar12"></a></span> (<i>Kara Amid</i> or Black Amid; the Roman
+<i>Amida</i>), the chief town of a vilayet of Asiatic Turkey, situated
+on a basaltic plateau on the right bank of the Tigris, which here
+flows in a deep open valley. The town is still surrounded by the
+masonry walls of black basalt which give it the name of <i>Kara</i>
+or Black Amid; they are well built and imposing on the west
+facing the open country, but almost in ruins where they overlook
+the river. A mass of gardens and orchards cover the slope down
+to the river on the S.W., but there are no suburbs outside the
+walls. The houses are rather crowded but only partially fill
+the walled area. The population numbers 38,000, nearly half
+being Christian, comprising Turks, Kurds, Arabs, Turkomans,
+Armenians, Chaldeans, Jacobites and a few Greeks. The streets
+are 10 ft. to 15 ft. wide, badly paved and dirty; the houses and
+shops are low, mostly of stone, and some of stone and mud.
+The bazaar is a good one, and gold and silver filigree work is
+made, peculiar in character and design. The cotton industry is
+declining, but manufacture of silk is increasing. Fruit is good and
+abundant as the rich volcanic soil is well watered from the town
+springs. The size of the melons is specially famous. To the
+south, the walls are some 40 ft. high, faced with large cut stone
+blocks of very solid construction, with towers and square bastions
+rising to 500 ft. There are four gates: on the north the Kharput
+gate, on the west the Rum, on the south the Mardin, and on the
+east the Yeni Kapu or new gate. A citadel enclosure stands
+at the N. E. corner and is now partly in ruins, but the interior
+space is occupied by the government konak. The summer
+climate in the confined space within the town is excessively hot
+and unhealthy. Epidemics of typhus are not unknown, as well
+as ophthalmia. The Diarbekr boil is like the &ldquo;Aleppo button,&rdquo;
+lasting a long time and leaving a deep scar. Winters are frequently
+severe but do not last long. Snow sometimes lies, and
+ice is stored for summer use. Scorpions noted for the virulence of
+their poison abound as well as horse leeches in the tanks. The
+town is supplied with water both by springs inside the town
+and by aqueducts from fountains at Ali Punar and Hamervat.
+The principal exports are wool, mohair and copper ore, and
+imports are cotton and woollen goods, indigo, coffee, sugar,
+petroleum, &amp;c.</p>
+
+<p>The Great Mosque, Ulu Jami, formerly a Christian church,
+occupies the site of a Sassanian palace and was built with
+materials from an older palace, probably that of Tigranes II.
+The remains consist of the façades of two palaces 400 ft. apart,
+each formed by a row of Corinthian columns surmounted by an
+equal number of a Byzantine type. Kufic inscriptions run across
+the fronts under the entablature. The court of the mosque
+is entered by a gateway on which lions and other animals are
+sculptured. The churches of greatest interest are those of SS.
+Cosmas and Damian (Jacobite) and the church of St James
+(Greek). In the 19th century Diarbekr was one of the largest
+and most flourishing cities of Asia, and as a commercial centre it
+now stands at the meeting-point of several important routes. It
+is at the head of the navigation of the Tigris, which is traversed
+down stream by <i>keleks</i> or rafts supported by inflated skins.
+There is a good road to Aleppo and Alexandretta on the Mediterranean,
+and to Samsun on the Black Sea by Kharput, Malatia
+and Sivas. There are also routes to Mosul and Bitlis.</p>
+
+<p>Diarbekr became a Roman colony in <span class="sc">a.d.</span> 230 under the name
+of Amida, and received a Christian bishop in <span class="sc">a.d.</span> 325. It was
+enlarged and strengthened by Constantius II., in whose reign it
+was taken after a long siege by Shapur (Sapor) II., king of Persia.
+The historian Ammianus Marcellinus, who took part in the
+defence, gives a detailed account of it. In the later wars between
+the Persians and Romans it more than once changed hands.
+Though ceded by Jovian to the Persians it again became annexed
+to the Roman empire, and in the reign of Anastasius (<span class="sc">a.d.</span> 502)
+was once more taken by the Persians, when 80,000 of its inhabitants
+were slain. It was taken c. 638 by the Arabs, and
+afterwards passed into the hands of the Seljuks and Persians,
+from whom it was finally captured by Selim I. in 1515; and
+since that date it has remained under Ottoman rule. About 2 m.
+below the town is a masonry bridge over the Tigris; the older
+portion being probably Roman, and the western part, which bears
+a Kufic inscription, being Arab.</p>
+
+<p>The vilayet of Diarbekr extends south from Palu on the
+Euphrates to Mardin and Nisibin on the edge of the Mesopotamian
+plain, and is divided into three sanjaks&mdash;Arghana, Diarbekr and
+Mardin. The headwaters of the main arm of the Tigris have
+their source in the vilayet.</p>
+
+<p>Cereals, cotton, tobacco, rice and silk are produced, but most of
+the fertile lands have been abandoned to semi-nomads, who raise
+large quantities of live stock. The richest portion of the vilayet
+lies east of the capital in the rolling plains watered by tributaries
+of the Tigris. An exceptionally rich copper mine exists at
+Arghana Maden, but it is very imperfectly worked; galena
+mineral oil and silicious sand are also found.</p>
+<div class="author">(C. W. W.; F. R. M.)</div>
+
+<hr class="foot" /> <div class="note">
+
+<p><a name="ft1b" id="ft1b" href="#fa1b"><span class="fn">1</span></a> From <i>Diar</i>, land, and Bekr (<i>i.e.</i> Abu Bekr, the caliph).</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIARRHOEA<a name="ar13" id="ar13"></a></span> (from Gr. <span class="grk" title="dia">&#948;&#953;&#940;</span>, through, <span class="grk" title="rheô">&#8165;&#941;&#969;</span>, flow), an excessive
+looseness of the bowels, a symptom of irritation which
+may be due to various causes, or may be associated with
+some specific disease. The treatment in such latter cases
+necessarily varies, since the symptom itself may be remedial,
+but in ordinary cases depends on the removal of the cause of
+irritation by the use of aperients, various sedatives being also
+prescribed. In chronic diarrhoea careful attention to the diet is
+necessary.</p>
+
+<p><span class="pagenum"><a name="page168" id="page168"></a>168</span></p>
+
+
+<hr class="art" />
+<p><span class="bold">DIARY,<a name="ar14" id="ar14"></a></span> the Lat. <i>diarium</i> (from <i>dies</i>, a day), the book in which
+are preserved the daily memoranda regarding events and actions
+which come under the writer&rsquo;s personal observation, or are
+related to him by others. The person who keeps this record is
+called a diarist. It is not necessary that the entries in a diary
+should be made each day, since every life, however full, must
+contain absolutely empty intervals. But it is essential that the
+entry should be made during the course of the day to which it
+refers. When this has evidently not been done, as in the case of
+Evelyn&rsquo;s diary, there is nevertheless an effort made to give the
+memoranda the effect of being so recorded, and in point of fact,
+even in a case like that of Evelyn, it is probable that what we
+now read is an enlargement of brief notes jotted down on the day
+cited. When this is not approximately the case, the diary is a
+fraud, for its whole value depends on its instantaneous transcript
+of impressions.</p>
+
+<p>In its primitive form, the diary must always have existed; as
+soon as writing was invented, men and women must have wished
+to note down, in some almanac or journal, memoranda respecting
+their business, their engagements or their adventures. But
+the literary value of these would be extremely insignificant until
+the spirit of individualism had crept in, and human beings began
+to be interesting to other human beings for their own sake. It
+is not, therefore, until the close of the Renaissance that we find
+diaries beginning to have literary value, although, as the study of
+sociology extends, every scrap of genuine and unaffected record
+of early history possesses an ethical interest. In the 17th century,
+diaries began to be <span class="correction" title="amended from largly">largely</span> written in England, although in most
+cases without any idea of even eventual publication. Sir William
+Dugdale (1605-1686) had certainly no expectation that his slight
+diary would ever see the light. There is no surviving record of
+a journal kept by Clarendon, Richard Baxter, Lucy Hutchinson
+and other autobiographical writers of the middle of the century,
+but we may take it for granted that they possessed some such
+record, kept from day to day. Bulstrode Whitelocke (1605-1675),
+whose <i>Memorials of the English Affairs</i> covers the ground
+from 1625 to 1660, was a genuine diarist. So was the elder George
+Fox (1624-1690), who kept not merely &ldquo;a great journal,&rdquo; but
+&ldquo;the little journal books,&rdquo; and whose work was published in
+1694. The famous diary of John Evelyn (1620-1706) professes
+to be the record of seventy years, and, although large tracts of it
+are covered in a very perfunctory manner, while in others many of
+the entries have the air of having been written in long after the
+event, this is a very interesting and amusing work; it was not
+published until 1818. In spite of all its imperfections there is a
+great charm about the diary of Evelyn, and it would hold a still
+higher position in the history of literature than it does if it were
+not overshadowed by what is unquestionably the most illustrious
+of the diaries of the world, that of Samuel Pepys (1633-1703).
+This was begun on the 1st of January 1660 and was carried on
+until the 29th of May 1669. The extraordinary value of Pepys&rsquo;
+diary consists in its fidelity to the portraiture of its author&rsquo;s
+character. He feigns nothing, conceals nothing, sets nothing
+down in malice or insincerity. He wrote in a form of shorthand
+intelligible to no one but himself, and not a phrase betrays the
+smallest expectation that any eye but his own would ever
+investigate the pages of his confession. The importance of this
+wonderful document, in fact, lay unsuspected until 1819, when
+the Rev. John Smith of Baldock began to decipher the MS. in
+Magdalene College, Cambridge. It was not until 1825 that Lord
+Braybrooke published part of what was only fully edited, under
+the care of Mr Wheatley, in 1893-1896. In the age which succeeded
+that of Pepys, a diary of extraordinary emotional interest
+was kept by Swift from 1710 to 1713, and was sent to Ireland in
+the form of a &ldquo;Journal to Stella&rdquo;; it is a surprising amalgam
+of ambition, affection, wit and freakishness. John Byrom
+(1692-1763), the Manchester poet, kept a journal, which was
+published in 1854. The diary of the celebrated dissenting divine,
+Philip Doddridge (1702-1751), was printed in 1829. Of far
+greater interest are the admirably composed and vigorously
+written journals of John Wesley (1703-1791). But the most
+celebrated work of this kind produced in the latter half of the 18th
+century was the diary of Fanny Burney (Madame D&rsquo;Arblay),
+published in 1842-1846. It will be perceived that, without
+exception, these works were posthumously published, and the
+whole conception of the diary has been that it should be written
+for the writer alone, or, if for the public, for the public when all
+prejudice shall have passed away and all passion cooled down.
+Thus, and thus only, can the diary be written so as to impress
+upon its eventual readers a sense of its author&rsquo;s perfect sincerity
+and courage.</p>
+
+<p>Many of the diaries described above were first published in the
+opening years of the 19th century, and it is unquestionable that
+the interest which they awakened in the public led to their
+imitation. Diaries ceased to be rare, but as a rule the specimens
+which have hitherto appeared have not presented much literary
+interest. Exception must be made in favour of the journals of
+two minor politicians, Charles Greville (1794-1865) and Thomas
+Creevey (1768-1838), whose indiscretions have added much to the
+gaiety of nations; the papers of the former appeared in 1874-1887,
+those of the latter in 1903. The diary of Henry Crabb
+Robinson (1775-1867), printed in 1869, contains excellent
+biographical material. Tom Moore&rsquo;s journal, published in 1856
+by Lord John Russell, disappointed its readers. But it is
+probable, if we reason by the analogy of the past, that the most
+curious and original diaries of the 19th century are still unknown
+to us, and lie jealously guarded under lock and key by the
+descendants of those who compiled them.</p>
+
+<p>It was natural that the form of the diary should appeal to a
+people so sensitive to social peculiarities and so keen in the
+observation of them as the French. A medieval document of
+immense value is the diary kept by an anonymous <i>curé</i> during
+the reigns of Charles VI. and Charles VII. This <i>Journal d&rsquo;un
+bourgeois de Paris</i> was kept from 1409 to 1431, and was continued
+by another hand down to 1449. The marquis de Dangeau
+(1638-1720) kept a diary from 1684 till the year of his death;
+this although dull, and as Saint-Simon said &ldquo;of an insipidity to
+make you sick,&rdquo; is an inexhaustible storehouse of facts about
+the reign of Louis XIV. Saint-Simon&rsquo;s own brilliant memoirs,
+written from 1691 to 1723, may be considered as a sort of diary.
+The lawyer, Edmond Barbier (1689-1771), wrote a journal of the
+anecdotes and little facts which came to his knowledge from
+1718 to 1762. The studious care which he took to be correct, and
+his manifest candour, give a singular value to Barbier&rsquo;s record;
+his diary was not printed at all until 1847, nor, in its entirety,
+until 1857. The song-writer, Charles Collé (1709-1783), kept a
+<i>journal historique</i> from 1758 to 1782; it is full of vivacity, but very
+scandalous and spiteful. It saw the light in 1805, and surprised
+those to whom Collé, in his lifetime, had seemed the most placid
+and good-natured of men. Petit de Bachaumont (1690-1770)
+had access to remarkable sources of information, and his
+<i>Mémoires secrets</i> (a diary the publication of which began in
+1762 and was continued after Bachaumont&rsquo;s death, until 1787,
+by other persons) contains a valuable mass of documents. The
+marquis d&rsquo;Argenson (1694-1757) kept a diary, of which a comparatively
+full text was first published in 1859. In recent times the
+posthumous publication of the diaries of the Russian artist, Marie
+Bashkirtseff (1860-1884), produced a great sensation in 1887, and
+revealed a most remarkable temperament. The brothers Jules
+and Edmond de Goncourt kept a very minute diary of all that
+occurred around them in artistic and literary Paris; after
+the death of Jules, in 1870, this was continued by Edmond, who
+published the three first volumes in 1888. The publication of this
+work was continued, and it produced no little scandal. It is
+excessively ill-natured in parts, but of its vivid picturesqueness,
+and of its general accuracy as a transcript of conversation, there
+can be no two opinions.</p>
+<div class="author">(E. G.)</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIASPORE,<a name="ar15" id="ar15"></a></span> a native aluminium hydroxide, AlO(OH), crystallizing
+in the orthorhombic system and isomorphous with göthite
+and manganite. It occurs sometimes as flattened crystals, but
+usually as lamellar or scaly masses, the flattened surface being a
+direction of perfect cleavage on which the lustre is markedly
+pearly in character. It is colourless or greyish-white, yellowish,
+sometimes violet in colour, and varies from translucent to
+<span class="pagenum"><a name="page169" id="page169"></a>169</span>
+transparent. It may be readily distinguished from other colourless
+transparent minerals, with a perfect cleavage and pearly
+lustre&mdash;mica, talc, brucite, gypsum&mdash;by its greater hardness
+of 6½-7. The specific gravity is 3.4. When heated before the
+blowpipe it decrepitates violently, breaking up into white pearly
+scales; it was because of this property that the mineral was
+named diaspore by R. J. Hauy in 1801, from <span class="grk" title="diaspeirein">&#948;&#953;&#945;&#963;&#960;&#949;&#943;&#961;&#949;&#953;&#957;</span>, &ldquo;to
+scatter.&rdquo; The mineral occurs as an alteration product of
+corundum or emery, and is found in granular limestone and
+other crystalline rocks. Well-developed crystals are found in the
+emery deposits of the Urals and at Chester, Massachusetts, and
+in kaolin at Schemnitz in Hungary. If obtainable in large
+quantity it would be of economic importance as a source of
+alumina.</p>
+<div class="author">(L. J. S)</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIASTYLE<a name="ar16" id="ar16"></a></span> (from Gr. <span class="grk" title="dia">&#948;&#953;&#940;</span>, through, and <span class="grk" title="stylos">&#963;&#964;&#8166;&#955;&#959;&#962;</span>, column), in
+architecture, a term used to designate an intercolumniation of
+three or four diameters.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIATOMACEAE.<a name="ar17" id="ar17"></a></span> For the knowledge we possess of these
+beautiful plants, so minute as to be undiscernible by our unaided
+vision, we are indebted to the assistance of the microscope. It
+was not till towards the close of the 18th century that the first
+known forms of this group were discovered by O. F. Muller. And
+so slow was the process of discovery in this field of scientific research
+that in the course of half a century, when Agardh published
+his <i>Systema algarum</i> in 1824, only forty-nine species included
+under eight genera had been described. Since that time, however,
+with modern microscopes and microscopic methods, eminent
+botanists in all parts of the civilized world have studied these
+minute plants, with the result that the number of known genera
+and species has been greatly increased. Over 10,000 species of
+diatoms have been described, and about 1200 species and
+numerous varieties occur in the fresh waters and on the coasts
+of Great Britain and Ireland. Rabenhorst, in the index to his
+<i>Flora Europaea algarum</i> (1864) enumerated about 4000 forms
+which had up to that time been discovered throughout the
+continent of Europe.</p>
+
+<table class="nobctr" summary="Illustration">
+<tr><td class="figcenter" colspan="2"><img style="width:378px; height:328px" src="images/img169a.jpg" alt="" /></td></tr>
+<tr><td class="caption sc" colspan="2">Fig. 1.</td></tr>
+<tr><td class="tcl f90">A and B, <i>Melosira arenaria.</i></td> <td class="tcl f90">C-E, <i>Melosira varians.</i></td></tr>
+<tr><td class="tcl f90">E, showing formation of auxospore.</td> <td class="tcl">&nbsp;</td></tr></table>
+
+<table class="nobctr" style="float: left; width: 330px;" summary="Illustration">
+<tr><td class="figleft1"><img style="width:282px; height:29px" src="images/img169b.jpg" alt="" /></td></tr>
+<tr><td class="caption sc">Fig. 2.</td></tr></table>
+
+<p>The diatoms are more commonly known among systematic
+botanists as the Bacillarieae, particularly on the continent of
+Europe, and although such an immense number of very diverse
+forms are included in it, the group as a whole exhibits a remarkable
+uniformity of structure. The Bacillarieae is one of the
+large groups of Algae, placed by some in close proximity to the
+Conjugatae and by others as an order of the Brown Algae (or
+Phaeophyceae), but their characters are so distinctive and their
+structure is so uniform as to warrant the separation of the diatoms
+as a distinct class. The affinities of the group are doubtful.</p>
+
+<table class="nobctr" style="float: right; width: 410px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:85px; height:186px" src="images/img169c.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 3.</span>&mdash;<i>Podosphenia Lyngbyii.</i></td></tr>
+<tr><td class="figright1"><img style="width:363px; height:93px" src="images/img169d.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 4.</span>&mdash;<i>Pleurosigma balticum.</i></td></tr></table>
+
+<p>The diatoms exhibit great
+variety of form. While some
+species are circular and more
+or less disk-shaped, others are oval in outline. Some are
+linear, as <i>Synedra Ulna</i> (fig. 2), others more or less crescentic;
+others again are cuneate, as <i>Podosphenia Lyngbyii</i>
+(fig. 3); some few have a sigmoid outline, as <i>Pleurosigma
+balticum</i> (fig. 4); but the prevailing
+forms are naviculoid, as in the large family
+Naviculaceae, of which the genus <i>Navicula</i>
+embraces upwards of 1000 species. They vary
+also in their modes of growth,&mdash;some being
+free-floating, others attached to foreign bodies
+by simple or branched gelatinous stalks, which
+in some species are short and thick, while in
+others they are long and slender. In some
+genera the forms are simple, while in others the
+frustules are connected together in ribbon-like
+filaments, or form, as in other cases, zigzag
+chains. In some genera the individuals are
+naked, while in many others they are enclosed in a more or less
+definite gelatinous investment. The conditions necessary to
+their growth are
+moisture and
+light. Wherever
+these circumstances
+coexist,
+diatomaceous
+forms will almost invariably be found. They occur mixed
+with other organisms on the surface of moist rocks; in
+streamlets and pools, they form a brownish stratum on
+the surface of the mud, or cover the stems and leaves of
+water plants or floating twigs with a furry investment.
+Marine forms are usually attached to various sea-weeds, and
+many are found in the stomachs of molluscs, holothurians,
+ascidians and other denizens of the ocean. The fresh-water
+forms are specifically distinct from those incidental to salt or
+brackish water,&mdash;fresh-water species, however, are sometimes
+carried some distance into the sea by the force of the current, and
+in tidal rivers marine forms are carried up by the force of the tide.
+Some notion may be formed of the extreme minuteness of these
+forms from the fact that one the length of which is <span class="spp">1</span>&frasl;<span class="suu">200</span>th of an
+inch may be considered as beyond the medium size. Some few,
+indeed, are much larger, but by far the greater proportion are of
+very much smaller dimensions.</p>
+
+<table class="nobctr" summary="Illustration">
+<tr><td class="figcenter" colspan="2"><img style="width:411px; height:408px" src="images/img169e.jpg" alt="" /></td></tr>
+<tr><td class="caption sc" colspan="2">Fig. 5.</td></tr>
+<tr><td class="tcl f90">A-C, <i>Tetracyclus lacustris.</i></td> <td class="tcl f90">D and E, <i>Tabellaria fenestrata.</i></td></tr>
+<tr><td class="tcl f90">F and G, <i>Tabellaria flocculosa.</i></td> <td class="tcl">&nbsp;</td></tr></table>
+
+<p>Diatoms are unicellular plants distinguished from kindred
+forms by the fact of having their soft vegetative part covered by
+a siliceous case. Each individual is known as a frustule, and the
+cell-wall consists of two similar valves nearly parallel to each
+other, each valve being furnished with a rim (or connecting-band)
+projecting from it at a right angle.</p>
+
+<p>One of these valves with its rim is slightly smaller than the
+<span class="pagenum"><a name="page170" id="page170"></a>170</span>
+other, the smaller fitting into the larger pretty much as a pill-box
+fits into its cover. This peculiarity of structure affords ample
+scope for the growth of the protoplasmic cell-contents, for as the
+latter increase in volume the siliceous valves are pushed out, and
+their corresponding siliceous rims become broader. The connecting-bands
+although closely fitting their respective valves are
+distinct from them, and together the two bands form the girdle.</p>
+
+<p>An individual diatom is usually described from two aspects,
+one in which the surface of the valve is exposed to view&mdash;the
+valve view, and one in which the girdle side is exposed&mdash;the
+girdle view. The valves are thin and transparent, convex on the
+outside, and generally ornamented with a variety of sculptured
+markings. These sculptures often present the aspect of striae
+across the face of the valve, and the best lenses have shown them
+to consist of a series of small cavities within the siliceous wall of
+the cell. The valves of some of the marine genera exhibit a
+beautiful areolated structure due to the presence of larger
+chambers within the siliceous cell-wall. Many diatoms possess
+thickenings of the cell-wall, visible in the valve view, in the
+centre of the valve and at each extremity. These thickenings
+are known as the nodules, and they are generally connected by a
+long median line, the raphe, which is a cleft in the siliceous valve,
+extending at least some part of its length.</p>
+
+<p>The protoplasmic contents of this siliceous box-like unicell are
+very similar to the contents of many other algal cells. There is a
+living protoplasmic layer or primordial utricle, connected either
+by two broad bands or by a number of anastomosing threads with
+a central mass of protoplasm in which the nucleus is embedded.
+The greater part of the cavity of the cell is occupied by one
+or several fluid vacuoles. The characteristic brown colour of
+diatoms is due to the presence of chromatophores embedded in
+the lining layer of protoplasm. In number and form these
+chromatophores are variable. They contain chlorophyll, but the
+green colour is masked by the presence of diatomin, a brown
+pigment which resembles that which occurs in the Brown Algae
+or Phaeophyceae. The chromatophores contain a variable
+number of pyrenoids, colourless proteid bodies of a crystalloidal
+character.</p>
+
+<p>One of the first phenomena which comes under the notice of
+the observer is the extraordinary power of motion with which
+the frustules are endowed. Some species move slowly backwards
+and forwards in pretty much the same line, but in the case of
+<i>Bacillaria paradoxa</i> the motion is very rapid, the frustules darting
+through the water in a zigzag course. To account for this motion
+various theories have been suggested, none of which appear to be
+altogether satisfactory. There is little doubt that the movements
+are connected with the raphe, and in some diatoms there is much
+evidence to prove that they are due to an exudation of mucilage.</p>
+
+<p><i>Classification.</i>&mdash;The most natural system of classification of the
+Bacillarieae is the one put forward by Schütt (1896), and since
+generally followed by systematists. He separates them into two
+primary divisions, the &lsquo;Centricae&rsquo; and the &lsquo;Pennatae.&rsquo; The
+former includes all those diatoms which in the valve view possess
+a radial symmetry around a central point, and which are destitute
+of a raphe (or a pseudoraphe). The latter includes those which
+are zygomorphic or otherwise irregular, and in which the valve
+view is generally boat-shaped or needle-shaped, with the markings
+arranged in a sagittal manner on each side of a raphe or
+pseudoraphe.</p>
+
+<p><i>Reproduction.</i>&mdash;In the Diatomaceae, as well as in the Desmidieae,
+the ordinary mode of increase is by simple cell-division. The
+cell-contents within the enclosure of the siliceous case separate
+into two distinct masses. As these two daughter-masses become
+more and more developed, the valves of the mother-cell are pushed
+more and more widely apart. A new siliceous valve is secreted by
+each of the two masses on the side opposite to the original valve,
+the new valves being situated within the girdle of the original
+frustule. When this process has been completed the girdle of
+the mother frustule gives way, and two distinct frustules are
+formed, the siliceous valves in each of these new frustules being
+one of the valves of the mother-cell, and a newly formed valve
+similar and more or less parallel to it.</p>
+
+<p>During the life of the plant this process of self-division is
+continued with an almost incredible rapidity. On this subject
+the observation of Professor William Smith, writing in 1853, is
+worthy of special notice:&mdash;&ldquo;I have been unable to ascertain the
+time occupied in a single act of self-division, but supposing it to be
+completed in twenty-four hours we should have, as the progeny of
+a single frustule, the amazing number of 1,000,000,000 in a single
+month, a circumstance which will in some degree explain the
+sudden, or at least rapid, appearance of these organisms in
+localities where they were a
+short time previously either
+unrecognized or sparingly diffused&rdquo;
+(<i>British Diatomaceae</i>,
+vol. i. p. 25).</p>
+
+<table class="nobctr" style="float: right; width: 310px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:264px; height:445px" src="images/img170.jpg" alt="" /></td></tr>
+<tr><td class="caption1"><span class="sc">Fig. 6.</span>&mdash;Formation of Auxospores.<br />
+A. <i>Navicula limosa.</i><br />
+B. <i>Achnanthes flexella.</i><br />
+C. <i>Navicula Amphisbaena.</i><br />
+D. <i>Navicula viridis.</i><br /></td></tr></table>
+
+<p>Individual diatoms when
+once produced by cell-division
+are incapable of any increase
+in size owing to the rigidity of
+their siliceous cell-walls, and
+since the new valves are always
+formed <i>within</i> the girdle of the
+old ones, it would follow that
+every succeeding generation is
+reduced in size by the thickness
+of the girdle. In some diatoms,
+however, this is not strictly
+true as daughter-cells are sometimes
+produced of larger size
+than the parent-cells. Thus,
+the reduction in size of the
+individuals is not always
+proportionate to the number
+of cell-divisions.</p>
+
+<p>On the diminution in size
+having reached a limit in any
+species, the maximum size is
+regained by the formation of
+an auxospore. There are five
+known methods of reproduction by auxospores, but it is unnecessary
+here to enter into details of these methods. Suffice it to
+say that a normal auxospore is produced by the conjugation
+of two parent-cells, its distinguishing feature being a rejuvenescence
+accompanied by a marked increase in size. These
+auxospores formed without conjugation are parthenogenetic.</p>
+
+<p><i>Mode of Preparation.</i>&mdash;The Diatomaceae are usually gathered
+in small bottles, and special care should be taken to collect them
+as free as possible from extraneous matter. A small portion having
+been examined under the microscope, should the gathering be
+thought worthy of preservation, some of the material is boiled in
+acid for the purpose of cleaning it. The acids usually employed
+are hydrochloric, nitric or sulphuric, according as circumstances
+require. When the operator considers that by this process all
+foreign matter has been eliminated, the residuum is put into a
+precipitating jar of a conical shape, broader at the bottom than
+at the top, and covered to the brim with filtered or distilled water.
+When the diatoms have settled in the bottom of the jar, the
+supernatant fluid is carefully removed by a syringe or some
+similar instrument, so that the sediment be not disturbed. The
+jar is again filled with water, and the process repeated till the acid
+has been completely removed. It is desirable afterwards to boil
+the sediment for a short time with supercarbonate of soda, the
+alkali being removed in the same manner as the acid. A small
+portion may then be placed with a pipette upon a slip of glass,
+and, when the moisture has been thoroughly evaporated, the film
+that remains should be covered with dilute Canada balsam, and,
+a thin glass cover having been gently laid over the balsam, the
+preparation should be laid aside for a short time to harden, and
+then is ready for observation.</p>
+
+<p><i>General Remarks.</i>&mdash;Diatoms are most abundant in cold
+latitudes, having a general preference for cold water. In the
+pelagic waters of lakes and of the oceans they are often very
+abundant, and in the cold waters of the Arctic and Antarctic
+<span class="pagenum"><a name="page171" id="page171"></a>171</span>
+Oceans they exist in prodigious numbers. They thus form a large
+proportion of both the marine and the fresh-water plankton.</p>
+
+<p>Large numbers of fossil diatoms are known. Not only are
+these minute plants assisting at the present time in the accumulation
+of oceanic and lake deposits, but in former ages they have
+been sufficiently active to give rise to considerable deposits of
+diatomaceous earths. When the plant has fulfilled its natural
+course the siliceous covering sinks to the bottom of the water in
+which it had lived, and there forms part of the sediment. When
+in the process of ages, as it has often happened, the accumulated
+sediment has been hardened into solid rock, the siliceous frustules
+of the diatoms remain unaltered, and, if the rock be disintegrated
+by natural or artificial means, may be removed from the
+enveloping matrix and subjected to examination under the
+microscope. The forms found may from their character help in
+some degree to illustrate the conditions under which the stratum
+of rock had been originally deposited. These earths are generally
+of a white or grey colour. Some of them are hard, but most
+are soft and friable. Many of them are of economic importance,
+being used as polishing powders (&ldquo;Tripoli&rdquo;), as absorbents for
+nitroglycerin in the manufacture of dynamite (&ldquo;Kieselguhr&rdquo;),
+as a dentifrice, and more recently they have been used to a large
+extent in the manufacture of non-conducting and sound-proof
+materials. Most of these diatomaceous earths are associated
+with rocks of Tertiary formations, although it is generally
+regarded that the earliest appearance of diatoms is in the Upper
+Cretaceous (chalk).</p>
+
+<p>Vast deposits of Diatomaceous earths have been discovered in
+various parts of the world,&mdash;some the deposit of fresh, others of
+salt water. Of these deposits the most remarkable for extent,
+as well as for the number and beauty of the species contained in it,
+is that of Richmond, in Virginia, one of the United States of
+America. It extends for many miles, and is in some places at
+least 40 ft. deep. It is a remarkable fact that though the generations
+of a diatom in the space of a few months far exceed in
+number the generation of man during the period usually assigned
+to the existence of the race, the fossil genera and species are
+in most respects to the most minute details identical with the
+numerous living representatives of their class.</p>
+<div class="author">(E. O&rsquo;M.; G. S. W.*)</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIAULOS<a name="ar18" id="ar18"></a></span> (from Gr. <span class="grk" title="di-">&#948;&#953;-</span>, double, and <span class="grk" title="aulos">&#945;&#8016;&#955;&#972;&#962;</span>, pipe), in architecture,
+the peristyle round the great court of the palaestra,
+described by Vitruvius (v. II), which measured two stadia
+(1200 ft.) in length; on the south side this peristyle had two
+rows of columns, so that in stormy weather the rain might not
+be driven into the inner part. The word was also used in ancient
+Greece for a foot-race of twice the usual length.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIAVOLO, FRA<a name="ar19" id="ar19"></a></span> (1771-1806), the popular name given to a
+famous Italian brigand associated with the political revolutions
+of southern Italy at the time of the French invasion. His real
+name was Michele Pezza, and he was born of low parentage
+at Itri; he had committed many murders and robberies in the
+Terra di Lavoro, but by good luck combined with audacity he
+always escaped capture, whence his name of Fra Diavolo, popular
+superstition having invested him with the characters of a monk
+and a demon, and it seems that at one time he actually was a
+monk. When the kingdom of Naples was overrun by the French
+and the Parthenopaean Republic established (1799), Cardinal
+Ruffo, acting on behalf of the Bourbon king Ferdinand IV., who
+had fled to Sicily, undertook the reconquest of the country, and
+for this purpose he raised bands of peasants, gaol-birds, brigands,
+&amp;c., under the name of Sanfedisti or <i>bande della Santa Fede</i>
+(&ldquo;bands of the Holy Faith&rdquo;). Fra Diavolo was made leader
+of one of them, and waged untiring war against the French troops,
+cutting off isolated detachments and murdering stragglers and
+couriers. Owing to his unrivalled knowledge of the country, he
+succeeded in interrupting the enemy&rsquo;s communications between
+Rome and Naples. But although, like his fellow-brigands under
+Ruffo, he styled himself &ldquo;the faithful servant and subject of His
+Sicilian Majesty,&rdquo; wore a military uniform and held military rank,
+and was even created duke of Cassano, his atrocities were worthy
+of a bandit chief. On one occasion he threw some of his prisoners,
+men, women and children, over a precipice, and on another he
+had a party of seventy shot. His excesses while at Albano were
+such that the Neapolitan general Naselli had him arrested and
+imprisoned in the castle of St Angelo, but he was liberated soon
+after. When Joseph Bonaparte was made king of Naples, extraordinary
+tribunals were established to suppress brigandage, and
+a price was put on Fra Diavolo&rsquo;s head. After spreading terror
+through Calabria, he crossed over to Sicily, where he concerted
+further attacks on the French. He returned to the mainland at
+the head of 200 convicts, and committed further excesses in
+the Terra di Lavoro; but the French troops were everywhere
+on the alert to capture him and he had to take refuge in the woods
+of Lenola. For two months he evaded his pursuers, but at
+length, hungry and ill, he went in disguise to the village of
+Baronissi, where he was recognized and arrested, tried by an
+extraordinary tribunal, condemned to death and shot. In his
+last moments he cursed both the Bourbons and Admiral Sir
+Sidney Smith for having induced him to engage in this reckless
+adventure (1806). Although his cruelty was abominable, he
+was not altogether without generosity, and by his courage and
+audacity he acquired a certain romantic popularity. His name
+has gained a world-wide celebrity as the title of a famous opera
+by Auber.</p>
+
+<div class="condensed">
+<p>The best known account of Fra Diavolo is in Pietro Colletta&rsquo;s
+<i>Storia del reame di Napoli</i> (2nd ed., Florence, 1848); B. Amante&rsquo;s
+<i>Fra Diavolo e il suo tempo</i> (Florence, 1904) is an attempted rehabilitation;
+but A. Luzio, whose account in <i>Profili e bozzetti storici</i>
+(Milan, 1906) gives the latest information on the subject, has demolished
+Amante&rsquo;s arguments.</p>
+</div>
+<div class="author">(L. V.*)</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIAZ, NARCISSE VIRGILIO<a name="ar20" id="ar20"></a></span> (1808-1876), French painter, was
+born in Bordeaux of Spanish parents, on the 25th of August 1808.
+At first a figure-painter who indulged in strong colour, in his later
+life Diaz became a painter of the forest and a &ldquo;tone artist&rdquo; of
+the first order. He spent much time at Barbizon; and although
+he is the least exalted of the half-dozen great artists who are
+usually grouped round that name, he sometimes produced works
+of the highest quality. At the age of ten Diaz became an orphan,
+and misfortune dogged his earlier years. His foot was bitten by a
+reptile in Meudon wood, near Sèvres, where he had been taken to
+live with some friends of his mother. The bite was badly dressed,
+and ultimately it cost him his leg. Afterwards his wooden stump
+became famous. At fifteen he entered the studios at Sèvres,
+where the decoration of porcelain occupied him; but tiring of the
+restraint of fixed hours, he took to painting Eastern figures
+dressed in richly coloured garments. Turks and Oriental scenes
+attracted him, and many brilliant gems remain of this period.
+About 1831 Diaz encountered Théodore Rousseau, for whom he
+entertained a great veneration, although Rousseau was four years
+his junior; but it was not until ten years later that the remarkable
+incident took place of Rousseau teaching Diaz to paint trees.
+At Fontainebleau Diaz found Rousseau painting his wonderful
+forest pictures, and determined to paint in the same way if
+possible. Rousseau, then in poor health, worried at home, and
+embittered against the world, was difficult to approach. Diaz
+followed him surreptitiously to the forest,&mdash;wooden leg not
+hindering,&mdash;and he dodged round after the painter, trying to
+observe his method of work. After a time Diaz found a way
+to become friendly with Rousseau, and revealed his anxiety
+to understand his painting. Rousseau was touched with the
+passionate words of admiration, and finally taught Diaz all he
+knew. Diaz exhibited many pictures at the Paris Salon, and was
+decorated in 1851. During the Franco-German War he went to
+Brussels. After 1871 he became fashionable, his works gradually
+rose in the estimation of collectors, and he worked constantly and
+successfully. In 1876 he caught cold at his son&rsquo;s grave, and on
+the 18th of November of that year he died at Mentone, whither
+he had gone to recruit his health. Diaz&rsquo;s finest pictures are his
+forest scenes and storms, and it is on these, and not on his pretty
+figures, that his fame is likely to rest. There are several fairly
+good examples of the master in the Louvre, and three small figure
+pictures in the Wallace collection, Hertford House. Perhaps the
+most notable of Diaz&rsquo;s works are &ldquo;La Fée aux Perles&rdquo; (1857),
+in the Louvre; &ldquo;Sunset in the Forest&rdquo; (1868); &ldquo;The Storm,&rdquo;
+<span class="pagenum"><a name="page172" id="page172"></a>172</span>
+and &ldquo;The Forest of Fontainebleau&rdquo; (1870) at Leeds. Diaz
+had no well-known pupils, but Léon Richet followed markedly
+his methods of tree-painting, and J. F. Millet at one period
+painted small figures in avowed imitation of Diaz&rsquo;s then
+popular subjects.</p>
+
+<div class="condensed">
+<p>See A. Hustin, <i>Les Artistes célèbres: Diaz</i> (Paris); D. Croal
+Thomson, <i>The Barbizon School of Painters</i> (London, 1890);
+J. W. Mollett, <i>Diaz</i> (London, 1890); J. Claretie, <i>Peintres et sculpteurs
+contemporains: Diaz</i> (Paris, 1882); Albert Wolff, <i>La Capitale de
+l&rsquo;art: Narcisse Diaz</i> (Paris, 1886); Ph. Burty, <i>Maîtres et petit-maîtres:
+N. Diaz</i> (Paris, 1877).</p>
+</div>
+<div class="author">(D. C. T.)</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIAZ, PORFIRIO<a name="ar21" id="ar21"></a></span> (1830-&emsp;&emsp;), president of the republic of
+Mexico (<i>q.v.</i>), was born in the southern state of Oaxaca, on the
+15th of September 1830. His father was an innkeeper in the little
+capital of that province, and died three years after the birth of
+Porfirio, leaving a family of seven children. The boy, who had
+Indian blood in his veins, was educated for the Catholic Church,
+a body having immense influence in the country at that time and
+ordering and controlling revolutions by the strength of their filled
+coffers. Arrived at the age of sixteen Porfirio Diaz threw off the
+authority of the priests. Fired with enthusiasm by stories told by
+the revolutionary soldiers continually passing through Oaxaca,
+and hearing about the war with the United States, a year later
+he determined to set out for Mexico city and join the National
+Guard. There being no trains, and he being too poor to ride, he
+walked the greater part of the 250 m., but arrived there too late,
+as the treaty of Guadalupe-Hidalgo (1848) had been already
+signed, and Texas finally ceded to the United States. Thus
+his entering the army was for the time defeated. Thereupon he
+returned to his native town and began studying law. He took
+pupils in order to pay his own fees at the Law Institute, and help
+his mother. At this time he came under the notice and influence
+of Don Marcos Pérez and Benito Juárez, the first a judge, the
+second a governor of the state of Oaxaca, and soon to become
+famous as the deliverer of Mexico from the priesthood (War of
+Reform). Diaz continued in his native town until 1854, when,
+refusing to vote for the dictator, Santa Anna, he was stung by a
+taunt of cowardice, and hastily pushing his way to the voting
+place, he recorded his vote in favour of Alvarez and the revolutionists.
+Orders were given for his arrest, but seizing a rifle and
+mounting a horse he placed himself at the head of a few revolting
+peasants, and from that moment became one of the leading
+spirits in that long struggle for reform, known as the War of
+Reform, which, under the leadership of Juárez, followed the overthrow
+of Santa Anna. Promotion succeeded promotion, as Diaz
+led his troops from victory to victory, amid great privations and
+difficulties. He was made captain (1856), lieutenant-colonel and
+colonel (1859), brigadier-general (1861), and general of division
+for the army (1863). Closely following on civil war, political strife,
+open rebellion and the great War of Reform, came the French
+invasion of 1862, and the landing of the emperor Maximilian in
+1864. From the moment the French disclosed their intentions of
+settling in Mexico in 1862, Diaz took a prominent part against the
+foreign invasion. He was twice seriously wounded, imprisoned on
+three different occasions, had two hairbreadth escapes, and took
+part in many daring engagements. So important a personage did
+he become that both Marshal Bazaine and the emperor Maximilian
+made overtures to him. At the time of Maximilian&rsquo;s death (with
+which Diaz personally had nothing to do) he was carrying on the
+siege of Mexico city, which ended in the surrender of the town
+two days after the emperor was shot at Quérétaro between his
+two leading generals. Diaz at once set to work to pay up arrears
+due to his soldiers, proclaimed death as the penalty of plunder
+and theft, and in the few weeks that followed showed his great
+administrative powers, the officers as well as the rank and file
+receiving arrears of pay. On the very day that he occupied
+Mexico city, the great commander of the army of the east, to
+everyone&rsquo;s surprise, sent in his resignation. He was, indeed,
+appointed to the command of the second division of the army by
+President Juárez in his military reorganization, but Diaz, seeing
+men who had given great and loyal service to the state dismissed
+from their positions in the government, and disgusted at this
+course, retired to the little city of Oaxaca; there he lived, helping
+in the reorganization of the army but taking no active part in the
+government until 1871.</p>
+
+<p>On Juárez&rsquo; death Lerdo succeeded as president, in 1872. His
+term of office again brought discord, and when it was known that
+he was attempting to be re-elected in 1876, the storm broke.
+Diaz came from retirement, took up the leadership against Lerdo,
+and after desperate struggles and a daring escape finally made a
+triumphal entry into Mexico city on the 24th of November 1876,
+as provisional president, quickly followed by the full presidentship.
+His term of office marks a prominent change in the history
+of Mexico; from that date he at once forged ahead with financial
+and political reform, the scrupulous settlement of all national
+debts, the welding together of the peoples and tribes (there are
+150 different Indian tribes) of his country, the establishment
+of railroads and telegraphs, and all this in a land which had
+been upheaved for a century with revolutions and bloodshed,
+and which had had fifty-two dictators, presidents and rulers
+in fifty-nine years. In 1880 Diaz was succeeded by Gonzalez,
+the former minister of war, for four years (owing to the limit
+of the presidential office), but in 1884 he was unanimously
+re-elected. The government having set aside the above-mentioned
+limitation, Diaz was continually re-elected to the
+presidency. He married twice and had a son and two daughters.
+His gifted second wife (Carmelita), very popular in Mexico, was
+many years younger than himself. King Edward VII. made him
+an honorary grand commander of the Bath in June 1906, in
+recognition of his wonderful administration as perpetual president
+for over a quarter of a century.</p>
+
+<div class="condensed">
+<p>See also Mrs Alec Tweedie, <i>Porfirio Diaz, Seven Times President of
+Mexico</i> (1906), and <i>Mexico as I saw it</i> (1901); Dr Noll, <i>From Empire
+to Republic</i> (1890); Lieut. Seaton Schroeder, <i>Fall of Maximilian&rsquo;s
+Empire</i> (New York, 1887); R. de Z. Enriquez, <i>P. Diaz</i> (1908);
+and an article by Percy Martin in <i>Quarterly Review</i> for October
+1909.</p>
+</div>
+<div class="author">(E. A. T.)</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIAZ DE NOVAES, BARTHOLOMEU<a name="ar22" id="ar22"></a></span> (fl. 1481-1500),
+Portuguese explorer, discoverer of the Cape of Good Hope, was
+probably a kinsman of João Diaz, one of the first Portuguese to
+round Cape Bojador (1434), and of Diniz Diaz, the discoverer
+of Cape Verde (1445). In 1478 a Bartholomeu Diaz, probably
+identical with the discoverer, was exempted from certain
+customary payments on ivory brought from the Guinea coast.
+In 1481 he commanded one of the vessels sent by King John II.
+under Diogo d&rsquo;Azambuja to the Gold Coast. In 1486 he seems to
+have been a cavalier of the king&rsquo;s household, and superintendent
+of the royal warehouses; on the 10th of October in this year he
+received an annuity of 6000 reis from King John for &ldquo;services
+to come&rdquo;; and some time after this (probably about July or
+August 1487, rather than July 1486, the traditional date) he left
+Lisbon with three ships to carry on the work of African exploration
+so greatly advanced by Diogo Cão (1482-1486). Passing
+Cão&rsquo;s farthest point near Cape Cross (in the modern German
+South-west Africa and) in 21° 50&prime; S., he erected a pillar on what is
+now known as Diaz Point, south of Angra Pequena or Lüderitz
+Bay, in 26° 38&prime; S.; of this fragments still exist. From this point
+(according to De Barros) Diaz ran thirteen days southwards
+before strong winds, which freshened to dangerous stormy
+weather, in a comparatively high southern latitude, considerably
+south of the Cape. When the storm subsided the Portuguese
+stood east; and failing, after several days&rsquo; search, to find land,
+turned north, and so struck the south coast of Cape Colony at
+Mossel Bay (Diaz&rsquo; Bahia dos Vaqueiros), half way between the
+Cape of Good Hope and Port Elizabeth (February 3, 1488). Thence
+they coasted eastward, passing Algoa Bay (Diaz&rsquo; Bahia da Roca),
+erecting pillars (or perhaps wooden crosses), it is said, on one of the
+islands in this bay and at or near Cape Padrone farther east; of
+these no traces remain. The officers and men now began to insist
+on return, and Diaz could only persuade them to go as far as the
+estuary of the Great Fish River (Diaz&rsquo; Rio do Iffante, so named
+from his colleague, Captain João Iffante). Here, however, half way
+between Port Elizabeth and East London (and indeed from
+Cape Padrone), the north-easterly trend of the coast became
+unmistakable; the way round Africa had been laid open. On
+his return Diaz perhaps named Cape Agulhas after St Brandan;
+<span class="pagenum"><a name="page173" id="page173"></a>173</span>
+while on the southernmost projection of the modern Cape
+peninsula, whose remarkable highlands (Table Mountain, &amp;c.)
+doubtless impressed him as the practical termination of the
+continent, he bestowed, says De Barros, the name of Cape of
+Storms (<i>Cabo Tormentoso</i>) in memory of the storms he had
+experienced in these far southern waters; this name (in the
+ordinary tradition) was changed by King John to that of Good
+Hope (<i>Cabo da Boa Esperança</i>). Some excellent authorities,
+however, make Diaz himself give the Cape its present name.
+Hard by this &ldquo;so many ages unknown promontory&rdquo; the explorer
+probably erected his last pillar. After touching at the
+Ilha do Principe (Prince&rsquo;s Island, south-west of the Cameroons)
+as well as at the Gold Coast, he appeared at Lisbon in December
+1488. He had discovered 1260 m. of hitherto unknown coast;
+and his voyage, taken with the letters soon afterwards received
+from Pero de Covilhão (who by way of Cairo and Aden had
+reached Malabar on one side and the &ldquo;Zanzibar coast&rdquo; on the
+other as far south as Sofala, in 1487-1488) was rightly considered
+to have solved the question of an ocean route round Africa to the
+Indies and other lands of South and East Asia.</p>
+
+<p>No record has yet been found of any adequate reward for Diaz:
+on the contrary, when the great Indian expedition was being
+prepared (for Vasco da Gama&rsquo;s future leadership) Bartolomeu
+only superintended the building and outfit of the ships; when
+the fleet sailed in 1497, he only accompanied da Gama to the Cape
+Verde Islands, and after this was ordered to El Mina on the Gold
+Coast. On Cabral&rsquo;s voyage of 1500 he was indeed permitted
+to take part in the discovery of Brazil (April 22), and thence
+should have helped to guide the fleet to India; but he perished
+in a great storm off his own Cabo Tormentoso. Like Moses, as
+Galvano says, he was allowed to see the Promised Land, but not
+to enter in.</p>
+
+<div class="condensed">
+<p>See João de Barros, <i>Asia</i>, Dec. I. bk. iii. ch. 4; Duarte Pacheco
+Pereira, <i>Esmeraldo de situ orbis</i>, esp. pp. 15, 90, 92, 94 and Raphael
+Bastos&rsquo;s introduction to the edition of 1892 (Pacheco met Diaz,
+returning from his great voyage, at the Ilha do Principe); a marginal
+note, probably by Christopher Columbus himself, on fol. 13 of a copy
+of Pierre d&rsquo;Ailly&rsquo;s <i>Imago mundi</i>, now in the Colombina at Seville
+(the writer of this note fixes Diaz&rsquo;s return to Lisbon, December 1488,
+and says he was present at Diaz&rsquo;s interview with the king of Portugal,
+when the explorer described his voyage and showed his route upon
+the chart he had kept); a similar but briefer note in a copy of Pope
+Pius II.&rsquo;s <i>Historia rerum ubique gestarum</i>, from the same hand; the
+<i>Roteiro</i> of Vasco da Gama&rsquo;s First Voyage (<i>Journal of the First Voyage
+of ... Da Gama</i>, Hakluyt Soc., ed. E. G. Ravenstein (1898), pp. 9,
+14); Ramusio, <i>Navigationi</i> (3rd ed.), vol. i. fol. 144; Castanheda,
+<i>Historia</i>, bk. i. ch. 1; Galvano, <i>Descobrimentos (Discoveries of the
+World)</i>, Hakluyt Soc. (1862), p. 77; E. G. Ravenstein, &ldquo;Voyages of ...
+Cão and ... Dias,&rdquo; in <i>Geog. Journ.</i> (London, December 1900), vol. xvi.
+pp. 638-655), an excellent critical summary in the light of the most
+recent investigations of all the material. The fragments of Diaz&rsquo;s
+only remaining pillar (from Diaz Point) are now partly at the Cape
+Museum, partly at Lisbon: the latter are photographed in Ravenstein&rsquo;s
+paper in <i>Geog. Journ.</i> (December 1900, p. 642).</p>
+</div>
+<div class="author">(C. R. B.)</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIAZO COMPOUNDS,<a name="ar23" id="ar23"></a></span> in organic chemistry, compounds of the
+type R·N·<span class="su">2</span>·X (where R = a hydrocarbon radical, and X = an
+acid radical or a hydroxyl group). These compounds may be
+divided into two classes, namely, the true diazo compounds,
+characterized by the grouping &minus;N = N&minus;, and the diazonium
+compounds, characterized by the grouping N &#8758; N &lt;.</p>
+
+<p>The diazonium compounds were first discovered by P. Griess
+(<i>Ann.</i>, 1858, 106, pp. 123 et seq.), and may be prepared by the
+action of nitrous fumes on a well-cooled solution of a salt of a
+primary amine,</p>
+
+<p class="center">C<span class="su">6</span>H<span class="su">5</span>NH<span class="su">2</span>·HNO<span class="su">3</span> + HNO<span class="su">2</span> = C<span class="su">6</span>H<span class="su">5</span>N<span class="su">2</span>·NO<span class="su">3</span> + 2H<span class="su">2</span>O,</p>
+
+<p class="noind">or, as is more usually the case (since the diazonium salts
+themselves are generally used only in aqueous solution) by the
+addition of a well-cooled solution of potassium or sodium nitrite
+to a well-cooled dilute acid solution of the primary amine. In
+order to isolate the anhydrous diazonium salts, the method of
+E. Knoevenagel (<i>Ber.</i>, 1890, 23, p. 2094) may be employed. In
+this process the amine salt is dissolved in absolute alcohol and
+diazotized by the addition of amyl nitrite; a crystalline precipitate
+of the diazonium salt is formed on standing, or on the
+addition of a small quantity of ether. The diazonium salts are
+also formed by the action of zinc-dust and acids on the nitrates
+of primary amines (R. Mohlau, <i>Ber.</i>, 1883, 16, p. 3080), and by the
+action of hydroxylamine on nitrosobenzenes. They are colourless
+crystalline solids which turn brown on exposure. They dissolve
+easily in water, but only to a slight extent in alcohol and ether.
+They are very unstable, exploding violently when heated or
+rubbed. <i>Benzene diazonium nitrate</i>, C<span class="su">6</span>H<span class="su">5</span>N(NO<span class="su">3</span>)&#8758;N, crystallizes
+in long silky needles. The sulphate and chloride are similar,
+but they are not quite so unstable as the nitrate. The bromide
+may be prepared by the addition of bromine to an ethereal
+solution of diazo-amino-benzene (tribromaniline remaining in
+solution). By the addition of potassium bromide and bromine
+water to diazonium salts they are converted into a <i>perbromide</i>,
+<i>e.g.</i> C<span class="su">6</span>H<span class="su">5</span>N<span class="su">2</span>Br<span class="su">3</span>, which crystallizes in yellow plates.</p>
+
+<div class="condensed">
+<p>The diazonium salts are characterized by their great reactivity and
+consequently are important reagents in synthetical processes, since by
+their agency the amino group in a primary amine may be exchanged
+for other elements or radicals. The chief reactions are as follows:&mdash;</p>
+
+<p>1. <i>Replacement of -NH<span class="su">2</span> by -OH</i>:&mdash;The amine is diazotized and
+the aqueous solution of the diazonium salt is heated, nitrogen being
+eliminated and a phenol formed.</p>
+
+<p>2. <i>Replacement of -NH<span class="su">2</span> by halogens and by the -CN and -CNO
+groups</i>:&mdash;The diazonium salt is warmed with an acid solution of the
+corresponding cuprous salt (T. Sandmeyer, <i>Ber.</i>, 1884, 17, p. 2650), or
+with copper powder (L. Gattermann, Ber., 1890, 23, p. 1218; 1892,
+25, p. 1074). In the case of iodine, the substitution is effected by
+adding a warm solution of potassium iodide to the diazonium
+solution, no copper or cuprous salt being necessary; whilst for
+the production of nitriles a solution of potassium cuprous cyanide is
+used. This reaction (the so-called &ldquo;Sandmeyer&rdquo; reaction) has been
+investigated by A. Hantzsch and J. W. Blagden (<i>Ber.</i>, 1900, 33, p. 2544),
+who consider that three simultaneous reactions occur, namely, the
+formation of labile double salts which decompose in such a fashion
+that the radical attached to the copper atom wanders to the aromatic
+nucleus; a catalytic action, in which nitrogen is eliminated and the
+acid radical attaches itself to the aromatic nucleus; and finally, the
+formation of azo compounds.</p>
+
+<p>3. <i>Replacement of -NH<span class="su">2</span> by -NO<span class="su">2</span></i>:&mdash;A well-cooled concentrated
+solution of potassium mercuric nitrate is added to a cooled
+solution of benzene diazonium nitrate, when the crystalline salt
+2C<span class="su">6</span>H<span class="su">5</span>N<span class="su">2</span>·NO<span class="su">3</span>, Hg(NO<span class="su">2</span>)<span class="su">2</span> is precipitated. On warming this with
+copper powder, it gives a quantitative yield of nitrobenzene (A.
+Hantzsch, <i>Ber.</i>, 1900, 33, p. 2551).</p>
+
+<p>4. <i>Replacement of -NH<span class="su">2</span> by hydrogen</i>:&mdash;This exchange is brought
+about, in some cases, by boiling the diazonium salt with alcohol;
+but I. Remsen and his pupils (<i>Amer. Chem. Journ.</i>, 1888, 9, pp. 389
+et seq.) have shown that the main product of this reaction is usually
+a phenolic ether. This reaction has also been investigated by
+A. Hantzsch and E. Jochem (<i>Ber.</i>, 1901, 34, p. 3337), who arrived at
+the conclusion that the normal decomposition of diazonium salts
+by alcohols results in the formation of phenolic ethers, but that an
+increase in the molecular weight of the alcohol, or the accumulation
+of negative groups in the aromatic nucleus, diminishes the yield of
+the ether and increases the amount of the hydrocarbon formed. The
+replacement is more readily brought about by the use of sodium
+stannite (P. Friedlander, <i>Ber.</i>, 1889, 22, p. 587), or by the use of a
+concentrated solution of hypophosphorous acid (J. Mai, <i>Ber.</i>, 1902, 35,
+p. 162). A. Hantzsch (<i>Ber.</i>, 1896, 29, p. 947; 1898, 31, p. 1253) has shown
+that the chlor- and brom- diazoniumthiocyanates, when dissolved in
+alcohol containing a trace of hydrochloric acid, become converted
+into the isomeric thiocyanbenzene diazonium chlorides and bromides.
+This change only occurs when the halogen atom is in the ortho- or
+para- position to the -N<span class="su">2</span>- group.</p>
+
+<p><i>Metallic Diazo Derivatives.</i>&mdash;Benzene diazonium chloride is decomposed
+by silver oxide in aqueous solution, with the formation of
+<i>benzene diazonium hydroxide</i>, C<span class="su">6</span>H<span class="su">5</span>·N(OH)&#8758;N. This hydroxide,
+although possessing powerful basic properties, is unstable in the
+presence of alkalis and neutralizes them, being converted first into
+the isomeric benzene-diazotic acid, the potassium salt of which is
+obtained when the diazonium chloride is added to an excess of cold
+concentrated potash (A. Hantzsch and W. B. Davidson, <i>Ber.</i>, 1898,
+31, p. 1612). <i>Potassium benzene diazotate</i>, C<span class="su">6</span>H<span class="su">5</span>N<span class="su">2</span>·OK, crystallizes in
+colourless silky needles. The free acid is not known; by the addition
+of the potassium salt to 50% acetic acid at -20° C., the acid
+anhydride, <i>benzene diazo oxide</i>, (C<span class="su">6</span>H<span class="su">5</span>N<span class="su">2</span>)<span class="su">2</span>O, is obtained as a very
+unstable, yellow, insoluble compound, exploding spontaneously at
+0° C. Strong acids convert it into a diazonium salt, and potash
+converts it into the diazotate. On the constitution, of these anhydrides
+see E. Bamberger, <i>Ber.</i>, 1896, 29, p. 446, and A. Hantzsch, <i>Ber.</i>,
+1896, 29, p. 1067; 1898, 31, p. 636. By the addition of the diazonium
+salts to a hot concentrated solution of a caustic alkali, C. Schraube
+and C. Schmidt (<i>Ber.</i>, 1894, 27, p. 520) obtained an isomer of potassium
+benzene diazotate. These <i>iso-</i>diazotates are formed much more
+readily when the aromatic nucleus in the diazonium salt contains
+negative radicals. <i>Potassium benzene iso-diazotate</i> resembles the
+normal salt, but is more stable, and is more highly ionized. Carbon
+dioxide converts it into <i>phenyl nitrosamine</i>, C<span class="su">6</span>H<span class="su">5</span>NH·NO
+<span class="pagenum"><a name="page174" id="page174"></a>174</span>
+(A. Hantzsch). The potassium salt of the iso-diazo hydroxide yields
+on methylation a nitrogen ether, R·N(CH<span class="su">3</span>)·NO, whilst the silver salt
+yields an oxygen ether, R·N:N·OCH<span class="su">3</span>. These results point to the
+conclusion that the iso-diazo hydroxide is a tautomeric substance.
+The same oxygen ether is formed by the methylation of the silver salt
+of the normal diazo hydroxide; this points to the conclusion that the
+isomeric hydroxides, corresponding with the silver derivatives, have
+the same structural formulae, namely, R·N:N·OH. These oxygen
+ethers contain the grouping -N:N-, since they couple very readily
+with the phenols in alkaline solution to form azo compounds (<i>q.v.</i>)
+(E. Bamberger, <i>Ber.</i>, 1895, 28, p. 225); they are also explosive.</p>
+
+<p>By oxidizing potassium benzene iso-diazotate with alkaline
+potassium ferricyanide, E. Bamberger (<i>Ber.</i>, 1894, 27, p. 914) obtained
+the <i>diazoic acids</i>, R·NH·NO<span class="su">2</span>, substances which he had previously
+prepared by similarly oxidizing the diazonium salts, by dehydrating
+the nitrates of primary amines with acetic anhydride, and by the
+action of nitric anhydride on the primary amines. Concentrated
+acids convert them into the isomeric nitro-amines, the -NO<span class="su">2</span> group
+going into the nucleus in the ortho- or para- position to the amine
+nitrogen; this appears to indicate that the compounds are nitramines.
+They behave, however, as tautomeric substances, since
+their alkali salts on methylation give nitrogen ethers, whilst their
+silver salts yield oxygen ethers:</p>
+
+<div class="center"><img style="width:491px; height:52px" src="images/img174a.jpg" alt="" /></div>
+
+<p><i>Phenyl nitramine</i>, C<span class="su">6</span>H<span class="su">5</span>NH·NO<span class="su">2</span>, is a colourless crystalline solid,
+which melts at 46° C. Sodium amalgam in alkaline solution reduces
+it to phenylhydrazine.</p>
+
+<p><i>Constitution of the Diazo Compounds.</i>&mdash;P. Griess (<i>Ann.</i>, 1866, 137,
+p. 39) considered that the diazo compounds were formed by the addition
+of complex groupings of the type C<span class="su">6</span>H<span class="su">4</span>N<span class="su">2</span>- to the inorganic acids;
+whilst A. Kekulé (<i>Zeit. f. Chemie</i>, 1866, 2, p. 308), on account of their
+ready condensation to form azo compounds and their easy reduction
+to hydrazines, assumed that they were substances of the type
+R·N:N·Cl. The constitution of the diazonium group -N<span class="su">2</span>·X, may be
+inferred from the following facts:&mdash;The group C<span class="su">6</span>H<span class="su">5</span>N<span class="su">2</span>- behaves in
+many respects similarly to an alkali metal, and even more so to the
+ammonium group, since it is capable of forming colourless neutral
+salts with mineral acids, which in dilute aqueous solution are strongly
+ionized, but do not show any trace of hydrolytic dissociation
+(A. Hantzsch, <i>Ber.</i>, 1895, 28, p. 1734). Again, the diazonium chlorides
+combine with platinic chloride to form difficultly soluble double
+platinum salts, such as (C<span class="su">6</span>H<span class="su">5</span>N<span class="su">2</span>Cl)<span class="su">2</span>·PtCl<span class="su">4</span>; similar gold salts,
+C<span class="su">6</span>H<span class="su">5</span>N<span class="su">2</span>Cl·AuCl<span class="su">3</span>, are known. Determinations of the electrical conductivity
+of the diazonium chloride and nitrate also show that the
+diazonium radical is strictly comparable with other quaternary
+ammonium ions. For these reasons, one must assume the existence
+of pentavalent nitrogen in the diazonium salts, in order to account
+for their basic properties.</p>
+
+<p>The constitution of the isomeric diazo hydroxides has given rise
+to much discussion. E. Bamberger (<i>Ber.</i>, 1895, 28, pp. 444 et seq.) and
+C. W. Blomstrand (<i>Journ. prakt. Chem.</i>, 1896, 53, pp. 169 et seq.) hold
+that the compounds are structurally different, the normal diazo-hydroxide
+being a diazonium derivative of the type R·N(&#8758;N)·OH.
+The recent work of A. Hantzsch and his pupils seems to invalidate this
+view (<i>Ber.</i>, 1894, 27, pp. 1702 et seq.; see also A. Hantzsch, <i>Die Diazoverbindungen</i>).
+According to Hantzsch the isomeric diazo hydroxides
+are structurally identical, and the differences in behaviour are due
+to stereo-chemical relations, the isomerism being comparable with
+that of the oximes (<i>q.v.</i>). On such a hypothesis, the relatively
+unstable normal diazo hydroxides would be the <i>syn-</i>compounds,
+since here the nitrogen atoms would be more easily eliminated, whilst
+the stable iso-diazo derivatives would be the <i>anti-</i>compounds, thus:</p>
+
+<table class="nobctr" summary="Illustration">
+<tr><td class="figcenter" colspan="2"><img style="width:323px; height:39px" src="images/img174b.jpg" alt="" /></td></tr>
+<tr><td class="caption">Normal hydroxide<br />(Syn-compound)</td>
+<td class="caption">Iso hydroxide<br />(Anti-compound)</td></tr></table>
+
+<p class="noind">In support of this theory, Hantzsch has succeeded in isolating a series
+of syn- and anti-diazo-cyanides and -sulphonates (<i>Ber.</i>, 1895, 28, p. 666;
+1900, 33, p. 2161; 1901, 34, p. 4166). By diazotizing para-chloraniline
+and adding a cold solution of potassium cyanide, a salt (melting at
+29° C.) is obtained, which readily loses nitrogen, and forms para-chlorbenzonitrile
+on the addition of copper powder. By dissolving
+this diazocyanide in alcohol and reprecipitating it by water, it is
+converted into the isomeric diazocyanide (melting at 105-106° C.),
+which does not yield para-chlorbenzonitrile when treated with copper
+powder. Similar results have been obtained by using diazotized
+para-anisidine, a syn- and an anti- compound being formed, as well
+as a third isomeric cyanide, obtained by evaporating para-methoxy-benzenediazonium
+hydroxide in the presence of an excess of hydrocyanic
+acid at ordinary temperatures. This salt is a colourless
+crystalline substance of composition CH<span class="su">3</span>O·C<span class="su">6</span>H<span class="su">4</span>·N<span class="su">2</span>·CN·HCN·2H<span class="su">2</span>O,
+and has the properties of a metallic salt; it is very soluble in water
+and its solution is an electrolyte, whereas the solutions of the syn-
+and anti- compounds are not electrolytes. The isolation of these
+compounds is a powerful argument in favour of the Hantzsch
+hypothesis which requires the existence of these three different types,
+whilst the Bamberger-Blomstrand view only accounts for the formation
+of two isomeric cyanides, namely, one of the normal diazonium
+type and one of the iso-diazocyanide type.</p>
+
+<p>Benzene diazonium hydroxide, although a strong base, reacts with
+the alkaline hydroxides to form salts with the evolution of heat, and
+generally behaves as a weak acid. On mixing dilute solutions of the
+diazonium hydroxide and the alkali together, it is found that the
+molecular conductivity of the mixture is much less than the sum of
+the two electrical conductivities of the solutions separately, from
+which it follows that a portion of the ions present have changed to
+the non-ionized condition. This behaviour is explained by considering
+the non-ionized part of the diazonium hydroxide to exist in
+solution in a hydrated form, the equation of equilibrium being:</p>
+
+<div class="center"><img style="width:347px; height:52px" src="images/img174c.jpg" alt="" /></div>
+
+<p class="noind">On adding the alkaline hydroxide to the solution, this hydrate is
+supposed to lose water, yielding the syn-diazo hydroxide, which then
+gives rise to a certain amount of the sodium salt (A. Hantzsch, <i>Ber.</i>,
+1898, 31, p. 1612),</p>
+
+<div class="center"><img style="width:349px; height:58px" src="images/img174d.jpg" alt="" /></div>
+
+<p class="noind">This assumption also shows the relationship of the diazonium
+hydroxides to other quaternary ammonium compounds, for most of
+the quaternary ammonium hydroxides (except such as have the
+nitrogen atom attached to four saturated hydrocarbon radicals) are
+unstable, and readily pass over into compounds in which the hydroxyl
+group is no longer attached to the amine nitrogen; thus the syn-diazo
+hydroxides are to be regarded as pseudo-diazonium derivatives.
+(A. Hantzsch, <i>Ber.</i>, 1899, 32, p. 3109; 1900, 33, p. 278.) It is generally
+accepted that the iso-diazo hydroxides possess the oxime structure
+R·N:N·OH.</p>
+
+<p>Hantzsch explains the characteristic reactions of the diazonium
+compounds by the assumption that an addition compound is first
+formed, which breaks down with the elimination of the hydride of
+the acid radical, and the formation of an unstable syn-diazo compound,
+which, in its turn, decomposes with evolution of nitrogen
+(<i>Ber.</i>, 1897, 30, p. 2548; 1898, 31, p. 2053).</p>
+
+<div class="center"><img style="width:496px; height:52px" src="images/img174e.jpg" alt="" /></div>
+
+<p>J. Cain (<i>Jour. Chem. Soc.</i>, 1907, 91, p. 1049) suggested a quinonoid
+formula for diazonium salts, which has been combated by Hantzsch
+(<i>Ber.</i>, 1908, 41, pp. 3532 et seq.). G. T. Morgan and F. M. G. Micklethwaite
+(<i>Jour. Chem. Soc.</i>, 1908, 93, p. 617; 1909, 95, p. 1319) have
+pointed out that the salts may possess a dynamic formula, Cain&rsquo;s
+representing the middle stage, thus:</p>
+
+<div class="center"><img style="width:292px; height:78px" src="images/img174f.jpg" alt="" /></div>
+
+<p><i>Diazoamines.</i>&mdash;The diazoamines, R·N<span class="su">2</span>·NHR, may be prepared by
+the action of the primary and secondary amines on the diazonium
+salts, or by the action of nitrous acid on the free primary amine. In the
+latter reaction it is assumed that the isodiazohydroxide first formed
+is immediately attacked by a second molecule of the amine. They
+are yellow crystalline solids, which do not unite with acids. Nitrous
+acid converts them, in acid solution, into diazonium salts.</p>
+
+<p class="center">C<span class="su">6</span>H<span class="su">5</span>N<span class="su">2</span>·NHC<span class="su">6</span>H<span class="su">5</span> + 2HCl + HNO<span class="su">2</span> = 2C<span class="su">6</span>H<span class="su">5</span>N<span class="su">2</span>Cl + 2H<span class="su">2</span>O.</p>
+
+<p>They are readily converted into the isomeric aminoazo compounds,
+either by standing in alcoholic solution, or by warming with a
+mixture of the parent base and its hydrochloride; the diazo group
+preferably going into the para-position to the amino group. When
+the para-position is occupied, the diazo group takes the ortho-position.
+H. Goldschmidt and R. U. Reinders (<i>Ber.</i>, 1896, 29, p. 1369,
+1899) have shown that the transformation is a monomolecular
+reaction, the velocity of transformation in moderately dilute solution
+being independent of the concentration, but proportional to the
+amount of the catalyst present (amine hydrochloride) and to the
+temperature. It has also been shown that when different salts of the
+amine are used, their catalytic influence varies in amount and is
+almost proportional to their degree of ionization in aqueous solution.
+Diazoaminobenzene, C<span class="su">6</span>H<span class="su">5</span>N<span class="su">2</span>·NHC<span class="su">6</span>H<span class="su">5</span>, crystallizes in golden yellow
+laminae, which melt at 96° C. and explode at a slightly higher temperature.
+It is readily soluble in alcohol, ether and benzene. Concentrated
+hydrochloric acid converts it into chlorbenzene, aniline and
+nitrogen. Zinc dust and alcoholic acetic acid reduce it to aniline
+and phenylhydrazine.</p>
+
+<p><i>Diazoimino compounds</i>, R·N<span class="su">3</span>, may be regarded as derivatives of
+azoimide (<i>q.v.</i>); they are formed by the action of ammonia on the
+diazoperbromides, or by the action of hydroxylamine on the diazonium
+sulphates (J. Mai, <i>Ber.</i>, 1892, 25, p. 372; T. Curtius, <i>Ber.</i>, 1893, 26,
+p. 1271). Diazobenzeneimide, C<span class="su">6</span>H<span class="su">5</span>N<span class="su">3</span>, is a yellowish oil of stupefying
+odour. It boils at 59° C. (12 mm.), and explodes when heated.
+Concentrated hydrochloric acid decomposes it with formation of
+<span class="pagenum"><a name="page175" id="page175"></a>175</span>
+chloranilines and elimination of nitrogen, whilst on boiling with
+sulphuric acid it is converted into aminophenols.</p>
+
+<p><i>Aliphatic Diazo Compounds.</i>&mdash;The esters of the aliphatic amino
+acids may be diazotized in a manner similar to the primary aromatic
+amines, a fact discovered by T. Curtius (<i>Ber.</i>, 1833, 16, p. 2230). The
+first aliphatic diazo compound to be isolated was <i>diazoacetic ester</i>,
+CH·N<span class="su">2</span>·CO<span class="su">2</span>C<span class="su">2</span>H<span class="su">5</span>, which is prepared by the action of potassium nitrite
+on the ethyl ester of glycocoll hydrochloride, HCl·NH<span class="su">2</span>·CH<span class="su">2</span>·CO<span class="su">2</span>C<span class="su">2</span>H<span class="su">5</span>
+ + KNO<span class="su">2</span> = CHN<span class="su">2</span>·CO<span class="su">2</span>C<span class="su">2</span>H<span class="su">5</span> + KCl + 2H<span class="su">2</span>O. It is a yellowish oil which
+melts at -24° C.; it boils at 143-144° C., but cannot be distilled safely
+as it decomposes violently, giving nitrogen and ethyl fumarate. It
+explodes in contact with concentrated sulphuric acid. On reduction
+it yields ammonia and glycocoll (aminoacetic acid). When heated
+with water it forms ethyl hydroxy-acetate; with alcohol it yields
+ethyl ethoxyacetate. Halogen acids convert it into monohalogen
+fatty acids, and the halogens themselves convert it into dihalogen
+fatty acids. It unites with aldehydes to form esters of ketonic acids,
+and with aniline yields anilido-acetic acid. It forms an addition
+product with acrylic ester, which on heating loses nitrogen and leaves
+trimethylene dicarboxylic ester. Concentrated ammonia converts
+it into <i>diazoacetamide</i>, CHN<span class="su">2</span>·CONH<span class="su">2</span>, which crystallizes in golden
+yellow plates which melt at 114° C. For other reactions see
+<span class="sc"><a href="#artlinks">Hydrazine</a></span>. The constitution of the diazo fatty esters is inferred
+from the fact that the two nitrogen atoms, when split off, are
+replaced by two monovalent elements or groups, thus leading to
+the formula <img style="width:146px; height:42px" src="images/img175.jpg" alt="" /> for diazoacetic ester.</p>
+
+<p><i>Diazosuccinic ester</i>, N<span class="su">2</span>·C(CO<span class="su">2</span>C<span class="su">2</span>H<span class="su">5</span>)<span class="su">2</span>, is similarly prepared by the
+action of nitrous acid on the hydrochloride of aspartic ester. It is
+decomposed by boiling water and yields fumaric ester.</p>
+
+<p><i>Diazomethane</i>, CH<span class="su">2</span>N<span class="su">2</span>, was first obtained in 1894 by H. v. Pechmann
+(<i>Ber.</i>, 1894, 27, p. 1888; 1895, 28, p. 855). It is prepared by the
+action of aqueous or alcoholic solutions of the caustic alkalis on
+the nitroso-acidyl derivatives of methylamine (such, for example,
+as <i>nitrosomethyl urethane</i>, NO·N(CH<span class="su">3</span>)·CO<span class="su">2</span>C<span class="su">2</span>H<span class="su">5</span>, which is formed on
+passing nitrous fumes into an ethereal solution of methyl urethane).
+E. Bamberger (<i>Ber.</i>, 1895, 28, p. 1682) regards it as the anhydride of
+iso-diazomethane, CH<span class="su">3</span>·N:N·OH, and has prepared it by a method
+similar to that used for the preparation of iso-diazobenzene. By the
+action of bleaching powder on methylamine hydrochloride, there
+is obtained a volatile liquid (<i>methyldichloramine</i>, CH<span class="su">3</span>·N·Cl<span class="su">2</span>), boiling
+at 58-60° C., which explodes violently when heated with water,
+yielding hydrocyanic acid (CH<span class="su">3</span>NCl<span class="su">2</span> = HCN + 2HCl). Well-dried
+hydroxylamine hydrochloride is dissolved in methyl alcohol and
+mixed with sodium methylate; a solution of methyldichloramine in
+absolute ether is then added and an ethereal solution of diazomethane
+distils over. Diazomethane is a yellow inodorous gas, very poisonous
+and corrosive. It may be condensed to a liquid, which boils at about
+0° C. It is a powerful methylating agent, reacting with water to form
+methyl alcohol, and converting acetic acid into methylacetate, hydrochloric
+acid into methyl chloride, hydrocyanic acid into acetonitrile,
+and phenol into anisol, nitrogen being eliminated in each case. It is
+reduced by sodium amalgam (in alcoholic solution) to <i>methylhydrazine</i>,
+CH<span class="su">3</span>·NH·NH<span class="su">2</span>. It unites directly with acetylene to form pyrazole
+(H. v. Pechmann, <i>Ber.</i>, 1898, 31, p. 2950) and with fumaric methyl
+ester it forms pyrazolin dicarboxylic ester.</p>
+<div class="author">(F. G. P.*)</div>
+
+<p>See G. T. Morgan, <i>B.A. Rep.</i>, 1902; J. Cain, <i>Diazo Compounds</i>, 1908.</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIAZOMATA<a name="ar24" id="ar24"></a></span> (Gr. <span class="grk" title="diazôma">&#948;&#953;&#940;&#950;&#969;&#956;&#945;</span>, a girdle), in architecture, the
+landing places and passages which were carried round the semicircle
+and separated the upper and lower tiers in a Greek theatre.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIBDIN, CHARLES<a name="ar25" id="ar25"></a></span> (1745-1814), British musician, dramatist,
+novelist, actor and song-writer, the son of a parish clerk, was born
+at Southampton on or before the 4th of March 1745, and was the
+youngest of a family of eighteen. His parents designing him for
+the church, he was sent to Winchester; but his love of music
+early diverted his thoughts from the clerical profession. After
+receiving some instruction from the organist of Winchester
+cathedral, where he was a chorister from 1756 to 1759, he went
+to London at the age of fifteen. Here he was placed in a music
+warehouse in Cheapside, but he soon abandoned this employment
+to become a singing actor at Covent Garden. On the 21st of May
+1762 his first work, an operetta entitled <i>The Shepherd&rsquo;s Artifice</i>,
+with words and music by himself, was produced at this theatre.
+Other works followed, his reputation being firmly established
+by the music to the play of <i>The Padlock</i>, produced at Drury Lane
+under Garrick&rsquo;s management in 1768, the composer himself taking
+the part of Mungo with conspicuous success. He continued for
+some years to be connected with Drury Lane, both as composer
+and as actor, and produced during this period two of his best
+known works, <i>The Waterman</i> (1774) and <i>The Quaker</i> (1775). A
+quarrel with Garrick led to the termination of his engagement.
+In <i>The Comic Mirror</i> he ridiculed prominent contemporary figures
+through the medium of a puppet show. In 1782 he became joint
+manager of the Royal circus, afterwards known as the Surrey
+theatre. In three years he lost this position owing to a quarrel
+with his partner. His opera <i>Liberty Hall</i>, containing the successful
+songs &ldquo;Jock Ratlin,&rdquo; &ldquo;The Highmettled Racer,&rdquo; and
+&ldquo;The Bells of Aberdovey,&rdquo; was produced at Drury Lane theatre
+on the 8th of February 1785. In 1788 he sailed for the East
+Indies, but the vessel having put in to Torbay in stress of weather,
+he changed his mind and returned to London. In a musical
+variety entertainment called <i>The Oddities</i>, he succeeded in winning
+marked popularity with a number of songs that included
+&ldquo;&rsquo;Twas in the good ship &lsquo;Rover&rsquo;,&rdquo; &ldquo;Saturday Night at Sea,&rdquo; &ldquo;I
+sailed from the Downs in the &lsquo;Nancy,&rsquo;&rdquo; and the immortal &ldquo;Tom
+Bowling,&rdquo; written on the death of his eldest brother, Captain
+Thomas Dibdin, at whose invitation he had planned his visit
+to India. A series of monodramatic entertainments which he
+gave at his theatre, Sans Souci, in Leicester Square, brought his
+songs, music and recitations more prominently into notice, and
+permanently established his fame as a lyric poet. It was at these
+entertainments that he first introduced many of those sea-songs
+which so powerfully influenced the national spirit. The words
+breathe the simple loyalty and dauntless courage that are the
+cardinal virtues of the British sailor, and the music was appropriate
+and naturally melodious. Their effect in stimulating
+and ennobling the spirit of the navy during the war with France
+was so marked as to call for special acknowledgment. In 1803
+Dibdin was rewarded by government with a pension of £200 a
+year, of which he was only for a time deprived under the administration
+of Lord Grenville. During this period he opened a
+music shop in the Strand, but the venture was a failure. Dibdin
+died of paralysis in London on the 25th of July 1814. Besides his
+<i>Musical Tour through England</i> (1788), his <i>Professional Life</i>, an
+autobiography published in 1803, a <i>History of the Stage</i> (1795), and
+several smaller works, he wrote upwards of 1400 songs and about
+thirty dramatic pieces. He also wrote the following novels:&mdash;<i>The
+Devil</i> (1785); <i>Hannah Hewitt</i> (1792); <i>The Younger Brother</i>
+(1793). An edition of his songs by G. Hogarth (1843) contains
+a memoir of his life. His two sons, Charles and Thomas John
+Dibdin (<i>q.v.</i>), whose works are often confused with those of their
+father, were also popular dramatists in their day.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIBDIN, THOMAS FROGNALL<a name="ar26" id="ar26"></a></span> (1776-1847), English bibliographer,
+born at Calcutta in 1776, was the son of Thomas Dibdin,
+the sailor brother of Charles Dibdin. His father and mother both
+died on the way home to England in 1780, and Thomas was
+brought up by a maternal uncle. He was educated at St John&rsquo;s
+College, Oxford, and studied for a time at Lincoln&rsquo;s Inn. After
+an unsuccessful attempt to obtain practice as a provincial counsel
+at Worcester, he was ordained a clergyman at the close of 1804,
+being appointed to a curacy at Kensington. It was not until
+1823 that he received the living of Exning in Sussex. Soon afterwards
+he was appointed by Lord Liverpool to the rectory of St
+Mary&rsquo;s, Bryanston Square, which he held until his death on the
+18th of November 1847. The first of his numerous bibliographical
+works was his <i>Introduction to the Knowledge of Editions of the
+Classics</i> (1802), which brought him under the notice of the
+third Earl Spencer, to whom he owed much important aid in
+his bibliographical pursuits. The rich library at Althorp was
+thrown open to him; he spent much of his time in it, and in
+1814-1815 published his <i>Bibliotheca Spenceriana</i>. As the library
+was not open to the general public, the information given in the
+<i>Bibliotheca</i> was found very useful, but since its author was unable
+even to read the characters in which the books he described were
+written, the work was marred by the errors which more or less
+characterize all his productions. This fault of inaccuracy however
+was less obtrusive in his series of playful, discursive works in
+the form of dialogues on his favourite subject, the first of which,
+<i>Bibliomania</i> (1809), was republished with large additions in
+1811, and was very popular, passing through numerous editions.
+To the same class belonged the <i>Bibliographical Decameron</i>, a larger
+work, which appeared in 1817. In 1810 he began the publication
+of a new and much extended edition of Ames&rsquo;s <i>Typographical
+Antiquities</i>. The first volume was a great success, but the publication
+was checked by the failure of the fourth volume, and was
+<span class="pagenum"><a name="page176" id="page176"></a>176</span>
+never completed. In 1818 Dibdin was commissioned by Earl
+Spencer to purchase books for him on the continent, an expedition
+described in his sumptuous <i>Bibliographical, Antiquarian and
+Picturesque Tour in France and Germany</i> (1821). In 1824 he
+made an ambitious venture in his <i>Library Companion, or the
+Young Man&rsquo;s Guide and Old Man&rsquo;s Comfort in the Choice of a
+Library</i>, intended to point out the best works in all departments
+of literature. His culture was not broad enough, however, to
+render him competent for the task, and the work was severely
+criticized. For some years Dibdin gave himself up chiefly to
+religious literature. He returned to bibliography in his
+<i>Bibliophobia, or Remarks on the Present Depression in the State of
+Literature and the Book Trade</i> (1832), and the same subject
+furnishes the main interest of his <i>Reminiscences of a Literary Life</i>
+(1836), and his <i>Bibliographical, Antiquarian and Picturesque
+Tour in the Northern Counties of England and Scotland</i> (1838).
+Dibdin was the originator and vice-president, Lord Spencer
+being the president, of the Roxburghe Club, founded in 1812,&mdash;the
+first of the numerous book clubs which have done such
+service to literature.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIBDIN, THOMAS JOHN<a name="ar27" id="ar27"></a></span> (1771-1841), English dramatist and
+song-writer, son of Charles Dibdin, the song-writer, and of Mrs
+Davenet, an actress whose real name was Harriet Pitt, was born
+on the 21st of March 1771. He was apprenticed to his maternal
+uncle, a London upholsterer, and later to William Rawlins,
+afterwards sheriff of London. He summoned his second master
+unsuccessfully for rough treatment; and after a few years of
+service he ran away to join a company of country players. From
+1789 to 1795 he played in all sorts of parts; he acted as scene
+painter at Liverpool in 1791; and during this period he composed
+more than 1000 songs. He made his first attempt as a
+dramatic writer in <i>Something New</i>, followed by <i>The Mad Guardian</i>
+in 1795. He returned to London in 1795, having married two
+years before; and in the winter of 1798-1799 his <i>Jew and the
+Doctor</i> was produced at Covent Garden. From this time he
+contributed a very large number of comedies, operas, farces, &amp;c.,
+to the public entertainment. Some of these brought immense
+popularity to the writer and immense profits to the theatres. It is
+stated that the pantomime of <i>Mother Goose</i> (1807) produced more
+than £20,000 for the management at Covent Garden theatre, and
+the <i>High-mettled Racer</i>, adapted as a pantomime from his father&rsquo;s
+play, £18,000 at Astley&rsquo;s. Dibdin was prompter and pantomime
+writer at Drury Lane until 1816, when he took the Surrey theatre.
+This venture proved disastrous and he became bankrupt. After
+this he was manager of the Haymarket, but without his old
+success, and his last years were passed in comparative poverty.
+In 1827 he published two volumes of <i>Reminiscences</i>; and at the
+time of his death he was preparing an edition of his father&rsquo;s sea
+songs, for which a small sum was allowed him weekly by the lords
+of the admiralty. Of his own songs &ldquo;The Oak Table&rdquo; and
+&ldquo;The Snug Little Island&rdquo; are well-known examples. He died in
+London on the 16th of September 1841.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIBRA<a name="ar28" id="ar28"></a></span> (Slav. <i>Debra</i>), the capital of a sanjak bearing the same
+name, in the vilayet of Monastir, eastern Albania, Turkey. Pop.
+(1900) about 15,000. Dibra occupies a valley enclosed by
+mountains, and watered by the Tsrni Drin and Radika rivers,
+which meet 3 m. S. It is a fortified city, and the only episcopal
+see of the Bulgarian exarchate in Albania; most of the inhabitants
+are Albanians, but there is a strong Bulgarian colony. The
+local trade is almost entirely agricultural.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIBRUGARH,<a name="ar29" id="ar29"></a></span> a town of British India, in the Lakhimpur
+district of eastern Bengal and Assam, of which it is the headquarters,
+situated on the Dibru river about 4 m. above its
+confluence with the Brahmaputra. Pop. (1901) 11,227. It is the
+terminus of steamer navigation on the Brahmaputra, and also of
+a railway running to important coal-mines and petroleum wells,
+which connects with the Assam-Bengal system. Large quantities
+of coal and tea are exported. There are a military cantonment,
+the headquarters of the volunteer corps known as the Assam
+Valley Light Horse; a government high school, a training school
+for masters; and an aided school for girls. In 1900 a medical
+school for the province was established, out of a bequest left
+by Brigade-Surgeon J. Berry-White, which is maintained by
+the government, to train hospital assistants for the tea gardens.
+The Williamson artisan school is entirely supported by an
+endowment.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DICAEARCHUS,<a name="ar30" id="ar30"></a></span> of Messene in Sicily, Peripatetic philosopher
+and pupil of Aristotle, historian, and geographer, flourished about
+320 <span class="sc">b.c.</span> He was a friend of Theophrastus, to whom he dedicated
+the majority of his works. Of his writings, which comprised
+treatises on a great variety of subjects, only the titles and a few
+fragments survive. The most important of them was his
+<span class="grk" title="bios tês Hellados">&#946;&#943;&#959;&#962; &#964;&#8134;&#962; &#904;&#955;&#955;&#940;&#948;&#959;&#962;</span> (<i>Life in Greece</i>), in which the moral, political
+and social condition of the people was very fully discussed. In
+his <i>Tripoliticos</i> he described the best form of government as a
+mixture of monarchy, aristocracy and democracy, and illustrated
+it by the example of Sparta. Among the philosophical works of
+Dicaearchus may be mentioned the <i>Lesbiaci</i>, a dialogue in three
+books, in which the author endeavours to prove that the soul is
+mortal, to which he added a supplement called <i>Corinthiaci</i>. He
+also wrote a <i>Description of the World</i> illustrated by maps, in
+which was probably included his <i>Measurements of Mountains</i>.
+A description of Greece (150 iambics, in C. Müller, <i>Frag. hist.
+Graec</i>. i. 238-243) was formerly attributed to him, but, as the
+initial letters of the first twenty-three lines show, was really
+the work of Dionysius, son of Calliphon. Three considerable
+fragments of a prose description of Greece (Müller, i. 97-110)
+are now assigned to an unknown author named Heracleides. The
+<i>De re publica</i> of Cicero is supposed to be founded on one of
+Dicaearchus&rsquo;s works.</p>
+
+<div class="condensed">
+<p>The best edition of the fragments is by M. Fuhr (1841), a work of
+great learning; see also a dissertation by F. G. Osann, <i>Beiträge zur
+röm. und griech. Litteratur</i>, ii. pp. 1-117 (1839); Pauly-Wissowa,
+<i>Realencyclopädie der klass. Altertumswiss</i>. v. pt. 1 (1905).</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DICE<a name="ar31" id="ar31"></a></span> (plural of die, O. Fr. <i>de</i>, derived from Lat. <i>dare</i>, to give),
+small cubes of ivory, bone, wood or metal, used in gaming. The
+six sides of a die are each marked with a different number of
+incised dots in such a manner that the sum of the dots on any two
+opposite sides shall be 7. Dice seem always to have been
+employed, as is the case to-day, for gambling purposes, and they
+are also used in such games as backgammon. There are many
+methods of playing, from one to five dice being used, although
+two or three are the ordinary numbers employed in Great Britain
+and America. The dice are thrown upon a table or other smooth
+surface either from the hand or from a receptacle called a dice-box,
+the latter method having been in common use in Greece, Rome
+and the Orient in ancient times. Dice-boxes have been made in
+many shapes and of various materials, such as wood, leather,
+agate, crystal, metal or paper. Many contain bars within to ensure
+a proper agitation of the dice, and thus defeat trickery. Some,
+formerly used in England, were employed with unmarked dice,
+and allowed the cubes to fall through a kind of funnel upon a
+board marked off into six equal parts numbered from 1 to 6.
+It is a remarkable fact, that, wherever dice have been found,
+whether in the tombs of ancient Egypt, of classic Greece, or of
+the far East, they differ in no material respect from those in use
+to-day, the elongated ones with rounded ends found in Roman
+graves having been, not dice but <i>tali</i>, or knucklebones. Eight-sided
+dice have comparatively lately been introduced in France
+as aids to children in learning the multiplication table. The
+teetotum, or spinning die, used in many modern games, was
+known in ancient times in China and Japan. The increased
+popularity of the more elaborate forms of gaming has resulted in
+the decline of dicing. The usual method is to throw three times
+with three dice. If one or more sixes or fives are thrown the first
+time they may be reserved, the other throws being made with the
+dice that are left. The object is to throw three sixes = 18 or as
+near that number as possible, the highest throw winning, or, when
+drinks are to be paid for, the lowest throw losing. (For other
+methods of throwing consult the <i>Encyclopaedia of Indoor Games</i>,
+by R. F. Foster, 1903.) The most popular form of pure gambling
+with dice at the present day, particularly with the lower classes in
+America, is <i>Craps</i>, or <i>Crap-Shooting</i>, a simple form of <i>Hazard</i>, of
+French origin. Two dice are used. Each player puts up a stake
+<span class="pagenum"><a name="page177" id="page177"></a>177</span>
+and the first caster may cover any or all of the bets. He then
+<i>shoots</i>, <i>i.e.</i> throws the dice from his open hand upon the table.
+If the sum of the dice is 7 or 11 the throw is a <i>nick</i>, or <i>natural</i>, and
+the caster wins all stakes. If the throw is either 2, 3 or 12 it is
+a <i>crap</i>, and the caster loses all. If any other number is thrown
+it is a <i>point</i>, and the caster continues until he throws the same
+number again, in which case he wins, or a 7, in which case he
+loses. The now practically obsolete game of Hazard was much
+more complicated than <i>Craps</i>. (Consult <i>The Game of Hazard
+Investigated</i>, by George Lowbut.) <i>Poker dice</i> are marked with ace,
+king, queen, jack and ten-spot. Five are used and the object is,
+in three throws, to make pairs, triplets, full hands or fours and
+fives of a kind, five aces being the highest hand. Straights do
+not count. In throwing to decide the payment of drinks the
+usual method is called <i>horse and horse</i>, in which the highest
+throws retire, leaving the two lowest to decide the loser by the
+best two in three throws. Should each player win one throw
+both are said to be <i>horse and horse</i>, and the next throw determines
+the loser. The two last casters may also agree to <i>sudden death</i>, <i>i.e.</i>
+a single throw. <i>Loaded dice</i>, <i>i.e.</i> dice weighted slightly on the side
+of the lowest number, have been used by swindlers from the very
+earliest times to the present day, a fact proved by countless
+literary allusions. Modern dice are often rounded at the corners,
+which are otherwise apt to wear off irregularly.</p>
+
+<p><i>History.</i>&mdash;Dice were probably evolved from knucklebones.
+The antiquary Thomas Hyde, in his <i>Syntagma</i>, records his
+opinion that the game of &ldquo;odd or even,&rdquo; played with pebbles, is
+nearly coeval with the creation of man. It is almost impossible
+to trace clearly the development of dice as distinguished from
+knucklebones, on account of the confusing of the two games
+by the ancient writers. It is certain, however, that both were
+played in times antecedent to those of which we possess any
+written records. Sophocles, in a fragment, ascribed their invention
+to Palamedes, a Greek, who taught them to his countrymen
+during the siege of Troy, and who, according to Pausanias
+(on Corinth, xx.), made an offering of them on the altar of the
+temple of Fortune. Herodotus (<i>Clio</i>) relates that the Lydians,
+during a period of famine in the days of King Atys, invented dice,
+knucklebones and indeed all other games except chess. The fact
+that dice have been used throughout the Orient from time
+immemorial, as has been proved by excavations from ancient
+tombs, seems to point clearly to an Asiatic origin. Dicing is
+mentioned as an Indian game in the <i>Rig-veda</i>. In its primitive
+form knucklebones was essentially a game of skill, played by
+women and children, while dice were used for gambling, and
+it was doubtless the gambling spirit of the age which was
+responsible for the derivative form of knucklebones, in which
+four sides of the bones received different values, which were then
+counted, like dice. Gambling with three, sometimes two, dice
+(<span class="grk" title="kuboi">&#954;&#973;&#946;&#959;&#953;</span>) was a very popular form of amusement in Greece, especially
+with the upper classes, and was an almost invariable accompaniment
+to the symposium, or drinking banquet. The dice were cast
+from conical beakers, and the highest throw was three sixes,
+called <i>Aphrodite</i>, while the lowest, three aces, was called the <i>dog</i>.
+Both in Greece and Rome different modes of counting were in
+vogue. Roman dice were called <i>tesserae</i> from the Greek word for
+four, indicative of the four sides. The Romans were passionate
+gamblers, especially in the luxurious days of the Empire, and
+dicing was a favourite form, though it was forbidden except
+during the Saturnalia. The emperor Augustus wrote in a letter
+to Suetonius concerning a game that he had played with his
+friends: &ldquo;Whoever threw a <i>dog</i> or a six paid a <i>denarius</i> to the
+bank for every die, and whoever threw a <i>Venus</i> (the highest) won
+everything.&rdquo; In the houses of the rich the dice-beakers were
+of carved ivory and the dice of crystal inlaid with gold. Mark
+Antony wasted his time at Alexandria with dicing, while, according
+to Suetonius, the emperors Augustus, Nero and Claudius were
+passionately fond of it, the last named having written a book on
+the game. Caligula notoriously cheated at the game; Domitian
+played it, and Commodus set apart special rooms in his palace
+for it. The emperor Verus, adopted son of Antonine, is known
+to have thrown dice whole nights together. Fashionable society
+followed the lead of its emperors, and, in spite of the severity of
+the laws, fortunes were squandered at the dicing table. Horace
+derided the youth of the period, who wasted his time amid the
+dangers of dicing instead of taming his charger and giving himself
+up to the hardships of the chase. Throwing dice for money
+was the cause of many special laws in Rome, according to one of
+which no suit could be brought by a person who allowed gambling
+in his house, even if he had been cheated or assaulted. Professional
+gamblers were common, and some of their loaded dice
+are preserved in museums. The common public-houses were the
+resorts of gamblers, and a fresco is extant showing two quarrelling
+dicers being ejected by the indignant host. Virgil, in the <i>Copa</i>
+generally ascribed to him, characterizes the spirit of that age in
+verse, which has been Englished as follows:&mdash;</p>
+
+<table class="reg" summary="poem"><tr><td> <div class="poemr">
+<p>&ldquo;What ho! Bring dice and good wine!</p>
+<p class="i2">Who cares for the morrow?</p>
+<p class="i05">Live&mdash;so calls grinning Death&mdash;</p>
+<p class="i2">Live, for I come to you soon!&rdquo;</p>
+</div> </td></tr></table>
+
+<p class="noind">That the barbarians were also given to gaming, whether or
+not they learned it from their Roman conquerors, is proved by
+Tacitus, who states that the Germans were passionately fond
+of dicing, so much so, indeed, that, having lost everything, they
+would even stake their personal liberty. Centuries later, during
+the middle ages, dicing became the favourite pastime of the
+knights, and both dicing schools (<i>scholae deciorum</i>) and gilds
+of dicers existed. After the downfall of feudalism the famous
+German mercenaries called <i>landsknechts</i> established a reputation
+as the most notorious dicing gamblers of their time. Many of the
+dice of the period were curiously carved in the images of men and
+beasts. In France both knights and ladies were given to dicing,
+which repeated legislation, including interdictions on the part of
+St Louis in 1254 and 1256, did not abolish. In Japan, China,
+Korea, India and other Asiatic countries dice have always been
+popular and are so still.</p>
+
+<div class="condensed">
+<p>See Foster&rsquo;s <i>Encyclopaedia of Indoor Games</i> (1903); Raymond&rsquo;s
+<i>Illustriertes Knobelbrevier</i> (Oranienburg, 1888); <i>Les Jeux des Anciens</i>,
+by L. Becq de Fouquières (Paris, 1869); <i>Das Knöchelspiel der Alten</i>,
+by Bolle (Wismar, 1886); <i>Die Spiele der Griechen und Römer</i>, by
+W. Richter (Leipzig, 1887); Raymond&rsquo;s <i>Alte und neue Würfelspiele</i>;
+<i>Chinese Games with Dice</i>, by Stewart Culin (Philadelphia, 1889);
+<i>Korean Games</i>, by Stewart Culin (Philadelphia, 1895).</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DICETO, RALPH DE<a name="ar32" id="ar32"></a></span> (d. c. 1202), dean of St Paul&rsquo;s, London,
+and chronicler, is first mentioned in 1152, when he received the
+archdeaconry of Middlesex. He was probably born between
+1120 and 1130; of his parentage and nationality we know
+nothing. The common statement that he derived his surname
+from Diss in Norfolk is a mere conjecture; Dicetum may equally
+well be a Latinized form of Dissai, or Dicy, or Dizy, place names
+which are found in Maine, Picardy, Burgundy and Champagne.
+In 1152 Diceto was already a master of arts; presumably he had
+studied at Paris. His reputation for learning and integrity stood
+high; he was regarded with respect and favour by Arnulf of
+Lisieux and Gilbert Foliot of Hereford (afterwards of London),
+two of the most eminent bishops of their time. Quite naturally,
+the archdeacon took in the Becket question the same side as his
+friends. Although his narrative is colourless, and although he
+was one of those who showed some sympathy for Becket at the
+council of Northampton (1164), the correspondence of Diceto
+shows that he regarded the archbishop&rsquo;s conduct as ill-considered,
+and that he gave advice to those whom Becket regarded as his
+chief enemies. Diceto was selected, in 1166, as the envoy of the
+English bishops when they protested against the excommunications
+launched by Becket. But, apart from this episode, which he
+characteristically omits to record, he remained in the background.
+The natural impartiality of his intellect was accentuated by a
+certain timidity, which is apparent in his writings no less than
+in his life. About 1180 he became dean of St Paul&rsquo;s. In this
+office he distinguished himself by careful management of the
+estates, by restoring the discipline of the chapter, and by building
+at his own expense a deanery-house. A scholar and a man of
+considerable erudition, he showed a strong preference for historical
+studies; and about the time when he was preferred to
+the deanery he began to collect materials for the history of his
+<span class="pagenum"><a name="page178" id="page178"></a>178</span>
+own times. His friendships with Richard Fitz Nigel, who succeeded
+Foliot in the see of London, with William Longchamp, the
+chancellor of Richard I., and with Walter of Coutances, the archbishop
+of Rouen, gave him excellent opportunities of collecting
+information. His two chief works, the <i>Abbreviationes Chronicorum</i>
+and the <i>Ymagines Historiarum</i>, cover the history of the
+world from the birth of Christ to the year 1202. The former,
+which ends in 1147, is a work of learning and industry, but
+almost entirely based upon extant sources. The latter, beginning
+as a compilation from Robert de Monte and the letters of
+Foliot, becomes an original authority about 1172, and a contemporary
+record about 1181. In precision and fulness of detail the
+<i>Ymagines</i> are inferior to the chronicles of the so-called Benedict
+and of Hoveden. Though an annalist, Diceto is careless in his
+chronology; and the documents which he incorporates, while
+often important, are selected on no principle. He has little sense
+of style; but displays considerable insight when he ventures to
+discuss a political situation. For this reason, and on account of
+the details with which they supplement the more important
+chronicles of the period, the <i>Ymagines</i> are a valuable though a
+secondary source.</p>
+
+<div class="condensed">
+<p>See W. Stubbs&rsquo; edition of the <i>Historical Works</i> of Diceto (Rolls ed.
+1876, 2 vols.), and especially the introduction. The second volume
+contains minor works which are the barest compendia of facts taken
+from well-known sources. Diceto&rsquo;s fragmentary Domesday of the
+capitular estates has been edited by Archdeacon Hale in <i>The Domesday
+of St Paul&rsquo;s</i>, pp. 109 ff. (Camden Society, 1858).</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DICEY, EDWARD<a name="ar33" id="ar33"></a></span> (1832-&emsp;&emsp;), English writer, son of T. E.
+Dicey of Claybrook Hall, Leicestershire, was born in 1832. Educated
+at Trinity College, Cambridge, where he took mathematical
+and classical honours, he became an active journalist, contributing
+largely to the principal reviews. He was called to the bar
+in 1875, became a bencher of Gray&rsquo;s Inn in 1896, and was
+treasurer in 1903-1904. He was connected with the <i>Daily
+Telegraph</i> as leader writer and then as special correspondent, and
+after a short spell in 1870 as editor of the <i>Daily News</i> he became
+editor of the <i>Observer</i>, a position which he held until 1889. Of
+his many books on foreign affairs perhaps the most important are
+his <i>England and Egypt</i> (1884), <i>Bulgaria, the Peasant State</i> (1895),
+<i>The Story of the Khedivate</i> (1902), and <i>The Egypt of the Future</i>
+(1907). He was created C.B. in 1886.</p>
+
+<p>His brother <span class="sc">Albert Venn Dicey</span> (b. 1835), English jurist,
+was educated at Balliol College, Oxford, where he took a first
+class in the classical schools in 1858. He was called to the bar at
+the Inner Temple in 1863. He held fellowships successively
+at Balliol, Trinity and All Souls&rsquo;, and from 1882 to 1909 was
+Vinerian professor of law. He became Q.C. in 1890. His chief
+works are the <i>Introduction to the Study of the Law of the Constitution</i>
+(1885, 6th ed. 1902), which ranks as a standard work on
+the subject; <i>England&rsquo;s Case against Home Rule</i> (1886); <i>A Digest
+of the Law of England with Reference to the Conflict of Laws</i> (1896),
+and <i>Lectures on the Relation between Law and Public Opinion in
+England during the 19th century</i> (1905).</p>
+
+
+<hr class="art" />
+<p><span class="bold">DICHOTOMY<a name="ar34" id="ar34"></a></span> (Gr. <span class="grk" title="dicha">&#948;&#943;&#967;&#945;</span>, apart, <span class="grk" title="temnein">&#964;&#941;&#956;&#957;&#949;&#953;&#957;</span>, to cut), literally a
+cutting asunder, the technical term for a form of logical division,
+consisting in the separation of a genus into two species, one of
+which has and the other has not, a certain quality or attribute.
+Thus men may be thus divided into white men, and men who are
+not white; each of these may be subdivided similarly. On the
+principle of contradiction this division is both exhaustive and
+exclusive; there can be no overlapping, and no members of the
+original genus or the lower groups are omitted. This method of
+classification, though formally accurate, has slight value in the
+exact sciences, partly because at every step one of the two groups
+is merely negatively characterized and therefore incapable of real
+subdivision; it is useful, however, in setting forth clearly the
+gradual descent from the most inclusive genus (<i>summum genus</i>)
+through species to the lowest class (<i>infima species</i>), which is
+divisible only into individual persons or things. (See further
+<span class="sc"><a href="#artlinks">Division</a></span>.) In astronomy the term is used for the aspect of the
+moon or of a planet when apparently half illuminated, so that its
+disk has the form of a semicircle.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DICK, ROBERT<a name="ar35" id="ar35"></a></span> (1811-1866), Scottish geologist and botanist,
+was born at Tullibody, in Clackmannanshire, in January 1811.
+His father was an officer of excise. At the age of thirteen, after
+receiving a good elementary education at the parish school,
+Robert Dick was apprenticed to a baker, and served for three
+years. In these early days he became interested in wild flowers&mdash;he
+made a collection of plants and gradually acquired some
+knowledge of their names from an old encyclopaedia. When
+his time was out he left Tullibody and gained employment as a
+journeyman baker at Leith, Glasgow and Greenock. Meanwhile
+his father, who in 1826 had been removed to Thurso, as supervisor
+of excise, advised his son to set up a baker&rsquo;s shop in that
+town. Thither Robert Dick went in 1830, he started in business
+as a baker and worked laboriously until he died on the 24th of
+December 1866. Throughout this period he zealously devoted
+himself to studying and collecting the plants, mollusca and insects
+of a wide area of Caithness, and his attention was directed soon
+after he settled in Thurso to the rocks and fossils. In 1835 he first
+found remains of fossil fishes; but it was not till some years later
+that his interest became greatly stirred. Then he obtained a copy
+of Hugh Miller&rsquo;s <i>Old Red Sandstone</i> (published in 1841), and
+he began systematically to collect with hammer and chisel the
+fossils from the Caithness flags. In 1845 he found remains of
+<i>Holoptychius</i> and forwarded specimens to Hugh Miller, and he
+continued to send the best of his fossil fishes to that geologist, and
+to others after the death of Miller. In this way he largely contributed
+to the progress of geological knowledge, although he himself
+published nothing and was ever averse from publicity. His
+herbarium, which consisted of about 200 folios of mosses, ferns
+and flowering plants &ldquo;almost unique in its completeness,&rdquo; is now
+stored, with many of his fossils, in the museum at Thurso. Dick
+had a hard struggle for existence, especially through competition
+during his late years, when he was reduced almost to beggary:
+but of this few, if any, of his friends were aware until it was
+too late. A monument erected in the new cemetery at Thurso
+testifies to the respect which his life-work created, when the
+merits of this enthusiastic naturalist came to be appreciated.</p>
+
+<div class="condensed">
+<p>See <i>Robert Dick, Baker of Thurso, Geologist and Botanist</i>, by
+Samuel Smiles (1878).</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DICK, THOMAS<a name="ar36" id="ar36"></a></span> (1774-1857), Scottish writer on astronomy,
+was born at Dundee on the 24th of November 1774. The
+appearance of a brilliant meteor inspired him, when in his ninth
+year, with a passion for astronomy; and at the age of sixteen he
+forsook the loom, and supported himself by teaching. In 1794
+he entered the university of Edinburgh, and set up a school on the
+termination of his course; then, in 1801, took out a licence to
+preach, and officiated for some years as probationer in the
+United Presbyterian church. From about 1807 to 1817 he taught
+in the secession school at Methven in Perthshire, and during the
+ensuing decade in that of Perth, where he composed his first
+substantive book, <i>The Christian Philosopher</i> (1823, 8th ed. 1842).
+Its success determined his vocation as an author; he built
+himself, in 1827, a cottage at Broughty Ferry, near Dundee, and
+devoted himself wholly to literary and scientific pursuits. They
+proved, however, owing to his unpractical turn of mind, but
+slightly remunerative, and he was in 1847 relieved from actual
+poverty by a crown pension of £50 a year, eked out by a local
+subscription. He died on the 29th of July 1857. His best-known
+works are: <i>Celestial Scenery</i> (1837), <i>The Sidereal Heavens</i>
+(1840), and <i>The Practical Astronomer</i> (1845), in which is contained
+(p. 204) a remarkable forecast of the powers and uses of
+celestial photography. Written with competent knowledge, and
+in an agreeable style, they obtained deserved and widespread
+popularity.</p>
+
+<div class="condensed">
+<p>See R. Chambers&rsquo;s <i>Eminent Scotsmen</i> (ed. 1868); <i>Monthly Notices
+Roy. Astr. Society</i>, xviii. 98; <i>Athenaeum</i> (1857), p. 1008.</p>
+</div>
+<div class="author">(A. M. C.)</div>
+
+
+<hr class="art" />
+<p><span class="bold">DICKENS, CHARLES JOHN HUFFAM<a name="ar37" id="ar37"></a></span> (1812-1870), English
+novelist, was born on the 7th of February 1812 at a house in
+the Mile End Terrace, Commercial Road, Landport (Portsea)&mdash;a
+house which was opened as a Dickens Museum on 22nd July 1904.
+His father John Dickens (d. 1851), a clerk in the navy-pay office
+<span class="pagenum"><a name="page179" id="page179"></a>179</span>
+on a salary of £80 a year, and stationed for the time being at
+Portsmouth, had married in 1809 Elizabeth, daughter of Thomas
+Barrow, and she bore him a family of eight children, Charles
+being the second. In the winter of 1814 the family moved
+from Portsea in the snow, as he remembered, to London, and
+lodged for a time near the Middlesex hospital. The country
+of the novelist&rsquo;s childhood, however, was the kingdom of Kent,
+where the family was established in proximity to the dockyard
+at Chatham from 1816 to 1821. He looked upon himself in later
+years as a man of Kent, and his capital abode as that in Ordnance
+Terrace, or 18 St Mary&rsquo;s Place, Chatham, amid surroundings
+classified in Mr Pickwick&rsquo;s notes as &ldquo;appearing&rdquo; to be soldiers,
+sailors, Jews, chalk, shrimps, officers and dockyard men. He fell
+into a family the general tendency of which was to go down in
+the world, during one of its easier periods (John Dickens was
+now fifth clerk on £250 a year), and he always regarded himself
+as belonging by right to a comfortable, genteel, lower middle-class
+stratum of society. His mother taught him to read; to his
+father he appeared very early in the light of a young prodigy, and
+by him Charles was made to sit on a tall chair and warble popular
+ballads, or even to tell stories and anecdotes for the benefit of
+fellow-clerks in the office. John Dickens, however, had a small
+collection of books which were kept in a little room upstairs
+that led out of Charles&rsquo;s own, and in this attic the boy found
+his true literary instructors in <i>Roderick Random</i>, <i>Peregrine
+Pickle</i>, <i>Humphry Clinker</i>, <i>Tom Jones</i>, <i>The Vicar of Wakefield</i>,
+<i>Don Quixote</i>, <i>Gil Blas</i> and <i>Robinson Crusoe</i>. The story of how he
+played at the characters in these books and sustained his idea of
+Roderick Random for a month at a stretch is picturesquely told
+in <i>David Copperfield</i>. Here as well as in his first and last books
+and in what many regard as his best, <i>Great Expectations</i>, Dickens
+returns with unabated fondness and mastery to the surroundings
+of his childhood. From seven to nine years he was at a
+school kept in Clover Lane, Chatham, by a Baptist minister
+named William Giles, who gave him Goldsmith&rsquo;s <i>Bee</i> as a keepsake
+when the call to Somerset House necessitated the removal
+of the family from Rochester to a shabby house in Bayham Street,
+Camden Town. At the very moment when a consciousness of
+capacity was beginning to plump his youthful ambitions, the
+whole flattering dream vanished and left not a rack behind.
+Happiness and Chatham had been left behind together, and
+Charles was about to enter a school far sterner and also far
+more instructive than that in Clover Lane. The family income
+had been first decreased and then mortgaged; the creditors of
+the &ldquo;prodigal father&rdquo; would not give him time; John Dickens
+was consigned to the Marshalsea; Mrs Dickens started an
+&ldquo;Educational Establishment&rdquo; as a forlorn hope in Upper Gower
+Street; and Charles, who had helped his mother with the children,
+blacked the boots, carried things to the pawnshop and done
+other menial work, was now sent out to earn his own living as a
+young hand in a blacking warehouse, at Old Hungerford Stairs, on
+a salary of six shillings a week. He tied, trimmed and labelled
+blacking pots for over a year, dining off a saveloy and a slice of
+pudding, consorting with two very rough boys, Bob Fagin and
+Pol Green, and sleeping in an attic in Little College Street,
+Camden Town, in the house of Mrs Roylance (Pipchin), while on
+Sunday he spent the day with his parents in their comfortable
+prison, where they had the services of a &ldquo;marchioness&rdquo; imported
+from the Chatham workhouse.</p>
+
+<p>Already consumed by ambition, proud, sensitive and on his
+dignity to an extent not uncommon among boys of talent, he felt
+his position keenly, and in later years worked himself up into a
+passion of self-pity in connexion with the &ldquo;degradation&rdquo; and
+&ldquo;humiliation&rdquo; of this episode. The two years of childish hardship
+which ate like iron into his soul were obviously of supreme
+importance in the growth of the novelist. Recollections of the
+streets and the prison and its purlieus supplied him with a store
+of literary material upon which he drew through all the years of
+his best activity. And the bitterness of such an experience was
+not prolonged sufficiently to become sour. From 1824 to 1826,
+having been rescued by a family quarrel and by a windfall in the
+shape of a legacy to his father, from the warehouse, he spent two
+years at an academy known as Wellington House, at the corner
+of Granby Street and the Hampstead Road (the lighter traits of
+which are reproduced in Salem House), and was there known as
+a merry and rather mischievous boy. Fortunately he learned
+nothing there to compromise the results of previous instruction.
+His father had now emerged from the Marshalsea and was seeking
+employment as a parliamentary reporter. A Gray&rsquo;s Inn solicitor
+with whom he had had dealings was attracted by the bright,
+clever look of Charles, and took him into his office as a boy at
+a salary of thirteen and sixpence (rising to fifteen shillings) a
+week. He remained in Mr Blackmore&rsquo;s office from May 1827 to
+November 1828, but he had lost none of his eager thirst for distinction,
+and spent all his spare time mastering Gurney&rsquo;s shorthand
+and reading early and late at the British Museum. A more
+industrious apprentice in the lower grades of the literary profession
+has never been known, and the consciousness of opportunities
+used to the most splendid advantage can hardly have been absent
+from the man who was shortly to take his place at the head of it
+as if to the manner born. Lowten and Guppy, and Swiveller
+had been observed from this office lad&rsquo;s stool; he was now
+greatly to widen his area of study as a reporter in Doctors&rsquo;
+Commons and various police courts, including Bow Street,
+working all day at law and much of the night at shorthand. Some
+one asked John Dickens, during the first eager period of curiosity
+as to the man behind &ldquo;Pickwick,&rdquo; where his son Charles was
+educated. &ldquo;Well really,&rdquo; said the prodigal father, &ldquo;he may be
+said&mdash;haw&mdash;haw&mdash;to have educated himself.&rdquo; He was one of
+the most rapid and accurate reporters in London when, at nineteen
+years of age, in 1831, he realized his immediate ambition
+and &ldquo;entered the gallery&rdquo; as parliamentary reporter to the
+<i>True Sun</i>. Later he was reporter to the <i>Mirror of Parliament</i>
+and then to the <i>Morning Chronicle</i>. Several of his earliest letters
+are concerned with his exploits as a reporter, and allude to the
+experiences he had, travelling fifteen miles an hour and being
+upset in almost every description of known vehicle in various parts
+of Britain between 1831 and 1836. The family was now living in
+Bentwick Street, Manchester Square, but John Dickens was
+still no infrequent inmate of the sponging-houses. With all the
+accessories of these places of entertainment his son had grown to
+be excessively familiar. Writing about 1832 to his school friend
+Tom Mitton, Dickens tells him that his father has been arrested
+at the suit of a wine firm, and begs him go over to Cursitor Street
+and see what can be done. On another occasion of a paternal
+disappearance he observes: &ldquo;I own that his absence does not
+give me any great uneasiness, knowing how apt he is to get out
+of the way when anything goes wrong.&rdquo; In yet another letter
+he asks for a loan of four shillings.</p>
+
+<p>In the meanwhile, however, he had commenced author in a
+more creative sense by penning some sketches of contemporary
+London life, such as he had attempted in his school days in imitation
+of the sketches published in the <i>London</i> and other magazines
+of that day. The first of these appeared in the December number
+of the <i>Old Monthly Magazine</i> for 1833. By the following August,
+when the signature &ldquo;Boz&rdquo; was first given, five of these sketches
+had appeared. By the end of 1834 we find him settled in rooms
+in Furnival&rsquo;s Inn, and a little later his salary on the <i>Morning
+Chronicle</i> was raised, owing to the intervention of one of its chiefs,
+George Hogarth, the father of (in addition to six sons) eight
+charming daughters, to one of whom, Catherine, Charles was
+engaged to be married before the year was out. Clearly as his
+career now seemed designated, he was at this time or a little before
+it coquetting very seriously with the stage: but circumstances
+were rapidly to determine another stage in his career. A year
+before Queen Victoria&rsquo;s accession appeared in two volumes
+<i>Sketches by Boz</i>, <i>Illustrative of Everyday Life and Everyday
+People</i>. The book came from a prentice hand, but like the
+little tract on the Puritan abuse of the Sabbath entitled &ldquo;Sunday
+under three Heads&rdquo; which appeared a few months later, it
+contains in germ all, or almost all, the future Dickens. Glance
+at the headings of the pages. Here we have the Beadle and all
+connected with him, London streets, theatres, shows, the pawnshop,
+Doctors&rsquo; Commons, Christmas, Newgate, coaching, the
+<span class="pagenum"><a name="page180" id="page180"></a>180</span>
+river. Here comes a satirical picture of parliament, fun made of
+cheap snobbery, a rap on the knuckles of sectarianism. And what
+could be more prophetic than the title of the opening chapter&mdash;Our
+Parish? With the Parish&mdash;a large one indeed&mdash;Dickens
+to the end concerned himself; he began with a rapid survey of
+his whole field, hinting at all he might accomplish, indicating
+the limits he was not to pass. This year was to be still more
+momentous to Dickens, for, on the 2nd of April 1836, he was
+married to George Hogarth&rsquo;s eldest daughter Catherine. He
+seems to have fallen in love with the daughters collectively,
+and, judging by subsequent events, it has been suggested that
+perhaps he married the wrong one. His wife&rsquo;s sister Mary was
+the romance of his early married life, and another sister, Georgina,
+was the dearest friend of his last ten years.</p>
+
+<p>A few days before the marriage, just two months after the
+appearance of the <i>Sketches</i>, the first part of <i>The Posthumous Papers
+of the Pickwick Club</i> was announced. One of the chief vogues of
+the day was the issue of humorous, sporting or anecdotal novels
+in parts, with plates, and some of the best talent of the day, represented
+by Ainsworth, Bulwer, Marryat, Maxwell, Egan, Hook
+and Surtees, had been pressed into this kind of enterprise. The
+publishers of the day had not been slow to perceive Dickens&rsquo;s
+aptitude for this species of &ldquo;letterpress.&rdquo; A member of the
+firm of Chapman &amp; Hall called upon him at Furnival&rsquo;s Inn in
+December 1835 with a proposal that he should write about a
+Nimrod Club of amateur sportsmen, foredoomed to perpetual
+ignominies, while the comic illustrations were to be etched by
+Seymour, a well-known rival of Cruikshank (the illustrator of
+<i>Boz</i>). The offer was too tempting for Dickens to refuse, but he
+changed the idea from a club of Cockney sportsmen to that of a
+club of eccentric peripatetics, on the sensible grounds, first that
+sporting sketches were stale, and, secondly, that he knew nothing
+worth speaking of about sport. The first seven pictures appeared
+with the signature of Seymour and the letterpress of Dickens.
+Before the eighth picture appeared Seymour had blown his brains
+out. After a brief interval of Buss, Dickens obtained the services
+of Hablot K. Browne, known to all as &ldquo;Phiz.&rdquo; Author and
+illustrator were as well suited to one another and to the common
+creation of a unique thing as Gilbert and Sullivan. Having early
+got rid of the sporting element, Dickens found himself at once.
+The subject exactly suited his knowledge, his skill in arranging
+incidents&mdash;nay, his very limitations too. No modern book is
+so incalculable. We commence laughing heartily at Pickwick
+and his troupe. The laugh becomes kindlier. We are led on
+through a tangle of adventure, never dreaming what is before us.
+The landscape changes: Pickwick becomes the symbol of kindheartedness,
+simplicity and innocent levity. Suddenly in the Fleet
+Prison a deeper note is struck. The medley of human relationships,
+the loneliness, the mystery and sadness of human destinies
+are fathomed. The tragedy of human life is revealed to us amid
+its most farcical elements. The droll and laughable figure of the
+hero is transfigured by the kindliness of human sympathy into
+a beneficent and bespectacled angel in shorts and gaiters. By
+defying accepted rules, Dickens had transcended the limited
+sphere hitherto allotted to his art: he had produced a book to
+be enshrined henceforth in the inmost hearts of all sorts and
+conditions of his countrymen, and had definitely enlarged the
+boundaries of English humour and English fiction. As for Mr
+Pickwick, he is a fairy like Puck or Santa Claus, while his creator
+is &ldquo;the last of the mythologists and perhaps the greatest.&rdquo;</p>
+
+<p>When <i>The Pickwick Papers</i> appeared in book form at the close
+of 1837 Dickens&rsquo;s popular reputation was made. From the
+appearance of Sam Weller in part v. the universal hunger for the
+monthly parts had risen to a furore. The book was promptly
+translated into French and German. The author had received
+little assistance from press or critics, he had no influential connexions,
+his class of subjects was such as to &ldquo;expose him at the
+outset to the fatal objections of vulgarity,&rdquo; yet in less than six
+months from the appearance of the first number, as the <i>Quarterly
+Review</i> almost ruefully admits, the whole reading world was
+talking about the Pickwickians. The names of Winkle, Wardle,
+Weller, Jingle, Snodgrass, Dodson &amp; Fogg, were as familiar as
+household words. Pickwick chintzes figured in the linendrapers&rsquo;
+windows, and Pickwick cigars in every tobacconist&rsquo;s; Weller
+corduroys became the stock-in-trade of every breeches-maker;
+Boz cabs might be seen rattling through the streets, and the
+portrait of the author of <i>Pelham</i> and <i>Crichton</i> was scraped down
+to make way for that of the new popular favourite on the omnibuses.
+A new and original genius had suddenly sprung up, there
+was no denying it, even though, as the <i>Quarterly</i> concluded, &ldquo;it
+required no gift of prophecy to foretell his fate&mdash;he has risen like
+a rocket and he will come down like the stick.&rdquo; It would have
+needed a very emphatic gift of prophecy indeed to foretell that
+Dickens&rsquo;s reputation would have gone on rising until at the
+present day (after one sharp fall, which reached an extreme
+about 1887) it stands higher than it has ever stood before.</p>
+
+<p>Dickens&rsquo;s assumption of the literary purple was as amazing as
+anything else about him. Accepting the homage of the luminaries
+of the literary, artistic and polite worlds as if it had been his
+natural due, he arranges for the settlement of his family, decrees,
+like another Edmund Kean, that his son is to go to Eton, carries
+on the most complicated negotiations with his publishers and
+editors, presides and orates with incomparable force at innumerable
+banquets, public and private, arranges elaborate villegiatures
+in the country, at the seaside, in France or in Italy, arbitrates in
+public on every topic, political, ethical, artistic, social or literary,
+entertains and legislates for an increasingly large domestic circle,
+both juvenile and adult, rules himself and his time-table with
+a rod of iron. In his letter-writing alone, Dickens did a life&rsquo;s
+literary work. Nowadays no one thinks of writing such letters;
+that is to say, letters of such length and detail, for the quality is
+Dickens&rsquo;s own. He evidently enjoyed this use of the pen. Page
+after page of Forster&rsquo;s <i>Life</i> (750 pages in the <i>Letters</i> edited by
+his daughter and sister-in-law) is occupied with transcription from
+private correspondence, and never a line of this but is thoroughly
+worthy of print and preservation. If he makes a tour in any
+part of the British Isles, he writes a full description of all he
+sees, of everything that happens, and writes it with such gusto,
+such mirth, such strokes of fine picturing, as appear in no other
+private letters ever given to the public. Naturally buoyant in
+all circumstances, a holiday gave him the exhilaration of a schoolboy.
+See how he writes from Cornwall, when on a trip with two
+or three friends, in 1843. &ldquo;Heavens! if you could have seen the
+necks of bottles, distracting in their immense variety of shape,
+peering out of the carriage pockets! If you could have witnessed
+the deep devotion of the post-boys, the maniac glee of the waiters!
+If you could have followed us into the earthy old churches we
+visited, and into the strange caverns on the gloomy seashore, and
+down into the depths of mines, and up to the tops of giddy heights,
+where the unspeakably green water was roaring, I don&rsquo;t know
+how many hundred feet below.... I never laughed in my life
+as I did on this journey. It would have done you good to hear
+me. I was choking and gasping and bursting the buckles off the
+back of my stock, all the way. And Stanfield&rdquo;&mdash;the painter&mdash;&ldquo;got
+into such apoplectic entanglements that we were obliged
+to beat him on the back with portmanteaus before we could
+recover him.&rdquo;</p>
+
+<p>The animation of Dickens&rsquo;s look would attract the attention
+of any one, anywhere. His figure was not that of an Adonis, but
+his brightness made him the centre and pivot of every society
+he was in. The keenness and vivacity of his eye combined with
+his inordinate appetite for life to give the unique quality to all
+that he wrote. His instrument is that of the direct, sinewy
+English of Smollett, combined with much of the humorous grace
+of Goldsmith (his two favourite authors), but modernized to a
+certain extent under the influence of Washington Irving, Sydney
+Smith, Jeffrey, Lamb, and other writers of the <i>London Magazine</i>.
+He taught himself to speak French and Italian, but he could have
+read little in any language. His ideas were those of the inchoate
+and insular liberalism of the &rsquo;thirties. His unique force in
+literature he was to owe to no supreme artistic or intellectual
+quality, but almost entirely to his inordinate gift of observation,
+his sympathy with the humble, his power over the emotions
+and his incomparable endowment of unalloyed human fun. To
+<span class="pagenum"><a name="page181" id="page181"></a>181</span>
+contemporaries he was not so much a man as an institution, at
+the very mention of whose name faces were puckered with grins
+or wreathed in smiles. To many his work was a revelation, the
+revelation of a new world and one far better than their own.
+And his influence went further than this in the direction of
+revolution or revival. It gave what were then universally referred
+to as &ldquo;the lower orders&rdquo; a new sense of self-respect, a new
+feeling of citizenship. Like the defiance of another Luther, or the
+Declaration of a new Independence, it emitted a fresh ray of hope
+across the firmament. He did for the whole English-speaking
+race what Burns had done for Scotland&mdash;he gave it a new
+conceit of itself. He knew what a people wanted and he told
+what he knew. He could do this better than anybody else
+because his mind was theirs. He shared many of their &ldquo;great
+useless virtues,&rdquo; among which generosity ranks before justice, and
+sympathy before truth, even though, true to his middle-class vein,
+he exalts piety, chastity and honesty in a manner somewhat alien
+to the mind of the low-bred man. This is what makes Dickens
+such a demigod and his public success such a marvel, and this
+also is why any exclusively literary criticism of his work is bound
+to be so inadequate. It should also help us to make the necessary
+allowances for the man. Dickens, even the Dickens of legend
+that we know, is far from perfect. The Dickens of reality to
+which Time may furnish a nearer approximation is far less
+perfect. But when we consider the corroding influence of adulation,
+and the intoxication of unbridled success, we cannot but
+wonder at the relatively high level of moderation and self-control
+that Dickens almost invariably observed. Mr G. K. Chesterton
+remarks suggestively that Dickens had all his life the faults of
+the little boy who is kept up too late at night. He is overwrought
+by happiness to the verge of exasperation, and yet as a matter
+of fact he does keep on the right side of the breaking point. The
+specific and curative in his case was the work in which he took
+such anxious pride, and such unmitigated delight. He revelled
+in punctual and regular work; at his desk he was often in the
+highest spirits. Behold how he pictured himself, one day at
+Broadstairs, where he was writing <i>Chuzzlewit</i>. &ldquo;In a bay-window
+in a one-pair sits, from nine o&rsquo;clock to one, a gentleman
+with rather long hair and no neckcloth, who writes and grins, as
+if he thought he was very funny indeed. At one he disappears,
+presently emerges from a bathing-machine, and may be seen,
+a kind of salmon-colour porpoise, splashing about in the ocean.
+After that, he may be viewed in another bay-window on the
+ground-floor eating a strong lunch; and after that, walking a
+dozen miles or so, or lying on his back on the sand reading a book.
+Nobody bothers him, unless they know he is disposed to be
+talked to, and I am told he is very comfortable indeed. He&rsquo;s as
+brown as a berry, and they do say he is as good as a small fortune
+to the innkeeper, who sells beer and cold punch.&rdquo; Here is the
+secret of such work as that of Dickens; it is done with delight&mdash;done
+(in a sense) easily, done with the mechanism of mind and
+body in splendid order. Even so did Scott write; though more
+rapidly and with less conscious care: his chapter finished before
+the world had got up to breakfast. Later, Dickens produced
+novels less excellent with much more of mental strain. The
+effects of age could not have shown themselves so soon, but
+for the unfortunate loss of energy involved in his non-literary
+labours.</p>
+
+<p>While the public were still rejoicing in the first sprightly
+runnings of the &ldquo;new humour,&rdquo; the humorist set to work
+desperately on the grim scenes of <i>Oliver Twist</i>, the story of a
+parish orphan, the nucleus of which had already seen the light
+in his <i>Sketches</i>. The early scenes are of a harrowing reality,
+despite the germ of forced pathos which the observant reader may
+detect in the pitiful parting between Oliver and little Dick; but
+what will strike every reader at once in this book is the directness
+and power of the English style, so nervous and unadorned:
+from its unmistakable clearness and vigour Dickens was to travel
+far as time went on. But the full effect of the old simplicity is
+felt in such masterpieces of description as the drive of Oliver and
+Sikes to Chertsey, the condemned-cell ecstasy of Fagin, or the
+unforgettable first encounter between Oliver and the Artful
+Dodger. Before November 1837 had ended, Charles Dickens
+entered on an engagement to write a successor to <i>Pickwick</i> on
+similar lines of publication. <i>Oliver Twist</i> was then in mid-career;
+a <i>Life of Grimaldi</i> and <i>Barnaby Rudge</i> were already covenanted
+for. Dickens forged ahead with the new tale of <i>Nicholas Nickleby</i>
+and was justified by the results, for its sale far surpassed even
+that of <i>Pickwick</i>. As a conception it is one of his weakest. An
+unmistakably 18th-century character pervades it. Some of the
+vignettes are among the most piquant and besetting ever written.
+Large parts of it are totally unobserved conventional melodrama;
+but the Portsmouth Theatre and Dotheboys Hall and
+Mrs Nickleby (based to some extent, it is thought, upon Miss
+Bates in <i>Emma</i>, but also upon the author&rsquo;s Mamma) live for ever
+as Dickens conceived them in the pages of <i>Nicholas Nickleby</i>.</p>
+
+<p>Having got rid of <i>Nicholas Nickleby</i> and resigned his editorship
+of <i>Bentley&rsquo;s Miscellany</i>, in which <i>Oliver Twist</i> originally
+appeared, Dickens conceived the idea of a weekly periodical to
+be issued as <i>Master Humphrey&rsquo;s Clock</i>, to comprise short stories,
+essays and miscellaneous papers, after the model of Addison&rsquo;s
+<i>Spectator</i>. To make the weekly numbers &ldquo;go,&rdquo; he introduced
+Mr Pickwick, Sam Weller and his father in friendly intercourse.
+But the public requisitioned &ldquo;a story,&rdquo; and in No. 4 he had
+to brace himself up to give them one. Thus was commenced
+<i>The Old Curiosity Shop</i>, which was continued with slight interruptions,
+and followed by <i>Barnaby Rudge</i>. For the first time
+we find Dickens obsessed by a highly complicated plot. The
+tonality achieved in <i>The Old Curiosity Shop</i> surpassed anything
+he had attempted in this difficult vein, while the rich humour of
+Dick Swiveller and the Marchioness, and the vivid portraiture
+of the wandering Bohemians, attain the very highest level of
+Dickensian drollery; but in the lamentable tale of Little Nell
+(though Landor and Jeffrey thought the character-drawing of
+this infant comparable with that of Cordelia), it is generally
+admitted that he committed an indecent assault upon the
+emotions by exhibiting a veritable monster of piety and long-suffering
+in a child of tender years. In <i>Barnaby Rudge</i> he was
+manifestly affected by the influence of Scott, whose achievements
+he always regarded with a touching veneration. The plot, again,
+is of the utmost complexity, and Edgar Allan Poe (who predicted
+the conclusion) must be one of the few persons who ever really
+mastered it. But few of Dickens&rsquo;s books are written in a more
+admirable style.</p>
+
+<p><i>Master Humphrey&rsquo;s Clock</i> concluded, Dickens started in 1842
+on his first visit to America&mdash;an episode hitherto without parallel
+in English literary history, for he was received everywhere with
+popular acclamation as the representative of a grand triumph
+of the English language and imagination, without regard to
+distinctions of nationality. He offended the American public
+grievously by a few words of frank description and a few
+quotations of the advertisement columns of American papers
+illustrating the essential barbarity of the old slave system
+(<i>American Notes</i>). Dickens was soon pining for home&mdash;no English
+writer is more essentially and insularly English in inspiration
+and aspiration than he is. He still brooded over the perverseness
+of America on the copyright question, and in his next book he
+took the opportunity of uttering a few of his impressions about
+the objectionable sides of American democracy, the result being
+that &ldquo;all Yankee-doodle-dom blazed up like one universal soda
+bottle,&rdquo; as Carlyle said. <i>Martin Chuzzlewit</i> (1843-1844) is important
+as closing his great character period. His <i>sève originale</i>, as the
+French would say, was by this time to a considerable extent
+exhausted, and he had to depend more upon artistic elaboration,
+upon satires, upon <i>tours de force</i> of description, upon romantic
+and ingenious contrivances. But all these resources combined
+proved unequal to his powers as an original observer of popular
+types, until he reinforced himself by autobiographic reminiscence,
+as in <i>David Copperfield</i> and <i>Great Expectations</i>, the two great
+books remaining to his later career.</p>
+
+<p>After these two masterpieces and the three wonderful books
+with which he made his début, we are inclined to rank <i>Chuzzlewit</i>.
+Nothing in Dickens is more admirably seen and presented than
+Todgers&rsquo;s, a bit of London particular cut out with a knife. Mr
+<span class="pagenum"><a name="page182" id="page182"></a>182</span>
+Pecksniff and Mrs Gamp, Betsy Prig and &ldquo;Mrs Harris&rdquo; have
+passed into the national language and life. The coach journey,
+the windy autumn night, the stealthy trail of Jonas, the undertone
+of tragedy in the Charity and Mercy and Chuffey episodes
+suggest a blending of imaginative vision and physical penetration
+hardly seen elsewhere. Two things are specially notable about
+this novel&mdash;the exceptional care taken over it (as shown by the
+interlineations in the MS.) and the caprice or nonchalance of
+the purchasing public, its sales being far lower than those of
+any of its monthly predecessors.</p>
+
+<p>At the close of 1843, to pay outstanding debts of his now
+lavish housekeeping, he wrote that pioneer of Christmas numbers,
+that national benefit as Thackeray called it, <i>A Christmas Carol</i>.
+It failed to realize his pecuniary anticipations, and Dickens
+resolved upon a drastic policy of retrenchment and reform.
+He would save expense by living abroad and would punish his
+publishers by withdrawing his custom from them, at least for a
+time. Like everything else upon which he ever determined, this
+resolution was carried out with the greatest possible precision and
+despatch. In June 1844 he set out for Marseilles with his now
+rapidly increasing family (the journey cost him £200). In a villa
+on the outskirts of Genoa he wrote <i>The Chimes</i>, which, during a
+brief excursion to London before Christmas, he read to a select
+circle of friends (the germ of his subsequent lecture-audiences),
+including Forster, Carlyle, Stanfield, Dyce, Maclise and Jerrold.
+He was again in London in 1845, enjoying his favourite diversion
+of private theatricals; and in January 1846 he experimented
+briefly as the editor of a London morning paper&mdash;the <i>Daily
+News</i>. By early spring he was back at Lausanne, writing his
+customary vivid letters to his friends, craving as usual for
+London streets, commencing <i>Dombey and Son</i>, and walking his
+fourteen miles daily. The success of <i>Dombey and Son</i> completely
+rehabilitated the master&rsquo;s finances, enabled him to return to
+England, send his son to Eton and to begin to save money.
+Artistically it is less satisfactory; it contains some of Dickens&rsquo;s
+prime curios, such as Cuttle, Bunsby, Toots, Blimber, Pipchin,
+Mrs MacStinger and young Biler; it contains also that masterpiece
+of sentimentality which trembles upon the borderland
+of the sublime and the ridiculous, the death of Paul Dombey
+(&ldquo;that sweet Paul,&rdquo; as Jeffrey, the &ldquo;critic laureate,&rdquo; called him),
+and some grievous and unquestionable blemishes. As a narrative,
+moreover, it tails off into a highly complicated and exacting plot.
+It was followed by a long rest at Broadstairs before Dickens
+returned to the native home of his genius, and early in 1849
+&ldquo;began to prepare for <i>David Copperfield</i>.&rdquo;</p>
+
+<p>&ldquo;Of all my books,&rdquo; Dickens wrote, &ldquo;I like this the best; like
+many fond parents I have my favourite child, and his name is
+David Copperfield.&rdquo; In some respects it stands to Dickens in
+something of the same relation in which the contemporary
+<i>Pendennis</i> stands to Thackeray. As in that book, too, the earlier
+portions are the best. They gained in intensity by the autobiographical
+form into which they are thrown; as Thackeray
+observed, there was no writing against such power. The tragedy
+of Emily and the character of Rosa Dartle are stagey and unreal;
+Uriah Heep is bad art; Agnes, again, is far less convincing
+as a consolation than Dickens would have us believe; but these
+are more than compensated by the wonderful realization
+of early boyhood in the book, by the picture of Mr Creakle&rsquo;s
+school, the Peggottys, the inimitable Mr Micawber, Betsy Trotwood
+and that monument of selfish misery, Mrs Gummidge.</p>
+
+<p>At the end of March 1850 commenced the new twopenny
+weekly called <i>Household Words</i>, which Dickens planned to form
+a direct means of communication between himself and his
+readers, and as a means of collecting around him and encouraging
+the talents of the younger generation. No one was better qualified
+than he for this work, whether we consider his complete
+freedom from literary jealousy or his magical gift of inspiring
+young authors. Following the somewhat dreary and incoherent
+<i>Bleak House</i> of 1852, <i>Hard Times</i> (1854)&mdash;an anti-Manchester
+School tract, which Ruskin regarded as Dickens&rsquo;s best work&mdash;was
+the first long story written for <i>Household Words</i>. About this
+time Dickens made his final home at Gad&rsquo;s Hill, near Rochester,
+and put the finishing touch to another long novel published upon
+the old plan, <i>Little Dorrit</i> (1855-1857). In spite of the exquisite
+comedy of the master of the Marshalsea and the final tragedy
+of the central figure, <i>Little Dorrit</i> is sadly deficient in the old
+vitality, the humour is often a mock reality, and the repetition
+of comic catch-words and overstrung similes and metaphors is
+such as to affect the reader with nervous irritation. The plot
+and characters ruin each other in this amorphous production.
+The <i>Tale of Two Cities</i>, commenced in <i>All the Year Round</i> (the
+successor of <i>Household Words</i>) in 1859, is much better: the main
+characters are powerful, the story genuinely tragic, and the
+atmosphere lurid; but enormous labour was everywhere expended
+upon the construction of stylistic ornament.</p>
+
+<p>The <i>Tale of Two Cities</i> was followed by two finer efforts at
+atmospheric delineation, the best things he ever did of this kind:
+<i>Great Expectations</i> (1861), over which there broods the mournful
+impression of the foggy marshes of the Lower Thames; and <i>Our
+Mutual Friend</i> (1864-1865), in which the ooze and mud and
+slime of Rotherhithe, its boatmen and loafers, are made to pervade
+the whole book with cumulative effect. The general effect
+produced by the stories is, however, very different. In the first
+case, the foreground was supplied by autobiographical material
+of the most vivid interest, and the lucidity of the creative impulse
+impelled him to write upon this occasion with the old simplicity,
+though with an added power. Nothing therefore, in the whole
+range of Dickens surpassed the early chapters of <i>Great Expectations</i>
+in perfection of technique or in mastery of all the resources
+of the novelist&rsquo;s art. To have created Abel Magwitch alone is to
+be a god indeed, says Mr Swinburne, among the creators of deathless
+men. Pumblechook is actually better and droller and truer
+to imaginative life than Pecksniff; Joe Gargery is worthy to have
+been praised and loved at once by Fielding and by Sterne: Mr
+Jaggers and his clients, Mr Wemmick and his parent and his
+bride, are such figures as Shakespeare, when dropping out of
+poetry, might have created, if his lot had been cast in a later
+century. &ldquo;Can as much be said,&rdquo; Mr Swinburne boldly asks,
+&ldquo;for the creatures of any other man or god?&rdquo;</p>
+
+<p>In November 1867 Dickens made a second expedition to
+America, leaving all the writing that he was ever to complete behind
+him. He was to make a round sum of money, enough to free
+him from all embarrassments, by a long series of exhausting readings,
+commencing at the Tremont Temple, Boston, on the 2nd of
+December. The strain of Dickens&rsquo;s ordinary life was so tense and
+so continuous that it is, perhaps, rash to assume that he broke
+down eventually under this particular stress; for other reasons,
+however, his persistence in these readings, subsequent to his
+return, was strongly deprecated by his literary friends, led by
+the arbitrary and relentless Forster. It is a long testimony to
+Dickens&rsquo;s self-restraint, even in his most capricious and despotic
+moments, that he never broke the cord of obligation which bound
+him to his literary mentor, though sparring matches between them
+were latterly of frequent occurrence. His farewell reading was
+given on the 15th of March 1870, at St James&rsquo;s Hall. He then
+vanished from &ldquo;those garish lights,&rdquo; as he called them, &ldquo;for
+evermore.&rdquo; Of the three brief months that remained to him,
+his last book, <i>The Mystery of Edwin Drood</i>, was the chief occupation.
+It hardly promised to become a masterpiece (Longfellow&rsquo;s
+opinion) as did Thackeray&rsquo;s <i>Denis Duval</i>, but contained much fine
+descriptive technique, grouped round a scene of which Dickens
+had an unrivalled sympathetic knowledge.</p>
+
+<p>In March and April 1870 Dickens, as was his wont, was mixing
+in the best society; he dined with the prince at Lord Houghton&rsquo;s
+and was twice at court, once at a long deferred private interview
+with the queen, who had given him a presentation copy of her
+<i>Leaves from a Journal of our Life in the Highlands</i> with the
+inscription &ldquo;From one of the humblest of authors to one of the
+greatest&rdquo;; and who now begged him on his persistent refusal
+of any other title to accept the nominal distinction of a privy
+councillor. He took for four months the Milner Gibsons&rsquo; house
+at 5 Hyde Park Place, opposite the Marble Arch, where he gave
+a brilliant reception on the 7th of April. His last public appearance
+was made at the Royal Academy banquet early in May.
+<span class="pagenum"><a name="page183" id="page183"></a>183</span>
+He returned to his regular methodical routine of work at Gad&rsquo;s
+Hill on the 30th of May, and one of the last instalments he wrote
+of <i>Edwin Drood</i> contained an ominous speculation as to the next
+two people to die at Cloisterham: &ldquo;Curious to make a guess at
+the two, or say at one of the two.&rdquo; Two letters bearing the well-known
+superscription &ldquo;Gad&rsquo;s Hill Place, Higham by Rochester,
+Kent&rdquo; are dated the 8th of June, and, on the same Thursday, after
+a long spell of writing in the Châlet where he habitually wrote,
+he collapsed suddenly at dinner. Startled by the sudden change
+in the colour and expression of his face, his sister-in-law (Miss
+Hogarth) asked him if he was ill; he said &ldquo;Yes, very ill,&rdquo; but
+added that he would finish dinner and go on afterwards to London.
+&ldquo;Come and lie down,&rdquo; she entreated; &ldquo;Yes, on the ground,&rdquo;
+he said, very distinctly; these were the last words he spoke, and
+he slid from her arms and fell upon the floor. He died at 6-10 P.M.
+on Friday, the 9th of June, and was buried privately in Poets&rsquo;
+Corner, Westminster Abbey, in the early morning of the 14th of
+June. One of the most appealing memorials was the drawing
+by his &ldquo;new illustrator&rdquo; Luke Fildes in the <i>Graphic</i> of &ldquo;The
+Empty Chair; Gad&rsquo;s Hill: ninth of June, 1870.&rdquo; &ldquo;Statesmen,
+men of science, philanthropists, the acknowledged benefactors of
+their race, might pass away, and yet not leave the void which will
+be caused by the death of Charles Dickens&rdquo; (<i>The Times</i>). In
+his will he enjoined his friends to erect no monument in his
+honour, and directed his name and dates only to be inscribed on
+his tomb, adding this proud provision, &ldquo;I rest my claim to
+the remembrance of my country on my published works.&rdquo;</p>
+
+<p>Dickens had no artistic ideals worth speaking about. The
+sympathy of his readers was the one thing he cared about and,
+like Cobbett, he went straight for it through the avenue of the
+emotions. In personality, intensity and range of creative genius
+he can hardly be said to have any modern rival. His creations
+live, move and have their being about us constantly, like those
+of Homer, Virgil, Chaucer, Rabelais, Cervantes, Shakespeare,
+Bunyan, Molière and Sir Walter Scott. As to the books themselves,
+the backgrounds on which these mighty figures are projected,
+they are manifestly too vast, too chaotic and too unequal
+ever to become classics. Like most of the novels constructed upon
+the unreformed model of Smollett and Fielding, those of Dickens
+are enormous stock-pots into which the author casts every kind
+of autobiographical experience, emotion, pleasantry, anecdote,
+adage or apophthegm. The fusion is necessarily very incomplete
+and the hotch-potch is bound to fall to pieces with time.
+Dickens&rsquo;s plots, it must be admitted, are strangely unintelligible,
+the repetitions and stylistic decorations of his work exceed
+all bounds, the form is unmanageable and insignificant. The
+diffuseness of the English novel, in short, and its extravagant
+didacticism cannot fail to be most prejudicial to its perpetuation.
+In these circumstances there is very little fiction that will stand
+concentration and condensation so well as that of Dickens.</p>
+
+<p>For these reasons among others our interest in Dickens&rsquo;s novels
+as integers has diminished and is diminishing. But, on the other
+hand, our interest and pride in him as a man and as a representative
+author of his age and nation has been steadily augmented
+and is still mounting. Much of the old criticism of his work, that
+it was not up to a sufficiently high level of art, scholarship or
+gentility, that as an author he is given to caricature, redundancy
+and a shameless subservience to popular caprice, must now be
+discarded as irrelevant.</p>
+
+<p>As regards formal excellence it is plain that Dickens labours
+under the double disadvantage of writing in the least disciplined
+of all literary genres in the most lawless literary milieu of the
+modern world, that of Victorian England. In spite of these
+defects, which are those of masters such as Rabelais, Hugo and
+Tolstoy, the work of Dickens is more and more instinctively felt
+to be true, original and ennobling. It is already beginning to
+undergo a process of automatic sifting, segregation and crystallization,
+at the conclusion of which it will probably occupy a larger
+segment in the literary consciousness of the English-spoken race
+than ever before.</p>
+
+<p>Portraits of Dickens, from the gay and alert &ldquo;Boz&rdquo; of Samuel
+Lawrence, and the self-conscious, rather foppish portrait by
+Maclise which served as frontispiece to <i>Nicholas Nickleby</i>, to
+the sketch of him as Bobadil by C. R. Leslie, the Drummond and
+Ary Scheffer portraits of middle age and the haggard and drawn
+representations of him from photographs after his shattering
+experiences as a public entertainer from 1856 (the year of his
+separation from his wife) onwards, are reproduced in Kitton, in
+Forster and Gissing and in the other biographies. Sketches are
+also given in most of the books of his successive dwelling places
+at Ordnance Terrace and 18 St Mary&rsquo;s Place, Chatham; Bayham
+Street, Camden Town; 15 Furnival&rsquo;s Inn; 48 Doughty Street;
+1 Devonshire Terrace, Regent&rsquo;s Park; Tavistock House,
+Tavistock Square; and Gad&rsquo;s Hill Place. The manuscripts of all
+the novels, with the exception of the <i>Tale of Two Cities</i> and
+<i>Edwin Drood</i>, were given to Forster, and are now preserved in the
+Dyce and Forster Museum at South Kensington. The work of
+Dickens was a prize for which publishers naturally contended both
+before and after his death. The first collective edition of his
+works was begun in April 1847, and their number is now very
+great. The most complete is still that of Messrs Chapman &amp;
+Hall, the original publishers of <i>Pickwick</i>; others of special
+interest are the Harrap edition, originally edited by F. G. Kitton;
+Macmillan&rsquo;s edition with original illustrations and introduction
+by Charles Dickens the younger; and the edition in the World&rsquo;s
+Classics with introductions by G.K. Chesterton. Of the translations
+the best known is that done into French by Lorain, Pichot
+and others, with B.H. Gausseron&rsquo;s excellent <i>Pages Choisies</i> (1903).</p>
+
+<div class="condensed">
+<p><span class="sc">Bibliography.</span>&mdash;During his lifetime Dickens&rsquo;s biographer was
+clearly indicated in his guide, philosopher and friend, John Forster,
+who had known the novelist intimately since the days of his first
+triumph with <i>Pickwick</i>, who had constituted himself a veritable
+encyclopaedia of information about Dickens, and had clung to his
+subject (in spite of many rebuffs which his peremptory temper found
+it hard to digest) as tightly as ever Boswell had enveloped Johnson.
+Two volumes of Forster&rsquo;s <i>Life of Charles Dickens</i> appeared in 1872
+and a third in 1874. He relied much on Dickens&rsquo;s letters to himself
+and produced what must always remain the authoritative work.
+The first two volumes are put together with much art, the portrait
+as a whole has been regarded as truthful, and the immediate success
+was extraordinary. In the opinion of Carlyle, Forster&rsquo;s book was not
+unworthy to be named after that of Boswell. A useful abridgment
+was carried out in 1903 by the novelist George Gissing. Gissing also
+wrote <i>Charles Dickens: A Critical Study</i> (1898), which ranks with
+G.K. Chesterton&rsquo;s <i>Charles Dickens</i>(1906) as a commentary inspired by
+deep insight and adorned by great literary talent upon the genius of
+the master-novelist. The names of other lives, sketches, articles and
+estimates of Dickens and his works would occupy a large volume in
+the mere enumeration. See R.H. Shepherd, <i>The Bibliography of
+Dickens</i> (1880); <i>James Cooke&rsquo;s Bibliography of the Writings of Charles
+Dickens</i> (1879); <i>Dickensiana</i>, by F. G. Kitton (1886); and <i>Bibliography</i>
+by J.P. Anderson, appended to Sir F.T. Marzials&rsquo;s <i>Life of
+Charles Dickens</i> (1887). Among the earlier sketches may be specially
+cited the lives by J. C. Hotten and G. A. Sala (1870), the Anecdote-Biography
+edited by the American R. H. Stoddard (1874), Dr A. W.
+Ward in the English Men of Letters Series (1878), that by Sir Leslie
+Stephen in the <i>Dictionary of National Biography</i>, and that by Professor
+Minto in the eighth edition of the <i>Encyclopaedia Britannica</i>.
+The <i>Letters</i> were first issued in two volumes edited by his daughter
+and sister-in-law in 1880. For Dickens&rsquo;s connexion with Kent the
+following books are specially valuable:&mdash;Robert Langton&rsquo;s <i>Childhood
+and Youth of Charles Dickens</i> (1883); Langton&rsquo;s <i>Dickens and
+Rochester</i> (1880); Thomas Frost&rsquo;s <i>In Kent with Charles Dickens</i>
+(1880); F. G. Kitton&rsquo;s <i>The Dickens Country</i> (1905); H. S. Ward&rsquo;s
+<i>The Real Dickens Land</i> (1904); R. Allbut&rsquo;s <i>Rambles in Dickens Land</i>
+(1899 and 1903). For Dickens&rsquo;s reading tours see G. Dolby&rsquo;s
+<i>Charles Dickens as I knew him</i> (1884); J. T. Fields&rsquo;s <i>In and Out of
+Doors with Charles Dickens</i> (1876); Charles Kent&rsquo;s <i>Dickens as a
+Reader</i> (1872). And for other aspects of his life see M. Dickens&rsquo;s <i>My
+Father as I recall him</i> (1897); P. H. Fitzgerald&rsquo;s <i>Life of C. Dickens as
+revealed in his Writings</i> (1905), and <i>Bozland</i> (1895); F. G. Kitton&rsquo;s
+<i>Charles Dickens, his Life, Writings and Personality</i>, a useful compendium
+(1902); T. E. Pemberton&rsquo;s <i>Charles Dickens and the Stage</i>, and
+<i>Dickens&rsquo;s London</i> (1876); F. Miltoun&rsquo;s <i>Dickens&rsquo;s London</i> (1904);
+Kitton&rsquo;s <i>Dickens and his Illustrators</i>; W. Teignmouth Shore&rsquo;s <i>Charles
+Dickens and his Friends</i> (1904 and 1909); B. W. Matz, <i>Story of
+Dickens&rsquo;s Life and Work</i> (1904), and review of solutions to <i>Edwin
+Drood</i> in <i>The Bookman</i> for March 1908; the recollections of Edmund
+Yates, Trollope, James Payn, Lehmann, R. H. Horne, Lockwood
+and many others. <i>The Dickensian</i>, a magazine devoted to Dickensian
+subjects, was started in 1905; it is the organ of the Dickens Fellowship,
+and in a sense of the Boz Club. <i>A Dickens Dictionary</i> (by G. A.
+Pierce) appeared in 1872 and 1878; another (by A. J. Philip) in 1909;
+and a <i>Dickens Concordance</i> by Mary Williams in 1907.</p>
+</div>
+<div class="author">(T. Se.)</div>
+
+
+<hr class="art" />
+<p><span class="pagenum"><a name="page184" id="page184"></a>184</span></p>
+<p><span class="bold">DICKINSON, ANNA ELIZABETH<a name="ar38" id="ar38"></a></span> (1842-&emsp;&emsp;), American
+author and lecturer, was born, of Quaker parentage, at
+Philadelphia, Pennsylvania, on the 28th of October 1842. She
+was educated at the Friends&rsquo; Free School in Philadelphia, and
+was for a time a teacher. In 1861 she obtained a clerkship in the
+United States mint, but was removed for criticizing General
+McClellan at a public meeting. She had gradually become
+widely known as an eloquent and persuasive public speaker, one
+of the first of her sex to mount the platform to discuss the burning
+questions of the hour. Before the Civil War she lectured on
+anti-slavery topics, during the war she toured the country on behalf
+of the Sanitary Commission, and also lectured on reconstruction,
+temperance and woman&rsquo;s rights. She wrote several plays, including
+<i>The Crown of Thorns</i> (1876); <i>Mary Tudor</i> (1878), in which
+she appeared in the title rôle; <i>Aurelian</i> (1878); and <i>An American
+Girl</i> (1880), successfully acted by Fanny Davenport. She also
+published a novel, <i>Which Answer?</i> (1868); <i>A Paying Investment,
+a Plea for Education</i> (1876); and <i>A Ragged Register of People,
+Places and Opinions</i> (1879).</p>
+
+
+<hr class="art" />
+<p><span class="bold">DICKINSON, JOHN<a name="ar39" id="ar39"></a></span> (1732-1808), American statesman and
+pamphleteer, was born in Talbot county, Maryland, on the 8th
+of November 1732. He removed with his father to Kent county,
+Delaware, in 1740, studied under private tutors, read law, and in
+1753 entered the Middle Temple, London. Returning to America
+in 1757, he began the practice of law in Philadelphia, was speaker
+of the Delaware assembly in 1760, and was a member of the
+Pennsylvania assembly in 1762-1765 and again in 1770-1776.<a name="fa1c" id="fa1c" href="#ft1c"><span class="sp">1</span></a>
+He represented Pennsylvania in the Stamp Act Congress (1765)
+and in the Continental Congress from 1774 to 1776, when he
+was defeated owing to his opposition to the Declaration of
+Independence. He then retired to Delaware, served for a time
+as private and later as brigader-general in the state militia, and
+was again a member of the Continental Congress (from Delaware)
+in 1779-1780. He was president of the executive council, or chief
+executive officer, of Delaware in 1781-1782, and of Pennsylvania
+in 1782-1785, and was a delegate from Delaware to the Annapolis
+convention of 1786 and the Federal Constitutional convention
+of 1787. Dickinson has aptly been called the &ldquo;Penman of the
+Revolution.&rdquo; No other writer of the day presented arguments so
+numerous, so timely and so popular. He drafted the &ldquo;Declaration
+of Rights&rdquo; of the Stamp Act Congress, the &ldquo;Petition to the
+King&rdquo; and the &ldquo;Address to the Inhabitants of Quebec&rdquo; of the
+Congress of 1774, and the second &ldquo;Petition to the King&rdquo;<a name="fa2c" id="fa2c" href="#ft2c"><span class="sp">2</span></a> and
+the &ldquo;Articles of Confederation&rdquo; of the second Congress. Most
+influential of all, however, were <i>The Letters of a Farmer in
+Pennsylvania</i>, written in 1767-1768 in condemnation of the
+Townshend Acts of 1767, in which he rejected speculative
+natural rights theories and appealed to the common sense of
+the people through simple legal arguments. By opposing the
+Declaration of Independence, he lost his popularity and was never
+able entirely to regain it. As the representative of a small state,
+he championed the principle of state equality in the constitutional
+convention, but was one of the first to advocate the
+compromise, which was finally adopted, providing for equal
+representation, in one house and proportional representation in
+the other. He was probably influenced by Delaware prejudice
+against Pennsylvania when he drafted the clause which forbids
+the creation of a new state by the junction of two or more states
+or parts of states without the consent of the states concerned as
+well as of congress. After the adjournment of the convention he
+defended its work in a series of letters signed &ldquo;Fabius,&rdquo; which
+will bear comparison with the best of the Federalist productions.
+It was largely through his influence that Delaware and
+Pennsylvania were the first two states to ratify the Constitution.
+Dickinson&rsquo;s interests were not exclusively political. He helped
+to found Dickinson College (named in his honour) at Carlisle,
+Pennsylvania, in 1783, was the first president of its board of
+trustees, and was for many years its chief benefactor. He died
+on the 14th of February 1808 and was buried in the Friends&rsquo;
+burial ground in Wilmington, Del.</p>
+
+<div class="condensed">
+<p>See C. J. Stillé, <i>Life and Times of John Dickinson</i>, and P. L. Ford
+(editor), <i>The Writings of John Dickinson</i>, in vols. xiii. and xiv.
+respectively of the <i>Memoirs of the Historical Society of Pennsylvania</i>
+(Philadelphia, 1891 and 1895).</p>
+</div>
+
+<hr class="foot" /> <div class="note">
+
+<p><a name="ft1c" id="ft1c" href="#fa1c"><span class="fn">1</span></a> Being under the same proprietor and the same governor,
+Pennsylvania and Delaware were so closely connected before the
+Revolution that there was an interchange of public men.</p>
+
+<p><a name="ft2c" id="ft2c" href="#fa2c"><span class="fn">2</span></a> The &ldquo;Declaration of the United Colonies of North America ...
+setting forth the Causes and the Necessity of their Taking up Arms&rdquo;
+(often erroneously attributed to Thomas Jefferson).</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DICKSON, SIR ALEXANDER<a name="ar40" id="ar40"></a></span> (1777-1840), British artillerist,
+entered the Royal Military Academy in 1793, passing out as
+second lieutenant in the Royal Artillery in the following year.
+As a subaltern he saw service in Minorca in 1798 and at Malta in
+1800. As a captain he took part in the unfortunate Montevideo
+Expedition of 1806-07, and in 1809 he accompanied Howorth
+to the Peninsular War as brigade-major of the artillery. He soon
+obtained a command in the Portuguese artillery, and as a
+lieutenant-colonel of the Portuguese service took part in the
+various battles of 1810-11. At the two sieges of Budazoz,
+Ciudad Rodrigo, the Salamanca forts and Burgos, he was
+entrusted by Wellington (who had the highest opinion of him)
+with most of the detailed artillery work, and at Salamanca battle
+he commanded the reserve artillery. In the end he became
+commander of the whole of the artillery of the allied army, and
+though still only a substantive captain in the British service he
+had under his orders some 8000 men. At Vitoria, the Pyrenees
+battles and Toulouse he directed the movements of the artillery
+engaged, and at the end of the war received handsome presents
+from the officers who had served under him, many of whom were
+his seniors in the army list. He was at the disastrous affair of
+New Orleans, but returned to Europe in time for the Waterloo
+campaign. He was present at Quatre Bras and Waterloo on the
+artillery staff of Wellington&rsquo;s army, and subsequently commanded
+the British battering train at the sieges of the French fortresses
+left behind the advancing allies. For the rest of his life he was on
+home service, principally as a staff officer of artillery. He died,
+a major-general and G.C.B., in 1840. A memorial was erected at
+Woolwich in 1847. Dickson was one of the earliest fellows of the
+Royal Geographical Society.</p>
+
+<div class="condensed">
+<p>His diaries kept in the Peninsula were the main source of information
+used in Duncan&rsquo;s <i>History of the Royal Artillery</i>.</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DICKSON, SIR JAMES ROBERT<a name="ar41" id="ar41"></a></span> (1832-1901), Australian
+statesman, was born in Plymouth on the 30th of November 1832.
+He was brought up in Glasgow, receiving his education at the
+high school, and became a clerk in the City of Glasgow Bank.
+In 1854 he emigrated to Victoria, but after some years spent
+in that colony and in New South Wales, he settled in 1862 in
+Queensland, where he was connected with many important
+business enterprises, among them the Royal Bank of Queensland.
+He entered the Queensland House of Assembly in 1872, and
+became minister of works (1876), treasurer (1876-1879, and 1883-1887),
+acting premier (1884), but resigned in 1887 on the question
+of taxing land. In 1889 he retired from business, and spent three
+years in Europe before resuming political life. He fought for
+the introduction of Polynesian labour on the Queensland sugar
+plantations at the general election of 1892, and was elected to the
+House of Assembly in that year and again at the elections of 1893
+and 1896. He became secretary for railways in 1897, minister for
+home affairs in 1898, represented Queensland in the federal
+council of Australia in 1896 and at the postal conference at
+Hobart in 1898, and in 1898 became premier. His energies were
+now devoted to the formation of an Australian commonwealth.
+He secured the reference of the question to a plebiscite, the result
+of which justified his anticipations. He resigned the premiership
+in November 1899, but in the ministry of Robert Philp, formed
+in the next month, he was reappointed to the offices of chief
+secretary and vice-president of the executive council which he had
+combined with the office of premier. He represented Queensland
+in 1900 at the conference held in London to consider the question
+of Australian unity, and on his return was appointed minister of
+defence in the first government of the Australian Commonwealth.
+He did not long survive the accomplishment of his political aims,
+dying at Sydney on the 10th of January 1901, in the midst of
+the festivities attending the inauguration of the new state.</p>
+
+<p><span class="pagenum"><a name="page185" id="page185"></a>185</span></p>
+
+
+<hr class="art" />
+<p><span class="bold">DICOTYLEDONS,<a name="ar42" id="ar42"></a></span> in botany, the larger of the two great classes
+of angiosperms, embracing most of the common flower-bearing
+plants. The name expresses the most universal character of the
+class, the importance of which was first noticed by John Ray,
+namely, the presence of a pair of seed-leaves or cotyledons, in
+the plantlet or embryo contained in the seed. The embryo is
+generally surrounded by a larger or smaller amount of foodstuff
+(endosperm) which serves to nourish it in its development to
+form a seedling when the seed germinates; frequently, however,
+as in pea or bean and their allies, the whole of the nourishment for
+future use is stored up in the cotyledons themselves, which then
+become thick and fleshy. In germination of the seed the root of
+the embryo (radicle) grows out to get a holdfast for the plant;
+this is generally followed by the growth of the short stem
+immediately above the root, the so-called &ldquo;hypocotyl,&rdquo; which
+carries up the cotyledons above the ground, where they spread
+to the light and become the first green leaves of the plant.
+Protected between the cotyledons and terminating the axis of the
+plant is the first stem-bud (the plumule of the embryo), by the
+further growth and development of which the aerial portion of
+the plant, consisting of stem, leaves and branches, is formed,
+while the development of the radicle forms the root-system.
+The size and manner of growth of the adult plant show a great
+variety, from the small herb lasting for one season only, to the
+forest tree living for centuries. The arrangement of the conducting
+tissue in the stem is characteristic; a transverse section of
+the very young stem shows a <span class="correction" title="amended from nunber">number</span> of distinct conducting
+strands&mdash;vascular bundles&mdash;arranged in a ring round the pith;
+these soon become united to form a closed ring of bast and
+wood, separated by a layer of formative tissue (cambium). In
+perennials the stem shows a regular increase in thickness each
+year by the addition of a new ring of wood outside the old one&mdash;for
+details of structure see <span class="sc"><a href="#artlinks">Plants</a></span>: <i>Anatomy</i>. A similar growth
+occurs in the root. This increase in the diameter of stem and root
+is correlated with the increase in leaf-area each season, due to the
+continued production of new leaf-bearing branches. A characteristic
+of the class is afforded by the complicated network formed
+by the leaf-veins,&mdash;well seen in a skeleton leaf, from which the soft
+parts have been removed by maceration. The parts of the
+flower are most frequently arranged in fives, or multiples of fives;
+for instance, a common arrangement is as follows,&mdash;five sepals,
+succeeded by five petals, ten stamens in two sets of five, and five
+or fewer carpels; an arrangement in fours is less frequent, while
+the arrangement in threes, so common in monocotyledons, is rare
+in dicotyledons. In some orders the parts are numerous, chiefly
+in the case of the stamens and the carpels, as in the buttercup and
+other members of the order Ranunculaceae. There is a very wide
+range in the general structure and arrangement of the parts of the
+flower, associated with the means for ensuring the transference of
+pollen; in the simplest cases the flower consists only of a few
+stamens or carpels, with no enveloping sepals or petals, as in the
+willow, while in the more elaborate type each series is represented,
+the whole forming a complicated structure closely correlated
+with the size, form and habits of the pollinating agent (see
+<span class="sc"><a href="#artlinks">Flower</a></span>). The characters of the fruit and seed and the means
+for ensuring the dispersal of the seeds are also very varied (see
+<span class="sc"><a href="#artlinks">Fruit</a></span>).</p>
+
+
+<hr class="art" />
+<p><span class="bold">DICTATOR<a name="ar43" id="ar43"></a></span> (from the Lat. <i>dictare</i>, frequentative of <i>dicere</i>, to
+speak). In modern usage this term is loosely used for a personal
+ruler enjoying extraordinary and extra-constitutional power.
+The etymological sense of one who &ldquo;dictates&rdquo;&mdash;<i>i.e.</i> one whose
+word (<i>dictum</i>) is law (from which that of one who &ldquo;dictates,&rdquo; <i>i.e.</i>
+speaks for some writer to record, is to be distinguished)&mdash;has
+been assisted by the historical use of the term, in ancient times,
+for an extraordinary magistrate in the Roman commonwealth.
+It is unknown precisely how the Roman word came into use,
+though an explanation of the earlier official title, magister populi,
+throws some light on the subject. That designation may mean
+&ldquo;head of the (infantry) host&rdquo; as opposed to his subordinate, the
+magister equitum, who was &ldquo;head of the cavalry.&rdquo; If this explanation
+be accepted, emphasis was thus laid in early times on the
+military aspect of the dictatorship, and in fact the office seems to
+have been instituted for the purpose of meeting a military crisis
+such as might have proved too serious for the annual consuls with
+their divided command. Later constitutional theory held that
+the repression of civil discord was also one of the motives for the
+institution of a dictatorship. Such is the view expressed by
+Cicero in the <i>De legibus</i> (iii. 3, 9) and by the emperor Claudius
+in his extant <i>Oratio</i> (i. 28). This function of the office, although
+it may not have been contemplated at first, is attested by
+the internal history of Rome. In the crisis of the agitation that
+gathered round the Licinian laws (367 <span class="sc">b.c.</span>) a dictator was appointed,
+and in 314 <span class="sc">b.c.</span> we have the notice of a dictator created
+for purposes of criminal jurisdiction (<i>quaestionibus exercendis</i>).
+The dictator appointed to meet the dangers of war, sedition or
+crime was technically described as &ldquo;the administrative dictator&rdquo;
+(<i>rei gerundae causa</i>). Minor, or merely formal, needs of the state
+might lead to the creation of other types of this office. Thus we
+find dictators destined to hold the elections, to make out the list
+of the senate, to celebrate games, to establish festivals, and to
+drive the nail into the temple of Jupiter&mdash;an act of natural
+magic which was believed to avert pestilence. These dictators
+appointed for minor purposes were expected to retire from office
+as soon as their function was completed. The &ldquo;administrative
+dictator&rdquo; held office for at least six months.</p>
+
+<p>The powers of a dictator were a temporary revival of those
+of the kings; but there were some limitations to his authority.
+He was never concerned with civil jurisdiction, and was
+dependent on the senate for supplies of money. His military
+authority was confined to Italy; and his power of life and death
+over the citizens was at an early period limited by law. It was
+probably the <i>lex Valeria</i> of 300 <span class="sc">b.c.</span> that made him subject to the
+right of criminal appeal (<i>provocatio</i>) within the limits of the city.
+But during his tenure of power all the magistrates of the people
+were regarded as his subordinates; and it was even held that
+the right of assistance (<i>auxilium</i>), furnished by the tribunes of the
+plebs to members of the citizen body, should not be effectively
+exercised when the state was under this type of martial law. The
+dictator was nominated by one of the consuls. But here as elsewhere
+the senate asserted its authority over the magistrates, and
+the view was finally held that the senate should not only suggest
+the need of nomination but also the name of the nominee. After
+the nomination, the imperium of the dictator was confirmed by
+a <i>lex curiata</i> (see <span class="sc"><a href="#artlinks">Comitia</a></span>). To emphasize the superiority of this
+imperium over that of the consuls, the dictator might be preceded
+by twenty-four lictors, not by the usual twelve; and, at least in
+the earlier period of the office, these lictors bore the axes, the
+symbols of life and death, within the city walls.</p>
+
+<p>Tradition represents the dictatorship as having a life of three
+centuries in the history of the Roman state. The first dictator
+is said to have been created in 501 <span class="sc">b.c.</span>; the last of the
+&ldquo;administrative&rdquo; dictators belongs to the year 216 <span class="sc">b.c.</span> It was
+an office that was incompatible both with the growing spirit of
+constitutionalism and with the greater security of the city; and
+the epoch of the Second Punic War was marked by experiments
+with the office, such as the election of Q. Fabius Maximus by the
+people, and the co-dictatorship of M. Minucius with Fabius, which
+heralded its disuse (see <span class="sc"><a href="#artlinks">Punic Wars</a></span>). The emergency office of
+the early and middle Republic has few points of contact, except
+those of the extraordinary position and almost unfettered
+authority of its holder, with the dictatorship as revised by Sulla
+and by Caesar. Sulla&rsquo;s dictatorship was the form taken by a
+provisional government. He was created &ldquo;for the establishment
+of the Republic.&rdquo; It is less certain whether the dictatorships held
+by Caesar were of a consciously provisional character. Since the
+office represented the only supreme <i>Imperium</i> in Rome, it was
+the natural resort of the founder of a monarchy (see <span class="sc"><a href="#artlinks">Sulla</a></span> and
+<span class="sc"><a href="#artlinks">Caesar</a></span>). Ostensibly to prevent its further use for such a purpose,
+M. Antonius in 44 <span class="sc">b.c.</span> carried a law abolishing the dictatorship as
+a part of the constitution.</p>
+
+<div class="condensed">
+<p><span class="sc">Bibliography.</span>&mdash;Mommsen, <i>Römisches Staatsrecht</i>, ii. 141 foll.
+(3rd ed., Leipzig, 1887); Herzog, <i>Geschichte und System der römischen
+Staatsverfassung</i>, i. 718 foll. (Leipzig, 1884); Pauly-Wissowa,
+<i>Realencyclopädie</i>, v. 370 foll. (new edition, Stuttgart. 1893, &amp;c.);
+<span class="pagenum"><a name="page186" id="page186"></a>186</span>
+Lange, <i>Römische Alterthümer</i>, i. 542 foll. (Berlin, 1856, &amp;c.); Daremberg-Saglio,
+<i>Dictionnaire des antiquités grecques et romaines</i>, ii. 161
+foll. (1875, &amp;c.); Haverfield, &ldquo;The Abolition of the Dictatorship,&rdquo;
+in <i>Classical Review</i>, iii. 77.</p>
+</div>
+<div class="author">(A. H. J. G.)</div>
+
+
+<hr class="art" />
+<p><span class="bold">DICTIONARY.<a name="ar44" id="ar44"></a></span> In its proper and most usual meaning a
+dictionary is a book containing a collection of the words of a
+language, dialect or subject, arranged alphabetically or
+in some other definite order, and with explanations in the
+<span class="sidenote">Definition and history.</span>
+same or some other language. When the words are few in
+number, being only a small part of those belonging to
+the subject, or when they are given without explanation, or some
+only are explained, or the explanations are partial, the work is
+called a <i>vocabulary</i>; and when there is merely a list of explanations
+of the technical words and expressions in some particular
+subject, a <i>glossary</i>. An alphabetical arrangement of the words
+of some book or author with references to the places where
+they occur is called an index (<i>q.v.</i>). When under each word
+the phrases containing it are added to the references, the work is
+called a <i>concordance</i>. Sometimes, however, these names are given
+to true dictionaries; thus the great Italian dictionary of the
+<i>Accademia della Crusca</i>, in six volumes folio, is called <i>Vocabolario</i>,
+and Ernesti&rsquo;s dictionary to Cicero is called <i>Index</i>. When the
+words are arranged according to a definite system of classification
+under heads and subdivisions, according to their nature or their
+meaning, the book is usually called a classed vocabulary; but
+when sufficient explanations are given it is often accepted as a
+dictionary, like the <i>Onomasticon</i> of Julius Pollux, or the native
+dictionaries of Sanskrit, Manchu and many other languages.</p>
+
+<p>Dictionaries were originally books of reference explaining the
+words of a language or of some part of it. As the names of
+things, as well as those of persons and places, are words, and
+often require explanation even more than other classes of words,
+they were necessarily included in dictionaries, and often to a very
+great extent. In time, books were devoted to them alone, and
+were limited to special subjects, and these have so multiplied,
+that dictionaries of things now rival in number and variety those
+of words or of languages, while they often far surpass them in bulk.
+There are dictionaries of biography and history, real and fictitious,
+general and special, relating to men of all countries, characters
+and professions; the English <i>Dictionary of National Biography</i>
+(see <span class="sc"><a href="#artlinks">Biography</a></span>) is a great instance of one form of these;
+dictionaries of bibliography, relating to all books, or to those
+of some particular kind or country; dictionaries of geography
+(sometimes called <i>gazetteers</i>) of the whole world, of particular
+countries, or of small districts, of towns and of villages, of
+castles, monasteries and other buildings. There are dictionaries
+of philosophy; of the Bible; of mathematics; of natural history,
+zoology, botany; of birds, trees, plants and flowers; of
+chemistry, geology and mineralogy; of architecture, painting
+and music; of medicine, surgery, anatomy, pathology and
+physiology; of diplomacy; of law, canon, civil, statutory and
+criminal; of political and social sciences; of agriculture, rural
+economy and gardening; of commerce, navigation, horsemanship
+and the military arts; of mechanics, machines and
+the manual arts. There are dictionaries of antiquities, of
+chronology, of dates, of genealogy, of heraldry, of diplomatics, of
+abbreviations, of useful receipts, of monograms, of adulterations
+and of very many other subjects. These works are separately
+referred to in the bibliographies attached to the articles on the
+separate subjects. And lastly, there are dictionaries of the arts
+and sciences, and their comprehensive offspring, encyclopaedias
+(<i>q.v.</i>), which include in themselves every branch of knowledge.
+Neither under the heading of <i>dictionary</i> nor under that of
+<i>encyclopaedia</i> do we propose to include a mention of every work
+of its class, but many of these will be referred to in the separate
+articles on the subjects to which they pertain. And in this
+article we confine ourselves to an account of those dictionaries
+which are primarily word-books. This is practically the most
+convenient distinction from the subject-book or encyclopaedia;
+though the two characters are often combined in one work. Thus
+the <i>Century Dictionary</i> has encyclopaedic features, while the
+present edition of the <i>Encyclopaedia Britannica</i>, restoring its
+earlier tradition but carrying out the idea more systematically,
+also embodies dictionary features.</p>
+
+<p><i>Dictionarium</i> is a word of low or modern Latinity;<a name="fa1d" id="fa1d" href="#ft1d"><span class="sp">1</span></a> <i>dictio</i>,
+from which it was formed, was used in medieval Latin to mean
+a word. <i>Lexicon</i> is a corresponding word of Greek origin,
+meaning a book of or for words&mdash;a dictionary. A <i>glossary</i> is
+properly a collection of unusual or foreign words requiring
+explanation. It is the name frequently given to English
+dictionaries of dialects, which the Germans usually call <i>idioticon</i>,
+and the Italians <i>vocabolario</i>. <i>Wörterbuch</i>, a book of words, was
+first used among the Germans, according to Grimm, by Kramer
+(1719), imitated from the Dutch <i>woordenboek</i>. From the Germans
+the Swedes and Danes adopted <i>ordbok</i>, <i>ordbog</i>. The Icelandic
+<i>ordabôk</i>, like the German, contains the genitive plural. The
+Slavonic nations use <i>slovar</i>, <i>slovnik</i>, and the southern Slavs
+<i>ryetshnik</i>, from <i>slovo</i>, <i>ryetsh</i>, a word, formed, like dictionary
+and lexicon, without composition. Many other names have been
+given to dictionaries, as <i>thesaurus</i>, <i>Sprachschatz</i>, <i>cornucopia</i>,
+<i>gazophylacium</i>, <i>comprehensorium</i>, <i>catholicon</i>, to indicate their
+completeness; <i>manipulus predicantium</i>, <i>promptorium puerorum</i>,
+<i>liber memorialis</i>, <i>hortus vocabulorum</i>, <i>ionia</i> (a violet bed), <i>alveary</i>
+(a beehive), <i>kamoos</i> (the sea), <i>haft kulzum</i> (the seven seas), <i>tsze
+tien</i> (a standard of character), <i>onomasticon</i>, <i>nomenclator</i>, <i>bibliotheca</i>,
+<i>elucidario</i>, <i>Mundart-sammlung</i>, <i>clavis</i>, <i>scala</i>, <i>pharetra</i>,<a name="fa2d" id="fa2d" href="#ft2d"><span class="sp">2</span></a> <i>La
+Crusca</i> from the great Italian dictionary, and <i>Calepino</i> (in Spanish
+and Italian) from the Latin dictionary of Calepinus.</p>
+
+<p>The tendency of great dictionaries is to unite in themselves all
+the peculiar features of special dictionaries. A large dictionary
+is most useful when a word is to be thoroughly studied, or when
+there is difficulty in making out the meaning of a word or phrase.
+Special dictionaries are more useful for special purposes; for
+instance, synonyms are best studied in a dictionary of synonyms.
+And small dictionaries are more convenient for frequent use, as
+in translating from an unfamiliar language, for words may be
+found more quickly, and they present the words and their
+meanings in a concentrated and compact form, instead of being
+scattered over a large space, and separated by other matter.
+Dictionaries of several languages, called <i>polyglots</i>, are of different
+kinds. Some are polyglot in the vocabulary, but not in the
+explanation, like Johnson&rsquo;s dictionary of Persian and Arabic
+explained in English; some in the interpretation, but not in the
+vocabulary or explanation, like <i>Calepini octoglotton</i>, a Latin
+dictionary of Latin, with the meanings in seven languages.
+Many great dictionaries are now polyglot in this sense. Some are
+polyglot in the vocabulary and interpretation, but are explained
+in one language, like Jal&rsquo;s <i>Glossaire nautique</i>, a glossary of sea
+terms in many languages, giving the equivalents of each word in
+the other languages, but the explanation in French. Pauthier&rsquo;s
+<i>Annamese Dictionary</i> is polyglot in a peculiar way. It gives
+the Chinese characters with their pronunciation in Chinese and
+Annamese. Special dictionaries are of many kinds. There are
+technical dictionaries of etymology, foreign words, dialects,
+secret languages, slang, neology, barbarous words, faults of expression,
+choice words, prosody, pronunciation, spelling, orators,
+poets, law, music, proper names, particular authors, nouns, verbs,
+participles, particles, double forms, difficulties and many others.
+Fick&rsquo;s dictionary (Göttingen, 1868, 8vo; 1874-1876, 8vo, 4 vols.)
+is a remarkable attempt to ascertain the common language of
+the Indo-European nations before each of their great separations.
+In the second edition of his <i>Etymologische Forschungen</i> (Lemgo
+and Detmoldt, 1859-1873, 8vo, 7217 pages) Pott gives a
+comparative lexicon of Indo-European roots, 2226 in number,
+occupying 5140 pages.</p>
+
+<p><span class="pagenum"><a name="page187" id="page187"></a>187</span></p>
+
+<p>At no time was progress in the making of general dictionaries
+so rapid as during the second half of the 19th century. It is to
+be seen in three things: in the perfecting of the theory of what
+a general dictionary should be; in the elaboration
+<span class="sidenote">Methods.</span>
+of methods of collecting and editing lexicographic
+materials; and in the magnitude and improved quality of the
+work which has been accomplished or planned. Each of these
+can best be illustrated from English lexicography, in which the
+process of development has in all directions been carried farthest.
+The advance that has been made in theory began with a radical
+change of opinion with regard to the chief end of the general
+dictionary of a language. The older view of the matter was that
+the lexicographer should furnish a standard of usage&mdash;should
+register only those words which are, or at some period of the
+language have been, &ldquo;good&rdquo; from a literary point of view, with
+their &ldquo;proper&rdquo; senses and uses, or should at least furnish the
+means of determining what these are. In other words, his chief
+duty was conceived to be to sift and refine, to decide authoritatively
+questions with regard to good usage, and thus to fix the
+language as completely as might be possible within the limits
+determined by the literary taste of his time. Thus the Accademia
+della Crusca, founded near the close of the 16th century, was
+established for the purpose of purifying in this way the Italian
+tongue, and in 1612 the <i>Vocabolario degli Accademici della
+Crusca</i>, long the standard of that language, was published. The
+Académie Française, the first edition of whose dictionary
+appeared in 1694, had a similar origin. In England the idea of
+constructing a dictionary upon this principle arose during the
+second quarter of the 18th century. It was imagined by men of
+letters&mdash;among them Alexander Pope&mdash;that the English language
+had then attained such perfection that further improvement was
+hardly possible, and it was feared that if it were not fixed by
+lexicographic authority deterioration would soon begin. Since
+there was no English &ldquo;Academy,&rdquo; it was necessary that the task
+should fall to some one whose judgment would command respect,
+and the man who undertook it was Samuel Johnson. His dictionary,
+the first edition of which, in two folio volumes, appeared
+in 1755, was in many respects admirable, but it was inadequate
+even as a standard of the then existing literary usage.
+Johnson himself did not long entertain the belief that the natural
+development of a language can be arrested in that or in any
+other way. His work was, however, generally accepted as a final
+authority, and the ideas upon which it was founded dominated
+English lexicography for more than a century. The first effective
+protest in England against the supremacy of this literary view was
+made by Dean (later Archbishop) Trench, in a paper on &ldquo;Some
+Deficiencies in Existing English Dictionaries&rdquo; read before the
+Philological Society in 1857. &ldquo;A dictionary,&rdquo; he said, &ldquo;according
+to that idea of it which seems to me alone capable of being
+logically maintained, is an <i>inventory of the language</i>; much more,
+but this primarily.... It is no task of the maker of it to select
+the <i>good</i> words of the language.... The business which he has
+undertaken is to collect and arrange <i>all</i> words, whether good or
+bad, whether they commend themselves to his judgment or otherwise....
+<i>He is an historian of</i> [the language], <i>not a critic.</i>&rdquo;
+That is, for the literary view of the chief end of the general
+dictionary should be substituted the philological or scientific.
+In Germany this substitution had already been effected by Jacob
+and Wilhelm Grimm in their dictionary of the German language,
+the first volume of which appeared in 1854. In brief, then, the
+modern view is that the general dictionary of a language
+should be a record of all the words&mdash;current or obsolete&mdash;of
+that language, with all their meanings and uses, but should not
+attempt to be, except secondarily or indirectly, a guide to
+&ldquo;good&rdquo; usage. A &ldquo;standard&rdquo; dictionary has, in fact, been
+recognized to be an impossibility, if not an absurdity.</p>
+
+<p>This theoretical requirement must, of course, be modified
+considerably in practice. The date at which a modern language
+is to be regarded by the lexicographer as &ldquo;beginning&rdquo; must, as
+a rule, be somewhat arbitrarily chosen; while considerable
+portions of its earlier vocabulary cannot be recovered because
+of the incompleteness of the literary record. Moreover, not even
+the most complete dictionary can include all the words which the
+records&mdash;earlier and later&mdash;actually contain. Many words, that
+is to say, which are found in the literature of a language cannot
+be regarded as, for lexicographic purposes, belonging to that
+language; while many more may or may not be held to belong
+to it, according to the judgment&mdash;almost the whim&mdash;of the
+individual lexicographer. This is especially true of the English
+tongue. &ldquo;That vast aggregate of words and phrases which
+constitutes the vocabulary of English-speaking men presents, to
+the mind that endeavours to grasp it as a definite whole, the
+aspect of one of those nebulous masses familiar to the astronomer,
+in which a clear and unmistakable nucleus shades off on all sides,
+through zones of decreasing brightness, to a dim marginal film
+that seems to end nowhere, but to lose itself imperceptibly in
+the surrounding darkness&rdquo; (Dr J. A. H. Murray, <i>Oxford
+Dict.</i> General Explanations, p. xvii). This &ldquo;marginal film&rdquo; of
+words with more or less doubtful claims to recognition includes
+thousands of the terms of the natural sciences (the New-Latin
+classificatory names of zoology and botany, names of chemical
+compounds and of minerals, and the like); half-naturalized
+foreign words; dialectal words; slang terms; trade names
+(many of which have passed or are passing into common use);
+proper names and many more. Many of these even the most
+complete dictionary should exclude; others it should include;
+but where the line shall be drawn will always remain a vexed
+question.</p>
+
+<p>Another important principle upon which Trench insisted, and
+which also expresses a requirement of modern scientific philology,
+is that the dictionary shall be not merely a record, but also an
+<i>historical</i> record of words and their uses. From the literary point
+of view the most important thing is present usage. To that alone
+the idea of a &ldquo;standard&rdquo; has any application. Dictionaries of
+the older type, therefore, usually make the common, or &ldquo;proper&rdquo;
+or &ldquo;root&rdquo; meaning of a word the starting point of its definition,
+and arrange its other senses in a logical or accidental order
+commonly ignoring the historical order in which the various
+meanings arose. Still less do they attempt to give data from
+which the vocabulary of the language at any previous period may
+be determined. The philologist, however, for whom the growth,
+or progressive alteration, of a language is a fact of central
+importance, regards no record of a language as complete which
+does not exhibit this growth in its successive stages. He desires
+to know when and where each word, and each form and sense
+of it, are first found in the language; if the word or sense is
+obsolete, when it died; and any other fact that throws light upon
+its history. He requires, accordingly, of the lexicographer that,
+having ascertained these data, he shall make them the foundation
+of his exposition&mdash;in particular, of the division and arrangement
+of his definitions, that sense being placed first which appeared
+first in order of time. In other words, each article in the dictionary
+should furnish an orderly biography of the word of which it
+treats, each word and sense being so dated that the exact time
+of its appearance and the duration of its use may as nearly as
+possible be determined. This, in principle, is the method of the
+new lexicography. In practice it is subject to limitations similar
+to those of the vocabulary mentioned above. Incompleteness
+of the early record is here an even greater obstacle; and there
+are many words whose history is, for one reason or another, so
+unimportant that to treat it elaborately would be a waste of
+labour and space.</p>
+
+<p>The adoption of the historical principle involves a further noteworthy
+modification of older methods, namely, an important
+extension of the use of quotations. To Dr Johnson belongs the
+credit of showing how useful, when properly chosen, they may be,
+not only in corroborating the lexicographer&rsquo;s statements, but also
+in revealing special shades of meaning or variations of use which
+his definitions cannot well express. No part of Johnson&rsquo;s work
+is more valuable than this. This idea was more fully developed
+and applied by Dr Charles Richardson, whose <i>New Dictionary
+of the English Language ... Illustrated by Quotations from the
+Best Authors</i> (1835-1836) still remains a most valuable collection
+of literary illustrations. Lexicographers, however, have, with
+<span class="pagenum"><a name="page188" id="page188"></a>188</span>
+few exceptions, until a recent date, employed quotations chiefly
+for the ends just mentioned&mdash;as instances of use or as illustrations
+of correct usage&mdash;with scarcely any recognition of their
+value as historical evidence; and they have taken them almost
+exclusively from the works of the &ldquo;best&rdquo; authors. But since all
+the data upon which conclusions with regard to the history of
+a word can be based must be collected from the literature of
+the language, it is evident that, in so far as the lexicographer
+is required to furnish evidence for an historical inference, a
+quotation is the best form in which he can give it. In fact,
+extracts, properly selected and grouped, are generally sufficient to
+show the entire meaning and biography of a word without the aid
+of elaborate definitions. The latter simply save the reader the
+trouble of drawing the proper conclusions for himself. A further
+rule of the new lexicography, accordingly, is that quotations
+should be used, primarily, as historical evidence, and that the
+history of words and meanings should be exhibited by means of
+them. The earliest instance of use that can be found, and (if the
+word or sense is obsolete) the latest, are as a rule to be given;
+while in the case of an important word or sense, instances taken
+from successive periods of its currency also should be cited.
+Moreover, a quotation which contains an important bit of
+historical evidence must be used, whether its source is &ldquo;good,&rdquo;
+from the literary point of view, or not&mdash;whether it is a classic
+of the language or from a daily newspaper; though where choice
+is possible, preference should, of course, be given to quotations
+extracted from the works of the best writers. This rule does not
+do away with the illustrative use of quotations, which is still
+recognized as highly important, but it subordinates it to their
+historical use. It is necessary to add that it implies that the
+extracts must be given exactly, and in the original spelling and
+capitalization, accurately dated, and furnished with a precise
+reference to author, book, volume, page and edition; for
+insistence upon these requirements&mdash;which are obviously important,
+whatever the use of the quotation may be&mdash;is one of the
+most noteworthy of modern innovations. Johnson usually gave
+simply the author&rsquo;s name, and often quoted from memory and
+inaccurately; and many of his successors to this day have
+followed&mdash;altogether or to some extent&mdash;his example.</p>
+
+<p>The chief difficulty in the way of this use of quotations&mdash;after
+the difficulty of collection&mdash;is that of finding space for them in a
+dictionary of reasonable size. Preference must be given to those
+which are essential, the number of those which are cited merely
+on methodical grounds being made as small as possible. It is
+hardly necessary to add that the negative evidence furnished by
+quotations is generally of little value; one can seldom, that is,
+be certain that the lexicographer has actually found the earliest
+or the latest use, or that the word or sense has not been current
+during some intermediate period from which he has no quotations.</p>
+
+<p>Lastly, a much more important place in the scheme of the ideal
+dictionary is now assigned to the <i>etymology</i> of words. This may
+be attributed, in part, to the recent rapid development of etymology
+as a science, and to the greater abundance of trustworthy
+data; but it is chiefly due to the fact that from the historical
+point of view the connexion between that section of the biography
+of a word which lies within the language&mdash;subsequent, that is,
+to the time when the language may, for lexicographical purposes,
+be assumed to have begun, or to the time when the word was
+adopted or invented&mdash;and its antecedent history has become more
+vital and interesting. Etymology, in other words, is essentially
+the history of the <i>form</i> of a word up to the time when it became
+a part of the language, and is, in a measure, an extension of the
+history of the development of the word in the language. Moreover,
+it is the only means by which the exact relations of allied
+words can be ascertained, and the separation of words of the same
+form but of diverse origin (homonyms) can be effected, and is
+thus, for the dictionary, the foundation of all <i>family history</i> and
+correct <i>genealogy</i>. In fact, the attention that has been paid to
+these two points in the best recent lexicography is one of its
+distinguishing and most important characteristics. Related to
+the etymology of words are the changes in their form which may
+have occurred while they have been in use as parts of the language&mdash;modifications
+of their pronunciation, corruptions by popular
+etymology or false associations, and the like. The facts with
+regard to these things which the wide research necessitated
+by the historical method furnishes abundantly to the modern
+lexicographer are often among the most novel and interesting
+of his acquisitions.</p>
+
+<p>It should be added that even approximate conformity to the
+theoretical requirements of modern lexicography as above outlined
+is possible only under conditions similar to those under which
+the Oxford <i>New English Dictionary</i> was undertaken (see below).
+The labour demanded is too vast, and the necessary bulk of the
+dictionary too great. When, however, a language is recorded
+in one such dictionary, those of smaller size and more modest
+pretensions can rest upon it as an authority and conform to it
+as a model so far as their special limitations permit.</p>
+
+<p>The ideal thus developed is primarily that of the general
+dictionary of the purely philological type, but it applies also to
+the encyclopaedic dictionary. In so far as the latter is strictly
+lexicographic&mdash;deals with words as words, and not with the things
+they denote&mdash;it should be made after the model of the former,
+and is defective to the extent in which it deviates from it. The
+addition of encyclopaedic matter to the philological in no way
+affects the general principles involved. It may, however, for
+practical reasons, modify their application in various ways. For
+example, the number of obsolete and dialectal words included
+may be much diminished and the number of scientific terms (for
+instance, new Latin botanical and zoological names) be increased;
+and the relative amount of space devoted to etymologies and
+quotations may be lessened. In general, since books of this kind
+are designed to serve more or less as works of general reference,
+the making of them must be governed by considerations of
+practical utility which the compilers of a purely philological
+dictionary are not obliged to regard. The encyclopaedic type
+itself, although it has often been criticized as hybrid&mdash;as a mixture
+of two things which should be kept distinct&mdash;is entirely defensible.
+Between the dictionary and the encyclopaedia the dividing line
+cannot sharply be drawn. There are words the meaning of which
+cannot be explained fully without some description of things,
+and, on the other hand, the description of things and processes
+often involves the definition of names. To the combination of
+the two objection cannot justly be made, so long as it is effected
+in a way&mdash;with a selection of material&mdash;that leaves the dictionary
+essentially a dictionary and not an encyclopaedia. Moreover,
+the large vocabulary of the general dictionary makes it possible
+to present certain kinds of encyclopaedic matter with a degree of
+fulness and a convenience of arrangement which are possible in
+no single work of any other class. In fact, it may be said that if
+the encyclopaedic dictionary did not exist it would have to be
+invented; that its justification is its indispensableness. Not
+the least of its advantages is that it makes legitimate the use of
+diagrams and pictorial illustrations, which, if properly selected
+and executed, are often valuable aids to definition.</p>
+
+<p>On its practical side the advance in lexicography has consisted
+in the elaboration of methods long in use rather than in the invention
+of new ones. The only way to collect the data upon which
+the vocabulary, the definitions and the history are to be based
+is, of course, to search for them in the written monuments of the
+language, as all lexicographers who have not merely borrowed from
+their predecessors have done. But the wider scope and special
+aims of the new lexicography demand that the investigation shall
+be vastly more comprehensive, systematic and precise. It is
+necessary, in brief, that, as far as may be possible, the literature
+(of all kinds) of every period of the language shall be examined
+systematically, in order that all the words, and senses and forms
+of words, which have existed during any period may be found,
+and that enough excerpts (carefully verified, credited and dated) to
+cover all the essential facts shall be made. The books, pamphlets,
+journals, newspapers, and so on which must thus be searched will
+be numbered by thousands, and the quotations selected may (as
+in the case of the Oxford <i>New English Dictionary</i>) be counted by
+millions. This task is beyond the powers of any one man, even
+though he be a Johnson, or a Littré or a Grimm, and it is now
+<span class="pagenum"><a name="page189" id="page189"></a>189</span>
+assigned to a corps of readers whose number is limited only by the
+ability of the editor to obtain such assistance. The modern
+method of editing the material thus accumulated&mdash;the actual
+work of compilation&mdash;also is characterized by the application of
+the principle of the division of labour. Johnson boasted that his
+dictionary was written with but little assistance from the learned,
+and the same was in large measure true of that of Littré. Such
+attempts on the part of one man to write practically the whole of
+a general dictionary are no longer possible, not merely because of
+the vast labour and philological research necessitated by modern
+aims, but more especially because the immense development of
+the vocabulary of the special sciences renders indispensable the
+assistance, in the work of definition, of persons who are expert in
+those sciences. The tendency, accordingly, has been to enlarge
+greatly the editorial staff of the dictionary, scores of sub-editors
+and contributors being now employed where a dozen or fewer
+were formerly deemed sufficient. In other words, the making of
+a &ldquo;complete&rdquo; dictionary has become a co-operative enterprise,
+to the success of which workers in all the fields of literature and
+science contribute.</p>
+
+<p>The most complete exemplification of these principles and
+methods is the <i>Oxford New English Dictionary, on historical
+principles, founded mainly on materials collected by the Philological
+Society</i>. This monumental work originated in the suggestion
+of Trench that an attempt should be made, under the
+direction of the Philological Society, to complete the vocabulary
+of existing dictionaries and to supply the historical information
+which they lacked. The suggestion was adopted, considerable
+material was collected, and Mr Herbert Coleridge was appointed
+general editor. He died in 1861, and was succeeded by Dr F. J.
+Furnivall. Little, however, was done, beyond the collection of
+quotations&mdash;about 2,000,000 of which were gathered&mdash;until in
+1878 the expense of printing and publishing the proposed
+dictionary was assumed by the Delegates of the University Press,
+and the editorship was entrusted to Dr (afterwards Sir) J. A. H.
+Murray. As the historical point of beginning, the middle of the
+12th century was selected, all words that were obsolete at that
+date being excluded, though the history of words that were
+current both before and after that date is given in its entirety;
+and it was decided that the search for quotations&mdash;which, according
+to the original design, was to cover the entire literature down
+to the beginning of the 16th century and as much of the subsequent
+literature (especially the works of the more important
+writers and works on special subjects) as might be possible&mdash;should
+be made more thorough. More than 800 readers, in all
+parts of the world, offered their aid; and when the preface to the
+first volume appeared in 1888, the editor was able to announce
+that the readers had increased to 1300, and that 3,500,000 of
+quotations, taken from the writings of more than 5000 authors,
+had already been amassed. The whole work was planned to be
+completed in ten large volumes, each issued first in smaller parts.
+The first part was issued in 1884, and by the beginning of 1910
+the first part of the letter S had been reached.</p>
+
+<p>The historical method of exposition, particularly by quotations,
+is applied in the <i>New English Dictionary</i>, if not in all cases
+with entire success, yet, on the whole, with a regularity and a
+precision which leave little to be desired. A minor fault is that
+excerpts from second or third rate authors have occasionally been
+used where better ones from writers of the first class either must
+have been at hand or could have been found. As was said above,
+the literary quality of the question is highly important even
+in historical lexicography, and should not be neglected unnecessarily.
+Other special features of the book are the completeness
+with which variations of pronunciation and orthography
+(with dates) are given; the fulness and scientific excellence of the
+etymologies, which abound in new information and corrections
+of old errors; the phonetic precision with which the present
+(British) pronunciation is indicated; and the elaborate subdivision
+of meanings. The definitions as a whole are marked by
+a high degree of accuracy, though in a certain number of cases
+(not explicable by the date of the volumes) the lists of meanings
+are not so good as one would expect, as compared (say) with
+the <i>Century Dictionary</i>. Work of such magnitude and quality is
+possible, practically, only when the editor of the dictionary can command
+not merely the aid of a very large number of scholars and
+men of science, but their gratuitous aid. In this the <i>New English
+Dictionary</i> has been singularly fortunate. The conditions under
+which it originated, and its aim, have interested scholars everywhere,
+and led them to contribute to the perfecting of it their
+knowledge and time. The long list of names of such helpers in Sir
+J. A. H. Murray&rsquo;s preface is in curious contrast with their absence
+from Dr Johnson&rsquo;s and the few which are given in that of Littré.
+The editor&rsquo;s principal assistants were Dr Henry Bradley and
+Dr W. A. Craigie. Of the dictionary as a whole it may be said
+that it is one of the greatest achievements, whether in literature
+or science, of modern English scholarship and research.</p>
+
+<div class="condensed">
+<p>The <i>New English Dictionary</i> furnishes for the first time data from
+which the extent of the English word-store at any given period, and
+the direction and rapidity of its growth, can fairly be estimated.
+For this purpose the materials furnished by the older dictionaries are
+quite insufficient, on account of their incompleteness and unhistorical
+character. For example 100 pages of the <i>New English Dictionary</i>
+(from the letter H) contain 1002 words, of which, as the dated quotations
+show, 585 were current in 1750 (though some, of course, were
+very rare, some dialectal, and so on), 191 were obsolete at that date,
+and 226 have since come into use. But of the more than 700 words&mdash;current
+or obsolete&mdash;which Johnson might thus have recorded, he
+actually did record only about 300. Later dictionaries give more of
+them, but they in no way show their status at the date in question.
+It is worth noting that the figures given seem to indicate that not
+very many more words have been added to the vocabulary of the
+language during the past 150 years than had been lost by 1750. The
+pages selected, however, contain comparatively few recent scientific
+terms. A broader comparison would probably show that the gain
+has been more than twice as great as the loss.</p>
+</div>
+
+<p>In the <i>Deutsches Wörterbuch</i> of Jacob and Wilhelm Grimm
+the scientific spirit, as was said above, first found expression in
+general lexicography. The desirability of a complete inventory
+and investigation of German words was recognized by Leibnitz
+and by various 18th-century scholars, but the plan and methods
+of the Grimms were the direct product of the then new scientific
+philology. Their design, in brief, was to give an exhaustive
+account of the words of the literary language (New High German)
+from about the end of the 15th century, including their earlier
+etymological and later history, with references to important
+dialectal words and forms; and to illustrate their use and history
+abundantly by quotations. The first volume appeared in 1854.
+Jacob Grimm (died 1863) edited the first, second (with his
+brother, who died in 1859), third and a part of the fourth
+volumes; the others have been edited by various distinguished
+scholars. The scope and methods of this dictionary have been
+broadened somewhat as the work has advanced. In general it
+may be said that it differs from the <i>New English Dictionary</i>
+chiefly in its omission of pronunciations and other pedagogic
+matter; its irregular treatment of dates; its much less systematic
+and less lucid statement of etymologies; its less systematic and
+less fruitful use of quotations; and its less convenient and less
+intelligible arrangement of material and typography.</p>
+
+<p>These general principles lie also at the foundation of the
+scholarly <i>Dictionnaire de la langue française</i> of E. Littré, though
+they are there carried out less systematically and less completely.
+In the arrangement of the definitions the first place is given to
+the most primitive meaning of the word instead of to the most
+common one, as in the dictionary of the Academy; but the other
+meanings follow in an order that is often logical rather than
+historical. Quotations also are frequently used merely as literary
+illustrations, or are entirely omitted; in the special paragraphs
+on the history of words before the 16th century, however, they
+are put to a strictly historical use. This dictionary&mdash;perhaps the
+greatest ever compiled by one man&mdash;was published 1863-1872.
+(Supplement, 1878.)</p>
+
+<p>The <i>Thesaurus Linguae Latinae</i>, prepared under the auspices of
+the German Academies of Berlin, Göttingen, Leipzig, Munich
+and Vienna, is a notable application of the principles and
+practical co-operative method of modern lexicography to the
+classical tongues. The plan of the work is to collect quotations
+which shall register, with its full context, every word (except
+<span class="pagenum"><a name="page190" id="page190"></a>190</span>
+the most familiar particles) in the text of each Latin author
+down to the middle of the 2nd century <span class="sc">a.d.</span>, and to extract
+all important passages from all writers of the following
+centuries down to the 7th; and upon these materials to found
+a complete historical dictionary of the Latin language. The
+work of collecting quotations was begun in 1894, and the first
+part of the first volume has been published.</p>
+
+<p>In the making of all these great dictionaries (except, of course,
+the last) the needs of the general public as well as those of scholars
+have been kept in view. But the type to which the general
+dictionary designed for popular use has tended more and more
+to conform is the <i>encyclopaedic</i>. This combination of lexicon
+and encyclopaedia is exhibited in an extreme&mdash;and theoretically
+objectionable&mdash;form in the <i>Grand dictionnaire universel du XIX<span class="sp">e</span>
+siècle</i> of Pierre Larousse. Besides common words and their
+definitions, it contains a great many proper names, with a
+correspondingly large number of biographical, geographical,
+historical and other articles, the connexion of which with the
+strictly lexicographical part is purely mechanical. Its utility,
+which&mdash;notwithstanding its many defects&mdash;is very great, makes
+it, however, a model in many respects. Fifteen volumes were
+published (1866-1876), and supplements were brought out later
+(1878-1890). The <i>Nouveau Larousse illustré</i> started publication
+in 1901, and was completed in 1904 (7 vols.). This is not an
+abridgment or a fresh edition of the <i>Grand Dictionnaire</i> of Pierre
+Larousse, but a new and distinct publication.</p>
+
+<p>The most notable work of this class, in English, is the <i>Century
+Dictionary</i>, an American product, edited by Professor W. D.
+Whitney, and published 1889-1891 in six volumes, containing
+7046 pages (large quarto). It conforms to the philological mode
+in giving with great fulness the older as well as the present
+vocabulary of the language, and in the completeness of its
+etymologies; but it does not attempt to give the full history
+of every word within the language. Among its other more noteworthy
+characteristics are the inclusion of a great number of
+modern scientific and technical words, and the abundance of its
+quotations. The quotations are for the most part provided with
+references, but they are not dated. Even when compared with
+the much larger <i>New English Dictionary</i>, the <i>Century&rsquo;s</i> great
+merit is the excellent enumeration of meanings, and the accuracy
+of its explanations; in this respect it is often better and
+fuller than the <i>New English</i>. In the application of the encyclopaedic
+method this dictionary is conservative, excluding, with a
+few exceptions, proper names, and restricting, for the most part,
+the encyclopaedic matter to descriptive and other details which
+may legitimately be added to the definitions. Its pictorial
+illustrations are very numerous and well executed. In the
+manner of its compilation it is a good example of modern cooperative
+dictionary-making, being the joint product of a large
+number of specialists. Next to the <i>New English Dictionary</i> it
+is the most complete and scholarly of English lexicons.</p>
+
+<p><i>Bibliography.</i>&mdash;The following list of dictionaries (from the 9th
+edition of this work, with occasional corrections) is given for its
+historical interest, but in recent years dictionary-making has been
+so abundant that no attempt is made to be completely inclusive of
+later works; the various articles on languages may be consulted
+for these. The list is arranged geographically by families of
+languages, or by regions. In each group the order, when not
+alphabetical, is usually from north to south, extinct languages
+generally coming first, and dialects being placed under their
+language. Dictionaries forming parts of other works, such as
+travels, histories, transactions, periodicals, reading-books, &amp;c.,
+are generally excluded. The system here adopted was chosen
+as on the whole the one best calculated to keep together
+dictionaries naturally associated. The languages to be considered
+are too many for an alphabetical arrangement, which ignores all
+relations both natural and geographical, and too few to require a
+strict classification by affinities, by which the European languages,
+which for many reasons should be kept together, would be
+dispersed. Under either system, Arabic, Persian and Turkish,
+whose dictionaries are so closely connected, would be widely
+separated. A wholly geographical arrangement would be inconvenient,
+especially in Europe. Any system, however, which
+attempts to arrange in a consecutive series the great network of
+languages by which the whole world is enclosed, must be open
+to some objections; and the arrangement adopted in this list
+has produced some anomalies and dispersions which might cause
+inconvenience if not pointed out. The old Italic languages
+are placed under Latin, all dialects of France under French
+(but Provençal as a distinct language), and Wallachian among
+Romanic languages. Low German and its dialects are not
+separated from High German. Basque is placed after Celtic;
+Albanian, Gipsy and Turkish at the end of Europe, the last being
+thus separated from its dialects and congeners in Northern
+and Central Asia, among which are placed the Kazan dialect of
+Tatar, Samoyed and Ostiak. Accadian is placed after Assyrian
+among the Semitic languages, and Maltese as a dialect of Arabic;
+while the Ethiopic is among African languages as it seemed
+undesirable to separate it from the other Abyssinian languages,
+or these from their neighbours to the north and south. Circassian
+and Ossetic are joined to the first group of Aryan languages lying
+to the north-west of Persia, and containing Armenian, Georgian
+and Kurd. The following is the order of the groups, some of the
+more important languages, that is, of those best provided with
+dictionaries, standing alone:&mdash;</p>
+
+<p><span class="sc">Europe</span>: Greek, Latin, French, Romance, Teutonic (Scandinavian
+and German), Celtic, Basque, Baltic, Slavonic, Ugrian,
+Gipsy, Albanian.</p>
+
+<p><span class="sc">Asia</span>: Semitic, Armenian, Persian, Sanskrit, Indian, Indo-Chinese,
+Malay Archipelago, Philippines, Chinese, Japanese,
+Northern and Central Asia.</p>
+
+<p><span class="sc">Africa</span>: Egypt and Abyssinia, Eastern Africa, Southern,
+Western, Central, Berber.</p>
+
+<p><span class="sc">Australia and Polynesia.</span></p>
+
+<p><span class="sc">America</span>: North, Central (with Mexico), South.</p>
+
+<p class="center pt2">EUROPE</p>
+
+<div class="condensed">
+<p><span class="bold">Greek.</span>&mdash;Athenaeus quotes 35 writers of works, known or supposed
+to be dictionaries, for, as they are all lost, it is often difficult to
+decide on their nature. Of these, Anticlides, who lived after the reign
+of Alexander the Great, wrote <span class="grk" title="Exêgêtikos">&#904;&#958;&#951;&#947;&#951;&#964;&#953;&#954;&#972;&#962;</span>, which seems to have been a
+sort of dictionary, perhaps explaining the words and phrases occurring
+in ancient stories. Zenodotus, the first superintendent of the great
+library of Alexandria, who lived in the reigns of Ptolemy I. and Ptolemy
+II., wrote <span class="grk" title="Glôssai">&#915;&#955;&#8182;&#963;&#963;&#945;&#953;</span>, and also <span class="grk" title="Lexeis ethnikai">&#923;&#941;&#958;&#949;&#953;&#962; &#7952;&#952;&#957;&#953;&#954;&#945;&#7984;</span>, a dictionary of barbarous or
+foreign phrases. Aristophanes of Byzantium, son of Apelles the painter,
+who lived in the reigns of Ptolemy II. and Ptolemy III., and had the
+supreme management of the Alexandrian library, wrote a number
+of works, as <span class="grk" title="Attikai Lexeis, Lakônikai Glôssai">&#902;&#964;&#964;&#953;&#954;&#945;&#943; &#923;&#941;&#958;&#949;&#953;&#962;, &#923;&#945;&#954;&#969;&#957;&#953;&#954;&#945;&#943; &#915;&#955;&#8182;&#963;&#963;&#945;&#953;</span> which, from the titles,
+should be dictionaries, but a fragment of his <span class="grk" title="Lexeis">&#923;&#941;&#958;&#949;&#953;&#962;</span> printed by
+Boissonade, in his edition of Herodian (London, 1869, 8vo, pp.
+181-189), is not alphabetical. Artemidorus, a pupil of Aristophanes,
+wrote a dictionary of technical terms used in cookery. Nicander
+Colophonius, hereditary priest of Apollo Clarius, born at Claros,
+near Colophon in Ionia, in reputation for 50 years, from 181 to
+135, wrote <span class="grk" title="Glôssai">&#915;&#955;&#8182;&#963;&#963;&#945;&#953;</span> in at least three books. Parthenius, a pupil
+of the Alexandrian grammarian Dionysius (who lived in the 1st
+century before Christ), wrote on choice words used by historians.
+Didymus, called <span class="grk" title="chalkenteros">&#967;&#945;&#955;&#954;&#941;&#957;&#949;&#961;&#959;&#962;</span>, who, according to Athenaeus, wrote
+3500 books, and, according to Seneca, 4000, wrote lexicons of the
+tragic poets (of which book 28 is quoted), of the comic poets, of
+ambiguous words and of corrupt expressions. Glossaries of Attic
+words were written by Crates, Philemon, Philetas and Theodorus;
+of Cretan, by Hermon or Hermonax; of Phrygian, by Neoptolemus;
+of Rhodian, by Moschus; of Italian, by Diodorus of Tarsus; of
+foreign words, by Silenus; of synonyms, by Simaristus; of cookery,
+by Heracleon; and of drinking vessels, by Apollodorus of Cyrene.
+According to Suidas, the most ancient Greek lexicographer was
+Apollonius the sophist, son of Archibius. According to the common
+opinion, he lived in the time of Augustus at Alexandria. He composed
+a lexicon of words used by Homer, <span class="grk" title="Lexeis Homêrikai">&#923;&#941;&#958;&#949;&#953;&#962; &#908;&#956;&#951;&#961;&#953;&#954;&#945;&#943;</span>, a very
+valuable and useful work, though much interpolated, edited by
+Villoison, from a MS. of the 10th century, Paris, 1773, 4to, 2 vols.;
+and by Tollius, Leiden, 1788, 8vo; ed. Bekker, Berlin, 1833, 8vo.
+Erotian or Herodian, physician to Nero, wrote a lexicon on Hippocrates,
+arranged in alphabetical order, probably by some copyist,
+whom Klein calls &ldquo;homo sciolus.&rdquo; It was first published in
+Greek in H. Stephani <i>Dictionarium Medicum</i>, Paris, 1564, 8vo; ed.
+Klein, Lipsiae, 1865, 8vo, with additional fragments. Timaeus the
+sophist, who, according to Ruhnken, lived in the 3rd century, wrote
+a very short lexicon to Plato, which, though much interpolated, is of
+great value, 1st ed. Ruhnken, Leiden, 1754; ed. locupletior, Lugd.
+Bat. 1789, 8vo. Aelius Moeris, called the Atticist, lived about 190
+<span class="pagenum"><a name="page191" id="page191"></a>191</span>
+<span class="sc">a.d.</span>, and wrote an Attic lexicon, 1st ed. Hudson, Oxf. 1712, Bekker,
+1833. Julius Pollux (<span class="grk" title="Ioulios Polydeukês">&#906;&#959;&#973;&#955;&#953;&#959;&#962; &#928;&#959;&#955;&#965;&#948;&#949;&#973;&#954;&#951;&#962;</span>) of Naucratis, in Egypt, died,
+aged fifty-eight, in the reign of Commodus (180-192), who made him
+professor of rhetoric at Athens. He wrote, besides other lost works,
+an Onomasticon in ten books, being a classed vocabulary, intended to
+supply all the words required by each subject with the usage of the
+best authors. It is of the greatest value for the knowledge both of
+language and of antiquities. First printed by Aldus, Venice, 1500,
+fol.; often afterwards; ed. Lederlinus and Hemsterhuis, Amst. 1706,
+2 vols.; Dindorf, 1824, 5 vols., Bethe (1900 f.). Harpocration of
+Alexandria, probably of the 2nd century, wrote a lexicon on the ten
+Attic orators, first printed by Aldus, Ven. 1503, fol.; ed. Dindorf,
+Oxford, 1853, 8vo, 2 vols. from 14 MSS. Orion, a grammarian of
+Thebes, in Egypt, who lived between 390 and 460, wrote an etymological
+dictionary, printed by Sturz, Leipzig, 1820, 4to. Helladius
+a priest of Jupiter at Alexandria, when the heathen temples there
+were destroyed by Theophilus in 389 or 391 escaped to Constantinople,
+where he was living in 408. He wrote an alphabetical lexicon, now
+lost, chiefly of prose, called by Photius the largest (<span class="grk" title="polystichôtaton">&#960;&#959;&#955;&#965;&#963;&#964;&#953;&#967;&#974;&#964;&#945;&#964;&#959;&#957;</span>)
+which he knew. Ammonius, professor of grammar at Alexandria,
+and priest of the Egyptian ape, fled to Constantinople with Helladius,
+and wrote a dictionary of words similar in sound but different in
+meaning, which has been often printed in Greek lexicons, as Aldus,
+1497, Stephanus, and separately by Valckenaer, Lugd. Bat. 1739,
+4to, 2 vols., and by others. Zenodotus wrote on the cries of animals,
+printed in Valckenaer&rsquo;s <i>Ammonius</i>; with this may be compared
+the work of Vincentio Caralucci, <i>Lexicon vocum quae a brutis animalibus
+emittuntur</i>, Perusia, 1779, 12mo. Hesychius of Alexandria wrote a
+lexicon, important for the knowledge of the language and literature,
+containing many dialectic and local expressions and quotations from
+other authors, 1st ed. Aldus, Ven. 1514, fol.; the best is Alberti and
+Ruhnken, Lugd. Bat. 1746-1766, fol. 2 vols.; collated with the MS.
+in St Mark&rsquo;s library, Venice, the only MS. existing, by Niels Iversen
+Schow, Leipzig, 1792, 8vo; ed. Schmidt, Jena, 1867, 8vo. The
+foundation of this lexicon is supposed to have been that of Pamphilus,
+an Alexandrian grammarian, quoted by Athenaeus, which, according
+to Suidas, was in 95 books from &Epsilon; to &Omega;; &Alpha; to &Delta; had been compiled
+by Zopirion. Photius, consecrated patriarch of Constantinople, 25th
+December 857, living in 886, left a lexicon, partly extant, and printed
+with Zonaras, Lips. 1808, 4to, 3 vols., being vol. iii.; ed. Naber,
+Leidae, 1864-1865, 8vo, 2 vols. The most celebrated of the Greek
+glossaries is that of Suidas, of whom nothing is known. He probably
+lived in the 10th century. His lexicon is an alphabetical dictionary
+of words including the names of persons and places&mdash;a compilation
+of extracts from Greek writers, grammarians, scholiasts and lexicographers,
+very carelessly and unequally executed. It was first
+printed by Demetrius Chalcondylas, Milan, 1499, fol.; the best
+edition, Bernhardy, Halle, 1853, 4to, 2 vols. John Zonaras, a celebrated
+Byzantine historian and theologian, who lived in the 12th
+century, compiled a lexicon, first printed by Tittmann, Lips. 1808.
+4to, 2 vols. An anonymous Greek glossary, entitled <span class="grk" title="Etymologikon mega">&#904;&#964;&#965;&#956;&#959;&#955;&#959;&#947;&#953;&#954;&#8056;&#957; &#956;&#941;&#947;&#945;</span>,
+<i>Etymologicum magnum</i>, has been frequently printed. The first
+edition is by Musurus, Venitia, 1499, fol.; the best by Gaisford,
+Oxonii, 1848, fol. It contains many grammatical remarks by famous
+authorities, many passages of authors, and mythological and
+historical notices. The MSS. vary so much that they look like the
+works of different authors. To Eudocia Augusta of Makrembolis, wife
+of the emperors Constantine XI. and Romanus IV. (1059 to 1071),
+was ascribed a dictionary of history and mythology, <span class="grk" title="Iônia">&#906;&#969;&#957;&#953;&#940;</span> (bed
+of violets), first printed by D&rsquo;Ansse de Villoison, <i>Anecdota Graeca</i>,
+Venetiis, 1781, 4to, vol. i. pp. 1-442. It was supposed to have been
+of much value before it was published. Thomas, Magister Officiorum
+under Andronicus Palaeologus, afterward called as a monk Theodulus,
+wrote <span class="grk" title="Eklogai onomatôn Attikôn">&#904;&#954;&#955;&#959;&#947;&#945;&#8054; &#8000;&#957;&#959;&#956;&#940;&#964;&#969;&#957; &#902;&#964;&#964;&#953;&#954;&#8182;&#957;</span>, printed by Callierges, Romae,
+1517, 8vo: Papias, <i>Vocabularium</i>, Mediolani, 1476, fol.: Craston,
+an Italian Carmelite monk of Piacenza, compiled a Greek and Latin
+lexicon, edited by Bonus Accursius, printed at Milan, 1478, fol.:
+Aldus, Venetiis, 1497, fol.: Guarino, born about 1450 at Favora,
+near Camarino, who called himself both Phavorinus and Camers,
+published his <i>Thesaurus</i> in 1504. These three lexicons were frequently
+reprinted. Estienne, <i>Thesaurus</i>, Genevae, 1572, fol., 4 vols.; ed.
+Valpy, Lond. 1816-1826, 6 vols. fol.; Paris, 1831-1865, 9 vols. fol., 9902
+pages: <span class="grk" title="Kibôtos">&#922;&#953;&#946;&#969;&#964;&#972;&#962;</span>, the ark, was intended to give the whole language,
+ancient and modern, but vol. i., Constantinople, 1819, fol., 763 pages,
+&Alpha; to &Delta;, only appeared, as the publication was put an end to by the
+events of 1821. <span class="sc">English.</span>&mdash;Jones, London, 1823, 8vo: Dunbar,
+Edin. 3rd ed. 1850, 4to: Liddell and Scott, 8th ed. Oxford, 1897, 4to.
+<span class="sc">French.</span>&mdash;Alexandre, 12th ed. Paris, 1863, 8vo; 1869-1871, 2 vols:
+Chassang, ib. 1872, 8vo. <span class="sc">Italian.</span>&mdash;Camini, Torino, 1865, 8vo, 972
+pages: Müller, ib. 1871, 8vo. <span class="sc">Spanish.</span>&mdash;<i>Diccionario manual, por les
+padres Esculapios</i>, Madrid, 1859, 8vo. <span class="sc">German.</span>&mdash;Passow, 5th ed.
+Leipzig, 1841-1857, 4to: Jacobitz and Seiler, 4th ed. ib. 1856, 8vo:
+Benseler, ib. 1859, 8vo: Pape, Braunschweig, 1870-1874, 8vo, 4 vols.
+Prellwitz, <i>Etymologisches Wörterbuch der griechischen Sprache</i>, new
+edition, 1906: Herwerden, <i>Lexicon Graecum suppletorium et dialecticum</i>,
+1902. <span class="sc">Dialects.</span>&mdash;<i>Attic</i>: Moeris, ed. Pierson, Lugd. Bat.
+1759. 8vo. <i>Attic Orators</i>: Reiske, Oxon. 1828, 8vo, 2 vols. <i>Doric</i>:
+Portus, Franckof. 1605, 8vo. <i>Ionic</i>: Id. ib. 1603, 8vo; 1817; 1825.
+<span class="sc">Prosody.</span>&mdash;Morell, Etonae, 1762, 4to; ed. Maltby, Lond. 1830, 4to:
+Brasse, Lond. 1850, 8vo. <span class="sc">Rhetoric.</span>&mdash;Ernesti, Lips. 1795, 8vo.
+<span class="sc">Music</span>.&mdash;Drieberg, Berlin, 1855. <span class="sc">Etymology</span>.&mdash;Curtius, Leipzig,
+1858-1862: Lancelot, Paris, 1863, 8vo. <span class="sc">Synonyms</span>.&mdash;Peucer, Dresden,
+1766, 8vo: Pillon, Paris, 1847, 8vo. <span class="sc">Proper Names</span>.&mdash;Pape, ed.
+Sengebusch, 1866, 8vo, 969 pages. <span class="sc">Verbs</span>.&mdash;Veitch, 2nd ed. Oxf.
+1866. <span class="sc">Terminations</span>.&mdash;Hoogeveen, Cantab. 1810, 4to: Pape,
+Berlin, 1836, 8vo. <span class="sc">Particular Authors</span>.&mdash;<i>Aeschylus</i>: Wellauer,
+2 vols. Lips. 1830-1831, 8vo. <i>Aristophanes</i>: Caravella, Oxonii, 1822,
+8vo. <i>Demosthenes</i>: Reiske, Lips. 1775, 8vo. <i>Euripides</i>: Beck,
+Cantab. 1829, 8vo. <i>Herodotus</i>: Schweighäuser, Strassburg, 1824, 8vo,
+2 vols. <i>Hesiod</i>: Osoruis, Neapol. 1791, 8vo. <i>Homer</i>: Apollonius
+Sophista, ed. Tollius, Lugd. Bat., 1788, 8vo: Schaufelberger, Zürich,
+1761-1768, 8vo, 8 vols.: Crusius, Hanover, 1836, 8vo: Wittich,
+London, 1843, 8vo: Döderlein, Erlangen, 8vo, 3 vols.: Eberling,
+Lipsiae, 1875, 8vo: Autenrieth, Leipzig, 1873, 8vo; London, 1877,
+8vo. <i>Isocrates</i>: Mitchell, Oxon. 1828, 8vo. <i>Pindar</i>: Portus,
+Hanov. 1606, 8vo. <i>Plato</i>: Timaeus, ed. Koch, Lips. 1828, 8vo:
+Mitchell, Oxon. 1832, 8vo: Ast, Lips. 1835-1838, 8vo, 3 vols.
+<i>Plutarch</i>: Wyttenbach, Lips. 1835, 8vo, 2 vols. <i>Sophocles</i>: Ellendt,
+Regiomonti, 1834-1835, 8vo ed.; Genthe, Berlin, 1872, 8vo. <i>Thucydides</i>:
+Bétant, Geneva, 1843-1847, 8vo, 2 vols. <i>Xenophon</i>: Sturtz,
+Lips. 1801-1804, 8vo, 4 vols.: Cannesin (Anabasis, Gr.-Finnish), Helsirgissä,
+1868, 8vo: Sauppe, Lipsiae, 1869, 8vo. <i>Septuagint</i>: Hutter,
+Noribergae, 1598, 4to: Biel, Hagae, 1779-1780, 8vo. <i>New Testament</i>:
+Lithocomus, Colon, 1552, 8vo: Parkhurst, ed. Major, London, 1845,
+8vo: Schleusner (juxta ed. Lips. quartam), Glasguae, 1824, 4to.</p>
+
+<p><span class="bold">Medieval and Modern Greek.</span>&mdash;Meursius, Lugd. Bat. 1614, 4to:
+Critopulos, Stendaliae, 1787, 8vo: Portius, Par. 1635, 4to: Du
+Cange, Paris, 1682, fol., 2 vols.; Ludg. 1688, fol. <span class="sc">English</span>.&mdash;Polymera,
+Hermopolis, 1854, 8vo: Sophocles, Cambr. Mass.
+1860-1887: Contopoulos, Athens, 1867, 8vo; Smyrna, 1868-1870,
+8vo, 2 parts, 1042 pages. <span class="sc">French</span>.&mdash;Skarlatos, Athens, 1852, 4to:
+Byzantius, ib. 1856, 8vo, 2 vols.: Varvati, 4th ed. ib., 1860, 8vo.
+<span class="sc">Italian</span>.&mdash;Germano, Romae, 1622, 8vo: Somavera, Parigi, 1709,
+fol., 2 vols.: Pericles, Hermopolis, 1857, 8vo. <span class="sc">German</span>.&mdash;Schmidt,
+Lips. 1825-1827, 12mo, 2 vols.: Petraris, Leipz. 1897. <span class="sc">Polyglots</span>.&mdash;Koniaz
+(Russian and Fr.), Moscow, 1811, 4to; Schmidt (Fr.-Germ.),
+Leipzig, 1837-1840, 12mo, 3 vols.: Theocharopulas de Patras (Fr.-Eng.),
+Munich, 1840, 12mo.</p>
+
+<p><span class="bold">Latin.</span>&mdash;Johannes de Janua, <i>Catholicon</i> or <i>Summa</i>, finished in
+1286, printed Moguntiæ 1460, fol.; Venice, 1487; and about 20
+editions before 1500: Johannes, <i>Comprehensorium</i>, Valentia, 1475,
+fol.: Nestor Dionysius, <i>Onomasticon</i>, Milan, 1477, fol.: Stephanus,
+Paris, 1531, fol., 2 vols.: Gesner, Lips. 1749, fol., 4 vols.: Forcellini,
+Patavii, 1771, fol., 4 vols. <span class="sc">Polyglot</span>.&mdash;Calepinus, Reggio, 1502, fol.
+(Aldus printed 16 editions, with the Greek equivalents of the Latin
+words; Venetiis, 1575, fol., added Italian, French and Spanish;
+Basileae, 1590, fol., is in 11 languages; several editions, from 1609,
+are called Octolingue; many of the latter 2 vol. editions were edited
+by John Facciolati): Verantius (Ital., Germ., Dalmatian, Hungarian),
+Venetiis, 1595, 4to: Lodereckerus (Ital., Germ., Dalm., Hungar.,
+Bohem., Polish), Pragae, 1605, 4to. <span class="sc">English</span>.&mdash;<i>Promptorium
+parvulorum</i>, compiled in 1440 by Galfridus Grammaticus, a Dominican
+monk of Lynn Episcopi, in Norfolk, was printed by Pynson, 1499;
+8 editions, 1508-1528, ed. Way, Camden Society, 1843-1865, 3 vols.
+4to; <i>Medulla grammaticis</i>, probably by the same author, MS. written
+1483; printed as <i>Ortus vocabulorum</i>, by Wynkyn de Worde, 1500;
+13 editions 1509-1523; Sir Thomas Elyot, London, 1538, fol.; 2nd ed.
+1543; <i>Bibliotheca Eliotae</i>, ed. Cooper, ib. 1545, fol.: Huloet,
+<i>Abecedarium</i>, London, 1552, fol.; <i>Dictionarie</i>, 1572, fol.: Cooper,
+London, 1565, fol.; 4th edition, 1584, fol.: Baret, <i>Alvearie</i>, ib. 1575,
+fol.; 1580, fol.: Fleming, ib. 1583, fol.: Ainsworth, London, 1736,
+4to; ed. Morell, London, 1796, 4to, 2 vols.; ed. Beatson and Ellis,
+ib. 1860, 8vo: Scheller, translated by Riddle, Oxford, 1835, fol.:
+Smith, London, 1855, 8vo; 1870: Lewis and Short, Oxford, 1879.
+<span class="sc">Eng.-Latin</span>.&mdash;Levins, <i>Manipulus puerorum</i>, Lond. 1570, 4to: Riddle,
+ib. 1838, 8vo: Smith, ib. 1855, 8vo. <span class="sc">French</span>.&mdash;<i>Catholicon parvum</i>,
+Geneva, 1487: Estienne, <i>Dictionnaire</i>, Paris, 1539, fol. 675 pages;
+enlarged 1549; ed. Huggins, Lond. 1572: Id. <i>Dictionarium Latino-Gallicum</i>,
+Lutetiae, 1546, fol.; Paris, 1552; 1560: Id., <i>Dictionariolum
+puerorum</i>, Paris, 1542, 4to: <i>Les Mots français</i>, Paris, 1544, 4to; the
+copy in the British Museum has the autograph of Queen Catherine
+Parr: Thierry (Fr.-Lat.), Paris, 1564, fol.: Danet, Ad usum
+Delphini, Paris, 1700, 4to, 2 vols.; and frequently: Quicherat, 9th
+ed. Paris, 1857, 8vo: Theil, 3rd ed. Paris, 1863, 8vo: Freund, ib.
+1835-1865, 4to, 3 vols. <span class="sc">German</span>.&mdash;Joh. Melber, of Gerolzhofen,
+<i>Vocabularius praedicantium</i>, of which 26 editions are described by
+Hain (<i>Repertorium</i>, No. 11,022, &amp;c.), 15 undated, 7 dated 1480-1495,
+4to, and 3 after 1504: <i>Vocabularius gemma gemmarum</i>, Antwerp,
+1484, 4to; 1487; 12 editions, 1505-1518: Herman Torentinus, <i>Elucidarius
+carminum</i>, Daventri, 1501, 4to; 22 editions, 1504-1536: Binnart,
+Ant. 1649, 8vo: Id., <i>Biglotton</i>, ib. 1661; 4th ed. 1688: Faber, ed.
+Gesner, Hagae Com. 1735, fol., 2 vols.: Hederick, Lips. 1766, 8vo,
+2 vols.: Ingerslev, Braunschweig, 1835-1855, 8vo, 2 vols.: <i>Thesaurus
+linguae Latinae</i>, Leipzig, 1900: Walde, <i>Lateinisches etymologisches
+Wörterbuch</i>, 1906. <span class="sc">Italian</span>.&mdash;Seebar (Sicilian translation of
+Lebrixa), Venet. 1525, 8vo: Venuti, 1589, 8vo: Galesini, Venez.
+1605, 8vo: Bazzarini and Bellini, Torino, 1864, 4to, 2 vols. 3100
+pages. <span class="sc">Spanish</span>.&mdash;Salmanticae, 1494, fol.; Antonio de Lebrixa,
+Nebrissenis, Compluti, 1520, fol., 2 vols.: Sanchez de la Ballesta,
+Salamanca, 1587, 4to: Valbuena, Madrid, 1826, fol. <span class="sc">Portuguese</span>.&mdash;Bluteau,
+<span class="pagenum"><a name="page192" id="page192"></a>192</span>
+Lisbon, 1712-1728, fol., 10 vols: Fonseca, ib. 1771, fol.:
+Ferreira, Paris, 1834, 4to; 1852. <span class="sc">Romansch</span>.&mdash;<i>Promptuario di voci
+volgari</i>, Valgrisii, 1565, 4to. <span class="sc">Vlach</span>.&mdash;Divalitu, Bucuresci, 1852,
+8vo. <span class="sc">Swedish</span>.&mdash;<i>Vocabula</i>, Rostock, 1574, 8vo; Stockholm, 1579:
+Lindblom, Upsala, 1790, 4to. <span class="sc">Dutch</span>.&mdash;Binnart, Antw. 1649, 8vo:
+Scheller, Lugd. Bat. 1799, 4to, 2 vols. <span class="sc">Flemish</span>.&mdash;Paludanus,
+Gandavi, 1544, 4to. <span class="sc">Polish</span>.&mdash;Macinius, Königsberg, 1564, fol.:
+Garszynski, Breslau, 1823, 8vo, 2 vols. <span class="sc">Bohemian</span>.&mdash;Johannes
+Aquensis, Pilsnae, 1511, 4to: Reschel, Olmucii, 1560-1562, 4to, 2 vols.:
+Cnapius, Cracovia, 1661, fol., 3 vols. <span class="sc">Illyrian</span>.&mdash;Bellosztenecz,
+Zagrab, 1740, 4to: Jambresich (also Germ. and Hungar.), Zagrab,
+1742, 4to. <span class="sc">Servian</span>.&mdash;Swotlik, Budae, 1721, 8vo. <span class="sc">Hungarian</span>.&mdash;Molnar,
+Frankf. a. M. 1645, 8vo: Pariz-Papai, Leutschen, 1708, 8vo;
+1767. <span class="sc">Finnish</span>.&mdash;Rothsen, Helsingissä, 1864, 8vo. <span class="sc">Poetic</span>.&mdash;<i>Epithetorum
+et synonymorum thesaurus</i>, Paris, 1662, 8vo, attributed
+to Chatillon; reprinted by Paul Aler, a German Jesuit, as <i>Gradus ad
+Parnassum</i>, Paris, 1687, 8vo; many subsequent editions: <i>Schirach</i>,
+Hal. 1768, 8vo: Noel, Paris, 1810, 8vo; 1826: Quicherat, Paris,
+1852, 8vo: Young, London, 1856, 8vo. <span class="sc">Erotic</span>.&mdash;Rambach,
+Stuttgart, 1836, 8vo. <span class="sc">Rhetorical</span>.&mdash;Ernesti, Lips. 1797, 8vo.
+<span class="sc">Civil Law</span>.&mdash;Dirksen, Berolini, 1837, 4to. <span class="sc">Synonyms</span>.&mdash;Hill, Edinb.
+1804, 4to: Döderlein, Lips. 1826-1828, 8vo, 6 vols. <span class="sc">Etymology</span>.&mdash;Danet,
+Paris, 1677, 8vo: Vossius, Neap. 1762, fol., 2 vols.: Salmon,
+London, 1796, 8vo, 2 vols.: Nagel, Berlin, 1869, 8vo; Latin roots,
+with their French and English derivatives, explained in German:
+Zehetmayr, Vindobonae, 1873, 8vo: Vani&#269;ek, Leipz. 1874, 8vo.
+<span class="sc">Barbarous</span>.&mdash;Marchellus, Mediol. 1753, 4to; Krebs, Frankf. a. M.
+1834, 8vo; 1837. <span class="sc">Particular Authors</span>.&mdash;<i>Caesar</i>: Crusius, Hanov.
+1838, 8vo. <i>Cicero</i>: Nizzoli, Brescia, 1535, fol.; ed. Facciolati,
+Patavii, 1734, fol.; London, 1820, 8vo, 3 vols.: Ernesti, Lips. 1739,
+8vo; Halle, 1831. <i>Cornelius Nepos</i>: Schmieder, Halle, 1798, 8vo;
+1816: Billerbeck, Hanover, 1825, 8vo. <i>Curtius Rufus</i>: Crusius,
+Hanov. 1844, 8vo. <i>Horace</i>: Ernesti, Berlin, 1802-1804, 8vo, 3 vols.:
+Döring, Leipz. 1829, 8vo. <i>Justin</i>: Meinecke, Lemgo, 1793, 8vo; 2nd
+ed. 1818. <i>Livy</i>: Ernesti, Lips. 1784, 8vo; ed Schäfer, 1804. <i>Ovid</i>:
+Gierig, Leipz. 1814: (Metamorphoses) Meinecke, 2nd ed., Lemgo, 1825,
+8vo: Billerbeck (Do.), Hanover, 1831, 8vo. <i>Phaedrus</i>: Oertel,
+Nürnberg, 1798, 8vo: Hörstel, Leipz. 1803, 8vo: Billerbeck
+Hanover, 1828, 8vo. <i>Plautus</i>: Paraeus, Frankf. 1614, 8vo. <i>Pliny</i>:
+Denso, Rostock, 1766, 8vo<i>. Pliny, jun.</i>: Wensch, Wittenberg, 1837-1839,
+4to. <i>Quintilian</i>: Bonnellus, Leipz. 1834, 8vo. <i>Sallust</i>:
+Schneider, Leipz. 1834, 8vo: Crusius, Hanover, 1840, 8vo. <i>Tacitus</i>:
+Bötticher, Berlin, 1830, 8vo. <i>Velleius Paterculus</i>: Koch, Leipz.
+1857, 8vo. <i>Virgil</i>: <i>Clavis</i>, London, 1742, 8vo: Braunhard, Coburg,
+1834, 8vo. <i>Vitruvius</i>: Rode, Leipz. 1679, 4to, 2 vols.: Orsini,
+Perugia, 1801, 8vo.</p>
+
+<p><span class="sc">Old Italian Languages</span>.&mdash;Fabretti, Torini, 1858, 4to. <i>Umbrian</i>:
+Huschke, Leipz. 1860, 8vo. <i>Oscan and Sabellian</i>: Id. Elberfeld,
+1856, 8vo.</p>
+
+<p><span class="sc">Medieval Latin</span>.&mdash;Du Cange, <i>Glossarium</i>, Paris, 1733-1736, fol.,
+6 vols.; Carpentier, Suppl., Paris, 1766, fol., 4 vols.; ed. Adelung,
+Halae, 1772-1784, 8vo, 6 vols.; ed. Henschel, Paris, 1840-1850, 4to,
+7 vols. (vol. vii. contains a glossary of Old French): Brinckmeier,
+Gotha, 1850-1863, 8vo, 2 vols.: Hildebrand (<i>Glossarium saec. ix.</i>),
+Götting. 1854, 4to: Diefenbach, <i>Glossarium</i>, Frankf. 1857, 4to: Id.
+<i>Gloss. novum</i>, ib. 1867, 4to. <span class="sc">Ecclesiastical</span>.&mdash;Magri, Messina, 1644,
+4to; 8th ed. Venezia, 1732; Latin translation, <i>Magri Hierolexicon</i>,
+Romae, 1677, fol.; 6th ed. Bologna, 1765, 4to, 2 vols.</p>
+
+<p class="center pt2"><i>Romance Languages.  </i></p>
+
+<p><span class="bold">Romance Languages generally.</span>&mdash;Diez, Bonn, 1853, 8vo; 2nd ed.
+ib. 1861-1862, 8vo, 2 vols.; 3rd ed. ib. 1869-1870, 8vo, 2 vols.;
+transl. by Donkin, 1864, 8vo.</p>
+
+<p><span class="bold">French.</span>&mdash;Ranconet, <i>Thresor</i>, ed. Nicot, Paris, 1606, fol.; ib.
+1618, 4to: Richelet, Genève, 1680, fol., 2 vols.; ed. Gattel, Paris,
+1840, 8vo, 2 vols.</p>
+
+<p>The French Academy, after five years&rsquo; consideration, began their
+dictionary, on the 7th of February 1639, by examining the letter A,
+which took them nine months to go through. The word Académie was
+for some time omitted by oversight. They decided, on the 8th of March
+1638, not to cite authorities, and they have since always claimed the
+right of making their own examples. Olivier justifies them by saying
+that for eighty years all the best writers belonged to their body, and
+they could not be expected to cite each other. Their design was to
+raise the language to its last perfection, and to open a road to reach
+the highest eloquence. Antoine Furetière, one of their members,
+compiled a dictionary which he says cost him forty years&rsquo; labour for
+ten hours a day, and the manuscript filled fifteen chests. He gave
+words of all kinds, especially technical, names of persons and places,
+and phrases. As a specimen, he published his <i>Essai</i>, Paris, 1684,
+4to; Amst. 1685, 12mo. The Academy charged him with using the
+materials they had prepared for their dictionary, and expelled him, on
+the 22nd of January 1685, for plagiarism. He died on the 14th of May
+1688, in the midst of the consequent controversy and law suit. His
+complete work was published, with a preface by Bayle, La Haye and
+Rotterdam, 1690, fol., 3 vols.; again edited by Basnage de Beauval,
+1701; La Haye, 1707, fol., 4 vols. From the edition of 1701 the
+very popular so-called <i>Dictionnaire de Trevoux</i>, Trevoux, 1704, fol.,
+2 vols., was made by the Jesuits, who excluded everything that
+seemed to favour the Calvinism of Basnage. The last of its many
+editions is Paris, 1771, fol., 8 vols. The Academy&rsquo;s dictionary was
+first printed Paris, 1694, fol., 2 vols. They began the revision in 1700;
+second edition 1718, fol., 2 vols.; 3rd, 1740, fol., 2 vols.; 6th, 1835,
+2 vols. 4to, reprinted 1855; Supplément, by F. Raymond, 1836,
+4to; Complément, 1842, 4to, reprinted 1856; <i>Dictionnaire historique</i>,
+Paris, 1858-1865, 4to, 2 parts (A to Actu), 795 pages, published by the
+Institut: Dochez, Paris, 1859, 4to: Bescherelle, ib. 1844, 4to, 2 vols.;
+5th ed. Paris, 1857, 4to, 2 vols.; 1865; 1887: Landais, Paris, 1835;
+12th ed. ib. 1854, 4to, 2 vols.: Littré, Paris, 1863-1873, 4to, 4 vols.
+7118 pages: Supplément, Paris, 1877, 4to: Godefroy (with dialects
+from 9th to 15th cent.), Paris, 1881-1895, and <i>Complément</i>: Hatzfield,
+Darmesteter, and Thomas, Paris, 1890-1900: Larive and Fleury,
+(<i>mots et choses, illustré</i>), Paris, 1884-1891. <span class="sc">English</span>.&mdash;Palsgrave,
+<i>Lesclaircissement de la langue francoyse</i>, London, 1530, 4to, 2 parts;
+1852: Hollyband, London, 1533, 4to: Cotgrave, ib. 1611, fol.:
+Boyer, La Haye, 1702, 4to, 2 vols.; 37th ed. Paris, 1851, 8vo, 2 vols.:
+Fleming and Tibbins, Paris, 1846-1849, 4to, 2 vols.; ib. 1854, 4to,
+2 vols.; ib. 1870-1872, 4to, 2 vols.: Tarver, London, 1853-1854,
+8vo, 2 vols.; 1867-1872: Bellows, Gloucester, 1873, 16mo; ib.
+1876. <span class="sc">Ideological, or Analogical</span>.&mdash;Robertson, Paris, 1859, 8vo:
+Boissière, Paris, 1862, 8vo. <span class="sc">Etymology</span>.&mdash;Lebon, Paris, 1571, 8vo:
+Ménage, ib. 1650, 4to. Pougens projected a <i>Trésor des origines</i>, his
+extracts for which, filling nearly 100 volumes folio, are in the library
+of the Institut. He published a specimen, Paris, 1819, 4to. After
+his death, <i>Archéologie française</i>, Paris, 1821, 8vo, 2 vols., was compiled
+from his MSS., which were much used by Littré: Scheler,
+Bruxelles, 1862, 8vo; 1873: Brachet, 2nd ed. Paris, 1870, 12mo;
+English trans. Kitchin, Oxf. 1866, 8vo. <span class="sc">Greek Words</span>.&mdash;Trippault,
+Orleans, 1580, 8vo: Morin, Paris, 1809, 8vo. <span class="sc">German Words</span>.&mdash;Atzler,
+Cöthen, 1867, 8vo. <span class="sc">Oriental Words</span>.&mdash;Pihan, Paris, 1847,
+8vo; 1866: Devic, ib. 1876, 8vo. <span class="sc">Neology</span>.&mdash;Desfontaines, 3rd ed.
+Amst. 1728, 12mo: Mercier, Paris, 1801, 8vo, 2 vols.: Richard, ib.
+1842, 8vo; 2nd ed. 1845. <span class="sc">Poetic</span>.&mdash;<i>Dict. des rimes</i> (by La Noue),
+Geneve, 1596, 8vo; Cologny, 1624, 8vo: Carpentier, <i>Le Gradus
+français</i>, Paris, 1825, 8vo, 2 vols. <span class="sc">Erotic</span>.&mdash;De Landes, Bruxelles,
+1861, 12mo. <span class="sc">Oratory</span>.&mdash;Demandre and Fontenai, Paris, 1802, 8vo:
+Planche, ib. 1819-1820, 8vo, 3 vols. <span class="sc">Pronunciation</span>.&mdash;Féline, ib.
+1857, 8vo. <span class="sc">Double Forms</span>.&mdash;Brachet, ib. 1871, 8vo. <span class="sc">Epithets</span>.&mdash;Daire,
+ib. 1817, 8vo. <span class="sc">Verbs</span>.&mdash;Bescherelle, ib. 1855, 8vo, 2 vols.:
+3rd ed. 1858. <span class="sc">Participles</span>.&mdash;Id., ib. 1861, 12mo. <span class="sc">Difficulties</span>.&mdash;Boiste,
+London, 1828, 12mo: Laveaux, Paris, 1872, 8vo, 843 pages.
+<span class="sc">Synonyms</span>.&mdash;Boinvilliers, Paris, 1826, 8vo: Lafaye, ib. 1858,
+8vo; 1861; 1869: Guizot, ib. 1809, 8vo; 6th ed. 1863; 1873.
+<span class="sc">Homonyms</span>.&mdash;Zlatagorski (Germ., Russian, Eng.), Leipzig, 1862,
+8vo, 664 pages. <span class="sc">Imitative Words</span>.&mdash;Nodier, <i>Onomatopées</i>, ib. 1828,
+8vo. <span class="sc">Technology</span>.&mdash;D&rsquo;Hautel, ib. 1808, 8vo, 2 vols.: Desgranges,
+ib. 1821, 8vo: Tolhausen (Fr., Eng., Germ.), Leipz. 1873, 8vo, 3 vols.
+<span class="sc">Faults of Expression</span>.&mdash;Roland, Gap, 1823, 8vo: Blondin, Paris,
+1823, 8vo. <span class="sc">Particular Authors</span>.&mdash;<i>Corneille</i>: Godefroy, ib. 1862,
+8vo, 2 vols.: Marty-Laveaux, ib. 1868, 8vo, 2 vols. <i>La Fontaine</i>:
+Lorin, ib. 1852, 8vo. <i>Malherbe</i>: Regnier, ib. 1869, 8vo. <i>Molière</i>:
+Genin, ib. 1846, 8vo: Marty-Laveaux, ib. 8vo. <i>Racine</i>: Marty-Laveaux,
+ib. 1873, 8vo, 2 vols. <i>M<span class="sp">me</span> de Sévigné</i>: Sommer, ib. 1867,
+8vo, 2 vols. <span class="sc">Old French</span>.&mdash;La Curne de St Palaye prepared a
+dictionary, of which he only published <i>Projet d&rsquo;un glossaire</i>, Paris,
+1756, 4to. His MSS. in many volumes are in the National Library,
+and were much used by Littré. They were printed by L. Favre, and
+fasciculi 21-30 (tom. iii.), Niort, 4to, 484 pages, were published in
+February 1877. Lacombe (vieux langage), Paris, 1766, 2 vols. 4to:
+Kelham (Norman and Old French), London, 1779, 8vo: Roquefort
+(langue romane), Paris, 1808, 8vo; Supplément, ib. 1820, 8vo:
+Pougens, <i>Archéologie</i>, ib. 1821, 8vo, 2 vols.: Burguy, Berlin, 1851-1856,
+8vo, 3 vols.: Laborde (<i>Notice des émaux ... du Louvre</i>, part ii.),
+Paris, 1853, 8vo, 564 pages:<a name="fa3d" id="fa3d" href="#ft3d"><span class="sp">3</span></a> Gachet (rhymed chronicles), Bruxelles,
+1859, 4to: Le Héricher (Norman, English and French), Paris, 1862,
+3 vols. 8vo: Hippeau (12th and 13th centuries), Paris, 1875, 8vo.
+<span class="sc">Dialects</span>.&mdash;Jaubert (central), Paris, 1856-1857, 8vo, 2 vols.:
+Baumgarten (north and centre), Coblentz, 1870, 8vo: Azais, <i>Idiomes
+romans du midi</i>, Montpellier, 1877. <i>Austrasian</i>: François. Metz,
+1773, 8vo. <i>Auvergne</i>: Mège, Riom, 1861, 12mo. <i>Bearn</i>: Lespi, Pau,
+1858, 8vo. <i>Beaucaire</i>: Bonnet (Bouguirén), Nismes, 1840, 8vo.
+<i>Pays de Bray</i>: Decorde, Neufchâtel, 1852, 8vo. <i>Burgundy</i>:
+Mignard, Dijon, 1870, 8vo. <i>Pays de Castres</i>: Couzinié, Castres,
+1850, 4to. <i>Dauphiné</i>: Champollion-Figeac, Paris, 1809, 8vo: Jules,
+Valence, 1835, 8vo; Paris, 1840, 4to. <i>Dep. of Doubs</i>: Tissot
+(Patois des Fourg, arr. de Pontarlier) Besançon, 1865, 8vo.
+<i>Forez</i>: Gras, Paris, 1864, 8vo; Neolas, Lyon, 1865, 8vo. <i>Franche
+Comté</i>: Maisonforte, 2nd ed. Besançon, 1753, 8vo. <i>Gascony</i>: Desgrouais
+(Gasconismes corrigés), Toulouse, 1766, 8vo; 1769; 1812,
+12mo, 2 vols.; 1825, 8vo, 2 vols. <i>Dep. of Gers</i>: Cenac-Montaut, Paris,
+1863, 8vo. <i>Geneva</i>: Humbert, Geneve, 1820, 8vo. <i>Languedoc</i>: Odde,
+Tolose, 1578, 8vo: Doujat, Toulouse, 1638, 8vo: De S.[auvages],
+Nismes, 1756, 2 vols.; 1785; Alais, 1820: Azais, Beziers, 1876,
+&amp;c., 8vo: Hombres, Alais, 1872, 4to: Thomas (<i>Greek words</i>) Montpellier,
+1843, 4to. <i>Liége</i>: Forir, Liége, 1866, 8vo, vol i. 455 pages.
+<i>Lille</i>: Vermesse, Lille, 1861, 12mo: Debuire du Buc ib., 1867,
+8vo. <i>Limousin</i>: Beronie, ed. Vialle (Corrèze), Tulle, 1823, 4to.
+<span class="pagenum"><a name="page193" id="page193"></a>193</span>
+<i>Lyonnais, Forez, Beaujolais</i>: Onofrio, Lyon, 1864, 8vo. <i>Haut
+Maine</i>: R[aoul] de M.[ontesson], Paris, 1857; 1859, 503 pages.
+<i>Mentone</i>: Andrews, Nice, 1877, 12mo. <i>Dep. de la Meuse</i>: Cordier,
+Paris, 1853, 8vo. <i>Norman</i>: Edélestand and Alfred Duméril, Caen,
+1849, 8vo: Dubois, ib. 1857, 8vo: Le Héricher (<i>Philologie topographique</i>),
+Caen, 1863, 4to: Id. (éléments scandinaves), Avranches,
+1861, 12mo: Metivier (Guernsey), London, 1870, 8vo: Vasnier
+(arrond de Pont Audemer), Rouen, 1861, 8vo: Delboulle (Vallée
+d&rsquo;Yères), Le Havre, 1876. <i>Picardy</i>: Corblet, Amiens, 1851, 8vo.
+<i>Poitou, Saintonge, Aunis</i>: Favre, Niort, 1867, 8vo. <i>Poitou</i>:
+Beauchet-Filleau, Paris, 1864, 8vo: Levrier, Niort, 1867, 8vo:
+Lalanne, Poitiers, 1868, 8vo. <i>Saintonge</i>: Boucherie, Angoulême,
+1865, 8vo: Jonain, Royan, 1867, 8vo. <i>Savoy</i>: Pont (Terratzu de
+la Tarantaise), Chambery, 1869, 8vo. <i>La Suisse Romande</i>: Bridel,
+Lausanne, 1866, 8vo. <i>Dep. of Tarn</i>: Gary, Castre, 1845, 8vo. <i>Dep.
+of Vaucluse</i>: Barjavel, Carpentras, 1849, 8vo. <i>Walloon (Rouchi)</i>:
+Cambresier, Liége, 1787, 8vo: Grandgagnage, ib. 1845-1850, 8vo.
+2 vols.: Chavée, Paris, 1857, 18mo: Vermesse, Doudi, 1867, 8vo.
+Sigart (<i>Montois</i>), Bruxelles, 1870, 8vo. <span class="sc">Slang</span>.&mdash;Oudin, <i>Curiositez
+Françaises</i>, Paris, 1640, 8vo: Baudeau de Saumaise (Précieuses,
+Langue de Ruelles), Paris, 1660, 12mo; ed. Livet, ib. 1856: Le
+Roux, <i>Dict. Comique</i>, Amst. 1788, and 6 other editions: Carême
+Prenant [<i>i.e.</i> Taumaise], (argot réforme), Paris, 1829, 8vo: Larchey
+(excentricitées du langage), Paris, 1860, 12mo; 5th ed. 1865:
+Delvau (langue verte, Parisian), Paris, 1867, 8vo: Larchey, Paris,
+1873, 4to, 236 pages.</p>
+
+<p><span class="bold">Provençal.</span>&mdash;Pallas, Avignon, 1723, 4to: Bastero, <i>La Crusca Provenzale</i>,
+Roma, 1724, fol. vol. i. only: Raynouard, Paris, 1836-1844,
+8vo, 6 vols.: Garcin, Draguignand, 1841, 8vo, 2 vols.: Honnorat,
+Digne, 1846-1849, 4to, 4 vols. 107,201 words: Id., <i>Vocab. fr. prov.</i>,
+ib. 1848, 12mo, 1174 pages.</p>
+
+<p><span class="bold">Spanish.</span>&mdash;Covarruvias Orosco, Madrid, 1611, fol.: ib. 1673-1674,
+fol. 2 vols.; Academia Española, Madrid, 1726-1739, fol. 6 vols.; 8th
+ed. 1837: Caballero, Madrid, 1849, fol.; 8th ed. ib. 1860, 4to, 2 vols.:
+Cuesta, ib. 1872, fol. 2 vols.: Campano, Paris, 1876, 18mo, 1015 pages.
+Cuervo, 1886-1894; Monlau, 1881; Zerola, Toro y Gomes, and Isaza,
+1895; Serrano (encyclopaedic) 1876-1881. <span class="sc">English</span>.&mdash;Percivall,
+London, 1591, 4to: Pineda, London, 1740, fol.: Connelly and
+Higgins, Madrid, 1797-1798, 4to, 4 vols.: Neuman and Baretti, 9th ed.
+London, 1831, 8vo, 2 vols.; 1874. <span class="sc">French</span>.&mdash;Oudin, Paris, 1607, 4to,
+1660; Gattel, Lyon, 1803, 4to, 2 vols.: Dominguez, Madrid, 1846,
+8vo, 6 vols.: Blanc, Paris, 1862, 8vo, 2 vols. <span class="sc">German</span>.&mdash;Wagener,
+Hamb. 1801-1805, 8vo, 4 vols.: Seckendorp, ib. 1823, 8vo, 3 vols.:
+Franceson, 3rd ed. Leipzig, 1862, 8vo, 2 vols. <span class="sc">Italian</span>.&mdash;Franciosini,
+Venezia, 1735, 8vo, 2 vols.; Cormon y Manni, Leon, 1843, 16mo,
+2 vols.: Romero, Madrid, 1844, 4to. <span class="sc">Synonyms</span>.&mdash;<i>Diccionario de
+Sinonimos</i>, Paris, 1853, 4to. <span class="sc">Etymology</span>.&mdash;Aldrete, Madrid, 1682,
+fol.: Monlau y Roca, ib. 1856, 12mo; Barcia, 1881-1883. <span class="sc">Arabic
+Words</span>.&mdash;Hammer Purgstall, Wien, 1855, 8vo: Dozy and Engelmann,
+2d ed. Leiden, 1869, 8vo. <span class="sc">Ancient</span>.&mdash;Sanchez, Paris, 1842,
+8vo. <span class="sc">Rhyming</span>.&mdash;Garcia de Rengifo (consonancias) Salmantica,
+1592, 4to; 1876. <span class="sc">Don Quixote</span>.&mdash;Beneke (German), Leipzig, 1800,
+16mo; 4th ed. Berlin, 1841, 16mo. <span class="sc">Dialects</span>.&mdash;<i>Aragonese</i>: Peralta,
+Zaragoza, 1836, 8vo: Borao, ib. 1859, 4to. <i>Catalan</i>: Rocha de
+Girona (Latin), Barcinone, 1561, fol.: <i>Dictionari Catala</i> (Lat. Fr.
+Span.), Barcelona, 1642, 8vo: Lacavalleria (Cat.-Lat.), ib. 1696, fol.:
+Esteve, ed. Belvitges, &amp;c. (Catal.-Sp. Lat.), Barcelona, 1805-1835,
+fol. 2 vols.: Saura (Cat.-Span.), ib. 1851, 16mo; 2nd ed.(Span.-Cat.),
+ib. 1854; 3rd ed. (id.) ib. 1862, 8vo: Labernia, ib. 1844-1848, 8vo, 2
+vols. 1864. <i>Gallegan</i>: Rodriguez, Coruña, 1863, 4to: Cuveira y Piñol,
+Madrid, 1877, 8vo. Majorca: Figuera, Palma, 1840, 4to: Amengual,
+ib. 1845, 4to. <i>Minorca</i>: <i>Diccionario</i>, Madrid, 1848, 8vo. <i>Valencian</i>:
+Palmyreno, Valentiae, 1569: Ros, Valencia, 1764, 8vo: Fuster, ib.
+1827, 8vo: Lamarca, 2nd ed. ib. 1842, 16mo. <i>Cuba</i>: <i>Glossary of
+Creole Words</i>, London, 1840, 8vo: Pichardo, 1836; 2nd ed. Havana,
+1849, 8vo; 3rd ed. ib. 1862, 8vo; Madrid, 1860, 4to.</p>
+
+<p><span class="bold">Portuguese.</span>&mdash;Lima, Lisbon, 1783, 4to: Moraes da Silva, ib.
+1789, 4to, 2 vols.; 6th ed. 1858: Academia real das Sciencas, ib.
+1793, tom. i., ccvi. and 544 pages (A to Azurrar); Faria, ib. 1849,
+fol. 2 vols.; 3rd ed. ib. 1850-1857, fol. 2 vols. 2220 pages. <span class="sc">English</span>.&mdash;Vieyra,
+London, 1773, 2 vols. 4to: Lacerda, Lisboa, 1866-1871, 4to,
+2 vols. <span class="sc">French</span>.&mdash;Marquez, Lisboa, 1756-1761, fol. 2 vols.: Roquette,
+Paris, 1841, 8vo, 2 vols.; 4th ed. 1860: Marques, Lisbonne, 1875,
+fol. 2 vols.: Souza Pinto, Paris, 1877, 32mo, 1024 pages. <span class="sc">German</span>.&mdash;Wagener,
+Leipzig, 1811-1812, 8vo, 2 vols.: Wollheim, ib. 1844, 12mo,
+2 vols.: Bösche, Hamburg, 1858, 8vo, 2 vols. 1660 pages. <span class="sc">Italian</span>.&mdash;Costa
+e Sá, Lisboa, 1773-1774, fol. 2 vols. 1652 pages: Prefumo,
+Lisboa, 1853, 8vo, 1162 pages. <span class="sc">Ancient</span>.&mdash;Joaquim de Sancta Rosa
+de Viterbo, ib. 1798, fol. 2 vols.; 1824, 8vo. <span class="sc">Arabic Words</span>.&mdash;Souza,
+ib. 1789, 4to; 2nd ed. by S. Antonio Moura, ib. 1830, 224 pages.
+<span class="sc">Oriental and African Words, not Arabic</span>.&mdash;Saõ Luiz, ib. 1837,
+4to, 123 pages. <span class="sc">French Words</span>.&mdash;Id., ib. 1827, 4to; 2nd ed. Rio de
+Janeiro, 1835, 8vo. <span class="sc">Synonyms</span>.&mdash;Id., ib. 1821, 4to; 2nd ed. ib.
+1824-1828, 8vo. Fonseca, Paris, 1833, 8vo; 1859, 18mo, 863 pages.
+<span class="sc">Homonyms</span>.&mdash;De Couto, Lisboa, 1842, fol. <span class="sc">Poetic</span>.&mdash;Luzitano (<i>i.e.</i>
+Freire), ib. 1765, 8vo, 2 vols.; 3rd ed. ib. 1820, 4to, 2 vols. <span class="sc">Rhyming</span>.&mdash;Couto
+Guerreiro, Lisboa, 1763, 4to. <span class="sc">Naval</span>.&mdash;Tiberghien, Rio de
+Janeiro, 1870, 8vo. <span class="sc">Ceylon-Portuguese</span>.&mdash;Fox, Colombo, 1819,
+8vo: Callaway, ib. 1823, 8vo.</p>
+
+<p><span class="bold">Italian.</span>&mdash;Accarigi, <i>Vocabulario</i>, Cento, 1543, 4to: Alunno, <i>La</i>
+<i>fabrica del mundo</i>, Vinezia, 1548, fol.: Porccachi, Venetia, 1588, fol.:
+Accademici della Crusca, <i>Vocabulario</i>, Venez. 1612, fol.; 4th ed.
+Firenze, 1729-1738, fol. 6 vols.: Costa and Cardinali, Bologna, 1819-1826,
+4to, 7 vols.: Tommaseo and Bellini, Torino, 1861, &amp;c., 4to, 4
+vols.: Petrocchi, 1884-1891. <span class="sc">English.</span>&mdash;Thomas, London, 1598, 4to:
+Florio, London, 1598, 4to, 1611: Baretti, London, 1794, 2 vols.:
+1854, 8vo, 2 vols.: Petronj and Davenport, Londra, 1828, 8vo, 3 vols.:
+Grassi, Leipz. 1854, 12mo: Millhouse, Lond., 1868, 8vo, 2 vols. 1348
+pages. <span class="sc">French.</span>&mdash;Alberti, Paris, 1771, 4to, 2 vols.; Milan, 1862:
+Barberi, Paris, 1838, 4to, 2 vols.: Renzi, Paris, 1850, 8vo. <span class="sc">German.</span>&mdash;<i>Libro
+utilissimo</i>, Venetiis, 1499, 4to: Valentini, Leipzig, 1834-1836,
+4to, 4 vols. <span class="sc">Etymology.</span>&mdash;Menage, Geneva, 1685, fol.: Bolza, Vienna,
+1852, 4to. <span class="sc">Provençal Words.</span>&mdash;Nannucci, Firenze, 1840, 8vo.
+<span class="sc">Synonyms.</span>&mdash;Rabbi, Venezia, 1774, 4to; 10th ed. 1817; Tommaseo,
+Firenze, 1839-1840, 4to, 2 vols.: Milano, 1856, 8vo; 1867. <span class="sc">Verbs.</span>&mdash;Mastrofini,
+Roma, 1814, 4to, 2 vols. <span class="sc">Select Words and Phrases.</span>&mdash;Redi,
+Brescia, 1769, 8vo. <span class="sc">Incorrect Words and Phrases.</span>&mdash;Molassi,
+Parma, 1830-1841, 8vo, 854 pages. <span class="sc">Supposed Gallicisms.</span>&mdash;Viani,
+Firenze, 1858-1860, 8vo, 2 vols. <span class="sc">Additions To the Dictionaries.</span>&mdash;Gherardini,
+Milano, 1819-1821, 8vo, 2 vols.; ib. 1852-1857,
+8vo, 6 vols. <span class="sc">Rhyming.</span>&mdash;Falco, Napoli, 1535, 4to: Ruscelli, Venetia,
+1563, 8vo; 1827: Stigliani, Roma, 1658, 8vo: Rosasco, Padova,
+1763, 4to; Palermo, 1840, 8vo. <span class="sc">Technical.</span>&mdash;Bonavilla-Aquilino,
+Mil. 1819-1821, 8vo, 5 vols.; 2nd ed. 1829-1831, 4to, 2 vols.: Vogtberg
+(Germ.), Wein, 1831, 8vo. <span class="sc">Particular Authors.</span>&mdash;<i>Boccaccio</i>:
+Aluno, <i>Le ricchezze della lingua volgare</i>, Vinegia, 1543. fol. <i>Dante</i>:
+Blanc, Leipzig, 1852, 8vo; Firenze, 1859, 8vo. <span class="sc">Dialects.</span>&mdash;<i>Bergamo</i>:
+Gasparini, Mediol. 1565: Zappetini, Bergamo, 1859, 8vo:
+Tiraboschi (anc. and mod.), Turin, 1873, 8vo. <i>Bologna</i>: Bumaldi,
+Bologna, 1660, 12mo: Ferrari, ib. 1820, 8vo; 1838, 4to. <i>Brescia</i>:
+Gagliardi, Brescia, 1759, 8vo: Melchiori, ib. 1817-1820, 8vo: <i>Vocabularietto</i>,
+ib. 1872, 4to. <i>Como</i>: Monti, Milano, 1845, 8vo. <i>Ferrara</i>:
+Manini, Ferrara, 1805, 8vo: Azzi, ib. 1857, 8vo. <i>Friuli</i>: Scala,
+Pordenone, 1870, 8vo. <i>Genoa</i>: Casaccia, Gen. 1842-1851, 8vo; 1873,
+&amp;c.: Paganini, ib. 1857, 8vo. <i>Lombardy</i>: Margharini, Tuderti,
+1870, 8vo. <i>Mantua</i>: Cherubini, Milano, 1827, 4to. <i>Milan</i>: Varon,
+ib. 1606, 8vo: Cherubini, ib. 1814, 8vo, 2 vols.; 1841-1844, 8vo,
+4 vols.; 1851-1861, 8vo, 5 vols.: Banfi, ib. 1857, 8vo: 1870, 8vo.
+<i>Modena</i>: Galvani, Modena, 1868, 8vo. <i>Naples</i>: Galiani, Napoli,
+1789, 12mo, 2 vols. <i>Parma</i>: Peschieri, Parma, 1828-1831, 8vo, 3
+vols. 1840; Malespina, ib. 1856, 8vo, 2 vols. <i>Pavia</i>: <i>Dizionario domestico
+pavese</i>, Pavia, 1829, 8vo: Gambini, ib. 1850, 4to, 346 pages.
+<i>Piacenza</i>: Nicolli, Piacenza, 1832: Foresti, ib. 1837-1838, 8vo, 2 pts.
+<i>Piedmont</i>: Pino, Torino, 1784, 4to: Capello (Fr.), Turin, 1814, 8vo,
+2 pts.: Zalli (Ital. Lat. Fr.), Carmagnola, 1815, 8vo, 2 vols: Sant&rsquo;
+Albino, Torino, 1860, 4to. <i>Reggio</i>: <i>Vocabulario Reggiano</i>, 1832.
+<i>Romagna</i>: Morri, Fienza, 1840. <i>Rome</i>: <i>Raccolto di voci Romani e
+Marchiani</i>, Osimo, 1769, 8vo. <i>Roveretano and Trentino</i>: Azzolini,
+Venezia, 1856, 8vo. <i>Sardinia</i>: Porru, Casteddu, 1832, fol.: Spano,
+Cagliari, 1851-1852, fol. 3 vols. <i>Sicily</i>: Bono (It. Lat.), Palermo,
+1751-1754, 4to, 3 vols.; 1783-1785, 4to, 5 vols.: Pasqualino, ib. 1785-1795,
+4to, 5 vols.: Mortillaro, ib. 1853, 4to, 956 pages: Biundi, ib. 1857,
+12mo, 578 pages: Traina, ib. 1870, 8vo. <i>Siena</i>: Barbagli, Siena,
+1602, 4to. <i>Taranto</i>: Vincentiis, Taranto, 1872, 8vo. <i>Turin</i>:
+Somis di Chavrie, Torino, 1843, 8vo. <i>Tuscany</i>: Luna, Napoli, 1536,
+4to: Politi, Roma, 1604, 8vo; Venezia, 1615; 1628; 1665; Paulo,
+ib. 1740, 4to. <i>Vaudois</i>: Callet, Lausanne, 1862, 12mo. <i>Venetian</i>:
+Patriarchi (<i>Veneziano e padevano</i>), Padova, 1755, 4to; 1796, 1821:
+Boerio, Venezia, 1829, 4to; 1858-1859; 1861. <i>Verona</i>: Angeli,
+Verona, 1821, 8vo. <i>Vicenza</i>: Conti, Vicenza, 1871, 8vo. <span class="sc">Lingua
+Franca.</span>&mdash;<i>Dictionnaire de la langue Franque, ou Petit Mauresque</i>,
+Marseille, 1830, 16mo, 107 pages. <span class="sc">Slang.</span>&mdash;Sabio (lingua Zerga),
+Venetia, 1556, 8vo; 1575: <i>Trattato degli bianti</i>, Pisa, 1828, 8vo.</p>
+
+<p><span class="bold">Romansh.</span>&mdash;<i>Promptuario de voci volgari e Latine</i>, Valgrisii,
+1565, 4to: <i>Der, die, das, oder Nomenclatura</i> (German nouns
+explained in Rom.), Scoul, 1744, 8vo: Conradi, Zurich, 1820, 8vo;
+1826, 12mo, 2 vols.: Carisch, Chur, 1821, 8vo; 1852, 16mo.</p>
+
+<p><span class="bold">Vlach.</span>&mdash;<i>Lesicon Rumanese</i> (Lat. Hung. Germ.), Budae, 1825,
+4to: Bobb (Lat. Hung.), Clus, 1822-1823, 4to, 2 vols. <span class="sc">French.</span>&mdash;Vaillant,
+Boucoureshti, 1840, 8vo: Poyenar, Aaron and Hill,
+Boucourest, 1840-1841, 4to, 2 vols.; Jassi, 1852, 16mo, 2 vols.:
+De Pontbriant, Bucuresci, 1862, 8vo: Cihac, Frankf. 1870, 8vo:
+Costinescu, Bucuresci, 1870, 8vo, 724 pages: Antonescu, Bucharest,
+1874, 16mo, 2 vols. 919 pages. <span class="sc">German.</span>&mdash;Clemens, Hermanstadt,
+1823, 8vo: Isser, Kronstadt, 1850: Polyzu, ib. 1857, 8vo.</p>
+
+<p class="center pt2"><span class="sc">Teutonic</span>: (1) <i>Scandinavian.</i></p>
+
+<p><span class="bold">Icelandic.</span>&mdash;<span class="sc">Latin.</span>&mdash;Andreae, Havniae, 1683, 8vo: Halderson
+(Lat. Danish), ib. 1814, 4to, 2 vols. <span class="sc">English.</span>&mdash;Cleasby-Vigfusson,
+Oxford, 1874, 4to. <span class="sc">German.</span>&mdash;Dieterich, Stockholm, 1844, 8vo:
+Möbius, Leipzig, 1866, 8vo. <span class="sc">Danish.</span>&mdash;Jonssen, Kjöbenhavn, 1863,
+8vo. <span class="sc">Norwegian.</span>&mdash;Kraft, Christiania, 1863, 8vo: Fritzner,
+Kristiania, 1867, 8vo. <span class="sc">Poetic.</span>&mdash;Egilsson (Latin), Hafniae, 1860,
+8vo; 1864.</p>
+
+<p><span class="bold">Swedish.</span>&mdash;Kindblad, Stockholm, 1840, 4to: Almqvist, Örebro,
+1842-1844, 8vo: Dalin, <i>Ordbog.</i> Stockholm, 1850-1853, 8vo, 2 vols.
+1668 pages; 1867, &amp;c. 4to (vol. i. ii., A to Fjermare, 928 pages):
+Id., <i>Handordbog</i>, ib. 1868, 12mo, 804 pages; Svenska Academien.
+Stockholm, 1870, 4to (A) pp. 187. <span class="sc">Latin.</span>&mdash;Stjernhjelm, Holm,
+1643, 4to: Verelius, Upsala, 1691, 8vo: Ihre (Sueo-Gothicum),
+<span class="pagenum"><a name="page194" id="page194"></a>194</span>
+Upsala, 1769, fol. 2 vols. <span class="sc">English</span>.&mdash;Serenius, Nyköping, 1757,
+4to: Brisnon, Upsala, 1784, 4to: Widegren, Stockholm, 1788, 4to;
+Brisman, Upsala, 1801, 4to; 3rd ed. 1815, 2 vols.: Deleen Örebro,
+1829, 8vo: Granberg, ib. 1832, 12mo: Nilssen, Widmark, &amp;c.,
+Stockholm, 1875, 8vo. <span class="sc">French</span>.&mdash;Möller, Stockholm, 1745, 4to:
+Björkengren, ib. 1795, 2 vols.: Nordforss, ib. 1805, 8vo, 2 vols.: 2nd
+ed. Örebro, 1827, 12mo: West, Stockh. 1807, 8vo: Dalin, ib. 1842-1843,
+4to, 2 vols.; 1872. <span class="sc">German</span>.&mdash;Dähnert, Holmiae, 1746, 4to:
+Heinrich, Christiansund, 1814, 4to, 2 vols.; 4th ed. Örebro, 1841,
+12mo: Helms, Leipzig, 1858, 8vo; 1872. <span class="sc">Danish</span>.&mdash;Höst,
+Kjöbenhavn, 1799, 4to: Welander, Stockholm, 1844, 8vo: Dalin,
+ib. 1869, 16mo: Kaper, Kjöbenhavn, 1876, 16mo. <span class="sc">Etymology</span>.&mdash;Tamm,
+Upsala, 1874, &amp;c., 8vo (A and B), 200 pages. <span class="sc">Foreign
+Words</span>.&mdash;Sahlstedt, Wästerås, 1769, 8vo: Andersson (20,000),
+Stockholm, 1857, 16mo: Tullberg, ib. 1868, 8vo: Ekbohrn, ib. 1870,
+12mo: Dalin, ib. 1870, &amp;c., 8vo. <span class="sc">Synonyms</span>.&mdash;Id., ib. 1870, 12mo.
+<span class="sc">Naval</span>.&mdash;Ramsten, ib. 1866, 8vo. <span class="sc">Technical</span>.&mdash;Jungberg, ib. 1873,
+8vo. <span class="sc">Dialects</span>.&mdash;Ihre, Upsala, 1766, 4to: Rietz, Lund, 1862-1867,
+4to, 859 pages. <i>Bohuslän</i>: <i>Idioticon Bohusiense</i>, Götaborg, 1776,
+4to. <i>Dalecarlia</i>: Arborelius, Upsala, 1813, 4to. <i>Gothland</i>: Hof
+(Sven), Stockholmiae, 1772, 8vo: Rääf (Ydre), Örebro, 1859, 8vo.
+<i>Halland</i>: Möller, Lund, 158, 8vo. <i>Helsingland</i>: Lenström, ib.
+1841, 8vo: Fornminnessällskap, Hudikswall, 1870, 8vo.</p>
+
+<p><span class="bold">Norwegian.</span>&mdash;Jenssen, Kjöbenhavn, 1646, 8vo: Pontoppidan,
+Bergen, 1749, 8vo: Hanson (German), Christiania, 1840, 8vo:
+Aasen, ib. 1873, 8vo, 992 pages.</p>
+
+<p><span class="bold">Danish.</span>&mdash;Aphelen, Kopenh, 1764, 4to, 2 vols.; 1775, 4to, 3 vols.:
+Molbech, Kjöbenhavn, 1833, 8vo, 2 vols.: ib. 1859, 2 vols.: Videnskabernes
+Selskab, ib. 1793-1865, Kalkar. <span class="sc">English</span>.&mdash;Berthelson
+(Eng. Dan.), 1754, 4to: Wolff, London, 1779, 4to. Bay, ib. 1807,
+8vo, 2 vols.; 1824, 8vo: Hornbeck, ib. 1863, 8vo: Ferrall and Repp,
+ib. 1814, 16mo; 1873, 8vo: Rosing, Copenhagen, 1869, 8vo: Ancker,
+ib. 1874, 8vo. <span class="sc">French</span>.&mdash;Aphelen,  1754, 8vo: Id., ib. 1759,
+4to, 2 vols.; 2nd ed. 1772-1777, vol. i. ii. <span class="sc">German</span>.&mdash;Id., ib. 1764,
+4to, 2 vols.: Grönberg, 2nd ed. Kopenh. 1836-1839, 12mo, 2 vols.;
+1851, Helms, Leipzig, 1858, 8vo. <span class="sc">Synonyms</span>.&mdash;Müller, Kjöbenhavn,
+1853, 8vo. <span class="sc">Foreign Words</span>.&mdash;Hansen, Christiania, 1842, 12mo.
+<span class="sc">Naval</span>.&mdash;Wilsoet, Copenhagen, 1830, 8vo: Fisker (French),
+Kjöbenhavn, 1839, 8vo. <span class="sc">Old Danish</span>.&mdash;Molbech, ib. 1857-1868,
+8vo, 2 vols. <span class="sc">Dialects</span>.&mdash;Id., ib. 1841, 8vo. <i>Bornholm</i>: Adler, <i>ib.</i>
+1856, 8vo. <i>South Jutland</i>: Kok, 1867, 8vo. <span class="sc">Slang</span>.&mdash;Kristiansen
+(Gadesproget), ib. 1866, 8vo. p. 452.</p>
+
+<p class="center pt2">(2) <i>Germanic.</i></p>
+
+<p><span class="bold">Teutonic.</span>&mdash;<span class="sc">Comparative</span>.&mdash;Meidinger, Frankf. a. M. 1833, 8vo,
+2nd ed. 1836, 8vo.</p>
+
+<p><span class="bold">Gothic.</span>&mdash;Junius, Dortrecht, 1665, 4to: 1671; 1684, Diefenbach
+(comparative), Franckf. a. M. 1846-1851, 2 vols. 8vo: Schulze,
+Magdeburg, 1848, 4to: 1867, 8vo: Skeat, London, 1868, 4to:
+Balg (<i>Comparative Glossary</i>), Magvike, Wisconsin, 1887-1889.
+<span class="sc">Ulphilas</span> (editions with dictionaries).&mdash;Castilionaeus, Mediol, 1829,
+4to: Gabelentz and Löbe, Altenburg, 1836-1843, 4to, 2 vols.: Gaugengigl,
+Passau, 1848, 8vo: Stamm, Paderborn, 1857: Stamm and
+Heyne, ib. 1866, 8vo.</p>
+
+<p><span class="bold">Anglo-Saxon.</span>&mdash;<span class="sc">Latin</span>.&mdash;Somner (Lat. Eng.), Oxonii, 1659,
+fol.: Benson, ib. 1701, 8vo: Lye (A.-S. and Gothic), London, 1772,
+fol. 2 vols.: Ettmüller, Quedlinburg, 1851, 8vo. 838 pages. <span class="sc">English</span>.&mdash;Bosworth,
+London, 1838, 8vo, 721 pages: Id. (<i>Compendious</i>),
+1848, 278 pages. Corson (A.-S. and Early English), New York, 1871,
+8vo, 587 pages; Toller (based on Bosworth), Oxford, 1882-1898.
+<span class="sc">German</span>.&mdash;Bouterwek, Gütersloh, 1850, 8vo, 418 pages: Grein
+(Poets), Göttingen, 1861-1863, 8vo, 2 vols.: Leo, Halle, 1872, 8vo.</p>
+
+<p><span class="bold">English.</span>&mdash;Cockeram, London, 1623, 8vo: 9th ed. 1650: Blount,
+ib. 1656, 8vo: Philips, The new World of Words, London, 1658, fol.:
+Bailey, London, 1721, 8vo; 2nd ed. ib. 1736, fol.; 24th ed. ib. 1782,
+8vo: Johnson, ib. 1755, fol. 2 vols.; ed. Todd, London, 1818,
+4to, 4 vols.; ib. 1827. 4to, 3 vols.; ed. Latham, ib. 1866-1874, 4to,
+4 vols. (2 in 4 parts): Barclay, London, 1774, 4to; ed. Woodward,
+ib. 1848: Sheridan, ib. 1780, 4to, 2 vols.: Webster, New York, 1828,
+4to, 2 vols.; London, 1832, 4to, 2 vols.; ed. Goodrich and Porter,
+1865, 4to: Richardson, ib. 1836, 4to, 2 vols.; Supplement, 1856:
+Ogilvie, <i>Imperial Dictionary</i>, Glasgow, 1850-1855, 8vo, 3 vols. (the
+new edition of Ogilvie by Charles Annandale, 4 vols., 1882, was an
+encyclopaedic dictionary, which served to some extent as the foundation
+of the <i>Century Dictionary</i>); Boag, <i>Do.</i>, Edinburgh, 1852-1853,
+8vo, 2 vols.: Craik, ib. 1856, 8vo: Worcester, Boston, 1863, 4to.
+Stormouth and Bayne, 1885; Murray and Bradley, <i>The Oxford
+English Dictionary</i>, 1884-&emsp;&emsp;; Whitney, <i>The Century Dict.</i>, New
+York, 1889-1891; Porter, <i>Webster&rsquo;s Internat. Dict.</i>, Springfield,
+Massachusetts, 1890; Funk, <i>Standard Dict.</i>, New York, 1894; Hunter,
+<i>The Encyclopaedic Dict.</i>, 1879-1888. <span class="sc">Etymology</span>.&mdash;Skinner, Londini,
+1671, fol.: Junius, Oxonii, 1743, fol.: Wedgewood, London, 1859-1865,
+3 vols.; ib. 1872, 8vo. Skeat, Oxford, 1881; Fennell (Anglicized
+words), Camb. 1892. <span class="sc">Pronouncing</span>.&mdash;Walker, London, 1774, 4to:
+by Smart, 2nd ed. ib. 1846, 8vo. <span class="sc">Pronouncing in German</span>.&mdash;Hausner,
+Frankf. 1793, 8vo; 3rd ed. 1807; Winkelmann, Berlin, 1818,
+8vo: Voigtmann, Coburg, 1835, 8vo: Albert, Leipz. 1839, 8vo:
+Bassler, ib. 1840, 16mo. <span class="sc">Analytical</span>.&mdash;Booth, Bath, 1836, 4to:
+Roget, <i>Thesaurus</i>, London, 1852, 8vo; 6th ed. 1857; Boston, 1874.
+<span class="sc">Synonyms</span>.&mdash;Piozzi, London, 1794, 8vo, 2 vols.: L. [abarthe], Paris,
+1803, 8vo, 2 vols.: Crabb, London, 1823, 8vo; 11th ed. 1859:
+C. J. Smith, ib. 1871, 8vo, 610 pages. <span class="sc">Reduplicated Words</span>.&mdash;Wheatley,
+ib. 1866, 8vo. <span class="sc">Surnames</span>.&mdash;Arthur, New York, 1857,
+12mo, about 2600 names: Lower, ib. 1860, 4to. <span class="sc">Particles</span>.&mdash;Le
+Febure de Villebrune, Paris, 1774, 8vo. <span class="sc">Rhyming</span>.&mdash;Levins,
+<i>Manipulus Puerorum</i>, London, 1570, 4to; ed. Wheatley, ib. 1867,
+8vo: Walker, London, 1775, 8vo; 1865, 8vo. <span class="sc">Shakespeare</span>.&mdash;Nares,
+Berlin, 1822, 4to; ed. Halliwell and Wright, London, 1859,
+8vo: Schmidt, Berlin, 1874. <span class="sc">Old English</span>.&mdash;Spelman, London
+[1626], fol. (A to I only); 1664 (completed); 1687 (best ed.):
+Coleridge (1250-1300), ib. 1859, 8vo: Stratmann (Early Eng.),
+Krefeld, 1867, 8vo; 2nd ed. 1873, 4to: Bradley (new edition of
+Stratman), Oxford, 1891; Matzner and Bieling, Berlin, 1878-    .
+<span class="sc">Old and Provincial</span>.&mdash;Halliwell, London, 1844-1846, 8vo; 2nd ed.
+ib. 1850, 2 vols.: 6th ed. 1904: Wright, ib. 1857, 8vo, 2 vols.; 1862.
+<span class="sc">Dialects</span>.&mdash;Ray, ib. 1674, 12mo: Grose, ib. 1787, 8vo; 1790:
+Holloway, Lewes, 1840, 8vo; Wright, <i>Eng. Dialect Dict.</i>, London,
+1898-1905, 28 vols. <i>Scotch</i>: Jamieson, Edin. 1806, 4to, 2 vols.;
+Supplement, 1826, 2 vols.; abridged by Johnstone, ib. 1846, 8vo:
+Brown, Edin, 1845, 8vo: Motherby (German), Königsberg, 1826-1828,
+8vo: (<i>Shetland and Orkney</i>), Edmonston, London, 1866, 8vo:
+(<i>Banffshire</i>), Gregor, ib. 1866, 8vo. <i>North Country</i>: Brockett,
+London, 1839, 8vo, 2 vols. <i>Berkshire</i>: [Lousley] ib. 1852, 8vo,
+<i>Cheshire</i>: Wilbraham, ib. 1817, 4to; 1826, 12mo: Leigh, Chester,
+1877, 8vo. <i>Cumberland</i>: <i>Glossary</i>, ib. 1851, 12mo: Dickenson,
+Whitehaven, 1854, 12mo; Supplement, 1867: Ferguson (Scandinavian
+Words), London, 1856, 8vo. <i>Derbyshire</i>: Hooson (mining),
+Wrexham, 1747, 8vo: Sleigh, London, 1865, 8vo. <i>Dorset</i>: Barnes,
+Berlin, 1863, 8vo. <i>Durham</i>: [Dinsdale] (Teesdale), London, 1849,
+12mo. <i>Gloucestershire</i>: Huntley (Cotswold), ib. 1868, 8vo. <i>Herefordshire</i>:
+[Sir George Cornewall Lewis,] London, 1839, 12mo. <i>Lancashire</i>:
+Nodal and Milner, Manchester Literary Club, 1875, 8vo,
+Morris (Furness), London, 1869, 8vo: R. B. Peacock (Lonsdale,
+North and South of the Sands), ib. 1869, 8vo. <i>Leicestershire</i>:
+A. B. Evans, ib. 1848, 8vo. <i>Lincolnshire</i>: Brogden, ib. 1866, 12mo:
+Peacock (Manley &amp; Corringham), ib. 1877, 8vo. <i>Norfolk and Suffolk</i>;
+Forby, London, 1830, 8vo, 2 vols. <i>Northamptonshire</i>: Sternberg,
+ib. 1851, 8vo: Miss Anne E. Baker, ib. 1866, 8vo, 2 vols. 868 pages.
+<i>Somersetshire</i>: Jennings, ib. 1869, 8vo: W. P. Williams and W. A.
+Jones, Taunton, 1873, 8vo. <i>Suffolk</i>: Moor, Woodbridge, 1823, 12mo:
+Bowditch (Surnames), Boston, U.S., 1851, 8vo; 1858; 3rd ed.
+London, 1861, 8vo, 784 pages. <i>Sussex</i>: Cooper, Brighton, 1836,
+8vo: Parish, Farncombe, 1875, 8vo. <i>Wiltshire</i>: Akerman, London,
+1842, 12mo. <i>Yorkshire (North and East)</i>, Toone, ib. 1832, 8vo:
+(<i>Craven</i>), Carr, 2nd ed. London, 1828, 8vo, 2 vols.: (<i>Swaledale</i>),
+Harland, ib. 1873, 8vo: (<i>Cleveland</i>), Atkinson, ib. 1868, 4to, 653
+pages: (<i>Whitby</i>) [F. K. Robinson], ib. 1876, 8vo: (<i>Mid-Yorkshire
+and Lower Niddersdale</i>), C. Clough Robinson, ib. 1876, 8vo: (<i>Leeds</i>),
+Id., ib. 1861, 12mo: (<i>Wakefield</i>), Banks, ib. 1865, 16mo: (<i>Hallamshire</i>),
+Hunter, London, 1829, 8vo. <i>Ireland: (Forth and Bargy, Co.
+Wexford)</i>, Poole, London, 1867, 8vo. <i>America</i>: Pickering, Boston,
+1816, 8vo: Bartlett, New York, 1848, 8vo; 3rd ed. Boston, 1860.
+8vo; Dutch transl. by Keijzer, Gorinchen, 1854, 12mo; Germ.
+transl. by Köhler, Leipz. 1868, 8vo. Elwyn, Philadelphia, 1859.
+8vo. <i>Negro English</i>: Kingos, St Croix, 1770, 8vo: Focke (Dutch),
+Leiden, 1855, 8vo: Wullschlaegel, Löbau, 1856, 8vo. 350 pages.
+<span class="sc">Slang</span>.&mdash;Grose, London, 1785, 8vo; 1796: Hotten, ib. 1864, 8vo;
+1866; Farmer &amp; Henley (7 vols., 1890-1904).</p>
+
+<p><span class="bold">Frisic.</span>&mdash;Wassenbergh, Leeuwarden, 1802, 8vo: Franeker, 1806,
+8vo: Outzen, Kopenh. 1837, 4to: Hettema (Dutch), Leuwarden,
+1832, 8vo; 1874, 8vo, 607 pages: Winkler (Nederdeutsch en Friesch
+Dialectikon), &rsquo;s Gravenhage, 1874, 8vo, 2 vols. 1025 pages. <span class="sc">Old
+Frisic</span>.&mdash;Wiarda (Germ.), Aurich, 1786, 8vo: Richthofen, Göttingen,
+1840, 4to. <span class="sc">North Frisic</span>.&mdash;Bendson (Germ.), Leiden, 1860, 8vo:
+Johansen (Föhringer und Amrumer Mundart), Kiel, 1862, 8vo.
+<span class="sc">East Frisic</span>.&mdash;Stürenburg, Aurich, 1857, 8vo. <span class="sc">Heligoland</span>.&mdash;Oelrichs,
+s. l., 1836, 16mo.</p>
+
+<p><span class="bold">Dutch.</span>&mdash;Kok, 2nd ed. Amst. 1785-1798, 8vo, 38 vols.: Weiland,
+Amst. 1790-1811, 8vo, 11 vols.: Harrebomée, Utrecht, 1857, 4to;
+1862-1870, 8vo, 3 vols.: De Vries and Te Winkel, Gravenh. 1864, &amp;c.,
+4to (new ed. 1882-    ); Dale, ib. 4th ed. 1898; <span class="sc">English</span>.&mdash;Hexham,
+ed. Manley, Rotterdam, 1675-1678, 4to: Holtrop, Dortrecht,
+1823-1824, 8vo, 2 vols.: Bomhoff, Nimeguen, 1859, 8vo, 2 vols. 2323
+pages: Jaeger, Gouda, 1862, 16mo: Calisch, Tiel, 1871, &amp;c., 8vo.
+<span class="sc">French</span>.&mdash;Halma, Amst. 1710, 4to; 4th ed. 1761: Marin, ib. 1793,
+4to, 2 vols.: Winkelman, ib. 1793, 4to, 2 vols.: Mook, Zutphen,
+1824-1825, 8vo, 4 vols.; Gouda, 1857, 8vo, 2 vols. 2818 pages: Kramers,
+ib. 1859-1862, 2 vols. 16mo. <span class="sc">German</span>.&mdash;Kramer, Nürnb. 1719, fol.;
+1759, 4to, 2 vols.; ed. Titius, 1784, Weiland, Haag, 1812, 8vo:
+Terwen, Amst. 1844, 8vo. <span class="sc">Etymology</span>.&mdash;Franck, 1884-1892.
+<span class="sc">Oriental Words</span>.&mdash;Dozy, &rsquo;s Gravenhage, 1867, 8vo. <span class="sc">Genders of
+Nouns</span>.&mdash;Bilderdijk, Amst. 1822, 8vo, 2 vols. <span class="sc">Spelling</span>.&mdash;Id.,
+&rsquo;s Gravenhage, 1829, 8vo. <span class="sc">Frequentatives</span>.&mdash;De Jager, Gouda,
+1875, 8vo, vol. i. <span class="sc">Old Dutch</span>.&mdash;Suringer, Leyden, 1865, 8vo.
+<span class="sc">Middle Dutch</span>.&mdash;De Vries, &rsquo;s Gravenhage, 1864, &amp;c., 4to. Verwijs
+and Verdam, ib. 1885-&emsp;&emsp;.</p>
+
+<p><span class="bold">Flemish.</span>&mdash;Kilian, Antw. 1511, 8vo; ed. Hasselt, Utrecht, 1777,
+4to, 2 vols. <span class="sc">French</span>.&mdash;Berlemont, Anvers, 1511, 4to: Meurier, ib.
+1557, 8vo: Rouxell and Halma, Amst. 1708, 4to; 6th ed. 1821:
+Van de Velde and Sleeckx, Brux. 1848-1851, 8vo, 2440 pages; ib.
+<span class="pagenum"><a name="page195" id="page195"></a>195</span>
+1860, 8vo, 2 vols. <span class="sc">Ancient Names of Places</span>.&mdash;Grandgagnage
+(East Belgium), Bruxelles, 1859, 8vo.</p>
+
+<p><span class="bold">German.</span>&mdash;Josua Pictorius (Maaler), <i>Die teütsch Spraach</i>, Tiguri,
+1561, 8vo; Stieler, Nürnb. 1691, 4to: Adelung, Leipz. 1774-1786,
+4to, 5 vols.; 1793-1818, 5 vols.: Campe, Braunschweig, 1807-1811,
+4to, 5 vols.: Grimm, Leipzig, 1854, &amp;c., 4to: Sanders, ib. 1860-1865,
+4to, 3 vols. 1885: Diefenbach and Wülcker (High and Low
+German, to supplement Grimm), Frankf. a. M. 1874, 1885, 8vo.;
+Kluge, Strassburg, 1883; Heine, Leipzig, 1890-1895; Weigand,
+Giessen, 1873. <span class="sc">English</span>.&mdash;Adelung, 1783-1796, 8vo, 3 vols.: Hilpert,
+Karlsruhe, 1828-1829, 8vo, 2 vols.; 1845-1846, 4to, 2 vols.: Flügel,
+Leipz. 1830, 8vo, 2 vols.; London, 1857, 8vo; Leipzig, 1870:
+Müller, Cöthen, 1867, 8vo, 2 vols. <span class="sc">French</span>.&mdash;Laveaux, Strassburg,
+1812, 4to: Mozin, Stuttgard, 1811-1812, 4to, 4 vols.; 1842-1846, 8vo,
+4 vols., 3rd ed. 1850-1851, 8vo: Schuster, Strasb. 1859, 8vo: Daniel,
+Paris, 1877, 16mo. <span class="sc">Old High German</span>.&mdash;Haltaeus, Lipsiae, 1758,
+fol. 2 vols.: Graff, Berlin, 1834-1846, 4to, 7 vols.: Brinckmeier,
+Gotha, 1850-1863, 4to, 2 vols.: Kehrein (from Latin records), Nordhausen,
+1863, 8vo. Schade, Halle, 1872-1882. <span class="sc">Middle High German</span>.&mdash;Ziemann,
+Quedlinburg, 1838, 8vo: Benecke, Müller and Zarnche,
+Leipz. 1854-1866, 8vo, 3 vols.: Lexer, Leipzig, 1870, 8vo. <span class="sc">Middle
+Low German</span>.&mdash;Schiller and Lübben, Bremen, 1872, &amp;c., 8vo, in
+progress. <span class="sc">Low German</span>.&mdash;Vollbeding, Zerbst, 1806, 8vo: Kosegarten,
+Griefswald, 1839, 4to; 1856, &amp;c., 4to. <span class="sc">Etymology</span>.&mdash;Helvigius,
+Hanov. 1620, 8vo: Wachter, Lipsiae, 1737, fol. 2 vols.:
+Kaindl, Salzbach, 1815-1830, 8vo, 7 vols.: Heyse, Magdeburg, 1843-1849,
+8vo, 3 vols.: Kehrein, Wiesbaden, 1847-1852, 2 vols. <span class="sc">Synonyms</span>.&mdash;Eberhard,
+Maas, and Grüber, 4th ed. Leipzig, 1852-1863, 8vo, 4
+vols.: Aue (Engl.), Edinb. 1836, 8vo: Eberhard, 11th ed. Berlin, 1854,
+12mo: Sanders, Hamburg, 1872, 8vo, 743 pages. <span class="sc">Foreign Words</span>.&mdash;Campe,
+Braunschweig, 1813, 4to: Heyse, <i>Fremdwörterbuch</i>,
+Hannover, 1848, 8vo. <span class="sc">Names</span>.&mdash;Pott. Leipz. 1853, 8vo: Michaelis
+(Taufnamen), Berlin, 1856, 8vo: Förstemann (Old Germ.) Nordhausen,
+1856-1859, 4to, 2 vols. 1573 pages, 12,000 names: Steub
+(Oberdeutschen), München, 1871, 8vo. <span class="sc">Luther</span>.&mdash;Dietz, Leipzig,
+1869-1872, 8vo, 2 vols. <span class="sc">Dialects</span>.&mdash;Popowitsch, Wien, 1780, 8vo:
+Fulda, Berlin, 1788, 8vo: Klein, Frankf. 1792, 8vo, 2 vols.: Kaltschmidt,
+Nordlingen, 1851, 4to; 1854, 5th ed. 1865. <i>Aix-la-Chapelle</i>,
+Müller and Weitz, Aachen, 1836, 12mo. <i>Appenzell</i>: Tobler,
+Zürich, 1837, 8vo. <i>Austria</i>: Höfer, Linz, 1815, 8vo; Castelli, Wien,
+1847, 12mo: Scheuchenstül (mining), ib. 1856, 8vo. <i>Bavaria</i>:
+Zaupser, München, 1789, 8vo: Deling, ib. 1820, 2 vols.: Schmeller,
+Stuttg. 1827-1837, 8vo, 4 vols.; 2nd ed. München, 1872, 4to, vol.
+i. 1799 pages. <i>Berlin</i>: Trachsel. Berlin, 1873, 8vo. <i>Bremen</i>:
+Bremisch Deutsch Gesellschaft, Bremen, 1767-1771, 1869, 8vo, 6 vols.
+Oelrich (anc. statutes), Frankf. a. M. 1767, 8vo. <i>Carinthia</i>: Ueberfelder,
+Klagenfurt, 1862, 8vo: Lexe, Leipzig, 1862, 8vo. <i>Cleves</i>:
+De Schueren, <i>Teuthonista</i>, Colon, 1477, fol.; Leiden, 1804, 4to.
+<i>Göttingen</i>: Schambach, Hannover, 1838, 8vo. <i>Hamburg</i>: Richey,
+Hamb. 1873, 4to; 1755, 8vo. <i>Henneberg</i>: Reinwold, Berlin and
+Stettin, 1793, 1801, 8vo, 2 vols.: Brückner, Meiningen, 1843, 4to.
+<i>Hesse</i>: Vilmar, Marburg, 1868, 8vo, 488 pages. <i>Holstein</i>: Schütz
+Hamb. 1800-1806, 8vo, 4 vols. <i>Hungary</i>: Schoer, Wien, 1858.
+<i>Livonia</i>: Bergmann, Salisburg, 1785, 8vo: Gutzeit, Riga, 1859-1864,
+8vo, 2 parts. <i>Upper Lusatia</i>: Anton, Görlitz, 1825-1839, 13 parts.
+<i>Luxembourg</i>: Gangler, Lux. 1847, 8vo, 406 pages. <i>Mecklenburg and
+Western Pomerania</i>: M., Leipzig, 1876, 8vo, 114 pages. <i>Nassau</i>:
+Kehrein, Weilburg, 1860, 8vo. <i>Osnaburg</i>: Strodtmann, Leipz. 1756,
+8vo. <i>Pomerania and Rügen</i>: Dähnert, Stralsund, 1781, 4to. <i>Posen</i>:
+Bernd, Bonn, 1820, 8vo. <i>Prussia</i>: Bock, Königsb. 1759, 8vo:
+Hennig, ib. 1785, 8vo. <i>Saxony</i>: Schmeller (from Heliand, &amp;c.),
+Stuttg. 1840, 4to. <i>Silesia</i>: Berndt, Stendal, 1787, 8vo. <i>Swabia</i>:
+Schmid, Berlin, 1795, 8vo; Stuttg. 1831, 8vo. <i>Switzerland</i>:
+Stalder, Aarau, 1807-1813, 8vo, 2 vols. <i>Thuringia</i>: Keller, Jena,
+1819, 8vo. <i>Transylvania</i>: Schuller, Prag, 1865, 8vo. <i>Tirol</i>:
+Schöpf, Innspruck, 1866, 8vo. <i>Venetian Alps</i>: Schmeller, Wien,
+1854, 8vo. <i>Vienna</i>: Hugel, ib. 1873, 8vo. <span class="sc">Hunting</span>.&mdash;<i>Westerwald</i>:
+Schmidt, Hadamar, 1800, 8vo; Kehrein, Wiesbaden, 1871, 12mo.
+<span class="sc">Slang</span>.&mdash;<i>Gauner Sprache</i>: Schott, Erlangen, 1821, 8vo: Grolmann,
+Giessen, 1822, 8vo: Train, Meissen, 1833, 8vo: Anton, 2nd ed.
+Magdeburg, 1843, 8vo; 1859: Avé-Lallemant, <i>Das Deutsche
+Gaunerthun</i>, Leipzig, 1858-1862, 8vo, vol. iv. pp. 515-628. <i>Student
+Slang</i>: Vollmann (Burschicoses), Ragaz, 1846, 16mo, 562 pages.</p>
+
+<p class="center pt2"><i>Celtic.</i></p>
+
+<p><span class="bold">Celtic generally.</span>&mdash;Lluyd, Archaeologia Britannica, Oxford,
+1707, folio: Bullet, Besançon, 1754-1860, fol. 2 vols.</p>
+
+<p><span class="bold">Irish.</span>&mdash;Cormac, bishop of Cashel, born 831, slain in battle 903,
+wrote a Glossary, <i>Sanas Cormaic</i>, printed by Dr Whitley Stokes,
+London, 1862, 8vo, with another, finished in 1569, by O&rsquo;Davoren,
+a schoolmaster at Burren Castle, Co. Clare: O&rsquo;Clery, Lovanii, 1643,
+8vo: MacCuirtin (Eng.-Irish), Paris, 1732, 4to: O&rsquo;Brien, ib. 1768,
+4to; Dublin, 1832, 8vo: O&rsquo;Reilly, 1817, 4to: 1821; ed. O&rsquo;Donovan,
+ib. 1864, 4to, 725 pages: Foley (Eng.-Irish), ib. 1855, 8vo: Connellan
+(do.), 1863, 8vo.</p>
+
+<p><span class="bold">Gaelic.</span>&mdash;Macdonald, Edin. 1741, 8vo: Shaw, London, 1780,
+4to, 2 vols.: Allan, Edin. 1804, 4to: Armstrong, London, 1825,
+4to: Highland Society, ib. 1828, 4to, 2 vols.: Macleod and Dewar,
+Glasgow, 1853, 8vo.</p>
+
+<p><span class="bold">Manx.</span>&mdash;Cregeen, Douglas, 1835, 8vo: Kelly, ib. 1866, 8vo, 2 vols.</p>
+
+<p><span class="bold">Welsh.</span>&mdash;<span class="sc">Latin</span>.&mdash;Davies, London, 1632, fol.: Boxhornius,
+Amstelodami, 1654, 4to. <span class="sc">English</span>.&mdash;Salesbury, London, 1547, 4to:
+1551: Richards, Bristol, 1759, 8vo: Owen (W.), London, 1793-1794,
+8vo, 2 vols.; 1803, 4to, 3 vols.: Walters, ib. 1794, 4to: Owen-Pughe,
+Denbigh, 1832, 8vo; 3rd ed. Pryse, ib. 1866, 8vo: D. S. Evans
+(Eng.-Welsh), ib. 1852-1853, 8vo; 1887.</p>
+
+<p><span class="bold">Cornish.</span>&mdash;Pryce, <i>Archaeologia</i>, Sherborne, 1770, 4to: Williams,
+Llandovery, 1862-1865, 4to. <span class="sc">Names</span>.&mdash;Bannister (20,000), Truro,
+1869-1871, 8vo.</p>
+
+<p><span class="bold">Breton.</span>&mdash;Legadeuc, <i>Le Catholicon breton</i>, finished 1464, printed
+at Lantrequier, 1499, fol. 210 pages; 1501, 4to; L&rsquo;Orient, 1868,
+8vo: Quicquer de Roskoff, Morlaix, 1633, 8vo: Rostrenen, Rennes,
+1732, 4to, 978 pages; ed. Jolivet, Guingamps, 1834, 8vo, 2 vols.:
+l&rsquo;A.[rmerie], Leyde, 1744, 8vo; La Haye, 1756: Lepelletier, Paris,
+1752, fol.: Legonidec, Angouleme, 1821, 8vo; St Brieuc, 1847-1850,
+4to, 924 pages. <span class="sc">Dialect of Léon</span>.&mdash;Troude (Fr.-Bret.), Brest,
+1870, 8vo; Id. (Bret.-Fr.), ib. 1876, 8vo, 845 pages. <span class="sc">Diocese of
+Vannes</span>.&mdash;Armerie, Leyde, 1774, 8vo.</p>
+
+<p class="center pt2"><i>Basque.</i></p>
+
+<p><span class="bold">Basque.</span>&mdash;Larramendi, St Sebastian, 1745, fol. 2 vols.; ed.
+Zuazua, ib. 1854, fol.; Chaho, Bayonne, 1856, 4to, 1867: Fabre,
+ib. 1870, 8vo: Van Eys, Paris, 1873, 8vo: Egúren, Madrid, 1877.</p>
+
+<p class="center pt2"><i>Baltic.</i></p>
+
+<p><span class="bold">Lithuanian.</span>&mdash;Szyrwid, 3rd ed., Vilnae, 1642, 8vo; 5th ed. 1713:
+Schleicher, Prag, 1856-1857, 8vo, 2 vols.: Kurmin, Wilno, 1858, 8vo:
+Kurschat, Halle, 1870, &amp;c., 8vo.</p>
+
+<p><span class="bold">Lettic.</span>&mdash;Mancelius, Riga, 1638, 4to: Elvers, ib. 1748, 8vo:
+Lange, Mitau, 1777, 4to: Sjögren, Petersburg, 1861, 4to: Ulmann,
+ed. Bielenstein, Riga, 1872, &amp;c., 8vo.</p>
+
+<p><span class="bold">Prussian.</span>&mdash;Bock, Königsberg, 1759, 8vo: Hennig, ib. 1785, 8vo:
+Nesselmann, Berlin, 1873, 8vo: Pierson, ib. 1875, 8vo.</p>
+
+<p class="center pt2"><i>Slavonic</i>.</p>
+
+<p><span class="bold">Slavonic generally.</span>&mdash;Franta-Sumavski (Russ. Bulg. Old Slav.
+Boh. Polish), Praga, 1857, 8vo, Miklosich, Wien, 1886.</p>
+
+<p><span class="bold">Old Slavonic.</span>&mdash;Beruinda, Kiev, 1627, 8vo; Kuteinsk, 1653,
+4to: Polycarpi (Slav. Greek, Latin), Mosque, 1704, 4to: Alexyeev,
+St Petersb. 1773, 8vo; 4th ed. ib. 1817-1819, 8vo, 5 vols.: Russian
+Imp. Academy, ib. 1847, 4to, 4 vols.: Miklosich, Vindobonae, 1850:
+4to; 1862-1865, 8vo, Mikhailovski, St Petersb. 1875, 8vo: Charkovski,
+Warschaw, 1873, 8vo.</p>
+
+<p><span class="bold">Russian.</span>&mdash;Russian Academy, St Petersburg, 1789-1794, 4to, 6
+vols.; 1806-1822, ib. 1869, 8vo, 3 vols.: Dahl, Moskva, 1862-1866,
+fol. 4 vols.; d., ib. 1873, &amp;c., 4to; a 3rd edition, 1903, &amp;c.
+<span class="sc">French-Germ.-Eng</span>.&mdash;Reiff,
+ib. 1852-1854, 4to. <span class="sc">German, Latin</span>.&mdash;Holterhof,
+Moskva, 1778, 8vo, 2 vols.; 3rd ed. 1853-1855, 8vo, 2 vols.: Weismann,
+ib. 1731, 4to; 1782, and frequently. <span class="sc">French, German</span>.&mdash;Nordstet,
+ib. 1780-1782, 4to, 2 vols.: Heym, Moskau, 1796-1805, 4to, 4 vols.:
+Booch-Arkossi and Frey, Leipzig, 1871, &amp;c., 8vo. <span class="sc">English</span>.&mdash;Nordstet,
+London, 1780, 4to: Grammatin and Parenogo, Moskva,
+1808-1817, 4to, 4 vols. <span class="sc">French</span>.&mdash;Tatischeff, 2nd ed. St Petersb. 1798,
+8vo, 2 vols.; Moskau, 1816, 4to, 2 vols.: Reiff, St Petersb. 1835-1836,
+8vo, 2 vols.: Makaroff, ib. 1872, 8vo, 2 vols, 1110 pages; 1873-1874,
+12mo, 2 vols. <span class="sc">German</span>.&mdash;Pawlowski, Riga, 1859, 8vo: Lenström,
+Mitau, 1871, 8vo. <span class="sc">Swedish</span>.&mdash;Geitlin, Helsingfors, 1833, 12mo:
+Meurmann, ib. 1846, 8vo. <span class="sc">Polish</span>.&mdash;Jakubowicz, Warszawa, 1825-1828,
+8vo, 2 vols.: Amszejewicz, ib. 1866, 8vo: Szlezigier, ib. 1867, 8vo.
+<span class="sc">Technical</span>.&mdash;Grakov (Germ.), St Petersb. 1872, 8vo. <span class="sc">Naval</span>.&mdash;Butakov,
+ib. 1837. <span class="sc">Dialects</span>.&mdash;<i>North-west Russia</i>: Gorbachevski
+(old language, in Russian), Vilna, 1874, 8vo, 418 pages. <i>White
+Russia</i>: Nosovich (Russian), St Petersburg, 1870, 4to, 760 pages.
+<i>Red Russia</i>: Patritzkii (German), Lemberg, 1867, 8vo, 2 vols.
+842 pages. <i>Ukraine</i>: Piskanov (Russian), Odessa, 1873, 4to, 156
+pages.</p>
+
+<p><span class="bold">Polish.</span>&mdash;Linde (explained in Lat. Germ. and 13 Slav dialects),
+Warszawie, 1807-1814, 4to, 6 vols. 4574 pages. <span class="sc">English</span>.&mdash;[Rykaczewski],
+<i>Complete Dictionary</i>, Berlin, 1849-1851, 8vo, 2 vols.: Rykaczewski,
+Berlin, 1866, 16mo, 1161 pages. <span class="sc">French and German</span>.&mdash;Troc,
+Leipz. 1742-1764, 8vo, 4 vols.; 4th ed. ib. 1806-1822, 4to, 4 vols.:
+Bandtke, Breslau, 1806, 8vo, 2 vols.; 1833-1839, 8vo. <span class="sc">French</span>.&mdash;Schmidt,
+Leipzig, 1870, 16mo. <span class="sc">Russian and German</span>.&mdash;Schmidt
+(J. A. E.), Breslau, 1834, 8vo. <span class="sc">German</span>.&mdash;Mrongovius, Königsberg,
+1765; 1835, 4to; 1837: Troianski, Berlin, 1835-1838, 8vo, 2 vols.:
+Booch-Arkossi, Leipzig, 1864-1868, 8vo, 2 vols.: Jordan, ib. 1866,
+8vo. <span class="sc">Italian</span>.&mdash;Plazowski, Warszawa, 1860, 8vo. 2 vols. 730 pages.
+<span class="sc">Russian</span>.&mdash;Potocki, Lipsk, 1873, &amp;c., 12mo.</p>
+
+<p><span class="bold">Wendish.</span>&mdash;Matthäi, Budissen, 1721, 8vo: Bose, Grimma, 1840,
+8vo: Pfuhl, w Budzsinje, 1866, 8vo, 1210 pages. <span class="sc">Upper Lusatian</span>.&mdash;Pfuhl
+and Jordan, Leipz. 1844, 8vo. <span class="sc">Lower Lusatian</span>.&mdash;Zwahr,
+Spremberg, 1847, 8vo.</p>
+
+<p><span class="bold">Czech.</span>&mdash;Rohn (Germ. Lat.), Prag, 1780, 4to, 4 vols.:
+Dobrowski and Hanka, ib. 1802-1821, 4to, 2 vols. <span class="sc">Lat. Germ.
+Hungar</span>.&mdash;Jungmann, Praze, 1835-1839, 6 vols. 4to, 5316 pages.
+<span class="sc">German</span>.&mdash;Thàm, Prag. 1805-1807, 8vo, 2 vols.: Sumavski, ib. 1844-1846,
+8vo, 2 vols.: Koneney, ib. 1855, 18mo, 2 vols.: Rank (Germ.
+Boh.), ib. 1860, 16mo, 775 pages. <span class="sc">Technical</span>.&mdash;Spatny, ib. 1864,
+8vo: Kheil (names of goods, Germ. Boh.), ib. 1864, 8vo, 432 pages.
+<span class="sc">Hunting</span>.&mdash;Spatny, ib. 1870, 8vo, 137 pages.</p>
+
+<p><span class="pagenum"><a name="page196" id="page196"></a>196</span></p>
+
+<p><span class="bold">South Slavic.</span>&mdash;Richter and Ballman, Wien, 1839-1840, 8vo, 2 vols.
+<span class="sc">Servian</span>.&mdash;Karaji&#263; (Germ. Lat.), ib. 1818, 8vo; 1852: Lavrovski
+(Russian), St Petersb. 1870, 8vo, 814 pages. <span class="sc">Bosnian</span>.&mdash;Micalia,
+Laureti, 1649, 8vo. <span class="sc">Slovak</span>.&mdash;Bernolak (Lat. Germ.
+Hung.), Budae, 1825-1827, 8vo, 6 vols.: Loos (Hung. and Germ.),
+Pest, 1869, &amp;c., 3 vols. <span class="sc">Slovene</span>.&mdash;Gutsmann, Klagenfurt, 1789,
+4to: Relkovich, Wien, 1796, 4to, 2 vols.: Murko, Grätz, 1838, 8vo,
+2 vols.: Janezi&#263;, Klagenfurt, 1851, 12mo. <span class="sc">Dalmatian</span>.&mdash;Ardelio
+della Bella, Venezia, 1728, 8vo; 2nd ed. Ragusae, 1785, 4to: Stulli,
+ib. 1801-1810, 4to, 2 vols. <span class="sc">Croatian</span>.&mdash;Habdelich, Grätz, 1670, 8vo:
+Sulek, Agram, 1854-1860, 8vo, 2 vols. 1716 pages. <span class="sc">Carinthian</span>.&mdash;Lexer,
+Leipzig, 1862, 8vo. <span class="sc">Old Servian</span>.&mdash;Danitziye (Servian),
+Belgrad, 1864, 8vo, 3 vols.</p>
+
+<p><span class="bold">Bulgarian.</span>&mdash;Daniel (Romaic, Albanian, Rumanian, and Bulgarian),
+Moschopolis, 1770; Venice, 1802, 4to. <span class="sc">English</span>.&mdash;Morse and
+Vassiliev, Constantinople, 1860, 8vo. <span class="sc">Russian</span>.&mdash;Borogoff, Vienna,
+1872, &amp;c., 8vo.</p>
+
+<p class="center pt2"><i>Ugrian.</i></p>
+
+<p><span class="bold">Ugrian, Comparative.</span>&mdash;Donner, Helsingfors, 1874, 8vo, in progress:
+Budenz (Ugrian-Magyar), Budapest, 1872-1875, 8vo.</p>
+
+<p><span class="bold">Lappish.</span>&mdash;<i>Manuale</i>, Holmiae, 1648, 8vo: Fjellström, ib. 1738,
+8vo: Leem and Sandberg, Havn. 1768-1781, 4to, 2 parts: Lindahl
+and Oehrling, Holm. 1780, 8vo. <span class="sc">North Lappish</span>.&mdash;Stockfleht,
+Christiania, 1852, 8vo.</p>
+
+<p><span class="bold">Finnish.</span>&mdash;Juslenius, Holmiae, 1745, 4to, 567 pages: Renvall,
+Aboae, 1826, 4to, 2 vols.: Europaeus, Helsingissä, 1852-1853, 16mo,
+2 vols. 742 pages: Lunin, Derpt, 1853, 8vo: Eurén, Tavashuus, 1860,
+8vo: Ahlman, ib. 1864, 8vo: Wiedemann, St Petersb. 1869, 4to:
+Godenhjelm (Germ.), Helsingfors, 1871: Lönnrot, Helsingissä,
+1874. <span class="sc">Naval</span>.&mdash;Stjerncreutz, ib. 1863, 8vo.</p>
+
+<p><span class="bold">Esthonian.</span>&mdash;Hupel, Mitau, 1818, 8vo, 832 pages: Körber,
+Dorpat, 1860, 8vo: Wiedemann, St Petersb. 1869, 4to, 1002 pages:
+Aminoff (Esth.-Finnish), Helsingissä, 1869, 8vo: Meves (Russian),
+Riga, 1876, 12mo.</p>
+
+<p><span class="bold">Permian.</span>&mdash;Rogord (Russian), St Petersb. 1869, 8vo, 420 pages.</p>
+
+<p><span class="bold">Votiak.</span>&mdash;Wiedemann, Reval, 1847, 8vo: Ahlquist, Helsingfors,
+1856, 4to.</p>
+
+<p><span class="bold">Cheremiss.</span>&mdash;Budenz, Pest, 1866, 8vo.</p>
+
+<p><span class="bold">Ersa-Mordvine.</span>&mdash;Wiedemann, St Petersb. 1865, 4to.
+<span class="sc">Moksha-Mordvine</span>.&mdash;Ahlquist, ib. 1862, 8vo.</p>
+
+<p><span class="bold">Magyar.</span>&mdash;Szabo, Kassan, 1792, 8vo: Guczor and Fogarazi
+(Hung. Academy), Pesth, 1862, 8vo, in progress. <span class="sc">English</span>.&mdash;Dallos,
+Pesth, 1860, 8vo. <span class="sc">French</span>.&mdash;Kiss, ib. 1844, 12mo, 2 vols.:
+Karady, Leipz. 1848, 12mo: Mole, Pesth, 1865, 8vo, 2 vols. <span class="sc">German</span>.&mdash;Schuster,
+Wien, 1838, 8vo: Bloch, Pesth, 1857, 4to, 2 vols.: Ballagi,
+ib. 1857, 8vo; 6th ed. 1905, 8vo, 2 vols.: Loos, ib. 1870, 8vo, 914 pages.
+<span class="sc">Etymological</span>.&mdash;Dankovsky (Lat.-Germ.), Pressburg, 1853, 8vo:
+Kresznerics (under roots, in Hung.), Budân, 1831-1832, 4to, 2 vols.:
+Podhorsky (from Chinese roots, in Germ.), Budapest, 1877, 8vo.
+<span class="sc">New Words</span>.&mdash;Kunoss, Pesth, 1836, 8vo; 1844.</p>
+
+<p><span class="bold">Turkish.</span>&mdash;<span class="sc">Arab. Pers</span>.&mdash;Esaad Effendi, Constantinople, 1802,
+fol. <span class="sc">Romaic</span>.&mdash;Alexandrides, Vienna, 1812, 4to. <span class="sc">Polyglotts</span>.&mdash;Pianzola
+(Ital. Grec. volgare, e Turca), Padova, 1789, 4to: Ciakciak
+(Ital. Armeno, Turco), Venice, 1804, 4to; 2nd ed. 1829: Azarian
+(Ellenico, Ital. Arm. Turco), Vienna, 1848, 8vo: Mechitarist
+Congregation (Ital. Francese, Arm. Turco), ib. 1846, 8vo. <span class="sc">Latin</span>.&mdash;Mesgnien-Meninski,
+Viennae, 1680, fol. 3 vols.; ed. Jenisch and
+Klezl, ib. 1780-1802, fol. 4 vols. <span class="sc">English</span>.&mdash;Sauerwein, London,
+1855, 12mo: Redhouse, ib. 1856, 8vo, 1176 pages: Id., Eng. Turkish,
+ib. 860, 8vo. <span class="sc">French</span>.&mdash;Kieffer and Bianchi (Turk.-Fr.), Paris,
+1835-1837, 2 vols. 2118 pages: Bianchi (Fr.-Turk.) Paris, 1843-1846,
+8vo, 2 vols. 2287 pages; 1850, 8vo, 2 vols.: Mallouf, ib. 1863-1867,
+8vo, 2 vols. <span class="sc">French and German</span>.&mdash;Zenker (Arab. Pers.), Leipz,
+1862-1876, 4to, 2 vols, 982 pages. <span class="sc">German</span>.&mdash;Korabinsky, Pressburg,
+1788, 8vo: Vambéry, Constantinople, 1858, 8vo. <span class="sc">Italian</span>.&mdash;Molina,
+Roma, 1641, 8vo: Masais, Firenze, 1677, 8vo: Ciadyrgy,
+Milano, 1832-1834, 4to, 2 vols. <span class="sc">Russian</span>.&mdash;Budagov (Comparative
+lexicon of the Turkish-Tartar dialects), St Petersburg, 1869, 8vo,
+2 vols.</p>
+
+<p><span class="bold">Gipsy.</span>&mdash;Bischoff, Ilmenau, 1827, 8vo: Truxillo, Madrid, 1844,
+8vo: Jimenes, Sevilla, 1846, 16mo: Baudrimont, Bordeaux, 1862,
+8vo: Vaillant, Paris, 1868, 8vo: Paspati; Constantinople, 1870,
+4to: Borrow, <i>Romany Lavo Lil</i>, London, 1874, 8vo: Smart and
+Crofton, London, 1875, 8vo.</p>
+
+<p><span class="bold">Albanian.</span>&mdash;Blanchus, Romae, 1635, 8vo: Kaballioti (Romaic,
+Wallach. Alb.), Venice, 1770, 8vo: Xylander, Frankfurt a. M. 1835,
+8vo: Hahn, Jena, 1854, 4to: Rossi da Montalto, Roma, 1866, 8vo.</p>
+
+<p class="center pt2">ASIA</p>
+
+<p><span class="bold">Semitic.</span>&mdash;<span class="sc">Polyglotts</span>.&mdash;Thurneissius, Berolini, 1585, fol.:
+Thorndike, London, 1635, fol.: Schindler, Pentaglotton, Frankf,
+ad M. 1653, fol.: Hottinger, Heptaglotton, ib. 1661, fol.: Castellus,
+London, 1669, fol. 2 vols. (Hebrew, Chaldaic, Syriac, Samaritan,
+Aethiopic and Arabic in one alphabet; Persian separately. It
+occupied him for seventeen years, during which he worked sixteen
+to eighteen hours a day): Otho, Frankf. a. M. 1702, 4to (the same
+languages with Rabbinical).</p>
+
+<p><span class="bold">Hebrew.</span>&mdash;About 875, Zema&#7717;, head of the school of Pumbeditha,
+wrote a Talmudical dictionary of words and things, arranged
+in alphabetical order, which is lost. About 880, Jehudah ben
+&rsquo;Alan, of Tiberias, and Jehudah ibn Koreish, of Tahurt, in Morocco
+wrote Hebrew dictionaries. Saadia ben Joseph (born 892, died 942),
+of Fayum, in Upper Egypt, wrote <span title="Kefer Igaron">&#1503;&#1493;&#1512;&#1490;&#1488; &#1512;&#1508;&#1499;</span>, probably a Hebrew-Arabic
+dictionary. Mena&#7717;em ben Jacob Ibn Sar&#363;q (born 910, died
+about 970), of Tortosa and Cordova, wrote a copious Hebrew
+dictionary, first printed by Herschell F. Filipowski, Edinburgh, 1855,
+8vo, from five MSS. David ben Abraham, of F&#257;s, wrote, in Arabic,
+a large Hebrew dictionary, the MS. of which, a quarto of 313 leaves
+on cotton paper, was found about 1830 by A. Firkowitz, of Eupatoria,
+in the cellar of a Qaraite synagogue in Jerusalem. The age of this
+work cannot be ascertained. About 1050, Ali ben Suleiman wrote a
+dictionary in Arabic, on the plan of that of David ben Abraham. The
+MS. of 429 leaves belongs to Firkowitz. Haja ben Sherira, the
+famous teacher of the Academy of Pumbeditha, wrote a Hebrew
+dictionary in Arabic, called <i>al &#7716;&#257;vi</i> (The Gathering), arranged
+alphabetically in the order of the last radical letter. This dictionary
+is lost, as well as that of the Spaniard Isaac ben Saul, of Lucena.
+Iona ibn Gana&#7717;, of Cordova, born about 985, wrote a Hebrew
+dictionary in Arabic called <i>Kit&#257;b al Azul</i> (Book of Roots). This,
+as well as a Hebrew translation by Samuel ibn Tab&#333;n, is extant
+in MS., and was used by Gesenius in his <i>Thesaurus</i>. Rabbi David
+ben Joseph &#7731;im&#7717;i died soon after 1232. His lexicon of roots, called
+<span title="Shorashim">&#1501;&#1497;&#1513;&#1493;&#1513;</span>, was printed at Naples 1490, fol.; Constantinople, 1513, fol.;
+Naples, 1491, 8vo; Venice, 1552; Berolini, 1838, 4to. <i>Tishbi</i> (The
+Tishbite), by Elijah ben Asher, the Levite, so called because it contained
+712 roots, was printed at Isny 1541, 8vo and 4to, and often
+afterwards. <span class="sc">Latin</span>.&mdash;Münster, Basileae, 1523, 8vo; 5 editions to
+1564: Zamora, Compluti, 1526, fol.: Pellicanus, Argentorati, 1540,
+fol.: Reuchlin, Basil, 1556, fol.: Avenarius, Wittebergae, 1568, fol.;
+auctus, 1589: Pagnini, Lugd. Bat. 1575, fol.; 1577; Genevae, 1614;
+Buxtorf, Basil. 1607, 8vo; 1615, and many other editions: Frey
+(Lat.-Eng.), 2nd ed. London, 1815, 8vo: Gesenius, <i>Thesaurus</i>, Leipz.
+1829-1858, 4to, 3 vols. <span class="sc">English</span>.&mdash;Bale, London, 1767, 4to: Parkhurst,
+ib. 1792, 4to: Lee, ib. 1840, 8vo: Gesenius, translated by
+Robinson, ib. 1844, 8vo; by Tregelles, ib. 1846, 4to: Fuerst, 4th ed.
+transl. by Davidson, ib. 1866, 8vo: 1871, 8vo, 1547 pages. <span class="sc">French</span>.&mdash;Leigh,
+Amst. 1703, 4to: Glaire, Paris, 1830, 8vo; 1843. <span class="sc">German</span>.&mdash;Gesenius,
+Leipzig, 1810-1812, 8vo, 2 vols.: Fuerst, ib. 1842, 16mo:
+ib. 1876, 8vo, 2 vols. <span class="sc">Italian</span>.&mdash;Modena, Venetia, 1612, 4to; 1640;
+Coen, Reggio, 1811, 8vo: Fontanella, Venezia, 1824, 8vo. <span class="sc">Dutch</span>.&mdash;Waterman,
+Rotterdam, 1859, &amp;c., 8vo. <span class="sc">Hungarian</span>.&mdash;Ehrentheil
+(Pentateuch), Pest, 1868, 8vo. <span class="sc">Romaic</span>.&mdash;Loundes, Melité. 1845,
+8vo, 987 pages.</p>
+
+<p><span class="bold">Rabbinical and Chaldee.</span>&mdash;Nathan ben Ye&#7717;iel of Rome wrote in the
+beginning of the 12th century a Talmudic dictionary, <i>Aruch</i>, printed
+1480 (?), s. l., fol.; Pesaro, 1517, fol.; Venice, 1531; and often:
+Isaiah ben Loeb, Berlin, wrote a supplement to <i>Aruch</i>, vol. i. Breslau,
+1830, 8vo; vol. ii. (&#1500; to &#1514;), Wien, 1859, 8vo: Münster, Basil. 1527,
+4to, 1530, fol.: Elijah ben Asher, the Levite, transl. by Fagius,
+Isnae, 1541, fol.; Venet. 1560: David ben Isaac de Pomis, <i>Zama&#7717;
+David</i>, Venet. 1587, fol.: Buxtorf, Basileae, 1639, fol.: ed. Fischer,
+Leipz. 1866-1875, 4to: Otho, Geneva, 1675, 8vo; Altona, 1757, 8vo:
+Zanolini, Patavii, 1747, 8vo: Hornheim, Halle, 1807, 8vo: Landau,
+Prag, 1819-1824, 8vo, 5 vols.: Dessauer, Erlangen, 1838, 8vo: Nork
+(<i>i.e.</i> Korn), Grimma, 1842, 4to: Schönhak, Warschau, 1858, 8vo,
+2 vols. <span class="sc">Targums</span>.&mdash;Levy, Leipzig, 1866-68 4to, 2 vols.; 1875:
+Id. (Eng.), London, 1869, 8vo, 2 vols. <span class="sc">Talmud</span>.&mdash;Löwy (in Heb.),
+Wien, 1863, 8vo: Levy, Leipzig, 1876, &amp;c., 4to. <span class="sc">Prayer-Book</span>.&mdash;Hecht,
+Kreuznach, 1860, 8vo: Nathan, Berlin, 1854, 12mo.
+<span class="sc">Synonyms</span>.&mdash;Pantavitius, Lodevae, 1640, fol. <span class="sc">Foreign Words</span>.&mdash;Rabeini,
+Lemberg, 1857, 8vo, &amp;c. <span class="sc">Jewish-German</span>.&mdash;Callenberg,
+Halle, 1736, 8vo: Vollbeding, Hamburg, 1808, 8vo: Stern,
+München, 1833, 8vo, 2 vols.: Theile, Berlin, 1842-1843, 8vo, 2 vols.:
+Avé-Lallemant, <i>Das deutsche Gaunerthum</i>, Leipzig, 1858, 8vo, 4 vols.;
+vol. iv. pp. 321-512.</p>
+
+<p><span class="bold">Ph&oelig;nician.</span>&mdash;M. A. Levy, Breslau, 1864, 8vo.</p>
+
+<p><span class="bold">Samaritan.</span>&mdash;Crinesius, Altdorphi, 1613, 4to: Morini, Parisiis,
+1657, 12mo: Hilligerus, Wittebergae, 1679, 4to: Cellarius, Cizae,
+1682, 4to; Frankof. 1705: Uhlemann, Leipsiae, 1837, 8vo: Nicholls,
+London, 1859, 8vo.</p>
+
+<p><span class="bold">Assyrian.</span>&mdash;Norris, London, 1868, 8vo, 3 vols. <span class="sc">Proper Names</span>.&mdash;Menant,
+Paris, 1861, 8vo.</p>
+
+<p><span class="bold">Accadian.</span>&mdash;Lenormant, Paris, 1875, 8vo.</p>
+
+<p><span class="bold">Syriac.</span>&mdash;Joshua ben Ali, a physician, who lived about 885, made
+a Syro-Arabic lexicon, of which there is a MS. in the Vatican.
+Hoffmann printed this lexicon from Alif to Mim, from a Gotha MS.,
+Kiel, 1874, 4to. Joshua bar Bahlul, living 963, wrote another, great
+part of which Castelli put into his lexicon. His MS. is now at
+Cambridge, and, with those at Florence and Oxford, was used by
+Bernstein. Elias bar Shinaya, born 975, metropolitan of Nisibis,
+1009, wrote a Syriac and Arabic lexicon, entitled <i>Kit&#257;b &#363;t Tarjuman
+fi Taalem Loghat es S&#363;ri&#257;n</i> (Book called the Interpreter for teaching
+the Language of the Syrians), of which there is a MS. in the British
+Museum. It was translated into Latin by Thomas à Novaria, a
+Minorite friar, edited by Germanus, and published at Rome by
+Obicinus, 1636, 8vo. It is a classified vocabulary, divided in 30
+chapters, each containing several sections. Crinesius, Wittebergae,
+<span class="pagenum"><a name="page197" id="page197"></a>197</span>
+1612, 4to: Buxforf, Basileae, 1622, 4to: Ferrarius, Romae, 1622,
+4to: Trost, Cothenis Anhaltor, 1643, 4to: Gutbir, Hamburgi, 1667,
+8vo: Schaaf, Lugd. Bat, 1708, 4to: Zanolini, Patavii, 1742, 4to:
+Castellus, ed. Michaelis, Göttingen, 1788, 4to, 2 vols.: Bernstein,
+Berlin, 1857, &amp;c. fol.: Smith (Robt. Paine), Dean of Canterbury,
+Oxonii, 1868, &amp;c. fol.: fasc. 1-3 contain 538 pages: Zingerle,
+Romae, 1873, 8vo, 148 pages.</p>
+
+<p><span class="bold">Arabic.</span>&mdash;The native lexicons are very many, voluminous and
+copious. In the preface to his great Arabic-English lexicon, Lane
+describes 33, the most remarkable of which are-the <i>&rsquo;Ain</i>, so called
+from the letter which begins its alphabet, commonly ascribed to al
+Khalil (who died before A.H. 175 [<span class="sc">a.d.</span> 791], aged seventy-four): the
+<i>Sihah</i> of Jauhari (died 398 [1003]): the <i>Mohkam</i> of Ibn Sidah the
+Andalusian, who was blind, and died A.H. 458 [<span class="sc">a.d.</span> 1066], aged about
+sixty: the <i>Asas</i> of Zamakhshari (born 467 [1075], died 538 [1144]),
+&ldquo;a most excellent repertory of choice words and phrases&rdquo;: the
+<i>Lis&#257;n el &rsquo;Arab</i> of Ibn Mukarram (born 630 [1232], died 711 [1311]);
+Lane&rsquo;s copy is in 28 vols. 4to: the <i>Kamus</i> (The Sea) of Fairuzabadi
+(born 729 [1328], died 816 [1413]),: the <i>Taj el Arus</i>, by Murtada
+Ez Zebadi (born <span class="sc">a.d.</span> 1732, died 1791)&mdash;the copy made for Lane
+is in 24 vols. thick 4to. The <i>Sihah</i> was printed Hardervici Getorum,
+1774, 4to; Bulak, 1865, fol. 2 vols.: <i>Kamus</i>, Calcutta, 1817, fol.
+2 vols.; Bombay, 1855, fol. 920 pages: <i>Sirr el Lagal</i>, by Farish esh
+Shidiac, Tunis, fol. 609 pages: <i>Muh&#299;t al Muh&#299;t</i>, by Beitrus Al
+Bustani Beirut, 1867-1870, 2 vols. 4to, 2358 pages (abridged as
+<i>Katr Al Muhit</i>, ib. 1867-1869, 2 vols. 8vo, 2352 pages), is excellent for
+spoken Arabic. <span class="sc">Persian.</span>&mdash;The <i>Surah</i>, by Jumal, Calcutta, 1812-1815,
+2 vols. 4to: <i>Samachsharii Lexicon</i>, ed. Wetzstein, Leipz. 1845,
+4to; 1850: <i>Muntakhal al Loghat</i>, Calcutta, 1808; ib. 1836; Lucknow,
+1845; Bombay, 1862, 8vo, 2 vols.: <i>Muntaha l&rsquo;Arabi</i>, 4 vols. fol.
+1840: <i>Shams al Loghat</i>, Bombay, 1860, fol. 2 vols. 509 pages.
+<span class="sc">Turkish.</span>&mdash;<i>Achteri Kabir</i>, Constantinople. 1827, fol.: <i>El Kamus</i>,
+ib. 1816, fol. 3 vols.; translated by Açan Effendi, Bulak, fol.
+3 vols.; <i>El Sihah</i>, translated by Al Vani, Constantinople, 1728, fol.
+2 vols.: 1755-1756; Scutari, 1802, fol. 2 vols. <span class="sc">Latin.</span>&mdash;Raphelengius,
+Leiden, 1613, fol.: Giggeius, Mediolani, 1632, fol. 4 vols.: Golius
+Lugd. Bat. 1653, fol. (the best before Lane&rsquo;s): Jahn, Vindobonae,
+1802, 8vo: Freytag, Halle, 1830-1838, 4 vols. 4to; abridged, ib. 1837,
+4to. <span class="sc">English.</span>&mdash;Catafago (Arab.-Eng. and Eng.-Arab.), London,
+1858, 8vo, 2 vols.; 2nd ed. 1873, 8vo: Lane, London, 1863-1893
+(edited after Lane&rsquo;s death, from 1876, by his grandnephew, Stanley
+Lane-Poole. The Arabic title is <i>Medd el Kamoos</i>, meaning either the
+Flow of the Sea, or The Extension of the Kamus. It was undertaken
+in 1842, at the suggestion and at the cost of the 6th duke of
+Northumberland, then Lord Prudhoe, by Mr Lane, who returned to
+Egypt for the purpose, and lived in Cairo for seven years to study, and
+obtain copies of, the great MS. lexicons in the libraries of the mosques,
+few of which had ever been seen by a European, and which were so
+quickly disappearing through decay, carelessness and theft, that the
+means of composing such a work would not long have existed).
+Newman (modern), ib. 1872, 8vo, 2 vols. 856 pages. <span class="sc">French.</span>&mdash;Ruphy
+(Fr.-Ar.), Paris, 1802, 4to: Bochtor (do.), Paris, 1828, 4to,
+2 vols.; 2nd ed. ib. 1850: Roland de Bussy (Algiers, Fr.-Ar.), Alger,
+1835, 16mo: Id., 1836, 8vo; 1839: Berggren (Fr.-vulg. Ar., Syria
+and Egypt.), Upsala, 1844, 4to: Farhat (Germanos), revu par
+Rochaid ed Dahdah, Marseille, 1849, 4to: Biberstein Kasimirski,
+Paris, 1846, 8vo, 2 vols.; 1853-1856; 1860, 2 vols. 3032 pages: Marcel
+(vulgar dialects of Africa), Paris, 1830; 1835, 8vo; 1837; enlarged,
+1869, 8vo; Paulmier (Algeria), 2nd ed. Paris, 1860, 8vo, 931 pages;
+1872: Bernard (Egypt), Lyon, 1864, 18mo: Cuche, Beirut, 1862,
+8vo; 1867: Nar Bey (A. Calfa), 2nd ed. Paris, 1872, 12mo, 1042
+pages: Cherbonneau (written language), Paris, 1876, 2 vols. 8vo:
+Id. (Fr.-Ar.), Paris, 1872, 8vo: Beausier (Algiers, Tunis, legal,
+epistolary), Alger, 1871, 4to, 764 pages; 1873. <span class="sc">German.</span>&mdash;Seyfarth
+(Algeria), Grimma, 1849, 16mo: Wolff (Mod. Ar.), Leipzig, 1867,
+8vo: Wahrmund (do.), Giessen, 1870-1875, 8vo, 4 vols. <span class="sc">Italian.</span>&mdash;Germano,
+Roma, 1636, 8vo; (Ar. Lat. It.), Romae, 1639, fol.:
+<i>Dizionario</i>, Bulak. 1824, 4to: Schiaparelli, Firenze, 1871, 4to,
+641 pages. <span class="sc">Spanish.</span>&mdash;Alcala, Grenada, 1505, 4to: Cañes, Madrid,
+1787, fol. 3 vols. <span class="sc">Sufi Technical Terms.</span>&mdash;Abd Errahin, ed.
+Sprenger, Calcutta, 1845, 8vo. <span class="sc">Technical Terms of the Mussulman
+Sciences.</span>&mdash;Abd al Hagg and Gholam Kadir, Calcutta, 1853-1862,
+4to, 1593 pages. <span class="sc">Medical Terms.</span>&mdash;Pharaon and Bertherand,
+Paris, 1860, 12mo. <span class="sc">Materia Medica.</span>&mdash;Muhammed Abd Allah
+Shirazi, <i>Ulfaz Udwiyeh</i>, translated by Gladwin (Eng. Pers. Hindi),
+Calcutta, 1793, 4to, 1441 words. <span class="sc">Noms des Vêtements.</span>&mdash;Dozy,
+Amst. 1845, 8vo. <span class="sc">Wörter in entgegengesetzten Bedeutungen.</span>&mdash;Redslob,
+Göttingen, 1873, 8vo. <span class="sc">Koran.</span>&mdash;Willmet (also in
+Haririum et vitam Timuri), Lugd. Bat. 1784, 4to; Amst. 1790:
+Fluegel, <i>Concordantia</i>, Leipz. 1842, 4to: Penrice, <i>Dictionary and
+Glossary</i>, London, 1873, 4to. <span class="sc">El Tabrizi&rsquo;s Logic.</span>&mdash;Mir Abufeth
+(French), Bulak, 1842, 8vo. <span class="sc">Maltese.</span>&mdash;Vassali, Romae, 1796,
+4to: Falzon (Malt. Ital. Eng.), Malta, <i>s.a.</i> 8vo: Vella, Livorno,
+1843, 8vo.</p>
+
+<p><span class="bold">Armenian.</span>&mdash;Mechitar, Venice, 1749-1769, 4to, 2 vols.: Avedichiam,
+Sürmelian and Aucher (Aukerian), ib. 1836-1837, 4to, 2 vols.:
+Aucher, ib. 1846, 4to. <span class="sc">Polyglot.</span>&mdash;Villa (Arm.-vulg., litteralis, Lat.
+Indicae et Gallicae), Romae, 1780. <span class="sc">Greek and Latin.</span>&mdash;Lazarists,
+Venice, 1836-1837, 4to, 2 vols. 2217 pages. <span class="sc">Latin.</span>&mdash;Rivola, Mediolani,
+1621, fol.: Nierszesovicz, Romae, 1695, 4to; Villotte, ib. 1714,
+fol.: Mechitar, Venetiae, 1747-1763, 4to, 2 vols. <span class="sc">English.</span>&mdash;Aucher,
+Venice, 1821-1825, 4to, 2 vols. <span class="sc">French.</span>&mdash;Aucher, Venise, 1812-1817,
+8vo, 2 vols.; (Fr.-Arm. Turc.), ib. 1840, 4to: Eminian, Vienna, 1853,
+4to: Calfa, Paris, 1861, 8vo, 1016 pages; 1872. <span class="sc">Italian.</span>&mdash;Ciakciak,
+Venezia, 1837, 4to. <span class="sc">Russian.</span>&mdash;Khudobashev [Khutapashian],
+Moskva, 1838, 8vo, 2 vols. <span class="sc">Russ. Arm.</span>&mdash;Adamdarov, ib.
+1821, 8vo: Popov, ib. 1841, 8vo, 2 vols. <span class="sc">Modern Words.</span>&mdash;Riggs,
+Smyrna, 1847, 8vo.</p>
+
+<p><span class="bold">Georgian.</span>&mdash;Paolini (Ital.), Roma, 1629, 4to: Klaproth (Fr.),
+Paris, 1827, 8vo: Tshubinov (Russian, French), St Petersburg, 1840,
+4to; 1846, 8vo, 2 vols. 1187 pages.</p>
+
+<p><span class="bold">Circassian.</span>&mdash;Loewe, London, 1854, 8vo.</p>
+
+<p><span class="bold">Ossetic.</span>&mdash;Sjörgen, St Petersb. 1844, 4to.</p>
+
+<p><span class="bold">Kurd.</span>&mdash;Garzoni, Roma, 1787, 8vo: Lerch (German), St Petersburg,
+1857, 8vo: Id. (Russian), ib. 1856-1858, 8vo.</p>
+
+<p><span class="bold">Persian.</span>&mdash;<i>Burhani Qatiu</i>, arranged by J. Roebuck, Calcutta,
+1818, 4to: <i>Burhan i Kati</i>, Bulak, 1836, fol.: Muhammed Kazim,
+Tabriz, 1844, fol.: <i>Haft Kulzum</i> (The Seven Seas), by Ghazi ed din
+Haidar, King of Oude, Lucknow, 1822, fol. 7 vols. <span class="sc">Arabic.</span>&mdash;<i>Shums
+ul Loghat</i>, Calcutta, 1806, 4to, 2 vols. <span class="sc">Turkish.</span>&mdash;Ibrahim Effendi,
+<i>Farhangi Shu&rsquo;uri</i>, ib. 1742, fol. 2 vols. 22,530 words, and 22,450
+poetical quotations: <i>Burhan Kati</i>, by Ibn Kalif, translated by
+Ahmed Asin Aintabi, ib. 1799, fol.; Bulak, 1836, fol.: Hayret
+Effendi, ib. 1826, 8vo. <span class="sc">Armenian.</span>&mdash;Douzean, Constantinople,
+1826, fol. <span class="sc">Bengali.</span>&mdash;Jay Gopal, Serampore, 1818, 8vo. <span class="sc">Latin.</span>&mdash;Vullers
+(Zend appendix), Bonnae ad Rhen, 1855-1868, 4to, 2 vols.
+2544 pages; Supplement of Roots, 1867, 142 pages. <span class="sc">English.</span>&mdash;Gladwin,
+Malda in Bengal, 1780, 4to; Calcutta, 1797: Kirkpatrick,
+London, 1785, 4to: Moises, Newcastle, 1794, 4to: Rousseau,
+London, 1802, 8vo; 1810: Richardson (Arab, and Pers.), ib. 1780-1800,
+fol. 2 vols.; ed. Wilkins, ib. 1806-1810, 4to, 2 vols.; ed Johnson,
+ib. 1829, 4to: Ramdhen Sen, Calcutta, 1829, 8vo; 1831: Tucker
+(Eng.-Pers.), London, 1850, 4to: Johnson (Pers. and Arab.), ib.
+1852, 4to: Palmer, ib. 1876, 8vo, 726 pages. <span class="sc">French.</span>&mdash;Handjeri
+(Pers. Arab. and Turkish), Moscou, 1841, 4to, 3 vols. 2764 pages:
+Bergé, Leipzig, 1869, 12mo. <span class="sc">German.</span>&mdash;Richardson, translated by
+Wahl as <i>Orientalische Bibliotheque</i>, Lemg, 1788-1792, 8vo, 3 vols.
+<span class="sc">Italian.</span>&mdash;Angelus a S. Josepho [<i>i.e.</i> Labrosse] (Ital. Lat. Fr.), Amst.
+1684, fol.</p>
+
+<p><span class="bold">Old Persian.</span>&mdash;(Cuneiform), Benfey (German), Leipzig, 1847, 8vo:
+Spiegel (id.), ib. 1862, 8vo: Kossovich (Latin), Petropoli, 1872, 8vo.</p>
+
+<p><span class="bold">Zend.</span>&mdash;Justi, Leipzig, 1864, 4to: Vullers, Persian Lexicon,
+Appendix: Lagarde, Leipzig, 1868, 8vo.</p>
+
+<p><span class="bold">Pahlavi.</span>&mdash;<i>An old Pahlavi and Pazend Glossary</i>, translated by
+Destur Hoshengi Jamaspji, ed. Haug, London, 1867, 8vo; 1870,
+8vo: West, Bombay, 1874, 8vo.</p>
+
+<p><span class="sc pt2">Indian Terms.</span>&mdash;<i>The Indian Vocabulary</i>, London, 1788, 16mo:
+Gladwin, Calcutta, 1797, 4to: Roberts, London, 1800, 8vo: Rousseau,
+ib. 1802, 8vo: Roebuck (naval), ib. 1813, 12mo: C. P. Brown,
+<i>Zillah Dict.</i>, Madras, 1852, 8vo: Robinson (Bengal Courts), Calcutta,
+1854, 8vo; 1860: Wilson, London, 1855, 4to: Fallon, Calcutta,
+1858, 8vo.</p>
+
+<p><span class="bold">Sanskrit.</span>&mdash;Amarasimha (lived before <span class="sc">a.d.</span> 1000), <i>Amarakosha</i>
+Calcutta, 1807, 8vo; ib. 1834, 4to; Bombay, 1860, 4to; Lucknow,
+1863, 4to; Madras, 1870, 8vo, in Grantha characters; Cottayam,
+1873, 8vo, in Malaylim characters; Benares, 1867, fol. with
+<i>Amaraviveka</i>, a commentary by Mahesvara: Rajah Radhakanta
+Deva, <i>Sabdakalpadruma</i>, Calcutta, 1821-1857, 4to, 8 vols. 8730 pages:
+2nd ed. 1874, &amp;c.: Bhattachdrya, <i>Sabdastoma Mahanidhi</i>, Calcutta,
+1869-1870, 8vo, parts i.-vii. 528 pages: <i>Abhidhanaratnamala</i>, by
+Halayudha, ed. Aufrecht, London, 1861, 8vo: <span class="sc">Vachaspatya</span>, by
+Taranatha Tarkavachaspati, Calcutta, 1873, &amp;c., 4to (parts i.-vii.,
+1680 pages). <span class="sc">Bengali.</span>&mdash;<i>Sabdasindhu</i>, Calcutta, 1808: <i>Amarakosa</i>,
+translated by Ramodoyu Bidjalunker, Calcutta, 1831, 4to:
+Mathurana Tarkaratna, <i>Sabdasandarbhasindhu</i>, Calcutta, 1863, 4to.
+<span class="sc">Marathi.</span>&mdash;Ananta Sastri Talekar, Poona, 1853, 8vo, 495 pages:
+Madhava Chandora, Bombay, 1870, 4to, 695 pages. <span class="sc">Telugu.</span>&mdash;<i>Amarakosha</i>,
+Madras, 1861, ed. Kala, with <i>Gurubalala prabodhika</i>, a
+commentary, ib. 1861, 4to; with the same, ib. 1875, 4to, 516 pages;
+with <i>Amarapadaparijata</i> (Sans. and Tel.), by Vavilla Ramasvani
+Sastri, ib. 1862, 4to; ib. 1863, 8vo; 3rd ed. by Jaganmohana
+Tarkalankara and Khetramohana, 1872, &amp;c., parts i.-iv. 600 pages:
+Suria Pracasa Row, <i>Sarva-Sabda-Sambodhini</i>, ib. 1875, 4to, 1064
+pages. <span class="sc">Tibetan and Mongol.</span>&mdash;Schiefner, <i>Buddhistische Triglotte</i>,
+St Petersburg, 1859, fol., the <i>Vyupatti</i> or <i>Mahavyupatti</i> from the
+<i>Tanguir</i>, vol. 123 of the Sutra. <span class="sc">Latin.</span>&mdash;Paulinus a Sancto
+Bartholomeo, Amarasinha, sectio i. de coelo, Romae, 1798, 4to:
+Bopp. Berlin, 1828-1830, 4to; 2nd ed. 1840-1844; 3rd, 1866, 4to.
+<span class="sc">English.</span>&mdash;<i>Amarakosha</i>, trans. by Colebrooke, Serampore, 1808,
+4to; 1845, 8vo: Rousseau, London, 1812, 4to: Wilson, Calcutta,
+1819, 4to; 2nd ed. 1832: ed. Goldstücker, Berlin, 1862, &amp;c., folio,
+to be in 20 parts: Yates, Calcutta, 1846, 4to: Benfey, London, 1865,
+8vo: Ram Jasen, Benares, 1871, 8vo, 713 pages: Williams, Oxford,
+1872, 4to. <span class="sc">English-Sanskrit.</span>&mdash;Williams, London, 1851, 4to.
+<span class="sc">French.</span>&mdash;Amarakosha, transl. by Loiseleur Deslongchamps, Paris,
+1839-1845, 8vo, 2 vols. 796 pages: Burnouf and Leupol, Nancy,
+1863-1864, 8vo. <span class="sc">German.</span>&mdash;Böhtlingk and Roth, St Petersb. 1853,
+&amp;c., 4to, 7 vols. to 1875. <span class="sc">Italian.</span>&mdash;Gubernatis, Torino, 1856, &amp;c.
+8vo, unfinished, 2 parts. <span class="sc">Russian.</span>&mdash;Kossovich, St Petersburg, 1859,
+<span class="pagenum"><a name="page198" id="page198"></a>198</span>
+8vo. <span class="sc">Roots</span>.&mdash;Wilkins, London, 1815, 4to: Rosen, Berolini, 1827,
+8vo: Westergaard, Bonnae, 1840-1841, 8vo: Vishnu Parasurama
+Sastri Pandita (Sans. and Marathi), Bombay, 1865, 8vo: Taranatha
+Tarkavachaspati, <i>Dhatupadarsa</i>, Calcutta, 1869, 8vo: Leupol, Paris,
+1870, 8vo. <span class="sc">Synonyms</span>.&mdash;<i>Abhidhanacintamani</i>, by Hemachadra, ed.
+Colebrooke, Calcutta, 1807, 8vo; translated by Böhtlingk and Rieu
+(German), St Petersburg, 1847, 8vo. <span class="sc">Homonyms</span>.&mdash;Medinikara,
+<i>Medinikosha</i>, Benares, 1865, 4to; Calcutta, 1869, 8vo; ib. 1872,
+8vo. <span class="sc">Derivatives</span>.&mdash;Hirochand and Rooji Rangit, <i>Dhatumanjari</i>,
+Bombay, 1865, 8vo. <span class="sc">Technical Terms of the Nyâya Philosophy.</span>&mdash;<i>Nyâyakosa</i>,
+by Bhimachârya Jhalakîkar (Sanskrit),
+Bombay, 1875, 8vo, 183 pages. <span class="sc">Rig Veda</span>.&mdash;Grassmann, Leipzig,
+1873-1875, 8vo.</p>
+
+<p><span class="bold">Bengali.</span>&mdash;Manoel, Lisboa, 1743, 8vo: Forster, Calcutta, 1799-1802,
+4to, 2 vols. 893 pages: Carey, Serampore, 1815-1825, 4to, 2 vols.;
+ed. Marshman, ib. 1827-1828, 8vo, 2 vols.; 3rd ed. ib. 1864-1867,
+8vo; abridged by Marshman, ib. 1865, 8vo; ib. 1871, 8vo, 2 vols.
+936 pages: Morton, Calcutta, 1828, 8vo: Houghton, London, 1833,
+4to: Adea, <i>Shabdabudhi</i>, Calcutta, 1854, 604 pages. <span class="sc">English</span>.&mdash;Ram
+Comul Sen, ib. 1834, 4to, 2 vols.; London, 1835, 4to:
+D&rsquo;Rozario, Calcutta, 1837, 8vo: Adea, <i>Abhidan</i>, Calcutta, 1854,
+761 pages. <span class="sc">English Lat.</span>&mdash;Ramkissen Sen, ib. 1821, 4to. <span class="sc">Eng.-Beng.
+and Manipuri.</span>&mdash;[Gordon], Calcutta, 1837, 8vo.</p>
+
+<p><span class="bold">Canarese.</span>&mdash;Reeve, Madras, 1824-1832, 4to, 2 vols.; ed. Sanderson,
+Bangalore, 1858, 8vo, 1040 pages; abridged by the same, 1858,
+8vo, 276 pages: <i>Dictionarium Canarense</i>, Bengalori, 1855, 8vo:
+<i>School Dictionary</i>, Mangalore, 1876, 8vo, 575 pages.</p>
+
+<p><span class="bold">Dardic Languages.</span>&mdash;Leitner (Astori, Ghilghiti, Chilasi, and dialects
+of Shina, viz. Arnyia, Khajuna and Kalasha), Lahore, 1868, 4to.</p>
+
+<p><span class="bold">Guzarati.</span>&mdash;(English) Mirza Mohammed Cauzim, Bombay, 1846,
+4to; Shapurji Edalji, ib. 1868, 8vo, 896 pages: Karsandas Mulji,
+ib. 1868, 8vo, 643 pages.</p>
+
+<p><span class="bold">Hindi.</span>&mdash;Rousseau, London, 1812, 4to: Adam, Calcutta, 1829,
+8vo: Thompson, ib. 1846, 8vo: J. D. Bate, London, 1876, 8vo, 809
+pages. <span class="sc">English</span>.&mdash;Adam, Calcutta, 1833, 8vo. <span class="sc">English, Urdu
+and Hindi</span>.&mdash;Mathuraprasada Mirsa, Benares, 1865, 8vo, 1345
+pages.</p>
+
+<p><span class="bold">Hindustani.</span>&mdash;Ferguson, London, 1773, 4to: Gilchrist, Calcutta,
+1800, 8vo; ed. Hunter, Edinb. 1810; Lond. 1825: Taylor, Calcutta,
+1808, 4to, 2 vols.: Gladwin (Persian and Hind.), Calcutta, 1809,
+8vo, 2 vols.: Shakespeare, London, 1817, 4to; 1820; 1834; 1849:
+Forbes, London, 1847, 8vo; 1857: Bertrand (French), Paris, 1858,
+8vo: Brice, London, 1864, 12mo: Fallon, Banaras, 1876, &amp;c., to
+be in about 25 parts and 1200 pages. <span class="sc">English</span>.&mdash;Gilchrist, 1787-1780,
+4to, 2 parts: Thompson, Serampore, 1838, 8vo.</p>
+
+<p><span class="bold">Kashmiri.</span>&mdash;Elmslie, London, 1872, 12mo.</p>
+
+<p><span class="bold">Khassia.</span>&mdash;Roberts, Calcutta, 1875, 12mo.</p>
+
+<p><span class="bold">Malayalim.</span>&mdash;Fabricius and Breithaupt, Weperg, 1779, 4to: Bailey,
+Cottayam, 1846, 8vo: Gundert, Mangalore, 1871, 8vo, 1171 pages.</p>
+
+<p><span class="bold">Marathi.</span>&mdash;Carey, Serampore, 1810, 8vo: Kennedy, Bombay,
+1824, fol.: Jugunnauth Shastri Kramavant, Bombay, 1829-1831,
+4to, 3 vols.: Molesworth, ib. 1831, 4to; 2nd ed. 1847, 4to; ed. Candy,
+Bombay, 1857, 4to, 957 pages; abridged by Baba Padmanji,
+ib. 1863, 8vo; 2nd ed. (abridged), London, 1876, 8vo, 644 pages.
+<span class="sc">English</span>.&mdash;Molesworth, Bombay, 1847, 4to.</p>
+
+<p><span class="bold">Oriya.</span>&mdash;Mohunpersaud Takoor, Serampore, 1811, 8vo: Sutton,
+Cuttack, 1841-1848, 8vo, 3 vols. 856 pages.</p>
+
+<p><span class="bold">Pali.</span>&mdash;Clough, Colombo, 1824, 8vo: Moggallana Thero (a Sinhalese
+priest of the 12th century), <i>Abhidhanappika</i> (Pali, Eng.
+Sinhalese), ed. Waskeduwe Subheti, Colombo, 1865, 8vo: Childers,
+London, 1872-1875, 8vo, 658 pages. <span class="sc">Roots</span>.&mdash;Silavansa, <i>Dhatumanjusa</i>
+(Pali Sing. and Eng.), Colombo, 1872, 8vo.</p>
+
+<p><span class="bold">Prakrit.</span>&mdash;Delius, <i>Radices</i>, Bonnae ad Rh., 1839, 8vo.</p>
+
+<p><span class="bold">Punjabi.</span>&mdash;Starkey, 1850, 8vo; Lodiana Mission, Lodiana,
+1854-1860, 444 pages.</p>
+
+<p><span class="bold">Pushtu</span> or <b>Afghan.</b>&mdash;Dorn, St Petersb. 1845, 4to: Raverty,
+London, 1860, 4to; 2nd ed. ib. 1867, 4to: Bellew, 1867, 8vo.</p>
+
+<p><span class="bold">Sindhi.</span>&mdash;Eastwick, Bombay, 1843, fol. 73 pages: Stack, ib. 1855,
+8vo, 2 vols.</p>
+
+<p><span class="bold">Sinhalese.</span>&mdash;Clough, Colombo, 1821-1830, 8vo, 2 vols.: Callaway
+(Eng., Portuguese and Sinhalese), ib. 1818, 8vo: Id., <i>School
+Dictionary</i>, ib. 1821, 8vo: Bridgenell (Sinh.-Eng.), ib. 1847, 18mo:
+Nicholson (Eng.-Sinh.), 1864, 32mo, 646 pages.</p>
+
+<p><span class="bold">Tamil.</span>&mdash;Provenza (Portug.), Ambalacotae, 1679, 8vo: <i>Sadur
+Agurardi</i>, written by Beschi in 1732, Madras, 1827, fol.; Pondicherry,
+1875, 8vo: Blin (French), Paris, 1834, 8vo: Rottler, Madras, 1834-1841,
+4to, 4 vols.: Jaffna Book Society (Tamil), Jaffna, 1842, 8vo,
+about 58,500 words: Knight and Spaulding (Eng. Tam.), ib. 1844,
+8vo; <i>Dictionary</i>, ib. 1852, 4to: Pope, 2nd ed. ib. 1859, 8vo: Winslow,
+Madras, 1862, 4to, 992 pages, 67,452 words.</p>
+
+<p><span class="bold">Telugu.</span>&mdash;Campbell, Madras, 1821, 4to: C. P. Brown, Madras
+(Eng.-Tel.), 1852, 8vo, 1429 pages: Id. (Tel.-Eng.), ib. 1852, 8vo,
+1319 pages. <span class="sc">Mixed Telugu</span>.&mdash;Id., ib. 1854, 8vo.</p>
+
+<p><span class="bold">Thuggee.</span>&mdash;Sleeman, Calcutta, 1830, 8vo, 680 Ramasi words.</p>
+
+<p><span class="bold">Indo-Chinese Languages.</span>&mdash;Leyden, <i>Comparative Vocabulary of
+Barma, Malaya and Thai</i>, Serampore, 1810, 8vo. <i>Annamese</i>:
+Rhodes (Portug. and Lat.), Romae, 1651, 4to: Pigneaux and Taberd,
+Fredericinagori, 1838, 4to; Legrand de la Liraye, Paris, 1874, 8vo:
+Pauthier (Chin. Ann.-Fr. Lat.), Paris, 1867, &amp;c., 8vo. <i>Assamese</i>:
+Mrs Cutter, Saipur, 1840, 12 mo; Bronson, London, 1876, 8vo, 617
+pages. <i>Burmese</i>: Hough (Eng.-Burm.), Serampore, 1825, Moulmain,
+1845, 8vo, 2 vols. 955 pages: Judson, Calcutta, 1826, 8vo;
+(Eng. Burm.), Moulmain, 1849, 4to; (Burm. Eng.), ib. 1852, 8vo;
+2nd ed., Rangoon, 1866, 8vo, 2 vols. 968 pages: Lane, Calcutta, 1841,
+4to. <i>Cambodian</i>: Aymonier (Fr.-Camb.), Saigon, 1874, 4to; Id.
+(Camb.-Fr.), ib. 1875, fol. <i>Karen</i>: Sau-kau Too (Karen), Tavoy,
+1847, 12mo, 4 vols.: Mason, Tavoy, 1840, 4to. <i>Sgau-Karen</i>: Wade,
+ib. 1849, 8vo. <i>Siamese (Thai)</i>: Pallegoix (Lat. French, Eng.),
+Paris, 1854, 4to: <i>Dictionarium Latinum Thai</i>, Bangkok, 1850,
+4to, 498 pages.</p>
+
+<p><span class="bold">Malay.</span>&mdash;<span class="sc">Latin</span>.&mdash;Haex, Romae, 1631, 4to; Batavia, 1707.
+<span class="sc">Dutch</span>.&mdash;Houtmann (Malay and Malagasy), Amst. 1603, 4to;
+1673; 1680; 1687; 1703; Batavia, 1707: Wiltens and Dankaarts,
+Gravenhage, 1623, 4to; Amst. 1650; 1677; Batavia, 1708, 4to:
+Heurnius, Amst. 1640, 4to: Gueynier, Batavia, 1677, 4to; 1708:
+Loder, ib. 1707-1708, 4to: Van der Worm, ib. 1708, 4to: Roorda van
+Eysinga (Low), ib. 1824-1825, 8vo, 2 vols.; 12th ed. &rsquo;s Gravenhage,
+1863, 8vo; Id. (Hof, Volks en Lagen Taal), ib. 1855, 8vo: Dissel
+and Lucardie (High Malay), Leiden, 1860, 12mo: Pijnappel, Amst.
+1863, 8vo: Badings, Schoonhoven, 1873, 8vo. <span class="sc">English</span>.&mdash;Houtmann
+(Malay and Malagasy), translated by A. Spaulding, London,
+1614, 4to: Bowrey, ib. 1701, 4to: Howison, ib. 1801, 4to: Marsden,
+ib. 1812, 4to: Thomsen, Malacca, 1820, 8vo; 1827: Crawford,
+London, 1851, 8vo, 2 vols. <span class="sc">French</span>.&mdash;Boze, Paris, 1825, 16mo:
+Elout (Dutch-Malay and French-Malay), Harlem, 1826, 4to:
+Bougourd, Le Havre, 1856, 8vo: Richard, Paris, 1873, 8vo, 2 vols.:
+Favre, Vienna, 1875, 8vo, 2 vols.</p>
+
+<p><span class="bold">Malay Archipelago.</span>&mdash;<i>Batak</i>: Van der Tuuk, Amsterdam, 1861,
+8vo, 564 pages. <i>Bugis</i>: Mathes, Gravenh. 1874, 8vo, 1188 pages:
+Thomsen (Eng.-Bugis and Malay), Singapore, 1833, 8vo. <i>Dyak</i>:
+Hardeland (German), Amst. 1859, 8vo, 646 pages. <i>Javanese</i>: Senerpont
+Domis, Samarang, 1827, 4to, 2 vols.: Roorda van Eysinga,
+Kampen, 1834-1835, 8vo, 2 vols.: Gericke, Amst. 1847, 8vo; ed.
+Taco Roorda, ib. 1871, &amp;c. parts i.-v., 880 pages: Jansz and
+Klinkert, Samarang, 1851, 8vo; 1865: Favre (French), Vienne,
+1870, 8vo. <i>Macassar</i>: Matthes, Amst. 1859, 8vo, 951 pages.
+Sunda: De Wilde (Dutch, Malay and Sunda), Amsterdam,
+1841, 8vo: Rigg (Eng.), Batavia, 1862, 4to, 573 pages. <i>Formosa</i>:
+Happart (Favorlang dialect, written about 1650), Parrapattan,
+1840, 12mo.</p>
+
+<p><span class="bold">Philippines.</span>&mdash;<i>Bicol</i>: Marcos, Sampaloc, 1754, fol. <i>Bisaya</i>: Sanchez,
+Manila, 1711, fol.: Bergaño, ib. 1735, fol.: Noceda, ib. 1841:
+Mentrida (also Hiliguena and Haraya) ib. 1637, 4to; 1841, fol. 827
+pages: Felis de la Encarnacion, ib. 1851, 4to, 2 vols. 1217 pages.
+<i>Ibanac</i>: Bugarin, ib. 1854, 4to. <i>Ilocana</i>, Carro, ib. 1849, fol.
+<i>Pampanga</i>: Bergaño, ib. 1732, fol. <i>Tagala</i>: Santos, Toyabas, 1703,
+fol.; ib. 1835, 4to, 857 pages: Noceda and San Lucar, Manila, 1754,
+fol.; 1832.</p>
+
+<p><span class="bold">Chinese.</span>&mdash;Native Dictionaries are very numerous. Many are
+very copious and voluminous, and have passed through many
+editions. <i>Shwo wan</i>, by Hü Shin, is a collection of the ancient characters,
+about 10,000 in number, arranged under 540 radicals, published
+150 <span class="sc">b.c.</span>, usually in 12 vols.: <i>Yu pien</i>, by Ku Ye Wang, published
+<span class="sc">a.d.</span> 530, arranged under 542 radicals, is the basis of the Chinese
+Japanese Dictionaries used in Japan: <i>Ping tseu loui pien</i>, Peking,
+1726, 8vo, 130 vols.: <i>Pei wan yün fu</i> (Thesaurus of Literary Phrases),
+1711, 131 vols. 8vo, prepared by 66 doctors of the Han lin Academy
+in seven years. It contains 10,362 characters, and countless combinations
+of two, three or four characters, forming compound words
+and idioms, with numerous and copious quotations. According to
+Williams (<i>On the word Shin</i>, p. 79), an English translation would fill
+140 volumes octavo of 1000 pages each. <i>Kanghi tsze tien</i> (Kanghi&rsquo;s
+Standard or Canon of the Character), the dictionary of Kanghi, the
+first emperor of the present dynasty, was composed by 30 members
+of the Han lin, and published in 1716, 40 vols. 4to, with a preface by
+the emperor. It contains 49,030 characters, arranged under the 214
+radicals. It is generally in 12 vols., and is universally used in China,
+being the standard authority among native scholars for the readings
+as well as the meanings of characters. <span class="sc">Latin</span>.&mdash;De Guignes (French,
+Lat.), Paris, 1813, fol.; Klaproth, Supplément, 1819; ed. Bazil
+(Latin), Hong-Kong, 1853, 4to: Gonçalves (Lat.-Chin.), Macao,
+1841, fol.: Callery, <i>Systema phoneticum</i>, Macao, 1841, 8vo: Schott,
+<i>Vocabularium</i>, Berlin, 1844, 4to. <span class="sc">English</span>.&mdash;Raper, London, 1807,
+fol. 4 vols.: Morrison, Macao, 1815-1823, 4to, 3 parts in 6 vols.:
+Medhurst, Batavia, 1842-1843, 8vo, 2 vols.: Thom, Canton, 1843,
+8vo: Lobscheid, Hong-Kong, 1871, 4to: Williams, Shanghai, 1874,
+4to. <span class="sc">Eng. Chinese</span>.&mdash;Morrison, part iii.: Williams, Macao, 1844,
+8vo: Medhurst, Shanghai, 1847-1848, 8vo, 2 vols.: Hung Maou,
+<i>Tung yung fan hwa</i> (Common words of the Red-haired Foreigners),
+1850, 8vo. Doolittle, Foochow, 1872, 4to, vol. i. 550 pages. <span class="sc">French</span>,&mdash;Callery,
+<i>Dict. encyclopédique</i>, Macao and Paris, 1845 (radicals 1-20
+only): M. A. H., 1876, 8vo, autographié, 1730 pages. <span class="sc">French-Chin</span>.&mdash;Perny
+(Fr.-Latin, Spoken Mandarin), Paris, 1869, 4to;
+Appendice, 1770; Lemaire and Giguel, Shanghai, 1874, 16mo.
+<span class="sc">Portuguese</span>.&mdash;Gonçalves (Port.-Chin.), Macao, 1830, 8vo, 2 vols.:
+Id. (Chin.-Port.), ib. 1833, 8vo. <span class="sc">Idioms</span>.&mdash;Giles, Shanghai, 1873,
+4to. <span class="sc">Phrases</span>.&mdash;Yaou Pei-keen, <i>Luy yih</i>, 1742-1765, 8vo, 55 vols.:
+Tseen Ta-hin, <i>Shing luy</i>, 1853, 8vo, 4 vols. <span class="sc">Classical Expressions</span>.&mdash;Keang
+Yang and 30 others, <i>Sze Shoo teen Lin</i>, 1795, 8vo, 30 vols.
+<span class="sc">Elegant Expressions</span>.&mdash;Chang ting yuh, <i>Fun luy tsze kin</i>, 1722,
+<span class="pagenum"><a name="page199" id="page199"></a>199</span>
+8vo, 64 vols. <span class="sc">Phrases of Three Words.</span>&mdash;Julien (Latin), Paris,
+1864, 8vo. <span class="sc">Poetical.</span>&mdash;<i>Pei wan she yun</i>, 1800, 8vo, 5 vols. <span class="sc">Proper
+Names.</span>&mdash;F. Porter Smith (China, Japan, Corea, Annam, &amp;c.,
+Chinese-Eng.), Shanghai, 1870, 8vo. <span class="sc">Topography.</span>&mdash;Williams,
+Canton, 1841, 8vo. <span class="sc">Names of Towns.</span>&mdash;Biot, Paris, 1842, 8vo.
+<span class="sc">Ancient Characters.</span>&mdash;Foo Lwantseang, <i>Luh shoo fun luy</i>, 1800,
+8vo, 12 vols. <span class="sc">Seal Character.</span>&mdash;Heu Shin, <i>Shwo wan</i>, ed. Seu
+Heuen, 1527, 8vo, 12 vols. <span class="sc">Running Hand.</span>&mdash;St Aulaire and
+Groeneveld (Square Characters, Running Hand; Running, Square),
+Amst. 1861, 4to, 117 pages. <span class="sc">Technical Terms</span> (in Buddhist translations
+from Sanskrit)&mdash;Yuen Ying, <i>Yih &rsquo;see king pin e</i>, 1848, 8vo.
+<span class="sc">Dialects.</span>&mdash;<i>Amoy</i>: Douglas, London, 1873, 4to, 632 pages:
+Macgowan, Hong-Kong, 1869, 8vo. <i>Canton</i>: Yu Heo-poo and Wan
+ke-shih, <i>Keang hoo chih tuh fun yun tso yaou ho tseih</i>, Canton, 1772,
+8vo, 4 vols.; 1803, 8vo, 4 vols.; Fuh-shan, 1833, 8vo, 4 vols.:
+Morrison, Macao, 1828, 8vo: Wan ke shih, Canton, 1856, 8vo:
+Williams (tonic, Eng.-Chinese), Canton, 1856, 8vo: Chalmers, Hong-Kong,
+1859, 12mo; 3rd ed. 1873, 8vo. <i>Changchow in Fuhkeen</i>: Seay
+Sew-lin, <i>Ya suh tung shih woo yin</i>, 1818, 8vo, 8 vols.; 1820. <i>Foo-chow</i>:
+Tseih (a Japanese general) and Lin Peih shan, <i>Pa yin ho ting</i>,
+ed. Tsin Gan, 1841, 8vo: Maclay and Baldwin, Foochow, 1870, 8vo,
+1123 pages. <i>Hok-keen</i>: Medhurst, Macao, 1832, 4to: <i>Peking</i>,
+Stent, Shanghai, 1871, 8vo.</p>
+
+<p><span class="bold">Corean.</span>&mdash;<span class="sc">Chinese, Corean and Japanese.</span>&mdash;<i>Cham Seen Wo
+Kwo tsze mei</i>, translated by Medhurst, Batavia, 1835, 8vo. <span class="sc">Russian.</span>&mdash;Putzillo,
+St Petersburg, 1874, 12mo, 746 pages.</p>
+
+<p><span class="bold">Japanese.</span>&mdash;<i>Sio Ken Zi Ko</i> (Examination of Words and Characters),
+1608, 8vo, 10 vols.: <i>Wa Kan Won Se Ki Sio Gen Zi Ko</i>, lithographed
+by Siebold, Lugd. Bat., 1835, fol. <span class="sc">Jap.-Chinese.</span>&mdash;<i>Faga biki set yo
+siu</i>. <span class="sc">Chinese-Jap.</span>&mdash;<i>Kanghi Tse Tein</i>, 30 vols. 12mo: <i>Zi rin gioku
+ben</i>. <span class="sc">Dutch Dictionaries printed by Japanese.</span>&mdash;<i>Nieeu verzameld
+Japansch en Hollandsch Woordenbock</i>, by the interpreter, B. Sadayok,
+1810: Minamoto Masataka, Prince of Nakats (Jap. Chinese-Dutch),
+5 vols. 4to, printed at Kakats by his servants: <i>Jedo-Halma</i> (Dutch-Jap.),
+Jedo, 4to, 20 vols.: <i>Nederduitsche taal</i>, Dutch Chinese, for
+the use of interpreters. <span class="sc">Latin and Portuguese.</span>&mdash;Calepinus, <i>Dictionarium</i>,
+Amacusa, 1595, 4to. <span class="sc">Latin.</span>&mdash;Collado, <i>Compendium</i>,
+Romae, 1632, 4to: <i>Lexicon</i>, Romae, 1870, 4to, from Calepinus.
+<span class="sc">English.</span>&mdash;Medhurst, Batavia, 1830, 8vo: Hepburn, Shanghai,
+1867, 8vo; 1872. <span class="sc">Eng.-Jap.</span>&mdash;Hori Tatnoskoy, Yedo, 1862, 8vo;
+2nd ed. Yeddo, 1866, 8vo: Satow and Ishibashi Masakata (spoken
+language), London, 1876, 8vo. <span class="sc">French.</span>&mdash;Rosny (Jap. Fr. Eng.),
+Paris, 1857, 4to, vol. i.: Pagés, Paris, 1869, 4to, translated from
+Calepinus. <span class="sc">Fr.-Jap.</span>&mdash;Soutcovey, Paris, 1864, 8vo. <span class="sc">Fr. Eng. Jap.</span>&mdash;Mermet
+de Cachon, Paris, 1866, 8vo, unfinished. <span class="sc">German.</span>&mdash;Pfizmaier
+(Jap.-Ger., Eng.), Wien, 1851, 4to, unfinished. <span class="sc">Spanish.</span>&mdash;<i>Vocabulario
+del Japon</i>, Manila, 1630, 4to, translated from the next.
+<span class="sc">Portuguese.</span>&mdash;<i>Vocabulario da Lingua de Japam</i>, Nagasaki, 1603,
+4to. <span class="sc">Russian.</span>&mdash;Goshkevich, St Petersburg, 1857, 8vo, 487 pages.
+<span class="sc">Chinese Characters with Japanese Pronunciation.</span>&mdash;Rosny,
+Paris, 1867, 8vo. <span class="sc">Chinese and Japanese Names of Plants.</span>&mdash;Hoffmann,
+Leyde, 1864, 8vo.</p>
+
+<p><span class="bold">Aino.</span>&mdash;Pfizmaier, Wien, 1854, 4to.</p>
+
+<p><span class="bold">Northern and Central Asia.</span>&mdash;<i>Buriat</i>: Castrén, St Petersburg, 1857,
+8vo. <i>Calmuck</i>: Zwick, Villingen, 1853, 4to: Smirnov, Kazan,
+1857, 12mo: Jügl, <i>Siddhi Kur</i>, Leipzig, 1866, 8vo. <i>Chuvash</i>:
+Clergy of the school of the Kazan Eparchia, Kazan, 1836, 8vo, 2481
+words: Lyulé (Russ.-Chuv. French), Odessa, 1846, 8vo, 244 pages:
+Zolotnitski, Kazan, 1875, 8vo, 287 pages. <i>Jagatai</i>: Mir Ali Shir,
+<i>Abuska</i>, ed. Vámbéry, with Hungarian translation, Pesth, 1862, 8vo:
+Vámbéry, Leipzig, 1867, 8vo: Pavet de Courteille, Paris, 1870, 8vo.
+<i>Koibal and Karagas</i>: Castrén, St Petersburg, 1857, 8vo. <i>Manchu</i>:
+<i>Yutchi tseng ting tsing wen kian</i> (Manchu Chinese), 1771, 4to, 6 vols.:
+<i>Sze li hoh pik wen kian</i> (Manchu-Mongol, Tibetan, Chinese) 10 vols.
+4to, the Chinese pronunciation represented in Manchu: <i>San hoh
+pien lan</i> (Manchu-Chinese, Mongol), 1792, 8vo, 12 vols.;&mdash;all three
+classed vocabularies: Langlès (French), Paris, 1789-1790, 4to, 3 vols.:
+Gabelentz (German), Leipzig, 1864, 8vo: Zakharov (Russian), St
+Petersburg, 1875, 8vo, 1235 pages: <i>Mongol</i>: I. J. Schmidt (German,
+Russian), St Petersburg, 1835, 4to: Schergin, Kazan, 1841, 8vo:
+Kovalevski, Kasan, 1844-1849, 4to, 3 vols. 2703 pages. <i>Ostiak</i>:
+Castrén, St Petersb. 1858, 8vo. <i>Samoyed</i>: Castrén, St Petersb. 1855,
+8vo, 308 pages. Tartar: Giganov (Tobolsk), St Petersburg, 1804,
+4to; (Russ.-Tartar), ib. 1840, 4to: Troyanski (Karan), Kasan,
+1835-1855, 4to. <i>Tibetan</i>: <i>Minggi djamtoo</i> (Tibet-Mongol): <i>Bodschi
+dajig togpar lama</i>: <i>Kad shi schand scharwi melonggi jige</i>
+(Manchu-Mongol-Tibetan-Chinese), Kanghi&rsquo;s Dictionary with the Tibetan
+added in the reign of Khian lung (1736-1795); Csoma de Körös (Eng.),
+Calcutta, 1834, 4to: I. J. Schmidt (German), St Petersburg, 1841,
+4to: Id. (Russian), ib. 1843, 4to: Jaeschke (Eng.), London, 1870,
+8vo, 160 pages: Id. (Germ.), Gnadau, 1871, 658 pages: (Bhotanta),
+Schroeter, Serampore, 1826, 4to. <i>Tungusian</i>: Castrén, St Petersburg,
+1856, 8vo, 632 pages. <i>Uigur</i>: Vámbéry, Innspruck, 1870, 4to.
+<i>Yakut</i>: Böhtlingk, ib. 1854, 4to, 2 vols. <i>Yenissei Ostiak</i>: Castrén,
+ib. 1849, 8vo.</p>
+
+<p class="center pt2">AFRICA</p>
+
+<p><span class="bold">Egyptian.</span>&mdash;Young (enchorial), London, 1830-1831, 8vo: Sharpe,
+London, 1837, 4to: Birch, London. 1838, 4to: Champollion (died
+March 4, 1832), <i>Dictionnaire égyptien</i>, Paris, 1841, 4to: Brugsch,
+<i>Hieroglyphisch-Demotisches Wörterbuch</i>, Leipzig, 1867-1868, 4to,
+4 vols. 1775 pages, nearly 4700 words, arranged according to the
+hieroglyphic alphabet of 28 letters: Pierret, <i>Vocabulaire hiérog.</i>, Paris,
+1875, 8vo, containing also names of persons and places: Birch, in
+vol. v. pp. 337-580 of Bunsen&rsquo;s <i>Egypt&rsquo;s Place</i>, 2nd ed. London, 1867,
+&amp;c. 8vo, 5010 words. <span class="sc">Proper Names.</span>&mdash;Brugsch, Berlin, 1851, 8vo,
+726 names: Parthey, ib. 1864, 8vo, about 1500 names: Lieblein,
+Christiania, 1871, 8vo, about 3200 from hieroglyphic texts. <span class="sc">Book
+of the Dead.</span>&mdash;Id., Paris, 1875, 12mo.</p>
+
+<p><span class="bold">Coptic.</span>&mdash;Veyssière de la Croze, Oxon. 1775, 8vo: Rossi, Romae,
+1807, 4to: Tattam, Oxon. 1855, 8vo: Peyron, 1835, 4to (the
+standard): Parthey, Berolini, 1844, 8vo.</p>
+
+<p><span class="bold">Ethiopic.</span>&mdash;Wemmer, Romae, 1638, 4to: Ludolf, London, 1661,
+4to: Francof. ad M., 1699, fol.: Dillmann (Tigré appendix),
+Leipzig, 1863-1865, 4to, 828 pages.</p>
+
+<p><span class="bold">Amharic.</span>&mdash;Ludolphus, Franc. ad Maenum, 1698, fol.: Isenberg,
+London, 1841, 4to, 442 pages. <i>Tigré</i>: Munzinger, Leipzig, 1865,
+8vo: Beurmann, ib. 1868, 8vo.</p>
+
+<p><span class="bold">East Coast.</span>&mdash;<i>Dankali</i>: Isenberg, London, 1840, 12mo. <i>Galla</i>:
+Krapf, London, 1842, 8vo: Tutschek, München, 1844, 8vo. <i>Engutuk
+Iloigob</i>: Erhardt, Ludwigsberg, 1857, 8vo. <i>Kisuaheli</i>: <i>Vocabulary
+of the Soahili</i>, Cambridge, U.S. 1845, 8vo: Steere, London,
+1870, 8vo, about 5800 words. <i>Kisuaheli, Kinika, Kikamba, Kipokono,
+Kikian, Kigalla</i>: Krapf, Tübingen, 1850, 8vo.</p>
+
+<p><span class="bold">Malagasy.</span>&mdash;Houtmann (Malaysche en Madagask Talen), Amst.
+1603, 2nd ed. Matthysz, ib. 1680, 8vo: Huet de Froberville, Isle de
+France, fol. 2 vols.: Flacourt, Paris, 1658, 8vo: Challand (Southern),
+Isle de France, 1773, 4to: Freeman and Johns, London, 1835, 8vo,
+2 vols.: Dalmont (Malgache, Salalave, et Betsimara), 1842, 8vo:
+Kessler, London, 1870, 8vo.</p>
+
+<p><span class="bold">Southern Africa.</span>&mdash;Bleek, <i>The Languages of Mozambique</i>, London,
+1856, 8vo. <i>Kaffre</i>: Bennie, Lovedale, 1826, 16mo: Ayliffe,
+Graham&rsquo;s Town, 1846, 12mo: Appleyard, 1850, 8vo: Bleek, Bonn,
+1853, 4to, 646 pages. <i>Zulu-Kaffre</i>: Perrin (Kaffre-Eng.), London,
+1855, 24mo, 172 pages: Id. (Eng.-Kaffre), Pietermaritzburg, 1855,
+24mo, 227 pages: Id. (Eng.-Zulu), ib. 1865, 12mo, 226 pages:
+Dohne, Cape Town, 1857, 8vo, 428 pages: Colenso, Pietermaritzburg,
+1861, 8vo, 560 pages, about 8000 words. <i>Hottentot</i>: Bleek,
+Cape Town, 1857, 4to, 261 pages. <i>Namaqua</i>: Tindall, ib. 1852, 8vo:
+<i>Vocabulary</i>, Barmen, 1854, 8vo: Hahn, Leipzig, 1870, 12mo.
+Sechuana: Casalis, Paris, 1841, 8vo. <i>Herero</i>: Hahn, Berlin, 1857,
+8vo, 207 pages, 4300 words.</p>
+
+<p><span class="bold">Western Africa.</span>&mdash;<i>Akra</i> or <i>Ga</i>: Zimmermann, Stuttgart, 1858,
+8vo, 690 pages. <i>Ashantee</i>: Christaller (also Akra), Basel, 1874,
+8vo, 299 pages. <i>Bullom</i>: Nylander, London, 1814, 12mo. <i>Bunda
+or Angola</i>: Cannecatim, Lisboa, 1804, 4to, 722 pages. <i>Dualla
+Grammatical Elements</i>, &amp;c., Cameroons, 1855, 8vo. <i>Efik</i> or <i>Old
+Calabar</i>: Waddell, Old Calabar, 1846, 16mo, 126 pages; Edinb,
+1849, 8vo, 95 pages. <i>Eyo</i>: Raban, London, 1830-1831, 12mo, 2 parts.
+<i>Grebo</i>: <i>Vocabulary</i>, Cape Palmas, 1837, 8vo; <i>Dictionary</i>, ib. 1839,
+8vo, 119 pages. <i>Ifa</i>: Schlegel, Stuttgart, 1857, 8vo. <i>Mpongwe</i>:
+De Lorme (Franç.-Pongoué), Paris, 1876, 12mo, 354 pages. <i>Oji</i>:
+Riis, Basel, 1854, 8vo, 284 pages. <i>Sherbro&rsquo;</i>: Schön, <i>s. a. et l.</i>
+8vo, written in 1839, 42 pages. <i>Susu</i>: Brunton, Edinburgh, 1802,
+8vo, 145 pages. <i>Vei</i>: Koelle, London, 1854, 8vo, 266 pages.
+<i>Wolof and Bambarra</i>: Dard, Paris, 1825, 8vo. <i>Wolof</i>: Roger, ib.
+1829, 8vo: Missionnaires de S. Esprit, Dakar, 1855, &amp;c. 16mo.
+Faidherbe (French-Wolof, Poula and Soninke), St Louis, Senegambia,
+1860, 12mo. <i>Yoruba</i>: Crowther, London, 1843, 8vo;
+1852, 298 pages: Vidal, ib. 1852, 8vo: Bowen, Washington, 1858,
+4to.</p>
+
+<p><span class="bold">Central Africa.</span>&mdash;Barth, <i>Vocabularies</i>. Gotha, 1862-1866, 4to. <i>Bari</i>:
+Mitterreutzner, Brixen, 1867, 8vo: Reinisch, Vienna, 1874, 8vo.
+<i>Dinka</i>: Mitterreutzner, Brixen, 1866, 8vo. <i>Haussa</i>: Schön (Eng.),
+London, 1843, 8vo.</p>
+
+<p><span class="bold">Berber.</span>&mdash;Venture de Paradis, Paris, 1844, 8vo: Brosselard, ib.
+1844, 8vo: Delaporte, ib. 1844, 4to, by order of the Minister of
+War: Creusat, Franç.-Kabyle (Zouaoua), Alger, 1873, 8vo. <i>Siwah</i>:
+Minutoli, Berlin, 1827, 4to.</p>
+
+<p class="center pt2">AUSTRALIA AND POLYNESIA</p>
+
+<p><span class="bold">Australia.</span>&mdash;<i>New South Wales</i>: Threlkeld (Lake Macquarie
+Language), Sydney, 1834, 8vo. <i>Victoria</i>: Bunce, Melbourne, 1856,
+12mo, about 2200 words. <i>South Australia</i>: Williams, South
+Australia, 1839, 8vo: Teichelmann and Schürmann, Adelaide,
+1840, 8vo: Meyer, ib. 1843, 8vo. <i>Murray River</i>: Moorhouse, ib.
+1846, 8vo. <i>Parnkalla</i>: Schürmann, Adelaide, 1844, 8vo. <i>Woolner
+District</i>: <i>Vocabulary</i>, ib. 1869, 12mo. <i>Western Australia</i>: Sir
+George Grey, Perth, 1839, 4to; London, 1840, 8vo: Moore, ib. 1843:
+Brady, Roma, 1845, 24mo, 8vo, 187 pages. <i>Tasmania</i>: Millegan,
+Tasmania, 1857.</p>
+
+<p><span class="bold">Polynesia.</span>&mdash;Hale, <i>Grammars and Vocabularies of all the Polynesian
+Languages</i>, Philadelphia, 1846, 4to. <i>Marquesas, Sandwich
+Gambier</i>: Mosblech, Paris, 1843, 8vo. <i>Hawaiian</i>: Andrews,
+<i>Vocabulary</i>, Lahainaluna, 1636, 8vo: Id., <i>Dictionary</i>, Honolulu,
+1865, 8vo, 575 pages, about 15,500 words. <i>Marquesas</i>: Pierquin,
+de Gembloux, Bourges, 1843, 8vo: Buschmann, Berlin, 1843, 8vo.
+<i>Samoan</i>: <i>Dictionary</i>, Samoa, 1862, 8vo. <i>Tahitian</i>: <i>A Tahitian and
+English Dictionary</i>, Tahiti, 1851, 8vo, 314 pages. <i>Tonga</i>: Rabone,
+Vavau, 1845, 8vo. <i>Fijian</i>: Hazlewood (Fiji-Eng.), Vewa. 1850,
+<span class="pagenum"><a name="page200" id="page200"></a>200</span>
+12mo: Id. (Eng.-Fiji), ib. 1852, 12mo: Id., London, 1872, 8vo.
+<i>Maori</i>: Kendall, 1820, 12mo: Williams, Paihia, 1844, 8vo; 3rd ed.
+London, 1871, 8vo: Taylor, Auckland, 1870, 12mo.</p>
+
+<p class="center pt2">AMERICA</p>
+
+<p><span class="bold">North America.</span>&mdash;<i>Eskimo</i>: Washington, London, 1850, 8vo:
+Petitot (Mackenzie and Anderson Rivers), Paris, 1876, 4to.
+<i>Kinai</i>: Radloff, St Petersburg, 1874, 4to. <i>Greenland</i>: Egede (Gr.
+Dan. Lat., 3 parts), Hafn, 1750, 8vo; 1760, Fabricius, Kjöbenhavn,
+1804, 4to. <i>Hudson&rsquo;s Bay Indians</i>: Bowrey, London, 1701, fol.
+<i>Abnaki</i>: Rasles, Cambridge, U.S., 1833, 4to. <i>Chippewa</i>: Baraga,
+Cincinnati, 1853, 12mo, 622 pages: Petitot, Paris, 1876, 4to, 455
+pages. <i>Massachusetts</i> or <i>Natick</i>: Cotton, Cambridge, U.S. 1829, 8vo.
+<i>Onondaga</i>: Shea (French-Onon.), from a MS. (of 17th century),
+London, 1860, 4to, 109 pages. <i>Dacota</i>: Riggs, New York, 1851, 4to,
+424 pages: Williamson (Eng. Dac.), Santos Agency, Nebraska,
+12mo, 139 pages. <i>Mohawk</i>: Bruyas, New York, 1863, 8vo.
+<i>Hidatsa (Minnetarees, Gros Ventres of the Missouri)</i>: Matthews,
+ib. 1874, 8vo. <i>Choctaw</i>: Byington, ib. 1852, 16mo. <i>Clallam and
+Lummi</i>: Gibbs, ib. 1863, 8vo. <i>Yakama</i>: Pandosy, translated by
+Gibbs and Shea, ib. 1862, 8vo. <i>Chinook</i>: Gibbs, New York, 1863,
+4to. <i>Chinook Jargon, the trade language of Oregon</i>: Id., ib. 1863,
+8vo. <i>Tatche</i> or <i>Telamé</i>: Sitjar, ib. 1841, 8vo.</p>
+
+<p><span class="bold">Mexico and Central America.</span>&mdash;<i>Tepehuan</i>: Rinaldini, Mexico,
+1743, 4to. <i>Cora</i>: Ortega, Mexico, 1732, 4to. <i>Tarahumara</i>: Steffel,
+Brünn, 1791, 8vo. <i>Otomi</i>: Carochi, Mexico, 1645, 4to: Neve y
+Molina, ib. 1767, 8vo: Yepes, ib. 1826, 4to: Piccolomini, Roma,
+1841, 8vo. <i>Mexican</i> or <i>Aztec</i>: Molina, Mexico, 1555, 4to; 1571,
+fol. 2 vols.: Arenas, ib. 1583; 1611, 8vo; 1683; 1725; 1793,
+12mo: Biondelli, Milan, 1869, fol. <i>Mexican, Tontonacan, and
+Huastecan</i>: Olmos, Mexico, 1555-1560, 4to, 2 vols. <i>Huastecan</i>:
+Tapia Zenteno, ib. 1767, 4to, 128 pages. <i>Opata</i> or <i>Tequima</i>:
+Lombardo, ib. 1702, 4to. <i>Tarasca</i>: Gilberti, ib. 1559, 4to: Lagunas,
+ib. 1574, 8vo. <i>Mixtecan</i>: Alvarado, Mexico, 1593, 4to. <i>Zapoteca</i>:
+Cordova, ib. 1578, 4to. <i>Maya</i>: Beltran de Santa Rosa Maria, ib.
+1746, 4to; Merida de Yucatan, 1859, 4to, 250 pages: Brasseur de
+Bourbourg, Paris, 1874, 8vo, 745 pages. <i>Quiché</i>: Id. (also Cakchiquel
+and Trutuhil dialects), ib. 1862, 8vo.</p>
+
+<p><span class="bold">South America.</span>&mdash;<i>Chibcha</i>: Uricoechea, Paris, 1871, 8vo.
+<i>Chayma</i>: Tauste, Madrid, 1680, 4to: Yanguas, Burgos, 1683, 4to.
+<i>Carib</i>: Raymond, Auxerre, 1665-1666, 8vo. <i>Galibi</i>: D.[e]. L.[a]
+S.[auvage], Paris, 1763, 8vo. <i>Tupi</i>: Costa Rubim, Rio de Janeiro,
+1853, 8vo: Silva Guimaräes, Bahia, 1854, 8vo: Diaz, Lipsia, 1858,
+16mo. <i>Guarani</i>: Ruiz de Montoyo, Madrid, 1639, 4to; 1640;
+1722, 4to; ed. Platzmann, Leipzig, 1876, &amp;c., 8vo, to be in 4 vols.
+1850 pages. <i>Moxa</i>: Marban, Lima, 1701, 8vo. <i>Lule</i>: Machoni
+de Corderia, Madrid, 1732, 12mo. <i>Quichua</i>: Santo Thomas, Ciudad
+de los Reyes, 1586, 8vo: Torres Rubio, Sevilla, 1603, 8vo; Lima,
+1609, 8vo; ed. Figueredo, Lima, 1754, 8vo; Holguin, Ciudad de
+los Reyes, 1608, 8vo: Tschudi, Wien, 1853, 8vo, 2 vols.: Markham,
+London, 1864, 8vo: Lopez, <i>Les Races Aryennes de Perou</i>, Paris, 1871,
+8vo, comparative vocabulary, pp. 345-421. <i>Aymara</i>: Bertonio,
+Chicuyto, 1612, 4to, 2 vols. <i>Chileno</i>: Valdivia (also Allentiac
+and Milcocayac), Lima, 1607, 8vo: Febres, ib. 1765, 12mo; ed.
+Hernandez y Caluza, Santiago, 1846, 8vo, 2 vols. <i>Tsonecan</i>
+(Patagonian): Schmid, Bristol, 1860, 12mo.</p>
+
+<p>The above article incorporates the salient features of the 9th-edition
+article by the Rev. Ponsonby A. Lyons, and the 10th-edition
+article by Benjamin E. Smith.</p>
+</div>
+
+<hr class="foot" /> <div class="note">
+
+<p><a name="ft1d" id="ft1d" href="#fa1d"><span class="fn">1</span></a> Joannes de Garlandia (John Garland; fl. 1202-1252) gives
+the following explanation in his <i>Dictionarius</i>, which is a classed
+vocabulary:&mdash;&ldquo;Dictionarius dicitur libellus iste a dictionibus magis
+necessariis, quas tenetur quilibet scolaris, non tantum in scrinio de
+lignis facto, sed in cordis armariolo firmiter retinere.&rdquo; This has been
+supposed to be the first use of the word.</p>
+
+<p><a name="ft2d" id="ft2d" href="#fa2d"><span class="fn">2</span></a> An excellent dictionary of quotations, perhaps the first of the
+kind; a large folio volume printed in Strassburg about 1475 is
+entitled &ldquo;Pharetra auctoritates et dicta doctorum, philosophorum,
+et poetarum continens.&rdquo;</p>
+
+<p><a name="ft3d" id="ft3d" href="#fa3d"><span class="fn">3</span></a> This volume was issued with a new title-page as <i>Glossaire du
+moyen âge</i>, Paris, 1872.</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DICTYOGENS<a name="ar45" id="ar45"></a></span> (Gr. <span class="grk" title="diktyon">&#948;&#943;&#954;&#964;&#965;&#959;&#957;</span>, a net, and the termination <span class="grk" title="-genês">-&#947;&#949;&#957;&#951;&#962;</span>,
+produced), a botanical name proposed by John Lindley for a
+class including certain families of Monocotyledons which have
+net-veined leaves. The class was not generally recognized.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DICTYS CRETENSIS,<a name="ar46" id="ar46"></a></span> of Cnossus in Crete, the supposed companion
+of Idomeneus during the Trojan War, and author of a
+diary of its events. The MS. of this work, written in Phoenician
+characters, was said to have been found in his tomb (enclosed in a
+leaden box) at the time of an earthquake during the reign of Nero,
+by whose order it was translated into Greek. In the 4th century
+<span class="sc">a.d.</span> a certain Lucius Septimius brought out <i>Dictys Cretensis
+Ephemeris belli Trojani</i>, which professed to be a Latin translation
+of the Greek version. Scholars were not agreed whether any
+Greek original really existed; but all doubt on the point was
+removed by the discovery of a fragment in Greek amongst the
+papyri found by B. P. Grenfell and A. S. Hunt in 1905-1906.
+Possibly the Latin Ephemeris was the work of Septimius himself.
+Its chief interest lies in the fact that (together with Dares
+Phrygius&rsquo;s <i>De excidio Trojae</i>) it was the source from which the
+Homeric legends were introduced into the romantic literature
+of the middle ages.</p>
+
+<div class="condensed">
+<p>Best edition by F. Meister (1873), with short but useful introduction
+and index of Latinity; see also G. Körting, <i>Diktys und Dares</i>
+(1874), with concise bibliography; H. Dunger, <i>Die Sage vom trojanischen
+Kriege in den Bearbeitungen des Mittelalters und ihren
+antiken Quellen</i> (1869, with a literary genealogical table); E. Collilieux,
+<i>Étude sur Dictys de Crète et Darès de Phrygie</i> (1887), with bibliography;
+W. Greif, &ldquo;Die mittelalterlichen Bearbeitungen der Trojanersage,&rdquo;
+in E. M. Stengel&rsquo;s <i>Ausgaben und Abhandlungen aus dem
+Gebiete der romanischen Philologie</i>, No. 61 (1886, esp. sections 82, 83,
+168-172); F. Colagrosso, &ldquo;Ditte Cretese&rdquo; in <i>Atti della r. Accademia
+di Archeologia</i> (Naples, 1897, vol. 18, pt. ii. 2); F. Noack, &ldquo;Der
+griechische Dictys,&rdquo; in <i>Philologus</i>, supp. vi. 403 ff.; N. E. Griffin,
+<i>Dares and Dictys, Introduction to the Study of the Medieval Versions
+of the Story of Troy</i> (1907).</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DICUIL<a name="ar47" id="ar47"></a></span> (fl. 825), Irish monastic scholar, grammarian and
+geographer. He was the author of the <i>De mensura orbis terrae</i>,
+finished in 825, which contains the earliest clear notice of a
+European discovery of and settlement in Iceland and the most
+definite Western reference to the old freshwater canal between
+the Nile and the Red Sea, finally blocked up in 767. In 795
+(February 1-August 1) Irish hermits had visited Iceland; on
+their return they reported the marvel of the perpetual day at
+midsummer in &ldquo;Thule,&rdquo; where there was then &ldquo;no darkness to
+hinder one from doing what one would.&rdquo; These eremites also
+navigated the sea north of Iceland on their first arrival, and
+found it ice-free for one day&rsquo;s sail, after which they came to
+the ice-wall. Relics of this, and perhaps of other Irish religious
+settlements, were found by the permanent Scandinavian colonists
+of Iceland in the 9th century. Of the old Egyptian freshwater
+canal Dicuil learnt from one &ldquo;brother Fidelis,&rdquo; probably another
+Irish monk, who, on his way to Jerusalem, sailed along the
+&ldquo;Nile&rdquo; into the Red Sea&mdash;passing on his way the &ldquo;Barns of
+Joseph&rdquo; or Pyramids of Giza, which are well described. Dicuil&rsquo;s
+knowledge of the islands north and west of Britain is evidently
+intimate; his references to Irish exploration and colonization,
+and to (more recent) Scandinavian devastation of the same, as
+far as the Faeroes, are noteworthy, like his notice of the elephant
+sent by Harun al-Rashid (in 801) to Charles the Great, the most
+curious item in a political and diplomatic intercourse of high
+importance. Dicuil&rsquo;s reading was wide; he quotes from, or
+refers to, thirty Greek and Latin writers, including the classical
+Homer, Hecataeus, Herodotus, Thucydides, Virgil, Pliny and
+King Juba, the sub-classical Solinus, the patristic St Isidore and
+Orosius, and his contemporary the Irish poet Sedulius;&mdash;in
+particular, he professes to utilize the alleged surveys of the
+Roman world executed by order of Julius Caesar, Augustus and
+Theodosius (whether Theodosius the Great or Theodosius II.
+is uncertain). He probably did not know Greek; his references
+to Greek authors do not imply this. Though certainly Irish
+by birth, it has been conjectured (from his references to
+Sedulius and the caliph&rsquo;s elephant) that he was in later life
+in an Irish monastery in the Frankish empire. Letronne inclines
+to identify him with Dicuil or Dichull, abbot of Pahlacht,
+born about 760.</p>
+
+<div class="condensed">
+<p>There are seven chief MSS. of the <i>De mensura</i> (Dicuil&rsquo;s tract
+on grammar is lost); of these the earliest and best are (1) Paris,
+National Library, Lat. 4806; (2) Dresden, Regius D. 182; both
+are of the 10th century. Three editions exist: (1) C. A. Walckenaer&rsquo;s,
+Paris, 1807; (2) A. Letronne&rsquo;s, Paris, 1814, best as to commentary;
+(3) G. Parthey&rsquo;s, Berlin, 1870, best as to text. See also C. R. Beazley,
+<i>Dawn of Modern Geography</i> (London, 1897), i. 317-327, 522-523, 529;
+T. Wright, <i>Biographia Britannica literaria, Anglo-Saxon Period</i>
+(London, 1842), pp. 372-376.</p>
+</div>
+<div class="author">(C. R. B.)</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIDACH&#274;, THE,<a name="ar48" id="ar48"></a></span> or <i>Teaching of the (twelve) Apostles</i>,&mdash;the
+most important of the recent recoveries in the region of early
+Christian literature (see <span class="sc"><a href="#artlinks">Apocryphal Literature</a></span>). It was
+previously known by name from lists of canonical and extra-canonical
+books compiled by Eusebius and other writers. Moreover,
+it had come to be suspected by several scholars that a lost
+book, variously entitled <i>The Two Ways</i> or <i>The Judgment of Peter</i>,
+had been freely used in a number of works, of which mention
+must presently be made. In 1882 a critical reconstruction of
+this book was made by Adam Krawutzcky with marvellous
+accuracy, as was shown when in the very next year the Greek
+bishop and metropolitan, Philotheus Bryennius, published <i>The
+Teaching of the Twelve Apostles</i> from the same manuscript from
+<span class="pagenum"><a name="page201" id="page201"></a>201</span>
+which he had previously published the complete form of the
+Epistle of Clement.<a name="fa1e" id="fa1e" href="#ft1e"><span class="sp">1</span></a></p>
+
+<p><i>The Didach&#275;</i>, as we now have it in the Greek, falls into two
+marked divisions: (a) a book of moral precepts, opening with the
+words, &ldquo;There are two ways&rdquo;; (b) a manual of church ordinances,
+linked on to the foregoing by the words, &ldquo;Having first
+said all these things, baptize, &amp;c.&rdquo; Each of these must be
+considered separately before we approach the question of the
+locality and date of the whole book in its present form.</p>
+
+<p>1. <i>The Two Ways.</i>&mdash;The author of the complete work, as we
+now have it, has modified the original <i>Two Ways</i> by inserting near
+the beginning a considerable section containing, among other
+matter, passages from the Sermon on the Mount, in which the
+language of St Matthew&rsquo;s Gospel is blended with that of St
+Luke&rsquo;s. He has also added at the close a few sentences, beginning,
+&ldquo;If thou canst not bear (the whole yoke of the Lord), bear
+what thou canst&rdquo; (vi. 2); and among minor changes he has
+introduced, in dealing with confession, reference to &ldquo;the church&rdquo;
+(iv. 14). No part of this matter is to be found in the following
+documents, which present us in varying degrees of accuracy with
+<i>The Two Ways</i>: (i.) the Epistle of Barnabas, chaps. xix., xx. (in
+which the order of the book has been much broken up, and a
+good deal has been omitted); (ii.) the <i>Ecclesiastical Canons of the
+Holy Apostles</i>, usually called the <i>Apostolic Church Order</i>, a book
+which presents a parallel to the <i>Teaching</i>, in so far as it consists
+first of a form of <i>The Two Ways</i>, and secondly of a number of
+church ordinances (here, however, as in the Syriac <i>Didascalia</i>,
+which gives about the same amount of <i>The Two Ways</i>, various
+sections are ascribed to individual apostles, <i>e.g.</i> &ldquo;John said,
+There are two ways,&rdquo; &amp;c.); (iii.) a discourse of the Egyptian
+monk Schnudi (d. 451), preserved in Arabic (see Iselin, <i>Texte
+u. Unters.</i>, 1895); (iv.) a Latin version, of which a fragment
+was published by O. von Gebhardt in 1884, and the whole by
+J. Schlecht in 1900. When by the aid of this evidence <i>The Two
+Ways</i> is restored to us free of glosses, it has the appearance of
+being a Jewish manual which has been carried over into the
+use of the Christian church. This is of course only a probable
+inference; there is no prototype extant in Jewish literature, and,
+comparing the moral (non-doctrinal) instruction for Christian
+catechumens in Hermas, <i>Shepherd</i> (<i>Mand.</i> i.-ix.), no real need to
+assume one. There was a danger of admitting Gentile converts
+to the church on too easy moral terms; hence the need of such
+insistence on the ideal as in The Two Ways and the <i>Mandates</i>.
+The recent recovery of the Latin version is of singular interest,
+as showing that, even without the distinctively Christian
+additions and interpolations which our full form of the <i>Teaching</i>
+presents, it was circulating under the title <i>Doctrina apostolorum</i>.<a name="fa2e" id="fa2e" href="#ft2e"><span class="sp">2</span></a></p>
+
+<p>2. The second part of our <i>Teaching</i> might be called a church
+directory. It consists of precepts relating to church life, which
+are couched in the second person plural; whereas <i>The Two Ways</i>
+uses throughout the second person singular. It appears to be
+a composite work. First (vii. 1-xi. 2) is a short sacramental
+manual intended for the use of local elders or presbyters, though
+such are not named, for they were not yet a distinctive order or
+clergy. This section was probably added to <i>The Two Ways</i> before
+the addition of the remainder. It orders baptism in the threefold
+name, making a distinction as to waters which has Jewish
+parallels, and permitting a threefold pouring on the head, if
+sufficient water for immersion cannot be had. It prescribes a
+fast before baptism for the baptizer as well as the candidate.
+Fasts are to be kept on Wednesday and Friday, not Monday and
+Thursday, which are the fast days of &ldquo;the hypocrites,&rdquo; <i>i.e.</i> by
+a perversion of the Lord&rsquo;s words, the Jews. &ldquo;Neither pray ye as
+the hypocrites; but as the Lord commanded in His Gospel.&rdquo;
+Then follows the Lord&rsquo;s Prayer, almost exactly as in St Matthew,
+with a brief doxology&mdash;&ldquo;for Thine is the power and the glory
+forever.&rdquo; This is to be said three times a day. Next come three
+eucharistic prayers, the language of which is clearly marked off
+from that of the rest of the book, and shows parallels with the
+diction of St John&rsquo;s Gospel. They are probably founded on
+Jewish thanksgivings, and it is of interest to note that a portion
+of them is prescribed as a grace before meat in (pseudo-)
+Athanasius&rsquo; <i>De virginitate</i>. A trace of them is found in one of the
+liturgical prayers of Serapion, bishop of Thmui, in Egypt, but
+they have left little mark on the liturgies of the church. As in
+Ignatius and other early writers, the eucharist, a real meal (x. 1)
+of a family character, is regarded as producing immortality
+(cf. &ldquo;spiritual food and drink and eternal life&rdquo;). None are to
+partake of it save those who have been &ldquo;baptized in the name
+of the Lord&rdquo; (an expression which is of interest in a document
+which prescribes the threefold formula). The prophets are not
+to be confined to these forms, but may &ldquo;give thanks as much as
+they will.&rdquo; This appears to show that a prophet, if present,
+would naturally preside over the eucharist. The next section
+(xi. 3-xiii.) deals with the ministry of spiritual gifts as exercised
+by apostles, prophets and teachers. An apostle is to be &ldquo;received
+as the Lord&rdquo;; but he must follow the Gospel precepts,
+stay but one or two days, and take no money, but only bread
+enough for a day&rsquo;s journey. Here we have that wider use of the
+term &ldquo;apostle&rdquo; to which Lightfoot had already drawn attention.
+A prophet, on the contrary, may settle if he chooses, and in that
+case he is to receive tithes and first-fruits; &ldquo;for they are your
+high priests.&rdquo; If he be once approved as a true prophet, his
+words and acts are not to be criticized; for this is the sin that
+shall not be forgiven. Next comes a section (xiv., xv.) reflecting
+a somewhat later development concerning fixed services and
+ministry; the desire for a stated service, and the need of regular
+provision for it, is leading to a new order of things. The
+eucharist is to be celebrated every Lord&rsquo;s Day, and preceded by
+confession of sins, &ldquo;that your sacrifice may be pure ... for this
+is that sacrifice which was spoken of by the Lord, In every place
+and time to offer unto Me a pure sacrifice. Appoint therefore
+unto yourselves bishops and deacons, worthy of the Lord, men
+meek and uncovetous, and true and approved; for they also
+minister unto you the ministration of the prophets and teachers.
+Therefore despise them not; for they are your honoured ones,
+together with the prophets and teachers.&rdquo; This is an arrangement
+recommended by one who has tried it, and he reassures the
+old-fashioned believer who clings to the less formal régime (and
+whose protest was voiced in the Montanist movement), that there
+will be no spiritual loss under the new system. The book closes
+(chap. xvi.) with exhortations to steadfastness in the last days,
+and to the coming of the &ldquo;world-deceiver&rdquo; or Antichrist, which
+will precede the coming of the Lord. This section is perhaps the
+actual utterance of a Christian prophet, and may be of earlier
+origin than the two preceding sections.</p>
+
+<p>3. It will now be clear that indications of the locality and date
+of our present <i>Teaching</i> must be sought for only in the second
+part, and in the Christian interpolations in the first part. We
+have no ground for thinking that the second part ever existed
+independently as a separate book. The whole work was in the
+hands of the writer of the seventh book of the <i>Apostolic Constitutions</i>,
+who embodies almost every sentence of it, interspersing
+it with passages of Scripture, and modifying the precepts of the
+second part to suit a later (4th-century) stage of church development;
+this writer was also the interpolator of the Epistles of
+Ignatius, and belonged to the Syrian Church. Whether the
+second part was known to the writer of the <i>Apostolic Church
+Order</i> is not clear, as his only quotation of it comes from one of the
+eucharistic prayers. The allusions of early writers seem to point
+to Egypt, but their references are mostly to the first part, so that
+we must be careful how we argue from them as to the provenance
+of the book as a whole. Against Egypt has been urged the
+allusion in one of the eucharistic prayers to &ldquo;corn upon the
+mountains.&rdquo; This is found in the Prayer-book of Serapion
+<span class="pagenum"><a name="page202" id="page202"></a>202</span>
+(c. 350) but omitted in a later Egyptian prayer; the form as
+we have it in <i>The Didach&#275;</i> may have passed into Egypt with
+the authority of tradition which was afterwards weakened. The
+anti-Jewish tone of the second part suggests the neighbourhood
+of Jews, from whom the Christians were to be sharply distinguished.
+Either Egypt or Syria would satisfy this condition,
+and in favour of Syria is the fact that the presbyterate there was
+to a late date regarded as a rank rather than an office. If we can
+connect the injunctions (vi. 3) concerning (abstinence from certain)
+food and that which is offered to idols with the old trouble that
+arose at Antioch (Acts xv. 1) and was legislated for by the
+Jerusalem council, we have additional support for the Syrian
+claim. But all that we can safely say as to locality is that the
+community here represented seems to have been isolated, and
+out of touch with the larger centres of Christian life.</p>
+
+<p>This last consideration helps us in discussing the question of
+date. For such an isolated community may have preserved
+primitive customs for some time after they had generally disappeared.
+Certainly the stage of development is an early one, as
+is shown, <i>e.g.</i>, by the prominence of prophets, and the need that
+was felt for the vindication of the position of the bishops and
+deacons (there is no mention at all of presbyters); moreover,
+there is no reference to a canon of Scripture (though the written
+Gospel is expressly mentioned) or to a creed. On the other hand
+the &ldquo;apostles&rdquo; of the second part are obviously not &ldquo;the
+twelve apostles&rdquo; of the title; and the prophets seem in some
+instances to have proved unworthy of their high position. The
+ministry of enthusiasm which they represent is about to give way
+to the ministry of office, a transition which is reflected in the New
+Testament in the 3rd Epistle of John. Three of the Gospels have
+clearly been for some time in circulation; St Matthew&rsquo;s is used
+several times, and there are phrases which occur only in St Luke&rsquo;s,
+while St John&rsquo;s Gospel lies behind the eucharistic prayers which
+the writer has embodied in his work. There are no indications
+of any form of doctrinal heresy as needing rebuke; the warnings
+against false teaching are quite general. While the first part
+must be dated before the Epistle of Barnabas, <i>i.e.</i> before <span class="sc">a.d.</span> 90,
+it seems wisest not to place the complete work much earlier than
+<span class="sc">a.d.</span> 120, and there are passages which may well be later.</p>
+
+<div class="condensed">
+<p>A large literature has sprung up round The <i>Didach&#275;</i> since 1884.
+Harnack&rsquo;s edition in <i>Texte u. Unters.</i> vol. ii. (1884) is indispensable
+to the student; and his discussions in <i>Altchristl. Litteratur</i> and
+<i>Chronologie</i> give clear summaries of his work. Other editions of the
+text are those of F. X. Funk, <i>Patres Apostolici</i>, vol. i. (Tübingen,
+1901); H. Lietzmann (Bonn, 1903; with Latin version). Dr J. E.
+Odgers has published an English translation with introduction and
+notes (London, 1906). Dr C. Taylor in 1886 drew attention to some
+important parallels in Jewish literature; his edition contains an
+English translation. Dr Rendel Harris published in 1887 a complete
+facsimile, and gathered a great store of patristic illustration. Text
+and translation will also be found in Lightfoot&rsquo;s <i>Apostolic Fathers</i>
+(ed. min.) The fullest critical treatment in English is by Dr Vernon
+Bartlet in the extra volume of Hastings&rsquo;s <i>Dictionary of the Bible</i>;
+the most complete commentary on the text is by P. Drews in
+Hennecke&rsquo;s <i>Handbuch zu den N.T. Apocryphen</i> (1904). Other
+references to the literature may be found by consulting Harnack&rsquo;s
+<i>Altchristl. Litteratur</i>.</p>
+</div>
+
+<hr class="foot" /> <div class="note">
+
+<p><a name="ft1e" id="ft1e" href="#fa1e"><span class="fn">1</span></a> The MS. was found in the Library of the Jerusalem Monastery
+of the Most Holy Sepulchre, in Phanar, the Greek quarter of
+Constantinople. It is a small octavo volume of 120 parchment
+leaves, written throughout by Leo, &ldquo;notary and sinner,&rdquo; who
+finished his task on the 11th of June 1156. Besides The <i>Didach&#275;</i> and
+the Epistles of Clement it contains several spurious Ignatian epistles.</p>
+
+<p><a name="ft2e" id="ft2e" href="#fa2e"><span class="fn">2</span></a> The word <i>twelve</i> had no place in the original title and was inserted
+when the original <i>Didach&#275;</i> or <i>Teaching</i> (<i>e.g.</i> <i>The Two Ways</i>) was
+combined with the church manual which mentions apostles outside
+of the twelve. It may be noted that the division of the <i>Didach&#275;</i> into
+chapters is due to Bryennius, that into verses to A. Harnack.</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIDACTIC POETRY,<a name="ar49" id="ar49"></a></span> that form of verse the aim of which is,
+less to excite the hearer by passion or move him by pathos,
+than to instruct his mind and improve his morals. The Greek
+word <span class="grk" title="didaktikos">&#948;&#953;&#948;&#945;&#954;&#964;&#953;&#954;&#972;&#962;</span> signifies a teacher, from the verb <span class="grk" title="didaskein">&#948;&#953;&#948;&#940;&#963;&#954;&#949;&#953;&#957;</span>,
+and poetry of the class under discussion approaches us with the
+arts and graces of a schoolmaster. At no time was it found
+convenient to combine lyrical verse with instruction, and therefore
+from the beginning of literature the didactic poets have
+chosen a form approaching the epical. Modern criticism, which
+discourages the epic, and is increasingly anxious to limit the word
+&ldquo;poetry&rdquo; to lyric, is inclined to exclude the term &ldquo;didactic
+poetry&rdquo; from our nomenclature, as a phrase absurd in itself.
+It is indeed more than probable that didactic verse is hopelessly
+obsolete. Definite information is now to be found in a thousand
+shapes, directly and boldly presented in clear and technical prose.
+No farmer, however elegant, will, any longer choose to study
+agriculture in hexameters, or even in Tusser&rsquo;s shambling metre.
+The sciences and the professions will not waste their time on
+methods of instruction which must, from their very nature, be
+artless, inexact and vague. But in the morning of the world, those
+who taught with authority might well believe that verse was the
+proper, nay, the only serious vehicle of their instruction. What
+they knew was extremely limited, and in its nature it was
+simple and straightforward; it had little technical subtlety; it
+constantly lapsed into the fabulous and the conjectural. Not
+only could what early sages knew, or guessed, about astronomy
+and medicine and geography be conveniently put into rolling
+verse, but, in the absence of all written books, this was the
+easiest way in which information could be made attractive to the
+ear and be retained by the memory.</p>
+
+<p>In the prehistoric dawn of Greek civilization there appear
+to have been three classes of poetry, to which the literature of
+Europe looks back as to its triple fountain-head. There were
+romantic epics, dealing with the adventures of gods and heroes;
+these Homer represents. There were mystic chants and religious
+odes, purely lyrical in character, of which the best Orphic Hymns
+must have been the type. And lastly there was a great body of
+verse occupied entirely with increasing the knowledge of citizens in
+useful branches of art and observation; these were the beginnings
+of didactic poetry, and we class them together under the dim name
+of Hesiod. It is impossible to date these earliest didactic poems,
+which nevertheless set the fashion of form which has been
+preserved ever since. The <i>Works and Days</i>, which passes as the
+direct masterpiece of Hesiod (<i>q.v.</i>), is the type of all the poetry
+which has had education as its aim. Hesiod is supposed to have
+been a tiller of the ground in a Boeotian village, who determined
+to enrich his neighbours&rsquo; minds by putting his own ripe stores of
+useful information into sonorous metre. Historically examined,
+the legend of Hesiod becomes a shadow, but the substance of
+the poems attributed to him remains. The genuine parts of
+the <i>Works and Days</i>, which Professor Gilbert Murray has called
+&ldquo;a slow, lowly, simple poem,&rdquo; deal with rules for agriculture.
+The <i>Theogony</i> is an annotated catalogue of the gods. Other
+poems attributed to Hesiod, but now lost, were on astronomy, on
+auguries by birds, on the character of the physical world; still
+others seem to have been genealogies of famous women. All this
+mass of Boeotian verse was composed for educational purposes,
+in an age when even preposterous information was better than
+no knowledge at all. In slightly later times, as the Greek nation
+became better supplied with intellectual appliances, the stream
+of didactic poetry flowed more and more closely in one, and that
+a theological, channel. The great poem of Parmenides <i>On Nature</i>
+and those of Empedocles exist only in fragments, but enough
+remains to show that these poets carried on the didactic method
+in mythology. Cleostratus of Tenedos wrote an astronomical
+poem in the 6th century, and Periander a medical one in the
+4th, but didactic poetry did not flourish again in Greece until
+the 3rd century, when Aratus, in the Alexandrian age, wrote his
+famous <i>Phenomena</i>, a poem about things seen in the heavens.
+Other later Greek didactic poets were Nicander, and perhaps
+Euphorion.</p>
+
+<p>It was from the hands of these Alexandrian writers that the
+genius of didactic poetry passed over to Rome, since, although it
+is possible that some of the lost works of the early republic, and in
+particular those of Ennius, may have possessed an educational
+character, the first and by far the greatest didactic Latin poet
+known to us is Lucretius. A highly finished translation by
+Cicero into Latin hexameters of the principal works of Aratus is
+believed to have drawn the attention of Lucretius to this school
+of Greek poetry, and it was not without reference to the Greeks,
+although in a more archaic and far purer taste, that he composed,
+in the 1st century before Christ, his magnificent <i>De rerum
+natura</i>. By universal consent, this is the noblest didactic poem
+in the literature of the world. It was intended to instruct mankind
+in the interpretation and in the working of the system of
+philosophy revealed by Epicurus, which at that time was exciting
+the sympathetic attention of all classes of Roman society. What
+gave the poem of Lucretius its extraordinary interest, and what
+has prolonged and even increased its vitality, was the imaginative
+and illustrative insight of the author, piercing and lighting up the
+<span class="pagenum"><a name="page203" id="page203"></a>203</span>
+recesses of human experience. On a lower intellectual level, but
+of a still greater technical excellence, was the <i>Georgics</i> of Virgil,
+a poem on the processes of agriculture, published about 30 <span class="sc">b.c.</span>
+The brilliant execution of this famous work has justly made it the
+type and unapproachable standard of all poetry which desires
+to impart useful information in the guise of exquisite literature.
+Himself once a farmer on the banks of the Mincio, Virgil, at the
+apex of his genius, set himself in his Campanian villa to recall
+whatever had been essential in the agricultural life of his boyish
+home, and the result, in spite of the ardours of the subject, was
+what J. W. Mackail has called &ldquo;the most splendid literary production
+of the Empire.&rdquo; In the rest of surviving Latin didactic
+poetry, the influence and the imitation of Virgil and Lucretius
+are manifest. Manilius, turning again to Alexandria, produced
+a fine <i>Astronomica</i> towards the close of the reign of Augustus.
+Columella, regretting that Virgil had omitted to sing of gardens,
+composed a smooth poem on horticulture. Natural philosophy
+inspired Lucilius junior, of whom a didactic poem on Etna
+survives. Long afterwards, under Diocletian, a poet of Carthage,
+Nemesianus, wrote in the manner of Virgil the <i>Cynegetica</i>, a
+poem on hunting with dogs, which has had numerous imitations
+in later European literatures. These are the most important
+specimens of didactic poetry which ancient Rome has handed
+down to us.</p>
+
+<p>In Anglo-Saxon and early English poetic literature, and
+especially in the religious part of it, an element of didacticism is
+not to be overlooked. But it would be difficult to say that anything
+of importance was written in verse with the sole purpose of
+imparting information, until we reach the 16th century. Some of
+the later medieval allegories are didactic or nothing. The first
+poem, however, which we can in any reasonable way compare
+with the classic works of which we have been speaking is the
+<i>Hundreth Pointes of Good Husbandrie</i>, published in 1557 by
+Thomas Tusser; these humble Georgics aimed at a practical
+description of the whole art of English farming. Throughout the
+early part of the 17th century, when our national poetry was in
+its most vivid and brilliant condition, the last thing a poet
+thought of doing was the setting down of scientific facts in
+rhyme. We come across, however, one or two writers who were
+as didactic as the age would permit them to be, Samuel Daniel with
+his philosophy, Fulke Greville, Lord Brooke with his &ldquo;treatises&rdquo;
+of war and monarchy. After the Restoration, as the lyrical
+element rapidly died out of English poetry, there was more and
+more room left for educational rhetoric in verse. The poems
+about prosody, founded upon Horace, and signed by John
+Sheffield, 3rd earl of Mulgrave (1648-1721), and Lord Roscommon,
+were among the earliest purely didactic verse-studies in English.
+John Philips deserves a certain pre-eminence, as his poem called
+Cyder, in 1706, set the fashion which lasted all down the 18th
+century, of writing precisely in verse about definite branches of
+industry or employment. None of the greater poets of the age of
+Anne quite succumbed to the practice, but there is a very distinct
+flavour of the purely didactic about a great deal of the verse of
+Pope and Gay. In such productions as Gilbert West&rsquo;s (1703-1756)
+<i>Education</i>, Dyer&rsquo;s <i>Fleece</i>, and Somerville&rsquo;s <i>Chase</i>, we see
+technical information put forward as the central aim of the poet.
+Instead of a passionate pleasure, or at least an uplifted enthusiasm,
+being the poet&rsquo;s object, he frankly admits that, first and
+foremost, he has some facts about wool or dogs or schoolmasters
+which he wishes to bring home to his readers, and that, secondly,
+he consents to use verse, as brilliantly as he can, for the purpose
+of gilding the pill and attracting an unwilling attention. As we
+descend the 18th century, these works become more and more
+numerous, and more dry, especially when opposed by the descriptive
+and rural poets of the school of Thomson, the poet of
+<i>The Seasons</i>. But Thomson himself wrote a huge poem of
+<i>Liberty</i> (1732), for which we have no name if we must not call it
+didactic. Even Gray began, though he failed to finish, a work of
+this class, on <i>The Alliance of Education and Government</i>. These
+poems were discredited by the publication of <i>The Sugar-Cane</i>
+(1764), a long verse-treatise about the cultivation of sugar by
+negroes in the West Indies, by James Grainger (1721-1766), but,
+though liable to ridicule, such versified treatises continued to
+appear. Whether so great a writer as Cowper is to be counted
+among the didactic poets is a question on which readers of <i>The
+Task</i> may be divided; this poem belongs rather to the class of
+descriptive poetry, but a strong didactic tendency is visible in
+parts of it. Perhaps the latest frankly educational poem which
+enjoyed a great popularity was <i>The Course of Time</i> by Robert
+Pollok (1798-1827), in which a system of Calvinistic divinity is
+laid down with severity and in the pomp of blank verse. This
+kind of literature had already been exposed, and discouraged, by
+the teaching of Wordsworth, who had insisted on the imperative
+necessity of charging all poetry with imagination and passion.
+Oddly enough, <i>The Excursion</i> of Wordsworth himself is perhaps
+the most didactic poem of the 19th century, but it must be
+acknowledged that his influence, in this direction, was saner
+than his practice. Since the days of Coleridge and Shelley it
+has been almost impossible to conceive a poet of any value composing
+in verse a work written with the purpose of inculcating
+useful information.</p>
+
+<p>The history of didactic poetry in France repeats, in great
+measure, but in drearier language, that of England. Boileau, like
+Pope, but with a more definite purpose as a teacher, offered
+instruction in his <i>Art poétique</i> and in his <i>Epistles</i>. But his
+doctrine was always literary, not purely educational. At the
+beginning of the 18th century, the younger Racine (1692-1763)
+wrote sermons in verse, and at the close of it the Abbé Delille
+(1738-1813) tried to imitate Virgil in poems about horticulture.
+Between these two there lies a vast mass of verse written for the
+indulgence of intellect rather than at the dictates of the heart;
+wherever this aims at increasing knowledge, it at once becomes
+basely and flatly didactic. There is nothing in French literature
+of the transitional class that deserves mention beside <i>The Task</i> or
+<i>The Excursion</i>.</p>
+
+<p>During the century which preceded the Romantic revival of
+poetry in Germany, didactic verse was cultivated in that country
+on the lines of imitation of the French, but with a greater dryness
+and on a lower level of utility. Modern German literature
+began with Martin Opitz (1597-1639) and the Silesian School,
+who were in their essence rhetorical and educational, and who
+gave their tone to German verse. Albrecht von Haller (1708-1777)
+brought a very considerable intellectual force to bear on
+his huge poems, <i>The Origin of Evil</i>, which was theological, and
+<i>The Alps</i> (1729), botanical and topographical. Johann Peter Uz
+(1720-1796) wrote a <i>Theodicée</i>, which was very popular, and not
+without dignity. Johann Jacob Dusch (1725-1787) undertook to
+put <i>The Sciences</i> into the eight books of a great didactic poem.
+Tiedge (1752-1840) was the last of the school; in a once-famous
+<i>Urania</i>, he sang of God and Immortality and Liberty. These
+German pieces were the most unswervingly didactic that any
+modern European literature has produced. There was hardly
+the pretence of introducing into them descriptions of natural
+beauty, as the English poets did, or of grace and wit like the
+French. The German poets simply poured into a lumbering
+mould of verse as much solid information and direct instruction
+as the form would hold.</p>
+
+<p>Didactic poetry has, in modern times, been antipathetic to
+the spirit of the Latin peoples, and neither Italian nor Spanish
+literature has produced a really notable work in this class. An
+examination of the poems, ancient and modern, which have been
+mentioned above, will show that from primitive times there have
+been two classes of poetic work to which the epithet didactic has
+been given. It is desirable to distinguish these a little more
+exactly. One is the pure instrument of teaching, the poetry
+which desires to impart all that it knows about the growing of
+cabbages or the prevention of disasters at sea, the revolution of
+the planets or the blessings of inoculation. This is didactic poetry
+proper, and this, it is almost certain, became irrevocably obsolete
+at the close of the 18th century. No future Virgil will give the
+world a second <i>Georgics</i>. But there is another species which it
+is very improbable that criticism has entirely dislodged; that is
+the poetry which combines, with philosophical instruction, an impetus
+of imaginative movement, and a certain definite cultivation
+<span class="pagenum"><a name="page204" id="page204"></a>204</span>
+of fire and beauty. In hands so noble as those of Lucretius
+and Goethe this species of didactic poetry has enriched the world
+with durable masterpieces, and, although the circle of readers
+which will endure scientific disquisition in the bonds of verse
+grows narrower and narrower, it is probable that the great poet
+who is also a great thinker will now and again insist on being
+heard. In Sully-Prudhomme France has possessed an eminent
+writer whose methods are directly instructive, and both <i>La
+Justice</i> (1878) and <i>Le Bonheur</i> (1888) are typically didactic poems.
+Perhaps future historians may name these as the latest of their
+class.</p>
+<div class="author">(E. G.)</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIDEROT, DENIS<a name="ar50" id="ar50"></a></span> (1713-1784), French man of letters and
+encyclopaedist, was born at Langres on the 5th of October 1713.
+He was educated by the Jesuits, like most of those who afterwards
+became the bitterest enemies of Catholicism; and, when
+his education was at an end, he vexed his brave and worthy
+father&rsquo;s heart by turning away from respectable callings, like law
+or medicine, and throwing himself into the vagabond life of a
+bookseller&rsquo;s hack in Paris. An imprudent marriage (1743) did
+not better his position. His wife, Anne Toinette Champion, was
+a devout Catholic, but her piety did not restrain a narrow and
+fretful temper, and Diderot&rsquo;s domestic life was irregular and
+unhappy. He sought consolation for chagrins at home in attachments
+abroad, first with a Madame Puisieux, a fifth-rate female
+scribbler, and then with Sophie Voland, to whom he was constant
+for the rest of her life. His letters to her are among the most
+graphic of all the pictures that we have of the daily life of the
+philosophic circle in Paris. An interesting contrast may be
+made between the Bohemianism of the famous English literary
+set who supped at the Turk&rsquo;s Head with the Tory Johnson and
+the Conservative Burke for their oracles, and the Bohemianism of
+the French set who about the same time dined once a week at the
+baron D&rsquo;Holbach&rsquo;s, to listen to the wild sallies and the inspiring
+declamations of Diderot. For Diderot was not a great writer;
+he stands out as a fertile, suggestive and daring thinker, and a
+prodigious and most eloquent talker.</p>
+
+<p>Diderot&rsquo;s earliest writings were of as little importance as
+Goldsmith&rsquo;s <i>Enquiry into the State of Polite Learning</i> or Burke&rsquo;s
+<i>Abridgement of English History</i>. He earned 100 crowns by
+translating Stanyan&rsquo;s <i>History of Greece</i> (1743); with two
+colleagues he produced a translation of James&rsquo;s <i>Dictionary of
+Medicine</i> (1746-1748) and about the same date he published a
+free rendering of Shaftesbury&rsquo;s <i>Inquiry Concerning Virtue and
+Merit</i> (1745), with some original notes of his own. With strange
+and characteristic versatility, he turned from ethical speculation
+to the composition of a volume of stories, the <i>Bijoux indiscrets</i>
+(1748), gross without liveliness, and impure without wit. In later
+years he repented of this shameless work, just as Boccaccio is
+said in the day of his grey hairs to have thought of the sprightliness
+of the <i>Decameron</i> with strong remorse. From tales Diderot
+went back to the more congenial region of philosophy. Between
+the morning of Good Friday and the evening of Easter Monday he
+wrote the <i>Pensées philosophiques</i> (1746), and he presently added
+to this a short complementary essay on the sufficiency of natural
+religion. The gist of these performances is to press the ordinary
+rationalistic objections to a supernatural revelation; but though
+Diderot did not at this time pass out into the wilderness
+beyond natural religion, yet there are signs that he accepted that
+less as a positive doctrine, resting on grounds of its own, than as
+a convenient point of attack against Christianity. In 1747 he
+wrote the <i>Promenade du sceptique</i>, a rather poor allegory&mdash;pointing
+first to the extravagances of Catholicism; second, to the
+vanity of the pleasures of that world which is the rival of
+the church; and third, to the desperate and unfathomable
+uncertainty of the philosophy which professes to be so high
+above both church and world.</p>
+
+<p>Diderot&rsquo;s next piece was what first introduced him to the world
+as an original thinker, his famous <i>Lettre sur les aveugles</i> (1749).
+The immediate object of this short but pithy writing was to show
+the dependence of men&rsquo;s ideas on their five senses. It considers
+the case of the intellect deprived of the aid of one of the senses;
+and in a second piece, published afterwards, Diderot considered
+the case of a similar deprivation in the deaf and dumb. The
+<i>Lettre sur les sourds et muets</i>, however, is substantially a digressive
+examination of some points in aesthetics. The philosophic
+significance of the two essays is in the advance they make
+towards the principle of Relativity. But what interested the
+militant philosophers of that day was an episodic application
+of the principle of relativity to the master-conception of God.
+What makes the <i>Lettre sur les aveugles</i> interesting is its presentation,
+in a distinct though undigested form, of the modern theory
+of variability, and of survival by superior adaptation. It is worth
+noticing, too, as an illustration of the comprehensive freedom
+with which Diderot felt his way round any subject that he
+approached, that in this theoretic essay he suggests the possibility
+of teaching the blind to read through the sense of touch. If the
+<i>Lettre sur les aveugles</i> introduced Diderot into the worshipful
+company of the philosophers, it also introduced him to the
+penalties of philosophy. His speculation was too hardy for the
+authorities, and he was thrown into the prison of Vincennes.
+Here he remained for three months; then he was released, to
+enter upon the gigantic undertaking of his life.</p>
+
+<p>The bookseller Lebreton had applied to him with a project
+for the publication of a translation into French of Ephraim
+Chambers&rsquo;s <i>Cyclopaedia</i>, undertaken in the first instance by an
+Englishman, John Mills, and a German, Gottfried Sellius (for
+particulars see <span class="sc"><a href="#artlinks">Encyclopaedia</a></span>). Diderot accepted the proposal,
+but in his busy and pregnant intelligence the scheme became
+transformed. Instead of a mere reproduction of Chambers, he
+persuaded the bookseller to enter upon a new work, which should
+collect under one roof all the active writers, all the new ideas, all
+the new knowledge, that were then moving the cultivated class
+to its depths, but still were comparatively ineffectual by reason of
+their dispersion. His enthusiasm infected the publishers; they
+collected a sufficient capital for a vaster enterprise than they had
+at first planned; D&rsquo;Alembert was persuaded to become Diderot&rsquo;s
+colleague; the requisite permission was procured from the
+government; in 1750 an elaborate prospectus announced the
+project to a delighted public; and in 1751 the first volume was
+given to the world. The last of the letterpress was issued in
+1765, but it was 1772 before the subscribers received the final
+volumes of the plates. These twenty years were to Diderot years
+not merely of incessant drudgery, but of harassing persecution,
+of sufferings from the cabals of enemies, and of injury from the
+desertion of friends. The ecclesiastical party detested the
+<i>Encyclopaedia</i>, in which they saw a rising stronghold for their
+philosophic enemies. By 1757 they could endure the sight no
+longer. The subscribers had grown from 2000 to 4000, and this
+was a right measure of the growth of the work in popular influence
+and power. To any one who turns over the pages of these redoubtable
+volumes now, it seems surprising that their doctrines
+should have stirred such portentous alarm. There is no atheism,
+no overt attack on any of the cardinal mysteries of the faith, no
+direct denunciation even of the notorious abuses of the church.
+Yet we feel that the atmosphere of the book may well have been
+displeasing to authorities who had not yet learnt to encounter
+the modern spirit on equal terms. The <i>Encyclopaedia</i> takes for
+granted the justice of religious tolerance and speculative freedom.
+It asserts in distinct tones the democratic doctrine that it is
+the common people in a nation whose lot ought to be the main
+concern of the nation&rsquo;s government. From beginning to end
+it is one unbroken process of exaltation of scientific knowledge on
+the one hand, and pacific industry on the other. All these things
+were odious to the old governing classes of France; their spirit
+was absolutist, ecclesiastical and military. Perhaps the most
+alarming thought of all was the current belief that the <i>Encyclopaedia</i>
+was the work of an organized band of conspirators against
+society, and that a pestilent doctrine was now made truly
+formidable by the confederation of its preachers into an open
+league. When the seventh volume appeared, it contained an
+article on &ldquo;Geneva,&rdquo; written by D&rsquo;Alembert. The writer
+contrived a panegyric on the pastors of Geneva, of which every
+word was a stinging reproach to the abbés and prelates of
+Versailles. At the same moment Helvétius&rsquo;s book, <i>L&rsquo;Esprit</i>,
+<span class="pagenum"><a name="page205" id="page205"></a>205</span>
+appeared, and gave a still more profound and, let us add, a more
+reasonable shock to the ecclesiastical party. Authority could
+brook no more, and in 1759 the <i>Encyclopaedia</i> was formally
+suppressed.</p>
+
+<p>The decree, however, did not arrest the continuance of the
+work. The connivance of the authorities at the breach of their
+own official orders was common in those times of distracted
+government. The work went on, but with its difficulties increased
+by the necessity of being clandestine. And a worse thing
+than troublesome interference by the police now befell Diderot.
+D&rsquo;Alembert, wearied of shifts and indignities, withdrew from
+the enterprise. Other powerful colleagues, Turgot among them,
+declined to contribute further to a book which had acquired
+an evil fame. Diderot was left to bring the task to an end as he
+best could. For seven years he laboured like a slave at the oar.
+He wrote several hundred articles, some of them very slight, but
+many of them most laborious, comprehensive and ample. He
+wore out his eyesight in correcting proofs, and he wearied his soul
+in bringing the manuscript of less competent contributors into
+decent shape. He spent his days in the workshops, mastering the
+processes of manufactures, and his nights in reproducing on paper
+what he had learnt during the day. And he was incessantly
+harassed all the time by alarms of a descent from the police. At
+the last moment, when his immense work was just drawing to
+an end, he encountered one last and crowning mortification: he
+discovered that the bookseller, fearing the displeasure of the
+government, had struck out from the proof sheets, after they had
+left Diderot&rsquo;s hands, all passages that he chose to think too hardy.
+The monument to which Diderot had given the labour of twenty
+long and oppressive years was irreparably mutilated and defaced.
+It is calculated that the average annual salary received by
+Diderot for his share in the <i>Encyclopaedia</i> was about £120
+sterling. &ldquo;And then to think,&rdquo; said Voltaire, &ldquo;that an army
+contractor makes £800 in a day!&rdquo;</p>
+
+<p>Although the <i>Encyclopaedia</i> was Diderot&rsquo;s monumental work,
+he is the author of a shower of dispersed pieces that sowed nearly
+every field of intellectual interest with new and fruitful ideas.
+We find no masterpiece, but only thoughts for masterpieces; no
+creation, but a criticism with the quality to inspire and direct
+creation. He wrote plays&mdash;<i>Le Fils naturel</i> (1757) and <i>Le Père de
+famille</i> (1758)&mdash;and they are very insipid performances in the sentimental
+vein. But he accompanied them by essays on dramatic
+poetry, including especially the <i>Paradoxe sur le comédien</i>, in
+which he announced the principles of a new drama,&mdash;the serious,
+domestic, bourgeois drama of real life, in opposition to the stilted
+conventions of the classic French stage. It was Diderot&rsquo;s lessons
+and example that gave a decisive bias to the dramatic taste of
+Lessing, whose plays, and his <i>Hamburgische Dramaturgie</i> (1768),
+mark so important an epoch in the history of the modern theatre.
+In the pictorial art, Diderot&rsquo;s criticisms are no less rich, fertile
+and wide in their ideas. His article on &ldquo;Beauty&rdquo; in the
+<i>Encyclopaedia</i> shows that he had mastered and passed beyond
+the metaphysical theories on the subject, and the <i>Essai sur la
+peinture</i> was justly described by Goethe, who thought it worth
+translating, as &ldquo;a magnificent work, which speaks even more
+helpfully to the poet than to the painter, though to the painter
+too it is as a blazing torch.&rdquo; Diderot&rsquo;s most intimate friend was
+Grimm, one of the conspicuous figures of the philosophic body.
+Grimm wrote news-letters to various high personages in Germany,
+reporting what was going on in the world of art and literature
+in Paris, then without a rival as the capital of the intellectual
+activity of Europe. Diderot helped his friend at one time and
+another between 1759 and 1779, by writing for him an account
+of the annual exhibitions of paintings. These <i>Salons</i> are among
+the most readable of all pieces of art criticism. They have a
+freshness, a reality, a life, which take their readers into a different
+world from the dry and conceited pedantries of the ordinary
+virtuoso. As has been said by Sainte-Beuve, they initiated the
+French into a new sentiment, and introduced people to the
+mystery and purport of colour by ideas. &ldquo;Before Diderot,&rdquo;
+Madame Necker said, &ldquo;I had never seen anything in pictures
+except dull and lifeless colours; it was his imagination that gave
+them relief and life, and it is almost a new sense for which I am
+indebted to his genius.&rdquo;</p>
+
+<p>Greuze was Diderot&rsquo;s favourite among contemporary artists,
+and it is easy to see why. Greuze&rsquo;s most characteristic pictures
+were the rendering in colour of the same sentiment of domestic
+virtue and the pathos of common life, which Diderot attempted
+with inferior success to represent upon the stage. For Diderot
+was above all things interested in the life of men,&mdash;not the
+abstract life of the race, but the incidents of individual character,
+the fortunes of a particular family, the relations of real and
+concrete motives in this or that special case. He delighted with
+the enthusiasm of a born casuist in curious puzzles of right
+and wrong, and in devising a conflict between the generalities of
+ethics and the conditions of an ingeniously contrived practical
+dilemma. Mostly his interest expressed itself in didactic and
+sympathetic form; in two, however, of the most remarkable
+of all his pieces, it is not sympathetic, but ironical. <i>Jacques le
+fataliste</i> (written in 1773, but not published until 1796) is in
+manner an imitation of <i>Tristram Shandy</i> and <i>The Sentimental
+Journey</i>. Few modern readers will find in it any true diversion.
+In spite of some excellent criticisms dispersed here and there,
+and in spite of one or two stories that are not without a certain
+effective realism, it must as a whole be pronounced savourless,
+forced, and as leaving unmoved those springs of laughter and
+of tears which are the common fountain of humour. <i>Le Neveu
+de Rameau</i> is a far superior performance. If there were any inevitable
+compulsion to name a masterpiece for Diderot, one must
+select this singular &ldquo;farce-tragedy.&rdquo; Its intention has been
+matter of dispute; whether it was designed to be merely a satire
+on contemporary manners, or a reduction of the theory of self-interest
+to an absurdity, or the application of an ironical clincher
+to the ethics of ordinary convention, or a mere setting for a
+discussion about music, or a vigorous dramatic sketch of a
+parasite and a human original. There is no dispute as to its
+curious literary flavour, its mixed qualities of pungency, bitterness,
+pity and, in places, unflinching shamelessness. Goethe&rsquo;s
+translation (1805) was the first introduction of <i>Le Neveu de
+Rameau</i> to the European public. After executing it, he gave
+back the original French manuscript to Schiller, from whom he
+had it. No authentic French copy of it appeared until the writer
+had been nearly forty years in his grave (1823).</p>
+
+<p>It would take several pages merely to contain the list of
+Diderot&rsquo;s miscellaneous pieces, from an infinitely graceful trifle
+like the <i>Regrets sur ma vieille robe de chambre</i> up to <i>Le Rêve de
+D&rsquo;Alembert</i>, where he plunges into the depths of the controversy
+as to the ultimate constitution of matter and the meaning of life.
+It is a mistake to set down Diderot for a coherent and systematic
+materialist. We ought to look upon him &ldquo;as a philosopher in
+whom all the contradictions of the time struggle with one another&rdquo;
+(Rosenkranz). That is to say, he is critical and not dogmatic.
+There is no unity in Diderot, as there was in Voltaire or in
+Rousseau. Just as in cases of conduct he loves to make new
+ethical assumptions and argue them out as a professional sophist
+might have done, so in the speculative problems as to the organization
+of matter, the origin of life, the compatibility between
+physiological machinery and free will, he takes a certain standpoint,
+and follows it out more or less digressively to its consequences.
+He seizes a hypothesis and works it to its end, and
+this made him the inspirer in others of materialist doctrines
+which they held more definitely than he did. Just as Diderot
+could not attain to the concentration, the positiveness, the
+finality of aim needed for a masterpiece of literature, so he could
+not attain to those qualities in the way of dogma and system.
+Yet he drew at last to the conclusions of materialism, and contributed
+many of its most declamatory pages to the <i>Système de la
+nature</i> of his friend D&rsquo;Holbach,&mdash;the very Bible of atheism, as
+some one styled it. All that he saw, if we reduce his opinions to
+formulae, was motion in space: &ldquo;attraction and repulsion, the
+only truth.&rdquo; If matter produces life by spontaneous generation,
+and if man has no alternative but to obey the compulsion of
+nature, what remains for God to do?</p>
+
+<p>In proportion as these conclusions deepened in him, the more
+<span class="pagenum"><a name="page206" id="page206"></a>206</span>
+did Diderot turn for the hope of the race to virtue; in other
+words, to such a regulation of conduct and motive as shall make
+us tender, pitiful, simple, contented. Hence his one great literary
+passion, his enthusiasm for Richardson, the English novelist.
+Hence, also, his deepening aversion for the political system of
+France, which makes the realization of a natural and contented
+domestic life so hard. Diderot had almost as much to say
+against society as even Rousseau himself. The difference between
+them was that Rousseau was a fervent theist. The atheism of
+the Holbachians, as he called Diderot&rsquo;s group, was intolerable
+to him; and this feeling, aided by certain private perversities of
+humour, led to a breach of what had once been an intimate
+friendship between Rousseau and Diderot (1757). Diderot was
+still alive when Rousseau&rsquo;s <i>Confessions</i> appeared, and he was so
+exasperated by Rousseau&rsquo;s stories about Grimm, then and always
+Diderot&rsquo;s intimate, that in 1782 he transformed a life of Seneca,
+that he had written four years earlier, into an <i>Essai sur les règnes
+de Claude et de Néron</i> (1778-1782), which is much less an account
+of Seneca than a vindication of Diderot and Grimm, and is one of
+the most rambling and inept productions in literature. As for the
+merits of the old quarrel between Rousseau and Diderot, we may
+agree with the latter, that too many sensible people would be in
+the wrong if Jean Jacques was in the right.</p>
+
+<p>Varied and incessant as was Diderot&rsquo;s mental activity, it was
+not of a kind to bring him riches. He secured none of the posts
+that were occasionally given to needy men of letters; he could
+not even obtain that bare official recognition of merit which was
+implied by being chosen a member of the Academy. The time
+came for him to provide a dower for his daughter, and he saw
+no other alternative than to sell his library. When the empress
+Catherine of Russia heard of his straits, she commissioned an
+agent in Paris to buy the library at a price equal to about £1000
+of English money, and then handsomely requested the philosopher
+to retain the books in Paris until she required them, and to
+constitute himself her librarian, with a yearly salary. In 1773
+Diderot started on an expedition to thank his imperial benefactress
+in person, and he passed some months at St Petersburg.
+The empress received him cordially. The strange pair passed their
+afternoons in disputes on a thousand points of high philosophy,
+and they debated with a vivacity and freedom not usual in
+courts. &ldquo;<i>Fi, donc,</i>&rdquo; said Catherine one day, when Diderot
+hinted that he argued with her at a disadvantage, &ldquo;<i>is there any
+difference among men?</i>&rdquo; Diderot returned home in 1774. Ten
+years remained to him, and he spent them in the industrious
+acquisition of new knowledge, in the composition of a host of
+fragmentary pieces, some of them mentioned above, and in
+luminous declamations with his friends. All accounts agree that
+Diderot was seen at his best in conversation. &ldquo;He who only
+knows Diderot in his writings,&rdquo; says Marmontel, &ldquo;does not know
+him at all. When he grew animated in talk, and allowed his
+thoughts to flow in all their abundance, then he became truly
+ravishing. In his writings he had not the art of ensemble; the
+first operation which orders and places everything was too slow
+and too painful to him.&rdquo; Diderot himself was conscious of the
+want of literary merit in his pieces. In truth he set no high value
+on what he had done. It is doubtful whether he was ever alive to
+the waste that circumstance and temperament together made of
+an intelligence from which, if it had been free to work systematically,
+the world of thought had so much to hope. He was one
+of those simple, disinterested and intellectually sterling workers
+to whom their own personality is as nothing in presence of the
+vast subjects that engage the thoughts of their lives. He wrote
+what he found to write, and left the piece, as Carlyle has said,
+&ldquo;on the waste of accident, with an ostrich-like indifference.&rdquo;
+When he heard one day that a collected edition of his works was
+in the press at Amsterdam, he greeted the news with &ldquo;peals of
+laughter,&rdquo; so well did he know the haste and the little heed with
+which those works had been dashed off.</p>
+
+<p>Diderot died on the 30th of July 1784, six years after Voltaire
+and Rousseau, one year after his old colleague D&rsquo;Alembert, and
+five years before D&rsquo;Holbach, his host and intimate for a lifetime.
+Notwithstanding Diderot&rsquo;s peals of laughter at the thought, an
+elaborate and exhaustive collection of his writings in twenty
+stout volumes, edited by MM. Assézat and Tourneux, was completed
+in 1875-1877.</p>
+
+<div class="condensed">
+<p><span class="sc">Authorities.</span>&mdash;Studies on Diderot by Scherer (1880); by
+E. Faguet (1890); by Sainte-Beuve in the <i>Causeries du lundi</i>; by
+F. Brunetière in the <i>Études critiques</i>, 2nd series, may be consulted.
+In English, Diderot has been the subject of a biography by John
+Morley [Viscount Morley of Blackburn] (1878). See also Karl
+Rosenkranz, <i>Diderots Leben und Werke</i> (1866). For a discussion of
+the authenticity of the posthumous works of Diderot see R. Dominic
+in the <i>Revue des deux mondes</i> (October 15, 1902).</p>
+</div>
+<div class="author">(J. Mo.)</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIDIUS SALVIUS JULIANUS, MARCUS,<a name="ar51" id="ar51"></a></span> Roman emperor for
+two months (March 28-June 2) during the year <span class="sc">a.d.</span> 193. He
+was the grandson of the famous jurist Salvius Julianus (under
+Hadrian and the Antonines), and the son of a distinguished
+general, who might have ascended the throne after the death of
+Antoninus Pius, had not his loyalty to the ruling house prevented
+him. Didius filled several civil and military offices with distinguished
+success, but subsequently abandoned himself to
+dissipation. On the death of Pertinax, the praetorian guards
+offered the throne to the highest bidder. Flavius Sulpicianus,
+the father-in-law of Pertinax and praefect of the city, had already
+made an offer; Didius, urged on by the members of his family,
+his freedmen and parasites, hurried to the praetorian camp to
+contend for the prize. He and Sulpicianus bid against each
+other, and finally the throne was knocked down to Didius. The
+senate and nobles professed their loyalty; but the people
+made no attempt to conceal their indignation at this insult to
+the state, and the armies of Britain, Syria and Illyricum broke
+out into open revolt. Septimius Severus, the commander of
+the Pannonian legions, was declared emperor and hastened by
+forced marches to Italy. Didius, abandoned by the praetorians,
+was condemned and executed by order of the senate, which at
+once acknowledged Severus.</p>
+
+<div class="condensed">
+<p><span class="sc">Authorities.</span>&mdash;Dio Cassius lxxiii. 11-17, who was actually in
+Rome at the time; Aelius Spartianus, <i>Didius Julianus</i>; Julius
+Capitolinus, <i>Pertinax</i>; Herodian ii.; Aurelius Victor, <i>De Caesaribus</i>,
+19; Zosimus i. 7; Gibbon, <i>Decline and Fall</i>, chap. 5.</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIDO,<a name="ar52" id="ar52"></a></span> or <span class="sc">Elissa</span>, the reputed founder of Carthage (<i>q.v.</i>), in
+Africa, daughter of the Tyrian king Metten (Mutto, Methres,
+Belus), wife of Acerbas (more correctly Sicharbas; Sychaeus in
+Virgil), a priest of Hercules. Her husband having been slain by
+her brother Pygmalion, Dido fled to Cyprus, and thence to the
+coast of Africa, where she purchased from a local chieftain
+Iarbas a piece of land on which she built Carthage. The city
+soon began to prosper and Iarbas sought Dido&rsquo;s hand in marriage,
+threatening her with war in case of refusal. To escape from him,
+Dido constructed a funeral pile, on which she stabbed herself
+before the people (Justin xviii. 4-7). Virgil, in defiance of the
+usually accepted chronology, makes Dido a contemporary of
+Aeneas, with whom she fell in love after his landing in Africa, and
+attributes her suicide to her abandonment by him at the command
+of Jupiter (<i>Aeneid</i>, iv.). Dido was worshipped at Carthage as a
+divinity under the name of Caelestis, the Roman counterpart of
+Tanit, the tutelary goddess of Carthage. According to Timaeus,
+the oldest authority for the story, her name was Theiosso, in
+Phoenician Helissa, and she was called Dido from her wanderings,
+Dido being the Phoenician equivalent of <span class="grk" title="planêtis">&#960;&#955;&#945;&#957;&#8134;&#964;&#953;&#962;</span> (<i>Etymologicum
+Magnum</i>, <i>s.v.</i>); some modern scholars, however,
+translate the name by &ldquo;beloved.&rdquo; Timaeus makes no mention
+of Aeneas, who seems to have been introduced by Naevius in his
+<i>Bellum Poenicum</i>, followed by Ennius in his <i>Annales</i>.</p>
+
+<div class="condensed">
+<p>For the variations of the legend in earlier and later Latin authors,
+see O. Rossbach in Pauly-Wissowa&rsquo;s <i>Realencyclopädie</i>, v. pt. 1 (1905);
+O. Meltzer&rsquo;s <i>Geschichte der Karthager</i>, i. (1879), and his article in
+Roscher&rsquo;s <i>Lexikon der Mythologie</i>.</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIDON, HENRI<a name="ar53" id="ar53"></a></span> (1840-1900), French Dominican, was born
+at Trouvet, Isère, on the 17th of March 1840. He joined the
+Dominicans, under the influence of Lacordaire, in 1858, and
+completed his theological studies at the Minerva convent at
+Rome. The influence of Lacordaire was shown in the zeal displayed
+by Didon in favour of a reconciliation between philosophy
+and science. In 1871 his fame had so much grown that he was
+chosen to deliver the funeral oration over the murdered archbishop
+of Paris, Monseigneur G. Darboy. He also delivered some
+<span class="pagenum"><a name="page207" id="page207"></a>207</span>
+discourses at the church of St Jean de Beauvais in Paris on the
+relations between science and religion; but his utterances,
+especially on the question of divorce, were deemed suspicious by
+his superiors, and his intimacy with Claude Bernard the physiologist
+was disapproved. He was interdicted from preaching and
+sent into retirement at the convent of Corbara in Corsica. After
+eighteen months he emerged, and travelled in Germany, publishing
+an interesting work upon that country, entitled <i>Les Allemands</i>
+(English translation by R. Ledos de Beaufort, London, 1884).
+On his return to France in 1890 he produced his best known
+work, <i>Jésus-Christ</i> (2 vols., Paris), for which he had qualified
+himself by travel in the Holy Land. In the same year he became
+director of the Collège Albert-le-Grand at Arcueil, and founded
+three auxiliary institutions, École Lacordaire, École Laplace and
+École St Dominique. He wrote, in addition, several works on
+educational questions, and augmented his fame as an eloquent
+preacher by discourses preached during Lent and Advent. He
+died at Toulouse on the 13th of March 1900.</p>
+
+<div class="condensed">
+<p>See the biographies by J. de Romano (1891), and A. de Coulanges
+(Paris, 1900); and especially the work of Stanislas Reynaud,
+entitled <i>Le Père Didon, sa vie et son &oelig;uvre</i> (Paris, 1904).</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIDOT,<a name="ar54" id="ar54"></a></span> the name of a family of learned French printers and
+publishers. <span class="sc">François Didot</span> (1689-1757), founder of the
+family, was born at Paris. He began business as a bookseller and
+printer in 1713, and among his undertakings was a collection
+of the travels of his friend the Abbé Prévost, in twenty volumes
+(1747). It was remarkable for its typographical perfection,
+and was adorned with many engravings and maps. <span class="sc">François
+Ambroise Didot</span> (1730-1804), son of François, made important
+improvements in type-founding, and was the first to attempt
+printing on vellum paper. Among the works which he published
+was the famous collection of French classics prepared by order
+of Louis XVI. for the education of the Dauphin, and the folio
+edition of <i>L&rsquo;Art de vérifier les dates</i>. <span class="sc">Pierre François Didot</span>
+(1732-1795), his brother, devoted much attention to the art of
+type-founding and to paper-making. Among the works which
+issued from his press was an edition in folio of the <i>Imitatio
+Christi</i> (1788). <span class="sc">Henri Didot</span> (1765-1852), son of Pierre François,
+is celebrated for his &ldquo;microscopic&rdquo; editions of various standard
+works, for which he engraved the type when nearly seventy years
+of age. He was also the engraver of the <i>assignats</i> issued by the
+Constituent and Legislative Assemblies and the Convention.
+<span class="sc">Didot Saint-Léger</span>, second son of Pierre François, was the
+inventor of the paper-making machine known in England as
+the Didot machine. <span class="sc">Pierre Didot</span> (1760-1853), eldest son of
+François Ambroise, is celebrated as the publisher of the beautiful
+&ldquo;Louvre&rdquo; editions of Virgil, Horace and Racine. The Racine,
+in three volumes folio, was pronounced in 1801 to be &ldquo;the most
+perfect typographical production of all ages.&rdquo; <span class="sc">Firmin Didot</span>
+(1764-1836), his brother, second son of François Ambroise,
+sustained the reputation of the family both as printer and type-founder.
+He revived (if he did not invent&mdash;a distinction which
+in order of time belongs to William Ged) the process of stereotyping,
+and coined its name, and he first used the process in his
+edition of Callet&rsquo;s <i>Tables of Logarithms</i> (1795), in which he secured
+an accuracy till then unattainable. He published stereotyped
+editions of French, English and Italian classics at a very low
+price. He was the author of two tragedies&mdash;<i>La Reine de
+Portugal</i> and <i>La Mort d&rsquo;Annibal</i>; and he wrote metrical translations
+from Virgil, Tyrtaeus and Theocritus. <span class="sc">Ambroise Firmin
+Didot</span> (1790-1876) was his eldest son. After receiving a classical
+education, he spent three years in Greece and in the East; and on
+the retirement of his father in 1827 he undertook, in conjunction
+with his brother Hyacinthe, the direction of the publishing
+business. Their greatest undertaking was a new edition of the
+<i>Thesaurus Graecae linguae</i> of Henri Estienne, under the editorial
+care of the brothers Dindorf and M. Hase (9 vols., 1855-1859).
+Among the numerous important works published by the brothers,
+the 200 volumes forming the <i>Bibliothèque des auteurs grecs</i>,
+<i>Bibliothèque latine</i>, and <i>Bibliothèque française</i> deserve special
+mention. Ambroise Firmin Didot was the first to propose
+(1823) a subscription in favour of the Greeks, then in insurrection
+against Turkish tyranny. Besides a translation of Thucydides
+(1833), he wrote the articles &ldquo;Estienne&rdquo; in the <i>Nouvelle Biographie
+générale</i>, and &ldquo;Typographie&rdquo; in the <i>Ency. mod.</i>, as well
+as <i>Observations sur l&rsquo;orthographie française</i> (1867), &amp;c. In 1875
+he published a very learned and elaborate monograph on Aldus
+Manutius. His collection of MSS., the richest in France, was
+said to have been worth, at the time of his death, not less than
+2,000,000 francs.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIDRON, ADOLPHE NAPOLÉON<a name="ar55" id="ar55"></a></span> (1806-1867), French
+archaeologist, was born at Hautvillers, in the department of
+Marne, on the 13th of March 1806. At first a student of law,
+he began in 1830, by the advice of Victor Hugo, a study of the
+Christian archaeology of the middle ages. After visiting and
+examining the principal churches, first of Normandy, then of
+central and southern France, he was on his return appointed by
+Guizot secretary to the Historical Committee of Arts and Monuments
+(1835); and in the following years he delivered several
+courses of lectures on Christian iconography at the Bibliothèque
+Royale. In 1839 he visited Greece for the purpose of examining
+the art of the Eastern Church, both in its buildings and its
+manuscripts. In 1844 he originated the <i>Annales archéologiques</i>,
+a periodical devoted to his favourite subject, which he edited
+until his death. In 1845 he established at Paris a special archaeological
+library, and at the same time a manufactory of painted
+glass. In the same year he was admitted to the Legion of
+Honour. His most important work is the <i>Iconographie chrétienne</i>,
+of which, however, the first portion only, <i>Histoire de Dieu</i> (1843),
+was published. It was translated into English by E. J. Millington.
+Among his other works may be mentioned the <i>Manuel d&rsquo;iconographie
+chrétienne grecque et latine</i> (1845), the <i>Iconographie des
+chapiteaux du palais ducal de Venise</i> (1857), and the <i>Manuel des
+objets de bronze et d&rsquo;orfèvrerie</i> (1859). He died on the 13th of
+November 1867.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIDYMI,<a name="ar56" id="ar56"></a></span> or <span class="sc">Didyma</span> (mod. <i>Hieronta</i>), an ancient sanctuary
+of Apollo in Asia Minor situated in the territory of Miletus, from
+which it was distant about 10 m. S. and on the promontory
+Poseideion. It was sometimes called <i>Branchidae</i> from the name
+of its priestly caste which claimed descent from Branchus, a
+youth beloved by Apollo. As the seat of a famous oracle, the
+original temple attracted offerings from Pharaoh Necho (in whose
+army there was a contingent of Milesian mercenaries), and the
+Lydian Croesus, and was plundered by Darius of Persia. Xerxes
+finally sacked and burnt it (481 <span class="sc">b.c.</span>) and exiled the Branchidae
+to the far north-east of his empire. This exile was believed to
+be voluntary, the priests having betrayed their treasures to the
+Persian; and on this belief Alexander the Great acted 150 years
+later, when, finding the descendants of the Branchidae established
+in a city beyond the Oxus, he ordered them to be exterminated
+for the sin of their fathers (328). The celebrated cult-statue of
+Apollo by Canachus, familiar to us from reproductions on Milesian
+coins, was also carried to Persia, there to remain till restored by
+Seleucus I. in 295, and the oracle ceased to speak for a century
+and a half. The Milesians were not able to undertake the rebuilding
+till about 332 <span class="sc">b.c.</span>, when the oracle revived at the bidding
+of Alexander. The work proved too costly, and despite a special
+effort made by the Asian province nearly 400 years later, at the
+bidding of the emperor Caligula, the structure was never quite
+finished: but even as it was, Strabo ranked the Didymeum the
+greatest of Greek temples and Pliny placed it among the four
+most splendid and second only to the Artemisium at Ephesus.
+In point of fact it was a little smaller than the Samian Heraeum
+and the temple of Cybele at Sardis, and almost exactly the same
+size as the Artemisium. The area covered by the platform
+measures roughly 360 × 160 ft.</p>
+
+<p>When Cyriac of Ancona visited the spot in 1446, it seems that
+the temple was still standing in great part, although the <i>cella</i> had
+been converted into a fortress by the Byzantines: but when the
+next European visitor, the Englishman Dr Pickering, arrived
+in 1673, it had collapsed. It is conjectured that the cause was
+the great earthquake of 1493. The Society of Dilettanti sent two
+expeditions to explore the ruins, the first in 1764 under Richard
+Chandler, the second in 1812 under Sir Wm. Gell; and the French
+<span class="pagenum"><a name="page208" id="page208"></a>208</span>
+&ldquo;Rothschild Expedition&rdquo; of 1873 under MM. O. Rayet and
+A. Thomas sent a certain amount of architectural sculpture to
+the Louvre. But no excavation was attempted till MM. E.
+Pontremoli and B. Haussoullier were sent out by the French
+Schools of Rome and Athens in 1895. They cleared the western
+façade and the <i>prodomos</i>, and discovered inscriptions giving
+information about other parts which they left still buried.
+Finally the site was purchased by, and the French rights were
+ceded to, Dr Th. Wiegand, the German explorer of Miletus, who
+in 1905 began a thorough clearance of what is incomparably the
+finest temple ruin in Asia Minor.</p>
+
+<p>The temple was a decastyle peripteral structure of the Ionic
+order, standing on seven steps and possessing double rows of outer
+columns 60 ft. high, twenty-one in each row on the flanks. It
+is remarkable not only for its great size, but (<i>inter alia</i>) for (1) the
+rich ornament of its column bases, which show great variety of
+design; (2) its various developments of the Ionic capital, <i>e.g.</i>
+heads of gods, probably of Pergamene art, spring from the
+&ldquo;eyes&rdquo; of the volutes with bulls&rsquo; heads between them; (3) the
+massive building two storeys high at least, which served below
+for <i>prodomos</i>, and above for a dispensary of oracles (<span class="grk" title="chrêsmographia">&#967;&#961;&#951;&#963;&#956;&#959;&#947;&#961;&#940;&#966;&#953;&#945;</span>
+mentioned in the inscriptions) and a treasury; two flights of
+stairs called &ldquo;labyrinths&rdquo; in the inscriptions, led up to these
+chambers; (4) the pylon and staircase at the west; (5) the
+frieze of Medusa heads and foliage. Two outer columns are still
+erect on the north-east flank, carrying their entablature, and one
+of the inner order stands on the south-west. The fact that the
+temple was never finished is evident from the state in which some
+bases still remain at the west. There were probably no pedimental
+sculptures. A sacred way led from the temple to the sea
+at Panormus, which was flanked with rows of archaic statues, ten
+of which were excavated and sent to the British Museum in 1858
+by C. T. Newton. Fragments of architectural monuments, which
+once adorned this road, have also been found. Modern Hieronta
+is a large and growing Greek village, the only settlement within a
+radius of several miles. Its harbour is Kovella, distant about
+2½ m., and on the N. of the promontory.</p>
+
+<div class="condensed">
+<p>See Dilettanti Society, <i>Ionian Antiquities</i>, ii. (1821); C. T.
+Newton, <i>Hist. of Discoveries</i>, &amp;c. (1862) and <i>Travels in the Levant</i>,
+ii. (1865); O. Rayet and A. Thomas, <i>Milet et le Golfe Latmique</i>
+(1877); E. Pontremoli and B. Haussoullier, <i>Didymes</i> (1904).</p>
+</div>
+<div class="author">(D. G. H.)</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIDYMIUM<a name="ar57" id="ar57"></a></span> (from the Gr. <span class="grk" title="didymos">&#948;&#953;&#948;&#965;&#956;&#959;&#962;</span>, twin), the name given to
+the supposed element isolated by C. G. Mosander from cerite
+(1839-1841). In 1879, however, Lecoq de Boisbaudran showed
+that Mosander&rsquo;s &ldquo;didymium&rdquo; contained samarium; while the
+residual &ldquo;didymium,&rdquo; after removal of samarium, was split
+by Auer v. Welsbach (<i>Monats. f. Chemie</i>, 1885, 6, 477) into
+two components (known respectively as neodymium and
+praseodymium) by repeated fractional crystallization of the
+double nitrate of ammonium and didymium in nitric acid.
+<i>Neodymium</i> (Nd) forms the chief portion of the old &ldquo;didymium.&rdquo;
+Its salts are reddish violet in colour, and give a characteristic
+absorption spectrum. It forms oxides of composition Nd<span class="su">2</span>O<span class="su">3</span>
+and Nd<span class="su">2</span>O<span class="su">5</span>, the latter being obtained by ignition of the nitrate
+(B. Brauner). The atomic weight of neodymium is 143.6
+(B. Brauner, <i>Proc. Chem. Soc.</i>, 1897-1898, p. 70). <i>Praseodymium</i>
+(Pr) forms oxides of composition Pr<span class="su">2</span>O<span class="su">3</span>, Pr<span class="su">2</span>O<span class="su">5</span> ,xH<span class="su">2</span>O
+(B. Brauner), and Pr<span class="su">4</span>O<span class="su">7</span>. The peroxide, Pr<span class="su">4</span>O<span class="su">7</span>, forms a dark
+brown powder, and is obtained by ignition of the oxalate or
+nitrate. The sesquioxide, Pr<span class="su">2</span>O<span class="su">3</span>, is obtained as a greenish white
+mass by the reduction of the peroxide. The salts of praseodymium
+are green in colour, and give a characteristic spark spectrum.
+The atomic weight of praseodymium is 140.5.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIDYMUS<a name="ar58" id="ar58"></a></span> (?309-?394), surnamed &ldquo;the Blind,&rdquo; ecclesiastical
+writer of Alexandria, was born about the year 309. Although
+he became blind at the age of four, before he had learned to read;
+he succeeded in mastering the whole circle of the sciences then
+known; and on entering the service of the Church he was placed
+at the head of the Catechetical school in Alexandria, where he
+lived and worked till almost the close of the century. Among
+his pupils were Jerome and Rufinus. He was a loyal follower of
+Origen, though stoutly opposed to Arian and Macedonian teaching.
+Such of his writings as survive show a remarkable knowledge
+of scripture, and have distinct value as theological literature.
+Among them are the <i>De Trinitate</i>, <i>De Spiritu Sancto</i> (Jerome&rsquo;s
+Latin translation), <i>Adversus Manichaeos</i>, and notes and expositions
+of various books, especially the Psalms and the Catholic
+Epistles.</p>
+
+<div class="condensed">
+<p>See Migne, <i>Patrol. Graec.</i> xxxix.; O. Bardenhewer, <i>Patrologie</i>,
+pp. 290-293 (Freiburg, 1894).</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIDYMUS CHALCENTERUS<a name="ar59" id="ar59"></a></span> (c. 63 <span class="sc">b.c.</span>-<span class="sc">a.d.</span> 10), Greek
+scholar and grammarian, flourished in the time of Cicero and
+Augustus. His surname (Gr. <span class="grk" title="Chalkenteros">&#935;&#945;&#955;&#954;&#941;&#957;&#964;&#949;&#961;&#959;&#962;</span>, brazen-bowelled)
+came from his indefatigable industry; he was said to have
+written so many books (more than 3500) that he was unable to
+recollect their names (<span class="grk" title="bibliolathas">&#946;&#953;&#946;&#955;&#953;&#959;&#955;&#940;&#952;&#945;&#962;</span>). He lived and taught in
+Alexandria and Rome, where he became the friend of Varro.
+He is chiefly important as having introduced Alexandrian
+learning to the Romans. He was a follower of the school of
+Aristarchus, upon whose recension of Homer he wrote a treatise,
+fragments of which have been preserved in the Venetian Scholia.
+He also wrote commentaries on many other Greek poets and
+prose authors. In his work on the lyric poets he treated of the
+various classes of poetry and their chief representatives, and
+his lists of words and phrases (used in tragedy and comedy
+and by orators and historians), of words of doubtful meaning,
+and of corrupt expressions, furnished the later grammarians with
+valuable material. His activity extended to all kinds of subjects:
+grammar (orthography, inflexions), proverbs, wonderful stories,
+the law-tablets (<span class="grk" title="axones">&#7940;&#958;&#959;&#957;&#949;&#962;</span>) of Solon, stones, and different kinds of
+wood. His polemic against Cicero&rsquo;s <i>De republica</i> (Ammianus
+Marcellinus xxii. 16) provoked a reply from Suetonius. In spite
+of his stupendous industry, Didymus was little more than a
+compiler, of little critical judgment and doubtful accuracy, but
+he deserves recognition for having incorporated in his numerous
+writings the works of earlier critics and commentators.</p>
+
+<div class="condensed">
+<p>See M. W. Schmidt, <i>De Didymo Chalcentero</i> (1853) and <i>Didymi
+Chalcenteri fragmenta</i> (1854); also F. Susemihl, <i>Geschichte der griech.
+Literatur in der Alexandrinerzeit</i>, ii. (1891); J. E. Sandys, <i>History of
+Classical Scholarship</i>, i. (1906).</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIE,<a name="ar60" id="ar60"></a></span> a town of south-eastern France, capital of an arrondissement
+in the department of Drôme, 43 m. E.S.E. of Valence on the
+Paris-Lyon railway. Pop. (1906) 3090. The town is situated in a
+plain enclosed by mountains on the right bank of the Drôme
+below its confluence with the Meyrosse, which supplies power to
+some of the industries. The most interesting structures of Die
+are the old cathedral, with a porch of the 11th century supported
+on granite columns from an ancient temple of Cybele; and the
+Porte St Marcel, a Roman gateway flanked by massive towers.
+The Roman remains also include the ruins of aqueducts and altars.
+Die is the seat of a sub-prefect, and of a tribunal of first instance.
+The manufactures are silk, furniture, cloth, lime and cement, and
+there are flour and saw mills. Trade is in timber, especially
+walnut, and in white wine known as <i>clairette de Die</i>. The mulberry
+is largely grown for the rearing of silkworms. Under the Romans,
+Die (<i>Dea Augusta Vocontiorum</i>) was an important colony. It was
+formerly the seat of a bishopric, united to that of Valence from
+1276 to 1687 and suppressed in 1790. Previous to the revocation
+of the edict of Nantes in 1685 it had a Calvinistic university.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIE<a name="ar61" id="ar61"></a></span> (Fr. <i>dé</i>, from Lat. <i>datum</i>, given), a word used in various
+senses, for a small cube of ivory, &amp;c. (see <span class="sc"><a href="#artlinks">Dice</a></span>), for the engraved
+stamps used in coining money, &amp;c., and various mechanical
+appliances in engineering. In architecture a &ldquo;die&rdquo; is the term
+used for the square base of a column, and it is applied also to
+the vertical face of a pedestal or podium.</p>
+
+<table class="nobctr" summary="Illustration">
+<tr><td class="figcenter1"><img style="width:267px; height:269px" src="images/img209.jpg" alt="" /></td></tr></table>
+
+<p>The fabrics known as &ldquo;dice&rdquo; take their name from the
+rectangular form of the figure. The original figures would
+probably be perfectly square, but to-day the same principle of
+weaving is applied, and the name dice is given to all figures of
+rectangular form. The different effects in the adjacent squares or
+rectangles are due to precisely the same reasons as those explained
+in connexion with the ground and the figure of damasks. The
+same weaves are used in both damasks and dices, but simpler
+<span class="pagenum"><a name="page209" id="page209"></a>209</span>
+weaves are generally employed for the commoner classes of the
+latter. The effect is, in every case, obtained by what are technically
+called warp and weft float weaves. The illustration B shows
+the two double damask weaves
+arranged to form a dice pattern,
+while A shows a similar
+pattern made from two four-thread
+twill weaves. C and D
+represent respectively the disposition
+of the threads in A
+and B with the first pick,
+and the solid marks represent
+the floats of warp. The four
+squares, which are almost as
+pronounced in the cloth as
+those of a chess-board, may
+be made of any size by repeating
+each weave for the amount
+of surface required. It is only in the finest cloths that the double
+damask weaves B are used for dice patterns, the single damask
+weaves and the twill weaves being employed to a greater extent.
+This class of pattern is largely employed for the production of
+table-cloths of lower and medium qualities. The term damask
+is also often applied to cloths of this character, and especially so
+when the figure is formed by rectangles of different sizes.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIEBITSCH, HANS KARL FRIEDRICH ANTON,<a name="ar62" id="ar62"></a></span> count von
+Diebitsch and Narden, called by the Russians Ivan Ivanovich,
+Count Diebich-Zabalkansky (1785-1831), Russian field-marshal,
+was born in Silesia on the 13th of May 1785. He was educated
+at the Berlin cadet school, but by the desire of his father, a
+Prussian officer who had passed into the service of Russia, he also
+did the same in 1801. He served in the campaign of 1805, and
+was wounded at Austerlitz, fought at Eylau and Friedland, and
+after Friedland was promoted captain. During the next five
+years of peace he devoted himself to the study of military science,
+engaging once more in active service in the War of 1812. He
+distinguished himself very greatly in Wittgenstein&rsquo;s campaign,
+and in particular at Polotzk (October 18 and 19), after which
+combat he was raised to the rank of major-general. In the latter
+part of the campaign he served against the Prussian contingent
+of General Yorck (von Wartenburg), with whom, through
+Clausewitz, he negotiated the celebrated convention of Tauroggen,
+serving thereafter with Yorck in the early part of the War of
+Liberation. After the battle of Lützen he served in Silesia
+and took part in negotiating the secret treaty of Reichenbach.
+Having distinguished himself at the battles of Dresden and
+Leipzig he was promoted lieutenant-general. At the crisis of
+the campaign of 1814 he strongly urged the march of the allies on
+Paris; and after their entry the emperor Alexander conferred on
+him the order of St Alexander Nevsky. In 1815 he attended the
+congress of Vienna, and was afterwards made adjutant-general
+to the emperor, with whom, as also with his successor Nicholas,
+he had great influence. By Nicholas he was created baron, and
+later count. In 1820 he had become chief of the general staff,
+and in 1825 he assisted in suppressing the St Petersburg <i>émeute</i>.
+His greatest exploits were in the Russo-Turkish War of 1828-1829,
+which, after a period of doubtful contest, was decided by
+Diebitsch&rsquo;s brilliant campaign of Adrianople; this won him the
+rank of field-marshal and the honorary title of Zabalkanski
+to commemorate his crossing of the Balkans. In 1830 he was
+appointed to command the great army destined to suppress the
+insurrection in Poland. He won the terrible battle of Gróchow on
+the 25th of February, and was again victorious at Ostrolenka on
+the 26th of May, but soon afterwards he died of cholera (or by his
+own hand) at Klecksewo near Pultusk, on the 10th of June 1831.</p>
+
+<div class="condensed">
+<p>See Belmont (Schümberg), <i>Graf Diebitsch</i> (Dresden, 1830);
+Stürmer, <i>Der Tod des Grafen Diebitsch</i> (Berlin, 1832); Bantych-Kamenski,
+<i>Biographies of Russian Field-Marshals</i> (in Russian,
+St Petersburg, 1841).</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIEDENHOFEN<a name="ar63" id="ar63"></a></span> (Fr. <i>Thionville</i>), a fortified town of Germany,
+in Alsace-Lorraine, dist. Lorraine, on the Mosel, 22 m. N. from
+Metz by rail. Pop. (1905) 6047. It is a railway junction of
+some consequence, with cultivation of vines, fruit and vegetables,
+brewing, tanning, &amp;c. Diedenhofen is an ancient Frank town
+(Theudonevilla, Totonisvilla), in which imperial diets were held
+in the 8th century; was captured by Condé in 1643 and fortified
+by Vauban; capitulated to the Prussians, after a severe bombardment,
+on the 25th of November 1870.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIEKIRCH,<a name="ar64" id="ar64"></a></span> a small town in the grand duchy of Luxemburg,
+charmingly situated on the banks of the Sûre. Pop. (1905)
+3705. Its name is said to be derived from Dide or Dido, granddaughter
+of Odin and niece of Thor. The mountain at the foot of
+which the town lies, now called Herrenberg, was formerly known
+as Thorenberg, or Thor&rsquo;s mountain. On the summit of this rock
+rises a perennial stream which flows down into the town under the
+name of Bellenflesschen. Diekirch was an important Roman
+station, and in the 14th century John of Luxemburg, the blind
+king of Bohemia, fortified it, surrounding the place with a
+castellated wall and a ditch supplied by the stream mentioned.
+It remained more <span class="correction" title="amended from for">or</span> less fortified until the beginning of the 19th
+century when the French during their occupation levelled the old
+walls, and substituted the avenues of trees that now encircle the
+town. Diekirch is the administrative centre of one of the three
+provincial divisions of the grand duchy. It is visited during the
+summer by many thousand tourists and travellers from Holland,
+Belgium and Germany.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIELECTRIC,<a name="ar65" id="ar65"></a></span> in electricity, a non-conductor of electricity; it
+is the same as insulator. The &ldquo;dielectric constant&rdquo; of a medium
+is its specific inductive capacity, and on the electromagnetic
+theory of light it equals the square of its refractive index for light
+of infinite wave length (see <span class="sc"><a href="#artlinks">Electrostatics</a></span>; <span class="sc"><a href="#artlinks">Magneto-Optics</a></span>).</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIELMANN, FREDERICK<a name="ar66" id="ar66"></a></span> (1847-&emsp;&emsp;), American portrait
+and figure painter, was born at Hanover, Germany, on the 25th
+of December 1847. He was taken to the United States in
+early childhood; studied under Diez at the Royal Academy at
+Munich; was first an illustrator, and became a distinguished
+draughtsman and painter of genre pictures. His mural decorations
+and mosaic panels for the Congressional library, Washington,
+are notable. He was elected in 1899 president of the National
+Academy of Design.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIEMEN, ANTHONY VAN<a name="ar67" id="ar67"></a></span> (1593-1645), Dutch admiral and
+governor-general of the East Indian settlements, was born at
+Kuilenburg in 1593. He was educated in commerce, and on
+entering the service of the East India Company speedily attained
+high rank. In 1631 he led a Dutch fleet from the Indies to
+Holland, and in 1636 he was raised to the governor-generalship.
+He came into conflict with the Portuguese, and took their
+possessions in Ceylon and Malacca from them. He greatly
+extended the commercial relationships of the Dutch, opening up
+trade with Tong-king, China and Japan. As an administrator
+also he showed ability, and the foundation of a Latin school and
+several churches in Batavia is to be ascribed to him. Exploring
+expeditions were sent to Australia under his auspices in 1636 and
+1642, and Abel Tasman named after him (Van Diemen&rsquo;s Land)
+the island now called Tasmania. Van Diemen died at Batavia on
+the 19th of April 1645.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIEPENBECK, ABRAHAM VAN<a name="ar68" id="ar68"></a></span> (1599-1675), Flemish
+painter, was born at Herzogenbusch, and studied painting at
+Antwerp, where he became one of Rubens&rsquo;s &ldquo;hundred pupils.&rdquo;
+But he was not one of the cleverest of Rubens&rsquo;s followers, and
+he succeeded, at the best, in imitating the style and aping the
+peculiarities of his master. We see this in his earliest pictures&mdash;a
+portrait dated 1629 in the Munich Pinakothek, and a &ldquo;Distribution
+of Alms&rdquo; of the same period in the same collection. Yet even
+at this time there were moments when Diepenbeck probably
+fancied that he might take another path. A solitary copperplate
+executed with his own hand in 1630 represents a peasant sitting
+under a tree holding the bridle of an ass, and this is a minute and
+finished specimen of the engraver&rsquo;s art which shows that the
+master might at one time have hoped to rival the animal draughtsmen
+who flourished in the schools of Holland. However, large
+commissions now poured in upon him; he was asked for altarpieces,
+subject-pieces and pagan allegories. He was tempted to
+try the profession of a glass-painter, and at last he gave up every
+<span class="pagenum"><a name="page210" id="page210"></a>210</span>
+other occupation for the lucrative business of a draughtsman and
+designer for engravings. Most of Diepenbeck&rsquo;s important canvases
+are in continental galleries. The best are the &ldquo;Marriage of
+St Catherine&rdquo; at Berlin and &ldquo;Mary with Angels Wailing over the
+Dead Body of Christ&rdquo; in the Belvedere at Vienna, the first a very
+fair specimen of the artist&rsquo;s skill, the second a picture of more
+energy and feeling than might be expected from one who knew
+more of the outer form than of the spirit of Rubens. Then we
+have the fine &ldquo;Entombment&rdquo; at Brunswick, and &ldquo;St Francis Adoring
+the Sacrament&rdquo; at the museum at Brussels, &ldquo;Clelia and her
+Nymphs Flying from the Presence and Pursuit of Porsenna&rdquo; in
+two examples at Berlin and Paris, and &ldquo;Neptune and Amphitrite&rdquo;
+at Dresden. In all these compositions the drawing and execution
+are after the fashion of Rubens, though inferior to Rubens in
+harmony of tone and force of contrasted light and shade. Occasionally
+a tendency may be observed to imitate the style of Vandyck,
+for whom, in respect of pictures, Diepenbeck in his lifetime
+was frequently taken. But Diepenbeck spent much less of his
+leisure on canvases than on glass-painting. Though he failed to
+master the secrets of gorgeous tinting, which were lost, apparently
+for ever in the 16th century, he was constantly employed during
+the best years of his life in that branch of his profession. In 1635
+he finished forty scenes from the life of St Francis of Paula in the
+church of the Minimes at Antwerp. In 1644 he received payment
+for four windows in St Jacques of Antwerp, two of which are still
+preserved, and represent Virgins to whom Christ appears after
+the Resurrection. The windows ascribed to him at St Gudule
+of Brussels were executed from the cartoons of Theodore van
+Thulden. On the occasion of his matriculation at Antwerp in
+1638-1639, Diepenbeck was registered in the guild of St Luke as a
+glass-painter. He resigned his membership in the Artist Club of
+the Violette in 1542, apparently because he felt hurt by a valuation
+then made of drawings furnished for copperplates to the
+engraver Pieter de Jode. The earliest record of his residence at
+Antwerp is that of his election to the brotherhood (Sodalität)
+&ldquo;of the Bachelors&rdquo; in 1634. It is probable that before this time
+he had visited Rome and London, as noted in the work of
+Houbraken. In 1636 he was made a burgess of Antwerp. He
+married twice, in 1637 and 1652. He died in December 1675, and
+was buried at St Jacques of Antwerp.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIEPPE,<a name="ar69" id="ar69"></a></span> a seaport of northern France, capital of an arrondissement
+in the department of Seine-Inférieure, on the English
+Channel, 38 m. N. of Rouen, and 105 m. N.W. of Paris by the
+Western railway. Pop. (1906) 22,120. It is situated at the
+mouth of the river Arques in a valley bordered on each side
+by steep white cliffs. The main part of the town lies to the west,
+and the fishing suburb of Le Pollet to the east of the river and
+harbour. The sea-front of Dieppe, which in summer attracts
+large numbers of visitors, consists of a pebbly beach backed by a
+handsome marine promenade. Dieppe has a modern aspect; its
+streets are wide and its houses, in most cases, are built of brick.
+Two squares side by side and immediately to the west of the outer
+harbour form the nucleus of the town, the Place Nationale, overlooked
+by the statue of Admiral A. Duquesne, and the Place St
+Jacques, named after the beautiful Gothic church which stands
+in its centre. The Grande Rue, the busiest and handsomest
+street, leads westward from the Place Nationale. The church
+of St Jacques was founded in the 13th century, but consists in
+large measure of later workmanship and was in some portions
+restored in the 19th century. The castle, overlooking the beach
+from the summit of the western cliff, was erected in 1435. The
+church of Notre-Dame de Bon Secours on the opposite cliff, and
+the church of St Remy, of the 16th and 17th centuries, are other
+noteworthy buildings. A well-equipped casino stands at the
+west end of the sea-front. The public institutions include the subprefecture,
+tribunals of first instance and commerce, a chamber
+of commerce, a communal college and a school of navigation.</p>
+
+<p>Dieppe has one of the safest and deepest harbours on the
+English Channel. A curved passage cut in the bed of the Arques
+and protected by an eastern and a western jetty gives access to
+the outer harbour, which communicates at the east end by a lockgate
+with the Bassin Duquesne and the Bassin Bérigny, and at
+the west end by the New Channel, with an inner tidal harbour
+and two other basins. Vessels drawing 20 ft. can enter the new
+docks at neap tide. A dry-dock and a gridiron are included
+among the repairing facilities of the port. The harbour railway
+station is on the north-west quay of the outer harbour alongside
+which the steamers from Newhaven lie. The distance of Dieppe
+from Newhaven, with which there has long been daily communication,
+is 64 m. The imports include silk and cotton goods, thread,
+oil-seeds, timber, coal and mineral oil; leading exports are wine,
+silk, woollen and cotton fabrics, vegetables and fruit and flint-pebbles.
+The average annual value of imports for the five years
+1901-1905 was £4,916,000 (£4,301,000 for the years 1896-1900);
+the exports were valued at £9,206,000 (£7,023,000 for years
+1896-1900). The industries comprise shipbuilding, cotton-spinning,
+steam-sawing, the manufacture of machinery, porcelain,
+briquettes, lace, and articles in ivory and bone, the production
+of which dates from the 15th century. There is also a tobacco
+factory of some importance. The fishermen of Le Pollet, to
+whom tradition ascribes a Venetian origin, are among the main
+providers of the Parisian market. The sea-bathing attracts
+many visitors in the summer. Two miles to the north-east of
+the town is the ancient camp known as the Cité de Limes, which
+perhaps furnished the nucleus of the population of Dieppe.</p>
+
+<p>It is suggested on the authority of its name, that Dieppe owed
+its origin to a band of Norman adventurers, who found its &ldquo;diep&rdquo;
+or inlet suitable for their ships, but it was unimportant till the
+latter half of the 12th century. Its first castle was probably built
+in 1188 by Henry II. of England, and it was counted a place of
+some consideration when Philip Augustus attacked it in 1195.
+By Richard I. of England it was bestowed in 1197 on the archbishop
+of Rouen in return for certain territory in the neighbourhood
+of the episcopal city. In 1339 it was plundered by the
+English, but it soon recovered from the blow, and in spite of the
+opposition of the lords of Hantot managed to surround itself with
+fortifications. Its commercial activity was already great, and it
+is believed that its seamen visited the coast of Guinea in 1339,
+and founded there a Petit Dieppe in 1365. The town was
+occupied by the English from 1420 to 1435. A siege undertaken
+in 1442 by John Talbot, first earl of Shrewsbury, was raised by
+the dauphin, afterwards Louis XI., and the day of the deliverance
+continued for centuries to be celebrated by a great procession
+and miracle plays. In the beginning of the 16th century Jean
+Parmentier, a native of the town, made voyages to Brazil and
+Sumatra; and a little later its merchant prince, Jacques Ango,
+was able to blockade the Portuguese fleet in the Tagus. Francis
+I. began improvements which were continued under his successor.
+Its inhabitants in great number embraced the reformed religion;
+and they were among the first to acknowledge Henry IV., who
+fought one of his great battles at the neighbouring village of
+Arques. Few of the cities of France suffered more from the
+revocation of the edict of Nantes in 1685; and this blow was
+followed in 1694 by a terrible bombardment on the part of the
+English and Dutch. The town was rebuilt after the peace of
+Ryswick, but the decrease of its population and the deterioration
+of its port prevented the restoration of its commercial prosperity.
+During the 19th century it made rapid advances, partly owing to
+Marie Caroline, duchess of Berry, who brought it into fashion as a
+watering-place; and also because the establishment of railway
+communication with Paris gave an impetus to its trade. During
+the Franco-German War the town was occupied by the Germans
+from December 1870 till July 1871.</p>
+
+<div class="condensed">
+<p>See L. Vitet, <i>Histoire de Dieppe</i> (Paris, 1844); D. Asseline, <i>Les
+Antiquités et chroniques de la ville de Dieppe</i>, a 17th-century account
+published at Paris in 1874.</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIERX, LÉON<a name="ar70" id="ar70"></a></span> (1838-&emsp;&emsp;), French poet, was born in the
+island of Réunion in 1838. He came to Paris to study at the
+Central School of Arts and Manufactures, and subsequently
+settled there, taking up a post in the education office. He
+became a disciple of Leconte de Lisle and one of the most
+distinguished of the Parnassians. In the death of Stéphane
+Mallarmé in 1898 he was acclaimed &ldquo;prince of poets&rdquo;
+by &ldquo;les jeunes.&rdquo; His works include: <i>Poèmes et poésies</i> (1864);
+<span class="pagenum"><a name="page211" id="page211"></a>211</span>
+<i>Lèvres closes</i> (1867); <i>Paroles d&rsquo;un vaincu</i> (1871); <i>La Rencontre</i>, a
+dramatic scene (1875) and <i>Les Amants</i> (1879). His <i>Poésies
+complètes</i> (1872) were crowned by the French Academy. A complete
+edition of his works was published in 2 vols., 1894-1896.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIES, CHRISTOPH ALBERT<a name="ar71" id="ar71"></a></span> (1755-1822), German painter,
+was born at Hanover, and learned the rudiments of art in his
+native place. For one year he studied in the academy of Dusseldorf,
+and then he started at the age of twenty with thirty ducats
+in his pocket for Rome. There he lived a frugal life till 1796.
+Copying pictures, chiefly by Salvator Rosa, for a livelihood, his
+taste led him to draw and paint from nature in Tivoli, Albano
+and other picturesque places in the vicinity of Rome. Naples,
+the birthplace of his favourite master, he visited more than once
+for the same reasons. In this way he became a bold executant in
+water-colours and in oil, though he failed to acquire any originality
+of his own. Lord Bristol, who encouraged him as a copyist,
+predicted that he would be a second Salvator Rosa. But Dies
+was not of the wood which makes original artists. Besides other
+disqualifications, he had necessities which forced him to give
+up the great career of an independent painter. David, then
+composing his Horatii at Rome, wished to take him to Paris.
+But Dies had reasons for not accepting the offer. He was courting
+a young Roman whom he subsequently married. Meanwhile he
+had made the acquaintance of Volpato, for whom he executed
+numerous drawings, and this no doubt suggested the plan, which
+he afterwards carried out, of publishing, in partnership with
+Méchan, Reinhardt and Frauenholz, the series of plates known
+as the <i>Collection de vues pittoresques de l&rsquo;Italie</i>, published in
+seventy-two sheets at Nuremberg in 1799. With so many
+irons in the fire Dies naturally lost the power of concentration.
+Other causes combined to affect his talent. In 1787 he swallowed
+by mistake three-quarters of an ounce of sugar of lead. His recovery
+from this poison was slow and incomplete. He settled at
+Vienna, and lived there on the produce of his brush as a landscape
+painter, and on that of his pencil or graver as a draughtsman and
+etcher. But instead of getting better, his condition became
+worse, and he even lost the use of one of his hands. In this
+condition he turned from painting to music, and spent his leisure
+hours in the pleasures of authorship. He did not long survive,
+dying at Vienna in 1822, after long years of chronic suffering.
+From two pictures now in the Belvedere gallery, and from
+numerous engraved drawings from the neighbourhood of Tivoli,
+we gather that Dies was never destined to rise above a respectable
+mediocrity. He followed Salvator Rosa&rsquo;s example in imitating
+the manner of Claude Lorraine. But Salvator adapted the style
+of Claude, whilst Dies did no more than copy it.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIEST,<a name="ar72" id="ar72"></a></span> a small town in the province of Brabant, Belgium,
+situated on the Demer at its junction with the Bever. Pop.
+(1904) 8383. It lies about half-way between Hasselt and
+Louvain, and is still one of the five fortified places in Belgium.
+It contains many breweries, and is famous for the excellence of
+its beer.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIESTERWEG, FRIEDRICH ADOLF WILHELM<a name="ar73" id="ar73"></a></span> (1790-1866),
+German educationist, was born at Siegen on the 29th of October
+1790. Educated at Herborn and Tübingen universities, he took
+to the profession of teaching in 1811. In 1820 he was appointed
+director of the new school at Mörs, where he put in practice the
+methods of Pestalozzi. In 1832 he was summoned to Berlin to
+direct the new state-schools seminary in that city. Here he
+proved himself a strong supporter of unsectarian religious teaching.
+In 1846 he established the Pestalozzi institution at Pankow,
+and the Pestalozzi societies for the support of teachers&rsquo; widows
+and orphans. In 1850 he retired on a pension, but continued
+vigorously to advocate his educational views. In 1858 he was
+elected to the chamber of deputies as member for the city of
+Berlin, and voted with the Liberal opposition. He died in Berlin
+on the 7th of July 1866. Diesterweg was a voluminous writer
+on educational subjects, and was the author of various school
+text-books.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIET,<a name="ar74" id="ar74"></a></span> a term used in two senses, (1) food or the regulation
+of feeding (see <span class="sc"><a href="#artlinks">Dietary</a></span> and <span class="sc"><a href="#artlinks">Dietetics</a></span>), (2) an assembly
+or council (Fr. <i>diète</i>; It. <i>dieta</i>; Low Lat. <i>diaeta</i>; Ger. <i>Tag</i>).
+We are here concerned only with this second sense. In
+modern usage, though in Scotland the term is still sometimes
+applied to any assembly or session, it is practically confined to
+the sense of an assembly of estates or of national or federal
+representatives. The origin of the word in this connotation is
+somewhat complicated. It is undoubtedly ultimately derived
+from the Greek <span class="grk" title="diaita">&#948;&#943;&#945;&#953;&#964;&#945;</span> (Lat. <i>diaeta</i>), which meant &ldquo;mode of
+life&rdquo; and thence &ldquo;prescribed mode of life,&rdquo; the English &ldquo;diet&rdquo;
+or &ldquo;regimen.&rdquo; This was connected with the verb <span class="grk" title="diaitan">&#948;&#953;&#945;&#953;&#964;&#8118;&#957;</span>, in
+the sense of &ldquo;to rule,&rdquo; &ldquo;to regulate&rdquo;; compare the office of
+<span class="grk" title="diaitêtês">&#948;&#953;&#945;&#953;&#964;&#951;&#964;&#942;&#962;</span> at Athens, and <i>dieteta</i>, &ldquo;umpire,&rdquo; in Late Latin.
+In both Greek and Latin, too, the word meant &ldquo;a room,&rdquo; from
+which the transition to &ldquo;a place of assembly&rdquo; and so to &ldquo;an
+assembly&rdquo; would be easy. In the latter sense the word, however,
+actually occurs only in Low Latin, Du Cange (<i>Glossarium</i>, <i>s.v.</i>)
+deriving it from the late sense of &ldquo;meal&rdquo; or &ldquo;feast,&rdquo; the Germans
+being accustomed to combine their political assemblies with
+feasting. It is clear, too, that the word <i>diaeta</i> early became
+confused with Lat. <i>dies</i>, &ldquo;day&rdquo; (Ger. <i>Tag</i>), &ldquo;especially a set
+day, a day appointed for public business; whence, by extension,
+meeting for business, an assembly&rdquo; (Skeat). Instances of this
+confusion are given by Du Cange, <i>e.g.</i> <i>diaeta</i> for <i>dieta</i>, &ldquo;a day&rsquo;s
+journey&rdquo; (also an obsolete sense of &ldquo;diet&rdquo; in English), and
+<i>dieta</i> for &ldquo;the ordinary course of the church,&rdquo; <i>i.e.</i> &ldquo;the daily
+office,&rdquo; which suggests the original sense of <i>diaeta</i> as &ldquo;a prescribed
+mode of life.&rdquo;</p>
+
+<p>The word &ldquo;diet&rdquo; is now used in English for the <i>Reichstag</i>,
+&ldquo;imperial diet&rdquo; of the old Holy Roman Empire; for the
+<i>Bundestag</i>, &ldquo;federal diet,&rdquo; of the former Germanic confederation;
+sometimes for the <i>Reichstag</i> of the modern German empire; for
+the <i>Landtage</i>, &ldquo;territorial diets&rdquo; of the constituent states of the
+German and Austrian empires; as well as for the former or
+existing federal or national assemblies of Switzerland, Hungary,
+Poland, &amp;c. Although, however, the word is still sometimes used
+of all the above, the tendency is to confine it, so far as contemporary
+assemblies are concerned, to those of subordinate
+importance. Thus &ldquo;parliament&rdquo; is often used of the German
+<i>Reichstag</i> or of the Russian Landtag, while the <i>Landtag</i>, <i>e.g.</i> of
+Styria, would always be rendered &ldquo;diet.&rdquo; In what follows we
+confine ourselves to the diet of the Holy Roman Empire and its
+relation to its successors in modern Germany.</p>
+
+<p>The origin of the diet, or deliberative assembly, of the Holy
+Roman Empire must be sought in the <i>placitum</i> of the Frankish
+empire. This represented the tribal assembly of the Franks,
+meeting (originally in March, but after 755 in May, whence it is
+called the Campus Maii) partly for a military review on the eve
+of the summer campaign, partly for deliberation on important
+matters of politics and justice. By the side of this larger
+assembly, however, which contained in theory, if not in practice,
+the whole body of Franks available for war, there had developed,
+even before Carolingian times, a smaller body composed of the
+magnates of the Empire, both lay and ecclesiastical. The germ
+of this smaller body is to be found in the episcopal synods, which,
+afforced by the attendance of lay magnates, came to be used
+by the king for the settlement of national affairs. Under the
+Carolingians it was usual to combine the assembly of magnates
+with the <i>generalis conventus</i> of the &ldquo;field of May,&rdquo; and it was
+in this inner assembly, rather than in the general body (whose
+approval was merely formal, and confined to matters momentous
+enough to be referred to a general vote), that the centre of power
+really lay. It is from the assembly of magnates that the diet
+of medieval Germany springs. The general assembly became
+meaningless and unnecessary, as the feudal array gradually
+superseded the old levy <i>en masse</i>, in which each freeman had
+been liable to service; and after the close of the 10th century
+it no longer existed.</p>
+
+<p>The imperial diet (<i>Reichstag</i>) of the middle ages might sometimes
+contain representatives of Italy, the <i>regnum Italicum</i>; but
+it was practically always confined to the magnates of Germany,
+the <i>regnum Teutonicum</i>. Upon occasion a summons to the diet
+might be sent even to the knights, but the regular members were
+the princes (<i>Fürsten</i>), both lay and ecclesiastical. In the 13th
+<span class="pagenum"><a name="page212" id="page212"></a>212</span>
+century the seven electors began to disengage themselves from
+the prince as a separate element, and the Golden Bull (1356)
+made their separation complete; from the 14th century onwards
+the nobles (both counts and other lords) are regarded as regular
+members; while after 1250 the imperial and episcopal towns
+often appear through their representatives. By the 14th century,
+therefore, the originally homogeneous diet of princes is already,
+at any rate practically if not yet in legal form, divided into three
+colleges&mdash;the electors, the princes and nobles, and the representatives
+of the towns (though, as we shall see, the latter can
+hardly be reckoned as regular members until the century of the
+Reformation). Under the Hohenstaufen it is still the rule that
+every member of the diet must attend personally, or lose his vote;
+at a later date the principle of representation by proxy, which
+eventually made the diet into a mere congress of envoys, was
+introduced. By the end of the 13th century the vote of the
+majority had come to be regarded as decisive; but in accordance
+with the strong sense of social distinctions which marks German
+history, the quality as well as the quantity of votes was weighed,
+and if the most powerful of the princes were agreed, the opinion
+of the lesser magnates was not consulted. The powers of the
+medieval diet extended to matters like legislation, the decision
+upon expeditions (especially the <i>expeditio Romana</i>), taxation and
+changes in the constitution of the principalities or the Empire.
+The election of the king, which was originally regarded as one of
+the powers of the diet, had passed to the electors by the middle
+of the 13th century.</p>
+
+<p>A new era in the history of the diet begins with the Reformation.
+The division of the diet into three colleges becomes definite
+and precise; the right of the electors, for instance, to constitute
+a separate college is explicitly recognized as a matter of established
+custom in 1544. The representatives of the towns now become
+regular members. In the 15th century they had only attended
+when special business, such as imperial reform or taxation, fell
+under discussion; in 1500, however, they were recognized as a
+separate and regular estate, though it was not until 1648 that
+they were recognized as equal to the other estates of the diet.
+The estate of the towns, or college of municipal representatives,
+was divided into two benches, the Rhenish and the Swabian.
+The estate of the princes and counts, which stood midway
+between the electors and the towns, also attained, in the years
+that followed the Reformation, its final organization. The vote
+of the great princes ceased to be personal, and began to be
+territorial. This had two results. The division of a single
+territory among the different sons of a family no longer, as of old,
+multiplied the voting power of the family; while in the opposite
+case, the union of various territories in the hands of a single
+person no longer meant the extinction of several votes, since the
+new owner was now allowed to give a vote for each of his territories.
+The position of the counts and other lords, who joined
+with the princes in forming the middle estate, was finally fixed
+by the middle of the 17th century. While each of the princes
+enjoyed an individual vote, the counts and other lords were
+arranged in groups, each of which voted as a whole, though the
+whole of its vote (<i>Kuriatstimme</i>) only counted as equal to the
+vote of a single prince (<i>Virilstimme</i>). There were six of these
+groups; but as the votes of the whole college of princes and
+counts (at any rate in the 18th century) numbered 100, they
+could exercise but little weight.</p>
+
+<p>The last era in the history of the diet may be said to open with
+the treaty of Westphalia (1648). The treaty acknowledged that
+Germany was no longer a unitary state, but a loose confederation
+of sovereign princes; and the diet accordingly ceased to bear the
+character of a national assembly, and became a mere congress of
+envoys. The &ldquo;last diet&rdquo; which issued a regular recess (<i>Reichsabschied</i>&mdash;the
+term applied to the <i>acta</i> of the diet, as formally
+compiled and enunciated at its dissolution) was that of Regensburg
+in 1654. The next diet, which met at Regensburg in 1663,
+never issued a recess, and was never dissolved; it continued in
+permanent session, as it were, till the dissolution of the Empire
+in 1806. This result was achieved by the process of turning the
+diet from an assembly of principals into a congress of envoys.
+The emperor was represented by two <i>commissarii</i>; the electors,
+princes and towns were similarly represented by their accredited
+agents. Some legislation was occasionally done by this body; a
+<i>conclusum imperii</i> (so called in distinction from the old <i>recessus
+imperii</i> of the period before 1663) might slowly (very slowly&mdash;for
+the agents, imperfectly instructed, had constantly to refer
+matters back to their principals) be achieved; but it rested with
+the various princes to promulgate and enforce the <i>conclusum</i> in
+their territories, and they were sufficiently occupied in issuing
+and enforcing their own decrees. In practice the diet had
+nothing to do; and its members occupied themselves in
+&ldquo;wrangling about chairs&rdquo;&mdash;that is to say, in unending disputes
+about degrees and precedences.</p>
+
+<p>In the Germanic Confederation, which occupies the interval
+between the death of the Holy Roman Empire and the formation
+of the North German Confederation (1815-1866), a diet
+(<i>Bundestag</i>) existed, which was modelled on the old diet of the 18th
+century. It was a standing congress of envoys at Frankfort-on-Main.
+Austria presided in the diet, which, in the earlier years of
+its history, served, under the influence of Metternich, as an organ
+for the suppression of Liberal opinion. In the North German
+Confederation (1867-1870) a new departure was made, which has
+been followed in the constitution of the present German empire.
+Two bodies were instituted&mdash;a <i>Bundesrat</i>, which resembles the old
+diet in being a congress of envoys sent by the sovereigns of the
+different states of the confederation, and a <i>Reichstag</i>, which bears
+the name of the old diet, but differs entirely in composition. The
+new Reichstag is a popular representative assembly, based on
+wide suffrage and elected by ballot; and, above all, it is an
+assembly representing, not the several states, but the whole
+Empire, which is divided for this purpose into electoral districts.
+Both as a popular assembly, and as an assembly which represents
+the whole of a united Germany, the new Reichstag goes back, one
+may almost say, beyond the diet even of the middle ages, to the
+days of the old Teutonic folk-moot.</p>
+
+<div class="condensed">
+<p>See R. Schröder, <i>Lehrbuch der deutschen Rechtsgeschichte</i> (1902),
+pp. 149, 508, 820, 880. Schröder gives a bibliography of monographs
+bearing on the history of the medieval diet.</p>
+</div>
+<div class="author">(E. Br.)</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIETARY,<a name="ar75" id="ar75"></a></span> in a general sense, a system or course of diet, in the
+sense of food; more particularly, such an allowance and regulation
+of food as that supplied to workhouses, the army and navy,
+prisons, &amp;c. Lowest in the scale of such dietaries comes what
+is termed &ldquo;bare existence&rdquo; diet, administered to certain classes
+of the community who have a claim on their fellow-countrymen
+that their lives and health shall be preserved <i>in statu quo</i>, but
+nothing further. This applies particularly to the members of
+a temporarily famine-stricken community. Before the days of
+prison reform, too, the dietary scale of many prisons was to
+a certain extent penal, in that the food supplied to prisoners
+was barely sufficient for existence. Nowadays more humane
+principles apply; there is no longer the obvious injustice of
+applying the same scale of quantity and quality to all prisoners
+under varying circumstances of constitution and surroundings,
+and whether serving long or short periods of imprisonment.</p>
+
+<div class="condensed">
+<p>The system of dietary in force in the local and convict prisons of
+England and Wales is that recommended by the Home Office on the
+advice of a departmental committee. As to the local prison dietary,
+its application is based on (1) the principle of variation of diet with
+length of sentence; (2) the system of progressive dietary; (3) the
+distinction between hard labour diets and non-hard labour diets;
+(4) the differentiation of diet according to age and sex. There are
+three classes of diet, classes A, B and C. Class A diet is given
+to prisoners undergoing not more than seven days&rsquo; imprisonment.
+The food is good and wholesome, but sufficiently plain and unattractive,
+so as not to offer temptation to the loafer or mendicant.
+It is given in quantity sufficient to maintain health and strength
+during the single week. Prisoners sentenced to more than seven days
+and not more than fourteen days are given class A diet for the first
+seven days and class B for the remainder of the sentence. In most
+of the local prisons in England and Wales prisoners sentenced to
+hard labour received hard labour diet, although quite 60% were
+unable to perform the hardest forms of prison labour either through
+physical defect, age or infirmity. The departmental committee
+of 1899 in their report recommended that no distinction should be
+made between hard labour and non-hard labour diets. Class A diet
+is as follows:&mdash;<i>Breakfast</i>, Bread, 8 oz. daily (6 oz. for women and
+juveniles) with 1 pint of gruel. Juveniles (males and females under
+<span class="pagenum"><a name="page213" id="page213"></a>213</span>
+sixteen years of age) get, in addition, ½ pint of milk. <i>Dinner</i>, 8 oz. of
+bread daily, with 1 pint of porridge on three days of the week, 8 oz.
+of potatoes (representing the vegetable element) on two other days,
+and 8 oz. of suet pudding (representing the fatty element) on the
+other two days. <i>Supper</i>, the breakfast fare repeated.</p>
+
+<p>Class B diet, which is also given to (1) prisoners on remand or
+awaiting trial, (2) offenders of the 1st division who do not maintain
+themselves, (3) offenders of the 2nd division and (4) debtors, is as
+shown in Table I.</p>
+
+
+
+<p class="center pt2 sc">Table I.</p>
+
+<table class="ws" summary="Contents">
+
+<tr><td class="tcc allb">&nbsp;</td> <td class="tcc allb">&nbsp;</td> <td class="tcc allb">Men.</td> <td class="tcc allb">Women.</td> <td class="tcc allb">Juveniles.</td></tr>
+
+<tr><td class="tcl lb rb" rowspan="5">Breakfast.</td> <td class="tcl rb">Daily:&mdash;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td></tr>
+<tr><td class="tcl rb">&emsp;Bread</td> <td class="tcc rb">8 oz.</td> <td class="tcc rb">6 oz.</td> <td class="tcc rb">6 oz.</td></tr>
+<tr><td class="tcl rb">&emsp;Gruel</td> <td class="tcc rb">1 pt.</td> <td class="tcc rb">1 pt.</td> <td class="tcc rb">1 pt.</td></tr>
+<tr><td class="tcl rb">&emsp;Milk</td> <td class="tcc rb">· ·</td> <td class="tcc rb bb">· ·</td> <td class="tcc rb bb">½ pt.</td></tr>
+<tr><td class="tcl rb">&ensp;</td> <td class="tcc rb">&ensp;</td> <td class="tcc rb" colspan="2">&ensp;</td></tr>
+<tr><td class="tcl lb rb" rowspan="36">Dinner.</td> <td class="tcl rb">Sunday:&mdash;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb" colspan="2">&ensp;</td></tr>
+<tr><td class="tcl rb">&emsp;Bread</td> <td class="tcc rb">6 oz.</td> <td class="tcc rb" colspan="2">6 oz.</td></tr>
+<tr><td class="tcl rb">&emsp;Potatoes</td> <td class="tcc rb">8&emsp;&rdquo;</td> <td class="tcc rb" colspan="2">8&emsp;&rdquo;</td></tr>
+<tr><td class="tcl rb">&emsp;Cooked meat, preserved by heat</td> <td class="tcc rb">4&emsp;&rdquo;</td> <td class="tcc rb" colspan="2">3&emsp;&rdquo;</td></tr>
+<tr><td class="tcl rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb" colspan="2">&nbsp;</td></tr>
+<tr><td class="tcl rb">Monday:&mdash;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb" colspan="2">&nbsp;</td></tr>
+<tr><td class="tcl rb">&emsp;Bread</td> <td class="tcc rb">6 oz.</td> <td class="tcc rb" colspan="2">6 oz.</td></tr>
+<tr><td class="tcl rb">&emsp;Potatoes</td> <td class="tcc rb">8&emsp;&rdquo;</td> <td class="tcc rb" colspan="2">8&emsp;&rdquo;</td></tr>
+<tr><td class="tcl rb">&emsp;Beans</td> <td class="tcc rb">10&emsp;&rdquo;</td> <td class="tcc rb" colspan="2">8&emsp;&rdquo;</td></tr>
+<tr><td class="tcl rb">&emsp;Fat bacon</td> <td class="tcc rb">2&emsp;&rdquo;</td> <td class="tcc rb" colspan="2">1&emsp;&rdquo;</td></tr>
+<tr><td class="tcl rb">&ensp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb" colspan="2">&nbsp;</td></tr>
+<tr><td class="tcl rb">Tuesday:&mdash;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb" colspan="2">&nbsp;</td></tr>
+<tr><td class="tcl rb">&emsp;Bread</td> <td class="tcc rb">6 oz</td> <td class="tcc rb" colspan="2">6 oz.</td></tr>
+<tr><td class="tcl rb">&emsp;Potatoes</td> <td class="tcc rb">8&emsp;&rdquo;</td> <td class="tcc rb" colspan="2">8&emsp;&rdquo;</td></tr>
+<tr><td class="tcl rb">&emsp;Soup</td> <td class="tcc rb">1 pt.</td> <td class="tcc rb" colspan="2">1 pt.</td></tr>
+<tr><td class="tcl rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb" colspan="2">&nbsp;</td></tr>
+<tr><td class="tcl rb">Wednesday:&mdash;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb" colspan="2">&nbsp;</td></tr>
+<tr><td class="tcl rb">&emsp;Bread</td> <td class="tcc rb">6 oz.</td> <td class="tcc rb" colspan="2">6 oz.</td></tr>
+<tr><td class="tcl rb">&emsp;Potatoes</td> <td class="tcc rb">8&emsp;&rdquo;</td> <td class="tcc rb" colspan="2">8&emsp;&rdquo;</td></tr>
+<tr><td class="tcl rb">&emsp;Suet pudding</td> <td class="tcc rb">10&emsp;&rdquo;</td> <td class="tcc rb" colspan="2">8&emsp;&rdquo;</td></tr>
+<tr><td class="tcl rb">&ensp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb" colspan="2">&nbsp;</td></tr>
+<tr><td class="tcl rb">Thursday:&mdash;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb" colspan="2">&nbsp;</td></tr>
+<tr><td class="tcl rb">&emsp;Bread</td> <td class="tcc rb">6 oz.</td> <td class="tcc rb" colspan="2">6 oz.</td></tr>
+<tr><td class="tcl rb">&emsp;Potatoes</td> <td class="tcc rb">8&emsp;&rdquo;</td> <td class="tcc rb" colspan="2">8&emsp;&rdquo;</td></tr>
+<tr><td class="tcl rb">&emsp;Cooked beef, without bone</td> <td class="tcc rb">4&emsp;&rdquo;</td> <td class="tcc rb" colspan="2">3&emsp;&rdquo;</td></tr>
+<tr><td class="tcl rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb" colspan="2">&nbsp;</td></tr>
+<tr><td class="tcl rb">Friday:&mdash;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb" colspan="2">&nbsp;</td></tr>
+<tr><td class="tcl rb">&emsp;Bread</td> <td class="tcc rb">6 oz.</td> <td class="tcc rb" colspan="2">6 oz.</td></tr>
+<tr><td class="tcl rb">&emsp;Potatoes</td> <td class="tcc rb">8&emsp;&rdquo;</td> <td class="tcc rb" colspan="2">8&emsp;&rdquo;</td></tr>
+<tr><td class="tcl rb">&emsp;Soup</td> <td class="tcc rb">1 pt.</td> <td class="tcc rb" colspan="2">1 pt.</td></tr>
+<tr><td class="tcl rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb" colspan="2">&nbsp;</td></tr>
+<tr><td class="tcl rb">Saturday:&mdash;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb" colspan="2">&nbsp;</td></tr>
+<tr><td class="tcl rb">&emsp;Bread</td> <td class="tcc rb">6 oz.</td> <td class="tcc rb" colspan="2">6 oz.</td></tr>
+<tr><td class="tcl rb">&emsp;Potatoes</td> <td class="tcc rb">8&emsp;&rdquo;</td> <td class="tcc rb" colspan="2">8&emsp;&rdquo;</td></tr>
+<tr><td class="tcl rb">&emsp;Suet pudding</td> <td class="tcc rb">10&emsp;&rdquo;</td> <td class="tcc rb" colspan="2">8&emsp;&rdquo;</td></tr>
+<tr><td class="tcl rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb" colspan="2">&nbsp;</td></tr>
+<tr><td class="tcl lb rb bb" rowspan="5">Supper.</td> <td class="tcl rb">Daily:&mdash;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb tb">&nbsp;</td> <td class="tcc rb tb">&nbsp;</td></tr>
+<tr><td class="tcl rb">&emsp;Bread</td> <td class="tcc rb">8 oz.</td> <td class="tcc rb ">6 oz.</td> <td class="tcc rb ">6 oz.</td></tr>
+<tr><td class="tcl rb">&emsp;Porridge</td> <td class="tcc rb">1 pt.</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td></tr>
+<tr><td class="tcl rb">&emsp;Gruel</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">1 pt.</td> <td class="tcc rb">&nbsp;</td></tr>
+<tr><td class="tcl rb bb">&emsp;Cocoa</td> <td class="tcc rb bb">&nbsp;</td> <td class="tcc rb bb">&nbsp;</td> <td class="tcc rb bb">1 pt.</td></tr>
+</table>
+
+<p>Class C diet is class B amplified, and is given to those prisoners
+serving sentences of three months and over.</p>
+
+<p>The dietary of convict prisons, in which prisoners are all under long
+sentence, is divided into a diet for convicts employed at hard labour
+and a diet for convicts employed at sedentary, indoor and light
+labour. It will be found set forth in the Blue-book mentioned above.
+The sparest of all prison diets is called &ldquo;punishment diet,&rdquo; and is
+administered for offences against the internal discipline of the prison.
+It is limited to a period of three days. It consists of 1 &#8468; of bread
+and as much water as the prisoner chooses to drink.</p>
+
+<p>In French prisons the dietary is nearly two pounds weight of bread,
+with two meals of thin soup (breakfast and dinner) made from
+potatoes, beans or other vegetables, and on two days a week made
+from meat. In France the canteen system is in vogue, additional
+food, such as sausages, cheese, fruit, &amp;c., may be obtained by the
+prisoner, according to the wages he receives for his labours. The
+dietary of Austrian prisons is 1½ &#8468; of bread daily, a dinner of soup
+on four days of the week, and of meat on the other three days,
+with a supper of soup or vegetable stew. Additional food can be
+purchased by the prisoner out of his earnings.</p>
+
+<p>These dietaries may be taken as more or less typical of the ordinary
+prison fare in most civilized countries, though in some countries it
+may err on the side of severity, as in Sweden, prisoners being given
+only two meals a day, one at mid-day and one at seven p.m., porridge
+or gruel being the principal element in both meals. On the other
+hand, the prison dietaries of many of the United States prisons go
+to the other extreme, fresh fish, green vegetables, even coffee and
+fruit, figuring in the dietary.</p>
+
+<p>Another class of dietary is that given to paupers. In England,
+until 1900, almost every individual workhouse had its own special
+dietary, with the consequence that many erred on the side of scantiness
+and unsuitability, while others were too lavish. By an order of
+the Local Government Board of that year, acting on a report of a
+committee, all inmates of workhouses, with the exception of the sick,
+children under three years of age, and certain other special cases,
+are dieted in accordance with certain dietary tables as framed and
+settled by the board. The order contained a great number of different
+rations, it being left to the discretion of the guardians as to the final
+settlement of the tables. For adult inmates the dietary tables are
+for each sex respectively, two in number, one termed &ldquo;plain diet&rdquo;
+and the other &ldquo;infirm diet.&rdquo; All male inmates certified as healthy
+able-bodied persons receive plain diet only. All inmates, however,
+in workhouses are kept employed according to their capacity and
+ability, and this is taken into consideration in giving allowances of
+food. For instance, for work with sustained exertion, such as stone-breaking,
+digging, &amp;c., more food is given than for work without
+sustained exertion, such as wood-chopping, weeding or sewing.
+Table II. shows an example of a workhouse dietary.</p>
+
+<p class="center pt2 sc">Table II.</p>
+
+<table class="ws" summary="Contents">
+
+<tr><td class="tcc allb" colspan="3">&nbsp;</td> <td class="tcc allb">Sun.</td> <td class="tcc allb">Mon.</td> <td class="tcc allb">Tue.</td> <td class="tcc allb">Wed.</td> <td class="tcc allb">Thu.</td> <td class="tcc allb">Fri.</td> <td class="tcc allb">Sat.</td></tr>
+
+<tr><td class="tccm allb" rowspan="2">Breakfast.</td> <td class="tcl">Bread.</td> <td class="tcr rb">oz.</td> <td class="tcc rb">8</td> <td class="tcc rb">4</td> <td class="tcc rb">4</td> <td class="tcc rb">4</td> <td class="tcc rb">4</td> <td class="tcc rb">4</td> <td class="tcc rb">4</td></tr>
+<tr><td class="tcl bb">Porridge.</td> <td class="tcr rb bb">pt.</td> <td class="tcc rb bb">*</td> <td class="tcc rb bb">1½</td> <td class="tcc rb bb">1½</td> <td class="tcc rb bb">1½</td> <td class="tcc rb bb">1½</td> <td class="tcc rb bb">1½</td> <td class="tcc rb bb">1½</td></tr>
+
+<tr><td class="tccm allb" rowspan="10">Dinner.</td> <td class="tcl">Bread.</td> <td class="tcr rb">oz.</td> <td class="tcc rb">4</td> <td class="tcc rb">6</td> <td class="tcc rb">..</td> <td class="tcc rb">4</td> <td class="tcc rb">4</td> <td class="tcc rb">8</td> <td class="tcc rb">6</td></tr>
+<tr><td class="tcl">Beef.</td> <td class="tcr rb">oz.</td> <td class="tcc rb">4½</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">4½</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td></tr>
+<tr><td class="tcl">Vegetables.</td> <td class="tcr rb">oz.</td> <td class="tcc rb">12</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">12</td> <td class="tcc rb">12</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td></tr>
+<tr><td class="tcl">Barley Soup.</td> <td class="tcr rb">pt.</td> <td class="tcc rb">..</td> <td class="tcc rb">1½</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td></tr>
+<tr><td class="tcl">Pork.</td> <td class="tcr rb">oz.</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">4½</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td></tr>
+<tr><td class="tcl">Beans.</td> <td class="tcr rb">oz.</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">12</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td></tr>
+<tr><td class="tcl">Fish.</td> <td class="tcr rb">oz.</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">10</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td></tr>
+<tr><td class="tcl">Cheese.</td> <td class="tcr rb">oz.</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">3</td> <td class="tcc rb">..</td></tr>
+<tr><td class="tcl">Broth.</td> <td class="tcr rb">pt.</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">1</td> <td class="tcc rb">..</td></tr>
+<tr><td class="tcl bb">Irish Stew.</td> <td class="tcr rb bb">pt.</td> <td class="tcc rb bb">..</td> <td class="tcc rb bb">..</td> <td class="tcc rb bb">..</td> <td class="tcc rb bb">..</td> <td class="tcc rb bb">..</td> <td class="tcc rb bb">..</td> <td class="tcc rb bb">1</td></tr>
+
+<tr><td class="tccm allb" rowspan="6">Supper.</td> <td class="tcl">Bread.</td> <td class="tcr rb">oz.</td> <td class="tcc rb">8</td> <td class="tcc rb">6</td> <td class="tcc rb">6</td> <td class="tcc rb">6</td> <td class="tcc rb">8</td> <td class="tcc rb">6</td> <td class="tcc rb">6</td></tr>
+<tr><td class="tcl">Butter.</td> <td class="tcr rb">oz.</td> <td class="tcc rb">½</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td></tr>
+<tr><td class="tcl">Tea.</td> <td class="tcr rb">pt.</td> <td class="tcc rb">1</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td></tr>
+<tr><td class="tcl">Gruel.</td> <td class="tcr rb">pt.</td> <td class="tcc rb">..</td> <td class="tcc rb">1½</td> <td class="tcc rb">1½</td> <td class="tcc rb">1½</td> <td class="tcc rb">..</td> <td class="tcc rb">1½</td> <td class="tcc rb">1½</td></tr>
+<tr><td class="tcl">Broth.</td> <td class="tcr rb">pt.</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">1</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td></tr>
+<tr><td class="tcl bb">Cheese.</td> <td class="tcr rb bb">oz.</td> <td class="tcc rb bb">..</td> <td class="tcc rb bb">..</td> <td class="tcc rb bb">..</td> <td class="tcc rb bb">..</td> <td class="tcc rb bb">2</td> <td class="tcc rb bb">..</td> <td class="tcc rb bb">..</td></tr>
+
+<tr> <td class="tcc f90" colspan="10">* On Sundays 1 pint of tea and 2½ oz. of butter are given instead of porridge.</td></tr>
+</table>
+
+<p>In the casual wards of workhouses the dietary is plainer, consisting
+of 8 oz. of bread, or 6 oz. of bread and one pint of gruel or broth for
+breakfast; the same for supper; for dinner 8 oz. of bread and 1½ oz.
+of cheese or 6 oz. of bread and one pint of soup. The American poor
+law system is based broadly on that of England, and the methods
+of relief are much the same. Each state, however, makes its own
+regulations, and there is considerable diversity in workhouse dietaries
+in consequence. The German system of poor relief is more methodical
+than those of England and America. The really deserving are treated
+<span class="pagenum"><a name="page214" id="page214"></a>214</span>
+with more commiseration, and a larger amount of outdoor relief is
+given than in England. There is no casual ward, tramps and beggars
+being liable to penal treatment, but there are &ldquo;relief stations,&rdquo;
+somewhat corresponding to casual wards, where destitute persons
+tramping from one place to another can obtain food and lodging in
+return for work done.</p>
+
+<p>In the British navy certain staple articles of diet are supplied to
+the men to the value approximately of 6d. per diem&mdash;the standard
+government ration&mdash;and, in addition, a messing allowance of 4d. per
+diem, which may either be expended on luxuries in the canteen, or
+in taking up government provisions on board ship, in addition to
+the standard ration. The standard ration as recommended in 1907
+by a committee appointed to inquire into the question of victualling
+in the navy is as follows:&mdash;</p>
+
+<p class="center"><i>Service Afloat.</i></p>
+
+<table class="reg" summary="poem"><tr><td> <div class="poemr">
+<p>1 &#8468; bread (or ¾ &#8468; bread and ¼ &#8468; trade flour).</p>
+<p>½ &#8468; fresh meat.</p>
+<p>1 &#8468; fresh vegetables.</p>
+<p><span class="spp">1</span>&frasl;<span class="suu">8</span> pint spirit.</p>
+<p>4 oz. sugar.</p>
+<p>½ oz. tea (or 1 oz. coffee for every ¼ oz. tea).</p>
+<p>½ oz. ordinary or soluble chocolate (or 1 oz. coffee).</p>
+<p>¾ oz. condensed milk.</p>
+<p>1 oz. jam or marmalade.</p>
+<p>4 oz. preserved meat on <i>one</i> day of the week in harbour, or on <i>two</i> days at sea.</p>
+<p>Mustard, pepper, vinegar, and salt as required.</p>
+</div> </td></tr></table>
+
+<p>Substitute for soft bread when the latter is not available&mdash;</p>
+<p class="center">½ &#8468; biscuit (new type) or 1 &#8468; flour.</p>
+
+<p>Substitutes for fresh meat when the latter is not available:&mdash;</p>
+
+<table class="nobctr" summary="Contents">
+
+<tr><td class="tccm" style="background-color: #f5f5f5;" rowspan="11">On<br />alternate<br />days.</td> <td class="tcl" colspan="2">(1) Salt pork day:&mdash;</td></tr>
+<tr><td class="tcl" colspan="2">&emsp;&emsp;½ &#8468; salt pork.</td></tr>
+<tr><td class="tcl" colspan="2">&emsp;&emsp;¼ &#8468; split peas.</td></tr>
+<tr><td class="tcl" colspan="2">&emsp;&emsp;Celery seed, ½ oz. to every 8 &#8468; of split peas put into the coppers.</td></tr>
+<tr><td class="tcl" colspan="2">&emsp;&emsp;½ &#8468; potatoes (or 1 oz. compressed vegetables).</td></tr>
+
+<tr><td class="tcl" colspan="2">(2) Preserved meat day:&mdash;</td></tr>
+<tr><td class="tcl" colspan="2">&emsp;&emsp;6 oz. preserved meat.</td></tr>
+<tr><td class="tcl">&emsp;&emsp;8 oz. trade flour</td><td class="tcl" rowspan="3"><span style="font-size: 3em; font-family: 'Courier New'; color: #a0a0a0;">}</span> or 4 oz. rice.</td></tr>
+<tr><td class="tcl">&emsp;&emsp;¾ oz. refined suet</td></tr>
+<tr><td class="tcl">&emsp;&emsp;2 oz. raisins</td></tr>
+<tr><td class="tcl" colspan="2">&emsp;&emsp;½ &#8468; potatoes (or 1 oz. compressed vegetables).</td></tr>
+</table>
+
+<p>On shore establishments and depot ships ¼ pt. fresh milk is issued
+in lieu of the ¾ oz. of condensed milk.</p>
+
+<p>In the United States navy there is more liberality and variety of
+diet, the approximate daily cost of the rations supplied being 1s. 3d.
+per head. In the American mercantile marine, too, according to
+the scale sanctioned by act of Congress (December 21, 1898) for
+American ships, the seaman is better off than in the British merchant
+service. The scale is shown in Table III.</p>
+
+<p class="center pt2 sc">Table III.</p>
+
+<table class="ws" summary="Contents">
+
+<tr><td class="tccm allb">Weekly<br />Scale.</td> <td class="tccm lb rb2 tb bb">Articles.</td> <td class="tccm allb">Weekly<br />Scale.</td> <td class="tccm allb">Articles.</td></tr>
+
+<tr><td class="tcl lb rb">&ensp;3½  &#8468;</td> <td class="tcl rb2">Biscuits.</td> <td class="tcl rb">&emsp;<span class="spp">7</span>&frasl;<span class="suu">8</span> oz.</td> <td class="tcl rb">Tea.</td></tr>
+<tr><td class="tcl lb rb">&ensp;3¾   &rdquo;</td> <td class="tcl rb2">Salt beef.</td> <td class="tcl rb">21&emsp;&rdquo;</td> <td class="tcl rb">Sugar.</td></tr>
+<tr><td class="tcl lb rb">&ensp;3&emsp;&rdquo;</td> <td class="tcl rb2">&ldquo;    pork.</td> <td class="tcl rb">&ensp;1½  &#8468;</td> <td class="tcl rb">Molasses.</td></tr>
+<tr><td class="tcl lb rb">&ensp;1½   &rdquo;</td> <td class="tcl rb2">Flour.</td> <td class="tcl rb">&ensp;9&emsp;oz.</td> <td class="tcl rb">Fruits, dried.</td></tr>
+<tr><td class="tcl lb rb">&ensp;2&emsp;&rdquo;</td> <td class="tcl rb2">Meats, preserved.</td> <td class="tcl rb">&emsp;¾   pt.</td> <td class="tcl rb">Pickles.</td></tr>
+<tr><td class="tcl lb rb">10½  &rdquo;</td> <td class="tcl rb2">Bread, fresh (8 &#8468; flour in lieu).</td>  <td class="tcl rb">&ensp;1&emsp;&rdquo;</td> <td class="tcl rb">Vinegar.</td></tr>
+<tr><td class="tcl lb rb">&ensp;1&emsp;&rdquo;</td> <td class="tcl rb2">Fish, dried.</td> <td class="tcl rb">&ensp;8&emsp;oz.</td> <td class="tcl rb">Corn Meal.</td></tr>
+<tr><td class="tcl lb rb">&ensp;7&emsp;&rdquo;</td> <td class="tcl rb2">Potatoes or yams.</td> <td class="tcl rb">12&emsp;&rdquo;</td> <td class="tcl rb">Onions.</td></tr>
+<tr><td class="tcl lb rb">&ensp;1&emsp;&rdquo;</td> <td class="tcl rb2">Tomatoes, preserved.</td> <td class="tcl rb">&ensp;7&emsp;&rdquo;</td> <td class="tcl rb">Lard.</td></tr>
+<tr><td class="tcl lb rb">&emsp;<span class="spp">2</span>&frasl;<span class="suu">3</span> &rdquo;</td> <td class="tcl rb2">Peas.</td> <td class="tcl rb">&ensp;7&emsp;&rdquo;</td> <td class="tcl rb">Butter.</td></tr>
+<tr><td class="tcl lb rb">&emsp;<span class="spp">2</span>&frasl;<span class="suu">3</span> &rdquo;</td> <td class="tcl rb2">Calavances.</td> <td class="tcl rb">&emsp;¼   &rdquo;</td> <td class="tcl rb">Mustard.</td></tr>
+<tr><td class="tcl lb rb">&emsp;<span class="spp">2</span>&frasl;<span class="suu">3</span> &rdquo;</td> <td class="tcl rb2">Rice.</td> <td class="tcl rb">&emsp;¼   &rdquo;</td> <td class="tcl rb">Pepper.</td></tr>
+<tr><td class="tcl lb rb bb">&ensp;5¼ oz.</td> <td class="tcl rb2 bb">Coffee, green.</td> <td class="tcl rb bb">&emsp;¼   &rdquo;</td> <td class="tcl rb bb">Salt.</td></tr>
+</table>
+
+<p>In the British mercantile marine there is no scale of provisions
+prescribed by the Board of Trade; there is, however, a traditional
+scale very generally adopted, having the sanction of custom only
+and seldom adhered to. The following dietary scale for steerage
+passengers, laid down in the 12th schedule of the Merchant Shipping
+Act 1894, is of interest. See Table IV.</p>
+
+<p class="center pt2"><span class="sc">Table IV.</span>&mdash;<i>Weekly, per Statute Adult.</i></p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tccm allb">&nbsp;</td>
+
+<td class="tcl allb">&emsp;&emsp;Scale A.<br />For voyages not<br />&emsp;exceeding 84 days<br />&emsp;for sailing ships<br />
+&emsp;or 50 days<br />&emsp;for steamships.<br /></td>
+
+<td class="tcl allb">&emsp;&emsp;Scale B.<br />For voyages<br />&emsp;exceeding 84 days<br />&emsp;for sailing ships<br />
+&emsp;or 50 days<br />&emsp;for steamships.<br /></td></tr>
+
+<tr><td class="tcl lb rb">&nbsp;</td> <td class="tcc rb">&#8468;&emsp;oz.</td> <td class="tcc rb">&#8468;&emsp;oz.</td></tr>
+<tr><td class="tcl lb rb">Bread or biscuit, not inferior to navy biscuit</td> <td class="tcc rb">3&emsp;8</td> <td class="tcc rb">3&emsp;8</td></tr>
+<tr><td class="tcl lb rb">Wheaten flour</td> <td class="tcc rb">1&emsp;0</td> <td class="tcc rb">2&emsp;0</td></tr>
+<tr><td class="tcl lb rb">Oatmeal</td> <td class="tcc rb">1&emsp;8</td> <td class="tcc rb">1&emsp;0</td></tr>
+<tr><td class="tcl lb rb">Rice</td> <td class="tcc rb">1&emsp;8</td> <td class="tcc rb">0&emsp;8</td></tr>
+<tr><td class="tcl lb rb">Peas</td> <td class="tcc rb">1&emsp;8</td> <td class="tcc rb">1&emsp;8</td></tr>
+<tr><td class="tcl lb rb">Beef</td> <td class="tcc rb">1&emsp;4</td> <td class="tcc rb">1&emsp;4</td></tr>
+<tr><td class="tcl lb rb">Pork</td> <td class="tcc rb">1&emsp;0</td> <td class="tcc rb">1&emsp;0</td></tr>
+<tr><td class="tcl lb rb">Butter</td> <td class="tcc rb">· ·</td> <td class="tcc rb">0&emsp;4</td></tr>
+<tr><td class="tcl lb rb">Potatoes</td> <td class="tcc rb">2&emsp;0</td> <td class="tcc rb">2&emsp;0</td></tr>
+<tr><td class="tcl lb rb">Sugar</td> <td class="tcc rb">1&emsp;0</td> <td class="tcc rb">1&emsp;0</td></tr>
+<tr><td class="tcl lb rb">Tea</td> <td class="tcc rb">0&emsp;2</td> <td class="tcc rb">0&emsp;2</td></tr>
+<tr><td class="tcl lb rb">Salt</td> <td class="tcc rb">0&emsp;2</td> <td class="tcc rb">0&emsp;2</td></tr>
+<tr><td class="tcl lb rb">Pepper (white or black), ground</td> <td class="tcc rb">0&emsp;0½</td> <td class="tcc rb">0&emsp;0½</td></tr>
+<tr><td class="tcl lb rb">Vinegar</td> <td class="tcc rb">1 gill</td> <td class="tcc rb">1 gill</td></tr>
+<tr><td class="tcl lb rb">Preserved meat</td> <td class="tcc rb">· ·</td> <td class="tcc rb">1&emsp;0</td></tr>
+<tr><td class="tcl lb rb">Suet</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">0&emsp;6</td></tr>
+<tr><td class="tcl lb rb">Raisins</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">0&emsp;8</td></tr>
+<tr><td class="tcl lb rb bb">Lime juice</td> <td class="tcc rb bb">&nbsp;</td> <td class="tcc rb bb">0&emsp;6</td></tr>
+</table>
+
+<p>Certain substitutions may be made in this scale at the option
+of the master of any emigrant ship, provided that the substituted
+articles are set forth in the contract tickets of the steerage passengers.</p>
+
+<p>In the British army the soldier is fed partly by a system of co-operation.
+He gets a free ration from government of 1 &#8468; of bread and
+¾ &#8468; of meat; in addition there is a messing allowance of 3½d. per
+man per day. He is able to supplement his food by purchases from
+the canteen. Much depends on the individual management in each
+regiment as to the satisfactory expenditure of the messing allowance.
+In some regiments an allowance is made from the canteen funds
+towards messing in addition to that granted by the government.
+The ordinary <i>field</i> ration of the British soldier is 1½ &#8468; of bread or
+1 &#8468; of biscuit; 1 &#8468; of fresh, salt or preserved meat; ½ oz. of coffee;
+1/6 oz. of tea; 2 oz. of sugar; ½ oz. of salt, <span class="spp">1</span>&frasl;<span class="suu">36</span> oz. of pepper, the
+whole weighing something over 2 &#8468; 3 oz. This cannot be looked
+on as a fixed ration, as it varies in different campaigns, according to
+the country into which the troops may be sent. The Prussian soldier
+during peace gets weekly from his canteen 11 &#8468; 1 oz. of rye bread
+and not quite 2½ &#8468; of meat. This is obviously insufficient, but under</p>
+
+<p>the conscription system it is reckoned that he will be able to make
+up the deficiency out of his own private means, or obtain charitable
+contributions from his friends. In the French infantry of the line
+each man during peace gets weekly 15 &#8468; of bread, 3<span class="spp">3</span>&frasl;<span class="suu">10</span> &#8468; of meat,
+2½ &#8468; of haricot beans or other vegetables, with salt and pepper, and
+1¾ oz. of brandy.</p>
+
+<p>An Austrian under the same circumstances receives 13.9 &#8468; of
+bread, ½ &#8468; of flour and 3.3 &#8468; of meat.</p>
+
+<p>The Russian conscript is allowed weekly:&mdash;</p>
+
+<table class="nobctr" style="width: 60%;" summary="Contents">
+<tr><td class="tcl">Black bread</td> <td class="tcl">7 &#8468;.</td></tr>
+<tr><td class="tcl">Meat</td> <td class="tcl">7 &#8468;.</td></tr>
+<tr><td class="tcl">Kvass (beer)</td> <td class="tcl">7.7 quarts.</td></tr>
+<tr><td class="tcl">Sour cabbage</td> <td class="tcl">24½ gills = 122½ oz.</td></tr>
+<tr><td class="tcl">Barley</td> <td class="tcl">24½ gills = 122½ oz.</td></tr>
+<tr><td class="tcl">Salts</td> <td class="tcl">10½ oz.</td></tr>
+<tr><td class="tcl">Horse-radish</td> <td class="tcl">28 grains.</td></tr>
+<tr><td class="tcl">Pepper</td> <td class="tcl">28 grains.</td></tr>
+<tr><td class="tcl">Vinegar</td> <td class="tcl">5½ gills = 26½ oz.</td></tr>
+</table>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIETETICS,<a name="ar76" id="ar76"></a></span> the science of diet, <i>i.e.</i> the food and nutrition of
+man in health and disease (see <span class="sc"><a href="#artlinks">Nutrition</a></span>). This article deals
+mainly with that part of the subject which has to do with the
+composition and nutritive values of foods and their adaptation
+to the use of people in health. The principal topics considered
+are: (1) Food and its functions; (2) Metabolism of matter and
+energy; (3) Composition of food materials; (4) Digestibility of
+food; (5) Fuel value of food; (6) Food consumption; (7) Quantities
+of nutrients needed; (8) Hygienic economy of food; (9)
+Pecuniary economy of food.</p>
+
+<p>1. <i>Food and its Functions.</i>&mdash;For practical purposes, food may be
+defined as that which, when taken into the body, may be utilized
+for the formation and repair of body tissue, and the production
+of energy. More specifically, food meets the requirements of the
+body in several ways. It is used for the formation of the tissues
+and fluids of the body, and for the restoration of losses of substance
+due to bodily activity. The potential energy of the food
+is converted into heat or muscular work or other forms of energy.
+In being thus utilized, food protects body substance or previously
+acquired nutritive material from consumption. When the amount
+<span class="pagenum"><a name="page215" id="page215"></a>215</span>
+of food taken into the body is in excess of immediate needs, the
+surplus may be stored for future consumption.</p>
+
+<p>Ordinary food materials, such as meat, fish, eggs, vegetables,
+&amp;c., consist of inedible materials, or <i>refuse</i>, <i>e.g.</i> bone of meat
+and fish, shell of eggs, rind and seed of vegetables; and <i>edible
+material</i>, as flesh of meat and fish, white and yolk of eggs, wheat
+flour, &amp;c. The edible material is by no means a simple substance,
+but consists of <i>water</i>, and some or all of the compounds
+variously designated as food stuffs, proximate principles, nutritive
+ingredients or nutrients, which are classified as <i>protein</i>, <i>fats</i>,
+<i>carbohydrates</i> and <i>mineral matters</i>. These have various functions
+in the nourishment of the body.</p>
+
+<p>The <i>refuse</i> commonly contains compounds similar to those
+in the food from which it is derived, but since it cannot be eaten,
+it is usually considered as a non-nutrient. It is of importance
+chiefly in a consideration of the pecuniary economy of food.
+<i>Water</i> is also considered as a non-nutrient, because although it is a
+constituent of all the tissues and fluids of the body, the body may
+obtain the water it needs from that drunk; hence, that contained
+in the food materials is of no special significance as a nutrient.</p>
+
+<p><i>Mineral matters</i>, such as sulphates, chlorides, phosphates and
+carbonates of sodium, potassium, calcium, &amp;c., are found in
+different combinations and quantities in most food materials.
+These are used by the body in the formation of the various
+tissues, especially the skeletal and protective tissues, in digestion,
+and in metabolic processes within the body. They yield little
+or no energy, unless perhaps the very small amount involved in
+their chemical transformation.</p>
+
+<p>Protein<a name="fa1f" id="fa1f" href="#ft1f"><span class="sp">1</span></a> is a term used to designate the whole group of
+nitrogenous compounds of food except the nitrogenous fats. It
+includes the albuminoids, as albumin of egg-white, and of blood
+serum, myosin of meat (muscle), casein of milk, globulin of blood
+and of egg yolk, fibrin of blood, gluten of flour; the gelatinoids,
+as gelatin and allied substances of connective tissue, collagen of
+tendon, ossein of bone and the so-called extractives (<i>e.g.</i> creatin)
+of meats; and the amids (<i>e.g.</i> asparagin) and allied compounds of
+vegetables and fruits.</p>
+
+<p>The albuminoids and gelatinoids, classed together as proteids,
+are the most important constituents of food, because they alone
+can supply the nitrogenous material necessary for the formation
+of the body tissues. For this purpose, the albuminoids are most
+valuable. Both groups of compounds, however, supply the body
+with energy, and the gelatinoids in being thus utilized protect
+the albuminoids from consumption for this purpose. When their
+supply in the food is in excess of the needs of the body, the surplus
+proteids may be converted into body fat and stored.</p>
+
+<p>The so-called extractives, which are the principal constituents
+of meat extract, beef tea and the like, act principally as stimulants
+and appetizers. It has been believed that they serve neither
+to build tissue nor to yield energy, but recent investigations<a name="fa2f" id="fa2f" href="#ft2f"><span class="sp">2</span></a>
+indicate that creatin may be metabolized in the body.</p>
+
+<p>The <i>fats</i> of food include both the animal fats and the vegetable
+oils. The <i>carbohydrates</i> include such compounds as starches,
+sugars and the fibre of plants or cellulose, though the latter has
+but little value as food for man. The more important function
+of both these classes of nutrients is to supply energy to the body
+to meet its requirements above that which it may obtain from the
+proteids. It is not improbable that the atoms of their molecules
+as well as those from the proteids are built up into the protoplasmic
+substance of the tissues. In this sense, these nutrients
+may be considered as being utilized also for the formation of
+tissue; but they are rather the accessory ingredients, whereas the
+proteids are the essential ingredients for this purpose. The fats
+in the food in excess of the body requirements may be stored as
+body fat, and the surplus carbohydrates may also be converted
+into fat and stored.</p>
+
+<p>To a certain extent, then, the nutrients of the food may
+substitute each other. All may be incorporated into the protoplasmic
+structure of body tissue, though only the proteids can
+supply the essential nitrogenous ingredients; and apart from
+the portion of the proteid material that is indispensable for this
+purpose, all the nutrients are used as a source of energy. If the
+supply of energy in the food is not sufficient, the body will use
+its own proteid and fat for this purpose. The gelatinoids, fats
+and carbohydrates in being utilized for energy protect the body
+proteids from consumption. The fat stored in the body from the
+excess of food is a reserve of energy material, on which the body
+may draw when the quantity of energy in the food is insufficient
+for its immediate needs.</p>
+
+<p>What compounds are especially concerned in intellectual
+activity is not known. The belief that fish is especially rich in
+phosphorus and valuable as a brain food has no foundation in
+observed fact.</p>
+
+<p>2. <i>Metabolism of Matter and Energy.</i>&mdash;The processes of nutrition
+thus consist largely of the transformation of food into body
+material and the conversion of the potential energy of both food
+and body material into the kinetic energy of heat and muscular
+work and other forms of energy. These various processes are
+generally designated by the term metabolism. The metabolism
+of matter in the body is governed largely by the needs of the body
+for energy. The science of nutrition, of which the present subject
+forms a part, is based on the principle that the transformations
+of matter and energy in the body occur in accordance with the
+laws of the conservation of matter and of energy. That the body
+can neither create nor destroy matter has long been universally
+accepted. It would seem that the transformation of energy must
+likewise be governed by the law of the conservation of energy;
+indeed there is every reason a priori to believe that it must; but
+the experimental difficulties in the way of absolute demonstration
+of the principle are considerable. For such demonstration it is
+necessary to prove that the income and expenditure of energy
+are equal. Apparatus and methods of inquiry devised in recent
+years, however, afford means for a comparison of the amounts of
+both matter and energy received and expended by the body, and
+from the results obtained in a large amount of such research,
+it seems probable that the law obtains in the living organism in
+general.</p>
+
+<p>The first attempt at such demonstration was made by
+M. Rubner<a name="fa3f" id="fa3f" href="#ft3f"><span class="sp">3</span></a> in 1894, experimenting with dogs doing no external
+muscular work. The income of energy (as heat) was computed,
+but the heat eliminated was measured. In the average of eight
+experiments continuing forty-five days, the two quantities agreed
+within 0.47%, thus demonstrating what it was desired to prove&mdash;that
+the heat given off by the body came solely from the
+oxidation of food within it. Results in accordance with these
+were reported by Studenski<a name="fa4f" id="fa4f" href="#ft4f"><span class="sp">4</span></a> in 1897, and by Laulanie<a name="fa5f" id="fa5f" href="#ft5f"><span class="sp">5</span></a> in 1898.</p>
+
+<p>The most extensive and complete data yet available on the
+subject have been obtained by W. O. Atwater, F. G. Benedict and
+associates<a name="fa6f" id="fa6f" href="#ft6f"><span class="sp">6</span></a> in experiments with men in the respiration calorimeter,
+in which a subject may remain for several consecutive days
+and nights. These experiments involve actual weighing and
+analyses of the food and drink, and of the gaseous, liquid and
+solid excretory products; determinations of potential energy
+(heat of oxidation) of the oxidizable material received and given
+off by the body (including estimation of the energy of the material
+gained or lost by the body); and measurements of the amounts of
+energy expended as heat and as external muscular work. By
+October 1906 eighty-eight experiments with fifteen different subjects
+had been completed. The separate experiments continued
+from two to thirteen days, making a total of over 270 days.
+<span class="pagenum"><a name="page216" id="page216"></a>216</span>
+In some cases the subjects were at rest; in others they performed
+varying amounts of external muscular work on an
+apparatus by means of which the amount of work done was
+measured. In some cases they fasted, and in others they received
+diets generally not far from sufficient to maintain nitrogen, and
+usually carbon, equilibrium in the body. In these experiments
+the amount of energy expended by the body as heat and as
+external muscular work measured in terms of heat agreed on
+the average very closely with the amount of heat that would be
+produced by the oxidation of all the matter metabolized in the
+body. The variations for individual days, and in the average for
+individual experiments as well, were in some cases appreciable,
+amounting to as much as 6%, which is not strange in view of the
+uncertainties in physiological experimenting; but in the average
+of all the experiments the energy of the expenditure was above
+99.9% of the energy of the income,&mdash;an agreement within one
+part in 1000. While these results do not absolutely prove the
+application of the law of the conservation of energy in the human
+body, they certainly approximate very closely to such demonstration.
+It is of course possible that energy may have given off
+<span class="pagenum"><a name="page217" id="page217"></a>217</span>
+from the body in other forms than heat and external muscular
+work. It is conceivable, for example, that intellectual activity
+may involve the transformation of physical energy, and that the
+energy involved may be eliminated in some form now unknown.
+But if the body did give off energy which was not measured in
+these experiments, the quantity must have been extremely small.
+It seems fair to infer from the results obtained that the metabolism
+of energy in the body occurred in conformity with the law
+of the conservation of energy.</p>
+
+<p>3. <i>Composition of Food Materials.</i>&mdash;The composition of food
+is determined by chemical analyses, the results of which are
+conventionally expressed in terms of the nutritive ingredients
+previously described. As a result of an enormous amount of
+such investigation in recent years, the kinds and proportions of
+nutrients in our common sorts of food are well known. Average
+values for percentage composition of some ordinary food materials
+are shown in Table I. (Table I. also includes figures for fuel
+value.)</p>
+
+<p class="center pt2"><span class="sc">Table I.</span>&mdash;<i>Percentage Composition of some Common Food Materials.</i></p>
+
+<table class="ws" summary="Contents">
+
+<tr><td class="tccm allb">Food Material.</td> <td class="tccm allb">Refuse.</td> <td class="tccm allb">Water.</td> <td class="tccm allb">Protein.</td> <td class="tccm allb">Fat.</td> <td class="tccm allb">Carbo-<br />hydrates.</td> <td class="tccm allb">Mineral<br />Matter.</td> <td class="tccm allb">Fuel Value<br />per &#8468;</td></tr>
+
+<tr><td class="tcl lb rb">&nbsp;</td> <td class="tcc rb">%</td> <td class="tcc rb">%</td> <td class="tcc rb">%</td> <td class="tcc rb">%</td> <td class="tcc rb">%</td> <td class="tcc rb">%</td> <td class="tcr rb">Calories.</td></tr>
+<tr><td class="tcl lb rb">Beef, fresh (medium fat)&mdash;</td> <td class="tcr rb">&nbsp;</td> <td class="tcr rb">&nbsp;</td> <td class="tcr rb">&nbsp;</td> <td class="tcr rb">&nbsp;</td> <td class="tcr rb">&nbsp;</td> <td class="tcr rb">&nbsp;</td> <td class="tcr rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">&emsp;Chuck</td> <td class="tcr rb">16.3</td> <td class="tcr rb">52.6</td> <td class="tcr rb">15.5</td> <td class="tcr rb">15.0</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.8</td> <td class="tcr rb">910</td></tr>
+<tr><td class="tcl lb rb">&emsp;Loin</td> <td class="tcr rb">13.3</td> <td class="tcr rb">52.5</td> <td class="tcr rb">16.1</td> <td class="tcr rb">17.5</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.9</td> <td class="tcr rb">1025</td></tr>
+<tr><td class="tcl lb rb">&emsp;Ribs</td> <td class="tcr rb">20.8</td> <td class="tcr rb">43.8</td> <td class="tcr rb">13.9</td> <td class="tcr rb">21.2</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.7</td> <td class="tcr rb">1135</td></tr>
+<tr><td class="tcl lb rb">&emsp;Round</td> <td class="tcr rb">7.2</td> <td class="tcr rb">60.7</td> <td class="tcr rb">19.0</td> <td class="tcr rb">12.8</td> <td class="tcc rb">· ·</td> <td class="tcr rb">1.0</td> <td class="tcr rb">890</td></tr>
+<tr><td class="tcl lb rb">&emsp;Shoulder</td> <td class="tcr rb">16.4</td> <td class="tcr rb">56.8</td> <td class="tcr rb">16.4</td> <td class="tcr rb">9.8</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.9</td> <td class="tcr rb">715</td></tr>
+<tr><td class="tcl lb rb">Beef, dried and smoked</td> <td class="tcr rb">4.7</td> <td class="tcr rb">53.7</td> <td class="tcr rb">26.4</td> <td class="tcr rb">6.9</td> <td class="tcc rb">· ·</td> <td class="tcr rb">8.9</td> <td class="tcr rb">790</td></tr>
+<tr><td class="tcl lb rb">Veal&mdash;</td> <td class="tcr rb">&nbsp;</td> <td class="tcr rb">&nbsp;</td> <td class="tcr rb">&nbsp;</td> <td class="tcr rb">&nbsp;</td> <td class="tcr rb">&nbsp;</td> <td class="tcr rb">&nbsp;</td> <td class="tcr rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">&emsp;Leg</td> <td class="tcr rb">14.2</td> <td class="tcr rb">60.1</td> <td class="tcr rb">15.5</td> <td class="tcr rb">7.9</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.9</td> <td class="tcr rb">625</td></tr>
+<tr><td class="tcl lb rb">&emsp;Loin</td> <td class="tcr rb">16.5</td> <td class="tcr rb">57.6</td> <td class="tcr rb">16.6</td> <td class="tcr rb">9.0</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.9</td> <td class="tcr rb">685</td></tr>
+<tr><td class="tcl lb rb">&emsp;Breast</td> <td class="tcr rb">21.3</td> <td class="tcr rb">52.0</td> <td class="tcr rb">15.4</td> <td class="tcr rb">11.0</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.8</td> <td class="tcr rb">745</td></tr>
+<tr><td class="tcl lb rb">Mutton&mdash;</td> <td class="tcr rb">&nbsp;</td> <td class="tcr rb">&nbsp;</td> <td class="tcr rb">&nbsp;</td> <td class="tcr rb">&nbsp;</td> <td class="tcr rb">&nbsp;</td> <td class="tcr rb">&nbsp;</td> <td class="tcr rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">&emsp;Leg</td> <td class="tcr rb">18.4</td> <td class="tcr rb">51.2</td> <td class="tcr rb">15.1</td> <td class="tcr rb">14.7</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.8</td> <td class="tcr rb">890</td></tr>
+<tr><td class="tcl lb rb">&emsp;Loin</td> <td class="tcr rb">16.0</td> <td class="tcr rb">42.0</td> <td class="tcr rb">13.5</td> <td class="tcr rb">28.3</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.7</td> <td class="tcr rb">1415</td></tr>
+<tr><td class="tcl lb rb">&emsp;Flank</td> <td class="tcr rb">9.9</td> <td class="tcr rb">39.0</td> <td class="tcr rb">13.8</td> <td class="tcr rb">36.9</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.6</td> <td class="tcr rb">1770</td></tr>
+<tr><td class="tcl lb rb">Pork&mdash;</td> <td class="tcr rb">&nbsp;</td> <td class="tcr rb">&nbsp;</td> <td class="tcr rb">&nbsp;</td> <td class="tcr rb">&nbsp;</td> <td class="tcr rb">&nbsp;</td> <td class="tcr rb">&nbsp;</td> <td class="tcr rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">&emsp;Loin</td> <td class="tcr rb">19.7</td> <td class="tcr rb">41.8</td> <td class="tcr rb">13.4</td> <td class="tcr rb">24.2</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.8</td> <td class="tcr rb">1245</td></tr>
+<tr><td class="tcl lb rb">&emsp;Ham, fresh</td> <td class="tcr rb">10.7</td> <td class="tcr rb">48.0</td> <td class="tcr rb">13.5</td> <td class="tcr rb">25.9</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.8</td> <td class="tcr rb">1320</td></tr>
+<tr><td class="tcl lb rb">&emsp;Ham, smoked and salted</td> <td class="tcr rb">13.6</td> <td class="tcr rb">34.8</td> <td class="tcr rb">14.2</td> <td class="tcr rb">33.4</td> <td class="tcc rb">· ·</td> <td class="tcr rb">4.2</td> <td class="tcr rb">1635</td></tr>
+<tr><td class="tcl lb rb">&emsp;Fat, salt</td> <td class="tcc rb">· ·</td> <td class="tcr rb">7.9</td> <td class="tcr rb">1.9</td> <td class="tcr rb">86.2</td> <td class="tcc rb">· ·</td> <td class="tcr rb">3.9</td> <td class="tcr rb">3555</td></tr>
+<tr><td class="tcl lb rb">&emsp;Bacon</td> <td class="tcr rb">7.7</td> <td class="tcr rb">17.4</td> <td class="tcr rb">9.1</td> <td class="tcr rb">62.2</td> <td class="tcc rb">· ·</td> <td class="tcr rb">4.1</td> <td class="tcr rb">2715</td></tr>
+<tr><td class="tcl lb rb">&emsp;Lard, refined</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcr rb">100.0</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcr rb">4100</td></tr>
+<tr><td class="tcl lb rb">Chicken</td> <td class="tcr rb">25.9</td> <td class="tcr rb">47.1</td> <td class="tcr rb">13.7</td> <td class="tcr rb">12.3</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.7</td> <td class="tcr rb">765</td></tr>
+<tr><td class="tcl lb rb">Turkey</td> <td class="tcr rb">22.7</td> <td class="tcr rb">42.4</td> <td class="tcr rb">16.1</td> <td class="tcr rb">18.4</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.8</td> <td class="tcr rb">1060</td></tr>
+<tr><td class="tcl lb rb">Goose</td> <td class="tcr rb">17.6</td> <td class="tcr rb">38.5</td> <td class="tcr rb">13.4</td> <td class="tcr rb">29.8</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.7</td> <td class="tcr rb">1475</td></tr>
+<tr><td class="tcl lb rb">Eggs</td> <td class="tcr rb">11.2</td> <td class="tcr rb">65.5</td> <td class="tcr rb">13.1</td> <td class="tcr rb">9.3</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.9</td> <td class="tcr rb">635</td></tr>
+<tr><td class="tcl lb rb">Cod, fresh</td> <td class="tcr rb">29.9</td> <td class="tcr rb">58.5</td> <td class="tcr rb">11.1</td> <td class="tcr rb">0.2</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.8</td> <td class="tcr rb">220</td></tr>
+<tr><td class="tcl lb rb">Cod, salted</td> <td class="tcr rb">24.9</td> <td class="tcr rb">40.2</td> <td class="tcr rb">16.0</td> <td class="tcr rb">0.4</td> <td class="tcc rb">· ·</td> <td class="tcr rb">18.5</td> <td class="tcr rb">325</td></tr>
+<tr><td class="tcl lb rb">Mackerel, fresh</td> <td class="tcr rb">44.7</td> <td class="tcr rb">40.4</td> <td class="tcr rb">10.2</td> <td class="tcr rb">4.2</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.7</td> <td class="tcr rb">370</td></tr>
+<tr><td class="tcl lb rb">Herring, smoked</td> <td class="tcr rb">44.4</td> <td class="tcr rb">19.2</td> <td class="tcr rb">20.5</td> <td class="tcr rb">8.8</td> <td class="tcc rb">· ·</td> <td class="tcr rb">7.4</td> <td class="tcr rb">755</td></tr>
+<tr><td class="tcl lb rb">Salmon, tinned</td> <td class="tcc rb">· ·</td> <td class="tcr rb">63.5</td> <td class="tcr rb">21.8</td> <td class="tcr rb">12.1</td> <td class="tcc rb">· ·</td> <td class="tcr rb">2.6</td> <td class="tcr rb">915 </td></tr>
+<tr><td class="tcl lb rb">Oysters, shelled</td> <td class="tcc rb">· ·</td> <td class="tcr rb">88.3</td> <td class="tcr rb">6.0</td> <td class="tcr rb">1.3</td> <td class="tcr rb">3.3</td> <td class="tcr rb">1.1</td> <td class="tcr rb">225</td></tr>
+<tr><td class="tcl lb rb">Butter</td> <td class="tcc rb">· ·</td> <td class="tcr rb">11.0</td> <td class="tcr rb">1.0</td> <td class="tcr rb">85.0</td> <td class="tcc rb">· ·</td> <td class="tcr rb">3.0</td> <td class="tcr rb">3410</td></tr>
+<tr><td class="tcl lb rb">Cheese</td> <td class="tcc rb">· ·</td> <td class="tcr rb">34.2</td> <td class="tcr rb">25.9</td> <td class="tcr rb">33.7</td> <td class="tcr rb">2.4</td> <td class="tcr rb">3.8</td> <td class="tcr rb">1885</td></tr>
+<tr><td class="tcl lb rb">Milk, whole</td> <td class="tcc rb">· ·</td> <td class="tcr rb">87.0</td> <td class="tcr rb">3.3</td> <td class="tcr rb">4.0</td> <td class="tcr rb">5.0</td> <td class="tcr rb">0.7</td> <td class="tcr rb">310</td></tr>
+<tr><td class="tcl lb rb">Milk, skimmed</td> <td class="tcc rb">· ·</td> <td class="tcr rb">90.5</td> <td class="tcr rb">3.4</td> <td class="tcr rb">0.3</td> <td class="tcr rb">5.1</td> <td class="tcr rb">0.7</td> <td class="tcr rb">165</td></tr>
+<tr><td class="tcl lb rb">Oatmeal</td> <td class="tcc rb">· ·</td> <td class="tcr rb">7.7</td> <td class="tcr rb">16.7</td> <td class="tcr rb">7.3</td> <td class="tcr rb">66.2</td> <td class="tcr rb">2.1</td> <td class="tcr rb">1800</td></tr>
+<tr><td class="tcl lb rb">Corn (maize) meal</td> <td class="tcc rb">· ·</td> <td class="tcr rb">12.5</td> <td class="tcr rb">9.2</td> <td class="tcr rb">1.9</td> <td class="tcr rb">75.4</td> <td class="tcr rb">1.0</td> <td class="tcr rb">1635</td></tr>
+<tr><td class="tcl lb rb">Rye flour</td> <td class="tcc rb">· ·</td> <td class="tcr rb">12.9</td> <td class="tcr rb">6.8</td> <td class="tcr rb">0.9</td> <td class="tcr rb">78.7</td> <td class="tcr rb">0.7</td> <td class="tcr rb">1620</td></tr>
+<tr><td class="tcl lb rb">Buckwheat flour</td> <td class="tcc rb">· ·</td> <td class="tcr rb">13.6</td> <td class="tcr rb">6.4</td> <td class="tcr rb">1.2</td> <td class="tcr rb">77.9</td> <td class="tcr rb">0.9</td> <td class="tcr rb">1605</td></tr>
+<tr><td class="tcl lb rb">Rice</td> <td class="tcc rb">· ·</td> <td class="tcr rb">12.3</td> <td class="tcr rb">8.0</td> <td class="tcr rb">0.3</td> <td class="tcr rb">79.0</td> <td class="tcr rb">0.4</td> <td class="tcr rb">1620</td></tr>
+<tr><td class="tcl lb rb">Wheat flour, white</td> <td class="tcc rb">· ·</td> <td class="tcr rb">12.0</td> <td class="tcr rb">11.4</td> <td class="tcr rb">1.0</td> <td class="tcr rb">75.1</td> <td class="tcr rb">0.5</td> <td class="tcr rb">1635</td></tr>
+<tr><td class="tcl lb rb">Wheat flour, graham</td> <td class="tcc rb">· ·</td> <td class="tcr rb">11.3</td> <td class="tcr rb">13.3</td> <td class="tcr rb">2.2</td> <td class="tcr rb">71.4</td> <td class="tcr rb">1.8</td> <td class="tcr rb">1645</td></tr>
+<tr><td class="tcl lb rb">Wheat, breakfast food</td> <td class="tcc rb">· ·</td> <td class="tcr rb">9.6</td> <td class="tcr rb">12.1</td> <td class="tcr rb">1.8</td> <td class="tcr rb">75.2</td> <td class="tcr rb">1.3</td> <td class="tcr rb">1680</td></tr>
+<tr><td class="tcl lb rb">Wheat bread, white</td> <td class="tcc rb">· ·</td> <td class="tcr rb">35.3</td> <td class="tcr rb">9.2</td> <td class="tcr rb">1.3</td> <td class="tcr rb">53.1</td> <td class="tcr rb">1.1</td> <td class="tcr rb">1200</td></tr>
+<tr><td class="tcl lb rb">Wheat bread, graham</td> <td class="tcc rb">· ·</td> <td class="tcr rb">35.7</td> <td class="tcr rb">8.9</td> <td class="tcr rb">1.8</td> <td class="tcr rb">52.1</td> <td class="tcr rb">1.5</td> <td class="tcr rb">1195</td></tr>
+<tr><td class="tcl lb rb">Rye bread</td> <td class="tcc rb">· ·</td> <td class="tcr rb">35.7</td> <td class="tcr rb">9.0</td> <td class="tcr rb">0.6</td> <td class="tcr rb">53.2</td> <td class="tcr rb">1.5</td> <td class="tcr rb">1170</td></tr>
+<tr><td class="tcl lb rb">Biscuit (crackers)</td> <td class="tcc rb">· ·</td> <td class="tcr rb">6.8</td> <td class="tcr rb">9.7</td> <td class="tcr rb">12.1</td> <td class="tcr rb">69.7</td> <td class="tcr rb">1.7</td> <td class="tcr rb">1925</td></tr>
+<tr><td class="tcl lb rb">Macaroni</td> <td class="tcc rb">· ·</td> <td class="tcr rb">10.3</td> <td class="tcr rb">13.4</td> <td class="tcr rb">0.9</td> <td class="tcr rb">74.1</td> <td class="tcr rb">1.3</td> <td class="tcr rb">1645</td></tr>
+<tr><td class="tcl lb rb">Sugar</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcr rb">100.0</td> <td class="tcc rb">· ·</td> <td class="tcr rb">1750</td></tr>
+<tr><td class="tcl lb rb">Starch (corn starch)</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcr rb">90.0</td> <td class="tcc rb">· ·</td> <td class="tcr rb">1680</td></tr>
+<tr><td class="tcl lb rb">Beans, dried</td> <td class="tcc rb">· ·</td> <td class="tcr rb">12.6</td> <td class="tcr rb">22.5</td> <td class="tcr rb">1.8</td> <td class="tcr rb">59.6</td> <td class="tcr rb">3.5</td> <td class="tcr rb">1520</td></tr>
+<tr><td class="tcl lb rb">Peas, dried</td> <td class="tcc rb">· ·</td> <td class="tcr rb">9.5</td> <td class="tcr rb">24.6</td> <td class="tcr rb">1.0</td> <td class="tcr rb">62.0</td> <td class="tcr rb">2.9</td> <td class="tcr rb">1565</td></tr>
+<tr><td class="tcl lb rb">Beets</td> <td class="tcr rb">20.0</td> <td class="tcr rb">70.0</td> <td class="tcr rb">1.3</td> <td class="tcr rb">0.1</td> <td class="tcr rb">7.7</td> <td class="tcr rb">0.9</td> <td class="tcr rb">160</td></tr>
+<tr><td class="tcl lb rb">Cabbage</td> <td class="tcr rb">50.0</td> <td class="tcr rb">4.2</td> <td class="tcr rb">0.7</td> <td class="tcr rb">0.2</td> <td class="tcr rb">4.5</td> <td class="tcr rb">0.4</td> <td class="tcr rb">100</td></tr>
+<tr><td class="tcl lb rb">Potatoes</td> <td class="tcr rb">20.0</td> <td class="tcr rb">62.6</td> <td class="tcr rb">1.8</td> <td class="tcr rb">0.1</td> <td class="tcr rb">14.7</td> <td class="tcr rb">0.8</td> <td class="tcr rb">295</td></tr>
+<tr><td class="tcl lb rb">Sweet potatoes</td> <td class="tcr rb">20.0</td> <td class="tcr rb">55.2</td> <td class="tcr rb">1.4</td> <td class="tcr rb">0.6</td> <td class="tcr rb">21.9</td> <td class="tcr rb">0.9</td> <td class="tcr rb">440</td></tr>
+<tr><td class="tcl lb rb">Tomatoes</td> <td class="tcc rb">· ·</td> <td class="tcr rb">94.3</td> <td class="tcr rb">0.9</td> <td class="tcr rb">0.4</td> <td class="tcr rb">3.9</td> <td class="tcr rb">0.5</td> <td class="tcr rb">100</td></tr>
+<tr><td class="tcl lb rb">Apples</td> <td class="tcr rb">25.0</td> <td class="tcr rb">63.3</td> <td class="tcr rb">0.3</td> <td class="tcr rb">0.3</td> <td class="tcr rb">10.8</td> <td class="tcr rb">0.3</td> <td class="tcr rb">190</td></tr>
+<tr><td class="tcl lb rb">Bananas</td> <td class="tcr rb">35.0</td> <td class="tcr rb">48.9</td> <td class="tcr rb">0.8</td> <td class="tcr rb">0.4</td> <td class="tcr rb">14.3</td> <td class="tcr rb">0.6</td> <td class="tcr rb">260</td></tr>
+<tr><td class="tcl lb rb">Grapes</td> <td class="tcr rb">25.0</td> <td class="tcr rb">58.0</td> <td class="tcr rb">1.0</td> <td class="tcr rb">1.2</td> <td class="tcr rb">14.4</td> <td class="tcr rb">0.4</td> <td class="tcr rb">295</td></tr>
+<tr><td class="tcl lb rb">Strawberries</td> <td class="tcr rb">5.0</td> <td class="tcr rb">85.9</td> <td class="tcr rb">0.9</td> <td class="tcr rb">0.6</td> <td class="tcr rb">7.0</td> <td class="tcr rb">0.6</td> <td class="tcr rb">150</td></tr>
+<tr><td class="tcl lb rb">Almonds</td> <td class="tcr rb">45.0</td> <td class="tcr rb">2.7</td> <td class="tcr rb">11.5</td> <td class="tcr rb">30.2</td> <td class="tcr rb">9.5</td> <td class="tcr rb">1.1</td> <td class="tcr rb">1515</td></tr>
+<tr><td class="tcl lb rb">Brazil nuts</td> <td class="tcr rb">49.6</td> <td class="tcr rb">2.6</td> <td class="tcr rb">8.6</td> <td class="tcr rb">33.7</td> <td class="tcr rb">3.5</td> <td class="tcr rb">2.0</td> <td class="tcr rb">1485</td></tr>
+<tr><td class="tcl lb rb">Chestnuts</td> <td class="tcr rb">16.0</td> <td class="tcr rb">37.8</td> <td class="tcr rb">5.2</td> <td class="tcr rb">4.5</td> <td class="tcr rb">35.4</td> <td class="tcr rb">1.1</td> <td class="tcr rb">915</td></tr>
+<tr><td class="tcl lb rb bb">Walnuts</td> <td class="tcr rb bb">58.1</td> <td class="tcr rb bb">1.0</td> <td class="tcr rb bb">6.9</td> <td class="tcr rb bb">26.6</td> <td class="tcr rb bb">6.8</td> <td class="tcr rb bb">0.6</td> <td class="tcr rb bb">1250</td></tr>
+</table>
+
+<p>It will be observed that different kinds of food materials vary
+widely in their proportions of nutrients. In general the animal
+foods contain the most protein and fats, and vegetable foods are
+rich in carbohydrates. The chief nutrient of lean meat and fish is
+protein; but in medium fat meats the proportion of fat is as large
+as that of protein, and in the fatter meats it is larger. Cheese
+is rich in both protein and fat. Among the vegetable foods, dried
+beans and peas are especially rich in protein. The proportion in
+oatmeal is also fairly large, in wheat it is moderate, and in maize
+meal and rice it is rather small. Oats contain more oil than any
+of the common cereals, but in none of them is the proportion
+especially large. The most abundant nutrient in all the cereals is
+starch, which comprises from two-thirds to three-fourths or more
+of their total nutritive substance. Cotton-seed is rich in edible
+oil, and so are olives. Some of the nuts contain fairly large
+proportions of both protein and fat. The nutrient of potatoes is
+starch, present in fair proportion. Fruits contain considerable
+carbohydrates, chiefly sugar. Green vegetables are not of much
+account as sources of any of the nutrients or energy.</p>
+
+<p>Similar food materials from different sources may also differ
+considerably in composition. This is especially true of meats.
+Thus, the leaner portions from a fat animal may contain nearly as
+much fat as the fatter portions from a lean animal. The data
+here presented are largely those for American food products,
+but the available analyses of English food materials indicate
+that the latter differ but little from the former in composition.
+The analyses of meats produced in Europe imply that they
+commonly contain somewhat less fat and more water, and
+often more protein, than American meats. The meats of English
+production compare with the American more than with the
+European meats. Similar vegetable foods from the different
+countries do not differ so much in composition.</p>
+
+<p>4. <i>Digestibility or Availability of Food Materials.</i>&mdash;The value
+of any food material for nutriment depends not merely upon the
+kinds and amounts of nutrients it contains, but also upon the
+ease and convenience with which the nutrients may be digested,
+and especially upon the proportion of the nutrients that will be
+actually digested and absorbed. Thus, two foods may contain
+equal amounts of the same nutrient, but the one most easily
+digested will really be of most value to the body, because less
+effort is necessary to utilize it. Considerable study of this factor
+is being made, and much valuable information is accumulating,
+but it is of more especial importance in cases of disordered
+digestion.</p>
+
+<p>The digestibility of food in the sense of thoroughness of
+digestion, however, is of particular importance in the present
+discussion. Only that portion of the food that is digested
+and absorbed is available to the body for the building of tissue
+and the production of energy. Not all the food eaten is thus
+actually digested; undigested material is excreted in the faeces.
+The thoroughness of digestion is determined experimentally by
+weighing and analysing the food eaten and the faeces pertaining
+to it. The difference between the corresponding ingredients of
+the two is commonly considered to represent the amounts of
+the ingredients digested. Expressed in percentages, these are
+called coefficients of digestibility. See Table II.</p>
+
+<p class="center pt2"><span class="sc">Table II.</span>&mdash;<i>Coefficients of Digestibility (or Availability)
+of Nutrients in Different Classes of Food Materials.</i></p>
+
+<table class="ws" summary="Contents">
+
+<tr><td class="tcc allb">Kind of Food.</td> <td class="tcc allb">Protein.</td> <td class="tcc allb">Fat.</td> <td class="tcc allb">Carbohydrates.</td></tr>
+
+<tr><td class="tcl lb rb">&nbsp;</td> <td class="tcc rb">%</td> <td class="tcc rb">%</td> <td class="tcc rb">%</td></tr>
+<tr><td class="tcl lb rb">Meats</td> <td class="tcc rb">98</td> <td class="tcc rb">98</td> <td class="tcc rb">· ·</td></tr>
+<tr><td class="tcl lb rb">Fish</td> <td class="tcc rb">96</td> <td class="tcc rb">97</td> <td class="tcc rb">· ·</td></tr>
+<tr><td class="tcl lb rb">Poultry</td> <td class="tcc rb">96</td> <td class="tcc rb">97</td> <td class="tcc rb">· ·</td></tr>
+<tr><td class="tcl lb rb">Eggs</td> <td class="tcc rb">97</td> <td class="tcc rb">98</td> <td class="tcc rb">· ·</td></tr>
+<tr><td class="tcl lb rb">Dairy products</td> <td class="tcc rb">97</td> <td class="tcc rb">96</td> <td class="tcc rb">98</td></tr>
+<tr><td class="tcl lb rb">Total animal food of mixed diet</td> <td class="tcc rb">97</td> <td class="tcc rb">97</td> <td class="tcc rb">98</td></tr>
+<tr><td class="tcl lb rb">Potatoes</td> <td class="tcc rb">73</td> <td class="tcc rb">· ·</td> <td class="tcc rb">98</td></tr>
+<tr><td class="tcl lb rb">Beets, carrots, &amp;c.</td> <td class="tcc rb">72</td> <td class="tcc rb">· ·</td> <td class="tcc rb">97</td></tr>
+<tr><td class="tcl lb rb">Cabbage, lettuce, &amp;c.</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">83</td></tr>
+<tr><td class="tcl lb rb">Legumes</td> <td class="tcc rb">78</td> <td class="tcc rb">90</td> <td class="tcc rb">95</td></tr>
+<tr><td class="tcl lb rb">Oatmeal</td> <td class="tcc rb">78</td> <td class="tcc rb">90</td> <td class="tcc rb">97</td></tr>
+<tr><td class="tcl lb rb">Corn meal</td> <td class="tcc rb">80</td> <td class="tcc rb">· ·</td> <td class="tcc rb">99</td></tr>
+<tr><td class="tcl lb rb">Wheat meals without bran</td> <td class="tcc rb">83</td> <td class="tcc rb">· ·</td> <td class="tcc rb">93</td></tr>
+<tr><td class="tcl lb rb">Wheat meals with bran</td> <td class="tcc rb">75</td> <td class="tcc rb">· ·</td> <td class="tcc rb">92</td></tr>
+<tr><td class="tcl lb rb">White bread</td> <td class="tcc rb">88</td> <td class="tcc rb">· ·</td> <td class="tcc rb">98</td></tr>
+<tr><td class="tcl lb rb">Entire wheat bread</td> <td class="tcc rb">82</td> <td class="tcc rb">· ·</td> <td class="tcc rb">94</td></tr>
+<tr><td class="tcl lb rb">Graham bread</td> <td class="tcc rb">76</td> <td class="tcc rb">· ·</td> <td class="tcc rb">90</td></tr>
+<tr><td class="tcl lb rb">Rice</td> <td class="tcc rb">76</td> <td class="tcc rb">· ·</td> <td class="tcc rb">91</td></tr>
+<tr><td class="tcl lb rb">Fruits and nuts</td> <td class="tcc rb">80</td> <td class="tcc rb">86</td> <td class="tcc rb">96</td></tr>
+<tr><td class="tcl lb rb">Sugars and starches</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">98</td></tr>
+<tr><td class="tcl lb rb">Total vegetable food of mixed diet</td> <td class="tcc rb">85</td> <td class="tcc rb">90</td> <td class="tcc rb">97</td></tr>
+<tr><td class="tcl lb rb bb">Total food of mixed diet</td> <td class="tcc rb bb">92</td> <td class="tcc rb bb">95</td> <td class="tcc rb bb">97</td></tr>
+</table>
+
+<p>Such a method is not strictly accurate, because the faeces do
+not consist entirely of undigested food but contain in addition
+to this the so-called metabolic products, which include the residuum
+of digestive juices not resorbed, fragments of intestinal
+epithelium, &amp;c. Since there is as yet no satisfactory method of
+separating these constituents of the excreta, the actual digestibility
+of the food is not determined. It has been suggested that
+since these materials must originally come from food, they
+represent, when expressed in terms of food ingredients, the cost of
+digestion; hence that the values determined as above explained
+represent the portion of food available to the body for the building
+of tissue and the yielding of energy, and what is commonly
+designated as digestibility should be called availability. Other
+writers retain the term &ldquo;digestibility,&rdquo; but express the results
+as &ldquo;apparent digestibility,&rdquo; until more knowledge regarding
+the metabolic products of the excreta is available and the actual
+digestibility may be ascertained.</p>
+
+<p>Experimental inquiry of this nature has been very active in
+recent years, especially in Europe, the United States and Japan;
+and the results of considerably over 1000 digestion experiments
+with single foods or combinations of food materials are available.
+These were mostly with men, but some were with women
+and with children. The larger part of these have been taken
+into account in the following estimations of the digestibility
+of the nutrients in different classes of food materials. The
+figures here shown are subject to revision as experimental data
+accumulate. They are not to be taken as exact measures of
+the digestibility (or availability) of every kind of food in each
+given class, but they probably represent fairly well the average
+digestibility of the classes of food materials as ordinarily utilized
+in the mixed diet.</p>
+
+<p>5. <i>Fuel Value of Food.</i>&mdash;The potential energy of food is
+commonly measured as the amount of heat evolved when the
+food is completely oxidized. In the laboratory this is determined
+by burning the food in oxygen in a calorimeter. The results,
+which are known as the heat of combustion of the food, are
+<span class="pagenum"><a name="page218" id="page218"></a>218</span>
+expressed in calories, one calory being the amount of heat
+necessary to raise the temperature of one kilogram of water one
+degree centigrade. But it is to be observed that this unit is
+employed simply from convenience, and without implication
+as to what extent the energy of food is converted into heat in
+the body. The unit employed in the measurement of some other
+form of energy might be used instead, as, for example, the foot-ton,
+which represents the amount of energy necessary to raise
+one ton through one foot.</p>
+
+<p class="center pt2"><span class="sc">Table III.</span>&mdash;<i>Estimates of Heats of Combustion and of Fuel Value
+of Nutrients in Ordinary Mixed Diet.</i></p>
+
+<table class="ws" summary="Contents">
+
+<tr><td class="tccm allb">Nutrients.</td> <td class="tccm allb">Heat of<br />Combustion.</td> <td class="tccm allb">Fuel Value.</td></tr>
+
+<tr><td class="tcl lb rb">&nbsp;</td> <td class="tcc rb">Calories.</td> <td class="tcc rb">Calories.</td></tr>
+<tr><td class="tcl lb rb">One gram of protein</td> <td class="tcc rb">5.65</td> <td class="tcc rb">4.05</td></tr>
+<tr><td class="tcl lb rb">One gram of fats</td> <td class="tcc rb">9.40</td> <td class="tcc rb">8.93</td></tr>
+<tr><td class="tcl lb rb bb">One gram of carbohydrates</td> <td class="tcc rb bb">4.15</td> <td class="tcc rb bb">4.03</td></tr>
+</table>
+
+<p>The amount of energy which a given quantity of food will
+produce on complete oxidation outside the body, however, is
+greater than that which the body will actually derive from it.
+In the first place, as previously shown, part of the food will not
+be digested and absorbed. In the second place, the nitrogenous
+compounds absorbed are not completely oxidized in the body,
+the residuum being excreted in the urine as urea and other bodies
+that are capable of further oxidation in the calorimeter. The
+total heat of combustion of the food eaten must therefore be
+diminished by the heat of combustion of the oxidizable material
+rejected by the body, to find what amount of energy is actually
+available to the organism for the production of work and heat.
+The amount thus determined is commonly known as the fuel
+value of food.</p>
+
+<p>Rubner&rsquo;s<a name="fa7f" id="fa7f" href="#ft7f"><span class="sp">7</span></a> commonly quoted estimates for the fuel value of the
+nutrients of mixed diet are,&mdash;for protein and carbohydrates 4.1,
+and for fats 9.3 calories per gram. According to the method of
+deduction, however, these factors were more applicable to digested
+than to total nutrients. Atwater<a name="fa8f" id="fa8f" href="#ft8f"><span class="sp">8</span></a> and associates have deduced,
+from data much more extensive than those available to Rubner,
+factors for total nutrients somewhat lower than these, as shown
+in Table III. These estimates seem to represent the best
+average factors at present available, but are subject to revision
+as knowledge is extended.</p>
+
+<p class="center pt2"><span class="sc">Table IV.</span>&mdash;<i>Quantities of Available Nutrients and Energy in Daily Food Consumption of Persons in
+Different Circumstances.</i></p>
+
+<table class="ws" summary="Contents">
+
+<tr><td class="tcc allb" rowspan="2">&nbsp;</td> <td class="tccm allb" rowspan="2">Number of<br />Studies.</td> <td class="tccm allb" colspan="4">Nutrients and Energy per Man per Day.</td></tr>
+<tr><td class="tccm allb">Protein.</td> <td class="tccm allb">Fat.</td> <td class="tccm allb">Carbo-<br />hydrates.</td> <td class="tccm allb">Fuel Value.</td></tr>
+
+<tr><td class="tcc lb rb pt1"><i>Persons with Active Work.</i></td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">Grams.</td> <td class="tcc rb">Grams.</td> <td class="tcc rb">Grams.</td> <td class="tcc rb">Calories.</td></tr>
+<tr><td class="tcl lb rb">English royal engineers</td> <td class="tcc rb">&ensp;1</td> <td class="tcc rb">132</td> <td class="tcc rb">&ensp;79</td> <td class="tcc rb">612</td> <td class="tcc rb">3835</td></tr>
+<tr><td class="tcl lb rb">Prussian machinists</td> <td class="tcc rb">&ensp;1</td> <td class="tcc rb">129</td> <td class="tcc rb">107</td> <td class="tcc rb">657</td> <td class="tcc rb">4265</td></tr>
+<tr><td class="tcl lb rb">Swedish mechanics</td> <td class="tcc rb">&ensp;5</td> <td class="tcc rb">174</td> <td class="tcc rb">105</td> <td class="tcc rb">693</td> <td class="tcc rb">4590</td></tr>
+<tr><td class="tcl lb rb">Bavarian lumbermen</td> <td class="tcc rb">&ensp;3</td> <td class="tcc rb">120</td> <td class="tcc rb">277</td> <td class="tcc rb">702</td> <td class="tcc rb">6015</td></tr>
+<tr><td class="tcl lb rb">American lumbermen</td> <td class="tcc rb">&ensp;5</td> <td class="tcc rb">155</td> <td class="tcc rb">327</td> <td class="tcc rb">804</td> <td class="tcc rb">6745</td></tr>
+<tr><td class="tcl lb rb">Japanese rice cleaner</td> <td class="tcc rb">&ensp;1</td> <td class="tcc rb">103</td> <td class="tcc rb">&ensp;11</td> <td class="tcc rb">917</td> <td class="tcc rb">4415</td></tr>
+<tr><td class="tcl lb rb">Japanese jinrikshaw runner</td> <td class="tcc rb">&ensp;1</td> <td class="tcc rb">137</td> <td class="tcc rb">&ensp;22</td> <td class="tcc rb">1010</td> <td class="tcc rb">5050</td></tr>
+<tr><td class="tcl lb rb">Chinese farm labourers in California</td> <td class="tcc rb">&ensp;1</td> <td class="tcc rb">132</td> <td class="tcc rb">&ensp;90</td> <td class="tcc rb">621</td> <td class="tcc rb">3980</td></tr>
+<tr><td class="tcl lb rb">American athletes</td> <td class="tcc rb">19</td> <td class="tcc rb">178</td> <td class="tcc rb">192</td> <td class="tcc rb">525</td> <td class="tcc rb">4740</td></tr>
+<tr><td class="tcl lb rb">American working-men&rsquo;s families</td> <td class="tcc rb">13</td> <td class="tcc rb">156</td> <td class="tcc rb">226</td> <td class="tcc rb">694</td> <td class="tcc rb">5650</td></tr>
+
+<tr><td class="tcc lb rb pt1"><i>Persons with Ordinary Work.</i></td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Bavarian mechanics</td> <td class="tcc rb">11</td> <td class="tcc rb">112</td> <td class="tcc rb">&ensp;32</td> <td class="tcc rb">553</td> <td class="tcc rb">3060</td></tr>
+<tr><td class="tcl lb rb">Bavarian farm labourers</td> <td class="tcc rb">&ensp;5</td> <td class="tcc rb">126</td> <td class="tcc rb">&ensp;52</td> <td class="tcc rb">526</td> <td class="tcc rb">3200</td></tr>
+<tr><td class="tcl lb rb">Russian peasants</td> <td class="tcc rb">..</td> <td class="tcc rb">119</td> <td class="tcc rb">&ensp;31</td> <td class="tcc rb">571</td> <td class="tcc rb">3155</td></tr>
+<tr><td class="tcl lb rb">Prussian prisoners</td> <td class="tcc rb">&ensp;1</td> <td class="tcc rb">117</td> <td class="tcc rb">&ensp;28</td> <td class="tcc rb">620</td> <td class="tcc rb">3320</td></tr>
+<tr><td class="tcl lb rb">Swedish mechanics</td> <td class="tcc rb">&ensp;6</td> <td class="tcc rb">123</td> <td class="tcc rb">&ensp;75</td> <td class="tcc rb">507</td> <td class="tcc rb">3325</td></tr>
+<tr><td class="tcl lb rb">American working-men&rsquo;s families</td> <td class="tcc rb">69</td> <td class="tcc rb">105</td> <td class="tcc rb">135</td> <td class="tcc rb">426</td> <td class="tcc rb">3480</td></tr>
+
+<tr><td class="tcc lb rb pt1"><i>Persons with Light Work.</i></td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">American artisans&rsquo; families</td> <td class="tcc rb">21</td> <td class="tcc rb">&ensp;93</td> <td class="tcc rb">107</td> <td class="tcc rb">358</td> <td class="tcc rb">2880</td></tr>
+<tr><td class="tcl lb rb">English tailors (prisoners)</td> <td class="tcc rb">&ensp;1</td> <td class="tcc rb">121</td> <td class="tcc rb">&ensp;37</td> <td class="tcc rb">509</td> <td class="tcc rb">2970</td></tr>
+<tr><td class="tcl lb rb">German shoemakers</td> <td class="tcc rb">&ensp;1</td> <td class="tcc rb">&ensp;99</td> <td class="tcc rb">&ensp;73</td> <td class="tcc rb">367</td> <td class="tcc rb"> 2629</td></tr>
+<tr><td class="tcl lb rb">Japanese prisoners</td> <td class="tcc rb">&ensp;1</td> <td class="tcc rb">&ensp;43</td> <td class="tcc rb">&emsp;6</td> <td class="tcc rb">444</td> <td class="tcc rb">2110</td></tr>
+
+<tr><td class="tcc lb rb pt1"><i>Professional and Business Men.</i></td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Japanese professional men</td> <td class="tcc rb">13</td> <td class="tcc rb">&ensp;75</td> <td class="tcc rb">&ensp;15</td> <td class="tcc rb">408</td> <td class="tcc rb">2190</td></tr>
+<tr><td class="tcl lb rb">Japanese students</td> <td class="tcc rb">&ensp;8</td> <td class="tcc rb">&ensp;85</td> <td class="tcc rb">&ensp;18</td> <td class="tcc rb">537</td> <td class="tcc rb">2800</td></tr>
+<tr><td class="tcl lb rb">Japanese military cadets</td> <td class="tcc rb">11</td> <td class="tcc rb">&ensp;98</td> <td class="tcc rb">&ensp;20</td> <td class="tcc rb">611</td> <td class="tcc rb">3185</td></tr>
+<tr><td class="tcl lb rb">German physicians</td> <td class="tcc rb">&ensp;2</td> <td class="tcc rb">121</td> <td class="tcc rb">&ensp;90</td> <td class="tcc rb">317</td> <td class="tcc rb">2685</td></tr>
+<tr><td class="tcl lb rb">Swedish medical students</td> <td class="tcc rb">&ensp;5</td> <td class="tcc rb">117</td> <td class="tcc rb">108</td> <td class="tcc rb">291</td> <td class="tcc rb">2725</td></tr>
+<tr><td class="tcl lb rb">Danish physicians</td> <td class="tcc rb">&ensp;1</td> <td class="tcc rb">124</td> <td class="tcc rb">133</td> <td class="tcc rb">242</td> <td class="tcc rb">2790</td></tr>
+<tr><td class="tcl lb rb">American professional and business men and students</td> <td class="tcc rb">51</td> <td class="tcc rb">&ensp;98</td> <td class="tcc rb">125</td> <td class="tcc rb">411</td> <td class="tcc rb">3285</td></tr>
+
+<tr><td class="tcc lb rb pt1"><i>Persons with Little or no Exercise.</i></td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Prussian prisoners</td> <td class="tcc rb">&ensp;2</td> <td class="tcc rb">&ensp;90</td> <td class="tcc rb">&ensp;27</td> <td class="tcc rb">427</td> <td class="tcc rb">2400</td></tr>
+<tr><td class="tcl lb rb">Japanese prisoners</td> <td class="tcc rb">&ensp;1</td> <td class="tcc rb">&ensp;36</td> <td class="tcc rb">&emsp;6</td> <td class="tcc rb">360</td> <td class="tcc rb">1725</td></tr>
+<tr><td class="tcl lb rb">Inmates of home for aged&mdash;Germany</td> <td class="tcc rb">&ensp;1</td> <td class="tcc rb">&ensp;85</td> <td class="tcc rb">&ensp;43</td> <td class="tcc rb">322</td> <td class="tcc rb">2097</td></tr>
+<tr><td class="tcl lb rb">Inmates of hospitals for insane&mdash;America</td> <td class="tcc rb">49</td> <td class="tcc rb">&ensp;80</td> <td class="tcc rb">&ensp;86</td> <td class="tcc rb">353</td> <td class="tcc rb">2590</td></tr>
+
+<tr><td class="tcc lb rb pt1"><i>Persons in Destitute Circumstances.</i></td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Prussian working people</td> <td class="tcc rb">13</td> <td class="tcc rb">&ensp;63</td> <td class="tcc rb">&ensp;43</td> <td class="tcc rb">372</td> <td class="tcc rb">2215</td></tr>
+<tr><td class="tcl lb rb">Italian mechanics</td> <td class="tcc rb">&ensp;5</td> <td class="tcc rb">&ensp;70</td> <td class="tcc rb">&ensp;36</td> <td class="tcc rb">384</td> <td class="tcc rb">2225</td></tr>
+<tr><td class="tcl lb rb bb">American working-men&rsquo;s families</td> <td class="tcc rb bb">11</td> <td class="tcc rb bb">&ensp;69</td> <td class="tcc rb bb">&ensp;75</td> <td class="tcc rb bb">263</td> <td class="tcc rb bb">2085</td></tr>
+</table>
+
+<p><span class="pagenum"><a name="page219" id="page219"></a>219</span></p>
+
+<p>The heats of combustion of all the fats in an ordinary mixed
+diet would average about 9.40 calories per gram, but as only
+95% of the fat would be available to the body, the fuel value
+per gram would be (9.40 × 0.95 =) 8.93 calories. Similarly, the
+average heat of combustion of carbohydrates of the diet would be
+about 4.15 calories per gram, and as 97% of the total quantity
+is available to the body, the fuel value per gram would be 4.03.
+(It is commonly assumed that the resorbed fats and carbohydrates
+are completely oxidized in the body.) The heats of
+combustion of all the kinds of protein in the diet would average
+about 5.65 calories per gram. Since about 92% of the total
+protein would be available to the body, the potential energy of
+the available protein would be equivalent to (5.65 × 0.92 =) 5.20
+calories; but as the available protein is not completely oxidized
+allowance must be made for the potential energy of the incompletely
+oxidized residue. This is estimated as equivalent to 1.15
+calories for the 0.92 gram of available protein; hence, the fuel
+value of the total protein is (5.20 &minus; 1.15 =) 4.05 calories per gram.
+Nutrients of the same class, but from different food materials,
+vary both in digestibility and in heat of combustion, and hence
+in fuel value. These factors are therefore not so applicable to the
+nutrients of the separate articles in a diet as to those of the diet as
+a whole.</p>
+
+<p>6. <i>Food Consumption.</i>&mdash;Much information regarding the food
+consumption of people in various circumstances in different parts
+of the world has accumulated during the past twenty years, as a
+result of studies of actual dietaries in England, Germany, Italy,
+Russia, Sweden and elsewhere in Europe, in Japan and other
+oriental countries, and especially in the United States. These
+studies commonly consist in ascertaining the kinds, amounts
+and composition of the different food materials consumed by a
+group of persons during a given period and the number of meals
+taken by each member of the group, and computing the quantities
+of the different nutrients in the food on the basis of one man for
+one day. When the members of the group are of different age,
+sex, occupation, &amp;c., account must be taken of the effect of these
+factors on consumption in estimating the value &ldquo;per man.&rdquo;
+Men as a rule eat more than women under similar conditions,
+women more than children, and persons at active work more than
+those at sedentary occupation. The navvy, for example, who
+is constantly using up more nutritive material or body tissue to
+supply the energy required for his muscular work needs more
+protein and energy in his food than a bookkeeper who sits at his
+desk all day.</p>
+
+<p>In making allowance for these differences, the various individuals
+are commonly compared with a man at moderately active
+muscular work, who is taken as unity. A man at hard muscular
+work is reckoned at 1.2 times such an individual; a man with
+light muscular work or a boy 15-16 years old, .9; a man at
+sedentary occupation, woman at moderately active muscular
+work, boy 13-14 or girl 15-16 years old, .8; woman at light work,
+boy 12 or girl 13-14 years old, .7; boy 10-11 or girl 10-12
+years old, .6; child 6-9 years old, .5; child 2-5 years old, .4;
+child under 2 years, .3. These factors are by no means absolute
+or final, but are based in part upon experimental data and in
+part upon arbitrary assumption.</p>
+
+<p>The total number of dietary studies on record is very large,
+but not all of them are complete enough to furnish reliable
+data. Upwards of 1000 are sufficiently accurate to be included
+in statistical averages of food consumed by people in different
+circumstances, nearly half of which have been made in the United
+States in the past decade. The number of persons in the individual
+studies has ranged from one to several hundred. Some
+typical results are shown in Table IV.</p>
+
+<p>7. <i>Quantities of Nutrients needed.</i>&mdash;For the proper nourishment
+of the body, the important problem is how much protein,
+fats and carbohydrates, or more simply, what amounts of protein
+and potential energy are needed under varying circumstances,
+to build and repair muscular and other tissues and to supply
+energy for muscular work, heat and other forms of energy.
+The answer to the problem is sought in the data obtained in
+dietary studies with considerable numbers of people, and in
+metabolism experiments with individuals in which the income
+and expenditure of the body are measured. From the information
+thus derived, different investigators have proposed so-called
+dietary standards, such as are shown in the table below, but
+unfortunately the experimental data are still insufficient for
+entirely trustworthy figures of this sort; hence the term
+&ldquo;standard&rdquo; as here used is misleading. The figures given are
+not to be considered as exact and final as that would suggest;
+they are merely tentative estimates of the average daily amounts
+of nutrients and energy required. (It is to be especially noted
+that these are available nutrients and fuel value rather than
+total nutrients and energy.) Some of the values proposed by
+other investigators are slightly larger than these, and others
+are decidedly smaller, but these are the ones that have hitherto
+been most commonly accepted in Europe and America.</p>
+
+<p class="center pt2"><span class="sc">Table V.</span>&mdash;<i>Standards for Dietaries. Available Nutrients and
+Energy per Man per Day.</i></p>
+
+<table class="ws" summary="Contents">
+
+<tr><td class="tccm allb">&nbsp;</td> <td class="tccm allb">Protein.</td> <td class="tccm allb">Fat.</td> <td class="tccm allb">Carbo-<br />hydrates.</td> <td class="tccm allb">Fuel<br />Value.</td></tr>
+
+<tr><td class="tcc lb rb pt1"><i>Voit&rsquo;s Standards.</i></td> <td class="tcc rb">Grams.<a name="fa9f" id="fa9f" href="#ft9f"><span class="sp">9</span></a></td> <td class="tcc rb">Grams.</td> <td class="tcc rb">Grams.</td> <td class="tcc rb">Calories.</td></tr>
+<tr><td class="tcl lb rb">Man at hard work</td> <td class="tcc rb">133</td> <td class="tcc rb">95</td> <td class="tcc rb">437</td> <td class="tcc rb">3270</td></tr>
+<tr><td class="tcl lb rb">Man at moderate work</td> <td class="tcc rb">109</td> <td class="tcc rb">53</td> <td class="tcc rb">485</td> <td class="tcc rb">2965</td></tr>
+
+<tr><td class="tcc lb rb pt1"><i>Atwater&rsquo;s Standards.</i></td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Man at very hard muscular work</td> <td class="tcc rb">161</td> <td class="tcc rb">· ·<a name="fa10f" id="fa10f" href="#ft10f"><span class="sp">10</span></a></td> <td class="tcc rb">· ·<a href="#ft10f"><span class="sp">10</span></a></td> <td class="tcc rb">5500</td></tr>
+<tr><td class="tcl lb rb">Man at hard muscular work</td> <td class="tcc rb">138</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">4150</td></tr>
+<tr><td class="tcl lb rb">Man at moderately active muscular work</td> <td class="tcc rb">115</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">3400</td></tr>
+<tr><td class="tcl lb rb">Man at light to moderate muscular work</td> <td class="tcc rb">103</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">3050</td></tr>
+<tr><td class="tcl lb rb">Man at &ldquo;sedentary&rdquo; or woman at moderately active work</td> <td class="tcc rb">&ensp;92</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">2700</td></tr>
+<tr><td class="tcl lb rb bb">Woman at light muscular work, or man without muscular exercise</td> <td class="tcc rb bb">&ensp;83</td> <td class="tcc rb bb">· ·</td> <td class="tcc rb bb">· ·</td> <td class="tcc rb bb">2450</td></tr>
+</table>
+
+<p>8. <i>Hygienic Economy of Food.</i>&mdash;For people in good health, there
+are two important rules to be observed in the regulation of the
+diet. One is to choose the foods that &ldquo;agree&rdquo; with them, and
+to avoid those which they cannot digest and assimilate without
+harm; and the other is to use such sorts and quantities of foods
+as will supply the kinds and amounts of nutrients needed by the
+body and yet to avoid burdening it with superfluous material to
+be disposed of at the cost of health and strength.</p>
+
+<p>As for the first-mentioned rule, it is practically impossible to
+give information that may be of more than general application.
+There are people who, because of some individual peculiarity,
+cannot use foods which for people in general are wholesome
+and nutritious. Some persons cannot endure milk, others suffer
+if they eat eggs, others have to eschew certain kinds of meat, or
+are made uncomfortable by fruit; but such cases are exceptions.
+Very little is known regarding the cause of these conditions. It
+is possible that in the metabolic processes to which the ingredients
+of the food are subjected in the body, or even during digestion
+before the substances are actually taken into the body, compounds
+may be formed that are in one way or another injurious.
+Whatever the cause may be, it is literally true in this sense that
+&ldquo;what is one man&rsquo;s meat is another man&rsquo;s poison,&rdquo; and each
+must learn for himself what foods &ldquo;agree&rdquo; with him and what
+ones do not. But for the great majority of people in health,
+<span class="pagenum"><a name="page220" id="page220"></a>220</span>
+suitable combinations of the ordinary sorts of wholesome food
+materials make a healthful diet. On the other hand, some foods
+are of particular value at times, aside from their use for nourishment.
+Fruits and green vegetables often benefit people greatly,
+not as nutriment merely, for they may have very little actual
+nutritive material, but because of fruit or vegetable acids or
+other substances which they contain, and which sometimes
+serve a most useful purpose.</p>
+
+<p class="center pt2"><span class="sc">Table VI.</span>&mdash;<i>Amounts of Nutrients and Energy Furnished for One Shilling in Food Materials at Ordinary Prices.</i></p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tccm allb" rowspan="3">Food Materials as Purchased.</td> <td class="tccm allb" rowspan="3">Prices<br />per &#8468;</td> <td class="tccm allb" colspan="5">One Shilling will buy</td></tr>
+<tr><td class="tccm allb" rowspan="2">Total Food<br />materials.</td> <td class="tccm allb" colspan="3">Available Nutrients.</td> <td class="tccm allb" rowspan="2">Fuel<br />Value.</td></tr>
+<tr><td class="tccm allb">Protein.</td> <td class="tccm allb">Fat.</td> <td class="tccm allb">Carbo-<br />hydrates.</td></tr>
+
+<tr><td class="tcl lb rb">&nbsp;</td> <td class="tcl rb">s.&emsp;d.</td> <td class="tcc rb">&#8468;</td> <td class="tcc rb">&#8468;</td> <td class="tcc rb">&#8468;</td> <td class="tcc rb">&#8468;</td> <td class="tcc rb">Calories.</td></tr>
+  <tr><td class="rb lb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Beef, round</td> <td class="tcl rb">0&emsp;10</td> <td class="tcr rb">1.20</td> <td class="tcr rb">.22</td> <td class="tcr rb">.14</td> <td class="tcc rb">· ·</td> <td class="tcr rb">1,155</td></tr>
+<tr><td class="tcl lb rb">&nbsp;</td> <td class="tcl rb">0&emsp;8½</td> <td class="tcr rb">1.41</td> <td class="tcr rb">.26</td> <td class="tcr rb">.17</td> <td class="tcc rb">· ·</td> <td class="tcr rb">1,235</td></tr>
+<tr><td class="tcl lb rb">&nbsp;</td> <td class="tcl rb">0&emsp;5</td> <td class="tcr rb">2.40</td> <td class="tcr rb">.44</td> <td class="tcr rb">.29</td> <td class="tcc rb">· ·</td> <td class="tcr rb">2,105</td></tr>
+  <tr><td class="rb lb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Beef, sirloin</td> <td class="tcl rb">0&emsp;10</td> <td class="tcr rb">1.20</td> <td class="tcr rb">.19</td> <td class="tcr rb">.20</td> <td class="tcc rb">· ·</td> <td class="tcr rb">1,225</td></tr>
+<tr><td class="tcl lb rb">&nbsp;</td> <td class="tcl rb">0&emsp;9</td> <td class="tcr rb">1.33</td> <td class="tcr rb">.21</td> <td class="tcr rb">.22</td> <td class="tcc rb">· ·</td> <td class="tcr rb">1,360</td></tr>
+<tr><td class="tcl lb rb">&nbsp;</td> <td class="tcl rb">0&emsp;8</td> <td class="tcr rb">1.50</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td></tr>
+<tr><td class="tcl lb rb">&nbsp;</td> <td class="tcl rb">0&emsp;5</td> <td class="tcr rb">2.40</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td></tr>
+  <tr><td class="rb lb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Beef, rib</td> <td class="tcl rb">0&emsp;9</td> <td class="tcr rb">1.33</td> <td class="tcr rb">.19</td> <td class="tcr rb">.19</td> <td class="tcc rb">· ·</td> <td class="tcr rb">1,200</td></tr>
+<tr><td class="tcl lb rb">&nbsp;</td> <td class="tcl rb">0&emsp;7½</td> <td class="tcr rb">1.60</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td></tr>
+<tr><td class="tcl lb rb">&nbsp;</td> <td class="tcl rb">0&emsp;4½</td> <td class="tcr rb">2.67</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td></tr>
+  <tr><td class="rb lb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Mutton, leg</td> <td class="tcl rb">0&emsp;9</td> <td class="tcr rb">1.33</td> <td class="tcr rb">.20</td> <td class="tcr rb">.20</td> <td class="tcc rb">· ·</td> <td class="tcr rb">1,245</td></tr>
+<tr><td class="tcl lb rb">&nbsp;</td> <td class="tcl rb">0&emsp;5</td> <td class="tcr rb">2.40</td> <td class="tcr rb">.37</td> <td class="tcr rb">.35</td> <td class="tcc rb">· ·</td> <td class="tcr rb">2,245</td></tr>
+  <tr><td class="rb lb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Pork, spare-rib</td> <td class="tcl rb">0&emsp;9</td> <td class="tcr rb">1.33</td> <td class="tcr rb">.17</td> <td class="tcr rb">.31</td> <td class="tcc rb">· ·</td> <td class="tcr rb">1,645</td></tr>
+<tr><td class="tcl lb rb">&nbsp;</td> <td class="tcl rb">0&emsp;7</td> <td class="tcr rb">1.71</td> <td class="tcr rb">.22</td> <td class="tcr rb">.39</td> <td class="tcc rb">· ·</td> <td class="tcr rb">2,110</td></tr>
+  <tr><td class="rb lb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Pork, salt, fat</td> <td class="tcl rb">0&emsp;7</td> <td class="tcr rb">1.71</td> <td class="tcr rb">.03</td> <td class="tcr rb">1.40</td> <td class="tcc rb">· ·</td> <td class="tcr rb">6,025</td></tr>
+<tr><td class="tcl lb rb">&nbsp;</td> <td class="tcl rb">0&emsp;5</td> <td class="tcr rb">2.40</td> <td class="tcr rb">.04</td> <td class="tcr rb">1.97</td> <td class="tcc rb">· ·</td> <td class="tcr rb">8,460</td></tr>
+  <tr><td class="rb lb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Pork, smoked ham</td> <td class="tcl rb">0&emsp;8</td> <td class="tcr rb">1.50</td> <td class="tcr rb">.20</td> <td class="tcr rb">.48</td> <td class="tcc rb">· ·</td> <td class="tcr rb">2,435</td></tr>
+<tr><td class="tcl lb rb">&nbsp;</td> <td class="tcl rb">0&emsp;4½</td> <td class="tcr rb">2.67</td> <td class="tcr rb">.36</td> <td class="tcr rb">.85</td> <td class="tcc rb">· ·</td> <td class="tcr rb">4,330</td></tr>
+  <tr><td class="rb lb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Fresh cod</td> <td class="tcl rb">0&emsp;4</td> <td class="tcr rb">3.00</td> <td class="tcr rb">.34</td> <td class="tcr rb">.01</td> <td class="tcc rb">· ·</td> <td class="tcr rb">710</td></tr>
+<tr><td class="tcl lb rb">&nbsp;</td> <td class="tcl rb">0&emsp;3</td> <td class="tcr rb">4.00</td> <td class="tcr rb">.45</td> <td class="tcr rb">.01</td> <td class="tcc rb">· ·</td> <td class="tcr rb">945</td></tr>
+  <tr><td class="rb lb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Salt cod</td> <td class="tcl rb">0&emsp;3½</td> <td class="tcr rb">3.43</td> <td class="tcr rb">.54</td> <td class="tcr rb">.07</td> <td class="tcc rb">· ·</td> <td class="tcr rb">1,370</td></tr>
+<tr><td class="tcl lb rb">&nbsp;</td> <td class="tcl rb">0&emsp;10</td> <td class="tcr rb">1.20</td> <td class="tcr rb">.07</td> <td class="tcr rb">.01</td> <td class="tcr rb">.04</td> <td class="tcr rb">275</td></tr>
+  <tr><td class="rb lb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Milk, whole, 4d. a qt.</td> <td class="tcl rb">0&emsp;2</td> <td class="tcr rb">6.00</td> <td class="tcr rb">.19</td> <td class="tcr rb">.23</td> <td class="tcr rb">.30</td> <td class="tcr rb">1,915</td></tr>
+<tr><td class="tcl lb rb">&emsp;&emsp;&rdquo;&emsp;&emsp;3d. a qt.</td> <td class="tcl rb">0&emsp;1½</td> <td class="tcr rb">8.00</td> <td class="tcr rb">.26</td> <td class="tcr rb">.30</td> <td class="tcr rb">.40</td> <td class="tcr rb">2,550</td></tr>
+<tr><td class="tcl lb rb">&emsp;&emsp;&rdquo;&emsp;&emsp;2d. a qt.</td> <td class="tcl rb">0&emsp;1</td> <td class="tcr rb">12.00</td> <td class="tcr rb">.38</td> <td class="tcr rb">.46</td> <td class="tcr rb">.60</td> <td class="tcr rb">3,825</td></tr>
+  <tr><td class="rb lb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Milk, skimmed, 2d. a qt.</td> <td class="tcl rb">0&emsp;1</td> <td class="tcr rb">12.00</td> <td class="tcr rb">.40</td> <td class="tcr rb">.03</td> <td class="tcr rb">.61</td> <td class="tcr rb">2,085</td></tr>
+  <tr><td class="rb lb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Butter</td> <td class="tcl rb">1&emsp;6</td> <td class="tcr rb">.67</td> <td class="tcr rb">.01</td> <td class="tcr rb">.54</td> <td class="tcc rb">· ·</td> <td class="tcr rb">2,320</td></tr>
+<tr><td class="tcl lb rb">&nbsp;</td> <td class="tcl rb">1&emsp;3</td> <td class="tcr rb">.80</td> <td class="tcr rb">.01</td> <td class="tcr rb">.64</td> <td class="tcc rb">· ·</td> <td class="tcr rb">2,770</td></tr>
+<tr><td class="tcl lb rb">&nbsp;</td> <td class="tcl rb">1&emsp;0</td> <td class="tcr rb">1.00</td> <td class="tcr rb">.01</td> <td class="tcr rb">.81</td> <td class="tcc rb">· ·</td> <td class="tcr rb">3,460</td></tr>
+  <tr><td class="rb lb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Margarine</td> <td class="tcl rb">0&emsp;4</td> <td class="tcr rb">3.00</td> <td class="tcc rb">· ·</td> <td class="tcr rb">2.37</td> <td class="tcc rb">· ·</td> <td class="tcr rb">10,080</td></tr>
+  <tr><td class="rb lb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Eggs, 2s. a dozen</td> <td class="tcl rb">1&emsp;4</td> <td class="tcr rb">.75</td> <td class="tcr rb">.10</td> <td class="tcr rb">.07</td> <td class="tcc rb">· ·</td> <td class="tcr rb">475</td></tr>
+<tr><td class="tcl lb rb">&emsp;&rdquo;&emsp;1½s. a dozen</td> <td class="tcl rb">1&emsp;0</td> <td class="tcr rb">1.00</td> <td class="tcr rb">.13</td> <td class="tcr rb">.09</td> <td class="tcc rb">· ·</td> <td class="tcr rb">635</td></tr>
+<tr><td class="tcl lb rb">&emsp;&rdquo;&emsp;1s. a dozen</td> <td class="tcl rb">0&emsp;8</td> <td class="tcr rb">1.50</td> <td class="tcr rb">.19</td> <td class="tcr rb">.13</td> <td class="tcc rb">· ·</td> <td class="tcr rb">950</td></tr>
+  <tr><td class="rb lb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Cheese</td> <td class="tcl rb">0&emsp;8</td> <td class="tcr rb">1.50</td> <td class="tcr rb">.38</td> <td class="tcr rb">.48</td> <td class="tcr rb">.04</td> <td class="tcr rb">2,865</td></tr>
+<tr><td class="tcl lb rb">&nbsp;</td> <td class="tcl rb">0&emsp;7</td> <td class="tcr rb">1.71</td> <td class="tcr rb">.43</td> <td class="tcr rb">.55</td> <td class="tcr rb">.04</td> <td class="tcr rb">3,265</td></tr>
+<tr><td class="tcl lb rb">&nbsp;</td> <td class="tcl rb">0&emsp;5</td> <td class="tcr rb">2.40</td> <td class="tcr rb">.60</td> <td class="tcr rb">.77</td> <td class="tcr rb">.06</td> <td class="tcr rb">4,585</td></tr>
+  <tr><td class="rb lb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Wheat bread</td> <td class="tcl rb">0&emsp;1<span class="spp">1</span>&frasl;<span class="suu">8</span></td> <td class="tcr rb">10.67</td> <td class="tcr rb">.76</td> <td class="tcr rb">.13</td> <td class="tcr rb">5.57</td> <td class="tcr rb">12,421</td></tr>
+  <tr><td class="rb lb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Wheat flour</td> <td class="tcl rb">0&emsp;1<span class="spp">3</span>&frasl;<span class="suu">5</span></td> <td class="tcr rb">7.64</td> <td class="tcr rb">.67</td> <td class="tcr rb">.07</td> <td class="tcr rb">5.63</td> <td class="tcr rb">12,110</td></tr>
+<tr><td class="tcl lb rb">&nbsp;</td> <td class="tcl rb">0&emsp;1½</td> <td class="tcr rb">8.16</td> <td class="tcr rb">.72</td> <td class="tcr rb">.07</td> <td class="tcr rb">6.01</td> <td class="tcr rb">12,935</td></tr>
+  <tr><td class="rb lb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Oatmeal</td> <td class="tcl rb">0&emsp;1<span class="spp">2</span>&frasl;<span class="suu">5</span></td> <td class="tcr rb">8.39</td> <td class="tcr rb">1.11</td> <td class="tcr rb">.54</td> <td class="tcr rb">5.54</td> <td class="tcr rb">14,835</td></tr>
+<tr><td class="tcl lb rb">&nbsp;</td> <td class="tcl rb">0&emsp;1½</td> <td class="tcr rb">8.16</td> <td class="tcr rb">1.08</td> <td class="tcr rb">.53</td> <td class="tcr rb">5.39</td> <td class="tcr rb">14,430</td></tr>
+  <tr><td class="rb lb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Rice</td> <td class="tcl rb">0&emsp;1¾</td> <td class="tcr rb">6.86</td> <td class="tcr rb">.45</td> <td class="tcr rb">.02</td> <td class="tcr rb">5.27</td> <td class="tcr rb">10,795</td></tr>
+  <tr><td class="rb lb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Potatoes</td> <td class="tcl rb">0&emsp;0<span class="spp">2</span>&frasl;<span class="suu">3</span></td> <td class="tcr rb">18.00</td> <td class="tcr rb">.25</td> <td class="tcr rb">.02</td> <td class="tcr rb">2.70</td> <td class="tcr rb">5,605</td></tr>
+<tr><td class="tcl lb rb">&nbsp;</td> <td class="tcl rb">0&emsp;0½</td> <td class="tcr rb">24.00</td> <td class="tcr rb">.34</td> <td class="tcr rb">.02</td> <td class="tcr rb">3.60</td> <td class="tcr rb">7,470</td></tr>
+  <tr><td class="rb lb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb">Beans</td> <td class="tcl rb">0&emsp;2 </td> <td class="tcr rb">6.00</td> <td class="tcr rb">1.05</td> <td class="tcr rb">.10</td> <td class="tcr rb">3.47</td> <td class="tcr rb">8,960</td></tr>
+  <tr><td class="rb lb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td> <td class="rb">&nbsp;</td></tr>
+<tr><td class="tcl lb rb bb">Sugar</td> <td class="tcl rb bb">1&emsp;¾</td> <td class="tcr rb bb">6.86</td> <td class="tcc rb bb">· ·</td> <td class="tcc rb bb">· ·</td> <td class="tcr rb bb">6.86</td> <td class="tcr rb bb">12,760</td></tr>
+</table>
+
+<p>The proper observance of the second rule mentioned requires
+information regarding the demands of the body for food under
+different circumstances. To supply this information is one
+purpose of the effort to determine the so-called dietary standards
+<span class="pagenum"><a name="page221" id="page221"></a>221</span>
+mentioned above. It should be observed, however, that these
+are generally more applicable to the proper feeding of a group
+or class of people as a whole than for particular individuals
+in this class. The needs of individuals will vary largely from
+the average in accordance with the activity and individuality.
+Moreover, it is neither necessary nor desirable for the individual
+to follow any standard exactly from day to day. It is requisite
+only that the average supply shall be sufficient to meet the
+demands of the body during a given period.</p>
+
+<p>The cooking of food and other modes of preparing it for
+consumption have much to do with its nutritive value. Many
+materials which, owing to their mechanical condition or to
+some other cause, are not particularly desirable food materials
+in their natural state, are quite nutritious when cooked or otherwise
+prepared for consumption. It is also a matter of common
+experience that well-cooked food is wholesome and appetizing,
+whereas the same material poorly prepared is unpalatable.
+There are three chief purposes of cooking; the first is to change
+the mechanical condition of the food. Heating changes the
+structure of many food materials very materially, so that they
+may be more easily chewed and brought into a condition in which
+the digestive juices can act upon them more freely, and in this
+way probably influencing the ease and thoroughness of digestion.
+The second is to make the food more appetizing by improving
+the appearance or flavour or both. Food which is attractive to
+the eye and pleasing to the palate quickens the flow of saliva
+and other digestive juices and thus aids digestion. The third
+is to kill, by heat, disease germs, parasites or other dangerous
+organisms that may be contained in food. This is often a very
+important matter and applies to both animal and vegetable foods.
+Scrupulous neatness should always be observed in storing,
+handling and serving food. If ever cleanliness is desirable it
+must be in the things we eat, and every care should be taken to
+ensure it for the sake of health as well as of decency. Cleanliness
+in this connexion means not only absence of visible dirt, but
+freedom from undesirable bacteria and other minute organisms
+and from worms and other parasites. If food, raw or cooked, is
+kept in dirty places, peddled from dirty carts, prepared in dirty
+rooms and in dirty dishes, or exposed to foul air, disease germs
+and other offensive and dangerous substances may easily enter it.</p>
+
+<p>9. <i>Pecuniary Economy of Food.</i>&mdash;Statistics of economy and of
+cost of living in Great Britain, Germany and the United States
+show that at least half, and commonly more, of the income of
+wage-earners and other people in moderate circumstances is
+expended for subsistence. The relatively large cost of food, and
+the important influence of diet upon health and strength, make a
+more widespread understanding of the subject of dietetics very
+desirable. The maxim that &ldquo;the best is the cheapest&rdquo; does not
+apply to food. The &ldquo;best&rdquo; food, in the sense of that which is
+the finest in appearance and flavour and which is sold at the
+highest price, is not generally the most economical.</p>
+
+<p>The price of food is not regulated largely by its value for
+nutriment. Its agreeableness to the palate or to the buyer&rsquo;s
+fancy is a large factor in determining the current demand and
+market price. There is no more nutriment in an ounce of protein
+or fat from the tender-loin of beef than from the round or shoulder.
+The protein of animal food has, however, some advantage over
+that of vegetable foods in that it is more thoroughly, and perhaps
+more easily, digested, for which reason it would be economical to
+pay somewhat more for the same quantity of nutritive material
+in the animal food. Furthermore, animal foods such as meats,
+fish and the like, gratify the palate as most vegetable foods do
+not. For persons in good health, foods in which the nutrients
+are the most expensive are like costly articles of adornment.
+People who can well afford them may be justified in buying
+them, but they are not economical. The most economical food
+is that which is at the same time most healthful and cheapest.</p>
+
+<p>The variations in the cost of the actual nutriment in different
+food materials may be illustrated by comparison of the amounts
+of nutrients obtained for a given sum in the materials as bought
+at ordinary market prices. This is done in Table VI., which
+shows the amounts of available nutrients contained in the quantities
+of different food materials that may be purchased for one
+shilling at prices common in England.</p>
+
+<p>When proper attention is given to the needs of the body for
+food and the relation between cost and nutritive value of food
+materials, it will be found that with care in the purchase and skill
+in the preparation of food, considerable control may be had over
+the expensiveness of a palatable, nutritious and healthful diet.</p>
+
+<div class="condensed">
+<p><span class="sc">Authorities.</span>&mdash;<span class="sc">Composition of Foods</span>:&mdash;König, <i>Chemie der
+menschlichen Nahrungs- und Genussmittel</i>; Atwater and Bryant,
+&ldquo;Composition of American Food Materials,&rdquo; Bul. 28, Office of
+Experiment Stations, U.S. Department of Agriculture. <span class="sc">Nutrition
+and Dietetics</span>:&mdash;Armsby, <i>Principles of Animal Nutrition</i>; Lusk,
+<i>The Science of Nutrition</i>; Burney Yeo, <i>Food in Health and Disease</i>;
+Munk and Uffelmann, <i>Die Ernährung des gesunden und kranken
+Menschen</i>; Von Leyden, <i>Ernährungstherapie und Diätetik</i>; Dujardin-Beaumetz,
+Hygiène alimentaire; Hutchison, <i>Food and Dietetics</i>; R.
+H. Chittenden, <i>Physiological Economy in Nutrition</i> (1904), <i>Nutrition of
+Man</i> (1907); Atwater, &ldquo;Chemistry and Economy of Food,&rdquo; Bul. 21,
+Office of Experiment Stations, U.S. Department of Agriculture. See
+also other Bulletins of the same office on composition of food, results
+of dietary studies, metabolism experiments, &amp;c., in the United States.
+<span class="sc">General Metabolism</span>:&mdash;Voit, <i>Physiologie des allgemeinen Stoffwechsels
+und der Ernährung</i>; Hermann, <i>Handbuch der Physiologie</i>,
+Bd. vi.; Von Noorden, <i>Pathologie des Stoffwechsels</i>; Schäfer, <i>Text-Book
+of Physiology</i>, vol. i.; Atwater and Langworthy, &ldquo;Digest of
+Metabolism Experiments,&rdquo; Bull. 45, Office of Experiment Stations,
+U.S. Department of Agriculture.</p>
+</div>
+<div class="author">(W. O. A.; R. D. M.)</div>
+
+<hr class="foot" /> <div class="note">
+
+<p><a name="ft1f" id="ft1f" href="#fa1f"><span class="fn">1</span></a> The terms applied by different writers to these nitrogenous
+compounds are conflicting. For instance, the term &ldquo;proteid&rdquo; is
+sometimes used as protein is here used, and sometimes to designate
+the group here called albuminoids. The classification and terminology
+here followed are those tentatively recommended by the Association
+of American Agricultural Colleges and Experiment Stations.</p>
+
+<p><a name="ft2f" id="ft2f" href="#fa2f"><span class="fn">2</span></a> Folin, <i>Festschrift für Olaf Hammarsten</i>, iii. (Upsala, 1906).</p>
+
+<p><a name="ft3f" id="ft3f" href="#fa3f"><span class="fn">3</span></a> <i>Ztschr. Biol.</i> 30, 73.</p>
+
+<p><a name="ft4f" id="ft4f" href="#fa4f"><span class="fn">4</span></a> In Russian. Cited in United States Department of Agriculture,
+Office of Experiment Stations, Bul. No. 45, <i>A Digest of Metabolism
+Experiments</i>, by W. O. Atwater and C. F. Langworthy.</p>
+
+<p><a name="ft5f" id="ft5f" href="#fa5f"><span class="fn">5</span></a> <i>Arch. physiol. norm. et path.</i> (1894) 4.</p>
+
+<p><a name="ft6f" id="ft6f" href="#fa6f"><span class="fn">6</span></a> U.S. Department of Agriculture, Office of Experiment Stations,
+Bulletins Nos. 63, 69, 109, 136, 175. For a description of the respiration
+calorimeter here mentioned see also publication No. 42 of the
+Carnegie Institution of Washington.</p>
+
+<p><a name="ft7f" id="ft7f" href="#fa7f"><span class="fn">7</span></a> <i>Ztschr. Biol.</i> 21 (1885), p. 377.</p>
+
+<p><a name="ft8f" id="ft8f" href="#fa8f"><span class="fn">8</span></a> <i>Connecticut</i> (Storrs) <i>Agricultural Experiment Station Report</i>
+(1899), 73.</p>
+
+<p><a name="ft9f" id="ft9f" href="#fa9f"><span class="fn">9</span></a> One ounce equals 28.35 grams.</p>
+
+<p><a name="ft10f" id="ft10f" href="#fa10f"><span class="fn">10</span></a> As the chief function of both fats and carbohydrates is to furnish
+energy, their exact proportion in the diet is of small account. The
+amount of either may vary largely according to taste, available
+supply, or other condition, as long as the total amount of both is
+sufficient, together with the protein to furnish the required energy.</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIETRICH, CHRISTIAN WILHELM ERNST<a name="ar77" id="ar77"></a></span> (1712-1774),
+German painter, was born at Weimar, where he was brought up
+early to the profession of art by his father Johann George, then
+painter of miniatures to the court of the duke. Having been sent
+to Dresden to perfect himself under the care of Alexander Thiele,
+he had the good fortune to finish in two hours, at the age of
+eighteen, a picture which attracted the attention of the king of
+Saxony. Augustus II. was so pleased with Dietrich&rsquo;s readiness
+of hand that he gave him means to study abroad, and visit in
+succession the chief cities of Italy and the Netherlands. There
+he learnt to copy and to imitate masters of the previous century
+with a versatility truly surprising. Winckelmann, to whom he
+had been recommended, did not hesitate to call him the Raphael
+of landscape. Yet in this branch of his practice he merely
+imitated Salvator Rosa and Everdingen. He was more successful
+in aping the style of Rembrandt, and numerous examples of this
+habit may be found in the galleries of St Petersburg, Vienna and
+Dresden. At Dresden, indeed, there are pictures acknowledged
+to be his, bearing the fictitious dates of 1636 and 1638, and the
+name of Rembrandt. Among Dietrich&rsquo;s cleverest reproductions
+we may account that of Ostade&rsquo;s manner in the &ldquo;Itinerant
+Singers&rdquo; at the National Gallery. His skill in catching the
+character of the later masters of Holland is shown in candlelight
+scenes, such as the &ldquo;Squirrel and the Peep-Show&rdquo; at St
+Petersburg, where we are easily reminded of Godfried Schalcken.
+Dietrich tried every branch of art except portraits, painting
+Italian and Dutch views alternately with Scripture scenes and
+still life. In 1741 he was appointed court painter to Augustus III.
+at Dresden, with an annual salary of 400 thalers (£60), conditional
+on the production of four cabinet pictures a year. This condition,
+no doubt, accounts for the presence of fifty-two of the master&rsquo;s
+panels and canvases in one of the rooms at the Dresden museum.
+Dietrich, though popular and probably the busiest artist of his
+time, never produced anything of his own; and his imitations
+are necessarily inferior to the originals which he affected to copy.
+His best work is certainly that which he gave to engravings.
+A collection of these at the British Museum, produced on the
+general lines of earlier men, such as Ostade and Rembrandt,
+reveal both spirit and skill. Dietrich, after his return from the
+Peninsula, generally signed himself &ldquo;Dietericij,&rdquo; and with this
+signature most of his extant pictures are inscribed. He died at
+Dresden, after he had successively filled the important appointments
+of director of the school of painting at the Meissen porcelain
+factory and professor of the Dresden academy of arts.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIETRICH OF BERN,<a name="ar78" id="ar78"></a></span> the name given in German popular
+poetry to Theodoric the Great. The legendary history of Dietrich
+differs so widely from the life of Theodoric that it has been
+suggested that the two were originally unconnected. Medieval
+<span class="pagenum"><a name="page222" id="page222"></a>222</span>
+chroniclers, however, repeatedly asserted the identity of Dietrich
+and Theodoric, although the more critical noted the anachronisms
+involved in making Ermanaric (d. 376) and Attila (d. 453) contemporary
+with Theodoric (b. 455). That the legend is based
+on vague historical reminiscences is proved by the retention of
+the names of Theodoric (Thiuda-reiks, Dietrich) and his father
+Theudemir (Dietmar), by Dietrich&rsquo;s connexion with Bern
+(Verona) and Raben (Ravenna). Something of the Gothic king&rsquo;s
+character descended to Dietrich, familiarly called the Berner,
+the favourite of German medieval saga heroes, although his
+story did not leave the same mark on later German literature as
+did that of the Nibelungs. The cycle of songs connected with his
+name in South Germany is partially preserved in the Heldenbuch
+(<i>q.v.</i>) in <i>Dietrich&rsquo;s Flucht</i>, the <i>Rabenschlacht</i> and <i>Alpharts Tod</i>;
+but it was reserved for an Icelandic author, writing in Norway
+in the 13th century, to compile, with many romantic additions, a
+consecutive account of Dietrich. In this Norse prose redaction,
+known as the <i>Vilkina Saga</i>, or more correctly the <i>Thidrekssaga</i>,
+is incorporated much extraneous matter from the Nibelungen
+and Wayland legends, in fact practically the whole of south
+German heroic tradition.</p>
+
+<p>There are traces of a form of the Dietrich legend in which he
+was represented as starting out from Byzantium, in accordance
+with historical tradition, for his conquest of Italy. But this
+early disappeared, and was superseded by the existing legend,
+in which, perhaps by an &ldquo;epic fusion&rdquo; with his father Theudemir,
+he was associated with Attila, and then by an easy transition
+with Ermanaric. Dietrich was driven from his kingdom of
+Bern by his uncle Ermanaric. After years of exile at the court
+of Attila he returned with a Hunnish army to Italy, and defeated
+Ermanaric in the Rabenschlacht, or battle of Ravenna. Attila&rsquo;s
+two sons, with Dietrich&rsquo;s brother, fell in the fight, and Dietrich
+returned to Attila&rsquo;s court to answer for the death of the young
+princes. This very improbable renunciation of the advantages of
+his victory suggests that in the original version of the story the
+Rabenschlacht was a defeat. In the poem of <i>Ermenrichs Tod</i>
+he is represented as slaying Ermanaric, as in fact Theodoric slew
+Odoacer. &ldquo;Otacher&rdquo; replaces Ermanaric as his adversary in the
+<i>Hildebrandslied</i>, which relates how thirty years after the earlier
+attempt he reconquered his Lombard kingdom. Dietrich&rsquo;s long
+residence at Attila&rsquo;s court represents the youth and early manhood
+of Theodoric spent at the imperial court and fighting in the
+Balkan peninsula, and, in accordance with epic custom, the period
+of exile was adorned with war-like exploits, with fights with
+dragons and giants, most of which had no essential connexion
+with the cycle. The romantic poems of <i>König Laurin</i>, <i>Sigenot</i>,
+<i>Eckenlied</i> and <i>Virginal</i> are based largely on local traditions
+originally independent of Dietrich. The court of Attila (Etzel)
+was a ready bridge to the Nibelungen legend. In the final catastrophe
+he was at length compelled, after steadily holding aloof
+from the combat, to avenge the slaughter of his Amelungs by
+the Burgundians, and delivered Hagen bound into the hands of
+Kriemhild. The flame breath which anger induced from him
+shows the influence of pure myth, but the tales of his demonic
+origin and of his being carried off by the devil in the shape of a
+black horse may safely be put down to the clerical hostility to
+Theodoric&rsquo;s Arianism.</p>
+
+<p>Generally speaking, Dietrich of Bern was the wise and just
+monarch as opposed to Ermanaric, the typical tyrant of Germanic
+legend. He was invariably represented as slow of provocation
+and a friend of peace, but once roused to battle not even Siegfried
+could withstand his onslaught. But probably Dietrich&rsquo;s fight
+with Siegfried in Kriemhild&rsquo;s rose garden at Worms is a late
+addition to the Rosengarten myth. The chief heroes of the
+Dietrich cycle are his tutor and companion in arms, Hildebrand
+(see <span class="sc"><a href="#artlinks">Hildebrand, Lay of</a></span>), with his nephews the Wolfings
+Alphart and Wolfhart; Wittich, who renounced his allegiance
+to Dietrich and slew the sons of Attila; Heime and Biterolf.</p>
+
+<div class="condensed">
+<p>The contents of the poems dealing with the Dietrich cycle are
+summarized by Uhland in <i>Schriften zur Geschichte der Dichtung und
+Sage</i> (Stuttgart, 1873). The <i>Thidrekssaga</i> (ed. C. Unger, Christiania,
+1853) is translated into German by F. H. v. der Hagen in <i>Altdeutsche
+und altnordische Heldensagen</i> (vols. i. and ii. 3rd ed., Breslau, 1872).
+A summary of it forms the concluding chapter of T. Hodgkin&rsquo;s
+<i>Theodoric the Goth</i> (1891). The variations in the Dietrich legend in
+the Latin historians, in Old and Middle High German literature,
+and in the northern saga, can be studied in W. Grimm&rsquo;s <i>Deutsche
+Heldensage</i> (2nd ed., Berlin, 1867). There is a good account in English
+in F. E. Sandbach&rsquo;s <i>Heroic Saga-cycle of Dietrich of Bern</i> (1906),
+forming No. 15 of Alfred Nutt&rsquo;s <i>Popular Studies in Mythology</i>, and
+another in M. Bentinck Smith&rsquo;s translation of Dr O. L. Jiriczek&rsquo;s
+<i>Deutsche Heldensage</i> (<i>Northern Legends</i>, London, 1902). For modern
+German authorities and commentators see B. Symons, &ldquo;Deutsche
+Heldensage&rdquo; in H. Paul&rsquo;s <i>Grd. d. german. Phil.</i> (Strassburg, new ed.,
+1905); also Goedeke, <i>Geschichte der deutschen Dichtung</i> (i. 241-246).</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIEZ, FRIEDRICH CHRISTIAN<a name="ar79" id="ar79"></a></span> (1794-1876), German
+philologist, was born at Giessen, in Hesse-Darmstadt, on the 15th
+of March 1794. He was educated first at the gymnasium and
+then at the university of his native town. There he studied
+classics under Friedrich Gottlieb Welcker (1784-1868) who had
+just returned from a two years&rsquo; residence in Italy to fill the chair
+of archaeology and Greek literature. It was Welcker who
+kindled in him a love of Italian poetry, and thus gave the first
+bent to his genius. In 1813 he joined the Hesse corps as a
+volunteer and served in the French campaign. Next year he
+returned to his books, and this short taste of military service was
+the only break in a long and uneventful life of literary labours.
+By his parents&rsquo; desire he applied himself for a short time to law,
+but a visit to Goethe in 1818 gave a new direction to his studies,
+and determined his future career. Goethe had been reading
+Raynouard&rsquo;s <i>Selections from the Romance Poets</i>, and advised the
+young scholar to explore the rich mine of Provençal literature
+which the French savant had opened up. This advice was
+eagerly followed, and henceforth Diez devoted himself to Romance
+literature. He thus became the founder of Romance philology.
+After supporting himself for some years by private teaching, he
+removed in 1822 to Bonn, where he held the position of privatdocent.
+In 1823 he published his first work, <i>An Introduction
+to Romance Poetry</i>; in the following year appeared <i>The Poetry
+of the Troubadours</i>, and in 1829 <i>The Lives and Works of the
+Troubadours</i>. In 1830 he was called to the chair of modern
+literature. The rest of his life was mainly occupied with the
+composition of the two great works on which his fame rests, the
+<i>Grammar of the Romance Languages</i> (1836-1844), and the <i>Lexicon
+of the Romance Languages&mdash;Italian, Spanish and French</i> (1853);
+in these two works Diez did for the Romance group of languages
+what Jacob Grimm did for the Teutonic family. He died at
+Bonn on the 29th of May 1876.</p>
+
+<div class="condensed">
+<p>The earliest French philologists, such as Perion and Henri Estienne,
+had sought to discover the origin of French in Greek and even in
+Hebrew. For more than a century Ménage&rsquo;s <i>Etymological Dictionary</i>
+held the field without a rival. Considering the time at which it was
+written (1650), it was a meritorious work, but philology was then in
+the empirical stage, and many of Ménage&rsquo;s derivations (such as
+that of &ldquo;rat&rdquo; from the Latin &ldquo;mus,&rdquo; or of &ldquo;haricot&rdquo; from &ldquo;faba&rdquo;)
+have since become bywords among philologists. A great advance
+was made by Raynouard, who by his critical editions of the works
+of the Troubadours, published in the first years of the 19th century,
+laid the foundations on which Diez afterwards built. The difference
+between Diez&rsquo;s method and that of his predecessors is well stated by
+him in the preface to his dictionary. In sum it is the difference
+between science and guess-work. The scientific method is to follow
+implicitly the discovered principles and rules of phonology, and not
+to swerve a foot&rsquo;s breadth from them unless plain, actual exceptions
+shall justify it; to follow the genius of the language, and by cross-questioning
+to elicit its secrets; to gauge each letter and estimate
+the value which attaches to it in each position; and lastly to possess
+the true philosophic spirit which is prepared to welcome any new
+fact, though it may modify or upset the most cherished theory.
+Such is the historical method which Diez pursues in his grammar
+and dictionary. To collect and arrange facts is, as he tells us, the
+sole secret of his success, and he adds in other words the famous
+apophthegm of Newton, &ldquo;hypotheses non fingo.&rdquo; The introduction
+to the grammar consists of two parts:&mdash;the first discusses the Latin,
+Greek and Teutonic elements common to the Romance languages;
+the second treats of the six dialects separately, their origin and the
+elements peculiar to each. The grammar itself is divided into four
+books, on phonology, on flexion, on the formation of words by
+composition and derivation, and on syntax.</p>
+
+<p>His dictionary is divided into two parts. The first contains words
+common to two at least of the three principal groups of Romance:&mdash;Italian,
+Spanish and Portuguese, and Provençal and French. The
+Italian, as nearest the original, is placed at the head of each article.
+<span class="pagenum"><a name="page223" id="page223"></a>223</span>
+The second part treats of words peculiar to one group. There is no
+separate glossary of Wallachian.</p>
+
+<p>Of the introduction to the grammar there is an English translation
+by C. B. Cayley. The dictionary has been published in a remodelled
+form for English readers by T. C. Donkin.</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIEZ,<a name="ar80" id="ar80"></a></span> a town of Germany, in the Prussian province of Hesse-Nassau,
+romantically situated in the deep valley of the Lahn,
+here crossed by an old bridge, 30 m. E. from Coblenz on the
+railway to Wetzlar. Pop. 4500. It is overlooked by a former
+castle of the counts of Nassau-Dillenburg, now a prison. Close
+by, on an eminence above the river, lies the castle of Oranienstein,
+formerly a Benedictine nunnery and now a cadet school,
+with beautiful gardens. There are a Roman Catholic and two
+Evangelical churches. The new part of the town is well built
+and contains numerous pretty villa residences. In addition to
+extensive iron-works there are sawmills and tanneries. In the
+vicinity are Fachingen, celebrated for its mineral waters, and
+the majestic castle of Schaumburg belonging to the prince of
+Waldeck-Pyrmont.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIFFERENCES, CALCULUS OF<a name="ar81" id="ar81"></a></span> (<i>Theory of Finite Differences</i>),
+that branch of mathematics which deals with the successive
+differences of the terms of a series.</p>
+
+<p>1. The most important of the cases to which mathematical
+methods can be applied are those in which the terms of the series
+are the values, taken at stated intervals (regular or irregular), of
+a continuously varying quantity. In these cases the formulae
+of finite differences enable certain quantities, whose exact value
+depends on the law of variation (<i>i.e.</i> the law which governs the
+relative magnitude of these terms) to be calculated, often with
+great accuracy, from the given terms of the series, without
+explicit reference to the law of variation itself. The methods
+used may be extended to cases where the series is a double series
+(series of double entry), <i>i.e.</i> where the value of each term depends
+on the values of a pair of other quantities.</p>
+
+<p>2. The <i>first differences</i> of a series are obtained by subtracting
+from each term the term immediately preceding it. If these are
+treated as terms of a new series, the first differences of this series
+are the <i>second differences</i> of the original series; and so on.
+The successive differences are also called <i>differences of the first,
+second, ... order</i>. The differences of successive orders are most
+conveniently arranged in successive columns of a table thus:&mdash;</p>
+
+<table class="ws" summary="Contents">
+
+<tr><td class="tcc allb">Term.</td> <td class="tcc allb">1st Diff.</td> <td class="tcc allb">2nd Diff.</td> <td class="tcc allb">3rd Diff.</td> <td class="tcc allb">4th Diff.</td></tr>
+
+<tr><td class="tcc lb rb">a</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td></tr>
+<tr><td class="tcc lb rb">&nbsp;</td> <td class="tcc rb">b &minus; a</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td></tr>
+<tr><td class="tcc lb rb">b</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">c &minus; 2b + a</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td></tr>
+<tr><td class="tcc lb rb">&nbsp;</td> <td class="tcc rb">c &minus; b</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">d &minus; 3c + 3b &minus; a</td> <td class="tcc rb">&nbsp;</td></tr>
+<tr><td class="tcc lb rb">c</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">d &minus; 2c + b</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">e &minus; 4d + 6c &minus; 4b + a</td></tr>
+<tr><td class="tcc lb rb">&nbsp;</td> <td class="tcc rb">d &minus; c</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">e &minus; 3d + 3c &minus; b</td> <td class="tcc rb">&nbsp;</td></tr>
+<tr><td class="tcc lb rb">d</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">e &minus; 2d + c</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td></tr>
+<tr><td class="tcc lb rb">&nbsp;</td> <td class="tcc rb">e &minus; d</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td></tr>
+<tr><td class="tcc lb rb bb">e</td> <td class="tcc rb bb">&nbsp;</td> <td class="tcc rb bb">&nbsp;</td> <td class="tcc rb bb">&nbsp;</td> <td class="tcc rb bb">&nbsp;</td></tr>
+</table>
+
+<p class="center pt2"><i>Algebra of Differences and Sums.</i></p>
+
+<table class="nobctr" style="float: left; width: 340px;" summary="Illustration">
+<tr><td class="figleft1"><img style="width:288px; height:124px" src="images/img223.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 1.</span></td></tr></table>
+
+
+<p>3. The formal relations between the terms of the series and the
+differences may be seen by comparing the arrangements (A) and (B)
+in fig. 1. In (A) the various terms and differences are the same as in
+§ 2, but placed differently. In
+(B) we take a new series of
+terms &alpha;, &beta;, &gamma;, &delta;, commencing
+with the same term &alpha;, and take
+the successive sums of pairs of
+terms, instead of the successive
+differences, but place them to
+the left instead of to the right.
+It will be seen, in the first
+place, that the successive terms
+in (A), reading downwards to the right, and the successive
+terms in (B), reading downwards to the left, consist each of
+a series of terms whose coefficients follow the binomial law; <i>i.e.</i>
+the coefficients in b &minus; a, c &minus; 2b + a, d &minus; 3c + 3b &minus; a, ... and in
+&alpha; + &beta;, &alpha; + 2&beta; + &gamma;, &alpha; + 3&beta; + 3&gamma; + &delta;, ... are respectively the same as
+in y &minus; x, (y &minus; x)², (y &minus; x)³, ... and in x + y, (x + y)², (x + y)³,....
+In the second place, it will be seen that the relations between the
+various terms in (A) are identical with the relations between the
+similarly placed terms in (B); <i>e.g.</i> &beta; + &gamma; is the difference of &alpha; + 2&beta; + &gamma;
+and &alpha; + &beta;, just as c &minus; b is the difference of c and b: and d &minus; c is the sum
+of c &minus; b and d &minus; 2c + b, just as &beta; + 2&gamma; + &delta; is the sum of &beta; + &gamma; and &gamma; + &delta;.
+Hence if we take &beta;, &gamma;, &delta;, ... of (B) as being the same as b &minus; a,
+c &minus; 2b + a, d &minus; 3c + 3b &minus; a, ... of (A), all corresponding terms in the
+two diagrams will be the same.</p>
+
+<p>Thus we obtain the two principal formulae connecting terms and
+differences. If we provisionally describe b &minus; a, c &minus; 2b + a, ... as the
+first, second, ... differences of the particular term a (§ 7), then
+(i.) the nth difference of a is</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">l &minus; nk + ... + (-1)<span class="sp">n-2</span></td>
+    <td>n·n &minus; 1</td>
+    <td rowspan="2">c + (-1)<span class="sp">n-1</span> nb + (-1)<span class="sp">n</span> a,</td></tr>
+<tr><td class="denom">1·2</td></tr>
+</table>
+
+<p class="noind">where l, k ... are the (n + 1)th, nth, ... terms of the series a, b, c,
+...; the coefficients being those of the terms in the expansion of
+(y &minus; x)<span class="sp">n</span>: and (ii.) the (n + 1)th term of the series, <i>i.e.</i> the nth term
+after a, is</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">a + n&beta; +</td> <td>n·n &minus; 1</td> <td rowspan="2">&gamma; + ...</td></tr>
+<tr><td class="denom">1·2</td></tr></table>
+
+<p class="noind">where &beta;, &gamma;, ... are the first, second, ... differences of a; the
+coefficients being those of the terms in the expansion of (x + y)<span class="sp">n</span>.</p>
+
+<p>4. Now suppose we treat the terms a, b, c, ... as being themselves
+the first differences of another series. Then, if the first term
+of this series is N, the subsequent terms are N + a, N + a + b, N + a +
+b + c, ...; <i>i.e.</i> the difference between the (n + 1)th term and the
+first term is the sum of the first n terms of the original series. The
+term N, in the diagram (A), will come above and to the left of a; and
+we see, by (ii.) of § 3, that the sum of the first n terms of the original
+series is</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2" class="f200 np">(</td>
+ <td rowspan="2">N + na +</td> <td>n·n &minus; 1</td>
+ <td rowspan="2">&beta; + ...</td> <td rowspan="2" class="f200 np">)</td>
+ <td rowspan="2">&minus; N = na +</td> <td>n·n &minus; 1</td> <td rowspan="2">&beta; +</td>
+ <td>n·n &minus; 1·n &minus; 2</td> <td rowspan="2">&gamma; + ...</td></tr>
+<tr><td class="denom">1·2</td> <td class="denom">1·2</td> <td class="denom">1 · 2 · 3</td></tr>
+</table>
+
+<p>5. As an example, take the arithmetical series</p>
+
+<p class="center">a, a + p, a + 2p, ...</p>
+
+<p class="noind">The first differences are p, p, p, ... and the differences of any higher
+order are zero. Hence, by (ii.) of § 3, the (n + 1)th term is a + np, and,
+by § 4, the sum of the first n terms is na + ½n(n &minus; 1)p = ½n{2a + (n &minus; 1)p}.</p>
+
+<p>6 As another example, take the series 1, 8, 27, ... the terms of
+which are the cubes of 1, 2, 3, ... The first, second and third
+differences of the first term are 7, 12 and 6, and it may be shown
+(§ 14 (i.)) that all differences of a higher order are zero. Hence the
+sum of the first n terms is</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">n + 7</td> <td>n·n &minus; 1</td>
+ <td rowspan="2">+ 12</td> <td>n·n &minus; 1·n &minus; 2</td>
+ <td rowspan="2">+ 6</td> <td>n·n &minus; 1·n &minus; 2·n &minus; 3</td>
+ <td rowspan="2">= ¼n<span class="sp">4</span> + ½n³ + ¼n² = {½n (n + 1)}².</td></tr>
+<tr><td class="denom">1·2</td> <td class="denom">1·2·3</td> <td class="denom">1·2·3·4</td></tr>
+</table>
+
+<p>7. In § 3 we have described b &minus; a, c &minus; 2b + a, ... as the first,
+second, ... differences of a. This ascription of the differences
+to particular terms of the series is quite arbitrary. If we read the
+differences in the table of § 2 upwards to the right instead of downwards
+to the right, we might describe e &minus; d, e &minus; 2d + c, ... as the
+first, second, ... differences of e. On the other hand, the term of
+greatest weight in c &minus; 2b + a, <i>i.e.</i> the term which has the numerically
+greatest coefficient, is b, and therefore c &minus; 2b + a might properly be
+regarded as the second difference of b, and similarly e &minus; 4d + 6c &minus; 4b + a
+might be regarded as the fourth difference of c. These three
+methods of regarding the differences lead to three different systems
+of notation, which are described in §§ 9, 10 and 11.</p>
+
+<p class="center pt2"><i>Notation of Differences and Sums.</i></p>
+
+<p>8. It is convenient to denote the terms a, b, c, ... of the series
+by u<span class="su">0</span>, u<span class="su">1</span>, u<span class="su">2</span>, u<span class="su">3</span>, ... If we merely have the terms of the series, u<span class="su">n</span>
+may be regarded as meaning the (n + 1)th term. Usually, however,
+the terms are the values of a quantity u, which is a function of
+another quantity x, and the values of x, to which a, b, c, ... correspond,
+proceed by a constant difference h. If x<span class="su">0</span> and u<span class="su">0</span> are a pair
+of corresponding values of x and u, and if any other value x<span class="su">0</span> + mh of x
+and the corresponding value of u are denoted by x<span class="su">m</span> and u<span class="su">m</span>, then
+the terms of the series will be ... u<span class="su">n-2</span>, u<span class="su">n-1</span>, u<span class="su">n</span>, u<span class="su">n+1</span>, u<span class="su">n+2</span> ..., corresponding
+to values of x denoted by  ... x<span class="su">n-2</span>, x<span class="su">n-1</span>, x<span class="su">n</span>, x<span class="su">n+1</span>, x<span class="su">n+2</span>....</p>
+
+<p>9. In the <i>advancing-difference notation</i> u<span class="su">n+1</span> &minus; u<span class="su">n</span> is denoted by
+&Delta;u<span class="su">n</span>. The differences &Delta;u<span class="su">0</span>, &Delta;u<span class="su">1</span>, &Delta;u<span class="su">2</span> ... may then be regarded as
+values of a function &Delta;u corresponding to values of x proceeding by
+constant difference h; and therefore &Delta;u<span class="su">n+1</span> &minus; &Delta;u<span class="su">n</span> denoted by &Delta;&Delta;u<span class="su">n</span>,
+or, more briefly, &Delta;²u<span class="su">n</span>; and so on. Hence the table of differences in
+§ 2, with the corresponding values of x and of u placed opposite each
+other in the ordinary manner of mathematical tables, becomes</p>
+
+<table class="ws" summary="Contents">
+
+<tr><td class="tcc allb">x</td> <td class="tcc allb">u</td> <td class="tcc allb">1st Diff.</td> <td class="tcc allb">2nd Diff.</td> <td class="tcc allb">3rd Diff.</td> <td class="tcc allb">4th Diff.</td></tr>
+<tr><td class="tcc lb rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td></tr>
+<tr><td class="tcc lb rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td></tr>
+<tr><td class="tcc lb rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td></tr>
+<tr><td class="tcc lb rb">x<span class="su">n-2</span></td> <td class="tcc rb">u<span class="su">n-2</span></td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&Delta;²u<span class="su">n-3</span></td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&Delta;<span class="sp">4</span>u<span class="su">n-4</span> ...</td></tr>
+<tr><td class="tcc lb rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&Delta;u<span class="su">n-2</span></td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&Delta;³u<span class="su">n-3</span></td> <td class="tcc rb">&nbsp;</td></tr>
+<tr><td class="tcc lb rb">x<span class="su">n-1</span></td> <td class="tcc rb">u<span class="su">n-1</span></td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&Delta;²u<span class="su">n-2</span></td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&Delta;<span class="sp">4</span>u<span class="su">n-3</span> ...</td></tr>
+<tr><td class="tcc lb rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&Delta;u<span class="su">n-1</span></td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&Delta;³u<span class="su">n-2</span></td> <td class="tcc rb">&nbsp;</td></tr>
+<tr><td class="tcc lb rb">x<span class="su">n</span>&ensp;</td> <td class="tcc rb">u<span class="su">n</span>&ensp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&Delta;²u<span class="su">n-1</span></td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&Delta;<span class="sp">4</span>u<span class="su">n-2</span> ...</td></tr>
+<tr><td class="tcc lb rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&Delta;u<span class="su">n</span>&ensp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&Delta;³u<span class="su">n-1</span></td> <td class="tcc rb">&nbsp;</td></tr>
+<tr><td class="tcc lb rb">x<span class="su">n+1</span></td> <td class="tcc rb">u<span class="su">n+1</span></td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&Delta;²u<span class="su">n</span>&ensp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&Delta;<span class="sp">4</span>u<span class="su">n-1</span> ...</td></tr>
+<tr><td class="tcc lb rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&Delta;u<span class="su">n+1</span></td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&Delta;³u<span class="su">n</span>&ensp;</td> <td class="tcc rb">&nbsp;</td></tr>
+<tr><td class="tcc lb rb">x<span class="su">n+2</span></td> <td class="tcc rb">u<span class="su">n+2</span></td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&Delta;²u<span class="su">n+1</span></td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&Delta;<span class="sp">4</span>u<span class="su">n</span>&ensp;   ...</td></tr>
+<tr><td class="tcc lb rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td></tr>
+<tr><td class="tcc lb rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td></tr>
+<tr><td class="tcc lb rb bb">·</td> <td class="tcc rb bb">·</td> <td class="tcc rb bb">·</td> <td class="tcc rb bb">·</td> <td class="tcc rb bb">·</td> <td class="tcc rb bb">·</td></tr>
+</table>
+
+<p>The terms of the series of which ... u<span class="su">n-1</span>, u<span class="su">n</span>, u<span class="su">n+1</span>, ... are
+the first differences are denoted by &Sigma;u, with proper suffixes, so
+<span class="pagenum"><a name="page224" id="page224"></a>224</span>
+that this series is ... &Sigma;u<span class="su">n-1</span>, &Sigma;u<span class="su">n</span>, &Sigma;u<span class="su">n+1</span>.... The suffixes are
+chosen so that we may have &Delta;&Sigma;u<span class="su">n</span> = u<span class="su">n</span>, whatever n may be; and
+therefore (§ 4) &Sigma;u<span class="su">n</span> may be regarded as being the sum of the terms
+of the series up to and including u<span class="su">n-1</span>. Thus if we write &Sigma;u<span class="su">n-1</span> =
+C + u<span class="su">n-2</span>, where C is any constant, we shall have</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcl">&Sigma;u<span class="su">n</span> = &Sigma;u<span class="su">n-1</span> + &Delta;&Sigma;u<span class="su">n-1</span> = C + u<span class="su">n-2</span> + u<span class="su">n-1</span>,</td></tr>
+<tr><td class="tcl">&Sigma;u<span class="su">n+1</span> = C + u<span class="su">n-2</span> + u<span class="su">n-1</span> + u<span class="su">n</span>,</td></tr>
+</table>
+
+<p class="noind">and so on. This is true whatever C may be, so that the knowledge
+of ... u<span class="su">n-1</span>, u<span class="su">n</span>, ... gives us no knowledge of the exact value
+of &Sigma;u<span class="su">n</span>; in other words, C is an arbitrary constant, the value of
+which must be supposed to be the same throughout any operations in
+which we are concerned with values of &Sigma;u corresponding to different
+suffixes.</p>
+
+<p>There is another symbol E, used in conjunction with u to denote
+the next term in the series. Thus Eu<span class="su">n</span> means u<span class="su">n+1</span>, so that
+Eu<span class="su">n</span> = u<span class="su">n</span> + &Delta;u<span class="su">n</span>.</p>
+
+<p>10. Corresponding to the advancing-difference notation there is
+a <i>receding-difference</i> notation, in which u<span class="su">n+1</span> &minus; u<span class="su">n</span> is regarded as
+a difference of u<span class="su">n+1</span>, and may be denoted by &Delta;&prime;u<span class="su">n+1</span>, and similarly
+u<span class="su">n+1</span> &minus; 2u<span class="su">n</span> + u<span class="su">n-1</span> may be denoted by &Delta;&prime;²u<span class="su">n+1</span>. This notation is only
+required for certain special purposes, and the usage is not settled
+(§ 19 (ii.)).</p>
+
+<p>11. The <i>central-difference</i> notation depends on treating
+u<span class="su">n+1</span> &minus; 2u<span class="su">n</span> &minus; u<span class="su">n-1</span> as the second <span class="correction" title="amended from dfference">difference</span> of u<span class="su">n</span>, and therefore as
+corresponding to the value x<span class="su">n</span>; but there is no settled system of
+notation. The following seems to be the most convenient. Since u<span class="su">n</span> is
+a function of x<span class="su">n</span>, and the second difference u<span class="su">n+2</span> &minus; 2u<span class="su">n+1</span> + u<span class="su">n</span> is a function
+of x<span class="su">n+1</span>, the first difference u<span class="su">n+1</span> &minus; u<span class="su">n</span> must be regarded as a function
+of x<span class="su">n+1/2</span>, <i>i.e.</i> of ½(x<span class="su">n</span> + x<span class="su">n+1</span>). We therefore write u<span class="su">n+1</span> &minus; u<span class="su">n</span> = &delta;u<span class="su">n+1/2</span>,
+and each difference in the table in § 9 will have the same suffix
+as the value of x in the same horizontal line; or, if the difference
+is of an odd order, its suffix will be the means of those of the two
+nearest values of x. This is shown in the table below.</p>
+
+<p>In this notation, instead of using the symbol E, we use a symbol &mu;
+to denote the mean of two consecutive values of u, or of two consecutive
+differences of the same order, the suffixes being assigned on the
+same principle as in the case of the differences. Thus</p>
+
+<p class="center">&mu;u<span class="su">n+1/2</span> = ½(u<span class="su">n</span> + u<span class="su">n+1</span>, &mu;&delta;u<span class="su">n</span> = ½(&delta;u<span class="su">n-1/2</span> + &delta;u<span class="su">n+1/2</span>, &amp;c.</p>
+
+<p>If we take the means of the differences of odd order immediately
+above and below the horizontal line through any value of x, these
+means, with the differences of even order in that line, constitute the
+<i>central differences</i> of the corresponding value of u. Thus the table
+of central differences is as follows, the values obtained as means
+being placed in brackets to distinguish them from the actual
+differences:&mdash;</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcc allb">x</td> <td class="tcc allb">u</td> <td class="tcc allb">1st Diff.</td> <td class="tcc allb">2nd Diff.</td> <td class="tcc allb">3rd Diff.</td> <td class="tcc allb">4th Diff.</td></tr>
+
+<tr><td class="tcc lb rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td></tr>
+<tr><td class="tcc lb rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td></tr>
+<tr><td class="tcc lb rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td></tr>
+<tr><td class="tcc lb rb">x<span class="su">n-2</span></td> <td class="tcc rb">u<span class="su">n-2</span></td> <td class="tcc rb">(&mu;&delta;u<span class="su">n-2</span>)</td> <td class="tcc rb">&delta;²u<span class="su">n-2</span></td> <td class="tcc rb">(&mu;&delta;³u<span class="su">n-2</span>)</td> <td class="tcc rb">&delta;<span class="sp">4</span>u<span class="su">n-2</span> ...</td></tr>
+<tr><td class="tcc lb rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&delta;u<span class="su">n-3/2</span></td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&delta;³u<span class="su">n-3/2</span></td> <td class="tcc rb">&nbsp;</td></tr>
+<tr><td class="tcc lb rb">x<span class="su">n-1</span></td> <td class="tcc rb">u<span class="su">n-1</span></td> <td class="tcc rb">(&mu;&delta;u<span class="su">n-1</span>)</td> <td class="tcc rb">&delta;²u<span class="su">n-1</span></td> <td class="tcc rb">(&mu;&delta;³u<span class="su">n-1</span>)</td> <td class="tcc rb">&delta;<span class="sp">4</span>u<span class="su">n-1</span> ...</td></tr>
+<tr><td class="tcc lb rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&delta;u<span class="su">n-1/2</span></td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&delta;³u<span class="su">n-2</span></td> <td class="tcc rb">&nbsp;</td></tr>
+<tr><td class="tcc lb rb">x<span class="su">n</span></td> <td class="tcc rb">u<span class="su">n</span>&ensp;</td> <td class="tcc rb">(&mu;&delta;u<span class="su">n</span>)&ensp;</td> <td class="tcc rb">&delta;²u<span class="su">n</span>&ensp;</td> <td class="tcc rb">(&mu;&delta;³u<span class="su">n</span>)&ensp;</td> <td class="tcc rb">&delta;<span class="sp">4</span>u<span class="su">n</span>&ensp;   ...</td></tr>
+<tr><td class="tcc lb rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&delta;u<span class="su">n+1/2</span></td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&delta;³u<span class="su">n+1/2</span></td> <td class="tcc rb">&nbsp;</td></tr>
+<tr><td class="tcc lb rb">x<span class="su">n+1</span></td> <td class="tcc rb">u<span class="su">n+1</span></td> <td class="tcc rb">(&mu;&delta;u<span class="su">n+1</span>)</td> <td class="tcc rb">&delta;²u<span class="su">n+1</span></td> <td class="tcc rb">(&mu;&delta;³u<span class="su">n+1</span>)</td> <td class="tcc rb">&delta;<span class="sp">4</span>u<span class="su">n+1</span> ...</td></tr>
+<tr><td class="tcc lb rb">&nbsp;</td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&delta;u<span class="su">n+3/2</span></td> <td class="tcc rb">&nbsp;</td> <td class="tcc rb">&delta;³u<span class="su">n+3/2</span></td> <td class="tcc rb">&nbsp;</td></tr>
+<tr><td class="tcc lb rb">x<span class="su">n+2</span></td> <td class="tcc rb">u<span class="su">n+2</span></td> <td class="tcc rb">(&mu;&delta;u<span class="su">n+2</span>)</td> <td class="tcc rb">&delta;²u<span class="su">n+2</span></td> <td class="tcc rb">(&mu;&delta;³u<span class="su">n+2</span>)</td> <td class="tcc rb">&delta;<span class="sp">4</span>u<span class="su">n+2</span> ...</td></tr>
+<tr><td class="tcc lb rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td></tr>
+<tr><td class="tcc lb rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td></tr>
+<tr><td class="tcc lb rb bb">·</td> <td class="tcc rb bb">·</td> <td class="tcc rb bb">·</td> <td class="tcc rb bb">·</td> <td class="tcc rb bb">·</td> <td class="tcc rb bb">·</td></tr>
+</table>
+
+<p>Similarly, by taking the means of consecutive values of u and also
+of consecutive differences of even order, we should get a series of
+terms and differences central to the intervals x<span class="su">n-2</span> to x<span class="su">n-1</span>, x<span class="su">n-1</span> to
+x<span class="su">n</span>, ....</p>
+
+<p>The terms of the series of which the values of u are the first differences
+are denoted by &sigma;u, with suffixes on the same principle; the
+suffixes being chosen so that &delta;&sigma;u<span class="su">n</span> shall be equal to u<span class="su">n</span>. Thus, if</p>
+
+<p class="center">&sigma;u<span class="su">n-3/2</span> = C + u<span class="su">n-2</span>,</p>
+
+<p class="noind">then</p>
+
+<p class="center">&sigma;u<span class="su">n-1/2</span> = C + u<span class="su">n-2</span> + u<span class="su">n-1</span>, &sigma;<span class="su">n+1/2</span> = C + u<span class="su">n-2</span> + u<span class="su">n-1</span> + u<span class="su">n</span>, &amp;c.,</p>
+
+<p class="noind">and also</p>
+
+<p class="center">&mu;&sigma;u<span class="su">n-1</span> = C + u<span class="su">n-2</span> + ½u<span class="su">n-1</span>, &mu;&sigma;u<span class="su">n</span> = C + u<span class="su">n-2</span> + u<span class="su">n-1</span> + ½u<span class="su">n</span>, &amp;c.,</p>
+
+<p class="noind">C being an arbitrary constant which must remain the same throughout
+any series of operations.</p>
+
+<p class="center pt2"><i>Operators and Symbolic Methods.</i></p>
+
+<p>12. There are two further stages in the use of the symbols &Delta;, &Sigma;,
+&delta;, &sigma;, &amp;c., which are not essential for elementary treatment but
+lead to powerful methods of deduction.</p>
+
+<p>(i.) Instead of treating &Delta;u as a function of x, so that &Delta;u<span class="su">n</span> means
+(&Delta;u)<span class="su">n</span>, we may regard &Delta; as denoting an <i>operation</i> performed on u,
+and take &Delta;u<span class="su">n</span> as meaning &Delta;.u<span class="su">n</span>. This applies to the other symbols
+E, &delta;, &amp;c., whether taken simply or in combination. Thus &Delta;Eu<span class="su">n</span>
+means that we first replace u<span class="su">n</span> by u<span class="su">n+1</span>, and then replace this by
+u<span class="su">n+2</span> &minus; u<span class="su">n+1</span>.</p>
+
+<p>(ii.) The operations &Delta;, E, &delta;, and &mu;, whether performed separately
+or in combination, or in combination also with numerical multipliers
+and with the operation of differentiation denoted by D (&equiv; d/dx),
+follow the ordinary rules of algebra: <i>e.g.</i> &Delta;(u<span class="su">n</span> + v<span class="su">n</span>) = &Delta;u<span class="su">n</span> + &Delta;v<span class="su">n</span>,
+&Delta;Du<span class="su">n</span> = D&Delta;u<span class="su">n</span>, &amp;c. Hence the symbols can be separated from the
+functions on which the operations are performed, and treated as
+if they were algebraical quantities. For instance, we have</p>
+
+<p class="center">E·u<span class="su">n</span> = u<span class="su">n+1</span> = u<span class="su">n</span> + &Delta;u<span class="su">n</span> = 1·u<span class="su">n</span> + &Delta;·u<span class="su">n</span>,</p>
+
+<p class="noind">so that we may write E = 1 + &Delta;, or &Delta; = E &minus; 1. The first of these is
+nothing more than a statement, in concise form, that if we take two
+quantities, subtract the first from the second, and add the result to
+the first, we get the second. This seems almost a truism. But, if
+we deduce E<span class="sp">n</span> = (1 + &Delta;)<span class="sp">n</span>, &Delta;<span class="sp">n</span> = (E-1)<span class="sp">n</span>, and expand by the binomial
+theorem and then operate on u<span class="su">0</span>, we get the general formulae</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">u<span class="su">n</span> = u<span class="su">0</span> + n&Delta;u<span class="su">0</span> +</td> <td>n·n &minus; 1</td>
+ <td rowspan="2">&Delta;<span class="sp">2</span>u<span class="su">0</span> + ... + &Delta;<span class="sp">n</span>u<span class="su">0</span>,</td></tr>
+<tr><td class="denom">1·2</td></tr></table>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&Delta;<span class="sp">n</span>u<span class="su">0</span> = u<span class="su">n</span> &minus; nu<span class="su">n-1</span> +</td> <td>n·n &minus; 1</td> <td rowspan="2">u<span class="su">n-2</span> + ... + (-1)<span class="sp">n</span>u<span class="su">0</span>,</td></tr>
+<tr><td class="denom">1·2</td></tr></table>
+
+<p class="noind">which are identical with the formulae in (ii.) and (i.) of § 3.</p>
+
+<p>(iii.) What has been said under (ii.) applies, with certain reservations,
+to the operations &Sigma; and &sigma;, and to the operation which represents
+integration. The latter is sometimes denoted by D<span class="sp">-1</span>; and, since
+&Delta;&Sigma;u<span class="su">n</span> = u<span class="su">n</span>, and &delta;&sigma;u<span class="su">n</span> = u<span class="su">n</span>, we might similarly replace &Sigma; and &sigma; by
+&Delta;<span class="sp">-1</span> and &delta;<span class="sp">-1</span>. These symbols can be combined with &Delta;, E, &amp;c.
+according to the ordinary laws of algebra, provided that proper
+account is taken of the arbitrary constants introduced by the
+operations D<span class="sp">-1</span>, &Delta;<span class="sp">-1</span>, &delta;<span class="sp">-1</span>.</p>
+
+<p class="center pt2"><i>Applications to Algebraical Series.</i></p>
+
+<p>13. <i>Summation of Series.</i>&mdash;If u<span class="su">r</span>, denotes the (r + 1)th term of a
+series, and if v<span class="su">r</span> is a function of r such that &Delta;v<span class="su">r</span> = u<span class="su">r</span> for all integral
+values of r, then the sum of the terms u<span class="su">m</span>, u<span class="su">m+1</span>, ... u<span class="su">n</span> is
+v<span class="su">n+1</span> &minus; v<span class="su">m</span>. Thus the sum of a number of terms of a series may often
+be found by inspection, in the same kind of way that an integral
+is found.</p>
+
+<p>14. <i>Rational Integral Functions.</i>&mdash;(i.) If u<span class="su">r</span> is a rational integral
+function of r of degree p, then &Delta;u<span class="su">r</span>, is a rational integral function of r
+of degree p &minus; 1.</p>
+
+<p>(ii.) A particular case is that of a <i>factorial</i>, <i>i.e.</i> a product of the
+form (r + a + 1) (r + a + 2) ... (r + b), each factor exceeding the preceding
+factor by 1. We have</p>
+
+<p class="center">&Delta; · (r + a + 1) (r + a + 2) ... (r + b) = (b &minus; a)·(r + a + 2) ... (r + b),</p>
+
+<p class="noind">whence, changing a into a-1,</p>
+
+<p class="center">&Sigma;(r + a + 1) (r + a + 2) ... (r + b) = <i>const.</i> + (r + a)(r + a + 1) ... (r + b)/(b &minus; a + 1).</p>
+
+<p class="noind">A similar method can be applied to the series whose (r + 1)th
+term is of the form 1/(r + a + 1) (r + a + 2) ... (r + b).</p>
+
+<p>(iii.) Any rational integral function can be converted into the sum
+of a number of factorials; and thus the sum of a series of which such
+a function is the general term can be found. For example, it may
+be shown in this way that the sum of the pth powers of the first n
+natural numbers is a rational integral function of n of degree p + 1,
+the coefficient of n<span class="sp">p+1</span> being 1/(p + 1).</p>
+
+<p>15. <i>Difference-equations.</i>&mdash;The summation of the series ...
++ u<span class="su">n+2</span> + u<span class="su">n-1</span> + u<span class="su">n</span> is a solution of the <i>difference-equation</i> &Delta;v<span class="su">n</span> = u<span class="su">n+1</span>,
+which may also be written (E &minus; 1)v<span class="su">n</span> = u<span class="su">n+1</span>. This is a simple form
+of difference-equation. There are several forms which have been
+investigated; a simple form, more general than the above, is the
+<i>linear equation</i> with <i>constant coefficients</i>&mdash;</p>
+
+<p class="center">v<span class="su">n+m</span> + a<span class="su">1</span>v<span class="su">n+m-1</span> + a<span class="su">2</span>v<span class="su">n+m-2</span> + ... + a<span class="su">m</span>v<span class="su">n</span> = N,</p>
+
+<p class="noind">where a<span class="su">1</span>, a<span class="su">2</span>, ... a<span class="su">m</span> are constants, and N is a given function of n.
+This may be written</p>
+
+<p class="center">(E<span class="sp">m</span> + a<span class="su">1</span>E<span class="sp">m-1</span> + ... + a<span class="su">m</span>)v<span class="su">n</span> = N</p>
+
+<p class="noind">or</p>
+
+<p class="center">(E &minus; p<span class="su">1</span>)(E &minus; p<span class="su">2</span>) ... (E &minus; p<span class="su">m</span>)v<span class="su">n</span> = N.</p>
+
+<p class="noind">The solution, if p<span class="su">1</span>, p<span class="su">2</span>, ... p<span class="su">m</span> are all different, is v<span class="su">n</span> = C<span class="su">1</span>p<span class="su">1</span><span class="sp">n</span> +
+C<span class="su">2</span>p<span class="su">2</span><span class="sp">n</span> + ... + C<span class="su">m</span>p<span class="su">m</span><span class="sp">n</span> + V<span class="su">n</span>, where C<span class="su">1</span>, C<span class="su">2</span> ... are constants, and
+v<span class="su">n</span> = V<span class="su">n</span> is any one solution of the equation. The method of finding
+a value for V<span class="su">n</span> depends on the form of N. Certain modifications are
+required when two or more of the p&rsquo;s are equal.</p>
+
+<p>It should be observed, in all cases of this kind, that, in describing
+C<span class="su">1</span>, C<span class="su">2</span> as &ldquo;constants,&rdquo; it is meant that the value of any one, as C<span class="su">1</span>, is
+the same for all values of n occurring in the series. A &ldquo;constant&rdquo;
+may, however, be a periodic function of n.</p>
+
+<p class="center pt2"><i>Applications to Continuous Functions.</i></p>
+
+<p>16. The cases of greatest practical importance are those in which
+u is a continuous function of x. The terms u<span class="su">1</span>, u<span class="su">2</span> ... of the series
+then represent the successive values of u corresponding to x = x<span class="su">1</span>, x<span class="su">2</span>....
+The important applications of the theory in these cases are to
+(i.) relations between differences and differential coefficients, (ii.)
+<span class="pagenum"><a name="page225" id="page225"></a>225</span>
+interpolation, or the determination of intermediate values of u, and
+(iii.) relations between sums and integrals.</p>
+
+<p>17. Starting from any pair of values x<span class="su">0</span> and u<span class="su">0</span>, we may suppose
+the interval h from x<span class="su">0</span> to x<span class="su">1</span> to be divided into q equal portions. If
+we suppose the corresponding values of u to be obtained, and their
+differences taken, the successive advancing differences of u<span class="su">0</span> being
+denoted by &part;u<span class="su">0</span>, &part;²u<span class="su">0</span> ..., we have (§ 3 (ii.))</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">u<span class="su">1</span> = u<span class="su">0</span> + q&part;u<span class="su">0</span> +</td> <td>q·q &minus; 1</td>
+ <td rowspan="2">&part;²u<span class="su">0</span> + ....</td></tr>
+<tr><td class="denom">1·2</td></tr></table>
+
+<p class="noind">When q is made indefinitely great, this (writing &fnof;(x) for u) becomes
+Taylor&rsquo;s Theorem (<span class="sc"><a href="#artlinks">Infinitesimal Calculus</a></span>)</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&fnof;(x + h) = &fnof;(x) + h&fnof;&prime;(x) +</td> <td>h²</td>
+ <td rowspan="2">&fnof;&Prime;(x) + ...,</td></tr>
+<tr><td class="denom">1·2</td></tr></table>
+
+<p class="noind">which, expressed in terms of operators, is</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">E = 1 + hD +</td> <td>h²</td>
+ <td rowspan="2">D² +</td> <td>h³</td>
+ <td rowspan="2">D³ + ... = e<span class="sp">hD</span>.</td></tr>
+<tr><td class="denom">1·2</td> <td class="denom">1·2·3</td></tr></table>
+
+<p class="noind">This gives the relation between &Delta; and D. Also we have</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">u<span class="su">2</span> = u<span class="su">0</span> + 2q&part;u<span class="su">0</span> +</td> <td>2q·2q &minus; 1</td> <td rowspan="2">&part;²u<span class="su">0</span> + ...</td></tr>
+<tr><td class="denom">1·2</td></tr>
+
+<tr><td rowspan="2">u<span class="su">3</span> = u<span class="su">0</span> + 3q&part;u<span class="su">0</span> +</td> <td>3q·3q &minus; 1</td> <td rowspan="2">&part;²u<span class="su">0</span> + ...</td></tr>
+<tr><td class="denom">1·2</td></tr>
+
+<tr><td colspan="3">·&emsp;&emsp;&emsp;·</td></tr>
+<tr><td colspan="3">·&emsp;&emsp;&emsp;·</td></tr>
+<tr><td colspan="3">·&emsp;&emsp;&emsp;·</td></tr>
+</table>
+
+<p class="noind">and, if p is any integer,</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">u<span class="su">p/q</span> = u<span class="su">0</span> + p&part;u<span class="su">0</span> +</td> <td>p·p &minus; 1</td>
+ <td rowspan="2">&part;²u<span class="su">0</span> + ....</td></tr>
+<tr><td class="denom">1·2</td></tr></table>
+
+<p class="noind">From these equations u<span class="su">p/q</span> could be expressed in terms of u<span class="su">0</span>, u<span class="su">1</span>,
+u<span class="su">2</span>, ...; this is a particular case of interpolation (<i>q.v.</i>).</p>
+
+<p>18. <i>Differences and Differential Coefficients.</i>&mdash;The various formulae
+are most quickly obtained by symbolical methods; <i>i.e.</i> by dealing
+with the operators &Delta;, E, D, ... as if they were algebraical
+quantities. Thus the relation E = e<span class="sp">hD</span> (§ 17) gives</p>
+
+<p class="center">hD = log<span class="su">e</span> (1 + &Delta;) = &Delta; &minus; ½&Delta;² + <span class="spp">1</span>&frasl;<span class="suu">3</span>&Delta;³ ...</p>
+
+<p class="noind">or</p>
+
+<p class="center">h(du/dx)<span class="su">0</span> = &Delta;u<span class="su">0</span> &minus; ½&Delta;²u<span class="su">0</span> + <span class="spp">1</span>&frasl;<span class="suu">3</span>&Delta;³u<span class="su">0</span> ....</p>
+
+<p>The formulae connecting central differences with differential
+coefficients are based on the relations
+&mu; = cosh ½hD = ½(e<span class="sp">1/2hD</span> + e<span class="sp">-1/2hD</span>),
+&delta; = 2 sinh ½hD &minus; e<span class="sp">1/2hD</span> &minus; e<span class="sp">-1/2hD</span>,
+and may be grouped as follows:&mdash;</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcr">u<span class="su">0</span></td>
+<td class="tcl">= u<span class="su">0</span></td>
+<td class="tcrm" rowspan="5"><span style="font-size: 8em; font-family: 'Courier New'; color: #a0a0a0;">}</span></td></tr>
+
+<tr><td class="tcr">&mu;&delta;u<span class="su">0</span></td>
+<td class="tcl">= (hD + <span class="spp">1</span>&frasl;<span class="suu">6</span> h³D³ + <span class="spp">1</span>&frasl;<span class="suu">120</span> h<span class="sp">5</span>D<span class="sp">5</span> + ...)u<span class="su">0</span></td></tr>
+
+<tr><td class="tcr">&delta;²u<span class="su">0</span></td>
+<td class="tcl">= (h²D² + <span class="spp">1</span>&frasl;<span class="suu">12</span> h<span class="sp">4</span>D<span class="sp">4</span> + <span class="spp">1</span>&frasl;<span class="suu">360</span> h<span class="sp">6</span>D<span class="sp">6</span> + ...)u<span class="su">0</span></td></tr>
+
+<tr><td class="tcr">&mu;&delta;³u<span class="su">0</span></td>
+<td class="tcl">= (h³D³ + <span class="spp">1</span>&frasl;<span class="suu">4</span> h<span class="sp">5</span>D<span class="sp">5</span> + ...)u<span class="su">0</span></td></tr>
+
+<tr><td class="tcr">&delta;<span class="sp">4</span>u<span class="su">0</span></td>
+<td class="tcl">= (h<span class="sp">4</span>D<span class="sp">4</span> + <span class="spp">1</span>&frasl;<span class="suu">6</span> h<span class="sp">6</span>D<span class="sp">6</span> + ...)u<span class="su">0</span></td></tr>
+
+
+<tr><td>&nbsp;</td> <td class="tcl" colspan="2">·&emsp;&emsp;&emsp;·&emsp;&emsp;&emsp;·</td></tr>
+<tr><td>&nbsp;</td> <td class="tcl" colspan="2">·&emsp;&emsp;&emsp;·&emsp;&emsp;&emsp;·</td></tr>
+<tr><td>&nbsp;</td> <td class="tcl" colspan="2">·&emsp;&emsp;&emsp;·&emsp;&emsp;&emsp;·</td></tr>
+
+
+<tr><td class="tcr">&mu;u<span class="su">1/2</span></td>
+<td class="tcl">= (1 + <span class="spp">1</span>&frasl;<span class="suu">8</span> h²D² + <span class="spp">1</span>&frasl;<span class="suu">384</span> h<span class="sp">4</span>D<span class="sp">4</span> + <span class="spp">1</span>&frasl;<span class="suu">46080</span> h<span class="sp">6</span>D<span class="sp">6</span> + ...)u<span class="su">1/2</span></td>
+<td class="tcrm" rowspan="5"><span style="font-size: 8em; font-family: 'Courier New'; color: #a0a0a0;">}</span></td></tr>
+
+<tr><td class="tcr">&delta;u<span class="su">1/2</span></td>
+<td class="tcl">= (hD + <span class="spp">1</span>&frasl;<span class="suu">24</span> h³D³ + <span class="spp">1</span>&frasl;<span class="suu">1920</span> h<span class="sp">5</span>D<span class="sp">5</span> + ...)u<span class="su">1/2</span></td></tr>
+
+<tr><td class="tcr">&mu;&delta;²u<span class="su">1/2</span></td>
+<td class="tcl">= (h²D² + <span class="spp">5</span>&frasl;<span class="suu">24</span> h<span class="sp">4</span>D<span class="sp">4</span> + <span class="spp">91</span>&frasl;<span class="suu">5760</span> h<span class="sp">6</span>D<span class="sp">6</span> + ...)u<span class="su">1/2</span></td></tr>
+
+<tr><td class="tcr">&delta;³u<span class="su">1/2</span></td>
+<td class="tcl">= (h³D³ + <span class="spp">1</span>&frasl;<span class="suu">8</span> h<span class="sp">5</span>D<span class="sp">5</span> + ...)u<span class="su">1/2</span></td></tr>
+
+<tr><td class="tcr">&mu;&delta;<span class="sp">4</span> u<span class="su">1/2</span></td>
+<td class="tcl">= (h<span class="sp">4</span>D<span class="sp">4</span> + <span class="spp">7</span>&frasl;<span class="suu">24</span> h<span class="sp">6</span>D<span class="sp">6</span> + ...)u<span class="su">1/2</span></td></tr>
+
+
+<tr><td>&nbsp;</td> <td class="tcl" colspan="2">·&emsp;&emsp;&emsp;·&emsp;&emsp;&emsp;·</td></tr>
+<tr><td>&nbsp;</td> <td class="tcl" colspan="2">·&emsp;&emsp;&emsp;·&emsp;&emsp;&emsp;·</td></tr>
+<tr><td>&nbsp;</td> <td class="tcl" colspan="2">·&emsp;&emsp;&emsp;·&emsp;&emsp;&emsp;·</td></tr>
+
+<tr><td class="tcr">u<span class="su">0</span></td>
+<td class="tcl">= u<span class="su">0</span></td>
+<td class="tcrm" rowspan="5"><span style="font-size: 8em; font-family: 'Courier New'; color: #a0a0a0;">}</span></td></tr>
+
+<tr><td class="tcr">hDu<span class="su">0</span></td>
+<td class="tcl">= (&mu;&delta; &minus; <span class="spp">1</span>&frasl;<span class="suu">6</span> &mu;&delta;³ + <span class="spp">1</span>&frasl;<span class="suu">30</span> &mu;&delta;<span class="sp">5</span> &minus; ...)u<span class="su">0</span></td></tr>
+
+<tr><td class="tcr">h²D²u<span class="su">0</span></td>
+<td class="tcl">= (&delta;² &minus; <span class="spp">1</span>&frasl;<span class="suu">12</span> &delta;<span class="sp">4</span> + <span class="spp">1</span>&frasl;<span class="suu">90</span> &delta;<span class="sp">6</span> &minus; ...)u<span class="su">0</span></td></tr>
+
+<tr><td class="tcr">h³D³u<span class="su">0</span></td>
+<td class="tcl">= (&mu;&delta;³ &minus; <span class="spp">1</span>&frasl;<span class="suu">4</span> &mu;&delta;<span class="sp">5</span> + ...)u<span class="su">0</span></td></tr>
+
+<tr><td class="tcr">h<span class="sp">4</span>D<span class="sp">4</span>u<span class="su">0</span></td>
+<td class="tcl">= (&delta;<span class="sp">4</span> &minus; <span class="spp">1</span>&frasl;<span class="suu">6</span> &delta;<span class="sp">6</span> + ...)u<span class="su">0</span></td></tr>
+
+
+<tr><td>&nbsp;</td> <td class="tcl" colspan="2">·&emsp;&emsp;&emsp;·&emsp;&emsp;&emsp;·</td></tr>
+<tr><td>&nbsp;</td> <td class="tcl" colspan="2">·&emsp;&emsp;&emsp;·&emsp;&emsp;&emsp;·</td></tr>
+<tr><td>&nbsp;</td> <td class="tcl" colspan="2">·&emsp;&emsp;&emsp;·&emsp;&emsp;&emsp;·</td></tr>
+
+
+<tr><td class="tcr">u<span class="su">1/2</span></td>
+<td class="tcl">= (&mu; &minus; <span class="spp">1</span>&frasl;<span class="suu">8</span> &mu;&delta;² + <span class="spp">3</span>&frasl;<span class="suu">128</span> &mu;&delta;<span class="sp">4</span> &minus; <span class="spp">5</span>&frasl;<span class="suu">1024</span> &mu;&delta;<span class="sp">6</span> + ...)u<span class="su">1/2</span></td>
+<td class="tcrm" rowspan="5"><span style="font-size: 8em; font-family: 'Courier New'; color: #a0a0a0;">}</span></td></tr>
+
+<tr><td class="tcr">hDu<span class="su">1/2</span></td>
+<td class="tcl">= (&delta; &minus; <span class="spp">1</span>&frasl;<span class="suu">24</span> &delta;³ + <span class="spp">3</span>&frasl;<span class="suu">640</span> &delta;<span class="sp">5</span> &minus; ...)u<span class="su">1/2</span></td></tr>
+
+<tr><td class="tcr">h²D²u<span class="su">1/2</span></td>
+<td class="tcl">= (&mu;&delta;² &minus; <span class="spp">5</span>&frasl;<span class="suu">24</span> &mu;&delta;<span class="sp">4</span> + <span class="spp">259</span>&frasl;<span class="suu">5760</span> &mu;&delta;<span class="sp">6</span> &minus; ...)u<span class="su">1/2</span></td></tr>
+
+<tr><td class="tcr">h³D³u<span class="su">1/2</span></td>
+<td class="tcl">= (&delta;³ &minus; <span class="spp">1</span>&frasl;<span class="suu">8</span> &delta;<span class="sp">5</span> + ...)u<span class="su">1/2</span></td></tr>
+
+<tr><td class="tcr">h<span class="sp">4</span>D<span class="sp">4</span> u<span class="su">1/2</span></td>
+<td class="tcl">= (&mu;&delta;<span class="sp">4</span> &minus; <span class="spp">7</span>&frasl;<span class="suu">24</span> &mu;&delta;<span class="sp">6</span> + ...)u<span class="su">1/2</span></td></tr>
+
+<tr><td>&nbsp;</td> <td class="tcl" colspan="2">·&emsp;&emsp;&emsp;·&emsp;&emsp;&emsp;·</td></tr>
+<tr><td>&nbsp;</td> <td class="tcl" colspan="2">·&emsp;&emsp;&emsp;·&emsp;&emsp;&emsp;·</td></tr>
+<tr><td>&nbsp;</td> <td class="tcl" colspan="2">·&emsp;&emsp;&emsp;·&emsp;&emsp;&emsp;·</td></tr>
+</table>
+
+<p>When u is a rational integral function of x, each of the above series
+is a terminating series. In other cases the series will be an infinite
+one, and may be divergent; but it may be used for purposes of
+approximation up to a certain point, and there will be a &ldquo;remainder,&rdquo;
+the limits of whose magnitude will be determinate.</p>
+
+<p>19. <i>Sums and Integrals.</i>&mdash;The relation between a sum and an
+integral is usually expressed by the <i>Euler-Maclaurin formula</i>. The
+principle of this formula is that, if u<span class="su">m</span> and u<span class="su">m+1</span>, are ordinates of a
+curve, distant h from one another, then for a first approximation to
+the area of the curve between u<span class="su">m</span> and u<span class="su">m+1</span> we have ½h(u<span class="su">m</span> + u<span class="su">m+1</span>),
+and the difference between this and the true value of the area can
+be expressed as the difference of two expressions, one of which is a
+function of x<span class="su">m</span>, and the other is the same function of x<span class="su">m+1</span>.
+Denoting these by &phi;(x<span class="su">m</span>) and &phi;(x<span class="su">m+1</span>), we have</p>
+
+<table class="math0" summary="math">
+<tr><td class="vb" rowspan="2"><span class="f200">&int;</span></td> <td>x<span class="su">m+1</span></td>
+  <td rowspan="2">udx = ½h(u<span class="su">m</span> + u<span class="su">m+1</span>) + &phi;(x<span class="su">m+1</span>) &minus; &phi;(x<span class="su">m</span>).</td></tr>
+<tr><td>x<span class="su">m</span></td></tr></table>
+
+<p class="noind">Adding a series of similar expressions, we find</p>
+
+<table class="math0" summary="math">
+<tr><td class="vb" rowspan="2"><span class="f200">&int;</span></td> <td>x<span class="su">n</span></td>
+  <td rowspan="2">udx = h{½u<span class="su">m</span> + u<span class="su">m+1</span> + u<span class="su">m+2</span> + ... + u<span class="su">n-1</span> + ½u<span class="su">n</span>} + &phi;(x<span class="su">n</span>) &minus; &phi;(x<span class="su">m</span>).</td></tr>
+<tr><td>x<span class="su">m</span></td></tr></table>
+
+<p>The function &phi;(x) can be expressed in terms either of differential
+coefficients of u or of advancing or central differences; thus there
+are three formulae.</p>
+
+<p>(i.) The Euler-Maclaurin formula, properly so called, (due independently
+to Euler and Maclaurin) is</p>
+
+<table class="math0" summary="math">
+<tr><td class="vb" rowspan="2"><span class="f200">&int;</span></td> <td>x<span class="su">n</span></td>
+ <td rowspan="2">udx = h·&mu;&sigma;u<span class="su">n</span> &minus; <span class="spp">1</span>&frasl;<span class="suu">12</span> h²</td> <td>du<span class="su">n</span></td>
+ <td rowspan="2">½ + <span class="spp">1</span>&frasl;<span class="suu">720</span> h<span class="sp">4</span></td> <td>d³u<span class="su">n</span></td>
+ <td rowspan="2">&minus; <span class="spp">1</span>&frasl;<span class="suu">30240</span> h<span class="sp">6</span></td> <td>d<span class="sp">5</span>u<span class="su">n</span></td>
+ <td rowspan="2">+ ... = h·&mu;&sigma;u<span class="su">n</span> &minus; </td> <td>B<span class="su">1</span></td>
+ <td rowspan="2">h<span class="su">2</span></td> <td>du<span class="su">n</span></td>
+ <td rowspan="2"> + </td> <td>B<span class="su">2</span></td>
+ <td rowspan="2">h<span class="sp">4</span></td> <td>d<span class="sp">3</span>u<span class="su">n</span></td>
+ <td rowspan="2">&minus; </td> <td>B<span class="su">3</span></td>
+ <td rowspan="2">h<span class="sp">6</span></td> <td>d<span class="sp">5</span>u<span class="su">n</span></td>
+ <td rowspan="2">+ ...</td></tr>
+<tr><td>&nbsp;</td> <td class="denom">dx</td> <td class="denom">dx<span class="sp">3</span></td>
+    <td class="denom">dx<span class="sp">5</span></td> <td class="denom">2!</td>
+    <td class="denom">dx</td> <td class="denom">4!</td>
+    <td class="denom">dx<span class="sp">3</span></td> <td class="denom">6!</td>
+    <td class="denom">dx<span class="sp">5</span></td></tr></table>
+
+<p class="noind">where B<span class="su">1</span>, B<span class="su">2</span>, B<span class="su">3</span> ... are <i>Bernoulli&rsquo;s numbers</i>.</p>
+
+<p>(ii.) If we express differential coefficients in terms of advancing
+differences, we get a theorem which is due to Laplace:&mdash;</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">1/h </td> <td class="vb" rowspan="2"><span class="f200">&int;</span></td> <td>x<span class="su">n</span></td>
+ <td rowspan="2">udx = &mu;&sigma;(u<span class="su">n</span> &minus; u<span class="su">0</span>) &minus; <span class="spp">1</span>&frasl;<span class="suu">12</span>(&Delta;u<span class="su">n</span> &minus;
+  &Delta;u<span class="su">0</span>) + <span class="spp">1</span>&frasl;<span class="suu">24</span>(&Delta;²u<span class="su">n</span> &minus; &Delta;²u<span class="su">0</span>) &minus;
+  <span class="spp">19</span>&frasl;<span class="suu">720</span>(&Delta;³u<span class="su">n</span> &minus; &Delta;³u<span class="su">0</span>) + <span class="spp">3</span>&frasl;<span class="suu">160</span>(&Delta;<span class="sp">4</span>u<span class="su">n</span> &minus; &Delta;<span class="sp">4</span>u<span class="su">0</span>) &minus; ...</td></tr>
+<tr><td>x<span class="su">0</span></td></tr></table>
+
+<p class="noind">For practical calculations this may more conveniently be written</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">1/h </td> <td class="vb" rowspan="2"><span class="f200">&int;</span></td> <td>x<span class="su">n</span></td>
+<td rowspan="2">udx = &mu;&sigma;(u<span class="su">n</span> &minus; u<span class="su">0</span>) + <span class="spp">1</span>&frasl;<span class="suu">12</span>(&Delta;u<span class="su">0</span> &minus; ½&Delta;²u<span class="su">0</span> + <span class="spp">19</span>&frasl;<span class="suu">60</span>&Delta;³u<span class="su">0</span> &minus; ...) +
+ <span class="spp">1</span>&frasl;<span class="suu">12</span>(&Delta;&prime; u<span class="su">n</span> &minus; ½&Delta;&prime; ²u<span class="su">n</span> + <span class="spp">19</span>&frasl;<span class="suu">60</span>&Delta;&prime; ³u<span class="su">n</span> &minus; ...),</td></tr>
+<tr><td>x<span class="su">0</span></td></tr></table>
+
+<p class="noind">where accented differences denote that the values of u are read backwards
+from u<span class="su">n</span>; <i>i.e.</i> &Delta;&prime;u<span class="su">n</span> denotes u<span class="su">n-1</span> &minus; u<span class="su">n</span>, not (as in § 10) u<span class="su">n</span> &minus; u<span class="su">n-1</span>.</p>
+
+<p>(iii.) Expressed in terms of central differences this becomes</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">1/h </td> <td class="vb" rowspan="2"><span class="f200">&int;</span></td> <td>x<span class="su">n</span></td>
+<td rowspan="2">udx = &mu;&sigma;(u<span class="su">n</span> &minus; u<span class="su">0</span>) &minus; <span class="spp">1</span>&frasl;<span class="suu">12</span>&mu;&delta;u<span class="su">n</span> + <span class="spp">11</span>&frasl;<span class="suu">720</span> &mu;&delta;³u<span class="su">n</span> &minus; ... +
+ <span class="spp">1</span>&frasl;<span class="suu">12</span>&mu;&delta;u<span class="su">0</span> &minus; <span class="spp">11</span>&frasl;<span class="suu">720</span> &mu;&delta;³u<span class="su">0</span> + ... =
+ &mu;(&sigma; &minus; <span class="spp">1</span>&frasl;<span class="suu">12</span>&delta; +  <span class="spp">11</span>&frasl;<span class="suu">720</span>&delta;³ &minus; <span class="spp">191</span>&frasl;<span class="suu">60480</span>&delta;<span class="sp">5</span> +
+ <span class="spp">2497</span>&frasl;<span class="suu">3628800</span>&delta;<span class="sp">7</span> &minus; ...)(u<span class="su">n</span> &minus; u<span class="su">0</span>).</td></tr>
+<tr><td>x<span class="su">0</span></td></tr></table>
+
+<p>(iv.) There are variants of these formulae, due to taking hu<span class="su">m+1/2</span> as
+the first approximation to the area of the curve between u<span class="su">m</span> and
+u<span class="su">m+1</span>; the formulae involve the sum
+u<span class="su">1/2</span> + u<span class="su">3/2</span> + ... + u<span class="su">n-1/2</span> &equiv; &sigma;(u<span class="su">n</span> &minus; u<span class="su">0</span>)
+(see <span class="sc"><a href="#artlinks">Mensuration</a></span>).</p>
+
+<p>20. The formulae in the last section can be obtained by symbolical
+methods from the relation</p>
+
+<table class="math0" summary="math">
+<tr><td>1/h </td> <td class="vb"><span class="f200">&int;</span></td>
+ <td>udx = 1/h D<span class="sp">-1</span>u = 1/hD · u.</td></tr></table>
+
+<p class="noind">Thus for central differences, if we write &theta; &equiv; ½hD, we have &mu; = cosh &theta;,
+&delta; = 2 sinh &theta;, &sigma; = &delta;<span class="sp">-1</span>, and the result in (iii.) corresponds to the formula</p>
+
+<p class="center">
+sinh &theta; = &theta; cosh &theta;/(1 + <span class="spp">1</span>&frasl;<span class="suu">3</span> sinh² &theta; &minus; <span class="spp">2</span>&frasl;<span class="suu">3·5</span> sinh<span class="sp">4</span> &theta; + <span class="spp">2·4</span>&frasl;<span class="suu">3·5·7</span> sinh<span class="sp">6</span> &theta; &minus; ...).</p>
+
+<div class="condensed">
+<p><span class="sc">References.</span>&mdash;There is no recent English work on the theory of
+finite differences as a whole. G. Boole&rsquo;s <i>Finite Differences</i> (1st ed.,
+1860, 2nd ed., edited by J. F. Moulton, 1872) is a comprehensive
+treatise, in which symbolical methods are employed very early.
+A. A. Markoff&rsquo;s <i>Differenzenrechnung</i> (German trans., 1896) contains
+general formulae. (Both these works ignore central differences.)
+<i>Encycl. der math. Wiss.</i> vol. i. pt. 2, pp. 919-935, may also be consulted.
+An elementary treatment of the subject will be found in
+many text-books, <i>e.g.</i> G. Chrystal&rsquo;s <i>Algebra</i> (pt. 2, ch. xxxi.).
+A. W. Sunderland, <i>Notes on Finite Differences</i> (1885), is intended for
+actuarial students. Various central-difference formulae with references
+are given in <i>Proc. Lond. Math. Soc.</i> xxxi. pp. 449-488. For
+other references see <span class="sc"><a href="#artlinks">Interpolation</a></span>.</p>
+</div>
+<div class="author">(W. F. Sh.)</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIFFERENTIAL EQUATION,<a name="ar82" id="ar82"></a></span> in mathematics, a relation between
+one or more functions and their differential coefficients.
+The subject is treated here in two parts: (1) an elementary
+introduction dealing with the more commonly recognized types
+of differential equations which can be solved by rule; and (2) the
+general theory.</p>
+
+<p class="center pt2"><i>Part I.&mdash;Elementary Introduction.</i></p>
+
+<p>Of equations involving only one independent variable, x (known
+as <i>ordinary</i> differential equations), and one dependent variable, y,
+and containing only the first differential coefficient dy/dx (and therefore
+said to be of the first <i>order</i>), the simplest form is that reducible
+to the type</p>
+
+<p class="center">dy/dx = &fnof;(x)/F(y),</p>
+
+<p class="noind">leading to the result &fnof;F(y)dy &minus; &fnof;&fnof;(x)dx = A, where A is an arbitrary
+constant; this result is said to solve the differential equation, the
+problem of evaluating the integrals belonging to the integral calculus.</p>
+
+<p><span class="pagenum"><a name="page226" id="page226"></a>226</span></p>
+
+<p>Another simple form is</p>
+
+<p class="center">dy/dx + yP = Q,</p>
+
+<p class="noind">where P, Q are functions of x only; this is known as the linear equation,
+since it contains y and dy/dx only to the first degree. If &fnof;Pdx = u, we clearly have</p>
+
+<table class="math0" summary="math">
+<tr><td>d</td> <td rowspan="2">(ye<span class="sp">u</span>) = e<span class="sp">u</span></td> <td rowspan="2" class="f200 np">(</td>
+<td>dy</td> <td rowspan="2">+ Py</td> <td rowspan="2" class="f200 np">)</td>
+<td rowspan="2">= e<span class="sp">u</span>Q,</td></tr>
+<tr><td class="denom">dx</td> <td class="denom">dx</td> </tr></table>
+
+<p class="noind">so that y = e<span class="sp">-u</span>(&fnof;e<span class="sp">u</span>Qdx + A) solves the equation, and is the only
+possible solution, A being an arbitrary constant. The rule for the
+solution of the linear equation is thus to multiply the equation by
+e<span class="sp">u</span>, where u = &fnof;Pdx.</p>
+
+<p>A third simple and important form is that denoted by</p>
+
+<p class="center">y = px + &fnof;(p),</p>
+
+<p class="noind">where p is an abbreviation for dy/dx; this is known as Clairaut&rsquo;s
+form. By differentiation in regard to x it gives</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">p = p + x</td> <td>dp</td>
+ <td rowspan="2">+ &fnof;&prime;(p)</td> <td>dp</td> <td rowspan="2">,</td></tr>
+<tr><td class="denom">dx</td> <td class="denom">dx</td></tr></table>
+
+<p class="noind">where</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&fnof;&prime;(p) =</td> <td>d</td>
+ <td rowspan="2">&fnof;(p);</td></tr>
+<tr><td class="denom">dp</td></tr></table>
+
+<p class="noind">thus, either (i.) dp/dx = 0, that is, p is constant on the curve satisfying
+the differential equation, which curve is thus any one of the
+straight lines y = cx = &fnof;(c), where c is an arbitrary constant, or else,
+(ii.) x + &fnof;&prime;(p) = 0; if this latter hypothesis be taken, and p be eliminated
+between x + &fnof;&prime;(p) = 0 and y = px + &fnof;(p), a relation connecting x and y,
+not containing an arbitrary constant, will be found, which obviously
+represents the envelope of the straight lines y = cx + &fnof;(c).</p>
+
+<p>In general if a differential equation &phi;(x, y, dy/dx) = 0 be satisfied
+by any one of the curves F(x, y, c) = 0, where c is an arbitrary constant,
+it is clear that the envelope of these curves, when existent, must
+also satisfy the differential equation; for this equation prescribes
+a relation connecting only the co-ordinates x, y and the differential
+coefficient dy/dx, and these three quantities are the same at any
+point of the envelope for the envelope and for the particular curve
+of the family which there touches the envelope. The relation expressing
+the equation of the envelope is called a <i>singular</i> solution of
+the differential equation, meaning an <i>isolated</i> solution, as not being
+one of a family of curves depending upon an arbitrary parameter.</p>
+
+<p>An extended form of Clairaut&rsquo;s equation expressed by</p>
+
+<p class="center">y = xF(p) + &fnof;(p)</p>
+
+<p class="noind">may be similarly solved by first differentiating in regard to p, when
+it reduces to a linear equation of which x is the dependent and p the
+independent variable; from the integral of this linear equation, and the
+original differential equation, the quantity p is then to be eliminated.</p>
+
+<p>Other types of solvable differential equations of the first order
+are (1)</p>
+
+<p class="center">M dy/dx = N,</p>
+
+<p class="noind">where M, N are homogeneous polynomials in x and y, of the same
+order; by putting v = y/x and eliminating y, the equation becomes
+of the first type considered above, in v and x. An equation (aB &#8823; bA)</p>
+
+<p class="center">(ax + by + c)dy/dx = Ax + By + C</p>
+
+<p class="noind">may be reduced to this rule by first putting x + h, y + k for x and y,
+and determining h, k so that ah + bk + c = 0, Ah + Bk + C = 0.</p>
+
+<p>(2) An equation in which y does not explicitly occur,</p>
+
+<p class="center">&fnof;(x, dy/dx) = 0,</p>
+
+<p class="noind">may, theoretically, be reduced to the type dy/dx = F(x); similarly
+an equation F(y, dy/dx) = 0.</p>
+
+<p>(3) An equation</p>
+
+<p class="center">&fnof;(dy/dx, x, y) = 0,</p>
+
+<p class="noind">which is an integral polynomial in dy/dx, may, theoretically, be
+solved for dy/dx, as an algebraic equation; to any root dy/dx = F<span class="su">1</span>(x, y)
+corresponds, suppose, a solution &phi;<span class="su">1</span>(x, y, c) = 0, where c is an arbitrary
+constant; the product equation &phi;<span class="su">1</span>(x, y, c)&phi;<span class="su">2</span>(x, y, c) ... = 0,
+consisting of as many factors as there were values of dy/dx, is
+effectively as general as if we wrote &phi;<span class="su">1</span>(x, y, c<span class="su">1</span>)&phi;<span class="su">2</span>(x, y, c<span class="su">2</span>) ... = 0;
+for, to evaluate the first form, we must necessarily consider the
+factors separately, and nothing is then gained by the multiple
+notation for the various arbitrary constants. The equation
+&phi;<span class="su">1</span>(x, y, c)&phi;<span class="su">2</span>(x, y, c) ... = 0 is thus the solution of the given differential
+equation.</p>
+
+<p>In all these cases there is, except for cases of singular solutions, one
+and only one arbitrary constant in the most general solution of the
+differential equation; that this must necessarily be so we may take
+as obvious, the differential equation being supposed to arise by
+elimination of this constant from the equation expressing its solution
+and the equation obtainable from this by differentiation in regard
+to x.</p>
+
+<p>A further type of differential equation of the first order, of the form</p>
+
+<p class="center">dy/dx = A + By + Cy²</p>
+
+<p class="noind">in which A, B, C are functions of x, will be briefly considered below
+under differential equations of the second order.</p>
+
+<p>When we pass to ordinary differential equations of the second order,
+that is, those expressing a relation between x, y, dy/dx and d²y/dx²,
+the number of types for which the solution can be found by a known
+procedure is very considerably reduced. Consider the general linear
+equation</p>
+
+<table class="math0" summary="math">
+<tr><td>d²y</td> <td rowspan="2">+ P</td> <td>dy</td> <td rowspan="2">+ Qy = R,</td></tr>
+<tr><td class="denom">dx²</td> <td class="denom">dx</td></tr></table>
+
+<p class="noind">where P, Q, R are functions of x only. There is no method always
+effective; the main general result for such a linear equation is that
+if any particular function of x, say y<span class="su">1</span>, can be discovered, for which</p>
+
+<table class="math0" summary="math">
+<tr><td>d²y<span class="su">1</span></td> <td rowspan="2">+ P</td> <td>dy<span class="su">1</span></td> <td rowspan="2">+ Qy<span class="su">1</span> = 0,</td></tr>
+<tr><td class="denom">dx²</td> <td class="denom">dx</td></tr></table>
+
+<p class="noind">then the substitution y = y<span class="su">1</span>&eta; in the original equation, with R on
+the right side, reduces this to a linear equation of the first order with
+the dependent variable d&eta;/dx. In fact, if y = y<span class="su">1</span>&eta; we have</p>
+
+<table class="math0" summary="math">
+<tr><td>dy</td>
+ <td rowspan="2">= y<span class="su">1</span></td> <td>d&eta;</td>
+ <td rowspan="2">+ &eta;</td> <td>dy<span class="su">1</span></td>
+ <td rowspan="2">and</td> <td>d²y</td>
+ <td rowspan="2">= y<span class="su">1</span></td> <td>d²&eta;</td>
+ <td rowspan="2">+ 2</td> <td>dy<span class="su">1</span></td> <td>d&eta;</td>
+ <td rowspan="2">+ &eta;</td> <td>d²y<span class="su">1</span></td> <td rowspan="2">,</td></tr>
+<tr><td class="denom">dx</td> <td class="denom">dx</td>
+ <td class="denom">dx</td> <td class="denom">dx²</td>
+ <td class="denom">dx²</td> <td class="denom">dx</td>
+ <td class="denom">dx</td> <td class="denom">dx²</td></tr></table>
+
+<p class="noind">and thus</p>
+
+<table class="math0" summary="math">
+<tr> <td>d²y</td> <td rowspan="2">+ P</td> <td>dy</td>
+ <td rowspan="2">+ Qy = y<span class="su">1</span></td> <td>d²&eta;</td>
+ <td rowspan="2">+ <span class="f200">(</span>2</td> <td>dy<span class="su">1</span></td>
+ <td rowspan="2">+ Py<span class="su">1</span><span class="f200">)</span></td> <td>d&eta;</td>
+ <td rowspan="2">+ <span class="f200">(</span></td> <td>d²y<span class="su">1</span></td>
+ <td rowspan="2">+ P</td> <td>dy<span class="su">1</span></td>
+ <td rowspan="2">+ Qy<span class="su">1</span><span class="f200">)</span>&eta;;</td> <td></td></tr>
+<tr><td class="denom">dx²</td> <td class="denom">dx</td>
+ <td class="denom">dx²</td> <td class="denom">dx</td>
+ <td class="denom">dx</td> <td class="denom">dx²</td>
+ <td class="denom">dx</td></tr></table>
+
+<p class="noind">if then</p>
+
+<table class="math0" summary="math">
+<tr> <td>d²y<span class="su">1</span></td> <td rowspan="2">+ P</td> <td>dy<span class="su">1</span></td>
+ <td rowspan="2">+ Qy<span class="su">1</span> = 0,</td></tr>
+<tr><td class="denom">dx²</td> <td class="denom">dx</td></tr></table>
+
+<p class="noind">and z denote d&eta;/dx, the original differential equation becomes</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">y<span class="su">1</span></td> <td>dz</td>
+<td rowspan="2">+ <span class="f200">(</span> 2</td> <td>dy<span class="su">1</span></td>
+<td rowspan="2">Py<span class="su">1</span><span class="f200">)</span> z = R.</td></tr>
+<tr><td class="denom">dx</td> <td class="denom">dx</td></tr></table>
+
+<p class="noind">From this equation z can be found by the rule given above for
+the linear equation of the first order, and will involve one arbitrary
+constant; thence y = y<span class="su">1</span> &eta; = y<span class="su">1</span> &int; zdx + Ay<span class="su">1</span>,
+where A is another arbitrary constant, will be the general solution of the original
+equation, and, as was to be expected, involves two arbitrary
+constants.</p>
+
+<p>The case of most frequent occurrence is that in which the coefficients
+P, Q are constants; we consider this case in some detail.
+If &theta; be a root of the quadratic equation &theta;² + &theta;P + Q = 0, it can be at
+once seen that a particular integral of the differential equation with
+zero on the right side is y<span class="su">1</span> = e<span class="sp">&theta;x</span>. Supposing first the roots of the
+quadratic equation to be different, and &phi; to be the other root, so that
+&phi; + &theta; = -P, the auxiliary differential equation for z, referred to above,
+becomes dz/dx + (&theta; &minus; &phi;)z = Re<span class="sp">-&theta;x</span>
+which leads to ze<span class="sp">(&theta;-&phi;)</span> = B + &int; Re<span class="sp">-&theta;x</span>dx,
+where B is an arbitrary constant, and hence to</p>
+
+<p class="center">
+y = Ae<span class="sp">&theta;<span class="sp"> x</span></span> + e<span class="sp">&theta;<span class="sp"> x</span></span> <span class="f150">&int;</span> Be<span class="sp">(&phi;-&theta;)<span class="sp"> x</span></span> dx + e<span class="sp">&theta;<span class="sp"> x</span></span> <span class="f150">&int;</span> e<span class="sp">(&phi;-&theta;)<span class="sp"> x</span></span> <span class="f150">&int;</span> Re<span class="sp">-&theta;<span class="sp"> x</span></span> dxdx,</p>
+
+<p class="noind">or say to y = Ae<span class="sp">&theta;<span class="sp"> x</span></span> + Ce<span class="sp">&theta;<span class="sp"> x</span></span> + U,
+where A, C are arbitrary constants and
+U is a function of x, not present at all when R = 0. If the quadratic
+equation &theta;² + P&theta; + Q = 0 has equal roots, so that 2&theta; = -P, the
+auxiliary equation in z becomes dz/dx = Re<span class="sp">&theta;<span class="sp"> x</span></span>
+giving z = B + &int; Re<span class="sp">&theta;<span class="sp"> x</span></span> dx,
+where B is an arbitrary constant, and hence</p>
+
+<p class="center">
+y = (A + Bx)e<span class="sp">&theta;<span class="sp"> x</span></span> + e<span class="sp">&theta;<span class="sp"> x</span></span> <span class="f150">&int; &int;</span> Re<span class="sp">-&theta;<span class="sp"> x</span></span> dxdx,</p>
+
+<p class="noind">or, say, y = (A + Bx)e<span class="sp">&theta;<span class="sp"> x</span></span> + U, where A, B are arbitrary constants, and
+U is a function of x not present at all when R = 0. The portion
+Ae<span class="sp">&theta;<span class="sp"> x</span></span> + Be<span class="sp">&theta;<span class="sp"> x</span></span> or (A + Bx)e<span class="sp">&theta;<span class="sp"> x</span></span> of the solution, which is known as the <i>complementary
+function</i>, can clearly be written down at once by inspection
+of the given differential equation. The remaining portion U
+may, by taking the constants in the complementary function
+properly, be replaced by any particular solution whatever of the
+differential equation</p>
+
+<table class="math0" summary="math">
+<tr><td>d²v</td> <td rowspan="2">+ P</td> <td>dy</td>
+ <td rowspan="2">+ Qy = R;</td></tr>
+<tr><td class="denom">dx²</td> <td class="denom">dx</td> </tr></table>
+
+<p class="noind">for if u be any particular solution, this has a form</p>
+
+<p class="center">u = A<span class="su">0</span> e<span class="sp">&theta;<span class="sp"> x</span></span> + B<span class="su">0</span> e<span class="sp">&phi;<span class="sp"> x</span></span> + U,</p>
+
+<p class="noind">or a form</p>
+
+<p class="center">u = (A<span class="su">0</span> + B<span class="su">0</span>x) e<span class="sp">&theta;<span class="sp"> x</span></span> + U;</p>
+
+<p class="noind">thus the general solution can be written</p>
+
+<p class="center">
+(A &minus; A<span class="su">0</span>)e<span class="sp">&theta;<span class="sp"> x</span></span> + (B &minus; B<span class="su">0</span>)e<span class="sp">&theta;<span class="sp"> x</span></span> + u,
+or {A &minus; A<span class="su">0</span> + (B &minus; B<span class="su">0</span>)x} e<span class="sp">&theta;<span class="sp"> x</span></span> + u,</p>
+
+<p class="noind">where A &minus; A<span class="su">0</span>, B &minus; B<span class="su">0</span>, like A, B, are arbitrary constants.</p>
+
+<p>A similar result holds for a linear differential equation of any order,
+say</p>
+
+<table class="math0" summary="math">
+<tr><td>d<span class="sp">n</span>y</td> <td rowspan="2">+ P<span class="su">1</span></td>
+  <td>d<span class="sp">n-1</span>y</td> <td rowspan="2">+ ... + P<span class="su">n</span>y = R,</td></tr>
+<tr><td class="denom">dx<span class="sp">n</span></td> <td class="denom">dx<span class="sp">n-1</span></td></tr></table>
+
+<p class="noind">where P<span class="su">1</span>, P<span class="su">2</span>, ... P<span class="su">n</span> are constants, and R is a function of x. If
+we form the algebraic equation &theta;<span class="sp">n</span> + P<span class="su">1</span>&theta;<span class="sp">n-1</span> + ... + P<span class="su">n</span> = 0, and all the
+roots of this equation be different, say they are &theta;<span class="su">1</span>, &theta;<span class="su">2</span>, ... &theta;<span class="su">n</span>, the
+general solution of the differential equation is</p>
+
+<p class="center">
+y = A<span class="su">1</span> e<span class="sp">&theta;1 x</span> + A<span class="su">2</span> e<span class="sp">&theta;2 x</span> + ... + A<span class="su">n</span> e<span class="sp">&theta;n x</span> + u,</p>
+
+<p class="noind">where A<span class="su">1</span>, A<span class="su">2</span>, ... A<span class="su">n</span> are arbitrary constants, and u is any
+<span class="pagenum"><a name="page227" id="page227"></a>227</span>
+particular solution whatever; but if there be one root &theta;<span class="su">1</span> repeated
+r times, the terms A<span class="su">1</span> e<span class="sp">&theta;1<span class="sp"> x</span></span> + ... + A<span class="su">r</span> e<span class="sp">&theta;r<span class="sp"> x</span></span>
+must be replaced by (A<span class="su">1</span> + A<span class="su">2</span>x + ... + A<span class="su">r</span>x<span class="sp">r-1</span>)e<span class="sp">&theta;1<span class="sp"> x</span></span>
+where A<span class="su">1</span>, ... A<span class="su">n</span> are arbitrary constants;
+the remaining terms in the complementary function will
+similarly need alteration of form if there be other repeated roots.</p>
+
+<p>To complete the solution of the differential equation we need some
+method of determining a particular integral u; we explain a procedure
+which is effective for this purpose in the cases in which R is
+a sum of terms of the form e<span class="sp">ax</span>&phi;(x), where &phi;(x) is an integral polynomial
+in x; this includes cases in which R contains terms of the
+form cos bx·&phi;(x) or sin bx·&phi;(x). Denote d/dx by D; it is clear that if
+u be any function of x,
+D(e<span class="sp">ax</span>u) = e<span class="sp">ax</span>Du + ae<span class="sp">ax</span>u,
+or say, D(e<span class="sp">ax</span>u) = e<span class="sp">ax</span>(D + a)u;
+hence D²(e<span class="sp">ax</span>u), <i>i.e.</i> d²/dx² (e<span class="sp">ax</span>u), being equal to D(e<span class="sp">ax</span>v),
+where v = (D + a)u, is equal to e<span class="sp">ax</span>(D + a)v, that is to e<span class="sp">ax</span>(D + a)²u.
+In this way we find D<span class="sp">n</span>(e<span class="sp">ax</span>u) = e<span class="sp">ax</span>(D + a)<span class="sp">n</span>u, where n is any positive
+integer. Hence if &psi;(D) be any polynomial in D with constant coefficients,
+&psi;(D) (e<span class="sp">ax</span>u) = e<span class="sp">ax</span>&psi;(D + a)u.
+Next, denoting &int; udx by  D<span class="sp">-1</span>u,
+and any solution of the differential equation dz/dx + az = u by z = (d + a)<span class="sp">-1</span>u,
+we have D[e<span class="sp">ax</span>(D + a)<span class="sp">-1</span>u] = D(e<span class="sp">ax</span>z) = e<span class="sp">ax</span>(D + a)z = e<span class="sp">ax</span>u,
+so that we may write D<span class="sp">-1</span>(e<span class="sp">ax</span>u) = e<span class="sp">ax</span>(D + a)<span class="sp">-1</span>u,
+where the meaning is that one value of the left side is equal to one value of the
+right side; from this, the expression D<span class="sp">-2</span>(e<span class="sp">ax</span>u), which means
+D<span class="sp">-1</span>[D<span class="sp">-1</span>(e<span class="sp">ax</span>u)],
+is equal to D<span class="sp">-1</span>(e<span class="sp">ax</span>z) and hence to e<span class="sp">ax</span>(D + a)<span class="sp">-1</span>z,
+which we write e<span class="sp">ax</span>(D + a)<span class="sp">-2</span>u; proceeding thus we obtain</p>
+
+<p class="center">D<span class="sp">-n</span>(e<span class="sp">ax</span>u) = e<span class="sp">ax</span>(D + a)<span class="sp">-n</span>u,</p>
+
+<p class="noind">where n is any positive integer, and the meaning, as before, is that
+one value of the first expression is equal to one value of the second.
+More generally, if &psi;(D) be any polynomial in D with constant coefficients,
+and we agree to denote by [1/&psi;(D)]u any solution z of the
+differential equation &psi;(D)z = u, we have, if v = [1/&psi;(D + a)]u, the identity
+&psi;(D)(e<span class="sp">ax</span>v) = e<span class="sp">ax</span>&psi;(D + a)v = e<span class="sp">ax</span>u,
+which we write in the form</p>
+
+<table class="math0" summary="math">
+<tr><td>1</td> <td rowspan="2">(e<span class="sp">ax</span>u) = e<span class="sp">ax</span></td> <td>1</td>
+ <td rowspan="2">u.</td></tr>
+<tr><td class="denom">&psi;(D)</td> <td class="denom">&psi;(D + a)</td></tr></table>
+
+<p class="noind">This gives us the first step in the method we are explaining,
+namely that a solution of the differential equation
+&psi;(D)y = e<span class="sp">ax</span>u + e<span class="sp">bx</span>v + ...
+where u, v, ... are any functions of x, is any function
+denoted by the expression</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">e<span class="sp">ax</span></td> <td>1</td>
+ <td rowspan="2">u + e<span class="sp">bx</span></td> <td>1</td>
+ <td rowspan="2">v + ....</td></tr>
+<tr><td class="denom">&psi;(D + a)</td> <td class="denom">&psi;(D + b)</td></tr></table>
+
+<p>It is now to be shown how to obtain one value of [1/&psi;(D + a)]u,
+when u is a polynomial in x, namely one solution of the differential equation
+&psi;(D + a)z = u. Let the highest power of x entering in u be x<span class="sp">m</span>; if t
+were a variable quantity, the rational fraction in t, 1/&psi;(t + a) by first
+writing it as a sum of partial fractions, or otherwise, could be identically
+written in the form</p>
+
+<p class="center">
+K<span class="su">r</span>t<span class="sp">-r</span> + K<span class="su">r-1</span>t<span class="sp">-r+1</span> + ... + K<span class="su">1</span>t<span class="sp">-1</span> + H + H<span class="su">1</span>t + ... + H<span class="su">m</span>t<span class="sp">m</span> + t<span class="sp">m+1</span>&phi;(t)/&psi;(t + a),</p>
+
+<p class="noind">where &phi;(t) is a polynomial in t; this shows that there exists an
+identity of the form</p>
+
+<p class="center">
+1 = &psi;(t + a)(K<span class="su">r</span>t<span class="sp">&minus;r</span>  + ... + K<span class="su">1</span>t<span class="sp">&minus;1</span> + H + H<span class="su">1</span>t + ... + H<span class="su">m</span>t<span class="sp">m</span>) + &phi;(t)t<span class="sp">m+1</span>,</p>
+
+<p class="noind">and hence an identity</p>
+
+<p class="center">
+u = &psi;(D + a) [K<span class="su">r</span>D<span class="sp">&minus;r</span> + ... + K<span class="su">1</span>D<span class="sp">&minus;1</span> + H + H<span class="su">1</span>D + ... + H<span class="su">m</span>D<span class="sp">m</span>] u + &phi;(D) D<span class="sp">m+1</span>u;</p>
+
+<p class="noind">in this, since u contains no power of x higher than x<span class="sp">m</span>, the second
+term on the right may be omitted. We thus reach the conclusion
+that a solution of the differential equation &psi;(D + a)z = u is given by</p>
+
+<p class="center">
+z = (K<span class="su">r</span>D<span class="sp">&minus;r</span> + ... + K<span class="su">1</span>D<span class="sp">&minus;1</span> + H + H<span class="su">1</span>D + ... + H<span class="su">m</span>D<span class="sp">m</span>)u,</p>
+
+<p class="noind">of which the operator on the right is obtained simply by expanding
+1/&psi;(D + a) in ascending powers of D, as if D were a numerical
+quantity, the expansion being carried as far as the highest power of
+D which, operating upon u, does not give zero. In this form every
+term in z is capable of immediate calculation.</p>
+
+<p><i>Example.</i>&mdash;For the equation</p>
+
+<table class="math0" summary="math">
+<tr><td>d<span class="sp">4</span>v</td> <td rowspan="2">+ 2</td> <td>d²y</td>
+ <td rowspan="2">+ y = x³ cos x&emsp;or (D² + 1)²y = x³ cos x,</td></tr>
+<tr><td class="denom">dx<span class="sp">4</span></td> <td class="denom">dx<span class="sp">3</span></td></tr></table>
+
+<p class="noind">the roots of the associated algebraic equation (&theta;² + 1)² = 0 are &theta; = ±i,
+each repeated; the complementary function is thus</p>
+
+<p class="center">(A + Bx)e<span class="sp">ix</span> + (C + Dx)e<span class="sp">&minus;ix</span>,</p>
+
+<p class="noind">where A, B, C, D are arbitrary constants; this is the same as</p>
+
+<p class="center">(H + Kx) cos x + (M + Nx) sin x,</p>
+
+<p class="noind">where H, K, M, N are arbitrary constants. To obtain a particular
+integral we must find a value of (1 + D²)&minus;²x³ cos x;
+this is the real part of (1 + D²)&minus;² e<span class="sp">ix</span>x³ and hence of e<span class="sp">ix</span> [1 + (D + i)²]&minus;² x³</p>
+
+<p class="noind">or</p>
+
+<p class="center">e<span class="sp">ix</span> [2iD(1 + ½iD)]&minus;² x³,</p>
+
+<p class="noind">or</p>
+
+<p class="center">&minus;¼e<span class="sp">ix</span> D&minus;² (1 + iD &minus; ¾D² &minus; ½iD³ + <span class="spp">5</span>&frasl;<span class="suu">16</span>D<span class="sp">4</span> + <span class="spp">3</span>&frasl;<span class="suu">16</span>iD<span class="sp">5</span> ...)x³,</p>
+
+<p class="noind">or</p>
+
+<p class="center">&minus;¼e<span class="sp">ix</span> (<span class="spp">1</span>&frasl;<span class="suu">20</span>x<span class="sp">5</span> + ¼ix<span class="sp">4</span> &minus; ¾x³ &minus; <span class="spp">3</span>&frasl;<span class="suu">2</span> ix² + <span class="spp">15</span>&frasl;<span class="suu">8</span> x + <span class="spp">9</span>&frasl;<span class="suu">8</span> i);</p>
+
+<p class="noind">the real part of this is</p>
+
+<p class="center">&minus;¼ (<span class="spp">1</span>&frasl;<span class="suu">20</span> x<span class="sp">5</span> &minus; ¾x² + <span class="spp">15</span>&frasl;<span class="suu">8</span>x) cos x + ¼ (¼x<span class="sp">4</span> &minus; <span class="spp">3</span>&frasl;<span class="suu">4</span>x² + <span class="spp">9</span>&frasl;<span class="suu">8</span>) sin x.</p>
+
+<p class="noind">This expression added to the complementary function found above
+gives the complete integral; and no generality is lost by omitting
+from the particular integral the terms &minus;<span class="spp">15</span>&frasl;<span class="suu">32</span>x cos x + <span class="spp">9</span>&frasl;<span class="suu">32</span> sin x, which
+are of the types of terms already occurring in the complementary
+function.</p>
+
+<p>The symbolical method which has been explained has wider applications
+than that to which we have, for simplicity of explanation,
+restricted it. For example, if &psi;(x) be any function of x, and
+a<span class="su">1</span>, a<span class="su">2</span>, ... a<span class="su">n</span> be different constants, and [(t + a<span class="su">1</span>) (t + a<span class="su">2</span>) ... (t + a<span class="su">n</span>)]<span class="sp">&minus;1</span>
+when expressed in partial fractions be written &Sigma;c<span class="su">m</span>(t + a<span class="su">m</span>)<span class="sp">&minus;1</span>, a particular
+integral of the differential equation (D + a<span class="su">1</span>)(D + a<span class="su">2</span>) ...
+(D + a<span class="su">n</span>)y = &psi;(x) is given by</p>
+
+<p class="center">
+y = &Sigma;c<span class="su">m</span>(D + a<span class="su">m</span>)<span class="sp">&minus;1</span> &psi;(x) = &Sigma;c<span class="su">m</span> (D + a<span class="su">m</span>)<span class="sp">&minus;1</span> e<span class="sp">&minus;a</span>m<span class="sp">x</span>e<span class="sp">a</span>m<span class="sp">x</span> &psi;(x) =
+&Sigma;c<span class="su">m</span>e<span class="sp">&minus;a</span>m<span class="sp">x</span>D<span class="sp">&minus;1</span> <span class="f150">(</span>e<span class="sp">a</span>m<span class="sp">x</span>&psi;(x) <span class="f150">)</span> = &Sigma;c<span class="su">m</span>e<span class="sp">&minus;a</span>m<span class="sp">x</span> <span class="f150">&int;</span> e<span class="sp">a</span>m<span class="sp">x</span>&psi;(x)dx.</p>
+
+<p class="noind">The particular integral is thus expressed as a sum of n integrals.
+A linear differential equation of which the left side has the form</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">x<span class="sp">n</span></td> <td>d<span class="sp">n</span>y</td>
+<td rowspan="2">+ P<span class="su">1</span>x<span class="sp">n&minus;1</span></td> <td>d<span class="sp">n&minus;1</span>y</td>
+<td rowspan="2">+ ... + P<span class="su">n&minus;1</span>x</td> <td>dy</td>
+<td rowspan="2">+ P<span class="su">n</span>y,</td></tr>
+<tr><td class="denom">dx<span class="sp">n</span></td> <td class="denom">dx<span class="sp">n&minus;1</span></td> <td class="denom">dx</td></tr></table>
+
+<p class="noind">where P<span class="su">1</span>, ... P<span class="su">n</span> are constants, can be reduced to the case considered
+above. Writing x = e<span class="sp">t</span> we have the identity</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">x<span class="sp">m</span></td> <td>d<span class="sp">m</span>u</td>
+<td rowspan="2">= &theta;(&theta; &minus; 1)(&theta; &minus; 2) ... (&theta; &minus; m + 1)u, where &theta; = d/dt.</td></tr>
+<tr><td class="denom">dx<span class="sp">m</span></td></tr></table>
+
+<p>When the linear differential equation, which we take to be of the
+second order, has variable coefficients, though there is no general rule
+for obtaining a solution in finite terms, there are some results which
+it is of advantage to have in mind. We have seen that if one solution
+of the equation obtained by putting the right side zero, say y<span class="su">1</span>, be
+known, the equation can be solved. If y<span class="su">2</span> be another solution of</p>
+
+<table class="math0" summary="math">
+<tr><td>d²y</td> <td rowspan="2">+ P</td> <td>dy</td>
+<td rowspan="2">+ Qy = 0,</td></tr>
+<tr><td class="denom">dx²</td> <td class="denom">dx</td></tr></table>
+
+<p class="noind">there being no relation of the form my<span class="su">1</span> + ny<span class="su">2</span> = k, where m, n, k are
+constants, it is easy to see that</p>
+
+<table class="math0" summary="math">
+<tr><td>d</td> <td rowspan="2">(y<span class="su">1</span>&prime;y<span class="su">2</span> &minus; y<span class="su">1</span>y<span class="su">2</span>&prime;) = P(y<span class="su">1</span>&prime;y<span class="su">2</span> &minus; y<span class="su">1</span>y<span class="su">2</span>&prime;),</td></tr>
+<tr><td class="denom">dx</td></tr></table>
+
+<p class="noind">so that we have</p>
+
+<p class="center">y<span class="su">1</span>&prime;y<span class="su">2</span> &minus; y<span class="su">1</span>y<span class="su">2</span>&prime;
+= A exp. <span class="f150">(&int;</span> Pdx<span class="f150">)</span>,</p>
+
+<p class="noind">where A is a suitably chosen constant, and exp. z denotes e<span class="sp">z</span>. In terms
+of the two solutions y<span class="su">1</span>, y<span class="su">2</span> of the differential equation having zero on
+the right side, the general solution of the equation with R = &phi;(x) on
+the right side can at once be verified to be Ay<span class="su">1</span> + By<span class="su">2</span> + y<span class="su">1</span>u &minus; y<span class="su">2</span>v,
+where u, v respectively denote the integrals</p>
+
+<p class="center">
+u = <span class="f150">&int;</span> y<span class="su">2</span>&phi;(x) (y<span class="su">1</span>&prime;y<span class="su">2</span> &minus; y<span class="su">2</span>&prime;y<span class="su">1</span>)<span class="sp">&minus;1</span>dx, v = <span class="f150">&int;</span> y<span class="su">1</span>&phi;(x) (y<span class="su">1</span>&prime;y<span class="su">2</span> &minus; y<span class="su">2</span>&prime;y<span class="su">1</span>)<span class="sp">&minus;1</span>dx.</p>
+
+<p>The equation</p>
+
+<table class="math0" summary="math">
+<tr><td>d²y</td> <td rowspan="2">+ P</td> <td>dy</td>
+ <td rowspan="2">+ Qy = 0,</td></tr>
+<tr><td class="denom">dx²</td> <td class="denom">dx</td></tr></table>
+
+<p class="noind">by writing y = v exp. (-½ &int; Pdx), is at once seen to be reduced to
+d²v/dx² + Iv = 0, where I = Q &minus; ½dP/dx &minus; ¼P². If &eta; = &minus; 1/v dv/dx, the equation
+d²v/dx² + Iv = 0 becomes d&eta;/dx = I + &eta;², a non-linear equation of the first
+order.</p>
+
+<p>More generally the equation</p>
+
+<table class="math0" summary="math">
+<tr><td>d&eta;</td> <td rowspan="2">= A + B&eta; + C&eta;²,</td></tr>
+<tr><td class="denom">dx</td></tr></table>
+
+<p class="noind">where A, B, C are functions of x, is, by the substitution</p>
+
+<table class="math0" style="border-collapse: separate;" summary="math">
+<tr><td rowspan="2">&eta; = &minus;</td> <td>1</td> <td>dy</td></tr>
+<tr><td class="denom">Cy</td> <td class="denom">dx</td></tr></table>
+
+<p class="noind">reduced to the linear equation</p>
+
+<table class="math0" style="border-collapse: separate;" summary="math">
+<tr><td>d²y</td> <td rowspan="2">&minus; <span class="f150">(</span>B + </td> <td>1</td> <td>dC</td>
+ <td rowspan="2"><span class="f150">)</span></td> <td>dy</td>
+ <td rowspan="2">+ ACy = 0.</td></tr>
+<tr><td class="denom">dx²</td> <td class="denom">C</td>
+ <td class="denom">dx</td> <td class="denom">dx</td></tr></table>
+
+<p class="noind">The equation</p>
+
+<table class="math0" summary="math">
+<tr><td>d&eta;</td> <td rowspan="2">= A + B&eta; + C&eta;²,</td></tr>
+<tr><td class="denom">dx</td></tr></table>
+
+<p class="noind">known as Riccati&rsquo;s equation, is transformed into an equation of
+the same form by a substitution of the form &eta; = (aY + b)/(cY + d),
+where a, b, c, d are any functions of x, and this fact may be utilized
+to obtain a solution when A, B, C have special forms; in particular
+if any particular solution of the equation be known, say &eta;<span class="su">0</span>, the
+<span class="pagenum"><a name="page228" id="page228"></a>228</span>
+substitution &eta; = &eta;<span class="su">0</span> &minus; 1/Y enables us at once to obtain the general
+solution; for instance, when</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">2B =</td> <td>d</td>
+ <td rowspan="2">log <span class="f150">(</span></td> <td>A</td>
+ <td rowspan="2"><span class="f150">)</span>,</td></tr>
+<tr><td class="denom">dx</td> <td class="denom">C</td></tr></table>
+
+<p class="noind">a particular solution is &eta;<span class="su">0</span> = &radic;(-A/C). This is a case of the remark,
+often useful in practice, that the linear equation</p>
+
+<table class="math0" style="border-collapse: separate;" summary="math">
+<tr><td rowspan="2">&phi;(x)</td> <td>d²y</td>
+ <td rowspan="2">+ ½</td> <td>d&phi;</td> <td>dy</td>
+ <td rowspan="2">+ &mu;y = 0,</td></tr>
+<tr><td class="denom">dx²</td> <td class="denom">dx</td> <td class="denom">dx</td></tr></table>
+
+<p class="noind">where &mu; is a constant, is reducible to a standard form by taking a new
+independent variable z = &int; dx[&phi;(x)]<span class="sp">-½</span>.</p>
+
+<p>We pass to other types of equations of which the solution can be
+obtained by rule. We may have cases in which there are two
+dependent variables, x and y, and one independent variable t, the
+differential coefficients dx/dt, dy/dt being given as functions of x, y
+and t. Of such equations a simple case is expressed by the pair</p>
+
+<table class="math0" summary="math">
+<tr><td>dx</td> <td rowspan="2">= ax + by + c,</td>
+<td>dy</td> <td rowspan="2">a&prime;x + b&prime;y + c&prime;,</td></tr>
+<tr><td class="denom">dt</td> <td class="denom">dt</td></tr></table>
+
+<p class="noind">wherein the coefficients a, b, c, a&prime;, b&prime;, c&prime;, are constants. To integrate
+these, form with the constant &lambda; the differential coefficient of
+z = x + &lambda;y, that is dz/dt = (a + &lambda;a&prime;)x + (b + &lambda;b&prime;)y + c + &lambda;c&prime;, the quantity
+&lambda; being so chosen that b + &lambda;b&prime; = &lambda;(a + &lambda;a&prime;), so that we have
+dz/dt = (a + &lambda;a&prime;)z + c + &lambda;c&prime;; this last equation is at once integrable
+in the form z(a + &lambda;a&prime;) + c + &lambda;c&prime; = Ae<span class="sp">(a + &lambda;a&prime;)t</span>, where A is an arbitrary
+constant. In general, the condition b + &lambda;b&prime; = &lambda;(a + &lambda;a&prime;) is satisfied by
+two different values of &lambda;, say &lambda;<span class="su">1</span>, &lambda;<span class="su">2</span>; the solutions corresponding to
+these give the values of x +&lambda;<span class="su">1</span>y and x + &lambda;<span class="su">2</span>y, from which x and y can
+be found as functions of t, involving two arbitrary constants. If,
+however, the two roots of the quadratic equation for &lambda; are equal,
+that is, if (a &minus; b&prime;)² + 4a&prime;b = 0, the method described gives only one
+equation, expressing x + &lambda;y in terms of t; by means of this equation
+y can be eliminated from dx/dt = ax + by + c, leading to an equation
+of the form dx/dt = Px + Q + Re<span class="sp">(a + &lambda;a&prime;)t</span>, where P, Q, R are constants.
+The integration of this gives x, and thence y can be found.</p>
+
+<p>A similar process is applicable when we have three or more
+dependent variables whose differential coefficients in regard to the
+single independent variables are given as linear functions of the
+dependent variables with constant coefficients.</p>
+
+<p>Another method of solution of the equations</p>
+
+<p class="center">dx/dt = ax + by + c, dy/dt = a&prime;x + b&prime;y + c&prime;,</p>
+
+<p class="noind">consists in differentiating the first equation, thereby obtaining</p>
+
+<table class="math0" summary="math">
+<tr><td>d²x</td> <td rowspan="2">= a</td> <td>dx</td>
+<td rowspan="2">+ b</td> <td>dy</td> <td rowspan="2">;</td></tr>
+<tr><td class="denom">dt²</td> <td class="denom">dt</td> <td class="denom">dx</td></tr></table>
+
+<p class="noind">from the two given equations, by elimination of y, we can express
+dy/dt as a linear function of x and dx/dt; we can thus form an
+equation of the shape d²x/dt² = P + Qx + Rdx/dt, where P, Q, R are
+constants; this can be integrated by methods previously explained,
+and the integral, involving two arbitrary constants, gives,
+by the equation dx/dt = ax + by + c, the corresponding value of y.
+Conversely it should be noticed that any single linear differential
+equation</p>
+
+<table class="math0" summary="math">
+<tr><td>d²x</td> <td rowspan="2">= u + vx + w</td> <td>dx</td> <td rowspan="2">,</td></tr>
+<tr><td class="denom">dt²</td> <td class="denom">dt</td></tr></table>
+
+<p class="noind">where u, v, w are functions of t, by writing y for dx/dt, is equivalent
+with the two equations dx/dt = y, dy/dt = u + vx + wy. In fact a
+similar reduction is possible for any system of differential equations
+with one independent variable.</p>
+
+<p>Equations occur to be integrated of the form</p>
+
+<p class="center">Xdx + Ydy + Zdz = 0,</p>
+
+<p class="noind">where X, Y, Z are functions of x, y, z. We consider only the case in
+which there exists an equation &phi;(x, y, z) = C whose differential</p>
+
+<table class="math0" summary="math">
+<tr><td>&part;&phi;</td> <td rowspan="2">dx +</td> <td>&part;&phi;</td>
+<td rowspan="2">dy +</td> <td>&part;&phi;</td> <td rowspan="2">dz = 0</td></tr>
+<tr><td class="denom">&part;x</td> <td class="denom">&part;y</td> <td class="denom">&part;z</td></tr></table>
+
+<p class="noind">is equivalent with the given differential equation; that is, &mu; being
+a proper function of x, y, z, we assume that there exist equations</p>
+
+<table class="math0" summary="math">
+<tr><td>&part;&phi;</td> <td rowspan="2">= &mu;X,</td> <td>&part;&phi;</td>
+<td rowspan="2">= &mu;Y,</td> <td>&part;&phi;</td>
+<td rowspan="2">= &mu;Z;</td></tr>
+<tr><td class="denom">&part;x</td> <td class="denom">&part;y</td> <td class="denom">&part;z</td></tr></table>
+
+<p class="noind">these equations require</p>
+
+<table class="math0" summary="math">
+<tr><td>&part;</td> <td rowspan="2">(&mu;Y) &asymp;</td> <td>&part;</td>
+<td rowspan="2">(&mu;Z), &amp;c.,</td></tr>
+<tr><td class="denom">&part;z</td> <td class="denom">&part;y</td></tr></table>
+
+<p class="noind">and hence</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">X<span class="f200">(</span></td> <td>&part;Z</td>
+<td rowspan="2">&minus;</td> <td>&part;Y</td>
+<td rowspan="2"><span class="f200">)</span> + Y<span class="f200">(</span></td> <td>&part;X</td>
+<td rowspan="2">&minus;</td> <td>&part;Z</td>
+<td rowspan="2"><span class="f200">)</span> + Z<span class="f200">(</span></td> <td>&part;Y</td>
+<td rowspan="2">&minus;</td> <td>&part;X</td> <td rowspan="2"><span class="f200">)</span> = 0;</td></tr>
+<tr><td class="denom">&part;y</td> <td class="denom">&part;z</td>
+ <td class="denom">&part;z</td> <td class="denom">&part;x</td>
+ <td class="denom">&part;x</td> <td class="denom">&part;y</td></tr></table>
+
+<p class="noind">conversely it can be proved that this is sufficient in order that &mu;
+may exist to render &mu;(Xdx + Ydy + Zdz) a perfect differential; in
+particular it may be satisfied in virtue of the three equations such as</p>
+
+<table class="math0" summary="math">
+<tr><td>&part;Z</td> <td rowspan="2">&minus;</td> <td>&part;Y</td>
+<td rowspan="2">= 0;</td></tr>
+<tr><td class="denom">&part;y</td> <td class="denom">&part;z</td></tr></table>
+
+<p class="noind">in which case we may take &mu; = 1. Assuming the condition in
+its general form, take in the given differential equation a plane
+section of the surface &phi; = C parallel to the plane z, viz. put z constant,
+and consider the resulting differential equation in the two
+variables x, y, namely Xdx + Ydy = 0; let &psi;(x, y, z) = constant, be its
+integral, the constant z entering, as a rule, in &psi; because it enters in
+X and Y. Now differentiate the relation &psi;(x, y, z) = &fnof;(z), where &fnof;
+is a function to be determined, so obtaining</p>
+
+<table class="math0" summary="math">
+<tr><td>&part;&psi;</td> <td rowspan="2">dx +</td> <td>&part;&psi;</td>
+<td rowspan="2">dy + <span class="f200">(</span></td> <td>&part;&psi;</td>
+<td rowspan="2">&minus;</td> <td>d&fnof;</td> <td rowspan="2"><span class="f200">)</span> dz = 0;</td></tr>
+<tr><td class="denom">&part;x </td> <td class="denom">&part;y</td>
+ <td class="denom">&part;z</td> <td class="denom">dz</td></tr></table>
+
+<p class="noind">there exists a function &sigma; of x, y, z such that</p>
+
+<table class="math0" summary="math">
+<tr> <td>&part;&psi;</td> <td rowspan="2">= &sigma;X,</td> <td>&part;&psi;</td>
+<td rowspan="2">= &sigma;Y,</td></tr>
+<tr><td class="denom">&part;x</td> <td class="denom">&part;y</td></tr></table>
+
+<p class="noind">because &psi; = constant, is the integral of Xdx + Ydy = 0; we desire to
+prove that &fnof; can be chosen so that also, in virtue of &psi;(x, y, z) = &fnof;(z),
+we have</p>
+
+<table class="math0" summary="math">
+<tr><td>&part;&psi;</td> <td rowspan="2">&minus;</td> <td>d&fnof;</td>
+<td rowspan="2">= &sigma;Z, namely</td> <td>d&fnof;</td>
+<td rowspan="2">=</td> <td>&part;&psi;</td> <td rowspan="2">&minus; &sigma;Z;</td></tr>
+<tr><td class="denom">&part;z</td> <td class="denom">dz</td>
+ <td class="denom">dz</td> <td class="denom">&part;z</td></tr></table>
+
+<p class="noind">if this can be proved the relation &psi;(x, y, z) &minus; &fnof;(z) = constant, will be
+the integral of the given differential equation. To prove this it is
+enough to show that, in virtue of &psi;(x, y, z) = &fnof;(z), the function
+&part;&psi;/&part;x &minus; &sigma;Z can be expressed in terms of z only. Now in consequence
+of the originally assumed relations,</p>
+
+<table class="math0" summary="math">
+<tr><td>&part;&psi;</td> <td rowspan="2">= &mu;X,</td> <td>&part;&phi;</td>
+<td rowspan="2">= &mu;Y,</td> <td>&part;&phi;</td> <td rowspan="2">= &mu;Z,</td></tr>
+<tr><td class="denom">&part;x</td> <td class="denom">&part;y</td> <td class="denom">&part;z</td></tr></table>
+
+<p class="noind">we have</p>
+
+<table class="math0" summary="math">
+<tr><td>&part;&psi;</td> <td rowspan="2"><span class="f200">/</span></td> <td>&part;&phi;</td>
+<td rowspan="2">=</td> <td>&sigma;</td>
+<td rowspan="2">=</td> <td>&part;&psi;</td>
+<td rowspan="2"><span class="f200">/</span></td> <td>&part;&phi;</td> <td rowspan="2">,</td></tr>
+<tr><td class="denom">&part;x</td> <td class="denom">&part;x</td>
+ <td class="denom">&mu;</td> <td class="denom">&part;y</td> <td class="denom">&part;y</td></tr></table>
+
+<p class="noind">and hence</p>
+
+<table class="math0" style="border-collapse: separate;" summary="math">
+<tr><td>&part;&psi;</td> <td>&part;&phi;</td> <td rowspan="2">&minus;</td>
+ <td>&part;&psi;</td> <td>&part;&phi;</td> <td rowspan="2">= 0;</td></tr>
+<tr><td class="denom">&part;x</td> <td class="denom">&part;y</td>
+ <td class="denom">&part;y</td> <td class="denom">&part;x</td></tr></table>
+
+<p class="noind">this shows that, as functions of x and y, &psi; is a function of &phi; (see the
+note at the end of part i. of this article, on Jacobian determinants),
+so that we may write &psi; = F(z, &phi;), from which</p>
+
+<table class="math0" style="border-collapse: separate;" summary="math">
+<tr><td>&sigma;</td> <td rowspan="2">=</td> <td>&part;F</td>
+<td rowspan="2">; then</td> <td>&part;&psi;</td>
+<td rowspan="2">=</td> <td>&part;F</td>
+<td rowspan="2">+</td> <td>&part;F</td> <td>&part;&phi;</td>
+<td rowspan="2">=</td> <td>&part;F</td>
+<td rowspan="2">+</td> <td>&sigma;</td>
+<td rowspan="2">· &mu;Z =</td> <td>&part;F</td>
+<td rowspan="2">+ &sigma;Z or</td> <td>&part;&psi;</td>
+<td rowspan="2">&minus; &sigma;Z =</td> <td>&part;F</td> <td rowspan="2">;</td></tr>
+<tr><td class="denom">&mu;</td> <td class="denom">&part;&phi;</td>
+<td class="denom">&part;z</td> <td class="denom">&part;z</td>
+<td class="denom">&part;&phi;</td> <td class="denom">&part;z</td>
+<td class="denom">&part;z</td> <td class="denom">&mu;</td>
+<td class="denom">&part;z</td> <td class="denom">&part;z</td> <td class="denom">&part;z</td></tr></table>
+
+<p class="noind">in virtue of &psi;(x, y, z) = &fnof;(z), and &psi; = F(z, &phi;), the function &phi; can be
+written in terms of z only, thus &part;F/&part;z can be written in terms of z only,
+and what we required to prove is proved.</p>
+
+<p>Consider lastly a simple type of differential equation containing
+<i>two</i> independent variables, say x and y, and one dependent variable
+z, namely the equation</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">P</td> <td>&part;z</td>
+<td rowspan="2">+ Q</td> <td>&part;z</td> <td rowspan="2">= R,</td></tr>
+<tr><td class="denom">&part;x</td> <td class="denom">&part;y</td></tr></table>
+
+<p class="noind">where P, Q, R are functions of x, y, z. This is known as Lagrange&rsquo;s
+linear partial differential equation of the first order. To integrate
+this, consider first the ordinary differential equations dx/dz = P/R,
+dy/dz = Q/R, and suppose that two functions u, v, of x, y, z can be
+determined, independent of one another, such that the equations
+u = a, v = b, where a, b are arbitrary constants, lead to these ordinary
+differential equations, namely such that</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">P</td> <td>&part;u</td>
+<td rowspan="2">+ Q</td> <td>&part;u</td>
+<td rowspan="2">+ R</td> <td>&part;u</td>
+<td rowspan="2">= 0 and P</td> <td>&part;v</td>
+<td rowspan="2">+ Q</td> <td>&part;v</td>
+<td rowspan="2">+ R</td> <td>&part;v</td> <td rowspan="2">= 0.</td></tr>
+<tr><td class="denom">&part;x</td> <td class="denom">&part;y</td>
+<td class="denom">&part;z</td> <td class="denom">&part;x</td>
+<td class="denom">&part;y</td> <td class="denom">&part;z</td></tr></table>
+
+<p class="noind">Then if F(x, y, z) = 0 be a relation satisfying the original differential
+equations, this relation giving rise to</p>
+
+<table class="math0" style="border-collapse: separate;" summary="math">
+<tr><td>&part;F</td> <td rowspan="2">+</td> <td>&part;F</td> <td>&part;z</td>
+<td rowspan="2">= 0 and</td> <td>&part;F</td>
+<td rowspan="2">+</td> <td>&part;F</td> <td>&part;z</td>
+<td rowspan="2">= 0, we have P</td> <td>&part;F</td>
+<td rowspan="2">+ Q</td> <td>&part;F</td>
+<td rowspan="2">+ R</td> <td>&part;F</td> <td rowspan="2">= 0.</td></tr>
+<tr><td class="denom">&part;x</td> <td class="denom">&part;z</td> <td class="denom">&part;x</td>
+<td class="denom">&part;y</td> <td class="denom">&part;z</td> <td class="denom">&part;y</td>
+<td class="denom">&part;x</td> <td class="denom">&part;y</td> <td class="denom">&part;z</td></tr></table>
+
+<p class="noind">It follows that the determinant of three rows and columns vanishes
+whose first row consists of the three quantities &part;F/&part;x, &part;F/&part;y, &part;F/&part;z,
+whose second row consists of the three quantities &part;u/&part;x, &part;u/&part;y, &part;u/&part;z,
+whose third row consists similarly of the partial derivatives of v.
+The vanishing of this so-called Jacobian determinant is known to
+imply that F is expressible as a function of u and v, unless these are
+themselves functionally related, which is contrary to hypothesis
+(see the note below on Jacobian determinants). Conversely, any
+relation &phi;(u, v) = 0 can easily be proved, in virtue of the equations
+satisfied by u and v, to lead to</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">P</td> <td>dz</td>
+<td rowspan="2">+ Q</td> <td>dz</td> <td rowspan="2">= R.</td></tr>
+<tr><td class="denom">dx</td> <td class="denom">dx</td></tr></table>
+
+<p class="noind">The solution of this partial equation is thus reduced to the solution
+of the two ordinary differential equations expressed by
+dx/P = dy/Q = dz/R. In regard to this problem one remark may be
+made which is often of use in practice: when one equation u = a
+has been found to satisfy the differential equations, we may utilize
+this to obtain the second equation v = b; for instance, we may, by
+means of u = a, eliminate z&mdash;when then from the resulting equations
+in x and y a relation v = b has been found containing x and y and a,
+the substitution a = u will give a relation involving x, y, z.</p>
+
+<p><i>Note on Jacobian Determinants.</i>&mdash;The fact assumed above that the
+vanishing of the Jacobian determinant whose elements are the partial
+derivatives of three functions F, u, v, of three variables x, y, z,
+<span class="pagenum"><a name="page229" id="page229"></a>229</span>
+involves that there exists a functional relation connecting the three
+functions F, u, v, may be proved somewhat roughly as follows:&mdash;</p>
+
+<p>The corresponding theorem is true for any number of variables.
+Consider first the case of two functions p, q, of two variables x, y.
+The function p, not being constant, must contain one of the variables,
+say x; we can then suppose x expressed in terms of y and the function
+p; thus the function q can be expressed in terms of y and the function
+p, say q = Q(p, y). This is clear enough in the simplest cases which
+arise, when the functions are rational. Hence we have</p>
+
+<table class="math0" style="border-collapse: separate;" summary="math">
+<tr><td>&part;q</td> <td rowspan="2">=</td> <td>&part;Q</td> <td>&part;p</td>
+<td rowspan="2">and</td> <td>&part;q</td>
+<td rowspan="2">=</td> <td>&part;Q</td> <td>&part;p</td>
+<td rowspan="2">+</td> <td>&part;Q</td> <td rowspan="2">;</td></tr>
+<tr><td class="denom">&part;x</td> <td class="denom">&part;p</td> <td class="denom">&part;x</td>
+<td class="denom">&part;y</td> <td class="denom">&part;p</td> <td class="denom">&part;y</td>
+<td class="denom">&part;y</td></tr></table>
+
+<p class="noind">these give</p>
+
+<table class="math0" summary="math">
+<tr><td>&part;p</td> <td>&part;q</td> <td rowspan="2">&minus;</td>
+<td>&part;p</td> <td>&part;q</td> <td rowspan="2">=</td>
+<td>&part;p</td> <td>&part;Q</td> <td rowspan="2">;</td></tr>
+ <tr><td class="ov">&part;x</td> <td class="ov">&part;y</td>
+ <td class="ov">&part;y</td> <td class="ov">&part;x</td>
+ <td class="ov">&part;x</td> <td class="ov">&part;y</td></tr></table>
+
+<p class="noind">by hypothesis &part;p/&part;x is not identically zero; therefore if the Jacobian
+determinant of p and q in regard to x and y is zero identically, so is
+&part;Q/&part;y, or Q does not contain y, so that q is expressible as a function
+of p only. Conversely, such an expression can be seen at once to
+make the Jacobian of p and q vanish identically.</p>
+
+<p>Passing now to the case of three variables, suppose that the
+Jacobian determinant of the three functions F, u, v in regard to
+x, y, z is identically zero. We prove that if u, v are not themselves
+functionally connected, F is expressible as a function of u and v.
+Suppose first that the minors of the elements of &part;F/&part;x, &part;F/&part;y, &part;F/&part;z
+in the determinant are all identically zero, namely the three determinants
+such as</p>
+
+<table class="math0" summary="math">
+<tr><td>&part;u</td> <td>&part;v</td> <td rowspan="2">&minus;</td>
+<td>&part;u</td> <td>&part;v</td> <td rowspan="2">;</td></tr>
+<tr><td class="ov">&part;y</td> <td class="ov">&part;z</td>
+<td class="ov">&part;z</td> <td class="ov">&part;y</td></tr></table>
+
+<p class="noind">then by the case of two variables considered above there exist three
+functional relations. &psi;<span class="su">1</span>(u, v, x) = 0, &psi;<span class="su">2</span>(u, v, y) = 0, &psi;<span class="su">3</span>(u, v, z) = 0, of which
+the first, for example, follows from the vanishing of</p>
+
+<table class="math0" summary="math">
+<tr><td>&part;u</td> <td>&part;v</td> <td rowspan="2">&minus;</td>
+<td>&part;u</td> <td>&part;v</td> <td rowspan="2">.</td></tr>
+<tr><td class="ov">&part;y</td> <td class="ov">&part;z</td>
+<td class="ov">&part;z</td> <td class="ov">&part;y</td></tr></table>
+
+<p class="noind">We cannot assume that x is absent from &psi;<span class="su">1</span>, or y from &psi;<span class="su">2</span>, or z from &psi;<span class="su">3</span>;
+but conversely we cannot simultaneously have x entering in &psi;<span class="su">1</span>, and
+y in &psi;<span class="su">2</span>, and z in &psi;<span class="su">3</span>, or else by elimination of u and v from the three
+equations &psi;<span class="su">1</span> = 0, &psi;<span class="su">2</span> = 0, &psi;<span class="su">3</span> = 0, we should find a necessary relation
+connecting the three independent quantities x, y, z; which is absurd.
+Thus when the three minors of &part;F/&part;x, &part;F/&part;y, &part;F/&part;z in the Jacobian
+determinant are all zero, there exists a functional relation connecting
+u and v only. Suppose no such relation to exist; we can then
+suppose, for example, that</p>
+
+<table class="math0" summary="math">
+<tr><td>&part;u</td> <td>&part;v</td> <td rowspan="2">&minus;</td>
+<td>&part;u</td> <td>&part;v</td></tr>
+<tr><td class="ov">&part;y</td> <td class="ov">&part;z</td>
+<td class="ov">&part;z</td> <td class="ov">&part;y</td></tr></table>
+
+<p class="noind">is not zero. Then from the equations u(x, y, z) = u, v(x, y, z) = v we can
+express y and z in terms of u, v, and x (the attempt to do this could
+only fail by leading to a relation connecting u, v and x, and the
+existence of such a relation would involve that the determinant</p>
+
+<table class="math0" summary="math">
+<tr><td>&part;u</td> <td>&part;v</td> <td rowspan="2">&minus;</td>
+<td>&part;u</td> <td>&part;v</td></tr>
+<tr><td class="ov">&part;y</td> <td class="ov">&part;z</td>
+<td class="ov">&part;z</td> <td class="ov">&part;y</td></tr></table>
+
+<p class="noind">was zero), and so write F in the form F(x, y, z) = &Phi;(u, v, x). We then
+have</p>
+
+<table class="math0" summary="math">
+<tr><td>&part;F</td> <td rowspan="2">=</td> <td>&part;&Phi;</td> <td>&part;u</td>
+<td rowspan="2">+</td> <td>&part;&Phi;</td> <td>&part;v</td>
+<td rowspan="2">+</td> <td>&part;&Phi;</td> <td rowspan="2">,</td> <td>&part;F</td>
+<td rowspan="2">=</td> <td>&part;&Phi;</td> <td>&part;u</td> <td rowspan="2">+</td> <td>&part;&Phi;</td> <td>&part;v</td>
+<td rowspan="2">,</td> <td>&part;F</td>
+<td rowspan="2">=</td> <td>&part;&Phi;</td> <td>&part;u</td> <td rowspan="2">+</td> <td>&part;&Phi;</td> <td>&part;v</td>
+<td rowspan="2">;</td>
+</tr>
+<tr><td class="ov">&part;x</td> <td class="ov">&part;u</td> <td class="ov">&part;x</td> <td class="ov">&part;v</td>
+<td class="ov">&part;x</td> <td class="ov">&part;x</td> <td class="ov">&part;y</td> <td class="ov">&part;u</td>
+<td class="ov">&part;y</td> <td class="ov">&part;v</td> <td class="ov">&part;y</td> <td class="ov">&part;z</td>
+<td class="ov">&part;u</td> <td class="ov">&part;z</td> <td class="ov">&part;v</td> <td class="ov">&part;z</td></tr></table>
+
+<p class="noind">thereby the Jacobian determinant of F, u, v is reduced to</p>
+
+<table class="math0" summary="math">
+<tr><td>&part;&Phi;</td> <td rowspan="2"><span class="f150">(</span></td>
+<td>&part;u</td> <td>&part;v</td> <td rowspan="2">&minus;</td>
+<td>&part;u</td> <td>&part;v</td> <td rowspan="2"><span class="f150">)</span>;</td></tr>
+<tr><td class="ov">&part;x</td> <td class="ov">&part;y</td> <td class="ov">&part;z</td>
+<td class="ov">&part;z</td> <td class="ov">&part;y</td></tr></table>
+
+<p class="noind">by hypothesis the second factor of this does not vanish identically;
+hence &part;&Phi;/&part;x = 0 identically, and &Phi; does not contain x; so that F
+is expressible in terms of u, v only; as was to be proved.</p>
+
+<p class="center pt2"><i>Part II.&mdash;General Theory.</i></p>
+
+<p>Differential equations arise in the expression of the relations
+between quantities by the elimination of details, either unknown
+or regarded as unessential to the formulation of the relations in
+question. They give rise, therefore, to the two closely connected
+problems of determining what arrangement of details is consistent
+with them, and of developing, apart from these details, the general
+properties expressed by them. Very roughly, two methods of
+study can be distinguished, with the names Transformation-theories,
+Function-theories; the former is concerned with the
+reduction of the algebraical relations to the fewest and simplest
+forms, eventually with the hope of obtaining explicit expressions
+of the dependent variables in terms of the independent variables;
+the latter is concerned with the determination of the general
+descriptive relations among the quantities which are involved by
+the differential equations, with as little use of algebraical calculations
+as may be possible. Under the former heading we may,
+with the assumption of a few theorems belonging to the latter,
+arrange the theory of partial differential equations and Pfaff&rsquo;s
+problem, with their geometrical interpretations, as at present
+developed, and the applications of Lie&rsquo;s theory of transformation-groups
+to partial and to ordinary equations; under the
+latter, the study of linear differential equations in the manner
+initiated by Riemann, the applications of discontinuous groups,
+the theory of the singularities of integrals, and the study of
+potential equations with existence-theorems arising therefrom.
+In order to be clear we shall enter into some detail in regard
+to partial differential equations of the first order, both those
+which are linear in any number of variables and those not
+linear in two independent variables, and also in regard to the
+function-theory of linear differential equations of the second
+order. Space renders impossible anything further than the
+briefest account of many other matters; in particular, the theories
+of partial equations of higher than the first order, the function-theory
+of the singularities of ordinary equations not linear and the
+applications to differential geometry, are taken account of only in
+the bibliography. It is believed that on the whole the article will
+be more useful to the reader than if explanations of method had
+been further curtailed to include more facts.</p>
+
+<p>When we speak of a function without qualification, it is to be
+understood that in the immediate neighbourhood of a particular
+set x<span class="su">0</span>, y<span class="su">0</span>, ... of values of the independent variables x, y, ...
+of the function, at whatever point of the range of values for
+x, y, ... under consideration x<span class="su">0</span>, y<span class="su">0</span>, ... may be chosen, the
+function can be expressed as a series of positive integral powers
+of the differences x &minus; x<span class="su">0</span>, y &minus; y<span class="su">0</span>, ..., convergent when these are
+sufficiently small (see <span class="sc"><a href="#artlinks">Function: Functions of Complex Variables</a></span>).
+Without this condition, which we express by saying that
+the function is developable about x<span class="su">0</span>, y<span class="su">0</span>, ..., many results
+provisionally stated in the transformation theories would be
+unmeaning or incorrect. If, then, we have a set of k functions,
+&fnof;<span class="su">1</span> ... &fnof;<span class="su">k</span> of n independent variables x<span class="su">1</span> ... x<span class="su">n</span>, we say that
+they are independent when n &ge; k and not every determinant of
+k rows and columns vanishes of the matrix of k rows and n
+columns whose r-th row has the constituents d&fnof;<span class="su">r</span>/dx<span class="su">1</span>, ... d&fnof;<span class="su">r</span>/dx<span class="su">n</span>;
+the justification being in the theorem, which we assume, that if
+the determinant involving, for instance, the first k columns be not
+zero for x<span class="su">1</span> = xº<span class="su">1</span> ... x<span class="su">n</span> = xº<span class="su">n</span>, and the functions be developable
+about this point, then from the equations &fnof;<span class="su">1</span> = c<span class="su">1</span>, ... &fnof;<span class="su">k</span> = c<span class="su">k</span> we
+can express x<span class="su">1</span>, ... x<span class="su">k</span> by convergent power series in the
+differences x<span class="su">k+1</span> &minus; xº<span class="su">k+1</span>, ... x<span class="su">n</span> &minus; x<span class="su">nº</span>, and so regard x<span class="su">1</span>, ... x<span class="su">k</span>
+as functions of the remaining variables. This we often express by
+saying that the equations &fnof;<span class="su">1</span> = c<span class="su">1</span>, ... &fnof;<span class="su">k</span> = c<span class="su">k</span> can be solved for
+x<span class="su">1</span>, ... x<span class="su">k</span>. The explanation is given as a type of explanation
+often understood in what follows.</p>
+
+<p>We may conveniently begin by stating the theorem: If each of
+the n functions &phi;<span class="su">1</span>, ... &phi;<span class="su">n</span> of the (n + 1) variables x<span class="su">1</span>, ... x<span class="su">n</span>t be developable
+<span class="sidenote">Ordinary equations of the first order.</span>
+about the values xº<span class="su">1</span>, ... x<span class="su">n</span><span class="sp">0</span>t<span class="sp">0</span>, the n differential
+equations of the form dx<span class="su">1</span>/dt = &phi;<span class="su">1</span>(tx<span class="su">1</span>, ... x<span class="su">n</span>) are satisfied
+by convergent power series</p>
+
+<p class="center">x<span class="su">r</span> = xº<span class="su">r</span> + (t &minus; t<span class="sp">0</span>) A<span class="su">r1</span> + (t &minus; t<span class="su">0</span>)² A<span class="su">r2</span> + ...</p>
+
+<p class="noind">reducing respectively to xº<span class="su">1</span>, ... xº<span class="su">n</span> when t = t<span class="sp">0</span>; and the
+only functions satisfying the equations and reducing respectively to
+xº<span class="su">1</span>, ... xº<span class="su">n</span> when t = t<span class="sp">0</span>, are those determined by continuation of these
+series. If the result of solving these n equations for xº<span class="su">1</span>, ... xº<span class="su">n</span> be
+written in the form &omega;<span class="su">1</span>(x<span class="su">1</span>, ... x<span class="su">n</span>t) = xº<span class="su">1</span>, ... &omega;<span class="su">n</span>(x<span class="su">1</span>, ... x<span class="su">n</span>t) = xº<span class="su">n</span>,
+<span class="sidenote">Single homogeneous partial equation of the first order.</span>
+it is at once evident that the differential equation</p>
+
+<p class="center">d&fnof;/dt + &phi;<span class="su">1</span>d&fnof;/dx<span class="su">1</span> + ... + &phi;<span class="su">n</span>d&fnof;/dx<span class="su">n</span> = 0</p>
+
+<p class="noind">possesses n integrals, namely, the functions &omega;<span class="su">1</span>, ... &omega;<span class="su">n</span>,
+which are developable about the values (xº<span class="su">1</span> ... x<span class="su">n</span><span class="sp">0</span>t<span class="sp">0</span>) and
+reduce respectively to x<span class="su">1</span>, ... x<span class="su">n</span> when t = t<span class="sp">0</span>. And in fact it
+has no other integrals so reducing. Thus this equation
+also possesses a unique integral reducing when t = t<span class="sp">0</span> to an arbitrary
+function &psi;(x<span class="su">1</span>, ... x<span class="su">n</span>), this integral being. &psi;(&omega;<span class="su">1</span>, ... &omega;<span class="su">n</span>). Conversely
+the existence of these <i>principal</i> integrals &omega;<span class="su">1</span>, ... &omega;<span class="su">n</span> of the partial
+equation establishes the existence of the specified solutions of the
+ordinary equations dx<span class="su">i</span>/dt = &phi;<span class="su">i</span>. The following sketch of the proof of
+the existence of these principal integrals for the case n = 2 will show
+the character of more general investigations. Put x for x &minus; x<span class="sp">0</span>, &amp;c.,
+and consider the equation a(xyt) d&fnof;/dx + b(xyt) d&fnof;/dy = d&fnof;/dt, wherein
+the functions a, b are developable about x = 0, y = 0, t = 0; say</p>
+
+<p class="center">a(xyt) = a<span class="su">0</span> + ta<span class="su">1</span> + t²a<span class="su">2</span>/2! + ..., b(xyt) = b<span class="su">0</span> + tb<span class="su">1</span> + t²b<span class="su">2</span>/2! + ...,</p>
+
+<p class="noind">so that</p>
+
+<p class="center">ad/dx + bd/dy = &delta;<span class="su">0</span> + t&delta;<span class="su">1</span> + ½t²&delta;<span class="su">2</span> + ...,</p>
+
+<p class="noind">where &delta; = a<span class="su">r</span>d/dx + b<span class="su">r</span>d/dy. In order that</p>
+
+<p class="center">&fnof; = p<span class="su">0</span> + tp<span class="su">1</span> + t²p<span class="su">2</span>/2! + ...</p>
+
+<p><span class="pagenum"><a name="page230" id="page230"></a>230</span></p>
+
+<p>wherein p<span class="su">0</span>, p<span class="su">1</span> ... are power series in x, y, should satisfy the equation,
+it is necessary, as we find by equating like terms, that</p>
+
+<p class="center">p<span class="su">1</span> = &delta;<span class="su">0</span>p<span class="su">0</span>, p<span class="su">2</span> = &delta;<span class="su">0</span>p<span class="su">1</span> + &delta;<span class="su">1</span>p<span class="su">0</span>, &amp;c.</p>
+
+<p class="noind">and in general<span class="sidenote">Proof of the existence of integrals.</span></p>
+
+<p class="center">
+p<span class="su">s+1</span> = &delta;<span class="su">0</span>p<span class="su">s</span> + s<span class="su">1</span>&delta;<span class="su">1</span>p<span class="su">s-1</span> + s<span class="su">2</span>&delta;<span class="su">2</span>p<span class="su">s-2</span> +... + &delta;<span class="su">s</span>p<span class="su">0</span>,</p>
+
+<p class="noind">where</p>
+
+<p class="center">s<span class="su">r</span> = (s!)/(r!) (s &minus; r)!</p>
+
+<p>Now compare with the given equation another equation</p>
+
+<p class="center">A(xyt)dF/dx + B(xyt)dF/dy = dF/dt,</p>
+
+<p class="noind">wherein each coefficient in the expansion of either A or B is real and
+positive, and not less than the absolute value of the corresponding
+coefficient in the expansion of a or b. In the second equation let us
+substitute a series</p>
+
+<p class="center">
+F = P<span class="su">0</span> + tP<span class="su">1</span> + t²P<span class="su">2</span>/2! + ...,</p>
+
+<p class="noind">wherein the coefficients in P<span class="su">0</span> are real and positive, and each not less
+than the absolute value of the corresponding coefficient in p<span class="su">0</span>; then
+putting &Delta;<span class="su">r</span> = A<span class="su">r</span>d/dx + B<span class="su">r</span>d/dy we obtain necessary equations of the
+same form as before, namely,</p>
+
+<p class="center">P<span class="su">1</span> = &Delta;<span class="su">0</span>P<span class="su">0</span>, P<span class="su">2</span> = &Delta;<span class="su">0</span>P<span class="su">1</span> + &Delta;<span class="su">1</span>P<span class="su">0</span>, ...</p>
+
+<p class="noind">and in general P<span class="su">s+1</span> = &Delta;<span class="su">0</span>P<span class="su">s</span>, + s<span class="su">1</span>&Delta;<span class="su">1</span>P<span class="su">s-1</span> + ... + &Delta;<span class="su">s</span>P<span class="su">0</span>. These give for
+every coefficient in P<span class="su">s+1</span> an integral aggregate with real positive
+coefficients of the coefficients in P<span class="su">s</span>, P<span class="su">s-1</span>, ..., P<span class="su">0</span> and the coefficients
+in A and B; and they are the same aggregates as would be given by
+the previously obtained equations for the corresponding coefficients
+in p<span class="su">s+1</span> in terms of the coefficients in p<span class="su">s</span>, p<span class="su">s-1</span>, ..., p<span class="su">0</span> and the coefficients
+in a and b. Hence as the coefficients in P<span class="su">0</span> and also in A, B
+are real and positive, it follows that the values obtained in succession
+for the coefficients in P<span class="su">1</span>, P<span class="su">2</span>, ... are real and positive; and further,
+taking account of the fact that the absolute value of a sum of terms
+is not greater than the sum of the absolute values of the terms, it
+follows, for each value of s, that every coefficient in p<span class="su">s+1</span> is, in absolute
+value, not greater than the corresponding coefficient in P<span class="su">s+1</span>. Thus
+if the series for F be convergent, the series for &fnof; will also be; and we
+are thus reduced to (1), specifying functions A, B with real positive
+coefficients, each in absolute value not less than the corresponding
+coefficient in a, b; (2) proving that the equation</p>
+
+<p class="center">AdF/dx + BdF/dy = dF/dt</p>
+
+<p class="noind">possesses an integral P<span class="su">0</span> + tP<span class="su">1</span> + t²P<span class="su">2</span>/2! + ... in which the coefficients
+in P<span class="su">0</span> are real and positive, and each not less than the absolute value
+of the corresponding coefficient in p<span class="su">0</span>. If a, b be developable for x, y
+both in absolute value less than r and for t less in absolute value than
+R, and for such values a, b be both less in absolute value than the
+real positive constant M, it is not difficult to verify that we may
+take A = B = M[1 &minus; (x + y)/r]<span class="sp">-1</span> (1 &minus; t/R)<span class="sp">-1</span>,
+and obtain</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">F = r &minus; (r &minus; x &minus; y) <span class="f150">[</span> 1 &minus;</td> <td>4MR</td>
+<td rowspan="2"><span class="f150">(</span>1 &minus;</td> <td>x + y</td>
+<td rowspan="2"><span class="f150">)</span></td> <td><span class="sp bk1">&minus;2</span></td>
+<td rowspan="2">log <span class="f150">(</span> 1 &minus;</td> <td>t</td>
+<td rowspan="2"><span class="f150">)</span></td> <td><span class="sp bk1">&minus;1</span></td>
+<td rowspan="2"><span class="f150">]</span></td> <td><span class="sp bk1">1/2</span></td> <td rowspan="2">,</td></tr>
+<tr><td class="denom">r</td> <td class="denom">r</td>
+<td>&nbsp;</td> <td class="denom">R</td>
+<td>&nbsp;</td> <td>&nbsp;</td></tr></table>
+
+<p class="noind">and that this solves the problem when x, y, t are sufficiently small
+for the two cases p<span class="su">0</span> = x, p<span class="su">0</span> = y. One obvious application of the
+general theorem is to the proof of the existence of an integral of
+an ordinary linear differential equation given by the n equations
+dy/dx = y<span class="su">1</span>, dy<span class="su">1</span>/dx = y<span class="su">2</span>, ...,</p>
+
+<p class="center">dy<span class="su">n-1</span>/dx = p &minus; p<span class="su">1</span>y<span class="su">n-1</span> &minus; ... &minus; p<span class="su">n</span>y;</p>
+
+<p class="noind">but in fact any simultaneous system of ordinary equations is reducible
+to a system of the form</p>
+
+<p class="center">dx<span class="su">i</span>/dt = &phi;<span class="su">i</span>(tx<span class="su">1</span>, ... x<span class="su">n</span>).</p>
+
+<p>Suppose we have k homogeneous linear partial equations of the
+first order in n independent variables, the general equation being
+a<span class="su">&sigma;1</span>d&fnof;/dx<span class="su">1</span> + ... + a<span class="su">&sigma;n</span>d&fnof;/dx<span class="su">n</span> = 0,
+where &sigma; = 1, ... k, and that
+<span class="sidenote">Simultaneous linear partial equations.</span>
+we desire to know whether the equations have common
+solutions, and if so, how many. It is to be understood
+that the equations are linearly independent, which implies
+that k &le; n and not every determinant of k rows and columns
+is identically zero in the matrix in which the i-th element of the &sigma;-th
+row is a<span class="su">&sigma;i</span>}(i = 1, ... n, &sigma; = 1, ... k). Denoting the left side of the
+&sigma;-th equation by P&sigma;&fnof;, it is clear that every common solution of the
+two equations P<span class="su">&sigma;</span>&fnof; = 0, P<span class="su">&rho;</span>&fnof; = 0,
+is also a solution of the equation
+P<span class="su">&rho;</span>(p<span class="su">&sigma;</span>&fnof;), P<span class="su">&sigma;</span>(p<span class="su">&rho;</span>&fnof;),
+We immediately find, however, that this is
+also a linear equation, namely, &Sigma;H<span class="su">i</span>d&fnof;/dx<span class="su">i</span> = 0
+where H<span class="su">i</span> = P<span class="su">&rho;</span>a<span class="su">&sigma;</span> &minus; P<span class="su">&sigma;</span>a<span class="su">&rho;</span>,
+and if it be not already contained among the given equations, or be
+linearly deducible from them, it may be added to them, as not introducing
+any additional limitation of the possibility of their having
+common solutions. Proceeding thus with every pair of the original
+equations, and then with every pair of the possibly augmented
+system so obtained, and so on continually, we shall arrive at a
+system of equations, linearly independent of each other and therefore
+not more than n in number, such that the combination, in the way
+described, of every pair of them, leads to an equation which is
+linearly deducible from them. If the number of this so-called
+<i>complete system</i> is n, the equations give d&fnof;/dx<span class="su">1</span> = 0 ... d&fnof;/dx<span class="su">n</span> = 0,
+leading to the nugatory result &fnof; = a constant. Suppose, then, the
+number of this system to be r &lt; n; suppose, further, that from the
+<span class="sidenote">Complete systems of linear partial equations.</span>
+matrix of the coefficients a determinant of r rows and
+columns not vanishing identically is that formed by the
+coefficients of the differential coefficients of &fnof; in regard
+to x<span class="su">1</span> ... x<span class="su">r</span>; also that the coefficients are all developable
+about the values x<span class="su">1</span> = xº<span class="su">1</span>, ... x<span class="su">n</span>= xº<span class="su">n</span>, and that for these
+values the determinant just spoken of is not zero.
+Then the main theorem is that the complete system of r equations,
+and therefore the originally given set of k equations,
+have in common n &minus; r solutions, say &omega;<span class="su">r+1</span>, ... &omega;<span class="su">n</span>, which reduce
+respectively to x<span class="su">r+1</span>, ... x<span class="su">n</span> when in them for x<span class="su">1</span>, ... x<span class="su">r</span> are respectively
+put xº<span class="su">1</span>, ... xº<span class="su">r</span>; so that also the equations have in common a
+solution reducing when x<span class="su">1</span> = xº<span class="su">1</span>, ... x<span class="su">r</span> = xº<span class="su">r</span> to an arbitrary function
+&psi;(x<span class="su">r+1</span>, ... x<span class="su">n</span>) which is developable about xº<span class="su">r+1</span>, ... xº<span class="su">n</span>, namely,
+this common solution is &psi;(&omega;<span class="su">r+1</span>, ... &omega;<span class="su">n</span>). It is seen at once
+that this result is a generalization of the theorem for r = 1, and its
+proof is conveniently given by induction from that case. It can be
+verified without difficulty (1) that if from the r equations of the
+complete system we form r independent linear aggregates, with
+coefficients not necessarily constants, the new system is also a complete
+system; (2) that if in place of the independent variables
+x<span class="su">1</span>, ... x<span class="su">n</span> we introduce any other variables which are independent
+functions of the former, the new equations also form a complete
+system. It is convenient, then, from the complete system of r
+equations to form r new equations by solving separately for d&fnof;/dx<span class="su">1</span>, ...,
+d&fnof;/dx<span class="su">r</span>; suppose the general equation of the new system to be</p>
+
+<p class="center">Q<span class="su">&sigma;</span>&fnof; = d&fnof;/dx<span class="su">&sigma;</span> + c<span class="su">&sigma;jr+1</span>d&fnof;/dx<span class="su">r+1</span> + ... + c<span class="su">&sigma;n</span>d&fnof;/dx<span class="su">n</span> = 0 (&sigma; = 1, ... r).</p>
+
+<p class="noind">Then it is easily obvious that the equation Q<span class="su">&rho;</span>Q<span class="su">&sigma;</span>&fnof; &minus; Q<span class="su">&sigma;</span>Q<span class="su">&rho;</span>&fnof; = 0
+contains only the differential coefficients of &fnof; in regard to x<span class="su">r+1</span> ... x<span class="su">n</span>; as
+it is at most a linear function of Q<span class="su">1</span>&fnof;, ... Q<span class="su">r</span>&fnof;, it must be identically
+zero. So reduced the system is called a Jacobian system. Of this
+system Q<span class="su">1</span>&fnof;=0 has n &minus; 1 principal solutions reducing respectively
+<span class="sidenote">Jacobian systems.</span>
+to x<span class="su">2</span>, ... x<span class="su">n</span> when</p>
+
+<p class="center">x<span class="su">1</span> = xº<span class="su">1</span>,</p>
+
+<p class="noind">and its form shows that of these the first r &minus; 1 are exactly x<span class="su">2</span> ... x<span class="su">r</span>.
+Let these n &minus; 1 functions together with x<span class="su">1</span> be introduced as n new
+independent variables in all the r equations. Since the first equation
+is satisfied by n &minus; 1 of the new independent variables, it will contain
+no differential coefficients in regard to them, and will reduce therefore
+simply to d&fnof;/dx<span class="su">1</span> = 0, expressing that any common solution of the r
+equations is a function only of the n &minus; 1 remaining variables. Thereby
+the investigation of the common solutions is reduced to the same
+problem for r &minus; 1 equations in n &minus; 1 variables. Proceeding thus, we
+reach at length one equation in n &minus; r + 1 variables, from which, by
+retracing the analysis, the proposition stated is seen to follow.</p>
+
+<p>The analogy with the case of one equation is, however, still closer.
+With the coefficients c<span class="su">&sigma;j</span>, of the equations Q<span class="su">&sigma;</span>&fnof; = 0 in transposed
+array (&sigma; = 1, ... r, j = r + 1, ... n) we can put down the
+(n &minus; r) equations, dx<span class="su">j</span> = c<span class="su">1j</span>dx<span class="su">1</span> + ... + c<span class="su">rj</span>dx<span class="su">r</span>, equivalent to
+<span class="sidenote">System of total differential equations.</span>
+the r(n &minus; r) equations dx<span class="su">j</span>/dx<span class="su">&sigma;</span> = c<span class="su">&sigma;r</span>. That consistent
+with them we may be able to regard x<span class="su">r+1</span>, ... x<span class="su">n</span> as
+functions of x<span class="su">1</span>, ... x<span class="su">r</span>, these being regarded as independent
+variables, it is clearly necessary that when we differentiate c<span class="su">&sigma;j</span> in
+regard to x<span class="su">&rho;</span> on this hypothesis the result should be the same as when
+we differentiate c<span class="su">&rho;j</span>, in regard to x<span class="su">&sigma;</span> on this hypothesis. The differential
+coefficient of a function &fnof; of x<span class="su">1</span>, ... x<span class="su">n</span> on this hypothesis, in
+regard to x<span class="su">&rho;j</span> is, however,</p>
+
+<p class="center">d&fnof;/dx<span class="su">&rho;</span> + c<span class="su">&rho;jr+1</span>d&fnof;/dx<span class="su">r+1</span> + ... + c<span class="su">&rho;n</span>d&fnof;/dx<span class="su">n</span>,</p>
+
+<p class="noind">namely, is Q<span class="su">&rho;</span>&fnof;. Thus the consistence of the n &minus; r total equations
+requires the conditions Q<span class="su">&rho;</span>c<span class="su">&sigma;j</span> &minus; Q<span class="su">&sigma;</span>c<span class="su">&rho;j</span> = 0, which are, however,
+verified in virtue of Q<span class="su">&rho;</span>(Q<span class="su">&sigma;</span>&fnof;) &minus; Q<span class="su">&sigma;</span>(Q<span class="su">&rho;</span>&fnof;) = 0. And it can in fact be
+easily verified that if &omega;<span class="su">r+1</span>, ... &omega;<span class="su">n</span> be the principal solutions of the
+Jacobian system, Q<span class="su">&sigma;</span>&fnof; = 0, reducing respectively to x<span class="su">r+1</span>, ... x<span class="su">n</span> when
+x<span class="su">1</span> = xº<span class="su">1</span>, ... x<span class="su">r</span> = xº<span class="su">r</span>, and the equations &omega;<span class="su">r+1</span> = x<span class="sp">0</span><span class="su">r+1</span>, ... &omega;<span class="su">n</span> = xº<span class="su">n</span>
+be solved for x<span class="su">r+1</span>, ... x<span class="su">n</span> to give x<span class="su">j</span> = &psi;<span class="su">j</span>(x<span class="su">1</span>, ... x<span class="su">r</span>, x<span class="sp">0</span><span class="su">r+1</span>, ... xº<span class="su">n</span>), these
+values solve the total equations and reduce respectively to x<span class="sp">0</span><span class="su">r+1</span>, ... xº<span class="su">n</span>
+when x<span class="su">1</span> = xº<span class="su">1</span> ... x<span class="su">r</span> = xº<span class="su">r</span>. And the total equations have no
+other solutions with these initial values. Conversely, the existence
+of these solutions of the total equations can be deduced a priori
+and the theory of the Jacobian system based upon them. The
+theory of such total equations, in general, finds its natural place
+under the heading <i>Pfaffian Expressions</i>, below.</p>
+
+<p>A practical method of reducing the solution of the r equations
+of a Jacobian system to that of a single equation in n &minus; r + 1 variables
+may be explained in connexion with a geometrical interpretation
+which will perhaps be clearer in a particular
+<span class="sidenote">Geometrical interpretation and solution.</span>
+case, say n = 3, r = 2. There is then only one total
+equation, say dz = adz + bdy; if we do not take account
+of the condition of integrability, which is in this case
+da/dy + bda/dz = db/dx + adb/dz, this equation may be regarded
+as defining through an arbitrary point (x<span class="su">0</span>, y<span class="su">0</span>, z<span class="su">0</span>) of three-dimensioned
+space (about which a, b are developable) a plane, namely,
+z &minus; z<span class="su">0</span> = a<span class="su">0</span>(x &minus; x<span class="su">0</span>) + b<span class="su">0</span>(y &minus; y<span class="su">0</span>), and therefore, through this arbitrary
+point &infin;<span class="sp">2</span> directions, namely, all those in the plane. If now there be
+a surface z = &psi;(x, y), satisfying dz = adz + bdy and passing through
+(x<span class="su">0</span>, y<span class="su">0</span>, z<span class="su">0</span>), this plane will touch the surface, and the operations of
+passing along the surface from (x<span class="su">0</span>, y<span class="su">0</span>, z<span class="su">0</span>) to</p>
+
+<p class="center">(x<span class="su">0</span> + dx<span class="su">0</span>, y<span class="su">0</span>, z<span class="su">0</span> + dz<span class="su">0</span>)</p>
+
+<p class="noind">and then to (x<span class="su">0</span> + dx<span class="su">0</span>, y<span class="su">0</span> + dy<span class="su">0</span>, Z<span class="su">0</span> + d<span class="sp">1</span>z<span class="su">0</span>), ought to lead to the same
+value of d<span class="sp">1</span>z<span class="su">0</span> as do the operations of passing along the surface from
+(x<span class="su">0</span>, y<span class="su">0</span>, z<span class="su">0</span>) to (x<span class="su">0</span>, y<span class="su">0</span> + dy<span class="su">0</span>, z<span class="su">0</span> + &delta;z<span class="su">0</span>), and then to</p>
+
+<p class="center">(x<span class="su">0</span> + dx<span class="su">0</span>, y<span class="su">0</span> + dy<span class="su">0</span>, z<span class="su">0</span> + &delta;<span class="sp">1</span>z<span class="su">0</span>),</p>
+
+<p class="noind">namely, &delta;<span class="sp">1</span>z<span class="su">0</span> ought to be equal to d<span class="sp">1</span>z<span class="su">0</span>. But we find</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">a<span class="su">0</span>dx<span class="su">0</span> + b<span class="su">0</span>dy<span class="su">0</span> + dx<span class="su">0</span>dy<span class="su">0</span><span class="f150">(</span></td> <td>db</td>
+<td rowspan="2">+ a<span class="su">0</span></td> <td>db</td>
+<td rowspan="2"><span class="f150">)</span>,</td></tr>
+<tr><td class="denom">dx<span class="su">0</span></td> <td class="denom">dz<span class="su">0</span></td></tr></table>
+
+<p class="noind">and so at once reach the condition of integrability. If now we put
+<span class="pagenum"><a name="page231" id="page231"></a>231</span>
+x = x<span class="su">0</span> + t, y = y<span class="su">0</span> + mt, and regard m as constant, we shall in fact be
+considering the section of the surface by a fixed plane y &minus; y<span class="su">0</span> = m(x &minus; x<span class="su">0</span>);
+along this section dz = dt(a + bm); if we then integrate the equation
+dx/dt = a + bm, where a, b are expressed as functions of m and t, with
+m kept constant, finding the solution which reduces to z<span class="su">0</span> for t = 0,
+and in the result again replace m by (y &minus; y<span class="su">0</span>)/(x &minus; x<span class="su">0</span>), we shall have the
+surface in question. In the general case the equations</p>
+
+<p class="center">dx<span class="su">j</span> = c<span class="su">ij</span>dx<span class="su">1</span> + ... c<span class="su">rj</span>dx<span class="su">r</span></p>
+
+<p class="noind">similarly determine through an arbitrary point xº<span class="su">1</span>, ... xº<span class="su">n</span>
+<span class="sidenote">Mayer&rsquo;s method of integration.</span>
+a planar manifold of r dimensions in space of n dimensions,
+and when the conditions of integrability are satisfied,
+every direction in this manifold through this point is
+tangent to the manifold of r dimensions, expressed by
+&omega;<span class="su">r+1</span> = x<span class="sp">0</span><span class="su">r+1</span>, ... &omega;_  =  xº<span class="su">n</span>,
+which satisfies the equations and passes through this
+point. If we put
+x<span class="su">1</span> &minus; xº<span class="su">1</span> = t, x<span class="su">2</span> &minus; xº<span class="su">2</span> = m<span class="su">2</span>t, ...  x<span class="su">r</span> &minus; xº<span class="su">r</span> = m<span class="su">r</span>t,
+and regard m<span class="su">2</span>, ... m<span class="su">r</span> as fixed, the (n &minus; r) total equations take the form
+dx<span class="su">j</span>/dt = c<span class="su">1j</span> + m<span class="su">2</span>c<span class="su">2j</span> + ... + m<span class="su">r</span>c<span class="su">rj</span>,
+and their integration is equivalent to that of the single partial equation</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">d&fnof;/dt + <span class="f150">&Sigma;</span></td> <td class="bk">n</td>
+<td rowspan="2">(c<span class="su">1j</span> + m<span class="su">2</span>c<span class="su">2j</span> + ... + m<span class="su">r</span>c<span class="su">rj</span>) d&fnof;/dx<span class="su">j</span> = 0</td></tr>
+<tr><td class="bk">j=r+1</td></tr></table>
+
+<p class="noind">in the n &minus; r + 1 variables t, x<span class="su">r+1</span>, ... x<span class="su">n</span>. Determining the solutions
+&Omega;<span class="su">r+1</span>, ... &Omega;<span class="su">n</span> which reduce to respectively x<span class="su">r+1</span>, ... x<span class="su">n</span> when t = 0, and substituting
+t = x<span class="su">1</span> &minus; xº<span class="su">1</span>, m<span class="su">2</span> = (x<span class="su">2</span> &minus; xº<span class="su">2</span>)/(x<span class="su">1</span> &minus; xº<span class="su">1</span>), ... m<span class="su">r</span> = (x<span class="su">r</span> &minus; xº<span class="su">r</span>)/(x<span class="su">1</span> &minus; xº<span class="su">1</span>),
+we obtain the solutions of the original system of partial equations
+previously denoted by &omega;<span class="su">r+1</span>, ... &omega;<span class="su">n</span>. It is to be remarked,
+however, that the presence of the fixed parameters m<span class="su">2</span>, ... m<span class="su">r</span> in
+the single integration may frequently render it more difficult than if
+they were assigned numerical quantities.</p>
+
+<p>We have above considered the integration of an equation</p>
+
+<p class="center">dz = adz + bdy</p>
+
+<p class="noind">on the hypothesis that the condition</p>
+
+<p class="center">da/dy + bda/dz = db/dz + adb/dz.</p>
+
+<p>It is natural to inquire what relations among x, y, z, if any,
+<span class="sidenote">Pfaffian Expressions.</span>
+are implied by, or are consistent with, a differential relation
+adx + bdy + cdx = 0, when a, b, c are unrestricted functions
+of x, y, z. This problem leads to the consideration of the
+so-called <i>Pfaffian Expression</i> adx + bdy + cdz. It can be shown (1) if
+each of the quantities db/dz &minus; dc/dy, dc/dx &minus; da/dz, da/dy &minus; db/dz, which
+we shall denote respectively by u<span class="su">23</span>, u<span class="su">31</span>, u<span class="su">12</span>, be identically zero, the
+expression is the differential of a function of x, y, z, equal to dt say;
+(2) that if the quantity au<span class="su">23</span> + bu<span class="su">31</span> + cu<span class="su">12</span> is identically zero, the expression
+is of the form udt, <i>i.e.</i> it can be made a perfect differential
+by multiplication by the factor 1/u; (3) that in general the expression
+is of the form dt + u<span class="su">1</span>dt<span class="su">1</span>. Consider the matrix of four
+rows and three columns, in which the elements of the first row are
+a, b, c, and the elements of the (r + 1)-th row, for r = 1, 2, 3, are the
+quantities u<span class="su">r1</span>, u<span class="su">r2</span>, u<span class="su">r3</span>, where u<span class="su">11</span> = u<span class="su">22</span> = u<span class="su">33</span> = 0. Then it is easily
+seen that the cases (1), (2), (3) above correspond respectively to the
+cases when (1) every determinant of this matrix of two rows and
+columns is zero, (2) every determinant of three rows and columns
+is zero, (3) when no condition is assumed. This result can be generalized
+as follows: if a<span class="su">1</span>, ... a<span class="su">n</span> be any functions of x<span class="su">1</span>, ... x<span class="su">n</span>, the so-called
+Pfaffian expression a<span class="su">1</span>dx<span class="su">1</span> + ... + a<span class="su">n</span>dx<span class="su">n</span> can be reduced to one
+or other of the two forms</p>
+
+<p class="center">u<span class="su">1</span>dt<span class="su">1</span> + ... + u<span class="su">k</span>dt<span class="su">k</span>, dt + u<span class="su">1</span>dt<span class="su">1</span> +  ... + u<span class="su">k-1</span>dt<span class="su">k-1</span>,</p>
+
+<p class="noind">wherein t, u<span class="su">1</span> ..., t<span class="su">1</span>, ... are independent functions of x<span class="su">1</span>, ... x<span class="su">n</span>, and k
+is such that in these two cases respectively 2k or 2k &minus; 1 is the rank of
+a certain matrix of n + 1 rows and n columns, that is, the greatest
+number of rows and columns in a non-vanishing determinant of the
+matrix; the matrix is that whose first row is constituted by the
+quantities a<span class="su">1</span>, ... a<span class="su">n</span>, whose s-th element in the (r + 1)-th row is the
+quantity da<span class="su">r</span>/dx<span class="su">s</span> &minus; da<span class="su">s</span>/dx<span class="su">r</span>. The proof of such a reduced form can
+be obtained from the two results: (1) If t be any given function
+of the 2m independent variables u<span class="su">1</span>, ... u<span class="su">m</span>, t<span class="su">1</span>, ... t<span class="su">m</span>, the expression
+dt + u<span class="su">1</span>dt<span class="su">1</span> +  ... + u<span class="su">m</span>dt<span class="su">m</span> can be put into the form u&prime;<span class="su">1</span>dt&prime;<span class="su">1</span> + ...  + u&prime;<span class="su">m</span>dt&prime;<span class="su">m</span>.
+(2) If the quantities u<span class="su">1</span>, ..., u<span class="su">1</span>, t<span class="su">1</span>, ... t<span class="su">m</span> be connected by a relation,
+the expression n<span class="su">1</span>dt<span class="su">1</span> +  ... + u<span class="su">m</span>dt<span class="su">m</span> can be put into the format dt&prime; + u&prime;<span class="su">1</span>dt&prime;<span class="su">1</span>
+ + ... + u&prime;<span class="su">m-1</span>dt&prime;<span class="su">m-1</span>; and if the relation connecting u<span class="su">1</span>,  u<span class="su">m</span>, t<span class="su">1</span>, ... t<span class="su">m</span>
+be homogeneous in u<span class="su">1</span>, ... u<span class="su">m</span>, then t&prime; can be taken to be zero. These
+two results are deductions from the theory of <i>contact transformations</i>
+(see below), and their demonstration requires, beside elementary
+algebraical considerations, only the theory of complete systems of
+linear homogeneous partial differential equations of the first order.
+When the existence of the reduced form of the Pfaffian expression
+containing only independent quantities is thus once assured, the
+identification of the number k with that defined by the specified
+matrix may, with some difficulty, be made <i>a posteriori</i>.</p>
+
+<p>In all cases of a single Pfaffian equation we are thus led to consider
+what is implied by a relation dt &minus; u<span class="su">1</span>dt<span class="su">1</span> &minus; ... &minus; u<span class="su">m</span>dt<span class="su">m</span> = 0, in which
+t, u<span class="su">1</span>, ... u<span class="su">m</span>, t<span class="su">1</span> ..., t<span class="su">m</span> are, except for this equation,
+independent variables. This is to be satisfied in virtue of
+<span class="sidenote">Single linear Pfaffian equation.</span>
+one or several relations connecting the variables; these
+must involve relations connecting t, t<span class="su">1</span>, ... t<span class="su">m</span> only, and
+in one of these at least t must actually enter. We can
+then suppose that in one actual system of relations in virtue of which
+the Pfaffian equation is satisfied, all the relations connecting t, t<span class="su">1</span> ...
+t<span class="su">m</span> only are given by</p>
+
+<p class="center">t = &psi;(t<span class="su">s+1</span> ... t<span class="su">m</span>), t<span class="su">1</span> = &psi;<span class="su">1</span>(t<span class="su">s+1</span> ... t<span class="su">m</span>), ... t<span class="su">s</span> = &psi;<span class="su">s</span>(t<span class="su">s+1</span> ... t<span class="su">m</span>);</p>
+
+<p class="noind">so that the equation</p>
+
+<p class="center">d&psi; &minus; u<span class="su">1</span>d&psi;<span class="su">1</span> &minus; ... &minus; u<span class="su">s</span>d&psi;<span class="su">s</span> &minus; u<span class="su">s+1</span>dt<span class="su">s+1</span> &minus; ... &minus; u<span class="su">m</span>dt<span class="su">m</span> = 0</p>
+
+<p class="noind">is identically true in regard to u<span class="su">1</span>, ... u<span class="su">m</span>, t<span class="su">s+1</span> ..., t<span class="su">m</span>; equating to
+zero the coefficients of the differentials of these variables, we thus
+obtain m &minus; s relations of the form</p>
+
+<p class="center">d&psi;/dt<span class="su">j</span> &minus; u<span class="su">1</span>d&psi;<span class="su">1</span>/dt<span class="su">j</span> &minus; ... &minus; u<span class="su">s</span>d&psi;<span class="su">s</span>/dt<span class="su">j</span> &minus; u<span class="su">j</span> = 0;</p>
+
+<p class="noind">these m &minus; s relations, with the previous s + 1 relations, constitute a set
+of m + 1 relations connecting the 2m + 1 variables in virtue of which
+the Pfaffian equation is satisfied independently of the form of the
+functions &psi;,&psi;<span class="su">1</span>, ... &psi;<span class="su">s</span>. There is clearly such a set for each of the
+values s = 0, s = 1, ..., s = m &minus; 1, s = m. And for any value of s there
+may exist relations additional to the specified m + 1 relations, provided
+they do not involve any relation connecting t, t<span class="su">1</span>, ... t<span class="su">m</span> only,
+and are consistent with the m &minus; s relations connecting u<span class="su">1</span>, ... u<span class="su">m</span>. It
+is now evident that, essentially, the integration of a Pfaffian equation</p>
+
+<p class="center">a<span class="su">1</span>dx<span class="su">1</span> + ... + a<span class="su">n</span>dx<span class="su">n</span> = 0,</p>
+
+<p class="noind">wherein a<span class="su">1</span>, ... a<span class="su">n</span> are functions of x<span class="su">1</span>, ... x<span class="su">n</span>, is effected by the
+processes necessary to bring it to its reduced form, involving only
+independent variables. And it is easy to see that if we suppose this
+reduction to be carried out in all possible ways, there is no need to
+distinguish the classes of integrals corresponding to the various
+values of s; for it can be verified without difficulty that by putting
+t&prime; = t &minus; u<span class="su">1</span>t<span class="su">1</span> &minus; ... &minus; u<span class="su">s</span>t<span class="su">s</span>, t&prime;<span class="su">1</span> = u<span class="su">1</span>, ... t&prime;<span class="su">s</span> = u<span class="su">s</span>, u&prime;<span class="su">1</span> = &minus;t<span class="su">1</span>, ..., u&prime;<span class="su">s</span> = &minus;t<span class="su">s</span>,
+t&prime;<span class="su">s+1</span> = t<span class="su">s+1</span>, ... t&prime;<span class="su">m</span> = t<span class="su">m</span>, u&prime;<span class="su">s+1</span> = u<span class="su">s+1</span>, ... u&prime;<span class="su">m</span> = u<span class="su">m</span>,
+the reduced equation
+becomes changed to dt&prime; &minus; u&prime;<span class="su">1</span>dt&prime;<span class="su">1</span> &minus;  ... &minus; u&prime;<span class="su">m</span>dt&prime;<span class="su">m</span> = 0, and the general
+relations changed to</p>
+
+<p class="center">t&prime; = &psi;(t&prime;<span class="su">s+1</span>, ... t&prime;<span class="su">m</span>) &minus; t&prime;<span class="su">1</span>&psi;<span class="su">1</span>(t&prime;<span class="su">s+1</span>, ... t&prime;<span class="su">m</span>) &minus;  ... &minus; t&prime;<span class="su">s</span>&psi;<span class="su">s</span>(t&prime;<span class="su">s+1</span>, ... t&prime;<span class="su">m</span>), = &phi;,</p>
+
+<p class="noind">say, together with u&prime;<span class="su">1</span> = d&phi;/dt&prime;<span class="su">1</span>, ..., u&prime;<span class="su">m</span> = d&phi;/dt&prime;<span class="su">m</span>, which contain only
+one relation connecting the variables t&prime;, t&prime;<span class="su">1</span>, ... t&prime;<span class="su">m</span> only.</p>
+
+<p>This method for a single Pfaffian equation can, strictly speaking,
+be generalized to a simultaneous system of (n &minus; r) Pfaffian equations
+dx<span class="su">j</span> = c<span class="su">1j</span>dx<span class="su">1</span> +  ...  + c<span class="su">rj</span>dx<span class="su">r</span> only in the case already treated,
+<span class="sidenote">Simultaneous Pfaffian equations.</span>
+when this system is satisfied by regarding x<span class="su">r+1</span>, ... x<span class="su">n</span> as
+suitable functions of the independent variables x<span class="su">1</span>, ... x<span class="su">r</span>;
+in that case the integral manifolds are of r dimensions.
+When these are non-existent, there may be integral manifolds
+of higher dimensions; for if</p>
+
+<p class="center">d&phi; = &phi;<span class="su">1</span>dx<span class="su">1</span> +  ... + &phi;<span class="su">r</span>dx<span class="su">r</span> + &phi;<span class="su">r+1</span>(c<span class="su">1,r+1</span>dx<span class="su">1</span> +  ...  + c<span class="su">r,r+1</span>dx<span class="su">r</span>) + &phi;<span class="su">r+2</span>(&emsp;) +  ...</p>
+
+<p class="noind">be identically zero, then &phi;&sigma; + c&sigma;,<span class="su">r+1</span>&phi;<span class="su">r+1</span> +  ... + c&sigma;,<span class="su">n</span>&phi;<span class="su">n</span> &asymp; 0, or &phi; satisfies
+the r partial differential equations previously associated with the
+total equations; when these are not a complete system, but included
+in a complete system of r &minus; &mu; equations, having therefore
+n &minus; r &minus; &mu; independent integrals, the total equations are satisfied over
+a manifold of r + &mu; dimensions (see E. v. Weber, <i>Math. Annal.</i> 1v.
+(1901), p. 386).</p>
+
+<p>It seems desirable to add here certain results, largely of algebraic
+character, which naturally arise in connexion with the theory of
+contact transformations. For any two functions of the 2n
+<span class="sidenote">Contact transformations.</span>
+independent variables x<span class="su">1</span>, ... x<span class="su">n</span>, p<span class="su">1</span>, ... p<span class="su">n</span> we denote by (&phi;&psi;)
+the sum of the n terms such as d&phi;d&psi;/dp<span class="su">i</span>dx<span class="su">i</span> &minus; d&psi;d&phi;/dp<span class="su">i</span>dx<span class="su">i</span> For two
+functions of the (2n + 1) independent variables z, x<span class="su">1</span>, ... x<span class="su">n</span>, p<span class="su">1</span>, ... p<span class="su">n</span>
+we denote by &phi;&psi; the sum of the n terms such as</p>
+
+<table class="math0" summary="math">
+<tr><td>d&phi;</td> <td rowspan="2"><span class="f150">(</span></td> <td>d&psi;</td>
+<td rowspan="2">+ p<span class="su">i</span></td> <td>d&psi;</td>
+<td rowspan="2"><span class="f150">)</span> &minus;</td> <td>d&psi;</td>
+<td rowspan="2"><span class="f150">(</span></td> <td>d&phi;</td>
+<td rowspan="2">p<span class="su">i</span></td> <td>d&phi;</td> <td rowspan="2"><span class="f150">)</span>.</td></tr>
+<tr><td class="denom">dp<span class="su">i</span></td> <td class="denom">dx<span class="su">i</span></td>
+<td class="denom">dz</td> <td class="denom">dp<span class="su">i</span></td>
+<td class="denom">dx<span class="su">i</span></td> <td class="denom">dz</td></tr></table>
+
+<p class="noind">It can at once be verified that for any three functions [&fnof;[&phi;&psi;]] + [&phi;[&psi;&fnof;]]
++ [psi[&fnof;&phi;]] = d&fnof;/dz [&phi;&psi;] + d&phi;/dz [&psi;&fnof;] + d&psi;/dz [&fnof;&phi;],
+which when &fnof;, &phi;,&psi; do not contain z
+becomes the identity (&fnof;(&phi;&psi;)) + (phi(&psi;&fnof;)) + (&psi;(&fnof;&phi;)) = 0.Then, if X<span class="su">1</span>, ... X<span class="su">n</span>,
+P<span class="su">1</span>, ... P<span class="su">n</span> be such functions Of x<span class="su">1</span>, ... x<span class="su">n</span>, p<span class="su">1</span> ... p<span class="su">n</span> that P<span class="su">1</span>dX<span class="su">1</span>
++ ...  + P<span class="su">n</span>dX<span class="su">n</span> is identically equal to p<span class="su">1</span>dx<span class="su">1</span> +  ...  + p<span class="su">n</span>dx<span class="su">n</span>, it can be
+shown by elementary algebra, after equating coefficients of independent
+differentials, (1) that the functions X<span class="su">1</span>, ... P<span class="su">n</span> are independent
+functions of the 2n variables x<span class="su">1</span>, ... p<span class="su">n</span>, so that the equations
+x&prime;<span class="su">i</span> = X<span class="su">i</span>, p&prime;<span class="su">i</span> = P<span class="su">i</span> can be solved for x<span class="su">1</span>, ... x<span class="su">n</span>, p<span class="su">1</span>, ... p<span class="su">n</span>, and represent
+therefore a transformation, which we call a homogeneous contact
+transformation; (2) that the X<span class="su">1</span>, ... X<span class="su">n</span> are homogeneous functions of
+p<span class="su">1</span>, ... p<span class="su">n</span> of zero dimensions, the P<span class="su">1</span>, ... P<span class="su">n</span> are homogeneous functions
+of p<span class="su">1</span>, ... p<span class="su">n</span> of dimension one, and the ½n(n &minus; 1) relations (X<span class="su">i</span>X<span class="su">j</span>) = 0
+are verified. So also are the n² relations (P<span class="su">i</span>X<span class="su">i</span> = 1, (P<span class="su">i</span>X<span class="su">j</span>) = 0,
+(P<span class="su">i</span>P<span class="su">j</span>) = 0. Conversely, if X<span class="su">1</span>, ... X<span class="su">n</span> be independent functions, each
+homogeneous of zero dimension in p<span class="su">1</span>, ... p<span class="su">n</span> satisfying the ½n(n &minus; 1)
+relations (X<span class="su">i</span>X<span class="su">j</span>) = 0, then P<span class="su">1</span>, ... P<span class="su">n</span> can be uniquely determined, by
+solving linear algebraic equations, such that P<span class="su">1</span>dX<span class="su">1</span> +  ...  + P<span class="su">n</span>dX<span class="su">n</span>
+= p<span class="su">1</span>dx<span class="su">1</span> +  ...  + p<span class="su">n</span>dx<span class="su">n</span>. If now we put n + 1 for n, put z for x<span class="su">n+1</span>,
+Z for X<span class="su">n+1</span>, Q<span class="su">i</span> for -P<span class="su">i</span>/P<span class="su">n+1</span>, for i = 1, ... n, put q<span class="su">i</span> for -p<span class="su">i</span>/p<span class="su">n+1</span> and &sigma;
+for q<span class="su">n+1</span>/Q<span class="su">n+1</span>, and then finally write P<span class="su">1</span>, ... P<span class="su">n</span>, p<span class="su">1</span>, ... p<span class="su">n</span> for Q<span class="su">1</span>, ... Q<span class="su">n</span>,
+q<span class="su">1</span>, ... q<span class="su">n</span>, we obtain the following results: If ZX<span class="su">1</span> ... X<span class="su">n</span>, P<span class="su">1</span>, ... P<span class="su">n</span>
+be functions of z, x<span class="su">1</span>, ... x<span class="su">n</span>, p<span class="su">1</span>, ... p<span class="su">n</span>, such that the expression
+dZ &minus; P<span class="su">1</span>dX<span class="su">1</span> &minus; ... &minus; P<span class="su">n</span>dX<span class="su">n</span> is identically equal to &sigma;(dz &minus; p<span class="su">1</span>dx<span class="su">1</span> &minus; ... &minus; p<span class="su">n</span>dx<span class="su">n</span>),
+and &sigma; not zero, then (1) the functions Z, X<span class="su">1</span>, ... X<span class="su">n</span>, P<span class="su">1</span>, ... P<span class="su">n</span>
+are independent functions of z, x<span class="su">1</span>, ... x<span class="su">n</span>, p<span class="su">1</span>, ... p<span class="su">n</span>, so that the
+equations z&prime; = Z, x&prime;<span class="su">i</span> = X<span class="su">i</span>, p&prime;<span class="su">i</span> = P<span class="su">i</span> can be solved for z, x<span class="su">1</span>, ... x<span class="su">n</span>, p<span class="su">1</span>, ... p<span class="su">n</span>
+and determine a transformation which we call a (non-homogeneous)
+contact transformation; (2) the Z, X<span class="su">1</span>, ... X<span class="su">n</span> verify the ½n(n + 1)
+<span class="pagenum"><a name="page232" id="page232"></a>232</span>
+identities [ZX<span class="su">i</span>] = 0, [X<span class="su">i</span>X<span class="su">j</span>] = 0. And the further identities</p>
+
+<p class="center">[P<span class="su">i</span>X<span class="su">i</span>] = &sigma;, [P<span class="su">i</span>X<span class="su">j</span>] = 0, [P<span class="su">i</span>Z] = &sigma;P<span class="su">i</span>, [P<span class="su">i</span>P<span class="su">j</span>] = 0,</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">[Z&sigma;] = &sigma;</td> <td>dZ</td>
+<td rowspan="2">&minus; &sigma;², [X<span class="su">i</span>&sigma;] = &sigma;</td> <td>dX<span class="su">i</span></td>
+<td rowspan="2">, [P<span class="su">i</span>&sigma;] =</td> <td>dP<span class="su">i</span></td></tr>
+<tr><td class="denom">dz</td> <td class="denom">dz</td> <td class="denom">dz</td></tr></table>
+
+<p class="noind">are also verified. Conversely, if Z, x<span class="su">1</span>, ... X<span class="su">n</span> be independent functions
+satisfying the identities [ZX<span class="su">i</span>] = 0, [X<span class="su">i</span>X<span class="su">j</span>] = 0, then &sigma;, other than
+zero, and P<span class="su">1</span>, ... P<span class="su">n</span> can be uniquely determined, by solution of
+algebraic equations, such that</p>
+
+<p class="center">dZ &minus; P<span class="su">1</span>dX<span class="su">1</span> &minus; ... &minus; P<span class="su">n</span>dX<span class="su">n</span> = &sigma;(dz &minus; p<span class="su">1</span>dx<span class="su">1</span> &minus; ... &minus; p<span class="su">n</span>dx<span class="su">n</span>).</p>
+
+<p>Finally, there is a particular case of great importance arising when
+&sigma; = 1, which gives the results: (1) If U, X<span class="su">1</span>, ... X<span class="su">n</span>, P<span class="su">1</span>, ... P<span class="su">n</span> be
+2n + 1 functions of the 2n independent variables x<span class="su">1</span>, ... x<span class="su">n</span>, p<span class="su">1</span>,
+... p<span class="su">n</span>, satisfying the identity</p>
+
+<p class="center">dU + P<span class="su">1</span>dx<span class="su">1</span> + ... + P<span class="su">n</span>dX<span class="su">n</span> = p<span class="su">1</span>dx<span class="su">1</span> + ... + p<span class="su">n</span>dx<span class="su">n</span>,</p>
+
+<p class="noind">then the 2n functions P<span class="su">1</span>, ... P<span class="su">n</span>, X<span class="su">1</span>, ... X<span class="su">n</span> are independent,
+and we have</p>
+
+<p class="center">(X<span class="su">i</span>X<span class="su">j</span>) = 0, (X<span class="su">i</span>U) = &delta;X<span class="su">i</span>, (P<span class="su">i</span>X<span class="su">i</span>) = 1,  (P<span class="su">i</span>X<span class="su">j</span>) = 0, (P<span class="su">i</span>P<span class="su">j</span>) = 0, (P<span class="su">i</span>U) + P<span class="su">i</span> = &delta;P<span class="su">i</span>,</p>
+
+<p class="noind">where &delta; denotes the operator p<span class="su">1</span>d/dp<span class="su">1</span> + ... + p<span class="su">n</span>d/dp<span class="su">n</span>; (2) If
+X<span class="su">1</span>, ... X<span class="su">n</span> be independent functions of x<span class="su">1</span>, ... x<span class="su">n</span>, p<span class="su">1</span>, ... p<span class="su">n</span>,
+such that (X<span class="su">i</span>X<span class="su">j</span>) = 0, then U can be found by a quadrature, such that</p>
+
+<p class="center">(X<span class="su">i</span>U) = &delta;X<span class="su">i</span>;</p>
+
+<p class="noind">and when X<span class="su">i</span>, ... X<span class="su">n</span>, U satisfy these ½n(n + 1) conditions, then
+P<span class="su">1</span>, ... P<span class="su">n</span> can be found, by solution of linear algebraic equations, to
+render true the identity dU + P<span class="su">1</span>dX<span class="su">1</span> + ... + P<span class="su">n</span>dX<span class="su">n</span> = p<span class="su">1</span>dx<span class="su">1</span> + ... + p<span class="su">n</span>dx<span class="su">n</span>;
+(3) Functions X<span class="su">1</span>, ... X<span class="su">n</span>, P<span class="su">1</span>, ... P<span class="su">n</span> can be found to satisfy
+this differential identity when U is an arbitrary given function of
+x<span class="su">1</span>, ... x<span class="su">n</span>, p<span class="su">1</span>, ... p<span class="su">n</span>; but this requires integrations. In order
+to see what integrations, it is only necessary to verify the statement
+that if U be an arbitrary given function of x<span class="su">1</span>, ... x<span class="su">n</span>, p<span class="su">1</span>, ... p<span class="su">n</span>,
+and, for r &lt; n, X<span class="su">1</span>, ... X<span class="su">r</span> be independent functions of these variables,
+such that (X&sigma;U) = &delta;X&sigma;, (X&rho;X&sigma;) = 0, for &rho;, &sigma; = 1 ... r, then
+the r + 1 homogeneous linear partial differential equations of the
+first order (U&fnof;) + &delta;&fnof; = 0, (X&rho;&fnof;) = 0, form a complete system. It will
+be seen that the assumptions above made for the reduction of
+Pfaffian expressions follow from the results here enunciated for
+contact transformations.</p>
+
+<p>We pass on now to consider the solution of any partial
+differential equation of the first order; we attempt to explain
+certain ideas relatively to a single equation with any
+number of independent variables (in particular, an
+<span class="sidenote">Partial differential equation of the first order.</span>
+ordinary equation of the first order with one independent
+variable) by speaking of a single equation with
+two independent variables x, y, and one dependent
+variable z. It will be seen that we are naturally led to
+consider systems of such simultaneous equations, which we
+consider below. The central discovery of the transformation
+theory of the solution of an equation F(x, y, z, dz/dx, dz/dy) = 0
+is that its solution can always be reduced to the solution of
+partial equations which are <i>linear</i>. For this, however, we must
+regard dz/dx, dz/dy, during the process of integration, not as the
+differential coefficients of a function z in regard to x and y, but as
+variables independent of x, y, z, the too great indefiniteness that
+might thus appear to be introduced being provided for in another
+way. We notice that if z = &psi;(x, y) be a solution of the differential
+equation, then dz = dxd&psi;/dx + dyd&psi;/dy; thus if we denote
+the equation by F(x, y, z, p, q,) = 0, and prescribe the condition
+dz = pdx + qdy for every solution, any solution such as z = &psi;(x, y)
+will necessarily be associated with the equations p = dz/dx,
+q = dz/dy, and z will satisfy the equation in its original form. We
+have previously seen (under <i>Pfaffian Expressions</i>) that if five
+variables x, y, z, p, q, otherwise independent, be subject to
+dz &minus; pdx &minus; qdy = 0, they must in fact be subject to at least three
+mutual relations. If we associate with a point (x, y, z) the plane</p>
+
+<p class="center">Z &minus; z = p(X &minus; x) + q(Y &minus; y)</p>
+
+<p class="noind">passing through it, where X, Y, Z are current co-ordinates, and
+call this association a surface-element; and if two consecutive
+elements of which the point(x + dx, y + dy, z + dz) of one lies on the
+plane of the other, for which, that is, the condition dz = pdx + qdy
+is satisfied, be said to be <i>connected,</i> and an infinity of connected
+elements following one another continuously be called a <i>connectivity</i>,
+then our statement is that a connectivity consists of not
+more than &infin;² elements, the whole number of elements (x, y, z, p, q)
+that are possible being called &infin;<span class="sp">5</span>. The solution of an equation
+F(x, y, z, dz/dx, dz/dy) = 0 is then to be understood to mean finding
+in all possible ways, from the &infin;<span class="sp">4</span> elements (x, y, z, p, q) which
+satisfy F(x, y, z, p, q) = 0 a set of &infin;² elements forming a connectivity;
+or, more analytically, finding in all possible ways two
+relations G = 0, H = 0 connecting x, y, z, p, q and independent of
+F = 0, so that the three relations together may involve</p>
+
+<p class="center">dz = pdx + qdy.</p>
+
+<p class="noind">Such a set of three relations may, for example, be of the form
+z = &psi;(x, y), p = d&psi;/dx, q = d&psi;/dy; but it may also, as another
+case, involve two relations z = &psi;(y), x = &psi;<span class="su">1</span>(y) connecting x, y, z,
+the third relation being</p>
+
+<p class="center">&psi;&prime;(y) = p&psi;&prime;<span class="su">1</span>(y) + q,</p>
+
+<p class="noind">the connectivity consisting in that case, geometrically, of a curve
+in space taken with &infin;¹ of its tangent planes; or, finally, a
+connectivity is constituted by a fixed point and all the planes
+passing through that point. This generalized view of the meaning
+of a solution of F = 0 is of advantage, moreover, in view of
+anomalies otherwise arising from special forms of the equation
+<span class="sidenote">Meaning of a solution of the equation.</span>
+itself. For instance, we may include the case, sometimes
+arising when the equation to be solved is obtained
+by transformation from another equation, in which F
+does not contain either p or q. Then the equation has
+&infin;² solutions, each consisting of an arbitrary point of the surface
+F = 0 and all the &infin;² planes passing through this point; it also
+has &infin;² solutions, each consisting of a curve drawn on the surface
+F = 0 and all the tangent planes of this curve, the whole consisting
+of &infin;² elements; finally, it has also an isolated (or singular)
+solution consisting of the points of the surface, each associated
+with the tangent plane of the surface thereat, also &infin;² elements in
+all. Or again, a linear equation F = Pp + Qq &minus; R = 0, wherein
+P, Q, R are functions of x, y, z only, has &infin;² solutions, each
+consisting of one of the curves defined by</p>
+
+<p class="center">dx/P = dy/Q = dz/R</p>
+
+<p class="noind">taken with all the tangent planes of this curve; and the same
+equation has &infin;² solutions, each consisting of the points of a
+surface containing &infin;¹ of these curves and the tangent planes of
+this surface. And for the case of n variables there is similarly
+the possibility of n + 1 kinds of solution of an equation
+F(x<span class="su">1</span>, ... x<span class="su">n</span>, z, p<span class="su">1</span>, ... p<span class="su">n</span>) = 0; these can, however, by a
+simple contact transformation be reduced to one kind, in which
+there is only one relation z&prime; = &psi;(x&prime;<span class="su">1</span>, ... x&prime;<span class="su">n</span>) connecting the
+new variables x&rsquo;<span class="su">1</span>, ... x&prime;<span class="su">n</span>, z&prime; (see under <span class="sc"><a href="#artlinks">Pfaffian Expressions</a></span>);
+just as in the case of the solution</p>
+
+<p class="center">z = &psi;(y), x = &psi;<span class="su">1</span>(y), &psi;&prime;(y) = p&psi;&prime;<span class="su">1</span>(y) + q</p>
+
+<p class="noind">of the equation Pp + Qq = R the transformation z&rsquo; = z &minus; px,
+x&prime; = p, p&prime; = &minus;x, y&prime; = y, q&prime; = q gives the solution</p>
+
+<p class="center">z&prime; = &psi;(y&prime;) + x&prime;&psi;<span class="su">1</span>(y&prime;), p&prime; = dz&prime;/dx&prime;, q&prime; = dz&prime;/dy&prime;</p>
+
+<p class="noind">of the transformed equation. These explanations take no
+account of the possibility of p and q being infinite; this can be
+dealt with by writing p = -u/w, q = -v/w, and considering
+homogeneous equations in u, v, w, with udx + vdy + wdz = 0 as the
+differential relation necessary for a connectivity; in practice we
+use the ideas associated with such a procedure more often without
+the appropriate notation.</p>
+
+<p>In utilizing these general notions we shall first consider
+the theory of characteristic chains, initiated by Cauchy, which
+shows well the nature of the relations implied by the given
+differential equation; the alternative ways of carrying
+<span class="sidenote">Order of the ideas.</span>
+out the necessary integrations are suggested by considering
+the method of Jacobi and Mayer, while a good
+summary is obtained by the formulation in terms of a Pfaffian
+expression.</p>
+
+<p>Consider a solution of F = 0 expressed by the three independent
+equations F = 0, G = 0, H = 0. If it be a solution in which there is
+more than one relation connecting x, y, z, let new variables x&prime;, y&prime;, z&prime;, p&prime;, q&prime;
+be introduced, as before explained under <span class="sc"><a href="#artlinks">Pfaffian Expressions</a></span>,
+<span class="sidenote">Characteristic chains.</span>
+in which z&rsquo; is of the form</p>
+
+<p class="center">z&prime; = z &minus; p<span class="su">1</span>x<span class="su">1</span> &minus; ... &minus; p<span class="su">s</span>x<span class="su">s</span> (s = 1 or 2),</p>
+
+<p class="noind">so that the solution becomes of a form z&rsquo; = &psi;(x&prime;y&prime;),
+p&prime; = d&psi;/dx&prime;, q&prime; = d&psi;/dy&prime;, which then will identically satisfy the transformed
+equations F&prime; = 0, G&prime; = 0, H&prime; = 0. The equation F&prime; = 0, if x&prime;, y&prime;, z&prime;
+be regarded as fixed, states that the plane Z &minus; z&prime; = p&prime;(X &minus; x&prime;)  + q&prime;(Y &minus; y&prime;)
+is tangent to a certain cone whose vertex is (x&prime;, y&prime;, z&prime;), the consecutive
+point (x&prime; + dx&prime;, y&prime; + dy&prime;, z&prime; + dz&prime;) of the generator of contact being such
+that</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">dx&prime;<span class="f150">/</span></td> <td>dF&prime;</td>
+<td rowspan="2">= dy&prime;<span class="f150">/</span></td> <td>dF&prime;</td>
+<td rowspan="2">= dz&prime;<span class="f150">/ (</span> p&prime;</td> <td>dF&prime;</td>
+<td rowspan="2">+ q&prime;</td> <td>dF&prime;</td> <td rowspan="2"><span class="f150">)</span>.</td></tr>
+<tr><td class="denom">dp&prime;</td> <td class="denom">dq&prime;</td>
+<td class="denom">dp&prime;</td> <td class="denom">dq&prime;</td></tr></table>
+
+<p class="noind">Passing in this direction on the surface z&prime; = &psi;(x&prime;, y&prime;) the tangent
+<span class="pagenum"><a name="page233" id="page233"></a>233</span>
+plane of the surface at this consecutive point is (p&prime; + dp&prime;, q&prime; + dq&prime;),
+where, since F&prime;(x&prime;, y&prime;, &psi;, d&psi;/dx&prime;, d&psi;/dy&prime;) = 0 is identical, we have
+dx&prime; (dF&prime;/dx&prime; + p&prime;dF&prime;/dz&prime;) + dp&prime;dF&prime;/dp&prime; = 0. Thus the equations, which
+we shall call the characteristic equations,</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">dx&prime;<span class="f150">/</span></td> <td>dF&prime;</td>
+<td rowspan="2">= dy&prime;<span class="f150">/</span></td> <td>dF&prime;</td>
+<td rowspan="2">= dz&prime;<span class="f150">/ (</span> p&prime;</td> <td>dF&prime;</td>
+<td rowspan="2">+ q&prime;</td> <td>dF&prime;</td>
+<td rowspan="2"><span class="f150">)</span> = dp&prime;<span class="f150">/ (</span> &minus;</td> <td>dF&prime;</td>
+<td rowspan="2">&minus; p&prime;</td> <td>dF&prime;</td>
+<td rowspan="2"><span class="f150">)</span> = dq&prime;<span class="f150">/ (</span> &minus;</td> <td>dF&prime;</td>
+<td rowspan="2">&minus; q&prime;</td> <td>dF&prime;</td> <td rowspan="2"><span class="f150">)</span></td></tr>
+<tr><td class="denom">dp&prime;</td> <td class="denom">dq&prime;</td>
+<td class="denom">dp&prime;</td> <td class="denom">dq&prime;</td>
+<td class="denom">dx&prime;</td> <td class="denom">dz&prime;</td>
+<td class="denom">dy&prime;</td> <td class="denom">dz&prime;</td></tr></table>
+
+<p class="noind">are satisfied along a connectivity of &infin;¹ elements consisting of a curve
+on z&prime; = &psi;(x&prime;, y&prime;) and the tangent planes of the surface along this curve.
+The equation F&prime; = 0, when p&prime;, q&prime; are fixed, represents a curve in the
+plane Z &minus; z&prime; = p&prime;(X &minus; x&prime;) + q&prime;(Y &minus; y&prime;) passing through (x&prime;, y&prime;, z&prime;); if
+(x&prime; + &delta;x&prime;, y&prime; <span class="correction" title="amended from =">+</span> &delta;y&prime;, z&prime; + &delta;z&prime;) be a consecutive point of this curve, we
+find at once</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&delta;x&prime;<span class="f150">(</span></td> <td>dF&prime;</td>
+<td rowspan="2">+ p&prime;</td> <td>dF&prime;</td>
+<td rowspan="2"><span class="f150">)</span> + &delta;y&prime;<span class="f150">(</span></td> <td>dF&prime;</td>
+<td rowspan="2">q&prime;</td> <td>dF&prime;</td> <td rowspan="2"><span class="f150">)</span> = 0;</td></tr>
+<tr><td class="denom">dx&prime;</td> <td class="denom">dz&prime;</td>
+<td class="denom">dy&prime;</td> <td class="denom">dz&prime;</td></tr></table>
+
+<p class="noind">thus the equations above give &delta;x&prime;dp&prime; + &delta;y&prime;dq&prime; = 0, or the tangent line
+of the plane curve, is, on the surface z&prime; = &psi;(x&prime;, y&prime;), in a direction conjugate
+to that of the generator of the cone. Putting each of the
+fractions in the characteristic equations equal to dt, the equations
+enable us, starting from an arbitrary element x&prime;<span class="su">0</span>, y&prime;<span class="su">0</span>, z&prime;<span class="su">0</span>, p&prime;<span class="su">0</span>, q&prime;<span class="su">0</span>,
+about which all the quantities F&prime;, dF&prime;/dp&prime;, &amp;c., occurring in the
+denominators, are developable, to define, from the differential
+equation F&prime; = 0 alone, a connectivity of &infin;¹ elements, which we call
+a <i>characteristic chain</i>; and it is remarkable that when we transform
+again to the original variables (x, y, z, p, q), the form of the differential
+equations for the chain is unaltered, so that they can be written
+down at once from the equation F = 0. Thus we have proved that
+the characteristic chain starting from any ordinary element of any
+integral of this equation F = 0 consists only of elements belonging
+to this integral. For instance, if the equation do not contain p, q,
+the characteristic chain, starting from an arbitrary plane through
+an arbitrary point of the surface F = 0, consists of a pencil of planes
+whose axis is a tangent line of the surface F = 0. Or if F = 0 be of
+the form Pp + Qq = R, the chain consists of a curve satisfying
+dx/P = dy/Q = dz/R and a single infinity of tangent planes of this
+curve, determined by the tangent plane chosen at the initial point.
+In all cases there are &infin;³ characteristic chains, whose aggregate may
+therefore be expected to exhaust the &infin;<span class="sp">4</span> elements satisfying F = 0.</p>
+
+
+
+<p>Consider, in fact, a single infinity of connected elements each
+satisfying F = 0, say a chain connectivity T, consisting of elements
+specified by x<span class="su">0</span>, y<span class="su">0</span>, z<span class="su">0</span>, p<span class="su">0</span>, q<span class="su">0</span>, which we suppose expressed as
+<span class="sidenote">Complete integral constructed with characteristic chains.</span>
+functions of a parameter u, so that</p>
+
+<p class="center">U<span class="su">0</span> = dz<span class="su">0</span>/du &minus; p<span class="su">0</span>dx<span class="su">0</span>/du &minus; q<span class="su">0</span>dy<span class="su">0</span>/du</p>
+
+<p class="noind">is everywhere zero on this chain; further, suppose that
+each of F, dF/dp, ... , dF/dx + pdF/dz is developable
+about each element of this chain T, and that T is <i>not</i> a
+characteristic chain. Then consider the aggregate of the
+characteristic chains issuing from all the elements of T.
+The &infin;² elements, consisting of the aggregate of these
+characteristic chains, satisfy F = 0, provided the chain
+connectivity T consists of elements satisfying F = 0; for each
+characteristic chain satisfies dF = 0. It can be shown that these
+chains are connected; in other words, that if x, y, z, p, q, be any
+element of one of these characteristic chains, not only is</p>
+
+<p class="center">dz/dt &minus; pdx/dt &minus; qdy/dt = 0,</p>
+
+<p class="noind">as we know, but also U = dz/du &minus; pdx/du &minus; qdy/du is also zero. For
+we have</p>
+
+<table class="math0" summary="math">
+<tr><td>dU</td> <td rowspan="2">=</td> <td>d</td>
+<td rowspan="2"><span class="f150">(</span></td> <td>dz</td>
+<td rowspan="2">&minus; p</td> <td>dx</td>
+<td rowspan="2">&minus; q</td> <td>dy</td>
+<td rowspan="2"><span class="f150">)</span> &minus;</td> <td>d</td>
+<td rowspan="2"><span class="f150">(</span></td> <td>dz</td>
+<td rowspan="2">&minus; p</td> <td>dx</td>
+<td rowspan="2">&minus; q</td> <td>dy</td> <td rowspan="2"><span class="f150">)</span></td></tr>
+<tr><td class="denom">dt</td> <td class="denom">dt</td> <td class="denom">du</td>
+<td class="denom">du</td> <td class="denom">du</td> <td class="denom">du</td>
+<td class="denom">dt</td> <td class="denom">dt</td> <td class="denom">dt</td></tr></table>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">=</td> <td>dp</td> <td>dx</td>
+<td rowspan="2">&minus;</td> <td>dp</td> <td>dx</td>
+<td rowspan="2">+</td> <td>dq</td> <td>dy</td>
+<td rowspan="2">&minus;</td> <td>dq</td> <td>dy</td> <td rowspan="2">,</td></tr>
+<tr><td class="ov">du</td> <td class="ov">dt</td> <td class="ov">dt</td>
+<td class="ov">du</td> <td class="ov">du</td> <td class="ov">dt</td>
+<td class="ov">dt</td> <td class="ov">du</td></tr></table>
+
+<p class="noind">which is equal to</p>
+
+<table class="math0" summary="math">
+<tr><td>dp</td> <td>dF</td> <td rowspan="2">+</td> <td>dx</td>
+ <td rowspan="2"><span class="f150">(</span></td> <td>dF</td>
+ <td rowspan="2">+ p</td> <td>dF</td> <td rowspan="2"><span class="f150">)</span> +</td>
+ <td>dq</td> <td>dF</td> <td rowspan="2">+</td> <td>dy</td>
+ <td rowspan="2"><span class="f150">(</span></td> <td>dF</td>
+ <td rowspan="2">+ q</td> <td>dF</td> <td rowspan="2"><span class="f150">)</span>  = &minus;</td>
+ <td>dF</td> <td rowspan="2">U.</td></tr>
+<tr><td class="ov">du</td> <td class="ov">dp</td> <td class="ov">du</td>
+ <td class="ov">dx</td> <td class="ov">dz</td> <td class="ov">du</td>
+ <td class="ov">dq</td> <td class="ov">du</td> <td class="ov">dy</td>
+ <td class="ov">dz</td> <td class="ov">dz</td></tr></table>
+
+<p class="noind">As dF/dz is a developable function of t, this, giving</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">U = U<span class="su">0</span> exp<span class="f150">(</span> &minus; <span class="f150">&int;</span></td> <td class="bk">t</td> <td>dF</td>
+<td rowspan="2">dt <span class="f150">)</span>,</td></tr>
+<tr><td class="bk">t<span class="su">0</span></td> <td class="ov">dz</td></tr></table>
+
+<p class="noind">shows that U is everywhere zero. Thus integrals of F = 0 are
+obtainable by considering the aggregate of characteristic chains
+issuing from arbitrary chain connectivities T satisfying F = 0; and
+such connectivities T are, it is seen at once, determinable without
+integration. Conversely, as such a chain connectivity T can be taken
+out from the elements of any given integral all possible integrals
+are obtainable in this way. For instance, an arbitrary curve in
+space, given by x<span class="su">0</span> = &theta;(u), y<span class="su">0</span> = &phi;(u), z<span class="su">0</span> = &psi;(u), determines by the two
+equations F(x<span class="su">0</span>, y<span class="su">0</span>, z<span class="su">0</span>, p<span class="su">0</span>, q<span class="su">0</span>) = 0, &psi;&prime;(u) = p<span class="su">0</span>&theta;&prime;(u) + q<span class="su">0</span>&phi;&prime;(u), such a
+chain connectivity T, through which there passes a perfectly
+definite integral of the equation F = 0. By taking &infin;² initial chain
+connectivities T, as for instance by taking the curves x<span class="su">0</span> = &theta;, y<span class="su">0</span>  = &phi;,
+z<span class="su">0</span>  = &psi; to be the &infin;² curves upon an arbitrary surface, we thus obtain
+&infin;² integrals, and so &infin;<span class="sp">4</span> elements satisfying F = 0. In general, if
+functions G, H, independent of F, be obtained, such that the
+equations F = 0, G = b, H = c represent an integral for all values of the
+constants b, c, these equations are said to constitute a <i>complete
+integral</i>. Then &infin;<span class="sp">4</span> elements satisfying F = 0 are known, and in fact
+every other form of integral can be obtained without further integrations.</p>
+
+<p>In the foregoing discussion of the differential equations of a
+characteristic chain, the denominators dF/dp, ... may be supposed
+to be modified in form by means of F = 0 in any way conducive to
+a simple integration. In the immediately following explanation of
+ideas, however, we consider indifferently all equations F = constant;
+when a function of x, y, z, p, q is said to be zero, it is meant that this
+is so identically, not in virtue of F = 0; in other words, we consider
+the integration of F = a, where a is an arbitrary constant. In the
+theory of linear partial equations we have seen that the integration
+<span class="sidenote">Operations necessary for integration of F = a.</span>
+of the equations of the characteristic chains, from which,
+as has just been seen, that of the equation F = a follows
+at once, would be involved in completely integrating
+the single linear homogeneous partial differential equation
+of the first order [F&fnof;] = 0 where the notation is that
+explained above under <span class="sc"><a href="#artlinks">Contact Transformations</a></span>. One
+obvious integral is &fnof; = F. Putting F = a, where a is arbitrary,
+and eliminating one of the independent variables, we can reduce
+this equation [F&fnof;] = 0 to one in four variables; and so on. Calling, then,
+the determination of a single integral of a single homogeneous partial
+differential equation of the first order in n independent variables, <i>an
+operation of order</i> n &minus; 1, the characteristic chains, and therefore the
+most general integral of F = a, can be obtained by successive operations
+of orders 3, 2, 1. If, however, an integral of F = a be represented
+by F = a, G = b, H = c, where b and c are arbitrary constants,
+the expression of the fact that a characteristic chain of F = a satisfies
+dG = 0, gives [FG] = 0; similarly, [FH] = 0 and [GH] = 0, these
+three relations being identically true. Conversely, suppose that an
+integral G, independent of F, has been obtained of the equation
+[F&fnof;] = 0, which is an operation of order three. Then it follows from
+the identity [&fnof;[&phi;&psi;]] + [&phi;[&psi;&fnof;]] + [&psi;[&fnof;&phi;]] = d&fnof;/dz [&psi;&phi;] + d&phi;/dz [&psi;&fnof;] + d&psi;/dz [&fnof;&phi;] before
+remarked, by putting &phi; = F, &psi; = G, and then [F&fnof;] = A(&fnof;), [G&fnof;] = B(&fnof;),
+that AB(&fnof;) &minus; BA(&fnof;) = dF/dz B(&fnof;) &minus; dG/dz A(&fnof;), so that the two linear equations
+[F&fnof;] = 0, [G&fnof;] = 0 form a complete system; as two integrals F, G are
+known, they have a common integral H, independent of F, G, determinable
+by an operation of order one only. The three functions
+F, G, H thus identically satisfy the relations [FG] = [GH] = [FH] = 0.
+The &infin;² elements satisfying F = a, G = b, H = c, wherein a, b, c are
+assigned constants, can then be seen to constitute an integral of F = a.
+For the conditions that a characteristic chain of G = b issuing from
+an element satisfying F = a, G = b, H = c should consist only of
+elements satisfying these three equations are simply [FG] = 0, [GH] = 0.
+Thus, starting from an arbitrary element of (F = a, G = b, H = c), we
+can single out a connectivity of elements of (F = a, G = b, H = c)
+forming a characteristic chain of G = b; then the aggregate of the
+characteristic chains of F = a issuing from the elements of this
+characteristic chain of G = b will be a connectivity consisting only of
+elements of</p>
+
+<p class="center">(F = a, G = b, H = c),</p>
+
+<p class="noind">and will therefore constitute an integral of F = a; further, it will
+include all elements of (F = a, G = b, H = c). This result follows also
+from a theorem given under <i>Contact Transformations</i>, which shows,
+moreover, that though the characteristic chains of F = a are not
+determined by the three equations F = a, G = b, H = c, no further
+integration is now necessary to find them. By this theorem, since
+identically [FG] = [GH] = [FH] = 0, we can find, by the solution of
+linear algebraic equations only, a non-vanishing function &sigma; and two
+functions A, C, such that</p>
+
+<p class="center">dG &minus; AdF &minus; CdH = &sigma;(dz &minus; pdz &minus; qdy);</p>
+
+<p class="noind">thus all the elements satisfying F = a, G = b, H = c, satisfy dz = pdx + qdy
+and constitute a connectivity, which is therefore an integral of
+F = a. While, further, from the associated theorems, F, G, H, A, C
+are independent functions and [FC] = 0. Thus C may be taken to
+be the remaining integral independent of G, H, of the equation
+[F&fnof;] = 0, whereby the characteristic chains are entirely determined.</p>
+
+<p>When we consider the particular equation F = 0, neglecting the
+case when neither p nor q enters, and supposing p to enter, we may
+express p from F = 0 in terms of x, y, z, q, and then eliminate it from
+all other equations. Then instead of the equation [F&fnof;] = 0, we
+have, if F = 0 give p = &psi;(x, y, z, q), the equation</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&Sigma;&fnof; = &minus; <span class="f150">(</span></td> <td>d&fnof;</td>
+<td rowspan="2">+ &psi;</td> <td>d&fnof;</td>
+<td rowspan="2"><span class="f150">)</span> +</td> <td>d&psi;</td>
+<td rowspan="2"><span class="f150">(</span></td> <td>d&fnof;</td>
+<td rowspan="2">+ q</td> <td>d&fnof;</td>
+<td rowspan="2"><span class="f150">)</span> &minus; <span class="f150">(</span></td> <td>d&psi;</td>
+<td rowspan="2">+ q</td> <td>d&psi;</td>
+<td rowspan="2"><span class="f150">)</span></td> <td>d&fnof;</td> <td rowspan="2">= 0,</td></tr>
+<tr><td class="denom">dx</td> <td class="denom">dz</td> <td class="denom">dq</td>
+<td class="denom">dy</td> <td class="denom">dz</td> <td class="denom">dy</td>
+<td class="denom">dz</td> <td class="denom">dq</td></tr></table>
+
+<p class="noind">moreover obtainable by omitting the term in d&fnof;/dp in [p &minus; &psi;, &fnof;] = 0.
+Let x<span class="su">0</span>, y<span class="su">0</span>, z<span class="su">0</span>,  q<span class="su">0</span>, be values about which the coefficients in
+<span class="sidenote">The single equation F = 0 and Pfaffian formulations.</span>
+this equation are developable, and let &zeta;, &eta;, &omega; be the
+principal solutions reducing respectively to z, y and q
+when x = x<span class="su">0</span>. Then the equations p = &psi;, &zeta; = z<span class="su">0</span>, &eta; = y<span class="su">0</span>, &omega; = q<span class="su">0</span>
+represent a characteristic chain issuing from the element
+x<span class="su">0</span>, y<span class="su">0</span>, z<span class="su">0</span>, &psi;<span class="su">0</span>, q<span class="su">0</span>; we have seen that the aggregate of
+such chains issuing from the elements of an arbitrary
+chain satisfying</p>
+
+<p class="center">dz<span class="su">0</span> = p<span class="su">0</span>dx<span class="su">0</span> &minus; q<span class="su">0</span>dy<span class="su">0</span>  = 0</p>
+
+<p class="noind">constitute an integral of the equation p = &psi;. Let this arbitrary
+<span class="pagenum"><a name="page234" id="page234"></a>234</span>
+chain be taken so that x<span class="su">0</span> is constant; then the condition for initial
+values is only</p>
+
+<p class="center">dz<span class="su">0</span> &minus; q<span class="su">0</span>dy<span class="su">0</span> = 0,</p>
+
+<p class="noind">and the elements of the integral constituted by the characteristic
+chains issuing therefrom satisfy</p>
+
+<p class="center">d&zeta; &minus; &omega;d&eta; = 0.</p>
+
+<p class="noind">Hence this equation involves dz &minus; &psi;dx &minus; qdy = 0, or we have</p>
+
+<p class="center">dz &minus; &psi;dx &minus; qdy = &sigma;(d&zeta; &minus; &omega;d&eta;),</p>
+
+<p class="noind">where &sigma; is not zero. Conversely, the integration of p = &psi; is, essentially,
+the problem of writing the expression dz &minus; &psi;dx &minus; qdy in the form
+&sigma;(d&zeta; &minus; &omega;d&eta;), as must be possible (from what was said under <span class="sc"><a href="#artlinks">Pfaffian
+Expressions</a></span>).</p>
+
+<p>To integrate a system of simultaneous equations of the first
+order X<span class="su">1</span> = a<span class="su">1</span>, ... X<span class="su">r</span> = a<span class="su">r</span> in n independent variables x<span class="su">1</span>, ... x<span class="su">n</span>
+and one dependent variable z, we write p<span class="su">1</span> for dz/dx<span class="su">1</span>, &amp;c.,
+<span class="sidenote">System of equations of the first order.</span>
+and attempt to find n + 1 &minus; r further functions Z, X<span class="su">r+1</span>
+... X<span class="su">n</span>, such that the equations Z = a, X<span class="su">i</span> = a<span class="su">i</span>,(i = 1, ... n)
+involve dz &minus; p<span class="su">1</span>dx<span class="su">1</span> &minus; ... &minus; p<span class="su">n</span>dx<span class="su">n</span> = 0. By an argument
+already given, the common integral, if existent, must be satisfied
+by the equations of the characteristic chains of any one equation
+X<span class="su">i</span> = a<span class="su">i</span>; thus each of the expressions [X<span class="su">i</span>X<span class="su">j</span>] must vanish in virtue
+of the equations expressing the integral, and we may without loss of
+generality assume that each of the corresponding ½r(r &minus; 1) expressions
+formed from the r given differential equations vanishes in virtue of
+these equations. The determination of the remaining n + 1 &minus; r
+functions may, as before, be made to depend on characteristic chains,
+which in this case, however, are manifolds of r dimensions obtained
+by integrating the equations [X<span class="su">1</span>&fnof;] = 0, ... [X<span class="su">r</span>&fnof;] = 0; or having
+obtained one integral of this system other than X<span class="su">1</span>, ... X<span class="su">r</span>, say
+X<span class="su">r+1</span>, we may consider the system [X<span class="su">1</span>&fnof;] = 0, ... [X<span class="su">r+1</span>&fnof;] = 0, for
+which, again, we have a choice; and at any stage we may use Mayer&rsquo;s
+method and reduce the simultaneous linear equations to one equation
+involving parameters; while if at any stage of the process we find
+some but not all of the integrals of the simultaneous system, they
+can be used to simplify the remaining work; this can only be clearly
+explained in connexion with the theory of so-called function groups
+for which we have no space. One result arising is that the simultaneous
+system p<span class="su">1</span> = &phi;<span class="su">1</span>, ... p<span class="su">r</span> = &phi;<span class="su">r</span>, wherein p<span class="su">1</span>, ... p<span class="su">r</span> are not involved in
+&phi;<span class="su">1</span>, ... &phi;<span class="su">r</span>, if it satisfies the ½r(r &minus; 1) relations [p<span class="su">i</span> &minus; &phi;<span class="su">i</span>, p<span class="su">j</span> &minus; &phi;<span class="su">j</span>] = 0,
+has a solution z = &psi;(x<span class="su">1</span>, ... x<span class="su">n</span>), p<span class="su">1</span> = d&psi;/dx<span class="su">1</span>, ... p<span class="su">n</span> = d&psi;/dx<span class="su">n</span>,
+reducing to an arbitrary function of x<span class="su">r+1</span>, ... x<span class="su">n</span> only, when x<span class="su">1</span> = xº<span class="su">1</span>,
+... x<span class="su">r</span> = xº<span class="su">r</span> under certain conditions as to developability; a
+generalization of the theorem for linear equations. The problem of
+integration of this system is, as before, to put</p>
+
+<p class="center">dz &minus; &phi;<span class="su">1</span>dx<span class="su">1</span> &minus; ... &minus; &phi;<span class="su">r</span>dx<span class="su">r</span> &minus; p<span class="su">r+1</span>dx<span class="su">r+1</span> &minus; ... &minus; p<span class="su">n</span>dx<span class="su">n</span></p>
+
+<p class="noind">into the form &sigma;(d&zeta; &minus; &omega;<span class="su">r+1</span> + d&xi;<span class="su">r+1</span> &minus; ... &minus; &omega;<span class="su">n</span>d&xi;<span class="su">n</span>); and here &zeta;, &xi;<span class="su">r+1</span>, ... &xi;<span class="su">n</span>,
+&omega;<span class="su">r+1</span>, ... &omega;<span class="su">n</span> may be taken, as before, to be principal integrals
+of a certain complete system of linear equations; those, namely,
+determining the characteristic chains.</p>
+
+<p>If L be a function of t and of the 2n quantities x<span class="su">1</span>, ... x<span class="su">n</span>, &#7819;<span class="su">1</span>, ...
+&#7819;<span class="su">n</span>, where &#7819;<span class="su">i</span>, denotes dx<span class="su">i</span>/dt, &amp;c., and if in the n equations</p>
+
+<table class="math0" summary="math">
+<tr><td>d</td> <td rowspan="2"><span class="f150">(</span></td> <td>dL</td>
+<td rowspan="2"><span class="f150">)</span> =</td> <td>dL</td></tr>
+<tr><td class="denom">dt</td> <td class="denom">dx<span class="su">i</span></td> <td class="denom">dx<span class="su">i</span></td></tr></table>
+
+<p class="noind">we put p<span class="su">i</span> = dL/d&#7819;<span class="su">i</span>, and so express &#7819;<span class="su">i</span>, ... &#7819;<span class="su">n</span> in terms of t, x<span class="su">i</span>, ...
+x<span class="su">n</span>, p<span class="su">1</span>, ... p<span class="su">n</span>, assuming that the determinant of the quantities
+d²L/dx<span class="su">i</span>d&#7819;<span class="su">j</span> is not zero; if, further, H denote the function of t, x<span class="su">1</span>, ...
+x<span class="su">n</span>, p<span class="su">1</span>, ... p<span class="su">n</span>, numerically equal to p<span class="su">1</span>&#7819;<span class="su">1</span> + ... + p<span class="su">n</span>&#7819;<span class="su">n</span> &minus; L, it is easy
+<span class="sidenote">Equations of dynamics.</span>
+to prove that dp<span class="su">i</span>/dt = &minus;dH/dx<span class="su">i</span>, dx<span class="su">i</span>/dt = dH/dp<span class="su">i</span>. These
+so-called <i>canonical</i> equations form part of those for
+the characteristic chains of the single partial equation
+dz/dt + H(t, x<span class="su">1</span>, ... x<span class="su">n</span>, dz/dx<span class="su">1</span>, ..., dz/dx<span class="su">n</span>) = 0, to which
+then the solution of the original equations for x<span class="su">1</span> ... x<span class="su">n</span> can be
+reduced. It may be shown (1) that if z = &psi;(t, x<span class="su">1</span>, ... x<span class="su">n</span>, c<span class="su">1</span>, .. c<span class="su">n</span>) + c
+be a complete integral of this equation, then p<span class="su">i</span> = d&psi;/dx<span class="su">i</span>, d&psi;/dc<span class="su">i</span> = e<span class="su">i</span>
+are 2n equations giving the solution of the canonical equations
+referred to, where c<span class="su">1</span> ... c<span class="su">n</span> and e<span class="su">1</span>, ... e<span class="su">n</span> are arbitrary constants;
+(2) that if x<span class="su">i</span> = X<span class="su">i</span>(t, x<span class="sp">0</span><span class="su">1</span>, ... pº<span class="su">n</span>), p<span class="su">i</span> = P<span class="su">i</span>(t, xº<span class="su">1</span>, ... p<span class="sp">0</span><span class="su">n</span>) be the principal
+solutions of the canonical equations for t = t<span class="sp">0</span>, and &omega; denote the result
+of substituting these values in p<span class="su">1</span>dH/dp<span class="su">1</span> + ... + p<span class="su">n</span>dH/dp<span class="su">n</span> &minus; H, and
+&Omega; = &int;<span class="sp1">t</span><span class="su">t0</span> &omega;dt, where, after integration, &Omega; is to be expressed as a function
+of t, x<span class="su">1</span>, ... x<span class="su">n</span>, xº<span class="su">1</span>, ... xº<span class="su">n</span>, then z = &Omega; + z<span class="sp">0</span> is a complete integral of
+the partial equation.</p>
+
+<p>A system of differential equations is said to allow a certain
+continuous group of transformations (see <span class="sc"><a href="#artlinks">Groups, Theory of</a></span>)
+when the introduction for the variables in the differential
+equations of the new variables given by the
+<span class="sidenote">Application of theory of continuous groups to formal theories.</span>
+equations of the group leads, for all values of the
+parameters of the group, to the same differential equations
+in the new variables. It would be interesting
+to verify in examples that this is the case in at least
+the majority of the differential equations which are
+known to be integrable in finite terms. We give a theorem of
+very general application for the case of a simultaneous complete
+system of linear partial homogeneous differential equations of the
+first order, to the solution of which the various differential equations
+discussed have been reduced. It will be enough to consider
+whether the given differential equations allow the infinitesimal
+transformations of the group.</p>
+
+<p>It can be shown easily that sufficient conditions in order that a
+complete system &Pi;<span class="su">1</span>&fnof; = 0 ... &Pi;<span class="su">k</span>&fnof; = 0, in n independent variables,
+should allow the infinitesimal transformation P&fnof; = 0 are expressed
+by k equations &Pi;<span class="su">i</span>P&fnof; &minus; P&Pi;<span class="su">i</span>&fnof; = &lambda;<span class="su">i1</span>&Pi;<span class="su">1</span>&fnof; + ... + &lambda;<span class="su">ik</span>&Pi;<span class="su">k</span>&fnof;. Suppose now
+a complete system of n &minus; r equations in n variables to allow a
+group of r infinitesimal transformations (P<span class="su">1</span>f, ..., P<span class="su">r</span>&fnof;) which has
+an invariant subgroup of r &minus; 1 parameters (P<span class="su">1</span>&fnof;, ..., P<span class="su">r-1</span>&fnof;), it
+being supposed that the n quantities &Pi;<span class="su">1</span>&fnof;, ..., &Pi;<span class="su">n-r</span>&fnof;, P<span class="su">1</span>&fnof;, ...,
+P<span class="su">r</span>&fnof; are not connected by an identical linear equation (with coefficients
+even depending on the independent variables). Then
+it can be shown that one solution of the complete system is determinable
+by a quadrature. For each of &Pi;<span class="su">i</span>P&sigma;&fnof; &minus; P&sigma;&Pi;<span class="su">i</span>f is a linear
+function of &Pi;<span class="su">1</span>&fnof;, ..., &Pi;<span class="su">n-r</span>&fnof; and the simultaneous system of independent
+equations &Pi;<span class="su">1</span>&fnof; = 0, ... &Pi;<span class="su">n-r</span>&fnof; = 0, P<span class="su">1</span>&fnof; = 0, ... P<span class="su">r-1</span>&fnof; = 0
+is therefore a complete system, allowing the infinitesimal transformation
+P<span class="su">r</span>&fnof;. This complete system of n &minus; 1 equations has therefore
+one common solution &omega;, and P<span class="su">r</span>(&omega;) is a function of &omega;. By
+choosing &omega; suitably, we can then make P<span class="su">r</span>(&omega;) = 1. From this
+equation and the n &minus; 1 equations &Pi;<span class="su">i</span>&omega; = 0, P<span class="su">&sigma;&omega;</span> = 0, we can determine
+&omega; by a quadrature only. Hence can be deduced a much more
+general result, <i>that if the group of r parameters be integrable, the
+complete system can be entirety solved by quadratures</i>; it is only
+necessary to introduce the solution found by the first quadrature as
+an independent variable, whereby we obtain a complete system of
+n &minus; r equations in n &minus; 1 variables, subject to an integrable group of
+r &minus; 1 parameters, and to continue this process. We give some
+examples of the application of the theorem. (1) If an equation of
+the first order y&prime; = &psi;(x, y) allow the infinitesimal transformation
+&xi;d&fnof;/dx + &eta;d&fnof;/dy, the integral curves &omega;(x, y) = y<span class="sp">0</span>, wherein &omega;(x, y) is
+the solution of d&fnof;/dx + &psi;(x, y) d&fnof;/dy = 0 reducing to y for x = x<span class="sp">0</span>, are
+interchanged among themselves by the infinitesimal transformation,
+or &omega;(x, y) can be chosen to make &xi;d<span class="su">&omega;</span>/dx + &eta;d<span class="su">&omega;</span>/dy = 1; this, with
+d&omega;/dx + &psi;d&omega;/dy = 0, determines &omega; as the integral of the complete
+differential (dy &minus; &psi;dx)/(&eta; &minus; &psi;&xi;). This result itself shows that every
+ordinary differential equation of the first order is subject to an
+infinite number of infinitesimal transformations. But every infinitesimal
+transformation &xi;d&fnof;/dx + &eta;d&fnof;/dy can by change of variables
+(after integration) be brought to the form d&fnof;/dy, and all differential
+equations of the first order allowing this group can then be reduced
+to the form F(x, dy/dx) = 0. (2) In an ordinary equation of the
+second order y&rdquo; = &psi;(x, y, y&prime;), equivalent to dy/dx = y<span class="su">1</span>, dy<span class="su">1</span>/dx = &psi;(x, y, y<span class="su">1</span>),
+if H, H<span class="su">1</span> be the solutions for y and y<span class="su">1</span> chosen to reduce to y<span class="sp">0</span> and
+yº<span class="su">1</span> when x = x<span class="sp">0</span>, and the equations H = y, H<span class="su">1</span>= y<span class="su">1</span> be equivalent
+to &omega; = y<span class="sp">0</span>, &omega;<span class="su">1</span> = yº<span class="su">1</span>, then &omega;, &omega;<span class="su">1</span> are the principal solutions of
+&Pi;&fnof; = d&fnof;/dx + y<span class="su">1</span>d&fnof;/dy + &psi;d&fnof;/dy<span class="su">1</span> = 0. If the original equation allow
+an infinitesimal transformation whose first <i>extended</i> form (see
+<span class="sc"><a href="#artlinks">Groups</a></span>) is P&fnof; = &xi;d&fnof;/dx + &eta;d&fnof;/dy  + &eta;<span class="su">1</span>d&fnof;/dy<span class="su">1</span>, where &eta;<span class="su">1</span>&delta;t is the increment
+of dy/dx when &xi;&delta;t, &eta;&delta;t are the increments of x, y, and is to be
+expressed in terms of x, y, y<span class="su">1</span>, then each of P&omega; and P&omega;<span class="su">1</span> must
+be functions of &omega; and &omega;<span class="su">1</span>, or the partial differential equation &Pi;&fnof;
+must allow the group P&fnof;. Thus by our general theorem, if the
+differential equation allow a group of two parameters (and such
+a group is always integrable), it can be solved by quadratures, our
+explanation sufficing, however, only provided the form &Pi;&fnof; and the
+two infinitesimal transformations are not linearly connected. It
+can be shown, from the fact that &eta;<span class="su">1</span> is a quadratic polynomial in y<span class="su">1</span>,
+that no differential equation of the second order can allow more
+than 8 really independent infinitesimal transformations, and that
+every homogeneous linear differential equation of the second order
+allows just 8, being in fact reducible to d²y/dx² = 0. Since every
+group of more than two parameters has subgroups of two parameters,
+a differential equation of the second order allowing a group
+of more than two parameters can, as a rule, be solved by quadratures.
+By transforming the group we see that if a differential equation of
+the second order allows a single infinitesimal transformation, it can
+be transformed to the form F(x, d&gamma;/dx, d²&gamma;/dx²); this is not the case
+for every differential equation of the second order. (3) For an
+ordinary differential equation of the third order, allowing an integrable
+group of three parameters whose infinitesimal transformations
+are not linearly connected with the partial equation to which the
+solution of the given ordinary equation is reducible, the similar
+result follows that it can be integrated by quadratures. But if the
+group of three parameters be simple, this result must be replaced
+by the statement that the integration is reducible to quadratures
+and that of a so-called Riccati equation of the first order, of the
+form dy/dx = A + By + Cy², where A, B, C are functions of x. (4) Similarly
+for the integration by quadratures of an ordinary equation
+y<span class="su">n</span> = &psi;(x, y, y<span class="su">1</span>, ... y<span class="su">n-1</span>) of any order. Moreover, the group allowed
+by the equation may quite well consist of extended contact transformations.
+An important application is to the case where the differential
+equation is the resolvent equation defining the group of
+<span class="pagenum"><a name="page235" id="page235"></a>235</span>
+transformations or rationality group of another differential equation
+(see below); in particular, when the rationality group of an ordinary
+linear differential equation is integrable, the equation can be solved
+by quadratures.</p>
+
+<p>Following the practical and provisional division of theories
+of differential equations, to which we alluded at starting, into
+transformation theories and function theories, we pass
+now to give some account of the latter. These are both
+<span class="sidenote">Consideration of function theories of differential equations.</span>
+a necessary logical complement of the former, and the
+only remaining resource when the expedients of the
+former have been exhausted. While in the former
+investigations we have dealt only with values of the
+independent variables about which the functions are
+developable, the leading idea now becomes, as was long ago
+remarked by G. Green, the consideration of the neighbourhood of
+the values of the variables for which this developable character
+ceases. Beginning, as before, with existence theorems applicable
+for ordinary values of the variables, we are to consider the cases of
+failure of such theorems.</p>
+
+<p>When in a given set of differential equations the number of
+equations is greater than the number of dependent variables, the
+equations cannot be expected to have common solutions unless
+certain conditions of compatibility, obtainable by equating
+different forms of the same differential coefficients deducible from
+the equations, are satisfied. We have had examples in systems
+of linear equations, and in the case of a set of equations
+p<span class="su">1</span> = &phi;<span class="su">1</span>, ..., p<span class="su">r</span> = &phi;<span class="su">r</span>. For the case when the number of equations
+is the same as that of dependent variables, the following is a
+general theorem which should be referred to: Let there be r
+equations in r dependent variables z<span class="su">1</span>, ... z<span class="su">r</span> and n independent
+<span class="sidenote">A general existence theorem.</span>
+variables x<span class="su">1</span>, ... x<span class="su">n</span>; let the differential coefficient of
+z<span class="su">&sigma;</span> of highest order which enters be of order h<span class="su">&sigma;</span>, and
+suppose d<span class="sp">h&sigma;</span>z<span class="su">&sigma;</span> / dx<span class="su">1</span><span class="sp">h&sigma;</span> to enter, so that the equations can be
+written d<span class="sp">h&sigma;</span>z<span class="su">&sigma;</span> / dx<span class="su">1</span><span class="sp">h&sigma;</span> = &Phi;<span class="su">&sigma;</span>, where in the general differential
+coefficient of z<span class="su">&rho;</span> which enters in &Phi;<span class="su">&sigma;</span>, say</p>
+
+<p class="center">d<span class="sp">k1 + ... + kn</span> z<span class="su">&rho;</span> / dx<span class="su">1</span><span class="sp">k1</span> ... dx<span class="su">n</span><span class="sp">kn</span>,</p>
+
+<p class="noind">we have k<span class="su">1</span> &lt; h<span class="su">&rho;</span> and k<span class="su">1</span> + ... + k<span class="su">n</span> &le; h<span class="su">&rho;</span>. Let a<span class="su">1</span>, ... a<span class="su">n</span>,
+b<span class="su">1</span>, ... b<span class="su">r</span>, and b&rho;<span class="su">k1 ... kn</span> be a set of values of</p>
+
+<p class="center">x<span class="su">1</span>, ... x<span class="su">n</span>, z<span class="su">1</span>, ... z<span class="su">r</span></p>
+
+<p class="noind">and of the differential coefficients entering in &Phi;<span class="su">&sigma;</span> about which
+all the functions &Phi;<span class="su">1</span>, ... &Phi;<span class="su">r</span>, are developable. Corresponding
+to each dependent variable z<span class="su">&sigma;</span>, we take now a set of h<span class="su">&sigma;</span> functions of
+x<span class="su">2</span>, ... x<span class="su">n</span>, say &phi;<span class="su">&sigma;</span>, &phi;<span class="su">&sigma;</span>;<span class="sp">(1)</span>, ... ,&phi;<span class="su">&sigma;</span><span class="sp">h&minus;1</span> arbitrary save that they must
+be developable about a<span class="su">2</span>, a<span class="su">3</span>, ... a<span class="su">n</span>, and such that for these
+values of x<span class="su">2</span>, ... x<span class="su">n</span>, the function &phi;<span class="su">&rho;</span> reduces to b<span class="su">&rho;</span>, and the
+differential coefficient</p>
+
+<p class="center">d<span class="sp">k2 + ... + kn</span> &phi;<span class="su">&rho;</span><span class="sp">k1</span> / dx<span class="su">2</span><span class="sp">k2</span> ... dx<span class="su">n</span><span class="sp">kn</span></p>
+
+<p class="noind">reduces to b<span class="sp">&rho;</span><span class="su">k1 ... kn</span>. Then the theorem is that there exists
+one, and only one, set of functions z<span class="su">1</span>, ... z<span class="su">r</span>, of x<span class="su">2</span>, ... x<span class="su">n</span>
+developable about a<span class="su">1</span>, ... a<span class="su">n</span> satisfying the given differential
+equations, and such that for x<span class="su">1</span> = a<span class="su">1</span> we have</p>
+
+<p class="center">z<span class="su">&sigma;</span> = &phi;<span class="su">&sigma;</span>, dz<span class="su">&sigma;</span> / dx<span class="su">1</span> = &phi;<span class="su">&sigma;</span><span class="sp">(1)</span>, ... d<span class="sp">h&sigma;&minus;1</span>z<span class="su">&sigma;</span> / d<span class="sp">h&sigma;&minus;1</span>x<span class="su">1</span> = &phi;<span class="su">&sigma;</span><span class="sp">h&sigma;&minus;1</span>.</p>
+
+<p class="noind">And, moreover, if the arbitrary functions &phi;<span class="su">&sigma;</span>, &phi;<span class="su">&sigma;</span><span class="sp">(1)</span> ... contain a
+certain number of arbitrary variables t<span class="su">1</span>, ... t<span class="su">m</span>, and be developable
+about the values tº<span class="su">1</span>, ... tº<span class="su">m</span> of these variables, the
+solutions z<span class="su">1</span>, ... z<span class="su">r</span> will contain t<span class="su">1</span>, ... t<span class="su">m</span>, and be developable
+about tº<span class="su">1</span>, ... tº<span class="su">m</span>.</p>
+
+<p>The proof of this theorem may be given by showing that if
+ordinary power series in x<span class="su">1</span> &minus; a<span class="su">1</span>, ... x<span class="su">n</span> &minus; a<span class="su">n</span>, t<span class="su">1</span> &minus; tº<span class="su">1</span>, ... t<span class="su">m</span> &minus; tº<span class="su">m</span>
+be substituted in the equations wherein in z<span class="su">&sigma;</span> the coefficients of
+(x<span class="su">1</span> &minus; a<span class="su">1</span>)º, x<span class="su">1</span> &minus; a<span class="su">1</span>, ..., (x<span class="su">1</span> &minus; a<span class="su">1</span>)<span class="sp">h&sigma;&minus;1</span> are the arbitrary functions
+&phi;<span class="su">&sigma;</span>, &phi;<span class="su">&sigma;</span><span class="sp">(1)</span>, ..., &phi;<span class="su">&sigma;</span><span class="sp">h&minus;1</span>, divided respectively by 1, 1!, 2!, &amp;c., then the
+differential equations determine uniquely all the other coefficients,
+and that the resulting series are convergent. We rely, in fact,
+upon the theory of monogenic analytical functions (see <span class="sc"><a href="#artlinks">Function</a></span>),
+a function being determined entirely by its development in the
+neighbourhood of one set of values of the independent variables,
+from which all its other values arise by <i>continuation</i>; it being of
+course understood that the coefficients in the differential equations
+are to be continued at the same time. But it is to be remarked that
+there is no ground for believing, if this method of continuation be
+utilized, that the function is single-valued; we may quite well return
+to the same values of the independent variables with a different
+<span class="sidenote">Singular points of solutions.</span>
+value of the function; belonging, as we say, to a different
+branch of the function; and there is even no reason for
+assuming that the number of branches is finite, or that
+different branches have the same singular points and
+regions of existence. Moreover, and this is the most difficult consideration
+of all, all these circumstances may be dependent upon the
+values supposed given to the arbitrary constants of the integral; in
+other words, the singular points may be either <i>fixed</i>, being determined
+by the differential equations themselves, or they may be
+<i>movable</i> with the variation of the arbitrary constants of integration.
+Such difficulties arise even in establishing the reversion of an elliptic
+integral, in solving the equation</p>
+
+<p class="center">(dx/ds)² = (x &minus; a<span class="su">1</span>)(x &minus; a<span class="su">2</span>)(x &minus; a<span class="su">3</span>)(x &minus; a<span class="su">4</span>);</p>
+
+<p class="noind">about an ordinary value the right side is developable; if we put
+x &minus; a<span class="su">1</span> = t<span class="su">1</span>², the right side becomes developable about t<span class="su">1</span> = 0; if we
+put x = 1/t, the right side of the changed equation is developable
+about t = 0; it is quite easy to show that the integral reducing to a
+definite value x<span class="su">0</span> for a value s<span class="su">0</span> is obtainable by a series in integral
+powers; this, however, must be supplemented by showing that for
+no value of s does the value of x become entirely undetermined.</p>
+
+<p>These remarks will show the place of the theory now to be
+sketched of a particular class of ordinary linear homogeneous
+<span class="sidenote">Linear differential equations with rational coefficients.</span>
+differential equations whose importance arises from
+the completeness and generality with which they can
+be discussed. We have seen that if in the equations</p>
+
+<p class="center">dy/dx = y<span class="su">1</span>, dy<span class="su">1</span>/dx = y<span class="su">2</span>, ..., dy<span class="su">n&minus;2</span>/dx = y<span class="su">n&minus;1</span>,<br />
+dy<span class="su">n&minus;1</span>/dx = a<span class="su">n</span>y + a<span class="su">n&minus;1</span>y<span class="su">1</span> + ... + a<span class="su">1</span>y<span class="su">n&minus;1</span>,</p>
+
+<p class="noind">where a<span class="su">1</span>, a<span class="su">2</span>, ..., a<span class="su">n</span> are now to be taken to be rational
+functions of x, the value x  = xº be one for which no one of
+these rational functions is infinite, and yº, yº<span class="su">1</span>, ..., yº<span class="su">n&minus;1</span> be quite
+arbitrary finite values, then the equations are satisfied by</p>
+
+<p class="center">y = yºu + yº<span class="su">1</span>u<span class="su">1</span> + ... + yº<span class="su">n&minus;1</span>u<span class="su">n&minus;1</span>,</p>
+
+<p class="noind">where u, u<span class="su">1</span>, ..., u<span class="su">n&minus;1</span> are functions of x, independent of yº, ...
+yº<span class="su">n&minus;1</span>, developable about x = xº; this value of y is such that for
+x = xº the functions y, y<span class="su">1</span> ... y<span class="su">n&minus;1</span> reduce respectively to yº, yº<span class="su">1</span>,
+... yº<span class="su">n&minus;1</span>; it can be proved that the region of existence of these
+series extends within a circle centre xº and radius equal to the
+distance from xº of the nearest point at which one of a<span class="su">1</span>, ... a<span class="su">n</span>
+becomes infinite. Now consider a region enclosing xº and only one
+of the places, say &Sigma;, at which one of a<span class="su">1</span>, ... a<span class="su">n</span> becomes infinite.
+When x is made to describe a closed curve in this region, including
+this point &Sigma; in its interior, it may well happen that the continuations
+of the functions u, u<span class="su">1</span>, ..., u<span class="su">n&minus;1</span> give, when we have returned to
+the point x, values v, v<span class="su">1</span>, ..., v<span class="su">n&minus;1</span>, so that the integral under consideration
+becomes changed to yº + yº<span class="su">1</span>v<span class="su">1</span> + ... + yº<span class="su">n&minus;1</span>v<span class="su">n&minus;1</span>. At
+xº let this branch and the corresponding values of y<span class="su">1</span>, ... y<span class="su">n&minus;1</span> be
+&eta;º, &eta;º<span class="su">1</span>, ... &eta;º<span class="su">n&minus;1</span>; then, as there is only one series satisfying the
+equation and reducing to (&eta;º, &eta;º<span class="su">1</span>, ... &eta;º<span class="su">n&minus;1</span>) for x = xº and the
+coefficients in the differential equation are single-valued functions,
+we must have &eta;ºu + &eta;º<span class="su">1</span>u<span class="su">1</span> + ... + &eta;º<span class="su">n&minus;1</span>u<span class="su">n&minus;1</span> = yºv + yº<span class="su">1</span>v<span class="su">1</span> + ...
++ yº<span class="su">n&minus;1</span>v<span class="su">n&minus;1</span>; as this holds for arbitrary values of yº ... yº<span class="su">n&minus;1</span>, upon
+which u, ... u<span class="su">n&minus;1</span> and v, ... v<span class="su">n&minus;1</span> do not depend, it follows that
+each of v, ... v<span class="su">n&minus;1</span> is a linear function of u, ... u<span class="su">n&minus;1</span> with constant
+coefficients, say v<span class="su">i</span> = A<span class="su">i1</span>u + ... + A<span class="su">in</span>u<span class="su">n&minus;1</span>. Then</p>
+
+<p class="center">yºv + ... + yº<span class="su">n&minus;1</span>v<span class="su">n&minus;1</span> = (&Sigma;<span class="su">i</span> A<span class="su">i1</span> yº<span class="su">i</span>)u + ... + (&Sigma;<span class="su">i</span> A<span class="su">in</span> yº<span class="su">i</span>) u<span class="su">n&minus;1</span>;</p>
+
+<p class="noind">this is equal to &mu;(yºu + ... + yº<span class="su">n&minus;1</span>u<span class="su">n&minus;1</span>) if &Sigma;<span class="su">i</span> A<span class="su">ir</span> yº<span class="su">i</span> = &mu;yº<span class="su">r&minus;1</span>;
+eliminating yº ... yº<span class="su">n&minus;1</span> from these linear equations, we have a
+determinantal equation of order n for &mu;; let &mu;<span class="su">1</span> be one of its roots;
+determining the ratios of yº, y<span class="su">1</span>º, ... yº<span class="su">n&minus;1</span> to satisfy the linear
+equations, we have thus proved that there exists an integral,
+H, of the equation, which when continued round the point &Sigma; and
+back to the starting-point, becomes changed to H<span class="su">1</span> = &mu;<span class="su">1</span>H. Let now
+&xi; be the value of x at &Sigma; and r<span class="su">1</span> one of the values of (½&pi;i) log &mu;<span class="su">1</span>;
+consider the function (x &minus; &xi;)<span class="sp">&minus;r1</span>H; when x makes a circuit round x = &xi;,
+this becomes changed to</p>
+
+<p class="center">exp (-2&pi;ir<span class="su">1</span>) (x &minus; &xi;)<span class="sp">&minus;r1</span> &mu;H,</p>
+
+<p class="noind">that is, is unchanged; thus we may put H = (x &minus; &xi;)<span class="sp">r1</span>&phi;<span class="su">1</span>, &phi;<span class="su">1</span> being a
+function single-valued for paths in the region considered described
+about &Sigma;, and therefore, by Laurent&rsquo;s Theorem (see <span class="sc"><a href="#artlinks">Function</a></span>),
+capable of expression in the annular region about this point by a
+series of positive and negative integral powers of x &minus; &xi;, which in
+general may contain an infinite number of negative powers; there is,
+however, no reason to suppose r<span class="su">1</span> to be an integer, or even real.
+Thus, if all the roots of the determinantal equation in &mu; are different,
+we obtain n integrals of the forms (x &minus; &xi;)<span class="sp">r1</span>&phi;<span class="su">1</span>, ..., (x &minus; &xi;)<span class="sp">rn</span>&phi;<span class="su">n</span>.
+In general we obtain as many integrals of this form as there are
+really different roots; and the problem arises to discover, in case a
+root be k times repeated, k &minus; 1 equations of as simple a form as
+possible to replace the k &minus; 1 equations of the form yº + ... +
+yº<span class="su">n&minus;1</span>v<span class="su">n&minus;1</span> = &mu;(yº + ... + yº<span class="su">n&minus;1</span>u<span class="su">n&minus;1</span>) which would have existed had
+the roots been different. The most natural method of obtaining
+a suggestion lies probably in remarking that if r<span class="su">2</span> = r<span class="su">1</span> + h, there is an
+integral [(x &minus; &xi;)<span class="sp">r1 + h</span>&phi;<span class="su">2</span> &minus; (x &minus; &xi;)<span class="sp">r1</span>&phi;<span class="su">1</span>] / h, where the coefficients in &phi;<span class="su">2</span> are
+<span class="pagenum"><a name="page236" id="page236"></a>236</span>
+the same functions of r<span class="su">1</span> + h as are the coefficients in &phi;<span class="su">1</span> of r<span class="su">1</span>; when
+h vanishes, this integral takes the form</p>
+
+<p class="center">(x &minus; &xi;)<span class="sp">r1</span> [d&phi;<span class="su">1</span>/dr<span class="su">1</span> + &phi;<span class="su">1</span> log (x &minus; &xi;)],</p>
+
+<p class="noind">or say</p>
+
+<p class="center">(x &minus; &xi;)<span class="sp">r1</span> [&phi;<span class="su">1</span> + &psi;<span class="su">1</span> log (x &minus; &xi;)];</p>
+
+<p class="noind">denoting this by 2&pi;i&mu;<span class="su">1</span>K, and (x &minus; &xi;)<span class="sp">r1</span> &phi;<span class="su">1</span> by H, a circuit of the point
+&xi; changes K into</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">K&prime; =</td> <td>1</td>
+<td rowspan="2"><span class="f150">[</span>e<span class="sp">2&pi;ir1</span> (x &minus; &xi;)<span class="sp">r1</span> &psi;<span class="su">1</span> + e<span class="sp">2&pi;ir j</span> (x &minus; &xi;)<span class="sp">r1</span>  &phi;<span class="su">1</span> (2&pi;i + log(x &minus; &xi;) )<span class="f150">]</span> = &mu;<span class="su">1</span>K + H.</td></tr>
+<tr><td class="denom">2&pi;i&mu;<span class="su">1</span></td></tr></table>
+
+<p class="noind">A similar artifice suggests itself when three of the roots of the determinantal
+equation are the same, and so on. We are thus led to the
+result, which is justified by an examination of the algebraic conditions,
+that whatever may be the circumstances as to the roots of the
+determinantal equation, n integrals exist, breaking up into
+batches, the values of the constituents H<span class="su">1</span>, H<span class="su">2</span>, ... of a batch after
+circuit about x = &xi; being H<span class="su">1</span>&prime; = &mu;<span class="su">1</span>H<span class="su">1</span>, H<span class="su">2</span>&prime; = &mu;<span class="su">1</span>H<span class="su">2</span> + H<span class="su">1</span>, H<span class="su">3</span>&prime; = &mu;<span class="su">1</span>H<span class="su">3</span> + H<span class="su">2</span>,
+and so on. And this is found to lead to the forms (x &minus; &xi;)<span class="sp">r1</span>&phi;<span class="su">1</span>,
+(x &minus; &xi;)<span class="sp">r1</span> [&psi;<span class="su">1</span> + &phi;<span class="su">1</span> log (x &minus; &xi;)],
+(x &minus; &xi;)<span class="sp">r1</span> [&chi;<span class="su">1</span> + &chi;<span class="su">2</span> log (x &minus; &xi;) + &phi;<span class="su">1</span>(log(x &minus; &xi;) )²],
+and so on. Here each of &phi;<span class="su">1</span>, &psi;<span class="su">1</span>, &chi;<span class="su">1</span>, &chi;<span class="su">2</span>, ... is a series of positive
+and negative integral powers of x &minus; &xi; in which the number of negative
+powers may be infinite.</p>
+
+<p>It appears natural enough now to inquire whether, under proper
+conditions for the forms of the rational functions a<span class="su">1</span>, ... a<span class="su">n</span>, it may
+be possible to ensure that in each of the series &phi;<span class="su">1</span>, &psi;<span class="su">1</span>, [chi]<span class="su">1</span>, ...
+the number of negative powers shall be finite. Herein
+<span class="sidenote">Regular equations.</span>
+lies, in fact, the limitation which experience has shown
+to be justified by the completeness of the results obtained. Assuming
+n integrals in which in each of &phi;<span class="su">1</span>, &psi;<span class="su">1</span>, &chi;<span class="su">1</span> ... the number of
+negative powers is finite, there is a definite homogeneous linear
+differential equation having these integrals; this is found by
+forming it to have the form</p>
+
+<p class="center">y&prime; <span class="sp">n</span> = (x &minus; &xi;)<span class="sp">&minus;1</span> b<span class="su">1</span>y&prime; <span class="sp">(n&minus;1)</span> + (x &minus; &xi;)<span class="sp">&minus;2</span> b<span class="su">2</span>y&prime; <span class="sp">(n&minus;2)</span> + ... + (x &minus; &xi;)<span class="sp">&minus;n</span> b<span class="su">n</span>y,</p>
+
+<p class="noind">where b<span class="su">1</span>, ... b<span class="su">n</span> are finite for x = &xi;. Conversely, assume the
+equation to have this form. Then on substituting a series of
+the form (x &minus; &xi;)<span class="sp">r</span> [1 + A<span class="su">1</span>(x &minus; &xi;) + A<span class="su">2</span>(x &minus; &xi;)² +  ... ] and equating the
+coefficients of like powers of x &minus; &xi;, it is found that r must be a root of
+an algebraic equation of order n; this equation, which we shall call
+the index equation, can be obtained at once by substituting for y
+only (x &minus; &xi;)<span class="sp">r</span> and replacing each of b<span class="su">1</span>, ... b<span class="su">n</span> by their values at
+x = &xi;; arrange the roots r<span class="su">1</span>, r<span class="su">2</span>, ... of this equation so that the
+real part of r<span class="su">i</span> is equal to, or greater than, the real part of r<span class="su">i+1</span>,
+and take r equal to r<span class="su">1</span>; it is found that the coefficients A<span class="su">1</span>, A<span class="su">2</span> ...
+are uniquely determinate, and that the series converges within a
+circle about x = &xi; which includes no other of the points at which
+the rational functions a<span class="su">1</span> ... a<span class="su">n</span> become infinite. We have thus a
+solution H<span class="su">1</span> = (x &minus; &xi;)<span class="sp">r1</span>&phi;<span class="su">1</span> of the differential equation. If we now
+substitute in the equation y = H<span class="su">1</span>&int;&eta;dx, it is found to reduce to an
+equation of order n &minus; 1 for &eta; of the form</p>
+
+<p class="center">&eta;&prime; <span class="sp">(n&minus;1)</span> = (x &minus; &xi;)<span class="sp">&minus;1</span> c<span class="su">1</span>&eta;&prime; <span class="sp">(n&minus;2)</span> + ... + (x &minus; &xi;)<span class="sp">(n&minus;1)</span> c<span class="su">n&minus;1</span>&eta;,</p>
+
+<p class="noind">where c<span class="su">1</span>, ... c<span class="su">n&minus;1</span> are not infinite at x = &xi;. To this equation
+precisely similar reasoning can then be applied; its index equation
+has in fact the roots r<span class="su">2</span> &minus; r<span class="su">1</span> &minus; 1, ..., r<span class="su">n</span> &minus; r<span class="su">1</span> &minus; 1; if r<span class="su">2</span> &minus; r<span class="su">1</span> be zero,
+the integral (x &minus; &xi;)<span class="sp">&minus;1</span>&psi;<span class="su">1</span> of the &eta; equation will give an integral of the
+original equation containing log (x &minus; &xi;); if r<span class="su">2</span> &minus; r<span class="su">1</span> be an integer, and
+therefore a negative integer, the same will be true, unless in &psi;<span class="su">1</span> the
+term in (x &minus; &xi;)<span class="sp">r1 &minus; r2</span> be absent; if neither of these arise, the original
+equation will have an integral (x &minus; &xi;)<span class="sp">r2</span>&phi;<span class="su">2</span>. The &eta; equation can now,
+by means of the one integral of it belonging to the index r<span class="su">2</span> &minus; r<span class="su">1</span> &minus; 1,
+be similarly reduced to one of order n &minus; 2, and so on. The result will
+be that stated above. We shall say that an equation of the form in
+question is <i>regular</i> about x = &xi;.</p>
+
+<p>We may examine in this way the behaviour of the integrals at
+all the points at which any one of the rational functions a<span class="su">1</span> ... a<span class="su">n</span>
+becomes infinite; in general we must expect that beside
+these the value x = &infin; will be a singular point for the
+<span class="sidenote">Fuchsian equations.</span>
+solutions of the differential equation. To test this we
+put x = 1/t throughout, and examine as before at t = 0. For instance,
+the ordinary linear equation with constant coefficients has no singular
+point for finite values of x; at x = &infin; it has a singular point and is not
+regular; or again, Bessel&rsquo;s equation x²y&Prime; + xy&prime; + (x² &minus; n²)y = 0 is
+regular about x = 0, but not about x = &infin;. An equation regular at all
+the finite singularities and also at x = &infin; is called a Fuchsian equation.
+We proceed to examine particularly the case of an equation of the
+second order</p>
+
+<p class="center">y&Prime; + ay&prime; + by = 0.</p>
+
+<p class="noind">Putting x = 1/t, it becomes</p>
+
+<p class="center">d²y/dt² + (2t<span class="sp">&minus;1</span> &minus; at<span class="sp">&minus;2</span>) dy/dt + bt<span class="sp">&minus;4</span> y = 0,</p>
+
+<p class="noind">which is not regular about t = 0 unless 2 &minus; at<span class="sp">&minus;1</span> and bt<span class="sp">&minus;2</span>, that is,
+unless ax and bx² are finite at x = &infin;; which we thus assume; putting
+y = t<span class="sp">r</span>(1 + A<span class="su">1</span>t +  ... ), we find for the index equation at x = &infin;
+the equation r(r &minus; 1) + r(2 &minus; ax)<span class="su">0</span> + (bx²)<span class="su">0</span> = 0. If there be
+<span class="sidenote">Equation of the second order.</span>
+finite singular points at &xi;<span class="su">1</span>, ... &xi;<span class="su">m</span>, where we assume
+m &gt; 1, the cases m = 0, m = 1 being easily dealt with, and
+if &phi;(x) = (x &minus; &xi;<span class="su">1</span>) ... (x &minus; &xi;<span class="su">m</span>), we must have a·&phi;(x)
+and b·[&phi;(x)]² finite for all finite values of x, equal say to the respective
+polynomials &psi;(x) and &theta;(x), of which by the conditions at
+x = &infin; the highest respective orders possible are m &minus; 1 and 2(m &minus; 1).
+The index equation at x = &xi;<span class="su">1</span> is r(r &minus; 1) + r&psi;(&xi;<span class="su">1</span>) / &phi;&prime; (&xi;<span class="su">1</span>) + &theta;(&xi;)<span class="su">1</span> / [&phi;&prime;(&xi;<span class="su">1</span>)]² = 0,
+and if &alpha;<span class="su">1</span>, &beta;<span class="su">1</span> be its roots, we have &alpha;<span class="su">1</span> + &beta;<span class="su">1</span> = 1 &minus; &psi;(&xi;<span class="su">1</span>) / &phi;&prime; (&xi;<span class="su">1</span>) and
+&alpha;<span class="su">1</span>&beta;<span class="su">1</span> = &theta;(&xi;)<span class="su">1</span> / [&phi;&prime;(&xi;<span class="su">1</span>)]². Thus by an elementary theorem of algebra,
+the sum &Sigma;(1 &minus; &alpha;<span class="su">i</span> &minus; &beta;<span class="su">i</span>) / (x &minus; &xi;<span class="su">i</span>), extended to the m finite singular
+points, is equal to &psi;(x) / &phi;(x), and the sum &Sigma;(1 &minus; &alpha;<span class="su">i</span> &minus; &beta;<span class="su">i</span>) is equal to
+the ratio of the coefficients of the highest powers of x in &psi;(x) and
+&phi;(x), and therefore equal to 1 + &alpha; + &beta;, where &alpha;, &beta; are the indices at
+x = &infin;. Further, if (x, 1)<span class="su">m&minus;2</span> denote the integral part of the quotient
+&theta;(x) / &phi;(x), we have &Sigma; &alpha;<span class="su">i</span>&beta;<span class="su">i</span>&phi;&prime; (&xi;<span class="su">i</span>) / (x = &xi;<span class="su">i</span>) equal to &minus;(x, 1)<span class="su">m&minus;2</span> + &theta;(x)/&phi;(x),
+and the coefficient of x<span class="sp">m&minus;2</span> in (x, 1)<span class="su">m&minus;2</span> is &alpha;&beta;. Thus the differential
+equation has the form</p>
+
+<p class="center">y&Prime; + y&prime;&Sigma; (1 &minus; &alpha;<span class="su">i</span> &minus; &beta;<span class="su">i</span>) / (x &minus; &xi;<span class="su">i</span>) + y[(x, 1)<span class="su">m-2</span> + &Sigma; &alpha;<span class="su">i</span>&beta;<span class="su">i</span>&phi;&prime;(&xi;<span class="su">i</span>) / (x &minus; &xi;<span class="su">i</span>)]/&phi;(x) = 0.</p>
+
+<p class="noind">If, however, we make a change in the dependent variable, putting
+y = (x &minus; &xi;<span class="su">1</span>)<span class="sp">&alpha;1</span> ... (x &minus; &xi;<span class="su">m</span>)<span class="sp">&alpha; m&eta;</span>, it is easy to see that the equation
+changes into one having the same singular points about each of
+which it is regular, and that the indices at x = &xi;<span class="su">i</span> become 0 and &beta;<span class="su">i</span> &minus; &alpha;<span class="su">i</span>,
+which we shall denote by &lambda;<span class="su">i</span>, for (x &minus; &xi;<span class="su">i</span>)<span class="sp">&alpha;j</span> can be developed in positive
+integral powers of x &minus; &xi;<span class="su">i</span> about x = &xi;<span class="su">i</span>; by this transformation the
+indices at x = &infin; are changed to</p>
+
+<p class="center">&alpha; + &alpha;<span class="su">1</span> + ... + &alpha;<span class="su">m</span>, &beta; + &beta;<span class="su">1</span> + ... + &beta;<span class="su">m</span></p>
+
+<p class="noind">which we shall denote by &lambda;, &mu;. If we suppose this change to have
+been introduced, and still denote the independent variable by y,
+the equation has the form</p>
+
+<p class="center">y&Prime; + y&prime;&Sigma; (1 &minus; &lambda;<span class="su">i</span>) / (x &minus; &xi;<span class="su">i</span>) + y(x, 1)<span class="su">m&minus;2</span> / &phi;(x) = 0,</p>
+
+<p class="noind">while &lambda; + &mu; + &lambda;<span class="su">1</span> +  ...  + &lambda;<span class="su">m</span> = m &minus; 1. Conversely, it is easy to verify
+that if &lambda;&mu; be the coefficient of x<span class="sp">m&minus;2</span> in (x, 1)<span class="su">m&minus;2</span>, this equation has
+the specified singular points and indices whatever be the other
+coefficients in (x, 1)<span class="su">m&minus;2</span>.</p>
+
+<p>Thus we see that (beside the cases m = 0, m = 1) the &ldquo;Fuchsian
+equation&rdquo; of the second order with <i>two</i> finite singular points is
+distinguished by the fact that it has a definite form
+when the singular points and the indices are assigned.
+<span class="sidenote">Hypergeometric equation.</span>
+In that case, putting (x &minus; &xi;<span class="su">1</span>) / (x &minus; &xi;<span class="su">2</span>)  = t / (t &minus; 1), the singular
+points are transformed to 0, 1, &infin;, and, as is clear, without
+change of indices. Still denoting the independent variable by x,
+the equation then has the form</p>
+
+<p class="center">x(1 &minus; x)y&Prime; + y&prime;[1 &minus; &lambda;<span class="su">1</span> &minus; x(1  + &lambda; + &mu;)] &minus; &lambda;&mu;y = 0,</p>
+
+<p class="noind">which is the ordinary hypergeometric equation. Provided none
+of &lambda;<span class="su">1</span>, &lambda;<span class="su">2</span>, &lambda; &minus; &mu; be zero or integral about x = 0, it has the solutions</p>
+
+<p class="center">F(&lambda;, &mu;, 1 &minus; &lambda;<span class="su">1</span>, x), x<span class="sp">&lambda;1</span> F(&lambda; + &lambda;<span class="su">1</span>, &mu; + &lambda;<span class="su">1</span>, 1 + &lambda;<span class="su">1</span>, x);</p>
+
+<p class="noind">about x = 1 it has the solutions</p>
+
+<p class="center">F(&lambda;, &mu;, 1 &minus; &lambda;<span class="su">2</span>, 1 &minus; x), (1 &minus; x)<span class="sp">&lambda;2</span> F(&lambda; + &lambda;<span class="su">2</span>, &mu; + &lambda;<span class="su">2</span>, 1 + &lambda;<span class="su">2</span>, 1 &minus; x),</p>
+
+<p class="noind">where &lambda; + &mu; + &lambda;<span class="su">1</span> + &lambda;<span class="su">2</span> = 1; about x = &infin; it has the solutions</p>
+
+<p class="center">x<span class="sp">&minus;&lambda;</span> F(&lambda;, &lambda; + &lambda;<span class="su">1</span>, &lambda; &minus; &mu; + 1, x<span class="sp">&minus;1</span>), x<span class="sp">&minus;&mu;</span> F(&mu;, &mu; + &lambda;<span class="su">1</span>, &mu; &minus; &lambda; + 1, x<span class="sp">&minus;1</span>),</p>
+
+<p class="noind">where F(&alpha;, &beta;, &gamma;, x) is the series</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">1 +</td> <td>&alpha;&beta;x</td>
+<td rowspan="2">+</td> <td>&alpha;(&alpha; + 1)&beta;(&beta; + 1)x²</td> <td rowspan="2">...,</td></tr>
+<tr><td class="denom">&gamma;</td> <td class="denom">1·2·&gamma;(&gamma; + 1)</td></tr></table>
+
+<p class="noind">which converges when |x| &lt; 1, whatever &alpha;, &beta;, &gamma; may be, converges
+for all values of x for which |x| = 1 provided the real part of &gamma; &minus; &alpha; &minus; &beta; &lt; 0
+algebraically, and converges for all these values except x = 1 provided
+the real part of &gamma; &minus; &alpha; &minus; &beta; &gt; &minus;1 algebraically.</p>
+
+<p>In accordance with our general theory, logarithms are to be expected
+in the solution when one of &lambda;<span class="su">1</span>, &lambda;<span class="su">2</span>, &lambda; &minus; &mu; is zero or integral.
+Indeed when &lambda;<span class="su">1</span> is a negative integer, not zero, the second solution
+about x = 0 would contain vanishing factors in the denominators
+of its coefficients; in case &lambda; or &mu; be one of the positive integers
+1, 2, ... (&minus;&lambda;<span class="su">1</span>), vanishing factors occur also in the numerators;
+and then, in fact, the second solution about x = 0 becomes x<span class="sp">&lambda;1</span> times
+an integral polynomial of degree (&minus;&lambda;<span class="su">1</span>) &minus; &lambda; or of degree (&minus;&lambda;<span class="su">1</span>) &minus; &mu;.
+But when &lambda;<span class="su">1</span> is a negative integer including zero, and neither &lambda; nor &mu;
+is one of the positive integers 1, 2 ... (&minus;&lambda;<span class="su">1</span>), the second solution
+about x = 0 involves a term having the factor log x. When &lambda;<span class="su">1</span> is a
+positive integer, not zero, the second solution about x = 0 persists as
+a solution, in accordance with the order of arrangement of the roots
+of the index equation in our theory; the first solution is then
+replaced by an integral polynomial of degree -&lambda; or &minus;&mu;<span class="su">1</span>, when &lambda; or &mu;
+is one of the negative integers 0, &minus;1, &minus;2, ..., 1 &minus; &lambda;<span class="su">1</span>, but otherwise
+contains a logarithm. Similarly for the solutions about x = 1 or
+x = &infin;; it will be seen below how the results are deducible from
+those for x = 0.</p>
+
+<p>Denote now the solutions about x = 0 by u<span class="su">1</span>, u<span class="su">2</span>; those about x = 1
+by v<span class="su">1</span>, v<span class="su">2</span>; and those about x = &infin; by w<span class="su">1</span>, w<span class="su">2</span>; in the region (S<span class="su">0</span>S<span class="su">1</span>)
+common to the circles S<span class="su">0</span>, S<span class="su">1</span> of radius 1 whose centres
+are the points x = 0, x = 1, all the first four are valid,
+<span class="sidenote">March of the Integral.</span>
+and there exist equations u<span class="su">1</span> =Av<span class="su">1</span> + Bv<span class="su">2</span>, u<span class="su">2</span> = Cv<span class="su">1</span> + Dv<span class="su">2</span>
+where A, B, C, D are constants; in the region (S<span class="su">1</span>S)
+lying inside the circle S<span class="su">1</span> and outside the circle S<span class="su">0</span>, those that are
+valid are v<span class="su">1</span>, v<span class="su">2</span>, w<span class="su">1</span>, w<span class="su">2</span>, and there exist equations v<span class="su">1</span> = Pw<span class="su">1</span> + Qw<span class="su">2</span>,
+v<span class="su">2</span> = Rw<span class="su">1</span> + Tw<span class="su">2</span>, where P, Q, R, T are constants; thus considering
+any integral whose expression within the circle S<span class="su">0</span> is au<span class="su">1</span> + bu<span class="su">2</span>, where
+a, b are constants, the same integral will be represented within the
+circle S<span class="su">1</span> by (aA + bC)v<span class="su">1</span> + (aB + bD)v<span class="su">2</span>, and outside these circles will be
+represented by</p>
+
+<p class="center">[aA + bC)P + (aB + bD)R]w<span class="su">1</span> + [(aA + bC)Q + (aB + bD)T]w<span class="su">2</span>.</p>
+
+<p class="noind">A single-valued branch of such integral can be obtained by making
+a barrier in the plane joining &infin; to 0 and 1 to &infin;; for instance, by
+excluding the consideration of real negative values of x and of real
+<span class="pagenum"><a name="page237" id="page237"></a>237</span>
+positive values greater than 1, and defining the phase of x and x &minus; 1
+for real values between 0 and 1 as respectively 0 and &pi;.</p>
+
+<p>We can form the Fuchsian equation of the second order with
+three arbitrary singular points &xi;<span class="su">1</span>, &xi;<span class="su">2</span>, &xi;<span class="su">3</span>, and no singular point
+at x = &infin;, and with respective indices &alpha;<span class="su">1</span>, &beta;<span class="su">1</span>, &alpha;<span class="su">2</span>, &beta;<span class="su">2</span>, &alpha;<span class="su">3</span>, &beta;<span class="su">3</span> such
+that &alpha;<span class="su">1</span> + &beta;<span class="su">1</span> + &alpha;<span class="su">2</span> + &beta;<span class="su">2</span> + &alpha;<span class="su">3</span> + &beta;<span class="su">3</span> = 1. This equation can then be
+<span class="sidenote">Transformation of the equation into itself.</span>
+transformed into the hypergeometric equation in 24 ways;
+for out of &xi;<span class="su">1</span>, &xi;<span class="su">2</span>, &xi;<span class="su">3</span> we can in six ways choose two, say
+&xi;<span class="su">1</span>, &xi;<span class="su">2</span>, which are to be transformed respectively into
+0 and 1, by (x &minus; &xi;<span class="su">1</span>)/(x &minus; &xi;<span class="su">2</span>) = t(t &minus; 1); and then there
+are four possible transformations of the dependent variable which
+will reduce one of the indices at t = 0 to zero and one of the indices
+at t = 1 also to zero, namely, we may reduce either &alpha;<span class="su">1</span> or &beta;<span class="su">1</span> at t = 0,
+and simultaneously either &alpha;<span class="su">2</span> or &beta;<span class="su">2</span> at t = 1. Thus the hypergeometric
+equation itself can be transformed into itself in 24 ways,
+and from the expression F(&lambda;, &mu;, 1 &minus; &lambda;<span class="su">1</span>, x) which satisfies it follow 23
+other forms of solution; they involve four series in each of the arguments,
+x, x &minus; 1, 1/x, 1/(1 &minus; x), (x &minus; 1)/x, x/(x &minus; 1). Five of the 23
+solutions agree with the fundamental solutions already described
+about x = 0, x = 1, x = &infin;; and from the principles by which these
+were obtained it is immediately clear that the 24 forms are, in value,
+equal in fours.</p>
+
+<p>The quarter periods K, K&prime; of Jacobi&rsquo;s theory of elliptic functions,
+of which K = &int;<span class="sp1">&pi;/2</span><span class="su1">0</span> (1 &minus; h sin ²&theta;)<span class="sp">&minus;½</span>d&theta;, and K&prime; is the same function of
+1-h, can easily be proved to be the solutions of a hypergeometric
+<span class="sidenote">Inversion. Modular functions.</span>
+equation of which h is the independent variable. When K, K&prime; are
+regarded as defined in terms of h by the differential
+equation, the ratio K&prime;/K is an infinitely many valued
+function of h. But it is remarkable that Jacobi&rsquo;s own
+theory of theta functions leads to an expression for h in
+terms of K&prime;/K (see <span class="sc"><a href="#artlinks">Function</a></span>) in terms of single-valued functions.
+We may then attempt to investigate, in general, in what cases the
+independent variable x of a hypergeometric equation is a single-valued
+function of the ratio s of two independent integrals of the equation.
+The same inquiry is suggested by the problem of ascertaining in what
+cases the hypergeometric series F(&alpha;, &beta;, &gamma;, x) is the expansion of an
+algebraic (irrational) function of x. In order to explain the meaning
+of the question, suppose that the plane of x is divided along the real
+axis from -&infin; to 0 and from 1 to +&infin;, and, supposing logarithms
+not to enter about x = 0, choose two quite definite integrals y<span class="su">1</span>, y<span class="su">2</span> of
+the equation, say</p>
+
+<p class="center">y<span class="su">1</span> = F(&lambda;, &mu;, 1 &minus; &lambda;<span class="su">1</span>, x),
+y<span class="su">2</span> = x<span class="sp">&lambda;1</span> F(&lambda; + &lambda;<span class="su">1</span>, &mu; + &lambda;<span class="su">1</span>, 1 + &lambda;<span class="su">1</span>, x),</p>
+
+<p class="noind">with the condition that the phase of x is zero when x is real
+and between 0 and 1. Then the value of &sigmaf; = y<span class="su">2</span>/y<span class="su">1</span> is definite for all
+values of x in the divided plane, &sigmaf; being a single-valued monogenic
+branch of an analytical function existing and without singularities
+all over this region. If, now, the values of &sigmaf; that so arise be plotted
+on to another plane, a value p + iq of &sigma; being represented by a point
+(p, q) of this &sigmaf;-plane, and the value of x from which it arose being
+mentally associated with this point of the &sigma;-plane, these points will
+fill a connected region therein, with a continuous boundary formed
+of four portions corresponding to the two sides of the two barriers
+of the x-plane. The question is then, firstly, whether the same value
+of s can arise for two different values of x, that is, whether the same
+point (p, q) of the &sigmaf;-plane can arise twice, or in other words, whether
+the region of the &sigmaf;-plane overlaps itself or not. Supposing this is not
+so, a second part of the question presents itself. If in the x-plane the
+barrier joining -&infin; to 0 be momentarily removed, and x describe a
+small circle with centre at x = 0 starting from a point x = &minus;h &minus; ik,
+where h, k are small, real, and positive and coming back to this point,
+the original value s at this point will be changed to a value &sigma;, which in
+the original case did not arise for this value of x, and possibly not
+at all. If, now, after restoring the barrier the values arising by
+continuation from &sigma; be similarly plotted on the &sigmaf;-plane, we shall
+again obtain a region which, while not overlapping itself, may quite
+possibly overlap the former region. In that case two values of x
+would arise for the same value or values of the quotient y<span class="su">2</span>/y<span class="su">1</span>, arising
+from two different branches of this quotient. We shall understand
+then, by the condition that x is to be a single-valued function of x,
+that the region in the &sigmaf;-plane corresponding to any branch is not to
+overlap itself, and that no two of the regions corresponding to the
+different branches are to overlap. Now in describing the circle
+about x = 0 from x =  &minus;h &minus; ik to &minus;h + ik, where h is small and k
+evanescent,</p>
+
+<p class="center">&sigmaf; = x<span class="sp">&lambda;1</span> F(&lambda; + &lambda;<span class="su">1</span>, &mu; + &lambda;<span class="su">1</span>, 1 + &lambda;<span class="su">1</span>, x) / F(&lambda;, &mu;, 1 &minus; &lambda;<span class="su">1</span>, x)</p>
+
+<p class="noind">is changed to &sigma; = &sigmaf;e<span class="sp">2&pi;i&lambda;1</span>. Thus the two portions of boundary of the
+s-region corresponding to the two sides of the barrier (&minus;&infin;, 0) meet
+(at &sigmaf; = 0 if the real part of &lambda;<span class="su">1</span> be positive) at an angle 2&pi;L<span class="su">1</span>, where L<span class="su">1</span>
+is the absolute value of the real part of &lambda;<span class="su">1</span>; the same is true for the
+&sigma;-region representing the branch &sigma;. The condition that the s-region
+shall not overlap itself requires, then, L<span class="su">1</span> = 1. But, further, we may
+form an infinite number of branches &sigma; = &sigmaf;e<span class="sp">2&pi;i&lambda;1</span>, &sigma;<span class="su">1</span> = e<span class="sp">2&pi;i&lambda;1</span>, ...
+in the same way, and the corresponding regions in the plane upon which
+y<span class="su">2</span>/y<span class="su">1</span> is represented will have a common point and each have an
+angle 2&pi;L<span class="su">1</span>; if neither overlaps the preceding, it will happen, if L<span class="su">1</span>
+is not zero, that at length one is reached overlapping the first, unless
+for some positive integer &alpha; we have 2&pi;&alpha;L<span class="su">1</span> = 2&pi;, in other words
+L<span class="su">1</span> = 1/&alpha;. If this be so, the branch &sigma;<span class="su">&alpha;&minus;1</span> = &sigmaf;e<span class="sp">2&pi;i&alpha;&lambda;1</span> will be represented
+by a region having the angle at the common point common with the
+region for the branch &sigmaf;; but not altogether coinciding with this last
+region unless &lambda;<span class="su">1</span> be real, and therefore = ±1/&alpha;; then there is only
+a finite number, &alpha;, of branches obtainable in this way by crossing
+the barrier (&minus;&infin;, 0). In precisely the same way, if we had begun
+by taking the quotient</p>
+
+<p class="center">&sigmaf;&prime; = (x &minus; 1)<span class="sp">&lambda;2</span> F(&lambda; + &lambda;<span class="su">2</span>, &mu; + &lambda;<span class="su">2</span>, 1 + &lambda;<span class="su">2</span>, 1 &minus; x) / F(&lambda;, &mu;, 1 &minus; &lambda;<span class="su">2</span>, 1 &minus; x)</p>
+
+<p class="noind">of the two solutions about x = 1, we should have found that x is not
+a single-valued function of &sigmaf;&prime; unless &lambda;<span class="su">2</span> is the inverse of an integer, or
+is zero; as &sigmaf;&prime; is of the form (A<span class="su">&sigma;</span> + B)/(C<span class="su">&sigmaf;</span> + D), A, B, C, D constants,
+the same is true in our case; equally, by considering the integrals
+about x = &infin; we find, as a third condition necessary in order that x
+may be a single-valued function of &sigmaf;, that &lambda; &minus; &mu; must be the inverse
+of an integer or be zero. These three differences of the indices,
+namely, &lambda;<span class="su">1</span>, &lambda;<span class="su">2</span>, &lambda; &minus; &mu;, are the quantities which enter in the differential
+equation satisfied by x as a function of &sigmaf;, which is easily found to be</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&minus;</td> <td>x<span class="su">111</span></td>
+<td rowspan="2">+</td> <td>3x²<span class="su">11</span></td>
+<td rowspan="2">½(h &minus; h<span class="su">1</span> &minus; h<span class="su">2</span>)x<span class="sp">&minus;1</span>(x &minus; 1)<span class="sp">&minus;1</span> + ½h<span class="su">1</span>x<span class="sp">&minus;2</span> + ½h<span class="su">2</span>(x &minus; 1)<span class="sp">&minus;2</span>,</td></tr>
+<tr><td class="denom">x<span class="su">1</span>³</td> <td class="denom">2x<span class="su">1</span><span class="sp">4</span></td></tr></table>
+
+<p class="noind">where x<span class="su">1</span> = dx/d&sigmaf;, &amp;c.; and h<span class="su">1</span> = 1 &minus; y<span class="su">1</span>², h<span class="su">2</span> = 1 &minus; &lambda;<span class="su">2</span>², h<span class="su">3</span> = 1 &minus; (&lambda; &minus; &mu;)². Into
+the converse question whether the three conditions are sufficient
+to ensure (1) that the &sigma; region corresponding to any branch does
+not overlap itself, (2) that no two such regions overlap, we have no
+space to enter. The second question clearly requires the inquiry
+whether the group (that is, the monodromy group) of the differential
+equation is properly discontinuous. (See <span class="sc"><a href="#artlinks">Groups, Theory of</a></span>.)</p>
+
+<p>The foregoing account will give an idea of the nature of the
+function theories of differential equations; it appears essential
+not to exclude some explanation of a theory intimately related
+both to such theories and to transformation theories, which is a
+generalization of Galois&rsquo;s theory of algebraic equations. We deal
+only with the application to homogeneous linear differential
+equations.</p>
+
+<p>In general a function of variables x<span class="su">1</span>, x<span class="su">2</span> ... is said to be rational
+when it can be formed from them and the integers 1, 2, 3, ... by a
+finite number of additions, subtractions, multiplications
+and divisions. We generalize this definition. Assume that
+<span class="sidenote">Rationality group of a linear equation.</span>
+we have assigned a fundamental series of quantities and
+functions of x, in which x itself is included, such that all
+quantities formed by a finite number of additions, subtractions,
+multiplications, divisions <i>and differentiations in regard to x</i>,
+of the terms of this series, are themselves members of this series.
+Then the quantities of this series, and only these, are called <i>rational</i>.
+By a rational function of quantities p, q, r, ... is meant a function
+formed from them and any of the fundamental rational quantities
+by a finite number of the five fundamental operations. Thus it is a
+function which would be called, simply, rational if the fundamental
+series were widened by the addition to it of the quantities p, q, r, ...
+and those derivable from them by the five fundamental operations.
+A rational ordinary differential equation, with x as independent and
+y as dependent variable, is then one which equates to zero a rational
+function of y, the order k of the differential equation being that of the
+highest differential coefficient y<span class="sp">(k)</span> which enters; only such equations
+are here discussed. Such an equation P = 0 is called <i>irreducible</i> when,
+firstly, being arranged as an integral polynomial in y<span class="sp">(k)</span>, this polynomial
+<span class="sidenote">Irreducibility of a rational equation.</span>
+is not the product of other polynomials in y<span class="sp">(k)</span> also
+of rational form; and, secondly, the equation has no
+solution satisfying also a rational equation of lower order.
+From this it follows that if an irreducible equation P = 0
+have one solution satisfying another rational equation Q = 0
+of the same or higher order, then all the solutions of P = 0 also satisfy
+Q = 0. For from the equation P = 0 we can by differentiation express
+y<span class="sp">(k+1)</span>, y<span class="sp">(k+2)</span>, ... in terms of x, y, y<span class="sp">(1)</span>, ... , y<span class="sp">(k)</span>, and so put the
+function Q rationally in terms of these quantities only. It is
+sufficient, then, to prove the result when the equation Q = 0 is of the
+same order as P = 0. Let both the equations be arranged as integral
+polynomials in y<span class="sp">(k)</span>; their algebraic eliminant in regard to y<span class="sp">(k)</span> must
+then vanish identically, for they are known to have one common
+solution not satisfying an equation of lower order; thus the equation
+P = 0 involves Q = 0 for all solutions of P = 0.</p>
+
+<p>Now let y<span class="sp">(n)</span> = a<span class="su">1</span>y<span class="sp">(n&minus;1)</span> + ... + a<span class="su">n</span>y be a given rational homogeneous
+linear differential equation; let y<span class="su">1</span>, ... y<span class="su">n</span> be n particular
+functions of x, unconnected by any equation with constant coefficients
+of the form c<span class="su">1</span>y<span class="su">1</span> + ... + c<span class="su">n</span>y<span class="su">n</span> = 0, all satisfying
+<span class="sidenote">The variant function for a linear equation.</span>
+the differential equation; let &eta;<span class="su">1</span>, ... &eta;<span class="su">n</span> be linear functions
+of y<span class="su">1</span>, ... y<span class="su">n</span>, say &eta;<span class="su">i</span> = A<span class="su">i1</span>y<span class="su">1</span> + ... + A<span class="su">in</span>y<span class="su">n</span>, where the
+constant coefficients A<span class="su">ij</span> have a non-vanishing determinant;
+write (&eta;) = A(y), these being the equations of a
+general linear homogeneous group whose transformations
+may be denoted by A, B, .... We desire to form a
+rational function &phi;(&eta;), or say &phi;(A(y)), of &eta;<span class="su">1</span>, ... &eta;, in which the
+&eta;² constants A<span class="su">ij</span> shall all be essential, and not reduce effectively to a
+fewer number, as they would, for instance, if the y<span class="su">1</span>, ... y<span class="su">n</span> were
+connected by a linear equation with constant coefficients. Such a
+function is in fact given, if the solutions y<span class="su">1</span>, ... y<span class="su">n</span> be developable
+<span class="pagenum"><a name="page238" id="page238"></a>238</span>
+in positive integral powers about x = a, by &phi;(&eta;) = &eta;<span class="su">1</span> + (x &minus; a)<span class="sp">n</span> &eta;<span class="su">2</span> + ... +
+(x &minus; a)<span class="sp">(n&minus;1)n</span> &eta;<span class="su">n</span>. Such a function, V, we call a <i>variant</i>.</p>
+
+<p>Then differentiating V in regard to x, and replacing &eta;<span class="su">i</span><span class="sp">(n)</span> by its
+value a<span class="su">1</span>&eta;<span class="sp">(n&minus;1)</span> + ... + a<span class="su">n</span>&eta;, we can arrange dV/dx, and similarly each
+of d²/dx² ... d<span class="sp">N</span>V/dx<span class="sp">N</span>, where N = n², as a linear function of
+the N quantities &eta;<span class="su">1</span>, ... &eta;<span class="su">n</span>, ... &eta;<span class="su">1</span><span class="sp">(n&minus;1)</span>, ... &eta;<span class="su">n</span><span class="sp">(n&minus;1)</span>, and
+<span class="sidenote">The resolvent eqution.</span>
+thence by elimination obtain a linear differential equation
+for V of order N with rational coefficients. This we
+denote by F = 0. Further, each of &eta;<span class="su">1</span> ... &eta;<span class="su">n</span> is expressible
+as a linear function of V, dV/dx, ... d<span class="sp">N&minus;1</span>V / dx<span class="sp">N&minus;1</span>, with rational coefficients
+not involving any of the n² coefficients A<span class="su">ij</span>, since otherwise
+V would satisfy a linear equation of order less than N, which is
+impossible, as it involves (linearly) the n² arbitrary coefficients A<span class="su">ij</span>,
+which would not enter into the coefficients of the supposed equation.
+In particular, y<span class="su">1</span> ,.. y<span class="su">n</span> are expressible rationally as linear functions
+of &omega;, d&omega;/dx, ... d<span class="sp">N&minus;1</span>&omega; / dx<span class="sp">N&minus;1</span>, where &omega; is the particular function
+&phi;(y). Any solution W of the equation F = 0 is derivable from
+functions &zeta;<span class="su">1</span>, ... &zeta;<span class="su">n</span>, which are linear functions of y<span class="su">1</span>, ... y<span class="su">n</span>, just
+as V was derived from &eta;<span class="su">1</span>, ... &eta;<span class="su">n</span>; but it does not follow that these
+functions &zeta;<span class="su">i</span>, ... &zeta;<span class="su">n</span> are obtained from y<span class="su">1</span>, ... y<span class="su">n</span> by a transformation
+of the linear group A, B, ... ; for it may happen that the
+determinant d(&zeta;<span class="su">1</span>, ... &zeta;<span class="su">n</span>) / (dy<span class="su">1</span>, ... y<span class="su">n</span>) is zero. In that case
+&zeta;<span class="su">1</span>, ... &zeta;<span class="su">n</span> may be called a singular set, and W a singular solution; it
+satisfies an equation of lower than the N-th order. But every solution
+V, W, ordinary or singular, of the equation F = 0, is expressible
+rationally in terms of &omega;, d&omega; / dx, ... d<span class="sp">N&minus;1</span>&omega; / dx<span class="sp">N&minus;1</span>; we shall write,
+simply, V = r(&omega;). Consider now the rational irreducible equation
+of lowest order, not necessarily a linear equation, which is satisfied
+by &omega;; as y<span class="su">1</span>, ... y<span class="su">n</span> are particular functions, it may quite well
+be of order less than N; we call it the <i>resolvent equation</i>, suppose it
+of order p, and denote it by &gamma;(v). Upon it the whole theory turns.
+In the first place, as &gamma;(v)  = 0 is satisfied by the solution &omega; of F  = 0, all
+the solutions of &gamma;(v) are solutions F = 0, and are therefore rationally
+expressible by &omega;; any one may then be denoted by r(&omega;). If this
+solution of F = 0 be not singular, it corresponds to a transformation
+A of the linear group (A, B, ...), effected upon y<span class="su">1</span>, ... y<span class="su">n</span>. The
+coefficients A<span class="su">ij</span> of this transformation follow from the expressions
+before mentioned for &eta;<span class="su">1</span> ... &eta;<span class="su">n</span> in terms of V, dV/dx, d²V/dx², ... by
+substituting V = r(&omega;); thus they depend on the p arbitrary parameters
+which enter into the general expression for the integral of
+the equation &gamma;(v)  = 0. Without going into further details, it is then
+clear enough that the resolvent equation, being irreducible and such
+that any solution is expressible rationally, with p parameters, in
+terms of the solution &omega;, enables us to define a linear homogeneous
+group of transformations of y<span class="su">1</span> ... y<span class="su">n</span> depending on p parameters;
+and every operation of this (continuous) group corresponds to a
+rational transformation of the solution of the resolvent equation.
+This is the group called the <i>rationality group</i>, or the <i>group of transformations</i>
+of the original homogeneous linear differential equation.</p>
+
+<p>The group must not be confounded with a subgroup of itself,
+the <i>monodromy group</i> of the equation, often called simply the group
+of the equation, which is a set of transformations, not depending
+on arbitrary variable parameters, arising for one particular
+fundamental set of solutions of the linear equation (see <span class="sc"><a href="#artlinks">Groups,
+Theory of</a></span>).</p>
+
+<p>The importance of the rationality group consists in three propositions.
+(1) Any rational function of y<span class="su">1</span>, ... y<span class="su">n</span> which is unaltered in
+value by the transformations of the group can be written
+in rational form. (2) If any rational function be changed
+<span class="sidenote">The fundamental theorem in regard to the rationality group.</span>
+in form, becoming a rational function of y<span class="su">1</span>, ... y<span class="su">n</span>, a
+transformation of the group applied to its new form will
+leave its value unaltered. (3) Any homogeneous linear
+transformation leaving unaltered the value of every
+rational function of y<span class="su">1</span>, ... y<span class="su">n</span> which has a rational value,
+belongs to the group. It follows from these that any
+group of linear homogeneous transformations having the
+properties (1) (2) is identical with the group in question. It is clear
+that with these properties the group must be of the greatest importance
+in attempting to discover what functions of x must be regarded as
+rational in order that the values of y<span class="su">1</span> ... y<span class="su">n</span> may be expressed.
+And this is the problem of solving the equation from another point
+of view.</p>
+
+<div class="condensed">
+<p><span class="sc">Literature.</span>&mdash;(&alpha;) <i>Formal or Transformation Theories for Equations
+of the First Order</i>:&mdash;E. Goursat, <i>Leçons sur l&rsquo;intégration des équations
+aux dérivées partielles du premier ordre</i> (Paris, 1891); E. v.
+Weber, <i>Vorlesungen über das Pfaff&rsquo;sche Problem und die Theorie der
+partiellen Differentialgleichungen erster Ordnung</i> (Leipzig, 1900);
+S. Lie und G. Scheffers, <i>Geometrie der Berührungstransformationen</i>,
+Bd. i. (Leipzig, 1896); Forsyth, <i>Theory of Differential Equations,
+Part i., Exact Equations and Pfaff&rsquo;s Problem</i> (Cambridge, 1890);
+S. Lie, &ldquo;Allgemeine Untersuchungen über Differentialgleichungen, die
+eine continuirliche endliche Gruppe gestatten&rdquo; (Memoir), <i>Mathem.
+Annal.</i>xxv. (1885), pp. 71-151; S. Lie und G. Scheffers, <i>Vorlesungen
+über Differentialgleichungen mit bekannten infinitesimalen Transformationen</i>
+(Leipzig, 1891). A very full bibliography is given in the book
+of E. v. Weber referred to; those here named are perhaps sufficiently
+representative of modern works. Of classical works may be named:
+Jacobi, <i>Vorlesungen über Dynamik</i> (von A. Clebsch, Berlin, 1866);
+<i>Werke, Supplementband</i>; G Monge, <i>Application de l&rsquo;analyse à la
+géométrie</i> (par M. Liouville, Paris, 1850); J. L. Lagrange, <i>Leçons
+sur le calcul des fonctions</i> (Paris, 1806), and <i>Théorie des fonctions
+analytiques</i> (Paris, Prairial, an V); G. Boole, <i>A Treatise on Differential
+Equations</i> (London, 1859); and <i>Supplementary Volume</i>
+(London, 1865); Darboux, <i>Leçons sur la théorie générale des
+surfaces</i>, tt. i.-iv. (Paris, 1887-1896); S. Lie, <i>Théorie der transformationsgruppen</i>
+ii. (on Contact Transformations) (Leipzig, 1890).</p>
+
+<p>(&beta;) <i>Quantitative or Function Theories for Linear Equations</i>:&mdash;C.
+Jordan, <i>Cours d&rsquo;analyse</i>, t. iii. (Paris, 1896); E. Picard, <i>Traité
+d&rsquo;analyse</i>, tt. ii. and iii. (Paris, 1893, 1896); Fuchs, <i>Various
+Memoirs, beginning with that in Crelle&rsquo;s Journal</i>, Bd. lxvi. p. 121;
+Riemann, <i>Werke</i>, 2<span class="sp">r</span> Aufl. (1892); Schlesinger, <i>Handbuch der
+Theorie der linearen Differentialgleichungen</i>, Bde. i.-ii. (Leipzig,
+1895-1898); Heffter, <i>Einleitung in die Theorie der linearen Differentialgleichungen
+mit einer unabhängigen Variablen</i> (Leipzig, 1894);
+Klein, <i>Vorlesungen über lineare Differentialgleichungen der zweiten
+Ordnung</i> (Autographed, Göttingen, 1894); and <i>Vorlesungen über
+die hypergeometrische Function</i> (Autographed, Göttingen, 1894);
+Forsyth, <i>Theory of Differential Equations, Linear Equations</i>.</p>
+
+<p>(&gamma;) <i>Rationality Group (of Linear Differential Equations)</i>:&mdash;Picard,
+<i>Traité d&rsquo;Analyse</i>, as above, t. iii.; Vessiot, <i>Annales de
+l&rsquo;École Normale</i>, série III. t. ix. p. 199 (Memoir); S. Lie,
+<i>Transformationsgruppen</i>, as above, iii. A connected account is
+given in Schlesinger, as above, Bd. ii., erstes Theil.</p>
+
+<p>(&delta;) <i>Function Theories of Non-Linear Ordinary Equations</i>:&mdash;Painlevé,
+<i>Leçons sur la théorie analytique des équations différentielles</i>
+(Paris, 1897, Autographed); Forsyth, <i>Theory of Differential Equations,
+Part ii., Ordinary Equations not Linear</i> (two volumes, ii. and iii.)
+(Cambridge, 1900); Königsberger, <i>Lehrbuch der Theorie der Differentialgleichungen</i>
+(Leipzig, 1889); Painlevé, <i>Leçons sur l&rsquo;intégration
+des équations differentielles de la mécanique et applications</i> (Paris,
+1895).</p>
+
+<p>(&epsilon;) <i>Formal Theories of Partial Equations of the Second and Higher
+Orders</i>:&mdash;E. Goursat, <i>Leçons sur l&rsquo;intégration des équations aux
+dérivées partielles du second ordre</i>, tt. i. and ii. (Paris, 1896, 1898);
+Forsyth, <i>Treatise on Differential Equations</i> (London, 1889); and
+<i>Phil. Trans. Roy. Soc.</i> (A.), vol. cxci. (1898), pp. 1-86.</p>
+
+<p>(&zeta;) See also the six extensive articles in the second volume of
+the German <i>Encyclopaedia of Mathematics</i>.</p>
+</div>
+<div class="author">(H. F. Ba.)</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIFFLUGIA<a name="ar83" id="ar83"></a></span> (L. Leclerc), a genus of lobose Rhizopoda, characterized
+by a shell formed of sand granules cemented together;
+these are swallowed by the animal, and during the process of
+bud-fission they pass to the surface of the daughter-bud and
+are cemented there. <i>Centropyxis</i> (Steia) and <i>Lecqueureuxia</i>
+(Schlumberg) differ only in minor points.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIFFRACTION OF LIGHT.<a name="ar84" id="ar84"></a></span>&mdash;1. When light proceeding from
+a small source falls upon an opaque object, a shadow is cast upon
+a screen situated behind the obstacle, and this shadow is found to
+be bordered by alternations of brightness and darkness, known
+as &ldquo;diffraction bands.&rdquo; The phenomena thus presented were
+described by Grimaldi and by Newton. Subsequently T. Young
+showed that in their formation interference plays an important
+part, but the complete explanation was reserved for A. J. Fresnel.
+Later investigations by Fraunhofer, Airy and others have
+greatly widened the field, and under the head of &ldquo;diffraction&rdquo;
+are now usually treated all the effects dependent upon the
+limitation of a beam of light, as well as those which arise from
+irregularities of any kind at surfaces through which it is transmitted,
+or at which it is reflected.</p>
+
+<p>2. <i>Shadows.</i>&mdash;In the infancy of the undulatory theory the
+objection most frequently urged against it was the difficulty of
+explaining the very existence of shadows. Thanks to Fresnel
+and his followers, this department of optics is now precisely the
+one in which the theory has gained its greatest triumphs. The
+principle employed in these investigations is due to C. Huygens,
+and may be thus formulated. If round the origin of waves an
+ideal closed surface be drawn, the whole action of the waves in the
+region beyond may be regarded as due to the motion continually
+propagated across the various elements of this surface. The wave
+motion due to any element of the surface is called a <i>secondary</i>
+wave, and in estimating the total effect regard must be paid to the
+phases as well as the amplitudes of the components. It is usually
+convenient to choose as the surface of resolution a <i>wave-front</i>, <i>i.e.</i>
+a surface at which the primary vibrations are in one phase. Any
+obscurity that may hang over Huygens&rsquo;s principle is due mainly to
+the indefiniteness of thought and expression which we must be
+content to put up with if we wish to avoid pledging ourselves as
+to the character of the vibrations. In the application to sound,
+where we know what we are dealing with, the matter is simple
+enough in principle, although mathematical difficulties would
+often stand in the way of the calculations we might wish to make.
+<span class="pagenum"><a name="page239" id="page239"></a>239</span>
+The ideal surface of resolution may be there regarded as a flexible
+lamina; and we know that, if by forces locally applied every
+element of the lamina be made to move normally to itself exactly
+as the air at that place does, the external aerial motion is fully
+determined. By the principle of superposition the whole effect
+may be found by integration of the partial effects due to each
+element of the surface, the other elements remaining at rest.</p>
+
+<table class="nobctr" style="float: left; width: 220px;" summary="Illustration">
+<tr><td class="figleft1"><img style="width:169px; height:178px" src="images/img239.jpg" alt="" /></td></tr>
+<tr><td class="caption sc">Fig. 1.</td></tr></table>
+
+<p>We will now consider in detail the important case in which uniform
+plane waves are resolved at a surface coincident with a wave-front
+(OQ). We imagine a wave-front divided
+into elementary rings or zones&mdash;often named
+after Huygens, but better after Fresnel&mdash;by
+spheres described round P (the point at
+which the aggregate effect is to be estimated),
+the first sphere, touching the plane at O, with
+a radius equal to PO, and the succeeding
+spheres with radii increasing at each step
+by ½&lambda;. There are thus marked out a series
+of circles, whose radii x are given by
+x² + r² = (r +  ½n&lambda;)², or x² = n&lambda;r nearly; so that
+the rings are at first of nearly equal area.
+Now the effect upon P of each element of the
+plane is proportional to its area; but it
+depends also upon the distance from P, and possibly upon the
+inclination of the secondary ray to the direction of vibration and
+to the wave-front.</p>
+
+<p>The latter question can only be treated in connexion with the
+dynamical theory (see below, § 11); but under all ordinary circumstances
+the result is independent of the precise answer that may be
+given. All that it is necessary to assume is that the effects of the
+successive zones gradually diminish, whether from the increasing
+obliquity of the secondary ray or because (on account of the limitation
+of the region of integration) the zones become at last more and
+more incomplete. The component vibrations at P due to the
+successive zones are thus nearly equal in amplitude and opposite in
+phase (the phase of each corresponding to that of the infinitesimal
+circle midway between the boundaries), and the series which we have
+to sum is one in which the terms are alternately opposite in sign
+and, while at first nearly constant in numerical magnitude, gradually
+diminish to zero. In such a series each term may be regarded as very
+nearly indeed destroyed by the halves of its immediate neighbours,
+and thus the sum of the whole series is represented by half the first
+term, which stands over uncompensated. The question is thus
+reduced to that of finding the effect of the first zone, or central
+circle, of which the area is &pi;&lambda;r.</p>
+
+<p>We have seen that the problem before us is independent of the
+law of the secondary wave as regards obliquity; but the result of
+the integration necessarily involves the law of the intensity and
+phase of a secondary wave as a function of r, the distance from the
+origin. And we may in fact, as was done by A. Smith (<i>Camb. Math.
+Journ.</i>, 1843, 3, p. 46), determine the law of the secondary wave, by
+comparing the result of the integration with that obtained by supposing
+the primary wave to pass on to P without resolution.</p>
+
+<p>Now as to the phase of the secondary wave, it might appear
+natural to suppose that it starts from any point Q with the phase
+of the primary wave, so that on arrival at P, it is retarded by the
+amount corresponding to QP. But a little consideration will prove
+that in that case the series of secondary waves could not reconstitute
+the primary wave. For the aggregate effect of the secondary waves
+is the half of that of the first Fresnel zone, and it is the central
+element only of that zone for which the distance to be travelled is
+equal to r. Let us conceive the zone in question to be divided
+into infinitesimal rings of equal area. The effects due to each of
+these rings are equal in amplitude and of phase ranging uniformly
+over half a complete period. The phase of the resultant is midway
+between those of the extreme elements, that is to say, a quarter of
+a period behind that due to the element at the centre of the circle.
+It is accordingly necessary to suppose that the secondary waves
+start with a phase one-quarter of a period in advance of that of the
+primary wave at the surface of resolution.</p>
+
+<p>Further, it is evident that account must be taken of the variation
+of phase in estimating the magnitude of the effect at P of the first
+zone. The middle element alone contributes without deduction;
+the effect of every other must be found by introduction of a resolving
+factor, equal to cos &theta;, if &theta; represent the difference of phase
+between this element and the resultant. Accordingly, the amplitude
+of the resultant will be less than if all its components had the same
+phase, in the ratio</p>
+
+<table class="math0" summary="math">
+<tr><td class="np" rowspan="2"><span class="f200">&int;</span></td> <td>+½&pi;</td>
+<td rowspan="2">cos &theta;d&theta; : &pi;,</td></tr>
+<tr><td>-½&pi;</td></tr></table>
+
+<p class="noind">or 2 : &pi;. Now 2 area /&pi; = 2&lambda;r; so that, in order to reconcile the
+amplitude of the primary wave (taken as unity) with the half effect
+of the first zone, the amplitude, at distance r, of the secondary wave
+emitted from the element of area dS must be taken to be</p>
+
+<p class="center">dS/&lambda;r &emsp; &emsp; (1).</p>
+
+<p class="noind">By this expression, in conjunction with the quarter-period acceleration
+of phase, the law of the secondary wave is determined.</p>
+
+<p>That the amplitude of the secondary wave should vary as r<span class="sp">-1</span> was
+to be expected from considerations respecting energy; but the
+occurrence of the factor &lambda;<span class="sp">-1</span>, and the acceleration of phase, have
+sometimes been regarded as mysterious. It may be well therefore
+to remember that precisely these laws apply to a secondary wave
+of sound, which can be investigated upon the strictest mechanical
+principles.</p>
+
+<p>The recomposition of the secondary waves may also be treated
+analytically. If the primary wave at O be cos kat, the effect of the
+secondary wave proceeding from the element dS at Q is</p>
+
+<table class="math0" summary="math">
+<tr><td>dS</td> <td rowspan="2">cos k(at &minus; &rho; + ¼&lambda;) = &minus;</td> <td>dS</td>
+<td rowspan="2">sin k(at &minus; &rho;).</td></tr>
+<tr><td class="denom">&lambda;&rho;</td> <td class="denom">&lambda;&rho;</td></tr></table>
+
+<p class="noind">If dS = 2&pi;xdx, we have for the whole effect</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&minus;</td> <td>2&pi;</td>
+<td rowspan="2" class="f200 np">&int;</td> <td>&infin;</td> <td>sin k(at &minus; &rho;)x dx</td>
+<td rowspan="2">,</td></tr>
+<tr><td class="denom">&lambda;</td> <td>0</td> <td class="denom">&rho;</td></tr></table>
+
+<p class="noind">or, since xdx = &rho;d&rho;, k = 2&pi;/&lambda;,</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&minus;k <span class="f150">&int;</span></td> <td class="bk">&infin;</td>
+<td rowspan="2"> sin k(at &minus; &rho;)d&rho; = <span class="f150">[</span> &minus;cos k(at &minus; &rho;) <span class="f150">]</span></td> <td class="bk">&infin;</td> <td rowspan="2"></td>
+<td rowspan="2">.</td></tr>
+<tr><td class="bk">r</td> <td class="bk">r</td></tr></table>
+
+<p class="noind">In order to obtain the effect of the primary wave, as retarded by
+traversing the distance r, viz. cos k(at &minus; r), it is necessary to suppose
+that the integrated term vanishes at the upper limit. And it is important
+to notice that without some further understanding the
+integral is really ambiguous. According to the assumed law of
+the secondary wave, the result must actually depend upon the
+precise radius of the outer boundary of the region of integration,
+supposed to be exactly circular. This case is, however, at most
+very special and exceptional. We may usually suppose that a large
+number of the outer rings are incomplete, so that the integrated term
+at the upper limit may properly be taken to vanish. If a formal
+proof be desired, it may be obtained by introducing into the integral
+a factor such as e<span class="sp">-h&rho;</span>, in which h is ultimately made to diminish
+without limit.</p>
+
+<p>When the primary wave is plane, the area of the first Fresnel
+zone is &pi;&lambda;r, and, since the secondary waves vary as r<span class="sp">-1</span>, the intensity
+is independent of r, as of course it should be. If, however, the
+primary wave be spherical, and of radius a at the wave-front of
+resolution, then we know that at a distance r further on the
+amplitude of the primary wave will be diminished in the ratio
+a : (r + a). This may be regarded as a consequence of the altered
+area of the first Fresnel zone. For, if x be its radius, we have</p>
+
+<p class="center">{(r + ½&lambda;)² &minus; x²} + &radic; {a² &minus; x²} = r + a,</p>
+
+<p class="noind">so that</p>
+
+<p class="center">x² = &lambda;ar/(a + r) nearly.</p>
+
+<p class="noind">Since the distance to be travelled by the secondary waves is still
+r, we see how the effect of the first zone, and therefore of the whole
+series is proportional to a/(a + r). In like manner may be treated
+other cases, such as that of a primary wave-front of unequal principal
+curvatures.</p>
+
+<p>The general explanation of the formation of shadows may also
+be conveniently based upon Fresnel&rsquo;s zones. If the point under
+consideration be so far away from the geometrical shadow that a
+large number of the earlier zones are complete, then the illumination,
+determined sensibly by the first zone, is the same as if there
+were no obstruction at all. If, on the other hand, the point be well
+immersed in the geometrical shadow, the earlier zones are altogether
+missing, and, instead of a series of terms beginning with finite
+numerical magnitude and gradually diminishing to zero, we have
+now to deal with one of which the terms diminish to zero <i>at both
+ends</i>. The sum of such a series is very approximately zero, each term
+being neutralized by the halves of its immediate neighbours, which
+are of the opposite sign. The question of light or darkness then
+depends upon whether the series begins or ends abruptly. With few
+exceptions, abruptness can occur only in the presence of the first
+term, viz. when the secondary wave of least retardation is unobstructed,
+or when a <i>ray</i> passes through the point under consideration.
+According to the undulatory theory the light cannot be regarded
+strictly as travelling along a ray; but the existence of an unobstructed
+ray implies that the system of Fresnel&rsquo;s zones can be commenced,
+and, if a large number of these zones are fully developed and do not
+terminate abruptly, the illumination is unaffected by the neighbourhood
+of obstacles. Intermediate cases in which a few zones only are
+formed belong especially to the province of diffraction.</p>
+
+<p>An interesting exception to the general rule that full brightness
+requires the existence of the first zone occurs when the obstacle
+assumes the form of a small circular disk parallel to the plane of
+the incident waves. In the earlier half of the 18th century R. Delisle
+found that the centre of the circular shadow was occupied by a
+bright point of light, but the observation passed into oblivion
+until S. D. Poisson brought forward as an objection to Fresnel&rsquo;s
+theory that it required at the centre of a circular shadow a point as
+bright as if no obstacle were intervening. If we conceive the primary
+wave to be broken up at the plane of the disk, a system of Fresnel&rsquo;s
+zones can be constructed which begin from the circumference;
+and the first zone external to the disk plays the part ordinarily
+taken by the centre of the entire system. The whole effect is the
+<span class="pagenum"><a name="page240" id="page240"></a>240</span>
+half of that of the first existing zone, and this is sensibly the same
+as if there were no obstruction.</p>
+
+<p>When light passes through a small circular or annular aperture,
+the illumination at any point along the axis depends upon the
+precise relation between the aperture and the distance from it at
+which the point is taken. If, as in the last paragraph, we imagine
+a system of zones to be drawn commencing from the inner circular
+boundary of the aperture, the question turns upon the manner in
+which the series terminates at the outer boundary. If the aperture
+be such as to fit exactly an integral number of zones, the aggregate
+effect may be regarded as the half of those due to the first and last
+zones. If the number of zones be even, the action of the first and last
+zones are antagonistic, and there is complete darkness at the point.
+If on the other hand the number of zones be odd, the effects conspire;
+and the illumination (proportional to the square of the amplitude)
+is four times as great as if there were no obstruction at all.</p>
+
+<p>The process of augmenting the resultant illumination at a particular
+point by stopping some of the secondary rays may be carried
+much further (Soret, <i>Pogg. Ann.</i>, 1875, 156, p. 99). By the aid of
+photography it is easy to prepare a plate, transparent where the zones
+of odd order fall, and opaque where those of even order fall. Such
+a plate has the power of a condensing lens, and gives an illumination
+out of all proportion to what could be obtained without it. An even
+greater effect (fourfold) can be attained by providing that the
+stoppage of the light from the alternate zones is replaced by a
+phase-reversal without loss of amplitude. R. W. Wood (<i>Phil. Mag.</i>,
+1898, 45, p 513) has succeeded in constructing zone plates upon this
+principle.</p>
+
+<p>In such experiments the narrowness of the zones renders necessary
+a pretty close approximation to the geometrical conditions. Thus
+in the case of the circular disk, equidistant (r) from the source of
+light and from the screen upon which the shadow is observed, the
+width of the first exterior zone is given by</p>
+
+<p class="center">dx = &lambda;(2r)/4(2x),</p>
+
+<p>2x being the diameter of the disk. If 2r = 1000 cm., 2x = 1 cm.,
+&lambda; = 6 × 10<span class="sp">-5</span> cm., then dx = .0015 cm. Hence, in order that this
+zone may be perfectly formed, there should be no error in the circumference
+of the order of .001 cm. (It is easy to see that the radius of
+the bright spot is of the same order of magnitude.) The experiment
+succeeds in a dark room of the length above mentioned, with a
+threepenny bit (supported by three threads) as obstacle, the origin
+of light being a small needle hole in a plate of tin, through which the
+sun&rsquo;s rays shine horizontally after reflection from an external mirror.
+In the absence of a heliostat it is more convenient to obtain a point of
+light with the aid of a lens of short focus.</p>
+
+<p>The amplitude of the light at any point in the axis, when plane
+waves are incident perpendicularly upon an annular aperture, is,
+as above,</p>
+
+<p class="center">cos k(at &minus; r<span class="su">1</span>) &minus; cos k(at &minus; r<span class="su">2</span>) = 2 sin kat sin k(r<span class="su">1</span> &minus; r<span class="su">2</span>),</p>
+
+<p class="noind">r<span class="su">2</span>, r<span class="su">1</span> being the distances of the outer and inner boundaries
+from the point in question. It is scarcely necessary to remark
+that in all such cases the calculation applies in the first instance
+to homogeneous light, and that, in accordance with Fourier&rsquo;s
+theorem, each homogeneous component of a mixture may be treated
+separately. When the original light is white, the presence of some
+components and the absence of others will
+usually give rise to coloured effects, variable
+with the precise circumstances of the case.</p>
+
+<table class="nobctr" style="float: left; width: 220px;" summary="Illustration">
+<tr><td class="figleft1"><img style="width:168px; height:205px" src="images/img240.jpg" alt="" /></td></tr>
+<tr><td class="caption sc">Fig. 2.</td></tr></table>
+
+<p>Although the matter can be fully treated
+only upon the basis of a dynamical theory, it
+is proper to point out at once that there is an
+element of assumption in the application of
+Huygens&rsquo;s principle to the calculation of the
+effects produced by opaque screens of limited
+extent. Properly applied, the principle could
+not fail; but, as may readily be proved in
+the case of sonorous waves, it is not in strictness
+sufficient to assume the expression for
+a secondary wave suitable when the primary
+wave is undisturbed, with mere limitation of
+the integration to the transparent parts of the screen. But, except
+perhaps in the case of very fine gratings, it is probable that the error
+thus caused is insignificant; for the incorrect estimation of the
+secondary waves will be limited to distances of a few wave-lengths
+only from the boundary of opaque and transparent parts.</p>
+
+<p>3. <i>Fraunhofer&rsquo;s Diffraction Phenomena.</i>&mdash;A very general
+problem in diffraction is the investigation of the distribution
+of light over a screen upon which impinge divergent or convergent
+spherical waves after passage through various diffracting
+apertures. When the waves are convergent and the recipient
+screen is placed so as to contain the centre of convergency&mdash;the
+image of the original radiant point, the calculation assumes a less
+complicated form. This class of phenomena was investigated
+by J. von Fraunhofer (upon principles laid down by Fresnel),
+and are sometimes called after his name. We may conveniently
+commence with them on account of their simplicity and great
+importance in respect to the theory of optical instruments.</p>
+
+<p>If &fnof; be the radius of the spherical wave at the place of resolution,
+where the vibration is represented by cos kat, then at any point
+M (fig. 2) in the recipient screen the vibration due to an element dS
+of the wave-front is (§ 2)</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&minus;</td> <td>dS</td>
+<td rowspan="2">sin k(at &minus; &rho;),</td></tr>
+<tr><td class="denom">&lambda;&rho;</td></tr></table>
+
+<p class="noind">&rho; being the distance between M and the element dS.</p>
+
+<p>Taking co-ordinates in the plane of the screen with the centre of
+the wave as origin, let us represent M by &xi;, &eta;, and P (where dS is
+situated) by x, y, z.</p>
+
+<p class="noind">Then</p>
+
+<p class="center">&rho;² = (x &minus; &xi;)² + (y &minus; &eta;)² + z², &fnof;² = x² + y² + z²;</p>
+
+<p class="noind">so that</p>
+
+<p class="center">&rho;² = &fnof;² &minus; 2x&xi; &minus; 2y&eta; + &xi;² + &eta;².</p>
+
+<p class="noind">In the applications with which we are concerned, &xi;, &eta; are very
+small quantities; and we may take</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&rho; = &fnof;<span class="f150">{</span> 1 &minus;</td> <td>x&xi; + y&eta;</td>
+<td rowspan="2"><span class="f150">}</span>.</td></tr>
+<tr><td class="denom">&fnof;²</td></tr></table>
+
+<p class="noind">At the same time dS may be identified with dxdy, and in the denominator
+&rho; may be treated as constant and equal to &fnof;. Thus the
+expression for the vibration at M becomes</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&minus;</td> <td>1</td>
+<td rowspan="2"><span class="f150">&int;&int;</span>sin k <span class="f150">{</span> at &minus; &fnof; +</td> <td>x&xi; + y&eta;</td>
+<td rowspan="2"><span class="f150">}</span> dxdy &emsp; &emsp;(1);</td></tr>
+<tr><td class="denom">&lambda;&fnof;</td> <td class="denom">&fnof;</td></tr></table>
+
+<p class="noind">and for the intensity, represented by the square of the amplitude,</p>
+
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">I² =</td> <td>1</td>
+<td rowspan="2"><span class="f150">[ &int;&int;</span> sin k</td> <td>x&xi; + y&eta;</td>
+<td rowspan="2">dxdy <span class="f150">]</span></td> <td>²</td>
+<td rowspan="2">&nbsp;</td></tr>
+<tr><td class="denom">&lambda;²&fnof;²</td> <td class="denom">&fnof;</td> <td>&nbsp;</td></tr>
+
+<tr><td rowspan="2">&emsp; +</td> <td>1</td>
+<td rowspan="2"><span class="f150">[ &int;&int;</span> cos k</td> <td>x&xi; + y&eta;</td>
+<td rowspan="2">dxdy <span class="f150">]</span></td> <td>²</td>
+<td rowspan="2"> &emsp; &emsp; (2).</td></tr>
+<tr><td class="denom">&lambda;²&fnof;²</td> <td class="denom">&fnof;</td> <td>&nbsp;</td></tr>
+</table>
+
+<p class="noind">This expression for the intensity becomes rigorously applicable when
+&fnof; is indefinitely great, so that ordinary optical aberration disappears.
+The incident waves are thus plane, and are limited to a plane aperture
+coincident with a wave-front. The integrals are then properly
+functions of the <i>direction</i> in which the light is to be estimated.</p>
+
+<p>In experiment under ordinary circumstances it makes no difference
+whether the collecting lens is in front of or behind the diffracting
+aperture. It is usually most convenient to employ a telescope
+focused upon the radiant point, and to place the diffracting apertures
+immediately in front of the object-glass. What is seen through the
+eye-piece in any case is the same as would be depicted upon a screen
+in the focal plane.</p>
+
+<p>Before proceeding to special cases it may be well to call attention
+to some general properties of the solution expressed by (2) (see
+Bridge, <i>Phil. Mag.</i>, 1858).</p>
+
+<p>If when the aperture is given, the wave-length (proportional to
+k<span class="sp">-1</span>) varies, the composition of the integrals is unaltered, provided
+&xi; and &eta; are taken universely proportional to &lambda;. A diminution of
+&lambda; thus leads to a simple proportional shrinkage of the diffraction
+pattern, attended by an augmentation of brilliancy in proportion
+to &lambda;<span class="sp">-2</span>.</p>
+
+<p>If the wave-length remains unchanged, similar effects are produced
+by an increase in the scale of the aperture. The linear
+dimension of the diffraction pattern is inversely as that of the
+aperture, and the brightness at corresponding points is as the
+<i>square</i> of the area of aperture.</p>
+
+<p>If the aperture and wave-length increase in the same proportion,
+the size and shape of the diffraction pattern undergo no change.</p>
+
+<p>We will now apply the integrals (2) to the case of a rectangular
+aperture of width a parallel to x and of width b parallel to y. The
+limits of integration for x may thus be taken to be &minus;½a and +½a,
+and for y to be &minus;½b, +½b. We readily find (with substitution for
+k of 2&pi;/&lambda;)</p>
+
+<table class="math0" summary="math">
+<tr><td colspan="2">&nbsp;</td> <td rowspan="4">·</td>
+ <td rowspan="2">sin²</td> <td>&pi;a&xi;</td>
+ <td rowspan="4">·</td> <td rowspan="2">sin²</td>
+ <td>&pi;b&eta;</td> <td rowspan="4">&emsp; &emsp; (3),</td></tr>
+<tr><td rowspan="2">I² =</td> <td>a²b²</td> <td class="denom">&fnof;&lambda;</td>
+ <td class="denom">&fnof;&lambda;</td></tr>
+<tr><td class="denom">&fnof;²&lambda;²</td>
+ <td class="denom" colspan="2">&pi;²a²&xi;²</td> <td class="denom" colspan="2">&pi;²b²&eta;²</td></tr>
+<tr><td colspan="2">&nbsp;</td>
+ <td class="ov" colspan="2">&fnof;²&lambda;²</td> <td class="ov" colspan="2">&fnof;²&lambda;²</td></tr>
+</table>
+
+<p class="noind">as representing the distribution of light in the image of a mathematical
+point when the aperture is rectangular, as is often the case
+in spectroscopes.</p>
+
+<p>The second and third factors of (3) being each of the form sin²u/u²,
+we have to examine the character of this function. It vanishes
+when u = m&pi;, m being any whole number other than zero. When
+u = 0, it takes the value unity. The maxima occur when</p>
+
+<p class="center">u = tan u,&emsp; &emsp; (4),</p>
+
+<p class="noind">and then</p>
+
+<p class="center">sin²u / u² = cos²u &emsp; &emsp; (5).</p>
+
+<p class="noind">To calculate the roots of (5) we may assume</p>
+
+<p class="center">u = (m + ½)&pi; &minus; y = U &minus; y,</p>
+
+<p><span class="pagenum"><a name="page241" id="page241"></a>241</span></p>
+
+<p>where y is a positive quantity which is small when u is large. Substituting
+this, we find cot y = U &minus; y, whence</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">y =</td> <td>1</td>
+<td rowspan="2"><span class="f200">(</span>1 +</td> <td>y</td>
+<td rowspan="2">+</td> <td>y-</td>
+<td rowspan="2">+ ...<span class="f200">)</span> &minus;</td> <td>y³</td>
+<td rowspan="2">&minus;</td> <td>2y<span class="sp">5</span></td>
+<td rowspan="2">&minus;</td> <td>17y<span class="sp">7</span></td> <td rowspan="2">.</td></tr>
+<tr><td class="denom">U</td> <td class="denom">U</td> <td class="denom">U²</td>
+ <td class="denom">3</td> <td class="denom">15</td> <td class="denom">315</td></tr></table>
+
+<p class="noind">This equation is to be solved by successive approximation. It will
+readily be found that</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">u = U &minus; y = U &minus; U<span class="sp">&minus;1</span> &minus;</td> <td>2</td>
+<td rowspan="2">U<span class="sp">&minus;3</span> &minus;</td> <td>13</td>
+<td rowspan="2">U<span class="sp">&minus;5</span> &minus;</td> <td>146</td> <td rowspan="2">U<span class="sp">&minus;7</span> &minus; ... &emsp; (6).</td></tr>
+<tr><td class="denom">3</td> <td class="denom">15</td> <td class="denom">105</td></tr></table>
+
+<p>In the first quadrant there is no root after zero, since tan u &gt; u,
+and in the second quadrant there is none because the signs of u and
+tan u are opposite. The first root after zero is thus in the third
+quadrant, corresponding to m = 1. Even in this case the series
+converges sufficiently to give the value of the root with considerable
+accuracy, while for higher values of m it is all that could be desired.
+The actual values of u/&pi; (calculated in another manner by F. M.
+Schwerd) are 1.4303, 2.4590, 3.4709, 4.4747, 5.4818, 6.4844, &amp;c.</p>
+
+<p>Since the maxima occur when u = (m + ½)&pi; nearly, the successive
+values are not very different from</p>
+
+<table class="math0" summary="math">
+<tr><td>4</td> <td rowspan="2">,</td> <td>4</td>
+<td rowspan="2">,</td> <td>4</td>
+<td rowspan="2">, &amp;c.</td></tr>
+<tr><td class="denom">9&pi;²</td> <td class="denom">25&pi;</td> <td class="denom">49&pi;²</td></tr></table>
+
+<p>The application of these results to (3) shows that the field is
+brightest at the centre &xi; = 0, &eta; = 0, viz. at the geometrical image
+of the radiant point. It is traversed by dark lines whose equations
+are</p>
+
+<p class="center">&xi; = mf&lambda; / a, &eta; = mf&lambda; / b.</p>
+
+<p class="noind">Within the rectangle formed by pairs of consecutive dark lines,
+and not far from its centre, the brightness rises to a maximum;
+but these subsequent maxima are in all cases much inferior to the
+brightness at the centre of the entire pattern (&xi; = 0, &eta; = 0).</p>
+
+<p>By the principle of energy the illumination over the entire focal
+plane must be equal to that over the diffracting area; and thus, in
+accordance with the suppositions by which (3) was obtained, its
+value when integrated from &xi; = &infin; to &xi; = +&infin;, and from &eta; = &minus;&infin;
+to &eta; = +&infin; should be equal to ab. This integration, employed
+originally by P. Kelland (<i>Edin. Trans.</i> 15, p. 315) to determine the
+absolute intensity of a secondary wave, may be at once effected by
+means of the known formula</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2"><span class="f150">&int;</span></td> <td class="bk">+&infin;</td> <td>sin²u</td>
+<td rowspan="2">du = <span class="f150">&int;</span></td> <td class="bk">+&infin;</td> <td>sin u</td>
+<td rowspan="2">du = &pi;.</td></tr>
+<tr><td class="bk">&minus;&infin;</td> <td class="denom">u²</td>
+<td class="bk">&minus;&infin;</td> <td class="denom">u</td></tr></table>
+
+<p class="noind">It will be observed that, while the total intensity is proportional to
+ab, the intensity at the focal point is proportional to a²b². If the
+aperture be increased, not only is the total brightness over the focal
+plane increased with it, but there is also a concentration of the
+diffraction pattern. The form of (3) shows immediately that, if
+a and b be altered, the co-ordinates of any characteristic point in the
+pattern vary as a<span class="sp">&minus;1</span> and b<span class="sp">&minus;1</span>.</p>
+
+<p>The contraction of the diffraction pattern with increase of aperture
+is of fundamental importance in connexion with the resolving power
+of optical instruments. According to common optics, where images
+are absolute, the diffraction pattern is supposed to be infinitely
+small, and two radiant points, however near together, form separated
+images. This is tantamount to an assumption that &lambda; is infinitely
+small. The actual finiteness of &lambda; imposes a limit upon the separating
+or resolving power of an optical instrument.</p>
+
+<p>This indefiniteness of images is sometimes said to be due to
+diffraction by the edge of the aperture, and proposals have even been
+made for curing it by causing the transition between the interrupted
+and transmitted parts of the primary wave to be less abrupt. Such
+a view of the matter is altogether misleading. What requires
+explanation is not the imperfection of actual images so much as the
+possibility of their being as good as we find them.</p>
+
+<p>At the focal point (&xi; = 0, &eta; = 0) all the secondary waves agree in
+phase, and the intensity is easily expressed, whatever be the form
+of the aperture. From the general formula (2), if A be the <i>area</i> of
+aperture,</p>
+
+<p class="center">I<span class="su">0</span>² = A² / &lambda;²&fnof;² &emsp; &emsp; (7).</p>
+
+<p>The formation of a sharp image of the radiant point requires
+that the illumination become insignificant when &xi;, &eta; attain small
+values, and this insignificance can only arise as a consequence of
+discrepancies of phase among the secondary waves from various
+parts of the aperture. So long as there is no sensible discrepancy
+of phase there can be no sensible diminution of brightness as compared
+with that to be found at the focal point itself. We may go
+further, and lay it down that there can be no considerable loss of
+brightness until the difference of phase of the waves proceeding from
+the nearest and farthest parts of the aperture amounts to ¼&lambda;.</p>
+
+<p>When the difference of phase amounts to &lambda;, we may expect the
+resultant illumination to be very much reduced. In the particular
+case of a rectangular aperture the course of things can be readily
+followed, especially if we conceive &fnof; to be infinite. In the direction
+(suppose horizontal) for which &eta; = 0, &xi;/&fnof; = sin &theta;, the phases of the
+secondary waves range over a complete period when sin &theta; = &lambda;/a, and,
+since all parts of the horizontal aperture are equally effective, there
+is in this direction a complete compensation and consequent absence
+of illumination. When sin &theta; = <span class="spp">3</span>&frasl;<span class="suu">2</span>&lambda;/a, the phases range one and a half
+periods, and there is revival of illumination. We may compare
+the brightness with that in the direction &theta; = 0. The phase of the
+resultant amplitude is the same as that due to the central secondary
+wave, and the discrepancies of phase among the components reduce
+the amplitude in the proportion</p>
+
+<table class="math0" summary="math">
+<tr><td>1</td> <td rowspan="2"><span class="f150">&int;</span></td> <td class="bk">+<span class="spp">3</span>&frasl;<span class="suu">2</span>&pi;</td>
+<td rowspan="2">cos &phi; d&phi;: 1</td></tr>
+<tr><td class="denom">3&pi;</td> <td class="bk">&minus;<span class="spp">3</span>&frasl;<span class="suu">2</span>&pi;</td></tr></table>
+
+<p class="noind">or -<span class="spp">2</span>&frasl;<span class="suu">3</span>&pi; : 1; so that the brightness in this direction is <span class="spp">4</span>&frasl;<span class="suu">9</span>&pi;² of the
+maximum at &theta; = 0. In like manner we may find the illumination
+in any other direction, and it is obvious that it vanishes when sin &theta;
+is any multiple of &lambda;/a.</p>
+
+<p>The reason of the augmentation of resolving power with aperture
+will now be evident. The larger the aperture the smaller are the
+angles through which it is necessary to deviate from the principal
+direction in order to bring in specified discrepancies of phase&mdash;the
+more concentrated is the image.</p>
+
+<p>In many cases the subject of examination is a luminous line of
+uniform intensity, the various points of which are to be treated as
+independent sources of light. If the image of the line be &xi; = 0,
+the intensity at any point &xi;, &eta; of the diffraction pattern may be
+represented by</p>
+
+<table class="math0" summary="math">
+<tr><td colspan="4">&nbsp;</td> <td rowspan="2">sin²</td>
+ <td>&pi;a&xi;</td> <td rowspan="4"> &emsp; &emsp; (8),</td></tr>
+<tr><td rowspan="2"><span class="f150">&int;</span></td>
+ <td class="bk">+&infin;</td> <td rowspan="2">I²d&eta; =</td> <td>a²b</td> <td class="ov">&lambda;f</td></tr>
+<tr><td class="bk">&minus;&infin;</td> <td class="ov">&lambda;&fnof;</td>
+ <td class="denom" colspan="2">&pi;²a²&xi;²</td></tr>
+<tr><td colspan="4">&nbsp;</td> <td class="ov" colspan="2">&lambda;²f²</td></tr>
+</table>
+
+<p class="noind">the same law as obtains for a luminous point when horizontal
+directions are alone considered. The definition of a fine vertical
+line, and consequently the resolving power for contiguous vertical
+lines, is thus <i>independent of the vertical aperture of the instrument</i>,
+a law of great importance in the theory of the spectroscope.</p>
+
+<p>The distribution of illumination in the image of a luminous line
+is shown by the curve ABC (fig. 3), representing the value of the
+function sin²u/u² from u = 0 to u = 2&pi;. The part corresponding to
+negative values of u is similar, OA being a line of symmetry.</p>
+
+<table class="nobctr" style="float: right; width: 230px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:185px; height:181px" src="images/img241.jpg" alt="" /></td></tr>
+<tr><td class="caption sc">Fig. 3.</td></tr></table>
+
+<p>Let us now consider the distribution of brightness in the image
+of a double line whose components are of equal strength, and at
+such an angular interval that the central line in the image of one
+coincides with the first zero of brightness in the image of the other.
+In fig. 3 the curve of brightness for one component is ABC, and
+for the other OA&prime;C&prime;; and the curve representing half the combined
+brightnesses is E&prime;BE. The brightness (corresponding
+to B) midway between the two
+central points AA&rsquo; is .8106 of the brightness
+at the central points themselves. We
+may consider this to be about the limit of
+closeness at which there could be any
+decided appearance of resolution, though
+doubtless an observer accustomed to his
+instrument would recognize the duplicity
+with certainty. The obliquity, corresponding
+to u = &pi;, is such that the phases
+of the secondary waves range over a complete
+period, <i>i.e.</i> such that the projection of
+the horizontal aperture upon this direction
+is one wave-length. We conclude that a <i>double line cannot be
+fairly resolved unless its components subtend an angle exceeding that
+subtended by the wave-length of light at a distance equal to the horizontal
+aperture</i>. This rule is convenient on account of its simplicity; and
+it is sufficiently accurate in view of the necessary uncertainty as to
+what exactly is meant by resolution.</p>
+
+<p>If the angular interval between the components of a double line
+be half as great again as that supposed in the figure, the brightness
+midway between is .1802 as against 1.0450 at the central lines of
+each image. Such a falling off in the middle must be more than
+sufficient for resolution. If the angle subtended by the components
+of a double line be twice that subtended by the wave-length at a
+distance equal to the horizontal aperture, the central bands are
+just clear of one another, and there is a line of absolute blackness
+in the middle of the combined images.</p>
+
+<p>The resolving power of a telescope with circular or rectangular
+aperture is easily investigated experimentally. The best object for
+examination is a grating of fine wires, about fifty to the inch, backed
+by a sodium flame. The object-glass is provided with diaphragms
+pierced with round holes or slits. One of these, of width equal, say,
+to one-tenth of an inch, is inserted in front of the object-glass, and
+the telescope, carefully focused all the while, is drawn gradually back
+from the grating until the lines are no longer seen. From a measurement
+of the maximum distance the least angle between consecutive
+lines consistent with resolution may be deduced, and a comparison
+made with the rule stated above.</p>
+
+<p>Merely to show the dependence of resolving power on aperture it is
+not necessary to use a telescope at all. It is sufficient to look at wire
+gauze backed by the sky or by a flame, through a piece of blackened
+cardboard, pierced by a needle and held close to the eye. By
+varying the distance the point is easily found at which resolution
+ceases; and the observation is as sharp as with a telescope. The
+<span class="pagenum"><a name="page242" id="page242"></a>242</span>
+function of the telescope is in fact to allow the use of a wider, and
+therefore more easily measurable, aperture. An interesting modification
+of the experiment may be made by using light of various
+wave-lengths.</p>
+
+<p>Since the limitation of the width of the central band in the image
+of a luminous line depends upon discrepancies of phase among the
+secondary waves, and since the discrepancy is greatest for the waves
+which come from the edges of the aperture, the question arises
+how far the operation of the central parts of the aperture is advantageous.
+If we imagine the aperture reduced to two equal
+narrow slits bordering its edges, compensation will evidently be
+complete when the projection on an oblique direction is equal to
+½&lambda;, instead of &lambda; as for the complete aperture. By this procedure
+the width of the central band in the diffraction pattern is halved,
+and so far an advantage is attained. But, as will be evident, the
+bright bands bordering the central band are now not inferior to it
+in brightness; in fact, a band similar to the central band is reproduced
+an indefinite number of times, so long as there is no sensible
+discrepancy of phase in the secondary waves proceeding from the
+various parts of the <i>same</i> slit. Under these circumstances the
+narrowing of the band is paid for at a ruinous price, and the arrangement
+must be condemned altogether.</p>
+
+<p>A more moderate suppression of the central parts is, however,
+sometimes advantageous. Theory and experiment alike prove that
+a double line, of which the components are equally strong, is better
+resolved when, for example, one-sixth of the horizontal aperture is
+blocked off by a central screen; or the rays quite at the centre may
+be allowed to pass, while others a little farther removed are blocked
+off. Stops, each occupying one-eighth of the width, and with centres
+situated at the points of trisection, answer well the required purpose.</p>
+
+<p>It has already been suggested that the principle of energy requires
+that the general expression for I² in (2) when integrated over the
+whole of the plane &xi;, &eta; should be equal to A, where A is the area of
+the aperture. A general analytical verification has been given by
+Sir G. G. Stokes (<i>Edin. Trans.</i>, 1853, 20, p. 317). Analytically
+expressed&mdash;</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2"><span class="f150">&int;&int;</span></td> <td class="bk">+&infin;</td>
+<td rowspan="2">I² d&xi;d&eta; = <span class="f150">&int;&int;</span> dxdy = A &emsp; &emsp; (9).</td></tr>
+<tr><td class="bk">&minus;&infin;</td></tr></table>
+
+<p>We have seen that I<span class="su">0</span>² (the intensity at the focal point) was equal to
+A²/&lambda;²f². If A&prime; be the area over which the intensity must be I<span class="su">0</span>² in
+order to give the actual total intensity in accordance with</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">A&prime; I<span class="su">0</span>² = <span class="f150">&int;&int;</span></td> <td class="bk">+&infin;</td>
+ <td rowspan="2">I² d&xi;d&eta;,</td></tr>
+<tr><td class="bk">&minus;&infin;</td></tr></table>
+
+<p class="noind">the relation between A and A&prime; is AA&prime; = &lambda;²f². Since A&prime; is in some
+sense the area of the diffraction pattern, it may be considered to be a
+rough criterion of the definition, and we infer that the definition of a
+point depends principally upon the area of the aperture, and only in
+a very secondary degree upon the shape when the area is maintained
+constant.</p>
+
+<p>4. <i>Theory of Circular Aperture.</i>&mdash;We will now consider the
+important case where the form of the aperture is circular.</p>
+
+<p>Writing for brevity</p>
+
+<p class="center">k&xi;/f = p, k&eta;/f = q, &emsp; &emsp; (1),</p>
+
+<p class="noind">we have for the general expression (§ 11) of the intensity</p>
+
+<p class="center">&lambda;²f²I² = S² + C² &emsp; &emsp; (2),</p>
+
+<p class="noind">where</p>
+
+<p class="center">S = <span class="f150">&int;&int;</span> sin(px + qy)dx dy, &emsp; &emsp; (3),</p>
+
+<p class="center">C = <span class="f150">&int;&int;</span> cos(px + qy)dx dy, &emsp; &emsp; (4).</p>
+
+<p class="noind">When, as in the application to rectangular or circular apertures,
+the form is symmetrical with respect to the axes both of x and y,
+S = 0, and C reduces to</p>
+
+<p class="center">C = <span class="f150">&int;&int;</span> cos px cos qy dx dy, &emsp; &emsp; (5).</p>
+
+<p class="noind">In the case of the circular aperture the distribution of light is of
+course symmetrical with respect to the focal point p = 0, q = 0; and
+C is a function of p and q only through  &radic;(p² + q²). It is thus
+sufficient to determine the intensity along the axis of p. Putting
+q = 0, we get</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">C = <span class="f150">&int;&int;</span> cos px dx dy = 2 <span class="f150">&int;</span></td> <td class="bk">+R</td>
+<td rowspan="2">cos px &radic;(R² &minus; x²) dx,</td></tr>
+<tr><td class="bk">&minus;R</td></tr></table>
+
+<p class="noind">R being the radius of the aperture. This integral is the Bessel&rsquo;s
+function of order unity, defined by</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">J<span class="su">1</span>(z) =</td> <td>z</td>
+<td rowspan="2"><span class="f150">&int;</span></td> <td class="bk">&pi;</td>
+<td rowspan="2">cos (z cos &phi;) sin² &phi; d&phi; &emsp; &emsp; (6).</td></tr>
+<tr><td class="denom">&pi;</td> <td class="bk">0</td></tr></table>
+
+<p class="noind">Thus, if x = R cos &phi;,</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">C = &pi;²R</td> <td>2J<span class="su">1</span>(pR)</td>
+<td rowspan="2"> &emsp; &emsp; (7);</td></tr>
+<tr><td class="denom">pR</td></tr></table>
+
+<p class="noind">and the illumination at distance r from the focal point is</p>
+
+<table class="math0" summary="math">
+<tr><td colspan="2">&nbsp;</td> <td rowspan="4">·</td>
+ <td rowspan="2">4J<span class="su">1</span>² <span class="f150">(</span></td>
+ <td>2&pi;Rr</td> <td rowspan="2"><span class="f150">)</span></td>
+ <td rowspan="4"> &emsp; &emsp; (8).</td></tr>
+<tr><td rowspan="2">I² =</td> <td>&pi;²R<span class="sp">4</span></td> <td class="ov">&fnof;&lambda;</td></tr>
+<tr><td class="denom">&lambda;²f²</td> <td class="denom" rowspan="2"> &emsp; <span class="f150">(</span></td>
+ <td class="denom">2&pi;Rr</td> <td class="denom" rowspan="2"><span class="f150">)</span> ²</td></tr>
+<tr><td colspan="2">&nbsp;</td> <td class="ov">&fnof;&lambda;</td></tr>
+</table>
+
+<p>The ascending series for J<span class="su">1</span>(z), used by Sir G. B. Airy (<i>Camb. Trans.</i>,
+1834) in his original investigation of the diffraction of a circular
+object-glass, and readily obtained from (6), is</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">J<span class="su">1</span>(z) =</td> <td>z</td>
+<td rowspan="2">&minus;</td> <td>z³</td>
+<td rowspan="2">+</td> <td>z<span class="sp">5</span></td>
+<td rowspan="2">&minus;</td> <td>z<span class="sp">7</span></td> <td rowspan="2">+ ... &emsp; &emsp; (9).</td></tr>
+<tr><td class="denom">2</td> <td class="denom">2²·4</td>
+<td class="denom">2²·4²·6</td> <td class="denom">2²·4²·6²·8</td></tr></table>
+
+<p class="noind">When z is great, we may employ the semi-convergent series</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">J<span class="su">1</span>(z) =  <span class="f150">&radic;(</span></td> <td>2</td>
+<td rowspan="2"><span class="f150">)</span> sin (z &minus; ¼&pi;) <span class="f150">{</span> 1 +</td> <td>3·5·1</td>
+<td rowspan="2"><span class="f150">(</span></td> <td>1</td>
+<td rowspan="2"><span class="f150">)</span></td> <td>²</td>
+<td rowspan="2">&minus;</td> <td>3·5·7·9·1·3·5</td>
+<td rowspan="2"><span class="f150">(</span></td> <td>1</td>
+<td rowspan="2"><span class="f150">)</span></td> <td><span class="sp np">4</span></td>
+<td rowspan="2">+ ... <span class="f150">}</span></td></tr>
+<tr><td class="denom">&pi;z</td> <td class="denom">8·16</td>
+<td class="denom">z</td> <td>&nbsp;</td>
+<td class="denom">8·16·24·32</td> <td class="denom">z</td> <td>&nbsp;</td></tr></table>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">+ <span class="f150">&radic;(</span></td> <td>2</td>
+<td rowspan="2"><span class="f150">)</span> cos (z &minus; ¼&pi;) <span class="f150">{</span>3/8 · 1/z &minus;</td>
+<td>3·5·7·1·3</td> <td rowspan="2"><span class="f150">(</span></td> <td>1</td>
+<td rowspan="2"><span class="f150">)</span></td> <td>³</td>
+<td rowspan="2">+</td> <td>3·5·7·9·11·1·3·5·7</td>
+<td rowspan="2"><span class="f150">(</span></td> <td>1</td> <td rowspan="2"><span class="f150">)</span></td>
+<td><span class="sp">5</span></td> <td rowspan="2">&minus; ... <span class="f150">}</span> &emsp; &emsp; (10).</td></tr>
+<tr><td class="denom">&pi;z</td> <td class="denom">8·16·24</td>
+<td class="denom">z</td> <td>&nbsp;</td>
+<td class="denom">8·16·24·32·40</td> <td class="denom">z</td> <td>&nbsp;</td></tr></table>
+
+<p class="noind">A table of the values of 2z<span class="sp">-1</span>J<span class="su">1</span>(z) has been given by E. C. J. Lommel
+(<i>Schlömilch</i>, 1870, 15, p. 166), to whom is due the first systematic
+application of Bessel&rsquo;s functions to the diffraction integrals.</p>
+
+<p>The illumination vanishes in correspondence with the roots of the
+equation J<span class="su">1</span>(z) = 0. If these be called z<span class="su">1</span> z<span class="su">2</span>, z<span class="su">3</span>, ... the radii of the
+dark rings in the diffraction pattern are</p>
+
+<table class="math0" summary="math">
+<tr><td>&fnof;&lambda;z<span class="su">1</span></td> <td rowspan="2"></td>
+<td>&fnof;&lambda;z<span class="su">2</span></td> <td rowspan="2">, ...</td></tr>
+<tr><td class="denom">2&pi;R</td> <td class="denom">2&pi;R</td></tr></table>
+
+<p class="noind">being thus <i>inversely</i> proportional to R.</p>
+
+<p>The integrations may also be effected by means of polar co-ordinates,
+taking first the integration with respect to &phi; so as to
+obtain the result for an infinitely thin annular aperture. Thus, if</p>
+
+<p class="center">x = &rho; cos &phi;, y = &rho; sin &phi;,</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">C = <span class="f150">&int;&int;</span> cos px dx dy = <span class="f150">&int;</span></td> <td class="bk">R</td>
+<td rowspan="2"><span class="f150">&int;</span></td> <td class="bk">2&pi;</td> <td rowspan="2">cos (p&rho; cos &theta;) &rho;d&rho; d&theta;.</td></tr>
+<tr><td class="bk">0</td> <td class="bk">0</td></tr></table>
+
+<p class="noind">Now by definition</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">J<span class="su">0</span>(z) =</td> <td>2</td>
+<td rowspan="2"><span class="f150">&int;</span></td> <td class="bk">½&pi;</td>
+<td rowspan="2">cos (z cos &theta;) d&theta; = 1 &minus;</td> <td>z²</td>
+<td rowspan="2">+</td> <td>z<span class="sp">4</span></td>
+<td rowspan="2">&minus;</td> <td>z<span class="sp">6</span></td>
+<td rowspan="2">+ ... &emsp; &emsp; (11).</td></tr>
+<tr><td class="denom">&pi;</td> <td class="bk">0</td>
+<td class="denom">2²</td> <td class="denom">2²·4²</td> <td class="denom">2²·4²·6²</td></tr></table>
+
+<p class="noind">The value of C for an annular aperture of radius r and width dr is
+thus</p>
+
+<p class="center">dC = 2 &pi; J<span class="su">0</span> (p&rho;) &rho; d&rho;, &emsp; &emsp; (12).</p>
+
+<p class="noind">For the complete circle,</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">C =</td> <td>2&pi;</td>
+<td rowspan="2"><span class="f150">&int;</span></td> <td class="bk">pr</td>
+<td rowspan="2">J<span class="su">0</span>(z) zdz =</td> <td>2&pi;</td>
+<td rowspan="2"><span class="f150">{</span></td> <td>p²R²</td>
+<td rowspan="2">&minus;</td> <td>p<span class="sp">4</span>R<span class="sp">4</span></td>
+<td rowspan="2">+</td> <td>p<span class="sp">6</span>R<span class="sp">6</span></td>
+<td rowspan="2">&minus; ...<span class="f150">}</span>= &pi;R² ·</td> <td>2J<span class="su">1</span>(pR)</td>
+<td rowspan="2">as before.</td></tr>
+<tr><td class="denom">p²</td> <td class="bk">0</td>
+<td class="denom">p²</td> <td class="denom">2</td>
+<td class="denom">2²·4²</td> <td class="denom">2²·4²·6²</td> <td class="denom">pR</td></tr></table>
+
+<p>In these expressions we are to replace p by k&xi;/&fnof;, or rather, since
+the diffraction pattern is symmetrical, by kr/&fnof;, where r is the distance
+of any point in the focal plane from the centre of the system.</p>
+
+<p>The roots of J<span class="su">0</span>(z) after the first may be found from</p>
+
+<table class="math0" summary="math">
+<tr><td>z</td> <td rowspan="2">= i &minus; .25 +</td> <td>.050561</td>
+<td rowspan="2">&minus;</td> <td>.053041</td>
+<td rowspan="2">+</td> <td>.262051</td> <td rowspan="2"> &emsp; &emsp; (13),</td></tr>
+<tr><td class="denom">&pi;</td> <td class="denom">4i &minus; 1</td>
+<td class="denom">(4i &minus; 1)³</td> <td class="denom">(4i &minus; 1)<span class="sp">5</span></td></tr></table>
+
+<p class="noind">and those of J<span class="su">1</span>(z) from</p>
+
+<table class="math0" summary="math">
+<tr><td>z</td> <td rowspan="2">= i + .25 &minus;</td> <td>.151982</td>
+<td rowspan="2">+</td> <td>.015399</td>
+<td rowspan="2">&minus;</td> <td>.245835</td> <td rowspan="2"> &emsp; &emsp; (14),</td></tr>
+<tr><td class="denom">&pi;</td> <td class="denom">4i + 1</td>
+<td class="denom">(4i + 1)³</td> <td class="denom">(4i + 1)<span class="sp">5</span></td></tr></table>
+
+<p class="noind">formulae derived by Stokes (<i>Camb. Trans.</i>, 1850, vol. ix.) from the
+descending series.<a name="fa1g" id="fa1g" href="#ft1g"><span class="sp">1</span></a> The following table gives the actual values:&mdash;</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcc allb">i</td> <td class="tcc allb">z/&pi; for J<span class="su">0</span>(z) = 0</td> <td class="tcc allb">z/&pi; for J<span class="su">1</span>(z) = 0</td></tr>
+<tr><td class="tcr lb rb">1</td> <td class="tcr rb">7655</td> <td class="tcr rb">1 2197</td></tr>
+<tr><td class="tcr lb rb">2</td> <td class="tcr rb">1 7571</td> <td class="tcr rb">2 2330</td></tr>
+<tr><td class="tcr lb rb">3</td> <td class="tcr rb">2 7546</td> <td class="tcr rb">3 2383</td></tr>
+<tr><td class="tcr lb rb">4</td> <td class="tcr rb">3 7534</td> <td class="tcr rb">4 2411</td></tr>
+<tr><td class="tcr lb rb">5</td> <td class="tcr rb">4 7527</td> <td class="tcr rb">5 2428</td></tr>
+<tr><td class="tcr lb rb">6</td> <td class="tcr rb">5 7522</td> <td class="tcr rb">6 2439</td></tr>
+<tr><td class="tcr lb rb">7</td> <td class="tcr rb">6 7519</td> <td class="tcr rb">7 2448</td></tr>
+<tr><td class="tcr lb rb">8</td> <td class="tcr rb">7 7516</td> <td class="tcr rb">8 2454</td></tr>
+<tr><td class="tcr lb rb">9</td> <td class="tcr rb">8 7514</td> <td class="tcr rb">9 2459</td></tr>
+<tr><td class="tcr lb rb bb">10</td> <td class="tcr rb bb">9 7513</td> <td class="tcr rb bb">10 2463</td></tr>
+</table>
+
+<p>In both cases the image of a mathematical point is thus a
+symmetrical ring system. The greatest brightness is at the centre,
+where</p>
+
+<p class="center">dC = 2&pi;&rho; d&rho;, C = &pi; R².</p>
+
+<p class="noind">For a certain distance outwards this remains sensibly unimpaired
+and then gradually diminishes to zero, as the secondary waves
+become discrepant in phase. The subsequent revivals of brightness
+forming the bright rings are necessarily of inferior brilliancy as
+compared with the central disk.</p>
+
+<p>The first dark ring in the diffraction pattern of the complete
+circular aperture occurs when</p>
+
+<p class="center">r/&fnof; = 1.2197 × &lambda;/2R &emsp; &emsp; (15).</p>
+
+<p><span class="pagenum"><a name="page243" id="page243"></a>243</span></p>
+
+<p class="noind">We may compare this with the corresponding result for a rectangular
+aperture of width a,</p>
+
+<p class="center">&xi;/&fnof; =&lambda;/a;</p>
+
+<p class="noind">and it appears that in consequence of the preponderance of the
+central parts, the compensation in the case of the circle does not
+set in at so small an obliquity as when the circle is replaced by a
+rectangular aperture, whose side is equal to the diameter of the
+circle.</p>
+
+<p>Again, if we compare the complete circle with a narrow annular
+aperture of the same radius, we see that in the latter case the first
+dark ring occurs at a much smaller obliquity, viz.</p>
+
+<p class="center">r/&fnof; = .7655 × &lambda;/2R.</p>
+
+<p>It has been found by Sir William Herschel and others that the
+definition of a telescope is often improved by stopping off a part of
+the central area of the object-glass; but the advantage to be obtained
+in this way is in no case great, and anything like a reduction of the
+aperture to a narrow annulus is attended by a development of the
+external luminous rings sufficient to outweigh any improvement
+due to the diminished diameter of the central area.<a name="fa2g" id="fa2g" href="#ft2g"><span class="sp">2</span></a></p>
+
+<p>The maximum brightnesses and the places at which they occur
+are easily determined with the aid of certain properties of the
+Bessel&rsquo;s functions. It is known (see <span class="sc"><a href="#artlinks">Spherical Harmonics</a></span>) that</p>
+
+<p class="center">J<span class="su">0</span>&prime;(z) = &minus;J<span class="su">1</span>(z), &emsp; &emsp; (16);</p>
+
+<p class="center">J<span class="su">2</span>(z) = (1/z) J<span class="su">1</span>(z) &minus; J<span class="su">1</span>&prime;(z) &emsp; &emsp; (17);</p>
+
+<p class="center">J<span class="su">0</span>(z) + J<span class="su">2</span>(z) = (2/z) J<span class="su">1</span>(z) &emsp; &emsp; (18).</p>
+
+<p class="noind">The maxima of C occur when</p>
+
+<table class="math0" summary="math">
+<tr><td>d</td> <td rowspan="2"><span class="f150">(</span></td> <td>J<span class="su">1</span>(z)</td>
+<td rowspan="2"><span class="f150">)</span> =</td> <td>J<span class="su">1</span>&prime;(z)</td>
+<td rowspan="2">&minus;</td> <td>J<span class="su">1</span>(z)</td> <td rowspan="2"> = 0;</td></tr>
+<tr><td class="denom">dz</td> <td class="denom">z</td> <td class="denom">z</td> <td class="denom">z²</td></tr></table>
+
+<p class="noind">or by 17 when J<span class="su">2</span>(z) = 0. When z has one of the values thus
+determined,</p>
+
+<table class="math0" summary="math">
+<tr><td>2</td><td rowspan="2">J<span class="su">1</span>(z) = J<span class="su">0</span>(z).</td></tr>
+<tr><td class="denom">z</td></tr></table>
+
+<p class="noind">The accompanying table is given by Lommel, in which the first
+column gives the roots of J<span class="su">2</span>(z) = 0, and the second and third columns
+the corresponding values of the functions specified. If appears that
+the maximum brightness in the first ring is only about <span class="spp">1</span>&frasl;<span class="suu">57</span> of the
+brightness at the centre.</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcc lb tb">z</td> <td class="tcc tb">2z<span class="sp">&minus;1</span>J<span class="su">1</span>(z)</td> <td class="tcc rb tb">4z<span class="sp">&minus;2</span>J<span class="su">1</span>²(z)</td></tr>
+
+<tr><td class="tcr lb">.000000</td> <td class="tcr">+1.000000</td> <td class="tcr rb">1.000000</td></tr>
+<tr><td class="tcr lb">5.135630</td> <td class="tcr">&minus; .132279</td> <td class="tcr rb">.017498</td></tr>
+<tr><td class="tcr lb">8.417236</td> <td class="tcr">+ .064482</td> <td class="tcr rb">.004158</td></tr>
+<tr><td class="tcr lb">11.619857</td> <td class="tcr">&minus; .040008</td> <td class="tcr rb">.001601</td></tr>
+<tr><td class="tcr lb">14.795938</td> <td class="tcr">+ .027919</td> <td class="tcr rb">.000779</td></tr>
+<tr><td class="tcr lb bb">17.959820</td> <td class="tcr bb">&minus; .020905</td> <td class="tcr rb bb">.000437</td></tr>
+</table>
+
+<p>We will now investigate the total illumination distributed over
+the area of the circle of radius r. We have</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">I² =</td> <td>&pi;²R<span class="sp">4</span></td>
+<td rowspan="2">·</td> <td>4J<span class="su">1</span>²(z)</td> <td rowspan="2"> &emsp; &emsp; (19),</td></tr>
+<tr><td class="denom">&lambda;²&fnof;²</td> <td class="denom">z²</td></tr></table>
+
+<p class="noind">where</p>
+
+<p class="center">z = 2&pi;Rr/&lambda;f &emsp; &emsp; (20).</p>
+
+<p class="noind">Thus</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">2&pi; <span class="f150">&int;</span> I²rdr =</td> <td>&lambda;²f²</td>
+<td rowspan="2"><span class="f150">&int;</span> I²zdz = &pi;R²· 2 <span class="f150">&int;</span> z<span class="sp">&minus;1</span>J<span class="su">1</span>²(z)dz.</td></tr>
+<tr><td class="denom">2&pi;R²</td></tr></table>
+
+<p class="noind">Now by (17), (18)</p>
+
+<p class="center">z<span class="sp">-1</span>J<span class="su">1</span>(z) = J<span class="su">0</span>(z) &minus; J<span class="su">1</span>&prime;(z);</p>
+
+<p class="noind">so that</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">z<span class="sp">-1</span>J<span class="su">1</span>²(z) = &minus; ½</td> <td>d</td>
+<td rowspan="2">J<span class="su">0</span>² &minus; ½</td> <td>d</td>
+<td rowspan="2">J<span class="su">1</span>²(z),</td></tr>
+<tr><td class="denom">dz</td> <td class="denom">dz</td></tr></table>
+
+<p class="noind">and</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">2 <span class="f150">&int;</span></td> <td class="bk">z</td>
+<td rowspan="2">z<span class="sp">-1</span>J<span class="su">1</span>²(z)dz = 1 &minus; J<span class="su">0</span>²(z) &minus; J<span class="su">1</span>²(z) &emsp; &emsp; (21).</td></tr>
+<tr><td class="bk">0</td></tr></table>
+
+<p class="noind">If r, or z, be infinite, J<span class="su">0</span>(z), J<span class="su">1</span>(z) vanish, and the whole illumination
+is expressed by &pi;R², in accordance with the general principle. In
+any case the proportion of the whole illumination to be found outside
+the circle of radius r is given by</p>
+
+<p class="center">J<span class="su">0</span>²(z) + J<span class="su">1</span>²(z).</p>
+
+<p class="noind">For the dark rings J<span class="su">1</span>(z) = 0; so that the fraction of illumination
+outside any dark ring is simply J<span class="su">0</span>²(z). Thus for the first, second,
+third and fourth dark rings we get respectively .161, .090, .062, .047,
+showing that more than <span class="spp">9</span>&frasl;<span class="suu">10</span>ths of the whole light is concentrated
+within the area of the second dark ring (<i>Phil. Mag.</i>, 1881).</p>
+
+<p>When z is great, the descending series (10) gives</p>
+
+<table class="math0" summary="math">
+<tr><td>2J<span class="su">1</span>(z)</td> <td rowspan="2">=</td> <td>2</td>
+<td rowspan="2"><span class="f150">&radic;(</span></td> <td>2</td> <td rowspan="2"><span class="f150">)</span> sin(z &minus; ¼&pi;) &emsp; &emsp; (22);</td></tr>
+<tr><td class="denom">z</td> <td class="denom">z</td> <td class="denom">&pi;z</td></tr></table>
+
+<p class="noind">so that the places of maxima and minima occur at equal intervals.</p>
+
+<p>The mean brightness varies as z<span class="sp">-3</span> (or as r<span class="sp">-3</span>), and the integral
+found by multiplying it by zdz and integrating between 0 and &infin;
+converges.</p>
+
+<p>It may be instructive to contrast this with the case of an infinitely
+narrow annular aperture, where the brightness is proportional to
+J<span class="su">0</span>²(z). When z is great,</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">J<span class="su">0</span>(z) = <span class="f150">&radic;(</span></td> <td>2</td>
+<td rowspan="2"><span class="f150">)</span> cos(z<span class="sp">-1/4</span>&pi;).</td></tr>
+<tr><td class="denom">&pi;z</td></tr></table>
+
+<p class="noind">The mean brightness varies as z<span class="sp">-1</span>; and the integral
+&int;<span class="sp1">&infin;</span><span class="su1">0</span> J<span class="su">0</span>²(z)z dz
+is not convergent.</p>
+
+<p>5. <i>Resolving Power of Telescopes.</i>&mdash;The efficiency of a telescope
+is of course intimately connected with the size of the disk
+by which it represents a mathematical point. In estimating
+theoretically the resolving power on a double star we have to
+consider the illumination of the field due to the superposition of
+the two independent images. If the angular interval between the
+components of a double star were equal to twice that expressed
+in equation (15) above, the central disks of the diffraction patterns
+would be just in contact. Under these conditions there is no
+doubt that the star would appear to be fairly resolved, since the
+brightness of its external ring system is too small to produce any
+material confusion, unless indeed the components are of very
+unequal magnitude. The diminution of the star disks with
+increasing aperture was observed by Sir William Herschel, and in
+1823 Fraunhofer formulated the law of inverse proportionality.
+In investigations extending over a long series of years, the
+advantage of a large aperture in separating the components of
+close double stars was fully examined by W. R. Dawes.</p>
+
+<p>The resolving power of telescopes was investigated also by
+J. B. L. Foucault, who employed a scale of equal bright and dark
+alternate parts; it was found to be proportional to the aperture
+and independent of the focal length. In telescopes of the best
+construction and of moderate aperture the performance is not
+sensibly prejudiced by outstanding aberration, and the limit
+imposed by the finiteness of the waves of light is practically
+reached. M. E. Verdet has compared Foucault&rsquo;s results with
+theory, and has drawn the conclusion that the radius of the
+visible part of the image of a luminous point was equal to half the
+radius of the first dark ring.</p>
+
+<p>The application, unaccountably long delayed, of this principle
+to the microscope by H. L. F. Helmholtz in 1871 is the foundation
+of the important doctrine of the <i>microscopic limit</i>. It is true that
+in 1823 Fraunhofer, inspired by his observations upon gratings,
+had very nearly hit the mark.<a name="fa3g" id="fa3g" href="#ft3g"><span class="sp">3</span></a> And a little before Helmholtz,
+E. Abbe published a somewhat more complete investigation, also
+founded upon the phenomena presented by gratings. But
+although the argument from gratings is instructive and convenient
+in some respects, its use has tended to obscure the essential unity
+of the principle of the limit of resolution whether applied to
+telescopes or microscopes.</p>
+
+<table class="nobctr" style="float: right; width: 340px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:288px; height:145px" src="images/img243.jpg" alt="" /></td></tr>
+<tr><td class="caption sc">Fig. 4.</td></tr></table>
+
+<p>In fig. 4, AB represents the axis of an optical instrument (telescope
+or microscope), A being a point of the object and B a point
+of the image. By the operation of the object-glass LL&prime; all the rays
+issuing from A arrive in the same phase at B. Thus if A be self-luminous,
+the illumination is
+a maximum at B, where all
+the secondary waves agree in
+phase. B is in fact the centre
+of the diffraction disk which
+constitutes the image of A.
+At neighbouring points the
+illumination is less, in consequence
+of the discrepancies of
+phase which there enter. In
+like manner if we take a neighbouring
+point P, also self-luminous,
+in the plane of the object, the waves which issue from
+it will arrive at B with phases no longer absolutely concordant,
+and the discrepancy of phase will increase as the interval AP
+<span class="pagenum"><a name="page244" id="page244"></a>244</span>
+increases. When the interval is very small the discrepancy,
+though mathematically existent, produces no practical effect; and
+the illumination at B due to P is as important as that due to A,
+the intensities of the two luminous sources being supposed equal.
+Under these conditions it is clear that A and P are not separated in
+the image. The question is to what amount must the distance AP
+be increased in order that the difference of situation may make itself
+felt in the image. This is necessarily a question of degree; but it
+does not require detailed calculations in order to show that the
+discrepancy first becomes conspicuous when the phases corresponding
+to the various secondary waves which travel from P to B range over
+a complete period. The illumination at B due to P then becomes
+comparatively small, indeed for some forms of aperture evanescent.
+The extreme discrepancy is that between the waves which travel
+through the outermost parts of the object-glass at L and L&prime;; so that
+if we adopt the above standard of resolution, the question is where
+must P be situated in order that the relative retardation of the rays
+PL and PL&rsquo; may on their arrival at B amount to a wave-length (&lambda;).
+In virtue of the general law that the reduced optical path is stationary
+in value, this retardation may be calculated without allowance for
+the different paths pursued on the farther side of L, L&prime;, so that the
+value is simply PL &minus; PL&prime;. Now since AP is very small, AL&prime; &minus; PL&prime; =
+AP sin &alpha;, where &alpha; is the angular semi-aperture L&prime;AB. In like manner
+PL &minus; AL has the same value, so that</p>
+
+<p class="center">PL &minus; PL&prime; = 2AP sin &alpha;.</p>
+
+<p>According to the standard adopted, the condition of resolution is
+therefore that AP, or &epsilon;, should exceed ½&lambda;/sin &alpha;. If &epsilon; be less than this,
+the images overlap too much; while if &epsilon; greatly exceed the above
+value the images become unnecessarily separated.</p>
+
+<p>In the above argument the whole space between the object and
+the lens is supposed to be occupied by matter of one refractive index,
+and &lambda; represents the wave-length <i>in this medium</i> of the kind of light
+employed. If the restriction as to uniformity be violated, what we
+have ultimately to deal with is the wave-length in the medium
+immediately surrounding the object.</p>
+
+<p>Calling the refractive index &mu;, we have as the critical value of &epsilon;,</p>
+
+<p class="center">&epsilon; = ½&lambda;<span class="su">0</span> /&mu; sin &alpha;, &emsp; &emsp; (1),</p>
+
+<p class="noind">&lambda;<span class="su">0</span> being the wave-length <i>in vacuo</i>. The denominator &mu; sin &alpha; is the
+quantity well known (after Abbe) as the &ldquo;numerical aperture.&rdquo;</p>
+
+<p>The extreme value possible for &alpha; is a right angle, so that for the
+microscopic limit we have</p>
+
+<p class="center">&epsilon; = ½&lambda;<span class="su">0</span>/&mu; &emsp; &emsp; (2).</p>
+
+<p class="noind">The limit can be depressed only by a diminution in &lambda;<span class="su">0</span>, such as
+photography makes possible, or by an increase in &mu;, the refractive
+index of the medium in which the object is situated.</p>
+
+<p>The statement of the law of resolving power has been made in a
+form appropriate to the microscope, but it admits also of immediate
+application to the telescope. If 2R be the diameter of the object-glass
+and D the distance of the object, the angle subtended by AP is
+&epsilon;/D, and the angular resolving power is given by</p>
+
+<p class="center">&lambda;/2D sin &alpha; = &lambda;/2R &emsp; &emsp; (3).</p>
+
+<p class="noind">This method of derivation (substantially due to Helmholtz) makes
+it obvious that there is no essential difference of principle between
+the two cases, although the results are conveniently stated in different
+forms. In the case of the telescope we have to deal with a linear
+measure of aperture and an angular limit of resolution, whereas in
+the case of the microscope the limit of resolution is linear, and it is
+expressed in terms of angular aperture.</p>
+
+<p>It must be understood that the above argument distinctly assumes
+that the different parts of the object are self-luminous, or at least
+that the light proceeding from the various points is without phase
+relations. As has been emphasized by G. J. Stoney, the restriction
+is often, perhaps usually, violated in the microscope. A different
+treatment is then necessary, and for some of the problems which
+arise under this head the method of Abbe is convenient.</p>
+
+<p>The importance of the general conclusions above formulated, as
+imposing a limit upon our powers of direct observation, can hardly
+be overestimated; but there has been in some quarters a tendency
+to ascribe to it a more precise character than it can bear, or even to
+mistake its meaning altogether. A few words of further explanation
+may therefore be desirable. The first point to be emphasized is that
+nothing whatever is said as to the smallness of a single object that
+may be made visible. The eye, unaided or armed with a telescope,
+is able to see, as points of light, stars subtending no sensible angle.
+The visibility of a star is a question of brightness simply, and has
+nothing to do with resolving power. The latter element enters only
+when it is a question of recognizing the duplicity of a double star,
+or of distinguishing detail upon the surface of a planet. So in the
+microscope there is nothing except lack of light to hinder the visibility
+of an object however small. But if its dimensions be much
+less than the half wave-length, it can only be seen as a whole, and its
+parts cannot be distinctly separated, although in cases near the border
+line some inference may be possible, founded upon experience of what
+appearances are presented in various cases. Interesting observations
+upon particles, <i>ultra-microscopic</i> in the above sense, have been
+recorded by H. F. W. Siedentopf and R. A. Zsigmondy (<i>Drude&rsquo;s Ann.</i>,
+1903, 10, p. 1).</p>
+
+<p>In a somewhat similar way a dark linear interruption in a bright
+ground may be visible, although its actual width is much inferior
+to the half wave-length. In illustration of this fact a simple experiment
+may be mentioned. In front of the naked eye was held a piece
+of copper foil perforated by a fine needle hole. Observed through
+this the structure of some wire gauze just disappeared at a distance
+from the eye equal to 17 in., the gauze containing 46 meshes to
+the inch. On the other hand, a single wire 0.034 in. in diameter
+remained fairly visible up to a distance of 20 ft. The ratio between
+the limiting angles subtended by the periodic structure of the gauze
+and the diameter of the wire was (.022/.034) × (240/17) = 9.1. For
+further information upon this subject reference may be made to
+<i>Phil. Mag.</i>, 1896, 42, p. 167; <i>Journ. R. Micr. Soc.</i>, 1903, p. 447.</p>
+
+<p>6. <i>Coronas or Glories.</i>&mdash;The results of the theory of the diffraction
+patterns due to circular apertures admit of an interesting
+application to <i>coronas</i>, such as are often seen encircling the sun
+and moon. They are due to the interposition of small spherules
+of water, which act the part of diffracting obstacles. In order to
+the formation of a well-defined corona it is essential that the
+particles be exclusively, or preponderatingly, of one size.</p>
+
+<p>If the origin of light be treated as infinitely small, and be seen
+in focus, whether with the naked eye or with the aid of a telescope,
+the whole of the light in the absence of obstacles would be concentrated
+in the immediate neighbourhood of the focus. At other
+parts of the field the effect is the same, in accordance with the
+principle known as Babinet&rsquo;s, whether the imaginary screen in front
+of the object-glass is generally transparent but studded with a number
+of opaque circular disks, or is generally opaque but perforated with
+corresponding apertures. Since at these points the resultant due to
+the whole aperture is zero, any two portions into which the whole
+may be divided must give equal and opposite resultants. Consider
+now the light diffracted in a direction many times more oblique than
+any with which we should be concerned, were the whole aperture
+uninterrupted, and take first the effect of a single small aperture.
+The light in the proposed direction is that determined by the size of
+the small aperture in accordance with the laws already investigated,
+and its phase depends upon the position of the aperture. If we take
+a direction such that the light (of given wave-length) from a single
+aperture vanishes, the evanescence continues even when the whole
+series of apertures is brought into contemplation. Hence, whatever
+else may happen, there must be a system of dark rings formed,
+the same as from a single small aperture. In directions other than
+these it is a more delicate question how the partial effects should be
+compounded. If we make the extreme suppositions of an infinitely
+small source and absolutely homogeneous light, there is no escape
+from the conclusion that the light in a definite direction is arbitrary,
+that is, dependent upon the chance distribution of apertures. If,
+however, as in practice, the light be heterogeneous, the source of
+finite area, the obstacles in motion, and the discrimination of different
+directions imperfect, we are concerned merely with the mean brightness
+found by varying the arbitrary phase-relations, and this is
+obtained by simply multiplying the brightness due to a single
+aperture by the number of apertures (n) (see <span class="sc"><a href="#artlinks">Interference of
+Light</a></span>, § 4). The diffraction pattern is therefore that due to a single
+aperture, merely brightened n times.</p>
+
+<p>In his experiments upon this subject Fraunhofer employed plates
+of glass dusted over with lycopodium, or studded with small metallic
+disks of uniform size; and he found that the diameters of the rings
+were proportional to the length of the waves and inversely as the
+diameter of the disks.</p>
+
+<p>In another respect the observations of Fraunhofer appear at
+first sight to be in disaccord with theory; for his measures of the
+diameters of the red rings, visible when white light was employed,
+correspond with the law applicable to dark rings, and not to the
+different law applicable to the luminous maxima. Verdet has,
+however, pointed out that the observation in this form is essentially
+different from that in which homogeneous red light is employed,
+and that the position of the red rings would correspond to the
+<i>absence</i> of blue-green light rather than to the greatest abundance of
+red light. Verdet&rsquo;s own observations, conducted with great care,
+fully confirm this view, and exhibit a complete agreement with
+theory.</p>
+
+<p>By measurements of coronas it is possible to infer the size of the
+particles to which they are due, an application of considerable
+interest in the case of natural coronas&mdash;the general rule being the
+larger the corona the smaller the water spherules. Young employed
+this method not only to determine the diameters of cloud particles
+(<i>e.g.</i> <span class="spp">1</span>&frasl;<span class="suu">1000</span> in.), but also those of fibrous material, for which the
+theory is analogous. His instrument was called the <i>eriometer</i>
+(see &ldquo;Chromatics,&rdquo; vol. iii. of supp. to <i>Ency. Brit.</i>, 1817).</p>
+
+<p>7. <i>Influence of Aberration. Optical Power of Instruments.</i>&mdash;Our
+investigations and estimates of resolving power have thus
+far proceeded upon the supposition that there are no optical
+imperfections, whether of the nature of a regular aberration or
+dependent upon irregularities of material and workmanship. In
+<span class="pagenum"><a name="page245" id="page245"></a>245</span>
+practice there will always be a certain aberration or error of phase,
+which we may also regard as the deviation of the actual wave-surface
+from its intended position. In general, we may say that
+aberration is unimportant when it nowhere (or at any rate over a
+relatively small area only) exceeds a small fraction of the wave-length
+(&lambda;). Thus in estimating the intensity at a focal point,
+where, in the absence of aberration, all the secondary waves would
+have exactly the same phase, we see that an aberration nowhere
+exceeding ¼&lambda; can have but little effect.</p>
+
+<p>The only case in which the influence of small aberration upon
+the entire image has been calculated (<i>Phil. Mag.</i>, 1879) is that of a
+rectangular aperture, traversed by a cylindrical wave with aberration
+equal to cx³. The aberration is here unsymmetrical, the wave being
+in advance of its proper place in one half of the aperture, but behind
+in the other half. No terms in x or x² need be considered. The
+first would correspond to a general turning of the beam; and the
+second would imply imperfect focusing of the central parts. The
+effect of aberration may be considered in two ways. We may
+suppose the aperture (a) constant, and inquire into the operation
+of an increasing aberration; or we may take a given value of c (<i>i.e.</i>
+a given wave-surface) and examine the effect of a varying aperture.
+The results in the second case show that an increase of aperture
+up to that corresponding to an extreme aberration of half a period
+has no ill effect upon the central band (§ 3), but it increases unduly
+the intensity of one of the neighbouring lateral bands; and the
+practical conclusion is that the best results will be obtained from an
+aperture giving an extreme aberration of from a quarter to half a
+period, and that with an increased aperture aberration is not so
+much a direct cause of deterioration as an obstacle to the attainment
+of that improved definition which should accompany the increase
+of aperture.</p>
+
+<p>If, on the other hand, we suppose the aperture given, we find
+that aberration begins to be distinctly mischievous when it amounts
+to about a quarter period, <i>i.e.</i> when the wave-surface deviates at
+each end by a quarter wave-length from the true plane.</p>
+
+<p>As an application of this result, let us investigate what amount
+of temperature disturbance in the tube of a telescope may be expected
+to impair definition. According to J. B. Biot and F. J. D.
+Arago, the index &mu; for air at t° C. and at atmospheric pressure is given
+by</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&mu; &minus; 1 =</td> <td>.00029</td> <td rowspan="2">.</td></tr>
+<tr><td class="denom">1 + .0037 t</td></tr></table>
+
+<p class="noind">If we take 0° C. as standard temperature,</p>
+
+<p class="center"> &delta;&mu; = -1.1 × 10<span class="sp">-6</span>.</p>
+
+<p class="noind">Thus, on the supposition that the irregularity of temperature t
+extends through a length l, and produces an acceleration of a quarter
+of a wave-length,</p>
+
+<p class="center">¼&lambda; = 1.1 lt × 10<span class="sp">-6</span>;</p>
+
+<p class="noind">or, if we take &lambda; = 5.3 × 10<span class="sp">-5</span>,</p>
+
+<p class="center">lt = 12,</p>
+
+<p class="noind">the unit of length being the centimetre.</p>
+
+<p>We may infer that, in the case of a telescope tube 12 cm. long,
+a stratum of air heated 1° C. lying along the top of the tube, and
+occupying a moderate fraction of the whole volume, would produce
+a not insensible effect. If the change of temperature progressed
+uniformly from one side to the other, the result would be a lateral
+displacement of the image without loss of definition; but in general
+both effects would be observable. In longer tubes a similar disturbance
+would be caused by a proportionally less difference of
+temperature. S. P. Langley has proposed to obviate such ill-effects
+by stirring the air included within a telescope tube. It has long been
+known that the definition of a carbon bisulphide prism may be much
+improved by a vigorous shaking.</p>
+
+<p>We will now consider the application of the principle to the
+formation of images, unassisted by reflection or refraction (<i>Phil. Mag.</i>,
+1881). The function of a lens in forming an image is to compensate
+by its variable thickness the differences of phase which would otherwise
+exist between secondary waves arriving at the focal point from
+various parts of the aperture. If we suppose the diameter of the
+lens to be given (2R), and its focal length &fnof; gradually to increase, the
+original differences of phase at the image of an infinitely distant
+luminous point diminish without limit. When &fnof; attains a certain
+value, say &fnof;<span class="su">1</span>, the extreme error of phase to be compensated falls
+to ¼&lambda;. But, as we have seen, such an error of phase causes no sensible
+deterioration in the definition; so that from this point onwards
+the lens is useless, as only improving an image already sensibly as
+perfect as the aperture admits of. Throughout the operation of
+increasing the focal length, the resolving power of the instrument,
+which depends only upon the aperture, remains unchanged; and
+we thus arrive at the rather startling conclusion that a telescope
+of any degree of resolving power might be constructed without an
+object-glass, if only there were no limit to the admissible focal length.
+This last proviso, however, as we shall see, takes away almost all
+practical importance from the proposition.</p>
+
+<p>To get an idea of the magnitudes of the quantities involved, let us
+take the case of an aperture of <span class="spp">1</span>&frasl;<span class="suu">5</span> in., about that of the pupil of the
+eye. The distance &fnof;<span class="su">1</span>, which the actual focal length must exceed, is
+given by</p>
+
+<p class="center">&radic; (&fnof;<span class="su">1</span>² + R²) &minus; &fnof;<span class="su">1</span>  = ¼&lambda;;</p>
+
+<p class="noind">so that</p>
+
+<p class="center">&fnof;<span class="su">1</span>  = 2R²/&lambda; &emsp; &emsp; (1).</p>
+
+<p class="noind">Thus, if &lambda; = <span class="spp">1</span>&frasl;<span class="suu">40000</span>, R = <span class="spp">1</span>&frasl;<span class="suu">10</span>, we find</p>
+
+<p class="center">&fnof;<span class="su">1</span> = 800 inches.</p>
+
+<p>The image of the sun thrown upon a screen at a distance exceeding
+66 ft., through a hole <span class="spp">1</span>&frasl;<span class="suu">5</span> in. in diameter, is therefore at least as well
+defined as that seen direct.</p>
+
+<p>As the minimum focal length increases with the square of the
+aperture, a quite impracticable distance would be required to rival
+the resolving power of a modern telescope. Even for an aperture of
+4 in., &fnof;<span class="su">1</span> would have to be 5 miles.</p>
+
+<p>A similar argument may be applied to find at what point an
+achromatic lens becomes sensibly superior to a single one. The
+question is whether, when the adjustment of focus is correct for the
+central rays of the spectrum, the error of phase for the most extreme
+rays (which it is necessary to consider) amounts to a quarter of a
+wave-length. If not, the substitution of an achromatic lens will be
+of no advantage. Calculation shows that, if the aperture be <span class="spp">1</span>&frasl;<span class="suu">5</span> in.,
+an achromatic lens has no sensible advantage if the focal length
+be greater than about 11 in. If we suppose the focal length to be
+66 ft., a single lens is practically perfect up to an aperture of 1.7 in.</p>
+
+<p>Another obvious inference from the necessary imperfection of
+optical images is the uselessness of attempting anything like an
+absolute destruction of spherical aberration. An admissible error
+of phase of ¼&lambda; will correspond to an error of <span class="spp">1</span>&frasl;<span class="suu">8</span>&lambda; in a reflecting and ½&lambda;
+in a (glass) refracting surface, the incidence in both cases being
+perpendicular. If we inquire what is the greatest admissible longitudinal
+aberration (&delta;&fnof;) in an object-glass according to the above
+rule, we find</p>
+
+<p class="center">&delta;&fnof; = &lambda;&alpha;<span class="sp">-2</span> &emsp; &emsp; (2),</p>
+
+<p class="noind">&alpha; being the angular semi-aperture.</p>
+
+<p>In the case of a single lens of glass with the most favourable curvatures,
+&delta;&fnof; is about equal to &alpha;²&fnof;, so that &alpha;<span class="sp">4</span> must not exceed &lambda;/&fnof;. For
+a lens of 3 ft. focus this condition is satisfied if the aperture does
+not exceed 2 in.</p>
+
+<p>When parallel rays fall directly upon a spherical mirror the
+longitudinal aberration is only about one-eighth as great as for the
+most favourably shaped single lens of equal focal length and aperture.
+Hence a spherical mirror of 3 ft. focus might have an
+aperture of 2½ in., and the image would not suffer materially from
+aberration.</p>
+
+<p>On the same principle we may estimate the least visible displacement
+of the eye-piece of a telescope focused upon a distant object,
+a question of interest in connexion with range-finders. It appears
+(<i>Phil. Mag.</i>, 1885, 20, p. 354) that a displacement &delta;f from the true focus
+will not sensibly impair definition, provided</p>
+
+<p class="center">&delta;&fnof; &lt; &fnof;²&lambda;/R² &emsp; &emsp; (3),</p>
+
+<p class="noind">2R being the diameter of aperture. The linear accuracy required
+is thus a function of the <i>ratio</i> of aperture to focal length. The
+formula agrees well with experiment.</p>
+
+<p>The principle gives an instantaneous solution of the question of
+the ultimate optical efficiency in the method of &ldquo;mirror-reading,&rdquo;
+as commonly practised in various physical observations. A rotation
+by which one edge of the mirror advances ¼&lambda; (while the other edge
+retreats to a like amount) introduces a phase-discrepancy of a whole
+period where before the rotation there was complete agreement. A
+rotation of this amount should therefore be easily visible, but the
+limits of resolving power are being approached; and the conclusion
+is independent of the focal length of the mirror, and of the employment
+of a telescope, provided of course that the reflected image is
+seen in focus, and that the full width of the mirror is utilized.</p>
+
+<table class="nobctr" style="float: right; width: 270px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:220px; height:101px" src="images/img245.jpg" alt="" /></td></tr>
+<tr><td class="caption sc">Fig. 5.</td></tr></table>
+
+<p>A comparison with the method of a material pointer, attached to
+the parts whose rotation is under observation, and viewed through
+a microscope, is of interest. The
+limiting efficiency of the microscope
+is attained when the angular aperture
+amounts to 180°; and it is evident
+that a lateral displacement of the point
+under observation through ½&lambda; entails
+(at the old image) a phase-discrepancy
+of a whole period, one extreme ray
+being accelerated and the other retarded
+by half that amount. We may infer that the limits of
+efficiency in the two methods are the same when the length of the
+pointer is equal to the width of the mirror.</p>
+
+<p>We have seen that in perpendicular reflection a surface error not
+exceeding <span class="spp">1</span>&frasl;<span class="suu">8</span>&lambda; may be admissible. In the case of oblique reflection
+at an angle &phi;, the error of retardation due to an elevation BD (fig. 5)
+is</p>
+
+<p class="center">
+QQ&prime; &minus; QS = BD sec &phi;(1 &minus; cos SQQ&prime;) = BD sec &phi; (1 + cos 2&phi;) = 2BD cos &phi;;</p>
+
+<p><span class="pagenum"><a name="page246" id="page246"></a>246</span></p>
+
+<p class="noind">from which it follows that an error of given magnitude in the figure
+of a surface is less important in oblique than in perpendicular
+reflection. It must, however, be borne in mind that errors can
+sometimes be compensated by altering adjustments. If a surface
+intended to be flat is affected with a slight general curvature, a
+remedy may be found in an alteration of focus, and the remedy is
+the less complete as the reflection is more oblique.</p>
+
+<p>The formula expressing the optical power of prismatic spectroscopes
+may readily be investigated upon the principles of the wave
+theory. Let A<span class="su">0</span>B<span class="su">0</span> be a plane wave-surface of the light before it falls
+upon the prisms, AB the corresponding wave-surface for a particular
+part of the spectrum after the light has passed the prisms, or after it
+has passed the eye-piece of the observing telescope. The path of a
+ray from the wave-surface A<span class="su">0</span>B<span class="su">0</span> to A or B is determined by the condition
+that the optical distance, &int; &mu; ds, is a minimum; and, as AB
+is by supposition a wave-surface, this optical distance is the same
+for both points. Thus</p>
+
+<p class="center"><span class="f150">&int;</span> &mu; ds (for A) = <span class="f150">&int;</span> &mu; ds (for B) &emsp; &emsp; (4).</p>
+
+<p class="noind">We have now to consider the behaviour of light belonging to a
+neighbouring part of the spectrum. The path of a ray from the
+wave-surface A<span class="su">0</span>B<span class="su">0</span> to the point A is changed; but in virtue of the
+minimum property the change may be neglected in calculating the
+optical distance, as it influences the result by quantities of the second
+order only in the changes of refrangibility. Accordingly, the optical
+distance from A<span class="su">0</span>B<span class="su">0</span> to A is represented by &int;(&mu; + &delta;&mu;)ds, the integration
+being along the original path A<span class="su">0</span> ... A; and similarly the optical
+distance between A<span class="su">0</span>B<span class="su">0</span> and B is represented by &int; (&mu; + &delta;&mu;)ds, the
+integration being along B<span class="su">0</span> ... B. In virtue of (4) the difference
+of the optical distances to A and B is</p>
+
+<p class="center"><span class="f150">&int;</span> &delta;&mu; ds (along B<span class="su">0</span> ... B) &minus; <span class="f150">&int;</span> &delta;&mu; ds (along A<span class="su">0</span> ... A) &emsp; &emsp; (5).</p>
+
+<p class="noind">The new wave-surface is formed in such a position that the optical
+distance is constant; and therefore the <i>dispersion</i>, or the angle
+through which the wave-surface is turned by the change of refrangibility,
+is found simply by dividing (5) by the distance AB. If, as
+in common flint-glass spectroscopes, there is only one dispersing
+substance, &int; &delta;&mu; ds = &delta;&mu;·s, where s is simply the thickness traversed
+by the ray. If t<span class="su">2</span> and t<span class="su">1</span> be the thicknesses traversed by the extreme
+rays, and a denote the width of the emergent beam, the dispersion
+&theta; is given by</p>
+
+<p class="center">&theta; = &delta;&mu;(t<span class="su">2</span> &minus; t<span class="su">1</span>)/a,</p>
+
+<p class="noind">or, if t<span class="su">1</span> be negligible,</p>
+
+<p class="center">&theta; = &delta;&mu;t/a &emsp; &emsp; (6).</p>
+
+<p class="noind">The condition of resolution of a double line whose components
+subtend an angle &theta; is that &theta; must exceed &lambda;/a. Hence, in order
+that a double line may be resolved whose components have indices
+&mu; and &mu; + &delta;&mu;, it is necessary that t should exceed the value given
+by the following equation:&mdash;</p>
+
+<p class="center">t = &lambda;/&delta;&mu; &emsp; &emsp; (7).</p>
+
+<p>8. <i>Diffraction Gratings.</i>&mdash;Under the heading &ldquo;Colours of
+Striated Surfaces,&rdquo; Thomas Young (<i>Phil. Trans.</i>, 1802) in his
+usual summary fashion gave a general explanation of these
+colours, including the law of sines, the striations being supposed
+to be straight, parallel and equidistant. Later, in his article
+&ldquo;Chromatics&rdquo; in the supplement to the 5th edition of this
+encyclopaedia, he shows that the colours &ldquo;lose the mixed
+character of periodical colours, and resemble much more the
+ordinary prismatic spectrum, with intervals completely dark
+interposed,&rdquo; and explains it by the consideration that any phase-difference
+which may arise at neighbouring striae is multiplied in
+proportion to the total number of striae.</p>
+
+<p>The theory was further developed by A. J. Fresnel (1815), who
+gave a formula equivalent to (5) below. But it is to J. von
+Fraunhofer that we owe most of our knowledge upon this subject.
+His recent discovery of the &ldquo;fixed lines&rdquo; allowed a precision of
+observation previously impossible. He constructed gratings up
+to 340 periods to the inch by straining fine wire over screws.
+Subsequently he ruled gratings on a layer of gold-leaf attached to
+glass, or on a layer of grease similarly supported, and again by
+attacking the glass itself with a diamond point. The best gratings
+were obtained by the last method, but a suitable diamond point
+was hard to find, and to preserve. Observing through a telescope
+with light perpendicularly incident, he showed that the position
+of any ray was dependent only upon the grating interval, viz. the
+distance from the centre of one wire or line to the centre of the
+next, and not otherwise upon the thickness of the wire and the
+magnitude of the interspace. In different gratings the lengths
+of the spectra and their distances from the axis were inversely
+proportional to the grating interval, while with a given grating
+the distances of the various spectra from the axis were as 1, 2, 3,
+&amp;c. To Fraunhofer we owe the first accurate measurements
+of wave-lengths, and the method of separating the overlapping
+spectra by a prism dispersing in the perpendicular direction.
+He described also the complicated patterns seen when a point of
+light is viewed through two superposed gratings, whose lines cross
+one another perpendicularly or obliquely. The above observations
+relate to transmitted light, but Fraunhofer extended his
+inquiry to the light <i>reflected</i>. To eliminate the light returned
+from the hinder surface of an engraved grating, he covered it with
+a black varnish. It then appeared that under certain angles of
+incidence parts of the resulting spectra were <i>completely polarized</i>.
+These remarkable researches of Fraunhofer, carried out in the
+years 1817-1823, are republished in his <i>Collected Writings</i>
+(Munich, 1888).</p>
+
+<p>The principle underlying the action of gratings is identical with
+that discussed in § 2, and exemplified in J. L. Soret&rsquo;s &ldquo;zone plates.&rdquo;
+The alternate Fresnel&rsquo;s zones are blocked out or otherwise modified;
+in this way the original compensation is upset and a revival of light
+occurs in unusual directions. If the source be a point or a line, and
+a collimating lens be used, the incident waves may be regarded as
+plane. If, further, on leaving the grating the light be received by a
+focusing lens, <i>e.g.</i> the object-glass of a telescope, the Fresnel&rsquo;s zones
+are reduced to parallel and equidistant straight strips, which at
+certain angles coincide with the ruling. The directions of the lateral
+spectra are such that the passage from one element of the grating
+to the corresponding point of the next implies a retardation of
+an integral number of wave-lengths. If the grating be composed
+of alternate transparent and opaque parts, the question may be
+treated by means of the general integrals (§ 3) by merely limiting
+the integration to the transparent parts of the aperture. For an
+investigation upon these lines the reader is referred to Airy&rsquo;s
+<i>Tracts</i>, to Verdet&rsquo;s <i>Leçons</i>, or to R. W. Wood&rsquo;s <i>Physical Optics</i>. If,
+however, we assume the theory of a simple rectangular aperture
+(§ 3); the results of the ruling can be inferred by elementary methods,
+which are perhaps more instructive.</p>
+
+<p>Apart from the ruling, we know that the image of a mathematical
+line will be a series of narrow bands, of which the central one is
+by far the brightest. At the middle of this band there is complete
+agreement of phase among the secondary waves. The dark lines
+which separate the bands are the places at which the phases of the
+secondary wave range over an integral number of periods. If now
+we suppose the aperture AB to be covered by a great number of
+opaque strips or bars of width d, separated by transparent intervals
+of width a, the condition of things in the directions just spoken of
+is not materially changed. At the central point there is still complete
+agreement of phase; but the amplitude is diminished in the ratio of
+a : a + d. In another direction, making a small angle with the last,
+such that the projection of AB upon it amounts to a few wave-lengths,
+it is easy to see that the mode of interference is the same as
+if there were no ruling. For example, when the direction is such that
+the projection of AB upon it amounts to one wave-length, the
+elementary components neutralize one another, because their phases
+are distributed symmetrically, though discontinuously, round the
+entire period. The only effect of the ruling is to diminish the
+amplitude in the ratio a : a + d; and, except for the difference in
+illumination, the appearance of a line of light is the same as if the
+aperture were perfectly free.</p>
+
+<p>The lateral (spectral) images occur in such directions that the
+projection of the element (a + d) of the grating upon them is an exact
+multiple of &lambda;. The effect of each of the n elements of the grating
+is then the same; and, unless this vanishes on account of a particular
+adjustment of the ratio a : d, the resultant amplitude becomes comparatively
+very great. These directions, in which the retardation
+between A and B is exactly mn&lambda;, may be called the principal directions.
+On either side of any one of them the illumination is distributed
+according to the same law as for the central image (m = 0),
+vanishing, for example, when the retardation amounts to (mn ± 1)&lambda;.
+In considering the relative brightnesses of the different spectra, it
+is therefore sufficient to attend merely to the principal directions,
+provided that the whole deviation be not so great that its cosine
+differs considerably from unity.</p>
+
+<p>We have now to consider the amplitude due to a single element,
+which we may conveniently regard as composed of a transparent
+part a bounded by two opaque parts of width ½d. The phase of
+the resultant effect is by symmetry that of the component which
+comes from the middle of a. The fact that the other components
+have phases differing from this by amounts ranging between
+± am&pi;/(a + d) causes the resultant amplitude to be less than for
+the central image (where there is complete phase agreement).
+<span class="pagenum"><a name="page247" id="page247"></a>247</span>
+If B<span class="su">m</span> denote the brightness of the m<span class="sp">th</span> lateral image, and B<span class="su">0</span> that
+of the central image, we have</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">B<span class="su">m</span> : B <span class="su">0</span> = <span class="f150">[&int;</span></td> <td class="bk">+ am&pi;/(a+d)</td>
+<td rowspan="2">cosx dx ÷</td> <td>2am&pi;</td>
+<td rowspan="2"><span class="f150">]</span></td> <td>²</td>
+<td rowspan="2">= <span class="f150">(</span></td> <td>a + d</td>
+<td rowspan="2"><span class="f150">)</span></td> <td>²</td>
+<td rowspan="2">sin²</td> <td>am&pi;</td>
+<td rowspan="2"> &emsp; &emsp; (1).</td></tr>
+<tr><td class="bk">&minus; am&pi;/(a+d)</td> <td class="denom">a + d</td>
+ <td>&nbsp;</td> <td class="denom">am&pi;</td>
+ <td>&nbsp;</td> <td class="denom">a + d</td></tr></table>
+
+<p class="noind">If B denotes the brightness of the central image when the whole of
+the space occupied by the grating is transparent, we have</p>
+
+<p class="center">B<span class="su">0</span> : B = a² : (a + d)²,</p>
+
+<p class="noind">and thus</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">B<span class="su">m</span> : B =</td> <td>1</td>
+<td rowspan="2">sin²</td> <td>am&pi;</td> <td rowspan="2"> &emsp; &emsp; (2).</td></tr>
+<tr><td class="denom">m²&pi;²</td> <td class="denom">a + d</td></tr></table>
+
+<p>The sine of an angle can never be greater than unity; and consequently
+under the most favourable circumstances only 1/m²&pi;² of
+the original light can be obtained in the m<span class="sp">th</span> spectrum. We conclude
+that, with a grating composed of transparent and opaque
+parts, the utmost light obtainable in any one spectrum is in the first,
+and there amounts to 1/&pi;², or about <span class="spp">1</span>&frasl;<span class="suu">10</span>, and that for this purpose
+a and d must be equal. When d = a the general formula becomes</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">B<span class="su">m</span> : B =</td> <td>sin² ½m&pi;</td>
+<td rowspan="2"> &emsp; &emsp; (3),</td></tr>
+<tr><td class="denom">m²&pi;²</td></tr></table>
+
+<p class="noind">showing that, when m is even, B<span class="su">m</span> vanishes, and that, when m is odd,</p>
+
+<p class="center">B<span class="su">m</span> : B = 1/m²&pi;².</p>
+
+<p class="noind">The third spectrum has thus only <span class="spp">1</span>&frasl;<span class="suu">9</span> of the brilliancy of the first.</p>
+
+<p>Another particular case of interest is obtained by supposing a
+small relatively to (a + d). Unless the spectrum be of very high
+order, we have simply</p>
+
+<p class="center">B<span class="su">m</span> : B = {a/(a + d)}² &emsp; &emsp; (4);</p>
+
+<p class="noind">so that the brightnesses of all the spectra are the same.</p>
+
+<p>The light stopped by the opaque parts of the grating, together
+with that distributed in the central image and lateral spectra, ought
+to make up the brightness that would be found in the central image,
+were all the apertures transparent. Thus, if a = d, we should have</p>
+
+<p class="center">1 = ½ + ¼ + 2/&pi;² (1 + <span class="spp">1</span>&frasl;<span class="suu">9</span> + <span class="spp">1</span>&frasl;<span class="suu">25</span> + ...),</p>
+
+<p class="noind">which is true by a known theorem. In the general case</p>
+
+<table class="math0" summary="math">
+<tr><td>a</td> <td rowspan="2">= <span class="f150">(</span></td> <td>a</td>
+ <td rowspan="2"><span class="f150">)</span></td> <td>²</td>
+ <td rowspan="2">+</td> <td>2</td>
+ <td rowspan="2"><span class="f150">&Sigma;</span></td> <td class="bk">m=&infin;</td> <td>1</td>
+ <td rowspan="2">sin²<span class="f150">(</span></td> <td>m&pi;a</td>
+ <td rowspan="2"><span class="f150">)</span>,</td></tr>
+<tr><td class="denom">a + d</td> <td class="denom">a + d</td>
+ <td>&nbsp;</td> <td class="denom">&pi;²</td>
+ <td class="bk">m=1</td> <td class="denom">m²</td> <td class="denom">a + d</td></tr></table>
+
+<p class="noind">a formula which may be verified by Fourier&rsquo;s theorem.</p>
+
+<p>According to a general principle formulated by J. Babinet, the
+brightness of a lateral spectrum is not affected by an interchange
+of the transparent and opaque parts of the grating. The vibrations
+corresponding to the two parts are precisely antagonistic, since if
+both were operative the resultant would be zero. So far as the
+application to gratings is concerned, the same conclusion may be
+derived from (2).</p>
+
+<table class="nobctr" style="float: left; width: 190px;" summary="Illustration">
+<tr><td class="figleft1"><img style="width:136px; height:178px" src="images/img247a.jpg" alt="" /></td></tr>
+<tr><td class="caption sc">Fig. 6.</td></tr></table>
+
+<p>From the value of B<span class="su">m</span> : B<span class="su">0</span> we see that no lateral spectrum can
+surpass the central image in brightness; but this result depends upon
+the hypothesis that the ruling acts by opacity, which is generally
+very far from being the case in practice. In an engraved glass
+grating there is no opaque material present by which light could be
+absorbed, and the effect depends upon a difference of retardation in
+passing the alternate parts. It is possible to prepare gratings which
+give a lateral spectrum brighter than the central image, and the explanation
+is easy. For if the alternate parts were equal and alike
+transparent, but so constituted as to give a relative retardation of
+½&lambda;, it is evident that the central image would be entirely extinguished,
+while the first spectrum would be four times as bright as if the
+alternate parts were opaque. If it were possible to introduce at
+every part of the aperture of the grating an arbitrary retardation,
+all the light might be concentrated in any desired spectrum. By
+supposing the retardation to vary uniformly and continuously we
+fall upon the case of an ordinary prism: but there
+is then no diffraction spectrum in the usual sense.
+To obtain such it would be necessary that the
+retardation should gradually alter by a wave-length
+in passing over any element of the grating,
+and then fall back to its previous value, thus
+springing suddenly over a wave-length (<i>Phil.
+Mag.</i>, 1874, 47, p. 193). It is not likely that such
+a result will ever be fully attained in practice; but
+the case is worth stating, in order to show that
+there is no theoretical limit to the concentration
+of light of assigned wave-length in one spectrum,
+and as illustrating the frequently observed unsymmetrical
+character of the spectra on the two sides of the central
+image.<a name="fa4g" id="fa4g" href="#ft4g"><span class="sp">4</span></a></p>
+
+<p>We have hitherto supposed that the light is incident perpendicularly
+upon the grating; but the theory is easily extended. If
+the incident rays make an angle &theta; with the normal (fig. 6), and the
+diffracted rays make an angle &phi; (upon the same side), the relative
+retardation from each element of width (a + d) to the next is
+(a + d) (sin&theta; + sin&phi;); and this is the quantity which is to be equated
+to m&lambda;. Thus</p>
+
+<p class="center">sin&theta; + sin&phi; = 2sin ½(&theta; + &phi;) cos ½ (&theta; &minus; &phi;) = m&lambda;/(a + d) &emsp; &emsp; (5).</p>
+
+<p>The &ldquo;deviation&rdquo; is (&theta; + &phi;), and is therefore a minimum when
+&theta; = &phi;, <i>i.e.</i> when the grating is so situated that the angles of incidence
+and diffraction are equal.</p>
+
+<table class="nobctr" style="float: right; width: 220px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:165px; height:102px" src="images/img247b.jpg" alt="" /></td></tr>
+<tr><td class="caption sc">Fig. 7.</td></tr></table>
+
+<p>In the case of a reflection grating the same method applies. If
+&theta; and &phi; denote the angles with the normal made by the incident
+and diffracted rays, the formula (5) still holds,
+and, if the deviation be reckoned from the
+direction of the regularly reflected rays, it is
+expressed as before by (&theta; + &phi;), and is a minimum
+when &theta; = &phi;, that is, when the diffracted
+rays return upon the course of the incident
+rays.</p>
+
+<p>In either case (as also with a prism) the
+position of minimum deviation leaves the
+width of the beam unaltered, <i>i.e.</i> neither magnifies nor diminishes the
+angular width of the object under view.</p>
+
+<p>From (5) we see that, when the light falls perpendicularly upon
+a grating (&theta; = 0), there is no spectrum formed (the image corresponding
+to m = 0 not being counted as a spectrum), if the grating
+interval &sigma; or (a + d) is less than &lambda;. Under these circumstances,
+if the material of the grating be completely transparent, the whole
+of the light must appear in the direct image, and the ruling is not
+perceptible. From the absence of spectra Fraunhofer argued that
+there must be a microscopic limit represented by &lambda;; and the inference
+is plausible, to say the least (<i>Phil. Mag.</i>, 1886). Fraunhofer
+should, however, have fixed the microscopic limit at ½&lambda;, as appears
+from (5), when we suppose &theta; = ½&pi;, &phi; = ½&pi;.</p>
+
+<table class="nobctr" style="float: right; width: 210px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:163px; height:114px" src="images/img247c.jpg" alt="" /></td></tr>
+<tr><td class="caption sc">Fig. 8.</td></tr></table>
+
+<p>We will now consider the important subject of the resolving
+power of gratings, as dependent upon the
+number of lines (n) and the order of the spectrum
+observed (m). Let BP (fig. 8) be the
+direction of the principal maximum (middle
+of central band) for the wave-length &lambda; in the
+m<span class="sp">th</span> spectrum. Then the relative retardation
+of the extreme rays (corresponding to the
+edges A, B of the grating) is mn&lambda;. If BQ
+be the direction for the first minimum (the
+darkness between the central and first lateral
+band), the relative retardation of the extreme rays is (mn + 1)&lambda;.
+Suppose now that &lambda; + &delta;&lambda; is the wave-length for which BQ gives the
+principal maximum, then</p>
+
+<p class="center">(mn + 1)&lambda; = mn(&lambda; + &delta;&lambda;);</p>
+
+<p class="noind">whence</p>
+
+<p class="center">&delta;&lambda;/&lambda; = 1/mn &emsp; &emsp; (6).</p>
+
+<p class="noind">According to our former standard, this gives the smallest difference
+of wave-lengths in a double line which can be just resolved; and
+we conclude that the resolving power of a grating depends only
+upon the total number of lines, and upon the order of the spectrum,
+without regard to any other considerations. It is here of course
+assumed that the n lines are really utilized.</p>
+
+<p>In the case of the D lines the value of &delta;&lambda;/&lambda; is about 1/1000; so
+that to resolve this double line in the first spectrum requires 1000
+lines, in the second spectrum 500, and so on.</p>
+
+<p>It is especially to be noticed that the resolving power does not
+depend directly upon the closeness of the ruling. Let us take the
+case of a grating 1 in. broad, and containing 1000 lines, and consider
+the effect of interpolating an additional 1000 lines, so as to bisect
+the former intervals. There will be destruction by interference of
+the first, third and odd spectra generally; while the advantage
+gained in the spectra of even order is not in dispersion, nor in
+resolving power, but simply in brilliancy, which is increased four
+times. If we now suppose half the grating cut away, so as to leave
+1000 lines in half an inch, the dispersion will not be altered, while the
+brightness and resolving power are halved.</p>
+
+<p>There is clearly no theoretical limit to the resolving power of
+gratings, even in spectra of given order. But it is possible that,
+as suggested by Rowland,<a name="fa5g" id="fa5g" href="#ft5g"><span class="sp">5</span></a> the structure of natural spectra may
+be too coarse to give opportunity for resolving powers much higher
+than those now in use. However this may be, it would always
+be possible, with the aid of a grating of given resolving power, to
+construct artificially from white light mixtures of slightly different
+wave-length whose resolution or otherwise would discriminate
+between powers inferior and superior to the given one.<a name="fa6g" id="fa6g" href="#ft6g"><span class="sp">6</span></a></p>
+
+<p><span class="pagenum"><a name="page248" id="page248"></a>248</span></p>
+
+<p>If we define as the &ldquo;dispersion&rdquo; in a particular part of the
+spectrum the ratio of the angular interval d&theta; to the corresponding
+increment of wave-length d&lambda;, we may express it by a very simple
+formula. For the alteration of wave-length entails, at the two
+limits of a diffracted wave-front, a relative retardation equal to
+mnd&lambda;. Hence, if a be the width of the diffracted beam, and d&theta; the
+angle through which the wave-front is turned,</p>
+
+<p class="center">ad&theta; = mn d&lambda;,</p>
+
+<p class="noind">or</p>
+
+<p class="center">dispersion = mn/a &emsp; &emsp; (7).</p>
+
+<p>The resolving power and the width of the emergent beam fix
+the optical character of the instrument. The latter element must
+eventually be decreased until less than the diameter of the pupil
+of the eye. Hence a wide beam demands treatment with further
+apparatus (usually a telescope) of high magnifying power.</p>
+
+<p>In the above discussion it has been supposed that the ruling is
+accurate, and we have seen that by increase of m a high resolving
+power is attainable with a moderate number of lines. But this
+procedure (apart from the question of illumination) is open to the
+objection that it makes excessive demands upon accuracy. According
+to the principle already laid down it can make but little difference
+in the principal direction corresponding to the first spectrum,
+provided each line lie within a quarter of an interval (a + d) from its
+theoretical position. But, to obtain an equally good result in the
+m<span class="sp">th</span> spectrum, the error must be less than 1/m of the above amount.<a name="fa7g" id="fa7g" href="#ft7g"><span class="sp">7</span></a></p>
+
+<p>There are certain errors of a systematic character which demand
+special consideration. The spacing is usually effected by means of
+a screw, to each revolution of which corresponds a large number
+(<i>e.g.</i> one hundred) of lines. In this way it may happen that although
+there is almost perfect periodicity with each revolution of the screw
+after (say) 100 lines, yet the 100 lines themselves are not equally
+spaced. The &ldquo;ghosts&rdquo; thus arising were first described by G. H.
+Quincke (<i>Pogg. Ann.</i>, 1872, 146, p. 1), and have been elaborately
+investigated by C. S. Peirce (<i>Ann. Journ. Math.</i>, 1879, 2, p. 330), both
+theoretically and experimentally. The general nature of the effects
+to be expected in such a case may be made clear by means of an illustration
+already employed for another purpose. Suppose two similar
+and accurately ruled transparent gratings to be superposed in such
+a manner that the lines are parallel. If the one set of lines exactly
+bisect the intervals between the others, the grating interval is
+practically halved, and the previously existing spectra of odd order
+vanish. But a very slight relative displacement will cause the
+apparition of the odd spectra. In this case there is approximate
+periodicity in the half interval, but complete periodicity only after
+the whole interval. The advantage of approximate bisection lies
+in the superior brilliancy of the surviving spectra; but in any case
+the compound grating may be considered to be perfect in the
+longer interval, and the definition is as good as if the bisection were
+accurate.</p>
+
+<table class="nobctr" summary="Illustration">
+<tr><td class="figcenter" colspan="4"><img style="width:473px; height:87px" src="images/img248a.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 9.</span>&mdash;x².</td> <td class="caption"><span class="sc">Fig. 10.</span>&mdash;y².</td>
+<td class="caption"><span class="sc">Fig. 11.</span>&mdash;x³.</td> <td class="caption"><span class="sc">Fig. 12.</span>&mdash;xy².</td></tr></table>
+
+<table class="nobctr" style="float: left; width: 400px;" summary="Illustration">
+<tr><td class="figleft1"><img style="width:347px; height:82px" src="images/img248b.jpg" alt="" /></td></tr>
+<tr><td class="caption"><span class="sc">Fig. 13.</span>&mdash;xy. &emsp; &emsp; &emsp; <span class="sc">Fig. 14.</span>&mdash;x²y. &emsp; &emsp; &emsp; <span class="sc">Fig. 15.</span>&mdash;y³.</td></tr></table>
+
+<p>The effect of a gradual increase in the interval (fig. 9) as we
+pass across the grating has been investigated by M. A. Cornu
+(<i>C.R.</i>, 1875, 80, p. 655), who thus explains an anomaly observed by
+E. E. N. Mascart. The latter found that certain gratings exercised
+a converging power upon the spectra formed upon one side, and a
+corresponding diverging power upon the spectra on the other side.
+Let us suppose that the light is incident perpendicularly, and that
+the grating interval increases from the centre towards that edge
+which lies nearest to the spectrum under observation, and decreases
+towards the hinder edge. It is evident that the waves from <i>both</i>
+halves of the
+grating are accelerated
+in an
+increasing degree,
+as we pass from
+the centre outwards,
+as compared
+with the
+phase they would possess were the central value of the grating
+interval maintained throughout. The irregularity of spacing has
+thus the effect of a convex lens, which accelerates the marginal
+relatively to the central rays. On the other side the effect is
+reversed. This kind of irregularity may clearly be present in a
+degree surpassing the usual limits, without loss of definition, when
+the telescope is focused so as to secure the best effect.</p>
+
+<p>It may be worth while to examine further the other variations
+from correct ruling which correspond to the various terms expressing
+the deviation of the wave-surface from a perfect plane. If x and y
+be co-ordinates in the plane of the wave-surface, the axis of y being
+parallel to the lines of the grating, and the origin corresponding
+to the centre of the beam, we may take as an approximate equation
+to the wave-surface</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">z =</td> <td>x²</td>
+<td rowspan="2">+ Bxy +</td> <td>y²</td>
+<td rowspan="2">+ &alpha;x³ + &beta;x²y + &gamma;xy² + &delta;y³ + ... &emsp; &emsp; (8);</td></tr>
+<tr><td class="denom">2&rho;</td> <td class="denom">2&rho;&prime;</td></tr></table>
+
+<p class="noind">and, as we have just seen, the term in x² corresponds to a linear
+error in the spacing. In like manner, the term in y² corresponds
+to a general <i>curvature</i> of the lines (fig. 10), and does not influence
+the definition at the (primary) focus, although it may introduce
+astigmatism.<a name="fa8g" id="fa8g" href="#ft8g"><span class="sp">8</span></a> If we suppose that everything is symmetrical on
+the two sides of the primary plane y = 0, the coefficients B, &beta;, &delta;
+vanish. In spite of any inequality between &rho; and &rho;&rsquo;, the definition
+will be good to this order of approximation, provided &alpha; and &gamma; vanish.
+The former measures the <i>thickness</i> of the primary focal line, and the
+latter measures its <i>curvature</i>. The error of ruling giving rise to &alpha; is
+one in which the intervals increase or decrease in <i>both</i> directions
+from the centre outwards (fig. 11), and it may often be compensated
+by a slight rotation in azimuth of the object-glass of the observing
+telescope. The term in &gamma; corresponds to a <i>variation</i> of curvature
+in crossing the grating (fig. 12).</p>
+
+<p>When the plane zx is not a plane of symmetry, we have to consider
+the terms in xy, x²y, and y³. The first of these corresponds to a deviation
+from parallelism, causing the interval to alter gradually as we pass
+<i>along</i> the lines (fig. 13). The error thus arising may be compensated
+by a rotation of the object-glass about one of the diameters y = ± x.
+The term in x²y corresponds to a deviation from parallelism in the
+same direction on both sides of the central line (fig. 14); and that in
+y³ would be caused by a curvature such that there is a point of
+inflection at the middle of each line (fig. 15).</p>
+
+<p>All the errors, except that depending on &alpha;, and especially those
+depending on &gamma; and &delta;, can be diminished, without loss of resolving
+power, by contracting the <i>vertical</i> aperture. A linear error in the
+spacing, and a general curvature of the lines, are eliminated in the
+ordinary use of a grating.</p>
+
+<p>The explanation of the difference of focus upon the two sides as
+due to unequal spacing was verified by Cornu upon gratings purposely
+constructed with an increasing interval. He has also shown how to
+rule a plane surface with lines so disposed that the grating shall of
+itself give well-focused spectra.</p>
+
+<table class="nobctr" style="float: right; width: 230px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:184px; height:190px" src="images/img248c.jpg" alt="" /></td></tr>
+<tr><td class="caption1 sc">Fig. 16.</td></tr></table>
+
+<p>A similar idea appears to have guided H. A. Rowland to his
+brilliant invention of concave gratings, by
+which spectra can be photographed without
+any further optical appliance. In these
+instruments the lines are ruled upon a
+spherical surface of speculum metal, and
+mark the intersections of the surface by a
+system of parallel and equidistant planes,
+of which the middle member passes through
+the centre of the sphere. If we consider for
+the present only the primary plane of symmetry,
+the figure is reduced to two dimensions.
+Let AP (fig. 16) represent the surface
+of the grating, O being the centre of the
+circle. Then, if Q be any radiant point and
+Q&rsquo; its image (primary focus) in the spherical mirror AP, we have</p>
+
+<table class="math0" summary="math">
+<tr><td>1</td> <td rowspan="2">+</td> <td>1</td>
+<td rowspan="2">=</td> <td>2cos&phi;</td> <td rowspan="2">,</td></tr>
+<tr><td class="denom">v<span class="su">1</span></td> <td class="denom">u</td> <td class="denom">a</td></tr></table>
+
+<p class="noind">where v<span class="su">1</span> = AQ&prime;, u = AQ, a = OA, &phi; = angle of incidence QAO, equal to
+the angle of reflection Q&prime;AO. If Q be on the circle described upon
+OA as diameter, so that u = a cos &phi;, then Q&prime; lies also upon the same
+circle; and in this case it follows from the symmetry that the
+unsymmetrical aberration (depending upon a) vanishes.</p>
+
+<p>This disposition is adopted in Rowland&prime;s instrument; only, in
+addition to the central image formed at the angle &phi;&prime; = &phi;, there are
+a series of spectra with various values of &phi;&rsquo;, but all disposed upon
+the same circle. Rowland&rsquo;s investigation is contained in the paper
+already referred to; but the following account of the theory is in
+the form adopted by R. T. Glazebrook (<i>Phil. Mag.</i>, 1883).</p>
+
+<p>In order to find the difference of optical distances between the
+courses QAQ&prime;, QPQ&prime;, we have to express QP &minus; QA, PQ&prime; &minus; AQ&prime;. To
+find the former, we have, if OAQ = &phi;, AOP = &omega;,</p>
+
+<p class="center">QP² = u² + 4a²sin²½&omega; &minus; 4au sin ½&omega; sin (½&omega; &minus; &phi;)<br />
+= (u + a sin &phi; sin &omega;)² &minus; a² sin²&phi; sin²&omega; + 4a sin² ½&omega;(a &minus; u cos&phi;).</p>
+
+<p><span class="pagenum"><a name="page249" id="page249"></a>249</span></p>
+
+<p class="noind">Now as far as &omega;<span class="sp">4</span></p>
+
+<p class="center">4 sin² ½&omega; = sin²&omega; + ¼sin<span class="sp">4</span>&omega;,</p>
+
+<p class="noind">and thus to the same order</p>
+
+<p class="center">QP² = (u + a sin &phi; sin &omega;)²<br />
+&minus; a cos &phi;(u &minus; a cos &phi;) sin²&omega; + ¼ a(a &minus; u cos &phi;) sin<span class="sp">4</span> &omega;.</p>
+
+<p class="noind">pose that Q lies on the circle u = a cos &phi;, the
+middle term vanishes, and we get, correct as far as &omega;<span class="sp">4</span>,</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">QP = (u + a sin &phi; sin &omega;) <span class="f150">&radic; {</span>1 +</td> <td>a² sin² &phi; sin<span class="sp">4</span>&omega;</td>
+<td rowspan="2"><span class="f150">}</span>;</td></tr>
+<tr><td class="denom">4u</td></tr></table>
+
+<p class="noind">so that</p>
+
+<p class="center">QP &minus; u = a sin &phi; sin &omega; + <span class="spp">1</span>&frasl;<span class="suu">8</span>a sin &phi; tan &phi; sin<span class="sp">4</span> &omega; &emsp; &emsp; (9),</p>
+
+<p class="noind">in which it is to be noticed that the adjustment necessary to secure
+the disappearance of sin²&omega; is sufficient also to destroy the term in
+sin³&omega;.</p>
+
+<p>A similar expression can be found for Q&rsquo;P &minus; Q&prime;A; and thus, if
+Q&prime;A = v, Q&prime;AO = &phi;&prime;, where v = a cos &phi;&prime;, we get</p>
+
+<p class="center">QP + PQ&prime; &minus; QA -AQ&prime; = a sin &omega; (sin &phi; &minus; sin &phi;&prime;)<br />
++ <span class="spp">1</span>&frasl;<span class="suu">8</span>a sin<span class="sp">4</span> &omega; (sin &phi; tan &phi; + sin &phi;&prime; tan &phi;&prime;) &emsp; &emsp; (10).</p>
+
+<p class="noind">If &phi;&prime; = &phi;, the term of the first order vanishes, and the reduction of
+the difference of path <i>via</i> P and <i>via</i> A to a term of the fourth order
+proves not only that Q and Q&prime; are conjugate foci, but also that the
+foci are exempt from the most important term in the aberration.
+In the present application &phi;&prime; is not necessarily equal to &phi;; but if
+P correspond to a line upon the grating, the difference of retardations
+for consecutive positions of P, so far as expressed by the term
+of the first order, will be equal to &plusmn; m&lambda; (m integral), and therefore
+without influence, provided</p>
+
+<p class="center">&sigma; (sin &phi; &minus; sin&phi;&prime;) = &plusmn; m&lambda; &emsp; &emsp; (11),</p>
+
+<p class="noind">where &sigma; denotes the constant interval between the planes containing
+the lines. This is the ordinary formula for a reflecting plane
+grating, and it shows that the spectra are formed in the usual
+directions. They are here focused (so far as the rays in the primary
+plane are concerned) upon the circle OQ&prime;A, and the outstanding
+aberration is of the fourth order.</p>
+
+<p>In order that a large part of the field of view may be in focus at
+once, it is desirable that the locus of the focused spectrum should
+be nearly perpendicular to the line of vision. For this purpose
+Rowland places the eye-piece at O, so that &phi; = 0, and then by (11)
+the value of &phi;&prime; in the m<span class="sp">th</span> spectrum is</p>
+
+<p class="center">&sigma; sin &phi;&rsquo; = ± m&lambda; &emsp; &emsp; (12).</p>
+
+<p>If &omega; now relate to the edge of the grating, on which there are
+altogether n lines,</p>
+
+<p class="center">n&sigma; = 2a sin &omega;,</p>
+
+<p class="noind">and the value of the last term in (10) becomes</p>
+
+<p class="center"><span class="spp">1</span>&frasl;<span class="suu">16 </span>n&sigma;sin³ &omega; sin &phi;&prime; tan &phi;&prime;,</p>
+
+<p class="noind">or</p>
+
+<p class="center"><span class="spp">1</span>&frasl;<span class="suu">16 </span>mn&lambda; sin³&omega; tan &phi;&prime; &emsp; &emsp; (13).</p>
+
+<p>This expresses the retardation of the extreme relatively to the
+central ray, and is to be reckoned positive, whatever may be the
+signs of &omega;, and &phi;&prime;. If the semi-angular aperture (&omega;) be <span class="spp">1</span>&frasl;<span class="suu">100</span>, and
+tan &phi;&prime; = 1, mn might be as great as four millions before the error of
+phase would reach ¼&lambda;. If it were desired to use an angular aperture
+so large that the aberration according to (13) would be injurious,
+Rowland points out that on his machine there would be no difficulty
+in applying a remedy by making &sigma; slightly variable towards the
+edges. Or, retaining &sigma; constant, we might attain compensation by so
+polishing the surface as to bring the circumference slightly forward
+in comparison with the position it would occupy upon a true sphere.</p>
+
+<p>It may be remarked that these calculations apply to the rays in
+the primary plane only. The image is greatly affected with astigmatism;
+but this is of little consequence, if &gamma; in (8) be small enough.
+Curvature of the primary focal line having a very injurious effect
+upon definition, it may be inferred from the excellent performance
+of these gratings that &gamma; is in fact small. Its value does not appear
+to have been calculated. The other coefficients in (8) vanish in
+virtue of the symmetry.</p>
+
+<p>The mechanical arrangements for maintaining the focus are of
+great simplicity. The grating at A and the eye-piece at O are
+rigidly attached to a bar AO, whose ends rest on carriages, moving
+on rails OQ, AQ at right angles to each other. A tie between the
+middle point of the rod OA and Q can be used if thought desirable.</p>
+
+<p>The absence of chromatic aberration gives a great advantage in
+the comparison of overlapping spectra, which Rowland has turned
+to excellent account in his determinations of the relative wave-lengths
+of lines in the solar spectrum (<i>Phil. Mag.</i>, 1887).</p>
+
+<p>For absolute determinations of wave-lengths plane gratings are
+used. It is found (Bell, <i>Phil. Mag.</i>, 1887) that the angular
+measurements present less difficulty than the comparison of the
+grating interval with the standard metre. There is also some
+uncertainty as to the actual temperature of the grating when in
+use. In order to minimize the heating action of the light, it might
+be submitted to a preliminary prismatic analysis before it reaches
+the slit of the spectrometer, after the manner of Helmholtz.</p>
+
+<p>In spite of the many improvements introduced by Rowland and
+of the care with which his observations were made, recent workers
+have come to the conclusion that errors of unexpected amount
+have crept into his measurements of wave-lengths, and there is
+even a disposition to discard the grating altogether for fundamental
+work in favour of the so-called &ldquo;interference methods,&rdquo;
+as developed by A. A. Michelson, and by C. Fabry and J. B. Pérot.
+The grating would in any case retain its utility for the reference of
+new lines to standards otherwise fixed. For such standards
+a relative accuracy of at least one part in a million seems now
+to be attainable.</p>
+
+<p>Since the time of Fraunhofer many skilled mechanicians have
+given their attention to the ruling of gratings. Those of Nobert
+were employed by A. J. Ångström in his celebrated researches
+upon wave-lengths.  L. M. Rutherfurd introduced into common
+use the reflection grating, finding that speculum metal was less
+trying than glass to the diamond point, upon the permanence of
+which so much depends. In Rowland&rsquo;s dividing engine the
+screws were prepared by a special process devised by him, and
+the resulting gratings, plane and concave, have supplied the
+means for much of the best modern optical work. It would
+seem, however, that further improvements are not excluded.</p>
+
+<p>There are various copying processes by which it is possible
+to reproduce an original ruling in more or less perfection. The
+earliest is that of Quincke, who coated a glass grating with a
+chemical silver deposit, subsequently thickened with copper in
+an electrolytic bath. The metallic plate thus produced formed,
+when stripped from its support, a reflection grating reproducing
+many of the characteristics of the original. It is best to commence
+the electrolytic thickening in a silver acetate bath. At
+the present time excellent reproductions of Rowland&rsquo;s speculum
+gratings are on the market (Thorp, Ives, Wallace), prepared, after
+a suggestion of Sir David Brewster, by coating the original with a
+varnish, <i>e.g.</i> of celluloid. Much skill is required to secure that
+the film when stripped shall remain undeformed.</p>
+
+<p>A much easier method, applicable to glass originals, is that
+of photographic reproduction by contact printing. In several
+papers dating from 1872, Lord Rayleigh (see <i>Collected Papers</i>,
+i. 157, 160, 199, 504; iv. 226) has shown that success may
+be attained by a variety of processes, including bichromated
+gelatin and the old bitumen process, and has investigated the
+effect of imperfect approximation during the exposure between the
+prepared plate and the original. For many purposes the copies,
+containing lines up to 10,000 to the inch, are not inferior. It is
+to be desired that transparent gratings should be obtained from
+first-class ruling machines. To save the diamond point it might
+be possible to use something softer than ordinary glass as the
+material of the plate.</p>
+
+<p>9. <i>Talbot&rsquo;s Bands.</i>&mdash;These very remarkable bands are seen
+under certain conditions when a tolerably pure spectrum is regarded
+with the naked eye, or with a telescope, <i>half the aperture
+being covered by a thin plate</i>, <i>e.g.</i> <i>of glass or mica</i>. The view of the
+matter taken by the discoverer (<i>Phil. Mag.</i>, 1837, 10, p. 364) was
+that any ray which suffered in traversing the plate a retardation
+of an odd number of half wave-lengths would be extinguished,
+and that thus the spectrum would be seen interrupted by a
+number of dark bars. But this explanation cannot be accepted as
+it stands, being open to the same objection as Arago&rsquo;s theory of
+stellar scintillation.<a name="fa9g" id="fa9g" href="#ft9g"><span class="sp">9</span></a> It is as far as possible from being true that
+a body emitting homogeneous light would disappear on merely
+covering half the aperture of vision with a half-wave plate.
+Such a conclusion would be in the face of the principle of energy,
+which teaches plainly that the retardation in question leaves
+the aggregate brightness unaltered. The actual formation of
+<span class="pagenum"><a name="page250" id="page250"></a>250</span>
+the bands comes about in a very curious way, as is shown by a
+circumstance first observed by Brewster. When the retarding
+plate is held on the side towards the red of the spectrum, <i>the bands
+are not seen</i>. Even in the contrary case, the thickness of the plate
+must not exceed a certain limit, dependent upon the purity of
+the spectrum. A satisfactory explanation of these bands was first
+given by Airy (<i>Phil. Trans.</i>, 1840, 225; 1841, 1), but we shall here
+follow the investigation of Sir G. G. Stokes (<i>Phil. Trans.</i>, 1848,
+227), limiting ourselves, however, to the case where the retarded
+and unretarded beams are contiguous and of equal width.</p>
+
+<p>The aperture of the unretarded beam may thus be taken to be
+limited by x = -h, x = 0, y = -l, y= +l; and that of the beam retarded
+by R to be given by x = 0, x = h, y= -l, y = +l. For the
+former (1) § 3 gives</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&minus;</td> <td>1</td>
+<td rowspan="2"><span class="f150">&int;</span></td> <td class="bk">0</td>
+<td rowspan="2"><span class="f150">&int;</span></td> <td class="bk">+l</td>
+<td rowspan="2">sin k<span class="f150">{</span> at &minus; &fnof; +</td> <td>x&xi; + y&eta;</td>
+<td rowspan="2"><span class="f150">}</span>dxdy</td></tr>
+<tr><td class="denom">&lambda;&fnof;</td> <td class="bk">&minus;h</td>
+<td class="bk">&minus;l</td> <td class="denom">&fnof;</td></tr></table>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">= &minus;</td> <td>2lh</td>
+<td rowspan="2">·</td> <td>&fnof;</td>
+<td rowspan="2">sin</td> <td>k&eta;l</td>
+<td rowspan="2">·</td> <td>2&fnof;</td>
+<td rowspan="2">sin</td> <td>k&xi;h</td>
+<td rowspan="2">· sin k<span class="f150">{</span>at &minus; &fnof; &minus;</td> <td>&xi;h</td>
+<td rowspan="2"><span class="f150">}</span> &emsp; &emsp; (1),</td></tr>
+<tr><td class="denom">&lambda;&fnof;</td> <td class="denom">k&eta;l</td>
+<td class="denom">&fnof;</td> <td class="denom">k&xi;h</td>
+<td class="denom">2&fnof;</td> <td class="denom">2&fnof;</td></tr></table>
+
+<p class="noind">on integration and reduction.</p>
+
+<p>For the retarded stream the only difference is that we must subtract
+R from at, and that the limits of x are 0 and +h. We thus
+get for the disturbance at &xi;, &eta;, due to this stream</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&minus;</td> <td>2lh</td>
+<td rowspan="2">·</td> <td>&fnof;</td>
+<td rowspan="2">sin</td> <td>k&eta;l</td>
+<td rowspan="2">·</td> <td>2&fnof;</td>
+<td rowspan="2">sin</td> <td>k&xi;h</td>
+<td rowspan="2">· sin k<span class="f150">{</span>at &minus; &fnof; &minus; R +</td> <td>&xi;h</td>
+<td rowspan="2"><span class="f150">}</span> &emsp; &emsp; (2)</td></tr>
+<tr><td class="denom">&lambda;&fnof;</td> <td class="denom">k&eta;l</td>
+<td class="denom">&fnof;</td> <td class="denom">k&xi;h</td>
+<td class="denom">2&fnof;</td> <td class="denom">2&fnof;</td></tr></table>
+
+<p class="noind">If we put for shortness &pi; for the quantity under the last circular
+function in (1), the expressions (1), (2) may be put under the forms
+u sin &tau;, v sin (&tau; &minus; &alpha;) respectively; and, if I be the intensity, I will be
+measured by the sum of the squares of the coefficients of sin &tau; and
+cos &tau; in the expression</p>
+
+<p class="center">u sin &tau; + v sin (&tau; &minus; &alpha;),</p>
+
+<p class="noind">so that</p>
+
+<p class="center">I = u² + v² + 2uv cos &alpha;,</p>
+
+<p class="noind">which becomes on putting for u, v, and &alpha; their values, and putting</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2"><span class="f150">{</span></td> <td>&fnof;</td>
+<td rowspan="2">sin</td> <td>k&eta;l</td>
+<td rowspan="2"><span class="f150">}</span></td> <td>²</td>
+<td rowspan="2">= Q &emsp; &emsp; (3),</td></tr>
+<tr><td class="denom">k&eta;l</td> <td class="denom">&fnof;</td>
+<td>&nbsp;</td></tr></table>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">I = Q ·</td> <td>4l²</td>
+<td rowspan="2">sin²</td> <td>&pi;&xi;h</td>
+<td rowspan="2"><span class="f150">{</span>2 + 2 cos<span class="f150">(</span></td> <td>2&pi;R</td>
+<td rowspan="2">&minus;</td> <td>2&pi;&xi;h</td>
+<td rowspan="2"><span class="f150">)}</span> &emsp; &emsp; (4).</td></tr>
+<tr><td class="denom">&pi;²&xi;²</td> <td class="denom">&lambda;&fnof;</td>
+<td class="denom">&lambda;</td> <td class="denom">&lambda;&fnof;</td></tr></table>
+
+<p class="noind">If the subject of examination be a luminous line parallel to &eta;, we
+shall obtain what we require by integrating (4) with respect to &eta;
+from &minus;&infin; to +&infin;. The constant multiplier is of no especial interest
+so that we may take as applicable to the image of a line</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">I =</td> <td>2</td>
+<td rowspan="2">sin²</td> <td>&pi;&xi;h</td>
+<td rowspan="2"><span class="f150">{</span>1 + cos<span class="f150">(</span></td> <td>2&pi;R</td>
+<td rowspan="2">&minus;</td> <td>2&pi;&xi;h</td>
+<td rowspan="2"><span class="f150">)}</span> &emsp; &emsp; (5).</td></tr>
+<tr><td class="denom">&xi;²</td> <td class="denom">&lambda;&fnof;</td>
+<td class="denom">&lambda;</td> <td class="denom">&lambda;&fnof;</td></tr></table>
+
+<p class="noind">If R = ½&lambda;, I vanishes at &xi;= 0; but the whole illumination, represented
+by &int;<span class="sp1">+&infin;</span><span class="su1">&minus;&infin;</span> I d&xi;, is independent of the value of R. If R = 0,
+I = (1/&xi;²) sin² (2&pi;&xi;h/&lambda;&fnof;), in agreement with § 3, where a has the meaning
+here attached to 2h.</p>
+
+<p>The expression (5) gives the illumination at &xi; due to that part
+of the complete image whose geometrical focus is at &xi; = 0, the
+retardation for this component being R. Since we have now to
+integrate for the whole illumination at a particular point O due to
+all the components which have their foci in its neighbourhood, we
+may conveniently regard O as origin. &xi; is then the co-ordinate
+relatively to O of any focal point O&prime; for which the retardation is R;
+and the required result is obtained by simply integrating (5) with
+respect to &xi; from &minus;&infin; to +&infin;. To each value of &xi; corresponds
+a different value of &lambda;, and (in consequence of the dispersing power
+of the plate) of R. The variation of &lambda; may, however, be neglected
+in the integration, except in 2&pi;R/&lambda;, where a small variation of &lambda;
+entails a comparatively large alteration of phase. If we write</p>
+
+<p class="center">&rho; = 2&pi;R/&lambda; &emsp; &emsp; (6),</p>
+
+<p class="noind">we must regard &rho; as a function of &xi;, and we may take with sufficient
+approximation under any ordinary circumstances</p>
+
+<p class="center">&rho; = &rho;&prime; + <span class="ov">&omega;</span>&xi; &emsp; &emsp; (7),</p>
+
+<p class="noind">where &rho;&prime; denotes the value of &rho; at O, and <span class="ov">&omega;</span> is a constant, which is
+positive when the retarding plate is held at the side on which the
+lue of the spectrum <i>is seen</i>. The possibility of dark bands depends
+upon <span class="ov">&omega;</span> being positive. Only in this case can</p>
+
+<p class="center">cos {&rho;&prime; + (<span class="ov">&omega;</span> &minus; 2&pi;h/&lambda;&fnof;) &xi;}</p>
+
+<p class="noind">retain the constant value -1 throughout the integration, and then
+only when</p>
+
+<p class="center"><span class="ov">&omega;</span> = 2&pi;h / &lambda;&fnof; &emsp; &emsp; (8)</p>
+
+<p class="noind">and</p>
+
+<p class="center">cos &rho;&prime; = &minus;1 &emsp; &emsp; (9).</p>
+
+<p class="noind">The first of these equations is the condition for the formation of
+dark bands, and the second marks their situation, which is the
+same as that determined by the imperfect theory.</p>
+
+<p>The integration can be effected without much difficulty. For
+the first term in (5) the evaluation is effected at once by a known
+formula. In the second term if we observe that</p>
+
+<p class="center">cos {&rho;&prime; +(<span class="ov">&omega;</span> &minus; 2&pi;h/&lambda;&fnof;) &xi;} = cos {&rho;&prime; &minus; g<span class="su">1</span>&xi;}<br />
+= cos &rho;&prime; cos g<span class="su">1</span>&xi; + sin &rho;&prime; sin g<span class="su">1</span>&xi;,</p>
+
+<p class="noind">we see that the second part vanishes when integrated, and that
+the remaining integral is of the form</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">w = <span class="f150">&int;</span></td> <td class="bk">+&infin;</td>
+<td rowspan="2">sin² h<span class="su">1</span>&xi; cos g<span class="su">1</span>&xi;</td>
+<td>d&xi;</td> <td rowspan="2">,</td></tr>
+<tr><td class="bk">&minus;&infin;</td> <td class="denom">&xi;²</td></tr></table>
+
+<p class="noind">where</p>
+
+<p class="center">h<span class="su">1</span> = &pi;h/&lambda;&fnof;, &emsp; g<span class="su">1</span> = &omega; &minus; 2&pi;h/&lambda;&fnof; &emsp; &emsp; (10).</p>
+
+<p class="noind">By differentiation with respect to g<span class="su">1</span> it may be proved that</p>
+
+<table class="ws" summary="Contents">
+
+<tr><td class="tcl">w = 0</td> <td class="tcl">from g<span class="su">1</span> = &minus;&infin;</td> <td class="tcl">to g<span class="su">1</span> = &minus;2h<span class="su">1</span>,</td></tr>
+<tr><td class="tcl">w = ½&pi;(2h<span class="su">1</span> + g<span class="su">1</span>)</td> <td class="tcl">from g<span class="su">1</span> = &minus;2h<span class="su">1</span></td> <td class="tcl">to g<span class="su">1</span> = 0,</td></tr>
+<tr><td class="tcl">w = ½&pi;(2h<span class="su">1</span> &minus; g<span class="su">1</span>)</td> <td class="tcl">from g<span class="su">1</span> =  0</td> <td class="tcl">to g<span class="su">1</span> = 2h<span class="su">1</span>,</td></tr>
+<tr><td class="tcl">w = 0</td> <td class="tcl">from g<span class="su">1</span> = 2h<span class="su">1</span></td> <td class="tcl">to g<span class="su">1</span> = &infin;.</td></tr>
+</table>
+
+<p class="noind">The integrated intensity, I&prime;, or</p>
+
+<p class="center">2&pi;h<span class="su">1</span> + 2 cos &rho;w,</p>
+
+<p class="noind">is thus</p>
+
+<p class="center">I&prime; = 2&pi;h<span class="su">1</span> &emsp; &emsp; (11),</p>
+
+<p class="noind">when g<span class="su">1</span> numerically exceeds 2h<span class="su">1</span>; and, when g<span class="su">1</span> lies between ±2h<span class="su">1</span>,</p>
+
+<p class="center">I = &pi;{2h<span class="su">1</span> + (2h<span class="su">1</span> &minus; &radic; g<span class="su">1</span>²) cos &rho;&prime;} &emsp; &emsp; (12).</p>
+
+<p>It appears therefore that there are no bands at all unless &omega; lies
+between 0 and +4h<span class="su">1</span>, and that within these limits the best bands are
+formed at the middle of the range when &omega; = 2h<span class="su">1</span>. The formation
+of bands thus requires that the retarding plate be held upon the
+side already specified, so that &omega; be positive; and that the thickness
+of the plate (to which &omega; is proportional) do not exceed a certain
+limit, which we may call 2T<span class="su">0</span>. At the best thickness T<span class="su">0</span> the bands
+are black, and not otherwise.</p>
+
+<p>The linear width of the band (e) is the increment of &xi; which alters
+&rho; by 2&pi;, so that</p>
+
+<p class="center">e = 2&pi; / <span class="ov">&omega;</span> &emsp; &emsp; (13).</p>
+
+<p class="noind">With the best thickness</p>
+
+<p class="center"><span class="ov">&omega;</span> = 2&pi;h/&lambda;&fnof; &emsp; &emsp; (14),</p>
+
+<p class="noind">so that in this case</p>
+
+<p class="center">e = &lambda;&fnof; / h  &emsp; &emsp; (15).</p>
+
+<p class="noind">The bands are thus of the same width as those due to two infinitely
+narrow apertures coincident with the central lines of the retarded
+and unretarded streams, the subject of examination being itself a
+fine luminous line.</p>
+
+<p>If it be desired to see a given number of bands in the whole or
+in any part of the spectrum, the thickness of the retarding plate
+is thereby determined, independently of all other considerations.
+But in order that the bands may be really visible, and still more in
+order that they may be black, another condition must be satisfied.
+It is necessary that the aperture of the pupil be accommodated
+to the angular extent of the spectrum, or reciprocally. Black
+bands will be too fine to be well seen unless the aperture (2h) of
+the pupil be somewhat contracted. One-twentieth to one-fiftieth
+of an inch is suitable. The aperture and the number of bands being
+both fixed, the condition of blackness determines the angular magnitude
+of a band and of the spectrum. The use of a grating is very
+convenient, for not only are there several spectra in view at the same
+time, but the dispersion can be varied continuously by sloping the
+grating. The slits may be cut out of tin-plate, and half covered by
+mica or &ldquo;microscopic glass,&rdquo; held in position by a little cement.</p>
+
+<p>If a telescope be employed there is a distinction to be observed,
+according as the half-covered aperture is between the eye and the
+ocular, or in front of the object-glass. In the former case the
+function of the telescope is simply to increase the dispersion, and
+the formation of the bands is of course independent of the particular
+manner in which the dispersion arises. If, however, the
+half-covered aperture be in front of the object-glass, the phenomenon
+is magnified as a whole, and the desirable relation between
+the (unmagnified) dispersion and the aperture is the same as without
+the telescope. There appears to be no further advantage in the
+use of a telescope than the increased facility of accommodation,
+and for this of course a very low power suffices.</p>
+
+<p>The original investigation of Stokes, here briefly sketched, extends
+also to the case where the streams are of unequal width h, k,
+and are separated by an interval 2g. In the case of unequal width
+the bands cannot be black; but if h = k, the finiteness of 2g does
+not preclude the formation of black bands.</p>
+
+<p>The theory of Talbot&rsquo;s bands with a half-covered <i>circular</i> aperture
+has been considered by H. Struve (<i>St Peters. Trans.</i>, 1883, 31, No. 1).</p>
+
+<p>The subject of &ldquo;Talbot&rsquo;s bands&rdquo; has been treated in a very
+instructive manner by A. Schuster (<i>Phil. Mag.</i>, 1904), whose point
+of view offers the great advantage of affording an instantaneous
+explanation of the peculiarity noticed by Brewster. A plane
+<i>pulse</i>, <i>i.e.</i> a disturbance limited to an infinitely thin slice of the
+medium, is supposed to fall upon a parallel grating, which again may
+<span class="pagenum"><a name="page251" id="page251"></a>251</span>
+be regarded as formed of infinitely thin wires, or infinitely narrow
+lines traced upon glass. The secondary pulses diverted by the ruling
+fall upon an object-glass as usual, and on arrival at the focus
+constitute a procession equally spaced in time, the interval between
+consecutive members depending upon the obliquity. If a retarding
+plate be now inserted so as to operate upon the pulses which come
+from one side of the grating, while leaving the remainder unaffected,
+we have to consider what happens at the focal point chosen. A full
+discussion would call for the formal application of Fourier&rsquo;s theorem,
+but some conclusions of importance are almost obvious.</p>
+
+<p>Previously to the introduction of the plate we have an effect
+corresponding to wave-lengths closely grouped around the principal
+wave-length, viz. &sigma; sin &phi;, where &sigma; is the grating-interval and &phi; the
+obliquity, the closeness of the grouping increasing with the number
+of intervals. In addition to these wave-lengths there are other groups
+centred round the wave-lengths which are submultiples of the
+principal one&mdash;the overlapping spectra of the second and higher
+orders. Suppose now that the plate is introduced so as to cover naif
+the aperture and that it retards those pulses which would otherwise
+arrive first. The consequences must depend upon the amount of the
+retardation. As this increases from zero, the two processions which
+correspond to the two halves of the aperture begin to overlap, and
+the overlapping gradually increases until there is almost complete
+superposition. The stage upon which we will fix our attention is
+that where the one procession bisects the intervals between the
+other, so that a new simple procession is constituted, containing the
+same number of members as before the insertion of the plate, but
+now spaced at intervals only half as great. It is evident that the
+effect at the focal point is the obliteration of the first and other
+spectra of odd order, so that as regards the spectrum of the first order
+we may consider that the two beams <i>interfere</i>. The formation of
+black bands is thus explained, and it requires that the plate be
+introduced upon one particular side, and that the amount of the
+retardation be adjusted to a particular value. If the retardation
+be too little, the overlapping of the processions is incomplete, so that
+besides the procession of half period there are residues of the original
+processions of full period. The same thing occurs if the retardation
+be too great. If it exceed the double of the value necessary for
+black bands, there is again no overlapping and consequently no
+interference. If the plate be introduced upon the other side, so as
+to retard the procession originally in arrear, there is no overlapping,
+whatever may be the amount of retardation. In this way the
+principal features of the phenomenon are accounted for, and
+Schuster has shown further how to extend the results to spectra
+having their origin in prisms instead of gratings.</p>
+
+<p>10. <i>Diffraction when the Source of Light is not seen in Focus.</i>&mdash;The
+phenomena to be considered under this head are of less
+importance than those investigated by Fraunhofer, and will be
+treated in less detail; but in view of their historical interest and
+of the ease with which many of the experiments may be tried,
+some account of their theory cannot be omitted. One or two
+examples have already attracted our attention when considering
+Fresnel&rsquo;s zones, viz. the shadow of a circular disk and of a screen
+circularly perforated.</p>
+
+<p>Fresnel commenced his researches with an examination of the
+fringes, external and internal, which accompany the shadow of a
+narrow opaque strip, such as a wire. As a source of light he used
+sunshine passing through a very small hole perforated in a metal
+plate, or condensed by a lens of short focus. In the absence of a
+heliostat the latter was the more convenient. Following, unknown
+to himself, in the footsteps of Young, he deduced the
+principle of interference from the circumstance that the darkness
+of the interior bands requires the co-operation of light from both
+sides of the obstacle. At first, too, he followed Young in the view
+that the exterior bands are the result of interference between the
+direct light and that reflected from the edge of the obstacle, but
+he soon discovered that the character of the edge&mdash;<i>e.g.</i> whether
+it was the cutting edge or the back of a razor&mdash;made no material
+difference, and was thus led to the conclusion that the explanation
+of these phenomena requires nothing more than the application of
+Huygens&rsquo;s principle to the unobstructed parts of the wave. In
+observing the bands he received them at first upon a screen of
+finely ground glass, upon which a magnifying lens was focused;
+but it soon appeared that the ground glass could be dispensed with,
+the diffraction pattern being viewed in the same way as the image
+formed by the object-glass of a telescope is viewed through the
+eye-piece. This simplification was attended by a great saving of
+light, allowing measures to be taken such as would otherwise have
+presented great difficulties.</p>
+
+<table class="nobctr" style="float: right; width: 230px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:183px; height:125px" src="images/img251.jpg" alt="" /></td></tr>
+<tr><td class="caption sc">Fig. 17.</td></tr></table>
+
+<p>In theoretical investigations these problems are usually treated
+as of two dimensions only, everything being referred to the plane
+passing through the luminous point and perpendicular to the diffracting
+edges, supposed to be straight and parallel. In strictness this
+idea is appropriate only when the source is a luminous line, emitting
+cylindrical waves, such as might be obtained from a luminous point
+with the aid of a cylindrical lens. When, in order to apply Huygens&rsquo;s
+principle, the wave is supposed to be broken up, the phase is the same
+at every element of the surface of resolution which lies upon a line
+perpendicular to the plane of reference, and
+thus the effect of the whole line, or rather
+infinitesimal strip, is related in a constant
+manner to that of the element which lies
+in the plane of reference, and may be
+considered to be represented thereby. The
+same method of representation is applicable
+to spherical waves, issuing from a <i>point</i>, if
+the radius of curvature be large; for, although
+there is variation of phase along the
+length of the infinitesimal strip, the whole effect depends practically
+upon that of the central parts where the phase is sensibly constant.<a name="fa10g" id="fa10g" href="#ft10g"><span class="sp">10</span></a></p>
+
+<p>In fig. 17 APQ is the arc of the circle representative of the wave-front
+of resolution, the centre being at O, and the radius QA being
+equal to a. B is the point at which the effect is required, distant
+a + b from O, so that AB = b, AP = s, PQ = ds.</p>
+
+<p>Taking as the standard phase that of the secondary wave from
+A, we may represent the effect of PQ by</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">cos 2&pi;<span class="f150">(</span></td> <td>t</td>
+<td rowspan="2">&minus;</td> <td>&delta;</td>
+<td rowspan="2"><span class="f150">)</span>· ds,</td></tr>
+<tr><td class="denom">r</td> <td class="denom">&lambda;</td></tr></table>
+
+<p class="noind">where &delta; = BP &minus; AP is the retardation at B of the wave from P
+relatively to that from A.</p>
+
+<p class="noind">Now</p>
+
+<p class="center">&delta; = (a + b) s²/2ab &emsp; &emsp; (1),</p>
+
+<p class="noind">so that, if we write</p>
+
+<table class="math0" summary="math">
+<tr><td>2&pi;&delta;</td><td rowspan="2">=</td> <td>&pi;(a + b)s²</td>
+<td rowspan="2">=</td> <td>&pi;</td>
+<td rowspan="2">v² &emsp; &emsp; (2),</td></tr>
+<tr><td class="denom">&lambda;</td> <td class="denom">ab&lambda;</td> <td class="denom">2</td></tr></table>
+
+<p class="noind">the effect at B is</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2"><span class="f150">{</span></td> <td>ab&lambda;</td>
+<td rowspan="2"><span class="f150">}</span></td> <td>½</td>
+<td rowspan="2"><span class="f150">{</span> cos</td> <td>2&pi;t</td>
+<td rowspan="2"><span class="f150">&int;</span> cos ½&pi;v²·dv + sin</td> <td>2&pi;t</td>
+<td rowspan="2"><span class="f150">&int;</span> sin ½&pi;v²·dv <span class="f150">}</span> &emsp; &emsp; (3)</td></tr>
+<tr><td class="denom">2(a + b)</td> <td>&nbsp;</td>
+<td class="denom">&tau;</td> <td class="denom">&tau;</td></tr></table>
+
+<p class="noind">the limits of integration depending upon the disposition of the
+diffracting edges. When a, b, &lambda; are regarded as constant, the first
+factor may be omitted,&mdash;as indeed should be done for consistency&rsquo;s
+sake, inasmuch as other factors of the same nature have been
+omitted already.</p>
+
+<p>The intensity I², the quantity with which we are principally
+concerned, may thus be expressed</p>
+
+<p class="center">I²= <span class="f150">{ &int;</span> cos ½&pi;v²·dv<span class="f150">}</span>² + <span class="f150">{ &int;</span> sin ½&pi;v²·dv <span class="f150">}</span>² &emsp; &emsp; (4).</p>
+
+<p class="noind">These integrals, taken from v = 0, are known as Fresnel&rsquo;s integrals;
+we will denote them by C and S, so that</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">C = <span class="f150">&int;</span></td> <td class="bk">v</td>
+<td rowspan="2">cos ½&pi;v²·dv, &emsp; &emsp; S = <span class="f150">&int;</span></td> <td class="bk">v</td>
+<td rowspan="2">sin ½&pi;v²·dv &emsp; &emsp; (5).</td></tr>
+<tr><td class="bk">0</td> <td class="bk">0</td></tr></table>
+
+<p class="noind">When the upper limit is infinity, so that the limits correspond to
+the inclusion of half the primary wave, C and S are both equal to
+½, by a known formula; and on account of the rapid fluctuation
+of sign the parts of the range beyond very moderate values of v
+contribute but little to the result.</p>
+
+<p>Ascending series for C and S were given by K. W. Knockenhauer,
+and are readily investigated. Integrating by parts, we find</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">C + iS = <span class="f150">&int;</span></td> <td class="bk">v</td>
+<td rowspan="2">e</td> <td class="bk">i·½&pi;v²</td>
+<td rowspan="2">dv = e</td> <td class="bk">i·½&pi;v²</td>
+<td rowspan="2">· v &minus; <span class="spp">1</span>&frasl;<span class="suu">3</span> i&pi; <span class="f150">&int;</span></td> <td class="bk">v</td>
+<td rowspan="2">e</td> <td class="bk">i·½&pi;v²</td>
+<td rowspan="2">dv³;</td></tr>
+<tr><td class="bk">0</td> <td>&nbsp;</td>
+<td>&nbsp;</td> <td class="bk">0</td> <td>&nbsp;</td></tr></table>
+
+<p class="noind">and, by continuing this process,</p>
+
+<table class="math0" style="border-collapse: separate;" summary="math">
+<tr><td rowspan="2">C + iS = e</td> <td class="bk">i·½&pi;v²</td>
+<td rowspan="2"><span class="f150">{</span> v &minus;</td> <td>i&pi;</td>
+<td rowspan="2">v³ +</td> <td>i&pi;</td> <td>i&pi;</td>
+<td rowspan="2">v<span class="sp">5</span> &minus;</td> <td>i&pi;</td> <td>i&pi;</td> <td>i&pi;</td>
+<td rowspan="2">v<span class="sp">7</span> + ... <span class="f150">}</span>.</td></tr>
+<tr><td>&nbsp;</td> <td class="denom">3</td> <td class="denom">3</td> <td class="denom">5</td>
+<td class="denom">3</td> <td class="denom">5</td> <td class="denom">7</td></tr></table>
+
+<p class="noind">By separation of real and imaginary parts,</p>
+
+<table class="math0" summary="math">
+<tr><td>C = M cos ½&pi;v² &minus; N sin ½&pi;v²</td>
+<td rowspan="2"><span class="f200">}</span> &emsp; &emsp; (6),</td></tr>
+<tr><td>S = M sin ½&pi;v² &minus; N cos ½&pi;v²</td></tr></table>
+
+<p class="noind">where</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">M =</td> <td>v</td>
+<td rowspan="2">&minus;</td> <td>&pi;²v<span class="sp">5</span></td>
+<td rowspan="2">+</td> <td>&pi;<span class="sp">4</span>v<span class="sp">9</span></td>
+<td rowspan="2">&minus; ... &emsp; &emsp; (7),</td></tr>
+<tr><td class="denom">1</td> <td class="denom">3·5</td> <td class="denom">3·5·7·9</td></tr></table>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">N =</td> <td>&pi;v³</td>
+<td rowspan="2">&minus;</td> <td>&pi;<span class="sp">3</span>v<span class="sp">7</span></td>
+<td rowspan="2">+</td> <td>&pi;<span class="sp">5</span>v<span class="sp">11</span></td>
+<td rowspan="2">... &emsp; &emsp; (8).</td></tr>
+<tr><td class="denom">1·3</td> <td class="denom">1·3·5·7</td> <td class="denom">1·3·5·7·9·11</td></tr></table>
+
+<p class="noind">These series are convergent for all values of v, but are practically
+useful only when v is small.</p>
+
+<p>Expressions suitable for discussion when v is large were obtained
+<span class="pagenum"><a name="page252" id="page252"></a>252</span>
+by L. P. Gilbert (<i>Mem. cour. de l&rsquo;Acad. de Bruxelles</i>, 31, p. 1). Taking</p>
+
+<p class="center">½&pi;v² = u &emsp; &emsp; (9),</p>
+
+<p class="noind">we may write</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">C + iS =</td> <td>1</td>
+<td rowspan="2"><span class="f150">&int;</span></td> <td class="bk">u</td> <td>e<span class="sp">iu</span>du</td>
+<td rowspan="2"> &emsp; &emsp; (10).</td></tr>
+<tr><td class="denom">&radic;(2&pi;)</td> <td class="bk">0</td>
+<td class="denom">&radic;u</td></tr></table>
+
+<p class="noind">Again, by a known formula,</p>
+
+<table class="math0" summary="math">
+<tr><td>1</td> <td rowspan="2">=</td> <td>1</td>
+<td rowspan="2"><span class="f150">&int;</span></td> <td class="bk">&infin;</td> <td>e<span class="sp">&minus;ux</span>dx</td>
+<td rowspan="2"> &emsp; &emsp; (11).</td></tr>
+<tr><td class="denom">&radic; u</td> <td class="denom">&radic;&pi;</td>
+<td class="bk">0</td> <td class="denom">&radic;x</td></tr></table>
+
+<p class="noind">Substituting this in (10), and inverting the order of integration, we
+get</p>
+
+<table class="math0" style="border-collapse: separate;" summary="math">
+<tr><td rowspan="2">C + iS =</td> <td>1</td>
+<td rowspan="2"><span class="f150">&int;</span></td> <td class="bk">&infin;</td> <td>dx</td>
+<td rowspan="2"><span class="f150">&int;</span></td> <td class="bk">u</td>
+<td rowspan="2">e</td> <td class="bk">u(i &minus; x)</td>
+<td rowspan="2">dx =</td> <td>1</td>
+<td rowspan="2"><span class="f150">&int;</span></td> <td class="bk">&infin;</td>
+<td>dx</td> <td>e<span class="sp">u(i &minus; x)</span> &minus; 1</td>
+<td rowspan="2"> &emsp; &emsp; (12).</td></tr>
+<tr><td class="denom">&pi;&radic;2</td> <td class="bk">0</td>
+<td class="denom">&radic;x</td> <td class="bk">0</td>
+<td>&nbsp;</td> <td class="denom">&pi;&radic;2</td>
+<td>&nbsp;</td> <td class="denom">&radic;x</td> <td class="denom">i &minus; x</td></tr></table>
+
+<p class="noind">Thus, if we take</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">G =</td> <td>1</td>
+<td rowspan="2"><span class="f150">&int;</span></td> <td class="bk">&infin;</td> <td>e<span class="sp">&minus;ux</span> &radic;x · dx</td>
+<td rowspan="2">, H  =</td> <td>1</td>
+<td rowspan="2"><span class="f150">&int;</span></td> <td class="bk">&infin;</td> <td>e<span class="sp">&minus;ux</span> dx</td>
+<td rowspan="2"> &emsp; &emsp; (13),</td></tr>
+<tr><td class="denom">&pi;&radic;2</td> <td class="bk">0</td>
+<td class="denom">1 + x²</td> <td class="denom">&pi;&radic;2</td>
+<td class="bk">0</td> <td class="denom">&radic;x · (1 + x²)</td></tr></table>
+
+<p class="center">C = ½ &minus; G cos u + H sin u, &emsp; S = ½ &minus; G sin u &minus; H cos u &emsp; &emsp; (14).</p>
+
+<p class="noind">The constant parts in (14), viz. ½, may be determined by direct
+integration of (12), or from the observation that by their constitution
+G and H vanish when u = &infin;, coupled with the fact that C and
+S then assume the value ½.</p>
+
+<p>Comparing the expressions for C, S in terms of M, N, and in terms
+of G, H, we find that</p>
+
+<p class="center">G = ½ (cos u + sin u) &minus; M, &emsp; H = ½ (cos u &minus; sin u) + N &emsp; &emsp; &emsp; &emsp; (15),</p>
+
+<p class="noind">formulae which may be utilized for the calculation of G, H when
+u (or v) is small. For example, when u = 0, M = 0, N = 0, and consequently
+G = H = ½.</p>
+
+<p>Descending series of the semi-convergent class, available for
+numerical calculation when u is moderately large, can be obtained
+from (12) by writing x = uy, and expanding the denominator in
+powers of y. The integration of the several terms may then be
+effected by the formula</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2"><span class="f150">&int;</span></td> <td class="bk">&infin;</td>
+<td rowspan="2">e<span class="sp">&minus;y</span> y<span class="sp">q&minus;½</span>dy = &Gamma;(q + ½) = (q &minus; ½)(q &minus; <span class="spp">3</span>&frasl;<span class="suu">2</span>) ... ½ &radic;&pi;;</td></tr>
+<tr><td class="bk">0</td></tr></table>
+
+<p class="noind">and we get in terms of v</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">G =</td> <td>1</td>
+<td rowspan="2">&minus;</td> <td>1·3·5</td>
+<td rowspan="2">+</td> <td>1·3·5·9</td>
+<td rowspan="2">&minus; ... &emsp; &emsp; (16),</td></tr>
+<tr><td class="denom">&pi;²v³</td> <td class="denom">&pi;<span class="sp">4</span>v<span class="sp">7</span></td> <td class="denom">&pi;<span class="sp">6</span>v<span class="sp">11</span></td></tr></table>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">H =</td> <td>1</td>
+<td rowspan="2">&minus;</td> <td>1·3</td>
+<td rowspan="2">+</td> <td>1·3·5·7</td>
+<td rowspan="2">&minus; ... &emsp; &emsp; (17).</td></tr>
+<tr><td class="denom">&pi;v</td> <td class="denom">&pi;³v<span class="sp">5</span></td> <td class="denom">&pi;<span class="sp">5</span>v<span class="sp">9</span></td></tr></table>
+
+<p class="noind">The corresponding values of C and S were originally derived by
+A. L. Cauchy, without the use of Gilbert&rsquo;s integrals, by direct
+integration by parts.</p>
+
+<p>From the series for G and H just obtained it is easy to verify that</p>
+
+<table class="math0" summary="math">
+<tr><td>dH</td> <td rowspan="2">= &minus; &pi;vG, &emsp; &emsp; </td> <td>dG</td>
+<td rowspan="2">= &pi;vH &minus; 1 &emsp; &emsp; (18).</td></tr>
+<tr><td class="denom">dv</td> <td class="denom">dv</td></tr></table>
+
+<p>We now proceed to consider more particularly the distribution of
+light upon a screen PBQ near the shadow of a straight edge A.
+At a point P within the geometrical shadow of the obstacle, the
+half of the wave to the right of C (fig. 18), the nearest point on the
+wave-front, is wholly intercepted, and on the left the integration
+is to be taken from s = CA to s = &infin;. If V be the value of v corresponding
+to CA, viz.</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">V = <span class="f150">&radic;{</span></td> <td>2(a + b)</td>
+<td rowspan="2"><span class="f150">}</span> · CA, &emsp; &emsp; (19),</td></tr>
+<tr><td class="denom">ab&lambda;</td></tr></table>
+
+<p class="noind">we may write</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">I² = <span class="f150">( &int;</span></td> <td class="bk">&infin;</td>
+<td rowspan="2">cos ½&pi;v² · dv <span class="f150">)</span></td> <td>²</td>
+<td rowspan="2">+ <span class="f150">( &int;</span></td> <td class="bk">&infin;</td>
+<td rowspan="2">sin ½&pi;v² · dv <span class="f150">)</span></td> <td>²</td>
+<td rowspan="2"> &emsp; &emsp; (20),</td></tr>
+<tr><td class="bk">v</td> <td>&nbsp;</td>
+<td class="bk">v</td> <td>&nbsp;</td></tr></table>
+
+<p class="noind">or, according to our previous notation,</p>
+
+<p class="center">I²=(½ &minus; C<span class="su">v</span>)² + (½ &minus; S<span class="su">v</span>)² = G² + H² &emsp; &emsp; (21).</p>
+
+<table class="nobctr" style="float: left; width: 230px;" summary="Illustration">
+<tr><td class="figleft1"><img style="width:183px; height:182px" src="images/img252.jpg" alt="" /></td></tr>
+<tr><td class="caption sc">Fig. 18.</td></tr></table>
+
+<p>Now in the integrals represented by G and H every element
+diminishes as V increases from zero. Hence,
+as CA increases, viz. as the point P is more
+and more deeply immersed in the shadow,
+the illumination <i>continuously</i> decreases, and
+that without limit. It has long been known
+from observation that there are no bands
+on the interior side of the shadow of the
+edge.</p>
+
+<p>The law of diminution when V is moderately
+large is easily expressed with the aid
+of the series (16), (17) for G, H. We have
+ultimately G = 0, H = (&pi;V)<span class="sp">&minus;1</span>, so that</p>
+
+<p class="center">I² = 1/&pi;²V²,</p>
+
+<p class="noind">or the illumination is inversely as the square
+of the distance from the shadow of the edge.</p>
+
+<p>For a point Q outside the shadow the integration extends over
+<i>more</i> than half the primary wave. The intensity may be expressed by</p>
+
+<p class="center">I² = (½ + C<span class="su">v</span>)² + (½ + S<span class="su">v</span>)² &emsp; &emsp; (22);</p>
+
+<p class="noind">and the maxima and minima occur when</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">(½ + C<span class="su">v</span>)</td> <td>dC</td>
+<td rowspan="2">+ (½ + S<span class="su">v</span>)</td> <td>dS</td>
+<td rowspan="2">= 0,</td></tr>
+<tr><td class="denom">dV</td> <td class="denom">dV</td></tr></table>
+
+<p class="noind">whence</p>
+
+<p class="center">sin ½&pi;V² + cos ½&pi;V² = G &emsp; &emsp; (23).</p>
+
+<p class="noind">When V = 0, viz. at the edge of the shadow, I² = ½; when V = &infin;,
+I² = 2, on the scale adopted. The latter is the intensity due to the
+uninterrupted wave. The quadrupling of the intensity in passing
+outwards from the edge of the shadow is, however, accompanied by
+fluctuations giving rise to bright and dark bands. The position
+of these bands determined by (23) may be very simply expressed
+when V is large, for then sensibly G = 0, and</p>
+
+<p class="center">½&pi;V² = ¾&pi; + n&pi; &emsp; &emsp; (24),</p>
+
+<p class="noind">n being an integer. In terms of &delta;, we have from (2)</p>
+
+<p class="center">&delta; = (<span class="spp">3</span>&frasl;<span class="suu">8</span> + ½n)&lambda; &emsp; &emsp; (25).</p>
+
+<p class="noind">The first maximum in fact occurs when &delta; = <span class="spp">3</span>&frasl;<span class="suu">8</span>&lambda; &minus; .0046&lambda;, and the
+first minimum when &delta; = <span class="spp">7</span>&frasl;<span class="suu">8</span>&lambda; &minus; .0016&lambda;, the corrections being readily
+obtainable from a table of G by substitution of the approximate
+value of V.</p>
+
+<p>The position of Q corresponding to a given value of V, that is,
+to a band of given order, is by (19)</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">BQ =</td> <td>a + b</td>
+<td rowspan="2">AD = V <span class="f150">&radic; {</span></td> <td>b&lambda;(a + b)</td>
+<td rowspan="2"><span class="f150">}</span> &emsp; &emsp; (26).</td></tr>
+<tr><td class="denom">a</td> <td class="denom">2a</td></tr></table>
+
+<p class="noind">By means of this expression we may trace the locus of a band of
+given order as b varies. With sufficient approximation we may
+regard BQ and b as rectangular co-ordinates of Q. Denoting them
+by x, y, so that AB is axis of y and a perpendicular through A the
+axis of x, and rationalizing (26), we have</p>
+
+<p class="center">2ax² &minus; V²&lambda;y² &minus; V²a&lambda;y = 0,</p>
+
+<p class="noind">which represents a hyperbola with vertices at O and A.</p>
+
+<p>From (24), (26) we see that the width of the bands is of the order
+&radic;{b&lambda;(a + b)/a}. From this we may infer the limitation upon the
+width of the source of light, in order that the bands may be properly
+formed. If &omega; be the apparent magnitude of the source seen from A,
+&omega;b should be much smaller than the above quantity, or</p>
+
+<p class="center">&omega; &lt; &radic;{&lambda;(a + b)/ab} &emsp; &emsp; (27).</p>
+
+<p class="noind">If a be very great in relation to b, the condition becomes</p>
+
+<p class="center">&omega; &lt; &radic;(&lambda; / b) &emsp; &emsp; (28).</p>
+
+<p class="noind">so that if b is to be moderately great (1 metre), the apparent magnitude
+of the sun must be greatly reduced before it can be used as a
+source. The values of V for the maxima and minima of intensity,
+and the magnitudes of the latter, were calculated by Fresnel. An
+extract from his results is given in the accompanying table.</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcc allb">&nbsp;</td> <td class="tcc allb">V</td> <td class="tcc allb">I²</td></tr>
+
+<tr><td class="tcl lb rb">First maximum</td> <td class="tcc rb">1.2172</td> <td class="tcc rb">2.7413</td></tr>
+<tr><td class="tcl lb rb">First minimum</td> <td class="tcc rb">1.8726</td> <td class="tcc rb">1.5570</td></tr>
+<tr><td class="tcl lb rb">Second maximum</td> <td class="tcc rb">2.3449</td> <td class="tcc rb">2.3990</td></tr>
+<tr><td class="tcl lb rb">Second minimum </td> <td class="tcc rb">2.7392</td> <td class="tcc rb">1.6867</td></tr>
+<tr><td class="tcl lb rb">Third maximum.</td> <td class="tcc rb">3.0820</td> <td class="tcc rb">2.3022</td></tr>
+<tr><td class="tcl lb rb bb">Third minimum</td> <td class="tcc rb bb">3.3913</td> <td class="tcc rb bb">1.7440</td></tr>
+</table>
+
+<p>A very thorough investigation of this and other related questions,
+accompanied by fully worked-out tables of the functions concerned,
+will be found in a paper by E. Lommel (<i>Abh. bayer. Akad. d. Wiss.</i>
+II. CI., 15, Bd., iii. Abth., 1886).</p>
+
+<p>When the functions C and S have once been calculated, the
+discussion of various diffraction problems is much facilitated by
+the idea, due to M. A. Cornu (<i>Journ. de Phys.</i>, 1874, 3, p. 1; a similar
+suggestion was made independently by G. F. Fitzgerald), of exhibiting
+as a curve the relationship between C and S, considered as the
+rectangular co-ordinates (x, y) of a point. Such a curve is shown in
+fig. 19, where, according to the definition (5) of C, S,</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">x = <span class="f150">&int;</span></td> <td class="bk">v</td>
+<td rowspan="2">cos ½&pi;v²·dv, &emsp; y = <span class="f150">&int;</span></td> <td class="bk">v</td>
+<td rowspan="2">sin ½&pi;v²·dv &emsp; &emsp; (29).</td></tr>
+<tr><td class="bk">0</td> <td class="bk">0</td></tr></table>
+
+<p class="noind">The origin of co-ordinates O corresponds to v = 0; and the asymptotic
+points J, J&prime;, round which the curve revolves in an ever-closing spiral,
+correspond to v = ±&infin;.</p>
+
+<p>The intrinsic equation, expressing the relation between the arc
+&sigma; (measured from O) and the inclination &phi; of the tangent at any
+points to the axis of x, assumes a very simple form. For</p>
+
+<p class="center">dx = cos ½&pi;v²·dv, &emsp; dy = sin ½&pi;v²·dv;</p>
+
+<p class="noind">so that</p>
+
+<p class="center">&sigma; = <span class="f150">&int;</span> &radic;(dx² + dy²) = v, &emsp; &emsp; (30),</p>
+
+<p class="center">&phi; = tan<span class="sp">&minus;1</span>(dy/dx) = ½&pi;v² &emsp; &emsp; (31).</p>
+
+<p><span class="pagenum"><a name="page253" id="page253"></a>253</span></p>
+
+<p class="noind">Accordingly,</p>
+
+<p class="center">&phi; = ½&pi;&sigma;² &emsp; &emsp; (32);</p>
+
+<p class="noind">and for the curvature,</p>
+
+<p class="center">d&phi; / d&sigma; = &pi;&sigma; &emsp; &emsp; (33).</p>
+
+<table class="nobctr" style="float: left; width: 360px;" summary="Illustration">
+<tr><td class="figleft1"><img style="width:311px; height:290px" src="images/img253.jpg" alt="" /></td></tr>
+<tr><td class="caption sc">Fig. 19.</td></tr></table>
+
+<p>Cornu remarks that this equation suffices to determine the general
+character of the curve. For the osculating circle at any point
+includes the whole of the
+curve which lies beyond;
+and the successive convolutions
+envelop one another
+without intersection.</p>
+
+<p>The utility of the curve
+depends upon the fact that
+the elements of arc represent,
+in amplitude and
+phase, the component vibrations
+due to the corresponding
+portions of the
+primary wave-front. For
+by (30) d&sigma; = dv, and by
+(2) dv is proportional to ds.
+Moreover by (2) and (31)
+the retardation of phase of
+the elementary vibration
+from PQ (fig. 17) is 2&pi;&delta;/&lambda;,
+or &phi;. Hence, in accordance
+with the rule for compounding vector quantities, the resultant
+vibration at B, due to any finite part of the primary wave, is
+represented in amplitude and phase by the chord joining the extremities
+of the corresponding arc (&sigma;<span class="su">2</span> &minus; &sigma;<span class="su">1</span>).</p>
+
+<p>In applying the curve in special cases of diffraction to exhibit
+the effect at any point P (fig. 18) the centre of the curve O is to be
+considered to correspond to that point C of the primary wave-front
+which lies nearest to P. The operative part, or parts, of the curve
+are of course those which represent the unobstructed portions of
+the primary wave.</p>
+
+<p>Let us reconsider, following Cornu, the diffraction of a screen
+unlimited on one side, and on the other terminated by a straight
+edge. On the illuminated side, at a distance from the shadow, the
+vibration is represented by JJ&prime;. The co-ordinates of J, J&prime; being
+(½, ½), (&minus;½, &minus;½), I² is 2; and the phase is <span class="spp">1</span>&frasl;<span class="suu">8</span> period in arrear of
+that of the element at O. As the point under contemplation is
+supposed to approach the shadow, the vibration is represented by the
+chord drawn from J to a point on the other half of the curve, which
+travels inwards from J&prime; towards O. The amplitude is thus subject
+to fluctuations, which increase as the shadow is approached. At
+the point O the intensity is one-quarter of that of the entire wave,
+and after this point is passed, that is, when we have entered the
+geometrical shadow, the intensity falls off gradually to zero, <i>without
+fluctuations</i>. The whole progress of the phenomenon is thus exhibited
+to the eye in a very instructive manner.</p>
+
+<p>We will next suppose that the light is transmitted by a slit, and
+inquire what is the effect of varying the width of the slit upon the
+illumination at the projection of its centre. Under these circumstances
+the arc to be considered is bisected at O, and its length is
+proportional to the width of the slit. It is easy to see that the
+length of the chord (which passes in all cases through O) increases
+to a maximum near the place where the phase-retardation is <span class="spp">3</span>&frasl;<span class="suu">8</span> of
+a period, then diminishes to a minimum when the retardation is
+about <span class="spp">7</span>&frasl;<span class="suu">8</span> of a period, and so on.</p>
+
+<p>If the slit is of constant width and we require the illumination
+at various points on the screen behind it, we must regard the arc
+of the curve as of <i>constant length</i>. The intensity is then, as always,
+represented by the square of the length of the chord. If the slit
+be narrow, so that the arc is short, the intensity is constant over
+a wide range, and does not fall off to an important extent until
+the discrepancy of the extreme phases reaches about a quarter of a
+period.</p>
+
+<p>We have hitherto supposed that the shadow of a diffracting
+obstacle is received upon a diffusing screen, or, which comes to
+nearly the same thing, is observed with an eye-piece. If the eye,
+provided if necessary with a perforated plate in order to reduce the
+aperture, be situated inside the shadow at a place where the illumination
+is still sensible, and be focused upon the diffracting edge, the
+light which it receives will appear to come from the neighbourhood
+of the edge, and will present the effect of a silver lining. This is
+doubtless the explanation of a &ldquo;pretty optical phenomenon, seen
+in Switzerland, when the sun rises from behind distant trees standing
+on the summit of a mountain.&rdquo;<a name="fa11g" id="fa11g" href="#ft11g"><span class="sp">11</span></a></p>
+
+<p>II. <i>Dynamical Theory of Diffraction.</i>&mdash;The explanation of
+diffraction phenomena given by Fresnel and his followers is
+independent of special views as to the nature of the aether, at least
+in its main features; for in the absence of a more complete
+foundation it is impossible to treat rigorously the mode of action
+of a solid obstacle such as a screen. But, without entering upon
+matters of this kind, we may inquire in what manner a primary
+wave may be resolved into elementary secondary waves, and
+in particular as to the law of intensity and polarization in a
+secondary wave as dependent upon its direction of propagation,
+and upon the character as regards polarization of the primary
+wave. This question was treated by Stokes in his &ldquo;Dynamical
+Theory of Diffraction&rdquo; (<i>Camb. Phil. Trans.</i>, 1849) on the basis
+of the elastic solid theory.</p>
+
+<p>Let x, y, z be the co-ordinates of any particle of the medium in
+its natural state, and &chi;, &eta;, &zeta; the displacements of the same particle
+at the end of time t, measured in the directions of the three axes
+respectively. Then the first of the equations of motion may be put
+under the form</p>
+
+<table class="math0" summary="math">
+<tr><td>d²&xi;</td> <td rowspan="2">= b² <span class="f150">(</span></td> <td>d²&xi;</td>
+<td rowspan="2">+</td> <td>d²&xi;</td>
+<td rowspan="2">+</td> <td>d²&xi;</td>
+<td rowspan="2"><span class="f150">)</span> + (a² &minus; b²)</td> <td>d²</td>
+<td rowspan="2"><span class="f150">(</span></td> <td>d²&xi;</td>
+<td rowspan="2">+</td> <td>d²&eta;</td>
+<td rowspan="2">+</td> <td>d²&zeta;</td>
+<td rowspan="2"><span class="f150">)</span>,</td></tr>
+<tr><td class="denom">dt²</td> <td class="denom">dx²</td>
+<td class="denom">dy²</td> <td class="denom">dz²</td>
+<td class="denom">dx</td> <td class="denom">dx²</td>
+<td class="denom">dy²</td> <td class="denom">dz²</td></tr></table>
+
+<p class="noind">where a² and b² denote the two arbitrary constants. Put for shortness</p>
+
+<table class="math0" summary="math">
+<tr><td>d²&xi;</td> <td rowspan="2">+</td> <td>d²&eta;</td>
+<td rowspan="2">+</td> <td>d²&zeta;</td>
+<td rowspan="2">&asymp; &delta;&emsp;&emsp;(1),</td></tr>
+<tr><td class="denom">dx²</td> <td class="denom">dy²</td> <td class="denom">dz²</td></tr></table>
+
+<p class="noind">and represent by &Delta;²&chi; the quantity multiplied by b². According to
+this notation, the three equations of motion are</p>
+
+<table class="math0" summary="math">
+<tr><td>d²&xi;</td> <td rowspan="2">b²&Delta;²&xi; + (a² &minus; b²)</td> <td>d&delta;</td>
+<td rowspan="6"><span style="font-size: 8em; font-family: 'Courier New'; color: #a0a0a0;">}</span>&emsp;&emsp;(2).</td></tr>
+<tr><td class="denom">dt²</td> <td class="denom">dx</td></tr>
+
+<tr><td>d²&eta;</td> <td rowspan="2">b²&Delta;²&eta; + (a² &minus; b²)</td> <td>d&delta;</td></tr>
+<tr><td class="denom">dt²</td> <td class="denom">dy</td></tr>
+
+<tr><td>d²&zeta;</td> <td rowspan="2">b²&Delta;²&zeta; + (a² &minus; b²)</td> <td>d&delta;</td></tr>
+<tr><td class="denom">dt²</td> <td class="denom">dz</td></tr></table>
+
+<p class="noind">It is to be observed that S denotes the dilatation of volume of the
+element situated at (x, y, z). In the limiting case in which the
+medium is regarded as absolutely incompressible &delta; vanishes; but,
+in order that equations (2) may preserve their generality, we must
+suppose a at the same time to become infinite, and replace a²&delta; by
+a new function of the co-ordinates.</p>
+
+<p>These equations simplify very much in their application to plane
+waves. If the ray be parallel to OX, and the direction of vibration
+parallel to OZ, we have &xi; = 0, &eta; = 0, while &zeta; is a function of x and
+t only. Equation (1) and the first pair of equations (2) are thus
+satisfied identically. The third equation gives</p>
+
+<table class="math0" summary="math">
+<tr><td>d²&zeta;</td> <td rowspan="2">= b²</td> <td>d²&zeta;</td>
+<td rowspan="2">&emsp;&emsp;(3),</td></tr>
+<tr><td class="denom">dt²</td> <td class="denom">dx²</td></tr></table>
+
+<p class="noind">of which the solution is</p>
+
+<p class="center">&zeta; = &fnof;(bt &minus; x)&emsp;&emsp;(4),</p>
+
+<p class="noind">where &fnof; is an arbitrary function.</p>
+
+<p>The question as to the law of the secondary waves is thus answered
+by Stokes. &ldquo;Let &xi; = 0, &eta; = 0, &zeta; = &fnof;(bt &minus; x) be the displacements
+corresponding to the incident light; let O<span class="su">1</span> be any point in the plane
+P (of the wave-front), dS an element of that plane adjacent to O<span class="su">1</span>,
+and consider the disturbance due to that portion only of the incident
+disturbance which passes continually across dS. Let O be any point
+in the medium situated at a distance from the point O<span class="su">1</span> which is
+large in comparison with the length of a wave; let O<span class="su">1</span>O = r, and let
+this line make an angle &theta; with the direction of propagation of the
+incident light, or the axis of x, and &phi; with the direction of vibration,
+or axis of z. Then the displacement at O will take place in a direction
+perpendicular to O<span class="su">1</span>O, and lying in the plane ZO<span class="su">1</span>O; and, if &zeta;&prime; be the
+displacement at O, reckoned positive in the direction nearest to
+that in which the incident vibrations are reckoned positive,</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&zeta;&prime; =</td> <td>dS</td>
+<td rowspan="2">(1 + cos &theta;) sin &phi; &fnof;&prime;(bt &minus; r).</td></tr>
+<tr><td class="denom">4&pi;r</td></tr></table>
+
+<p class="noind">In particular, if</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&fnof;(bt &minus; x) = c sin</td> <td>2&pi;</td>
+<td rowspan="2">(bt &minus; x)&emsp;&emsp;(5),</td></tr>
+<tr><td class="denom">&lambda;</td></tr></table>
+
+<p class="noind">we shall have</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&zeta;&prime; =</td> <td>cdS</td>
+<td rowspan="2">(1 + cos &theta;) sin &phi;cos</td> <td>2&pi;</td>
+<td rowspan="2">(bt &minus; r)&emsp;&emsp;(6).&rdquo;</td></tr>
+<tr><td class="denom">2&lambda;r</td> <td class="denom">&lambda;</td></tr></table>
+
+<p class="noind">It is then verified that, after integration with respect to dS, (6)
+gives the same disturbance as if the primary wave had been supposed
+to pass on unbroken.</p>
+
+<p>The occurrence of sin &phi; as a factor in (6) shows that the relative
+intensities of the primary light and of that diffracted in the direction
+&theta; depend upon the condition of the former as regards polarization.
+If the direction of primary vibration be perpendicular to
+the plane of diffraction (containing both primary and secondary
+rays), sin &phi; = 1; but, if the primary vibration be in the plane of
+diffraction, sin &phi; = cos &theta;. This result was employed by Stokes as
+a criterion of the direction of vibration; and his experiments, conducted
+with gratings, led him to the conclusion that the vibrations
+<span class="pagenum"><a name="page254" id="page254"></a>254</span>
+of polarized light are executed in a direction <i>perpendicular</i> to the
+plane of polarization.</p>
+
+<p>The factor (1 + cos &theta;) shows in what manner the secondary disturbance
+depends upon the direction in which it is propagated with
+respect to the front of the primary wave.</p>
+
+<p>If, as suffices for all practical purposes, we limit the application
+of the formulae to points in advance of the plane at which the wave
+is supposed to be broken up, we may use simpler methods of resolution
+than that above considered. It appears indeed that the purely
+mathematical question has no definite answer. In illustration of
+this the analogous problem for sound may be referred to. Imagine
+a flexible lamina to be introduced so as to coincide with the plane
+at which resolution is to be effected. The introduction of the lamina
+(supposed to be devoid of inertia) will make no difference to the
+propagation of plane parallel sonorous waves through the position
+which it occupies. At every point the motion of the lamina will be
+the same as would have occurred in its absence, the pressure of the
+waves impinging from behind being just what is required to generate
+the waves in front. Now it is evident that the aerial motion in front
+of the lamina is determined by what happens at the lamina without
+regard to the cause of the motion there existing. Whether the
+necessary forces are due to aerial pressures acting on the rear, or to
+forces directly impressed from without, is a matter of indifference.
+The conception of the lamina leads immediately to two schemes,
+according to which a primary wave may be supposed to be broken
+up. In the first of these the element dS, the effect of which is to be
+estimated, is supposed to execute its actual motion, while every other
+element of the plane lamina is maintained at rest. The resulting
+aerial motion in front is readily calculated (see Rayleigh, <i>Theory of
+Sound</i>, § 278); it is symmetrical with respect to the origin, <i>i.e.</i> independent
+of &theta;. When the secondary disturbance thus obtained is
+integrated with respect to dS over the entire plane of the lamina, the
+result is necessarily the same as would have been obtained had the
+primary wave been supposed to pass on without resolution, for this
+is precisely the motion generated when every element of the lamina
+vibrates with a common motion, equal to that attributed to dS.
+The only assumption here involved is the evidently legitimate one
+that, when two systems of variously distributed motion at the
+lamina are superposed, the corresponding motions in front are
+superposed also.</p>
+
+<p>The method of resolution just described is the simplest, but it is
+only one of an indefinite number that might be proposed, and which
+are all equally legitimate, so long as the question is regarded as a
+merely mathematical one, without reference to the physical properties
+of actual screens. If, instead of supposing the <i>motion</i> at dS
+to be that of the primary wave, and to be zero elsewhere, we suppose
+the <i>force</i> operative over the element dS of the lamina to be that
+corresponding to the primary wave, and to vanish elsewhere, we
+obtain a secondary wave following quite a different law. In this
+case the motion in different directions varies as cos&theta;, vanishing at
+right angles to the direction of propagation of the primary wave.
+Here again, on integration over the entire lamina, the aggregate
+effect of the secondary waves is necessarily the same as that of the
+primary.</p>
+
+<p>In order to apply these ideas to the investigation of the secondary
+wave of light, we require the solution of a problem, first treated
+by Stokes, viz. the determination of the motion in an infinitely
+extended elastic solid due to a locally applied periodic force. If
+we suppose that the force impressed upon the element of mass
+D dx dy dz is</p>
+
+<p class="center">DZ dx dy dz,</p>
+
+<p class="noind">being everywhere parallel to the axis of Z, the only change required
+in our equations (1), (2) is the addition of the term Z to the second
+member of the third equation (2). In the forced vibration, now
+under consideration, Z, and the quantities &xi;, &eta;, &zeta;, &delta; expressing the
+resulting motion, are to be supposed proportional to e<span class="sp">int</span>, where
+i = &radic;(-1),   and n = 2&pi;/&tau;, &tau; being the periodic time. Under these
+circumstances the double differentiation with respect to t of any
+quantity is equivalent to multiplication by the factor -n², and thus
+our equations take the form</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">(b²&Delta;² + n²)&xi; + (a² &minus; b²)</td> <td>d&delta;</td>
+<td rowspan="2">= 0</td> <td rowspan="6"><span style="font-size: 8em; font-family: 'Courier New'; color: #a0a0a0;">}</span>&emsp;&emsp;(7).</td></tr>
+<tr><td class="denom">dx</td></tr>
+
+<tr><td rowspan="2">(b²&Delta;² + n²)&eta; + (a² &minus; b²)</td> <td>d&delta;</td>
+<td rowspan="2">= 0</td></tr>
+<tr><td class="denom">dy</td></tr>
+
+<tr><td rowspan="2">(b²&Delta;² + n²)&zeta; + (a² &minus; b²)</td> <td>d&delta;</td>
+<td rowspan="2">= &minus;Z</td></tr>
+<tr><td class="denom">dz</td></tr></table>
+
+<p class="noind">It will now be convenient to introduce the quantities.<span class="ov">&omega;</span><span class="su">1</span>, <span class="ov">&omega;</span><span class="su">2</span>, <span class="ov">&omega;</span><span class="su">3</span>
+which express the <i>rotations</i> of the elements of the medium round axes
+parallel to those of co-ordinates, in accordance with the equations</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2"><span class="ov">&omega;</span><span class="su">3</span> =</td> <td>d&xi;</td>
+<td rowspan="2">&minus;</td> <td>d&eta;</td>
+<td rowspan="2">,&emsp;<span class="ov">&omega;</span><span class="su">1</span> =</td> <td>d&eta;</td>
+<td rowspan="2">&minus;</td> <td>d&zeta;</td>
+<td rowspan="2">,&emsp;<span class="ov">&omega;</span><span class="su">2</span></td> <td>d&zeta;</td>
+<td rowspan="2">&minus;</td> <td>d&xi;</td>
+<td rowspan="2">&emsp;&emsp;(8).</td></tr>
+<tr><td class="denom">dy</td> <td class="denom">dx&prime;</td>
+<td class="denom">dz</td> <td class="denom">dy&prime;</td>
+<td class="denom">dx</td> <td class="denom">dz&prime;</td></tr></table>
+
+<p class="noind">In terms of these we obtain from (7), by differentiation and subtraction,</p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tcl">(b²&Delta;² + n²) <span class="ov">&omega;</span><span class="su">3</span> = 0</td> <td class="tcl" rowspan="3"><span style="font-size: 4em; font-family: 'Courier New'; color: #a0a0a0;">}</span>&emsp;&emsp;(9).</td></tr>
+<tr><td class="tcl">(b²&Delta;² + n²) <span class="ov">&omega;</span><span class="su">1</span> = dZ/dy</td></tr>
+<tr><td class="tcl">(b²&Delta;² + n²) <span class="ov">&omega;</span><span class="su">2</span> = &minus;dZ/dx</td></tr></table>
+
+<p class="noind">The first of equations (9) gives</p>
+
+<p class="center"><span class="ov">&omega;</span><span class="su">3</span> = 0&emsp;&emsp;(10).</p>
+
+<p class="noind">For <span class="ov">&omega;</span><span class="su">1</span>, we have</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2"><span class="ov">&omega;</span><span class="su">1</span> =</td> <td>1</td>
+<td rowspan="2"><span class="f150">&int;&int;&int;</span></td> <td>dZ</td> <td>e<span class="sp">&minus;ikr</span></td>
+<td rowspan="2">dx dy dz&emsp;&emsp;(11),</td></tr>
+<tr><td class="denom">4&pi;b²</td> <td class="ov">dy</td> <td class="denom">r</td></tr></table>
+
+<p class="noind">where r is the distance between the element dx dy dz and the point
+where <span class="ov">&omega;</span><span class="su">1</span> is estimated, and</p>
+
+<p class="center">k = n/b = 2&pi;/&lambda;&emsp;&emsp;(12),</p>
+
+<p class="noind">&lambda; being the wave-length.</p>
+
+<p>(This solution may be verified in the same manner as Poisson&rsquo;s
+theorem, in which k = 0.)</p>
+
+<p>We will now introduce the supposition that the force Z acts
+only within a small space of volume T, situated at (x, y, z), and for
+simplicity suppose that it is at the origin of co-ordinates that the
+rotations are to be estimated. Integrating by parts in (11), we get</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2"><span class="f150">&int;</span></td> <td>e<span class="sp">&minus;ikr</span></td> <td>dZ</td>
+<td rowspan="2">dy = <span class="f150">[</span> Z</td> <td>e<span class="sp">&minus;ikr</span></td>
+<td rowspan="2"><span class="f150">]</span> &minus; <span class="f150">&int;</span> Z</td> <td>d</td>
+<td rowspan="2"><span class="f150">(</span></td> <td>e<span class="sp">&minus;ikr</span></td>
+<td rowspan="2"><span class="f150">)</span> dy,</td></tr>
+<tr><td class="denom">r</td> <td class="ov">dy</td>
+<td class="denom">r</td> <td class="denom">dy</td> <td class="denom">r</td></tr></table>
+
+<p class="noind">in which the integrated terms at the limits vanish, Z being finite
+only within the region T. Thus</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2"><span class="ov">&omega;</span><span class="su">1</span> =</td> <td>1</td>
+<td rowspan="2"><span class="f150">&int;&int;&int;</span> Z</td> <td>d</td>
+<td rowspan="2"><span class="f150">(</span></td> <td>e<span class="sp">&minus;ikr</span></td>
+<td rowspan="2"><span class="f150">)</span> dx dy dz.</td></tr>
+<tr><td class="denom">4&pi;b²</td> <td class="denom">dy</td> <td class="denom">r</td></tr></table>
+
+<p class="noind">Since the dimensions of T are supposed to be very small in comparison
+with &lambda;, the factor d/dy (e<span class="sp">&minus;ikr</span>/r) is sensibly constant; so that,
+if Z stand for the mean value of Z over the volume T, we may write</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2"><span class="ov">&omega;</span><span class="su">1</span> =</td> <td>TZ</td>
+<td rowspan="2">·</td> <td>y</td>
+<td rowspan="2">·</td> <td>d</td>
+<td rowspan="2"><span class="f150">(</span></td> <td>e<span class="sp">&minus;ikr</span></td>
+<td rowspan="2"><span class="f150">)</span>&emsp;&emsp;(13).</td></tr>
+<tr><td class="denom">4&pi;b²</td> <td class="denom">r</td>
+<td class="denom">dr</td> <td class="denom">r</td></tr></table>
+
+<p class="noind">In like manner we find</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2"><span class="ov">&omega;</span><span class="su">2</span> = &minus;</td> <td>TZ</td>
+<td rowspan="2">·</td> <td>x</td>
+<td rowspan="2">·</td> <td>d</td>
+<td rowspan="2"><span class="f150">(</span></td> <td>e<span class="sp">&minus;ikr</span></td>
+<td rowspan="2"><span class="f150">)</span>&emsp;&emsp;(14).</td></tr>
+<tr><td class="denom">4&pi;b²</td> <td class="denom">r</td>
+<td class="denom">dr</td> <td class="denom">r</td></tr></table>
+
+<p class="noind">From (10), (13), (14) we see that, as might have been expected,
+the rotation at any point is about an axis perpendicular both to
+the direction of the force and to the line joining the point to the
+source of disturbance. If the resultant rotation be &omega;, we have</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2"><span class="ov">&omega;</span> =</td> <td>TZ</td>
+<td rowspan="2">·</td> <td>&radic;(x² + y²)</td>
+<td rowspan="2">·</td> <td>d</td>
+<td rowspan="2"><span class="f150">(</span></td> <td>e<span class="sp">&minus;ikr</span></td>
+<td rowspan="2"><span class="f150">)</span> =</td> <td>TZ sin &phi;</td> <td>d</td>
+<td rowspan="2"><span class="f150">(</span></td> <td>e<span class="sp">&minus;ikr</span></td>
+<td rowspan="2"><span class="f150">)</span>,</td></tr>
+<tr><td class="denom">4&pi;b²</td> <td class="denom">r</td>
+<td class="denom">dr</td> <td class="denom">r</td>
+<td class="denom">4&pi;b²</td> <td class="ov">dr</td>
+<td class="denom">r</td></tr></table>
+
+<p class="noind">&phi; denoting the angle between r and z. In differentiating e<span class="sp">&minus;ikr</span>/r
+with respect to r, we may neglect the term divided by r² as altogether
+insensible, kr being an exceedingly great quantity at any moderate
+distance from the origin of disturbance. Thus</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2"><span class="ov">&omega;</span> =</td> <td>&minus;ik · TZ sin &phi;</td>
+<td rowspan="2">·</td> <td>e<span class="sp">&minus;ikr</span></td>
+<td rowspan="2">&emsp;&emsp;(15),</td></tr>
+<tr><td class="denom">4&pi;b²</td> <td class="denom">r</td></tr></table>
+
+<p class="noind">which completely determines the rotation at any point. For a disturbing
+force of given integral magnitude it is seen to be everywhere
+about an axis perpendicular to r and the direction of the force, and
+in magnitude dependent only upon the angle (&phi;) between these two
+directions and upon the distance (r).</p>
+
+<p>The intensity of light is, however, more usually expressed in
+terms of the actual displacement in the plane of the wave. This
+displacement, which we may denote by &zeta;&prime;, is in the plane containing
+z and r, and perpendicular to the latter. Its connexion with <span class="ov">&omega;</span>is
+expressed by <span class="ov">&omega;</span> = d&zeta;&prime;/dr; so that</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&zeta;&prime; =</td> <td>TZ sin &phi;</td>
+<td rowspan="2">·</td> <td>e&prime; <span class="sp">(at&minus;kr)</span></td>
+<td rowspan="2">&emsp;&emsp;(16),</td></tr>
+<tr><td class="denom">4&pi;b²</td> <td class="denom">r</td></tr></table>
+
+<p class="noind">where the factor e<span class="sp">int</span> is restored.</p>
+
+<p>Retaining only the real part of (16), we find, as the result of a
+local application of force equal to</p>
+
+<p class="center">DTZ cos nt&emsp;&emsp;(17),</p>
+
+<p class="noind">the disturbance expressed by</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&zeta;&prime; =</td> <td>TZ sin &phi;</td>
+<td rowspan="2">·</td> <td>cos (nt &minus; kr)</td>
+<td rowspan="2">&emsp;&emsp;(18).</td></tr>
+<tr><td class="denom">4&pi;b²</td> <td class="denom">r</td></tr></table>
+
+<p>The occurrence of sin &phi; shows that there is no disturbance
+radiated in the direction of the force, a feature which might have
+been anticipated from considerations of symmetry.</p>
+
+<p>We will now apply (18) to the investigation of a law of secondary
+disturbance, when a primary wave</p>
+
+<p class="center">&zeta; = sin(nt &minus; kx)&emsp;&emsp;(19)</p>
+
+<p class="noind">is supposed to be broken up in passing the plane x = 0. The first step
+is to calculate the force which represents the reaction between the
+parts of the medium separated by x = 0. The force operative upon
+the positive half is parallel to OZ, and of amount per unit of area
+equal to</p>
+
+<p class="center">&minus;b²D d&zeta;/dx = b²kD cos nt;</p>
+
+<p class="noind">and to this force acting over the whole of the plane the actual
+motion on the positive side may be conceived to be due. The
+<span class="pagenum"><a name="page255" id="page255"></a>255</span>
+secondary disturbance corresponding to the element dS of the plane
+may be supposed to be that caused by a force of the above magnitude
+acting over dS and vanishing elsewhere; and it only remains to
+examine what the result of such a force would be.</p>
+
+<p>Now it is evident that the force in question, supposed to act
+upon the positive half only of the medium, produces just double of
+the effect that would be caused by the same force if the medium
+were undivided, and on the latter supposition (being also localized
+at a point) it comes under the head already considered. According
+to (18), the effect of the force acting at dS parallel to OZ, and of
+amount equal to</p>
+
+<p class="center">2b²kD dS cos nt,</p>
+
+<p class="noind">will be a disturbance</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&zeta;&prime; =</td> <td>dS sin &phi;</td>
+<td rowspan="2">cos (nt &minus; kr)&emsp;&emsp;(20),</td></tr>
+<tr><td class="denom">&lambda;r</td></tr></table>
+
+<p class="noind">regard being had to (12). This therefore expresses the secondary
+disturbance at a distance r and in a direction making an angle &phi;
+with OZ (the direction of primary vibration) due to the element dS
+of the wave-front.</p>
+
+<p>The proportionality of the secondary disturbance to sin &phi; is
+common to the present law and to that given by Stokes, but here
+there is no dependence upon the angle &theta; between the primary and
+secondary rays. The occurrence of the factor &lambda;r<span class="sp">&minus;1</span>, and the
+necessity of supposing the phase of the secondary wave accelerated
+by a quarter of an undulation, were first established by Archibald
+Smith, as the result of a comparison between the primary wave,
+supposed to pass on without resolution, and the integrated effect
+of all the secondary waves (§ 2). The occurrence of factors such
+as sin &phi;, or ½(1 + cos &theta;), in the expression of the secondary wave
+has no influence upon the result of the integration, the effects of
+all the elements for which the factors differ appreciably from unity
+being destroyed by mutual interference.</p>
+
+<p>The choice between various methods of resolution, all mathematically
+admissible, would be guided by physical considerations
+respecting the mode of action of obstacles. Thus, to refer again to
+the acoustical analogue in which plane waves are incident upon
+a perforated rigid screen, the circumstances of the case are best
+represented by the first method of resolution, leading to symmetrical
+secondary waves, in which the normal motion is supposed to be zero
+over the unperforated parts. Indeed, if the aperture is very small,
+this method gives the correct result, save as to a constant factor. In
+like manner our present law (20) would apply to the kind of obstruction
+that would be caused by an actual physical division of the elastic
+medium, extending over the whole of the area supposed to be occupied
+by the intercepting screen, but of course not extending to the parts
+supposed to be perforated.</p>
+
+<p>On the electromagnetic theory, the problem of diffraction becomes
+definite when the properties of the obstacle are laid down. The
+simplest supposition is that the material composing the obstacle
+is perfectly conducting, <i>i.e.</i> perfectly reflecting. On this basis
+A. J. W. Sommerfeld (<i>Math. Ann.</i>, 1895, 47, p. 317), with great mathematical
+skill, has solved the problem of the shadow thrown by a
+semi-infinite plane screen. A simplified exposition has been given by
+Horace Lamb (<i>Proc. Lond. Math. Soc.</i>, 1906, 4, p. 190). It appears that
+Fresnel&rsquo;s results, although based on an imperfect theory, require only
+insignificant corrections. Problems not limited to two dimensions,
+such for example as the shadow of a circular disk, present great
+difficulties, and have not hitherto been treated by a rigorous method;
+but there is no reason to suppose that Fresnel&rsquo;s results would be
+departed from materially.</p>
+<div class="author">(R.)</div>
+
+<hr class="foot" /> <div class="note">
+
+<p><a name="ft1g" id="ft1g" href="#fa1g"><span class="fn">1</span></a> The descending series for J<span class="su">0</span>(z) appears to have been first given
+by Sir W. Hamilton in a memoir on &ldquo;Fluctuating Functions,&rdquo;
+<i>Roy. Irish Trans.</i>, 1840.</p>
+
+<p><a name="ft2g" id="ft2g" href="#fa2g"><span class="fn">2</span></a> Airy, loc. cit. &ldquo;Thus the magnitude of the central spot is
+diminished, and the brightness of the rings increased, by covering
+the central parts of the object-glass.&rdquo;</p>
+
+<p><a name="ft3g" id="ft3g" href="#fa3g"><span class="fn">3</span></a> <i>&rdquo;Man kann daraus schliessen, was moglicher Weise durch Mikroskope
+noch zu sehen ist. Ein mikroskopischer Gegenstand z. B, dessen
+Durchmesser = (&lambda;) ist, und der aus zwei Theilen besteht, kann nicht
+mehr als aus zwei Theilen bestehend erkannt werden. Dieses zeigt uns
+eine Grenze des Sehvermogens durch Mikroskope&rdquo;</i> (<i>Gilbert&rsquo;s Ann.</i>
+74, 337). Lord Rayleigh has recorded that he was himself convinced
+by Fraunhofer&rsquo;s reasoning at a date antecedent to the writings of
+Helmholtz and Abbe.</p>
+
+<p><a name="ft4g" id="ft4g" href="#fa4g"><span class="fn">4</span></a> The last sentence is repeated from the writer&rsquo;s article &ldquo;Wave
+Theory&rdquo; in the 9th edition of this work, but A. A. Michelson&rsquo;s
+ingenious échelon grating constitutes a realization in an unexpected
+manner of what was thought to be impracticable.&mdash;[R.]</p>
+
+<p><a name="ft5g" id="ft5g" href="#fa5g"><span class="fn">5</span></a> Compare also F. F. Lippich, <i>Pogg. Ann.</i> cxxxix. p. 465, 1870;
+Rayleigh, <i>Nature</i> (October 2, 1873).</p>
+
+<p><a name="ft6g" id="ft6g" href="#fa6g"><span class="fn">6</span></a> The power of a grating to construct light of nearly definite wave-length
+is well illustrated by Young&rsquo;s comparison with the production
+of a musical note by reflection of a sudden sound from a row of
+palings. The objection raised by Herschel (<i>Light</i>, § 703) to this
+comparison depends on a misconception.</p>
+
+<p><a name="ft7g" id="ft7g" href="#fa7g"><span class="fn">7</span></a> It must not be supposed that errors of this order of magnitude are
+unobjectionable in all cases. The position of the middle of the bright
+band representative of a mathematical line can be fixed with a
+spider-line micrometer within a small fraction of the width of the
+band, just as the accuracy of astronomical observations far transcends
+the separating power of the instrument.</p>
+
+<p><a name="ft8g" id="ft8g" href="#fa8g"><span class="fn">8</span></a> &ldquo;In the same way we may conclude that in flat gratings any
+departure from a straight line has the effect of causing the dust in
+the slit and the spectrum to have different foci&mdash;a fact sometimes
+observed.&rdquo; (Rowland, &ldquo;On Concave Gratings for Optical Purposes,&rdquo;
+<i>Phil. Mag.</i>, September 1883).</p>
+
+<p><a name="ft9g" id="ft9g" href="#fa9g"><span class="fn">9</span></a> On account of inequalities in the atmosphere giving a variable
+refraction, the light from a star would be irregularly distributed over
+a screen. The experiment is easily made on a laboratory scale, with
+a small source of light, the rays from which, in their course
+towards a rather distant screen, are disturbed by the neighbourhood
+of a heated body. At a moment when the eye, or object-glass of a
+telescope, occupies a dark position, the star vanishes. A fraction
+of a second later the aperture occupies a bright place, and the star
+reappears. According to this view the chromatic effects depend
+entirely upon atmospheric dispersion.</p>
+
+<p><a name="ft10g" id="ft10g" href="#fa10g"><span class="fn">10</span></a> In experiment a line of light is sometimes substituted for a point
+in order to increase the illumination. The various parts of the line
+are here <i>independent</i> sources, and should be treated accordingly.
+To assume a cylindrical form of primary wave would be justifiable
+only when there is synchronism among the secondary waves issuing
+from the various centres.</p>
+
+<p><a name="ft11g" id="ft11g" href="#fa11g"><span class="fn">11</span></a> H. Necker (<i>Phil. Mag.</i>, November 1832); Fox Talbot (<i>Phil. Mag.</i>,
+June 1833). &ldquo;When the sun is about to emerge ... every branch
+and leaf is lighted up with a silvery lustre of indescribable beauty....
+The birds, as Mr Necker very truly describes, appear like flying
+brilliant sparks.&rdquo; Talbot ascribes the appearance to diffraction;
+and he recommends the use of a telescope.</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIFFUSION<a name="ar85" id="ar85"></a></span> (from the Lat. <i>diffundere; dis-</i>, asunder, and
+<i>fundere</i>, to pour out), in general, a spreading out, scattering
+or circulation; in physics the term is applied to a special
+phenomenon, treated below.</p>
+
+<p>1. <i>General Description.</i>&mdash;When two different substances are
+placed in contact with each other they sometimes remain
+separate, but in many cases a gradual mixing takes place. In the
+case where both the substances are gases the process of mixing
+continues until the result is a uniform mixture. In other cases
+the proportions in which two different substances can mix
+lie between certain fixed limits, but the mixture is distinguished
+from a chemical compound by the fact that between these limits
+the composition of the mixture is capable of continuous variation,
+while in chemical compounds, the proportions of the different
+constituents can only have a discrete series of numerical values,
+each different ratio representing a different compound. If we
+take, for example, air and water in the presence of each other, air
+will become dissolved in the water, and water will evaporate into
+the air, and the proportions of either constituent absorbed by the
+other will vary continuously. But a limit will come when the air
+will absorb no more water, and the water will absorb no more air,
+and throughout the change a definite surface of separation will
+exist between the liquid and the gaseous parts. When no surface
+of separation ever exists between two substances they must
+necessarily be capable of mixing in all proportions. If they are
+not capable of mixing in all proportions a discontinuous change
+must occur somewhere between the regions where the substances
+are still unmixed, thus giving rise to a surface of separation.</p>
+
+<p>The phenomena of mixing thus involves the following processes:&mdash;(1)
+A motion of the substances relative to one another
+throughout a definite <i>region</i> of space in which mixing is taking
+place. This relative motion is called &ldquo;diffusion.&rdquo; (2) The passage
+of portions of the mixing substances across the <i>surface</i> of
+separation when such a surface exists. These surface actions
+are described under various terms such as solution, evaporation,
+condensation and so forth. For example, when a soluble salt is
+placed in a liquid, the process which occurs at the surface of the
+salt is called &ldquo;solution,&rdquo; but the salt which enters the liquid by
+solution is transported from the surface into the interior of the
+liquid by &ldquo;diffusion.&rdquo;</p>
+
+<p>Diffusion may take place in solids, that is, in regions occupied
+by matter which continues to exhibit the properties of the solid
+state. Thus if two liquids which can mix are separated by a
+membrane or partition, the mixing may take place through the
+membrane. If a solution of salt is separated from pure water by
+a sheet of parchment, part of the salt will pass through the parchment
+into the water. If water and glycerin are separated in this
+way most of the water will pass into the glycerin and a little
+glycerin will pass through in the opposite direction, a property
+frequently used by microscopists for the purpose of gradually
+transferring minute algae from water into glycerin. A still more
+interesting series of examples is afforded by the passage of gases
+through partitions of metal, notably the passage of hydrogen
+through platinum and palladium at high temperatures. When
+the process is considered with reference to a membrane or partition
+taken as a whole, the passage of a substance from one side to the
+other is commonly known as &ldquo;osmosis&rdquo; or &ldquo;transpiration&rdquo;
+(see <span class="sc"><a href="#artlinks">Solution</a></span>), but what occurs in the material of the membrane
+itself is correctly described as diffusion.</p>
+
+<p>Simple cases of diffusion are easily observed qualitatively. If a
+solution of a coloured salt is carefully introduced by a funnel into
+the bottom of a jar containing water, the two portions will at first
+be fairly well defined, but if the mixture can exist in all proportions,
+the surface of separation will gradually disappear; and the
+rise of the colour into the upper part and its gradual weakening
+in the lower part, may be watched for days, weeks or even longer
+intervals. The diffusion of a strong aniline colouring matter into
+the interior of gelatine is easily observed, and is commonly seen in
+copying apparatus. Diffusion of gases may be shown to exist by
+taking glass jars containing vapours of hydrochloric acid and
+ammonia, and placing them in communication with the heavier
+gas downmost. The precipitation of ammonium chloride shows
+that diffusion exists, though the chemical action prevents this
+example from forming a typical case of diffusion. Again, when
+a film of Canada balsam is enclosed between glass plates, the
+disappearance during a few weeks of small air bubbles enclosed
+in the balsam can be watched under the microscope.</p>
+
+<p>In fluid media, whether liquids or gases, the process of mixing
+is greatly accelerated by stirring or agitating the fluids, and
+liquids which might take years to mix if left to themselves
+can thus be mixed in a few seconds. It is necessary to carefully
+distinguish the effects of agitation from those of diffusion proper.
+By shaking up two liquids which do not mix we split them up
+into a large number of different portions, and so greatly increase
+the area of the surface of separation, besides decreasing the
+thicknesses of the various portions. But even when we produce
+the appearance of a uniform turbid mixture, the small portions
+remain quite distinct. If however the fluids can really mix, the
+final process must in every case depend on diffusion, and all we
+do by shaking is to increase the sectional area, and decrease the
+thickness of the diffusing portions, thus rendering the completion
+of the operation more rapid. If a gas is shaken up in a liquid
+the process of absorption of the bubbles is also accelerated by
+capillary action, as occurs in an ordinary sparklet bottle. To
+state the matter precisely, however finely two fluids have been
+<span class="pagenum"><a name="page256" id="page256"></a>256</span>
+subdivided by agitation, the molecular constitution of the
+different portions remains unchanged. The ultimate process
+by which the individual molecules of two different substances
+become mixed, producing finally a homogeneous mixture, is in
+every case diffusion. In other words, diffusion is that relative
+motion of the molecules of two different substances by which the
+proportions of the molecules in any region containing a finite
+number of molecules are changed.</p>
+
+<div class="condensed">
+<p>In order, therefore, to make accurate observations of diffusion in
+fluids it is necessary to guard against any cause which may set up
+currents; and in some cases this is exceedingly difficult. Thus, if
+gas is absorbed at the upper surface of a liquid, and if the gaseous
+solution is heavier than the pure liquid, currents may be set up, and
+a steady state of diffusion may cease to exist. This has been tested
+experimentally by C. G. von Hüfner and W. E. Adney. The same
+thing may happen when a gas is evolved into a liquid at the surface
+of a solid even if no bubbles are formed; thus if pieces of aluminium
+are placed in caustic soda, the currents set up by the evolution of
+hydrogen are sufficient to set the aluminium pieces in motion, and
+it is probable that the motions of the Diatomaceae are similarly
+caused by the evolution of oxygen. In some pairs of substances
+diffusion may take place more rapidly than in others. Of course the
+progress of events in any experiment necessarily depends on various
+causes, such as the size of the containing vessels, but it is easy to see
+that when experiments with different substances are carried out under
+similar conditions, however these &ldquo;similar conditions&rdquo; be defined,
+the rates of diffusion must be capable of numerical comparison, and
+the results must be expressible in terms of at least one physical
+quantity, which for any two substances can be called their coefficient
+of diffusion. How to select this quantity we shall see later.</p>
+</div>
+
+<p>2 <i>Quantitative Methods of observing Diffusion.</i>&mdash;The simplest
+plan of determining the progress of diffusion between two liquids
+would be to draw off and examine portions from different strata
+at some stage in the process; the disturbance produced would,
+however, interfere with the subsequent process of diffusion, and
+the observations could not be continued. By placing in the
+liquid column hollow glass beads of different average densities,
+and observing at what height they remain suspended, it is
+possible to trace the variations of density of the liquid column
+at different depths, and different times. In this method, which
+was originally introduced by Lord Kelvin, difficulties were
+caused by the adherence of small air bubbles to the beads.</p>
+
+<p>In general, optical methods are the most capable of giving
+exact results, and the following may be distinguished, (a) <i>By
+refraction in a horizontal plane.</i> If the containing vessel is in
+the form of a prism, the deviation of a horizontal ray of light in
+passing through the prism determines the index of refraction,
+and consequently the density of the stratum through which the
+ray passes, (b) <i>By refraction in a vertical plane.</i> Owing to the
+density varying with the depth, a horizontal ray entering the
+liquid also undergoes a small vertical deviation, being bent
+downwards towards the layers of greater density. The observation
+of this vertical deviation determines not the actual density,
+but its rate of variation with the depth, <i>i.e.</i> the &ldquo;density gradient&rdquo;
+at any point, (c) <i>By the saccharimeter.</i> In the cases of solutions
+of sugar, which cause rotation of the plane of polarized light,
+the density of the sugar at any depth may be determined by
+observing the corresponding angle of rotation, this was done
+originally by W. Voigt.</p>
+
+<p>3. <i>Elementary Definitions of Coefficient of Diffusion.</i>&mdash;The
+simplest case of diffusion is that of a substance, say a gas, diffusing
+in the interior of a homogeneous solid medium, which remains at
+rest, when no external forces act on the system. We may regard
+it as the result of experience that: (1) if the density of the diffusing
+substance is everywhere the same no diffusion takes place, and
+(2) if the density of the diffusing substance is different at different
+points, diffusion will take place from places of greater to those of
+lesser density, and will not cease until the density is everywhere
+the same. It follows that the rate of flow of the diffusing substance
+at any point in any direction must depend on the density
+gradient at that point in that direction, <i>i.e.</i> on the rate at which
+the density of the diffusing substance decreases as we move in
+that direction. We may define the <i>coefficient of diffusion</i> as the
+ratio of the total mass per unit area which flows across any
+small section, to the rate of decrease of the density per unit
+distance in a direction perpendicular to that section.</p>
+
+<p>In the case of steady diffusion parallel to the axis of x, if &rho; be the
+density of the diffusing substance, and q the mass which flows across
+a unit of area in a plane perpendicular to the axis of x, then the density
+gradient is -d&rho;/dx and the ratio of q to this is called the &ldquo;coefficient
+of diffusion.&rdquo; By what has been said this ratio remains finite, however
+small the actual gradient and flow may be., and it is natural
+to assume, at any rate as a first approximation, that it is constant
+as far as the quantities in question are concerned. Thus if the
+coefficient of diffusion be denoted by K we have q= -K(d&rho;/dx).</p>
+
+<p>Further, the rate at which the quantity of substance is increasing
+in an element between the distances x and x+dx is equal to the
+difference of the rates of flow in and out of the two faces, whence as
+in hydrodynamics, we have d&rho;/dt =-dq/dx.</p>
+
+<p>It follows that the equation of diffusion in this case assumes the
+form</p>
+
+<table class="math0" summary="math">
+<tr><td>d&rho;</td> <td rowspan="2">=</td> <td>d</td>
+<td rowspan="2"><span class="f150">(</span> K</td> <td>d&rho;</td>
+<td rowspan="2"><span class="f150">)</span>,</td></tr>
+<tr><td class="denom">dt</td> <td class="denom">dx</td> <td class="denom">dx</td></tr></table>
+
+<p class="noind">which is identical with the equations representing conduction of
+heat, flow of electricity and other physical phenomena. For motion
+in three dimensions we have in like manner</p>
+
+<table class="math0" summary="math">
+<tr><td>d&rho;</td> <td rowspan="2">=</td> <td>d</td>
+<td rowspan="2"><span class="f150">(</span> K</td> <td>d&rho;</td>
+<td rowspan="2"><span class="f150">)</span> +</td> <td>d</td>
+<td rowspan="2"><span class="f150">(</span> K</td> <td>d&rho;</td>
+<td rowspan="2"><span class="f150">)</span> +</td> <td>d</td>
+<td rowspan="2"><span class="f150">(</span> K</td> <td>d&rho;</td>
+<td rowspan="2"><span class="f150">)</span>;</td></tr>
+<tr><td class="denom">dt</td> <td class="denom">dx </td>
+<td class="denom">dx </td> <td class="denom">dy</td>
+<td class="denom">dy</td> <td class="denom">dz</td> <td class="denom">dz</td></tr></table>
+
+<p class="noind">and the corresponding equations in electricity and heat for anisotropic
+substances would be available to account for any parallel
+phenomena, which may arise, or might be conceived, to exist in
+connexion with diffusion through a crystalline solid.</p>
+
+<p>In the case of a very dilute solution, the coefficient of diffusion
+of the dissolved substance can be defined in the same way as
+when the diffusion takes place in a solid, because the effects of
+diffusion will not have any perceptible influence on the solvent,
+and the latter may therefore be regarded as remaining practically
+at rest. But in most cases of diffusion between two fluids, both
+of the fluids are in motion, and hence there is far greater difficulty
+in determining the motion, and even in defining the coefficient of
+diffusion. It is important to notice in the first instance, that it
+is only the relative motion of the two substances which constitutes
+diffusion. Thus when a current of air is blowing, under
+ordinary circumstances the changes which take place are purely
+mechanical, and do not depend on the separate diffusions of the
+oxygen and nitrogen of which the air is mainly composed. It is
+only when two gases are flowing with unequal velocity, that
+is, when they have a relative motion, that these changes of
+relative distribution, which are called diffusion, take place. The
+best way out of the difficulty is to investigate the separate motions
+of the two fluids, taking account of the mechanical actions
+exerted on them, and supposing that the mutual action of the
+fluids causes either fluid to resist the relative motion of the other.</p>
+
+<p>4. <i>The Coefficient of Resistance.</i>&mdash;Let us call the two diffusing
+fluids A and B. If B were absent, the motion of the fluid A
+would be determined entirely by the variations of pressure of the
+fluid A, and by the external forces, such as that due to gravity
+acting on A. Similarly if A were absent, the motion of B would
+be determined entirely by the variations of pressure due to the
+fluid B, and by the external forces acting on B. When both
+fluids are mixed together, each fluid tends to resist the relative
+motion of the other, and by the law of equality of action and
+reaction, the resistance which A experiences from B is everywhere
+equal and opposite to the resistance which B experiences
+from A. If the amount of this resistance per unit volume be
+divided by the relative velocity of the two fluids, and also by the
+product of their densities, the quotient is called the &ldquo;coefficient of
+resistance.&rdquo; If then &rho;<span class="su">1</span>, &rho;<span class="su">2</span> are the densities cf the two fluids,
+u<span class="su">1</span>, u<span class="su">2</span> their velocities, C the coefficient of resistance, then the
+portion of the fluid A contained in a small element of volume v
+will experience from the fluid B a resistance C&rho;<span class="su">1</span>&rho;<span class="su">2</span>v(u<span class="su">1</span> &minus; u<span class="su">2</span>), and
+the fluid B contained in the same volume element will experience
+from the fluid A an equal and opposite resistance, C&rho;<span class="su">1</span>&rho;<span class="su">2</span>v(u<span class="su">2</span> &minus; u<span class="su">1</span>).</p>
+
+<p>This definition implies the following laws of resistance to
+diffusion, which must be regarded as based on experience, and
+not as self-evident truths: (1) each fluid tends to assume, so far
+as diffusion is concerned, the same equüibrium distribution that
+it would assume if its motion were unresisted by the presence of
+the other fluid. (Of course, the mutual attraction of gravitation
+of the two fluids might affect the final distribution, but this is
+practically negligible. Leaving such actions as this out of
+<span class="pagenum"><a name="page257" id="page257"></a>257</span>
+account the following statement is correct.) In a state of
+equilibrium, the density of each fluid at any point thus depends
+only on the partial pressure of that fluid alone, and is the same
+as if the other fluids were absent. It does not depend on the
+partial pressures of the other fluids. If this were not the case,
+the resistance to diffusion would be analogous to friction, and
+would contain terms which were independent of the relative
+velocity u<span class="su">2</span> &minus; u<span class="su">1</span>. (2) For slow motions the resistance to diffusion
+is (approximately at any rate) proportional to the relative
+velocity. (3) The coefficient of resistance C is not necessarily
+always constant; it may, for example, and, in general, does,
+depend on the temperature.</p>
+
+<p>If we form the equations of hydrodynamics for the different fluids
+occurring in any mixture, taking account of diffusion, but neglecting
+viscosity, and using suffixes 1, 2 to denote the separate fluids, these
+assume the form given by James Clerk Maxwell (&ldquo;Diffusion,&rdquo; in
+<i>Ency. Brit.</i>, 9th ed.):&mdash;</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&rho;<span class="su">1</span></td> <td>Du<span class="su">1</span></td>
+<td rowspan="2">+</td> <td>dp<span class="su">1</span></td>
+<td rowspan="2">&minus; X<span class="su">1</span>&rho;<span class="su">1</span> + C<span class="su">12</span>&rho;<span class="su">1</span>&rho;<span class="su">2</span>(u<span class="su">1</span> &minus; u<span class="su">2</span>) + &amp;c. = 0,</td></tr>
+<tr><td class="denom">Dt</td> <td class="denom">dx</td></tr></table>
+
+<p class="noind">where</p>
+
+<table class="math0" summary="math">
+<tr><td>Du<span class="su">1</span></td> <td rowspan="2">=</td> <td>du<span class="su">1</span></td>
+<td rowspan="2">+ u<span class="su">1</span></td> <td>du<span class="su">1</span></td>
+<td rowspan="2">+ v<span class="su">1</span></td> <td>du<span class="su">1</span></td>
+<td rowspan="2">+ w<span class="su">1</span></td> <td>du<span class="su">1</span></td>
+<td rowspan="2">,</td></tr>
+<tr><td class="denom">Dt</td> <td class="denom">dt</td>
+<td class="denom">dx</td> <td class="denom">dy</td> <td class="denom">dz</td></tr></table>
+
+<p class="noind">and these equations imply that when diffusion and other motions
+cease, the fluids satisfy the separate conditions of equilibrium
+dp<span class="su">1</span>/dx &minus; X<span class="su">1</span>&rho;<span class="su">1</span> = 0. The assumption made in the following account is
+that terms such as Du<span class="su">1</span>/Dt may be neglected in the cases considered.</p>
+
+<p>A further property based on experience is that the motions set
+up in a mixture by diffusion are very slow compared with those
+set up by mechanical actions, such as differences of pressure.
+Thus, if two gases at equal temperature and pressure be allowed
+to mix by diffusion, the heavier gas being below the lighter, the
+process will take a long time; on the other hand, if two gases,
+or parts of the same gas, at different pressures be connected,
+equalization of pressure will take place almost immediately.
+It follows from this property that the forces required to overcome
+the &ldquo;inertia&rdquo; of the fluids in the motions due to diffusion are
+quite imperceptible. At any stage of the process, therefore, any
+one of the diffusing fluids may be regarded as in equilibrium under
+the action of its own partial pressure, the external forces to which
+it is subjected and the resistance to diffusion of the other fluids.</p>
+
+<p>5. <i>Slow Diffusion of two Gases. Relation between the Coefficients
+of Resistance and of Diffusion.</i>&mdash;We now suppose the
+diffusing substances to be two gases which obey Boyle&rsquo;s law, and
+that diffusion takes place in a closed cylinder or tube of unit
+sectional area at constant temperature, the surfaces of equal
+density being perpendicular to the axis of the cylinder, so that the
+direction of diffusion is along the length of the cylinder, and we
+suppose no external forces, such as gravity, to act on the system.</p>
+
+<p>The densities of the gases are denoted by &rho;<span class="su">1</span>, &rho;<span class="su">2</span>, their velocities of
+diffusion by u<span class="su">1</span>, u<span class="su">2</span>,  and if their partial pressures are p<span class="su">1</span>, p<span class="su">2</span>,  we have by
+Boyle&rsquo;s law p<span class="su">1</span> = k<span class="su">1</span>&rho;<span class="su">1</span>, p<span class="su">2</span> = k<span class="su">2</span>&rho;<span class="su">2</span>, where k<span class="su">1</span>,k<span class="su">2</span> are constants for the two
+gases, the temperature being constant. The axis of the cylinder is
+taken as the axis of x.</p>
+
+<p>From the considerations of the preceding section, the effects of
+inertia of the diffusing gases may be neglected, and at any instant of
+the process either of the gases is to be treated as kept in equilibrium
+by its partial pressure and the resistance to diffusion produced by
+the other gas. Calling this resistance per unit volume R, and putting
+R = C&rho;<span class="su">1</span>&rho;<span class="su">2</span>(u<span class="su">1</span> &minus; u<span class="su">2</span>), where C is the coefficient of resistance, the equations
+of equilibrium give</p>
+
+<table class="math0" summary="math">
+<tr><td>dp<span class="su">1</span></td> <td rowspan="2">+ C&rho;<span class="su">1</span>&rho;<span class="su">2</span>(u<span class="su">1</span> &minus; u<span class="su">2</span>) = 0, and</td> <td>dp<span class="su">2</span></td>
+<td rowspan="2">+ C&rho;<span class="su">1</span>&rho;<span class="su">2</span>(u<span class="su">2</span> &minus; u<span class="su">1</span>) = 0&emsp;&emsp;(1)</td></tr>
+<tr><td class="denom">dx</td> <td class="denom">dx</td></tr></table>
+
+<p class="noind">These involve</p>
+
+<table class="math0" summary="math">
+<tr><td>dp<span class="su">1</span></td> <td rowspan="2">+</td> <td>dp<span class="su">2</span></td>
+<td rowspan="2">= 0 or p<span class="su">1</span> + p<span class="su">2</span> = P&emsp;&emsp;(2)</td></tr>
+<tr><td class="denom">dx</td> <td class="denom">dx</td></tr></table>
+
+<p class="noind">where P is the total pressure of the mixture, and is everywhere
+constant, consistently with the conditions of mechanical equilibrium.</p>
+
+<p>Now dp<span class="su">1</span>/dx is the pressure-gradient of the first gas, and is, by
+Boyle&rsquo;s law, equal to k<span class="su">1</span> times the corresponding density-gradient.
+Again &rho;<span class="su">1</span>u<span class="su">1</span> is the mass of gas flowing across any section per unit
+time, and k<span class="su">1</span>&rho;<span class="su">1</span>u<span class="su">1</span> or p<span class="su">1</span>u<span class="su">1</span> can be regarded as representing the flux of
+partial pressure produced by the motion of the gas. Since the total
+pressure is everywhere constant, and the ends of the cylinder are
+supposed fixed, the fluxes of partial pressure due to the two gases
+are equal and opposite, so that</p>
+
+<p class="center">p<span class="su">1</span>u<span class="su">1</span> + p<span class="su">2</span>u<span class="su">2</span> = 0 or k<span class="su">1</span>&rho;<span class="su">1</span>u<span class="su">1</span> + k<span class="su">2</span>&rho;<span class="su">2</span>u<span class="su">2</span> = 0&emsp;&emsp;(3).</p>
+
+<p class="noind">From (2) (3) we find by elementary algebra</p>
+
+<p class="center">u<span class="su">1</span>/p<span class="su">2</span> = &minus;u<span class="su">2</span>/p<span class="su">1</span> = (u<span class="su">1</span> &minus; u<span class="su">2</span>)/(p<span class="su">1</span> + p<span class="su">2</span>) = (u<span class="su">1</span> &minus; u<span class="su">2</span>)/P,</p>
+
+<p class="noind">and therefore</p>
+
+<p class="center">p<span class="su">2</span>u<span class="su">1</span> = &minus;p<span class="su">2</span>u<span class="su">2</span> = p<span class="su">1</span>p<span class="su">2</span>(u<span class="su">1</span> &minus; u<span class="su">2</span>)/P = k<span class="su">1</span>k<span class="su">2</span>&rho;<span class="su">1</span>&rho;<span class="su">2</span>(u<span class="su">1</span> &minus; u<span class="su">2</span>)/P</p>
+
+<p class="noind">Hence equations (1) (2) gives</p>
+
+<table class="math0" summary="math">
+<tr><td>dp<span class="su">1</span></td> <td rowspan="2">+</td> <td>CP</td>
+<td rowspan="2">(p<span class="su">1</span>u<span class="su">1</span>) = 0, and</td> <td>dp<span class="su">2</span></td>
+<td rowspan="2">+</td> <td>CP</td>
+<td rowspan="2">(p<span class="su">2</span>u<span class="su">2</span>) = 0;</td></tr>
+<tr><td class="denom">dx</td> <td class="denom">k<span class="su">1</span>k<span class="su">2</span></td>
+<td class="denom">dx</td> <td class="denom">k<span class="su">1</span>k<span class="su">2</span></td></tr></table>
+
+<p class="noind">whence also substituting p<span class="su">1</span> = k<span class="su">1</span>&rho;<span class="su">1</span>, p<span class="su">2</span> = k<span class="su">2</span>&rho;<span class="su">2</span>, and by transposing</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&rho;<span class="su">1</span>u<span class="su">1</span> = &minus;</td> <td>k<span class="su">1</span>k<span class="su">2</span></td> <td>d&rho;<span class="su">1</span></td>
+<td rowspan="2">, and &rho;<span class="su">2</span>u<span class="su">2</span> = &minus;</td> <td>k<span class="su">1</span>k<span class="su">2</span></td> <td>d&rho;<span class="su">2</span></td>
+<td rowspan="2">.</td></tr>
+<tr><td class="denom">CP</td> <td class="ov">dx</td>
+<td class="denom">CP</td> <td class="ov">dx</td></tr></table>
+
+<p>We may now define the &ldquo;coefficient of diffusion&rdquo; of either gas as
+the ratio of the rate of flow of that gas to its density-gradient. With
+this definition, the coefficients of diffusion of both the gases in a
+mixture are equal, each being equal to k<span class="su">1</span>k<span class="su">2</span>/CP. The ratios of the
+fluxes of partial pressure to the corresponding pressure-gradients are
+also equal to the same coefficient. Calling this coefficient K, we also
+observe that the equations of continuity for the two gases are</p>
+
+<table class="math0" summary="math">
+<tr><td>d&rho;<span class="su">1</span></td> <td rowspan="2">+</td> <td>d(&rho;<span class="su">1</span>u<span class="su">1</span>)</td>
+<td rowspan="2">= 0, and</td> <td>d&rho;<span class="su">2</span></td>
+<td rowspan="2">+</td> <td>d(&rho;<span class="su">2</span>u<span class="su">2</span>)</td>
+<td rowspan="2">= 0,</td></tr>
+<tr><td class="denom">dt</td> <td class="denom">dx</td>
+<td class="denom">dt</td> <td class="denom">dx</td></tr></table>
+
+<p class="noind">leading to the equations of diffusion</p>
+
+<table class="math0" summary="math">
+<tr><td>d&rho;<span class="su">1</span></td> <td rowspan="2">=</td> <td>d</td>
+<td rowspan="2"><span class="f150">(</span>K</td> <td>d&rho;<span class="su">1</span></td>
+<td rowspan="2"><span class="f150">)</span>, and</td> <td>d&rho;<span class="su">2</span></td>
+<td rowspan="2">=</td> <td>d</td>
+<td rowspan="2"><span class="f150">(</span>K</td> <td>d&rho;<span class="su">2</span></td>
+<td rowspan="2"><span class="f150">)</span>,</td></tr>
+<tr><td class="denom">dt</td> <td class="denom">dx</td>
+<td class="denom">dx</td> <td class="denom">dt</td>
+<td class="denom">dx</td> <td class="denom">dx</td></tr></table>
+
+<p class="noind">exactly as in the case of diffusion through a solid.</p>
+
+<p>If we attempt to treat diffusion in liquids by a similar method,
+it is, in the first place, necessary to define the &ldquo;partial pressure&rdquo;
+of the components occurring in a liquid mixture. This leads to
+the conception of &ldquo;osmotic pressure,&rdquo; which is dealt with in the
+article <span class="sc"><a href="#artlinks">Solution</a></span>. For dilute solutions at constant temperature,
+the assumption that the osmotic pressure is proportional to the
+density, leads to results agreeing fairly closely with experience,
+and this fact may be represented by the statement that a substance
+occurring in a dilute solution behaves like a perfect gas.</p>
+
+<p>6. <i>Relation of the Coefficient of Diffusion to the Units of Length
+and Time.</i>&mdash;We may write the equation defining K in the form</p>
+
+<table class="math0" summary="math">
+<tr><td rowspan="2">&minus;K ×</td> <td>I</td> <td>d&rho;</td>
+<td rowspan="2">.</td></tr>
+<tr><td class="ov">&rho;</td> <td class="ov">dx</td></tr></table>
+
+<p>Here &minus;d&rho;/&rho;dx represents the &ldquo;percentage rate&rdquo; at which the
+density decreases with the distance x; and we thus see that the
+coefficient of diffusion represents the ratio of the velocity of flow
+to the percentage rate at which the density decreases with the
+distance measured in the direction of flow. This percentage rate
+being of the nature of a number divided by a length, and the
+velocity being of the nature of a length divided by a time, we may
+state that K is of two dimensions in length and &minus;1 in time, <i>i.e.</i>
+dimensions L²/T.</p>
+
+<div class="condensed">
+<p><i>Example 1.</i> Taking K = 0.1423 for carbon dioxide and air (at
+temperature 0° C. and pressure 76 cm. of mercury) referred to a
+centimetre and a second as units, we may interpret the result as
+follows:&mdash;Supposing in a mixture of carbon dioxide and air, the
+density of the carbon dioxide decreases by, say, 1, 2 or 3% of
+itself in a distance of 1 cm., then the corresponding velocities
+of the diffusing carbon dioxide will be respectively 0.01, 0.02 and
+0.03 times 0.1423, that is, 0.001423, 0.002846 and 0.004269 cm.
+per second in the three cases.</p>
+
+<p><i>Example 2.</i> If we wished to take a foot and a second as our units,
+we should have to divide the value of the coefficient of diffusion in
+Example 1 by the square of the number of centimetres in 1 ft., that
+is, roughly speaking, by 900, giving the new value of K = 0.00016
+roughly.</p>
+</div>
+
+<p>7. <i>Numerical Values of the Coefficient of Diffusion.</i>&mdash;The
+table on p. 258 gives the values of the coefficient of diffusion of
+several of the principal pairs of gases at a pressure of 76 cm. of
+mercury, and also of a number of other substances. In the gases
+the centimetre and second are taken as fundamental units, in
+other cases the centimetre and day.</p>
+
+<p>8. <i>Irreversible Changes accompanying Diffusion.</i>&mdash;The diffusion
+of two gases at constant pressure and temperature is a good
+example of an &ldquo;irreversible process.&rdquo; The gases always tend to
+mix, never to separate. In order to separate the gases a change
+must be effected in the external conditions to which the mixture
+is subjected, either by liquefying one of the gases, or by separating
+them by diffusion through a membrane, or by bringing other outside
+influences to bear on them. In the case of liquids, electrolysis
+affords a means of separating the constituents of a mixture.
+Every such method involves some change taking place outside the
+mixture, and this change may be regarded as a &ldquo;compensating
+<span class="pagenum"><a name="page258" id="page258"></a>258</span>
+transformation.&rdquo; We thus have an instance of the property
+that every irreversible change leaves an indelible imprint somewhere
+or other on the progress of events in the universe. That
+the process of diffusion obeys the laws of irreversible thermodynamics
+(if these laws are properly stated) is proved by the fact
+that the compensating transformations required to separate
+mixed gases do not essentially involve anything but transformation
+of energy. The process of allowing gases to mix by diffusion,
+and then separating them by a compensating transformation,
+thus constitutes an irreversible cycle, the outside effects of which
+are that energy somewhere or other must be less capable of transformation
+than it was before the change. We express this fact by
+stating that an irreversible process essentially implies a loss of
+availability. To measure this loss we make use of the laws of
+thermodynamics, and in particular of Lord Kelvin&rsquo;s statement
+that &ldquo;It is impossible by means of inanimate material agency to
+derive mechanical effect from any portion of matter by cooling it
+below the temperature of the coldest of the surrounding objects.&rdquo;</p>
+
+<table class="ws" summary="Contents">
+
+<tr><td class="tcc allb">Substances.</td> <td class="tcc allb">Temp.</td> <td class="tcc allb">K.</td> <td class="tcc allb">Author.</td></tr>
+
+<tr><td class="tcl lb rb">Carbon dioxide and air</td> <td class="tcr rb">0°C.</td> <td class="tcl rb">0.1423 cm²/sec.</td> <td class="tcl rb">J. Loschmidt.</td></tr>
+<tr><td class="tcl lb rb">&emsp;&rdquo;&emsp;&emsp;&emsp;&rdquo;&emsp;&emsp;&emsp;hydrogen</td> <td class="tcr rb">0°C.</td> <td class="tcl rb">0.5558&emsp;&emsp;&rdquo;</td> <td class="tcc rb">&rdquo;</td></tr>
+<tr><td class="tcl lb rb">&emsp;&rdquo;&emsp;&emsp;&emsp;&rdquo;&emsp;&emsp;&emsp;oxygen</td> <td class="tcr rb">0°C.</td> <td class="tcl rb">0.1409&emsp;&emsp;&rdquo;</td> <td class="tcc rb">&rdquo;</td></tr>
+<tr><td class="tcl lb rb">&emsp;&rdquo;&emsp;&emsp;&emsp;&rdquo;&emsp;&emsp;&emsp;carbon monoxide</td> <td class="tcr rb">0°C.</td> <td class="tcl rb">0.1406&emsp;&emsp;&rdquo;</td> <td class="tcc rb">&rdquo;</td></tr>
+<tr><td class="tcl lb rb">&emsp;&rdquo;&emsp;&emsp;&emsp;&rdquo;&emsp;&emsp;&emsp;marsh gas (methane)</td> <td class="tcr rb">0°C.</td> <td class="tcl rb">0.1586&emsp;&emsp;&rdquo;</td> <td class="tcc rb">&rdquo;</td></tr>
+<tr><td class="tcl lb rb">&emsp;&rdquo;&emsp;&emsp;&emsp;&rdquo;&emsp;&emsp;&emsp;nitrous oxide</td> <td class="tcr rb">0°C.</td> <td class="tcl rb">0.0983&emsp;&emsp;&rdquo;</td> <td class="tcc rb">&rdquo;</td></tr>
+<tr><td class="tcl lb rb">Hydrogen and oxygen</td> <td class="tcr rb">0°C.</td> <td class="tcl rb">0.7214&emsp;&emsp;&rdquo;</td> <td class="tcc rb">&rdquo;</td></tr>
+<tr><td class="tcl lb rb">&emsp;&rdquo;&emsp;&emsp;&emsp;&rdquo;&emsp;carbon monoxide</td> <td class="tcr rb">0°C.</td> <td class="tcl rb">0.6422&emsp;&emsp;&rdquo;</td> <td class="tcc rb">&rdquo;</td></tr>
+<tr><td class="tcl lb rb">&emsp;&rdquo;&emsp;&emsp;&emsp;&rdquo;&emsp;sulphur dioxide</td> <td class="tcr rb">0°C.</td> <td class="tcl rb">0.4800&emsp;&emsp;&rdquo;</td> <td class="tcc rb">&rdquo;</td></tr>
+<tr><td class="tcl lb rb">Oxygen and carbon monoxide</td> <td class="tcr rb">0°C.</td> <td class="tcl rb">0.1802&emsp;&emsp;&rdquo;</td> <td class="tcc rb">&rdquo;</td></tr>
+<tr><td class="tcl lb rb">Water and ammonia</td> <td class="tcr rb">20°C.</td> <td class="tcl rb">1.250&ensp;&emsp;&emsp;&rdquo;</td> <td class="tcl rb">G. Hüfner.</td></tr>
+<tr><td class="tcl lb rb">&emsp;&rdquo;&emsp;&emsp;&emsp;&rdquo;</td> <td class="tcr rb">5°C.</td> <td class="tcl rb">0.822&ensp;&emsp;&emsp;&rdquo;</td> <td class="tcc rb">&rdquo;</td></tr>
+<tr><td class="tcl lb rb">&emsp;&rdquo;&emsp;&emsp;&emsp;common salt (density 1.0269)</td> <td class="tcr rb">&nbsp;</td> <td class="tcl rb">0.355&ensp;&emsp;&emsp;&rdquo;</td> <td class="tcl rb">J. Graham.</td></tr>
+<tr><td class="tcl lb rb">&emsp;&rdquo;&emsp;&emsp;&emsp;&emsp;&emsp;&rdquo;&emsp;&emsp;&emsp;&rdquo;</td> <td class="tcr rb">14.33°C.</td> <td class="tcl rb">1.020, 0.996, 0.972, 0.932 cm²/day.</td> <td class="tcl rb"> F. Heimbrodt.</td></tr>
+<tr><td class="tcl lb rb">&emsp;&rdquo;&emsp;&emsp;&emsp;zinc sulphate (0.312 gm/cm³)</td> <td class="tcr rb">&nbsp;</td> <td class="tcl rb">0.1162 cm²/day.</td> <td class="tcl rb">W. Seitz.</td></tr>
+<tr><td class="tcl lb rb">&emsp;&rdquo;&emsp;&emsp;&emsp;zinc sulphate (normal)</td> <td class="tcr rb">&nbsp;</td> <td class="tcl rb">0.2355&emsp;&emsp;&rdquo;</td> <td class="tcc rb">&rdquo;</td></tr>
+<tr><td class="tcl lb rb">&emsp;&rdquo;&emsp;&emsp;&emsp;zinc acetate (double normal)</td> <td class="tcr rb">&nbsp;</td> <td class="tcl rb">0.1195&emsp;&emsp;&rdquo;</td> <td class="tcc rb">&rdquo;</td></tr>
+<tr><td class="tcl lb rb">&emsp;&rdquo;&emsp;&emsp;&emsp;zinc formate (half normal)</td> <td class="tcr rb">&nbsp;</td> <td class="tcl rb">0.4654&emsp;&emsp;&rdquo;</td> <td class="tcc rb">&rdquo;</td></tr>
+<tr><td class="tcl lb rb">&emsp;&rdquo;&emsp;&emsp;&emsp;cadmium sulphate (double normal)</td> <td class="tcc rb">· ·</td> <td class="tcl rb">0.2456&emsp;&emsp;&rdquo;</td> <td class="tcc rb">&rdquo;</td></tr>
+<tr><td class="tcl lb rb">&emsp;&rdquo;&emsp;&emsp;&emsp;glycerin (<span class="spp">1</span>&frasl;<span class="suu">8</span>n, ½n, <span class="spp">7</span>&frasl;<span class="suu">8</span>n, 1.5n)</td> <td class="tcr rb">10.14°C.</td> <td class="tcl rb">0.356, 0.350, 0.342, 0.315 cm²/day.</td> <td class="tcl rb">F. Heimbrodt.</td></tr>
+<tr><td class="tcl lb rb">&emsp;&rdquo;&emsp;&emsp;&emsp;urea&emsp;&emsp;&emsp;&rdquo;&emsp;&emsp;&emsp;&rdquo;</td> <td class="tcr rb">14.83°C.</td> <td class="tcl rb">0.973, 0.946, 0.926, 0.883  cm²/day.</td> <td class="tcc rb">&rdquo;</td></tr>
+<tr><td class="tcl lb rb">&emsp;&rdquo;&emsp;&emsp;&emsp;hydrochloric acid</td> <td class="tcr rb">14.30°C.</td> <td class="tcl rb">2.208, 2.331, 2.480  cm²/day.</td> <td class="tcc rb">&rdquo;</td></tr>
+<tr><td class="tcl lb rb">Gelatin 20% and ammonia</td> <td class="tcr rb">17°C.</td> <td class="tcl rb">127.1 cm²/day.</td> <td class="tcl rb">A. Hagenbach.</td></tr>
+<tr><td class="tcl lb rb">&emsp;&rdquo;&emsp;&emsp;&rdquo;&emsp;&emsp;carbon dioxide</td> <td class="tcc rb">· ·</td> <td class="tcl rb">0.845&ensp;&emsp;&emsp;&rdquo;</td> <td class="tcc rb">&rdquo;</td></tr>
+<tr><td class="tcl lb rb">&emsp;&rdquo;&emsp;&emsp;&rdquo;&emsp;&emsp;nitrous oxide</td> <td class="tcc rb">· ·</td> <td class="tcl rb">0.509&ensp;&emsp;&emsp;&rdquo;</td> <td class="tcc rb">&rdquo;</td></tr>
+<tr><td class="tcl lb rb">&emsp;&rdquo;&emsp;&emsp;&rdquo;&emsp;&emsp;oxygen</td> <td class="tcc rb">· ·</td> <td class="tcl rb">0.230&ensp;&emsp;&emsp;&rdquo;</td> <td class="tcc rb">&rdquo;</td></tr>
+<tr><td class="tcl lb rb bb">&emsp;&rdquo;&emsp;&emsp;&rdquo;&emsp;&emsp;hydrogen</td> <td class="tcc rb bb">· ·</td> <td class="tcl rb bb">0.0565&emsp;&emsp;&rdquo;</td> <td class="tcc rb bb">&rdquo;</td></tr>
+</table>
+
+<div class="condensed">
+<p>Let us now assume that we have any syste m such as the gases
+above considered, and that it is in the  presence of an indefinitely
+extended medium which we shall call the  &ldquo;auxiliary medium.&rdquo; If
+heat be taken from any part of the system, only part of this heat can
+be converted into work by means of thermodynamic engines; and
+the rest will be given to the auxiliary medium, and will constitute
+unavailable energy or waste. To understand what this means, we
+may consider the case of a condensing steam engine. Only part of
+the energy liberated by the combustion of the coal is available for
+driving the engine, the rest takes the form of heat imparted to
+the condenser. The colder the condenser the more efficient is the
+engine, and the smaller is the quantity of waste.</p>
+
+<p>The amount of unavailable energy associated with any given
+transformation is proportional to the absolute temperature of the
+auxiliary medium. When divided by that temperature the quotient
+is called the change of &ldquo;entropy&rdquo; associated with the given change
+(see <span class="sc"><a href="#artlinks">Thermodynamics</a></span>). Thus if a body at temperature T receives
+a quantity of heat Q, and if T<span class="su">0</span> is the temperature of the auxiliary
+medium, the quantity of work which could be obtained from Q by
+means of ideal thermodynamic engines would be Q(1 &minus; T<span class="su">0</span>/T), and
+the balance, which is QT<span class="su">0</span>/T, would take the form of unavailable
+or waste energy given to the medium. The quotient of this, when
+divided by T<span class="su">0</span>, is Q/T, and this represents the quantity of entropy
+associated with Q units of heat at temperature T.</p>
+
+<p>Any irreversible change for which a compensating transformation
+of energy exists represents, therefore, an increase of unavailable
+energy, which is measurable in terms of entropy. The increase of
+entropy is independent of the temperature of the auxiliary medium.
+It thus affords a measure of the extent to which energy has run
+to waste during the change. Moreover, when a body is heated, the
+increase of entropy is the factor which determines how much of the
+energy imparted to the body is unavailable for conversion into work
+under given conditions. In all cases we have</p>
+
+<table class="math0" summary="math">
+<tr><td>increase of unavailable energy</td> <td rowspan="2">= increase of entropy.</td></tr>
+<tr><td class="denom">temperature of auxiliary medium</td></tr></table>
+
+<p>When diffusion takes place between two gases inside a closed
+vessel at uniform pressure and temperature no energy in the form
+of heat or work is received from without, and hence the entropy
+gained by the gases from without is zero. But the irreversible
+processes inside the vessel may involve a gain of entropy, and this
+can only be estimated by examining
+by what means mixed
+gases can be separated, and, in
+particular, under what conditions
+the process of mixing
+and separating the gases could
+(theoretically) be made reversible.</p>
+</div>
+
+<p>9. <i>Evidence derived from
+Liquefaction of one or both of
+the Gases.</i>&mdash;The gases in a
+mixture can often be separated
+by liquefying, or even solidifying,
+one or both of the components.
+In connexion with
+this property we have the
+important law according to
+which &ldquo;The pressure of a
+vapour in equilibrium with its
+liquid depends only on the
+temperature and is independent
+of the pressures of any
+other gases or vapours which
+may be mixed with it.&rdquo; Thus
+if two closed vessels be taken
+containing some water and
+one be exhausted, the other
+containing air, and if the temperatures
+be equal, evaporation
+will go on until the
+pressure of the vapour in the
+exhausted vessel is equal to
+its <i>partial</i> pressure in the other vessel, notwithstanding the fact
+that the <i>total</i> pressure in the latter vessel is greater by the
+pressure of the air.</p>
+
+<div class="condensed">
+<p>To separate mixed gases by liquefaction, they must be compressed
+and cooled till one separates in the form of a liquid. If no changes are
+to take place outside the system, the separate components must be
+allowed to expand until the work of expansion is equal to the work
+of compression, and the heat given out in compression is reabsorbed
+in expansion. The process may be made as nearly reversible as we
+like by performing the operations so slowly that the substances
+are practically in a state of equilibrium at every stage. This is a
+consequence of an important axiom in thermodynamics according
+to which &ldquo;any small change in the neighbourhood of a state of
+equilibrium is to a first approximation reversible.&rdquo;</p>
+
+<p>Suppose now that at any stage of the compression the partial
+pressures of the two gases are p<span class="su">1</span> and p<span class="su">2</span>, and that the volume is
+changed from V to V &minus; dV. The work of compression is (p<span class="su">1</span> + p<span class="su">2</span>)dV,
+and this work will be restored at the corresponding stage if each
+of the separated gases increases in volume from V &minus; dV to V. The
+ultimate state of the separated gases will thus be one in which
+each gas occupies the volume V originally occupied by the mixture.</p>
+
+<p>We may now obtain an estimate of the amount of energy rendered
+unavailable by diffusion. We suppose two gases occupying volumes
+V<span class="su">1</span> and V<span class="su">2</span> at equal pressure p to mix by diffusion, so that the final
+volume is V<span class="su">1</span> + V<span class="su">2</span>. Then if before mixing each gas had been allowed
+to expand till its volume was V<span class="su">1</span> + V<span class="su">2</span>, work would have been done
+in the expansion, and the gases could still have been mixed by a
+reversal of the process above described. In the actual diffusion this
+work of expansion is lost, and represents energy rendered unavailable
+at the temperature at which diffusion takes place. When divided
+by that temperature the quotient gives the increase of entropy.
+Thus the irreversible processes, and, in particular, the entropy
+changes associated with diffusion of two gases at uniform pressure,
+are the same as would take place if each of the gases in turn were to
+expand by rushing into a vacuum, till it occupied the whole volume
+of the mixture. A more rigorous proof involves considerations of
+the thermodynamic potentials, following the methods of J. Willard
+Gibbs (see <span class="sc"><a href="#artlinks">Energetics</a></span>).</p>
+
+<p><span class="pagenum"><a name="page259" id="page259"></a>259</span></p>
+
+<p>Another way in which two or more mixed gases can be separated
+is by placing them in the presence of a liquid which can freely absorb
+one of the gases, but in which the other gas or gases are insoluble.
+Here again it is found by experience that when equilibrium exists
+at a given temperature between the dissolved and undissolved
+portions of the first gas, the partial pressure of that gas in the
+mixture depends on the temperature alone, and is independent of
+the partial pressures of the insoluble gases with which it is mixed,
+so that the conclusions are the same as before.</p>
+</div>
+
+<p>10. <i>Diffusion through a Membrane or Partition. Theory of the
+semi-permeable Membrane.</i>&mdash;It has been pointed out that diffusion
+of gases frequently takes place in the interior of solids; moreover,
+different gases behave differently with respect to the same solid at
+the same temperature. A membrane or partition formed of such
+a solid can therefore be used to effect a more or less complete
+separation of gases from a mixture. This method is employed
+commercially for extracting oxygen from the atmosphere, in
+particular for use in projection lanterns where a high degree of
+purity is not required. A similar method is often applied to
+liquids and solutions and is known as &ldquo;dialysis.&rdquo;</p>
+
+<p>In such cases as can be tested experimentally it has been found
+that a gas always tends to pass through a membrane from the side
+where its density, and therefore its partial pressure, is greater
+to the side where it is less; so that for equilibrium the partial
+pressures on the two sides must be equal. This result is unaffected
+by the presence of other gases on one or both sides of the
+membrane. For example, if different gases at the same pressure
+are separated by a partition through which one gas can pass more
+rapidly than the other, the diffusion will give rise to a difference of
+pressure on the two sides, which is capable of doing mechanical
+work in moving the partition. In evidence of this conclusion
+Max Planck quotes a test experiment made by him in the Physical
+Institute of the university of Munich in 1883, depending on the
+fact that platinum foil at white heat is permeable to hydrogen but
+impermeable to air, so that if a platinum tube filled with hydrogen
+be heated the hydrogen will diffuse out, leaving a vacuum.</p>
+
+<div class="condensed">
+<p>The details of the experiment may be quoted here:&mdash;&ldquo;A glass
+tube of about 5 mm. internal diameter, blown out to a bulb at the
+middle, was provided with a stop-cock at one end. To the other a
+platinum tube 10 cm. long was fastened, and closed at the end. The
+whole tube was exhausted by a mercury pump, filled with hydrogen
+at ordinary atmospheric pressure, and then closed. The closed end
+of the platinum portion was then heated in a horizontal position by
+a Bunsen burner. The connexion between the glass and platinum
+tubes, having been made by means of sealing-wax, had to be kept
+cool by a continuous current of water to prevent the softening of the
+wax. After four hours the tube was taken from the flame, cooled
+to the temperature of the room, and the stop-cock opened under
+mercury. The mercury rose rapidly, almost completely filling the
+tube, proving that the tube had been very nearly exhausted.&rdquo;</p>
+</div>
+
+<table class="nobctr" style="float: left; width: 370px;" summary="Illustration">
+<tr><td class="figleft1"><img style="width:317px; height:156px" src="images/img259.jpg" alt="" /></td></tr></table>
+
+<p>In order that diffusion through a membrane may be reversible
+so far as a particular gas is concerned, the process must take place
+so slowly that equilibrium is set up at every stage (see § 9 above).
+In order to separate one
+gas from another consistently
+with this condition
+it is necessary
+that no diffusion of the
+latter gas should accompany
+the process.
+The name &ldquo;semi-permeable&rdquo;
+is applied to
+an ideal membrane or partition through which one gas can
+pass, and which offers an insuperable barrier to any diffusion
+whatever of a second gas. By means of two semi-permeable
+partitions acting oppositely with respect to two different gases
+A and B these gases could be mixed or separated by reversible
+methods. The annexed figure shows a diagrammatic representation
+of the process.</p>
+
+<div class="condensed">
+<p>We suppose the gases contained in a cylindrical tube; P, Q, R, S
+are four pistons, of which P and R are joined to one connecting rod,
+Q and S to another. P, S are impermeable to both gases; Q is
+semi-permeable, allowing the gas A to pass through but not B, similarly
+R allows the gas B to pass through but not A. The distance PR
+is equal to the distance QS, so that if the rods are pushed towards each
+other as far as they will go, P and Q will be in contact, as also R and
+S. Imagine the space RQ filled with a mixture of the two gases
+under these conditions. Then by slowly drawing the connecting
+rods apart until R, Q touch, the gas A will pass into the space PQ,
+and B will pass into the space RS, and the gases will finally be completely
+separated; similarly, by pushing the connecting rods together,
+the two gases will be remixed in the space RQ. By performing the
+operations slowly enough we may make the processes as nearly
+reversible as we please, so that no available energy is lost in either
+change. The gas A being at every instant in equilibrium on the two
+sides of the piston Q, its density, and therefore its partial pressure,
+is the same on both sides, and the same is true regarding the gas B
+on the two sides of R. Also <i>no work is done in moving the pistons</i>, for
+the partial pressures of B on the two sides of R balance each other,
+consequently, the resultant thrust on R is due to the gas A alone,
+and is equal and opposite to its resultant thrust on P, so that the
+connecting rods are at every instant in a state of mechanical equilibrium
+so far as the pressures of the gases A and B are concerned. We
+conclude that in the reversible separation of the gases by this method
+at constant temperature without the production or absorption of
+mechanical work, the densities and the partial pressures of the two
+separated gases are the same as they were in the mixture. These
+conclusions are in entire agreement with those of the preceding
+section. If this agreement did not exist it would be possible, theoretically,
+to obtain perpetual motion from the gases in a way that
+would be inconsistent with the second law of thermodynamics.</p>
+</div>
+
+<p>Most physicists admit, as Planck does, that it is impossible to
+obtain an ideal semi-permeable substance; indeed such a substance
+would necessarily have to possess an infinitely great resistance
+to diffusion for such gases as could not penetrate it. But in
+an experiment performed under actual conditions the losses of
+available energy arising from this cause would be attributable
+to the imperfect efficiency of the partitions and not to the gases
+themselves; moreover, these losses are, in every case, found to be
+completely in accordance with the laws of irreversible thermodynamics.
+The reasoning in this article being somewhat condensed
+the reader must necessarily be referred to treatises on
+thermodynamics for further information on points of detail
+connected with the argument. Even when he consults these
+treatises he may find some points omitted which have been
+examined in full detail at some time or other, but are not sufficiently
+often raised to require mention in print.</p>
+
+<p>II. <i>Kinetic Models of Diffusion.</i>&mdash;Imagine in the first instance
+that a very large number of red balls are distributed over one half
+of a billiard table, and an equal number of white balls over the
+other half. If the balls are set in motion with different velocities
+in various directions, diffusion will take place, the red balls finding
+their way among the white ones, and vice versa; and the
+process will be retarded by collisions between the balls. The
+simplest model of a perfect gas studied in the kinetic theory of
+gases (see <span class="sc"><a href="#artlinks">Molecule</a></span>) differs from the above illustration in that
+the bodies representing the molecules move in space instead of in
+a plane, and, unlike billiard balls, their motion is unresisted,
+and they are perfectly elastic, so that no kinetic energy is lost
+either during their free motions, or at a collision.</p>
+
+<div class="condensed">
+<p>The mathematical analysis connected with the application of the
+kinetic theory to diffusion is very long and cumbersome. We shall
+therefore confine our attention to regarding a medium formed of
+elastic spheres as a mechanical model, by which the most important
+features of diffusion can be illustrated. We shall assume the results
+of the kinetic theory, according to which:&mdash;(1) In a dynamical
+model of a perfect gas the mean kinetic energy of translation of the
+molecules represents the absolute temperature of the gas. (2) The
+pressure at any point is proportional to the product of the number
+of molecules in unit volume about that point into the mean square
+of the velocity. (The mean square of the velocity is different from
+but proportional to the square of the mean velocity, and in the
+subsequent arguments either of these two quantities can generally
+be taken.) (3) In a gas mixture represented by a mixture of molecules
+of unequal masses, the mean kinetic energies of the different
+kinds are equal.</p>
+
+<p>Consider now the problem of diffusion in a region containing two
+kinds of molecules A and B of unequal mass. The molecules of A
+in the neighbourhood of any point will, by their motion, spread out
+in every direction until they come into collision with other molecules
+of either kind, and this spreading out from every point of the medium
+will give rise to diffusion. If we imagine the velocities of the A
+molecules to be equally distributed in all directions, as they would
+be in a homogeneous mixture, it is obvious that the process of diffusion
+will be greater, <i>ceteris paribus</i>, the greater the velocity of the molecules,
+and the greater the length of the free path before a collision
+takes place. If we assume consistently with this, that the coefficient
+of diffusion of the gas A is proportional to the mean value of
+W{a}l{a}, where w{a} is the velocity and l{a} is the length of the path of a
+<span class="pagenum"><a name="page260" id="page260"></a>260</span>
+molecule of A, this expression for the coefficient of diffusion is of the
+right dimensions in length and time. If, moreover, we observe that
+when diffusion takes place in a fixed direction, say that of the axis
+of x, it depends only on the resolved part of the velocity and length
+of path in that direction: this hypothesis readily leads to our taking
+the mean value of <span class="spp">1</span>&frasl;<span class="suu">3</span>w<span class="su">a</span>l<span class="su">a</span> as the coefficient of diffusion for the gas A.
+This value was obtained by O. E. Meyer and others.</p>
+
+<p>Unfortunately, however, it makes the coefficients of diffusion
+unequal for the two gases, a result inconsistent with that obtained
+above from considerations of the coefficient of resistance, and
+leading to the consequence that differences of pressure would be
+set up in different parts of the gas. To equalize these differences of
+pressure, Meyer assumed that a counter current is set up, this current
+being, of course, very slow in practice; and J. Stefan assumed that
+the diffusion of one gas was not affected by collisions between molecules
+of the <i>same gas</i>. When the molecules are mixed in equal
+proportions both hypotheses lead to the value <span class="spp">1</span>&frasl;<span class="suu">6</span>([w<span class="su">a</span>l<span class="su">a</span>] + [w<span class="su">b</span>l<span class="su">b</span>]),
+(square brackets denoting mean values). When one gas preponderates
+largely over the other, the phenomena of diffusion are too
+difficult of observation to allow of accurate experimental tests
+being made. Moreover, in this case no difference exists unless the
+molecules are different in size or mass.</p>
+
+<p>Instead of supposing a velocity of translation added after the
+mathematical calculations have been performed, a better plan is to
+assume from the outset that the molecules of the two gases have
+small velocities of translation in opposite directions, superposed on
+the distribution of velocity, which would occur in a medium representing
+a gas at rest. When a collision occurs between molecules
+of different gases a transference of momentum takes place between
+them, and the quantity of momentum so transferred in one second
+in a unit of volume gives a dynamical measure of the resistance to
+diffusion. It is to be observed that, however small the relative
+velocity of the gases A and B, it plays an all-important part in
+determining the coefficient of resistance; for without such relative
+motion, and with the velocities evenly distributed in all directions, no
+transference of momentum could take place. The coefficient of
+resistance being found, the motion of each of the two gases may be
+discussed separately.</p>
+</div>
+
+<p>One of the most important consequences of the kinetic theory
+is that if the volume be kept constant the coefficient of diffusion
+varies as the square root of the absolute temperature. To prove
+this, we merely have to imagine the velocity of each molecule to
+be suddenly increased n fold; the subsequent processes, including
+diffusion, will then go on n times as fast; and the temperature
+T, being proportional to the kinetic energy, and therefore to the
+square of the velocity, will be increased n² fold. Thus K, the
+coefficient of diffusion, varies as &radic;T.</p>
+
+<p>The relation of K to the density when the temperature remains
+constant is more difficult to discuss, but it may be sufficient to
+notice that if the number of molecules is increased n fold, the
+chances of a collision are n times as great, and the distance
+traversed between collisions is (not <i>therefore</i> but as the result of
+more detailed reasoning) on the average 1/n of what it was before.
+Thus the free path, and therefore the coefficient of diffusion,
+varies inversely as the density, or directly as the volume. If the
+pressure p and temperature T be taken as variables, K varies
+inversely as p and directly as &radic;T³.</p>
+
+<p>Now according to the experiments first made by J. C. Maxwell
+and J. Loschmidt, it appeared that with constant density K
+was proportional to T more nearly than to &radic;T. The inference is
+that in this respect a medium formed of colliding spheres fails to
+give a correct mechanical model of gases. It has been found by
+L. Boltzmann, Maxwell and others that a system of particles
+whose mutual actions vary according to the inverse fifth power of
+the distance between them represents more correctly the relation
+between the coefficient of diffusion and temperature in actual
+gases. Other recent theories of diffusion have been advanced
+by M. Thiesen, P. Langevin and W. Sutherland. On the other
+hand, J. Thovert finds experimental evidence that the coefficient
+of diffusion <i>is</i> proportional to molecular velocity in the cases
+examined of non-electrolytes dissolved in water at 18° at 2.5
+grams per litre.</p>
+
+<div class="condensed">
+<p><span class="sc">Bibliography.</span>&mdash;The best introduction to the study of theories
+of diffusion is afforded by O. E. Meyer&rsquo;s Kinetic <i>Theory of Gases</i>,
+translated by Robert E. Baynes (London, 1899). The mathematical
+portion, though sufficient for ordinary purposes, is mostly of the
+simplest possible character. Another useful treatise is R. Ruhlmann&rsquo;s
+<i>Handbuch der mechanischen Wärmetheorie</i> (Brunswick, 1885). For
+a shorter sketch the reader may refer to J. C. Maxwell&rsquo;s <i>Theory of
+Heat</i>, chaps, xix. and xxii., or numerous other treatises on physics.
+The theory of the semi-permeable membrane is discussed by
+M. Planck in his <i>Treatise on Thermodynamics</i>, English translation
+by A. Ogg (1903), also in treatises on thermodynamics by W. Voigt
+and other writers. For a more detailed study of diffusion in general
+the following papers may be consulted:&mdash;L. Boltzmann, &ldquo;Zur
+Integration der Diffusionsgleichung,&rdquo; <i>Sitzung. der k. bayer. Akad math.-phys.
+Klasse</i> (May 1894); T. des Coudres, &ldquo;Diffusionsvorgänge in
+einem Zylinder,&rdquo; <i>Wied. Ann.</i> lv. (1895), p. 213; J. Loschmidt,
+&ldquo;Experimentaluntersuchungen über Diffusion,&rdquo; <i>Wien. Sitz.</i> lxi.,
+lxii. (1870); J. Stefan, &ldquo;Gleichgewicht und ... Diffusion von Gasmengen,&rdquo;
+<i>Wien. Sitz.</i> lxiii., &ldquo;Dynamische Theorie der Diffusion,&rdquo;
+<i>Wien. Sitz.</i> lxv. (April 1872); M. Toepler, &ldquo;Gas-diffusion,&rdquo; <i>Wied.
+Ann.</i> lviii. (1896), p. 599; A. Wretschko, &ldquo;Experimentaluntersuchungen
+über die Diffusion von Gasmengen,&rdquo; <i>Wien. Sitz.</i> lxii.
+The mathematical theory of diffusion, according to the kinetic
+theory of gases, has been treated by a number of different methods,
+and for the study of these the reader may consult L. Boltzmann,
+<i>Vorlesungen über Gastheorie</i> (Leipzig, 1896-1898); S. H. Burbury,
+<i>Kinetic Theory of Gases</i> (Cambridge, 1899), and papers by L. Boltzmann
+in <i>Wien. Sitz.</i> lxxxvi. (1882), lxxxvii. (1883); P. G. Tait,
+&ldquo;Foundations of the Kinetic Theory of Gases,&rdquo; <i>Trans. R.S.E.</i>
+xxxiii., xxxv., xxvi., or <i>Scientific Papers</i>, ii. (Cambridge, 1900).
+For recent work reference should be made to the current issues
+of <i>Science Abstracts</i> (London), and entries under the heading
+&ldquo;Diffusion&rdquo; will be found in the general index at the end of each
+volume.</p>
+</div>
+<div class="author">(G. H. Br.)</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIGBY, SIR EVERARD<a name="ar86" id="ar86"></a></span> (1578-1606), English conspirator, son
+of Everard Digby of Stoke Dry, Rutland, was born on the 16th
+of May 1578. He inherited a large estate at his father&rsquo;s death
+in 1592, and acquired a considerable increase by his marriage in
+1596 to Mary, daughter and heir of William Mulsho of Gothurst
+(now Gayhurst), in Buckinghamshire. He obtained a place in
+Queen Elizabeth&rsquo;s household and as a ward of the crown was
+brought up a Protestant; but about 1599 he came under the
+influence of the Jesuit, John Gerard, and soon afterwards joined
+the Roman Catholics. He supported James&rsquo;s accession and was
+knighted by the latter on the 23rd of April 1603. In a letter to
+Salisbury, the date of which has been ascribed to May 1605,
+Digby offered to go on a mission to the pope to obtain from
+the latter a promise to prevent Romanist attempts against the
+government in return for concessions to the Roman Catholics;
+adding that if severe measures were again taken against them
+&ldquo;within brief there will be massacres, rebellions and desperate
+attempts against the king and state.&rdquo; Digby had suffered no
+personal injury or persecution on account of his religion, but he
+sympathized with his co-religionists; and when at Michaelmas,
+1605, the government had fully decided to return to the policy of
+repression, the authors of the Gunpowder Plot (<i>q.v.</i>) sought his
+financial support, and he joined eagerly in the conspiracy. His
+particular share in the plan was the organization of a rising in the
+Midlands; and on the pretence of a hunting party he assembled a
+body of gentlemen together at Danchurch in Warwickshire on the
+5th of November, who were to take action immediately the news
+arrived from London of the successful destruction of the king
+and the House of Lords, and to seize the person of the princess
+Elizabeth, who was residing in the neighbourhood. The conspirators
+arrived late on the evening of the 6th to tell their story
+of failure and disaster, and Digby, who possibly might have
+escaped the more serious charge of high treason, was persuaded by
+Catesby, with a false tale that the king and Salisbury were dead,
+to further implicate himself in the plot and join the small band of
+conspirators in their hopeless endeavour to raise the country. He
+accompanied them, the same day, to Huddington in Worcestershire
+and on the 7th to Holbeche in Staffordshire. The following
+morning, however, he abandoned his companions, dismissed his
+servants except two, who declared &ldquo;they would never leave him
+but against their will,&rdquo; and attempted with these to conceal himself
+in a pit. He was, however, soon discovered and surrounded.
+He made a last effort to break through his captors on horseback,
+but was taken and conveyed a prisoner to the Tower. His trial
+took place in Westminster Hall, on the 27th of January 1606, and
+alone among the conspirators he pleaded guilty, declaring that
+the motives of his crime had been his friendship for Catesby
+and his devotion to his religion. He was condemned to death,
+and his execution, which took place on the 31st, in St Paul&rsquo;s
+Churchyard, was accompanied by all the brutalities exacted by
+the law.</p>
+
+<p>Digby was a handsome man, of fine presence. Father Gerard
+<span class="pagenum"><a name="page261" id="page261"></a>261</span>
+extols his skill in sport, his &ldquo;riding of great horses,&rdquo; as well as his
+skill in music, his gifts of mind and his religious devotion, and
+concludes &ldquo;he was as complete a man in all things, that deserved
+estimation or might win affection as one should see in a kingdom.&rdquo;
+Some of Digby&rsquo;s letters and papers, which include a poem
+before his execution, a last letter to his infant sons and correspondence
+with his wife from the Tower, were published in <i>The
+Gunpowder Treason</i> by Thomas Barlow, bishop of Lincoln, in
+1679. He left two sons, of whom the elder, Sir Kenelm Digby,
+was the well-known author and diplomatist.</p>
+
+<div class="condensed">
+<p>See works on the Gunpowder Plot; Narrative of Father Gerard,
+in <i>Condition of the Catholics under James I.</i> by J. Morris (1872),
+&amp;c. A life of Digby under the title of <i>A Life of a Conspirator</i>,
+by a Romish Recusant (Thomas Longueville), was published in
+1895.</p>
+</div>
+<div class="author">(P. C. Y.)</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIGBY, SIR KENELM<a name="ar87" id="ar87"></a></span> (1603-1665), English author, diplomatist
+and naval commander, son of Sir Everard Digby (<i>q.v.</i>),
+was born on the 11th of July 1603, and after his father&rsquo;s execution
+in 1606 resided with his mother at Gayhurst, being brought up
+apparently as a Roman Catholic. In 1617 he accompanied his
+cousin, Sir John Digby, afterwards 1st earl of Bristol, and then
+ambassador in Spain, to Madrid. On his return in April 1618 he
+entered Gloucester Hall (now Worcester College), Oxford, and
+studied under Thomas Allen (1542-1632), the celebrated mathematician,
+who was much impressed with his abilities and called
+him the <i>Mirandula</i>, <i>i.e.</i> the infant prodigy, of his age.<a name="fa1h" id="fa1h" href="#ft1h"><span class="sp">1</span></a> He left
+the university without taking a degree in 1620, and travelled
+in France, where, according to his own account, he inspired an
+uncontrollable passion in the queen-mother, Marie de&rsquo; Medici,
+now a lady of more than mature age and charms; he visited
+Florence, and in March 1623 joined Sir John Digby again at
+Madrid, at the time when Prince Charles and Buckingham arrived
+on their adventurous expedition. He joined the prince&rsquo;s household
+and returned with him to England on the 5th of October
+1623, being knighted by James I. on the 23rd of October and
+receiving the appointment of gentleman of the privy chamber to
+Prince Charles. In 1625 he married secretly Venetia, daughter of
+Sir Edward Hanley of Tonge Castle, Shropshire, a lady of extraordinary
+beauty and intellectual attainments, but of doubtful
+virtue. Digby was a man of great stature and bodily strength.
+Edward Hyde, afterwards earl of Clarendon, who with Ben
+Jonson was included among his most intimate friends, describes
+him as &ldquo;a man of very extraordinary person and presence which
+drew the eyes of all men upon him, a wonderful graceful
+behaviour, a flowing courtesy and civility, and such a volubility
+of language as surprised and delighted.&rdquo;<a name="fa2h" id="fa2h" href="#ft2h"><span class="sp">2</span></a> Digby for some time
+was excluded from public employment by Buckingham&rsquo;s jealousy
+of his cousin, Lord Bristol. At length in 1627, on the latter&rsquo;s
+advice, Digby determined to attempt &ldquo;some generous action,&rdquo;
+and on the 22nd of December, with the approval of the king,
+embarked as a privateer with two ships, with the object of attacking
+the French ships in the Venetian harbour of Scanderoon. On
+the 18th of January he arrived off Gibraltar and captured several
+Spanish and Flemish vessels. From the 15th of February to the
+27th of March he remained at anchor off Algiers on account of the
+sickness of his men, and extracted a promise from the authorities
+of better treatment of the English ships. He seized a rich Dutch
+vessel near Majorca, and after other adventures gained a complete
+victory over the French and Venetian ships in the harbour of
+Scanderoon on the 11th of June. His successes, however, brought
+upon the English merchants the risk of reprisals, and he was urged
+to depart. He returned home in triumph in February 1629, and
+was well received by the king, and was made a commissioner of
+the navy in October 1630, but his proceedings were disavowed on
+account of the complaints of the Venetian ambassador. In 1633
+Lady Digby died, and her memory was celebrated by Ben Jonson
+in a series of poems entitled <i>Eupheme</i>, and by other poets of
+the day. Digby retired to Gresham College, and exhibited extravagant
+grief, maintaining a seclusion for two years. About
+this time Digby professed himself a Protestant, but by October
+1635, while in France, he had already returned to the Roman
+Catholic faith.<a name="fa3h" id="fa3h" href="#ft3h"><span class="sp">3</span></a> In a letter dated the 27th of March 1636 Laud
+remonstrates with him, but assures him of the continuance of his
+friendship.<a name="fa4h" id="fa4h" href="#ft4h"><span class="sp">4</span></a> In 1638 he published <i>A Conference with a Lady about
+choice of a Religion</i>, in which he argues that the Roman Church,
+possessing alone the qualifications of universality, unity of
+doctrine and uninterrupted apostolic succession, is the only true
+church, and that the intrusion of error into it is impossible. The
+same subject is treated in letters to George Digby, afterwards
+2nd earl of Bristol, dated the 2nd of November 1638 and the 29th
+of November 1639, which were published in 1651, as well as in
+a further <i>Discourse concerning Infallibility in Religion</i> in 1652.
+Returning to England he associated himself with the queen and
+her Roman Catholic friends, and joined in the appeal to the
+English Romanists for money to support the king&rsquo;s Scottish
+expedition.<a name="fa5h" id="fa5h" href="#ft5h"><span class="sp">5</span></a> In consequence he was summoned to the bar of
+the House of Commons on the 27th of January 1641, and the
+king was petitioned to remove him with other recusants from his
+councils. He left England, and while at Paris killed in a duel a
+French lord who had insulted Charles I. in his presence. Louis
+XIII. took his part, and furnished him with a military escort into
+Flanders. Returning home he was imprisoned, by order of the
+House of Commons, early in 1642, successively in the &ldquo;Three
+Tobacco Pipes nigh Charing Cross,&rdquo; where his delightful conversation
+is said to have transformed the prison into &ldquo;a place of
+delight,&rdquo;<a name="fa6h" id="fa6h" href="#ft6h"><span class="sp">6</span></a> and at Winchester House. He was finally released and
+allowed to go to France on the 30th of July 1643, through the
+intervention of the queen of France, Anne of Austria, on condition
+that he would neither promote nor conceal any plots abroad
+against the English government.</p>
+
+<p>Before leaving England an attempt was made to draw from
+him an admission that Laud, with whom he had been intimate,
+had desired to be made a cardinal, but Digby denied that the
+archbishop had any leanings towards Rome. On the 1st of
+November 1643 it was resolved by the Commons to confiscate his
+property. He published in London the same year <i>Observations
+on the 22nd stanza in the 9th canto of the 2nd book of Spenser&rsquo;s
+&ldquo;Faërie Queene,&rdquo;</i> the MS. of which is in the Egerton collection
+(British Museum, No. 2725 f. 117 b), and <i>Observations</i> on a
+surreptitious and unauthorized edition of the <i>Religio Medici</i>, by
+Sir Thomas Browne, from the Roman Catholic point of view,
+which drew a severe rebuke from the author. After his arrival
+in Paris he published his chief philosophical works, <i>Of Bodies</i>
+and <i>Of the Immortality of Man&rsquo;s Soul</i> (1644), autograph MSS. of
+which are in the Bibliothèque Ste Geneviève at Paris, and made
+the acquaintance of Descartes. He was appointed by Queen
+Henrietta Maria her chancellor, and in the summer of 1645 he was
+despatched by her to Rome to obtain assistance. Digby promised
+the conversion of Charles and of his chief supporters. At first his
+eloquence made a great impression. Pope Innocent X. declared
+that he spoke not merely as a Catholic but as an ecclesiastic.
+But the absence of any warrant from Charles himself roused
+suspicions as to the solidity of his assurances, and he obtained
+nothing but a grant of 20,000 crowns. A violent quarrel with the
+pope followed, and he returned in 1646, having consented in the
+queen&rsquo;s name to complete religious freedom for the Roman
+Catholics, both in England and Ireland, to an independent parliament
+in Ireland, and to the surrender of Dublin and all the Irish
+fortresses into the hands of the Roman Catholics, the king&rsquo;s
+troops to be employed in enforcing the articles and the pope
+granting about £36,000 with a promise of further payments in
+obtaining direct assistance. In February 1649 Digby was invited
+to come to England to arrange a proposed toleration of the Roman
+Catholics, but on his arrival in May the scheme had already been
+abandoned. He was again banished on the 31st of August, and
+it was not till 1654 that he was allowed by the council of state to
+return. He now entered into close relations with Cromwell, from
+whom he hoped to obtain toleration for the Roman Catholics, and
+whose alliance he desired to secure for France rather than for
+<span class="pagenum"><a name="page262" id="page262"></a>262</span>
+Spain, and was engaged by Cromwell, much to the scandal of both
+Royalists and Roundheads, in negotiations abroad, of which the
+aim was probably to prevent a union between those two foreign
+powers. He visited Germany, in 1660 was in Paris, and at the
+Restoration returned to England. He was well received in spite
+of his former relations with Cromwell, and was confirmed in his
+post as Queen Henrietta Maria&rsquo;s chancellor. In January 1661
+he delivered a lecture, which was published the same month, at
+Gresham College, on the vegetation of plants, and became an
+original member of the Royal Society in 1663. In January 1664
+he was forbidden to appear at court, the cause assigned being that
+he had interposed too far in favour of the 2nd earl of Bristol,
+disgraced by the king on account of the charge of high treason
+brought by him against Clarendon into the House of Lords. The
+rest of his life was spent in the enjoyment of literary and scientific
+society at his house in Covent Garden. He died on the 11th of
+June 1665. He had five children, of whom two, a son and one
+daughter, survived him.</p>
+
+<p>Digby, though he possessed for the time a considerable knowledge
+of natural science, and is said to have been the first to
+explain the necessity of oxygen to the existence of plants, bears
+no high place in the history of science. He was a firm believer in
+astrology and alchemy, and the extraordinary fables which he
+circulated on the subject of his discoveries are evidence of anything
+rather than of the scientific spirit. In 1656 he made public
+a marvellous account of a city in Tripoli, petrified in a few hours,
+which he printed in the <i>Mercurius Politicus</i>. Malicious reports
+had been current that his wife had been poisoned by one of his
+prescriptions, viper wine, taken to preserve her beauty. Evelyn,
+who visited him in Paris in 1651, describes him as an &ldquo;errant
+mountebank.&rdquo; Henry Stubbes characterizes him as &ldquo;the very
+Pliny of our age for lying,&rdquo; and Lady Fanshawe refers to the same
+&ldquo;infirmity.&rdquo;<a name="fa7h" id="fa7h" href="#ft7h"><span class="sp">7</span></a> His famous &ldquo;powder of sympathy,&rdquo; which seems
+to have been only powder of &ldquo;vitriol,&rdquo; healed without any
+contact, by being merely applied to a rag or bandage taken from
+the wound, and Digby records a miraculous cure by this means in
+a lecture given by him at Montpellier on this subject in 1658,
+published in French and English the same year, in German in
+1660 and in Dutch in 1663; but Digby&rsquo;s claim to its original
+discovery is doubtful, Nathaniel Highmore in his <i>History of
+Generation</i> (1651, p. 113) calling the powder &ldquo;Talbot&rsquo;s powder,&rdquo;
+and ascribing its invention to Sir Gilbert Talbot. Some of Digby&rsquo;s
+pills and preparations, however, described in <i>The Closet of the
+Eminently Learned Sir Kenelm Digby Knt. Opened</i> (publ. 1677),
+are said to make less demand upon the faith of patients, and his
+injunction on the subject of the making of tea, to let the water
+&ldquo;remain upon it no longer than you can say the Miserere Psalm
+very leisurely,&rdquo; is one by no means to be ridiculed. As a philosopher
+and an Aristotelian Digby shows little originality and
+followed the methods of the schoolmen. His Roman Catholic
+orthodoxy mixed with rationalism, and his political opinions,
+according to which any existing authority should receive support,
+were evidently derived from Thomas White (1582-1676), the
+Roman Catholic philosopher, who lived with him in France.
+White published in 1651 <i>Institutionum Peripateticorum libri
+quinque</i>, purporting to expound Digby&rsquo;s &ldquo;peripatetic philosophy,&rdquo;
+but going far beyond Digby&rsquo;s published treatises.
+Digby&rsquo;s <i>Memoirs</i> are composed in the high-flown fantastic manner
+then usual when recounting incidents of love and adventure,
+but the style of his more sober works is excellent. In 1632 he
+presented to the Bodleian library a collection of 236 MSS., bequeathed
+to him by his former tutor Thomas Allen, and described
+in <i>Catalogi codicum manuscriptorum bibliothecae Bodleianae</i>, by
+W. D. Macray, part ix. Besides the works already mentioned
+Digby translated <i>A Treatise of adhering to God written by Albert
+the Great, Bishop of Ratisbon</i> (1653); and he was the author of
+<i>Private Memoirs</i>, published by Sir N. H. Nicholas from <i>Harleian
+MS. 6758</i> with introduction (1827); <i>Journal of the Scanderoon
+Voyage in 1628</i>, printed by J. Bruce with preface (Camden
+Society, 1868); <i>Poems from Sir Kenelm Digby&rsquo;s Papers</i>... with
+preface and notes (Roxburghe Club, 1877); in the <i>Add. MSS.</i>
+34,362 f. 66 is a poem <i>Of the Miserys of Man</i>, probably by Digby;
+<i>Choice of Experimental Receipts in Physick and Chirurgery</i> ...
+<i>collected by Sir K. Digby</i> (1668), and <i>Chymical Secrets and Rare
+Experiments</i> (1683), were published by G. Hartman, who describes
+himself as Digby&rsquo;s steward and laboratory assistant.</p>
+
+<div class="condensed">
+<p>See the <i>Life of Sir Kenelm Digby by one of his Descendants</i>
+(T. Longueville), 1896.</p>
+</div>
+<div class="author">(P. C. Y.)</div>
+
+<hr class="foot" /> <div class="note">
+
+<p><a name="ft1h" id="ft1h" href="#fa1h"><span class="fn">1</span></a> <i>Letters by Eminent Persons</i> (Aubrey&rsquo;s Lives), ii. 324.</p>
+
+<p><a name="ft2h" id="ft2h" href="#fa2h"><span class="fn">2</span></a> <i>Life and Continuation.</i></p>
+
+<p><a name="ft3h" id="ft3h" href="#fa3h"><span class="fn">3</span></a> Strafford&rsquo;s <i>Letters</i>, i. 474.</p>
+
+<p><a name="ft4h" id="ft4h" href="#fa4h"><span class="fn">4</span></a> Laud&rsquo;s <i>Works</i>, vi. 447.</p>
+
+<p><a name="ft5h" id="ft5h" href="#fa5h"><span class="fn">5</span></a> <i>Thomason Tracts</i>, Brit. Mus. E 164 (15).</p>
+
+<p><a name="ft6h" id="ft6h" href="#fa6h"><span class="fn">6</span></a> <i>Archaeologia Cantiana</i>, ii. 190.</p>
+
+<p><a name="ft7h" id="ft7h" href="#fa7h"><span class="fn">7</span></a> <i>Dict. of Nat. Biog.</i> sub &ldquo;Digby.&rdquo; See also Robert Boyle&rsquo;s
+<i>Works</i> (1744), v. 302.</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIGBY, KENELM HENRY<a name="ar88" id="ar88"></a></span> (1800-1880), English writer,
+youngest son of William Digby, dean of Clonfert, was born at
+Clonfert, Ireland, in 1800. He was educated at Trinity College,
+Cambridge, and soon after taking his B.A. degree there in 1819
+became a Roman Catholic. He spent most of his life, which was
+mainly devoted to literary pursuits, in London, where he died on
+the 22nd of March 1880. Digby&rsquo;s reputation rests chiefly on his
+earliest publication, <i>The Broadstone of Honour, or Rules for the
+Gentlemen of England</i> (1822), which contains an exhaustive survey
+of medieval customs, full of quotations from varied sources. The
+work was subsequently enlarged and issued (1826-1827) in four
+volumes entitled: <i>Godefridus</i>, <i>Tancredus</i>, <i>Morus</i> and <i>Orlandus</i>
+(numerous re-impressions, the best of which is the edition
+brought out by B. Quaritch in five volumes, 1876-1877).</p>
+
+<div class="condensed">
+<p>Among Digby&rsquo;s other works are: <i>Mores Catholici, or Ages of
+Faith</i> (11 vols., London, 1831-1840); <i>Compitum; or the Meeting of
+the Ways at the Catholic Church</i> (7 vols., London, 1848-1854); <i>The
+Lovers&rsquo; Seat, Kathemérina; or Common Things in relation to Beauty,
+Virtue and Faith</i> (2 vols., London, 1856). A complete list is given
+in J. Gillow&rsquo;s <i>Bibliographical Dictionary of English Catholics</i>, ii.
+81-83.</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIGENES ACRITAS, BASILIUS,<a name="ar89" id="ar89"></a></span> Byzantine national hero,
+probably lived in the 10th century. He is named Digenes (of
+double birth) as the son of a Moslem father and a Christian
+mother; Acritas (<span class="grk" title="akra">&#7940;&#954;&#961;&#945;</span>, frontier, boundary), as one of the frontier
+guards of the empire, corresponding to the Roman <i>milites
+limitanei</i>. The chief duty of these <i>acritae</i> consisted in repelling
+Moslem inroads and the raids of the <i>apelatae</i> (cattle-lifters),
+brigands who may be compared with the more modern Klephts.
+The original Digenes epic is lost, but four poems are extant, in
+which the different incidents of the legend have been worked
+up by different hands. The first of these consists of about 4000
+lines, written in the so-called &ldquo;political&rdquo; metre, and was discovered
+in the latter part of the 19th century, in a 16th-century
+MS., at Trebizond; the other three MSS. were found at Grotta
+Ferrata, Andros and Oxford. The poem, which has been compared
+with the <i>Chanson de Roland</i> and the <i>Romance of the Cid</i>,
+undoubtedly contains a kernel of fact, although it cannot be
+regarded as in any sense an historical record. The scene of action
+is laid in Cappadocia and the district of the Euphrates.</p>
+
+<div class="condensed">
+<p>Editions of the Trebizond MS. by C. Sathas and E. Legrand in
+the <i>Collection des monuments pour servir à l&rsquo;étude de la langue néohellénique</i>,
+new series, vi. (1875), and by S. Joannides (Constantinople,
+1887). See monographs by A. Luber (Salzburg, 1885) and G.
+Wartenberg (Berlin, 1897). Full information will be found in
+C. Krumbacher&rsquo;s <i>Geschichte der byzantinischen Litteratur</i>, p. 827
+(2nd ed., 1897); see also G. Schlumberger, <i>L&rsquo;Épopée Byzantine à
+la fin du dixième siècle</i> (1897).</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIGEST,<a name="ar90" id="ar90"></a></span> a term used generally of any digested or carefully
+arranged collection or compendium of written matter, but more
+particularly in law of a compilation in condensed form of a body
+of law digested in a systematical method; <i>e.g.</i> the Digest (<i>Digesta</i>)
+or Pandects (<span class="grk" title="Pandektai">&#928;&#940;&#957;&#948;&#949;&#954;&#964;&#945;&#953;</span>) of Justinian, a collection of extracts
+from the earlier jurists compiled by order of the emperor
+Justinian. The word is also given to the compilations of the
+main points (marginal or hand-notes) of decided cases, usually
+arranged in alphabetical and subject order, and published under
+such titles as &ldquo;Common Law Digest,&rdquo; &ldquo;Annual Digest,&rdquo; &amp;c.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIGESTIVE ORGANS<a name="ar91" id="ar91"></a></span> (<span class="sc">Pathology</span>). Several facts of importance
+have to be borne in mind for a proper appreciation of
+the pathology of the organs concerned in digestive processes (for
+the anatomy see <span class="sc"><a href="#artlinks">Alimentary Canal</a></span> and allied articles). In
+the first place, more than all other systems, the digestive comprises
+greater range of structure and exhibits wider diversity of function
+within its domain. Each separate structure and each different
+function presents special pathological signs and symptoms.
+Again, the duties imposed upon the system have to be performed
+<span class="pagenum"><a name="page263" id="page263"></a>263</span>
+notwithstanding constant variations in the work set them. The
+crude articles of diet offered them vary immensely in nature, bulk
+and utility, from which they must elaborate simple food-elements
+for absorption, incorporate them after absorption into complex
+organic substances properly designed to supply the constant needs
+of cellular activity, of growth and repair, and fitly harmonized
+to fulfil the many requirements of very divergent processes and
+functions. Any form of unphysiological diet, each failure to
+cater for the wants of any special tissue engaged in, or of any
+processes of, metabolism, carry with them pathological signs.
+Perhaps in greater degree than elsewhere are the individual
+sections of the digestive system dependent upon, and closely
+correlated with, one another. The lungs can only yield oxygen
+to the blood when the oxygen is uncombined; no compounds
+are of use. The digestive organs have to deal with an enormous
+variety of compound bodies, from which to obtain the elements
+necessary for protoplasmic upkeep and activity. Morbid lesions
+of the respiratory and circulatory systems are frequently capable
+of compensation through increased activity elsewhere, and the
+symptoms they give rise to follow chiefly along one line; diseases
+of the digestive organs are more liable to occasion disorders
+elsewhere than to excite compensatory actions. The digestive
+system includes every organ, function and process concerned
+with the utilization of food-stuffs, from the moment of their
+entrance into the mouth, their preparation in the canal, assimilation
+with the tissues, their employment therein, up to their
+excretion or expulsion in the form of waste. Each portion
+resembles a link of a continuous chain; each link depends upon
+the integrity of the others, the weakening or breaking of one
+straining or making impotent the chain as a whole.</p>
+
+<p>The mucous membrane lining the alimentary tract is the part
+most subject to pathological alterations, and in this connexion
+it should be remembered that this membrane differs both in
+structure and functions throughout the tract. Chiefly protective
+from the mouth to the cardia, it is secretory and absorbent in the
+stomach and bowel; while the glandular cells forming part of it
+secrete both acid and alkaline fluids, several ferments or mucus.
+Over the dorsum of the tongue its modified cells subserve the
+sense of taste. Without, connected with it by the submucous
+connective tissue, is placed the muscular coat, and externally over
+the greater portion of its length the peritoneal serous membrane.
+All parts are supplied with blood-vessels, lymph-ducts and
+nerves, the last belonging either to local or to central circuits.
+Associated with the tract are the salivary glands, the liver and
+the pancreas; while, in addition, lymphoid tissue is met with
+diffusely scattered throughout the lining membranes in the tonsils,
+appendix, solitary glands and Peyer&rsquo;s patches, and the mesenteric
+glands. The functions of the various parts of the system in whose
+lesions we are here interested are many in number, and can only
+be summarized here. (For the physiology of digestion see
+<span class="sc"><a href="#artlinks">Nutrition</a></span>.) Broadly, they maybe given as: (1) Ingestion and
+swallowing of food, transmission of it through the tract, and
+expulsion of the waste material; (2) secretion of acids and
+alkalis for the performance of digestive processes, aided by (3)
+elaboration and addition of complex bodies, termed enzymes
+or ferments; (4) secretion of mucus; (5) protection of the body
+against organismal infection, and against toxic products; (6)
+absorption of food elements and reconstitution of them into
+complex substances fitted for metabolic application; and (7)
+excretion of the waste products of protoplasmic action. These
+functions may be altered by disease, singly or in conjunction; it
+is rare, however, to find but one affected, while an apparently
+identical disturbance of function may often arise from totally
+different organic lesions. Another point of importance is seen in
+the close interdependence which exists between the secretions of
+acid and those of alkaline reaction. The difference in reaction
+seems to act <i>mutatis mutandis</i> as a stimulant in each instance.</p>
+
+<p class="center pt1"><i>General Diseases.</i></p>
+
+<p>In all sections of the alimentary canal actively engaged in the
+digestion of food, a well-marked local engorgement of the blood-vessels
+supplying the walls occurs. The hyperaemia abates soon
+after completion of the special duties of the individual sections.
+<span class="sidenote">Vascular lesions.</span>
+This normal condition may be abnormally exaggerated by overstimulation
+from irritant poisons introduced into the
+canal; from too rich, too copious or indigestible
+articles of diet; or from too prolonged an experience
+of some unvaried kind of food-stuff, especially if large quantities
+of it are necessary for metabolic needs; entering into the first
+stage of inflammation, acute hyperaemia. More important,
+because productive of less tractable lesions, is passive congestion
+of the digestive organs. Whenever the flow of blood into the
+right side of the heart is hindered, whether it arise from disease
+of the heart itself, or of the lungs, or proceed from obstruction in
+some part of the portal system, the damming-back of the venous
+circulation speedily produces a more or less pronounced stasis of
+the blood in the walls of the alimentary canal and in the associated
+abdominal glands. The lack of a sufficiently vigorous flow of
+blood is followed by deficient secretion of digestive agents from
+the glandular elements involved, by decreased motility of the
+muscular coats of the stomach and bowel, and lessened adaptability
+throughout for dealing with even slight irregular demands
+on their powers. The mucous membrane of the stomach and
+bowel, less able to withstand the effects of irritation, even of a
+minor character, readily passes into a condition of chronic
+catarrh, while it frequently is the seat of small abrasions,
+haemorrhagic erosions, which may cause vomiting of blood and
+the appearance of blood in the stools. Obstruction to the flow
+of blood from the liver leads to dilatation of its blood-vessels,
+consequent pressure upon the hepatic cells adjoining them, and
+their gradual loss of function, or even atrophy and degeneration.
+In addition to the results of such passive congestion exhibited
+by the stomach and bowel as noted above, passive congestion
+of the liver is often accompanied by varicose enlargement of the
+abdominal veins, in particular of those which surround the lower
+end of the oesophagus, the lowest part of the rectum and anus.
+In the latter position these dilated veins constitute what are
+known as haemorrhoids or piles, internal or external as their
+site lies within or outside the anal aperture.</p>
+
+<p>The mucous and serous membranes of the canal and the
+glandular elements of the associated organs are the parts most
+subject to inflammatory affections. Among the several sections
+of the digestive tract itself, the oesophagus and jejunum are
+singularly exempt from inflammatory processes; the fauces,
+stomach, caecum and appendix, ileum, mouth and duodenum
+(including the opening of the common bile-duct), are more
+commonly involved. <i>Stomatitis</i>, or inflammation of the mouth,
+<span class="sidenote">Inflammatory lesions.</span>
+has many predisposing factors, but it has now been
+definitely determined that its exciting cause is always
+some form of micro-organism. Any condition favouring
+oral sepsis, as carious teeth, pyorrhoea alveolaris (a discharge
+of pus due to inflamed granulations round carious teeth),
+granulations beneath thick crusts of tartar, or an irritating tooth
+plate, favours the growth of pyogenic organisms and hence of
+stomatitis. Many varieties of this disease have been described,
+but all are forms of &ldquo;pyogenic&rdquo; or &ldquo;septic stomatitis.&rdquo; This in
+its mildest form is catarrhal or erythematous, and is attended
+only by slight swelling tenderness and salivation. In its next
+stage of acuteness it is known as &ldquo;membranous,&rdquo; as a false
+membrane is produced somewhat resembling that due to
+diphtheria, though caused by a staphylococcus only. A still
+more acute form is &ldquo;ulcerative,&rdquo; which may go on to the formation
+of an abscess beneath the tongue. Scarlet fever usually
+gives rise to a slight inflammation of the mouth followed by
+desquamation, but more rarely it is accompanied by a most
+severe oedematous stomatitis with glossitis and tonsillitis.
+Erysipelas on the face may infect the mouth, and an acute
+stomatitis due to the diphtheria bacillus, Klebs-Loeffler bacillus,
+has been described. A distinct and very dangerous form of
+stomatitis in infants and young children is known as &ldquo;aphthous
+stomatitis&rdquo; or &ldquo;thrush.&rdquo; This is caused by the growth of
+<i>Oidium albicans</i>. It is always preceded by a gastro-enteritis and
+dry mouth, and if this is not attended to, soon attracts attention
+by the little white raised patches surrounded by a dusky red zone
+<span class="pagenum"><a name="page264" id="page264"></a>264</span>
+scattered on tongue and cheeks. Epidemics have occurred in
+hospitals and orphanages. Mouth breathing is the cause of many
+ills. As a result of this, the mucous membrane of the tongue, &amp;c.,
+becomes dry, micro-organisms multiply and the mouth becomes
+foul. Also from disease of the nose, the upper jaw, palate and
+teeth do not make proper progress in development. There is
+overgrowth of tonsils, and adenoids, with resulting deafness, and
+the child&rsquo;s mental development suffers. An ordinary &ldquo;sore
+throat&rdquo; usually signifies acute catarrh of the fauces, and is of
+purely organismal origin, &ldquo;catching cold&rdquo; being only a secondary
+and minor cause. In &ldquo;relaxed throats&rdquo; there is a chronic
+catarrhal state of the lining membrane, with some passive congestion.
+The tonsils are peculiarly liable to catarrhal attacks,
+as might a priori be expected by reason of their Cerberus-like
+function with regard to bacterial intruders. Still, acute attacks
+of tonsillitis appear on good evidence to be more common among
+individuals predisposed constitutionally to rheumatic manifestations.
+Cases of acute tonsillitis may or may not go on to suppuration
+or quinsy; in all there is great congestion of the glands,
+increased mucus secretion, and often secondary involvement of
+the lymphatic glands of the neck. Repeated acute attacks often
+lead to chronic inflammation, in which the glands are enlarged,
+and often hypertrophied in the true sense of the term. The
+oesophagus is the seat of inflammation but seldom. In infants
+and young children thrush due to <i>Oidium albicans</i> may spread
+from the mouth, and also a diphtheritic inflammation spreads
+from the fauces into the oesophagus. A catarrhal oesophagitis
+is rarely seen, but the commonest form is traumatic, due to the
+swallowing of boiling water, corrosive or irritant substances, &amp;c.
+A non-malignant ulceration may result which later leads on to
+an oesophageal stricture. The physical changes presented by the
+coats of the stomach and the intestine, the subjects of catarrhal
+attacks, closely resemble one another, but differ symptomatically.
+Acute catarrh of the stomach is associated with intense
+hyperaemia of its lining coats, with visible engorgement and
+swelling of the mucous membrane, and an excessive secretion of
+mucus. The formation of active gastric juice is arrested, digestion
+ceases, peristaltic movements are sluggish or absent, unless so
+over-stimulated that they act in a direction the reverse of the
+normal, and induce expulsion of the gastric contents by vomiting.
+The gastric contents, in whatever degree of dilution or concentration
+they may have been ingested, when ejected are of porridge-thick
+consistency, and often but slightly digested. Such
+conditions may succeed a severe alcoholic bout, be caused by
+irritant substances taken in by the mouth or arise from fermentative
+processes in the stomach contents themselves. Should
+the irritating material succeed in passing from the stomach into
+the bowel, similar physical signs are present; but as the quickest
+path offered for the expulsion of the offending substances from
+the body is downwards, peristalsis is increased, the flow of fluid
+from the intestinal glands is larger in bulk, though of less potency
+as regards its normal actions, than in health, and diarrhoea, with
+removal of the irritant, follows. As a general rule, the more
+marked the involvement of the large bowel, the severer and more
+fluid is the resultant diarrhoea. Inflammation of the stomach
+may be due to mechanical injury, thermal or chemical irritants
+or invasion by micro-organisms. Also all the symptoms of
+gastric catarrh may be brought on by any acute emotion. The
+commonest mechanical injury is that due to an excess of food,
+especially when following on a fast; poisons act as irritants, and
+also the weevils of cheese and the larvae of insects.</p>
+
+<p>Inflammatory affections of the caecum and its attached
+appendix vermiformis are very common, and give rise to several
+special symptoms and signs. Acute inflammatory appendicitis
+appears to be increasing in frequency, and is associated by many
+with the modern deterioration in the teeth. Constipation
+certainly predisposes to it, and it appears to be more prevalent
+among medical men, commercial travellers, or any engaged in
+arduous callings, subjected to irregular meals, fatigue and
+exposure. A foreign body is the exciting cause in many cases,
+though less commonly so than was formerly imagined. The
+inflammation in the appendix varies in intensity from a very
+slight catarrhal or simple form to an ulcerative variety, and much
+more rarely to the acute fulminating appendicitis in which
+necrosis of the appendix with abscess formation occurs. It is
+always accompanied by more or less peritonitis, which is protective
+in nature, shutting in the inflammatory process. Very
+similar symptomatically is the condition termed perityphlitis,
+doubtless in former days frequently due to the appendix, an acute
+or chronic inflammation of the walls of the caecum often leading
+to abscess formation outside the gut, with or without direct
+communication with the canal. The colon is subject to three
+main forms of inflammation. In simple <i>colitis</i> the mucous
+membrane of the colon is intensely injected, bright red in colour,
+and secreting a thick mucus, but there is no accompanying
+ulceration. It is often found in association with some constitutional
+disease, as Bright&rsquo;s disease, and also with cancer of the
+bowel. But when it has no association with other trouble it is
+probably bacterial in origin, the <i>Bacillus enteritidis spirogenes</i>
+having been isolated in many cases. The motions always contain
+large quantities of mucus and more or less blood. A second very
+severe form of inflammation of the colon is known as &ldquo;membranous
+colitis,&rdquo; and this may be either dyspeptic, or secondary to
+other diseases. In this trouble membranes are passed <i>per anum</i>,
+accompanied by a pain so intense as often to cause fainting. In
+severe cases complete tubular casts of the intestine have been
+found. Often the motions contain very little faecal matter, but
+consist only of membranes, mucus and a little blood. A third
+form is that known as &ldquo;ulcerative colitis.&rdquo; Any part of the large
+intestine may be affected, and the ulceration shows no special
+distribution. In severe cases the muscular coat is exposed, and
+perforation may ensue. The number of ulcers varies from a few
+to many dozen, and in size from a pea to a five-shilling piece.
+Like all chronic intestinal ulcers they show a tendency to become
+transverse.</p>
+
+<p>Chronic catarrhal affections of the stomach are very common,
+and often follow upon repeated acute attacks. In them the
+connective tissue increases at the expense of the glandular
+elements; the mucous membrane becomes thickened and less
+active in function. Should the muscular coat be involved, the
+elasticity and contractility of the organ suffer; peristaltic movement
+is weakened; expulsion of the contents through the pylorus
+hindered; and, aggravated by these effects, the condition
+becomes worse, atonic dyspepsia in its most pronounced form
+results, with or without dilatation. Chronic vascular congestion
+may occasion in process of time similar signs and symptoms.</p>
+
+<p>Duodenal catarrh is constantly associated with jaundice, indeed
+is most probably the commonest cause of catarrhal jaundice; often
+it is accompanied by catarrh of the common bile-duct. Chronic
+inflammation of the small intestine gives rise to less prominent
+symptoms than in the stomach. It generally arises from more than
+one cause; or rather secondary causes rapidly become as important
+as the primary in its incidence. Chronic congestion and prolonged
+irritation lead to deficient secretion and sluggish peristalsis;
+these effects encourage intestinal putrefaction and auto-intoxication;
+and these latter, in turn, increase the local unrest.</p>
+
+<p>The intestinal mucous membrane, the peritoneum and the
+mesenteric glands are the chief sites of tubercular infection in
+the digestive organs. Rarely met with in the gullet and
+stomach, and comparatively seldom in the mouth and
+<span class="sidenote">Infective lesions.</span>
+lips, tubercular inflammation of the small intestine
+and peritoneum is common. Tubercular enteritis is a frequent
+accompaniment of phthisis, but may occur apart from tubercle
+of other organs. Children are especially subject to the primary
+form. Tubercular peritonitis often is present also. The inflammatory
+process readily tends towards ulcer formation, with
+haemorrhage and sometimes perforation. If in the large bowel,
+the symptoms are usually less acute than those characterizing
+tubercular inflammation of the small intestine. The appendix
+has been found to be the seat of tubercular processes; in the
+rectum they form the general cause of the fistulae and abscesses so
+commonly met with here. Tubercular peritonitis may be primary
+or secondary, acute or chronic; occasionally very acute cases are
+seen running a rapid course; the majority are chronic in type.
+<span class="pagenum"><a name="page265" id="page265"></a>265</span>
+The tubercles spread over the surface of the serous membrane,
+and if small and not very numerous may give rise in chronic
+cases to few symptoms; if larger, and especially when they
+involve and obstruct the lymph- and blood-vessels, ascites follows.
+It is hardly possible that tubercular invasion of the mesenteric
+glands can ever occur unaccompanied by peritoneal infection;
+but when the infection of the glands constitutes the most prominent
+sign, the term <i>tabes mesenterica</i> is sometimes employed.
+Here the glands, enlarged, form a doughy mass in the abdomen,
+leading to marked protrusion of the abdominal walls, with
+wasting elsewhere and diarrhoea.</p>
+
+<p>The liver is seldom attacked by tubercle, unless in cases of
+general miliary tuberculosis. Now and then it contains large
+caseous tubercular masses in its substance.</p>
+
+<p>An important fact with regard to the tubercular processes in
+the digestive organs lies in the ready response to treatment shown
+by many cases of peritoneal or mesenteric invasion, particularly
+in the young.</p>
+
+<p>The later sequelae of syphilis display a predilection for the
+rectum and the liver, usually leading to the development of a
+stricture in the former, to a diffuse hepatitis or the formation
+of gummata in the second. In inherited syphilis the temporary
+teeth usually appear early, are discoloured and soon crumble
+away. The permanent teeth may be sound and healthy, but are
+often&mdash;especially the upper incisors&mdash;notched and stunted, when
+they are known as &ldquo;Hutchinson&rsquo;s teeth.&rdquo; As the result both of
+syphilis and of tubercle, the tissues of the liver and bowel may
+present a peculiar alteration; they become amyloid, or lardaceous,
+a condition in which they appear &ldquo;waxy,&rdquo; are coloured
+dark mahogany brown with dilute iodine solutions, and show
+degenerative changes in the connective tissue.</p>
+
+<p>The <i>Bacillus typhosus</i> discovered by Eberth is the causal agent
+of typhoid fever, and has its chief seat of activity in the small
+intestine, more especially in the lower half of the ileum. Attacking
+the lymphoid follicles in the mucous membrane, it causes first
+inflammatory enlargement, then necrosis and ulceration. The
+adjacent portions of the mucous membrane show acute catarrhal
+changes. Diarrhoea, of a special &ldquo;pea-soup&rdquo; type, may or may
+not be present; while haemorrhage from the bowel, if ulcers have
+formed, is common. As the ulcers frequently extend down to the
+peritoneal coat of the bowel, perforation of this membrane and
+extravasation into the peritoneal cavity is easily induced by
+irritants introduced into or elaborated in the bowel, acting
+physically or by the excitation of hyper-peristalsis.</p>
+
+<p>True Asiatic cholera is due to the comma-bacillus or spirillum
+of cholera, which is found in the rice-water evacuations, in the
+contents of the intestine after death, and in the mucous membrane
+of the intestine just beneath the epithelium. It has not been
+found in the blood. It produces an intense irritation of the bowel,
+seldom of the stomach, without giving rise locally to any marked
+physical change; it causes violent diarrhoea and copious discharges
+of &ldquo;rice-water&rdquo; stools, consisting largely of serum
+swarming with the organism.</p>
+
+<p>Dysentery gives rise to an inflammation of the large intestine
+and sometimes of the lower part of the ileum, resulting in extensive
+ulceration and accompanied by faecal discharges of mucus,
+muco-pus or blood. In some forms a protozoan, the <i>Amoeba
+dysenteriae</i>, is found in the stools&mdash;this is the amoebic dysentery;
+in other cases a bacillus, <i>Bacillus dysenteriae</i>, is found&mdash;the
+bacillary dysentery.</p>
+
+<p>Acute parotitis, or mumps, is an infectious disease of the parotid
+glands, chiefly interesting because of the association between it
+and the testes in males, inflammation of these glands occasionally
+following or replacing the affection of the parotids. The causal
+agent is probably organismal, but has as yet escaped detection.</p>
+
+<p>The relative frequency with which malignant growths occur in
+the different organs of the digestive system may be gathered from
+the tabular analysis, on p. 266, of 1768 cases recorded in
+the books of the Edinburgh Royal Infirmary as having
+<span class="sidenote">New growths.</span>
+been treated in the medical and surgical wards between
+the years 1892 and 1899 inclusive. Of these, 1263, or 71.44%,
+were males; 505, or 28.56%, females. (See Table I. p. 266.)</p>
+
+<p>If the figures there given be classified upon broader lines, the
+results are as given in Table II. p. 266, and speak for themselves.</p>
+
+<p>The digestive organs are peculiarly subject to malignant
+disease, a result of the incessant changes from passive to active
+conditions, and vice versa, called for by repeated introduction
+of food; while the comparative frequency with which different
+parts are attacked depends, in part, upon the degree of irritation
+or changes of function imposed upon them. Scirrhous, encephaloid
+and colloid forms of carcinoma occur. In the stomach
+and oesophagus the scirrhous form is most common, the soft
+encephaloid form coming next. The most common situation for
+cancerous growth in the stomach is the pyloric region. Walsh out
+of 1300 cases found 60.8% near the pylorus, 11.4% over the
+lesser curvature, and 4.7% more or less over the whole organ.
+The small intestine is rarely attacked by cancer; the large
+intestine frequently. The rectum, sigmoid flexure, caecum and
+colon are affected, and in this order, the cylindrical-celled form
+being the most common. Carcinoma of the peritoneum is
+generally colloid in character, and is often secondary to growths in
+other organs. Cancer of the liver follows cancer of the stomach
+and rectum in frequency of occurrence, and is relatively more
+common in females than males. Secondary invasion of the liver
+is a frequent sequel to gastric cancer. The pancreas occasionally
+is the seat of cancerous growth.</p>
+
+<p>Sarcomata are not so often met with in the digestive organs.
+When present, they generally involve the peritoneum or the
+mesenteric glands. The liver is sometimes attacked, the stomach
+rarely.</p>
+
+<p>Benign tumours are not of common occurrence in the digestive
+organs. Simple growths of the salivary glands, cysts of the
+pancreas and polypoid tumours of the rectum are the most
+frequent.</p>
+
+<p>The intestinal canal is the habitat of the majority of animal
+parasites found in man. Frequently their presence leads to no
+morbid symptoms, local or general; nor are the symptoms, when
+they do arise, always characteristic of the presence of
+<span class="sidenote">Animal parasites.</span>
+parasites alone.  Discovery of their bodies, or of their
+eggs, in the stools is in most instances the only satisfactory
+proof of their presence. The parasites found in the bowel
+belong principally to two natural groups, Protozoa and Metazoa.
+The great class of the Protozoa furnish amoebae, members of
+Sporozoa and Infusoria. The amoebae are almost invariably
+found in the large intestine; one species, indeed, is termed <i>Amoeba
+coli</i>. The frequently observed relation between attacks of
+dysentery and the presence of amoebae in the stools has led to the
+proposition that an <i>Amoeba dysenterica</i> exists, causing the disease&mdash;a
+theory supported by the detection of amoebae in the contents
+of dysenteric abscesses of the liver. No symptoms of injury to
+health appear to accompany the presence of Sporozoa in the
+bowel, while the species of Infusoria found in it, the <i>Cercomonas</i>,
+and <i>Trichomonas intestinalis</i>, and the <i>Balantidium coli</i>, may or
+may not be guilty of prolonging conditions within the bowel
+as have previously set up diarrhoea.</p>
+
+<p>The Metazoa supply examples of intestinal parasites from the
+classes Annuloida and Nematoidea. To the former class belong
+the various tapeworms found in the small intestine of man.
+They, like other intestinal parasites, are destitute of any power
+of active digestion, simply absorbing the nutritious proceeds of
+the digestive processes of their hosts. Nematode worms infest
+both the small and large intestine; <i>Ascaris lumbricoides</i>, the
+common round worm, and the male <i>Oxyuris vermicularis</i> are
+found in the small bowel, the adult female <i>Oxyuris vermicularis</i>
+and the <i>Tricocephalus dispar</i> in the large.</p>
+
+<p>The eggs of the <i>Trichina spiralis</i>, when introduced with the
+food, develop in the bowel into larval forms which invade the
+tissues of the body, to find in the muscles congenial spots wherein
+to reach maturity. Similarly, the eggs of the Echinococcus
+are hatched in the bowel, and the embryos proceed to take
+up their abode in the tissues of the body, developing into cysts
+capable of growth into mature worms after their ingestion by
+dogs.</p>
+
+<p><span class="pagenum"><a name="page266" id="page266"></a>266</span></p>
+
+<p>Numbers of bacterial forms habitually infest the alimentary
+canal. Many of them are non-pathogenic; some develop pathogenic
+characters only under provocation or when a
+suitable environment induces them to act in such a
+<span class="sidenote">Vegetable parasites.</span>
+manner; others may form the <i>materies morbi</i> of special
+lesions, or be casual visitors capable of originating disease if
+opportunity occurs. Apart from those organisms associated with
+acute infective diseases, disturbances of function and physical
+lesions may be the result of abnormal bacterial activity in the
+canal; and these disturbances may be both local and general.
+Many of the bacteria commonly present produce putrefactive
+changes in the contents of the tract by their metabolic processes.
+They render the medium they grow in alkaline, produce different
+gases and elaborate more or less virulent toxins. Other species
+set up an acid fermentation, seldom accompanied by gas or toxin
+formation. The products of either class are inimical to the free
+growth of members of the other. The species which produce acids
+are more resistant to the action of acids. Thus, when the contents
+of the stomach possess a normal or excessive proportion of free
+hydrochloric acid, a much larger number of putrefactive and
+pathogenic organisms in the food are destroyed or inhibited than
+of the bacteria of acid fermentation. Diminished gastric acidity
+allows of the entry of a greater number of putrefactive (and
+pathogenic) types, with, as a consequence, increased facilities
+for their growth and activity, and the appearance of intestinal
+derangements.</p>
+
+<p class="center pt2"><span class="sc">Table I.</span></p>
+
+<table class="ws" summary="Contents">
+
+<tr><td class="tccm allb" colspan="2">Males.</td> <td class="tccm allb" colspan="2">Females.</td> <td class="tccm allb" colspan="2">Both Sexes.</td></tr>
+<tr><td class="tccm allb">Organ or Tissue in<br />Order of Frequency.</td> <td class="tccm allb">Per-<br />centage</td>
+<td class="tccm allb">Organ or Tissue in<br />Order of Frequency.</td> <td class="tccm allb">Per-<br />centage</td>
+<td class="tccm allb">Organ or Tissue in<br />Order of Frequency.</td> <td class="tccm allb">Per-<br />centage</td></tr>
+
+<tr><td class="tcl lb rb">&ensp;1 Stomach</td> <td class="tcr rb">22.56</td> <td class="tcl rb">&ensp;1 Stomach</td> <td class="tcr rb">22.37</td> <td class="tcl rb">&ensp;1 Stomach</td> <td class="tcr rb">22.49</td></tr>
+<tr><td class="tcl lb rb">&ensp;2 Lip</td> <td class="tcr rb">12.94</td> <td class="tcl rb">&ensp;2 Rectum</td> <td class="tcr rb">17.24</td> <td class="tcl rb">&ensp;2 Rectum</td> <td class="tcr rb">13.12</td></tr>
+<tr><td class="tcl lb rb">&ensp;3 Rectum</td> <td class="tcr rb">11.57</td> <td class="tcl rb">&ensp;3 Liver</td> <td class="tcr rb">15.50</td> <td class="tcl rb">&ensp;3 Liver</td> <td class="tcr rb">10.02</td></tr>
+<tr><td class="tcl lb rb">&ensp;4 Tongue</td> <td class="tcr rb">11.36</td> <td class="tcl rb">&ensp;4 Peritoneum</td> <td class="tcr rb">7.86</td> <td class="tcl rb">&ensp;4 Lip</td> <td class="tcr rb">9.89</td></tr>
+<tr><td class="tcl lb rb">&ensp;5 Oesophagus</td> <td class="tcr rb">10.90</td> <td class="tcl rb">&ensp;5 Oesophagus</td> <td class="tcr rb">5.33</td> <td class="tcl rb">&ensp;5 Oesophagus</td> <td class="tcr rb">9.29</td></tr>
+<tr><td class="tcl lb rb">&ensp;6 Liver</td> <td class="tcr rb">7.80</td> <td class="tcl rb">&ensp;6 Sigmoid</td> <td class="tcr rb">4.53</td> <td class="tcl rb">&ensp;6 Tongue</td> <td class="tcr rb">8.96</td></tr>
+<tr><td class="tcl lb rb">&ensp;7 Jaw</td> <td class="tcr rb">6.38</td> <td class="tcl rb">&ensp;7 Pancreas</td> <td class="tcr rb">3.52</td> <td class="tcl rb">&ensp;7 Jaw</td> <td class="tcr rb">5.65</td></tr>
+<tr><td class="tcl lb rb">&ensp;8 Mouth</td> <td class="tcr rb">2.88</td> <td class="tcl rb">&ensp;8 Tongue</td> <td class="tcr rb">3.12</td> <td class="tcl rb">&ensp;8 Peritoneum</td> <td class="tcr rb">2.94</td></tr>
+<tr><td class="tcl lb rb">&ensp;9 Tonsils</td> <td class="tcr rb">2.09</td> <td class="tcl rb">&ensp;9 Omentum</td> <td class="tcr rb">2.98</td> <td class="tcl rb">&ensp;9 Sigmoid</td> <td class="tcr rb">2.56</td></tr>
+<tr><td class="tcl lb rb">10 Sigmoid flexure</td> <td class="tcr rb">1.77</td> <td class="tcl rb">10 Lip</td> <td class="tcr rb">2.57</td> <td class="tcl rb">10 Mouth</td> <td class="tcr rb">2.40</td></tr>
+<tr><td class="tcl lb rb">11 Parotid</td> <td class="tcr rb">1.10</td> <td class="tcl rb">11 Jaw</td> <td class="tcr rb">1.97</td> <td class="tcl rb">11 Pancreas</td> <td class="tcr rb">1.80</td></tr>
+<tr><td class="tcl lb rb">12 Pancreas</td> <td class="tcc rb">&rdquo;</td> <td class="tcl rb">12 Colon</td> <td class="tcr rb">1.84</td> <td class="tcl rb">12 Tonsils</td> <td class="tcr rb">1.35</td></tr>
+<tr><td class="tcl lb rb">13 Caecum</td> <td class="tcr rb">0.94 </td> <td class="tcl rb">13 Abdomen</td> <td class="tcc rb">&rdquo;</td> <td class="tcl rb">13 Omentum</td> <td class="tcr rb">1.25</td></tr>
+<tr><td class="tcl lb rb">14 Peritoneum</td> <td class="tcc rb">&rdquo;</td> <td class="tcl rb">14 Intestine</td> <td class="tcr rb">1.56</td> <td class="tcl rb">14 Parotid</td> <td class="tcr rb">1.12</td></tr>
+<tr><td class="tcl lb rb">15 Colon</td> <td class="tcr rb">0.89 </td> <td class="tcl rb">15 Caecum</td> <td class="tcr rb">1.37</td> <td class="tcl rb">15 Colon</td> <td class="tcc rb">&rdquo;</td></tr>
+<tr><td class="tcl lb rb">16 Pharynx</td> <td class="tcr rb">0.79</td> <td class="tcl rb">16 Mouth</td> <td class="tcr rb">1.18</td> <td class="tcl rb">16 Caecum</td> <td class="tcr rb">1.08</td></tr>
+<tr><td class="tcl lb rb">17 Intestine (site unknown)</td> <td class="tcc rb">&rdquo;</td> <td class="tcl rb">17 Parotid</td> <td class="tcc rb">&rdquo;</td> <td class="tcl rb">17 Intestine</td> <td class="tcr rb">1.00</td></tr>
+<tr><td class="tcl lb rb">18 Abdomen</td> <td class="tcr rb">0.71</td> <td class="tcl rb">18 Splenic flexure</td> <td class="tcr rb">0.98</td> <td class="tcl rb">18 Abdomen</td> <td class="tcc rb">&rdquo;</td></tr>
+<tr><td class="tcl lb rb">19 Mesentery</td> <td class="tcr rb">0.55</td> <td class="tcl rb">19 Jejunum and ileum</td> <td class="tcr rb">0.78</td> <td class="tcl rb">19 Pharynx</td> <td class="tcr rb">0.62</td></tr>
+<tr><td class="tcl lb rb">20 Omentum</td> <td class="tcc rb">&rdquo;</td> <td class="tcl rb">20 Tonsils</td> <td class="tcr rb">0.68</td> <td class="tcl rb">20 Mesentery</td> <td class="tcr rb">0.52</td></tr>
+<tr><td class="tcl lb rb">21 Hepatic flexure</td> <td class="tcr rb">0.39</td> <td class="tcl rb">21 Pharynx</td> <td class="tcr rb">0.40</td> <td class="tcl rb">21 Jejunum and ileum</td> <td class="tcr rb">0.44</td></tr>
+<tr><td class="tcl lb rb">22 Submaxillary gland</td> <td class="tcr rb">0.31</td> <td class="tcl rb">22 Hepatic flexure</td> <td class="tcc rb">&rdquo;</td> <td class="tcl rb">22 Hepatic flexure</td> <td class="tcc rb">&rdquo;</td></tr>
+<tr><td class="tcl lb rb">23 Jejunum and ileum</td> <td class="tcc rb">&rdquo;</td> <td class="tcl rb">23 Mesentery</td> <td class="tcc rb">&rdquo;</td> <td class="tcl rb">23 Splenic flexure</td> <td class="tcc rb">&rdquo;</td></tr>
+<tr><td class="tcl lb rb">24 Duodenum</td> <td class="tcr rb">0.23</td> <td class="tcl rb">24 Submaxillary</td> <td class="tcr rb">0.20</td> <td class="tcl rb">24 Submaxillary</td> <td class="tcr rb">0.28</td></tr>
+<tr><td class="tcl lb rb bb">25 Splenic flexure</td> <td class="tcr rb bb">0.15</td> <td class="tcl rb bb">25 Duodenum</td> <td class="tcc rb bb">&rdquo;</td> <td class="tcl rb bb">25 Duodenum</td> <td class="tcr rb bb">0.22</td></tr>
+<tr><td class="tcc f90" colspan="6"><i>Note.</i>&mdash;The figures where several organs are bracketed apply to each organ separately.</td></tr>
+</table>
+
+<p>In a healthy new-born infant the mouth is free from micro-organisms,
+and very few are found in a breast-fed baby, but
+<i>Bacillus lactis</i> may be found where the child is bottle fed.
+If there is trouble with the first dentition and food is allowed
+to collect, staphylococci, streptococci, pneumococci and colon
+bacilli may be present. Even in healthy babies <i>Oidium albicans</i>
+may be present, and in older children the pseudo-diphtheria
+bacillus. From carious teeth may be isolated streptothrix,
+leptothrix, spirilla and fusiform bacilli. Under conditions of
+health these micro-organisms live in the mouth as saprophytes,
+and show no virulence when cultivated
+and injected into animals.
+The two common pyogenetic organisms,
+<i>Staphylococcus albus</i> and
+<i>brevis</i>, show no virulence. Also
+the pneumococcus, though often
+present, must be raised in virulence
+before it can produce untoward
+results. The foulness of the mouth
+is supposed to be due to the colon
+bacillus and its allies, but those
+obtained from the mouth are innocuous.
+Also to enable the <i>Oidium
+albicans</i> to attack the mucous membrane
+there must be some slight
+inflammation or injury. The micro-organisms
+found in the stomach
+gain access to that organ in the
+food or by regurgitation from the
+small intestine. Most are relatively
+inert, but some have a special fermentative
+action on the food (see
+<span class="sc"><a href="#artlinks">Nutrition</a></span>). Abelous isolated sixteen
+distinct species of organism
+from a healthy stomach, including
+Sarcinae, <i>B. lactis</i>, <i>pyocyaneus</i>,
+<i>subtilis</i>, <i>lactis erythrogenes</i>, <i>amylobacter</i>,
+<i>megatherium</i>, and <i>Vibrio
+rugula</i>.</p>
+
+<p>Hare-lip, cleft palate, hernia
+and imperforate anus are physical
+abnormalities which are interesting to the surgeon rather than to
+the pathologist. The oesophagus may be the seat of a diverticulum,
+or blind pouch, usually situated in its lower half, which in
+<span class="sidenote">Physical abnormalities</span>
+most instances is probably partly acquired and partly
+congenital; a local weakness succumbing to pressure.
+Hypertrophy of the muscular coat of the pyloric region
+is an infrequent congenital gastric anomaly in infants,
+preventing the passage of food into the bowel, and causing death
+in a short time. Incomplete closure
+of the vitelline duct results in
+the presence of a diverticulum&mdash;Meckel&rsquo;s&mdash;generally
+connected with
+the ileum, mainly important by
+reason of the readiness with which
+it occasions intestinal obstruction.
+Idiopathic congenital dilatation of
+the colon has been described.</p>
+
+<p class="center pt2"><span class="sc">Table II.</span></p>
+
+<table class="ws" summary="Contents">
+<tr><td class="tccm allb">Males.</td> <td class="tccm allb">Per-<br />centage.</td> <td class="tccm allb">Females.</td> <td class="tccm allb">Per-<br />centage.</td> <td class="tccm allb">Total.</td> <td class="tccm allb">Per-<br />centage.</td></tr>
+
+<tr><td class="tcl lb rb">1 Mouth and pharynx</td> <td class="tcr rb">37.85</td> <td class="tcl rb">1 Intestines</td> <td class="tcr rb">28.9</td> <td class="tcl rb">1 Oesophagus and stomach</td> <td class="tcr rb">31.78</td></tr>
+<tr><td class="tcl lb rb">2 Oesophagus and stomach</td> <td class="tcr rb">33.46</td> <td class="tcl rb">2 Oesophagus and stomach</td> <td class="tcr rb">27.7</td> <td class="tcl rb">2 Mouth and pharynx</td> <td class="tcr rb">30.27</td></tr>
+<tr><td class="tcl lb rb">3 Intestines </td> <td class="tcr rb">17.04</td> <td class="tcl rb">3 Liver</td> <td class="tcr rb">15.5</td> <td class="tcl rb">3 Intestines</td> <td class="tcr rb">20.42</td></tr>
+<tr><td class="tcl lb rb">4 Liver</td> <td class="tcr rb">7.8&ensp;</td> <td class="tcl rb">4 Peritoneum</td> <td class="tcr rb">13.1</td> <td class="tcl rb">4 Liver</td> <td class="tcr rb">10.02</td></tr>
+<tr><td class="tcl lb rb">5 Peritoneum</td> <td class="tcr rb">2.75</td> <td class="tcl rb">5 Mouth and pharynx</td> <td class="tcr rb">11.3</td> <td class="tcl rb">5 Peritoneum</td> <td class="tcr rb">5.71</td></tr>
+<tr><td class="tcl lb rb bb">6 Pancreas </td> <td class="tcr rb bb">1.1&ensp;</td> <td class="tcl rb bb">6 Pancreas</td> <td class="tcr rb bb">3.5</td> <td class="tcl rb bb">6 Pancreas</td> <td class="tcr rb bb">1.80</td></tr>
+</table>
+
+<p>Traction diverticula of the oesophagus
+not uncommonly occur as
+sequels to suppurative inflammation
+of cervical lymphatic glands.
+More frequently dilatation of a section is met with, due as a
+rule to the presence of a stricture. The stomach often diverges
+from the normal in size, shape and position. Normally capable
+in the adult of containing from fifty to sixty ounces, either by
+reason of organic disease, or as the result of functional disturbance,
+its capacity may vary enormously. The writer has seen
+post mortem a stomach which held a gallon (160 ounces), and
+again one holding only two ounces. Cancer spread over a large
+area and cirrhosis of the stomach wall cause diminution in
+capacity; pyloric obstruction, weakness of the muscular coat,
+and nervous influences are associated with dilatation. A peculiar
+distortion of the shape of the stomach follows cicatrization of
+<span class="pagenum"><a name="page267" id="page267"></a>267</span>
+ulcers of greater or lesser curvature; the gastric cavity becomes
+&ldquo;hour-glass&rdquo; in shape. In addition, the stomach may be displaced
+downwards as a whole, a condition known as gastroptosis:
+if the pyloric portion only be displaced, the lesion is termed
+pyloroptosis. Ptoses of other abdominal organs are described;
+the liver, transverse colon, spleen and kidneys may be involved.
+Displacements downwards of the stomach and transverse colon,
+along with a movable right kidney and associated with dyspepsia
+and neurasthenia, form the malady termed by Glénard enteroptosis.
+A general visceroptosis often occurs in those patients
+who have some tuberculous lesion of the lungs or elsewhere,
+this disease causing a general weakening and subsequent
+stretching of all ligaments. Displacements of the abdominal
+viscera are almost invariably accompanied by symptoms of
+dyspepsia of a neurotic type. The rectum is liable to prolapse,
+consequent upon constipation and straining at stool, or following
+local injuries of the perineal floor.</p>
+
+<p>Every pathological lesion shown by digestive organs is closely
+associated with the state of the nervous system, general or local;
+so stoppage of active gastric digestive processes after
+profound nervous shock, and occurrence of nervous
+<span class="sidenote">Influence of the nervous system.</span>
+diarrhoea from the same cause. Gastric dyspepsia
+of nervous origin presents most varied and contradictory
+symptoms: diminished acidity of the gastric juice,
+hyper-acidity, over-production, arrest of secretion, lessened or
+increased movements, greater sensitiveness to the presence of
+contents, dilatation or spasm. Often the nervous cause can
+be traced back farther,&mdash;in females, frequently to the pelvic
+organs; in both sexes, to the condition of the blood, the brain or
+the bowel. Unhealthy conditions related to evacuation of the
+bowel-contents commonly induce reflex nervous manifestations of
+abnormal character referred to the stomach and liver. Gastric
+disturbances similarly react upon the proper conduct of intestinal
+functions.</p>
+
+<p class="center pt2"><i>Local Diseases.</i></p>
+
+<p><i>The Mouth.</i>&mdash;The lining membrane of the cheeks inside the
+mouth, of the gums and the under-surface and edges of the
+tongue, is often the seat of small irritable ulcers, usually associated
+with some digestive derangement. A crop of minute vesicles
+known as Koplik&rsquo;s spots over these parts has been lately stated
+by Koplik to be an early symptom of measles. Xerostomia, or
+dry mouth, is a rare condition, connected with lack of salivary
+secretion. Gangrenous stomatitis, cancrum oris, or noma,
+occasionally attacks debilitated children, or patients convalescing
+from acute fevers, more especially after measles. It commences
+in the gums or cheeks, and causes widespread sloughing of the
+adjacent soft parts&mdash;it may be of the bones.</p>
+
+<p><i>The Stomach.</i>&mdash;It were futile to attempt to enumerate all the
+protean manifestations of disturbance which proceed from a disordered
+stomach. The possible permutations and combinations
+of the causes of gastric vagaries almost reach infinity. Idiosyncrasy,
+past and present gastric education, penury or plethora,
+actual digestive power, motility, bodily requirements and conditions,
+environment, mental influences, local or adjacent organic
+lesions, and, not least, reflex impressions from other organs, all
+contribute to the variance.</p>
+
+<p>Ulcer of the stomach, however&mdash;the perforating gastric ulcer&mdash;occupies
+a unique position among diseases of this organ.
+Gastric ulcers are circumscribed, punched out, rarely larger than
+a sixpenny-bit, funnel-shaped, the narrower end towards the
+peritoneal coat, and distributed in those regions of the stomach
+wall which are most exposed to the action of the gastric contents.
+They occur most frequently in females, especially if anaemic, and
+are usually accompanied by excess of acid, actual or relative
+to the state of the blood, in the stomach contents. Local pain,
+dorsal pain, generally to the left of the eighth or ninth dorsal
+spinous process, and haematernesis and melaena, are symptomatic
+of it. The amount of blood lost varies with the rapidity of
+ulcer formation and the size of vessel opened into. Fatal results
+arise from ulceration into large blood-vessels, followed by copious
+haemorrhage, or by perforation of the ulcer into the peritoneal
+cavity. Scars of such ulcers may be found post mortem, although
+no symptoms of gastric disease have been exhibited during life;
+gastric ulcers, therefore, may be latent.</p>
+
+<p>Irritation of the sensory nerve-endings in the stomach wall
+from the presence of an increased proportion of acid, organic or
+mineral, in the stomach contents is accountable for the well known
+symptom heartburn. Water-brash is a term applied to
+eructation of a colourless, almost tasteless fluid, probably saliva,
+which has collected in the lower part of the oesophagus from
+failure of the cardiac sphincter of the stomach to relax; reversed
+oesophageal peristalsis causing regurgitation. A similar reversed
+action serves in merycism, or rumination, occasionally found in
+man, to raise part of the food, lately ingested, from the stomach to
+the mouth. Vomiting also is aided by reversed peristaltic action,
+both of the stomach and the oesophagus, with the help of the
+diaphragm and the muscles of the anterior abdominal wall.
+Emesis may be caused both by local nervous influence, and
+through the central nervous mechanism either reflexly or from
+the direct action of substances circulating in the blood. Further,
+the causal agent acting on the central nervous apparatus may be
+organic or functional, as well as medicinal. Vomiting without
+any apparent cause suggests nervous lesions, organic or reflex.
+The obstinate vomiting of pregnancy is a case in point. Here the
+primary cause proceeds reflexly from the pelvis. In females the
+pelvic organs are often the true source of emesis. Haematemesis
+accompanies gastric ulcer, cancer, chronic congestion with
+haemorrhagic erosion, congestion of the liver, or may follow
+violent acts of vomiting. In cases of ulcer the blood is usually
+bright and in considerable amount; in cancer, darker, like coffee-grounds;
+and in cases of erosion, in smaller quantity and of bright
+colour. The reaction of the stomach contents, if the cause be
+doubtful, yields valuable aid towards a diagnosis. Of increased
+acidity in gastric ulcer, normal in hepatic congestion, it is
+diminished in cancer; but as the acid present in cancer is largely
+lactic, analysis of the gastric contents must often be a <i>sine qua
+non</i>, because hyperacidity from lactic may obscure hypoacidity of
+hydrochloric acid.</p>
+
+<p>Flatulence usually results from fermentative processes in
+the stomach and bowel, as the outcome of bacterial activity. A
+different form of flatulence is common in neurotic individuals:
+in such the gas evolved consists simply in carbonic acid liberated
+from the blood, and its evolution is generally characterized by
+rapid development and by lack of all fermentative signs.</p>
+
+<p><i>The Liver.</i>&mdash;The liver is an organ frequently libelled for the
+delinquencies of other organs, and regarded as a common source of
+ill. In catarrhal jaundice it is in most cases the bowel that is at
+fault, the liver acting properly, but unable to get rid of all the bile
+produced. The liver suffers, however, from several diseases of its
+own. Its fibrous or connective tissue is very apt to increase
+at the expense of the cellular elements, destroying their functions.
+This cirrhotic process usually follows long-continued irritation,
+such as is produced by too much alcohol absorbed from the bowel
+habitually, the organ gradually becoming harder in texture and
+smaller in bulk. Hypertrophic cirrhosis of the liver is not uncommonly
+met with, in which the liver is much increased in size,
+the &ldquo;unilobular&rdquo; form, also of alcoholic origin. In still-born
+children and in some infants a form of hypertrophic cirrhosis is
+occasionally seen, probably of hereditary syphilitic origin. Acute
+congestion of the liver forms an important symptom of malarial
+fever, and often leads in time to establishment of cirrhotic changes;
+here the liver is generally enlarged, but not invariably so, and the
+part played by alcohol in its causation has still to be investigated.
+Acute yellow atrophy of the liver is a disease <i>sui generis</i>. Of rare
+occurrence, possibly of toxic origin, it is marked by jaundice, at
+first of usual type, later becoming most intense; by vomiting;
+haemorrhages widely distributed; rapid diminution in the size of
+the liver; the appearance of leucin and tyrosin in the urine, with
+lessened urea; and in two or three days, death. The liver after
+death is soft, of a reddish colour dotted with yellow patches, and
+weighs only about a third part of the normal&mdash;about 1½ lb in
+place of 3¾ lb. A closely analogous affection of the liver, known
+as Weil&rsquo;s disease, is of infectious type, and has been noted in
+<span class="pagenum"><a name="page268" id="page268"></a>268</span>
+epidemic form. In this the spleen and liver are commonly but
+not always swollen, and the liver is often tender on pressure. As
+a large proportion of the sufferers from this disease have been
+butchers, and the epidemics have occurred in the hot season of
+the year, it probably arises from contact with decomposing
+animal matter. Hepatic abscess may follow on an attack of
+amoebic dysentery, and is produced either by infection through
+the portal vein, or by direct infection from the adjacent colon.
+In general pyaemia multiple small abscesses may occur in the
+liver.</p>
+
+<p><i>The Gall-Bladder.</i>&mdash;The formation of biliary calculi in the gall-bladder
+is the chief point of interest here. At least 75% of such
+cases occur in women, especially in those who have borne children.
+Tight-lacing has been stated to act as an exciting cause, owing to
+the consequent retardation of the flow of bile. Gall-stones may
+number from one to many thousands. They are largely composed
+of cholesterin, combined with small amounts of bile-pigments
+and acids, lime and magnesium salts. Their presence
+may give rise to no symptoms, or may cause violent biliary colic,
+and, if the bile-stream be obstructed, to jaundice. Inflammatory
+processes may be initiated in the gall-bladder or the bile-ducts,
+catarrhal or suppurative in character.</p>
+
+<p><i>The Pancreas.</i>&mdash;Haemorrhages into the body of the pancreas,
+acute and chronic inflammation, calculi, cysts and tumours,
+among which cancer is by far the most common, are recognized as
+occurring in this organ; the point of greatest interest regarding
+them lies in the relations established between pancreatic disease
+and diabetes mellitus, affections of the gland frequently being
+complicated by, and probably causing, the appearance of sugar in
+the urine.</p>
+
+<p><i>The Small Intestine.</i>&mdash;Little remains to be added to the account
+of inflammatory lesions in connexion with the small intestine. It
+offers but few conditions peculiar to itself, save in typhoid fever,
+and the ease with which it contrives to become kinked, or intussuscepted,
+producing obstruction, or to take part in hernial
+protrusions. The first section, the duodenum, is subject to
+development of ulcers very similar to those of the gastric mucous
+membrane. For long duodenal ulceration has been regarded as a
+complication of extensive burns of the skin, but the relationship
+between them has not yet been quite satisfactorily explained.
+The condition of colic in the bowel usually arises from overdistension
+of some part of the small gut with gas, the frequent
+sharp turns of the gut facilitating temporary closure of its lumen
+by pressure of the dilated gut near a curve against the part
+beyond. In the large bowel accumulations of gas seldom cause
+such acute symptoms, having a readier exit.</p>
+
+<p><i>The Large Intestine.</i>&mdash;The colon, especially the ascending
+portion, may become immensely dilated, usually after prolonged
+constipation and paralysis of the gut; occasionally the condition
+is congenital. Straining efforts made in defaecation may often
+account for prolapse of the lower end of the rectum through the
+anus. Haemorrhage from the bowel is usually a sign of disease
+situated in the large intestine: if bright in colour, the source is
+probably low down; if dark, from the caecum or from above the
+ileo-caecal valve. Blood after a short stay in any section of the
+alimentary canal darkens, and eventually becomes almost black
+in colour.</p>
+<div class="author">(A. L. G.; M. F.*)</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIGGES, WEST<a name="ar92" id="ar92"></a></span> (1720-1786), English actor, made his first stage
+appearance in Dublin in 1749 as Jaffier in <i>Venice Preserved</i>; and
+both there and in Edinburgh until 1764 he acted in many tragic
+rôles with success. He was the original &ldquo;young Norval&rdquo; in
+Home&rsquo;s <i>Douglas</i> (1756). His first London appearance was as
+Cato in the Haymarket in 1777, and he afterwards played Lear,
+Macbeth, Shylock and Wolsey. In 1881 he returned to Dublin
+and retired in 1784.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIGIT<a name="ar93" id="ar93"></a></span> (Lat. <i>digitus</i>, finger), literally a finger or toe, and so used
+to mean, from counting on the fingers, a single numeral, or, from
+measuring, a finger&rsquo;s breadth. In astronomy a digit is the twelfth
+part of the diameter of the sun or moon; it is used to express the
+magnitude of an eclipse.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIGITALIS.<a name="ar94" id="ar94"></a></span> The leaves of the foxglove (<i>q.v.</i>), gathered from
+wild plants when about two-thirds of their flowers are expanded,
+deprived usually of the petiole and the thicker part of the midrib,
+bitter taste; and to preserve their properties they must be kept
+excluded from light in stoppered bottles. They are occasionally
+adulterated with the leaves of <i>Inula Conyza</i>, ploughman&rsquo;s
+spikenard, which may be distinguished by their greater roughness,
+their less divided margins, and their odour when rubbed;
+also with the leaves of <i>Symphytum officinale</i>, comfrey, and of
+<i>Verbascum Thapsus</i>, great mullein, which unlike those of the
+foxglove have woolly upper and under surfaces. The earliest
+known descriptions of the foxglove are those given by Leonhard
+Fuchs and Tragus about the middle of the 16th century, but its
+virtues were doubtless known to herbalists at a much remoter
+period. J. Gerarde, in his <i>Herbal</i> (1597), advocates the use of
+foxglove for a variety of complaints; and John Parkinson, in the
+<i>Theatrum Botanicum</i>, or <i>Theater of Plants</i> (1640), and later W.
+Salmon, in <i>The New London Dispensatory</i>, similarly praised the
+remedy. Digitalis was first brought prominently under the
+notice of the medical profession by Dr W. Withering, who, in his
+<i>Account of the Foxglove</i> (1785), gave details of upwards of 200
+cases chiefly dropsical, in which it was used.</p>
+
+<p>Digitalis contains four important glucosides, of which three are
+cardiac stimulants. The most powerful is <i>digitoxin</i> C<span class="su">34</span>H<span class="su">54</span>O<span class="su">11</span>,
+an extremely poisonous and cumulative drug, insoluble in water.
+<i>Digitalin</i>, C<span class="su">35</span>H<span class="su">56</span>O<span class="su">14</span>, is crystalline and is also insoluble in water.
+<i>Digitalein</i> is amorphous but readily soluble in water. It can
+therefore be administered subcutaneously, in doses of about one-hundredth
+of a grain. <i>Digitonin</i>, on the other hand, is a cardiac
+depressant, and has been found to be identical with saponin,
+the chief constituent of senega root. There are numerous preparations,
+patent and pharmacopeial, their composition being
+extremely varied, so that, unless one has reason to be certain of
+any particular preparation, it is almost better to use only the
+dried leaves themselves in the form of a powder (dose ½-2 grains).
+The pharmacopeial tincture may be given in doses of five to
+fifteen minims, and the infusion has the unusually small dose of
+two to four drachms&mdash;the dose of other infusions being an ounce
+or more. The tincture contains a fair proportion of both digitalin
+and digitoxin.</p>
+
+<p>Digitalis leaves have no definite external action. Taken by the
+mouth, the drug is apt to cause considerable digestive disturbance,
+varying in different cases and sometimes so severe as to cause
+serious difficulty. This action is probably due to the digitonin,
+which is thus a constituent in every way undesirable. The all-important
+property of the drug is its action on the circulation.
+Its first action on any of the body-tissues is upon unstriped
+muscle, so that the first consequence of its absorption is a contraction
+of the arteries and arterioles. No other known drug has
+an equally marked action in contracting the arterioles. As the
+vaso-motor centre in the medulla oblongata is also stimulated, as
+well as the contractions of the heart, there is thus trebly caused a
+very great rise in the blood-pressure.</p>
+
+<p>The clinical influence of digitalis upon the heart is very well
+defined. After the taking of a moderate dose the pulse is
+markedly slowed. This is due to a very definite influence upon
+the different portions of the cardiac cycle. The systole is not
+altered in length, but the diastole is very much prolonged, and
+since this is the period not only of cardiac rest but also of cardiac
+&ldquo;feeding&rdquo;&mdash;the coronary vessels being compressed and occluded
+during systole&mdash;the result is greatly to benefit the nutrition of the
+cardiac muscle. So definite is this that, despite a great increase
+in the force of the contractions and despite experimental proof
+that the heart does more work in a given time under the influence
+of digitalis, the organ subsequently displays all the signs of having
+rested, its improved vigour being really due to its obtaining a
+larger supply of the nutrient blood. Almost equally striking is
+the fact that digitalis causes an irregular pulse to become regular.
+Added to the greater force of cardiac contraction is a permanent
+tonic contraction of the organ, so that its internal capacity is
+reduced. The bearing of this fact on cases of cardiac dilatation
+is evident. In larger doses a remarkable sequel to these actions
+<span class="pagenum"><a name="page269" id="page269"></a>269</span>
+may be observed. The cardiac contractions become irregular, the
+ventricle assumes curious shapes&mdash;&ldquo;hour-glass,&rdquo; &amp;c.&mdash;becomes
+very pale and bloodless, and finally the heart stops in a state of
+spasm, which shortly afterwards becomes rigor-mortis. Before
+this final change the heart may be started again by the application
+of a soluble potassium salt, or by raising the fluid pressure
+within it. Clinically it is to be observed that the drug is cumulative,
+being very slowly excreted, and that after it has been taken
+for some time the pulse may become irregular, the blood-pressure
+low, and the cardiac pulsations rapid and feeble. These
+symptoms with more or less gastro-intestinal irritation and
+decrease in the quantity of urine passed indicate digitalis poisoning.
+The initial action of digitalis is a stimulation of the cardiac
+terminals of the vagus nerves, so that the heart&rsquo;s action is slowed.
+Thereafter follows the most important effect of the drug, which is
+a direct stimulation of the cardiac muscle. This can be proved to
+occur in a heart so embryonic that no nerves can be recognized in
+it, and in portions of cardiac muscle that contain neither nervecells
+nor nerve-fibres.</p>
+
+<p>The action of this drug on the kidney is of importance only
+second to its action on the circulation. In small or moderate
+doses it is a powerful diuretic. Though Heidenhain asserts that
+rise in the renal blood-pressure has not a diuretic action per se,
+it seems probable that this influence of the drug is due to a rise
+in the general blood-pressure associated with a relatively dilated
+condition of the renal vessels. In large doses, on the other hand,
+the renal vessels also are constricted and the amount of urine falls.
+It is probable that digitalis increases the amount of water rather
+than that of the urinary solids. In large doses the action of
+digitalis on the circulation causes various cerebral symptoms,
+such as seeing all objects blue, and various other disturbances of
+the special senses. There appears also to be a specific action of
+lowering the reflex excitability of the spinal cord.</p>
+
+<p>Digitalis is used in therapeutics exclusively for its action on the
+circulation. In prescribing this drug it must be remembered that
+fully three days elapse before it gets into the system, and thus it
+must always be combined with other remedies to tide the patient
+over this period. It must never be prescribed in large doses to
+begin with, as some patients are quite unable to take it, intractable
+vomiting being caused. The three days that must pass before
+any clinical effect is obtained renders it useless in an emergency.
+A certain consequence of its use is to cause or increase cardiac
+hypertrophy&mdash;a condition which has its own dangers and
+ultimately disastrous consequences, and must never be provoked
+beyond the positive needs of the case. But digitalis is indicated
+whenever the heart shows itself unequal to the work it has to
+perform. This formula includes the vast majority of cardiac
+cases. The drug is contra-indicated in all cases where the heart is
+already beating too slowly; in aortic incompetence&mdash;where the
+prolongation of diastole increases the amount of the blood that
+regurgitates through the incompetent valve; in chronic Bright&rsquo;s
+disease and in fatty degeneration of the heart&mdash;since nothing can
+cause fat to become contractile.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIGNE,<a name="ar95" id="ar95"></a></span> the chief town of the department of the Basses Alpes,
+in S.E. France, 14 m. by a branch line from the main railway
+line between Grenoble and Avignon. Pop. (1906), town, 4628;
+commune, 7456. The Ville Haute is built on a mountain spur
+running down to the left bank of the Bléone river, and is composed
+of a labyrinth of narrow winding streets, above which towers the
+present cathedral church, dating from the end of the 15th century,
+but largely reconstructed in modern times, and the former
+bishop&rsquo;s palace (now the prison). The fine Boulevard Gassendi
+separates the Ville Haute from the Ville Basse, which is of modern
+date. The old cathedral (Notre Dame du Bourg) is a building of
+the 13th century, but is now disused except for funerals: it
+stands at the east end of the Ville Basse. The neighbourhood of
+Digne is rich in orchards, which have long made the town famous
+in France for its preserved fruits and confections. It is the <i>Dinia</i>
+of the Romans, and was the capital of the Bodiontii. From the
+early 6th century at least it has been an episcopal see, which till
+1790 was in the ecclesiastical province of Embrun, but since 1802
+in that of Aix en Provence. The history of Digne in the middle
+ages is bound up with that of its bishops, under whom it prospered
+greatly. But it suffered much during the religious wars of the
+16th and 17th centuries, when it was sacked several times. A
+little way off, above the right bank of the Bléone, is Champtercier,
+the birthplace of the astronomer Gassendi (1592-1655), whose
+name has been given to the principal thoroughfare of the little
+town.</p>
+
+<div class="condensed">
+<p>See F. Guichard, <i>Souvenirs historiques sur la ville de Digne et ses
+environs</i> (Digne, 1847).</p>
+</div>
+<div class="author">(W. A. B. C.)</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIGOIN,<a name="ar96" id="ar96"></a></span> a town of east-central France, in the department of
+Saône-et-Loire, on the right bank of the Loire, 55 m. W.N.W.
+of Mâcon on the Paris-Lyon railway. Pop. (1906) 5321. It is
+situated at the meeting places of the Loire, the Lateral canal of the
+Loire and the Canal du Centre, which here crosses the Loire by a
+fine aqueduct. The town carries on considerable manufactures of
+faience, pottery and porcelain. The port on the Canal du Centre
+has considerable traffic in timber, sand, iron, coal and stone.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIJON,<a name="ar97" id="ar97"></a></span> a town of eastern France, capital of the department of
+Côte d&rsquo;Or and formerly capital of the province of Burgundy,
+195 m. S.E. of Paris on the Paris-Lyon railway. Pop. (1906)
+65,516. It is situated on the western border of the fertile plain of
+Burgundy, at the foot of Mont Afrique, the north-eastern summit
+of the Côte d&rsquo;Or range, and at the confluence of the Ouche and the
+Suzon; it also has a port on the canal of Burgundy. The great
+strategic importance of Dijon as a centre of railways and roads,
+and its position with reference to an invasion of France from the
+Rhine, have led to the creation of a fortress forming part of the
+Langres group. There is no <i>enceinte</i>, but on the east side detached
+forts, 3 to 4 m. distant from the centre, command all the great
+roads, while the hilly ground to the west is protected by Fort
+Hauteville to the N.W. and the &ldquo;groups&rdquo; of Motte Giron and
+Mont Afrique to the S.W., these latter being very formidable
+works. Including a fort near Saussy (about 8 m. to the N.W.)
+protecting the water-supply of Dijon, there are eight forts,
+besides the groups above mentioned. The fortifications which
+partly surrounded the old and central portion of the city have
+disappeared to make way for tree-lined boulevards with fine
+squares at intervals. The old churches and historic buildings of
+Dijon are to be found in the irregular streets of the old town, but
+industrial and commercial activity has been transferred to the
+new quarters beyond its limits. A fine park more than 80 acres
+in extent lies to the south of the city, which is rich in open spaces
+and promenades, the latter including the botanical garden and
+the Promenade de l&rsquo;Arquebuse, in which there is a black poplar
+famous for its size and age.</p>
+
+<p>The cathedral of St Bénigne, originally an abbey church,
+was built in the latter half of the 13th century on the site of a
+Romanesque basilica, of which the crypt remains. The west
+front is flanked by two towers and the crossing is surmounted by
+a slender timber spire. The plan consists of three naves, short
+transepts and a small choir, without ambulatory, terminating in
+three apses. In the interior there is a fine organ and a quantity of
+statuary, and the vaults contain the remains of Philip the Bold,
+duke of Burgundy, and Anne of Burgundy, daughter of John
+the Fearless. The site of the abbey buildings is occupied by
+the bishop&rsquo;s palace and an ecclesiastical seminary. The church
+of Notre-Dame, typical of the Gothic style of Burgundy, was
+erected from 1252 to 1334, and is distinguished for the grace of
+its interior and the beauty of the western façade. The portal
+consists of three arched openings, above which are two stages of
+arcades, open to the light and supported on slender columns.
+A row of gargoyles surmounts each storey of the façade, which is
+also ornamented by sculptured friezes. A turret to the right of
+the portal carries a clock called the Jaquemart, on which the hours
+are struck by two figures. The church of St Michel belongs to the
+15th century. The west façade, the most remarkable feature of
+the church, is, however, of the Renaissance period. The vaulting
+of the three portals is of exceptional depth owing to the projection
+of the lower storey of the façade. Above this storey rise two
+towers of five stages, the fifth stage being formed by an octagonal
+cupola. The columns decorating the façade represent all the four
+orders. The design of this façade is wrongly attributed to Hugues
+<span class="pagenum"><a name="page270" id="page270"></a>270</span>
+Sambin (fl. c. 1540), a native of Dijon, and pupil of Leonardo da
+Vinci, but the sculpture of the portals, including &ldquo;The Last
+Judgment&rdquo; on the tympanum of the main portal, is probably
+from his hand. St Jean (15th century) and St Étienne (15th,
+16th and 17th centuries), now used as the exchange, are the other
+chief churches. Of the ancient palace of the dukes of Burgundy
+there remain two towers, the Tour de la Terrasse and the Tour
+de Bar, the guard-room and the kitchens; these now form part
+of the hôtel de ville, the rest of which belongs to the 17th and
+18th centuries. This building contains an archaeological museum
+with a collection of Roman stone monuments; the archives of
+the town; and the principal museum, which, besides valuable
+paintings and other works of art, contains the magnificent tombs
+of Philip the Bold and John the Fearless, dukes of Burgundy.
+These were transferred from the Chartreuse of Dijon (or of
+Champmol), built by Philip the Bold as a mausoleum, now replaced
+by a lunatic asylum. Relics of it survive in the old Gothic
+entrance, the portal of the church, a tower and the well of Moses,
+which is adorned with statues of Moses and the prophets by
+Claux Sluter (fl. end of 14th century), the Dutch sculptor, who
+also designed the tomb of Philip the Bold. The Palais de
+Justice, which belongs to the reign of Louis XII., is of interest as
+the former seat of the <i>parlement</i> of Burgundy. Dijon possesses
+several houses of the 15th, 16th and 17th centuries, notably the
+Maison Richard in the Gothic, and the Hôtel Vogüé in the
+Renaissance style. St Bernard, the composer J. P. Rameau and
+the sculptor François Rude have statues in the town, of which
+they were natives. There are also monuments to those inhabitants
+of Dijon who fell in the engagement before the town
+in 1870, and to President Carnot and Garibaldi.</p>
+
+<p>The town is important as the seat of a prefecture, a bishopric, a
+court of appeal and a court of assizes, and as centre of an académie
+(educational district). There are tribunals of first instance and of
+commerce, a board of trade-arbitrators, a chamber of commerce,
+an exchange (occupying the former cathedral of St Étienne), and
+an important branch of the Bank of France. Its educational
+establishments include faculties of law, of science and of letters, a
+preparatory school of medicine and pharmacy, a higher school of
+commerce, a school of fine art, a conservatoire of music, <i>lycées</i> and
+training colleges, and there is a public library with about 100,000
+volumes.</p>
+
+<p>Dijon is well known for its mustard, and for the black currant
+liqueur called <i>cassis de Dijon</i>; its industries include the manufacture
+of machinery, automobiles, bicycles, soap, biscuits,
+brandy, leather, boots and shoes, candles and hosiery. There
+are also flour mills, breweries, important printing works, vinegar
+works and, in the vicinity, nursery gardens. The state has a
+large tobacco manufactory in the town. Dijon has considerable
+trade in cereals and wool, and is the second market for the wines
+of Burgundy.</p>
+
+<p>Under the Romans Dijon (<i>Divonense castrum</i>) was a <i>vicus</i> in
+the <i>civitas</i> of Langres. In the 2nd century it was the scene of
+the martyrdom of St Benignus (Bénigne, vulg. Berin, Berain),
+the apostle of Burgundy. About 274 the emperor Aurelian
+surrounded it with ramparts. Gregory of Tours, in the 6th
+century, comments on the strength and pleasant situation
+of the place, expressing surprise that it does not rank as
+a <i>civitas</i>. During the middle ages the fortunes of Dijon
+followed those of Burgundy, the dukes of which acquired it
+early in the 11th century. The communal privileges, conferred
+on the town in 1182 by Hugh III., duke of Burgundy, were
+confirmed by Philip Augustus in 1183, and in the 13th century
+the dukes took up their residence there. For the decoration of the
+palace and other monuments built by them, eminent artists were
+gathered from northern France and Flanders, and during this
+period the town became one of the great intellectual centres of
+France. The union of the duchy with the crown in 1477 deprived
+Dijon of the splendour of the ducal court; but to counterbalance
+this loss it was made the capital of the province and seat of a
+<i>parlement</i>. Its fidelity to the monarchy was tested in 1513,
+when the citizens were besieged by 50,000 Swiss and Germans,
+and forced to agree to a treaty so disadvantageous that Louis XII.
+refused to ratify it. In the wars of religion Dijon sided with the
+League, and only opened its gates to Henry IV. in 1595. The
+18th century was a brilliant period for the city; it became the
+seat of a bishopric, its streets were improved, its commerce
+developed, and an academy of science and letters founded;
+while its literary salons were hardly less celebrated than those of
+Paris. The neighbourhood was the scene of considerable fighting
+during the Franco-German War, which was, however, indirectly
+of some advantage to the city owing to the impetus given to its
+industries by the immigrants from Alsace.</p>
+
+<div class="condensed">
+<p>See H. Chabeuf, <i>Dijon à travers les âges</i> (Dijon, 1897), and <i>Dijon,
+monuments et souvenirs</i> (Dijon, 1894).</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DIKE,<a name="ar98" id="ar98"></a></span> or <span class="sc">Dyke</span> (Old Eng. <i>dic</i>, a word which appears in various
+forms in many Teutonic languages, cf. Dutch <i>dijk</i>, German <i>Teich</i>,
+Danish <i>dige</i>, and in French, derived from Teutonic, <i>digue</i>; it is
+the same word as &ldquo;ditch&rdquo; and is ultimately connected with the
+root of &ldquo;dig&rdquo;), properly a trench dug out of the earth for defensive
+and other purposes. Water naturally collects in such
+trenches, and hence the word is applied to natural and artificial
+channels filled with water, as appears in the proverbial expression
+&ldquo;February fill-dyke,&rdquo; and in the names of many narrow waterways
+in East Anglia. &ldquo;Dike&rdquo; also is naturally used of the bank
+of earth thrown up out of the ditch, and so of any embankment,
+dam or causeway, particularly the defensive works in Holland,
+the Fen district of England, and other low-lying districts which
+are liable to flooding by the sea or rivers (see <span class="sc"><a href="#artlinks">Holland</a></span> and <span class="sc"><a href="#artlinks">Fens</a></span>).
+In Scotland any wall, fence or even hedge, used as a boundary is
+called a dyke. In geology the term is applied to wall-like masses
+of rock (sometimes projecting beyond the surrounding surface)
+which fill up vertical or highly inclined fissures in the strata.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIKKA,<a name="ar99" id="ar99"></a></span> a term in Mahommedan architecture for the tribune
+raised upon columns, from which the Koran is recited and the
+prayers intoned by the Imam of the mosque.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DILAPIDATION<a name="ar100" id="ar100"></a></span> (Lat. for &ldquo;scattering the stones,&rdquo; <i>lapides</i>, of a
+building), a term meaning in general a falling into decay, but more
+particularly used in the plural in English law for (1) the waste
+committed by the incumbent of an ecclesiastical living; (2) the
+disrepair for which a tenant is usually liable when he has agreed
+to give up his premises in good repair (see <span class="sc"><a href="#artlinks">Easement</a></span>; <span class="sc"><a href="#artlinks">Flat</a></span>;
+<span class="sc"><a href="#artlinks">Landlord and Tenant</a></span>). By the general law a tenant for
+life has no power to cut down timber, destroy buildings, &amp;c.,
+(voluntary waste), or to let buildings fall into disrepair (permissive
+waste). In the eye of the law an incumbent of a living is
+a tenant for life of his benefice, and any waste, voluntary or permissive,
+on his part must be made good by his administrators to
+his successor in office. The principles on which such dilapidations
+are to be ascertained, and the application of the money payable in
+respect thereof, depend partly on old ecclesiastical law and partly
+on acts of parliament. Questions as to ecclesiastical dilapidations
+usually arise in respect of the residence house and other buildings
+belonging to the living. Inclosures, hedges, ditches and the like
+are included in things &ldquo;of which the beneficed person hath the
+burden and charge of reparation.&rdquo; In a leading case (<i>Ross</i> v.
+<i>Adcock</i>, 1868, L.R. 3 C.P. 657) it was said that the court was
+acquainted with no precedent or decision extending the liability
+of the executors of a deceased incumbent to any species of waste
+beyond dilapidation of the house, chancel or other buildings or
+fences of the benefice. And it has been held that the mere mismanagement
+or miscultivation of the ecclesiastical lands will not
+give rise to an action for dilapidations. To place the law relating
+to dilapidations on a more satisfactory footing, the Ecclesiastical
+Dilapidations Act 1871 was passed. The buildings to which the
+act applies are defined to be such houses of residence, chancels,
+walls, fences and other buildings and things as the incumbent of
+the benefice is by law and custom bound to maintain in repair.
+In each diocese a surveyor is appointed by the archdeacons and
+rural deans subject to the approval of the bishop; and such
+surveyor shall by the direction of the bishop examine the buildings
+on the following occasions&mdash;viz. (1) when the benefice is
+sequestrated; (2) when it is vacant; (3) at the request of the
+incumbent or on complaint by the archdeacon, rural dean or
+patron. The surveyor specifies the works required, and gives an
+<span class="pagenum"><a name="page271" id="page271"></a>271</span>
+estimate of their probable cost. In the case of a vacant benefice,
+the new incumbent and the old incumbent or his representatives
+may lodge objections to the surveyor&rsquo;s report on any grounds of
+fact or law, and the bishop, after consideration, may make an
+order for the repairs and their cost, for which the late incumbent
+or his representatives are liable. The sum so stated becomes a
+debt due from the late incumbent or his representatives to the
+new incumbent, who shall pay over the money when recovered
+to the governors of Queen Anne&rsquo;s Bounty. The governors pay
+for the works on execution on receipt of a certificate from the
+surveyor; and the surveyor, when the works have been completed
+to his satisfaction, gives a certificate to that effect, the effect of
+which, so far as regards the incumbent, is to protect him from
+liability for dilapidations for the next five years. Unnecessary
+buildings belonging to a residence house may, by the authority
+of the bishop and with the consent of the patron, be removed.
+An amending statute of 1872 (Ecclesiastical Dilapidations Act
+(1871) Amendment) relates chiefly to advances by the governors
+of Queen Anne&rsquo;s Bounty for the purposes of the act.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DILATATION<a name="ar101" id="ar101"></a></span> (from Lat. <i>dis-</i>, distributive, and <i>latus</i>, wide), a
+widening or enlarging; a term used in physiology, &amp;c.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DILATORY<a name="ar102" id="ar102"></a></span> (from Lat. <i>dilatus</i>, from <i>differre</i>, to put off or
+delay), delaying, or slow; in law a &ldquo;dilatory plea&rdquo; is one
+made merely for delaying the suit.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DILEMMA<a name="ar103" id="ar103"></a></span> (Gr. <span class="grk" title="dilêmma">&#948;&#943;&#955;&#951;&#956;&#956;&#945;</span>, a double proposition, from <span class="grk" title="di-">&#948;&#943;-</span> and
+<span class="grk" title="lambanein">&#955;&#945;&#956;&#946;&#940;&#957;&#949;&#953;&#957;</span>), a term used technically in logic, and popularly
+in common parlance and rhetoric. (1) The latter use has no
+exact definition, but in general it describes a situation wherein
+from either of two (or more) possible alternatives an unsatisfactory
+conclusion results. The alternatives are called the
+&ldquo;horns&rdquo; of the dilemma. Thus a nation which has to choose
+between bankruptcy and the repudiation of its debts is on the
+horns of a dilemma. (2) In logic there is considerable divergence
+of opinion as to the best definition. Whately defined it as &ldquo;a
+conditional syllogism with two or more antecedents in the major
+and a disjunctive minor.&rdquo; Aulus Gellius gives an example as
+follows:&mdash;&ldquo;Women are either fair or ugly; if you marry a fair
+woman, she will attract other men; if an ugly woman she will
+not please you; therefore marriage is absurd.&rdquo; From either
+alternative, an unpleasant result follows. Four kinds of dilemma
+are admitted:&mdash;(a) <i>Simple Constructive</i>: If A, then C; if B,
+then C, but either B or A; therefore C. (b) <i>Simple Destructive</i>:
+If A is true, B is true; if A is true, C is true; B and C are not both
+true; therefore A is not true. (c) <i>Complex Constructive</i>: If A,
+then B; if C, then D; but either A or C; therefore either B or D.
+(d) <i>Complex Destructive</i>: If A is true, B is true; if C is true, D is
+true; but B and D are not both true; hence A and C are not
+both true. The soundness of the dilemmatic argument in general
+depends on the alternative possibilities. Unless the alternatives
+produced exhaust the possibilities of the case, the conclusion is
+invalid. The logical form of the argument makes it especially
+valuable in public speaking, before uncritical audiences. It is, in
+fact, important rather as a rhetorical subtlety than as a serious
+argument.</p>
+
+<p><i>Dilemmist</i> is also a term used to translate <i>Vaibhashikas</i>, the
+name of a Buddhist school of philosophy.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DILETTANTE,<a name="ar104" id="ar104"></a></span> an Italian word for one who delights in the fine
+arts, especially in music and painting, so a lover of the fine arts
+in general. The Ital. <i>dilettare</i> is from Lat. <i>delectare</i>, to delight.
+Properly the word refers to an &ldquo;amateur&rdquo; as opposed to a
+&ldquo;professional&rdquo; cultivation of the arts, but like &ldquo;amateur&rdquo; it is
+often used in a depreciatory sense for one who is only a dabbler,
+or who only has a superficial knowledge or interest in art. The
+Dilettanti Society founded in 1733-1734 still exists in England.
+A history of the society, by Lionel Cust, was published in 1898.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DILIGENCE,<a name="ar105" id="ar105"></a></span> in law, the care which a person is bound to
+exercise in his relations with others. The possible degrees of
+diligence are of course numerous, and the same degree is not
+required in all cases. Thus a mere depositary would not be held
+bound to the same degree of diligence as a person borrowing an
+article for his own use and benefit. Jurists, following the divisions
+of the civil law, have concurred in fixing three approximate
+standards of diligence&mdash;viz. ordinary (<i>diligentia</i>), less than
+ordinary (<i>levissima diligentia</i>) and more than ordinary
+(<i>exactissima diligentia</i>). Ordinary or common diligence is defined
+by Story (<i>On Bailments</i>) as &ldquo;that degree of diligence which men
+in general exert in respect of their own concerns.&rdquo; So Sir William
+Jones:&mdash;&ldquo;This care, which every person of common prudence
+and capable of governing a family takes of his own concerns, is
+a proper measure of that which would uniformly be required in
+performing every contract, if there were not strong reasons for
+exacting in some of them a greater and permitting in others a less
+degree of attention&rdquo; (<i>Essay on Bailments</i>). The highest degree of
+diligence would be that which only very prudent persons bestow
+on their own concerns; the lowest, that which even careless
+persons bestow on their own concerns. The want of these various
+degrees of diligence is negligence in corresponding degrees. These
+approximations indicate roughly the greater or less severity with
+which the law will judge the performance of different classes of
+contracts; but English judges have been inclined to repudiate
+the distinction as a useless refinement of the jurists. Thus Baron
+Rolfe could see no difference between negligence and gross
+negligence; it was the same thing with the addition of a vituperative
+epithet. See <span class="sc"><a href="#artlinks">Negligence</a></span>.</p>
+
+<p><i>Diligence</i>, in Scots law, is a general term for the process by
+which persons, lands or effects are attached on execution, or in
+security for debt.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DILKE, SIR CHARLES WENTWORTH,<a name="ar106" id="ar106"></a></span> Bart. (1810-1869),
+English politician, son of Charles Wentworth Dilke, proprietor
+and editor of <i>The Athenaeum</i>, was born in London on the 18th
+of February 1810, and was educated at Westminster school and
+Trinity Hall, Cambridge. He studied law, and in 1834 took his
+degree of LL.B., but did not practise. He assisted his father in
+his literary work, and was for some years chairman of the council
+of the Society of Arts, besides taking a prominent part in the
+affairs of the Royal Horticultural Society and other bodies. He
+was one of the most zealous promoters of the Great Exhibition
+(1851), and a member of the executive committee. At the close
+of the exhibition he was honoured by foreign sovereigns, and the
+queen offered him knighthood, which, however, he did not accept;
+he also declined a large remuneration offered by the royal commission.
+In 1853 Dilke was one of the English commissioners at
+the New York Industrial Exhibition, and prepared a report on it.
+He again declined to receive any money reward for his services.
+He was appointed one of the five royal commissioners for the
+Great Exhibition of 1862; and soon after the death of the prince
+consort he was created a baronet. In 1865 he entered parliament
+as member for Wallingford. In 1869 he was sent to Russia as
+representative of England at the horticultural exhibition held
+at St Petersburg. His health, however, had been for some time
+failing, and he died suddenly in that city, on the 10th of May 1869.
+A selection from his writings, <i>Papers of a Critic</i> (2 vols., 1875),
+contains a biographical sketch by his son.</p>
+
+<p>His son, <span class="sc">Sir Charles Wentworth Dilke, Bart.</span> (1843-&emsp;&emsp;),
+became a prominent Liberal politician, as M.P. for Chelsea
+(1868-1886), under-secretary for foreign affairs (1880-1882), and
+president of the local government board (1882-1885); and he
+was then marked out as one of the best-informed and ablest of the
+advanced Radicals. He was chairman of the royal commission
+on the housing of the working classes in 1884-1885. But his
+sensational appearance as co-respondent in a divorce case of a
+peculiarly unpleasant character in 1885 cast a cloud over his
+career. He was defeated in Chelsea in 1886, and did not return
+to parliament till 1892, when he was elected for the Forest of
+Dean; and though his knowledge of foreign affairs and his
+powers as a critic and writer on military and naval questions were
+admittedly of the highest order, his official position in public life
+could not again be recovered. His military writings are <i>The
+British Army</i> (1888); <i>Army Reform</i> (1898) and, with Mr Spenser
+Wilkinson, <i>Imperial Defence</i> (1892). On colonial questions he
+wrote with equal authority. His <i>Greater Britain</i> (2 vols., 1866-1867)
+reached a fourth edition in 1868, and was followed by
+<i>Problems of Greater Britain</i> (2 vols., 1890) and <i>The British
+Empire</i> (1899). He was twice married, his second wife (née
+<span class="pagenum"><a name="page272" id="page272"></a>272</span>
+Emilia Frances Strong), the widow of Mark Pattison, being
+an accomplished art critic and collector. She died in 1904. The
+most important of her books were the studies on <i>French Painters
+of the Eighteenth Century</i> (1899) and three subsequent volumes on
+the architects and sculptors, furniture and decoration, engravers
+and draughtsmen of the same period, the last of which appeared
+in 1902. A posthumous volume, <i>The Book of the Spiritual Life</i>
+(1905), contains a memoir of her by Sir Charles Dilke.</p>
+
+
+<hr class="art" />
+<table class="nobctr" style="float: right; width: 360px;" summary="Illustration">
+<tr><td class="figright1"><img style="width:310px; height:260px" src="images/img272.jpg" alt="" /></td></tr>
+<tr><td class="caption1"> Dill (<i>Anethum</i> or <i>Peucedanum graveolens</i>),
+leaf and inflorescence.</td></tr></table>
+
+<p><span class="bold">DILL<a name="ar107" id="ar107"></a></span> (<i>Anethum</i> or <i>Peucedanum graveolens</i>), a member of the
+natural botanical order Umbelliferae, indigenous to the south of
+Europe, Egypt and the Cape of Good Hope. It resembles fennel
+in appearance. Its root is long and fusiform; the stem is round,
+jointed and about a yard high; the leaves have fragrant leaflets;
+and the fruits are brown,
+oval and concavo-convex.
+The plant flowers
+from June till August in
+England. The seeds are
+sown, preferably as soon
+as ripe, either broadcast
+or in drills between
+6 and 12 in. asunder.
+The young plants should
+be thinned when 3 or 4
+weeks old, so as to be
+at distances of about
+10 in. A sheltered spot
+and dry soil are needed
+for the production of the
+seed in the climate of England. The leaves of the dill are used in
+soups and sauces, and, as well as the umbels, for flavouring
+pickles. The seeds are employed for the preparation of dill-water
+and oil of dill; they are largely consumed in the manufacture of
+gin, and, when ground, are eaten in the East as a condiment.
+The British Pharmacopoeia contains the Aqua Anethi or dill-water
+(dose 1-2 oz.), and the Oleum Anethi, almost identical in
+composition with caraway oil, and given in doses of ½-3 minims.
+Dill-water is largely used as a carminative for children, and as a
+vehicle for the exhibition of nauseous drugs.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DILLEN<a name="ar108" id="ar108"></a></span> [<span class="sc">Dillenius</span>], <b>JOHANN JAKOB</b> (1684-1747), English
+botanist, was born at Darmstadt in 1684, and was educated at the
+university of Giessen, where he wrote several botanical papers for
+the <i>Ephemerides naturae curiosorum</i>, and printed, in 1719, his
+<i>Catalogus plantarum sponte circa Gissam nascentium</i>, illustrated
+with figures drawn and engraved by his own hand, and containing
+descriptions of many new species. In 1721, at the instance of the
+botanist William Sherard (1659-1728), he came to England, and
+in 1724 he published a new edition of Ray&rsquo;s <i>Synopsis stirpium
+Britannicarum</i>. In 1732 he published <i>Hortus Elthamensis</i>, a
+catalogue of the rare plants growing at Eltham, Kent, in the
+collection of Sherard&rsquo;s younger brother, James (1666-1738), who,
+after making a fortune as an apothecary, devoted himself to
+gardening and music. For this work Dillen himself executed 324
+plates, and it was described by Linnaeus, who spent a month
+with him at Oxford in 1736, and afterwards dedicated his <i>Critica
+botanica</i> to him, as &ldquo;opus botanicum quo absolutius mundus non
+vidit.&rdquo; In 1734 he was appointed Sherardian professor of botany
+at Oxford, in accordance with the will of W. Sherard, who at his
+death in 1728 left the university £3000 for the endowment of the
+chair, as well as his library and herbarium. Dillen, who was also
+the author of an <i>Historia muscorum</i> (1741), died at Oxford, of
+apoplexy, on the 2nd of April 1747. His manuscripts, books and
+collections of dried plants, with many drawings, were bought by
+his successor at Oxford, Dr Humphry Sibthorp (1713-1797), and
+ultimately passed into the possession of the university.</p>
+
+<div class="condensed">
+<p>For an account of his collections preserved at Oxford, see <i>The
+Dillenian Herbaria</i>, by G. Claridge Druce (Oxford, 1907).</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DILLENBURG,<a name="ar109" id="ar109"></a></span> a town of Germany, in the Prussian province of
+Hesse-Nassau, delightfully situated in the midst of a well-wooded
+country, on the Dill, 25 m. N.W. from. Giessen on the railway to
+Troisdorf. Pop. 4500. On an eminence above it lie the ruins
+of the castle of Dillenburg, founded by Count Henry the Rich
+of Nassau, about the year 1255, and the birthplace of Prince
+William of Orange (1533). It has an Evangelical church, with
+the vault of the princes of Nassau-Dillenburg, a Roman Catholic
+church, a classical school, a teachers&rsquo; seminary and a chamber
+of commerce. Its industries embrace iron-works, tanneries and
+the manufacture of cigars. Owing to its beautiful surroundings
+Dillenburg has become a favourite summer resort.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DILLENS, JULIEN<a name="ar110" id="ar110"></a></span> (1849-1904), Belgian sculptor, was born at
+Antwerp on the 8th of June 1849, son of a painter. He studied
+under Eugène Simonis at the Brussels Academy of Fine Arts.
+In 1877 he received the <i>prix de Rome</i> for &ldquo;A Gaulish Chief taken
+Prisoner by the Romans.&rdquo; At Brussels, in 1881, he executed
+the groups entitled &ldquo;Justice&rdquo; and &ldquo;Herkenbald, the Brussels
+Brutus.&rdquo; For the pediment of the orphanage at Uccle, &ldquo;Figure
+Kneeling&rdquo; (Brussels Gallery), and the statue of the lawyer
+Metdepenningen in front of the Palais de Justice at Ghent, he was
+awarded the medal of honour in 1889 at the Paris Universal
+Exhibition, where, in 1900, his &ldquo;Two Statues of the Anspach
+Monument&rdquo; gained him a similar distinction. For the town of
+Brussels he executed &ldquo;The Four Continents&rdquo; (Maison du Renard,
+Grand&rsquo; Place), &ldquo;The Lansquenets&rdquo; crowning the lucarnes of
+the Maison de Roi, and the &ldquo;Monument t&rsquo; Serclaes&rdquo; under the
+arcades of the Maison de l&rsquo;Etoile, and, for the Belgian government,
+&ldquo;Flemish Art,&rdquo; &ldquo;German Art,&rdquo; &ldquo;Classic Art&rdquo; and &ldquo;Art
+applied to Industry&rdquo; (all in the Palais des Beaux Arts, Brussels),
+&ldquo;The Laurel&rdquo; (Botanic Garden, Brussels), and the statue of
+&ldquo;Bernard van Orley&rdquo; (Place du petit Sablon, Brussels). Mention
+must also be made of &ldquo;An Enigma&rdquo; (1876), the bronze busts of
+&ldquo;Rogier de la Pasture&rdquo; and &ldquo;P. P. Rubens&rdquo; (1879), &ldquo;Etruria&rdquo;
+(1880), &ldquo;The Painter Leon Frederic&rdquo; (1888), &ldquo;Madame Leon
+Herbo,&rdquo; &ldquo;Hermes,&rdquo; a scheme of decoration for the ogival façade
+of the hôtel de ville at Ghent (1893), &ldquo;The Genius of the Funeral
+Monument of the Moselli Family,&rdquo; &ldquo;The Silence of Death&rdquo; (for
+the entrance of the cemetery of St Gilles), two caryatides for the
+town hall of St Gilles, presentation plaquette to Dr Heger, medals
+of MM. Godefroid and Vanderkindere and of &ldquo;The Three
+Burgomasters of Brussels,&rdquo; and the ivories &ldquo;Allegretto,&rdquo;
+&ldquo;Minerva&rdquo; and the &ldquo;Jamaer Memorial.&rdquo; Dillens died at
+Brussels in November 1904.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DILLINGEN,<a name="ar111" id="ar111"></a></span> a town of Germany, in the kingdom of Bavaria,
+on the left bank of the Danube, 25 m. N.E. from Ulm, on the
+railway to Ingolstadt. Pop. (1905) 6078. Its principal buildings
+are an old palace, formerly the residence of the bishops of
+Augsburg and now government offices, a royal gymnasium, a
+Latin school with a library of 75,000 volumes, seven churches
+(six Roman Catholic), two episcopal seminaries, a Capuchin
+monastery, a Franciscan convent and a deaf and dumb asylum.
+The university, founded in 1549, was abolished in 1804, being
+converted into a lyceum. The inhabitants are engaged in cattle-rearing,
+the cultivation of corn, hops and fruit, shipbuilding and
+the shipping trade, and the manufacture of cloth, paper and
+cutlery. In the vicinity is the Karolinen canal, which cuts off a
+bend in the Danube between Lauingen and Dillingen. In 1488
+Dillingen became the residence of the bishops of Augsburg; was
+taken by the Swedes in 1632 and 1648, by the Austrians in 1702,
+and on the 17th of June 1800 by the French. In 1803 it passed
+to Bavaria.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DILLMANN, CHRISTIAN FRIEDRICH AUGUST<a name="ar112" id="ar112"></a></span> (1823-1894),
+German orientalist and biblical scholar, the son of a Württemberg
+schoolmaster, was born at Illingen on the 25th of April 1823. He
+was educated at Tübingen, where he became a pupil and friend of
+Heinrich Ewald, and studied under F. C. Baur, though he did not
+join the new Tübingen school. For a short time he worked as
+pastor at Gersheim, near his native place, but he soon came to
+feel that his studies demanded his whole time. He devoted himself
+to the study of Ethiopic MSS. in the libraries of Paris, London
+and Oxford, and this work caused a revival of Ethiopic study in
+the 19th century. In 1847 and 1848 he prepared catalogues of
+the Ethiopic MSS. in the British Museum and the Bodleian
+library at Oxford. He then set to work upon an edition of the
+Ethiopic bible. Returning to Tübingen in 1848, in 1853 he was
+appointed professor extraordinarius. Subsequently he became
+<span class="pagenum"><a name="page273" id="page273"></a>273</span>
+professor of philosophy at Kiel (1854), and of theology at Giessen
+(1864) and Berlin (1869). He died on the 4th of July 1894.</p>
+
+<p>In 1851 he had published the <i>Book of Enoch</i> in Ethiopian
+(German, 1853), and at Kiel he completed the first part of the
+Ethiopic bible, <i>Octateuchus Aethiopicus</i> (1853-1855). In 1857
+appeared his <i>Grammatik der äthiopischen Sprache</i> (2nd ed. by
+C. Bezold, 1899); in 1859 the <i>Book of Jubilees</i>; in 1861 and 1871
+another part of the Ethiopic bible, <i>Libri Regum</i>; in 1865 his
+great <i>Lexicon linguae aethiopicae</i>; in 1866 his <i>Chrestomathia
+aethiopica</i>. Always a theologian at heart, however, he returned
+to theology in 1864. His Giessen lectures were published under
+the titles, <i>Ursprung der alttestamentlichen Religion</i> (1865) and
+<i>Die Propheten des alten Bundes nach ihrer politischen Wirksamkeit</i>
+(1868). In 1869 appeared his <i>Commentar zum Hiob</i> (4th ed. 1891)
+which stamped him as one of the foremost Old Testament
+exegetes. His renown as a theologian, however, was mainly
+founded by the series of commentaries, based on those of August
+Wilhelm Knobels&rsquo; <i>Genesis</i> (Leipzig, 1875; 6th ed. 1892; Eng.
+trans, by W. B. Stevenson, Edinburgh, 1897); <i>Exodus und
+Leviticus</i>, 1880, revised edition by V. Ryssel, 1897; <i>Numeri,
+Deuteronomium und Josua</i>, with a dissertation on the origin of
+the Hexateuch, 1886; <i>Jesaja</i>, 1890 (revised edition by Rudolf
+Kittel in 1898). In 1877 he published the <i>Ascension of Isaiah</i>
+in Ethiopian and Latin. He was also a contributor to D.
+Schenkel&rsquo;s <i>Bibellexikon</i>, Brockhaus&rsquo;s <i>Conversationslexikon</i>, and
+Herzog&rsquo;s <i>Realencyklopädie</i>. His lectures on Old Testament
+theology, <i>Vorlesungen über Theologie des Allen Testamentes</i>, were
+published by Kittel in 1895.</p>
+
+<div class="condensed">
+<p>See the articles in Herzog-Hauck, <i>Realencyklopädie</i>, and the
+<i>Allgemeine deutsche Biographie</i>; F. Lichtenberger, <i>History of
+German Theology in the Nineteenth Century</i> (1889); Wolf Baudissin,
+<i>A. Dillmann</i> (Leipzig, 1895).</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DILLON, ARTHUR RICHARD<a name="ar113" id="ar113"></a></span> (1721-1807), French archbishop,
+was the son of Arthur Dillon (1670-1733), an Irish
+gentleman who became general in the French service. He was
+born at St Germain, entered the priesthood and was successively
+curé of Elan near Mezières, vicar-general of Pontoise (1747),
+bishop of Evreux (1753) and archbishop of Toulouse (1758),
+archbishop of Narbonne in 1763, and in that capacity, president
+of the estates of Languedoc. He devoted himself much less to
+the spiritual direction of his diocese than to its temporal welfare,
+carrying out many works of public utility, bridges, canals, roads,
+harbours, &amp;c.; had chairs of chemistry and of physics created at
+Montpellier and at Toulouse, and tried to reduce the poverty,
+especially in Narbonne. In 1787 and in 1788 he was a member of
+the Assembly of Notables called together by Louis XVI., and in
+1788 presided over the assembly of the clergy. Having refused
+to accept the civil constitution of the clergy, Dillon had to leave
+Narbonne in 1790, then to emigrate to Coblenz in 1791. Soon
+afterwards he went to London, where he lived until his death in
+1807, never accepting the Concordat, which had suppressed his
+archiepiscopal see.</p>
+
+<div class="condensed">
+<p>See L. Audibret, <i>Le Dernier Président des États du Languedoc, Mgr.
+Arthur Richard Dillon, archevêque de Narbonne</i> (Bordeaux, 1868);
+L. de Lavergne, <i>Les Assemblées provinciales sous Louis XVI</i>
+(Paris, 1864).</p>
+</div>
+
+
+<hr class="art" />
+<p><span class="bold">DILLON, JOHN<a name="ar114" id="ar114"></a></span> (1851-&emsp;&emsp;), Irish nationalist politician, was
+the son of John Blake Dillon (1816-1866), who sat in parliament
+for Tipperary, and was one of the leaders of &ldquo;Young Ireland.&rdquo;
+John Dillon was educated at the Roman Catholic university of
+Dublin, and afterwards studied medicine. He entered parliament
+in 1880 as member for Tipperary, and was at first an ardent
+supporter of C. S. Parnell. In August he delivered a speech on
+the Land League at Kildare which was characterized as &ldquo;wicked
+and cowardly&rdquo; by W. E. Forster; he advocated boycotting, and
+was arrested in May 1881 under the Coercion Act, and again after
+two months of freedom in October. In 1883 he resigned his seat
+for reasons of health, but was returned unopposed in 1885 for
+East Mayo, which he continued to represent. He was one of the
+prime movers in the famous &ldquo;plan of campaign,&rdquo; which provided
+that the tenant should pay his rent to the National League instead
+of the landlord, and in case of eviction be supported by the general
+fund. Mr Dillon was compelled by the court of queen&rsquo;s bench on
+the 14th of December 1886 to find securities for good behaviour,
+but two days later he was arrested while receiving rents on Lord
+Clanricarde&rsquo;s estates. In this instance the jury disagreed, but
+in June 1888 under the provisions of the new Criminal Law
+Procedure Bill he was condemned to six months&rsquo; imprisonment.
+He was, however, released in September, and in the spring of 1889
+sailed for Australia and New Zealand, where he collected funds
+for the Nationalist party. On his return to Ireland he was again
+arrested, but, being allowed bail, sailed to America, and failed to
+appear at the trial. He returned to Ireland by way of Boulogne,
+where he and Mr W. O&rsquo;Brien held long and indecisive conferences
+with Parnell. They surrendered to the police in February, and
+on their release from Galway gaol in July declared their opposition
+to Parnell. After the expulsion of Mr T. M. Healy and others
+from the Irish National Federation, Mr Dillon became the chairman
+(February 1896). His early friendship with Mr O&rsquo;Brien
+gave place to considerable hostility, but the various sections of
+the party were ostensibly reconciled in 1900 under the leadership
+of Mr Redmond. In the autumn of 1896 he arranged a convention
+of the Irish race, which included 2000 delegates from various
+parts of the world. In 1897 Mr Dillon opposed in the House
+the Address to Queen Victoria on the occasion of the Diamond
+Jubilee, on the ground that her reign had not been a blessing to
+Ireland, and he showed the same uncompromising attitude in
+1901 when a grant to Lord Roberts was under discussion, accusing
+him of &ldquo;systematized inhumanity.&rdquo; He was suspended on the
+20th of March for violent language addressed to Mr Chamberlain.
+He married in 1895 Elizabeth (d. 1907), daughter of Lord justice
+J. C. Mathew.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DILUVIUM<a name="ar115" id="ar115"></a></span> (Lat. for &ldquo;deluge,&rdquo; from <i>diluere</i>, to wash away),
+a term in geology for superficial deposits formed by flood-like
+operations of water, and so contrasted with alluvium (<i>q.v.</i>) or
+alluvial deposits formed by slow and steady aqueous agencies.
+The term was formerly given to the &ldquo;boulder clay&rdquo; deposits,
+supposed to have been caused by the Noachian deluge.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIME<a name="ar116" id="ar116"></a></span> (from the Lat. <i>decima</i>, a tenth, through the O. Fr.
+<i>disme</i>), the tenth part, the tithe paid as church dues, or as tribute
+to a temporal power. In this sense it is obsolete, but is found in
+Wycliffe&rsquo;s translation of the Bible&mdash;&ldquo;He gave him dymes of alle
+thingis&rdquo; (Gen. xiv. 20). A dime is a silver coin of the United
+States, in value 10 cents (English equivalent about 5d.) or one-tenth
+of a dollar; hence &ldquo;dime-novel,&rdquo; a cheap sensational
+novel, a &ldquo;penny dreadful&rdquo;; also &ldquo;dime-museum.&rdquo;</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIMENSION<a name="ar117" id="ar117"></a></span> (from Lat. <i>dimensio</i>, a measuring), in geometry, a
+magnitude measured in a specified direction, <i>i.e.</i> length, breadth
+and thickness; thus a line has only length and is said to be of
+one dimension, a surface has length and breadth, and has two
+dimensions, a solid has length, breadth and thickness, and has
+three dimensions. This concept is extended to algebra: since
+a line, surface and solid are represented by linear, quadratic and
+cubic equations, and are of one, two and three dimensions; a
+biquadratic equation has its highest terms of four dimensions,
+and, in general, an equation in any number of variables which has
+the greatest sum of the indices of any term equal to n is said to
+have n dimensions. The &ldquo;fourth dimension&rdquo; is a type of non-Euclidean
+geometry, in which it is conceived that a &ldquo;solid&rdquo; has
+one dimension more than the solids of experience. For the
+dimensions of units see <span class="sc"><a href="#artlinks">Units, Dimensions of</a></span>.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DIMITY,<a name="ar118" id="ar118"></a></span> derived from the Gr. <span class="grk" title="dimitos">&#948;&#943;&#956;&#953;&#964;&#959;&#962;</span> &ldquo;double thread,&rdquo;
+through the Ital. <i>dimito</i>, &ldquo;a kind of course linzie-wolzie&rdquo;
+(Florio, 1611); a cloth commonly employed for bed upholstery
+and curtains, and usually white, though sometimes a pattern is
+printed on it in colours. It is stout in texture, and woven in
+raised patterns.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DINAJPUR,<a name="ar119" id="ar119"></a></span> a town (with a population in 1901 of 13,430) and
+district of <span class="correction" title="amended from Britsh">British</span> India, in the Rajshahi division of Eastern
+Bengal and Assam. The earthquake of the 12th of June 1897
+caused serious damage to most of the public buildings of the town.
+There is a railway station and a government high school. The
+district comprises an area of 3946 sq. m. It is traversed in every
+direction by a network of channels and water courses. Along the
+banks of the Kulik river, the undulating ridges and long lines of
+<span class="pagenum"><a name="page274" id="page274"></a>274</span>
+mango-trees give the landscape a beauty which is not found elsewhere.
+Dinajpur forms part of the rich arable tract lying between
+the Ganges and the southern slopes of the Himalayas. Although
+essentially a fluvial district, it does not possess any river navigable
+throughout the year by boats of 4 tons burden. Rice forms the
+staple agricultural product. The climate of the district, although
+cooler than that of Calcutta, is very unhealthy, and the people
+have a sickly appearance. The worst part of the year is at the
+close of the rains in September and October, during which months
+few of the natives escape fever. The average maximum temperature
+is 92.3° F., and the minimum 74.8°. The average rainfall
+is 85.54 in. In 1901 the population was 1,567,080, showing an
+increase of 6% in the decade. The district is partly traversed
+by the main line of the Eastern Bengal railway and by two branch
+lines. Save between 1404 and 1442, when it was the seat of
+an independent <i>raj</i>, founded by Raja Ganesh, a Hindu turned
+Mussulman, Dinajpur has no separate history. Pillars and
+copper-plate inscriptions have yielded numerous records of the
+Pal kings who ruled the country from the 9th century onwards,
+and the district is famous for many other antiquities, some of
+which are connected by legend with an immemorial past (see
+<i>Reports, Arch. Survey of India</i>, xv.; <i>Epigraphia Indica</i>, ii.).</p>
+
+
+<hr class="art" />
+<p><span class="bold">DINAN,<a name="ar120" id="ar120"></a></span> a town of north-western France, capital of an
+arrondissement in the department of Côtes-du-Nord, 37 m. E. of
+St Brieuc on the Western railway. Pop. (1906) 8588. Dinan is
+situated on a height on the left bank of the Ranee (here canalized),
+some 17 m. above its mouth at St Malo, with which it communicates
+by means of small steamers. It is united to the village
+of Lanvallay on the right bank of the river by a granite viaduct
+130 ft. in height. The town is almost entirely encircled by the
+ramparts of the middle ages, strengthened at intervals by towers
+and defended on the south by a castle of the late 14th century,
+which now serves as prison. Three old gateways are also preserved.
+Dinan has two interesting churches; that of St Malo, of
+late Gothic architecture, and St Sauveur, in which the Romanesque
+and Gothic styles are intermingled. In the latter church a
+granite monument contains the heart of Bertrand Du Guesclin,
+whose connexion with the town is also commemorated by an
+equestrian statue. The quaint winding streets of Dinan are often
+bordered by medieval houses. Its picturesqueness attracts large
+numbers of visitors and there are many English residents in the
+town and its vicinity. About three-quarters of a mile from the
+town are the ruins of the château and the Benedictine abbey at
+Léhon; near the neighbouring village of St Esprit stands the
+large lunatic asylum of Les Bas Foins, founded in 1836; and at
+no great distance is the now dismantled château of La Garaye,
+which was rendered famous in the 18th century by the philanthropic
+devotion of the count and countess whose story is told
+in Mrs Norton&rsquo;s <i>Lady of La Garaye</i>. Dinan is the seat of a subprefect
+and has a tribunal of first instance, and a communal
+college. There is trade in grain, cider, wax, butter and other
+agricultural products. The industries include the manufacture
+of leather, farm-implements and canvas.</p>
+
+<p>The principal event in the history of Dinan, which was a stronghold
+of the dukes of Brittany, is the siege by the English under the
+duke of Lancaster in 1359, during which Du Guesclin and an
+English knight called Thomas of Canterbury engaged in single
+combat.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DINANT,<a name="ar121" id="ar121"></a></span> an ancient town on the right bank of the Meuse in
+the province of Namur, Belgium, connected by a bridge with the
+left bank, on which are the station and the suburb of St Medard.
+Pop. (1904) 7674. The name is supposed to be derived from
+Diana, and as early as the 7th century it was named as one of the
+dependencies of the bishopric of Tongres. In the 10th century it
+passed under the titular sway of Liége, and remained the fief of the
+prince-bishopric till the French revolution put an end to that
+survival of feudalism. In the middle of the 15th century Dinant
+reached the height of its prosperity. With a population of
+60,000, and 8000 workers in copper, it was one of the most
+flourishing cities in Walloon Belgium, until it incurred the wrath
+of Charles the Bold. Belief in the strength of its walls and of the
+castle that occupied the centre bridge, thus effectually commanding
+navigation by the river, engendered arrogance and overconfidence,
+and the people of Dinant thought they could defy the
+full power of Burgundy. Perhaps they also expected aid from France or Liége. In 1466 Charles, in his father&rsquo;s name, laid siege
+to Dinant, and on the 27th of August carried the place by storm.
+He razed the walls and allowed the women, children and priests
+to retire in safety to Liége, but the male prisoners he either
+hanged or drowned in the river by causing them to be cast from
+the projecting cliff of Bouvignes. In 1675 the capture of Dinant
+formed one of the early military achievements of Louis XIV., and
+it remained in the hands of the French for nearly thirty years
+after that date. The citadel on the cliff, 300 ft. or 408 steps above
+the town, was fortified by the Dutch in 1818. It is now dismantled,
+but forms the chief curiosity of the place. The views
+of the river valley from this eminence are exceedingly fine. Half
+way up the cliff, but some distance south of the citadel, is the
+grotto of Montfat, alleged to be the site of Diana&rsquo;s shrine. The
+church of Notre Dame, dating from the 13th century, stands
+immediately under the citadel and flanking the bridge. It has
+been restored, and is considered by some authorities, although
+others make the same claim on behalf of Huy, the most complete
+specimen in Belgium of pointed Gothic architecture. The
+baptismal fonts date from the 12th century, and the curious spire
+in the form of an elongated pumpkin and covered with slates
+gives a fantastic and original appearance to the whole edifice.
+The present prosperity of Dinant is chiefly derived from its being
+a favourite summer resort for Belgians as well as foreigners. It
+has facilities for beating and bathing as well as for trips by
+steamer up and down the river Meuse. It is also a convenient
+central point for excursions into the Ardennes. Although there
+are some indications of increased industrial activity in recent
+years, the population of Dinant is not one-eighth of what it was
+at the time of the Burgundians.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DINAPUR,<a name="ar122" id="ar122"></a></span> a town and military station of British India, in the
+Patna district of Bengal, on the right bank of the Ganges, 12 m.
+W. of Patna city by rail. Pop. (1901) 33,699. It is the largest
+military cantonment in Bengal, with accommodation for two
+batteries of artillery, a European and a native infantry regiment.
+In 1857 the sepoy garrison of the place initiated the mutiny of
+that year in Patna district, but after a conflict with the European
+troops were forced to retire from the town, and subsequently laid
+siege to Arrah.</p>
+
+
+<hr class="art" />
+<p><span class="bold">DINARCHUS,<a name="ar123" id="ar123"></a></span> last of the &ldquo;ten&rdquo; Attic orators, son of Sostratus
+(or, according to Suidas, Socrates), born at Corinth about 361
+<span class="sc">b.c.</span> He settled at Athens early in life, and when not more than
+twenty-five was already active as a writer of speeches for the law
+courts. As an alien, he was unable to take part in the debates.
+He had been the pupil both of Theophrastus and of Demetrius
+Phalereus, and had early acquired a certain fluency and versatility
+of style. In 324 the Areopagus, after inquiry, reported
+that nine men had taken bribes from Harpalus, the fugitive
+treasurer of Alexander. Ten public prosecutors were appointed.
+Dinarchus wrote, for one or more of these prosecutors, the three
+speeches which are still extant&mdash;<i>Against Demosthenes</i>, <i>Against
+Aristogeiton</i>, <i>Against Philocles</i>. The sympathies of Dinarchus
+were in favour of an Athenian oligarchy under Macedonian
+control; but it should be remembered that he was not an
+Athenian citizen. Aeschines and Demades had no such excuse.
+In the Harpalus affair, Demosthenes was doubtless innocent,
+and so, probably, were others of the accused. Yet Hypereides,
+the most fiery of the patriots, was on the same side as Dinarchus.</p>
+
+<p>Under the regency of his old master, Demetrius Phalereus,
+Dinarchus exercised much political influence. The years 317-307
+were the most prosperous of his life. On the fall of Demetrius
+Phalereus and the restoration of the democracy by Demetrius
+Poliorcetes, Dinarchus was condemned to death and withdrew
+into exile at Chalcis in Euboea. About 292, thanks to his friend
+Theophrastus, he was able to return to Attica, and took up his
+abode in the country with a former associate, Proxenus. He
+afterwards brought an action against Proxenus on the ground
+that he had robbed him of some money and plate. Dinarchus
+died at Athens about 291.</p>
+
+<p>According to Suidas, Dinarchus wrote 160 speeches; and Dionysius held
+that, out of 85 extant speeches bearing his name, 58 were genuine,&mdash;28
+relating to public, 30 to private causes. Although the authenticity of
+the three speeches mentioned above is generally admitted, Demetrius of
+Magnesia doubted that of the speech <i>Against Demosthenes</i>, while A.
+Westermann rejected all three. Dinarchus had little individual style and
+imitated by turns Lysias, Hypereides and Demosthenes. He is called by
+Hermogenes <span class="grk" title="o critinos demostenes">&#8001; &#954;&#961;&#953;&#952;&#953;&#957;&#8056;&#962; &#916;&#951;&#956;&#959;&#963;&#952;&#941;&#957;&#951;&#962;</span>, a metaphor taken from barley compared
+with wheat, or beer compared with wine,&mdash;a Demosthenes whose strength
+is rougher, without flavour or sparkle.</p>
+
+<div class="condensed">
+<p>Editions: (text and exhaustive commentary) E. Mätzner (1842); (text) T.
+Thalheim (1887), F. Blass (1888); see L.L. Forman, <i>Index Andocideus,
+Lycurgeus, Dinarcheus</i> (1897); and, in general, F. Blass, <i>Attische
+Beredsamkeit</i>, iii. There is a valuable treatise on the life and speeches
+of Dinarchus by Dionysius of Halicarnassus.</p>
+</div>
+
+
+<hr class="art" />
+
+
+
+
+
+
+
+
+
+<pre>
+
+
+
+
+
+End of the Project Gutenberg EBook of Encyclopaedia Britannica, 11th
+Edition, Volume 8, Slice 4, by Various
+
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+</body>
+</html>
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@@ -0,0 +1,20493 @@
+The Project Gutenberg EBook of Encyclopaedia Britannica, 11th Edition,
+Volume 8, Slice 4, by Various
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+Title: Encyclopaedia Britannica, 11th Edition, Volume 8, Slice 4
+       "Diameter" to "Dinarchus"
+
+Author: Various
+
+Release Date: May 30, 2010 [EBook #32607]
+
+Language: English
+
+Character set encoding: ASCII
+
+*** START OF THIS PROJECT GUTENBERG EBOOK ENCYC. BRITANNICA, VOL 8 SL 4 ***
+
+
+
+
+Produced by Marius Masi, Don Kretz and the Online
+Distributed Proofreading Team at https://www.pgdp.net
+
+
+
+
+
+
+
+
+
+Transcriber's notes:
+
+(1) Numbers following letters (without space) like C2 were originally
+      printed in subscript. Letter subscripts are preceded by an
+      underscore, like C_n.
+
+(2) Characters following a carat (^) were printed in superscript.
+
+(3) Side-notes were relocated to function as titles of their respective
+      paragraphs.
+
+(4) Letters topped by Macron are represented as [=x].
+
+(5) dP stands for the partial-derivative symbol, or curled 'd'.
+
+(6) [oo] stands for the infinity symbol, and [int] for the integral
+       symbol.
+
+(7) The following typographical errors have been corrected:
+
+    Article DIAMOND: "If this be so the form of the diamond is really
+      the tetrahedron (and the various figures derived symmetrically from
+      it) and not the octahedron". 'octahedron' amended from
+      'octadehron'.
+
+    Article DIARY: "diaries began to be largely written in England,
+      although in most cases without any idea of even eventual
+      publication". 'largely' amended from 'largly'.
+
+    Article DICOTYLEDONS: "The arrangement of the conducting tissue in
+      the stem is characteristic; a transverse section of the very young
+      stem shows a number of distinct conducting strands". 'number'
+      amended from 'nunber'.
+
+    Article DIEKIRCH: "It remained more or less fortified until the
+      beginning of the 19th century when the French during their
+      occupation levelled the old walls, and substituted the avenues of
+      trees that now encircle the town". 'or' amended from 'for'.
+
+    Article DIFFERENCES, CALCULUS OF: "as the second difference of u_n,
+      and therefore as corresponding to the value x_n"; 'difference'
+      amended from 'dfference'.
+
+    Article DINAJPUR: "a town (with a population in 1901 of 13,430) and
+     district of British India, in the Rajshahi division of Eastern
+     Bengal and Assam". 'British' amended from 'Britsh'.
+
+
+
+
+          ENCYCLOPAEDIA BRITANNICA
+
+  A DICTIONARY OF ARTS, SCIENCES, LITERATURE
+           AND GENERAL INFORMATION
+
+              ELEVENTH EDITION
+
+
+           VOLUME VIII, SLICE IV
+
+           Diameter to Dinarchus
+
+
+
+
+ARTICLES IN THIS SLICE:
+
+
+  DIAMETER                             DIEDENHOFEN
+  DIAMOND                              DIEKIRCH
+  DIAMOND NECKLACE, THE AFFAIR OF THE  DIELECTRIC
+  DIANA                                DIELMANN, FREDERICK
+  DIANA MONKEY                         DIEMEN, ANTHONY VAN
+  DIANE DE FRANCE                      DIEPENBECK, ABRAHAM VAN
+  DIANE DE POITIERS                    DIEPPE
+  DIAPASON                             DIERX, LEON
+  DIAPER                               DIES, CHRISTOPH ALBERT
+  DIAPHORETICS                         DIEST
+  DIAPHRAGM                            DIESTERWEG, FRIEDRICH ADOLF WILHELM
+  DIARBEKR                             DIET
+  DIARRHOEA                            DIETARY
+  DIARY                                DIETETICS
+  DIASPORE                             DIETRICH, CHRISTIAN WILHELM ERNST
+  DIASTYLE                             DIETRICH OF BERN
+  DIATOMACEAE                          DIEZ, FRIEDRICH CHRISTIAN
+  DIAULOS                              DIEZ
+  DIAVOLO, FRA                         DIFFERENCES, CALCULUS OF
+  DIAZ, NARCISSE VIRGILIO              DIFFERENTIAL EQUATION
+  DIAZ, PORFIRIO                       DIFFLUGIA
+  DIAZ DE NOVAES, BARTHOLOMEU          DIFFRACTION OF LIGHT
+  DIAZO COMPOUNDS                      DIFFUSION
+  DIAZOMATA                            DIGBY, SIR EVERARD
+  DIBDIN, CHARLES                      DIGBY, SIR KENELM
+  DIBDIN, THOMAS FROGNALL              DIGBY, KENELM HENRY
+  DIBDIN, THOMAS JOHN                  DIGENES ACRITAS, BASILIUS
+  DIBRA                                DIGEST
+  DIBRUGARH                            DIGESTIVE ORGANS
+  DICAEARCHUS                          DIGGES, WEST
+  DICE                                 DIGIT
+  DICETO, RALPH DE                     DIGITALIS
+  DICEY, EDWARD                        DIGNE
+  DICHOTOMY                            DIGOIN
+  DICK, ROBERT                         DIJON
+  DICK, THOMAS                         DIKE
+  DICKENS, CHARLES JOHN HUFFAM         DIKKA
+  DICKINSON, ANNA ELIZABETH            DILAPIDATION
+  DICKINSON, JOHN                      DILATATION
+  DICKSON, SIR ALEXANDER               DILATORY
+  DICKSON, SIR JAMES ROBERT            DILEMMA
+  DICOTYLEDONS                         DILETTANTE
+  DICTATOR                             DILIGENCE
+  DICTIONARY                           DILKE, SIR CHARLES WENTWORTH
+  DICTYOGENS                           DILL
+  DICTYS CRETENSIS                     DILLEN, JOHANN JAKOB
+  DICUIL                               DILLENBURG
+  DIDACHE, THE                         DILLENS, JULIEN
+  DIDACTIC POETRY                      DILLINGEN
+  DIDEROT, DENIS                       DILLMANN, CHRISTIAN AUGUST
+  DIDIUS SALVIUS JULIANUS, MARCUS      DILLON, ARTHUR RICHARD
+  DIDO                                 DILLON, JOHN
+  DIDON, HENRI                         DILUVIUM
+  DIDOT                                DIME
+  DIDRON, ADOLPHE NAPOLEON             DIMENSION
+  DIDYMI                               DIMITY
+  DIDYMIUM                             DINAJPUR
+  DIDYMUS                              DINAN
+  DIDYMUS CHALCENTERUS                 DINANT
+  DIE (town of France)                 DINAPUR
+  DIE (datum)                          DINARCHUS
+  DIEBITSCH, HANS KARL FRIEDRICH ANTON
+
+
+
+
+DIAMETER (from the Gr. [Greek: dia], through, [Greek: metron],
+measure), in geometry, a line passing through the centre of a circle or
+conic section and terminated by the curve; the "principal diameters" of
+the ellipse and hyperbola coincide with the "axes" and are at right
+angles; "conjugate diameters" are such that each bisects chords parallel
+to the other. The diameter of a quadric surface is a line at the
+extremities of which the tangent planes are parallel. Newton defined the
+diameter of a curve of any order as the locus of the centres of the mean
+distances of the points of intersection of a system of parallel chords
+with the curve; this locus may be shown to be a straight line. The word
+is also used as a unit of linear measurement of the magnifying power of
+a lens or microscope.
+
+In architecture, the term is used to express the measure of the lower
+part of the shaft of a column. It is employed by Vitruvius (iii. 2) to
+determine the height of a column, which should vary from eight to ten
+diameters according to the intercolumniation: and it is generally the
+custom to fix the lower diameter of the shaft by the height required and
+the Order employed. Thus the diameter of the Roman Doric should be about
+one-eighth of the height, that of the Ionic one-ninth, and of the
+Corinthian one-tenth (see ORDER).
+
+
+
+
+DIAMOND, a mineral universally recognized as chief among precious
+stones; it is the hardest, the most imperishable, and also the most
+brilliant of minerals.[1] These qualities alone have made it supreme as
+a jewel since early times, and yet the real brilliancy of the stone is
+not displayed until it has been faceted by the art of the lapidary
+(q.v.); and this was scarcely developed before the year 1746. The
+consummate hardness of the diamond, in spite of its high price, has made
+it most useful for purposes of grinding, polishing and drilling.
+Numerous attempts have been made to manufacture the diamond by
+artificial means, and these attempts have a high scientific interest on
+account of the mystery which surrounds the natural origin of this
+remarkable mineral. Its physical and chemical properties have been the
+subject of much study, and have a special interest in view of the
+extraordinary difference between the physical characters of the diamond
+and those of graphite (blacklead) or charcoal, with which it is
+chemically identical, and into which it can be converted by the action
+of heat or electricity. Again, on account of the great value of the
+diamond, much of the romance of precious stones has centred round this
+mineral; and the history of some of the great diamonds of historic times
+has been traced through many extraordinary vicissitudes.
+
+The name [Greek: Adamas], "the invincible," was probably applied by the
+Greeks to hard metals, and thence to corundum (emery) and other hard
+stones. According to Charles William King, the first undoubted
+application of the name to the diamond is found in Manilius (A.D.
+16),--_Sic Adamas_, _punctum lapidis_, _pretiosior auro_,--and Pliny
+(A.D. 100) speaks of the rarity of the stone, "the most valuable of
+gems, known only to kings." Pliny described six varieties, among which
+the Indian, having six pointed angles, and also resembling two pyramids
+(_turbines_, whip-tops) placed base to base, may probably be identified
+as the ordinary octahedral crystal (fig. 1). The "diamond" (_Yahalom_)
+in the breastplate of the high priest (Ex. xxxix. 11) was certainly some
+other stone, for it bore the name of a tribe, and methods of engraving
+the true diamond cannot have been known so early. The stone can hardly
+have become familiar to the Romans until introduced from India, where it
+was probably mined at a very early period. But one or other of the
+remaining varieties mentioned by Pliny (the Macedonian, the Arabian, the
+Cyprian, &c.) may be the true diamond, which was in great request for
+the tool of the gem-engraver. Later Roman authors mentioned various
+rivers in India as yielding the _Adamas_ among their sands. The name
+_Adamas_ became corrupted into the forms _adamant_, _diamaunt_,
+_diamant_, _diamond_; but the same word, owing to a medieval
+misinterpretation which derived it from _adamare_ (compare the French
+word _aimant_), was also applied to the lodestone.
+
+Like all the precious stones, the diamond was credited with many
+marvellous virtues; among others the power of averting insanity, and of
+rendering poison harmless; and in the middle ages it was known as the
+"pietra della reconciliazione," as the peacemaker between husband and
+wife.
+
+_Scientific Characters._--The majority of minerals are found most
+commonly in masses which can with difficulty be recognized as aggregates
+of crystalline grains, and occur comparatively seldom as distinct
+crystals; but the diamond is almost always found in single crystals,
+which show no signs of previous attachment to any matrix; the stones
+were, until the discovery of the South African mines, almost entirely
+derived from sands or gravels, but owing to the hardness of the mineral
+it is rarely, if ever, water-worn, and the crystals are often very
+perfect. The crystals belong to the cubic system, generally assuming the
+form of the octahedron (fig. 1), but they may, in accordance with the
+principles of crystallography, also occur in other forms symmetrically
+derived from the octahedron,--for example, the cube, the 12-faced figure
+known as the rhombic dodecahedron (fig. 2), or the 48-faced figure known
+as the hexakis-octahedron (fig. 3), or in combinations of these. The
+octahedron faces are usually smooth; most of the other faces are rounded
+(fig. 4). The cube faces are rough with protruding points. The cube is
+sometimes found in Brazil, but is very rare among the S. African stones;
+and the dodecahedron is perhaps more common in Brazil than elsewhere.
+There is often a furrow running along the edges of the octahedron, or
+across the edges of the cube, and this indicates that the apparently
+simple crystal may really consist of eight individuals meeting at the
+centre; or, what comes to the same thing, of two individuals
+interpenetrating and projecting through each other. If this be so the
+form of the diamond is really the tetrahedron (and the various figures
+derived symmetrically from it) and not the octahedron. Fig. 5 shows how
+the octahedron with furrowed edge may be constructed from two
+interpenetrating tetrahedra (shown in dotted lines). If the grooves be
+left out of account, the large faces which have replaced each
+tetrahedron corner then make up a figure which has the aspect of a
+simple octahedron. Such regular interpenetrations are known in
+crystallography as "twins." There are also twins of diamond in which two
+octahedra (fig. 6) are united by contact along a surface parallel to an
+octahedron face without interpenetration. On account of their
+resemblance to the twins of the mineral spinel (which crystallizes in
+octahedra) these are known as "spinel twins." They are generally
+flattened along the plane of union. The crystals often display
+triangular markings, either elevations or pits, upon the octahedron
+faces; the latter are particularly well defined and have the form of
+equilateral triangles (fig. 7). They are similar to the "etched figures"
+produced by moistening an octahedron of alum, and have probably been
+produced, like them, by the action of some solvent. Similar, but
+somewhat different markings are produced by the combustion of diamond in
+oxygen, unaccompanied by any rounding of the edges.
+
+[Illustration: FIG. 1.]
+
+[Illustration: FIG. 2.]
+
+[Illustration: FIG. 3.]
+
+[Illustration: FIG. 4.]
+
+[Illustration: FIG. 5.]
+
+[Illustration: FIG. 6.]
+
+[Illustration: FIG. 7.]
+
+Diamond possesses a brilliant "adamantine" lustre, but this tends to be
+greasy on the surface of the natural stones and gives the rounded
+crystals somewhat the appearance of drops of gum. Absolutely colourless
+stones are not so common as cloudy and faintly coloured specimens; the
+usual tints are grey, brown, yellow or white; and as rarities, red,
+green, blue and black stones have been found. The colour can sometimes
+be removed or changed at a high temperature, but generally returns on
+cooling. It is therefore more probably due to metallic oxides than to
+hydrocarbons. Sir William Crookes has, however, changed a pale yellow
+diamond to a bluish-green colour by keeping it embedded in radium
+bromide for eleven weeks. The black coloration upon the surface produced
+by this process, as also by the electric bombardment in a vacuum tube,
+appears to be due to a conversion of the surface film into graphite.
+Diamond may break with a conchoidal fracture, but the crystals always
+cleave readily along planes parallel to the octahedron faces: of this
+property the diamond cutters avail themselves when reducing the stone to
+the most convenient form for cutting; a sawing process, has, however,
+now been introduced, which is preferable to that of cleavage. It is the
+hardest known substance (though tantalum, or an alloy of tantalum now
+competes with it) and is chosen as 10 in the mineralogist's scale of
+hardness; but the difference in hardness between diamond (10) and
+corundum (9) is really greater than that between corundum (9) and talc
+(1); there is a difference in the hardness of the different faces; the
+Borneo stones are also said to be harder than those of Australia, and
+the Australian harder than the African, but this is by no means certain.
+The specific gravity ranges from 3.56 to 3.50, generally about 3.52. The
+coefficient of expansion increases very rapidly above 750 deg., and
+diminishes very rapidly at low temperatures; the maximum density is
+attained about -42 deg. C.
+
+The very high refractive power (index = 2.417 for sodium light) gives
+the stone its extraordinary brilliancy; for light incident within a
+diamond at a greater angle than 24-1/2 deg. is reflected back into the
+stone instead of passing through it; the corresponding angle for glass
+is 40-1/2 deg. The very high dispersion (index for red light = 2.402,
+for blue light = 2.460) gives it the wonderful "fire" or display of
+spectral colours. Certain absorption bands at the blue end of the
+spectrum are supposed to be due to rare elements such as samarium.
+Unlike other cubic crystals, diamond experiences a diminution of
+refractive index with increase of temperature. It is very transparent
+for Rontgen rays, whereas paste imitations are opaque. It is a good
+conductor of heat, and therefore feels colder to the touch than glass
+and imitation stones. The diamond has also a somewhat greasy feel. The
+specific heat increases rapidly with rising temperature up to 60 deg.
+C., and then more slowly. Crystals belonging to the cubic system should
+not be birefringent unless strained; diamond often displays double
+refraction particularly in the neighbourhood of inclusions, both liquid
+and solid; this is probably due to strain, and the spontaneous explosion
+of diamonds has often been observed. Diamond differs from graphite in
+being a bad conductor of electricity: it becomes positively electrified
+by friction. The electrical resistance is about that of ordinary glass,
+and is diminished by one-half during exposure by Rontgen rays; the
+dielectric constant (16) is greater than that which should correspond to
+the specific gravity.
+
+The phosphorescence produced by friction has been known since the time
+of Robert Boyle (1663); the diamond becomes luminous in a dark room
+after exposure to sunlight or in the presence of radium; and many stones
+phosphoresce beautifully (generally with a pale green light) when
+subjected to the electric discharge in a vacuum tube. Some diamonds are
+more phosphorescent than others, and different faces of a crystal may
+display different tints. The combustibility of the diamond was predicted
+by Sir Isaac Newton on account of its high refractive power; it was
+first established experimentally by the Florentine Academicians in 1694.
+In oxygen or air diamond burns at about 850 deg., and only continues to do
+so if maintained at a high temperature; but in the absence of oxidising
+agents it may be raised to a much higher temperature. It is, however,
+infusible at the temperature of the electric arc, but becomes converted
+superficially into graphite. Experiments on the combustion of diamond
+were made by Smithson Tennant (1797) and Sir Humphry Davy (1816), with
+the object of proving that it is pure carbon; they showed that burnt in
+oxygen it yields exactly the same amount of carbon dioxide as that
+produced by burning the same weight of carbon. Still more convincing
+experiments were made by A. Krause in 1890. Similarly Guyton de Morveau
+showed that, like charcoal, diamond converts soft iron into steel.
+Diamond is insoluble in acid and alkalis, but is oxidised on heating
+with potassium bichromate and sulphuric acid.
+
+Bort (or Boart) is the name given to impure crystals or fragments
+useless for jewels; it is also applied to the rounded crystalline
+aggregates, which generally have a grey colour, a rough surface, often a
+radial structure, and are devoid of good cleavage. They are sometimes
+spherical ("shot bort"). Carbonado or "black diamond," found in Bahia
+(also recently in Minas Geraes), is a black material with a minutely
+crystalline structure somewhat porous, opaque, resembling charcoal in
+appearance, devoid of cleavage, rather harder than diamond, but of less
+specific gravity; it sometimes displays a rude cubic crystalline form.
+The largest specimen found (1895) weighed 3078 carats. Both bort and
+carbonado seem to be really aggregates of crystallized diamond, but the
+carbonado is so nearly structureless that it was till recently regarded
+as an amorphous modification of carbon.
+
+_Uses of the Diamond._--The use of the diamond for other purposes than
+jewelry depends upon its extreme hardness: it has always been the only
+material used for cutting or engraving the diamond itself. The
+employment of powdered bort and the lapidary's wheel for faceting
+diamonds was introduced by L. von Berquen of Bruges in 1476. Diamonds
+are now employed not only for faceting precious stones, but also for
+cutting and drilling glass, porcelain, &c,; for fine engraving such as
+scales; in dentistry for drilling; as a turning tool for electric-light
+carbons, hard rubber, &c.; and occasionally for finishing accurate
+turning work such as the axle of a transit instrument. For these tools
+the stone is actually shaped to the best form: it is now electroplated
+before being set in its metal mount in order to secure a firm fastening.
+It is also used for bearings in watches and electric meters. The best
+glaziers' diamonds are chosen from crystals such that a natural curved
+edge can be used. For rock drills, and revolving saws for stone cutting,
+either diamond, bort or carbonado is employed, set in steel tubes, disks
+or bands. Rock drilling is the most important industrial application;
+and for this, owing to its freedom from cleavage, the carbonado is more
+highly prized than diamond; it is broken into fragments about 3 carats
+in weight; and in 1905 the value of carbonado was no less than from L10
+to L14 a carat. It has been found that the "carbons" in drills can
+safely be subjected to a pressure of over 60 kilograms per square
+millimetre, and a speed of 25 metres per second. A recent application of
+the diamond is for wire drawing; a hole tapering towards the centre is
+drilled through a diamond, and the metal is drawn through this. No other
+tool is so endurable, or gives such uniform thickness of wire.
+
+_Distribution and Mining._--The most important localities for diamonds
+have been: (1) India, where they were mined from the earliest times till
+the close of the 19th century; (2) South America, where they have been
+mined since the middle of the 18th century; and (3) South Africa, to
+which almost the whole of the diamond-mining industry has been
+transferred since 1870.
+
+  _India._--The diamond is here found in ancient sandstones and
+  conglomerates, and in the river gravels and sands derived from them.
+  The sandstones and conglomerates belong to the Vindhyan formation and
+  overlie the old crystalline rocks: the diamantiferous beds are well
+  defined, often not more than 1 ft. in thickness, and contain pebbles
+  of quartzite, jasper, sandstone, slate, &c. The mines fall into five
+  groups situated on the eastern side of the Deccan plateau about the
+  following places (beginning from the south), the first three being in
+  Madras. (1) Chennur near Cuddapah on the river Pennar. (2) Kurnool
+  near Baneganapalle between the rivers Pennar and Kistna. (3) Kollar
+  near Bezwada on the river Kistna. (4) Sambalpur on the river Mahanadi
+  in the Central Provinces. (5) Panna near Allahabad, in Bundelkhand.
+  The mining has always been carried on by natives of low caste, and by
+  primitive methods which do not differ much from those described by the
+  French merchant Jean Baptiste Tavernier (1605-1689), who paid a
+  prolonged visit to most of the mines between 1638 and 1665 as a
+  dealer in precious stones. According to his description shallow pits
+  were sunk, and the gravel excavated was gathered into a walled
+  enclosure where it was crushed and water was poured over it, and it
+  was finally sifted in baskets and sorted by hand. The buying and
+  selling was at that period conducted by young children. In more modern
+  times there has been the same excavation of shallow pits, and
+  sluicing, sifting and sorting, by hand labour, the only machinery used
+  being chain pumps made of earthen bowls to remove the water from the
+  deeper pits.
+
+  At some of the Indian localities spasmodic mining has been carried on
+  at different periods for centuries, at some the work which had been
+  long abandoned was revived in recent times, at others it has long been
+  abandoned altogether. Many of the large stones of antiquity were
+  probably found in the Kollar group, where Tavernier found 60,000
+  workers in 1645 (?), the mines having, according to native accounts,
+  been discovered about 100 years previously. Golconda was the fortress
+  and the market for the diamond industry at this group of mines, and so
+  gave its name to them. The old mines have now been completely
+  abandoned, but in 1891 about 1000 carats were being raised annually in
+  the neighbourhood of Hyderabad. The Sambalpur group appear to have
+  been the most ancient mines of all, but they were not worked later
+  than 1850. The Panna group were the most productive during the 19th
+  century. India was no doubt the source of all the large stones of
+  antiquity; a stone of 67-3/8 carats was found at Wajra Karur in the
+  Chennur group in 1881, and one of 210-1/2 carats at Hira Khund in 1809.
+  Other Indian localities besides those mentioned above are Simla, in
+  the N.W. Provinces, where a few stones have been found, and a district
+  on the Gouel and the Sunk rivers in Bengal, which V. Ball has
+  identified with the Soumelpour mentioned by Tavernier. The mines of
+  Golconda and Kurnool were described as early as 1677 in the twelfth
+  volume of the _Philosophical Transactions_ of the Royal Society. At
+  the present time very few Indian diamonds find their way out of the
+  country, and, so far as the world's supply is concerned, Indian mining
+  of diamonds may be considered extinct. The first blow to this industry
+  was the discovery of the Brazilian mines in Minas Geraes and Bahia.
+
+  _Brazil._---Diamonds were found about 1725 at Tejuco (now Diamantina)
+  in Minas Geraes, and the mining became important about 1740. The chief
+  districts in Minas Geraes are (1) Bagagem on the W. side of the Serra
+  da Mata da Corda; (2) Rio Abaete on the E. side of the same range;
+  these two districts being among the head waters of the Rio de San
+  Francisco and its tributaries; (3) Diamantina, on and about the
+  watershed separating the Rio de San Francisco from the Rio
+  Jequitinhonha; and (4) Grao Mogul, nearly 200 m. to the N.E. of
+  Diamantina on the latter river.
+
+  The Rio Abaete district was worked on a considerable scale between
+  1785 and 1807, but is now abandoned. Diamantina is at present the most
+  important district; it occupies a mountainous plateau, and the
+  diamonds are found both on the plateau and in the river valleys below
+  it. The mountains consist here of an ancient laminated micaceous
+  quartzite, which is in parts a flexible sandstone known as
+  itacolumite, and in parts a conglomerate; it is interbedded with
+  clay-slate, mica-schist, hornblende-schist and haematite-schist, and
+  intersected by veins of quartz. This series is overlain unconformably
+  by a younger quartzite of similar character, and itself rests upon the
+  crystalline schists. The diamond is found under three conditions: (1)
+  in the gravels of the present rivers, embedded in a ferruginous
+  clay-cemented conglomerate known as _cascalho_; (2) in terraces
+  (gupiarras) in a similar conglomerate occupying higher levels in the
+  present valleys; (3) in plateau deposits in a coarse surface
+  conglomerate known as _gurgulho_, the diamond and other heavy minerals
+  being embedded in the red clay which cements the larger blocks. Under
+  all these three conditions the diamond is associated with fragments of
+  the rocks of the country and the minerals derived from them,
+  especially quartz, hornstone, jasper, the polymorphous oxide of
+  titanium (rutile, anatase and brookite), oxides and hydrates of iron
+  (magnetite, ilmenite, haematite, limonite), oxide of tin, iron
+  pyrites, tourmaline, garnet, xenotime, monazite, kyanite, diaspore,
+  sphene, topaz, and several phosphates, and also gold. Since the heavy
+  minerals of the _cascalho_ in the river beds are more worn than those
+  of the terraces, it is highly probable that they have been derived by
+  the cutting down of the older river gravels represented by the
+  terraces; and since in both deposits the heavy minerals are more
+  abundant near the heads of the valleys in the plateau, it is also
+  highly probable that both have really been derived from the plateau
+  deposit. In the latter, especially at Sao Joao da Chapada, the
+  minerals accompanying the diamond are scarcely worn at all; in the
+  terraces and the river beds they are more worn and more abundant; the
+  terraces, therefore, are to be regarded as a first concentration of
+  the plateau material by the old rivers; and the _cascalho_ as a second
+  concentration by the modern rivers. The mining is carried on by
+  negroes under the supervision of overseers; the _cascalho_ is dug out
+  in the dry season and removed to a higher level, and is afterwards
+  washed out by hand in running water in shallow wooden basins
+  (_bateas_). The terraces can be worked at all seasons, and the
+  material is partly washed out by leading streams on to it. The washing
+  of the plateau material is effected in reservoirs of rain water.
+
+  It is difficult to obtain an estimate of the actual production of the
+  Minas Geraes mines, for no official returns have been published, but
+  in recent years it has certainly been rivalled by the yield in Bahia.
+  The diamond here occurs in river gravels and sands associated with
+  the same minerals as in Minas Geraes; since 1844 the richest mines
+  have been worked in the Serra de Cincora, where the mountains are
+  intersected by the river Paraguassu and its tributaries; it is said
+  that there were as many as 20,000 miners working here in 1845, and it
+  was estimated that 54,000 carats were produced in Bahia in 1858. The
+  earlier workings were in the Serra de Chapada to the N.W. of the mines
+  just mentioned. In 1901 there were about 5000 negroes employed in the
+  Bahia mines; methods were still primitive; the _cascalho_ was dug out
+  from the river beds or tunnelled out from the valley side, and washed
+  once a week in sluices of running water, where it was turned over with
+  the hoe, and finally washed in wooden basins and picked over by hand;
+  sometimes also the diamantiferous material is scooped out of the bed
+  of the shallow rivers by divers, and by men working under water in
+  caissons. It is almost exclusively in the mines of Bahia, and in
+  particular in the Cincora district, that the valuable carbonado is
+  found. The carbonado and the diamond have been traced to an extensive
+  hard conglomerate which occurs in the middle of the sandstone
+  formation. Diamonds are also mined at Salobro on the river Pardo not
+  far inland from the port of Canavieras in the S.E. corner of Bahia.
+  The enormous development of the South African mines, which supplied in
+  1906, about 90% of the world's produce, has thrown into the shade the
+  Brazilian production; but the _Bulletin_ for Feb. 1909 of the
+  International Bureau of American Republics gave a very confident
+  account of its future, under improved methods.
+
+  _South Africa._---The first discovery was made in 1867 by Dr W. G.
+  Atherstone, who identified as diamond a pebble obtained from a child
+  in a farm on the banks of the Orange river and brought by a trader to
+  Grahamstown; it was bought for L500 and displayed in the Paris
+  Exhibition of that year. In 1869 a stone weighing 83-1/2 carats was
+  found near the Orange river; this was purchased by the earl of Dudley
+  for L25,000 and became famous as the "Star of South Africa." A rush of
+  prospectors at once took place to the banks of the Orange and Vaal
+  rivers, and resulted in considerable discoveries, so that in 1870
+  there was a mining camp of no less than 10,000 persons on the "River
+  Diggings." In the River Diggings the mining was carried on in the
+  coarse river gravels, and by the methods of the Brazilian negroes and
+  of gold placer-miners. A diggers' committee limited the size of claims
+  to 30 ft. square, with free access to the river bank; the gravel and
+  sand were washed in cradles provided with screens of perforated metal,
+  and the concentrates were sorted by hand on tables by means of an iron
+  scraper.
+
+  But towards the close of 1870 stones were found at Jagersfontein and
+  at Dutoitspan, far from the Vaal river, and led to a second great rush
+  of prospectors, especially to Dutoitspan, and in 1871 to what is now
+  the Kimberley mine in the neighbourhood of the latter. At each of
+  these spots the diamantiferous area was a roughly circular patch of
+  considerable size, and in some occupied the position of one of those
+  depressions or "pans" so frequent in S. Africa. These "dry diggings"
+  were therefore at first supposed to be alluvial in origin like the
+  river gravels; but it was soon discovered that, below the red surface
+  soil and the underlying calcareous deposit, diamonds were also found
+  in a layer of yellowish clay about 50 ft. thick known as "yellow
+  ground." Below this again was a hard bluish-green serpentinous rock
+  which was at first supposed to be barren bed-rock; but this also
+  contained the precious stone, and has become famous, under the name of
+  "blue ground," as the matrix of the S. African diamonds. The yellow
+  ground is merely decomposed blue ground. In the Kimberley district
+  five of these round patches of blue ground were found within an area
+  little more than 3 m. in diameter; that at Kimberley occupying 10
+  acres, that at Dutoitspan 23 acres. There were soon 50,000 workers on
+  this field, the canvas camp was replaced by a town of brick and iron
+  surrounded by the wooden huts of the natives, and Kimberley became an
+  important centre.
+
+  It was soon found that each mine was in reality a huge vertical funnel
+  or crater descending to an unknown depth, and filled with
+  diamantiferous blue ground. At first each claim was an independent pit
+  31 ft. square sunk into the blue ground; the diamantiferous rock was
+  hoisted by bucket and windlass, and roadways were left across the pit
+  to provide access to the claims. But the roadways soon fell in, and
+  ultimately haulage from the claims could only be provided by means of
+  a vast system of wire ropes extending from a triple staging of
+  windlasses erected round the entire edge of the mine, which had by
+  this time become a huge open pit; the ropes from the upper windlasses
+  extended to the centre, and those from the lower tier to the sides of
+  the pit; covering the whole mass like a gigantic cobweb. (See Plate
+  II. fig. 12.) The buckets of blue ground were hauled up these ropes by
+  means of horse whims, and in 1875 steam winding engines began to be
+  employed. By this time also improved methods in the treatment of the
+  blue ground were introduced. It was carried off in carts to open
+  spaces, where an exposure of some weeks to the air was found to
+  pulverize the hard rock far more efficiently than the old method of
+  crushing with mallets. The placer-miner's cradle and rocking-trough
+  were replaced by puddling troughs stirred by a revolving comb worked
+  by horse power; reservoirs were constructed for the scanty
+  water-supply, bucket elevators were introduced to carry away the
+  tailings; and the natives were confined in compounds. For these
+  improvements co-operation was necessary; the better claims, which in
+  1872 had risen from L100 to more than L4000 in value, began to be
+  consolidated, and a Mining Board was introduced.
+
+
+  PLATE I.
+
+  [Illustration: FIG. 9.--DE BEERS MINE, 1874.]
+
+  [Illustration: FIG. 10.--KIMBERLEY MINE, 1874.]
+
+  [Illustration: FIG. 11.--DE BEERS MINE, 1873. (From photographs by C.
+  Evans.)]
+
+
+  PLATE II.
+
+  [Illustration: _Fig. 12._--KIMBERLEY MINE, 1874.]
+
+  [Illustration: _Fig. 13._--KIMBERLEY MINE, 1902. (From Photographs by
+  C. Evans.)]
+
+
+  In a very few years, however, the open pit mining was rendered
+  impossible by the mud rushes, by the falls of the masses of barren
+  rock known as "reef," which were left standing in the mine, and by
+  landslips from the sides, so that in 1883, when the pit had reached a
+  depth of about 400 ft., mining in the Kimberley crater had become
+  almost impossible. By 1889, in the whole group of mines, Kimberley,
+  Dutoitspan, De Beers and Bultfontein, open pit working was practically
+  abandoned. Meanwhile mining below the bottom of the pits by means of
+  shafts and underground tunnels had been commenced; but the full
+  development of modern methods dates from the year 1889 when Cecil
+  Rhodes and Alfred Beit, who had already secured control of the De
+  Beers mine, acquired also the control of the Kimberley mine, and
+  shortly afterwards consolidated the entire group in the hands of the
+  De Beers Company. (See KIMBERLEY.)
+
+  The scene of native mining was now transferred from the open pit to
+  underground tunnels; the vast network of wire ropes (Plate II. fig.
+  12) with their ascending and descending buckets disappeared, and with
+  it the cosmopolitan crowd of busy miners working like ants at the
+  bottom of the pit. In place of all this, the visitor to Kimberley
+  encounters at the edge of the town only a huge crater, silent and
+  apparently deserted, with no visible sign of the great mining
+  operations which are conducted nearly half a mile below the surface.
+  The aspect of the Kimberley pit in 1906 is shown in fig. 13 of Plate
+  II., which may be compared with the section of fig. 8.
+
+  In fig. 13, Plate II., the sequence of the basalt, shale and melaphyre
+  is clearly visible on the sides of the pit; and fig. 8 shows how the
+  crater or "pipe" of blue ground has penetrated these rocks and also
+  the underlying quartzite. The workings at De Beers had extended into
+  the still more deeply seated granite in 1906. Figure 9, Plate I.,
+  shows the top of the De Beers' crater with basalt overlying the shale.
+  Figure 8 also explains the modern system of mining introduced by
+  Gardner Williams. A vertical shaft is sunk in the vicinity of the
+  mine, and from this horizontal tunnels are driven into the pipe at
+  different levels separated by intervals of 40 ft. Through the blue
+  ground itself on each level a series of parallel tunnels about 120 ft.
+  apart are driven to the opposite side of the pipe, and at right angles
+  to these, and 36 ft. apart, another series of tunnels. When the
+  tunnels reach the side of the mine they are opened upwards and
+  sideways so as to form a large chamber, and the overlying mass of blue
+  ground and debris is allowed to settle down and fill up the gallery.
+  On each level this process is carried somewhat farther back than on
+  the level below (fig. 8); material is thus continually withdrawn from
+  one side of the mine and extracted by means of the rock shaft on the
+  opposite side, while the superincumbent debris is continually sinking,
+  and is allowed to fall deeper on the side farthest from the shaft as
+  the blue ground is withdrawn from beneath it. In 1905 the main shaft
+  had been sunk to a depth of 2600 ft. at the Kimberley mine.
+
+  For the extraction and treatment of the blue ground the De Beers
+  Company in its great winding and washing plant employs labour-saving
+  machinery on a gigantic scale. The ground is transferred in trucks to
+  the shaft where it is automatically tipped into skips holding 96 cubic
+  ft. (six truck loads); these are rapidly hoisted to the surface, where
+  their contents are automatically dumped into side-tipping trucks, and
+  these in turn are drawn away in a continual procession by an endless
+  wire rope along the tram lines leading to the vast "distributing
+  floors." These are open tracts upon which the blue ground is spread
+  out and left exposed to sun and rain until it crumbles and
+  disintegrates, the process being hastened by harrowing with steam
+  ploughs; this may require a period of three or six months, or even a
+  year. The stock of blue ground on the floors at one time in 1905 was
+  nearly 4,500,000 loads. The disintegrated ground is then brought back
+  in the trucks and fed through perforated cylinders into the washing
+  pans; the hard blue which has resisted disintegration on the floors,
+  and the lumps which are too big to pass the cylindrical sieves, are
+  crushed before going to the pans. These are shallow cylindrical
+  troughs containing muddy water in which the diamonds and other heavy
+  minerals (concentrates) are swept to the rim by revolving toothed
+  arms, while the lighter stuff escapes near the centre of the pan. The
+  concentrates are then passed over sloping tables (pulsator) and shaken
+  to and fro under a stream of water which effects a second
+  concentration of the heaviest material.
+
+  Until recently the final separation of the diamond from the
+  concentrates was made by hand picking, but even this has now been
+  replaced by machinery, owing to the remarkable discovery that a
+  greased surface will hold a diamond while allowing the other heavy
+  minerals to pass over it. The concentrates are washed down a sloping
+  table of corrugated iron which is smeared with grease, and it is found
+  that practically all the diamonds adhere to the table, and the other
+  minerals are washed away. At the large and important Premier mine in
+  the Transvaal the Elmore process, used in British Columbia and in
+  Wales for the separation of metallic ores, has been also introduced.
+  In the Elmore process oil is employed to float off the materials which
+  adhere to it, while the other materials remain in the water, the oil
+  being separated from the water by centrifugal action. The other
+  minerals found in the concentrates are pebbles and fragments of
+  pyrope, zircon, cyanite, chrome-diopside, enstatite, a green pyroxene,
+  mica, ilmenite, magnetite, chromite, hornblende, olivine, barytes,
+  calcite and pyrites.
+
+  In all the S. African mines the diamonds are not only crystals of
+  various weights from fractions of a carat to 150 carats, but also
+  occur as microscopic crystals disseminated through the blue ground. In
+  spite of this, however, the average yield in the profitable mines is
+  only from 0.2 carat to 0.6 carat per load of 1600 lb., or on an
+  average about 1-1/2 grs. per ton. The annual output of diamonds from
+  the De Beers mines was valued in 1906 at nearly L5,000,000; the value
+  per carat ranging from about 35s. to 70s.
+
+  [Illustration: SECTION OF KIMBERLEY MINE LOOKING EAST
+
+  From Gardner Williams's _Diamond Mines of South Africa_. FIG. 8.]
+
+  Pipes similar to those which surround Kimberley have been found in
+  other parts of S. Africa. One of the best known is that of
+  Jagersfontein, which was really the first of the dry diggings
+  (discovered in 1870). This large mine is near Fauresmith and 80 m. to
+  the south of Kimberley. In 1905 the year's production from the Orange
+  River Colony mines was more than 320,000 carats, valued at L938,000.
+  But by far the largest of all the pipes hitherto discovered is the
+  Premier mine in the Transvaal, about 300 m. to the east of Kimberley.
+  This was discovered in 1902 and occupies an area of about 75 acres. In
+  1906 it was being worked as a shallow open mine; but the description
+  of the Kimberley methods given above is applicable to the washing
+  plant at that time being introduced into the Premier mine upon a very
+  large scale. Comparatively few of the pipes which have been discovered
+  are at all rich in diamonds, and many are quite barren; some are
+  filled with "hard blue" which even if diamantiferous may be too
+  expensive to work.
+
+  The most competent S. African geologists believe all these remarkable
+  pipes to be connected with volcanic outbursts which occurred over the
+  whole of S. Africa during the Cretaceous period (after the deposition
+  of the Stormberg beds), and drilled these enormous craters through all
+  the later formations. With the true pipes are associated dykes and
+  fissures also filled with diamantiferous blue ground. It is only in
+  the more northerly part of the country that the pipes are filled with
+  blue ground (or "kimberlite"), and that they are diamantiferous; but
+  over a great part of Cape Colony have been discovered what are
+  probably similar pipes filled with agglomerates, breccias and tuffs,
+  and some with basic lavas; one, in particular, in the Riversdale
+  Division near the southern coast, being occupied by a melilite-basalt.
+  It is quite clear that the occurrence of the diamond in the S. African
+  pipes is quite different from the occurrences in alluvial deposits
+  which have been described above. The question of the origin of the
+  diamond in S. Africa and elsewhere is discussed below.
+
+  The River Diggings on the Vaal river are still worked upon a small
+  scale, but the production from this source is so limited that they are
+  of little account in comparison with the mines in the blue ground. The
+  stones, however, are good; since they differ somewhat from the
+  Kimberley crystals it is probable that they were not derived from the
+  present pipes. Another S. African locality must be mentioned;
+  considerable finds were reported in 1905 and 1906 from gravels at
+  Somabula near Gwelo in Rhodesia where the diamond is associated with
+  chrysoberyl, corundum (both sapphire and ruby), topaz, garnet,
+  ilmenite, staurolite, rutile, with pebbles of quartz, granite,
+  chlorite-schist, &c. Diamond has also been reported from kimberlite
+  "pipes" in Rhodesia.
+
+  _Other Localities._--In addition to the South American localities
+  mentioned above, small diamonds have also been mined since their
+  discovery in 1890 on the river Mazaruni in British Guiana, and finds
+  have been reported in the gold washings of Dutch Guiana. Borneo has
+  possessed a diamond industry since the island was first settled by the
+  Malays; the references in the works of Garcia de Orta, Linschoten, De
+  Boot, De Laet and others, to Malacca as a locality relate to Borneo.
+  The large Borneo stone, over 360 carats in weight, known as the Matan,
+  is in all probability not a diamond. The chief mines are situated on
+  the river Kapuas in the west and near Bandjarmassin in the south-east
+  of the island, and the alluvial deposits in which they occur are
+  worked by a small number of Chinese and Malays. Australia has yielded
+  diamonds in alluvial deposits near Bathurst (where the first discovery
+  was made in 1851) and Mudgee in New South Wales, and also near Bingara
+  and Inverell in the north of the colony. At Mount Werong a stone
+  weighing 29 carats was found in 1905. At Ruby Hill near Bingara they
+  were found in a breccia filling a volcanic pipe. At Ballina, in New
+  England, diamonds have been found in the sea sand. Other Australian
+  localities are Echunga in South Australia; Beechworth, Arena and
+  Melbourne in Victoria; Freemantle and Nullagine in Western Australia;
+  the Palmer and Gilbert rivers in Queensland. These have been for the
+  most part discoveries in alluvial deposits of the goldfields, and the
+  stones were small. In Tasmania also diamonds have been found in the
+  Corinna goldfields. Europe has produced few diamonds. Humboldt
+  searched for them in the Urals on account of the similarity of the
+  gold and platinum deposits to those of Brazil, and small diamonds were
+  ultimately found (1829) in the gold washings of Bissersk, and later at
+  Ekaterinburg and other spots in the Urals. In Lapland they have been
+  found in the sands of the Pasevig river. Siberia has yielded isolated
+  diamonds from the gold washings of Yenisei. In North America a few
+  small stones have been found in alluvial deposits, mostly auriferous,
+  in Georgia, N. and S. Carolina, Kentucky, Virginia, Tennessee,
+  Wisconsin, California, Oregon and Indiana. A crystal weighing 23-3/4
+  carats was found in Virginia in 1855, and one of 21-1/4 carats in
+  Wisconsin in 1886. In 1906 a number of small diamonds were discovered
+  in an altered peridotite somewhat resembling the S. African blue
+  ground, at Murfreesboro, Pike county, Arkansas. Considerable interest
+  attaches to the diamonds found in Wisconsin, Michigan and Ohio near
+  the Great Lakes, for they are here found in the terminal moraines of
+  the great glacial sheet which is supposed to have spread southwards
+  from the region of Hudson Bay; several of the drift minerals of the
+  diamantiferous region of Indiana have been identified as probably of
+  Canadian origin; no diamonds have however yet been found in the
+  intervening country of Ontario. A rock similar to the blue ground of
+  Kimberley has been found in the states of Kentucky and New York. The
+  occurrence of diamond in meteorites is described below.
+
+  _Origin of the Diamond in Nature._--It appears from the foregoing
+  account that at most localities the diamond is found in alluvial
+  deposits probably far from the place where it originated. The minerals
+  associated with it do not afford much clue to the original conditions;
+  they are mostly heavy minerals derived from the neighbouring rocks, in
+  which the diamond itself has not been observed. Among the commonest
+  associates of the diamond are quartz, topaz, tourmaline, rutile,
+  zircon, magnetite, garnet, spinel and other minerals which are common
+  accessory constituents of granite, gneiss and the crystalline schists.
+  Gold (also platinum) is a not infrequent associate, but this may only
+  mean that the sands in which the diamond is found have been searched
+  because they were known to be auriferous; also that both gold and
+  diamond are among the most durable of minerals and may have survived
+  from ancient rocks of which other traces have been lost.
+
+  The localities at which the diamond has been supposed to occur in its
+  original matrix are the following:--at Wajra Karur, in the Cuddapah
+  district, India, M. Chaper found diamond with corundum in a decomposed
+  red pegmatite vein in gneiss. At S[=a]o Joao da Chapada, in Minas
+  Geraes, diamonds occur in a clay interstratified with the itacolumite,
+  and are accompanied by sharp crystals of rutile and haematite in the
+  neighbourhood of decomposed quartz veins which intersect the
+  itacolumite. It has been suggested that these three minerals were
+  originally formed in the quartz veins. In both these occurrences the
+  evidence is certainly not sufficient to establish the presence of an
+  original matrix. At Inverell in New South Wales a diamond (1906) has
+  been found embedded in a hornblende diabase which is described as a
+  dyke intersecting the granite. Finally there is the remarkable
+  occurrence in the blue ground of the African pipes.
+
+  There has been much controversy concerning the nature and origin of
+  the blue ground itself; and even granted that (as is generally
+  believed) the blue ground is a much serpentinized volcanic breccia
+  consisting originally of an olivine-bronzite-biotite rock (the
+  so-called kimberlite), it contains so many rounded and angular
+  fragments of various rocks and minerals that it is difficult to say
+  which of them may have belonged to the original rock, and whether any
+  were formed _in situ_, or were brought up from below as inclusions.
+  Carvill Lewis believed the blue ground to be true eruptive rock, and
+  the carbon to have been derived from the bituminous shales of which it
+  contains fragments. The Kimberley shales, which are penetrated by the
+  De Beers group of pipes, were, however, certainly not the source of
+  the carbon at the Premier (Transvaal) mine, for at this locality the
+  shales do not exist. The view that the diamond may have crystallized
+  out from solution in its present matrix receives some support from the
+  experiments of W. Luzi, who found that it can be corroded by the
+  solvent action of fused blue ground; from the experiments of J.
+  Friedlander, who obtained diamond by dissolving graphite in fused
+  olivine; and still more from the experiments of R. von Hasslinger and
+  J. Wolff, who have obtained it by dissolving graphite in a fused
+  mixture of silicates having approximately the composition of the blue
+  ground. E. Cohen, who regarded the pipes as of the nature of a mud
+  volcano, and the blue ground as a kimberlite breccia altered by
+  hydrothermal action, thought that the diamond and accompanying
+  minerals had been brought up from deep-seated crystalline schists.
+  Other authors have sought the origin of the diamond in the action of
+  the hydrated magnesian silicates on hydrocarbons derived from
+  bituminous schists, or in the decomposition of metallic carbides.
+
+  Of great scientific interest in this connexion is the discovery of
+  small diamonds in certain meteorites, both stones and irons; for
+  example, in the stone which fell at Novo-Urei in Penza, Russia, in
+  1886, in a stone found at Carcote in Chile, and in the iron found at
+  Canon Diablo in Arizona. Graphitic carbon in cubic form (cliftonite)
+  has also been found in certain meteoric "irons," for example in those
+  from Magura in Szepes county, Hungary, and Youndegin near York in
+  Western Australia. The latter is now generally believed to be altered
+  diamond. The fact that H. Moissan has produced the diamond
+  artificially, by allowing dissolved carbon to crystallize out at a
+  high temperature and pressure from molten iron, coupled with the
+  occurrence in meteoric iron, has led Sir William Crookes and others to
+  conclude that the mineral may have been derived from deep-seated iron
+  containing carbon in solution (see the article GEM, ARTIFICIAL). Adolf
+  Knop suggested that this may have first yielded hydrocarbons by
+  contact with water, and that from these the crystalline diamond has
+  been formed. The meteoric occurrence has even suggested the fanciful
+  notion that all diamonds were originally derived from meteorites. The
+  meteoric iron of Arizona, some of which contains diamond, is actually
+  found in and about a huge crater which is supposed by some to have
+  been formed by an immense meteorite penetrating the earth's crust.
+
+  It is, at any rate, established that carbon can crystallize as diamond
+  from solution in iron, and other metals; and it seems that high
+  temperature and pressure and the absence of oxidizing agents are
+  necessary conditions. The presence of sulphur, nickel, &c., in the
+  iron appears to favour the production of the diamond. On the other
+  hand, the occurrence in meteoric stones, and the experiments mentioned
+  above, show that the diamond may also crystallize from a basic magma,
+  capable of yielding some of the metallic oxides and ferro-magnesian
+  silicates; a magma, therefore, which is not devoid of oxygen. This is
+  still more forcibly suggested by the remarkable eclogite boulder found
+  in the blue ground of the Newlands mine, not far from the Vaal river,
+  and described by T. G. Bonney. The boulder is a crystalline rock
+  consisting of pyroxene (chrome-diopside), garnet, and a little
+  olivine, and is studded with diamond crystals; a portion of it is
+  preserved in the British Museum (Natural History). In another eclogite
+  boulder, diamond was found partly embedded in pyrope. Similar boulders
+  have also been found in the blue ground elsewhere. Specimens of pyrope
+  with attached or embedded diamond had previously been found in the
+  blue ground of the De Beers mines. In the Newlands boulder the
+  diamonds have the appearance of being an original constituent of the
+  eclogite. It seems therefore that a holocrystalline pyroxene-garnet
+  rock may be one source of the diamond found in blue ground. On the
+  other hand many tons of the somewhat similar eclogite in the De Beers
+  mine have been crushed and have not yielded diamond. Further, the
+  ilmenite, which is the most characteristic associate of the diamond in
+  blue ground, and other of the accompanying minerals, may have come
+  from basic rocks of a different nature.
+
+  The Inverell occurrence may prove to be another example of diamond
+  crystallized from a basic rock.
+
+  In both occurrences, however, there is still the possibility that the
+  eclogite or the basalt is not the original matrix, but may have caught
+  up the already formed diamond from some other matrix. Some regard the
+  eclogite boulders as derived from deep-seated crystalline rocks,
+  others as concretions in the blue ground.
+
+  None of the inclusions in the diamond gives any clue to its origin;
+  diamond itself has been found as an inclusion, as have also black
+  specks of some carbonaceous materials. Other black specks have been
+  identified as haematite and ilmenite; gold has also been found; other
+  included minerals recorded are rutile, topaz, quartz, pyrites,
+  apophyllite, and green scales of chlorite (?). Some of these are of
+  very doubtful identification; others (e.g. apophyllite and chlorite)
+  may have been introduced along cracks. Some of the fibrous inclusions
+  were identified by H. R. Goppert as vegetable structures and were
+  supposed to point to an organic origin, but this view is no longer
+  held. Liquid inclusions, some of which are certainly carbon dioxide,
+  have also been observed.
+
+  Finally, then, both experiment and the natural occurrence in rocks and
+  meteorites suggest that diamond may crystallize not only from iron but
+  also from a basic silicate magma, possibly from various rocks
+  consisting of basic silicates. The blue ground of S. Africa may be
+  the result of the serpentinization of several such rocks, and
+  although now both brecciated and serpentinized some of these may have
+  been the original matrix. A circumstance often mentioned in support of
+  this view is the fact that the diamonds in one pipe generally differ
+  somewhat in character from those of another, even though they be near
+  neighbours.
+
+_History._--All the famous diamonds of antiquity must have been Indian
+stones. The first author who described the Indian mines at all fully was
+the Portuguese, Garcia de Orta (1565), who was physician to the viceroy
+of Goa. Before that time there were only legendary accounts like that of
+Sindbad's "Valley of the Diamonds," or the tale of the stones found in
+the brains of serpents. V. Ball thinks that the former legend originated
+in the Indian practice of sacrificing cattle to the evil spirits when a
+new mine is opened; birds of prey would naturally carry off the flesh,
+and might give rise to the tale of the eagles carrying diamonds adhering
+to the meat.
+
+The following are some of the most famous diamonds of the world:--
+
+A large stone found in the Golconda mines and said to have weighed 787
+carats in the rough, before being cut by a Venetian lapidary, was seen
+in the treasury of Aurangzeb in 1665 by Tavernier, who estimated its
+weight after cutting as 280 (?) carats, and described it as a rounded
+rose-cut-stone, tall on one side. The name _Great Mogul_ has been
+frequently applied to this stone. Tavernier states that it was the
+famous stone given to Shah Jahan by the emir Jumla. The _Orloff_, stolen
+by a French soldier from the eye of an idol in a Brahmin temple, stolen
+again from him by a ship's captain, was bought by Prince Orloff for
+L90,000, and given to the empress Catharine II. It weighs 194-3/4 carats,
+is of a somewhat yellow tinge, and is among the Russian crown jewels.
+The _Koh-i-nor_, which was in 1739 in the possession of Nadir Shah, the
+Persian conqueror, and in 1813 in that of the raja of Lahore, passed
+into the hands of the East India Company and was by them presented to
+Queen Victoria in 1850. It then weighed 186-1/16 carats, but was recut in
+London by Amsterdam workmen, and now weighs 106-1/16 carats. There has
+been much discussion concerning the possibility of this stone and the
+Orloff being both fragments of the Great Mogul. The Mogul Baber in his
+memoirs (1526) relates how in his conquest of India he captured at Agra
+the great stone weighing 8 mishkals, or 320 ratis, which may be
+equivalent to about 187 carats. The Koh-i-nor has been identified by
+some authors with this stone and by others with the stone seen by
+Tavernier. Tavernier, however, subsequently described and sketched the
+diamond which he saw as shaped like a bisected egg, quite different
+therefore from the Koh-i-nor. Nevil Story Maskelyne has shown reason for
+believing that the stone which Tavernier saw was really the Koh-i-nor
+and that it is identical with the great diamond of Baber; and that the
+280 carats of Tavernier is a misinterpretation on his part of the Indian
+weights. He suggests that the other and larger diamond of antiquity
+which was given to Shah Jahan may be one which is now in the treasury of
+Teheran, and that this is the true Great Mogul which was confused by
+Tavernier with the one he saw. (See Ball, Appendix I. to Tavernier's
+_Travels_ (1889); and Maskelyne, _Nature_, 1891, 44, p. 555.).
+
+The _Regent_ or _Pitt_ diamond is a magnificent stone found in either
+India or Borneo; it weighed 410 carats and was bought for L20,400 by
+Pitt, the governor of Madras; it was subsequently, in 1717, bought for
+L80,000 (or, according to some authorities, L135,000) by the duke of
+Orleans, regent of France; it was reduced by cutting to 1361-4/16 carats;
+was stolen with the other crown jewels during the Revolution, but was
+recovered and is still in France. The _Akbar Shah_ was originally a
+stone of 116 carats with Arabic inscriptions engraved upon it; after
+being cut down to 71 carats it was bought by the gaikwar of Baroda for
+L35,000. The _Nizam_, now in the possession of the nizam of Hyderabad,
+is supposed to weigh 277 carats; but it is only a portion of a stone
+which is said to have weighed 440 carats before it was broken. The
+_Great Table_, a rectangular stone seen by Tavernier in 1642 at
+Golconda, was found by him to weigh 242-3/16 carats; Maskelyne regards it
+as identical with the _Darya-i-nur_, which is also a rectangular stone
+weighing about 186 carats in the possession of the shah of Persia.
+Another stone, the _Taj-e-mah_, belonging to the shah, is a pale rose
+pear-shaped stone and is said to weigh 146 carats.
+
+Other famous Indian diamonds are the following:--The _Sancy_, weighing
+531-2/16 carats, which is said to have been successively the property of
+Charles the Bold, de Sancy, Queen Elizabeth, Henrietta Maria, Cardinal
+Mazarin, Louis XIV.; to have been stolen with the Pitt during the French
+Revolution; and subsequently to have been the property of the king of
+Spain, Prince Demidoff and an Indian prince. The _Nassak_, 78-5/8 carats,
+the property of the duke of Westminster. The _Empress Eugenie_, 51
+carats, the property of the gaikwar of Baroda. The _Pigott_, 49
+carats(?), which cannot now be traced. The _Pasha_, 40 carats. The
+_White Saxon_, 48-3/4 carats. The _Star of Este_, 251-3/32 carats.
+
+Coloured Indian diamonds of large size are rare; the most famous are:--a
+beautiful blue brilliant, 67-2/16 carats, cut from a stone weighing
+112-3/16 carats brought to Europe by Tavernier. It was stolen from the
+French crown jewels with the Regent and was never recovered. The _Hope_,
+44-1/4 carats, has the same colour and is probably a portion of the missing
+stone: it was so-called as forming part of the collection of H. T. Hope
+(bought for L18,000), and was sold again in 1906 (resold 1909). Two
+other blue diamonds are known, weighing 13-3/4 and 1-3/4 carats, which may
+also be portions of the French diamond. The _Dresden Green_, one of the
+Saxon crown jewels, 40 carats, has a fine apple-green colour. The
+_Florentine_, 133-1/5 carats, one of the Austrian crown jewels, is a very
+pale yellow.
+
+The most famous Brazilian stones are:--The _Star of the South_, found in
+1853, when it weighed 254-1/2 carats and was sold for L40,000; when cut it
+weighed 125 carats and was bought by the gaikwar of Baroda for L80,000.
+Also a diamond belonging to Mr Dresden, 119 carats before, and 76-1/2
+carats after cutting.
+
+Many large stones have been found in South Africa; some are yellow but
+some are as colourless as the best Indian or Brazilian stones. The most
+famous are the following:--the _Star of South Africa_, or _Dudley_,
+mentioned above, 83-1/2 carats rough, 46-1/2 carats cut. The _Stewart_,
+288-3/8 carats rough, 120 carats cut. Both these were found in the river
+diggings. The _Porter Rhodes_ from Kimberley, of the finest water,
+weighed about 150 carats. The _Victoria_, 180 carats, was cut from an
+octahedron weighing 457-1/2 carats, and was sold to the nizam of
+Hyderabad for L400,000. The _Tiffany_, a magnificent orange-yellow
+stone, weighs 125-1/2 carats cut. A yellowish octahedron found at De
+Beers weighed 428-1/2 carats, and yielded a brilliant of 288-1/2 carats.
+Some of the finest and largest stones have come from the Jagersfontein
+mine; one, the _Jubilee_, found in 1895, weighed 640 carats in the rough
+and 239 carats when cut. Until 1905 the largest known diamond in the
+world was the _Excelsior_, found in 1893 at Jagersfontein by a native
+while loading a truck. It weighed 971 carats, and was ultimately cut
+into ten stones weighing from 68 to 13 carats. But all previous records
+were surpassed in 1905 by a magnificent stone more than three times the
+size of any known diamond, which was found in the yellow ground at the
+newly discovered Premier mine in the Transvaal. This extraordinary
+diamond weighed 3025-3/4 carats (11/3 lb.) and was clear and water
+white; the largest of its surfaces appeared to be a cleavage plane, so
+that it might be only a portion of a much larger stone. It was known as
+the _Cullinan Diamond_. This stone was purchased by the Transvaal
+government in 1907 and presented to King Edward VII. It was sent to
+Amsterdam to be cut, and in 1908 was divided into nine large stones and
+a number of small brilliants. The four largest stones weigh 516-1/2
+carats, 309-3/16 carats, 92 carats and 62 carats respectively. Of these
+the first and second are the largest brilliants in existence. All the
+stones are flawless and of the finest quality.
+
+  BIBLIOGRAPHY.--Boetius de Boot, _Gemmarum et lapidum historia_ (1609);
+  D. Jeffries, _A Treatise on Diamonds and Pearls_ (1757); J. Mawe,
+  _Travels in the Interior of Brazil_ (1812); _Treatise on Diamonds and
+  Precious Stones_ (1813): Pinder, _De adamante_ (1829); Murray, _Memoir
+  on the Nature of the Diamond_ (1831); C. Zerenner, _De adamante
+  dissertatio_ (1850); H. Emanuel, _Diamonds and Precious Stones_
+  (1865); A. Schrauf, _Edelsteinkunde_ (1869); N. Jacobs and N.
+  Chatrian, _Monographie du diamant_ (1880); V. Ball, _Geology of India_
+  (1881); C. W. King, _The Natural History of Precious Stones_ _and
+  Precious Metals_ (1883); M. E. Boutan, _Le Diamant_ (1886); S. M.
+  Burnham, _Precious Stones in Nature, Art and Literature_ (1887); P.
+  Groth, _Grundriss der Edelsteinkunde_ (1887); A. Liversidge, _The
+  Minerals of New South Wales_ (1888); _Tavernier's Travels in India_,
+  translated by V. Ball (1889); E. W. Streeter, _The Great Diamonds of
+  the World_ (1896); H. C. Lewis, _The Genesis and Matrix of the
+  Diamond_ (1897); L. de Launay, _Les Diamants du Cap_ (1897); C.
+  Hintze, _Handbuch der Mineralogie_ (1898); E. W. Streeter, _Precious
+  Stones and Gems_ (6th ed., 1898); Dana, _System of Mineralogy_ (1899);
+  Kunz and others, _The Production of Precious Stones_ (in annual,
+  _Mineral Resources of the United States_); M. Bauer, _Precious Stones_
+  (trans. L. J. Spencer, 1904); A. W. Rogers, _An Introduction to the
+  Geology of Cape Colony_ (1905); Gardner F. Williams, _The Diamond
+  Mines of South Africa_ (revised edition, 1906); George F. Kunz,
+  "Diamonds, a study of their occurrence in the United States, with
+  descriptions and comparisons of those from all known localities" (U.S.
+  Geol. Survey, 1909); P. A. Wagner, _Die Diamantfuhrenden Gesteine
+  Sudafrikas_ (1909).
+
+  Among papers in scientific periodicals may be mentioned articles by
+  Adler, Ball, Baumhauer, Beck, Bonney, Brewster, Chaper, Cohen,
+  Crookes, Daubree, Derby, Des Cloizeaux, Doelter, Dunn, Flight,
+  Friedel, Gorceix, Gurich, Goeppert, Harger, Hudleston, Hussak,
+  Jannettaz, Jeremejew, de Launay, Lewis, Maskelyne, Meunier, Moissan,
+  Molengraaff, Moulle, Rose, Sadebeck, Scheibe, Stelzner, Stow. See
+  generally Hintze's _Handbuch der Mineralogie_. (H. A. MI.)
+
+
+FOOTNOTE:
+
+  [1] Diamonds are invariably weighed in carats and in 1/2, 1/4, 1/8,
+    1/16, 1/32, 1/64 of a carat. One (English) carat = 3.17 grains = .2054
+    gram. One ounce = 151-1/2 carats. (See CARAT.)
+
+
+
+
+DIAMOND NECKLACE, THE AFFAIR OF THE, a mysterious incident at the court
+of Louis XVI. of France, which involved the queen Marie Antoinette. The
+Parisian jewellers Boehmer and Bassenge had spent some years collecting
+stones for a necklace which they hoped to sell to Madame Du Barry, the
+favourite of Louis XV., and after his death to Marie Antoinette. In 1778
+Louis XVI. proposed to the queen to make her a present of the necklace,
+which cost 1,600,000 livres. But the queen is said to have refused it,
+saying that the money would be better spent equipping a man-of-war.
+According to others, Louis XVI. himself changed his mind. After having
+vainly tried to place the necklace outside of France, the jewellers
+attempted again in 1781 to sell it to Marie Antoinette after the birth
+of the dauphin. It was again refused, but it was evident that the queen
+regretted not being able to acquire it.
+
+At that time there was a personage at the court whom Marie Antoinette
+particularly detested. It was the cardinal Louis de Rohan, formerly
+ambassador at Vienna, whence he had been recalled in 1774, having
+incurred the queen's displeasure by revealing to the empress Maria
+Theresa the frivolous actions of her daughter, a disclosure which
+brought a maternal reprimand, and for having spoken lightly of Maria
+Theresa in a letter of which Marie Antoinette learned the contents.
+After his return to France the cardinal was anxious to regain the favour
+of the queen in order to obtain the position of prime minister. In March
+1784 he entered into relations with a certain Jeanne de St Remy de
+Valois, a descendant of a bastard of Henry II., who after many
+adventures had married a _soi-disant_ comte de Lamotte, and lived on a
+small pension which the king granted her. This adventuress soon gained
+the greatest ascendancy over the cardinal, with whom she had intimate
+relations. She persuaded him that she had been received by the queen and
+enjoyed her favour; and Rohan resolved to use her to regain the queen's
+good will. The comtesse de Lamotte assured the cardinal that she was
+making efforts on his behalf, and soon announced to him that he might
+send his justification to Marie Antoinette. This was the beginning of a
+pretended correspondence between Rohan and the queen, the adventuress
+duly returning replies to Rohan's notes, which she affirmed to come from
+the queen. The tone of the letters became very warm, and the cardinal,
+convinced that Marie Antoinette was in love with him, became ardently
+enamoured of her. He begged the countess to obtain a secret interview
+for him with the queen, and a meeting took place in August 1784 in a
+grove in the garden at Versailles between him and a lady whom the
+cardinal believed to be the queen herself. Rohan offered her a rose, and
+she promised him that she would forget the past. Later a certain Marie
+Lejay (renamed by the comtesse "Baronne Gay d'Oliva," the last word
+being apparently an anagram of Valoi), who resembled Marie Antoinette,
+stated that she had been engaged to play the role of queen in this
+comedy. In any case the countess profited by the cardinal's conviction
+to borrow from him sums of money destined ostensibly for the queen's
+works of charity. Enriched by these, the countess was able to take an
+honourable place in society, and many persons believed her relations
+with Marie Antoinette, of which she boasted openly and unreservedly, to
+be genuine. It is still an unsettled question whether she simply
+mystified people, or whether she was really employed by the queen for
+some unknown purpose, perhaps to ruin the cardinal. In any case the
+jewellers believed in the relations of the countess with the queen, and
+they resolved to use her to sell their necklace. She at first refused
+their commission, then accepted it. On the 21st of January 1785 she
+announced that the queen would buy the necklace, but that not wishing to
+treat directly, she left the affair to a high personage. A little while
+later Rohan came to negotiate the purchase of the famous necklace for
+the 1,600,000 livres, payable in instalments. He said that he was
+authorized by the queen, and showed the jewellers the conditions of the
+bargain approved in the handwriting of Marie Antoinette. The necklace
+was given up. Rohan took it to the countess's house, where a man, in
+whom Rohan believed he recognized a valet of the queen, came to fetch
+it. Madame de Lamotte had told the cardinal that Marie Antoinette would
+make him a sign to indicate her thanks, and Rohan believed that she did
+make him a sign. Whether it was so, or merely chance or illusion, no one
+knows. But it is certain that the cardinal, convinced that he was acting
+for the queen, had engaged the jewellers to thank her; that Boehmer and
+Bassenge, before the sale, in order to be doubly sure, had sent word to
+the queen of the negotiations in her name; that Marie Antoinette had
+allowed the bargain to be concluded, and that after she had received a
+letter of thanks from Boehmer, she had burned it. Meanwhile the "comte
+de Lamotte" appears to have started at once for London, it is said with
+the necklace, which he broke up in order to sell the stones.
+
+When the time came to pay, the comtesse de Lamotte presented the
+cardinal's notes; but these were insufficient, and Boehmer complained to
+the queen, who told him that she had received no necklace and had never
+ordered it. She had the story of the negotiations repeated for her. Then
+followed a _coup de theatre_. On the 15th of August 1785, Assumption
+day, when the whole court was awaiting the king and queen in order to go
+to the chapel, the cardinal de Rohan, who was preparing to officiate,
+was arrested and taken to the Bastille. He was able, however, to destroy
+the correspondence exchanged, as he thought, with the queen, and it is
+not known whether there was any connivance of the officials, who did not
+prevent this, or not. The comtesse de Lamotte was not arrested until the
+18th of August, after having destroyed her papers. The police set to
+work to find all her accomplices, and arrested the girl Oliva and a
+certain Reteaux de Villette, a friend of the countess, who confessed
+that he had written the letters given to Rohan in the queen's name, and
+had imitated her signature on the conditions of the bargain. The famous
+charlatan Cagliostro was also arrested, but it was recognized that he
+had taken no part in the affair. The cardinal de Rohan accepted the
+parlement of Paris as judges. A sensational trial resulted (May 31,
+1786) in the acquittal of the cardinal, of the girl Oliva and of
+Cagliostro. The comtesse de Lamotte was condemned to be whipped, branded
+and shut up in the Salpetriere. Her husband was condemned, in his
+absence, to the galleys for life. Villette was banished.
+
+Public opinion was much excited by this trial. It is generally believed
+that Marie Antoinette was stainless in the matter, that Rohan was an
+innocent dupe, and that the Lamottes deceived both for their own ends.
+People, however, persisted in the belief that the queen had used the
+countess as an instrument to satisfy her hatred of the cardinal de
+Rohan. Various circumstances fortified this belief, which contributed to
+render Marie Antoinette very unpopular--her disappointment at Rohan's
+acquittal, the fact that he was deprived of his charges and exiled to
+the abbey of la Chaise-Dieu, and finally the escape of the comtesse de
+Lamotte from the Salpetriere, with the connivance, as people believed,
+of the court. The adventuress, having taken refuge abroad, published
+_Memoires_ in which she accused the queen. Her husband also wrote
+_Memoires_, and lived until 1831, after having, it is said, received
+subsidies from Louis XVIII.
+
+  See M. Tourneux, _Marie Antoinette devant l'histoire: Essai
+  bibliographique_ (2nd ed., Paris, 1901); Emile Campardon, _Marie
+  Antoinette et le proces du collier_ (Paris, 1863); P. Audebert,
+  _L'Affaire du collier de la reine, d'apres la correspondance inedite
+  du chevalier de Pujol_ (Rouen, 1901); F. d'Albini, _Marie Antoinette
+  and the Diamond Necklace from another Point of View_ (London, 1900);
+  Funck-Brentano, _L'Affaire du collier_ (1903); A. Lang, _Historical
+  Mysteries_ (1904). Carlyle's essay on _The Diamond Necklace_ (first
+  published in 1837 in _Fraser's Magazine_) is of historical literary
+  interest.
+
+
+
+
+DIANA, in Roman mythology, an old Italian goddess, in later times
+identified with the Greek Artemis (q.v.). That she was originally an
+independent Italian deity is shown by her name, which is the feminine
+form of Janus (= Dianus). She is essentially the goddess of the moon and
+light generally, and presides over wood, plain and water, the chase and
+war. As the goddess of childbirth, she was known, like Juno, by the name
+of Lucina, the "bringer to light." As the moon-goddess she was also
+identified with Hecate, and invoked as "three-formed" in reference to
+the phases of the moon. Her most celebrated shrine was in a grove at
+Aricia (whence her title of Nemorensis) near the modern lake of Nemi.
+Here she was worshipped side by side with a male deity Virbius, a god of
+the forest and the chase. This Virbius was subsequently identified with
+Hippolytus, the favourite of Artemis, who was said to have been brought
+to life by Aesculapius and conducted by Diana to Aricia (Ovid, _Fasti_,
+iii. 263, vi. 731, _Metam._ xv. 497; Virgil, _Aeneid_, vii. 761). A
+barbarous custom, perhaps reminiscent of human sacrifice once offered to
+her, prevailed in connexion with her ritual here; her priest, called
+_Rex Nemorensis_, who was a runaway slave, was obliged to qualify for
+office by slaying his predecessor in single combat (Strabo v. p. 239;
+Suetonius, _Caligula_, 35). This led to the identification of Diana with
+the Tauric Artemis, whose image was said to have been removed by Orestes
+to the grove of Aricia (see ARICINI).
+
+After the destruction of Alba Longa this grove was for a long time the
+united sanctuary of the neighbouring Latin and Rutulian cities, until at
+last it was extinguished beneath the supremacy of Rome. The festival of
+the goddess was on the ides (13th) of August, the full moon of the hot
+season. She was worshipped with torches, her aid was sought by women
+seeking a happy deliverance in childbirth, and many votive offerings
+have been found on the site. The worship of Diana was brought to Rome by
+Latin plebeians, and hence she was regarded as the protectress of the
+lower classes, and especially of slaves. In accordance with this, her
+most important temple was that on the Aventine, the chief seat of the
+plebeians, founded by Servius Tullius, originally as a sanctuary of the
+Latin league (Dion. Halic. iv. 26). No man was allowed to enter the
+temple, and on the day of its dedication (August 13) the slaves kept
+holiday (Plutarch, _Quaest. Rom._ 100). This Diana was identified with
+the sister of Apollo, and at the secular games she was worshipped simply
+as Artemis. Another celebrated sanctuary of Diana was that on the slopes
+of Mount Tifata near Capua (where she was worshipped under the name of
+Tifatina), a sanctuary specially favoured by Sulla and Vespasian. As
+Noctiluca ("giving light by night") she had a sanctuary on the Palatine
+which was kept illuminated throughout the night (Varro, _L.L._ v. 68).
+On the Nemi priesthood see J. G. Frazer, _Golden Bough_.
+
+
+
+
+DIANA MONKEY, a West African representative of the guenon monkeys taking
+its name, _Cercopithecus diana_, from the presence of a white crescent
+on the forehead; another characteristic feature being the pointed white
+beard. The general colour of the fur is greyish, with a deep tinge of
+chestnut from the middle of the back to the root of the tail. Together
+with _C. neglectus_ of East and Central Africa, _C. ignitus_ of Liberia,
+and _C. roloway_ of the Gold Coast, the diana represents the special
+subgenus of guenons known as _Pogonocebus_. Although the diana monkey is
+commonly seen in menageries, little is known of its habits in the wild
+state.
+
+
+
+
+DIANE DE FRANCE (1538-1619), duchess of Montmorency and Angouleme, was
+the natural daughter of Henry II. of France and a young Piedmontese,
+Filippe Duc. The constable de Montmorency went so far as to assert that
+of all the children of Henry II. Diane was the only one who resembled
+him. Catherine de' Medici was greatly incensed at this affront, and took
+her revenge by having the constable disgraced on the death of Henry II.
+Brantome is loud in praise of Diane. She was a perfect horsewoman and
+dancer, played several musical instruments, knew Spanish and Italian,
+and "estoit tres belle de visage et de taille." Legitimated in 1547, she
+was married in 1553 to Horace Farnese, second son of the duke of Parma,
+but her husband was killed soon afterwards at the siege of Hesdin. In
+order to assure his position, the constable de Montmorency wished to
+marry her to his eldest son, Francis. This was a romantic adventure, for
+Francis had clandestinely married Mademoiselle de Piennes. The constable
+dissolved this union, and after lengthy negotiations obtained the
+dispensation of the pope. On the 3rd of May 1559 Francis married Diane.
+A wise and moderate woman, Diane undoubtedly helped to make Francis de
+Montmorency one of the leaders of the party of the _politiques_. Again a
+widow in 1579, she had some influence at the court of Henry III., and
+negotiated his reconciliation with Henry of Navarre (1588). She retained
+her influence in the reign of Henry IV., conveyed the bodies of
+Catherine de' Medici and Henry III. to St Denis, and died in 1619 at her
+hotel of Angouleme.
+
+  See Brantome, ed. by Lalanne, in the _Coll de la societe d'histoire de
+  France_, vol. viii. (1875); J. de Thou, _Historia sui temporis..._
+  (1733); Matthieu de Morgues, _Oraison funebre de Diane de France_
+  (Paris, 1619).
+
+
+
+
+DIANE DE POITIERS (1499-1566), duchess of Valentinois, and mistress of
+Henry II. of France, was the daughter of Jean de Poitiers, seigneur de
+St Vallier, who came of an old family of Dauphine. In 1515 she married
+Louis de Breze, grand seneschal of Normandy, by whom she had two
+daughters. She became a widow in 1533, but soon replaced her husband by
+a more illustrious lover, the king's second son, Henry, who became
+dauphin in 1536. Although he was ten years younger than Diane, she
+inspired the young prince with a profound passion, which lasted until
+his death. The accession of Henry II. in 1547 was also the accession of
+Diane: she was virtual queen, while Henry's lawful wife, Catherine de'
+Medici, lived in comparative obscurity. The part Diane played, however,
+must not be exaggerated. More rapacious than ambitious, she concerned
+herself little with government, but devoted her energies chiefly to
+augmenting her income, and providing for her family and friends. Henry
+was the most prodigal of lovers, and gave her all rights over the duchy
+of Valentinois. Although she showed great tact in her dealings with the
+queen, Catherine drove her from the court after Henry's death, and
+forced her to restore the crown jewels and to accept Chaumont in
+exchange for Chenonceaux. Diane retired to her chateau at Anet, where
+she died in 1566.
+
+Several historians relate that she had been the mistress of Francis I.
+before she became the dauphin's mistress, and that she gave herself to
+the king in order to obtain the pardon of her father, who had been
+condemned to death as an accomplice of the constable de Bourbon. This
+rumour, however, has no serious foundation. Men vied with each other in
+celebrating Diane's beauty, which, if we may judge from her portraits,
+has been slightly exaggerated. She was a healthy, vigorous woman, and,
+by dint of great pains, succeeded in retaining her beauty late into
+life. It is said that even on the coldest mornings she would wash her
+face with well water. Diane was a patroness of the arts. She entrusted
+to Philibert de l'Orme the building of her chateau at Anet, and it was
+for her that Jean Goujon executed his masterpiece, the statue of Diana,
+now in the Louvre.
+
+  See G. Guiffrey, _Lettres inedites de Diane de Poytiers_ (Paris, 1866)
+  and _Proces criminel de Jehan de Poytiers_ (Paris, 1867); Capefigue,
+  _Diane de Poitiers_ (Paris, 1860); Hay, _Madame Dianne de Poytiers_
+  (London, 1900).
+
+
+
+
+DIAPASON (Gr. [Greek: dia pason], through all), a term in music,
+originally for an interval of an octave. The Greek is an abbreviation of
+[Greek: he dia pason chordon symphonia], a consonance through all the
+tones of the scale. In this sense it is only used now, loosely, for the
+compass of an instrument or voice, or for a harmonious melody. The name
+is given to the two foundation stops of an organ, the open and the
+stopped diapason (see ORGAN), and to a standard of musical pitch, as in
+the French _diapason normal_ (see PITCH, MUSICAL).
+
+
+
+
+DIAPER (derived through the Fr, from the Gr. [Greek: dia], through, and
+[Greek: aspros], white; the derivation from the town of Ypres,
+"d'Ypres," in Belgium is unhistorical, as diapers were known for
+centuries before its existence), the name given to a textile fabric,
+formerly of a rich and costly nature with embroidered ornament, but now
+of linen or cotton, with a simple woven pattern; and particularly
+restricted to small napkins. In architecture, the term "diaper" is given
+to any small pattern of a conventional nature repeated continuously and
+uniformly over a surface; the designs may be purely geometrical, or
+based on floral forms, and in early examples were regulated by the
+process of their textile origin. Subsequently, similar patterns were
+employed in the middle ages for the surface decoration of stone, as in
+Westminster Abbey and Bayeux cathedral in the spandrils of the arcades
+of the choir and nave; also in mural painting, stained glass, incised
+brasses, encaustic tiles, &c. Probably in most cases the pattern was
+copied, so far as the general design is concerned, from the tissues and
+stuffs of Byzantine manufacture, which came over to Europe and were
+highly prized as ecclesiastical vestments.
+
+[Illustration: A B C]
+
+  In its textile use, the term diaper was originally applied to silk
+  patterns of a geometrical pattern; it is now almost exclusively used
+  for diamond patterns made from linen or cotton yarns. An illustration
+  of two patterns of this nature is shown in the figure. The floats of
+  the warp and the weft are mostly in three; indeed the patterns are
+  made from a base weave which is composed entirely of floats of this
+  number. It will be seen that both designs are formed of what may be
+  termed concentric figures--alternately black and white. Pattern B
+  differs from pattern A only in that more of these concentric figures
+  are used for the complete figure. If pattern B, which shows only one
+  unit, were extended, the effect would be similar to A, except for the
+  size of the unit. In A there are four complete units, and hence the
+  pattern appears more striking. Again, the repeating of B would cause
+  the four corner pieces to join and to form a diamond similar to the
+  one in the centre. The two diamonds in B would then alternate
+  diagonally to left and right. Special names are given to certain kinds
+  of diapers, e.g. "bird's-eye," "pheasant's-eye"; these terms indicate,
+  to a certain extent, the size of the complete diamond in the
+  cloth--the smaller kind taking the name "bird's-eye." The size of the
+  pattern on paper has little connexion with the size of the pattern in
+  the cloth, for it is clearly the number of threads and picks per inch
+  which determine the size of the pattern in the cloth from any given
+  design. Although A is larger than what is usually termed the
+  "bird's-eye" pattern, it is evident that it may be made to appear as
+  such, provided that the cloth is fine enough. These designs, although
+  adapted mostly for cloths such as nursery-diapers, for pinafores, &c.,
+  are sometimes used in the production of towels and table-cloths. In
+  the figure, the first pick in A is identical with the first pick in B,
+  and the part C shows how each interweaves with the twenty-four
+  threads.
+
+
+
+
+DIAPHORETICS (from Gr. [Greek: diaphorein], to carry through), the name
+given to those remedies which promote perspiration. In health there is
+constantly taking place an exhalation of watery vapour from the skin, by
+which not only are many of the effete products of nutrition eliminated,
+but the body is kept cool. Under exertion or in a heated atmosphere this
+natural function of the skin is increased, sweating more or less profuse
+follows, and, evaporation going on rapidly over the whole surface,
+little or no rise in the temperature of the body takes place. In many
+forms of disease, such as fevers and inflammatory affections, the
+action of the skin is arrested, and the surface of the body feels harsh
+and dry, while the temperature is greatly elevated. The occurrence of
+perspiration not unfrequently marks a crisis in such diseases, and is in
+general regarded as a favourable event. In some chronic diseases, such
+as diabetes and some cases of Bright's disease, the absence of
+perspiration is a marked feature; while, on the other hand, in many
+wasting diseases, such as phthisis, the action of the skin is increased,
+and copious exhausting sweating occurs. Many means can be used to induce
+perspiration, among the best known being baths, either in the form of
+hot vapour or hot water baths, or in that part of the process of the
+Turkish bath which consists in exposing the body to a dry and hot
+atmosphere. Such measures, particularly if followed by the drinking of
+hot liquids and the wrapping of the body in warm clothing, seldom fail
+to excite copious perspiration. Numerous medicinal substances have the
+same effect.
+
+
+
+
+DIAPHRAGM (Gr. [Greek: diaphragma], a partition). The diaphragm or
+midriff (Anglo-Saxon, _mid_, middle, _hrif_, belly) in human anatomy is
+a large fibro-muscular partition between the cavities of the thorax and
+abdomen; it is convex toward the thorax, concave toward the abdomen, and
+consists of a central tendon and a muscular margin. The _central tendon_
+(q, fig. 1) is trefoil in shape, its leaflets being right, left and
+anterior; of these the right is the largest and the left the smallest.
+The fleshy fibres rise, in front from the back of the xiphoid cartilage
+of the sternum (d), laterally by six serrations, from the inner surfaces
+of the lower six ribs, interdigitating with the transversalis,
+posteriorly from the arcuate ligaments, of which there are five, a pair
+of external, a pair of internal, and a single median one. The _external
+arcuate ligament_ (h) stretches from the tip of the twelfth rib (b) to
+the costal process of the first lumbar vertebra in front of the
+quadratus lumborum muscle (o), the _internal_ and _middle_ are
+continuations of the _crura_ which rise from the ventro-lateral aspects
+of the bodies of the lumbar vertebrae, the right (e) coming from three,
+the left (f) from two. On reaching the level of the twelfth thoracic
+vertebra each crus spreads out into a fan-shaped mass of fibres, of
+which the innermost join their fellows from the opposite crus, in front
+of the aortic opening (k), to form the _middle arcuate ligament_; the
+outer ones (g) arch in front of the psoas muscle (n) to the tip of the
+costal process of the first lumbar vertebra to form the _internal
+arcuate ligament_, while the intermediate ones pass to the central
+tendon. There are three large openings in the diaphragm; the _aortic_
+(k) is behind the middle arcuate ligament and transmits the aorta, the
+vena azygos major, and the thoracic duct. In the right leaflet is an
+opening (sometimes called the _hiatus quadratus_) for the inferior vena
+cava and a branch of the right phrenic nerve (m), while in front and a
+little to the left of the aortic opening is one for the oesophagus and
+the two pneumogastric nerves (l), the left being in front and the right
+behind. The fleshy fibres on each side of this opening act as a
+sphincter. Passing between the xiphoid and costal origins in front are
+the superior epigastric arteries, while the other terminal branches of
+the internal mammaries, the musculo-phrenics, pass through between two
+costal origins.
+
+[Illustration: FIG. 1.--Abdominal Surface of the Diaphragm.]
+
+Through the crura pass the splanchnic nerves, and in addition to these
+the left crus is pierced by the vena azygos minor. The sympathetic
+nerves usually enter the abdomen behind the internal arcuate ligaments.
+The phrenic nerves, which are the main supply of the diaphragm, divide
+before reaching the muscle and pierce it in a number of places to enter
+its abdominal surface, but some of the lower intercostal nerves assist
+in the supply. The last thoracic or subcostal nerves pass behind the
+external arcuate ligament.
+
+For the action of the diaphragm see RESPIRATORY SYSTEM.
+
+  _Embryology._--The diaphragm is at first developed in the neck region
+  of the embryo, and this accounts for the phrenic nerves, which supply
+  it, rising from the fourth and fifth cervical. From the mesoderm on
+  the caudal side of the pericardium is developed the _septum
+  transversum_, and in this the central tendon is formed. The fleshy
+  portion is developed on each side in two parts, an anterior or
+  sterno-costal which is derived from the longitudinal neck musculature,
+  probably the same layer from which the sternothyroid comes, and a
+  spinal part which is a derivative of the transversalis sheet of the
+  trunk. Between these two parts is at one time a gap, the _spino-costal
+  hiatus_, and this is obliterated by the growth of the
+  pleuro-peritoneal membrane, which may occasionally fail to close and
+  so may form the site of a phrenic hernia. With the growth of the body
+  and the development of the lungs the diaphragm shifts its position
+  until it becomes the septum between the thoracic and abdominal
+  cavities. (See A. Keith, "On the Development of the Diaphragm," _Jour.
+  of Anat. and Phys._ vol. 39.) A. Paterson has recorded cases in which
+  the left half of the diaphragm is wanting (_Proceedings_ of the
+  Anatomical Society of Gt. Britain, June 1900; _Jour. of Anat. and
+  Phys._ vol. 34), and occasionally deficiencies are found elsewhere,
+  especially in the sternal portion. For further details see Quain's
+  _Anatomy_, vol. i. (London, 1908).
+
+  _Comparative Anatomy._--A complete diaphragm, separating the thoracic
+  from the abdominal parts of the coelom, is characteristic of the
+  Mammalia; it usually has the human structure and relations except that
+  below the Anthropoids it is separated from the pericardium by the
+  azygous lobe of the lung. In some Mammals, e.g. Echidna and Phocoena,
+  it is entirely muscular. In the Cetacea it is remarkable for its
+  obliquity; its vertebral attachment is much nearer the tail than its
+  sternal or ventral one; this allows a much larger lung space in the
+  dorsal than in the ventral part of the thorax, and may be concerned
+  with the equipoise of the animal. (Otto Muller, "Untersuchungen uber
+  die Veranderung, welche die Respirationsorgane der Saugetiere durch
+  die Anpassung an das Leben im Wasser erlitten haben," _Jen. Zeitschr.
+  f. Naturwiss._, 1898, p. 93.) In the Ungulata only one crus is found
+  (Windle and Parsons, "Muscles of the Ungulata," _Proc. Zool. Soc._,
+  1903, p. 287). Below the Mammals incomplete partitions between the
+  pleural and peritoneal cavities are found in Chelonians, Crocodiles
+  and Birds, and also in Amphibians (Xenopus and Pipa).     (F. G. P.)
+
+
+
+
+DIARBEKR[1] (_Kara Amid_ or Black Amid; the Roman _Amida_), the chief
+town of a vilayet of Asiatic Turkey, situated on a basaltic plateau on
+the right bank of the Tigris, which here flows in a deep open valley.
+The town is still surrounded by the masonry walls of black basalt which
+give it the name of _Kara_ or Black Amid; they are well built and
+imposing on the west facing the open country, but almost in ruins where
+they overlook the river. A mass of gardens and orchards cover the slope
+down to the river on the S.W., but there are no suburbs outside the
+walls. The houses are rather crowded but only partially fill the walled
+area. The population numbers 38,000, nearly half being Christian,
+comprising Turks, Kurds, Arabs, Turkomans, Armenians, Chaldeans,
+Jacobites and a few Greeks. The streets are 10 ft. to 15 ft. wide, badly
+paved and dirty; the houses and shops are low, mostly of stone, and some
+of stone and mud. The bazaar is a good one, and gold and silver filigree
+work is made, peculiar in character and design. The cotton industry is
+declining, but manufacture of silk is increasing. Fruit is good and
+abundant as the rich volcanic soil is well watered from the town
+springs. The size of the melons is specially famous. To the south, the
+walls are some 40 ft. high, faced with large cut stone blocks of very
+solid construction, with towers and square bastions rising to 500 ft.
+There are four gates: on the north the Kharput gate, on the west the
+Rum, on the south the Mardin, and on the east the Yeni Kapu or new
+gate. A citadel enclosure stands at the N. E. corner and is now partly
+in ruins, but the interior space is occupied by the government konak.
+The summer climate in the confined space within the town is excessively
+hot and unhealthy. Epidemics of typhus are not unknown, as well as
+ophthalmia. The Diarbekr boil is like the "Aleppo button," lasting a
+long time and leaving a deep scar. Winters are frequently severe but do
+not last long. Snow sometimes lies, and ice is stored for summer use.
+Scorpions noted for the virulence of their poison abound as well as
+horse leeches in the tanks. The town is supplied with water both by
+springs inside the town and by aqueducts from fountains at Ali Punar and
+Hamervat. The principal exports are wool, mohair and copper ore, and
+imports are cotton and woollen goods, indigo, coffee, sugar, petroleum,
+&c.
+
+The Great Mosque, Ulu Jami, formerly a Christian church, occupies the
+site of a Sassanian palace and was built with materials from an older
+palace, probably that of Tigranes II. The remains consist of the facades
+of two palaces 400 ft. apart, each formed by a row of Corinthian columns
+surmounted by an equal number of a Byzantine type. Kufic inscriptions
+run across the fronts under the entablature. The court of the mosque is
+entered by a gateway on which lions and other animals are sculptured.
+The churches of greatest interest are those of SS. Cosmas and Damian
+(Jacobite) and the church of St James (Greek). In the 19th century
+Diarbekr was one of the largest and most flourishing cities of Asia, and
+as a commercial centre it now stands at the meeting-point of several
+important routes. It is at the head of the navigation of the Tigris,
+which is traversed down stream by _keleks_ or rafts supported by
+inflated skins. There is a good road to Aleppo and Alexandretta on the
+Mediterranean, and to Samsun on the Black Sea by Kharput, Malatia and
+Sivas. There are also routes to Mosul and Bitlis.
+
+Diarbekr became a Roman colony in A.D. 230 under the name of Amida, and
+received a Christian bishop in A.D. 325. It was enlarged and
+strengthened by Constantius II., in whose reign it was taken after a
+long siege by Shapur (Sapor) II., king of Persia. The historian Ammianus
+Marcellinus, who took part in the defence, gives a detailed account of
+it. In the later wars between the Persians and Romans it more than once
+changed hands. Though ceded by Jovian to the Persians it again became
+annexed to the Roman empire, and in the reign of Anastasius (A.D. 502)
+was once more taken by the Persians, when 80,000 of its inhabitants were
+slain. It was taken c. 638 by the Arabs, and afterwards passed into the
+hands of the Seljuks and Persians, from whom it was finally captured by
+Selim I. in 1515; and since that date it has remained under Ottoman
+rule. About 2 m. below the town is a masonry bridge over the Tigris; the
+older portion being probably Roman, and the western part, which bears a
+Kufic inscription, being Arab.
+
+The vilayet of Diarbekr extends south from Palu on the Euphrates to
+Mardin and Nisibin on the edge of the Mesopotamian plain, and is divided
+into three sanjaks--Arghana, Diarbekr and Mardin. The headwaters of the
+main arm of the Tigris have their source in the vilayet.
+
+Cereals, cotton, tobacco, rice and silk are produced, but most of the
+fertile lands have been abandoned to semi-nomads, who raise large
+quantities of live stock. The richest portion of the vilayet lies east
+of the capital in the rolling plains watered by tributaries of the
+Tigris. An exceptionally rich copper mine exists at Arghana Maden, but
+it is very imperfectly worked; galena mineral oil and silicious sand are
+also found.     (C. W. W.; F. R. M.)
+
+
+FOOTNOTE:
+
+  [1] From _Diar_, land, and Bekr (i.e. Abu Bekr, the caliph).
+
+
+
+
+DIARRHOEA (from Gr. [Greek: dia], through, [Greek: rheo], flow), an
+excessive looseness of the bowels, a symptom of irritation which may be
+due to various causes, or may be associated with some specific disease.
+The treatment in such latter cases necessarily varies, since the symptom
+itself may be remedial, but in ordinary cases depends on the removal of
+the cause of irritation by the use of aperients, various sedatives being
+also prescribed. In chronic diarrhoea careful attention to the diet is
+necessary.
+
+
+
+
+DIARY, the Lat. _diarium_ (from _dies_, a day), the book in which are
+preserved the daily memoranda regarding events and actions which come
+under the writer's personal observation, or are related to him by
+others. The person who keeps this record is called a diarist. It is not
+necessary that the entries in a diary should be made each day, since
+every life, however full, must contain absolutely empty intervals. But
+it is essential that the entry should be made during the course of the
+day to which it refers. When this has evidently not been done, as in the
+case of Evelyn's diary, there is nevertheless an effort made to give the
+memoranda the effect of being so recorded, and in point of fact, even in
+a case like that of Evelyn, it is probable that what we now read is an
+enlargement of brief notes jotted down on the day cited. When this is
+not approximately the case, the diary is a fraud, for its whole value
+depends on its instantaneous transcript of impressions.
+
+In its primitive form, the diary must always have existed; as soon as
+writing was invented, men and women must have wished to note down, in
+some almanac or journal, memoranda respecting their business, their
+engagements or their adventures. But the literary value of these would
+be extremely insignificant until the spirit of individualism had crept
+in, and human beings began to be interesting to other human beings for
+their own sake. It is not, therefore, until the close of the Renaissance
+that we find diaries beginning to have literary value, although, as the
+study of sociology extends, every scrap of genuine and unaffected record
+of early history possesses an ethical interest. In the 17th century,
+diaries began to be largely written in England, although in most cases
+without any idea of even eventual publication. Sir William Dugdale
+(1605-1686) had certainly no expectation that his slight diary would
+ever see the light. There is no surviving record of a journal kept by
+Clarendon, Richard Baxter, Lucy Hutchinson and other autobiographical
+writers of the middle of the century, but we may take it for granted
+that they possessed some such record, kept from day to day. Bulstrode
+Whitelocke (1605-1675), whose _Memorials of the English Affairs_ covers
+the ground from 1625 to 1660, was a genuine diarist. So was the elder
+George Fox (1624-1690), who kept not merely "a great journal," but "the
+little journal books," and whose work was published in 1694. The famous
+diary of John Evelyn (1620-1706) professes to be the record of seventy
+years, and, although large tracts of it are covered in a very
+perfunctory manner, while in others many of the entries have the air of
+having been written in long after the event, this is a very interesting
+and amusing work; it was not published until 1818. In spite of all its
+imperfections there is a great charm about the diary of Evelyn, and it
+would hold a still higher position in the history of literature than it
+does if it were not overshadowed by what is unquestionably the most
+illustrious of the diaries of the world, that of Samuel Pepys
+(1633-1703). This was begun on the 1st of January 1660 and was carried
+on until the 29th of May 1669. The extraordinary value of Pepys' diary
+consists in its fidelity to the portraiture of its author's character.
+He feigns nothing, conceals nothing, sets nothing down in malice or
+insincerity. He wrote in a form of shorthand intelligible to no one but
+himself, and not a phrase betrays the smallest expectation that any eye
+but his own would ever investigate the pages of his confession. The
+importance of this wonderful document, in fact, lay unsuspected until
+1819, when the Rev. John Smith of Baldock began to decipher the MS. in
+Magdalene College, Cambridge. It was not until 1825 that Lord Braybrooke
+published part of what was only fully edited, under the care of Mr
+Wheatley, in 1893-1896. In the age which succeeded that of Pepys, a
+diary of extraordinary emotional interest was kept by Swift from 1710 to
+1713, and was sent to Ireland in the form of a "Journal to Stella"; it
+is a surprising amalgam of ambition, affection, wit and freakishness.
+John Byrom (1692-1763), the Manchester poet, kept a journal, which was
+published in 1854. The diary of the celebrated dissenting divine, Philip
+Doddridge (1702-1751), was printed in 1829. Of far greater interest are
+the admirably composed and vigorously written journals of John Wesley
+(1703-1791). But the most celebrated work of this kind produced in the
+latter half of the 18th century was the diary of Fanny Burney (Madame
+D'Arblay), published in 1842-1846. It will be perceived that, without
+exception, these works were posthumously published, and the whole
+conception of the diary has been that it should be written for the
+writer alone, or, if for the public, for the public when all prejudice
+shall have passed away and all passion cooled down. Thus, and thus only,
+can the diary be written so as to impress upon its eventual readers a
+sense of its author's perfect sincerity and courage.
+
+Many of the diaries described above were first published in the opening
+years of the 19th century, and it is unquestionable that the interest
+which they awakened in the public led to their imitation. Diaries ceased
+to be rare, but as a rule the specimens which have hitherto appeared
+have not presented much literary interest. Exception must be made in
+favour of the journals of two minor politicians, Charles Greville
+(1794-1865) and Thomas Creevey (1768-1838), whose indiscretions have
+added much to the gaiety of nations; the papers of the former appeared
+in 1874-1887, those of the latter in 1903. The diary of Henry Crabb
+Robinson (1775-1867), printed in 1869, contains excellent biographical
+material. Tom Moore's journal, published in 1856 by Lord John Russell,
+disappointed its readers. But it is probable, if we reason by the
+analogy of the past, that the most curious and original diaries of the
+19th century are still unknown to us, and lie jealously guarded under
+lock and key by the descendants of those who compiled them.
+
+It was natural that the form of the diary should appeal to a people so
+sensitive to social peculiarities and so keen in the observation of them
+as the French. A medieval document of immense value is the diary kept by
+an anonymous _cure_ during the reigns of Charles VI. and Charles VII.
+This _Journal d'un bourgeois de Paris_ was kept from 1409 to 1431, and
+was continued by another hand down to 1449. The marquis de Dangeau
+(1638-1720) kept a diary from 1684 till the year of his death; this
+although dull, and as Saint-Simon said "of an insipidity to make you
+sick," is an inexhaustible storehouse of facts about the reign of Louis
+XIV. Saint-Simon's own brilliant memoirs, written from 1691 to 1723, may
+be considered as a sort of diary. The lawyer, Edmond Barbier
+(1689-1771), wrote a journal of the anecdotes and little facts which
+came to his knowledge from 1718 to 1762. The studious care which he took
+to be correct, and his manifest candour, give a singular value to
+Barbier's record; his diary was not printed at all until 1847, nor, in
+its entirety, until 1857. The song-writer, Charles Colle (1709-1783),
+kept a _journal historique_ from 1758 to 1782; it is full of vivacity,
+but very scandalous and spiteful. It saw the light in 1805, and
+surprised those to whom Colle, in his lifetime, had seemed the most
+placid and good-natured of men. Petit de Bachaumont (1690-1770) had
+access to remarkable sources of information, and his _Memoires secrets_
+(a diary the publication of which began in 1762 and was continued after
+Bachaumont's death, until 1787, by other persons) contains a valuable
+mass of documents. The marquis d'Argenson (1694-1757) kept a diary, of
+which a comparatively full text was first published in 1859. In recent
+times the posthumous publication of the diaries of the Russian artist,
+Marie Bashkirtseff (1860-1884), produced a great sensation in 1887, and
+revealed a most remarkable temperament. The brothers Jules and Edmond de
+Goncourt kept a very minute diary of all that occurred around them in
+artistic and literary Paris; after the death of Jules, in 1870, this was
+continued by Edmond, who published the three first volumes in 1888. The
+publication of this work was continued, and it produced no little
+scandal. It is excessively ill-natured in parts, but of its vivid
+picturesqueness, and of its general accuracy as a transcript of
+conversation, there can be no two opinions.     (E. G.)
+
+
+
+
+DIASPORE, a native aluminium hydroxide, AlO(OH), crystallizing in the
+orthorhombic system and isomorphous with gothite and manganite. It
+occurs sometimes as flattened crystals, but usually as lamellar or scaly
+masses, the flattened surface being a direction of perfect cleavage on
+which the lustre is markedly pearly in character. It is colourless or
+greyish-white, yellowish, sometimes violet in colour, and varies from
+translucent to transparent. It may be readily distinguished from other
+colourless transparent minerals, with a perfect cleavage and pearly
+lustre--mica, talc, brucite, gypsum--by its greater hardness of 6-1/2-7.
+The specific gravity is 3.4. When heated before the blowpipe it
+decrepitates violently, breaking up into white pearly scales; it was
+because of this property that the mineral was named diaspore by R. J.
+Hauy in 1801, from [Greek: diaspeirein], "to scatter." The mineral
+occurs as an alteration product of corundum or emery, and is found in
+granular limestone and other crystalline rocks. Well-developed crystals
+are found in the emery deposits of the Urals and at Chester,
+Massachusetts, and in kaolin at Schemnitz in Hungary. If obtainable in
+large quantity it would be of economic importance as a source of
+alumina.     (L. J. S)
+
+
+
+
+DIASTYLE (from Gr. [Greek: dia], through, and [Greek: stylos], column),
+in architecture, a term used to designate an intercolumniation of three
+or four diameters.
+
+
+
+
+DIATOMACEAE. For the knowledge we possess of these beautiful plants, so
+minute as to be undiscernible by our unaided vision, we are indebted to
+the assistance of the microscope. It was not till towards the close of
+the 18th century that the first known forms of this group were
+discovered by O. F. Muller. And so slow was the process of discovery in
+this field of scientific research that in the course of half a century,
+when Agardh published his _Systema algarum_ in 1824, only forty-nine
+species included under eight genera had been described. Since that time,
+however, with modern microscopes and microscopic methods, eminent
+botanists in all parts of the civilized world have studied these minute
+plants, with the result that the number of known genera and species has
+been greatly increased. Over 10,000 species of diatoms have been
+described, and about 1200 species and numerous varieties occur in the
+fresh waters and on the coasts of Great Britain and Ireland. Rabenhorst,
+in the index to his _Flora Europaea algarum_ (1864) enumerated about
+4000 forms which had up to that time been discovered throughout the
+continent of Europe.
+
+[Illustration: FIG. 1. A and B, _Melosira arenaria._ C-E, _Melosira
+varians._ E, showing formation of auxospore.]
+
+[Illustration: FIG. 2.--_Synedra Ulna._]
+
+The diatoms are more commonly known among systematic botanists as the
+Bacillarieae, particularly on the continent of Europe, and although such
+an immense number of very diverse forms are included in it, the group as
+a whole exhibits a remarkable uniformity of structure. The Bacillarieae
+is one of the large groups of Algae, placed by some in close proximity
+to the Conjugatae and by others as an order of the Brown Algae (or
+Phaeophyceae), but their characters are so distinctive and their
+structure is so uniform as to warrant the separation of the diatoms as a
+distinct class. The affinities of the group are doubtful.
+
+The diatoms exhibit great variety of form. While some species are
+circular and more or less disk-shaped, others are oval in outline. Some
+are linear, as _Synedra Ulna_ (fig. 2), others more or less crescentic;
+others again are cuneate, as _Podosphenia Lyngbyii_ (fig. 3); some few
+have a sigmoid outline, as _Pleurosigma balticum_ (fig. 4); but the
+prevailing forms are naviculoid, as in the large family Naviculaceae, of
+which the genus _Navicula_ embraces upwards of 1000 species. They vary
+also in their modes of growth,--some being free-floating, others
+attached to foreign bodies by simple or branched gelatinous stalks,
+which in some species are short and thick, while in others they are long
+and slender. In some genera the forms are simple, while in others the
+frustules are connected together in ribbon-like filaments, or form, as
+in other cases, zigzag chains. In some genera the individuals are naked,
+while in many others they are enclosed in a more or less definite
+gelatinous investment. The conditions necessary to their growth are
+moisture and light. Wherever these circumstances coexist, diatomaceous
+forms will almost invariably be found. They occur mixed with other
+organisms on the surface of moist rocks; in streamlets and pools, they
+form a brownish stratum on the surface of the mud, or cover the stems
+and leaves of water plants or floating twigs with a furry investment.
+Marine forms are usually attached to various sea-weeds, and many are
+found in the stomachs of molluscs, holothurians, ascidians and other
+denizens of the ocean. The fresh-water forms are specifically distinct
+from those incidental to salt or brackish water,--fresh-water species,
+however, are sometimes carried some distance into the sea by the force
+of the current, and in tidal rivers marine forms are carried up by the
+force of the tide. Some notion may be formed of the extreme minuteness
+of these forms from the fact that one the length of which is 1/200th of
+an inch may be considered as beyond the medium size. Some few, indeed,
+are much larger, but by far the greater proportion are of very much
+smaller dimensions.
+
+[Illustration: FIG. 3.--_Podosphenia Lyngbyii._]
+
+[Illustration: FIG. 4.--_Pleurosigma balticum._]
+
+[Illustration: FIG. 5. A-C, _Tetracyclus lacustris._ D and E,
+_Tabellaria fenestrata._ F and G, _Tabellaria flocculosa._]
+
+Diatoms are unicellular plants distinguished from kindred forms by the
+fact of having their soft vegetative part covered by a siliceous case.
+Each individual is known as a frustule, and the cell-wall consists of
+two similar valves nearly parallel to each other, each valve being
+furnished with a rim (or connecting-band) projecting from it at a right
+angle.
+
+One of these valves with its rim is slightly smaller than the other,
+the smaller fitting into the larger pretty much as a pill-box fits into
+its cover. This peculiarity of structure affords ample scope for the
+growth of the protoplasmic cell-contents, for as the latter increase in
+volume the siliceous valves are pushed out, and their corresponding
+siliceous rims become broader. The connecting-bands although closely
+fitting their respective valves are distinct from them, and together the
+two bands form the girdle.
+
+An individual diatom is usually described from two aspects, one in which
+the surface of the valve is exposed to view--the valve view, and one in
+which the girdle side is exposed--the girdle view. The valves are thin
+and transparent, convex on the outside, and generally ornamented with a
+variety of sculptured markings. These sculptures often present the
+aspect of striae across the face of the valve, and the best lenses have
+shown them to consist of a series of small cavities within the siliceous
+wall of the cell. The valves of some of the marine genera exhibit a
+beautiful areolated structure due to the presence of larger chambers
+within the siliceous cell-wall. Many diatoms possess thickenings of the
+cell-wall, visible in the valve view, in the centre of the valve and at
+each extremity. These thickenings are known as the nodules, and they are
+generally connected by a long median line, the raphe, which is a cleft
+in the siliceous valve, extending at least some part of its length.
+
+The protoplasmic contents of this siliceous box-like unicell are very
+similar to the contents of many other algal cells. There is a living
+protoplasmic layer or primordial utricle, connected either by two broad
+bands or by a number of anastomosing threads with a central mass of
+protoplasm in which the nucleus is embedded. The greater part of the
+cavity of the cell is occupied by one or several fluid vacuoles. The
+characteristic brown colour of diatoms is due to the presence of
+chromatophores embedded in the lining layer of protoplasm. In number and
+form these chromatophores are variable. They contain chlorophyll, but
+the green colour is masked by the presence of diatomin, a brown pigment
+which resembles that which occurs in the Brown Algae or Phaeophyceae.
+The chromatophores contain a variable number of pyrenoids, colourless
+proteid bodies of a crystalloidal character.
+
+One of the first phenomena which comes under the notice of the observer
+is the extraordinary power of motion with which the frustules are
+endowed. Some species move slowly backwards and forwards in pretty much
+the same line, but in the case of _Bacillaria paradoxa_ the motion is
+very rapid, the frustules darting through the water in a zigzag course.
+To account for this motion various theories have been suggested, none of
+which appear to be altogether satisfactory. There is little doubt that
+the movements are connected with the raphe, and in some diatoms there is
+much evidence to prove that they are due to an exudation of mucilage.
+
+_Classification._--The most natural system of classification of the
+Bacillarieae is the one put forward by Schutt (1896), and since
+generally followed by systematists. He separates them into two primary
+divisions, the 'Centricae' and the 'Pennatae.' The former includes all
+those diatoms which in the valve view possess a radial symmetry around a
+central point, and which are destitute of a raphe (or a pseudoraphe).
+The latter includes those which are zygomorphic or otherwise irregular,
+and in which the valve view is generally boat-shaped or needle-shaped,
+with the markings arranged in a sagittal manner on each side of a raphe
+or pseudoraphe.
+
+_Reproduction._--In the Diatomaceae, as well as in the Desmidieae, the
+ordinary mode of increase is by simple cell-division. The cell-contents
+within the enclosure of the siliceous case separate into two distinct
+masses. As these two daughter-masses become more and more developed, the
+valves of the mother-cell are pushed more and more widely apart. A new
+siliceous valve is secreted by each of the two masses on the side
+opposite to the original valve, the new valves being situated within the
+girdle of the original frustule. When this process has been completed
+the girdle of the mother frustule gives way, and two distinct frustules
+are formed, the siliceous valves in each of these new frustules being
+one of the valves of the mother-cell, and a newly formed valve similar
+and more or less parallel to it.
+
+During the life of the plant this process of self-division is continued
+with an almost incredible rapidity. On this subject the observation of
+Professor William Smith, writing in 1853, is worthy of special
+notice:--"I have been unable to ascertain the time occupied in a single
+act of self-division, but supposing it to be completed in twenty-four
+hours we should have, as the progeny of a single frustule, the amazing
+number of 1,000,000,000 in a single month, a circumstance which will in
+some degree explain the sudden, or at least rapid, appearance of these
+organisms in localities where they were a short time previously either
+unrecognized or sparingly diffused" (_British Diatomaceae_, vol. i. p.
+25).
+
+[Illustration: FIG. 6.--Formation of Auxospores.
+
+  A. _Navicula limosa._
+  B. _Achnanthes flexella._
+  C. _Navicula Amphisbaena._
+  D. _Navicula viridis._]
+
+Individual diatoms when once produced by cell-division are incapable of
+any increase in size owing to the rigidity of their siliceous
+cell-walls, and since the new valves are always formed _within_ the
+girdle of the old ones, it would follow that every succeeding generation
+is reduced in size by the thickness of the girdle. In some diatoms,
+however, this is not strictly true as daughter-cells are sometimes
+produced of larger size than the parent-cells. Thus, the reduction in
+size of the individuals is not always proportionate to the number of
+cell-divisions.
+
+On the diminution in size having reached a limit in any species, the
+maximum size is regained by the formation of an auxospore. There are
+five known methods of reproduction by auxospores, but it is unnecessary
+here to enter into details of these methods. Suffice it to say that a
+normal auxospore is produced by the conjugation of two parent-cells, its
+distinguishing feature being a rejuvenescence accompanied by a marked
+increase in size. These auxospores formed without conjugation are
+parthenogenetic.
+
+_Mode of Preparation._--The Diatomaceae are usually gathered in small
+bottles, and special care should be taken to collect them as free as
+possible from extraneous matter. A small portion having been examined
+under the microscope, should the gathering be thought worthy of
+preservation, some of the material is boiled in acid for the purpose of
+cleaning it. The acids usually employed are hydrochloric, nitric or
+sulphuric, according as circumstances require. When the operator
+considers that by this process all foreign matter has been eliminated,
+the residuum is put into a precipitating jar of a conical shape, broader
+at the bottom than at the top, and covered to the brim with filtered or
+distilled water. When the diatoms have settled in the bottom of the jar,
+the supernatant fluid is carefully removed by a syringe or some similar
+instrument, so that the sediment be not disturbed. The jar is again
+filled with water, and the process repeated till the acid has been
+completely removed. It is desirable afterwards to boil the sediment for
+a short time with supercarbonate of soda, the alkali being removed in
+the same manner as the acid. A small portion may then be placed with a
+pipette upon a slip of glass, and, when the moisture has been thoroughly
+evaporated, the film that remains should be covered with dilute Canada
+balsam, and, a thin glass cover having been gently laid over the balsam,
+the preparation should be laid aside for a short time to harden, and
+then is ready for observation.
+
+_General Remarks._--Diatoms are most abundant in cold latitudes, having
+a general preference for cold water. In the pelagic waters of lakes and
+of the oceans they are often very abundant, and in the cold waters of
+the Arctic and Antarctic Oceans they exist in prodigious numbers. They
+thus form a large proportion of both the marine and the fresh-water
+plankton.
+
+Large numbers of fossil diatoms are known. Not only are these minute
+plants assisting at the present time in the accumulation of oceanic and
+lake deposits, but in former ages they have been sufficiently active to
+give rise to considerable deposits of diatomaceous earths. When the
+plant has fulfilled its natural course the siliceous covering sinks to
+the bottom of the water in which it had lived, and there forms part of
+the sediment. When in the process of ages, as it has often happened, the
+accumulated sediment has been hardened into solid rock, the siliceous
+frustules of the diatoms remain unaltered, and, if the rock be
+disintegrated by natural or artificial means, may be removed from the
+enveloping matrix and subjected to examination under the microscope. The
+forms found may from their character help in some degree to illustrate
+the conditions under which the stratum of rock had been originally
+deposited. These earths are generally of a white or grey colour. Some of
+them are hard, but most are soft and friable. Many of them are of
+economic importance, being used as polishing powders ("Tripoli"), as
+absorbents for nitroglycerin in the manufacture of dynamite
+("Kieselguhr"), as a dentifrice, and more recently they have been used
+to a large extent in the manufacture of non-conducting and sound-proof
+materials. Most of these diatomaceous earths are associated with rocks
+of Tertiary formations, although it is generally regarded that the
+earliest appearance of diatoms is in the Upper Cretaceous (chalk).
+
+Vast deposits of Diatomaceous earths have been discovered in various
+parts of the world,--some the deposit of fresh, others of salt water. Of
+these deposits the most remarkable for extent, as well as for the number
+and beauty of the species contained in it, is that of Richmond, in
+Virginia, one of the United States of America. It extends for many
+miles, and is in some places at least 40 ft. deep. It is a remarkable
+fact that though the generations of a diatom in the space of a few
+months far exceed in number the generation of man during the period
+usually assigned to the existence of the race, the fossil genera and
+species are in most respects to the most minute details identical with
+the numerous living representatives of their class.
+     (E. O'M.; G. S. W.*)
+
+
+
+
+DIAULOS (from Gr. [Greek: di-], double, and [Greek: aulos], pipe), in
+architecture, the peristyle round the great court of the palaestra,
+described by Vitruvius (v. II), which measured two stadia (1200 ft.) in
+length; on the south side this peristyle had two rows of columns, so
+that in stormy weather the rain might not be driven into the inner part.
+The word was also used in ancient Greece for a foot-race of twice the
+usual length.
+
+
+
+
+DIAVOLO, FRA (1771-1806), the popular name given to a famous Italian
+brigand associated with the political revolutions of southern Italy at
+the time of the French invasion. His real name was Michele Pezza, and he
+was born of low parentage at Itri; he had committed many murders and
+robberies in the Terra di Lavoro, but by good luck combined with
+audacity he always escaped capture, whence his name of Fra Diavolo,
+popular superstition having invested him with the characters of a monk
+and a demon, and it seems that at one time he actually was a monk. When
+the kingdom of Naples was overrun by the French and the Parthenopaean
+Republic established (1799), Cardinal Ruffo, acting on behalf of the
+Bourbon king Ferdinand IV., who had fled to Sicily, undertook the
+reconquest of the country, and for this purpose he raised bands of
+peasants, gaol-birds, brigands, &c., under the name of Sanfedisti or
+_bande della Santa Fede_ ("bands of the Holy Faith"). Fra Diavolo was
+made leader of one of them, and waged untiring war against the French
+troops, cutting off isolated detachments and murdering stragglers and
+couriers. Owing to his unrivalled knowledge of the country, he succeeded
+in interrupting the enemy's communications between Rome and Naples. But
+although, like his fellow-brigands under Ruffo, he styled himself "the
+faithful servant and subject of His Sicilian Majesty," wore a military
+uniform and held military rank, and was even created duke of Cassano,
+his atrocities were worthy of a bandit chief. On one occasion he threw
+some of his prisoners, men, women and children, over a precipice, and
+on another he had a party of seventy shot. His excesses while at Albano
+were such that the Neapolitan general Naselli had him arrested and
+imprisoned in the castle of St Angelo, but he was liberated soon after.
+When Joseph Bonaparte was made king of Naples, extraordinary tribunals
+were established to suppress brigandage, and a price was put on Fra
+Diavolo's head. After spreading terror through Calabria, he crossed over
+to Sicily, where he concerted further attacks on the French. He returned
+to the mainland at the head of 200 convicts, and committed further
+excesses in the Terra di Lavoro; but the French troops were everywhere
+on the alert to capture him and he had to take refuge in the woods of
+Lenola. For two months he evaded his pursuers, but at length, hungry and
+ill, he went in disguise to the village of Baronissi, where he was
+recognized and arrested, tried by an extraordinary tribunal, condemned
+to death and shot. In his last moments he cursed both the Bourbons and
+Admiral Sir Sidney Smith for having induced him to engage in this
+reckless adventure (1806). Although his cruelty was abominable, he was
+not altogether without generosity, and by his courage and audacity he
+acquired a certain romantic popularity. His name has gained a world-wide
+celebrity as the title of a famous opera by Auber.
+
+  The best known account of Fra Diavolo is in Pietro Colletta's _Storia
+  del reame di Napoli_ (2nd ed., Florence, 1848); B. Amante's _Fra
+  Diavolo e il suo tempo_ (Florence, 1904) is an attempted
+  rehabilitation; but A. Luzio, whose account in _Profili e bozzetti
+  storici_ (Milan, 1906) gives the latest information on the subject,
+  has demolished Amante's arguments.     (L. V.*)
+
+
+
+
+DIAZ, NARCISSE VIRGILIO (1808-1876), French painter, was born in
+Bordeaux of Spanish parents, on the 25th of August 1808. At first a
+figure-painter who indulged in strong colour, in his later life Diaz
+became a painter of the forest and a "tone artist" of the first order.
+He spent much time at Barbizon; and although he is the least exalted of
+the half-dozen great artists who are usually grouped round that name, he
+sometimes produced works of the highest quality. At the age of ten Diaz
+became an orphan, and misfortune dogged his earlier years. His foot was
+bitten by a reptile in Meudon wood, near Sevres, where he had been taken
+to live with some friends of his mother. The bite was badly dressed, and
+ultimately it cost him his leg. Afterwards his wooden stump became
+famous. At fifteen he entered the studios at Sevres, where the
+decoration of porcelain occupied him; but tiring of the restraint of
+fixed hours, he took to painting Eastern figures dressed in richly
+coloured garments. Turks and Oriental scenes attracted him, and many
+brilliant gems remain of this period. About 1831 Diaz encountered
+Theodore Rousseau, for whom he entertained a great veneration, although
+Rousseau was four years his junior; but it was not until ten years later
+that the remarkable incident took place of Rousseau teaching Diaz to
+paint trees. At Fontainebleau Diaz found Rousseau painting his wonderful
+forest pictures, and determined to paint in the same way if possible.
+Rousseau, then in poor health, worried at home, and embittered against
+the world, was difficult to approach. Diaz followed him surreptitiously
+to the forest,--wooden leg not hindering,--and he dodged round after the
+painter, trying to observe his method of work. After a time Diaz found a
+way to become friendly with Rousseau, and revealed his anxiety to
+understand his painting. Rousseau was touched with the passionate words
+of admiration, and finally taught Diaz all he knew. Diaz exhibited many
+pictures at the Paris Salon, and was decorated in 1851. During the
+Franco-German War he went to Brussels. After 1871 he became fashionable,
+his works gradually rose in the estimation of collectors, and he worked
+constantly and successfully. In 1876 he caught cold at his son's grave,
+and on the 18th of November of that year he died at Mentone, whither he
+had gone to recruit his health. Diaz's finest pictures are his forest
+scenes and storms, and it is on these, and not on his pretty figures,
+that his fame is likely to rest. There are several fairly good examples
+of the master in the Louvre, and three small figure pictures in the
+Wallace collection, Hertford House. Perhaps the most notable of Diaz's
+works are "La Fee aux Perles" (1857), in the Louvre; "Sunset in the
+Forest" (1868); "The Storm," and "The Forest of Fontainebleau" (1870)
+at Leeds. Diaz had no well-known pupils, but Leon Richet followed
+markedly his methods of tree-painting, and J. F. Millet at one period
+painted small figures in avowed imitation of Diaz's then popular
+subjects.
+
+  See A. Hustin, _Les Artistes celebres: Diaz_ (Paris); D. Croal
+  Thomson, _The Barbizon School of Painters_ (London, 1890); J. W.
+  Mollett, _Diaz_ (London, 1890); J. Claretie, _Peintres et sculpteurs
+  contemporains: Diaz_ (Paris, 1882); Albert Wolff, _La Capitale de
+  l'art: Narcisse Diaz_ (Paris, 1886); Ph. Burty, _Maitres et
+  petit-maitres: N. Diaz_ (Paris, 1877). (D.C.T.)
+
+
+
+
+DIAZ, PORFIRIO (1830-   ), president of the republic of Mexico (q.v.),
+was born in the southern state of Oaxaca, on the 15th of September 1830.
+His father was an innkeeper in the little capital of that province, and
+died three years after the birth of Porfirio, leaving a family of seven
+children. The boy, who had Indian blood in his veins, was educated for
+the Catholic Church, a body having immense influence in the country at
+that time and ordering and controlling revolutions by the strength of
+their filled coffers. Arrived at the age of sixteen Porfirio Diaz threw
+off the authority of the priests. Fired with enthusiasm by stories told
+by the revolutionary soldiers continually passing through Oaxaca, and
+hearing about the war with the United States, a year later he determined
+to set out for Mexico city and join the National Guard. There being no
+trains, and he being too poor to ride, he walked the greater part of the
+250 m., but arrived there too late, as the treaty of Guadalupe-Hidalgo
+(1848) had been already signed, and Texas finally ceded to the United
+States. Thus his entering the army was for the time defeated. Thereupon
+he returned to his native town and began studying law. He took pupils in
+order to pay his own fees at the Law Institute, and help his mother. At
+this time he came under the notice and influence of Don Marcos Perez and
+Benito Juarez, the first a judge, the second a governor of the state of
+Oaxaca, and soon to become famous as the deliverer of Mexico from the
+priesthood (War of Reform). Diaz continued in his native town until
+1854, when, refusing to vote for the dictator, Santa Anna, he was stung
+by a taunt of cowardice, and hastily pushing his way to the voting
+place, he recorded his vote in favour of Alvarez and the revolutionists.
+Orders were given for his arrest, but seizing a rifle and mounting a
+horse he placed himself at the head of a few revolting peasants, and
+from that moment became one of the leading spirits in that long struggle
+for reform, known as the War of Reform, which, under the leadership of
+Juarez, followed the overthrow of Santa Anna. Promotion succeeded
+promotion, as Diaz led his troops from victory to victory, amid great
+privations and difficulties. He was made captain (1856),
+lieutenant-colonel and colonel (1859), brigadier-general (1861), and
+general of division for the army (1863). Closely following on civil war,
+political strife, open rebellion and the great War of Reform, came the
+French invasion of 1862, and the landing of the emperor Maximilian in
+1864. From the moment the French disclosed their intentions of settling
+in Mexico in 1862, Diaz took a prominent part against the foreign
+invasion. He was twice seriously wounded, imprisoned on three different
+occasions, had two hairbreadth escapes, and took part in many daring
+engagements. So important a personage did he become that both Marshal
+Bazaine and the emperor Maximilian made overtures to him. At the time of
+Maximilian's death (with which Diaz personally had nothing to do) he was
+carrying on the siege of Mexico city, which ended in the surrender of
+the town two days after the emperor was shot at Queretaro between his
+two leading generals. Diaz at once set to work to pay up arrears due to
+his soldiers, proclaimed death as the penalty of plunder and theft, and
+in the few weeks that followed showed his great administrative powers,
+the officers as well as the rank and file receiving arrears of pay. On
+the very day that he occupied Mexico city, the great commander of the
+army of the east, to everyone's surprise, sent in his resignation. He
+was, indeed, appointed to the command of the second division of the army
+by President Juarez in his military reorganization, but Diaz, seeing men
+who had given great and loyal service to the state dismissed from their
+positions in the government, and disgusted at this course, retired to
+the little city of Oaxaca; there he lived, helping in the
+reorganization of the army but taking no active part in the government
+until 1871.
+
+On Juarez' death Lerdo succeeded as president, in 1872. His term of
+office again brought discord, and when it was known that he was
+attempting to be re-elected in 1876, the storm broke. Diaz came from
+retirement, took up the leadership against Lerdo, and after desperate
+struggles and a daring escape finally made a triumphal entry into Mexico
+city on the 24th of November 1876, as provisional president, quickly
+followed by the full presidentship. His term of office marks a prominent
+change in the history of Mexico; from that date he at once forged ahead
+with financial and political reform, the scrupulous settlement of all
+national debts, the welding together of the peoples and tribes (there
+are 150 different Indian tribes) of his country, the establishment of
+railroads and telegraphs, and all this in a land which had been upheaved
+for a century with revolutions and bloodshed, and which had had
+fifty-two dictators, presidents and rulers in fifty-nine years. In 1880
+Diaz was succeeded by Gonzalez, the former minister of war, for four
+years (owing to the limit of the presidential office), but in 1884 he
+was unanimously re-elected. The government having set aside the
+above-mentioned limitation, Diaz was continually re-elected to the
+presidency. He married twice and had a son and two daughters. His gifted
+second wife (Carmelita), very popular in Mexico, was many years younger
+than himself. King Edward VII. made him an honorary grand commander of
+the Bath in June 1906, in recognition of his wonderful administration as
+perpetual president for over a quarter of a century.
+
+  See also Mrs Alec Tweedie, _Porfirio Diaz, Seven Times President of
+  Mexico_ (1906), and _Mexico as I saw it_ (1901); Dr Noll, _From Empire
+  to Republic_ (1890); Lieut. Seaton Schroeder, _Fall of Maximilian's
+  Empire_ (New York, 1887); R. de Z. Enriquez, _P. Diaz_ (1908); and an
+  article by Percy Martin in _Quarterly Review_ for October 1909.
+       (E. A. T.)
+
+
+
+
+DIAZ DE NOVAES, BARTHOLOMEU (fl. 1481-1500), Portuguese explorer,
+discoverer of the Cape of Good Hope, was probably a kinsman of Joao
+Diaz, one of the first Portuguese to round Cape Bojador (1434), and of
+Diniz Diaz, the discoverer of Cape Verde (1445). In 1478 a Bartholomeu
+Diaz, probably identical with the discoverer, was exempted from certain
+customary payments on ivory brought from the Guinea coast. In 1481 he
+commanded one of the vessels sent by King John II. under Diogo
+d'Azambuja to the Gold Coast. In 1486 he seems to have been a cavalier
+of the king's household, and superintendent of the royal warehouses; on
+the 10th of October in this year he received an annuity of 6000 reis
+from King John for "services to come"; and some time after this
+(probably about July or August 1487, rather than July 1486, the
+traditional date) he left Lisbon with three ships to carry on the work
+of African exploration so greatly advanced by Diogo Cao (1482-1486).
+Passing Cao's farthest point near Cape Cross (in the modern German
+South-west Africa and) in 21 deg. 50' S., he erected a pillar on what is
+now known as Diaz Point, south of Angra Pequena or Luderitz Bay, in 26
+deg. 38' S.; of this fragments still exist. From this point (according
+to De Barros) Diaz ran thirteen days southwards before strong winds,
+which freshened to dangerous stormy weather, in a comparatively high
+southern latitude, considerably south of the Cape. When the storm
+subsided the Portuguese stood east; and failing, after several days'
+search, to find land, turned north, and so struck the south coast of
+Cape Colony at Mossel Bay (Diaz' Bahia dos Vaqueiros), half way between
+the Cape of Good Hope and Port Elizabeth (February 3, 1488). Thence they
+coasted eastward, passing Algoa Bay (Diaz' Bahia da Roca), erecting
+pillars (or perhaps wooden crosses), it is said, on one of the islands
+in this bay and at or near Cape Padrone farther east; of these no traces
+remain. The officers and men now began to insist on return, and Diaz
+could only persuade them to go as far as the estuary of the Great Fish
+River (Diaz' Rio do Iffante, so named from his colleague, Captain Joao
+Iffante). Here, however, half way between Port Elizabeth and East London
+(and indeed from Cape Padrone), the north-easterly trend of the coast
+became unmistakable; the way round Africa had been laid open. On his
+return Diaz perhaps named Cape Agulhas after St Brandan; while on the
+southernmost projection of the modern Cape peninsula, whose remarkable
+highlands (Table Mountain, &c.) doubtless impressed him as the practical
+termination of the continent, he bestowed, says De Barros, the name of
+Cape of Storms (_Cabo Tormentoso_) in memory of the storms he had
+experienced in these far southern waters; this name (in the ordinary
+tradition) was changed by King John to that of Good Hope (_Cabo da Boa
+Esperanca_). Some excellent authorities, however, make Diaz himself give
+the Cape its present name. Hard by this "so many ages unknown
+promontory" the explorer probably erected his last pillar. After
+touching at the Ilha do Principe (Prince's Island, south-west of the
+Cameroons) as well as at the Gold Coast, he appeared at Lisbon in
+December 1488. He had discovered 1260 m. of hitherto unknown coast; and
+his voyage, taken with the letters soon afterwards received from Pero de
+Covilhao (who by way of Cairo and Aden had reached Malabar on one side
+and the "Zanzibar coast" on the other as far south as Sofala, in
+1487-1488) was rightly considered to have solved the question of an
+ocean route round Afr ica to the Indies and other lands of South and
+East Asia.
+
+No record has yet been found of any adequate reward for Diaz: on the
+contrary, when the great Indian expedition was being prepared (for Vasco
+da Gama's future leadership) Bartolomeu only superintended the building
+and outfit of the ships; when the fleet sailed in 1497, he only
+accompanied da Gama to the Cape Verde Islands, and after this was
+ordered to El Mina on the Gold Coast. On Cabral's voyage of 1500 he was
+indeed permitted to take part in the discovery of Brazil (April 22), and
+thence should have helped to guide the fleet to India; but he perished
+in a great storm off his own Cabo Tormentoso. Like Moses, as Galvano
+says, he was allowed to see the Promised Land, but not to enter in.
+
+  See Joao de Barros, _Asia_, Dec. I. bk. iii. ch. 4; Duarte Pacheco
+  Pereira, _Esmeraldo de situ orbis_, esp. pp. 15, 90, 92, 94 and
+  Raphael Bastos's introduction to the edition of 1892 (Pacheco met
+  Diaz, returning from his great voyage, at the Ilha do Principe); a
+  marginal note, probably by Christopher Columbus himself, on fol. 13 of
+  a copy of Pierre d'Ailly's _Imago mundi_, now in the Colombina at
+  Seville (the writer of this note fixes Diaz's return to Lisbon,
+  December 1488, and says he was present at Diaz's interview with the
+  king of Portugal, when the explorer described his voyage and showed
+  his route upon the chart he had kept); a similar but briefer note in a
+  copy of Pope Pius II.'s _Historia rerum ubique gestarum_, from the
+  same hand; the _Roteiro_ of Vasco da Gama's First Voyage (_Journal of
+  the First Voyage of ... Da Gama_, Hakluyt Soc., ed. E. G. Ravenstein
+  (1898), pp. 9, 14); Ramusio, _Navigationi_ (3rd ed.), vol. i. fol.
+  144; Castanheda, _Historia_, bk. i. ch. 1; Galvano, _Descobrimentos
+  (Discoveries of the World)_, Hakluyt Soc. (1862), p. 77; E. G.
+  Ravenstein, "Voyages of ... Cao and ... Dias," in _Geog. Journ._
+  (London, December 1900), vol. xvi. pp. 638-655), an excellent critical
+  summary in the light of the most recent investigations of all the
+  material. The fragments of Diaz's only remaining pillar (from Diaz
+  Point) are now partly at the Cape Museum, partly at Lisbon: the latter
+  are photographed in Ravenstein's paper in _Geog. Journ._ (December
+  1900, p. 642).     (C. R. B.)
+
+
+
+
+DIAZO COMPOUNDS, in organic chemistry, compounds of the type R.N.2.X
+(where R = a hydrocarbon radical, and X = an acid radical or a hydroxyl
+group). These compounds may be divided into two classes, namely, the
+true diazo compounds, characterized by the grouping - N = N -, and the
+diazonium compounds, characterized by the grouping N:.N<.
+
+The diazonium compounds were first discovered by P. Griess (_Ann._,
+1858, 106, pp. 123 et seq.), and may be prepared by the action of
+nitrous fumes on a well-cooled solution of a salt of a primary amine,
+
+  C6H5NH2.HNO3 + HNO2 = C6H5N2.NO3 + 2H2O,
+
+or, as is more usually the case (since the diazonium salts themselves
+are generally used only in aqueous solution) by the addition of a
+well-cooled solution of potassium or sodium nitrite to a well-cooled
+dilute acid solution of the primary amine. In order to isolate the
+anhydrous diazonium salts, the method of E. Knoevenagel (_Ber._, 1890,
+23, p. 2094) may be employed. In this process the amine salt is
+dissolved in absolute alcohol and diazotized by the addition of amyl
+nitrite; a crystalline precipitate of the diazonium salt is formed on
+standing, or on the addition of a small quantity of ether. The diazonium
+salts are also formed by the action of zinc-dust and acids on the
+nitrates of primary amines (R. Mohlau, _Ber._, 1883, 16, p. 3080), and
+by the action of hydroxylamine on nitrosobenzenes. They are colourless
+crystalline solids which turn brown on exposure. They dissolve easily in
+water, but only to a slight extent in alcohol and ether. They are very
+unstable, exploding violently when heated or rubbed. _Benzene diazonium
+nitrate_, C6H5N(NO3):.N, crystallizes in long silky needles. The
+sulphate and chloride are similar, but they are not quite so unstable as
+the nitrate. The bromide may be prepared by the addition of bromine to
+an ethereal solution of diazo-amino-benzene (tribromaniline remaining in
+solution). By the addition of potassium bromide and bromine water to
+diazonium salts they are converted into a _perbromide_, e.g. C6H5N2Br3,
+which crystallizes in yellow plates.
+
+  The diazonium salts are characterized by their great reactivity and
+  consequently are important reagents in synthetical processes, since by
+  their agency the amino group in a primary amine may be exchanged for
+  other elements or radicals. The chief reactions are as follows:--
+
+  1. _Replacement of -NH2 by -OH_:--The amine is diazotized and the
+  aqueous solution of the diazonium salt is heated, nitrogen being
+  eliminated and a phenol formed.
+
+  2. _Replacement of -NH2 by halogens and by the -CN and -CNO
+  groups_:--The diazonium salt is warmed with an acid solution of the
+  corresponding cuprous salt (T. Sandmeyer, _Ber._, 1884, 17, p. 2650),
+  or with copper powder (L. Gattermann, Ber., 1890, 23, p. 1218; 1892,
+  25, p. 1074). In the case of iodine, the substitution is effected by
+  adding a warm solution of potassium iodide to the diazonium solution,
+  no copper or cuprous salt being necessary; whilst for the production
+  of nitriles a solution of potassium cuprous cyanide is used. This
+  reaction (the so-called "Sandmeyer" reaction) has been investigated by
+  A. Hantzsch and J. W. Blagden (_Ber._, 1900, 33, p. 2544), who
+  consider that three simultaneous reactions occur, namely, the
+  formation of labile double salts which decompose in such a fashion
+  that the radical attached to the copper atom wanders to the aromatic
+  nucleus; a catalytic action, in which nitrogen is eliminated and the
+  acid radical attaches itself to the aromatic nucleus; and finally, the
+  formation of azo compounds.
+
+  3. _Replacement of -NH2 by -NO2_:--A well-cooled concentrated solution
+  of potassium mercuric nitrate is added to a cooled solution of benzene
+  diazonium nitrate, when the crystalline salt 2C6H5N2.NO3, Hg(NO2)2 is
+  precipitated. On warming this with copper powder, it gives a
+  quantitative yield of nitrobenzene (A. Hantzsch, _Ber._, 1900, 33, p.
+  2551).
+
+  4. _Replacement of -NH2 by hydrogen_:--This exchange is brought about,
+  in some cases, by boiling the diazonium salt with alcohol; but I.
+  Remsen and his pupils (_Amer. Chem. Journ._, 1888, 9, pp. 389 et seq.)
+  have shown that the main product of this reaction is usually a
+  phenolic ether. This reaction has also been investigated by A.
+  Hantzsch and E. Jochem (_Ber._, 1901, 34, p. 3337), who arrived at the
+  conclusion that the normal decomposition of diazonium salts by
+  alcohols results in the formation of phenolic ethers, but that an
+  increase in the molecular weight of the alcohol, or the accumulation
+  of negative groups in the aromatic nucleus, diminishes the yield of
+  the ether and increases the amount of the hydrocarbon formed. The
+  replacement is more readily brought about by the use of sodium
+  stannite (P. Friedlander, _Ber._, 1889, 22, p. 587), or by the use of
+  a concentrated solution of hypophosphorous acid (J. Mai, _Ber._, 1902,
+  35, p. 162). A. Hantzsch (_Ber._, 1896, 29, p. 947; 1898, 31, p. 1253)
+  has shown that the chlor- and brom- diazoniumthiocyanates, when
+  dissolved in alcohol containing a trace of hydrochloric acid, become
+  converted into the isomeric thiocyanbenzene diazonium chlorides and
+  bromides. This change only occurs when the halogen atom is in the
+  ortho- or para- position to the -N2- group.
+
+  _Metallic Diazo Derivatives._--Benzene diazonium chloride is
+  decomposed by silver oxide in aqueous solution, with the formation of
+  _benzene diazonium hydroxide_, C6H5.N(OH):.N. This hydroxide, although
+  possessing powerful basic properties, is unstable in the presence of
+  alkalis and neutralizes them, being converted first into the isomeric
+  benzene-diazotic acid, the potassium salt of which is obtained when
+  the diazonium chloride is added to an excess of cold concentrated
+  potash (A. Hantzsch and W. B. Davidson, _Ber._, 1898, 31, p. 1612).
+  _Potassium benzene diazotate_, C6H5N2.OK, crystallizes in colourless
+  silky needles. The free acid is not known; by the addition of the
+  potassium salt to 50% acetic acid at -20 deg. C., the acid anhydride,
+  _benzene diazo oxide_, (C6H5N2)2O, is obtained as a very unstable,
+  yellow, insoluble compound, exploding spontaneously at 0 deg. C. Strong
+  acids convert it into a diazonium salt, and potash converts it into
+  the diazotate. On the constitution, of these anhydrides see E.
+  Bamberger, _Ber._, 1896, 29, p. 446, and A. Hantzsch, _Ber._, 1896,
+  29, p. 1067; 1898, 31, p. 636. By the addition of the diazonium salts
+  to a hot concentrated solution of a caustic alkali, C. Schraube and C.
+  Schmidt (_Ber._, 1894, 27, p. 520) obtained an isomer of potassium
+  benzene diazotate. These _iso-_diazotates are formed much more readily
+  when the aromatic nucleus in the diazonium salt contains negative
+  radicals. _Potassium benzene iso-diazotate_ resembles the normal salt,
+  but is more stable, and is more highly ionized. Carbon dioxide
+  converts it into _phenyl nitrosamine_, C6H5NH.NO (A. Hantzsch). The
+  potassium salt of the iso-diazo hydroxide yields on methylation a
+  nitrogen ether, R.N(CH3).NO, whilst the silver salt yields an oxygen
+  ether, R.N:N.OCH3. These results point to the conclusion that the
+  iso-diazo hydroxide is a tautomeric substance. The same oxygen ether
+  is formed by the methylation of the silver salt of the normal diazo
+  hydroxide; this points to the conclusion that the isomeric hydroxides,
+  corresponding with the silver derivatives, have the same structural
+  formulae, namely, R.N:N.OH. These oxygen ethers contain the grouping
+  -N:N-, since they couple very readily with the phenols in alkaline
+  solution to form azo compounds (q.v.) (E. Bamberger, _Ber._, 1895, 28,
+  p. 225); they are also explosive.
+
+  By oxidizing potassium benzene iso-diazotate with alkaline potassium
+  ferricyanide, E. Bamberger (_Ber._, 1894, 27, p. 914) obtained the
+  _diazoic acids_, R.NH.NO2, substances which he had previously prepared
+  by similarly oxidizing the diazonium salts, by dehydrating the
+  nitrates of primary amines with acetic anhydride, and by the action of
+  nitric anhydride on the primary amines. Concentrated acids convert
+  them into the isomeric nitro-amines, the -NO2 group going into the
+  nucleus in the ortho- or para- position to the amine nitrogen; this
+  appears to indicate that the compounds are nitramines. They behave,
+  however, as tautomeric substances, since their alkali salts on
+  methylation give nitrogen ethers, whilst their silver salts yield
+  oxygen ethers:
+
+            /-->  potassium salt --> R.N(CH3).NO2 nitramine.
+  R.NH.NO2<
+            \-->  silver salt    --> R.N:N.O.OCH3 diazoate.
+
+  _Phenyl nitramine_, C6H5NH.NO2, is a colourless crystalline solid,
+  which melts at 46 deg. C. Sodium amalgam in alkaline solution reduces it
+  to phenylhydrazine.
+
+  _Constitution of the Diazo Compounds._--P. Griess (_Ann._, 1866, 137,
+  p. 39) considered that the diazo compounds were formed by the addition
+  of complex groupings of the type C6H4N2- to the inorganic acids;
+  whilst A. Kekule (_Zeit. f. Chemie_, 1866, 2, p. 308), on account of
+  their ready condensation to form azo compounds and their easy
+  reduction to hydrazines, assumed that they were substances of the type
+  R.N:N.Cl. The constitution of the diazonium group -N2.X, may be
+  inferred from the following facts:--The group C6H5N2- behaves in many
+  respects similarly to an alkali metal, and even more so to the
+  ammonium group, since it is capable of forming colourless neutral
+  salts with mineral acids, which in dilute aqueous solution are
+  strongly ionized, but do not show any trace of hydrolytic dissociation
+  (A. Hantzsch, _Ber._, 1895, 28, p. 1734). Again, the diazonium
+  chlorides combine with platinic chloride to form difficultly soluble
+  double platinum salts, such as (C6H5N2Cl)2.PtCl4; similar gold salts,
+  C6H5N2Cl.AuCl3, are known. Determinations of the electrical
+  conductivity of the diazonium chloride and nitrate also show that the
+  diazonium radical is strictly comparable with other quaternary
+  ammonium ions. For these reasons, one must assume the existence of
+  pentavalent nitrogen in the diazonium salts, in order to account for
+  their basic properties.
+
+  The constitution of the isomeric diazo hydroxides has given rise to
+  much discussion. E. Bamberger (_Ber._, 1895, 28, pp. 444 et seq.) and
+  C. W. Blomstrand (_Journ. prakt. Chem._, 1896, 53, pp. 169 et seq.)
+  hold that the compounds are structurally different, the normal
+  diazo-hydroxide being a diazonium derivative of the type
+  R.N([3:]N).OH. The recent work of A. Hantzsch and his pupils seems to
+  invalidate this view (_Ber._, 1894, 27, pp. 1702 et seq.; see also A.
+  Hantzsch, _Die Diazoverbindungen_). According to Hantzsch the isomeric
+  diazo hydroxides are structurally identical, and the differences in
+  behaviour are due to stereo-chemical relations, the isomerism being
+  comparable with that of the oximes (q.v.). On such a hypothesis, the
+  relatively unstable normal diazo hydroxides would be the
+  _syn-_compounds, since here the nitrogen atoms would be more easily
+  eliminated, whilst the stable iso-diazo derivatives would be the
+  _anti-_compounds, thus:
+
+         R . N             R . N
+             ..                ..
+        HO . N                 N . OH
+    Normal hydroxide      Iso hydroxide
+    (Syn-compound)       (Anti-compound)
+
+  In support of this theory, Hantzsch has succeeded in isolating a
+  series of syn- and anti-diazo-cyanides and -sulphonates (_Ber._, 1895,
+  28, p. 666; 1900, 33, p. 2161; 1901, 34, p. 4166). By diazotizing
+  para-chloraniline and adding a cold solution of potassium cyanide, a
+  salt (melting at 29 deg. C.) is obtained, which readily loses
+  nitrogen, and forms para-chlorbenzonitrile on the addition of copper
+  powder. By dissolving this diazocyanide in alcohol and reprecipitating
+  it by water, it is converted into the isomeric diazocyanide (melting
+  at 105-106 deg. C.), which does not yield para-chlorbenzonitrile when
+  treated with copper powder. Similar results have been obtained by
+  using diazotized para-anisidine, a syn- and an anti- compound being
+  formed, as well as a third isomeric cyanide, obtained by evaporating
+  para-methoxy-benzenediazonium hydroxide in the presence of an excess
+  of hydrocyanic acid at ordinary temperatures. This salt is a
+  colourless crystalline substance of composition
+  CH3O.C6H4.N2.CN.HCN.2H2O, and has the properties of a metallic salt;
+  it is very soluble in water and its solution is an electrolyte,
+  whereas the solutions of the syn-and anti- compounds are not
+  electrolytes. The isolation of these compounds is a powerful argument
+  in favour of the Hantzsch hypothesis which requires the existence of
+  these three different types, whilst the Bamberger-Blomstrand view only
+  accounts for the formation of two isomeric cyanides, namely, one of
+  the normal diazonium type and one of the iso-diazocyanide type.
+
+  Benzene diazonium hydroxide, although a strong base, reacts with the
+  alkaline hydroxides to form salts with the evolution of heat, and
+  generally behaves as a weak acid. On mixing dilute solutions of the
+  diazonium hydroxide and the alkali together, it is found that the
+  molecular conductivity of the mixture is much less than the sum of the
+  two electrical conductivities of the solutions separately, from which
+  it follows that a portion of the ions present have changed to the
+  non-ionized condition. This behaviour is explained by considering the
+  non-ionized part of the diazonium hydroxide to exist in solution in a
+  hydrated form, the equation of equilibrium being:
+
+          C6H5.N.       --> C6H5.N.OH
+    H2O +     ... + OH'          |
+               N        <--   HO.N.H
+
+  On adding the alkaline hydroxide to the solution, this hydrate is
+  supposed to lose water, yielding the syn-diazo hydroxide, which then
+  gives rise to a certain amount of the sodium salt (A. Hantzsch,
+  _Ber._, 1898, 31, p. 1612),
+
+           ....
+    C6H5.N.:OH:  -->  C6H5.N  -->  C6H5.N
+         | :  :            ||           ||
+      HO.N.:H :  <--    HO.N  <--   NaO.N
+           ....
+
+  This assumption also shows the relationship of the diazonium
+  hydroxides to other quaternary ammonium compounds, for most of the
+  quaternary ammonium hydroxides (except such as have the nitrogen atom
+  attached to four saturated hydrocarbon radicals) are unstable, and
+  readily pass over into compounds in which the hydroxyl group is no
+  longer attached to the amine nitrogen; thus the syn-diazo hydroxides
+  are to be regarded as pseudo-diazonium derivatives. (A. Hantzsch,
+  _Ber._, 1899, 32, p. 3109; 1900, 33, p. 278.) It is generally accepted
+  that the iso-diazo hydroxides possess the oxime structure R.N:N.OH.
+
+  Hantzsch explains the characteristic reactions of the diazonium
+  compounds by the assumption that an addition compound is first formed,
+  which breaks down with the elimination of the hydride of the acid
+  radical, and the formation of an unstable syn-diazo compound, which,
+  in its turn, decomposes with evolution of nitrogen (_Ber._, 1897, 30,
+  p. 2548; 1898, 31, p. 2053).
+
+     R         X     R      X     R   X
+      \        |      \    /      |   |
+        N.:N + | -->   N.:N   --> |   | + HCl --> R.X + N2.
+      /        |      /    \      |   |
+    Cl         H    Cl      H     N = N
+
+  J. Cain (_Jour. Chem. Soc._, 1907, 91, p. 1049) suggested a quinonoid
+  formula for diazonium salts, which has been combated by Hantzsch
+  (_Ber._, 1908, 41, pp. 3532 et seq.). G. T. Morgan and F. M. G.
+  Micklethwaite (_Jour. Chem. Soc._, 1908, 93, p. 617; 1909, 95, p.
+  1319) have pointed out that the salts may possess a dynamic formula,
+  Cain's representing the middle stage, thus:
+
+        /||\                /||\                /||\
+       / || \              / || \              / || \
+    H / N.Cl \ H        H / N.Cl \ H        H / N.Cl \ H
+     |\  ||  ||  --->    ||  ||   |  --->    ||  ||  /|
+     | \ ||  ||  <---    ||  ||   |  <---    ||  || / |
+     |  \N   ||          ||  N    |          ||   N   |
+    H\\      / H        H\\  |   / H        H \      //H
+      \\    /             \\ |  /              \    //
+       \\  /               \\| /                \  //
+         H                   H                   H
+
+  _Diazoamines._--The diazoamines, R.N2.NHR, may be prepared by the
+  action of the primary and secondary amines on the diazonium salts, or
+  by the action of nitrous acid on the free primary amine. In the latter
+  reaction it is assumed that the isodiazohydroxide first formed is
+  immediately attacked by a second molecule of the amine. They are
+  yellow crystalline solids, which do not unite with acids. Nitrous acid
+  converts them, in acid solution, into diazonium salts.
+
+    C6H5N2.NHC6H5 + 2HCl + HNO2 = 2C6H5N2Cl + 2H2O.
+
+  They are readily converted into the isomeric aminoazo compounds,
+  either by standing in alcoholic solution, or by warming with a mixture
+  of the parent base and its hydrochloride; the diazo group preferably
+  going into the para-position to the amino group. When the
+  para-position is occupied, the diazo group takes the ortho-position.
+  H. Goldschmidt and R. U. Reinders (_Ber._, 1896, 29, p. 1369, 1899)
+  have shown that the transformation is a monomolecular reaction, the
+  velocity of transformation in moderately dilute solution being
+  independent of the concentration, but proportional to the amount of
+  the catalyst present (amine hydrochloride) and to the temperature. It
+  has also been shown that when different salts of the amine are used,
+  their catalytic influence varies in amount and is almost proportional
+  to their degree of ionization in aqueous solution. Diazoaminobenzene,
+  C6H5N2.NHC6H5, crystallizes in golden yellow laminae, which melt at
+  96 deg. C. and explode at a slightly higher temperature. It is readily
+  soluble in alcohol, ether and benzene. Concentrated hydrochloric acid
+  converts it into chlorbenzene, aniline and nitrogen. Zinc dust and
+  alcoholic acetic acid reduce it to aniline and phenylhydrazine.
+
+  _Diazoimino compounds_, R.N3, may be regarded as derivatives of
+  azoimide (q.v.); they are formed by the action of ammonia on the
+  diazoperbromides, or by the action of hydroxylamine on the diazonium
+  sulphates (J. Mai, _Ber._, 1892, 25, p. 372; T. Curtius, _Ber._, 1893,
+  26, p. 1271). Diazobenzeneimide, C6H5N3, is a yellowish oil of
+  stupefying odour. It boils at 59 deg. C. (12 mm.), and explodes when
+  heated. Concentrated hydrochloric acid decomposes it with formation of
+  chloranilines and elimination of nitrogen, whilst on boiling with
+  sulphuric acid it is converted into aminophenols.
+
+  _Aliphatic Diazo Compounds._--The esters of the aliphatic amino acids
+  may be diazotized in a manner similar to the primary aromatic amines,
+  a fact discovered by T. Curtius (_Ber._, 1833, 16, p. 2230). The first
+  aliphatic diazo compound to be isolated was _diazoacetic ester_,
+  CH.N2.CO2C2H5, which is prepared by the action of potassium nitrite on
+  the ethyl ester of glycocoll hydrochloride, HCl.NH2.CH2.CO2C2H5 + KNO2
+  = CHN2.CO2C2H5 + KCl + 2H2O. It is a yellowish oil which melts at -24
+  deg. C.; it boils at 143-144 deg. C., but cannot be distilled safely
+  as it decomposes violently, giving nitrogen and ethyl fumarate. It
+  explodes in contact with concentrated sulphuric acid. On reduction it
+  yields ammonia and glycocoll (aminoacetic acid). When heated with
+  water it forms ethyl hydroxy-acetate; with alcohol it yields ethyl
+  ethoxyacetate. Halogen acids convert it into monohalogen fatty acids,
+  and the halogens themselves convert it into dihalogen fatty acids. It
+  unites with aldehydes to form esters of ketonic acids, and with
+  aniline yields anilido-acetic acid. It forms an addition product with
+  acrylic ester, which on heating loses nitrogen and leaves trimethylene
+  dicarboxylic ester. Concentrated ammonia converts it into
+  _diazoacetamide_, CHN2.CONH2, which crystallizes in golden yellow
+  plates which melt at 114 deg. C. For other reactions see HYDRAZINE.
+  The constitution of the diazo fatty esters is inferred from the fact
+  that the two nitrogen atoms, when split off, are replaced by two
+  monovalent elements or groups, thus leading to the formula
+
+     N \
+    ..  > CH.CO2C2H5, for diazoacetic ester.
+     N /
+
+  _Diazosuccinic ester_, N2.C(CO2C2H5)2, is similarly prepared by the
+  action of nitrous acid on the hydrochloride of aspartic ester. It is
+  decomposed by boiling water and yields fumaric ester.
+
+  _Diazomethane_, CH2N2, was first obtained in 1894 by H. v. Pechmann
+  (_Ber._, 1894, 27, p. 1888; 1895, 28, p. 855). It is prepared by the
+  action of aqueous or alcoholic solutions of the caustic alkalis on the
+  nitroso-acidyl derivatives of methylamine (such, for example, as
+  _nitrosomethyl urethane_, NO.N(CH3).CO2C2H5, which is formed on
+  passing nitrous fumes into an ethereal solution of methyl urethane).
+  E. Bamberger (_Ber._, 1895, 28, p. 1682) regards it as the anhydride
+  of iso-diazomethane, CH3.N:N.OH, and has prepared it by a method
+  similar to that used for the preparation of iso-diazobenzene. By the
+  action of bleaching powder on methylamine hydrochloride, there is
+  obtained a volatile liquid (_methyldichloramine_, CH3.N.Cl2), boiling
+  at 58-60 deg. C., which explodes violently when heated with water,
+  yielding hydrocyanic acid (CH3NCl2 = HCN + 2HCl). Well-dried
+  hydroxylamine hydrochloride is dissolved in methyl alcohol and mixed
+  with sodium methylate; a solution of methyldichloramine in absolute
+  ether is then added and an ethereal solution of diazomethane distils
+  over. Diazomethane is a yellow inodorous gas, very poisonous and
+  corrosive. It may be condensed to a liquid, which boils at about 0
+  deg. C. It is a powerful methylating agent, reacting with water to
+  form methyl alcohol, and converting acetic acid into methylacetate,
+  hydrochloric acid into methyl chloride, hydrocyanic acid into
+  acetonitrile, and phenol into anisol, nitrogen being eliminated in
+  each case. It is reduced by sodium amalgam (in alcoholic solution) to
+  _methylhydrazine_, CH3.NH.NH2. It unites directly with acetylene to
+  form pyrazole (H. v. Pechmann, _Ber._, 1898, 31, p. 2950) and with
+  fumaric methyl ester it forms pyrazolin dicarboxylic ester.
+       (F. G. P.*)
+
+    See G. T. Morgan, _B.A. Rep._, 1902; J. Cain, _Diazo Compounds_,
+    1908.
+
+
+
+
+DIAZOMATA (Gr. [Greek: diazoma], a girdle), in architecture, the landing
+places and passages which were carried round the semicircle and
+separated the upper and lower tiers in a Greek theatre.
+
+
+
+
+DIBDIN, CHARLES (1745-1814), British musician, dramatist, novelist,
+actor and song-writer, the son of a parish clerk, was born at
+Southampton on or before the 4th of March 1745, and was the youngest of
+a family of eighteen. His parents designing him for the church, he was
+sent to Winchester; but his love of music early diverted his thoughts
+from the clerical profession. After receiving some instruction from the
+organist of Winchester cathedral, where he was a chorister from 1756 to
+1759, he went to London at the age of fifteen. Here he was placed in a
+music warehouse in Cheapside, but he soon abandoned this employment to
+become a singing actor at Covent Garden. On the 21st of May 1762 his
+first work, an operetta entitled _The Shepherd's Artifice_, with words
+and music by himself, was produced at this theatre. Other works
+followed, his reputation being firmly established by the music to the
+play of _The Padlock_, produced at Drury Lane under Garrick's management
+in 1768, the composer himself taking the part of Mungo with conspicuous
+success. He continued for some years to be connected with Drury Lane,
+both as composer and as actor, and produced during this period two of
+his best known works, _The Waterman_ (1774) and _The Quaker_ (1775). A
+quarrel with Garrick led to the termination of his engagement. In _The
+Comic Mirror_ he ridiculed prominent contemporary figures through the
+medium of a puppet show. In 1782 he became joint manager of the Royal
+circus, afterwards known as the Surrey theatre. In three years he lost
+this position owing to a quarrel with his partner. His opera _Liberty
+Hall_, containing the successful songs "Jock Ratlin," "The Highmettled
+Racer," and "The Bells of Aberdovey," was produced at Drury Lane theatre
+on the 8th of February 1785. In 1788 he sailed for the East Indies, but
+the vessel having put in to Torbay in stress of weather, he changed his
+mind and returned to London. In a musical variety entertainment called
+_The Oddities_, he succeeded in winning marked popularity with a number
+of songs that included "'Twas in the good ship 'Rover'," "Saturday Night
+at Sea," "I sailed from the Downs in the 'Nancy,'" and the immortal "Tom
+Bowling," written on the death of his eldest brother, Captain Thomas
+Dibdin, at whose invitation he had planned his visit to India. A series
+of monodramatic entertainments which he gave at his theatre, Sans Souci,
+in Leicester Square, brought his songs, music and recitations more
+prominently into notice, and permanently established his fame as a lyric
+poet. It was at these entertainments that he first introduced many of
+those sea-songs which so powerfully influenced the national spirit. The
+words breathe the simple loyalty and dauntless courage that are the
+cardinal virtues of the British sailor, and the music was appropriate
+and naturally melodious. Their effect in stimulating and ennobling the
+spirit of the navy during the war with France was so marked as to call
+for special acknowledgment. In 1803 Dibdin was rewarded by government
+with a pension of L200 a year, of which he was only for a time deprived
+under the administration of Lord Grenville. During this period he opened
+a music shop in the Strand, but the venture was a failure. Dibdin died
+of paralysis in London on the 25th of July 1814. Besides his _Musical
+Tour through England_ (1788), his _Professional Life_, an autobiography
+published in 1803, a _History of the Stage_ (1795), and several smaller
+works, he wrote upwards of 1400 songs and about thirty dramatic pieces.
+He also wrote the following novels:--_The Devil_ (1785); _Hannah Hewitt_
+(1792); _The Younger Brother_ (1793). An edition of his songs by G.
+Hogarth (1843) contains a memoir of his life. His two sons, Charles and
+Thomas John Dibdin (q.v.), whose works are often confused with those of
+their father, were also popular dramatists in their day.
+
+
+
+
+DIBDIN, THOMAS FROGNALL (1776-1847), English bibliographer, born at
+Calcutta in 1776, was the son of Thomas Dibdin, the sailor brother of
+Charles Dibdin. His father and mother both died on the way home to
+England in 1780, and Thomas was brought up by a maternal uncle. He was
+educated at St John's College, Oxford, and studied for a time at
+Lincoln's Inn. After an unsuccessful attempt to obtain practice as a
+provincial counsel at Worcester, he was ordained a clergyman at the
+close of 1804, being appointed to a curacy at Kensington. It was not
+until 1823 that he received the living of Exning in Sussex. Soon
+afterwards he was appointed by Lord Liverpool to the rectory of St
+Mary's, Bryanston Square, which he held until his death on the 18th of
+November 1847. The first of his numerous bibliographical works was his
+_Introduction to the Knowledge of Editions of the Classics_ (1802),
+which brought him under the notice of the third Earl Spencer, to whom he
+owed much important aid in his bibliographical pursuits. The rich
+library at Althorp was thrown open to him; he spent much of his time in
+it, and in 1814-1815 published his _Bibliotheca Spenceriana_. As the
+library was not open to the general public, the information given in the
+_Bibliotheca_ was found very useful, but since its author was unable
+even to read the characters in which the books he described were
+written, the work was marred by the errors which more or less
+characterize all his productions. This fault of inaccuracy however was
+less obtrusive in his series of playful, discursive works in the form of
+dialogues on his favourite subject, the first of which, _Bibliomania_
+(1809), was republished with large additions in 1811, and was very
+popular, passing through numerous editions. To the same class belonged
+the _Bibliographical Decameron_, a larger work, which appeared in 1817.
+In 1810 he began the publication of a new and much extended edition of
+Ames's _Typographical Antiquities_. The first volume was a great
+success, but the publication was checked by the failure of the fourth
+volume, and was never completed. In 1818 Dibdin was commissioned by
+Earl Spencer to purchase books for him on the continent, an expedition
+described in his sumptuous _Bibliographical, Antiquarian and Picturesque
+Tour in France and Germany_ (1821). In 1824 he made an ambitious venture
+in his _Library Companion, or the Young Man's Guide and Old Man's
+Comfort in the Choice of a Library_, intended to point out the best
+works in all departments of literature. His culture was not broad
+enough, however, to render him competent for the task, and the work was
+severely criticized. For some years Dibdin gave himself up chiefly to
+religious literature. He returned to bibliography in his _Bibliophobia,
+or Remarks on the Present Depression in the State of Literature and the
+Book Trade_ (1832), and the same subject furnishes the main interest of
+his _Reminiscences of a Literary Life_ (1836), and his _Bibliographical,
+Antiquarian and Picturesque Tour in the Northern Counties of England and
+Scotland_ (1838). Dibdin was the originator and vice-president, Lord
+Spencer being the president, of the Roxburghe Club, founded in
+1812,--the first of the numerous book clubs which have done such service
+to literature.
+
+
+
+
+DIBDIN, THOMAS JOHN (1771-1841), English dramatist and song-writer, son
+of Charles Dibdin, the song-writer, and of Mrs Davenet, an actress whose
+real name was Harriet Pitt, was born on the 21st of March 1771. He was
+apprenticed to his maternal uncle, a London upholsterer, and later to
+William Rawlins, afterwards sheriff of London. He summoned his second
+master unsuccessfully for rough treatment; and after a few years of
+service he ran away to join a company of country players. From 1789 to
+1795 he played in all sorts of parts; he acted as scene painter at
+Liverpool in 1791; and during this period he composed more than 1000
+songs. He made his first attempt as a dramatic writer in _Something
+New_, followed by _The Mad Guardian_ in 1795. He returned to London in
+1795, having married two years before; and in the winter of 1798-1799
+his _Jew and the Doctor_ was produced at Covent Garden. From this time
+he contributed a very large number of comedies, operas, farces, &c., to
+the public entertainment. Some of these brought immense popularity to
+the writer and immense profits to the theatres. It is stated that the
+pantomime of _Mother Goose_ (1807) produced more than L20,000 for the
+management at Covent Garden theatre, and the _High-mettled Racer_,
+adapted as a pantomime from his father's play, L18,000 at Astley's.
+Dibdin was prompter and pantomime writer at Drury Lane until 1816, when
+he took the Surrey theatre. This venture proved disastrous and he became
+bankrupt. After this he was manager of the Haymarket, but without his
+old success, and his last years were passed in comparative poverty. In
+1827 he published two volumes of _Reminiscences_; and at the time of his
+death he was preparing an edition of his father's sea songs, for which a
+small sum was allowed him weekly by the lords of the admiralty. Of his
+own songs "The Oak Table" and "The Snug Little Island" are well-known
+examples. He died in London on the 16th of September 1841.
+
+
+
+
+DIBRA (Slav. _Debra_), the capital of a sanjak bearing the same name, in
+the vilayet of Monastir, eastern Albania, Turkey. Pop. (1900) about
+15,000. Dibra occupies a valley enclosed by mountains, and watered by
+the Tsrni Drin and Radika rivers, which meet 3 m. S. It is a fortified
+city, and the only episcopal see of the Bulgarian exarchate in Albania;
+most of the inhabitants are Albanians, but there is a strong Bulgarian
+colony. The local trade is almost entirely agricultural.
+
+
+
+
+DIBRUGARH, a town of British India, in the Lakhimpur district of eastern
+Bengal and Assam, of which it is the headquarters, situated on the Dibru
+river about 4 m. above its confluence with the Brahmaputra. Pop. (1901)
+11,227. It is the terminus of steamer navigation on the Brahmaputra, and
+also of a railway running to important coal-mines and petroleum wells,
+which connects with the Assam-Bengal system. Large quantities of coal
+and tea are exported. There are a military cantonment, the headquarters
+of the volunteer corps known as the Assam Valley Light Horse; a
+government high school, a training school for masters; and an aided
+school for girls. In 1900 a medical school for the province was
+established, out of a bequest left by Brigade-Surgeon J. Berry-White,
+which is maintained by the government, to train hospital assistants for
+the tea gardens. The Williamson artisan school is entirely supported by
+an endowment.
+
+
+
+
+DICAEARCHUS, of Messene in Sicily, Peripatetic philosopher and pupil of
+Aristotle, historian, and geographer, flourished about 320 B.C. He was a
+friend of Theophrastus, to whom he dedicated the majority of his works.
+Of his writings, which comprised treatises on a great variety of
+subjects, only the titles and a few fragments survive. The most
+important of them was his [Greek: bios tes Hellados] (_Life in Greece_),
+in which the moral, political and social condition of the people was
+very fully discussed. In his _Tripoliticos_ he described the best form
+of government as a mixture of monarchy, aristocracy and democracy, and
+illustrated it by the example of Sparta. Among the philosophical works
+of Dicaearchus may be mentioned the _Lesbiaci_, a dialogue in three
+books, in which the author endeavours to prove that the soul is mortal,
+to which he added a supplement called _Corinthiaci_. He also wrote a
+_Description of the World_ illustrated by maps, in which was probably
+included his _Measurements of Mountains_. A description of Greece (150
+iambics, in C. Muller, _Frag. hist. Graec_. i. 238-243) was formerly
+attributed to him, but, as the initial letters of the first twenty-three
+lines show, was really the work of Dionysius, son of Calliphon. Three
+considerable fragments of a prose description of Greece (Muller, i.
+97-110) are now assigned to an unknown author named Heracleides. The _De
+re publica_ of Cicero is supposed to be founded on one of Dicaearchus's
+works.
+
+  The best edition of the fragments is by M. Fuhr (1841), a work of
+  great learning; see also a dissertation by F. G. Osann, _Beitrage zur
+  rom. und griech. Litteratur_, ii. pp. 1-117 (1839); Pauly-Wissowa,
+  _Realencyclopadie der klass. Altertumswiss_. v. pt. 1 (1905).
+
+
+
+
+DICE (plural of die, O. Fr. _de_, derived from Lat. _dare_, to give),
+small cubes of ivory, bone, wood or metal, used in gaming. The six sides
+of a die are each marked with a different number of incised dots in such
+a manner that the sum of the dots on any two opposite sides shall be 7.
+Dice seem always to have been employed, as is the case to-day, for
+gambling purposes, and they are also used in such games as backgammon.
+There are many methods of playing, from one to five dice being used,
+although two or three are the ordinary numbers employed in Great Britain
+and America. The dice are thrown upon a table or other smooth surface
+either from the hand or from a receptacle called a dice-box, the latter
+method having been in common use in Greece, Rome and the Orient in
+ancient times. Dice-boxes have been made in many shapes and of various
+materials, such as wood, leather, agate, crystal, metal or paper. Many
+contain bars within to ensure a proper agitation of the dice, and thus
+defeat trickery. Some, formerly used in England, were employed with
+unmarked dice, and allowed the cubes to fall through a kind of funnel
+upon a board marked off into six equal parts numbered from 1 to 6. It is
+a remarkable fact, that, wherever dice have been found, whether in the
+tombs of ancient Egypt, of classic Greece, or of the far East, they
+differ in no material respect from those in use to-day, the elongated
+ones with rounded ends found in Roman graves having been, not dice but
+_tali_, or knucklebones. Eight-sided dice have comparatively lately been
+introduced in France as aids to children in learning the multiplication
+table. The teetotum, or spinning die, used in many modern games, was
+known in ancient times in China and Japan. The increased popularity of
+the more elaborate forms of gaming has resulted in the decline of
+dicing. The usual method is to throw three times with three dice. If one
+or more sixes or fives are thrown the first time they may be reserved,
+the other throws being made with the dice that are left. The object is
+to throw three sixes = 18 or as near that number as possible, the
+highest throw winning, or, when drinks are to be paid for, the lowest
+throw losing. (For other methods of throwing consult the _Encyclopaedia
+of Indoor Games_, by R. F. Foster, 1903.) The most popular form of pure
+gambling with dice at the present day, particularly with the lower
+classes in America, is _Craps_, or _Crap-Shooting_, a simple form of
+_Hazard_, of French origin. Two dice are used. Each player puts up a
+stake and the first caster may cover any or all of the bets. He then
+_shoots_, i.e. throws the dice from his open hand upon the table. If the
+sum of the dice is 7 or 11 the throw is a _nick_, or _natural_, and the
+caster wins all stakes. If the throw is either 2, 3 or 12 it is a
+_crap_, and the caster loses all. If any other number is thrown it is a
+_point_, and the caster continues until he throws the same number again,
+in which case he wins, or a 7, in which case he loses. The now
+practically obsolete game of Hazard was much more complicated than
+_Craps_. (Consult _The Game of Hazard Investigated_, by George Lowbut.)
+_Poker dice_ are marked with ace, king, queen, jack and ten-spot. Five
+are used and the object is, in three throws, to make pairs, triplets,
+full hands or fours and fives of a kind, five aces being the highest
+hand. Straights do not count. In throwing to decide the payment of
+drinks the usual method is called _horse and horse_, in which the
+highest throws retire, leaving the two lowest to decide the loser by the
+best two in three throws. Should each player win one throw both are said
+to be _horse and horse_, and the next throw determines the loser. The
+two last casters may also agree to _sudden death_, i.e. a single throw.
+_Loaded dice_, i.e. dice weighted slightly on the side of the lowest
+number, have been used by swindlers from the very earliest times to the
+present day, a fact proved by countless literary allusions. Modern dice
+are often rounded at the corners, which are otherwise apt to wear off
+irregularly.
+
+_History._--Dice were probably evolved from knucklebones. The antiquary
+Thomas Hyde, in his _Syntagma_, records his opinion that the game of
+"odd or even," played with pebbles, is nearly coeval with the creation
+of man. It is almost impossible to trace clearly the development of dice
+as distinguished from knucklebones, on account of the confusing of the
+two games by the ancient writers. It is certain, however, that both were
+played in times antecedent to those of which we possess any written
+records. Sophocles, in a fragment, ascribed their invention to
+Palamedes, a Greek, who taught them to his countrymen during the siege
+of Troy, and who, according to Pausanias (on Corinth, xx.), made an
+offering of them on the altar of the temple of Fortune. Herodotus
+(_Clio_) relates that the Lydians, during a period of famine in the days
+of King Atys, invented dice, knucklebones and indeed all other games
+except chess. The fact that dice have been used throughout the Orient
+from time immemorial, as has been proved by excavations from ancient
+tombs, seems to point clearly to an Asiatic origin. Dicing is mentioned
+as an Indian game in the _Rig-veda_. In its primitive form knucklebones
+was essentially a game of skill, played by women and children, while
+dice were used for gambling, and it was doubtless the gambling spirit of
+the age which was responsible for the derivative form of knucklebones,
+in which four sides of the bones received different values, which were
+then counted, like dice. Gambling with three, sometimes two, dice
+([Greek: kuboi]) was a very popular form of amusement in Greece,
+especially with the upper classes, and was an almost invariable
+accompaniment to the symposium, or drinking banquet. The dice were cast
+from conical beakers, and the highest throw was three sixes, called
+_Aphrodite_, while the lowest, three aces, was called the _dog_. Both in
+Greece and Rome different modes of counting were in vogue. Roman dice
+were called _tesserae_ from the Greek word for four, indicative of the
+four sides. The Romans were passionate gamblers, especially in the
+luxurious days of the Empire, and dicing was a favourite form, though it
+was forbidden except during the Saturnalia. The emperor Augustus wrote
+in a letter to Suetonius concerning a game that he had played with his
+friends: "Whoever threw a _dog_ or a six paid a _denarius_ to the bank
+for every die, and whoever threw a _Venus_ (the highest) won
+everything." In the houses of the rich the dice-beakers were of carved
+ivory and the dice of crystal inlaid with gold. Mark Antony wasted his
+time at Alexandria with dicing, while, according to Suetonius, the
+emperors Augustus, Nero and Claudius were passionately fond of it, the
+last named having written a book on the game. Caligula notoriously
+cheated at the game; Domitian played it, and Commodus set apart special
+rooms in his palace for it. The emperor Verus, adopted son of Antonine,
+is known to have thrown dice whole nights together. Fashionable society
+followed the lead of its emperors, and, in spite of the severity of the
+laws, fortunes were squandered at the dicing table. Horace derided the
+youth of the period, who wasted his time amid the dangers of dicing
+instead of taming his charger and giving himself up to the hardships of
+the chase. Throwing dice for money was the cause of many special laws in
+Rome, according to one of which no suit could be brought by a person who
+allowed gambling in his house, even if he had been cheated or assaulted.
+Professional gamblers were common, and some of their loaded dice are
+preserved in museums. The common public-houses were the resorts of
+gamblers, and a fresco is extant showing two quarrelling dicers being
+ejected by the indignant host. Virgil, in the _Copa_ generally ascribed
+to him, characterizes the spirit of that age in verse, which has been
+Englished as follows:--
+
+  "What ho! Bring dice and good wine!
+     Who cares for the morrow?
+   Live--so calls grinning Death--
+     Live, for I come to you soon!"
+
+That the barbarians were also given to gaming, whether or not they
+learned it from their Roman conquerors, is proved by Tacitus, who states
+that the Germans were passionately fond of dicing, so much so, indeed,
+that, having lost everything, they would even stake their personal
+liberty. Centuries later, during the middle ages, dicing became the
+favourite pastime of the knights, and both dicing schools (_scholae
+deciorum_) and gilds of dicers existed. After the downfall of feudalism
+the famous German mercenaries called _landsknechts_ established a
+reputation as the most notorious dicing gamblers of their time. Many of
+the dice of the period were curiously carved in the images of men and
+beasts. In France both knights and ladies were given to dicing, which
+repeated legislation, including interdictions on the part of St Louis in
+1254 and 1256, did not abolish. In Japan, China, Korea, India and other
+Asiatic countries dice have always been popular and are so still.
+
+  See Foster's _Encyclopaedia of Indoor Games_ (1903); Raymond's
+  _Illustriertes Knobelbrevier_ (Oranienburg, 1888); _Les Jeux des
+  Anciens_, by L. Becq de Fouquieres (Paris, 1869); _Das Knochelspiel
+  der Alten_, by Bolle (Wismar, 1886); _Die Spiele der Griechen und
+  Romer_, by W. Richter (Leipzig, 1887); Raymond's _Alte und neue
+  Wurfelspiele_; _Chinese Games with Dice_, by Stewart Culin
+  (Philadelphia, 1889); _Korean Games_, by Stewart Culin (Philadelphia,
+  1895).
+
+
+
+
+DICETO, RALPH DE (d. c. 1202), dean of St Paul's, London, and
+chronicler, is first mentioned in 1152, when he received the
+archdeaconry of Middlesex. He was probably born between 1120 and 1130;
+of his parentage and nationality we know nothing. The common statement
+that he derived his surname from Diss in Norfolk is a mere conjecture;
+Dicetum may equally well be a Latinized form of Dissai, or Dicy, or
+Dizy, place names which are found in Maine, Picardy, Burgundy and
+Champagne. In 1152 Diceto was already a master of arts; presumably he
+had studied at Paris. His reputation for learning and integrity stood
+high; he was regarded with respect and favour by Arnulf of Lisieux and
+Gilbert Foliot of Hereford (afterwards of London), two of the most
+eminent bishops of their time. Quite naturally, the archdeacon took in
+the Becket question the same side as his friends. Although his narrative
+is colourless, and although he was one of those who showed some sympathy
+for Becket at the council of Northampton (1164), the correspondence of
+Diceto shows that he regarded the archbishop's conduct as
+ill-considered, and that he gave advice to those whom Becket regarded as
+his chief enemies. Diceto was selected, in 1166, as the envoy of the
+English bishops when they protested against the excommunications
+launched by Becket. But, apart from this episode, which he
+characteristically omits to record, he remained in the background. The
+natural impartiality of his intellect was accentuated by a certain
+timidity, which is apparent in his writings no less than in his life.
+About 1180 he became dean of St Paul's. In this office he distinguished
+himself by careful management of the estates, by restoring the
+discipline of the chapter, and by building at his own expense a
+deanery-house. A scholar and a man of considerable erudition, he showed
+a strong preference for historical studies; and about the time when he
+was preferred to the deanery he began to collect materials for the
+history of his own times. His friendships with Richard Fitz Nigel, who
+succeeded Foliot in the see of London, with William Longchamp, the
+chancellor of Richard I., and with Walter of Coutances, the archbishop
+of Rouen, gave him excellent opportunities of collecting information.
+His two chief works, the _Abbreviationes Chronicorum_ and the _Ymagines
+Historiarum_, cover the history of the world from the birth of Christ to
+the year 1202. The former, which ends in 1147, is a work of learning and
+industry, but almost entirely based upon extant sources. The latter,
+beginning as a compilation from Robert de Monte and the letters of
+Foliot, becomes an original authority about 1172, and a contemporary
+record about 1181. In precision and fulness of detail the _Ymagines_ are
+inferior to the chronicles of the so-called Benedict and of Hoveden.
+Though an annalist, Diceto is careless in his chronology; and the
+documents which he incorporates, while often important, are selected on
+no principle. He has little sense of style; but displays considerable
+insight when he ventures to discuss a political situation. For this
+reason, and on account of the details with which they supplement the
+more important chronicles of the period, the _Ymagines_ are a valuable
+though a secondary source.
+
+  See W. Stubbs' edition of the _Historical Works_ of Diceto (Rolls ed.
+  1876, 2 vols.), and especially the introduction. The second volume
+  contains minor works which are the barest compendia of facts taken
+  from well-known sources. Diceto's fragmentary Domesday of the
+  capitular estates has been edited by Archdeacon Hale in _The Domesday
+  of St Paul's_, pp. 109 ff. (Camden Society, 1858).
+
+
+
+
+DICEY, EDWARD (1832- ), English writer, son of T. E. Dicey of Claybrook
+Hall, Leicestershire, was born in 1832. Educated at Trinity College,
+Cambridge, where he took mathematical and classical honours, he became
+an active journalist, contributing largely to the principal reviews. He
+was called to the bar in 1875, became a bencher of Gray's Inn in 1896,
+and was treasurer in 1903-1904. He was connected with the _Daily
+Telegraph_ as leader writer and then as special correspondent, and after
+a short spell in 1870 as editor of the _Daily News_ he became editor of
+the _Observer_, a position which he held until 1889. Of his many books
+on foreign affairs perhaps the most important are his _England and
+Egypt_ (1884), _Bulgaria, the Peasant State_ (1895), _The Story of the
+Khedivate_ (1902), and _The Egypt of the Future_ (1907). He was created
+C.B. in 1886.
+
+His brother ALBERT VENN DICEY (b. 1835), English jurist, was educated at
+Balliol College, Oxford, where he took a first class in the classical
+schools in 1858. He was called to the bar at the Inner Temple in 1863.
+He held fellowships successively at Balliol, Trinity and All Souls', and
+from 1882 to 1909 was Vinerian professor of law. He became Q.C. in 1890.
+His chief works are the _Introduction to the Study of the Law of the
+Constitution_ (1885, 6th ed. 1902), which ranks as a standard work on
+the subject; _England's Case against Home Rule_ (1886); _A Digest of the
+Law of England with Reference to the Conflict of Laws_ (1896), and
+_Lectures on the Relation between Law and Public Opinion in England
+during the 19th century_ (1905).
+
+
+
+
+DICHOTOMY (Gr. [Greek: dicha], apart, [Greek: temnein], to cut),
+literally a cutting asunder, the technical term for a form of logical
+division, consisting in the separation of a genus into two species, one
+of which has and the other has not, a certain quality or attribute. Thus
+men may be thus divided into white men, and men who are not white; each
+of these may be subdivided similarly. On the principle of contradiction
+this division is both exhaustive and exclusive; there can be no
+overlapping, and no members of the original genus or the lower groups
+are omitted. This method of classification, though formally accurate,
+has slight value in the exact sciences, partly because at every step one
+of the two groups is merely negatively characterized and therefore
+incapable of real subdivision; it is useful, however, in setting forth
+clearly the gradual descent from the most inclusive genus (_summum
+genus_) through species to the lowest class (_infima species_), which is
+divisible only into individual persons or things. (See further
+DIVISION.) In astronomy the term is used for the aspect of the moon or
+of a planet when apparently half illuminated, so that its disk has the
+form of a semicircle.
+
+
+
+
+DICK, ROBERT (1811-1866), Scottish geologist and botanist, was born at
+Tullibody, in Clackmannanshire, in January 1811. His father was an
+officer of excise. At the age of thirteen, after receiving a good
+elementary education at the parish school, Robert Dick was apprenticed
+to a baker, and served for three years. In these early days he became
+interested in wild flowers--he made a collection of plants and gradually
+acquired some knowledge of their names from an old encyclopaedia. When
+his time was out he left Tullibody and gained employment as a journeyman
+baker at Leith, Glasgow and Greenock. Meanwhile his father, who in 1826
+had been removed to Thurso, as supervisor of excise, advised his son to
+set up a baker's shop in that town. Thither Robert Dick went in 1830, he
+started in business as a baker and worked laboriously until he died on
+the 24th of December 1866. Throughout this period he zealously devoted
+himself to studying and collecting the plants, mollusca and insects of a
+wide area of Caithness, and his attention was directed soon after he
+settled in Thurso to the rocks and fossils. In 1835 he first found
+remains of fossil fishes; but it was not till some years later that his
+interest became greatly stirred. Then he obtained a copy of Hugh
+Miller's _Old Red Sandstone_ (published in 1841), and he began
+systematically to collect with hammer and chisel the fossils from the
+Caithness flags. In 1845 he found remains of _Holoptychius_ and
+forwarded specimens to Hugh Miller, and he continued to send the best of
+his fossil fishes to that geologist, and to others after the death of
+Miller. In this way he largely contributed to the progress of geological
+knowledge, although he himself published nothing and was ever averse
+from publicity. His herbarium, which consisted of about 200 folios of
+mosses, ferns and flowering plants "almost unique in its completeness,"
+is now stored, with many of his fossils, in the museum at Thurso. Dick
+had a hard struggle for existence, especially through competition during
+his late years, when he was reduced almost to beggary: but of this few,
+if any, of his friends were aware until it was too late. A monument
+erected in the new cemetery at Thurso testifies to the respect which his
+life-work created, when the merits of this enthusiastic naturalist came
+to be appreciated.
+
+  See _Robert Dick, Baker of Thurso, Geologist and Botanist_, by Samuel
+  Smiles (1878).
+
+
+
+
+DICK, THOMAS (1774-1857), Scottish writer on astronomy, was born at
+Dundee on the 24th of November 1774. The appearance of a brilliant
+meteor inspired him, when in his ninth year, with a passion for
+astronomy; and at the age of sixteen he forsook the loom, and supported
+himself by teaching. In 1794 he entered the university of Edinburgh, and
+set up a school on the termination of his course; then, in 1801, took
+out a licence to preach, and officiated for some years as probationer in
+the United Presbyterian church. From about 1807 to 1817 he taught in the
+secession school at Methven in Perthshire, and during the ensuing decade
+in that of Perth, where he composed his first substantive book, _The
+Christian Philosopher_ (1823, 8th ed. 1842). Its success determined his
+vocation as an author; he built himself, in 1827, a cottage at Broughty
+Ferry, near Dundee, and devoted himself wholly to literary and
+scientific pursuits. They proved, however, owing to his unpractical turn
+of mind, but slightly remunerative, and he was in 1847 relieved from
+actual poverty by a crown pension of L50 a year, eked out by a local
+subscription. He died on the 29th of July 1857. His best-known works
+are: _Celestial Scenery_ (1837), _The Sidereal Heavens_ (1840), and _The
+Practical Astronomer_ (1845), in which is contained (p. 204) a
+remarkable forecast of the powers and uses of celestial photography.
+Written with competent knowledge, and in an agreeable style, they
+obtained deserved and widespread popularity.
+
+  See R. Chambers's _Eminent Scotsmen_ (ed. 1868); _Monthly Notices Roy.
+  Astr. Society_, xviii. 98; _Athenaeum_ (1857), p. 1008. (A. M. C.)
+
+
+
+
+DICKENS, CHARLES JOHN HUFFAM (1812-1870), English novelist, was born on
+the 7th of February 1812 at a house in the Mile End Terrace, Commercial
+Road, Landport (Portsea)--a house which was opened as a Dickens Museum
+on 22nd July 1904. His father John Dickens (d. 1851), a clerk in the
+navy-pay office on a salary of L80 a year, and stationed for the time
+being at Portsmouth, had married in 1809 Elizabeth, daughter of Thomas
+Barrow, and she bore him a family of eight children, Charles being the
+second. In the winter of 1814 the family moved from Portsea in the snow,
+as he remembered, to London, and lodged for a time near the Middlesex
+hospital. The country of the novelist's childhood, however, was the
+kingdom of Kent, where the family was established in proximity to the
+dockyard at Chatham from 1816 to 1821. He looked upon himself in later
+years as a man of Kent, and his capital abode as that in Ordnance
+Terrace, or 18 St Mary's Place, Chatham, amid surroundings classified in
+Mr Pickwick's notes as "appearing" to be soldiers, sailors, Jews, chalk,
+shrimps, officers and dockyard men. He fell into a family the general
+tendency of which was to go down in the world, during one of its easier
+periods (John Dickens was now fifth clerk on L250 a year), and he always
+regarded himself as belonging by right to a comfortable, genteel, lower
+middle-class stratum of society. His mother taught him to read; to his
+father he appeared very early in the light of a young prodigy, and by
+him Charles was made to sit on a tall chair and warble popular ballads,
+or even to tell stories and anecdotes for the benefit of fellow-clerks
+in the office. John Dickens, however, had a small collection of books
+which were kept in a little room upstairs that led out of Charles's own,
+and in this attic the boy found his true literary instructors in
+_Roderick Random_, _Peregrine Pickle_, _Humphry Clinker_, _Tom Jones_,
+_The Vicar of Wakefield_, _Don Quixote_, _Gil Blas_ and _Robinson
+Crusoe_. The story of how he played at the characters in these books and
+sustained his idea of Roderick Random for a month at a stretch is
+picturesquely told in _David Copperfield_. Here as well as in his first
+and last books and in what many regard as his best, _Great
+Expectations_, Dickens returns with unabated fondness and mastery to the
+surroundings of his childhood. From seven to nine years he was at a
+school kept in Clover Lane, Chatham, by a Baptist minister named William
+Giles, who gave him Goldsmith's _Bee_ as a keepsake when the call to
+Somerset House necessitated the removal of the family from Rochester to
+a shabby house in Bayham Street, Camden Town. At the very moment when a
+consciousness of capacity was beginning to plump his youthful ambitions,
+the whole flattering dream vanished and left not a rack behind.
+Happiness and Chatham had been left behind together, and Charles was
+about to enter a school far sterner and also far more instructive than
+that in Clover Lane. The family income had been first decreased and then
+mortgaged; the creditors of the "prodigal father" would not give him
+time; John Dickens was consigned to the Marshalsea; Mrs Dickens started
+an "Educational Establishment" as a forlorn hope in Upper Gower Street;
+and Charles, who had helped his mother with the children, blacked the
+boots, carried things to the pawnshop and done other menial work, was
+now sent out to earn his own living as a young hand in a blacking
+warehouse, at Old Hungerford Stairs, on a salary of six shillings a
+week. He tied, trimmed and labelled blacking pots for over a year,
+dining off a saveloy and a slice of pudding, consorting with two very
+rough boys, Bob Fagin and Pol Green, and sleeping in an attic in Little
+College Street, Camden Town, in the house of Mrs Roylance (Pipchin),
+while on Sunday he spent the day with his parents in their comfortable
+prison, where they had the services of a "marchioness" imported from the
+Chatham workhouse.
+
+Already consumed by ambition, proud, sensitive and on his dignity to an
+extent not uncommon among boys of talent, he felt his position keenly,
+and in later years worked himself up into a passion of self-pity in
+connexion with the "degradation" and "humiliation" of this episode. The
+two years of childish hardship which ate like iron into his soul were
+obviously of supreme importance in the growth of the novelist.
+Recollections of the streets and the prison and its purlieus supplied
+him with a store of literary material upon which he drew through all the
+years of his best activity. And the bitterness of such an experience was
+not prolonged sufficiently to become sour. From 1824 to 1826, having
+been rescued by a family quarrel and by a windfall in the shape of a
+legacy to his father, from the warehouse, he spent two years at an
+academy known as Wellington House, at the corner of Granby Street and
+the Hampstead Road (the lighter traits of which are reproduced in Salem
+House), and was there known as a merry and rather mischievous boy.
+Fortunately he learned nothing there to compromise the results of
+previous instruction. His father had now emerged from the Marshalsea and
+was seeking employment as a parliamentary reporter. A Gray's Inn
+solicitor with whom he had had dealings was attracted by the bright,
+clever look of Charles, and took him into his office as a boy at a
+salary of thirteen and sixpence (rising to fifteen shillings) a week. He
+remained in Mr Blackmore's office from May 1827 to November 1828, but he
+had lost none of his eager thirst for distinction, and spent all his
+spare time mastering Gurney's shorthand and reading early and late at
+the British Museum. A more industrious apprentice in the lower grades of
+the literary profession has never been known, and the consciousness of
+opportunities used to the most splendid advantage can hardly have been
+absent from the man who was shortly to take his place at the head of it
+as if to the manner born. Lowten and Guppy, and Swiveller had been
+observed from this office lad's stool; he was now greatly to widen his
+area of study as a reporter in Doctors' Commons and various police
+courts, including Bow Street, working all day at law and much of the
+night at shorthand. Some one asked John Dickens, during the first eager
+period of curiosity as to the man behind "Pickwick," where his son
+Charles was educated. "Well really," said the prodigal father, "he may
+be said--haw--haw--to have educated himself." He was one of the most
+rapid and accurate reporters in London when, at nineteen years of age,
+in 1831, he realized his immediate ambition and "entered the gallery" as
+parliamentary reporter to the _True Sun_. Later he was reporter to the
+_Mirror of Parliament_ and then to the _Morning Chronicle_. Several of
+his earliest letters are concerned with his exploits as a reporter, and
+allude to the experiences he had, travelling fifteen miles an hour and
+being upset in almost every description of known vehicle in various
+parts of Britain between 1831 and 1836. The family was now living in
+Bentwick Street, Manchester Square, but John Dickens was still no
+infrequent inmate of the sponging-houses. With all the accessories of
+these places of entertainment his son had grown to be excessively
+familiar. Writing about 1832 to his school friend Tom Mitton, Dickens
+tells him that his father has been arrested at the suit of a wine firm,
+and begs him go over to Cursitor Street and see what can be done. On
+another occasion of a paternal disappearance he observes: "I own that
+his absence does not give me any great uneasiness, knowing how apt he is
+to get out of the way when anything goes wrong." In yet another letter
+he asks for a loan of four shillings.
+
+In the meanwhile, however, he had commenced author in a more creative
+sense by penning some sketches of contemporary London life, such as he
+had attempted in his school days in imitation of the sketches published
+in the _London_ and other magazines of that day. The first of these
+appeared in the December number of the _Old Monthly Magazine_ for 1833.
+By the following August, when the signature "Boz" was first given, five
+of these sketches had appeared. By the end of 1834 we find him settled
+in rooms in Furnival's Inn, and a little later his salary on the
+_Morning Chronicle_ was raised, owing to the intervention of one of its
+chiefs, George Hogarth, the father of (in addition to six sons) eight
+charming daughters, to one of whom, Catherine, Charles was engaged to be
+married before the year was out. Clearly as his career now seemed
+designated, he was at this time or a little before it coquetting very
+seriously with the stage: but circumstances were rapidly to determine
+another stage in his career. A year before Queen Victoria's accession
+appeared in two volumes _Sketches by Boz_, _Illustrative of Everyday
+Life and Everyday People_. The book came from a prentice hand, but like
+the little tract on the Puritan abuse of the Sabbath entitled "Sunday
+under three Heads" which appeared a few months later, it contains in
+germ all, or almost all, the future Dickens. Glance at the headings of
+the pages. Here we have the Beadle and all connected with him, London
+streets, theatres, shows, the pawnshop, Doctors' Commons, Christmas,
+Newgate, coaching, the river. Here comes a satirical picture of
+parliament, fun made of cheap snobbery, a rap on the knuckles of
+sectarianism. And what could be more prophetic than the title of the
+opening chapter--Our Parish? With the Parish--a large one
+indeed--Dickens to the end concerned himself; he began with a rapid
+survey of his whole field, hinting at all he might accomplish,
+indicating the limits he was not to pass. This year was to be still more
+momentous to Dickens, for, on the 2nd of April 1836, he was married to
+George Hogarth's eldest daughter Catherine. He seems to have fallen in
+love with the daughters collectively, and, judging by subsequent events,
+it has been suggested that perhaps he married the wrong one. His wife's
+sister Mary was the romance of his early married life, and another
+sister, Georgina, was the dearest friend of his last ten years.
+
+A few days before the marriage, just two months after the appearance of
+the _Sketches_, the first part of _The Posthumous Papers of the Pickwick
+Club_ was announced. One of the chief vogues of the day was the issue of
+humorous, sporting or anecdotal novels in parts, with plates, and some
+of the best talent of the day, represented by Ainsworth, Bulwer,
+Marryat, Maxwell, Egan, Hook and Surtees, had been pressed into this
+kind of enterprise. The publishers of the day had not been slow to
+perceive Dickens's aptitude for this species of "letterpress." A member
+of the firm of Chapman & Hall called upon him at Furnival's Inn in
+December 1835 with a proposal that he should write about a Nimrod Club
+of amateur sportsmen, foredoomed to perpetual ignominies, while the
+comic illustrations were to be etched by Seymour, a well-known rival of
+Cruikshank (the illustrator of _Boz_). The offer was too tempting for
+Dickens to refuse, but he changed the idea from a club of Cockney
+sportsmen to that of a club of eccentric peripatetics, on the sensible
+grounds, first that sporting sketches were stale, and, secondly, that he
+knew nothing worth speaking of about sport. The first seven pictures
+appeared with the signature of Seymour and the letterpress of Dickens.
+Before the eighth picture appeared Seymour had blown his brains out.
+After a brief interval of Buss, Dickens obtained the services of Hablot
+K. Browne, known to all as "Phiz." Author and illustrator were as well
+suited to one another and to the common creation of a unique thing as
+Gilbert and Sullivan. Having early got rid of the sporting element,
+Dickens found himself at once. The subject exactly suited his knowledge,
+his skill in arranging incidents--nay, his very limitations too. No
+modern book is so incalculable. We commence laughing heartily at
+Pickwick and his troupe. The laugh becomes kindlier. We are led on
+through a tangle of adventure, never dreaming what is before us. The
+landscape changes: Pickwick becomes the symbol of kindheartedness,
+simplicity and innocent levity. Suddenly in the Fleet Prison a deeper
+note is struck. The medley of human relationships, the loneliness, the
+mystery and sadness of human destinies are fathomed. The tragedy of
+human life is revealed to us amid its most farcical elements. The droll
+and laughable figure of the hero is transfigured by the kindliness of
+human sympathy into a beneficent and bespectacled angel in shorts and
+gaiters. By defying accepted rules, Dickens had transcended the limited
+sphere hitherto allotted to his art: he had produced a book to be
+enshrined henceforth in the inmost hearts of all sorts and conditions of
+his countrymen, and had definitely enlarged the boundaries of English
+humour and English fiction. As for Mr Pickwick, he is a fairy like Puck
+or Santa Claus, while his creator is "the last of the mythologists and
+perhaps the greatest."
+
+When _The Pickwick Papers_ appeared in book form at the close of 1837
+Dickens's popular reputation was made. From the appearance of Sam Weller
+in part v. the universal hunger for the monthly parts had risen to a
+furore. The book was promptly translated into French and German. The
+author had received little assistance from press or critics, he had no
+influential connexions, his class of subjects was such as to "expose him
+at the outset to the fatal objections of vulgarity," yet in less than
+six months from the appearance of the first number, as the _Quarterly
+Review_ almost ruefully admits, the whole reading world was talking
+about the Pickwickians. The names of Winkle, Wardle, Weller, Jingle,
+Snodgrass, Dodson & Fogg, were as familiar as household words. Pickwick
+chintzes figured in the linendrapers' windows, and Pickwick cigars in
+every tobacconist's; Weller corduroys became the stock-in-trade of every
+breeches-maker; Boz cabs might be seen rattling through the streets, and
+the portrait of the author of _Pelham_ and _Crichton_ was scraped down
+to make way for that of the new popular favourite on the omnibuses. A
+new and original genius had suddenly sprung up, there was no denying it,
+even though, as the _Quarterly_ concluded, "it required no gift of
+prophecy to foretell his fate--he has risen like a rocket and he will
+come down like the stick." It would have needed a very emphatic gift of
+prophecy indeed to foretell that Dickens's reputation would have gone on
+rising until at the present day (after one sharp fall, which reached an
+extreme about 1887) it stands higher than it has ever stood before.
+
+Dickens's assumption of the literary purple was as amazing as anything
+else about him. Accepting the homage of the luminaries of the literary,
+artistic and polite worlds as if it had been his natural due, he
+arranges for the settlement of his family, decrees, like another Edmund
+Kean, that his son is to go to Eton, carries on the most complicated
+negotiations with his publishers and editors, presides and orates with
+incomparable force at innumerable banquets, public and private, arranges
+elaborate villegiatures in the country, at the seaside, in France or in
+Italy, arbitrates in public on every topic, political, ethical,
+artistic, social or literary, entertains and legislates for an
+increasingly large domestic circle, both juvenile and adult, rules
+himself and his time-table with a rod of iron. In his letter-writing
+alone, Dickens did a life's literary work. Nowadays no one thinks of
+writing such letters; that is to say, letters of such length and detail,
+for the quality is Dickens's own. He evidently enjoyed this use of the
+pen. Page after page of Forster's _Life_ (750 pages in the _Letters_
+edited by his daughter and sister-in-law) is occupied with transcription
+from private correspondence, and never a line of this but is thoroughly
+worthy of print and preservation. If he makes a tour in any part of the
+British Isles, he writes a full description of all he sees, of
+everything that happens, and writes it with such gusto, such mirth, such
+strokes of fine picturing, as appear in no other private letters ever
+given to the public. Naturally buoyant in all circumstances, a holiday
+gave him the exhilaration of a schoolboy. See how he writes from
+Cornwall, when on a trip with two or three friends, in 1843. "Heavens!
+if you could have seen the necks of bottles, distracting in their
+immense variety of shape, peering out of the carriage pockets! If you
+could have witnessed the deep devotion of the post-boys, the maniac glee
+of the waiters! If you could have followed us into the earthy old
+churches we visited, and into the strange caverns on the gloomy
+seashore, and down into the depths of mines, and up to the tops of giddy
+heights, where the unspeakably green water was roaring, I don't know how
+many hundred feet below.... I never laughed in my life as I did on this
+journey. It would have done you good to hear me. I was choking and
+gasping and bursting the buckles off the back of my stock, all the way.
+And Stanfield"--the painter--"got into such apoplectic entanglements
+that we were obliged to beat him on the back with portmanteaus before we
+could recover him."
+
+The animation of Dickens's look would attract the attention of any one,
+anywhere. His figure was not that of an Adonis, but his brightness made
+him the centre and pivot of every society he was in. The keenness and
+vivacity of his eye combined with his inordinate appetite for life to
+give the unique quality to all that he wrote. His instrument is that of
+the direct, sinewy English of Smollett, combined with much of the
+humorous grace of Goldsmith (his two favourite authors), but modernized
+to a certain extent under the influence of Washington Irving, Sydney
+Smith, Jeffrey, Lamb, and other writers of the _London Magazine_. He
+taught himself to speak French and Italian, but he could have read
+little in any language. His ideas were those of the inchoate and insular
+liberalism of the 'thirties. His unique force in literature he was to
+owe to no supreme artistic or intellectual quality, but almost entirely
+to his inordinate gift of observation, his sympathy with the humble, his
+power over the emotions and his incomparable endowment of unalloyed
+human fun. To contemporaries he was not so much a man as an
+institution, at the very mention of whose name faces were puckered with
+grins or wreathed in smiles. To many his work was a revelation, the
+revelation of a new world and one far better than their own. And his
+influence went further than this in the direction of revolution or
+revival. It gave what were then universally referred to as "the lower
+orders" a new sense of self-respect, a new feeling of citizenship. Like
+the defiance of another Luther, or the Declaration of a new
+Independence, it emitted a fresh ray of hope across the firmament. He
+did for the whole English-speaking race what Burns had done for
+Scotland--he gave it a new conceit of itself. He knew what a people
+wanted and he told what he knew. He could do this better than anybody
+else because his mind was theirs. He shared many of their "great useless
+virtues," among which generosity ranks before justice, and sympathy
+before truth, even though, true to his middle-class vein, he exalts
+piety, chastity and honesty in a manner somewhat alien to the mind of
+the low-bred man. This is what makes Dickens such a demigod and his
+public success such a marvel, and this also is why any exclusively
+literary criticism of his work is bound to be so inadequate. It should
+also help us to make the necessary allowances for the man. Dickens, even
+the Dickens of legend that we know, is far from perfect. The Dickens of
+reality to which Time may furnish a nearer approximation is far less
+perfect. But when we consider the corroding influence of adulation, and
+the intoxication of unbridled success, we cannot but wonder at the
+relatively high level of moderation and self-control that Dickens almost
+invariably observed. Mr G. K. Chesterton remarks suggestively that
+Dickens had all his life the faults of the little boy who is kept up too
+late at night. He is overwrought by happiness to the verge of
+exasperation, and yet as a matter of fact he does keep on the right side
+of the breaking point. The specific and curative in his case was the
+work in which he took such anxious pride, and such unmitigated delight.
+He revelled in punctual and regular work; at his desk he was often in
+the highest spirits. Behold how he pictured himself, one day at
+Broadstairs, where he was writing _Chuzzlewit_. "In a bay-window in a
+one-pair sits, from nine o'clock to one, a gentleman with rather long
+hair and no neckcloth, who writes and grins, as if he thought he was
+very funny indeed. At one he disappears, presently emerges from a
+bathing-machine, and may be seen, a kind of salmon-colour porpoise,
+splashing about in the ocean. After that, he may be viewed in another
+bay-window on the ground-floor eating a strong lunch; and after that,
+walking a dozen miles or so, or lying on his back on the sand reading a
+book. Nobody bothers him, unless they know he is disposed to be talked
+to, and I am told he is very comfortable indeed. He's as brown as a
+berry, and they do say he is as good as a small fortune to the
+innkeeper, who sells beer and cold punch." Here is the secret of such
+work as that of Dickens; it is done with delight--done (in a sense)
+easily, done with the mechanism of mind and body in splendid order. Even
+so did Scott write; though more rapidly and with less conscious care:
+his chapter finished before the world had got up to breakfast. Later,
+Dickens produced novels less excellent with much more of mental strain.
+The effects of age could not have shown themselves so soon, but for the
+unfortunate loss of energy involved in his non-literary labours.
+
+While the public were still rejoicing in the first sprightly runnings of
+the "new humour," the humorist set to work desperately on the grim
+scenes of _Oliver Twist_, the story of a parish orphan, the nucleus of
+which had already seen the light in his _Sketches_. The early scenes are
+of a harrowing reality, despite the germ of forced pathos which the
+observant reader may detect in the pitiful parting between Oliver and
+little Dick; but what will strike every reader at once in this book is
+the directness and power of the English style, so nervous and unadorned:
+from its unmistakable clearness and vigour Dickens was to travel far as
+time went on. But the full effect of the old simplicity is felt in such
+masterpieces of description as the drive of Oliver and Sikes to
+Chertsey, the condemned-cell ecstasy of Fagin, or the unforgettable
+first encounter between Oliver and the Artful Dodger. Before November
+1837 had ended, Charles Dickens entered on an engagement to write a
+successor to _Pickwick_ on similar lines of publication. _Oliver Twist_
+was then in mid-career; a _Life of Grimaldi_ and _Barnaby Rudge_ were
+already covenanted for. Dickens forged ahead with the new tale of
+_Nicholas Nickleby_ and was justified by the results, for its sale far
+surpassed even that of _Pickwick_. As a conception it is one of his
+weakest. An unmistakably 18th-century character pervades it. Some of the
+vignettes are among the most piquant and besetting ever written. Large
+parts of it are totally unobserved conventional melodrama; but the
+Portsmouth Theatre and Dotheboys Hall and Mrs Nickleby (based to some
+extent, it is thought, upon Miss Bates in _Emma_, but also upon the
+author's Mamma) live for ever as Dickens conceived them in the pages of
+_Nicholas Nickleby_.
+
+Having got rid of _Nicholas Nickleby_ and resigned his editorship of
+_Bentley's Miscellany_, in which _Oliver Twist_ originally appeared,
+Dickens conceived the idea of a weekly periodical to be issued as
+_Master Humphrey's Clock_, to comprise short stories, essays and
+miscellaneous papers, after the model of Addison's _Spectator_. To make
+the weekly numbers "go," he introduced Mr Pickwick, Sam Weller and his
+father in friendly intercourse. But the public requisitioned "a story,"
+and in No. 4 he had to brace himself up to give them one. Thus was
+commenced _The Old Curiosity Shop_, which was continued with slight
+interruptions, and followed by _Barnaby Rudge_. For the first time we
+find Dickens obsessed by a highly complicated plot. The tonality
+achieved in _The Old Curiosity Shop_ surpassed anything he had attempted
+in this difficult vein, while the rich humour of Dick Swiveller and the
+Marchioness, and the vivid portraiture of the wandering Bohemians,
+attain the very highest level of Dickensian drollery; but in the
+lamentable tale of Little Nell (though Landor and Jeffrey thought the
+character-drawing of this infant comparable with that of Cordelia), it
+is generally admitted that he committed an indecent assault upon the
+emotions by exhibiting a veritable monster of piety and long-suffering
+in a child of tender years. In _Barnaby Rudge_ he was manifestly
+affected by the influence of Scott, whose achievements he always
+regarded with a touching veneration. The plot, again, is of the utmost
+complexity, and Edgar Allan Poe (who predicted the conclusion) must be
+one of the few persons who ever really mastered it. But few of Dickens's
+books are written in a more admirable style.
+
+_Master Humphrey's Clock_ concluded, Dickens started in 1842 on his
+first visit to America--an episode hitherto without parallel in English
+literary history, for he was received everywhere with popular
+acclamation as the representative of a grand triumph of the English
+language and imagination, without regard to distinctions of nationality.
+He offended the American public grievously by a few words of frank
+description and a few quotations of the advertisement columns of
+American papers illustrating the essential barbarity of the old slave
+system (_American Notes_). Dickens was soon pining for home--no English
+writer is more essentially and insularly English in inspiration and
+aspiration than he is. He still brooded over the perverseness of America
+on the copyright question, and in his next book he took the opportunity
+of uttering a few of his impressions about the objectionable sides of
+American democracy, the result being that "all Yankee-doodle-dom blazed
+up like one universal soda bottle," as Carlyle said. _Martin Chuzzlewit_
+(1843-1844) is important as closing his great character period. His
+_seve originale_, as the French would say, was by this time to a
+considerable extent exhausted, and he had to depend more upon artistic
+elaboration, upon satires, upon _tours de force_ of description, upon
+romantic and ingenious contrivances. But all these resources combined
+proved unequal to his powers as an original observer of popular types,
+until he reinforced himself by autobiographic reminiscence, as in _David
+Copperfield_ and _Great Expectations_, the two great books remaining to
+his later career.
+
+After these two masterpieces and the three wonderful books with which he
+made his debut, we are inclined to rank _Chuzzlewit_. Nothing in Dickens
+is more admirably seen and presented than Todgers's, a bit of London
+particular cut out with a knife. Mr Pecksniff and Mrs Gamp, Betsy Prig
+and "Mrs Harris" have passed into the national language and life. The
+coach journey, the windy autumn night, the stealthy trail of Jonas, the
+undertone of tragedy in the Charity and Mercy and Chuffey episodes
+suggest a blending of imaginative vision and physical penetration hardly
+seen elsewhere. Two things are specially notable about this novel--the
+exceptional care taken over it (as shown by the interlineations in the
+MS.) and the caprice or nonchalance of the purchasing public, its sales
+being far lower than those of any of its monthly predecessors.
+
+At the close of 1843, to pay outstanding debts of his now lavish
+housekeeping, he wrote that pioneer of Christmas numbers, that national
+benefit as Thackeray called it, _A Christmas Carol_. It failed to
+realize his pecuniary anticipations, and Dickens resolved upon a drastic
+policy of retrenchment and reform. He would save expense by living
+abroad and would punish his publishers by withdrawing his custom from
+them, at least for a time. Like everything else upon which he ever
+determined, this resolution was carried out with the greatest possible
+precision and despatch. In June 1844 he set out for Marseilles with his
+now rapidly increasing family (the journey cost him L200). In a villa on
+the outskirts of Genoa he wrote _The Chimes_, which, during a brief
+excursion to London before Christmas, he read to a select circle of
+friends (the germ of his subsequent lecture-audiences), including
+Forster, Carlyle, Stanfield, Dyce, Maclise and Jerrold. He was again in
+London in 1845, enjoying his favourite diversion of private theatricals;
+and in January 1846 he experimented briefly as the editor of a London
+morning paper--the _Daily News_. By early spring he was back at
+Lausanne, writing his customary vivid letters to his friends, craving as
+usual for London streets, commencing _Dombey and Son_, and walking his
+fourteen miles daily. The success of _Dombey and Son_ completely
+rehabilitated the master's finances, enabled him to return to England,
+send his son to Eton and to begin to save money. Artistically it is less
+satisfactory; it contains some of Dickens's prime curios, such as
+Cuttle, Bunsby, Toots, Blimber, Pipchin, Mrs MacStinger and young Biler;
+it contains also that masterpiece of sentimentality which trembles upon
+the borderland of the sublime and the ridiculous, the death of Paul
+Dombey ("that sweet Paul," as Jeffrey, the "critic laureate," called
+him), and some grievous and unquestionable blemishes. As a narrative,
+moreover, it tails off into a highly complicated and exacting plot. It
+was followed by a long rest at Broadstairs before Dickens returned to
+the native home of his genius, and early in 1849 "began to prepare for
+_David Copperfield_."
+
+"Of all my books," Dickens wrote, "I like this the best; like many fond
+parents I have my favourite child, and his name is David Copperfield."
+In some respects it stands to Dickens in something of the same relation
+in which the contemporary _Pendennis_ stands to Thackeray. As in that
+book, too, the earlier portions are the best. They gained in intensity
+by the autobiographical form into which they are thrown; as Thackeray
+observed, there was no writing against such power. The tragedy of Emily
+and the character of Rosa Dartle are stagey and unreal; Uriah Heep is
+bad art; Agnes, again, is far less convincing as a consolation than
+Dickens would have us believe; but these are more than compensated by
+the wonderful realization of early boyhood in the book, by the picture
+of Mr Creakle's school, the Peggottys, the inimitable Mr Micawber, Betsy
+Trotwood and that monument of selfish misery, Mrs Gummidge.
+
+At the end of March 1850 commenced the new twopenny weekly called
+_Household Words_, which Dickens planned to form a direct means of
+communication between himself and his readers, and as a means of
+collecting around him and encouraging the talents of the younger
+generation. No one was better qualified than he for this work, whether
+we consider his complete freedom from literary jealousy or his magical
+gift of inspiring young authors. Following the somewhat dreary and
+incoherent _Bleak House_ of 1852, _Hard Times_ (1854)--an
+anti-Manchester School tract, which Ruskin regarded as Dickens's best
+work--was the first long story written for _Household Words_. About this
+time Dickens made his final home at Gad's Hill, near Rochester, and put
+the finishing touch to another long novel published upon the old plan,
+_Little Dorrit_ (1855-1857). In spite of the exquisite comedy of the
+master of the Marshalsea and the final tragedy of the central figure,
+_Little Dorrit_ is sadly deficient in the old vitality, the humour is
+often a mock reality, and the repetition of comic catch-words and
+overstrung similes and metaphors is such as to affect the reader with
+nervous irritation. The plot and characters ruin each other in this
+amorphous production. The _Tale of Two Cities_, commenced in _All the
+Year Round_ (the successor of _Household Words_) in 1859, is much
+better: the main characters are powerful, the story genuinely tragic,
+and the atmosphere lurid; but enormous labour was everywhere expended
+upon the construction of stylistic ornament.
+
+The _Tale of Two Cities_ was followed by two finer efforts at
+atmospheric delineation, the best things he ever did of this kind:
+_Great Expectations_ (1861), over which there broods the mournful
+impression of the foggy marshes of the Lower Thames; and _Our Mutual
+Friend_ (1864-1865), in which the ooze and mud and slime of Rotherhithe,
+its boatmen and loafers, are made to pervade the whole book with
+cumulative effect. The general effect produced by the stories is,
+however, very different. In the first case, the foreground was supplied
+by autobiographical material of the most vivid interest, and the
+lucidity of the creative impulse impelled him to write upon this
+occasion with the old simplicity, though with an added power. Nothing
+therefore, in the whole range of Dickens surpassed the early chapters of
+_Great Expectations_ in perfection of technique or in mastery of all the
+resources of the novelist's art. To have created Abel Magwitch alone is
+to be a god indeed, says Mr Swinburne, among the creators of deathless
+men. Pumblechook is actually better and droller and truer to imaginative
+life than Pecksniff; Joe Gargery is worthy to have been praised and
+loved at once by Fielding and by Sterne: Mr Jaggers and his clients, Mr
+Wemmick and his parent and his bride, are such figures as Shakespeare,
+when dropping out of poetry, might have created, if his lot had been
+cast in a later century. "Can as much be said," Mr Swinburne boldly
+asks, "for the creatures of any other man or god?"
+
+In November 1867 Dickens made a second expedition to America, leaving
+all the writing that he was ever to complete behind him. He was to make
+a round sum of money, enough to free him from all embarrassments, by a
+long series of exhausting readings, commencing at the Tremont Temple,
+Boston, on the 2nd of December. The strain of Dickens's ordinary life
+was so tense and so continuous that it is, perhaps, rash to assume that
+he broke down eventually under this particular stress; for other
+reasons, however, his persistence in these readings, subsequent to his
+return, was strongly deprecated by his literary friends, led by the
+arbitrary and relentless Forster. It is a long testimony to Dickens's
+self-restraint, even in his most capricious and despotic moments, that
+he never broke the cord of obligation which bound him to his literary
+mentor, though sparring matches between them were latterly of frequent
+occurrence. His farewell reading was given on the 15th of March 1870, at
+St James's Hall. He then vanished from "those garish lights," as he
+called them, "for evermore." Of the three brief months that remained to
+him, his last book, _The Mystery of Edwin Drood_, was the chief
+occupation. It hardly promised to become a masterpiece (Longfellow's
+opinion) as did Thackeray's _Denis Duval_, but contained much fine
+descriptive technique, grouped round a scene of which Dickens had an
+unrivalled sympathetic knowledge.
+
+In March and April 1870 Dickens, as was his wont, was mixing in the best
+society; he dined with the prince at Lord Houghton's and was twice at
+court, once at a long deferred private interview with the queen, who had
+given him a presentation copy of her _Leaves from a Journal of our Life
+in the Highlands_ with the inscription "From one of the humblest of
+authors to one of the greatest"; and who now begged him on his
+persistent refusal of any other title to accept the nominal distinction
+of a privy councillor. He took for four months the Milner Gibsons' house
+at 5 Hyde Park Place, opposite the Marble Arch, where he gave a
+brilliant reception on the 7th of April. His last public appearance was
+made at the Royal Academy banquet early in May. He returned to his
+regular methodical routine of work at Gad's Hill on the 30th of May, and
+one of the last instalments he wrote of _Edwin Drood_ contained an
+ominous speculation as to the next two people to die at Cloisterham:
+"Curious to make a guess at the two, or say at one of the two." Two
+letters bearing the well-known superscription "Gad's Hill Place, Higham
+by Rochester, Kent" are dated the 8th of June, and, on the same
+Thursday, after a long spell of writing in the Chalet where he
+habitually wrote, he collapsed suddenly at dinner. Startled by the
+sudden change in the colour and expression of his face, his
+sister-in-law (Miss Hogarth) asked him if he was ill; he said "Yes, very
+ill," but added that he would finish dinner and go on afterwards to
+London. "Come and lie down," she entreated; "Yes, on the ground," he
+said, very distinctly; these were the last words he spoke, and he slid
+from her arms and fell upon the floor. He died at 6-10 P.M. on Friday,
+the 9th of June, and was buried privately in Poets' Corner, Westminster
+Abbey, in the early morning of the 14th of June. One of the most
+appealing memorials was the drawing by his "new illustrator" Luke Fildes
+in the _Graphic_ of "The Empty Chair; Gad's Hill: ninth of June, 1870."
+"Statesmen, men of science, philanthropists, the acknowledged
+benefactors of their race, might pass away, and yet not leave the void
+which will be caused by the death of Charles Dickens" (_The Times_). In
+his will he enjoined his friends to erect no monument in his honour, and
+directed his name and dates only to be inscribed on his tomb, adding
+this proud provision, "I rest my claim to the remembrance of my country
+on my published works."
+
+Dickens had no artistic ideals worth speaking about. The sympathy of his
+readers was the one thing he cared about and, like Cobbett, he went
+straight for it through the avenue of the emotions. In personality,
+intensity and range of creative genius he can hardly be said to have any
+modern rival. His creations live, move and have their being about us
+constantly, like those of Homer, Virgil, Chaucer, Rabelais, Cervantes,
+Shakespeare, Bunyan, Moliere and Sir Walter Scott. As to the books
+themselves, the backgrounds on which these mighty figures are projected,
+they are manifestly too vast, too chaotic and too unequal ever to become
+classics. Like most of the novels constructed upon the unreformed model
+of Smollett and Fielding, those of Dickens are enormous stock-pots into
+which the author casts every kind of autobiographical experience,
+emotion, pleasantry, anecdote, adage or apophthegm. The fusion is
+necessarily very incomplete and the hotch-potch is bound to fall to
+pieces with time. Dickens's plots, it must be admitted, are strangely
+unintelligible, the repetitions and stylistic decorations of his work
+exceed all bounds, the form is unmanageable and insignificant. The
+diffuseness of the English novel, in short, and its extravagant
+didacticism cannot fail to be most prejudicial to its perpetuation. In
+these circumstances there is very little fiction that will stand
+concentration and condensation so well as that of Dickens.
+
+For these reasons among others our interest in Dickens's novels as
+integers has diminished and is diminishing. But, on the other hand, our
+interest and pride in him as a man and as a representative author of his
+age and nation has been steadily augmented and is still mounting. Much
+of the old criticism of his work, that it was not up to a sufficiently
+high level of art, scholarship or gentility, that as an author he is
+given to caricature, redundancy and a shameless subservience to popular
+caprice, must now be discarded as irrelevant.
+
+As regards formal excellence it is plain that Dickens labours under the
+double disadvantage of writing in the least disciplined of all literary
+genres in the most lawless literary milieu of the modern world, that of
+Victorian England. In spite of these defects, which are those of masters
+such as Rabelais, Hugo and Tolstoy, the work of Dickens is more and more
+instinctively felt to be true, original and ennobling. It is already
+beginning to undergo a process of automatic sifting, segregation and
+crystallization, at the conclusion of which it will probably occupy a
+larger segment in the literary consciousness of the English-spoken race
+than ever before.
+
+Portraits of Dickens, from the gay and alert "Boz" of Samuel Lawrence,
+and the self-conscious, rather foppish portrait by Maclise which served
+as frontispiece to _Nicholas Nickleby_, to the sketch of him as Bobadil
+by C. R. Leslie, the Drummond and Ary Scheffer portraits of middle age
+and the haggard and drawn representations of him from photographs after
+his shattering experiences as a public entertainer from 1856 (the year
+of his separation from his wife) onwards, are reproduced in Kitton, in
+Forster and Gissing and in the other biographies. Sketches are also
+given in most of the books of his successive dwelling places at Ordnance
+Terrace and 18 St Mary's Place, Chatham; Bayham Street, Camden Town; 15
+Furnival's Inn; 48 Doughty Street; 1 Devonshire Terrace, Regent's Park;
+Tavistock House, Tavistock Square; and Gad's Hill Place. The manuscripts
+of all the novels, with the exception of the _Tale of Two Cities_ and
+_Edwin Drood_, were given to Forster, and are now preserved in the Dyce
+and Forster Museum at South Kensington. The work of Dickens was a prize
+for which publishers naturally contended both before and after his
+death. The first collective edition of his works was begun in April
+1847, and their number is now very great. The most complete is still
+that of Messrs Chapman & Hall, the original publishers of _Pickwick_;
+others of special interest are the Harrap edition, originally edited by
+F. G. Kitton; Macmillan's edition with original illustrations and
+introduction by Charles Dickens the younger; and the edition in the
+World's Classics with introductions by G.K. Chesterton. Of the
+translations the best known is that done into French by Lorain, Pichot
+and others, with B.H. Gausseron's excellent _Pages Choisies_ (1903).
+
+  BIBLIOGRAPHY.--During his lifetime Dickens's biographer was clearly
+  indicated in his guide, philosopher and friend, John Forster, who had
+  known the novelist intimately since the days of his first triumph with
+  _Pickwick_, who had constituted himself a veritable encyclopaedia of
+  information about Dickens, and had clung to his subject (in spite of
+  many rebuffs which his peremptory temper found it hard to digest) as
+  tightly as ever Boswell had enveloped Johnson. Two volumes of
+  Forster's _Life of Charles Dickens_ appeared in 1872 and a third in
+  1874. He relied much on Dickens's letters to himself and produced what
+  must always remain the authoritative work. The first two volumes are
+  put together with much art, the portrait as a whole has been regarded
+  as truthful, and the immediate success was extraordinary. In the
+  opinion of Carlyle, Forster's book was not unworthy to be named after
+  that of Boswell. A useful abridgment was carried out in 1903 by the
+  novelist George Gissing. Gissing also wrote _Charles Dickens: A
+  Critical Study_ (1898), which ranks with G.K. Chesterton's _Charles
+  Dickens_(1906) as a commentary inspired by deep insight and adorned by
+  great literary talent upon the genius of the master-novelist. The
+  names of other lives, sketches, articles and estimates of Dickens and
+  his works would occupy a large volume in the mere enumeration. See
+  R.H. Shepherd, _The Bibliography of Dickens_ (1880); _James Cooke's
+  Bibliography of the Writings of Charles Dickens_ (1879);
+  _Dickensiana_, by F. G. Kitton (1886); and _Bibliography_ by J.P.
+  Anderson, appended to Sir F.T. Marzials's _Life of Charles Dickens_
+  (1887). Among the earlier sketches may be specially cited the lives by
+  J. C. Hotten and G. A. Sala (1870), the Anecdote-Biography edited by
+  the American R. H. Stoddard (1874), Dr A. W. Ward in the English Men
+  of Letters Series (1878), that by Sir Leslie Stephen in the
+  _Dictionary of National Biography_, and that by Professor Minto in the
+  eighth edition of the _Encyclopaedia Britannica_. The _Letters_ were
+  first issued in two volumes edited by his daughter and sister-in-law
+  in 1880. For Dickens's connexion with Kent the following books are
+  specially valuable:--Robert Langton's _Childhood and Youth of Charles
+  Dickens_ (1883); Langton's _Dickens and Rochester_ (1880); Thomas
+  Frost's _In Kent with Charles Dickens_ (1880); F. G. Kitton's _The
+  Dickens Country_ (1905); H. S. Ward's _The Real Dickens Land_ (1904);
+  R. Allbut's _Rambles in Dickens Land_ (1899 and 1903). For Dickens's
+  reading tours see G. Dolby's _Charles Dickens as I knew him_ (1884);
+  J. T. Fields's _In and Out of Doors with Charles Dickens_ (1876);
+  Charles Kent's _Dickens as a Reader_ (1872). And for other aspects of
+  his life see M. Dickens's _My Father as I recall him_ (1897); P. H.
+  Fitzgerald's _Life of C. Dickens as revealed in his Writings_ (1905),
+  and _Bozland_ (1895); F. G. Kitton's _Charles Dickens, his Life,
+  Writings and Personality_, a useful compendium (1902); T. E.
+  Pemberton's _Charles Dickens and the Stage_, and _Dickens's London_
+  (1876); F. Miltoun's _Dickens's London_ (1904); Kitton's _Dickens and
+  his Illustrators_; W. Teignmouth Shore's _Charles Dickens and his
+  Friends_ (1904 and 1909); B. W. Matz, _Story of Dickens's Life and
+  Work_ (1904), and review of solutions to _Edwin Drood_ in _The
+  Bookman_ for March 1908; the recollections of Edmund Yates, Trollope,
+  James Payn, Lehmann, R. H. Horne, Lockwood and many others. _The
+  Dickensian_, a magazine devoted to Dickensian subjects, was started in
+  1905; it is the organ of the Dickens Fellowship, and in a sense of the
+  Boz Club. _A Dickens Dictionary_ (by G. A. Pierce) appeared in 1872
+  and 1878; another (by A. J. Philip) in 1909; and a _Dickens
+  Concordance_ by Mary Williams in 1907.     (T. SE.)
+
+
+
+
+DICKINSON, ANNA ELIZABETH (1842-   ), American author and lecturer, was
+born, of Quaker parentage, at Philadelphia, Pennsylvania, on the 28th of
+October 1842. She was educated at the Friends' Free School in
+Philadelphia, and was for a time a teacher. In 1861 she obtained a
+clerkship in the United States mint, but was removed for criticizing
+General McClellan at a public meeting. She had gradually become widely
+known as an eloquent and persuasive public speaker, one of the first of
+her sex to mount the platform to discuss the burning questions of the
+hour. Before the Civil War she lectured on anti-slavery topics, during
+the war she toured the country on behalf of the Sanitary Commission, and
+also lectured on reconstruction, temperance and woman's rights. She
+wrote several plays, including _The Crown of Thorns_ (1876); _Mary
+Tudor_ (1878), in which she appeared in the title role; _Aurelian_
+(1878); and _An American Girl_ (1880), successfully acted by Fanny
+Davenport. She also published a novel, _Which Answer?_ (1868); _A Paying
+Investment, a Plea for Education_ (1876); and _A Ragged Register of
+People, Places and Opinions_ (1879).
+
+
+
+
+DICKINSON, JOHN (1732-1808), American statesman and pamphleteer, was
+born in Talbot county, Maryland, on the 8th of November 1732. He removed
+with his father to Kent county, Delaware, in 1740, studied under private
+tutors, read law, and in 1753 entered the Middle Temple, London.
+Returning to America in 1757, he began the practice of law in
+Philadelphia, was speaker of the Delaware assembly in 1760, and was a
+member of the Pennsylvania assembly in 1762-1765 and again in
+1770-1776.[1] He represented Pennsylvania in the Stamp Act Congress
+(1765) and in the Continental Congress from 1774 to 1776, when he was
+defeated owing to his opposition to the Declaration of Independence. He
+then retired to Delaware, served for a time as private and later as
+brigader-general in the state militia, and was again a member of the
+Continental Congress (from Delaware) in 1779-1780. He was president of
+the executive council, or chief executive officer, of Delaware in
+1781-1782, and of Pennsylvania in 1782-1785, and was a delegate from
+Delaware to the Annapolis convention of 1786 and the Federal
+Constitutional convention of 1787. Dickinson has aptly been called the
+"Penman of the Revolution." No other writer of the day presented
+arguments so numerous, so timely and so popular. He drafted the
+"Declaration of Rights" of the Stamp Act Congress, the "Petition to the
+King" and the "Address to the Inhabitants of Quebec" of the Congress of
+1774, and the second "Petition to the King"[2] and the "Articles of
+Confederation" of the second Congress. Most influential of all, however,
+were _The Letters of a Farmer in Pennsylvania_, written in 1767-1768 in
+condemnation of the Townshend Acts of 1767, in which he rejected
+speculative natural rights theories and appealed to the common sense of
+the people through simple legal arguments. By opposing the Declaration
+of Independence, he lost his popularity and was never able entirely to
+regain it. As the representative of a small state, he championed the
+principle of state equality in the constitutional convention, but was
+one of the first to advocate the compromise, which was finally adopted,
+providing for equal representation, in one house and proportional
+representation in the other. He was probably influenced by Delaware
+prejudice against Pennsylvania when he drafted the clause which forbids
+the creation of a new state by the junction of two or more states or
+parts of states without the consent of the states concerned as well as
+of congress. After the adjournment of the convention he defended its
+work in a series of letters signed "Fabius," which will bear comparison
+with the best of the Federalist productions. It was largely through his
+influence that Delaware and Pennsylvania were the first two states to
+ratify the Constitution. Dickinson's interests were not exclusively
+political. He helped to found Dickinson College (named in his honour) at
+Carlisle, Pennsylvania, in 1783, was the first president of its board of
+trustees, and was for many years its chief benefactor. He died on the
+14th of February 1808 and was buried in the Friends' burial ground in
+Wilmington, Del.
+
+  See C. J. Stille, _Life and Times of John Dickinson_, and P. L. Ford
+  (editor), _The Writings of John Dickinson_, in vols. xiii. and xiv.
+  respectively of the _Memoirs of the Historical Society of
+  Pennsylvania_ (Philadelphia, 1891 and 1895).
+
+
+FOOTNOTES:
+
+  [1] Being under the same proprietor and the same governor,
+    Pennsylvania and Delaware were so closely connected before the
+    Revolution that there was an interchange of public men.
+
+  [2] The "Declaration of the United Colonies of North America ...
+    setting forth the Causes and the Necessity of their Taking up Arms"
+    (often erroneously attributed to Thomas Jefferson).
+
+
+
+
+DICKSON, SIR ALEXANDER (1777-1840), British artillerist, entered the
+Royal Military Academy in 1793, passing out as second lieutenant in the
+Royal Artillery in the following year. As a subaltern he saw service in
+Minorca in 1798 and at Malta in 1800. As a captain he took part in the
+unfortunate Montevideo Expedition of 1806-07, and in 1809 he accompanied
+Howorth to the Peninsular War as brigade-major of the artillery. He soon
+obtained a command in the Portuguese artillery, and as a
+lieutenant-colonel of the Portuguese service took part in the various
+battles of 1810-11. At the two sieges of Budazoz, Ciudad Rodrigo, the
+Salamanca forts and Burgos, he was entrusted by Wellington (who had the
+highest opinion of him) with most of the detailed artillery work, and at
+Salamanca battle he commanded the reserve artillery. In the end he
+became commander of the whole of the artillery of the allied army, and
+though still only a substantive captain in the British service he had
+under his orders some 8000 men. At Vitoria, the Pyrenees battles and
+Toulouse he directed the movements of the artillery engaged, and at the
+end of the war received handsome presents from the officers who had
+served under him, many of whom were his seniors in the army list. He was
+at the disastrous affair of New Orleans, but returned to Europe in time
+for the Waterloo campaign. He was present at Quatre Bras and Waterloo on
+the artillery staff of Wellington's army, and subsequently commanded the
+British battering train at the sieges of the French fortresses left
+behind the advancing allies. For the rest of his life he was on home
+service, principally as a staff officer of artillery. He died, a
+major-general and G.C.B., in 1840. A memorial was erected at Woolwich in
+1847. Dickson was one of the earliest fellows of the Royal Geographical
+Society.
+
+  His diaries kept in the Peninsula were the main source of information
+  used in Duncan's _History of the Royal Artillery_.
+
+
+
+
+DICKSON, SIR JAMES ROBERT (1832-1901), Australian statesman, was born in
+Plymouth on the 30th of November 1832. He was brought up in Glasgow,
+receiving his education at the high school, and became a clerk in the
+City of Glasgow Bank. In 1854 he emigrated to Victoria, but after some
+years spent in that colony and in New South Wales, he settled in 1862 in
+Queensland, where he was connected with many important business
+enterprises, among them the Royal Bank of Queensland. He entered the
+Queensland House of Assembly in 1872, and became minister of works
+(1876), treasurer (1876-1879, and 1883-1887), acting premier (1884), but
+resigned in 1887 on the question of taxing land. In 1889 he retired from
+business, and spent three years in Europe before resuming political
+life. He fought for the introduction of Polynesian labour on the
+Queensland sugar plantations at the general election of 1892, and was
+elected to the House of Assembly in that year and again at the elections
+of 1893 and 1896. He became secretary for railways in 1897, minister for
+home affairs in 1898, represented Queensland in the federal council of
+Australia in 1896 and at the postal conference at Hobart in 1898, and in
+1898 became premier. His energies were now devoted to the formation of
+an Australian commonwealth. He secured the reference of the question to
+a plebiscite, the result of which justified his anticipations. He
+resigned the premiership in November 1899, but in the ministry of Robert
+Philp, formed in the next month, he was reappointed to the offices of
+chief secretary and vice-president of the executive council which he had
+combined with the office of premier. He represented Queensland in 1900
+at the conference held in London to consider the question of Australian
+unity, and on his return was appointed minister of defence in the first
+government of the Australian Commonwealth. He did not long survive the
+accomplishment of his political aims, dying at Sydney on the 10th of
+January 1901, in the midst of the festivities attending the inauguration
+of the new state.
+
+
+
+
+DICOTYLEDONS, in botany, the larger of the two great classes of
+angiosperms, embracing most of the common flower-bearing plants. The
+name expresses the most universal character of the class, the importance
+of which was first noticed by John Ray, namely, the presence of a pair
+of seed-leaves or cotyledons, in the plantlet or embryo contained in the
+seed. The embryo is generally surrounded by a larger or smaller amount
+of foodstuff (endosperm) which serves to nourish it in its development
+to form a seedling when the seed germinates; frequently, however, as in
+pea or bean and their allies, the whole of the nourishment for future
+use is stored up in the cotyledons themselves, which then become thick
+and fleshy. In germination of the seed the root of the embryo (radicle)
+grows out to get a holdfast for the plant; this is generally followed by
+the growth of the short stem immediately above the root, the so-called
+"hypocotyl," which carries up the cotyledons above the ground, where
+they spread to the light and become the first green leaves of the plant.
+Protected between the cotyledons and terminating the axis of the plant
+is the first stem-bud (the plumule of the embryo), by the further growth
+and development of which the aerial portion of the plant, consisting of
+stem, leaves and branches, is formed, while the development of the
+radicle forms the root-system. The size and manner of growth of the
+adult plant show a great variety, from the small herb lasting for one
+season only, to the forest tree living for centuries. The arrangement of
+the conducting tissue in the stem is characteristic; a transverse
+section of the very young stem shows a number of distinct conducting
+strands--vascular bundles--arranged in a ring round the pith; these soon
+become united to form a closed ring of bast and wood, separated by a
+layer of formative tissue (cambium). In perennials the stem shows a
+regular increase in thickness each year by the addition of a new ring of
+wood outside the old one--for details of structure see PLANTS: Anatomy.
+A similar growth occurs in the root. This increase in the diameter of
+stem and root is correlated with the increase in leaf-area each season,
+due to the continued production of new leaf-bearing branches. A
+characteristic of the class is afforded by the complicated network
+formed by the leaf-veins,--well seen in a skeleton leaf, from which the
+soft parts have been removed by maceration. The parts of the flower are
+most frequently arranged in fives, or multiples of fives; for instance,
+a common arrangement is as follows,--five sepals, succeeded by five
+petals, ten stamens in two sets of five, and five or fewer carpels; an
+arrangement in fours is less frequent, while the arrangement in threes,
+so common in monocotyledons, is rare in dicotyledons. In some orders the
+parts are numerous, chiefly in the case of the stamens and the carpels,
+as in the buttercup and other members of the order Ranunculaceae. There
+is a very wide range in the general structure and arrangement of the
+parts of the flower, associated with the means for ensuring the
+transference of pollen; in the simplest cases the flower consists only
+of a few stamens or carpels, with no enveloping sepals or petals, as in
+the willow, while in the more elaborate type each series is represented,
+the whole forming a complicated structure closely correlated with the
+size, form and habits of the pollinating agent (see FLOWER). The
+characters of the fruit and seed and the means for ensuring the
+dispersal of the seeds are also very varied (see FRUIT).
+
+
+
+
+DICTATOR (from the Lat. _dictare_, frequentative of _dicere_, to speak).
+In modern usage this term is loosely used for a personal ruler enjoying
+extraordinary and extra-constitutional power. The etymological sense of
+one who "dictates"--i.e. one whose word (_dictum_) is law (from which
+that of one who "dictates," i.e. speaks for some writer to record, is to
+be distinguished)--has been assisted by the historical use of the term,
+in ancient times, for an extraordinary magistrate in the Roman
+commonwealth. It is unknown precisely how the Roman word came into use,
+though an explanation of the earlier official title, magister populi,
+throws some light on the subject. That designation may mean "head of the
+(infantry) host" as opposed to his subordinate, the magister equitum,
+who was "head of the cavalry." If this explanation be accepted, emphasis
+was thus laid in early times on the military aspect of the dictatorship,
+and in fact the office seems to have been instituted for the purpose of
+meeting a military crisis such as might have proved too serious for the
+annual consuls with their divided command. Later constitutional theory
+held that the repression of civil discord was also one of the motives
+for the institution of a dictatorship. Such is the view expressed by
+Cicero in the _De legibus_ (iii. 3, 9) and by the emperor Claudius in
+his extant _Oratio_ (i. 28). This function of the office, although it
+may not have been contemplated at first, is attested by the internal
+history of Rome. In the crisis of the agitation that gathered round the
+Licinian laws (367 B.C.) a dictator was appointed, and in 314 B.C. we
+have the notice of a dictator created for purposes of criminal
+jurisdiction (_quaestionibus exercendis_). The dictator appointed to
+meet the dangers of war, sedition or crime was technically described as
+"the administrative dictator" (_rei gerundae causa_). Minor, or merely
+formal, needs of the state might lead to the creation of other types of
+this office. Thus we find dictators destined to hold the elections, to
+make out the list of the senate, to celebrate games, to establish
+festivals, and to drive the nail into the temple of Jupiter--an act of
+natural magic which was believed to avert pestilence. These dictators
+appointed for minor purposes were expected to retire from office as soon
+as their function was completed. The "administrative dictator" held
+office for at least six months.
+
+The powers of a dictator were a temporary revival of those of the kings;
+but there were some limitations to his authority. He was never concerned
+with civil jurisdiction, and was dependent on the senate for supplies of
+money. His military authority was confined to Italy; and his power of
+life and death over the citizens was at an early period limited by law.
+It was probably the _lex Valeria_ of 300 B.C. that made him subject to
+the right of criminal appeal (_provocatio_) within the limits of the
+city. But during his tenure of power all the magistrates of the people
+were regarded as his subordinates; and it was even held that the right
+of assistance (_auxilium_), furnished by the tribunes of the plebs to
+members of the citizen body, should not be effectively exercised when
+the state was under this type of martial law. The dictator was nominated
+by one of the consuls. But here as elsewhere the senate asserted its
+authority over the magistrates, and the view was finally held that the
+senate should not only suggest the need of nomination but also the name
+of the nominee. After the nomination, the imperium of the dictator was
+confirmed by a _lex curiata_ (see COMITIA). To emphasize the superiority
+of this imperium over that of the consuls, the dictator might be
+preceded by twenty-four lictors, not by the usual twelve; and, at least
+in the earlier period of the office, these lictors bore the axes, the
+symbols of life and death, within the city walls.
+
+Tradition represents the dictatorship as having a life of three
+centuries in the history of the Roman state. The first dictator is said
+to have been created in 501 B.C.; the last of the "administrative"
+dictators belongs to the year 216 B.C. It was an office that was
+incompatible both with the growing spirit of constitutionalism and with
+the greater security of the city; and the epoch of the Second Punic War
+was marked by experiments with the office, such as the election of Q.
+Fabius Maximus by the people, and the co-dictatorship of M. Minucius
+with Fabius, which heralded its disuse (see PUNIC WARS). The emergency
+office of the early and middle Republic has few points of contact,
+except those of the extraordinary position and almost unfettered
+authority of its holder, with the dictatorship as revised by Sulla and
+by Caesar. Sulla's dictatorship was the form taken by a provisional
+government. He was created "for the establishment of the Republic." It
+is less certain whether the dictatorships held by Caesar were of a
+consciously provisional character. Since the office represented the only
+supreme _Imperium_ in Rome, it was the natural resort of the founder of
+a monarchy (see SULLA and CAESAR). Ostensibly to prevent its further use
+for such a purpose, M. Antonius in 44 B.C. carried a law abolishing the
+dictatorship as a part of the constitution.
+
+  BIBLIOGRAPHY.--Mommsen, _Romisches Staatsrecht_, ii. 141 foll. (3rd
+  ed., Leipzig, 1887); Herzog, _Geschichte und System der romischen
+  Staatsverfassung_, i. 718 foll. (Leipzig, 1884); Pauly-Wissowa,
+  _Realencyclopadie_, v. 370 foll. (new edition, Stuttgart. 1893, &c.);
+  Lange, _Romische Alterthumer_, i. 542 foll. (Berlin, 1856, &c.);
+  Daremberg-Saglio, _Dictionnaire des antiquites grecques et romaines_,
+  ii. 161 foll. (1875, &c.); Haverfield, "The Abolition of the
+  Dictatorship," in _Classical Review_, iii. 77.     (A. H. J. G.)
+
+
+
+
+DICTIONARY.
+
+  Definition and history.
+
+In its proper and most usual meaning a dictionary is a book containing a
+collection of the words of a language, dialect or subject, arranged
+alphabetically or in some other definite order, and with explanations in
+the same or some other language. When the words are few in number, being
+only a small part of those belonging to the subject, or when they are
+given without explanation, or some only are explained, or the
+explanations are partial, the work is called a _vocabulary_; and when
+there is merely a list of explanations of the technical words and
+expressions in some particular subject, a _glossary_. An alphabetical
+arrangement of the words of some book or author with references to the
+places where they occur is called an index (q.v.). When under each word
+the phrases containing it are added to the references, the work is
+called a _concordance_. Sometimes, however, these names are given to
+true dictionaries; thus the great Italian dictionary of the _Accademia
+della Crusca_, in six volumes folio, is called _Vocabolario_, and
+Ernesti's dictionary to Cicero is called _Index_. When the words are
+arranged according to a definite system of classification under heads
+and subdivisions, according to their nature or their meaning, the book
+is usually called a classed vocabulary; but when sufficient explanations
+are given it is often accepted as a dictionary, like the _Onomasticon_
+of Julius Pollux, or the native dictionaries of Sanskrit, Manchu and
+many other languages.
+
+Dictionaries were originally books of reference explaining the words of
+a language or of some part of it. As the names of things, as well as
+those of persons and places, are words, and often require explanation
+even more than other classes of words, they were necessarily included in
+dictionaries, and often to a very great extent. In time, books were
+devoted to them alone, and were limited to special subjects, and these
+have so multiplied, that dictionaries of things now rival in number and
+variety those of words or of languages, while they often far surpass
+them in bulk. There are dictionaries of biography and history, real and
+fictitious, general and special, relating to men of all countries,
+characters and professions; the English _Dictionary of National
+Biography_ (see BIOGRAPHY) is a great instance of one form of these;
+dictionaries of bibliography, relating to all books, or to those of some
+particular kind or country; dictionaries of geography (sometimes called
+_gazetteers_) of the whole world, of particular countries, or of small
+districts, of towns and of villages, of castles, monasteries and other
+buildings. There are dictionaries of philosophy; of the Bible; of
+mathematics; of natural history, zoology, botany; of birds, trees,
+plants and flowers; of chemistry, geology and mineralogy; of
+architecture, painting and music; of medicine, surgery, anatomy,
+pathology and physiology; of diplomacy; of law, canon, civil, statutory
+and criminal; of political and social sciences; of agriculture, rural
+economy and gardening; of commerce, navigation, horsemanship and the
+military arts; of mechanics, machines and the manual arts. There are
+dictionaries of antiquities, of chronology, of dates, of genealogy, of
+heraldry, of diplomatics, of abbreviations, of useful receipts, of
+monograms, of adulterations and of very many other subjects. These works
+are separately referred to in the bibliographies attached to the
+articles on the separate subjects. And lastly, there are dictionaries of
+the arts and sciences, and their comprehensive offspring, encyclopaedias
+(q.v.), which include in themselves every branch of knowledge. Neither
+under the heading of _dictionary_ nor under that of _encyclopaedia_ do
+we propose to include a mention of every work of its class, but many of
+these will be referred to in the separate articles on the subjects to
+which they pertain. And in this article we confine ourselves to an
+account of those dictionaries which are primarily word-books. This is
+practically the most convenient distinction from the subject-book or
+encyclopaedia; though the two characters are often combined in one work.
+Thus the _Century Dictionary_ has encyclopaedic features, while the
+present edition of the _Encyclopaedia Britannica_, restoring its
+earlier tradition but carrying out the idea more systematically, also
+embodies dictionary features.
+
+_Dictionarium_ is a word of low or modern Latinity;[1] _dictio_, from
+which it was formed, was used in medieval Latin to mean a word.
+_Lexicon_ is a corresponding word of Greek origin, meaning a book of or
+for words--a dictionary. A _glossary_ is properly a collection of
+unusual or foreign words requiring explanation. It is the name
+frequently given to English dictionaries of dialects, which the Germans
+usually call _idioticon_, and the Italians _vocabolario_. _Worterbuch_,
+a book of words, was first used among the Germans, according to Grimm,
+by Kramer (1719), imitated from the Dutch _woordenboek_. From the
+Germans the Swedes and Danes adopted _ordbok_, _ordbog_. The Icelandic
+_ordabok_, like the German, contains the genitive plural. The Slavonic
+nations use _slovar_, _slovnik_, and the southern Slavs _ryetshnik_,
+from _slovo_, _ryetsh_, a word, formed, like dictionary and lexicon,
+without composition. Many other names have been given to dictionaries,
+as _thesaurus_, _Sprachschatz_, _cornucopia_, _gazophylacium_,
+_comprehensorium_, _catholicon_, to indicate their completeness;
+_manipulus predicantium_, _promptorium puerorum_, _liber memorialis_,
+_hortus vocabulorum_, _ionia_ (a violet bed), _alveary_ (a beehive),
+_kamoos_ (the sea), _haft kulzum_ (the seven seas), _tsze tien_ (a
+standard of character), _onomasticon_, _nomenclator_, _bibliotheca_,
+_elucidario_, _Mundart-sammlung_, _clavis_, _scala_, _pharetra_,[2] _La
+Crusca_ from the great Italian dictionary, and _Calepino_ (in Spanish
+and Italian) from the Latin dictionary of Calepinus.
+
+The tendency of great dictionaries is to unite in themselves all the
+peculiar features of special dictionaries. A large dictionary is most
+useful when a word is to be thoroughly studied, or when there is
+difficulty in making out the meaning of a word or phrase. Special
+dictionaries are more useful for special purposes; for instance,
+synonyms are best studied in a dictionary of synonyms. And small
+dictionaries are more convenient for frequent use, as in translating
+from an unfamiliar language, for words may be found more quickly, and
+they present the words and their meanings in a concentrated and compact
+form, instead of being scattered over a large space, and separated by
+other matter. Dictionaries of several languages, called _polyglots_, are
+of different kinds. Some are polyglot in the vocabulary, but not in the
+explanation, like Johnson's dictionary of Persian and Arabic explained
+in English; some in the interpretation, but not in the vocabulary or
+explanation, like _Calepini octoglotton_, a Latin dictionary of Latin,
+with the meanings in seven languages. Many great dictionaries are now
+polyglot in this sense. Some are polyglot in the vocabulary and
+interpretation, but are explained in one language, like Jal's _Glossaire
+nautique_, a glossary of sea terms in many languages, giving the
+equivalents of each word in the other languages, but the explanation in
+French. Pauthier's _Annamese Dictionary_ is polyglot in a peculiar way.
+It gives the Chinese characters with their pronunciation in Chinese and
+Annamese. Special dictionaries are of many kinds. There are technical
+dictionaries of etymology, foreign words, dialects, secret languages,
+slang, neology, barbarous words, faults of expression, choice words,
+prosody, pronunciation, spelling, orators, poets, law, music, proper
+names, particular authors, nouns, verbs, participles, particles, double
+forms, difficulties and many others. Fick's dictionary (Gottingen, 1868,
+8vo; 1874-1876, 8vo, 4 vols.) is a remarkable attempt to ascertain the
+common language of the Indo-European nations before each of their great
+separations. In the second edition of his _Etymologische Forschungen_
+(Lemgo and Detmoldt, 1859-1873, 8vo, 7217 pages) Pott gives a
+comparative lexicon of Indo-European roots, 2226 in number, occupying
+5140 pages.
+
+
+  Methods.
+
+At no time was progress in the making of general dictionaries so rapid
+as during the second half of the 19th century. It is to be seen in three
+things: in the perfecting of the theory of what a general dictionary
+should be; in the elaboration of methods of collecting and editing
+lexicographic materials; and in the magnitude and improved quality of
+the work which has been accomplished or planned. Each of these can best
+be illustrated from English lexicography, in which the process of
+development has in all directions been carried farthest. The advance
+that has been made in theory began with a radical change of opinion with
+regard to the chief end of the general dictionary of a language. The
+older view of the matter was that the lexicographer should furnish a
+standard of usage--should register only those words which are, or at
+some period of the language have been, "good" from a literary point of
+view, with their "proper" senses and uses, or should at least furnish
+the means of determining what these are. In other words, his chief duty
+was conceived to be to sift and refine, to decide authoritatively
+questions with regard to good usage, and thus to fix the language as
+completely as might be possible within the limits determined by the
+literary taste of his time. Thus the Accademia della Crusca, founded
+near the close of the 16th century, was established for the purpose of
+purifying in this way the Italian tongue, and in 1612 the _Vocabolario
+degli Accademici della Crusca_, long the standard of that language, was
+published. The Academie Francaise, the first edition of whose dictionary
+appeared in 1694, had a similar origin. In England the idea of
+constructing a dictionary upon this principle arose during the second
+quarter of the 18th century. It was imagined by men of letters--among
+them Alexander Pope--that the English language had then attained such
+perfection that further improvement was hardly possible, and it was
+feared that if it were not fixed by lexicographic authority
+deterioration would soon begin. Since there was no English "Academy," it
+was necessary that the task should fall to some one whose judgment would
+command respect, and the man who undertook it was Samuel Johnson. His
+dictionary, the first edition of which, in two folio volumes, appeared
+in 1755, was in many respects admirable, but it was inadequate even as a
+standard of the then existing literary usage. Johnson himself did not
+long entertain the belief that the natural development of a language can
+be arrested in that or in any other way. His work was, however,
+generally accepted as a final authority, and the ideas upon which it was
+founded dominated English lexicography for more than a century. The
+first effective protest in England against the supremacy of this
+literary view was made by Dean (later Archbishop) Trench, in a paper on
+"Some Deficiencies in Existing English Dictionaries" read before the
+Philological Society in 1857. "A dictionary," he said, "according to
+that idea of it which seems to me alone capable of being logically
+maintained, is an _inventory of the language_; much more, but this
+primarily.... It is no task of the maker of it to select the _good_
+words of the language.... The business which he has undertaken is to
+collect and arrange _all_ words, whether good or bad, whether they
+commend themselves to his judgment or otherwise.... _He is an historian
+of_ [the language], _not a critic._" That is, for the literary view of
+the chief end of the general dictionary should be substituted the
+philological or scientific. In Germany this substitution had already
+been effected by Jacob and Wilhelm Grimm in their dictionary of the
+German language, the first volume of which appeared in 1854. In brief,
+then, the modern view is that the general dictionary of a language
+should be a record of all the words--current or obsolete--of that
+language, with all their meanings and uses, but should not attempt to
+be, except secondarily or indirectly, a guide to "good" usage. A
+"standard" dictionary has, in fact, been recognized to be an
+impossibility, if not an absurdity.
+
+This theoretical requirement must, of course, be modified considerably
+in practice. The date at which a modern language is to be regarded by
+the lexicographer as "beginning" must, as a rule, be somewhat
+arbitrarily chosen; while considerable portions of its earlier
+vocabulary cannot be recovered because of the incompleteness of the
+literary record. Moreover, not even the most complete dictionary can
+include all the words which the records--earlier and later--actually
+contain. Many words, that is to say, which are found in the literature
+of a language cannot be regarded as, for lexicographic purposes,
+belonging to that language; while many more may or may not be held to
+belong to it, according to the judgment--almost the whim--of the
+individual lexicographer. This is especially true of the English tongue.
+"That vast aggregate of words and phrases which constitutes the
+vocabulary of English-speaking men presents, to the mind that endeavours
+to grasp it as a definite whole, the aspect of one of those nebulous
+masses familiar to the astronomer, in which a clear and unmistakable
+nucleus shades off on all sides, through zones of decreasing brightness,
+to a dim marginal film that seems to end nowhere, but to lose itself
+imperceptibly in the surrounding darkness" (Dr J. A. H. Murray, _Oxford
+Dict._ General Explanations, p. xvii). This "marginal film" of words
+with more or less doubtful claims to recognition includes thousands of
+the terms of the natural sciences (the New-Latin classificatory names of
+zoology and botany, names of chemical compounds and of minerals, and the
+like); half-naturalized foreign words; dialectal words; slang terms;
+trade names (many of which have passed or are passing into common use);
+proper names and many more. Many of these even the most complete
+dictionary should exclude; others it should include; but where the line
+shall be drawn will always remain a vexed question.
+
+Another important principle upon which Trench insisted, and which also
+expresses a requirement of modern scientific philology, is that the
+dictionary shall be not merely a record, but also an _historical_ record
+of words and their uses. From the literary point of view the most
+important thing is present usage. To that alone the idea of a "standard"
+has any application. Dictionaries of the older type, therefore, usually
+make the common, or "proper" or "root" meaning of a word the starting
+point of its definition, and arrange its other senses in a logical or
+accidental order commonly ignoring the historical order in which the
+various meanings arose. Still less do they attempt to give data from
+which the vocabulary of the language at any previous period may be
+determined. The philologist, however, for whom the growth, or
+progressive alteration, of a language is a fact of central importance,
+regards no record of a language as complete which does not exhibit this
+growth in its successive stages. He desires to know when and where each
+word, and each form and sense of it, are first found in the language; if
+the word or sense is obsolete, when it died; and any other fact that
+throws light upon its history. He requires, accordingly, of the
+lexicographer that, having ascertained these data, he shall make them
+the foundation of his exposition--in particular, of the division and
+arrangement of his definitions, that sense being placed first which
+appeared first in order of time. In other words, each article in the
+dictionary should furnish an orderly biography of the word of which it
+treats, each word and sense being so dated that the exact time of its
+appearance and the duration of its use may as nearly as possible be
+determined. This, in principle, is the method of the new lexicography.
+In practice it is subject to limitations similar to those of the
+vocabulary mentioned above. Incompleteness of the early record is here
+an even greater obstacle; and there are many words whose history is, for
+one reason or another, so unimportant that to treat it elaborately would
+be a waste of labour and space.
+
+The adoption of the historical principle involves a further noteworthy
+modification of older methods, namely, an important extension of the use
+of quotations. To Dr Johnson belongs the credit of showing how useful,
+when properly chosen, they may be, not only in corroborating the
+lexicographer's statements, but also in revealing special shades of
+meaning or variations of use which his definitions cannot well express.
+No part of Johnson's work is more valuable than this. This idea was more
+fully developed and applied by Dr Charles Richardson, whose _New
+Dictionary of the English Language ... Illustrated by Quotations from
+the Best Authors_ (1835-1836) still remains a most valuable collection
+of literary illustrations. Lexicographers, however, have, with few
+exceptions, until a recent date, employed quotations chiefly for the
+ends just mentioned--as instances of use or as illustrations of correct
+usage--with scarcely any recognition of their value as historical
+evidence; and they have taken them almost exclusively from the works of
+the "best" authors. But since all the data upon which conclusions with
+regard to the history of a word can be based must be collected from the
+literature of the language, it is evident that, in so far as the
+lexicographer is required to furnish evidence for an historical
+inference, a quotation is the best form in which he can give it. In
+fact, extracts, properly selected and grouped, are generally sufficient
+to show the entire meaning and biography of a word without the aid of
+elaborate definitions. The latter simply save the reader the trouble of
+drawing the proper conclusions for himself. A further rule of the new
+lexicography, accordingly, is that quotations should be used, primarily,
+as historical evidence, and that the history of words and meanings
+should be exhibited by means of them. The earliest instance of use that
+can be found, and (if the word or sense is obsolete) the latest, are as
+a rule to be given; while in the case of an important word or sense,
+instances taken from successive periods of its currency also should be
+cited. Moreover, a quotation which contains an important bit of
+historical evidence must be used, whether its source is "good," from the
+literary point of view, or not--whether it is a classic of the language
+or from a daily newspaper; though where choice is possible, preference
+should, of course, be given to quotations extracted from the works of
+the best writers. This rule does not do away with the illustrative use
+of quotations, which is still recognized as highly important, but it
+subordinates it to their historical use. It is necessary to add that it
+implies that the extracts must be given exactly, and in the original
+spelling and capitalization, accurately dated, and furnished with a
+precise reference to author, book, volume, page and edition; for
+insistence upon these requirements--which are obviously important,
+whatever the use of the quotation may be--is one of the most noteworthy
+of modern innovations. Johnson usually gave simply the author's name,
+and often quoted from memory and inaccurately; and many of his
+successors to this day have followed--altogether or to some extent--his
+example.
+
+The chief difficulty in the way of this use of quotations--after the
+difficulty of collection--is that of finding space for them in a
+dictionary of reasonable size. Preference must be given to those which
+are essential, the number of those which are cited merely on methodical
+grounds being made as small as possible. It is hardly necessary to add
+that the negative evidence furnished by quotations is generally of
+little value; one can seldom, that is, be certain that the lexicographer
+has actually found the earliest or the latest use, or that the word or
+sense has not been current during some intermediate period from which he
+has no quotations.
+
+Lastly, a much more important place in the scheme of the ideal
+dictionary is now assigned to the _etymology_ of words. This may be
+attributed, in part, to the recent rapid development of etymology as a
+science, and to the greater abundance of trustworthy data; but it is
+chiefly due to the fact that from the historical point of view the
+connexion between that section of the biography of a word which lies
+within the language--subsequent, that is, to the time when the language
+may, for lexicographical purposes, be assumed to have begun, or to the
+time when the word was adopted or invented--and its antecedent history
+has become more vital and interesting. Etymology, in other words, is
+essentially the history of the _form_ of a word up to the time when it
+became a part of the language, and is, in a measure, an extension of the
+history of the development of the word in the language. Moreover, it is
+the only means by which the exact relations of allied words can be
+ascertained, and the separation of words of the same form but of diverse
+origin (homonyms) can be effected, and is thus, for the dictionary, the
+foundation of all _family history_ and correct _genealogy_. In fact, the
+attention that has been paid to these two points in the best recent
+lexicography is one of its distinguishing and most important
+characteristics. Related to the etymology of words are the changes in
+their form which may have occurred while they have been in use as parts
+of the language--modifications of their pronunciation, corruptions by
+popular etymology or false associations, and the like. The facts with
+regard to these things which the wide research necessitated by the
+historical method furnishes abundantly to the modern lexicographer are
+often among the most novel and interesting of his acquisitions.
+
+It should be added that even approximate conformity to the theoretical
+requirements of modern lexicography as above outlined is possible only
+under conditions similar to those under which the Oxford _New English
+Dictionary_ was undertaken (see below). The labour demanded is too vast,
+and the necessary bulk of the dictionary too great. When, however, a
+language is recorded in one such dictionary, those of smaller size and
+more modest pretensions can rest upon it as an authority and conform to
+it as a model so far as their special limitations permit.
+
+The ideal thus developed is primarily that of the general dictionary of
+the purely philological type, but it applies also to the encyclopaedic
+dictionary. In so far as the latter is strictly lexicographic--deals
+with words as words, and not with the things they denote--it should be
+made after the model of the former, and is defective to the extent in
+which it deviates from it. The addition of encyclopaedic matter to the
+philological in no way affects the general principles involved. It may,
+however, for practical reasons, modify their application in various
+ways. For example, the number of obsolete and dialectal words included
+may be much diminished and the number of scientific terms (for instance,
+new Latin botanical and zoological names) be increased; and the relative
+amount of space devoted to etymologies and quotations may be lessened.
+In general, since books of this kind are designed to serve more or less
+as works of general reference, the making of them must be governed by
+considerations of practical utility which the compilers of a purely
+philological dictionary are not obliged to regard. The encyclopaedic
+type itself, although it has often been criticized as hybrid--as a
+mixture of two things which should be kept distinct--is entirely
+defensible. Between the dictionary and the encyclopaedia the dividing
+line cannot sharply be drawn. There are words the meaning of which
+cannot be explained fully without some description of things, and, on
+the other hand, the description of things and processes often involves
+the definition of names. To the combination of the two objection cannot
+justly be made, so long as it is effected in a way--with a selection of
+material--that leaves the dictionary essentially a dictionary and not an
+encyclopaedia. Moreover, the large vocabulary of the general dictionary
+makes it possible to present certain kinds of encyclopaedic matter with
+a degree of fulness and a convenience of arrangement which are possible
+in no single work of any other class. In fact, it may be said that if
+the encyclopaedic dictionary did not exist it would have to be invented;
+that its justification is its indispensableness. Not the least of its
+advantages is that it makes legitimate the use of diagrams and pictorial
+illustrations, which, if properly selected and executed, are often
+valuable aids to definition.
+
+On its practical side the advance in lexicography has consisted in the
+elaboration of methods long in use rather than in the invention of new
+ones. The only way to collect the data upon which the vocabulary, the
+definitions and the history are to be based is, of course, to search for
+them in the written monuments of the language, as all lexicographers who
+have not merely borrowed from their predecessors have done. But the
+wider scope and special aims of the new lexicography demand that the
+investigation shall be vastly more comprehensive, systematic and
+precise. It is necessary, in brief, that, as far as may be possible, the
+literature (of all kinds) of every period of the language shall be
+examined systematically, in order that all the words, and senses and
+forms of words, which have existed during any period may be found, and
+that enough excerpts (carefully verified, credited and dated) to cover
+all the essential facts shall be made. The books, pamphlets, journals,
+newspapers, and so on which must thus be searched will be numbered by
+thousands, and the quotations selected may (as in the case of the Oxford
+_New English Dictionary_) be counted by millions. This task is beyond
+the powers of any one man, even though he be a Johnson, or a Littre or a
+Grimm, and it is now assigned to a corps of readers whose number is
+limited only by the ability of the editor to obtain such assistance. The
+modern method of editing the material thus accumulated--the actual work
+of compilation--also is characterized by the application of the
+principle of the division of labour. Johnson boasted that his dictionary
+was written with but little assistance from the learned, and the same
+was in large measure true of that of Littre. Such attempts on the part
+of one man to write practically the whole of a general dictionary are no
+longer possible, not merely because of the vast labour and philological
+research necessitated by modern aims, but more especially because the
+immense development of the vocabulary of the special sciences renders
+indispensable the assistance, in the work of definition, of persons who
+are expert in those sciences. The tendency, accordingly, has been to
+enlarge greatly the editorial staff of the dictionary, scores of
+sub-editors and contributors being now employed where a dozen or fewer
+were formerly deemed sufficient. In other words, the making of a
+"complete" dictionary has become a co-operative enterprise, to the
+success of which workers in all the fields of literature and science
+contribute.
+
+The most complete exemplification of these principles and methods is the
+_Oxford New English Dictionary, on historical principles, founded mainly
+on materials collected by the Philological Society_. This monumental
+work originated in the suggestion of Trench that an attempt should be
+made, under the direction of the Philological Society, to complete the
+vocabulary of existing dictionaries and to supply the historical
+information which they lacked. The suggestion was adopted, considerable
+material was collected, and Mr Herbert Coleridge was appointed general
+editor. He died in 1861, and was succeeded by Dr F. J. Furnivall.
+Little, however, was done, beyond the collection of quotations--about
+2,000,000 of which were gathered--until in 1878 the expense of printing
+and publishing the proposed dictionary was assumed by the Delegates of
+the University Press, and the editorship was entrusted to Dr (afterwards
+Sir) J. A. H. Murray. As the historical point of beginning, the middle
+of the 12th century was selected, all words that were obsolete at that
+date being excluded, though the history of words that were current both
+before and after that date is given in its entirety; and it was decided
+that the search for quotations--which, according to the original design,
+was to cover the entire literature down to the beginning of the 16th
+century and as much of the subsequent literature (especially the works
+of the more important writers and works on special subjects) as might be
+possible--should be made more thorough. More than 800 readers, in all
+parts of the world, offered their aid; and when the preface to the first
+volume appeared in 1888, the editor was able to announce that the
+readers had increased to 1300, and that 3,500,000 of quotations, taken
+from the writings of more than 5000 authors, had already been amassed.
+The whole work was planned to be completed in ten large volumes, each
+issued first in smaller parts. The first part was issued in 1884, and by
+the beginning of 1910 the first part of the letter S had been reached.
+
+The historical method of exposition, particularly by quotations, is
+applied in the _New English Dictionary_, if not in all cases with entire
+success, yet, on the whole, with a regularity and a precision which
+leave little to be desired. A minor fault is that excerpts from second
+or third rate authors have occasionally been used where better ones from
+writers of the first class either must have been at hand or could have
+been found. As was said above, the literary quality of the question is
+highly important even in historical lexicography, and should not be
+neglected unnecessarily. Other special features of the book are the
+completeness with which variations of pronunciation and orthography
+(with dates) are given; the fulness and scientific excellence of the
+etymologies, which abound in new information and corrections of old
+errors; the phonetic precision with which the present (British)
+pronunciation is indicated; and the elaborate subdivision of meanings.
+The definitions as a whole are marked by a high degree of accuracy,
+though in a certain number of cases (not explicable by the date of the
+volumes) the lists of meanings are not so good as one would expect, as
+compared (say) with the _Century Dictionary_. Work of such magnitude
+and quality is possible, practically, only when the editor of the
+dictionary can command not merely the aid of a very large number of
+scholars and men of science, but their gratuitous aid. In this the _New
+English Dictionary_ has been singularly fortunate. The conditions under
+which it originated, and its aim, have interested scholars everywhere,
+and led them to contribute to the perfecting of it their knowledge and
+time. The long list of names of such helpers in Sir J. A. H. Murray's
+preface is in curious contrast with their absence from Dr Johnson's and
+the few which are given in that of Littre. The editor's principal
+assistants were Dr Henry Bradley and Dr W. A. Craigie. Of the dictionary
+as a whole it may be said that it is one of the greatest achievements,
+whether in literature or science, of modern English scholarship and
+research.
+
+  The _New English Dictionary_ furnishes for the first time data from
+  which the extent of the English word-store at any given period, and
+  the direction and rapidity of its growth, can fairly be estimated. For
+  this purpose the materials furnished by the older dictionaries are
+  quite insufficient, on account of their incompleteness and
+  unhistorical character. For example 100 pages of the _New English
+  Dictionary_ (from the letter H) contain 1002 words, of which, as the
+  dated quotations show, 585 were current in 1750 (though some, of
+  course, were very rare, some dialectal, and so on), 191 were obsolete
+  at that date, and 226 have since come into use. But of the more than
+  700 words--current or obsolete--which Johnson might thus have
+  recorded, he actually did record only about 300. Later dictionaries
+  give more of them, but they in no way show their status at the date in
+  question. It is worth noting that the figures given seem to indicate
+  that not very many more words have been added to the vocabulary of the
+  language during the past 150 years than had been lost by 1750. The
+  pages selected, however, contain comparatively few recent scientific
+  terms. A broader comparison would probably show that the gain has been
+  more than twice as great as the loss.
+
+In the _Deutsches Worterbuch_ of Jacob and Wilhelm Grimm the scientific
+spirit, as was said above, first found expression in general
+lexicography. The desirability of a complete inventory and investigation
+of German words was recognized by Leibnitz and by various 18th-century
+scholars, but the plan and methods of the Grimms were the direct product
+of the then new scientific philology. Their design, in brief, was to
+give an exhaustive account of the words of the literary language (New
+High German) from about the end of the 15th century, including their
+earlier etymological and later history, with references to important
+dialectal words and forms; and to illustrate their use and history
+abundantly by quotations. The first volume appeared in 1854. Jacob Grimm
+(died 1863) edited the first, second (with his brother, who died in
+1859), third and a part of the fourth volumes; the others have been
+edited by various distinguished scholars. The scope and methods of this
+dictionary have been broadened somewhat as the work has advanced. In
+general it may be said that it differs from the _New English Dictionary_
+chiefly in its omission of pronunciations and other pedagogic matter;
+its irregular treatment of dates; its much less systematic and less
+lucid statement of etymologies; its less systematic and less fruitful
+use of quotations; and its less convenient and less intelligible
+arrangement of material and typography.
+
+These general principles lie also at the foundation of the scholarly
+_Dictionnaire de la langue francaise_ of E. Littre, though they are
+there carried out less systematically and less completely. In the
+arrangement of the definitions the first place is given to the most
+primitive meaning of the word instead of to the most common one, as in
+the dictionary of the Academy; but the other meanings follow in an order
+that is often logical rather than historical. Quotations also are
+frequently used merely as literary illustrations, or are entirely
+omitted; in the special paragraphs on the history of words before the
+16th century, however, they are put to a strictly historical use. This
+dictionary--perhaps the greatest ever compiled by one man--was published
+1863-1872. (Supplement, 1878.)
+
+The _Thesaurus Linguae Latinae_, prepared under the auspices of the
+German Academies of Berlin, Gottingen, Leipzig, Munich and Vienna, is a
+notable application of the principles and practical co-operative method
+of modern lexicography to the classical tongues. The plan of the work is
+to collect quotations which shall register, with its full context, every
+word (except the most familiar particles) in the text of each Latin
+author down to the middle of the 2nd century A.D., and to extract all
+important passages from all writers of the following centuries down to
+the 7th; and upon these materials to found a complete historical
+dictionary of the Latin language. The work of collecting quotations was
+begun in 1894, and the first part of the first volume has been
+published.
+
+In the making of all these great dictionaries (except, of course, the
+last) the needs of the general public as well as those of scholars have
+been kept in view. But the type to which the general dictionary designed
+for popular use has tended more and more to conform is the
+_encyclopaedic_. This combination of lexicon and encyclopaedia is
+exhibited in an extreme--and theoretically objectionable--form in the
+_Grand dictionnaire universel du XIX^e siecle_ of Pierre Larousse.
+Besides common words and their definitions, it contains a great many
+proper names, with a correspondingly large number of biographical,
+geographical, historical and other articles, the connexion of which with
+the strictly lexicographical part is purely mechanical. Its utility,
+which--notwithstanding its many defects--is very great, makes it,
+however, a model in many respects. Fifteen volumes were published
+(1866-1876), and supplements were brought out later (1878-1890). The
+_Nouveau Larousse illustre_ started publication in 1901, and was
+completed in 1904 (7 vols.). This is not an abridgment or a fresh
+edition of the _Grand Dictionnaire_ of Pierre Larousse, but a new and
+distinct publication.
+
+The most notable work of this class, in English, is the _Century
+Dictionary_, an American product, edited by Professor W. D. Whitney, and
+published 1889-1891 in six volumes, containing 7046 pages (large
+quarto). It conforms to the philological mode in giving with great
+fulness the older as well as the present vocabulary of the language, and
+in the completeness of its etymologies; but it does not attempt to give
+the full history of every word within the language. Among its other more
+noteworthy characteristics are the inclusion of a great number of modern
+scientific and technical words, and the abundance of its quotations. The
+quotations are for the most part provided with references, but they are
+not dated. Even when compared with the much larger _New English
+Dictionary_, the _Century's_ great merit is the excellent enumeration of
+meanings, and the accuracy of its explanations; in this respect it is
+often better and fuller than the _New English_. In the application of
+the encyclopaedic method this dictionary is conservative, excluding,
+with a few exceptions, proper names, and restricting, for the most part,
+the encyclopaedic matter to descriptive and other details which may
+legitimately be added to the definitions. Its pictorial illustrations
+are very numerous and well executed. In the manner of its compilation it
+is a good example of modern cooperative dictionary-making, being the
+joint product of a large number of specialists. Next to the _New English
+Dictionary_ it is the most complete and scholarly of English lexicons.
+
+_Bibliography._--The following list of dictionaries (from the 9th
+edition of this work, with occasional corrections) is given for its
+historical interest, but in recent years dictionary-making has been so
+abundant that no attempt is made to be completely inclusive of later
+works; the various articles on languages may be consulted for these. The
+list is arranged geographically by families of languages, or by regions.
+In each group the order, when not alphabetical, is usually from north to
+south, extinct languages generally coming first, and dialects being
+placed under their language. Dictionaries forming parts of other works,
+such as travels, histories, transactions, periodicals, reading-books,
+&c., are generally excluded. The system here adopted was chosen as on
+the whole the one best calculated to keep together dictionaries
+naturally associated. The languages to be considered are too many for an
+alphabetical arrangement, which ignores all relations both natural and
+geographical, and too few to require a strict classification by
+affinities, by which the European languages, which for many reasons
+should be kept together, would be dispersed. Under either system,
+Arabic, Persian and Turkish, whose dictionaries are so closely
+connected, would be widely separated. A wholly geographical arrangement
+would be inconvenient, especially in Europe. Any system, however, which
+attempts to arrange in a consecutive series the great network of
+languages by which the whole world is enclosed, must be open to some
+objections; and the arrangement adopted in this list has produced some
+anomalies and dispersions which might cause inconvenience if not pointed
+out. The old Italic languages are placed under Latin, all dialects of
+France under French (but Provencal as a distinct language), and
+Wallachian among Romanic languages. Low German and its dialects are not
+separated from High German. Basque is placed after Celtic; Albanian,
+Gipsy and Turkish at the end of Europe, the last being thus separated
+from its dialects and congeners in Northern and Central Asia, among
+which are placed the Kazan dialect of Tatar, Samoyed and Ostiak.
+Accadian is placed after Assyrian among the Semitic languages, and
+Maltese as a dialect of Arabic; while the Ethiopic is among African
+languages as it seemed undesirable to separate it from the other
+Abyssinian languages, or these from their neighbours to the north and
+south. Circassian and Ossetic are joined to the first group of Aryan
+languages lying to the north-west of Persia, and containing Armenian,
+Georgian and Kurd. The following is the order of the groups, some of the
+more important languages, that is, of those best provided with
+dictionaries, standing alone:--
+
+EUROPE: Greek, Latin, French, Romance, Teutonic (Scandinavian and
+German), Celtic, Basque, Baltic, Slavonic, Ugrian, Gipsy, Albanian.
+
+ASIA: Semitic, Armenian, Persian, Sanskrit, Indian, Indo-Chinese, Malay
+Archipelago, Philippines, Chinese, Japanese, Northern and Central Asia.
+
+AFRICA: Egypt and Abyssinia, Eastern Africa, Southern, Western, Central,
+Berber.
+
+AUSTRALIA AND POLYNESIA.
+
+AMERICA: North, Central (with Mexico), South.
+
+
+EUROPE
+
+ Greek.---Athenaeus quotes 35 writers of works, known or supposed to be
+  dictionaries, for, as they are all lost, it is often difficult to
+  decide on their nature. Of these, Anticlides, who lived after the
+  reign of Alexander the Great, wrote [Greek: Exegetikos], which seems
+  to have been a sort of dictionary, perhaps explaining the words and
+  phrases occurring in ancient stories. Zenodotus, the first
+  superintendent of the great library of Alexandria, who lived in the
+  reigns of Ptolemy I. and Ptolemy II., wrote [Greek: Glossai], and also
+  [Greek: Lexeis ethnikai], a dictionary of barbarous or foreign
+  phrases. Aristophanes of Byzantium, son of Apelles the painter, who
+  lived in the reigns of Ptolemy II. and Ptolemy III., and had the
+  supreme management of the Alexandrian library, wrote a number of
+  works, as [Greek: Attikai Lexeis, Lakonikai Glossai] which, from the
+  titles, should be dictionaries, but a fragment of his [Greek: Lexeis]
+  printed by Boissonade, in his edition of Herodian (London, 1869, 8vo,
+  pp. 181-189), is not alphabetical. Artemidorus, a pupil of
+  Aristophanes, wrote a dictionary of technical terms used in cookery.
+  Nicander Colophonius, hereditary priest of Apollo Clarius, born at
+  Claros, near Colophon in Ionia, in reputation for 50 years, from 181
+  to 135, wrote [Greek: Glossai] in at least three books. Parthenius, a
+  pupil of the Alexandrian grammarian Dionysius (who lived in the 1st
+  century before Christ), wrote on choice words used by historians.
+  Didymus, called [Greek: chalkenteros], who, according to Athenaeus,
+  wrote 3500 books, and, according to Seneca, 4000, wrote lexicons of
+  the tragic poets (of which book 28 is quoted), of the comic poets, of
+  ambiguous words and of corrupt expressions. Glossaries of Attic words
+  were written by Crates, Philemon, Philetas and Theodorus; of Cretan,
+  by Hermon or Hermonax; of Phrygian, by Neoptolemus; of Rhodian, by
+  Moschus; of Italian, by Diodorus of Tarsus; of foreign words, by
+  Silenus; of synonyms, by Simaristus; of cookery, by Heracleon; and of
+  drinking vessels, by Apollodorus of Cyrene. According to Suidas, the
+  most ancient Greek lexicographer was Apollonius the sophist, son of
+  Archibius. According to the common opinion, he lived in the time of
+  Augustus at Alexandria. He composed a lexicon of words used by Homer,
+  [Greek: Lexeis Homerikai], a very valuable and useful work, though
+  much interpolated, edited by Villoison, from a MS. of the 10th
+  century, Paris, 1773, 4to, 2 vols.; and by Tollius, Leiden, 1788, 8vo;
+  ed. Bekker, Berlin, 1833, 8vo. Erotian or Herodian, physician to Nero,
+  wrote a lexicon on Hippocrates, arranged in alphabetical order,
+  probably by some copyist, whom Klein calls "homo sciolus." It was
+  first published in Greek in H. Stephani _Dictionarium Medicum_, Paris,
+  1564, 8vo; ed. Klein, Lipsiae, 1865, 8vo, with additional fragments.
+  Timaeus the sophist, who, according to Ruhnken, lived in the 3rd
+  century, wrote a very short lexicon to Plato, which, though much
+  interpolated, is of great value, 1st ed. Ruhnken, Leiden, 1754; ed.
+  locupletior, Lugd. Bat. 1789, 8vo. Aelius Moeris, called the Atticist,
+  lived about 190 A.D., and wrote an Attic lexicon, 1st ed. Hudson,
+  Oxf. 1712, Bekker, 1833. Julius Pollux ([Greek: Ioulios Polydeukes])
+  of Naucratis, in Egypt, died, aged fifty-eight, in the reign of
+  Commodus (180-192), who made him professor of rhetoric at Athens. He
+  wrote, besides other lost works, an Onomasticon in ten books, being a
+  classed vocabulary, intended to supply all the words required by each
+  subject with the usage of the best authors. It is of the greatest
+  value for the knowledge both of language and of antiquities. First
+  printed by Aldus, Venice, 1500, fol.; often afterwards; ed. Lederlinus
+  and Hemsterhuis, Amst. 1706, 2 vols.; Dindorf, 1824, 5 vols., Bethe
+  (1900 f.). Harpocration of Alexandria, probably of the 2nd century,
+  wrote a lexicon on the ten Attic orators, first printed by Aldus, Ven.
+  1503, fol.; ed. Dindorf, Oxford, 1853, 8vo, 2 vols. from 14 MSS.
+  Orion, a grammarian of Thebes, in Egypt, who lived between 390 and
+  460, wrote an etymological dictionary, printed by Sturz, Leipzig,
+  1820, 4to. Helladius a priest of Jupiter at Alexandria, when the
+  heathen temples there were destroyed by Theophilus in 389 or 391
+  escaped to Constantinople, where he was living in 408. He wrote an
+  alphabetical lexicon, now lost, chiefly of prose, called by Photius
+  the largest ([Greek: polystichotaton]) which he knew. Ammonius,
+  professor of grammar at Alexandria, and priest of the Egyptian ape,
+  fled to Constantinople with Helladius, and wrote a dictionary of words
+  similar in sound but different in meaning, which has been often
+  printed in Greek lexicons, as Aldus, 1497, Stephanus, and separately
+  by Valckenaer, Lugd. Bat. 1739, 4to, 2 vols., and by others. Zenodotus
+  wrote on the cries of animals, printed in Valckenaer's _Ammonius_;
+  with this may be compared the work of Vincentio Caralucci, _Lexicon
+  vocum quae a brutis animalibus emittuntur_, Perusia, 1779, 12mo.
+  Hesychius of Alexandria wrote a lexicon, important for the knowledge
+  of the language and literature, containing many dialectic and local
+  expressions and quotations from other authors, 1st ed. Aldus, Ven.
+  1514, fol.; the best is Alberti and Ruhnken, Lugd. Bat. 1746-1766,
+  fol. 2 vols.; collated with the MS. in St Mark's library, Venice, the
+  only MS. existing, by Niels Iversen Schow, Leipzig, 1792, 8vo; ed.
+  Schmidt, Jena, 1867, 8vo. The foundation of this lexicon is supposed
+  to have been that of Pamphilus, an Alexandrian grammarian, quoted by
+  Athenaeus, which, according to Suidas, was in 95 books from [Epsilon]
+  to [Omega]; [Alpha] to [Delta] had been compiled by Zopirion. Photius,
+  consecrated patriarch of Constantinople, 25th December 857, living in
+  886, left a lexicon, partly extant, and printed with Zonaras, Lips.
+  1808, 4to, 3 vols., being vol. iii.; ed. Naber, Leidae, 1864-1865,
+  8vo, 2 vols. The most celebrated of the Greek glossaries is that of
+  Suidas, of whom nothing is known. He probably lived in the 10th
+  century. His lexicon is an alphabetical dictionary of words including
+  the names of persons and places--a compilation of extracts from Greek
+  writers, grammarians, scholiasts and lexicographers, very carelessly
+  and unequally executed. It was first printed by Demetrius
+  Chalcondylas, Milan, 1499, fol.; the best edition, Bernhardy, Halle,
+  1853, 4to, 2 vols. John Zonaras, a celebrated Byzantine historian and
+  theologian, who lived in the 12th century, compiled a lexicon, first
+  printed by Tittmann, Lips. 1808. 4to, 2 vols. An anonymous Greek
+  glossary, entitled [Greek: Etymologikon mega], _Etymologicum magnum_,
+  has been frequently printed. The first edition is by Musurus, Venitia,
+  1499, fol.; the best by Gaisford, Oxonii, 1848, fol. It contains many
+  grammatical remarks by famous authorities, many passages of authors,
+  and mythological and historical notices. The MSS. vary so much that
+  they look like the works of different authors. To Eudocia Augusta of
+  Makrembolis, wife of the emperors Constantine XI. and Romanus IV.
+  (1059 to 1071), was ascribed a dictionary of history and mythology,
+  [Greek: Ionia] (bed of violets), first printed by D'Ansse de
+  Villoison, _Anecdota Graeca_, Venetiis, 1781, 4to, vol. i. pp. 1-442.
+  It was supposed to have been of much value before it was published.
+  Thomas, Magister Officiorum under Andronicus Palaeologus, afterward
+  called as a monk Theodulus, wrote [Greek: Eklogai onomaton Attikon],
+  printed by Callierges, Romae, 1517, 8vo: Papias, _Vocabularium_,
+  Mediolani, 1476, fol.: Craston, an Italian Carmelite monk of Piacenza,
+  compiled a Greek and Latin lexicon, edited by Bonus Accursius, printed
+  at Milan, 1478, fol.: Aldus, Venetiis, 1497, fol.: Guarino, born about
+  1450 at Favora, near Camarino, who called himself both Phavorinus and
+  Camers, published his _Thesaurus_ in 1504. These three lexicons were
+  frequently reprinted. Estienne, _Thesaurus_, Genevae, 1572, fol., 4
+  vols.; ed. Valpy, Lond. 1816-1826, 6 vols. fol.; Paris, 1831-1865, 9
+  vols. fol., 9902 pages: [Greek: Kibotos], the ark, was intended to
+  give the whole language, ancient and modern, but vol. i.,
+  Constantinople, 1819, fol., 763 pages, [Alpha] to [Delta], only
+  appeared, as the publication was put an end to by the events of 1821.
+  ENGLISH.--Jones, London, 1823, 8vo: Dunbar, Edin. 3rd ed. 1850, 4to:
+  Liddell and Scott, 8th ed. Oxford, 1897, 4to. FRENCH.--Alexandre, 12th
+  ed. Paris, 1863, 8vo; 1869-1871, 2 vols: Chassang, ib. 1872, 8vo.
+  ITALIAN.--Camini, Torino, 1865, 8vo, 972 pages: Muller, ib. 1871, 8vo.
+  SPANISH.--_Diccionario manual, por les padres Esculapios_, Madrid,
+  1859, 8vo. GERMAN.--Passow, 5th ed. Leipzig, 1841-1857, 4to: Jacobitz
+  and Seiler, 4th ed. ib. 1856, 8vo: Benseler, ib. 1859, 8vo: Pape,
+  Braunschweig, 1870-1874, 8vo, 4 vols. Prellwitz, _Etymologisches
+  Worterbuch der griechischen Sprache_, new edition, 1906: Herwerden,
+  _Lexicon Graecum suppletorium et dialecticum_, 1902.
+  DIALECTS.--_Attic_: Moeris, ed. Pierson, Lugd. Bat. 1759. 8vo. _Attic
+  Orators_: Reiske, Oxon. 1828, 8vo, 2 vols. _Doric_: Portus, Franckof.
+  1605, 8vo. _Ionic_: Id. ib. 1603, 8vo; 1817; 1825. PROSODY.--Morell,
+  Etonae, 1762, 4to; ed. Maltby, Lond. 1830, 4to: Brasse, Lond. 1850,
+  8vo. RHETORIC.--Ernesti, Lips. 1795, 8vo. MUSIC.--Drieberg, Berlin,
+  1855. ETYMOLOGY.--Curtius, Leipzig, 1858-1862: Lancelot, Paris, 1863,
+  8vo. SYNONYMS.--Peucer, Dresden, 1766, 8vo: Pillon, Paris, 1847, 8vo.
+  PROPER NAMES.--Pape, ed. Sengebusch, 1866, 8vo, 969 pages.
+  VERBS.--Veitch, 2nd ed. Oxf. 1866. TERMINATIONS.--Hoogeveen, Cantab.
+  1810, 4to: Pape, Berlin, 1836, 8vo. PARTICULAR AUTHORS.--_Aeschylus_:
+  Wellauer, 2 vols. Lips. 1830-1831, 8vo. _Aristophanes_: Caravella,
+  Oxonii, 1822, 8vo. _Demosthenes_: Reiske, Lips. 1775, 8vo.
+  _Euripides_: Beck, Cantab. 1829, 8vo. _Herodotus_: Schweighauser,
+  Strassburg, 1824, 8vo, 2 vols. _Hesiod_: Osoruis, Neapol. 1791, 8vo.
+  _Homer_: Apollonius Sophista, ed. Tollius, Lugd. Bat., 1788, 8vo:
+  Schaufelberger, Zurich, 1761-1768, 8vo, 8 vols.: Crusius, Hanover,
+  1836, 8vo: Wittich, London, 1843, 8vo: Doderlein, Erlangen, 8vo, 3
+  vols.: Eberling, Lipsiae, 1875, 8vo: Autenrieth, Leipzig, 1873, 8vo;
+  London, 1877, 8vo. _Isocrates_: Mitchell, Oxon. 1828, 8vo. _Pindar_:
+  Portus, Hanov. 1606, 8vo. _Plato_: Timaeus, ed. Koch, Lips. 1828, 8vo:
+  Mitchell, Oxon. 1832, 8vo: Ast, Lips. 1835-1838, 8vo, 3 vols.
+  _Plutarch_: Wyttenbach, Lips. 1835, 8vo, 2 vols. _Sophocles_: Ellendt,
+  Regiomonti, 1834-1835, 8vo ed.; Genthe, Berlin, 1872, 8vo.
+  _Thucydides_: Betant, Geneva, 1843-1847, 8vo, 2 vols. _Xenophon_:
+  Sturtz, Lips. 1801-1804, 8vo, 4 vols.: Cannesin (Anabasis,
+  Gr.-Finnish), Helsirgissa, 1868, 8vo: Sauppe, Lipsiae, 1869, 8vo.
+  _Septuagint_: Hutter, Noribergae, 1598, 4to: Biel, Hagae, 1779-1780,
+  8vo. _New Testament_: Lithocomus, Colon, 1552, 8vo: Parkhurst, ed.
+  Major, London, 1845, 8vo: Schleusner (juxta ed. Lips. quartam),
+  Glasguae, 1824, 4to.
+
+
+ Medieval and Modern Greek.--Meursius, Lugd. Bat. 1614, 4to:
+  Critopulos, Stendaliae, 1787, 8vo: Portius, Par. 1635, 4to: Du Cange,
+  Paris, 1682, fol., 2 vols.; Ludg. 1688, fol. ENGLISH.--Polymera,
+  Hermopolis, 1854, 8vo: Sophocles, Cambr. Mass. 1860-1887: Contopoulos,
+  Athens, 1867, 8vo; Smyrna, 1868-1870, 8vo, 2 parts, 1042 pages.
+  FRENCH.--Skarlatos, Athens, 1852, 4to: Byzantius, ib. 1856, 8vo, 2
+  vols.: Varvati, 4th ed. ib., 1860, 8vo. ITALIAN.--Germano, Romae,
+  1622, 8vo: Somavera, Parigi, 1709, fol., 2 vols.: Pericles,
+  Hermopolis, 1857, 8vo. GERMAN.--Schmidt, Lips. 1825-1827, 12mo, 2
+  vols.: Petraris, Leipz. 1897. POLYGLOTS.--Koniaz (Russian and Fr.),
+  Moscow, 1811, 4to; Schmidt (Fr.-Germ.), Leipzig, 1837-1840, 12mo, 3
+  vols.: Theocharopulas de Patras (Fr.-Eng.), Munich, 1840, 12mo.
+
+
+ Latin.--Johannes de Janua, _Catholicon_ or _Summa_, finished in 1286,
+  printed Moguntiae 1460, fol.; Venice, 1487; and about 20 editions
+  before 1500: Johannes, _Comprehensorium_, Valentia, 1475, fol.: Nestor
+  Dionysius, _Onomasticon_, Milan, 1477, fol.: Stephanus, Paris, 1531,
+  fol., 2 vols.: Gesner, Lips. 1749, fol., 4 vols.: Forcellini, Patavii,
+  1771, fol., 4 vols. POLYGLOT.--Calepinus, Reggio, 1502, fol. (Aldus
+  printed 16 editions, with the Greek equivalents of the Latin words;
+  Venetiis, 1575, fol., added Italian, French and Spanish; Basileae,
+  1590, fol., is in 11 languages; several editions, from 1609, are
+  called Octolingue; many of the latter 2 vol. editions were edited by
+  John Facciolati): Verantius (Ital., Germ., Dalmatian, Hungarian),
+  Venetiis, 1595, 4to: Lodereckerus (Ital., Germ., Dalm., Hungar.,
+  Bohem., Polish), Pragae, 1605, 4to. ENGLISH.--_Promptorium
+  parvulorum_, compiled in 1440 by Galfridus Grammaticus, a Dominican
+  monk of Lynn Episcopi, in Norfolk, was printed by Pynson, 1499; 8
+  editions, 1508-1528, ed. Way, Camden Society, 1843-1865, 3 vols. 4to;
+  _Medulla grammaticis_, probably by the same author, MS. written 1483;
+  printed as _Ortus vocabulorum_, by Wynkyn de Worde, 1500; 13 editions
+  1509-1523; Sir Thomas Elyot, London, 1538, fol.; 2nd ed. 1543;
+  _Bibliotheca Eliotae_, ed. Cooper, ib. 1545, fol.: Huloet,
+  _Abecedarium_, London, 1552, fol.; _Dictionarie_, 1572, fol.: Cooper,
+  London, 1565, fol.; 4th edition, 1584, fol.: Baret, _Alvearie_, ib.
+  1575, fol.; 1580, fol.: Fleming, ib. 1583, fol.: Ainsworth, London,
+  1736, 4to; ed. Morell, London, 1796, 4to, 2 vols.; ed. Beatson and
+  Ellis, ib. 1860, 8vo: Scheller, translated by Riddle, Oxford, 1835,
+  fol.: Smith, London, 1855, 8vo; 1870: Lewis and Short, Oxford, 1879.
+  ENG.-LATIN.--Levins, _Manipulus puerorum_, Lond. 1570, 4to: Riddle,
+  ib. 1838, 8vo: Smith, ib. 1855, 8vo. FRENCH.--_Catholicon parvum_,
+  Geneva, 1487: Estienne, _Dictionnaire_, Paris, 1539, fol. 675 pages;
+  enlarged 1549; ed. Huggins, Lond. 1572: Id. _Dictionarium
+  Latino-Gallicum_, Lutetiae, 1546, fol.; Paris, 1552; 1560: Id.,
+  _Dictionariolum puerorum_, Paris, 1542, 4to: _Les Mots francais_,
+  Paris, 1544, 4to; the copy in the British Museum has the autograph of
+  Queen Catherine Parr: Thierry (Fr.-Lat.), Paris, 1564, fol.: Danet, Ad
+  usum Delphini, Paris, 1700, 4to, 2 vols.; and frequently: Quicherat,
+  9th ed. Paris, 1857, 8vo: Theil, 3rd ed. Paris, 1863, 8vo: Freund, ib.
+  1835-1865, 4to, 3 vols. GERMAN.--Joh. Melber, of Gerolzhofen,
+  _Vocabularius praedicantium_, of which 26 editions are described by
+  Hain (_Repertorium_, No. 11,022, &c.), 15 undated, 7 dated 1480-1495,
+  4to, and 3 after 1504: _Vocabularius gemma gemmarum_, Antwerp, 1484,
+  4to; 1487; 12 editions, 1505-1518: Herman Torentinus, _Elucidarius
+  carminum_, Daventri, 1501, 4to; 22 editions, 1504-1536: Binnart, Ant.
+  1649, 8vo: Id., _Biglotton_, ib. 1661; 4th ed. 1688: Faber, ed.
+  Gesner, Hagae Com. 1735, fol., 2 vols.: Hederick, Lips. 1766, 8vo, 2
+  vols.: Ingerslev, Braunschweig, 1835-1855, 8vo, 2 vols.: _Thesaurus
+  linguae Latinae_, Leipzig, 1900: Walde, _Lateinisches etymologisches
+  Worterbuch_, 1906. ITALIAN.--Seebar (Sicilian translation of Lebrixa),
+  Venet. 1525, 8vo: Venuti, 1589, 8vo: Galesini, Venez. 1605, 8vo:
+  Bazzarini and Bellini, Torino, 1864, 4to, 2 vols. 3100 pages.
+  SPANISH.--Salmanticae, 1494, fol.; Antonio de Lebrixa, Nebrissenis,
+  Compluti, 1520, fol., 2 vols.: Sanchez de la Ballesta, Salamanca,
+  1587, 4to: Valbuena, Madrid, 1826, fol. PORTUGUESE.--Bluteau, Lisbon,
+  1712-1728, fol., 10 vols: Fonseca, ib. 1771, fol.: Ferreira, Paris,
+  1834, 4to; 1852. ROMANSCH.--_Promptuario di voci volgari_, Valgrisii,
+  1565, 4to. VLACH.--Divalitu, Bucuresci, 1852, 8vo.
+  SWEDISH.--_Vocabula_, Rostock, 1574, 8vo; Stockholm, 1579: Lindblom,
+  Upsala, 1790, 4to. DUTCH.--Binnart, Antw. 1649, 8vo: Scheller, Lugd.
+  Bat. 1799, 4to, 2 vols. FLEMISH.--Paludanus, Gandavi, 1544, 4to.
+  POLISH.--Macinius, Konigsberg, 1564, fol.: Garszynski, Breslau, 1823,
+  8vo, 2 vols. BOHEMIAN.--Johannes Aquensis, Pilsnae, 1511, 4to:
+  Reschel, Olmucii, 1560-1562, 4to, 2 vols.: Cnapius, Cracovia, 1661,
+  fol., 3 vols. ILLYRIAN.--Bellosztenecz, Zagrab, 1740, 4to: Jambresich
+  (also Germ. and Hungar.), Zagrab, 1742, 4to. SERVIAN.--Swotlik, Budae,
+  1721, 8vo. HUNGARIAN.--Molnar, Frankf. a. M. 1645, 8vo: Pariz-Papai,
+  Leutschen, 1708, 8vo; 1767. FINNISH.--Rothsen, Helsingissa, 1864, 8vo.
+  POETIC.--_Epithetorum et synonymorum thesaurus_, Paris, 1662, 8vo,
+  attributed to Chatillon; reprinted by Paul Aler, a German Jesuit, as
+  _Gradus ad Parnassum_, Paris, 1687, 8vo; many subsequent editions:
+  _Schirach_, Hal. 1768, 8vo: Noel, Paris, 1810, 8vo; 1826: Quicherat,
+  Paris, 1852, 8vo: Young, London, 1856, 8vo. EROTIC.--Rambach,
+  Stuttgart, 1836, 8vo. RHETORICAL.--Ernesti, Lips. 1797, 8vo. CIVIL
+  LAW.--Dirksen, Berolini, 1837, 4to. SYNONYMS.--Hill, Edinb. 1804, 4to:
+  Doderlein, Lips. 1826-1828, 8vo, 6 vols. ETYMOLOGY.--Danet, Paris,
+  1677, 8vo: Vossius, Neap. 1762, fol., 2 vols.: Salmon, London, 1796,
+  8vo, 2 vols.: Nagel, Berlin, 1869, 8vo; Latin roots, with their French
+  and English derivatives, explained in German: Zehetmayr, Vindobonae,
+  1873, 8vo: Vani[vc]ek, Leipz. 1874, 8vo. BARBAROUS.--Marchellus,
+  Mediol. 1753, 4to; Krebs, Frankf. a. M. 1834, 8vo; 1837. PARTICULAR
+  AUTHORS.--_Caesar_: Crusius, Hanov. 1838, 8vo. _Cicero_: Nizzoli,
+  Brescia, 1535, fol.; ed. Facciolati, Patavii, 1734, fol.; London,
+  1820, 8vo, 3 vols.: Ernesti, Lips. 1739, 8vo; Halle, 1831. _Cornelius
+  Nepos_: Schmieder, Halle, 1798, 8vo; 1816: Billerbeck, Hanover, 1825,
+  8vo. _Curtius Rufus_: Crusius, Hanov. 1844, 8vo. _Horace_: Ernesti,
+  Berlin, 1802-1804, 8vo, 3 vols.: Doring, Leipz. 1829, 8vo. _Justin_:
+  Meinecke, Lemgo, 1793, 8vo; 2nd ed. 1818. _Livy_: Ernesti, Lips. 1784,
+  8vo; ed Schafer, 1804. _Ovid_: Gierig, Leipz. 1814: (Metamorphoses)
+  Meinecke, 2nd ed., Lemgo, 1825, 8vo: Billerbeck (Do.), Hanover, 1831,
+  8vo. _Phaedrus_: Oertel, Nurnberg, 1798, 8vo: Horstel, Leipz. 1803,
+  8vo: Billerbeck Hanover, 1828, 8vo. _Plautus_: Paraeus, Frankf. 1614,
+  8vo. _Pliny_: Denso, Rostock, 1766, 8vo_. Pliny, jun._: Wensch,
+  Wittenberg, 1837-1839, 4to. _Quintilian_: Bonnellus, Leipz. 1834, 8vo.
+  _Sallust_: Schneider, Leipz. 1834, 8vo: Crusius, Hanover, 1840, 8vo.
+  _Tacitus_: Botticher, Berlin, 1830, 8vo. _Velleius Paterculus_: Koch,
+  Leipz. 1857, 8vo. _Virgil_: _Clavis_, London, 1742, 8vo: Braunhard,
+  Coburg, 1834, 8vo. _Vitruvius_: Rode, Leipz. 1679, 4to, 2 vols.:
+  Orsini, Perugia, 1801, 8vo.
+
+  OLD ITALIAN LANGUAGES.--Fabretti, Torini, 1858, 4to. _Umbrian_:
+  Huschke, Leipz. 1860, 8vo. _Oscan and Sabellian_: Id. Elberfeld, 1856,
+  8vo.
+
+  MEDIEVAL LATIN.--Du Cange, _Glossarium_, Paris, 1733-1736, fol., 6
+  vols.; Carpentier, Suppl., Paris, 1766, fol., 4 vols.; ed. Adelung,
+  Halae, 1772-1784, 8vo, 6 vols.; ed. Henschel, Paris, 1840-1850, 4to, 7
+  vols. (vol. vii. contains a glossary of Old French): Brinckmeier,
+  Gotha, 1850-1863, 8vo, 2 vols.: Hildebrand (_Glossarium saec. ix._),
+  Gotting. 1854, 4to: Diefenbach, _Glossarium_, Frankf. 1857, 4to: Id.
+  _Gloss. novum_, ib. 1867, 4to. ECCLESIASTICAL.--Magri, Messina, 1644,
+  4to; 8th ed. Venezia, 1732; Latin translation, _Magri Hierolexicon_,
+  Romae, 1677, fol.; 6th ed. Bologna, 1765, 4to, 2 vols.
+
+
+_Romance Languages. _
+
+ Romance Languages generally.--Diez, Bonn, 1853, 8vo; 2nd ed. ib.
+  1861-1862, 8vo, 2 vols.; 3rd ed. ib. 1869-1870, 8vo, 2 vols.; transl.
+  by Donkin, 1864, 8vo.
+
+
+ French.--Ranconet, _Thresor_, ed. Nicot, Paris, 1606, fol.; ib. 1618,
+  4to: Richelet, Geneve, 1680, fol., 2 vols.; ed. Gattel, Paris, 1840,
+  8vo, 2 vols.
+
+  The French Academy, after five years' consideration, began their
+  dictionary, on the 7th of February 1639, by examining the letter A,
+  which took them nine months to go through. The word Academie was for
+  some time omitted by oversight. They decided, on the 8th of March
+  1638, not to cite authorities, and they have since always claimed the
+  right of making their own examples. Olivier justifies them by saying
+  that for eighty years all the best writers belonged to their body, and
+  they could not be expected to cite each other. Their design was to
+  raise the language to its last perfection, and to open a road to reach
+  the highest eloquence. Antoine Furetiere, one of their members,
+  compiled a dictionary which he says cost him forty years' labour for
+  ten hours a day, and the manuscript filled fifteen chests. He gave
+  words of all kinds, especially technical, names of persons and places,
+  and phrases. As a specimen, he published his _Essai_, Paris, 1684,
+  4to; Amst. 1685, 12mo. The Academy charged him with using the
+  materials they had prepared for their dictionary, and expelled him, on
+  the 22nd of January 1685, for plagiarism. He died on the 14th of May
+  1688, in the midst of the consequent controversy and law suit. His
+  complete work was published, with a preface by Bayle, La Haye and
+  Rotterdam, 1690, fol., 3 vols.; again edited by Basnage de Beauval,
+  1701; La Haye, 1707, fol., 4 vols. From the edition of 1701 the very
+  popular so-called _Dictionnaire de Trevoux_, Trevoux, 1704, fol., 2
+  vols., was made by the Jesuits, who excluded everything that seemed to
+  favour the Calvinism of Basnage. The last of its many editions is
+  Paris, 1771, fol., 8 vols. The Academy's dictionary was first printed
+  Paris, 1694, fol., 2 vols. They began the revision in 1700; second
+  edition 1718, fol., 2 vols.; 3rd, 1740, fol., 2 vols.; 6th, 1835, 2
+  vols. 4to, reprinted 1855; Supplement, by F. Raymond, 1836, 4to;
+  Complement, 1842, 4to, reprinted 1856; _Dictionnaire historique_,
+  Paris, 1858-1865, 4to, 2 parts (A to Actu), 795 pages, published by
+  the Institut: Dochez, Paris, 1859, 4to: Bescherelle, ib. 1844, 4to, 2
+  vols.; 5th ed. Paris, 1857, 4to, 2 vols.; 1865; 1887: Landais, Paris,
+  1835; 12th ed. ib. 1854, 4to, 2 vols.: Littre, Paris, 1863-1873, 4to,
+  4 vols. 7118 pages: Supplement, Paris, 1877, 4to: Godefroy (with
+  dialects from 9th to 15th cent.), Paris, 1881-1895, and _Complement_:
+  Hatzfield, Darmesteter, and Thomas, Paris, 1890-1900: Larive and
+  Fleury, (_mots et choses, illustre_), Paris, 1884-1891.
+  ENGLISH.--Palsgrave, _Lesclaircissement de la langue francoyse_,
+  London, 1530, 4to, 2 parts; 1852: Hollyband, London, 1533, 4to:
+  Cotgrave, ib. 1611, fol.: Boyer, La Haye, 1702, 4to, 2 vols.; 37th ed.
+  Paris, 1851, 8vo, 2 vols.: Fleming and Tibbins, Paris, 1846-1849, 4to,
+  2 vols.; ib. 1854, 4to, 2 vols.; ib. 1870-1872, 4to, 2 vols.: Tarver,
+  London, 1853-1854, 8vo, 2 vols.; 1867-1872: Bellows, Gloucester, 1873,
+  16mo; ib. 1876. IDEOLOGICAL, OR ANALOGICAL.--Robertson, Paris, 1859,
+  8vo: Boissiere, Paris, 1862, 8vo. ETYMOLOGY.--Lebon, Paris, 1571, 8vo:
+  Menage, ib. 1650, 4to. Pougens projected a _Tresor des origines_, his
+  extracts for which, filling nearly 100 volumes folio, are in the
+  library of the Institut. He published a specimen, Paris, 1819, 4to.
+  After his death, _Archeologie francaise_, Paris, 1821, 8vo, 2 vols.,
+  was compiled from his MSS., which were much used by Littre: Scheler,
+  Bruxelles, 1862, 8vo; 1873: Brachet, 2nd ed. Paris, 1870, 12mo;
+  English trans. Kitchin, Oxf. 1866, 8vo. GREEK WORDS.--Trippault,
+  Orleans, 1580, 8vo: Morin, Paris, 1809, 8vo. GERMAN WORDS.--Atzler,
+  Cothen, 1867, 8vo. ORIENTAL WORDS.--Pihan, Paris, 1847, 8vo; 1866:
+  Devic, ib. 1876, 8vo. NEOLOGY.--Desfontaines, 3rd ed. Amst. 1728,
+  12mo: Mercier, Paris, 1801, 8vo, 2 vols.: Richard, ib. 1842, 8vo; 2nd
+  ed. 1845. POETIC.--_Dict. des rimes_ (by La Noue), Geneve, 1596, 8vo;
+  Cologny, 1624, 8vo: Carpentier, _Le Gradus francais_, Paris, 1825,
+  8vo, 2 vols. EROTIC.--De Landes, Bruxelles, 1861, 12mo.
+  ORATORY.--Demandre and Fontenai, Paris, 1802, 8vo: Planche, ib.
+  1819-1820, 8vo, 3 vols. PRONUNCIATION.--Feline, ib. 1857, 8vo. DOUBLE
+  FORMS.--Brachet, ib. 1871, 8vo. EPITHETS.--Daire, ib. 1817, 8vo.
+  VERBS.--Bescherelle, ib. 1855, 8vo, 2 vols.: 3rd ed. 1858.
+  PARTICIPLES.--Id., ib. 1861, 12mo. DIFFICULTIES.--Boiste, London,
+  1828, 12mo: Laveaux, Paris, 1872, 8vo, 843 pages.
+  SYNONYMS.--Boinvilliers, Paris, 1826, 8vo: Lafaye, ib. 1858, 8vo;
+  1861; 1869: Guizot, ib. 1809, 8vo; 6th ed. 1863; 1873.
+  HOMONYMS.--Zlatagorski (Germ., Russian, Eng.), Leipzig, 1862, 8vo, 664
+  pages. IMITATIVE WORDS.--Nodier, _Onomatopees_, ib. 1828, 8vo.
+  TECHNOLOGY.--D'Hautel, ib. 1808, 8vo, 2 vols.: Desgranges, ib. 1821,
+  8vo: Tolhausen (Fr., Eng., Germ.), Leipz. 1873, 8vo, 3 vols. FAULTS OF
+  EXPRESSION.--Roland, Gap, 1823, 8vo: Blondin, Paris, 1823, 8vo.
+  PARTICULAR AUTHORS.--_Corneille_: Godefroy, ib. 1862, 8vo, 2 vols.:
+  Marty-Laveaux, ib. 1868, 8vo, 2 vols. _La Fontaine_: Lorin, ib. 1852,
+  8vo. _Malherbe_: Regnier, ib. 1869, 8vo. _Moliere_: Genin, ib. 1846,
+  8vo: Marty-Laveaux, ib. 8vo. _Racine_: Marty-Laveaux, ib. 1873, 8vo, 2
+  vols. _M^me de Sevigne_: Sommer, ib. 1867, 8vo, 2 vols. OLD
+  FRENCH.--La Curne de St Palaye prepared a dictionary, of which he only
+  published _Projet d'un glossaire_, Paris, 1756, 4to. His MSS. in many
+  volumes are in the National Library, and were much used by Littre.
+  They were printed by L. Favre, and fasciculi 21-30 (tom. iii.), Niort,
+  4to, 484 pages, were published in February 1877. Lacombe (vieux
+  langage), Paris, 1766, 2 vols. 4to: Kelham (Norman and Old French),
+  London, 1779, 8vo: Roquefort (langue romane), Paris, 1808, 8vo;
+  Supplement, ib. 1820, 8vo: Pougens, _Archeologie_, ib. 1821, 8vo, 2
+  vols.: Burguy, Berlin, 1851-1856, 8vo, 3 vols.: Laborde (_Notice des
+  emaux ... du Louvre_, part ii.), Paris, 1853, 8vo, 564 pages:[3]
+  Gachet (rhymed chronicles), Bruxelles, 1859, 4to: Le Hericher (Norman,
+  English and French), Paris, 1862, 3 vols. 8vo: Hippeau (12th and 13th
+  centuries), Paris, 1875, 8vo. DIALECTS.--Jaubert (central), Paris,
+  1856-1857, 8vo, 2 vols.: Baumgarten (north and centre), Coblentz,
+  1870, 8vo: Azais, _Idiomes romans du midi_, Montpellier, 1877.
+  _Austrasian_: Francois. Metz, 1773, 8vo. _Auvergne_: Mege, Riom, 1861,
+  12mo. _Bearn_: Lespi, Pau, 1858, 8vo. _Beaucaire_: Bonnet (Bouguiren),
+  Nismes, 1840, 8vo. _Pays de Bray_: Decorde, Neufchatel, 1852, 8vo.
+  _Burgundy_: Mignard, Dijon, 1870, 8vo. _Pays de Castres_: Couzinie,
+  Castres, 1850, 4to. _Dauphine_: Champollion-Figeac, Paris, 1809, 8vo:
+  Jules, Valence, 1835, 8vo; Paris, 1840, 4to. _Dep. of Doubs_: Tissot
+  (Patois des Fourg, arr. de Pontarlier) Besancon, 1865, 8vo. _Forez_:
+  Gras, Paris, 1864, 8vo; Neolas, Lyon, 1865, 8vo. _Franche Comte_:
+  Maisonforte, 2nd ed. Besancon, 1753, 8vo. _Gascony_: Desgrouais
+  (Gasconismes corriges), Toulouse, 1766, 8vo; 1769; 1812, 12mo, 2
+  vols.; 1825, 8vo, 2 vols. _Dep. of Gers_: Cenac-Montaut, Paris, 1863,
+  8vo. _Geneva_: Humbert, Geneve, 1820, 8vo. _Languedoc_: Odde, Tolose,
+  1578, 8vo: Doujat, Toulouse, 1638, 8vo: De S.[auvages], Nismes, 1756,
+  2 vols.; 1785; Alais, 1820: Azais, Beziers, 1876, &c., 8vo: Hombres,
+  Alais, 1872, 4to: Thomas (_Greek words_) Montpellier, 1843, 4to.
+  _Liege_: Forir, Liege, 1866, 8vo, vol i. 455 pages. _Lille_: Vermesse,
+  Lille, 1861, 12mo: Debuire du Buc ib., 1867, 8vo. _Limousin_: Beronie,
+  ed. Vialle (Correze), Tulle, 1823, 4to. _Lyonnais, Forez,
+  Beaujolais_: Onofrio, Lyon, 1864, 8vo. _Haut Maine_: R[aoul] de
+  M.[ontesson], Paris, 1857; 1859, 503 pages. _Mentone_: Andrews, Nice,
+  1877, 12mo. _Dep. de la Meuse_: Cordier, Paris, 1853, 8vo. _Norman_:
+  Edelestand and Alfred Dumeril, Caen, 1849, 8vo: Dubois, ib. 1857, 8vo:
+  Le Hericher (_Philologie topographique_), Caen, 1863, 4to: Id.
+  (elements scandinaves), Avranches, 1861, 12mo: Metivier (Guernsey),
+  London, 1870, 8vo: Vasnier (arrond de Pont Audemer), Rouen, 1861, 8vo:
+  Delboulle (Vallee d'Yeres), Le Havre, 1876. _Picardy_: Corblet,
+  Amiens, 1851, 8vo. _Poitou, Saintonge, Aunis_: Favre, Niort, 1867,
+  8vo. _Poitou_: Beauchet-Filleau, Paris, 1864, 8vo: Levrier, Niort,
+  1867, 8vo: Lalanne, Poitiers, 1868, 8vo. _Saintonge_: Boucherie,
+  Angouleme, 1865, 8vo: Jonain, Royan, 1867, 8vo. _Savoy_: Pont
+  (Terratzu de la Tarantaise), Chambery, 1869, 8vo. _La Suisse Romande_:
+  Bridel, Lausanne, 1866, 8vo. _Dep. of Tarn_: Gary, Castre, 1845, 8vo.
+  _Dep. of Vaucluse_: Barjavel, Carpentras, 1849, 8vo. _Walloon
+  (Rouchi)_: Cambresier, Liege, 1787, 8vo: Grandgagnage, ib. 1845-1850,
+  8vo. 2 vols.: Chavee, Paris, 1857, 18mo: Vermesse, Doudi, 1867, 8vo.
+  Sigart (_Montois_), Bruxelles, 1870, 8vo. SLANG.--Oudin, _Curiositez
+  Francaises_, Paris, 1640, 8vo: Baudeau de Saumaise (Precieuses, Langue
+  de Ruelles), Paris, 1660, 12mo; ed. Livet, ib. 1856: Le Roux, _Dict.
+  Comique_, Amst. 1788, and 6 other editions: Careme Prenant [i.e.
+  Taumaise], (argot reforme), Paris, 1829, 8vo: Larchey (excentricitees
+  du langage), Paris, 1860, 12mo; 5th ed. 1865: Delvau (langue verte,
+  Parisian), Paris, 1867, 8vo: Larchey, Paris, 1873, 4to, 236 pages.
+
+
+ Provencal.--Pallas, Avignon, 1723, 4to: Bastero, _La Crusca
+  Provenzale_, Roma, 1724, fol. vol. i. only: Raynouard, Paris,
+  1836-1844, 8vo, 6 vols.: Garcin, Draguignand, 1841, 8vo, 2 vols.:
+  Honnorat, Digne, 1846-1849, 4to, 4 vols. 107,201 words: Id., _Vocab.
+  fr. prov._, ib. 1848, 12mo, 1174 pages.
+
+
+ Spanish.--Covarruvias Orosco, Madrid, 1611, fol.: ib. 1673-1674, fol.
+  2 vols.; Academia Espanola, Madrid, 1726-1739, fol. 6 vols.; 8th ed.
+  1837: Caballero, Madrid, 1849, fol.; 8th ed. ib. 1860, 4to, 2 vols.:
+  Cuesta, ib. 1872, fol. 2 vols.: Campano, Paris, 1876, 18mo, 1015
+  pages. Cuervo, 1886-1894; Monlau, 1881; Zerola, Toro y Gomes, and
+  Isaza, 1895; Serrano (encyclopaedic) 1876-1881. ENGLISH.--Percivall,
+  London, 1591, 4to: Pineda, London, 1740, fol.: Connelly and Higgins,
+  Madrid, 1797-1798, 4to, 4 vols.: Neuman and Baretti, 9th ed. London,
+  1831, 8vo, 2 vols.; 1874. FRENCH.--Oudin, Paris, 1607, 4to, 1660;
+  Gattel, Lyon, 1803, 4to, 2 vols.: Dominguez, Madrid, 1846, 8vo, 6
+  vols.: Blanc, Paris, 1862, 8vo, 2 vols. GERMAN.--Wagener, Hamb.
+  1801-1805, 8vo, 4 vols.: Seckendorp, ib. 1823, 8vo, 3 vols.:
+  Franceson, 3rd ed. Leipzig, 1862, 8vo, 2 vols. ITALIAN.--Franciosini,
+  Venezia, 1735, 8vo, 2 vols.; Cormon y Manni, Leon, 1843, 16mo, 2
+  vols.: Romero, Madrid, 1844, 4to. SYNONYMS.--_Diccionario de
+  Sinonimos_, Paris, 1853, 4to. ETYMOLOGY.--Aldrete, Madrid, 1682, fol.:
+  Monlau y Roca, ib. 1856, 12mo; Barcia, 1881-1883. ARABIC
+  WORDS.--Hammer Purgstall, Wien, 1855, 8vo: Dozy and Engelmann, 2d ed.
+  Leiden, 1869, 8vo. ANCIENT.--Sanchez, Paris, 1842, 8vo.
+  RHYMING.--Garcia de Rengifo (consonancias) Salmantica, 1592, 4to;
+  1876. DON QUIXOTE.--Beneke (German), Leipzig, 1800, 16mo; 4th ed.
+  Berlin, 1841, 16mo. DIALECTS.--_Aragonese_: Peralta, Zaragoza, 1836,
+  8vo: Borao, ib. 1859, 4to. _Catalan_: Rocha de Girona (Latin),
+  Barcinone, 1561, fol.: _Dictionari Catala_ (Lat. Fr. Span.),
+  Barcelona, 1642, 8vo: Lacavalleria (Cat.-Lat.), ib. 1696, fol.:
+  Esteve, ed. Belvitges, &c. (Catal.-Sp. Lat.), Barcelona, 1805-1835,
+  fol. 2 vols.: Saura (Cat.-Span.), ib. 1851, 16mo; 2nd ed.(Span.-Cat.),
+  ib. 1854; 3rd ed. (id.) ib. 1862, 8vo: Labernia, ib. 1844-1848, 8vo, 2
+  vols. 1864. _Gallegan_: Rodriguez, Coruna, 1863, 4to: Cuveira y Pinol,
+  Madrid, 1877, 8vo. Majorca: Figuera, Palma, 1840, 4to: Amengual, ib.
+  1845, 4to. _Minorca_: _Diccionario_, Madrid, 1848, 8vo. _Valencian_:
+  Palmyreno, Valentiae, 1569: Ros, Valencia, 1764, 8vo: Fuster, ib.
+  1827, 8vo: Lamarca, 2nd ed. ib. 1842, 16mo. _Cuba_: _Glossary of
+  Creole Words_, London, 1840, 8vo: Pichardo, 1836; 2nd ed. Havana,
+  1849, 8vo; 3rd ed. ib. 1862, 8vo; Madrid, 1860, 4to.
+
+
+ Portuguese.--Lima, Lisbon, 1783, 4to: Moraes da Silva, ib. 1789, 4to,
+  2 vols.; 6th ed. 1858: Academia real das Sciencas, ib. 1793, tom. i.,
+  ccvi. and 544 pages (A to Azurrar); Faria, ib. 1849, fol. 2 vols.; 3rd
+  ed. ib. 1850-1857, fol. 2 vols. 2220 pages. ENGLISH.--Vieyra, London,
+  1773, 2 vols. 4to: Lacerda, Lisboa, 1866-1871, 4to, 2 vols.
+  FRENCH.--Marquez, Lisboa, 1756-1761, fol. 2 vols.: Roquette, Paris,
+  1841, 8vo, 2 vols.; 4th ed. 1860: Marques, Lisbonne, 1875, fol. 2
+  vols.: Souza Pinto, Paris, 1877, 32mo, 1024 pages. GERMAN.--Wagener,
+  Leipzig, 1811-1812, 8vo, 2 vols.: Wollheim, ib. 1844, 12mo, 2 vols.:
+  Bosche, Hamburg, 1858, 8vo, 2 vols. 1660 pages. ITALIAN.--Costa e Sa,
+  Lisboa, 1773-1774, fol. 2 vols. 1652 pages: Prefumo, Lisboa, 1853,
+  8vo, 1162 pages. ANCIENT.--Joaquim de Sancta Rosa de Viterbo, ib.
+  1798, fol. 2 vols.; 1824, 8vo. ARABIC WORDS.--Souza, ib. 1789, 4to;
+  2nd ed. by S. Antonio Moura, ib. 1830, 224 pages. ORIENTAL AND AFRICAN
+  WORDS, NOT ARABIC.--Sao Luiz, ib. 1837, 4to, 123 pages. FRENCH
+  WORDS.--Id., ib. 1827, 4to; 2nd ed. Rio de Janeiro, 1835, 8vo.
+  SYNONYMS.--Id., ib. 1821, 4to; 2nd ed. ib. 1824-1828, 8vo. Fonseca,
+  Paris, 1833, 8vo; 1859, 18mo, 863 pages. HOMONYMS.--De Couto, Lisboa,
+  1842, fol. POETIC.--Luzitano (i.e. Freire), ib. 1765, 8vo, 2 vols.;
+  3rd ed. ib. 1820, 4to, 2 vols. RHYMING.--Couto Guerreiro, Lisboa,
+  1763, 4to. NAVAL.--Tiberghien, Rio de Janeiro, 1870, 8vo.
+  CEYLON-PORTUGUESE.--Fox, Colombo, 1819, 8vo: Callaway, ib. 1823, 8vo.
+
+
+ Italian.--Accarigi, _Vocabulario_, Cento, 1543, 4to: Alunno, _La_
+  _fabrica del mundo_, Vinezia, 1548, fol.: Porccachi, Venetia, 1588,
+  fol.: Accademici della Crusca, _Vocabulario_, Venez. 1612, fol.; 4th
+  ed. Firenze, 1729-1738, fol. 6 vols.: Costa and Cardinali, Bologna,
+  1819-1826, 4to, 7 vols.: Tommaseo and Bellini, Torino, 1861, &c., 4to,
+  4 vols.: Petrocchi, 1884-1891. ENGLISH.--Thomas, London, 1598, 4to:
+  Florio, London, 1598, 4to, 1611: Baretti, London, 1794, 2 vols.: 1854,
+  8vo, 2 vols.: Petronj and Davenport, Londra, 1828, 8vo, 3 vols.:
+  Grassi, Leipz. 1854, 12mo: Millhouse, Lond., 1868, 8vo, 2 vols. 1348
+  pages. FRENCH.--Alberti, Paris, 1771, 4to, 2 vols.; Milan, 1862:
+  Barberi, Paris, 1838, 4to, 2 vols.: Renzi, Paris, 1850, 8vo.
+  GERMAN.--_Libro utilissimo_, Venetiis, 1499, 4to: Valentini, Leipzig,
+  1834-1836, 4to, 4 vols. ETYMOLOGY.--Menage, Geneva, 1685, fol.: Bolza,
+  Vienna, 1852, 4to. PROVENCAL WORDS.--Nannucci, Firenze, 1840, 8vo.
+  SYNONYMS.--Rabbi, Venezia, 1774, 4to; 10th ed. 1817; Tommaseo,
+  Firenze, 1839-1840, 4to, 2 vols.: Milano, 1856, 8vo; 1867.
+  VERBS.--Mastrofini, Roma, 1814, 4to, 2 vols. SELECT WORDS AND
+  PHRASES.--Redi, Brescia, 1769, 8vo. INCORRECT WORDS AND
+  PHRASES.--Molassi, Parma, 1830-1841, 8vo, 854 pages. SUPPOSED
+  GALLICISMS.--Viani, Firenze, 1858-1860, 8vo, 2 vols. ADDITIONS TO THE
+  DICTIONARIES.--Gherardini, Milano, 1819-1821, 8vo, 2 vols.; ib.
+  1852-1857, 8vo, 6 vols. RHYMING.--Falco, Napoli, 1535, 4to: Ruscelli,
+  Venetia, 1563, 8vo; 1827: Stigliani, Roma, 1658, 8vo: Rosasco, Padova,
+  1763, 4to; Palermo, 1840, 8vo. TECHNICAL.--Bonavilla-Aquilino, Mil.
+  1819-1821, 8vo, 5 vols.; 2nd ed. 1829-1831, 4to, 2 vols.: Vogtberg
+  (Germ.), Wein, 1831, 8vo. PARTICULAR AUTHORS.--_Boccaccio_: Aluno, _Le
+  ricchezze della lingua volgare_, Vinegia, 1543. fol. _Dante_: Blanc,
+  Leipzig, 1852, 8vo; Firenze, 1859, 8vo. DIALECTS.--_Bergamo_:
+  Gasparini, Mediol. 1565: Zappetini, Bergamo, 1859, 8vo: Tiraboschi
+  (anc. and mod.), Turin, 1873, 8vo. _Bologna_: Bumaldi, Bologna, 1660,
+  12mo: Ferrari, ib. 1820, 8vo; 1838, 4to. _Brescia_: Gagliardi,
+  Brescia, 1759, 8vo: Melchiori, ib. 1817-1820, 8vo: _Vocabularietto_,
+  ib. 1872, 4to. _Como_: Monti, Milano, 1845, 8vo. _Ferrara_: Manini,
+  Ferrara, 1805, 8vo: Azzi, ib. 1857, 8vo. _Friuli_: Scala, Pordenone,
+  1870, 8vo. _Genoa_: Casaccia, Gen. 1842-1851, 8vo; 1873, &c.:
+  Paganini, ib. 1857, 8vo. _Lombardy_: Margharini, Tuderti, 1870, 8vo.
+  _Mantua_: Cherubini, Milano, 1827, 4to. _Milan_: Varon, ib. 1606, 8vo:
+  Cherubini, ib. 1814, 8vo, 2 vols.; 1841-1844, 8vo, 4 vols.; 1851-1861,
+  8vo, 5 vols.: Banfi, ib. 1857, 8vo: 1870, 8vo. _Modena_: Galvani,
+  Modena, 1868, 8vo. _Naples_: Galiani, Napoli, 1789, 12mo, 2 vols.
+  _Parma_: Peschieri, Parma, 1828-1831, 8vo, 3 vols. 1840; Malespina,
+  ib. 1856, 8vo, 2 vols. _Pavia_: _Dizionario domestico pavese_, Pavia,
+  1829, 8vo: Gambini, ib. 1850, 4to, 346 pages. _Piacenza_: Nicolli,
+  Piacenza, 1832: Foresti, ib. 1837-1838, 8vo, 2 pts. _Piedmont_: Pino,
+  Torino, 1784, 4to: Capello (Fr.), Turin, 1814, 8vo, 2 pts.: Zalli
+  (Ital. Lat. Fr.), Carmagnola, 1815, 8vo, 2 vols: Sant' Albino, Torino,
+  1860, 4to. _Reggio_: _Vocabulario Reggiano_, 1832. _Romagna_: Morri,
+  Fienza, 1840. _Rome_: _Raccolto di voci Romani e Marchiani_, Osimo,
+  1769, 8vo. _Roveretano and Trentino_: Azzolini, Venezia, 1856, 8vo.
+  _Sardinia_: Porru, Casteddu, 1832, fol.: Spano, Cagliari, 1851-1852,
+  fol. 3 vols. _Sicily_: Bono (It. Lat.), Palermo, 1751-1754, 4to, 3
+  vols.; 1783-1785, 4to, 5 vols.: Pasqualino, ib. 1785-1795, 4to, 5
+  vols.: Mortillaro, ib. 1853, 4to, 956 pages: Biundi, ib. 1857, 12mo,
+  578 pages: Traina, ib. 1870, 8vo. _Siena_: Barbagli, Siena, 1602, 4to.
+  _Taranto_: Vincentiis, Taranto, 1872, 8vo. _Turin_: Somis di Chavrie,
+  Torino, 1843, 8vo. _Tuscany_: Luna, Napoli, 1536, 4to: Politi, Roma,
+  1604, 8vo; Venezia, 1615; 1628; 1665; Paulo, ib. 1740, 4to. _Vaudois_:
+  Callet, Lausanne, 1862, 12mo. _Venetian_: Patriarchi (_Veneziano e
+  padevano_), Padova, 1755, 4to; 1796, 1821: Boerio, Venezia, 1829, 4to;
+  1858-1859; 1861. _Verona_: Angeli, Verona, 1821, 8vo. _Vicenza_:
+  Conti, Vicenza, 1871, 8vo. LINGUA FRANCA.--_Dictionnaire de la langue
+  Franque, ou Petit Mauresque_, Marseille, 1830, 16mo, 107 pages.
+  SLANG.--Sabio (lingua Zerga), Venetia, 1556, 8vo; 1575: _Trattato
+  degli bianti_, Pisa, 1828, 8vo.
+
+
+ Romansh.--_Promptuario de voci volgari e Latine_, Valgrisii, 1565,
+  4to: _Der, die, das, oder Nomenclatura_ (German nouns explained in
+  Rom.), Scoul, 1744, 8vo: Conradi, Zurich, 1820, 8vo; 1826, 12mo, 2
+  vols.: Carisch, Chur, 1821, 8vo; 1852, 16mo.
+
+
+ Vlach.--_Lesicon Rumanese_ (Lat. Hung. Germ.), Budae, 1825, 4to: Bobb
+  (Lat. Hung.), Clus, 1822-1823, 4to, 2 vols. FRENCH.--Vaillant,
+  Boucoureshti, 1840, 8vo: Poyenar, Aaron and Hill, Boucourest,
+  1840-1841, 4to, 2 vols.; Jassi, 1852, 16mo, 2 vols.: De Pontbriant,
+  Bucuresci, 1862, 8vo: Cihac, Frankf. 1870, 8vo: Costinescu, Bucuresci,
+  1870, 8vo, 724 pages: Antonescu, Bucharest, 1874, 16mo, 2 vols. 919
+  pages. GERMAN.--Clemens, Hermanstadt, 1823, 8vo: Isser, Kronstadt,
+  1850: Polyzu, ib. 1857, 8vo.
+
+
+TEUTONIC: (1) _Scandinavian._
+
+ Icelandic.--LATIN.--Andreae, Havniae, 1683, 8vo: Halderson (Lat.
+  Danish), ib. 1814, 4to, 2 vols. ENGLISH.--Cleasby-Vigfusson, Oxford,
+  1874, 4to. GERMAN.--Dieterich, Stockholm, 1844, 8vo: Mobius, Leipzig,
+  1866, 8vo. DANISH.--Jonssen, Kjobenhavn, 1863, 8vo. NORWEGIAN.--Kraft,
+  Christiania, 1863, 8vo: Fritzner, Kristiania, 1867, 8vo.
+  POETIC.--Egilsson (Latin), Hafniae, 1860, 8vo; 1864.
+
+
+ Swedish.--Kindblad, Stockholm, 1840, 4to: Almqvist, Orebro, 1842-1844,
+  8vo: Dalin, _Ordbog._ Stockholm, 1850-1853, 8vo, 2 vols. 1668 pages;
+  1867, &c. 4to (vol. i. ii., A to Fjermare, 928 pages): Id.,
+  _Handordbog_, ib. 1868, 12mo, 804 pages; Svenska Academien. Stockholm,
+  1870, 4to (A) pp. 187. LATIN.--Stjernhjelm, Holm, 1643, 4to: Verelius,
+  Upsala, 1691, 8vo: Ihre (Sueo-Gothicum), Upsala, 1769, fol. 2 vols.
+  ENGLISH.--Serenius, Nykoping, 1757, 4to: Brisnon, Upsala, 1784, 4to:
+  Widegren, Stockholm, 1788, 4to; Brisman, Upsala, 1801, 4to; 3rd ed.
+  1815, 2 vols.: Deleen Orebro, 1829, 8vo: Granberg, ib. 1832, 12mo:
+  Nilssen, Widmark, &c., Stockholm, 1875, 8vo. FRENCH.--Moller,
+  Stockholm, 1745, 4to: Bjorkengren, ib. 1795, 2 vols.: Nordforss, ib.
+  1805, 8vo, 2 vols.: 2nd ed. Orebro, 1827, 12mo: West, Stockh. 1807,
+  8vo: Dalin, ib. 1842-1843, 4to, 2 vols.; 1872. GERMAN.--Dahnert,
+  Holmiae, 1746, 4to: Heinrich, Christiansund, 1814, 4to, 2 vols.; 4th
+  ed. Orebro, 1841, 12mo: Helms, Leipzig, 1858, 8vo; 1872.
+  DANISH.--Host, Kjobenhavn, 1799, 4to: Welander, Stockholm, 1844, 8vo:
+  Dalin, ib. 1869, 16mo: Kaper, Kjobenhavn, 1876, 16mo.
+  ETYMOLOGY.--Tamm, Upsala, 1874, &c., 8vo (A and B), 200 pages. FOREIGN
+  WORDS.--Sahlstedt, Wasteras, 1769, 8vo: Andersson (20,000), Stockholm,
+  1857, 16mo: Tullberg, ib. 1868, 8vo: Ekbohrn, ib. 1870, 12mo: Dalin,
+  ib. 1870, &c., 8vo. SYNONYMS.--Id., ib. 1870, 12mo. NAVAL.--Ramsten,
+  ib. 1866, 8vo. TECHNICAL.--Jungberg, ib. 1873, 8vo. DIALECTS.--Ihre,
+  Upsala, 1766, 4to: Rietz, Lund, 1862-1867, 4to, 859 pages. _Bohuslan_:
+  _Idioticon Bohusiense_, Gotaborg, 1776, 4to. _Dalecarlia_: Arborelius,
+  Upsala, 1813, 4to. _Gothland_: Hof (Sven), Stockholmiae, 1772, 8vo:
+  Raaf (Ydre), Orebro, 1859, 8vo. _Halland_: Moller, Lund, 158, 8vo.
+  _Helsingland_: Lenstrom, ib. 1841, 8vo: Fornminnessallskap,
+  Hudikswall, 1870, 8vo.
+
+
+ Norwegian.--Jenssen, Kjobenhavn, 1646, 8vo: Pontoppidan, Bergen, 1749,
+  8vo: Hanson (German), Christiania, 1840, 8vo: Aasen, ib. 1873, 8vo,
+  992 pages.
+
+
+ Danish.--Aphelen, Kopenh, 1764, 4to, 2 vols.; 1775, 4to, 3 vols.:
+  Molbech, Kjobenhavn, 1833, 8vo, 2 vols.: ib. 1859, 2 vols.:
+  Videnskabernes Selskab, ib. 1793-1865, Kalkar. ENGLISH.--Berthelson
+  (Eng. Dan.), 1754, 4to: Wolff, London, 1779, 4to. Bay, ib. 1807, 8vo,
+  2 vols.; 1824, 8vo: Hornbeck, ib. 1863, 8vo: Ferrall and Repp, ib.
+  1814, 16mo; 1873, 8vo: Rosing, Copenhagen, 1869, 8vo: Ancker, ib.
+  1874, 8vo. FRENCH.--Aphelen, 1754, 8vo: Id., ib. 1759, 4to, 2 vols.;
+  2nd ed. 1772-1777, vol. i. ii. GERMAN.--Id., ib. 1764, 4to, 2 vols.:
+  Gronberg, 2nd ed. Kopenh. 1836-1839, 12mo, 2 vols.; 1851, Helms,
+  Leipzig, 1858, 8vo. SYNONYMS.--Muller, Kjobenhavn, 1853, 8vo. FOREIGN
+  WORDS.--Hansen, Christiania, 1842, 12mo. NAVAL.--Wilsoet, Copenhagen,
+  1830, 8vo: Fisker (French), Kjobenhavn, 1839, 8vo. OLD
+  DANISH.--Molbech, ib. 1857-1868, 8vo, 2 vols. DIALECTS.--Id., ib.
+  1841, 8vo. _Bornholm_: Adler, _ib._ 1856, 8vo. _South Jutland_: Kok,
+  1867, 8vo. SLANG.--Kristiansen (Gadesproget), ib. 1866, 8vo. p. 452.
+
+
+(2) _Germanic._
+
+ Teutonic.--COMPARATIVE.--Meidinger, Frankf. a. M. 1833, 8vo, 2nd ed.
+  1836, 8vo.
+
+
+ Gothic.--Junius, Dortrecht, 1665, 4to: 1671; 1684, Diefenbach
+  (comparative), Franckf. a. M. 1846-1851, 2 vols. 8vo: Schulze,
+  Magdeburg, 1848, 4to: 1867, 8vo: Skeat, London, 1868, 4to: Balg
+  (_Comparative Glossary_), Magvike, Wisconsin, 1887-1889. ULPHILAS
+  (editions with dictionaries).--Castilionaeus, Mediol, 1829, 4to:
+  Gabelentz and Lobe, Altenburg, 1836-1843, 4to, 2 vols.: Gaugengigl,
+  Passau, 1848, 8vo: Stamm, Paderborn, 1857: Stamm and Heyne, ib. 1866,
+  8vo.
+
+
+ Anglo-Saxon.--LATIN.--Somner (Lat. Eng.), Oxonii, 1659, fol.: Benson,
+  ib. 1701, 8vo: Lye (A.-S. and Gothic), London, 1772, fol. 2 vols.:
+  Ettmuller, Quedlinburg, 1851, 8vo. 838 pages. ENGLISH.--Bosworth,
+  London, 1838, 8vo, 721 pages: Id. (_Compendious_), 1848, 278 pages.
+  Corson (A.-S. and Early English), New York, 1871, 8vo, 587 pages;
+  Toller (based on Bosworth), Oxford, 1882-1898. GERMAN.--Bouterwek,
+  Gutersloh, 1850, 8vo, 418 pages: Grein (Poets), Gottingen, 1861-1863,
+  8vo, 2 vols.: Leo, Halle, 1872, 8vo.
+
+
+ English.--Cockeram, London, 1623, 8vo: 9th ed. 1650: Blount, ib. 1656,
+  8vo: Philips, The new World of Words, London, 1658, fol.: Bailey,
+  London, 1721, 8vo; 2nd ed. ib. 1736, fol.; 24th ed. ib. 1782, 8vo:
+  Johnson, ib. 1755, fol. 2 vols.; ed. Todd, London, 1818, 4to, 4 vols.;
+  ib. 1827. 4to, 3 vols.; ed. Latham, ib. 1866-1874, 4to, 4 vols. (2 in
+  4 parts): Barclay, London, 1774, 4to; ed. Woodward, ib. 1848:
+  Sheridan, ib. 1780, 4to, 2 vols.: Webster, New York, 1828, 4to, 2
+  vols.; London, 1832, 4to, 2 vols.; ed. Goodrich and Porter, 1865, 4to:
+  Richardson, ib. 1836, 4to, 2 vols.; Supplement, 1856: Ogilvie,
+  _Imperial Dictionary_, Glasgow, 1850-1855, 8vo, 3 vols. (the new
+  edition of Ogilvie by Charles Annandale, 4 vols., 1882, was an
+  encyclopaedic dictionary, which served to some extent as the
+  foundation of the _Century Dictionary_); Boag, _Do._, Edinburgh,
+  1852-1853, 8vo, 2 vols.: Craik, ib. 1856, 8vo: Worcester, Boston,
+  1863, 4to. Stormouth and Bayne, 1885; Murray and Bradley, _The Oxford
+  English Dictionary_, 1884- ; Whitney, _The Century Dict._, New York,
+  1889-1891; Porter, _Webster's Internat. Dict._, Springfield,
+  Massachusetts, 1890; Funk, _Standard Dict._, New York, 1894; Hunter,
+  _The Encyclopaedic Dict._, 1879-1888. ETYMOLOGY.--Skinner, Londini,
+  1671, fol.: Junius, Oxonii, 1743, fol.: Wedgewood, London, 1859-1865,
+  3 vols.; ib. 1872, 8vo. Skeat, Oxford, 1881; Fennell (Anglicized
+  words), Camb. 1892. PRONOUNCING.--Walker, London, 1774, 4to: by Smart,
+  2nd ed. ib. 1846, 8vo. PRONOUNCING IN GERMAN.--Hausner, Frankf. 1793,
+  8vo; 3rd ed. 1807; Winkelmann, Berlin, 1818, 8vo: Voigtmann, Coburg,
+  1835, 8vo: Albert, Leipz. 1839, 8vo: Bassler, ib. 1840, 16mo.
+  ANALYTICAL.--Booth, Bath, 1836, 4to: Roget, _Thesaurus_, London, 1852,
+  8vo; 6th ed. 1857; Boston, 1874. SYNONYMS.--Piozzi, London, 1794, 8vo,
+  2 vols.: L. [abarthe], Paris, 1803, 8vo, 2 vols.: Crabb, London,
+  1823, 8vo; 11th ed. 1859: C. J. Smith, ib. 1871, 8vo, 610 pages.
+  REDUPLICATED WORDS.--Wheatley, ib. 1866, 8vo. SURNAMES.--Arthur, New
+  York, 1857, 12mo, about 2600 names: Lower, ib. 1860, 4to.
+  PARTICLES.--Le Febure de Villebrune, Paris, 1774, 8vo.
+  RHYMING.--Levins, _Manipulus Puerorum_, London, 1570, 4to; ed.
+  Wheatley, ib. 1867, 8vo: Walker, London, 1775, 8vo; 1865, 8vo.
+  SHAKESPEARE.--Nares, Berlin, 1822, 4to; ed. Halliwell and Wright,
+  London, 1859, 8vo: Schmidt, Berlin, 1874. OLD ENGLISH.--Spelman,
+  London [1626], fol. (A to I only); 1664 (completed); 1687 (best ed.):
+  Coleridge (1250-1300), ib. 1859, 8vo: Stratmann (Early Eng.), Krefeld,
+  1867, 8vo; 2nd ed. 1873, 4to: Bradley (new edition of Stratman),
+  Oxford, 1891; Matzner and Bieling, Berlin, 1878- . OLD AND
+  PROVINCIAL.--Halliwell, London, 1844-1846, 8vo; 2nd ed. ib. 1850, 2
+  vols.: 6th ed. 1904: Wright, ib. 1857, 8vo, 2 vols.; 1862.
+  DIALECTS.--Ray, ib. 1674, 12mo: Grose, ib. 1787, 8vo; 1790: Holloway,
+  Lewes, 1840, 8vo; Wright, _Eng. Dialect Dict._, London, 1898-1905, 28
+  vols. _Scotch_: Jamieson, Edin. 1806, 4to, 2 vols.; Supplement, 1826,
+  2 vols.; abridged by Johnstone, ib. 1846, 8vo: Brown, Edin, 1845, 8vo:
+  Motherby (German), Konigsberg, 1826-1828, 8vo: (_Shetland and
+  Orkney_), Edmonston, London, 1866, 8vo: (_Banffshire_), Gregor, ib.
+  1866, 8vo. _North Country_: Brockett, London, 1839, 8vo, 2 vols.
+  _Berkshire_: [Lousley] ib. 1852, 8vo, _Cheshire_: Wilbraham, ib. 1817,
+  4to; 1826, 12mo: Leigh, Chester, 1877, 8vo. _Cumberland_: _Glossary_,
+  ib. 1851, 12mo: Dickenson, Whitehaven, 1854, 12mo; Supplement, 1867:
+  Ferguson (Scandinavian Words), London, 1856, 8vo. _Derbyshire_: Hooson
+  (mining), Wrexham, 1747, 8vo: Sleigh, London, 1865, 8vo. _Dorset_:
+  Barnes, Berlin, 1863, 8vo. _Durham_: [Dinsdale] (Teesdale), London,
+  1849, 12mo. _Gloucestershire_: Huntley (Cotswold), ib. 1868, 8vo.
+  _Herefordshire_: [Sir George Cornewall Lewis,] London, 1839, 12mo.
+  _Lancashire_: Nodal and Milner, Manchester Literary Club, 1875, 8vo,
+  Morris (Furness), London, 1869, 8vo: R. B. Peacock (Lonsdale, North
+  and South of the Sands), ib. 1869, 8vo. _Leicestershire_: A. B. Evans,
+  ib. 1848, 8vo. _Lincolnshire_: Brogden, ib. 1866, 12mo: Peacock
+  (Manley & Corringham), ib. 1877, 8vo. _Norfolk and Suffolk_; Forby,
+  London, 1830, 8vo, 2 vols. _Northamptonshire_: Sternberg, ib. 1851,
+  8vo: Miss Anne E. Baker, ib. 1866, 8vo, 2 vols. 868 pages.
+  _Somersetshire_: Jennings, ib. 1869, 8vo: W. P. Williams and W. A.
+  Jones, Taunton, 1873, 8vo. _Suffolk_: Moor, Woodbridge, 1823, 12mo:
+  Bowditch (Surnames), Boston, U.S., 1851, 8vo; 1858; 3rd ed. London,
+  1861, 8vo, 784 pages. _Sussex_: Cooper, Brighton, 1836, 8vo: Parish,
+  Farncombe, 1875, 8vo. _Wiltshire_: Akerman, London, 1842, 12mo.
+  _Yorkshire (North and East)_, Toone, ib. 1832, 8vo: (_Craven_), Carr,
+  2nd ed. London, 1828, 8vo, 2 vols.: (_Swaledale_), Harland, ib. 1873,
+  8vo: (_Cleveland_), Atkinson, ib. 1868, 4to, 653 pages: (_Whitby_) [F.
+  K. Robinson], ib. 1876, 8vo: (_Mid-Yorkshire and Lower Niddersdale_),
+  C. Clough Robinson, ib. 1876, 8vo: (_Leeds_), Id., ib. 1861, 12mo:
+  (_Wakefield_), Banks, ib. 1865, 16mo: (_Hallamshire_), Hunter, London,
+  1829, 8vo. _Ireland: (Forth and Bargy, Co. Wexford)_, Poole, London,
+  1867, 8vo. _America_: Pickering, Boston, 1816, 8vo: Bartlett, New
+  York, 1848, 8vo; 3rd ed. Boston, 1860. 8vo; Dutch transl. by Keijzer,
+  Gorinchen, 1854, 12mo; Germ. transl. by Kohler, Leipz. 1868, 8vo.
+  Elwyn, Philadelphia, 1859. 8vo. _Negro English_: Kingos, St Croix,
+  1770, 8vo: Focke (Dutch), Leiden, 1855, 8vo: Wullschlaegel, Lobau,
+  1856, 8vo. 350 pages. SLANG.--Grose, London, 1785, 8vo; 1796: Hotten,
+  ib. 1864, 8vo; 1866; Farmer & Henley (7 vols., 1890-1904).
+
+
+ Frisic.--Wassenbergh, Leeuwarden, 1802, 8vo: Franeker, 1806, 8vo:
+  Outzen, Kopenh. 1837, 4to: Hettema (Dutch), Leuwarden, 1832, 8vo;
+  1874, 8vo, 607 pages: Winkler (Nederdeutsch en Friesch Dialectikon),
+  's Gravenhage, 1874, 8vo, 2 vols. 1025 pages. OLD FRISIC.--Wiarda
+  (Germ.), Aurich, 1786, 8vo: Richthofen, Gottingen, 1840, 4to. NORTH
+  FRISIC.--Bendson (Germ.), Leiden, 1860, 8vo: Johansen (Fohringer und
+  Amrumer Mundart), Kiel, 1862, 8vo. EAST FRISIC.--Sturenburg, Aurich,
+  1857, 8vo. HELIGOLAND.--Oelrichs, s. l., 1836, 16mo.
+
+
+ Dutch.--Kok, 2nd ed. Amst. 1785-1798, 8vo, 38 vols.: Weiland, Amst.
+  1790-1811, 8vo, 11 vols.: Harrebomee, Utrecht, 1857, 4to; 1862-1870,
+  8vo, 3 vols.: De Vries and Te Winkel, Gravenh. 1864, &c., 4to (new ed.
+  1882- ); Dale, ib. 4th ed. 1898; ENGLISH.--Hexham, ed. Manley,
+  Rotterdam, 1675-1678, 4to: Holtrop, Dortrecht, 1823-1824, 8vo, 2
+  vols.: Bomhoff, Nimeguen, 1859, 8vo, 2 vols. 2323 pages: Jaeger,
+  Gouda, 1862, 16mo: Calisch, Tiel, 1871, &c., 8vo. FRENCH.--Halma,
+  Amst. 1710, 4to; 4th ed. 1761: Marin, ib. 1793, 4to, 2 vols.:
+  Winkelman, ib. 1793, 4to, 2 vols.: Mook, Zutphen, 1824-1825, 8vo, 4
+  vols.; Gouda, 1857, 8vo, 2 vols. 2818 pages: Kramers, ib. 1859-1862, 2
+  vols. 16mo. GERMAN.--Kramer, Nurnb. 1719, fol.; 1759, 4to, 2 vols.;
+  ed. Titius, 1784, Weiland, Haag, 1812, 8vo: Terwen, Amst. 1844, 8vo.
+  ETYMOLOGY.--Franck, 1884-1892. ORIENTAL WORDS.--Dozy, 's Gravenhage,
+  1867, 8vo. GENDERS OF NOUNS.--Bilderdijk, Amst. 1822, 8vo, 2 vols.
+  SPELLING.--Id., 's Gravenhage, 1829, 8vo. FREQUENTATIVES.--De Jager,
+  Gouda, 1875, 8vo, vol. i. OLD DUTCH.--Suringer, Leyden, 1865, 8vo.
+  MIDDLE DUTCH.--De Vries, 's Gravenhage, 1864, &c., 4to. Verwijs and
+  Verdam, ib. 1885- .
+
+
+ Flemish.--Kilian, Antw. 1511, 8vo; ed. Hasselt, Utrecht, 1777, 4to, 2
+  vols. FRENCH.--Berlemont, Anvers, 1511, 4to: Meurier, ib. 1557, 8vo:
+  Rouxell and Halma, Amst. 1708, 4to; 6th ed. 1821: Van de Velde and
+  Sleeckx, Brux. 1848-1851, 8vo, 2440 pages; ib. 1860, 8vo, 2 vols.
+  ANCIENT NAMES OF PLACES.--Grandgagnage (East Belgium), Bruxelles,
+  1859, 8vo.
+
+
+ German.--Josua Pictorius (Maaler), _Die teutsch Spraach_, Tiguri,
+  1561, 8vo; Stieler, Nurnb. 1691, 4to: Adelung, Leipz. 1774-1786, 4to,
+  5 vols.; 1793-1818, 5 vols.: Campe, Braunschweig, 1807-1811, 4to, 5
+  vols.: Grimm, Leipzig, 1854, &c., 4to: Sanders, ib. 1860-1865, 4to, 3
+  vols. 1885: Diefenbach and Wulcker (High and Low German, to supplement
+  Grimm), Frankf. a. M. 1874, 1885, 8vo.; Kluge, Strassburg, 1883;
+  Heine, Leipzig, 1890-1895; Weigand, Giessen, 1873. ENGLISH.--Adelung,
+  1783-1796, 8vo, 3 vols.: Hilpert, Karlsruhe, 1828-1829, 8vo, 2 vols.;
+  1845-1846, 4to, 2 vols.: Flugel, Leipz. 1830, 8vo, 2 vols.; London,
+  1857, 8vo; Leipzig, 1870: Muller, Cothen, 1867, 8vo, 2 vols.
+  FRENCH.--Laveaux, Strassburg, 1812, 4to: Mozin, Stuttgard, 1811-1812,
+  4to, 4 vols.; 1842-1846, 8vo, 4 vols., 3rd ed. 1850-1851, 8vo:
+  Schuster, Strasb. 1859, 8vo: Daniel, Paris, 1877, 16mo. OLD HIGH
+  GERMAN.--Haltaeus, Lipsiae, 1758, fol. 2 vols.: Graff, Berlin,
+  1834-1846, 4to, 7 vols.: Brinckmeier, Gotha, 1850-1863, 4to, 2 vols.:
+  Kehrein (from Latin records), Nordhausen, 1863, 8vo. Schade, Halle,
+  1872-1882. MIDDLE HIGH GERMAN.--Ziemann, Quedlinburg, 1838, 8vo:
+  Benecke, Muller and Zarnche, Leipz. 1854-1866, 8vo, 3 vols.: Lexer,
+  Leipzig, 1870, 8vo. MIDDLE LOW GERMAN.--Schiller and Lubben, Bremen,
+  1872, &c., 8vo, in progress. LOW GERMAN.--Vollbeding, Zerbst, 1806,
+  8vo: Kosegarten, Griefswald, 1839, 4to; 1856, &c., 4to.
+  ETYMOLOGY.--Helvigius, Hanov. 1620, 8vo: Wachter, Lipsiae, 1737, fol.
+  2 vols.: Kaindl, Salzbach, 1815-1830, 8vo, 7 vols.: Heyse, Magdeburg,
+  1843-1849, 8vo, 3 vols.: Kehrein, Wiesbaden, 1847-1852, 2 vols.
+  SYNONYMS.--Eberhard, Maas, and Gruber, 4th ed. Leipzig, 1852-1863,
+  8vo, 4 vols.: Aue (Engl.), Edinb. 1836, 8vo: Eberhard, 11th ed.
+  Berlin, 1854, 12mo: Sanders, Hamburg, 1872, 8vo, 743 pages. FOREIGN
+  WORDS.--Campe, Braunschweig, 1813, 4to: Heyse, _Fremdworterbuch_,
+  Hannover, 1848, 8vo. NAMES.--Pott. Leipz. 1853, 8vo: Michaelis
+  (Taufnamen), Berlin, 1856, 8vo: Forstemann (Old Germ.) Nordhausen,
+  1856-1859, 4to, 2 vols. 1573 pages, 12,000 names: Steub
+  (Oberdeutschen), Munchen, 1871, 8vo. LUTHER.--Dietz, Leipzig,
+  1869-1872, 8vo, 2 vols. DIALECTS.--Popowitsch, Wien, 1780, 8vo: Fulda,
+  Berlin, 1788, 8vo: Klein, Frankf. 1792, 8vo, 2 vols.: Kaltschmidt,
+  Nordlingen, 1851, 4to; 1854, 5th ed. 1865. _Aix-la-Chapelle_, Muller
+  and Weitz, Aachen, 1836, 12mo. _Appenzell_: Tobler, Zurich, 1837, 8vo.
+  _Austria_: Hofer, Linz, 1815, 8vo; Castelli, Wien, 1847, 12mo:
+  Scheuchenstul (mining), ib. 1856, 8vo. _Bavaria_: Zaupser, Munchen,
+  1789, 8vo: Deling, ib. 1820, 2 vols.: Schmeller, Stuttg. 1827-1837,
+  8vo, 4 vols.; 2nd ed. Munchen, 1872, 4to, vol. i. 1799 pages.
+  _Berlin_: Trachsel. Berlin, 1873, 8vo. _Bremen_: Bremisch Deutsch
+  Gesellschaft, Bremen, 1767-1771, 1869, 8vo, 6 vols. Oelrich (anc.
+  statutes), Frankf. a. M. 1767, 8vo. _Carinthia_: Ueberfelder,
+  Klagenfurt, 1862, 8vo: Lexe, Leipzig, 1862, 8vo. _Cleves_: De
+  Schueren, _Teuthonista_, Colon, 1477, fol.; Leiden, 1804, 4to.
+  _Gottingen_: Schambach, Hannover, 1838, 8vo. _Hamburg_: Richey, Hamb.
+  1873, 4to; 1755, 8vo. _Henneberg_: Reinwold, Berlin and Stettin, 1793,
+  1801, 8vo, 2 vols.: Bruckner, Meiningen, 1843, 4to. _Hesse_: Vilmar,
+  Marburg, 1868, 8vo, 488 pages. _Holstein_: Schutz Hamb. 1800-1806,
+  8vo, 4 vols. _Hungary_: Schoer, Wien, 1858. _Livonia_: Bergmann,
+  Salisburg, 1785, 8vo: Gutzeit, Riga, 1859-1864, 8vo, 2 parts. _Upper
+  Lusatia_: Anton, Gorlitz, 1825-1839, 13 parts. _Luxembourg_: Gangler,
+  Lux. 1847, 8vo, 406 pages. _Mecklenburg and Western Pomerania_: M.,
+  Leipzig, 1876, 8vo, 114 pages. _Nassau_: Kehrein, Weilburg, 1860, 8vo.
+  _Osnaburg_: Strodtmann, Leipz. 1756, 8vo. _Pomerania and Rugen_:
+  Dahnert, Stralsund, 1781, 4to. _Posen_: Bernd, Bonn, 1820, 8vo.
+  _Prussia_: Bock, Konigsb. 1759, 8vo: Hennig, ib. 1785, 8vo. _Saxony_:
+  Schmeller (from Heliand, &c.), Stuttg. 1840, 4to. _Silesia_: Berndt,
+  Stendal, 1787, 8vo. _Swabia_: Schmid, Berlin, 1795, 8vo; Stuttg. 1831,
+  8vo. _Switzerland_: Stalder, Aarau, 1807-1813, 8vo, 2 vols.
+  _Thuringia_: Keller, Jena, 1819, 8vo. _Transylvania_: Schuller, Prag,
+  1865, 8vo. _Tirol_: Schopf, Innspruck, 1866, 8vo. _Venetian Alps_:
+  Schmeller, Wien, 1854, 8vo. _Vienna_: Hugel, ib. 1873, 8vo.
+  HUNTING.--_Westerwald_: Schmidt, Hadamar, 1800, 8vo; Kehrein,
+  Wiesbaden, 1871, 12mo. SLANG.--_Gauner Sprache_: Schott, Erlangen,
+  1821, 8vo: Grolmann, Giessen, 1822, 8vo: Train, Meissen, 1833, 8vo:
+  Anton, 2nd ed. Magdeburg, 1843, 8vo; 1859: Ave-Lallemant, _Das
+  Deutsche Gaunerthun_, Leipzig, 1858-1862, 8vo, vol. iv. pp. 515-628.
+  _Student Slang_: Vollmann (Burschicoses), Ragaz, 1846, 16mo, 562
+  pages.
+
+
+_Celtic._
+
+ Celtic generally.--Lluyd, Archaeologia Britannica, Oxford, 1707,
+  folio: Bullet, Besancon, 1754-1860, fol. 2 vols.
+
+
+ Irish.--Cormac, bishop of Cashel, born 831, slain in battle 903, wrote
+  a Glossary, _Sanas Cormaic_, printed by Dr Whitley Stokes, London,
+  1862, 8vo, with another, finished in 1569, by O'Davoren, a
+  schoolmaster at Burren Castle, Co. Clare: O'Clery, Lovanii, 1643, 8vo:
+  MacCuirtin (Eng.-Irish), Paris, 1732, 4to: O'Brien, ib. 1768, 4to;
+  Dublin, 1832, 8vo: O'Reilly, 1817, 4to: 1821; ed. O'Donovan, ib. 1864,
+  4to, 725 pages: Foley (Eng.-Irish), ib. 1855, 8vo: Connellan (do.),
+  1863, 8vo.
+
+
+ Gaelic.--Macdonald, Edin. 1741, 8vo: Shaw, London, 1780, 4to, 2 vols.:
+  Allan, Edin. 1804, 4to: Armstrong, London, 1825, 4to: Highland
+  Society, ib. 1828, 4to, 2 vols.: Macleod and Dewar, Glasgow, 1853,
+  8vo.
+
+
+ Manx.--Cregeen, Douglas, 1835, 8vo: Kelly, ib. 1866, 8vo, 2 vols.
+
+
+ Welsh.--LATIN.--Davies, London, 1632, fol.: Boxhornius, Amstelodami,
+  1654, 4to. ENGLISH.--Salesbury, London, 1547, 4to: 1551: Richards,
+  Bristol, 1759, 8vo: Owen (W.), London, 1793-1794, 8vo, 2 vols.; 1803,
+  4to, 3 vols.: Walters, ib. 1794, 4to: Owen-Pughe, Denbigh, 1832, 8vo;
+  3rd ed. Pryse, ib. 1866, 8vo: D. S. Evans (Eng.-Welsh), ib. 1852-1853,
+  8vo; 1887.
+
+
+ Cornish.--Pryce, _Archaeologia_, Sherborne, 1770, 4to: Williams,
+  Llandovery, 1862-1865, 4to. NAMES.--Bannister (20,000), Truro,
+  1869-1871, 8vo.
+
+
+ Breton.--Legadeuc, _Le Catholicon breton_, finished 1464, printed at
+  Lantrequier, 1499, fol. 210 pages; 1501, 4to; L'Orient, 1868, 8vo:
+  Quicquer de Roskoff, Morlaix, 1633, 8vo: Rostrenen, Rennes, 1732, 4to,
+  978 pages; ed. Jolivet, Guingamps, 1834, 8vo, 2 vols.: l'A.[rmerie],
+  Leyde, 1744, 8vo; La Haye, 1756: Lepelletier, Paris, 1752, fol.:
+  Legonidec, Angouleme, 1821, 8vo; St Brieuc, 1847-1850, 4to, 924 pages.
+  DIALECT OF LEON.--Troude (Fr.-Bret.), Brest, 1870, 8vo; Id.
+  (Bret.-Fr.), ib. 1876, 8vo, 845 pages. DIOCESE OF VANNES.--Armerie,
+  Leyde, 1774, 8vo.
+
+
+_Basque._
+
+ Basque.--Larramendi, St Sebastian, 1745, fol. 2 vols.; ed. Zuazua, ib.
+  1854, fol.; Chaho, Bayonne, 1856, 4to, 1867: Fabre, ib. 1870, 8vo: Van
+  Eys, Paris, 1873, 8vo: Eguren, Madrid, 1877.
+
+
+_Baltic._
+
+ Lithuanian.--Szyrwid, 3rd ed., Vilnae, 1642, 8vo; 5th ed. 1713:
+  Schleicher, Prag, 1856-1857, 8vo, 2 vols.: Kurmin, Wilno, 1858, 8vo:
+  Kurschat, Halle, 1870, &c., 8vo.
+
+
+ Lettic.--Mancelius, Riga, 1638, 4to: Elvers, ib. 1748, 8vo: Lange,
+  Mitau, 1777, 4to: Sjogren, Petersburg, 1861, 4to: Ulmann, ed.
+  Bielenstein, Riga, 1872, &c., 8vo.
+
+
+ Prussian.--Bock, Konigsberg, 1759, 8vo: Hennig, ib. 1785, 8vo:
+  Nesselmann, Berlin, 1873, 8vo: Pierson, ib. 1875, 8vo.
+
+
+_Slavonic_.
+
+ Slavonic generally.--Franta-Sumavski (Russ. Bulg. Old Slav. Boh.
+  Polish), Praga, 1857, 8vo, Miklosich, Wien, 1886.
+
+
+ Old Slavonic.--Beruinda, Kiev, 1627, 8vo; Kuteinsk, 1653, 4to:
+  Polycarpi (Slav. Greek, Latin), Mosque, 1704, 4to: Alexyeev, St
+  Petersb. 1773, 8vo; 4th ed. ib. 1817-1819, 8vo, 5 vols.: Russian Imp.
+  Academy, ib. 1847, 4to, 4 vols.: Miklosich, Vindobonae, 1850: 4to;
+  1862-1865, 8vo, Mikhailovski, St Petersb. 1875, 8vo: Charkovski,
+  Warschaw, 1873, 8vo.
+
+
+ Russian.--Russian Academy, St Petersburg, 1789-1794, 4to, 6 vols.;
+  1806-1822, ib. 1869, 8vo, 3 vols.: Dahl, Moskva, 1862-1866, fol. 4
+  vols.; d., ib. 1873, &c., 4to; a 3rd edition, 1903, &c.
+  FRENCH-GERM.-ENG.--Reiff, ib. 1852-1854, 4to. GERMAN,
+  LATIN.--Holterhof, Moskva, 1778, 8vo, 2 vols.; 3rd ed. 1853-1855, 8vo,
+  2 vols.: Weismann, ib. 1731, 4to; 1782, and frequently. FRENCH,
+  GERMAN.--Nordstet, ib. 1780-1782, 4to, 2 vols.: Heym, Moskau,
+  1796-1805, 4to, 4 vols.: Booch-Arkossi and Frey, Leipzig, 1871, &c.,
+  8vo. ENGLISH.--Nordstet, London, 1780, 4to: Grammatin and Parenogo,
+  Moskva, 1808-1817, 4to, 4 vols. FRENCH.--Tatischeff, 2nd ed. St
+  Petersb. 1798, 8vo, 2 vols.; Moskau, 1816, 4to, 2 vols.: Reiff, St
+  Petersb. 1835-1836, 8vo, 2 vols.: Makaroff, ib. 1872, 8vo, 2 vols,
+  1110 pages; 1873-1874, 12mo, 2 vols. GERMAN.--Pawlowski, Riga, 1859,
+  8vo: Lenstrom, Mitau, 1871, 8vo. SWEDISH.--Geitlin, Helsingfors, 1833,
+  12mo: Meurmann, ib. 1846, 8vo. POLISH.--Jakubowicz, Warszawa,
+  1825-1828, 8vo, 2 vols.: Amszejewicz, ib. 1866, 8vo: Szlezigier, ib.
+  1867, 8vo. TECHNICAL.--Grakov (Germ.), St Petersb. 1872, 8vo.
+  NAVAL.--Butakov, ib. 1837. DIALECTS.--_North-west Russia_:
+  Gorbachevski (old language, in Russian), Vilna, 1874, 8vo, 418 pages.
+  _White Russia_: Nosovich (Russian), St Petersburg, 1870, 4to, 760
+  pages. _Red Russia_: Patritzkii (German), Lemberg, 1867, 8vo, 2 vols.
+  842 pages. _Ukraine_: Piskanov (Russian), Odessa, 1873, 4to, 156
+  pages.
+
+
+ Polish.--Linde (explained in Lat. Germ. and 13 Slav dialects),
+  Warszawie, 1807-1814, 4to, 6 vols. 4574 pages.
+  ENGLISH.--[Rykaczewski], _Complete Dictionary_, Berlin, 1849-1851,
+  8vo, 2 vols.: Rykaczewski, Berlin, 1866, 16mo, 1161 pages. FRENCH AND
+  GERMAN.--Troc, Leipz. 1742-1764, 8vo, 4 vols.; 4th ed. ib. 1806-1822,
+  4to, 4 vols.: Bandtke, Breslau, 1806, 8vo, 2 vols.; 1833-1839, 8vo.
+  FRENCH.--Schmidt, Leipzig, 1870, 16mo. RUSSIAN AND GERMAN.--Schmidt
+  (J. A. E.), Breslau, 1834, 8vo. GERMAN.--Mrongovius, Konigsberg, 1765;
+  1835, 4to; 1837: Troianski, Berlin, 1835-1838, 8vo, 2 vols.:
+  Booch-Arkossi, Leipzig, 1864-1868, 8vo, 2 vols.: Jordan, ib. 1866,
+  8vo. ITALIAN.--Plazowski, Warszawa, 1860, 8vo. 2 vols. 730 pages.
+  RUSSIAN.--Potocki, Lipsk, 1873, &c., 12mo.
+
+
+ Wendish.--Matthai, Budissen, 1721, 8vo: Bose, Grimma, 1840, 8vo:
+  Pfuhl, w Budzsinje, 1866, 8vo, 1210 pages. UPPER LUSATIAN.--Pfuhl and
+  Jordan, Leipz. 1844, 8vo. LOWER LUSATIAN.--Zwahr, Spremberg, 1847,
+  8vo.
+
+
+ Czech.--Rohn (Germ. Lat.), Prag, 1780, 4to, 4 vols.: Dobrowski and
+  Hanka, ib. 1802-1821, 4to, 2 vols. LAT. GERM. HUNGAR.--Jungmann,
+  Praze, 1835-1839, 6 vols. 4to, 5316 pages. GERMAN.--Tham, Prag.
+  1805-1807, 8vo, 2 vols.: Sumavski, ib. 1844-1846, 8vo, 2 vols.:
+  Koneney, ib. 1855, 18mo, 2 vols.: Rank (Germ. Boh.), ib. 1860, 16mo,
+  775 pages. TECHNICAL.--Spatny, ib. 1864, 8vo: Kheil (names of goods,
+  Germ. Boh.), ib. 1864, 8vo, 432 pages. HUNTING.--Spatny, ib. 1870,
+  8vo, 137 pages.
+
+
+ South Slavic.--Richter and Ballman, Wien, 1839-1840, 8vo, 2 vols.
+  SERVIAN.--Karajic (Germ. Lat.), ib. 1818, 8vo; 1852: Lavrovski
+  (Russian), St Petersb. 1870, 8vo, 814 pages. BOSNIAN.--Micalia,
+  Laureti, 1649, 8vo. SLOVAK.--Bernolak (Lat. Germ. Hung.), Budae,
+  1825-1827, 8vo, 6 vols.: Loos (Hung. and Germ.), Pest, 1869, &c., 3
+  vols. SLOVENE.--Gutsmann, Klagenfurt, 1789, 4to: Relkovich, Wien,
+  1796, 4to, 2 vols.: Murko, Gratz, 1838, 8vo, 2 vols.: Janezic,
+  Klagenfurt, 1851, 12mo. DALMATIAN.--Ardelio della Bella, Venezia,
+  1728, 8vo; 2nd ed. Ragusae, 1785, 4to: Stulli, ib. 1801-1810, 4to, 2
+  vols. CROATIAN.--Habdelich, Gratz, 1670, 8vo: Sulek, Agram, 1854-1860,
+  8vo, 2 vols. 1716 pages. CARINTHIAN.--Lexer, Leipzig, 1862, 8vo. OLD
+  SERVIAN.--Danitziye (Servian), Belgrad, 1864, 8vo, 3 vols.
+
+
+ Bulgarian.--Daniel (Romaic, Albanian, Rumanian, and Bulgarian),
+  Moschopolis, 1770; Venice, 1802, 4to. ENGLISH.--Morse and Vassiliev,
+  Constantinople, 1860, 8vo. RUSSIAN.--Borogoff, Vienna, 1872, &c., 8vo.
+
+
+_Ugrian._
+
+ Ugrian, Comparative.--Donner, Helsingfors, 1874, 8vo, in progress:
+  Budenz (Ugrian-Magyar), Budapest, 1872-1875, 8vo.
+
+
+ Lappish.--_Manuale_, Holmiae, 1648, 8vo: Fjellstrom, ib. 1738, 8vo:
+  Leem and Sandberg, Havn. 1768-1781, 4to, 2 parts: Lindahl and
+  Oehrling, Holm. 1780, 8vo. NORTH LAPPISH.--Stockfleht, Christiania,
+  1852, 8vo.
+
+
+ Finnish.--Juslenius, Holmiae, 1745, 4to, 567 pages: Renvall, Aboae,
+  1826, 4to, 2 vols.: Europaeus, Helsingissa, 1852-1853, 16mo, 2 vols.
+  742 pages: Lunin, Derpt, 1853, 8vo: Euren, Tavashuus, 1860, 8vo:
+  Ahlman, ib. 1864, 8vo: Wiedemann, St Petersb. 1869, 4to: Godenhjelm
+  (Germ.), Helsingfors, 1871: Lonnrot, Helsingissa, 1874.
+  NAVAL.--Stjerncreutz, ib. 1863, 8vo.
+
+
+ Esthonian.--Hupel, Mitau, 1818, 8vo, 832 pages: Korber, Dorpat, 1860,
+  8vo: Wiedemann, St Petersb. 1869, 4to, 1002 pages: Aminoff
+  (Esth.-Finnish), Helsingissa, 1869, 8vo: Meves (Russian), Riga, 1876,
+  12mo.
+
+
+ Permian.--Rogord (Russian), St Petersb. 1869, 8vo, 420 pages.
+
+
+ Votiak.--Wiedemann, Reval, 1847, 8vo: Ahlquist, Helsingfors, 1856,
+  4to.
+
+
+ Cheremiss.--Budenz, Pest, 1866, 8vo.
+
+
+ Ersa-Mordvine.--Wiedemann, St Petersb. 1865, 4to.
+  MOKSHA-MORDVINE.--Ahlquist, ib. 1862, 8vo.
+
+
+ Magyar.--Szabo, Kassan, 1792, 8vo: Guczor and Fogarazi (Hung.
+  Academy), Pesth, 1862, 8vo, in progress. ENGLISH.--Dallos, Pesth,
+  1860, 8vo. FRENCH.--Kiss, ib. 1844, 12mo, 2 vols.: Karady, Leipz.
+  1848, 12mo: Mole, Pesth, 1865, 8vo, 2 vols. GERMAN.--Schuster, Wien,
+  1838, 8vo: Bloch, Pesth, 1857, 4to, 2 vols.: Ballagi, ib. 1857, 8vo;
+  6th ed. 1905, 8vo, 2 vols.: Loos, ib. 1870, 8vo, 914 pages.
+  ETYMOLOGICAL.--Dankovsky (Lat.-Germ.), Pressburg, 1853, 8vo:
+  Kresznerics (under roots, in Hung.), Budan, 1831-1832, 4to, 2 vols.:
+  Podhorsky (from Chinese roots, in Germ.), Budapest, 1877, 8vo. NEW
+  WORDS.--Kunoss, Pesth, 1836, 8vo; 1844.
+
+
+ Turkish.--ARAB. PERS.--Esaad Effendi, Constantinople, 1802, fol.
+  ROMAIC.--Alexandrides, Vienna, 1812, 4to. POLYGLOTTS.--Pianzola (Ital.
+  Grec. volgare, e Turca), Padova, 1789, 4to: Ciakciak (Ital. Armeno,
+  Turco), Venice, 1804, 4to; 2nd ed. 1829: Azarian (Ellenico, Ital. Arm.
+  Turco), Vienna, 1848, 8vo: Mechitarist Congregation (Ital. Francese,
+  Arm. Turco), ib. 1846, 8vo. LATIN.--Mesgnien-Meninski, Viennae, 1680,
+  fol. 3 vols.; ed. Jenisch and Klezl, ib. 1780-1802, fol. 4 vols.
+  ENGLISH.--Sauerwein, London, 1855, 12mo: Redhouse, ib. 1856, 8vo, 1176
+  pages: Id., Eng. Turkish, ib. 860, 8vo. FRENCH.--Kieffer and Bianchi
+  (Turk.-Fr.), Paris, 1835-1837, 2 vols. 2118 pages: Bianchi (Fr.-Turk.)
+  Paris, 1843-1846, 8vo, 2 vols. 2287 pages; 1850, 8vo, 2 vols.:
+  Mallouf, ib. 1863-1867, 8vo, 2 vols. FRENCH AND GERMAN.--Zenker (Arab.
+  Pers.), Leipz, 1862-1876, 4to, 2 vols, 982 pages. GERMAN.--Korabinsky,
+  Pressburg, 1788, 8vo: Vambery, Constantinople, 1858, 8vo.
+  ITALIAN.--Molina, Roma, 1641, 8vo: Masais, Firenze, 1677, 8vo:
+  Ciadyrgy, Milano, 1832-1834, 4to, 2 vols. RUSSIAN.--Budagov
+  (Comparative lexicon of the Turkish-Tartar dialects), St Petersburg,
+  1869, 8vo, 2 vols.
+
+
+ Gipsy.--Bischoff, Ilmenau, 1827, 8vo: Truxillo, Madrid, 1844, 8vo:
+  Jimenes, Sevilla, 1846, 16mo: Baudrimont, Bordeaux, 1862, 8vo:
+  Vaillant, Paris, 1868, 8vo: Paspati; Constantinople, 1870, 4to:
+  Borrow, _Romany Lavo Lil_, London, 1874, 8vo: Smart and Crofton,
+  London, 1875, 8vo.
+
+
+ Albanian.--Blanchus, Romae, 1635, 8vo: Kaballioti (Romaic, Wallach.
+  Alb.), Venice, 1770, 8vo: Xylander, Frankfurt a. M. 1835, 8vo: Hahn,
+  Jena, 1854, 4to: Rossi da Montalto, Roma, 1866, 8vo.
+
+
+ASIA
+
+ Semitic.--POLYGLOTTS.--Thurneissius, Berolini, 1585, fol.: Thorndike,
+  London, 1635, fol.: Schindler, Pentaglotton, Frankf, ad M. 1653, fol.:
+  Hottinger, Heptaglotton, ib. 1661, fol.: Castellus, London, 1669, fol.
+  2 vols. (Hebrew, Chaldaic, Syriac, Samaritan, Aethiopic and Arabic in
+  one alphabet; Persian separately. It occupied him for seventeen years,
+  during which he worked sixteen to eighteen hours a day): Otho, Frankf.
+  a. M. 1702, 4to (the same languages with Rabbinical).
+
+
+ Hebrew.--About 875, Zemah, head of the school of Pumbeditha, wrote a
+  Talmudical dictionary of words and things, arranged in alphabetical
+  order, which is lost. About 880, Jehudah ben 'Alan, of Tiberias, and
+  Jehudah ibn Koreish, of Tahurt, in Morocco wrote Hebrew dictionaries.
+  Saadia ben Joseph (born 892, died 942), of Fayum, in Upper Egypt,
+  wrote [Hebrew: Kefer Igaron], probably a Hebrew-Arabic dictionary.
+  Menahem ben Jacob Ibn Sar[=u]q (born 910, died about 970), of Tortosa
+  and Cordova, wrote a copious Hebrew dictionary, first printed by
+  Herschell F. Filipowski, Edinburgh, 1855, 8vo, from five MSS. David
+  ben Abraham, of F[=a]s, wrote, in Arabic, a large Hebrew dictionary,
+  the MS. of which, a quarto of 313 leaves on cotton paper, was found
+  about 1830 by A. Firkowitz, of Eupatoria, in the cellar of a Qaraite
+  synagogue in Jerusalem. The age of this work cannot be ascertained.
+  About 1050, Ali ben Suleiman wrote a dictionary in Arabic, on the plan
+  of that of David ben Abraham. The MS. of 429 leaves belongs to
+  Firkowitz. Haja ben Sherira, the famous teacher of the Academy of
+  Pumbeditha, wrote a Hebrew dictionary in Arabic, called _al H[=a]vi_
+  (The Gathering), arranged alphabetically in the order of the last
+  radical letter. This dictionary is lost, as well as that of the
+  Spaniard Isaac ben Saul, of Lucena. Iona ibn Ganah, of Cordova, born
+  about 985, wrote a Hebrew dictionary in Arabic called _Kit[=a]b al
+  Azul_ (Book of Roots). This, as well as a Hebrew translation by Samuel
+  ibn Tab[=o]n, is extant in MS., and was used by Gesenius in his
+  _Thesaurus_. Rabbi David ben Joseph Kimhi died soon after 1232. His
+  lexicon of roots, called [Hebrew: Shorashim], was printed at Naples
+  1490, fol.; Constantinople, 1513, fol.; Naples, 1491, 8vo; Venice,
+  1552; Berolini, 1838, 4to. _Tishbi_ (The Tishbite), by Elijah ben
+  Asher, the Levite, so called because it contained 712 roots, was
+  printed at Isny 1541, 8vo and 4to, and often afterwards.
+  LATIN.--Munster, Basileae, 1523, 8vo; 5 editions to 1564: Zamora,
+  Compluti, 1526, fol.: Pellicanus, Argentorati, 1540, fol.: Reuchlin,
+  Basil, 1556, fol.: Avenarius, Wittebergae, 1568, fol.; auctus, 1589:
+  Pagnini, Lugd. Bat. 1575, fol.; 1577; Genevae, 1614; Buxtorf, Basil.
+  1607, 8vo; 1615, and many other editions: Frey (Lat.-Eng.), 2nd ed.
+  London, 1815, 8vo: Gesenius, _Thesaurus_, Leipz. 1829-1858, 4to, 3
+  vols. ENGLISH.--Bale, London, 1767, 4to: Parkhurst, ib. 1792, 4to:
+  Lee, ib. 1840, 8vo: Gesenius, translated by Robinson, ib. 1844, 8vo;
+  by Tregelles, ib. 1846, 4to: Fuerst, 4th ed. transl. by Davidson, ib.
+  1866, 8vo: 1871, 8vo, 1547 pages. FRENCH.--Leigh, Amst. 1703, 4to:
+  Glaire, Paris, 1830, 8vo; 1843. GERMAN.--Gesenius, Leipzig, 1810-1812,
+  8vo, 2 vols.: Fuerst, ib. 1842, 16mo: ib. 1876, 8vo, 2 vols.
+  ITALIAN.--Modena, Venetia, 1612, 4to; 1640; Coen, Reggio, 1811, 8vo:
+  Fontanella, Venezia, 1824, 8vo. DUTCH.--Waterman, Rotterdam, 1859,
+  &c., 8vo. HUNGARIAN.--Ehrentheil (Pentateuch), Pest, 1868, 8vo.
+  ROMAIC.--Loundes, Melite. 1845, 8vo, 987 pages.
+
+
+ Rabbinical and Chaldee.--Nathan ben Yehiel of Rome wrote in the
+  beginning of the 12th century a Talmudic dictionary, _Aruch_, printed
+  1480 (?), s. l., fol.; Pesaro, 1517, fol.; Venice, 1531; and often:
+  Isaiah ben Loeb, Berlin, wrote a supplement to _Aruch_, vol. i.
+  Breslau, 1830, 8vo; vol. ii. ([Hebrew: L] to [Hebrew: T]), Wien, 1859,
+  8vo: Munster, Basil. 1527, 4to, 1530, fol.: Elijah ben Asher, the
+  Levite, transl. by Fagius, Isnae, 1541, fol.; Venet. 1560: David ben
+  Isaac de Pomis, _Zamah David_, Venet. 1587, fol.: Buxtorf,
+  Basileae, 1639, fol.: ed. Fischer, Leipz. 1866-1875, 4to: Otho,
+  Geneva, 1675, 8vo; Altona, 1757, 8vo: Zanolini, Patavii, 1747, 8vo:
+  Hornheim, Halle, 1807, 8vo: Landau, Prag, 1819-1824, 8vo, 5 vols.:
+  Dessauer, Erlangen, 1838, 8vo: Nork (i.e. Korn), Grimma, 1842, 4to:
+  Schonhak, Warschau, 1858, 8vo, 2 vols. TARGUMS.--Levy, Leipzig,
+  1866-68 4to, 2 vols.; 1875: Id. (Eng.), London, 1869, 8vo, 2 vols.
+  TALMUD.--Lowy (in Heb.), Wien, 1863, 8vo: Levy, Leipzig, 1876, &c.,
+  4to. PRAYER-BOOK.--Hecht, Kreuznach, 1860, 8vo: Nathan, Berlin, 1854,
+  12mo. SYNONYMS.--Pantavitius, Lodevae, 1640, fol. FOREIGN
+  WORDS.--Rabeini, Lemberg, 1857, 8vo, &c. JEWISH-GERMAN.--Callenberg,
+  Halle, 1736, 8vo: Vollbeding, Hamburg, 1808, 8vo: Stern, Munchen,
+  1833, 8vo, 2 vols.: Theile, Berlin, 1842-1843, 8vo, 2 vols.:
+  Ave-Lallemant, _Das deutsche Gaunerthum_, Leipzig, 1858, 8vo, 4 vols.;
+  vol. iv. pp. 321-512.
+
+
+ Ph[oe]nician.--M. A. Levy, Breslau, 1864, 8vo.
+
+
+ Samaritan.--Crinesius, Altdorphi, 1613, 4to: Morini, Parisiis, 1657,
+  12mo: Hilligerus, Wittebergae, 1679, 4to: Cellarius, Cizae, 1682, 4to;
+  Frankof. 1705: Uhlemann, Leipsiae, 1837, 8vo: Nicholls, London, 1859,
+  8vo.
+
+
+ Assyrian.--Norris, London, 1868, 8vo, 3 vols. PROPER NAMES.--Menant,
+  Paris, 1861, 8vo.
+
+
+ Accadian.--Lenormant, Paris, 1875, 8vo.
+
+
+ Syriac.--Joshua ben Ali, a physician, who lived about 885, made a
+  Syro-Arabic lexicon, of which there is a MS. in the Vatican. Hoffmann
+  printed this lexicon from Alif to Mim, from a Gotha MS., Kiel, 1874,
+  4to. Joshua bar Bahlul, living 963, wrote another, great part of which
+  Castelli put into his lexicon. His MS. is now at Cambridge, and, with
+  those at Florence and Oxford, was used by Bernstein. Elias bar
+  Shinaya, born 975, metropolitan of Nisibis, 1009, wrote a Syriac and
+  Arabic lexicon, entitled _Kit[=a]b [=u]t Tarjuman fi Taalem Loghat es
+  S[=u]ri[=a]n_ (Book called the Interpreter for teaching the Language
+  of the Syrians), of which there is a MS. in the British Museum. It was
+  translated into Latin by Thomas a Novaria, a Minorite friar, edited by
+  Germanus, and published at Rome by Obicinus, 1636, 8vo. It is a
+  classified vocabulary, divided in 30 chapters, each containing several
+  sections. Crinesius, Wittebergae, 1612, 4to: Buxforf, Basileae, 1622,
+  4to: Ferrarius, Romae, 1622, 4to: Trost, Cothenis Anhaltor, 1643, 4to:
+  Gutbir, Hamburgi, 1667, 8vo: Schaaf, Lugd. Bat, 1708, 4to: Zanolini,
+  Patavii, 1742, 4to: Castellus, ed. Michaelis, Gottingen, 1788, 4to, 2
+  vols.: Bernstein, Berlin, 1857, &c. fol.: Smith (Robt. Paine), Dean of
+  Canterbury, Oxonii, 1868, &c. fol.: fasc. 1-3 contain 538 pages:
+  Zingerle, Romae, 1873, 8vo, 148 pages.
+
+
+ Arabic.--The native lexicons are very many, voluminous and copious. In
+  the preface to his great Arabic-English lexicon, Lane describes 33,
+  the most remarkable of which are-the _'Ain_, so called from the letter
+  which begins its alphabet, commonly ascribed to al Khalil (who died
+  before A.H. 175 [A.D. 791], aged seventy-four): the _Sihah_ of Jauhari
+  (died 398 [1003]): the _Mohkam_ of Ibn Sidah the Andalusian, who was
+  blind, and died A.H. 458 [A.D. 1066], aged about sixty: the _Asas_ of
+  Zamakhshari (born 467 [1075], died 538 [1144]), "a most excellent
+  repertory of choice words and phrases": the _Lis[=a]n el 'Arab_ of Ibn
+  Mukarram (born 630 [1232], died 711 [1311]); Lane's copy is in 28
+  vols. 4to: the _Kamus_ (The Sea) of Fairuzabadi (born 729 [1328], died
+  816 [1413]),: the _Taj el Arus_, by Murtada Ez Zebadi (born A.D. 1732,
+  died 1791)--the copy made for Lane is in 24 vols. thick 4to. The
+  _Sihah_ was printed Hardervici Getorum, 1774, 4to; Bulak, 1865, fol. 2
+  vols.: _Kamus_, Calcutta, 1817, fol. 2 vols.; Bombay, 1855, fol. 920
+  pages: _Sirr el Lagal_, by Farish esh Shidiac, Tunis, fol. 609 pages:
+  _Muh[=i]t al Muh[=i]t_, by Beitrus Al Bustani Beirut, 1867-1870, 2
+  vols. 4to, 2358 pages (abridged as _Katr Al Muhit_, ib. 1867-1869, 2
+  vols. 8vo, 2352 pages), is excellent for spoken Arabic. PERSIAN.--The
+  _Surah_, by Jumal, Calcutta, 1812-1815, 2 vols. 4to: _Samachsharii
+  Lexicon_, ed. Wetzstein, Leipz. 1845, 4to; 1850: _Muntakhal al
+  Loghat_, Calcutta, 1808; ib. 1836; Lucknow, 1845; Bombay, 1862, 8vo, 2
+  vols.: _Muntaha l'Arabi_, 4 vols. fol. 1840: _Shams al Loghat_,
+  Bombay, 1860, fol. 2 vols. 509 pages. TURKISH.--_Achteri Kabir_,
+  Constantinople. 1827, fol.: _El Kamus_, ib. 1816, fol. 3 vols.;
+  translated by Acan Effendi, Bulak, fol. 3 vols.; _El Sihah_,
+  translated by Al Vani, Constantinople, 1728, fol. 2 vols.: 1755-1756;
+  Scutari, 1802, fol. 2 vols. LATIN.--Raphelengius, Leiden, 1613, fol.:
+  Giggeius, Mediolani, 1632, fol. 4 vols.: Golius Lugd. Bat. 1653, fol.
+  (the best before Lane's): Jahn, Vindobonae, 1802, 8vo: Freytag, Halle,
+  1830-1838, 4 vols. 4to; abridged, ib. 1837, 4to. ENGLISH.--Catafago
+  (Arab.-Eng. and Eng.-Arab.), London, 1858, 8vo, 2 vols.; 2nd ed. 1873,
+  8vo: Lane, London, 1863-1893 (edited after Lane's death, from 1876, by
+  his grandnephew, Stanley Lane-Poole. The Arabic title is _Medd el
+  Kamoos_, meaning either the Flow of the Sea, or The Extension of the
+  Kamus. It was undertaken in 1842, at the suggestion and at the cost of
+  the 6th duke of Northumberland, then Lord Prudhoe, by Mr Lane, who
+  returned to Egypt for the purpose, and lived in Cairo for seven years
+  to study, and obtain copies of, the great MS. lexicons in the
+  libraries of the mosques, few of which had ever been seen by a
+  European, and which were so quickly disappearing through decay,
+  carelessness and theft, that the means of composing such a work would
+  not long have existed). Newman (modern), ib. 1872, 8vo, 2 vols. 856
+  pages. FRENCH.--Ruphy (Fr.-Ar.), Paris, 1802, 4to: Bochtor (do.),
+  Paris, 1828, 4to, 2 vols.; 2nd ed. ib. 1850: Roland de Bussy (Algiers,
+  Fr.-Ar.), Alger, 1835, 16mo: Id., 1836, 8vo; 1839: Berggren (Fr.-vulg.
+  Ar., Syria and Egypt.), Upsala, 1844, 4to: Farhat (Germanos), revu par
+  Rochaid ed Dahdah, Marseille, 1849, 4to: Biberstein Kasimirski, Paris,
+  1846, 8vo, 2 vols.; 1853-1856; 1860, 2 vols. 3032 pages: Marcel
+  (vulgar dialects of Africa), Paris, 1830; 1835, 8vo; 1837; enlarged,
+  1869, 8vo; Paulmier (Algeria), 2nd ed. Paris, 1860, 8vo, 931 pages;
+  1872: Bernard (Egypt), Lyon, 1864, 18mo: Cuche, Beirut, 1862, 8vo;
+  1867: Nar Bey (A. Calfa), 2nd ed. Paris, 1872, 12mo, 1042 pages:
+  Cherbonneau (written language), Paris, 1876, 2 vols. 8vo: Id.
+  (Fr.-Ar.), Paris, 1872, 8vo: Beausier (Algiers, Tunis, legal,
+  epistolary), Alger, 1871, 4to, 764 pages; 1873. GERMAN.--Seyfarth
+  (Algeria), Grimma, 1849, 16mo: Wolff (Mod. Ar.), Leipzig, 1867, 8vo:
+  Wahrmund (do.), Giessen, 1870-1875, 8vo, 4 vols. ITALIAN.--Germano,
+  Roma, 1636, 8vo; (Ar. Lat. It.), Romae, 1639, fol.: _Dizionario_,
+  Bulak. 1824, 4to: Schiaparelli, Firenze, 1871, 4to, 641 pages.
+  SPANISH.--Alcala, Grenada, 1505, 4to: Canes, Madrid, 1787, fol. 3
+  vols. SUFI TECHNICAL TERMS.--Abd Errahin, ed. Sprenger, Calcutta,
+  1845, 8vo. TECHNICAL TERMS OF THE MUSSULMAN SCIENCES.--Abd al Hagg and
+  Gholam Kadir, Calcutta, 1853-1862, 4to, 1593 pages. MEDICAL
+  TERMS.--Pharaon and Bertherand, Paris, 1860, 12mo. MATERIA
+  MEDICA.--Muhammed Abd Allah Shirazi, _Ulfaz Udwiyeh_, translated by
+  Gladwin (Eng. Pers. Hindi), Calcutta, 1793, 4to, 1441 words. NOMS DES
+  VETEMENTS.--Dozy, Amst. 1845, 8vo. WORTER IN ENTGEGENGESETZTEN
+  BEDEUTUNGEN.--Redslob, Gottingen, 1873, 8vo. KORAN.--Willmet (also in
+  Haririum et vitam Timuri), Lugd. Bat. 1784, 4to; Amst. 1790: Fluegel,
+  _Concordantia_, Leipz. 1842, 4to: Penrice, _Dictionary and Glossary_,
+  London, 1873, 4to. EL TABRIZI'S LOGIC.--Mir Abufeth (French), Bulak,
+  1842, 8vo. MALTESE.--Vassali, Romae, 1796, 4to: Falzon (Malt. Ital.
+  Eng.), Malta, _s.a._ 8vo: Vella, Livorno, 1843, 8vo.
+
+
+ Armenian.--Mechitar, Venice, 1749-1769, 4to, 2 vols.: Avedichiam,
+  Surmelian and Aucher (Aukerian), ib. 1836-1837, 4to, 2 vols.: Aucher,
+  ib. 1846, 4to. POLYGLOT.--Villa (Arm.-vulg., litteralis, Lat. Indicae
+  et Gallicae), Romae, 1780. GREEK AND LATIN.--Lazarists, Venice,
+  1836-1837, 4to, 2 vols. 2217 pages. LATIN.--Rivola, Mediolani, 1621,
+  fol.: Nierszesovicz, Romae, 1695, 4to; Villotte, ib. 1714, fol.:
+  Mechitar, Venetiae, 1747-1763, 4to, 2 vols. ENGLISH.--Aucher, Venice,
+  1821-1825, 4to, 2 vols. FRENCH.--Aucher, Venise, 1812-1817, 8vo, 2
+  vols.; (Fr.-Arm. Turc.), ib. 1840, 4to: Eminian, Vienna, 1853, 4to:
+  Calfa, Paris, 1861, 8vo, 1016 pages; 1872. ITALIAN.--Ciakciak,
+  Venezia, 1837, 4to. RUSSIAN.--Khudobashev [Khutapashian], Moskva,
+  1838, 8vo, 2 vols. RUSS. ARM.--Adamdarov, ib. 1821, 8vo: Popov, ib.
+  1841, 8vo, 2 vols. MODERN WORDS.--Riggs, Smyrna, 1847, 8vo.
+
+
+ Georgian.--Paolini (Ital.), Roma, 1629, 4to: Klaproth (Fr.), Paris,
+  1827, 8vo: Tshubinov (Russian, French), St Petersburg, 1840, 4to;
+  1846, 8vo, 2 vols. 1187 pages.
+
+
+ Circassian.--Loewe, London, 1854, 8vo.
+
+
+ Ossetic.--Sjorgen, St Petersb. 1844, 4to.
+
+
+ Kurd.--Garzoni, Roma, 1787, 8vo: Lerch (German), St Petersburg, 1857,
+  8vo: Id. (Russian), ib. 1856-1858, 8vo.
+
+
+ Persian.--_Burhani Qatiu_, arranged by J. Roebuck, Calcutta, 1818,
+  4to: _Burhan i Kati_, Bulak, 1836, fol.: Muhammed Kazim, Tabriz, 1844,
+  fol.: _Haft Kulzum_ (The Seven Seas), by Ghazi ed din Haidar, King of
+  Oude, Lucknow, 1822, fol. 7 vols. ARABIC.--_Shums ul Loghat_,
+  Calcutta, 1806, 4to, 2 vols. TURKISH.--Ibrahim Effendi, _Farhangi
+  Shu'uri_, ib. 1742, fol. 2 vols. 22,530 words, and 22,450 poetical
+  quotations: _Burhan Kati_, by Ibn Kalif, translated by Ahmed Asin
+  Aintabi, ib. 1799, fol.; Bulak, 1836, fol.: Hayret Effendi, ib. 1826,
+  8vo. ARMENIAN.--Douzean, Constantinople, 1826, fol. BENGALI.--Jay
+  Gopal, Serampore, 1818, 8vo. LATIN.--Vullers (Zend appendix), Bonnae
+  ad Rhen, 1855-1868, 4to, 2 vols. 2544 pages; Supplement of Roots,
+  1867, 142 pages. ENGLISH.--Gladwin, Malda in Bengal, 1780, 4to;
+  Calcutta, 1797: Kirkpatrick, London, 1785, 4to: Moises, Newcastle,
+  1794, 4to: Rousseau, London, 1802, 8vo; 1810: Richardson (Arab, and
+  Pers.), ib. 1780-1800, fol. 2 vols.; ed. Wilkins, ib. 1806-1810, 4to,
+  2 vols.; ed Johnson, ib. 1829, 4to: Ramdhen Sen, Calcutta, 1829, 8vo;
+  1831: Tucker (Eng.-Pers.), London, 1850, 4to: Johnson (Pers. and
+  Arab.), ib. 1852, 4to: Palmer, ib. 1876, 8vo, 726 pages.
+  FRENCH.--Handjeri (Pers. Arab. and Turkish), Moscou, 1841, 4to, 3
+  vols. 2764 pages: Berge, Leipzig, 1869, 12mo. GERMAN.--Richardson,
+  translated by Wahl as _Orientalische Bibliotheque_, Lemg, 1788-1792,
+  8vo, 3 vols. ITALIAN.--Angelus a S. Josepho [i.e. Labrosse] (Ital.
+  Lat. Fr.), Amst. 1684, fol.
+
+
+ Old Persian.--(Cuneiform), Benfey (German), Leipzig, 1847, 8vo:
+  Spiegel (id.), ib. 1862, 8vo: Kossovich (Latin), Petropoli, 1872, 8vo.
+
+
+ Zend.--Justi, Leipzig, 1864, 4to: Vullers, Persian Lexicon, Appendix:
+  Lagarde, Leipzig, 1868, 8vo.
+
+
+ Pahlavi.--_An old Pahlavi and Pazend Glossary_, translated by Destur
+  Hoshengi Jamaspji, ed. Haug, London, 1867, 8vo; 1870, 8vo: West,
+  Bombay, 1874, 8vo.
+
+
+  INDIAN TERMS.--_The Indian Vocabulary_, London, 1788, 16mo: Gladwin,
+  Calcutta, 1797, 4to: Roberts, London, 1800, 8vo: Rousseau, ib. 1802,
+  8vo: Roebuck (naval), ib. 1813, 12mo: C. P. Brown, _Zillah Dict._,
+  Madras, 1852, 8vo: Robinson (Bengal Courts), Calcutta, 1854, 8vo;
+  1860: Wilson, London, 1855, 4to: Fallon, Calcutta, 1858, 8vo.
+
+
+ Sanskrit.--Amarasimha (lived before A.D. 1000), _Amarakosha_ Calcutta,
+  1807, 8vo; ib. 1834, 4to; Bombay, 1860, 4to; Lucknow, 1863, 4to;
+  Madras, 1870, 8vo, in Grantha characters; Cottayam, 1873, 8vo, in
+  Malaylim characters; Benares, 1867, fol. with _Amaraviveka_, a
+  commentary by Mahesvara: Rajah Radhakanta Deva, _Sabdakalpadruma_,
+  Calcutta, 1821-1857, 4to, 8 vols. 8730 pages: 2nd ed. 1874, &c.:
+  Bhattachdrya, _Sabdastoma Mahanidhi_, Calcutta, 1869-1870, 8vo, parts
+  i.-vii. 528 pages: _Abhidhanaratnamala_, by Halayudha, ed. Aufrecht,
+  London, 1861, 8vo: VACHASPATYA, by Taranatha Tarkavachaspati,
+  Calcutta, 1873, &c., 4to (parts i.-vii., 1680 pages).
+  BENGALI.--_Sabdasindhu_, Calcutta, 1808: _Amarakosa_, translated by
+  Ramodoyu Bidjalunker, Calcutta, 1831, 4to: Mathurana Tarkaratna,
+  _Sabdasandarbhasindhu_, Calcutta, 1863, 4to. MARATHI.--Ananta Sastri
+  Talekar, Poona, 1853, 8vo, 495 pages: Madhava Chandora, Bombay, 1870,
+  4to, 695 pages. TELUGU.--_Amarakosha_, Madras, 1861, ed. Kala, with
+  _Gurubalala prabodhika_, a commentary, ib. 1861, 4to; with the same,
+  ib. 1875, 4to, 516 pages; with _Amarapadaparijata_ (Sans. and Tel.),
+  by Vavilla Ramasvani Sastri, ib. 1862, 4to; ib. 1863, 8vo; 3rd ed. by
+  Jaganmohana Tarkalankara and Khetramohana, 1872, &c., parts i.-iv. 600
+  pages: Suria Pracasa Row, _Sarva-Sabda-Sambodhini_, ib. 1875, 4to,
+  1064 pages. TIBETAN AND MONGOL.--Schiefner, _Buddhistische Triglotte_,
+  St Petersburg, 1859, fol., the _Vyupatti_ or _Mahavyupatti_ from the
+  _Tanguir_, vol. 123 of the Sutra. LATIN.--Paulinus a Sancto
+  Bartholomeo, Amarasinha, sectio i. de coelo, Romae, 1798, 4to: Bopp.
+  Berlin, 1828-1830, 4to; 2nd ed. 1840-1844; 3rd, 1866, 4to.
+  ENGLISH.--_Amarakosha_, trans. by Colebrooke, Serampore, 1808, 4to;
+  1845, 8vo: Rousseau, London, 1812, 4to: Wilson, Calcutta, 1819, 4to;
+  2nd ed. 1832: ed. Goldstucker, Berlin, 1862, &c., folio, to be in 20
+  parts: Yates, Calcutta, 1846, 4to: Benfey, London, 1865, 8vo: Ram
+  Jasen, Benares, 1871, 8vo, 713 pages: Williams, Oxford, 1872, 4to.
+  ENGLISH-SANSKRIT.--Williams, London, 1851, 4to. FRENCH.--Amarakosha,
+  transl. by Loiseleur Deslongchamps, Paris, 1839-1845, 8vo, 2 vols. 796
+  pages: Burnouf and Leupol, Nancy, 1863-1864, 8vo. GERMAN.--Bohtlingk
+  and Roth, St Petersb. 1853, &c., 4to, 7 vols. to 1875.
+  ITALIAN.--Gubernatis, Torino, 1856, &c. 8vo, unfinished, 2 parts.
+  RUSSIAN.--Kossovich, St Petersburg, 1859, 8vo. ROOTS.--Wilkins,
+  London, 1815, 4to: Rosen, Berolini, 1827, 8vo: Westergaard, Bonnae,
+  1840-1841, 8vo: Vishnu Parasurama Sastri Pandita (Sans. and Marathi),
+  Bombay, 1865, 8vo: Taranatha Tarkavachaspati, _Dhatupadarsa_,
+  Calcutta, 1869, 8vo: Leupol, Paris, 1870, 8vo.
+  SYNONYMS.--_Abhidhanacintamani_, by Hemachadra, ed. Colebrooke,
+  Calcutta, 1807, 8vo; translated by Bohtlingk and Rieu (German), St
+  Petersburg, 1847, 8vo. HOMONYMS.--Medinikara, _Medinikosha_, Benares,
+  1865, 4to; Calcutta, 1869, 8vo; ib. 1872, 8vo. DERIVATIVES.--Hirochand
+  and Rooji Rangit, _Dhatumanjari_, Bombay, 1865, 8vo. TECHNICAL TERMS
+  OF THE NYAYA PHILOSOPHY.--_Nyayakosa_, by Bhimacharya Jhalakikar
+  (Sanskrit), Bombay, 1875, 8vo, 183 pages. RIG VEDA.--Grassmann,
+  Leipzig, 1873-1875, 8vo.
+
+
+ Bengali.--Manoel, Lisboa, 1743, 8vo: Forster, Calcutta, 1799-1802,
+  4to, 2 vols. 893 pages: Carey, Serampore, 1815-1825, 4to, 2 vols.; ed.
+  Marshman, ib. 1827-1828, 8vo, 2 vols.; 3rd ed. ib. 1864-1867, 8vo;
+  abridged by Marshman, ib. 1865, 8vo; ib. 1871, 8vo, 2 vols. 936 pages:
+  Morton, Calcutta, 1828, 8vo: Houghton, London, 1833, 4to: Adea,
+  _Shabdabudhi_, Calcutta, 1854, 604 pages. ENGLISH.--Ram Comul Sen, ib.
+  1834, 4to, 2 vols.; London, 1835, 4to: D'Rozario, Calcutta, 1837, 8vo:
+  Adea, _Abhidan_, Calcutta, 1854, 761 pages. ENGLISH LAT.--Ramkissen
+  Sen, ib. 1821, 4to. ENG.-BENG. AND MANIPURI.--[Gordon], Calcutta,
+  1837, 8vo.
+
+
+ Canarese.--Reeve, Madras, 1824-1832, 4to, 2 vols.; ed. Sanderson,
+  Bangalore, 1858, 8vo, 1040 pages; abridged by the same, 1858, 8vo, 276
+  pages: _Dictionarium Canarense_, Bengalori, 1855, 8vo: _School
+  Dictionary_, Mangalore, 1876, 8vo, 575 pages.
+
+
+ Dardic Languages.--Leitner (Astori, Ghilghiti, Chilasi, and dialects
+  of Shina, viz. Arnyia, Khajuna and Kalasha), Lahore, 1868, 4to.
+
+
+ Guzarati.--(English) Mirza Mohammed Cauzim, Bombay, 1846, 4to;
+  Shapurji Edalji, ib. 1868, 8vo, 896 pages: Karsandas Mulji, ib. 1868,
+  8vo, 643 pages.
+
+
+ Hindi.--Rousseau, London, 1812, 4to: Adam, Calcutta, 1829, 8vo:
+  Thompson, ib. 1846, 8vo: J. D. Bate, London, 1876, 8vo, 809 pages.
+  ENGLISH.--Adam, Calcutta, 1833, 8vo. ENGLISH, URDU AND
+  HINDI.--Mathuraprasada Mirsa, Benares, 1865, 8vo, 1345 pages.
+
+
+ Hindustani.--Ferguson, London, 1773, 4to: Gilchrist, Calcutta, 1800,
+  8vo; ed. Hunter, Edinb. 1810; Lond. 1825: Taylor, Calcutta, 1808, 4to,
+  2 vols.: Gladwin (Persian and Hind.), Calcutta, 1809, 8vo, 2 vols.:
+  Shakespeare, London, 1817, 4to; 1820; 1834; 1849: Forbes, London,
+  1847, 8vo; 1857: Bertrand (French), Paris, 1858, 8vo: Brice, London,
+  1864, 12mo: Fallon, Banaras, 1876, &c., to be in about 25 parts and
+  1200 pages. ENGLISH.--Gilchrist, 1787-1780, 4to, 2 parts: Thompson,
+  Serampore, 1838, 8vo.
+
+
+ Kashmiri.--Elmslie, London, 1872, 12mo.
+
+
+ Khassia.--Roberts, Calcutta, 1875, 12mo.
+
+
+ Malayalim.--Fabricius and Breithaupt, Weperg, 1779, 4to: Bailey,
+  Cottayam, 1846, 8vo: Gundert, Mangalore, 1871, 8vo, 1171 pages.
+
+
+ Marathi.--Carey, Serampore, 1810, 8vo: Kennedy, Bombay, 1824, fol.:
+  Jugunnauth Shastri Kramavant, Bombay, 1829-1831, 4to, 3 vols.:
+  Molesworth, ib. 1831, 4to; 2nd ed. 1847, 4to; ed. Candy, Bombay, 1857,
+  4to, 957 pages; abridged by Baba Padmanji, ib. 1863, 8vo; 2nd ed.
+  (abridged), London, 1876, 8vo, 644 pages. ENGLISH.--Molesworth,
+  Bombay, 1847, 4to.
+
+
+ Oriya.--Mohunpersaud Takoor, Serampore, 1811, 8vo: Sutton, Cuttack,
+  1841-1848, 8vo, 3 vols. 856 pages.
+
+
+ Pali.--Clough, Colombo, 1824, 8vo: Moggallana Thero (a Sinhalese
+  priest of the 12th century), _Abhidhanappika_ (Pali, Eng. Sinhalese),
+  ed. Waskeduwe Subheti, Colombo, 1865, 8vo: Childers, London,
+  1872-1875, 8vo, 658 pages. ROOTS.--Silavansa, _Dhatumanjusa_ (Pali
+  Sing. and Eng.), Colombo, 1872, 8vo.
+
+
+ Prakrit.--Delius, _Radices_, Bonnae ad Rh., 1839, 8vo.
+
+
+ Punjabi.--Starkey, 1850, 8vo; Lodiana Mission, Lodiana, 1854-1860, 444
+  pages.
+
+
+ Pushtu or Afghan.--Dorn, St Petersb. 1845, 4to: Raverty, London, 1860,
+  4to; 2nd ed. ib. 1867, 4to: Bellew, 1867, 8vo.
+
+
+ Sindhi.--Eastwick, Bombay, 1843, fol. 73 pages: Stack, ib. 1855, 8vo,
+  2 vols.
+
+
+ Sinhalese.--Clough, Colombo, 1821-1830, 8vo, 2 vols.: Callaway (Eng.,
+  Portuguese and Sinhalese), ib. 1818, 8vo: Id., _School Dictionary_,
+  ib. 1821, 8vo: Bridgenell (Sinh.-Eng.), ib. 1847, 18mo: Nicholson
+  (Eng.-Sinh.), 1864, 32mo, 646 pages.
+
+
+ Tamil.--Provenza (Portug.), Ambalacotae, 1679, 8vo: _Sadur Agurardi_,
+  written by Beschi in 1732, Madras, 1827, fol.; Pondicherry, 1875, 8vo:
+  Blin (French), Paris, 1834, 8vo: Rottler, Madras, 1834-1841, 4to, 4
+  vols.: Jaffna Book Society (Tamil), Jaffna, 1842, 8vo, about 58,500
+  words: Knight and Spaulding (Eng. Tam.), ib. 1844, 8vo; _Dictionary_,
+  ib. 1852, 4to: Pope, 2nd ed. ib. 1859, 8vo: Winslow, Madras, 1862,
+  4to, 992 pages, 67,452 words.
+
+
+ Telugu.--Campbell, Madras, 1821, 4to: C. P. Brown, Madras (Eng.-Tel.),
+  1852, 8vo, 1429 pages: Id. (Tel.-Eng.), ib. 1852, 8vo, 1319 pages.
+  MIXED TELUGU.--Id., ib. 1854, 8vo.
+
+
+ Thuggee.--Sleeman, Calcutta, 1830, 8vo, 680 Ramasi words.
+
+
+ Indo-Chinese Languages.--Leyden, _Comparative Vocabulary of Barma,
+  Malaya and Thai_, Serampore, 1810, 8vo. _Annamese_: Rhodes (Portug.
+  and Lat.), Romae, 1651, 4to: Pigneaux and Taberd, Fredericinagori,
+  1838, 4to; Legrand de la Liraye, Paris, 1874, 8vo: Pauthier (Chin.
+  Ann.-Fr. Lat.), Paris, 1867, &c., 8vo. _Assamese_: Mrs Cutter, Saipur,
+  1840, 12 mo; Bronson, London, 1876, 8vo, 617 pages. _Burmese_: Hough
+  (Eng.-Burm.), Serampore, 1825, Moulmain, 1845, 8vo, 2 vols. 955 pages:
+  Judson, Calcutta, 1826, 8vo; (Eng. Burm.), Moulmain, 1849, 4to; (Burm.
+  Eng.), ib. 1852, 8vo; 2nd ed., Rangoon, 1866, 8vo, 2 vols. 968 pages:
+  Lane, Calcutta, 1841, 4to. _Cambodian_: Aymonier (Fr.-Camb.), Saigon,
+  1874, 4to; Id. (Camb.-Fr.), ib. 1875, fol. _Karen_: Sau-kau Too
+  (Karen), Tavoy, 1847, 12mo, 4 vols.: Mason, Tavoy, 1840, 4to.
+  _Sgau-Karen_: Wade, ib. 1849, 8vo. _Siamese (Thai)_: Pallegoix (Lat.
+  French, Eng.), Paris, 1854, 4to: _Dictionarium Latinum Thai_, Bangkok,
+  1850, 4to, 498 pages.
+
+
+ Malay.--LATIN.--Haex, Romae, 1631, 4to; Batavia, 1707.
+  DUTCH.--Houtmann (Malay and Malagasy), Amst. 1603, 4to; 1673; 1680;
+  1687; 1703; Batavia, 1707: Wiltens and Dankaarts, Gravenhage, 1623,
+  4to; Amst. 1650; 1677; Batavia, 1708, 4to: Heurnius, Amst. 1640, 4to:
+  Gueynier, Batavia, 1677, 4to; 1708: Loder, ib. 1707-1708, 4to: Van der
+  Worm, ib. 1708, 4to: Roorda van Eysinga (Low), ib. 1824-1825, 8vo, 2
+  vols.; 12th ed. 's Gravenhage, 1863, 8vo; Id. (Hof, Volks en Lagen
+  Taal), ib. 1855, 8vo: Dissel and Lucardie (High Malay), Leiden, 1860,
+  12mo: Pijnappel, Amst. 1863, 8vo: Badings, Schoonhoven, 1873, 8vo.
+  ENGLISH.--Houtmann (Malay and Malagasy), translated by A. Spaulding,
+  London, 1614, 4to: Bowrey, ib. 1701, 4to: Howison, ib. 1801, 4to:
+  Marsden, ib. 1812, 4to: Thomsen, Malacca, 1820, 8vo; 1827: Crawford,
+  London, 1851, 8vo, 2 vols. FRENCH.--Boze, Paris, 1825, 16mo: Elout
+  (Dutch-Malay and French-Malay), Harlem, 1826, 4to: Bougourd, Le Havre,
+  1856, 8vo: Richard, Paris, 1873, 8vo, 2 vols.: Favre, Vienna, 1875,
+  8vo, 2 vols.
+
+
+ Malay Archipelago.--_Batak_: Van der Tuuk, Amsterdam, 1861, 8vo, 564
+  pages. _Bugis_: Mathes, Gravenh. 1874, 8vo, 1188 pages: Thomsen
+  (Eng.-Bugis and Malay), Singapore, 1833, 8vo. _Dyak_: Hardeland
+  (German), Amst. 1859, 8vo, 646 pages. _Javanese_: Senerpont Domis,
+  Samarang, 1827, 4to, 2 vols.: Roorda van Eysinga, Kampen, 1834-1835,
+  8vo, 2 vols.: Gericke, Amst. 1847, 8vo; ed. Taco Roorda, ib. 1871, &c.
+  parts i.-v., 880 pages: Jansz and Klinkert, Samarang, 1851, 8vo; 1865:
+  Favre (French), Vienne, 1870, 8vo. _Macassar_: Matthes, Amst. 1859,
+  8vo, 951 pages. Sunda: De Wilde (Dutch, Malay and Sunda), Amsterdam,
+  1841, 8vo: Rigg (Eng.), Batavia, 1862, 4to, 573 pages. _Formosa_:
+  Happart (Favorlang dialect, written about 1650), Parrapattan, 1840,
+  12mo.
+
+
+ Philippines.--_Bicol_: Marcos, Sampaloc, 1754, fol. _Bisaya_: Sanchez,
+  Manila, 1711, fol.: Bergano, ib. 1735, fol.: Noceda, ib. 1841:
+  Mentrida (also Hiliguena and Haraya) ib. 1637, 4to; 1841, fol. 827
+  pages: Felis de la Encarnacion, ib. 1851, 4to, 2 vols. 1217 pages.
+  _Ibanac_: Bugarin, ib. 1854, 4to. _Ilocana_, Carro, ib. 1849, fol.
+  _Pampanga_: Bergano, ib. 1732, fol. _Tagala_: Santos, Toyabas, 1703,
+  fol.; ib. 1835, 4to, 857 pages: Noceda and San Lucar, Manila, 1754,
+  fol.; 1832.
+
+
+ Chinese.--Native Dictionaries are very numerous. Many are very copious
+  and voluminous, and have passed through many editions. _Shwo wan_, by
+  Hu Shin, is a collection of the ancient characters, about 10,000 in
+  number, arranged under 540 radicals, published 150 B.C., usually in 12
+  vols.: _Yu pien_, by Ku Ye Wang, published A.D. 530, arranged under
+  542 radicals, is the basis of the Chinese Japanese Dictionaries used
+  in Japan: _Ping tseu loui pien_, Peking, 1726, 8vo, 130 vols.: _Pei
+  wan yun fu_ (Thesaurus of Literary Phrases), 1711, 131 vols. 8vo,
+  prepared by 66 doctors of the Han lin Academy in seven years. It
+  contains 10,362 characters, and countless combinations of two, three
+  or four characters, forming compound words and idioms, with numerous
+  and copious quotations. According to Williams (_On the word Shin_, p.
+  79), an English translation would fill 140 volumes octavo of 1000
+  pages each. _Kanghi tsze tien_ (Kanghi's Standard or Canon of the
+  Character), the dictionary of Kanghi, the first emperor of the present
+  dynasty, was composed by 30 members of the Han lin, and published in
+  1716, 40 vols. 4to, with a preface by the emperor. It contains 49,030
+  characters, arranged under the 214 radicals. It is generally in 12
+  vols., and is universally used in China, being the standard authority
+  among native scholars for the readings as well as the meanings of
+  characters. LATIN.--De Guignes (French, Lat.), Paris, 1813, fol.;
+  Klaproth, Supplement, 1819; ed. Bazil (Latin), Hong-Kong, 1853, 4to:
+  Goncalves (Lat.-Chin.), Macao, 1841, fol.: Callery, _Systema
+  phoneticum_, Macao, 1841, 8vo: Schott, _Vocabularium_, Berlin, 1844,
+  4to. ENGLISH.--Raper, London, 1807, fol. 4 vols.: Morrison, Macao,
+  1815-1823, 4to, 3 parts in 6 vols.: Medhurst, Batavia, 1842-1843, 8vo,
+  2 vols.: Thom, Canton, 1843, 8vo: Lobscheid, Hong-Kong, 1871, 4to:
+  Williams, Shanghai, 1874, 4to. ENG. CHINESE.--Morrison, part iii.:
+  Williams, Macao, 1844, 8vo: Medhurst, Shanghai, 1847-1848, 8vo, 2
+  vols.: Hung Maou, _Tung yung fan hwa_ (Common words of the Red-haired
+  Foreigners), 1850, 8vo. Doolittle, Foochow, 1872, 4to, vol. i. 550
+  pages. FRENCH,--Callery, _Dict. encyclopedique_, Macao and Paris, 1845
+  (radicals 1-20 only): M. A. H., 1876, 8vo, autographie, 1730 pages.
+  FRENCH-CHIN.--Perny (Fr.-Latin, Spoken Mandarin), Paris, 1869, 4to;
+  Appendice, 1770; Lemaire and Giguel, Shanghai, 1874, 16mo.
+  PORTUGUESE.--Goncalves (Port.-Chin.), Macao, 1830, 8vo, 2 vols.: Id.
+  (Chin.-Port.), ib. 1833, 8vo. IDIOMS.--Giles, Shanghai, 1873, 4to.
+  PHRASES.--Yaou Pei-keen, _Luy yih_, 1742-1765, 8vo, 55 vols.: Tseen
+  Ta-hin, _Shing luy_, 1853, 8vo, 4 vols. CLASSICAL EXPRESSIONS.--Keang
+  Yang and 30 others, _Sze Shoo teen Lin_, 1795, 8vo, 30 vols. ELEGANT
+  EXPRESSIONS.--Chang ting yuh, _Fun luy tsze kin_, 1722, 8vo, 64 vols.
+  PHRASES OF THREE WORDS.--Julien (Latin), Paris, 1864, 8vo.
+  POETICAL.--_Pei wan she yun_, 1800, 8vo, 5 vols. PROPER NAMES.--F.
+  Porter Smith (China, Japan, Corea, Annam, &c., Chinese-Eng.),
+  Shanghai, 1870, 8vo. TOPOGRAPHY.--Williams, Canton, 1841, 8vo. NAMES
+  OF TOWNS.--Biot, Paris, 1842, 8vo. ANCIENT CHARACTERS.--Foo
+  Lwantseang, _Luh shoo fun luy_, 1800, 8vo, 12 vols. SEAL
+  CHARACTER.--Heu Shin, _Shwo wan_, ed. Seu Heuen, 1527, 8vo, 12 vols.
+  RUNNING HAND.--St Aulaire and Groeneveld (Square Characters, Running
+  Hand; Running, Square), Amst. 1861, 4to, 117 pages. TECHNICAL TERMS
+  (in Buddhist translations from Sanskrit)--Yuen Ying, _Yih 'see king
+  pin e_, 1848, 8vo. DIALECTS.--_Amoy_: Douglas, London, 1873, 4to, 632
+  pages: Macgowan, Hong-Kong, 1869, 8vo. _Canton_: Yu Heo-poo and Wan
+  ke-shih, _Keang hoo chih tuh fun yun tso yaou ho tseih_, Canton, 1772,
+  8vo, 4 vols.; 1803, 8vo, 4 vols.; Fuh-shan, 1833, 8vo, 4 vols.:
+  Morrison, Macao, 1828, 8vo: Wan ke shih, Canton, 1856, 8vo: Williams
+  (tonic, Eng.-Chinese), Canton, 1856, 8vo: Chalmers, Hong-Kong, 1859,
+  12mo; 3rd ed. 1873, 8vo. _Changchow in Fuhkeen_: Seay Sew-lin, _Ya suh
+  tung shih woo yin_, 1818, 8vo, 8 vols.; 1820. _Foo-chow_: Tseih (a
+  Japanese general) and Lin Peih shan, _Pa yin ho ting_, ed. Tsin Gan,
+  1841, 8vo: Maclay and Baldwin, Foochow, 1870, 8vo, 1123 pages.
+  _Hok-keen_: Medhurst, Macao, 1832, 4to: _Peking_, Stent, Shanghai,
+  1871, 8vo.
+
+
+ Corean.--CHINESE, COREAN AND JAPANESE.--_Cham Seen Wo Kwo tsze mei_,
+  translated by Medhurst, Batavia, 1835, 8vo. RUSSIAN.--Putzillo, St
+  Petersburg, 1874, 12mo, 746 pages.
+
+
+ Japanese.--_Sio Ken Zi Ko_ (Examination of Words and Characters),
+  1608, 8vo, 10 vols.: _Wa Kan Won Se Ki Sio Gen Zi Ko_, lithographed by
+  Siebold, Lugd. Bat., 1835, fol. JAP.-CHINESE.--_Faga biki set yo siu_.
+  CHINESE-JAP.--_Kanghi Tse Tein_, 30 vols. 12mo: _Zi rin gioku ben_.
+  DUTCH DICTIONARIES PRINTED BY JAPANESE.--_Nieeu verzameld Japansch en
+  Hollandsch Woordenbock_, by the interpreter, B. Sadayok, 1810:
+  Minamoto Masataka, Prince of Nakats (Jap. Chinese-Dutch), 5 vols. 4to,
+  printed at Kakats by his servants: _Jedo-Halma_ (Dutch-Jap.), Jedo,
+  4to, 20 vols.: _Nederduitsche taal_, Dutch Chinese, for the use of
+  interpreters. LATIN AND PORTUGUESE.--Calepinus, _Dictionarium_,
+  Amacusa, 1595, 4to. LATIN.--Collado, _Compendium_, Romae, 1632, 4to:
+  _Lexicon_, Romae, 1870, 4to, from Calepinus. ENGLISH.--Medhurst,
+  Batavia, 1830, 8vo: Hepburn, Shanghai, 1867, 8vo; 1872.
+  ENG.-JAP.--Hori Tatnoskoy, Yedo, 1862, 8vo; 2nd ed. Yeddo, 1866, 8vo:
+  Satow and Ishibashi Masakata (spoken language), London, 1876, 8vo.
+  FRENCH.--Rosny (Jap. Fr. Eng.), Paris, 1857, 4to, vol. i.: Pages,
+  Paris, 1869, 4to, translated from Calepinus. FR.-JAP.--Soutcovey,
+  Paris, 1864, 8vo. FR. ENG. JAP.--Mermet de Cachon, Paris, 1866, 8vo,
+  unfinished. GERMAN.--Pfizmaier (Jap.-Ger., Eng.), Wien, 1851, 4to,
+  unfinished. SPANISH.--_Vocabulario del Japon_, Manila, 1630, 4to,
+  translated from the next. PORTUGUESE.--_Vocabulario da Lingua de
+  Japam_, Nagasaki, 1603, 4to. RUSSIAN.--Goshkevich, St Petersburg,
+  1857, 8vo, 487 pages. CHINESE CHARACTERS WITH JAPANESE
+  PRONUNCIATION.--Rosny, Paris, 1867, 8vo. CHINESE AND JAPANESE NAMES OF
+  PLANTS.--Hoffmann, Leyde, 1864, 8vo.
+
+
+ Aino.--Pfizmaier, Wien, 1854, 4to.
+
+
+ Northern and Central Asia.--_Buriat_: Castren, St Petersburg, 1857,
+  8vo. _Calmuck_: Zwick, Villingen, 1853, 4to: Smirnov, Kazan, 1857,
+  12mo: Jugl, _Siddhi Kur_, Leipzig, 1866, 8vo. _Chuvash_: Clergy of the
+  school of the Kazan Eparchia, Kazan, 1836, 8vo, 2481 words: Lyule
+  (Russ.-Chuv. French), Odessa, 1846, 8vo, 244 pages: Zolotnitski,
+  Kazan, 1875, 8vo, 287 pages. _Jagatai_: Mir Ali Shir, _Abuska_, ed.
+  Vambery, with Hungarian translation, Pesth, 1862, 8vo: Vambery,
+  Leipzig, 1867, 8vo: Pavet de Courteille, Paris, 1870, 8vo. _Koibal and
+  Karagas_: Castren, St Petersburg, 1857, 8vo. _Manchu_: _Yutchi tseng
+  ting tsing wen kian_ (Manchu Chinese), 1771, 4to, 6 vols.: _Sze li hoh
+  pik wen kian_ (Manchu-Mongol, Tibetan, Chinese) 10 vols. 4to, the
+  Chinese pronunciation represented in Manchu: _San hoh pien lan_
+  (Manchu-Chinese, Mongol), 1792, 8vo, 12 vols.;--all three classed
+  vocabularies: Langles (French), Paris, 1789-1790, 4to, 3 vols.:
+  Gabelentz (German), Leipzig, 1864, 8vo: Zakharov (Russian), St
+  Petersburg, 1875, 8vo, 1235 pages: _Mongol_: I. J. Schmidt (German,
+  Russian), St Petersburg, 1835, 4to: Schergin, Kazan, 1841, 8vo:
+  Kovalevski, Kasan, 1844-1849, 4to, 3 vols. 2703 pages. _Ostiak_:
+  Castren, St Petersb. 1858, 8vo. _Samoyed_: Castren, St Petersb. 1855,
+  8vo, 308 pages. Tartar: Giganov (Tobolsk), St Petersburg, 1804, 4to;
+  (Russ.-Tartar), ib. 1840, 4to: Troyanski (Karan), Kasan, 1835-1855,
+  4to. _Tibetan_: _Minggi djamtoo_ (Tibet-Mongol): _Bodschi dajig togpar
+  lama_: _Kad shi schand scharwi melonggi jige_
+  (Manchu-Mongol-Tibetan-Chinese), Kanghi's Dictionary with the Tibetan
+  added in the reign of Khian lung (1736-1795); Csoma de Koros (Eng.),
+  Calcutta, 1834, 4to: I. J. Schmidt (German), St Petersburg, 1841, 4to:
+  Id. (Russian), ib. 1843, 4to: Jaeschke (Eng.), London, 1870, 8vo, 160
+  pages: Id. (Germ.), Gnadau, 1871, 658 pages: (Bhotanta), Schroeter,
+  Serampore, 1826, 4to. _Tungusian_: Castren, St Petersburg, 1856, 8vo,
+  632 pages. _Uigur_: Vambery, Innspruck, 1870, 4to. _Yakut_: Bohtlingk,
+  ib. 1854, 4to, 2 vols. _Yenissei Ostiak_: Castren, ib. 1849, 8vo.
+
+
+AFRICA
+
+ Egyptian.--Young (enchorial), London, 1830-1831, 8vo: Sharpe, London,
+  1837, 4to: Birch, London. 1838, 4to: Champollion (died March 4, 1832),
+  _Dictionnaire egyptien_, Paris, 1841, 4to: Brugsch,
+  _Hieroglyphisch-Demotisches Worterbuch_, Leipzig, 1867-1868, 4to, 4
+  vols. 1775 pages, nearly 4700 words, arranged according to the
+  hieroglyphic alphabet of 28 letters: Pierret, _Vocabulaire hierog._,
+  Paris, 1875, 8vo, containing also names of persons and places: Birch,
+  in vol. v. pp. 337-580 of Bunsen's _Egypt's Place_, 2nd ed. London,
+  1867, &c. 8vo, 5010 words. PROPER NAMES.--Brugsch, Berlin, 1851, 8vo,
+  726 names: Parthey, ib. 1864, 8vo, about 1500 names: Lieblein,
+  Christiania, 1871, 8vo, about 3200 from hieroglyphic texts. BOOK OF
+  THE DEAD.--Id., Paris, 1875, 12mo.
+
+
+ Coptic.--Veyssiere de la Croze, Oxon. 1775, 8vo: Rossi, Romae, 1807,
+  4to: Tattam, Oxon. 1855, 8vo: Peyron, 1835, 4to (the standard):
+  Parthey, Berolini, 1844, 8vo.
+
+
+ Ethiopic.--Wemmer, Romae, 1638, 4to: Ludolf, London, 1661, 4to:
+  Francof. ad M., 1699, fol.: Dillmann (Tigre appendix), Leipzig,
+  1863-1865, 4to, 828 pages.
+
+
+ Amharic.--Ludolphus, Franc. ad Maenum, 1698, fol.: Isenberg, London,
+  1841, 4to, 442 pages. _Tigre_: Munzinger, Leipzig, 1865, 8vo:
+  Beurmann, ib. 1868, 8vo.
+
+
+ East Coast.--_Dankali_: Isenberg, London, 1840, 12mo. _Galla_: Krapf,
+  London, 1842, 8vo: Tutschek, Munchen, 1844, 8vo. _Engutuk Iloigob_:
+  Erhardt, Ludwigsberg, 1857, 8vo. _Kisuaheli_: _Vocabulary of the
+  Soahili_, Cambridge, U.S. 1845, 8vo: Steere, London, 1870, 8vo, about
+  5800 words. _Kisuaheli, Kinika, Kikamba, Kipokono, Kikian, Kigalla_:
+  Krapf, Tubingen, 1850, 8vo.
+
+
+ Malagasy.--Houtmann (Malaysche en Madagask Talen), Amst. 1603, 2nd ed.
+  Matthysz, ib. 1680, 8vo: Huet de Froberville, Isle de France, fol. 2
+  vols.: Flacourt, Paris, 1658, 8vo: Challand (Southern), Isle de
+  France, 1773, 4to: Freeman and Johns, London, 1835, 8vo, 2 vols.:
+  Dalmont (Malgache, Salalave, et Betsimara), 1842, 8vo: Kessler,
+  London, 1870, 8vo.
+
+
+ Southern Africa.--Bleek, _The Languages of Mozambique_, London, 1856,
+  8vo. _Kaffre_: Bennie, Lovedale, 1826, 16mo: Ayliffe, Graham's Town,
+  1846, 12mo: Appleyard, 1850, 8vo: Bleek, Bonn, 1853, 4to, 646 pages.
+  _Zulu-Kaffre_: Perrin (Kaffre-Eng.), London, 1855, 24mo, 172 pages:
+  Id. (Eng.-Kaffre), Pietermaritzburg, 1855, 24mo, 227 pages: Id.
+  (Eng.-Zulu), ib. 1865, 12mo, 226 pages: Dohne, Cape Town, 1857, 8vo,
+  428 pages: Colenso, Pietermaritzburg, 1861, 8vo, 560 pages, about 8000
+  words. _Hottentot_: Bleek, Cape Town, 1857, 4to, 261 pages. _Namaqua_:
+  Tindall, ib. 1852, 8vo: _Vocabulary_, Barmen, 1854, 8vo: Hahn,
+  Leipzig, 1870, 12mo. Sechuana: Casalis, Paris, 1841, 8vo. _Herero_:
+  Hahn, Berlin, 1857, 8vo, 207 pages, 4300 words.
+
+
+ Western Africa.--_Akra_ or _Ga_: Zimmermann, Stuttgart, 1858, 8vo, 690
+  pages. _Ashantee_: Christaller (also Akra), Basel, 1874, 8vo, 299
+  pages. _Bullom_: Nylander, London, 1814, 12mo. _Bunda or Angola_:
+  Cannecatim, Lisboa, 1804, 4to, 722 pages. _Dualla Grammatical
+  Elements_, &c., Cameroons, 1855, 8vo. _Efik_ or _Old Calabar_:
+  Waddell, Old Calabar, 1846, 16mo, 126 pages; Edinb, 1849, 8vo, 95
+  pages. _Eyo_: Raban, London, 1830-1831, 12mo, 2 parts. _Grebo_:
+  _Vocabulary_, Cape Palmas, 1837, 8vo; _Dictionary_, ib. 1839, 8vo, 119
+  pages. _Ifa_: Schlegel, Stuttgart, 1857, 8vo. _Mpongwe_: De Lorme
+  (Franc.-Pongoue), Paris, 1876, 12mo, 354 pages. _Oji_: Riis, Basel,
+  1854, 8vo, 284 pages. _Sherbro'_: Schon, _s. a. et l._ 8vo, written in
+  1839, 42 pages. _Susu_: Brunton, Edinburgh, 1802, 8vo, 145 pages.
+  _Vei_: Koelle, London, 1854, 8vo, 266 pages. _Wolof and Bambarra_:
+  Dard, Paris, 1825, 8vo. _Wolof_: Roger, ib. 1829, 8vo: Missionnaires
+  de S. Esprit, Dakar, 1855, &c. 16mo. Faidherbe (French-Wolof, Poula
+  and Soninke), St Louis, Senegambia, 1860, 12mo. _Yoruba_: Crowther,
+  London, 1843, 8vo; 1852, 298 pages: Vidal, ib. 1852, 8vo: Bowen,
+  Washington, 1858, 4to.
+
+
+ Central Africa.--Barth, _Vocabularies_. Gotha, 1862-1866, 4to. _Bari_:
+  Mitterreutzner, Brixen, 1867, 8vo: Reinisch, Vienna, 1874, 8vo.
+  _Dinka_: Mitterreutzner, Brixen, 1866, 8vo. _Haussa_: Schon (Eng.),
+  London, 1843, 8vo.
+
+
+ Berber.--Venture de Paradis, Paris, 1844, 8vo: Brosselard, ib. 1844,
+  8vo: Delaporte, ib. 1844, 4to, by order of the Minister of War:
+  Creusat, Franc.-Kabyle (Zouaoua), Alger, 1873, 8vo. _Siwah_: Minutoli,
+  Berlin, 1827, 4to.
+
+
+AUSTRALIA AND POLYNESIA
+
+ Australia.--_New South Wales_: Threlkeld (Lake Macquarie Language),
+  Sydney, 1834, 8vo. _Victoria_: Bunce, Melbourne, 1856, 12mo, about
+  2200 words. _South Australia_: Williams, South Australia, 1839, 8vo:
+  Teichelmann and Schurmann, Adelaide, 1840, 8vo: Meyer, ib. 1843, 8vo.
+  _Murray River_: Moorhouse, ib. 1846, 8vo. _Parnkalla_: Schurmann,
+  Adelaide, 1844, 8vo. _Woolner District_: _Vocabulary_, ib. 1869, 12mo.
+  _Western Australia_: Sir George Grey, Perth, 1839, 4to; London, 1840,
+  8vo: Moore, ib. 1843: Brady, Roma, 1845, 24mo, 8vo, 187 pages.
+  _Tasmania_: Millegan, Tasmania, 1857.
+
+
+ Polynesia.--Hale, _Grammars and Vocabularies of all the Polynesian
+  Languages_, Philadelphia, 1846, 4to. _Marquesas, Sandwich Gambier_:
+  Mosblech, Paris, 1843, 8vo. _Hawaiian_: Andrews, _Vocabulary_,
+  Lahainaluna, 1636, 8vo: Id., _Dictionary_, Honolulu, 1865, 8vo, 575
+  pages, about 15,500 words. _Marquesas_: Pierquin, de Gembloux,
+  Bourges, 1843, 8vo: Buschmann, Berlin, 1843, 8vo. _Samoan_:
+  _Dictionary_, Samoa, 1862, 8vo. _Tahitian_: _A Tahitian and English
+  Dictionary_, Tahiti, 1851, 8vo, 314 pages. _Tonga_: Rabone, Vavau,
+  1845, 8vo. _Fijian_: Hazlewood (Fiji-Eng.), Vewa. 1850, 12mo: Id.
+  (Eng.-Fiji), ib. 1852, 12mo: Id., London, 1872, 8vo. _Maori_: Kendall,
+  1820, 12mo: Williams, Paihia, 1844, 8vo; 3rd ed. London, 1871, 8vo:
+  Taylor, Auckland, 1870, 12mo.
+
+
+AMERICA
+
+ North America.--_Eskimo_: Washington, London, 1850, 8vo: Petitot
+  (Mackenzie and Anderson Rivers), Paris, 1876, 4to. _Kinai_: Radloff,
+  St Petersburg, 1874, 4to. _Greenland_: Egede (Gr. Dan. Lat., 3 parts),
+  Hafn, 1750, 8vo; 1760, Fabricius, Kjobenhavn, 1804, 4to. _Hudson's Bay
+  Indians_: Bowrey, London, 1701, fol. _Abnaki_: Rasles, Cambridge,
+  U.S., 1833, 4to. _Chippewa_: Baraga, Cincinnati, 1853, 12mo, 622
+  pages: Petitot, Paris, 1876, 4to, 455 pages. _Massachusetts_ or
+  _Natick_: Cotton, Cambridge, U.S. 1829, 8vo. _Onondaga_: Shea
+  (French-Onon.), from a MS. (of 17th century), London, 1860, 4to, 109
+  pages. _Dacota_: Riggs, New York, 1851, 4to, 424 pages: Williamson
+  (Eng. Dac.), Santos Agency, Nebraska, 12mo, 139 pages. _Mohawk_:
+  Bruyas, New York, 1863, 8vo. _Hidatsa (Minnetarees, Gros Ventres of
+  the Missouri)_: Matthews, ib. 1874, 8vo. _Choctaw_: Byington, ib.
+  1852, 16mo. _Clallam and Lummi_: Gibbs, ib. 1863, 8vo. _Yakama_:
+  Pandosy, translated by Gibbs and Shea, ib. 1862, 8vo. _Chinook_:
+  Gibbs, New York, 1863, 4to. _Chinook Jargon, the trade language of
+  Oregon_: Id., ib. 1863, 8vo. _Tatche_ or _Telame_: Sitjar, ib. 1841,
+  8vo.
+
+
+ Mexico and Central America.--_Tepehuan_: Rinaldini, Mexico, 1743, 4to.
+  _Cora_: Ortega, Mexico, 1732, 4to. _Tarahumara_: Steffel, Brunn, 1791,
+  8vo. _Otomi_: Carochi, Mexico, 1645, 4to: Neve y Molina, ib. 1767,
+  8vo: Yepes, ib. 1826, 4to: Piccolomini, Roma, 1841, 8vo. _Mexican_ or
+  _Aztec_: Molina, Mexico, 1555, 4to; 1571, fol. 2 vols.: Arenas, ib.
+  1583; 1611, 8vo; 1683; 1725; 1793, 12mo: Biondelli, Milan, 1869, fol.
+  _Mexican, Tontonacan, and Huastecan_: Olmos, Mexico, 1555-1560, 4to, 2
+  vols. _Huastecan_: Tapia Zenteno, ib. 1767, 4to, 128 pages. _Opata_ or
+  _Tequima_: Lombardo, ib. 1702, 4to. _Tarasca_: Gilberti, ib. 1559,
+  4to: Lagunas, ib. 1574, 8vo. _Mixtecan_: Alvarado, Mexico, 1593, 4to.
+  _Zapoteca_: Cordova, ib. 1578, 4to. _Maya_: Beltran de Santa Rosa
+  Maria, ib. 1746, 4to; Merida de Yucatan, 1859, 4to, 250 pages:
+  Brasseur de Bourbourg, Paris, 1874, 8vo, 745 pages. _Quiche_: Id.
+  (also Cakchiquel and Trutuhil dialects), ib. 1862, 8vo.
+
+
+ South America.--_Chibcha_: Uricoechea, Paris, 1871, 8vo. _Chayma_:
+  Tauste, Madrid, 1680, 4to: Yanguas, Burgos, 1683, 4to. _Carib_:
+  Raymond, Auxerre, 1665-1666, 8vo. _Galibi_: D.[e]. L.[a] S.[auvage],
+  Paris, 1763, 8vo. _Tupi_: Costa Rubim, Rio de Janeiro, 1853, 8vo:
+  Silva Guimaraes, Bahia, 1854, 8vo: Diaz, Lipsia, 1858, 16mo.
+  _Guarani_: Ruiz de Montoyo, Madrid, 1639, 4to; 1640; 1722, 4to; ed.
+  Platzmann, Leipzig, 1876, &c., 8vo, to be in 4 vols. 1850 pages.
+  _Moxa_: Marban, Lima, 1701, 8vo. _Lule_: Machoni de Corderia, Madrid,
+  1732, 12mo. _Quichua_: Santo Thomas, Ciudad de los Reyes, 1586, 8vo:
+  Torres Rubio, Sevilla, 1603, 8vo; Lima, 1609, 8vo; ed. Figueredo,
+  Lima, 1754, 8vo; Holguin, Ciudad de los Reyes, 1608, 8vo: Tschudi,
+  Wien, 1853, 8vo, 2 vols.: Markham, London, 1864, 8vo: Lopez, _Les
+  Races Aryennes de Perou_, Paris, 1871, 8vo, comparative vocabulary,
+  pp. 345-421. _Aymara_: Bertonio, Chicuyto, 1612, 4to, 2 vols.
+  _Chileno_: Valdivia (also Allentiac and Milcocayac), Lima, 1607, 8vo:
+  Febres, ib. 1765, 12mo; ed. Hernandez y Caluza, Santiago, 1846, 8vo, 2
+  vols. _Tsonecan_ (Patagonian): Schmid, Bristol, 1860, 12mo.
+
+The above article incorporates the salient features of the 9th-edition
+article by the Rev. Ponsonby A. Lyons, and the 10th-edition article by
+Benjamin E. Smith.
+
+
+FOOTNOTES:
+
+  [1] Joannes de Garlandia (John Garland; fl. 1202-1252) gives the
+    following explanation in his _Dictionarius_, which is a classed
+    vocabulary:--"Dictionarius dicitur libellus iste a dictionibus magis
+    necessariis, quas tenetur quilibet scolaris, non tantum in scrinio de
+    lignis facto, sed in cordis armariolo firmiter retinere." This has
+    been supposed to be the first use of the word.
+
+  [2] An excellent dictionary of quotations, perhaps the first of the
+    kind; a large folio volume printed in Strassburg about 1475 is
+    entitled "Pharetra auctoritates et dicta doctorum, philosophorum, et
+    poetarum continens."
+
+  [3] This volume was issued with a new title-page as _Glossaire du
+    moyen age_, Paris, 1872.
+
+
+
+
+DICTYOGENS (Gr. [Greek: diktyon], a net, and the termination [Greek:
+-genes], produced), a botanical name proposed by John Lindley for a
+class including certain families of Monocotyledons which have net-veined
+leaves. The class was not generally recognized.
+
+
+
+
+DICTYS CRETENSIS, of Cnossus in Crete, the supposed companion of
+Idomeneus during the Trojan War, and author of a diary of its events.
+The MS. of this work, written in Phoenician characters, was said to have
+been found in his tomb (enclosed in a leaden box) at the time of an
+earthquake during the reign of Nero, by whose order it was translated
+into Greek. In the 4th century A.D. a certain Lucius Septimius brought
+out _Dictys Cretensis Ephemeris belli Trojani_, which professed to be a
+Latin translation of the Greek version. Scholars were not agreed whether
+any Greek original really existed; but all doubt on the point was
+removed by the discovery of a fragment in Greek amongst the papyri found
+by B. P. Grenfell and A. S. Hunt in 1905-1906. Possibly the Latin
+Ephemeris was the work of Septimius himself. Its chief interest lies in
+the fact that (together with Dares Phrygius's _De excidio Trojae_) it
+was the source from which the Homeric legends were introduced into the
+romantic literature of the middle ages.
+
+  Best edition by F. Meister (1873), with short but useful introduction
+  and index of Latinity; see also G. Korting, _Diktys und Dares_
+  (1874), with concise bibliography; H. Dunger, _Die Sage vom
+  trojanischen Kriege in den Bearbeitungen des Mittelalters und ihren
+  antiken Quellen_ (1869, with a literary genealogical table); E.
+  Collilieux, _Etude sur Dictys de Crete et Dares de Phrygie_ (1887),
+  with bibliography; W. Greif, "Die mittelalterlichen Bearbeitungen der
+  Trojanersage," in E. M. Stengel's _Ausgaben und Abhandlungen aus dem
+  Gebiete der romanischen Philologie_, No. 61 (1886, esp. sections 82,
+  83, 168-172); F. Colagrosso, "Ditte Cretese" in _Atti della r.
+  Accademia di Archeologia_ (Naples, 1897, vol. 18, pt. ii. 2); F.
+  Noack, "Der griechische Dictys," in _Philologus_, supp. vi. 403 ff.;
+  N. E. Griffin, _Dares and Dictys, Introduction to the Study of the
+  Medieval Versions of the Story of Troy_ (1907).
+
+
+
+
+DICUIL (fl. 825), Irish monastic scholar, grammarian and geographer. He
+was the author of the _De mensura orbis terrae_, finished in 825, which
+contains the earliest clear notice of a European discovery of and
+settlement in Iceland and the most definite Western reference to the old
+freshwater canal between the Nile and the Red Sea, finally blocked up in
+767. In 795 (February 1-August 1) Irish hermits had visited Iceland; on
+their return they reported the marvel of the perpetual day at midsummer
+in "Thule," where there was then "no darkness to hinder one from doing
+what one would." These eremites also navigated the sea north of Iceland
+on their first arrival, and found it ice-free for one day's sail, after
+which they came to the ice-wall. Relics of this, and perhaps of other
+Irish religious settlements, were found by the permanent Scandinavian
+colonists of Iceland in the 9th century. Of the old Egyptian freshwater
+canal Dicuil learnt from one "brother Fidelis," probably another Irish
+monk, who, on his way to Jerusalem, sailed along the "Nile" into the Red
+Sea--passing on his way the "Barns of Joseph" or Pyramids of Giza, which
+are well described. Dicuil's knowledge of the islands north and west of
+Britain is evidently intimate; his references to Irish exploration and
+colonization, and to (more recent) Scandinavian devastation of the same,
+as far as the Faeroes, are noteworthy, like his notice of the elephant
+sent by Harun al-Rashid (in 801) to Charles the Great, the most curious
+item in a political and diplomatic intercourse of high importance.
+Dicuil's reading was wide; he quotes from, or refers to, thirty Greek
+and Latin writers, including the classical Homer, Hecataeus, Herodotus,
+Thucydides, Virgil, Pliny and King Juba, the sub-classical Solinus, the
+patristic St Isidore and Orosius, and his contemporary the Irish poet
+Sedulius;--in particular, he professes to utilize the alleged surveys of
+the Roman world executed by order of Julius Caesar, Augustus and
+Theodosius (whether Theodosius the Great or Theodosius II. is
+uncertain). He probably did not know Greek; his references to Greek
+authors do not imply this. Though certainly Irish by birth, it has been
+conjectured (from his references to Sedulius and the caliph's elephant)
+that he was in later life in an Irish monastery in the Frankish empire.
+Letronne inclines to identify him with Dicuil or Dichull, abbot of
+Pahlacht, born about 760.
+
+  There are seven chief MSS. of the _De mensura_ (Dicuil's tract on
+  grammar is lost); of these the earliest and best are (1) Paris,
+  National Library, Lat. 4806; (2) Dresden, Regius D. 182; both are of
+  the 10th century. Three editions exist: (1) C. A. Walckenaer's, Paris,
+  1807; (2) A. Letronne's, Paris, 1814, best as to commentary; (3) G.
+  Parthey's, Berlin, 1870, best as to text. See also C. R. Beazley,
+  _Dawn of Modern Geography_ (London, 1897), i. 317-327, 522-523, 529;
+  T. Wright, _Biographia Britannica literaria, Anglo-Saxon Period_
+  (London, 1842), pp. 372-376.     (C. R. B.)
+
+
+
+
+DIDACHE, THE, or _Teaching of the (twelve) Apostles_,--the most
+important of the recent recoveries in the region of early Christian
+literature (see APOCRYPHAL LITERATURE). It was previously known by name
+from lists of canonical and extra-canonical books compiled by Eusebius
+and other writers. Moreover, it had come to be suspected by several
+scholars that a lost book, variously entitled _The Two Ways_ or _The
+Judgment of Peter_, had been freely used in a number of works, of which
+mention must presently be made. In 1882 a critical reconstruction of
+this book was made by Adam Krawutzcky with marvellous accuracy, as was
+shown when in the very next year the Greek bishop and metropolitan,
+Philotheus Bryennius, published _The Teaching of the Twelve Apostles_
+from the same manuscript from which he had previously published the
+complete form of the Epistle of Clement.[1]
+
+_The Didach[=e]_, as we now have it in the Greek, falls into two marked
+divisions: (a) a book of moral precepts, opening with the words, "There
+are two ways"; (b) a manual of church ordinances, linked on to the
+foregoing by the words, "Having first said all these things, baptize,
+&c." Each of these must be considered separately before we approach the
+question of the locality and date of the whole book in its present form.
+
+1. _The Two Ways._--The author of the complete work, as we now have it,
+has modified the original _Two Ways_ by inserting near the beginning a
+considerable section containing, among other matter, passages from the
+Sermon on the Mount, in which the language of St Matthew's Gospel is
+blended with that of St Luke's. He has also added at the close a few
+sentences, beginning, "If thou canst not bear (the whole yoke of the
+Lord), bear what thou canst" (vi. 2); and among minor changes he has
+introduced, in dealing with confession, reference to "the church" (iv.
+14). No part of this matter is to be found in the following documents,
+which present us in varying degrees of accuracy with _The Two Ways_:
+(i.) the Epistle of Barnabas, chaps. xix., xx. (in which the order of
+the book has been much broken up, and a good deal has been omitted);
+(ii.) the _Ecclesiastical Canons of the Holy Apostles_, usually called
+the _Apostolic Church Order_, a book which presents a parallel to the
+_Teaching_, in so far as it consists first of a form of _The Two Ways_,
+and secondly of a number of church ordinances (here, however, as in the
+Syriac _Didascalia_, which gives about the same amount of _The Two
+Ways_, various sections are ascribed to individual apostles, e.g. "John
+said, There are two ways," &c.); (iii.) a discourse of the Egyptian monk
+Schnudi (d. 451), preserved in Arabic (see Iselin, _Texte u. Unters._,
+1895); (iv.) a Latin version, of which a fragment was published by O.
+von Gebhardt in 1884, and the whole by J. Schlecht in 1900. When by the
+aid of this evidence _The Two Ways_ is restored to us free of glosses,
+it has the appearance of being a Jewish manual which has been carried
+over into the use of the Christian church. This is of course only a
+probable inference; there is no prototype extant in Jewish literature,
+and, comparing the moral (non-doctrinal) instruction for Christian
+catechumens in Hermas, _Shepherd_ (_Mand._ i.-ix.), no real need to
+assume one. There was a danger of admitting Gentile converts to the
+church on too easy moral terms; hence the need of such insistence on the
+ideal as in The Two Ways and the _Mandates_. The recent recovery of the
+Latin version is of singular interest, as showing that, even without the
+distinctively Christian additions and interpolations which our full form
+of the _Teaching_ presents, it was circulating under the title _Doctrina
+apostolorum_.[2]
+
+2. The second part of our _Teaching_ might be called a church directory.
+It consists of precepts relating to church life, which are couched in
+the second person plural; whereas _The Two Ways_ uses throughout the
+second person singular. It appears to be a composite work. First (vii.
+1-xi. 2) is a short sacramental manual intended for the use of local
+elders or presbyters, though such are not named, for they were not yet a
+distinctive order or clergy. This section was probably added to _The Two
+Ways_ before the addition of the remainder. It orders baptism in the
+threefold name, making a distinction as to waters which has Jewish
+parallels, and permitting a threefold pouring on the head, if sufficient
+water for immersion cannot be had. It prescribes a fast before baptism
+for the baptizer as well as the candidate. Fasts are to be kept on
+Wednesday and Friday, not Monday and Thursday, which are the fast days
+of "the hypocrites," i.e. by a perversion of the Lord's words, the Jews.
+"Neither pray ye as the hypocrites; but as the Lord commanded in His
+Gospel." Then follows the Lord's Prayer, almost exactly as in St
+Matthew, with a brief doxology--"for Thine is the power and the glory
+forever." This is to be said three times a day. Next come three
+eucharistic prayers, the language of which is clearly marked off from
+that of the rest of the book, and shows parallels with the diction of St
+John's Gospel. They are probably founded on Jewish thanksgivings, and it
+is of interest to note that a portion of them is prescribed as a grace
+before meat in (pseudo-) Athanasius' _De virginitate_. A trace of them
+is found in one of the liturgical prayers of Serapion, bishop of Thmui,
+in Egypt, but they have left little mark on the liturgies of the church.
+As in Ignatius and other early writers, the eucharist, a real meal (x.
+1) of a family character, is regarded as producing immortality (cf.
+"spiritual food and drink and eternal life"). None are to partake of it
+save those who have been "baptized in the name of the Lord" (an
+expression which is of interest in a document which prescribes the
+threefold formula). The prophets are not to be confined to these forms,
+but may "give thanks as much as they will." This appears to show that a
+prophet, if present, would naturally preside over the eucharist. The
+next section (xi. 3-xiii.) deals with the ministry of spiritual gifts as
+exercised by apostles, prophets and teachers. An apostle is to be
+"received as the Lord"; but he must follow the Gospel precepts, stay but
+one or two days, and take no money, but only bread enough for a day's
+journey. Here we have that wider use of the term "apostle" to which
+Lightfoot had already drawn attention. A prophet, on the contrary, may
+settle if he chooses, and in that case he is to receive tithes and
+first-fruits; "for they are your high priests." If he be once approved
+as a true prophet, his words and acts are not to be criticized; for this
+is the sin that shall not be forgiven. Next comes a section (xiv., xv.)
+reflecting a somewhat later development concerning fixed services and
+ministry; the desire for a stated service, and the need of regular
+provision for it, is leading to a new order of things. The eucharist is
+to be celebrated every Lord's Day, and preceded by confession of sins,
+"that your sacrifice may be pure ... for this is that sacrifice which
+was spoken of by the Lord, In every place and time to offer unto Me a
+pure sacrifice. Appoint therefore unto yourselves bishops and deacons,
+worthy of the Lord, men meek and uncovetous, and true and approved; for
+they also minister unto you the ministration of the prophets and
+teachers. Therefore despise them not; for they are your honoured ones,
+together with the prophets and teachers." This is an arrangement
+recommended by one who has tried it, and he reassures the old-fashioned
+believer who clings to the less formal regime (and whose protest was
+voiced in the Montanist movement), that there will be no spiritual loss
+under the new system. The book closes (chap. xvi.) with exhortations to
+steadfastness in the last days, and to the coming of the
+"world-deceiver" or Antichrist, which will precede the coming of the
+Lord. This section is perhaps the actual utterance of a Christian
+prophet, and may be of earlier origin than the two preceding sections.
+
+3. It will now be clear that indications of the locality and date of our
+present _Teaching_ must be sought for only in the second part, and in
+the Christian interpolations in the first part. We have no ground for
+thinking that the second part ever existed independently as a separate
+book. The whole work was in the hands of the writer of the seventh book
+of the _Apostolic Constitutions_, who embodies almost every sentence of
+it, interspersing it with passages of Scripture, and modifying the
+precepts of the second part to suit a later (4th-century) stage of
+church development; this writer was also the interpolator of the
+Epistles of Ignatius, and belonged to the Syrian Church. Whether the
+second part was known to the writer of the _Apostolic Church Order_ is
+not clear, as his only quotation of it comes from one of the eucharistic
+prayers. The allusions of early writers seem to point to Egypt, but
+their references are mostly to the first part, so that we must be
+careful how we argue from them as to the provenance of the book as a
+whole. Against Egypt has been urged the allusion in one of the
+eucharistic prayers to "corn upon the mountains." This is found in the
+Prayer-book of Serapion (c. 350) but omitted in a later Egyptian
+prayer; the form as we have it in _The Didach[=e]_ may have passed into
+Egypt with the authority of tradition which was afterwards weakened. The
+anti-Jewish tone of the second part suggests the neighbourhood of Jews,
+from whom the Christians were to be sharply distinguished. Either Egypt
+or Syria would satisfy this condition, and in favour of Syria is the
+fact that the presbyterate there was to a late date regarded as a rank
+rather than an office. If we can connect the injunctions (vi. 3)
+concerning (abstinence from certain) food and that which is offered to
+idols with the old trouble that arose at Antioch (Acts xv. 1) and was
+legislated for by the Jerusalem council, we have additional support for
+the Syrian claim. But all that we can safely say as to locality is that
+the community here represented seems to have been isolated, and out of
+touch with the larger centres of Christian life.
+
+This last consideration helps us in discussing the question of date. For
+such an isolated community may have preserved primitive customs for some
+time after they had generally disappeared. Certainly the stage of
+development is an early one, as is shown, e.g., by the prominence of
+prophets, and the need that was felt for the vindication of the position
+of the bishops and deacons (there is no mention at all of presbyters);
+moreover, there is no reference to a canon of Scripture (though the
+written Gospel is expressly mentioned) or to a creed. On the other hand
+the "apostles" of the second part are obviously not "the twelve
+apostles" of the title; and the prophets seem in some instances to have
+proved unworthy of their high position. The ministry of enthusiasm which
+they represent is about to give way to the ministry of office, a
+transition which is reflected in the New Testament in the 3rd Epistle of
+John. Three of the Gospels have clearly been for some time in
+circulation; St Matthew's is used several times, and there are phrases
+which occur only in St Luke's, while St John's Gospel lies behind the
+eucharistic prayers which the writer has embodied in his work. There are
+no indications of any form of doctrinal heresy as needing rebuke; the
+warnings against false teaching are quite general. While the first part
+must be dated before the Epistle of Barnabas, i.e. before A.D. 90, it
+seems wisest not to place the complete work much earlier than A.D. 120,
+and there are passages which may well be later.
+
+  A large literature has sprung up round The _Didach[=e]_ since 1884.
+  Harnack's edition in _Texte u. Unters._ vol. ii. (1884) is
+  indispensable to the student; and his discussions in _Altchristl.
+  Litteratur_ and _Chronologie_ give clear summaries of his work. Other
+  editions of the text are those of F. X. Funk, _Patres Apostolici_,
+  vol. i. (Tubingen, 1901); H. Lietzmann (Bonn, 1903; with Latin
+  version). Dr J. E. Odgers has published an English translation with
+  introduction and notes (London, 1906). Dr C. Taylor in 1886 drew
+  attention to some important parallels in Jewish literature; his
+  edition contains an English translation. Dr Rendel Harris published in
+  1887 a complete facsimile, and gathered a great store of patristic
+  illustration. Text and translation will also be found in Lightfoot's
+  _Apostolic Fathers_ (ed. min.) The fullest critical treatment in
+  English is by Dr Vernon Bartlet in the extra volume of Hastings's
+  _Dictionary of the Bible_; the most complete commentary on the text is
+  by P. Drews in Hennecke's _Handbuch zu den N.T. Apocryphen_ (1904).
+  Other references to the literature may be found by consulting
+  Harnack's _Altchristl. Litteratur_.
+
+
+FOOTNOTES:
+
+  [1] The MS. was found in the Library of the Jerusalem Monastery of
+    the Most Holy Sepulchre, in Phanar, the Greek quarter of
+    Constantinople. It is a small octavo volume of 120 parchment leaves,
+    written throughout by Leo, "notary and sinner," who finished his task
+    on the 11th of June 1156. Besides The _Didach[=e]_ and the Epistles
+    of Clement it contains several spurious Ignatian epistles.
+
+  [2] The word _twelve_ had no place in the original title and was
+    inserted when the original _Didach[=e]_ or _Teaching_ (e.g. _The Two
+    Ways_) was combined with the church manual which mentions apostles
+    outside of the twelve. It may be noted that the division of the
+    _Didach[=e]_ into chapters is due to Bryennius, that into verses to
+    A. Harnack.
+
+
+
+
+DIDACTIC POETRY, that form of verse the aim of which is, less to excite
+the hearer by passion or move him by pathos, than to instruct his mind
+and improve his morals. The Greek word [Greek: didaktikos] signifies a
+teacher, from the verb [Greek: didaskein], and poetry of the class under
+discussion approaches us with the arts and graces of a schoolmaster. At
+no time was it found convenient to combine lyrical verse with
+instruction, and therefore from the beginning of literature the didactic
+poets have chosen a form approaching the epical. Modern criticism, which
+discourages the epic, and is increasingly anxious to limit the word
+"poetry" to lyric, is inclined to exclude the term "didactic poetry"
+from our nomenclature, as a phrase absurd in itself. It is indeed more
+than probable that didactic verse is hopelessly obsolete. Definite
+information is now to be found in a thousand shapes, directly and boldly
+presented in clear and technical prose. No farmer, however elegant,
+will, any longer choose to study agriculture in hexameters, or even in
+Tusser's shambling metre. The sciences and the professions will not
+waste their time on methods of instruction which must, from their very
+nature, be artless, inexact and vague. But in the morning of the world,
+those who taught with authority might well believe that verse was the
+proper, nay, the only serious vehicle of their instruction. What they
+knew was extremely limited, and in its nature it was simple and
+straightforward; it had little technical subtlety; it constantly lapsed
+into the fabulous and the conjectural. Not only could what early sages
+knew, or guessed, about astronomy and medicine and geography be
+conveniently put into rolling verse, but, in the absence of all written
+books, this was the easiest way in which information could be made
+attractive to the ear and be retained by the memory.
+
+In the prehistoric dawn of Greek civilization there appear to have been
+three classes of poetry, to which the literature of Europe looks back as
+to its triple fountain-head. There were romantic epics, dealing with the
+adventures of gods and heroes; these Homer represents. There were mystic
+chants and religious odes, purely lyrical in character, of which the
+best Orphic Hymns must have been the type. And lastly there was a great
+body of verse occupied entirely with increasing the knowledge of
+citizens in useful branches of art and observation; these were the
+beginnings of didactic poetry, and we class them together under the dim
+name of Hesiod. It is impossible to date these earliest didactic poems,
+which nevertheless set the fashion of form which has been preserved ever
+since. The _Works and Days_, which passes as the direct masterpiece of
+Hesiod (q.v.), is the type of all the poetry which has had education as
+its aim. Hesiod is supposed to have been a tiller of the ground in a
+Boeotian village, who determined to enrich his neighbours' minds by
+putting his own ripe stores of useful information into sonorous metre.
+Historically examined, the legend of Hesiod becomes a shadow, but the
+substance of the poems attributed to him remains. The genuine parts of
+the _Works and Days_, which Professor Gilbert Murray has called "a slow,
+lowly, simple poem," deal with rules for agriculture. The _Theogony_ is
+an annotated catalogue of the gods. Other poems attributed to Hesiod,
+but now lost, were on astronomy, on auguries by birds, on the character
+of the physical world; still others seem to have been genealogies of
+famous women. All this mass of Boeotian verse was composed for
+educational purposes, in an age when even preposterous information was
+better than no knowledge at all. In slightly later times, as the Greek
+nation became better supplied with intellectual appliances, the stream
+of didactic poetry flowed more and more closely in one, and that a
+theological, channel. The great poem of Parmenides _On Nature_ and those
+of Empedocles exist only in fragments, but enough remains to show that
+these poets carried on the didactic method in mythology. Cleostratus of
+Tenedos wrote an astronomical poem in the 6th century, and Periander a
+medical one in the 4th, but didactic poetry did not flourish again in
+Greece until the 3rd century, when Aratus, in the Alexandrian age, wrote
+his famous _Phenomena_, a poem about things seen in the heavens. Other
+later Greek didactic poets were Nicander, and perhaps Euphorion.
+
+It was from the hands of these Alexandrian writers that the genius of
+didactic poetry passed over to Rome, since, although it is possible that
+some of the lost works of the early republic, and in particular those of
+Ennius, may have possessed an educational character, the first and by
+far the greatest didactic Latin poet known to us is Lucretius. A highly
+finished translation by Cicero into Latin hexameters of the principal
+works of Aratus is believed to have drawn the attention of Lucretius to
+this school of Greek poetry, and it was not without reference to the
+Greeks, although in a more archaic and far purer taste, that he
+composed, in the 1st century before Christ, his magnificent _De rerum
+natura_. By universal consent, this is the noblest didactic poem in the
+literature of the world. It was intended to instruct mankind in the
+interpretation and in the working of the system of philosophy revealed
+by Epicurus, which at that time was exciting the sympathetic attention
+of all classes of Roman society. What gave the poem of Lucretius its
+extraordinary interest, and what has prolonged and even increased its
+vitality, was the imaginative and illustrative insight of the author,
+piercing and lighting up the recesses of human experience. On a lower
+intellectual level, but of a still greater technical excellence, was the
+_Georgics_ of Virgil, a poem on the processes of agriculture, published
+about 30 B.C. The brilliant execution of this famous work has justly
+made it the type and unapproachable standard of all poetry which desires
+to impart useful information in the guise of exquisite literature.
+Himself once a farmer on the banks of the Mincio, Virgil, at the apex of
+his genius, set himself in his Campanian villa to recall whatever had
+been essential in the agricultural life of his boyish home, and the
+result, in spite of the ardours of the subject, was what J. W. Mackail
+has called "the most splendid literary production of the Empire." In the
+rest of surviving Latin didactic poetry, the influence and the imitation
+of Virgil and Lucretius are manifest. Manilius, turning again to
+Alexandria, produced a fine _Astronomica_ towards the close of the reign
+of Augustus. Columella, regretting that Virgil had omitted to sing of
+gardens, composed a smooth poem on horticulture. Natural philosophy
+inspired Lucilius junior, of whom a didactic poem on Etna survives. Long
+afterwards, under Diocletian, a poet of Carthage, Nemesianus, wrote in
+the manner of Virgil the _Cynegetica_, a poem on hunting with dogs,
+which has had numerous imitations in later European literatures. These
+are the most important specimens of didactic poetry which ancient Rome
+has handed down to us.
+
+In Anglo-Saxon and early English poetic literature, and especially in
+the religious part of it, an element of didacticism is not to be
+overlooked. But it would be difficult to say that anything of importance
+was written in verse with the sole purpose of imparting information,
+until we reach the 16th century. Some of the later medieval allegories
+are didactic or nothing. The first poem, however, which we can in any
+reasonable way compare with the classic works of which we have been
+speaking is the _Hundreth Pointes of Good Husbandrie_, published in 1557
+by Thomas Tusser; these humble Georgics aimed at a practical description
+of the whole art of English farming. Throughout the early part of the
+17th century, when our national poetry was in its most vivid and
+brilliant condition, the last thing a poet thought of doing was the
+setting down of scientific facts in rhyme. We come across, however, one
+or two writers who were as didactic as the age would permit them to be,
+Samuel Daniel with his philosophy, Fulke Greville, Lord Brooke with his
+"treatises" of war and monarchy. After the Restoration, as the lyrical
+element rapidly died out of English poetry, there was more and more room
+left for educational rhetoric in verse. The poems about prosody, founded
+upon Horace, and signed by John Sheffield, 3rd earl of Mulgrave
+(1648-1721), and Lord Roscommon, were among the earliest purely didactic
+verse-studies in English. John Philips deserves a certain pre-eminence,
+as his poem called Cyder, in 1706, set the fashion which lasted all down
+the 18th century, of writing precisely in verse about definite branches
+of industry or employment. None of the greater poets of the age of Anne
+quite succumbed to the practice, but there is a very distinct flavour of
+the purely didactic about a great deal of the verse of Pope and Gay. In
+such productions as Gilbert West's (1703-1756) _Education_, Dyer's
+_Fleece_, and Somerville's _Chase_, we see technical information put
+forward as the central aim of the poet. Instead of a passionate
+pleasure, or at least an uplifted enthusiasm, being the poet's object,
+he frankly admits that, first and foremost, he has some facts about wool
+or dogs or schoolmasters which he wishes to bring home to his readers,
+and that, secondly, he consents to use verse, as brilliantly as he can,
+for the purpose of gilding the pill and attracting an unwilling
+attention. As we descend the 18th century, these works become more and
+more numerous, and more dry, especially when opposed by the descriptive
+and rural poets of the school of Thomson, the poet of _The Seasons_. But
+Thomson himself wrote a huge poem of _Liberty_ (1732), for which we have
+no name if we must not call it didactic. Even Gray began, though he
+failed to finish, a work of this class, on _The Alliance of Education
+and Government_. These poems were discredited by the publication of _The
+Sugar-Cane_ (1764), a long verse-treatise about the cultivation of sugar
+by negroes in the West Indies, by James Grainger (1721-1766), but,
+though liable to ridicule, such versified treatises continued to
+appear. Whether so great a writer as Cowper is to be counted among the
+didactic poets is a question on which readers of _The Task_ may be
+divided; this poem belongs rather to the class of descriptive poetry,
+but a strong didactic tendency is visible in parts of it. Perhaps the
+latest frankly educational poem which enjoyed a great popularity was
+_The Course of Time_ by Robert Pollok (1798-1827), in which a system of
+Calvinistic divinity is laid down with severity and in the pomp of blank
+verse. This kind of literature had already been exposed, and
+discouraged, by the teaching of Wordsworth, who had insisted on the
+imperative necessity of charging all poetry with imagination and
+passion. Oddly enough, _The Excursion_ of Wordsworth himself is perhaps
+the most didactic poem of the 19th century, but it must be acknowledged
+that his influence, in this direction, was saner than his practice.
+Since the days of Coleridge and Shelley it has been almost impossible to
+conceive a poet of any value composing in verse a work written with the
+purpose of inculcating useful information.
+
+The history of didactic poetry in France repeats, in great measure, but
+in drearier language, that of England. Boileau, like Pope, but with a
+more definite purpose as a teacher, offered instruction in his _Art
+poetique_ and in his _Epistles_. But his doctrine was always literary,
+not purely educational. At the beginning of the 18th century, the
+younger Racine (1692-1763) wrote sermons in verse, and at the close of
+it the Abbe Delille (1738-1813) tried to imitate Virgil in poems about
+horticulture. Between these two there lies a vast mass of verse written
+for the indulgence of intellect rather than at the dictates of the
+heart; wherever this aims at increasing knowledge, it at once becomes
+basely and flatly didactic. There is nothing in French literature of the
+transitional class that deserves mention beside _The Task_ or _The
+Excursion_.
+
+During the century which preceded the Romantic revival of poetry in
+Germany, didactic verse was cultivated in that country on the lines of
+imitation of the French, but with a greater dryness and on a lower level
+of utility. Modern German literature began with Martin Opitz (1597-1639)
+and the Silesian School, who were in their essence rhetorical and
+educational, and who gave their tone to German verse. Albrecht von
+Haller (1708-1777) brought a very considerable intellectual force to
+bear on his huge poems, _The Origin of Evil_, which was theological, and
+_The Alps_ (1729), botanical and topographical. Johann Peter Uz
+(1720-1796) wrote a _Theodicee_, which was very popular, and not without
+dignity. Johann Jacob Dusch (1725-1787) undertook to put _The Sciences_
+into the eight books of a great didactic poem. Tiedge (1752-1840) was
+the last of the school; in a once-famous _Urania_, he sang of God and
+Immortality and Liberty. These German pieces were the most unswervingly
+didactic that any modern European literature has produced. There was
+hardly the pretence of introducing into them descriptions of natural
+beauty, as the English poets did, or of grace and wit like the French.
+The German poets simply poured into a lumbering mould of verse as much
+solid information and direct instruction as the form would hold.
+
+Didactic poetry has, in modern times, been antipathetic to the spirit of
+the Latin peoples, and neither Italian nor Spanish literature has
+produced a really notable work in this class. An examination of the
+poems, ancient and modern, which have been mentioned above, will show
+that from primitive times there have been two classes of poetic work to
+which the epithet didactic has been given. It is desirable to
+distinguish these a little more exactly. One is the pure instrument of
+teaching, the poetry which desires to impart all that it knows about the
+growing of cabbages or the prevention of disasters at sea, the
+revolution of the planets or the blessings of inoculation. This is
+didactic poetry proper, and this, it is almost certain, became
+irrevocably obsolete at the close of the 18th century. No future Virgil
+will give the world a second _Georgics_. But there is another species
+which it is very improbable that criticism has entirely dislodged; that
+is the poetry which combines, with philosophical instruction, an impetus
+of imaginative movement, and a certain definite cultivation of fire and
+beauty. In hands so noble as those of Lucretius and Goethe this species
+of didactic poetry has enriched the world with durable masterpieces,
+and, although the circle of readers which will endure scientific
+disquisition in the bonds of verse grows narrower and narrower, it is
+probable that the great poet who is also a great thinker will now and
+again insist on being heard. In Sully-Prudhomme France has possessed an
+eminent writer whose methods are directly instructive, and both _La
+Justice_ (1878) and _Le Bonheur_ (1888) are typically didactic poems.
+Perhaps future historians may name these as the latest of their class.
+     (E. G.)
+
+
+
+
+DIDEROT, DENIS (1713-1784), French man of letters and encyclopaedist,
+was born at Langres on the 5th of October 1713. He was educated by the
+Jesuits, like most of those who afterwards became the bitterest enemies
+of Catholicism; and, when his education was at an end, he vexed his
+brave and worthy father's heart by turning away from respectable
+callings, like law or medicine, and throwing himself into the vagabond
+life of a bookseller's hack in Paris. An imprudent marriage (1743) did
+not better his position. His wife, Anne Toinette Champion, was a devout
+Catholic, but her piety did not restrain a narrow and fretful temper,
+and Diderot's domestic life was irregular and unhappy. He sought
+consolation for chagrins at home in attachments abroad, first with a
+Madame Puisieux, a fifth-rate female scribbler, and then with Sophie
+Voland, to whom he was constant for the rest of her life. His letters to
+her are among the most graphic of all the pictures that we have of the
+daily life of the philosophic circle in Paris. An interesting contrast
+may be made between the Bohemianism of the famous English literary set
+who supped at the Turk's Head with the Tory Johnson and the Conservative
+Burke for their oracles, and the Bohemianism of the French set who about
+the same time dined once a week at the baron D'Holbach's, to listen to
+the wild sallies and the inspiring declamations of Diderot. For Diderot
+was not a great writer; he stands out as a fertile, suggestive and
+daring thinker, and a prodigious and most eloquent talker.
+
+Diderot's earliest writings were of as little importance as Goldsmith's
+_Enquiry into the State of Polite Learning_ or Burke's _Abridgement of
+English History_. He earned 100 crowns by translating Stanyan's _History
+of Greece_ (1743); with two colleagues he produced a translation of
+James's _Dictionary of Medicine_ (1746-1748) and about the same date he
+published a free rendering of Shaftesbury's _Inquiry Concerning Virtue
+and Merit_ (1745), with some original notes of his own. With strange and
+characteristic versatility, he turned from ethical speculation to the
+composition of a volume of stories, the _Bijoux indiscrets_ (1748),
+gross without liveliness, and impure without wit. In later years he
+repented of this shameless work, just as Boccaccio is said in the day of
+his grey hairs to have thought of the sprightliness of the _Decameron_
+with strong remorse. From tales Diderot went back to the more congenial
+region of philosophy. Between the morning of Good Friday and the evening
+of Easter Monday he wrote the _Pensees philosophiques_ (1746), and he
+presently added to this a short complementary essay on the sufficiency
+of natural religion. The gist of these performances is to press the
+ordinary rationalistic objections to a supernatural revelation; but
+though Diderot did not at this time pass out into the wilderness beyond
+natural religion, yet there are signs that he accepted that less as a
+positive doctrine, resting on grounds of its own, than as a convenient
+point of attack against Christianity. In 1747 he wrote the _Promenade du
+sceptique_, a rather poor allegory--pointing first to the extravagances
+of Catholicism; second, to the vanity of the pleasures of that world
+which is the rival of the church; and third, to the desperate and
+unfathomable uncertainty of the philosophy which professes to be so high
+above both church and world.
+
+Diderot's next piece was what first introduced him to the world as an
+original thinker, his famous _Lettre sur les aveugles_ (1749). The
+immediate object of this short but pithy writing was to show the
+dependence of men's ideas on their five senses. It considers the case of
+the intellect deprived of the aid of one of the senses; and in a second
+piece, published afterwards, Diderot considered the case of a similar
+deprivation in the deaf and dumb. The _Lettre sur les sourds et muets_,
+however, is substantially a digressive examination of some points in
+aesthetics. The philosophic significance of the two essays is in the
+advance they make towards the principle of Relativity. But what
+interested the militant philosophers of that day was an episodic
+application of the principle of relativity to the master-conception of
+God. What makes the _Lettre sur les aveugles_ interesting is its
+presentation, in a distinct though undigested form, of the modern theory
+of variability, and of survival by superior adaptation. It is worth
+noticing, too, as an illustration of the comprehensive freedom with
+which Diderot felt his way round any subject that he approached, that in
+this theoretic essay he suggests the possibility of teaching the blind
+to read through the sense of touch. If the _Lettre sur les aveugles_
+introduced Diderot into the worshipful company of the philosophers, it
+also introduced him to the penalties of philosophy. His speculation was
+too hardy for the authorities, and he was thrown into the prison of
+Vincennes. Here he remained for three months; then he was released, to
+enter upon the gigantic undertaking of his life.
+
+The bookseller Lebreton had applied to him with a project for the
+publication of a translation into French of Ephraim Chambers's
+_Cyclopaedia_, undertaken in the first instance by an Englishman, John
+Mills, and a German, Gottfried Sellius (for particulars see
+ENCYCLOPAEDIA). Diderot accepted the proposal, but in his busy and
+pregnant intelligence the scheme became transformed. Instead of a mere
+reproduction of Chambers, he persuaded the bookseller to enter upon a
+new work, which should collect under one roof all the active writers,
+all the new ideas, all the new knowledge, that were then moving the
+cultivated class to its depths, but still were comparatively ineffectual
+by reason of their dispersion. His enthusiasm infected the publishers;
+they collected a sufficient capital for a vaster enterprise than they
+had at first planned; D'Alembert was persuaded to become Diderot's
+colleague; the requisite permission was procured from the government; in
+1750 an elaborate prospectus announced the project to a delighted
+public; and in 1751 the first volume was given to the world. The last of
+the letterpress was issued in 1765, but it was 1772 before the
+subscribers received the final volumes of the plates. These twenty years
+were to Diderot years not merely of incessant drudgery, but of harassing
+persecution, of sufferings from the cabals of enemies, and of injury
+from the desertion of friends. The ecclesiastical party detested the
+_Encyclopaedia_, in which they saw a rising stronghold for their
+philosophic enemies. By 1757 they could endure the sight no longer. The
+subscribers had grown from 2000 to 4000, and this was a right measure of
+the growth of the work in popular influence and power. To any one who
+turns over the pages of these redoubtable volumes now, it seems
+surprising that their doctrines should have stirred such portentous
+alarm. There is no atheism, no overt attack on any of the cardinal
+mysteries of the faith, no direct denunciation even of the notorious
+abuses of the church. Yet we feel that the atmosphere of the book may
+well have been displeasing to authorities who had not yet learnt to
+encounter the modern spirit on equal terms. The _Encyclopaedia_ takes
+for granted the justice of religious tolerance and speculative freedom.
+It asserts in distinct tones the democratic doctrine that it is the
+common people in a nation whose lot ought to be the main concern of the
+nation's government. From beginning to end it is one unbroken process of
+exaltation of scientific knowledge on the one hand, and pacific industry
+on the other. All these things were odious to the old governing classes
+of France; their spirit was absolutist, ecclesiastical and military.
+Perhaps the most alarming thought of all was the current belief that the
+_Encyclopaedia_ was the work of an organized band of conspirators
+against society, and that a pestilent doctrine was now made truly
+formidable by the confederation of its preachers into an open league.
+When the seventh volume appeared, it contained an article on "Geneva,"
+written by D'Alembert. The writer contrived a panegyric on the pastors
+of Geneva, of which every word was a stinging reproach to the abbes and
+prelates of Versailles. At the same moment Helvetius's book, _L'Esprit_,
+appeared, and gave a still more profound and, let us add, a more
+reasonable shock to the ecclesiastical party. Authority could brook no
+more, and in 1759 the _Encyclopaedia_ was formally suppressed.
+
+The decree, however, did not arrest the continuance of the work. The
+connivance of the authorities at the breach of their own official orders
+was common in those times of distracted government. The work went on,
+but with its difficulties increased by the necessity of being
+clandestine. And a worse thing than troublesome interference by the
+police now befell Diderot. D'Alembert, wearied of shifts and
+indignities, withdrew from the enterprise. Other powerful colleagues,
+Turgot among them, declined to contribute further to a book which had
+acquired an evil fame. Diderot was left to bring the task to an end as
+he best could. For seven years he laboured like a slave at the oar. He
+wrote several hundred articles, some of them very slight, but many of
+them most laborious, comprehensive and ample. He wore out his eyesight
+in correcting proofs, and he wearied his soul in bringing the manuscript
+of less competent contributors into decent shape. He spent his days in
+the workshops, mastering the processes of manufactures, and his nights
+in reproducing on paper what he had learnt during the day. And he was
+incessantly harassed all the time by alarms of a descent from the
+police. At the last moment, when his immense work was just drawing to an
+end, he encountered one last and crowning mortification: he discovered
+that the bookseller, fearing the displeasure of the government, had
+struck out from the proof sheets, after they had left Diderot's hands,
+all passages that he chose to think too hardy. The monument to which
+Diderot had given the labour of twenty long and oppressive years was
+irreparably mutilated and defaced. It is calculated that the average
+annual salary received by Diderot for his share in the _Encyclopaedia_
+was about L120 sterling. "And then to think," said Voltaire, "that an
+army contractor makes L800 in a day!"
+
+Although the _Encyclopaedia_ was Diderot's monumental work, he is the
+author of a shower of dispersed pieces that sowed nearly every field of
+intellectual interest with new and fruitful ideas. We find no
+masterpiece, but only thoughts for masterpieces; no creation, but a
+criticism with the quality to inspire and direct creation. He wrote
+plays--_Le Fils naturel_ (1757) and _Le Pere de famille_ (1758)--and
+they are very insipid performances in the sentimental vein. But he
+accompanied them by essays on dramatic poetry, including especially the
+_Paradoxe sur le comedien_, in which he announced the principles of a
+new drama,--the serious, domestic, bourgeois drama of real life, in
+opposition to the stilted conventions of the classic French stage. It
+was Diderot's lessons and example that gave a decisive bias to the
+dramatic taste of Lessing, whose plays, and his _Hamburgische
+Dramaturgie_ (1768), mark so important an epoch in the history of the
+modern theatre. In the pictorial art, Diderot's criticisms are no less
+rich, fertile and wide in their ideas. His article on "Beauty" in the
+_Encyclopaedia_ shows that he had mastered and passed beyond the
+metaphysical theories on the subject, and the _Essai sur la peinture_
+was justly described by Goethe, who thought it worth translating, as "a
+magnificent work, which speaks even more helpfully to the poet than to
+the painter, though to the painter too it is as a blazing torch."
+Diderot's most intimate friend was Grimm, one of the conspicuous figures
+of the philosophic body. Grimm wrote news-letters to various high
+personages in Germany, reporting what was going on in the world of art
+and literature in Paris, then without a rival as the capital of the
+intellectual activity of Europe. Diderot helped his friend at one time
+and another between 1759 and 1779, by writing for him an account of the
+annual exhibitions of paintings. These _Salons_ are among the most
+readable of all pieces of art criticism. They have a freshness, a
+reality, a life, which take their readers into a different world from
+the dry and conceited pedantries of the ordinary virtuoso. As has been
+said by Sainte-Beuve, they initiated the French into a new sentiment,
+and introduced people to the mystery and purport of colour by ideas.
+"Before Diderot," Madame Necker said, "I had never seen anything in
+pictures except dull and lifeless colours; it was his imagination that
+gave them relief and life, and it is almost a new sense for which I am
+indebted to his genius."
+
+Greuze was Diderot's favourite among contemporary artists, and it is
+easy to see why. Greuze's most characteristic pictures were the
+rendering in colour of the same sentiment of domestic virtue and the
+pathos of common life, which Diderot attempted with inferior success to
+represent upon the stage. For Diderot was above all things interested in
+the life of men,--not the abstract life of the race, but the incidents
+of individual character, the fortunes of a particular family, the
+relations of real and concrete motives in this or that special case. He
+delighted with the enthusiasm of a born casuist in curious puzzles of
+right and wrong, and in devising a conflict between the generalities of
+ethics and the conditions of an ingeniously contrived practical dilemma.
+Mostly his interest expressed itself in didactic and sympathetic form;
+in two, however, of the most remarkable of all his pieces, it is not
+sympathetic, but ironical. _Jacques le fataliste_ (written in 1773, but
+not published until 1796) is in manner an imitation of _Tristram Shandy_
+and _The Sentimental Journey_. Few modern readers will find in it any
+true diversion. In spite of some excellent criticisms dispersed here and
+there, and in spite of one or two stories that are not without a certain
+effective realism, it must as a whole be pronounced savourless, forced,
+and as leaving unmoved those springs of laughter and of tears which are
+the common fountain of humour. _Le Neveu de Rameau_ is a far superior
+performance. If there were any inevitable compulsion to name a
+masterpiece for Diderot, one must select this singular "farce-tragedy."
+Its intention has been matter of dispute; whether it was designed to be
+merely a satire on contemporary manners, or a reduction of the theory of
+self-interest to an absurdity, or the application of an ironical
+clincher to the ethics of ordinary convention, or a mere setting for a
+discussion about music, or a vigorous dramatic sketch of a parasite and
+a human original. There is no dispute as to its curious literary
+flavour, its mixed qualities of pungency, bitterness, pity and, in
+places, unflinching shamelessness. Goethe's translation (1805) was the
+first introduction of _Le Neveu de Rameau_ to the European public. After
+executing it, he gave back the original French manuscript to Schiller,
+from whom he had it. No authentic French copy of it appeared until the
+writer had been nearly forty years in his grave (1823).
+
+It would take several pages merely to contain the list of Diderot's
+miscellaneous pieces, from an infinitely graceful trifle like the
+_Regrets sur ma vieille robe de chambre_ up to _Le Reve de D'Alembert_,
+where he plunges into the depths of the controversy as to the ultimate
+constitution of matter and the meaning of life. It is a mistake to set
+down Diderot for a coherent and systematic materialist. We ought to look
+upon him "as a philosopher in whom all the contradictions of the time
+struggle with one another" (Rosenkranz). That is to say, he is critical
+and not dogmatic. There is no unity in Diderot, as there was in Voltaire
+or in Rousseau. Just as in cases of conduct he loves to make new ethical
+assumptions and argue them out as a professional sophist might have
+done, so in the speculative problems as to the organization of matter,
+the origin of life, the compatibility between physiological machinery
+and free will, he takes a certain standpoint, and follows it out more or
+less digressively to its consequences. He seizes a hypothesis and works
+it to its end, and this made him the inspirer in others of materialist
+doctrines which they held more definitely than he did. Just as Diderot
+could not attain to the concentration, the positiveness, the finality of
+aim needed for a masterpiece of literature, so he could not attain to
+those qualities in the way of dogma and system. Yet he drew at last to
+the conclusions of materialism, and contributed many of its most
+declamatory pages to the _Systeme de la nature_ of his friend
+D'Holbach,--the very Bible of atheism, as some one styled it. All that
+he saw, if we reduce his opinions to formulae, was motion in space:
+"attraction and repulsion, the only truth." If matter produces life by
+spontaneous generation, and if man has no alternative but to obey the
+compulsion of nature, what remains for God to do?
+
+In proportion as these conclusions deepened in him, the more did
+Diderot turn for the hope of the race to virtue; in other words, to such
+a regulation of conduct and motive as shall make us tender, pitiful,
+simple, contented. Hence his one great literary passion, his enthusiasm
+for Richardson, the English novelist. Hence, also, his deepening
+aversion for the political system of France, which makes the realization
+of a natural and contented domestic life so hard. Diderot had almost as
+much to say against society as even Rousseau himself. The difference
+between them was that Rousseau was a fervent theist. The atheism of the
+Holbachians, as he called Diderot's group, was intolerable to him; and
+this feeling, aided by certain private perversities of humour, led to a
+breach of what had once been an intimate friendship between Rousseau and
+Diderot (1757). Diderot was still alive when Rousseau's _Confessions_
+appeared, and he was so exasperated by Rousseau's stories about Grimm,
+then and always Diderot's intimate, that in 1782 he transformed a life
+of Seneca, that he had written four years earlier, into an _Essai sur
+les regnes de Claude et de Neron_ (1778-1782), which is much less an
+account of Seneca than a vindication of Diderot and Grimm, and is one of
+the most rambling and inept productions in literature. As for the merits
+of the old quarrel between Rousseau and Diderot, we may agree with the
+latter, that too many sensible people would be in the wrong if Jean
+Jacques was in the right.
+
+Varied and incessant as was Diderot's mental activity, it was not of a
+kind to bring him riches. He secured none of the posts that were
+occasionally given to needy men of letters; he could not even obtain
+that bare official recognition of merit which was implied by being
+chosen a member of the Academy. The time came for him to provide a dower
+for his daughter, and he saw no other alternative than to sell his
+library. When the empress Catherine of Russia heard of his straits, she
+commissioned an agent in Paris to buy the library at a price equal to
+about L1000 of English money, and then handsomely requested the
+philosopher to retain the books in Paris until she required them, and to
+constitute himself her librarian, with a yearly salary. In 1773 Diderot
+started on an expedition to thank his imperial benefactress in person,
+and he passed some months at St Petersburg. The empress received him
+cordially. The strange pair passed their afternoons in disputes on a
+thousand points of high philosophy, and they debated with a vivacity and
+freedom not usual in courts. "_Fi, donc,_" said Catherine one day, when
+Diderot hinted that he argued with her at a disadvantage, "_is there any
+difference among men?_" Diderot returned home in 1774. Ten years
+remained to him, and he spent them in the industrious acquisition of new
+knowledge, in the composition of a host of fragmentary pieces, some of
+them mentioned above, and in luminous declamations with his friends. All
+accounts agree that Diderot was seen at his best in conversation. "He
+who only knows Diderot in his writings," says Marmontel, "does not know
+him at all. When he grew animated in talk, and allowed his thoughts to
+flow in all their abundance, then he became truly ravishing. In his
+writings he had not the art of ensemble; the first operation which
+orders and places everything was too slow and too painful to him."
+Diderot himself was conscious of the want of literary merit in his
+pieces. In truth he set no high value on what he had done. It is
+doubtful whether he was ever alive to the waste that circumstance and
+temperament together made of an intelligence from which, if it had been
+free to work systematically, the world of thought had so much to hope.
+He was one of those simple, disinterested and intellectually sterling
+workers to whom their own personality is as nothing in presence of the
+vast subjects that engage the thoughts of their lives. He wrote what he
+found to write, and left the piece, as Carlyle has said, "on the waste
+of accident, with an ostrich-like indifference." When he heard one day
+that a collected edition of his works was in the press at Amsterdam, he
+greeted the news with "peals of laughter," so well did he know the haste
+and the little heed with which those works had been dashed off.
+
+Diderot died on the 30th of July 1784, six years after Voltaire and
+Rousseau, one year after his old colleague D'Alembert, and five years
+before D'Holbach, his host and intimate for a lifetime. Notwithstanding
+Diderot's peals of laughter at the thought, an elaborate and exhaustive
+collection of his writings in twenty stout volumes, edited by MM.
+Assezat and Tourneux, was completed in 1875-1877.
+
+  AUTHORITIES.--Studies on Diderot by Scherer (1880); by E. Faguet
+  (1890); by Sainte-Beuve in the _Causeries du lundi_; by F. Brunetiere
+  in the _Etudes critiques_, 2nd series, may be consulted. In English,
+  Diderot has been the subject of a biography by John Morley [Viscount
+  Morley of Blackburn] (1878). See also Karl Rosenkranz, _Diderots Leben
+  und Werke_ (1866). For a discussion of the authenticity of the
+  posthumous works of Diderot see R. Dominic in the _Revue des deux
+  mondes_ (October 15, 1902).     (J. Mo.)
+
+
+
+
+DIDIUS SALVIUS JULIANUS, MARCUS, Roman emperor for two months (March
+28-June 2) during the year A.D. 193. He was the grandson of the famous
+jurist Salvius Julianus (under Hadrian and the Antonines), and the son
+of a distinguished general, who might have ascended the throne after the
+death of Antoninus Pius, had not his loyalty to the ruling house
+prevented him. Didius filled several civil and military offices with
+distinguished success, but subsequently abandoned himself to
+dissipation. On the death of Pertinax, the praetorian guards offered the
+throne to the highest bidder. Flavius Sulpicianus, the father-in-law of
+Pertinax and praefect of the city, had already made an offer; Didius,
+urged on by the members of his family, his freedmen and parasites,
+hurried to the praetorian camp to contend for the prize. He and
+Sulpicianus bid against each other, and finally the throne was knocked
+down to Didius. The senate and nobles professed their loyalty; but the
+people made no attempt to conceal their indignation at this insult to
+the state, and the armies of Britain, Syria and Illyricum broke out into
+open revolt. Septimius Severus, the commander of the Pannonian legions,
+was declared emperor and hastened by forced marches to Italy. Didius,
+abandoned by the praetorians, was condemned and executed by order of the
+senate, which at once acknowledged Severus.
+
+  AUTHORITIES.--Dio Cassius lxxiii. 11-17, who was actually in Rome at
+  the time; Aelius Spartianus, _Didius Julianus_; Julius Capitolinus,
+  _Pertinax_; Herodian ii.; Aurelius Victor, _De Caesaribus_, 19;
+  Zosimus i. 7; Gibbon, _Decline and Fall_, chap. 5.
+
+
+
+
+DIDO, or ELISSA, the reputed founder of Carthage (q.v.), in Africa,
+daughter of the Tyrian king Metten (Mutto, Methres, Belus), wife of
+Acerbas (more correctly Sicharbas; Sychaeus in Virgil), a priest of
+Hercules. Her husband having been slain by her brother Pygmalion, Dido
+fled to Cyprus, and thence to the coast of Africa, where she purchased
+from a local chieftain Iarbas a piece of land on which she built
+Carthage. The city soon began to prosper and Iarbas sought Dido's hand
+in marriage, threatening her with war in case of refusal. To escape from
+him, Dido constructed a funeral pile, on which she stabbed herself
+before the people (Justin xviii. 4-7). Virgil, in defiance of the
+usually accepted chronology, makes Dido a contemporary of Aeneas, with
+whom she fell in love after his landing in Africa, and attributes her
+suicide to her abandonment by him at the command of Jupiter (_Aeneid_,
+iv.). Dido was worshipped at Carthage as a divinity under the name of
+Caelestis, the Roman counterpart of Tanit, the tutelary goddess of
+Carthage. According to Timaeus, the oldest authority for the story, her
+name was Theiosso, in Phoenician Helissa, and she was called Dido from
+her wanderings, Dido being the Phoenician equivalent of [Greek:
+planetis] (_Etymologicum Magnum_, s.v.); some modern scholars, however,
+translate the name by "beloved." Timaeus makes no mention of Aeneas, who
+seems to have been introduced by Naevius in his _Bellum Poenicum_,
+followed by Ennius in his _Annales_.
+
+  For the variations of the legend in earlier and later Latin authors,
+  see O. Rossbach in Pauly-Wissowa's _Realencyclopadie_, v. pt. 1
+  (1905); O. Meltzer's _Geschichte der Karthager_, i. (1879), and his
+  article in Roscher's _Lexikon der Mythologie_.
+
+
+
+
+DIDON, HENRI (1840-1900), French Dominican, was born at Trouvet, Isere,
+on the 17th of March 1840. He joined the Dominicans, under the influence
+of Lacordaire, in 1858, and completed his theological studies at the
+Minerva convent at Rome. The influence of Lacordaire was shown in the
+zeal displayed by Didon in favour of a reconciliation between philosophy
+and science. In 1871 his fame had so much grown that he was chosen to
+deliver the funeral oration over the murdered archbishop of Paris,
+Monseigneur G. Darboy. He also delivered some discourses at the church
+of St Jean de Beauvais in Paris on the relations between science and
+religion; but his utterances, especially on the question of divorce,
+were deemed suspicious by his superiors, and his intimacy with Claude
+Bernard the physiologist was disapproved. He was interdicted from
+preaching and sent into retirement at the convent of Corbara in Corsica.
+After eighteen months he emerged, and travelled in Germany, publishing
+an interesting work upon that country, entitled _Les Allemands_ (English
+translation by R. Ledos de Beaufort, London, 1884). On his return to
+France in 1890 he produced his best known work, _Jesus-Christ_ (2 vols.,
+Paris), for which he had qualified himself by travel in the Holy Land.
+In the same year he became director of the College Albert-le-Grand at
+Arcueil, and founded three auxiliary institutions, Ecole Lacordaire,
+Ecole Laplace and Ecole St Dominique. He wrote, in addition, several
+works on educational questions, and augmented his fame as an eloquent
+preacher by discourses preached during Lent and Advent. He died at
+Toulouse on the 13th of March 1900.
+
+  See the biographies by J. de Romano (1891), and A. de Coulanges
+  (Paris, 1900); and especially the work of Stanislas Reynaud, entitled
+  _Le Pere Didon, sa vie et son oeuvre_ (Paris, 1904).
+
+
+
+
+DIDOT, the name of a family of learned French printers and publishers.
+FRANCOIS DIDOT (1689-1757), founder of the family, was born at Paris. He
+began business as a bookseller and printer in 1713, and among his
+undertakings was a collection of the travels of his friend the Abbe
+Prevost, in twenty volumes (1747). It was remarkable for its
+typographical perfection, and was adorned with many engravings and maps.
+FRANCOIS AMBROISE DIDOT (1730-1804), son of Francois, made important
+improvements in type-founding, and was the first to attempt printing on
+vellum paper. Among the works which he published was the famous
+collection of French classics prepared by order of Louis XVI. for the
+education of the Dauphin, and the folio edition of _L'Art de verifier
+les dates_. PIERRE FRANCOIS DIDOT (1732-1795), his brother, devoted much
+attention to the art of type-founding and to paper-making. Among the
+works which issued from his press was an edition in folio of the
+_Imitatio Christi_ (1788). HENRI DIDOT (1765-1852), son of Pierre
+Francois, is celebrated for his "microscopic" editions of various
+standard works, for which he engraved the type when nearly seventy years
+of age. He was also the engraver of the _assignats_ issued by the
+Constituent and Legislative Assemblies and the Convention. DIDOT
+SAINT-LEGER, second son of Pierre Francois, was the inventor of the
+paper-making machine known in England as the Didot machine. PIERRE DIDOT
+(1760-1853), eldest son of Francois Ambroise, is celebrated as the
+publisher of the beautiful "Louvre" editions of Virgil, Horace and
+Racine. The Racine, in three volumes folio, was pronounced in 1801 to be
+"the most perfect typographical production of all ages." FIRMIN DIDOT
+(1764-1836), his brother, second son of Francois Ambroise, sustained the
+reputation of the family both as printer and type-founder. He revived
+(if he did not invent--a distinction which in order of time belongs to
+William Ged) the process of stereotyping, and coined its name, and he
+first used the process in his edition of Callet's _Tables of Logarithms_
+(1795), in which he secured an accuracy till then unattainable. He
+published stereotyped editions of French, English and Italian classics
+at a very low price. He was the author of two tragedies--_La Reine de
+Portugal_ and _La Mort d'Annibal_; and he wrote metrical translations
+from Virgil, Tyrtaeus and Theocritus. AMBROISE FIRMIN DIDOT (1790-1876)
+was his eldest son. After receiving a classical education, he spent
+three years in Greece and in the East; and on the retirement of his
+father in 1827 he undertook, in conjunction with his brother Hyacinthe,
+the direction of the publishing business. Their greatest undertaking was
+a new edition of the _Thesaurus Graecae linguae_ of Henri Estienne,
+under the editorial care of the brothers Dindorf and M. Hase (9 vols.,
+1855-1859). Among the numerous important works published by the
+brothers, the 200 volumes forming the _Bibliotheque des auteurs grecs_,
+_Bibliotheque latine_, and _Bibliotheque francaise_ deserve special
+mention. Ambroise Firmin Didot was the first to propose (1823) a
+subscription in favour of the Greeks, then in insurrection against
+Turkish tyranny. Besides a translation of Thucydides (1833), he wrote
+the articles "Estienne" in the _Nouvelle Biographie generale_, and
+"Typographie" in the _Ency. mod._, as well as _Observations sur
+l'orthographie francaise_ (1867), &c. In 1875 he published a very
+learned and elaborate monograph on Aldus Manutius. His collection of
+MSS., the richest in France, was said to have been worth, at the time of
+his death, not less than 2,000,000 francs.
+
+
+
+
+DIDRON, ADOLPHE NAPOLEON (1806-1867), French archaeologist, was born at
+Hautvillers, in the department of Marne, on the 13th of March 1806. At
+first a student of law, he began in 1830, by the advice of Victor Hugo,
+a study of the Christian archaeology of the middle ages. After visiting
+and examining the principal churches, first of Normandy, then of central
+and southern France, he was on his return appointed by Guizot secretary
+to the Historical Committee of Arts and Monuments (1835); and in the
+following years he delivered several courses of lectures on Christian
+iconography at the Bibliotheque Royale. In 1839 he visited Greece for
+the purpose of examining the art of the Eastern Church, both in its
+buildings and its manuscripts. In 1844 he originated the _Annales
+archeologiques_, a periodical devoted to his favourite subject, which he
+edited until his death. In 1845 he established at Paris a special
+archaeological library, and at the same time a manufactory of painted
+glass. In the same year he was admitted to the Legion of Honour. His
+most important work is the _Iconographie chretienne_, of which, however,
+the first portion only, _Histoire de Dieu_ (1843), was published. It was
+translated into English by E. J. Millington. Among his other works may
+be mentioned the _Manuel d'iconographie chretienne grecque et latine_
+(1845), the _Iconographie des chapiteaux du palais ducal de Venise_
+(1857), and the _Manuel des objets de bronze et d'orfevrerie_ (1859). He
+died on the 13th of November 1867.
+
+
+
+
+DIDYMI, or DIDYMA (mod. _Hieronta_), an ancient sanctuary of Apollo in
+Asia Minor situated in the territory of Miletus, from which it was
+distant about 10 m. S. and on the promontory Poseideion. It was
+sometimes called _Branchidae_ from the name of its priestly caste which
+claimed descent from Branchus, a youth beloved by Apollo. As the seat of
+a famous oracle, the original temple attracted offerings from Pharaoh
+Necho (in whose army there was a contingent of Milesian mercenaries),
+and the Lydian Croesus, and was plundered by Darius of Persia. Xerxes
+finally sacked and burnt it (481 B.C.) and exiled the Branchidae to the
+far north-east of his empire. This exile was believed to be voluntary,
+the priests having betrayed their treasures to the Persian; and on this
+belief Alexander the Great acted 150 years later, when, finding the
+descendants of the Branchidae established in a city beyond the Oxus, he
+ordered them to be exterminated for the sin of their fathers (328). The
+celebrated cult-statue of Apollo by Canachus, familiar to us from
+reproductions on Milesian coins, was also carried to Persia, there to
+remain till restored by Seleucus I. in 295, and the oracle ceased to
+speak for a century and a half. The Milesians were not able to undertake
+the rebuilding till about 332 B.C., when the oracle revived at the
+bidding of Alexander. The work proved too costly, and despite a special
+effort made by the Asian province nearly 400 years later, at the bidding
+of the emperor Caligula, the structure was never quite finished: but
+even as it was, Strabo ranked the Didymeum the greatest of Greek temples
+and Pliny placed it among the four most splendid and second only to the
+Artemisium at Ephesus. In point of fact it was a little smaller than the
+Samian Heraeum and the temple of Cybele at Sardis, and almost exactly
+the same size as the Artemisium. The area covered by the platform
+measures roughly 360 X 160 ft.
+
+When Cyriac of Ancona visited the spot in 1446, it seems that the temple
+was still standing in great part, although the _cella_ had been
+converted into a fortress by the Byzantines: but when the next European
+visitor, the Englishman Dr Pickering, arrived in 1673, it had collapsed.
+It is conjectured that the cause was the great earthquake of 1493. The
+Society of Dilettanti sent two expeditions to explore the ruins, the
+first in 1764 under Richard Chandler, the second in 1812 under Sir Wm.
+Gell; and the French "Rothschild Expedition" of 1873 under MM. O. Rayet
+and A. Thomas sent a certain amount of architectural sculpture to the
+Louvre. But no excavation was attempted till MM. E. Pontremoli and B.
+Haussoullier were sent out by the French Schools of Rome and Athens in
+1895. They cleared the western facade and the _prodomos_, and discovered
+inscriptions giving information about other parts which they left still
+buried. Finally the site was purchased by, and the French rights were
+ceded to, Dr Th. Wiegand, the German explorer of Miletus, who in 1905
+began a thorough clearance of what is incomparably the finest temple
+ruin in Asia Minor.
+
+The temple was a decastyle peripteral structure of the Ionic order,
+standing on seven steps and possessing double rows of outer columns 60
+ft. high, twenty-one in each row on the flanks. It is remarkable not
+only for its great size, but (_inter alia_) for (1) the rich ornament of
+its column bases, which show great variety of design; (2) its various
+developments of the Ionic capital, e.g. heads of gods, probably of
+Pergamene art, spring from the "eyes" of the volutes with bulls' heads
+between them; (3) the massive building two storeys high at least, which
+served below for _prodomos_, and above for a dispensary of oracles
+([Greek: chresmographia] mentioned in the inscriptions) and a treasury;
+two flights of stairs called "labyrinths" in the inscriptions, led up to
+these chambers; (4) the pylon and staircase at the west; (5) the frieze
+of Medusa heads and foliage. Two outer columns are still erect on the
+north-east flank, carrying their entablature, and one of the inner order
+stands on the south-west. The fact that the temple was never finished is
+evident from the state in which some bases still remain at the west.
+There were probably no pedimental sculptures. A sacred way led from the
+temple to the sea at Panormus, which was flanked with rows of archaic
+statues, ten of which were excavated and sent to the British Museum in
+1858 by C. T. Newton. Fragments of architectural monuments, which once
+adorned this road, have also been found. Modern Hieronta is a large and
+growing Greek village, the only settlement within a radius of several
+miles. Its harbour is Kovella, distant about 2-1/2 m., and on the N. of
+the promontory.
+
+  See Dilettanti Society, _Ionian Antiquities_, ii. (1821); C. T.
+  Newton, _Hist. of Discoveries_, &c. (1862) and _Travels in the
+  Levant_, ii. (1865); O. Rayet and A. Thomas, _Milet et le Golfe
+  Latmique_ (1877); E. Pontremoli and B. Haussoullier, _Didymes_ (1904).
+       (D. G. H.)
+
+
+
+
+DIDYMIUM (from the Gr. [Greek: didymos], twin), the name given to the
+supposed element isolated by C. G. Mosander from cerite (1839-1841). In
+1879, however, Lecoq de Boisbaudran showed that Mosander's "didymium"
+contained samarium; while the residual "didymium," after removal of
+samarium, was split by Auer v. Welsbach (_Monats. f. Chemie_, 1885, 6,
+477) into two components (known respectively as neodymium and
+praseodymium) by repeated fractional crystallization of the double
+nitrate of ammonium and didymium in nitric acid. _Neodymium_ (Nd) forms
+the chief portion of the old "didymium." Its salts are reddish violet in
+colour, and give a characteristic absorption spectrum. It forms oxides
+of composition Nd2O3 and Nd2O5, the latter being obtained by ignition of
+the nitrate (B. Brauner). The atomic weight of neodymium is 143.6 (B.
+Brauner, _Proc. Chem. Soc._, 1897-1898, p. 70). _Praseodymium_ (Pr)
+forms oxides of composition Pr2O3, Pr2O5, xH2O (B. Brauner), and Pr4O7.
+The peroxide, Pr4O7, forms a dark brown powder, and is obtained by
+ignition of the oxalate or nitrate. The sesquioxide, Pr2O3, is obtained
+as a greenish white mass by the reduction of the peroxide. The salts of
+praseodymium are green in colour, and give a characteristic spark
+spectrum. The atomic weight of praseodymium is 140.5.
+
+
+
+
+DIDYMUS (?309-?394), surnamed "the Blind," ecclesiastical writer of
+Alexandria, was born about the year 309. Although he became blind at the
+age of four, before he had learned to read; he succeeded in mastering
+the whole circle of the sciences then known; and on entering the service
+of the Church he was placed at the head of the Catechetical school in
+Alexandria, where he lived and worked till almost the close of the
+century. Among his pupils were Jerome and Rufinus. He was a loyal
+follower of Origen, though stoutly opposed to Arian and Macedonian
+teaching. Such of his writings as survive show a remarkable knowledge of
+scripture, and have distinct value as theological literature. Among them
+are the _De Trinitate_, _De Spiritu Sancto_ (Jerome's Latin
+translation), _Adversus Manichaeos_, and notes and expositions of
+various books, especially the Psalms and the Catholic Epistles.
+
+  See Migne, _Patrol. Graec._ xxxix.; O. Bardenhewer, _Patrologie_, pp.
+  290-293 (Freiburg, 1894).
+
+
+
+
+DIDYMUS CHALCENTERUS (c. 63 B.C.-A.D. 10), Greek scholar and grammarian,
+flourished in the time of Cicero and Augustus. His surname (Gr. [Greek:
+Chalkenteros], brazen-bowelled) came from his indefatigable industry; he
+was said to have written so many books (more than 3500) that he was
+unable to recollect their names ([Greek: bibliolathas]). He lived and
+taught in Alexandria and Rome, where he became the friend of Varro. He
+is chiefly important as having introduced Alexandrian learning to the
+Romans. He was a follower of the school of Aristarchus, upon whose
+recension of Homer he wrote a treatise, fragments of which have been
+preserved in the Venetian Scholia. He also wrote commentaries on many
+other Greek poets and prose authors. In his work on the lyric poets he
+treated of the various classes of poetry and their chief
+representatives, and his lists of words and phrases (used in tragedy and
+comedy and by orators and historians), of words of doubtful meaning, and
+of corrupt expressions, furnished the later grammarians with valuable
+material. His activity extended to all kinds of subjects: grammar
+(orthography, inflexions), proverbs, wonderful stories, the law-tablets
+([Greek: axones]) of Solon, stones, and different kinds of wood. His
+polemic against Cicero's _De republica_ (Ammianus Marcellinus xxii. 16)
+provoked a reply from Suetonius. In spite of his stupendous industry,
+Didymus was little more than a compiler, of little critical judgment and
+doubtful accuracy, but he deserves recognition for having incorporated
+in his numerous writings the works of earlier critics and commentators.
+
+  See M. W. Schmidt, _De Didymo Chalcentero_ (1853) and _Didymi
+  Chalcenteri fragmenta_ (1854); also F. Susemihl, _Geschichte der
+  griech. Literatur in der Alexandrinerzeit_, ii. (1891); J. E. Sandys,
+  _History of Classical Scholarship_, i. (1906).
+
+
+
+
+DIE, a town of south-eastern France, capital of an arrondissement in the
+department of Drome, 43 m. E.S.E. of Valence on the Paris-Lyon railway.
+Pop. (1906) 3090. The town is situated in a plain enclosed by mountains
+on the right bank of the Drome below its confluence with the Meyrosse,
+which supplies power to some of the industries. The most interesting
+structures of Die are the old cathedral, with a porch of the 11th
+century supported on granite columns from an ancient temple of Cybele;
+and the Porte St Marcel, a Roman gateway flanked by massive towers. The
+Roman remains also include the ruins of aqueducts and altars. Die is the
+seat of a sub-prefect, and of a tribunal of first instance. The
+manufactures are silk, furniture, cloth, lime and cement, and there are
+flour and saw mills. Trade is in timber, especially walnut, and in white
+wine known as _clairette de Die_. The mulberry is largely grown for the
+rearing of silkworms. Under the Romans, Die (_Dea Augusta Vocontiorum_)
+was an important colony. It was formerly the seat of a bishopric, united
+to that of Valence from 1276 to 1687 and suppressed in 1790. Previous to
+the revocation of the edict of Nantes in 1685 it had a Calvinistic
+university.
+
+
+
+
+DIE (Fr. _de_, from Lat. _datum_, given), a word used in various senses,
+for a small cube of ivory, &c. (see DICE), for the engraved stamps used
+in coining money, &c., and various mechanical appliances in engineering.
+In architecture a "die" is the term used for the square base of a
+column, and it is applied also to the vertical face of a pedestal or
+podium.
+
+The fabrics known as "dice" take their name from the rectangular form of
+the figure. The original figures would probably be perfectly square, but
+to-day the same principle of weaving is applied, and the name dice is
+given to all figures of rectangular form. The different effects in the
+adjacent squares or rectangles are due to precisely the same reasons as
+those explained in connexion with the ground and the figure of damasks.
+The same weaves are used in both damasks and dices, but simpler weaves
+are generally employed for the commoner classes of the latter. The
+effect is, in every case, obtained by what are technically called warp
+and weft float weaves. The illustration B shows the two double damask
+weaves arranged to form a dice pattern, while A shows a similar pattern
+made from two four-thread twill weaves. C and D represent respectively
+the disposition of the threads in A and B with the first pick, and the
+solid marks represent the floats of warp. The four squares, which are
+almost as pronounced in the cloth as those of a chess-board, may be made
+of any size by repeating each weave for the amount of surface required.
+It is only in the finest cloths that the double damask weaves B are used
+for dice patterns, the single damask weaves and the twill weaves being
+employed to a greater extent. This class of pattern is largely employed
+for the production of table-cloths of lower and medium qualities. The
+term damask is also often applied to cloths of this character, and
+especially so when the figure is formed by rectangles of different
+sizes.
+
+[Illustration: A B C D]
+
+
+
+
+DIEBITSCH, HANS KARL FRIEDRICH ANTON, count von Diebitsch and Narden,
+called by the Russians Ivan Ivanovich, Count Diebich-Zabalkansky
+(1785-1831), Russian field-marshal, was born in Silesia on the 13th of
+May 1785. He was educated at the Berlin cadet school, but by the desire
+of his father, a Prussian officer who had passed into the service of
+Russia, he also did the same in 1801. He served in the campaign of 1805,
+and was wounded at Austerlitz, fought at Eylau and Friedland, and after
+Friedland was promoted captain. During the next five years of peace he
+devoted himself to the study of military science, engaging once more in
+active service in the War of 1812. He distinguished himself very greatly
+in Wittgenstein's campaign, and in particular at Polotzk (October 18 and
+19), after which combat he was raised to the rank of major-general. In
+the latter part of the campaign he served against the Prussian
+contingent of General Yorck (von Wartenburg), with whom, through
+Clausewitz, he negotiated the celebrated convention of Tauroggen,
+serving thereafter with Yorck in the early part of the War of
+Liberation. After the battle of Lutzen he served in Silesia and took
+part in negotiating the secret treaty of Reichenbach. Having
+distinguished himself at the battles of Dresden and Leipzig he was
+promoted lieutenant-general. At the crisis of the campaign of 1814 he
+strongly urged the march of the allies on Paris; and after their entry
+the emperor Alexander conferred on him the order of St Alexander Nevsky.
+In 1815 he attended the congress of Vienna, and was afterwards made
+adjutant-general to the emperor, with whom, as also with his successor
+Nicholas, he had great influence. By Nicholas he was created baron, and
+later count. In 1820 he had become chief of the general staff, and in
+1825 he assisted in suppressing the St Petersburg _emeute_. His greatest
+exploits were in the Russo-Turkish War of 1828-1829, which, after a
+period of doubtful contest, was decided by Diebitsch's brilliant
+campaign of Adrianople; this won him the rank of field-marshal and the
+honorary title of Zabalkanski to commemorate his crossing of the
+Balkans. In 1830 he was appointed to command the great army destined to
+suppress the insurrection in Poland. He won the terrible battle of
+Grochow on the 25th of February, and was again victorious at Ostrolenka
+on the 26th of May, but soon afterwards he died of cholera (or by his
+own hand) at Klecksewo near Pultusk, on the 10th of June 1831.
+
+  See Belmont (Schumberg), _Graf Diebitsch_ (Dresden, 1830); Sturmer,
+  _Der Tod des Grafen Diebitsch_ (Berlin, 1832); Bantych-Kamenski,
+  _Biographies of Russian Field-Marshals_ (in Russian, St Petersburg,
+  1841).
+
+
+
+
+DIEDENHOFEN (Fr. _Thionville_), a fortified town of Germany, in
+Alsace-Lorraine, dist. Lorraine, on the Mosel, 22 m. N. from Metz by
+rail. Pop. (1905) 6047. It is a railway junction of some consequence,
+with cultivation of vines, fruit and vegetables, brewing, tanning, &c.
+Diedenhofen is an ancient Frank town (Theudonevilla, Totonisvilla), in
+which imperial diets were held in the 8th century; was captured by Conde
+in 1643 and fortified by Vauban; capitulated to the Prussians, after a
+severe bombardment, on the 25th of November 1870.
+
+
+
+
+DIEKIRCH, a small town in the grand duchy of Luxemburg, charmingly
+situated on the banks of the Sure. Pop. (1905) 3705. Its name is said to
+be derived from Dide or Dido, granddaughter of Odin and niece of Thor.
+The mountain at the foot of which the town lies, now called Herrenberg,
+was formerly known as Thorenberg, or Thor's mountain. On the summit of
+this rock rises a perennial stream which flows down into the town under
+the name of Bellenflesschen. Diekirch was an important Roman station,
+and in the 14th century John of Luxemburg, the blind king of Bohemia,
+fortified it, surrounding the place with a castellated wall and a ditch
+supplied by the stream mentioned. It remained more or less fortified
+until the beginning of the 19th century when the French during their
+occupation levelled the old walls, and substituted the avenues of trees
+that now encircle the town. Diekirch is the administrative centre of one
+of the three provincial divisions of the grand duchy. It is visited
+during the summer by many thousand tourists and travellers from Holland,
+Belgium and Germany.
+
+
+
+
+DIELECTRIC, in electricity, a non-conductor of electricity; it is the
+same as insulator. The "dielectric constant" of a medium is its specific
+inductive capacity, and on the electromagnetic theory of light it equals
+the square of its refractive index for light of infinite wave length
+(see ELECTROSTATICS; MAGNETO-OPTICS).
+
+
+
+
+DIELMANN, FREDERICK (1847- ), American portrait and figure painter, was
+born at Hanover, Germany, on the 25th of December 1847. He was taken to
+the United States in early childhood; studied under Diez at the Royal
+Academy at Munich; was first an illustrator, and became a distinguished
+draughtsman and painter of genre pictures. His mural decorations and
+mosaic panels for the Congressional library, Washington, are notable. He
+was elected in 1899 president of the National Academy of Design.
+
+
+
+
+DIEMEN, ANTHONY VAN (1593-1645), Dutch admiral and governor-general of
+the East Indian settlements, was born at Kuilenburg in 1593. He was
+educated in commerce, and on entering the service of the East India
+Company speedily attained high rank. In 1631 he led a Dutch fleet from
+the Indies to Holland, and in 1636 he was raised to the
+governor-generalship. He came into conflict with the Portuguese, and
+took their possessions in Ceylon and Malacca from them. He greatly
+extended the commercial relationships of the Dutch, opening up trade
+with Tong-king, China and Japan. As an administrator also he showed
+ability, and the foundation of a Latin school and several churches in
+Batavia is to be ascribed to him. Exploring expeditions were sent to
+Australia under his auspices in 1636 and 1642, and Abel Tasman named
+after him (Van Diemen's Land) the island now called Tasmania. Van Diemen
+died at Batavia on the 19th of April 1645.
+
+
+
+
+DIEPENBECK, ABRAHAM VAN (1599-1675), Flemish painter, was born at
+Herzogenbusch, and studied painting at Antwerp, where he became one of
+Rubens's "hundred pupils." But he was not one of the cleverest of
+Rubens's followers, and he succeeded, at the best, in imitating the
+style and aping the peculiarities of his master. We see this in his
+earliest pictures--a portrait dated 1629 in the Munich Pinakothek, and a
+"Distribution of Alms" of the same period in the same collection. Yet
+even at this time there were moments when Diepenbeck probably fancied
+that he might take another path. A solitary copperplate executed with
+his own hand in 1630 represents a peasant sitting under a tree holding
+the bridle of an ass, and this is a minute and finished specimen of the
+engraver's art which shows that the master might at one time have hoped
+to rival the animal draughtsmen who flourished in the schools of
+Holland. However, large commissions now poured in upon him; he was asked
+for altarpieces, subject-pieces and pagan allegories. He was tempted to
+try the profession of a glass-painter, and at last he gave up every
+other occupation for the lucrative business of a draughtsman and
+designer for engravings. Most of Diepenbeck's important canvases are in
+continental galleries. The best are the "Marriage of St Catherine" at
+Berlin and "Mary with Angels Wailing over the Dead Body of Christ" in
+the Belvedere at Vienna, the first a very fair specimen of the artist's
+skill, the second a picture of more energy and feeling than might be
+expected from one who knew more of the outer form than of the spirit of
+Rubens. Then we have the fine "Entombment" at Brunswick, and "St Francis
+Adoring the Sacrament" at the museum at Brussels, "Clelia and her Nymphs
+Flying from the Presence and Pursuit of Porsenna" in two examples at
+Berlin and Paris, and "Neptune and Amphitrite" at Dresden. In all these
+compositions the drawing and execution are after the fashion of Rubens,
+though inferior to Rubens in harmony of tone and force of contrasted
+light and shade. Occasionally a tendency may be observed to imitate the
+style of Vandyck, for whom, in respect of pictures, Diepenbeck in his
+lifetime was frequently taken. But Diepenbeck spent much less of his
+leisure on canvases than on glass-painting. Though he failed to master
+the secrets of gorgeous tinting, which were lost, apparently for ever in
+the 16th century, he was constantly employed during the best years of
+his life in that branch of his profession. In 1635 he finished forty
+scenes from the life of St Francis of Paula in the church of the Minimes
+at Antwerp. In 1644 he received payment for four windows in St Jacques
+of Antwerp, two of which are still preserved, and represent Virgins to
+whom Christ appears after the Resurrection. The windows ascribed to him
+at St Gudule of Brussels were executed from the cartoons of Theodore van
+Thulden. On the occasion of his matriculation at Antwerp in 1638-1639,
+Diepenbeck was registered in the guild of St Luke as a glass-painter. He
+resigned his membership in the Artist Club of the Violette in 1542,
+apparently because he felt hurt by a valuation then made of drawings
+furnished for copperplates to the engraver Pieter de Jode. The earliest
+record of his residence at Antwerp is that of his election to the
+brotherhood (Sodalitat) "of the Bachelors" in 1634. It is probable that
+before this time he had visited Rome and London, as noted in the work of
+Houbraken. In 1636 he was made a burgess of Antwerp. He married twice,
+in 1637 and 1652. He died in December 1675, and was buried at St Jacques
+of Antwerp.
+
+
+
+
+DIEPPE, a seaport of northern France, capital of an arrondissement in
+the department of Seine-Inferieure, on the English Channel, 38 m. N. of
+Rouen, and 105 m. N.W. of Paris by the Western railway. Pop. (1906)
+22,120. It is situated at the mouth of the river Arques in a valley
+bordered on each side by steep white cliffs. The main part of the town
+lies to the west, and the fishing suburb of Le Pollet to the east of the
+river and harbour. The sea-front of Dieppe, which in summer attracts
+large numbers of visitors, consists of a pebbly beach backed by a
+handsome marine promenade. Dieppe has a modern aspect; its streets are
+wide and its houses, in most cases, are built of brick. Two squares side
+by side and immediately to the west of the outer harbour form the
+nucleus of the town, the Place Nationale, overlooked by the statue of
+Admiral A. Duquesne, and the Place St Jacques, named after the beautiful
+Gothic church which stands in its centre. The Grande Rue, the busiest
+and handsomest street, leads westward from the Place Nationale. The
+church of St Jacques was founded in the 13th century, but consists in
+large measure of later workmanship and was in some portions restored in
+the 19th century. The castle, overlooking the beach from the summit of
+the western cliff, was erected in 1435. The church of Notre-Dame de Bon
+Secours on the opposite cliff, and the church of St Remy, of the 16th
+and 17th centuries, are other noteworthy buildings. A well-equipped
+casino stands at the west end of the sea-front. The public institutions
+include the subprefecture, tribunals of first instance and commerce, a
+chamber of commerce, a communal college and a school of navigation.
+
+Dieppe has one of the safest and deepest harbours on the English
+Channel. A curved passage cut in the bed of the Arques and protected by
+an eastern and a western jetty gives access to the outer harbour, which
+communicates at the east end by a lockgate with the Bassin Duquesne and
+the Bassin Berigny, and at the west end by the New Channel, with an
+inner tidal harbour and two other basins. Vessels drawing 20 ft. can
+enter the new docks at neap tide. A dry-dock and a gridiron are included
+among the repairing facilities of the port. The harbour railway station
+is on the north-west quay of the outer harbour alongside which the
+steamers from Newhaven lie. The distance of Dieppe from Newhaven, with
+which there has long been daily communication, is 64 m. The imports
+include silk and cotton goods, thread, oil-seeds, timber, coal and
+mineral oil; leading exports are wine, silk, woollen and cotton fabrics,
+vegetables and fruit and flint-pebbles. The average annual value of
+imports for the five years 1901-1905 was L4,916,000 (L4,301,000 for the
+years 1896-1900); the exports were valued at L9,206,000 (L7,023,000 for
+years 1896-1900). The industries comprise shipbuilding, cotton-spinning,
+steam-sawing, the manufacture of machinery, porcelain, briquettes, lace,
+and articles in ivory and bone, the production of which dates from the
+15th century. There is also a tobacco factory of some importance. The
+fishermen of Le Pollet, to whom tradition ascribes a Venetian origin,
+are among the main providers of the Parisian market. The sea-bathing
+attracts many visitors in the summer. Two miles to the north-east of the
+town is the ancient camp known as the Cite de Limes, which perhaps
+furnished the nucleus of the population of Dieppe.
+
+It is suggested on the authority of its name, that Dieppe owed its
+origin to a band of Norman adventurers, who found its "diep" or inlet
+suitable for their ships, but it was unimportant till the latter half of
+the 12th century. Its first castle was probably built in 1188 by Henry
+II. of England, and it was counted a place of some consideration when
+Philip Augustus attacked it in 1195. By Richard I. of England it was
+bestowed in 1197 on the archbishop of Rouen in return for certain
+territory in the neighbourhood of the episcopal city. In 1339 it was
+plundered by the English, but it soon recovered from the blow, and in
+spite of the opposition of the lords of Hantot managed to surround
+itself with fortifications. Its commercial activity was already great,
+and it is believed that its seamen visited the coast of Guinea in 1339,
+and founded there a Petit Dieppe in 1365. The town was occupied by the
+English from 1420 to 1435. A siege undertaken in 1442 by John Talbot,
+first earl of Shrewsbury, was raised by the dauphin, afterwards Louis
+XI., and the day of the deliverance continued for centuries to be
+celebrated by a great procession and miracle plays. In the beginning of
+the 16th century Jean Parmentier, a native of the town, made voyages to
+Brazil and Sumatra; and a little later its merchant prince, Jacques
+Ango, was able to blockade the Portuguese fleet in the Tagus. Francis I.
+began improvements which were continued under his successor. Its
+inhabitants in great number embraced the reformed religion; and they
+were among the first to acknowledge Henry IV., who fought one of his
+great battles at the neighbouring village of Arques. Few of the cities
+of France suffered more from the revocation of the edict of Nantes in
+1685; and this blow was followed in 1694 by a terrible bombardment on
+the part of the English and Dutch. The town was rebuilt after the peace
+of Ryswick, but the decrease of its population and the deterioration of
+its port prevented the restoration of its commercial prosperity. During
+the 19th century it made rapid advances, partly owing to Marie Caroline,
+duchess of Berry, who brought it into fashion as a watering-place; and
+also because the establishment of railway communication with Paris gave
+an impetus to its trade. During the Franco-German War the town was
+occupied by the Germans from December 1870 till July 1871.
+
+  See L. Vitet, _Histoire de Dieppe_ (Paris, 1844); D. Asseline, _Les
+  Antiquites et chroniques de la ville de Dieppe_, a 17th-century
+  account published at Paris in 1874.
+
+
+
+
+DIERX, LEON (1838-   ), French poet, was born in the island of Reunion
+in 1838. He came to Paris to study at the Central School of Arts and
+Manufactures, and subsequently settled there, taking up a post in the
+education office. He became a disciple of Leconte de Lisle and one of
+the most distinguished of the Parnassians. In the death of Stephane
+Mallarme in 1898 he was acclaimed "prince of poets" by "les jeunes." His
+works include: _Poemes et poesies_ (1864); _Levres closes_ (1867);
+_Paroles d'un vaincu_ (1871); _La Rencontre_, a dramatic scene (1875)
+and _Les Amants_ (1879). His _Poesies completes_ (1872) were crowned by
+the French Academy. A complete edition of his works was published in 2
+vols., 1894-1896.
+
+
+
+
+DIES, CHRISTOPH ALBERT (1755-1822), German painter, was born at Hanover,
+and learned the rudiments of art in his native place. For one year he
+studied in the academy of Dusseldorf, and then he started at the age of
+twenty with thirty ducats in his pocket for Rome. There he lived a
+frugal life till 1796. Copying pictures, chiefly by Salvator Rosa, for a
+livelihood, his taste led him to draw and paint from nature in Tivoli,
+Albano and other picturesque places in the vicinity of Rome. Naples, the
+birthplace of his favourite master, he visited more than once for the
+same reasons. In this way he became a bold executant in water-colours
+and in oil, though he failed to acquire any originality of his own. Lord
+Bristol, who encouraged him as a copyist, predicted that he would be a
+second Salvator Rosa. But Dies was not of the wood which makes original
+artists. Besides other disqualifications, he had necessities which
+forced him to give up the great career of an independent painter. David,
+then composing his Horatii at Rome, wished to take him to Paris. But
+Dies had reasons for not accepting the offer. He was courting a young
+Roman whom he subsequently married. Meanwhile he had made the
+acquaintance of Volpato, for whom he executed numerous drawings, and
+this no doubt suggested the plan, which he afterwards carried out, of
+publishing, in partnership with Mechan, Reinhardt and Frauenholz, the
+series of plates known as the _Collection de vues pittoresques de
+l'Italie_, published in seventy-two sheets at Nuremberg in 1799. With so
+many irons in the fire Dies naturally lost the power of concentration.
+Other causes combined to affect his talent. In 1787 he swallowed by
+mistake three-quarters of an ounce of sugar of lead. His recovery from
+this poison was slow and incomplete. He settled at Vienna, and lived
+there on the produce of his brush as a landscape painter, and on that of
+his pencil or graver as a draughtsman and etcher. But instead of getting
+better, his condition became worse, and he even lost the use of one of
+his hands. In this condition he turned from painting to music, and spent
+his leisure hours in the pleasures of authorship. He did not long
+survive, dying at Vienna in 1822, after long years of chronic suffering.
+From two pictures now in the Belvedere gallery, and from numerous
+engraved drawings from the neighbourhood of Tivoli, we gather that Dies
+was never destined to rise above a respectable mediocrity. He followed
+Salvator Rosa's example in imitating the manner of Claude Lorraine. But
+Salvator adapted the style of Claude, whilst Dies did no more than copy
+it.
+
+
+
+
+DIEST, a small town in the province of Brabant, Belgium, situated on the
+Demer at its junction with the Bever. Pop. (1904) 8383. It lies about
+half-way between Hasselt and Louvain, and is still one of the five
+fortified places in Belgium. It contains many breweries, and is famous
+for the excellence of its beer.
+
+
+
+
+DIESTERWEG, FRIEDRICH ADOLF WILHELM (1790-1866), German educationist,
+was born at Siegen on the 29th of October 1790. Educated at Herborn and
+Tubingen universities, he took to the profession of teaching in 1811. In
+1820 he was appointed director of the new school at Mors, where he put
+in practice the methods of Pestalozzi. In 1832 he was summoned to Berlin
+to direct the new state-schools seminary in that city. Here he proved
+himself a strong supporter of unsectarian religious teaching. In 1846 he
+established the Pestalozzi institution at Pankow, and the Pestalozzi
+societies for the support of teachers' widows and orphans. In 1850 he
+retired on a pension, but continued vigorously to advocate his
+educational views. In 1858 he was elected to the chamber of deputies as
+member for the city of Berlin, and voted with the Liberal opposition. He
+died in Berlin on the 7th of July 1866. Diesterweg was a voluminous
+writer on educational subjects, and was the author of various school
+text-books.
+
+
+
+
+DIET, a term used in two senses, (1) food or the regulation of feeding
+(see DIETARY and DIETETICS), (2) an assembly or council (Fr. _diete_;
+It. _dieta_; Low Lat. _diaeta_; Ger. _Tag_). We are here concerned only
+with this second sense. In modern usage, though in Scotland the term is
+still sometimes applied to any assembly or session, it is practically
+confined to the sense of an assembly of estates or of national or
+federal representatives. The origin of the word in this connotation is
+somewhat complicated. It is undoubtedly ultimately derived from the
+Greek [Greek: diaita] (Lat. _diaeta_), which meant "mode of life" and
+thence "prescribed mode of life," the English "diet" or "regimen." This
+was connected with the verb [Greek: diaitan], in the sense of "to rule,"
+"to regulate"; compare the office of [Greek: diaitetes] at Athens, and
+_dieteta_, "umpire," in Late Latin. In both Greek and Latin, too, the
+word meant "a room," from which the transition to "a place of assembly"
+and so to "an assembly" would be easy. In the latter sense the word,
+however, actually occurs only in Low Latin, Du Cange (_Glossarium_,
+s.v.) deriving it from the late sense of "meal" or "feast," the Germans
+being accustomed to combine their political assemblies with feasting. It
+is clear, too, that the word _diaeta_ early became confused with Lat.
+_dies_, "day" (Ger. _Tag_), "especially a set day, a day appointed for
+public business; whence, by extension, meeting for business, an
+assembly" (Skeat). Instances of this confusion are given by Du Cange,
+e.g. _diaeta_ for _dieta_, "a day's journey" (also an obsolete sense of
+"diet" in English), and _dieta_ for "the ordinary course of the church,"
+i.e. "the daily office," which suggests the original sense of _diaeta_
+as "a prescribed mode of life."
+
+The word "diet" is now used in English for the _Reichstag_, "imperial
+diet" of the old Holy Roman Empire; for the _Bundestag_, "federal diet,"
+of the former Germanic confederation; sometimes for the _Reichstag_ of
+the modern German empire; for the _Landtage_, "territorial diets" of the
+constituent states of the German and Austrian empires; as well as for
+the former or existing federal or national assemblies of Switzerland,
+Hungary, Poland, &c. Although, however, the word is still sometimes used
+of all the above, the tendency is to confine it, so far as contemporary
+assemblies are concerned, to those of subordinate importance. Thus
+"parliament" is often used of the German _Reichstag_ or of the Russian
+Landtag, while the _Landtag_, e.g. of Styria, would always be rendered
+"diet." In what follows we confine ourselves to the diet of the Holy
+Roman Empire and its relation to its successors in modern Germany.
+
+The origin of the diet, or deliberative assembly, of the Holy Roman
+Empire must be sought in the _placitum_ of the Frankish empire. This
+represented the tribal assembly of the Franks, meeting (originally in
+March, but after 755 in May, whence it is called the Campus Maii) partly
+for a military review on the eve of the summer campaign, partly for
+deliberation on important matters of politics and justice. By the side
+of this larger assembly, however, which contained in theory, if not in
+practice, the whole body of Franks available for war, there had
+developed, even before Carolingian times, a smaller body composed of the
+magnates of the Empire, both lay and ecclesiastical. The germ of this
+smaller body is to be found in the episcopal synods, which, afforced by
+the attendance of lay magnates, came to be used by the king for the
+settlement of national affairs. Under the Carolingians it was usual to
+combine the assembly of magnates with the _generalis conventus_ of the
+"field of May," and it was in this inner assembly, rather than in the
+general body (whose approval was merely formal, and confined to matters
+momentous enough to be referred to a general vote), that the centre of
+power really lay. It is from the assembly of magnates that the diet of
+medieval Germany springs. The general assembly became meaningless and
+unnecessary, as the feudal array gradually superseded the old levy _en
+masse_, in which each freeman had been liable to service; and after the
+close of the 10th century it no longer existed.
+
+The imperial diet (_Reichstag_) of the middle ages might sometimes
+contain representatives of Italy, the _regnum Italicum_; but it was
+practically always confined to the magnates of Germany, the _regnum
+Teutonicum_. Upon occasion a summons to the diet might be sent even to
+the knights, but the regular members were the princes (_Fursten_), both
+lay and ecclesiastical. In the 13th century the seven electors began to
+disengage themselves from the prince as a separate element, and the
+Golden Bull (1356) made their separation complete; from the 14th century
+onwards the nobles (both counts and other lords) are regarded as regular
+members; while after 1250 the imperial and episcopal towns often appear
+through their representatives. By the 14th century, therefore, the
+originally homogeneous diet of princes is already, at any rate
+practically if not yet in legal form, divided into three colleges--the
+electors, the princes and nobles, and the representatives of the towns
+(though, as we shall see, the latter can hardly be reckoned as regular
+members until the century of the Reformation). Under the Hohenstaufen it
+is still the rule that every member of the diet must attend personally,
+or lose his vote; at a later date the principle of representation by
+proxy, which eventually made the diet into a mere congress of envoys,
+was introduced. By the end of the 13th century the vote of the majority
+had come to be regarded as decisive; but in accordance with the strong
+sense of social distinctions which marks German history, the quality as
+well as the quantity of votes was weighed, and if the most powerful of
+the princes were agreed, the opinion of the lesser magnates was not
+consulted. The powers of the medieval diet extended to matters like
+legislation, the decision upon expeditions (especially the _expeditio
+Romana_), taxation and changes in the constitution of the principalities
+or the Empire. The election of the king, which was originally regarded
+as one of the powers of the diet, had passed to the electors by the
+middle of the 13th century.
+
+A new era in the history of the diet begins with the Reformation. The
+division of the diet into three colleges becomes definite and precise;
+the right of the electors, for instance, to constitute a separate
+college is explicitly recognized as a matter of established custom in
+1544. The representatives of the towns now become regular members. In
+the 15th century they had only attended when special business, such as
+imperial reform or taxation, fell under discussion; in 1500, however,
+they were recognized as a separate and regular estate, though it was not
+until 1648 that they were recognized as equal to the other estates of
+the diet. The estate of the towns, or college of municipal
+representatives, was divided into two benches, the Rhenish and the
+Swabian. The estate of the princes and counts, which stood midway
+between the electors and the towns, also attained, in the years that
+followed the Reformation, its final organization. The vote of the great
+princes ceased to be personal, and began to be territorial. This had two
+results. The division of a single territory among the different sons of
+a family no longer, as of old, multiplied the voting power of the
+family; while in the opposite case, the union of various territories in
+the hands of a single person no longer meant the extinction of several
+votes, since the new owner was now allowed to give a vote for each of
+his territories. The position of the counts and other lords, who joined
+with the princes in forming the middle estate, was finally fixed by the
+middle of the 17th century. While each of the princes enjoyed an
+individual vote, the counts and other lords were arranged in groups,
+each of which voted as a whole, though the whole of its vote
+(_Kuriatstimme_) only counted as equal to the vote of a single prince
+(_Virilstimme_). There were six of these groups; but as the votes of the
+whole college of princes and counts (at any rate in the 18th century)
+numbered 100, they could exercise but little weight.
+
+The last era in the history of the diet may be said to open with the
+treaty of Westphalia (1648). The treaty acknowledged that Germany was no
+longer a unitary state, but a loose confederation of sovereign princes;
+and the diet accordingly ceased to bear the character of a national
+assembly, and became a mere congress of envoys. The "last diet" which
+issued a regular recess (_Reichsabschied_--the term applied to the
+_acta_ of the diet, as formally compiled and enunciated at its
+dissolution) was that of Regensburg in 1654. The next diet, which met at
+Regensburg in 1663, never issued a recess, and was never dissolved; it
+continued in permanent session, as it were, till the dissolution of the
+Empire in 1806. This result was achieved by the process of turning the
+diet from an assembly of principals into a congress of envoys. The
+emperor was represented by two _commissarii_; the electors, princes and
+towns were similarly represented by their accredited agents. Some
+legislation was occasionally done by this body; a _conclusum imperii_
+(so called in distinction from the old _recessus imperii_ of the period
+before 1663) might slowly (very slowly--for the agents, imperfectly
+instructed, had constantly to refer matters back to their principals) be
+achieved; but it rested with the various princes to promulgate and
+enforce the _conclusum_ in their territories, and they were sufficiently
+occupied in issuing and enforcing their own decrees. In practice the
+diet had nothing to do; and its members occupied themselves in
+"wrangling about chairs"--that is to say, in unending disputes about
+degrees and precedences.
+
+In the Germanic Confederation, which occupies the interval between the
+death of the Holy Roman Empire and the formation of the North German
+Confederation (1815-1866), a diet (_Bundestag_) existed, which was
+modelled on the old diet of the 18th century. It was a standing congress
+of envoys at Frankfort-on-Main. Austria presided in the diet, which, in
+the earlier years of its history, served, under the influence of
+Metternich, as an organ for the suppression of Liberal opinion. In the
+North German Confederation (1867-1870) a new departure was made, which
+has been followed in the constitution of the present German empire. Two
+bodies were instituted--a _Bundesrat_, which resembles the old diet in
+being a congress of envoys sent by the sovereigns of the different
+states of the confederation, and a _Reichstag_, which bears the name of
+the old diet, but differs entirely in composition. The new Reichstag is
+a popular representative assembly, based on wide suffrage and elected by
+ballot; and, above all, it is an assembly representing, not the several
+states, but the whole Empire, which is divided for this purpose into
+electoral districts. Both as a popular assembly, and as an assembly
+which represents the whole of a united Germany, the new Reichstag goes
+back, one may almost say, beyond the diet even of the middle ages, to
+the days of the old Teutonic folk-moot.
+
+  See R. Schroder, _Lehrbuch der deutschen Rechtsgeschichte_ (1902), pp.
+  149, 508, 820, 880. Schroder gives a bibliography of monographs
+  bearing on the history of the medieval diet.     (E. BR.)
+
+
+
+
+DIETARY, in a general sense, a system or course of diet, in the sense of
+food; more particularly, such an allowance and regulation of food as
+that supplied to workhouses, the army and navy, prisons, &c. Lowest in
+the scale of such dietaries comes what is termed "bare existence" diet,
+administered to certain classes of the community who have a claim on
+their fellow-countrymen that their lives and health shall be preserved
+_in statu quo_, but nothing further. This applies particularly to the
+members of a temporarily famine-stricken community. Before the days of
+prison reform, too, the dietary scale of many prisons was to a certain
+extent penal, in that the food supplied to prisoners was barely
+sufficient for existence. Nowadays more humane principles apply; there
+is no longer the obvious injustice of applying the same scale of
+quantity and quality to all prisoners under varying circumstances of
+constitution and surroundings, and whether serving long or short periods
+of imprisonment.
+
+  The system of dietary in force in the local and convict prisons of
+  England and Wales is that recommended by the Home Office on the advice
+  of a departmental committee. As to the local prison dietary, its
+  application is based on (1) the principle of variation of diet with
+  length of sentence; (2) the system of progressive dietary; (3) the
+  distinction between hard labour diets and non-hard labour diets; (4)
+  the differentiation of diet according to age and sex. There are three
+  classes of diet, classes A, B and C. Class A diet is given to
+  prisoners undergoing not more than seven days' imprisonment. The food
+  is good and wholesome, but sufficiently plain and unattractive, so as
+  not to offer temptation to the loafer or mendicant. It is given in
+  quantity sufficient to maintain health and strength during the single
+  week. Prisoners sentenced to more than seven days and not more than
+  fourteen days are given class A diet for the first seven days and
+  class B for the remainder of the sentence. In most of the local
+  prisons in England and Wales prisoners sentenced to hard labour
+  received hard labour diet, although quite 60% were unable to perform
+  the hardest forms of prison labour either through physical defect, age
+  or infirmity. The departmental committee of 1899 in their report
+  recommended that no distinction should be made between hard labour and
+  non-hard labour diets. Class A diet is as follows:--_Breakfast_,
+  Bread, 8 oz. daily (6 oz. for women and juveniles) with 1 pint of
+  gruel. Juveniles (males and females under sixteen years of age) get,
+  in addition, 1/2 pint of milk. _Dinner_, 8 oz. of bread daily, with 1
+  pint of porridge on three days of the week, 8 oz. of potatoes
+  (representing the vegetable element) on two other days, and 8 oz. of
+  suet pudding (representing the fatty element) on the other two days.
+  _Supper_, the breakfast fare repeated.
+
+  Class B diet, which is also given to (1) prisoners on remand or
+  awaiting trial, (2) offenders of the 1st division who do not maintain
+  themselves, (3) offenders of the 2nd division and (4) debtors, is as
+  shown in Table I.
+
+  Class C diet is class B amplified, and is given to those prisoners
+  serving sentences of three months and over.
+
+
+      TABLE I.
+
+    +------------+------------------+--------+--------+-----------+
+    |            |                  |  Men.  | Women. | Juveniles.|
+    +------------+------------------+--------+--------+-----------+
+    |            |                  |        |        |           |
+    | Breakfast. | Daily:--         |        |        |           |
+    |            |    Bread         |  8 oz. |  6 oz. |   6 oz.   |
+    |            |    Gruel         |  1 pt. |  1 pt. |   1 pt.   |
+    |            |    Milk          |    ..  |    ..  |   1/2 pt. |
+    |            |                  |        |        |           |
+    | Dinner.    | Sunday:--        +--------+--------+-----------+
+    |            |    Bread         |  6 oz. |       6 oz.        |
+    |            |    Potatoes      |  8  "  |       8  "         |
+    |            |    Cooked meat,  |        |                    |
+    |            |      preserved   |  4  "  |       3  "         |
+    |            |      by heat     |        |                    |
+    |            | Monday:--        |        |                    |
+    |            |    Bread         |  6 oz. |       6 oz.        |
+    |            |    Potatoes      |  8  "  |       8  "         |
+    |            |    Beans         | 10  "  |       8  "         |
+    |            |    Fat bacon     |  2  "  |       1  "         |
+    |            |                  |        |                    |
+    |            | Tuesday:--       |        |                    |
+    |            |    Bread         |  6 oz  |       6 oz.        |
+    |            |    Potatoes      |  8  "  |       8  "         |
+    |            |    Soup          |  1 pt. |       1 pt.        |
+    |            |                  |        |                    |
+    |            | Wednesday:--     |        |                    |
+    |            |    Bread         |  6 oz. |       6 oz.        |
+    |            |    Potatoes      |  8  "  |       8  "         |
+    |            |    Suet pudding  | 10  "  |       8  "         |
+    |            |                  |        |                    |
+    |            | Thursday:--      |        |                    |
+    |            |    Bread         |  6 oz. |       6 oz.        |
+    |            |    Potatoes      |  8  "  |       8  "         |
+    |            |    Cooked beef,  |        |                    |
+    |            |      without bone|  4  "  |       3  "         |
+    |            |                  |        |                    |
+    |            | Friday:--        |        |                    |
+    |            |    Bread         |  6 oz. |       6 oz.        |
+    |            |    Potatoes      |  8  "  |       8  "         |
+    |            |    Soup          |  1 pt. |       1 pt.        |
+    |            |                  |        |                    |
+    |            | Saturday:--      |        |                    |
+    |            |    Bread         |  6 oz. |       6 oz.        |
+    |            |    Potatoes      |  8  "  |       8  "         |
+    |            |    Suet pudding  | 10  "  |       8  "         |
+    |            |                  |        |                    |
+    | Supper.    | Daily:--         |        +---------+----------+
+    |            |    Bread         |  8 oz. |   6 oz. |   6 oz.  |
+    |            |    Porridge      |  1 pt. |         |          |
+    |            |    Gruel         |        |   1 pt. |          |
+    |            |    Cocoa         |        |         |   1 pt.  |
+    +------------+------------------+--------+---------+----------+
+
+
+      TABLE II.
+    +------------------+-----+------+------+------+------+------+------+
+    |                  |Sun. | Mon. | Tue. | Wed. | Thu. | Fri. | Sat. |
+    +------------------+-----+------+------+------+------+------+------+
+    | Breakfast.       |     |      |      |      |      |      |      |
+    |  Bread.       oz.|  8  |   4  |   4  |   4  |   4  |   4  |   4  |
+    |  Porridge.    pt.|  *  |1-1/2 |1-1/2 |1-1/2 |1-1/2 |1-1/2 |1-1/2 |
+    +------------------+-----+------+------+------+------+------+------+
+    | Dinner.          |     |      |      |      |      |      |      |
+    |  Bread.       oz.|  4  |   6  |  ..  |   4  |   4  |   8  |   6  |
+    |  Beef.        oz.|4-1/2|  ..  |  ..  |  ..  | 4-1/2|  ..  |  ..  |
+    |  Vegetables.  oz.| 12  |  ..  |  ..  |  12  |  12  |  ..  |  ..  |
+    |  Barley Soup. pt.| ..  | 1-1/2|  ..  |  ..  |  ..  |  ..  |  ..  |
+    |  Pork.        oz.| ..  |  ..  | 4-1/2|  ..  |  ..  |  ..  |  ..  |
+    |  Beans.       oz.| ..  |  ..  |  12  |  ..  |  ..  |  ..  |  ..  |
+    |  Fish.        oz.| ..  |  ..  |  ..  |  10  |  ..  |  ..  |  ..  |
+    |  Cheese.      oz.| ..  |  ..  |  ..  |  ..  |  ..  |   3  |  ..  |
+    |  Broth.       pt.| ..  |  ..  |  ..  |  ..  |  ..  |   1  |  ..  |
+    |  Irish Stew.  pt.| ..  |  ..  |  ..  |  ..  |  ..  |  ..  |   1  |
+    +------------------+-----+------+------+------+------+------+------+
+    | Supper.          |     |      |      |      |      |      |      |
+    |  Bread.       oz.|  8  |   6  |   6  |   6  |   8  |   6  |   6  |
+    |  Butter.      oz.| 1/2 |  ..  |  ..  |  ..  |  ..  |  ..  |  ..  |
+    |  Tea.         pt.|  1  |  ..  |  ..  |  ..  |  ..  |  ..  |  ..  |
+    |  Gruel.       pt.| ..  | 1-1/2| 1-1/2| 1-1/2|  ..  | 1-1/2| 1-1/2|
+    |  Broth.       pt.| ..  |  ..  |  ..  |  ..  |   1  |  ..  |  ..  |
+    |  Cheese.      oz.| ..  |  ..  |  ..  |  ..  |   2  |  ..  |  ..  |
+    +------------------+-----+------+------+------+------+------+------+
+        * On Sundays 1 pint of tea and 2-1/2 oz. of butter are given instead
+            of porridge.
+
+  The dietary of convict prisons, in which prisoners are all under long
+  sentence, is divided into a diet for convicts employed at hard labour
+  and a diet for convicts employed at sedentary, indoor and light
+  labour. It will be found set forth in the Blue-book mentioned above.
+  The sparest of all prison diets is called "punishment diet," and is
+  administered for offences against the internal discipline of the
+  prison. It is limited to a period of three days. It consists of 1 lb.
+  of bread and as much water as the prisoner chooses to drink.
+
+  In French prisons the dietary is nearly two pounds weight of bread,
+  with two meals of thin soup (breakfast and dinner) made from potatoes,
+  beans or other vegetables, and on two days a week made from meat. In
+  France the canteen system is in vogue, additional food, such as
+  sausages, cheese, fruit, &c., may be obtained by the prisoner,
+  according to the wages he receives for his labours. The dietary of
+  Austrian prisons is 1-1/2 lb. of bread daily, a dinner of soup on four
+  days of the week, and of meat on the other three days, with a supper
+  of soup or vegetable stew. Additional food can be purchased by the
+  prisoner out of his earnings.
+
+  These dietaries may be taken as more or less typical of the ordinary
+  prison fare in most civilized countries, though in some countries it
+  may err on the side of severity, as in Sweden, prisoners being given
+  only two meals a day, one at mid-day and one at seven p.m., porridge
+  or gruel being the principal element in both meals. On the other hand,
+  the prison dietaries of many of the United States prisons go to the
+  other extreme, fresh fish, green vegetables, even coffee and fruit,
+  figuring in the dietary.
+
+  Another class of dietary is that given to paupers. In England, until
+  1900, almost every individual workhouse had its own special dietary,
+  with the consequence that many erred on the side of scantiness and
+  unsuitability, while others were too lavish. By an order of the Local
+  Government Board of that year, acting on a report of a committee, all
+  inmates of workhouses, with the exception of the sick, children under
+  three years of age, and certain other special cases, are dieted in
+  accordance with certain dietary tables as framed and settled by the
+  board. The order contained a great number of different rations, it
+  being left to the discretion of the guardians as to the final
+  settlement of the tables. For adult inmates the dietary tables are for
+  each sex respectively, two in number, one termed "plain diet" and the
+  other "infirm diet." All male inmates certified as healthy able-bodied
+  persons receive plain diet only. All inmates, however, in workhouses
+  are kept employed according to their capacity and ability, and this is
+  taken into consideration in giving allowances of food. For instance,
+  for work with sustained exertion, such as stone-breaking, digging,
+  &c., more food is given than for work without sustained exertion, such
+  as wood-chopping, weeding or sewing. Table II. shows an example of a
+  workhouse dietary.
+
+  In the casual wards of workhouses the dietary is plainer, consisting
+  of 8 oz. of bread, or 6 oz. of bread and one pint of gruel or broth
+  for breakfast; the same for supper; for dinner 8 oz. of bread and 1-1/2
+  oz. of cheese or 6 oz. of bread and one pint of soup. The American
+  poor law system is based broadly on that of England, and the methods
+  of relief are much the same. Each state, however, makes its own
+  regulations, and there is considerable diversity in workhouse
+  dietaries in consequence. The German system of poor relief is more
+  methodical than those of England and America. The really deserving are
+  treated with more commiseration, and a larger amount of outdoor
+  relief is given than in England. There is no casual ward, tramps and
+  beggars being liable to penal treatment, but there are "relief
+  stations," somewhat corresponding to casual wards, where destitute
+  persons tramping from one place to another can obtain food and lodging
+  in return for work done.
+
+  In the British navy certain staple articles of diet are supplied to
+  the men to the value approximately of 6d. per diem--the standard
+  government ration--and, in addition, a messing allowance of 4d. per
+  diem, which may either be expended on luxuries in the canteen, or in
+  taking up government provisions on board ship, in addition to the
+  standard ration. The standard ration as recommended in 1907 by a
+  committee appointed to inquire into the question of victualling in the
+  navy is as follows:--
+
+  _Service Afloat._
+
+    1 lb. bread (or 3/4 lb. bread and 1/4 lb. trade flour).
+    1/2 lb. fresh meat.
+    1 lb. fresh vegetables.
+    1/8 pint spirit.
+    4 oz. sugar.
+    1/2 oz. tea (or 1 oz. coffee for every 1/4 oz. tea).
+    1/2 oz. ordinary or soluble chocolate (or 1 oz. coffee).
+    3/4 oz. condensed milk.
+    1 oz. jam or marmalade.
+    4 oz. preserved meat on _one_ day of the week in harbour, or on
+       _two_ days at sea.
+
+  Mustard, pepper, vinegar, and salt as required.
+
+  Substitute for soft bread when the latter is not available--
+      1/2 lb. biscuit (new type) or 1 lb. flour.
+
+  Substitutes for fresh meat when the latter is not available:--
+
+              /(1) Salt pork day:--
+             |     1/2 lb. salt pork.
+             |     1/4 lb. split peas.
+             |     Celery seed, 1/2 oz. to every 8 lb. of split peas put
+             |       into the coppers.
+      On     |     1/2 lb. potatoes (or 1 oz. compressed vegetables).
+   alternate<
+     days    | (2) Preserved meat day:--
+             |     6 oz. preserved meat.
+             |     8 oz. trade flour   \
+             |     3/4 oz. refined suet   > or 4 oz. rice.
+             |     2 oz. raisins       /
+              \    1/2 lb. potatoes (or 1 oz. compressed vegetables).
+
+  On shore establishments and depot ships 1/4 pt. fresh milk is issued in
+  lieu of the 3/4 oz. of condensed milk.
+
+  In the United States navy there is more liberality and variety of
+  diet, the approximate daily cost of the rations supplied being 1s. 3d.
+  per head. In the American mercantile marine, too, according to the
+  scale sanctioned by act of Congress (December 21, 1898) for American
+  ships, the seaman is better off than in the British merchant service.
+  The scale is shown in Table III.
+
+
+      TABLE III.
+
+    +---------+-----------------------++-----------+----------------+
+    | Weekly  |       Articles.       ||  Weekly   |    Articles.   |
+    | Scale.  |                       ||  Scale.   |                |
+    +---------+-----------------------++-----------+----------------+
+    |3-1/2 lb.| Biscuits.             || 7/8  oz.  | Tea.           |
+    |3-3/4  " | Salt beef.            || 21    "   | Sugar.         |
+    | 3     " |  "   pork.            || 1-1/2  lb.| Molasses.      |
+    |1-1/2  " | Flour.                ||  9   oz.  | Fruits, dried. |
+    | 2     " | Meats, preserved.     ||  3/4  pt. | Pickles.       |
+    |10-1/2 " | Bread, fresh (8 lb.   ||   1   "   | Vinegar.       |
+    |         |   flour in lieu).     ||  8   oz.  | Corn Meal.     |
+    | 1    "  | Fish, dried.          || 12    "   | Onions.        |
+    | 7    "  | Potatoes or yams.     ||  7    "   | Lard.          |
+    | 1    "  | Tomatoes, preserved.  ||  7    "   | Butter.        |
+    |2/3   "  | Peas.                 ||  1/4  "   | Mustard.       |
+    |2/3   "  | Calavances.           ||  1/4  "   | Pepper.        |
+    |2/3   "  | Rice.                 ||  1/4  "   | Salt.          |
+    |5-1/4 oz.| Coffee, green.        ||           |                |
+    +---------+-----------------------++-----------+----------------+
+
+  In the British mercantile marine there is no scale of provisions
+  prescribed by the Board of Trade; there is, however, a traditional
+  scale very generally adopted, having the sanction of custom only and
+  seldom adhered to. The following dietary scale for steerage
+  passengers, laid down in the 12th schedule of the Merchant Shipping
+  Act 1894, is of interest. See Table IV.
+
+  Certain substitutions may be made in this scale at the option of the
+  master of any emigrant ship, provided that the substituted articles
+  are set forth in the contract tickets of the steerage passengers.
+
+  In the British army the soldier is fed partly by a system of
+  co-operation. He gets a free ration from government of 1 lb. of bread
+  and 3/4 lb. of meat; in addition there is a messing allowance of 3-1/2d.
+  per man per day. He is able to supplement his food by purchases from
+  the canteen. Much depends on the individual management in each
+  regiment as to the satisfactory expenditure of the messing allowance.
+  In some regiments an allowance is made from the canteen funds towards
+  messing in addition to that granted by the government. The ordinary
+  _field_ ration of the British soldier is 1-1/2 lb. of bread or 1 lb. of
+  biscuit; 1 lb. of fresh, salt or preserved meat; 1/2 oz. of coffee; 1/6
+  oz. of tea; 2 oz. of sugar; 1/2 oz. of salt, 1/36 oz. of pepper, the
+  whole weighing something over 2 lb. 3 oz. This cannot be looked on as
+  a fixed ration, as it varies in different campaigns, according to the
+  country into which the troops may be sent. The Prussian soldier during
+  peace gets weekly from his canteen 11 lb. 1 oz. of rye bread and not
+  quite 2-1/2 lb. of meat. This is obviously insufficient, but under
+
+
+      TABLE IV.--_Weekly, per Statute Adult._
+
+    +---------------------------+-------------------+-------------------+
+    |                           |      Scale A.     |      Scale B.     |
+    |                           |For voyages not    |For voyages        |
+    |                           | exceeding 84 days | exceeding 84 days |
+    |                           | for sailing ships | for sailing ships |
+    |                           | or 50 days        | or 50 days        |
+    |                           | for steamships.   | for steamships.   |
+    +---------------------------+-------------------+-------------------+
+    |                           |     lb.   oz.     |     lb.   oz.     |
+    | Bread or biscuit, not     |                   |                   |
+    |  inferior to navy biscuit |      3    8       |      3    8       |
+    | Wheaten flour             |      1    0       |      2    0       |
+    | Oatmeal                   |      1    8       |      1    0       |
+    | Rice                      |      1    8       |      0    8       |
+    | Peas                      |      1    8       |      1    8       |
+    | Beef                      |      1    4       |      1    4       |
+    | Pork                      |      1    0       |      1    0       |
+    | Butter                    |        ..         |      0    4       |
+    | Potatoes                  |      2    0       |      2    0       |
+    | Sugar                     |      1    0       |      1    0       |
+    | Tea                       |      0    2       |      0    2       |
+    | Salt                      |      0    2       |      0    2       |
+    | Pepper (white or          |                   |                   |
+    |  black), ground           |      0    0-1/2   |      0    0-1/2   |
+    | Vinegar                   |      1 gill       |      1 gill       |
+    | Preserved meat            |        ..         |      1    0       |
+    | Suet                      |                   |      0    6       |
+    | Raisins                   |                   |      0    8       |
+    | Lime juice                |                   |      0    6       |
+    +---------------------------+-------------------+-------------------+
+
+  the conscription system it is reckoned that he will be able to make up
+  the deficiency out of his own private means, or obtain charitable
+  contributions from his friends. In the French infantry of the line
+  each man during peace gets weekly 15 lb. of bread, 3-3/10 lb. of meat,
+  2-1/2 lb. of haricot beans or other vegetables, with salt and pepper, and
+  1-3/4 oz. of brandy.
+
+  An Austrian under the same circumstances receives 13.9 lb. of bread, 1/2
+  lb. of flour and 3.3 lb. of meat.
+
+  The Russian conscript is allowed weekly:--
+
+    Black bread              7 lb.
+    Meat                     7 lb.
+    Kvass (beer)             7.7 quarts.
+    Sour cabbage             24-1/2 gills = 122-1/2 oz.
+    Barley                   24-1/2 gills = 122-1/2 oz.
+    Salts                    10-1/2 oz.
+    Horse-radish             28 grains.
+    Pepper                   28 grains.
+    Vinegar                  5-1/2 gills = 26-1/2 oz.
+
+
+
+
+DIETETICS, the science of diet, i.e. the food and nutrition of man in
+health and disease (see NUTRITION). This article deals mainly with that
+part of the subject which has to do with the composition and nutritive
+values of foods and their adaptation to the use of people in health. The
+principal topics considered are: (1) Food and its functions; (2)
+Metabolism of matter and energy; (3) Composition of food materials; (4)
+Digestibility of food; (5) Fuel value of food; (6) Food consumption; (7)
+Quantities of nutrients needed; (8) Hygienic economy of food; (9)
+Pecuniary economy of food.
+
+1. _Food and its Functions._--For practical purposes, food may be
+defined as that which, when taken into the body, may be utilized for the
+formation and repair of body tissue, and the production of energy. More
+specifically, food meets the requirements of the body in several ways.
+It is used for the formation of the tissues and fluids of the body, and
+for the restoration of losses of substance due to bodily activity. The
+potential energy of the food is converted into heat or muscular work or
+other forms of energy. In being thus utilized, food protects body
+substance or previously acquired nutritive material from consumption.
+When the amount of food taken into the body is in excess of immediate
+needs, the surplus may be stored for future consumption.
+
+Ordinary food materials, such as meat, fish, eggs, vegetables, &c.,
+consist of inedible materials, or _refuse_, e.g. bone of meat and fish,
+shell of eggs, rind and seed of vegetables; and _edible material_, as
+flesh of meat and fish, white and yolk of eggs, wheat flour, &c. The
+edible material is by no means a simple substance, but consists of
+_water_, and some or all of the compounds variously designated as food
+stuffs, proximate principles, nutritive ingredients or nutrients, which
+are classified as _protein_, _fats_, _carbohydrates_ and _mineral
+matters_. These have various functions in the nourishment of the body.
+
+The _refuse_ commonly contains compounds similar to those in the food
+from which it is derived, but since it cannot be eaten, it is usually
+considered as a non-nutrient. It is of importance chiefly in a
+consideration of the pecuniary economy of food. _Water_ is also
+considered as a non-nutrient, because although it is a constituent of
+all the tissues and fluids of the body, the body may obtain the water it
+needs from that drunk; hence, that contained in the food materials is of
+no special significance as a nutrient.
+
+_Mineral matters_, such as sulphates, chlorides, phosphates and
+carbonates of sodium, potassium, calcium, &c., are found in different
+combinations and quantities in most food materials. These are used by
+the body in the formation of the various tissues, especially the
+skeletal and protective tissues, in digestion, and in metabolic
+processes within the body. They yield little or no energy, unless
+perhaps the very small amount involved in their chemical transformation.
+
+Protein[1] is a term used to designate the whole group of nitrogenous
+compounds of food except the nitrogenous fats. It includes the
+albuminoids, as albumin of egg-white, and of blood serum, myosin of meat
+(muscle), casein of milk, globulin of blood and of egg yolk, fibrin of
+blood, gluten of flour; the gelatinoids, as gelatin and allied
+substances of connective tissue, collagen of tendon, ossein of bone and
+the so-called extractives (e.g. creatin) of meats; and the amids (e.g.
+asparagin) and allied compounds of vegetables and fruits.
+
+The albuminoids and gelatinoids, classed together as proteids, are the
+most important constituents of food, because they alone can supply the
+nitrogenous material necessary for the formation of the body tissues.
+For this purpose, the albuminoids are most valuable. Both groups of
+compounds, however, supply the body with energy, and the gelatinoids in
+being thus utilized protect the albuminoids from consumption for this
+purpose. When their supply in the food is in excess of the needs of the
+body, the surplus proteids may be converted into body fat and stored.
+
+The so-called extractives, which are the principal constituents of meat
+extract, beef tea and the like, act principally as stimulants and
+appetizers. It has been believed that they serve neither to build tissue
+nor to yield energy, but recent investigations[2] indicate that creatin
+may be metabolized in the body.
+
+The _fats_ of food include both the animal fats and the vegetable oils.
+The _carbohydrates_ include such compounds as starches, sugars and the
+fibre of plants or cellulose, though the latter has but little value as
+food for man. The more important function of both these classes of
+nutrients is to supply energy to the body to meet its requirements above
+that which it may obtain from the proteids. It is not improbable that
+the atoms of their molecules as well as those from the proteids are
+built up into the protoplasmic substance of the tissues. In this sense,
+these nutrients may be considered as being utilized also for the
+formation of tissue; but they are rather the accessory ingredients,
+whereas the proteids are the essential ingredients for this purpose. The
+fats in the food in excess of the body requirements may be stored as
+body fat, and the surplus carbohydrates may also be converted into fat
+and stored.
+
+To a certain extent, then, the nutrients of the food may substitute each
+other. All may be incorporated into the protoplasmic structure of body
+tissue, though only the proteids can supply the essential nitrogenous
+ingredients; and apart from the portion of the proteid material that is
+indispensable for this purpose, all the nutrients are used as a source
+of energy. If the supply of energy in the food is not sufficient, the
+body will use its own proteid and fat for this purpose. The gelatinoids,
+fats and carbohydrates in being utilized for energy protect the body
+proteids from consumption. The fat stored in the body from the excess of
+food is a reserve of energy material, on which the body may draw when
+the quantity of energy in the food is insufficient for its immediate
+needs.
+
+What compounds are especially concerned in intellectual activity is not
+known. The belief that fish is especially rich in phosphorus and
+valuable as a brain food has no foundation in observed fact.
+
+2. _Metabolism of Matter and Energy._--The processes of nutrition thus
+consist largely of the transformation of food into body material and the
+conversion of the potential energy of both food and body material into
+the kinetic energy of heat and muscular work and other forms of energy.
+These various processes are generally designated by the term metabolism.
+The metabolism of matter in the body is governed largely by the needs of
+the body for energy. The science of nutrition, of which the present
+subject forms a part, is based on the principle that the transformations
+of matter and energy in the body occur in accordance with the laws of
+the conservation of matter and of energy. That the body can neither
+create nor destroy matter has long been universally accepted. It would
+seem that the transformation of energy must likewise be governed by the
+law of the conservation of energy; indeed there is every reason a priori
+to believe that it must; but the experimental difficulties in the way of
+absolute demonstration of the principle are considerable. For such
+demonstration it is necessary to prove that the income and expenditure
+of energy are equal. Apparatus and methods of inquiry devised in recent
+years, however, afford means for a comparison of the amounts of both
+matter and energy received and expended by the body, and from the
+results obtained in a large amount of such research, it seems probable
+that the law obtains in the living organism in general.
+
+The first attempt at such demonstration was made by M. Rubner[3] in
+1894, experimenting with dogs doing no external muscular work. The
+income of energy (as heat) was computed, but the heat eliminated was
+measured. In the average of eight experiments continuing forty-five
+days, the two quantities agreed within 0.47%, thus demonstrating what it
+was desired to prove--that the heat given off by the body came solely
+from the oxidation of food within it. Results in accordance with these
+were reported by Studenski[4] in 1897, and by Laulanie[5] in 1898.
+
+The most extensive and complete data yet available on the subject have
+been obtained by W. O. Atwater, F. G. Benedict and associates[6] in
+experiments with men in the respiration calorimeter, in which a subject
+may remain for several consecutive days and nights. These experiments
+involve actual weighing and analyses of the food and drink, and of the
+gaseous, liquid and solid excretory products; determinations of
+potential energy (heat of oxidation) of the oxidizable material received
+and given off by the body (including estimation of the energy of the
+material gained or lost by the body); and measurements of the amounts of
+energy expended as heat and as external muscular work. By October 1906
+eighty-eight experiments with fifteen different subjects had been
+completed. The separate experiments continued from two to thirteen days,
+making a total of over 270 days.
+
+
+    TABLE I.--_Percentage Composition of some Common Food Materials._
+
+  +--------------------------+--------+--------+---------+------+---------+--------+-----------+
+  |     Food Material.       | Refuse.|  Water.| Protein.| Fat. | Carbo-  | Mineral| Fuel Value|
+  |                          |        |        |         |      |hydrates.| Matter.| per lb.   |
+  +--------------------------+--------+--------+---------+------+---------+--------+-----------+
+  |                          |        |        |         |      |         |        |           |
+  |                          |    %   |    %   |    %    |   %  |    %    |    %   | Calories. |
+  |Beef, fresh (medium fat)--|        |        |         |      |         |        |           |
+  |  Chuck                   |  16.3  |  52.6  |  15.5   | 15.0 |   . .   |   0.8  |    910    |
+  |  Loin                    |  13.3  |  52.5  |  16.1   | 17.5 |   . .   |   0.9  |   1025    |
+  |  Ribs                    |  20.8  |  43.8  |  13.9   | 21.2 |   . .   |   0.7  |   1135    |
+  |  Round                   |   7.2  |  60.7  |  19.0   | 12.8 |   . .   |   1.0  |    890    |
+  |  Shoulder                |  16.4  |  56.8  |  16.4   |  9.8 |   . .   |   0.9  |    715    |
+  |Beef, dried and smoked    |   4.7  |  53.7  |  26.4   |  6.9 |   . .   |   8.9  |    790    |
+  |Veal--                    |        |        |         |      |         |        |           |
+  |  Leg                     |  14.2  |  60.1  |  15.5   |  7.9 |   . .   |   0.9  |    625    |
+  |  Loin                    |  16.5  |  57.6  |  16.6   |  9.0 |   . .   |   0.9  |    685    |
+  |  Breast                  |  21.3  |  52.0  |  15.4   | 11.0 |   . .   |   0.8  |    745    |
+  |Mutton--                  |        |        |         |      |         |        |           |
+  |  Leg                     |  18.4  |  51.2  |  15.1   | 14.7 |   . .   |   0.8  |    890    |
+  |  Loin                    |  16.0  |  42.0  |  13.5   | 28.3 |   . .   |   0.7  |   1415    |
+  |  Flank                   |   9.9  |  39.0  |  13.8   | 36.9 |   . .   |   0.6  |   1770    |
+  |Pork--                    |        |        |         |      |         |        |           |
+  |  Loin                    |  19.7  |  41.8  |  13.4   | 24.2 |   . .   |   0.8  |   1245    |
+  |  Ham, fresh              |  10.7  |  48.0  |  13.5   | 25.9 |   . .   |   0.8  |   1320    |
+  |  Ham, smoked and salted  |  13.6  |  34.8  |  14.2   | 33.4 |   . .   |   4.2  |   1635    |
+  |  Fat, salt               |  . .   |   7.9  |   1.9   | 86.2 |   . .   |   3.9  |   3555    |
+  |  Bacon                   |   7.7  |  17.4  |   9.1   | 62.2 |   . .   |   4.1  |   2715    |
+  |  Lard, refined           |  . .   |  . .   |  . .    |100.0 |   . .   |  . .   |   4100    |
+  |Chicken                   |  25.9  |  47.1  |  13.7   | 12.3 |   . .   |   0.7  |    765    |
+  |Turkey                    |  22.7  |  42.4  |  16.1   | 18.4 |   . .   |   0.8  |   1060    |
+  |Goose                     |  17.6  |  38.5  |  13.4   | 29.8 |   . .   |   0.7  |   1475    |
+  |Eggs                      |  11.2  |  65.5  |  13.1   |  9.3 |   . .   |   0.9  |    635    |
+  |Cod, fresh                |  29.9  |  58.5  |  11.1   |  0.2 |   . .   |   0.8  |    220    |
+  |Cod, salted               |  24.9  |  40.2  |  16.0   |  0.4 |   . .   |  18.5  |    325    |
+  |Mackerel, fresh           |  44.7  |  40.4  |  10.2   |  4.2 |   . .   |   0.7  |    370    |
+  |Herring, smoked           |  44.4  |  19.2  |  20.5   |  8.8 |   . .   |   7.4  |    755    |
+  |Salmon, tinned            |  . .   |  63.5  |  21.8   | 12.1 |   . .   |   2.6  |    915    |
+  |Oysters, shelled          |  . .   |  88.3  |   6.0   |  1.3 |    3.3  |   1.1  |    225    |
+  |Butter                    |  . .   |  11.0  |   1.0   | 85.0 |   . .   |   3.0  |   3410    |
+  |Cheese                    |  . .   |  34.2  |  25.9   | 33.7 |    2.4  |   3.8  |   1885    |
+  |Milk, whole               |  . .   |  87.0  |   3.3   |  4.0 |    5.0  |   0.7  |    310    |
+  |Milk, skimmed             |  . .   |  90.5  |   3.4   |  0.3 |    5.1  |   0.7  |    165    |
+  |Oatmeal                   |  . .   |   7.7  |  16.7   |  7.3 |   66.2  |   2.1  |   1800    |
+  |Corn (maize) meal         |  . .   |  12.5  |   9.2   |  1.9 |   75.4  |   1.0  |   1635    |
+  |Rye flour                 |  . .   |  12.9  |   6.8   |  0.9 |   78.7  |   0.7  |   1620    |
+  |Buckwheat flour           |  . .   |  13.6  |   6.4   |  1.2 |   77.9  |   0.9  |   1605    |
+  |Rice                      |  . .   |  12.3  |   8.0   |  0.3 |   79.0  |   0.4  |   1620    |
+  |Wheat flour, white        |  . .   |  12.0  |  11.4   |  1.0 |   75.1  |   0.5  |   1635    |
+  |Wheat flour, graham       |  . .   |  11.3  |  13.3   |  2.2 |   71.4  |   1.8  |   1645    |
+  |Wheat, breakfast food     |  . .   |   9.6  |  12.1   |  1.8 |   75.2  |   1.3  |   1680    |
+  |Wheat bread, white        |  . .   |  35.3  |   9.2   |  1.3 |   53.1  |   1.1  |   1200    |
+  |Wheat bread, graham       |  . .   |  35.7  |   8.9   |  1.8 |   52.1  |   1.5  |   1195    |
+  |Rye bread                 |  . .   |  35.7  |   9.0   |  0.6 |   53.2  |   1.5  |   1170    |
+  |Biscuit (crackers)        |  . .   |   6.8  |   9.7   | 12.1 |   69.7  |   1.7  |   1925    |
+  |Macaroni                  |  . .   |  10.3  |  13.4   |  0.9 |   74.1  |   1.3  |   1645    |
+  |Sugar                     |  . .   |  . .   |  . .    | . .  |  100.0  |  . .   |   1750    |
+  |Starch (corn starch)      |  . .   |  . .   |  . .    | . .  |   90.0  |  . .   |   1680    |
+  |Beans, dried              |  . .   |  12.6  |  22.5   |  1.8 |   59.6  |   3.5  |   1520    |
+  |Peas, dried               |  . .   |  9.5   |  24.6   |  1.0 |   62.0  |   2.9  |   1565    |
+  |Beets                     |  20.0  |  70.0  |   1.3   |  0.1 |    7.7  |   0.9  |    160    |
+  |Cabbage                   |  50.0  |  44.2  |   0.7   |  0.2 |    4.5  |   0.4  |    100    |
+  |Potatoes                  |  20.0  |  62.6  |   1.8   |  0.1 |   14.7  |   0.8  |    295    |
+  |Sweet potatoes            |  20.0  |  55.2  |   1.4   |  0.6 |   21.9  |   0.9  |    440    |
+  |Tomatoes                  |  . .   |  94.3  |   0.9   |  0.4 |    3.9  |   0.5  |    100    |
+  |Apples                    |  25.0  |  63.3  |   0.3   |  0.3 |   10.8  |   0.3  |    190    |
+  |Bananas                   |  35.0  |  48.9  |   0.8   |  0.4 |   14.3  |   0.6  |    260    |
+  |Grapes                    |  25.0  |  58.0  |   1.0   |  1.2 |   14.4  |   0.4  |    295    |
+  |Strawberries              |   5.0  |  85.9  |   0.9   |  0.6 |    7.0  |   0.6  |    150    |
+  |Almonds                   |  45.0  |   2.7  |  11.5   | 30.2 |    9.5  |   1.1  |   1515    |
+  |Brazil nuts               |  49.6  |   2.6  |   8.6   | 33.7 |    3.5  |   2.0  |   1485    |
+  |Chestnuts                 |  16.0  |  37.8  |   5.2   |  4.5 |   35.4  |   1.1  |    915    |
+  |Walnuts                   |  58.1  |   1.0  |   6.9   | 26.6 |    6.8  |   0.6  |   1250    |
+  +--------------------------+--------+--------+---------+------+---------+--------+-----------+
+
+In some cases the subjects were at rest; in others they performed
+varying amounts of external muscular work on an apparatus by means of
+which the amount of work done was measured. In some cases they fasted,
+and in others they received diets generally not far from sufficient to
+maintain nitrogen, and usually carbon, equilibrium in the body. In these
+experiments the amount of energy expended by the body as heat and as
+external muscular work measured in terms of heat agreed on the average
+very closely with the amount of heat that would be produced by the
+oxidation of all the matter metabolized in the body. The variations for
+individual days, and in the average for individual experiments as well,
+were in some cases appreciable, amounting to as much as 6%, which is not
+strange in view of the uncertainties in physiological experimenting; but
+in the average of all the experiments the energy of the expenditure was
+above 99.9% of the energy of the income,--an agreement within one part
+in 1000. While these results do not absolutely prove the application of
+the law of the conservation of energy in the human body, they certainly
+approximate very closely to such demonstration. It is of course possible
+that energy may have given off from the body in other forms than heat
+and external muscular work. It is conceivable, for example, that
+intellectual activity may involve the transformation of physical energy,
+and that the energy involved may be eliminated in some form now unknown.
+But if the body did give off energy which was not measured in these
+experiments, the quantity must have been extremely small. It seems fair
+to infer from the results obtained that the metabolism of energy in the
+body occurred in conformity with the law of the conservation of energy.
+
+3. _Composition of Food Materials._--The composition of food is
+determined by chemical analyses, the results of which are conventionally
+expressed in terms of the nutritive ingredients previously described. As
+a result of an enormous amount of such investigation in recent years,
+the kinds and proportions of nutrients in our common sorts of food are
+well known. Average values for percentage composition of some ordinary
+food materials are shown in Table I. (Table I. also includes figures for
+fuel value.)
+
+It will be observed that different kinds of food materials vary widely
+in their proportions of nutrients. In general the animal foods contain
+the most protein and fats, and vegetable foods are rich in
+carbohydrates. The chief nutrient of lean meat and fish is protein; but
+in medium fat meats the proportion of fat is as large as that of
+protein, and in the fatter meats it is larger. Cheese is rich in both
+protein and fat. Among the vegetable foods, dried beans and peas are
+especially rich in protein. The proportion in oatmeal is also fairly
+large, in wheat it is moderate, and in maize meal and rice it is rather
+small. Oats contain more oil than any of the common cereals, but in none
+of them is the proportion especially large. The most abundant nutrient
+in all the cereals is starch, which comprises from two-thirds to
+three-fourths or more of their total nutritive substance. Cotton-seed is
+rich in edible oil, and so are olives. Some of the nuts contain fairly
+large proportions of both protein and fat. The nutrient of potatoes is
+starch, present in fair proportion. Fruits contain considerable
+carbohydrates, chiefly sugar. Green vegetables are not of much account
+as sources of any of the nutrients or energy.
+
+Similar food materials from different sources may also differ
+considerably in composition. This is especially true of meats. Thus, the
+leaner portions from a fat animal may contain nearly as much fat as the
+fatter portions from a lean animal. The data here presented are largely
+those for American food products, but the available analyses of English
+food materials indicate that the latter differ but little from the
+former in composition. The analyses of meats produced in Europe imply
+that they commonly contain somewhat less fat and more water, and often
+more protein, than American meats. The meats of English production
+compare with the American more than with the European meats. Similar
+vegetable foods from the different countries do not differ so much in
+composition.
+
+4. _Digestibility or Availability of Food Materials._--The value of any
+food material for nutriment depends not merely upon the kinds and
+amounts of nutrients it contains, but also upon the ease and convenience
+with which the nutrients may be digested, and especially upon the
+proportion of the nutrients that will be actually digested and
+absorbed. Thus, two foods may contain equal amounts of the same
+nutrient, but the one most easily digested will really be of most value
+to the body, because less effort is necessary to utilize it.
+Considerable study of this factor is being made, and much valuable
+information is accumulating, but it is of more especial importance in
+cases of disordered digestion.
+
+    TABLE II.--_Coefficients of Digestibility (or Availability) of
+    Nutrients in Different Classes of Food Materials._
+
+  +--------------------------+----------+----------+----------------+
+  |       Kind of Food.      | Protein. |   Fat.   | Carbohydrates. |
+  +--------------------------+----------+----------+----------------+
+  |                          |          |          |                |
+  |                          |     %    |     %    |        %       |
+  | Meats                    |    98    |    98    |       ..       |
+  | Fish                     |    96    |    97    |       ..       |
+  | Poultry                  |    96    |    97    |       ..       |
+  | Eggs                     |    97    |    98    |       ..       |
+  | Dairy products           |    97    |    96    |       98       |
+  | Total animal food of     |          |          |                |
+  |   mixed diet             |    97    |    97    |       98       |
+  | Potatoes                 |    73    |    ..    |       98       |
+  | Beets, carrots, &c.      |    72    |    ..    |       97       |
+  | Cabbage, lettuce, &c.    |    ..    |    ..    |       83       |
+  | Legumes                  |    78    |    90    |       95       |
+  | Oatmeal                  |    78    |    90    |       97       |
+  | Corn meal                |    80    |    ..    |       99       |
+  | Wheat meals without bran |    83    |    ..    |       93       |
+  | Wheat meals with bran    |    75    |    ..    |       92       |
+  | White bread              |    88    |    ..    |       98       |
+  | Entire wheat bread       |    82    |    ..    |       94       |
+  | Graham bread             |    76    |    ..    |       90       |
+  | Rice                     |    76    |    ..    |       91       |
+  | Fruits and nuts          |    80    |    86    |       96       |
+  | Sugars and starches      |    ..    |    ..    |       98       |
+  | Total vegetable food of  |          |          |                |
+  |   mixed diet             |    85    |    90    |       97       |
+  | Total food of mixed diet |    92    |    95    |       97       |
+  +--------------------------+----------+----------+----------------+
+
+The digestibility of food in the sense of thoroughness of digestion,
+however, is of particular importance in the present discussion. Only
+that portion of the food that is digested and absorbed is available to
+the body for the building of tissue and the production of energy. Not
+all the food eaten is thus actually digested; undigested material is
+excreted in the faeces. The thoroughness of digestion is determined
+experimentally by weighing and analysing the food eaten and the faeces
+pertaining to it. The difference between the corresponding ingredients
+of the two is commonly considered to represent the amounts of the
+ingredients digested. Expressed in percentages, these are called
+coefficients of digestibility. See Table II.
+
+Such a method is not strictly accurate, because the faeces do not
+consist entirely of undigested food but contain in addition to this the
+so-called metabolic products, which include the residuum of digestive
+juices not resorbed, fragments of intestinal epithelium, &c. Since there
+is as yet no satisfactory method of separating these constituents of the
+excreta, the actual digestibility of the food is not determined. It has
+been suggested that since these materials must originally come from
+food, they represent, when expressed in terms of food ingredients, the
+cost of digestion; hence that the values determined as above explained
+represent the portion of food available to the body for the building of
+tissue and the yielding of energy, and what is commonly designated as
+digestibility should be called availability. Other writers retain the
+term "digestibility," but express the results as "apparent
+digestibility," until more knowledge regarding the metabolic products of
+the excreta is available and the actual digestibility may be
+ascertained.
+
+Experimental inquiry of this nature has been very active in recent
+years, especially in Europe, the United States and Japan; and the
+results of considerably over 1000 digestion experiments with single
+foods or combinations of food materials are available. These were mostly
+with men, but some were with women and with children. The larger part of
+these have been taken into account in the following estimations of the
+digestibility of the nutrients in different classes of food materials.
+The figures here shown are subject to revision as experimental data
+accumulate. They are not to be taken as exact measures of the
+digestibility (or availability) of every kind of food in each given
+class, but they probably represent fairly well the average digestibility
+of the classes of food materials as ordinarily utilized in the mixed
+diet.
+
+5. _Fuel Value of Food._--The potential energy of food is commonly
+measured as the amount of heat evolved when the food is completely
+oxidized. In the laboratory this is determined by burning the food in
+oxygen in a calorimeter. The results, which are known as the heat of
+combustion of the food, are expressed in calories, one calory being the
+amount of heat necessary to raise the temperature of one kilogram of
+water one degree centigrade. But it is to be observed that this unit is
+employed simply from convenience, and without implication as to what
+extent the energy of food is converted into heat in the body. The unit
+employed in the measurement of some other form of energy might be used
+instead, as, for example, the foot-ton, which represents the amount of
+energy necessary to raise one ton through one foot.
+
+  TABLE III.--_Estimates of Heats of Combustion and of Fuel Value of
+  Nutrients in Ordinary Mixed Diet._
+
+  +---------------------------+-------------+-------------+
+  |         Nutrients.        |   Heat of   | Fuel Value. |
+  |                           | Combustion. |             |
+  +---------------------------+-------------+-------------+
+  |                           |             |             |
+  |                           |  Calories.  |  Calories.  |
+  |                           |             |             |
+  | One gram of protein       |    5.65     |    4.05     |
+  | One gram of fats          |    9.40     |    8.93     |
+  | One gram of carbohydrates |    4.15     |    4.03     |
+  |                           |             |             |
+  +---------------------------+-------------+-------------+
+
+  The amount of energy which a given quantity of food will produce on
+  complete oxidation outside the body, however, is greater than that
+  which the body will actually derive from it. In the first place, as
+  previously shown, part of the food will not be digested and absorbed.
+  In the second place, the nitrogenous compounds absorbed are not
+  completely oxidized in the body, the residuum being excreted in the
+  urine as urea and other bodies that are capable of further oxidation
+  in the calorimeter. The total heat of combustion of the food eaten
+  must therefore be diminished by the heat of combustion of the
+  oxidizable material rejected by the body, to find what amount of
+  energy is actually available to the organism for the production of
+  work and heat. The amount thus determined is commonly known as the
+  fuel value of food.
+
+  Rubner's[7] commonly quoted estimates for the fuel value of the
+  nutrients of mixed diet are,--for protein and carbohydrates 4.1, and
+  for fats 9.3 calories per gram. According to the method of deduction,
+  however, these factors were more applicable to digested than to total
+  nutrients. Atwater[8] and associates have deduced, from data much more
+  extensive than those available to Rubner, factors for total nutrients
+  somewhat lower than these, as shown in Table III. These estimates seem
+  to represent the best average factors at present available, but are
+  subject to revision as knowledge is extended.
+
+
+      TABLE IV.--_Quantities of Available Nutrients and Energy in Daily
+      Food Consumption of Persons in Different Circumstances._
+
+    +------------------------------------------+--------+--------------------------------+
+    |                                          |        |     Nutrients and Energy       |
+    |                                          | Number |        per Man per Day.        |                                |
+    |                                          |   of   +------+------+--------+---------+
+    |                                          |Studies.| Pro- | Fat. |Carbohy-|  Fuel   |
+    |                                          |        | tein.|      | drates.| Value.  |
+    +------------------------------------------+--------+------+------+--------+---------+
+    |                                          |        |      |      |        |         |
+    |     _Persons with Active Work._          |        |Grams.|Grams.|  Grams.|Calories.|
+    | English royal engineers                  |    1   |  132 |   79 |   612  |   3835  |
+    | Prussian machinists                      |    1   |  129 |  107 |   657  |   4265  |
+    | Swedish mechanics                        |    5   |  174 |  105 |   693  |   4590  |
+    | Bavarian lumbermen                       |    3   |  120 |  277 |   702  |   6015  |
+    | American lumbermen                       |    5   |  155 |  327 |   804  |   6745  |
+    | Japanese rice cleaner                    |    1   |  103 |   11 |   917  |   4415  |
+    | Japanese jinrikshaw runner               |    1   |  137 |   22 |  1010  |   5050  |
+    | Chinese farm labourers in California     |    1   |  132 |   90 |   621  |   3980  |
+    | American athletes                        |   19   |  178 |  192 |   525  |   4740  |
+    | American working-men's families          |   13   |  156 |  226 |   694  |   5650  |
+    |                                          |        |      |      |        |         |
+    |                                          |        |      |      |        |         |
+    |     _Persons with Ordinary Work._        |        |      |      |        |         |
+    | Bavarian mechanics.                      |   11   |  112 |   32 |   553  |   3060  |
+    | Bavarian farm labourers                  |    5   |  126 |   52 |   526  |   3200  |
+    | Russian peasants                         |   ..   |  119 |   31 |   571  |   3155  |
+    | Prussian prisoners                       |    1   |  117 |   28 |   620  |   3320  |
+    | Swedish mechanics.                       |    6   |  123 |   75 |   507  |   3325  |
+    | American working-men's families          |   69   |  105 |  135 |   426  |   3480  |
+    |                                          |        |      |      |        |         |
+    |     _Persons with Light Work._           |        |      |      |        |         |
+    | American artisans' families              |   21   |   93 |  107 |   358  |   2880  |
+    | English tailors (prisoners)              |    1   |  121 |   37 |   509  |   2970  |
+    | German shoemakers                        |    1   |   99 |   73 |   367  |   2629  |
+    | Japanese prisoners                       |    1   |   43 |    6 |   444  |   2110  |
+    |                                          |        |      |      |        |         |
+    |     _Professional and Business Men._     |        |      |      |        |         |
+    | Japanese professional men.               |   13   |   75 |   15 |   408  |   2190  |
+    | Japanese students                        |    8   |   85 |   18 |   537  |   2800  |
+    | Japanese military cadets                 |   11   |   98 |   20 |   611  |   3185  |
+    | German physicians                        |    2   |  121 |   90 |   317  |   2685  |
+    | Swedish medical students                 |    5   |  117 |  108 |   291  |   2725  |
+    | Danish physicians                        |    1   |  124 |  133 |   242  |   2790  |
+    | American professional and business       |        |      |      |        |         |
+    |   men and students                       |   51   |   98 |  125 |   411  |   3285  |
+    |                                          |        |      |      |        |         |
+    |     _Persons with Little or no Exercise._|        |      |      |        |         |
+    | Prussian prisoners                       |    2   |   90 |   27 |   427  |   2400  |
+    | Japanese prisoners                       |    1   |   36 |    6 |   360  |   1725  |
+    | Inmates of home for aged--Germany        |    1   |   85 |   43 |   322  |   2097  |
+    | Inmates of hospitals for insane--America |   49   |   80 |   86 |   353  |   2590  |
+    |                                          |        |      |      |        |         |
+    |     _Persons in Destitute Circumstances._|        |      |      |        |         |
+    | Prussian working people                  |   13   |   63 |   43 |   372  |   2215  |
+    | Italian mechanics                        |    5   |   70 |   36 |   384  |   2225  |
+    | American working-men's families          |   11   |   69 |   75 |   263  |   2085  |
+    +------------------------------------------+--------+------+------+--------+---------+
+
+The heats of combustion of all the fats in an ordinary mixed diet would
+average about 9.40 calories per gram, but as only 95% of the fat would
+be available to the body, the fuel value per gram would be (9.40 X 0.95
+=) 8.93 calories. Similarly, the average heat of combustion of
+carbohydrates of the diet would be about 4.15 calories per gram, and as
+97% of the total quantity is available to the body, the fuel value per
+gram would be 4.03. (It is commonly assumed that the resorbed fats and
+carbohydrates are completely oxidized in the body.) The heats of
+combustion of all the kinds of protein in the diet would average about
+5.65 calories per gram. Since about 92% of the total protein would be
+available to the body, the potential energy of the available protein
+would be equivalent to (5.65 X 0.92 =) 5.20 calories; but as the
+available protein is not completely oxidized allowance must be made for
+the potential energy of the incompletely oxidized residue. This is
+estimated as equivalent to 1.15 calories for the 0.92 gram of available
+protein; hence, the fuel value of the total protein is (5.20 - 1.15 =)
+4.05 calories per gram. Nutrients of the same class, but from different
+food materials, vary both in digestibility and in heat of combustion,
+and hence in fuel value. These factors are therefore not so applicable
+to the nutrients of the separate articles in a diet as to those of the
+diet as a whole.
+
+6. _Food Consumption._--Much information regarding the food consumption
+of people in various circumstances in different parts of the world has
+accumulated during the past twenty years, as a result of studies of
+actual dietaries in England, Germany, Italy, Russia, Sweden and
+elsewhere in Europe, in Japan and other oriental countries, and
+especially in the United States. These studies commonly consist in
+ascertaining the kinds, amounts and composition of the different food
+materials consumed by a group of persons during a given period and the
+number of meals taken by each member of the group, and computing the
+quantities of the different nutrients in the food on the basis of one
+man for one day. When the members of the group are of different age,
+sex, occupation, &c., account must be taken of the effect of these
+factors on consumption in estimating the value "per man." Men as a rule
+eat more than women under similar conditions, women more than children,
+and persons at active work more than those at sedentary occupation. The
+navvy, for example, who is constantly using up more nutritive material
+or body tissue to supply the energy required for his muscular work needs
+more protein and energy in his food than a bookkeeper who sits at his
+desk all day.
+
+In making allowance for these differences, the various individuals are
+commonly compared with a man at moderately active muscular work, who is
+taken as unity. A man at hard muscular work is reckoned at 1.2 times
+such an individual; a man with light muscular work or a boy 15-16 years
+old, .9; a man at sedentary occupation, woman at moderately active
+muscular work, boy 13-14 or girl 15-16 years old, .8; woman at light
+work, boy 12 or girl 13-14 years old, .7; boy 10-11 or girl 10-12 years
+old, .6; child 6-9 years old, .5; child 2-5 years old, .4; child under 2
+years, .3. These factors are by no means absolute or final, but are
+based in part upon experimental data and in part upon arbitrary
+assumption.
+
+The total number of dietary studies on record is very large, but not all
+of them are complete enough to furnish reliable data. Upwards of 1000
+are sufficiently accurate to be included in statistical averages of food
+consumed by people in different circumstances, nearly half of which have
+been made in the United States in the past decade. The number of persons
+in the individual studies has ranged from one to several hundred. Some
+typical results are shown in Table IV.
+
+7. _Quantities of Nutrients needed._--For the proper nourishment of the
+body, the important problem is how much protein, fats and carbohydrates,
+or more simply, what amounts of protein and potential energy are needed
+under varying circumstances, to build and repair muscular and other
+tissues and to supply energy for muscular work, heat and other forms of
+energy. The answer to the problem is sought in the data obtained in
+dietary studies with considerable numbers of people, and in metabolism
+experiments with individuals in which the income and expenditure of the
+body are measured. From the information thus derived, different
+investigators have proposed so-called dietary standards, such as are
+shown in the table below, but unfortunately the experimental data are
+still insufficient for entirely trustworthy figures of this sort; hence
+the term "standard" as here used is misleading. The figures given are
+not to be considered as exact and final as that would suggest; they are
+merely tentative estimates of the average daily amounts of nutrients and
+energy required. (It is to be especially noted that these are available
+nutrients and fuel value rather than total nutrients and energy.) Some
+of the values proposed by other investigators are slightly larger than
+these, and others are decidedly smaller, but these are the ones that
+have hitherto been most commonly accepted in Europe and America.
+
+
+    TABLE V.--_Standards for Dietaries. Available Nutrients and Energy
+    per Man per Day._
+
+  +---------------------------+---------+--------+---------+---------+
+  |                           | Protein.|  Fat.  | Carbo-  |  Fuel   |
+  |                           |         |        |hydrates.|  Value. |
+  +---------------------------+---------+--------+---------+---------+
+  |                           |         |        |         |         |
+  |   _Voit's Standards._     |Grams.[9]| Grams. |  Grams. |Calories.|
+  | Man at hard work          |   133   |   95   |   437   |  3270   |
+  | Man at moderate work      |   109   |   53   |   485   |  2965   |
+  |    _Atwater's Standards._ |         |        |         |         |
+  | Man at very hard          |         |        |         |         |
+  |   muscular work           |   161   |  ..[10]|   ..[10]|  5500   |
+  | Man at hard muscular work |   138   |   ..   |   ..    |  4150   |
+  | Man at moderately         |         |        |         |         |
+  |   active muscular work    |   115   |   ..   |   ..    |  3400   |
+  | Man at light to moderate  |         |        |         |         |
+  |    muscular work          |   103   |   ..   |   ..    |  3050   |
+  | Man at "sedentary"        |         |        |         |         |
+  |    or woman at moderately |         |        |         |         |
+  |    active work            |    92   |   ..   |   ..    |  2700   |
+  | Woman at light muscular   |         |        |         |         |
+  |    work, or man without   |         |        |         |         |
+  |    muscular exercise      |    83   |   ..   |   ..    |  2450   |
+  |                           |         |        |         |         |
+  +---------------------------+---------+--------+---------+---------+
+
+8. _Hygienic Economy of Food._--For people in good health, there are two
+important rules to be observed in the regulation of the diet. One is to
+choose the foods that "agree" with them, and to avoid those which they
+cannot digest and assimilate without harm; and the other is to use such
+sorts and quantities of foods as will supply the kinds and amounts of
+nutrients needed by the body and yet to avoid burdening it with
+superfluous material to be disposed of at the cost of health and
+strength.
+
+As for the first-mentioned rule, it is practically impossible to give
+information that may be of more than general application. There are
+people who, because of some individual peculiarity, cannot use foods
+which for people in general are wholesome and nutritious. Some persons
+cannot endure milk, others suffer if they eat eggs, others have to
+eschew certain kinds of meat, or are made uncomfortable by fruit; but
+such cases are exceptions. Very little is known regarding the cause of
+these conditions. It is possible that in the metabolic processes to
+which the ingredients of the food are subjected in the body, or even
+during digestion before the substances are actually taken into the body,
+compounds may be formed that are in one way or another injurious.
+Whatever the cause may be, it is literally true in this sense that "what
+is one man's meat is another man's poison," and each must learn for
+himself what foods "agree" with him and what ones do not. But for the
+great majority of people in health, suitable combinations of the
+ordinary sorts of wholesome food materials make a healthful diet. On the
+other hand, some foods are of particular value at times, aside from
+their use for nourishment. Fruits and green vegetables often benefit
+people greatly, not as nutriment merely, for they may have very little
+actual nutritive material, but because of fruit or vegetable acids or
+other substances which they contain, and which sometimes serve a most
+useful purpose.
+
+
+    TABLE VI.--_Amounts of Nutrients and Energy Furnished for One
+    Shilling in Food Materials at Ordinary Prices._
+
+  +----------------------+-------+----------------------------------------------+
+  |                      |       |            One Shilling will buy             |
+  |                      |       +----------+-------------------------+---------+
+  |   Food Materials     |Prices |          |   Available Nutrients.  |         |
+  |    as Purchased.     |  per  |Total Food+-------------------------+   Fuel  |
+  |                      |  lb.  |materials.|        |      | Carbo-  |  Value. |
+  |                      |       |          |Protein.| Fat. |hydrates.|         |
+  +----------------------+-------+----------+--------+------+---------+---------+
+  |                      | s. d. |    lb.   |    lb. |  lb. |    lb.  |Calories.|
+  | Beef, round          | 0 10  |   1.20   |   .22  |  .14 |    ..   |  1,155  |
+  |                      |0 8-1/2|   1.41   |   .26  |  .17 |    ..   |  1,235  |
+  |                      | 0  5  |   2.40   |   .44  |  .29 |    ..   |  2,105  |
+  |                      |       |          |        |      |         |         |
+  | Beef, sirloin        | 0 10  |   1.20   |   .19  |  .20 |    ..   |  1,225  |
+  |                      | 0  9  |   1.33   |   .21  |  .22 |    ..   |  1,360  |
+  |                      | 0  8  |   1.50   |    ..  |   .. |    ..   |    ..   |
+  |                      | 0  5  |   2.40   |    ..  |   .. |    ..   |    ..   |
+  |                      |       |          |        |      |         |         |
+  | Beef, rib            | 0  9  |   1.33   |   .19  |  .19 |    ..   |  1,200  |
+  |                      |0 7-1/2|   1.60   |    ..  |   .. |    ..   |    ..   |
+  |                      |0 4-1/2|   2.67   |    ..  |   .. |    ..   |    ..   |
+  |                      |       |          |        |      |         |         |
+  | Mutton, leg          | 0  9  |   1.33   |   .20  |  .20 |    ..   |  1,245  |
+  |                      | 0  5  |   2.40   |   .37  |  .35 |    ..   |  2,245  |
+  |                      |       |          |        |      |         |         |
+  | Pork, spare-rib      | 0  9  |   1.33   |   .17  |  .31 |    ..   |  1,645  |
+  |                      | 0  7  |   1.71   |   .22  |  .39 |    ..   |  2,110  |
+  |                      |       |          |        |      |         |         |
+  | Pork, salt, fat      | 0  7  |   1.71   |   .03  | 1.40 |    ..   |  6,025  |
+  |                      | 0  5  |   2.40   |   .04  | 1.97 |    ..   |  8,460  |
+  |                      |       |          |        |      |         |         |
+  | Pork, smoked ham     | 0  8  |   1.50   |   .20  |  .48 |    ..   |  2,435  |
+  |                      |0 4-1/2|   2.67   |   .36  |  .85 |    ..   |  4,330  |
+  |                      |       |          |        |      |         |         |
+  | Fresh cod            | 0  4  |   3.00   |   .34  |  .01 |    ..   |    710  |
+  |                      | 0  3  |   4.00   |   .45  |  .01 |    ..   |    945  |
+  |                      |       |          |        |      |         |         |
+  | Salt cod             |0 3-1/2|   3.43   |   .54  |  .07 |    ..   |  1,370  |
+  |                      | 0 10  |   1.20   |   .07  |  .01 |   .04   |    275  |
+  |                      |       |          |        |      |         |         |
+  |Milk, whole, 4d. a qt.| 0  2  |   6.00   |   .19  |  .23 |   .30   |  1,915  |
+  |    "        3d. a qt.|0 1-1/2|   8.00   |   .26  |  .30 |   .40   |  2,550  |
+  |    "        2d. a qt.| 0  1  |  12.00   |   .38  |  .46 |   .60   |  3,825  |
+  |                      |       |          |        |      |         |         |
+  | Milk, skimmed, 2d. a | 0  1  |  12.00   |   .40  |  .03 |   .61   |  2,085  |
+  |  qt.                 |       |          |        |      |         |         |
+  | Butter               | 1  6  |    .67   |   .01  |  .54 |    ..   |  2,320  |
+  |                      | 1  3  |    .80   |   .01  |  .64 |    ..   |  2,770  |
+  |                      | 1  0  |   1.00   |   .01  |  .81 |    ..   |  3,460  |
+  |                      |       |          |        |      |         |         |
+  | Margarine            | 0  4  |   3.00   |    ..  | 2.37 |    ..   | 10,080  |
+  |                      |       |          |        |      |         |         |
+  | Eggs, 2s. a dozen    | 1  4  |    .75   |   .10  |  .07 |    ..   |    475  |
+  |   "  1-1/2s. a dozen | 1  0  |   1.00   |   .13  |  .09 |    ..   |    635  |
+  |   "    1s. a dozen   | 0  8  |   1.50   |   .19  |  .13 |    ..   |    950  |
+  |                      |       |          |        |      |         |         |
+  | Cheese               | 0  8  |   1.50   |   .38  |  .48 |   .04   |  2,865  |
+  |                      | 0  7  |   1.71   |   .43  |  .55 |   .04   |  3,265  |
+  |                      | 0  5  |   2.40   |   .60  |  .77 |   .06   |  4,585  |
+  |                      |       |          |        |      |         |         |
+  | Wheat bread          |0 1-1/8|  10.67   |   .76  |  .13 |  5.57   | 12,421  |
+  |                      |       |          |        |      |         |         |
+  | Wheat flour          |0 1-3/5|   7.64   |   .67  |  .07 |  5.63   | 12,110  |
+  |                      |0 1-1/2|   8.16   |   .72  |  .07 |  6.01   | 12,935  |
+  |                      |       |          |        |      |         |         |
+  | Oatmeal              |0 1-2/5|   8.39   |  1.11  |  .54 |  5.54   | 14,835  |
+  |                      |0 1-1/2|   8.16   |  1.08  |  .53 |  5.39   | 14,430  |
+  |                      |       |          |        |      |         |         |
+  | Rice                 |0 1-3/4|   6.86   |   .45  |  .02 |  5.27   | 10,795  |
+  |                      |       |          |        |      |         |         |
+  | Potatoes             |0 0-2/3|  18.00   |   .25  |  .02 |  2.70   |  5,605  |
+  |                      |0 0-1/2|  24.00   |   .34  |  .02 |  3.60   |  7,470  |
+  |                      |       |          |        |      |         |         |
+  | Beans                | 0 2   |   6.00   |  1.05  |  .10 |  3.47   |  8,960  |
+  |                      |       |          |        |      |         |         |
+  | Sugar                | 1 3/4 |   6.86   |   ..   |  ..  |  6.86   | 12,760  |
+  +----------------------+-------+----------+--------+------+---------+---------+
+
+The proper observance of the second rule mentioned requires information
+regarding the demands of the body for food under different
+circumstances. To supply this information is one purpose of the effort
+to determine the so-called dietary standards mentioned above. It should
+be observed, however, that these are generally more applicable to the
+proper feeding of a group or class of people as a whole than for
+particular individuals in this class. The needs of individuals will vary
+largely from the average in accordance with the activity and
+individuality. Moreover, it is neither necessary nor desirable for the
+individual to follow any standard exactly from day to day. It is
+requisite only that the average supply shall be sufficient to meet the
+demands of the body during a given period.
+
+The cooking of food and other modes of preparing it for consumption have
+much to do with its nutritive value. Many materials which, owing to
+their mechanical condition or to some other cause, are not particularly
+desirable food materials in their natural state, are quite nutritious
+when cooked or otherwise prepared for consumption. It is also a matter
+of common experience that well-cooked food is wholesome and appetizing,
+whereas the same material poorly prepared is unpalatable. There are
+three chief purposes of cooking; the first is to change the mechanical
+condition of the food. Heating changes the structure of many food
+materials very materially, so that they may be more easily chewed and
+brought into a condition in which the digestive juices can act upon them
+more freely, and in this way probably influencing the ease and
+thoroughness of digestion. The second is to make the food more
+appetizing by improving the appearance or flavour or both. Food which is
+attractive to the eye and pleasing to the palate quickens the flow of
+saliva and other digestive juices and thus aids digestion. The third is
+to kill, by heat, disease germs, parasites or other dangerous organisms
+that may be contained in food. This is often a very important matter and
+applies to both animal and vegetable foods. Scrupulous neatness should
+always be observed in storing, handling and serving food. If ever
+cleanliness is desirable it must be in the things we eat, and every care
+should be taken to ensure it for the sake of health as well as of
+decency. Cleanliness in this connexion means not only absence of visible
+dirt, but freedom from undesirable bacteria and other minute organisms
+and from worms and other parasites. If food, raw or cooked, is kept in
+dirty places, peddled from dirty carts, prepared in dirty rooms and in
+dirty dishes, or exposed to foul air, disease germs and other offensive
+and dangerous substances may easily enter it.
+
+9. _Pecuniary Economy of Food._--Statistics of economy and of cost of
+living in Great Britain, Germany and the United States show that at
+least half, and commonly more, of the income of wage-earners and other
+people in moderate circumstances is expended for subsistence. The
+relatively large cost of food, and the important influence of diet upon
+health and strength, make a more widespread understanding of the subject
+of dietetics very desirable. The maxim that "the best is the cheapest"
+does not apply to food. The "best" food, in the sense of that which is
+the finest in appearance and flavour and which is sold at the highest
+price, is not generally the most economical.
+
+The price of food is not regulated largely by its value for nutriment.
+Its agreeableness to the palate or to the buyer's fancy is a large
+factor in determining the current demand and market price. There is no
+more nutriment in an ounce of protein or fat from the tender-loin of
+beef than from the round or shoulder. The protein of animal food has,
+however, some advantage over that of vegetable foods in that it is more
+thoroughly, and perhaps more easily, digested, for which reason it would
+be economical to pay somewhat more for the same quantity of nutritive
+material in the animal food. Furthermore, animal foods such as meats,
+fish and the like, gratify the palate as most vegetable foods do not.
+For persons in good health, foods in which the nutrients are the most
+expensive are like costly articles of adornment. People who can well
+afford them may be justified in buying them, but they are not
+economical. The most economical food is that which is at the same time
+most healthful and cheapest.
+
+The variations in the cost of the actual nutriment in different food
+materials may be illustrated by comparison of the amounts of nutrients
+obtained for a given sum in the materials as bought at ordinary market
+prices. This is done in Table VI., which shows the amounts of available
+nutrients contained in the quantities of different food materials that
+may be purchased for one shilling at prices common in England.
+
+When proper attention is given to the needs of the body for food and the
+relation between cost and nutritive value of food materials, it will be
+found that with care in the purchase and skill in the preparation of
+food, considerable control may be had over the expensiveness of a
+palatable, nutritious and healthful diet.
+
+  AUTHORITIES.--COMPOSITION OF FOODS:--Konig, _Chemie der menschlichen
+  Nahrungs- und Genussmittel_; Atwater and Bryant, "Composition of
+  American Food Materials," Bul. 28, Office of Experiment Stations, U.S.
+  Department of Agriculture. NUTRITION AND DIETETICS:--Armsby,
+  _Principles of Animal Nutrition_; Lusk, _The Science of Nutrition_;
+  Burney Yeo, _Food in Health and Disease_; Munk and Uffelmann, _Die
+  Ernahrung des gesunden und kranken Menschen_; Von Leyden,
+  _Ernahrungstherapie und Diatetik_; Dujardin-Beaumetz, Hygiene
+  alimentaire; Hutchison, _Food and Dietetics_; R. H. Chittenden,
+  _Physiological Economy in Nutrition_ (1904), _Nutrition of Man_
+  (1907); Atwater, "Chemistry and Economy of Food," Bul. 21, Office of
+  Experiment Stations, U.S. Department of Agriculture. See also other
+  Bulletins of the same office on composition of food, results of
+  dietary studies, metabolism experiments, &c., in the United States.
+  GENERAL METABOLISM:--Voit, _Physiologie des allgemeinen Stoffwechsels
+  und der Ernahrung_; Hermann, _Handbuch der Physiologie_, Bd. vi.; Von
+  Noorden, _Pathologie des Stoffwechsels_; Schafer, _Text-Book of
+  Physiology_, vol. i.; Atwater and Langworthy, "Digest of Metabolism
+  Experiments," Bull. 45, Office of Experiment Stations, U.S. Department
+  of Agriculture.     (W. O. A.; R. D. M.)
+
+
+FOOTNOTES:
+
+  [1] The terms applied by different writers to these nitrogenous
+    compounds are conflicting. For instance, the term "proteid" is
+    sometimes used as protein is here used, and sometimes to designate
+    the group here called albuminoids. The classification and terminology
+    here followed are those tentatively recommended by the Association of
+    American Agricultural Colleges and Experiment Stations.
+
+  [2] Folin, _Festschrift fur Olaf Hammarsten_, iii. (Upsala, 1906).
+
+  [3] _Ztschr. Biol._ 30, 73.
+
+  [4] In Russian. Cited in United States Department of Agriculture,
+    Office of Experiment Stations, Bul. No. 45, _A Digest of Metabolism
+    Experiments_, by W. O. Atwater and C. F. Langworthy.
+
+  [5] _Arch. physiol. norm. et path._ (1894) 4.
+
+  [6] U.S. Department of Agriculture, Office of Experiment Stations,
+    Bulletins Nos. 63, 69, 109, 136, 175. For a description of the
+    respiration calorimeter here mentioned see also publication No. 42 of
+    the Carnegie Institution of Washington.
+
+  [7] _Ztschr. Biol._ 21 (1885), p. 377.
+
+  [8] _Connecticut_ (Storrs) _Agricultural Experiment Station Report_
+    (1899), 73.
+
+  [9] One ounce equals 28.35 grams.
+
+  [10] As the chief function of both fats and carbohydrates is to
+    furnish energy, their exact proportion in the diet is of small
+    account. The amount of either may vary largely according to taste,
+    available supply, or other condition, as long as the total amount of
+    both is sufficient, together with the protein to furnish the required
+    energy.
+
+
+
+
+DIETRICH, CHRISTIAN WILHELM ERNST (1712-1774), German painter, was born
+at Weimar, where he was brought up early to the profession of art by his
+father Johann George, then painter of miniatures to the court of the
+duke. Having been sent to Dresden to perfect himself under the care of
+Alexander Thiele, he had the good fortune to finish in two hours, at the
+age of eighteen, a picture which attracted the attention of the king of
+Saxony. Augustus II. was so pleased with Dietrich's readiness of hand
+that he gave him means to study abroad, and visit in succession the
+chief cities of Italy and the Netherlands. There he learnt to copy and
+to imitate masters of the previous century with a versatility truly
+surprising. Winckelmann, to whom he had been recommended, did not
+hesitate to call him the Raphael of landscape. Yet in this branch of his
+practice he merely imitated Salvator Rosa and Everdingen. He was more
+successful in aping the style of Rembrandt, and numerous examples of
+this habit may be found in the galleries of St Petersburg, Vienna and
+Dresden. At Dresden, indeed, there are pictures acknowledged to be his,
+bearing the fictitious dates of 1636 and 1638, and the name of
+Rembrandt. Among Dietrich's cleverest reproductions we may account that
+of Ostade's manner in the "Itinerant Singers" at the National Gallery.
+His skill in catching the character of the later masters of Holland is
+shown in candlelight scenes, such as the "Squirrel and the Peep-Show" at
+St Petersburg, where we are easily reminded of Godfried Schalcken.
+Dietrich tried every branch of art except portraits, painting Italian
+and Dutch views alternately with Scripture scenes and still life. In
+1741 he was appointed court painter to Augustus III. at Dresden, with an
+annual salary of 400 thalers (L60), conditional on the production of
+four cabinet pictures a year. This condition, no doubt, accounts for the
+presence of fifty-two of the master's panels and canvases in one of the
+rooms at the Dresden museum. Dietrich, though popular and probably the
+busiest artist of his time, never produced anything of his own; and his
+imitations are necessarily inferior to the originals which he affected
+to copy. His best work is certainly that which he gave to engravings. A
+collection of these at the British Museum, produced on the general lines
+of earlier men, such as Ostade and Rembrandt, reveal both spirit and
+skill. Dietrich, after his return from the Peninsula, generally signed
+himself "Dietericij," and with this signature most of his extant
+pictures are inscribed. He died at Dresden, after he had successively
+filled the important appointments of director of the school of painting
+at the Meissen porcelain factory and professor of the Dresden academy of
+arts.
+
+
+
+
+DIETRICH OF BERN, the name given in German popular poetry to Theodoric
+the Great. The legendary history of Dietrich differs so widely from the
+life of Theodoric that it has been suggested that the two were
+originally unconnected. Medieval chroniclers, however, repeatedly
+asserted the identity of Dietrich and Theodoric, although the more
+critical noted the anachronisms involved in making Ermanaric (d. 376)
+and Attila (d. 453) contemporary with Theodoric (b. 455). That the
+legend is based on vague historical reminiscences is proved by the
+retention of the names of Theodoric (Thiuda-reiks, Dietrich) and his
+father Theudemir (Dietmar), by Dietrich's connexion with Bern (Verona)
+and Raben (Ravenna). Something of the Gothic king's character descended
+to Dietrich, familiarly called the Berner, the favourite of German
+medieval saga heroes, although his story did not leave the same mark on
+later German literature as did that of the Nibelungs. The cycle of songs
+connected with his name in South Germany is partially preserved in the
+Heldenbuch (q.v.) in _Dietrich's Flucht_, the _Rabenschlacht_ and
+_Alpharts Tod_; but it was reserved for an Icelandic author, writing in
+Norway in the 13th century, to compile, with many romantic additions, a
+consecutive account of Dietrich. In this Norse prose redaction, known as
+the _Vilkina Saga_, or more correctly the _Thidrekssaga_, is
+incorporated much extraneous matter from the Nibelungen and Wayland
+legends, in fact practically the whole of south German heroic tradition.
+
+There are traces of a form of the Dietrich legend in which he was
+represented as starting out from Byzantium, in accordance with
+historical tradition, for his conquest of Italy. But this early
+disappeared, and was superseded by the existing legend, in which,
+perhaps by an "epic fusion" with his father Theudemir, he was associated
+with Attila, and then by an easy transition with Ermanaric. Dietrich was
+driven from his kingdom of Bern by his uncle Ermanaric. After years of
+exile at the court of Attila he returned with a Hunnish army to Italy,
+and defeated Ermanaric in the Rabenschlacht, or battle of Ravenna.
+Attila's two sons, with Dietrich's brother, fell in the fight, and
+Dietrich returned to Attila's court to answer for the death of the young
+princes. This very improbable renunciation of the advantages of his
+victory suggests that in the original version of the story the
+Rabenschlacht was a defeat. In the poem of _Ermenrichs Tod_ he is
+represented as slaying Ermanaric, as in fact Theodoric slew Odoacer.
+"Otacher" replaces Ermanaric as his adversary in the _Hildebrandslied_,
+which relates how thirty years after the earlier attempt he reconquered
+his Lombard kingdom. Dietrich's long residence at Attila's court
+represents the youth and early manhood of Theodoric spent at the
+imperial court and fighting in the Balkan peninsula, and, in accordance
+with epic custom, the period of exile was adorned with war-like
+exploits, with fights with dragons and giants, most of which had no
+essential connexion with the cycle. The romantic poems of _Konig
+Laurin_, _Sigenot_, _Eckenlied_ and _Virginal_ are based largely on
+local traditions originally independent of Dietrich. The court of Attila
+(Etzel) was a ready bridge to the Nibelungen legend. In the final
+catastrophe he was at length compelled, after steadily holding aloof
+from the combat, to avenge the slaughter of his Amelungs by the
+Burgundians, and delivered Hagen bound into the hands of Kriemhild. The
+flame breath which anger induced from him shows the influence of pure
+myth, but the tales of his demonic origin and of his being carried off
+by the devil in the shape of a black horse may safely be put down to the
+clerical hostility to Theodoric's Arianism.
+
+Generally speaking, Dietrich of Bern was the wise and just monarch as
+opposed to Ermanaric, the typical tyrant of Germanic legend. He was
+invariably represented as slow of provocation and a friend of peace, but
+once roused to battle not even Siegfried could withstand his onslaught.
+But probably Dietrich's fight with Siegfried in Kriemhild's rose garden
+at Worms is a late addition to the Rosengarten myth. The chief heroes of
+the Dietrich cycle are his tutor and companion in arms, Hildebrand (see
+HILDEBRAND, lay of), with his nephews the Wolfings Alphart and Wolfhart;
+Wittich, who renounced his allegiance to Dietrich and slew the sons of
+Attila; Heime and Biterolf.
+
+  The contents of the poems dealing with the Dietrich cycle are
+  summarized by Uhland in _Schriften zur Geschichte der Dichtung und
+  Sage_ (Stuttgart, 1873). The _Thidrekssaga_ (ed. C. Unger,
+  Christiania, 1853) is translated into German by F. H. v. der Hagen in
+  _Altdeutsche und altnordische Heldensagen_ (vols. i. and ii. 3rd ed.,
+  Breslau, 1872). A summary of it forms the concluding chapter of T.
+  Hodgkin's _Theodoric the Goth_ (1891). The variations in the Dietrich
+  legend in the Latin historians, in Old and Middle High German
+  literature, and in the northern saga, can be studied in W. Grimm's
+  _Deutsche Heldensage_ (2nd ed., Berlin, 1867). There is a good account
+  in English in F. E. Sandbach's _Heroic Saga-cycle of Dietrich of Bern_
+  (1906), forming No. 15 of Alfred Nutt's _Popular Studies in
+  Mythology_, and another in M. Bentinck Smith's translation of Dr O. L.
+  Jiriczek's _Deutsche Heldensage_ (_Northern Legends_, London, 1902).
+  For modern German authorities and commentators see B. Symons,
+  "Deutsche Heldensage" in H. Paul's _Grd. d. german. Phil._
+  (Strassburg, new ed., 1905); also Goedeke, _Geschichte der deutschen
+  Dichtung_ (i. 241-246).
+
+
+
+
+DIEZ, FRIEDRICH CHRISTIAN (1794-1876), German philologist, was born at
+Giessen, in Hesse-Darmstadt, on the 15th of March 1794. He was educated
+first at the gymnasium and then at the university of his native town.
+There he studied classics under Friedrich Gottlieb Welcker (1784-1868)
+who had just returned from a two years' residence in Italy to fill the
+chair of archaeology and Greek literature. It was Welcker who kindled in
+him a love of Italian poetry, and thus gave the first bent to his
+genius. In 1813 he joined the Hesse corps as a volunteer and served in
+the French campaign. Next year he returned to his books, and this short
+taste of military service was the only break in a long and uneventful
+life of literary labours. By his parents' desire he applied himself for
+a short time to law, but a visit to Goethe in 1818 gave a new direction
+to his studies, and determined his future career. Goethe had been
+reading Raynouard's _Selections from the Romance Poets_, and advised the
+young scholar to explore the rich mine of Provencal literature which the
+French savant had opened up. This advice was eagerly followed, and
+henceforth Diez devoted himself to Romance literature. He thus became
+the founder of Romance philology. After supporting himself for some
+years by private teaching, he removed in 1822 to Bonn, where he held the
+position of privatdocent. In 1823 he published his first work, _An
+Introduction to Romance Poetry_; in the following year appeared _The
+Poetry of the Troubadours_, and in 1829 _The Lives and Works of the
+Troubadours_. In 1830 he was called to the chair of modern literature.
+The rest of his life was mainly occupied with the composition of the two
+great works on which his fame rests, the _Grammar of the Romance
+Languages_ (1836-1844), and the _Lexicon of the Romance
+Languages--Italian, Spanish and French_ (1853); in these two works Diez
+did for the Romance group of languages what Jacob Grimm did for the
+Teutonic family. He died at Bonn on the 29th of May 1876.
+
+  The earliest French philologists, such as Perion and Henri Estienne,
+  had sought to discover the origin of French in Greek and even in
+  Hebrew. For more than a century Menage's _Etymological Dictionary_
+  held the field without a rival. Considering the time at which it was
+  written (1650), it was a meritorious work, but philology was then in
+  the empirical stage, and many of Menage's derivations (such as that of
+  "rat" from the Latin "mus," or of "haricot" from "faba") have since
+  become bywords among philologists. A great advance was made by
+  Raynouard, who by his critical editions of the works of the
+  Troubadours, published in the first years of the 19th century, laid
+  the foundations on which Diez afterwards built. The difference between
+  Diez's method and that of his predecessors is well stated by him in
+  the preface to his dictionary. In sum it is the difference between
+  science and guess-work. The scientific method is to follow implicitly
+  the discovered principles and rules of phonology, and not to swerve a
+  foot's breadth from them unless plain, actual exceptions shall justify
+  it; to follow the genius of the language, and by cross-questioning to
+  elicit its secrets; to gauge each letter and estimate the value which
+  attaches to it in each position; and lastly to possess the true
+  philosophic spirit which is prepared to welcome any new fact, though
+  it may modify or upset the most cherished theory. Such is the
+  historical method which Diez pursues in his grammar and dictionary. To
+  collect and arrange facts is, as he tells us, the sole secret of his
+  success, and he adds in other words the famous apophthegm of Newton,
+  "hypotheses non fingo." The introduction to the grammar consists of
+  two parts:--the first discusses the Latin, Greek and Teutonic elements
+  common to the Romance languages; the second treats of the six dialects
+  separately, their origin and the elements peculiar to each. The
+  grammar itself is divided into four books, on phonology, on flexion,
+  on the formation of words by composition and derivation, and on
+  syntax.
+
+  His dictionary is divided into two parts. The first contains words
+  common to two at least of the three principal groups of
+  Romance:--Italian, Spanish and Portuguese, and Provencal and French.
+  The Italian, as nearest the original, is placed at the head of each
+  article. The second part treats of words peculiar to one group. There
+  is no separate glossary of Wallachian.
+
+  Of the introduction to the grammar there is an English translation by
+  C. B. Cayley. The dictionary has been published in a remodelled form
+  for English readers by T. C. Donkin.
+
+
+
+
+DIEZ, a town of Germany, in the Prussian province of Hesse-Nassau,
+romantically situated in the deep valley of the Lahn, here crossed by an
+old bridge, 30 m. E. from Coblenz on the railway to Wetzlar. Pop. 4500.
+It is overlooked by a former castle of the counts of Nassau-Dillenburg,
+now a prison. Close by, on an eminence above the river, lies the castle
+of Oranienstein, formerly a Benedictine nunnery and now a cadet school,
+with beautiful gardens. There are a Roman Catholic and two Evangelical
+churches. The new part of the town is well built and contains numerous
+pretty villa residences. In addition to extensive iron-works there are
+sawmills and tanneries. In the vicinity are Fachingen, celebrated for
+its mineral waters, and the majestic castle of Schaumburg belonging to
+the prince of Waldeck-Pyrmont.
+
+
+
+
+DIFFERENCES, CALCULUS OF (_Theory of Finite Differences_), that branch
+of mathematics which deals with the successive differences of the terms
+of a series.
+
+1. The most important of the cases to which mathematical methods can be
+applied are those in which the terms of the series are the values, taken
+at stated intervals (regular or irregular), of a continuously varying
+quantity. In these cases the formulae of finite differences enable
+certain quantities, whose exact value depends on the law of variation
+(i.e. the law which governs the relative magnitude of these terms) to be
+calculated, often with great accuracy, from the given terms of the
+series, without explicit reference to the law of variation itself. The
+methods used may be extended to cases where the series is a double
+series (series of double entry), i.e. where the value of each term
+depends on the values of a pair of other quantities.
+
+2. The _first differences_ of a series are obtained by subtracting from
+each term the term immediately preceding it. If these are treated as
+terms of a new series, the first differences of this series are the
+_second differences_ of the original series; and so on. The successive
+differences are also called _differences of the first, second, ...
+order_. The differences of successive orders are most conveniently
+arranged in successive columns of a table thus:--
+
+  +-----+----------+-----------+-----------------+----------------------+
+  |Term.| 1st Diff.| 2nd Diff. |    3rd Diff.    |      4th Diff.       |
+  +-----+----------+-----------+-----------------+----------------------+
+  |     |          |           |                 |                      |
+  |  a  |          |           |                 |                      |
+  |     |   b - a  |           |                 |                      |
+  |  b  |          | c - 2b +a |                 |                      |
+  |     |   c - b  |           | d - 3c + 3b - a |                      |
+  |  c  |          | d - 2c +b |                 | e - 4d + 6c - 4b + a |
+  |     |   d - c  |           | e - 3d + 3c - b |                      |
+  |  d  |          | e - 2d +c |                 |                      |
+  |     |   e - d  |           |                 |                      |
+  |  e  |          |           |                 |                      |
+  +-----+----------+-----------+-----------------+----------------------+
+
+    _Algebra of Differences and Sums._
+
+  [Illustration: FIG. 1.]
+
+  3. The formal relations between the terms of the series and the
+  differences may be seen by comparing the arrangements (A) and (B) in
+  fig. 1. In (A) the various terms and differences are the same as in S
+  2, but placed differently. In (B) we take a new series of terms
+  [alpha], [beta], [gamma], [delta], commencing with the same term
+  [alpha], and take the successive sums of pairs of terms, instead of
+  the successive differences, but place them to the left instead of to
+  the right. It will be seen, in the first place, that the successive
+  terms in (A), reading downwards to the right, and the successive terms
+  in (B), reading downwards to the left, consist each of a series of
+  terms whose coefficients follow the binomial law; i.e. the
+  coefficients in b - a, c - 2b + a, d - 3c + 3b - a, ... and in [alpha]
+  + [beta], [alpha] + 2[beta] + [gamma], [alpha] + 3[beta] + 3[gamma] +
+  [delta], ... are respectively the same as in y - x, (y - x)^2, (y -
+  x)^3, ... and in x + y, (x + y)^2, (x + y)^3,.... In the second place,
+  it will be seen that the relations between the various terms in (A)
+  are identical with the relations between the similarly placed terms in
+  (B); e.g. [beta] + [gamma] is the difference of [alpha] + 2[beta] +
+  [gamma] and [alpha] + [beta], just as c - b is the difference of c and
+  b: and d - c is the sum of c - b and d - 2c + b, just as [beta] +
+  2[gamma] + [delta] is the sum of [beta] + [gamma] and [gamma] +
+  [delta]. Hence if we take [beta], [gamma], [delta], ... of (B) as
+  being the same as b - a, c - 2b + a, d -3c + 3b - a, ... of (A), all
+  corresponding terms in the two diagrams will be the same.
+
+  Thus we obtain the two principal formulae connecting terms and
+  differences. If we provisionally describe b - a, c - 2b + a, ... as
+  the first, second, ... differences of the particular term a (S 7),
+  then (i.) the nth difference of a is
+
+                              n.n - 1
+    l - nk + ... + (-1)^(n-2) ------- c + (-1)^(n-1) nb + (-1)^n a,
+                                1.2
+
+  where l, k ... are the (n + 1)th, nth, ... terms of the series a, b,
+  c, ...; the coefficients being those of the terms in the expansion of
+  (y -x)^n: and (ii.) the (n + 1)th term of the series, i.e. the nth
+  term after a, is
+
+                  n.n - 1
+    a + n[beta] + ------- [gamma] + ...
+                    1.2
+
+  where [beta], [gamma], ... are the first, second, ... differences of
+  a; the coefficients being those of the terms in the expansion of (x +
+  y)^n.
+
+  4. Now suppose we treat the terms a, b, c, ... as being themselves the
+  first differences of another series. Then, if the first term of this
+  series is N, the subsequent terms are N + a, N + a + b, N + a + b + c,
+  ...; i.e. the difference between the (n + 1)th term and the first term
+  is the sum of the first n terms of the original series. The term N, in
+  the diagram (A), will come above and to the left of a; and we see, by
+  (ii.) of S 3, that the sum of the first n terms of the original series
+  is
+
+     /         n.n - 1            \             n.n - 1          n.n - 1.n - 2
+    ( N + na + ------- [beta] + ...) - N = na + ------- [beta] + ------------- [gamma] + ...
+     \           1.2              /               1.2              1 . 2 . 3
+
+  5. As an example, take the arithmetical series
+
+    a, a + p, a + 2p, ...
+
+  The first differences are p, p, p, ... and the differences of any
+  higher order are zero. Hence, by (ii.) of S 3, the (n + 1)th term is a
+  + np, and, by S 4, the sum of the first n terms is na + 1/2n(n - 1)p =
+  1/2n{2a + (n - 1)p}.
+
+  6 As another example, take the series 1, 8, 27, ... the terms of which
+  are the cubes of 1, 2, 3, ... The first, second and third differences
+  of the first term are 7, 12 and 6, and it may be shown (S 14 (i.))
+  that all differences of a higher order are zero. Hence the sum of the
+  first n terms is
+
+          n.n - 1      n.n - 1.n - 2     n.n - 1.n - 2.n - 3
+    n + 7 ------- + 12 ------------- + 6 ------------------- =
+            1.2            1.2.3              1.2.3.4
+
+      1/4n^4 + 1/2n^3 + 1/4n^2 = {1/2n(n + 1)}^2.
+
+  7. In S 3 we have described b - a, c - 2b + a, ... as the first,
+  second, ... differences of a. This ascription of the differences to
+  particular terms of the series is quite arbitrary. If we read the
+  differences in the table of S 2 upwards to the right instead of
+  downwards to the right, we might describe e - d, e - 2d + c, ... as
+  the first, second, ... differences of e. On the other hand, the term
+  of greatest weight in c -2b + a, i.e. the term which has the
+  numerically greatest coefficient, is b, and therefore c - 2b + a might
+  properly be regarded as the second difference of b, and similarly e -
+  4d + 6c - 4b + a might be regarded as the fourth difference of c.
+  These three methods of regarding the differences lead to three
+  different systems of notation, which are described in SS 9, 10 and 11.
+
+
+  _Notation of Differences and Sums._
+
+  8. It is convenient to denote the terms a, b, c, ... of the series by
+  u0, u1, u2, u3, ... If we merely have the terms of the series, un may
+  be regarded as meaning the (n + 1)th term. Usually, however, the terms
+  are the values of a quantity u, which is a function of another
+  quantity x, and the values of x, to which a, b, c, ... correspond,
+  proceed by a constant difference h. If x0 and u0 are a pair of
+  corresponding values of x and u, and if any other value x0 + mh of x
+  and the corresponding value of u are denoted by xm and um, then the
+  terms of the series will be ... u_(n-2), u_(n-1), u_n, u_(n+1),
+  u_(n+2) ..., corresponding to values Of x denoted by ... x_(n-2),
+  x_(n-1), x_n, x_(n+1), x_(n+2)....
+
+  9. In the _advancing-difference notation_ u_(n+1) - u_n is denoted by
+  [Delta]un. The differences [Delta]u0, [Delta]u1, [Delta]u2 ... may
+  then be regarded as values of a function [Delta]u corresponding to
+  values of x proceeding by constant difference h; and therefore
+  [Delta]u_(n+1) -[Delta]u_n denoted by [Delta][Delta]u_n, or, more
+  briefly, [Delta]^2u_n; and so on. Hence the table of differences in S
+  2, with the corresponding values of x and of u placed opposite each
+  other in the ordinary manner of mathematical tables, becomes
+
+    +---------+---------+----------------+-----------------+-----------------+----------------------+
+    |    x    |    u    |    1st Diff.   |    2nd Diff.    |     3rd Diff.   |       4th Diff.      |
+    +---------+---------+----------------+-----------------+-----------------+----------------------+
+    |    .    |    .    |        .       |        .        |        .        |          .           |
+    |    .    |    .    |        .       |        .        |        .        |          .           |
+    |    .    |    .    |        .       |        .        |        .        |          .           |
+    |         |         |                |                 |                 |                      |
+    | x_(n-2) | u_(n-2) |                | [Delta]^2u_(n-3)|                 | [Delta]^4u_(n-4) ... |
+    |         |         | [Delta]u_(n-2) |                 | [Delta]^3u_(n-3)|                      |
+    | x_(n-1) | u_(n-1) |                | [Delta]^2u_(n-2)|                 | [Delta]^4u_(n-3) ... |
+    |         |         | [Delta]u_(n-1) |                 | [Delta]^3u_(n-2)|                      |
+    | xn      | u_n     |                | [Delta]^2u_(n-1)|                 | [Delta]^4u_(n-2) ... |
+    |         |         | [Delta]u_n     |                 | [Delta]^3u_(n-1)|                      |
+    | x_(n+1) | u_(n+1) |                | [Delta]^2u_n    |                 | [Delta]^4u_(n-1) ... |
+    |         |         | [Delta]u_(n+1) |                 | [Delta]^3u_n    |                      |
+    | x_(n+2) | u_(n+2) |                | [Delta]^2u_(n+1)|                 | [Delta]^4u_n     ... |
+    |    .    |    .    |        .       |        .        |        .        |          .           |
+    |    .    |    .    |        .       |        .        |        .        |          .           |
+    |    .    |    .    |        .       |        .        |        .        |          .           |
+    +---------+---------+----------------+-----------------+-----------------+----------------------+
+
+  The terms of the series of which ... u_(n-1), u_n, u_(n+1), ... are
+  the first differences are denoted by [Sigma]u, with proper suffixes,
+  so that this series is ... [Sigma]u_(n-1), [Sigma]u_n,
+  [Sigma]u_(n+1).... The suffixes are chosen so that we may have
+  [Delta][Sigma]un = un, whatever n may be; and therefore (S 4)
+  [Sigma]un may be regarded as being the sum of the terms of the series
+  up to and including un-1. Thus if we write [Sigma]u_(n-1) = C + un-2,
+  where C is any constant, we shall have
+
+    [Sigma]u_n = [Sigma]u_(n-1) + [Delta][Sigma]u_(n-1) = C + u_(n-2) + u_(n-1),
+    [Sigma]u_(n+1) = C + u_(n-2) + u_(n-1) + u_n,
+
+  and so on. This is true whatever C may be, so that the knowledge of
+  ... u_n-1, u_n, ... gives us no knowledge of the exact value of
+  [Sigma]u_n; in other words, C is an arbitrary constant, the value of
+  which must be supposed to be the same throughout any operations in
+  which we are concerned with values of [Sigma]_u corresponding to
+  different suffixes.
+
+  There is another symbol E, used in conjunction with u to denote the
+  next term in the series. Thus Eun means u_(n+1), so that Eun = u_n +
+  [Delta]u_n.
+
+  10. Corresponding to the advancing-difference notation there is a
+  _receding-difference_ notation, in which u_(n+1) - u_n is regarded as
+  a difference of u_(n+1), and may be denoted by [Delta]'u_(n+1), and
+  similarly u_(n+1) - 2u_n + u_(n-1) may be denoted by [Delta]'^2u_(n+1).
+  This notation is only required for certain special purposes, and the
+  usage is not settled (S 19 (ii.)).
+
+  11. The _central-difference_ notation depends on treating u_(n+1) -
+  2u_n -u_(n-1) as the second difference of un, and therefore as
+  corresponding to the value x_n; but there is no settled system of
+  notation. The following seems to be the most convenient. Since un is a
+  function of x_n, and the second difference u_(n+2) - 2u_(n+1) + u_n is
+  a function of x_(n+1), the first difference u_(n+1) - u_n must be
+  regarded as a function of x_(n+1/2), i.e. of 1/2{x_n + x_(n+1)}. We
+  therefore write u_(n+1) - u_n = [delta]u_(n+1/2), and each difference in
+  the table in S 9 will have the same suffix as the value of x in the
+  same horizontal line; or, if the difference is of an odd order, its
+  suffix will be the means of those of the two nearest values of x. This
+  is shown in the table below.
+
+  In this notation, instead of using the symbol E, we use a symbol [mu]
+  to denote the mean of two consecutive values of u, or of two
+  consecutive differences of the same order, the suffixes being assigned
+  on the same principle as in the case of the differences. Thus
+
+    [mu]u_(n+1/2) = 1/2{u_n + u_(n+1)}, [mu][delta]u_n = 1/2{[delta]u_(n-1/2)} + [delta]u_(n+1/2), &c.
+
+  If we take the means of the differences of odd order immediately above
+  and below the horizontal line through any value of x, these means,
+  with the differences of even order in that line, constitute the
+  _central differences_ of the corresponding value of u. Thus the table
+  of central differences is as follows, the values obtained as means
+  being placed in brackets to distinguish them from the actual
+  differences:--
+
+    +-------+-------+---------------------+----------------+----------------------+----------------------+
+    |   x   |   u   |      1st Diff.      |    2nd Diff.   |       3rd Diff.      |      4th Diff.       |
+    +-------+-------+---------------------+----------------+----------------------+----------------------+
+    |   .   |   .   |          .          |        .       |           .          |           .          |
+    |   .   |   .   |          .          |        .       |           .          |           .          |
+    |   .   |   .   |          .          |        .       |           .          |           .          |
+    |x_(n-2)|u_(n-2)| {[mu][delta]u_(n-2)}|[delta]^2u_(n-2)|{[mu][delta]^3u_(n-2)}| [delta]^4u_(n-2) ... |
+    |       |       |   [delta]u_(n-3/2)  |                | [delta]^3u_(n-3/2)   |                      |
+    |x_(n-1)|u_(n-1)| {[mu][delta]u_(n-1)}|[delta]^2u_(n-1)|{[mu][delta]^3u_(n-1)}| [delta]^4u_(n-1) ... |
+    |       |       |   [delta]u_(n-1/2)  |                |   [delta]^3u_(n-2    |                      |
+    |x_n    |u_n    |  ([mu][delta]u_n)   |[delta]^2u_n    | ([mu][delta]^3u_n)   | [delta]^4u_n     ... |
+    |       |       |   [delta]u_(n+1/2)  |                |  [delta]^3u_(n+1/2)  |                      |
+    |x_(n+1)|u_(n+1)| {[mu][delta]u_(n+1)}|[delta]^2u_(n+1)|{[mu][delta]^3u_(n+1)}| [delta]^4u_(n+1) ... |
+    |       |       |   [delta]u_(n+3/2)  |                | [delta]^3u_(n+3/2)   |                      |
+    |x_(n+2)|u_(n+2)| {[mu][delta]u_(n+2)}|[delta]^2u_(n+2)|{[mu][delta]^3u_(n+2)}| [delta]^4u_(n+2) ... |
+    |   .   |   .   |          .          |        .       |           .          |           .          |
+    |   .   |   .   |          .          |        .       |           .          |           .          |
+    |   .   |   .   |          .          |        .       |           .          |           .          |
+    +-------+-------+---------------------+----------------+----------------------+----------------------+
+
+  Similarly, by taking the means of consecutive values of u and also of
+  consecutive differences of even order, we should get a series of terms
+  and differences central to the intervals x_(n-2) to x_(n-1), x_(n-1)
+  to x_n, ....
+
+  The terms of the series of which the values of u are the first
+  differences are denoted by [sigma]u, with suffixes on the same
+  principle; the suffixes being chosen so that [delta][sigma]un shall be
+  equal to un. Thus, if
+
+    [sigma]u_(n-3/2) = C + u_(n-2),
+
+  then
+
+    [sigma]u_(n-1/2) = C + u_(n-2) + u_(n-1), [sigma]_(n+1/2)
+       = C + u_(n-2) + u_(n-1) + u_n, &c.,
+
+  and also
+
+    [mu][sigma]u_(n-1) = C + u_(n-2) + 1/2u_(n-1), [mu][sigma]u_n
+       = C + u_(n-2) + u_(n-1) + 1/2u_n, &c.,
+
+  C being an arbitrary constant which must remain the same throughout
+  any series of operations.
+
+
+  _Operators and Symbolic Methods._
+
+  12. There are two further stages in the use of the symbols [Delta],
+  [Sigma], [delta], [sigma], &c., which are not essential for elementary
+  treatment but lead to powerful methods of deduction.
+
+  (i.) Instead of treating [Delta]u as a function of x, so that
+  [Delta]u_n means ([Delta]u)_n, we may regard [Delta] as denoting an
+  _operation_ performed on u, and take [Delta]un as meaning [Delta].u_n.
+  This applies to the other symbols E, [delta], &c., whether taken
+  simply or in combination. Thus [Delta]Eu_n means that we first replace
+  un by un+1, and then replace this by u_(n+2) - u_(n+1).
+
+  (ii.) The operations [Delta], E, [delta], and [mu], whether performed
+  separately or in combination, or in combination also with numerical
+  multipliers and with the operation of differentiation denoted by D (:=
+  d/dx), follow the ordinary rules of algebra: e.g. [Delta](u_n + v_n) =
+  [Delta]u_n + [Delta]v_n, [Delta]Du_n = D[Delta]u_n, &c. Hence the
+  symbols can be separated from the functions on which the operations
+  are performed, and treated as if they were algebraical quantities. For
+  instance, we have
+
+    E.u_n = u_(n+1) = u_n + [Delta]u_n = 1.u_n + [Delta].u_n,
+
+  so that we may write E = 1 + [Delta], or [Delta] = E - 1. The first of
+  these is nothing more than a statement, in concise form, that if we
+  take two quantities, subtract the first from the second, and add the
+  result to the first, we get the second. This seems almost a truism.
+  But, if we deduce E^n = (1 + [Delta])^n, [Delta]^n = (E-1)^n, and
+  expand by the binomial theorem and then operate on u0, we get the
+  general formulae
+
+                           n.n - 1
+    un = u0 + n[Delta]u0 + ------- [Delta]^2u0 + ... + [Delta]^nu0,
+                             1.2
+                                   n.n - 1
+    [Delta]^nu0 = u_n - nu_(n-1) + ------- u_(n-2) + ... + (-1)^nu0,
+                                     1.2
+
+  which are identical with the formulae in (ii.) and (i.) of S 3.
+
+  (iii.) What has been said under (ii.) applies, with certain
+  reservations, to the operations [Sigma] and [sigma], and to the
+  operation which represents integration. The latter is sometimes
+  denoted by D^-1; and, since [Delta][Sigma]un = un, and
+  [delta][sigma]u_n = u_n, we might similarly replace [Sigma] and
+  [sigma] by [Delta]^-1 and [delta]^-1. These symbols can be combined
+  with [Delta], E, &c. according to the ordinary laws of algebra,
+  provided that proper account is taken of the arbitrary constants
+  introduced by the operations D^-1, [Delta]^-1, [delta]^-1.
+
+
+  _Applications to Algebraical Series._
+
+  13. _Summation of Series._--If ur, denotes the (r+1)th term of a
+  series, and if vr is a function of r such that [Delta]v_r = u_r for
+  all integral values of r, then the sum of the terms u_m, u_(m+1), ...
+  un is v_(n+1) -v_m. Thus the sum of a number of terms of a series may
+  often be found by inspection, in the same kind of way that an integral
+  is found.
+
+  14. _Rational Integral Functions._--(i.) If u_r is a rational integral
+  function of r of degree p, then [Delta]ur, is a rational integral
+  function of r of degree p-1.
+
+  (ii.) A particular case is that of a _factorial_, i.e. a product of
+  the form (r+a+1) (r+a+2) ... (r+b), each factor exceeding the
+  preceding factor by 1. We have
+
+    [Delta].(r+a+1) (r+a+2) ... (r+b) = (b-a).(r+a+2) ... (r+b),
+
+  whence, changing a into a-1,
+
+    [Sigma](r+a+1)(r+a+2) ... (r+b) = _const._ + (r+a)(r+a+1) ...
+      (r+b)/(b-a+1).
+
+  A similar method can be applied to the series whose (r+1)th term is of
+  the form 1/(r+a+1) (r+a+2) ... (r+b).
+
+  (iii.) Any rational integral function can be converted into the sum of
+  a number of factorials; and thus the sum of a series of which such a
+  function is the general term can be found. For example, it may be
+  shown in this way that the sum of the pth powers of the first n
+  natural numbers is a rational integral function of n of degree p+1,
+  the coefficient of n^p+1 being 1/(p+1).
+
+  15. _Difference-equations._--The summation of the series ... + u_(n+2)
+  + u_(n-1) + u_n is a solution of the _difference-equation_ [Delta]v_n
+  = u_(n+1), which may also be written (E-1)v_n = u_(n+1). This is a
+  simple form of difference-equation. There are several forms which have
+  been investigated; a simple form, more general than the above, is the
+  _linear equation_ with _constant coefficients_--
+
+    v_(n+m) + a1v_(n+m-1) + a2v_(n+m-2) + ... + a_mv_n = N,
+
+  where a1, a2, ... am are constants, and N is a given function of n.
+  This may be written
+
+    (E^m + a1E^(m-1) + ... + a_m)v_n = N
+
+  or
+
+    (E-p1)(E-p2) ... (E-p_m)v_n = N.
+
+  The solution, if p1, p2, ... pm are all different, is vn = C1p1^n +
+  C2p2^n + ... + C_mp_m^n + V_n, where C1, C2 ... are constants, and v_n
+  = V_n is any one solution of the equation. The method of finding a
+  value for Vn depends on the form of N. Certain modifications are
+  required when two or more of the p's are equal.
+
+  It should be observed, in all cases of this kind, that, in describing
+  C1, C2 as "constants," it is meant that the value of any one, as C1,
+  is the same for all values of n occurring in the series. A "constant"
+  may, however, be a periodic function of n.
+
+
+  _Applications to Continuous Functions._
+
+  16. The cases of greatest practical importance are those in which u is
+  a continuous function of x. The terms u1, u2 ... of the series then
+  represent the successive values of u corresponding to x = x1, x2....
+  The important applications of the theory in these cases are to (i.)
+  relations between differences and differential coefficients, (ii.)
+  interpolation, or the determination of intermediate values of u, and
+  (iii.) relations between sums and integrals.
+
+  17. Starting from any pair of values x0 and u0, we may suppose the
+  interval h from x0 to x1 to be divided into q equal portions. If we
+  suppose the corresponding values of u to be obtained, and their
+  differences taken, the successive advancing differences of u0 being
+  denoted by dPu0, dP^2u0 ..., we have (S 3 (ii.))
+
+                         q.q - 1
+    u1  = u0  + qdPu0  + ------- dP^2u0  + ....
+                           1.2
+
+  When q is made indefinitely great, this (writing f(x) for u) becomes
+  Taylor's Theorem (INFINITESIMAL CALCULUS)
+
+                               h^2
+    f(x + h) = f(x) + hf'(x) + --- f"(x) + ...,
+                               1.2
+
+  which, expressed in terms of operators, is
+
+                 h^2       h^3
+    E = 1 + hD + ---D^2 + ----- D^3 + ... = e^(hD).
+                 1.2      1.2.3
+
+  This gives the relation between [Delta] and D. Also we have
+
+                         2q.2q - 1
+    u2 = u0  + 2qdPu0  + --------- dP^2u0  + ...
+                            1.2
+
+                         3q.3q - 1
+    u3 = u0  + 3qdPu0  + --------- dP^2u0  + ...
+                            1.2
+    .            .
+    .            .
+    .            .
+
+  and, if p is any integer,
+
+                           p.p - 1
+    u_(p/q) = u0 + pdPu0 + ------- dP^2u0 + ....
+                             1.2
+
+  From these equations up/q could be expressed in terms of u0, u1, u2,
+  ...; this is a particular case of interpolation (q.v.).
+
+  18. _Differences and Differential Coefficients._--The various formulae
+  are most quickly obtained by symbolical methods; i.e. by dealing with
+  the operators [Delta], E, D, ... as if they were algebraical
+  quantities. Thus the relation E = e^(hD) (S 17) gives
+
+    hD = log_e  (1 + [Delta]) = [Delta] - 1/2[Delta]^2 + 1/3 [Delta]^3 ...
+
+         /du\
+    or h( -- ) = [Delta]u0 - 1/2[Delta]^2u0 + 1/3 [Delta]^3u0  ....
+         \dx/0
+
+  The formulae connecting central differences with differential
+  coefficients are based on the relations [mu] = cosh 1/2hD = 1/2(e^1/2hD
+  + e^ -1/2hD), [delta] = 2 sinh 1/2hD - e^ 1/2hD - e^ -1/2hD, and
+  may be grouped as follows:--
+
+              u0  = u0                                               \
+                                                                     |
+    [mu][delta]u0 = (hD + 1/6 h^3D^3 + 1/120 h^5 D^5 + ...)u0        |
+                                                                     |
+      [delta]^2u0 = (h^2D^2 + 1/12 h^4 D^4 + 1/360 h^6 D^6 + ...)u0  >
+                                                                     |
+  [mu][delta]^3u0 = (h^3D^3 + 1/4 h^5 D^5 + ...)u0                   |
+                                                                     |
+     [delta]^4 u0 = (h^4 D^4 + 1/6 h^6 D^6 + ...)u0                  /
+
+                  .                .             .
+                  .                .             .
+                  .                .             .
+
+            [mu]u_1/2 = (1 + 1/8 h^2D^2 + 1/384 h^4 D^4 + 1/46080 h^6 D^6 + ...)u_1/2 \
+                                                                                      |
+         [delta]u_1/2 = (hD + 1/24 h^3D^3 + 1/1920 h^5 D^5 + ...)u_1/2                |
+                                                                                      |
+   [mu][delta]^2u_1/2 = (h^2D^2 + 5/24 h^4 D^4 + 91/5760 h^6 D^6 + ...)u_1/2           >
+                                                                                      |
+       [delta]^3u_1/2 = (h^3D^3 + 1/8 h^5 D^5 + ...)u_1/2                             |
+                                                                                      |
+  [mu][delta]^4 u_1/2 = (h^4 D^4 + 7/24 h^6 D^6 + ...)u_1/2                          /
+
+                  .                .             .
+                  .                .             .
+                  .                .             .
+
+               u0 = u0                                                              \
+                                                                                    |
+            hDu0  = ([mu][delta] - 1/6 [mu][delta]^3 + 1/30 [mu][delta]^5  - ...)u0 |
+                                                                                    |
+         h^2D^2u0 = ([delta]^2 - 1/12 [delta]^4 + 1/90 [delta]^6 - ...)u0            >
+                                                                                    |
+        h^3D^3u0  = ([mu][delta]^3 1/4 [mu][delta]^5  + ...)u0                      |
+                                                                                    |
+      h^4 D^4 u_0 = ([delta]^4 - 1/6 [delta]^6 + ...)u0                            /
+
+                  .                .             .
+                  .                .             .
+                  .                .             .
+
+            u_1/2 = ([mu] - 1/8 [mu][delta]^2 + 3/128 [mu][delta]^4 - 5/1024 [mu][delta]^6 + ...)u_1/2 \
+                                                                                                       |
+         hDu_1/2  = ([delta] - 1/24 [delta]^3 + 3/640 [delta]^5 - ...)u_1/2                            |
+                                                                                                       |
+      h^2D^2u_1/2 = ([mu][delta]^2 - 5/24 [mu][delta]^ + 259/5760 [mu][delta]^6 - ...)u_1/2             >
+                                                                                                       |
+     h^3D^3u_1/2  = ([delta]^3 - 1/8 [delta]^5 + ...)u_1/2                                             |
+                                                                                                       |
+    h^4 D^4 u_1/2 = ([mu][delta]^4 - 7/24 [mu][delta]^6 + ...)u_1/2                                   /
+
+                  .                .             .
+                  .                .             .
+                  .                .             .
+
+  When u is a rational integral function of x, each of the above series
+  is a terminating series. In other cases the series will be an infinite
+  one, and may be divergent; but it may be used for purposes of
+  approximation up to a certain point, and there will be a "remainder,"
+  the limits of whose magnitude will be determinate.
+
+  19. _Sums and Integrals._--The relation between a sum and an integral
+  is usually expressed by the _Euler-Maclaurin formula_. The principle
+  of this formula is that, if um and um+1, are ordinates of a curve,
+  distant h from one another, then for a first approximation to the area
+  of the curve between um and um+1 we have 1/2h(u_m + u_m+1), and the
+  difference between this and the true value of the area can be
+  expressed as the difference of two expressions, one of which is a
+  function of x_m, and the other is the same function of x_m+1. Denoting
+  these by [phi](x_m) and [phi](xm+1), we have
+
+      _ x_m+1
+     /
+     |   udx = 1/2h(u_m  + u_m+1) + [phi](x_m+1 ) - [phi](x_m).
+    _/x_m
+
+  Adding a series of similar expressions, we find
+
+      _ x_n
+     /
+     |  udx = h{1/2u_m + u_m+1 + u_m+2 + ... + u_n-1 + 1/2u_n} + [phi](x_n) - [phi](x_m).
+    _/x_m
+
+  The function [phi](x) can be expressed in terms either of differential
+  coefficients of u or of advancing or central differences; thus there
+  are three formulae.
+
+  (i.) The Euler-Maclaurin formula, properly so called, (due
+  independently to Euler and Maclaurin) is
+
+      _ x_n
+     /                             1     du_n    1       d^3u_n     1       d^5 u_n
+     |   udx = h.[mu][sigma]u_n - -- h^2 ---- + --- h^4  ------ - ----- h^6 ------- + ...
+    _/x_m                         12      dx    720       dx^3    30240       dx^5
+
+                                  B1    du_n   B2     d^3u_n   B3     d^5u_n
+             = h.[mu][sigma]u_n - -- h2 ---- + -- h^4 ------ - -- h^6 ------ + ...,
+                                  2!     dx    4!      dx^3    6!      dx^5
+
+  where B1, B2, B3 ... are _Bernoulli's numbers_.
+
+  (ii.) If we express differential coefficients in terms of advancing
+  differences, we get a theorem which is due to Laplace:--
+
+        _ x_n
+    1  /
+    -  |  udx = [mu][sigma](u_n - u0) - 1/12 ([Delta]u_n - [Delta]u0) + 1/24 ( [Delta]^2u_n - [Delta]^2u0)
+    h _/x0
+
+           - 19/720 ([Delta]^3u_n - [Delta]^3u_0) + 3/160 ([Delta]^4 u_n - [Delta]^4 u0) - ...
+
+
+  For practical calculations this may more conveniently be written
+
+        _ x_n
+    1  /
+    -  |  udx = [mu][sigma](u_n - u0) + 1/12 ([Delta]u0 - 1/2[Delta]^2u0 + 19/60 [Delta]^3u0 - ...)
+    h _/x0
+
+                 + 1/12 ([Delta]'u_n - 1/2[Delta]'^2u_n + 19/60 [Delta]'^3u_n - ...),
+
+  where accented differences denote that the values of u are read
+  backwards from un; i.e. [Delta]'un denotes u_n-1 - u_n, not (as in S
+  10) u_n - u_n-1.
+
+  (iii.) Expressed in terms of central differences this becomes
+
+        _ x_n
+    1  /
+    -  |   udx = [mu][sigma](u_n - u0) - 1/12 [mu][delta]u_n + 11/720 [mu][delta]^3u_n - ...
+    h _/x0
+                            + 1/12 [mu][delta]u0  - 11/720 [mu][delta]^3u0 + ...
+
+           /          1            11              191               2497                 \  /      \
+    = [mu]([sigma] - -- [delta] + --- [delta]^3 - ----- [delta]^5 + ------- [delta]^7 - ...)(u_n - u0).
+           \         12           720             60480             3628800               /  \      /
+
+  (iv.) There are variants of these formulae, due to taking hum+1/2 as the
+  first approximation to the area of the curve between um and um+1; the
+  formulae involve the sum u_1/2 + u_3/2 + ... + u_n-1/2 := [sigma](u_n -
+  u0) (see MENSURATION).
+
+  20. The formulae in the last section can be obtained by symbolical
+  methods from the relation
+
+        _
+    1  /       1         1
+    -  | udx = - D^1 u = --.u.
+    h _/       h         hD
+
+  Thus for central differences, if we write [theta] := 1/2hD, we have [mu]
+  = cosh [theta], [delta] = 2 sinh [theta], [sigma] = [delta]^-1, and
+  the result in (iii.) corresponds to the formula
+
+                                        / /   1                  2                   2.4                    \
+    sinh [theta] = [theta] cosh [theta]/ (1 + - sinh^2[theta] - --- sinh^4[theta] + ----- sinh^6[theta] - ...).
+                                      /   \   3                 3.5                 3.5.7                   /
+
+  REFERENCES.--There is no recent English work on the theory of finite
+  differences as a whole. G. Boole's _Finite Differences_ (1st ed.,
+  1860, 2nd ed., edited by J. F. Moulton, 1872) is a comprehensive
+  treatise, in which symbolical methods are employed very early. A. A.
+  Markoff's _Differenzenrechnung_ (German trans., 1896) contains general
+  formulae. (Both these works ignore central differences.) _Encycl. der
+  math. Wiss._ vol. i. pt. 2, pp. 919-935, may also be consulted. An
+  elementary treatment of the subject will be found in many text-books,
+  e.g. G. Chrystal's _Algebra_ (pt. 2, ch. xxxi.). A. W. Sunderland,
+  _Notes on Finite Differences_ (1885), is intended for actuarial
+  students. Various central-difference formulae with references are
+  given in _Proc. Lond. Math. Soc._ xxxi. pp. 449-488. For other
+  references see INTERPOLATION.     (W. F. SH.)
+
+
+
+
+DIFFERENTIAL EQUATION, in mathematics, a relation between one or more
+functions and their differential coefficients. The subject is treated
+here in two parts: (1) an elementary introduction dealing with the more
+commonly recognized types of differential equations which can be solved
+by rule; and (2) the general theory.
+
+
+  _Part I.--Elementary Introduction._
+
+  Of equations involving only one independent variable, x (known as
+  _ordinary_ differential equations), and one dependent variable, y, and
+  containing only the first differential coefficient dy/dx (and
+  therefore said to be of the first _order_), the simplest form is that
+  reducible to the type
+
+    dy/dx = f(x)/F(y),
+
+  leading to the result fF(y)dy - ff(x)dx = A, where A is an arbitrary
+  constant; this result is said to solve the differential equation, the
+  problem of evaluating the integrals belonging to the integral
+  calculus.
+
+
+  Another simple form is
+
+    dy/dx + yP = Q,
+
+  where P, Q are functions of x only; this is known as the linear
+  equation, since it contains y and dy/dx only to the first degree. If
+
+    fPdx = u, we clearly have
+
+     d             /dy    \
+    --(ye^u) =e^u ( -- + Py) = e^u Q,
+    dx             \dx    /
+
+  so that y = e^-u(fe^u Qdx + A) solves the equation, and is the only
+  possible solution, A being an arbitrary constant. The rule for the
+  solution of the linear equation is thus to multiply the equation by
+  e^u, where u = fPdx.
+
+  A third simple and important form is that denoted by
+
+    y = px + f(p),
+
+  where p is an abbreviation for dy/dx; this is known as Clairaut's
+  form. By differentiation in regard to x it gives
+
+             dp        dp
+    p = p + x-- + f'(p)--,
+             dx        dx
+
+  where
+
+             d
+    f'(p) = -- f(p);
+            dp
+
+  thus, either (i.) dp/dx = 0, that is, p is constant on the curve
+  satisfying the differential equation, which curve is thus any one of
+  the straight lines y = cx = f(c), where c is an arbitrary constant, or
+  else, (ii.) x + [f]'(p) = 0; if this latter hypothesis be taken, and p
+  be eliminated between x + f'(p) = 0 and y = px + f(p), a relation
+  connecting x and y, not containing an arbitrary constant, will be
+  found, which obviously represents the envelope of the straight lines y
+  = cx + f(c).
+
+  In general if a differential equation [phi](x, y, dy/dx) = 0 be
+  satisfied by any one of the curves F(x, y, c) = 0, where c is an
+  arbitrary constant, it is clear that the envelope of these curves,
+  when existent, must also satisfy the differential equation; for this
+  equation prescribes a relation connecting only the co-ordinates x, y
+  and the differential coefficient dy/dx, and these three quantities are
+  the same at any point of the envelope for the envelope and for the
+  particular curve of the family which there touches the envelope. The
+  relation expressing the equation of the envelope is called a
+  _singular_ solution of the differential equation, meaning an
+  _isolated_ solution, as not being one of a family of curves depending
+  upon an arbitrary parameter.
+
+  An extended form of Clairaut's equation expressed by
+
+    y = xF(p) + f(p)
+
+  may be similarly solved by first differentiating in regard to p, when
+  it reduces to a linear equation of which x is the dependent and p the
+  independent variable; from the integral of this linear equation, and
+  the original differential equation, the quantity p is then to be
+  eliminated.
+
+  Other types of solvable differential equations of the first order are
+  (1)
+
+    M dy/dx = N,
+
+  where M, N are homogeneous polynomials in x and y, of the same order;
+  by putting v = y/x and eliminating y, the equation becomes of the
+  first type considered above, in v and x. An equation (aB <> bA)
+
+    (ax + by + c)dy/dx = Ax + By + C
+
+  may be reduced to this rule by first putting x + h, y + k for x and y,
+  and determining h, k so that ah + bk + c = 0, Ah + Bk + C = 0.
+
+  (2) An equation in which y does not explicitly occur,
+
+    f(x, dy/dx) = 0,
+
+  may, theoretically, be reduced to the type dy/dx = F(x); similarly an
+  equation F(y, dy/dx) = 0.
+
+  (3) An equation
+
+    f(dy/dx, x, y) = 0,
+
+  which is an integral polynomial in dy/dx, may, theoretically, be
+  solved for dy/dx, as an algebraic equation; to any root dy/dx = F1(x,
+  y) corresponds, suppose, a solution [phi]1(x, y, c) = 0, where c is an
+  arbitrary constant; the product equation [phi]1(x, y, c)[phi]2(x, y,
+  c) ... = 0, consisting of as many factors as there were values of
+  dy/dx, is effectively as general as if we wrote [phi]1(x, y, c1)
+  [phi]2(x, y, c2) ... = 0; for, to evaluate the first form, we must
+  necessarily consider the factors separately, and nothing is then
+  gained by the multiple notation for the various arbitrary constants.
+  The equation [phi]1(x, y, c)[phi]2(x, y, c) ... = 0 is thus the
+  solution of the given differential equation.
+
+  In all these cases there is, except for cases of singular solutions,
+  one and only one arbitrary constant in the most general solution of
+  the differential equation; that this must necessarily be so we may
+  take as obvious, the differential equation being supposed to arise by
+  elimination of this constant from the equation expressing its solution
+  and the equation obtainable from this by differentiation in regard to
+  x.
+
+  A further type of differential equation of the first order, of the
+  form
+
+    dy/dx = A + By + Cy^2
+
+  in which A, B, C are functions of x, will be briefly considered below
+  under differential equations of the second order.
+
+  When we pass to ordinary differential equations of the second order,
+  that is, those expressing a relation between x, y, dy/dx and d^2y/dx^2,
+  the number of types for which the solution can be found by a known
+  procedure is very considerably reduced. Consider the general linear
+  equation
+
+    d^2y   dy
+    --- + P-- + Qy = R,
+    dx^2   dx
+
+  where P, Q, R are functions of x only. There is no method always
+  effective; the main general result for such a linear equation is that
+  if any particular function of x, say y1, can be discovered, for which
+
+    d^2y1   dy1
+    ---- + P--- + Qy1 = 0,
+    dx^2    dx
+
+  then the substitution y = y1[eta] in the original equation, with R on
+  the right side, reduces this to a linear equation of the first order
+  with the dependent variable d[eta]/dx. In fact, if y = y1[eta] we have
+
+    dy     d[eta]        dy1     d^2y    d^2[eta]   dy1 d[eta]        d^2y1
+    -- = y1------ + [eta]--- and --- = y1------- + 2--- ------ + [eta]-----,
+    dx       dx          dx      dx^2      dx^2     dx    dx           dx^2
+
+  and thus
+
+    d^2y     dy         d^2[eta]   / dy1     \  d[eta]    /d^2y1    dy1     \
+    --- + P -- + Qy = y1------- + ( 2--- + Py1) ------ + ( ----- + P--- + Qy1)[eta];
+    dx^2     dx           dx^2     \ dx      /    dx      \ dx^2    dx      /
+
+  if then
+
+    d^2y1    dy1
+    ---- + P --- + Qy1 = 0,
+    dx^2     dx
+
+  and z denote d[eta]/dx, the original differential equation becomes
+
+      dz    / dy1     \
+    y1-- + ( 2--- + Py1)z = R.
+      dx    \ dx      /
+
+  From this equation z can be found by the rule given above for the
+  linear equation of the first order, and will involve one arbitrary
+  constant; thence y = y1 [eta] = y1 [int] zdx + Ay1, where A is another
+  arbitrary constant, will be the general solution of the original
+  equation, and, as was to be expected, involves two arbitrary
+  constants.
+
+  The case of most frequent occurrence is that in which the coefficients
+  P, Q are constants; we consider this case in some detail. If [t]*
+  be a root of the quadratic equation [t]^2 + [t]P + Q = 0, it
+  can be at once seen that a particular integral of the differential
+  equation with zero on the right side is y1 = e^[theta]x. Supposing
+  first the roots of the quadratic equation to be different, and [phi]
+  to be the other root, so that [p] + [t] = -P, the auxiliary
+  differential equation for z, referred to above, becomes dz/dx +
+  ([t] - [p])z = Re^(-[t]^x), which leads to
+  ze^{([t]-[p])^x} = B + [int] Re^(-[p]^x)dx, where B is an
+  arbitrary constant, and hence to
+
+(*) [t] = [theta]; [p] = [phi].
+                               _                             _               _
+                              /                             /               /
+    y = Ae^([t]^x) + e^([t]^x)| Be^([p]-[t])^x dx + e^[t]^x | e^([p]-[t])^x | Re^-[p]^x dxdx,
+                             _/                            _/              _/
+
+  or say to y = Ae^[t]^x + Ce^[p]^x + U, where A, C are arbitrary
+  constants and U is a function of x, not present at all when R = 0. If
+  the quadratic equation [t]^2 + P[t] + Q = 0 has equal roots, so that
+  2[t] = -P, the auxiliary equation in z becomes dz/dx = Re^-[t]^x,
+  giving z = B + [int] Re^-[t]^x dx, where B is an arbitrary constant,
+  and hence
+                                   _  _
+                                  /  /
+    y = (A + Bx)e^[t]^x + e^[t]^x |  | Re^-[t]^x dxdx,
+                                 _/ _/
+
+  or, say, y = (A + Bx)e^[t]^x + U, where A, B are arbitrary constants,
+  and U is a function of x not present at all when R = 0. The portion
+  Ae^[t]^x + Be^[p]^x or (A + Bx)e^[t]^x of the solution, which is known
+  as the _complementary function_, can clearly be written down at once
+  by inspection of the given differential equation. The remaining
+  portion U may, by taking the constants in the complementary function
+  properly, be replaced by any particular solution whatever of the
+  differential equation
+
+    d^2v     dy
+    ---- + P -- + Qy = R;
+    dx^2     dx
+
+  for if u be any particular solution, this has a form
+
+    u = A0  e^[t]^x + B0 e^[p]^x + U,
+
+  or a form
+
+    u = (A0 + B0x)e^[t]^x + U;
+
+  thus the general solution can be written
+
+    (A - A0)e^[t]^x + (B - B0)e^[p]^x + u,
+
+  or
+
+    {A - A0  + (B - B0)x}e^[t]^x + u,
+
+  where A - A0, B - B0, like A, B, are arbitrary constants.
+
+  A similar result holds for a linear differential equation of any
+  order, say
+
+    d^n y      d^n-1 y
+    ----- + P1 ------- + ... + P_n y = R,
+    dx_n       dx^n-1
+
+  where P1, P2, ... Pn are constants, and R is a function of x. If we
+  form the algebraic equation [t]^n + P1[t]^n-1 + ... + P_n = 0, and all
+  the roots of this equation be different, say they are [t]1, [t]2, ...
+  [t]n, the general solution of the differential equation is
+
+    y = A1 e^[t]1^x + A2 e^[t]2^x + ... + A_n e^[t]_n^x + u,
+
+  where A1, A2, ... An are arbitrary constants, and u is any particular
+  solution whatever; but if there be one root [t]1 repeated r times, the
+  terms A1 e^[t]1^x + ... + A_r e^[t]_r^x must be replaced by (A1 + A2x
+  + ... + A_r x^r-1)e^[t]1x where A1, ... An are arbitrary constants;
+  the remaining terms in the complementary function will similarly need
+  alteration of form if there be other repeated roots.
+
+  To complete the solution of the differential equation we need some
+  method of determining a particular integral u; we explain a procedure
+  which is effective for this purpose in the cases in which R is a sum
+  of terms of the form e^ax[p](x), where [p](x) is an integral
+  polynomial in x; this includes cases in which R contains terms of the
+  form cos bx.[p](x) or sin bx.[p](x). Denote d/dx by D; it is clear
+  that if u be any function of x, D(e^ax u) = e^ax Du + ae^ax u, or say,
+  D(e^ax u) = e^ax (D + a)u; hence D^2(e^ax u), i.e. d^2/dx^2 (e^ax u),
+  being equal to D(e^ax v), where v=(D + a)u, is equal to e^ax(D + a)v,
+  that is to e^ax(D + a)^2u. In this way we find D^n(e^ax u) = e^ax(D +
+  a)^n u, where n is any positive integer. Hence if [psi](D) be any
+  polynomial in D with constant coefficients, [psi](D)(e^ax u) = e^ax
+  [psi](D + a)u. Next, denoting [int] udx by D^-1 u, and any solution of
+  the differential equation dz/dx + az = u by z = (d + a)^-1 u, we have
+  D[e^ax(D + a)^-1 u] = D(e^ax z) = e^ax(D + a)z = e^ax u, so that we
+  may write D^-1(e^ax u) = e^ax(D+a)^-1 u, where the meaning is that one
+  value of the left side is equal to one value of the right side; from
+  this, the expression D^-2(e^axu), which means D^-1[D^-1(e^ax u)], is
+  equal to D^-1(e^ax z) and hence to e^ax(D + a)^-1 z, which we write
+  e^ax(D + a)^-2 u; proceeding thus we obtain
+
+    D^-n(e^ax u) = e^ax(D + a)^-n u,
+
+  where n is any positive integer, and the meaning, as before, is that
+  one value of the first expression is equal to one value of the second.
+  More generally, if [psi](D) be any polynomial in D with constant
+  coefficients, and we agree to denote by 1/[psi](D) u any solution z of
+  the differential equation [psi](D)z = u, we have, if v = 1/[psi](D +
+  a) u, the identity [psi](D)(e^ax v) = e^ax [psi](D + a)v = e^ax u,
+  which we write in the form
+
+        1                        1
+    --------(e^ax u) = e^ax ------------ u.
+    [psi](D)                [psi](D + a)
+
+  This gives us the first step in the method we are explaining, namely
+  that a solution of the differential equation [psi](D)y = e^ax u + e^bx
+  v + ... where u, v, ... are any functions of x, is any function
+  denoted by the expression
+
+              1                     1
+    e^ax ------------ u + e^ax ------------ v + ....
+         [psi](D + a)          [psi](D + b)
+
+  It is now to be shown how to obtain one value of 1/[psi](D + a) u,
+  when u is a polynomial in x, namely one solution of the differential
+  equation [psi](D + a)z = u. Let the highest power of x entering in u
+  be x^m; if t were a variable quantity, the rational fraction in t,
+  1/[psi](t + a), by first writing it as a sum of partial fractions, or
+  otherwise, could be identically written in the form
+
+    K_r t^-r + K_r-1 t^-r+1 + ... + K1 t^-1 + H + H1t + ... + H_m t^m + t^m+1 [p](t)/[psi](t + a),
+
+  where [p](t) is a polynomial in t; this shows that there exists an
+  identity of the form
+
+    1 = [psi](t + a)(K_r t^-r + ... + K1t^-1 + H + H1t + ... + H_m t^m) + [p](t)t^m+1,
+
+  and hence an identity
+
+    u = [psi](D + a)[K_r D^-r + ... + K1D^-1 + H + H1D + ... + H_m D^m]u + [p](D)D^m+1 u;
+
+  in this, since u contains no power of x higher than x^m, the second
+  term on the right may be omitted. We thus reach the conclusion that a
+  solution of the differential equation [psi](D + a)z = u is given by
+
+    z = (K_r D^-r + ... + K1D^-1 + H + H1D + ... + H_m D^m)u,
+
+  of which the operator on the right is obtained simply by expanding
+  1/[psi](D + a) in ascending powers of D, as if D were a numerical
+  quantity, the expansion being carried as far as the highest power of D
+  which, operating upon u, does not give zero. In this form every term
+  in z is capable of immediate calculation.
+
+  _Example._--For the equation
+
+    d^4v    d^2y
+    ---- + 2--- + y = x^3 cos x or (D^2 + 1)^2y = x^3 cos x,
+    dx^4    dx^3
+
+  the roots of the associated algebraic equation ([t]^2+1)^2 = 0 are [t] =
+  [+-]i, each repeated; the complementary function is thus
+
+    (A + Bx)e^ix + (C + Dx)e^ix,
+
+  where A, B, C, D are arbitrary constants; this is the same as
+
+    (H + Kx) cos x + (M + Nx) sin x,
+
+  where H, K, M, N are arbitrary constants. To obtain a particular
+  integral we must find a value of (1 + D^2)^-2 x^3 cos x; this is the
+  real part of (1 + D^2)^-2 e^ix x^3 and hence of e^ix [1 + (D + i)^2]^-2
+  x^3
+
+  or    e^ix [2iD(1 + 1/2iD)]^-2 x^3,
+
+  or    -1/4e^ix D^-2 (1 + iD - 3/4D^2 - 1/2iD^3 + 5/16 D^4 + 3/16 iD^5 ...)x^3,
+
+  or    -1/4e^ix(1/20 x^5 + 1/4ix^4 - 3/4x^3 - 3/2 ix^2 + 15/8 x + 9/8 i);
+
+  the real part of this is
+
+    -1/4(1/20 x^5 - 3/4x^2 + 15/8 x) cos x + 1/4(1/4x^4 - 3/2 x^2 + 9/8) sin x.
+
+  This expression added to the complementary function found above gives
+  the complete integral; and no generality is lost by omitting from the
+  particular integral the terms -15/32 x cos x + 9/32 sin x, which are
+  of the types of terms already occurring in the complementary function.
+
+  The symbolical method which has been explained has wider applications
+  than that to which we have, for simplicity of explanation, restricted
+  it. For example, if [psi](x) be any function of x, and a1, a2, ... an
+  be different constants, and [(t + a1) (t + a2) ... (t + an)]^-1 when
+  expressed in partial fractions be written [Sigma]c_m(t + a_m)^-1, a
+  particular integral of the differential equation (D + a1)(D + a2) ...
+  (D + a_n)y = [psi](x) is given by
+
+    y = [Sigma]c_m(D + a_m)^-1 [psi](x) = [Sigma]c_m(D + a_m)^-1 e^-a m^x e^a m^x [psi](x) =
+
+    [Sigma]c_m e^-a m^x D^-1 (e^a m^x [psi](x)) = [Sigma]c_m e^-a m^x [int] e^a m^x [psi](x)dx.
+
+  The particular integral is thus expressed as a sum of n integrals.
+
+  A linear differential equation of which the left side has the form
+
+        d^ny           d^n-1 y                dy
+    x^n ---- + P1x^n-1 ------- + ... + P_n-1 x-- + P_n y,
+        dx^n           dx^n-1                 dx
+
+  where P1, ... Pn are constants, can be reduced to the case considered
+  above. Writing x = e^t we have the identity
+
+        d^mu
+    x^m ----  = [t]([t] - 1)([t] - 2) ... ([t] - m + 1)u, where [t] = d/dt.
+        dx^m
+
+  When the linear differential equation, which we take to be of the
+  second order, has variable coefficients, though there is no general
+  rule for obtaining a solution in finite terms, there are some results
+  which it is of advantage to have in mind. We have seen that if one
+  solution of the equation obtained by putting the right side zero, say
+  y1, be known, the equation can be solved. If y2 be another solution of
+
+    d^2y    dy
+    ---- + P-- + Qy = 0,
+    dx^2    dx
+
+  there being no relation of the form my1 + ny2 = k, where m, n, k are
+  constants, it is easy to see that
+
+    d/dx(y1'y2 - y1y2') = P(y1'y2 - y1y2'),
+
+  so that we have
+
+    y1'y2 - y1y2' = A exp.([int] Pdx),
+
+  where A is a suitably chosen constant, and exp. z denotes e^z. In
+  terms of the two solutions y1, y2 of the differential equation having
+  zero on the right side, the general solution of the equation with R =
+  [phi](x) on the right side can at once be verified to be Ay1 + By2 +
+  y1u - y2v, where u, v respectively denote the integrals
+         _                                     _
+        /                                     /
+    u = |y2[phi](x)(y1'y2 - y2'y1)^-1 dx, v = |y1[phi](x)(y1'y2 - y2'y1)^-1 dx.
+       _/                                    _/
+
+  The equation
+
+    d^2y    dy
+    ---- + P-- + Qy = 0,
+    dx^2    dx
+
+  by writing y = v exp. (-1/2 [int] Pdx), is at once seen to be reduced to
+  d^2v/dx^2 + 1v = 0, where 1 = Q - 1/2dP/dx - 1/4P^2. If [eta] = - 1/v dv/dx,
+  the equation d^2v/dx^2 + 1v = 0 becomes d[eta]/dx = 1 + [eta]^2, a
+  non-linear equation of the first order.
+
+  More generally the equation
+
+    d[eta]
+    ------ = A + B[eta] + C[eta]^2,
+      dx
+
+  where A, B, C are functions of x, is, by the substitution
+
+               1 dy
+    [eta] = - -- --,
+              Cy dx
+
+  reduced to the linear equation
+
+  d^2y    /    1 dC\ dy
+  ---- - ( B + - -- )-- + ACy = 0.
+  dx^2    \    C dx/ dx
+
+  The equation
+
+    d[eta]
+    ------ = A + B[eta] + C[eta]^2,
+      dx
+
+  known as Riccati's equation, is transformed into an equation of the
+  same form by a substitution of the form [eta] = (aY + b)/(cY + d),
+  where a, b, c, d are any functions of x, and this fact may be utilized
+  to obtain a solution when A, B, C have special forms; in particular if
+  any particular solution of the equation be known, say [eta]0, the
+  substitution [eta] = [eta]0 - 1/Y enables us at once to obtain the
+  general solution; for instance, when
+
+         d      /A\
+    2B = -- log( - ),
+         dx     \C/
+
+  a particular solution is [eta]0 = [root](-A/C). This is a case of the
+  remark, often useful in practice, that the linear equation
+
+            d^2y      d[phi] dy
+    [phi](x)---- + 1/2------ -- + [mu]y = 0,
+            dx^2        dx   dx
+
+  where [mu] is a constant, is reducible to a standard form by taking a
+  new independent variable
+         _
+        /
+    z = | dx[[p](x)]^-1/2.
+       _/
+
+  We pass to other types of equations of which the solution can be
+  obtained by rule. We may have cases in which there are two dependent
+  variables, x and y, and one independent variable t, the differential
+  coefficients dx/dt, dy/dt being given as functions of x, y and t. Of
+  such equations a simple case is expressed by the pair
+
+    dx                dy
+    -- = ax + by + c, -- = a'x + b'y + c',
+    dt                dt
+
+  wherein the coefficients a, b, c, a', b', c', are constants. To
+  integrate these, form with the constant [lambda] the differential
+  coefficient of z = x + [lambda]y, that is dz/dt = (a + [lambda]a')x +
+  (b + [lambda]b')y + c + [lambda]c', the quantity [lambda] being so
+  chosen that b + [lambda]b' = [lambda](a + [lambda]a'), so that we have
+  dz/dt = (a + [lambda]a')z + c + [lambda]c'; this last equation is at
+  once integrable in the form z(a + [lambda]a') + c + [lambda]c' = Ae^(a
+  + [lambda]a')t, where A is an arbitrary constant. In general, the
+  condition b + [lambda]b' = [lambda](a + [lambda]a') is satisfied by
+  two different values of [lambda], say [lambda]1, [lambda]2; the
+  solutions corresponding to these give the values of x +[lambda]1y and
+  x + [lambda]2y, from which x and y can be found as functions of t,
+  involving two arbitrary constants. If, however, the two roots of the
+  quadratic equation for [lambda] are equal, that is, if (a - b')^2 +
+  4a'b = 0, the method described gives only one equation, expressing x +
+  [lambda]y in terms of t; by means of this equation y can be eliminated
+  from dx/dt = ax + by + c, leading to an equation of the form dx/dt =
+  Px + Q + Re^(a + [lambda]a')t, where P, Q, R are constants. The
+  integration of this gives x, and thence y can be found.
+
+  A similar process is applicable when we have three or more dependent
+  variables whose differential coefficients in regard to the single
+  independent variables are given as linear functions of the dependent
+  variables with constant coefficients.
+
+  Another method of solution of the equations
+
+    dx/dt = ax + by + c, dy/dt = a'x + b'y + c',
+
+  consists in differentiating the first equation, thereby obtaining
+
+    d^2x    dx    dy
+    ---- = a-- + b--;
+    dt^2    dt    dx
+
+  from the two given equations, by elimination of y, we can express
+  dy/dt as a linear function of x and dx/dt; we can thus form an
+  equation of the shape d^2x/dt^2 = P + Qx + Rdx/dt, where P, Q, R are
+  constants; this can be integrated by methods previously explained, and
+  the integral, involving two arbitrary constants, gives, by the
+  equation dx/dt = ax + by + c, the corresponding value of y. Conversely
+  it should be noticed that any single linear differential equation
+
+    d^2x             dx
+    ---- = u + vx + w--,
+    dt^2             dt
+
+  where u, v, w are functions of t, by writing y for dx/dt, is
+  equivalent with the two equations dx/dt = y, dy/dt = u + vx + wy. In
+  fact a similar reduction is possible for any system of differential
+  equations with one independent variable.
+
+  Equations occur to be integrated of the form
+
+    Xdx + Ydy + Zdz = 0,
+
+  where X, Y, Z are functions of x, y, z. We consider only the case in
+  which there exists an equation [phi](x, y, z) = C whose differential
+
+    dP[phi]     dP[phi]     dP[phi]
+    -------dx + -------dy + -------dz = 0
+      dPx         dPy         dPz
+
+  is equivalent with the given differential equation; that is, [mu]
+  being a proper function of x, y, z, we assume that there exist
+  equations
+
+    dP[phi]          dP[phi]          v[phi]
+    ------- = [mu]X, ------- = [mu]Y, ------ = [mu]Z;
+      dPx              vy               vz
+
+  these equations require
+
+    dP           dP
+    ---([mu]Y) = ---([mu]Z), &c.,
+    dPz          dPy
+
+  and hence
+
+      /dPZ   dPY\      /dPX   dPZ\      /dPY   dPX\
+    X( --- - --- ) + Y( --- - --- ) + Z( --- - --- ) = 0;
+      \dPy   dPz/      \dPz   dPx/      \dPx   dPy/
+
+  conversely it can be proved that this is sufficient in order that [mu]
+  may exist to render [mu](Xdx + Ydy + Zdz) a perfect differential; in
+  particular it may be satisfied in virtue of the three equations such
+  as
+
+    dPZ   dPY
+    --- - --- = 0;
+    dPy   dPz
+
+  in which case we may take [mu] = 1. Assuming the condition in its
+  general form, take in the given differential equation a plane section
+  of the surface [phi] = C parallel to the plane z, viz. put z constant,
+  and consider the resulting differential equation in the two variables
+  x, y, namely Xdx + Ydy = 0; let [psi](x, y, z) = constant, be its
+  integral, the constant z entering, as a rule, in [psi] because it
+  enters in X and Y. Now differentiate the relation [psi](x, y, z) =
+  [f](z), where [f] is a function to be determined, so obtaining
+
+    dP[psi]     dP[psi]      /dP[psi]   df\
+    -------dx + -------dy + ( ------- - -- )dz = 0;
+      dPx         dPy        \  dPz     dz/
+
+  there exists a function [sigma] of x, y, z such that
+
+    dP[psi]              dP[psi]
+    -------- = [sigma]X, ------- = [sigma]Y,
+      dPx                  dPy
+
+  because [psi] = constant, is the integral of Xdx + Ydy = 0; we desire
+  to prove that [f] can be chosen so that also, in virtue of [psi](x, y,
+  z) = f(z), we have
+
+    dP[psi]   df                    df   dP[psi]
+    ------- - -- = [sigma]Z, namely -- = ------- - [sigma]Z;
+      dPz     dz                    dz     dPz
+
+  if this can be proved the relation [psi](x, y, z) - f(z) = constant,
+  will be the integral of the given differential equation. To prove this
+  it is enough to show that, in virtue of [psi](x, y, z) = [f](z), the
+  function dP[psi]/dPx - [sigma]Z can be expressed in terms of z only.
+  Now in consequence of the originally assumed relations,
+
+    dP[psi]          dP[phi]          dP[phi]
+    ------- = [mu]X, ------- = [mu]Y, ------- = [mu]Z,
+      dPx              dPy              dPz
+
+  we have
+
+    dP[psi]  /dP[phi]   [sigma]   dP[psi]  /dP[phi]
+    ------- / ------- = ------- = ------- / -------,
+      dPx  /    dPx      [mu]       dPy  /    dPy
+
+  and hence
+
+    dP[psi] dP[phi]   dP[psi] dP[phi]
+    ------- ------- - ------- ------- = 0;
+      dPx     dPy       dPy     dPx
+
+  this shows that, as functions of x and y, [psi] is a function of [phi]
+  (see the note at the end of part i. of this article, on Jacobian
+  determinants), so that we may write [psi] = F(z, [phi]), from which
+
+    [sigma]     dPF         dP[psi]   dPF     dPF   dP[phi]   dPF   [sigma]           dPF
+    ------- = -------; then ------- = --- + ------- ------- = --- + ------- . [mu]Z = --- + [sigma]Z
+     [mu]     dP[phi]         dPz     dPz   dP[phi]   dPz     dPz    [mu]             dPz
+
+         dP[psi]              dPF
+      or ------- - [sigma]Z = ---;
+           dPz                dPz
+
+  in virtue of [psi](x, y, z) = f(z), and [psi] = F(z, [phi]), the
+  function [phi] can be written in terms of z only, thus dPF/dPz can be
+  written in terms of z only, and what we required to prove is proved.
+
+  Consider lastly a simple type of differential equation containing
+  _two_ independent variables, say x and y, and one dependent variable
+  z, namely the equation
+
+     dPz    dPz
+    P--- + Q--- = R,
+     dPx    dPy
+
+  where P, Q, R are functions of x, y, z. This is known as Lagrange's
+  linear partial differential equation of the first order. To integrate
+  this, consider first the ordinary differential equations dx/dz = P/R,
+  dy/dz = Q/R, and suppose that two functions u, v, of x, y, z can be
+  determined, independent of one another, such that the equations u = a,
+  v = b, where a, b are arbitrary constants, lead to these ordinary
+  differential equations, namely such that
+
+     dPu    dPu    dPu          dPv    dPv    dPv
+    P--- + Q--- = R--- = 0 and P--- + Q--- = R--- = 0.
+     dPx    dPy    dPz          dPx    dPy    dPz
+
+  Then if F(x, y, z) = 0 be a relation satisfying the original
+  differential equations, this relation giving rise to
+
+    dPF   dPF dPz         dPF   dPF dPz               dPF    dPF    dPF
+    --- + --- --- = 0 and --- + --- --- = 0, we have P--- + Q--- = R--- = 0.
+    dPx   dPz dPx         dPy   dPz dPy               dPx    dPy    dPz
+
+  It follows that the determinant of three rows and columns vanishes
+  whose first row consists of the three quantities dPF/dPx, dPF/dPy,
+  dPF/dPz, whose second row consists of the three quantities dPu/dPx,
+  dPu/dPy, dPu/dPz, whose third row consists similarly of the partial
+  derivatives of v. The vanishing of this so-called Jacobian determinant
+  is known to imply that F is expressible as a function of u and v,
+  unless these are themselves functionally related, which is contrary to
+  hypothesis (see the note below on Jacobian determinants). Conversely,
+  any relation [phi](u, v) = 0 can easily be proved, in virtue of the
+  equations satisfied by u and v, to lead to
+
+     dz    dz
+    P-- + Q-- = R.
+     dx    dx
+
+  The solution of this partial equation is thus reduced to the solution
+  of the two ordinary differential equations expressed by dx/P = dy/Q =
+  dz/R. In regard to this problem one remark may be made which is often
+  of use in practice: when one equation u = a has been found to satisfy
+  the differential equations, we may utilize this to obtain the second
+  equation v = b; for instance, we may, by means of u = a, eliminate
+  z--when then from the resulting equations in x and y a relation v = b
+  has been found containing x and y and a, the substitution a = u will
+  give a relation involving x, y, z.
+
+  _Note on Jacobian Determinants._--The fact assumed above that the
+  vanishing of the Jacobian determinant whose elements are the partial
+  derivatives of three functions F, u, v, of three variables x, y, z,
+  involves that there exists a functional relation connecting the three
+  functions F, u, v, may be proved somewhat roughly as follows:--
+
+  The corresponding theorem is true for any number of variables.
+  Consider first the case of two functions p, q, of two variables x, y.
+  The function p, not being constant, must contain one of the variables,
+  say x; we can then suppose x expressed in terms of y and the function
+  p; thus the function q can be expressed in terms of y and the function
+  p, say q = Q(p, y). This is clear enough in the simplest cases which
+  arise, when the functions are rational. Hence we have
+
+    dPq   dPQ dPp     dPq   dPQ dPp   dPQ
+    --- = --- --- and --- = --- --- + ---;
+    dPx   dPp dPx     dPy   dPp dPy   dPy
+
+  these give
+
+    dPp dPq   dPp dPq   dPp dPQ
+    --- --- - --- --- = --- ---;
+    dPx dPy   dPy dPx   dPx dPy
+
+  by hypothesis dPp/dPx is not identically zero; therefore if the
+  Jacobian determinant of p and q in regard to x and y is zero
+  identically, so is dPQ/dPy, or Q does not contain y, so that q is
+  expressible as a function of p only. Conversely, such an expression
+  can be seen at once to make the Jacobian of p and q vanish
+  identically.
+
+  Passing now to the case of three variables, suppose that the Jacobian
+  determinant of the three functions F, u, v in regard to x, y, z is
+  identically zero. We prove that if u, v are not themselves
+  functionally connected, F is expressible as a function of u and v.
+  Suppose first that the minors of the elements of dPF/dPx, dPF/dPy,
+  dPF/dPz in the determinant are all identically zero, namely the three
+  determinants such as
+
+    dPu dPv   dPu dPv
+    --- --- - --- ---;
+    dPy dPz   dPz dPy
+
+  then by the case of two variables considered above there exist three
+  functional relations. [psi]1(u, v, x) = 0, [psi]2(u, v, y) = 0,
+  [psi]3(u, v, z) = 0, of which the first, for example, follows from the
+  vanishing of
+
+    dPu dPv   dPu dPv
+    --- --- - --- ---.
+    dPy dPz   dPz dPy
+
+  We cannot assume that x is absent from [psi]1, or y from [psi]2, or z
+  from [psi]3; but conversely we cannot simultaneously have x entering
+  in [psi]1, and y in [psi]2, and z in [psi]3, or else by elimination of
+  u and v from the three equations [psi]1 = 0, [psi]2 = 0, [psi]3 = 0,
+  we should find a necessary relation connecting the three independent
+  quantities x, y, z; which is absurd. Thus when the three minors of
+  dPF/dPx, dPF/dPy, dPF/dPz in the Jacobian determinant are all zero,
+  there exists a functional relation connecting u and v only. Suppose no
+  such relation to exist; we can then suppose, for example, that
+
+    dPu dPv   dPu dPv
+    --- --- - --- ---
+    dPy dPz   dPz dPy
+
+  is not zero. Then from the equations u(x, y, z) = u, v(x, y, z) = v we
+  can express y and z in terms of u, v, and x (the attempt to do this
+  could only fail by leading to a relation connecting u, v and x, and
+  the existence of such a relation would involve that the determinant
+
+    dPu dPv   dPu dPv
+    --- --- - --- ---
+    dPy dPz   dPz dPy
+
+  was zero), and so write F in the form F(x, y, z) = [Phi](u, v, x). We
+  then have
+
+    dPF   dP[Phi] dPu   dP[Phi] dPv   dP[Phi]  dPF   dP[Phi] dPu   dP[Phi] dPv  dPF   dP[Phi] dPu   dP[Phi] dPv
+    --- = ------- --- + ------- --- + -------, --- = ------- --- + ------- ---, --- = ------- --- + ------- ---;
+    dPx     dPu   dPx     dPv   dPx     dPx    dPy     dPu   dPy     dPv   dPy  dPz     dPu   dPz     dPv   dPz
+
+  thereby the Jacobian determinant of F, u, v is reduced to
+
+    dP[Phi] /dPu dPv   dPu dPv\
+    -------( --- --- - --- --- );
+      dPx   \dPy dPz   dPz dPy/
+
+  by hypothesis the second factor of this does not vanish identically;
+  hence dP[Phi]/dPx = 0 identically, and [Phi] does not contain x; so
+  that F is expressible in terms of u, v only; as was to be proved.
+
+
+_Part II.--General Theory._
+
+Differential equations arise in the expression of the relations between
+quantities by the elimination of details, either unknown or regarded as
+unessential to the formulation of the relations in question. They give
+rise, therefore, to the two closely connected problems of determining
+what arrangement of details is consistent with them, and of developing,
+apart from these details, the general properties expressed by them. Very
+roughly, two methods of study can be distinguished, with the names
+Transformation-theories, Function-theories; the former is concerned with
+the reduction of the algebraical relations to the fewest and simplest
+forms, eventually with the hope of obtaining explicit expressions of the
+dependent variables in terms of the independent variables; the latter is
+concerned with the determination of the general descriptive relations
+among the quantities which are involved by the differential equations,
+with as little use of algebraical calculations as may be possible. Under
+the former heading we may, with the assumption of a few theorems
+belonging to the latter, arrange the theory of partial differential
+equations and Pfaff's problem, with their geometrical interpretations,
+as at present developed, and the applications of Lie's theory of
+transformation-groups to partial and to ordinary equations; under the
+latter, the study of linear differential equations in the manner
+initiated by Riemann, the applications of discontinuous groups, the
+theory of the singularities of integrals, and the study of potential
+equations with existence-theorems arising therefrom. In order to be
+clear we shall enter into some detail in regard to partial differential
+equations of the first order, both those which are linear in any number
+of variables and those not linear in two independent variables, and also
+in regard to the function-theory of linear differential equations of the
+second order. Space renders impossible anything further than the
+briefest account of many other matters; in particular, the theories of
+partial equations of higher than the first order, the function-theory of
+the singularities of ordinary equations not linear and the applications
+to differential geometry, are taken account of only in the bibliography.
+It is believed that on the whole the article will be more useful to the
+reader than if explanations of method had been further curtailed to
+include more facts.
+
+When we speak of a function without qualification, it is to be
+understood that in the immediate neighbourhood of a particular set x0,
+y0, ... of values of the independent variables x, y, ... of the
+function, at whatever point of the range of values for x, y, ... under
+consideration x0, y0, ... may be chosen, the function can be expressed
+as a series of positive integral powers of the differences x - x0, y
+-y0, ..., convergent when these are sufficiently small (see FUNCTION:
+Functions of Complex Variables). Without this condition, which we
+express by saying that the function is developable about x0, y0, ...,
+many results provisionally stated in the transformation theories would
+be unmeaning or incorrect. If, then, we have a set of k functions, f1
+... fk of n independent variables x1 ... xn, we say that they are
+independent when n >= k and not every determinant of k rows and columns
+vanishes of the matrix of k rows and n columns whose r-th row has the
+constituents dfr/dx1, ... dfr/dxn; the justification being in the
+theorem, which we assume, that if the determinant involving, for
+instance, the first k columns be not zero for x1 = x1^0 ... xn = xn^0,
+and the functions be developable about this point, then from the
+equations f1 = c1, ... fk = ck we can express x1, ... xk by convergent
+power series in the differences x_k+1 - x_k+1^0, ... x_n - x_n^0, and so
+regard x1, ... xk as functions of the remaining variables. This we often
+express by saying that the equations f1 = c1, ... fk = ck can be solved
+for x1, ... xk. The explanation is given as a type of explanation often
+understood in what follows.
+
+
+    Ordinary equations of the first order.
+
+    Single homogeneous partial equation of the first order.
+
+    Proof of the existence of integrals.
+
+  We may conveniently begin by stating the theorem: If each of the n
+  functions [phi]1, ... [phi]n of the (n + 1) variables x1, ... x_nt be
+  developable about the values x1^0, ... x_n^0t^0, the n differential
+  equations of the form dx1/dt = [phi]1(tx1, ... xn) are satisfied by
+  convergent power series
+
+    x_r = x_r^0 + (t - t^0 ) A_r1 + (t - t0 )^2A_r2 + ...
+
+  reducing respectively to x1^0, ... xn^0 when t = t^0; and the only
+  functions satisfying the equations and reducing respectively to x1^0,
+  ... xn^0 when t = t^0, are those determined by continuation of these
+  series. If the result of solving these n equations for x1^0, ... xn^0
+  be written in the form [omega]1(x1, ... xnt) = x1^0, ... [omega]n(x1,
+  ... xnt) = xn^0, it is at once evident that the differential equation
+
+    df/dt + [phi]1 df/dx1 + ... + [phi]n df/dxn = 0
+
+  possesses n integrals, namely, the functions [omega]1, ... [omega]n,
+  which are developable about the values (x1^0 ... xn^0t^0) and reduce
+  respectively to x1, ... xn when t = t^0. And in fact it has no other
+  integrals so reducing. Thus this equation also possesses a unique
+  integral reducing when t = t^0 to an arbitrary function [psi](x1, ...
+  xn), this integral being. [psi]([omega]1, ... [omega]n). Conversely
+  the existence of these _principal_ integrals [omega]1, ... [omega]n of
+  the partial equation establishes the existence of the specified
+  solutions of the ordinary equations dxi/dt = [phi]i. The following
+  sketch of the proof of the existence of these principal integrals for
+  the case n = 2 will show the character of more general investigations.
+  Put x for x - x^0, &c., and consider the equation a(xyt) df/dx +
+  b(xyt) df/dy = df/dt, wherein the functions a, b are developable about
+  x = 0, y = 0, t = 0; say
+
+    a(xyt) = a0 + ta1 + t^2a2/2! + ..., b(xyt) = b0 + tb1 + t^2b2/2! + ...,
+
+  so that
+
+    ad/dx + bd/dy = [delta]0 + t[delta]1 + 1/2t^2[delta]2 + ...,
+
+  where [delta] = a_r d/dx + b_r d/dy. In order that
+
+    f = p0 + tp1 + t^2p2/2! + ...
+
+  wherein p0, p1 ... are power series in x, y, should satisfy the
+  equation, it is necessary, as we find by equating like terms, that
+
+    p1 = [delta]0 p0, p2 = [delta]0 p1 + [delta]1 p0, &c.
+
+  and in general
+
+    p_s+1 = [delta]0 p_s + s1 [delta]1 p_s-1 + ... + [delta]_s p0,
+
+  where s_r = (s!)/(r!) (s - r)!
+
+  Now compare with the given equation another equation
+
+    A(xyt)dF/dx + B(xyt)dF/dy = dF/dt,
+
+  wherein each coefficient in the expansion of either A or B is real and
+  positive, and not less than the absolute value of the corresponding
+  coefficient in the expansion of a or b. In the second equation let us
+  substitute a series
+
+    F = P0 + tP1 + t^2P2/2! + ...,
+
+  wherein the coefficients in P0 are real and positive, and each not
+  less than the absolute value of the corresponding coefficient in p0;
+  then putting [Delta]r = A_r d/dx + B_r d/dy we obtain necessary
+  equations of the same form as before, namely,
+
+    P1 = [Delta]0 P0, P2= [Delta]0 P1 + [Delta]1 P0, ...
+
+  and in general P_s+1 = [Delta]0 P_s, + s1[Delta]1 P_s-1 + ... +
+  [Delta]_s P0. These give for every coefficient in Ps+1 an integral
+  aggregate with real positive coefficients of the coefficients in P_s,
+  P_s-1, ..., P0 and the coefficients in A and B; and they are the same
+  aggregates as would be given by the previously obtained equations for
+  the corresponding coefficients in p_s+1 in terms of the coefficients
+  in ps, p_s-1, ..., p0 and the coefficients in a and b. Hence as the
+  coefficients in P0 and also in A, B are real and positive, it follows
+  that the values obtained in succession for the coefficients in P1, P2,
+  ... are real and positive; and further, taking account of the fact
+  that the absolute value of a sum of terms is not greater than the sum
+  of the absolute values of the terms, it follows, for each value of s,
+  that every coefficient in p_s+1 is, in absolute value, not greater
+  than the corresponding coefficient in P_s+1. Thus if the series for F
+  be convergent, the series for f will also be; and we are thus reduced
+  to (1), specifying functions A, B with real positive coefficients,
+  each in absolute value not less than the corresponding coefficient in
+  a, b; (2) proving that the equation
+
+    AdF/dx + BdF/dy = dF/dt
+
+  possesses an integral P0 + tP1 + t^2P2/2! + ... in which the
+  coefficients in P0 are real and positive, and each not less than the
+  absolute value of the corresponding coefficient in p0. If a, b be
+  developable for x, y both in absolute value less than r and for t less
+  in absolute value than R, and for such values a, b be both less in
+  absolute value than the real positive constant M, it is not difficult
+  to verify that we may take
+
+              /    x + y\-1 /    t\-1
+    A = B = M( 1 - ----- ) ( 1 - - ),
+              \      r  /   \    R/
+
+  and obtain
+                         _                                 _
+                        |     4MR /   x + y\-2     /   t\-1 |1/2
+    F = r - (r - x - y) | 1 - ---(1 - ------) log (1 - - )  |,
+                        |_     r  \     r  /       \   R/  _|
+
+  and that this solves the problem when x, y, t are sufficiently small
+  for the two cases p0 = x, p0 = y. One obvious application of the
+  general theorem is to the proof of the existence of an integral of an
+  ordinary linear differential equation given by the n equations dy/dx =
+  y1, dy1/dx = y2, ...,
+
+    dy_n-1/dx = p - p1 y_n-1 - ... - p_n y;
+
+  but in fact any simultaneous system of ordinary equations is reducible
+  to a system of the form
+
+    dx1/dt = [phi](tx1, ... x_n).
+
+
+    Simultaneous linear partial equations.
+
+    Complete systems of linear partial equations.
+
+    Jacobian systems.
+
+  Suppose we have k homogeneous linear partial equations of the first
+  order in n independent variables, the general equation being
+  a_[sigma]1 df/dx1 + ... + a_[sigma]n df/dx_n = 0, where [sigma] = 1,
+  ... k, and that we desire to know whether the equations have common
+  solutions, and if so, how many. It is to be understood that the
+  equations are linearly independent, which implies that k <= n and not
+  every determinant of k rows and columns is identically zero in the
+  matrix in which the i-th element of the [sigma]-th row is a[sigma]_i(i
+  = 1, ... n, [sigma] = 1, ... k). Denoting the left side of the
+  [sigma]-th equation by P[sigma]f, it is clear that every common
+  solution of the two equations P_[sigma]f = 0, P_[rho]f = 0, is also a
+  solution of the equation P_[rho](P_[sigma]f), P_[sigma](P_[rho]f), We
+  immediately find, however, that this is also a linear equation,
+  namely, [Sigma]H_i df/dx_i = 0 where H_i = P[rho]a[sigma]_i -
+  P[sigma]a[rho]_i, and if it be not already contained among the given
+  equations, or be linearly deducible from them, it may be added to
+  them, as not introducing any additional limitation of the possibility
+  of their having common solutions. Proceeding thus with every pair of
+  the original equations, and then with every pair of the possibly
+  augmented system so obtained, and so on continually, we shall arrive
+  at a system of equations, linearly independent of each other and
+  therefore not more than n in number, such that the combination, in the
+  way described, of every pair of them, leads to an equation which is
+  linearly deducible from them. If the number of this so-called
+  _complete system_ is n, the equations give df/dx1 = 0 ... df/dxn = 0,
+  leading to the nugatory result f = a constant. Suppose, then, the
+  number of this system to be r < n; suppose, further, that from the
+  matrix of the coefficients a determinant of r rows and columns not
+  vanishing identically is that formed by the coefficients of the
+  differential coefficients of f in regard to x1 ... x_r; also that the
+  coefficients are all developable about the values x1 = x1^0, ... xn=
+  xn^0, and that for these values the determinant just spoken of is not
+  zero. Then the main theorem is that the complete system of r
+  equations, and therefore the originally given set of k equations,
+  have in common n - r solutions, say [omega]r+1, ... [omega]n, which
+  reduce respectively to x_r+1, ... x_n when in them for x1, ... x_r are
+  respectively put x1^0, ... x_r^0; so that also the equations have in
+  common a solution reducing when x1 = x1^0, ... x_r = x_r^0 to an
+  arbitrary function [psi](x_r+1, ... x_n) which is developable about
+  x_r+1^0, ... x_n^0, namely, this common solution is [psi]([omega]_r+1,
+  ... [omega]_n). It is seen at once that this result is a
+  generalization of the theorem for r = 1, and its proof is conveniently
+  given by induction from that case. It can be verified without
+  difficulty (1) that if from the r equations of the complete system we
+  form r independent linear aggregates, with coefficients not
+  necessarily constants, the new system is also a complete system; (2)
+  that if in place of the independent variables x1, ... xn we introduce
+  any other variables which are independent functions of the former, the
+  new equations also form a complete system. It is convenient, then,
+  from the complete system of r equations to form r new equations by
+  solving separately for df/dx1, ..., df/dx_r; suppose the general
+  equation of the new system to be
+
+    Q_[sigma]f = df/dx_[sigma] + c_[sigma],r+1 df/dx_r+1 + ... + c_[sigma]n df/dx_n = 0 ([sigma] = 1, ... r).
+
+  Then it is easily obvious that the equation Q_[rho]Q_[sigma]f -
+  Q_[sigma]Q_[rho]f = 0 contains only the differential coefficients of f
+  in regard to x_r+1 ... xn; as it is at most a linear function of Q1f,
+  ... Qrf, it must be identically zero. So reduced the system is called
+  a Jacobian system. Of this system Q1f=0 has n - 1 principal solutions
+  reducing respectively to x2, ... xn when
+
+    x1 = x1^0,
+
+  and its form shows that of these the first r - 1 are exactly x2 ...
+  xr. Let these n - 1 functions together with x1 be introduced as n new
+  independent variables in all the r equations. Since the first equation
+  is satisfied by n - 1 of the new independent variables, it will
+  contain no differential coefficients in regard to them, and will
+  reduce therefore simply to df/dx1 = 0, expressing that any common
+  solution of the r equations is a function only of the n - 1 remaining
+  variables. Thereby the investigation of the common solutions is
+  reduced to the same problem for r - 1 equations in n - 1 variables.
+  Proceeding thus, we reach at length one equation in n - r + 1
+  variables, from which, by retracing the analysis, the proposition
+  stated is seen to follow.
+
+
+    System of total differential equations.
+
+  The analogy with the case of one equation is, however, still closer.
+  With the coefficients c_[sigma]j, of the equations Q_[sigma]f = 0 in
+  transposed array ([sigma] = 1, ... r, j = r + 1, ... n) we can put
+  down the (n - r) equations, dx_j = c1_j dx1 + ... + c_rj dx_r,
+  equivalent to the r(n - r) equations dx_j/dx_[sigma] = c_[sigma]r.
+  That consistent with them we may be able to regard x_r+1, ... x_n as
+  functions of x1, ... x_r, these being regarded as independent
+  variables, it is clearly necessary that when we differentiate
+  c_[sigma]j in regard to x_[rho] on this hypothesis the result should
+  be the same as when we differentiate c[rho]j, in regard to x[sigma] on
+  this hypothesis. The differential coefficient of a function f of x1,
+  ... xn on this hypothesis, in regard to x_[rho]j is, however,
+
+    df/dx_[rho] + c_[rho],r+1 df/dx_r+1 + ... + c_[rho]n df/dx_n,
+
+  namely, is Q_[rho]f. Thus the consistence of the n - r total equations
+  requires the conditions Q_[rho]c_[sigma]j - Q_[sigma]c_[rho]j = 0,
+  which are, however, verified in virtue of Q[rho](Q[sigma][f]) -
+  Q_[sigma](Q_[rho]f) = 0. And it can in fact be easily verified that if
+  [omega]_r+1, ... [omega]_n be the principal solutions of the Jacobian
+  system, Q_[sigma]f = 0, reducing respectively to x_r+1, ... xn when x1
+  = x1^0, ... x_r = x_r^0, and the equations [omega]_r+1 = x_r+1^0, ...
+  [omega]_n = x_n^0 be solved for x_r+1, ... x_n to give x_j =
+  [psi]_j(x1, ... x_r, x_r+1^0, ... x_n^0), these values solve the total
+  equations and reduce respectively to x_r+1^0, ... x_n^0 when x1 = x1^0
+  ... x_r = x_r^0. And the total equations have no other solutions with
+  these initial values. Conversely, the existence of these solutions of
+  the total equations can be deduced a priori and the theory of the
+  Jacobian system based upon them. The theory of such total equations,
+  in general, finds its natural place under the heading _Pfaffian
+  Expressions_, below.
+
+
+    Geometrical interpretation and solution.
+
+    Mayer's method of integration.
+
+  A practical method of reducing the solution of the r equations of a
+  Jacobian system to that of a single equation in n - r + 1 variables
+  may be explained in connexion with a geometrical interpretation which
+  will perhaps be clearer in a particular case, say n = 3, r = 2. There
+  is then only one total equation, say dz = adz + bdy; if we do not take
+  account of the condition of integrability, which is in this case da/dy
+  + bda/dz = db/dx + adb/dz, this equation may be regarded as defining
+  through an arbitrary point (x0, y0, z0) of three-dimensioned space
+  (about which a, b are developable) a plane, namely, z - z0 = a0(x -
+  x0) + b0(y - y0), and therefore, through this arbitrary point [oo]^2
+  directions, namely, all those in the plane. If now there be a surface
+  z = [psi](x, y), satisfying dz = adz + bdy and passing through (x0,
+  y0, z0), this plane will touch the surface, and the operations of
+  passing along the surface from (x0, y0, z0) to
+
+    (x0 + dx0, y0, z0 + dz0)
+
+  and then to (x0 + dx0, y0 + dy0, Z0 + d^1z0), ought to lead to the same
+  value of d^1z0 as do the operations of passing along the surface from
+  (x0, y0, z0) to (x0, y0 + dy0, z0 + [delta]z0), and then to
+
+  (x_  + dx_  , y_  + dy_  , Z_  + [delta]^1z_  ),
+    0    0   0    0   0           0
+
+  namely, [delta]^1z0 ought to be equal to d^1z0. But we find
+
+    d^1z0 = a0dx0 + b(x0 + dx0 , y0, z0 + a0dx0)dy0 =
+
+                           /db      db \
+    a0dx0 + b0dy0 + dx0dy0( --- + a0--- ),
+                           \dx0     dz0/
+
+  and so at once reach the condition of integrability. If now we put x
+  = x0 + t, y = y0 + mt, and regard m as constant, we shall in fact be
+  considering the section of the surface by a fixed plane y - y0 = m(x -
+  x0); along this section dz = dt(a + bm); if we then integrate the
+  equation dx/dt = a + bm, where a, b are expressed as functions of m
+  and t, with m kept constant, finding the solution which reduces to z0
+  for t = 0, and in the result again replace m by (y - y0)/(x - x0), we
+  shall have the surface in question. In the general case the equations
+
+    dx_j - c_1j dx1 + ... c_rj dx_r
+
+  similarly determine through an arbitrary point x1^0, ... xn^0 a planar
+  manifold of r dimensions in space of n dimensions, and when the
+  conditions of integrability are satisfied, every direction in this
+  manifold through this point is tangent to the manifold of r
+  dimensions, expressed by [omega]_r+1 = x_r+1^0, ... [omega]_n = x_n^0,
+  which satisfies the equations and passes through this point. If we put
+  x1 = x1^0 = t, x2 = x2^0 = m2t, ... xr = xr^0 = mrt, and regard m2,
+  ... mr as fixed, the (n-r) total equations take the form dx_j/dt =
+  c_1j + m2c_2j + ... + m_rc_rj, and their integration is equivalent to
+  that of the single partial equation
+
+                n
+    df/dt + [Sigma](c_1j + m2c_2j + ... + m_rc_rj)df/dx_j = 0
+              j=r+1
+
+  in the n - r + 1 variables t, xr+1, ... xn. Determining the solutions
+  [Omega]_r+1, ... [Omega]_n which reduce to respectively x_r+1, ... x_n
+  when t = 0, and substituting t = x1 - x1^0, m2 = (x2 - x2^0)/(x1 -
+  x1^0), ... mr = (xr - xr^0)/(x1 - x1^0), we obtain the solutions of
+  the original system of partial equations previously denoted by
+  [omega]_r+1, ... [omega]_n. It is to be remarked, however, that the
+  presence of the fixed parameters m2, ... mr in the single integration
+  may frequently render it more difficult than if they were assigned
+  numerical quantities.
+
+
+    Pfaffian Expressions.
+
+  We have above considered the integration of an equation
+
+    dz = adz + bdy
+
+  on the hypothesis that the condition
+
+    da/dy + bda/dz = db/dz + adb/dz.
+
+  It is natural to inquire what relations among x, y, z, if any, are
+  implied by, or are consistent with, a differential relation adx + bdy
+  + cdx = 0, when a, b, c are unrestricted functions of x, y, z. This
+  problem leads to the consideration of the so-called _Pfaffian
+  Expression_ adx + bdy + cdz. It can be shown (1) if each of the
+  quantities db/dz - dc/dy, dc/dx - da/dz, da/dy - db/dz, which we shall
+  denote respectively by u23, u31, u12, be identically zero, the
+  expression is the differential of a function of x, y, z, equal to dt
+  say; (2) that if the quantity au23 + bu31 + cu12 is identically zero,
+  the expression is of the form udt, i.e. it can be made a perfect
+  differential by multiplication by the factor 1/u; (3) that in general
+  the expression is of the form dt + u1dt1. Consider the matrix of four
+  rows and three columns, in which the elements of the first row are a,
+  b, c, and the elements of the (r+1)-th row, for r = 1, 2, 3, are the
+  quantities u_r1, u_r2, u_r3, where u11 = u22 = u33 = 0. Then it is
+  easily seen that the cases (1), (2), (3) above correspond respectively
+  to the cases when (1) every determinant of this matrix of two rows and
+  columns is zero, (2) every determinant of three rows and columns is
+  zero, (3) when no condition is assumed. This result can be generalized
+  as follows: if a1, ... an be any functions of x1, ... xn, the
+  so-called Pfaffian expression a1dx1 + ... + a_ndx_n can be reduced to
+  one or other of the two forms
+
+    u1dt1 + ... + u_kdt_k, dt + u1dt1 +  ... + u_k-1 dt_k-1,
+
+  wherein t, u1 ..., t1, ... are independent functions of x1, ... xn,
+  and k is such that in these two cases respectively 2k or 2k - 1 is the
+  rank of a certain matrix of n + 1 rows and n columns, that is, the
+  greatest number of rows and columns in a non-vanishing determinant of
+  the matrix; the matrix is that whose first row is constituted by the
+  quantities a1, ... an, whose s-th element in the (r+1)-th row is the
+  quantity da_r/dx_s - da_s/dx_r. The proof of such a reduced form can
+  be obtained from the two results: (1) If t be any given function of
+  the 2m independent variables u1, ... um, t1, ... tm, the expression dt
+  + u1 dt1 + ... + u_m dt_m can be put into the form u'1 dt'1 + ... +
+  u'_mdt'_m. (2) If the quantities u1, ..., u1, t1, ... tm be connected
+  by a relation, the expression n1dt1 + ... + umdtm can be put into the
+  format dt' + u'1 dt'1 + ... + u'_m-1 dt'_m-1; and if the relation
+  connecting u1, um, t1, ... tm be homogeneous in u1, ... um, then t'
+  can be taken to be zero. These two results are deductions from the
+  theory of _contact transformations_ (see below), and their
+  demonstration requires, beside elementary algebraical considerations,
+  only the theory of complete systems of linear homogeneous partial
+  differential equations of the first order. When the existence of the
+  reduced form of the Pfaffian expression containing only independent
+  quantities is thus once assured, the identification of the number k
+  with that defined by the specified matrix may, with some difficulty,
+  be made _a posteriori_.
+
+
+    Single linear Pfaffian equation.
+
+  In all cases of a single Pfaffian equation we are thus led to consider
+  what is implied by a relation dt - u1dt1 - ... - umdtm = 0, in which
+  t, u1, ... um, t1 ..., tm are, except for this equation, independent
+  variables. This is to be satisfied in virtue of one or several
+  relations connecting the variables; these must involve relations
+  connecting t, t1, ... tm only, and in one of these at least t must
+  actually enter. We can then suppose that in one actual system of
+  relations in virtue of which the Pfaffian equation is satisfied, all
+  the relations connecting t, t1 ... tm only are given by
+
+    t = [psi](t_s+1 ... t_m), t1 = [psi]1(t_s+1 ... t_m), ... t_s = [psi]_s(t_s+1 ... t_m);
+
+  so that the equation
+
+    d[psi] - u1d[psi]1 - ... - u_s d[psi]_s - u_s+1 dt_s+1 - ... - u_m dt_m = 0
+
+  is identically true in regard to u1, ... um, t_s+1 ..., t_m; equating
+  to zero the coefficients of the differentials of these variables, we
+  thus obtain m - s relations of the form
+
+    d[psi]/dt_j - u1 d[psi]1/dt_j - ... - u_s d[psi]_s/dt_j - u_j = 0;
+
+  these m - s relations, with the previous s + 1 relations, constitute a
+  set of m + 1 relations connecting the 2m + 1 variables in virtue of
+  which the Pfaffian equation is satisfied independently of the form of
+  the functions [psi],[psi]1, ... [psi]s. There is clearly such a set
+  for each of the values s = 0, s = 1, ..., s = m - 1, s = m. And for
+  any value of s there may exist relations additional to the specified m
+  + 1 relations, provided they do not involve any relation connecting t,
+  t1, ... tm only, and are consistent with the m - s relations
+  connecting u1, ... um. It is now evident that, essentially, the
+  integration of a Pfaffian equation
+
+    a1dx1 + ... + a_n dx_n = 0,
+
+  wherein a1, ... an are functions of x1, ... xn, is effected by the
+  processes necessary to bring it to its reduced form, involving only
+  independent variables. And it is easy to see that if we suppose this
+  reduction to be carried out in all possible ways, there is no need to
+  distinguish the classes of integrals corresponding to the various
+  values of s; for it can be verified without difficulty that by putting
+  t' = t - u1t1 - ... - u_s t_s, t'1 = u1, ... t'_s = u_s, u'1 = -t1,
+  ..., u'_s = -t_s, t'_s+1 = t_s+1, ... t'_m = t_m, u'_s+1 = u_s+1, ...
+  u'_m = u_m, the reduced equation becomes changed to dt' - u'1 dt'1 -
+  ... - u'_m dt'_m = 0, and the general relations changed to
+
+    t' = [psi](t'_s+l, ... t'_m) - t'1[psi]1(t'_s+1, ... t'_m) -  ... -t'_s[psi]_s(t'_s+1, ... t'_m), = [phi],
+
+  say, together with u'1 = d[phi]/dt'1, ..., u'm = d[phi]/dt'm, which
+  contain only one relation connecting the variables t', t'1, ... t'm
+  only.
+
+
+    Simultaneous Pfaffian equations.
+
+  This method for a single Pfaffian equation can, strictly speaking, be
+  generalized to a simultaneous system of (n - r) Pfaffian equations dxj
+  = c_1j dx1 + ... + c_rj dxr only in the case already treated, when
+  this system is satisfied by regarding x_r+1, ... x_n as suitable
+  functions of the independent variables x1, ... xr; in that case the
+  integral manifolds are of r dimensions. When these are non-existent,
+  there may be integral manifolds of higher dimensions; for if
+
+  d[phi] = [phi]1 dx_r +  ... + [phi]_r dx_r + [phi]_r+1(c_1,r+1 dx1 +  ...  + c_r,r+1 dx_r) + [phi]_r+2 ( ) +  ...
+
+  be identically zero, then [phi][sigma] + c[sigma]_,r+1 [phi]_r+1 + ...
+  + c[sigma]_,n [phi]_n = 0, or [phi] satisfies the r partial
+  differential equations previously associated with the total equations;
+  when these are not a complete system, but included in a complete
+  system of r - [mu] equations, having therefore n - r - [mu]
+  independent integrals, the total equations are satisfied over a
+  manifold of r + [mu] dimensions (see E. v. Weber, _Math. Annal._ 1v.
+  (1901), p. 386).
+
+
+    Contact transformations.
+
+  It seems desirable to add here certain results, largely of algebraic
+  character, which naturally arise in connexion with the theory of
+  contact transformations. For any two functions of the 2n independent
+  variables x1, ... xn, p1, ... pn we denote by ([phi][psi]) the sum of
+  the n terms such as d[phi]d[psi]/dp_idx_i - d[psi]d[phi]/dp_idx_i. For
+  two functions of the (2n + 1) independent variables z, x1, ... xn, p1,
+  ... pn we denote by [phi][psi] the sum of the n terms such as
+
+    d[phi] /d[psi]      d[psi]\    d[psi] /d[phi]      d[phi]\
+    ------( ------ + p_i------ ) - ------( ------ + p_i------ ).
+     dpi   \ dxi          dz  /     dpi   \ dxi          dz  /
+
+  It can at once be verified that for any three functions
+  [f[[phi][psi]]] + [[phi][psi]f]] + [[psi][f[phi]]] = df/dz
+  [[phi][psi]] + d[phi]/dz [[psi]f] + d[psi]/dz [f[phi]], which when f,
+  [phi],[psi] do not contain z becomes the identity (f([phi][psi])) +
+  (phi([psi]f)) + ([psi](f[phi])) = 0. Then, if X1, ... Xn, P1, ... Pn
+  be such functions Of x1, ... xn, p1 ... pn that P1 dX1 + ... + Pn dXn
+  is identically equal to p1dx1 + ... + pn dxn, it can be shown by
+  elementary algebra, after equating coefficients of independent
+  differentials, (1) that the functions X1, ... Pn are independent
+  functions of the 2n variables x1, ... pn, so that the equations x'i =
+  Xi, p'i = Pi can be solved for x1, ... xn, p1, ... pn, and represent
+  therefore a transformation, which we call a homogeneous contact
+  transformation; (2) that the X1, ... Xn are homogeneous functions of
+  p1, ... pn of zero dimensions, the P1, ... Pn are homogeneous
+  functions of p1, ... pn of dimension one, and the 1/2n(n - 1) relations
+  (Xi Xj) = 0 are verified. So also are the n^2 relations (Pi Xi) = 1,
+  (Pi Xj) = 0, (Pi Pj) = 0. Conversely, if X1, ... Xn be independent
+  functions, each homogeneous of zero dimension in p1, ... pn satisfying
+  the 1/2n(n - 1) relations (Xi Xj) = 0, then P1, ... Pn can be uniquely
+  determined, by solving linear algebraic equations, such that P1 dX1 +
+  ... + Pn dXn = p1 dx1 + ... + pn dxn. If now we put n + 1 for n, put z
+  for x_n+1, Z for X_n+1, Qi for -Pi/P_n+1, for i = 1, ... n, put qi for
+  -p_i/p_n+1 and [sigma] for q_n+1/Q_n+1, and then finally write P1, ...
+  Pn, p1, ... pn for Q1, ... Qn, q1, ... qn, we obtain the following
+  results: If ZX1 ... Xn, P1, ... Pn be functions of z, x1, ... xn, p1,
+  ... pn, such that the expression dZ - P1 dX1 - ... - Pn dXn is
+  identically equal to [sigma](dz - p1 dx1 - ... - pn dxn), and [sigma]
+  not zero, then (1) the functions Z, X1, ... Xn, P1, ... Pn are
+  independent functions of z, x1, ... xn, p1, ... pn, so that the
+  equations z' = Z, x'i = Xi, p'i = Pi can be solved for z, x1, ... xn,
+  p1, ... pn and determine a transformation which we call a
+  (non-homogeneous) contact transformation; (2) the Z, X1, ... Xn verify
+  the 1/2n(n + 1) identities [Z Xi] = 0, [Xi Xj] = 0. And the further
+  identities
+
+    [Pi Xi] = [sigma], [Pi Xj] = 0, [Pi Z] = [sigma]Pi, [Pi Pj] = 0,
+
+                        dZ                                    dXi                        dPi
+    [Z[sigma]] = [sigma]-- - [sigma]^2, [Xi [sigma]] = [sigma]---, [Pi [sigma]] = [sigma]---
+                        dz                                    dz                         dz
+
+  are also verified. Conversely, if Z, x1, ... Xn be independent
+  functions satisfying the identities [Z Xi] = 0, [Xi Xj] = 0, then
+  [sigma], other than zero, and P1, ... Pn can be uniquely determined,
+  by solution of algebraic equations, such that
+
+    dZ - P1 dX1 - ... - Pn dXn = [sigma](dz - p1 dx1 - ... - p_n dx_n).
+
+  Finally, there is a particular case of great importance arising when
+  [sigma] = 1, which gives the results: (1) If U, X1, ... Xn, P1, ... Pn
+  be 2n + 1 functions of the 2n independent variables x1, ... xn, p1,
+  ... pn, satisfying the identity
+
+    dU + P1 dx1 + ... + Pn dXn = p1 dx1 + ... + p_n dx_n,
+
+  then the 2n functions P1, ... Pn, X1, ... Xn are independent, and we
+  have
+
+    (Xi Xj) = 0, (Xi U) = [delta]Xi, (Pi Xi) = 1, (Pi Xj) = 0, (Pi Pj ) = 0, (Pi U) + Pi = [delta]Pi,
+
+  where [delta] denotes the operator p1d/dp1 + ... + pnd/dpn; (2) If X1,
+  ... Xn be independent functions of x1, ... xn, p1, ... pn, such that
+  (Xi Xj) = 0, then U can be found by a quadrature, such that
+
+    (Xi U) = [delta]Xi;
+
+  and when Xi, ... Xn, U satisfy these 1/2n(n + 1) conditions, then P1,
+  ... Pn can be found, by solution of linear algebraic equations, to
+  render true the identity dU + P1 dX1 + ... + Pn dXn = p1 dx1 + ... +
+  pn dxn; (3) Functions X1, ... Xn, P1, ... Pn can be found to satisfy
+  this differential identity when U is an arbitrary given function of
+  x1, ... xn, p1, ... pn; but this requires integrations. In order to
+  see what integrations, it is only necessary to verify the statement
+  that if U be an arbitrary given function of x1, ... xn, p1, ... pn,
+  and, for r < n, X1, ... Xr be independent functions of these
+  variables, such that (X_[sigma] U) = [delta]X_[sigma], (X_[rho]
+  X_[sigma]) = 0, for [rho], [sigma] = 1 ... r, then the r + 1
+  homogeneous linear partial differential equations of the first order
+  (Uf) + [delta]f = 0, (X[rho]f) = 0, form a complete system. It will be
+  seen that the assumptions above made for the reduction of Pfaffian
+  expressions follow from the results here enunciated for contact
+  transformations.
+
+
+  Partial differential equation of the first order.
+
+  Meaning of a solution of the equation.
+
+We pass on now to consider the solution of any partial differential
+equation of the first order; we attempt to explain certain ideas
+relatively to a single equation with any number of independent variables
+(in particular, an ordinary equation of the first order with one
+independent variable) by speaking of a single equation with two
+independent variables x, y, and one dependent variable z. It will be
+seen that we are naturally led to consider systems of such simultaneous
+equations, which we consider below. The central discovery of the
+transformation theory of the solution of an equation F(x, y, z, dz/dx,
+dz/dy) = 0 is that its solution can always be reduced to the solution of
+partial equations which are _linear_. For this, however, we must regard
+dz/dx, dz/dy, during the process of integration, not as the differential
+coefficients of a function z in regard to x and y, but as variables
+independent of x, y, z, the too great indefiniteness that might thus
+appear to be introduced being provided for in another way. We notice
+that if z = [psi](x, y) be a solution of the differential equation, then
+dz = dxd[psi]/dx + dyd[psi]/dy; thus if we denote the equation by F(x,
+y, z, p, q,) = 0, and prescribe the condition dz = pdx + qdy for every
+solution, any solution such as z = [psi](x, y) will necessarily be
+associated with the equations p = dz/dx, q = dz/dy, and z will satisfy
+the equation in its original form. We have previously seen (under
+_Pfaffian Expressions_) that if five variables x, y, z, p, q, otherwise
+independent, be subject to dz - pdx - qdy = 0, they must in fact be
+subject to at least three mutual relations. If we associate with a point
+(x, y, z) the plane
+
+  Z - z = p(X - x) + q(Y - y)
+
+passing through it, where X, Y, Z are current co-ordinates, and call
+this association a surface-element; and if two consecutive elements of
+which the point(x + dx, y + dy, z + dz) of one lies on the plane of the
+other, for which, that is, the condition dz = pdx + qdy is satisfied, be
+said to be _connected,_ and an infinity of connected elements following
+one another continuously be called a _connectivity_, then our statement
+is that a connectivity consists of not more than [oo]^2 elements, the
+whole number of elements (x, y, z, p, q) that are possible being called
+[oo]^5. The solution of an equation F(x, y, z, dz/dx, dz/dy) = 0 is then
+to be understood to mean finding in all possible ways, from the [oo]^4
+elements (x, y, z, p, q) which satisfy F(x, y, z, p, q) = 0 a set of
+[oo]^2 elements forming a connectivity; or, more analytically, finding in
+all possible ways two relations G = 0, H = 0 connecting x, y, z, p, q
+and independent of F = 0, so that the three relations together may
+involve
+
+  dz = pdx + qdy.
+
+Such a set of three relations may, for example, be of the form z =
+[psi](x, y), p = d[psi]/dx, q = d[psi]/dy; but it may also, as another
+case, involve two relations z = [psi](y), x = [psi]1(y) connecting x, y,
+z, the third relation being
+
+  [psi]'(y) = p[psi]'1(y) + q,
+
+the connectivity consisting in that case, geometrically, of a curve in
+space taken with [oo]^1 of its tangent planes; or, finally, a
+connectivity is constituted by a fixed point and all the planes passing
+through that point. This generalized view of the meaning of a solution
+of F = 0 is of advantage, moreover, in view of anomalies otherwise
+arising from special forms of the equation itself. For instance, we may
+include the case, sometimes arising when the equation to be solved is
+obtained by transformation from another equation, in which F does not
+contain either p or q. Then the equation has [oo]^2 solutions, each
+consisting of an arbitrary point of the surface F = 0 and all the [oo]^2
+planes passing through this point; it also has [oo]^2 solutions, each
+consisting of a curve drawn on the surface F = 0 and all the tangent
+planes of this curve, the whole consisting of [oo]^2 elements; finally,
+it has also an isolated (or singular) solution consisting of the points
+of the surface, each associated with the tangent plane of the surface
+thereat, also [oo]^2 elements in all. Or again, a linear equation F = Pp
++ Qq - R = 0, wherein P, Q, R are functions of x, y, z only, has [oo]^2
+solutions, each consisting of one of the curves defined by
+
+  dx/P = dy/Q = dz/R
+
+taken with all the tangent planes of this curve; and the same equation
+has [oo]^2 solutions, each consisting of the points of a surface
+containing [oo]^1 of these curves and the tangent planes of this surface.
+And for the case of n variables there is similarly the possibility of n
++ 1 kinds of solution of an equation F(x1, ... xn, z, p1, ... pn) = 0;
+these can, however, by a simple contact transformation be reduced to one
+kind, in which there is only one relation z' = [psi](x'1, ... x'n)
+connecting the new variables x'1, ... x'n, z' (see under PFAFFIAN
+EXPRESSIONS); just as in the case of the solution
+
+  z = [psi](y), x = [psi]1(y), [psi]'(y) = p[psi]'1(y) + q
+
+of the equation Pp + Qq = R the transformation z' = z - px, x' = p, p' =
+-x, y' = y, q' = q gives the solution
+
+  z' = [psi](y') + x'[psi]1(y'), p' = dz'/dx', q' = dz'/dy'
+
+of the transformed equation. These explanations take no account of the
+possibility of p and q being infinite; this can be dealt with by writing
+p = -u/w, q = -v/w, and considering homogeneous equations in u, v, w,
+with udx + vdy + wdz = 0 as the differential relation necessary for a
+connectivity; in practice we use the ideas associated with such a
+procedure more often without the appropriate notation.
+
+
+  Order of the ideas.
+
+In utilizing these general notions we shall first consider the theory of
+characteristic chains, initiated by Cauchy, which shows well the nature
+of the relations implied by the given differential equation; the
+alternative ways of carrying out the necessary integrations are
+suggested by considering the method of Jacobi and Mayer, while a good
+summary is obtained by the formulation in terms of a Pfaffian
+expression.
+
+
+    Characteristic chains.
+
+  Consider a solution of F = 0 expressed by the three independent
+  equations F = 0, G = 0, H = 0. If it be a solution in which there is
+  more than one relation connecting x, y, z, let new variables x', y',
+  z', p', q' be introduced, as before explained under PFAFFIAN
+  EXPRESSIONS, in which z' is of the form
+
+    z' = z - p1x1 - ... - p_s x_s (s = 1 or 2),
+
+  so that the solution becomes of a form z' = [psi](x'y'), p' =
+  d[psi]/dx', q' = d[psi]/dy', which then will identically satisfy the
+  transformed equations F' = 0, G' = 0, H' = 0. The equation F' = 0, if
+  x', y', z' be regarded as fixed, states that the plane Z - z' = p'(X -
+  x') + q'(Y - y') is tangent to a certain cone whose vertex is (x', y',
+  z'), the consecutive point (x' + dx', y' + dy', z' + dz') of the
+  generator of contact being such that
+
+        /dF'       /dF'       / /  dF'      dF'\
+    dx'/ --  = dy'/ --  = dz'/ ( p'--- + q' --- ).
+      /  dp'     /  dq'     /   \  dp'      dq'/
+
+
+  Passing in this direction on the surface z' = [psi](x', y') the
+  tangent plane of the surface at this consecutive point is (p' + dp',
+  q' + dq'), where, since F'(x', y', [psi], d[psi]/dx', d[psi]/dy') = 0
+  is identical, we have dx' (dF'/dx' + p'dF'/dz') + dp'dF'/dp' = 0. Thus
+  the equations, which we shall call the characteristic equations,
+
+        /dF'       /dF'       //   dF'     dF'\        //  dF'     dF'\
+    dx'/ --- = dy'/ --- = dz'/( p' --- + q'--- ) = dp'/( - --- - p'--- )
+      /  dp'     /  dq'     /  \   dp'     dq'/      /  \  dx'     dz'/
+
+                                                       //  dF'     dF'\
+                                                 = dq'/( - --- - q'--- )
+                                                     /  \  dy'     dz'/
+
+  are satisfied along a connectivity of [oo]^1 elements consisting of a
+  curve on z' = [psi](x', y') and the tangent planes of the surface
+  along this curve. The equation F' = 0, when p', q' are fixed,
+  represents a curve in the plane Z - z' = p'(X - x') + q'(Y - y')
+  passing through (x', y', z'); if (x' + [delta]x', y' + [delta]y', z' +
+  [delta]z') be a consecutive point of this curve, we find at once
+
+              /dF'     dF'\              /dF'     dF'\
+    [delta]x'( --- + p'--- ) + [delta]y'( --- + q'--- ) = 0;
+              \dx'     dz'/              \dy'     dz'/
+
+  thus the equations above give [delta]x'dp' + [delta]y'dq' = 0, or the
+  tangent line of the plane curve, is, on the surface z' = [psi](x',
+  y'), in a direction conjugate to that of the generator of the cone.
+  Putting each of the fractions in the characteristic equations equal to
+  dt, the equations enable us, starting from an arbitrary element x'0,
+  y'0, z'0, p'0, q'0, about which all the quantities F', dF'/dp', &c.,
+  occurring in the denominators, are developable, to define, from the
+  differential equation F' = 0 alone, a connectivity of [oo]^1 elements,
+  which we call a _characteristic chain_; and it is remarkable that when
+  we transform again to the original variables (x, y, z, p, q), the form
+  of the differential equations for the chain is unaltered, so that they
+  can be written down at once from the equation F = 0. Thus we have
+  proved that the characteristic chain starting from any ordinary
+  element of any integral of this equation F = 0 consists only of
+  elements belonging to this integral. For instance, if the equation do
+  not contain p, q, the characteristic chain, starting from an arbitrary
+  plane through an arbitrary point of the surface F = 0, consists of a
+  pencil of planes whose axis is a tangent line of the surface F = 0. Or
+  if F = 0 be of the form Pp + Qq = R, the chain consists of a curve
+  satisfying dx/P = dy/Q = dz/R and a single infinity of tangent planes
+  of this curve, determined by the tangent plane chosen at the initial
+  point. In all cases there are [oo]^3 characteristic chains, whose
+  aggregate may therefore be expected to exhaust the [oo]^4 elements
+  satisfying F = 0.
+
+
+    Complete integral constructed with characteristic chains.
+
+  Consider, in fact, a single infinity of connected elements each
+  satisfying F = 0, say a chain connectivity T, consisting of elements
+  specified by x0, y0, z0, p0, q0, which we suppose expressed as
+  functions of a parameter u, so that
+
+    U0 = dz0/du - p0dx0/du - q0dy0/du
+
+  is everywhere zero on this chain; further, suppose that each of F,
+  dF/dp, ... , dF/dx + pdF/dz is developable about each element of this
+  chain T, and that T is _not_ a characteristic chain. Then consider the
+  aggregate of the characteristic chains issuing from all the elements
+  of T. The [oo]^2 elements, consisting of the aggregate of these
+  characteristic chains, satisfy F = 0, provided the chain connectivity
+  T consists of elements satisfying F = 0; for each characteristic chain
+  satisfies dF = 0. It can be shown that these chains are connected; in
+  other words, that if x, y, z, p, q, be any element of one of these
+  characteristic chains, not only is
+
+    dz/dt - pdx/dt - qdy/dt = 0,
+
+  as we know, but also U = dz/du - pdx/du - qdy/du is also zero. For we
+  have
+
+    dU   d  /dz    dx    dy\    d  /dz    dx    dy\
+    -- = --( -- - p-- - q-- ) - --( -- - p-- - q-- )
+    dt   dt \du    du    du/    du \dt    dt    dt/
+
+         dp dx   dp dx   dq dy   dq dy
+       = -- -- - -- -- + -- -- - -- -- ,
+         du dt   dt du   du dt   dt du
+
+  which is equal to
+
+    dp dF   dx /dF    dF\    dq dF   dy /dF    dF\      dF
+    -- -- + --( -- + p-- ) + -- -- + --( -- + q-- ) = - -- U.
+    du dp   du \dx    dz/    du dq   du \dy    dz/      dz
+
+     dF
+  As -- is a developable function of t, this, giving
+     dz
+                       _
+                  /   / t  dF  \
+    U = U_{0} exp( -  |    --dt ),
+                  \  _/t0  dz  /
+
+  shows that U is everywhere zero. Thus integrals of F = 0 are
+  obtainable by considering the aggregate of characteristic chains
+  issuing from arbitrary chain connectivities T satisfying F = 0; and
+  such connectivities T are, it is seen at once, determinable without
+  integration. Conversely, as such a chain connectivity T can be taken
+  out from the elements of any given integral all possible integrals are
+  obtainable in this way. For instance, an arbitrary curve in space,
+  given by x0 = [theta](u), y0 = [phi](u), z0 = [psi](u), determines by
+  the two equations F(x0, y0, z0, p0, q0) = 0, [psi]'(u) = p0[theta]'(u)
+  + q0[phi]'(u), such a chain connectivity T, through which there passes
+  a perfectly definite integral of the equation F = 0. By taking [oo]^2
+  initial chain connectivities T, as for instance by taking the curves
+  x0 = [theta], y0 = [phi], z0 = [psi] to be the [oo]^2 curves upon an
+  arbitrary surface, we thus obtain [oo]^2 integrals, and so [oo]^4
+  elements satisfying F = 0. In general, if functions G, H, independent
+  of F, be obtained, such that the equations F = 0, G = b, H = c
+  represent an integral for all values of the constants b, c, these
+  equations are said to constitute a _complete integral_. Then [oo]^4
+  elements satisfying F = 0 are known, and in fact every other form of
+  integral can be obtained without further integrations.
+
+
+    Operations necessary for integration of F = a.
+
+  In the foregoing discussion of the differential equations of a
+  characteristic chain, the denominators dF/dp, ... may be supposed to
+  be modified in form by means of F = 0 in any way conducive to a simple
+  integration. In the immediately following explanation of ideas,
+  however, we consider indifferently all equations F = constant; when a
+  function of x, y, z, p, q is said to be zero, it is meant that this is
+  so identically, not in virtue of F = 0; in other words, we consider
+  the integration of F = a, where a is an arbitrary constant. In the
+  theory of linear partial equations we have seen that the integration
+  of the equations of the characteristic chains, from which, as has just
+  been seen, that of the equation F = a follows at once, would be
+  involved in completely integrating the single linear homogeneous
+  partial differential equation of the first order [Ff] = 0 where the
+  notation is that explained above under CONTACT TRANSFORMATIONS. One
+  obvious integral is f = F. Putting F = a, where a is arbitrary, and
+  eliminating one of the independent variables, we can reduce this
+  equation [Ff] = 0 to one in four variables; and so on. Calling, then,
+  the determination of a single integral of a single homogeneous partial
+  differential equation of the first order in n independent variables,
+  _an operation of order_ n - 1, the characteristic chains, and
+  therefore the most general integral of F = a, can be obtained by
+  successive operations of orders 3, 2, 1. If, however, an integral of F
+  = a be represented by F = a, G = b, H = c, where b and c are arbitrary
+  constants, the expression of the fact that a characteristic chain of F
+  = a satisfies dG = 0, gives [FG] = 0; similarly, [FH] = 0 and [GH] =
+  0, these three relations being identically true. Conversely, suppose
+  that an integral G, independent of F, has been obtained of the
+  equation [Ff] = 0, which is an operation of order three. Then it
+  follows from the identity [f[[phi][psi]]] + [[phi][[psi]f]] +
+  [[psi][f[phi]]] = df/dz [[psi][phi]] + d[phi]/dz [psif] + d[psi]/dz
+  [f[phi]] before remarked, by putting [phi] = F, [psi] = G, and then
+  [Ff] = A(f), [Gf] = B(f), that AB(f) - BA(f) = dF/dz B(f) - dG/dz
+  A(f), so that the two linear equations [Ff] = 0, [Gf] = 0 form a
+  complete system; as two integrals F, G are known, they have a common
+  integral H, independent of F, G, determinable by an operation of order
+  one only. The three functions F, G, H thus identically satisfy the
+  relations [FG] = [GH] = [FH] = 0. The [oo]^2 elements satisfying F = a,
+  G = b, H = c, wherein a, b, c are assigned constants, can then be seen
+  to constitute an integral of F = a. For the conditions that a
+  characteristic chain of G = b issuing from an element satisfying F =
+  a, G = b, H = c should consist only of elements satisfying these three
+  equations are simply [FG] = 0, [GH] = 0. Thus, starting from an
+  arbitrary element of (F = a, G = b, H = c), we can single out a
+  connectivity of elements of (F = a, G = b, H = c) forming a
+  characteristic chain of G = b; then the aggregate of the
+  characteristic chains of F = a issuing from the elements of this
+  characteristic chain of G = b will be a connectivity consisting only
+  of elements of
+
+    (F = a, G = b, H = c),
+
+  and will therefore constitute an integral of F = a; further, it will
+  include all elements of (F = a, G = b, H = c). This result follows
+  also from a theorem given under CONTACT TRANSFORMATIONS, which shows,
+  moreover, that though the characteristic chains of F = a are not
+  determined by the three equations F = a, G = b, H = c, no further
+  integration is now necessary to find them. By this theorem, since
+  identically [FG] = [GH] = [FH] = 0, we can find, by the solution of
+  linear algebraic equations only, a non-vanishing function [sigma] and
+  two functions A, C, such that
+
+    dG - AdF - CdH = [sigma](dz - pdz - qdy);
+
+  thus all the elements satisfying F = a, G = b, H = c, satisfy dz = pdx
+  + qdy and constitute a connectivity, which is therefore an integral of
+  F = a. While, further, from the associated theorems, F, G, H, A, C are
+  independent functions and [FC] = 0. Thus C may be taken to be the
+  remaining integral independent of G, H, of the equation [Ff] = 0,
+  whereby the characteristic chains are entirely determined.
+
+
+    The single equation F = 0 and Pfaffian formulations.
+
+  When we consider the particular equation F = 0, neglecting the case
+  when neither p nor q enters, and supposing p to enter, we may express
+  p from F = 0 in terms of x, y, z, q, and then eliminate it from all
+  other equations. Then instead of the equation [Ff] = 0, we have, if F
+  = 0 give p = [psi](x, y, z, q), the equation
+
+                  /df         df\    d[psi]  /df     df\     /d[psi]     d[psi]\  df
+    [Sigma]f = - ( -- + [psi] -- ) + ------ ( -- + q -- ) - ( ------ + q ------ ) -- = 0,
+                  \dx         dz/      dq    \dy     dz/     \  dy         dz  /  dq
+
+  moreover obtainable by omitting the term in df/dp in [p-[psi], f] = 0.
+  Let x0, y0, z0, q0, be values about which the coefficients in this
+  equation are developable, and let [zeta], [eta], [omega] be the
+  principal solutions reducing respectively to z, y and q when x = x0.
+  Then the equations p = [psi], [zeta] = z0, [eta] = y0, [omega] = q0
+  represent a characteristic chain issuing from the element x0, y0, z0,
+  [psi]0, q0; we have seen that the aggregate of such chains issuing
+  from the elements of an arbitrary chain satisfying
+
+    dz0 = p0dx0 - q0dy0 = 0
+
+  constitute an integral of the equation p = [psi]. Let this arbitrary
+  chain be taken so that x0 is constant; then the condition for initial
+  values is only
+
+    dz0 - q0dy0 = 0,
+
+  and the elements of the integral constituted by the characteristic
+  chains issuing therefrom satisfy
+
+    d[zeta] - [omega]d[eta] = 0.
+
+  Hence this equation involves dz - [psi]dx - qdy = 0, or we have
+
+    dz - [psi]dx - qdy = [sigma](d[zeta] - [omega]d[eta]),
+
+  where [sigma] is not zero. Conversely, the integration of p = [psi]
+  is, essentially, the problem of writing the expression dz - [psi]dx -
+  qdy in the form [sigma](d[zeta] - [omega]d[eta]), as must be possible
+  (from what was said under _Pfaffian Expressions_).
+
+
+    System of equations of the first order.
+
+  To integrate a system of simultaneous equations of the first order X1
+  = a1, ... Xr = ar in n independent variables x1, ... xn and one
+  dependent variable z, we write p1 for dz/dx1, &c., and attempt to find
+  n + 1 - r further functions Z, X_r+1 ... Xn, such that the equations Z
+  = a, Xi = ai,(i = 1, ... n) involve dz - p1dx1 - ... - pndxn = 0. By
+  an argument already given, the common integral, if existent, must be
+  satisfied by the equations of the characteristic chains of any one
+  equation Xi = ai; thus each of the expressions [Xi Xj] must vanish in
+  virtue of the equations expressing the integral, and we may without
+  loss of generality assume that each of the corresponding 1/2r(r - 1)
+  expressions formed from the r given differential equations vanishes in
+  virtue of these equations. The determination of the remaining n + 1 -
+  r functions may, as before, be made to depend on characteristic
+  chains, which in this case, however, are manifolds of r dimensions
+  obtained by integrating the equations [X1f] = 0, ... [Xrf] = 0; or
+  having obtained one integral of this system other than X1, ... Xr, say
+  Xr+1, we may consider the system [X1f] = 0, ... [X_r+1 f] = 0, for
+  which, again, we have a choice; and at any stage we may use Mayer's
+  method and reduce the simultaneous linear equations to one equation
+  involving parameters; while if at any stage of the process we find
+  some but not all of the integrals of the simultaneous system, they can
+  be used to simplify the remaining work; this can only be clearly
+  explained in connexion with the theory of so-called function groups
+  for which we have no space. One result arising is that the
+  simultaneous system p1 = [phi]1, ... pr = [phi]r, wherein p1, ... pr
+  are not involved in [phi]1, ... [phi]r, if it satisfies the 1/2r(r - 1)
+  relations [pi - [phi]i, pj - [phi]j] = 0, has a solution z = [psi](x1,
+  ... xn), p1 = d[psi]/dx1, ... pn = d[psi]/dxn, reducing to an
+  arbitrary function of x_r+1, ... xn only, when x1 = x1^0, ... xr =
+  xr^0 under certain conditions as to developability; a generalization
+  of the theorem for linear equations. The problem of integration of
+  this system is, as before, to put
+
+    dz - [phi]1dx1 - ... - [phi]_r dx_r - p_r+1 dx_r+1 - ... - p_n dx_n
+
+  into the form [sigma](d[zeta] - [omega]_r+1 + d[xi]_r+1 - ... -
+  [omega]_n d[xi]_n); and here [zeta], [xi]_r+1, ... [xi]_n,
+  [omega]_r+1, ... [omega]_n may be taken, as before, to be principal
+  integrals of a certain complete system of linear equations; those,
+  namely, determining the characteristic chains.
+
+
+    Equations of dynamics.
+
+  If L be a function of t and of the 2n quantities x1, ... xn, [.x]1,
+  ... [.x]n, where [.x]i, denotes dxi/dt, &c., and if in the n equations
+
+     d   /  dL  \     dL
+    --- (--------) = ----
+    dt   \ dx_i /    dx_i
+
+  we put p_i = dL/d[.x]_i, and so express [.x]1 , ... [.x]_n in terms of
+  t, x_i, ... x_n, p1, ... p_n, assuming that the determinant of the
+  quantities d^2L/dx_i d[.x]_j is not zero; if, further, H denote the
+  function of t, x1, ... xn, p1, ... pn, numerically equal to p1[.x]1 +
+  ... + pn[.x]n - L, it is easy to prove that dpi/dt = -dH/dxi, dxi/dt =
+  dH/dp_i. These so-called _canonical_ equations form part of those for
+  the characteristic chains of the single partial equation dz/dt + H(t,
+  x1, ... xn, dz/dx1, ..., dz/dx_n) = 0, to which then the solution of
+  the original equations for x1 ... xn can be reduced. It may be shown
+  (1) that if z = [psi](t, x1, ... xn, c1, .. cn) + c be a complete
+  integral of this equation, then pi = d[psi]/dx_i, d[psi]/dc_i = e_i are
+  2n equations giving the solution of the canonical equations referred
+  to, where c1 ... cn and e1, ... en are arbitrary constants; (2) that
+  if xi = Xi(t, x^01, ... pn^0), pi=Pi(t, x1^0, ... p^0n) be the
+  principal solutions of the canonical equations for t = t^0, and
+  [omega] denote the result of substituting these values in p1dH/dp1 +
+  ... + pndH/dpn - H, and [Omega] = [int] [t0 to t] [omega]dt, where,
+  after integration, [Omega] is to be expressed as a function of t, x1,
+  ... xn, x1^0, ... xn^0, then z = [Omega] + z^0 is a complete integral
+  of the partial equation.
+
+
+  Application of theory of continuous groups to formal theories.
+
+A system of differential equations is said to allow a certain continuous
+group of transformations (see GROUPS, THEORY OF) when the introduction
+for the variables in the differential equations of the new variables
+given by the equations of the group leads, for all values of the
+parameters of the group, to the same differential equations in the new
+variables. It would be interesting to verify in examples that this is
+the case in at least the majority of the differential equations which
+are known to be integrable in finite terms. We give a theorem of very
+general application for the case of a simultaneous complete system of
+linear partial homogeneous differential equations of the first order, to
+the solution of which the various differential equations discussed have
+been reduced. It will be enough to consider whether the given
+differential equations allow the infinitesimal transformations of the
+group.
+
+  It can be shown easily that sufficient conditions in order that a
+  complete system [Pi]1f = 0 ... [Pi]kf = 0, in n independent variables,
+  should allow the infinitesimal transformation Pf = 0 are expressed by
+  k equations [Pi]_i Pf - P[Pi]_i f = [lambda]_i1 [Pi]1f + ... +
+  [lambda]_ik [Pi]_kf. Suppose now a complete system of n - r equations
+  in n variables to allow a group of r infinitesimal transformations
+  (P1f, ..., Prf) which has an invariant subgroup of r - 1 parameters
+  (P1f, ..., Pr-1f), it being supposed that the n quantities [Pi]1f,
+  ..., [Pi]_n-r f, P1 f, ..., P_r f are not connected by an identical
+  linear equation (with coefficients even depending on the independent
+  variables). Then it can be shown that one solution of the complete
+  system is determinable by a quadrature. For each of [Pi]_i P_[sigma] f
+  - P_[sigma] [Pi]_i f is a linear function of [Pi]1f, ..., [Pi]_n-r f
+  and the simultaneous system of independent equations [Pi]1f = 0, ...
+  [Pi]_n-r f = 0, P1f = 0, ... P_r-1 f = 0 is therefore a complete
+  system, allowing the infinitesimal transformation Prf. This complete
+  system of n - 1 equations has therefore one common solution [omega],
+  and P_r([omega]) is a function of [omega]. By choosing [omega]
+  suitably, we can then make Pr([omega]) = 1. From this equation and the
+  n - 1 equations [Pi]_i[omega] = 0, P_[sigma][omega] = 0, we can
+  determine [omega] by a quadrature only. Hence can be deduced a much
+  more general result, _that if the group of r parameters be integrable,
+  the complete system can be entirety solved by quadratures_; it is only
+  necessary to introduce the solution found by the first quadrature as
+  an independent variable, whereby we obtain a complete system of n - r
+  equations in n - 1 variables, subject to an integrable group of r - 1
+  parameters, and to continue this process. We give some examples of the
+  application of the theorem. (1) If an equation of the first order y' =
+  [psi](x, y) allow the infinitesimal transformation [xi]df/dx +
+  [eta]df/dy, the integral curves [omega](x, y) = y^0, wherein [omega](x,
+  y) is the solution of df/dx + [psi](x, y) df/dy = 0 reducing to y for
+  x = x^0, are interchanged among themselves by the infinitesimal
+  transformation, or [omega](x, y) can be chosen to make [xi]d[omega]/dx
+  + [eta]d[omega]/dy = 1; this, with d[omega]/dx + [psi]d[omega]/dy = 0,
+  determines [omega] as the integral of the complete differential (dy -
+  [psi]dx)/([eta] - [psi][xi]). This result itself shows that every
+  ordinary differential equation of the first order is subject to an
+  infinite number of infinitesimal transformations. But every
+  infinitesimal transformation [xi]df/dx + [eta]df/dy can by change of
+  variables (after integration) be brought to the form df/dy, and all
+  differential equations of the first order allowing this group can then
+  be reduced to the form F(x, dy/dx) = 0. (2) In an ordinary equation of
+  the second order y" = [psi](x, y, y'), equivalent to dy/dx = y1,
+  dy1/dx = [psi](x, y, y1), if H, H1 be the solutions for y and y1
+  chosen to reduce to y^0 and y1^0 when x = x^0, and the equations H = y,
+  H1= y1 be equivalent to [omega] = y^0, [omega]1 = y1^0, then [omega],
+  [omega]1 are the principal solutions of [Pi]f = df/dx + y1df/dy +
+  [psi]df/dy1 = 0. If the original equation allow an infinitesimal
+  transformation whose first _extended_ form (see GROUPS) is Pf =
+  [xi]df/dx + [eta]df/dy + [eta]1df/dy1, where [eta]1[delta]t is the
+  increment of dy/dx when [xi][delta]t, [eta][delta]t are the increments
+  of x, y, and is to be expressed in terms of x, y, y1, then each of
+  P[omega] and P[omega]1 must be functions of [omega] and [omega]1, or
+  the partial differential equation [Pi]f must allow the group Pf. Thus
+  by our general theorem, if the differential equation allow a group of
+  two parameters (and such a group is always integrable), it can be
+  solved by quadratures, our explanation sufficing, however, only
+  provided the form [Pi]f and the two infinitesimal transformations are
+  not linearly connected. It can be shown, from the fact that [eta]1 is
+  a quadratic polynomial in y1, that no differential equation of the
+  second order can allow more than 8 really independent infinitesimal
+  transformations, and that every homogeneous linear differential
+  equation of the second order allows just 8, being in fact reducible to
+  d^2y/dx^2 = 0. Since every group of more than two parameters has
+  subgroups of two parameters, a differential equation of the second
+  order allowing a group of more than two parameters can, as a rule, be
+  solved by quadratures. By transforming the group we see that if a
+  differential equation of the second order allows a single
+  infinitesimal transformation, it can be transformed to the form F(x,
+  d[gamma]/dx, d^2[gamma]/dx^2); this is not the case for every
+  differential equation of the second order. (3) For an ordinary
+  differential equation of the third order, allowing an integrable group
+  of three parameters whose infinitesimal transformations are not
+  linearly connected with the partial equation to which the solution of
+  the given ordinary equation is reducible, the similar result follows
+  that it can be integrated by quadratures. But if the group of three
+  parameters be simple, this result must be replaced by the statement
+  that the integration is reducible to quadratures and that of a
+  so-called Riccati equation of the first order, of the form dy/dx = A +
+  By + Cy^2, where A, B, C are functions of x. (4) Similarly for the
+  integration by quadratures of an ordinary equation yn = [psi](x, y,
+  y1, ... yn-1) of any order. Moreover, the group allowed by the
+  equation may quite well consist of extended contact transformations.
+  An important application is to the case where the differential
+  equation is the resolvent equation defining the group of
+  transformations or rationality group of another differential equation
+  (see below); in particular, when the rationality group of an ordinary
+  linear differential equation is integrable, the equation can be solved
+  by quadratures.
+
+
+  Consideration of function theories of differential equations.
+
+Following the practical and provisional division of theories of
+differential equations, to which we alluded at starting, into
+transformation theories and function theories, we pass now to give some
+account of the latter. These are both a necessary logical complement of
+the former, and the only remaining resource when the expedients of the
+former have been exhausted. While in the former investigations we have
+dealt only with values of the independent variables about which the
+functions are developable, the leading idea now becomes, as was long ago
+remarked by G. Green, the consideration of the neighbourhood of the
+values of the variables for which this developable character ceases.
+Beginning, as before, with existence theorems applicable for ordinary
+values of the variables, we are to consider the cases of failure of such
+theorems.
+
+
+  A general existence theorem.
+
+When in a given set of differential equations the number of equations is
+greater than the number of dependent variables, the equations cannot be
+expected to have common solutions unless certain conditions of
+compatibility, obtainable by equating different forms of the same
+differential coefficients deducible from the equations, are satisfied.
+We have had examples in systems of linear equations, and in the case of
+a set of equations p1 = [phi]1, ..., pr = [phi]r. For the case when the
+number of equations is the same as that of dependent variables, the
+following is a general theorem which should be referred to: Let there be
+r equations in r dependent variables z1, ... zr and n independent
+variables x1, ... xn; let the differential coefficient of z[sigma] of
+highest order which enters be of order h[sigma], and suppose d^h_[sigma]
+z_[sigma]/dx1^h_[sigma] to enter, so that the equations can be written
+d^h_[sigma] z_[sigma]/dx1^h_[sigma] = [Phi]_[sigma], where in the
+general differential coefficient of z_[rho] which enters in
+[Phi]_[sigma], say
+
+  d^(k1 + ... + kn) z_[rho]/dx1^k1 ... dx_n^k_n,
+
+we have k1 < h_[rho] and k1 + ... + k_n <= h_[rho]. Let a1, ... an, b1,
+... br, and b[rho]_(k1 ... kn) be a set of values of
+
+  x1, ... x_n, z1, ... z_r
+
+and of the differential coefficients entering in [Phi]_[sigma] about
+which all the functions [Phi]1, ... [Phi]_r, are developable.
+Corresponding to each dependent variable z_[sigma], we take now a set of
+h_[sigma] functions of x2, ... xn, say [phi][sigma], [phi][sigma]^(1),
+..., [phi][sigma]^(h-1) arbitrary save that they must be developable
+about a2, a3, ... an, and such that for these values of x2, ... xn, the
+function [phi]_[rho] reduces to b_[rho], and the differential
+coefficient
+
+  d^(k2 + ... + kn) [phi]_[rho]^(k1)/dx2^k2 ... dx_n^kn
+
+reduces to b^kn_(k1 ... kn). Then the theorem is that there exists one,
+and only one, set of functions z1, ... z_r, of x2, ... x_n developable
+about a1, ... an satisfying the given differential equations, and such
+that for x1 = a1 we have
+
+  z_[sigma] = [phi]_[sigma], dz_[sigma]/dx1 = [phi]_[sigma]^(1), ...
+         d^(h_[sigma]-1) z_[sigma]/d^(h_[sigma]-1) x1 = [phi][sigma]^(h_[sigma]-1).
+
+And, moreover, if the arbitrary functions [phi]_[sigma],
+[phi]_[sigma]^(1) ... contain a certain number of arbitrary variables
+t1, ... tm, and be developable about the values t1^0, ... tm^0 of these
+variables, the solutions z1, ... zr will contain t1, ... tm, and be
+developable about t1^0, ... tm^0.
+
+
+    Singular points of solutions.
+
+  The proof of this theorem may be given by showing that if ordinary
+  power series in x1 - -a1, ... xn - an, t1 - t1^0, ... tm - tm^0 be
+  substituted in the equations wherein in z[sigma] the coefficients of
+  (x1 - a1)^0, x1 - a1, ..., (x1 - a1)^(h_[sigma]-1) are the arbitrary
+  functions [phi]_[sigma], [phi]_[sigma]^(1), ..., [phi]_[sigma]^h-1,
+  divided respectively by 1, 1!, 2!, &c., then the differential
+  equations determine uniquely all the other coefficients, and that the
+  resulting series are convergent. We rely, in fact, upon the theory of
+  monogenic analytical functions (see FUNCTION), a function being
+  determined entirely by its development in the neighbourhood of one set
+  of values of the independent variables, from which all its other
+  values arise by _continuation_; it being of course understood that the
+  coefficients in the differential equations are to be continued at the
+  same time. But it is to be remarked that there is no ground for
+  believing, if this method of continuation be utilized, that the
+  function is single-valued; we may quite well return to the same values
+  of the independent variables with a different value of the function;
+  belonging, as we say, to a different branch of the function; and there
+  is even no reason for assuming that the number of branches is finite,
+  or that different branches have the same singular points and regions
+  of existence. Moreover, and this is the most difficult consideration
+  of all, all these circumstances may be dependent upon the values
+  supposed given to the arbitrary constants of the integral; in other
+  words, the singular points may be either _fixed_, being determined by
+  the differential equations themselves, or they may be _movable_ with
+  the variation of the arbitrary constants of integration. Such
+  difficulties arise even in establishing the reversion of an elliptic
+  integral, in solving the equation
+
+     /dx\^2
+    ( -- ) = (x-a1)(x - a2)(x - a3)(x - a4);
+     \ds/
+
+  about an ordinary value the right side is developable; if we put x -
+  a1 = t1^2, the right side becomes developable about t1 = 0; if we put x
+  = 1/t, the right side of the changed equation is developable about t =
+  0; it is quite easy to show that the integral reducing to a definite
+  value x0 for a value s0 is obtainable by a series in integral powers;
+  this, however, must be supplemented by showing that for no value of s
+  does the value of x become entirely undetermined.
+
+
+    Linear differential equations with rational coefficients.
+
+  These remarks will show the place of the theory now to be sketched of
+  a particular class of ordinary linear homogeneous differential
+  equations whose importance arises from the completeness and generality
+  with which they can be discussed. We have seen that if in the
+  equations dy/dx = y1, dy1/dx = y2, ..., dy_n-2/dx = y_n-1,
+
+    dy_n-1/dx = a_n y + a_n-1 y1 + ... + a1 y_n-1,
+
+  where a1, a2, ..., an are now to be taken to be rational functions of
+  x, the value x = x^0 be one for which no one of these rational
+  functions is infinite, and y^0, y^01, ..., y^0_n-1 be quite arbitrary
+  finite values, then the equations are satisfied by
+
+    y = y^0u + y^01u1 + ... + y^0_n-1 u_n-1,
+
+  where u, u1, ..., un-1 are functions of x, independent of y^0, ...
+  y^0_n-1, developable about x = x^0; this value of y is such that for x =
+  x^0 the functions y, y1 ... y_n-1 reduce respectively to y^0, y^01, ...
+  y^0_n-1; it can be proved that the region of existence of these series
+  extends within a circle centre x^0 and radius equal to the distance
+  from x^0 of the nearest point at which one of a1, ... an becomes
+  infinite. Now consider a region enclosing x^0 and only one of the
+  places, say [Sigma], at which one of a1, ... an becomes infinite. When
+  x is made to describe a closed curve in this region, including this
+  point [Sigma] in its interior, it may well happen that the
+  continuations of the functions u, u1, ..., u_n-1 give, when we have
+  returned to the point x, values v, v1, ..., v_n-1, so that the
+  integral under consideration becomes changed to y^0 + y^01v1 + ... +
+  y^0_n-1 v_n-1. At x^0 let this branch and the corresponding values of
+  y1, ... y_n-1 be [eta]^0, [eta]^0_1, ... [eta]^0_n-1; then, as there is
+  only one series satisfying the equation and reducing to ([eta]^0,
+  [eta]^0_1, ... [eta]^0_n-1) for x = x^0 and the coefficients in the
+  differential equation are single-valued functions, we must have
+  [eta]^0_u + [eta]^0_1u1 + ... + [eta]^0_n-1 u_n-1 = y^0v + y^01v1 + ... +
+  y^0_n-1 v_n-1; as this holds for arbitrary values of y^0 ... y^0_n-1,
+  upon which u, ... u_n-1 and v, ... v_n-1 do not depend, it follows
+  that each of v, ... v_n-1 is a linear function of u, ... u_n-1 with
+  constant coefficients, say v_i = A_i1 u + ... + A_in u_n-1. Then
+
+    y^0v + ... + y^0_n-1 v_n-1 = ([Sigma]_i A_i1 y_i^0)u + ... + ([Sigma]_i A_in y^0_i)u_n-1;
+
+  this is equal to [mu](y^0u + ... + y^0_n-1 u_n-1) if [Sigma]_i A_ir y^0_i
+  = [mu]y^0_r-1; eliminating y^0 ... y^0_n-1 from these linear equations,
+  we have a determinantal equation of order n for [mu]; let [mu]1 be one
+  of its roots; determining the ratios of y^0, y1^0, ... y^0_n-1 to satisfy
+  the linear equations, we have thus proved that there exists an
+  integral, H, of the equation, which when continued round the point
+  [Sigma] and back to the starting-point, becomes changed to H1 =
+  [mu]1H. Let now [xi] be the value of x at [Sigma] and r1 one of the
+  values of (1/2[pi]i) log [mu]1; consider the function (x - [xi])^r1 H;
+  when x makes a circuit round x = [xi], this becomes changed to
+
+    exp(-2[pi]ir1) (x - [xi])^-r1 [mu]H,
+
+  that is, is unchanged; thus we may put H = (x - [xi])^r1 [phi]1,
+  [phi]1 being a function single-valued for paths in the region
+  considered described about [Sigma], and therefore, by Laurent's
+  Theorem (see FUNCTION), capable of expression in the annular region
+  about this point by a series of positive and negative integral powers
+  of x - [xi], which in general may contain an infinite number of
+  negative powers; there is, however, no reason to suppose r1 to be an
+  integer, or even real. Thus, if all the roots of the determinantal
+  equation in [mu] are different, we obtain n integrals of the forms (x
+  -[xi])^r1 phi1, ..., (x - [xi])^rn [phi]_n. In general we obtain as
+  many integrals of this form as there are really different roots; and
+  the problem arises to discover, in case a root be k times repeated, k
+  - 1 equations of as simple a form as possible to replace the k - 1
+  equations of the form y^0 + ... + y^0_n-1 v_n-1 = [mu](y^0 + ... + y^0_n-1
+  u_n-1) which would have existed had the roots been different. The most
+  natural method of obtaining a suggestion lies probably in remarking
+  that if r2 = r1 + h, there is an integral [(x - [xi])^(r1 + h) [phi]2
+  - (x -[xi])^r1 [phi]1]/h, where the coefficients in [phi]2 are the
+  same functions of r1 + h as are the coefficients in [phi]1 of r1; when
+  h vanishes, this integral takes the form
+                    _                              _
+                   | d[phi]1                        |
+    (x - [xi])^r1  | ------- + [phi]1 log (x - [xi])|,
+                   |_  dr1                         _|
+
+  or say       (x-[xi])^r1 [[phi]1 + [psi]1 log (x - [xi])];
+
+  denoting this by 2[pi]i[mu]1K, and (x-[xi])^r1 [phi]1 by H, a circuit
+  of the point [xi] changes K into
+
+              1
+    K' = ----------- [e^(2[pi]ir1) (x - [xi])^r1 [psi]1 + e^(2[pi]ir1) (x - [xi])^r1 [phi]1 (2[pi]i + log(x - [xi]))]
+         2[pi]i[mu]1
+
+       = [mu]1K + H.
+
+  A similar artifice suggests itself when three of the roots of the
+  determinantal equation are the same, and so on. We are thus led to the
+  result, which is justified by an examination of the algebraic
+  conditions, that whatever may be the circumstances as to the roots of
+  the determinantal equation, n integrals exist, breaking up into
+  batches, the values of the constituents H1, H2, ... of a batch after
+  circuit about x = [xi] being H1' = [mu]1H1, H2' = [mu]1H2 + H1, H3' =
+  [mu]1H3 + H2, and so on. And this is found to lead to the forms (x -
+  [xi])^r1 [phi]1, (x - [xi])^r1 [[psi]1 + [phi]1 log (x - [xi])], (x -
+  [xi])^r1 [[chi]1 + [chi]2 log (x - [xi]) + [phi]1(log(x - [xi]))^2],
+  and so on. Here each of [phi]1, [psi]1, [chi]1, [chi]2, ... is a
+  series of positive and negative integral powers of x - [xi] in which
+  the number of negative powers may be infinite.
+
+
+    Regular equations.
+
+  It appears natural enough now to inquire whether, under proper
+  conditions for the forms of the rational functions a1, ... an, it may
+  be possible to ensure that in each of the series [phi]1, [psi]1,
+  [chi]1, ... the number of negative powers shall be finite. Herein
+  lies, in fact, the limitation which experience has shown to be
+  justified by the completeness of the results obtained. Assuming n
+  integrals in which in each of [phi]1, [psi]1, [chi]1 ... the number of
+  negative powers is finite, there is a definite homogeneous linear
+  differential equation having these integrals; this is found by forming
+  it to have the form
+
+    y'^n = (x - [xi])^-1 b1y'^(n-1) + (x - [xi])^-2 b2y'^(n-2) + ... +(x - [xi])^-n b_n y,
+
+  where b1, ... bn are finite for x = [xi]. Conversely, assume the
+  equation to have this form. Then on substituting a series of the form
+  (x - [xi])^r [1 + A1(x - [xi]) + A2(x - [xi])^2 + ... ] and equating
+  the coefficients of like powers of x-[xi], it is found that r must be
+  a root of an algebraic equation of order n; this equation, which we
+  shall call the index equation, can be obtained at once by substituting
+  for y only (x - [xi])^r and replacing each of b1, ... bn by their
+  values at x = [xi]; arrange the roots r1, r2, ... of this equation so
+  that the real part of ri is equal to, or greater than, the real part
+  of r_i+1, and take r equal to r1; it is found that the coefficients
+  A1, A2 ... are uniquely determinate, and that the series converges
+  within a circle about x = [xi] which includes no other of the points
+  at which the rational functions a1 ... an become infinite. We have
+  thus a solution H1 = (x -[xi])^r1 [phi]1 of the differential equation.
+  If we now substitute in the equation y = H1 f[eta]dx, it is found to
+  reduce to an equation of order n - 1 for [eta] of the form
+
+    [eta]'^(n-1) = (x - [xi])^-1 c1[eta]'^(n-2) + ... + (x-[xi])^(n-1) c_n-1 [eta],
+
+  where c1, ... c_n-1 are not infinite at x = [xi]. To this equation
+  precisely similar reasoning can then be applied; its index equation
+  has in fact the roots r2 - r1 - 1, ... , rn - r1 - 1; if r2 - r1 be
+  zero, the integral (x - [xi])^-1 [psi]1 of the [eta] equation will
+  give an integral of the original equation containing log (x - [xi]);
+  if r2 - r1 be an integer, and therefore a negative integer, the same
+  will be true, unless in [psi]1 the term in (x - [xi])^(r1 - r2) be
+  absent; if neither of these arise, the original equation will have an
+  integral (x -[xi])^r2 [phi]2. The [eta] equation can now, by means of
+  the one integral of it belonging to the index r2 - r1 - 1, be
+  similarly reduced to one of order n - 2, and so on. The result will be
+  that stated above. We shall say that an equation of the form in
+  question is _regular_ about x = [xi].
+
+
+    Fuchsian equations.
+
+    Equation of the second order.
+
+  We may examine in this way the behaviour of the integrals at all the
+  points at which any one of the rational functions a1 ... an becomes
+  infinite; in general we must expect that beside these the value x =
+  [oo] will be a singular point for the solutions of the differential
+  equation. To test this we put x = 1/t throughout, and examine as
+  before at t = 0. For instance, the ordinary linear equation with
+  constant coefficients has no singular point for finite values of x; at
+  x = [oo] it has a singular point and is not regular; or again,
+  Bessel's equation x^2 + xy' + (x^2 - n^2)y = 0 is regular about x = 0,
+  but not about x = [oo]. An equation regular at all the finite
+  singularities and also at x = [oo] is called a Fuchsian equation. We
+  proceed to examine particularly the case of an equation of the second
+  order
+
+    y" + ay' + by = 0.
+
+  Putting x = 1/t, it becomes
+
+    d^2y/dt^2 + (2t^-1 - at^-2)dy/dt + bt^-4 y = 0,
+
+  which is not regular about t = 0 unless 2 - at^-1 and bt^-2, that is,
+  unless ax and bx^2 are finite at x =[oo]; which we thus assume; putting
+  y = t^r(1 + A1t + ... ), we find for the index equation at x =
+  [inifinity] the equation r(r - 1) + r(2 - ax)_0 + (bx^2)_0 = 0. If
+  there be finite singular points at [xi]1, ... [xi]m, where we assume
+  m>1, the cases m = 0, m = 1 being easily dealt with, and if [phi](x) =
+  (x - [xi]1) ... (x -[xi]m), we must have a.[phi](x) and b.[[phi](x)]^2
+  finite for all finite values of x, equal say to the respective
+  polynomials [psi](x) and [theta](x), of which by the conditions at x =
+  [oo] the highest respective orders possible are m - 1 and 2(m - 1).
+  The index equation at x = [xi]1 is r(r - 1) +
+  r[psi]([xi]1)/[phi]'([xi]1) + [theta]([xi])1/[[phi]'([xi]1)]^2 = 0, and
+  if [alpha]1, [beta]1 be its roots, we have [alpha]1 + [beta]1 = 1 -
+  [psi]([xi]1)/[phi]'([xi]1) and [alpha]1[beta]1 =
+  [theta]([xi])1/[[phi]'([xi]1)]^2. Thus by an elementary theorem of
+  algebra, the sum [Sigma](1 - [alpha]i - [beta]i)/(x - [xi]i), extended
+  to the m finite singular points, is equal to [psi](x)/[phi](x), and
+  the sum [Sigma](1 - [alpha]i - [beta]i) is equal to the ratio of the
+  coefficients of the highest powers of x in [psi](x) and [phi](x), and
+  therefore equal to 1 + [alpha] + [beta], where [alpha], [beta] are the
+  indices at x = [oo]. Further, if (x, 1)m-2 denote the integral part of
+  the quotient [theta](x)/[phi](x), we have
+  [Sigma][alpha]_i[beta]_i[phi]'([xi]_i)/(x - [xi]_i) equal to -(x,
+  1)_m-2 + [theta](x)/[phi](x), and the coefficient of x^m-2 in (x,
+  1)_m-2 is [alpha][beta]. Thus the differential equation has the form
+
+    y" + y'[Sigma](1 - [alpha]_i - [beta]_i)/(x - [xi]_i) + y[(x, 1)_m-2 +
+         [Sigma][alpha]_i[beta]_i[phi]'([xi]_i)/(x - [xi]_i)]/[phi](x) = 0.
+
+  If, however, we make a change in the dependent variable, putting y =
+  (x - [xi]1)^[alpha]1 ... (x - [xi]_m)^[alpha] m[eta], it is easy to
+  see that the equation changes into one having the same singular points
+  about each of which it is regular, and that the indices at x = [xi]_i
+  become 0 and [beta]_i - [alpha]_i, which we shall denote by [lambda]i,
+  for (x -[xi]_i)^[alpha]j can be developed in positive integral powers
+  of x -[xi]_i about x = [xi]_i; by this transformation the indices at x
+  = [oo] are changed to
+
+    [alpha] + [alpha]1 + ... + [alpha]m, [beta] + [beta]1 + ... + [beta]m
+
+  which we shall denote by [lambda], [mu]. If we suppose this change to
+  have been introduced, and still denote the independent variable by y,
+  the equation has the form
+
+    y" + y'[Sigma](1 - [lambda]_i)/(x - [xi]_i) + y(x, 1)_m-2/[phi](x) = 0,
+
+  while [lambda] + [mu] + [lambda]1 + ... + [lambda]_m = m - 1.
+  Conversely, it is easy to verify that if [lambda][mu] be the
+  coefficient of x^m-2 in (x, 1)_m-2, this equation has the specified
+  singular points and indices whatever be the other coefficients in (x,
+  1)_m-2.
+
+
+    Hypergeometric equation.
+
+  Thus we see that (beside the cases m = 0, m = 1) the "Fuchsian
+  equation" of the second order with _two_ finite singular points is
+  distinguished by the fact that it has a definite form when the
+  singular points and the indices are assigned. In that case, putting (x
+  - [xi]1)/(x - [xi]2) = t/(t - 1), the singular points are transformed
+  to 0, 1, [oo], and, as is clear, without change of indices. Still
+  denoting the independent variable by x, the equation then has the form
+
+    x(1 - x)y" + y'[1 - [lambda]1 - x(1  + [lambda] + [mu])] - [lambda][mu]y = 0,
+
+  which is the ordinary hypergeometric equation. Provided none of
+  [lambda]1, [lambda]2, [lambda] - [mu] be zero or integral about x = 0,
+  it has the solutions
+
+    F([lambda], [mu], 1 - [lambda]1, x), x^[lambda]1 F([lambda] + [lambda]1, [mu] + [lambda]1, 1 + [lambda]1, x);
+
+  about x = 1 it has the solutions
+
+    F([lambda], [mu], 1 - [lambda]2, 1 - x), (1 - x)^[lambda]1 F([lambda] + [lambda]2, [mu] + [lambda]2, 1 + [lambda]2, 1 - x),
+
+  where [lambda] + [mu] + [lambda]1 + [lambda]2 = 1; about x = [oo] it
+  has the solutions
+
+    x^-[lambda] F([lambda], [lambda] + [lambda]1, [lambda] - [mu] + 1, x^-1),
+      x^-[mu] F([mu], [mu] + [lambda]1, [mu] - [lambda] + 1, x^-1),
+
+  where F([alpha], [beta], [gamma], x) is the series
+
+        [alpha][beta]x   [alpha]([alpha] + 1)[beta]([beta] + 1)x^2
+    1 + -------------- + ----------------------------------------- ...,
+           [gamma]               1.2.[gamma]([gamma] + 1)
+
+  which converges when |x| < 1, whatever [alpha], [beta], [gamma] may
+  be, converges for all values of x for which |x| = 1 provided the real
+  part of [gamma] - [alpha] - [beta] < 0 algebraically, and converges
+  for all these values except x = 1 provided the real part of [gamma] -
+  [alpha] -[beta] > -1 algebraically.
+
+  In accordance with our general theory, logarithms are to be expected
+  in the solution when one of [lambda]1, [lambda]2, [lambda] - [mu] is
+  zero or integral. Indeed when [lambda]1 is a negative integer, not
+  zero, the second solution about x = 0 would contain vanishing factors
+  in the denominators of its coefficients; in case [lambda] or [mu] be
+  one of the positive integers 1, 2, ... (-[lambda]1), vanishing factors
+  occur also in the numerators; and then, in fact, the second solution
+  about x = 0 becomes x^[lambda]1 times an integral polynomial of degree
+  (-[lambda]1) - [lambda] or of degree (-[lambda]1) - [mu]. But when
+  [lambda]1 is a negative integer including zero, and neither [lambda]
+  nor [mu] is one of the positive integers 1, 2 ... (-[lambda]1), the
+  second solution about x = 0 involves a term having the factor log x.
+  When [lambda]1 is a positive integer, not zero, the second solution
+  about x = 0 persists as a solution, in accordance with the order of
+  arrangement of the roots of the index equation in our theory; the
+  first solution is then replaced by an integral polynomial of degree
+  -[lambda] or -[mu]1, when [lambda] or [mu] is one of the negative
+  integers 0, -1, -2, ..., 1 - [lambda]1, but otherwise contains a
+  logarithm. Similarly for the solutions about x = 1 or x = [oo]; it
+  will be seen below how the results are deducible from those for x = 0.
+
+
+    March of the Integral.
+
+  Denote now the solutions about x = 0 by u1, u2; those about x = 1 by
+  v1, v2; and those about x = [oo] by w1, w2; in the region (S0S1)
+  common to the circles S0, S1 of radius 1 whose centres are the points
+  x = 0, x = 1, all the first four are valid, and there exist equations
+  u1 =Av1 + Bv2, u2 = Cv1 + Dv2 where A, B, C, D are constants; in the
+  region (S1S) lying inside the circle S1 and outside the circle S0,
+  those that are valid are v1, v2, w1, w2, and there exist equations v1
+  = Pw1 + Qw2, v2 = Rw1 + Tw2, where P, Q, R, T are constants; thus
+  considering any integral whose expression within the circle S0 is au1
+  + bu2, where a, b are constants, the same integral will be represented
+  within the circle S1 by (aA + bC)v1 + (aB + bD)v2, and outside these
+  circles will be represented by
+
+    [(aA + bC)P + (aB + bD)R]w1 + [(aA + bC)Q + (aB + bD)T]w2.
+
+  A single-valued branch of such integral can be obtained by making a
+  barrier in the plane joining [oo] to 0 and 1 to [oo]; for instance, by
+  excluding the consideration of real negative values of x and of real
+  positive values greater than 1, and defining the phase of x and x - 1
+  for real values between 0 and 1 as respectively 0 and [pi].
+
+
+    Transformation of the equation into itself.
+
+  We can form the Fuchsian equation of the second order with three
+  arbitrary singular points [xi]1, [xi]2, [xi]3, and no singular point
+  at x = [oo], and with respective indices [alpha]1, [beta]1, [alpha]2,
+  [beta]2, [alpha]3, [beta]3 such that [alpha]1 + [beta]1 + [alpha]2 +
+  [beta]2 + [alpha]3 + [beta]3 = 1. This equation can then be
+  transformed into the hypergeometric equation in 24 ways; for out of
+  [xi]1, [xi]2, [xi]3 we can in six ways choose two, say [xi]1, [xi]2,
+  which are to be transformed respectively into 0 and 1, by (x -
+  [xi]1)/(x - [xi]2) = t(t - 1); and then there are four possible
+  transformations of the dependent variable which will reduce one of the
+  indices at t = 0 to zero and one of the indices at t = 1 also to zero,
+  namely, we may reduce either [alpha]1 or [beta]1 at t = 0, and
+  simultaneously either [alpha]2 or [beta]2 at t = 1. Thus the
+  hypergeometric equation itself can be transformed into itself in 24
+  ways, and from the expression F([lambda], [mu], 1 - [lambda]1, x)
+  which satisfies it follow 23 other forms of solution; they involve
+  four series in each of the arguments, x, x-1, 1/x, 1/(1-x), (x-1)/x,
+  x/(x-1). Five of the 23 solutions agree with the fundamental solutions
+  already described about x = 0, x = 1, x = [oo]; and from the
+  principles by which these were obtained it is immediately clear that
+  the 24 forms are, in value, equal in fours.
+
+
+    Inversion. Modular functions.
+
+  The quarter periods K, K' of Jacobi's theory of elliptic functions, of
+  which K = [int] [0 to [pi]/2] (1 - h sin^2[theta])^-1/2 d[theta], and K'
+  is the same function of 1-h, can easily be proved to be the solutions
+  of a hypergeometric equation of which h is the independent variable.
+  When K, K' are regarded as defined in terms of h by the differential
+  equation, the ratio K'/K is an infinitely many valued function of h.
+  But it is remarkable that Jacobi's own theory of theta functions leads
+  to an expression for h in terms of K'/K (see FUNCTION) in terms of
+  single-valued functions. We may then attempt to investigate, in
+  general, in what cases the independent variable x of a hypergeometric
+  equation is a single-valued function of the ratio s of two independent
+  integrals of the equation. The same inquiry is suggested by the
+  problem of ascertaining in what cases the hypergeometric series
+  F([alpha], [beta], [gamma], x) is the expansion of an algebraic
+  (irrational) function of x. In order to explain the meaning of the
+  question, suppose that the plane of x is divided along the real axis
+  from -[oo] to 0 and from 1 to +[oo], and, supposing logarithms not to
+  enter about x = 0, choose two quite definite integrals y1, y2 of the
+  equation, say
+
+    y1 = F([lambda], [mu], 1-[lambda]1, x),
+    y2 = x^[lambda]1 F([lambda] + [lambda]1, [mu] + [lambda]1, 1 + [lambda]1, x),
+
+  with the condition that the phase of x is zero when x is real and
+  between 0 and 1. Then the value of [sigma] = y2/y1 is definite for all
+  values of x in the divided plane, [sigma] being a single-valued
+  monogenic branch of an analytical function existing and without
+  singularities all over this region. If, now, the values of [sigma]
+  that so arise be plotted on to another plane, a value p + iq of
+  [sigma] being represented by a point (p, q) of this [stigma]-plane, and
+  the value of x from which it arose being mentally associated with this
+  point of the [sigma]-plane, these points will fill a connected region
+  therein, with a continuous boundary formed of four portions
+  corresponding to the two sides of the two barriers of the x-plane. The
+  question is then, firstly, whether the same value of s can arise for
+  two different values of x, that is, whether the same point (p, q) of
+  the [sigma]-plane can arise twice, or in other words, whether the
+  region of the [sigma]-plane overlaps itself or not. Supposing this is
+  not so, a second part of the question presents itself. If in the
+  x-plane the barrier joining -[oo] to 0 be momentarily removed, and x
+  describe a small circle with centre at x = 0 starting from a point x =
+  -h - ik, where h, k are small, real, and positive and coming back to
+  this point, the original value s at this point will be changed to a
+  value [sigma], which in the original case did not arise for this value
+  of x, and possibly not at all. If, now, after restoring the barrier
+  the values arising by continuation from [sigma] be similarly plotted
+  on the s-plane, we shall again obtain a region which, while not
+  overlapping itself, may quite possibly overlap the former region. In
+  that case two values of x would arise for the same value or values of
+  the quotient y2/y1, arising from two different branches of this
+  quotient. We shall understand then, by the condition that x is to be a
+  single-valued function of x, that the region in the [stimga]-plane
+  corresponding to any branch is not to overlap itself, and that no two
+  of the regions corresponding to the different branches are to overlap.
+  Now in describing the circle about x = 0 from x = -h - ik to -h + ik,
+  where h is small and k evanescent,
+
+    [stigma] = x^[lambda]1 F([lambda] + [lambda]1, [mu] + [lambda]1, 1 + [lambda]1, x)/F([lambda], [mu], 1 - [lambda]1, x)
+
+  is changed to [sigma] = [stigma]e^(2[pi]i[lambda])1. Thus the two
+  portions of boundary of the s-region corresponding to the two sides of
+  the barrier (-[oo], 0) meet (at [sigmaf] = 0 if the real part of
+  [lambda]1 be positive) at an angle 2[pi]L1, where L1 is the absolute
+  value of the real part of [lambda]1; the same is true for the
+  [sigma]-region representing the branch [sigma]. The condition that the
+  s-region shall not overlap itself requires, then, L1 = 1. But,
+  further, we may form an infinite number of branches [sigma] =
+  [stigma]e^(2[pi]i[lambda])1, [sigma]1 = e^(2[pi]i[lambda])1, ... in
+  the same way, and the corresponding regions in the plane upon which
+  y2/y1 is represented will have a common point and each have an angle
+  2[pi]L1; if neither overlaps the preceding, it will happen, if L1 is
+  not zero, that at length one is reached overlapping the first, unless
+  for some positive integer [alpha] we have 2[pi][alpha]L1 = 2[pi], in
+  other words L1 = 1/a. If this be so, the branch [sigma]_a-1 =
+  [stigma]e^(2[pi]ia[lambda])1 will be represented by a region having
+  the angle at the common point common with the region for the branch
+  [stigma]; but not altogether coinciding with this last region unless
+  [lambda]1 be real, and therefore = [+-]1/a; then there is only a finite
+  number, a, of branches obtainable in this way by crossing the barrier
+  (-[oo], 0). In precisely the same way, if we had begun by taking the
+  quotient
+
+    [stigma]' = (x - 1)^[lambda]2 F([lambda] + [lambda]2, [mu] + [lambda]2, 1 + [lambda]2, 1 - x)/F([lambda], [mu], 1 - [lambda]2, 1 - x)
+
+  of the two solutions about x = 1, we should have found that x is not a
+  single-valued function of [stigma]' unless [lambda]2 is the inverse of
+  an integer, or is zero; as [stigma]' is of the form (A[stigma] +
+  B)/(C[stigma] + D), A, B, C, D constants, the same is true in our
+  case; equally, by considering the integrals about x = [oo] we find, as
+  a third condition necessary in order that x may be a single-valued
+  function of [stigma], that [lambda] - [mu] must be the inverse of an
+  integer or be zero. These three differences of the indices, namely,
+  [lambda]1, [lambda]2, [lambda] - [mu], are the quantities which enter
+  in the differential equation satisfied by x as a function of [stigma],
+  which is easily found to be
+
+       x111   3^2x^211
+    -  ---- + -------- = 1/2(h - h1  - h2)x^-1 (x - 1)^-1 + 1/2h1 x^-2 + 1/2h2(x - 1)^-2,
+       x1^3     2x1^4
+
+  where x1 = dx/d[stigma], &c.; and h1 = 1 - y1^2, h2 = 1 - [lambda]2^2,
+  h3 = 1 - ([lambda] - [mu])^2. Into the converse question whether the
+  three conditions are sufficient to ensure (1) that the [stigma] region
+  corresponding to any branch does not overlap itself, (2) that no two
+  such regions overlap, we have no space to enter. The second question
+  clearly requires the inquiry whether the group (that is, the monodromy
+  group) of the differential equation is properly discontinuous. (See
+  GROUPS, THEORY OF.)
+
+The foregoing account will give an idea of the nature of the function
+theories of differential equations; it appears essential not to exclude
+some explanation of a theory intimately related both to such theories
+and to transformation theories, which is a generalization of Galois's
+theory of algebraic equations. We deal only with the application to
+homogeneous linear differential equations.
+
+
+    Rationality group of a linear equation.
+
+    Irreducibility of a rational equation.
+
+  In general a function of variables x1, x2 ... is said to be rational
+  when it can be formed from them and the integers 1, 2, 3, ... by a
+  finite number of additions, subtractions, multiplications and
+  divisions. We generalize this definition. Assume that we have assigned
+  a fundamental series of quantities and functions of x, in which x
+  itself is included, such that all quantities formed by a finite number
+  of additions, subtractions, multiplications, divisions _and
+  differentiations in regard to x_, of the terms of this series, are
+  themselves members of this series. Then the quantities of this series,
+  and only these, are called _rational_. By a rational function of
+  quantities p, q, r, ... is meant a function formed from them and any
+  of the fundamental rational quantities by a finite number of the five
+  fundamental operations. Thus it is a function which would be called,
+  simply, rational if the fundamental series were widened by the
+  addition to it of the quantities p, q, r, ... and those derivable from
+  them by the five fundamental operations. A rational ordinary
+  differential equation, with x as independent and y as dependent
+  variable, is then one which equates to zero a rational function of y,
+  the order k of the differential equation being that of the highest
+  differential coefficient y^(k) which enters; only such equations are
+  here discussed. Such an equation P = 0 is called _irreducible_ when,
+  firstly, being arranged as an integral polynomial in y^(k), this
+  polynomial is not the product of other polynomials in y^(k) also of
+  rational form; and, secondly, the equation has no solution satisfying
+  also a rational equation of lower order. From this it follows that if
+  an irreducible equation P = 0 have one solution satisfying another
+  rational equation Q = 0 of the same or higher order, then all the
+  solutions of P = 0 also satisfy Q = 0. For from the equation P = 0 we
+  can by differentiation express y^(k+1), y^(k+2), ... in terms of x, y,
+  y^(1), ... , y^(k), and so put the function Q rationally in terms of
+  these quantities only. It is sufficient, then, to prove the result
+  when the equation Q = 0 is of the same order as P = 0. Let both the
+  equations be arranged as integral polynomials in y^(k); their
+  algebraic eliminant in regard to y^(k) must then vanish identically,
+  for they are known to have one common solution not satisfying an
+  equation of lower order; thus the equation P = 0 involves Q = 0 for
+  all solutions of P = 0.
+
+
+    The variant function for a linear equation.
+
+  Now let y^(n) = [alpha]1y^(n-1) + ... + [alpha]_n y be a given
+  rational homogeneous linear differential equation; let y1, ... yn be n
+  particular functions of x, unconnected by any equation with constant
+  coefficients of the form c1y1 + ... + cnyn = 0, all satisfying the
+  differential equation; let [eta]1, ... [eta]n be linear functions of
+  y1, ... yn, say [eta]i = A_i1 y1 + ... + A_in yn, where the constant
+  coefficients Aij have a non-vanishing determinant; write ([eta]) =
+  A(y), these being the equations of a general linear homogeneous group
+  whose transformations may be denoted by A, B, .... We desire to form a
+  rational function [phi]([eta]), or say [phi](A(y)), of [eta]1, ...
+  [eta], in which the [eta]^2 constants Aij shall all be essential, and
+  not reduce effectively to a fewer number, as they would, for instance,
+  if the y1, ... yn were connected by a linear equation with constant
+  coefficients. Such a function is in fact given, if the solutions y1,
+  ... yn be developable in positive integral powers about x = a, by
+  [phi]([eta]) = [eta]1 + (x - a)^n[eta]2 + ... + (x - a)^(n-1)n[eta]n.
+  Such a function, V, we call a _variant_.
+
+
+    The resolvent eqution.
+
+  Then differentiating V in regard to x, and replacing [eta]i^(n) by
+  its value a1[eta]^(n-1) + ... + an[eta], we can arrange dV/dx, and
+  similarly each of d^2/dx^2 ... d^NV/dx^N, where N = n^2, as a linear
+  function of the N quantities [eta]1, ... [eta]n, ... [eta]1^(n-1), ...
+  [eta]n^(n-1), and thence by elimination obtain a linear differential
+  equation for V of order N with rational coefficients. This we denote
+  by F = 0. Further, each of [eta]1 ... [eta]n is expressible as a
+  linear function of V, dV/dx, ... d^(N-1)V/dx^(N-1), with rational
+  coefficients not involving any of the n^2 coefficients A_ij, since
+  otherwise V would satisfy a linear equation of order less than N,
+  which is impossible, as it involves (linearly) the n^2 arbitrary
+  coefficients Aij, which would not enter into the coefficients of the
+  supposed equation. In particular, y1 ,.. yn are expressible rationally
+  as linear functions of [omega], d[omega]/dx, ...
+  d^(N-1)[omega]/dx^(N-1), where [omega] is the particular function
+  [phi](y). Any solution W of the equation F = 0 is derivable from
+  functions [zeta]1, ... [zeta]n, which are linear functions of y1, ...
+  yn, just as V was derived from [eta]1, ... [eta]n; but it does not
+  follow that these functions [zeta]i, ... [zeta]n are obtained from y1,
+  ... yn by a transformation of the linear group A, B, ... ; for it may
+  happen that the determinant d([zeta]1, ... [zeta]n)/(dy1, ... yn) is
+  zero. In that case [zeta]1, ... [zeta]n may be called a singular set,
+  and W a singular solution; it satisfies an equation of lower than the
+  N-th order. But every solution V, W, ordinary or singular, of the
+  equation F = 0, is expressible rationally in terms of [omega],
+  d[omega]/dx, ... d^(N-1)[omega]/dx^(N-1); we shall write, simply, V =
+  r([omega]). Consider now the rational irreducible equation of lowest
+  order, not necessarily a linear equation, which is satisfied by
+  [omega]; as y1, ... yn are particular functions, it may quite well be
+  of order less than N; we call it the _resolvent equation_, suppose it
+  of order p, and denote it by [gamma](v). Upon it the whole theory
+  turns. In the first place, as [gamma](v) = 0 is satisfied by the
+  solution [omega] of F = 0, all the solutions of [gamma](v) are
+  solutions F = 0, and are therefore rationally expressible by [omega];
+  any one may then be denoted by r([omega]). If this solution of F = 0
+  be not singular, it corresponds to a transformation A of the linear
+  group (A, B, ...), effected upon y1, ... yn. The coefficients Aij of
+  this transformation follow from the expressions before mentioned for
+  [eta]1 ... [eta]n in terms of V, dV/dx, d^2V/dx^2, ... by substituting V
+  = r([omega]); thus they depend on the p arbitrary parameters which
+  enter into the general expression for the integral of the equation
+  [gamma](v) = 0. Without going into further details, it is then clear
+  enough that the resolvent equation, being irreducible and such that
+  any solution is expressible rationally, with p parameters, in terms of
+  the solution [omega], enables us to define a linear homogeneous group
+  of transformations of y1 ... yn depending on p parameters; and every
+  operation of this (continuous) group corresponds to a rational
+  transformation of the solution of the resolvent equation. This is the
+  group called the _rationality group_, or the _group of
+  transformations_ of the original homogeneous linear differential
+  equation.
+
+  The group must not be confounded with a subgroup of itself, the
+  _monodromy group_ of the equation, often called simply the group of
+  the equation, which is a set of transformations, not depending on
+  arbitrary variable parameters, arising for one particular fundamental
+  set of solutions of the linear equation (see GROUPS, THEORY OF).
+
+
+    The fundamental theorem in regard to the rationality group.
+
+  The importance of the rationality group consists in three
+  propositions. (1) Any rational function of y1, ... yn which is
+  unaltered in value by the transformations of the group can be written
+  in rational form. (2) If any rational function be changed in form,
+  becoming a rational function of y1, ... yn, a transformation of the
+  group applied to its new form will leave its value unaltered. (3) Any
+  homogeneous linear transformation leaving unaltered the value of every
+  rational function of y1, ... yn which has a rational value, belongs to
+  the group. It follows from these that any group of linear homogeneous
+  transformations having the properties (1) (2) is identical with the
+  group in question. It is clear that with these properties the group
+  must be of the greatest importance in attempting to discover what
+  functions of x must be regarded as rational in order that the values
+  of y1 ... yn may be expressed. And this is the problem of solving the
+  equation from another point of view.
+
+  LITERATURE.--([alpha]) _Formal or Transformation Theories for
+  Equations of the First Order_:--E. Goursat, _Lecons sur l'integration
+  des equations aux derivees partielles du premier ordre_ (Paris, 1891);
+  E. v. Weber, _Vorlesungen uber das Pfaff'sche Problem und die Theorie
+  der partiellen Differentialgleichungen erster Ordnung_ (Leipzig,
+  1900); S. Lie und G. Scheffers, _Geometrie der
+  Beruhrungstransformationen_, Bd. i. (Leipzig, 1896); Forsyth, _Theory
+  of Differential Equations, Part i., Exact Equations and Pfaff's
+  Problem_ (Cambridge, 1890); S. Lie, "Allgemeine Untersuchungen uber
+  Differentialgleichungen, die eine continuirliche endliche Gruppe
+  gestatten" (Memoir), _Mathem. Annal._xxv. (1885), pp. 71-151; S. Lie
+  und G. Scheffers, _Vorlesungen uber Differentialgleichungen mit
+  bekannten infinitesimalen Transformationen_ (Leipzig, 1891). A very
+  full bibliography is given in the book of E. v. Weber referred to;
+  those here named are perhaps sufficiently representative of modern
+  works. Of classical works may be named: Jacobi, _Vorlesungen uber
+  Dynamik_ (von A. Clebsch, Berlin, 1866); _Werke, Supplementband_; G
+  Monge, _Application de l'analyse a la geometrie_ (par M. Liouville,
+  Paris, 1850); J. L. Lagrange, _Lecons sur le calcul des fonctions_
+  (Paris, 1806), and _Theorie des fonctions analytiques_ (Paris,
+  Prairial, an V); G. Boole, _A Treatise on Differential Equations_
+  (London, 1859); and _Supplementary Volume_ (London, 1865); Darboux,
+  _Lecons sur la theorie generale des surfaces_, tt. i.-iv. (Paris,
+  1887-1896); S. Lie, _Theorie der transformationsgruppen_ ii. (on
+  Contact Transformations) (Leipzig, 1890).
+
+  ([beta]) _Quantitative or Function Theories for Linear Equations_:--C.
+  Jordan, _Cours d'analyse_, t. iii. (Paris, 1896); E. Picard, _Traite
+  d'analyse_, tt. ii. and iii. (Paris, 1893, 1896); Fuchs, _Various
+  Memoirs, beginning with that in Crelle's Journal_, Bd. lxvi. p. 121;
+  Riemann, _Werke_, 2^r Aufl. (1892); Schlesinger, _Handbuch der Theorie
+  der linearen Differentialgleichungen_, Bde. i.-ii. (Leipzig,
+  1895-1898); Heffter, _Einleitung in die Theorie der linearen
+  Differentialgleichungen mit einer unabhangigen Variablen_ (Leipzig,
+  1894); Klein, _Vorlesungen uber lineare Differentialgleichungen der
+  zweiten Ordnung_ (Autographed, Gottingen, 1894); and _Vorlesungen uber
+  die hypergeometrische Function_ (Autographed, Gottingen, 1894);
+  Forsyth, _Theory of Differential Equations, Linear Equations_.
+
+  ([gamma]) _Rationality Group (of Linear Differential
+  Equations)_:--Picard, _Traite d'Analyse_, as above, t. iii.; Vessiot,
+  _Annales de l'Ecole Normale_, serie III. t. ix. p. 199 (Memoir); S.
+  Lie, _Transformationsgruppen_, as above, iii. A connected account is
+  given in Schlesinger, as above, Bd. ii., erstes Theil.
+
+  ([delta]) _Function Theories of Non-Linear Ordinary
+  Equations_:--Painleve, _Lecons sur la theorie analytique des equations
+  differentielles_ (Paris, 1897, Autographed); Forsyth, _Theory of
+  Differential Equations, Part ii., Ordinary Equations not Linear_ (two
+  volumes, ii. and iii.) (Cambridge, 1900); Konigsberger, _Lehrbuch der
+  Theorie der Differentialgleichungen_ (Leipzig, 1889); Painleve,
+  _Lecons sur l'integration des equations differentielles de la
+  mecanique et applications_ (Paris, 1895).
+
+  ([epsilon]) _Formal Theories of Partial Equations of the Second and
+  Higher Orders_:--E. Goursat, _Lecons sur l'integration des equations
+  aux derivees partielles du second ordre_, tt. i. and ii. (Paris, 1896,
+  1898); Forsyth, _Treatise on Differential Equations_ (London, 1889);
+  and _Phil. Trans. Roy. Soc._ (A.), vol. cxci. (1898), pp. 1-86.
+
+  ([zeta]) See also the six extensive articles in the second volume of
+  the German _Encyclopaedia of Mathematics_.     (H. F. BA.)
+
+
+
+
+DIFFLUGIA (L. Leclerc), a genus of lobose Rhizopoda, characterized by a
+shell formed of sand granules cemented together; these are swallowed by
+the animal, and during the process of bud-fission they pass to the
+surface of the daughter-bud and are cemented there. _Centropyxis_
+(Steia) and _Lecqueureuxia_ (Schlumberg) differ only in minor points.
+
+
+
+
+DIFFRACTION OF LIGHT.--1. When light proceeding from a small source
+falls upon an opaque object, a shadow is cast upon a screen situated
+behind the obstacle, and this shadow is found to be bordered by
+alternations of brightness and darkness, known as "diffraction bands."
+The phenomena thus presented were described by Grimaldi and by Newton.
+Subsequently T. Young showed that in their formation interference plays
+an important part, but the complete explanation was reserved for A. J.
+Fresnel. Later investigations by Fraunhofer, Airy and others have
+greatly widened the field, and under the head of "diffraction" are now
+usually treated all the effects dependent upon the limitation of a beam
+of light, as well as those which arise from irregularities of any kind
+at surfaces through which it is transmitted, or at which it is
+reflected.
+
+2. _Shadows._--In the infancy of the undulatory theory the objection
+most frequently urged against it was the difficulty of explaining the
+very existence of shadows. Thanks to Fresnel and his followers, this
+department of optics is now precisely the one in which the theory has
+gained its greatest triumphs. The principle employed in these
+investigations is due to C. Huygens, and may be thus formulated. If
+round the origin of waves an ideal closed surface be drawn, the whole
+action of the waves in the region beyond may be regarded as due to the
+motion continually propagated across the various elements of this
+surface. The wave motion due to any element of the surface is called a
+_secondary_ wave, and in estimating the total effect regard must be paid
+to the phases as well as the amplitudes of the components. It is usually
+convenient to choose as the surface of resolution a _wave-front_, i.e. a
+surface at which the primary vibrations are in one phase. Any obscurity
+that may hang over Huygens's principle is due mainly to the
+indefiniteness of thought and expression which we must be content to put
+up with if we wish to avoid pledging ourselves as to the character of
+the vibrations. In the application to sound, where we know what we are
+dealing with, the matter is simple enough in principle, although
+mathematical difficulties would often stand in the way of the
+calculations we might wish to make. The ideal surface of resolution may
+be there regarded as a flexible lamina; and we know that, if by forces
+locally applied every element of the lamina be made to move normally to
+itself exactly as the air at that place does, the external aerial motion
+is fully determined. By the principle of superposition the whole effect
+may be found by integration of the partial effects due to each element
+of the surface, the other elements remaining at rest.
+
+  We will now consider in detail the important case in which uniform
+  plane waves are resolved at a surface coincident with a wave-front
+  (OQ). We imagine a wave-front divided into elementary rings or
+  zones--often named after Huygens, but better after Fresnel--by spheres
+  described round P (the point at which the aggregate effect is to be
+  estimated), the first sphere, touching the plane at O, with a radius
+  equal to PO, and the succeeding spheres with radii increasing at each
+  step by 1/2[lambda]. There are thus marked out a series of circles,
+  whose radii x are given by x^2 + r^2 = (r + 1/2n[lambda])^2, or x^2 =
+  n[lambda]r nearly; so that the rings are at first of nearly equal
+  area. Now the effect upon P of each element of the plane is
+  proportional to its area; but it depends also upon the distance from
+  P, and possibly upon the inclination of the secondary ray to the
+  direction of vibration and to the wave-front.
+
+        O     x          Q
+    ---------------------------
+        |                /
+        |               /
+        |              /
+        |             /
+        |            /
+        |           /
+        |          /
+       r|         /
+        |        /
+        |       /
+        |      /
+        |     /
+        |    /
+        |   /
+        |  /
+        | /
+       P|/
+
+    FIG. 1.
+
+  The latter question can only be treated in connexion with the
+  dynamical theory (see below, S 11); but under all ordinary
+  circumstances the result is independent of the precise answer that may
+  be given. All that it is necessary to assume is that the effects of
+  the successive zones gradually diminish, whether from the increasing
+  obliquity of the secondary ray or because (on account of the
+  limitation of the region of integration) the zones become at last more
+  and more incomplete. The component vibrations at P due to the
+  successive zones are thus nearly equal in amplitude and opposite in
+  phase (the phase of each corresponding to that of the infinitesimal
+  circle midway between the boundaries), and the series which we have to
+  sum is one in which the terms are alternately opposite in sign and,
+  while at first nearly constant in numerical magnitude, gradually
+  diminish to zero. In such a series each term may be regarded as very
+  nearly indeed destroyed by the halves of its immediate neighbours, and
+  thus the sum of the whole series is represented by half the first
+  term, which stands over uncompensated. The question is thus reduced to
+  that of finding the effect of the first zone, or central circle, of
+  which the area is [pi][lambda]r.
+
+  We have seen that the problem before us is independent of the law of
+  the secondary wave as regards obliquity; but the result of the
+  integration necessarily involves the law of the intensity and phase of
+  a secondary wave as a function of r, the distance from the origin. And
+  we may in fact, as was done by A. Smith (_Camb. Math. Journ._, 1843,
+  3, p. 46), determine the law of the secondary wave, by comparing the
+  result of the integration with that obtained by supposing the primary
+  wave to pass on to P without resolution.
+
+  Now as to the phase of the secondary wave, it might appear natural to
+  suppose that it starts from any point Q with the phase of the primary
+  wave, so that on arrival at P, it is retarded by the amount
+  corresponding to QP. But a little consideration will prove that in
+  that case the series of secondary waves could not reconstitute the
+  primary wave. For the aggregate effect of the secondary waves is the
+  half of that of the first Fresnel zone, and it is the central element
+  only of that zone for which the distance to be travelled is equal to
+  r. Let us conceive the zone in question to be divided into
+  infinitesimal rings of equal area. The effects due to each of these
+  rings are equal in amplitude and of phase ranging uniformly over half
+  a complete period. The phase of the resultant is midway between those
+  of the extreme elements, that is to say, a quarter of a period behind
+  that due to the element at the centre of the circle. It is accordingly
+  necessary to suppose that the secondary waves start with a phase
+  one-quarter of a period in advance of that of the primary wave at the
+  surface of resolution.
+
+  Further, it is evident that account must be taken of the variation of
+  phase in estimating the magnitude of the effect at P of the first
+  zone. The middle element alone contributes without deduction; the
+  effect of every other must be found by introduction of a resolving
+  factor, equal to cos [theta], if [theta] represent the difference of
+  phase between this element and the resultant. Accordingly, the
+  amplitude of the resultant will be less than if all its components had
+  the same phase, in the ratio
+
+      _ +1/2[pi]
+     /
+     |      cos [theta]d[theta] : [pi],
+    _/-1/2[pi]
+
+  or 2 : [pi]. Now 2 area /[pi] = 2[lambda]r; so that, in order to
+  reconcile the amplitude of the primary wave (taken as unity) with the
+  half effect of the first zone, the amplitude, at distance r, of the
+  secondary wave emitted from the element of area dS must be taken to be
+
+    dS/[lambda]r                   (1).
+
+
+  By this expression, in conjunction with the quarter-period
+  acceleration of phase, the law of the secondary wave is determined.
+
+  That the amplitude of the secondary wave should vary as r^-1 was to be
+  expected from considerations respecting energy; but the occurrence of
+  the factor [lambda]^-1, and the acceleration of phase, have sometimes
+  been regarded as mysterious. It may be well therefore to remember that
+  precisely these laws apply to a secondary wave of sound, which can be
+  investigated upon the strictest mechanical principles.
+
+  The recomposition of the secondary waves may also be treated
+  analytically. If the primary wave at O be cos kat, the effect of the
+  secondary wave proceeding from the element dS at Q is
+
+         dS                                              dS
+    ------------- cos k(at - [rho] + 1/4[lambda]) = ------------- sin k(at - [rho]).
+    [lambda][rho]                                   [lambda][rho]
+
+
+  If dS = 2[pi]xdx, we have for the whole effect
+
+                  _[oo]
+       2[pi]     /   sin k(at - [rho])x dx
+    - --------   |   ---------------------,
+      [lambda]  _/ 0         [rho]
+
+  or, since xdx = [rho]d[rho], k = 2[pi]/[lambda],
+
+        _[oo]                       _                 _
+       /                           |                   |[oo]
+    -k | sin k(at - [rho])d[rho] = | -cos k(at - [rho])|    .
+      _/r                          |_                 _|r
+
+  In order to obtain the effect of the primary wave, as retarded by
+  traversing the distance r, viz. cos k(at - r), it is necessary to
+  suppose that the integrated term vanishes at the upper limit. And it
+  is important to notice that without some further understanding the
+  integral is really ambiguous. According to the assumed law of the
+  secondary wave, the result must actually depend upon the precise
+  radius of the outer boundary of the region of integration, supposed to
+  be exactly circular. This case is, however, at most very special and
+  exceptional. We may usually suppose that a large number of the outer
+  rings are incomplete, so that the integrated term at the upper limit
+  may properly be taken to vanish. If a formal proof be desired, it may
+  be obtained by introducing into the integral a factor such as
+  e^-h[rho], in which h is ultimately made to diminish without limit.
+
+  When the primary wave is plane, the area of the first Fresnel zone is
+  [pi][lambda]r, and, since the secondary waves vary as r^-1, the
+  intensity is independent of r, as of course it should be. If, however,
+  the primary wave be spherical, and of radius a at the wave-front of
+  resolution, then we know that at a distance r further on the amplitude
+  of the primary wave will be diminished in the ratio a:(r + a). This
+  may be regarded as a consequence of the altered area of the first
+  Fresnel zone. For, if x be its radius, we have
+
+                                    /
+    {(r + 1/2[lambda])^2 - x^2} + \/ {a^2 - x^2} = r + a,
+
+  so that
+
+    x^2 = [lambda]ar/(a + r) nearly.
+
+  Since the distance to be travelled by the secondary waves is still r,
+  we see how the effect of the first zone, and therefore of the whole
+  series is proportional to a/(a + r). In like manner may be treated
+  other cases, such as that of a primary wave-front of unequal principal
+  curvatures.
+
+  The general explanation of the formation of shadows may also be
+  conveniently based upon Fresnel's zones. If the point under
+  consideration be so far away from the geometrical shadow that a large
+  number of the earlier zones are complete, then the illumination,
+  determined sensibly by the first zone, is the same as if there were no
+  obstruction at all. If, on the other hand, the point be well immersed
+  in the geometrical shadow, the earlier zones are altogether missing,
+  and, instead of a series of terms beginning with finite numerical
+  magnitude and gradually diminishing to zero, we have now to deal with
+  one of which the terms diminish to zero _at both ends_. The sum of
+  such a series is very approximately zero, each term being neutralized
+  by the halves of its immediate neighbours, which are of the opposite
+  sign. The question of light or darkness then depends upon whether the
+  series begins or ends abruptly. With few exceptions, abruptness can
+  occur only in the presence of the first term, viz. when the secondary
+  wave of least retardation is unobstructed, or when a _ray_ passes
+  through the point under consideration. According to the undulatory
+  theory the light cannot be regarded strictly as travelling along a
+  ray; but the existence of an unobstructed ray implies that the system
+  of Fresnel's zones can be commenced, and, if a large number of these
+  zones are fully developed and do not terminate abruptly, the
+  illumination is unaffected by the neighbourhood of obstacles.
+  Intermediate cases in which a few zones only are formed belong
+  especially to the province of diffraction.
+
+  An interesting exception to the general rule that full brightness
+  requires the existence of the first zone occurs when the obstacle
+  assumes the form of a small circular disk parallel to the plane of the
+  incident waves. In the earlier half of the 18th century R. Delisle
+  found that the centre of the circular shadow was occupied by a bright
+  point of light, but the observation passed into oblivion until S. D.
+  Poisson brought forward as an objection to Fresnel's theory that it
+  required at the centre of a circular shadow a point as bright as if no
+  obstacle were intervening. If we conceive the primary wave to be
+  broken up at the plane of the disk, a system of Fresnel's zones can be
+  constructed which begin from the circumference; and the first zone
+  external to the disk plays the part ordinarily taken by the centre of
+  the entire system. The whole effect is the half of that of the first
+  existing zone, and this is sensibly the same as if there were no
+  obstruction.
+
+  When light passes through a small circular or annular aperture, the
+  illumination at any point along the axis depends upon the precise
+  relation between the aperture and the distance from it at which the
+  point is taken. If, as in the last paragraph, we imagine a system of
+  zones to be drawn commencing from the inner circular boundary of the
+  aperture, the question turns upon the manner in which the series
+  terminates at the outer boundary. If the aperture be such as to fit
+  exactly an integral number of zones, the aggregate effect may be
+  regarded as the half of those due to the first and last zones. If the
+  number of zones be even, the action of the first and last zones are
+  antagonistic, and there is complete darkness at the point. If on the
+  other hand the number of zones be odd, the effects conspire; and the
+  illumination (proportional to the square of the amplitude) is four
+  times as great as if there were no obstruction at all.
+
+  The process of augmenting the resultant illumination at a particular
+  point by stopping some of the secondary rays may be carried much
+  further (Soret, _Pogg. Ann._, 1875, 156, p. 99). By the aid of
+  photography it is easy to prepare a plate, transparent where the zones
+  of odd order fall, and opaque where those of even order fall. Such a
+  plate has the power of a condensing lens, and gives an illumination
+  out of all proportion to what could be obtained without it. An even
+  greater effect (fourfold) can be attained by providing that the
+  stoppage of the light from the alternate zones is replaced by a
+  phase-reversal without loss of amplitude. R. W. Wood (_Phil. Mag._,
+  1898, 45, p 513) has succeeded in constructing zone plates upon this
+  principle.
+
+  In such experiments the narrowness of the zones renders necessary a
+  pretty close approximation to the geometrical conditions. Thus in the
+  case of the circular disk, equidistant (r) from the source of light
+  and from the screen upon which the shadow is observed, the width of
+  the first exterior zone is given by
+
+    dx = [lambda](2r)/4(2x),
+
+  2x being the diameter of the disk. If 2r = 1000 cm., 2x = 1 cm.,
+  [lambda] = 6 X 10^-5 cm., then dx = .0015 cm. Hence, in order that
+  this zone may be perfectly formed, there should be no error in the
+  circumference of the order of .001 cm. (It is easy to see that the
+  radius of the bright spot is of the same order of magnitude.) The
+  experiment succeeds in a dark room of the length above mentioned, with
+  a threepenny bit (supported by three threads) as obstacle, the origin
+  of light being a small needle hole in a plate of tin, through which
+  the sun's rays shine horizontally after reflection from an external
+  mirror. In the absence of a heliostat it is more convenient to obtain
+  a point of light with the aid of a lens of short focus.
+
+  The amplitude of the light at any point in the axis, when plane waves
+  are incident perpendicularly upon an annular aperture, is, as above,
+
+    cos k(at - r1) - cos k(at - r2) = 2 sin kat sin k(r1 - r2),
+
+  r2, r1 being the distances of the outer and inner boundaries from the
+  point in question. It is scarcely necessary to remark that in all such
+  cases the calculation applies in the first instance to homogeneous
+  light, and that, in accordance with Fourier's theorem, each
+  homogeneous component of a mixture may be treated separately. When the
+  original light is white, the presence of some components and the
+  absence of others will usually give rise to coloured effects, variable
+  with the precise circumstances of the case.
+
+  [Illustration: FIG. 2.]
+
+  Although the matter can be fully treated only upon the basis of a
+  dynamical theory, it is proper to point out at once that there is an
+  element of assumption in the application of Huygens's principle to the
+  calculation of the effects produced by opaque screens of limited
+  extent. Properly applied, the principle could not fail; but, as may
+  readily be proved in the case of sonorous waves, it is not in
+  strictness sufficient to assume the expression for a secondary wave
+  suitable when the primary wave is undisturbed, with mere limitation of
+  the integration to the transparent parts of the screen. But, except
+  perhaps in the case of very fine gratings, it is probable that the
+  error thus caused is insignificant; for the incorrect estimation of
+  the secondary waves will be limited to distances of a few wave-lengths
+  only from the boundary of opaque and transparent parts.
+
+3. _Fraunhofer's Diffraction Phenomena._--A very general problem in
+diffraction is the investigation of the distribution of light over a
+screen upon which impinge divergent or convergent spherical waves after
+passage through various diffracting apertures. When the waves are
+convergent and the recipient screen is placed so as to contain the
+centre of convergency--the image of the original radiant point, the
+calculation assumes a less complicated form. This class of phenomena was
+investigated by J. von Fraunhofer (upon principles laid down by
+Fresnel), and are sometimes called after his name. We may conveniently
+commence with them on account of their simplicity and great importance
+in respect to the theory of optical instruments.
+
+  If f be the radius of the spherical wave at the place of resolution,
+  where the vibration is represented by cos kat, then at any point M
+  (fig. 2) in the recipient screen the vibration due to an element dS of
+  the wave-front is (S 2)
+
+           dS
+    - ------------- sin k(at - [rho]),
+      [lambda][rho]
+
+  [rho] being the distance between M and the element dS.
+
+  Taking co-ordinates in the plane of the screen with the centre of the
+  wave as origin, let us represent M by [xi], [eta], and P (where dS is
+  situated) by x, y, z. Then
+
+    [rho]^2 = (x - [xi])^2 + (y - [eta])^2 + z^2, f^2 = x^2 + y^2 + z^2;
+
+  so that
+
+    [rho]^2 = f^2 - 2x[xi] - 2y[eta] + [xi]^2 + [eta]^2.
+
+  In the applications with which we are concerned, [xi], [eta] are very
+  small quantities; and we may take
+
+               /    x[xi] + y[eta]\
+    [rho] = f ( 1 - -------------- ).
+               \          f^2     /
+
+  At the same time dS may be identified with dxdy, and in the
+  denominator [rho] may be treated as constant and equal to f. Thus the
+  expression for the vibration at M becomes
+
+                        _ _
+             1         / /        /         x[xi] + y[eta]\
+    - ---------------  | | sin k ( at - f + -------------- ) dxdy    (1);
+      [lambda]^2[f]^2 _/_/        \               f       /
+
+  and for the intensity, represented by the square of the amplitude,
+
+                         _   _ _                         _
+               1        |   / /       x[xi] + y[eta]      |^2
+    I^2 = ------------- |   | | sin k -------------- dxdy |
+          [lambda]^2f^2 |_ _/_/             f            _|
+                        _   _ _                          _
+               1       |   / /        x[xi] + y[eta]      |^2
+       + ------------- |   | |  cos k -------------- dxdy |         (2).
+         [lambda]^2f^2 |_ _/_/              f            _|
+
+  This expression for the intensity becomes rigorously applicable when f
+  is indefinitely great, so that ordinary optical aberration disappears.
+  The incident waves are thus plane, and are limited to a plane aperture
+  coincident with a wave-front. The integrals are then properly
+  functions of the _direction_ in which the light is to be estimated.
+
+  In experiment under ordinary circumstances it makes no difference
+  whether the collecting lens is in front of or behind the diffracting
+  aperture. It is usually most convenient to employ a telescope focused
+  upon the radiant point, and to place the diffracting apertures
+  immediately in front of the object-glass. What is seen through the
+  eye-piece in any case is the same as would be depicted upon a screen
+  in the focal plane.
+
+  Before proceeding to special cases it may be well to call attention to
+  some general properties of the solution expressed by (2) (see Bridge,
+  _Phil. Mag._, 1858).
+
+  If when the aperture is given, the wave-length (proportional to k^-1)
+  varies, the composition of the integrals is unaltered, provided [xi]
+  and [eta] are taken universely proportional to [lambda]. A diminution
+  of [lambda] thus leads to a simple proportional shrinkage of the
+  diffraction pattern, attended by an augmentation of brilliancy in
+  proportion to [lambda]^-2.
+
+  If the wave-length remains unchanged, similar effects are produced by
+  an increase in the scale of the aperture. The linear dimension of the
+  diffraction pattern is inversely as that of the aperture, and the
+  brightness at corresponding points is as the _square_ of the area of
+  aperture.
+
+  If the aperture and wave-length increase in the same proportion, the
+  size and shape of the diffraction pattern undergo no change.
+
+  We will now apply the integrals (2) to the case of a rectangular
+  aperture of width a parallel to x and of width b parallel to y. The
+  limits of integration for x may thus be taken to be -1/2a and +1/2a,
+  and for y to be -1/2b, +1/2b. We readily find (with substitution for k
+  of 2[pi]/[lambda])
+
+                              [pi]a[xi]           [pi]b[eta]
+                        sin^2 ---------     sin^2 ----------
+             a^2b^2           f[lambda]            f[lambda]
+    I^2 = ------------ . ----------------- . -----------------   (3),
+         f^2[lambda]^2    [pi]^2a^2[xi]^2     [pi]^2b^2[eta]^2
+                          --------------        -------------
+                          f^2[lambda]^2         f^2[lambda]^2
+
+  as representing the distribution of light in the image of a
+  mathematical point when the aperture is rectangular, as is often the
+  case in spectroscopes.
+
+  The second and third factors of (3) being each of the form sin^2u/u^2,
+  we have to examine the character of this function. It vanishes when u
+  = m[pi], m being any whole number other than zero. When u = 0, it
+  takes the value unity. The maxima occur when
+
+    u = tan u,                                                 (4),
+
+  and then
+
+    sin^2u/u^2 = cos^2u                                           (5).
+
+  To calculate the roots of (5) we may assume
+
+    u = (m + 1/2)[pi] - y = U - y,
+
+
+
+  where y is a positive quantity which is small when u is large.
+  Substituting this, we find cot y = U - y, whence
+
+        1  /    y   y-       \    y^3 2y^5   17y^7
+    y = - ( 1 + - + --- + ... ) - --- ---- - -----.
+        U  \    U   U^2      /     3   15     315
+
+  This equation is to be solved by successive approximation. It will
+  readily be found that
+
+                           2         13        146
+    u = U - y = U - U^-1 - -- U^-3 - -- U^-5 - --- U^-7 - ...   (6).
+                           3         15        105
+
+  In the first quadrant there is no root after zero, since tan u > u,
+  and in the second quadrant there is none because the signs of u and
+  tan u are opposite. The first root after zero is thus in the third
+  quadrant, corresponding to m = 1. Even in this case the series
+  converges sufficiently to give the value of the root with considerable
+  accuracy, while for higher values of m it is all that could be
+  desired. The actual values of u/[pi] (calculated in another manner by
+  F. M. Schwerd) are 1.4303, 2.4590, 3.4709, 4.4747, 5.4818, 6.4844, &c.
+
+  Since the maxima occur when u = (m + 1/2)[pi] nearly, the successive
+  values are not very different from
+
+       4        4       4
+    -------, ------, --------, &c.
+    9[pi]^2  25[pi]  49[pi]^2
+
+  The application of these results to (3) shows that the field is
+  brightest at the centre [xi] = 0, [eta] = 0, viz. at the geometrical
+  image of the radiant point. It is traversed by dark lines whose
+  equations are
+
+    [xi] = mf[lambda]/a, [eta] = mf[lambda]/b.
+
+  Within the rectangle formed by pairs of consecutive dark lines, and
+  not far from its centre, the brightness rises to a maximum; but these
+  subsequent maxima are in all cases much inferior to the brightness at
+  the centre of the entire pattern ([xi] = 0, [eta] = 0).
+
+  By the principle of energy the illumination over the entire focal
+  plane must be equal to that over the diffracting area; and thus, in
+  accordance with the suppositions by which (3) was obtained, its value
+  when integrated from [xi] = [oo] to [xi] = +[oo], and from [eta] =
+  -[oo] to [eta] = +[oo] should be equal to ab. This integration,
+  employed originally by P. Kelland (_Edin. Trans._ 15, p. 315) to
+  determine the absolute intensity of a secondary wave, may be at once
+  effected by means of the known formula
+
+      _+[oo]        _+[oo]
+     / sin^2u      / sin u
+     | ----- du =  | ----- du = [pi].
+    _/   u^2      _/   u
+     -[oo]         -[oo]
+
+  It will be observed that, while the total intensity is proportional to
+  ab, the intensity at the focal point is proportional to a^2b^2. If the
+  aperture be increased, not only is the total brightness over the focal
+  plane increased with it, but there is also a concentration of the
+  diffraction pattern. The form of (3) shows immediately that, if a and
+  b be altered, the co-ordinates of any characteristic point in the
+  pattern vary as a^-1 and b^-1.
+
+  The contraction of the diffraction pattern with increase of aperture
+  is of fundamental importance in connexion with the resolving power of
+  optical instruments. According to common optics, where images are
+  absolute, the diffraction pattern is supposed to be infinitely small,
+  and two radiant points, however near together, form separated images.
+  This is tantamount to an assumption that [lambda] is infinitely small.
+  The actual finiteness of [lambda] imposes a limit upon the separating
+  or resolving power of an optical instrument.
+
+  This indefiniteness of images is sometimes said to be due to
+  diffraction by the edge of the aperture, and proposals have even been
+  made for curing it by causing the transition between the interrupted
+  and transmitted parts of the primary wave to be less abrupt. Such a
+  view of the matter is altogether misleading. What requires explanation
+  is not the imperfection of actual images so much as the possibility of
+  their being as good as we find them.
+
+  At the focal point ([xi] = 0, [eta] = 0) all the secondary waves agree
+  in phase, and the intensity is easily expressed, whatever be the form
+  of the aperture. From the general formula (2), if A be the _area_ of
+  aperture,
+
+    I0^2 = A^2/[lambda]^2f^2           (7).
+
+  The formation of a sharp image of the radiant point requires that the
+  illumination become insignificant when [xi], [eta] attain small
+  values, and this insignificance can only arise as a consequence of
+  discrepancies of phase among the secondary waves from various parts of
+  the aperture. So long as there is no sensible discrepancy of phase
+  there can be no sensible diminution of brightness as compared with
+  that to be found at the focal point itself. We may go further, and lay
+  it down that there can be no considerable loss of brightness until the
+  difference of phase of the waves proceeding from the nearest and
+  farthest parts of the aperture amounts to 1/4[lambda].
+
+  When the difference of phase amounts to [lambda], we may expect the
+  resultant illumination to be very much reduced. In the particular case
+  of a rectangular aperture the course of things can be readily
+  followed, especially if we conceive f to be infinite. In the direction
+  (suppose horizontal) for which [eta] = 0, [xi]/f = sin [theta], the
+  phases of the secondary waves range over a complete period when sin
+  [theta] = [lambda]/a, and, since all parts of the horizontal aperture
+  are equally effective, there is in this direction a complete
+  compensation and consequent absence of illumination. When sin [theta]
+  = 3/2[lambda]/a, the phases range one and a half periods, and there
+  is revival of illumination. We may compare the brightness with that in
+  the direction [theta] = 0. The phase of the resultant amplitude is the
+  same as that due to the central secondary wave, and the discrepancies
+  of phase among the components reduce the amplitude in the proportion
+
+            _+3/2[pi]
+      1    /
+    -----  | cos [phi] d[phi]: 1,
+    3[pi] _/-3/2[pi]
+
+  or -2/3[pi]:1; so that the brightness in this direction is 4/9[pi]^2 of
+  the maximum at [theta] = 0. In like manner we may find the
+  illumination in any other direction, and it is obvious that it
+  vanishes when sin [theta] is any multiple of [lamba]/a.
+
+  The reason of the augmentation of resolving power with aperture will
+  now be evident. The larger the aperture the smaller are the angles
+  through which it is necessary to deviate from the principal direction
+  in order to bring in specified discrepancies of phase--the more
+  concentrated is the image.
+
+  In many cases the subject of examination is a luminous line of uniform
+  intensity, the various points of which are to be treated as
+  independent sources of light. If the image of the line be [xi] = 0,
+  the intensity at any point [xi], [eta] of the diffraction pattern may
+  be represented by
+
+                                   [pi]a[xi]
+      _+[oo]                  sin^2---------
+     /               a^2b          [lambda]f
+     | I^2d[eta] = ---------   -------------     (8),
+    _/             [lambda]f  [pi]^2a^2[xi]^2
+     -[oo]                    ---------------
+                               [lambda]^2f^2
+
+  the same law as obtains for a luminous point when horizontal
+  directions are alone considered. The definition of a fine vertical
+  line, and consequently the resolving power for contiguous vertical
+  lines, is thus _independent of the vertical aperture of the
+  instrument_, a law of great importance in the theory of the
+  spectroscope.
+
+  The distribution of illumination in the image of a luminous line is
+  shown by the curve ABC (fig. 3), representing the value of the
+  function sin^2u/u^2 from u = 0 to u = 2[pi]. The part corresponding to
+  negative values of u is similar, OA being a line of symmetry.
+
+  [Illustration: Fig. 3.]
+
+  Let us now consider the distribution of brightness in the image of a
+  double line whose components are of equal strength, and at such an
+  angular interval that the central line in the image of one coincides
+  with the first zero of brightness in the image of the other. In fig. 3
+  the curve of brightness for one component is ABC, and for the other
+  OA'C'; and the curve representing half the combined brightnesses is
+  E'BE. The brightness (corresponding to B) midway between the two
+  central points AA' is .8106 of the brightness at the central points
+  themselves. We may consider this to be about the limit of closeness at
+  which there could be any decided appearance of resolution, though
+  doubtless an observer accustomed to his instrument would recognize the
+  duplicity with certainty. The obliquity, corresponding to u = [pi], is
+  such that the phases of the secondary waves range over a complete
+  period, i.e. such that the projection of the horizontal aperture upon
+  this direction is one wave-length. We conclude that a _double line
+  cannot be fairly resolved unless its components subtend an angle
+  exceeding that subtended by the wave-length of light at a distance
+  equal to the horizontal aperture_. This rule is convenient on account
+  of its simplicity; and it is sufficiently accurate in view of the
+  necessary uncertainty as to what exactly is meant by resolution.
+
+  If the angular interval between the components of a double line be
+  half as great again as that supposed in the figure, the brightness
+  midway between is .1802 as against 1.0450 at the central lines of each
+  image. Such a falling off in the middle must be more than sufficient
+  for resolution. If the angle subtended by the components of a double
+  line be twice that subtended by the wave-length at a distance equal to
+  the horizontal aperture, the central bands are just clear of one
+  another, and there is a line of absolute blackness in the middle of
+  the combined images.
+
+  The resolving power of a telescope with circular or rectangular
+  aperture is easily investigated experimentally. The best object for
+  examination is a grating of fine wires, about fifty to the inch,
+  backed by a sodium flame. The object-glass is provided with diaphragms
+  pierced with round holes or slits. One of these, of width equal, say,
+  to one-tenth of an inch, is inserted in front of the object-glass, and
+  the telescope, carefully focused all the while, is drawn gradually
+  back from the grating until the lines are no longer seen. From a
+  measurement of the maximum distance the least angle between
+  consecutive lines consistent with resolution may be deduced, and a
+  comparison made with the rule stated above.
+
+  Merely to show the dependence of resolving power on aperture it is not
+  necessary to use a telescope at all. It is sufficient to look at wire
+  gauze backed by the sky or by a flame, through a piece of blackened
+  cardboard, pierced by a needle and held close to the eye. By varying
+  the distance the point is easily found at which resolution ceases; and
+  the observation is as sharp as with a telescope. The function of the
+  telescope is in fact to allow the use of a wider, and therefore more
+  easily measurable, aperture. An interesting modification of the
+  experiment may be made by using light of various wave-lengths.
+
+  Since the limitation of the width of the central band in the image of
+  a luminous line depends upon discrepancies of phase among the
+  secondary waves, and since the discrepancy is greatest for the waves
+  which come from the edges of the aperture, the question arises how far
+  the operation of the central parts of the aperture is advantageous. If
+  we imagine the aperture reduced to two equal narrow slits bordering
+  its edges, compensation will evidently be complete when the projection
+  on an oblique direction is equal to 1/2[lambda], instead of [lambda] as
+  for the complete aperture. By this procedure the width of the central
+  band in the diffraction pattern is halved, and so far an advantage is
+  attained. But, as will be evident, the bright bands bordering the
+  central band are now not inferior to it in brightness; in fact, a band
+  similar to the central band is reproduced an indefinite number of
+  times, so long as there is no sensible discrepancy of phase in the
+  secondary waves proceeding from the various parts of the _same_ slit.
+  Under these circumstances the narrowing of the band is paid for at a
+  ruinous price, and the arrangement must be condemned altogether.
+
+  A more moderate suppression of the central parts is, however,
+  sometimes advantageous. Theory and experiment alike prove that a
+  double line, of which the components are equally strong, is better
+  resolved when, for example, one-sixth of the horizontal aperture is
+  blocked off by a central screen; or the rays quite at the centre may
+  be allowed to pass, while others a little farther removed are blocked
+  off. Stops, each occupying one-eighth of the width, and with centres
+  situated at the points of trisection, answer well the required
+  purpose.
+
+  It has already been suggested that the principle of energy requires
+  that the general expression for I^2 in (2) when integrated over the
+  whole of the plane [xi], [eta] should be equal to A, where A is the
+  area of the aperture. A general analytical verification has been given
+  by Sir G. G. Stokes (_Edin. Trans._, 1853, 20, p. 317). Analytically
+  expressed--
+
+      _ _+[oo]              _ _
+     / /                   / /
+     | | I^2 d[xi]d[eta] = | | dxdy  = A        (9).
+    _/_/-[oo]             _/_/
+
+  We have seen that I0^2 (the intensity at the focal point) was equal to
+  A^2/[lambda]^2f^2. If A' be the area over which the intensity must be I0^2
+  in order to give the actual total intensity in accordance with
+
+              _ _+[oo]
+             / /
+    A'I0^2 = | |  I^2 d[xi]d[eta],
+            _/_/-[oo]
+
+  the relation between A and A' is AA' = [lambda]^2f^2. Since A' is in
+  some sense the area of the diffraction pattern, it may be considered
+  to be a rough criterion of the definition, and we infer that the
+  definition of a point depends principally upon the area of the
+  aperture, and only in a very secondary degree upon the shape when the
+  area is maintained constant.
+
+4. _Theory of Circular Aperture._--We will now consider the important
+case where the form of the aperture is circular.
+
+  Writing for brevity
+
+    k[xi]/f = p, k[eta]/f = q,        (1),
+
+  we have for the general expression (S 11) of the intensity
+
+    [lambda]^2f^2I^2 = S^2 + C^2      (2),
+
+  where
+          _ _
+         / /
+    S =  | | sin(px + qy)dx dy,       (3),
+        _/_/
+          _ _
+         / /
+    C =  | | cos(px + qy)dx dy,       (4).
+        _/_/
+
+  When, as in the application to rectangular or circular apertures, the
+  form is symmetrical with respect to the axes both of x and y, S = 0,
+  and C reduces to
+          _ _
+         / /
+    C =  | | cos px cos qy dx dy,     (5).
+        _/_/
+
+  In the case of the circular aperture the distribution of light is of
+  course symmetrical with respect to the focal point p = 0, q = 0; and C
+  is a function of p and q only through [sqrt](p^2 + q^2). It is thus
+  sufficient to determine the intensity along the axis of p. Putting q =
+  0, we get
+          _ _                  _+R
+         / /                  /          /
+    C =  | | cos px dx dy = 2 | cos px \/(R^2 - x^2) dx,
+        _/_/                 _/-R
+
+  R being the radius of the aperture. This integral is the Bessel's
+  function of order unity, defined by
+
+                   _[pi]
+             z    /
+    J1(z) = ----  | cos(z cos [phi]) sin^2 [phi] d[phi]    (6).
+            [pi] _/0
+
+  Thus, if x = R cos [phi],
+
+               2J1(pR)
+    C = [pi]^2R -------                                    (7);
+                 pR
+
+  and the illumination at distance r from the focal point is
+
+                             / 2[pi]Rr \
+                       4J1^2( --------- )
+          [pi]^2R^4          \f[lambda]/
+    I^2 = ----------- . -----------------                  (8).
+         [lambda]^2f^2    / 2[pi]Rr \^2
+                         ( --------- )
+                          \f[lambda]/
+
+  The ascending series for J1(z), used by Sir G. B. Airy (_Camb.
+  Trans._, 1834) in his original investigation of the diffraction of a
+  circular object-glass, and readily obtained from (6), is
+
+            z    z^3       z^5           z^7
+    J1(z) = - - ----- + --------- - ------------- + ...         (9).
+            2   2^2.4   2^2.4^2.6   2^2.4^2.6^2.8
+
+  When z is great, we may employ the semi-convergent series
+                                            _
+              / /  2  \                    |    3.5.1   /1\^2
+    J1(z) =  / ( ----- ) sin (z - 1/4[pi]) |1 + ------ ( - )
+           \/   \[pi]z/                    |_    8.16   \z/
+                                          _
+               3.5.7.9.1.3.5  /1\^4        |
+             - ------------- ( - )   + ... |
+                 8.16.24.32   \z/         _|
+                                         _
+           / /  2  \                    | 3   1   3.5.7.1.3  /1\ ^3
+       +  / ( ----- ) cos (z - 1/4[pi]) | - . - - --------- ( - )
+        \/   \[pi]z/                    |_8   z    8.16.24   \z/
+                                         _
+         3.5.7.9.11.1.3.5.7  /1\^5        |
+       + ------------------ ( - )   - ... | ...          (10).
+           8.16.24.32.40     \z/         _|
+
+  A table of the values of 2z^-1J1(z) has been given by E. C. J. Lommel
+  (_Schlomilch_, 1870, 15, p. 166), to whom is due the first systematic
+  application of Bessel's functions to the diffraction integrals.
+
+  The illumination vanishes in correspondence with the roots of the
+  equation J1(z) = 0. If these be called z1 z2, z3, ... the radii of the
+  dark rings in the diffraction pattern are
+
+    f[lambda]z1  f[lambda]z2
+    -----------, -----------, ...
+      2[pi]R       2[pi]R
+
+  being thus _inversely_ proportional to R.
+
+  The integrations may also be effected by means of polar co-ordinates,
+  taking first the integration with respect to [phi] so as to obtain the
+  result for an infinitely thin annular aperture. Thus, if
+
+    x = [rho] cos [phi], y = [rho] sin [phi],
+
+         _ _                _R  _2[pi]
+        / /                /   /
+    C = | | cos px dx dy = |   |   cos (p[rho] cos [theta]) [rho]d[rho] d[theta].
+       _/_/               _/0 _/0
+
+  Now by definition
+
+                   _1/2[pi]
+              2   /                              z^2     z^4         z^6
+    J0(z) = ----  |  cos(z cos[theta])d[theta] = --- + ------- - ----------- + ...  (11).
+            [pi] _/0                             2^2   2^2.4^2   2^2.4^2.6^2
+
+  The value of C for an annular aperture of radius r and width dr is thus
+
+    dC = 2 [pi]J0 (p[rho]) [rho] d[rho],                                       (12).
+
+  For the complete circle,
+
+               _ pR
+        2[pi] /
+    C = ----- | J0(z) zdz
+         p^2 _/0
+
+         2[pi]  /p^2R^2   p^4 R^4     p^6 R^6        \
+      = ------ ( ------ - ------- + ----------- - ... )
+          p^2   \  2      2^2.4^2   2^2.4^2.6^2      /
+
+                  2J1(pR)
+      = [pi]R^2 . ------- as before.
+                    pR
+
+  In these expressions we are to replace p by k[xi]/f, or rather, since
+  the diffraction pattern is symmetrical, by kr/f, where r is the
+  distance of any point in the focal plane from the centre of the
+  system.
+
+  The roots of J0(z) after the first may be found from
+
+     z               .050561    .053041      .262051
+    ---- = i - .25 + ------- - ---------- + ---------- ...  (13),
+    [pi]             4i - 1    (4i - 1)^3   (4i - 1)^5
+
+  and those of J1(z) from
+
+     z               .151982    .015399      .245835
+    ---- = i + .25 - ------- + ---------- + ---------- ...  (14),
+    [pi]             4i + 1    (4i + 1)^3   (4i + 1)^5
+
+  formulae derived by Stokes (_Camb. Trans._, 1850, vol. ix.) from the
+  descending series.[1] The following table gives the actual values:--
+
+    +---+--------------------+--------------------+
+    |   |   z                |  z                 |
+    | i | ---- for J0(z) = 0 | ---- for J1(z) = 0 |
+    |   | [pi]               | [pi]               |
+    +---+--------------------+--------------------+
+    | 1 |        7655        |       1 2197       |
+    | 2 |      1 7571        |       2 2330       |
+    | 3 |      2 7546        |       3 2383       |
+    | 4 |      3 7534        |       4 2411       |
+    | 5 |      4 7527        |       5 2428       |
+    | 6 |      5 7522        |       6 2439       |
+    | 7 |      6 7519        |       7 2448       |
+    | 8 |      7 7516        |       8 2454       |
+    | 9 |      8 7514        |       9 2459       |
+    |10 |      9 7513        |      10 2463       |
+    +---+--------------------+--------------------+
+
+  In both cases the image of a mathematical point is thus a symmetrical
+  ring system. The greatest brightness is at the centre, where
+
+    dC = 2[pi][rho] d[rho], C = [pi]R^2.
+
+  For a certain distance outwards this remains sensibly unimpaired and
+  then gradually diminishes to zero, as the secondary waves become
+  discrepant in phase. The subsequent revivals of brightness forming the
+  bright rings are necessarily of inferior brilliancy as compared with
+  the central disk.
+
+  The first dark ring in the diffraction pattern of the complete
+  circular aperture occurs when
+
+    r/f = 1.2197 X [lambda]/2R                  (15).
+
+  We may compare this with the corresponding result for a rectangular
+  aperture of width a,
+
+    [xi]/f =[lambda]/a;
+
+  and it appears that in consequence of the preponderance of the central
+  parts, the compensation in the case of the circle does not set in at
+  so small an obliquity as when the circle is replaced by a rectangular
+  aperture, whose side is equal to the diameter of the circle.
+
+  Again, if we compare the complete circle with a narrow annular
+  aperture of the same radius, we see that in the latter case the first
+  dark ring occurs at a much smaller obliquity, viz.
+
+    r/f = .7655 X [lambda]/2R.
+
+  It has been found by Sir William Herschel and others that the
+  definition of a telescope is often improved by stopping off a part of
+  the central area of the object-glass; but the advantage to be obtained
+  in this way is in no case great, and anything like a reduction of the
+  aperture to a narrow annulus is attended by a development of the
+  external luminous rings sufficient to outweigh any improvement due to
+  the diminished diameter of the central area.[2]
+
+  The maximum brightnesses and the places at which they occur are easily
+  determined with the aid of certain properties of the Bessel's
+  functions. It is known (see SPHERICAL HARMONICS) that
+
+    J0'(z) = -J1(z),                    (16);
+
+            1
+    J2(z) = - J1(z) - J1'(z)            (17);
+            z
+
+                    2
+    J0(z) + J2(z) = - J1(z)             (18).
+                    z
+
+  The maxima of C occur when
+
+     d  /J1(z)\    J1'(z)   J1(z)
+    -- (-------) = ------ - ----- = 0;
+    dz  \  z  /      z       z^2
+
+  or by 17 when J2(z) = 0. When z has one of the values thus determined,
+
+    2
+    - J1(z) = J0(z).
+    z
+
+  The accompanying table is given by Lommel, in which the first column
+  gives the roots of J2(z) = 0, and the second and third columns the
+  corresponding values of the functions specified. If appears that the
+  maximum brightness in the first ring is only about 1/57 of the
+  brightness at the centre.
+
+    +-------------------------------------------+
+    |       z       2z^-1 J1(z)   4z^-2 J1^2(z) |
+    +-------------------------------------------+
+    |                                           |
+    |    .000000     +1.000000      1.000000    |
+    |   5.135630     - .132279       .017498    |
+    |   8.417236     + .064482       .004158    |
+    |  11.619857     - .040008       .001601    |
+    |  14.795938     + .027919       .000779    |
+    |  17.959820     - .020905       .000437    |
+    +-------------------------------------------+
+
+  We will now investigate the total illumination distributed over the
+  area of the circle of radius r. We have
+
+            [pi]^2R^4     4J1^2(z)
+    I^2 = ------------- . -------                (19),
+          [lambda]^2f^2     z^2
+
+  where
+
+    z = 2[pi]Rr/[lambda]f                      (20).
+
+  Thus
+           _                        _                    _
+          /          [lambda]^2f^2 /                    /
+    2[pi] | I^2rdr = ------------- | I^2zdz = [pi]R^2.2 | z^-1 J1^2(z)dz.
+         _/             2[pi]R^2  _/                   _/
+
+  Now by (17), (18)
+
+    z^-1 J1(z) = J0(z) - J1'(z);
+
+  so that
+
+                      d             d
+    z^-1J1^2(z) = 1/2 -- J0^2 - 1/2 -- J1^2(z),
+                      dz            dz
+
+  and
+
+       _z
+      /
+    2 | z^-1 J1^2(z)dz = 1 - J0^2(z) - J1^2(z)    (21).
+     _/0
+
+  If r, or z, be infinite, J0(z), J1(z) vanish, and the whole
+  illumination is expressed by [pi]R^2, in accordance with the general
+  principle. In any case the proportion of the whole illumination to be
+  found outside the circle of radius r is given by
+
+    J0^2(z) + J1^2(z).
+
+  For the dark rings J1(z) = 0; so that the fraction of illumination
+  outside any dark ring is simply J0^2(z). Thus for the first, second,
+  third and fourth dark rings we get respectively .161, .090, .062,
+  .047, showing that more than 9/10ths of the whole light is
+  concentrated within the area of the second dark ring (_Phil. Mag._,
+  1881).
+
+  When z is great, the descending series (10) gives
+
+    2J1(z)   2    / /  2  \
+    ------ = -   / ( ----- ) sin(z - 1/4[pi])    (22);
+      z      z \/   \[pi]z/
+
+  so that the places of maxima and minima occur at equal intervals.
+
+  The mean brightness varies as z^-3 (or as r^-3), and the integral
+  found by multiplying it by zdz and integrating between 0 and [oo]
+  converges.
+
+  It may be instructive to contrast this with the case of an infinitely
+  narrow annular aperture, where the brightness is proportional to
+  J0^2(z). When z is great,
+
+                /  2
+    J0(z) = \  / ----- cos(z^-1/4 [pi]).
+             \/  [pi]z
+
+  The mean brightness varies as z^-1; and the integral
+      _
+     / [oo]
+     |   J0^2(z)z dz  is not convergent.
+    _/ 0
+
+5. _Resolving Power of Telescopes._--The efficiency of a telescope is of
+course intimately connected with the size of the disk by which it
+represents a mathematical point. In estimating theoretically the
+resolving power on a double star we have to consider the illumination of
+the field due to the superposition of the two independent images. If the
+angular interval between the components of a double star were equal to
+twice that expressed in equation (15) above, the central disks of the
+diffraction patterns would be just in contact. Under these conditions
+there is no doubt that the star would appear to be fairly resolved,
+since the brightness of its external ring system is too small to produce
+any material confusion, unless indeed the components are of very unequal
+magnitude. The diminution of the star disks with increasing aperture was
+observed by Sir William Herschel, and in 1823 Fraunhofer formulated the
+law of inverse proportionality. In investigations extending over a long
+series of years, the advantage of a large aperture in separating the
+components of close double stars was fully examined by W. R. Dawes.
+
+The resolving power of telescopes was investigated also by J. B. L.
+Foucault, who employed a scale of equal bright and dark alternate parts;
+it was found to be proportional to the aperture and independent of the
+focal length. In telescopes of the best construction and of moderate
+aperture the performance is not sensibly prejudiced by outstanding
+aberration, and the limit imposed by the finiteness of the waves of
+light is practically reached. M. E. Verdet has compared Foucault's
+results with theory, and has drawn the conclusion that the radius of the
+visible part of the image of a luminous point was equal to half the
+radius of the first dark ring.
+
+The application, unaccountably long delayed, of this principle to the
+microscope by H. L. F. Helmholtz in 1871 is the foundation of the
+important doctrine of the _microscopic limit_. It is true that in 1823
+Fraunhofer, inspired by his observations upon gratings, had very nearly
+hit the mark.[3] And a little before Helmholtz, E. Abbe published a
+somewhat more complete investigation, also founded upon the phenomena
+presented by gratings. But although the argument from gratings is
+instructive and convenient in some respects, its use has tended to
+obscure the essential unity of the principle of the limit of resolution
+whether applied to telescopes or microscopes.
+
+  [Illustration: Fig. 4.]
+
+  In fig. 4, AB represents the axis of an optical instrument (telescope
+  or microscope), A being a point of the object and B a point of the
+  image. By the operation of the object-glass LL' all the rays issuing
+  from A arrive in the same phase at B. Thus if A be self-luminous, the
+  illumination is a maximum at B, where all the secondary waves agree in
+  phase. B is in fact the centre of the diffraction disk which
+  constitutes the image of A. At neighbouring points the illumination is
+  less, in consequence of the discrepancies of phase which there enter.
+  In like manner if we take a neighbouring point P, also self-luminous,
+  in the plane of the object, the waves which issue from it will arrive
+  at B with phases no longer absolutely concordant, and the discrepancy
+  of phase will increase as the interval AP increases. When the
+  interval is very small the discrepancy, though mathematically
+  existent, produces no practical effect; and the illumination at B due
+  to P is as important as that due to A, the intensities of the two
+  luminous sources being supposed equal. Under these conditions it is
+  clear that A and P are not separated in the image. The question is to
+  what amount must the distance AP be increased in order that the
+  difference of situation may make itself felt in the image. This is
+  necessarily a question of degree; but it does not require detailed
+  calculations in order to show that the discrepancy first becomes
+  conspicuous when the phases corresponding to the various secondary
+  waves which travel from P to B range over a complete period. The
+  illumination at B due to P then becomes comparatively small, indeed
+  for some forms of aperture evanescent. The extreme discrepancy is that
+  between the waves which travel through the outermost parts of the
+  object-glass at L and L'; so that if we adopt the above standard of
+  resolution, the question is where must P be situated in order that the
+  relative retardation of the rays PL and PL' may on their arrival at B
+  amount to a wave-length ([lambda]). In virtue of the general law that
+  the reduced optical path is stationary in value, this retardation may
+  be calculated without allowance for the different paths pursued on the
+  farther side of L, L', so that the value is simply PL - PL'. Now since
+  AP is very small, AL' - PL' = AP sin [alpha], where [alpha] is the
+  angular semi-aperture L'AB. In like manner PL - AL has the same value,
+  so that
+
+    PL - PL' = 2AP sin [alpha].
+
+  According to the standard adopted, the condition of resolution is
+  therefore that AP, or [epsilon], should exceed 1/2[lambda]/sin [alpha].
+  If [epsilon] be less than this, the images overlap too much; while if
+  [epsilon] greatly exceed the above value the images become
+  unnecessarily separated.
+
+  In the above argument the whole space between the object and the lens
+  is supposed to be occupied by matter of one refractive index, and
+  [lambda] represents the wave-length _in this medium_ of the kind of
+  light employed. If the restriction as to uniformity be violated, what
+  we have ultimately to deal with is the wave-length in the medium
+  immediately surrounding the object.
+
+  Calling the refractive index [mu], we have as the critical value of
+  [epsilon],
+
+    [epsilon] = 1/2[lambda]0/[mu] sin[alpha],      (1),
+
+  [lambda]0 being the wave-length _in vacuo_. The denominator [mu] sin
+  [alpha] is the quantity well known (after Abbe) as the "numerical
+  aperture."
+
+  The extreme value possible for [alpha] is a right angle, so that for
+  the microscopic limit we have
+
+    [epsilon] = 1/2[lambda]0/[mu]                  (2).
+
+  The limit can be depressed only by a diminution in [lambda]0, such as
+  photography makes possible, or by an increase in [mu], the refractive
+  index of the medium in which the object is situated.
+
+  The statement of the law of resolving power has been made in a form
+  appropriate to the microscope, but it admits also of immediate
+  application to the telescope. If 2R be the diameter of the
+  object-glass and D the distance of the object, the angle subtended by
+  AP is [epsilon]/D, and the angular resolving power is given by
+
+    [lambda]/2D sin[alpha] = [lambda]/2R         (3).
+
+  This method of derivation (substantially due to Helmholtz) makes it
+  obvious that there is no essential difference of principle between the
+  two cases, although the results are conveniently stated in different
+  forms. In the case of the telescope we have to deal with a linear
+  measure of aperture and an angular limit of resolution, whereas in the
+  case of the microscope the limit of resolution is linear, and it is
+  expressed in terms of angular aperture.
+
+  It must be understood that the above argument distinctly assumes that
+  the different parts of the object are self-luminous, or at least that
+  the light proceeding from the various points is without phase
+  relations. As has been emphasized by G. J. Stoney, the restriction is
+  often, perhaps usually, violated in the microscope. A different
+  treatment is then necessary, and for some of the problems which arise
+  under this head the method of Abbe is convenient.
+
+  The importance of the general conclusions above formulated, as
+  imposing a limit upon our powers of direct observation, can hardly be
+  overestimated; but there has been in some quarters a tendency to
+  ascribe to it a more precise character than it can bear, or even to
+  mistake its meaning altogether. A few words of further explanation may
+  therefore be desirable. The first point to be emphasized is that
+  nothing whatever is said as to the smallness of a single object that
+  may be made visible. The eye, unaided or armed with a telescope, is
+  able to see, as points of light, stars subtending no sensible angle.
+  The visibility of a star is a question of brightness simply, and has
+  nothing to do with resolving power. The latter element enters only
+  when it is a question of recognizing the duplicity of a double star,
+  or of distinguishing detail upon the surface of a planet. So in the
+  microscope there is nothing except lack of light to hinder the
+  visibility of an object however small. But if its dimensions be much
+  less than the half wave-length, it can only be seen as a whole, and
+  its parts cannot be distinctly separated, although in cases near the
+  border line some inference may be possible, founded upon experience of
+  what appearances are presented in various cases. Interesting
+  observations upon particles, _ultra-microscopic_ in the above sense,
+  have been recorded by H. F. W. Siedentopf and R. A. Zsigmondy
+  (_Drude's Ann._, 1903, 10, p. 1).
+
+  In a somewhat similar way a dark linear interruption in a bright
+  ground may be visible, although its actual width is much inferior to
+  the half wave-length. In illustration of this fact a simple experiment
+  may be mentioned. In front of the naked eye was held a piece of copper
+  foil perforated by a fine needle hole. Observed through this the
+  structure of some wire gauze just disappeared at a distance from the
+  eye equal to 17 in., the gauze containing 46 meshes to the inch. On
+  the other hand, a single wire 0.034 in. in diameter remained fairly
+  visible up to a distance of 20 ft. The ratio between the limiting
+  angles subtended by the periodic structure of the gauze and the
+  diameter of the wire was (.022/.034) X (240/17) = 9.1. For further
+  information upon this subject reference may be made to _Phil. Mag._,
+  1896, 42, p. 167; _Journ. R. Micr. Soc._, 1903, p. 447.
+
+6. _Coronas or Glories._--The results of the theory of the diffraction
+patterns due to circular apertures admit of an interesting application
+to _coronas_, such as are often seen encircling the sun and moon. They
+are due to the interposition of small spherules of water, which act the
+part of diffracting obstacles. In order to the formation of a
+well-defined corona it is essential that the particles be exclusively,
+or preponderatingly, of one size.
+
+  If the origin of light be treated as infinitely small, and be seen in
+  focus, whether with the naked eye or with the aid of a telescope, the
+  whole of the light in the absence of obstacles would be concentrated
+  in the immediate neighbourhood of the focus. At other parts of the
+  field the effect is the same, in accordance with the principle known
+  as Babinet's, whether the imaginary screen in front of the
+  object-glass is generally transparent but studded with a number of
+  opaque circular disks, or is generally opaque but perforated with
+  corresponding apertures. Since at these points the resultant due to
+  the whole aperture is zero, any two portions into which the whole may
+  be divided must give equal and opposite resultants. Consider now the
+  light diffracted in a direction many times more oblique than any with
+  which we should be concerned, were the whole aperture uninterrupted,
+  and take first the effect of a single small aperture. The light in the
+  proposed direction is that determined by the size of the small
+  aperture in accordance with the laws already investigated, and its
+  phase depends upon the position of the aperture. If we take a
+  direction such that the light (of given wave-length) from a single
+  aperture vanishes, the evanescence continues even when the whole
+  series of apertures is brought into contemplation. Hence, whatever
+  else may happen, there must be a system of dark rings formed, the same
+  as from a single small aperture. In directions other than these it is
+  a more delicate question how the partial effects should be compounded.
+  If we make the extreme suppositions of an infinitely small source and
+  absolutely homogeneous light, there is no escape from the conclusion
+  that the light in a definite direction is arbitrary, that is,
+  dependent upon the chance distribution of apertures. If, however, as
+  in practice, the light be heterogeneous, the source of finite area,
+  the obstacles in motion, and the discrimination of different
+  directions imperfect, we are concerned merely with the mean brightness
+  found by varying the arbitrary phase-relations, and this is obtained
+  by simply multiplying the brightness due to a single aperture by the
+  number of apertures (n) (see INTERFERENCE OF LIGHT, S 4). The
+  diffraction pattern is therefore that due to a single aperture, merely
+  brightened n times.
+
+  In his experiments upon this subject Fraunhofer employed plates of
+  glass dusted over with lycopodium, or studded with small metallic
+  disks of uniform size; and he found that the diameters of the rings
+  were proportional to the length of the waves and inversely as the
+  diameter of the disks.
+
+  In another respect the observations of Fraunhofer appear at first
+  sight to be in disaccord with theory; for his measures of the
+  diameters of the red rings, visible when white light was employed,
+  correspond with the law applicable to dark rings, and not to the
+  different law applicable to the luminous maxima. Verdet has, however,
+  pointed out that the observation in this form is essentially different
+  from that in which homogeneous red light is employed, and that the
+  position of the red rings would correspond to the _absence_ of
+  blue-green light rather than to the greatest abundance of red light.
+  Verdet's own observations, conducted with great care, fully confirm
+  this view, and exhibit a complete agreement with theory.
+
+  By measurements of coronas it is possible to infer the size of the
+  particles to which they are due, an application of considerable
+  interest in the case of natural coronas--the general rule being the
+  larger the corona the smaller the water spherules. Young employed this
+  method not only to determine the diameters of cloud particles (e.g.
+  1/1000 in.), but also those of fibrous material, for which the theory
+  is analogous. His instrument was called the _eriometer_ (see
+  "Chromatics," vol. iii. of supp. to _Ency. Brit._, 1817).
+
+7. _Influence of Aberration. Optical Power of Instruments._--Our
+investigations and estimates of resolving power have thus far proceeded
+upon the supposition that there are no optical imperfections, whether of
+the nature of a regular aberration or dependent upon irregularities of
+material and workmanship. In practice there will always be a certain
+aberration or error of phase, which we may also regard as the deviation
+of the actual wave-surface from its intended position. In general, we
+may say that aberration is unimportant when it nowhere (or at any rate
+over a relatively small area only) exceeds a small fraction of the
+wave-length ([lamda]). Thus in estimating the intensity at a focal point,
+where, in the absence of aberration, all the secondary waves would have
+exactly the same phase, we see that an aberration nowhere exceeding
+1/4[lambda] can have but little effect.
+
+  The only case in which the influence of small aberration upon the
+  entire image has been calculated (_Phil. Mag._, 1879) is that of a
+  rectangular aperture, traversed by a cylindrical wave with aberration
+  equal to cx^3. The aberration is here unsymmetrical, the wave being in
+  advance of its proper place in one half of the aperture, but behind in
+  the other half. No terms in x or x^2 need be considered. The first
+  would correspond to a general turning of the beam; and the second
+  would imply imperfect focusing of the central parts. The effect of
+  aberration may be considered in two ways. We may suppose the aperture
+  (a) constant, and inquire into the operation of an increasing
+  aberration; or we may take a given value of c (i.e. a given
+  wave-surface) and examine the effect of a varying aperture. The
+  results in the second case show that an increase of aperture up to
+  that corresponding to an extreme aberration of half a period has no
+  ill effect upon the central band (S 3), but it increases unduly the
+  intensity of one of the neighbouring lateral bands; and the practical
+  conclusion is that the best results will be obtained from an aperture
+  giving an extreme aberration of from a quarter to half a period, and
+  that with an increased aperture aberration is not so much a direct
+  cause of deterioration as an obstacle to the attainment of that
+  improved definition which should accompany the increase of aperture.
+
+  If, on the other hand, we suppose the aperture given, we find that
+  aberration begins to be distinctly mischievous when it amounts to
+  about a quarter period, i.e. when the wave-surface deviates at each
+  end by a quarter wave-length from the true plane.
+
+  As an application of this result, let us investigate what amount of
+  temperature disturbance in the tube of a telescope may be expected to
+  impair definition. According to J. B. Biot and F. J. D. Arago, the
+  index [mu] for air at t deg. C. and at atmospheric pressure is given by
+
+                .00029
+    [mu] - 1 = -----------.
+               1 + .0037 t
+
+  If we take 0 deg. C. as standard temperature,
+
+    [delta][mu] = -1.1 X 10^-6.
+
+  Thus, on the supposition that the irregularity of temperature t
+  extends through a length l, and produces an acceleration of a quarter
+  of a wave-length,
+
+    1/4[lambda] = 1.1 lt X 10^-6;
+
+  or, if we take [lambda] = 5.3 X 10^-5,
+
+    lt = 12,
+
+  the unit of length being the centimetre.
+
+  We may infer that, in the case of a telescope tube 12 cm. long, a
+  stratum of air heated 1 deg. C. lying along the top of the tube, and
+  occupying a moderate fraction of the whole volume, would produce a not
+  insensible effect. If the change of temperature progressed uniformly
+  from one side to the other, the result would be a lateral displacement
+  of the image without loss of definition; but in general both effects
+  would be observable. In longer tubes a similar disturbance would be
+  caused by a proportionally less difference of temperature. S. P.
+  Langley has proposed to obviate such ill-effects by stirring the air
+  included within a telescope tube. It has long been known that the
+  definition of a carbon bisulphide prism may be much improved by a
+  vigorous shaking.
+
+  We will now consider the application of the principle to the formation
+  of images, unassisted by reflection or refraction (_Phil. Mag._,
+  1881). The function of a lens in forming an image is to compensate by
+  its variable thickness the differences of phase which would otherwise
+  exist between secondary waves arriving at the focal point from various
+  parts of the aperture. If we suppose the diameter of the lens to be
+  given (2R), and its focal length f gradually to increase, the original
+  differences of phase at the image of an infinitely distant luminous
+  point diminish without limit. When f attains a certain value, say f1,
+  the extreme error of phase to be compensated falls to 1/4[lambda]. But,
+  as we have seen, such an error of phase causes no sensible
+  deterioration in the definition; so that from this point onwards the
+  lens is useless, as only improving an image already sensibly as
+  perfect as the aperture admits of. Throughout the operation of
+  increasing the focal length, the resolving power of the instrument,
+  which depends only upon the aperture, remains unchanged; and we thus
+  arrive at the rather startling conclusion that a telescope of any
+  degree of resolving power might be constructed without an
+  object-glass, if only there were no limit to the admissible focal
+  length. This last proviso, however, as we shall see, takes away almost
+  all practical importance from the proposition.
+
+  To get an idea of the magnitudes of the quantities involved, let us
+  take the case of an aperture of 1/5 in., about that of the pupil of
+  the eye. The distance f1, which the actual focal length must exceed,
+  is given by
+
+      /
+    \/ (f1^2 + R^2) - f1  = 1/4[lambda];
+
+  so that
+
+    f1  = 2R^2/[lambda]                        (1).
+
+  Thus, if [lambda] = 1/4000, R = 1/10, we find
+
+    f1 = 800 inches.
+
+  The image of the sun thrown upon a screen at a distance exceeding 66
+  ft., through a hole 1/5 in. in diameter, is therefore at least as well
+  defined as that seen direct.
+
+  As the minimum focal length increases with the square of the aperture,
+  a quite impracticable distance would be required to rival the
+  resolving power of a modern telescope. Even for an aperture of 4 in.,
+  f1 would have to be 5 miles.
+
+  A similar argument may be applied to find at what point an achromatic
+  lens becomes sensibly superior to a single one. The question is
+  whether, when the adjustment of focus is correct for the central rays
+  of the spectrum, the error of phase for the most extreme rays (which
+  it is necessary to consider) amounts to a quarter of a wave-length. If
+  not, the substitution of an achromatic lens will be of no advantage.
+  Calculation shows that, if the aperture be 1/5 in., an achromatic lens
+  has no sensible advantage if the focal length be greater than about 11
+  in. If we suppose the focal length to be 66 ft., a single lens is
+  practically perfect up to an aperture of 1.7 in.
+
+  Another obvious inference from the necessary imperfection of optical
+  images is the uselessness of attempting anything like an absolute
+  destruction of spherical aberration. An admissible error of phase of
+  1/4[lambda] will correspond to an error of 1/8[lambda] in a reflecting
+  and 1/2[lambda] in a (glass) refracting surface, the incidence in both
+  cases being perpendicular. If we inquire what is the greatest
+  admissible longitudinal aberration ([delta]f) in an object-glass
+  according to the above rule, we find
+
+    [delta]f = [lambda][alpha]^-2               (2),
+
+  [alpha] being the angular semi-aperture.
+
+  In the case of a single lens of glass with the most favourable
+  curvatures, [delta]f is about equal to [alpha]^2f, so that [alpha]^4
+  must not exceed [lambda]/f. For a lens of 3 ft. focus this condition
+  is satisfied if the aperture does not exceed 2 in.
+
+  When parallel rays fall directly upon a spherical mirror the
+  longitudinal aberration is only about one-eighth as great as for the
+  most favourably shaped single lens of equal focal length and aperture.
+  Hence a spherical mirror of 3 ft. focus might have an aperture of 21/2
+  in., and the image would not suffer materially from aberration.
+
+  On the same principle we may estimate the least visible displacement
+  of the eye-piece of a telescope focused upon a distant object, a
+  question of interest in connexion with range-finders. It appears
+  (_Phil. Mag._, 1885, 20, p. 354) that a displacement [delta]f from the
+  true focus will not sensibly impair definition, provided
+
+    [delta]f < f^2[lambda]/R^2                    (3),
+
+  2R being the diameter of aperture. The linear accuracy required is
+  thus a function of the _ratio_ of aperture to focal length. The
+  formula agrees well with experiment.
+
+  The principle gives an instantaneous solution of the question of the
+  ultimate optical efficiency in the method of "mirror-reading," as
+  commonly practised in various physical observations. A rotation by
+  which one edge of the mirror advances 1/4[lambda] (while the other edge
+  retreats to a like amount) introduces a phase-discrepancy of a whole
+  period where before the rotation there was complete agreement. A
+  rotation of this amount should therefore be easily visible, but the
+  limits of resolving power are being approached; and the conclusion is
+  independent of the focal length of the mirror, and of the employment
+  of a telescope, provided of course that the reflected image is seen in
+  focus, and that the full width of the mirror is utilized.
+
+  A comparison with the method of a material pointer, attached to the
+  parts whose rotation is under observation, and viewed through a
+  microscope, is of interest. The limiting efficiency of the microscope
+  is attained when the angular aperture amounts to 180 deg.; and it is
+  evident that a lateral displacement of the point under observation
+  through 1/2[lambda] entails (at the old image) a phase-discrepancy of a
+  whole period, one extreme ray being accelerated and the other retarded
+  by half that amount. We may infer that the limits of efficiency in the
+  two methods are the same when the length of the pointer is equal to
+  the width of the mirror.
+
+  [Illustraton: FIG. 5.]
+
+  We have seen that in perpendicular reflection a surface error not
+  exceeding 1/8[lambda] may be admissible. In the case of oblique
+  reflection at an angle [phi], the error of retardation due to an
+  elevation BD (fig. 5) is
+
+    QQ' - QS = BD sec [phi](1 - cos SQQ') = BD sec [phi] (1 + cos 2[phi]) = 2BD cos [phi];
+
+  from which it follows that an error of given magnitude in the figure
+  of a surface is less important in oblique than in perpendicular
+  reflection. It must, however, be borne in mind that errors can
+  sometimes be compensated by altering adjustments. If a surface
+  intended to be flat is affected with a slight general curvature, a
+  remedy may be found in an alteration of focus, and the remedy is the
+  less complete as the reflection is more oblique.
+
+  The formula expressing the optical power of prismatic spectroscopes
+  may readily be investigated upon the principles of the wave theory.
+  Let A0B0 be a plane wave-surface of the light before it falls upon the
+  prisms, AB the corresponding wave-surface for a particular part of the
+  spectrum after the light has passed the prisms, or after it has passed
+  the eye-piece of the observing telescope. The path of a ray from the
+  wave-surface A0B0 to A or B is determined by the condition that the
+  optical distance, [int] [mu]ds, is a minimum; and, as AB is by
+  supposition a wave-surface, this optical distance is the same for both
+  points. Thus
+      _                  _
+     /                  /
+     | [mu]ds (for A) = | [mu]ds (for B)          (4).
+    _/                 _/
+
+  We have now to consider the behaviour of light belonging to a
+  neighbouring part of the spectrum. The path of a ray from the
+  wave-surface A0B0 to the point A is changed; but in virtue of the
+  minimum property the change may be neglected in calculating the
+  optical distance, as it influences the result by quantities of the
+  second order only in the changes of refrangibility. Accordingly, the
+  optical distance from A0B0 to A is represented by [int]([mu] +
+  [delta][mu])ds, the integration being along the original path A0 ...
+  A; and similarly the optical distance between A0B0 and B is
+  represented by [int] ([mu] + [delta][mu])ds, the integration being
+  along B0 ... B. In virtue of (4) the difference of the optical
+  distances to A and B is
+      _                                  _
+     /                                  /
+     | [delta][mu]ds (along B0 ... B) - | [delta][mu]ds (along A0 ... A)  (5).
+    _/                                 _/
+
+  The new wave-surface is formed in such a position that the optical
+  distance is constant; and therefore the _dispersion_, or the angle
+  through which the wave-surface is turned by the change of
+  refrangibility, is found simply by dividing (5) by the distance AB.
+  If, as in common flint-glass spectroscopes, there is only one
+  dispersing substance, [int] [delta][mu] ds = [delta][mu].s, where s is
+  simply the thickness traversed by the ray. If t2 and t1 be the
+  thicknesses traversed by the extreme rays, and a denote the width of
+  the emergent beam, the dispersion [theta] is given by
+
+    [theta] = [delta][mu](t2 - t1)/a,
+
+  or, if t1 be negligible,
+
+    [theta] = [delta][mu]t/a                 (6).
+
+  The condition of resolution of a double line whose components subtend
+  an angle [theta] is that [theta] must exceed [lambda]/a. Hence, in
+  order that a double line may be resolved whose components have indices
+  [mu] and [mu] + [delta][mu], it is necessary that t should exceed the
+  value given by the following equation:--
+
+    t = [lambda]/[delta][mu]                 (7).
+
+8. _Diffraction Gratings._--Under the heading "Colours of Striated
+Surfaces," Thomas Young (_Phil. Trans._, 1802) in his usual summary
+fashion gave a general explanation of these colours, including the law
+of sines, the striations being supposed to be straight, parallel and
+equidistant. Later, in his article "Chromatics" in the supplement to the
+5th edition of this encyclopaedia, he shows that the colours "lose the
+mixed character of periodical colours, and resemble much more the
+ordinary prismatic spectrum, with intervals completely dark interposed,"
+and explains it by the consideration that any phase-difference which may
+arise at neighbouring striae is multiplied in proportion to the total
+number of striae.
+
+The theory was further developed by A. J. Fresnel (1815), who gave a
+formula equivalent to (5) below. But it is to J. von Fraunhofer that we
+owe most of our knowledge upon this subject. His recent discovery of the
+"fixed lines" allowed a precision of observation previously impossible.
+He constructed gratings up to 340 periods to the inch by straining fine
+wire over screws. Subsequently he ruled gratings on a layer of gold-leaf
+attached to glass, or on a layer of grease similarly supported, and
+again by attacking the glass itself with a diamond point. The best
+gratings were obtained by the last method, but a suitable diamond point
+was hard to find, and to preserve. Observing through a telescope with
+light perpendicularly incident, he showed that the position of any ray
+was dependent only upon the grating interval, viz. the distance from the
+centre of one wire or line to the centre of the next, and not otherwise
+upon the thickness of the wire and the magnitude of the interspace. In
+different gratings the lengths of the spectra and their distances from
+the axis were inversely proportional to the grating interval, while with
+a given grating the distances of the various spectra from the axis were
+as 1, 2, 3, &c. To Fraunhofer we owe the first accurate measurements of
+wave-lengths, and the method of separating the overlapping spectra by a
+prism dispersing in the perpendicular direction. He described also the
+complicated patterns seen when a point of light is viewed through two
+superposed gratings, whose lines cross one another perpendicularly or
+obliquely. The above observations relate to transmitted light, but
+Fraunhofer extended his inquiry to the light _reflected_. To eliminate
+the light returned from the hinder surface of an engraved grating, he
+covered it with a black varnish. It then appeared that under certain
+angles of incidence parts of the resulting spectra were _completely
+polarized_. These remarkable researches of Fraunhofer, carried out in
+the years 1817-1823, are republished in his _Collected Writings_
+(Munich, 1888).
+
+  The principle underlying the action of gratings is identical with that
+  discussed in S 2, and exemplified in J. L. Soret's "zone plates." The
+  alternate Fresnel's zones are blocked out or otherwise modified; in
+  this way the original compensation is upset and a revival of light
+  occurs in unusual directions. If the source be a point or a line, and
+  a collimating lens be used, the incident waves may be regarded as
+  plane. If, further, on leaving the grating the light be received by a
+  focusing lens, e.g. the object-glass of a telescope, the Fresnel's
+  zones are reduced to parallel and equidistant straight strips, which
+  at certain angles coincide with the ruling. The directions of the
+  lateral spectra are such that the passage from one element of the
+  grating to the corresponding point of the next implies a retardation
+  of an integral number of wave-lengths. If the grating be composed of
+  alternate transparent and opaque parts, the question may be treated by
+  means of the general integrals (S 3) by merely limiting the
+  integration to the transparent parts of the aperture. For an
+  investigation upon these lines the reader is referred to Airy's
+  _Tracts_, to Verdet's _Lecons_, or to R. W. Wood's _Physical Optics_.
+  If, however, we assume the theory of a simple rectangular aperture (S
+  3); the results of the ruling can be inferred by elementary methods,
+  which are perhaps more instructive.
+
+  Apart from the ruling, we know that the image of a mathematical line
+  will be a series of narrow bands, of which the central one is by far
+  the brightest. At the middle of this band there is complete agreement
+  of phase among the secondary waves. The dark lines which separate the
+  bands are the places at which the phases of the secondary wave range
+  over an integral number of periods. If now we suppose the aperture AB
+  to be covered by a great number of opaque strips or bars of width d,
+  separated by transparent intervals of width a, the condition of things
+  in the directions just spoken of is not materially changed. At the
+  central point there is still complete agreement of phase; but the
+  amplitude is diminished in the ratio of a : a + d. In another
+  direction, making a small angle with the last, such that the
+  projection of AB upon it amounts to a few wave-lengths, it is easy to
+  see that the mode of interference is the same as if there were no
+  ruling. For example, when the direction is such that the projection of
+  AB upon it amounts to one wave-length, the elementary components
+  neutralize one another, because their phases are distributed
+  symmetrically, though discontinuously, round the entire period. The
+  only effect of the ruling is to diminish the amplitude in the ratio a
+  : a + d; and, except for the difference in illumination, the
+  appearance of a line of light is the same as if the aperture were
+  perfectly free.
+
+  The lateral (spectral) images occur in such directions that the
+  projection of the element (a + d) of the grating upon them is an exact
+  multiple of [lambda]. The effect of each of the n elements of the
+  grating is then the same; and, unless this vanishes on account of a
+  particular adjustment of the ratio a : d, the resultant amplitude
+  becomes comparatively very great. These directions, in which the
+  retardation between A and B is exactly mn[lambda], may be called the
+  principal directions. On either side of any one of them the
+  illumination is distributed according to the same law as for the
+  central image (m = 0), vanishing, for example, when the retardation
+  amounts to (mn [+-] 1)[lambda]. In considering the relative brightnesses
+  of the different spectra, it is therefore sufficient to attend merely
+  to the principal directions, provided that the whole deviation be not
+  so great that its cosine differs considerably from unity.
+
+  We have now to consider the amplitude due to a single element, which
+  we may conveniently regard as composed of a transparent part a bounded
+  by two opaque parts of width 1/2d. The phase of the resultant effect is
+  by symmetry that of the component which comes from the middle of a.
+  The fact that the other components have phases differing from this by
+  amounts ranging between [+-] am[pi]/(a + d) causes the resultant
+  amplitude to be less than for the central image (where there is
+  complete phase agreement). If Bm denote the brightness of the m^th
+  lateral image, and B0 that of the central image, we have
+
+                _    _+ am[pi]/(a + d)         _
+               |    /                  2am[pi]  |^2  /a + d \^2      am[pi]
+    B_m : B0 = |    |       cosx dx :- -------  | = ( ------ ) sin^2 ------ (1).
+               |_  _/                   a + d  _|    \am[pi]/        a + d
+                     -am[pi]/(a + d)
+
+  If B denotes the brightness of the central image when the whole of the
+  space occupied by the grating is transparent, we have
+
+    B0 : B = a^2 : (a + d)^2,
+
+  and thus
+
+                 1           am[pi]
+    Bm : B = --------- sin^2 ------          (2).
+             m^2[pi]^2       a + d
+
+  The sine of an angle can never be greater than unity; and consequently
+  under the most favourable circumstances only 1/m^2[pi]^2 of the original
+  light can be obtained in the m^th spectrum. We conclude that, with a
+  grating composed of transparent and opaque parts, the utmost light
+  obtainable in any one spectrum is in the first, and there amounts to
+  1/[pi]^2, or about 1/10, and that for this purpose a and d must be
+  equal. When d = a the general formula becomes
+
+              sin^2 1/2m[pi]
+    Bm : B =  -------------               (3),
+                m^2[pi]^2
+
+  showing that, when m is even, Bm vanishes, and that, when m is odd,
+
+    Bm : B = 1/m^2[pi]^2.
+
+  The third spectrum has thus only 1/9 of the brilliancy of the first.
+
+  Another particular case of interest is obtained by supposing a small
+  relatively to (a + d). Unless the spectrum be of very high order, we
+  have simply
+
+    Bm : B = a/(a + d)^2                   (4);
+
+  so that the brightnesses of all the spectra are the same.
+
+  The light stopped by the opaque parts of the grating, together with
+  that distributed in the central image and lateral spectra, ought to
+  make up the brightness that would be found in the central image, were
+  all the apertures transparent. Thus, if a = d, we should have
+
+        1   1     2     /    1    1      \
+    1 = - + - + ------ ( 1 + - + -- + ... ),
+        2   4   [pi]^2  \    9   25      /
+
+  which is true by a known theorem. In the general case
+
+                               ___m=[oo]
+      a      /  a  \^2     2   \   1        /m[pi]a\
+    ----- = ( ----- ) + ------  >  -- sin^2( ------ ),
+    a + d    \a + d/    [pi]^2 /__ m^2      \ a + d/
+                               m=1
+
+  a formula which may be verified by Fourier's theorem.
+
+  According to a general principle formulated by J. Babinet, the
+  brightness of a lateral spectrum is not affected by an interchange of
+  the transparent and opaque parts of the grating. The vibrations
+  corresponding to the two parts are precisely antagonistic, since if
+  both were operative the resultant would be zero. So far as the
+  application to gratings is concerned, the same conclusion may be
+  derived from (2).
+
+  [Illustration: FIG. 6.]
+
+  From the value of Bm : B0 we see that no lateral spectrum can surpass
+  the central image in brightness; but this result depends upon the
+  hypothesis that the ruling acts by opacity, which is generally very
+  far from being the case in practice. In an engraved glass grating
+  there is no opaque material present by which light could be absorbed,
+  and the effect depends upon a difference of retardation in passing the
+  alternate parts. It is possible to prepare gratings which give a
+  lateral spectrum brighter than the central image, and the explanation
+  is easy. For if the alternate parts were equal and alike transparent,
+  but so constituted as to give a relative retardation of 1/2[lambda], it
+  is evident that the central image would be entirely extinguished,
+  while the first spectrum would be four times as bright as if the
+  alternate parts were opaque. If it were possible to introduce at every
+  part of the aperture of the grating an arbitrary retardation, all the
+  light might be concentrated in any desired spectrum. By supposing the
+  retardation to vary uniformly and continuously we fall upon the case
+  of an ordinary prism: but there is then no diffraction spectrum in the
+  usual sense. To obtain such it would be necessary that the retardation
+  should gradually alter by a wave-length in passing over any element of
+  the grating, and then fall back to its previous value, thus springing
+  suddenly over a wave-length (_Phil. Mag._, 1874, 47, p. 193). It is
+  not likely that such a result will ever be fully attained in practice;
+  but the case is worth stating, in order to show that there is no
+  theoretical limit to the concentration of light of assigned
+  wave-length in one spectrum, and as illustrating the frequently
+  observed unsymmetrical character of the spectra on the two sides of
+  the central image.[4]
+
+  We have hitherto supposed that the light is incident perpendicularly
+  upon the grating; but the theory is easily extended. If the incident
+  rays make an angle [theta] with the normal (fig. 6), and the
+  diffracted rays make an angle [phi] (upon the same side), the relative
+  retardation from each element of width (a + d) to the next is (a + d)
+  (sin[theta] + sin[phi]); and this is the quantity which is to be
+  equated to m[lambda]. Thus
+
+    sin[theta] + sin[phi] = 2 sin 1/2([theta] + [phi]) cos 1/2([theta] - [phi]) = m[lambda]/(a + d) (5).
+
+  The "deviation" is ([theta] + [phi]), and is therefore a minimum when
+  [theta] = [phi], i.e. when the grating is so situated that the angles
+  of incidence and diffraction are equal.
+
+  In the case of a reflection grating the same method applies. If
+  [theta] and [phi] denote the angles with the normal made by the
+  incident and diffracted rays, the formula (5) still holds, and, if the
+  deviation be reckoned from the direction of the regularly reflected
+  rays, it is expressed as before by ([theta] + [phi]), and is a minimum
+  when [theta] = [phi], that is, when the diffracted rays return upon
+  the course of the incident rays.
+
+  [Illustration: FIG. 7.]
+
+  In either case (as also with a prism) the position of minimum
+  deviation leaves the width of the beam unaltered, i.e. neither
+  magnifies nor diminishes the angular width of the object under view.
+
+  From (5) we see that, when the light falls perpendicularly upon a
+  grating ([theta] = 0), there is no spectrum formed (the image
+  corresponding to m = 0 not being counted as a spectrum), if the
+  grating interval [sigma] or (a + d) is less than [lambda]. Under these
+  circumstances, if the material of the grating be completely
+  transparent, the whole of the light must appear in the direct image,
+  and the ruling is not perceptible. From the absence of spectra
+  Fraunhofer argued that there must be a microscopic limit represented
+  by [lambda]; and the inference is plausible, to say the least (_Phil.
+  Mag._, 1886). Fraunhofer should, however, have fixed the microscopic
+  limit at 1/2[lambda], as appears from (5), when we suppose [theta] =
+  1/2[pi], [phi] = 1/2[pi].
+
+  [Illustration: FIG. 8.]
+
+  We will now consider the important subject of the resolving power of
+  gratings, as dependent upon the number of lines (n) and the order of
+  the spectrum observed (m). Let BP (fig. 8) be the direction of the
+  principal maximum (middle of central band) for the wave-length
+  [lambda] in the m^th spectrum. Then the relative retardation of the
+  extreme rays (corresponding to the edges A, B of the grating) is
+  mn[lambda]. If BQ be the direction for the first minimum (the darkness
+  between the central and first lateral band), the relative retardation
+  of the extreme rays is (mn + 1)[lambda]. Suppose now that [lambda] +
+  [delta][lambda] is the wave-length for which BQ gives the principal
+  maximum, then
+
+    (mn + 1)[lambda] = mn([lambda] + [delta][lambda]);
+
+  whence
+
+    [delta][lambda]/[lambda] = 1/mn            (6).
+
+  According to our former standard, this gives the smallest difference
+  of wave-lengths in a double line which can be just resolved; and we
+  conclude that the resolving power of a grating depends only upon the
+  total number of lines, and upon the order of the spectrum, without
+  regard to any other considerations. It is here of course assumed that
+  the n lines are really utilized.
+
+  In the case of the D lines the value of [delta][lambda]/[lambda] is
+  about 1/1000; so that to resolve this double line in the first
+  spectrum requires 1000 lines, in the second spectrum 500, and so on.
+
+  It is especially to be noticed that the resolving power does not
+  depend directly upon the closeness of the ruling. Let us take the case
+  of a grating 1 in. broad, and containing 1000 lines, and consider the
+  effect of interpolating an additional 1000 lines, so as to bisect the
+  former intervals. There will be destruction by interference of the
+  first, third and odd spectra generally; while the advantage gained in
+  the spectra of even order is not in dispersion, nor in resolving
+  power, but simply in brilliancy, which is increased four times. If we
+  now suppose half the grating cut away, so as to leave 1000 lines in
+  half an inch, the dispersion will not be altered, while the brightness
+  and resolving power are halved.
+
+  There is clearly no theoretical limit to the resolving power of
+  gratings, even in spectra of given order. But it is possible that, as
+  suggested by Rowland,[5] the structure of natural spectra may be too
+  coarse to give opportunity for resolving powers much higher than those
+  now in use. However this may be, it would always be possible, with the
+  aid of a grating of given resolving power, to construct artificially
+  from white light mixtures of slightly different wave-length whose
+  resolution or otherwise would discriminate between powers inferior and
+  superior to the given one.[6]
+
+  If we define as the "dispersion" in a particular part of the spectrum
+  the ratio of the angular interval d[theta] to the corresponding
+  increment of wave-length d[lambda], we may express it by a very simple
+  formula. For the alteration of wave-length entails, at the two limits
+  of a diffracted wave-front, a relative retardation equal to
+  mnd[lambda]. Hence, if a be the width of the diffracted beam, and
+  d[theta] the angle through which the wave-front is turned,
+
+    ad[theta] = mn d[lambda],
+
+  or  dispersion = mn/a                       (7).
+
+  The resolving power and the width of the emergent beam fix the optical
+  character of the instrument. The latter element must eventually be
+  decreased until less than the diameter of the pupil of the eye. Hence
+  a wide beam demands treatment with further apparatus (usually a
+  telescope) of high magnifying power.
+
+  In the above discussion it has been supposed that the ruling is
+  accurate, and we have seen that by increase of m a high resolving
+  power is attainable with a moderate number of lines. But this
+  procedure (apart from the question of illumination) is open to the
+  objection that it makes excessive demands upon accuracy. According to
+  the principle already laid down it can make but little difference in
+  the principal direction corresponding to the first spectrum, provided
+  each line lie within a quarter of an interval (a + d) from its
+  theoretical position. But, to obtain an equally good result in the
+  m^th spectrum, the error must be less than 1/m of the above amount.[7]
+
+  There are certain errors of a systematic character which demand
+  special consideration. The spacing is usually effected by means of a
+  screw, to each revolution of which corresponds a large number (e.g.
+  one hundred) of lines. In this way it may happen that although there
+  is almost perfect periodicity with each revolution of the screw after
+  (say) 100 lines, yet the 100 lines themselves are not equally spaced.
+  The "ghosts" thus arising were first described by G. H. Quincke
+  (_Pogg. Ann._, 1872, 146, p. 1), and have been elaborately
+  investigated by C. S. Peirce (_Ann. Journ. Math._, 1879, 2, p. 330),
+  both theoretically and experimentally. The general nature of the
+  effects to be expected in such a case may be made clear by means of an
+  illustration already employed for another purpose. Suppose two similar
+  and accurately ruled transparent gratings to be superposed in such a
+  manner that the lines are parallel. If the one set of lines exactly
+  bisect the intervals between the others, the grating interval is
+  practically halved, and the previously existing spectra of odd order
+  vanish. But a very slight relative displacement will cause the
+  apparition of the odd spectra. In this case there is approximate
+  periodicity in the half interval, but complete periodicity only after
+  the whole interval. The advantage of approximate bisection lies in the
+  superior brilliancy of the surviving spectra; but in any case the
+  compound grating may be considered to be perfect in the longer
+  interval, and the definition is as good as if the bisection were
+  accurate.
+
+  [Illustration:
+
+      | | | | |         ( ( (            | |  |   |         ) | (
+
+    FIG. 9.--x^2.   FIG. 10.--y^2.    FIG. 11.--x^3.   FIG. 12.--xy^2.
+
+                                        /    /    /
+       \ | | /         |  \  |          |    |    |
+                                        /    /    /
+
+    FIG. 13.--xy.   FIG. 14.--x^2y.    FIG. 15.--y^3.]
+
+  The effect of a gradual increase in the interval (fig. 9) as we pass
+  across the grating has been investigated by M. A. Cornu (_C.R._, 1875,
+  80, p. 655), who thus explains an anomaly observed by E. E. N.
+  Mascart. The latter found that certain gratings exercised a converging
+  power upon the spectra formed upon one side, and a corresponding
+  diverging power upon the spectra on the other side. Let us suppose
+  that the light is incident perpendicularly, and that the grating
+  interval increases from the centre towards that edge which lies
+  nearest to the spectrum under observation, and decreases towards the
+  hinder edge. It is evident that the waves from _both_ halves of the
+  grating are accelerated in an increasing degree, as we pass from the
+  centre outwards, as compared with the phase they would possess were
+  the central value of the grating interval maintained throughout. The
+  irregularity of spacing has thus the effect of a convex lens, which
+  accelerates the marginal relatively to the central rays. On the other
+  side the effect is reversed. This kind of irregularity may clearly be
+  present in a degree surpassing the usual limits, without loss of
+  definition, when the telescope is focused so as to secure the best
+  effect.
+
+  It may be worth while to examine further the other variations from
+  correct ruling which correspond to the various terms expressing the
+  deviation of the wave-surface from a perfect plane. If x and y be
+  co-ordinates in the plane of the wave-surface, the axis of y being
+  parallel to the lines of the grating, and the origin corresponding to
+  the centre of the beam, we may take as an approximate equation to the
+  wave-surface
+
+         x^2             y^2
+    z = ------ + Bxy + ------- + [alpha]x^3 + [beta]x^2y + [gamma]xy^2 + [delta]y^3 + ... (8);
+        2[rho]         2[rho]'
+
+  and, as we have just seen, the term in x^2 corresponds to a linear
+  error in the spacing. In like manner, the term in y^2 corresponds to a
+  general _curvature_ of the lines (fig. 10), and does not influence the
+  definition at the (primary) focus, although it may introduce
+  astigmatism.[8] If we suppose that everything is symmetrical on the
+  two sides of the primary plane y = 0, the coefficients B, [beta],
+  [delta] vanish. In spite of any inequality between [rho] and [rho]',
+  the definition will be good to this order of approximation, provided
+  [alpha] and [gamma] vanish. The former measures the _thickness_ of the
+  primary focal line, and the latter measures its _curvature_. The error
+  of ruling giving rise to [alpha] is one in which the intervals
+  increase or decrease in _both_ directions from the centre outwards
+  (fig. 11), and it may often be compensated by a slight rotation in
+  azimuth of the object-glass of the observing telescope. The term in
+  [gamma] corresponds to a _variation_ of curvature in crossing the
+  grating (fig. 12).
+
+  When the plane zx is not a plane of symmetry, we have to consider the
+  terms in xy, x^2y, and y^3. The first of these corresponds to a
+  deviation from parallelism, causing the interval to alter gradually as
+  we pass _along_ the lines (fig. 13). The error thus arising may be
+  compensated by a rotation of the object-glass about one of the
+  diameters y = [+-] x. The term in x^2y corresponds to a deviation from
+  parallelism in the same direction on both sides of the central line
+  (fig. 14); and that in y^3 would be caused by a curvature such that
+  there is a point of inflection at the middle of each line (fig. 15).
+
+  All the errors, except that depending on [alpha], and especially those
+  depending on [gamma] and [delta], can be diminished, without loss of
+  resolving power, by contracting the _vertical_ aperture. A linear
+  error in the spacing, and a general curvature of the lines, are
+  eliminated in the ordinary use of a grating.
+
+  The explanation of the difference of focus upon the two sides as due
+  to unequal spacing was verified by Cornu upon gratings purposely
+  constructed with an increasing interval. He has also shown how to rule
+  a plane surface with lines so disposed that the grating shall of
+  itself give well-focused spectra.
+
+  [Illustration: FIG. 16.]
+
+  A similar idea appears to have guided H. A. Rowland to his brilliant
+  invention of concave gratings, by which spectra can be photographed
+  without any further optical appliance. In these instruments the lines
+  are ruled upon a spherical surface of speculum metal, and mark the
+  intersections of the surface by a system of parallel and equidistant
+  planes, of which the middle member passes through the centre of the
+  sphere. If we consider for the present only the primary plane of
+  symmetry, the figure is reduced to two dimensions. Let AP (fig. 16)
+  represent the surface of the grating, O being the centre of the
+  circle. Then, if Q be any radiant point and Q' its image (primary
+  focus) in the spherical mirror AP, we have
+
+    1    1   2cos[phi]
+    -- + - = ---------,
+    v1   u       a
+
+  where v1 = AQ', u = AQ, a = OA, [phi] = angle of incidence QAO, equal
+  to the angle of reflection Q'AO. If Q be on the circle described upon
+  OA as diameter, so that u = a cos [phi], then Q' lies also upon the
+  same circle; and in this case it follows from the symmetry that the
+  unsymmetrical aberration (depending upon a) vanishes.
+
+  This disposition is adopted in Rowland's instrument; only, in addition
+  to the central image formed at the angle [phi]' = [phi], there are a
+  series of spectra with various values of [phi]', but all disposed upon
+  the same circle. Rowland's investigation is contained in the paper
+  already referred to; but the following account of the theory is in the
+  form adopted by R. T. Glazebrook (_Phil. Mag._, 1883).
+
+  In order to find the difference of optical distances between the
+  courses QAQ', QPQ', we have to express QP - QA, PQ' - AQ'. To find the
+  former, we have, if OAQ = [phi], AOP = [omega],
+
+    QP^2 = u^2 + 4a^2sin^21/2[omega] - 4au sin 1/2[omega] sin (1/2[omega] - [phi])
+   = (u + a sin[phi] sin[omega])^2 - a^2 sin^2[phi] sin^2[omega] + 4a sin^2 1/2[omega](a - u cos[phi]).
+
+  Now as far as [omega]^4
+
+    4 sin^2 1/2[omega] = sin^2[omega] + 1/4sin^4[omega],
+
+  and thus to the same order
+
+      QP^2 = (u + a sin [phi] sin [omega])^2
+    -a cos [phi](u - a cos [phi]) sin^2[omega] + 1/4 a(a - u cos[phi]) sin^4 [omega].
+
+  But if we now suppose that Q lies on the circle u = a cos [phi], the
+  middle term vanishes, and we get, correct as far as [omega]^4,
+
+                                       / /    a^2 sin^2[phi] sin^4[omega]\
+    QP = (u + a sin[phi] sin[omega])  / ( 1 + --------------------------- );
+                                    \/   \                4u             /
+  so that
+
+    QP - u = a sin [phi] sin [omega] + 1/8 a sin[phi] tan[phi] sin^4 [omega] (9),
+
+  in which it is to be noticed that the adjustment necessary to secure
+  the disappearance of sin^2[omega] is sufficient also to destroy the
+  term in sin^3[omega].
+
+  A similar expression can be found for Q'P - Q'A; and thus, if Q'A = v,
+  Q'AO = [phi]', where v = a cos [phi]', we get
+
+    QP + PQ' - QA -AQ' = a sin[omega] (sin[phi] - sin[phi]')
+       + 1/8 a sin^4 [omega] (sin[phi] tan[phi] + sin[phi]' tan[phi]')         (10).
+
+  If [phi]' = [phi], the term of the first order vanishes, and the
+  reduction of the difference of path _via_ P and _via_ A to a term of
+  the fourth order proves not only that Q and Q' are conjugate foci, but
+  also that the foci are exempt from the most important term in the
+  aberration. In the present application [phi]' is not necessarily equal
+  to [phi]; but if P correspond to a line upon the grating, the
+  difference of retardations for consecutive positions of P, so far as
+  expressed by the term of the first order, will be equal to [-+]
+  m[lambda] (m integral), and therefore without influence, provided
+
+    [sigma] (sin[phi] - sin[phi]') = [+-] m[lambda]      (11),
+
+  where [sigma] denotes the constant interval between the planes
+  containing the lines. This is the ordinary formula for a reflecting
+  plane grating, and it shows that the spectra are formed in the usual
+  directions. They are here focused (so far as the rays in the primary
+  plane are concerned) upon the circle OQ'A, and the outstanding
+  aberration is of the fourth order.
+
+  In order that a large part of the field of view may be in focus at
+  once, it is desirable that the locus of the focused spectrum should be
+  nearly perpendicular to the line of vision. For this purpose Rowland
+  places the eye-piece at O, so that [phi] = 0, and then by (11) the
+  value of [phi]' in the m^th spectrum is
+
+    [sigma] sin [phi]' = [+-] m[lambda]                  (12).
+
+  If [omega] now relate to the edge of the grating, on which there are
+  altogether n lines,
+
+  n[sigma] = 2a sin [omega],
+
+  and the value of the last term in (10) becomes
+
+    1/16 n[sigma] sin^3[omega] sin[phi]' tan[phi]',
+
+  or
+
+    1/16 mn[lambda] sin^3[omega] tan [phi]'            (13).
+
+  This expresses the retardation of the extreme relatively to the
+  central ray, and is to be reckoned positive, whatever may be the signs
+  of [omega], and [phi]'. If the semi-angular aperture ([omega]) be
+  1/100, and tan [phi]' = 1, mn might be as great as four millions
+  before the error of phase would reach 1/4[lambda]. If it were desired to
+  use an angular aperture so large that the aberration according to (13)
+  would be injurious, Rowland points out that on his machine there would
+  be no difficulty in applying a remedy by making [sigma] slightly
+  variable towards the edges. Or, retaining [sigma] constant, we might
+  attain compensation by so polishing the surface as to bring the
+  circumference slightly forward in comparison with the position it
+  would occupy upon a true sphere.
+
+  It may be remarked that these calculations apply to the rays in the
+  primary plane only. The image is greatly affected with astigmatism;
+  but this is of little consequence, if [gamma] in (8) be small enough.
+  Curvature of the primary focal line having a very injurious effect
+  upon definition, it may be inferred from the excellent performance of
+  these gratings that [gamma] is in fact small. Its value does not
+  appear to have been calculated. The other coefficients in (8) vanish
+  in virtue of the symmetry.
+
+  The mechanical arrangements for maintaining the focus are of great
+  simplicity. The grating at A and the eye-piece at O are rigidly
+  attached to a bar AO, whose ends rest on carriages, moving on rails
+  OQ, AQ at right angles to each other. A tie between the middle point
+  of the rod OA and Q can be used if thought desirable.
+
+  The absence of chromatic aberration gives a great advantage in the
+  comparison of overlapping spectra, which Rowland has turned to
+  excellent account in his determinations of the relative wave-lengths of
+  lines in the solar spectrum (_Phil. Mag._, 1887).
+
+  For absolute determinations of wave-lengths plane gratings are used.
+  It is found (Bell, _Phil. Mag._, 1887) that the angular measurements
+  present less difficulty than the comparison of the grating interval
+  with the standard metre. There is also some uncertainty as to the
+  actual temperature of the grating when in use. In order to minimize
+  the heating action of the light, it might be submitted to a
+  preliminary prismatic analysis before it reaches the slit of the
+  spectrometer, after the manner of Helmholtz.
+
+In spite of the many improvements introduced by Rowland and of the care
+with which his observations were made, recent workers have come to the
+conclusion that errors of unexpected amount have crept into his
+measurements of wave-lengths, and there is even a disposition to discard
+the grating altogether for fundamental work in favour of the so-called
+"interference methods," as developed by A. A. Michelson, and by C. Fabry
+and J. B. Perot. The grating would in any case retain its utility for
+the reference of new lines to standards otherwise fixed. For such
+standards a relative accuracy of at least one part in a million seems
+now to be attainable.
+
+Since the time of Fraunhofer many skilled mechanicians have given their
+attention to the ruling of gratings. Those of Nobert were employed by A.
+J. Angstrom in his celebrated researches upon wave-lengths. L. M.
+Rutherfurd introduced into common use the reflection grating, finding
+that speculum metal was less trying than glass to the diamond point,
+upon the permanence of which so much depends. In Rowland's dividing
+engine the screws were prepared by a special process devised by him, and
+the resulting gratings, plane and concave, have supplied the means for
+much of the best modern optical work. It would seem, however, that
+further improvements are not excluded.
+
+There are various copying processes by which it is possible to reproduce
+an original ruling in more or less perfection. The earliest is that of
+Quincke, who coated a glass grating with a chemical silver deposit,
+subsequently thickened with copper in an electrolytic bath. The metallic
+plate thus produced formed, when stripped from its support, a reflection
+grating reproducing many of the characteristics of the original. It is
+best to commence the electrolytic thickening in a silver acetate bath.
+At the present time excellent reproductions of Rowland's speculum
+gratings are on the market (Thorp, Ives, Wallace), prepared, after a
+suggestion of Sir David Brewster, by coating the original with a
+varnish, e.g. of celluloid. Much skill is required to secure that the
+film when stripped shall remain undeformed.
+
+A much easier method, applicable to glass originals, is that of
+photographic reproduction by contact printing. In several papers dating
+from 1872, Lord Rayleigh (see _Collected Papers_, i. 157, 160, 199, 504;
+iv. 226) has shown that success may be attained by a variety of
+processes, including bichromated gelatin and the old bitumen process,
+and has investigated the effect of imperfect approximation during the
+exposure between the prepared plate and the original. For many purposes
+the copies, containing lines up to 10,000 to the inch, are not inferior.
+It is to be desired that transparent gratings should be obtained from
+first-class ruling machines. To save the diamond point it might be
+possible to use something softer than ordinary glass as the material of
+the plate.
+
+9. _Talbot's Bands._--These very remarkable bands are seen under certain
+conditions when a tolerably pure spectrum is regarded with the naked
+eye, or with a telescope, _half the aperture being covered by a thin
+plate_, e.g. _of glass or mica_. The view of the matter taken by the
+discoverer (_Phil. Mag._, 1837, 10, p. 364) was that any ray which
+suffered in traversing the plate a retardation of an odd number of half
+wave-lengths would be extinguished, and that thus the spectrum would be
+seen interrupted by a number of dark bars. But this explanation cannot
+be accepted as it stands, being open to the same objection as Arago's
+theory of stellar scintillation.[9] It is as far as possible from being
+true that a body emitting homogeneous light would disappear on merely
+covering half the aperture of vision with a half-wave plate. Such a
+conclusion would be in the face of the principle of energy, which
+teaches plainly that the retardation in question leaves the aggregate
+brightness unaltered. The actual formation of the bands comes about in
+a very curious way, as is shown by a circumstance first observed by
+Brewster. When the retarding plate is held on the side towards the red
+of the spectrum, _the bands are not seen_. Even in the contrary case,
+the thickness of the plate must not exceed a certain limit, dependent
+upon the purity of the spectrum. A satisfactory explanation of these
+bands was first given by Airy (_Phil. Trans._, 1840, 225; 1841, 1), but
+we shall here follow the investigation of Sir G. G. Stokes (_Phil.
+Trans._, 1848, 227), limiting ourselves, however, to the case where the
+retarded and unretarded beams are contiguous and of equal width.
+
+  The aperture of the unretarded beam may thus be taken to be limited by
+  x = -h, x = 0, y = -l, y= +l; and that of the beam retarded by R to be
+  given by x = 0, x = h, y= -l, y = +l. For the former (1) S 3 gives
+                   _     _
+          1       / 0   / +l       /        x[xi] + y[eta]\
+    - ---------   |     |   sin k (at - f + -------------- )dxdy
+      [lambda]f  _/-h  _/-l        \              f       /
+
+           2lh         f        k[eta]l     2f       k[xi]h          /       [xi]h \
+    = - --------- . ------- sin ------- . ------ sin ------ . sin k (at - f - ----- ) (1),
+        [lambda]f   k[eta]l        f      k[xi]h       2f            \         2f  /
+
+  on integration and reduction.
+
+  For the retarded stream the only difference is that we must subtract R
+  from at, and that the limits of x are 0 and +h. We thus get for the
+  disturbance at [xi], [eta], due to this stream
+
+         2lh         f        k[eta]l     2f       k[xi]h          /           [xi]h \
+    - --------- . ------- sin ------- . ------ sin ------ . sin k (at - f - R + ----- ) (2).
+      [lambda]f   k[eta]l        f      k[xi]h       2f            \            2f   /
+
+  If we put for shortness [pi] for the quantity under the last circular
+  function in (1), the expressions (1), (2) may be put under the forms u
+  sin [tau], v sin ([tau] - [alpha]) respectively; and, if I be the
+  intensity, I will be measured by the sum of the squares of the
+  coefficients of sin [tau] and cos [tau] in the expression
+
+    u sin[tau] + v sin([tau] - [alpha]),
+
+  so that
+
+    I = u^2 + v^2 + 2uv cos[alpha],
+
+  which becomes on putting for u, v, and [alpha] their values, and
+  putting
+
+     /   f        k[eta]l \^2
+    ( ------- sin -------  )  = Q           (3),
+     \k[eta]l        f    /
+                                          _                                  _
+                4l^2           [pi][xi]h |           / 2[pi]R    2[pi][xi]h\  |
+    I = Q . ------------ sin^2 --------- |2 + 2 cos ( -------- - ---------- ) | (4).
+            [pi]^2[xi]^2       [lambda]f |_          \[lambda]   [lambda]f / _|
+
+  If the subject of examination be a luminous line parallel to [eta], we
+  shall obtain what we require by integrating (4) with respect to [eta]
+  from -[oo] to +[oo]. The constant multiplier is of no especial
+  interest so that we may take as applicable to the image of a line
+                                _                                 _
+           2         [pi][xi]h |         / 2[pi]R    2[pi][xi]h \  |
+    I = ------ sin^2 --------- |1 + cos ( -------- - ----------  ) | (5).
+        [xi]^2       [lambda]f |_        \[lambda]   [lambda]f  / _|
+
+  If R = 1/2[lambda], I vanishes at [xi]= 0; but the whole illumination,
+  represented by
+      _
+     / +[oo]
+     |    I d[xi], is independent of the value of R. If R = 0,
+    _/-[oo]
+
+          1          2[pi][xi]h
+    I = ------ sin^2 ----------,
+        [xi]^2       [lambda]f
+
+  in agreement with S 3, where a has the meaning here attached to 2h.
+
+  The expression (5) gives the illumination at [xi] due to that part of
+  the complete image whose geometrical focus is at [xi] = 0, the
+  retardation for this component being R. Since we have now to integrate
+  for the whole illumination at a particular point O due to all the
+  components which have their foci in its neighbourhood, we may
+  conveniently regard O as origin. [xi] is then the co-ordinate
+  relatively to O of any focal point O' for which the retardation is R;
+  and the required result is obtained by simply integrating (5) with
+  respect to [xi] from -[oo] to +[oo]. To each value of [xi] corresponds
+  a different value of [lambda], and (in consequence of the dispersing
+  power of the plate) of R. The variation of [lambda] may, however, be
+  neglected in the integration, except in 2[pi]R/[lambda], where a small
+  variation of [lambda] entails a comparatively large alteration of
+  phase. If we write
+
+    [rho] = 2[pi]R/[lambda]                                                     (6),
+
+  we must regard [rho] as a function of [xi], and we may take with
+  sufficient approximation under any ordinary circumstances
+
+    [rho] = [rho]' + [=omega][xi]                                               (7),
+
+  where [rho]' denotes the value of [rho] at O, and [=omega] is a
+  constant, which is positive when the retarding plate is held at the
+  side on which the lue of the spectrum _is seen_. The possibility of
+  dark bands depends upon [=omega] being positive. Only in this case can
+
+    cos {[rho]' + ([=omega] - 2[pi]h/[lambda]f)[xi]}
+
+  retain the constant value -1 throughout the integration, and then only
+  when
+
+    [=omega] = 2[pi]h / [lambda]f                                               (8)
+
+  and
+
+    cos [rho]' = -1                                                             (9).
+
+  The first of these equations is the condition for the formation of
+  dark bands, and the second marks their situation, which is the same
+  as that determined by the imperfect theory.
+
+  The integration can be effected without much difficulty. For the first
+  term in (5) the evaluation is effected at once by a known formula. In
+  the second term if we observe that
+
+    cos {[rho]' +([=omega] - 2[pi]h/[lambda]f)[xi]} = cos {[rho]'- g1[xi]}
+      = cos [rho]' cos g1[xi] + sin [rho]' sin g1[xi],
+
+  we see that the second part vanishes when integrated, and that the
+  remaining integral is of the form
+
+          _+[oo]
+         /                          d[xi]
+    w =  | sin^2 h1[xi] cos g1[xi] ------,
+        _/-[oo]                    [xi]^2
+
+  where
+
+    h1 = [pi]h/[lambda]f,  g1 = [omega] - 2[pi]h/[lambda]f    (10).
+
+  By differentiation with respect to g1 it may be proved that
+
+    w = 0                  from g1 = -[oo] to g1 = -2h1,
+    w = 1/2[pi](2h1 + g1)  from g1 = -2h1  to g1 = 0,
+    w = 1/2[pi](2h1 - g1)  from g1 =  0    to g1 = 2h1,
+    w = 0                  from g1 = 2h1   to g1 = [oo].
+
+  The integrated intensity, I', or
+
+    2[pi]h1 + 2 cos[rho]w,
+
+  is thus
+
+    I' = 2[pi]h1              (11),
+
+  when g1 numerically exceeds 2h1; and, when g1 lies between [+-]2h1,
+
+    I = [pi]2h1 + (2h1 - [sqrt] g1^2) cos[rho]'     (12).
+
+  It appears therefore that there are no bands at all unless [omega]
+  lies between 0 and +4h1, and that within these limits the best bands
+  are formed at the middle of the range when [omega] = 2h1. The
+  formation of bands thus requires that the retarding plate be held upon
+  the side already specified, so that [omega] be positive; and that the
+  thickness of the plate (to which [omega] is proportional) do not
+  exceed a certain limit, which we may call 2T0. At the best thickness
+  T0 the bands are black, and not otherwise.
+
+  The linear width of the band (e) is the increment of [xi] which alters
+  [rho] by 2[pi], so that
+
+    e = 2[pi]/[=omega]                             (13).
+
+  With the best thickness
+
+    [=omega] = 2[pi]h/[lambda]f                    (14),
+
+  so that in this case
+
+    e = [lambda]f/h                                (15).
+
+  The bands are thus of the same width as those due to two infinitely
+  narrow apertures coincident with the central lines of the retarded and
+  unretarded streams, the subject of examination being itself a fine
+  luminous line.
+
+  If it be desired to see a given number of bands in the whole or in any
+  part of the spectrum, the thickness of the retarding plate is thereby
+  determined, independently of all other considerations. But in order
+  that the bands may be really visible, and still more in order that
+  they may be black, another condition must be satisfied. It is
+  necessary that the aperture of the pupil be accommodated to the
+  angular extent of the spectrum, or reciprocally. Black bands will be
+  too fine to be well seen unless the aperture (2h) of the pupil be
+  somewhat contracted. One-twentieth to one-fiftieth of an inch is
+  suitable. The aperture and the number of bands being both fixed, the
+  condition of blackness determines the angular magnitude of a band and
+  of the spectrum. The use of a grating is very convenient, for not only
+  are there several spectra in view at the same time, but the dispersion
+  can be varied continuously by sloping the grating. The slits may be
+  cut out of tin-plate, and half covered by mica or "microscopic glass,"
+  held in position by a little cement.
+
+  If a telescope be employed there is a distinction to be observed,
+  according as the half-covered aperture is between the eye and the
+  ocular, or in front of the object-glass. In the former case the
+  function of the telescope is simply to increase the dispersion, and
+  the formation of the bands is of course independent of the particular
+  manner in which the dispersion arises. If, however, the half-covered
+  aperture be in front of the object-glass, the phenomenon is magnified
+  as a whole, and the desirable relation between the (unmagnified)
+  dispersion and the aperture is the same as without the telescope.
+  There appears to be no further advantage in the use of a telescope
+  than the increased facility of accommodation, and for this of course a
+  very low power suffices.
+
+  The original investigation of Stokes, here briefly sketched, extends
+  also to the case where the streams are of unequal width h, k, and are
+  separated by an interval 2g. In the case of unequal width the bands
+  cannot be black; but if h = k, the finiteness of 2g does not preclude
+  the formation of black bands.
+
+  The theory of Talbot's bands with a half-covered _circular_ aperture
+  has been considered by H. Struve (_St Peters. Trans._, 1883, 31, No.
+  1).
+
+  The subject of "Talbot's bands" has been treated in a very instructive
+  manner by A. Schuster (_Phil. Mag._, 1904), whose point of view offers
+  the great advantage of affording an instantaneous explanation of the
+  peculiarity noticed by Brewster. A plane _pulse_, i.e. a disturbance
+  limited to an infinitely thin slice of the medium, is supposed to fall
+  upon a parallel grating, which again may be regarded as formed of
+  infinitely thin wires, or infinitely narrow lines traced upon glass.
+  The secondary pulses diverted by the ruling fall upon an object-glass
+  as usual, and on arrival at the focus constitute a procession equally
+  spaced in time, the interval between consecutive members depending
+  upon the obliquity. If a retarding plate be now inserted so as to
+  operate upon the pulses which come from one side of the grating, while
+  leaving the remainder unaffected, we have to consider what happens at
+  the focal point chosen. A full discussion would call for the formal
+  application of Fourier's theorem, but some conclusions of importance
+  are almost obvious.
+
+  Previously to the introduction of the plate we have an effect
+  corresponding to wave-lengths closely grouped around the principal
+  wave-length, viz. [sigma] sin [phi], where [sigma] is the
+  grating-interval and [phi] the obliquity, the closeness of the
+  grouping increasing with the number of intervals. In addition to these
+  wave-lengths there are other groups centred round the wave-lengths
+  which are submultiples of the principal one--the overlapping spectra
+  of the second and higher orders. Suppose now that the plate is
+  introduced so as to cover naif the aperture and that it retards those
+  pulses which would otherwise arrive first. The consequences must
+  depend upon the amount of the retardation. As this increases from
+  zero, the two processions which correspond to the two halves of the
+  aperture begin to overlap, and the overlapping gradually increases
+  until there is almost complete superposition. The stage upon which we
+  will fix our attention is that where the one procession bisects the
+  intervals between the other, so that a new simple procession is
+  constituted, containing the same number of members as before the
+  insertion of the plate, but now spaced at intervals only half as
+  great. It is evident that the effect at the focal point is the
+  obliteration of the first and other spectra of odd order, so that as
+  regards the spectrum of the first order we may consider that the two
+  beams _interfere_. The formation of black bands is thus explained, and
+  it requires that the plate be introduced upon one particular side, and
+  that the amount of the retardation be adjusted to a particular value.
+  If the retardation be too little, the overlapping of the processions
+  is incomplete, so that besides the procession of half period there are
+  residues of the original processions of full period. The same thing
+  occurs if the retardation be too great. If it exceed the double of the
+  value necessary for black bands, there is again no overlapping and
+  consequently no interference. If the plate be introduced upon the
+  other side, so as to retard the procession originally in arrear, there
+  is no overlapping, whatever may be the amount of retardation. In this
+  way the principal features of the phenomenon are accounted for, and
+  Schuster has shown further how to extend the results to spectra having
+  their origin in prisms instead of gratings.
+
+10. _Diffraction when the Source of Light is not seen in Focus._--The
+phenomena to be considered under this head are of less importance than
+those investigated by Fraunhofer, and will be treated in less detail;
+but in view of their historical interest and of the ease with which many
+of the experiments may be tried, some account of their theory cannot be
+omitted. One or two examples have already attracted our attention when
+considering Fresnel's zones, viz. the shadow of a circular disk and of a
+screen circularly perforated.
+
+Fresnel commenced his researches with an examination of the fringes,
+external and internal, which accompany the shadow of a narrow opaque
+strip, such as a wire. As a source of light he used sunshine passing
+through a very small hole perforated in a metal plate, or condensed by a
+lens of short focus. In the absence of a heliostat the latter was the
+more convenient. Following, unknown to himself, in the footsteps of
+Young, he deduced the principle of interference from the circumstance
+that the darkness of the interior bands requires the co-operation of
+light from both sides of the obstacle. At first, too, he followed Young
+in the view that the exterior bands are the result of interference
+between the direct light and that reflected from the edge of the
+obstacle, but he soon discovered that the character of the edge--e.g.
+whether it was the cutting edge or the back of a razor--made no material
+difference, and was thus led to the conclusion that the explanation of
+these phenomena requires nothing more than the application of Huygens's
+principle to the unobstructed parts of the wave. In observing the bands
+he received them at first upon a screen of finely ground glass, upon
+which a magnifying lens was focused; but it soon appeared that the
+ground glass could be dispensed with, the diffraction pattern being
+viewed in the same way as the image formed by the object-glass of a
+telescope is viewed through the eye-piece. This simplification was
+attended by a great saving of light, allowing measures to be taken such
+as would otherwise have presented great difficulties.
+
+  In theoretical investigations these problems are usually treated as of
+  two dimensions only, everything being referred to the plane passing
+  through the luminous point and perpendicular to the diffracting edges,
+  supposed to be straight and parallel. In strictness this idea is
+  appropriate only when the source is a luminous line, emitting
+  cylindrical waves, such as might be obtained from a luminous point
+  with the aid of a cylindrical lens. When, in order to apply Huygens's
+  principle, the wave is supposed to be broken up, the phase is the same
+  at every element of the surface of resolution which lies upon a line
+  perpendicular to the plane of reference, and thus the effect of the
+  whole line, or rather infinitesimal strip, is related in a constant
+  manner to that of the element which lies in the plane of reference,
+  and may be considered to be represented thereby. The same method of
+  representation is applicable to spherical waves, issuing from a
+  _point_, if the radius of curvature be large; for, although there is
+  variation of phase along the length of the infinitesimal strip, the
+  whole effect depends practically upon that of the central parts where
+  the phase is sensibly constant.[10]
+
+  [Illustration: FIG. 17.]
+
+  In fig. 17 APQ is the arc of the circle representative of the
+  wave-front of resolution, the centre being at O, and the radius QA
+  being equal to a. B is the point at which the effect is required,
+  distant a + b from O, so that AB = b, AP = s, PQ = ds.
+
+  Taking as the standard phase that of the secondary wave from A, we may
+  represent the effect of PQ by
+
+               /t   [delta] \
+    cos 2[pi] ( - - -------- ).ds,
+               \r   [lambda]/
+
+  where [delta] = BP - AP is the retardation at B of the wave from P
+  relatively to that from A.
+
+  Now
+
+    [delta] = (a + b) s^2/2ab                    (1),
+
+  so that, if we write
+
+    2[pi][delta] =  [pi](a + b)s^2     [pi]v^2
+    ------------   ---------------  = ------     (2),
+      [lambda]       ab[lambda]         2
+
+  the effect at B is
+                                  _                                _
+     /ab[lambda]\1/2 /    2[pi]t /                         2[pi]t /                  \
+    ( ---------- )  ( cos ------ | cos 1/2[pi]v^2.dv + sin ------ | sin 1/2[pi]v^2.dv ) (3),
+     \2(a + b)  /    \    [tau] _/                         [tau] _/                  /
+
+  the limits of integration depending upon the disposition of the
+  diffracting edges. When a, b, [lambda] are regarded as constant, the
+  first factor may be omitted,--as indeed should be done for
+  consistency's sake, inasmuch as other factors of the same nature have
+  been omitted already.
+
+  The intensity I^2, the quantity with which we are principally
+  concerned, may thus be expressed
+
+              _                           _
+          /  /                  \^2   /  /                  \^2
+    I^2= (   | cos 1/2[pi]v^2.dv ) + (   | sin 1/2[pi]v^2.dv ) (4).
+          \ _/                  /     \ _/                  /
+
+  These integrals, taken from v = 0, are known as Fresnel's integrals;
+  we will denote them by C and S, so that
+         _                          _
+        / v                        / v
+    C = | cos 1/2[pi]v^2.dv,   S = | cos 1/2[pi]v^2.dv               (5).
+       _/0                        _/0
+
+  When the upper limit is infinity, so that the limits correspond to the
+  inclusion of half the primary wave, C and S are both equal to 1/2, by a
+  known formula; and on account of the rapid fluctuation of sign the
+  parts of the range beyond very moderate values of v contribute but
+  little to the result.
+
+  Ascending series for C and S were given by K. W. Knockenhauer, and are
+  readily investigated. Integrating by parts, we find
+
+              _v                                        _v
+             /   i.1/2[pi]v^2   i.1/2[pi]v^2   1       /   i.1/2[pi]v^2
+    C + iS = |  e         dv = e         . v - - i[pi] |  e         dv^3;
+            _/0                                3      _/0
+
+  and, by continuing this process,
+
+              i.1/2[pi]v^2 /    i[pi]       i[pi] i[pi]       i[pi] i[pi] i[pi]          \
+    C + iS = e            ( v - ----- v^3 + ----- ----- v^5 - ----- ----- ----- v^7 + ... ).
+                           \      3           3     5           3     5     7            /
+
+  By separation of real and imaginary parts,
+
+    C = M cos 1/2[pi]v^2 - N sin 1/2[pi]v^2  \
+    S = M sin 1/2[pi]v^2 - N cos 1/2[pi]v^2  /          (6)
+
+  where
+
+        v   [pi]^2v^5   [pi]^4v^9
+    M = - - --------- + --------- - ...           (7)
+        1     3.5        3.5.7.9
+
+        [pi]v^3  [pi]^3v^7    [pi]^5v^11
+    N = ------ - --------- + ------------ ...     (8)
+         1.3      1.3.5.7    1.3.5.7.9.11
+
+  These series are convergent for all values of v, but are practically
+  useful only when v is small.
+
+  Expressions suitable for discussion when v is large were obtained by
+  L. P. Gilbert (_Mem. cour. de l'Acad. de Bruxelles_, 31, p. 1). Taking
+
+    1/2[pi]v^2 = u                                (9),
+
+  we may write
+                                _
+                    1          /u  e^iu du
+    C + iS =  -------------    |   --------      (10).
+              [sqrt](2[pi])   _/0  [sqrt] u
+
+Again, by a known formula,
+
+                             _[oo]
+        1           1       /  e^-ux dx
+    --------  = ----------  |  --------         (11).
+    [sqrt] u    [sqrt][pi] _/0  [sqrt]x
+
+  Substituting this in (10), and inverting the order of integration, we
+  get
+
+                       _[oo]         _u
+                1     /     dx      /  e^u(i - x)
+    C + iS = -------  |  --------   |  ----------- dx
+             [sqrt]2 _/0 [sqrt] x  _/0   [sqrt]x
+
+                   _[oo]
+           1     /     dx    e^u(i - x) - 1
+      = -------  |  -------- -------------- dx     (12).
+        [sqrt]2 _/0 [sqrt] x     i - x
+
+  Thus, if we take
+
+                      _[oo]
+             1       /  e^-ux [sqrt](x).dx
+    G = -----------  |  ------------------,
+        [pi][sqrt]2 _/0       1 + x^2
+
+                      _[oo]
+             1       /      e^-ux dx
+    H = -----------  | ------------------         (13).
+        [pi][sqrt]2 _/ [sqrt]x . (1 + x^2)
+                     0
+
+    C = 1/2 - G cos u + H sin u,  S = 1/2 - G sin u - H cos u   (14).
+
+  The constant parts in (14), viz. 1/2, may be determined by direct
+  integration of (12), or from the observation that by their
+  constitution G and H vanish when u = [oo], coupled with the fact that
+  C and S then assume the value 1/2.
+
+  Comparing the expressions for C, S in terms of M, N, and in terms of
+  G, H, we find that
+
+    G = 1/2 (cos u + sin u) - M,  H = 1/2 (cos u - sin u) + N  (15),
+
+  formulae which may be utilized for the calculation of G, H when u (or
+  v) is small. For example, when u = 0, M = 0, N = 0, and consequently G
+  = H = 1/2.
+
+  Descending series of the semi-convergent class, available for
+  numerical calculation when u is moderately large, can be obtained from
+  (12) by writing x = uy, and expanding the denominator in powers of y.
+  The integration of the several terms may then be effected by the
+  formula
+
+      _ [oo]
+     /  -y   q-1/2
+     | e    y    dy = [Gamma](q + 1/2) = (q - 1/2)(q - 3/2) ... 1/2[sqrt][pi];
+    _/0
+
+  and we get in terms of v
+
+            1          1.3.5       1.3.5.9
+    G = --------- - ---------- + ----------- -          (16),
+        [pi]^2v^3   [pi]^4 v^7   [pi]^6 v^11
+
+          1         1.3       1.3.5.7
+    H = ----- - ---------- + ---------- -               (17).
+        [pi]v   [pi]^3 v^5   [pi]^5 v^9
+
+  The corresponding values of C and S were originally derived by A. L.
+  Cauchy, without the use of Gilbert's integrals, by direct integration
+  by parts.
+
+  From the series for G and H just obtained it is easy to verify that
+
+    dH             dG
+    -- = - [pi]vG, -- = [pi]vH - 1                     (18).
+    dv             dv
+
+  We now proceed to consider more particularly the distribution of light
+  upon a screen PBQ near the shadow of a straight edge A. At a point P
+  within the geometrical shadow of the obstacle, the half of the wave to
+  the right of C (fig. 18), the nearest point on the wave-front, is
+  wholly intercepted, and on the left the integration is to be taken
+  from s = CA to s = [oo]. If V be the value of v corresponding to CA,
+  viz.
+
+         / /  2(a + b)  \
+    V=  / (  ----------  ).CA,                       (19),
+      \/   \ ab[lambda] /
+
+  we may write
+
+               _[oo]                       _[oo]
+           /  /                  \^2   /  /                  \^2
+    I^2 = (   | cos 1/2[pi]v^2.dv ) + (   | sin 1/2[pi]v^2.dv ) (20),
+           \ _/v                 /     \ _/v                 /
+
+  or, according to our previous notation,
+
+    I^2 = (1/2 - Cv)^2 + (1/2 - Sv)^2 = G^2 + H^2               (21).
+
+  Now in the integrals represented by G and H every element diminishes
+  as V increases from zero. Hence, as CA increases, viz. as the point P
+  is more and more deeply immersed in the shadow, the illumination
+  _continuously_ decreases, and that without limit. It has long been
+  known from observation that there are no bands on the interior side of
+  the shadow of the edge.
+
+  [Illustration: FIG. 18.]
+
+  The law of diminution when V is moderately large is easily expressed
+  with the aid of the series (16), (17) for G, H. We have ultimately G =
+  0, H = ([pi]V)^-1, so that
+
+    I^2 = 1/[pi]^2V^2,
+
+  or the illumination is inversely as the square of the distance from
+  the shadow of the edge.
+
+  For a point Q outside the shadow the integration extends over _more_
+  than half the primary wave. The intensity may be expressed by
+
+    I^2 = (1/2 + Cv)^2 + (1/2 + Sv)^2                 (22);
+
+  and the maxima and minima occur when
+
+                dC               dS
+    (1/2 + C_v) -- + (1/2 + S_v) -- = 0,
+                dV               dV
+
+  whence
+
+    sin 1/2[pi]V^2 + cos 1/2[pi]V^2 = G               (23).
+
+  When V = 0, viz. at the edge of the shadow, I^2 = 1/2; when V = [oo],
+  I^2 = 2, on the scale adopted. The latter is the intensity due to the
+  uninterrupted wave. The quadrupling of the intensity in passing
+  outwards from the edge of the shadow is, however, accompanied by
+  fluctuations giving rise to bright and dark bands. The position of
+  these bands determined by (23) may be very simply expressed when V is
+  large, for then sensibly G = 0, and
+
+    1/2[pi]V^2 = 3/4[pi] + n[pi]                      (24),
+
+  n being an integer. In terms of [delta], we have from (2)
+
+    [delta] = (3/8 + 1/2n)[lambda]                    (25).
+
+  The first maximum in fact occurs when [delta] = 3/8[lambda]
+  -.0046[lambda], and the first minimum when [delta] = 7/8[lambda]
+  -.0016[lambda], the corrections being readily obtainable from a table
+  of G by substitution of the approximate value of V.
+
+  The position of Q corresponding to a given value of V, that is, to a
+  band of given order, is by (19)
+
+         a + b          / / b[lambda](a + b) \
+    BQ = ----- AD = V  / (  ----------------- )                               (26).
+           a         \/   \        2a        /
+
+  By means of this expression we may trace the locus of a band of given
+  order as b varies. With sufficient approximation we may regard BQ and
+  b as rectangular co-ordinates of Q. Denoting them by x, y, so that AB
+  is axis of y and a perpendicular through A the axis of x, and
+  rationalizing (26), we have
+
+    2ax^2 - V^2[lambda]y^2 - V^2a[lambda]y = 0,
+
+  which represents a hyperbola with vertices at O and A.
+
+  From (24), (26) we see that the width of the bands is of the order
+  [sqrt] {b[lambda](a + b)/a}. From this we may infer the limitation
+  upon the width of the source of light, in order that the bands may be
+  properly formed. If [omega] be the apparent magnitude of the source
+  seen from A, [omega]b should be much smaller than the above quantity,
+  or
+
+    [omega] < [sqrt] {[lambda](a + b)/ab}        (27).
+
+  If a be very great in relation to b, the condition becomes
+
+    [omega] < [sqrt] ([lambda]/b)                (28).
+
+  so that if b is to be moderately great (1 metre), the apparent
+  magnitude of the sun must be greatly reduced before it can be used as
+  a source. The values of V for the maxima and minima of intensity, and
+  the magnitudes of the latter, were calculated by Fresnel. An extract
+  from his results is given in the accompanying table.
+
+    +--------------------+----------+------------+
+    |                    |     V    |    I^2     |
+    +--------------------+----------+------------+
+    | First maximum      |  1.2172  |   2.7413   |
+    | First minimum      |  1.8726  |   1.5570   |
+    | Second maximum     |  2.3449  |   2.3990   |
+    | Second minimum     |  2.7392  |   1.6867   |
+    | Third maximum.     |  3.0820  |   2.3022   |
+    | Third minimum      |  3.3913  |   1.7440   |
+    +--------------------+----------+------------+
+
+  A very thorough investigation of this and other related questions,
+  accompanied by fully worked-out tables of the functions concerned,
+  will be found in a paper by E. Lommel (_Abh. bayer. Akad. d. Wiss._
+  II. CI., 15, Bd., iii. Abth., 1886).
+
+  When the functions C and S have once been calculated, the discussion
+  of various diffraction problems is much facilitated by the idea, due
+  to M. A. Cornu (_Journ. de Phys._, 1874, 3, p. 1; a similar suggestion
+  was made independently by G. F. Fitzgerald), of exhibiting as a curve
+  the relationship between C and S, considered as the rectangular
+  co-ordinates (x, y) of a point. Such a curve is shown in fig. 19,
+  where, according to the definition (5) of C, S,
+
+          _ v                           _ v
+         /                             /
+    x =  | cos 1/2[pi]v^2.dv,      y = | sin 1/2[pi]v^2.dv   (29).
+        _/0                           _/0
+
+  The origin of co-ordinates O corresponds to v = 0; and the asymptotic
+  points J, J', round which the curve revolves in an ever-closing
+  spiral, correspond to v = [+-][oo].
+
+  The intrinsic equation, expressing the relation between the arc
+  [sigma] (measured from O) and the inclination [phi] of the tangent at
+  any points to the axis of x, assumes a very simple form. For
+
+    dx = cos 1/2[pi]v^2.dv,   dy = sin 1/2[pi]v^2.dv;
+
+  so that
+               _
+              /
+    [sigma] = | [sqrt] (dx^2 + dy^2) = v,                (30),
+             _/
+
+    [phi] = tan^-1 (dy/dx) = 1/2[pi]v^2                  (31).
+
+
+
+  Accordingly,
+
+    [phi] = 1/2[pi][sigma]^2                             (32);
+
+  and for the curvature,
+
+    d[phi]/d[sigma] = [pi][sigma]                        (33).
+
+  Cornu remarks that this equation suffices to determine the general
+  character of the curve. For the osculating circle at any point
+  includes the whole of the curve which lies beyond; and the successive
+  convolutions envelop one another without intersection.
+
+  [Illustration: Fig. 19.]
+
+  The utility of the curve depends upon the fact that the elements of
+  arc represent, in amplitude and phase, the component vibrations due to
+  the corresponding portions of the primary wave-front. For by (30)
+  d[sigma] = dv, and by (2) dv is proportional to ds. Moreover by (2)
+  and (31) the retardation of phase of the elementary vibration from PQ
+  (fig. 17) is 2[pi][delta]/[lambda], or [phi]. Hence, in accordance
+  with the rule for compounding vector quantities, the resultant
+  vibration at B, due to any finite part of the primary wave, is
+  represented in amplitude and phase by the chord joining the
+  extremities of the corresponding arc ([sigma]2 - [sigma]1).
+
+  In applying the curve in special cases of diffraction to exhibit the
+  effect at any point P (fig. 18) the centre of the curve O is to be
+  considered to correspond to that point C of the primary wave-front
+  which lies nearest to P. The operative part, or parts, of the curve
+  are of course those which represent the unobstructed portions of the
+  primary wave.
+
+  Let us reconsider, following Cornu, the diffraction of a screen
+  unlimited on one side, and on the other terminated by a straight edge.
+  On the illuminated side, at a distance from the shadow, the vibration
+  is represented by JJ'. The co-ordinates oi J, J' being (1/2, 1/2),
+  (-1/2, -1/2), I^2 is 2; and the phase is 1/8 period in arrear of that
+  of the element at O. As the point under contemplation is supposed to
+  approach the shadow, the vibration is represented by the chord drawn
+  from J to a point on the other half of the curve, which travels
+  inwards from J' towards O. The amplitude is thus subject to
+  fluctuations, which increase as the shadow is approached. At the point
+  O the intensity is one-quarter of that of the entire wave, and after
+  this point is passed, that is, when we have entered the geometrical
+  shadow, the intensity falls off gradually to zero, _without
+  fluctuations_. The whole progress of the phenomenon is thus exhibited
+  to the eye in a very instructive manner.
+
+  We will next suppose that the light is transmitted by a slit, and
+  inquire what is the effect of varying the width of the slit upon the
+  illumination at the projection of its centre. Under these
+  circumstances the arc to be considered is bisected at O, and its
+  length is proportional to the width of the slit. It is easy to see
+  that the length of the chord (which passes in all cases through O)
+  increases to a maximum near the place where the phase-retardation is
+  3/8 of a period, then diminishes to a minimum when the retardation is
+  about 7/8 of a period, and so on.
+
+  If the slit is of constant width and we require the illumination at
+  various points on the screen behind it, we must regard the arc of the
+  curve as of _constant length_. The intensity is then, as always,
+  represented by the square of the length of the chord. If the slit be
+  narrow, so that the arc is short, the intensity is constant over a
+  wide range, and does not fall off to an important extent until the
+  discrepancy of the extreme phases reaches about a quarter of a period.
+
+  We have hitherto supposed that the shadow of a diffracting obstacle is
+  received upon a diffusing screen, or, which comes to nearly the same
+  thing, is observed with an eye-piece. If the eye, provided if
+  necessary with a perforated plate in order to reduce the aperture, be
+  situated inside the shadow at a place where the illumination is still
+  sensible, and be focused upon the diffracting edge, the light which it
+  receives will appear to come from the neighbourhood of the edge, and
+  will present the effect of a silver lining. This is doubtless the
+  explanation of a "pretty optical phenomenon, seen in Switzerland, when
+  the sun rises from behind distant trees standing on the summit of a
+  mountain."[11]
+
+II. _Dynamical Theory of Diffraction._--The explanation of diffraction
+phenomena given by Fresnel and his followers is independent of special
+views as to the nature of the aether, at least in its main features; for
+in the absence of a more complete foundation it is impossible to treat
+rigorously the mode of action of a solid obstacle such as a screen. But,
+without entering upon matters of this kind, we may inquire in what
+manner a primary wave may be resolved into elementary secondary waves,
+and in particular as to the law of intensity and polarization in a
+secondary wave as dependent upon its direction of propagation, and upon
+the character as regards polarization of the primary wave. This question
+was treated by Stokes in his "Dynamical Theory of Diffraction" (_Camb.
+Phil. Trans._, 1849) on the basis of the elastic solid theory.
+
+  Let x, y, z be the co-ordinates of any particle of the medium in its
+  natural state, and [chi], [eta], [zeta] the displacements of the same
+  particle at the end of time t, measured in the directions of the three
+  axes respectively. Then the first of the equations of motion may be
+  put under the form
+
+    d^2[xi]      /d^2[xi]   d^2[xi]   d^2[xi]\               d^2 /d^2[xi]   d^2[eta]   d^2[zeta]\
+    ------ = b^2( ------ + -------- + ------- ) + (a^2 - b^2)---( ------- + -------- + --------- ),
+     dt^2        \ dx^2      dy^2      dz^2  /               dx  \  dx^2      dy^2        dz^2  /
+
+  where a2 and b2 denote the two arbitrary constants. Put for shortness
+
+    d^2[xi]   d^2[eta]   d^2[zeta]
+    ------- + -------- + --------- = [delta]    (1),
+     dx^2       dy^2        dz^2
+
+  and represent by [Delta]^2[chi] the quantity multiplied by b^2.
+  According to this notation, the three equations of motion are
+
+    d^2[xi]                                      d[delta] \
+    -------   = b^2[Delta]^2[xi]   + (a^2 - b^2) -------- |
+      dt^2                                          dx    |
+                                                          |
+    d^2[eta]                                     d[delta] |
+    --------  = b^2[Delta]^2[eta]  + (a^2 - b^2) --------  >   (2).
+      dt^2                                          dy    |
+                                                          |
+    d^2[zeta]                                    d[delta] |
+    --------- = b^2[Delta]^2[zeta] + (a^2 - b^2) -------- |
+      dt^2                                          dz    /
+
+  It is to be observed that S denotes the dilatation of volume of the
+  element situated at (x, y, z). In the limiting case in which the
+  medium is regarded as absolutely incompressible [delta] vanishes; but,
+  in order that equations (2) may preserve their generality, we must
+  suppose a at the same time to become infinite, and replace a^2[delta]
+  by a new function of the co-ordinates.
+
+  These equations simplify very much in their application to plane
+  waves. If the ray be parallel to OX, and the direction of vibration
+  parallel to OZ, we have [xi] = 0, [eta] = 0, while [zeta] is a
+  function of x and t only. Equation (1) and the first pair of equations
+  (2) are thus satisfied identically. The third equation gives
+
+    d^2[zeta]   d^2[zeta]
+    --------- = ---------                   (3),
+      dt^2        dx^2
+
+  of which the solution is
+
+    [zeta] = f(bt - x)                      (4),
+
+  where f is an arbitrary function.
+
+  The question as to the law of the secondary waves is thus answered by
+  Stokes. "Let [xi] = 0, [eta] = 0, [zeta] = f(bt-x) be the
+  displacements corresponding to the incident light; let O1 be any point
+  in the plane P (of the wave-front), dS an element of that plane
+  adjacent to O1, and consider the disturbance due to that portion only
+  of the incident disturbance which passes continually across dS. Let O
+  be any point in the medium situated at a distance from the point O1
+  which is large in comparison with the length of a wave; let O1O = r,
+  and let this line make an angle [theta] with the direction of
+  propagation of the incident light, or the axis of x, and [phi] with
+  the direction of vibration, or axis of z. Then the displacement at O
+  will take place in a direction perpendicular to O1O, and lying in the
+  plane ZO1O; and, if [zeta]' be the displacement at O, reckoned
+  positive in the direction nearest to that in which the incident
+  vibrations are reckoned positive,
+
+                dS
+    [zeta]' = ------ ( 1 + cos[theta]) sin[phi] f'(bt - r).
+              4[pi]r
+
+  In particular, if
+
+                        2[pi]
+    f(bt - x) = c sin -------- (bt - x)        (5),
+                      [lambda]
+
+  we shall have
+
+                   cdS                                  2[pi]
+    [zeta]' =  ---------- (1 + cos[theta]) sin[phi]cos -------- (bt - r)  (6)."
+               2[lambda]r                              [lambda]
+
+  It is then verified that, after integration with respect to dS, (6)
+  gives the same disturbance as if the primary wave had been supposed to
+  pass on unbroken.
+
+  The occurrence of sin [phi] as a factor in (6) shows that the relative
+  intensities of the primary light and of that diffracted in the
+  direction [theta] depend upon the condition of the former as regards
+  polarization. If the direction of primary vibration be perpendicular
+  to the plane of diffraction (containing both primary and secondary
+  rays), sin [phi] = 1; but, if the primary vibration be in the plane of
+  diffraction, sin [phi] = cos [theta]. This result was employed by
+  Stokes as a criterion of the direction of vibration; and his
+  experiments, conducted with gratings, led him to the conclusion that
+  the vibrations of polarized light are executed in a direction
+  _perpendicular_ to the plane of polarization.
+
+  The factor (1 + cos [theta]) shows in what manner the secondary
+  disturbance depends upon the direction in which it is propagated with
+  respect to the front of the primary wave.
+
+  If, as suffices for all practical purposes, we limit the application
+  of the formulae to points in advance of the plane at which the wave is
+  supposed to be broken up, we may use simpler methods of resolution
+  than that above considered. It appears indeed that the purely
+  mathematical question has no definite answer. In illustration of this
+  the analogous problem for sound may be referred to. Imagine a flexible
+  lamina to be introduced so as to coincide with the plane at which
+  resolution is to be effected. The introduction of the lamina (supposed
+  to be devoid of inertia) will make no difference to the propagation of
+  plane parallel sonorous waves through the position which it occupies.
+  At every point the motion of the lamina will be the same as would have
+  occurred in its absence, the pressure of the waves impinging from
+  behind being just what is required to generate the waves in front. Now
+  it is evident that the aerial motion in front of the lamina is
+  determined by what happens at the lamina without regard to the cause
+  of the motion there existing. Whether the necessary forces are due to
+  aerial pressures acting on the rear, or to forces directly impressed
+  from without, is a matter of indifference. The conception of the
+  lamina leads immediately to two schemes, according to which a primary
+  wave may be supposed to be broken up. In the first of these the
+  element dS, the effect of which is to be estimated, is supposed to
+  execute its actual motion, while every other element of the plane
+  lamina is maintained at rest. The resulting aerial motion in front is
+  readily calculated (see Rayleigh, _Theory of Sound_, S 278); it is
+  symmetrical with respect to the origin, i.e. independent of [theta].
+  When the secondary disturbance thus obtained is integrated with
+  respect to dS over the entire plane of the lamina, the result is
+  necessarily the same as would have been obtained had the primary wave
+  been supposed to pass on without resolution, for this is precisely the
+  motion generated when every element of the lamina vibrates with a
+  common motion, equal to that attributed to dS. The only assumption
+  here involved is the evidently legitimate one that, when two systems
+  of variously distributed motion at the lamina are superposed, the
+  corresponding motions in front are superposed also.
+
+  The method of resolution just described is the simplest, but it is
+  only one of an indefinite number that might be proposed, and which are
+  all equally legitimate, so long as the question is regarded as a
+  merely mathematical one, without reference to the physical properties
+  of actual screens. If, instead of supposing the _motion_ at dS to be
+  that of the primary wave, and to be zero elsewhere, we suppose the
+  _force_ operative over the element dS of the lamina to be that
+  corresponding to the primary wave, and to vanish elsewhere, we obtain
+  a secondary wave following quite a different law. In this case the
+  motion in different directions varies as cos[theta], vanishing at
+  right angles to the direction of propagation of the primary wave. Here
+  again, on integration over the entire lamina, the aggregate effect of
+  the secondary waves is necessarily the same as that of the primary.
+
+  In order to apply these ideas to the investigation of the secondary
+  wave of light, we require the solution of a problem, first treated by
+  Stokes, viz. the determination of the motion in an infinitely extended
+  elastic solid due to a locally applied periodic force. If we suppose
+  that the force impressed upon the element of mass D dx dy dz is
+
+    DZ dx dy dz,
+
+  being everywhere parallel to the axis of Z, the only change required
+  in our equations (1), (2) is the addition of the term Z to the second
+  member of the third equation (2). In the forced vibration, now under
+  consideration, Z, and the quantities [xi], [eta], [zeta], [delta]
+  expressing the resulting motion, are to be supposed proportional to
+  e^int, where i = [sqrt](-1), and n = 2[pi]/[tau], [tau] being the
+  periodic time. Under these circumstances the double differentiation
+  with respect to t of any quantity is equivalent to multiplication by
+  the factor -n^2, and thus our equations take the form
+
+                                             d[delta]      \
+    (b^2[Delta]^2 + n^2)[xi]   + (a^2 - b^2) -------- = 0  |
+                                                dx         |
+                                                           |
+                                             d[delta]      |
+    (b^2[Delta]^2 + n^2)[eta]  + (a^2 - b^2) -------- = 0   >   (7).
+                                                dx         |
+                                                           |
+                                             d[delta]      |
+    (b^2[Delta]^2 + n^2)[zeta] + (a^2 - b^2) -------- = -Z |
+                                                dx         /
+
+  It will now be convenient to introduce the quantities.[=omega]1,
+  [=omega]2, [=omega]3 which express the _rotations_ of the elements of
+  the medium round axes parallel to those of co-ordinates, in accordance
+  with the equations
+
+                d[xi]   d[eta]              d[eta]   d[zeta]
+    [=omega]3 = ----- - ------, [=omega]1 = ------ - -------,
+                 dy       dx'                 dz       dy
+
+                d[zeta]   d[xi]
+    [=omega]2 = ------- - -----                (8).
+                  dx       dz
+
+  In terms of these we obtain from (7), by differentiation and
+  subtraction,
+
+    (b^2[Delta]^2 + n^2) [=omega]3 = 0       \
+    (b^2[Delta]^2 + n^2) [=omega]1 = dZ/dy    >     (9).
+    (b^2[Delta]^2 + n^2) [=omega]2 = -dZ/dx  /
+
+
+  The first of equations (9) gives
+
+    [=omega]3 = 0                                  (10).
+
+  For =[omega]1, we have
+                           _ _ _    -ikr
+                   1      / / / dZ e
+    [=omega]1 = --------  | | | -- ----- dx dy dz  (11),
+                4[pi]b^2 _/_/_/ dy   r
+
+  where r is the distance between the element dx dy dz and the point
+  where [=omega]1 is estimated, and
+
+    k = n/b = 2[pi]/[lambda]                       (12),
+
+  [lambda] being the wave-length.
+
+  (This solution may be verified in the same manner as Poisson's
+  theorem, in which k = 0.)
+
+  We will now introduce the supposition that the force Z acts only
+  within a small space of volume T, situated at (x, y, z), and for
+  simplicity suppose that it is at the origin of co-ordinates that the
+  rotations are to be estimated. Integrating by parts in (11), we get
+
+      _ -ikr           _       _     _
+     / e      dZ      | Ze^-ikr |   /   d   / e^-ikr\
+     | ------ -- dy = | ------- | - | Z -- ( ------- ) dy,
+    _/    r   dy      |_   r   _|  _/   dy  \    r  /
+
+  in which the integrated terms at the limits vanish, Z being finite
+  only within the region T. Thus
+
+                           _ _ _         -ikr
+                   1      / / /   d   /e^      \
+    [=omega]1 = -------   | | | Z -- ( -------- ) dx dy dz.
+                4[pi]b^2 _/_/_/   dy  \    r   /
+
+
+  Since the dimensions of T are supposed to be very small in comparison
+  with [lambda], the factor d/dy (e^-ikr / r) is sensibly constant; so
+  that, if Z stand for the mean value of Z over the volume T, we may
+  write
+
+                  TZ       y   d   / e^-ikr \
+    [=omega]1 = -------- . - . -- (  ------  )        (13).
+                4[pi]b^2   r   dr  \   r    /
+
+  In like manner we find
+
+                   TZ       x   d   / e^-ikr \
+    [=omega]2 = --------  . - . -- ( -------  )       (14).
+                4[pi]b^2    r   dr  \     r  /
+
+
+  From (10), (13), (14) we see that, as might have been expected, the
+  rotation at any point is about an axis perpendicular both to the
+  direction of the force and to the line joining the point to the source
+  of disturbance. If the resultant rotation be [omega], we have
+
+                 TZ      [sqrt](x^2 + y^2)   d   /e^-ikr\
+    [=omega] = ------- . ----------------- . -- ( ------ ) =
+               4[pi]b^2          r           dr  \   r  /
+
+                TZ sin[phi] d   /e^-ikr\
+             =  ----------- -- ( ------ ),
+                  4[pi]b^2  dr  \   r  /
+
+  [phi] denoting the angle between r and z. In differentiating
+  e^(-ikr)/r with respect to r, we may neglect the term divided by r^2 as
+  altogether insensible, kr being an exceedingly great quantity at any
+  moderate distance from the origin of disturbance. Thus
+
+               -ik.TZ sin[phi]    /e^-ikr\
+    [=omega] = --------------- . ( ------ )         (15),
+                  4[pi]b^2        \  r   /
+
+  which completely determines the rotation at any point. For a
+  disturbing force of given integral magnitude it is seen to be
+  everywhere about an axis perpendicular to r and the direction of the
+  force, and in magnitude dependent only upon the angle ([phi]) between
+  these two directions and upon the distance (r).
+
+  The intensity of light is, however, more usually expressed in terms of
+  the actual displacement in the plane of the wave. This displacement,
+  which we may denote by [zeta]', is in the plane containing z and r,
+  and perpendicular to the latter. Its connexion with [=omega]is
+  expressed by [=omega] = d[zeta]'/dr; so that
+
+              TZ sin [phi]   /e^-ikr\
+    [zeta]' = ----------- . ( ------ )              (16),
+                4[pi]b^2     \  r   /
+
+  where the factor e^int is restored.
+
+  Retaining only the real part of (16), we find, as the result of a
+  local application of force equal to
+
+    DTZ cos nt                                      (17),
+
+  the disturbance expressed by
+
+              TZ sin [phi]    /cos(nt - kr)\
+    [zeta]' = ------------ . ( ------------ )       (18).
+                4[pi]b^2      \     r      /
+
+  The occurrence of sin [phi] shows that there is no disturbance
+  radiated in the direction of the force, a feature which might have
+  been anticipated from considerations of symmetry.
+
+  We will now apply (18) to the investigation of a law of secondary
+  disturbance, when a primary wave
+
+    [zeta] = sin(nt - kx)                           (19)
+
+  is supposed to be broken up in passing the plane x = 0. The first step
+  is to calculate the force which represents the reaction between the
+  parts of the medium separated by x = 0. The force operative upon the
+  positive half is parallel to OZ, and of amount per unit of area equal
+  to
+
+    -b^2D d[zeta]/dx = b^2kD cos nt;
+
+  and to this force acting over the whole of the plane the actual motion
+  on the positive side may be conceived to be due. The secondary
+  disturbance corresponding to the element dS of the plane may be
+  supposed to be that caused by a force of the above magnitude acting
+  over dS and vanishing elsewhere; and it only remains to examine what
+  the result of such a force would be.
+
+  Now it is evident that the force in question, supposed to act upon the
+  positive half only of the medium, produces just double of the effect
+  that would be caused by the same force if the medium were undivided,
+  and on the latter supposition (being also localized at a point) it
+  comes under the head already considered. According to (18), the effect
+  of the force acting at dS parallel to OZ, and of amount equal to
+
+    2b^2kD dS cos nt,
+
+  will be a disturbance
+
+              dS sin [phi]
+    [zeta]' = ------------ cos(nt - kr)            (20),
+               [lambda]r
+
+  regard being had to (12). This therefore expresses the secondary
+  disturbance at a distance r and in a direction making an angle [phi]
+  with OZ (the direction of primary vibration) due to the element dS of
+  the wave-front.
+
+  The proportionality of the secondary disturbance to sin [phi] is
+  common to the present law and to that given by Stokes, but here there
+  is no dependence upon the angle [theta] between the primary and
+  secondary rays. The occurrence of the factor [lambda]r^-1, and the
+  necessity of supposing the phase of the secondary wave accelerated by
+  a quarter of an undulation, were first established by Archibald Smith,
+  as the result of a comparison between the primary wave, supposed to
+  pass on without resolution, and the integrated effect of all the
+  secondary waves (S 2). The occurrence of factors such as sin [phi], or
+  1/2(1 + cos [theta]), in the expression of the secondary wave has no
+  influence upon the result of the integration, the effects of all the
+  elements for which the factors differ appreciably from unity being
+  destroyed by mutual interference.
+
+  The choice between various methods of resolution, all mathematically
+  admissible, would be guided by physical considerations respecting the
+  mode of action of obstacles. Thus, to refer again to the acoustical
+  analogue in which plane waves are incident upon a perforated rigid
+  screen, the circumstances of the case are best represented by the
+  first method of resolution, leading to symmetrical secondary waves, in
+  which the normal motion is supposed to be zero over the unperforated
+  parts. Indeed, if the aperture is very small, this method gives the
+  correct result, save as to a constant factor. In like manner our
+  present law (20) would apply to the kind of obstruction that would be
+  caused by an actual physical division of the elastic medium, extending
+  over the whole of the area supposed to be occupied by the intercepting
+  screen, but of course not extending to the parts supposed to be
+  perforated.
+
+  On the electromagnetic theory, the problem of diffraction becomes
+  definite when the properties of the obstacle are laid down. The
+  simplest supposition is that the material composing the obstacle is
+  perfectly conducting, i.e. perfectly reflecting. On this basis A. J.
+  W. Sommerfeld (_Math. Ann._, 1895, 47, p. 317), with great
+  mathematical skill, has solved the problem of the shadow thrown by a
+  semi-infinite plane screen. A simplified exposition has been given by
+  Horace Lamb (_Proc. Lond. Math. Soc._, 1906, 4, p. 190). It appears
+  that Fresnel's results, although based on an imperfect theory, require
+  only insignificant corrections. Problems not limited to two
+  dimensions, such for example as the shadow of a circular disk, present
+  great difficulties, and have not hitherto been treated by a rigorous
+  method; but there is no reason to suppose that Fresnel's results would
+  be departed from materially.     (R.)
+
+
+FOOTNOTES:
+
+  [1] The descending series for J0(z) appears to have been first given
+    by Sir W. Hamilton in a memoir on "Fluctuating Functions," _Roy.
+    Irish Trans._, 1840.
+
+  [2] Airy, loc. cit. "Thus the magnitude of the central spot is
+    diminished, and the brightness of the rings increased, by covering
+    the central parts of the object-glass."
+
+  [3] _"Man kann daraus schliessen, was moglicher Weise durch
+    Mikroskope noch zu sehen ist. Ein mikroskopischer Gegenstand z. B,
+    dessen Durchmesser = ([lambda]) ist, und der aus zwei Theilen
+    besteht, kann nicht mehr als aus zwei Theilen bestehend erkannt
+    werden. Dieses zeigt uns eine Grenze des Sehvermogens durch
+    Mikroskope"_ (_Gilbert's Ann._ 74, 337). Lord Rayleigh has recorded
+    that he was himself convinced by Fraunhofer's reasoning at a date
+    antecedent to the writings of Helmholtz and Abbe.
+
+  [4] The last sentence is repeated from the writer's article "Wave
+    Theory" in the 9th edition of this work, but A. A. Michelson's
+    ingenious echelon grating constitutes a realization in an unexpected
+    manner of what was thought to be impracticable.--[R.]
+
+  [5] Compare also F. F. Lippich, _Pogg. Ann._ cxxxix. p. 465, 1870;
+    Rayleigh, _Nature_ (October 2, 1873).
+
+  [6] The power of a grating to construct light of nearly definite
+   wave-length is well illustrated by Young's comparison with the
+   production of a musical note by reflection of a sudden sound from a
+   row of palings. The objection raised by Herschel (_Light_, S 703) to
+   this comparison depends on a misconception.
+
+  [7] It must not be supposed that errors of this order of magnitude
+    are unobjectionable in all cases. The position of the middle of the
+    bright band representative of a mathematical line can be fixed with a
+    spider-line micrometer within a small fraction of the width of the
+    band, just as the accuracy of astronomical observations far
+    transcends the separating power of the instrument.
+
+  [8] "In the same way we may conclude that in flat gratings any
+    departure from a straight line has the effect of causing the dust in
+    the slit and the spectrum to have different foci--a fact sometimes
+    observed." (Rowland, "On Concave Gratings for Optical Purposes,"
+    _Phil. Mag._, September 1883).
+
+  [9] On account of inequalities in the atmosphere giving a variable
+    refraction, the light from a star would be irregularly distributed
+    over a screen. The experiment is easily made on a laboratory scale,
+    with a small source of light, the rays from which, in their course
+    towards a rather distant screen, are disturbed by the neighbourhood
+    of a heated body. At a moment when the eye, or object-glass of a
+    telescope, occupies a dark position, the star vanishes. A fraction of
+    a second later the aperture occupies a bright place, and the star
+    reappears. According to this view the chromatic effects depend
+    entirely upon atmospheric dispersion.
+
+  [10] In experiment a line of light is sometimes substituted for a
+    point in order to increase the illumination. The various parts of the
+    line are here _independent_ sources, and should be treated
+    accordingly. To assume a cylindrical form of primary wave would be
+    justifiable only when there is synchronism among the secondary waves
+    issuing from the various centres.
+
+  [11] H. Necker (_Phil. Mag._, November 1832); Fox Talbot (_Phil.
+    Mag._, June 1833). "When the sun is about to emerge ... every branch
+    and leaf is lighted up with a silvery lustre of indescribable
+    beauty.... The birds, as Mr Necker very truly describes, appear like
+    flying brilliant sparks." Talbot ascribes the appearance to
+    diffraction; and he recommends the use of a telescope.
+
+
+
+
+DIFFUSION (from the Lat. _diffundere; dis-_, asunder, and _fundere_, to
+pour out), in general, a spreading out, scattering or circulation; in
+physics the term is applied to a special phenomenon, treated below.
+
+1. _General Description._--When two different substances are placed in
+contact with each other they sometimes remain separate, but in many
+cases a gradual mixing takes place. In the case where both the
+substances are gases the process of mixing continues until the result is
+a uniform mixture. In other cases the proportions in which two different
+substances can mix lie between certain fixed limits, but the mixture is
+distinguished from a chemical compound by the fact that between these
+limits the composition of the mixture is capable of continuous
+variation, while in chemical compounds, the proportions of the different
+constituents can only have a discrete series of numerical values, each
+different ratio representing a different compound. If we take, for
+example, air and water in the presence of each other, air will become
+dissolved in the water, and water will evaporate into the air, and the
+proportions of either constituent absorbed by the other will vary
+continuously. But a limit will come when the air will absorb no more
+water, and the water will absorb no more air, and throughout the change
+a definite surface of separation will exist between the liquid and the
+gaseous parts. When no surface of separation ever exists between two
+substances they must necessarily be capable of mixing in all
+proportions. If they are not capable of mixing in all proportions a
+discontinuous change must occur somewhere between the regions where the
+substances are still unmixed, thus giving rise to a surface of
+separation.
+
+The phenomena of mixing thus involves the following processes:--(1) A
+motion of the substances relative to one another throughout a definite
+_region_ of space in which mixing is taking place. This relative motion
+is called "diffusion." (2) The passage of portions of the mixing
+substances across the _surface_ of separation when such a surface
+exists. These surface actions are described under various terms such as
+solution, evaporation, condensation and so forth. For example, when a
+soluble salt is placed in a liquid, the process which occurs at the
+surface of the salt is called "solution," but the salt which enters the
+liquid by solution is transported from the surface into the interior of
+the liquid by "diffusion."
+
+Diffusion may take place in solids, that is, in regions occupied by
+matter which continues to exhibit the properties of the solid state.
+Thus if two liquids which can mix are separated by a membrane or
+partition, the mixing may take place through the membrane. If a solution
+of salt is separated from pure water by a sheet of parchment, part of
+the salt will pass through the parchment into the water. If water and
+glycerin are separated in this way most of the water will pass into the
+glycerin and a little glycerin will pass through in the opposite
+direction, a property frequently used by microscopists for the purpose
+of gradually transferring minute algae from water into glycerin. A still
+more interesting series of examples is afforded by the passage of gases
+through partitions of metal, notably the passage of hydrogen through
+platinum and palladium at high temperatures. When the process is
+considered with reference to a membrane or partition taken as a whole,
+the passage of a substance from one side to the other is commonly known
+as "osmosis" or "transpiration" (see SOLUTION), but what occurs in the
+material of the membrane itself is correctly described as diffusion.
+
+Simple cases of diffusion are easily observed qualitatively. If a
+solution of a coloured salt is carefully introduced by a funnel into the
+bottom of a jar containing water, the two portions will at first be
+fairly well defined, but if the mixture can exist in all proportions,
+the surface of separation will gradually disappear; and the rise of the
+colour into the upper part and its gradual weakening in the lower part,
+may be watched for days, weeks or even longer intervals. The diffusion
+of a strong aniline colouring matter into the interior of gelatine is
+easily observed, and is commonly seen in copying apparatus. Diffusion of
+gases may be shown to exist by taking glass jars containing vapours of
+hydrochloric acid and ammonia, and placing them in communication with
+the heavier gas downmost. The precipitation of ammonium chloride shows
+that diffusion exists, though the chemical action prevents this example
+from forming a typical case of diffusion. Again, when a film of Canada
+balsam is enclosed between glass plates, the disappearance during a few
+weeks of small air bubbles enclosed in the balsam can be watched under
+the microscope.
+
+In fluid media, whether liquids or gases, the process of mixing is
+greatly accelerated by stirring or agitating the fluids, and liquids
+which might take years to mix if left to themselves can thus be mixed in
+a few seconds. It is necessary to carefully distinguish the effects of
+agitation from those of diffusion proper. By shaking up two liquids
+which do not mix we split them up into a large number of different
+portions, and so greatly increase the area of the surface of separation,
+besides decreasing the thicknesses of the various portions. But even
+when we produce the appearance of a uniform turbid mixture, the small
+portions remain quite distinct. If however the fluids can really mix,
+the final process must in every case depend on diffusion, and all we do
+by shaking is to increase the sectional area, and decrease the thickness
+of the diffusing portions, thus rendering the completion of the
+operation more rapid. If a gas is shaken up in a liquid the process of
+absorption of the bubbles is also accelerated by capillary action, as
+occurs in an ordinary sparklet bottle. To state the matter precisely,
+however finely two fluids have been subdivided by agitation, the
+molecular constitution of the different portions remains unchanged. The
+ultimate process by which the individual molecules of two different
+substances become mixed, producing finally a homogeneous mixture, is in
+every case diffusion. In other words, diffusion is that relative motion
+of the molecules of two different substances by which the proportions of
+the molecules in any region containing a finite number of molecules are
+changed.
+
+  In order, therefore, to make accurate observations of diffusion in
+  fluids it is necessary to guard against any cause which may set up
+  currents; and in some cases this is exceedingly difficult. Thus, if
+  gas is absorbed at the upper surface of a liquid, and if the gaseous
+  solution is heavier than the pure liquid, currents may be set up, and
+  a steady state of diffusion may cease to exist. This has been tested
+  experimentally by C. G. von Hufner and W. E. Adney. The same thing may
+  happen when a gas is evolved into a liquid at the surface of a solid
+  even if no bubbles are formed; thus if pieces of aluminium are placed
+  in caustic soda, the currents set up by the evolution of hydrogen are
+  sufficient to set the aluminium pieces in motion, and it is probable
+  that the motions of the Diatomaceae are similarly caused by the
+  evolution of oxygen. In some pairs of substances diffusion may take
+  place more rapidly than in others. Of course the progress of events in
+  any experiment necessarily depends on various causes, such as the size
+  of the containing vessels, but it is easy to see that when experiments
+  with different substances are carried out under similar conditions,
+  however these "similar conditions" be defined, the rates of diffusion
+  must be capable of numerical comparison, and the results must be
+  expressible in terms of at least one physical quantity, which for any
+  two substances can be called their coefficient of diffusion. How to
+  select this quantity we shall see later.
+
+2 _Quantitative Methods of observing Diffusion._--The simplest plan of
+determining the progress of diffusion between two liquids would be to
+draw off and examine portions from different strata at some stage in the
+process; the disturbance produced would, however, interfere with the
+subsequent process of diffusion, and the observations could not be
+continued. By placing in the liquid column hollow glass beads of
+different average densities, and observing at what height they remain
+suspended, it is possible to trace the variations of density of the
+liquid column at different depths, and different times. In this method,
+which was originally introduced by Lord Kelvin, difficulties were caused
+by the adherence of small air bubbles to the beads.
+
+In general, optical methods are the most capable of giving exact
+results, and the following may be distinguished, (a) _By refraction in a
+horizontal plane._ If the containing vessel is in the form of a prism,
+the deviation of a horizontal ray of light in passing through the prism
+determines the index of refraction, and consequently the density of the
+stratum through which the ray passes, (b) _By refraction in a vertical
+plane._ Owing to the density varying with the depth, a horizontal ray
+entering the liquid also undergoes a small vertical deviation, being
+bent downwards towards the layers of greater density. The observation of
+this vertical deviation determines not the actual density, but its rate
+of variation with the depth, i.e. the "density gradient" at any point,
+(c) _By the saccharimeter._ In the cases of solutions of sugar, which
+cause rotation of the plane of polarized light, the density of the sugar
+at any depth may be determined by observing the corresponding angle of
+rotation, this was done originally by W. Voigt.
+
+3. _Elementary Definitions of Coefficient of Diffusion._--The simplest
+case of diffusion is that of a substance, say a gas, diffusing in the
+interior of a homogeneous solid medium, which remains at rest, when no
+external forces act on the system. We may regard it as the result of
+experience that: (1) if the density of the diffusing substance is
+everywhere the same no diffusion takes place, and (2) if the density of
+the diffusing substance is different at different points, diffusion will
+take place from places of greater to those of lesser density, and will
+not cease until the density is everywhere the same. It follows that the
+rate of flow of the diffusing substance at any point in any direction
+must depend on the density gradient at that point in that direction,
+i.e. on the rate at which the density of the diffusing substance
+decreases as we move in that direction. We may define the _coefficient
+of diffusion_ as the ratio of the total mass per unit area which flows
+across any small section, to the rate of decrease of the density per
+unit distance in a direction perpendicular to that section.
+
+  In the case of steady diffusion parallel to the axis of x, if [rho] be
+  the density of the diffusing substance, and q the mass which flows
+  across a unit of area in a plane perpendicular to the axis of x, then
+  the density gradient is -d[rho]/dx and the ratio of q to this is
+  called the "coefficient of diffusion." By what has been said this
+  ratio remains finite, however small the actual gradient and flow may
+  be., and it is natural to assume, at any rate as a first
+  approximation, that it is constant as far as the quantities in
+  question are concerned. Thus if the coefficient of diffusion be
+  denoted by K we have q= -K(d[rho]/dx).
+
+  Further, the rate at which the quantity of substance is increasing in
+  an element between the distances x and x+dx is equal to the difference
+  of the rates of flow in and out of the two faces, whence as in
+  hydrodynamics, we have d[rho]/dt =-dq/dx.
+
+  It follows that the equation of diffusion in this case assumes the
+  form
+
+    d[rho]   d   /  d[rho] \
+    ------ = -- ( K ------  ),
+      dt     dx  \    dx   /
+
+  which is identical with the equations representing conduction of heat,
+  flow of electricity and other physical phenomena. For motion in three
+  dimensions we have in like manner
+
+    d[rho]   d   /  d[rho]\    d   /  d[rho]\    d   /  d[rho]\
+    ------ = -- ( K ------ ) + -- ( K ------ ) + -- ( K ------ );
+      dt     dx  \   dx   /    dy  \   dy   /    dz  \   dz   /
+
+  and the corresponding equations in electricity and heat for
+  anisotropic substances would be available to account for any parallel
+  phenomena, which may arise, or might be conceived, to exist in
+  connexion with diffusion through a crystalline solid.
+
+In the case of a very dilute solution, the coefficient of diffusion of
+the dissolved substance can be defined in the same way as when the
+diffusion takes place in a solid, because the effects of diffusion will
+not have any perceptible influence on the solvent, and the latter may
+therefore be regarded as remaining practically at rest. But in most
+cases of diffusion between two fluids, both of the fluids are in motion,
+and hence there is far greater difficulty in determining the motion, and
+even in defining the coefficient of diffusion. It is important to notice
+in the first instance, that it is only the relative motion of the two
+substances which constitutes diffusion. Thus when a current of air is
+blowing, under ordinary circumstances the changes which take place are
+purely mechanical, and do not depend on the separate diffusions of the
+oxygen and nitrogen of which the air is mainly composed. It is only when
+two gases are flowing with unequal velocity, that is, when they have a
+relative motion, that these changes of relative distribution, which are
+called diffusion, take place. The best way out of the difficulty is to
+investigate the separate motions of the two fluids, taking account of
+the mechanical actions exerted on them, and supposing that the mutual
+action of the fluids causes either fluid to resist the relative motion
+of the other.
+
+4. _The Coefficient of Resistance._--Let us call the two diffusing
+fluids A and B. If B were absent, the motion of the fluid A would be
+determined entirely by the variations of pressure of the fluid A, and by
+the external forces, such as that due to gravity acting on A. Similarly
+if A were absent, the motion of B would be determined entirely by the
+variations of pressure due to the fluid B, and by the external forces
+acting on B. When both fluids are mixed together, each fluid tends to
+resist the relative motion of the other, and by the law of equality of
+action and reaction, the resistance which A experiences from B is
+everywhere equal and opposite to the resistance which B experiences from
+A. If the amount of this resistance per unit volume be divided by the
+relative velocity of the two fluids, and also by the product of their
+densities, the quotient is called the "coefficient of resistance." If
+then [rho]1, [rho]2 are the densities cf the two fluids, u1, u2 their
+velocities, C the coefficient of resistance, then the portion of the
+fluid A contained in a small element of volume v will experience from
+the fluid B a resistance C[rho]1[rho]2v(u1- u2), and the fluid B
+contained in the same volume element will experience from the fluid A an
+equal and opposite resistance, C[rho]1[rho]2v(u2 - u1).
+
+This definition implies the following laws of resistance to diffusion,
+which must be regarded as based on experience, and not as self-evident
+truths: (1) each fluid tends to assume, so far as diffusion is
+concerned, the same equuibrium distribution that it would assume if its
+motion were unresisted by the presence of the other fluid. (Of course,
+the mutual attraction of gravitation of the two fluids might affect the
+final distribution, but this is practically negligible. Leaving such
+actions as this out of account the following statement is correct.) In
+a state of equilibrium, the density of each fluid at any point thus
+depends only on the partial pressure of that fluid alone, and is the
+same as if the other fluids were absent. It does not depend on the
+partial pressures of the other fluids. If this were not the case, the
+resistance to diffusion would be analogous to friction, and would
+contain terms which were independent of the relative velocity u2 - u1.
+(2) For slow motions the resistance to diffusion is (approximately at
+any rate) proportional to the relative velocity. (3) The coefficient of
+resistance C is not necessarily always constant; it may, for example,
+and, in general, does, depend on the temperature.
+
+  If we form the equations of hydrodynamics for the different fluids
+  occurring in any mixture, taking account of diffusion, but neglecting
+  viscosity, and using suffixes 1, 2 to denote the separate fluids,
+  these assume the form given by James Clerk Maxwell ("Diffusion," in
+  _Ency. Brit._, 9th ed.):--
+
+          Du1   dp1
+    [rho] --- + --- - X1[rho]1 + C12[rho]1[rho]2(u1 - u2) + &c. = 0,
+          Dt    dx
+
+  where
+
+    Du1   du1      du1      du1      du1
+    --- = --- + u1 --- + v1 --- + w1 ---,
+    Dt    dt       dx       dy       dz
+
+  and these equations imply that when diffusion and other motions cease,
+  the fluids satisfy the separate conditions of equilibrium dp1/dx -
+  X1[rho]1 = 0. The assumption made in the following account is that
+  terms such as Du1/Dt may be neglected in the cases considered.
+
+A further property based on experience is that the motions set up in a
+mixture by diffusion are very slow compared with those set up by
+mechanical actions, such as differences of pressure. Thus, if two gases
+at equal temperature and pressure be allowed to mix by diffusion, the
+heavier gas being below the lighter, the process will take a long time;
+on the other hand, if two gases, or parts of the same gas, at different
+pressures be connected, equalization of pressure will take place almost
+immediately. It follows from this property that the forces required to
+overcome the "inertia" of the fluids in the motions due to diffusion are
+quite imperceptible. At any stage of the process, therefore, any one of
+the diffusing fluids may be regarded as in equilibrium under the action
+of its own partial pressure, the external forces to which it is
+subjected and the resistance to diffusion of the other fluids.
+
+5. _Slow Diffusion of two Gases. Relation between the Coefficients of
+Resistance and of Diffusion._--We now suppose the diffusing substances
+to be two gases which obey Boyle's law, and that diffusion takes place
+in a closed cylinder or tube of unit sectional area at constant
+temperature, the surfaces of equal density being perpendicular to the
+axis of the cylinder, so that the direction of diffusion is along the
+length of the cylinder, and we suppose no external forces, such as
+gravity, to act on the system.
+
+  The densities of the gases are denoted by [rho]1, [rho]2, their
+  velocities of diffusion by u1, u2, and if their partial pressures are
+  p1, p2, we have by Boyle's law p1 = k1[rho]1, p2 = k2[rho]2, where
+  k1, k2 are constants for the two gases, the temperature being constant.
+  The axis of the cylinder is taken as the axis of x.
+
+  From the considerations of the preceding section, the effects of
+  inertia of the diffusing gases may be neglected, and at any instant of
+  the process either of the gases is to be treated as kept in
+  equilibrium by its partial pressure and the resistance to diffusion
+  produced by the other gas. Calling this resistance per unit volume R,
+  and putting R = C[rho]1[rho]2(u1 - u2), where C is the coefficient of
+  resistance, the equations of equilibrium give
+
+    dp1                                  dp2
+    --- + C[rho]1[rho]2(u1 - u2)= 0, and --- + C[rho]1[rho]2(u2 - u1)= 0 (1).
+    dx                                   dx
+
+  These involve
+
+    dp1   dp2
+    --- + --- = 0 or p1 + p2 = P                      (2)
+    dx    dx
+
+  where P is the total pressure of the mixture, and is everywhere
+  constant, consistently with the conditions of mechanical equilibrium.
+
+  Now dp1/dx is the pressure-gradient of the first gas, and is, by
+  Boyle's law, equal to k1 times the corresponding density-gradient.
+  Again [rho]1u1 is the mass of gas flowing across any section per unit
+  time, and k1[rho]1u1 or p1u1 can be regarded as representing the flux
+  of partial pressure produced by the motion of the gas. Since the total
+  pressure is everywhere constant, and the ends of the cylinder are
+  supposed fixed, the fluxes of partial pressure due to the two gases
+  are equal and opposite, so that
+
+    p1u1 + p2u2 = 0 or k1[rho]1u1 + k2[rho]2u2 = 0    (3).
+
+  From (2) (3) we find by elementary algebra
+
+    u1/p2 = - u2/p1 = (u1 - u2)/(p1 + p2) = (u1 - u2)/P,
+
+  and therefore
+
+    p2u1 = - p2u2 = p1p2(u1 - u2)/P = k1k2[rho]1[rho]2(u1 - u2)/P
+
+  Hence equations (1) (2) gives
+
+    dp1    CP                   dp2    CP
+    --- + ---- (p1u1) = 0, and  --- + ---- (p2u2) = 0;
+    dx    k1k2                  dx    k1k2
+
+  whence also substituting p1 = k1[rho]1, p2 = k2[rho]2, and by
+  transposing
+
+                 k1k2 d[rho]1                   k1k2 d[rho]2
+    [rho]1u1 = - ---- -------, and [rho]2u2 = - ---- -------.
+                  CP    dx                       CP     dx
+
+  We may now define the "coefficient of diffusion" of either gas as the
+  ratio of the rate of flow of that gas to its density-gradient. With
+  this definition, the coefficients of diffusion of both the gases in a
+  mixture are equal, each being equal to k1k2/CP. The ratios of the
+  fluxes of partial pressure to the corresponding pressure-gradients are
+  also equal to the same coefficient. Calling this coefficient K, we
+  also observe that the equations of continuity for the two gases are
+
+    d[rho]1   d([rho]1u1)          d[rho]2   d([rho]2u2)
+    ------- + ----------- = 0, and ------- + ----------- = 0,
+       dt         dx                  dt         dx
+
+  leading to the equations of diffusion
+
+    d[rho]1   d   /  d[rho]1\        d[rho]2   d   /  d[rho]2\
+    ------- = -- ( K ------- ) , and ------- = -- ( K ------- ),
+      dt      dx  \    dx   /          dt      dx  \    dx   /
+
+  exactly as in the case of diffusion through a solid.
+
+If we attempt to treat diffusion in liquids by a similar method, it is,
+in the first place, necessary to define the "partial pressure" of the
+components occurring in a liquid mixture. This leads to the conception
+of "osmotic pressure," which is dealt with in the article SOLUTION. For
+dilute solutions at constant temperature, the assumption that the
+osmotic pressure is proportional to the density, leads to results
+agreeing fairly closely with experience, and this fact may be
+represented by the statement that a substance occurring in a dilute
+solution behaves like a perfect gas.
+
+6. _Relation of the Coefficient of Diffusion to the Units of Length and
+Time._--We may write the equation defining K in the form
+
+             I   d[rho]
+  u = -K X ----- ------.
+           [rho]   dx
+
+Here -d[rho]/[rho]dx represents the "percentage rate" at which the
+density decreases with the distance x; and we thus see that the
+coefficient of diffusion represents the ratio of the velocity of flow to
+the percentage rate at which the density decreases with the distance
+measured in the direction of flow. This percentage rate being of the
+nature of a number divided by a length, and the velocity being of the
+nature of a length divided by a time, we may state that K is of two
+dimensions in length and - 1 in time, i.e. dimensions L^2/T.
+
+  _Example 1._ Taking K = 0.1423 for carbon dioxide and air (at
+  temperature 0 deg. C. and pressure 76 cm. of mercury) referred to a
+  centimetre and a second as units, we may interpret the result as
+  follows:--Supposing in a mixture of carbon dioxide and air, the
+  density of the carbon dioxide decreases by, say, 1, 2 or 3% of itself
+  in a distance of 1 cm., then the corresponding velocities of the
+  diffusing carbon dioxide will be respectively 0.01, 0.02 and 0.03
+  times 0.1423, that is, 0.001423, 0.002846 and 0.004269 cm. per second
+  in the three cases.
+
+  _Example 2._ If we wished to take a foot and a second as our units, we
+  should have to divide the value of the coefficient of diffusion in
+  Example 1 by the square of the number of centimetres in 1 ft., that
+  is, roughly speaking, by 900, giving the new value of K = 0.00016
+  roughly.
+
+7. _Numerical Values of the Coefficient of Diffusion._--The table on p.
+258 gives the values of the coefficient of diffusion of several of the
+principal pairs of gases at a pressure of 76 cm. of mercury, and also of
+a number of other substances. In the gases the centimetre and second are
+taken as fundamental units, in other cases the centimetre and day.
+
+8. _Irreversible Changes accompanying Diffusion._--The diffusion of two
+gases at constant pressure and temperature is a good example of an
+"irreversible process." The gases always tend to mix, never to separate.
+In order to separate the gases a change must be effected in the external
+conditions to which the mixture is subjected, either by liquefying one
+of the gases, or by separating them by diffusion through a membrane, or
+by bringing other outside influences to bear on them. In the case of
+liquids, electrolysis affords a means of separating the constituents of
+a mixture. Every such method involves some change taking place outside
+the mixture, and this change may be regarded as a "compensating
+transformation." We thus have an instance of the property that every
+irreversible change leaves an indelible imprint somewhere or other on
+the progress of events in the universe. That the process of diffusion
+obeys the laws of irreversible thermodynamics (if these laws are
+properly stated) is proved by the fact that the compensating
+transformations required to separate mixed gases do not essentially
+involve anything but transformation of energy. The process of allowing
+gases to mix by diffusion, and then separating them by a compensating
+transformation, thus constitutes an irreversible cycle, the outside
+effects of which are that energy somewhere or other must be less capable
+of transformation than it was before the change. We express this fact by
+stating that an irreversible process essentially implies a loss of
+availability. To measure this loss we make use of the laws of
+thermodynamics, and in particular of Lord Kelvin's statement that "It is
+impossible by means of inanimate material agency to derive mechanical
+effect from any portion of matter by cooling it below the temperature of
+the coldest of the surrounding objects."
+
+  +-------------------------------------------+------------+---------------------+--------------+
+  |                Substances.                |    Temp.   |         K.          |   Author.    |
+  +-------------------------------------------+------------+---------------------+--------------+
+  | Carbon dioxide and air                    |    0 deg.C.| 0.1423 cm^2/sec.    | J. Loschmidt.|
+  |    "      "        hydrogen               |    0 deg.C.| 0.5558    "         |      "       |
+  |    "      "        oxygen                 |    0 deg.C.| 0.1409    "         |      "       |
+  |    "      "        carbon monoxide        |    0 deg.C.| 0.1406    "         |      "       |
+  |    "      "        marsh gas (methane)    |    0 deg.C.| 0.1586    "         |      "       |
+  |    "      "        nitrous oxide          |    0 deg.C.| 0.0983    "         |      "       |
+  | Hydrogen and oxygen                       |    0 deg.C.| 0.7214    "         |      "       |
+  |    "      "  carbon monoxide              |    0 deg.C.| 0.6422    "         |      "       |
+  |    "      "  sulphur dioxide              |    0 deg.C.| 0.4800    "         |      "       |
+  | Oxygen and carbon monoxide                |    0 deg.C.| 0.1802    "         |      "       |
+  | Water and ammonia                         |   20 deg.C.| 1.250     "         | G. Hufner.   |
+  |    "        "                             |    5 deg.C.| 0.822     "         |      "       |
+  |    "      common salt (density 1.0269)    |            | 0.355 cm^2/hour.    | J. Graham.   |
+  |    "         "          "        "        |14.33 deg.C.| 1.020, 0.996, 0.972,|      "       |
+  |                                           |            | 0.932 cm^2/day.     | F. Heimbrodt.|
+  |    "      zinc sulphate (0.312 gm/cm^3)   |            | 0.1162    "         | W. Seitz.    |
+  |    "      zinc sulphate (normal)          |            | 0.2355    "         |      "       |
+  |    "      zinc acetate (double normal)    |            | 0.1195    "         |      "       |
+  |    "      zinc formate (half normal)      |            | 0.4654    "         |      "       |
+  |    "      cadmium sulphate (double normal)|            | 0.2456    "         |      "       |
+  |    "      glycerin (1/8n, 1/2n,           |10.14 deg.C.| 0.356, 0.350, 0.342,| F. Heimbrodt.|
+  |               7/8n, 7/8n)                 |            | 0.315 cm^2/day.     |      "       |
+  |    "      urea          "         "       |14.83 deg.C.| 0.973, 0.946, 0.926,|      "       |
+  |                                           |            | 0.883 cm^2/day.     |      "       |
+  |    "      hydrochloric acid               |14.30 deg.C.| 2.208, 2.331,       |      "       |
+  |                                           |            | 2.480 cm^2/day      |      "       |
+  | Gelatin 20% and ammonia                   |   17 deg.C.| 127.1     "         | A. Hagenbach.|
+  |    "     "      carbon dioxide            |            | 0.845     "         |      "       |
+  |    "     "      nitrous oxide             |            | 0.509     "         |      "       |
+  |    "     "      oxygen                    |            | 0.230     "         |      "       |
+  |    "     "      hydrogen                  |            | 0.0565    "         |      "       |
+  +-------------------------------------------+------------+---------------------+--------------+
+
+  Let us now assume that we have any syste m such as the gases above
+  considered, and that it is in the presence of an indefinitely extended
+  medium which we shall call the "auxiliary medium." If heat be taken
+  from any part of the system, only part of this heat can be converted
+  into work by means of thermodynamic engines; and the rest will be
+  given to the auxiliary medium, and will constitute unavailable energy
+  or waste. To understand what this means, we may consider the case of a
+  condensing steam engine. Only part of the energy liberated by the
+  combustion of the coal is available for driving the engine, the rest
+  takes the form of heat imparted to the condenser. The colder the
+  condenser the more efficient is the engine, and the smaller is the
+  quantity of waste.
+
+  The amount of unavailable energy associated with any given
+  transformation is proportional to the absolute temperature of the
+  auxiliary medium. When divided by that temperature the quotient is
+  called the change of "entropy" associated with the given change (see
+  THERMODYNAMICS). Thus if a body at temperature T receives a quantity
+  of heat Q, and if T0 is the temperature of the auxiliary medium, the
+  quantity of work which could be obtained from Q by means of ideal
+  thermodynamic engines would be Q(1 - T0/T), and the balance, which is
+  QT0/T, would take the form of unavailable or waste energy given to the
+  medium. The quotient of this, when divided by T0, is Q/T, and this
+  represents the quantity of entropy associated with Q units of heat at
+  temperature T.
+
+  Any irreversible change for which a compensating transformation of
+  energy exists represents, therefore, an increase of unavailable
+  energy, which is measurable in terms of entropy. The increase of
+  entropy is independent of the temperature of the auxiliary medium. It
+  thus affords a measure of the extent to which energy has run to waste
+  during the change. Moreover, when a body is heated, the increase of
+  entropy is the factor which determines how much of the energy imparted
+  to the body is unavailable for conversion into work under given
+  conditions. In all cases we have
+
+    increase of unavailable energy
+    ------------------------------- = increase of entropy.
+    temperature of auxiliary medium
+
+  When diffusion takes place between two gases inside a closed vessel at
+  uniform pressure and temperature no energy in the form of heat or work
+  is received from without, and hence the entropy gained by the gases
+  from without is zero. But the irreversible processes inside the vessel
+  may involve a gain of entropy, and this can only be estimated by
+  examining by what means mixed gases can be separated, and, in
+  particular, under what conditions the process of mixing and separating
+  the gases could (theoretically) be made reversible.
+
+9. _Evidence derived from Liquefaction of one or both of the
+Gases._--The gases in a mixture can often be separated by liquefying, or
+even solidifying, one or both of the components. In connexion with this
+property we have the important law according to which "The pressure of a
+vapour in equilibrium with its liquid depends only on the temperature
+and is independent of the pressures of any other gases or vapours which
+may be mixed with it." Thus if two closed vessels be taken containing
+some water and one be exhausted, the other containing air, and if the
+temperatures be equal, evaporation will go on until the pressure of the
+vapour in the exhausted vessel is equal to its _partial_ pressure in the
+other vessel, notwithstanding the fact that the _total_ pressure in the
+latter vessel is greater by the pressure of the air.
+
+  To separate mixed gases by liquefaction, they must be compressed and
+  cooled till one separates in the form of a liquid. If no changes are
+  to take place outside the system, the separate components must be
+  allowed to expand until the work of expansion is equal to the work of
+  compression, and the heat given out in compression is reabsorbed in
+  expansion. The process may be made as nearly reversible as we like by
+  performing the operations so slowly that the substances are
+  practically in a state of equilibrium at every stage. This is a
+  consequence of an important axiom in thermodynamics according to which
+  "any small change in the neighbourhood of a state of equilibrium is to
+  a first approximation reversible."
+
+  Suppose now that at any stage of the compression the partial pressures
+  of the two gases are p1 and p2, and that the volume is changed from V
+  to V - dV. The work of compression is (p1 + p2)dV, and this work will
+  be restored at the corresponding stage if each of the separated gases
+  increases in volume from V - dV to V. The ultimate state of the
+  separated gases will thus be one in which each gas occupies the volume
+  V originally occupied by the mixture.
+
+  We may now obtain an estimate of the amount of energy rendered
+  unavailable by diffusion. We suppose two gases occupying volumes V1
+  and V2 at equal pressure p to mix by diffusion, so that the final
+  volume is V1 + V2. Then if before mixing each gas had been allowed to
+  expand till its volume was V1 + V2, work would have been done in the
+  expansion, and the gases could still have been mixed by a reversal of
+  the process above described. In the actual diffusion this work of
+  expansion is lost, and represents energy rendered unavailable at the
+  temperature at which diffusion takes place. When divided by that
+  temperature the quotient gives the increase of entropy. Thus the
+  irreversible processes, and, in particular, the entropy changes
+  associated with diffusion of two gases at uniform pressure, are the
+  same as would take place if each of the gases in turn were to expand
+  by rushing into a vacuum, till it occupied the whole volume of the
+  mixture. A more rigorous proof involves considerations of the
+  thermodynamic potentials, following the methods of J. Willard Gibbs
+  (see ENERGETICS).
+
+  Another way in which two or more mixed gases can be separated is by
+  placing them in the presence of a liquid which can freely absorb one
+  of the gases, but in which the other gas or gases are insoluble. Here
+  again it is found by experience that when equilibrium exists at a
+  given temperature between the dissolved and undissolved portions of
+  the first gas, the partial pressure of that gas in the mixture depends
+  on the temperature alone, and is independent of the partial pressures
+  of the insoluble gases with which it is mixed, so that the conclusions
+  are the same as before.
+
+10. _Diffusion through a Membrane or Partition. Theory of the
+semi-permeable Membrane._--It has been pointed out that diffusion of
+gases frequently takes place in the interior of solids; moreover,
+different gases behave differently with respect to the same solid at the
+same temperature. A membrane or partition formed of such a solid can
+therefore be used to effect a more or less complete separation of gases
+from a mixture. This method is employed commercially for extracting
+oxygen from the atmosphere, in particular for use in projection lanterns
+where a high degree of purity is not required. A similar method is often
+applied to liquids and solutions and is known as "dialysis."
+
+In such cases as can be tested experimentally it has been found that a
+gas always tends to pass through a membrane from the side where its
+density, and therefore its partial pressure, is greater to the side
+where it is less; so that for equilibrium the partial pressures on the
+two sides must be equal. This result is unaffected by the presence of
+other gases on one or both sides of the membrane. For example, if
+different gases at the same pressure are separated by a partition
+through which one gas can pass more rapidly than the other, the
+diffusion will give rise to a difference of pressure on the two sides,
+which is capable of doing mechanical work in moving the partition. In
+evidence of this conclusion Max Planck quotes a test experiment made by
+him in the Physical Institute of the university of Munich in 1883,
+depending on the fact that platinum foil at white heat is permeable to
+hydrogen but impermeable to air, so that if a platinum tube filled with
+hydrogen be heated the hydrogen will diffuse out, leaving a vacuum.
+
+  The details of the experiment may be quoted here:--"A glass tube of
+  about 5 mm. internal diameter, blown out to a bulb at the middle, was
+  provided with a stop-cock at one end. To the other a platinum tube 10
+  cm. long was fastened, and closed at the end. The whole tube was
+  exhausted by a mercury pump, filled with hydrogen at ordinary
+  atmospheric pressure, and then closed. The closed end of the platinum
+  portion was then heated in a horizontal position by a Bunsen burner.
+  The connexion between the glass and platinum tubes, having been made
+  by means of sealing-wax, had to be kept cool by a continuous current
+  of water to prevent the softening of the wax. After four hours the
+  tube was taken from the flame, cooled to the temperature of the room,
+  and the stop-cock opened under mercury. The mercury rose rapidly,
+  almost completely filling the tube, proving that the tube had been
+  very nearly exhausted."
+
+[Illustration]
+
+In order that diffusion through a membrane may be reversible so far as a
+particular gas is concerned, the process must take place so slowly that
+equilibrium is set up at every stage (see S 9 above). In order to
+separate one gas from another consistently with this condition it is
+necessary that no diffusion of the latter gas should accompany the
+process. The name "semi-permeable" is applied to an ideal membrane or
+partition through which one gas can pass, and which offers an
+insuperable barrier to any diffusion whatever of a second gas. By means
+of two semi-permeable partitions acting oppositely with respect to two
+different gases A and B these gases could be mixed or separated by
+reversible methods. The annexed figure shows a diagrammatic
+representation of the process.
+
+  We suppose the gases contained in a cylindrical tube; P, Q, R, S are
+  four pistons, of which P and R are joined to one connecting rod, Q and
+  S to another. P, S are impermeable to both gases; Q is semi-permeable,
+  allowing the gas A to pass through but not B, similarly R allows the
+  gas B to pass through but not A. The distance PR is equal to the
+  distance QS, so that if the rods are pushed towards each other as far
+  as they will go, P and Q will be in contact, as also R and S. Imagine
+  the space RQ filled with a mixture of the two gases under these
+  conditions. Then by slowly drawing the connecting rods apart until R,
+  Q touch, the gas A will pass into the space PQ, and B will pass into
+  the space RS, and the gases will finally be completely separated;
+  similarly, by pushing the connecting rods together, the two gases will
+  be remixed in the space RQ. By performing the operations slowly enough
+  we may make the processes as nearly reversible as we please, so that
+  no available energy is lost in either change. The gas A being at every
+  instant in equilibrium on the two sides of the piston Q, its density,
+  and therefore its partial pressure, is the same on both sides, and the
+  same is true regarding the gas B on the two sides of R. Also _no work
+  is done in moving the pistons_, for the partial pressures of B on the
+  two sides of R balance each other, consequently, the resultant thrust
+  on R is due to the gas A alone, and is equal and opposite to its
+  resultant thrust on P, so that the connecting rods are at every
+  instant in a state of mechanical equilibrium so far as the pressures
+  of the gases A and B are concerned. We conclude that in the reversible
+  separation of the gases by this method at constant temperature without
+  the production or absorption of mechanical work, the densities and the
+  partial pressures of the two separated gases are the same as they were
+  in the mixture. These conclusions are in entire agreement with those
+  of the preceding section. If this agreement did not exist it would be
+  possible, theoretically, to obtain perpetual motion from the gases in
+  a way that would be inconsistent with the second law of
+  thermodynamics.
+
+Most physicists admit, as Planck does, that it is impossible to obtain
+an ideal semi-permeable substance; indeed such a substance would
+necessarily have to possess an infinitely great resistance to diffusion
+for such gases as could not penetrate it. But in an experiment performed
+under actual conditions the losses of available energy arising from this
+cause would be attributable to the imperfect efficiency of the
+partitions and not to the gases themselves; moreover, these losses are,
+in every case, found to be completely in accordance with the laws of
+irreversible thermodynamics. The reasoning in this article being
+somewhat condensed the reader must necessarily be referred to treatises
+on thermodynamics for further information on points of detail connected
+with the argument. Even when he consults these treatises he may find
+some points omitted which have been examined in full detail at some time
+or other, but are not sufficiently often raised to require mention in
+print.
+
+II. _Kinetic Models of Diffusion._--Imagine in the first instance that a
+very large number of red balls are distributed over one half of a
+billiard table, and an equal number of white balls over the other half.
+If the balls are set in motion with different velocities in various
+directions, diffusion will take place, the red balls finding their way
+among the white ones, and vice versa; and the process will be retarded
+by collisions between the balls. The simplest model of a perfect gas
+studied in the kinetic theory of gases (see MOLECULE) differs from the
+above illustration in that the bodies representing the molecules move in
+space instead of in a plane, and, unlike billiard balls, their motion is
+unresisted, and they are perfectly elastic, so that no kinetic energy is
+lost either during their free motions, or at a collision.
+
+  The mathematical analysis connected with the application of the
+  kinetic theory to diffusion is very long and cumbersome. We shall
+  therefore confine our attention to regarding a medium formed of
+  elastic spheres as a mechanical model, by which the most important
+  features of diffusion can be illustrated. We shall assume the results
+  of the kinetic theory, according to which:--(1) In a dynamical model
+  of a perfect gas the mean kinetic energy of translation of the
+  molecules represents the absolute temperature of the gas. (2) The
+  pressure at any point is proportional to the product of the number of
+  molecules in unit volume about that point into the mean square of the
+  velocity. (The mean square of the velocity is different from but
+  proportional to the square of the mean velocity, and in the subsequent
+  arguments either of these two quantities can generally be taken.) (3)
+  In a gas mixture represented by a mixture of molecules of unequal
+  masses, the mean kinetic energies of the different kinds are equal.
+
+  Consider now the problem of diffusion in a region containing two kinds
+  of molecules A and B of unequal mass. The molecules of A in the
+  neighbourhood of any point will, by their motion, spread out in every
+  direction until they come into collision with other molecules of
+  either kind, and this spreading out from every point of the medium
+  will give rise to diffusion. If we imagine the velocities of the A
+  molecules to be equally distributed in all directions, as they would
+  be in a homogeneous mixture, it is obvious that the process of
+  diffusion will be greater, _ceteris paribus_, the greater the velocity
+  of the molecules, and the greater the length of the free path before a
+  collision takes place. If we assume consistently with this, that the
+  coefficient of diffusion of the gas A is proportional to the mean
+  value of Wala, where wa is the velocity and la is the length of the
+  path of a molecule of A, this expression for the coefficient of
+  diffusion is of the right dimensions in length and time. If, moreover,
+  we observe that when diffusion takes place in a fixed direction, say
+  that of the axis of x, it depends only on the resolved part of the
+  velocity and length of path in that direction: this hypothesis readily
+  leads to our taking the mean value of 1/3w_a l_a as the coefficient of
+  diffusion for the gas A. This value was obtained by O. E. Meyer and
+  others.
+
+  Unfortunately, however, it makes the coefficients of diffusion unequal
+  for the two gases, a result inconsistent with that obtained above from
+  considerations of the coefficient of resistance, and leading to the
+  consequence that differences of pressure would be set up in different
+  parts of the gas. To equalize these differences of pressure, Meyer
+  assumed that a counter current is set up, this current being, of
+  course, very slow in practice; and J. Stefan assumed that the
+  diffusion of one gas was not affected by collisions between molecules
+  of the _same gas_. When the molecules are mixed in equal proportions
+  both hypotheses lead to the value 1/6([w_a l_a] + [w_b l_b]), (square
+  brackets denoting mean values). When one gas preponderates largely
+  over the other, the phenomena of diffusion are too difficult of
+  observation to allow of accurate experimental tests being made.
+  Moreover, in this case no difference exists unless the molecules are
+  different in size or mass.
+
+  Instead of supposing a velocity of translation added after the
+  mathematical calculations have been performed, a better plan is to
+  assume from the outset that the molecules of the two gases have small
+  velocities of translation in opposite directions, superposed on the
+  distribution of velocity, which would occur in a medium representing a
+  gas at rest. When a collision occurs between molecules of different
+  gases a transference of momentum takes place between them, and the
+  quantity of momentum so transferred in one second in a unit of volume
+  gives a dynamical measure of the resistance to diffusion. It is to be
+  observed that, however small the relative velocity of the gases A and
+  B, it plays an all-important part in determining the coefficient of
+  resistance; for without such relative motion, and with the velocities
+  evenly distributed in all directions, no transference of momentum
+  could take place. The coefficient of resistance being found, the
+  motion of each of the two gases may be discussed separately.
+
+One of the most important consequences of the kinetic theory is that if
+the volume be kept constant the coefficient of diffusion varies as the
+square root of the absolute temperature. To prove this, we merely have
+to imagine the velocity of each molecule to be suddenly increased n
+fold; the subsequent processes, including diffusion, will then go on n
+times as fast; and the temperature T, being proportional to the kinetic
+energy, and therefore to the square of the velocity, will be increased
+n^2 fold. Thus K, the coefficient of diffusion, varies as [sqrt]T.
+
+The relation of K to the density when the temperature remains constant
+is more difficult to discuss, but it may be sufficient to notice that if
+the number of molecules is increased n fold, the chances of a collision
+are n times as great, and the distance traversed between collisions is
+(not _therefore_ but as the result of more detailed reasoning) on the
+average 1/n of what it was before. Thus the free path, and therefore the
+coefficient of diffusion, varies inversely as the density, or directly
+as the volume. If the pressure p and temperature T be taken as
+variables, K varies inversely as p and directly as [sqrt]T^3.
+
+Now according to the experiments first made by J. C. Maxwell and J.
+Loschmidt, it appeared that with constant density K was proportional to
+T more nearly than to [sqrt]T. The inference is that in this respect a
+medium formed of colliding spheres fails to give a correct mechanical
+model of gases. It has been found by L. Boltzmann, Maxwell and others
+that a system of particles whose mutual actions vary according to the
+inverse fifth power of the distance between them represents more
+correctly the relation between the coefficient of diffusion and
+temperature in actual gases. Other recent theories of diffusion have
+been advanced by M. Thiesen, P. Langevin and W. Sutherland. On the other
+hand, J. Thovert finds experimental evidence that the coefficient of
+diffusion is proportional to molecular velocity in the cases examined of
+non-electrolytes dissolved in water at 18 deg. at 2.5 grams per litre.
+
+  BIBLIOGRAPHY.--The best introduction to the study of theories of
+  diffusion is afforded by O. E. Meyer's Kinetic _Theory of Gases_,
+  translated by Robert E. Baynes (London, 1899). The mathematical
+  portion, though sufficient for ordinary purposes, is mostly of the
+  simplest possible character. Another useful treatise is R. Ruhlmann's
+  _Handbuch der mechanischen Warmetheorie_ (Brunswick, 1885). For a
+  shorter sketch the reader may refer to J. C. Maxwell's _Theory of
+  Heat_, chaps, xix. and xxii., or numerous other treatises on physics.
+  The theory of the semi-permeable membrane is discussed by M. Planck
+  in his _Treatise on Thermodynamics_, English translation by A. Ogg
+  (1903), also in treatises on thermodynamics by W. Voigt and other
+  writers. For a more detailed study of diffusion in general the
+  following papers may be consulted:--L. Boltzmann, "Zur Integration der
+  Diffusionsgleichung," _Sitzung. der k. bayer. Akad math.-phys. Klasse_
+  (May 1894); T. des Coudres, "Diffusionsvorgange in einem Zylinder,"
+  _Wied. Ann._ lv. (1895), p. 213; J. Loschmidt,
+  "Experimentaluntersuchungen uber Diffusion," _Wien. Sitz._ lxi., lxii.
+  (1870); J. Stefan, "Gleichgewicht und ... Diffusion von Gasmengen,"
+  _Wien. Sitz._ lxiii., "Dynamische Theorie der Diffusion," _Wien.
+  Sitz._ lxv. (April 1872); M. Toepler, "Gas-diffusion," _Wied. Ann._
+  lviii. (1896), p. 599; A. Wretschko, "Experimentaluntersuchungen uber
+  die Diffusion von Gasmengen," _Wien. Sitz._ lxii. The mathematical
+  theory of diffusion, according to the kinetic theory of gases, has
+  been treated by a number of different methods, and for the study of
+  these the reader may consult L. Boltzmann, _Vorlesungen uber
+  Gastheorie_ (Leipzig, 1896-1898); S. H. Burbury, _Kinetic Theory of
+  Gases_ (Cambridge, 1899), and papers by L. Boltzmann in _Wien. Sitz._
+  lxxxvi. (1882), lxxxvii. (1883); P. G. Tait, "Foundations of the
+  Kinetic Theory of Gases," _Trans. R.S.E._ xxxiii., xxxv., xxvi., or
+  _Scientific Papers_, ii. (Cambridge, 1900). For recent work reference
+  should be made to the current issues of _Science Abstracts_ (London),
+  and entries under the heading "Diffusion" will be found in the general
+  index at the end of each volume.     (G. H. BR.)
+
+
+
+
+DIGBY, SIR EVERARD (1578-1606), English conspirator, son of Everard
+Digby of Stoke Dry, Rutland, was born on the 16th of May 1578. He
+inherited a large estate at his father's death in 1592, and acquired a
+considerable increase by his marriage in 1596 to Mary, daughter and heir
+of William Mulsho of Gothurst (now Gayhurst), in Buckinghamshire. He
+obtained a place in Queen Elizabeth's household and as a ward of the
+crown was brought up a Protestant; but about 1599 he came under the
+influence of the Jesuit, John Gerard, and soon afterwards joined the
+Roman Catholics. He supported James's accession and was knighted by the
+latter on the 23rd of April 1603. In a letter to Salisbury, the date of
+which has been ascribed to May 1605, Digby offered to go on a mission to
+the pope to obtain from the latter a promise to prevent Romanist
+attempts against the government in return for concessions to the Roman
+Catholics; adding that if severe measures were again taken against them
+"within brief there will be massacres, rebellions and desperate attempts
+against the king and state." Digby had suffered no personal injury or
+persecution on account of his religion, but he sympathized with his
+co-religionists; and when at Michaelmas, 1605, the government had fully
+decided to return to the policy of repression, the authors of the
+Gunpowder Plot (q.v.) sought his financial support, and he joined
+eagerly in the conspiracy. His particular share in the plan was the
+organization of a rising in the Midlands; and on the pretence of a
+hunting party he assembled a body of gentlemen together at Danchurch in
+Warwickshire on the 5th of November, who were to take action immediately
+the news arrived from London of the successful destruction of the king
+and the House of Lords, and to seize the person of the princess
+Elizabeth, who was residing in the neighbourhood. The conspirators
+arrived late on the evening of the 6th to tell their story of failure
+and disaster, and Digby, who possibly might have escaped the more
+serious charge of high treason, was persuaded by Catesby, with a false
+tale that the king and Salisbury were dead, to further implicate himself
+in the plot and join the small band of conspirators in their hopeless
+endeavour to raise the country. He accompanied them, the same day, to
+Huddington in Worcestershire and on the 7th to Holbeche in
+Staffordshire. The following morning, however, he abandoned his
+companions, dismissed his servants except two, who declared "they would
+never leave him but against their will," and attempted with these to
+conceal himself in a pit. He was, however, soon discovered and
+surrounded. He made a last effort to break through his captors on
+horseback, but was taken and conveyed a prisoner to the Tower. His trial
+took place in Westminster Hall, on the 27th of January 1606, and alone
+among the conspirators he pleaded guilty, declaring that the motives of
+his crime had been his friendship for Catesby and his devotion to his
+religion. He was condemned to death, and his execution, which took place
+on the 31st, in St Paul's Churchyard, was accompanied by all the
+brutalities exacted by the law.
+
+Digby was a handsome man, of fine presence. Father Gerard extols his
+skill in sport, his "riding of great horses," as well as his skill in
+music, his gifts of mind and his religious devotion, and concludes "he
+was as complete a man in all things, that deserved estimation or might
+win affection as one should see in a kingdom." Some of Digby's letters
+and papers, which include a poem before his execution, a last letter to
+his infant sons and correspondence with his wife from the Tower, were
+published in _The Gunpowder Treason_ by Thomas Barlow, bishop of
+Lincoln, in 1679. He left two sons, of whom the elder, Sir Kenelm Digby,
+was the well-known author and diplomatist.
+
+  See works on the Gunpowder Plot; Narrative of Father Gerard, in
+  _Condition of the Catholics under James I._ by J. Morris (1872), &c. A
+  life of Digby under the title of _A Life of a Conspirator_, by a
+  Romish Recusant (Thomas Longueville), was published in 1895.
+       (P. C. Y.)
+
+
+
+
+DIGBY, SIR KENELM (1603-1665), English author, diplomatist and naval
+commander, son of Sir Everard Digby (q.v.), was born on the 11th of July
+1603, and after his father's execution in 1606 resided with his mother
+at Gayhurst, being brought up apparently as a Roman Catholic. In 1617 he
+accompanied his cousin, Sir John Digby, afterwards 1st earl of Bristol,
+and then ambassador in Spain, to Madrid. On his return in April 1618 he
+entered Gloucester Hall (now Worcester College), Oxford, and studied
+under Thomas Allen (1542-1632), the celebrated mathematician, who was
+much impressed with his abilities and called him the _Mirandula_, i.e.
+the infant prodigy, of his age.[1] He left the university without taking
+a degree in 1620, and travelled in France, where, according to his own
+account, he inspired an uncontrollable passion in the queen-mother,
+Marie de' Medici, now a lady of more than mature age and charms; he
+visited Florence, and in March 1623 joined Sir John Digby again at
+Madrid, at the time when Prince Charles and Buckingham arrived on their
+adventurous expedition. He joined the prince's household and returned
+with him to England on the 5th of October 1623, being knighted by James
+I. on the 23rd of October and receiving the appointment of gentleman of
+the privy chamber to Prince Charles. In 1625 he married secretly
+Venetia, daughter of Sir Edward Hanley of Tonge Castle, Shropshire, a
+lady of extraordinary beauty and intellectual attainments, but of
+doubtful virtue. Digby was a man of great stature and bodily strength.
+Edward Hyde, afterwards earl of Clarendon, who with Ben Jonson was
+included among his most intimate friends, describes him as "a man of
+very extraordinary person and presence which drew the eyes of all men
+upon him, a wonderful graceful behaviour, a flowing courtesy and
+civility, and such a volubility of language as surprised and
+delighted."[2] Digby for some time was excluded from public employment
+by Buckingham's jealousy of his cousin, Lord Bristol. At length in 1627,
+on the latter's advice, Digby determined to attempt "some generous
+action," and on the 22nd of December, with the approval of the king,
+embarked as a privateer with two ships, with the object of attacking the
+French ships in the Venetian harbour of Scanderoon. On the 18th of
+January he arrived off Gibraltar and captured several Spanish and
+Flemish vessels. From the 15th of February to the 27th of March he
+remained at anchor off Algiers on account of the sickness of his men,
+and extracted a promise from the authorities of better treatment of the
+English ships. He seized a rich Dutch vessel near Majorca, and after
+other adventures gained a complete victory over the French and Venetian
+ships in the harbour of Scanderoon on the 11th of June. His successes,
+however, brought upon the English merchants the risk of reprisals, and
+he was urged to depart. He returned home in triumph in February 1629,
+and was well received by the king, and was made a commissioner of the
+navy in October 1630, but his proceedings were disavowed on account of
+the complaints of the Venetian ambassador. In 1633 Lady Digby died, and
+her memory was celebrated by Ben Jonson in a series of poems entitled
+_Eupheme_, and by other poets of the day. Digby retired to Gresham
+College, and exhibited extravagant grief, maintaining a seclusion for
+two years. About this time Digby professed himself a Protestant, but by
+October 1635, while in France, he had already returned to the Roman
+Catholic faith.[3] In a letter dated the 27th of March 1636 Laud
+remonstrates with him, but assures him of the continuance of his
+friendship.[4] In 1638 he published _A Conference with a Lady about
+choice of a Religion_, in which he argues that the Roman Church,
+possessing alone the qualifications of universality, unity of doctrine
+and uninterrupted apostolic succession, is the only true church, and
+that the intrusion of error into it is impossible. The same subject is
+treated in letters to George Digby, afterwards 2nd earl of Bristol,
+dated the 2nd of November 1638 and the 29th of November 1639, which were
+published in 1651, as well as in a further _Discourse concerning
+Infallibility in Religion_ in 1652. Returning to England he associated
+himself with the queen and her Roman Catholic friends, and joined in the
+appeal to the English Romanists for money to support the king's Scottish
+expedition.[5] In consequence he was summoned to the bar of the House of
+Commons on the 27th of January 1641, and the king was petitioned to
+remove him with other recusants from his councils. He left England, and
+while at Paris killed in a duel a French lord who had insulted Charles
+I. in his presence. Louis XIII. took his part, and furnished him with a
+military escort into Flanders. Returning home he was imprisoned, by
+order of the House of Commons, early in 1642, successively in the "Three
+Tobacco Pipes nigh Charing Cross," where his delightful conversation is
+said to have transformed the prison into "a place of delight,"[6] and at
+Winchester House. He was finally released and allowed to go to France on
+the 30th of July 1643, through the intervention of the queen of France,
+Anne of Austria, on condition that he would neither promote nor conceal
+any plots abroad against the English government.
+
+Before leaving England an attempt was made to draw from him an admission
+that Laud, with whom he had been intimate, had desired to be made a
+cardinal, but Digby denied that the archbishop had any leanings towards
+Rome. On the 1st of November 1643 it was resolved by the Commons to
+confiscate his property. He published in London the same year
+_Observations on the 22nd stanza in the 9th canto of the 2nd book of
+Spenser's "Faerie Queene,"_ the MS. of which is in the Egerton
+collection (British Museum, No. 2725 f. 117 b), and _Observations_ on a
+surreptitious and unauthorized edition of the _Religio Medici_, by Sir
+Thomas Browne, from the Roman Catholic point of view, which drew a
+severe rebuke from the author. After his arrival in Paris he published
+his chief philosophical works, _Of Bodies_ and _Of the Immortality of
+Man's Soul_ (1644), autograph MSS. of which are in the Bibliotheque Ste
+Genevieve at Paris, and made the acquaintance of Descartes. He was
+appointed by Queen Henrietta Maria her chancellor, and in the summer of
+1645 he was despatched by her to Rome to obtain assistance. Digby
+promised the conversion of Charles and of his chief supporters. At first
+his eloquence made a great impression. Pope Innocent X. declared that he
+spoke not merely as a Catholic but as an ecclesiastic. But the absence
+of any warrant from Charles himself roused suspicions as to the solidity
+of his assurances, and he obtained nothing but a grant of 20,000 crowns.
+A violent quarrel with the pope followed, and he returned in 1646,
+having consented in the queen's name to complete religious freedom for
+the Roman Catholics, both in England and Ireland, to an independent
+parliament in Ireland, and to the surrender of Dublin and all the Irish
+fortresses into the hands of the Roman Catholics, the king's troops to
+be employed in enforcing the articles and the pope granting about
+L36,000 with a promise of further payments in obtaining direct
+assistance. In February 1649 Digby was invited to come to England to
+arrange a proposed toleration of the Roman Catholics, but on his arrival
+in May the scheme had already been abandoned. He was again banished on
+the 31st of August, and it was not till 1654 that he was allowed by the
+council of state to return. He now entered into close relations with
+Cromwell, from whom he hoped to obtain toleration for the Roman
+Catholics, and whose alliance he desired to secure for France rather
+than for Spain, and was engaged by Cromwell, much to the scandal of
+both Royalists and Roundheads, in negotiations abroad, of which the aim
+was probably to prevent a union between those two foreign powers. He
+visited Germany, in 1660 was in Paris, and at the Restoration returned
+to England. He was well received in spite of his former relations with
+Cromwell, and was confirmed in his post as Queen Henrietta Maria's
+chancellor. In January 1661 he delivered a lecture, which was published
+the same month, at Gresham College, on the vegetation of plants, and
+became an original member of the Royal Society in 1663. In January 1664
+he was forbidden to appear at court, the cause assigned being that he
+had interposed too far in favour of the 2nd earl of Bristol, disgraced
+by the king on account of the charge of high treason brought by him
+against Clarendon into the House of Lords. The rest of his life was
+spent in the enjoyment of literary and scientific society at his house
+in Covent Garden. He died on the 11th of June 1665. He had five
+children, of whom two, a son and one daughter, survived him.
+
+Digby, though he possessed for the time a considerable knowledge of
+natural science, and is said to have been the first to explain the
+necessity of oxygen to the existence of plants, bears no high place in
+the history of science. He was a firm believer in astrology and alchemy,
+and the extraordinary fables which he circulated on the subject of his
+discoveries are evidence of anything rather than of the scientific
+spirit. In 1656 he made public a marvellous account of a city in
+Tripoli, petrified in a few hours, which he printed in the _Mercurius
+Politicus_. Malicious reports had been current that his wife had been
+poisoned by one of his prescriptions, viper wine, taken to preserve her
+beauty. Evelyn, who visited him in Paris in 1651, describes him as an
+"errant mountebank." Henry Stubbes characterizes him as "the very Pliny
+of our age for lying," and Lady Fanshawe refers to the same
+"infirmity."[7] His famous "powder of sympathy," which seems to have
+been only powder of "vitriol," healed without any contact, by being
+merely applied to a rag or bandage taken from the wound, and Digby
+records a miraculous cure by this means in a lecture given by him at
+Montpellier on this subject in 1658, published in French and English the
+same year, in German in 1660 and in Dutch in 1663; but Digby's claim to
+its original discovery is doubtful, Nathaniel Highmore in his _History
+of Generation_ (1651, p. 113) calling the powder "Talbot's powder," and
+ascribing its invention to Sir Gilbert Talbot. Some of Digby's pills and
+preparations, however, described in _The Closet of the Eminently Learned
+Sir Kenelm Digby Knt. Opened_ (publ. 1677), are said to make less demand
+upon the faith of patients, and his injunction on the subject of the
+making of tea, to let the water "remain upon it no longer than you can
+say the Miserere Psalm very leisurely," is one by no means to be
+ridiculed. As a philosopher and an Aristotelian Digby shows little
+originality and followed the methods of the schoolmen. His Roman
+Catholic orthodoxy mixed with rationalism, and his political opinions,
+according to which any existing authority should receive support, were
+evidently derived from Thomas White (1582-1676), the Roman Catholic
+philosopher, who lived with him in France. White published in 1651
+_Institutionum Peripateticorum libri quinque_, purporting to expound
+Digby's "peripatetic philosophy," but going far beyond Digby's published
+treatises. Digby's _Memoirs_ are composed in the high-flown fantastic
+manner then usual when recounting incidents of love and adventure, but
+the style of his more sober works is excellent. In 1632 he presented to
+the Bodleian library a collection of 236 MSS., bequeathed to him by his
+former tutor Thomas Allen, and described in _Catalogi codicum
+manuscriptorum bibliothecae Bodleianae_, by W. D. Macray, part ix.
+Besides the works already mentioned Digby translated _A Treatise of
+adhering to God written by Albert the Great, Bishop of Ratisbon_ (1653);
+and he was the author of _Private Memoirs_, published by Sir N. H.
+Nicholas from _Harleian MS. 6758_ with introduction (1827); _Journal of
+the Scanderoon Voyage in 1628_, printed by J. Bruce with preface (Camden
+Society, 1868); _Poems from Sir Kenelm Digby's Papers_... with preface
+and notes (Roxburghe Club, 1877); in the _Add. MSS._ 34,362 f. 66 is a
+poem _Of the Miserys of Man_, probably by Digby; _Choice of Experimental
+Receipts in Physick and Chirurgery_ ... _collected by Sir K. Digby_
+(1668), and _Chymical Secrets and Rare Experiments_ (1683), were
+published by G. Hartman, who describes himself as Digby's steward and
+laboratory assistant.
+
+  See the _Life of Sir Kenelm Digby by one of his Descendants_ (T.
+  Longueville), 1896.     (P. C. Y.)
+
+
+FOOTNOTES:
+
+  [1] _Letters by Eminent Persons_ (Aubrey's Lives), ii. 324.
+
+  [2] _Life and Continuation._
+
+  [3] Strafford's _Letters_, i. 474.
+
+  [4] Laud's _Works_, vi. 447.
+
+  [5] _Thomason Tracts_, Brit. Mus. E 164 (15).
+
+  [6] _Archaeologia Cantiana_, ii. 190.
+
+  [7] _Dict. of Nat. Biog._ sub "Digby." See also Robert Boyle's _Works_
+    (1744), v. 302.
+
+
+
+
+DIGBY, KENELM HENRY (1800-1880), English writer, youngest son of William
+Digby, dean of Clonfert, was born at Clonfert, Ireland, in 1800. He was
+educated at Trinity College, Cambridge, and soon after taking his B.A.
+degree there in 1819 became a Roman Catholic. He spent most of his life,
+which was mainly devoted to literary pursuits, in London, where he died
+on the 22nd of March 1880. Digby's reputation rests chiefly on his
+earliest publication, _The Broadstone of Honour, or Rules for the
+Gentlemen of England_ (1822), which contains an exhaustive survey of
+medieval customs, full of quotations from varied sources. The work was
+subsequently enlarged and issued (1826-1827) in four volumes entitled:
+_Godefridus_, _Tancredus_, _Morus_ and _Orlandus_ (numerous
+re-impressions, the best of which is the edition brought out by B.
+Quaritch in five volumes, 1876-1877).
+
+  Among Digby's other works are: _Mores Catholici, or Ages of Faith_ (11
+  vols., London, 1831-1840); _Compitum; or the Meeting of the Ways at
+  the Catholic Church_ (7 vols., London, 1848-1854); _The Lovers' Seat,
+  Kathemerina; or Common Things in relation to Beauty, Virtue and Faith_
+  (2 vols., London, 1856). A complete list is given in J. Gillow's
+  _Bibliographical Dictionary of English Catholics_, ii. 81-83.
+
+
+
+
+DIGENES ACRITAS, BASILIUS, Byzantine national hero, probably lived in
+the 10th century. He is named Digenes (of double birth) as the son of a
+Moslem father and a Christian mother; Acritas ([Greek: akra], frontier,
+boundary), as one of the frontier guards of the empire, corresponding to
+the Roman _milites limitanei_. The chief duty of these _acritae_
+consisted in repelling Moslem inroads and the raids of the _apelatae_
+(cattle-lifters), brigands who may be compared with the more modern
+Klephts. The original Digenes epic is lost, but four poems are extant,
+in which the different incidents of the legend have been worked up by
+different hands. The first of these consists of about 4000 lines,
+written in the so-called "political" metre, and was discovered in the
+latter part of the 19th century, in a 16th-century MS., at Trebizond;
+the other three MSS. were found at Grotta Ferrata, Andros and Oxford.
+The poem, which has been compared with the _Chanson de Roland_ and the
+_Romance of the Cid_, undoubtedly contains a kernel of fact, although it
+cannot be regarded as in any sense an historical record. The scene of
+action is laid in Cappadocia and the district of the Euphrates.
+
+  Editions of the Trebizond MS. by C. Sathas and E. Legrand in the
+  _Collection des monuments pour servir a l'etude de la langue
+  neohellenique_, new series, vi. (1875), and by S. Joannides
+  (Constantinople, 1887). See monographs by A. Luber (Salzburg, 1885)
+  and G. Wartenberg (Berlin, 1897). Full information will be found in C.
+  Krumbacher's _Geschichte der byzantinischen Litteratur_, p. 827 (2nd
+  ed., 1897); see also G. Schlumberger, _L'Epopee Byzantine a la fin du
+  dixieme siecle_ (1897).
+
+
+
+
+DIGEST, a term used generally of any digested or carefully arranged
+collection or compendium of written matter, but more particularly in law
+of a compilation in condensed form of a body of law digested in a
+systematical method; e.g. the Digest (_Digesta_) or Pandects ([Greek:
+Pandektai]) of Justinian, a collection of extracts from the earlier
+jurists compiled by order of the emperor Justinian. The word is also
+given to the compilations of the main points (marginal or hand-notes) of
+decided cases, usually arranged in alphabetical and subject order, and
+published under such titles as "Common Law Digest," "Annual Digest," &c.
+
+
+
+
+DIGESTIVE ORGANS (PATHOLOGY). Several facts of importance have to be
+borne in mind for a proper appreciation of the pathology of the organs
+concerned in digestive processes (for the anatomy see ALIMENTARY CANAL
+and allied articles). In the first place, more than all other systems,
+the digestive comprises greater range of structure and exhibits wider
+diversity of function within its domain. Each separate structure and
+each different function presents special pathological signs and
+symptoms. Again, the duties imposed upon the system have to be performed
+notwithstanding constant variations in the work set them. The crude
+articles of diet offered them vary immensely in nature, bulk and
+utility, from which they must elaborate simple food-elements for
+absorption, incorporate them after absorption into complex organic
+substances properly designed to supply the constant needs of cellular
+activity, of growth and repair, and fitly harmonized to fulfil the many
+requirements of very divergent processes and functions. Any form of
+unphysiological diet, each failure to cater for the wants of any special
+tissue engaged in, or of any processes of, metabolism, carry with them
+pathological signs. Perhaps in greater degree than elsewhere are the
+individual sections of the digestive system dependent upon, and closely
+correlated with, one another. The lungs can only yield oxygen to the
+blood when the oxygen is uncombined; no compounds are of use. The
+digestive organs have to deal with an enormous variety of compound
+bodies, from which to obtain the elements necessary for protoplasmic
+upkeep and activity. Morbid lesions of the respiratory and circulatory
+systems are frequently capable of compensation through increased
+activity elsewhere, and the symptoms they give rise to follow chiefly
+along one line; diseases of the digestive organs are more liable to
+occasion disorders elsewhere than to excite compensatory actions. The
+digestive system includes every organ, function and process concerned
+with the utilization of food-stuffs, from the moment of their entrance
+into the mouth, their preparation in the canal, assimilation with the
+tissues, their employment therein, up to their excretion or expulsion in
+the form of waste. Each portion resembles a link of a continuous chain;
+each link depends upon the integrity of the others, the weakening or
+breaking of one straining or making impotent the chain as a whole.
+
+The mucous membrane lining the alimentary tract is the part most subject
+to pathological alterations, and in this connexion it should be
+remembered that this membrane differs both in structure and functions
+throughout the tract. Chiefly protective from the mouth to the cardia,
+it is secretory and absorbent in the stomach and bowel; while the
+glandular cells forming part of it secrete both acid and alkaline
+fluids, several ferments or mucus. Over the dorsum of the tongue its
+modified cells subserve the sense of taste. Without, connected with it
+by the submucous connective tissue, is placed the muscular coat, and
+externally over the greater portion of its length the peritoneal serous
+membrane. All parts are supplied with blood-vessels, lymph-ducts and
+nerves, the last belonging either to local or to central circuits.
+Associated with the tract are the salivary glands, the liver and the
+pancreas; while, in addition, lymphoid tissue is met with diffusely
+scattered throughout the lining membranes in the tonsils, appendix,
+solitary glands and Peyer's patches, and the mesenteric glands. The
+functions of the various parts of the system in whose lesions we are
+here interested are many in number, and can only be summarized here.
+(For the physiology of digestion see NUTRITION.) Broadly, they maybe
+given as: (1) Ingestion and swallowing of food, transmission of it
+through the tract, and expulsion of the waste material; (2) secretion of
+acids and alkalis for the performance of digestive processes, aided by
+(3) elaboration and addition of complex bodies, termed enzymes or
+ferments; (4) secretion of mucus; (5) protection of the body against
+organismal infection, and against toxic products; (6) absorption of food
+elements and reconstitution of them into complex substances fitted for
+metabolic application; and (7) excretion of the waste products of
+protoplasmic action. These functions may be altered by disease, singly
+or in conjunction; it is rare, however, to find but one affected, while
+an apparently identical disturbance of function may often arise from
+totally different organic lesions. Another point of importance is seen
+in the close interdependence which exists between the secretions of acid
+and those of alkaline reaction. The difference in reaction seems to act
+_mutatis mutandis_ as a stimulant in each instance.
+
+
+_General Diseases._
+
+  Vascular lesions.
+
+In all sections of the alimentary canal actively engaged in the
+digestion of food, a well-marked local engorgement of the blood-vessels
+supplying the walls occurs. The hyperaemia abates soon after completion
+of the special duties of the individual sections. This normal condition
+may be abnormally exaggerated by overstimulation from irritant poisons
+introduced into the canal; from too rich, too copious or indigestible
+articles of diet; or from too prolonged an experience of some unvaried
+kind of food-stuff, especially if large quantities of it are necessary
+for metabolic needs; entering into the first stage of inflammation,
+acute hyperaemia. More important, because productive of less tractable
+lesions, is passive congestion of the digestive organs. Whenever the
+flow of blood into the right side of the heart is hindered, whether it
+arise from disease of the heart itself, or of the lungs, or proceed from
+obstruction in some part of the portal system, the damming-back of the
+venous circulation speedily produces a more or less pronounced stasis of
+the blood in the walls of the alimentary canal and in the associated
+abdominal glands. The lack of a sufficiently vigorous flow of blood is
+followed by deficient secretion of digestive agents from the glandular
+elements involved, by decreased motility of the muscular coats of the
+stomach and bowel, and lessened adaptability throughout for dealing with
+even slight irregular demands on their powers. The mucous membrane of
+the stomach and bowel, less able to withstand the effects of irritation,
+even of a minor character, readily passes into a condition of chronic
+catarrh, while it frequently is the seat of small abrasions,
+haemorrhagic erosions, which may cause vomiting of blood and the
+appearance of blood in the stools. Obstruction to the flow of blood from
+the liver leads to dilatation of its blood-vessels, consequent pressure
+upon the hepatic cells adjoining them, and their gradual loss of
+function, or even atrophy and degeneration. In addition to the results
+of such passive congestion exhibited by the stomach and bowel as noted
+above, passive congestion of the liver is often accompanied by varicose
+enlargement of the abdominal veins, in particular of those which
+surround the lower end of the oesophagus, the lowest part of the rectum
+and anus. In the latter position these dilated veins constitute what are
+known as haemorrhoids or piles, internal or external as their site lies
+within or outside the anal aperture.
+
+
+  Inflammatory lesions.
+
+The mucous and serous membranes of the canal and the glandular elements
+of the associated organs are the parts most subject to inflammatory
+affections. Among the several sections of the digestive tract itself,
+the oesophagus and jejunum are singularly exempt from inflammatory
+processes; the fauces, stomach, caecum and appendix, ileum, mouth and
+duodenum (including the opening of the common bile-duct), are more
+commonly involved. _Stomatitis_, or inflammation of the mouth, has many
+predisposing factors, but it has now been definitely determined that its
+exciting cause is always some form of micro-organism. Any condition
+favouring oral sepsis, as carious teeth, pyorrhoea alveolaris (a
+discharge of pus due to inflamed granulations round carious teeth),
+granulations beneath thick crusts of tartar, or an irritating tooth
+plate, favours the growth of pyogenic organisms and hence of stomatitis.
+Many varieties of this disease have been described, but all are forms of
+"pyogenic" or "septic stomatitis." This in its mildest form is catarrhal
+or erythematous, and is attended only by slight swelling tenderness and
+salivation. In its next stage of acuteness it is known as "membranous,"
+as a false membrane is produced somewhat resembling that due to
+diphtheria, though caused by a staphylococcus only. A still more acute
+form is "ulcerative," which may go on to the formation of an abscess
+beneath the tongue. Scarlet fever usually gives rise to a slight
+inflammation of the mouth followed by desquamation, but more rarely it
+is accompanied by a most severe oedematous stomatitis with glossitis and
+tonsillitis. Erysipelas on the face may infect the mouth, and an acute
+stomatitis due to the diphtheria bacillus, Klebs-Loeffler bacillus, has
+been described. A distinct and very dangerous form of stomatitis in
+infants and young children is known as "aphthous stomatitis" or
+"thrush." This is caused by the growth of _Oidium albicans_. It is
+always preceded by a gastro-enteritis and dry mouth, and if this is not
+attended to, soon attracts attention by the little white raised patches
+surrounded by a dusky red zone scattered on tongue and cheeks.
+Epidemics have occurred in hospitals and orphanages. Mouth breathing is
+the cause of many ills. As a result of this, the mucous membrane of the
+tongue, &c., becomes dry, micro-organisms multiply and the mouth becomes
+foul. Also from disease of the nose, the upper jaw, palate and teeth do
+not make proper progress in development. There is overgrowth of tonsils,
+and adenoids, with resulting deafness, and the child's mental
+development suffers. An ordinary "sore throat" usually signifies acute
+catarrh of the fauces, and is of purely organismal origin, "catching
+cold" being only a secondary and minor cause. In "relaxed throats" there
+is a chronic catarrhal state of the lining membrane, with some passive
+congestion. The tonsils are peculiarly liable to catarrhal attacks, as
+might a priori be expected by reason of their Cerberus-like function
+with regard to bacterial intruders. Still, acute attacks of tonsillitis
+appear on good evidence to be more common among individuals predisposed
+constitutionally to rheumatic manifestations. Cases of acute tonsillitis
+may or may not go on to suppuration or quinsy; in all there is great
+congestion of the glands, increased mucus secretion, and often secondary
+involvement of the lymphatic glands of the neck. Repeated acute attacks
+often lead to chronic inflammation, in which the glands are enlarged,
+and often hypertrophied in the true sense of the term. The oesophagus is
+the seat of inflammation but seldom. In infants and young children
+thrush due to _Oidium albicans_ may spread from the mouth, and also a
+diphtheritic inflammation spreads from the fauces into the oesophagus. A
+catarrhal oesophagitis is rarely seen, but the commonest form is
+traumatic, due to the swallowing of boiling water, corrosive or irritant
+substances, &c. A non-malignant ulceration may result which later leads
+on to an oesophageal stricture. The physical changes presented by the
+coats of the stomach and the intestine, the subjects of catarrhal
+attacks, closely resemble one another, but differ symptomatically. Acute
+catarrh of the stomach is associated with intense hyperaemia of its
+lining coats, with visible engorgement and swelling of the mucous
+membrane, and an excessive secretion of mucus. The formation of active
+gastric juice is arrested, digestion ceases, peristaltic movements are
+sluggish or absent, unless so over-stimulated that they act in a
+direction the reverse of the normal, and induce expulsion of the gastric
+contents by vomiting. The gastric contents, in whatever degree of
+dilution or concentration they may have been ingested, when ejected are
+of porridge-thick consistency, and often but slightly digested. Such
+conditions may succeed a severe alcoholic bout, be caused by irritant
+substances taken in by the mouth or arise from fermentative processes in
+the stomach contents themselves. Should the irritating material succeed
+in passing from the stomach into the bowel, similar physical signs are
+present; but as the quickest path offered for the expulsion of the
+offending substances from the body is downwards, peristalsis is
+increased, the flow of fluid from the intestinal glands is larger in
+bulk, though of less potency as regards its normal actions, than in
+health, and diarrhoea, with removal of the irritant, follows. As a
+general rule, the more marked the involvement of the large bowel, the
+severer and more fluid is the resultant diarrhoea. Inflammation of the
+stomach may be due to mechanical injury, thermal or chemical irritants
+or invasion by micro-organisms. Also all the symptoms of gastric catarrh
+may be brought on by any acute emotion. The commonest mechanical injury
+is that due to an excess of food, especially when following on a fast;
+poisons act as irritants, and also the weevils of cheese and the larvae
+of insects.
+
+Inflammatory affections of the caecum and its attached appendix
+vermiformis are very common, and give rise to several special symptoms
+and signs. Acute inflammatory appendicitis appears to be increasing in
+frequency, and is associated by many with the modern deterioration in
+the teeth. Constipation certainly predisposes to it, and it appears to
+be more prevalent among medical men, commercial travellers, or any
+engaged in arduous callings, subjected to irregular meals, fatigue and
+exposure. A foreign body is the exciting cause in many cases, though
+less commonly so than was formerly imagined. The inflammation in the
+appendix varies in intensity from a very slight catarrhal or simple
+form to an ulcerative variety, and much more rarely to the acute
+fulminating appendicitis in which necrosis of the appendix with abscess
+formation occurs. It is always accompanied by more or less peritonitis,
+which is protective in nature, shutting in the inflammatory process.
+Very similar symptomatically is the condition termed perityphlitis,
+doubtless in former days frequently due to the appendix, an acute or
+chronic inflammation of the walls of the caecum often leading to abscess
+formation outside the gut, with or without direct communication with the
+canal. The colon is subject to three main forms of inflammation. In
+simple _colitis_ the mucous membrane of the colon is intensely injected,
+bright red in colour, and secreting a thick mucus, but there is no
+accompanying ulceration. It is often found in association with some
+constitutional disease, as Bright's disease, and also with cancer of the
+bowel. But when it has no association with other trouble it is probably
+bacterial in origin, the _Bacillus enteritidis spirogenes_ having been
+isolated in many cases. The motions always contain large quantities of
+mucus and more or less blood. A second very severe form of inflammation
+of the colon is known as "membranous colitis," and this may be either
+dyspeptic, or secondary to other diseases. In this trouble membranes are
+passed _per anum_, accompanied by a pain so intense as often to cause
+fainting. In severe cases complete tubular casts of the intestine have
+been found. Often the motions contain very little faecal matter, but
+consist only of membranes, mucus and a little blood. A third form is
+that known as "ulcerative colitis." Any part of the large intestine may
+be affected, and the ulceration shows no special distribution. In severe
+cases the muscular coat is exposed, and perforation may ensue. The
+number of ulcers varies from a few to many dozen, and in size from a pea
+to a five-shilling piece. Like all chronic intestinal ulcers they show a
+tendency to become transverse.
+
+Chronic catarrhal affections of the stomach are very common, and often
+follow upon repeated acute attacks. In them the connective tissue
+increases at the expense of the glandular elements; the mucous membrane
+becomes thickened and less active in function. Should the muscular coat
+be involved, the elasticity and contractility of the organ suffer;
+peristaltic movement is weakened; expulsion of the contents through the
+pylorus hindered; and, aggravated by these effects, the condition
+becomes worse, atonic dyspepsia in its most pronounced form results,
+with or without dilatation. Chronic vascular congestion may occasion in
+process of time similar signs and symptoms.
+
+Duodenal catarrh is constantly associated with jaundice, indeed is most
+probably the commonest cause of catarrhal jaundice; often it is
+accompanied by catarrh of the common bile-duct. Chronic inflammation of
+the small intestine gives rise to less prominent symptoms than in the
+stomach. It generally arises from more than one cause; or rather
+secondary causes rapidly become as important as the primary in its
+incidence. Chronic congestion and prolonged irritation lead to deficient
+secretion and sluggish peristalsis; these effects encourage intestinal
+putrefaction and auto-intoxication; and these latter, in turn, increase
+the local unrest.
+
+
+  Infective lesions.
+
+The intestinal mucous membrane, the peritoneum and the mesenteric glands
+are the chief sites of tubercular infection in the digestive organs.
+Rarely met with in the gullet and stomach, and comparatively seldom in
+the mouth and lips, tubercular inflammation of the small intestine and
+peritoneum is common. Tubercular enteritis is a frequent accompaniment
+of phthisis, but may occur apart from tubercle of other organs. Children
+are especially subject to the primary form. Tubercular peritonitis often
+is present also. The inflammatory process readily tends towards ulcer
+formation, with haemorrhage and sometimes perforation. If in the large
+bowel, the symptoms are usually less acute than those characterizing
+tubercular inflammation of the small intestine. The appendix has been
+found to be the seat of tubercular processes; in the rectum they form
+the general cause of the fistulae and abscesses so commonly met with
+here. Tubercular peritonitis may be primary or secondary, acute or
+chronic; occasionally very acute cases are seen running a rapid course;
+the majority are chronic in type. The tubercles spread over the surface
+of the serous membrane, and if small and not very numerous may give rise
+in chronic cases to few symptoms; if larger, and especially when they
+involve and obstruct the lymph- and blood-vessels, ascites follows. It
+is hardly possible that tubercular invasion of the mesenteric glands can
+ever occur unaccompanied by peritoneal infection; but when the infection
+of the glands constitutes the most prominent sign, the term _tabes
+mesenterica_ is sometimes employed. Here the glands, enlarged, form a
+doughy mass in the abdomen, leading to marked protrusion of the
+abdominal walls, with wasting elsewhere and diarrhoea.
+
+The liver is seldom attacked by tubercle, unless in cases of general
+miliary tuberculosis. Now and then it contains large caseous tubercular
+masses in its substance.
+
+An important fact with regard to the tubercular processes in the
+digestive organs lies in the ready response to treatment shown by many
+cases of peritoneal or mesenteric invasion, particularly in the young.
+
+The later sequelae of syphilis display a predilection for the rectum and
+the liver, usually leading to the development of a stricture in the
+former, to a diffuse hepatitis or the formation of gummata in the
+second. In inherited syphilis the temporary teeth usually appear early,
+are discoloured and soon crumble away. The permanent teeth may be sound
+and healthy, but are often--especially the upper incisors--notched and
+stunted, when they are known as "Hutchinson's teeth." As the result both
+of syphilis and of tubercle, the tissues of the liver and bowel may
+present a peculiar alteration; they become amyloid, or lardaceous, a
+condition in which they appear "waxy," are coloured dark mahogany brown
+with dilute iodine solutions, and show degenerative changes in the
+connective tissue.
+
+The _Bacillus typhosus_ discovered by Eberth is the causal agent of
+typhoid fever, and has its chief seat of activity in the small
+intestine, more especially in the lower half of the ileum. Attacking the
+lymphoid follicles in the mucous membrane, it causes first inflammatory
+enlargement, then necrosis and ulceration. The adjacent portions of the
+mucous membrane show acute catarrhal changes. Diarrhoea, of a special
+"pea-soup" type, may or may not be present; while haemorrhage from the
+bowel, if ulcers have formed, is common. As the ulcers frequently extend
+down to the peritoneal coat of the bowel, perforation of this membrane
+and extravasation into the peritoneal cavity is easily induced by
+irritants introduced into or elaborated in the bowel, acting physically
+or by the excitation of hyper-peristalsis.
+
+True Asiatic cholera is due to the comma-bacillus or spirillum of
+cholera, which is found in the rice-water evacuations, in the contents
+of the intestine after death, and in the mucous membrane of the
+intestine just beneath the epithelium. It has not been found in the
+blood. It produces an intense irritation of the bowel, seldom of the
+stomach, without giving rise locally to any marked physical change; it
+causes violent diarrhoea and copious discharges of "rice-water" stools,
+consisting largely of serum swarming with the organism.
+
+Dysentery gives rise to an inflammation of the large intestine and
+sometimes of the lower part of the ileum, resulting in extensive
+ulceration and accompanied by faecal discharges of mucus, muco-pus or
+blood. In some forms a protozoan, the _Amoeba dysenteriae_, is found in
+the stools--this is the amoebic dysentery; in other cases a bacillus,
+_Bacillus dysenteriae_, is found--the bacillary dysentery.
+
+Acute parotitis, or mumps, is an infectious disease of the parotid
+glands, chiefly interesting because of the association between it and
+the testes in males, inflammation of these glands occasionally following
+or replacing the affection of the parotids. The causal agent is probably
+organismal, but has as yet escaped detection.
+
+
+  New growths.
+
+The relative frequency with which malignant growths occur in the
+different organs of the digestive system may be gathered from the
+tabular analysis, on p. 266, of 1768 cases recorded in the books of the
+Edinburgh Royal Infirmary as having been treated in the medical and
+surgical wards between the years 1892 and 1899 inclusive. Of these,
+1263, or 71.44%, were males; 505, or 28.56%, females. (See Table I. p.
+266.)
+
+If the figures there given be classified upon broader lines, the results
+are as given in Table II. p. 266, and speak for themselves.
+
+The digestive organs are peculiarly subject to malignant disease, a
+result of the incessant changes from passive to active conditions, and
+vice versa, called for by repeated introduction of food; while the
+comparative frequency with which different parts are attacked depends,
+in part, upon the degree of irritation or changes of function imposed
+upon them. Scirrhous, encephaloid and colloid forms of carcinoma occur.
+In the stomach and oesophagus the scirrhous form is most common, the
+soft encephaloid form coming next. The most common situation for
+cancerous growth in the stomach is the pyloric region. Walsh out of 1300
+cases found 60.8% near the pylorus, 11.4% over the lesser curvature, and
+4.7% more or less over the whole organ. The small intestine is rarely
+attacked by cancer; the large intestine frequently. The rectum, sigmoid
+flexure, caecum and colon are affected, and in this order, the
+cylindrical-celled form being the most common. Carcinoma of the
+peritoneum is generally colloid in character, and is often secondary to
+growths in other organs. Cancer of the liver follows cancer of the
+stomach and rectum in frequency of occurrence, and is relatively more
+common in females than males. Secondary invasion of the liver is a
+frequent sequel to gastric cancer. The pancreas occasionally is the seat
+of cancerous growth.
+
+Sarcomata are not so often met with in the digestive organs. When
+present, they generally involve the peritoneum or the mesenteric glands.
+The liver is sometimes attacked, the stomach rarely.
+
+Benign tumours are not of common occurrence in the digestive organs.
+Simple growths of the salivary glands, cysts of the pancreas and
+polypoid tumours of the rectum are the most frequent.
+
+
+  Animal parasites.
+
+The intestinal canal is the habitat of the majority of animal parasites
+found in man. Frequently their presence leads to no morbid symptoms,
+local or general; nor are the symptoms, when they do arise, always
+characteristic of the presence of parasites alone. Discovery of their
+bodies, or of their eggs, in the stools is in most instances the only
+satisfactory proof of their presence. The parasites found in the bowel
+belong principally to two natural groups, Protozoa and Metazoa. The
+great class of the Protozoa furnish amoebae, members of Sporozoa and
+Infusoria. The amoebae are almost invariably found in the large
+intestine; one species, indeed, is termed _Amoeba coli_. The frequently
+observed relation between attacks of dysentery and the presence of
+amoebae in the stools has led to the proposition that an _Amoeba
+dysenterica_ exists, causing the disease--a theory supported by the
+detection of amoebae in the contents of dysenteric abscesses of the
+liver. No symptoms of injury to health appear to accompany the presence
+of Sporozoa in the bowel, while the species of Infusoria found in it,
+the _Cercomonas_, and _Trichomonas intestinalis_, and the _Balantidium
+coli_, may or may not be guilty of prolonging conditions within the
+bowel as have previously set up diarrhoea.
+
+The Metazoa supply examples of intestinal parasites from the classes
+Annuloida and Nematoidea. To the former class belong the various
+tapeworms found in the small intestine of man. They, like other
+intestinal parasites, are destitute of any power of active digestion,
+simply absorbing the nutritious proceeds of the digestive processes of
+their hosts. Nematode worms infest both the small and large intestine;
+_Ascaris lumbricoides_, the common round worm, and the male _Oxyuris
+vermicularis_ are found in the small bowel, the adult female _Oxyuris
+vermicularis_ and the _Tricocephalus dispar_ in the large.
+
+The eggs of the _Trichina spiralis_, when introduced with the food,
+develop in the bowel into larval forms which invade the tissues of the
+body, to find in the muscles congenial spots wherein to reach maturity.
+Similarly, the eggs of the Echinococcus are hatched in the bowel, and
+the embryos proceed to take up their abode in the tissues of the body,
+developing into cysts capable of growth into mature worms after their
+ingestion by dogs.
+
+
+  Vegetable parasites.
+
+Numbers of bacterial forms habitually infest the alimentary canal. Many
+of them are non-pathogenic; some develop pathogenic characters only
+under provocation or when a suitable environment induces them to act in
+such a manner; others may form the _materies morbi_ of special lesions,
+or be casual visitors capable of originating disease if opportunity
+occurs. Apart from those organisms associated with acute infective
+diseases, disturbances of function and physical lesions may be the
+result of abnormal bacterial activity in the canal; and these
+disturbances may be both local and general. Many of the bacteria
+commonly present produce putrefactive changes in the contents of the
+tract by their metabolic processes. They render the medium they grow in
+alkaline, produce different gases and elaborate more or less virulent
+toxins. Other species set up an acid fermentation, seldom accompanied by
+gas or toxin formation. The products of either class are inimical to the
+free growth of members of the other. The species which produce acids are
+more resistant to the action of acids. Thus, when the contents of the
+stomach possess a normal or excessive proportion of free hydrochloric
+acid, a much larger number of putrefactive and pathogenic organisms in
+the food are destroyed or inhibited than of the bacteria of acid
+fermentation. Diminished gastric acidity allows of the entry of a
+greater number of putrefactive (and pathogenic) types, with, as a
+consequence, increased facilities for their growth and activity, and the
+appearance of intestinal derangements.
+
+    TABLE I.
+
+  +-----------------------------+--------------------------+--------------------------+
+  |        Males.               |        Females.          |        Both Sexes.       |
+  +---------------------+-------+------------------+-------+------------------+-------+
+  |Organ or Tissue in   | Per-  |Organ or Tissue in| Per-  |Organ or Tissue in| Per-  |
+  |Order of Frequency.  |centage|Order of Frequency|centage|Order of Frequency|centage|
+  +---------------------+-------+------------------+-------+------------------+-------+
+  | 1 Stomach           | 22.56 | 1 Stomach        | 22.37 | 1 Stomach        | 22.49 |
+  | 2 Lip               | 12.94 | 2 Rectum         | 17.24 | 2 Rectum         | 13.12 |
+  | 3 Rectum            | 11.57 | 3 Liver          | 15.50 | 3 Liver          | 10.02 |
+  | 4 Tongue            | 11.36 | 4 Peritoneum     |  7.86 | 4 Lip            |  9.89 |
+  | 5 Oesophagus        | 10.90 | 5 Oesophagus     |  5.33 | 5 Oesophagus     |  9.29 |
+  | 6 Liver             |  7.80 | 6 Sigmoid        |  4.53 | 6 Tongue         |  8.96 |
+  | 7 Jaw               |  6.38 | 7 Pancreas       |  3.52 | 7 Jaw            |  5.65 |
+  | 8 Mouth             |  2.88 | 8 Tongue         |  3.12 | 8 Peritoneum     |  2.94 |
+  | 9 Tonsils           |  2.09 | 9 Omentum        |  2.98 | 9 Sigmoid        |  2.56 |
+  |10 Sigmoid flexure   |  1.77 |10 Lip            |  2.57 |10 Mouth          |  2.40 |
+  |11 Parotid           |  1.10 |11 Jaw            |  1.97 |11 Pancreas       |  1.80 |
+  |12 Pancreas          |   "   |12 Colon          |  1.84 |12 Tonsils        |  1.35 |
+  |13 Caecum            |  0.94 |13 Abdomen        |   "   |13 Omentum        |  1.25 |
+  |14 Peritoneum        |   "   |14 Intestine      |  1.56 |14 Parotid        |  1.12 |
+  |15 Colon             |  0.89 |15 Caecum         |  1.37 |15 Colon          |   "   |
+  |16 Pharynx           |       |16 Mouth          |  1.18 |16 Caecum         |  1.08 |
+  |17 Intestine (site   |       |17 Parotid        |   "   |17 Intestine      |  1.00 |
+  |     unknown)        |  0.79 |18 Splenic flexure|  0.98 |18 Abdomen        |   "   |
+  |18 Abdomen           |  0.71 |19 Jejunum and    |       |19 Pharynx        |  0.62 |
+  |19 Mesentery         |  0.55 |     ileum        |  0.68 |20 Mesentery      |  0.52 |
+  |20 Omentum           |   "   |20 Tonsils        |  0.68 |21 Jejunum and    |       |
+  |21 Hepatic flexure   |  0.39 |21 Pharynx        |  0.40 |     ileum        |  0.44 |
+  |22 Submaxillary gland|  0.31 |22 Hepatic flexure|   "   |22 Hepatic flexure|   "   |
+  |23 Jejunum and ileum |   "   |23 Mesentery      |   "   |23 Splenic flexure|  0.28 |
+  |24 Duodenum          |  0.23 |24 Submaxillary   |  0.20 |24 Submaxillary   |  0.22 |
+  |25 Splenic flexure   |  0.15 |25 Duodenum       |   "   |25 Duodenum       |       |
+  +---------------------+-------+------------------+-------+------------------+-------+
+
+  _Note._--The figures where several organs are bracketed apply to each organ separately.
+
+In a healthy new-born infant the mouth is free from micro-organisms, and
+very few are found in a breast-fed baby, but _Bacillus lactis_ may be
+found where the child is bottle fed. If there is trouble with the first
+dentition and food is allowed to collect, staphylococci, streptococci,
+pneumococci and colon bacilli may be present. Even in healthy babies
+_Oidium albicans_ may be present, and in older children the
+pseudo-diphtheria bacillus. From carious teeth may be isolated
+streptothrix, leptothrix, spirilla and fusiform bacilli. Under
+conditions of health these micro-organisms live in the mouth as
+saprophytes, and show no virulence when cultivated and injected into
+animals. The two common pyogenetic organisms, _Staphylococcus albus_ and
+_brevis_, show no virulence. Also the pneumococcus, though often
+present, must be raised in virulence before it can produce untoward
+results. The foulness of the mouth is supposed to be due to the colon
+bacillus and its allies, but those obtained from the mouth are
+innocuous. Also to enable the _Oidium albicans_ to attack the mucous
+membrane there must be some slight inflammation or injury. The
+micro-organisms found in the stomach gain access to that organ in the
+food or by regurgitation from the small intestine. Most are relatively
+inert, but some have a special fermentative action on the food (see
+NUTRITION). Abelous isolated sixteen distinct species of organism from a
+healthy stomach, including Sarcinae, _B. lactis_, _pyocyaneus_,
+_subtilis_, _lactis erythrogenes_, _amylobacter_, _megatherium_, and
+_Vibrio rugula_.
+
+
+  Physical abnormalities
+
+Hare-lip, cleft palate, hernia and imperforate anus are physical
+abnormalities which are interesting to the surgeon rather than to the
+pathologist. The oesophagus may be the seat of a diverticulum, or blind
+pouch, usually situated in its lower half, which in most instances is
+probably partly acquired and partly congenital; a local weakness
+succumbing to pressure. Hypertrophy of the muscular coat of the pyloric
+region is an infrequent congenital gastric anomaly in infants,
+preventing the passage of food into the bowel, and causing death in a
+short time. Incomplete closure of the vitelline duct results in the
+presence of a diverticulum--Meckel's--generally connected with the
+ileum, mainly important by reason of the readiness with which it
+occasions intestinal obstruction. Idiopathic congenital dilatation of
+the colon has been described.
+
+    TABLE II.
+
+  +------------------+--------+------------------+--------+------------------+--------+
+  |      Males.      |  Per-  |     Females.     |  Per-  |      Total.      |  Per-  |
+  |                  |centage.|                  |centage.|                  |centage.|
+  +------------------+--------+------------------+--------+------------------+--------+
+  | 1 Mouth and      |        | 1 Intestines     |  28.9  | 1 Oesophagus and |        |
+  |     pharynx      | 37.85  | 2 Oesophagus and |        |     stomach      | 31.78  |
+  | 2 Oesophagus and |        |     stomach      |  27.7  | 2 Mouth and      |        |
+  |     stomach      | 33.46  | 3 Liver          |  15.5  |     pharynx      | 30.27  |
+  | 3 Intestines     | 17.04  | 4 Peritoneum     |  13.1  | 3 Intestines     | 20.42  |
+  | 4 Liver          |   7.8  | 5 Mouth and      |        | 4 Liver          | 10.02  |
+  | 5 Peritoneum     |   2.75 |     pharynx      |  11.3  | 5 Peritoneum     |  5.71  |
+  | 6 Pancreas       |   1.1  | 6 Pancreas       |   3.5  | 6 Pancreas       |  1.80  |
+  +------------------+--------+------------------+--------+------------------+--------+
+
+Traction diverticula of the oesophagus not uncommonly occur as sequels
+to suppurative inflammation of cervical lymphatic glands. More
+frequently dilatation of a section is met with, due as a rule to the
+presence of a stricture. The stomach often diverges from the normal in
+size, shape and position. Normally capable in the adult of containing
+from fifty to sixty ounces, either by reason of organic disease, or as
+the result of functional disturbance, its capacity may vary enormously.
+The writer has seen post mortem a stomach which held a gallon (160
+ounces), and again one holding only two ounces. Cancer spread over a
+large area and cirrhosis of the stomach wall cause diminution in
+capacity; pyloric obstruction, weakness of the muscular coat, and
+nervous influences are associated with dilatation. A peculiar distortion
+of the shape of the stomach follows cicatrization of ulcers of greater
+or lesser curvature; the gastric cavity becomes "hour-glass" in shape.
+In addition, the stomach may be displaced downwards as a whole, a
+condition known as gastroptosis: if the pyloric portion only be
+displaced, the lesion is termed pyloroptosis. Ptoses of other abdominal
+organs are described; the liver, transverse colon, spleen and kidneys
+may be involved. Displacements downwards of the stomach and transverse
+colon, along with a movable right kidney and associated with dyspepsia
+and neurasthenia, form the malady termed by Glenard enteroptosis. A
+general visceroptosis often occurs in those patients who have some
+tuberculous lesion of the lungs or elsewhere, this disease causing a
+general weakening and subsequent stretching of all ligaments.
+Displacements of the abdominal viscera are almost invariably accompanied
+by symptoms of dyspepsia of a neurotic type. The rectum is liable to
+prolapse, consequent upon constipation and straining at stool, or
+following local injuries of the perineal floor.
+
+
+  Influence of the nervous system.
+
+Every pathological lesion shown by digestive organs is closely
+associated with the state of the nervous system, general or local; so
+stoppage of active gastric digestive processes after profound nervous
+shock, and occurrence of nervous diarrhoea from the same cause. Gastric
+dyspepsia of nervous origin presents most varied and contradictory
+symptoms: diminished acidity of the gastric juice, hyper-acidity,
+over-production, arrest of secretion, lessened or increased movements,
+greater sensitiveness to the presence of contents, dilatation or spasm.
+Often the nervous cause can be traced back farther,--in females,
+frequently to the pelvic organs; in both sexes, to the condition of the
+blood, the brain or the bowel. Unhealthy conditions related to
+evacuation of the bowel-contents commonly induce reflex nervous
+manifestations of abnormal character referred to the stomach and liver.
+Gastric disturbances similarly react upon the proper conduct of
+intestinal functions.
+
+
+_Local Diseases._
+
+_The Mouth._--The lining membrane of the cheeks inside the mouth, of the
+gums and the under-surface and edges of the tongue, is often the seat of
+small irritable ulcers, usually associated with some digestive
+derangement. A crop of minute vesicles known as Koplik's spots over
+these parts has been lately stated by Koplik to be an early symptom of
+measles. Xerostomia, or dry mouth, is a rare condition, connected with
+lack of salivary secretion. Gangrenous stomatitis, cancrum oris, or
+noma, occasionally attacks debilitated children, or patients
+convalescing from acute fevers, more especially after measles. It
+commences in the gums or cheeks, and causes widespread sloughing of the
+adjacent soft parts--it may be of the bones.
+
+_The Stomach._--It were futile to attempt to enumerate all the protean
+manifestations of disturbance which proceed from a disordered stomach.
+The possible permutations and combinations of the causes of gastric
+vagaries almost reach infinity. Idiosyncrasy, past and present gastric
+education, penury or plethora, actual digestive power, motility, bodily
+requirements and conditions, environment, mental influences, local or
+adjacent organic lesions, and, not least, reflex impressions from other
+organs, all contribute to the variance.
+
+Ulcer of the stomach, however--the perforating gastric ulcer--occupies a
+unique position among diseases of this organ. Gastric ulcers are
+circumscribed, punched out, rarely larger than a sixpenny-bit,
+funnel-shaped, the narrower end towards the peritoneal coat, and
+distributed in those regions of the stomach wall which are most exposed
+to the action of the gastric contents. They occur most frequently in
+females, especially if anaemic, and are usually accompanied by excess of
+acid, actual or relative to the state of the blood, in the stomach
+contents. Local pain, dorsal pain, generally to the left of the eighth
+or ninth dorsal spinous process, and haematernesis and melaena, are
+symptomatic of it. The amount of blood lost varies with the rapidity of
+ulcer formation and the size of vessel opened into. Fatal results arise
+from ulceration into large blood-vessels, followed by copious
+haemorrhage, or by perforation of the ulcer into the peritoneal cavity.
+Scars of such ulcers may be found post mortem, although no symptoms of
+gastric disease have been exhibited during life; gastric ulcers,
+therefore, may be latent.
+
+Irritation of the sensory nerve-endings in the stomach wall from the
+presence of an increased proportion of acid, organic or mineral, in the
+stomach contents is accountable for the well known symptom heartburn.
+Water-brash is a term applied to eructation of a colourless, almost
+tasteless fluid, probably saliva, which has collected in the lower part
+of the oesophagus from failure of the cardiac sphincter of the stomach
+to relax; reversed oesophageal peristalsis causing regurgitation. A
+similar reversed action serves in merycism, or rumination, occasionally
+found in man, to raise part of the food, lately ingested, from the
+stomach to the mouth. Vomiting also is aided by reversed peristaltic
+action, both of the stomach and the oesophagus, with the help of the
+diaphragm and the muscles of the anterior abdominal wall. Emesis may be
+caused both by local nervous influence, and through the central nervous
+mechanism either reflexly or from the direct action of substances
+circulating in the blood. Further, the causal agent acting on the
+central nervous apparatus may be organic or functional, as well as
+medicinal. Vomiting without any apparent cause suggests nervous lesions,
+organic or reflex. The obstinate vomiting of pregnancy is a case in
+point. Here the primary cause proceeds reflexly from the pelvis. In
+females the pelvic organs are often the true source of emesis.
+Haematemesis accompanies gastric ulcer, cancer, chronic congestion with
+haemorrhagic erosion, congestion of the liver, or may follow violent
+acts of vomiting. In cases of ulcer the blood is usually bright and in
+considerable amount; in cancer, darker, like coffee-grounds; and in
+cases of erosion, in smaller quantity and of bright colour. The reaction
+of the stomach contents, if the cause be doubtful, yields valuable aid
+towards a diagnosis. Of increased acidity in gastric ulcer, normal in
+hepatic congestion, it is diminished in cancer; but as the acid present
+in cancer is largely lactic, analysis of the gastric contents must often
+be a _sine qua non_, because hyperacidity from lactic may obscure
+hypoacidity of hydrochloric acid.
+
+Flatulence usually results from fermentative processes in the stomach
+and bowel, as the outcome of bacterial activity. A different form of
+flatulence is common in neurotic individuals: in such the gas evolved
+consists simply in carbonic acid liberated from the blood, and its
+evolution is generally characterized by rapid development and by lack of
+all fermentative signs.
+
+_The Liver._--The liver is an organ frequently libelled for the
+delinquencies of other organs, and regarded as a common source of ill.
+In catarrhal jaundice it is in most cases the bowel that is at fault,
+the liver acting properly, but unable to get rid of all the bile
+produced. The liver suffers, however, from several diseases of its own.
+Its fibrous or connective tissue is very apt to increase at the expense
+of the cellular elements, destroying their functions. This cirrhotic
+process usually follows long-continued irritation, such as is produced
+by too much alcohol absorbed from the bowel habitually, the organ
+gradually becoming harder in texture and smaller in bulk. Hypertrophic
+cirrhosis of the liver is not uncommonly met with, in which the liver is
+much increased in size, the "unilobular" form, also of alcoholic origin.
+In still-born children and in some infants a form of hypertrophic
+cirrhosis is occasionally seen, probably of hereditary syphilitic
+origin. Acute congestion of the liver forms an important symptom of
+malarial fever, and often leads in time to establishment of cirrhotic
+changes; here the liver is generally enlarged, but not invariably so,
+and the part played by alcohol in its causation has still to be
+investigated. Acute yellow atrophy of the liver is a disease _sui
+generis_. Of rare occurrence, possibly of toxic origin, it is marked by
+jaundice, at first of usual type, later becoming most intense; by
+vomiting; haemorrhages widely distributed; rapid diminution in the size
+of the liver; the appearance of leucin and tyrosin in the urine, with
+lessened urea; and in two or three days, death. The liver after death is
+soft, of a reddish colour dotted with yellow patches, and weighs only
+about a third part of the normal--about 11/2 lb in place of 3-3/4 lb. A
+closely analogous affection of the liver, known as Weil's disease, is of
+infectious type, and has been noted in epidemic form. In this the
+spleen and liver are commonly but not always swollen, and the liver is
+often tender on pressure. As a large proportion of the sufferers from
+this disease have been butchers, and the epidemics have occurred in the
+hot season of the year, it probably arises from contact with decomposing
+animal matter. Hepatic abscess may follow on an attack of amoebic
+dysentery, and is produced either by infection through the portal vein,
+or by direct infection from the adjacent colon. In general pyaemia
+multiple small abscesses may occur in the liver.
+
+_The Gall-Bladder._--The formation of biliary calculi in the
+gall-bladder is the chief point of interest here. At least 75% of such
+cases occur in women, especially in those who have borne children.
+Tight-lacing has been stated to act as an exciting cause, owing to the
+consequent retardation of the flow of bile. Gall-stones may number from
+one to many thousands. They are largely composed of cholesterin,
+combined with small amounts of bile-pigments and acids, lime and
+magnesium salts. Their presence may give rise to no symptoms, or may
+cause violent biliary colic, and, if the bile-stream be obstructed, to
+jaundice. Inflammatory processes may be initiated in the gall-bladder or
+the bile-ducts, catarrhal or suppurative in character.
+
+_The Pancreas._--Haemorrhages into the body of the pancreas, acute and
+chronic inflammation, calculi, cysts and tumours, among which cancer is
+by far the most common, are recognized as occurring in this organ; the
+point of greatest interest regarding them lies in the relations
+established between pancreatic disease and diabetes mellitus, affections
+of the gland frequently being complicated by, and probably causing, the
+appearance of sugar in the urine.
+
+_The Small Intestine._--Little remains to be added to the account of
+inflammatory lesions in connexion with the small intestine. It offers
+but few conditions peculiar to itself, save in typhoid fever, and the
+ease with which it contrives to become kinked, or intussuscepted,
+producing obstruction, or to take part in hernial protrusions. The first
+section, the duodenum, is subject to development of ulcers very similar
+to those of the gastric mucous membrane. For long duodenal ulceration
+has been regarded as a complication of extensive burns of the skin, but
+the relationship between them has not yet been quite satisfactorily
+explained. The condition of colic in the bowel usually arises from
+overdistension of some part of the small gut with gas, the frequent
+sharp turns of the gut facilitating temporary closure of its lumen by
+pressure of the dilated gut near a curve against the part beyond. In the
+large bowel accumulations of gas seldom cause such acute symptoms,
+having a readier exit.
+
+_The Large Intestine._--The colon, especially the ascending portion, may
+become immensely dilated, usually after prolonged constipation and
+paralysis of the gut; occasionally the condition is congenital.
+Straining efforts made in defaecation may often account for prolapse of
+the lower end of the rectum through the anus. Haemorrhage from the bowel
+is usually a sign of disease situated in the large intestine: if bright
+in colour, the source is probably low down; if dark, from the caecum or
+from above the ileo-caecal valve. Blood after a short stay in any
+section of the alimentary canal darkens, and eventually becomes almost
+black in colour.     (A. L. G.; M. F.*)
+
+
+
+
+DIGGES, WEST (1720-1786), English actor, made his first stage appearance
+in Dublin in 1749 as Jaffier in _Venice Preserved_; and both there and
+in Edinburgh until 1764 he acted in many tragic roles with success. He
+was the original "young Norval" in Home's _Douglas_ (1756). His first
+London appearance was as Cato in the Haymarket in 1777, and he
+afterwards played Lear, Macbeth, Shylock and Wolsey. In 1881 he returned
+to Dublin and retired in 1784.
+
+
+
+
+DIGIT (Lat. _digitus_, finger), literally a finger or toe, and so used
+to mean, from counting on the fingers, a single numeral, or, from
+measuring, a finger's breadth. In astronomy a digit is the twelfth part
+of the diameter of the sun or moon; it is used to express the magnitude
+of an eclipse.
+
+
+
+
+DIGITALIS. The leaves of the foxglove (q.v.), gathered from wild plants
+when about two-thirds of their flowers are expanded, deprived usually
+of the petiole and the thicker part of the midrib, bitter taste; and to
+preserve their properties they must be kept excluded from light in
+stoppered bottles. They are occasionally adulterated with the leaves of
+_Inula Conyza_, ploughman's spikenard, which may be distinguished by
+their greater roughness, their less divided margins, and their odour
+when rubbed; also with the leaves of _Symphytum officinale_, comfrey,
+and of _Verbascum Thapsus_, great mullein, which unlike those of the
+foxglove have woolly upper and under surfaces. The earliest known
+descriptions of the foxglove are those given by Leonhard Fuchs and
+Tragus about the middle of the 16th century, but its virtues were
+doubtless known to herbalists at a much remoter period. J. Gerarde, in
+his _Herbal_ (1597), advocates the use of foxglove for a variety of
+complaints; and John Parkinson, in the _Theatrum Botanicum_, or _Theater
+of Plants_ (1640), and later W. Salmon, in _The New London
+Dispensatory_, similarly praised the remedy. Digitalis was first brought
+prominently under the notice of the medical profession by Dr W.
+Withering, who, in his _Account of the Foxglove_ (1785), gave details of
+upwards of 200 cases chiefly dropsical, in which it was used.
+
+Digitalis contains four important glucosides, of which three are cardiac
+stimulants. The most powerful is _digitoxin_ C34H54O11, an extremely
+poisonous and cumulative drug, insoluble in water. _Digitalin_,
+C35H56O14, is crystalline and is also insoluble in water. _Digitalein_
+is amorphous but readily soluble in water. It can therefore be
+administered subcutaneously, in doses of about one-hundredth of a grain.
+_Digitonin_, on the other hand, is a cardiac depressant, and has been
+found to be identical with saponin, the chief constituent of senega
+root. There are numerous preparations, patent and pharmacopeial, their
+composition being extremely varied, so that, unless one has reason to be
+certain of any particular preparation, it is almost better to use only
+the dried leaves themselves in the form of a powder (dose 1/2-2 grains).
+The pharmacopeial tincture may be given in doses of five to fifteen
+minims, and the infusion has the unusually small dose of two to four
+drachms--the dose of other infusions being an ounce or more. The
+tincture contains a fair proportion of both digitalin and digitoxin.
+
+Digitalis leaves have no definite external action. Taken by the mouth,
+the drug is apt to cause considerable digestive disturbance, varying in
+different cases and sometimes so severe as to cause serious difficulty.
+This action is probably due to the digitonin, which is thus a
+constituent in every way undesirable. The all-important property of the
+drug is its action on the circulation. Its first action on any of the
+body-tissues is upon unstriped muscle, so that the first consequence of
+its absorption is a contraction of the arteries and arterioles. No other
+known drug has an equally marked action in contracting the arterioles.
+As the vaso-motor centre in the medulla oblongata is also stimulated, as
+well as the contractions of the heart, there is thus trebly caused a
+very great rise in the blood-pressure.
+
+The clinical influence of digitalis upon the heart is very well defined.
+After the taking of a moderate dose the pulse is markedly slowed. This
+is due to a very definite influence upon the different portions of the
+cardiac cycle. The systole is not altered in length, but the diastole is
+very much prolonged, and since this is the period not only of cardiac
+rest but also of cardiac "feeding"--the coronary vessels being
+compressed and occluded during systole--the result is greatly to benefit
+the nutrition of the cardiac muscle. So definite is this that, despite a
+great increase in the force of the contractions and despite experimental
+proof that the heart does more work in a given time under the influence
+of digitalis, the organ subsequently displays all the signs of having
+rested, its improved vigour being really due to its obtaining a larger
+supply of the nutrient blood. Almost equally striking is the fact that
+digitalis causes an irregular pulse to become regular. Added to the
+greater force of cardiac contraction is a permanent tonic contraction of
+the organ, so that its internal capacity is reduced. The bearing of this
+fact on cases of cardiac dilatation is evident. In larger doses a
+remarkable sequel to these actions may be observed. The cardiac
+contractions become irregular, the ventricle assumes curious
+shapes--"hour-glass," &c.--becomes very pale and bloodless, and finally
+the heart stops in a state of spasm, which shortly afterwards becomes
+rigor-mortis. Before this final change the heart may be started again by
+the application of a soluble potassium salt, or by raising the fluid
+pressure within it. Clinically it is to be observed that the drug is
+cumulative, being very slowly excreted, and that after it has been taken
+for some time the pulse may become irregular, the blood-pressure low,
+and the cardiac pulsations rapid and feeble. These symptoms with more or
+less gastro-intestinal irritation and decrease in the quantity of urine
+passed indicate digitalis poisoning. The initial action of digitalis is
+a stimulation of the cardiac terminals of the vagus nerves, so that the
+heart's action is slowed. Thereafter follows the most important effect
+of the drug, which is a direct stimulation of the cardiac muscle. This
+can be proved to occur in a heart so embryonic that no nerves can be
+recognized in it, and in portions of cardiac muscle that contain neither
+nervecells nor nerve-fibres.
+
+The action of this drug on the kidney is of importance only second to
+its action on the circulation. In small or moderate doses it is a
+powerful diuretic. Though Heidenhain asserts that rise in the renal
+blood-pressure has not a diuretic action per se, it seems probable that
+this influence of the drug is due to a rise in the general
+blood-pressure associated with a relatively dilated condition of the
+renal vessels. In large doses, on the other hand, the renal vessels also
+are constricted and the amount of urine falls. It is probable that
+digitalis increases the amount of water rather than that of the urinary
+solids. In large doses the action of digitalis on the circulation causes
+various cerebral symptoms, such as seeing all objects blue, and various
+other disturbances of the special senses. There appears also to be a
+specific action of lowering the reflex excitability of the spinal cord.
+
+Digitalis is used in therapeutics exclusively for its action on the
+circulation. In prescribing this drug it must be remembered that fully
+three days elapse before it gets into the system, and thus it must
+always be combined with other remedies to tide the patient over this
+period. It must never be prescribed in large doses to begin with, as
+some patients are quite unable to take it, intractable vomiting being
+caused. The three days that must pass before any clinical effect is
+obtained renders it useless in an emergency. A certain consequence of
+its use is to cause or increase cardiac hypertrophy--a condition which
+has its own dangers and ultimately disastrous consequences, and must
+never be provoked beyond the positive needs of the case. But digitalis
+is indicated whenever the heart shows itself unequal to the work it has
+to perform. This formula includes the vast majority of cardiac cases.
+The drug is contra-indicated in all cases where the heart is already
+beating too slowly; in aortic incompetence--where the prolongation of
+diastole increases the amount of the blood that regurgitates through the
+incompetent valve; in chronic Bright's disease and in fatty degeneration
+of the heart--since nothing can cause fat to become contractile.
+
+
+
+
+DIGNE, the chief town of the department of the Basses Alpes, in S.E.
+France, 14 m. by a branch line from the main railway line between
+Grenoble and Avignon. Pop. (1906), town, 4628; commune, 7456. The Ville
+Haute is built on a mountain spur running down to the left bank of the
+Bleone river, and is composed of a labyrinth of narrow winding streets,
+above which towers the present cathedral church, dating from the end of
+the 15th century, but largely reconstructed in modern times, and the
+former bishop's palace (now the prison). The fine Boulevard Gassendi
+separates the Ville Haute from the Ville Basse, which is of modern date.
+The old cathedral (Notre Dame du Bourg) is a building of the 13th
+century, but is now disused except for funerals: it stands at the east
+end of the Ville Basse. The neighbourhood of Digne is rich in orchards,
+which have long made the town famous in France for its preserved fruits
+and confections. It is the _Dinia_ of the Romans, and was the capital of
+the Bodiontii. From the early 6th century at least it has been an
+episcopal see, which till 1790 was in the ecclesiastical province of
+Embrun, but since 1802 in that of Aix en Provence. The history of Digne
+in the middle ages is bound up with that of its bishops, under whom it
+prospered greatly. But it suffered much during the religious wars of the
+16th and 17th centuries, when it was sacked several times. A little way
+off, above the right bank of the Bleone, is Champtercier, the birthplace
+of the astronomer Gassendi (1592-1655), whose name has been given to the
+principal thoroughfare of the little town.
+
+  See F. Guichard, _Souvenirs historiques sur la ville de Digne et ses
+  environs_ (Digne, 1847).     (W. A. B. C.)
+
+
+
+
+DIGOIN, a town of east-central France, in the department of
+Saone-et-Loire, on the right bank of the Loire, 55 m. W.N.W. of Macon on
+the Paris-Lyon railway. Pop. (1906) 5321. It is situated at the meeting
+places of the Loire, the Lateral canal of the Loire and the Canal du
+Centre, which here crosses the Loire by a fine aqueduct. The town
+carries on considerable manufactures of faience, pottery and porcelain.
+The port on the Canal du Centre has considerable traffic in timber,
+sand, iron, coal and stone.
+
+
+
+
+DIJON, a town of eastern France, capital of the department of Cote d'Or
+and formerly capital of the province of Burgundy, 195 m. S.E. of Paris
+on the Paris-Lyon railway. Pop. (1906) 65,516. It is situated on the
+western border of the fertile plain of Burgundy, at the foot of Mont
+Afrique, the north-eastern summit of the Cote d'Or range, and at the
+confluence of the Ouche and the Suzon; it also has a port on the canal
+of Burgundy. The great strategic importance of Dijon as a centre of
+railways and roads, and its position with reference to an invasion of
+France from the Rhine, have led to the creation of a fortress forming
+part of the Langres group. There is no _enceinte_, but on the east side
+detached forts, 3 to 4 m. distant from the centre, command all the great
+roads, while the hilly ground to the west is protected by Fort
+Hauteville to the N.W. and the "groups" of Motte Giron and Mont Afrique
+to the S.W., these latter being very formidable works. Including a fort
+near Saussy (about 8 m. to the N.W.) protecting the water-supply of
+Dijon, there are eight forts, besides the groups above mentioned. The
+fortifications which partly surrounded the old and central portion of
+the city have disappeared to make way for tree-lined boulevards with
+fine squares at intervals. The old churches and historic buildings of
+Dijon are to be found in the irregular streets of the old town, but
+industrial and commercial activity has been transferred to the new
+quarters beyond its limits. A fine park more than 80 acres in extent
+lies to the south of the city, which is rich in open spaces and
+promenades, the latter including the botanical garden and the Promenade
+de l'Arquebuse, in which there is a black poplar famous for its size and
+age.
+
+The cathedral of St Benigne, originally an abbey church, was built in
+the latter half of the 13th century on the site of a Romanesque
+basilica, of which the crypt remains. The west front is flanked by two
+towers and the crossing is surmounted by a slender timber spire. The
+plan consists of three naves, short transepts and a small choir, without
+ambulatory, terminating in three apses. In the interior there is a fine
+organ and a quantity of statuary, and the vaults contain the remains of
+Philip the Bold, duke of Burgundy, and Anne of Burgundy, daughter of
+John the Fearless. The site of the abbey buildings is occupied by the
+bishop's palace and an ecclesiastical seminary. The church of
+Notre-Dame, typical of the Gothic style of Burgundy, was erected from
+1252 to 1334, and is distinguished for the grace of its interior and the
+beauty of the western facade. The portal consists of three arched
+openings, above which are two stages of arcades, open to the light and
+supported on slender columns. A row of gargoyles surmounts each storey
+of the facade, which is also ornamented by sculptured friezes. A turret
+to the right of the portal carries a clock called the Jaquemart, on
+which the hours are struck by two figures. The church of St Michel
+belongs to the 15th century. The west facade, the most remarkable
+feature of the church, is, however, of the Renaissance period. The
+vaulting of the three portals is of exceptional depth owing to the
+projection of the lower storey of the facade. Above this storey rise two
+towers of five stages, the fifth stage being formed by an octagonal
+cupola. The columns decorating the facade represent all the four orders.
+The design of this facade is wrongly attributed to Hugues Sambin (fl.
+c. 1540), a native of Dijon, and pupil of Leonardo da Vinci, but the
+sculpture of the portals, including "The Last Judgment" on the tympanum
+of the main portal, is probably from his hand. St Jean (15th century)
+and St Etienne (15th, 16th and 17th centuries), now used as the
+exchange, are the other chief churches. Of the ancient palace of the
+dukes of Burgundy there remain two towers, the Tour de la Terrasse and
+the Tour de Bar, the guard-room and the kitchens; these now form part of
+the hotel de ville, the rest of which belongs to the 17th and 18th
+centuries. This building contains an archaeological museum with a
+collection of Roman stone monuments; the archives of the town; and the
+principal museum, which, besides valuable paintings and other works of
+art, contains the magnificent tombs of Philip the Bold and John the
+Fearless, dukes of Burgundy. These were transferred from the Chartreuse
+of Dijon (or of Champmol), built by Philip the Bold as a mausoleum, now
+replaced by a lunatic asylum. Relics of it survive in the old Gothic
+entrance, the portal of the church, a tower and the well of Moses, which
+is adorned with statues of Moses and the prophets by Claux Sluter (fl.
+end of 14th century), the Dutch sculptor, who also designed the tomb of
+Philip the Bold. The Palais de Justice, which belongs to the reign of
+Louis XII., is of interest as the former seat of the _parlement_ of
+Burgundy. Dijon possesses several houses of the 15th, 16th and 17th
+centuries, notably the Maison Richard in the Gothic, and the Hotel Vogue
+in the Renaissance style. St Bernard, the composer J. P. Rameau and the
+sculptor Francois Rude have statues in the town, of which they were
+natives. There are also monuments to those inhabitants of Dijon who fell
+in the engagement before the town in 1870, and to President Carnot and
+Garibaldi.
+
+The town is important as the seat of a prefecture, a bishopric, a court
+of appeal and a court of assizes, and as centre of an academie
+(educational district). There are tribunals of first instance and of
+commerce, a board of trade-arbitrators, a chamber of commerce, an
+exchange (occupying the former cathedral of St Etienne), and an
+important branch of the Bank of France. Its educational establishments
+include faculties of law, of science and of letters, a preparatory
+school of medicine and pharmacy, a higher school of commerce, a school
+of fine art, a conservatoire of music, _lycees_ and training colleges,
+and there is a public library with about 100,000 volumes.
+
+Dijon is well known for its mustard, and for the black currant liqueur
+called _cassis de Dijon_; its industries include the manufacture of
+machinery, automobiles, bicycles, soap, biscuits, brandy, leather, boots
+and shoes, candles and hosiery. There are also flour mills, breweries,
+important printing works, vinegar works and, in the vicinity, nursery
+gardens. The state has a large tobacco manufactory in the town. Dijon
+has considerable trade in cereals and wool, and is the second market for
+the wines of Burgundy.
+
+Under the Romans Dijon (_Divonense castrum_) was a _vicus_ in the
+_civitas_ of Langres. In the 2nd century it was the scene of the
+martyrdom of St Benignus (Benigne, vulg. Berin, Berain), the apostle of
+Burgundy. About 274 the emperor Aurelian surrounded it with ramparts.
+Gregory of Tours, in the 6th century, comments on the strength and
+pleasant situation of the place, expressing surprise that it does not
+rank as a _civitas_. During the middle ages the fortunes of Dijon
+followed those of Burgundy, the dukes of which acquired it early in the
+11th century. The communal privileges, conferred on the town in 1182 by
+Hugh III., duke of Burgundy, were confirmed by Philip Augustus in 1183,
+and in the 13th century the dukes took up their residence there. For the
+decoration of the palace and other monuments built by them, eminent
+artists were gathered from northern France and Flanders, and during this
+period the town became one of the great intellectual centres of France.
+The union of the duchy with the crown in 1477 deprived Dijon of the
+splendour of the ducal court; but to counterbalance this loss it was
+made the capital of the province and seat of a _parlement_. Its fidelity
+to the monarchy was tested in 1513, when the citizens were besieged by
+50,000 Swiss and Germans, and forced to agree to a treaty so
+disadvantageous that Louis XII. refused to ratify it. In the wars of
+religion Dijon sided with the League, and only opened its gates to Henry
+IV. in 1595. The 18th century was a brilliant period for the city; it
+became the seat of a bishopric, its streets were improved, its commerce
+developed, and an academy of science and letters founded; while its
+literary salons were hardly less celebrated than those of Paris. The
+neighbourhood was the scene of considerable fighting during the
+Franco-German War, which was, however, indirectly of some advantage to
+the city owing to the impetus given to its industries by the immigrants
+from Alsace.
+
+  See H. Chabeuf, _Dijon a travers les ages_ (Dijon, 1897), and _Dijon,
+  monuments et souvenirs_ (Dijon, 1894).
+
+
+
+
+DIKE, or DYKE (Old Eng. _dic_, a word which appears in various forms in
+many Teutonic languages, cf. Dutch _dijk_, German _Teich_, Danish
+_dige_, and in French, derived from Teutonic, _digue_; it is the same
+word as "ditch" and is ultimately connected with the root of "dig"),
+properly a trench dug out of the earth for defensive and other purposes.
+Water naturally collects in such trenches, and hence the word is applied
+to natural and artificial channels filled with water, as appears in the
+proverbial expression "February fill-dyke," and in the names of many
+narrow waterways in East Anglia. "Dike" also is naturally used of the
+bank of earth thrown up out of the ditch, and so of any embankment, dam
+or causeway, particularly the defensive works in Holland, the Fen
+district of England, and other low-lying districts which are liable to
+flooding by the sea or rivers (see HOLLAND and FENS). In Scotland any
+wall, fence or even hedge, used as a boundary is called a dyke. In
+geology the term is applied to wall-like masses of rock (sometimes
+projecting beyond the surrounding surface) which fill up vertical or
+highly inclined fissures in the strata.
+
+
+
+
+DIKKA, a term in Mahommedan architecture for the tribune raised upon
+columns, from which the Koran is recited and the prayers intoned by the
+Imam of the mosque.
+
+
+
+
+DILAPIDATION (Lat. for "scattering the stones," _lapides_, of a
+building), a term meaning in general a falling into decay, but more
+particularly used in the plural in English law for (1) the waste
+committed by the incumbent of an ecclesiastical living; (2) the
+disrepair for which a tenant is usually liable when he has agreed to
+give up his premises in good repair (see EASEMENT; FLAT; LANDLORD AND
+TENANT). By the general law a tenant for life has no power to cut down
+timber, destroy buildings, &c., (voluntary waste), or to let buildings
+fall into disrepair (permissive waste). In the eye of the law an
+incumbent of a living is a tenant for life of his benefice, and any
+waste, voluntary or permissive, on his part must be made good by his
+administrators to his successor in office. The principles on which such
+dilapidations are to be ascertained, and the application of the money
+payable in respect thereof, depend partly on old ecclesiastical law and
+partly on acts of parliament. Questions as to ecclesiastical
+dilapidations usually arise in respect of the residence house and other
+buildings belonging to the living. Inclosures, hedges, ditches and the
+like are included in things "of which the beneficed person hath the
+burden and charge of reparation." In a leading case (_Ross_ v. _Adcock_,
+1868, L.R. 3 C.P. 657) it was said that the court was acquainted with no
+precedent or decision extending the liability of the executors of a
+deceased incumbent to any species of waste beyond dilapidation of the
+house, chancel or other buildings or fences of the benefice. And it has
+been held that the mere mismanagement or miscultivation of the
+ecclesiastical lands will not give rise to an action for dilapidations.
+To place the law relating to dilapidations on a more satisfactory
+footing, the Ecclesiastical Dilapidations Act 1871 was passed. The
+buildings to which the act applies are defined to be such houses of
+residence, chancels, walls, fences and other buildings and things as the
+incumbent of the benefice is by law and custom bound to maintain in
+repair. In each diocese a surveyor is appointed by the archdeacons and
+rural deans subject to the approval of the bishop; and such surveyor
+shall by the direction of the bishop examine the buildings on the
+following occasions--viz. (1) when the benefice is sequestrated; (2)
+when it is vacant; (3) at the request of the incumbent or on complaint
+by the archdeacon, rural dean or patron. The surveyor specifies the
+works required, and gives an estimate of their probable cost. In the
+case of a vacant benefice, the new incumbent and the old incumbent or
+his representatives may lodge objections to the surveyor's report on any
+grounds of fact or law, and the bishop, after consideration, may make an
+order for the repairs and their cost, for which the late incumbent or
+his representatives are liable. The sum so stated becomes a debt due
+from the late incumbent or his representatives to the new incumbent, who
+shall pay over the money when recovered to the governors of Queen Anne's
+Bounty. The governors pay for the works on execution on receipt of a
+certificate from the surveyor; and the surveyor, when the works have
+been completed to his satisfaction, gives a certificate to that effect,
+the effect of which, so far as regards the incumbent, is to protect him
+from liability for dilapidations for the next five years. Unnecessary
+buildings belonging to a residence house may, by the authority of the
+bishop and with the consent of the patron, be removed. An amending
+statute of 1872 (Ecclesiastical Dilapidations Act (1871) Amendment)
+relates chiefly to advances by the governors of Queen Anne's Bounty for
+the purposes of the act.
+
+
+
+
+DILATATION (from Lat. _dis-_, distributive, and _latus_, wide), a
+widening or enlarging; a term used in physiology, &c.
+
+
+
+
+DILATORY (from Lat. _dilatus_, from _differre_, to put off or delay),
+delaying, or slow; in law a "dilatory plea" is one made merely for
+delaying the suit.
+
+
+
+
+DILEMMA (Gr. [Greek: dilemma], a double proposition, from [Greek: di-]
+and [Greek: lambanein]), a term used technically in logic, and popularly
+in common parlance and rhetoric. (1) The latter use has no exact
+definition, but in general it describes a situation wherein from either
+of two (or more) possible alternatives an unsatisfactory conclusion
+results. The alternatives are called the "horns" of the dilemma. Thus a
+nation which has to choose between bankruptcy and the repudiation of its
+debts is on the horns of a dilemma. (2) In logic there is considerable
+divergence of opinion as to the best definition. Whately defined it as
+"a conditional syllogism with two or more antecedents in the major and a
+disjunctive minor." Aulus Gellius gives an example as follows:--"Women
+are either fair or ugly; if you marry a fair woman, she will attract
+other men; if an ugly woman she will not please you; therefore marriage
+is absurd." From either alternative, an unpleasant result follows. Four
+kinds of dilemma are admitted:--(a) _Simple Constructive_: If A, then C;
+if B, then C, but either B or A; therefore C. (b) _Simple Destructive_:
+If A is true, B is true; if A is true, C is true; B and C are not both
+true; therefore A is not true. (c) _Complex Constructive_: If A, then B;
+if C, then D; but either A or C; therefore either B or D. (d) _Complex
+Destructive_: If A is true, B is true; if C is true, D is true; but B
+and D are not both true; hence A and C are not both true. The soundness
+of the dilemmatic argument in general depends on the alternative
+possibilities. Unless the alternatives produced exhaust the
+possibilities of the case, the conclusion is invalid. The logical form
+of the argument makes it especially valuable in public speaking, before
+uncritical audiences. It is, in fact, important rather as a rhetorical
+subtlety than as a serious argument.
+
+_Dilemmist_ is also a term used to translate _Vaibhashikas_, the name of
+a Buddhist school of philosophy.
+
+
+
+
+DILETTANTE, an Italian word for one who delights in the fine arts,
+especially in music and painting, so a lover of the fine arts in
+general. The Ital. _dilettare_ is from Lat. _delectare_, to delight.
+Properly the word refers to an "amateur" as opposed to a "professional"
+cultivation of the arts, but like "amateur" it is often used in a
+depreciatory sense for one who is only a dabbler, or who only has a
+superficial knowledge or interest in art. The Dilettanti Society founded
+in 1733-1734 still exists in England. A history of the society, by
+Lionel Cust, was published in 1898.
+
+
+
+
+DILIGENCE, in law, the care which a person is bound to exercise in his
+relations with others. The possible degrees of diligence are of course
+numerous, and the same degree is not required in all cases. Thus a mere
+depositary would not be held bound to the same degree of diligence as a
+person borrowing an article for his own use and benefit. Jurists,
+following the divisions of the civil law, have concurred in fixing three
+approximate standards of diligence--viz. ordinary (_diligentia_), less
+than ordinary (_levissima diligentia_) and more than ordinary
+(_exactissima diligentia_). Ordinary or common diligence is defined by
+Story (_On Bailments_) as "that degree of diligence which men in general
+exert in respect of their own concerns." So Sir William Jones:--"This
+care, which every person of common prudence and capable of governing a
+family takes of his own concerns, is a proper measure of that which
+would uniformly be required in performing every contract, if there were
+not strong reasons for exacting in some of them a greater and permitting
+in others a less degree of attention" (_Essay on Bailments_). The
+highest degree of diligence would be that which only very prudent
+persons bestow on their own concerns; the lowest, that which even
+careless persons bestow on their own concerns. The want of these various
+degrees of diligence is negligence in corresponding degrees. These
+approximations indicate roughly the greater or less severity with which
+the law will judge the performance of different classes of contracts;
+but English judges have been inclined to repudiate the distinction as a
+useless refinement of the jurists. Thus Baron Rolfe could see no
+difference between negligence and gross negligence; it was the same
+thing with the addition of a vituperative epithet. See NEGLIGENCE.
+
+_Diligence_, in Scots law, is a general term for the process by which
+persons, lands or effects are attached on execution, or in security for
+debt.
+
+
+
+
+DILKE, SIR CHARLES WENTWORTH, Bart. (1810-1869), English politician, son
+of Charles Wentworth Dilke, proprietor and editor of _The Athenaeum_,
+was born in London on the 18th of February 1810, and was educated at
+Westminster school and Trinity Hall, Cambridge. He studied law, and in
+1834 took his degree of LL.B., but did not practise. He assisted his
+father in his literary work, and was for some years chairman of the
+council of the Society of Arts, besides taking a prominent part in the
+affairs of the Royal Horticultural Society and other bodies. He was one
+of the most zealous promoters of the Great Exhibition (1851), and a
+member of the executive committee. At the close of the exhibition he was
+honoured by foreign sovereigns, and the queen offered him knighthood,
+which, however, he did not accept; he also declined a large remuneration
+offered by the royal commission. In 1853 Dilke was one of the English
+commissioners at the New York Industrial Exhibition, and prepared a
+report on it. He again declined to receive any money reward for his
+services. He was appointed one of the five royal commissioners for the
+Great Exhibition of 1862; and soon after the death of the prince consort
+he was created a baronet. In 1865 he entered parliament as member for
+Wallingford. In 1869 he was sent to Russia as representative of England
+at the horticultural exhibition held at St Petersburg. His health,
+however, had been for some time failing, and he died suddenly in that
+city, on the 10th of May 1869. A selection from his writings, _Papers of
+a Critic_ (2 vols., 1875), contains a biographical sketch by his son.
+
+His son, SIR CHARLES WENTWORTH DILKE, BART. (1843-   ), became a
+prominent Liberal politician, as M.P. for Chelsea (1868-1886),
+under-secretary for foreign affairs (1880-1882), and president of the
+local government board (1882-1885); and he was then marked out as one of
+the best-informed and ablest of the advanced Radicals. He was chairman
+of the royal commission on the housing of the working classes in
+1884-1885. But his sensational appearance as co-respondent in a divorce
+case of a peculiarly unpleasant character in 1885 cast a cloud over his
+career. He was defeated in Chelsea in 1886, and did not return to
+parliament till 1892, when he was elected for the Forest of Dean; and
+though his knowledge of foreign affairs and his powers as a critic and
+writer on military and naval questions were admittedly of the highest
+order, his official position in public life could not again be
+recovered. His military writings are _The British Army_ (1888); _Army
+Reform_ (1898) and, with Mr Spenser Wilkinson, _Imperial Defence_
+(1892). On colonial questions he wrote with equal authority. His
+_Greater Britain_ (2 vols., 1866-1867) reached a fourth edition in 1868,
+and was followed by _Problems of Greater Britain_ (2 vols., 1890) and
+_The British Empire_ (1899). He was twice married, his second wife (nee
+Emilia Frances Strong), the widow of Mark Pattison, being an
+accomplished art critic and collector. She died in 1904. The most
+important of her books were the studies on _French Painters of the
+Eighteenth Century_ (1899) and three subsequent volumes on the
+architects and sculptors, furniture and decoration, engravers and
+draughtsmen of the same period, the last of which appeared in 1902. A
+posthumous volume, _The Book of the Spiritual Life_ (1905), contains a
+memoir of her by Sir Charles Dilke.
+
+[Illustration: Dill (_Anethum_ or _Peucedanum graveolens_), leaf and
+inflorescence.]
+
+
+
+
+DILL (_Anethum_ or _Peucedanum graveolens_), a member of the natural
+botanical order Umbelliferae, indigenous to the south of Europe, Egypt
+and the Cape of Good Hope. It resembles fennel in appearance. Its root
+is long and fusiform; the stem is round, jointed and about a yard high;
+the leaves have fragrant leaflets; and the fruits are brown, oval and
+concavo-convex. The plant flowers from June till August in England. The
+seeds are sown, preferably as soon as ripe, either broadcast or in
+drills between 6 and 12 in. asunder. The young plants should be thinned
+when 3 or 4 weeks old, so as to be at distances of about 10 in. A
+sheltered spot and dry soil are needed for the production of the seed in
+the climate of England. The leaves of the dill are used in soups and
+sauces, and, as well as the umbels, for flavouring pickles. The seeds
+are employed for the preparation of dill-water and oil of dill; they are
+largely consumed in the manufacture of gin, and, when ground, are eaten
+in the East as a condiment. The British Pharmacopoeia contains the Aqua
+Anethi or dill-water (dose 1-2 oz.), and the Oleum Anethi, almost
+identical in composition with caraway oil, and given in doses of 1/2-3
+minims. Dill-water is largely used as a carminative for children, and as
+a vehicle for the exhibition of nauseous drugs.
+
+
+
+
+DILLEN [DILLENIUS], JOHANN JAKOB (1684-1747), English botanist, was born
+at Darmstadt in 1684, and was educated at the university of Giessen,
+where he wrote several botanical papers for the _Ephemerides naturae
+curiosorum_, and printed, in 1719, his _Catalogus plantarum sponte circa
+Gissam nascentium_, illustrated with figures drawn and engraved by his
+own hand, and containing descriptions of many new species. In 1721, at
+the instance of the botanist William Sherard (1659-1728), he came to
+England, and in 1724 he published a new edition of Ray's _Synopsis
+stirpium Britannicarum_. In 1732 he published _Hortus Elthamensis_, a
+catalogue of the rare plants growing at Eltham, Kent, in the collection
+of Sherard's younger brother, James (1666-1738), who, after making a
+fortune as an apothecary, devoted himself to gardening and music. For
+this work Dillen himself executed 324 plates, and it was described by
+Linnaeus, who spent a month with him at Oxford in 1736, and afterwards
+dedicated his _Critica botanica_ to him, as "opus botanicum quo
+absolutius mundus non vidit." In 1734 he was appointed Sherardian
+professor of botany at Oxford, in accordance with the will of W.
+Sherard, who at his death in 1728 left the university L3000 for the
+endowment of the chair, as well as his library and herbarium. Dillen,
+who was also the author of an _Historia muscorum_ (1741), died at
+Oxford, of apoplexy, on the 2nd of April 1747. His manuscripts, books
+and collections of dried plants, with many drawings, were bought by his
+successor at Oxford, Dr Humphry Sibthorp (1713-1797), and ultimately
+passed into the possession of the university.
+
+  For an account of his collections preserved at Oxford, see _The
+  Dillenian Herbaria_, by G. Claridge Druce (Oxford, 1907).
+
+
+
+
+DILLENBURG, a town of Germany, in the Prussian province of Hesse-Nassau,
+delightfully situated in the midst of a well-wooded country, on the
+Dill, 25 m. N.W. from. Giessen on the railway to Troisdorf. Pop. 4500.
+On an eminence above it lie the ruins of the castle of Dillenburg,
+founded by Count Henry the Rich of Nassau, about the year 1255, and the
+birthplace of Prince William of Orange (1533). It has an Evangelical
+church, with the vault of the princes of Nassau-Dillenburg, a Roman
+Catholic church, a classical school, a teachers' seminary and a chamber
+of commerce. Its industries embrace iron-works, tanneries and the
+manufacture of cigars. Owing to its beautiful surroundings Dillenburg
+has become a favourite summer resort.
+
+
+
+
+DILLENS, JULIEN (1849-1904), Belgian sculptor, was born at Antwerp on
+the 8th of June 1849, son of a painter. He studied under Eugene Simonis
+at the Brussels Academy of Fine Arts. In 1877 he received the _prix de
+Rome_ for "A Gaulish Chief taken Prisoner by the Romans." At Brussels,
+in 1881, he executed the groups entitled "Justice" and "Herkenbald, the
+Brussels Brutus." For the pediment of the orphanage at Uccle, "Figure
+Kneeling" (Brussels Gallery), and the statue of the lawyer
+Metdepenningen in front of the Palais de Justice at Ghent, he was
+awarded the medal of honour in 1889 at the Paris Universal Exhibition,
+where, in 1900, his "Two Statues of the Anspach Monument" gained him a
+similar distinction. For the town of Brussels he executed "The Four
+Continents" (Maison du Renard, Grand' Place), "The Lansquenets" crowning
+the lucarnes of the Maison de Roi, and the "Monument t' Serclaes" under
+the arcades of the Maison de l'Etoile, and, for the Belgian government,
+"Flemish Art," "German Art," "Classic Art" and "Art applied to Industry"
+(all in the Palais des Beaux Arts, Brussels), "The Laurel" (Botanic
+Garden, Brussels), and the statue of "Bernard van Orley" (Place du petit
+Sablon, Brussels). Mention must also be made of "An Enigma" (1876), the
+bronze busts of "Rogier de la Pasture" and "P. P. Rubens" (1879),
+"Etruria" (1880), "The Painter Leon Frederic" (1888), "Madame Leon
+Herbo," "Hermes," a scheme of decoration for the ogival facade of the
+hotel de ville at Ghent (1893), "The Genius of the Funeral Monument of
+the Moselli Family," "The Silence of Death" (for the entrance of the
+cemetery of St Gilles), two caryatides for the town hall of St Gilles,
+presentation plaquette to Dr Heger, medals of MM. Godefroid and
+Vanderkindere and of "The Three Burgomasters of Brussels," and the
+ivories "Allegretto," "Minerva" and the "Jamaer Memorial." Dillens died
+at Brussels in November 1904.
+
+
+
+
+DILLINGEN, a town of Germany, in the kingdom of Bavaria, on the left
+bank of the Danube, 25 m. N.E. from Ulm, on the railway to Ingolstadt.
+Pop. (1905) 6078. Its principal buildings are an old palace, formerly
+the residence of the bishops of Augsburg and now government offices, a
+royal gymnasium, a Latin school with a library of 75,000 volumes, seven
+churches (six Roman Catholic), two episcopal seminaries, a Capuchin
+monastery, a Franciscan convent and a deaf and dumb asylum. The
+university, founded in 1549, was abolished in 1804, being converted into
+a lyceum. The inhabitants are engaged in cattle-rearing, the cultivation
+of corn, hops and fruit, shipbuilding and the shipping trade, and the
+manufacture of cloth, paper and cutlery. In the vicinity is the
+Karolinen canal, which cuts off a bend in the Danube between Lauingen
+and Dillingen. In 1488 Dillingen became the residence of the bishops of
+Augsburg; was taken by the Swedes in 1632 and 1648, by the Austrians in
+1702, and on the 17th of June 1800 by the French. In 1803 it passed to
+Bavaria.
+
+
+
+
+DILLMANN, CHRISTIAN FRIEDRICH AUGUST (1823-1894), German orientalist and
+biblical scholar, the son of a Wurttemberg schoolmaster, was born at
+Illingen on the 25th of April 1823. He was educated at Tubingen, where
+he became a pupil and friend of Heinrich Ewald, and studied under F. C.
+Baur, though he did not join the new Tubingen school. For a short time
+he worked as pastor at Gersheim, near his native place, but he soon came
+to feel that his studies demanded his whole time. He devoted himself to
+the study of Ethiopic MSS. in the libraries of Paris, London and Oxford,
+and this work caused a revival of Ethiopic study in the 19th century. In
+1847 and 1848 he prepared catalogues of the Ethiopic MSS. in the British
+Museum and the Bodleian library at Oxford. He then set to work upon an
+edition of the Ethiopic bible. Returning to Tubingen in 1848, in 1853 he
+was appointed professor extraordinarius. Subsequently he became
+professor of philosophy at Kiel (1854), and of theology at Giessen
+(1864) and Berlin (1869). He died on the 4th of July 1894.
+
+In 1851 he had published the _Book of Enoch_ in Ethiopian (German,
+1853), and at Kiel he completed the first part of the Ethiopic bible,
+_Octateuchus Aethiopicus_ (1853-1855). In 1857 appeared his _Grammatik
+der athiopischen Sprache_ (2nd ed. by C. Bezold, 1899); in 1859 the
+_Book of Jubilees_; in 1861 and 1871 another part of the Ethiopic bible,
+_Libri Regum_; in 1865 his great _Lexicon linguae aethiopicae_; in 1866
+his _Chrestomathia aethiopica_. Always a theologian at heart, however,
+he returned to theology in 1864. His Giessen lectures were published
+under the titles, _Ursprung der alttestamentlichen Religion_ (1865) and
+_Die Propheten des alten Bundes nach ihrer politischen Wirksamkeit_
+(1868). In 1869 appeared his _Commentar zum Hiob_ (4th ed. 1891) which
+stamped him as one of the foremost Old Testament exegetes. His renown as
+a theologian, however, was mainly founded by the series of commentaries,
+based on those of August Wilhelm Knobels' _Genesis_ (Leipzig, 1875; 6th
+ed. 1892; Eng. trans, by W. B. Stevenson, Edinburgh, 1897); _Exodus und
+Leviticus_, 1880, revised edition by V. Ryssel, 1897; _Numeri,
+Deuteronomium und Josua_, with a dissertation on the origin of the
+Hexateuch, 1886; _Jesaja_, 1890 (revised edition by Rudolf Kittel in
+1898). In 1877 he published the _Ascension of Isaiah_ in Ethiopian and
+Latin. He was also a contributor to D. Schenkel's _Bibellexikon_,
+Brockhaus's _Conversationslexikon_, and Herzog's _Realencyklopadie_. His
+lectures on Old Testament theology, _Vorlesungen uber Theologie des
+Allen Testamentes_, were published by Kittel in 1895.
+
+  See the articles in Herzog-Hauck, _Realencyklopadie_, and the
+  _Allgemeine deutsche Biographie_; F. Lichtenberger, _History of German
+  Theology in the Nineteenth Century_ (1889); Wolf Baudissin, _A.
+  Dillmann_ (Leipzig, 1895).
+
+
+
+
+DILLON, ARTHUR RICHARD (1721-1807), French archbishop, was the son of
+Arthur Dillon (1670-1733), an Irish gentleman who became general in the
+French service. He was born at St Germain, entered the priesthood and
+was successively cure of Elan near Mezieres, vicar-general of Pontoise
+(1747), bishop of Evreux (1753) and archbishop of Toulouse (1758),
+archbishop of Narbonne in 1763, and in that capacity, president of the
+estates of Languedoc. He devoted himself much less to the spiritual
+direction of his diocese than to its temporal welfare, carrying out many
+works of public utility, bridges, canals, roads, harbours, &c.; had
+chairs of chemistry and of physics created at Montpellier and at
+Toulouse, and tried to reduce the poverty, especially in Narbonne. In
+1787 and in 1788 he was a member of the Assembly of Notables called
+together by Louis XVI., and in 1788 presided over the assembly of the
+clergy. Having refused to accept the civil constitution of the clergy,
+Dillon had to leave Narbonne in 1790, then to emigrate to Coblenz in
+1791. Soon afterwards he went to London, where he lived until his death
+in 1807, never accepting the Concordat, which had suppressed his
+archiepiscopal see.
+
+  See L. Audibret, _Le Dernier President des Etats du Languedoc, Mgr.
+  Arthur Richard Dillon, archeveque de Narbonne_ (Bordeaux, 1868); L. de
+  Lavergne, _Les Assemblees provinciales sous Louis XVI_ (Paris, 1864).
+
+
+
+
+DILLON, JOHN (1851-   ), Irish nationalist politician, was the son of
+John Blake Dillon (1816-1866), who sat in parliament for Tipperary, and
+was one of the leaders of "Young Ireland." John Dillon was educated at
+the Roman Catholic university of Dublin, and afterwards studied
+medicine. He entered parliament in 1880 as member for Tipperary, and was
+at first an ardent supporter of C. S. Parnell. In August he delivered a
+speech on the Land League at Kildare which was characterized as "wicked
+and cowardly" by W. E. Forster; he advocated boycotting, and was
+arrested in May 1881 under the Coercion Act, and again after two months
+of freedom in October. In 1883 he resigned his seat for reasons of
+health, but was returned unopposed in 1885 for East Mayo, which he
+continued to represent. He was one of the prime movers in the famous
+"plan of campaign," which provided that the tenant should pay his rent
+to the National League instead of the landlord, and in case of eviction
+be supported by the general fund. Mr Dillon was compelled by the court
+of queen's bench on the 14th of December 1886 to find securities for
+good behaviour, but two days later he was arrested while receiving rents
+on Lord Clanricarde's estates. In this instance the jury disagreed, but
+in June 1888 under the provisions of the new Criminal Law Procedure Bill
+he was condemned to six months' imprisonment. He was, however, released
+in September, and in the spring of 1889 sailed for Australia and New
+Zealand, where he collected funds for the Nationalist party. On his
+return to Ireland he was again arrested, but, being allowed bail, sailed
+to America, and failed to appear at the trial. He returned to Ireland by
+way of Boulogne, where he and Mr W. O'Brien held long and indecisive
+conferences with Parnell. They surrendered to the police in February,
+and on their release from Galway gaol in July declared their opposition
+to Parnell. After the expulsion of Mr T. M. Healy and others from the
+Irish National Federation, Mr Dillon became the chairman (February
+1896). His early friendship with Mr O'Brien gave place to considerable
+hostility, but the various sections of the party were ostensibly
+reconciled in 1900 under the leadership of Mr Redmond. In the autumn of
+1896 he arranged a convention of the Irish race, which included 2000
+delegates from various parts of the world. In 1897 Mr Dillon opposed in
+the House the Address to Queen Victoria on the occasion of the Diamond
+Jubilee, on the ground that her reign had not been a blessing to
+Ireland, and he showed the same uncompromising attitude in 1901 when a
+grant to Lord Roberts was under discussion, accusing him of
+"systematized inhumanity." He was suspended on the 20th of March for
+violent language addressed to Mr Chamberlain. He married in 1895
+Elizabeth (d. 1907), daughter of Lord justice J. C. Mathew.
+
+
+
+
+DILUVIUM (Lat. for "deluge," from _diluere_, to wash away), a term in
+geology for superficial deposits formed by flood-like operations of
+water, and so contrasted with alluvium (q.v.) or alluvial deposits
+formed by slow and steady aqueous agencies. The term was formerly given
+to the "boulder clay" deposits, supposed to have been caused by the
+Noachian deluge.
+
+
+
+
+DIME (from the Lat. _decima_, a tenth, through the O. Fr. _disme_), the
+tenth part, the tithe paid as church dues, or as tribute to a temporal
+power. In this sense it is obsolete, but is found in Wycliffe's
+translation of the Bible--"He gave him dymes of alle thingis" (Gen. xiv.
+20). A dime is a silver coin of the United States, in value 10 cents
+(English equivalent about 5d.) or one-tenth of a dollar; hence
+"dime-novel," a cheap sensational novel, a "penny dreadful"; also
+"dime-museum."
+
+
+
+
+DIMENSION (from Lat. _dimensio_, a measuring), in geometry, a magnitude
+measured in a specified direction, i.e. length, breadth and thickness;
+thus a line has only length and is said to be of one dimension, a
+surface has length and breadth, and has two dimensions, a solid has
+length, breadth and thickness, and has three dimensions. This concept is
+extended to algebra: since a line, surface and solid are represented by
+linear, quadratic and cubic equations, and are of one, two and three
+dimensions; a biquadratic equation has its highest terms of four
+dimensions, and, in general, an equation in any number of variables
+which has the greatest sum of the indices of any term equal to n is said
+to have n dimensions. The "fourth dimension" is a type of non-Euclidean
+geometry, in which it is conceived that a "solid" has one dimension more
+than the solids of experience. For the dimensions of units see UNITS,
+DIMENSIONS OF.
+
+
+
+
+DIMITY, derived from the Gr. [Greek: dimitos] "double thread," through
+the Ital. _dimito_, "a kind of course linzie-wolzie" (Florio, 1611); a
+cloth commonly employed for bed upholstery and curtains, and usually
+white, though sometimes a pattern is printed on it in colours. It is
+stout in texture, and woven in raised patterns.
+
+
+
+
+DINAJPUR, a town (with a population in 1901 of 13,430) and district of
+British India, in the Rajshahi division of Eastern Bengal and Assam. The
+earthquake of the 12th of June 1897 caused serious damage to most of the
+public buildings of the town. There is a railway station and a
+government high school. The district comprises an area of 3946 sq. m. It
+is traversed in every direction by a network of channels and water
+courses. Along the banks of the Kulik river, the undulating ridges and
+long lines of mango-trees give the landscape a beauty which is not
+found elsewhere. Dinajpur forms part of the rich arable tract lying
+between the Ganges and the southern slopes of the Himalayas. Although
+essentially a fluvial district, it does not possess any river navigable
+throughout the year by boats of 4 tons burden. Rice forms the staple
+agricultural product. The climate of the district, although cooler than
+that of Calcutta, is very unhealthy, and the people have a sickly
+appearance. The worst part of the year is at the close of the rains in
+September and October, during which months few of the natives escape
+fever. The average maximum temperature is 92.3 deg. F., and the minimum
+74.8 deg.. The average rainfall is 85.54 in. In 1901 the population was
+1,567,080, showing an increase of 6% in the decade. The district is
+partly traversed by the main line of the Eastern Bengal railway and by
+two branch lines. Save between 1404 and 1442, when it was the seat of an
+independent _raj_, founded by Raja Ganesh, a Hindu turned Mussulman,
+Dinajpur has no separate history. Pillars and copper-plate inscriptions
+have yielded numerous records of the Pal kings who ruled the country
+from the 9th century onwards, and the district is famous for many other
+antiquities, some of which are connected by legend with an immemorial
+past (see _Reports, Arch. Survey of India_, xv.; _Epigraphia Indica_,
+ii.).
+
+
+
+
+DINAN, a town of north-western France, capital of an arrondissement in
+the department of Cotes-du-Nord, 37 m. E. of St Brieuc on the Western
+railway. Pop. (1906) 8588. Dinan is situated on a height on the left
+bank of the Ranee (here canalized), some 17 m. above its mouth at St
+Malo, with which it communicates by means of small steamers. It is
+united to the village of Lanvallay on the right bank of the river by a
+granite viaduct 130 ft. in height. The town is almost entirely encircled
+by the ramparts of the middle ages, strengthened at intervals by towers
+and defended on the south by a castle of the late 14th century, which
+now serves as prison. Three old gateways are also preserved. Dinan has
+two interesting churches; that of St Malo, of late Gothic architecture,
+and St Sauveur, in which the Romanesque and Gothic styles are
+intermingled. In the latter church a granite monument contains the heart
+of Bertrand Du Guesclin, whose connexion with the town is also
+commemorated by an equestrian statue. The quaint winding streets of
+Dinan are often bordered by medieval houses. Its picturesqueness
+attracts large numbers of visitors and there are many English residents
+in the town and its vicinity. About three-quarters of a mile from the
+town are the ruins of the chateau and the Benedictine abbey at Lehon;
+near the neighbouring village of St Esprit stands the large lunatic
+asylum of Les Bas Foins, founded in 1836; and at no great distance is
+the now dismantled chateau of La Garaye, which was rendered famous in
+the 18th century by the philanthropic devotion of the count and countess
+whose story is told in Mrs Norton's _Lady of La Garaye_. Dinan is the
+seat of a subprefect and has a tribunal of first instance, and a
+communal college. There is trade in grain, cider, wax, butter and other
+agricultural products. The industries include the manufacture of
+leather, farm-implements and canvas.
+
+The principal event in the history of Dinan, which was a stronghold of
+the dukes of Brittany, is the siege by the English under the duke of
+Lancaster in 1359, during which Du Guesclin and an English knight called
+Thomas of Canterbury engaged in single combat.
+
+
+
+
+DINANT, an ancient town on the right bank of the Meuse in the province
+of Namur, Belgium, connected by a bridge with the left bank, on which
+are the station and the suburb of St Medard. Pop. (1904) 7674. The name
+is supposed to be derived from Diana, and as early as the 7th century it
+was named as one of the dependencies of the bishopric of Tongres. In the
+10th century it passed under the titular sway of Liege, and remained the
+fief of the prince-bishopric till the French revolution put an end to
+that survival of feudalism. In the middle of the 15th century Dinant
+reached the height of its prosperity. With a population of 60,000, and
+8000 workers in copper, it was one of the most flourishing cities in
+Walloon Belgium, until it incurred the wrath of Charles the Bold. Belief
+in the strength of its walls and of the castle that occupied the centre
+bridge, thus effectually commanding navigation by the river, engendered
+arrogance and overconfidence, and the people of Dinant thought they
+could defy the full power of Burgundy. Perhaps they also expected aid
+from France or Liege. In 1466 Charles, in his father's name, laid siege
+to Dinant, and on the 27th of August carried the place by storm. He
+razed the walls and allowed the women, children and priests to retire in
+safety to Liege, but the male prisoners he either hanged or drowned in
+the river by causing them to be cast from the projecting cliff of
+Bouvignes. In 1675 the capture of Dinant formed one of the early
+military achievements of Louis XIV., and it remained in the hands of the
+French for nearly thirty years after that date. The citadel on the
+cliff, 300 ft. or 408 steps above the town, was fortified by the Dutch
+in 1818. It is now dismantled, but forms the chief curiosity of the
+place. The views of the river valley from this eminence are exceedingly
+fine. Half way up the cliff, but some distance south of the citadel, is
+the grotto of Montfat, alleged to be the site of Diana's shrine. The
+church of Notre Dame, dating from the 13th century, stands immediately
+under the citadel and flanking the bridge. It has been restored, and is
+considered by some authorities, although others make the same claim on
+behalf of Huy, the most complete specimen in Belgium of pointed Gothic
+architecture. The baptismal fonts date from the 12th century, and the
+curious spire in the form of an elongated pumpkin and covered with
+slates gives a fantastic and original appearance to the whole edifice.
+The present prosperity of Dinant is chiefly derived from its being a
+favourite summer resort for Belgians as well as foreigners. It has
+facilities for beating and bathing as well as for trips by steamer up
+and down the river Meuse. It is also a convenient central point for
+excursions into the Ardennes. Although there are some indications of
+increased industrial activity in recent years, the population of Dinant
+is not one-eighth of what it was at the time of the Burgundians.
+
+
+
+
+DINAPUR, a town and military station of British India, in the Patna
+district of Bengal, on the right bank of the Ganges, 12 m. W. of Patna
+city by rail. Pop. (1901) 33,699. It is the largest military cantonment
+in Bengal, with accommodation for two batteries of artillery, a European
+and a native infantry regiment. In 1857 the sepoy garrison of the place
+initiated the mutiny of that year in Patna district, but after a
+conflict with the European troops were forced to retire from the town,
+and subsequently laid siege to Arrah.
+
+
+
+
+DINARCHUS, last of the "ten" Attic orators, son of Sostratus (or,
+according to Suidas, Socrates), born at Corinth about 361 B.C. He
+settled at Athens early in life, and when not more than twenty-five was
+already active as a writer of speeches for the law courts. As an alien,
+he was unable to take part in the debates. He had been the pupil both of
+Theophrastus and of Demetrius Phalereus, and had early acquired a
+certain fluency and versatility of style. In 324 the Areopagus, after
+inquiry, reported that nine men had taken bribes from Harpalus, the
+fugitive treasurer of Alexander. Ten public prosecutors were appointed.
+Dinarchus wrote, for one or more of these prosecutors, the three
+speeches which are still extant--_Against Demosthenes_, _Against
+Aristogeiton_, _Against Philocles_. The sympathies of Dinarchus were in
+favour of an Athenian oligarchy under Macedonian control; but it should
+be remembered that he was not an Athenian citizen. Aeschines and Demades
+had no such excuse. In the Harpalus affair, Demosthenes was doubtless
+innocent, and so, probably, were others of the accused. Yet Hypereides,
+the most fiery of the patriots, was on the same side as Dinarchus.
+
+Under the regency of his old master, Demetrius Phalereus, Dinarchus
+exercised much political influence. The years 317-307 were the most
+prosperous of his life. On the fall of Demetrius Phalereus and the
+restoration of the democracy by Demetrius Poliorcetes, Dinarchus was
+condemned to death and withdrew into exile at Chalcis in Euboea. About
+292, thanks to his friend Theophrastus, he was able to return to Attica,
+and took up his abode in the country with a former associate, Proxenus.
+He afterwards brought an action against Proxenus on the ground that he
+had robbed him of some money and plate. Dinarchus died at Athens about
+291.
+
+According to Suidas, Dinarchus wrote 160 speeches; and Dionysius held
+that, out of 85 extant speeches bearing his name, 58 were genuine,--28
+relating to public, 30 to private causes. Although the authenticity of
+the three speeches mentioned above is generally admitted, Demetrius of
+Magnesia doubted that of the speech Against Demosthenes, while A.
+Westermann rejected all three. Dinarchus had little individual style and
+imitated by turns Lysias, Hypereides and Demosthenes. He is called by
+Hermogenes [Greek: o critinos Demosthenes], a metaphor taken from barley compared
+with wheat, or beer compared with wine,--a Demosthenes whose strength
+is rougher, without flavour or sparkle.
+
+  Editions: (text and exhaustive commentary) E. Metzner (1842); (text)
+  T. Thalheim (1887), F. Blass (1888); see L. L. Forman, _Index
+  Andocideus, Lycurgeus, Dinarcheus_ (1897); and, in general, F. Blass,
+  _Attische Beredsamkeit_, iii. There is a valuable treatise on the life
+  and speeches of Dinarchus by Dionysius of Halicarnassus.
+
+
+
+
+
+End of the Project Gutenberg EBook of Encyclopaedia Britannica, 11th
+Edition, Volume 8, Slice 4, by Various
+
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