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diff --git a/32607-8.txt b/32607-8.txt new file mode 100644 index 0000000..7b02af5 --- /dev/null +++ b/32607-8.txt @@ -0,0 +1,20492 @@ +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. 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