diff options
| author | Roger Frank <rfrank@pglaf.org> | 2025-10-14 19:57:54 -0700 |
|---|---|---|
| committer | Roger Frank <rfrank@pglaf.org> | 2025-10-14 19:57:54 -0700 |
| commit | 209e8a224387cbbd58ec63cf2ca1ad856c6e8c5f (patch) | |
| tree | ab5f0f908537c462dcf46ce925acddfb80c334b6 /32607-h | |
Diffstat (limited to '32607-h')
| -rw-r--r-- | 32607-h/32607-h.htm | 22001 | ||||
| -rw-r--r-- | 32607-h/images/img158a.jpg | bin | 0 -> 14117 bytes | |||
| -rw-r--r-- | 32607-h/images/img158b.jpg | bin | 0 -> 8188 bytes | |||
| -rw-r--r-- | 32607-h/images/img158c.jpg | bin | 0 -> 5046 bytes | |||
| -rw-r--r-- | 32607-h/images/img158d.jpg | bin | 0 -> 11031 bytes | |||
| -rw-r--r-- | 32607-h/images/img160a1.jpg | bin | 0 -> 34627 bytes | |||
| -rw-r--r-- | 32607-h/images/img160a2.jpg | bin | 0 -> 25607 bytes | |||
| -rw-r--r-- | 32607-h/images/img160a3.jpg | bin | 0 -> 70106 bytes | |||
| -rw-r--r-- | 32607-h/images/img160b1.jpg | bin | 0 -> 73728 bytes | |||
| -rw-r--r-- | 32607-h/images/img160b2.jpg | bin | 0 -> 66562 bytes | |||
| -rw-r--r-- | 32607-h/images/img161.jpg | bin | 0 -> 87136 bytes | |||
| -rw-r--r-- | 32607-h/images/img166a.jpg | bin | 0 -> 30807 bytes | |||
| -rw-r--r-- | 32607-h/images/img166b.jpg | bin | 0 -> 57222 bytes | |||
| -rw-r--r-- | 32607-h/images/img169a.jpg | bin | 0 -> 31021 bytes | |||
| -rw-r--r-- | 32607-h/images/img169b.jpg | bin | 0 -> 3131 bytes | |||
| -rw-r--r-- | 32607-h/images/img169c.jpg | bin | 0 -> 3615 bytes | |||
| -rw-r--r-- | 32607-h/images/img169d.jpg | bin | 0 -> 8777 bytes | |||
| -rw-r--r-- | 32607-h/images/img169e.jpg | bin | 0 -> 40775 bytes | |||
| -rw-r--r-- | 32607-h/images/img170.jpg | bin | 0 -> 33065 bytes | |||
| -rw-r--r-- | 32607-h/images/img174a.jpg | bin | 0 -> 6623 bytes | |||
| -rw-r--r-- | 32607-h/images/img174b.jpg | bin | 0 -> 2124 bytes | |||
| -rw-r--r-- | 32607-h/images/img174c.jpg | bin | 0 -> 4355 bytes | |||
| -rw-r--r-- | 32607-h/images/img174d.jpg | bin | 0 -> 5126 bytes | |||
| -rw-r--r-- | 32607-h/images/img174e.jpg | bin | 0 -> 6356 bytes | |||
| -rw-r--r-- | 32607-h/images/img174f.jpg | bin | 0 -> 5747 bytes | |||
| -rw-r--r-- | 32607-h/images/img175.jpg | bin | 0 -> 1974 bytes | |||
| -rw-r--r-- | 32607-h/images/img209.jpg | bin | 0 -> 17829 bytes | |||
| -rw-r--r-- | 32607-h/images/img223.jpg | bin | 0 -> 9164 bytes | |||
| -rw-r--r-- | 32607-h/images/img239.jpg | bin | 0 -> 3202 bytes | |||
| -rw-r--r-- | 32607-h/images/img240.jpg | bin | 0 -> 3563 bytes | |||
| -rw-r--r-- | 32607-h/images/img241.jpg | bin | 0 -> 5373 bytes | |||
| -rw-r--r-- | 32607-h/images/img243.jpg | bin | 0 -> 3096 bytes | |||
| -rw-r--r-- | 32607-h/images/img245.jpg | bin | 0 -> 4134 bytes | |||
| -rw-r--r-- | 32607-h/images/img247a.jpg | bin | 0 -> 4303 bytes | |||
| -rw-r--r-- | 32607-h/images/img247b.jpg | bin | 0 -> 3698 bytes | |||
| -rw-r--r-- | 32607-h/images/img247c.jpg | bin | 0 -> 3317 bytes | |||
| -rw-r--r-- | 32607-h/images/img248a.jpg | bin | 0 -> 4058 bytes | |||
| -rw-r--r-- | 32607-h/images/img248b.jpg | bin | 0 -> 3421 bytes | |||
| -rw-r--r-- | 32607-h/images/img248c.jpg | bin | 0 -> 6788 bytes | |||
| -rw-r--r-- | 32607-h/images/img251.jpg | bin | 0 -> 2764 bytes | |||
| -rw-r--r-- | 32607-h/images/img252.jpg | bin | 0 -> 3782 bytes | |||
| -rw-r--r-- | 32607-h/images/img253.jpg | bin | 0 -> 8521 bytes | |||
| -rw-r--r-- | 32607-h/images/img259.jpg | bin | 0 -> 13911 bytes | |||
| -rw-r--r-- | 32607-h/images/img272.jpg | bin | 0 -> 17303 bytes |
44 files changed, 22001 insertions, 0 deletions
diff --git a/32607-h/32607-h.htm b/32607-h/32607-h.htm new file mode 100644 index 0000000..99b975e --- /dev/null +++ b/32607-h/32607-h.htm @@ -0,0 +1,22001 @@ +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> +<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en"> + + <head> + <meta http-equiv="Content-Type" content= + "text/html; charset=iso-8859-1" /> + + <title> + The Project Gutenberg eBook of Encyclopædia Britannica, Volume VIII Slice IV - Diameter to Dinarchus. + </title> + + <style type="text/css"> + + body { margin-left: 12%; margin-right: 12%; text-align: justify; } + p { margin-top: .75em; margin-bottom: .75em; text-indent: 1em; line-height: 1.4em;} + p.c { margin-top: .25em; margin-bottom: .25em; text-indent: 1em; padding-left: 1em; line-height: 1.4em;} + p.noind { margin-top: .75em; margin-bottom: .75em; text-indent: 0; } + + h2,h3 { text-align: center; } + hr { margin-left: auto; margin-right: auto; text-align: center; width: 70%; height: 5px; background-color: #dcdcdc; border:none; } + hr.art { margin-left: auto; margin-right: auto; width: 40%; height: 5px; background-color: #778899; + margin-top: 2em; margin-bottom: 6em } + hr.foot {margin-left: 2em; width: 16%; background-color: black; margin-top: 1em; margin-bottom: 0; height: 1px; } + hr.full {width: 100%} + + table.ws {white-space: nowrap; border-collapse: collapse; margin-left: auto; margin-right: auto; + margin-top: 2em; margin-bottom: 2em;} + table.reg { margin-left: auto; margin-right: auto; clear: both; } + table.nobctr { margin-left: auto; margin-right: auto; border-collapse: collapse; } + table.pic { margin-left: auto; margin-right: auto; } + table.math0 { vertical-align: middle; margin-left: auto; margin-right: auto; border-collapse: collapse;} + table.math0 td {text-align: center;} + table.math0 td.np {text-align: center; padding-left: 0; padding-right: 0;} + + table.reg td { padding-left: 3em; text-indent: -2em; white-space: normal;} + table.reg td.tc5p { padding-left: 2em; text-indent: 0em; white-space: normal;} + table.nobctr td { white-space: normal; } + table.pic td { white-space: normal; text-indent: 1em; padding-left: 2em; padding-right: 1em;} + table.nobctr p {text-indent: -1em; margin-left: 1em;} + + td { white-space: nowrap; padding-right: 0.3em; padding-left: 0.3em;} + td.norm { white-space: normal; } + td.denom { border-top: 1px solid black; text-align: center; padding-right: 0.3em; padding-left: 0.3em;} + + td.tcc { padding-right: 0.5em; padding-left: 0.5em; text-align: center; vertical-align: top;} + td.tccm { padding-right: 0.5em; padding-left: 0.5em; text-align: center; vertical-align: middle;} + td.tccb { padding-right: 0.5em; padding-left: 0.5em; text-align: center; vertical-align: bottom;} + td.tcr { padding-right: 0.5em; padding-left: 0.5em; text-align: right; vertical-align: top;} + td.tcrm { padding-right: 0.5em; padding-left: 0.5em; text-align: right; vertical-align: middle;} + td.tcl { padding-right: 0.5em; padding-left: 0.5em; text-align: left; vertical-align: top;} + td.tclm { padding-right: 0.5em; padding-left: 0.5em; text-align: left; vertical-align: middle;} + td.vb { vertical-align: bottom; } + + .caption { font-size: 0.9em; text-align: center; padding-bottom: 1em; padding-left: 1em; padding-right: 1em;} + .caption1 { font-size: 0.9em; text-align: left; padding-bottom: 1em; padding-left: 3em; padding-right: 2em;} + + td.lb {border-left: black 1px solid;} + td.ltb {border-left: black 1px solid; border-top: black 1px solid;} + td.rb {border-right: black 1px solid;} + td.rb2 {border-right: black 2px solid;} + td.tb, span.tb {border-top: black 1px solid;} + td.bb {border-bottom: black 1px solid;} + td.rlb {border-right: black 1px solid; border-left : black 1px solid;} + td.allb {border: black 1px solid;} + + table p { margin: 0;} + + a:link, a:visited, link {text-decoration:none} + + .author {text-align: right; margin-top: -1em; margin-right: 1em; font-variant: small-caps;} + .center {text-align: center; text-indent: 0;} + .center1 {text-align: center; text-indent: 0; margin-top: 1em; margin-bottom: 1em;} + .grk {font-style: normal; font-family:"Palatino Linotype","New Athena Unicode",Gentium,"Lucida Grande", Galilee, "Arial Unicode MS", sans-serif;} + + .f80 {font-size: 80%} + .f90 {font-size: 90%} + .f150 {font-size: 150%} + .f200 {font-size: 200%} + + .sp {position: relative; bottom: 0.5em; font-size: 0.75em;} + .sp1 {position: relative; bottom: 0.6em; font-size: 0.75em;} + .su {position: relative; top: 0.3em; font-size: 0.75em;} + .su1 {position: relative; top: 0.5em; font-size: 0.75em; margin-left: -1.2ex;} + .spp {position: relative; bottom: 0.5em; font-size: 0.6em;} + .suu {position: relative; top: 0.2em; font-size: 0.6em;} + .sc {font-variant: small-caps;} + .scs {font-variant: small-caps; font-size: 70%;} + .ov {text-decoration: overline} + .cl {background-color: #f5f5f5;} + .bk {padding-left: 0; font-size: 80%;} + .bk1 {margin-left: -1em;} + + .pagenum {position: absolute; right: 5%; text-align: right; font-size: 10pt; + background-color: #f5f5f5; color: #778899; text-indent: 0; + padding-left: 0.5em; padding-right: 0.5em; font-style: normal; } + span.sidenote {width: 8em; margin-bottom: 1em; margin-top: 1.7em; margin-right: 2em; + font-size: 85%; float: left; clear: left; font-weight: bold; + font-style: italic; text-align: left; text-indent: 0; + background-color: #f5f5f5; color: black; } + .note {margin-left: 2em; margin-right: 2em; font-size: 0.9em; } + .fn { position: absolute; left: 12%; text-align: left; background-color: #f5f5f5; + text-indent: 0; padding-left: 0.2em; padding-right: 0.2em; } + span.correction {border-bottom: 1px dashed red;} + + div.poemr { margin-top: .75em; margin-bottom: .75em;} + div.poemr p { margin-left: 0; padding-left: 3em; text-indent: -3em; margin-top: 0em; margin-bottom: 0em; } + div.poemr p.s { margin-top: 1.5em; } + div.poemr p.i05 { margin-left: 0.4em; } + div.poemr p.i2 { margin-left: 2em; } + div.poemr p.i4 { margin-left: 4em; } + + .figright1 { padding-right: 1em; padding-left: 2em; padding-top: 1.5em; text-align: center; } + .figleft1 { padding-right: 2em; padding-left: 1em; padding-top: 1.5em; text-align: center; } + .figcenter {text-align: center; margin: auto; margin-left: auto; margin-right: auto; padding-top: 1.5em;} + .figcenter1 {text-align: center; margin-left: auto; margin-right: auto; padding-top: 2em; padding-bottom: 2em;} + .figure {text-align: center; padding-left: 1.5em; padding-right: 1.5em; padding-top: 1.5em; padding-bottom: 0;} + .bold {font-weight: bold; } + + div.minind {text-align: justify;} + div.condensed, div.condensed1 { line-height: 1.3em; margin-left: 3%; margin-right: 3%; font-size: 95%; } + div.condensed1 p {margin-left: 0; padding-left: 2em; text-indent: -2em;} + div.condensed span.sidenote {font-size: 90%} + + div.list {margin-left: 0;} + div.list p {padding-left: 6em; text-indent: -3em;} + + .pt05 {padding-top: 0.5em;} + .pt1 {padding-top: 1em;} + .pt2 {padding-top: 2em;} + .ptb1 {padding-top: 1em; padding-bottom: 1em;} + + </style> + </head> +<body> + + +<pre> + +The Project Gutenberg EBook of Encyclopaedia Britannica, 11th Edition, +Volume 8, Slice 4, by Various + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: Encyclopaedia Britannica, 11th Edition, Volume 8, Slice 4 + "Diameter" to "Dinarchus" + +Author: Various + +Release Date: May 30, 2010 [EBook #32607] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK ENCYC. BRITANNICA, VOL 8 SL 4 *** + + + + +Produced by Marius Masi, Don Kretz and the Online +Distributed Proofreading Team at https://www.pgdp.net + + + + + + +</pre> + + +<table border="0" cellpadding="10" style="background-color: #dcdcdc; color: #696969; " summary="Transcriber's note"> +<tr> +<td style="width:25%; vertical-align:top"> +Transcriber's note: +</td> +<td class="norm"> +A few typographical errors have been corrected. They +appear in the text <span class="correction" title="explanation will pop up">like this</span>, and the +explanation will appear when the mouse pointer is moved over the marked +passage. Sections in Greek will yield a transliteration +when the pointer is moved over them, and words using diacritic characters in the +Latin Extended Additional block, which may not display in some fonts or browsers, will +display an unaccented version. <br /><br /> +<a name="artlinks">Links to other EB articles:</a> Links to articles residing in other EB volumes will +be made available when the respective volumes are introduced online. +</td> +</tr> +</table> +<div style="padding-top: 3em; "> </div> + + +<h2>THE ENCYCLOPÆDIA BRITANNICA</h2> + +<h2>A DICTIONARY OF ARTS, SCIENCES, LITERATURE AND GENERAL INFORMATION</h2> + +<h3>ELEVENTH EDITION</h3> +<div style="padding-top: 3em; "> </div> + +<hr class="full" /> +<h3>VOLUME VIII SLICE IV<br /><br /> +Diameter to Dinarchus</h3> +<hr class="full" /> +<div style="padding-top: 3em; "> </div> + +<p class="center1" style="font-size: 150%; font-family: 'verdana';">Articles in This Slice</p> +<table class="reg" style="width: 90%; font-size: 90%; border: gray 2px solid;" summary="Contents"> + +<tr><td class="tcl"><a href="#ar1">DIAMETER</a></td> <td class="tcl"><a href="#ar63">DIEDENHOFEN</a></td></tr> +<tr><td class="tcl"><a href="#ar2">DIAMOND</a></td> <td class="tcl"><a href="#ar64">DIEKIRCH</a></td></tr> +<tr><td class="tcl"><a href="#ar3">DIAMOND NECKLACE, THE AFFAIR OF THE</a></td> <td class="tcl"><a href="#ar65">DIELECTRIC</a></td></tr> +<tr><td class="tcl"><a href="#ar4">DIANA</a></td> <td class="tcl"><a href="#ar66">DIELMANN, FREDERICK</a></td></tr> +<tr><td class="tcl"><a href="#ar5">DIANA MONKEY</a></td> <td class="tcl"><a href="#ar67">DIEMEN, ANTHONY VAN</a></td></tr> +<tr><td class="tcl"><a href="#ar6">DIANE DE FRANCE</a></td> <td class="tcl"><a href="#ar68">DIEPENBECK, ABRAHAM VAN</a></td></tr> +<tr><td class="tcl"><a href="#ar7">DIANE DE POITIERS</a></td> <td class="tcl"><a href="#ar69">DIEPPE</a></td></tr> +<tr><td class="tcl"><a href="#ar8">DIAPASON</a></td> <td class="tcl"><a href="#ar70">DIERX, LÉON</a></td></tr> +<tr><td class="tcl"><a href="#ar9">DIAPER</a></td> <td class="tcl"><a href="#ar71">DIES, CHRISTOPH ALBERT</a></td></tr> +<tr><td class="tcl"><a href="#ar10">DIAPHORETICS</a></td> <td class="tcl"><a href="#ar72">DIEST</a></td></tr> +<tr><td class="tcl"><a href="#ar11">DIAPHRAGM </a></td> <td class="tcl"><a href="#ar73">DIESTERWEG, FRIEDRICH ADOLF WILHELM</a></td></tr> +<tr><td class="tcl"><a href="#ar12">DIARBEKR</a></td> <td class="tcl"><a href="#ar74">DIET</a></td></tr> +<tr><td class="tcl"><a href="#ar13">DIARRHOEA</a></td> <td class="tcl"><a href="#ar75">DIETARY</a></td></tr> +<tr><td class="tcl"><a href="#ar14">DIARY</a></td> <td class="tcl"><a href="#ar76">DIETETICS</a></td></tr> +<tr><td class="tcl"><a href="#ar15">DIASPORE</a></td> <td class="tcl"><a href="#ar77">DIETRICH, CHRISTIAN WILHELM ERNST</a></td></tr> +<tr><td class="tcl"><a href="#ar16">DIASTYLE</a></td> <td class="tcl"><a href="#ar78">DIETRICH OF BERN</a></td></tr> +<tr><td class="tcl"><a href="#ar17">DIATOMACEAE</a></td> <td class="tcl"><a href="#ar79">DIEZ, FRIEDRICH CHRISTIAN</a></td></tr> +<tr><td class="tcl"><a href="#ar18">DIAULOS</a></td> <td class="tcl"><a href="#ar80">DIEZ</a></td></tr> +<tr><td class="tcl"><a href="#ar19">DIAVOLO, FRA</a></td> <td class="tcl"><a href="#ar81">DIFFERENCES, CALCULUS OF</a></td></tr> +<tr><td class="tcl"><a href="#ar20">DIAZ, NARCISSE VIRGILIO</a></td> <td class="tcl"><a href="#ar82">DIFFERENTIAL EQUATION</a></td></tr> +<tr><td class="tcl"><a href="#ar21">DIAZ, PORFIRIO</a></td> <td class="tcl"><a href="#ar83">DIFFLUGIA</a></td></tr> +<tr><td class="tcl"><a href="#ar22">DIAZ DE NOVAES, BARTHOLOMEU</a></td> <td class="tcl"><a href="#ar84">DIFFRACTION OF LIGHT</a></td></tr> +<tr><td class="tcl"><a href="#ar23">DIAZO COMPOUNDS</a></td> <td class="tcl"><a href="#ar85">DIFFUSION</a></td></tr> +<tr><td class="tcl"><a href="#ar24">DIAZOMATA</a></td> <td class="tcl"><a href="#ar86">DIGBY, SIR EVERARD</a></td></tr> +<tr><td class="tcl"><a href="#ar25">DIBDIN, CHARLES</a></td> <td class="tcl"><a href="#ar87">DIGBY, SIR KENELM</a></td></tr> +<tr><td class="tcl"><a href="#ar26">DIBDIN, THOMAS FROGNALL</a></td> <td class="tcl"><a href="#ar88">DIGBY, KENELM HENRY</a></td></tr> +<tr><td class="tcl"><a href="#ar27">DIBDIN, THOMAS JOHN</a></td> <td class="tcl"><a href="#ar89">DIGENES ACRITAS, BASILIUS</a></td></tr> +<tr><td class="tcl"><a href="#ar28">DIBRA</a></td> <td class="tcl"><a href="#ar90">DIGEST</a></td></tr> +<tr><td class="tcl"><a href="#ar29">DIBRUGARH</a></td> <td class="tcl"><a href="#ar91">DIGESTIVE ORGANS</a></td></tr> +<tr><td class="tcl"><a href="#ar30">DICAEARCHUS</a></td> <td class="tcl"><a href="#ar92">DIGGES, WEST</a></td></tr> +<tr><td class="tcl"><a href="#ar31">DICE </a></td> <td class="tcl"><a href="#ar93">DIGIT</a></td></tr> +<tr><td class="tcl"><a href="#ar32">DICETO, RALPH DE</a></td> <td class="tcl"><a href="#ar94">DIGITALIS</a></td></tr> +<tr><td class="tcl"><a href="#ar33">DICEY, EDWARD</a></td> <td class="tcl"><a href="#ar95">DIGNE</a></td></tr> +<tr><td class="tcl"><a href="#ar34">DICHOTOMY</a></td> <td class="tcl"><a href="#ar96">DIGOIN</a></td></tr> +<tr><td class="tcl"><a href="#ar35">DICK, ROBERT</a></td> <td class="tcl"><a href="#ar97">DIJON</a></td></tr> +<tr><td class="tcl"><a href="#ar36">DICK, THOMAS</a></td> <td class="tcl"><a href="#ar98">DIKE</a></td></tr> +<tr><td class="tcl"><a href="#ar37">DICKENS, CHARLES JOHN HUFFAM</a></td> <td class="tcl"><a href="#ar99">DIKKA</a></td></tr> +<tr><td class="tcl"><a href="#ar38">DICKINSON, ANNA ELIZABETH</a></td> <td class="tcl"><a href="#ar100">DILAPIDATION</a></td></tr> +<tr><td class="tcl"><a href="#ar39">DICKINSON, JOHN</a></td> <td class="tcl"><a href="#ar101">DILATATION</a></td></tr> +<tr><td class="tcl"><a href="#ar40">DICKSON, SIR ALEXANDER</a></td> <td class="tcl"><a href="#ar102">DILATORY</a></td></tr> +<tr><td class="tcl"><a href="#ar41">DICKSON, SIR JAMES ROBERT</a></td> <td class="tcl"><a href="#ar103">DILEMMA</a></td></tr> +<tr><td class="tcl"><a href="#ar42">DICOTYLEDONS</a></td> <td class="tcl"><a href="#ar104">DILETTANTE</a></td></tr> +<tr><td class="tcl"><a href="#ar43">DICTATOR</a></td> <td class="tcl"><a href="#ar105">DILIGENCE</a></td></tr> +<tr><td class="tcl"><a href="#ar44">DICTIONARY</a></td> <td class="tcl"><a href="#ar106">DILKE, SIR CHARLES WENTWORTH</a></td></tr> +<tr><td class="tcl"><a href="#ar45">DICTYOGENS</a></td> <td class="tcl"><a href="#ar107">DILL</a></td></tr> +<tr><td class="tcl"><a href="#ar46">DICTYS CRETENSIS</a></td> <td class="tcl"><a href="#ar108">DILLEN, JOHANN JAKOB</a></td></tr> +<tr><td class="tcl"><a href="#ar47">DICUIL</a></td> <td class="tcl"><a href="#ar109">DILLENBURG</a></td></tr> +<tr><td class="tcl"><a href="#ar48">DIDACHĒ, THE</a></td> <td class="tcl"><a href="#ar110">DILLENS, JULIEN</a></td></tr> +<tr><td class="tcl"><a href="#ar49">DIDACTIC POETRY</a></td> <td class="tcl"><a href="#ar111">DILLINGEN</a></td></tr> +<tr><td class="tcl"><a href="#ar50">DIDEROT, DENIS</a></td> <td class="tcl"><a href="#ar112">DILLMANN, CHRISTIAN FRIEDRICH AUGUST</a></td></tr> +<tr><td class="tcl"><a href="#ar51">DIDIUS SALVIUS JULIANUS, MARCUS</a></td> <td class="tcl"><a href="#ar113">DILLON, ARTHUR RICHARD</a></td></tr> +<tr><td class="tcl"><a href="#ar52">DIDO</a></td> <td class="tcl"><a href="#ar114">DILLON, JOHN</a></td></tr> +<tr><td class="tcl"><a href="#ar53">DIDON, HENRI</a></td> <td class="tcl"><a href="#ar115">DILUVIUM</a></td></tr> +<tr><td class="tcl"><a href="#ar54">DIDOT</a></td> <td class="tcl"><a href="#ar116">DIME</a></td></tr> +<tr><td class="tcl"><a href="#ar55">DIDRON, ADOLPHE NAPOLÉON</a></td> <td class="tcl"><a href="#ar117">DIMENSION</a></td></tr> +<tr><td class="tcl"><a href="#ar56">DIDYMI</a></td> <td class="tcl"><a href="#ar118">DIMITY</a></td></tr> +<tr><td class="tcl"><a href="#ar57">DIDYMIUM</a></td> <td class="tcl"><a href="#ar119">DINAJPUR</a></td></tr> +<tr><td class="tcl"><a href="#ar58">DIDYMUS</a></td> <td class="tcl"><a href="#ar120">DINAN</a></td></tr> +<tr><td class="tcl"><a href="#ar59">DIDYMUS CHALCENTERUS</a></td> <td class="tcl"><a href="#ar121">DINANT</a></td></tr> +<tr><td class="tcl"><a href="#ar60">DIE</a> (town of France)</td> <td class="tcl"><a href="#ar122">DINAPUR</a></td></tr> +<tr><td class="tcl"><a href="#ar61">DIE</a> (datum)</td> <td class="tcl"><a href="#ar123">DINARCHUS</a></td></tr> +<tr><td class="tcl"><a href="#ar62">DIEBITSCH, HANS KARL FRIEDRICH ANTON</a></td> <td> </td></tr> +</table> + + +<hr class="art" /> +<p><span class="pagenum"><a name="page158" id="page158"></a>158</span></p> +<p><span class="bold">DIAMETER<a name="ar1" id="ar1"></a></span> (from the Gr. <span class="grk" title="dia">διά</span>, through, <span class="grk" title="metron">μέτρον</span>, 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.</p> + +<p>In architecture, the term is used to express the measure of the +lower part of the shaft of a column. It is employed by Vitruvius +(iii. 2) to determine the height of a column, which should vary +from eight to ten diameters according to the intercolumniation: +and it is generally the custom to fix the lower diameter of the +shaft by the height required and the Order employed. Thus +the diameter of the Roman Doric should be about one-eighth of +the height, that of the Ionic one-ninth, and of the Corinthian +one-tenth (see <span class="sc"><a href="#artlinks">Order</a></span>).</p> + + +<hr class="art" /> +<p><span class="bold">DIAMOND,<a name="ar2" id="ar2"></a></span> a mineral universally recognized as chief among +precious stones; it is the hardest, the most imperishable, and +also the most brilliant of minerals.<a name="fa1a" id="fa1a" href="#ft1a"><span class="sp">1</span></a> These qualities alone +have made it supreme as a jewel since early times, and yet the +real brilliancy of the stone is not displayed until it has been +faceted by the art of the lapidary (<i>q.v.</i>); and this was scarcely +developed before the year 1746. The consummate hardness of +the diamond, in spite of its high price, has made it most useful +for purposes of grinding, polishing and drilling. Numerous +attempts have been made to manufacture the diamond by artificial +means, and these attempts have a high scientific interest on +account of the mystery which surrounds the natural origin of this +remarkable mineral. Its physical and chemical properties have +been the subject of much study, and have a special interest +in view of the extraordinary difference between the physical +characters of the diamond and those of graphite (blacklead) or +charcoal, with which it is chemically identical, and into which it +can be converted by the action of heat or electricity. Again, on +account of the great value of the diamond, much of the romance +of precious stones has centred round this mineral; and the +history of some of the great diamonds of historic times has been +traced through many extraordinary vicissitudes.</p> + +<p>The name <span class="grk" title="Adamas">Άδάμας</span>, “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 (<span class="sc">a.d.</span> 16),—<i>Sic Adamas</i>, <i>punctum lapidis</i>, <i>pretiosior +auro</i>,—and Pliny (<span class="sc">a.d.</span> 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 (<i>turbines</i>, whip-tops) +placed base to base, may probably be identified as the ordinary +octahedral crystal (fig. 1). The “diamond” (<i>Yahalom</i>) in the +breastplate of the high priest (Ex. xxxix. 11) was certainly some +other stone, for it bore the name of a tribe, and methods of +engraving the true diamond cannot have been known so early. +The stone can hardly have become familiar to the Romans until +introduced from India, where it was probably mined at a very +early period. But one or other of the remaining varieties +mentioned by Pliny (the Macedonian, the Arabian, the Cyprian, +&c.) may be the true diamond, which was in great request for +the tool of the gem-engraver. Later Roman authors mentioned +various rivers in India as yielding the <i>Adamas</i> among their sands. +The name <i>Adamas</i> became corrupted into the forms <i>adamant</i>, +<i>diamaunt</i>, <i>diamant</i>, <i>diamond</i>; but the same word, owing to +a medieval misinterpretation which derived it from <i>adamare</i> +(compare the French word <i>aimant</i>), was also applied to the +lodestone.</p> + +<p>Like all the precious stones, the diamond was credited with +many marvellous virtues; among others the power of averting +insanity, and of rendering poison harmless; and in the middle +ages it was known as the “pietra della reconciliazione,” as the +peacemaker between husband and wife.</p> + +<table class="nobctr" summary="Illustration"> +<tr><td class="figcenter" colspan="4"><img style="width:519px; height:132px" src="images/img158a.jpg" alt="" /></td></tr> +<tr><td class="caption sc">Fig. 1.</td> <td class="caption sc">Fig. 2.</td> +<td class="caption sc">Fig. 3.</td> <td class="caption sc">Fig. 4.</td></tr></table> + +<table class="nobctr" style="float: right; width: 240px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:192px; height:162px" src="images/img158b.jpg" alt="" /></td></tr> +<tr><td class="caption sc">Fig. 5.</td></tr> +<tr><td class="figright1"><img style="width:163px; height:148px" src="images/img158c.jpg" alt="" /></td></tr> +<tr><td class="caption sc">Fig. 6.</td></tr> +<tr><td class="figright1"><img style="width:177px; height:172px" src="images/img158d.jpg" alt="" /></td></tr> +<tr><td class="caption sc">Fig. 7.</td></tr></table> + +<p><i>Scientific Characters.</i>—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 <span class="correction" title="amended from octadehron">octahedron</span> Fig. 5 shows +how the octahedron with furrowed edge +may be constructed from two interpenetrating +tetrahedra (shown in dotted lines). +If the grooves be left out of account, the +large faces which have replaced each tetrahedron +corner then make up a figure which +has the aspect of a simple octahedron. +Such regular interpenetrations are known +in crystallography as “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.</p> + +<p>Diamond possesses a brilliant “adamantine” lustre, but this +tends to be greasy on the surface of the natural stones and gives +<span class="pagenum"><a name="page159" id="page159"></a>159</span> +the rounded crystals somewhat the appearance of drops of gum. +Absolutely colourless stones are not so common as cloudy and +faintly coloured specimens; the usual tints are grey, brown, +yellow or white; and as rarities, red, green, blue and black +stones have been found. The colour can sometimes be removed +or changed at a high temperature, but generally returns on +cooling. It is therefore more probably due to metallic oxides than +to hydrocarbons. Sir William Crookes has, however, changed +a pale yellow diamond to a bluish-green colour by keeping it +embedded in radium bromide for eleven weeks. The black +coloration upon the surface produced by this process, as also by +the electric bombardment in a vacuum tube, appears to be due +to a conversion of the surface film into graphite. Diamond may +break with a conchoidal fracture, but the crystals always cleave +readily along planes parallel to the octahedron faces: of this +property the diamond cutters avail themselves when reducing +the stone to the most convenient form for cutting; a sawing +process, has, however, now been introduced, which is preferable +to that of cleavage. It is the hardest known substance (though +tantalum, or an alloy of tantalum now competes with it) and is +chosen as 10 in the mineralogist’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.</p> + +<p>The very high refractive power (index = 2.417 for sodium light) +gives the stone its extraordinary brilliancy; for light incident +within a diamond at a greater angle than 24½° is reflected back +into the stone instead of passing through it; the corresponding +angle for glass is 40½°. The very high dispersion (index for red +light = 2.402, for blue light = 2.460) gives it the wonderful “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.</p> + +<p>The phosphorescence produced by friction has been known +since the time of Robert Boyle (1663); the diamond becomes +luminous in a dark room after exposure to sunlight or in the +presence of radium; and many stones phosphoresce beautifully +(generally with a pale green light) when subjected to the electric +discharge in a vacuum tube. Some diamonds are more phosphorescent +than others, and different faces of a crystal may display +different tints. The combustibility of the diamond was predicted +by Sir Isaac Newton on account of its high refractive +power; it was first established experimentally by the Florentine +Academicians in 1694. In oxygen or air diamond burns at about +850°, and only continues to do so if maintained at a high temperature; +but in the absence of oxidising agents it may be raised +to a much higher temperature. It is, however, infusible at +the temperature of the electric arc, but becomes converted +superficially into graphite. Experiments on the combustion of +diamond were made by Smithson Tennant (1797) and Sir +Humphry Davy (1816), with the object of proving that it is pure +carbon; they showed that burnt in oxygen it yields exactly the +same amount of carbon dioxide as that produced by burning the +same weight of carbon. Still more convincing experiments were +made by A. Krause in 1890. Similarly Guyton de Morveau +showed that, like charcoal, diamond converts soft iron into steel. +Diamond is insoluble in acid and alkalis, but is oxidised on +heating with potassium bichromate and sulphuric acid.</p> + +<p>Bort (or Boart) is the name given to impure crystals or fragments +useless for jewels; it is also applied to the rounded +crystalline aggregates, which generally have a grey colour, +a rough surface, often a radial structure, and are devoid of +good cleavage. They are sometimes spherical (“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.</p> + +<p><i>Uses of the Diamond.</i>—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.</p> + +<p><i>Distribution and Mining.</i>—The most important localities for +diamonds have been: (1) India, where they were mined from +the earliest times till the close of the 19th century; (2) South +America, where they have been mined since the middle of the +18th century; and (3) South Africa, to which almost the whole +of the diamond-mining industry has been transferred since 1870.</p> + +<div class="condensed"> +<p><i>India.</i>—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 +<span class="pagenum"><a name="page160" id="page160"></a>160</span> +of the mines between 1638 and 1665 as a dealer in precious stones. +According to his description shallow pits were sunk, and the gravel +excavated was gathered into a walled enclosure where it was crushed +and water was poured over it, and it was finally sifted in baskets and +sorted by hand. The buying and selling was at that period conducted +by young children. In more modern times there has been the same +excavation of shallow pits, and sluicing, sifting and sorting, by hand +labour, the only machinery used being chain pumps made of earthen +bowls to remove the water from the deeper pits.</p> + +<p>At some of the Indian localities spasmodic mining has been carried +on at different periods for centuries, at some the work which had been +long abandoned was revived in recent times, at others it has long been +abandoned altogether. Many of the large stones of antiquity were +probably found in the Kollar group, where Tavernier found 60,000 +workers in 1645 (?), the mines having, according to native accounts, +been discovered about 100 years previously. Golconda was the +fortress and the market for the diamond industry at this group of +mines, and so gave its name to them. The old mines have now been +completely abandoned, but in 1891 about 1000 carats were being +raised annually in the neighbourhood of Hyderabad. The Sambalpur +group appear to have been the most ancient mines of all, but they +were not worked later than 1850. The Panna group were the most +productive during the 19th century. India was no doubt the source +of all the large stones of antiquity; a stone of 67<span class="spp">3</span>⁄<span class="suu">8</span> carats was found +at Wajra Karur in the Chennur group in 1881, and one of 210½ +carats at Hira Khund in 1809. Other Indian localities besides those +mentioned above are Simla, in the N.W. Provinces, where a few +stones have been found, and a district on the Gouel and the Sunk +rivers in Bengal, which V. Ball has identified with the Soumelpour +mentioned by Tavernier. The mines of Golconda and Kurnool were +described as early as 1677 in the twelfth volume of the <i>Philosophical +Transactions</i> of the Royal Society. At the present time very few +Indian diamonds find their way out of the country, and, so far as +the world’s supply is concerned, Indian mining of diamonds may be +considered extinct. The first blow to this industry was the discovery +of the Brazilian mines in Minas Geraes and Bahia.</p> + +<p><i>Brazil.</i>—-Diamonds were found about 1725 at Tejuco (now Diamantina) +in Minas Geraes, and the mining became important about +1740. The chief districts in Minas Geraes are (1) Bagagem on the W. +side of the Serra da Mata da Corda; (2) Rio Abaete on the E. side of +the same range; these two districts being among the head waters of +the Rio de San Francisco and its tributaries; (3) Diamantina, on and +about the watershed separating the Rio de San Francisco from the +Rio Jequitinhonha; and (4) Grao Mogul, nearly 200 m. to the N.E. +of Diamantina on the latter river.</p> + +<p>The Rio Abaete district was worked on a considerable scale between +1785 and 1807, but is now abandoned. Diamantina is at present the +most important district; it occupies a mountainous plateau, and +the diamonds are found both on the plateau and in the river valleys +below it. The mountains consist here of an ancient laminated +micaceous quartzite, which is in parts a flexible sandstone known as +itacolumite, and in parts a conglomerate; it is interbedded with +clay-slate, mica-schist, hornblende-schist and haematite-schist, and +intersected by veins of quartz. This series is overlain unconformably +by a younger quartzite of similar character, and itself rests upon the +crystalline schists. The diamond is found under three conditions: +(1) in the gravels of the present rivers, embedded in a ferruginous clay-cemented +conglomerate known as <i>cascalho</i>; (2) in terraces (gupiarras) +in a similar conglomerate occupying higher levels in the present +valleys; (3) in plateau deposits in a coarse surface conglomerate +known as <i>gurgulho</i>, the diamond and other heavy minerals being +embedded in the red clay which cements the larger blocks. Under +all these three conditions the diamond is associated with fragments +of the rocks of the country and the minerals derived from them, +especially quartz, hornstone, jasper, the polymorphous oxide of +titanium (rutile, anatase and brookite), oxides and hydrates of iron +(magnetite, ilmenite, haematite, limonite), oxide of tin, iron pyrites, +tourmaline, garnet, xenotime, monazite, kyanite, diaspore, sphene, +topaz, and several phosphates, and also gold. Since the heavy +minerals of the <i>cascalho</i> in the river beds are more worn than those of +the terraces, it is highly probable that they have been derived by the +cutting down of the older river gravels represented by the terraces; +and since in both deposits the heavy minerals are more abundant +near the heads of the valleys in the plateau, it is also highly probable +that both have really been derived from the plateau deposit. In the +latter, especially at São João da Chapada, the minerals accompanying +the diamond are scarcely worn at all; in the terraces and the river +beds they are more worn and more abundant; the terraces, therefore, +are to be regarded as a first concentration of the plateau material by +the old rivers; and the <i>cascalho</i> as a second concentration by the +modern rivers. The mining is carried on by negroes under the supervision +of overseers; the <i>cascalho</i> is dug out in the dry season and +removed to a higher level, and is afterwards washed out by hand in +running water in shallow wooden basins (<i>bateas</i>). The terraces can +be worked at all seasons, and the material is partly washed out +by leading streams on to it. The washing of the plateau material is +effected in reservoirs of rain water.</p> + +<p>It is difficult to obtain an estimate of the actual production of the +Minas Geraes mines, for no official returns have been published, but +in recent years it has certainly been rivalled by the yield in Bahia. +The diamond here occurs in river gravels and sands associated with +the same minerals as in Minas Geraes; since 1844 the richest mines +have been worked in the Serra de Cincora, where the mountains are +intersected by the river Paraguassu and its tributaries; it is said +that there were as many as 20,000 miners working here in 1845, and +it was estimated that 54,000 carats were produced in Bahia in 1858. +The earlier workings were in the Serra de Chapada to the N.W. of +the mines just mentioned. In 1901 there were about 5000 negroes +employed in the Bahia mines; methods were still primitive; the +<i>cascalho</i> was dug out from the river beds or tunnelled out from the +valley side, and washed once a week in sluices of running water, +where it was turned over with the hoe, and finally washed in wooden +basins and picked over by hand; sometimes also the diamantiferous +material is scooped out of the bed of the shallow rivers by divers, and +by men working under water in caissons. It is almost exclusively in +the mines of Bahia, and in particular in the Cincora district, that the +valuable carbonado is found. The carbonado and the diamond have +been traced to an extensive hard conglomerate which occurs in the +middle of the sandstone formation. Diamonds are also mined at +Salobro on the river Pardo not far inland from the port of Canavieras +in the S.E. corner of Bahia. The enormous development of the South +African mines, which supplied in 1906, about 90% of the world’s produce, +has thrown into the shade the Brazilian production; but the <i>Bulletin</i> +for Feb. 1909 of the International Bureau of American Republics gave +a very confident account of its future, under improved methods.</p> + +<p><i>South Africa.</i>—-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.</p> + +<p>But towards the close of 1870 stones were found at Jagersfontein +and at Dutoitspan, far from the Vaal river, and led to a second great +rush of prospectors, especially to Dutoitspan, and in 1871 to what +is now the Kimberley mine in the neighbourhood of the latter. At +each of these spots the diamantiferous area was a roughly circular +patch of considerable size, and in some occupied the position of +one of those depressions or “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.</p> + +<p>It was soon found that each mine was in reality a huge vertical +funnel or crater descending to an unknown depth, and filled with +diamantiferous blue ground. At first each claim was an independent +pit 31 ft. square sunk into the blue ground; the diamantiferous rock +was hoisted by bucket and windlass, and roadways were left across +the pit to provide access to the claims. But the roadways soon fell +in, and ultimately haulage from the claims could only be provided by +means of a vast system of wire ropes extending from a triple staging +of windlasses erected round the entire edge of the mine, which had by +this time become a huge open pit; the ropes from the upper windlasses +extended to the centre, and those from the lower tier to the +sides of the pit; covering the whole mass like a gigantic cobweb. +(See Plate II. fig. 12.) The buckets of blue ground were hauled up +these ropes by means of horse whims, and in 1875 steam winding +engines began to be employed. By this time also improved methods +in the treatment of the blue ground were introduced. It was carried +off in carts to open spaces, where an exposure of some weeks to the air +was found to pulverize the hard rock far more efficiently than the +old method of crushing with mallets. The placer-miner’s cradle and +rocking-trough were replaced by puddling troughs stirred by a +revolving comb worked by horse power; reservoirs were constructed +for the scanty water-supply, bucket elevators were introduced to +carry away the tailings; and the natives were confined in compounds. +For these improvements co-operation was necessary; the better +claims, which in 1872 had risen from £100 to more than £4000 in +value, began to be consolidated, and a Mining Board was introduced.</p> +</div> + +<p class="noind pt2 sc f80">Plate I.</p> + +<table class="nobctr" summary="Illustration"> +<tr><td class="figcenter"><img style="width:320px; height:408px" src="images/img160a1.jpg" alt="" /></td> +<td class="figcenter"><img style="width:320px; height:407px" src="images/img160a2.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 9.</span>—DE BEERS MINE, 1874.</td> +<td class="caption"><span class="sc">Fig. 10.</span>—KIMBERLEY MINE, 1874.</td></tr> + +<tr><td class="figcenter" colspan="2"><img style="width:700px; height:432px" src="images/img160a3.jpg" alt="" /></td></tr> +<tr><td class="caption" colspan="2"><span class="sc">Fig. 11.</span>—DE BEERS MINE, 1873.<br /> +<span class="f80">(From photographs by C. Evans.)</span></td></tr></table> + + +<p class="noind pt2 sc f80">Plate II.</p> + +<table class="nobctr" summary="Illustration"> +<tr><td class="figcenter"><img style="width:700px; height:439px" src="images/img160b1.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 12.</span>—KIMBERLEY MINE, 1874.</td></tr> + +<tr><td class="figcenter"><img style="width:700px; height:443px" src="images/img160b2.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 13.</span>—KIMBERLEY MINE, 1902.<br /> +<span class="f80">(From photographs by C. Evans.)</span></td></tr></table> + +<p><span class="pagenum"><a name="page161" id="page161"></a>161</span></p> + +<div class="condensed"> +<p>In a very few years, however, the open pit mining was rendered +impossible by the mud rushes, by the falls of the masses of barren +rock known as “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 <span class="sc"><a href="#artlinks">Kimberley</a></span>.)</p> + +<p>The scene of native mining was now transferred from the open pit +to underground tunnels; the vast network of wire ropes (Plate II. +fig. 12) with their ascending and descending buckets disappeared, and +with it the cosmopolitan crowd of busy miners working like ants at +the bottom of the pit. In place of all this, the visitor to Kimberley +encounters at the edge of the town only a huge crater, +silent and apparently deserted, with no visible sign of the +great mining operations which are conducted nearly half +a mile below the surface. The aspect of the Kimberley +pit in 1906 is shown in fig. 13 of Plate II., which may +be compared with the section of fig. 8.</p> + +<p>In fig. 13, Plate II., the sequence of the basalt, shale and +melaphyre is clearly visible on the sides of the pit; and +fig. 8 shows how the crater or “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.</p> + +<p>For the extraction and treatment of the blue ground +the De Beers Company in its great winding and washing plant employs +labour-saving machinery on a gigantic scale. The ground is +transferred in trucks to the shaft where it is automatically tipped into +skips holding 96 cubic ft. (six truck loads); these are rapidly hoisted +to the surface, where their contents are automatically dumped into +side-tipping trucks, and these in turn are drawn away in a continual +procession by an endless wire rope along the tram lines leading to the +vast “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.</p> + +<p>Until recently the final separation of the diamond from the concentrates +was made by hand picking, but even this has now been +replaced by machinery, owing to the remarkable discovery that a +greased surface will hold a diamond while allowing the other heavy +minerals to pass over it. The concentrates are washed down a sloping +table of corrugated iron which is smeared with grease, and it is found +that practically all the diamonds adhere to the table, and the other +minerals are washed away. At the large and important Premier mine +in the Transvaal the Elmore process, used in British Columbia and +in Wales for the separation of metallic ores, has been also introduced. +In the Elmore process oil is employed to float off the materials which +adhere to it, while the other materials remain in the water, the oil +being separated from the water by centrifugal action. The other +minerals found in the concentrates are pebbles and fragments of +pyrope, zircon, cyanite, chrome-diopside, enstatite, a green pyroxene, +mica, ilmenite, magnetite, chromite, hornblende, olivine, barytes, +calcite and pyrites.</p> + +<p>In all the S. African mines the diamonds are not only crystals of +various weights from fractions of a carat to 150 carats, but also occur +as microscopic crystals disseminated through the blue ground. In +spite of this, however, the average yield in the profitable mines is +only from 0.2 carat to 0.6 carat per load of 1600 lb, or on an average +about 1½ grs. per ton. The annual output of diamonds from the De +Beers mines was valued in 1906 at nearly £5,000,000; the value per +carat ranging from about 35s. to 70s.</p> + +<table class="nobctr" summary="Illustration"> +<tr><td class="figcenter"><img style="width:603px; height:489px" src="images/img161.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="f90">From Gardner Williams’s <i>Diamond Mines of South Africa</i>.</span><br /> +<span class="sc">Fig. 8.</span></td></tr></table> + +<p>Pipes similar to those which surround Kimberley have been found +in other parts of S. Africa. One of the best known is that of Jagersfontein, +which was really the first of the dry diggings (discovered in +1870). This large mine is near Fauresmith and 80 m. to the south +of Kimberley. In 1905 the year’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.</p> + +<p>The most competent S. African geologists believe all these remarkable +pipes to be connected with volcanic outbursts which occurred +over the whole of S. Africa during the Cretaceous period (after the +deposition of the Stormberg beds), and drilled these enormous craters +through all the later formations. With the true pipes are associated +dykes and fissures also filled with diamantiferous blue ground. It +is only in the more northerly part of the country that the pipes +are filled with blue ground (or “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.</p> + +<p>The River Diggings on the Vaal river are still worked upon a small +scale, but the production from this source is so limited that they are +of little account in comparison with the mines in the blue ground. +The stones, however, are good; since they differ somewhat from the +Kimberley crystals it is probable that they were not derived from +the present pipes. Another S. African locality must be mentioned; +considerable finds were reported in 1905 and 1906 from gravels +at Somabula near Gwelo in Rhodesia where the diamond is associated +with chrysoberyl, corundum (both sapphire and ruby), topaz, +garnet, ilmenite, staurolite, rutile, with pebbles of quartz, granite, +<span class="pagenum"><a name="page162" id="page162"></a>162</span> +chlorite-schist, &c. Diamond has also been reported from kimberlite +“pipes” in Rhodesia.</p> + +<p><i>Other Localities.</i>—In addition to the South American localities +mentioned above, small diamonds have also been mined since their +discovery in 1890 on the river Mazaruni in British Guiana, and +finds have been reported in the gold washings of Dutch Guiana. +Borneo has possessed a diamond industry since the island was first +settled by the Malays; the references in the works of Garcia de Orta, +Linschoten, De Boot, De Laet and others, to Malacca as a locality +relate to Borneo. The large Borneo stone, over 360 carats in weight, +known as the Matan, is in all probability not a diamond. The chief +mines are situated on the river Kapuas in the west and near +Bandjarmassin in the south-east of the island, and the alluvial +deposits in which they occur are worked by a small number of Chinese +and Malays. Australia has yielded diamonds in alluvial deposits +near Bathurst (where the first discovery was made in 1851) and +Mudgee in New South Wales, and also near Bingara and Inverell +in the north of the colony. At Mount Werong a stone weighing +29 carats was found in 1905. At Ruby Hill near Bingara they were +found in a breccia filling a volcanic pipe. At Ballina, in New England, +diamonds have been found in the sea sand. Other Australian +localities are Echunga in South Australia; Beechworth, Arena and +Melbourne in Victoria; Freemantle and Nullagine in Western +Australia; the Palmer and Gilbert rivers in Queensland. These have +been for the most part discoveries in alluvial deposits of the goldfields, +and the stones were small. In Tasmania also diamonds have +been found in the Corinna goldfields. Europe has produced few +diamonds. Humboldt searched for them in the Urals on account of +the similarity of the gold and platinum deposits to those of Brazil, +and small diamonds were ultimately found (1829) in the gold washings +of Bissersk, and later at Ekaterinburg and other spots in the Urals. +In Lapland they have been found in the sands of the Pasevig river. +Siberia has yielded isolated diamonds from the gold washings of +Yenisei. In North America a few small stones have been found in +alluvial deposits, mostly auriferous, in Georgia, N. and S. Carolina, +Kentucky, Virginia, Tennessee, Wisconsin, California, Oregon and +Indiana. A crystal weighing 23¾ carats was found in Virginia in +1855, and one of 21¼ carats in Wisconsin in 1886. In 1906 a number +of small diamonds were discovered in an altered peridotite somewhat +resembling the S. African blue ground, at Murfreesboro, Pike +county, Arkansas. Considerable interest attaches to the diamonds +found in Wisconsin, Michigan and Ohio near the Great Lakes, for they +are here found in the terminal moraines of the great glacial sheet which +is supposed to have spread southwards from the region of Hudson +Bay; several of the drift minerals of the diamantiferous region of +Indiana have been identified as probably of Canadian origin; no +diamonds have however yet been found in the intervening country of +Ontario. A rock similar to the blue ground of Kimberley has been +found in the states of Kentucky and New York. The occurrence of +diamond in meteorites is described below.</p> + +<p><i>Origin of the Diamond in Nature.</i>—It appears from the foregoing +account that at most localities the diamond is found in alluvial deposits +probably far from the place where it originated. The minerals +associated with it do not afford much clue to the original conditions; +they are mostly heavy minerals derived from the neighbouring rocks, +in which the diamond itself has not been observed. Among the +commonest associates of the diamond are quartz, topaz, tourmaline, +rutile, zircon, magnetite, garnet, spinel and other minerals which are +common accessory constituents of granite, gneiss and the crystalline +schists. Gold (also platinum) is a not infrequent associate, but this +may only mean that the sands in which the diamond is found have +been searched because they were known to be auriferous; also that +both gold and diamond are among the most durable of minerals and +may have survived from ancient rocks of which other traces have been +lost.</p> + +<p>The localities at which the diamond has been supposed to occur +in its original matrix are the following:—at Wajra Karur, in the +Cuddapah district, India, M. Chaper found diamond with corundum +in a decomposed red pegmatite vein in gneiss. At Sāo João da +Chapada, in Minas Geraes, diamonds occur in a clay interstratified +with the itacolumite, and are accompanied by sharp crystals of rutile +and haematite in the neighbourhood of decomposed quartz veins +which intersect the itacolumite. It has been suggested that these +three minerals were originally formed in the quartz veins. In both +these occurrences the evidence is certainly not sufficient to establish +the presence of an original matrix. At Inverell in New South Wales +a diamond (1906) has been found embedded in a hornblende diabase +which is described as a dyke intersecting the granite. Finally there is +the remarkable occurrence in the blue ground of the African pipes.</p> + +<p>There has been much controversy concerning the nature and origin +of the blue ground itself; and even granted that (as is generally +believed) the blue ground is a much serpentinized volcanic breccia +consisting originally of an olivine-bronzite-biotite rock (the so-called +kimberlite), it contains so many rounded and angular fragments of +various rocks and minerals that it is difficult to say which of them +may have belonged to the original rock, and whether any were formed +<i>in situ</i>, or were brought up from below as inclusions. Carvill Lewis +believed the blue ground to be true eruptive rock, and the carbon to +have been derived from the bituminous shales of which it contains +fragments. The Kimberley shales, which are penetrated by the De +Beers group of pipes, were, however, certainly not the source of the +carbon at the Premier (Transvaal) mine, for at this locality the shales +do not exist. The view that the diamond may have crystallized out +from solution in its present matrix receives some support from the +experiments of W. Luzi, who found that it can be corroded by the +solvent action of fused blue ground; from the experiments of +J. Friedländer, who obtained diamond by dissolving graphite in fused +olivine; and still more from the experiments of R. von Hasslinger +and J. Wolff, who have obtained it by dissolving graphite in a fused +mixture of silicates having approximately the composition of the +blue ground. E. Cohen, who regarded the pipes as of the nature of a +mud volcano, and the blue ground as a kimberlite breccia altered by +hydrothermal action, thought that the diamond and accompanying +minerals had been brought up from deep-seated crystalline schists. +Other authors have sought the origin of the diamond in the action +of the hydrated magnesian silicates on hydrocarbons derived from +bituminous schists, or in the decomposition of metallic carbides.</p> + +<p>Of great scientific interest in this connexion is the discovery of +small diamonds in certain meteorites, both stones and irons; for +example, in the stone which fell at Novo-Urei in Penza, Russia, in +1886, in a stone found at Carcote in Chile, and in the iron found at +Cañon Diablo in Arizona. Graphitic carbon in cubic form (cliftonite) +has also been found in certain meteoric “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 <span class="sc"><a href="#artlinks">Gem, Artificial</a></span>). +Adolf Knop suggested that this may have first yielded hydrocarbons +by contact with water, and that from these the crystalline diamond +has been formed. The meteoric occurrence has even suggested the +fanciful notion that all diamonds were originally derived from +meteorites. The meteoric iron of Arizona, some of which contains +diamond, is actually found in and about a huge crater which is +supposed by some to have been formed by an immense meteorite +penetrating the earth’s crust.</p> + +<p>It is, at any rate, established that carbon can crystallize as diamond +from solution in iron, and other metals; and it seems that high +temperature and pressure and the absence of oxidizing agents are +necessary conditions. The presence of sulphur, nickel, &c., in the +iron appears to favour the production of the diamond. On the other +hand, the occurrence in meteoric stones, and the experiments +mentioned above, show that the diamond may also crystallize from +a basic magma, capable of yielding some of the metallic oxides and +ferro-magnesian silicates; a magma, therefore, which is not devoid +of oxygen. This is still more forcibly suggested by the remarkable +eclogite boulder found in the blue ground of the Newlands mine, not +far from the Vaal river, and described by T. G. Bonney. The boulder +is a crystalline rock consisting of pyroxene (chrome-diopside), garnet, +and a little olivine, and is studded with diamond crystals; a portion +of it is preserved in the British Museum (Natural History). In +another eclogite boulder, diamond was found partly embedded in +pyrope. Similar boulders have also been found in the blue ground +elsewhere. Specimens of pyrope with attached or embedded diamond +had previously been found in the blue ground of the De Beers mines. +In the Newlands boulder the diamonds have the appearance of being +an original constituent of the eclogite. It seems therefore that a holocrystalline +pyroxene-garnet rock may be one source of the diamond +found in blue ground. On the other hand many tons of the somewhat +similar eclogite in the De Beers mine have been crushed and have not +yielded diamond. Further, the ilmenite, which is the most characteristic +associate of the diamond in blue ground, and other of the +accompanying minerals, may have come from basic rocks of a +different nature.</p> + +<p>The Inverell occurrence may prove to be another example of +diamond crystallized from a basic rock.</p> + +<p>In both occurrences, however, there is still the possibility that the +eclogite or the basalt is not the original matrix, but may have caught +up the already formed diamond from some other matrix. Some +regard the eclogite boulders as derived from deep-seated crystalline +rocks, others as concretions in the blue ground.</p> + +<p>None of the inclusions in the diamond gives any clue to its origin; +diamond itself has been found as an inclusion, as have also black +specks of some carbonaceous materials. Other black specks have been +identified as haematite and ilmenite; gold has also been found; +other included minerals recorded are rutile, topaz, quartz, pyrites, +apophyllite, and green scales of chlorite (?). Some of these are of very +doubtful identification; others (<i>e.g.</i> apophyllite and chlorite) may +have been introduced along cracks. Some of the fibrous inclusions +were identified by H. R. Göppert as vegetable structures and were +supposed to point to an organic origin, but this view is no longer held. +Liquid inclusions, some of which are certainly carbon dioxide, have +also been observed.</p> + +<p>Finally, then, both experiment and the natural occurrence in rocks +and meteorites suggest that diamond may crystallize not only from +iron but also from a basic silicate magma, possibly from various rocks +consisting of basic silicates. The blue ground of S. Africa may be +<span class="pagenum"><a name="page163" id="page163"></a>163</span> +the result of the serpentinization of several such rocks, and although +now both brecciated and serpentinized some of these may have been +the original matrix. A circumstance often mentioned in support of +this view is the fact that the diamonds in one pipe generally differ +somewhat in character from those of another, even though they be +near neighbours.</p> +</div> + +<p><i>History.</i>—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.</p> + +<p>The following are some of the most famous diamonds of the +world:—</p> + +<p>A large stone found in the Golconda mines and said to have +weighed 787 carats in the rough, before being cut by a Venetian +lapidary, was seen in the treasury of Aurangzeb in 1665 by +Tavernier, who estimated its weight after cutting as 280 (?) +carats, and described it as a rounded rose-cut-stone, tall on one +side. The name <i>Great Mogul</i> has been frequently applied to this +stone. Tavernier states that it was the famous stone given to +Shah Jahan by the emir Jumla. The <i>Orloff</i>, stolen by a French +soldier from the eye of an idol in a Brahmin temple, stolen again +from him by a ship’s captain, was bought by Prince Orloff for +£90,000, and given to the empress Catharine II. It weighs +194¾ carats, is of a somewhat yellow tinge, and is among the +Russian crown jewels. The <i>Koh-i-nor</i>, which was in 1739 in the +possession of Nadir Shah, the Persian conqueror, and in 1813 in +that of the raja of Lahore, passed into the hands of the East +India Company and was by them presented to Queen Victoria +in 1850. It then weighed 186<span class="spp">1</span>⁄<span class="suu">16</span> carats, but was recut in London +by Amsterdam workmen, and now weighs 106<span class="spp">1</span>⁄<span class="suu">16</span> carats. There +has been much discussion concerning the possibility of this stone +and the Orloff being both fragments of the Great Mogul. The +Mogul Baber in his memoirs (1526) relates how in his conquest of +India he captured at Agra the great stone weighing 8 mishkals, +or 320 ratis, which may be equivalent to about 187 carats. The +Koh-i-nor has been identified by some authors with this stone and +by others with the stone seen by Tavernier. Tavernier, however, +subsequently described and sketched the diamond which he saw +as shaped like a bisected egg, quite different therefore from the +Koh-i-nor. Nevil Story Maskelyne has shown reason for believing +that the stone which Tavernier saw was really the Koh-i-nor +and that it is identical with the great diamond of Baber; and +that the 280 carats of Tavernier is a misinterpretation on his part +of the Indian weights. He suggests that the other and larger +diamond of antiquity which was given to Shah Jahan may +be one which is now in the treasury of Teheran, and that this is +the true Great Mogul which was confused by Tavernier with the +one he saw. (See Ball, Appendix I. to Tavernier’s <i>Travels</i> (1889); +and Maskelyne, <i>Nature</i>, 1891, 44, p. 555.).</p> + +<p>The <i>Regent</i> or <i>Pitt</i> diamond is a magnificent stone found in +either India or Borneo; it weighed 410 carats and was bought for +£20,400 by Pitt, the governor of Madras; it was subsequently, +in 1717, bought for £80,000 (or, according to some authorities, +£135,000) by the duke of Orleans, regent of France; it was reduced +by cutting to 136<span class="spp">14</span>⁄<span class="suu">16</span> carats; was stolen with the other crown +jewels during the Revolution, but was recovered and is still in +France. The <i>Akbar Shah</i> was originally a stone of 116 carats with +Arabic inscriptions engraved upon it; after being cut down to +71 carats it was bought by the gaikwar of Baroda for £35,000. +The <i>Nizam</i>, now in the possession of the nizam of Hyderabad, is +supposed to weigh 277 carats; but it is only a portion of a stone +which is said to have weighed 440 carats before it was broken. +The <i>Great Table</i>, a rectangular stone seen by Tavernier in 1642 +at Golconda, was found by him to weigh 242<span class="spp">3</span>⁄<span class="suu">16</span> carats; Maskelyne +regards it as identical with the <i>Darya-i-nur</i>, which is also a +rectangular stone weighing about 186 carats in the possession of +the shah of Persia. Another stone, the <i>Taj-e-mah</i>, belonging to +the shah, is a pale rose pear-shaped stone and is said to weigh +146 carats.</p> + +<p>Other famous Indian diamonds are the following:—The <i>Sancy</i>, +weighing 53<span class="spp">12</span>⁄<span class="suu">16</span> carats, which is said to have been successively the +property of Charles the Bold, de Sancy, Queen Elizabeth, +Henrietta Maria, Cardinal Mazarin, Louis XIV.; to have been +stolen with the Pitt during the French Revolution; and subsequently +to have been the property of the king of Spain, Prince +Demidoff and an Indian prince. The <i>Nassak</i>, 78<span class="spp">5</span>⁄<span class="suu">8</span> carats, the +property of the duke of Westminster. The <i>Empress Eugénie</i>, +51 carats, the property of the gaikwar of Baroda. The <i>Pigott</i>, +49 carats(?), which cannot now be traced. The <i>Pasha</i>, 40 carats. +The <i>White Saxon</i>, 48¾ carats. The <i>Star of Este</i>, 25<span class="spp">13</span>⁄<span class="suu">32</span> carats.</p> + +<p>Coloured Indian diamonds of large size are rare; the most +famous are:—a beautiful blue brilliant, 67<span class="spp">2</span>⁄<span class="suu">16</span> carats, cut from a +stone weighing 112<span class="spp">3</span>⁄<span class="suu">16</span> carats brought to Europe by Tavernier. +It was stolen from the French crown jewels with the Regent and +was never recovered. The <i>Hope</i>, 44¼ carats, has the same colour +and is probably a portion of the missing stone: it was so-called +as forming part of the collection of H. T. Hope (bought for +£18,000), and was sold again in 1906 (resold 1909). Two other +blue diamonds are known, weighing 13¾ and 1¾ carats, which may +also be portions of the French diamond. The <i>Dresden Green</i>, one +of the Saxon crown jewels, 40 carats, has a fine apple-green +colour. The <i>Florentine</i>, 133<span class="spp">1</span>⁄<span class="suu">5</span> carats, one of the Austrian crown +jewels, is a very pale yellow.</p> + +<p>The most famous Brazilian stones are:—The <i>Star of the South</i>, +found in 1853, when it weighed 254½ carats and was sold for +£40,000; when cut it weighed 125 carats and was bought by the +gaikwar of Baroda for £80,000. Also a diamond belonging to +Mr Dresden, 119 carats before, and 76½ carats after cutting.</p> + +<p>Many large stones have been found in South Africa; some are +yellow but some are as colourless as the best Indian or Brazilian +stones. The most famous are the following:—the <i>Star of South +Africa</i>, or <i>Dudley</i>, mentioned above, 83½ carats rough, 46½ carats +cut. The <i>Stewart</i>, 288<span class="spp">3</span>⁄<span class="suu">8</span> carats rough, 120 carats cut. Both these +were found in the river diggings. The <i>Porter Rhodes</i> from +Kimberley, of the finest water, weighed about 150 carats. The +<i>Victoria</i>, 180 carats, was cut from an octahedron weighing 457½ +carats, and was sold to the nizam of Hyderabad for £400,000. +The <i>Tiffany</i>, a magnificent orange-yellow stone, weighs 125½ +carats cut. A yellowish octahedron found at De Beers weighed +428½ carats, and yielded a brilliant of 288½ carats. Some of the +finest and largest stones have come from the Jagersfontein mine; +one, the <i>Jubilee</i>, found in 1895, weighed 640 carats in the rough +and 239 carats when cut. Until 1905 the largest known diamond +in the world was the <i>Excelsior</i>, found in 1893 at Jagersfontein by +a native while loading a truck. It weighed 971 carats, and was +ultimately cut into ten stones weighing from 68 to 13 carats. +But all previous records were surpassed in 1905 by a magnificent +stone more than three times the size of any known diamond, +which was found in the yellow ground at the newly discovered +Premier mine in the Transvaal. This extraordinary diamond +weighed 3025¾ carats (1<span class="spp">1</span>⁄<span class="suu">3</span> ℔) and was clear and water white; the +largest of its surfaces appeared to be a cleavage plane, so that it +might be only a portion of a much larger stone. It was known +as the <i>Cullinan Diamond</i>. This stone was purchased by the +Transvaal government in 1907 and presented to King Edward VII. +It was sent to Amsterdam to be cut, and in 1908 was divided into +nine large stones and a number of small brilliants. The four +largest stones weigh 516½ carats, 309<span class="spp">3</span>⁄<span class="suu">16</span> carats, 92 carats and 62 +carats respectively. Of these the first and second are the largest +brilliants in existence. All the stones are flawless and of the +finest quality.</p> + +<div class="condensed"> +<p><span class="sc">Bibliography.</span>—Boetius de Boot, <i>Gemmarum et lapidum +historia</i> (1609); D. Jeffries, <i>A Treatise on Diamonds and Pearls</i> +(1757); J. Mawe, <i>Travels in the Interior of Brazil</i> (1812); <i>Treatise on +Diamonds and Precious Stones</i> (1813): Pinder, <i>De adamante</i> (1829); +Murray, <i>Memoir on the Nature of the Diamond</i> (1831); C. Zerenner, +<i>De adamante dissertatio</i> (1850); H. Emanuel, <i>Diamonds and +Precious Stones</i> (1865); A. Schrauf, <i>Edelsteinkunde</i> (1869); N. Jacobs +and N. Chatrian, <i>Monographie du diamant</i> (1880); V. Ball, <i>Geology +of India</i> (1881); C. W. King, <i>The Natural History of Precious Stones</i> +<span class="pagenum"><a name="page164" id="page164"></a>164</span> +<i>and Precious Metals</i> (1883); M. E. Boutan, <i>Le Diamant</i> (1886); +S. M. Burnham, <i>Precious Stones in Nature, Art and Literature</i> (1887); +P. Groth, <i>Grundriss der Edelsteinkunde</i> (1887); A. Liversidge, <i>The +Minerals of New South Wales</i> (1888); <i>Tavernier’s Travels in India</i>, +translated by V. Ball (1889); E. W. Streeter, <i>The Great Diamonds +of the World</i> (1896); H. C. Lewis, <i>The Genesis and Matrix of the +Diamond</i> (1897); L. de Launay, <i>Les Diamants du Cap</i> (1897); +C. Hintze, <i>Handbuch der Mineralogie</i> (1898); E. W. Streeter, +<i>Precious Stones and Gems</i> (6th ed., 1898); Dana, <i>System of Mineralogy</i> +(1899); Kunz and others, <i>The Production of Precious Stones</i> (in +annual, <i>Mineral Resources of the United States</i>); M. Bauer, <i>Precious +Stones</i> (trans. L. J. Spencer, 1904); A. W. Rogers, <i>An Introduction +to the Geology of Cape Colony</i> (1905); Gardner F. Williams, <i>The +Diamond Mines of South Africa</i> (revised edition, 1906); George F. +Kunz, “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, <i>Die Diamantführenden +Gesteine Südafrikas</i> (1909).</p> + +<p>Among papers in scientific periodicals may be mentioned articles +by Adler, Ball, Baumhauer, Beck, Bonney, Brewster, Chaper, Cohen, +Crookes, Daubrée, Derby, Des Cloizeaux, Doelter, Dunn, Flight, +Friedel, Gorceix, Gürich, Goeppert, Harger, Hudleston, Hussak, +Jannettaz, Jeremejew, de Launay, Lewis, Maskelyne, Meunier, +Moissan, Molengraaff, Moulle, Rose, Sadebeck, Scheibe, Stelzner, +Stow. See generally Hintze’s <i>Handbuch der Mineralogie</i>.</p> +</div> +<div class="author">(H. A. Mi.)</div> + +<hr class="foot" /> <div class="note"> + +<p><a name="ft1a" id="ft1a" href="#fa1a"><span class="fn">1</span></a> Diamonds are invariably weighed in carats and in ½, ¼, <span class="spp">1</span>⁄<span class="suu">8</span>, <span class="spp">1</span>⁄<span class="suu">16</span>, <span class="spp">1</span>⁄<span class="suu">32</span>, <span class="spp">1</span>⁄<span class="suu">64</span> +of a carat. One (English) carat = 3.17 grains = .2054 gram. One +ounce = 151½ carats. (See <span class="sc"><a href="#artlinks">Carat</a></span>.)</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DIAMOND NECKLACE, THE AFFAIR OF THE,<a name="ar3" id="ar3"></a></span> a mysterious +incident at the court of Louis XVI. of France, which involved +the queen Marie Antoinette. The Parisian jewellers Boehmer and +Bassenge had spent some years collecting stones for a necklace +which they hoped to sell to Madame Du Barry, the favourite of +Louis XV., and after his death to Marie Antoinette. In 1778 +Louis XVI. proposed to the queen to make her a present of +the necklace, which cost 1,600,000 livres. But the queen is +said to have refused it, saying that the money would be better +spent equipping a man-of-war. According to others, Louis XVI. +himself changed his mind. After having vainly tried to place the +necklace outside of France, the jewellers attempted again in 1781 +to sell it to Marie Antoinette after the birth of the dauphin. It +was again refused, but it was evident that the queen regretted +not being able to acquire it.</p> + +<p>At that time there was a personage at the court whom Marie +Antoinette particularly detested. It was the cardinal Louis de +Rohan, formerly ambassador at Vienna, whence he had been +recalled in 1774, having incurred the queen’s displeasure by +revealing to the empress Maria Theresa the frivolous actions of +her daughter, a disclosure which brought a maternal reprimand, +and for having spoken lightly of Maria Theresa in a letter of +which Marie Antoinette learned the contents. After his return +to France the cardinal was anxious to regain the favour of the +queen in order to obtain the position of prime minister. In March +1784 he entered into relations with a certain Jeanne de St Remy +de Valois, a descendant of a bastard of Henry II., who after many +adventures had married a <i>soi-disant</i> comte de Lamotte, and lived +on a small pension which the king granted her. This adventuress +soon gained the greatest ascendancy over the cardinal, with whom +she had intimate relations. She persuaded him that she had been +received by the queen and enjoyed her favour; and Rohan +resolved to use her to regain the queen’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.</p> + +<p>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 <i>coup +de théâtre</i>. On the 15th of August 1785, Assumption day, when +the whole court was awaiting the king and queen in order to go to +the chapel, the cardinal de Rohan, who was preparing to officiate, +was arrested and taken to the Bastille. He was able, however, to +destroy the correspondence exchanged, as he thought, with the +queen, and it is not known whether there was any connivance of +the officials, who did not prevent this, or not. The comtesse de +Lamotte was not arrested until the 18th of August, after having +destroyed her papers. The police set to work to find all her +accomplices, and arrested the girl Oliva and a certain Reteaux +de Villette, a friend of the countess, who confessed that he had +written the letters given to Rohan in the queen’s name, and +had imitated her signature on the conditions of the bargain. The +famous charlatan Cagliostro was also arrested, but it was recognized +that he had taken no part in the affair. The cardinal de +Rohan accepted the parlement of Paris as judges. A sensational +trial resulted (May 31, 1786) in the acquittal of the cardinal, of +the girl Oliva and of Cagliostro. The comtesse de Lamotte was +condemned to be whipped, branded and shut up in the +Salpetrière. Her husband was condemned, in his absence, to the +galleys for life. Villette was banished.</p> + +<p>Public opinion was much excited by this trial. It is generally +believed that Marie Antoinette was stainless in the matter, that +Rohan was an innocent dupe, and that the Lamottes deceived +both for their own ends. People, however, persisted in the belief +that the queen had used the countess as an instrument to satisfy +her hatred of the cardinal de Rohan. Various circumstances +fortified this belief, which contributed to render Marie Antoinette +very unpopular—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 <i>Mémoires</i> in which she accused the queen. Her +<span class="pagenum"><a name="page165" id="page165"></a>165</span> +husband also wrote <i>Mémoires</i>, and lived until 1831, after having, +it is said, received subsidies from Louis XVIII.</p> + +<div class="condensed"> +<p>See M. Tourneux, <i>Marie Antoinette devant l’histoire: Essai bibliographique</i> +(2nd ed., Paris, 1901); Émile Campardon, <i>Marie Antoinette +et le procès du collier</i> (Paris, 1863); P. Audebert, <i>L’Affaire du collier +de la reine, d’après la correspondance inédite du chevalier de Pujol</i> +(Rouen, 1901); F. d’Albini, <i>Marie Antoinette and the Diamond Necklace +from another Point of View</i> (London, 1900); Funck-Brentano, +<i>L’Affaire du collier</i> (1903); A. Lang, <i>Historical Mysteries</i> (1904). +Carlyle’s essay on <i>The Diamond Necklace</i> (first published in 1837 in +<i>Fraser’s Magazine</i>) is of historical literary interest.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DIANA,<a name="ar4" id="ar4"></a></span> in Roman mythology, an old Italian goddess, in later +times identified with the Greek Artemis (<i>q.v.</i>). That she was +originally an independent Italian deity is shown by her name, +which is the feminine form of Janus (= Dianus). She is essentially +the goddess of the moon and light generally, and presides over +wood, plain and water, the chase and war. As the goddess of +childbirth, she was known, like Juno, by the name of Lucina, the +“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, <i>Fasti</i>, iii. 263, vi. 731, +<i>Metam.</i> xv. 497; Virgil, <i>Aeneid</i>, vii. 761). A barbarous custom, +perhaps reminiscent of human sacrifice once offered to her, +prevailed in connexion with her ritual here; her priest, called +<i>Rex Nemorensis</i>, who was a runaway slave, was obliged to qualify +for office by slaying his predecessor in single combat (Strabo v. +p. 239; Suetonius, <i>Caligula</i>, 35). This led to the identification of +Diana with the Tauric Artemis, whose image was said to have been +removed by Orestes to the grove of Aricia (see <span class="sc"><a href="#artlinks">Aricini</a></span>).</p> + +<p>After the destruction of Alba Longa this grove was for a long time +the united sanctuary of the neighbouring Latin and Rutulian cities, +until at last it was extinguished beneath the supremacy of Rome. +The festival of the goddess was on the ides (13th) of August, the +full moon of the hot season. She was worshipped with torches, +her aid was sought by women seeking a happy deliverance in +childbirth, and many votive offerings have been found on the site. +The worship of Diana was brought to Rome by Latin plebeians, +and hence she was regarded as the protectress of the lower +classes, and especially of slaves. In accordance with this, her +most important temple was that on the Aventine, the chief seat +of the plebeians, founded by Servius Tullius, originally as a +sanctuary of the Latin league (Dion. Halic. iv. 26). No man was +allowed to enter the temple, and on the day of its dedication +(August 13) the slaves kept holiday (Plutarch, <i>Quaest. Rom.</i> 100). +This Diana was identified with the sister of Apollo, and at the +secular games she was worshipped simply as Artemis. Another +celebrated sanctuary of Diana was that on the slopes of Mount +Tifata near Capua (where she was worshipped under the name of +Tifatina), a sanctuary specially favoured by Sulla and Vespasian. +As Noctiluca (“giving light by night”) she had a sanctuary on +the Palatine which was kept illuminated throughout the night +(Varro, <i>L.L.</i> v. 68). On the Nemi priesthood see J. G. Frazer, +<i>Golden Bough</i>.</p> + + +<hr class="art" /> +<p><span class="bold">DIANA MONKEY,<a name="ar5" id="ar5"></a></span> a West African representative of the +guenon monkeys taking its name, <i>Cercopithecus diana</i>, from the +presence of a white crescent on the forehead; another characteristic +feature being the pointed white beard. The general colour +of the fur is greyish, with a deep tinge of chestnut from the +middle of the back to the root of the tail. Together with +<i>C. neglectus</i> of East and Central Africa, <i>C. ignitus</i> of Liberia, and +<i>C. roloway</i> of the Gold Coast, the diana represents the special +subgenus of guenons known as <i>Pogonocebus</i>. Although the diana +monkey is commonly seen in menageries, little is known of its +habits in the wild state.</p> + + +<hr class="art" /> +<p><span class="bold">DIANE DE FRANCE<a name="ar6" id="ar6"></a></span> (1538-1619), duchess of Montmorency +and Angoulême, was the natural daughter of Henry II. of France +and a young Piedmontese, Filippe Duc. The constable de +Montmorency went so far as to assert that of all the children of +Henry II. Diane was the only one who resembled him. Catherine +de’ 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 <i>politiques</i>. Again a widow in 1579, she had some +influence at the court of Henry III., and negotiated his reconciliation +with Henry of Navarre (1588). She retained her influence +in the reign of Henry IV., conveyed the bodies of Catherine +de’ Medici and Henry III. to St Denis, and died in 1619 at her +hôtel of Angoulême.</p> + +<div class="condensed"> +<p>See Brantôme, ed. by Lalanne, in the <i>Coll de la société d’histoire +de France</i>, vol. viii. (1875); J. de Thou, <i>Historia sui temporis...</i> +(1733); Matthieu de Morgues, <i>Oraison funèbre de Diane de France</i> +(Paris, 1619).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DIANE DE POITIERS<a name="ar7" id="ar7"></a></span> (1499-1566), duchess of Valentinois, +and mistress of Henry II. of France, was the daughter of Jean +de Poitiers, seigneur de St Vallier, who came of an old family of +Dauphiné. In 1515 she married Louis de Brézé, grand seneschal +of Normandy, by whom she had two daughters. She became a +widow in 1533, but soon replaced her husband by a more illustrious +lover, the king’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.</p> + +<p>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.</p> + +<div class="condensed"> +<p>See G. Guiffrey, <i>Lettres inédites de Diane de Poytiers</i> (Paris, 1866) +and <i>Procès criminel de Jehan de Poytiers</i> (Paris, 1867); Capefigue, +<i>Diane de Poitiers</i> (Paris, 1860); Hay, <i>Madame Dianne de Poytiers</i> +(London, 1900).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DIAPASON<a name="ar8" id="ar8"></a></span> (Gr. <span class="grk" title="dia pasôn">διὰ πασῶν</span>, through all), a term in music, +originally for an interval of an octave. The Greek is an abbreviation +of <span class="grk" title="hê dia pasôn chordôn symphônia">ἡ διὰ πασῶν χορδῶν συμφωνία</span>, a consonance +through all the tones of the scale. In this sense it is only +used now, loosely, for the compass of an instrument or voice, +or for a harmonious melody. The name is given to the two +<span class="pagenum"><a name="page166" id="page166"></a>166</span> +foundation stops of an organ, the open and the stopped diapason +(see <span class="sc"><a href="#artlinks">Organ</a></span>), and to a standard of musical pitch, as in the French +<i>diapason normal</i> (see <span class="sc"><a href="#artlinks">Pitch, Musical</a></span>).</p> + + +<hr class="art" /> +<p><span class="bold">DIAPER<a name="ar9" id="ar9"></a></span> (derived through the Fr, from the Gr. <span class="grk" title="dia">διά</span>, through, +and <span class="grk" title="aspros">ἄσπρος</span>, 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.</p> + +<table class="nobctr" summary="Illustration"> +<tr><td class="figcenter1"><img style="width:367px; height:261px" src="images/img166a.jpg" alt="" /></td></tr></table> + +<div class="condensed"> +<p>In its textile use, the term diaper was originally applied to silk +patterns of a geometrical pattern; it is now almost exclusively used +for diamond patterns made from linen or cotton yarns. An illustration +of two patterns of this nature is shown in the figure. The floats +of the warp and the weft are mostly in three; indeed the patterns +are made from a base weave which is composed entirely of +floats of this number. It will be seen that both designs are formed +of what may be termed concentric figures—alternately black and +white. Pattern B differs from pattern A only in that more of these +concentric figures are used for the complete figure. If pattern B, +which shows only one unit, were extended, the effect would be similar +to A, except for the size of the unit. In A there are four complete +units, and hence the pattern appears more striking. Again, the +repeating of B would cause the four corner pieces to join and to form +a diamond similar to the one in the centre. The two diamonds in B +would then alternate diagonally to left and right. Special names are +given to certain kinds of diapers, <i>e.g.</i> “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.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DIAPHORETICS<a name="ar10" id="ar10"></a></span> (from Gr. <span class="grk" title="diaphorein">διαφορεῖν</span>, to carry through), +the name given to those remedies which promote perspiration. +In health there is constantly taking place an exhalation of +watery vapour from the skin, by which not only are many of the +effete products of nutrition eliminated, but the body is kept cool. +Under exertion or in a heated atmosphere this natural function +of the skin is increased, sweating more or less profuse follows, +and, evaporation going on rapidly over the whole surface, little +or no rise in the temperature of the body takes place. In many +forms of disease, such as fevers and inflammatory affections, the +action of the skin is arrested, and the surface of the body feels +harsh and dry, while the temperature is greatly elevated. The +occurrence of perspiration not unfrequently marks a crisis in such +diseases, and is in general regarded as a favourable event. In +some chronic diseases, such as diabetes and some cases of +Bright’s disease, the absence of perspiration is a marked feature; +while, on the other hand, in many wasting diseases, such as +phthisis, the action of the skin is increased, and copious exhausting +sweating occurs. Many means can be used to induce perspiration, +among the best known being baths, either in the form of hot +vapour or hot water baths, or in that part of the process of +the Turkish bath which consists in exposing the body to a dry and +hot atmosphere. Such measures, particularly if followed by the +drinking of hot liquids and the wrapping of the body in warm +clothing, seldom fail to excite copious perspiration. Numerous +medicinal substances have the same effect.</p> + + +<hr class="art" /> +<p><span class="bold">DIAPHRAGM<a name="ar11" id="ar11"></a></span> (Gr. <span class="grk" title="diaphragma">διάφραγμα</span>, a partition). The diaphragm +or midriff (Anglo-Saxon, <i>mid</i>, middle, <i>hrif</i>, belly) in +human anatomy is a large fibro-muscular partition between the +cavities of the thorax and abdomen; it is convex toward the +thorax, concave toward the abdomen, and consists of a central +tendon and a muscular margin. The <i>central tendon</i> (q, fig. 1) is trefoil +in shape, its leaflets being right, left and anterior; of these the right +is the largest and the left the smallest. The fleshy fibres rise, in +front from the back of the xiphoid cartilage of the sternum (d), +laterally by six serrations, from the inner surfaces of the lower six +ribs, interdigitating with the transversalis, posteriorly from the +arcuate ligaments, of which there are five, a pair of external, a +pair of internal, and a single median one. The <i>external arcuate +ligament</i> (h) stretches from the tip of the twelfth rib (b) to the +costal process of the first lumbar vertebra in front of the quadratus +lumborum muscle (o), the <i>internal</i> and <i>middle</i> are continuations +of the <i>crura</i> which rise from the ventro-lateral aspects of +the bodies of the lumbar vertebrae, the right (e) coming from +three, the left (f) from two. On reaching the level of the twelfth +thoracic vertebra each crus spreads out into a fan-shaped mass of +fibres, of which the innermost join their fellows from the opposite +crus, in front of the aortic opening (k), to form the <i>middle arcuate +ligament</i>; the outer ones (g) arch in front of the psoas muscle (n) +to the tip of the costal process of the first lumbar vertebra to +form the <i>internal arcuate ligament</i>, while the intermediate ones +pass to the central tendon. There are three large openings in the +diaphragm; the <i>aortic</i> (k) is behind the middle arcuate ligament +and transmits the aorta, the vena azygos major, and the thoracic +duct. In the right leaflet is an opening (sometimes called the +<i>hiatus quadratus</i>) for the inferior vena cava and a branch of the +right phrenic nerve (m), while in front and a little to the left of +the aortic opening is one for the oesophagus and the two pneumogastric +nerves (l), the left being in front and the right behind. +<span class="pagenum"><a name="page167" id="page167"></a>167</span> +The fleshy fibres on each side of this opening act as a sphincter. +Passing between the xiphoid and costal origins in front are the +superior epigastric arteries, while the other terminal branches of +the internal mammaries, the musculo-phrenics, pass through +between two costal origins.</p> + +<table class="nobctr" summary="Illustration"> +<tr><td class="figcenter"><img style="width:413px; height:430px" src="images/img166b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 1.</span>—Abdominal Surface of the Diaphragm.</td></tr></table> + +<p>Through the crura pass the splanchnic nerves, and in addition +to these the left crus is pierced by the vena azygos minor. The +sympathetic nerves usually enter the abdomen behind the internal +arcuate ligaments. The phrenic nerves, which are the main +supply of the diaphragm, divide before reaching the muscle and +pierce it in a number of places to enter its abdominal surface, but +some of the lower intercostal nerves assist in the supply. The last +thoracic or subcostal nerves pass behind the external arcuate +ligament.</p> + +<p>For the action of the diaphragm see <span class="sc"><a href="#artlinks">Respiratory System</a></span>.</p> + +<div class="condensed"> +<p><i>Embryology.</i>—The diaphragm is at first developed in the neck region +of the embryo, and this accounts for the phrenic nerves, which supply +it, rising from the fourth and fifth cervical. From the mesoderm on +the caudal side of the pericardium is developed the <i>septum transversum</i>, +and in this the central tendon is formed. The fleshy portion is +developed on each side in two parts, an anterior or sterno-costal +which is derived from the longitudinal neck musculature, probably +the same layer from which the sternothyroid comes, and a spinal part +which is a derivative of the transversalis sheet of the trunk. Between +these two parts is at one time a gap, the <i>spino-costal hiatus</i>, and this +is obliterated by the growth of the pleuro-peritoneal membrane, which +may occasionally fail to close and so may form the site of a phrenic +hernia. With the growth of the body and the development of the +lungs the diaphragm shifts its position until it becomes the septum +between the thoracic and abdominal cavities. (See A. Keith, “On the +Development of the Diaphragm,” <i>Jour. of Anat. and Phys.</i> vol. 39.) +A. Paterson has recorded cases in which the left half of the diaphragm +is wanting (<i>Proceedings</i> of the Anatomical Society of Gt. Britain, +June 1900; <i>Jour. of Anat. and Phys.</i> vol. 34), and occasionally +deficiencies are found elsewhere, especially in the sternal portion. +For further details see Quain’s <i>Anatomy</i>, vol. i. (London, 1908).</p> + +<p><i>Comparative Anatomy.</i>—A complete diaphragm, separating the +thoracic from the abdominal parts of the coelom, is characteristic of +the Mammalia; it usually has the human structure and relations +except that below the Anthropoids it is separated from the pericardium +by the azygous lobe of the lung. In some Mammals, <i>e.g.</i> Echidna +and Phocoena, it is entirely muscular. In the Cetacea it is remarkable +for its obliquity; its vertebral attachment is much nearer the tail +than its sternal or ventral one; this allows a much larger lung space +in the dorsal than in the ventral part of the thorax, and may be +concerned with the equipoise of the animal. (Otto Müller, “Untersuchungen +über die Veränderung, welche die Respirationsorgane der +Säugetiere durch die Anpassung an das Leben im Wasser erlitten +haben,” <i>Jen. Zeitschr. f. Naturwiss.</i>, 1898, p. 93.) In the Ungulata +only one crus is found (Windle and Parsons, “Muscles of the +Ungulata,” <i>Proc. Zool. Soc.</i>, 1903, p. 287). Below the Mammals +incomplete partitions between the pleural and peritoneal cavities +are found in Chelonians, Crocodiles and Birds, and also in Amphibians +(Xenopus and Pipa).</p> +</div> +<div class="author">(F. G. P.)</div> + + +<hr class="art" /> +<p><span class="bold">DIARBEKR<a name="fa1b" id="fa1b" href="#ft1b"><span class="sp">1</span></a><a name="ar12" id="ar12"></a></span> (<i>Kara Amid</i> or Black Amid; the Roman +<i>Amida</i>), the chief town of a vilayet of Asiatic Turkey, situated +on a basaltic plateau on the right bank of the Tigris, which here +flows in a deep open valley. The town is still surrounded by the +masonry walls of black basalt which give it the name of <i>Kara</i> +or Black Amid; they are well built and imposing on the west +facing the open country, but almost in ruins where they overlook +the river. A mass of gardens and orchards cover the slope down +to the river on the S.W., but there are no suburbs outside the +walls. The houses are rather crowded but only partially fill +the walled area. The population numbers 38,000, nearly half +being Christian, comprising Turks, Kurds, Arabs, Turkomans, +Armenians, Chaldeans, Jacobites and a few Greeks. The streets +are 10 ft. to 15 ft. wide, badly paved and dirty; the houses and +shops are low, mostly of stone, and some of stone and mud. +The bazaar is a good one, and gold and silver filigree work is +made, peculiar in character and design. The cotton industry is +declining, but manufacture of silk is increasing. Fruit is good and +abundant as the rich volcanic soil is well watered from the town +springs. The size of the melons is specially famous. To the +south, the walls are some 40 ft. high, faced with large cut stone +blocks of very solid construction, with towers and square bastions +rising to 500 ft. There are four gates: on the north the Kharput +gate, on the west the Rum, on the south the Mardin, and on the +east the Yeni Kapu or new gate. A citadel enclosure stands +at the N. E. corner and is now partly in ruins, but the interior +space is occupied by the government konak. The summer +climate in the confined space within the town is excessively hot +and unhealthy. Epidemics of typhus are not unknown, as well +as ophthalmia. The Diarbekr boil is like the “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.</p> + +<p>The Great Mosque, Ulu Jami, formerly a Christian church, +occupies the site of a Sassanian palace and was built with +materials from an older palace, probably that of Tigranes II. +The remains consist of the façades of two palaces 400 ft. apart, +each formed by a row of Corinthian columns surmounted by an +equal number of a Byzantine type. Kufic inscriptions run across +the fronts under the entablature. The court of the mosque +is entered by a gateway on which lions and other animals are +sculptured. The churches of greatest interest are those of SS. +Cosmas and Damian (Jacobite) and the church of St James +(Greek). In the 19th century Diarbekr was one of the largest +and most flourishing cities of Asia, and as a commercial centre it +now stands at the meeting-point of several important routes. It +is at the head of the navigation of the Tigris, which is traversed +down stream by <i>keleks</i> or rafts supported by inflated skins. +There is a good road to Aleppo and Alexandretta on the Mediterranean, +and to Samsun on the Black Sea by Kharput, Malatia +and Sivas. There are also routes to Mosul and Bitlis.</p> + +<p>Diarbekr became a Roman colony in <span class="sc">a.d.</span> 230 under the name +of Amida, and received a Christian bishop in <span class="sc">a.d.</span> 325. It was +enlarged and strengthened by Constantius II., in whose reign it +was taken after a long siege by Shapur (Sapor) II., king of Persia. +The historian Ammianus Marcellinus, who took part in the +defence, gives a detailed account of it. In the later wars between +the Persians and Romans it more than once changed hands. +Though ceded by Jovian to the Persians it again became annexed +to the Roman empire, and in the reign of Anastasius (<span class="sc">a.d.</span> 502) +was once more taken by the Persians, when 80,000 of its inhabitants +were slain. It was taken c. 638 by the Arabs, and +afterwards passed into the hands of the Seljuks and Persians, +from whom it was finally captured by Selim I. in 1515; and +since that date it has remained under Ottoman rule. About 2 m. +below the town is a masonry bridge over the Tigris; the older +portion being probably Roman, and the western part, which bears +a Kufic inscription, being Arab.</p> + +<p>The vilayet of Diarbekr extends south from Palu on the +Euphrates to Mardin and Nisibin on the edge of the Mesopotamian +plain, and is divided into three sanjaks—Arghana, Diarbekr and +Mardin. The headwaters of the main arm of the Tigris have +their source in the vilayet.</p> + +<p>Cereals, cotton, tobacco, rice and silk are produced, but most of +the fertile lands have been abandoned to semi-nomads, who raise +large quantities of live stock. The richest portion of the vilayet +lies east of the capital in the rolling plains watered by tributaries +of the Tigris. An exceptionally rich copper mine exists at +Arghana Maden, but it is very imperfectly worked; galena +mineral oil and silicious sand are also found.</p> +<div class="author">(C. W. W.; F. R. M.)</div> + +<hr class="foot" /> <div class="note"> + +<p><a name="ft1b" id="ft1b" href="#fa1b"><span class="fn">1</span></a> From <i>Diar</i>, land, and Bekr (<i>i.e.</i> Abu Bekr, the caliph).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DIARRHOEA<a name="ar13" id="ar13"></a></span> (from Gr. <span class="grk" title="dia">διά</span>, through, <span class="grk" title="rheô">ῥέω</span>, flow), an excessive +looseness of the bowels, a symptom of irritation which +may be due to various causes, or may be associated with +some specific disease. The treatment in such latter cases +necessarily varies, since the symptom itself may be remedial, +but in ordinary cases depends on the removal of the cause of +irritation by the use of aperients, various sedatives being also +prescribed. In chronic diarrhoea careful attention to the diet is +necessary.</p> + +<p><span class="pagenum"><a name="page168" id="page168"></a>168</span></p> + + +<hr class="art" /> +<p><span class="bold">DIARY,<a name="ar14" id="ar14"></a></span> the Lat. <i>diarium</i> (from <i>dies</i>, a day), the book in which +are preserved the daily memoranda regarding events and actions +which come under the writer’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.</p> + +<p>In its primitive form, the diary must always have existed; as +soon as writing was invented, men and women must have wished +to note down, in some almanac or journal, memoranda respecting +their business, their engagements or their adventures. But +the literary value of these would be extremely insignificant until +the spirit of individualism had crept in, and human beings began +to be interesting to other human beings for their own sake. It +is not, therefore, until the close of the Renaissance that we find +diaries beginning to have literary value, although, as the study of +sociology extends, every scrap of genuine and unaffected record +of early history possesses an ethical interest. In the 17th century, +diaries began to be <span class="correction" title="amended from largly">largely</span> written in England, although in most +cases without any idea of even eventual publication. Sir William +Dugdale (1605-1686) had certainly no expectation that his slight +diary would ever see the light. There is no surviving record of +a journal kept by Clarendon, Richard Baxter, Lucy Hutchinson +and other autobiographical writers of the middle of the century, +but we may take it for granted that they possessed some such +record, kept from day to day. Bulstrode Whitelocke (1605-1675), +whose <i>Memorials of the English Affairs</i> covers the ground +from 1625 to 1660, was a genuine diarist. So was the elder George +Fox (1624-1690), who kept not merely “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.</p> + +<p>Many of the diaries described above were first published in the +opening years of the 19th century, and it is unquestionable that +the interest which they awakened in the public led to their +imitation. Diaries ceased to be rare, but as a rule the specimens +which have hitherto appeared have not presented much literary +interest. Exception must be made in favour of the journals of +two minor politicians, Charles Greville (1794-1865) and Thomas +Creevey (1768-1838), whose indiscretions have added much to the +gaiety of nations; the papers of the former appeared in 1874-1887, +those of the latter in 1903. The diary of Henry Crabb +Robinson (1775-1867), printed in 1869, contains excellent +biographical material. Tom Moore’s journal, published in 1856 +by Lord John Russell, disappointed its readers. But it is +probable, if we reason by the analogy of the past, that the most +curious and original diaries of the 19th century are still unknown +to us, and lie jealously guarded under lock and key by the +descendants of those who compiled them.</p> + +<p>It was natural that the form of the diary should appeal to a +people so sensitive to social peculiarities and so keen in the +observation of them as the French. A medieval document of +immense value is the diary kept by an anonymous <i>curé</i> during +the reigns of Charles VI. and Charles VII. This <i>Journal d’un +bourgeois de Paris</i> was kept from 1409 to 1431, and was continued +by another hand down to 1449. The marquis de Dangeau +(1638-1720) kept a diary from 1684 till the year of his death; +this although dull, and as Saint-Simon said “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 +<i>journal historique</i> from 1758 to 1782; it is full of vivacity, but very +scandalous and spiteful. It saw the light in 1805, and surprised +those to whom Collé, in his lifetime, had seemed the most placid +and good-natured of men. Petit de Bachaumont (1690-1770) +had access to remarkable sources of information, and his +<i>Mémoires secrets</i> (a diary the publication of which began in +1762 and was continued after Bachaumont’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.</p> +<div class="author">(E. G.)</div> + + +<hr class="art" /> +<p><span class="bold">DIASPORE,<a name="ar15" id="ar15"></a></span> a native aluminium hydroxide, AlO(OH), crystallizing +in the orthorhombic system and isomorphous with göthite +and manganite. It occurs sometimes as flattened crystals, but +usually as lamellar or scaly masses, the flattened surface being a +direction of perfect cleavage on which the lustre is markedly +pearly in character. It is colourless or greyish-white, yellowish, +sometimes violet in colour, and varies from translucent to +<span class="pagenum"><a name="page169" id="page169"></a>169</span> +transparent. It may be readily distinguished from other colourless +transparent minerals, with a perfect cleavage and pearly +lustre—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 <span class="grk" title="diaspeirein">διασπείρειν</span>, “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.</p> +<div class="author">(L. J. S)</div> + + +<hr class="art" /> +<p><span class="bold">DIASTYLE<a name="ar16" id="ar16"></a></span> (from Gr. <span class="grk" title="dia">διά</span>, through, and <span class="grk" title="stylos">στῦλος</span>, column), in +architecture, a term used to designate an intercolumniation of +three or four diameters.</p> + + +<hr class="art" /> +<p><span class="bold">DIATOMACEAE.<a name="ar17" id="ar17"></a></span> For the knowledge we possess of these +beautiful plants, so minute as to be undiscernible by our unaided +vision, we are indebted to the assistance of the microscope. It +was not till towards the close of the 18th century that the first +known forms of this group were discovered by O. F. Muller. And +so slow was the process of discovery in this field of scientific research +that in the course of half a century, when Agardh published +his <i>Systema algarum</i> in 1824, only forty-nine species included +under eight genera had been described. Since that time, however, +with modern microscopes and microscopic methods, eminent +botanists in all parts of the civilized world have studied these +minute plants, with the result that the number of known genera +and species has been greatly increased. Over 10,000 species of +diatoms have been described, and about 1200 species and +numerous varieties occur in the fresh waters and on the coasts +of Great Britain and Ireland. Rabenhorst, in the index to his +<i>Flora Europaea algarum</i> (1864) enumerated about 4000 forms +which had up to that time been discovered throughout the +continent of Europe.</p> + +<table class="nobctr" summary="Illustration"> +<tr><td class="figcenter" colspan="2"><img style="width:378px; height:328px" src="images/img169a.jpg" alt="" /></td></tr> +<tr><td class="caption sc" colspan="2">Fig. 1.</td></tr> +<tr><td class="tcl f90">A and B, <i>Melosira arenaria.</i></td> <td class="tcl f90">C-E, <i>Melosira varians.</i></td></tr> +<tr><td class="tcl f90">E, showing formation of auxospore.</td> <td class="tcl"> </td></tr></table> + +<table class="nobctr" style="float: left; width: 330px;" summary="Illustration"> +<tr><td class="figleft1"><img style="width:282px; height:29px" src="images/img169b.jpg" alt="" /></td></tr> +<tr><td class="caption sc">Fig. 2.</td></tr></table> + +<p>The diatoms are more commonly known among systematic +botanists as the Bacillarieae, particularly on the continent of +Europe, and although such an immense number of very diverse +forms are included in it, the group as a whole exhibits a remarkable +uniformity of structure. The Bacillarieae is one of the +large groups of Algae, placed by some in close proximity to the +Conjugatae and by others as an order of the Brown Algae (or +Phaeophyceae), but their characters are so distinctive and their +structure is so uniform as to warrant the separation of the diatoms +as a distinct class. The affinities of the group are doubtful.</p> + +<table class="nobctr" style="float: right; width: 410px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:85px; height:186px" src="images/img169c.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 3.</span>—<i>Podosphenia Lyngbyii.</i></td></tr> +<tr><td class="figright1"><img style="width:363px; height:93px" src="images/img169d.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 4.</span>—<i>Pleurosigma balticum.</i></td></tr></table> + +<p>The diatoms exhibit great +variety of form. While some +species are circular and more +or less disk-shaped, others are oval in outline. Some are +linear, as <i>Synedra Ulna</i> (fig. 2), others more or less crescentic; +others again are cuneate, as <i>Podosphenia Lyngbyii</i> +(fig. 3); some few have a sigmoid outline, as <i>Pleurosigma +balticum</i> (fig. 4); but the prevailing +forms are naviculoid, as in the large family +Naviculaceae, of which the genus <i>Navicula</i> +embraces upwards of 1000 species. They vary +also in their modes of growth,—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 <span class="spp">1</span>⁄<span class="suu">200</span>th of an +inch may be considered as beyond the medium size. Some few, +indeed, are much larger, but by far the greater proportion are of +very much smaller dimensions.</p> + +<table class="nobctr" summary="Illustration"> +<tr><td class="figcenter" colspan="2"><img style="width:411px; height:408px" src="images/img169e.jpg" alt="" /></td></tr> +<tr><td class="caption sc" colspan="2">Fig. 5.</td></tr> +<tr><td class="tcl f90">A-C, <i>Tetracyclus lacustris.</i></td> <td class="tcl f90">D and E, <i>Tabellaria fenestrata.</i></td></tr> +<tr><td class="tcl f90">F and G, <i>Tabellaria flocculosa.</i></td> <td class="tcl"> </td></tr></table> + +<p>Diatoms are unicellular plants distinguished from kindred +forms by the fact of having their soft vegetative part covered by +a siliceous case. Each individual is known as a frustule, and the +cell-wall consists of two similar valves nearly parallel to each +other, each valve being furnished with a rim (or connecting-band) +projecting from it at a right angle.</p> + +<p>One of these valves with its rim is slightly smaller than the +<span class="pagenum"><a name="page170" id="page170"></a>170</span> +other, the smaller fitting into the larger pretty much as a pill-box +fits into its cover. This peculiarity of structure affords ample +scope for the growth of the protoplasmic cell-contents, for as the +latter increase in volume the siliceous valves are pushed out, and +their corresponding siliceous rims become broader. The connecting-bands +although closely fitting their respective valves are +distinct from them, and together the two bands form the girdle.</p> + +<p>An individual diatom is usually described from two aspects, +one in which the surface of the valve is exposed to view—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.</p> + +<p>The protoplasmic contents of this siliceous box-like unicell are +very similar to the contents of many other algal cells. There is a +living protoplasmic layer or primordial utricle, connected either +by two broad bands or by a number of anastomosing threads with +a central mass of protoplasm in which the nucleus is embedded. +The greater part of the cavity of the cell is occupied by one +or several fluid vacuoles. The characteristic brown colour of +diatoms is due to the presence of chromatophores embedded in +the lining layer of protoplasm. In number and form these +chromatophores are variable. They contain chlorophyll, but the +green colour is masked by the presence of diatomin, a brown +pigment which resembles that which occurs in the Brown Algae +or Phaeophyceae. The chromatophores contain a variable +number of pyrenoids, colourless proteid bodies of a crystalloidal +character.</p> + +<p>One of the first phenomena which comes under the notice of +the observer is the extraordinary power of motion with which +the frustules are endowed. Some species move slowly backwards +and forwards in pretty much the same line, but in the case of +<i>Bacillaria paradoxa</i> the motion is very rapid, the frustules darting +through the water in a zigzag course. To account for this motion +various theories have been suggested, none of which appear to be +altogether satisfactory. There is little doubt that the movements +are connected with the raphe, and in some diatoms there is much +evidence to prove that they are due to an exudation of mucilage.</p> + +<p><i>Classification.</i>—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.</p> + +<p><i>Reproduction.</i>—In the Diatomaceae, as well as in the Desmidieae, +the ordinary mode of increase is by simple cell-division. The +cell-contents within the enclosure of the siliceous case separate +into two distinct masses. As these two daughter-masses become +more and more developed, the valves of the mother-cell are pushed +more and more widely apart. A new siliceous valve is secreted by +each of the two masses on the side opposite to the original valve, +the new valves being situated within the girdle of the original +frustule. When this process has been completed the girdle of +the mother frustule gives way, and two distinct frustules are +formed, the siliceous valves in each of these new frustules being +one of the valves of the mother-cell, and a newly formed valve +similar and more or less parallel to it.</p> + +<p>During the life of the plant this process of self-division is +continued with an almost incredible rapidity. On this subject +the observation of Professor William Smith, writing in 1853, is +worthy of special notice:—“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” +(<i>British Diatomaceae</i>, +vol. i. p. 25).</p> + +<table class="nobctr" style="float: right; width: 310px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:264px; height:445px" src="images/img170.jpg" alt="" /></td></tr> +<tr><td class="caption1"><span class="sc">Fig. 6.</span>—Formation of Auxospores.<br /> +A. <i>Navicula limosa.</i><br /> +B. <i>Achnanthes flexella.</i><br /> +C. <i>Navicula Amphisbaena.</i><br /> +D. <i>Navicula viridis.</i><br /></td></tr></table> + +<p>Individual diatoms when +once produced by cell-division +are incapable of any increase +in size owing to the rigidity of +their siliceous cell-walls, and +since the new valves are always +formed <i>within</i> the girdle of the +old ones, it would follow that +every succeeding generation is +reduced in size by the thickness +of the girdle. In some diatoms, +however, this is not strictly +true as daughter-cells are sometimes +produced of larger size +than the parent-cells. Thus, +the reduction in size of the +individuals is not always +proportionate to the number +of cell-divisions.</p> + +<p>On the diminution in size +having reached a limit in any +species, the maximum size is +regained by the formation of +an auxospore. There are five +known methods of reproduction by auxospores, but it is unnecessary +here to enter into details of these methods. Suffice it to +say that a normal auxospore is produced by the conjugation +of two parent-cells, its distinguishing feature being a rejuvenescence +accompanied by a marked increase in size. These +auxospores formed without conjugation are parthenogenetic.</p> + +<p><i>Mode of Preparation.</i>—The Diatomaceae are usually gathered +in small bottles, and special care should be taken to collect them +as free as possible from extraneous matter. A small portion having +been examined under the microscope, should the gathering be +thought worthy of preservation, some of the material is boiled in +acid for the purpose of cleaning it. The acids usually employed +are hydrochloric, nitric or sulphuric, according as circumstances +require. When the operator considers that by this process all +foreign matter has been eliminated, the residuum is put into a +precipitating jar of a conical shape, broader at the bottom than +at the top, and covered to the brim with filtered or distilled water. +When the diatoms have settled in the bottom of the jar, the +supernatant fluid is carefully removed by a syringe or some +similar instrument, so that the sediment be not disturbed. The +jar is again filled with water, and the process repeated till the acid +has been completely removed. It is desirable afterwards to boil +the sediment for a short time with supercarbonate of soda, the +alkali being removed in the same manner as the acid. A small +portion may then be placed with a pipette upon a slip of glass, +and, when the moisture has been thoroughly evaporated, the film +that remains should be covered with dilute Canada balsam, and, +a thin glass cover having been gently laid over the balsam, the +preparation should be laid aside for a short time to harden, and +then is ready for observation.</p> + +<p><i>General Remarks.</i>—Diatoms are most abundant in cold +latitudes, having a general preference for cold water. In the +pelagic waters of lakes and of the oceans they are often very +abundant, and in the cold waters of the Arctic and Antarctic +<span class="pagenum"><a name="page171" id="page171"></a>171</span> +Oceans they exist in prodigious numbers. They thus form a large +proportion of both the marine and the fresh-water plankton.</p> + +<p>Large numbers of fossil diatoms are known. Not only are +these minute plants assisting at the present time in the accumulation +of oceanic and lake deposits, but in former ages they have +been sufficiently active to give rise to considerable deposits of +diatomaceous earths. When the plant has fulfilled its natural +course the siliceous covering sinks to the bottom of the water in +which it had lived, and there forms part of the sediment. When +in the process of ages, as it has often happened, the accumulated +sediment has been hardened into solid rock, the siliceous frustules +of the diatoms remain unaltered, and, if the rock be disintegrated +by natural or artificial means, may be removed from the +enveloping matrix and subjected to examination under the +microscope. The forms found may from their character help in +some degree to illustrate the conditions under which the stratum +of rock had been originally deposited. These earths are generally +of a white or grey colour. Some of them are hard, but most +are soft and friable. Many of them are of economic importance, +being used as polishing powders (“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).</p> + +<p>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.</p> +<div class="author">(E. O’M.; G. S. W.*)</div> + + +<hr class="art" /> +<p><span class="bold">DIAULOS<a name="ar18" id="ar18"></a></span> (from Gr. <span class="grk" title="di-">δι-</span>, double, and <span class="grk" title="aulos">αὐλός</span>, pipe), in architecture, +the peristyle round the great court of the palaestra, +described by Vitruvius (v. II), which measured two stadia +(1200 ft.) in length; on the south side this peristyle had two +rows of columns, so that in stormy weather the rain might not +be driven into the inner part. The word was also used in ancient +Greece for a foot-race of twice the usual length.</p> + + +<hr class="art" /> +<p><span class="bold">DIAVOLO, FRA<a name="ar19" id="ar19"></a></span> (1771-1806), the popular name given to a +famous Italian brigand associated with the political revolutions +of southern Italy at the time of the French invasion. His real +name was Michele Pezza, and he was born of low parentage +at Itri; he had committed many murders and robberies in the +Terra di Lavoro, but by good luck combined with audacity he +always escaped capture, whence his name of Fra Diavolo, popular +superstition having invested him with the characters of a monk +and a demon, and it seems that at one time he actually was a +monk. When the kingdom of Naples was overrun by the French +and the Parthenopaean Republic established (1799), Cardinal +Ruffo, acting on behalf of the Bourbon king Ferdinand IV., who +had fled to Sicily, undertook the reconquest of the country, and +for this purpose he raised bands of peasants, gaol-birds, brigands, +&c., under the name of Sanfedisti or <i>bande della Santa Fede</i> +(“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.</p> + +<div class="condensed"> +<p>The best known account of Fra Diavolo is in Pietro Colletta’s +<i>Storia del reame di Napoli</i> (2nd ed., Florence, 1848); B. Amante’s +<i>Fra Diavolo e il suo tempo</i> (Florence, 1904) is an attempted rehabilitation; +but A. Luzio, whose account in <i>Profili e bozzetti storici</i> +(Milan, 1906) gives the latest information on the subject, has demolished +Amante’s arguments.</p> +</div> +<div class="author">(L. V.*)</div> + + +<hr class="art" /> +<p><span class="bold">DIAZ, NARCISSE VIRGILIO<a name="ar20" id="ar20"></a></span> (1808-1876), French painter, was +born in Bordeaux of Spanish parents, on the 25th of August 1808. +At first a figure-painter who indulged in strong colour, in his later +life Diaz became a painter of the forest and a “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,” +<span class="pagenum"><a name="page172" id="page172"></a>172</span> +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.</p> + +<div class="condensed"> +<p>See A. Hustin, <i>Les Artistes célèbres: Diaz</i> (Paris); D. Croal +Thomson, <i>The Barbizon School of Painters</i> (London, 1890); +J. W. Mollett, <i>Diaz</i> (London, 1890); J. Claretie, <i>Peintres et sculpteurs +contemporains: Diaz</i> (Paris, 1882); Albert Wolff, <i>La Capitale de +l’art: Narcisse Diaz</i> (Paris, 1886); Ph. Burty, <i>Maîtres et petit-maîtres: +N. Diaz</i> (Paris, 1877).</p> +</div> +<div class="author">(D. C. T.)</div> + + +<hr class="art" /> +<p><span class="bold">DIAZ, PORFIRIO<a name="ar21" id="ar21"></a></span> (1830-  ), president of the republic of +Mexico (<i>q.v.</i>), was born in the southern state of Oaxaca, on the +15th of September 1830. His father was an innkeeper in the little +capital of that province, and died three years after the birth of +Porfirio, leaving a family of seven children. The boy, who had +Indian blood in his veins, was educated for the Catholic Church, +a body having immense influence in the country at that time and +ordering and controlling revolutions by the strength of their filled +coffers. Arrived at the age of sixteen Porfirio Diaz threw off the +authority of the priests. Fired with enthusiasm by stories told by +the revolutionary soldiers continually passing through Oaxaca, +and hearing about the war with the United States, a year later +he determined to set out for Mexico city and join the National +Guard. There being no trains, and he being too poor to ride, he +walked the greater part of the 250 m., but arrived there too late, +as the treaty of Guadalupe-Hidalgo (1848) had been already +signed, and Texas finally ceded to the United States. Thus +his entering the army was for the time defeated. Thereupon he +returned to his native town and began studying law. He took +pupils in order to pay his own fees at the Law Institute, and help +his mother. At this time he came under the notice and influence +of Don Marcos Pérez and Benito Juárez, the first a judge, the +second a governor of the state of Oaxaca, and soon to become +famous as the deliverer of Mexico from the priesthood (War of +Reform). Diaz continued in his native town until 1854, when, +refusing to vote for the dictator, Santa Anna, he was stung by a +taunt of cowardice, and hastily pushing his way to the voting +place, he recorded his vote in favour of Alvarez and the revolutionists. +Orders were given for his arrest, but seizing a rifle and +mounting a horse he placed himself at the head of a few revolting +peasants, and from that moment became one of the leading +spirits in that long struggle for reform, known as the War of +Reform, which, under the leadership of Juárez, followed the overthrow +of Santa Anna. Promotion succeeded promotion, as Diaz +led his troops from victory to victory, amid great privations and +difficulties. He was made captain (1856), lieutenant-colonel and +colonel (1859), brigadier-general (1861), and general of division +for the army (1863). Closely following on civil war, political strife, +open rebellion and the great War of Reform, came the French +invasion of 1862, and the landing of the emperor Maximilian in +1864. From the moment the French disclosed their intentions of +settling in Mexico in 1862, Diaz took a prominent part against the +foreign invasion. He was twice seriously wounded, imprisoned on +three different occasions, had two hairbreadth escapes, and took +part in many daring engagements. So important a personage did +he become that both Marshal Bazaine and the emperor Maximilian +made overtures to him. At the time of Maximilian’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.</p> + +<p>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.</p> + +<div class="condensed"> +<p>See also Mrs Alec Tweedie, <i>Porfirio Diaz, Seven Times President of +Mexico</i> (1906), and <i>Mexico as I saw it</i> (1901); Dr Noll, <i>From Empire +to Republic</i> (1890); Lieut. Seaton Schroeder, <i>Fall of Maximilian’s +Empire</i> (New York, 1887); R. de Z. Enriquez, <i>P. Diaz</i> (1908); +and an article by Percy Martin in <i>Quarterly Review</i> for October +1909.</p> +</div> +<div class="author">(E. A. T.)</div> + + +<hr class="art" /> +<p><span class="bold">DIAZ DE NOVAES, BARTHOLOMEU<a name="ar22" id="ar22"></a></span> (fl. 1481-1500), +Portuguese explorer, discoverer of the Cape of Good Hope, was +probably a kinsman of João Diaz, one of the first Portuguese to +round Cape Bojador (1434), and of Diniz Diaz, the discoverer +of Cape Verde (1445). In 1478 a Bartholomeu Diaz, probably +identical with the discoverer, was exempted from certain +customary payments on ivory brought from the Guinea coast. +In 1481 he commanded one of the vessels sent by King John II. +under Diogo d’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; +<span class="pagenum"><a name="page173" id="page173"></a>173</span> +while on the southernmost projection of the modern Cape +peninsula, whose remarkable highlands (Table Mountain, &c.) +doubtless impressed him as the practical termination of the +continent, he bestowed, says De Barros, the name of Cape of +Storms (<i>Cabo Tormentoso</i>) in memory of the storms he had +experienced in these far southern waters; this name (in the +ordinary tradition) was changed by King John to that of Good +Hope (<i>Cabo da Boa Esperança</i>). Some excellent authorities, +however, make Diaz himself give the Cape its present name. +Hard by this “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.</p> + +<p>No record has yet been found of any adequate reward for Diaz: +on the contrary, when the great Indian expedition was being +prepared (for Vasco da Gama’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.</p> + +<div class="condensed"> +<p>See João de Barros, <i>Asia</i>, Dec. I. bk. iii. ch. 4; Duarte Pacheco +Pereira, <i>Esmeraldo de situ orbis</i>, esp. pp. 15, 90, 92, 94 and Raphael +Bastos’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 <i>Imago mundi</i>, 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 <i>Historia rerum ubique gestarum</i>, from the same hand; the +<i>Roteiro</i> of Vasco da Gama’s First Voyage (<i>Journal of the First Voyage +of ... Da Gama</i>, Hakluyt Soc., ed. E. G. Ravenstein (1898), pp. 9, +14); Ramusio, <i>Navigationi</i> (3rd ed.), vol. i. fol. 144; Castanheda, +<i>Historia</i>, bk. i. ch. 1; Galvano, <i>Descobrimentos (Discoveries of the +World)</i>, Hakluyt Soc. (1862), p. 77; E. G. Ravenstein, “Voyages of ... +Cão and ... Dias,” in <i>Geog. Journ.</i> (London, December 1900), vol. xvi. +pp. 638-655), an excellent critical summary in the light of the most +recent investigations of all the material. The fragments of Diaz’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 <i>Geog. Journ.</i> (December 1900, p. 642).</p> +</div> +<div class="author">(C. R. B.)</div> + + +<hr class="art" /> +<p><span class="bold">DIAZO COMPOUNDS,<a name="ar23" id="ar23"></a></span> in organic chemistry, compounds of the +type R·N·<span class="su">2</span>·X (where R = a hydrocarbon radical, and X = an +acid radical or a hydroxyl group). These compounds may be +divided into two classes, namely, the true diazo compounds, +characterized by the grouping −N = N−, and the diazonium +compounds, characterized by the grouping N ∶ N <.</p> + +<p>The diazonium compounds were first discovered by P. Griess +(<i>Ann.</i>, 1858, 106, pp. 123 et seq.), and may be prepared by the +action of nitrous fumes on a well-cooled solution of a salt of a +primary amine,</p> + +<p class="center">C<span class="su">6</span>H<span class="su">5</span>NH<span class="su">2</span>·HNO<span class="su">3</span> + HNO<span class="su">2</span> = C<span class="su">6</span>H<span class="su">5</span>N<span class="su">2</span>·NO<span class="su">3</span> + 2H<span class="su">2</span>O,</p> + +<p class="noind">or, as is more usually the case (since the diazonium salts +themselves are generally used only in aqueous solution) by the +addition of a well-cooled solution of potassium or sodium nitrite +to a well-cooled dilute acid solution of the primary amine. In +order to isolate the anhydrous diazonium salts, the method of +E. Knoevenagel (<i>Ber.</i>, 1890, 23, p. 2094) may be employed. In +this process the amine salt is dissolved in absolute alcohol and +diazotized by the addition of amyl nitrite; a crystalline precipitate +of the diazonium salt is formed on standing, or on the +addition of a small quantity of ether. The diazonium salts are +also formed by the action of zinc-dust and acids on the nitrates +of primary amines (R. Mohlau, <i>Ber.</i>, 1883, 16, p. 3080), and by the +action of hydroxylamine on nitrosobenzenes. They are colourless +crystalline solids which turn brown on exposure. They dissolve +easily in water, but only to a slight extent in alcohol and ether. +They are very unstable, exploding violently when heated or +rubbed. <i>Benzene diazonium nitrate</i>, C<span class="su">6</span>H<span class="su">5</span>N(NO<span class="su">3</span>)∶N, crystallizes +in long silky needles. The sulphate and chloride are similar, +but they are not quite so unstable as the nitrate. The bromide +may be prepared by the addition of bromine to an ethereal +solution of diazo-amino-benzene (tribromaniline remaining in +solution). By the addition of potassium bromide and bromine +water to diazonium salts they are converted into a <i>perbromide</i>, +<i>e.g.</i> C<span class="su">6</span>H<span class="su">5</span>N<span class="su">2</span>Br<span class="su">3</span>, which crystallizes in yellow plates.</p> + +<div class="condensed"> +<p>The diazonium salts are characterized by their great reactivity and +consequently are important reagents in synthetical processes, since by +their agency the amino group in a primary amine may be exchanged +for other elements or radicals. The chief reactions are as follows:—</p> + +<p>1. <i>Replacement of -NH<span class="su">2</span> by -OH</i>:—The amine is diazotized and +the aqueous solution of the diazonium salt is heated, nitrogen being +eliminated and a phenol formed.</p> + +<p>2. <i>Replacement of -NH<span class="su">2</span> by halogens and by the -CN and -CNO +groups</i>:—The diazonium salt is warmed with an acid solution of the +corresponding cuprous salt (T. Sandmeyer, <i>Ber.</i>, 1884, 17, p. 2650), or +with copper powder (L. Gattermann, Ber., 1890, 23, p. 1218; 1892, +25, p. 1074). In the case of iodine, the substitution is effected by +adding a warm solution of potassium iodide to the diazonium +solution, no copper or cuprous salt being necessary; whilst for +the production of nitriles a solution of potassium cuprous cyanide is +used. This reaction (the so-called “Sandmeyer” reaction) has been +investigated by A. Hantzsch and J. W. Blagden (<i>Ber.</i>, 1900, 33, p. 2544), +who consider that three simultaneous reactions occur, namely, the +formation of labile double salts which decompose in such a fashion +that the radical attached to the copper atom wanders to the aromatic +nucleus; a catalytic action, in which nitrogen is eliminated and the +acid radical attaches itself to the aromatic nucleus; and finally, the +formation of azo compounds.</p> + +<p>3. <i>Replacement of -NH<span class="su">2</span> by -NO<span class="su">2</span></i>:—A well-cooled concentrated +solution of potassium mercuric nitrate is added to a cooled +solution of benzene diazonium nitrate, when the crystalline salt +2C<span class="su">6</span>H<span class="su">5</span>N<span class="su">2</span>·NO<span class="su">3</span>, Hg(NO<span class="su">2</span>)<span class="su">2</span> is precipitated. On warming this with +copper powder, it gives a quantitative yield of nitrobenzene (A. +Hantzsch, <i>Ber.</i>, 1900, 33, p. 2551).</p> + +<p>4. <i>Replacement of -NH<span class="su">2</span> by hydrogen</i>:—This exchange is brought +about, in some cases, by boiling the diazonium salt with alcohol; +but I. Remsen and his pupils (<i>Amer. Chem. Journ.</i>, 1888, 9, pp. 389 +et seq.) have shown that the main product of this reaction is usually +a phenolic ether. This reaction has also been investigated by +A. Hantzsch and E. Jochem (<i>Ber.</i>, 1901, 34, p. 3337), who arrived at +the conclusion that the normal decomposition of diazonium salts +by alcohols results in the formation of phenolic ethers, but that an +increase in the molecular weight of the alcohol, or the accumulation +of negative groups in the aromatic nucleus, diminishes the yield of +the ether and increases the amount of the hydrocarbon formed. The +replacement is more readily brought about by the use of sodium +stannite (P. Friedlander, <i>Ber.</i>, 1889, 22, p. 587), or by the use of a +concentrated solution of hypophosphorous acid (J. Mai, <i>Ber.</i>, 1902, 35, +p. 162). A. Hantzsch (<i>Ber.</i>, 1896, 29, p. 947; 1898, 31, p. 1253) has shown +that the chlor- and brom- diazoniumthiocyanates, when dissolved in +alcohol containing a trace of hydrochloric acid, become converted +into the isomeric thiocyanbenzene diazonium chlorides and bromides. +This change only occurs when the halogen atom is in the ortho- or +para- position to the -N<span class="su">2</span>- group.</p> + +<p><i>Metallic Diazo Derivatives.</i>—Benzene diazonium chloride is decomposed +by silver oxide in aqueous solution, with the formation of +<i>benzene diazonium hydroxide</i>, C<span class="su">6</span>H<span class="su">5</span>·N(OH)∶N. This hydroxide, +although possessing powerful basic properties, is unstable in the +presence of alkalis and neutralizes them, being converted first into +the isomeric benzene-diazotic acid, the potassium salt of which is +obtained when the diazonium chloride is added to an excess of cold +concentrated potash (A. Hantzsch and W. B. Davidson, <i>Ber.</i>, 1898, +31, p. 1612). <i>Potassium benzene diazotate</i>, C<span class="su">6</span>H<span class="su">5</span>N<span class="su">2</span>·OK, crystallizes in +colourless silky needles. The free acid is not known; by the addition +of the potassium salt to 50% acetic acid at -20° C., the acid +anhydride, <i>benzene diazo oxide</i>, (C<span class="su">6</span>H<span class="su">5</span>N<span class="su">2</span>)<span class="su">2</span>O, is obtained as a very +unstable, yellow, insoluble compound, exploding spontaneously at +0° C. Strong acids convert it into a diazonium salt, and potash +converts it into the diazotate. On the constitution, of these anhydrides +see E. Bamberger, <i>Ber.</i>, 1896, 29, p. 446, and A. Hantzsch, <i>Ber.</i>, +1896, 29, p. 1067; 1898, 31, p. 636. By the addition of the diazonium +salts to a hot concentrated solution of a caustic alkali, C. Schraube +and C. Schmidt (<i>Ber.</i>, 1894, 27, p. 520) obtained an isomer of potassium +benzene diazotate. These <i>iso-</i>diazotates are formed much more +readily when the aromatic nucleus in the diazonium salt contains +negative radicals. <i>Potassium benzene iso-diazotate</i> resembles the +normal salt, but is more stable, and is more highly ionized. Carbon +dioxide converts it into <i>phenyl nitrosamine</i>, C<span class="su">6</span>H<span class="su">5</span>NH·NO +<span class="pagenum"><a name="page174" id="page174"></a>174</span> +(A. Hantzsch). The potassium salt of the iso-diazo hydroxide yields +on methylation a nitrogen ether, R·N(CH<span class="su">3</span>)·NO, whilst the silver salt +yields an oxygen ether, R·N:N·OCH<span class="su">3</span>. These results point to the +conclusion that the iso-diazo hydroxide is a tautomeric substance. +The same oxygen ether is formed by the methylation of the silver salt +of the normal diazo hydroxide; this points to the conclusion that the +isomeric hydroxides, corresponding with the silver derivatives, have +the same structural formulae, namely, R·N:N·OH. These oxygen +ethers contain the grouping -N:N-, since they couple very readily +with the phenols in alkaline solution to form azo compounds (<i>q.v.</i>) +(E. Bamberger, <i>Ber.</i>, 1895, 28, p. 225); they are also explosive.</p> + +<p>By oxidizing potassium benzene iso-diazotate with alkaline +potassium ferricyanide, E. Bamberger (<i>Ber.</i>, 1894, 27, p. 914) obtained +the <i>diazoic acids</i>, R·NH·NO<span class="su">2</span>, substances which he had previously +prepared by similarly oxidizing the diazonium salts, by dehydrating +the nitrates of primary amines with acetic anhydride, and by the +action of nitric anhydride on the primary amines. Concentrated +acids convert them into the isomeric nitro-amines, the -NO<span class="su">2</span> group +going into the nucleus in the ortho- or para- position to the amine +nitrogen; this appears to indicate that the compounds are nitramines. +They behave, however, as tautomeric substances, since +their alkali salts on methylation give nitrogen ethers, whilst their +silver salts yield oxygen ethers:</p> + +<div class="center"><img style="width:491px; height:52px" src="images/img174a.jpg" alt="" /></div> + +<p><i>Phenyl nitramine</i>, C<span class="su">6</span>H<span class="su">5</span>NH·NO<span class="su">2</span>, is a colourless crystalline solid, +which melts at 46° C. Sodium amalgam in alkaline solution reduces +it to phenylhydrazine.</p> + +<p><i>Constitution of the Diazo Compounds.</i>—P. Griess (<i>Ann.</i>, 1866, 137, +p. 39) considered that the diazo compounds were formed by the addition +of complex groupings of the type C<span class="su">6</span>H<span class="su">4</span>N<span class="su">2</span>- to the inorganic acids; +whilst A. Kekulé (<i>Zeit. f. Chemie</i>, 1866, 2, p. 308), on account of their +ready condensation to form azo compounds and their easy reduction +to hydrazines, assumed that they were substances of the type +R·N:N·Cl. The constitution of the diazonium group -N<span class="su">2</span>·X, may be +inferred from the following facts:—The group C<span class="su">6</span>H<span class="su">5</span>N<span class="su">2</span>- behaves in +many respects similarly to an alkali metal, and even more so to the +ammonium group, since it is capable of forming colourless neutral +salts with mineral acids, which in dilute aqueous solution are strongly +ionized, but do not show any trace of hydrolytic dissociation +(A. Hantzsch, <i>Ber.</i>, 1895, 28, p. 1734). Again, the diazonium chlorides +combine with platinic chloride to form difficultly soluble double +platinum salts, such as (C<span class="su">6</span>H<span class="su">5</span>N<span class="su">2</span>Cl)<span class="su">2</span>·PtCl<span class="su">4</span>; similar gold salts, +C<span class="su">6</span>H<span class="su">5</span>N<span class="su">2</span>Cl·AuCl<span class="su">3</span>, are known. Determinations of the electrical conductivity +of the diazonium chloride and nitrate also show that the +diazonium radical is strictly comparable with other quaternary +ammonium ions. For these reasons, one must assume the existence +of pentavalent nitrogen in the diazonium salts, in order to account +for their basic properties.</p> + +<p>The constitution of the isomeric diazo hydroxides has given rise +to much discussion. E. Bamberger (<i>Ber.</i>, 1895, 28, pp. 444 et seq.) and +C. W. Blomstrand (<i>Journ. prakt. Chem.</i>, 1896, 53, pp. 169 et seq.) hold +that the compounds are structurally different, the normal diazo-hydroxide +being a diazonium derivative of the type R·N(∶N)·OH. +The recent work of A. Hantzsch and his pupils seems to invalidate this +view (<i>Ber.</i>, 1894, 27, pp. 1702 et seq.; see also A. Hantzsch, <i>Die Diazoverbindungen</i>). +According to Hantzsch the isomeric diazo hydroxides +are structurally identical, and the differences in behaviour are due +to stereo-chemical relations, the isomerism being comparable with +that of the oximes (<i>q.v.</i>). On such a hypothesis, the relatively +unstable normal diazo hydroxides would be the <i>syn-</i>compounds, +since here the nitrogen atoms would be more easily eliminated, whilst +the stable iso-diazo derivatives would be the <i>anti-</i>compounds, thus:</p> + +<table class="nobctr" summary="Illustration"> +<tr><td class="figcenter" colspan="2"><img style="width:323px; height:39px" src="images/img174b.jpg" alt="" /></td></tr> +<tr><td class="caption">Normal hydroxide<br />(Syn-compound)</td> +<td class="caption">Iso hydroxide<br />(Anti-compound)</td></tr></table> + +<p class="noind">In support of this theory, Hantzsch has succeeded in isolating a series +of syn- and anti-diazo-cyanides and -sulphonates (<i>Ber.</i>, 1895, 28, p. 666; +1900, 33, p. 2161; 1901, 34, p. 4166). By diazotizing para-chloraniline +and adding a cold solution of potassium cyanide, a salt (melting at +29° C.) is obtained, which readily loses nitrogen, and forms para-chlorbenzonitrile +on the addition of copper powder. By dissolving +this diazocyanide in alcohol and reprecipitating it by water, it is +converted into the isomeric diazocyanide (melting at 105-106° C.), +which does not yield para-chlorbenzonitrile when treated with copper +powder. Similar results have been obtained by using diazotized +para-anisidine, a syn- and an anti- compound being formed, as well +as a third isomeric cyanide, obtained by evaporating para-methoxy-benzenediazonium +hydroxide in the presence of an excess of hydrocyanic +acid at ordinary temperatures. This salt is a colourless +crystalline substance of composition CH<span class="su">3</span>O·C<span class="su">6</span>H<span class="su">4</span>·N<span class="su">2</span>·CN·HCN·2H<span class="su">2</span>O, +and has the properties of a metallic salt; it is very soluble in water +and its solution is an electrolyte, whereas the solutions of the syn- +and anti- compounds are not electrolytes. The isolation of these +compounds is a powerful argument in favour of the Hantzsch +hypothesis which requires the existence of these three different types, +whilst the Bamberger-Blomstrand view only accounts for the formation +of two isomeric cyanides, namely, one of the normal diazonium +type and one of the iso-diazocyanide type.</p> + +<p>Benzene diazonium hydroxide, although a strong base, reacts with +the alkaline hydroxides to form salts with the evolution of heat, and +generally behaves as a weak acid. On mixing dilute solutions of the +diazonium hydroxide and the alkali together, it is found that the +molecular conductivity of the mixture is much less than the sum of +the two electrical conductivities of the solutions separately, from +which it follows that a portion of the ions present have changed to +the non-ionized condition. This behaviour is explained by considering +the non-ionized part of the diazonium hydroxide to exist in +solution in a hydrated form, the equation of equilibrium being:</p> + +<div class="center"><img style="width:347px; height:52px" src="images/img174c.jpg" alt="" /></div> + +<p class="noind">On adding the alkaline hydroxide to the solution, this hydrate is +supposed to lose water, yielding the syn-diazo hydroxide, which then +gives rise to a certain amount of the sodium salt (A. Hantzsch, <i>Ber.</i>, +1898, 31, p. 1612),</p> + +<div class="center"><img style="width:349px; height:58px" src="images/img174d.jpg" alt="" /></div> + +<p class="noind">This assumption also shows the relationship of the diazonium +hydroxides to other quaternary ammonium compounds, for most of +the quaternary ammonium hydroxides (except such as have the +nitrogen atom attached to four saturated hydrocarbon radicals) are +unstable, and readily pass over into compounds in which the hydroxyl +group is no longer attached to the amine nitrogen; thus the syn-diazo +hydroxides are to be regarded as pseudo-diazonium derivatives. +(A. Hantzsch, <i>Ber.</i>, 1899, 32, p. 3109; 1900, 33, p. 278.) It is generally +accepted that the iso-diazo hydroxides possess the oxime structure +R·N:N·OH.</p> + +<p>Hantzsch explains the characteristic reactions of the diazonium +compounds by the assumption that an addition compound is first +formed, which breaks down with the elimination of the hydride of +the acid radical, and the formation of an unstable syn-diazo compound, +which, in its turn, decomposes with evolution of nitrogen +(<i>Ber.</i>, 1897, 30, p. 2548; 1898, 31, p. 2053).</p> + +<div class="center"><img style="width:496px; height:52px" src="images/img174e.jpg" alt="" /></div> + +<p>J. Cain (<i>Jour. Chem. Soc.</i>, 1907, 91, p. 1049) suggested a quinonoid +formula for diazonium salts, which has been combated by Hantzsch +(<i>Ber.</i>, 1908, 41, pp. 3532 et seq.). G. T. Morgan and F. M. G. Micklethwaite +(<i>Jour. Chem. Soc.</i>, 1908, 93, p. 617; 1909, 95, p. 1319) have +pointed out that the salts may possess a dynamic formula, Cain’s +representing the middle stage, thus:</p> + +<div class="center"><img style="width:292px; height:78px" src="images/img174f.jpg" alt="" /></div> + +<p><i>Diazoamines.</i>—The diazoamines, R·N<span class="su">2</span>·NHR, may be prepared by +the action of the primary and secondary amines on the diazonium +salts, or by the action of nitrous acid on the free primary amine. In the +latter reaction it is assumed that the isodiazohydroxide first formed +is immediately attacked by a second molecule of the amine. They +are yellow crystalline solids, which do not unite with acids. Nitrous +acid converts them, in acid solution, into diazonium salts.</p> + +<p class="center">C<span class="su">6</span>H<span class="su">5</span>N<span class="su">2</span>·NHC<span class="su">6</span>H<span class="su">5</span> + 2HCl + HNO<span class="su">2</span> = 2C<span class="su">6</span>H<span class="su">5</span>N<span class="su">2</span>Cl + 2H<span class="su">2</span>O.</p> + +<p>They are readily converted into the isomeric aminoazo compounds, +either by standing in alcoholic solution, or by warming with a +mixture of the parent base and its hydrochloride; the diazo group +preferably going into the para-position to the amino group. When +the para-position is occupied, the diazo group takes the ortho-position. +H. Goldschmidt and R. U. Reinders (<i>Ber.</i>, 1896, 29, p. 1369, +1899) have shown that the transformation is a monomolecular +reaction, the velocity of transformation in moderately dilute solution +being independent of the concentration, but proportional to the +amount of the catalyst present (amine hydrochloride) and to the +temperature. It has also been shown that when different salts of the +amine are used, their catalytic influence varies in amount and is +almost proportional to their degree of ionization in aqueous solution. +Diazoaminobenzene, C<span class="su">6</span>H<span class="su">5</span>N<span class="su">2</span>·NHC<span class="su">6</span>H<span class="su">5</span>, crystallizes in golden yellow +laminae, which melt at 96° C. and explode at a slightly higher temperature. +It is readily soluble in alcohol, ether and benzene. Concentrated +hydrochloric acid converts it into chlorbenzene, aniline and +nitrogen. Zinc dust and alcoholic acetic acid reduce it to aniline +and phenylhydrazine.</p> + +<p><i>Diazoimino compounds</i>, R·N<span class="su">3</span>, may be regarded as derivatives of +azoimide (<i>q.v.</i>); they are formed by the action of ammonia on the +diazoperbromides, or by the action of hydroxylamine on the diazonium +sulphates (J. Mai, <i>Ber.</i>, 1892, 25, p. 372; T. Curtius, <i>Ber.</i>, 1893, 26, +p. 1271). Diazobenzeneimide, C<span class="su">6</span>H<span class="su">5</span>N<span class="su">3</span>, is a yellowish oil of stupefying +odour. It boils at 59° C. (12 mm.), and explodes when heated. +Concentrated hydrochloric acid decomposes it with formation of +<span class="pagenum"><a name="page175" id="page175"></a>175</span> +chloranilines and elimination of nitrogen, whilst on boiling with +sulphuric acid it is converted into aminophenols.</p> + +<p><i>Aliphatic Diazo Compounds.</i>—The esters of the aliphatic amino +acids may be diazotized in a manner similar to the primary aromatic +amines, a fact discovered by T. Curtius (<i>Ber.</i>, 1833, 16, p. 2230). The +first aliphatic diazo compound to be isolated was <i>diazoacetic ester</i>, +CH·N<span class="su">2</span>·CO<span class="su">2</span>C<span class="su">2</span>H<span class="su">5</span>, which is prepared by the action of potassium nitrite +on the ethyl ester of glycocoll hydrochloride, HCl·NH<span class="su">2</span>·CH<span class="su">2</span>·CO<span class="su">2</span>C<span class="su">2</span>H<span class="su">5</span> + + KNO<span class="su">2</span> = CHN<span class="su">2</span>·CO<span class="su">2</span>C<span class="su">2</span>H<span class="su">5</span> + KCl + 2H<span class="su">2</span>O. It is a yellowish oil which +melts at -24° C.; it boils at 143-144° C., but cannot be distilled safely +as it decomposes violently, giving nitrogen and ethyl fumarate. It +explodes in contact with concentrated sulphuric acid. On reduction +it yields ammonia and glycocoll (aminoacetic acid). When heated +with water it forms ethyl hydroxy-acetate; with alcohol it yields +ethyl ethoxyacetate. Halogen acids convert it into monohalogen +fatty acids, and the halogens themselves convert it into dihalogen +fatty acids. It unites with aldehydes to form esters of ketonic acids, +and with aniline yields anilido-acetic acid. It forms an addition +product with acrylic ester, which on heating loses nitrogen and leaves +trimethylene dicarboxylic ester. Concentrated ammonia converts +it into <i>diazoacetamide</i>, CHN<span class="su">2</span>·CONH<span class="su">2</span>, which crystallizes in golden +yellow plates which melt at 114° C. For other reactions see +<span class="sc"><a href="#artlinks">Hydrazine</a></span>. The constitution of the diazo fatty esters is inferred +from the fact that the two nitrogen atoms, when split off, are +replaced by two monovalent elements or groups, thus leading to +the formula <img style="width:146px; height:42px" src="images/img175.jpg" alt="" /> for diazoacetic ester.</p> + +<p><i>Diazosuccinic ester</i>, N<span class="su">2</span>·C(CO<span class="su">2</span>C<span class="su">2</span>H<span class="su">5</span>)<span class="su">2</span>, is similarly prepared by the +action of nitrous acid on the hydrochloride of aspartic ester. It is +decomposed by boiling water and yields fumaric ester.</p> + +<p><i>Diazomethane</i>, CH<span class="su">2</span>N<span class="su">2</span>, was first obtained in 1894 by H. v. Pechmann +(<i>Ber.</i>, 1894, 27, p. 1888; 1895, 28, p. 855). It is prepared by the +action of aqueous or alcoholic solutions of the caustic alkalis on +the nitroso-acidyl derivatives of methylamine (such, for example, +as <i>nitrosomethyl urethane</i>, NO·N(CH<span class="su">3</span>)·CO<span class="su">2</span>C<span class="su">2</span>H<span class="su">5</span>, which is formed on +passing nitrous fumes into an ethereal solution of methyl urethane). +E. Bamberger (<i>Ber.</i>, 1895, 28, p. 1682) regards it as the anhydride of +iso-diazomethane, CH<span class="su">3</span>·N:N·OH, and has prepared it by a method +similar to that used for the preparation of iso-diazobenzene. By the +action of bleaching powder on methylamine hydrochloride, there +is obtained a volatile liquid (<i>methyldichloramine</i>, CH<span class="su">3</span>·N·Cl<span class="su">2</span>), boiling +at 58-60° C., which explodes violently when heated with water, +yielding hydrocyanic acid (CH<span class="su">3</span>NCl<span class="su">2</span> = HCN + 2HCl). Well-dried +hydroxylamine hydrochloride is dissolved in methyl alcohol and +mixed with sodium methylate; a solution of methyldichloramine in +absolute ether is then added and an ethereal solution of diazomethane +distils over. Diazomethane is a yellow inodorous gas, very poisonous +and corrosive. It may be condensed to a liquid, which boils at about +0° C. It is a powerful methylating agent, reacting with water to form +methyl alcohol, and converting acetic acid into methylacetate, hydrochloric +acid into methyl chloride, hydrocyanic acid into acetonitrile, +and phenol into anisol, nitrogen being eliminated in each case. It is +reduced by sodium amalgam (in alcoholic solution) to <i>methylhydrazine</i>, +CH<span class="su">3</span>·NH·NH<span class="su">2</span>. It unites directly with acetylene to form pyrazole +(H. v. Pechmann, <i>Ber.</i>, 1898, 31, p. 2950) and with fumaric methyl +ester it forms pyrazolin dicarboxylic ester.</p> +<div class="author">(F. G. P.*)</div> + +<p>See G. T. Morgan, <i>B.A. Rep.</i>, 1902; J. Cain, <i>Diazo Compounds</i>, 1908.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DIAZOMATA<a name="ar24" id="ar24"></a></span> (Gr. <span class="grk" title="diazôma">διάζωμα</span>, a girdle), in architecture, the +landing places and passages which were carried round the semicircle +and separated the upper and lower tiers in a Greek theatre.</p> + + +<hr class="art" /> +<p><span class="bold">DIBDIN, CHARLES<a name="ar25" id="ar25"></a></span> (1745-1814), British musician, dramatist, +novelist, actor and song-writer, the son of a parish clerk, was born +at Southampton on or before the 4th of March 1745, and was the +youngest of a family of eighteen. His parents designing him for +the church, he was sent to Winchester; but his love of music +early diverted his thoughts from the clerical profession. After +receiving some instruction from the organist of Winchester +cathedral, where he was a chorister from 1756 to 1759, he went +to London at the age of fifteen. Here he was placed in a music +warehouse in Cheapside, but he soon abandoned this employment +to become a singing actor at Covent Garden. On the 21st of May +1762 his first work, an operetta entitled <i>The Shepherd’s Artifice</i>, +with words and music by himself, was produced at this theatre. +Other works followed, his reputation being firmly established +by the music to the play of <i>The Padlock</i>, produced at Drury Lane +under Garrick’s management in 1768, the composer himself taking +the part of Mungo with conspicuous success. He continued for +some years to be connected with Drury Lane, both as composer +and as actor, and produced during this period two of his best +known works, <i>The Waterman</i> (1774) and <i>The Quaker</i> (1775). A +quarrel with Garrick led to the termination of his engagement. +In <i>The Comic Mirror</i> he ridiculed prominent contemporary figures +through the medium of a puppet show. In 1782 he became joint +manager of the Royal circus, afterwards known as the Surrey +theatre. In three years he lost this position owing to a quarrel +with his partner. His opera <i>Liberty Hall</i>, containing the successful +songs “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 <i>The Oddities</i>, 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 +<i>Musical Tour through England</i> (1788), his <i>Professional Life</i>, an +autobiography published in 1803, a <i>History of the Stage</i> (1795), and +several smaller works, he wrote upwards of 1400 songs and about +thirty dramatic pieces. He also wrote the following novels:—<i>The +Devil</i> (1785); <i>Hannah Hewitt</i> (1792); <i>The Younger Brother</i> +(1793). An edition of his songs by G. Hogarth (1843) contains +a memoir of his life. His two sons, Charles and Thomas John +Dibdin (<i>q.v.</i>), whose works are often confused with those of their +father, were also popular dramatists in their day.</p> + + +<hr class="art" /> +<p><span class="bold">DIBDIN, THOMAS FROGNALL<a name="ar26" id="ar26"></a></span> (1776-1847), English bibliographer, +born at Calcutta in 1776, was the son of Thomas Dibdin, +the sailor brother of Charles Dibdin. His father and mother both +died on the way home to England in 1780, and Thomas was +brought up by a maternal uncle. He was educated at St John’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 <i>Introduction to the Knowledge of Editions of the +Classics</i> (1802), which brought him under the notice of the +third Earl Spencer, to whom he owed much important aid in +his bibliographical pursuits. The rich library at Althorp was +thrown open to him; he spent much of his time in it, and in +1814-1815 published his <i>Bibliotheca Spenceriana</i>. As the library +was not open to the general public, the information given in the +<i>Bibliotheca</i> was found very useful, but since its author was unable +even to read the characters in which the books he described were +written, the work was marred by the errors which more or less +characterize all his productions. This fault of inaccuracy however +was less obtrusive in his series of playful, discursive works in +the form of dialogues on his favourite subject, the first of which, +<i>Bibliomania</i> (1809), was republished with large additions in +1811, and was very popular, passing through numerous editions. +To the same class belonged the <i>Bibliographical Decameron</i>, a larger +work, which appeared in 1817. In 1810 he began the publication +of a new and much extended edition of Ames’s <i>Typographical +Antiquities</i>. The first volume was a great success, but the publication +was checked by the failure of the fourth volume, and was +<span class="pagenum"><a name="page176" id="page176"></a>176</span> +never completed. In 1818 Dibdin was commissioned by Earl +Spencer to purchase books for him on the continent, an expedition +described in his sumptuous <i>Bibliographical, Antiquarian and +Picturesque Tour in France and Germany</i> (1821). In 1824 he +made an ambitious venture in his <i>Library Companion, or the +Young Man’s Guide and Old Man’s Comfort in the Choice of a +Library</i>, intended to point out the best works in all departments +of literature. His culture was not broad enough, however, to +render him competent for the task, and the work was severely +criticized. For some years Dibdin gave himself up chiefly to +religious literature. He returned to bibliography in his +<i>Bibliophobia, or Remarks on the Present Depression in the State of +Literature and the Book Trade</i> (1832), and the same subject +furnishes the main interest of his <i>Reminiscences of a Literary Life</i> +(1836), and his <i>Bibliographical, Antiquarian and Picturesque +Tour in the Northern Counties of England and Scotland</i> (1838). +Dibdin was the originator and vice-president, Lord Spencer +being the president, of the Roxburghe Club, founded in 1812,—the +first of the numerous book clubs which have done such +service to literature.</p> + + +<hr class="art" /> +<p><span class="bold">DIBDIN, THOMAS JOHN<a name="ar27" id="ar27"></a></span> (1771-1841), English dramatist and +song-writer, son of Charles Dibdin, the song-writer, and of Mrs +Davenet, an actress whose real name was Harriet Pitt, was born +on the 21st of March 1771. He was apprenticed to his maternal +uncle, a London upholsterer, and later to William Rawlins, +afterwards sheriff of London. He summoned his second master +unsuccessfully for rough treatment; and after a few years of +service he ran away to join a company of country players. From +1789 to 1795 he played in all sorts of parts; he acted as scene +painter at Liverpool in 1791; and during this period he composed +more than 1000 songs. He made his first attempt as a +dramatic writer in <i>Something New</i>, followed by <i>The Mad Guardian</i> +in 1795. He returned to London in 1795, having married two +years before; and in the winter of 1798-1799 his <i>Jew and the +Doctor</i> was produced at Covent Garden. From this time he +contributed a very large number of comedies, operas, farces, &c., +to the public entertainment. Some of these brought immense +popularity to the writer and immense profits to the theatres. It is +stated that the pantomime of <i>Mother Goose</i> (1807) produced more +than £20,000 for the management at Covent Garden theatre, and +the <i>High-mettled Racer</i>, adapted as a pantomime from his father’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 <i>Reminiscences</i>; 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.</p> + + +<hr class="art" /> +<p><span class="bold">DIBRA<a name="ar28" id="ar28"></a></span> (Slav. <i>Debra</i>), the capital of a sanjak bearing the same +name, in the vilayet of Monastir, eastern Albania, Turkey. Pop. +(1900) about 15,000. Dibra occupies a valley enclosed by +mountains, and watered by the Tsrni Drin and Radika rivers, +which meet 3 m. S. It is a fortified city, and the only episcopal +see of the Bulgarian exarchate in Albania; most of the inhabitants +are Albanians, but there is a strong Bulgarian colony. The +local trade is almost entirely agricultural.</p> + + +<hr class="art" /> +<p><span class="bold">DIBRUGARH,<a name="ar29" id="ar29"></a></span> a town of British India, in the Lakhimpur +district of eastern Bengal and Assam, of which it is the headquarters, +situated on the Dibru river about 4 m. above its +confluence with the Brahmaputra. Pop. (1901) 11,227. It is the +terminus of steamer navigation on the Brahmaputra, and also of +a railway running to important coal-mines and petroleum wells, +which connects with the Assam-Bengal system. Large quantities +of coal and tea are exported. There are a military cantonment, +the headquarters of the volunteer corps known as the Assam +Valley Light Horse; a government high school, a training school +for masters; and an aided school for girls. In 1900 a medical +school for the province was established, out of a bequest left +by Brigade-Surgeon J. Berry-White, which is maintained by +the government, to train hospital assistants for the tea gardens. +The Williamson artisan school is entirely supported by an +endowment.</p> + + +<hr class="art" /> +<p><span class="bold">DICAEARCHUS,<a name="ar30" id="ar30"></a></span> of Messene in Sicily, Peripatetic philosopher +and pupil of Aristotle, historian, and geographer, flourished about +320 <span class="sc">b.c.</span> He was a friend of Theophrastus, to whom he dedicated +the majority of his works. Of his writings, which comprised +treatises on a great variety of subjects, only the titles and a few +fragments survive. The most important of them was his +<span class="grk" title="bios tês Hellados">βίος τῆς Έλλάδος</span> (<i>Life in Greece</i>), in which the moral, political +and social condition of the people was very fully discussed. In +his <i>Tripoliticos</i> he described the best form of government as a +mixture of monarchy, aristocracy and democracy, and illustrated +it by the example of Sparta. Among the philosophical works of +Dicaearchus may be mentioned the <i>Lesbiaci</i>, a dialogue in three +books, in which the author endeavours to prove that the soul is +mortal, to which he added a supplement called <i>Corinthiaci</i>. He +also wrote a <i>Description of the World</i> illustrated by maps, in +which was probably included his <i>Measurements of Mountains</i>. +A description of Greece (150 iambics, in C. Müller, <i>Frag. hist. +Graec</i>. i. 238-243) was formerly attributed to him, but, as the +initial letters of the first twenty-three lines show, was really +the work of Dionysius, son of Calliphon. Three considerable +fragments of a prose description of Greece (Müller, i. 97-110) +are now assigned to an unknown author named Heracleides. The +<i>De re publica</i> of Cicero is supposed to be founded on one of +Dicaearchus’s works.</p> + +<div class="condensed"> +<p>The best edition of the fragments is by M. Fuhr (1841), a work of +great learning; see also a dissertation by F. G. Osann, <i>Beiträge zur +röm. und griech. Litteratur</i>, ii. pp. 1-117 (1839); Pauly-Wissowa, +<i>Realencyclopädie der klass. Altertumswiss</i>. v. pt. 1 (1905).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DICE<a name="ar31" id="ar31"></a></span> (plural of die, O. Fr. <i>de</i>, derived from Lat. <i>dare</i>, to give), +small cubes of ivory, bone, wood or metal, used in gaming. The +six sides of a die are each marked with a different number of +incised dots in such a manner that the sum of the dots on any two +opposite sides shall be 7. Dice seem always to have been +employed, as is the case to-day, for gambling purposes, and they +are also used in such games as backgammon. There are many +methods of playing, from one to five dice being used, although +two or three are the ordinary numbers employed in Great Britain +and America. The dice are thrown upon a table or other smooth +surface either from the hand or from a receptacle called a dice-box, +the latter method having been in common use in Greece, Rome +and the Orient in ancient times. Dice-boxes have been made in +many shapes and of various materials, such as wood, leather, +agate, crystal, metal or paper. Many contain bars within to ensure +a proper agitation of the dice, and thus defeat trickery. Some, +formerly used in England, were employed with unmarked dice, +and allowed the cubes to fall through a kind of funnel upon a +board marked off into six equal parts numbered from 1 to 6. +It is a remarkable fact, that, wherever dice have been found, +whether in the tombs of ancient Egypt, of classic Greece, or of +the far East, they differ in no material respect from those in use +to-day, the elongated ones with rounded ends found in Roman +graves having been, not dice but <i>tali</i>, or knucklebones. Eight-sided +dice have comparatively lately been introduced in France +as aids to children in learning the multiplication table. The +teetotum, or spinning die, used in many modern games, was +known in ancient times in China and Japan. The increased +popularity of the more elaborate forms of gaming has resulted in +the decline of dicing. The usual method is to throw three times +with three dice. If one or more sixes or fives are thrown the first +time they may be reserved, the other throws being made with the +dice that are left. The object is to throw three sixes = 18 or as +near that number as possible, the highest throw winning, or, when +drinks are to be paid for, the lowest throw losing. (For other +methods of throwing consult the <i>Encyclopaedia of Indoor Games</i>, +by R. F. Foster, 1903.) The most popular form of pure gambling +with dice at the present day, particularly with the lower classes in +America, is <i>Craps</i>, or <i>Crap-Shooting</i>, a simple form of <i>Hazard</i>, of +French origin. Two dice are used. Each player puts up a stake +<span class="pagenum"><a name="page177" id="page177"></a>177</span> +and the first caster may cover any or all of the bets. He then +<i>shoots</i>, <i>i.e.</i> throws the dice from his open hand upon the table. +If the sum of the dice is 7 or 11 the throw is a <i>nick</i>, or <i>natural</i>, and +the caster wins all stakes. If the throw is either 2, 3 or 12 it is +a <i>crap</i>, and the caster loses all. If any other number is thrown +it is a <i>point</i>, and the caster continues until he throws the same +number again, in which case he wins, or a 7, in which case he +loses. The now practically obsolete game of Hazard was much +more complicated than <i>Craps</i>. (Consult <i>The Game of Hazard +Investigated</i>, by George Lowbut.) <i>Poker dice</i> are marked with ace, +king, queen, jack and ten-spot. Five are used and the object is, +in three throws, to make pairs, triplets, full hands or fours and +fives of a kind, five aces being the highest hand. Straights do +not count. In throwing to decide the payment of drinks the +usual method is called <i>horse and horse</i>, in which the highest +throws retire, leaving the two lowest to decide the loser by the +best two in three throws. Should each player win one throw +both are said to be <i>horse and horse</i>, and the next throw determines +the loser. The two last casters may also agree to <i>sudden death</i>, <i>i.e.</i> +a single throw. <i>Loaded dice</i>, <i>i.e.</i> dice weighted slightly on the side +of the lowest number, have been used by swindlers from the very +earliest times to the present day, a fact proved by countless +literary allusions. Modern dice are often rounded at the corners, +which are otherwise apt to wear off irregularly.</p> + +<p><i>History.</i>—Dice were probably evolved from knucklebones. +The antiquary Thomas Hyde, in his <i>Syntagma</i>, 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 (<i>Clio</i>) relates that the Lydians, +during a period of famine in the days of King Atys, invented dice, +knucklebones and indeed all other games except chess. The fact +that dice have been used throughout the Orient from time +immemorial, as has been proved by excavations from ancient +tombs, seems to point clearly to an Asiatic origin. Dicing is +mentioned as an Indian game in the <i>Rig-veda</i>. In its primitive +form knucklebones was essentially a game of skill, played by +women and children, while dice were used for gambling, and +it was doubtless the gambling spirit of the age which was +responsible for the derivative form of knucklebones, in which +four sides of the bones received different values, which were then +counted, like dice. Gambling with three, sometimes two, dice +(<span class="grk" title="kuboi">κύβοι</span>) was a very popular form of amusement in Greece, especially +with the upper classes, and was an almost invariable accompaniment +to the symposium, or drinking banquet. The dice were cast +from conical beakers, and the highest throw was three sixes, +called <i>Aphrodite</i>, while the lowest, three aces, was called the <i>dog</i>. +Both in Greece and Rome different modes of counting were in +vogue. Roman dice were called <i>tesserae</i> from the Greek word for +four, indicative of the four sides. The Romans were passionate +gamblers, especially in the luxurious days of the Empire, and +dicing was a favourite form, though it was forbidden except +during the Saturnalia. The emperor Augustus wrote in a letter +to Suetonius concerning a game that he had played with his +friends: “Whoever threw a <i>dog</i> or a six paid a <i>denarius</i> to the +bank for every die, and whoever threw a <i>Venus</i> (the highest) won +everything.” In the houses of the rich the dice-beakers were +of carved ivory and the dice of crystal inlaid with gold. Mark +Antony wasted his time at Alexandria with dicing, while, according +to Suetonius, the emperors Augustus, Nero and Claudius were +passionately fond of it, the last named having written a book on +the game. Caligula notoriously cheated at the game; Domitian +played it, and Commodus set apart special rooms in his palace +for it. The emperor Verus, adopted son of Antonine, is known +to have thrown dice whole nights together. Fashionable society +followed the lead of its emperors, and, in spite of the severity of +the laws, fortunes were squandered at the dicing table. Horace +derided the youth of the period, who wasted his time amid the +dangers of dicing instead of taming his charger and giving himself +up to the hardships of the chase. Throwing dice for money +was the cause of many special laws in Rome, according to one of +which no suit could be brought by a person who allowed gambling +in his house, even if he had been cheated or assaulted. Professional +gamblers were common, and some of their loaded dice +are preserved in museums. The common public-houses were the +resorts of gamblers, and a fresco is extant showing two quarrelling +dicers being ejected by the indignant host. Virgil, in the <i>Copa</i> +generally ascribed to him, characterizes the spirit of that age in +verse, which has been Englished as follows:—</p> + +<table class="reg" summary="poem"><tr><td> <div class="poemr"> +<p>“What ho! Bring dice and good wine!</p> +<p class="i2">Who cares for the morrow?</p> +<p class="i05">Live—so calls grinning Death—</p> +<p class="i2">Live, for I come to you soon!”</p> +</div> </td></tr></table> + +<p class="noind">That the barbarians were also given to gaming, whether or +not they learned it from their Roman conquerors, is proved by +Tacitus, who states that the Germans were passionately fond +of dicing, so much so, indeed, that, having lost everything, they +would even stake their personal liberty. Centuries later, during +the middle ages, dicing became the favourite pastime of the +knights, and both dicing schools (<i>scholae deciorum</i>) and gilds +of dicers existed. After the downfall of feudalism the famous +German mercenaries called <i>landsknechts</i> established a reputation +as the most notorious dicing gamblers of their time. Many of the +dice of the period were curiously carved in the images of men and +beasts. In France both knights and ladies were given to dicing, +which repeated legislation, including interdictions on the part of +St Louis in 1254 and 1256, did not abolish. In Japan, China, +Korea, India and other Asiatic countries dice have always been +popular and are so still.</p> + +<div class="condensed"> +<p>See Foster’s <i>Encyclopaedia of Indoor Games</i> (1903); Raymond’s +<i>Illustriertes Knobelbrevier</i> (Oranienburg, 1888); <i>Les Jeux des Anciens</i>, +by L. Becq de Fouquières (Paris, 1869); <i>Das Knöchelspiel der Alten</i>, +by Bolle (Wismar, 1886); <i>Die Spiele der Griechen und Römer</i>, by +W. Richter (Leipzig, 1887); Raymond’s <i>Alte und neue Würfelspiele</i>; +<i>Chinese Games with Dice</i>, by Stewart Culin (Philadelphia, 1889); +<i>Korean Games</i>, by Stewart Culin (Philadelphia, 1895).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DICETO, RALPH DE<a name="ar32" id="ar32"></a></span> (d. c. 1202), dean of St Paul’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 +<span class="pagenum"><a name="page178" id="page178"></a>178</span> +own times. His friendships with Richard Fitz Nigel, who succeeded +Foliot in the see of London, with William Longchamp, the +chancellor of Richard I., and with Walter of Coutances, the archbishop +of Rouen, gave him excellent opportunities of collecting +information. His two chief works, the <i>Abbreviationes Chronicorum</i> +and the <i>Ymagines Historiarum</i>, cover the history of the +world from the birth of Christ to the year 1202. The former, +which ends in 1147, is a work of learning and industry, but +almost entirely based upon extant sources. The latter, beginning +as a compilation from Robert de Monte and the letters of +Foliot, becomes an original authority about 1172, and a contemporary +record about 1181. In precision and fulness of detail the +<i>Ymagines</i> are inferior to the chronicles of the so-called Benedict +and of Hoveden. Though an annalist, Diceto is careless in his +chronology; and the documents which he incorporates, while +often important, are selected on no principle. He has little sense +of style; but displays considerable insight when he ventures to +discuss a political situation. For this reason, and on account of +the details with which they supplement the more important +chronicles of the period, the <i>Ymagines</i> are a valuable though a +secondary source.</p> + +<div class="condensed"> +<p>See W. Stubbs’ edition of the <i>Historical Works</i> of Diceto (Rolls ed. +1876, 2 vols.), and especially the introduction. The second volume +contains minor works which are the barest compendia of facts taken +from well-known sources. Diceto’s fragmentary Domesday of the +capitular estates has been edited by Archdeacon Hale in <i>The Domesday +of St Paul’s</i>, pp. 109 ff. (Camden Society, 1858).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DICEY, EDWARD<a name="ar33" id="ar33"></a></span> (1832-  ), 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 <i>Daily +Telegraph</i> as leader writer and then as special correspondent, and +after a short spell in 1870 as editor of the <i>Daily News</i> he became +editor of the <i>Observer</i>, a position which he held until 1889. Of +his many books on foreign affairs perhaps the most important are +his <i>England and Egypt</i> (1884), <i>Bulgaria, the Peasant State</i> (1895), +<i>The Story of the Khedivate</i> (1902), and <i>The Egypt of the Future</i> +(1907). He was created C.B. in 1886.</p> + +<p>His brother <span class="sc">Albert Venn Dicey</span> (b. 1835), English jurist, +was educated at Balliol College, Oxford, where he took a first +class in the classical schools in 1858. He was called to the bar at +the Inner Temple in 1863. He held fellowships successively +at Balliol, Trinity and All Souls’, and from 1882 to 1909 was +Vinerian professor of law. He became Q.C. in 1890. His chief +works are the <i>Introduction to the Study of the Law of the Constitution</i> +(1885, 6th ed. 1902), which ranks as a standard work on +the subject; <i>England’s Case against Home Rule</i> (1886); <i>A Digest +of the Law of England with Reference to the Conflict of Laws</i> (1896), +and <i>Lectures on the Relation between Law and Public Opinion in +England during the 19th century</i> (1905).</p> + + +<hr class="art" /> +<p><span class="bold">DICHOTOMY<a name="ar34" id="ar34"></a></span> (Gr. <span class="grk" title="dicha">δίχα</span>, apart, <span class="grk" title="temnein">τέμνειν</span>, to cut), literally a +cutting asunder, the technical term for a form of logical division, +consisting in the separation of a genus into two species, one of +which has and the other has not, a certain quality or attribute. +Thus men may be thus divided into white men, and men who are +not white; each of these may be subdivided similarly. On the +principle of contradiction this division is both exhaustive and +exclusive; there can be no overlapping, and no members of the +original genus or the lower groups are omitted. This method of +classification, though formally accurate, has slight value in the +exact sciences, partly because at every step one of the two groups +is merely negatively characterized and therefore incapable of real +subdivision; it is useful, however, in setting forth clearly the +gradual descent from the most inclusive genus (<i>summum genus</i>) +through species to the lowest class (<i>infima species</i>), which is +divisible only into individual persons or things. (See further +<span class="sc"><a href="#artlinks">Division</a></span>.) In astronomy the term is used for the aspect of the +moon or of a planet when apparently half illuminated, so that its +disk has the form of a semicircle.</p> + + +<hr class="art" /> +<p><span class="bold">DICK, ROBERT<a name="ar35" id="ar35"></a></span> (1811-1866), Scottish geologist and botanist, +was born at Tullibody, in Clackmannanshire, in January 1811. +His father was an officer of excise. At the age of thirteen, after +receiving a good elementary education at the parish school, +Robert Dick was apprenticed to a baker, and served for three +years. In these early days he became interested in wild flowers—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 <i>Old Red Sandstone</i> (published in 1841), and +he began systematically to collect with hammer and chisel the +fossils from the Caithness flags. In 1845 he found remains of +<i>Holoptychius</i> and forwarded specimens to Hugh Miller, and he +continued to send the best of his fossil fishes to that geologist, and +to others after the death of Miller. In this way he largely contributed +to the progress of geological knowledge, although he himself +published nothing and was ever averse from publicity. His +herbarium, which consisted of about 200 folios of mosses, ferns +and flowering plants “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.</p> + +<div class="condensed"> +<p>See <i>Robert Dick, Baker of Thurso, Geologist and Botanist</i>, by +Samuel Smiles (1878).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DICK, THOMAS<a name="ar36" id="ar36"></a></span> (1774-1857), Scottish writer on astronomy, +was born at Dundee on the 24th of November 1774. The +appearance of a brilliant meteor inspired him, when in his ninth +year, with a passion for astronomy; and at the age of sixteen he +forsook the loom, and supported himself by teaching. In 1794 +he entered the university of Edinburgh, and set up a school on the +termination of his course; then, in 1801, took out a licence to +preach, and officiated for some years as probationer in the +United Presbyterian church. From about 1807 to 1817 he taught +in the secession school at Methven in Perthshire, and during the +ensuing decade in that of Perth, where he composed his first +substantive book, <i>The Christian Philosopher</i> (1823, 8th ed. 1842). +Its success determined his vocation as an author; he built +himself, in 1827, a cottage at Broughty Ferry, near Dundee, and +devoted himself wholly to literary and scientific pursuits. They +proved, however, owing to his unpractical turn of mind, but +slightly remunerative, and he was in 1847 relieved from actual +poverty by a crown pension of £50 a year, eked out by a local +subscription. He died on the 29th of July 1857. His best-known +works are: <i>Celestial Scenery</i> (1837), <i>The Sidereal Heavens</i> +(1840), and <i>The Practical Astronomer</i> (1845), in which is contained +(p. 204) a remarkable forecast of the powers and uses of +celestial photography. Written with competent knowledge, and +in an agreeable style, they obtained deserved and widespread +popularity.</p> + +<div class="condensed"> +<p>See R. Chambers’s <i>Eminent Scotsmen</i> (ed. 1868); <i>Monthly Notices +Roy. Astr. Society</i>, xviii. 98; <i>Athenaeum</i> (1857), p. 1008.</p> +</div> +<div class="author">(A. M. C.)</div> + + +<hr class="art" /> +<p><span class="bold">DICKENS, CHARLES JOHN HUFFAM<a name="ar37" id="ar37"></a></span> (1812-1870), English +novelist, was born on the 7th of February 1812 at a house in +the Mile End Terrace, Commercial Road, Landport (Portsea)—a +house which was opened as a Dickens Museum on 22nd July 1904. +His father John Dickens (d. 1851), a clerk in the navy-pay office +<span class="pagenum"><a name="page179" id="page179"></a>179</span> +on a salary of £80 a year, and stationed for the time being at +Portsmouth, had married in 1809 Elizabeth, daughter of Thomas +Barrow, and she bore him a family of eight children, Charles +being the second. In the winter of 1814 the family moved +from Portsea in the snow, as he remembered, to London, and +lodged for a time near the Middlesex hospital. The country +of the novelist’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 <i>Roderick Random</i>, <i>Peregrine +Pickle</i>, <i>Humphry Clinker</i>, <i>Tom Jones</i>, <i>The Vicar of Wakefield</i>, +<i>Don Quixote</i>, <i>Gil Blas</i> and <i>Robinson Crusoe</i>. The story of how he +played at the characters in these books and sustained his idea of +Roderick Random for a month at a stretch is picturesquely told +in <i>David Copperfield</i>. Here as well as in his first and last books +and in what many regard as his best, <i>Great Expectations</i>, Dickens +returns with unabated fondness and mastery to the surroundings +of his childhood. From seven to nine years he was at a +school kept in Clover Lane, Chatham, by a Baptist minister +named William Giles, who gave him Goldsmith’s <i>Bee</i> as a keepsake +when the call to Somerset House necessitated the removal +of the family from Rochester to a shabby house in Bayham Street, +Camden Town. At the very moment when a consciousness of +capacity was beginning to plump his youthful ambitions, the +whole flattering dream vanished and left not a rack behind. +Happiness and Chatham had been left behind together, and +Charles was about to enter a school far sterner and also far +more instructive than that in Clover Lane. The family income +had been first decreased and then mortgaged; the creditors of +the “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.</p> + +<p>Already consumed by ambition, proud, sensitive and on his +dignity to an extent not uncommon among boys of talent, he felt +his position keenly, and in later years worked himself up into a +passion of self-pity in connexion with the “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 +<i>True Sun</i>. Later he was reporter to the <i>Mirror of Parliament</i> +and then to the <i>Morning Chronicle</i>. Several of his earliest letters +are concerned with his exploits as a reporter, and allude to the +experiences he had, travelling fifteen miles an hour and being +upset in almost every description of known vehicle in various parts +of Britain between 1831 and 1836. The family was now living in +Bentwick Street, Manchester Square, but John Dickens was +still no infrequent inmate of the sponging-houses. With all the +accessories of these places of entertainment his son had grown to +be excessively familiar. Writing about 1832 to his school friend +Tom Mitton, Dickens tells him that his father has been arrested +at the suit of a wine firm, and begs him go over to Cursitor Street +and see what can be done. On another occasion of a paternal +disappearance he observes: “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.</p> + +<p>In the meanwhile, however, he had commenced author in a +more creative sense by penning some sketches of contemporary +London life, such as he had attempted in his school days in imitation +of the sketches published in the <i>London</i> and other magazines +of that day. The first of these appeared in the December number +of the <i>Old Monthly Magazine</i> for 1833. By the following August, +when the signature “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 <i>Morning +Chronicle</i> was raised, owing to the intervention of one of its chiefs, +George Hogarth, the father of (in addition to six sons) eight +charming daughters, to one of whom, Catherine, Charles was +engaged to be married before the year was out. Clearly as his +career now seemed designated, he was at this time or a little before +it coquetting very seriously with the stage: but circumstances +were rapidly to determine another stage in his career. A year +before Queen Victoria’s accession appeared in two volumes +<i>Sketches by Boz</i>, <i>Illustrative of Everyday Life and Everyday +People</i>. The book came from a prentice hand, but like the +little tract on the Puritan abuse of the Sabbath entitled “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 +<span class="pagenum"><a name="page180" id="page180"></a>180</span> +river. Here comes a satirical picture of parliament, fun made of +cheap snobbery, a rap on the knuckles of sectarianism. And what +could be more prophetic than the title of the opening chapter—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.</p> + +<p>A few days before the marriage, just two months after the +appearance of the <i>Sketches</i>, the first part of <i>The Posthumous Papers +of the Pickwick Club</i> was announced. One of the chief vogues of +the day was the issue of humorous, sporting or anecdotal novels +in parts, with plates, and some of the best talent of the day, represented +by Ainsworth, Bulwer, Marryat, Maxwell, Egan, Hook +and Surtees, had been pressed into this kind of enterprise. The +publishers of the day had not been slow to perceive Dickens’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 +<i>Boz</i>). The offer was too tempting for Dickens to refuse, but he +changed the idea from a club of Cockney sportsmen to that of a +club of eccentric peripatetics, on the sensible grounds, first that +sporting sketches were stale, and, secondly, that he knew nothing +worth speaking of about sport. The first seven pictures appeared +with the signature of Seymour and the letterpress of Dickens. +Before the eighth picture appeared Seymour had blown his brains +out. After a brief interval of Buss, Dickens obtained the services +of Hablot K. Browne, known to all as “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.”</p> + +<p>When <i>The Pickwick Papers</i> 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 <i>Quarterly +Review</i> almost ruefully admits, the whole reading world was +talking about the Pickwickians. The names of Winkle, Wardle, +Weller, Jingle, Snodgrass, Dodson & 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 <i>Pelham</i> and <i>Crichton</i> was scraped down +to make way for that of the new popular favourite on the omnibuses. +A new and original genius had suddenly sprung up, there +was no denying it, even though, as the <i>Quarterly</i> concluded, “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.</p> + +<p>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 <i>Life</i> (750 pages in the <i>Letters</i> edited by +his daughter and sister-in-law) is occupied with transcription from +private correspondence, and never a line of this but is thoroughly +worthy of print and preservation. If he makes a tour in any +part of the British Isles, he writes a full description of all he +sees, of everything that happens, and writes it with such gusto, +such mirth, such strokes of fine picturing, as appear in no other +private letters ever given to the public. Naturally buoyant in +all circumstances, a holiday gave him the exhilaration of a schoolboy. +See how he writes from Cornwall, when on a trip with two +or three friends, in 1843. “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.”</p> + +<p>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 <i>London Magazine</i>. +He taught himself to speak French and Italian, but he could have +read little in any language. His ideas were those of the inchoate +and insular liberalism of the ’thirties. His unique force in +literature he was to owe to no supreme artistic or intellectual +quality, but almost entirely to his inordinate gift of observation, +his sympathy with the humble, his power over the emotions +and his incomparable endowment of unalloyed human fun. To +<span class="pagenum"><a name="page181" id="page181"></a>181</span> +contemporaries he was not so much a man as an institution, at +the very mention of whose name faces were puckered with grins +or wreathed in smiles. To many his work was a revelation, the +revelation of a new world and one far better than their own. +And his influence went further than this in the direction of +revolution or revival. It gave what were then universally referred +to as “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 <i>Chuzzlewit</i>. “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.</p> + +<p>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 <i>Oliver Twist</i>, the story of a +parish orphan, the nucleus of which had already seen the light +in his <i>Sketches</i>. The early scenes are of a harrowing reality, +despite the germ of forced pathos which the observant reader may +detect in the pitiful parting between Oliver and little Dick; but +what will strike every reader at once in this book is the directness +and power of the English style, so nervous and unadorned: +from its unmistakable clearness and vigour Dickens was to travel +far as time went on. But the full effect of the old simplicity is +felt in such masterpieces of description as the drive of Oliver and +Sikes to Chertsey, the condemned-cell ecstasy of Fagin, or the +unforgettable first encounter between Oliver and the Artful +Dodger. Before November 1837 had ended, Charles Dickens +entered on an engagement to write a successor to <i>Pickwick</i> on +similar lines of publication. <i>Oliver Twist</i> was then in mid-career; +a <i>Life of Grimaldi</i> and <i>Barnaby Rudge</i> were already covenanted +for. Dickens forged ahead with the new tale of <i>Nicholas Nickleby</i> +and was justified by the results, for its sale far surpassed even +that of <i>Pickwick</i>. As a conception it is one of his weakest. An +unmistakably 18th-century character pervades it. Some of the +vignettes are among the most piquant and besetting ever written. +Large parts of it are totally unobserved conventional melodrama; +but the Portsmouth Theatre and Dotheboys Hall and +Mrs Nickleby (based to some extent, it is thought, upon Miss +Bates in <i>Emma</i>, but also upon the author’s Mamma) live for ever +as Dickens conceived them in the pages of <i>Nicholas Nickleby</i>.</p> + +<p>Having got rid of <i>Nicholas Nickleby</i> and resigned his editorship +of <i>Bentley’s Miscellany</i>, in which <i>Oliver Twist</i> originally +appeared, Dickens conceived the idea of a weekly periodical to +be issued as <i>Master Humphrey’s Clock</i>, to comprise short stories, +essays and miscellaneous papers, after the model of Addison’s +<i>Spectator</i>. 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 +<i>The Old Curiosity Shop</i>, which was continued with slight interruptions, +and followed by <i>Barnaby Rudge</i>. For the first time +we find Dickens obsessed by a highly complicated plot. The +tonality achieved in <i>The Old Curiosity Shop</i> surpassed anything +he had attempted in this difficult vein, while the rich humour of +Dick Swiveller and the Marchioness, and the vivid portraiture +of the wandering Bohemians, attain the very highest level of +Dickensian drollery; but in the lamentable tale of Little Nell +(though Landor and Jeffrey thought the character-drawing of +this infant comparable with that of Cordelia), it is generally +admitted that he committed an indecent assault upon the +emotions by exhibiting a veritable monster of piety and long-suffering +in a child of tender years. In <i>Barnaby Rudge</i> he was +manifestly affected by the influence of Scott, whose achievements +he always regarded with a touching veneration. The plot, again, +is of the utmost complexity, and Edgar Allan Poe (who predicted +the conclusion) must be one of the few persons who ever really +mastered it. But few of Dickens’s books are written in a more +admirable style.</p> + +<p><i>Master Humphrey’s Clock</i> 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 +(<i>American Notes</i>). 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. <i>Martin Chuzzlewit</i> (1843-1844) is important +as closing his great character period. His <i>sève originale</i>, as the +French would say, was by this time to a considerable extent +exhausted, and he had to depend more upon artistic elaboration, +upon satires, upon <i>tours de force</i> of description, upon romantic +and ingenious contrivances. But all these resources combined +proved unequal to his powers as an original observer of popular +types, until he reinforced himself by autobiographic reminiscence, +as in <i>David Copperfield</i> and <i>Great Expectations</i>, the two great +books remaining to his later career.</p> + +<p>After these two masterpieces and the three wonderful books +with which he made his début, we are inclined to rank <i>Chuzzlewit</i>. +Nothing in Dickens is more admirably seen and presented than +Todgers’s, a bit of London particular cut out with a knife. Mr +<span class="pagenum"><a name="page182" id="page182"></a>182</span> +Pecksniff and Mrs Gamp, Betsy Prig and “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.</p> + +<p>At the close of 1843, to pay outstanding debts of his now +lavish housekeeping, he wrote that pioneer of Christmas numbers, +that national benefit as Thackeray called it, <i>A Christmas Carol</i>. +It failed to realize his pecuniary anticipations, and Dickens +resolved upon a drastic policy of retrenchment and reform. +He would save expense by living abroad and would punish his +publishers by withdrawing his custom from them, at least for a +time. Like everything else upon which he ever determined, this +resolution was carried out with the greatest possible precision and +despatch. In June 1844 he set out for Marseilles with his now +rapidly increasing family (the journey cost him £200). In a villa +on the outskirts of Genoa he wrote <i>The Chimes</i>, which, during a +brief excursion to London before Christmas, he read to a select +circle of friends (the germ of his subsequent lecture-audiences), +including Forster, Carlyle, Stanfield, Dyce, Maclise and Jerrold. +He was again in London in 1845, enjoying his favourite diversion +of private theatricals; and in January 1846 he experimented +briefly as the editor of a London morning paper—the <i>Daily +News</i>. By early spring he was back at Lausanne, writing his +customary vivid letters to his friends, craving as usual for +London streets, commencing <i>Dombey and Son</i>, and walking his +fourteen miles daily. The success of <i>Dombey and Son</i> completely +rehabilitated the master’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 <i>David Copperfield</i>.”</p> + +<p>“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 +<i>Pendennis</i> stands to Thackeray. As in that book, too, the earlier +portions are the best. They gained in intensity by the autobiographical +form into which they are thrown; as Thackeray +observed, there was no writing against such power. The tragedy +of Emily and the character of Rosa Dartle are stagey and unreal; +Uriah Heep is bad art; Agnes, again, is far less convincing +as a consolation than Dickens would have us believe; but these +are more than compensated by the wonderful realization +of early boyhood in the book, by the picture of Mr Creakle’s +school, the Peggottys, the inimitable Mr Micawber, Betsy Trotwood +and that monument of selfish misery, Mrs Gummidge.</p> + +<p>At the end of March 1850 commenced the new twopenny +weekly called <i>Household Words</i>, which Dickens planned to form +a direct means of communication between himself and his +readers, and as a means of collecting around him and encouraging +the talents of the younger generation. No one was better qualified +than he for this work, whether we consider his complete +freedom from literary jealousy or his magical gift of inspiring +young authors. Following the somewhat dreary and incoherent +<i>Bleak House</i> of 1852, <i>Hard Times</i> (1854)—an anti-Manchester +School tract, which Ruskin regarded as Dickens’s best work—was +the first long story written for <i>Household Words</i>. 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, <i>Little Dorrit</i> (1855-1857). In spite of the exquisite +comedy of the master of the Marshalsea and the final tragedy +of the central figure, <i>Little Dorrit</i> is sadly deficient in the old +vitality, the humour is often a mock reality, and the repetition +of comic catch-words and overstrung similes and metaphors is +such as to affect the reader with nervous irritation. The plot +and characters ruin each other in this amorphous production. +The <i>Tale of Two Cities</i>, commenced in <i>All the Year Round</i> (the +successor of <i>Household Words</i>) in 1859, is much better: the main +characters are powerful, the story genuinely tragic, and the +atmosphere lurid; but enormous labour was everywhere expended +upon the construction of stylistic ornament.</p> + +<p>The <i>Tale of Two Cities</i> was followed by two finer efforts at +atmospheric delineation, the best things he ever did of this kind: +<i>Great Expectations</i> (1861), over which there broods the mournful +impression of the foggy marshes of the Lower Thames; and <i>Our +Mutual Friend</i> (1864-1865), in which the ooze and mud and +slime of Rotherhithe, its boatmen and loafers, are made to pervade +the whole book with cumulative effect. The general effect +produced by the stories is, however, very different. In the first +case, the foreground was supplied by autobiographical material +of the most vivid interest, and the lucidity of the creative impulse +impelled him to write upon this occasion with the old simplicity, +though with an added power. Nothing therefore, in the whole +range of Dickens surpassed the early chapters of <i>Great Expectations</i> +in perfection of technique or in mastery of all the resources +of the novelist’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?”</p> + +<p>In November 1867 Dickens made a second expedition to +America, leaving all the writing that he was ever to complete behind +him. He was to make a round sum of money, enough to free +him from all embarrassments, by a long series of exhausting readings, +commencing at the Tremont Temple, Boston, on the 2nd of +December. The strain of Dickens’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, <i>The Mystery of Edwin Drood</i>, was the chief occupation. +It hardly promised to become a masterpiece (Longfellow’s +opinion) as did Thackeray’s <i>Denis Duval</i>, but contained much fine +descriptive technique, grouped round a scene of which Dickens +had an unrivalled sympathetic knowledge.</p> + +<p>In March and April 1870 Dickens, as was his wont, was mixing +in the best society; he dined with the prince at Lord Houghton’s +and was twice at court, once at a long deferred private interview +with the queen, who had given him a presentation copy of her +<i>Leaves from a Journal of our Life in the Highlands</i> with the +inscription “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. +<span class="pagenum"><a name="page183" id="page183"></a>183</span> +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 <i>Edwin Drood</i> 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 <i>Graphic</i> 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” (<i>The Times</i>). In +his will he enjoined his friends to erect no monument in his +honour, and directed his name and dates only to be inscribed on +his tomb, adding this proud provision, “I rest my claim to +the remembrance of my country on my published works.”</p> + +<p>Dickens had no artistic ideals worth speaking about. The +sympathy of his readers was the one thing he cared about and, +like Cobbett, he went straight for it through the avenue of the +emotions. In personality, intensity and range of creative genius +he can hardly be said to have any modern rival. His creations +live, move and have their being about us constantly, like those +of Homer, Virgil, Chaucer, Rabelais, Cervantes, Shakespeare, +Bunyan, Molière and Sir Walter Scott. As to the books themselves, +the backgrounds on which these mighty figures are projected, +they are manifestly too vast, too chaotic and too unequal +ever to become classics. Like most of the novels constructed upon +the unreformed model of Smollett and Fielding, those of Dickens +are enormous stock-pots into which the author casts every kind +of autobiographical experience, emotion, pleasantry, anecdote, +adage or apophthegm. The fusion is necessarily very incomplete +and the hotch-potch is bound to fall to pieces with time. +Dickens’s plots, it must be admitted, are strangely unintelligible, +the repetitions and stylistic decorations of his work exceed +all bounds, the form is unmanageable and insignificant. The +diffuseness of the English novel, in short, and its extravagant +didacticism cannot fail to be most prejudicial to its perpetuation. +In these circumstances there is very little fiction that will stand +concentration and condensation so well as that of Dickens.</p> + +<p>For these reasons among others our interest in Dickens’s novels +as integers has diminished and is diminishing. But, on the other +hand, our interest and pride in him as a man and as a representative +author of his age and nation has been steadily augmented +and is still mounting. Much of the old criticism of his work, that +it was not up to a sufficiently high level of art, scholarship or +gentility, that as an author he is given to caricature, redundancy +and a shameless subservience to popular caprice, must now be +discarded as irrelevant.</p> + +<p>As regards formal excellence it is plain that Dickens labours +under the double disadvantage of writing in the least disciplined +of all literary genres in the most lawless literary milieu of the +modern world, that of Victorian England. In spite of these +defects, which are those of masters such as Rabelais, Hugo and +Tolstoy, the work of Dickens is more and more instinctively felt +to be true, original and ennobling. It is already beginning to +undergo a process of automatic sifting, segregation and crystallization, +at the conclusion of which it will probably occupy a larger +segment in the literary consciousness of the English-spoken race +than ever before.</p> + +<p>Portraits of Dickens, from the gay and alert “Boz” of Samuel +Lawrence, and the self-conscious, rather foppish portrait by +Maclise which served as frontispiece to <i>Nicholas Nickleby</i>, to +the sketch of him as Bobadil by C. R. Leslie, the Drummond and +Ary Scheffer portraits of middle age and the haggard and drawn +representations of him from photographs after his shattering +experiences as a public entertainer from 1856 (the year of his +separation from his wife) onwards, are reproduced in Kitton, in +Forster and Gissing and in the other biographies. Sketches are +also given in most of the books of his successive dwelling places +at Ordnance Terrace and 18 St Mary’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 <i>Tale of Two Cities</i> and +<i>Edwin Drood</i>, were given to Forster, and are now preserved in the +Dyce and Forster Museum at South Kensington. The work of +Dickens was a prize for which publishers naturally contended both +before and after his death. The first collective edition of his +works was begun in April 1847, and their number is now very +great. The most complete is still that of Messrs Chapman & +Hall, the original publishers of <i>Pickwick</i>; 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 <i>Pages Choisies</i> (1903).</p> + +<div class="condensed"> +<p><span class="sc">Bibliography.</span>—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 <i>Pickwick</i>, who had constituted himself a veritable +encyclopaedia of information about Dickens, and had clung to his +subject (in spite of many rebuffs which his peremptory temper found +it hard to digest) as tightly as ever Boswell had enveloped Johnson. +Two volumes of Forster’s <i>Life of Charles Dickens</i> 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 <i>Charles Dickens: A Critical Study</i> (1898), which ranks with +G.K. Chesterton’s <i>Charles Dickens</i>(1906) as a commentary inspired by +deep insight and adorned by great literary talent upon the genius of +the master-novelist. The names of other lives, sketches, articles and +estimates of Dickens and his works would occupy a large volume in +the mere enumeration. See R.H. Shepherd, <i>The Bibliography of +Dickens</i> (1880); <i>James Cooke’s Bibliography of the Writings of Charles +Dickens</i> (1879); <i>Dickensiana</i>, by F. G. Kitton (1886); and <i>Bibliography</i> +by J.P. Anderson, appended to Sir F.T. Marzials’s <i>Life of +Charles Dickens</i> (1887). Among the earlier sketches may be specially +cited the lives by J. C. Hotten and G. A. Sala (1870), the Anecdote-Biography +edited by the American R. H. Stoddard (1874), Dr A. W. +Ward in the English Men of Letters Series (1878), that by Sir Leslie +Stephen in the <i>Dictionary of National Biography</i>, and that by Professor +Minto in the eighth edition of the <i>Encyclopaedia Britannica</i>. +The <i>Letters</i> were first issued in two volumes edited by his daughter +and sister-in-law in 1880. For Dickens’s connexion with Kent the +following books are specially valuable:—Robert Langton’s <i>Childhood +and Youth of Charles Dickens</i> (1883); Langton’s <i>Dickens and +Rochester</i> (1880); Thomas Frost’s <i>In Kent with Charles Dickens</i> +(1880); F. G. Kitton’s <i>The Dickens Country</i> (1905); H. S. Ward’s +<i>The Real Dickens Land</i> (1904); R. Allbut’s <i>Rambles in Dickens Land</i> +(1899 and 1903). For Dickens’s reading tours see G. Dolby’s +<i>Charles Dickens as I knew him</i> (1884); J. T. Fields’s <i>In and Out of +Doors with Charles Dickens</i> (1876); Charles Kent’s <i>Dickens as a +Reader</i> (1872). And for other aspects of his life see M. Dickens’s <i>My +Father as I recall him</i> (1897); P. H. Fitzgerald’s <i>Life of C. Dickens as +revealed in his Writings</i> (1905), and <i>Bozland</i> (1895); F. G. Kitton’s +<i>Charles Dickens, his Life, Writings and Personality</i>, a useful compendium +(1902); T. E. Pemberton’s <i>Charles Dickens and the Stage</i>, and +<i>Dickens’s London</i> (1876); F. Miltoun’s <i>Dickens’s London</i> (1904); +Kitton’s <i>Dickens and his Illustrators</i>; W. Teignmouth Shore’s <i>Charles +Dickens and his Friends</i> (1904 and 1909); B. W. Matz, <i>Story of +Dickens’s Life and Work</i> (1904), and review of solutions to <i>Edwin +Drood</i> in <i>The Bookman</i> for March 1908; the recollections of Edmund +Yates, Trollope, James Payn, Lehmann, R. H. Horne, Lockwood +and many others. <i>The Dickensian</i>, a magazine devoted to Dickensian +subjects, was started in 1905; it is the organ of the Dickens Fellowship, +and in a sense of the Boz Club. <i>A Dickens Dictionary</i> (by G. A. +Pierce) appeared in 1872 and 1878; another (by A. J. Philip) in 1909; +and a <i>Dickens Concordance</i> by Mary Williams in 1907.</p> +</div> +<div class="author">(T. Se.)</div> + + +<hr class="art" /> +<p><span class="pagenum"><a name="page184" id="page184"></a>184</span></p> +<p><span class="bold">DICKINSON, ANNA ELIZABETH<a name="ar38" id="ar38"></a></span> (1842-  ), 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 +<i>The Crown of Thorns</i> (1876); <i>Mary Tudor</i> (1878), in which +she appeared in the title rôle; <i>Aurelian</i> (1878); and <i>An American +Girl</i> (1880), successfully acted by Fanny Davenport. She also +published a novel, <i>Which Answer?</i> (1868); <i>A Paying Investment, +a Plea for Education</i> (1876); and <i>A Ragged Register of People, +Places and Opinions</i> (1879).</p> + + +<hr class="art" /> +<p><span class="bold">DICKINSON, JOHN<a name="ar39" id="ar39"></a></span> (1732-1808), American statesman and +pamphleteer, was born in Talbot county, Maryland, on the 8th +of November 1732. He removed with his father to Kent county, +Delaware, in 1740, studied under private tutors, read law, and in +1753 entered the Middle Temple, London. Returning to America +in 1757, he began the practice of law in Philadelphia, was speaker +of the Delaware assembly in 1760, and was a member of the +Pennsylvania assembly in 1762-1765 and again in 1770-1776.<a name="fa1c" id="fa1c" href="#ft1c"><span class="sp">1</span></a> +He represented Pennsylvania in the Stamp Act Congress (1765) +and in the Continental Congress from 1774 to 1776, when he +was defeated owing to his opposition to the Declaration of +Independence. He then retired to Delaware, served for a time +as private and later as brigader-general in the state militia, and +was again a member of the Continental Congress (from Delaware) +in 1779-1780. He was president of the executive council, or chief +executive officer, of Delaware in 1781-1782, and of Pennsylvania +in 1782-1785, and was a delegate from Delaware to the Annapolis +convention of 1786 and the Federal Constitutional convention +of 1787. Dickinson has aptly been called the “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”<a name="fa2c" id="fa2c" href="#ft2c"><span class="sp">2</span></a> and +the “Articles of Confederation” of the second Congress. Most +influential of all, however, were <i>The Letters of a Farmer in +Pennsylvania</i>, written in 1767-1768 in condemnation of the +Townshend Acts of 1767, in which he rejected speculative +natural rights theories and appealed to the common sense of +the people through simple legal arguments. By opposing the +Declaration of Independence, he lost his popularity and was never +able entirely to regain it. As the representative of a small state, +he championed the principle of state equality in the constitutional +convention, but was one of the first to advocate the +compromise, which was finally adopted, providing for equal +representation, in one house and proportional representation in +the other. He was probably influenced by Delaware prejudice +against Pennsylvania when he drafted the clause which forbids +the creation of a new state by the junction of two or more states +or parts of states without the consent of the states concerned as +well as of congress. After the adjournment of the convention he +defended its work in a series of letters signed “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.</p> + +<div class="condensed"> +<p>See C. J. Stillé, <i>Life and Times of John Dickinson</i>, and P. L. Ford +(editor), <i>The Writings of John Dickinson</i>, in vols. xiii. and xiv. +respectively of the <i>Memoirs of the Historical Society of Pennsylvania</i> +(Philadelphia, 1891 and 1895).</p> +</div> + +<hr class="foot" /> <div class="note"> + +<p><a name="ft1c" id="ft1c" href="#fa1c"><span class="fn">1</span></a> Being under the same proprietor and the same governor, +Pennsylvania and Delaware were so closely connected before the +Revolution that there was an interchange of public men.</p> + +<p><a name="ft2c" id="ft2c" href="#fa2c"><span class="fn">2</span></a> The “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).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DICKSON, SIR ALEXANDER<a name="ar40" id="ar40"></a></span> (1777-1840), British artillerist, +entered the Royal Military Academy in 1793, passing out as +second lieutenant in the Royal Artillery in the following year. +As a subaltern he saw service in Minorca in 1798 and at Malta in +1800. As a captain he took part in the unfortunate Montevideo +Expedition of 1806-07, and in 1809 he accompanied Howorth +to the Peninsular War as brigade-major of the artillery. He soon +obtained a command in the Portuguese artillery, and as a +lieutenant-colonel of the Portuguese service took part in the +various battles of 1810-11. At the two sieges of Budazoz, +Ciudad Rodrigo, the Salamanca forts and Burgos, he was +entrusted by Wellington (who had the highest opinion of him) +with most of the detailed artillery work, and at Salamanca battle +he commanded the reserve artillery. In the end he became +commander of the whole of the artillery of the allied army, and +though still only a substantive captain in the British service he +had under his orders some 8000 men. At Vitoria, the Pyrenees +battles and Toulouse he directed the movements of the artillery +engaged, and at the end of the war received handsome presents +from the officers who had served under him, many of whom were +his seniors in the army list. He was at the disastrous affair of +New Orleans, but returned to Europe in time for the Waterloo +campaign. He was present at Quatre Bras and Waterloo on the +artillery staff of Wellington’s army, and subsequently commanded +the British battering train at the sieges of the French fortresses +left behind the advancing allies. For the rest of his life he was on +home service, principally as a staff officer of artillery. He died, +a major-general and G.C.B., in 1840. A memorial was erected at +Woolwich in 1847. Dickson was one of the earliest fellows of the +Royal Geographical Society.</p> + +<div class="condensed"> +<p>His diaries kept in the Peninsula were the main source of information +used in Duncan’s <i>History of the Royal Artillery</i>.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DICKSON, SIR JAMES ROBERT<a name="ar41" id="ar41"></a></span> (1832-1901), Australian +statesman, was born in Plymouth on the 30th of November 1832. +He was brought up in Glasgow, receiving his education at the +high school, and became a clerk in the City of Glasgow Bank. +In 1854 he emigrated to Victoria, but after some years spent +in that colony and in New South Wales, he settled in 1862 in +Queensland, where he was connected with many important +business enterprises, among them the Royal Bank of Queensland. +He entered the Queensland House of Assembly in 1872, and +became minister of works (1876), treasurer (1876-1879, and 1883-1887), +acting premier (1884), but resigned in 1887 on the question +of taxing land. In 1889 he retired from business, and spent three +years in Europe before resuming political life. He fought for +the introduction of Polynesian labour on the Queensland sugar +plantations at the general election of 1892, and was elected to the +House of Assembly in that year and again at the elections of 1893 +and 1896. He became secretary for railways in 1897, minister for +home affairs in 1898, represented Queensland in the federal +council of Australia in 1896 and at the postal conference at +Hobart in 1898, and in 1898 became premier. His energies were +now devoted to the formation of an Australian commonwealth. +He secured the reference of the question to a plebiscite, the result +of which justified his anticipations. He resigned the premiership +in November 1899, but in the ministry of Robert Philp, formed +in the next month, he was reappointed to the offices of chief +secretary and vice-president of the executive council which he had +combined with the office of premier. He represented Queensland +in 1900 at the conference held in London to consider the question +of Australian unity, and on his return was appointed minister of +defence in the first government of the Australian Commonwealth. +He did not long survive the accomplishment of his political aims, +dying at Sydney on the 10th of January 1901, in the midst of +the festivities attending the inauguration of the new state.</p> + +<p><span class="pagenum"><a name="page185" id="page185"></a>185</span></p> + + +<hr class="art" /> +<p><span class="bold">DICOTYLEDONS,<a name="ar42" id="ar42"></a></span> in botany, the larger of the two great classes +of angiosperms, embracing most of the common flower-bearing +plants. The name expresses the most universal character of the +class, the importance of which was first noticed by John Ray, +namely, the presence of a pair of seed-leaves or cotyledons, in +the plantlet or embryo contained in the seed. The embryo is +generally surrounded by a larger or smaller amount of foodstuff +(endosperm) which serves to nourish it in its development to +form a seedling when the seed germinates; frequently, however, +as in pea or bean and their allies, the whole of the nourishment for +future use is stored up in the cotyledons themselves, which then +become thick and fleshy. In germination of the seed the root of +the embryo (radicle) grows out to get a holdfast for the plant; +this is generally followed by the growth of the short stem +immediately above the root, the so-called “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 <span class="correction" title="amended from nunber">number</span> 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 <span class="sc"><a href="#artlinks">Plants</a></span>: <i>Anatomy</i>. A similar growth +occurs in the root. This increase in the diameter of stem and root +is correlated with the increase in leaf-area each season, due to the +continued production of new leaf-bearing branches. A characteristic +of the class is afforded by the complicated network formed +by the leaf-veins,—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 +<span class="sc"><a href="#artlinks">Flower</a></span>). The characters of the fruit and seed and the means +for ensuring the dispersal of the seeds are also very varied (see +<span class="sc"><a href="#artlinks">Fruit</a></span>).</p> + + +<hr class="art" /> +<p><span class="bold">DICTATOR<a name="ar43" id="ar43"></a></span> (from the Lat. <i>dictare</i>, frequentative of <i>dicere</i>, to +speak). In modern usage this term is loosely used for a personal +ruler enjoying extraordinary and extra-constitutional power. +The etymological sense of one who “dictates”—<i>i.e.</i> one whose +word (<i>dictum</i>) is law (from which that of one who “dictates,” <i>i.e.</i> +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 <i>De legibus</i> (iii. 3, 9) and by the emperor Claudius +in his extant <i>Oratio</i> (i. 28). This function of the office, although +it may not have been contemplated at first, is attested by +the internal history of Rome. In the crisis of the agitation that +gathered round the Licinian laws (367 <span class="sc">b.c.</span>) a dictator was appointed, +and in 314 <span class="sc">b.c.</span> we have the notice of a dictator created +for purposes of criminal jurisdiction (<i>quaestionibus exercendis</i>). +The dictator appointed to meet the dangers of war, sedition or +crime was technically described as “the administrative dictator” +(<i>rei gerundae causa</i>). Minor, or merely formal, needs of the state +might lead to the creation of other types of this office. Thus we +find dictators destined to hold the elections, to make out the list +of the senate, to celebrate games, to establish festivals, and to +drive the nail into the temple of Jupiter—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.</p> + +<p>The powers of a dictator were a temporary revival of those +of the kings; but there were some limitations to his authority. +He was never concerned with civil jurisdiction, and was +dependent on the senate for supplies of money. His military +authority was confined to Italy; and his power of life and death +over the citizens was at an early period limited by law. It was +probably the <i>lex Valeria</i> of 300 <span class="sc">b.c.</span> that made him subject to the +right of criminal appeal (<i>provocatio</i>) within the limits of the city. +But during his tenure of power all the magistrates of the people +were regarded as his subordinates; and it was even held that +the right of assistance (<i>auxilium</i>), furnished by the tribunes of the +plebs to members of the citizen body, should not be effectively +exercised when the state was under this type of martial law. The +dictator was nominated by one of the consuls. But here as elsewhere +the senate asserted its authority over the magistrates, and +the view was finally held that the senate should not only suggest +the need of nomination but also the name of the nominee. After +the nomination, the imperium of the dictator was confirmed by +a <i>lex curiata</i> (see <span class="sc"><a href="#artlinks">Comitia</a></span>). To emphasize the superiority of this +imperium over that of the consuls, the dictator might be preceded +by twenty-four lictors, not by the usual twelve; and, at least in +the earlier period of the office, these lictors bore the axes, the +symbols of life and death, within the city walls.</p> + +<p>Tradition represents the dictatorship as having a life of three +centuries in the history of the Roman state. The first dictator +is said to have been created in 501 <span class="sc">b.c.</span>; the last of the +“administrative” dictators belongs to the year 216 <span class="sc">b.c.</span> It was +an office that was incompatible both with the growing spirit of +constitutionalism and with the greater security of the city; and +the epoch of the Second Punic War was marked by experiments +with the office, such as the election of Q. Fabius Maximus by the +people, and the co-dictatorship of M. Minucius with Fabius, which +heralded its disuse (see <span class="sc"><a href="#artlinks">Punic Wars</a></span>). The emergency office of +the early and middle Republic has few points of contact, except +those of the extraordinary position and almost unfettered +authority of its holder, with the dictatorship as revised by Sulla +and by Caesar. Sulla’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 <i>Imperium</i> in Rome, it was +the natural resort of the founder of a monarchy (see <span class="sc"><a href="#artlinks">Sulla</a></span> and +<span class="sc"><a href="#artlinks">Caesar</a></span>). Ostensibly to prevent its further use for such a purpose, +M. Antonius in 44 <span class="sc">b.c.</span> carried a law abolishing the dictatorship as +a part of the constitution.</p> + +<div class="condensed"> +<p><span class="sc">Bibliography.</span>—Mommsen, <i>Römisches Staatsrecht</i>, ii. 141 foll. +(3rd ed., Leipzig, 1887); Herzog, <i>Geschichte und System der römischen +Staatsverfassung</i>, i. 718 foll. (Leipzig, 1884); Pauly-Wissowa, +<i>Realencyclopädie</i>, v. 370 foll. (new edition, Stuttgart. 1893, &c.); +<span class="pagenum"><a name="page186" id="page186"></a>186</span> +Lange, <i>Römische Alterthümer</i>, i. 542 foll. (Berlin, 1856, &c.); Daremberg-Saglio, +<i>Dictionnaire des antiquités grecques et romaines</i>, ii. 161 +foll. (1875, &c.); Haverfield, “The Abolition of the Dictatorship,” +in <i>Classical Review</i>, iii. 77.</p> +</div> +<div class="author">(A. H. J. G.)</div> + + +<hr class="art" /> +<p><span class="bold">DICTIONARY.<a name="ar44" id="ar44"></a></span> In its proper and most usual meaning a +dictionary is a book containing a collection of the words of a +language, dialect or subject, arranged alphabetically or +in some other definite order, and with explanations in the +<span class="sidenote">Definition and history.</span> +same or some other language. When the words are few in +number, being only a small part of those belonging to +the subject, or when they are given without explanation, or some +only are explained, or the explanations are partial, the work is +called a <i>vocabulary</i>; and when there is merely a list of explanations +of the technical words and expressions in some particular +subject, a <i>glossary</i>. An alphabetical arrangement of the words +of some book or author with references to the places where +they occur is called an index (<i>q.v.</i>). When under each word +the phrases containing it are added to the references, the work is +called a <i>concordance</i>. Sometimes, however, these names are given +to true dictionaries; thus the great Italian dictionary of the +<i>Accademia della Crusca</i>, in six volumes folio, is called <i>Vocabolario</i>, +and Ernesti’s dictionary to Cicero is called <i>Index</i>. When the +words are arranged according to a definite system of classification +under heads and subdivisions, according to their nature or their +meaning, the book is usually called a classed vocabulary; but +when sufficient explanations are given it is often accepted as a +dictionary, like the <i>Onomasticon</i> of Julius Pollux, or the native +dictionaries of Sanskrit, Manchu and many other languages.</p> + +<p>Dictionaries were originally books of reference explaining the +words of a language or of some part of it. As the names of +things, as well as those of persons and places, are words, and +often require explanation even more than other classes of words, +they were necessarily included in dictionaries, and often to a very +great extent. In time, books were devoted to them alone, and +were limited to special subjects, and these have so multiplied, +that dictionaries of things now rival in number and variety those +of words or of languages, while they often far surpass them in bulk. +There are dictionaries of biography and history, real and fictitious, +general and special, relating to men of all countries, characters +and professions; the English <i>Dictionary of National Biography</i> +(see <span class="sc"><a href="#artlinks">Biography</a></span>) is a great instance of one form of these; +dictionaries of bibliography, relating to all books, or to those +of some particular kind or country; dictionaries of geography +(sometimes called <i>gazetteers</i>) of the whole world, of particular +countries, or of small districts, of towns and of villages, of +castles, monasteries and other buildings. There are dictionaries +of philosophy; of the Bible; of mathematics; of natural history, +zoology, botany; of birds, trees, plants and flowers; of +chemistry, geology and mineralogy; of architecture, painting +and music; of medicine, surgery, anatomy, pathology and +physiology; of diplomacy; of law, canon, civil, statutory and +criminal; of political and social sciences; of agriculture, rural +economy and gardening; of commerce, navigation, horsemanship +and the military arts; of mechanics, machines and +the manual arts. There are dictionaries of antiquities, of +chronology, of dates, of genealogy, of heraldry, of diplomatics, of +abbreviations, of useful receipts, of monograms, of adulterations +and of very many other subjects. These works are separately +referred to in the bibliographies attached to the articles on the +separate subjects. And lastly, there are dictionaries of the arts +and sciences, and their comprehensive offspring, encyclopaedias +(<i>q.v.</i>), which include in themselves every branch of knowledge. +Neither under the heading of <i>dictionary</i> nor under that of +<i>encyclopaedia</i> do we propose to include a mention of every work +of its class, but many of these will be referred to in the separate +articles on the subjects to which they pertain. And in this +article we confine ourselves to an account of those dictionaries +which are primarily word-books. This is practically the most +convenient distinction from the subject-book or encyclopaedia; +though the two characters are often combined in one work. Thus +the <i>Century Dictionary</i> has encyclopaedic features, while the +present edition of the <i>Encyclopaedia Britannica</i>, restoring its +earlier tradition but carrying out the idea more systematically, +also embodies dictionary features.</p> + +<p><i>Dictionarium</i> is a word of low or modern Latinity;<a name="fa1d" id="fa1d" href="#ft1d"><span class="sp">1</span></a> <i>dictio</i>, +from which it was formed, was used in medieval Latin to mean +a word. <i>Lexicon</i> is a corresponding word of Greek origin, +meaning a book of or for words—a dictionary. A <i>glossary</i> is +properly a collection of unusual or foreign words requiring +explanation. It is the name frequently given to English +dictionaries of dialects, which the Germans usually call <i>idioticon</i>, +and the Italians <i>vocabolario</i>. <i>Wörterbuch</i>, a book of words, was +first used among the Germans, according to Grimm, by Kramer +(1719), imitated from the Dutch <i>woordenboek</i>. From the Germans +the Swedes and Danes adopted <i>ordbok</i>, <i>ordbog</i>. The Icelandic +<i>ordabôk</i>, like the German, contains the genitive plural. The +Slavonic nations use <i>slovar</i>, <i>slovnik</i>, and the southern Slavs +<i>ryetshnik</i>, from <i>slovo</i>, <i>ryetsh</i>, a word, formed, like dictionary +and lexicon, without composition. Many other names have been +given to dictionaries, as <i>thesaurus</i>, <i>Sprachschatz</i>, <i>cornucopia</i>, +<i>gazophylacium</i>, <i>comprehensorium</i>, <i>catholicon</i>, to indicate their +completeness; <i>manipulus predicantium</i>, <i>promptorium puerorum</i>, +<i>liber memorialis</i>, <i>hortus vocabulorum</i>, <i>ionia</i> (a violet bed), <i>alveary</i> +(a beehive), <i>kamoos</i> (the sea), <i>haft kulzum</i> (the seven seas), <i>tsze +tien</i> (a standard of character), <i>onomasticon</i>, <i>nomenclator</i>, <i>bibliotheca</i>, +<i>elucidario</i>, <i>Mundart-sammlung</i>, <i>clavis</i>, <i>scala</i>, <i>pharetra</i>,<a name="fa2d" id="fa2d" href="#ft2d"><span class="sp">2</span></a> <i>La +Crusca</i> from the great Italian dictionary, and <i>Calepino</i> (in Spanish +and Italian) from the Latin dictionary of Calepinus.</p> + +<p>The tendency of great dictionaries is to unite in themselves all +the peculiar features of special dictionaries. A large dictionary +is most useful when a word is to be thoroughly studied, or when +there is difficulty in making out the meaning of a word or phrase. +Special dictionaries are more useful for special purposes; for +instance, synonyms are best studied in a dictionary of synonyms. +And small dictionaries are more convenient for frequent use, as +in translating from an unfamiliar language, for words may be +found more quickly, and they present the words and their +meanings in a concentrated and compact form, instead of being +scattered over a large space, and separated by other matter. +Dictionaries of several languages, called <i>polyglots</i>, are of different +kinds. Some are polyglot in the vocabulary, but not in the +explanation, like Johnson’s dictionary of Persian and Arabic +explained in English; some in the interpretation, but not in the +vocabulary or explanation, like <i>Calepini octoglotton</i>, a Latin +dictionary of Latin, with the meanings in seven languages. +Many great dictionaries are now polyglot in this sense. Some are +polyglot in the vocabulary and interpretation, but are explained +in one language, like Jal’s <i>Glossaire nautique</i>, a glossary of sea +terms in many languages, giving the equivalents of each word in +the other languages, but the explanation in French. Pauthier’s +<i>Annamese Dictionary</i> is polyglot in a peculiar way. It gives +the Chinese characters with their pronunciation in Chinese and +Annamese. Special dictionaries are of many kinds. There are +technical dictionaries of etymology, foreign words, dialects, +secret languages, slang, neology, barbarous words, faults of expression, +choice words, prosody, pronunciation, spelling, orators, +poets, law, music, proper names, particular authors, nouns, verbs, +participles, particles, double forms, difficulties and many others. +Fick’s dictionary (Göttingen, 1868, 8vo; 1874-1876, 8vo, 4 vols.) +is a remarkable attempt to ascertain the common language of +the Indo-European nations before each of their great separations. +In the second edition of his <i>Etymologische Forschungen</i> (Lemgo +and Detmoldt, 1859-1873, 8vo, 7217 pages) Pott gives a +comparative lexicon of Indo-European roots, 2226 in number, +occupying 5140 pages.</p> + +<p><span class="pagenum"><a name="page187" id="page187"></a>187</span></p> + +<p>At no time was progress in the making of general dictionaries +so rapid as during the second half of the 19th century. It is to +be seen in three things: in the perfecting of the theory of what +a general dictionary should be; in the elaboration +<span class="sidenote">Methods.</span> +of methods of collecting and editing lexicographic +materials; and in the magnitude and improved quality of the +work which has been accomplished or planned. Each of these +can best be illustrated from English lexicography, in which the +process of development has in all directions been carried farthest. +The advance that has been made in theory began with a radical +change of opinion with regard to the chief end of the general +dictionary of a language. The older view of the matter was that +the lexicographer should furnish a standard of usage—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 <i>Vocabolario degli Accademici della +Crusca</i>, long the standard of that language, was published. The +Académie Française, the first edition of whose dictionary +appeared in 1694, had a similar origin. In England the idea of +constructing a dictionary upon this principle arose during the +second quarter of the 18th century. It was imagined by men of +letters—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 <i>inventory of the language</i>; much more, +but this primarily.... It is no task of the maker of it to select +the <i>good</i> words of the language.... The business which he has +undertaken is to collect and arrange <i>all</i> words, whether good or +bad, whether they commend themselves to his judgment or otherwise.... +<i>He is an historian of</i> [the language], <i>not a critic.</i>” +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.</p> + +<p>This theoretical requirement must, of course, be modified +considerably in practice. The date at which a modern language +is to be regarded by the lexicographer as “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, <i>Oxford +Dict.</i> 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.</p> + +<p>Another important principle upon which Trench insisted, and +which also expresses a requirement of modern scientific philology, +is that the dictionary shall be not merely a record, but also an +<i>historical</i> record of words and their uses. From the literary point +of view the most important thing is present usage. To that alone +the idea of a “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.</p> + +<p>The adoption of the historical principle involves a further noteworthy +modification of older methods, namely, an important +extension of the use of quotations. To Dr Johnson belongs the +credit of showing how useful, when properly chosen, they may be, +not only in corroborating the lexicographer’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 <i>New Dictionary +of the English Language ... Illustrated by Quotations from the +Best Authors</i> (1835-1836) still remains a most valuable collection +of literary illustrations. Lexicographers, however, have, with +<span class="pagenum"><a name="page188" id="page188"></a>188</span> +few exceptions, until a recent date, employed quotations chiefly +for the ends just mentioned—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.</p> + +<p>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.</p> + +<p>Lastly, a much more important place in the scheme of the ideal +dictionary is now assigned to the <i>etymology</i> of words. This may +be attributed, in part, to the recent rapid development of etymology +as a science, and to the greater abundance of trustworthy +data; but it is chiefly due to the fact that from the historical +point of view the connexion between that section of the biography +of a word which lies within the language—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 <i>form</i> of a word up to the time when it became +a part of the language, and is, in a measure, an extension of the +history of the development of the word in the language. Moreover, +it is the only means by which the exact relations of allied +words can be ascertained, and the separation of words of the same +form but of diverse origin (homonyms) can be effected, and is +thus, for the dictionary, the foundation of all <i>family history</i> and +correct <i>genealogy</i>. In fact, the attention that has been paid to +these two points in the best recent lexicography is one of its +distinguishing and most important characteristics. Related to +the etymology of words are the changes in their form which may +have occurred while they have been in use as parts of the language—modifications +of their pronunciation, corruptions by popular +etymology or false associations, and the like. The facts with +regard to these things which the wide research necessitated +by the historical method furnishes abundantly to the modern +lexicographer are often among the most novel and interesting +of his acquisitions.</p> + +<p>It should be added that even approximate conformity to the +theoretical requirements of modern lexicography as above outlined +is possible only under conditions similar to those under which +the Oxford <i>New English Dictionary</i> was undertaken (see below). +The labour demanded is too vast, and the necessary bulk of the +dictionary too great. When, however, a language is recorded +in one such dictionary, those of smaller size and more modest +pretensions can rest upon it as an authority and conform to it +as a model so far as their special limitations permit.</p> + +<p>The ideal thus developed is primarily that of the general +dictionary of the purely philological type, but it applies also to +the encyclopaedic dictionary. In so far as the latter is strictly +lexicographic—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.</p> + +<p>On its practical side the advance in lexicography has consisted +in the elaboration of methods long in use rather than in the invention +of new ones. The only way to collect the data upon which +the vocabulary, the definitions and the history are to be based +is, of course, to search for them in the written monuments of the +language, as all lexicographers who have not merely borrowed from +their predecessors have done. But the wider scope and special +aims of the new lexicography demand that the investigation shall +be vastly more comprehensive, systematic and precise. It is +necessary, in brief, that, as far as may be possible, the literature +(of all kinds) of every period of the language shall be examined +systematically, in order that all the words, and senses and forms +of words, which have existed during any period may be found, +and that enough excerpts (carefully verified, credited and dated) to +cover all the essential facts shall be made. The books, pamphlets, +journals, newspapers, and so on which must thus be searched will +be numbered by thousands, and the quotations selected may (as +in the case of the Oxford <i>New English Dictionary</i>) be counted by +millions. This task is beyond the powers of any one man, even +though he be a Johnson, or a Littré or a Grimm, and it is now +<span class="pagenum"><a name="page189" id="page189"></a>189</span> +assigned to a corps of readers whose number is limited only by the +ability of the editor to obtain such assistance. The modern +method of editing the material thus accumulated—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.</p> + +<p>The most complete exemplification of these principles and +methods is the <i>Oxford New English Dictionary, on historical +principles, founded mainly on materials collected by the Philological +Society</i>. This monumental work originated in the suggestion +of Trench that an attempt should be made, under the +direction of the Philological Society, to complete the vocabulary +of existing dictionaries and to supply the historical information +which they lacked. The suggestion was adopted, considerable +material was collected, and Mr Herbert Coleridge was appointed +general editor. He died in 1861, and was succeeded by Dr F. J. +Furnivall. Little, however, was done, beyond the collection of +quotations—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.</p> + +<p>The historical method of exposition, particularly by quotations, +is applied in the <i>New English Dictionary</i>, if not in all cases +with entire success, yet, on the whole, with a regularity and a +precision which leave little to be desired. A minor fault is that +excerpts from second or third rate authors have occasionally been +used where better ones from writers of the first class either must +have been at hand or could have been found. As was said above, +the literary quality of the question is highly important even +in historical lexicography, and should not be neglected unnecessarily. +Other special features of the book are the completeness +with which variations of pronunciation and orthography +(with dates) are given; the fulness and scientific excellence of the +etymologies, which abound in new information and corrections +of old errors; the phonetic precision with which the present +(British) pronunciation is indicated; and the elaborate subdivision +of meanings. The definitions as a whole are marked by +a high degree of accuracy, though in a certain number of cases +(not explicable by the date of the volumes) the lists of meanings +are not so good as one would expect, as compared (say) with +the <i>Century Dictionary</i>. Work of such magnitude and quality is +possible, practically, only when the editor of the dictionary can command +not merely the aid of a very large number of scholars and +men of science, but their gratuitous aid. In this the <i>New English +Dictionary</i> has been singularly fortunate. The conditions under +which it originated, and its aim, have interested scholars everywhere, +and led them to contribute to the perfecting of it their +knowledge and time. The long list of names of such helpers in Sir +J. A. H. Murray’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.</p> + +<div class="condensed"> +<p>The <i>New English Dictionary</i> furnishes for the first time data from +which the extent of the English word-store at any given period, and +the direction and rapidity of its growth, can fairly be estimated. +For this purpose the materials furnished by the older dictionaries are +quite insufficient, on account of their incompleteness and unhistorical +character. For example 100 pages of the <i>New English Dictionary</i> +(from the letter H) contain 1002 words, of which, as the dated quotations +show, 585 were current in 1750 (though some, of course, were +very rare, some dialectal, and so on), 191 were obsolete at that date, +and 226 have since come into use. But of the more than 700 words—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.</p> +</div> + +<p>In the <i>Deutsches Wörterbuch</i> of Jacob and Wilhelm Grimm +the scientific spirit, as was said above, first found expression in +general lexicography. The desirability of a complete inventory +and investigation of German words was recognized by Leibnitz +and by various 18th-century scholars, but the plan and methods +of the Grimms were the direct product of the then new scientific +philology. Their design, in brief, was to give an exhaustive +account of the words of the literary language (New High German) +from about the end of the 15th century, including their earlier +etymological and later history, with references to important +dialectal words and forms; and to illustrate their use and history +abundantly by quotations. The first volume appeared in 1854. +Jacob Grimm (died 1863) edited the first, second (with his +brother, who died in 1859), third and a part of the fourth +volumes; the others have been edited by various distinguished +scholars. The scope and methods of this dictionary have been +broadened somewhat as the work has advanced. In general it +may be said that it differs from the <i>New English Dictionary</i> +chiefly in its omission of pronunciations and other pedagogic +matter; its irregular treatment of dates; its much less systematic +and less lucid statement of etymologies; its less systematic and +less fruitful use of quotations; and its less convenient and less +intelligible arrangement of material and typography.</p> + +<p>These general principles lie also at the foundation of the +scholarly <i>Dictionnaire de la langue française</i> of E. Littré, though +they are there carried out less systematically and less completely. +In the arrangement of the definitions the first place is given to +the most primitive meaning of the word instead of to the most +common one, as in the dictionary of the Academy; but the other +meanings follow in an order that is often logical rather than +historical. Quotations also are frequently used merely as literary +illustrations, or are entirely omitted; in the special paragraphs +on the history of words before the 16th century, however, they +are put to a strictly historical use. This dictionary—perhaps the +greatest ever compiled by one man—was published 1863-1872. +(Supplement, 1878.)</p> + +<p>The <i>Thesaurus Linguae Latinae</i>, prepared under the auspices of +the German Academies of Berlin, Göttingen, Leipzig, Munich +and Vienna, is a notable application of the principles and +practical co-operative method of modern lexicography to the +classical tongues. The plan of the work is to collect quotations +which shall register, with its full context, every word (except +<span class="pagenum"><a name="page190" id="page190"></a>190</span> +the most familiar particles) in the text of each Latin author +down to the middle of the 2nd century <span class="sc">a.d.</span>, and to extract +all important passages from all writers of the following +centuries down to the 7th; and upon these materials to found +a complete historical dictionary of the Latin language. The +work of collecting quotations was begun in 1894, and the first +part of the first volume has been published.</p> + +<p>In the making of all these great dictionaries (except, of course, +the last) the needs of the general public as well as those of scholars +have been kept in view. But the type to which the general +dictionary designed for popular use has tended more and more +to conform is the <i>encyclopaedic</i>. This combination of lexicon +and encyclopaedia is exhibited in an extreme—and theoretically +objectionable—form in the <i>Grand dictionnaire universel du XIX<span class="sp">e</span> +siècle</i> of Pierre Larousse. Besides common words and their +definitions, it contains a great many proper names, with a +correspondingly large number of biographical, geographical, +historical and other articles, the connexion of which with the +strictly lexicographical part is purely mechanical. Its utility, +which—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 <i>Nouveau Larousse illustré</i> started publication +in 1901, and was completed in 1904 (7 vols.). This is not an +abridgment or a fresh edition of the <i>Grand Dictionnaire</i> of Pierre +Larousse, but a new and distinct publication.</p> + +<p>The most notable work of this class, in English, is the <i>Century +Dictionary</i>, an American product, edited by Professor W. D. +Whitney, and published 1889-1891 in six volumes, containing +7046 pages (large quarto). It conforms to the philological mode +in giving with great fulness the older as well as the present +vocabulary of the language, and in the completeness of its +etymologies; but it does not attempt to give the full history +of every word within the language. Among its other more noteworthy +characteristics are the inclusion of a great number of +modern scientific and technical words, and the abundance of its +quotations. The quotations are for the most part provided with +references, but they are not dated. Even when compared with +the much larger <i>New English Dictionary</i>, the <i>Century’s</i> great +merit is the excellent enumeration of meanings, and the accuracy +of its explanations; in this respect it is often better and +fuller than the <i>New English</i>. In the application of the encyclopaedic +method this dictionary is conservative, excluding, with a +few exceptions, proper names, and restricting, for the most part, +the encyclopaedic matter to descriptive and other details which +may legitimately be added to the definitions. Its pictorial +illustrations are very numerous and well executed. In the +manner of its compilation it is a good example of modern cooperative +dictionary-making, being the joint product of a large +number of specialists. Next to the <i>New English Dictionary</i> it +is the most complete and scholarly of English lexicons.</p> + +<p><i>Bibliography.</i>—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:—</p> + +<p><span class="sc">Europe</span>: Greek, Latin, French, Romance, Teutonic (Scandinavian +and German), Celtic, Basque, Baltic, Slavonic, Ugrian, +Gipsy, Albanian.</p> + +<p><span class="sc">Asia</span>: Semitic, Armenian, Persian, Sanskrit, Indian, Indo-Chinese, +Malay Archipelago, Philippines, Chinese, Japanese, +Northern and Central Asia.</p> + +<p><span class="sc">Africa</span>: Egypt and Abyssinia, Eastern Africa, Southern, +Western, Central, Berber.</p> + +<p><span class="sc">Australia and Polynesia.</span></p> + +<p><span class="sc">America</span>: North, Central (with Mexico), South.</p> + +<p class="center pt2">EUROPE</p> + +<div class="condensed"> +<p><span class="bold">Greek.</span>—Athenaeus quotes 35 writers of works, known or supposed +to be dictionaries, for, as they are all lost, it is often difficult to +decide on their nature. Of these, Anticlides, who lived after the reign +of Alexander the Great, wrote <span class="grk" title="Exêgêtikos">Έξηγητικός</span>, which seems to have been a +sort of dictionary, perhaps explaining the words and phrases occurring +in ancient stories. Zenodotus, the first superintendent of the great +library of Alexandria, who lived in the reigns of Ptolemy I. and Ptolemy +II., wrote <span class="grk" title="Glôssai">Γλῶσσαι</span>, and also <span class="grk" title="Lexeis ethnikai">Λέξεις ἐθνικαἰ</span>, a dictionary of barbarous or +foreign phrases. Aristophanes of Byzantium, son of Apelles the painter, +who lived in the reigns of Ptolemy II. and Ptolemy III., and had the +supreme management of the Alexandrian library, wrote a number +of works, as <span class="grk" title="Attikai Lexeis, Lakônikai Glôssai">Άττικαί Λέξεις, Λακωνικαί Γλῶσσαι</span> which, from the titles, +should be dictionaries, but a fragment of his <span class="grk" title="Lexeis">Λέξεις</span> printed by +Boissonade, in his edition of Herodian (London, 1869, 8vo, pp. +181-189), is not alphabetical. Artemidorus, a pupil of Aristophanes, +wrote a dictionary of technical terms used in cookery. Nicander +Colophonius, hereditary priest of Apollo Clarius, born at Claros, +near Colophon in Ionia, in reputation for 50 years, from 181 to +135, wrote <span class="grk" title="Glôssai">Γλῶσσαι</span> in at least three books. Parthenius, a pupil +of the Alexandrian grammarian Dionysius (who lived in the 1st +century before Christ), wrote on choice words used by historians. +Didymus, called <span class="grk" title="chalkenteros">χαλκένερος</span>, who, according to Athenaeus, wrote +3500 books, and, according to Seneca, 4000, wrote lexicons of the +tragic poets (of which book 28 is quoted), of the comic poets, of +ambiguous words and of corrupt expressions. Glossaries of Attic +words were written by Crates, Philemon, Philetas and Theodorus; +of Cretan, by Hermon or Hermonax; of Phrygian, by Neoptolemus; +of Rhodian, by Moschus; of Italian, by Diodorus of Tarsus; of +foreign words, by Silenus; of synonyms, by Simaristus; of cookery, +by Heracleon; and of drinking vessels, by Apollodorus of Cyrene. +According to Suidas, the most ancient Greek lexicographer was +Apollonius the sophist, son of Archibius. According to the common +opinion, he lived in the time of Augustus at Alexandria. He composed +a lexicon of words used by Homer, <span class="grk" title="Lexeis Homêrikai">Λέξεις Όμηρικαί</span>, a very +valuable and useful work, though much interpolated, edited by +Villoison, from a MS. of the 10th century, Paris, 1773, 4to, 2 vols.; +and by Tollius, Leiden, 1788, 8vo; ed. Bekker, Berlin, 1833, 8vo. +Erotian or Herodian, physician to Nero, wrote a lexicon on Hippocrates, +arranged in alphabetical order, probably by some copyist, +whom Klein calls “homo sciolus.” It was first published in +Greek in H. Stephani <i>Dictionarium Medicum</i>, Paris, 1564, 8vo; ed. +Klein, Lipsiae, 1865, 8vo, with additional fragments. Timaeus the +sophist, who, according to Ruhnken, lived in the 3rd century, wrote +a very short lexicon to Plato, which, though much interpolated, is of +great value, 1st ed. Ruhnken, Leiden, 1754; ed. locupletior, Lugd. +Bat. 1789, 8vo. Aelius Moeris, called the Atticist, lived about 190 +<span class="pagenum"><a name="page191" id="page191"></a>191</span> +<span class="sc">a.d.</span>, and wrote an Attic lexicon, 1st ed. Hudson, Oxf. 1712, Bekker, +1833. Julius Pollux (<span class="grk" title="Ioulios Polydeukês">Ίούλιος Πολυδεύκης</span>) of Naucratis, in Egypt, died, +aged fifty-eight, in the reign of Commodus (180-192), who made him +professor of rhetoric at Athens. He wrote, besides other lost works, +an Onomasticon in ten books, being a classed vocabulary, intended to +supply all the words required by each subject with the usage of the +best authors. It is of the greatest value for the knowledge both of +language and of antiquities. First printed by Aldus, Venice, 1500, +fol.; often afterwards; ed. Lederlinus and Hemsterhuis, Amst. 1706, +2 vols.; Dindorf, 1824, 5 vols., Bethe (1900 f.). Harpocration of +Alexandria, probably of the 2nd century, wrote a lexicon on the ten +Attic orators, first printed by Aldus, Ven. 1503, fol.; ed. Dindorf, +Oxford, 1853, 8vo, 2 vols. from 14 MSS. Orion, a grammarian of +Thebes, in Egypt, who lived between 390 and 460, wrote an etymological +dictionary, printed by Sturz, Leipzig, 1820, 4to. Helladius +a priest of Jupiter at Alexandria, when the heathen temples there +were destroyed by Theophilus in 389 or 391 escaped to Constantinople, +where he was living in 408. He wrote an alphabetical lexicon, now +lost, chiefly of prose, called by Photius the largest (<span class="grk" title="polystichôtaton">πολυστιχώτατον</span>) +which he knew. Ammonius, professor of grammar at Alexandria, +and priest of the Egyptian ape, fled to Constantinople with Helladius, +and wrote a dictionary of words similar in sound but different in +meaning, which has been often printed in Greek lexicons, as Aldus, +1497, Stephanus, and separately by Valckenaer, Lugd. Bat. 1739, +4to, 2 vols., and by others. Zenodotus wrote on the cries of animals, +printed in Valckenaer’s <i>Ammonius</i>; with this may be compared +the work of Vincentio Caralucci, <i>Lexicon vocum quae a brutis animalibus +emittuntur</i>, Perusia, 1779, 12mo. Hesychius of Alexandria wrote a +lexicon, important for the knowledge of the language and literature, +containing many dialectic and local expressions and quotations from +other authors, 1st ed. Aldus, Ven. 1514, fol.; the best is Alberti and +Ruhnken, Lugd. Bat. 1746-1766, fol. 2 vols.; collated with the MS. +in St Mark’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 Ε to Ω; Α to Δ 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 <span class="grk" title="Etymologikon mega">Έτυμολογικὸν μέγα</span>, +<i>Etymologicum magnum</i>, has been frequently printed. The first +edition is by Musurus, Venitia, 1499, fol.; the best by Gaisford, +Oxonii, 1848, fol. It contains many grammatical remarks by famous +authorities, many passages of authors, and mythological and +historical notices. The MSS. vary so much that they look like the +works of different authors. To Eudocia Augusta of Makrembolis, wife +of the emperors Constantine XI. and Romanus IV. (1059 to 1071), +was ascribed a dictionary of history and mythology, <span class="grk" title="Iônia">Ίωνιά</span> (bed +of violets), first printed by D’Ansse de Villoison, <i>Anecdota Graeca</i>, +Venetiis, 1781, 4to, vol. i. pp. 1-442. It was supposed to have been +of much value before it was published. Thomas, Magister Officiorum +under Andronicus Palaeologus, afterward called as a monk Theodulus, +wrote <span class="grk" title="Eklogai onomatôn Attikôn">Έκλογαὶ ὀνομάτων Άττικῶν</span>, printed by Callierges, Romae, +1517, 8vo: Papias, <i>Vocabularium</i>, Mediolani, 1476, fol.: Craston, +an Italian Carmelite monk of Piacenza, compiled a Greek and Latin +lexicon, edited by Bonus Accursius, printed at Milan, 1478, fol.: +Aldus, Venetiis, 1497, fol.: Guarino, born about 1450 at Favora, +near Camarino, who called himself both Phavorinus and Camers, +published his <i>Thesaurus</i> in 1504. These three lexicons were frequently +reprinted. Estienne, <i>Thesaurus</i>, Genevae, 1572, fol., 4 vols.; ed. +Valpy, Lond. 1816-1826, 6 vols. fol.; Paris, 1831-1865, 9 vols. fol., 9902 +pages: <span class="grk" title="Kibôtos">Κιβωτός</span>, the ark, was intended to give the whole language, +ancient and modern, but vol. i., Constantinople, 1819, fol., 763 pages, +Α to Δ, only appeared, as the publication was put an end to by the +events of 1821. <span class="sc">English.</span>—Jones, London, 1823, 8vo: Dunbar, +Edin. 3rd ed. 1850, 4to: Liddell and Scott, 8th ed. Oxford, 1897, 4to. +<span class="sc">French.</span>—Alexandre, 12th ed. Paris, 1863, 8vo; 1869-1871, 2 vols: +Chassang, ib. 1872, 8vo. <span class="sc">Italian.</span>—Camini, Torino, 1865, 8vo, 972 +pages: Müller, ib. 1871, 8vo. <span class="sc">Spanish.</span>—<i>Diccionario manual, por les +padres Esculapios</i>, Madrid, 1859, 8vo. <span class="sc">German.</span>—Passow, 5th ed. +Leipzig, 1841-1857, 4to: Jacobitz and Seiler, 4th ed. ib. 1856, 8vo: +Benseler, ib. 1859, 8vo: Pape, Braunschweig, 1870-1874, 8vo, 4 vols. +Prellwitz, <i>Etymologisches Wörterbuch der griechischen Sprache</i>, new +edition, 1906: Herwerden, <i>Lexicon Graecum suppletorium et dialecticum</i>, +1902. <span class="sc">Dialects.</span>—<i>Attic</i>: Moeris, ed. Pierson, Lugd. Bat. +1759. 8vo. <i>Attic Orators</i>: Reiske, Oxon. 1828, 8vo, 2 vols. <i>Doric</i>: +Portus, Franckof. 1605, 8vo. <i>Ionic</i>: Id. ib. 1603, 8vo; 1817; 1825. +<span class="sc">Prosody.</span>—Morell, Etonae, 1762, 4to; ed. Maltby, Lond. 1830, 4to: +Brasse, Lond. 1850, 8vo. <span class="sc">Rhetoric.</span>—Ernesti, Lips. 1795, 8vo. +<span class="sc">Music</span>.—Drieberg, Berlin, 1855. <span class="sc">Etymology</span>.—Curtius, Leipzig, +1858-1862: Lancelot, Paris, 1863, 8vo. <span class="sc">Synonyms</span>.—Peucer, Dresden, +1766, 8vo: Pillon, Paris, 1847, 8vo. <span class="sc">Proper Names</span>.—Pape, ed. +Sengebusch, 1866, 8vo, 969 pages. <span class="sc">Verbs</span>.—Veitch, 2nd ed. Oxf. +1866. <span class="sc">Terminations</span>.—Hoogeveen, Cantab. 1810, 4to: Pape, +Berlin, 1836, 8vo. <span class="sc">Particular Authors</span>.—<i>Aeschylus</i>: Wellauer, +2 vols. Lips. 1830-1831, 8vo. <i>Aristophanes</i>: Caravella, Oxonii, 1822, +8vo. <i>Demosthenes</i>: Reiske, Lips. 1775, 8vo. <i>Euripides</i>: Beck, +Cantab. 1829, 8vo. <i>Herodotus</i>: Schweighäuser, Strassburg, 1824, 8vo, +2 vols. <i>Hesiod</i>: Osoruis, Neapol. 1791, 8vo. <i>Homer</i>: Apollonius +Sophista, ed. Tollius, Lugd. Bat., 1788, 8vo: Schaufelberger, Zürich, +1761-1768, 8vo, 8 vols.: Crusius, Hanover, 1836, 8vo: Wittich, +London, 1843, 8vo: Döderlein, Erlangen, 8vo, 3 vols.: Eberling, +Lipsiae, 1875, 8vo: Autenrieth, Leipzig, 1873, 8vo; London, 1877, +8vo. <i>Isocrates</i>: Mitchell, Oxon. 1828, 8vo. <i>Pindar</i>: Portus, +Hanov. 1606, 8vo. <i>Plato</i>: Timaeus, ed. Koch, Lips. 1828, 8vo: +Mitchell, Oxon. 1832, 8vo: Ast, Lips. 1835-1838, 8vo, 3 vols. +<i>Plutarch</i>: Wyttenbach, Lips. 1835, 8vo, 2 vols. <i>Sophocles</i>: Ellendt, +Regiomonti, 1834-1835, 8vo ed.; Genthe, Berlin, 1872, 8vo. <i>Thucydides</i>: +Bétant, Geneva, 1843-1847, 8vo, 2 vols. <i>Xenophon</i>: Sturtz, +Lips. 1801-1804, 8vo, 4 vols.: Cannesin (Anabasis, Gr.-Finnish), Helsirgissä, +1868, 8vo: Sauppe, Lipsiae, 1869, 8vo. <i>Septuagint</i>: Hutter, +Noribergae, 1598, 4to: Biel, Hagae, 1779-1780, 8vo. <i>New Testament</i>: +Lithocomus, Colon, 1552, 8vo: Parkhurst, ed. Major, London, 1845, +8vo: Schleusner (juxta ed. Lips. quartam), Glasguae, 1824, 4to.</p> + +<p><span class="bold">Medieval and Modern Greek.</span>—Meursius, Lugd. Bat. 1614, 4to: +Critopulos, Stendaliae, 1787, 8vo: Portius, Par. 1635, 4to: Du +Cange, Paris, 1682, fol., 2 vols.; Ludg. 1688, fol. <span class="sc">English</span>.—Polymera, +Hermopolis, 1854, 8vo: Sophocles, Cambr. Mass. +1860-1887: Contopoulos, Athens, 1867, 8vo; Smyrna, 1868-1870, +8vo, 2 parts, 1042 pages. <span class="sc">French</span>.—Skarlatos, Athens, 1852, 4to: +Byzantius, ib. 1856, 8vo, 2 vols.: Varvati, 4th ed. ib., 1860, 8vo. +<span class="sc">Italian</span>.—Germano, Romae, 1622, 8vo: Somavera, Parigi, 1709, +fol., 2 vols.: Pericles, Hermopolis, 1857, 8vo. <span class="sc">German</span>.—Schmidt, +Lips. 1825-1827, 12mo, 2 vols.: Petraris, Leipz. 1897. <span class="sc">Polyglots</span>.—Koniaz +(Russian and Fr.), Moscow, 1811, 4to; Schmidt (Fr.-Germ.), +Leipzig, 1837-1840, 12mo, 3 vols.: Theocharopulas de Patras (Fr.-Eng.), +Munich, 1840, 12mo.</p> + +<p><span class="bold">Latin.</span>—Johannes de Janua, <i>Catholicon</i> or <i>Summa</i>, finished in +1286, printed Moguntiæ 1460, fol.; Venice, 1487; and about 20 +editions before 1500: Johannes, <i>Comprehensorium</i>, Valentia, 1475, +fol.: Nestor Dionysius, <i>Onomasticon</i>, Milan, 1477, fol.: Stephanus, +Paris, 1531, fol., 2 vols.: Gesner, Lips. 1749, fol., 4 vols.: Forcellini, +Patavii, 1771, fol., 4 vols. <span class="sc">Polyglot</span>.—Calepinus, Reggio, 1502, fol. +(Aldus printed 16 editions, with the Greek equivalents of the Latin +words; Venetiis, 1575, fol., added Italian, French and Spanish; +Basileae, 1590, fol., is in 11 languages; several editions, from 1609, +are called Octolingue; many of the latter 2 vol. editions were edited +by John Facciolati): Verantius (Ital., Germ., Dalmatian, Hungarian), +Venetiis, 1595, 4to: Lodereckerus (Ital., Germ., Dalm., Hungar., +Bohem., Polish), Pragae, 1605, 4to. <span class="sc">English</span>.—<i>Promptorium +parvulorum</i>, compiled in 1440 by Galfridus Grammaticus, a Dominican +monk of Lynn Episcopi, in Norfolk, was printed by Pynson, 1499; +8 editions, 1508-1528, ed. Way, Camden Society, 1843-1865, 3 vols. +4to; <i>Medulla grammaticis</i>, probably by the same author, MS. written +1483; printed as <i>Ortus vocabulorum</i>, by Wynkyn de Worde, 1500; +13 editions 1509-1523; Sir Thomas Elyot, London, 1538, fol.; 2nd ed. +1543; <i>Bibliotheca Eliotae</i>, ed. Cooper, ib. 1545, fol.: Huloet, +<i>Abecedarium</i>, London, 1552, fol.; <i>Dictionarie</i>, 1572, fol.: Cooper, +London, 1565, fol.; 4th edition, 1584, fol.: Baret, <i>Alvearie</i>, ib. 1575, +fol.; 1580, fol.: Fleming, ib. 1583, fol.: Ainsworth, London, 1736, +4to; ed. Morell, London, 1796, 4to, 2 vols.; ed. Beatson and Ellis, +ib. 1860, 8vo: Scheller, translated by Riddle, Oxford, 1835, fol.: +Smith, London, 1855, 8vo; 1870: Lewis and Short, Oxford, 1879. +<span class="sc">Eng.-Latin</span>.—Levins, <i>Manipulus puerorum</i>, Lond. 1570, 4to: Riddle, +ib. 1838, 8vo: Smith, ib. 1855, 8vo. <span class="sc">French</span>.—<i>Catholicon parvum</i>, +Geneva, 1487: Estienne, <i>Dictionnaire</i>, Paris, 1539, fol. 675 pages; +enlarged 1549; ed. Huggins, Lond. 1572: Id. <i>Dictionarium Latino-Gallicum</i>, +Lutetiae, 1546, fol.; Paris, 1552; 1560: Id., <i>Dictionariolum +puerorum</i>, Paris, 1542, 4to: <i>Les Mots français</i>, Paris, 1544, 4to; the +copy in the British Museum has the autograph of Queen Catherine +Parr: Thierry (Fr.-Lat.), Paris, 1564, fol.: Danet, Ad usum +Delphini, Paris, 1700, 4to, 2 vols.; and frequently: Quicherat, 9th +ed. Paris, 1857, 8vo: Theil, 3rd ed. Paris, 1863, 8vo: Freund, ib. +1835-1865, 4to, 3 vols. <span class="sc">German</span>.—Joh. Melber, of Gerolzhofen, +<i>Vocabularius praedicantium</i>, of which 26 editions are described by +Hain (<i>Repertorium</i>, No. 11,022, &c.), 15 undated, 7 dated 1480-1495, +4to, and 3 after 1504: <i>Vocabularius gemma gemmarum</i>, Antwerp, +1484, 4to; 1487; 12 editions, 1505-1518: Herman Torentinus, <i>Elucidarius +carminum</i>, Daventri, 1501, 4to; 22 editions, 1504-1536: Binnart, +Ant. 1649, 8vo: Id., <i>Biglotton</i>, ib. 1661; 4th ed. 1688: Faber, ed. +Gesner, Hagae Com. 1735, fol., 2 vols.: Hederick, Lips. 1766, 8vo, +2 vols.: Ingerslev, Braunschweig, 1835-1855, 8vo, 2 vols.: <i>Thesaurus +linguae Latinae</i>, Leipzig, 1900: Walde, <i>Lateinisches etymologisches +Wörterbuch</i>, 1906. <span class="sc">Italian</span>.—Seebar (Sicilian translation of +Lebrixa), Venet. 1525, 8vo: Venuti, 1589, 8vo: Galesini, Venez. +1605, 8vo: Bazzarini and Bellini, Torino, 1864, 4to, 2 vols. 3100 +pages. <span class="sc">Spanish</span>.—Salmanticae, 1494, fol.; Antonio de Lebrixa, +Nebrissenis, Compluti, 1520, fol., 2 vols.: Sanchez de la Ballesta, +Salamanca, 1587, 4to: Valbuena, Madrid, 1826, fol. <span class="sc">Portuguese</span>.—Bluteau, +<span class="pagenum"><a name="page192" id="page192"></a>192</span> +Lisbon, 1712-1728, fol., 10 vols: Fonseca, ib. 1771, fol.: +Ferreira, Paris, 1834, 4to; 1852. <span class="sc">Romansch</span>.—<i>Promptuario di voci +volgari</i>, Valgrisii, 1565, 4to. <span class="sc">Vlach</span>.—Divalitu, Bucuresci, 1852, +8vo. <span class="sc">Swedish</span>.—<i>Vocabula</i>, Rostock, 1574, 8vo; Stockholm, 1579: +Lindblom, Upsala, 1790, 4to. <span class="sc">Dutch</span>.—Binnart, Antw. 1649, 8vo: +Scheller, Lugd. Bat. 1799, 4to, 2 vols. <span class="sc">Flemish</span>.—Paludanus, +Gandavi, 1544, 4to. <span class="sc">Polish</span>.—Macinius, Königsberg, 1564, fol.: +Garszynski, Breslau, 1823, 8vo, 2 vols. <span class="sc">Bohemian</span>.—Johannes +Aquensis, Pilsnae, 1511, 4to: Reschel, Olmucii, 1560-1562, 4to, 2 vols.: +Cnapius, Cracovia, 1661, fol., 3 vols. <span class="sc">Illyrian</span>.—Bellosztenecz, +Zagrab, 1740, 4to: Jambresich (also Germ. and Hungar.), Zagrab, +1742, 4to. <span class="sc">Servian</span>.—Swotlik, Budae, 1721, 8vo. <span class="sc">Hungarian</span>.—Molnar, +Frankf. a. M. 1645, 8vo: Pariz-Papai, Leutschen, 1708, 8vo; +1767. <span class="sc">Finnish</span>.—Rothsen, Helsingissä, 1864, 8vo. <span class="sc">Poetic</span>.—<i>Epithetorum +et synonymorum thesaurus</i>, Paris, 1662, 8vo, attributed +to Chatillon; reprinted by Paul Aler, a German Jesuit, as <i>Gradus ad +Parnassum</i>, Paris, 1687, 8vo; many subsequent editions: <i>Schirach</i>, +Hal. 1768, 8vo: Noel, Paris, 1810, 8vo; 1826: Quicherat, Paris, +1852, 8vo: Young, London, 1856, 8vo. <span class="sc">Erotic</span>.—Rambach, +Stuttgart, 1836, 8vo. <span class="sc">Rhetorical</span>.—Ernesti, Lips. 1797, 8vo. +<span class="sc">Civil Law</span>.—Dirksen, Berolini, 1837, 4to. <span class="sc">Synonyms</span>.—Hill, Edinb. +1804, 4to: Döderlein, Lips. 1826-1828, 8vo, 6 vols. <span class="sc">Etymology</span>.—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: Vaniček, Leipz. 1874, 8vo. +<span class="sc">Barbarous</span>.—Marchellus, Mediol. 1753, 4to; Krebs, Frankf. a. M. +1834, 8vo; 1837. <span class="sc">Particular Authors</span>.—<i>Caesar</i>: Crusius, Hanov. +1838, 8vo. <i>Cicero</i>: Nizzoli, Brescia, 1535, fol.; ed. Facciolati, +Patavii, 1734, fol.; London, 1820, 8vo, 3 vols.: Ernesti, Lips. 1739, +8vo; Halle, 1831. <i>Cornelius Nepos</i>: Schmieder, Halle, 1798, 8vo; +1816: Billerbeck, Hanover, 1825, 8vo. <i>Curtius Rufus</i>: Crusius, +Hanov. 1844, 8vo. <i>Horace</i>: Ernesti, Berlin, 1802-1804, 8vo, 3 vols.: +Döring, Leipz. 1829, 8vo. <i>Justin</i>: Meinecke, Lemgo, 1793, 8vo; 2nd +ed. 1818. <i>Livy</i>: Ernesti, Lips. 1784, 8vo; ed Schäfer, 1804. <i>Ovid</i>: +Gierig, Leipz. 1814: (Metamorphoses) Meinecke, 2nd ed., Lemgo, 1825, +8vo: Billerbeck (Do.), Hanover, 1831, 8vo. <i>Phaedrus</i>: Oertel, +Nürnberg, 1798, 8vo: Hörstel, Leipz. 1803, 8vo: Billerbeck +Hanover, 1828, 8vo. <i>Plautus</i>: Paraeus, Frankf. 1614, 8vo. <i>Pliny</i>: +Denso, Rostock, 1766, 8vo<i>. Pliny, jun.</i>: Wensch, Wittenberg, 1837-1839, +4to. <i>Quintilian</i>: Bonnellus, Leipz. 1834, 8vo. <i>Sallust</i>: +Schneider, Leipz. 1834, 8vo: Crusius, Hanover, 1840, 8vo. <i>Tacitus</i>: +Bötticher, Berlin, 1830, 8vo. <i>Velleius Paterculus</i>: Koch, Leipz. +1857, 8vo. <i>Virgil</i>: <i>Clavis</i>, London, 1742, 8vo: Braunhard, Coburg, +1834, 8vo. <i>Vitruvius</i>: Rode, Leipz. 1679, 4to, 2 vols.: Orsini, +Perugia, 1801, 8vo.</p> + +<p><span class="sc">Old Italian Languages</span>.—Fabretti, Torini, 1858, 4to. <i>Umbrian</i>: +Huschke, Leipz. 1860, 8vo. <i>Oscan and Sabellian</i>: Id. Elberfeld, +1856, 8vo.</p> + +<p><span class="sc">Medieval Latin</span>.—Du Cange, <i>Glossarium</i>, Paris, 1733-1736, fol., +6 vols.; Carpentier, Suppl., Paris, 1766, fol., 4 vols.; ed. Adelung, +Halae, 1772-1784, 8vo, 6 vols.; ed. Henschel, Paris, 1840-1850, 4to, +7 vols. (vol. vii. contains a glossary of Old French): Brinckmeier, +Gotha, 1850-1863, 8vo, 2 vols.: Hildebrand (<i>Glossarium saec. ix.</i>), +Götting. 1854, 4to: Diefenbach, <i>Glossarium</i>, Frankf. 1857, 4to: Id. +<i>Gloss. novum</i>, ib. 1867, 4to. <span class="sc">Ecclesiastical</span>.—Magri, Messina, 1644, +4to; 8th ed. Venezia, 1732; Latin translation, <i>Magri Hierolexicon</i>, +Romae, 1677, fol.; 6th ed. Bologna, 1765, 4to, 2 vols.</p> + +<p class="center pt2"><i>Romance Languages. </i></p> + +<p><span class="bold">Romance Languages generally.</span>—Diez, Bonn, 1853, 8vo; 2nd ed. +ib. 1861-1862, 8vo, 2 vols.; 3rd ed. ib. 1869-1870, 8vo, 2 vols.; +transl. by Donkin, 1864, 8vo.</p> + +<p><span class="bold">French.</span>—Ranconet, <i>Thresor</i>, ed. Nicot, Paris, 1606, fol.; ib. +1618, 4to: Richelet, Genève, 1680, fol., 2 vols.; ed. Gattel, Paris, +1840, 8vo, 2 vols.</p> + +<p>The French Academy, after five years’ 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 <i>Essai</i>, Paris, 1684, +4to; Amst. 1685, 12mo. The Academy charged him with using the +materials they had prepared for their dictionary, and expelled him, on +the 22nd of January 1685, for plagiarism. He died on the 14th of May +1688, in the midst of the consequent controversy and law suit. His +complete work was published, with a preface by Bayle, La Haye and +Rotterdam, 1690, fol., 3 vols.; again edited by Basnage de Beauval, +1701; La Haye, 1707, fol., 4 vols. From the edition of 1701 the +very popular so-called <i>Dictionnaire de Trevoux</i>, Trevoux, 1704, fol., +2 vols., was made by the Jesuits, who excluded everything that +seemed to favour the Calvinism of Basnage. The last of its many +editions is Paris, 1771, fol., 8 vols. The Academy’s dictionary was +first printed Paris, 1694, fol., 2 vols. They began the revision in 1700; +second edition 1718, fol., 2 vols.; 3rd, 1740, fol., 2 vols.; 6th, 1835, +2 vols. 4to, reprinted 1855; Supplément, by F. Raymond, 1836, +4to; Complément, 1842, 4to, reprinted 1856; <i>Dictionnaire historique</i>, +Paris, 1858-1865, 4to, 2 parts (A to Actu), 795 pages, published by the +Institut: Dochez, Paris, 1859, 4to: Bescherelle, ib. 1844, 4to, 2 vols.; +5th ed. Paris, 1857, 4to, 2 vols.; 1865; 1887: Landais, Paris, 1835; +12th ed. ib. 1854, 4to, 2 vols.: Littré, Paris, 1863-1873, 4to, 4 vols. +7118 pages: Supplément, Paris, 1877, 4to: Godefroy (with dialects +from 9th to 15th cent.), Paris, 1881-1895, and <i>Complément</i>: Hatzfield, +Darmesteter, and Thomas, Paris, 1890-1900: Larive and Fleury, +(<i>mots et choses, illustré</i>), Paris, 1884-1891. <span class="sc">English</span>.—Palsgrave, +<i>Lesclaircissement de la langue francoyse</i>, London, 1530, 4to, 2 parts; +1852: Hollyband, London, 1533, 4to: Cotgrave, ib. 1611, fol.: +Boyer, La Haye, 1702, 4to, 2 vols.; 37th ed. Paris, 1851, 8vo, 2 vols.: +Fleming and Tibbins, Paris, 1846-1849, 4to, 2 vols.; ib. 1854, 4to, +2 vols.; ib. 1870-1872, 4to, 2 vols.: Tarver, London, 1853-1854, +8vo, 2 vols.; 1867-1872: Bellows, Gloucester, 1873, 16mo; ib. +1876. <span class="sc">Ideological, or Analogical</span>.—Robertson, Paris, 1859, 8vo: +Boissière, Paris, 1862, 8vo. <span class="sc">Etymology</span>.—Lebon, Paris, 1571, 8vo: +Ménage, ib. 1650, 4to. Pougens projected a <i>Trésor des origines</i>, his +extracts for which, filling nearly 100 volumes folio, are in the library +of the Institut. He published a specimen, Paris, 1819, 4to. After +his death, <i>Archéologie française</i>, Paris, 1821, 8vo, 2 vols., was compiled +from his MSS., which were much used by Littré: Scheler, +Bruxelles, 1862, 8vo; 1873: Brachet, 2nd ed. Paris, 1870, 12mo; +English trans. Kitchin, Oxf. 1866, 8vo. <span class="sc">Greek Words</span>.—Trippault, +Orleans, 1580, 8vo: Morin, Paris, 1809, 8vo. <span class="sc">German Words</span>.—Atzler, +Cöthen, 1867, 8vo. <span class="sc">Oriental Words</span>.—Pihan, Paris, 1847, +8vo; 1866: Devic, ib. 1876, 8vo. <span class="sc">Neology</span>.—Desfontaines, 3rd ed. +Amst. 1728, 12mo: Mercier, Paris, 1801, 8vo, 2 vols.: Richard, ib. +1842, 8vo; 2nd ed. 1845. <span class="sc">Poetic</span>.—<i>Dict. des rimes</i> (by La Noue), +Geneve, 1596, 8vo; Cologny, 1624, 8vo: Carpentier, <i>Le Gradus +français</i>, Paris, 1825, 8vo, 2 vols. <span class="sc">Erotic</span>.—De Landes, Bruxelles, +1861, 12mo. <span class="sc">Oratory</span>.—Demandre and Fontenai, Paris, 1802, 8vo: +Planche, ib. 1819-1820, 8vo, 3 vols. <span class="sc">Pronunciation</span>.—Féline, ib. +1857, 8vo. <span class="sc">Double Forms</span>.—Brachet, ib. 1871, 8vo. <span class="sc">Epithets</span>.—Daire, +ib. 1817, 8vo. <span class="sc">Verbs</span>.—Bescherelle, ib. 1855, 8vo, 2 vols.: +3rd ed. 1858. <span class="sc">Participles</span>.—Id., ib. 1861, 12mo. <span class="sc">Difficulties</span>.—Boiste, +London, 1828, 12mo: Laveaux, Paris, 1872, 8vo, 843 pages. +<span class="sc">Synonyms</span>.—Boinvilliers, Paris, 1826, 8vo: Lafaye, ib. 1858, +8vo; 1861; 1869: Guizot, ib. 1809, 8vo; 6th ed. 1863; 1873. +<span class="sc">Homonyms</span>.—Zlatagorski (Germ., Russian, Eng.), Leipzig, 1862, +8vo, 664 pages. <span class="sc">Imitative Words</span>.—Nodier, <i>Onomatopées</i>, ib. 1828, +8vo. <span class="sc">Technology</span>.—D’Hautel, ib. 1808, 8vo, 2 vols.: Desgranges, +ib. 1821, 8vo: Tolhausen (Fr., Eng., Germ.), Leipz. 1873, 8vo, 3 vols. +<span class="sc">Faults of Expression</span>.—Roland, Gap, 1823, 8vo: Blondin, Paris, +1823, 8vo. <span class="sc">Particular Authors</span>.—<i>Corneille</i>: Godefroy, ib. 1862, +8vo, 2 vols.: Marty-Laveaux, ib. 1868, 8vo, 2 vols. <i>La Fontaine</i>: +Lorin, ib. 1852, 8vo. <i>Malherbe</i>: Regnier, ib. 1869, 8vo. <i>Molière</i>: +Genin, ib. 1846, 8vo: Marty-Laveaux, ib. 8vo. <i>Racine</i>: Marty-Laveaux, +ib. 1873, 8vo, 2 vols. <i>M<span class="sp">me</span> de Sévigné</i>: Sommer, ib. 1867, +8vo, 2 vols. <span class="sc">Old French</span>.—La Curne de St Palaye prepared a +dictionary, of which he only published <i>Projet d’un glossaire</i>, Paris, +1756, 4to. His MSS. in many volumes are in the National Library, +and were much used by Littré. They were printed by L. Favre, and +fasciculi 21-30 (tom. iii.), Niort, 4to, 484 pages, were published in +February 1877. Lacombe (vieux langage), Paris, 1766, 2 vols. 4to: +Kelham (Norman and Old French), London, 1779, 8vo: Roquefort +(langue romane), Paris, 1808, 8vo; Supplément, ib. 1820, 8vo: +Pougens, <i>Archéologie</i>, ib. 1821, 8vo, 2 vols.: Burguy, Berlin, 1851-1856, +8vo, 3 vols.: Laborde (<i>Notice des émaux ... du Louvre</i>, part ii.), +Paris, 1853, 8vo, 564 pages:<a name="fa3d" id="fa3d" href="#ft3d"><span class="sp">3</span></a> Gachet (rhymed chronicles), Bruxelles, +1859, 4to: Le Héricher (Norman, English and French), Paris, 1862, +3 vols. 8vo: Hippeau (12th and 13th centuries), Paris, 1875, 8vo. +<span class="sc">Dialects</span>.—Jaubert (central), Paris, 1856-1857, 8vo, 2 vols.: +Baumgarten (north and centre), Coblentz, 1870, 8vo: Azais, <i>Idiomes +romans du midi</i>, Montpellier, 1877. <i>Austrasian</i>: François. Metz, +1773, 8vo. <i>Auvergne</i>: Mège, Riom, 1861, 12mo. <i>Bearn</i>: Lespi, Pau, +1858, 8vo. <i>Beaucaire</i>: Bonnet (Bouguirén), Nismes, 1840, 8vo. +<i>Pays de Bray</i>: Decorde, Neufchâtel, 1852, 8vo. <i>Burgundy</i>: +Mignard, Dijon, 1870, 8vo. <i>Pays de Castres</i>: Couzinié, Castres, +1850, 4to. <i>Dauphiné</i>: Champollion-Figeac, Paris, 1809, 8vo: Jules, +Valence, 1835, 8vo; Paris, 1840, 4to. <i>Dep. of Doubs</i>: Tissot +(Patois des Fourg, arr. de Pontarlier) Besançon, 1865, 8vo. +<i>Forez</i>: Gras, Paris, 1864, 8vo; Neolas, Lyon, 1865, 8vo. <i>Franche +Comté</i>: Maisonforte, 2nd ed. Besançon, 1753, 8vo. <i>Gascony</i>: Desgrouais +(Gasconismes corrigés), Toulouse, 1766, 8vo; 1769; 1812, +12mo, 2 vols.; 1825, 8vo, 2 vols. <i>Dep. of Gers</i>: Cenac-Montaut, Paris, +1863, 8vo. <i>Geneva</i>: Humbert, Geneve, 1820, 8vo. <i>Languedoc</i>: Odde, +Tolose, 1578, 8vo: Doujat, Toulouse, 1638, 8vo: De S.[auvages], +Nismes, 1756, 2 vols.; 1785; Alais, 1820: Azais, Beziers, 1876, +&c., 8vo: Hombres, Alais, 1872, 4to: Thomas (<i>Greek words</i>) Montpellier, +1843, 4to. <i>Liége</i>: Forir, Liége, 1866, 8vo, vol i. 455 pages. +<i>Lille</i>: Vermesse, Lille, 1861, 12mo: Debuire du Buc ib., 1867, +8vo. <i>Limousin</i>: Beronie, ed. Vialle (Corrèze), Tulle, 1823, 4to. +<span class="pagenum"><a name="page193" id="page193"></a>193</span> +<i>Lyonnais, Forez, Beaujolais</i>: Onofrio, Lyon, 1864, 8vo. <i>Haut +Maine</i>: R[aoul] de M.[ontesson], Paris, 1857; 1859, 503 pages. +<i>Mentone</i>: Andrews, Nice, 1877, 12mo. <i>Dep. de la Meuse</i>: Cordier, +Paris, 1853, 8vo. <i>Norman</i>: Edélestand and Alfred Duméril, Caen, +1849, 8vo: Dubois, ib. 1857, 8vo: Le Héricher (<i>Philologie topographique</i>), +Caen, 1863, 4to: Id. (éléments scandinaves), Avranches, +1861, 12mo: Metivier (Guernsey), London, 1870, 8vo: Vasnier +(arrond de Pont Audemer), Rouen, 1861, 8vo: Delboulle (Vallée +d’Yères), Le Havre, 1876. <i>Picardy</i>: Corblet, Amiens, 1851, 8vo. +<i>Poitou, Saintonge, Aunis</i>: Favre, Niort, 1867, 8vo. <i>Poitou</i>: +Beauchet-Filleau, Paris, 1864, 8vo: Levrier, Niort, 1867, 8vo: +Lalanne, Poitiers, 1868, 8vo. <i>Saintonge</i>: Boucherie, Angoulême, +1865, 8vo: Jonain, Royan, 1867, 8vo. <i>Savoy</i>: Pont (Terratzu de +la Tarantaise), Chambery, 1869, 8vo. <i>La Suisse Romande</i>: Bridel, +Lausanne, 1866, 8vo. <i>Dep. of Tarn</i>: Gary, Castre, 1845, 8vo. <i>Dep. +of Vaucluse</i>: Barjavel, Carpentras, 1849, 8vo. <i>Walloon (Rouchi)</i>: +Cambresier, Liége, 1787, 8vo: Grandgagnage, ib. 1845-1850, 8vo. +2 vols.: Chavée, Paris, 1857, 18mo: Vermesse, Doudi, 1867, 8vo. +Sigart (<i>Montois</i>), Bruxelles, 1870, 8vo. <span class="sc">Slang</span>.—Oudin, <i>Curiositez +Françaises</i>, Paris, 1640, 8vo: Baudeau de Saumaise (Précieuses, +Langue de Ruelles), Paris, 1660, 12mo; ed. Livet, ib. 1856: Le +Roux, <i>Dict. Comique</i>, Amst. 1788, and 6 other editions: Carême +Prenant [<i>i.e.</i> Taumaise], (argot réforme), Paris, 1829, 8vo: Larchey +(excentricitées du langage), Paris, 1860, 12mo; 5th ed. 1865: +Delvau (langue verte, Parisian), Paris, 1867, 8vo: Larchey, Paris, +1873, 4to, 236 pages.</p> + +<p><span class="bold">Provençal.</span>—Pallas, Avignon, 1723, 4to: Bastero, <i>La Crusca Provenzale</i>, +Roma, 1724, fol. vol. i. only: Raynouard, Paris, 1836-1844, +8vo, 6 vols.: Garcin, Draguignand, 1841, 8vo, 2 vols.: Honnorat, +Digne, 1846-1849, 4to, 4 vols. 107,201 words: Id., <i>Vocab. fr. prov.</i>, +ib. 1848, 12mo, 1174 pages.</p> + +<p><span class="bold">Spanish.</span>—Covarruvias Orosco, Madrid, 1611, fol.: ib. 1673-1674, +fol. 2 vols.; Academia Española, Madrid, 1726-1739, fol. 6 vols.; 8th +ed. 1837: Caballero, Madrid, 1849, fol.; 8th ed. ib. 1860, 4to, 2 vols.: +Cuesta, ib. 1872, fol. 2 vols.: Campano, Paris, 1876, 18mo, 1015 pages. +Cuervo, 1886-1894; Monlau, 1881; Zerola, Toro y Gomes, and Isaza, +1895; Serrano (encyclopaedic) 1876-1881. <span class="sc">English</span>.—Percivall, +London, 1591, 4to: Pineda, London, 1740, fol.: Connelly and +Higgins, Madrid, 1797-1798, 4to, 4 vols.: Neuman and Baretti, 9th ed. +London, 1831, 8vo, 2 vols.; 1874. <span class="sc">French</span>.—Oudin, Paris, 1607, 4to, +1660; Gattel, Lyon, 1803, 4to, 2 vols.: Dominguez, Madrid, 1846, +8vo, 6 vols.: Blanc, Paris, 1862, 8vo, 2 vols. <span class="sc">German</span>.—Wagener, +Hamb. 1801-1805, 8vo, 4 vols.: Seckendorp, ib. 1823, 8vo, 3 vols.: +Franceson, 3rd ed. Leipzig, 1862, 8vo, 2 vols. <span class="sc">Italian</span>.—Franciosini, +Venezia, 1735, 8vo, 2 vols.; Cormon y Manni, Leon, 1843, 16mo, +2 vols.: Romero, Madrid, 1844, 4to. <span class="sc">Synonyms</span>.—<i>Diccionario de +Sinonimos</i>, Paris, 1853, 4to. <span class="sc">Etymology</span>.—Aldrete, Madrid, 1682, +fol.: Monlau y Roca, ib. 1856, 12mo; Barcia, 1881-1883. <span class="sc">Arabic +Words</span>.—Hammer Purgstall, Wien, 1855, 8vo: Dozy and Engelmann, +2d ed. Leiden, 1869, 8vo. <span class="sc">Ancient</span>.—Sanchez, Paris, 1842, +8vo. <span class="sc">Rhyming</span>.—Garcia de Rengifo (consonancias) Salmantica, +1592, 4to; 1876. <span class="sc">Don Quixote</span>.—Beneke (German), Leipzig, 1800, +16mo; 4th ed. Berlin, 1841, 16mo. <span class="sc">Dialects</span>.—<i>Aragonese</i>: Peralta, +Zaragoza, 1836, 8vo: Borao, ib. 1859, 4to. <i>Catalan</i>: Rocha de +Girona (Latin), Barcinone, 1561, fol.: <i>Dictionari Catala</i> (Lat. Fr. +Span.), Barcelona, 1642, 8vo: Lacavalleria (Cat.-Lat.), ib. 1696, fol.: +Esteve, ed. Belvitges, &c. (Catal.-Sp. Lat.), Barcelona, 1805-1835, +fol. 2 vols.: Saura (Cat.-Span.), ib. 1851, 16mo; 2nd ed.(Span.-Cat.), +ib. 1854; 3rd ed. (id.) ib. 1862, 8vo: Labernia, ib. 1844-1848, 8vo, 2 +vols. 1864. <i>Gallegan</i>: Rodriguez, Coruña, 1863, 4to: Cuveira y Piñol, +Madrid, 1877, 8vo. Majorca: Figuera, Palma, 1840, 4to: Amengual, +ib. 1845, 4to. <i>Minorca</i>: <i>Diccionario</i>, Madrid, 1848, 8vo. <i>Valencian</i>: +Palmyreno, Valentiae, 1569: Ros, Valencia, 1764, 8vo: Fuster, ib. +1827, 8vo: Lamarca, 2nd ed. ib. 1842, 16mo. <i>Cuba</i>: <i>Glossary of +Creole Words</i>, London, 1840, 8vo: Pichardo, 1836; 2nd ed. Havana, +1849, 8vo; 3rd ed. ib. 1862, 8vo; Madrid, 1860, 4to.</p> + +<p><span class="bold">Portuguese.</span>—Lima, Lisbon, 1783, 4to: Moraes da Silva, ib. +1789, 4to, 2 vols.; 6th ed. 1858: Academia real das Sciencas, ib. +1793, tom. i., ccvi. and 544 pages (A to Azurrar); Faria, ib. 1849, +fol. 2 vols.; 3rd ed. ib. 1850-1857, fol. 2 vols. 2220 pages. <span class="sc">English</span>.—Vieyra, +London, 1773, 2 vols. 4to: Lacerda, Lisboa, 1866-1871, 4to, +2 vols. <span class="sc">French</span>.—Marquez, Lisboa, 1756-1761, fol. 2 vols.: Roquette, +Paris, 1841, 8vo, 2 vols.; 4th ed. 1860: Marques, Lisbonne, 1875, +fol. 2 vols.: Souza Pinto, Paris, 1877, 32mo, 1024 pages. <span class="sc">German</span>.—Wagener, +Leipzig, 1811-1812, 8vo, 2 vols.: Wollheim, ib. 1844, 12mo, +2 vols.: Bösche, Hamburg, 1858, 8vo, 2 vols. 1660 pages. <span class="sc">Italian</span>.—Costa +e Sá, Lisboa, 1773-1774, fol. 2 vols. 1652 pages: Prefumo, +Lisboa, 1853, 8vo, 1162 pages. <span class="sc">Ancient</span>.—Joaquim de Sancta Rosa +de Viterbo, ib. 1798, fol. 2 vols.; 1824, 8vo. <span class="sc">Arabic Words</span>.—Souza, +ib. 1789, 4to; 2nd ed. by S. Antonio Moura, ib. 1830, 224 pages. +<span class="sc">Oriental and African Words, not Arabic</span>.—Saõ Luiz, ib. 1837, +4to, 123 pages. <span class="sc">French Words</span>.—Id., ib. 1827, 4to; 2nd ed. Rio de +Janeiro, 1835, 8vo. <span class="sc">Synonyms</span>.—Id., ib. 1821, 4to; 2nd ed. ib. +1824-1828, 8vo. Fonseca, Paris, 1833, 8vo; 1859, 18mo, 863 pages. +<span class="sc">Homonyms</span>.—De Couto, Lisboa, 1842, fol. <span class="sc">Poetic</span>.—Luzitano (<i>i.e.</i> +Freire), ib. 1765, 8vo, 2 vols.; 3rd ed. ib. 1820, 4to, 2 vols. <span class="sc">Rhyming</span>.—Couto +Guerreiro, Lisboa, 1763, 4to. <span class="sc">Naval</span>.—Tiberghien, Rio de +Janeiro, 1870, 8vo. <span class="sc">Ceylon-Portuguese</span>.—Fox, Colombo, 1819, +8vo: Callaway, ib. 1823, 8vo.</p> + +<p><span class="bold">Italian.</span>—Accarigi, <i>Vocabulario</i>, Cento, 1543, 4to: Alunno, <i>La</i> +<i>fabrica del mundo</i>, Vinezia, 1548, fol.: Porccachi, Venetia, 1588, fol.: +Accademici della Crusca, <i>Vocabulario</i>, Venez. 1612, fol.; 4th ed. +Firenze, 1729-1738, fol. 6 vols.: Costa and Cardinali, Bologna, 1819-1826, +4to, 7 vols.: Tommaseo and Bellini, Torino, 1861, &c., 4to, 4 +vols.: Petrocchi, 1884-1891. <span class="sc">English.</span>—Thomas, London, 1598, 4to: +Florio, London, 1598, 4to, 1611: Baretti, London, 1794, 2 vols.: +1854, 8vo, 2 vols.: Petronj and Davenport, Londra, 1828, 8vo, 3 vols.: +Grassi, Leipz. 1854, 12mo: Millhouse, Lond., 1868, 8vo, 2 vols. 1348 +pages. <span class="sc">French.</span>—Alberti, Paris, 1771, 4to, 2 vols.; Milan, 1862: +Barberi, Paris, 1838, 4to, 2 vols.: Renzi, Paris, 1850, 8vo. <span class="sc">German.</span>—<i>Libro +utilissimo</i>, Venetiis, 1499, 4to: Valentini, Leipzig, 1834-1836, +4to, 4 vols. <span class="sc">Etymology.</span>—Menage, Geneva, 1685, fol.: Bolza, Vienna, +1852, 4to. <span class="sc">Provençal Words.</span>—Nannucci, Firenze, 1840, 8vo. +<span class="sc">Synonyms.</span>—Rabbi, Venezia, 1774, 4to; 10th ed. 1817; Tommaseo, +Firenze, 1839-1840, 4to, 2 vols.: Milano, 1856, 8vo; 1867. <span class="sc">Verbs.</span>—Mastrofini, +Roma, 1814, 4to, 2 vols. <span class="sc">Select Words and Phrases.</span>—Redi, +Brescia, 1769, 8vo. <span class="sc">Incorrect Words and Phrases.</span>—Molassi, +Parma, 1830-1841, 8vo, 854 pages. <span class="sc">Supposed Gallicisms.</span>—Viani, +Firenze, 1858-1860, 8vo, 2 vols. <span class="sc">Additions To the Dictionaries.</span>—Gherardini, +Milano, 1819-1821, 8vo, 2 vols.; ib. 1852-1857, +8vo, 6 vols. <span class="sc">Rhyming.</span>—Falco, Napoli, 1535, 4to: Ruscelli, Venetia, +1563, 8vo; 1827: Stigliani, Roma, 1658, 8vo: Rosasco, Padova, +1763, 4to; Palermo, 1840, 8vo. <span class="sc">Technical.</span>—Bonavilla-Aquilino, +Mil. 1819-1821, 8vo, 5 vols.; 2nd ed. 1829-1831, 4to, 2 vols.: Vogtberg +(Germ.), Wein, 1831, 8vo. <span class="sc">Particular Authors.</span>—<i>Boccaccio</i>: +Aluno, <i>Le ricchezze della lingua volgare</i>, Vinegia, 1543. fol. <i>Dante</i>: +Blanc, Leipzig, 1852, 8vo; Firenze, 1859, 8vo. <span class="sc">Dialects.</span>—<i>Bergamo</i>: +Gasparini, Mediol. 1565: Zappetini, Bergamo, 1859, 8vo: +Tiraboschi (anc. and mod.), Turin, 1873, 8vo. <i>Bologna</i>: Bumaldi, +Bologna, 1660, 12mo: Ferrari, ib. 1820, 8vo; 1838, 4to. <i>Brescia</i>: +Gagliardi, Brescia, 1759, 8vo: Melchiori, ib. 1817-1820, 8vo: <i>Vocabularietto</i>, +ib. 1872, 4to. <i>Como</i>: Monti, Milano, 1845, 8vo. <i>Ferrara</i>: +Manini, Ferrara, 1805, 8vo: Azzi, ib. 1857, 8vo. <i>Friuli</i>: Scala, +Pordenone, 1870, 8vo. <i>Genoa</i>: Casaccia, Gen. 1842-1851, 8vo; 1873, +&c.: Paganini, ib. 1857, 8vo. <i>Lombardy</i>: Margharini, Tuderti, +1870, 8vo. <i>Mantua</i>: Cherubini, Milano, 1827, 4to. <i>Milan</i>: Varon, +ib. 1606, 8vo: Cherubini, ib. 1814, 8vo, 2 vols.; 1841-1844, 8vo, +4 vols.; 1851-1861, 8vo, 5 vols.: Banfi, ib. 1857, 8vo: 1870, 8vo. +<i>Modena</i>: Galvani, Modena, 1868, 8vo. <i>Naples</i>: Galiani, Napoli, +1789, 12mo, 2 vols. <i>Parma</i>: Peschieri, Parma, 1828-1831, 8vo, 3 +vols. 1840; Malespina, ib. 1856, 8vo, 2 vols. <i>Pavia</i>: <i>Dizionario domestico +pavese</i>, Pavia, 1829, 8vo: Gambini, ib. 1850, 4to, 346 pages. +<i>Piacenza</i>: Nicolli, Piacenza, 1832: Foresti, ib. 1837-1838, 8vo, 2 pts. +<i>Piedmont</i>: Pino, Torino, 1784, 4to: Capello (Fr.), Turin, 1814, 8vo, +2 pts.: Zalli (Ital. Lat. Fr.), Carmagnola, 1815, 8vo, 2 vols: Sant’ +Albino, Torino, 1860, 4to. <i>Reggio</i>: <i>Vocabulario Reggiano</i>, 1832. +<i>Romagna</i>: Morri, Fienza, 1840. <i>Rome</i>: <i>Raccolto di voci Romani e +Marchiani</i>, Osimo, 1769, 8vo. <i>Roveretano and Trentino</i>: Azzolini, +Venezia, 1856, 8vo. <i>Sardinia</i>: Porru, Casteddu, 1832, fol.: Spano, +Cagliari, 1851-1852, fol. 3 vols. <i>Sicily</i>: Bono (It. Lat.), Palermo, +1751-1754, 4to, 3 vols.; 1783-1785, 4to, 5 vols.: Pasqualino, ib. 1785-1795, +4to, 5 vols.: Mortillaro, ib. 1853, 4to, 956 pages: Biundi, ib. 1857, +12mo, 578 pages: Traina, ib. 1870, 8vo. <i>Siena</i>: Barbagli, Siena, +1602, 4to. <i>Taranto</i>: Vincentiis, Taranto, 1872, 8vo. <i>Turin</i>: +Somis di Chavrie, Torino, 1843, 8vo. <i>Tuscany</i>: Luna, Napoli, 1536, +4to: Politi, Roma, 1604, 8vo; Venezia, 1615; 1628; 1665; Paulo, +ib. 1740, 4to. <i>Vaudois</i>: Callet, Lausanne, 1862, 12mo. <i>Venetian</i>: +Patriarchi (<i>Veneziano e padevano</i>), Padova, 1755, 4to; 1796, 1821: +Boerio, Venezia, 1829, 4to; 1858-1859; 1861. <i>Verona</i>: Angeli, +Verona, 1821, 8vo. <i>Vicenza</i>: Conti, Vicenza, 1871, 8vo. <span class="sc">Lingua +Franca.</span>—<i>Dictionnaire de la langue Franque, ou Petit Mauresque</i>, +Marseille, 1830, 16mo, 107 pages. <span class="sc">Slang.</span>—Sabio (lingua Zerga), +Venetia, 1556, 8vo; 1575: <i>Trattato degli bianti</i>, Pisa, 1828, 8vo.</p> + +<p><span class="bold">Romansh.</span>—<i>Promptuario de voci volgari e Latine</i>, Valgrisii, +1565, 4to: <i>Der, die, das, oder Nomenclatura</i> (German nouns +explained in Rom.), Scoul, 1744, 8vo: Conradi, Zurich, 1820, 8vo; +1826, 12mo, 2 vols.: Carisch, Chur, 1821, 8vo; 1852, 16mo.</p> + +<p><span class="bold">Vlach.</span>—<i>Lesicon Rumanese</i> (Lat. Hung. Germ.), Budae, 1825, +4to: Bobb (Lat. Hung.), Clus, 1822-1823, 4to, 2 vols. <span class="sc">French.</span>—Vaillant, +Boucoureshti, 1840, 8vo: Poyenar, Aaron and Hill, +Boucourest, 1840-1841, 4to, 2 vols.; Jassi, 1852, 16mo, 2 vols.: +De Pontbriant, Bucuresci, 1862, 8vo: Cihac, Frankf. 1870, 8vo: +Costinescu, Bucuresci, 1870, 8vo, 724 pages: Antonescu, Bucharest, +1874, 16mo, 2 vols. 919 pages. <span class="sc">German.</span>—Clemens, Hermanstadt, +1823, 8vo: Isser, Kronstadt, 1850: Polyzu, ib. 1857, 8vo.</p> + +<p class="center pt2"><span class="sc">Teutonic</span>: (1) <i>Scandinavian.</i></p> + +<p><span class="bold">Icelandic.</span>—<span class="sc">Latin.</span>—Andreae, Havniae, 1683, 8vo: Halderson +(Lat. Danish), ib. 1814, 4to, 2 vols. <span class="sc">English.</span>—Cleasby-Vigfusson, +Oxford, 1874, 4to. <span class="sc">German.</span>—Dieterich, Stockholm, 1844, 8vo: +Möbius, Leipzig, 1866, 8vo. <span class="sc">Danish.</span>—Jonssen, Kjöbenhavn, 1863, +8vo. <span class="sc">Norwegian.</span>—Kraft, Christiania, 1863, 8vo: Fritzner, +Kristiania, 1867, 8vo. <span class="sc">Poetic.</span>—Egilsson (Latin), Hafniae, 1860, +8vo; 1864.</p> + +<p><span class="bold">Swedish.</span>—Kindblad, Stockholm, 1840, 4to: Almqvist, Örebro, +1842-1844, 8vo: Dalin, <i>Ordbog.</i> Stockholm, 1850-1853, 8vo, 2 vols. +1668 pages; 1867, &c. 4to (vol. i. ii., A to Fjermare, 928 pages): +Id., <i>Handordbog</i>, ib. 1868, 12mo, 804 pages; Svenska Academien. +Stockholm, 1870, 4to (A) pp. 187. <span class="sc">Latin.</span>—Stjernhjelm, Holm, +1643, 4to: Verelius, Upsala, 1691, 8vo: Ihre (Sueo-Gothicum), +<span class="pagenum"><a name="page194" id="page194"></a>194</span> +Upsala, 1769, fol. 2 vols. <span class="sc">English</span>.—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. <span class="sc">French</span>.—Möller, Stockholm, 1745, 4to: +Björkengren, ib. 1795, 2 vols.: Nordforss, ib. 1805, 8vo, 2 vols.: 2nd +ed. Örebro, 1827, 12mo: West, Stockh. 1807, 8vo: Dalin, ib. 1842-1843, +4to, 2 vols.; 1872. <span class="sc">German</span>.—Dähnert, Holmiae, 1746, 4to: +Heinrich, Christiansund, 1814, 4to, 2 vols.; 4th ed. Örebro, 1841, +12mo: Helms, Leipzig, 1858, 8vo; 1872. <span class="sc">Danish</span>.—Höst, +Kjöbenhavn, 1799, 4to: Welander, Stockholm, 1844, 8vo: Dalin, +ib. 1869, 16mo: Kaper, Kjöbenhavn, 1876, 16mo. <span class="sc">Etymology</span>.—Tamm, +Upsala, 1874, &c., 8vo (A and B), 200 pages. <span class="sc">Foreign +Words</span>.—Sahlstedt, Wästerås, 1769, 8vo: Andersson (20,000), +Stockholm, 1857, 16mo: Tullberg, ib. 1868, 8vo: Ekbohrn, ib. 1870, +12mo: Dalin, ib. 1870, &c., 8vo. <span class="sc">Synonyms</span>.—Id., ib. 1870, 12mo. +<span class="sc">Naval</span>.—Ramsten, ib. 1866, 8vo. <span class="sc">Technical</span>.—Jungberg, ib. 1873, +8vo. <span class="sc">Dialects</span>.—Ihre, Upsala, 1766, 4to: Rietz, Lund, 1862-1867, +4to, 859 pages. <i>Bohuslän</i>: <i>Idioticon Bohusiense</i>, Götaborg, 1776, +4to. <i>Dalecarlia</i>: Arborelius, Upsala, 1813, 4to. <i>Gothland</i>: Hof +(Sven), Stockholmiae, 1772, 8vo: Rääf (Ydre), Örebro, 1859, 8vo. +<i>Halland</i>: Möller, Lund, 158, 8vo. <i>Helsingland</i>: Lenström, ib. +1841, 8vo: Fornminnessällskap, Hudikswall, 1870, 8vo.</p> + +<p><span class="bold">Norwegian.</span>—Jenssen, Kjöbenhavn, 1646, 8vo: Pontoppidan, +Bergen, 1749, 8vo: Hanson (German), Christiania, 1840, 8vo: +Aasen, ib. 1873, 8vo, 992 pages.</p> + +<p><span class="bold">Danish.</span>—Aphelen, Kopenh, 1764, 4to, 2 vols.; 1775, 4to, 3 vols.: +Molbech, Kjöbenhavn, 1833, 8vo, 2 vols.: ib. 1859, 2 vols.: Videnskabernes +Selskab, ib. 1793-1865, Kalkar. <span class="sc">English</span>.—Berthelson +(Eng. Dan.), 1754, 4to: Wolff, London, 1779, 4to. Bay, ib. 1807, +8vo, 2 vols.; 1824, 8vo: Hornbeck, ib. 1863, 8vo: Ferrall and Repp, +ib. 1814, 16mo; 1873, 8vo: Rosing, Copenhagen, 1869, 8vo: Ancker, +ib. 1874, 8vo. <span class="sc">French</span>.—Aphelen, 1754, 8vo: Id., ib. 1759, +4to, 2 vols.; 2nd ed. 1772-1777, vol. i. ii. <span class="sc">German</span>.—Id., ib. 1764, +4to, 2 vols.: Grönberg, 2nd ed. Kopenh. 1836-1839, 12mo, 2 vols.; +1851, Helms, Leipzig, 1858, 8vo. <span class="sc">Synonyms</span>.—Müller, Kjöbenhavn, +1853, 8vo. <span class="sc">Foreign Words</span>.—Hansen, Christiania, 1842, 12mo. +<span class="sc">Naval</span>.—Wilsoet, Copenhagen, 1830, 8vo: Fisker (French), +Kjöbenhavn, 1839, 8vo. <span class="sc">Old Danish</span>.—Molbech, ib. 1857-1868, +8vo, 2 vols. <span class="sc">Dialects</span>.—Id., ib. 1841, 8vo. <i>Bornholm</i>: Adler, <i>ib.</i> +1856, 8vo. <i>South Jutland</i>: Kok, 1867, 8vo. <span class="sc">Slang</span>.—Kristiansen +(Gadesproget), ib. 1866, 8vo. p. 452.</p> + +<p class="center pt2">(2) <i>Germanic.</i></p> + +<p><span class="bold">Teutonic.</span>—<span class="sc">Comparative</span>.—Meidinger, Frankf. a. M. 1833, 8vo, +2nd ed. 1836, 8vo.</p> + +<p><span class="bold">Gothic.</span>—Junius, Dortrecht, 1665, 4to: 1671; 1684, Diefenbach +(comparative), Franckf. a. M. 1846-1851, 2 vols. 8vo: Schulze, +Magdeburg, 1848, 4to: 1867, 8vo: Skeat, London, 1868, 4to: +Balg (<i>Comparative Glossary</i>), Magvike, Wisconsin, 1887-1889. +<span class="sc">Ulphilas</span> (editions with dictionaries).—Castilionaeus, Mediol, 1829, +4to: Gabelentz and Löbe, Altenburg, 1836-1843, 4to, 2 vols.: Gaugengigl, +Passau, 1848, 8vo: Stamm, Paderborn, 1857: Stamm and +Heyne, ib. 1866, 8vo.</p> + +<p><span class="bold">Anglo-Saxon.</span>—<span class="sc">Latin</span>.—Somner (Lat. Eng.), Oxonii, 1659, +fol.: Benson, ib. 1701, 8vo: Lye (A.-S. and Gothic), London, 1772, +fol. 2 vols.: Ettmüller, Quedlinburg, 1851, 8vo. 838 pages. <span class="sc">English</span>.—Bosworth, +London, 1838, 8vo, 721 pages: Id. (<i>Compendious</i>), +1848, 278 pages. Corson (A.-S. and Early English), New York, 1871, +8vo, 587 pages; Toller (based on Bosworth), Oxford, 1882-1898. +<span class="sc">German</span>.—Bouterwek, Gütersloh, 1850, 8vo, 418 pages: Grein +(Poets), Göttingen, 1861-1863, 8vo, 2 vols.: Leo, Halle, 1872, 8vo.</p> + +<p><span class="bold">English.</span>—Cockeram, London, 1623, 8vo: 9th ed. 1650: Blount, +ib. 1656, 8vo: Philips, The new World of Words, London, 1658, fol.: +Bailey, London, 1721, 8vo; 2nd ed. ib. 1736, fol.; 24th ed. ib. 1782, +8vo: Johnson, ib. 1755, fol. 2 vols.; ed. Todd, London, 1818, +4to, 4 vols.; ib. 1827. 4to, 3 vols.; ed. Latham, ib. 1866-1874, 4to, +4 vols. (2 in 4 parts): Barclay, London, 1774, 4to; ed. Woodward, +ib. 1848: Sheridan, ib. 1780, 4to, 2 vols.: Webster, New York, 1828, +4to, 2 vols.; London, 1832, 4to, 2 vols.; ed. Goodrich and Porter, +1865, 4to: Richardson, ib. 1836, 4to, 2 vols.; Supplement, 1856: +Ogilvie, <i>Imperial Dictionary</i>, Glasgow, 1850-1855, 8vo, 3 vols. (the +new edition of Ogilvie by Charles Annandale, 4 vols., 1882, was an +encyclopaedic dictionary, which served to some extent as the foundation +of the <i>Century Dictionary</i>); Boag, <i>Do.</i>, Edinburgh, 1852-1853, +8vo, 2 vols.: Craik, ib. 1856, 8vo: Worcester, Boston, 1863, 4to. +Stormouth and Bayne, 1885; Murray and Bradley, <i>The Oxford +English Dictionary</i>, 1884-  ; Whitney, <i>The Century Dict.</i>, New +York, 1889-1891; Porter, <i>Webster’s Internat. Dict.</i>, Springfield, +Massachusetts, 1890; Funk, <i>Standard Dict.</i>, New York, 1894; Hunter, +<i>The Encyclopaedic Dict.</i>, 1879-1888. <span class="sc">Etymology</span>.—Skinner, Londini, +1671, fol.: Junius, Oxonii, 1743, fol.: Wedgewood, London, 1859-1865, +3 vols.; ib. 1872, 8vo. Skeat, Oxford, 1881; Fennell (Anglicized +words), Camb. 1892. <span class="sc">Pronouncing</span>.—Walker, London, 1774, 4to: +by Smart, 2nd ed. ib. 1846, 8vo. <span class="sc">Pronouncing in German</span>.—Hausner, +Frankf. 1793, 8vo; 3rd ed. 1807; Winkelmann, Berlin, 1818, +8vo: Voigtmann, Coburg, 1835, 8vo: Albert, Leipz. 1839, 8vo: +Bassler, ib. 1840, 16mo. <span class="sc">Analytical</span>.—Booth, Bath, 1836, 4to: +Roget, <i>Thesaurus</i>, London, 1852, 8vo; 6th ed. 1857; Boston, 1874. +<span class="sc">Synonyms</span>.—Piozzi, London, 1794, 8vo, 2 vols.: L. [abarthe], Paris, +1803, 8vo, 2 vols.: Crabb, London, 1823, 8vo; 11th ed. 1859: +C. J. Smith, ib. 1871, 8vo, 610 pages. <span class="sc">Reduplicated Words</span>.—Wheatley, +ib. 1866, 8vo. <span class="sc">Surnames</span>.—Arthur, New York, 1857, +12mo, about 2600 names: Lower, ib. 1860, 4to. <span class="sc">Particles</span>.—Le +Febure de Villebrune, Paris, 1774, 8vo. <span class="sc">Rhyming</span>.—Levins, +<i>Manipulus Puerorum</i>, London, 1570, 4to; ed. Wheatley, ib. 1867, +8vo: Walker, London, 1775, 8vo; 1865, 8vo. <span class="sc">Shakespeare</span>.—Nares, +Berlin, 1822, 4to; ed. Halliwell and Wright, London, 1859, +8vo: Schmidt, Berlin, 1874. <span class="sc">Old English</span>.—Spelman, London +[1626], fol. (A to I only); 1664 (completed); 1687 (best ed.): +Coleridge (1250-1300), ib. 1859, 8vo: Stratmann (Early Eng.), +Krefeld, 1867, 8vo; 2nd ed. 1873, 4to: Bradley (new edition of +Stratman), Oxford, 1891; Matzner and Bieling, Berlin, 1878- . +<span class="sc">Old and Provincial</span>.—Halliwell, London, 1844-1846, 8vo; 2nd ed. +ib. 1850, 2 vols.: 6th ed. 1904: Wright, ib. 1857, 8vo, 2 vols.; 1862. +<span class="sc">Dialects</span>.—Ray, ib. 1674, 12mo: Grose, ib. 1787, 8vo; 1790: +Holloway, Lewes, 1840, 8vo; Wright, <i>Eng. Dialect Dict.</i>, London, +1898-1905, 28 vols. <i>Scotch</i>: Jamieson, Edin. 1806, 4to, 2 vols.; +Supplement, 1826, 2 vols.; abridged by Johnstone, ib. 1846, 8vo: +Brown, Edin, 1845, 8vo: Motherby (German), Königsberg, 1826-1828, +8vo: (<i>Shetland and Orkney</i>), Edmonston, London, 1866, 8vo: +(<i>Banffshire</i>), Gregor, ib. 1866, 8vo. <i>North Country</i>: Brockett, +London, 1839, 8vo, 2 vols. <i>Berkshire</i>: [Lousley] ib. 1852, 8vo, +<i>Cheshire</i>: Wilbraham, ib. 1817, 4to; 1826, 12mo: Leigh, Chester, +1877, 8vo. <i>Cumberland</i>: <i>Glossary</i>, ib. 1851, 12mo: Dickenson, +Whitehaven, 1854, 12mo; Supplement, 1867: Ferguson (Scandinavian +Words), London, 1856, 8vo. <i>Derbyshire</i>: Hooson (mining), +Wrexham, 1747, 8vo: Sleigh, London, 1865, 8vo. <i>Dorset</i>: Barnes, +Berlin, 1863, 8vo. <i>Durham</i>: [Dinsdale] (Teesdale), London, 1849, +12mo. <i>Gloucestershire</i>: Huntley (Cotswold), ib. 1868, 8vo. <i>Herefordshire</i>: +[Sir George Cornewall Lewis,] London, 1839, 12mo. <i>Lancashire</i>: +Nodal and Milner, Manchester Literary Club, 1875, 8vo, +Morris (Furness), London, 1869, 8vo: R. B. Peacock (Lonsdale, +North and South of the Sands), ib. 1869, 8vo. <i>Leicestershire</i>: +A. B. Evans, ib. 1848, 8vo. <i>Lincolnshire</i>: Brogden, ib. 1866, 12mo: +Peacock (Manley & Corringham), ib. 1877, 8vo. <i>Norfolk and Suffolk</i>; +Forby, London, 1830, 8vo, 2 vols. <i>Northamptonshire</i>: Sternberg, +ib. 1851, 8vo: Miss Anne E. Baker, ib. 1866, 8vo, 2 vols. 868 pages. +<i>Somersetshire</i>: Jennings, ib. 1869, 8vo: W. P. Williams and W. A. +Jones, Taunton, 1873, 8vo. <i>Suffolk</i>: Moor, Woodbridge, 1823, 12mo: +Bowditch (Surnames), Boston, U.S., 1851, 8vo; 1858; 3rd ed. +London, 1861, 8vo, 784 pages. <i>Sussex</i>: Cooper, Brighton, 1836, +8vo: Parish, Farncombe, 1875, 8vo. <i>Wiltshire</i>: Akerman, London, +1842, 12mo. <i>Yorkshire (North and East)</i>, Toone, ib. 1832, 8vo: +(<i>Craven</i>), Carr, 2nd ed. London, 1828, 8vo, 2 vols.: (<i>Swaledale</i>), +Harland, ib. 1873, 8vo: (<i>Cleveland</i>), Atkinson, ib. 1868, 4to, 653 +pages: (<i>Whitby</i>) [F. K. Robinson], ib. 1876, 8vo: (<i>Mid-Yorkshire +and Lower Niddersdale</i>), C. Clough Robinson, ib. 1876, 8vo: (<i>Leeds</i>), +Id., ib. 1861, 12mo: (<i>Wakefield</i>), Banks, ib. 1865, 16mo: (<i>Hallamshire</i>), +Hunter, London, 1829, 8vo. <i>Ireland: (Forth and Bargy, Co. +Wexford)</i>, Poole, London, 1867, 8vo. <i>America</i>: Pickering, Boston, +1816, 8vo: Bartlett, New York, 1848, 8vo; 3rd ed. Boston, 1860. +8vo; Dutch transl. by Keijzer, Gorinchen, 1854, 12mo; Germ. +transl. by Köhler, Leipz. 1868, 8vo. Elwyn, Philadelphia, 1859. +8vo. <i>Negro English</i>: Kingos, St Croix, 1770, 8vo: Focke (Dutch), +Leiden, 1855, 8vo: Wullschlaegel, Löbau, 1856, 8vo. 350 pages. +<span class="sc">Slang</span>.—Grose, London, 1785, 8vo; 1796: Hotten, ib. 1864, 8vo; +1866; Farmer & Henley (7 vols., 1890-1904).</p> + +<p><span class="bold">Frisic.</span>—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. <span class="sc">Old +Frisic</span>.—Wiarda (Germ.), Aurich, 1786, 8vo: Richthofen, Göttingen, +1840, 4to. <span class="sc">North Frisic</span>.—Bendson (Germ.), Leiden, 1860, 8vo: +Johansen (Föhringer und Amrumer Mundart), Kiel, 1862, 8vo. +<span class="sc">East Frisic</span>.—Stürenburg, Aurich, 1857, 8vo. <span class="sc">Heligoland</span>.—Oelrichs, +s. l., 1836, 16mo.</p> + +<p><span class="bold">Dutch.</span>—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; <span class="sc">English</span>.—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. +<span class="sc">French</span>.—Halma, Amst. 1710, 4to; 4th ed. 1761: Marin, ib. 1793, +4to, 2 vols.: Winkelman, ib. 1793, 4to, 2 vols.: Mook, Zutphen, +1824-1825, 8vo, 4 vols.; Gouda, 1857, 8vo, 2 vols. 2818 pages: Kramers, +ib. 1859-1862, 2 vols. 16mo. <span class="sc">German</span>.—Kramer, Nürnb. 1719, fol.; +1759, 4to, 2 vols.; ed. Titius, 1784, Weiland, Haag, 1812, 8vo: +Terwen, Amst. 1844, 8vo. <span class="sc">Etymology</span>.—Franck, 1884-1892. +<span class="sc">Oriental Words</span>.—Dozy, ’s Gravenhage, 1867, 8vo. <span class="sc">Genders of +Nouns</span>.—Bilderdijk, Amst. 1822, 8vo, 2 vols. <span class="sc">Spelling</span>.—Id., +’s Gravenhage, 1829, 8vo. <span class="sc">Frequentatives</span>.—De Jager, Gouda, +1875, 8vo, vol. i. <span class="sc">Old Dutch</span>.—Suringer, Leyden, 1865, 8vo. +<span class="sc">Middle Dutch</span>.—De Vries, ’s Gravenhage, 1864, &c., 4to. Verwijs +and Verdam, ib. 1885-  .</p> + +<p><span class="bold">Flemish.</span>—Kilian, Antw. 1511, 8vo; ed. Hasselt, Utrecht, 1777, +4to, 2 vols. <span class="sc">French</span>.—Berlemont, Anvers, 1511, 4to: Meurier, ib. +1557, 8vo: Rouxell and Halma, Amst. 1708, 4to; 6th ed. 1821: +Van de Velde and Sleeckx, Brux. 1848-1851, 8vo, 2440 pages; ib. +<span class="pagenum"><a name="page195" id="page195"></a>195</span> +1860, 8vo, 2 vols. <span class="sc">Ancient Names of Places</span>.—Grandgagnage +(East Belgium), Bruxelles, 1859, 8vo.</p> + +<p><span class="bold">German.</span>—Josua Pictorius (Maaler), <i>Die teütsch Spraach</i>, Tiguri, +1561, 8vo; Stieler, Nürnb. 1691, 4to: Adelung, Leipz. 1774-1786, +4to, 5 vols.; 1793-1818, 5 vols.: Campe, Braunschweig, 1807-1811, +4to, 5 vols.: Grimm, Leipzig, 1854, &c., 4to: Sanders, ib. 1860-1865, +4to, 3 vols. 1885: Diefenbach and Wülcker (High and Low +German, to supplement Grimm), Frankf. a. M. 1874, 1885, 8vo.; +Kluge, Strassburg, 1883; Heine, Leipzig, 1890-1895; Weigand, +Giessen, 1873. <span class="sc">English</span>.—Adelung, 1783-1796, 8vo, 3 vols.: Hilpert, +Karlsruhe, 1828-1829, 8vo, 2 vols.; 1845-1846, 4to, 2 vols.: Flügel, +Leipz. 1830, 8vo, 2 vols.; London, 1857, 8vo; Leipzig, 1870: +Müller, Cöthen, 1867, 8vo, 2 vols. <span class="sc">French</span>.—Laveaux, Strassburg, +1812, 4to: Mozin, Stuttgard, 1811-1812, 4to, 4 vols.; 1842-1846, 8vo, +4 vols., 3rd ed. 1850-1851, 8vo: Schuster, Strasb. 1859, 8vo: Daniel, +Paris, 1877, 16mo. <span class="sc">Old High German</span>.—Haltaeus, Lipsiae, 1758, +fol. 2 vols.: Graff, Berlin, 1834-1846, 4to, 7 vols.: Brinckmeier, +Gotha, 1850-1863, 4to, 2 vols.: Kehrein (from Latin records), Nordhausen, +1863, 8vo. Schade, Halle, 1872-1882. <span class="sc">Middle High German</span>.—Ziemann, +Quedlinburg, 1838, 8vo: Benecke, Müller and Zarnche, +Leipz. 1854-1866, 8vo, 3 vols.: Lexer, Leipzig, 1870, 8vo. <span class="sc">Middle +Low German</span>.—Schiller and Lübben, Bremen, 1872, &c., 8vo, in +progress. <span class="sc">Low German</span>.—Vollbeding, Zerbst, 1806, 8vo: Kosegarten, +Griefswald, 1839, 4to; 1856, &c., 4to. <span class="sc">Etymology</span>.—Helvigius, +Hanov. 1620, 8vo: Wachter, Lipsiae, 1737, fol. 2 vols.: +Kaindl, Salzbach, 1815-1830, 8vo, 7 vols.: Heyse, Magdeburg, 1843-1849, +8vo, 3 vols.: Kehrein, Wiesbaden, 1847-1852, 2 vols. <span class="sc">Synonyms</span>.—Eberhard, +Maas, and Grüber, 4th ed. Leipzig, 1852-1863, 8vo, 4 +vols.: Aue (Engl.), Edinb. 1836, 8vo: Eberhard, 11th ed. Berlin, 1854, +12mo: Sanders, Hamburg, 1872, 8vo, 743 pages. <span class="sc">Foreign Words</span>.—Campe, +Braunschweig, 1813, 4to: Heyse, <i>Fremdwörterbuch</i>, +Hannover, 1848, 8vo. <span class="sc">Names</span>.—Pott. Leipz. 1853, 8vo: Michaelis +(Taufnamen), Berlin, 1856, 8vo: Förstemann (Old Germ.) Nordhausen, +1856-1859, 4to, 2 vols. 1573 pages, 12,000 names: Steub +(Oberdeutschen), München, 1871, 8vo. <span class="sc">Luther</span>.—Dietz, Leipzig, +1869-1872, 8vo, 2 vols. <span class="sc">Dialects</span>.—Popowitsch, Wien, 1780, 8vo: +Fulda, Berlin, 1788, 8vo: Klein, Frankf. 1792, 8vo, 2 vols.: Kaltschmidt, +Nordlingen, 1851, 4to; 1854, 5th ed. 1865. <i>Aix-la-Chapelle</i>, +Müller and Weitz, Aachen, 1836, 12mo. <i>Appenzell</i>: Tobler, +Zürich, 1837, 8vo. <i>Austria</i>: Höfer, Linz, 1815, 8vo; Castelli, Wien, +1847, 12mo: Scheuchenstül (mining), ib. 1856, 8vo. <i>Bavaria</i>: +Zaupser, München, 1789, 8vo: Deling, ib. 1820, 2 vols.: Schmeller, +Stuttg. 1827-1837, 8vo, 4 vols.; 2nd ed. München, 1872, 4to, vol. +i. 1799 pages. <i>Berlin</i>: Trachsel. Berlin, 1873, 8vo. <i>Bremen</i>: +Bremisch Deutsch Gesellschaft, Bremen, 1767-1771, 1869, 8vo, 6 vols. +Oelrich (anc. statutes), Frankf. a. M. 1767, 8vo. <i>Carinthia</i>: Ueberfelder, +Klagenfurt, 1862, 8vo: Lexe, Leipzig, 1862, 8vo. <i>Cleves</i>: +De Schueren, <i>Teuthonista</i>, Colon, 1477, fol.; Leiden, 1804, 4to. +<i>Göttingen</i>: Schambach, Hannover, 1838, 8vo. <i>Hamburg</i>: Richey, +Hamb. 1873, 4to; 1755, 8vo. <i>Henneberg</i>: Reinwold, Berlin and +Stettin, 1793, 1801, 8vo, 2 vols.: Brückner, Meiningen, 1843, 4to. +<i>Hesse</i>: Vilmar, Marburg, 1868, 8vo, 488 pages. <i>Holstein</i>: Schütz +Hamb. 1800-1806, 8vo, 4 vols. <i>Hungary</i>: Schoer, Wien, 1858. +<i>Livonia</i>: Bergmann, Salisburg, 1785, 8vo: Gutzeit, Riga, 1859-1864, +8vo, 2 parts. <i>Upper Lusatia</i>: Anton, Görlitz, 1825-1839, 13 parts. +<i>Luxembourg</i>: Gangler, Lux. 1847, 8vo, 406 pages. <i>Mecklenburg and +Western Pomerania</i>: M., Leipzig, 1876, 8vo, 114 pages. <i>Nassau</i>: +Kehrein, Weilburg, 1860, 8vo. <i>Osnaburg</i>: Strodtmann, Leipz. 1756, +8vo. <i>Pomerania and Rügen</i>: Dähnert, Stralsund, 1781, 4to. <i>Posen</i>: +Bernd, Bonn, 1820, 8vo. <i>Prussia</i>: Bock, Königsb. 1759, 8vo: +Hennig, ib. 1785, 8vo. <i>Saxony</i>: Schmeller (from Heliand, &c.), +Stuttg. 1840, 4to. <i>Silesia</i>: Berndt, Stendal, 1787, 8vo. <i>Swabia</i>: +Schmid, Berlin, 1795, 8vo; Stuttg. 1831, 8vo. <i>Switzerland</i>: +Stalder, Aarau, 1807-1813, 8vo, 2 vols. <i>Thuringia</i>: Keller, Jena, +1819, 8vo. <i>Transylvania</i>: Schuller, Prag, 1865, 8vo. <i>Tirol</i>: +Schöpf, Innspruck, 1866, 8vo. <i>Venetian Alps</i>: Schmeller, Wien, +1854, 8vo. <i>Vienna</i>: Hugel, ib. 1873, 8vo. <span class="sc">Hunting</span>.—<i>Westerwald</i>: +Schmidt, Hadamar, 1800, 8vo; Kehrein, Wiesbaden, 1871, 12mo. +<span class="sc">Slang</span>.—<i>Gauner Sprache</i>: Schott, Erlangen, 1821, 8vo: Grolmann, +Giessen, 1822, 8vo: Train, Meissen, 1833, 8vo: Anton, 2nd ed. +Magdeburg, 1843, 8vo; 1859: Avé-Lallemant, <i>Das Deutsche +Gaunerthun</i>, Leipzig, 1858-1862, 8vo, vol. iv. pp. 515-628. <i>Student +Slang</i>: Vollmann (Burschicoses), Ragaz, 1846, 16mo, 562 pages.</p> + +<p class="center pt2"><i>Celtic.</i></p> + +<p><span class="bold">Celtic generally.</span>—Lluyd, Archaeologia Britannica, Oxford, +1707, folio: Bullet, Besançon, 1754-1860, fol. 2 vols.</p> + +<p><span class="bold">Irish.</span>—Cormac, bishop of Cashel, born 831, slain in battle 903, +wrote a Glossary, <i>Sanas Cormaic</i>, printed by Dr Whitley Stokes, +London, 1862, 8vo, with another, finished in 1569, by O’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.</p> + +<p><span class="bold">Gaelic.</span>—Macdonald, Edin. 1741, 8vo: Shaw, London, 1780, +4to, 2 vols.: Allan, Edin. 1804, 4to: Armstrong, London, 1825, +4to: Highland Society, ib. 1828, 4to, 2 vols.: Macleod and Dewar, +Glasgow, 1853, 8vo.</p> + +<p><span class="bold">Manx.</span>—Cregeen, Douglas, 1835, 8vo: Kelly, ib. 1866, 8vo, 2 vols.</p> + +<p><span class="bold">Welsh.</span>—<span class="sc">Latin</span>.—Davies, London, 1632, fol.: Boxhornius, +Amstelodami, 1654, 4to. <span class="sc">English</span>.—Salesbury, London, 1547, 4to: +1551: Richards, Bristol, 1759, 8vo: Owen (W.), London, 1793-1794, +8vo, 2 vols.; 1803, 4to, 3 vols.: Walters, ib. 1794, 4to: Owen-Pughe, +Denbigh, 1832, 8vo; 3rd ed. Pryse, ib. 1866, 8vo: D. S. Evans +(Eng.-Welsh), ib. 1852-1853, 8vo; 1887.</p> + +<p><span class="bold">Cornish.</span>—Pryce, <i>Archaeologia</i>, Sherborne, 1770, 4to: Williams, +Llandovery, 1862-1865, 4to. <span class="sc">Names</span>.—Bannister (20,000), Truro, +1869-1871, 8vo.</p> + +<p><span class="bold">Breton.</span>—Legadeuc, <i>Le Catholicon breton</i>, 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. <span class="sc">Dialect of Léon</span>.—Troude (Fr.-Bret.), Brest, +1870, 8vo; Id. (Bret.-Fr.), ib. 1876, 8vo, 845 pages. <span class="sc">Diocese of +Vannes</span>.—Armerie, Leyde, 1774, 8vo.</p> + +<p class="center pt2"><i>Basque.</i></p> + +<p><span class="bold">Basque.</span>—Larramendi, St Sebastian, 1745, fol. 2 vols.; ed. +Zuazua, ib. 1854, fol.; Chaho, Bayonne, 1856, 4to, 1867: Fabre, +ib. 1870, 8vo: Van Eys, Paris, 1873, 8vo: Egúren, Madrid, 1877.</p> + +<p class="center pt2"><i>Baltic.</i></p> + +<p><span class="bold">Lithuanian.</span>—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.</p> + +<p><span class="bold">Lettic.</span>—Mancelius, Riga, 1638, 4to: Elvers, ib. 1748, 8vo: +Lange, Mitau, 1777, 4to: Sjögren, Petersburg, 1861, 4to: Ulmann, +ed. Bielenstein, Riga, 1872, &c., 8vo.</p> + +<p><span class="bold">Prussian.</span>—Bock, Königsberg, 1759, 8vo: Hennig, ib. 1785, 8vo: +Nesselmann, Berlin, 1873, 8vo: Pierson, ib. 1875, 8vo.</p> + +<p class="center pt2"><i>Slavonic</i>.</p> + +<p><span class="bold">Slavonic generally.</span>—Franta-Sumavski (Russ. Bulg. Old Slav. +Boh. Polish), Praga, 1857, 8vo, Miklosich, Wien, 1886.</p> + +<p><span class="bold">Old Slavonic.</span>—Beruinda, Kiev, 1627, 8vo; Kuteinsk, 1653, +4to: Polycarpi (Slav. Greek, Latin), Mosque, 1704, 4to: Alexyeev, +St Petersb. 1773, 8vo; 4th ed. ib. 1817-1819, 8vo, 5 vols.: Russian +Imp. Academy, ib. 1847, 4to, 4 vols.: Miklosich, Vindobonae, 1850: +4to; 1862-1865, 8vo, Mikhailovski, St Petersb. 1875, 8vo: Charkovski, +Warschaw, 1873, 8vo.</p> + +<p><span class="bold">Russian.</span>—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. +<span class="sc">French-Germ.-Eng</span>.—Reiff, +ib. 1852-1854, 4to. <span class="sc">German, Latin</span>.—Holterhof, +Moskva, 1778, 8vo, 2 vols.; 3rd ed. 1853-1855, 8vo, 2 vols.: Weismann, +ib. 1731, 4to; 1782, and frequently. <span class="sc">French, German</span>.—Nordstet, +ib. 1780-1782, 4to, 2 vols.: Heym, Moskau, 1796-1805, 4to, 4 vols.: +Booch-Arkossi and Frey, Leipzig, 1871, &c., 8vo. <span class="sc">English</span>.—Nordstet, +London, 1780, 4to: Grammatin and Parenogo, Moskva, +1808-1817, 4to, 4 vols. <span class="sc">French</span>.—Tatischeff, 2nd ed. St Petersb. 1798, +8vo, 2 vols.; Moskau, 1816, 4to, 2 vols.: Reiff, St Petersb. 1835-1836, +8vo, 2 vols.: Makaroff, ib. 1872, 8vo, 2 vols, 1110 pages; 1873-1874, +12mo, 2 vols. <span class="sc">German</span>.—Pawlowski, Riga, 1859, 8vo: Lenström, +Mitau, 1871, 8vo. <span class="sc">Swedish</span>.—Geitlin, Helsingfors, 1833, 12mo: +Meurmann, ib. 1846, 8vo. <span class="sc">Polish</span>.—Jakubowicz, Warszawa, 1825-1828, +8vo, 2 vols.: Amszejewicz, ib. 1866, 8vo: Szlezigier, ib. 1867, 8vo. +<span class="sc">Technical</span>.—Grakov (Germ.), St Petersb. 1872, 8vo. <span class="sc">Naval</span>.—Butakov, +ib. 1837. <span class="sc">Dialects</span>.—<i>North-west Russia</i>: Gorbachevski +(old language, in Russian), Vilna, 1874, 8vo, 418 pages. <i>White +Russia</i>: Nosovich (Russian), St Petersburg, 1870, 4to, 760 pages. +<i>Red Russia</i>: Patritzkii (German), Lemberg, 1867, 8vo, 2 vols. +842 pages. <i>Ukraine</i>: Piskanov (Russian), Odessa, 1873, 4to, 156 +pages.</p> + +<p><span class="bold">Polish.</span>—Linde (explained in Lat. Germ. and 13 Slav dialects), +Warszawie, 1807-1814, 4to, 6 vols. 4574 pages. <span class="sc">English</span>.—[Rykaczewski], +<i>Complete Dictionary</i>, Berlin, 1849-1851, 8vo, 2 vols.: Rykaczewski, +Berlin, 1866, 16mo, 1161 pages. <span class="sc">French and German</span>.—Troc, +Leipz. 1742-1764, 8vo, 4 vols.; 4th ed. ib. 1806-1822, 4to, 4 vols.: +Bandtke, Breslau, 1806, 8vo, 2 vols.; 1833-1839, 8vo. <span class="sc">French</span>.—Schmidt, +Leipzig, 1870, 16mo. <span class="sc">Russian and German</span>.—Schmidt +(J. A. E.), Breslau, 1834, 8vo. <span class="sc">German</span>.—Mrongovius, Königsberg, +1765; 1835, 4to; 1837: Troianski, Berlin, 1835-1838, 8vo, 2 vols.: +Booch-Arkossi, Leipzig, 1864-1868, 8vo, 2 vols.: Jordan, ib. 1866, +8vo. <span class="sc">Italian</span>.—Plazowski, Warszawa, 1860, 8vo. 2 vols. 730 pages. +<span class="sc">Russian</span>.—Potocki, Lipsk, 1873, &c., 12mo.</p> + +<p><span class="bold">Wendish.</span>—Matthäi, Budissen, 1721, 8vo: Bose, Grimma, 1840, +8vo: Pfuhl, w Budzsinje, 1866, 8vo, 1210 pages. <span class="sc">Upper Lusatian</span>.—Pfuhl +and Jordan, Leipz. 1844, 8vo. <span class="sc">Lower Lusatian</span>.—Zwahr, +Spremberg, 1847, 8vo.</p> + +<p><span class="bold">Czech.</span>—Rohn (Germ. Lat.), Prag, 1780, 4to, 4 vols.: +Dobrowski and Hanka, ib. 1802-1821, 4to, 2 vols. <span class="sc">Lat. Germ. +Hungar</span>.—Jungmann, Praze, 1835-1839, 6 vols. 4to, 5316 pages. +<span class="sc">German</span>.—Thàm, Prag. 1805-1807, 8vo, 2 vols.: Sumavski, ib. 1844-1846, +8vo, 2 vols.: Koneney, ib. 1855, 18mo, 2 vols.: Rank (Germ. +Boh.), ib. 1860, 16mo, 775 pages. <span class="sc">Technical</span>.—Spatny, ib. 1864, +8vo: Kheil (names of goods, Germ. Boh.), ib. 1864, 8vo, 432 pages. +<span class="sc">Hunting</span>.—Spatny, ib. 1870, 8vo, 137 pages.</p> + +<p><span class="pagenum"><a name="page196" id="page196"></a>196</span></p> + +<p><span class="bold">South Slavic.</span>—Richter and Ballman, Wien, 1839-1840, 8vo, 2 vols. +<span class="sc">Servian</span>.—Karajić (Germ. Lat.), ib. 1818, 8vo; 1852: Lavrovski +(Russian), St Petersb. 1870, 8vo, 814 pages. <span class="sc">Bosnian</span>.—Micalia, +Laureti, 1649, 8vo. <span class="sc">Slovak</span>.—Bernolak (Lat. Germ. +Hung.), Budae, 1825-1827, 8vo, 6 vols.: Loos (Hung. and Germ.), +Pest, 1869, &c., 3 vols. <span class="sc">Slovene</span>.—Gutsmann, Klagenfurt, 1789, +4to: Relkovich, Wien, 1796, 4to, 2 vols.: Murko, Grätz, 1838, 8vo, +2 vols.: Janezić, Klagenfurt, 1851, 12mo. <span class="sc">Dalmatian</span>.—Ardelio +della Bella, Venezia, 1728, 8vo; 2nd ed. Ragusae, 1785, 4to: Stulli, +ib. 1801-1810, 4to, 2 vols. <span class="sc">Croatian</span>.—Habdelich, Grätz, 1670, 8vo: +Sulek, Agram, 1854-1860, 8vo, 2 vols. 1716 pages. <span class="sc">Carinthian</span>.—Lexer, +Leipzig, 1862, 8vo. <span class="sc">Old Servian</span>.—Danitziye (Servian), +Belgrad, 1864, 8vo, 3 vols.</p> + +<p><span class="bold">Bulgarian.</span>—Daniel (Romaic, Albanian, Rumanian, and Bulgarian), +Moschopolis, 1770; Venice, 1802, 4to. <span class="sc">English</span>.—Morse and +Vassiliev, Constantinople, 1860, 8vo. <span class="sc">Russian</span>.—Borogoff, Vienna, +1872, &c., 8vo.</p> + +<p class="center pt2"><i>Ugrian.</i></p> + +<p><span class="bold">Ugrian, Comparative.</span>—Donner, Helsingfors, 1874, 8vo, in progress: +Budenz (Ugrian-Magyar), Budapest, 1872-1875, 8vo.</p> + +<p><span class="bold">Lappish.</span>—<i>Manuale</i>, Holmiae, 1648, 8vo: Fjellström, ib. 1738, +8vo: Leem and Sandberg, Havn. 1768-1781, 4to, 2 parts: Lindahl +and Oehrling, Holm. 1780, 8vo. <span class="sc">North Lappish</span>.—Stockfleht, +Christiania, 1852, 8vo.</p> + +<p><span class="bold">Finnish.</span>—Juslenius, Holmiae, 1745, 4to, 567 pages: Renvall, +Aboae, 1826, 4to, 2 vols.: Europaeus, Helsingissä, 1852-1853, 16mo, +2 vols. 742 pages: Lunin, Derpt, 1853, 8vo: Eurén, Tavashuus, 1860, +8vo: Ahlman, ib. 1864, 8vo: Wiedemann, St Petersb. 1869, 4to: +Godenhjelm (Germ.), Helsingfors, 1871: Lönnrot, Helsingissä, +1874. <span class="sc">Naval</span>.—Stjerncreutz, ib. 1863, 8vo.</p> + +<p><span class="bold">Esthonian.</span>—Hupel, Mitau, 1818, 8vo, 832 pages: Körber, +Dorpat, 1860, 8vo: Wiedemann, St Petersb. 1869, 4to, 1002 pages: +Aminoff (Esth.-Finnish), Helsingissä, 1869, 8vo: Meves (Russian), +Riga, 1876, 12mo.</p> + +<p><span class="bold">Permian.</span>—Rogord (Russian), St Petersb. 1869, 8vo, 420 pages.</p> + +<p><span class="bold">Votiak.</span>—Wiedemann, Reval, 1847, 8vo: Ahlquist, Helsingfors, +1856, 4to.</p> + +<p><span class="bold">Cheremiss.</span>—Budenz, Pest, 1866, 8vo.</p> + +<p><span class="bold">Ersa-Mordvine.</span>—Wiedemann, St Petersb. 1865, 4to. +<span class="sc">Moksha-Mordvine</span>.—Ahlquist, ib. 1862, 8vo.</p> + +<p><span class="bold">Magyar.</span>—Szabo, Kassan, 1792, 8vo: Guczor and Fogarazi +(Hung. Academy), Pesth, 1862, 8vo, in progress. <span class="sc">English</span>.—Dallos, +Pesth, 1860, 8vo. <span class="sc">French</span>.—Kiss, ib. 1844, 12mo, 2 vols.: +Karady, Leipz. 1848, 12mo: Mole, Pesth, 1865, 8vo, 2 vols. <span class="sc">German</span>.—Schuster, +Wien, 1838, 8vo: Bloch, Pesth, 1857, 4to, 2 vols.: Ballagi, +ib. 1857, 8vo; 6th ed. 1905, 8vo, 2 vols.: Loos, ib. 1870, 8vo, 914 pages. +<span class="sc">Etymological</span>.—Dankovsky (Lat.-Germ.), Pressburg, 1853, 8vo: +Kresznerics (under roots, in Hung.), Budân, 1831-1832, 4to, 2 vols.: +Podhorsky (from Chinese roots, in Germ.), Budapest, 1877, 8vo. +<span class="sc">New Words</span>.—Kunoss, Pesth, 1836, 8vo; 1844.</p> + +<p><span class="bold">Turkish.</span>—<span class="sc">Arab. Pers</span>.—Esaad Effendi, Constantinople, 1802, +fol. <span class="sc">Romaic</span>.—Alexandrides, Vienna, 1812, 4to. <span class="sc">Polyglotts</span>.—Pianzola +(Ital. Grec. volgare, e Turca), Padova, 1789, 4to: Ciakciak +(Ital. Armeno, Turco), Venice, 1804, 4to; 2nd ed. 1829: Azarian +(Ellenico, Ital. Arm. Turco), Vienna, 1848, 8vo: Mechitarist +Congregation (Ital. Francese, Arm. Turco), ib. 1846, 8vo. <span class="sc">Latin</span>.—Mesgnien-Meninski, +Viennae, 1680, fol. 3 vols.; ed. Jenisch and +Klezl, ib. 1780-1802, fol. 4 vols. <span class="sc">English</span>.—Sauerwein, London, +1855, 12mo: Redhouse, ib. 1856, 8vo, 1176 pages: Id., Eng. Turkish, +ib. 860, 8vo. <span class="sc">French</span>.—Kieffer and Bianchi (Turk.-Fr.), Paris, +1835-1837, 2 vols. 2118 pages: Bianchi (Fr.-Turk.) Paris, 1843-1846, +8vo, 2 vols. 2287 pages; 1850, 8vo, 2 vols.: Mallouf, ib. 1863-1867, +8vo, 2 vols. <span class="sc">French and German</span>.—Zenker (Arab. Pers.), Leipz, +1862-1876, 4to, 2 vols, 982 pages. <span class="sc">German</span>.—Korabinsky, Pressburg, +1788, 8vo: Vambéry, Constantinople, 1858, 8vo. <span class="sc">Italian</span>.—Molina, +Roma, 1641, 8vo: Masais, Firenze, 1677, 8vo: Ciadyrgy, +Milano, 1832-1834, 4to, 2 vols. <span class="sc">Russian</span>.—Budagov (Comparative +lexicon of the Turkish-Tartar dialects), St Petersburg, 1869, 8vo, +2 vols.</p> + +<p><span class="bold">Gipsy.</span>—Bischoff, Ilmenau, 1827, 8vo: Truxillo, Madrid, 1844, +8vo: Jimenes, Sevilla, 1846, 16mo: Baudrimont, Bordeaux, 1862, +8vo: Vaillant, Paris, 1868, 8vo: Paspati; Constantinople, 1870, +4to: Borrow, <i>Romany Lavo Lil</i>, London, 1874, 8vo: Smart and +Crofton, London, 1875, 8vo.</p> + +<p><span class="bold">Albanian.</span>—Blanchus, Romae, 1635, 8vo: Kaballioti (Romaic, +Wallach. Alb.), Venice, 1770, 8vo: Xylander, Frankfurt a. M. 1835, +8vo: Hahn, Jena, 1854, 4to: Rossi da Montalto, Roma, 1866, 8vo.</p> + +<p class="center pt2">ASIA</p> + +<p><span class="bold">Semitic.</span>—<span class="sc">Polyglotts</span>.—Thurneissius, Berolini, 1585, fol.: +Thorndike, London, 1635, fol.: Schindler, Pentaglotton, Frankf, +ad M. 1653, fol.: Hottinger, Heptaglotton, ib. 1661, fol.: Castellus, +London, 1669, fol. 2 vols. (Hebrew, Chaldaic, Syriac, Samaritan, +Aethiopic and Arabic in one alphabet; Persian separately. It +occupied him for seventeen years, during which he worked sixteen +to eighteen hours a day): Otho, Frankf. a. M. 1702, 4to (the same +languages with Rabbinical).</p> + +<p><span class="bold">Hebrew.</span>—About 875, Zemaḥ, 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 <span title="Kefer Igaron">ןורגא רפכ</span>, probably a Hebrew-Arabic +dictionary. Menaḥem ben Jacob Ibn Sarūq (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 Fās, wrote, in Arabic, +a large Hebrew dictionary, the MS. of which, a quarto of 313 leaves +on cotton paper, was found about 1830 by A. Firkowitz, of Eupatoria, +in the cellar of a Qaraite synagogue in Jerusalem. The age of this +work cannot be ascertained. About 1050, Ali ben Suleiman wrote a +dictionary in Arabic, on the plan of that of David ben Abraham. The +MS. of 429 leaves belongs to Firkowitz. Haja ben Sherira, the +famous teacher of the Academy of Pumbeditha, wrote a Hebrew +dictionary in Arabic, called <i>al Ḥāvi</i> (The Gathering), arranged +alphabetically in the order of the last radical letter. This dictionary +is lost, as well as that of the Spaniard Isaac ben Saul, of Lucena. +Iona ibn Ganaḥ, of Cordova, born about 985, wrote a Hebrew +dictionary in Arabic called <i>Kitāb al Azul</i> (Book of Roots). This, +as well as a Hebrew translation by Samuel ibn Tabōn, is extant +in MS., and was used by Gesenius in his <i>Thesaurus</i>. Rabbi David +ben Joseph ḳimḥi died soon after 1232. His lexicon of roots, called +<span title="Shorashim">םישוש</span>, was printed at Naples 1490, fol.; Constantinople, 1513, fol.; +Naples, 1491, 8vo; Venice, 1552; Berolini, 1838, 4to. <i>Tishbi</i> (The +Tishbite), by Elijah ben Asher, the Levite, so called because it contained +712 roots, was printed at Isny 1541, 8vo and 4to, and often +afterwards. <span class="sc">Latin</span>.—Münster, Basileae, 1523, 8vo; 5 editions to +1564: Zamora, Compluti, 1526, fol.: Pellicanus, Argentorati, 1540, +fol.: Reuchlin, Basil, 1556, fol.: Avenarius, Wittebergae, 1568, fol.; +auctus, 1589: Pagnini, Lugd. Bat. 1575, fol.; 1577; Genevae, 1614; +Buxtorf, Basil. 1607, 8vo; 1615, and many other editions: Frey +(Lat.-Eng.), 2nd ed. London, 1815, 8vo: Gesenius, <i>Thesaurus</i>, Leipz. +1829-1858, 4to, 3 vols. <span class="sc">English</span>.—Bale, London, 1767, 4to: Parkhurst, +ib. 1792, 4to: Lee, ib. 1840, 8vo: Gesenius, translated by +Robinson, ib. 1844, 8vo; by Tregelles, ib. 1846, 4to: Fuerst, 4th ed. +transl. by Davidson, ib. 1866, 8vo: 1871, 8vo, 1547 pages. <span class="sc">French</span>.—Leigh, +Amst. 1703, 4to: Glaire, Paris, 1830, 8vo; 1843. <span class="sc">German</span>.—Gesenius, +Leipzig, 1810-1812, 8vo, 2 vols.: Fuerst, ib. 1842, 16mo: +ib. 1876, 8vo, 2 vols. <span class="sc">Italian</span>.—Modena, Venetia, 1612, 4to; 1640; +Coen, Reggio, 1811, 8vo: Fontanella, Venezia, 1824, 8vo. <span class="sc">Dutch</span>.—Waterman, +Rotterdam, 1859, &c., 8vo. <span class="sc">Hungarian</span>.—Ehrentheil +(Pentateuch), Pest, 1868, 8vo. <span class="sc">Romaic</span>.—Loundes, Melité. 1845, +8vo, 987 pages.</p> + +<p><span class="bold">Rabbinical and Chaldee.</span>—Nathan ben Yeḥiel of Rome wrote in the +beginning of the 12th century a Talmudic dictionary, <i>Aruch</i>, printed +1480 (?), s. l., fol.; Pesaro, 1517, fol.; Venice, 1531; and often: +Isaiah ben Loeb, Berlin, wrote a supplement to <i>Aruch</i>, vol. i. Breslau, +1830, 8vo; vol. ii. (ל to ת), Wien, 1859, 8vo: Münster, Basil. 1527, +4to, 1530, fol.: Elijah ben Asher, the Levite, transl. by Fagius, +Isnae, 1541, fol.; Venet. 1560: David ben Isaac de Pomis, <i>Zamaḥ +David</i>, Venet. 1587, fol.: Buxtorf, Basileae, 1639, fol.: ed. Fischer, +Leipz. 1866-1875, 4to: Otho, Geneva, 1675, 8vo; Altona, 1757, 8vo: +Zanolini, Patavii, 1747, 8vo: Hornheim, Halle, 1807, 8vo: Landau, +Prag, 1819-1824, 8vo, 5 vols.: Dessauer, Erlangen, 1838, 8vo: Nork +(<i>i.e.</i> Korn), Grimma, 1842, 4to: Schönhak, Warschau, 1858, 8vo, +2 vols. <span class="sc">Targums</span>.—Levy, Leipzig, 1866-68 4to, 2 vols.; 1875: +Id. (Eng.), London, 1869, 8vo, 2 vols. <span class="sc">Talmud</span>.—Löwy (in Heb.), +Wien, 1863, 8vo: Levy, Leipzig, 1876, &c., 4to. <span class="sc">Prayer-Book</span>.—Hecht, +Kreuznach, 1860, 8vo: Nathan, Berlin, 1854, 12mo. +<span class="sc">Synonyms</span>.—Pantavitius, Lodevae, 1640, fol. <span class="sc">Foreign Words</span>.—Rabeini, +Lemberg, 1857, 8vo, &c. <span class="sc">Jewish-German</span>.—Callenberg, +Halle, 1736, 8vo: Vollbeding, Hamburg, 1808, 8vo: Stern, +München, 1833, 8vo, 2 vols.: Theile, Berlin, 1842-1843, 8vo, 2 vols.: +Avé-Lallemant, <i>Das deutsche Gaunerthum</i>, Leipzig, 1858, 8vo, 4 vols.; +vol. iv. pp. 321-512.</p> + +<p><span class="bold">Phœnician.</span>—M. A. Levy, Breslau, 1864, 8vo.</p> + +<p><span class="bold">Samaritan.</span>—Crinesius, Altdorphi, 1613, 4to: Morini, Parisiis, +1657, 12mo: Hilligerus, Wittebergae, 1679, 4to: Cellarius, Cizae, +1682, 4to; Frankof. 1705: Uhlemann, Leipsiae, 1837, 8vo: Nicholls, +London, 1859, 8vo.</p> + +<p><span class="bold">Assyrian.</span>—Norris, London, 1868, 8vo, 3 vols. <span class="sc">Proper Names</span>.—Menant, +Paris, 1861, 8vo.</p> + +<p><span class="bold">Accadian.</span>—Lenormant, Paris, 1875, 8vo.</p> + +<p><span class="bold">Syriac.</span>—Joshua ben Ali, a physician, who lived about 885, made +a Syro-Arabic lexicon, of which there is a MS. in the Vatican. +Hoffmann printed this lexicon from Alif to Mim, from a Gotha MS., +Kiel, 1874, 4to. Joshua bar Bahlul, living 963, wrote another, great +part of which Castelli put into his lexicon. His MS. is now at +Cambridge, and, with those at Florence and Oxford, was used by +Bernstein. Elias bar Shinaya, born 975, metropolitan of Nisibis, +1009, wrote a Syriac and Arabic lexicon, entitled <i>Kitāb ūt Tarjuman +fi Taalem Loghat es Sūriān</i> (Book called the Interpreter for teaching +the Language of the Syrians), of which there is a MS. in the British +Museum. It was translated into Latin by Thomas à Novaria, a +Minorite friar, edited by Germanus, and published at Rome by +Obicinus, 1636, 8vo. It is a classified vocabulary, divided in 30 +chapters, each containing several sections. Crinesius, Wittebergae, +<span class="pagenum"><a name="page197" id="page197"></a>197</span> +1612, 4to: Buxforf, Basileae, 1622, 4to: Ferrarius, Romae, 1622, +4to: Trost, Cothenis Anhaltor, 1643, 4to: Gutbir, Hamburgi, 1667, +8vo: Schaaf, Lugd. Bat, 1708, 4to: Zanolini, Patavii, 1742, 4to: +Castellus, ed. Michaelis, Göttingen, 1788, 4to, 2 vols.: Bernstein, +Berlin, 1857, &c. fol.: Smith (Robt. Paine), Dean of Canterbury, +Oxonii, 1868, &c. fol.: fasc. 1-3 contain 538 pages: Zingerle, +Romae, 1873, 8vo, 148 pages.</p> + +<p><span class="bold">Arabic.</span>—The native lexicons are very many, voluminous and +copious. In the preface to his great Arabic-English lexicon, Lane +describes 33, the most remarkable of which are-the <i>’Ain</i>, so called +from the letter which begins its alphabet, commonly ascribed to al +Khalil (who died before A.H. 175 [<span class="sc">a.d.</span> 791], aged seventy-four): the +<i>Sihah</i> of Jauhari (died 398 [1003]): the <i>Mohkam</i> of Ibn Sidah the +Andalusian, who was blind, and died A.H. 458 [<span class="sc">a.d.</span> 1066], aged about +sixty: the <i>Asas</i> of Zamakhshari (born 467 [1075], died 538 [1144]), +“a most excellent repertory of choice words and phrases”: the +<i>Lisān el ’Arab</i> of Ibn Mukarram (born 630 [1232], died 711 [1311]); +Lane’s copy is in 28 vols. 4to: the <i>Kamus</i> (The Sea) of Fairuzabadi +(born 729 [1328], died 816 [1413]),: the <i>Taj el Arus</i>, by Murtada +Ez Zebadi (born <span class="sc">a.d.</span> 1732, died 1791)—the copy made for Lane +is in 24 vols. thick 4to. The <i>Sihah</i> was printed Hardervici Getorum, +1774, 4to; Bulak, 1865, fol. 2 vols.: <i>Kamus</i>, Calcutta, 1817, fol. +2 vols.; Bombay, 1855, fol. 920 pages: <i>Sirr el Lagal</i>, by Farish esh +Shidiac, Tunis, fol. 609 pages: <i>Muhīt al Muhīt</i>, by Beitrus Al +Bustani Beirut, 1867-1870, 2 vols. 4to, 2358 pages (abridged as +<i>Katr Al Muhit</i>, ib. 1867-1869, 2 vols. 8vo, 2352 pages), is excellent for +spoken Arabic. <span class="sc">Persian.</span>—The <i>Surah</i>, by Jumal, Calcutta, 1812-1815, +2 vols. 4to: <i>Samachsharii Lexicon</i>, ed. Wetzstein, Leipz. 1845, +4to; 1850: <i>Muntakhal al Loghat</i>, Calcutta, 1808; ib. 1836; Lucknow, +1845; Bombay, 1862, 8vo, 2 vols.: <i>Muntaha l’Arabi</i>, 4 vols. fol. +1840: <i>Shams al Loghat</i>, Bombay, 1860, fol. 2 vols. 509 pages. +<span class="sc">Turkish.</span>—<i>Achteri Kabir</i>, Constantinople. 1827, fol.: <i>El Kamus</i>, +ib. 1816, fol. 3 vols.; translated by Açan Effendi, Bulak, fol. +3 vols.; <i>El Sihah</i>, translated by Al Vani, Constantinople, 1728, fol. +2 vols.: 1755-1756; Scutari, 1802, fol. 2 vols. <span class="sc">Latin.</span>—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. <span class="sc">English.</span>—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 <i>Medd el Kamoos</i>, meaning either the +Flow of the Sea, or The Extension of the Kamus. It was undertaken +in 1842, at the suggestion and at the cost of the 6th duke of +Northumberland, then Lord Prudhoe, by Mr Lane, who returned to +Egypt for the purpose, and lived in Cairo for seven years to study, and +obtain copies of, the great MS. lexicons in the libraries of the mosques, +few of which had ever been seen by a European, and which were so +quickly disappearing through decay, carelessness and theft, that the +means of composing such a work would not long have existed). +Newman (modern), ib. 1872, 8vo, 2 vols. 856 pages. <span class="sc">French.</span>—Ruphy +(Fr.-Ar.), Paris, 1802, 4to: Bochtor (do.), Paris, 1828, 4to, +2 vols.; 2nd ed. ib. 1850: Roland de Bussy (Algiers, Fr.-Ar.), Alger, +1835, 16mo: Id., 1836, 8vo; 1839: Berggren (Fr.-vulg. Ar., Syria +and Egypt.), Upsala, 1844, 4to: Farhat (Germanos), revu par +Rochaid ed Dahdah, Marseille, 1849, 4to: Biberstein Kasimirski, +Paris, 1846, 8vo, 2 vols.; 1853-1856; 1860, 2 vols. 3032 pages: Marcel +(vulgar dialects of Africa), Paris, 1830; 1835, 8vo; 1837; enlarged, +1869, 8vo; Paulmier (Algeria), 2nd ed. Paris, 1860, 8vo, 931 pages; +1872: Bernard (Egypt), Lyon, 1864, 18mo: Cuche, Beirut, 1862, +8vo; 1867: Nar Bey (A. Calfa), 2nd ed. Paris, 1872, 12mo, 1042 +pages: Cherbonneau (written language), Paris, 1876, 2 vols. 8vo: +Id. (Fr.-Ar.), Paris, 1872, 8vo: Beausier (Algiers, Tunis, legal, +epistolary), Alger, 1871, 4to, 764 pages; 1873. <span class="sc">German.</span>—Seyfarth +(Algeria), Grimma, 1849, 16mo: Wolff (Mod. Ar.), Leipzig, 1867, +8vo: Wahrmund (do.), Giessen, 1870-1875, 8vo, 4 vols. <span class="sc">Italian.</span>—Germano, +Roma, 1636, 8vo; (Ar. Lat. It.), Romae, 1639, fol.: +<i>Dizionario</i>, Bulak. 1824, 4to: Schiaparelli, Firenze, 1871, 4to, +641 pages. <span class="sc">Spanish.</span>—Alcala, Grenada, 1505, 4to: Cañes, Madrid, +1787, fol. 3 vols. <span class="sc">Sufi Technical Terms.</span>—Abd Errahin, ed. +Sprenger, Calcutta, 1845, 8vo. <span class="sc">Technical Terms of the Mussulman +Sciences.</span>—Abd al Hagg and Gholam Kadir, Calcutta, 1853-1862, +4to, 1593 pages. <span class="sc">Medical Terms.</span>—Pharaon and Bertherand, +Paris, 1860, 12mo. <span class="sc">Materia Medica.</span>—Muhammed Abd Allah +Shirazi, <i>Ulfaz Udwiyeh</i>, translated by Gladwin (Eng. Pers. Hindi), +Calcutta, 1793, 4to, 1441 words. <span class="sc">Noms des Vêtements.</span>—Dozy, +Amst. 1845, 8vo. <span class="sc">Wörter in entgegengesetzten Bedeutungen.</span>—Redslob, +Göttingen, 1873, 8vo. <span class="sc">Koran.</span>—Willmet (also in +Haririum et vitam Timuri), Lugd. Bat. 1784, 4to; Amst. 1790: +Fluegel, <i>Concordantia</i>, Leipz. 1842, 4to: Penrice, <i>Dictionary and +Glossary</i>, London, 1873, 4to. <span class="sc">El Tabrizi’s Logic.</span>—Mir Abufeth +(French), Bulak, 1842, 8vo. <span class="sc">Maltese.</span>—Vassali, Romae, 1796, +4to: Falzon (Malt. Ital. Eng.), Malta, <i>s.a.</i> 8vo: Vella, Livorno, +1843, 8vo.</p> + +<p><span class="bold">Armenian.</span>—Mechitar, Venice, 1749-1769, 4to, 2 vols.: Avedichiam, +Sürmelian and Aucher (Aukerian), ib. 1836-1837, 4to, 2 vols.: +Aucher, ib. 1846, 4to. <span class="sc">Polyglot.</span>—Villa (Arm.-vulg., litteralis, Lat. +Indicae et Gallicae), Romae, 1780. <span class="sc">Greek and Latin.</span>—Lazarists, +Venice, 1836-1837, 4to, 2 vols. 2217 pages. <span class="sc">Latin.</span>—Rivola, Mediolani, +1621, fol.: Nierszesovicz, Romae, 1695, 4to; Villotte, ib. 1714, +fol.: Mechitar, Venetiae, 1747-1763, 4to, 2 vols. <span class="sc">English.</span>—Aucher, +Venice, 1821-1825, 4to, 2 vols. <span class="sc">French.</span>—Aucher, Venise, 1812-1817, +8vo, 2 vols.; (Fr.-Arm. Turc.), ib. 1840, 4to: Eminian, Vienna, 1853, +4to: Calfa, Paris, 1861, 8vo, 1016 pages; 1872. <span class="sc">Italian.</span>—Ciakciak, +Venezia, 1837, 4to. <span class="sc">Russian.</span>—Khudobashev [Khutapashian], +Moskva, 1838, 8vo, 2 vols. <span class="sc">Russ. Arm.</span>—Adamdarov, ib. +1821, 8vo: Popov, ib. 1841, 8vo, 2 vols. <span class="sc">Modern Words.</span>—Riggs, +Smyrna, 1847, 8vo.</p> + +<p><span class="bold">Georgian.</span>—Paolini (Ital.), Roma, 1629, 4to: Klaproth (Fr.), +Paris, 1827, 8vo: Tshubinov (Russian, French), St Petersburg, 1840, +4to; 1846, 8vo, 2 vols. 1187 pages.</p> + +<p><span class="bold">Circassian.</span>—Loewe, London, 1854, 8vo.</p> + +<p><span class="bold">Ossetic.</span>—Sjörgen, St Petersb. 1844, 4to.</p> + +<p><span class="bold">Kurd.</span>—Garzoni, Roma, 1787, 8vo: Lerch (German), St Petersburg, +1857, 8vo: Id. (Russian), ib. 1856-1858, 8vo.</p> + +<p><span class="bold">Persian.</span>—<i>Burhani Qatiu</i>, arranged by J. Roebuck, Calcutta, +1818, 4to: <i>Burhan i Kati</i>, Bulak, 1836, fol.: Muhammed Kazim, +Tabriz, 1844, fol.: <i>Haft Kulzum</i> (The Seven Seas), by Ghazi ed din +Haidar, King of Oude, Lucknow, 1822, fol. 7 vols. <span class="sc">Arabic.</span>—<i>Shums +ul Loghat</i>, Calcutta, 1806, 4to, 2 vols. <span class="sc">Turkish.</span>—Ibrahim Effendi, +<i>Farhangi Shu’uri</i>, ib. 1742, fol. 2 vols. 22,530 words, and 22,450 +poetical quotations: <i>Burhan Kati</i>, by Ibn Kalif, translated by +Ahmed Asin Aintabi, ib. 1799, fol.; Bulak, 1836, fol.: Hayret +Effendi, ib. 1826, 8vo. <span class="sc">Armenian.</span>—Douzean, Constantinople, +1826, fol. <span class="sc">Bengali.</span>—Jay Gopal, Serampore, 1818, 8vo. <span class="sc">Latin.</span>—Vullers +(Zend appendix), Bonnae ad Rhen, 1855-1868, 4to, 2 vols. +2544 pages; Supplement of Roots, 1867, 142 pages. <span class="sc">English.</span>—Gladwin, +Malda in Bengal, 1780, 4to; Calcutta, 1797: Kirkpatrick, +London, 1785, 4to: Moises, Newcastle, 1794, 4to: Rousseau, +London, 1802, 8vo; 1810: Richardson (Arab, and Pers.), ib. 1780-1800, +fol. 2 vols.; ed. Wilkins, ib. 1806-1810, 4to, 2 vols.; ed Johnson, +ib. 1829, 4to: Ramdhen Sen, Calcutta, 1829, 8vo; 1831: Tucker +(Eng.-Pers.), London, 1850, 4to: Johnson (Pers. and Arab.), ib. +1852, 4to: Palmer, ib. 1876, 8vo, 726 pages. <span class="sc">French.</span>—Handjeri +(Pers. Arab. and Turkish), Moscou, 1841, 4to, 3 vols. 2764 pages: +Bergé, Leipzig, 1869, 12mo. <span class="sc">German.</span>—Richardson, translated by +Wahl as <i>Orientalische Bibliotheque</i>, Lemg, 1788-1792, 8vo, 3 vols. +<span class="sc">Italian.</span>—Angelus a S. Josepho [<i>i.e.</i> Labrosse] (Ital. Lat. Fr.), Amst. +1684, fol.</p> + +<p><span class="bold">Old Persian.</span>—(Cuneiform), Benfey (German), Leipzig, 1847, 8vo: +Spiegel (id.), ib. 1862, 8vo: Kossovich (Latin), Petropoli, 1872, 8vo.</p> + +<p><span class="bold">Zend.</span>—Justi, Leipzig, 1864, 4to: Vullers, Persian Lexicon, +Appendix: Lagarde, Leipzig, 1868, 8vo.</p> + +<p><span class="bold">Pahlavi.</span>—<i>An old Pahlavi and Pazend Glossary</i>, translated by +Destur Hoshengi Jamaspji, ed. Haug, London, 1867, 8vo; 1870, +8vo: West, Bombay, 1874, 8vo.</p> + +<p><span class="sc pt2">Indian Terms.</span>—<i>The Indian Vocabulary</i>, London, 1788, 16mo: +Gladwin, Calcutta, 1797, 4to: Roberts, London, 1800, 8vo: Rousseau, +ib. 1802, 8vo: Roebuck (naval), ib. 1813, 12mo: C. P. Brown, +<i>Zillah Dict.</i>, Madras, 1852, 8vo: Robinson (Bengal Courts), Calcutta, +1854, 8vo; 1860: Wilson, London, 1855, 4to: Fallon, Calcutta, +1858, 8vo.</p> + +<p><span class="bold">Sanskrit.</span>—Amarasimha (lived before <span class="sc">a.d.</span> 1000), <i>Amarakosha</i> +Calcutta, 1807, 8vo; ib. 1834, 4to; Bombay, 1860, 4to; Lucknow, +1863, 4to; Madras, 1870, 8vo, in Grantha characters; Cottayam, +1873, 8vo, in Malaylim characters; Benares, 1867, fol. with +<i>Amaraviveka</i>, a commentary by Mahesvara: Rajah Radhakanta +Deva, <i>Sabdakalpadruma</i>, Calcutta, 1821-1857, 4to, 8 vols. 8730 pages: +2nd ed. 1874, &c.: Bhattachdrya, <i>Sabdastoma Mahanidhi</i>, Calcutta, +1869-1870, 8vo, parts i.-vii. 528 pages: <i>Abhidhanaratnamala</i>, by +Halayudha, ed. Aufrecht, London, 1861, 8vo: <span class="sc">Vachaspatya</span>, by +Taranatha Tarkavachaspati, Calcutta, 1873, &c., 4to (parts i.-vii., +1680 pages). <span class="sc">Bengali.</span>—<i>Sabdasindhu</i>, Calcutta, 1808: <i>Amarakosa</i>, +translated by Ramodoyu Bidjalunker, Calcutta, 1831, 4to: +Mathurana Tarkaratna, <i>Sabdasandarbhasindhu</i>, Calcutta, 1863, 4to. +<span class="sc">Marathi.</span>—Ananta Sastri Talekar, Poona, 1853, 8vo, 495 pages: +Madhava Chandora, Bombay, 1870, 4to, 695 pages. <span class="sc">Telugu.</span>—<i>Amarakosha</i>, +Madras, 1861, ed. Kala, with <i>Gurubalala prabodhika</i>, a +commentary, ib. 1861, 4to; with the same, ib. 1875, 4to, 516 pages; +with <i>Amarapadaparijata</i> (Sans. and Tel.), by Vavilla Ramasvani +Sastri, ib. 1862, 4to; ib. 1863, 8vo; 3rd ed. by Jaganmohana +Tarkalankara and Khetramohana, 1872, &c., parts i.-iv. 600 pages: +Suria Pracasa Row, <i>Sarva-Sabda-Sambodhini</i>, ib. 1875, 4to, 1064 +pages. <span class="sc">Tibetan and Mongol.</span>—Schiefner, <i>Buddhistische Triglotte</i>, +St Petersburg, 1859, fol., the <i>Vyupatti</i> or <i>Mahavyupatti</i> from the +<i>Tanguir</i>, vol. 123 of the Sutra. <span class="sc">Latin.</span>—Paulinus a Sancto +Bartholomeo, Amarasinha, sectio i. de coelo, Romae, 1798, 4to: +Bopp. Berlin, 1828-1830, 4to; 2nd ed. 1840-1844; 3rd, 1866, 4to. +<span class="sc">English.</span>—<i>Amarakosha</i>, trans. by Colebrooke, Serampore, 1808, +4to; 1845, 8vo: Rousseau, London, 1812, 4to: Wilson, Calcutta, +1819, 4to; 2nd ed. 1832: ed. Goldstücker, Berlin, 1862, &c., folio, +to be in 20 parts: Yates, Calcutta, 1846, 4to: Benfey, London, 1865, +8vo: Ram Jasen, Benares, 1871, 8vo, 713 pages: Williams, Oxford, +1872, 4to. <span class="sc">English-Sanskrit.</span>—Williams, London, 1851, 4to. +<span class="sc">French.</span>—Amarakosha, transl. by Loiseleur Deslongchamps, Paris, +1839-1845, 8vo, 2 vols. 796 pages: Burnouf and Leupol, Nancy, +1863-1864, 8vo. <span class="sc">German.</span>—Böhtlingk and Roth, St Petersb. 1853, +&c., 4to, 7 vols. to 1875. <span class="sc">Italian.</span>—Gubernatis, Torino, 1856, &c. +8vo, unfinished, 2 parts. <span class="sc">Russian.</span>—Kossovich, St Petersburg, 1859, +<span class="pagenum"><a name="page198" id="page198"></a>198</span> +8vo. <span class="sc">Roots</span>.—Wilkins, London, 1815, 4to: Rosen, Berolini, 1827, +8vo: Westergaard, Bonnae, 1840-1841, 8vo: Vishnu Parasurama +Sastri Pandita (Sans. and Marathi), Bombay, 1865, 8vo: Taranatha +Tarkavachaspati, <i>Dhatupadarsa</i>, Calcutta, 1869, 8vo: Leupol, Paris, +1870, 8vo. <span class="sc">Synonyms</span>.—<i>Abhidhanacintamani</i>, by Hemachadra, ed. +Colebrooke, Calcutta, 1807, 8vo; translated by Böhtlingk and Rieu +(German), St Petersburg, 1847, 8vo. <span class="sc">Homonyms</span>.—Medinikara, +<i>Medinikosha</i>, Benares, 1865, 4to; Calcutta, 1869, 8vo; ib. 1872, +8vo. <span class="sc">Derivatives</span>.—Hirochand and Rooji Rangit, <i>Dhatumanjari</i>, +Bombay, 1865, 8vo. <span class="sc">Technical Terms of the Nyâya Philosophy.</span>—<i>Nyâyakosa</i>, +by Bhimachârya Jhalakîkar (Sanskrit), +Bombay, 1875, 8vo, 183 pages. <span class="sc">Rig Veda</span>.—Grassmann, Leipzig, +1873-1875, 8vo.</p> + +<p><span class="bold">Bengali.</span>—Manoel, Lisboa, 1743, 8vo: Forster, Calcutta, 1799-1802, +4to, 2 vols. 893 pages: Carey, Serampore, 1815-1825, 4to, 2 vols.; +ed. Marshman, ib. 1827-1828, 8vo, 2 vols.; 3rd ed. ib. 1864-1867, +8vo; abridged by Marshman, ib. 1865, 8vo; ib. 1871, 8vo, 2 vols. +936 pages: Morton, Calcutta, 1828, 8vo: Houghton, London, 1833, +4to: Adea, <i>Shabdabudhi</i>, Calcutta, 1854, 604 pages. <span class="sc">English</span>.—Ram +Comul Sen, ib. 1834, 4to, 2 vols.; London, 1835, 4to: +D’Rozario, Calcutta, 1837, 8vo: Adea, <i>Abhidan</i>, Calcutta, 1854, +761 pages. <span class="sc">English Lat.</span>—Ramkissen Sen, ib. 1821, 4to. <span class="sc">Eng.-Beng. +and Manipuri.</span>—[Gordon], Calcutta, 1837, 8vo.</p> + +<p><span class="bold">Canarese.</span>—Reeve, Madras, 1824-1832, 4to, 2 vols.; ed. Sanderson, +Bangalore, 1858, 8vo, 1040 pages; abridged by the same, 1858, +8vo, 276 pages: <i>Dictionarium Canarense</i>, Bengalori, 1855, 8vo: +<i>School Dictionary</i>, Mangalore, 1876, 8vo, 575 pages.</p> + +<p><span class="bold">Dardic Languages.</span>—Leitner (Astori, Ghilghiti, Chilasi, and dialects +of Shina, viz. Arnyia, Khajuna and Kalasha), Lahore, 1868, 4to.</p> + +<p><span class="bold">Guzarati.</span>—(English) Mirza Mohammed Cauzim, Bombay, 1846, +4to; Shapurji Edalji, ib. 1868, 8vo, 896 pages: Karsandas Mulji, +ib. 1868, 8vo, 643 pages.</p> + +<p><span class="bold">Hindi.</span>—Rousseau, London, 1812, 4to: Adam, Calcutta, 1829, +8vo: Thompson, ib. 1846, 8vo: J. D. Bate, London, 1876, 8vo, 809 +pages. <span class="sc">English</span>.—Adam, Calcutta, 1833, 8vo. <span class="sc">English, Urdu +and Hindi</span>.—Mathuraprasada Mirsa, Benares, 1865, 8vo, 1345 +pages.</p> + +<p><span class="bold">Hindustani.</span>—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. <span class="sc">English</span>.—Gilchrist, 1787-1780, +4to, 2 parts: Thompson, Serampore, 1838, 8vo.</p> + +<p><span class="bold">Kashmiri.</span>—Elmslie, London, 1872, 12mo.</p> + +<p><span class="bold">Khassia.</span>—Roberts, Calcutta, 1875, 12mo.</p> + +<p><span class="bold">Malayalim.</span>—Fabricius and Breithaupt, Weperg, 1779, 4to: Bailey, +Cottayam, 1846, 8vo: Gundert, Mangalore, 1871, 8vo, 1171 pages.</p> + +<p><span class="bold">Marathi.</span>—Carey, Serampore, 1810, 8vo: Kennedy, Bombay, +1824, fol.: Jugunnauth Shastri Kramavant, Bombay, 1829-1831, +4to, 3 vols.: Molesworth, ib. 1831, 4to; 2nd ed. 1847, 4to; ed. Candy, +Bombay, 1857, 4to, 957 pages; abridged by Baba Padmanji, +ib. 1863, 8vo; 2nd ed. (abridged), London, 1876, 8vo, 644 pages. +<span class="sc">English</span>.—Molesworth, Bombay, 1847, 4to.</p> + +<p><span class="bold">Oriya.</span>—Mohunpersaud Takoor, Serampore, 1811, 8vo: Sutton, +Cuttack, 1841-1848, 8vo, 3 vols. 856 pages.</p> + +<p><span class="bold">Pali.</span>—Clough, Colombo, 1824, 8vo: Moggallana Thero (a Sinhalese +priest of the 12th century), <i>Abhidhanappika</i> (Pali, Eng. +Sinhalese), ed. Waskeduwe Subheti, Colombo, 1865, 8vo: Childers, +London, 1872-1875, 8vo, 658 pages. <span class="sc">Roots</span>.—Silavansa, <i>Dhatumanjusa</i> +(Pali Sing. and Eng.), Colombo, 1872, 8vo.</p> + +<p><span class="bold">Prakrit.</span>—Delius, <i>Radices</i>, Bonnae ad Rh., 1839, 8vo.</p> + +<p><span class="bold">Punjabi.</span>—Starkey, 1850, 8vo; Lodiana Mission, Lodiana, +1854-1860, 444 pages.</p> + +<p><span class="bold">Pushtu</span> or <b>Afghan.</b>—Dorn, St Petersb. 1845, 4to: Raverty, +London, 1860, 4to; 2nd ed. ib. 1867, 4to: Bellew, 1867, 8vo.</p> + +<p><span class="bold">Sindhi.</span>—Eastwick, Bombay, 1843, fol. 73 pages: Stack, ib. 1855, +8vo, 2 vols.</p> + +<p><span class="bold">Sinhalese.</span>—Clough, Colombo, 1821-1830, 8vo, 2 vols.: Callaway +(Eng., Portuguese and Sinhalese), ib. 1818, 8vo: Id., <i>School +Dictionary</i>, ib. 1821, 8vo: Bridgenell (Sinh.-Eng.), ib. 1847, 18mo: +Nicholson (Eng.-Sinh.), 1864, 32mo, 646 pages.</p> + +<p><span class="bold">Tamil.</span>—Provenza (Portug.), Ambalacotae, 1679, 8vo: <i>Sadur +Agurardi</i>, written by Beschi in 1732, Madras, 1827, fol.; Pondicherry, +1875, 8vo: Blin (French), Paris, 1834, 8vo: Rottler, Madras, 1834-1841, +4to, 4 vols.: Jaffna Book Society (Tamil), Jaffna, 1842, 8vo, +about 58,500 words: Knight and Spaulding (Eng. Tam.), ib. 1844, +8vo; <i>Dictionary</i>, ib. 1852, 4to: Pope, 2nd ed. ib. 1859, 8vo: Winslow, +Madras, 1862, 4to, 992 pages, 67,452 words.</p> + +<p><span class="bold">Telugu.</span>—Campbell, Madras, 1821, 4to: C. P. Brown, Madras +(Eng.-Tel.), 1852, 8vo, 1429 pages: Id. (Tel.-Eng.), ib. 1852, 8vo, +1319 pages. <span class="sc">Mixed Telugu</span>.—Id., ib. 1854, 8vo.</p> + +<p><span class="bold">Thuggee.</span>—Sleeman, Calcutta, 1830, 8vo, 680 Ramasi words.</p> + +<p><span class="bold">Indo-Chinese Languages.</span>—Leyden, <i>Comparative Vocabulary of +Barma, Malaya and Thai</i>, Serampore, 1810, 8vo. <i>Annamese</i>: +Rhodes (Portug. and Lat.), Romae, 1651, 4to: Pigneaux and Taberd, +Fredericinagori, 1838, 4to; Legrand de la Liraye, Paris, 1874, 8vo: +Pauthier (Chin. Ann.-Fr. Lat.), Paris, 1867, &c., 8vo. <i>Assamese</i>: +Mrs Cutter, Saipur, 1840, 12 mo; Bronson, London, 1876, 8vo, 617 +pages. <i>Burmese</i>: Hough (Eng.-Burm.), Serampore, 1825, Moulmain, +1845, 8vo, 2 vols. 955 pages: Judson, Calcutta, 1826, 8vo; +(Eng. Burm.), Moulmain, 1849, 4to; (Burm. Eng.), ib. 1852, 8vo; +2nd ed., Rangoon, 1866, 8vo, 2 vols. 968 pages: Lane, Calcutta, 1841, +4to. <i>Cambodian</i>: Aymonier (Fr.-Camb.), Saigon, 1874, 4to; Id. +(Camb.-Fr.), ib. 1875, fol. <i>Karen</i>: Sau-kau Too (Karen), Tavoy, +1847, 12mo, 4 vols.: Mason, Tavoy, 1840, 4to. <i>Sgau-Karen</i>: Wade, +ib. 1849, 8vo. <i>Siamese (Thai)</i>: Pallegoix (Lat. French, Eng.), +Paris, 1854, 4to: <i>Dictionarium Latinum Thai</i>, Bangkok, 1850, +4to, 498 pages.</p> + +<p><span class="bold">Malay.</span>—<span class="sc">Latin</span>.—Haex, Romae, 1631, 4to; Batavia, 1707. +<span class="sc">Dutch</span>.—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. <span class="sc">English</span>.—Houtmann +(Malay and Malagasy), translated by A. Spaulding, London, +1614, 4to: Bowrey, ib. 1701, 4to: Howison, ib. 1801, 4to: Marsden, +ib. 1812, 4to: Thomsen, Malacca, 1820, 8vo; 1827: Crawford, +London, 1851, 8vo, 2 vols. <span class="sc">French</span>.—Boze, Paris, 1825, 16mo: +Elout (Dutch-Malay and French-Malay), Harlem, 1826, 4to: +Bougourd, Le Havre, 1856, 8vo: Richard, Paris, 1873, 8vo, 2 vols.: +Favre, Vienna, 1875, 8vo, 2 vols.</p> + +<p><span class="bold">Malay Archipelago.</span>—<i>Batak</i>: Van der Tuuk, Amsterdam, 1861, +8vo, 564 pages. <i>Bugis</i>: Mathes, Gravenh. 1874, 8vo, 1188 pages: +Thomsen (Eng.-Bugis and Malay), Singapore, 1833, 8vo. <i>Dyak</i>: +Hardeland (German), Amst. 1859, 8vo, 646 pages. <i>Javanese</i>: Senerpont +Domis, Samarang, 1827, 4to, 2 vols.: Roorda van Eysinga, +Kampen, 1834-1835, 8vo, 2 vols.: Gericke, Amst. 1847, 8vo; ed. +Taco Roorda, ib. 1871, &c. parts i.-v., 880 pages: Jansz and +Klinkert, Samarang, 1851, 8vo; 1865: Favre (French), Vienne, +1870, 8vo. <i>Macassar</i>: Matthes, Amst. 1859, 8vo, 951 pages. +Sunda: De Wilde (Dutch, Malay and Sunda), Amsterdam, +1841, 8vo: Rigg (Eng.), Batavia, 1862, 4to, 573 pages. <i>Formosa</i>: +Happart (Favorlang dialect, written about 1650), Parrapattan, +1840, 12mo.</p> + +<p><span class="bold">Philippines.</span>—<i>Bicol</i>: Marcos, Sampaloc, 1754, fol. <i>Bisaya</i>: Sanchez, +Manila, 1711, fol.: Bergaño, ib. 1735, fol.: Noceda, ib. 1841: +Mentrida (also Hiliguena and Haraya) ib. 1637, 4to; 1841, fol. 827 +pages: Felis de la Encarnacion, ib. 1851, 4to, 2 vols. 1217 pages. +<i>Ibanac</i>: Bugarin, ib. 1854, 4to. <i>Ilocana</i>, Carro, ib. 1849, fol. +<i>Pampanga</i>: Bergaño, ib. 1732, fol. <i>Tagala</i>: Santos, Toyabas, 1703, +fol.; ib. 1835, 4to, 857 pages: Noceda and San Lucar, Manila, 1754, +fol.; 1832.</p> + +<p><span class="bold">Chinese.</span>—Native Dictionaries are very numerous. Many are +very copious and voluminous, and have passed through many +editions. <i>Shwo wan</i>, by Hü Shin, is a collection of the ancient characters, +about 10,000 in number, arranged under 540 radicals, published +150 <span class="sc">b.c.</span>, usually in 12 vols.: <i>Yu pien</i>, by Ku Ye Wang, published +<span class="sc">a.d.</span> 530, arranged under 542 radicals, is the basis of the Chinese +Japanese Dictionaries used in Japan: <i>Ping tseu loui pien</i>, Peking, +1726, 8vo, 130 vols.: <i>Pei wan yün fu</i> (Thesaurus of Literary Phrases), +1711, 131 vols. 8vo, prepared by 66 doctors of the Han lin Academy +in seven years. It contains 10,362 characters, and countless combinations +of two, three or four characters, forming compound words +and idioms, with numerous and copious quotations. According to +Williams (<i>On the word Shin</i>, p. 79), an English translation would fill +140 volumes octavo of 1000 pages each. <i>Kanghi tsze tien</i> (Kanghi’s +Standard or Canon of the Character), the dictionary of Kanghi, the +first emperor of the present dynasty, was composed by 30 members +of the Han lin, and published in 1716, 40 vols. 4to, with a preface by +the emperor. It contains 49,030 characters, arranged under the 214 +radicals. It is generally in 12 vols., and is universally used in China, +being the standard authority among native scholars for the readings +as well as the meanings of characters. <span class="sc">Latin</span>.—De Guignes (French, +Lat.), Paris, 1813, fol.; Klaproth, Supplément, 1819; ed. Bazil +(Latin), Hong-Kong, 1853, 4to: Gonçalves (Lat.-Chin.), Macao, +1841, fol.: Callery, <i>Systema phoneticum</i>, Macao, 1841, 8vo: Schott, +<i>Vocabularium</i>, Berlin, 1844, 4to. <span class="sc">English</span>.—Raper, London, 1807, +fol. 4 vols.: Morrison, Macao, 1815-1823, 4to, 3 parts in 6 vols.: +Medhurst, Batavia, 1842-1843, 8vo, 2 vols.: Thom, Canton, 1843, +8vo: Lobscheid, Hong-Kong, 1871, 4to: Williams, Shanghai, 1874, +4to. <span class="sc">Eng. Chinese</span>.—Morrison, part iii.: Williams, Macao, 1844, +8vo: Medhurst, Shanghai, 1847-1848, 8vo, 2 vols.: Hung Maou, +<i>Tung yung fan hwa</i> (Common words of the Red-haired Foreigners), +1850, 8vo. Doolittle, Foochow, 1872, 4to, vol. i. 550 pages. <span class="sc">French</span>,—Callery, +<i>Dict. encyclopédique</i>, Macao and Paris, 1845 (radicals 1-20 +only): M. A. H., 1876, 8vo, autographié, 1730 pages. <span class="sc">French-Chin</span>.—Perny +(Fr.-Latin, Spoken Mandarin), Paris, 1869, 4to; +Appendice, 1770; Lemaire and Giguel, Shanghai, 1874, 16mo. +<span class="sc">Portuguese</span>.—Gonçalves (Port.-Chin.), Macao, 1830, 8vo, 2 vols.: +Id. (Chin.-Port.), ib. 1833, 8vo. <span class="sc">Idioms</span>.—Giles, Shanghai, 1873, +4to. <span class="sc">Phrases</span>.—Yaou Pei-keen, <i>Luy yih</i>, 1742-1765, 8vo, 55 vols.: +Tseen Ta-hin, <i>Shing luy</i>, 1853, 8vo, 4 vols. <span class="sc">Classical Expressions</span>.—Keang +Yang and 30 others, <i>Sze Shoo teen Lin</i>, 1795, 8vo, 30 vols. +<span class="sc">Elegant Expressions</span>.—Chang ting yuh, <i>Fun luy tsze kin</i>, 1722, +<span class="pagenum"><a name="page199" id="page199"></a>199</span> +8vo, 64 vols. <span class="sc">Phrases of Three Words.</span>—Julien (Latin), Paris, +1864, 8vo. <span class="sc">Poetical.</span>—<i>Pei wan she yun</i>, 1800, 8vo, 5 vols. <span class="sc">Proper +Names.</span>—F. Porter Smith (China, Japan, Corea, Annam, &c., +Chinese-Eng.), Shanghai, 1870, 8vo. <span class="sc">Topography.</span>—Williams, +Canton, 1841, 8vo. <span class="sc">Names of Towns.</span>—Biot, Paris, 1842, 8vo. +<span class="sc">Ancient Characters.</span>—Foo Lwantseang, <i>Luh shoo fun luy</i>, 1800, +8vo, 12 vols. <span class="sc">Seal Character.</span>—Heu Shin, <i>Shwo wan</i>, ed. Seu +Heuen, 1527, 8vo, 12 vols. <span class="sc">Running Hand.</span>—St Aulaire and +Groeneveld (Square Characters, Running Hand; Running, Square), +Amst. 1861, 4to, 117 pages. <span class="sc">Technical Terms</span> (in Buddhist translations +from Sanskrit)—Yuen Ying, <i>Yih ’see king pin e</i>, 1848, 8vo. +<span class="sc">Dialects.</span>—<i>Amoy</i>: Douglas, London, 1873, 4to, 632 pages: +Macgowan, Hong-Kong, 1869, 8vo. <i>Canton</i>: Yu Heo-poo and Wan +ke-shih, <i>Keang hoo chih tuh fun yun tso yaou ho tseih</i>, Canton, 1772, +8vo, 4 vols.; 1803, 8vo, 4 vols.; Fuh-shan, 1833, 8vo, 4 vols.: +Morrison, Macao, 1828, 8vo: Wan ke shih, Canton, 1856, 8vo: +Williams (tonic, Eng.-Chinese), Canton, 1856, 8vo: Chalmers, Hong-Kong, +1859, 12mo; 3rd ed. 1873, 8vo. <i>Changchow in Fuhkeen</i>: Seay +Sew-lin, <i>Ya suh tung shih woo yin</i>, 1818, 8vo, 8 vols.; 1820. <i>Foo-chow</i>: +Tseih (a Japanese general) and Lin Peih shan, <i>Pa yin ho ting</i>, +ed. Tsin Gan, 1841, 8vo: Maclay and Baldwin, Foochow, 1870, 8vo, +1123 pages. <i>Hok-keen</i>: Medhurst, Macao, 1832, 4to: <i>Peking</i>, +Stent, Shanghai, 1871, 8vo.</p> + +<p><span class="bold">Corean.</span>—<span class="sc">Chinese, Corean and Japanese.</span>—<i>Cham Seen Wo +Kwo tsze mei</i>, translated by Medhurst, Batavia, 1835, 8vo. <span class="sc">Russian.</span>—Putzillo, +St Petersburg, 1874, 12mo, 746 pages.</p> + +<p><span class="bold">Japanese.</span>—<i>Sio Ken Zi Ko</i> (Examination of Words and Characters), +1608, 8vo, 10 vols.: <i>Wa Kan Won Se Ki Sio Gen Zi Ko</i>, lithographed +by Siebold, Lugd. Bat., 1835, fol. <span class="sc">Jap.-Chinese.</span>—<i>Faga biki set yo +siu</i>. <span class="sc">Chinese-Jap.</span>—<i>Kanghi Tse Tein</i>, 30 vols. 12mo: <i>Zi rin gioku +ben</i>. <span class="sc">Dutch Dictionaries printed by Japanese.</span>—<i>Nieeu verzameld +Japansch en Hollandsch Woordenbock</i>, by the interpreter, B. Sadayok, +1810: Minamoto Masataka, Prince of Nakats (Jap. Chinese-Dutch), +5 vols. 4to, printed at Kakats by his servants: <i>Jedo-Halma</i> (Dutch-Jap.), +Jedo, 4to, 20 vols.: <i>Nederduitsche taal</i>, Dutch Chinese, for +the use of interpreters. <span class="sc">Latin and Portuguese.</span>—Calepinus, <i>Dictionarium</i>, +Amacusa, 1595, 4to. <span class="sc">Latin.</span>—Collado, <i>Compendium</i>, +Romae, 1632, 4to: <i>Lexicon</i>, Romae, 1870, 4to, from Calepinus. +<span class="sc">English.</span>—Medhurst, Batavia, 1830, 8vo: Hepburn, Shanghai, +1867, 8vo; 1872. <span class="sc">Eng.-Jap.</span>—Hori Tatnoskoy, Yedo, 1862, 8vo; +2nd ed. Yeddo, 1866, 8vo: Satow and Ishibashi Masakata (spoken +language), London, 1876, 8vo. <span class="sc">French.</span>—Rosny (Jap. Fr. Eng.), +Paris, 1857, 4to, vol. i.: Pagés, Paris, 1869, 4to, translated from +Calepinus. <span class="sc">Fr.-Jap.</span>—Soutcovey, Paris, 1864, 8vo. <span class="sc">Fr. Eng. Jap.</span>—Mermet +de Cachon, Paris, 1866, 8vo, unfinished. <span class="sc">German.</span>—Pfizmaier +(Jap.-Ger., Eng.), Wien, 1851, 4to, unfinished. <span class="sc">Spanish.</span>—<i>Vocabulario +del Japon</i>, Manila, 1630, 4to, translated from the next. +<span class="sc">Portuguese.</span>—<i>Vocabulario da Lingua de Japam</i>, Nagasaki, 1603, +4to. <span class="sc">Russian.</span>—Goshkevich, St Petersburg, 1857, 8vo, 487 pages. +<span class="sc">Chinese Characters with Japanese Pronunciation.</span>—Rosny, +Paris, 1867, 8vo. <span class="sc">Chinese and Japanese Names of Plants.</span>—Hoffmann, +Leyde, 1864, 8vo.</p> + +<p><span class="bold">Aino.</span>—Pfizmaier, Wien, 1854, 4to.</p> + +<p><span class="bold">Northern and Central Asia.</span>—<i>Buriat</i>: Castrén, St Petersburg, 1857, +8vo. <i>Calmuck</i>: Zwick, Villingen, 1853, 4to: Smirnov, Kazan, +1857, 12mo: Jügl, <i>Siddhi Kur</i>, Leipzig, 1866, 8vo. <i>Chuvash</i>: +Clergy of the school of the Kazan Eparchia, Kazan, 1836, 8vo, 2481 +words: Lyulé (Russ.-Chuv. French), Odessa, 1846, 8vo, 244 pages: +Zolotnitski, Kazan, 1875, 8vo, 287 pages. <i>Jagatai</i>: Mir Ali Shir, +<i>Abuska</i>, ed. Vámbéry, with Hungarian translation, Pesth, 1862, 8vo: +Vámbéry, Leipzig, 1867, 8vo: Pavet de Courteille, Paris, 1870, 8vo. +<i>Koibal and Karagas</i>: Castrén, St Petersburg, 1857, 8vo. <i>Manchu</i>: +<i>Yutchi tseng ting tsing wen kian</i> (Manchu Chinese), 1771, 4to, 6 vols.: +<i>Sze li hoh pik wen kian</i> (Manchu-Mongol, Tibetan, Chinese) 10 vols. +4to, the Chinese pronunciation represented in Manchu: <i>San hoh +pien lan</i> (Manchu-Chinese, Mongol), 1792, 8vo, 12 vols.;—all three +classed vocabularies: Langlès (French), Paris, 1789-1790, 4to, 3 vols.: +Gabelentz (German), Leipzig, 1864, 8vo: Zakharov (Russian), St +Petersburg, 1875, 8vo, 1235 pages: <i>Mongol</i>: I. J. Schmidt (German, +Russian), St Petersburg, 1835, 4to: Schergin, Kazan, 1841, 8vo: +Kovalevski, Kasan, 1844-1849, 4to, 3 vols. 2703 pages. <i>Ostiak</i>: +Castrén, St Petersb. 1858, 8vo. <i>Samoyed</i>: Castrén, St Petersb. 1855, +8vo, 308 pages. Tartar: Giganov (Tobolsk), St Petersburg, 1804, +4to; (Russ.-Tartar), ib. 1840, 4to: Troyanski (Karan), Kasan, +1835-1855, 4to. <i>Tibetan</i>: <i>Minggi djamtoo</i> (Tibet-Mongol): <i>Bodschi +dajig togpar lama</i>: <i>Kad shi schand scharwi melonggi jige</i> +(Manchu-Mongol-Tibetan-Chinese), Kanghi’s Dictionary with the Tibetan +added in the reign of Khian lung (1736-1795); Csoma de Körös (Eng.), +Calcutta, 1834, 4to: I. J. Schmidt (German), St Petersburg, 1841, +4to: Id. (Russian), ib. 1843, 4to: Jaeschke (Eng.), London, 1870, +8vo, 160 pages: Id. (Germ.), Gnadau, 1871, 658 pages: (Bhotanta), +Schroeter, Serampore, 1826, 4to. <i>Tungusian</i>: Castrén, St Petersburg, +1856, 8vo, 632 pages. <i>Uigur</i>: Vámbéry, Innspruck, 1870, 4to. +<i>Yakut</i>: Böhtlingk, ib. 1854, 4to, 2 vols. <i>Yenissei Ostiak</i>: Castrén, +ib. 1849, 8vo.</p> + +<p class="center pt2">AFRICA</p> + +<p><span class="bold">Egyptian.</span>—Young (enchorial), London, 1830-1831, 8vo: Sharpe, +London, 1837, 4to: Birch, London. 1838, 4to: Champollion (died +March 4, 1832), <i>Dictionnaire égyptien</i>, Paris, 1841, 4to: Brugsch, +<i>Hieroglyphisch-Demotisches Wörterbuch</i>, Leipzig, 1867-1868, 4to, +4 vols. 1775 pages, nearly 4700 words, arranged according to the +hieroglyphic alphabet of 28 letters: Pierret, <i>Vocabulaire hiérog.</i>, Paris, +1875, 8vo, containing also names of persons and places: Birch, in +vol. v. pp. 337-580 of Bunsen’s <i>Egypt’s Place</i>, 2nd ed. London, 1867, +&c. 8vo, 5010 words. <span class="sc">Proper Names.</span>—Brugsch, Berlin, 1851, 8vo, +726 names: Parthey, ib. 1864, 8vo, about 1500 names: Lieblein, +Christiania, 1871, 8vo, about 3200 from hieroglyphic texts. <span class="sc">Book +of the Dead.</span>—Id., Paris, 1875, 12mo.</p> + +<p><span class="bold">Coptic.</span>—Veyssière de la Croze, Oxon. 1775, 8vo: Rossi, Romae, +1807, 4to: Tattam, Oxon. 1855, 8vo: Peyron, 1835, 4to (the +standard): Parthey, Berolini, 1844, 8vo.</p> + +<p><span class="bold">Ethiopic.</span>—Wemmer, Romae, 1638, 4to: Ludolf, London, 1661, +4to: Francof. ad M., 1699, fol.: Dillmann (Tigré appendix), +Leipzig, 1863-1865, 4to, 828 pages.</p> + +<p><span class="bold">Amharic.</span>—Ludolphus, Franc. ad Maenum, 1698, fol.: Isenberg, +London, 1841, 4to, 442 pages. <i>Tigré</i>: Munzinger, Leipzig, 1865, +8vo: Beurmann, ib. 1868, 8vo.</p> + +<p><span class="bold">East Coast.</span>—<i>Dankali</i>: Isenberg, London, 1840, 12mo. <i>Galla</i>: +Krapf, London, 1842, 8vo: Tutschek, München, 1844, 8vo. <i>Engutuk +Iloigob</i>: Erhardt, Ludwigsberg, 1857, 8vo. <i>Kisuaheli</i>: <i>Vocabulary +of the Soahili</i>, Cambridge, U.S. 1845, 8vo: Steere, London, +1870, 8vo, about 5800 words. <i>Kisuaheli, Kinika, Kikamba, Kipokono, +Kikian, Kigalla</i>: Krapf, Tübingen, 1850, 8vo.</p> + +<p><span class="bold">Malagasy.</span>—Houtmann (Malaysche en Madagask Talen), Amst. +1603, 2nd ed. Matthysz, ib. 1680, 8vo: Huet de Froberville, Isle de +France, fol. 2 vols.: Flacourt, Paris, 1658, 8vo: Challand (Southern), +Isle de France, 1773, 4to: Freeman and Johns, London, 1835, 8vo, +2 vols.: Dalmont (Malgache, Salalave, et Betsimara), 1842, 8vo: +Kessler, London, 1870, 8vo.</p> + +<p><span class="bold">Southern Africa.</span>—Bleek, <i>The Languages of Mozambique</i>, London, +1856, 8vo. <i>Kaffre</i>: Bennie, Lovedale, 1826, 16mo: Ayliffe, +Graham’s Town, 1846, 12mo: Appleyard, 1850, 8vo: Bleek, Bonn, +1853, 4to, 646 pages. <i>Zulu-Kaffre</i>: Perrin (Kaffre-Eng.), London, +1855, 24mo, 172 pages: Id. (Eng.-Kaffre), Pietermaritzburg, 1855, +24mo, 227 pages: Id. (Eng.-Zulu), ib. 1865, 12mo, 226 pages: +Dohne, Cape Town, 1857, 8vo, 428 pages: Colenso, Pietermaritzburg, +1861, 8vo, 560 pages, about 8000 words. <i>Hottentot</i>: Bleek, +Cape Town, 1857, 4to, 261 pages. <i>Namaqua</i>: Tindall, ib. 1852, 8vo: +<i>Vocabulary</i>, Barmen, 1854, 8vo: Hahn, Leipzig, 1870, 12mo. +Sechuana: Casalis, Paris, 1841, 8vo. <i>Herero</i>: Hahn, Berlin, 1857, +8vo, 207 pages, 4300 words.</p> + +<p><span class="bold">Western Africa.</span>—<i>Akra</i> or <i>Ga</i>: Zimmermann, Stuttgart, 1858, +8vo, 690 pages. <i>Ashantee</i>: Christaller (also Akra), Basel, 1874, +8vo, 299 pages. <i>Bullom</i>: Nylander, London, 1814, 12mo. <i>Bunda +or Angola</i>: Cannecatim, Lisboa, 1804, 4to, 722 pages. <i>Dualla +Grammatical Elements</i>, &c., Cameroons, 1855, 8vo. <i>Efik</i> or <i>Old +Calabar</i>: Waddell, Old Calabar, 1846, 16mo, 126 pages; Edinb, +1849, 8vo, 95 pages. <i>Eyo</i>: Raban, London, 1830-1831, 12mo, 2 parts. +<i>Grebo</i>: <i>Vocabulary</i>, Cape Palmas, 1837, 8vo; <i>Dictionary</i>, ib. 1839, +8vo, 119 pages. <i>Ifa</i>: Schlegel, Stuttgart, 1857, 8vo. <i>Mpongwe</i>: +De Lorme (Franç.-Pongoué), Paris, 1876, 12mo, 354 pages. <i>Oji</i>: +Riis, Basel, 1854, 8vo, 284 pages. <i>Sherbro’</i>: Schön, <i>s. a. et l.</i> +8vo, written in 1839, 42 pages. <i>Susu</i>: Brunton, Edinburgh, 1802, +8vo, 145 pages. <i>Vei</i>: Koelle, London, 1854, 8vo, 266 pages. +<i>Wolof and Bambarra</i>: Dard, Paris, 1825, 8vo. <i>Wolof</i>: Roger, ib. +1829, 8vo: Missionnaires de S. Esprit, Dakar, 1855, &c. 16mo. +Faidherbe (French-Wolof, Poula and Soninke), St Louis, Senegambia, +1860, 12mo. <i>Yoruba</i>: Crowther, London, 1843, 8vo; +1852, 298 pages: Vidal, ib. 1852, 8vo: Bowen, Washington, 1858, +4to.</p> + +<p><span class="bold">Central Africa.</span>—Barth, <i>Vocabularies</i>. Gotha, 1862-1866, 4to. <i>Bari</i>: +Mitterreutzner, Brixen, 1867, 8vo: Reinisch, Vienna, 1874, 8vo. +<i>Dinka</i>: Mitterreutzner, Brixen, 1866, 8vo. <i>Haussa</i>: Schön (Eng.), +London, 1843, 8vo.</p> + +<p><span class="bold">Berber.</span>—Venture de Paradis, Paris, 1844, 8vo: Brosselard, ib. +1844, 8vo: Delaporte, ib. 1844, 4to, by order of the Minister of +War: Creusat, Franç.-Kabyle (Zouaoua), Alger, 1873, 8vo. <i>Siwah</i>: +Minutoli, Berlin, 1827, 4to.</p> + +<p class="center pt2">AUSTRALIA AND POLYNESIA</p> + +<p><span class="bold">Australia.</span>—<i>New South Wales</i>: Threlkeld (Lake Macquarie +Language), Sydney, 1834, 8vo. <i>Victoria</i>: Bunce, Melbourne, 1856, +12mo, about 2200 words. <i>South Australia</i>: Williams, South +Australia, 1839, 8vo: Teichelmann and Schürmann, Adelaide, +1840, 8vo: Meyer, ib. 1843, 8vo. <i>Murray River</i>: Moorhouse, ib. +1846, 8vo. <i>Parnkalla</i>: Schürmann, Adelaide, 1844, 8vo. <i>Woolner +District</i>: <i>Vocabulary</i>, ib. 1869, 12mo. <i>Western Australia</i>: Sir +George Grey, Perth, 1839, 4to; London, 1840, 8vo: Moore, ib. 1843: +Brady, Roma, 1845, 24mo, 8vo, 187 pages. <i>Tasmania</i>: Millegan, +Tasmania, 1857.</p> + +<p><span class="bold">Polynesia.</span>—Hale, <i>Grammars and Vocabularies of all the Polynesian +Languages</i>, Philadelphia, 1846, 4to. <i>Marquesas, Sandwich +Gambier</i>: Mosblech, Paris, 1843, 8vo. <i>Hawaiian</i>: Andrews, +<i>Vocabulary</i>, Lahainaluna, 1636, 8vo: Id., <i>Dictionary</i>, Honolulu, +1865, 8vo, 575 pages, about 15,500 words. <i>Marquesas</i>: Pierquin, +de Gembloux, Bourges, 1843, 8vo: Buschmann, Berlin, 1843, 8vo. +<i>Samoan</i>: <i>Dictionary</i>, Samoa, 1862, 8vo. <i>Tahitian</i>: <i>A Tahitian and +English Dictionary</i>, Tahiti, 1851, 8vo, 314 pages. <i>Tonga</i>: Rabone, +Vavau, 1845, 8vo. <i>Fijian</i>: Hazlewood (Fiji-Eng.), Vewa. 1850, +<span class="pagenum"><a name="page200" id="page200"></a>200</span> +12mo: Id. (Eng.-Fiji), ib. 1852, 12mo: Id., London, 1872, 8vo. +<i>Maori</i>: Kendall, 1820, 12mo: Williams, Paihia, 1844, 8vo; 3rd ed. +London, 1871, 8vo: Taylor, Auckland, 1870, 12mo.</p> + +<p class="center pt2">AMERICA</p> + +<p><span class="bold">North America.</span>—<i>Eskimo</i>: Washington, London, 1850, 8vo: +Petitot (Mackenzie and Anderson Rivers), Paris, 1876, 4to. +<i>Kinai</i>: Radloff, St Petersburg, 1874, 4to. <i>Greenland</i>: Egede (Gr. +Dan. Lat., 3 parts), Hafn, 1750, 8vo; 1760, Fabricius, Kjöbenhavn, +1804, 4to. <i>Hudson’s Bay Indians</i>: Bowrey, London, 1701, fol. +<i>Abnaki</i>: Rasles, Cambridge, U.S., 1833, 4to. <i>Chippewa</i>: Baraga, +Cincinnati, 1853, 12mo, 622 pages: Petitot, Paris, 1876, 4to, 455 +pages. <i>Massachusetts</i> or <i>Natick</i>: Cotton, Cambridge, U.S. 1829, 8vo. +<i>Onondaga</i>: Shea (French-Onon.), from a MS. (of 17th century), +London, 1860, 4to, 109 pages. <i>Dacota</i>: Riggs, New York, 1851, 4to, +424 pages: Williamson (Eng. Dac.), Santos Agency, Nebraska, +12mo, 139 pages. <i>Mohawk</i>: Bruyas, New York, 1863, 8vo. +<i>Hidatsa (Minnetarees, Gros Ventres of the Missouri)</i>: Matthews, +ib. 1874, 8vo. <i>Choctaw</i>: Byington, ib. 1852, 16mo. <i>Clallam and +Lummi</i>: Gibbs, ib. 1863, 8vo. <i>Yakama</i>: Pandosy, translated by +Gibbs and Shea, ib. 1862, 8vo. <i>Chinook</i>: Gibbs, New York, 1863, +4to. <i>Chinook Jargon, the trade language of Oregon</i>: Id., ib. 1863, +8vo. <i>Tatche</i> or <i>Telamé</i>: Sitjar, ib. 1841, 8vo.</p> + +<p><span class="bold">Mexico and Central America.</span>—<i>Tepehuan</i>: Rinaldini, Mexico, +1743, 4to. <i>Cora</i>: Ortega, Mexico, 1732, 4to. <i>Tarahumara</i>: Steffel, +Brünn, 1791, 8vo. <i>Otomi</i>: Carochi, Mexico, 1645, 4to: Neve y +Molina, ib. 1767, 8vo: Yepes, ib. 1826, 4to: Piccolomini, Roma, +1841, 8vo. <i>Mexican</i> or <i>Aztec</i>: Molina, Mexico, 1555, 4to; 1571, +fol. 2 vols.: Arenas, ib. 1583; 1611, 8vo; 1683; 1725; 1793, +12mo: Biondelli, Milan, 1869, fol. <i>Mexican, Tontonacan, and +Huastecan</i>: Olmos, Mexico, 1555-1560, 4to, 2 vols. <i>Huastecan</i>: +Tapia Zenteno, ib. 1767, 4to, 128 pages. <i>Opata</i> or <i>Tequima</i>: +Lombardo, ib. 1702, 4to. <i>Tarasca</i>: Gilberti, ib. 1559, 4to: Lagunas, +ib. 1574, 8vo. <i>Mixtecan</i>: Alvarado, Mexico, 1593, 4to. <i>Zapoteca</i>: +Cordova, ib. 1578, 4to. <i>Maya</i>: Beltran de Santa Rosa Maria, ib. +1746, 4to; Merida de Yucatan, 1859, 4to, 250 pages: Brasseur de +Bourbourg, Paris, 1874, 8vo, 745 pages. <i>Quiché</i>: Id. (also Cakchiquel +and Trutuhil dialects), ib. 1862, 8vo.</p> + +<p><span class="bold">South America.</span>—<i>Chibcha</i>: Uricoechea, Paris, 1871, 8vo. +<i>Chayma</i>: Tauste, Madrid, 1680, 4to: Yanguas, Burgos, 1683, 4to. +<i>Carib</i>: Raymond, Auxerre, 1665-1666, 8vo. <i>Galibi</i>: D.[e]. L.[a] +S.[auvage], Paris, 1763, 8vo. <i>Tupi</i>: Costa Rubim, Rio de Janeiro, +1853, 8vo: Silva Guimaräes, Bahia, 1854, 8vo: Diaz, Lipsia, 1858, +16mo. <i>Guarani</i>: Ruiz de Montoyo, Madrid, 1639, 4to; 1640; +1722, 4to; ed. Platzmann, Leipzig, 1876, &c., 8vo, to be in 4 vols. +1850 pages. <i>Moxa</i>: Marban, Lima, 1701, 8vo. <i>Lule</i>: Machoni +de Corderia, Madrid, 1732, 12mo. <i>Quichua</i>: Santo Thomas, Ciudad +de los Reyes, 1586, 8vo: Torres Rubio, Sevilla, 1603, 8vo; Lima, +1609, 8vo; ed. Figueredo, Lima, 1754, 8vo; Holguin, Ciudad de +los Reyes, 1608, 8vo: Tschudi, Wien, 1853, 8vo, 2 vols.: Markham, +London, 1864, 8vo: Lopez, <i>Les Races Aryennes de Perou</i>, Paris, 1871, +8vo, comparative vocabulary, pp. 345-421. <i>Aymara</i>: Bertonio, +Chicuyto, 1612, 4to, 2 vols. <i>Chileno</i>: Valdivia (also Allentiac +and Milcocayac), Lima, 1607, 8vo: Febres, ib. 1765, 12mo; ed. +Hernandez y Caluza, Santiago, 1846, 8vo, 2 vols. <i>Tsonecan</i> +(Patagonian): Schmid, Bristol, 1860, 12mo.</p> + +<p>The above article incorporates the salient features of the 9th-edition +article by the Rev. Ponsonby A. Lyons, and the 10th-edition +article by Benjamin E. Smith.</p> +</div> + +<hr class="foot" /> <div class="note"> + +<p><a name="ft1d" id="ft1d" href="#fa1d"><span class="fn">1</span></a> Joannes de Garlandia (John Garland; fl. 1202-1252) gives +the following explanation in his <i>Dictionarius</i>, which is a classed +vocabulary:—“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.</p> + +<p><a name="ft2d" id="ft2d" href="#fa2d"><span class="fn">2</span></a> An excellent dictionary of quotations, perhaps the first of the +kind; a large folio volume printed in Strassburg about 1475 is +entitled “Pharetra auctoritates et dicta doctorum, philosophorum, +et poetarum continens.”</p> + +<p><a name="ft3d" id="ft3d" href="#fa3d"><span class="fn">3</span></a> This volume was issued with a new title-page as <i>Glossaire du +moyen âge</i>, Paris, 1872.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DICTYOGENS<a name="ar45" id="ar45"></a></span> (Gr. <span class="grk" title="diktyon">δίκτυον</span>, a net, and the termination <span class="grk" title="-genês">-γενης</span>, +produced), a botanical name proposed by John Lindley for a +class including certain families of Monocotyledons which have +net-veined leaves. The class was not generally recognized.</p> + + +<hr class="art" /> +<p><span class="bold">DICTYS CRETENSIS,<a name="ar46" id="ar46"></a></span> of Cnossus in Crete, the supposed companion +of Idomeneus during the Trojan War, and author of a +diary of its events. The MS. of this work, written in Phoenician +characters, was said to have been found in his tomb (enclosed in a +leaden box) at the time of an earthquake during the reign of Nero, +by whose order it was translated into Greek. In the 4th century +<span class="sc">a.d.</span> a certain Lucius Septimius brought out <i>Dictys Cretensis +Ephemeris belli Trojani</i>, which professed to be a Latin translation +of the Greek version. Scholars were not agreed whether any +Greek original really existed; but all doubt on the point was +removed by the discovery of a fragment in Greek amongst the +papyri found by B. P. Grenfell and A. S. Hunt in 1905-1906. +Possibly the Latin Ephemeris was the work of Septimius himself. +Its chief interest lies in the fact that (together with Dares +Phrygius’s <i>De excidio Trojae</i>) it was the source from which the +Homeric legends were introduced into the romantic literature +of the middle ages.</p> + +<div class="condensed"> +<p>Best edition by F. Meister (1873), with short but useful introduction +and index of Latinity; see also G. Körting, <i>Diktys und Dares</i> +(1874), with concise bibliography; H. Dunger, <i>Die Sage vom trojanischen +Kriege in den Bearbeitungen des Mittelalters und ihren +antiken Quellen</i> (1869, with a literary genealogical table); E. Collilieux, +<i>Étude sur Dictys de Crète et Darès de Phrygie</i> (1887), with bibliography; +W. Greif, “Die mittelalterlichen Bearbeitungen der Trojanersage,” +in E. M. Stengel’s <i>Ausgaben und Abhandlungen aus dem +Gebiete der romanischen Philologie</i>, No. 61 (1886, esp. sections 82, 83, +168-172); F. Colagrosso, “Ditte Cretese” in <i>Atti della r. Accademia +di Archeologia</i> (Naples, 1897, vol. 18, pt. ii. 2); F. Noack, “Der +griechische Dictys,” in <i>Philologus</i>, supp. vi. 403 ff.; N. E. Griffin, +<i>Dares and Dictys, Introduction to the Study of the Medieval Versions +of the Story of Troy</i> (1907).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DICUIL<a name="ar47" id="ar47"></a></span> (fl. 825), Irish monastic scholar, grammarian and +geographer. He was the author of the <i>De mensura orbis terrae</i>, +finished in 825, which contains the earliest clear notice of a +European discovery of and settlement in Iceland and the most +definite Western reference to the old freshwater canal between +the Nile and the Red Sea, finally blocked up in 767. In 795 +(February 1-August 1) Irish hermits had visited Iceland; on +their return they reported the marvel of the perpetual day at +midsummer in “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.</p> + +<div class="condensed"> +<p>There are seven chief MSS. of the <i>De mensura</i> (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, +<i>Dawn of Modern Geography</i> (London, 1897), i. 317-327, 522-523, 529; +T. Wright, <i>Biographia Britannica literaria, Anglo-Saxon Period</i> +(London, 1842), pp. 372-376.</p> +</div> +<div class="author">(C. R. B.)</div> + + +<hr class="art" /> +<p><span class="bold">DIDACHĒ, THE,<a name="ar48" id="ar48"></a></span> or <i>Teaching of the (twelve) Apostles</i>,—the +most important of the recent recoveries in the region of early +Christian literature (see <span class="sc"><a href="#artlinks">Apocryphal Literature</a></span>). It was +previously known by name from lists of canonical and extra-canonical +books compiled by Eusebius and other writers. Moreover, +it had come to be suspected by several scholars that a lost +book, variously entitled <i>The Two Ways</i> or <i>The Judgment of Peter</i>, +had been freely used in a number of works, of which mention +must presently be made. In 1882 a critical reconstruction of +this book was made by Adam Krawutzcky with marvellous +accuracy, as was shown when in the very next year the Greek +bishop and metropolitan, Philotheus Bryennius, published <i>The +Teaching of the Twelve Apostles</i> from the same manuscript from +<span class="pagenum"><a name="page201" id="page201"></a>201</span> +which he had previously published the complete form of the +Epistle of Clement.<a name="fa1e" id="fa1e" href="#ft1e"><span class="sp">1</span></a></p> + +<p><i>The Didachē</i>, 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.</p> + +<p>1. <i>The Two Ways.</i>—The author of the complete work, as we +now have it, has modified the original <i>Two Ways</i> by inserting near +the beginning a considerable section containing, among other +matter, passages from the Sermon on the Mount, in which the +language of St Matthew’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 +<i>The Two Ways</i>: (i.) the Epistle of Barnabas, chaps. xix., xx. (in +which the order of the book has been much broken up, and a +good deal has been omitted); (ii.) the <i>Ecclesiastical Canons of the +Holy Apostles</i>, usually called the <i>Apostolic Church Order</i>, a book +which presents a parallel to the <i>Teaching</i>, in so far as it consists +first of a form of <i>The Two Ways</i>, and secondly of a number of +church ordinances (here, however, as in the Syriac <i>Didascalia</i>, +which gives about the same amount of <i>The Two Ways</i>, various +sections are ascribed to individual apostles, <i>e.g.</i> “John said, +There are two ways,” &c.); (iii.) a discourse of the Egyptian +monk Schnudi (d. 451), preserved in Arabic (see Iselin, <i>Texte +u. Unters.</i>, 1895); (iv.) a Latin version, of which a fragment +was published by O. von Gebhardt in 1884, and the whole by +J. Schlecht in 1900. When by the aid of this evidence <i>The Two +Ways</i> is restored to us free of glosses, it has the appearance of +being a Jewish manual which has been carried over into the +use of the Christian church. This is of course only a probable +inference; there is no prototype extant in Jewish literature, and, +comparing the moral (non-doctrinal) instruction for Christian +catechumens in Hermas, <i>Shepherd</i> (<i>Mand.</i> i.-ix.), no real need to +assume one. There was a danger of admitting Gentile converts +to the church on too easy moral terms; hence the need of such +insistence on the ideal as in The Two Ways and the <i>Mandates</i>. +The recent recovery of the Latin version is of singular interest, +as showing that, even without the distinctively Christian +additions and interpolations which our full form of the <i>Teaching</i> +presents, it was circulating under the title <i>Doctrina apostolorum</i>.<a name="fa2e" id="fa2e" href="#ft2e"><span class="sp">2</span></a></p> + +<p>2. The second part of our <i>Teaching</i> might be called a church +directory. It consists of precepts relating to church life, which +are couched in the second person plural; whereas <i>The Two Ways</i> +uses throughout the second person singular. It appears to be +a composite work. First (vii. 1-xi. 2) is a short sacramental +manual intended for the use of local elders or presbyters, though +such are not named, for they were not yet a distinctive order or +clergy. This section was probably added to <i>The Two Ways</i> before +the addition of the remainder. It orders baptism in the threefold +name, making a distinction as to waters which has Jewish +parallels, and permitting a threefold pouring on the head, if +sufficient water for immersion cannot be had. It prescribes a +fast before baptism for the baptizer as well as the candidate. +Fasts are to be kept on Wednesday and Friday, not Monday and +Thursday, which are the fast days of “the hypocrites,” <i>i.e.</i> 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’ <i>De virginitate</i>. A trace of them is found in one of the +liturgical prayers of Serapion, bishop of Thmui, in Egypt, but +they have left little mark on the liturgies of the church. As in +Ignatius and other early writers, the eucharist, a real meal (x. 1) +of a family character, is regarded as producing immortality +(cf. “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.</p> + +<p>3. It will now be clear that indications of the locality and date +of our present <i>Teaching</i> must be sought for only in the second +part, and in the Christian interpolations in the first part. We +have no ground for thinking that the second part ever existed +independently as a separate book. The whole work was in the +hands of the writer of the seventh book of the <i>Apostolic Constitutions</i>, +who embodies almost every sentence of it, interspersing +it with passages of Scripture, and modifying the precepts of the +second part to suit a later (4th-century) stage of church development; +this writer was also the interpolator of the Epistles of +Ignatius, and belonged to the Syrian Church. Whether the +second part was known to the writer of the <i>Apostolic Church +Order</i> is not clear, as his only quotation of it comes from one of the +eucharistic prayers. The allusions of early writers seem to point +to Egypt, but their references are mostly to the first part, so that +we must be careful how we argue from them as to the provenance +of the book as a whole. Against Egypt has been urged the +allusion in one of the eucharistic prayers to “corn upon the +mountains.” This is found in the Prayer-book of Serapion +<span class="pagenum"><a name="page202" id="page202"></a>202</span> +(c. 350) but omitted in a later Egyptian prayer; the form as +we have it in <i>The Didachē</i> may have passed into Egypt with +the authority of tradition which was afterwards weakened. The +anti-Jewish tone of the second part suggests the neighbourhood +of Jews, from whom the Christians were to be sharply distinguished. +Either Egypt or Syria would satisfy this condition, +and in favour of Syria is the fact that the presbyterate there was +to a late date regarded as a rank rather than an office. If we can +connect the injunctions (vi. 3) concerning (abstinence from certain) +food and that which is offered to idols with the old trouble that +arose at Antioch (Acts xv. 1) and was legislated for by the +Jerusalem council, we have additional support for the Syrian +claim. But all that we can safely say as to locality is that the +community here represented seems to have been isolated, and +out of touch with the larger centres of Christian life.</p> + +<p>This last consideration helps us in discussing the question of +date. For such an isolated community may have preserved +primitive customs for some time after they had generally disappeared. +Certainly the stage of development is an early one, as +is shown, <i>e.g.</i>, by the prominence of prophets, and the need that +was felt for the vindication of the position of the bishops and +deacons (there is no mention at all of presbyters); moreover, +there is no reference to a canon of Scripture (though the written +Gospel is expressly mentioned) or to a creed. On the other hand +the “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>i.e.</i> before <span class="sc">a.d.</span> 90, +it seems wisest not to place the complete work much earlier than +<span class="sc">a.d.</span> 120, and there are passages which may well be later.</p> + +<div class="condensed"> +<p>A large literature has sprung up round The <i>Didachē</i> since 1884. +Harnack’s edition in <i>Texte u. Unters.</i> vol. ii. (1884) is indispensable +to the student; and his discussions in <i>Altchristl. Litteratur</i> and +<i>Chronologie</i> give clear summaries of his work. Other editions of the +text are those of F. X. Funk, <i>Patres Apostolici</i>, vol. i. (Tübingen, +1901); H. Lietzmann (Bonn, 1903; with Latin version). Dr J. E. +Odgers has published an English translation with introduction and +notes (London, 1906). Dr C. Taylor in 1886 drew attention to some +important parallels in Jewish literature; his edition contains an +English translation. Dr Rendel Harris published in 1887 a complete +facsimile, and gathered a great store of patristic illustration. Text +and translation will also be found in Lightfoot’s <i>Apostolic Fathers</i> +(ed. min.) The fullest critical treatment in English is by Dr Vernon +Bartlet in the extra volume of Hastings’s <i>Dictionary of the Bible</i>; +the most complete commentary on the text is by P. Drews in +Hennecke’s <i>Handbuch zu den N.T. Apocryphen</i> (1904). Other +references to the literature may be found by consulting Harnack’s +<i>Altchristl. Litteratur</i>.</p> +</div> + +<hr class="foot" /> <div class="note"> + +<p><a name="ft1e" id="ft1e" href="#fa1e"><span class="fn">1</span></a> The MS. was found in the Library of the Jerusalem Monastery +of the Most Holy Sepulchre, in Phanar, the Greek quarter of +Constantinople. It is a small octavo volume of 120 parchment +leaves, written throughout by Leo, “notary and sinner,” who +finished his task on the 11th of June 1156. Besides The <i>Didachē</i> and +the Epistles of Clement it contains several spurious Ignatian epistles.</p> + +<p><a name="ft2e" id="ft2e" href="#fa2e"><span class="fn">2</span></a> The word <i>twelve</i> had no place in the original title and was inserted +when the original <i>Didachē</i> or <i>Teaching</i> (<i>e.g.</i> <i>The Two Ways</i>) was +combined with the church manual which mentions apostles outside +of the twelve. It may be noted that the division of the <i>Didachē</i> into +chapters is due to Bryennius, that into verses to A. Harnack.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DIDACTIC POETRY,<a name="ar49" id="ar49"></a></span> that form of verse the aim of which is, +less to excite the hearer by passion or move him by pathos, +than to instruct his mind and improve his morals. The Greek +word <span class="grk" title="didaktikos">διδακτικός</span> signifies a teacher, from the verb <span class="grk" title="didaskein">διδάσκειν</span>, +and poetry of the class under discussion approaches us with the +arts and graces of a schoolmaster. At no time was it found +convenient to combine lyrical verse with instruction, and therefore +from the beginning of literature the didactic poets have +chosen a form approaching the epical. Modern criticism, which +discourages the epic, and is increasingly anxious to limit the word +“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.</p> + +<p>In the prehistoric dawn of Greek civilization there appear +to have been three classes of poetry, to which the literature of +Europe looks back as to its triple fountain-head. There were +romantic epics, dealing with the adventures of gods and heroes; +these Homer represents. There were mystic chants and religious +odes, purely lyrical in character, of which the best Orphic Hymns +must have been the type. And lastly there was a great body of +verse occupied entirely with increasing the knowledge of citizens in +useful branches of art and observation; these were the beginnings +of didactic poetry, and we class them together under the dim name +of Hesiod. It is impossible to date these earliest didactic poems, +which nevertheless set the fashion of form which has been +preserved ever since. The <i>Works and Days</i>, which passes as the +direct masterpiece of Hesiod (<i>q.v.</i>), is the type of all the poetry +which has had education as its aim. Hesiod is supposed to have +been a tiller of the ground in a Boeotian village, who determined +to enrich his neighbours’ minds by putting his own ripe stores of +useful information into sonorous metre. Historically examined, +the legend of Hesiod becomes a shadow, but the substance of +the poems attributed to him remains. The genuine parts of +the <i>Works and Days</i>, which Professor Gilbert Murray has called +“a slow, lowly, simple poem,” deal with rules for agriculture. +The <i>Theogony</i> is an annotated catalogue of the gods. Other +poems attributed to Hesiod, but now lost, were on astronomy, on +auguries by birds, on the character of the physical world; still +others seem to have been genealogies of famous women. All this +mass of Boeotian verse was composed for educational purposes, +in an age when even preposterous information was better than +no knowledge at all. In slightly later times, as the Greek nation +became better supplied with intellectual appliances, the stream +of didactic poetry flowed more and more closely in one, and that +a theological, channel. The great poem of Parmenides <i>On Nature</i> +and those of Empedocles exist only in fragments, but enough +remains to show that these poets carried on the didactic method +in mythology. Cleostratus of Tenedos wrote an astronomical +poem in the 6th century, and Periander a medical one in the +4th, but didactic poetry did not flourish again in Greece until +the 3rd century, when Aratus, in the Alexandrian age, wrote his +famous <i>Phenomena</i>, a poem about things seen in the heavens. +Other later Greek didactic poets were Nicander, and perhaps +Euphorion.</p> + +<p>It was from the hands of these Alexandrian writers that the +genius of didactic poetry passed over to Rome, since, although it +is possible that some of the lost works of the early republic, and in +particular those of Ennius, may have possessed an educational +character, the first and by far the greatest didactic Latin poet +known to us is Lucretius. A highly finished translation by +Cicero into Latin hexameters of the principal works of Aratus is +believed to have drawn the attention of Lucretius to this school +of Greek poetry, and it was not without reference to the Greeks, +although in a more archaic and far purer taste, that he composed, +in the 1st century before Christ, his magnificent <i>De rerum +natura</i>. By universal consent, this is the noblest didactic poem +in the literature of the world. It was intended to instruct mankind +in the interpretation and in the working of the system of +philosophy revealed by Epicurus, which at that time was exciting +the sympathetic attention of all classes of Roman society. What +gave the poem of Lucretius its extraordinary interest, and what +has prolonged and even increased its vitality, was the imaginative +and illustrative insight of the author, piercing and lighting up the +<span class="pagenum"><a name="page203" id="page203"></a>203</span> +recesses of human experience. On a lower intellectual level, but +of a still greater technical excellence, was the <i>Georgics</i> of Virgil, +a poem on the processes of agriculture, published about 30 <span class="sc">b.c.</span> +The brilliant execution of this famous work has justly made it the +type and unapproachable standard of all poetry which desires +to impart useful information in the guise of exquisite literature. +Himself once a farmer on the banks of the Mincio, Virgil, at the +apex of his genius, set himself in his Campanian villa to recall +whatever had been essential in the agricultural life of his boyish +home, and the result, in spite of the ardours of the subject, was +what J. W. Mackail has called “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 <i>Astronomica</i> towards the close of the reign of Augustus. +Columella, regretting that Virgil had omitted to sing of gardens, +composed a smooth poem on horticulture. Natural philosophy +inspired Lucilius junior, of whom a didactic poem on Etna +survives. Long afterwards, under Diocletian, a poet of Carthage, +Nemesianus, wrote in the manner of Virgil the <i>Cynegetica</i>, a +poem on hunting with dogs, which has had numerous imitations +in later European literatures. These are the most important +specimens of didactic poetry which ancient Rome has handed +down to us.</p> + +<p>In Anglo-Saxon and early English poetic literature, and +especially in the religious part of it, an element of didacticism is +not to be overlooked. But it would be difficult to say that anything +of importance was written in verse with the sole purpose of +imparting information, until we reach the 16th century. Some of +the later medieval allegories are didactic or nothing. The first +poem, however, which we can in any reasonable way compare +with the classic works of which we have been speaking is the +<i>Hundreth Pointes of Good Husbandrie</i>, published in 1557 by +Thomas Tusser; these humble Georgics aimed at a practical +description of the whole art of English farming. Throughout the +early part of the 17th century, when our national poetry was in +its most vivid and brilliant condition, the last thing a poet +thought of doing was the setting down of scientific facts in +rhyme. We come across, however, one or two writers who were +as didactic as the age would permit them to be, Samuel Daniel with +his philosophy, Fulke Greville, Lord Brooke with his “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) +<i>Education</i>, Dyer’s <i>Fleece</i>, and Somerville’s <i>Chase</i>, we see +technical information put forward as the central aim of the poet. +Instead of a passionate pleasure, or at least an uplifted enthusiasm, +being the poet’s object, he frankly admits that, first and +foremost, he has some facts about wool or dogs or schoolmasters +which he wishes to bring home to his readers, and that, secondly, +he consents to use verse, as brilliantly as he can, for the purpose +of gilding the pill and attracting an unwilling attention. As we +descend the 18th century, these works become more and more +numerous, and more dry, especially when opposed by the descriptive +and rural poets of the school of Thomson, the poet of +<i>The Seasons</i>. But Thomson himself wrote a huge poem of +<i>Liberty</i> (1732), for which we have no name if we must not call it +didactic. Even Gray began, though he failed to finish, a work of +this class, on <i>The Alliance of Education and Government</i>. These +poems were discredited by the publication of <i>The Sugar-Cane</i> +(1764), a long verse-treatise about the cultivation of sugar by +negroes in the West Indies, by James Grainger (1721-1766), but, +though liable to ridicule, such versified treatises continued to +appear. Whether so great a writer as Cowper is to be counted +among the didactic poets is a question on which readers of <i>The +Task</i> may be divided; this poem belongs rather to the class of +descriptive poetry, but a strong didactic tendency is visible in +parts of it. Perhaps the latest frankly educational poem which +enjoyed a great popularity was <i>The Course of Time</i> by Robert +Pollok (1798-1827), in which a system of Calvinistic divinity is +laid down with severity and in the pomp of blank verse. This +kind of literature had already been exposed, and discouraged, by +the teaching of Wordsworth, who had insisted on the imperative +necessity of charging all poetry with imagination and passion. +Oddly enough, <i>The Excursion</i> of Wordsworth himself is perhaps +the most didactic poem of the 19th century, but it must be +acknowledged that his influence, in this direction, was saner +than his practice. Since the days of Coleridge and Shelley it +has been almost impossible to conceive a poet of any value composing +in verse a work written with the purpose of inculcating +useful information.</p> + +<p>The history of didactic poetry in France repeats, in great +measure, but in drearier language, that of England. Boileau, like +Pope, but with a more definite purpose as a teacher, offered +instruction in his <i>Art poétique</i> and in his <i>Epistles</i>. But his +doctrine was always literary, not purely educational. At the +beginning of the 18th century, the younger Racine (1692-1763) +wrote sermons in verse, and at the close of it the Abbé Delille +(1738-1813) tried to imitate Virgil in poems about horticulture. +Between these two there lies a vast mass of verse written for the +indulgence of intellect rather than at the dictates of the heart; +wherever this aims at increasing knowledge, it at once becomes +basely and flatly didactic. There is nothing in French literature +of the transitional class that deserves mention beside <i>The Task</i> or +<i>The Excursion</i>.</p> + +<p>During the century which preceded the Romantic revival of +poetry in Germany, didactic verse was cultivated in that country +on the lines of imitation of the French, but with a greater dryness +and on a lower level of utility. Modern German literature +began with Martin Opitz (1597-1639) and the Silesian School, +who were in their essence rhetorical and educational, and who +gave their tone to German verse. Albrecht von Haller (1708-1777) +brought a very considerable intellectual force to bear on +his huge poems, <i>The Origin of Evil</i>, which was theological, and +<i>The Alps</i> (1729), botanical and topographical. Johann Peter Uz +(1720-1796) wrote a <i>Theodicée</i>, which was very popular, and not +without dignity. Johann Jacob Dusch (1725-1787) undertook to +put <i>The Sciences</i> into the eight books of a great didactic poem. +Tiedge (1752-1840) was the last of the school; in a once-famous +<i>Urania</i>, he sang of God and Immortality and Liberty. These +German pieces were the most unswervingly didactic that any +modern European literature has produced. There was hardly +the pretence of introducing into them descriptions of natural +beauty, as the English poets did, or of grace and wit like the +French. The German poets simply poured into a lumbering +mould of verse as much solid information and direct instruction +as the form would hold.</p> + +<p>Didactic poetry has, in modern times, been antipathetic to +the spirit of the Latin peoples, and neither Italian nor Spanish +literature has produced a really notable work in this class. An +examination of the poems, ancient and modern, which have been +mentioned above, will show that from primitive times there have +been two classes of poetic work to which the epithet didactic has +been given. It is desirable to distinguish these a little more +exactly. One is the pure instrument of teaching, the poetry +which desires to impart all that it knows about the growing of +cabbages or the prevention of disasters at sea, the revolution of +the planets or the blessings of inoculation. This is didactic poetry +proper, and this, it is almost certain, became irrevocably obsolete +at the close of the 18th century. No future Virgil will give the +world a second <i>Georgics</i>. But there is another species which it +is very improbable that criticism has entirely dislodged; that is +the poetry which combines, with philosophical instruction, an impetus +of imaginative movement, and a certain definite cultivation +<span class="pagenum"><a name="page204" id="page204"></a>204</span> +of fire and beauty. In hands so noble as those of Lucretius +and Goethe this species of didactic poetry has enriched the world +with durable masterpieces, and, although the circle of readers +which will endure scientific disquisition in the bonds of verse +grows narrower and narrower, it is probable that the great poet +who is also a great thinker will now and again insist on being +heard. In Sully-Prudhomme France has possessed an eminent +writer whose methods are directly instructive, and both <i>La +Justice</i> (1878) and <i>Le Bonheur</i> (1888) are typically didactic poems. +Perhaps future historians may name these as the latest of their +class.</p> +<div class="author">(E. G.)</div> + + +<hr class="art" /> +<p><span class="bold">DIDEROT, DENIS<a name="ar50" id="ar50"></a></span> (1713-1784), French man of letters and +encyclopaedist, was born at Langres on the 5th of October 1713. +He was educated by the Jesuits, like most of those who afterwards +became the bitterest enemies of Catholicism; and, when +his education was at an end, he vexed his brave and worthy +father’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.</p> + +<p>Diderot’s earliest writings were of as little importance as +Goldsmith’s <i>Enquiry into the State of Polite Learning</i> or Burke’s +<i>Abridgement of English History</i>. He earned 100 crowns by +translating Stanyan’s <i>History of Greece</i> (1743); with two +colleagues he produced a translation of James’s <i>Dictionary of +Medicine</i> (1746-1748) and about the same date he published a +free rendering of Shaftesbury’s <i>Inquiry Concerning Virtue and +Merit</i> (1745), with some original notes of his own. With strange +and characteristic versatility, he turned from ethical speculation +to the composition of a volume of stories, the <i>Bijoux indiscrets</i> +(1748), gross without liveliness, and impure without wit. In later +years he repented of this shameless work, just as Boccaccio is +said in the day of his grey hairs to have thought of the sprightliness +of the <i>Decameron</i> with strong remorse. From tales Diderot +went back to the more congenial region of philosophy. Between +the morning of Good Friday and the evening of Easter Monday he +wrote the <i>Pensées philosophiques</i> (1746), and he presently added +to this a short complementary essay on the sufficiency of natural +religion. The gist of these performances is to press the ordinary +rationalistic objections to a supernatural revelation; but though +Diderot did not at this time pass out into the wilderness +beyond natural religion, yet there are signs that he accepted that +less as a positive doctrine, resting on grounds of its own, than as +a convenient point of attack against Christianity. In 1747 he +wrote the <i>Promenade du sceptique</i>, a rather poor allegory—pointing +first to the extravagances of Catholicism; second, to the +vanity of the pleasures of that world which is the rival of +the church; and third, to the desperate and unfathomable +uncertainty of the philosophy which professes to be so high +above both church and world.</p> + +<p>Diderot’s next piece was what first introduced him to the world +as an original thinker, his famous <i>Lettre sur les aveugles</i> (1749). +The immediate object of this short but pithy writing was to show +the dependence of men’s ideas on their five senses. It considers +the case of the intellect deprived of the aid of one of the senses; +and in a second piece, published afterwards, Diderot considered +the case of a similar deprivation in the deaf and dumb. The +<i>Lettre sur les sourds et muets</i>, however, is substantially a digressive +examination of some points in aesthetics. The philosophic +significance of the two essays is in the advance they make +towards the principle of Relativity. But what interested the +militant philosophers of that day was an episodic application +of the principle of relativity to the master-conception of God. +What makes the <i>Lettre sur les aveugles</i> interesting is its presentation, +in a distinct though undigested form, of the modern theory +of variability, and of survival by superior adaptation. It is worth +noticing, too, as an illustration of the comprehensive freedom +with which Diderot felt his way round any subject that he +approached, that in this theoretic essay he suggests the possibility +of teaching the blind to read through the sense of touch. If the +<i>Lettre sur les aveugles</i> introduced Diderot into the worshipful +company of the philosophers, it also introduced him to the +penalties of philosophy. His speculation was too hardy for the +authorities, and he was thrown into the prison of Vincennes. +Here he remained for three months; then he was released, to +enter upon the gigantic undertaking of his life.</p> + +<p>The bookseller Lebreton had applied to him with a project +for the publication of a translation into French of Ephraim +Chambers’s <i>Cyclopaedia</i>, undertaken in the first instance by an +Englishman, John Mills, and a German, Gottfried Sellius (for +particulars see <span class="sc"><a href="#artlinks">Encyclopaedia</a></span>). Diderot accepted the proposal, +but in his busy and pregnant intelligence the scheme became +transformed. Instead of a mere reproduction of Chambers, he +persuaded the bookseller to enter upon a new work, which should +collect under one roof all the active writers, all the new ideas, all +the new knowledge, that were then moving the cultivated class +to its depths, but still were comparatively ineffectual by reason of +their dispersion. His enthusiasm infected the publishers; they +collected a sufficient capital for a vaster enterprise than they had +at first planned; D’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 +<i>Encyclopaedia</i>, in which they saw a rising stronghold for their +philosophic enemies. By 1757 they could endure the sight no +longer. The subscribers had grown from 2000 to 4000, and this +was a right measure of the growth of the work in popular influence +and power. To any one who turns over the pages of these redoubtable +volumes now, it seems surprising that their doctrines +should have stirred such portentous alarm. There is no atheism, +no overt attack on any of the cardinal mysteries of the faith, no +direct denunciation even of the notorious abuses of the church. +Yet we feel that the atmosphere of the book may well have been +displeasing to authorities who had not yet learnt to encounter +the modern spirit on equal terms. The <i>Encyclopaedia</i> takes for +granted the justice of religious tolerance and speculative freedom. +It asserts in distinct tones the democratic doctrine that it is +the common people in a nation whose lot ought to be the main +concern of the nation’s government. From beginning to end +it is one unbroken process of exaltation of scientific knowledge on +the one hand, and pacific industry on the other. All these things +were odious to the old governing classes of France; their spirit +was absolutist, ecclesiastical and military. Perhaps the most +alarming thought of all was the current belief that the <i>Encyclopaedia</i> +was the work of an organized band of conspirators against +society, and that a pestilent doctrine was now made truly +formidable by the confederation of its preachers into an open +league. When the seventh volume appeared, it contained an +article on “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, <i>L’Esprit</i>, +<span class="pagenum"><a name="page205" id="page205"></a>205</span> +appeared, and gave a still more profound and, let us add, a more +reasonable shock to the ecclesiastical party. Authority could +brook no more, and in 1759 the <i>Encyclopaedia</i> was formally +suppressed.</p> + +<p>The decree, however, did not arrest the continuance of the +work. The connivance of the authorities at the breach of their +own official orders was common in those times of distracted +government. The work went on, but with its difficulties increased +by the necessity of being clandestine. And a worse thing +than troublesome interference by the police now befell Diderot. +D’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 <i>Encyclopaedia</i> was about £120 +sterling. “And then to think,” said Voltaire, “that an army +contractor makes £800 in a day!”</p> + +<p>Although the <i>Encyclopaedia</i> 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—<i>Le Fils naturel</i> (1757) and <i>Le Père de +famille</i> (1758)—and they are very insipid performances in the sentimental +vein. But he accompanied them by essays on dramatic +poetry, including especially the <i>Paradoxe sur le comédien</i>, in +which he announced the principles of a new drama,—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 <i>Hamburgische Dramaturgie</i> (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 +<i>Encyclopaedia</i> shows that he had mastered and passed beyond +the metaphysical theories on the subject, and the <i>Essai sur la +peinture</i> was justly described by Goethe, who thought it worth +translating, as “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 <i>Salons</i> are among +the most readable of all pieces of art criticism. They have a +freshness, a reality, a life, which take their readers into a different +world from the dry and conceited pedantries of the ordinary +virtuoso. As has been said by Sainte-Beuve, they initiated the +French into a new sentiment, and introduced people to the +mystery and purport of colour by ideas. “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.”</p> + +<p>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. <i>Jacques le +fataliste</i> (written in 1773, but not published until 1796) is in +manner an imitation of <i>Tristram Shandy</i> and <i>The Sentimental +Journey</i>. Few modern readers will find in it any true diversion. +In spite of some excellent criticisms dispersed here and there, +and in spite of one or two stories that are not without a certain +effective realism, it must as a whole be pronounced savourless, +forced, and as leaving unmoved those springs of laughter and +of tears which are the common fountain of humour. <i>Le Neveu +de Rameau</i> is a far superior performance. If there were any inevitable +compulsion to name a masterpiece for Diderot, one must +select this singular “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 <i>Le Neveu de +Rameau</i> to the European public. After executing it, he gave +back the original French manuscript to Schiller, from whom he +had it. No authentic French copy of it appeared until the writer +had been nearly forty years in his grave (1823).</p> + +<p>It would take several pages merely to contain the list of +Diderot’s miscellaneous pieces, from an infinitely graceful trifle +like the <i>Regrets sur ma vieille robe de chambre</i> up to <i>Le Rêve de +D’Alembert</i>, where he plunges into the depths of the controversy +as to the ultimate constitution of matter and the meaning of life. +It is a mistake to set down Diderot for a coherent and systematic +materialist. We ought to look upon him “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 <i>Système de la +nature</i> 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?</p> + +<p>In proportion as these conclusions deepened in him, the more +<span class="pagenum"><a name="page206" id="page206"></a>206</span> +did Diderot turn for the hope of the race to virtue; in other +words, to such a regulation of conduct and motive as shall make +us tender, pitiful, simple, contented. Hence his one great literary +passion, his enthusiasm for Richardson, the English novelist. +Hence, also, his deepening aversion for the political system of +France, which makes the realization of a natural and contented +domestic life so hard. Diderot had almost as much to say +against society as even Rousseau himself. The difference between +them was that Rousseau was a fervent theist. The atheism of +the Holbachians, as he called Diderot’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 <i>Confessions</i> 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 <i>Essai sur les règnes +de Claude et de Néron</i> (1778-1782), which is much less an account +of Seneca than a vindication of Diderot and Grimm, and is one of +the most rambling and inept productions in literature. As for the +merits of the old quarrel between Rousseau and Diderot, we may +agree with the latter, that too many sensible people would be in +the wrong if Jean Jacques was in the right.</p> + +<p>Varied and incessant as was Diderot’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. “<i>Fi, donc,</i>” said Catherine one day, when Diderot +hinted that he argued with her at a disadvantage, “<i>is there any +difference among men?</i>” 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.</p> + +<p>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.</p> + +<div class="condensed"> +<p><span class="sc">Authorities.</span>—Studies on Diderot by Scherer (1880); by +E. Faguet (1890); by Sainte-Beuve in the <i>Causeries du lundi</i>; by +F. Brunetière in the <i>Études critiques</i>, 2nd series, may be consulted. +In English, Diderot has been the subject of a biography by John +Morley [Viscount Morley of Blackburn] (1878). See also Karl +Rosenkranz, <i>Diderots Leben und Werke</i> (1866). For a discussion of +the authenticity of the posthumous works of Diderot see R. Dominic +in the <i>Revue des deux mondes</i> (October 15, 1902).</p> +</div> +<div class="author">(J. Mo.)</div> + + +<hr class="art" /> +<p><span class="bold">DIDIUS SALVIUS JULIANUS, MARCUS,<a name="ar51" id="ar51"></a></span> Roman emperor for +two months (March 28-June 2) during the year <span class="sc">a.d.</span> 193. He +was the grandson of the famous jurist Salvius Julianus (under +Hadrian and the Antonines), and the son of a distinguished +general, who might have ascended the throne after the death of +Antoninus Pius, had not his loyalty to the ruling house prevented +him. Didius filled several civil and military offices with distinguished +success, but subsequently abandoned himself to +dissipation. On the death of Pertinax, the praetorian guards +offered the throne to the highest bidder. Flavius Sulpicianus, +the father-in-law of Pertinax and praefect of the city, had already +made an offer; Didius, urged on by the members of his family, +his freedmen and parasites, hurried to the praetorian camp to +contend for the prize. He and Sulpicianus bid against each +other, and finally the throne was knocked down to Didius. The +senate and nobles professed their loyalty; but the people +made no attempt to conceal their indignation at this insult to +the state, and the armies of Britain, Syria and Illyricum broke +out into open revolt. Septimius Severus, the commander of +the Pannonian legions, was declared emperor and hastened by +forced marches to Italy. Didius, abandoned by the praetorians, +was condemned and executed by order of the senate, which at +once acknowledged Severus.</p> + +<div class="condensed"> +<p><span class="sc">Authorities.</span>—Dio Cassius lxxiii. 11-17, who was actually in +Rome at the time; Aelius Spartianus, <i>Didius Julianus</i>; Julius +Capitolinus, <i>Pertinax</i>; Herodian ii.; Aurelius Victor, <i>De Caesaribus</i>, +19; Zosimus i. 7; Gibbon, <i>Decline and Fall</i>, chap. 5.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DIDO,<a name="ar52" id="ar52"></a></span> or <span class="sc">Elissa</span>, the reputed founder of Carthage (<i>q.v.</i>), in +Africa, daughter of the Tyrian king Metten (Mutto, Methres, +Belus), wife of Acerbas (more correctly Sicharbas; Sychaeus in +Virgil), a priest of Hercules. Her husband having been slain by +her brother Pygmalion, Dido fled to Cyprus, and thence to the +coast of Africa, where she purchased from a local chieftain +Iarbas a piece of land on which she built Carthage. The city +soon began to prosper and Iarbas sought Dido’s hand in marriage, +threatening her with war in case of refusal. To escape from him, +Dido constructed a funeral pile, on which she stabbed herself +before the people (Justin xviii. 4-7). Virgil, in defiance of the +usually accepted chronology, makes Dido a contemporary of +Aeneas, with whom she fell in love after his landing in Africa, and +attributes her suicide to her abandonment by him at the command +of Jupiter (<i>Aeneid</i>, iv.). Dido was worshipped at Carthage as a +divinity under the name of Caelestis, the Roman counterpart of +Tanit, the tutelary goddess of Carthage. According to Timaeus, +the oldest authority for the story, her name was Theiosso, in +Phoenician Helissa, and she was called Dido from her wanderings, +Dido being the Phoenician equivalent of <span class="grk" title="planêtis">πλανῆτις</span> (<i>Etymologicum +Magnum</i>, <i>s.v.</i>); 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 +<i>Bellum Poenicum</i>, followed by Ennius in his <i>Annales</i>.</p> + +<div class="condensed"> +<p>For the variations of the legend in earlier and later Latin authors, +see O. Rossbach in Pauly-Wissowa’s <i>Realencyclopädie</i>, v. pt. 1 (1905); +O. Meltzer’s <i>Geschichte der Karthager</i>, i. (1879), and his article in +Roscher’s <i>Lexikon der Mythologie</i>.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DIDON, HENRI<a name="ar53" id="ar53"></a></span> (1840-1900), French Dominican, was born +at Trouvet, Isère, on the 17th of March 1840. He joined the +Dominicans, under the influence of Lacordaire, in 1858, and +completed his theological studies at the Minerva convent at +Rome. The influence of Lacordaire was shown in the zeal displayed +by Didon in favour of a reconciliation between philosophy +and science. In 1871 his fame had so much grown that he was +chosen to deliver the funeral oration over the murdered archbishop +of Paris, Monseigneur G. Darboy. He also delivered some +<span class="pagenum"><a name="page207" id="page207"></a>207</span> +discourses at the church of St Jean de Beauvais in Paris on the +relations between science and religion; but his utterances, +especially on the question of divorce, were deemed suspicious by +his superiors, and his intimacy with Claude Bernard the physiologist +was disapproved. He was interdicted from preaching and +sent into retirement at the convent of Corbara in Corsica. After +eighteen months he emerged, and travelled in Germany, publishing +an interesting work upon that country, entitled <i>Les Allemands</i> +(English translation by R. Ledos de Beaufort, London, 1884). +On his return to France in 1890 he produced his best known +work, <i>Jésus-Christ</i> (2 vols., Paris), for which he had qualified +himself by travel in the Holy Land. In the same year he became +director of the Collège Albert-le-Grand at Arcueil, and founded +three auxiliary institutions, École Lacordaire, École Laplace and +École St Dominique. He wrote, in addition, several works on +educational questions, and augmented his fame as an eloquent +preacher by discourses preached during Lent and Advent. He +died at Toulouse on the 13th of March 1900.</p> + +<div class="condensed"> +<p>See the biographies by J. de Romano (1891), and A. de Coulanges +(Paris, 1900); and especially the work of Stanislas Reynaud, +entitled <i>Le Père Didon, sa vie et son œuvre</i> (Paris, 1904).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DIDOT,<a name="ar54" id="ar54"></a></span> the name of a family of learned French printers and +publishers. <span class="sc">François Didot</span> (1689-1757), founder of the +family, was born at Paris. He began business as a bookseller and +printer in 1713, and among his undertakings was a collection +of the travels of his friend the Abbé Prévost, in twenty volumes +(1747). It was remarkable for its typographical perfection, +and was adorned with many engravings and maps. <span class="sc">François +Ambroise Didot</span> (1730-1804), son of François, made important +improvements in type-founding, and was the first to attempt +printing on vellum paper. Among the works which he published +was the famous collection of French classics prepared by order +of Louis XVI. for the education of the Dauphin, and the folio +edition of <i>L’Art de vérifier les dates</i>. <span class="sc">Pierre François Didot</span> +(1732-1795), his brother, devoted much attention to the art of +type-founding and to paper-making. Among the works which +issued from his press was an edition in folio of the <i>Imitatio +Christi</i> (1788). <span class="sc">Henri Didot</span> (1765-1852), son of Pierre François, +is celebrated for his “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 <i>assignats</i> issued by the +Constituent and Legislative Assemblies and the Convention. +<span class="sc">Didot Saint-Léger</span>, second son of Pierre François, was the +inventor of the paper-making machine known in England as +the Didot machine. <span class="sc">Pierre Didot</span> (1760-1853), eldest son of +François Ambroise, is celebrated as the publisher of the beautiful +“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.” <span class="sc">Firmin Didot</span> +(1764-1836), his brother, second son of François Ambroise, +sustained the reputation of the family both as printer and type-founder. +He revived (if he did not invent—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 <i>Tables of Logarithms</i> (1795), in which he secured +an accuracy till then unattainable. He published stereotyped +editions of French, English and Italian classics at a very low +price. He was the author of two tragedies—<i>La Reine de +Portugal</i> and <i>La Mort d’Annibal</i>; and he wrote metrical translations +from Virgil, Tyrtaeus and Theocritus. <span class="sc">Ambroise Firmin +Didot</span> (1790-1876) was his eldest son. After receiving a classical +education, he spent three years in Greece and in the East; and on +the retirement of his father in 1827 he undertook, in conjunction +with his brother Hyacinthe, the direction of the publishing +business. Their greatest undertaking was a new edition of the +<i>Thesaurus Graecae linguae</i> of Henri Estienne, under the editorial +care of the brothers Dindorf and M. Hase (9 vols., 1855-1859). +Among the numerous important works published by the brothers, +the 200 volumes forming the <i>Bibliothèque des auteurs grecs</i>, +<i>Bibliothèque latine</i>, and <i>Bibliothèque française</i> deserve special +mention. Ambroise Firmin Didot was the first to propose +(1823) a subscription in favour of the Greeks, then in insurrection +against Turkish tyranny. Besides a translation of Thucydides +(1833), he wrote the articles “Estienne” in the <i>Nouvelle Biographie +générale</i>, and “Typographie” in the <i>Ency. mod.</i>, as well +as <i>Observations sur l’orthographie française</i> (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.</p> + + +<hr class="art" /> +<p><span class="bold">DIDRON, ADOLPHE NAPOLÉON<a name="ar55" id="ar55"></a></span> (1806-1867), French +archaeologist, was born at Hautvillers, in the department of +Marne, on the 13th of March 1806. At first a student of law, +he began in 1830, by the advice of Victor Hugo, a study of the +Christian archaeology of the middle ages. After visiting and +examining the principal churches, first of Normandy, then of +central and southern France, he was on his return appointed by +Guizot secretary to the Historical Committee of Arts and Monuments +(1835); and in the following years he delivered several +courses of lectures on Christian iconography at the Bibliothèque +Royale. In 1839 he visited Greece for the purpose of examining +the art of the Eastern Church, both in its buildings and its +manuscripts. In 1844 he originated the <i>Annales archéologiques</i>, +a periodical devoted to his favourite subject, which he edited +until his death. In 1845 he established at Paris a special archaeological +library, and at the same time a manufactory of painted +glass. In the same year he was admitted to the Legion of +Honour. His most important work is the <i>Iconographie chrétienne</i>, +of which, however, the first portion only, <i>Histoire de Dieu</i> (1843), +was published. It was translated into English by E. J. Millington. +Among his other works may be mentioned the <i>Manuel d’iconographie +chrétienne grecque et latine</i> (1845), the <i>Iconographie des +chapiteaux du palais ducal de Venise</i> (1857), and the <i>Manuel des +objets de bronze et d’orfèvrerie</i> (1859). He died on the 13th of +November 1867.</p> + + +<hr class="art" /> +<p><span class="bold">DIDYMI,<a name="ar56" id="ar56"></a></span> or <span class="sc">Didyma</span> (mod. <i>Hieronta</i>), an ancient sanctuary +of Apollo in Asia Minor situated in the territory of Miletus, from +which it was distant about 10 m. S. and on the promontory +Poseideion. It was sometimes called <i>Branchidae</i> from the name +of its priestly caste which claimed descent from Branchus, a +youth beloved by Apollo. As the seat of a famous oracle, the +original temple attracted offerings from Pharaoh Necho (in whose +army there was a contingent of Milesian mercenaries), and the +Lydian Croesus, and was plundered by Darius of Persia. Xerxes +finally sacked and burnt it (481 <span class="sc">b.c.</span>) and exiled the Branchidae +to the far north-east of his empire. This exile was believed to +be voluntary, the priests having betrayed their treasures to the +Persian; and on this belief Alexander the Great acted 150 years +later, when, finding the descendants of the Branchidae established +in a city beyond the Oxus, he ordered them to be exterminated +for the sin of their fathers (328). The celebrated cult-statue of +Apollo by Canachus, familiar to us from reproductions on Milesian +coins, was also carried to Persia, there to remain till restored by +Seleucus I. in 295, and the oracle ceased to speak for a century +and a half. The Milesians were not able to undertake the rebuilding +till about 332 <span class="sc">b.c.</span>, when the oracle revived at the bidding +of Alexander. The work proved too costly, and despite a special +effort made by the Asian province nearly 400 years later, at the +bidding of the emperor Caligula, the structure was never quite +finished: but even as it was, Strabo ranked the Didymeum the +greatest of Greek temples and Pliny placed it among the four +most splendid and second only to the Artemisium at Ephesus. +In point of fact it was a little smaller than the Samian Heraeum +and the temple of Cybele at Sardis, and almost exactly the same +size as the Artemisium. The area covered by the platform +measures roughly 360 × 160 ft.</p> + +<p>When Cyriac of Ancona visited the spot in 1446, it seems that +the temple was still standing in great part, although the <i>cella</i> had +been converted into a fortress by the Byzantines: but when the +next European visitor, the Englishman Dr Pickering, arrived +in 1673, it had collapsed. It is conjectured that the cause was +the great earthquake of 1493. The Society of Dilettanti sent two +expeditions to explore the ruins, the first in 1764 under Richard +Chandler, the second in 1812 under Sir Wm. Gell; and the French +<span class="pagenum"><a name="page208" id="page208"></a>208</span> +“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 <i>prodomos</i>, and discovered inscriptions giving +information about other parts which they left still buried. +Finally the site was purchased by, and the French rights were +ceded to, Dr Th. Wiegand, the German explorer of Miletus, who +in 1905 began a thorough clearance of what is incomparably the +finest temple ruin in Asia Minor.</p> + +<p>The temple was a decastyle peripteral structure of the Ionic +order, standing on seven steps and possessing double rows of outer +columns 60 ft. high, twenty-one in each row on the flanks. It +is remarkable not only for its great size, but (<i>inter alia</i>) for (1) the +rich ornament of its column bases, which show great variety of +design; (2) its various developments of the Ionic capital, <i>e.g.</i> +heads of gods, probably of Pergamene art, spring from the +“eyes” of the volutes with bulls’ heads between them; (3) the +massive building two storeys high at least, which served below +for <i>prodomos</i>, and above for a dispensary of oracles (<span class="grk" title="chrêsmographia">χρησμογράφια</span> +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.</p> + +<div class="condensed"> +<p>See Dilettanti Society, <i>Ionian Antiquities</i>, ii. (1821); C. T. +Newton, <i>Hist. of Discoveries</i>, &c. (1862) and <i>Travels in the Levant</i>, +ii. (1865); O. Rayet and A. Thomas, <i>Milet et le Golfe Latmique</i> +(1877); E. Pontremoli and B. Haussoullier, <i>Didymes</i> (1904).</p> +</div> +<div class="author">(D. G. H.)</div> + + +<hr class="art" /> +<p><span class="bold">DIDYMIUM<a name="ar57" id="ar57"></a></span> (from the Gr. <span class="grk" title="didymos">διδυμος</span>, twin), the name given to +the supposed element isolated by C. G. Mosander from cerite +(1839-1841). In 1879, however, Lecoq de Boisbaudran showed +that Mosander’s “didymium” contained samarium; while the +residual “didymium,” after removal of samarium, was split +by Auer v. Welsbach (<i>Monats. f. Chemie</i>, 1885, 6, 477) into +two components (known respectively as neodymium and +praseodymium) by repeated fractional crystallization of the +double nitrate of ammonium and didymium in nitric acid. +<i>Neodymium</i> (Nd) forms the chief portion of the old “didymium.” +Its salts are reddish violet in colour, and give a characteristic +absorption spectrum. It forms oxides of composition Nd<span class="su">2</span>O<span class="su">3</span> +and Nd<span class="su">2</span>O<span class="su">5</span>, the latter being obtained by ignition of the nitrate +(B. Brauner). The atomic weight of neodymium is 143.6 +(B. Brauner, <i>Proc. Chem. Soc.</i>, 1897-1898, p. 70). <i>Praseodymium</i> +(Pr) forms oxides of composition Pr<span class="su">2</span>O<span class="su">3</span>, Pr<span class="su">2</span>O<span class="su">5</span> ,xH<span class="su">2</span>O +(B. Brauner), and Pr<span class="su">4</span>O<span class="su">7</span>. The peroxide, Pr<span class="su">4</span>O<span class="su">7</span>, forms a dark +brown powder, and is obtained by ignition of the oxalate or +nitrate. The sesquioxide, Pr<span class="su">2</span>O<span class="su">3</span>, is obtained as a greenish white +mass by the reduction of the peroxide. The salts of praseodymium +are green in colour, and give a characteristic spark spectrum. +The atomic weight of praseodymium is 140.5.</p> + + +<hr class="art" /> +<p><span class="bold">DIDYMUS<a name="ar58" id="ar58"></a></span> (?309-?394), surnamed “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 <i>De Trinitate</i>, <i>De Spiritu Sancto</i> (Jerome’s +Latin translation), <i>Adversus Manichaeos</i>, and notes and expositions +of various books, especially the Psalms and the Catholic +Epistles.</p> + +<div class="condensed"> +<p>See Migne, <i>Patrol. Graec.</i> xxxix.; O. Bardenhewer, <i>Patrologie</i>, +pp. 290-293 (Freiburg, 1894).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DIDYMUS CHALCENTERUS<a name="ar59" id="ar59"></a></span> (c. 63 <span class="sc">b.c.</span>-<span class="sc">a.d.</span> 10), Greek +scholar and grammarian, flourished in the time of Cicero and +Augustus. His surname (Gr. <span class="grk" title="Chalkenteros">Χαλκέντερος</span>, brazen-bowelled) +came from his indefatigable industry; he was said to have +written so many books (more than 3500) that he was unable to +recollect their names (<span class="grk" title="bibliolathas">βιβλιολάθας</span>). He lived and taught in +Alexandria and Rome, where he became the friend of Varro. +He is chiefly important as having introduced Alexandrian +learning to the Romans. He was a follower of the school of +Aristarchus, upon whose recension of Homer he wrote a treatise, +fragments of which have been preserved in the Venetian Scholia. +He also wrote commentaries on many other Greek poets and +prose authors. In his work on the lyric poets he treated of the +various classes of poetry and their chief representatives, and +his lists of words and phrases (used in tragedy and comedy +and by orators and historians), of words of doubtful meaning, +and of corrupt expressions, furnished the later grammarians with +valuable material. His activity extended to all kinds of subjects: +grammar (orthography, inflexions), proverbs, wonderful stories, +the law-tablets (<span class="grk" title="axones">ἄξονες</span>) of Solon, stones, and different kinds of +wood. His polemic against Cicero’s <i>De republica</i> (Ammianus +Marcellinus xxii. 16) provoked a reply from Suetonius. In spite +of his stupendous industry, Didymus was little more than a +compiler, of little critical judgment and doubtful accuracy, but +he deserves recognition for having incorporated in his numerous +writings the works of earlier critics and commentators.</p> + +<div class="condensed"> +<p>See M. W. Schmidt, <i>De Didymo Chalcentero</i> (1853) and <i>Didymi +Chalcenteri fragmenta</i> (1854); also F. Susemihl, <i>Geschichte der griech. +Literatur in der Alexandrinerzeit</i>, ii. (1891); J. E. Sandys, <i>History of +Classical Scholarship</i>, i. (1906).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DIE,<a name="ar60" id="ar60"></a></span> a town of south-eastern France, capital of an arrondissement +in the department of Drôme, 43 m. E.S.E. of Valence on the +Paris-Lyon railway. Pop. (1906) 3090. The town is situated in a +plain enclosed by mountains on the right bank of the Drôme +below its confluence with the Meyrosse, which supplies power to +some of the industries. The most interesting structures of Die +are the old cathedral, with a porch of the 11th century supported +on granite columns from an ancient temple of Cybele; and the +Porte St Marcel, a Roman gateway flanked by massive towers. +The Roman remains also include the ruins of aqueducts and altars. +Die is the seat of a sub-prefect, and of a tribunal of first instance. +The manufactures are silk, furniture, cloth, lime and cement, and +there are flour and saw mills. Trade is in timber, especially +walnut, and in white wine known as <i>clairette de Die</i>. The mulberry +is largely grown for the rearing of silkworms. Under the Romans, +Die (<i>Dea Augusta Vocontiorum</i>) was an important colony. It was +formerly the seat of a bishopric, united to that of Valence from +1276 to 1687 and suppressed in 1790. Previous to the revocation +of the edict of Nantes in 1685 it had a Calvinistic university.</p> + + +<hr class="art" /> +<p><span class="bold">DIE<a name="ar61" id="ar61"></a></span> (Fr. <i>dé</i>, from Lat. <i>datum</i>, given), a word used in various +senses, for a small cube of ivory, &c. (see <span class="sc"><a href="#artlinks">Dice</a></span>), 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.</p> + +<table class="nobctr" summary="Illustration"> +<tr><td class="figcenter1"><img style="width:267px; height:269px" src="images/img209.jpg" alt="" /></td></tr></table> + +<p>The fabrics known as “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 +<span class="pagenum"><a name="page209" id="page209"></a>209</span> +weaves are generally employed for the commoner classes of the +latter. The effect is, in every case, obtained by what are technically +called warp and weft float weaves. The illustration B shows +the two double damask weaves +arranged to form a dice pattern, +while A shows a similar +pattern made from two four-thread +twill weaves. C and D +represent respectively the disposition +of the threads in A +and B with the first pick, +and the solid marks represent +the floats of warp. The four +squares, which are almost as +pronounced in the cloth as +those of a chess-board, may +be made of any size by repeating +each weave for the amount +of surface required. It is only in the finest cloths that the double +damask weaves B are used for dice patterns, the single damask +weaves and the twill weaves being employed to a greater extent. +This class of pattern is largely employed for the production of +table-cloths of lower and medium qualities. The term damask +is also often applied to cloths of this character, and especially so +when the figure is formed by rectangles of different sizes.</p> + + +<hr class="art" /> +<p><span class="bold">DIEBITSCH, HANS KARL FRIEDRICH ANTON,<a name="ar62" id="ar62"></a></span> count von +Diebitsch and Narden, called by the Russians Ivan Ivanovich, +Count Diebich-Zabalkansky (1785-1831), Russian field-marshal, +was born in Silesia on the 13th of May 1785. He was educated +at the Berlin cadet school, but by the desire of his father, a +Prussian officer who had passed into the service of Russia, he also +did the same in 1801. He served in the campaign of 1805, and +was wounded at Austerlitz, fought at Eylau and Friedland, and +after Friedland was promoted captain. During the next five +years of peace he devoted himself to the study of military science, +engaging once more in active service in the War of 1812. He +distinguished himself very greatly in Wittgenstein’s campaign, +and in particular at Polotzk (October 18 and 19), after which +combat he was raised to the rank of major-general. In the latter +part of the campaign he served against the Prussian contingent +of General Yorck (von Wartenburg), with whom, through +Clausewitz, he negotiated the celebrated convention of Tauroggen, +serving thereafter with Yorck in the early part of the War of +Liberation. After the battle of Lützen he served in Silesia +and took part in negotiating the secret treaty of Reichenbach. +Having distinguished himself at the battles of Dresden and +Leipzig he was promoted lieutenant-general. At the crisis of +the campaign of 1814 he strongly urged the march of the allies on +Paris; and after their entry the emperor Alexander conferred on +him the order of St Alexander Nevsky. In 1815 he attended the +congress of Vienna, and was afterwards made adjutant-general +to the emperor, with whom, as also with his successor Nicholas, +he had great influence. By Nicholas he was created baron, and +later count. In 1820 he had become chief of the general staff, +and in 1825 he assisted in suppressing the St Petersburg <i>émeute</i>. +His greatest exploits were in the Russo-Turkish War of 1828-1829, +which, after a period of doubtful contest, was decided by +Diebitsch’s brilliant campaign of Adrianople; this won him the +rank of field-marshal and the honorary title of Zabalkanski +to commemorate his crossing of the Balkans. In 1830 he was +appointed to command the great army destined to suppress the +insurrection in Poland. He won the terrible battle of Gróchow on +the 25th of February, and was again victorious at Ostrolenka on +the 26th of May, but soon afterwards he died of cholera (or by his +own hand) at Klecksewo near Pultusk, on the 10th of June 1831.</p> + +<div class="condensed"> +<p>See Belmont (Schümberg), <i>Graf Diebitsch</i> (Dresden, 1830); +Stürmer, <i>Der Tod des Grafen Diebitsch</i> (Berlin, 1832); Bantych-Kamenski, +<i>Biographies of Russian Field-Marshals</i> (in Russian, +St Petersburg, 1841).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DIEDENHOFEN<a name="ar63" id="ar63"></a></span> (Fr. <i>Thionville</i>), a fortified town of Germany, +in Alsace-Lorraine, dist. Lorraine, on the Mosel, 22 m. N. from +Metz by rail. Pop. (1905) 6047. It is a railway junction of +some consequence, with cultivation of vines, fruit and vegetables, +brewing, tanning, &c. Diedenhofen is an ancient Frank town +(Theudonevilla, Totonisvilla), in which imperial diets were held +in the 8th century; was captured by Condé in 1643 and fortified +by Vauban; capitulated to the Prussians, after a severe bombardment, +on the 25th of November 1870.</p> + + +<hr class="art" /> +<p><span class="bold">DIEKIRCH,<a name="ar64" id="ar64"></a></span> a small town in the grand duchy of Luxemburg, +charmingly situated on the banks of the Sûre. Pop. (1905) +3705. Its name is said to be derived from Dide or Dido, granddaughter +of Odin and niece of Thor. The mountain at the foot of +which the town lies, now called Herrenberg, was formerly known +as Thorenberg, or Thor’s mountain. On the summit of this rock +rises a perennial stream which flows down into the town under the +name of Bellenflesschen. Diekirch was an important Roman +station, and in the 14th century John of Luxemburg, the blind +king of Bohemia, fortified it, surrounding the place with a +castellated wall and a ditch supplied by the stream mentioned. +It remained more <span class="correction" title="amended from for">or</span> less fortified until the beginning of the 19th +century when the French during their occupation levelled the old +walls, and substituted the avenues of trees that now encircle the +town. Diekirch is the administrative centre of one of the three +provincial divisions of the grand duchy. It is visited during the +summer by many thousand tourists and travellers from Holland, +Belgium and Germany.</p> + + +<hr class="art" /> +<p><span class="bold">DIELECTRIC,<a name="ar65" id="ar65"></a></span> in electricity, a non-conductor of electricity; it +is the same as insulator. The “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 <span class="sc"><a href="#artlinks">Electrostatics</a></span>; <span class="sc"><a href="#artlinks">Magneto-Optics</a></span>).</p> + + +<hr class="art" /> +<p><span class="bold">DIELMANN, FREDERICK<a name="ar66" id="ar66"></a></span> (1847-  ), American portrait +and figure painter, was born at Hanover, Germany, on the 25th +of December 1847. He was taken to the United States in +early childhood; studied under Diez at the Royal Academy at +Munich; was first an illustrator, and became a distinguished +draughtsman and painter of genre pictures. His mural decorations +and mosaic panels for the Congressional library, Washington, +are notable. He was elected in 1899 president of the National +Academy of Design.</p> + + +<hr class="art" /> +<p><span class="bold">DIEMEN, ANTHONY VAN<a name="ar67" id="ar67"></a></span> (1593-1645), Dutch admiral and +governor-general of the East Indian settlements, was born at +Kuilenburg in 1593. He was educated in commerce, and on +entering the service of the East India Company speedily attained +high rank. In 1631 he led a Dutch fleet from the Indies to +Holland, and in 1636 he was raised to the governor-generalship. +He came into conflict with the Portuguese, and took their +possessions in Ceylon and Malacca from them. He greatly +extended the commercial relationships of the Dutch, opening up +trade with Tong-king, China and Japan. As an administrator +also he showed ability, and the foundation of a Latin school and +several churches in Batavia is to be ascribed to him. Exploring +expeditions were sent to Australia under his auspices in 1636 and +1642, and Abel Tasman named after him (Van Diemen’s Land) +the island now called Tasmania. Van Diemen died at Batavia on +the 19th of April 1645.</p> + + +<hr class="art" /> +<p><span class="bold">DIEPENBECK, ABRAHAM VAN<a name="ar68" id="ar68"></a></span> (1599-1675), Flemish +painter, was born at Herzogenbusch, and studied painting at +Antwerp, where he became one of Rubens’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 +<span class="pagenum"><a name="page210" id="page210"></a>210</span> +other occupation for the lucrative business of a draughtsman and +designer for engravings. Most of Diepenbeck’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.</p> + + +<hr class="art" /> +<p><span class="bold">DIEPPE,<a name="ar69" id="ar69"></a></span> a seaport of northern France, capital of an arrondissement +in the department of Seine-Inférieure, on the English +Channel, 38 m. N. of Rouen, and 105 m. N.W. of Paris by the +Western railway. Pop. (1906) 22,120. It is situated at the +mouth of the river Arques in a valley bordered on each side +by steep white cliffs. The main part of the town lies to the west, +and the fishing suburb of Le Pollet to the east of the river and +harbour. The sea-front of Dieppe, which in summer attracts +large numbers of visitors, consists of a pebbly beach backed by a +handsome marine promenade. Dieppe has a modern aspect; its +streets are wide and its houses, in most cases, are built of brick. +Two squares side by side and immediately to the west of the outer +harbour form the nucleus of the town, the Place Nationale, overlooked +by the statue of Admiral A. Duquesne, and the Place St +Jacques, named after the beautiful Gothic church which stands +in its centre. The Grande Rue, the busiest and handsomest +street, leads westward from the Place Nationale. The church +of St Jacques was founded in the 13th century, but consists in +large measure of later workmanship and was in some portions +restored in the 19th century. The castle, overlooking the beach +from the summit of the western cliff, was erected in 1435. The +church of Notre-Dame de Bon Secours on the opposite cliff, and +the church of St Remy, of the 16th and 17th centuries, are other +noteworthy buildings. A well-equipped casino stands at the +west end of the sea-front. The public institutions include the subprefecture, +tribunals of first instance and commerce, a chamber +of commerce, a communal college and a school of navigation.</p> + +<p>Dieppe has one of the safest and deepest harbours on the +English Channel. A curved passage cut in the bed of the Arques +and protected by an eastern and a western jetty gives access to +the outer harbour, which communicates at the east end by a lockgate +with the Bassin Duquesne and the Bassin Bérigny, and at +the west end by the New Channel, with an inner tidal harbour +and two other basins. Vessels drawing 20 ft. can enter the new +docks at neap tide. A dry-dock and a gridiron are included +among the repairing facilities of the port. The harbour railway +station is on the north-west quay of the outer harbour alongside +which the steamers from Newhaven lie. The distance of Dieppe +from Newhaven, with which there has long been daily communication, +is 64 m. The imports include silk and cotton goods, thread, +oil-seeds, timber, coal and mineral oil; leading exports are wine, +silk, woollen and cotton fabrics, vegetables and fruit and flint-pebbles. +The average annual value of imports for the five years +1901-1905 was £4,916,000 (£4,301,000 for the years 1896-1900); +the exports were valued at £9,206,000 (£7,023,000 for years +1896-1900). The industries comprise shipbuilding, cotton-spinning, +steam-sawing, the manufacture of machinery, porcelain, +briquettes, lace, and articles in ivory and bone, the production +of which dates from the 15th century. There is also a tobacco +factory of some importance. The fishermen of Le Pollet, to +whom tradition ascribes a Venetian origin, are among the main +providers of the Parisian market. The sea-bathing attracts +many visitors in the summer. Two miles to the north-east of +the town is the ancient camp known as the Cité de Limes, which +perhaps furnished the nucleus of the population of Dieppe.</p> + +<p>It is suggested on the authority of its name, that Dieppe owed +its origin to a band of Norman adventurers, who found its “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.</p> + +<div class="condensed"> +<p>See L. Vitet, <i>Histoire de Dieppe</i> (Paris, 1844); D. Asseline, <i>Les +Antiquités et chroniques de la ville de Dieppe</i>, a 17th-century account +published at Paris in 1874.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DIERX, LÉON<a name="ar70" id="ar70"></a></span> (1838-  ), 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: <i>Poèmes et poésies</i> (1864); +<span class="pagenum"><a name="page211" id="page211"></a>211</span> +<i>Lèvres closes</i> (1867); <i>Paroles d’un vaincu</i> (1871); <i>La Rencontre</i>, a +dramatic scene (1875) and <i>Les Amants</i> (1879). His <i>Poésies +complètes</i> (1872) were crowned by the French Academy. A complete +edition of his works was published in 2 vols., 1894-1896.</p> + + +<hr class="art" /> +<p><span class="bold">DIES, CHRISTOPH ALBERT<a name="ar71" id="ar71"></a></span> (1755-1822), German painter, +was born at Hanover, and learned the rudiments of art in his +native place. For one year he studied in the academy of Dusseldorf, +and then he started at the age of twenty with thirty ducats +in his pocket for Rome. There he lived a frugal life till 1796. +Copying pictures, chiefly by Salvator Rosa, for a livelihood, his +taste led him to draw and paint from nature in Tivoli, Albano +and other picturesque places in the vicinity of Rome. Naples, +the birthplace of his favourite master, he visited more than once +for the same reasons. In this way he became a bold executant in +water-colours and in oil, though he failed to acquire any originality +of his own. Lord Bristol, who encouraged him as a copyist, +predicted that he would be a second Salvator Rosa. But Dies +was not of the wood which makes original artists. Besides other +disqualifications, he had necessities which forced him to give +up the great career of an independent painter. David, then +composing his Horatii at Rome, wished to take him to Paris. +But Dies had reasons for not accepting the offer. He was courting +a young Roman whom he subsequently married. Meanwhile he +had made the acquaintance of Volpato, for whom he executed +numerous drawings, and this no doubt suggested the plan, which +he afterwards carried out, of publishing, in partnership with +Méchan, Reinhardt and Frauenholz, the series of plates known +as the <i>Collection de vues pittoresques de l’Italie</i>, published in +seventy-two sheets at Nuremberg in 1799. With so many +irons in the fire Dies naturally lost the power of concentration. +Other causes combined to affect his talent. In 1787 he swallowed +by mistake three-quarters of an ounce of sugar of lead. His recovery +from this poison was slow and incomplete. He settled at +Vienna, and lived there on the produce of his brush as a landscape +painter, and on that of his pencil or graver as a draughtsman and +etcher. But instead of getting better, his condition became +worse, and he even lost the use of one of his hands. In this +condition he turned from painting to music, and spent his leisure +hours in the pleasures of authorship. He did not long survive, +dying at Vienna in 1822, after long years of chronic suffering. +From two pictures now in the Belvedere gallery, and from +numerous engraved drawings from the neighbourhood of Tivoli, +we gather that Dies was never destined to rise above a respectable +mediocrity. He followed Salvator Rosa’s example in imitating +the manner of Claude Lorraine. But Salvator adapted the style +of Claude, whilst Dies did no more than copy it.</p> + + +<hr class="art" /> +<p><span class="bold">DIEST,<a name="ar72" id="ar72"></a></span> a small town in the province of Brabant, Belgium, +situated on the Demer at its junction with the Bever. Pop. +(1904) 8383. It lies about half-way between Hasselt and +Louvain, and is still one of the five fortified places in Belgium. +It contains many breweries, and is famous for the excellence of +its beer.</p> + + +<hr class="art" /> +<p><span class="bold">DIESTERWEG, FRIEDRICH ADOLF WILHELM<a name="ar73" id="ar73"></a></span> (1790-1866), +German educationist, was born at Siegen on the 29th of October +1790. Educated at Herborn and Tübingen universities, he took +to the profession of teaching in 1811. In 1820 he was appointed +director of the new school at Mörs, where he put in practice the +methods of Pestalozzi. In 1832 he was summoned to Berlin to +direct the new state-schools seminary in that city. Here he +proved himself a strong supporter of unsectarian religious teaching. +In 1846 he established the Pestalozzi institution at Pankow, +and the Pestalozzi societies for the support of teachers’ widows +and orphans. In 1850 he retired on a pension, but continued +vigorously to advocate his educational views. In 1858 he was +elected to the chamber of deputies as member for the city of +Berlin, and voted with the Liberal opposition. He died in Berlin +on the 7th of July 1866. Diesterweg was a voluminous writer +on educational subjects, and was the author of various school +text-books.</p> + + +<hr class="art" /> +<p><span class="bold">DIET,<a name="ar74" id="ar74"></a></span> a term used in two senses, (1) food or the regulation +of feeding (see <span class="sc"><a href="#artlinks">Dietary</a></span> and <span class="sc"><a href="#artlinks">Dietetics</a></span>), (2) an assembly +or council (Fr. <i>diète</i>; It. <i>dieta</i>; Low Lat. <i>diaeta</i>; Ger. <i>Tag</i>). +We are here concerned only with this second sense. In +modern usage, though in Scotland the term is still sometimes +applied to any assembly or session, it is practically confined to +the sense of an assembly of estates or of national or federal +representatives. The origin of the word in this connotation is +somewhat complicated. It is undoubtedly ultimately derived +from the Greek <span class="grk" title="diaita">δίαιτα</span> (Lat. <i>diaeta</i>), which meant “mode of +life” and thence “prescribed mode of life,” the English “diet” +or “regimen.” This was connected with the verb <span class="grk" title="diaitan">διαιτᾶν</span>, in +the sense of “to rule,” “to regulate”; compare the office of +<span class="grk" title="diaitêtês">διαιτητής</span> at Athens, and <i>dieteta</i>, “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 (<i>Glossarium</i>, <i>s.v.</i>) +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 <i>diaeta</i> early became +confused with Lat. <i>dies</i>, “day” (Ger. <i>Tag</i>), “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, <i>e.g.</i> <i>diaeta</i> for <i>dieta</i>, “a day’s +journey” (also an obsolete sense of “diet” in English), and +<i>dieta</i> for “the ordinary course of the church,” <i>i.e.</i> “the daily +office,” which suggests the original sense of <i>diaeta</i> as “a prescribed +mode of life.”</p> + +<p>The word “diet” is now used in English for the <i>Reichstag</i>, +“imperial diet” of the old Holy Roman Empire; for the +<i>Bundestag</i>, “federal diet,” of the former Germanic confederation; +sometimes for the <i>Reichstag</i> of the modern German empire; for +the <i>Landtage</i>, “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 +<i>Reichstag</i> or of the Russian Landtag, while the <i>Landtag</i>, <i>e.g.</i> 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.</p> + +<p>The origin of the diet, or deliberative assembly, of the Holy +Roman Empire must be sought in the <i>placitum</i> of the Frankish +empire. This represented the tribal assembly of the Franks, +meeting (originally in March, but after 755 in May, whence it is +called the Campus Maii) partly for a military review on the eve +of the summer campaign, partly for deliberation on important +matters of politics and justice. By the side of this larger +assembly, however, which contained in theory, if not in practice, +the whole body of Franks available for war, there had developed, +even before Carolingian times, a smaller body composed of the +magnates of the Empire, both lay and ecclesiastical. The germ +of this smaller body is to be found in the episcopal synods, which, +afforced by the attendance of lay magnates, came to be used +by the king for the settlement of national affairs. Under the +Carolingians it was usual to combine the assembly of magnates +with the <i>generalis conventus</i> of the “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 <i>en masse</i>, in which each freeman had +been liable to service; and after the close of the 10th century +it no longer existed.</p> + +<p>The imperial diet (<i>Reichstag</i>) of the middle ages might sometimes +contain representatives of Italy, the <i>regnum Italicum</i>; but +it was practically always confined to the magnates of Germany, +the <i>regnum Teutonicum</i>. Upon occasion a summons to the diet +might be sent even to the knights, but the regular members were +the princes (<i>Fürsten</i>), both lay and ecclesiastical. In the 13th +<span class="pagenum"><a name="page212" id="page212"></a>212</span> +century the seven electors began to disengage themselves from +the prince as a separate element, and the Golden Bull (1356) +made their separation complete; from the 14th century onwards +the nobles (both counts and other lords) are regarded as regular +members; while after 1250 the imperial and episcopal towns +often appear through their representatives. By the 14th century, +therefore, the originally homogeneous diet of princes is already, +at any rate practically if not yet in legal form, divided into three +colleges—the electors, the princes and nobles, and the representatives +of the towns (though, as we shall see, the latter can +hardly be reckoned as regular members until the century of the +Reformation). Under the Hohenstaufen it is still the rule that +every member of the diet must attend personally, or lose his vote; +at a later date the principle of representation by proxy, which +eventually made the diet into a mere congress of envoys, was +introduced. By the end of the 13th century the vote of the +majority had come to be regarded as decisive; but in accordance +with the strong sense of social distinctions which marks German +history, the quality as well as the quantity of votes was weighed, +and if the most powerful of the princes were agreed, the opinion +of the lesser magnates was not consulted. The powers of the +medieval diet extended to matters like legislation, the decision +upon expeditions (especially the <i>expeditio Romana</i>), taxation and +changes in the constitution of the principalities or the Empire. +The election of the king, which was originally regarded as one of +the powers of the diet, had passed to the electors by the middle +of the 13th century.</p> + +<p>A new era in the history of the diet begins with the Reformation. +The division of the diet into three colleges becomes definite +and precise; the right of the electors, for instance, to constitute +a separate college is explicitly recognized as a matter of established +custom in 1544. The representatives of the towns now become +regular members. In the 15th century they had only attended +when special business, such as imperial reform or taxation, fell +under discussion; in 1500, however, they were recognized as a +separate and regular estate, though it was not until 1648 that +they were recognized as equal to the other estates of the diet. +The estate of the towns, or college of municipal representatives, +was divided into two benches, the Rhenish and the Swabian. +The estate of the princes and counts, which stood midway +between the electors and the towns, also attained, in the years +that followed the Reformation, its final organization. The vote +of the great princes ceased to be personal, and began to be +territorial. This had two results. The division of a single +territory among the different sons of a family no longer, as of old, +multiplied the voting power of the family; while in the opposite +case, the union of various territories in the hands of a single +person no longer meant the extinction of several votes, since the +new owner was now allowed to give a vote for each of his territories. +The position of the counts and other lords, who joined +with the princes in forming the middle estate, was finally fixed +by the middle of the 17th century. While each of the princes +enjoyed an individual vote, the counts and other lords were +arranged in groups, each of which voted as a whole, though the +whole of its vote (<i>Kuriatstimme</i>) only counted as equal to the +vote of a single prince (<i>Virilstimme</i>). There were six of these +groups; but as the votes of the whole college of princes and +counts (at any rate in the 18th century) numbered 100, they +could exercise but little weight.</p> + +<p>The last era in the history of the diet may be said to open with +the treaty of Westphalia (1648). The treaty acknowledged that +Germany was no longer a unitary state, but a loose confederation +of sovereign princes; and the diet accordingly ceased to bear the +character of a national assembly, and became a mere congress of +envoys. The “last diet” which issued a regular recess (<i>Reichsabschied</i>—the +term applied to the <i>acta</i> of the diet, as formally +compiled and enunciated at its dissolution) was that of Regensburg +in 1654. The next diet, which met at Regensburg in 1663, +never issued a recess, and was never dissolved; it continued in +permanent session, as it were, till the dissolution of the Empire +in 1806. This result was achieved by the process of turning the +diet from an assembly of principals into a congress of envoys. +The emperor was represented by two <i>commissarii</i>; the electors, +princes and towns were similarly represented by their accredited +agents. Some legislation was occasionally done by this body; a +<i>conclusum imperii</i> (so called in distinction from the old <i>recessus +imperii</i> of the period before 1663) might slowly (very slowly—for +the agents, imperfectly instructed, had constantly to refer +matters back to their principals) be achieved; but it rested with +the various princes to promulgate and enforce the <i>conclusum</i> in +their territories, and they were sufficiently occupied in issuing +and enforcing their own decrees. In practice the diet had +nothing to do; and its members occupied themselves in +“wrangling about chairs”—that is to say, in unending disputes +about degrees and precedences.</p> + +<p>In the Germanic Confederation, which occupies the interval +between the death of the Holy Roman Empire and the formation +of the North German Confederation (1815-1866), a diet +(<i>Bundestag</i>) existed, which was modelled on the old diet of the 18th +century. It was a standing congress of envoys at Frankfort-on-Main. +Austria presided in the diet, which, in the earlier years of +its history, served, under the influence of Metternich, as an organ +for the suppression of Liberal opinion. In the North German +Confederation (1867-1870) a new departure was made, which has +been followed in the constitution of the present German empire. +Two bodies were instituted—a <i>Bundesrat</i>, which resembles the old +diet in being a congress of envoys sent by the sovereigns of the +different states of the confederation, and a <i>Reichstag</i>, which bears +the name of the old diet, but differs entirely in composition. The +new Reichstag is a popular representative assembly, based on +wide suffrage and elected by ballot; and, above all, it is an +assembly representing, not the several states, but the whole +Empire, which is divided for this purpose into electoral districts. +Both as a popular assembly, and as an assembly which represents +the whole of a united Germany, the new Reichstag goes back, one +may almost say, beyond the diet even of the middle ages, to the +days of the old Teutonic folk-moot.</p> + +<div class="condensed"> +<p>See R. Schröder, <i>Lehrbuch der deutschen Rechtsgeschichte</i> (1902), +pp. 149, 508, 820, 880. Schröder gives a bibliography of monographs +bearing on the history of the medieval diet.</p> +</div> +<div class="author">(E. Br.)</div> + + +<hr class="art" /> +<p><span class="bold">DIETARY,<a name="ar75" id="ar75"></a></span> in a general sense, a system or course of diet, in the +sense of food; more particularly, such an allowance and regulation +of food as that supplied to workhouses, the army and navy, +prisons, &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 <i>in statu quo</i>, but +nothing further. This applies particularly to the members of +a temporarily famine-stricken community. Before the days of +prison reform, too, the dietary scale of many prisons was to +a certain extent penal, in that the food supplied to prisoners +was barely sufficient for existence. Nowadays more humane +principles apply; there is no longer the obvious injustice of +applying the same scale of quantity and quality to all prisoners +under varying circumstances of constitution and surroundings, +and whether serving long or short periods of imprisonment.</p> + +<div class="condensed"> +<p>The system of dietary in force in the local and convict prisons of +England and Wales is that recommended by the Home Office on the +advice of a departmental committee. As to the local prison dietary, +its application is based on (1) the principle of variation of diet with +length of sentence; (2) the system of progressive dietary; (3) the +distinction between hard labour diets and non-hard labour diets; +(4) the differentiation of diet according to age and sex. There are +three classes of diet, classes A, B and C. Class A diet is given +to prisoners undergoing not more than seven days’ 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:—<i>Breakfast</i>, Bread, 8 oz. daily (6 oz. for women and +juveniles) with 1 pint of gruel. Juveniles (males and females under +<span class="pagenum"><a name="page213" id="page213"></a>213</span> +sixteen years of age) get, in addition, ½ pint of milk. <i>Dinner</i>, 8 oz. of +bread daily, with 1 pint of porridge on three days of the week, 8 oz. +of potatoes (representing the vegetable element) on two other days, +and 8 oz. of suet pudding (representing the fatty element) on the +other two days. <i>Supper</i>, the breakfast fare repeated.</p> + +<p>Class B diet, which is also given to (1) prisoners on remand or +awaiting trial, (2) offenders of the 1st division who do not maintain +themselves, (3) offenders of the 2nd division and (4) debtors, is as +shown in Table I.</p> + + + +<p class="center pt2 sc">Table I.</p> + +<table class="ws" summary="Contents"> + +<tr><td class="tcc allb"> </td> <td class="tcc allb"> </td> <td class="tcc allb">Men.</td> <td class="tcc allb">Women.</td> <td class="tcc allb">Juveniles.</td></tr> + +<tr><td class="tcl lb rb" rowspan="5">Breakfast.</td> <td class="tcl rb">Daily:—</td> <td class="tcc rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb"> </td></tr> +<tr><td class="tcl rb"> Bread</td> <td class="tcc rb">8 oz.</td> <td class="tcc rb">6 oz.</td> <td class="tcc rb">6 oz.</td></tr> +<tr><td class="tcl rb"> Gruel</td> <td class="tcc rb">1 pt.</td> <td class="tcc rb">1 pt.</td> <td class="tcc rb">1 pt.</td></tr> +<tr><td class="tcl rb"> Milk</td> <td class="tcc rb">· ·</td> <td class="tcc rb bb">· ·</td> <td class="tcc rb bb">½ pt.</td></tr> +<tr><td class="tcl rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb" colspan="2"> </td></tr> +<tr><td class="tcl lb rb" rowspan="36">Dinner.</td> <td class="tcl rb">Sunday:—</td> <td class="tcc rb"> </td> <td class="tcc rb" colspan="2"> </td></tr> +<tr><td class="tcl rb"> Bread</td> <td class="tcc rb">6 oz.</td> <td class="tcc rb" colspan="2">6 oz.</td></tr> +<tr><td class="tcl rb"> Potatoes</td> <td class="tcc rb">8 ”</td> <td class="tcc rb" colspan="2">8 ”</td></tr> +<tr><td class="tcl rb"> Cooked meat, preserved by heat</td> <td class="tcc rb">4 ”</td> <td class="tcc rb" colspan="2">3 ”</td></tr> +<tr><td class="tcl rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb" colspan="2"> </td></tr> +<tr><td class="tcl rb">Monday:—</td> <td class="tcc rb"> </td> <td class="tcc rb" colspan="2"> </td></tr> +<tr><td class="tcl rb"> Bread</td> <td class="tcc rb">6 oz.</td> <td class="tcc rb" colspan="2">6 oz.</td></tr> +<tr><td class="tcl rb"> Potatoes</td> <td class="tcc rb">8 ”</td> <td class="tcc rb" colspan="2">8 ”</td></tr> +<tr><td class="tcl rb"> Beans</td> <td class="tcc rb">10 ”</td> <td class="tcc rb" colspan="2">8 ”</td></tr> +<tr><td class="tcl rb"> Fat bacon</td> <td class="tcc rb">2 ”</td> <td class="tcc rb" colspan="2">1 ”</td></tr> +<tr><td class="tcl rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb" colspan="2"> </td></tr> +<tr><td class="tcl rb">Tuesday:—</td> <td class="tcc rb"> </td> <td class="tcc rb" colspan="2"> </td></tr> +<tr><td class="tcl rb"> Bread</td> <td class="tcc rb">6 oz</td> <td class="tcc rb" colspan="2">6 oz.</td></tr> +<tr><td class="tcl rb"> Potatoes</td> <td class="tcc rb">8 ”</td> <td class="tcc rb" colspan="2">8 ”</td></tr> +<tr><td class="tcl rb"> Soup</td> <td class="tcc rb">1 pt.</td> <td class="tcc rb" colspan="2">1 pt.</td></tr> +<tr><td class="tcl rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb" colspan="2"> </td></tr> +<tr><td class="tcl rb">Wednesday:—</td> <td class="tcc rb"> </td> <td class="tcc rb" colspan="2"> </td></tr> +<tr><td class="tcl rb"> Bread</td> <td class="tcc rb">6 oz.</td> <td class="tcc rb" colspan="2">6 oz.</td></tr> +<tr><td class="tcl rb"> Potatoes</td> <td class="tcc rb">8 ”</td> <td class="tcc rb" colspan="2">8 ”</td></tr> +<tr><td class="tcl rb"> Suet pudding</td> <td class="tcc rb">10 ”</td> <td class="tcc rb" colspan="2">8 ”</td></tr> +<tr><td class="tcl rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb" colspan="2"> </td></tr> +<tr><td class="tcl rb">Thursday:—</td> <td class="tcc rb"> </td> <td class="tcc rb" colspan="2"> </td></tr> +<tr><td class="tcl rb"> Bread</td> <td class="tcc rb">6 oz.</td> <td class="tcc rb" colspan="2">6 oz.</td></tr> +<tr><td class="tcl rb"> Potatoes</td> <td class="tcc rb">8 ”</td> <td class="tcc rb" colspan="2">8 ”</td></tr> +<tr><td class="tcl rb"> Cooked beef, without bone</td> <td class="tcc rb">4 ”</td> <td class="tcc rb" colspan="2">3 ”</td></tr> +<tr><td class="tcl rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb" colspan="2"> </td></tr> +<tr><td class="tcl rb">Friday:—</td> <td class="tcc rb"> </td> <td class="tcc rb" colspan="2"> </td></tr> +<tr><td class="tcl rb"> Bread</td> <td class="tcc rb">6 oz.</td> <td class="tcc rb" colspan="2">6 oz.</td></tr> +<tr><td class="tcl rb"> Potatoes</td> <td class="tcc rb">8 ”</td> <td class="tcc rb" colspan="2">8 ”</td></tr> +<tr><td class="tcl rb"> Soup</td> <td class="tcc rb">1 pt.</td> <td class="tcc rb" colspan="2">1 pt.</td></tr> +<tr><td class="tcl rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb" colspan="2"> </td></tr> +<tr><td class="tcl rb">Saturday:—</td> <td class="tcc rb"> </td> <td class="tcc rb" colspan="2"> </td></tr> +<tr><td class="tcl rb"> Bread</td> <td class="tcc rb">6 oz.</td> <td class="tcc rb" colspan="2">6 oz.</td></tr> +<tr><td class="tcl rb"> Potatoes</td> <td class="tcc rb">8 ”</td> <td class="tcc rb" colspan="2">8 ”</td></tr> +<tr><td class="tcl rb"> Suet pudding</td> <td class="tcc rb">10 ”</td> <td class="tcc rb" colspan="2">8 ”</td></tr> +<tr><td class="tcl rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb" colspan="2"> </td></tr> +<tr><td class="tcl lb rb bb" rowspan="5">Supper.</td> <td class="tcl rb">Daily:—</td> <td class="tcc rb"> </td> <td class="tcc rb tb"> </td> <td class="tcc rb tb"> </td></tr> +<tr><td class="tcl rb"> Bread</td> <td class="tcc rb">8 oz.</td> <td class="tcc rb ">6 oz.</td> <td class="tcc rb ">6 oz.</td></tr> +<tr><td class="tcl rb"> Porridge</td> <td class="tcc rb">1 pt.</td> <td class="tcc rb"> </td> <td class="tcc rb"> </td></tr> +<tr><td class="tcl rb"> Gruel</td> <td class="tcc rb"> </td> <td class="tcc rb">1 pt.</td> <td class="tcc rb"> </td></tr> +<tr><td class="tcl rb bb"> Cocoa</td> <td class="tcc rb bb"> </td> <td class="tcc rb bb"> </td> <td class="tcc rb bb">1 pt.</td></tr> +</table> + +<p>Class C diet is class B amplified, and is given to those prisoners +serving sentences of three months and over.</p> + +<p>The dietary of convict prisons, in which prisoners are all under long +sentence, is divided into a diet for convicts employed at hard labour +and a diet for convicts employed at sedentary, indoor and light +labour. It will be found set forth in the Blue-book mentioned above. +The sparest of all prison diets is called “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 ℔ of bread +and as much water as the prisoner chooses to drink.</p> + +<p>In French prisons the dietary is nearly two pounds weight of bread, +with two meals of thin soup (breakfast and dinner) made from +potatoes, beans or other vegetables, and on two days a week made +from meat. In France the canteen system is in vogue, additional +food, such as sausages, cheese, fruit, &c., may be obtained by the +prisoner, according to the wages he receives for his labours. The +dietary of Austrian prisons is 1½ ℔ of bread daily, a dinner of soup +on four days of the week, and of meat on the other three days, +with a supper of soup or vegetable stew. Additional food can be +purchased by the prisoner out of his earnings.</p> + +<p>These dietaries may be taken as more or less typical of the ordinary +prison fare in most civilized countries, though in some countries it +may err on the side of severity, as in Sweden, prisoners being given +only two meals a day, one at mid-day and one at seven p.m., porridge +or gruel being the principal element in both meals. On the other +hand, the prison dietaries of many of the United States prisons go +to the other extreme, fresh fish, green vegetables, even coffee and +fruit, figuring in the dietary.</p> + +<p>Another class of dietary is that given to paupers. In England, +until 1900, almost every individual workhouse had its own special +dietary, with the consequence that many erred on the side of scantiness +and unsuitability, while others were too lavish. By an order of +the Local Government Board of that year, acting on a report of a +committee, all inmates of workhouses, with the exception of the sick, +children under three years of age, and certain other special cases, +are dieted in accordance with certain dietary tables as framed and +settled by the board. The order contained a great number of different +rations, it being left to the discretion of the guardians as to the final +settlement of the tables. For adult inmates the dietary tables are +for each sex respectively, two in number, one termed “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.</p> + +<p class="center pt2 sc">Table II.</p> + +<table class="ws" summary="Contents"> + +<tr><td class="tcc allb" colspan="3"> </td> <td class="tcc allb">Sun.</td> <td class="tcc allb">Mon.</td> <td class="tcc allb">Tue.</td> <td class="tcc allb">Wed.</td> <td class="tcc allb">Thu.</td> <td class="tcc allb">Fri.</td> <td class="tcc allb">Sat.</td></tr> + +<tr><td class="tccm allb" rowspan="2">Breakfast.</td> <td class="tcl">Bread.</td> <td class="tcr rb">oz.</td> <td class="tcc rb">8</td> <td class="tcc rb">4</td> <td class="tcc rb">4</td> <td class="tcc rb">4</td> <td class="tcc rb">4</td> <td class="tcc rb">4</td> <td class="tcc rb">4</td></tr> +<tr><td class="tcl bb">Porridge.</td> <td class="tcr rb bb">pt.</td> <td class="tcc rb bb">*</td> <td class="tcc rb bb">1½</td> <td class="tcc rb bb">1½</td> <td class="tcc rb bb">1½</td> <td class="tcc rb bb">1½</td> <td class="tcc rb bb">1½</td> <td class="tcc rb bb">1½</td></tr> + +<tr><td class="tccm allb" rowspan="10">Dinner.</td> <td class="tcl">Bread.</td> <td class="tcr rb">oz.</td> <td class="tcc rb">4</td> <td class="tcc rb">6</td> <td class="tcc rb">..</td> <td class="tcc rb">4</td> <td class="tcc rb">4</td> <td class="tcc rb">8</td> <td class="tcc rb">6</td></tr> +<tr><td class="tcl">Beef.</td> <td class="tcr rb">oz.</td> <td class="tcc rb">4½</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">4½</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td></tr> +<tr><td class="tcl">Vegetables.</td> <td class="tcr rb">oz.</td> <td class="tcc rb">12</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">12</td> <td class="tcc rb">12</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td></tr> +<tr><td class="tcl">Barley Soup.</td> <td class="tcr rb">pt.</td> <td class="tcc rb">..</td> <td class="tcc rb">1½</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td></tr> +<tr><td class="tcl">Pork.</td> <td class="tcr rb">oz.</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">4½</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td></tr> +<tr><td class="tcl">Beans.</td> <td class="tcr rb">oz.</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">12</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td></tr> +<tr><td class="tcl">Fish.</td> <td class="tcr rb">oz.</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">10</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td></tr> +<tr><td class="tcl">Cheese.</td> <td class="tcr rb">oz.</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">3</td> <td class="tcc rb">..</td></tr> +<tr><td class="tcl">Broth.</td> <td class="tcr rb">pt.</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">1</td> <td class="tcc rb">..</td></tr> +<tr><td class="tcl bb">Irish Stew.</td> <td class="tcr rb bb">pt.</td> <td class="tcc rb bb">..</td> <td class="tcc rb bb">..</td> <td class="tcc rb bb">..</td> <td class="tcc rb bb">..</td> <td class="tcc rb bb">..</td> <td class="tcc rb bb">..</td> <td class="tcc rb bb">1</td></tr> + +<tr><td class="tccm allb" rowspan="6">Supper.</td> <td class="tcl">Bread.</td> <td class="tcr rb">oz.</td> <td class="tcc rb">8</td> <td class="tcc rb">6</td> <td class="tcc rb">6</td> <td class="tcc rb">6</td> <td class="tcc rb">8</td> <td class="tcc rb">6</td> <td class="tcc rb">6</td></tr> +<tr><td class="tcl">Butter.</td> <td class="tcr rb">oz.</td> <td class="tcc rb">½</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td></tr> +<tr><td class="tcl">Tea.</td> <td class="tcr rb">pt.</td> <td class="tcc rb">1</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td></tr> +<tr><td class="tcl">Gruel.</td> <td class="tcr rb">pt.</td> <td class="tcc rb">..</td> <td class="tcc rb">1½</td> <td class="tcc rb">1½</td> <td class="tcc rb">1½</td> <td class="tcc rb">..</td> <td class="tcc rb">1½</td> <td class="tcc rb">1½</td></tr> +<tr><td class="tcl">Broth.</td> <td class="tcr rb">pt.</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">1</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td></tr> +<tr><td class="tcl bb">Cheese.</td> <td class="tcr rb bb">oz.</td> <td class="tcc rb bb">..</td> <td class="tcc rb bb">..</td> <td class="tcc rb bb">..</td> <td class="tcc rb bb">..</td> <td class="tcc rb bb">2</td> <td class="tcc rb bb">..</td> <td class="tcc rb bb">..</td></tr> + +<tr> <td class="tcc f90" colspan="10">* On Sundays 1 pint of tea and 2½ oz. of butter are given instead of porridge.</td></tr> +</table> + +<p>In the casual wards of workhouses the dietary is plainer, consisting +of 8 oz. of bread, or 6 oz. of bread and one pint of gruel or broth for +breakfast; the same for supper; for dinner 8 oz. of bread and 1½ oz. +of cheese or 6 oz. of bread and one pint of soup. The American poor +law system is based broadly on that of England, and the methods +of relief are much the same. Each state, however, makes its own +regulations, and there is considerable diversity in workhouse dietaries +in consequence. The German system of poor relief is more methodical +than those of England and America. The really deserving are treated +<span class="pagenum"><a name="page214" id="page214"></a>214</span> +with more commiseration, and a larger amount of outdoor relief is +given than in England. There is no casual ward, tramps and beggars +being liable to penal treatment, but there are “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.</p> + +<p>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:—</p> + +<p class="center"><i>Service Afloat.</i></p> + +<table class="reg" summary="poem"><tr><td> <div class="poemr"> +<p>1 ℔ bread (or ¾ ℔ bread and ¼ ℔ trade flour).</p> +<p>½ ℔ fresh meat.</p> +<p>1 ℔ fresh vegetables.</p> +<p><span class="spp">1</span>⁄<span class="suu">8</span> pint spirit.</p> +<p>4 oz. sugar.</p> +<p>½ oz. tea (or 1 oz. coffee for every ¼ oz. tea).</p> +<p>½ oz. ordinary or soluble chocolate (or 1 oz. coffee).</p> +<p>¾ oz. condensed milk.</p> +<p>1 oz. jam or marmalade.</p> +<p>4 oz. preserved meat on <i>one</i> day of the week in harbour, or on <i>two</i> days at sea.</p> +<p>Mustard, pepper, vinegar, and salt as required.</p> +</div> </td></tr></table> + +<p>Substitute for soft bread when the latter is not available—</p> +<p class="center">½ ℔ biscuit (new type) or 1 ℔ flour.</p> + +<p>Substitutes for fresh meat when the latter is not available:—</p> + +<table class="nobctr" summary="Contents"> + +<tr><td class="tccm" style="background-color: #f5f5f5;" rowspan="11">On<br />alternate<br />days.</td> <td class="tcl" colspan="2">(1) Salt pork day:—</td></tr> +<tr><td class="tcl" colspan="2">  ½ ℔ salt pork.</td></tr> +<tr><td class="tcl" colspan="2">  ¼ ℔ split peas.</td></tr> +<tr><td class="tcl" colspan="2">  Celery seed, ½ oz. to every 8 ℔ of split peas put into the coppers.</td></tr> +<tr><td class="tcl" colspan="2">  ½ ℔ potatoes (or 1 oz. compressed vegetables).</td></tr> + +<tr><td class="tcl" colspan="2">(2) Preserved meat day:—</td></tr> +<tr><td class="tcl" colspan="2">  6 oz. preserved meat.</td></tr> +<tr><td class="tcl">  8 oz. trade flour</td><td class="tcl" rowspan="3"><span style="font-size: 3em; font-family: 'Courier New'; color: #a0a0a0;">}</span> or 4 oz. rice.</td></tr> +<tr><td class="tcl">  ¾ oz. refined suet</td></tr> +<tr><td class="tcl">  2 oz. raisins</td></tr> +<tr><td class="tcl" colspan="2">  ½ ℔ potatoes (or 1 oz. compressed vegetables).</td></tr> +</table> + +<p>On shore establishments and depot ships ¼ pt. fresh milk is issued +in lieu of the ¾ oz. of condensed milk.</p> + +<p>In the United States navy there is more liberality and variety of +diet, the approximate daily cost of the rations supplied being 1s. 3d. +per head. In the American mercantile marine, too, according to +the scale sanctioned by act of Congress (December 21, 1898) for +American ships, the seaman is better off than in the British merchant +service. The scale is shown in Table III.</p> + +<p class="center pt2 sc">Table III.</p> + +<table class="ws" summary="Contents"> + +<tr><td class="tccm allb">Weekly<br />Scale.</td> <td class="tccm lb rb2 tb bb">Articles.</td> <td class="tccm allb">Weekly<br />Scale.</td> <td class="tccm allb">Articles.</td></tr> + +<tr><td class="tcl lb rb"> 3½ ℔</td> <td class="tcl rb2">Biscuits.</td> <td class="tcl rb"> <span class="spp">7</span>⁄<span class="suu">8</span> oz.</td> <td class="tcl rb">Tea.</td></tr> +<tr><td class="tcl lb rb"> 3¾ ”</td> <td class="tcl rb2">Salt beef.</td> <td class="tcl rb">21 ”</td> <td class="tcl rb">Sugar.</td></tr> +<tr><td class="tcl lb rb"> 3 ”</td> <td class="tcl rb2">“ pork.</td> <td class="tcl rb"> 1½ ℔</td> <td class="tcl rb">Molasses.</td></tr> +<tr><td class="tcl lb rb"> 1½ ”</td> <td class="tcl rb2">Flour.</td> <td class="tcl rb"> 9 oz.</td> <td class="tcl rb">Fruits, dried.</td></tr> +<tr><td class="tcl lb rb"> 2 ”</td> <td class="tcl rb2">Meats, preserved.</td> <td class="tcl rb"> ¾ pt.</td> <td class="tcl rb">Pickles.</td></tr> +<tr><td class="tcl lb rb">10½ ”</td> <td class="tcl rb2">Bread, fresh (8 ℔ flour in lieu).</td> <td class="tcl rb"> 1 ”</td> <td class="tcl rb">Vinegar.</td></tr> +<tr><td class="tcl lb rb"> 1 ”</td> <td class="tcl rb2">Fish, dried.</td> <td class="tcl rb"> 8 oz.</td> <td class="tcl rb">Corn Meal.</td></tr> +<tr><td class="tcl lb rb"> 7 ”</td> <td class="tcl rb2">Potatoes or yams.</td> <td class="tcl rb">12 ”</td> <td class="tcl rb">Onions.</td></tr> +<tr><td class="tcl lb rb"> 1 ”</td> <td class="tcl rb2">Tomatoes, preserved.</td> <td class="tcl rb"> 7 ”</td> <td class="tcl rb">Lard.</td></tr> +<tr><td class="tcl lb rb"> <span class="spp">2</span>⁄<span class="suu">3</span> ”</td> <td class="tcl rb2">Peas.</td> <td class="tcl rb"> 7 ”</td> <td class="tcl rb">Butter.</td></tr> +<tr><td class="tcl lb rb"> <span class="spp">2</span>⁄<span class="suu">3</span> ”</td> <td class="tcl rb2">Calavances.</td> <td class="tcl rb"> ¼ ”</td> <td class="tcl rb">Mustard.</td></tr> +<tr><td class="tcl lb rb"> <span class="spp">2</span>⁄<span class="suu">3</span> ”</td> <td class="tcl rb2">Rice.</td> <td class="tcl rb"> ¼ ”</td> <td class="tcl rb">Pepper.</td></tr> +<tr><td class="tcl lb rb bb"> 5¼ oz.</td> <td class="tcl rb2 bb">Coffee, green.</td> <td class="tcl rb bb"> ¼ ”</td> <td class="tcl rb bb">Salt.</td></tr> +</table> + +<p>In the British mercantile marine there is no scale of provisions +prescribed by the Board of Trade; there is, however, a traditional +scale very generally adopted, having the sanction of custom only +and seldom adhered to. The following dietary scale for steerage +passengers, laid down in the 12th schedule of the Merchant Shipping +Act 1894, is of interest. See Table IV.</p> + +<p class="center pt2"><span class="sc">Table IV.</span>—<i>Weekly, per Statute Adult.</i></p> + +<table class="ws" summary="Contents"> +<tr><td class="tccm allb"> </td> + +<td class="tcl allb">  Scale A.<br />For voyages not<br /> exceeding 84 days<br /> for sailing ships<br /> + or 50 days<br /> for steamships.<br /></td> + +<td class="tcl allb">  Scale B.<br />For voyages<br /> exceeding 84 days<br /> for sailing ships<br /> + or 50 days<br /> for steamships.<br /></td></tr> + +<tr><td class="tcl lb rb"> </td> <td class="tcc rb">℔ oz.</td> <td class="tcc rb">℔ oz.</td></tr> +<tr><td class="tcl lb rb">Bread or biscuit, not inferior to navy biscuit</td> <td class="tcc rb">3 8</td> <td class="tcc rb">3 8</td></tr> +<tr><td class="tcl lb rb">Wheaten flour</td> <td class="tcc rb">1 0</td> <td class="tcc rb">2 0</td></tr> +<tr><td class="tcl lb rb">Oatmeal</td> <td class="tcc rb">1 8</td> <td class="tcc rb">1 0</td></tr> +<tr><td class="tcl lb rb">Rice</td> <td class="tcc rb">1 8</td> <td class="tcc rb">0 8</td></tr> +<tr><td class="tcl lb rb">Peas</td> <td class="tcc rb">1 8</td> <td class="tcc rb">1 8</td></tr> +<tr><td class="tcl lb rb">Beef</td> <td class="tcc rb">1 4</td> <td class="tcc rb">1 4</td></tr> +<tr><td class="tcl lb rb">Pork</td> <td class="tcc rb">1 0</td> <td class="tcc rb">1 0</td></tr> +<tr><td class="tcl lb rb">Butter</td> <td class="tcc rb">· ·</td> <td class="tcc rb">0 4</td></tr> +<tr><td class="tcl lb rb">Potatoes</td> <td class="tcc rb">2 0</td> <td class="tcc rb">2 0</td></tr> +<tr><td class="tcl lb rb">Sugar</td> <td class="tcc rb">1 0</td> <td class="tcc rb">1 0</td></tr> +<tr><td class="tcl lb rb">Tea</td> <td class="tcc rb">0 2</td> <td class="tcc rb">0 2</td></tr> +<tr><td class="tcl lb rb">Salt</td> <td class="tcc rb">0 2</td> <td class="tcc rb">0 2</td></tr> +<tr><td class="tcl lb rb">Pepper (white or black), ground</td> <td class="tcc rb">0 0½</td> <td class="tcc rb">0 0½</td></tr> +<tr><td class="tcl lb rb">Vinegar</td> <td class="tcc rb">1 gill</td> <td class="tcc rb">1 gill</td></tr> +<tr><td class="tcl lb rb">Preserved meat</td> <td class="tcc rb">· ·</td> <td class="tcc rb">1 0</td></tr> +<tr><td class="tcl lb rb">Suet</td> <td class="tcc rb"> </td> <td class="tcc rb">0 6</td></tr> +<tr><td class="tcl lb rb">Raisins</td> <td class="tcc rb"> </td> <td class="tcc rb">0 8</td></tr> +<tr><td class="tcl lb rb bb">Lime juice</td> <td class="tcc rb bb"> </td> <td class="tcc rb bb">0 6</td></tr> +</table> + +<p>Certain substitutions may be made in this scale at the option +of the master of any emigrant ship, provided that the substituted +articles are set forth in the contract tickets of the steerage passengers.</p> + +<p>In the British army the soldier is fed partly by a system of co-operation. +He gets a free ration from government of 1 ℔ of bread and +¾ ℔ of meat; in addition there is a messing allowance of 3½d. per +man per day. He is able to supplement his food by purchases from +the canteen. Much depends on the individual management in each +regiment as to the satisfactory expenditure of the messing allowance. +In some regiments an allowance is made from the canteen funds +towards messing in addition to that granted by the government. +The ordinary <i>field</i> ration of the British soldier is 1½ ℔ of bread or +1 ℔ of biscuit; 1 ℔ of fresh, salt or preserved meat; ½ oz. of coffee; +1/6 oz. of tea; 2 oz. of sugar; ½ oz. of salt, <span class="spp">1</span>⁄<span class="suu">36</span> oz. of pepper, the +whole weighing something over 2 ℔ 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 ℔ 1 oz. of rye bread +and not quite 2½ ℔ of meat. This is obviously insufficient, but under</p> + +<p>the conscription system it is reckoned that he will be able to make +up the deficiency out of his own private means, or obtain charitable +contributions from his friends. In the French infantry of the line +each man during peace gets weekly 15 ℔ of bread, 3<span class="spp">3</span>⁄<span class="suu">10</span> ℔ of meat, +2½ ℔ of haricot beans or other vegetables, with salt and pepper, and +1¾ oz. of brandy.</p> + +<p>An Austrian under the same circumstances receives 13.9 ℔ of +bread, ½ ℔ of flour and 3.3 ℔ of meat.</p> + +<p>The Russian conscript is allowed weekly:—</p> + +<table class="nobctr" style="width: 60%;" summary="Contents"> +<tr><td class="tcl">Black bread</td> <td class="tcl">7 ℔.</td></tr> +<tr><td class="tcl">Meat</td> <td class="tcl">7 ℔.</td></tr> +<tr><td class="tcl">Kvass (beer)</td> <td class="tcl">7.7 quarts.</td></tr> +<tr><td class="tcl">Sour cabbage</td> <td class="tcl">24½ gills = 122½ oz.</td></tr> +<tr><td class="tcl">Barley</td> <td class="tcl">24½ gills = 122½ oz.</td></tr> +<tr><td class="tcl">Salts</td> <td class="tcl">10½ oz.</td></tr> +<tr><td class="tcl">Horse-radish</td> <td class="tcl">28 grains.</td></tr> +<tr><td class="tcl">Pepper</td> <td class="tcl">28 grains.</td></tr> +<tr><td class="tcl">Vinegar</td> <td class="tcl">5½ gills = 26½ oz.</td></tr> +</table> +</div> + + +<hr class="art" /> +<p><span class="bold">DIETETICS,<a name="ar76" id="ar76"></a></span> the science of diet, <i>i.e.</i> the food and nutrition of +man in health and disease (see <span class="sc"><a href="#artlinks">Nutrition</a></span>). This article deals +mainly with that part of the subject which has to do with the +composition and nutritive values of foods and their adaptation +to the use of people in health. The principal topics considered +are: (1) Food and its functions; (2) Metabolism of matter and +energy; (3) Composition of food materials; (4) Digestibility of +food; (5) Fuel value of food; (6) Food consumption; (7) Quantities +of nutrients needed; (8) Hygienic economy of food; (9) +Pecuniary economy of food.</p> + +<p>1. <i>Food and its Functions.</i>—For practical purposes, food may be +defined as that which, when taken into the body, may be utilized +for the formation and repair of body tissue, and the production +of energy. More specifically, food meets the requirements of the +body in several ways. It is used for the formation of the tissues +and fluids of the body, and for the restoration of losses of substance +due to bodily activity. The potential energy of the food +is converted into heat or muscular work or other forms of energy. +In being thus utilized, food protects body substance or previously +acquired nutritive material from consumption. When the amount +<span class="pagenum"><a name="page215" id="page215"></a>215</span> +of food taken into the body is in excess of immediate needs, the +surplus may be stored for future consumption.</p> + +<p>Ordinary food materials, such as meat, fish, eggs, vegetables, +&c., consist of inedible materials, or <i>refuse</i>, <i>e.g.</i> bone of meat +and fish, shell of eggs, rind and seed of vegetables; and <i>edible +material</i>, as flesh of meat and fish, white and yolk of eggs, wheat +flour, &c. The edible material is by no means a simple substance, +but consists of <i>water</i>, and some or all of the compounds +variously designated as food stuffs, proximate principles, nutritive +ingredients or nutrients, which are classified as <i>protein</i>, <i>fats</i>, +<i>carbohydrates</i> and <i>mineral matters</i>. These have various functions +in the nourishment of the body.</p> + +<p>The <i>refuse</i> commonly contains compounds similar to those +in the food from which it is derived, but since it cannot be eaten, +it is usually considered as a non-nutrient. It is of importance +chiefly in a consideration of the pecuniary economy of food. +<i>Water</i> is also considered as a non-nutrient, because although it is a +constituent of all the tissues and fluids of the body, the body may +obtain the water it needs from that drunk; hence, that contained +in the food materials is of no special significance as a nutrient.</p> + +<p><i>Mineral matters</i>, such as sulphates, chlorides, phosphates and +carbonates of sodium, potassium, calcium, &c., are found in +different combinations and quantities in most food materials. +These are used by the body in the formation of the various +tissues, especially the skeletal and protective tissues, in digestion, +and in metabolic processes within the body. They yield little +or no energy, unless perhaps the very small amount involved in +their chemical transformation.</p> + +<p>Protein<a name="fa1f" id="fa1f" href="#ft1f"><span class="sp">1</span></a> is a term used to designate the whole group of +nitrogenous compounds of food except the nitrogenous fats. It +includes the albuminoids, as albumin of egg-white, and of blood +serum, myosin of meat (muscle), casein of milk, globulin of blood +and of egg yolk, fibrin of blood, gluten of flour; the gelatinoids, +as gelatin and allied substances of connective tissue, collagen of +tendon, ossein of bone and the so-called extractives (<i>e.g.</i> creatin) +of meats; and the amids (<i>e.g.</i> asparagin) and allied compounds of +vegetables and fruits.</p> + +<p>The albuminoids and gelatinoids, classed together as proteids, +are the most important constituents of food, because they alone +can supply the nitrogenous material necessary for the formation +of the body tissues. For this purpose, the albuminoids are most +valuable. Both groups of compounds, however, supply the body +with energy, and the gelatinoids in being thus utilized protect +the albuminoids from consumption for this purpose. When their +supply in the food is in excess of the needs of the body, the surplus +proteids may be converted into body fat and stored.</p> + +<p>The so-called extractives, which are the principal constituents +of meat extract, beef tea and the like, act principally as stimulants +and appetizers. It has been believed that they serve neither +to build tissue nor to yield energy, but recent investigations<a name="fa2f" id="fa2f" href="#ft2f"><span class="sp">2</span></a> +indicate that creatin may be metabolized in the body.</p> + +<p>The <i>fats</i> of food include both the animal fats and the vegetable +oils. The <i>carbohydrates</i> include such compounds as starches, +sugars and the fibre of plants or cellulose, though the latter has +but little value as food for man. The more important function +of both these classes of nutrients is to supply energy to the body +to meet its requirements above that which it may obtain from the +proteids. It is not improbable that the atoms of their molecules +as well as those from the proteids are built up into the protoplasmic +substance of the tissues. In this sense, these nutrients +may be considered as being utilized also for the formation of +tissue; but they are rather the accessory ingredients, whereas the +proteids are the essential ingredients for this purpose. The fats +in the food in excess of the body requirements may be stored as +body fat, and the surplus carbohydrates may also be converted +into fat and stored.</p> + +<p>To a certain extent, then, the nutrients of the food may +substitute each other. All may be incorporated into the protoplasmic +structure of body tissue, though only the proteids can +supply the essential nitrogenous ingredients; and apart from +the portion of the proteid material that is indispensable for this +purpose, all the nutrients are used as a source of energy. If the +supply of energy in the food is not sufficient, the body will use +its own proteid and fat for this purpose. The gelatinoids, fats +and carbohydrates in being utilized for energy protect the body +proteids from consumption. The fat stored in the body from the +excess of food is a reserve of energy material, on which the body +may draw when the quantity of energy in the food is insufficient +for its immediate needs.</p> + +<p>What compounds are especially concerned in intellectual +activity is not known. The belief that fish is especially rich in +phosphorus and valuable as a brain food has no foundation in +observed fact.</p> + +<p>2. <i>Metabolism of Matter and Energy.</i>—The processes of nutrition +thus consist largely of the transformation of food into body +material and the conversion of the potential energy of both food +and body material into the kinetic energy of heat and muscular +work and other forms of energy. These various processes are +generally designated by the term metabolism. The metabolism +of matter in the body is governed largely by the needs of the body +for energy. The science of nutrition, of which the present subject +forms a part, is based on the principle that the transformations +of matter and energy in the body occur in accordance with the +laws of the conservation of matter and of energy. That the body +can neither create nor destroy matter has long been universally +accepted. It would seem that the transformation of energy must +likewise be governed by the law of the conservation of energy; +indeed there is every reason a priori to believe that it must; but +the experimental difficulties in the way of absolute demonstration +of the principle are considerable. For such demonstration it is +necessary to prove that the income and expenditure of energy +are equal. Apparatus and methods of inquiry devised in recent +years, however, afford means for a comparison of the amounts of +both matter and energy received and expended by the body, and +from the results obtained in a large amount of such research, +it seems probable that the law obtains in the living organism in +general.</p> + +<p>The first attempt at such demonstration was made by +M. Rubner<a name="fa3f" id="fa3f" href="#ft3f"><span class="sp">3</span></a> in 1894, experimenting with dogs doing no external +muscular work. The income of energy (as heat) was computed, +but the heat eliminated was measured. In the average of eight +experiments continuing forty-five days, the two quantities agreed +within 0.47%, thus demonstrating what it was desired to prove—that +the heat given off by the body came solely from the +oxidation of food within it. Results in accordance with these +were reported by Studenski<a name="fa4f" id="fa4f" href="#ft4f"><span class="sp">4</span></a> in 1897, and by Laulanie<a name="fa5f" id="fa5f" href="#ft5f"><span class="sp">5</span></a> in 1898.</p> + +<p>The most extensive and complete data yet available on the +subject have been obtained by W. O. Atwater, F. G. Benedict and +associates<a name="fa6f" id="fa6f" href="#ft6f"><span class="sp">6</span></a> in experiments with men in the respiration calorimeter, +in which a subject may remain for several consecutive days +and nights. These experiments involve actual weighing and +analyses of the food and drink, and of the gaseous, liquid and +solid excretory products; determinations of potential energy +(heat of oxidation) of the oxidizable material received and given +off by the body (including estimation of the energy of the material +gained or lost by the body); and measurements of the amounts of +energy expended as heat and as external muscular work. By +October 1906 eighty-eight experiments with fifteen different subjects +had been completed. The separate experiments continued +from two to thirteen days, making a total of over 270 days. +<span class="pagenum"><a name="page216" id="page216"></a>216</span> +In some cases the subjects were at rest; in others they performed +varying amounts of external muscular work on an +apparatus by means of which the amount of work done was +measured. In some cases they fasted, and in others they received +diets generally not far from sufficient to maintain nitrogen, and +usually carbon, equilibrium in the body. In these experiments +the amount of energy expended by the body as heat and as +external muscular work measured in terms of heat agreed on +the average very closely with the amount of heat that would be +produced by the oxidation of all the matter metabolized in the +body. The variations for individual days, and in the average for +individual experiments as well, were in some cases appreciable, +amounting to as much as 6%, which is not strange in view of the +uncertainties in physiological experimenting; but in the average +of all the experiments the energy of the expenditure was above +99.9% of the energy of the income,—an agreement within one +part in 1000. While these results do not absolutely prove the +application of the law of the conservation of energy in the human +body, they certainly approximate very closely to such demonstration. +It is of course possible that energy may have given off +<span class="pagenum"><a name="page217" id="page217"></a>217</span> +from the body in other forms than heat and external muscular +work. It is conceivable, for example, that intellectual activity +may involve the transformation of physical energy, and that the +energy involved may be eliminated in some form now unknown. +But if the body did give off energy which was not measured in +these experiments, the quantity must have been extremely small. +It seems fair to infer from the results obtained that the metabolism +of energy in the body occurred in conformity with the law +of the conservation of energy.</p> + +<p>3. <i>Composition of Food Materials.</i>—The composition of food +is determined by chemical analyses, the results of which are +conventionally expressed in terms of the nutritive ingredients +previously described. As a result of an enormous amount of +such investigation in recent years, the kinds and proportions of +nutrients in our common sorts of food are well known. Average +values for percentage composition of some ordinary food materials +are shown in Table I. (Table I. also includes figures for fuel +value.)</p> + +<p class="center pt2"><span class="sc">Table I.</span>—<i>Percentage Composition of some Common Food Materials.</i></p> + +<table class="ws" summary="Contents"> + +<tr><td class="tccm allb">Food Material.</td> <td class="tccm allb">Refuse.</td> <td class="tccm allb">Water.</td> <td class="tccm allb">Protein.</td> <td class="tccm allb">Fat.</td> <td class="tccm allb">Carbo-<br />hydrates.</td> <td class="tccm allb">Mineral<br />Matter.</td> <td class="tccm allb">Fuel Value<br />per ℔</td></tr> + +<tr><td class="tcl lb rb"> </td> <td class="tcc rb">%</td> <td class="tcc rb">%</td> <td class="tcc rb">%</td> <td class="tcc rb">%</td> <td class="tcc rb">%</td> <td class="tcc rb">%</td> <td class="tcr rb">Calories.</td></tr> +<tr><td class="tcl lb rb">Beef, fresh (medium fat)—</td> <td class="tcr rb"> </td> <td class="tcr rb"> </td> <td class="tcr rb"> </td> <td class="tcr rb"> </td> <td class="tcr rb"> </td> <td class="tcr rb"> </td> <td class="tcr rb"> </td></tr> +<tr><td class="tcl lb rb"> Chuck</td> <td class="tcr rb">16.3</td> <td class="tcr rb">52.6</td> <td class="tcr rb">15.5</td> <td class="tcr rb">15.0</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.8</td> <td class="tcr rb">910</td></tr> +<tr><td class="tcl lb rb"> Loin</td> <td class="tcr rb">13.3</td> <td class="tcr rb">52.5</td> <td class="tcr rb">16.1</td> <td class="tcr rb">17.5</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.9</td> <td class="tcr rb">1025</td></tr> +<tr><td class="tcl lb rb"> Ribs</td> <td class="tcr rb">20.8</td> <td class="tcr rb">43.8</td> <td class="tcr rb">13.9</td> <td class="tcr rb">21.2</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.7</td> <td class="tcr rb">1135</td></tr> +<tr><td class="tcl lb rb"> Round</td> <td class="tcr rb">7.2</td> <td class="tcr rb">60.7</td> <td class="tcr rb">19.0</td> <td class="tcr rb">12.8</td> <td class="tcc rb">· ·</td> <td class="tcr rb">1.0</td> <td class="tcr rb">890</td></tr> +<tr><td class="tcl lb rb"> Shoulder</td> <td class="tcr rb">16.4</td> <td class="tcr rb">56.8</td> <td class="tcr rb">16.4</td> <td class="tcr rb">9.8</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.9</td> <td class="tcr rb">715</td></tr> +<tr><td class="tcl lb rb">Beef, dried and smoked</td> <td class="tcr rb">4.7</td> <td class="tcr rb">53.7</td> <td class="tcr rb">26.4</td> <td class="tcr rb">6.9</td> <td class="tcc rb">· ·</td> <td class="tcr rb">8.9</td> <td class="tcr rb">790</td></tr> +<tr><td class="tcl lb rb">Veal—</td> <td class="tcr rb"> </td> <td class="tcr rb"> </td> <td class="tcr rb"> </td> <td class="tcr rb"> </td> <td class="tcr rb"> </td> <td class="tcr rb"> </td> <td class="tcr rb"> </td></tr> +<tr><td class="tcl lb rb"> Leg</td> <td class="tcr rb">14.2</td> <td class="tcr rb">60.1</td> <td class="tcr rb">15.5</td> <td class="tcr rb">7.9</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.9</td> <td class="tcr rb">625</td></tr> +<tr><td class="tcl lb rb"> Loin</td> <td class="tcr rb">16.5</td> <td class="tcr rb">57.6</td> <td class="tcr rb">16.6</td> <td class="tcr rb">9.0</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.9</td> <td class="tcr rb">685</td></tr> +<tr><td class="tcl lb rb"> Breast</td> <td class="tcr rb">21.3</td> <td class="tcr rb">52.0</td> <td class="tcr rb">15.4</td> <td class="tcr rb">11.0</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.8</td> <td class="tcr rb">745</td></tr> +<tr><td class="tcl lb rb">Mutton—</td> <td class="tcr rb"> </td> <td class="tcr rb"> </td> <td class="tcr rb"> </td> <td class="tcr rb"> </td> <td class="tcr rb"> </td> <td class="tcr rb"> </td> <td class="tcr rb"> </td></tr> +<tr><td class="tcl lb rb"> Leg</td> <td class="tcr rb">18.4</td> <td class="tcr rb">51.2</td> <td class="tcr rb">15.1</td> <td class="tcr rb">14.7</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.8</td> <td class="tcr rb">890</td></tr> +<tr><td class="tcl lb rb"> Loin</td> <td class="tcr rb">16.0</td> <td class="tcr rb">42.0</td> <td class="tcr rb">13.5</td> <td class="tcr rb">28.3</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.7</td> <td class="tcr rb">1415</td></tr> +<tr><td class="tcl lb rb"> Flank</td> <td class="tcr rb">9.9</td> <td class="tcr rb">39.0</td> <td class="tcr rb">13.8</td> <td class="tcr rb">36.9</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.6</td> <td class="tcr rb">1770</td></tr> +<tr><td class="tcl lb rb">Pork—</td> <td class="tcr rb"> </td> <td class="tcr rb"> </td> <td class="tcr rb"> </td> <td class="tcr rb"> </td> <td class="tcr rb"> </td> <td class="tcr rb"> </td> <td class="tcr rb"> </td></tr> +<tr><td class="tcl lb rb"> Loin</td> <td class="tcr rb">19.7</td> <td class="tcr rb">41.8</td> <td class="tcr rb">13.4</td> <td class="tcr rb">24.2</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.8</td> <td class="tcr rb">1245</td></tr> +<tr><td class="tcl lb rb"> Ham, fresh</td> <td class="tcr rb">10.7</td> <td class="tcr rb">48.0</td> <td class="tcr rb">13.5</td> <td class="tcr rb">25.9</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.8</td> <td class="tcr rb">1320</td></tr> +<tr><td class="tcl lb rb"> Ham, smoked and salted</td> <td class="tcr rb">13.6</td> <td class="tcr rb">34.8</td> <td class="tcr rb">14.2</td> <td class="tcr rb">33.4</td> <td class="tcc rb">· ·</td> <td class="tcr rb">4.2</td> <td class="tcr rb">1635</td></tr> +<tr><td class="tcl lb rb"> Fat, salt</td> <td class="tcc rb">· ·</td> <td class="tcr rb">7.9</td> <td class="tcr rb">1.9</td> <td class="tcr rb">86.2</td> <td class="tcc rb">· ·</td> <td class="tcr rb">3.9</td> <td class="tcr rb">3555</td></tr> +<tr><td class="tcl lb rb"> Bacon</td> <td class="tcr rb">7.7</td> <td class="tcr rb">17.4</td> <td class="tcr rb">9.1</td> <td class="tcr rb">62.2</td> <td class="tcc rb">· ·</td> <td class="tcr rb">4.1</td> <td class="tcr rb">2715</td></tr> +<tr><td class="tcl lb rb"> Lard, refined</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcr rb">100.0</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcr rb">4100</td></tr> +<tr><td class="tcl lb rb">Chicken</td> <td class="tcr rb">25.9</td> <td class="tcr rb">47.1</td> <td class="tcr rb">13.7</td> <td class="tcr rb">12.3</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.7</td> <td class="tcr rb">765</td></tr> +<tr><td class="tcl lb rb">Turkey</td> <td class="tcr rb">22.7</td> <td class="tcr rb">42.4</td> <td class="tcr rb">16.1</td> <td class="tcr rb">18.4</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.8</td> <td class="tcr rb">1060</td></tr> +<tr><td class="tcl lb rb">Goose</td> <td class="tcr rb">17.6</td> <td class="tcr rb">38.5</td> <td class="tcr rb">13.4</td> <td class="tcr rb">29.8</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.7</td> <td class="tcr rb">1475</td></tr> +<tr><td class="tcl lb rb">Eggs</td> <td class="tcr rb">11.2</td> <td class="tcr rb">65.5</td> <td class="tcr rb">13.1</td> <td class="tcr rb">9.3</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.9</td> <td class="tcr rb">635</td></tr> +<tr><td class="tcl lb rb">Cod, fresh</td> <td class="tcr rb">29.9</td> <td class="tcr rb">58.5</td> <td class="tcr rb">11.1</td> <td class="tcr rb">0.2</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.8</td> <td class="tcr rb">220</td></tr> +<tr><td class="tcl lb rb">Cod, salted</td> <td class="tcr rb">24.9</td> <td class="tcr rb">40.2</td> <td class="tcr rb">16.0</td> <td class="tcr rb">0.4</td> <td class="tcc rb">· ·</td> <td class="tcr rb">18.5</td> <td class="tcr rb">325</td></tr> +<tr><td class="tcl lb rb">Mackerel, fresh</td> <td class="tcr rb">44.7</td> <td class="tcr rb">40.4</td> <td class="tcr rb">10.2</td> <td class="tcr rb">4.2</td> <td class="tcc rb">· ·</td> <td class="tcr rb">0.7</td> <td class="tcr rb">370</td></tr> +<tr><td class="tcl lb rb">Herring, smoked</td> <td class="tcr rb">44.4</td> <td class="tcr rb">19.2</td> <td class="tcr rb">20.5</td> <td class="tcr rb">8.8</td> <td class="tcc rb">· ·</td> <td class="tcr rb">7.4</td> <td class="tcr rb">755</td></tr> +<tr><td class="tcl lb rb">Salmon, tinned</td> <td class="tcc rb">· ·</td> <td class="tcr rb">63.5</td> <td class="tcr rb">21.8</td> <td class="tcr rb">12.1</td> <td class="tcc rb">· ·</td> <td class="tcr rb">2.6</td> <td class="tcr rb">915 </td></tr> +<tr><td class="tcl lb rb">Oysters, shelled</td> <td class="tcc rb">· ·</td> <td class="tcr rb">88.3</td> <td class="tcr rb">6.0</td> <td class="tcr rb">1.3</td> <td class="tcr rb">3.3</td> <td class="tcr rb">1.1</td> <td class="tcr rb">225</td></tr> +<tr><td class="tcl lb rb">Butter</td> <td class="tcc rb">· ·</td> <td class="tcr rb">11.0</td> <td class="tcr rb">1.0</td> <td class="tcr rb">85.0</td> <td class="tcc rb">· ·</td> <td class="tcr rb">3.0</td> <td class="tcr rb">3410</td></tr> +<tr><td class="tcl lb rb">Cheese</td> <td class="tcc rb">· ·</td> <td class="tcr rb">34.2</td> <td class="tcr rb">25.9</td> <td class="tcr rb">33.7</td> <td class="tcr rb">2.4</td> <td class="tcr rb">3.8</td> <td class="tcr rb">1885</td></tr> +<tr><td class="tcl lb rb">Milk, whole</td> <td class="tcc rb">· ·</td> <td class="tcr rb">87.0</td> <td class="tcr rb">3.3</td> <td class="tcr rb">4.0</td> <td class="tcr rb">5.0</td> <td class="tcr rb">0.7</td> <td class="tcr rb">310</td></tr> +<tr><td class="tcl lb rb">Milk, skimmed</td> <td class="tcc rb">· ·</td> <td class="tcr rb">90.5</td> <td class="tcr rb">3.4</td> <td class="tcr rb">0.3</td> <td class="tcr rb">5.1</td> <td class="tcr rb">0.7</td> <td class="tcr rb">165</td></tr> +<tr><td class="tcl lb rb">Oatmeal</td> <td class="tcc rb">· ·</td> <td class="tcr rb">7.7</td> <td class="tcr rb">16.7</td> <td class="tcr rb">7.3</td> <td class="tcr rb">66.2</td> <td class="tcr rb">2.1</td> <td class="tcr rb">1800</td></tr> +<tr><td class="tcl lb rb">Corn (maize) meal</td> <td class="tcc rb">· ·</td> <td class="tcr rb">12.5</td> <td class="tcr rb">9.2</td> <td class="tcr rb">1.9</td> <td class="tcr rb">75.4</td> <td class="tcr rb">1.0</td> <td class="tcr rb">1635</td></tr> +<tr><td class="tcl lb rb">Rye flour</td> <td class="tcc rb">· ·</td> <td class="tcr rb">12.9</td> <td class="tcr rb">6.8</td> <td class="tcr rb">0.9</td> <td class="tcr rb">78.7</td> <td class="tcr rb">0.7</td> <td class="tcr rb">1620</td></tr> +<tr><td class="tcl lb rb">Buckwheat flour</td> <td class="tcc rb">· ·</td> <td class="tcr rb">13.6</td> <td class="tcr rb">6.4</td> <td class="tcr rb">1.2</td> <td class="tcr rb">77.9</td> <td class="tcr rb">0.9</td> <td class="tcr rb">1605</td></tr> +<tr><td class="tcl lb rb">Rice</td> <td class="tcc rb">· ·</td> <td class="tcr rb">12.3</td> <td class="tcr rb">8.0</td> <td class="tcr rb">0.3</td> <td class="tcr rb">79.0</td> <td class="tcr rb">0.4</td> <td class="tcr rb">1620</td></tr> +<tr><td class="tcl lb rb">Wheat flour, white</td> <td class="tcc rb">· ·</td> <td class="tcr rb">12.0</td> <td class="tcr rb">11.4</td> <td class="tcr rb">1.0</td> <td class="tcr rb">75.1</td> <td class="tcr rb">0.5</td> <td class="tcr rb">1635</td></tr> +<tr><td class="tcl lb rb">Wheat flour, graham</td> <td class="tcc rb">· ·</td> <td class="tcr rb">11.3</td> <td class="tcr rb">13.3</td> <td class="tcr rb">2.2</td> <td class="tcr rb">71.4</td> <td class="tcr rb">1.8</td> <td class="tcr rb">1645</td></tr> +<tr><td class="tcl lb rb">Wheat, breakfast food</td> <td class="tcc rb">· ·</td> <td class="tcr rb">9.6</td> <td class="tcr rb">12.1</td> <td class="tcr rb">1.8</td> <td class="tcr rb">75.2</td> <td class="tcr rb">1.3</td> <td class="tcr rb">1680</td></tr> +<tr><td class="tcl lb rb">Wheat bread, white</td> <td class="tcc rb">· ·</td> <td class="tcr rb">35.3</td> <td class="tcr rb">9.2</td> <td class="tcr rb">1.3</td> <td class="tcr rb">53.1</td> <td class="tcr rb">1.1</td> <td class="tcr rb">1200</td></tr> +<tr><td class="tcl lb rb">Wheat bread, graham</td> <td class="tcc rb">· ·</td> <td class="tcr rb">35.7</td> <td class="tcr rb">8.9</td> <td class="tcr rb">1.8</td> <td class="tcr rb">52.1</td> <td class="tcr rb">1.5</td> <td class="tcr rb">1195</td></tr> +<tr><td class="tcl lb rb">Rye bread</td> <td class="tcc rb">· ·</td> <td class="tcr rb">35.7</td> <td class="tcr rb">9.0</td> <td class="tcr rb">0.6</td> <td class="tcr rb">53.2</td> <td class="tcr rb">1.5</td> <td class="tcr rb">1170</td></tr> +<tr><td class="tcl lb rb">Biscuit (crackers)</td> <td class="tcc rb">· ·</td> <td class="tcr rb">6.8</td> <td class="tcr rb">9.7</td> <td class="tcr rb">12.1</td> <td class="tcr rb">69.7</td> <td class="tcr rb">1.7</td> <td class="tcr rb">1925</td></tr> +<tr><td class="tcl lb rb">Macaroni</td> <td class="tcc rb">· ·</td> <td class="tcr rb">10.3</td> <td class="tcr rb">13.4</td> <td class="tcr rb">0.9</td> <td class="tcr rb">74.1</td> <td class="tcr rb">1.3</td> <td class="tcr rb">1645</td></tr> +<tr><td class="tcl lb rb">Sugar</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcr rb">100.0</td> <td class="tcc rb">· ·</td> <td class="tcr rb">1750</td></tr> +<tr><td class="tcl lb rb">Starch (corn starch)</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcr rb">90.0</td> <td class="tcc rb">· ·</td> <td class="tcr rb">1680</td></tr> +<tr><td class="tcl lb rb">Beans, dried</td> <td class="tcc rb">· ·</td> <td class="tcr rb">12.6</td> <td class="tcr rb">22.5</td> <td class="tcr rb">1.8</td> <td class="tcr rb">59.6</td> <td class="tcr rb">3.5</td> <td class="tcr rb">1520</td></tr> +<tr><td class="tcl lb rb">Peas, dried</td> <td class="tcc rb">· ·</td> <td class="tcr rb">9.5</td> <td class="tcr rb">24.6</td> <td class="tcr rb">1.0</td> <td class="tcr rb">62.0</td> <td class="tcr rb">2.9</td> <td class="tcr rb">1565</td></tr> +<tr><td class="tcl lb rb">Beets</td> <td class="tcr rb">20.0</td> <td class="tcr rb">70.0</td> <td class="tcr rb">1.3</td> <td class="tcr rb">0.1</td> <td class="tcr rb">7.7</td> <td class="tcr rb">0.9</td> <td class="tcr rb">160</td></tr> +<tr><td class="tcl lb rb">Cabbage</td> <td class="tcr rb">50.0</td> <td class="tcr rb">4.2</td> <td class="tcr rb">0.7</td> <td class="tcr rb">0.2</td> <td class="tcr rb">4.5</td> <td class="tcr rb">0.4</td> <td class="tcr rb">100</td></tr> +<tr><td class="tcl lb rb">Potatoes</td> <td class="tcr rb">20.0</td> <td class="tcr rb">62.6</td> <td class="tcr rb">1.8</td> <td class="tcr rb">0.1</td> <td class="tcr rb">14.7</td> <td class="tcr rb">0.8</td> <td class="tcr rb">295</td></tr> +<tr><td class="tcl lb rb">Sweet potatoes</td> <td class="tcr rb">20.0</td> <td class="tcr rb">55.2</td> <td class="tcr rb">1.4</td> <td class="tcr rb">0.6</td> <td class="tcr rb">21.9</td> <td class="tcr rb">0.9</td> <td class="tcr rb">440</td></tr> +<tr><td class="tcl lb rb">Tomatoes</td> <td class="tcc rb">· ·</td> <td class="tcr rb">94.3</td> <td class="tcr rb">0.9</td> <td class="tcr rb">0.4</td> <td class="tcr rb">3.9</td> <td class="tcr rb">0.5</td> <td class="tcr rb">100</td></tr> +<tr><td class="tcl lb rb">Apples</td> <td class="tcr rb">25.0</td> <td class="tcr rb">63.3</td> <td class="tcr rb">0.3</td> <td class="tcr rb">0.3</td> <td class="tcr rb">10.8</td> <td class="tcr rb">0.3</td> <td class="tcr rb">190</td></tr> +<tr><td class="tcl lb rb">Bananas</td> <td class="tcr rb">35.0</td> <td class="tcr rb">48.9</td> <td class="tcr rb">0.8</td> <td class="tcr rb">0.4</td> <td class="tcr rb">14.3</td> <td class="tcr rb">0.6</td> <td class="tcr rb">260</td></tr> +<tr><td class="tcl lb rb">Grapes</td> <td class="tcr rb">25.0</td> <td class="tcr rb">58.0</td> <td class="tcr rb">1.0</td> <td class="tcr rb">1.2</td> <td class="tcr rb">14.4</td> <td class="tcr rb">0.4</td> <td class="tcr rb">295</td></tr> +<tr><td class="tcl lb rb">Strawberries</td> <td class="tcr rb">5.0</td> <td class="tcr rb">85.9</td> <td class="tcr rb">0.9</td> <td class="tcr rb">0.6</td> <td class="tcr rb">7.0</td> <td class="tcr rb">0.6</td> <td class="tcr rb">150</td></tr> +<tr><td class="tcl lb rb">Almonds</td> <td class="tcr rb">45.0</td> <td class="tcr rb">2.7</td> <td class="tcr rb">11.5</td> <td class="tcr rb">30.2</td> <td class="tcr rb">9.5</td> <td class="tcr rb">1.1</td> <td class="tcr rb">1515</td></tr> +<tr><td class="tcl lb rb">Brazil nuts</td> <td class="tcr rb">49.6</td> <td class="tcr rb">2.6</td> <td class="tcr rb">8.6</td> <td class="tcr rb">33.7</td> <td class="tcr rb">3.5</td> <td class="tcr rb">2.0</td> <td class="tcr rb">1485</td></tr> +<tr><td class="tcl lb rb">Chestnuts</td> <td class="tcr rb">16.0</td> <td class="tcr rb">37.8</td> <td class="tcr rb">5.2</td> <td class="tcr rb">4.5</td> <td class="tcr rb">35.4</td> <td class="tcr rb">1.1</td> <td class="tcr rb">915</td></tr> +<tr><td class="tcl lb rb bb">Walnuts</td> <td class="tcr rb bb">58.1</td> <td class="tcr rb bb">1.0</td> <td class="tcr rb bb">6.9</td> <td class="tcr rb bb">26.6</td> <td class="tcr rb bb">6.8</td> <td class="tcr rb bb">0.6</td> <td class="tcr rb bb">1250</td></tr> +</table> + +<p>It will be observed that different kinds of food materials vary +widely in their proportions of nutrients. In general the animal +foods contain the most protein and fats, and vegetable foods are +rich in carbohydrates. The chief nutrient of lean meat and fish is +protein; but in medium fat meats the proportion of fat is as large +as that of protein, and in the fatter meats it is larger. Cheese +is rich in both protein and fat. Among the vegetable foods, dried +beans and peas are especially rich in protein. The proportion in +oatmeal is also fairly large, in wheat it is moderate, and in maize +meal and rice it is rather small. Oats contain more oil than any +of the common cereals, but in none of them is the proportion +especially large. The most abundant nutrient in all the cereals is +starch, which comprises from two-thirds to three-fourths or more +of their total nutritive substance. Cotton-seed is rich in edible +oil, and so are olives. Some of the nuts contain fairly large +proportions of both protein and fat. The nutrient of potatoes is +starch, present in fair proportion. Fruits contain considerable +carbohydrates, chiefly sugar. Green vegetables are not of much +account as sources of any of the nutrients or energy.</p> + +<p>Similar food materials from different sources may also differ +considerably in composition. This is especially true of meats. +Thus, the leaner portions from a fat animal may contain nearly as +much fat as the fatter portions from a lean animal. The data +here presented are largely those for American food products, +but the available analyses of English food materials indicate +that the latter differ but little from the former in composition. +The analyses of meats produced in Europe imply that they +commonly contain somewhat less fat and more water, and +often more protein, than American meats. The meats of English +production compare with the American more than with the +European meats. Similar vegetable foods from the different +countries do not differ so much in composition.</p> + +<p>4. <i>Digestibility or Availability of Food Materials.</i>—The value +of any food material for nutriment depends not merely upon the +kinds and amounts of nutrients it contains, but also upon the +ease and convenience with which the nutrients may be digested, +and especially upon the proportion of the nutrients that will be +actually digested and absorbed. Thus, two foods may contain +equal amounts of the same nutrient, but the one most easily +digested will really be of most value to the body, because less +effort is necessary to utilize it. Considerable study of this factor +is being made, and much valuable information is accumulating, +but it is of more especial importance in cases of disordered +digestion.</p> + +<p>The digestibility of food in the sense of thoroughness of +digestion, however, is of particular importance in the present +discussion. Only that portion of the food that is digested +and absorbed is available to the body for the building of tissue +and the production of energy. Not all the food eaten is thus +actually digested; undigested material is excreted in the faeces. +The thoroughness of digestion is determined experimentally by +weighing and analysing the food eaten and the faeces pertaining +to it. The difference between the corresponding ingredients of +the two is commonly considered to represent the amounts of +the ingredients digested. Expressed in percentages, these are +called coefficients of digestibility. See Table II.</p> + +<p class="center pt2"><span class="sc">Table II.</span>—<i>Coefficients of Digestibility (or Availability) +of Nutrients in Different Classes of Food Materials.</i></p> + +<table class="ws" summary="Contents"> + +<tr><td class="tcc allb">Kind of Food.</td> <td class="tcc allb">Protein.</td> <td class="tcc allb">Fat.</td> <td class="tcc allb">Carbohydrates.</td></tr> + +<tr><td class="tcl lb rb"> </td> <td class="tcc rb">%</td> <td class="tcc rb">%</td> <td class="tcc rb">%</td></tr> +<tr><td class="tcl lb rb">Meats</td> <td class="tcc rb">98</td> <td class="tcc rb">98</td> <td class="tcc rb">· ·</td></tr> +<tr><td class="tcl lb rb">Fish</td> <td class="tcc rb">96</td> <td class="tcc rb">97</td> <td class="tcc rb">· ·</td></tr> +<tr><td class="tcl lb rb">Poultry</td> <td class="tcc rb">96</td> <td class="tcc rb">97</td> <td class="tcc rb">· ·</td></tr> +<tr><td class="tcl lb rb">Eggs</td> <td class="tcc rb">97</td> <td class="tcc rb">98</td> <td class="tcc rb">· ·</td></tr> +<tr><td class="tcl lb rb">Dairy products</td> <td class="tcc rb">97</td> <td class="tcc rb">96</td> <td class="tcc rb">98</td></tr> +<tr><td class="tcl lb rb">Total animal food of mixed diet</td> <td class="tcc rb">97</td> <td class="tcc rb">97</td> <td class="tcc rb">98</td></tr> +<tr><td class="tcl lb rb">Potatoes</td> <td class="tcc rb">73</td> <td class="tcc rb">· ·</td> <td class="tcc rb">98</td></tr> +<tr><td class="tcl lb rb">Beets, carrots, &c.</td> <td class="tcc rb">72</td> <td class="tcc rb">· ·</td> <td class="tcc rb">97</td></tr> +<tr><td class="tcl lb rb">Cabbage, lettuce, &c.</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">83</td></tr> +<tr><td class="tcl lb rb">Legumes</td> <td class="tcc rb">78</td> <td class="tcc rb">90</td> <td class="tcc rb">95</td></tr> +<tr><td class="tcl lb rb">Oatmeal</td> <td class="tcc rb">78</td> <td class="tcc rb">90</td> <td class="tcc rb">97</td></tr> +<tr><td class="tcl lb rb">Corn meal</td> <td class="tcc rb">80</td> <td class="tcc rb">· ·</td> <td class="tcc rb">99</td></tr> +<tr><td class="tcl lb rb">Wheat meals without bran</td> <td class="tcc rb">83</td> <td class="tcc rb">· ·</td> <td class="tcc rb">93</td></tr> +<tr><td class="tcl lb rb">Wheat meals with bran</td> <td class="tcc rb">75</td> <td class="tcc rb">· ·</td> <td class="tcc rb">92</td></tr> +<tr><td class="tcl lb rb">White bread</td> <td class="tcc rb">88</td> <td class="tcc rb">· ·</td> <td class="tcc rb">98</td></tr> +<tr><td class="tcl lb rb">Entire wheat bread</td> <td class="tcc rb">82</td> <td class="tcc rb">· ·</td> <td class="tcc rb">94</td></tr> +<tr><td class="tcl lb rb">Graham bread</td> <td class="tcc rb">76</td> <td class="tcc rb">· ·</td> <td class="tcc rb">90</td></tr> +<tr><td class="tcl lb rb">Rice</td> <td class="tcc rb">76</td> <td class="tcc rb">· ·</td> <td class="tcc rb">91</td></tr> +<tr><td class="tcl lb rb">Fruits and nuts</td> <td class="tcc rb">80</td> <td class="tcc rb">86</td> <td class="tcc rb">96</td></tr> +<tr><td class="tcl lb rb">Sugars and starches</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">98</td></tr> +<tr><td class="tcl lb rb">Total vegetable food of mixed diet</td> <td class="tcc rb">85</td> <td class="tcc rb">90</td> <td class="tcc rb">97</td></tr> +<tr><td class="tcl lb rb bb">Total food of mixed diet</td> <td class="tcc rb bb">92</td> <td class="tcc rb bb">95</td> <td class="tcc rb bb">97</td></tr> +</table> + +<p>Such a method is not strictly accurate, because the faeces do +not consist entirely of undigested food but contain in addition +to this the so-called metabolic products, which include the residuum +of digestive juices not resorbed, fragments of intestinal +epithelium, &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.</p> + +<p>Experimental inquiry of this nature has been very active in +recent years, especially in Europe, the United States and Japan; +and the results of considerably over 1000 digestion experiments +with single foods or combinations of food materials are available. +These were mostly with men, but some were with women +and with children. The larger part of these have been taken +into account in the following estimations of the digestibility +of the nutrients in different classes of food materials. The +figures here shown are subject to revision as experimental data +accumulate. They are not to be taken as exact measures of +the digestibility (or availability) of every kind of food in each +given class, but they probably represent fairly well the average +digestibility of the classes of food materials as ordinarily utilized +in the mixed diet.</p> + +<p>5. <i>Fuel Value of Food.</i>—The potential energy of food is +commonly measured as the amount of heat evolved when the +food is completely oxidized. In the laboratory this is determined +by burning the food in oxygen in a calorimeter. The results, +which are known as the heat of combustion of the food, are +<span class="pagenum"><a name="page218" id="page218"></a>218</span> +expressed in calories, one calory being the amount of heat +necessary to raise the temperature of one kilogram of water one +degree centigrade. But it is to be observed that this unit is +employed simply from convenience, and without implication +as to what extent the energy of food is converted into heat in +the body. The unit employed in the measurement of some other +form of energy might be used instead, as, for example, the foot-ton, +which represents the amount of energy necessary to raise +one ton through one foot.</p> + +<p class="center pt2"><span class="sc">Table III.</span>—<i>Estimates of Heats of Combustion and of Fuel Value +of Nutrients in Ordinary Mixed Diet.</i></p> + +<table class="ws" summary="Contents"> + +<tr><td class="tccm allb">Nutrients.</td> <td class="tccm allb">Heat of<br />Combustion.</td> <td class="tccm allb">Fuel Value.</td></tr> + +<tr><td class="tcl lb rb"> </td> <td class="tcc rb">Calories.</td> <td class="tcc rb">Calories.</td></tr> +<tr><td class="tcl lb rb">One gram of protein</td> <td class="tcc rb">5.65</td> <td class="tcc rb">4.05</td></tr> +<tr><td class="tcl lb rb">One gram of fats</td> <td class="tcc rb">9.40</td> <td class="tcc rb">8.93</td></tr> +<tr><td class="tcl lb rb bb">One gram of carbohydrates</td> <td class="tcc rb bb">4.15</td> <td class="tcc rb bb">4.03</td></tr> +</table> + +<p>The amount of energy which a given quantity of food will +produce on complete oxidation outside the body, however, is +greater than that which the body will actually derive from it. +In the first place, as previously shown, part of the food will not +be digested and absorbed. In the second place, the nitrogenous +compounds absorbed are not completely oxidized in the body, +the residuum being excreted in the urine as urea and other bodies +that are capable of further oxidation in the calorimeter. The +total heat of combustion of the food eaten must therefore be +diminished by the heat of combustion of the oxidizable material +rejected by the body, to find what amount of energy is actually +available to the organism for the production of work and heat. +The amount thus determined is commonly known as the fuel +value of food.</p> + +<p>Rubner’s<a name="fa7f" id="fa7f" href="#ft7f"><span class="sp">7</span></a> commonly quoted estimates for the fuel value of the +nutrients of mixed diet are,—for protein and carbohydrates 4.1, +and for fats 9.3 calories per gram. According to the method of +deduction, however, these factors were more applicable to digested +than to total nutrients. Atwater<a name="fa8f" id="fa8f" href="#ft8f"><span class="sp">8</span></a> and associates have deduced, +from data much more extensive than those available to Rubner, +factors for total nutrients somewhat lower than these, as shown +in Table III. These estimates seem to represent the best +average factors at present available, but are subject to revision +as knowledge is extended.</p> + +<p class="center pt2"><span class="sc">Table IV.</span>—<i>Quantities of Available Nutrients and Energy in Daily Food Consumption of Persons in +Different Circumstances.</i></p> + +<table class="ws" summary="Contents"> + +<tr><td class="tcc allb" rowspan="2"> </td> <td class="tccm allb" rowspan="2">Number of<br />Studies.</td> <td class="tccm allb" colspan="4">Nutrients and Energy per Man per Day.</td></tr> +<tr><td class="tccm allb">Protein.</td> <td class="tccm allb">Fat.</td> <td class="tccm allb">Carbo-<br />hydrates.</td> <td class="tccm allb">Fuel Value.</td></tr> + +<tr><td class="tcc lb rb pt1"><i>Persons with Active Work.</i></td> <td class="tcc rb"> </td> <td class="tcc rb">Grams.</td> <td class="tcc rb">Grams.</td> <td class="tcc rb">Grams.</td> <td class="tcc rb">Calories.</td></tr> +<tr><td class="tcl lb rb">English royal engineers</td> <td class="tcc rb"> 1</td> <td class="tcc rb">132</td> <td class="tcc rb"> 79</td> <td class="tcc rb">612</td> <td class="tcc rb">3835</td></tr> +<tr><td class="tcl lb rb">Prussian machinists</td> <td class="tcc rb"> 1</td> <td class="tcc rb">129</td> <td class="tcc rb">107</td> <td class="tcc rb">657</td> <td class="tcc rb">4265</td></tr> +<tr><td class="tcl lb rb">Swedish mechanics</td> <td class="tcc rb"> 5</td> <td class="tcc rb">174</td> <td class="tcc rb">105</td> <td class="tcc rb">693</td> <td class="tcc rb">4590</td></tr> +<tr><td class="tcl lb rb">Bavarian lumbermen</td> <td class="tcc rb"> 3</td> <td class="tcc rb">120</td> <td class="tcc rb">277</td> <td class="tcc rb">702</td> <td class="tcc rb">6015</td></tr> +<tr><td class="tcl lb rb">American lumbermen</td> <td class="tcc rb"> 5</td> <td class="tcc rb">155</td> <td class="tcc rb">327</td> <td class="tcc rb">804</td> <td class="tcc rb">6745</td></tr> +<tr><td class="tcl lb rb">Japanese rice cleaner</td> <td class="tcc rb"> 1</td> <td class="tcc rb">103</td> <td class="tcc rb"> 11</td> <td class="tcc rb">917</td> <td class="tcc rb">4415</td></tr> +<tr><td class="tcl lb rb">Japanese jinrikshaw runner</td> <td class="tcc rb"> 1</td> <td class="tcc rb">137</td> <td class="tcc rb"> 22</td> <td class="tcc rb">1010</td> <td class="tcc rb">5050</td></tr> +<tr><td class="tcl lb rb">Chinese farm labourers in California</td> <td class="tcc rb"> 1</td> <td class="tcc rb">132</td> <td class="tcc rb"> 90</td> <td class="tcc rb">621</td> <td class="tcc rb">3980</td></tr> +<tr><td class="tcl lb rb">American athletes</td> <td class="tcc rb">19</td> <td class="tcc rb">178</td> <td class="tcc rb">192</td> <td class="tcc rb">525</td> <td class="tcc rb">4740</td></tr> +<tr><td class="tcl lb rb">American working-men’s families</td> <td class="tcc rb">13</td> <td class="tcc rb">156</td> <td class="tcc rb">226</td> <td class="tcc rb">694</td> <td class="tcc rb">5650</td></tr> + +<tr><td class="tcc lb rb pt1"><i>Persons with Ordinary Work.</i></td> <td class="tcc rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb"> </td></tr> +<tr><td class="tcl lb rb">Bavarian mechanics</td> <td class="tcc rb">11</td> <td class="tcc rb">112</td> <td class="tcc rb"> 32</td> <td class="tcc rb">553</td> <td class="tcc rb">3060</td></tr> +<tr><td class="tcl lb rb">Bavarian farm labourers</td> <td class="tcc rb"> 5</td> <td class="tcc rb">126</td> <td class="tcc rb"> 52</td> <td class="tcc rb">526</td> <td class="tcc rb">3200</td></tr> +<tr><td class="tcl lb rb">Russian peasants</td> <td class="tcc rb">..</td> <td class="tcc rb">119</td> <td class="tcc rb"> 31</td> <td class="tcc rb">571</td> <td class="tcc rb">3155</td></tr> +<tr><td class="tcl lb rb">Prussian prisoners</td> <td class="tcc rb"> 1</td> <td class="tcc rb">117</td> <td class="tcc rb"> 28</td> <td class="tcc rb">620</td> <td class="tcc rb">3320</td></tr> +<tr><td class="tcl lb rb">Swedish mechanics</td> <td class="tcc rb"> 6</td> <td class="tcc rb">123</td> <td class="tcc rb"> 75</td> <td class="tcc rb">507</td> <td class="tcc rb">3325</td></tr> +<tr><td class="tcl lb rb">American working-men’s families</td> <td class="tcc rb">69</td> <td class="tcc rb">105</td> <td class="tcc rb">135</td> <td class="tcc rb">426</td> <td class="tcc rb">3480</td></tr> + +<tr><td class="tcc lb rb pt1"><i>Persons with Light Work.</i></td> <td class="tcc rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb"> </td></tr> +<tr><td class="tcl lb rb">American artisans’ families</td> <td class="tcc rb">21</td> <td class="tcc rb"> 93</td> <td class="tcc rb">107</td> <td class="tcc rb">358</td> <td class="tcc rb">2880</td></tr> +<tr><td class="tcl lb rb">English tailors (prisoners)</td> <td class="tcc rb"> 1</td> <td class="tcc rb">121</td> <td class="tcc rb"> 37</td> <td class="tcc rb">509</td> <td class="tcc rb">2970</td></tr> +<tr><td class="tcl lb rb">German shoemakers</td> <td class="tcc rb"> 1</td> <td class="tcc rb"> 99</td> <td class="tcc rb"> 73</td> <td class="tcc rb">367</td> <td class="tcc rb"> 2629</td></tr> +<tr><td class="tcl lb rb">Japanese prisoners</td> <td class="tcc rb"> 1</td> <td class="tcc rb"> 43</td> <td class="tcc rb"> 6</td> <td class="tcc rb">444</td> <td class="tcc rb">2110</td></tr> + +<tr><td class="tcc lb rb pt1"><i>Professional and Business Men.</i></td> <td class="tcc rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb"> </td></tr> +<tr><td class="tcl lb rb">Japanese professional men</td> <td class="tcc rb">13</td> <td class="tcc rb"> 75</td> <td class="tcc rb"> 15</td> <td class="tcc rb">408</td> <td class="tcc rb">2190</td></tr> +<tr><td class="tcl lb rb">Japanese students</td> <td class="tcc rb"> 8</td> <td class="tcc rb"> 85</td> <td class="tcc rb"> 18</td> <td class="tcc rb">537</td> <td class="tcc rb">2800</td></tr> +<tr><td class="tcl lb rb">Japanese military cadets</td> <td class="tcc rb">11</td> <td class="tcc rb"> 98</td> <td class="tcc rb"> 20</td> <td class="tcc rb">611</td> <td class="tcc rb">3185</td></tr> +<tr><td class="tcl lb rb">German physicians</td> <td class="tcc rb"> 2</td> <td class="tcc rb">121</td> <td class="tcc rb"> 90</td> <td class="tcc rb">317</td> <td class="tcc rb">2685</td></tr> +<tr><td class="tcl lb rb">Swedish medical students</td> <td class="tcc rb"> 5</td> <td class="tcc rb">117</td> <td class="tcc rb">108</td> <td class="tcc rb">291</td> <td class="tcc rb">2725</td></tr> +<tr><td class="tcl lb rb">Danish physicians</td> <td class="tcc rb"> 1</td> <td class="tcc rb">124</td> <td class="tcc rb">133</td> <td class="tcc rb">242</td> <td class="tcc rb">2790</td></tr> +<tr><td class="tcl lb rb">American professional and business men and students</td> <td class="tcc rb">51</td> <td class="tcc rb"> 98</td> <td class="tcc rb">125</td> <td class="tcc rb">411</td> <td class="tcc rb">3285</td></tr> + +<tr><td class="tcc lb rb pt1"><i>Persons with Little or no Exercise.</i></td> <td class="tcc rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb"> </td></tr> +<tr><td class="tcl lb rb">Prussian prisoners</td> <td class="tcc rb"> 2</td> <td class="tcc rb"> 90</td> <td class="tcc rb"> 27</td> <td class="tcc rb">427</td> <td class="tcc rb">2400</td></tr> +<tr><td class="tcl lb rb">Japanese prisoners</td> <td class="tcc rb"> 1</td> <td class="tcc rb"> 36</td> <td class="tcc rb"> 6</td> <td class="tcc rb">360</td> <td class="tcc rb">1725</td></tr> +<tr><td class="tcl lb rb">Inmates of home for aged—Germany</td> <td class="tcc rb"> 1</td> <td class="tcc rb"> 85</td> <td class="tcc rb"> 43</td> <td class="tcc rb">322</td> <td class="tcc rb">2097</td></tr> +<tr><td class="tcl lb rb">Inmates of hospitals for insane—America</td> <td class="tcc rb">49</td> <td class="tcc rb"> 80</td> <td class="tcc rb"> 86</td> <td class="tcc rb">353</td> <td class="tcc rb">2590</td></tr> + +<tr><td class="tcc lb rb pt1"><i>Persons in Destitute Circumstances.</i></td> <td class="tcc rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb"> </td></tr> +<tr><td class="tcl lb rb">Prussian working people</td> <td class="tcc rb">13</td> <td class="tcc rb"> 63</td> <td class="tcc rb"> 43</td> <td class="tcc rb">372</td> <td class="tcc rb">2215</td></tr> +<tr><td class="tcl lb rb">Italian mechanics</td> <td class="tcc rb"> 5</td> <td class="tcc rb"> 70</td> <td class="tcc rb"> 36</td> <td class="tcc rb">384</td> <td class="tcc rb">2225</td></tr> +<tr><td class="tcl lb rb bb">American working-men’s families</td> <td class="tcc rb bb">11</td> <td class="tcc rb bb"> 69</td> <td class="tcc rb bb"> 75</td> <td class="tcc rb bb">263</td> <td class="tcc rb bb">2085</td></tr> +</table> + +<p><span class="pagenum"><a name="page219" id="page219"></a>219</span></p> + +<p>The heats of combustion of all the fats in an ordinary mixed +diet would average about 9.40 calories per gram, but as only +95% of the fat would be available to the body, the fuel value +per gram would be (9.40 × 0.95 =) 8.93 calories. Similarly, the +average heat of combustion of carbohydrates of the diet would be +about 4.15 calories per gram, and as 97% of the total quantity +is available to the body, the fuel value per gram would be 4.03. +(It is commonly assumed that the resorbed fats and carbohydrates +are completely oxidized in the body.) The heats of +combustion of all the kinds of protein in the diet would average +about 5.65 calories per gram. Since about 92% of the total +protein would be available to the body, the potential energy of +the available protein would be equivalent to (5.65 × 0.92 =) 5.20 +calories; but as the available protein is not completely oxidized +allowance must be made for the potential energy of the incompletely +oxidized residue. This is estimated as equivalent to 1.15 +calories for the 0.92 gram of available protein; hence, the fuel +value of the total protein is (5.20 − 1.15 =) 4.05 calories per gram. +Nutrients of the same class, but from different food materials, +vary both in digestibility and in heat of combustion, and hence +in fuel value. These factors are therefore not so applicable to the +nutrients of the separate articles in a diet as to those of the diet as +a whole.</p> + +<p>6. <i>Food Consumption.</i>—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.</p> + +<p>In making allowance for these differences, the various individuals +are commonly compared with a man at moderately active +muscular work, who is taken as unity. A man at hard muscular +work is reckoned at 1.2 times such an individual; a man with +light muscular work or a boy 15-16 years old, .9; a man at +sedentary occupation, woman at moderately active muscular +work, boy 13-14 or girl 15-16 years old, .8; woman at light work, +boy 12 or girl 13-14 years old, .7; boy 10-11 or girl 10-12 +years old, .6; child 6-9 years old, .5; child 2-5 years old, .4; +child under 2 years, .3. These factors are by no means absolute +or final, but are based in part upon experimental data and in +part upon arbitrary assumption.</p> + +<p>The total number of dietary studies on record is very large, +but not all of them are complete enough to furnish reliable +data. Upwards of 1000 are sufficiently accurate to be included +in statistical averages of food consumed by people in different +circumstances, nearly half of which have been made in the United +States in the past decade. The number of persons in the individual +studies has ranged from one to several hundred. Some +typical results are shown in Table IV.</p> + +<p>7. <i>Quantities of Nutrients needed.</i>—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.</p> + +<p class="center pt2"><span class="sc">Table V.</span>—<i>Standards for Dietaries. Available Nutrients and +Energy per Man per Day.</i></p> + +<table class="ws" summary="Contents"> + +<tr><td class="tccm allb"> </td> <td class="tccm allb">Protein.</td> <td class="tccm allb">Fat.</td> <td class="tccm allb">Carbo-<br />hydrates.</td> <td class="tccm allb">Fuel<br />Value.</td></tr> + +<tr><td class="tcc lb rb pt1"><i>Voit’s Standards.</i></td> <td class="tcc rb">Grams.<a name="fa9f" id="fa9f" href="#ft9f"><span class="sp">9</span></a></td> <td class="tcc rb">Grams.</td> <td class="tcc rb">Grams.</td> <td class="tcc rb">Calories.</td></tr> +<tr><td class="tcl lb rb">Man at hard work</td> <td class="tcc rb">133</td> <td class="tcc rb">95</td> <td class="tcc rb">437</td> <td class="tcc rb">3270</td></tr> +<tr><td class="tcl lb rb">Man at moderate work</td> <td class="tcc rb">109</td> <td class="tcc rb">53</td> <td class="tcc rb">485</td> <td class="tcc rb">2965</td></tr> + +<tr><td class="tcc lb rb pt1"><i>Atwater’s Standards.</i></td> <td class="tcc rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb"> </td></tr> +<tr><td class="tcl lb rb">Man at very hard muscular work</td> <td class="tcc rb">161</td> <td class="tcc rb">· ·<a name="fa10f" id="fa10f" href="#ft10f"><span class="sp">10</span></a></td> <td class="tcc rb">· ·<a href="#ft10f"><span class="sp">10</span></a></td> <td class="tcc rb">5500</td></tr> +<tr><td class="tcl lb rb">Man at hard muscular work</td> <td class="tcc rb">138</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">4150</td></tr> +<tr><td class="tcl lb rb">Man at moderately active muscular work</td> <td class="tcc rb">115</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">3400</td></tr> +<tr><td class="tcl lb rb">Man at light to moderate muscular work</td> <td class="tcc rb">103</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">3050</td></tr> +<tr><td class="tcl lb rb">Man at “sedentary” or woman at moderately active work</td> <td class="tcc rb"> 92</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">2700</td></tr> +<tr><td class="tcl lb rb bb">Woman at light muscular work, or man without muscular exercise</td> <td class="tcc rb bb"> 83</td> <td class="tcc rb bb">· ·</td> <td class="tcc rb bb">· ·</td> <td class="tcc rb bb">2450</td></tr> +</table> + +<p>8. <i>Hygienic Economy of Food.</i>—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.</p> + +<p>As for the first-mentioned rule, it is practically impossible to +give information that may be of more than general application. +There are people who, because of some individual peculiarity, +cannot use foods which for people in general are wholesome +and nutritious. Some persons cannot endure milk, others suffer +if they eat eggs, others have to eschew certain kinds of meat, or +are made uncomfortable by fruit; but such cases are exceptions. +Very little is known regarding the cause of these conditions. It +is possible that in the metabolic processes to which the ingredients +of the food are subjected in the body, or even during digestion +before the substances are actually taken into the body, compounds +may be formed that are in one way or another injurious. +Whatever the cause may be, it is literally true in this sense that +“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, +<span class="pagenum"><a name="page220" id="page220"></a>220</span> +suitable combinations of the ordinary sorts of wholesome food +materials make a healthful diet. On the other hand, some foods +are of particular value at times, aside from their use for nourishment. +Fruits and green vegetables often benefit people greatly, +not as nutriment merely, for they may have very little actual +nutritive material, but because of fruit or vegetable acids or +other substances which they contain, and which sometimes +serve a most useful purpose.</p> + +<p class="center pt2"><span class="sc">Table VI.</span>—<i>Amounts of Nutrients and Energy Furnished for One Shilling in Food Materials at Ordinary Prices.</i></p> + +<table class="ws" summary="Contents"> +<tr><td class="tccm allb" rowspan="3">Food Materials as Purchased.</td> <td class="tccm allb" rowspan="3">Prices<br />per ℔</td> <td class="tccm allb" colspan="5">One Shilling will buy</td></tr> +<tr><td class="tccm allb" rowspan="2">Total Food<br />materials.</td> <td class="tccm allb" colspan="3">Available Nutrients.</td> <td class="tccm allb" rowspan="2">Fuel<br />Value.</td></tr> +<tr><td class="tccm allb">Protein.</td> <td class="tccm allb">Fat.</td> <td class="tccm allb">Carbo-<br />hydrates.</td></tr> + +<tr><td class="tcl lb rb"> </td> <td class="tcl rb">s. d.</td> <td class="tcc rb">℔</td> <td class="tcc rb">℔</td> <td class="tcc rb">℔</td> <td class="tcc rb">℔</td> <td class="tcc rb">Calories.</td></tr> + <tr><td class="rb lb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td></tr> +<tr><td class="tcl lb rb">Beef, round</td> <td class="tcl rb">0 10</td> <td class="tcr rb">1.20</td> <td class="tcr rb">.22</td> <td class="tcr rb">.14</td> <td class="tcc rb">· ·</td> <td class="tcr rb">1,155</td></tr> +<tr><td class="tcl lb rb"> </td> <td class="tcl rb">0 8½</td> <td class="tcr rb">1.41</td> <td class="tcr rb">.26</td> <td class="tcr rb">.17</td> <td class="tcc rb">· ·</td> <td class="tcr rb">1,235</td></tr> +<tr><td class="tcl lb rb"> </td> <td class="tcl rb">0 5</td> <td class="tcr rb">2.40</td> <td class="tcr rb">.44</td> <td class="tcr rb">.29</td> <td class="tcc rb">· ·</td> <td class="tcr rb">2,105</td></tr> + <tr><td class="rb lb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td></tr> +<tr><td class="tcl lb rb">Beef, sirloin</td> <td class="tcl rb">0 10</td> <td class="tcr rb">1.20</td> <td class="tcr rb">.19</td> <td class="tcr rb">.20</td> <td class="tcc rb">· ·</td> <td class="tcr rb">1,225</td></tr> +<tr><td class="tcl lb rb"> </td> <td class="tcl rb">0 9</td> <td class="tcr rb">1.33</td> <td class="tcr rb">.21</td> <td class="tcr rb">.22</td> <td class="tcc rb">· ·</td> <td class="tcr rb">1,360</td></tr> +<tr><td class="tcl lb rb"> </td> <td class="tcl rb">0 8</td> <td class="tcr rb">1.50</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td></tr> +<tr><td class="tcl lb rb"> </td> <td class="tcl rb">0 5</td> <td class="tcr rb">2.40</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td></tr> + <tr><td class="rb lb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td></tr> +<tr><td class="tcl lb rb">Beef, rib</td> <td class="tcl rb">0 9</td> <td class="tcr rb">1.33</td> <td class="tcr rb">.19</td> <td class="tcr rb">.19</td> <td class="tcc rb">· ·</td> <td class="tcr rb">1,200</td></tr> +<tr><td class="tcl lb rb"> </td> <td class="tcl rb">0 7½</td> <td class="tcr rb">1.60</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td></tr> +<tr><td class="tcl lb rb"> </td> <td class="tcl rb">0 4½</td> <td class="tcr rb">2.67</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td> <td class="tcc rb">· ·</td></tr> + <tr><td class="rb lb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td></tr> +<tr><td class="tcl lb rb">Mutton, leg</td> <td class="tcl rb">0 9</td> <td class="tcr rb">1.33</td> <td class="tcr rb">.20</td> <td class="tcr rb">.20</td> <td class="tcc rb">· ·</td> <td class="tcr rb">1,245</td></tr> +<tr><td class="tcl lb rb"> </td> <td class="tcl rb">0 5</td> <td class="tcr rb">2.40</td> <td class="tcr rb">.37</td> <td class="tcr rb">.35</td> <td class="tcc rb">· ·</td> <td class="tcr rb">2,245</td></tr> + <tr><td class="rb lb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td></tr> +<tr><td class="tcl lb rb">Pork, spare-rib</td> <td class="tcl rb">0 9</td> <td class="tcr rb">1.33</td> <td class="tcr rb">.17</td> <td class="tcr rb">.31</td> <td class="tcc rb">· ·</td> <td class="tcr rb">1,645</td></tr> +<tr><td class="tcl lb rb"> </td> <td class="tcl rb">0 7</td> <td class="tcr rb">1.71</td> <td class="tcr rb">.22</td> <td class="tcr rb">.39</td> <td class="tcc rb">· ·</td> <td class="tcr rb">2,110</td></tr> + <tr><td class="rb lb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td></tr> +<tr><td class="tcl lb rb">Pork, salt, fat</td> <td class="tcl rb">0 7</td> <td class="tcr rb">1.71</td> <td class="tcr rb">.03</td> <td class="tcr rb">1.40</td> <td class="tcc rb">· ·</td> <td class="tcr rb">6,025</td></tr> +<tr><td class="tcl lb rb"> </td> <td class="tcl rb">0 5</td> <td class="tcr rb">2.40</td> <td class="tcr rb">.04</td> <td class="tcr rb">1.97</td> <td class="tcc rb">· ·</td> <td class="tcr rb">8,460</td></tr> + <tr><td class="rb lb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td></tr> +<tr><td class="tcl lb rb">Pork, smoked ham</td> <td class="tcl rb">0 8</td> <td class="tcr rb">1.50</td> <td class="tcr rb">.20</td> <td class="tcr rb">.48</td> <td class="tcc rb">· ·</td> <td class="tcr rb">2,435</td></tr> +<tr><td class="tcl lb rb"> </td> <td class="tcl rb">0 4½</td> <td class="tcr rb">2.67</td> <td class="tcr rb">.36</td> <td class="tcr rb">.85</td> <td class="tcc rb">· ·</td> <td class="tcr rb">4,330</td></tr> + <tr><td class="rb lb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td></tr> +<tr><td class="tcl lb rb">Fresh cod</td> <td class="tcl rb">0 4</td> <td class="tcr rb">3.00</td> <td class="tcr rb">.34</td> <td class="tcr rb">.01</td> <td class="tcc rb">· ·</td> <td class="tcr rb">710</td></tr> +<tr><td class="tcl lb rb"> </td> <td class="tcl rb">0 3</td> <td class="tcr rb">4.00</td> <td class="tcr rb">.45</td> <td class="tcr rb">.01</td> <td class="tcc rb">· ·</td> <td class="tcr rb">945</td></tr> + <tr><td class="rb lb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td></tr> +<tr><td class="tcl lb rb">Salt cod</td> <td class="tcl rb">0 3½</td> <td class="tcr rb">3.43</td> <td class="tcr rb">.54</td> <td class="tcr rb">.07</td> <td class="tcc rb">· ·</td> <td class="tcr rb">1,370</td></tr> +<tr><td class="tcl lb rb"> </td> <td class="tcl rb">0 10</td> <td class="tcr rb">1.20</td> <td class="tcr rb">.07</td> <td class="tcr rb">.01</td> <td class="tcr rb">.04</td> <td class="tcr rb">275</td></tr> + <tr><td class="rb lb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td></tr> +<tr><td class="tcl lb rb">Milk, whole, 4d. a qt.</td> <td class="tcl rb">0 2</td> <td class="tcr rb">6.00</td> <td class="tcr rb">.19</td> <td class="tcr rb">.23</td> <td class="tcr rb">.30</td> <td class="tcr rb">1,915</td></tr> +<tr><td class="tcl lb rb">  ”  3d. a qt.</td> <td class="tcl rb">0 1½</td> <td class="tcr rb">8.00</td> <td class="tcr rb">.26</td> <td class="tcr rb">.30</td> <td class="tcr rb">.40</td> <td class="tcr rb">2,550</td></tr> +<tr><td class="tcl lb rb">  ”  2d. a qt.</td> <td class="tcl rb">0 1</td> <td class="tcr rb">12.00</td> <td class="tcr rb">.38</td> <td class="tcr rb">.46</td> <td class="tcr rb">.60</td> <td class="tcr rb">3,825</td></tr> + <tr><td class="rb lb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td></tr> +<tr><td class="tcl lb rb">Milk, skimmed, 2d. a qt.</td> <td class="tcl rb">0 1</td> <td class="tcr rb">12.00</td> <td class="tcr rb">.40</td> <td class="tcr rb">.03</td> <td class="tcr rb">.61</td> <td class="tcr rb">2,085</td></tr> + <tr><td class="rb lb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td></tr> +<tr><td class="tcl lb rb">Butter</td> <td class="tcl rb">1 6</td> <td class="tcr rb">.67</td> <td class="tcr rb">.01</td> <td class="tcr rb">.54</td> <td class="tcc rb">· ·</td> <td class="tcr rb">2,320</td></tr> +<tr><td class="tcl lb rb"> </td> <td class="tcl rb">1 3</td> <td class="tcr rb">.80</td> <td class="tcr rb">.01</td> <td class="tcr rb">.64</td> <td class="tcc rb">· ·</td> <td class="tcr rb">2,770</td></tr> +<tr><td class="tcl lb rb"> </td> <td class="tcl rb">1 0</td> <td class="tcr rb">1.00</td> <td class="tcr rb">.01</td> <td class="tcr rb">.81</td> <td class="tcc rb">· ·</td> <td class="tcr rb">3,460</td></tr> + <tr><td class="rb lb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td></tr> +<tr><td class="tcl lb rb">Margarine</td> <td class="tcl rb">0 4</td> <td class="tcr rb">3.00</td> <td class="tcc rb">· ·</td> <td class="tcr rb">2.37</td> <td class="tcc rb">· ·</td> <td class="tcr rb">10,080</td></tr> + <tr><td class="rb lb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td></tr> +<tr><td class="tcl lb rb">Eggs, 2s. a dozen</td> <td class="tcl rb">1 4</td> <td class="tcr rb">.75</td> <td class="tcr rb">.10</td> <td class="tcr rb">.07</td> <td class="tcc rb">· ·</td> <td class="tcr rb">475</td></tr> +<tr><td class="tcl lb rb"> ” 1½s. a dozen</td> <td class="tcl rb">1 0</td> <td class="tcr rb">1.00</td> <td class="tcr rb">.13</td> <td class="tcr rb">.09</td> <td class="tcc rb">· ·</td> <td class="tcr rb">635</td></tr> +<tr><td class="tcl lb rb"> ” 1s. a dozen</td> <td class="tcl rb">0 8</td> <td class="tcr rb">1.50</td> <td class="tcr rb">.19</td> <td class="tcr rb">.13</td> <td class="tcc rb">· ·</td> <td class="tcr rb">950</td></tr> + <tr><td class="rb lb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td></tr> +<tr><td class="tcl lb rb">Cheese</td> <td class="tcl rb">0 8</td> <td class="tcr rb">1.50</td> <td class="tcr rb">.38</td> <td class="tcr rb">.48</td> <td class="tcr rb">.04</td> <td class="tcr rb">2,865</td></tr> +<tr><td class="tcl lb rb"> </td> <td class="tcl rb">0 7</td> <td class="tcr rb">1.71</td> <td class="tcr rb">.43</td> <td class="tcr rb">.55</td> <td class="tcr rb">.04</td> <td class="tcr rb">3,265</td></tr> +<tr><td class="tcl lb rb"> </td> <td class="tcl rb">0 5</td> <td class="tcr rb">2.40</td> <td class="tcr rb">.60</td> <td class="tcr rb">.77</td> <td class="tcr rb">.06</td> <td class="tcr rb">4,585</td></tr> + <tr><td class="rb lb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td></tr> +<tr><td class="tcl lb rb">Wheat bread</td> <td class="tcl rb">0 1<span class="spp">1</span>⁄<span class="suu">8</span></td> <td class="tcr rb">10.67</td> <td class="tcr rb">.76</td> <td class="tcr rb">.13</td> <td class="tcr rb">5.57</td> <td class="tcr rb">12,421</td></tr> + <tr><td class="rb lb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td></tr> +<tr><td class="tcl lb rb">Wheat flour</td> <td class="tcl rb">0 1<span class="spp">3</span>⁄<span class="suu">5</span></td> <td class="tcr rb">7.64</td> <td class="tcr rb">.67</td> <td class="tcr rb">.07</td> <td class="tcr rb">5.63</td> <td class="tcr rb">12,110</td></tr> +<tr><td class="tcl lb rb"> </td> <td class="tcl rb">0 1½</td> <td class="tcr rb">8.16</td> <td class="tcr rb">.72</td> <td class="tcr rb">.07</td> <td class="tcr rb">6.01</td> <td class="tcr rb">12,935</td></tr> + <tr><td class="rb lb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td></tr> +<tr><td class="tcl lb rb">Oatmeal</td> <td class="tcl rb">0 1<span class="spp">2</span>⁄<span class="suu">5</span></td> <td class="tcr rb">8.39</td> <td class="tcr rb">1.11</td> <td class="tcr rb">.54</td> <td class="tcr rb">5.54</td> <td class="tcr rb">14,835</td></tr> +<tr><td class="tcl lb rb"> </td> <td class="tcl rb">0 1½</td> <td class="tcr rb">8.16</td> <td class="tcr rb">1.08</td> <td class="tcr rb">.53</td> <td class="tcr rb">5.39</td> <td class="tcr rb">14,430</td></tr> + <tr><td class="rb lb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td></tr> +<tr><td class="tcl lb rb">Rice</td> <td class="tcl rb">0 1¾</td> <td class="tcr rb">6.86</td> <td class="tcr rb">.45</td> <td class="tcr rb">.02</td> <td class="tcr rb">5.27</td> <td class="tcr rb">10,795</td></tr> + <tr><td class="rb lb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td></tr> +<tr><td class="tcl lb rb">Potatoes</td> <td class="tcl rb">0 0<span class="spp">2</span>⁄<span class="suu">3</span></td> <td class="tcr rb">18.00</td> <td class="tcr rb">.25</td> <td class="tcr rb">.02</td> <td class="tcr rb">2.70</td> <td class="tcr rb">5,605</td></tr> +<tr><td class="tcl lb rb"> </td> <td class="tcl rb">0 0½</td> <td class="tcr rb">24.00</td> <td class="tcr rb">.34</td> <td class="tcr rb">.02</td> <td class="tcr rb">3.60</td> <td class="tcr rb">7,470</td></tr> + <tr><td class="rb lb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td></tr> +<tr><td class="tcl lb rb">Beans</td> <td class="tcl rb">0 2 </td> <td class="tcr rb">6.00</td> <td class="tcr rb">1.05</td> <td class="tcr rb">.10</td> <td class="tcr rb">3.47</td> <td class="tcr rb">8,960</td></tr> + <tr><td class="rb lb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td></tr> +<tr><td class="tcl lb rb bb">Sugar</td> <td class="tcl rb bb">1 ¾</td> <td class="tcr rb bb">6.86</td> <td class="tcc rb bb">· ·</td> <td class="tcc rb bb">· ·</td> <td class="tcr rb bb">6.86</td> <td class="tcr rb bb">12,760</td></tr> +</table> + +<p>The proper observance of the second rule mentioned requires +information regarding the demands of the body for food under +different circumstances. To supply this information is one +purpose of the effort to determine the so-called dietary standards +<span class="pagenum"><a name="page221" id="page221"></a>221</span> +mentioned above. It should be observed, however, that these +are generally more applicable to the proper feeding of a group +or class of people as a whole than for particular individuals +in this class. The needs of individuals will vary largely from +the average in accordance with the activity and individuality. +Moreover, it is neither necessary nor desirable for the individual +to follow any standard exactly from day to day. It is requisite +only that the average supply shall be sufficient to meet the +demands of the body during a given period.</p> + +<p>The cooking of food and other modes of preparing it for +consumption have much to do with its nutritive value. Many +materials which, owing to their mechanical condition or to +some other cause, are not particularly desirable food materials +in their natural state, are quite nutritious when cooked or otherwise +prepared for consumption. It is also a matter of common +experience that well-cooked food is wholesome and appetizing, +whereas the same material poorly prepared is unpalatable. +There are three chief purposes of cooking; the first is to change +the mechanical condition of the food. Heating changes the +structure of many food materials very materially, so that they +may be more easily chewed and brought into a condition in which +the digestive juices can act upon them more freely, and in this +way probably influencing the ease and thoroughness of digestion. +The second is to make the food more appetizing by improving +the appearance or flavour or both. Food which is attractive to +the eye and pleasing to the palate quickens the flow of saliva +and other digestive juices and thus aids digestion. The third +is to kill, by heat, disease germs, parasites or other dangerous +organisms that may be contained in food. This is often a very +important matter and applies to both animal and vegetable foods. +Scrupulous neatness should always be observed in storing, +handling and serving food. If ever cleanliness is desirable it +must be in the things we eat, and every care should be taken to +ensure it for the sake of health as well as of decency. Cleanliness +in this connexion means not only absence of visible dirt, but +freedom from undesirable bacteria and other minute organisms +and from worms and other parasites. If food, raw or cooked, is +kept in dirty places, peddled from dirty carts, prepared in dirty +rooms and in dirty dishes, or exposed to foul air, disease germs +and other offensive and dangerous substances may easily enter it.</p> + +<p>9. <i>Pecuniary Economy of Food.</i>—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.</p> + +<p>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.</p> + +<p>The variations in the cost of the actual nutriment in different +food materials may be illustrated by comparison of the amounts +of nutrients obtained for a given sum in the materials as bought +at ordinary market prices. This is done in Table VI., which +shows the amounts of available nutrients contained in the quantities +of different food materials that may be purchased for one +shilling at prices common in England.</p> + +<p>When proper attention is given to the needs of the body for +food and the relation between cost and nutritive value of food +materials, it will be found that with care in the purchase and skill +in the preparation of food, considerable control may be had over +the expensiveness of a palatable, nutritious and healthful diet.</p> + +<div class="condensed"> +<p><span class="sc">Authorities.</span>—<span class="sc">Composition of Foods</span>:—König, <i>Chemie der +menschlichen Nahrungs- und Genussmittel</i>; Atwater and Bryant, +“Composition of American Food Materials,” Bul. 28, Office of +Experiment Stations, U.S. Department of Agriculture. <span class="sc">Nutrition +and Dietetics</span>:—Armsby, <i>Principles of Animal Nutrition</i>; Lusk, +<i>The Science of Nutrition</i>; Burney Yeo, <i>Food in Health and Disease</i>; +Munk and Uffelmann, <i>Die Ernährung des gesunden und kranken +Menschen</i>; Von Leyden, <i>Ernährungstherapie und Diätetik</i>; Dujardin-Beaumetz, +Hygiène alimentaire; Hutchison, <i>Food and Dietetics</i>; R. +H. Chittenden, <i>Physiological Economy in Nutrition</i> (1904), <i>Nutrition of +Man</i> (1907); Atwater, “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. +<span class="sc">General Metabolism</span>:—Voit, <i>Physiologie des allgemeinen Stoffwechsels +und der Ernährung</i>; Hermann, <i>Handbuch der Physiologie</i>, +Bd. vi.; Von Noorden, <i>Pathologie des Stoffwechsels</i>; Schäfer, <i>Text-Book +of Physiology</i>, vol. i.; Atwater and Langworthy, “Digest of +Metabolism Experiments,” Bull. 45, Office of Experiment Stations, +U.S. Department of Agriculture.</p> +</div> +<div class="author">(W. O. A.; R. D. M.)</div> + +<hr class="foot" /> <div class="note"> + +<p><a name="ft1f" id="ft1f" href="#fa1f"><span class="fn">1</span></a> The terms applied by different writers to these nitrogenous +compounds are conflicting. For instance, the term “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.</p> + +<p><a name="ft2f" id="ft2f" href="#fa2f"><span class="fn">2</span></a> Folin, <i>Festschrift für Olaf Hammarsten</i>, iii. (Upsala, 1906).</p> + +<p><a name="ft3f" id="ft3f" href="#fa3f"><span class="fn">3</span></a> <i>Ztschr. Biol.</i> 30, 73.</p> + +<p><a name="ft4f" id="ft4f" href="#fa4f"><span class="fn">4</span></a> In Russian. Cited in United States Department of Agriculture, +Office of Experiment Stations, Bul. No. 45, <i>A Digest of Metabolism +Experiments</i>, by W. O. Atwater and C. F. Langworthy.</p> + +<p><a name="ft5f" id="ft5f" href="#fa5f"><span class="fn">5</span></a> <i>Arch. physiol. norm. et path.</i> (1894) 4.</p> + +<p><a name="ft6f" id="ft6f" href="#fa6f"><span class="fn">6</span></a> U.S. Department of Agriculture, Office of Experiment Stations, +Bulletins Nos. 63, 69, 109, 136, 175. For a description of the respiration +calorimeter here mentioned see also publication No. 42 of the +Carnegie Institution of Washington.</p> + +<p><a name="ft7f" id="ft7f" href="#fa7f"><span class="fn">7</span></a> <i>Ztschr. Biol.</i> 21 (1885), p. 377.</p> + +<p><a name="ft8f" id="ft8f" href="#fa8f"><span class="fn">8</span></a> <i>Connecticut</i> (Storrs) <i>Agricultural Experiment Station Report</i> +(1899), 73.</p> + +<p><a name="ft9f" id="ft9f" href="#fa9f"><span class="fn">9</span></a> One ounce equals 28.35 grams.</p> + +<p><a name="ft10f" id="ft10f" href="#fa10f"><span class="fn">10</span></a> As the chief function of both fats and carbohydrates is to furnish +energy, their exact proportion in the diet is of small account. The +amount of either may vary largely according to taste, available +supply, or other condition, as long as the total amount of both is +sufficient, together with the protein to furnish the required energy.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DIETRICH, CHRISTIAN WILHELM ERNST<a name="ar77" id="ar77"></a></span> (1712-1774), +German painter, was born at Weimar, where he was brought up +early to the profession of art by his father Johann George, then +painter of miniatures to the court of the duke. Having been sent +to Dresden to perfect himself under the care of Alexander Thiele, +he had the good fortune to finish in two hours, at the age of +eighteen, a picture which attracted the attention of the king of +Saxony. Augustus II. was so pleased with Dietrich’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.</p> + + +<hr class="art" /> +<p><span class="bold">DIETRICH OF BERN,<a name="ar78" id="ar78"></a></span> the name given in German popular +poetry to Theodoric the Great. The legendary history of Dietrich +differs so widely from the life of Theodoric that it has been +suggested that the two were originally unconnected. Medieval +<span class="pagenum"><a name="page222" id="page222"></a>222</span> +chroniclers, however, repeatedly asserted the identity of Dietrich +and Theodoric, although the more critical noted the anachronisms +involved in making Ermanaric (d. 376) and Attila (d. 453) contemporary +with Theodoric (b. 455). That the legend is based +on vague historical reminiscences is proved by the retention of +the names of Theodoric (Thiuda-reiks, Dietrich) and his father +Theudemir (Dietmar), by Dietrich’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 +(<i>q.v.</i>) in <i>Dietrich’s Flucht</i>, the <i>Rabenschlacht</i> and <i>Alpharts Tod</i>; +but it was reserved for an Icelandic author, writing in Norway +in the 13th century, to compile, with many romantic additions, a +consecutive account of Dietrich. In this Norse prose redaction, +known as the <i>Vilkina Saga</i>, or more correctly the <i>Thidrekssaga</i>, +is incorporated much extraneous matter from the Nibelungen +and Wayland legends, in fact practically the whole of south +German heroic tradition.</p> + +<p>There are traces of a form of the Dietrich legend in which he +was represented as starting out from Byzantium, in accordance +with historical tradition, for his conquest of Italy. But this +early disappeared, and was superseded by the existing legend, +in which, perhaps by an “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 <i>Ermenrichs Tod</i> +he is represented as slaying Ermanaric, as in fact Theodoric slew +Odoacer. “Otacher” replaces Ermanaric as his adversary in the +<i>Hildebrandslied</i>, which relates how thirty years after the earlier +attempt he reconquered his Lombard kingdom. Dietrich’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 <i>König Laurin</i>, <i>Sigenot</i>, +<i>Eckenlied</i> and <i>Virginal</i> are based largely on local traditions +originally independent of Dietrich. The court of Attila (Etzel) +was a ready bridge to the Nibelungen legend. In the final catastrophe +he was at length compelled, after steadily holding aloof +from the combat, to avenge the slaughter of his Amelungs by +the Burgundians, and delivered Hagen bound into the hands of +Kriemhild. The flame breath which anger induced from him +shows the influence of pure myth, but the tales of his demonic +origin and of his being carried off by the devil in the shape of a +black horse may safely be put down to the clerical hostility to +Theodoric’s Arianism.</p> + +<p>Generally speaking, Dietrich of Bern was the wise and just +monarch as opposed to Ermanaric, the typical tyrant of Germanic +legend. He was invariably represented as slow of provocation +and a friend of peace, but once roused to battle not even Siegfried +could withstand his onslaught. But probably Dietrich’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 <span class="sc"><a href="#artlinks">Hildebrand, Lay of</a></span>), with his nephews the Wolfings +Alphart and Wolfhart; Wittich, who renounced his allegiance +to Dietrich and slew the sons of Attila; Heime and Biterolf.</p> + +<div class="condensed"> +<p>The contents of the poems dealing with the Dietrich cycle are +summarized by Uhland in <i>Schriften zur Geschichte der Dichtung und +Sage</i> (Stuttgart, 1873). The <i>Thidrekssaga</i> (ed. C. Unger, Christiania, +1853) is translated into German by F. H. v. der Hagen in <i>Altdeutsche +und altnordische Heldensagen</i> (vols. i. and ii. 3rd ed., Breslau, 1872). +A summary of it forms the concluding chapter of T. Hodgkin’s +<i>Theodoric the Goth</i> (1891). The variations in the Dietrich legend in +the Latin historians, in Old and Middle High German literature, +and in the northern saga, can be studied in W. Grimm’s <i>Deutsche +Heldensage</i> (2nd ed., Berlin, 1867). There is a good account in English +in F. E. Sandbach’s <i>Heroic Saga-cycle of Dietrich of Bern</i> (1906), +forming No. 15 of Alfred Nutt’s <i>Popular Studies in Mythology</i>, and +another in M. Bentinck Smith’s translation of Dr O. L. Jiriczek’s +<i>Deutsche Heldensage</i> (<i>Northern Legends</i>, London, 1902). For modern +German authorities and commentators see B. Symons, “Deutsche +Heldensage” in H. Paul’s <i>Grd. d. german. Phil.</i> (Strassburg, new ed., +1905); also Goedeke, <i>Geschichte der deutschen Dichtung</i> (i. 241-246).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DIEZ, FRIEDRICH CHRISTIAN<a name="ar79" id="ar79"></a></span> (1794-1876), German +philologist, was born at Giessen, in Hesse-Darmstadt, on the 15th +of March 1794. He was educated first at the gymnasium and +then at the university of his native town. There he studied +classics under Friedrich Gottlieb Welcker (1784-1868) who had +just returned from a two years’ 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 <i>Selections from the Romance Poets</i>, and advised the +young scholar to explore the rich mine of Provençal literature +which the French savant had opened up. This advice was +eagerly followed, and henceforth Diez devoted himself to Romance +literature. He thus became the founder of Romance philology. +After supporting himself for some years by private teaching, he +removed in 1822 to Bonn, where he held the position of privatdocent. +In 1823 he published his first work, <i>An Introduction +to Romance Poetry</i>; in the following year appeared <i>The Poetry +of the Troubadours</i>, and in 1829 <i>The Lives and Works of the +Troubadours</i>. In 1830 he was called to the chair of modern +literature. The rest of his life was mainly occupied with the +composition of the two great works on which his fame rests, the +<i>Grammar of the Romance Languages</i> (1836-1844), and the <i>Lexicon +of the Romance Languages—Italian, Spanish and French</i> (1853); +in these two works Diez did for the Romance group of languages +what Jacob Grimm did for the Teutonic family. He died at +Bonn on the 29th of May 1876.</p> + +<div class="condensed"> +<p>The earliest French philologists, such as Perion and Henri Estienne, +had sought to discover the origin of French in Greek and even in +Hebrew. For more than a century Ménage’s <i>Etymological Dictionary</i> +held the field without a rival. Considering the time at which it was +written (1650), it was a meritorious work, but philology was then in +the empirical stage, and many of Ménage’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.</p> + +<p>His dictionary is divided into two parts. The first contains words +common to two at least of the three principal groups of Romance:—Italian, +Spanish and Portuguese, and Provençal and French. The +Italian, as nearest the original, is placed at the head of each article. +<span class="pagenum"><a name="page223" id="page223"></a>223</span> +The second part treats of words peculiar to one group. There is no +separate glossary of Wallachian.</p> + +<p>Of the introduction to the grammar there is an English translation +by C. B. Cayley. The dictionary has been published in a remodelled +form for English readers by T. C. Donkin.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DIEZ,<a name="ar80" id="ar80"></a></span> a town of Germany, in the Prussian province of Hesse-Nassau, +romantically situated in the deep valley of the Lahn, +here crossed by an old bridge, 30 m. E. from Coblenz on the +railway to Wetzlar. Pop. 4500. It is overlooked by a former +castle of the counts of Nassau-Dillenburg, now a prison. Close +by, on an eminence above the river, lies the castle of Oranienstein, +formerly a Benedictine nunnery and now a cadet school, +with beautiful gardens. There are a Roman Catholic and two +Evangelical churches. The new part of the town is well built +and contains numerous pretty villa residences. In addition to +extensive iron-works there are sawmills and tanneries. In the +vicinity are Fachingen, celebrated for its mineral waters, and +the majestic castle of Schaumburg belonging to the prince of +Waldeck-Pyrmont.</p> + + +<hr class="art" /> +<p><span class="bold">DIFFERENCES, CALCULUS OF<a name="ar81" id="ar81"></a></span> (<i>Theory of Finite Differences</i>), +that branch of mathematics which deals with the successive +differences of the terms of a series.</p> + +<p>1. The most important of the cases to which mathematical +methods can be applied are those in which the terms of the series +are the values, taken at stated intervals (regular or irregular), of +a continuously varying quantity. In these cases the formulae +of finite differences enable certain quantities, whose exact value +depends on the law of variation (<i>i.e.</i> the law which governs the +relative magnitude of these terms) to be calculated, often with +great accuracy, from the given terms of the series, without +explicit reference to the law of variation itself. The methods +used may be extended to cases where the series is a double series +(series of double entry), <i>i.e.</i> where the value of each term depends +on the values of a pair of other quantities.</p> + +<p>2. The <i>first differences</i> of a series are obtained by subtracting +from each term the term immediately preceding it. If these are +treated as terms of a new series, the first differences of this series +are the <i>second differences</i> of the original series; and so on. +The successive differences are also called <i>differences of the first, +second, ... order</i>. The differences of successive orders are most +conveniently arranged in successive columns of a table thus:—</p> + +<table class="ws" summary="Contents"> + +<tr><td class="tcc allb">Term.</td> <td class="tcc allb">1st Diff.</td> <td class="tcc allb">2nd Diff.</td> <td class="tcc allb">3rd Diff.</td> <td class="tcc allb">4th Diff.</td></tr> + +<tr><td class="tcc lb rb">a</td> <td class="tcc rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb"> </td></tr> +<tr><td class="tcc lb rb"> </td> <td class="tcc rb">b − a</td> <td class="tcc rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb"> </td></tr> +<tr><td class="tcc lb rb">b</td> <td class="tcc rb"> </td> <td class="tcc rb">c − 2b + a</td> <td class="tcc rb"> </td> <td class="tcc rb"> </td></tr> +<tr><td class="tcc lb rb"> </td> <td class="tcc rb">c − b</td> <td class="tcc rb"> </td> <td class="tcc rb">d − 3c + 3b − a</td> <td class="tcc rb"> </td></tr> +<tr><td class="tcc lb rb">c</td> <td class="tcc rb"> </td> <td class="tcc rb">d − 2c + b</td> <td class="tcc rb"> </td> <td class="tcc rb">e − 4d + 6c − 4b + a</td></tr> +<tr><td class="tcc lb rb"> </td> <td class="tcc rb">d − c</td> <td class="tcc rb"> </td> <td class="tcc rb">e − 3d + 3c − b</td> <td class="tcc rb"> </td></tr> +<tr><td class="tcc lb rb">d</td> <td class="tcc rb"> </td> <td class="tcc rb">e − 2d + c</td> <td class="tcc rb"> </td> <td class="tcc rb"> </td></tr> +<tr><td class="tcc lb rb"> </td> <td class="tcc rb">e − d</td> <td class="tcc rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb"> </td></tr> +<tr><td class="tcc lb rb bb">e</td> <td class="tcc rb bb"> </td> <td class="tcc rb bb"> </td> <td class="tcc rb bb"> </td> <td class="tcc rb bb"> </td></tr> +</table> + +<p class="center pt2"><i>Algebra of Differences and Sums.</i></p> + +<table class="nobctr" style="float: left; width: 340px;" summary="Illustration"> +<tr><td class="figleft1"><img style="width:288px; height:124px" src="images/img223.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 1.</span></td></tr></table> + + +<p>3. The formal relations between the terms of the series and the +differences may be seen by comparing the arrangements (A) and (B) +in fig. 1. In (A) the various terms and differences are the same as in +§ 2, but placed differently. In +(B) we take a new series of +terms α, β, γ, δ, commencing +with the same term α, and take +the successive sums of pairs of +terms, instead of the successive +differences, but place them to +the left instead of to the right. +It will be seen, in the first +place, that the successive terms +in (A), reading downwards to the right, and the successive +terms in (B), reading downwards to the left, consist each of +a series of terms whose coefficients follow the binomial law; <i>i.e.</i> +the coefficients in b − a, c − 2b + a, d − 3c + 3b − a, ... and in +α + β, α + 2β + γ, α + 3β + 3γ + δ, ... 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); <i>e.g.</i> β + γ is the difference of α + 2β + γ +and α + β, 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 β + 2γ + δ is the sum of β + γ and γ + δ. +Hence if we take β, γ, δ, ... 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.</p> + +<p>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</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">l − nk + ... + (-1)<span class="sp">n-2</span></td> + <td>n·n − 1</td> + <td rowspan="2">c + (-1)<span class="sp">n-1</span> nb + (-1)<span class="sp">n</span> a,</td></tr> +<tr><td class="denom">1·2</td></tr> +</table> + +<p class="noind">where l, k ... are the (n + 1)th, nth, ... terms of the series a, b, c, +...; the coefficients being those of the terms in the expansion of +(y − x)<span class="sp">n</span>: and (ii.) the (n + 1)th term of the series, <i>i.e.</i> the nth term +after a, is</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">a + nβ +</td> <td>n·n − 1</td> <td rowspan="2">γ + ...</td></tr> +<tr><td class="denom">1·2</td></tr></table> + +<p class="noind">where β, γ, ... are the first, second, ... differences of a; the +coefficients being those of the terms in the expansion of (x + y)<span class="sp">n</span>.</p> + +<p>4. Now suppose we treat the terms a, b, c, ... as being themselves +the first differences of another series. Then, if the first term +of this series is N, the subsequent terms are N + a, N + a + b, N + a + +b + c, ...; <i>i.e.</i> the difference between the (n + 1)th term and the +first term is the sum of the first n terms of the original series. The +term N, in the diagram (A), will come above and to the left of a; and +we see, by (ii.) of § 3, that the sum of the first n terms of the original +series is</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2" class="f200 np">(</td> + <td rowspan="2">N + na +</td> <td>n·n − 1</td> + <td rowspan="2">β + ...</td> <td rowspan="2" class="f200 np">)</td> + <td rowspan="2">− N = na +</td> <td>n·n − 1</td> <td rowspan="2">β +</td> + <td>n·n − 1·n − 2</td> <td rowspan="2">γ + ...</td></tr> +<tr><td class="denom">1·2</td> <td class="denom">1·2</td> <td class="denom">1 · 2 · 3</td></tr> +</table> + +<p>5. As an example, take the arithmetical series</p> + +<p class="center">a, a + p, a + 2p, ...</p> + +<p class="noind">The first differences are p, p, p, ... and the differences of any higher +order are zero. Hence, by (ii.) of § 3, the (n + 1)th term is a + np, and, +by § 4, the sum of the first n terms is na + ½n(n − 1)p = ½n{2a + (n − 1)p}.</p> + +<p>6 As another example, take the series 1, 8, 27, ... the terms of +which are the cubes of 1, 2, 3, ... The first, second and third +differences of the first term are 7, 12 and 6, and it may be shown +(§ 14 (i.)) that all differences of a higher order are zero. Hence the +sum of the first n terms is</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">n + 7</td> <td>n·n − 1</td> + <td rowspan="2">+ 12</td> <td>n·n − 1·n − 2</td> + <td rowspan="2">+ 6</td> <td>n·n − 1·n − 2·n − 3</td> + <td rowspan="2">= ¼n<span class="sp">4</span> + ½n³ + ¼n² = {½n (n + 1)}².</td></tr> +<tr><td class="denom">1·2</td> <td class="denom">1·2·3</td> <td class="denom">1·2·3·4</td></tr> +</table> + +<p>7. In § 3 we have described b − 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>i.e.</i> 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.</p> + +<p class="center pt2"><i>Notation of Differences and Sums.</i></p> + +<p>8. It is convenient to denote the terms a, b, c, ... of the series +by u<span class="su">0</span>, u<span class="su">1</span>, u<span class="su">2</span>, u<span class="su">3</span>, ... If we merely have the terms of the series, u<span class="su">n</span> +may be regarded as meaning the (n + 1)th term. Usually, however, +the terms are the values of a quantity u, which is a function of +another quantity x, and the values of x, to which a, b, c, ... correspond, +proceed by a constant difference h. If x<span class="su">0</span> and u<span class="su">0</span> are a pair +of corresponding values of x and u, and if any other value x<span class="su">0</span> + mh of x +and the corresponding value of u are denoted by x<span class="su">m</span> and u<span class="su">m</span>, then +the terms of the series will be ... u<span class="su">n-2</span>, u<span class="su">n-1</span>, u<span class="su">n</span>, u<span class="su">n+1</span>, u<span class="su">n+2</span> ..., corresponding +to values of x denoted by ... x<span class="su">n-2</span>, x<span class="su">n-1</span>, x<span class="su">n</span>, x<span class="su">n+1</span>, x<span class="su">n+2</span>....</p> + +<p>9. In the <i>advancing-difference notation</i> u<span class="su">n+1</span> − u<span class="su">n</span> is denoted by +Δu<span class="su">n</span>. The differences Δu<span class="su">0</span>, Δu<span class="su">1</span>, Δu<span class="su">2</span> ... may then be regarded as +values of a function Δu corresponding to values of x proceeding by +constant difference h; and therefore Δu<span class="su">n+1</span> − Δu<span class="su">n</span> denoted by ΔΔu<span class="su">n</span>, +or, more briefly, Δ²u<span class="su">n</span>; and so on. Hence the table of differences in +§ 2, with the corresponding values of x and of u placed opposite each +other in the ordinary manner of mathematical tables, becomes</p> + +<table class="ws" summary="Contents"> + +<tr><td class="tcc allb">x</td> <td class="tcc allb">u</td> <td class="tcc allb">1st Diff.</td> <td class="tcc allb">2nd Diff.</td> <td class="tcc allb">3rd Diff.</td> <td class="tcc allb">4th Diff.</td></tr> +<tr><td class="tcc lb rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td></tr> +<tr><td class="tcc lb rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td></tr> +<tr><td class="tcc lb rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td></tr> +<tr><td class="tcc lb rb">x<span class="su">n-2</span></td> <td class="tcc rb">u<span class="su">n-2</span></td> <td class="tcc rb"> </td> <td class="tcc rb">Δ²u<span class="su">n-3</span></td> <td class="tcc rb"> </td> <td class="tcc rb">Δ<span class="sp">4</span>u<span class="su">n-4</span> ...</td></tr> +<tr><td class="tcc lb rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb">Δu<span class="su">n-2</span></td> <td class="tcc rb"> </td> <td class="tcc rb">Δ³u<span class="su">n-3</span></td> <td class="tcc rb"> </td></tr> +<tr><td class="tcc lb rb">x<span class="su">n-1</span></td> <td class="tcc rb">u<span class="su">n-1</span></td> <td class="tcc rb"> </td> <td class="tcc rb">Δ²u<span class="su">n-2</span></td> <td class="tcc rb"> </td> <td class="tcc rb">Δ<span class="sp">4</span>u<span class="su">n-3</span> ...</td></tr> +<tr><td class="tcc lb rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb">Δu<span class="su">n-1</span></td> <td class="tcc rb"> </td> <td class="tcc rb">Δ³u<span class="su">n-2</span></td> <td class="tcc rb"> </td></tr> +<tr><td class="tcc lb rb">x<span class="su">n</span> </td> <td class="tcc rb">u<span class="su">n</span> </td> <td class="tcc rb"> </td> <td class="tcc rb">Δ²u<span class="su">n-1</span></td> <td class="tcc rb"> </td> <td class="tcc rb">Δ<span class="sp">4</span>u<span class="su">n-2</span> ...</td></tr> +<tr><td class="tcc lb rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb">Δu<span class="su">n</span> </td> <td class="tcc rb"> </td> <td class="tcc rb">Δ³u<span class="su">n-1</span></td> <td class="tcc rb"> </td></tr> +<tr><td class="tcc lb rb">x<span class="su">n+1</span></td> <td class="tcc rb">u<span class="su">n+1</span></td> <td class="tcc rb"> </td> <td class="tcc rb">Δ²u<span class="su">n</span> </td> <td class="tcc rb"> </td> <td class="tcc rb">Δ<span class="sp">4</span>u<span class="su">n-1</span> ...</td></tr> +<tr><td class="tcc lb rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb">Δu<span class="su">n+1</span></td> <td class="tcc rb"> </td> <td class="tcc rb">Δ³u<span class="su">n</span> </td> <td class="tcc rb"> </td></tr> +<tr><td class="tcc lb rb">x<span class="su">n+2</span></td> <td class="tcc rb">u<span class="su">n+2</span></td> <td class="tcc rb"> </td> <td class="tcc rb">Δ²u<span class="su">n+1</span></td> <td class="tcc rb"> </td> <td class="tcc rb">Δ<span class="sp">4</span>u<span class="su">n</span>  ...</td></tr> +<tr><td class="tcc lb rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td></tr> +<tr><td class="tcc lb rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td></tr> +<tr><td class="tcc lb rb bb">·</td> <td class="tcc rb bb">·</td> <td class="tcc rb bb">·</td> <td class="tcc rb bb">·</td> <td class="tcc rb bb">·</td> <td class="tcc rb bb">·</td></tr> +</table> + +<p>The terms of the series of which ... u<span class="su">n-1</span>, u<span class="su">n</span>, u<span class="su">n+1</span>, ... are +the first differences are denoted by Σu, with proper suffixes, so +<span class="pagenum"><a name="page224" id="page224"></a>224</span> +that this series is ... Σu<span class="su">n-1</span>, Σu<span class="su">n</span>, Σu<span class="su">n+1</span>.... The suffixes are +chosen so that we may have ΔΣu<span class="su">n</span> = u<span class="su">n</span>, whatever n may be; and +therefore (§ 4) Σu<span class="su">n</span> may be regarded as being the sum of the terms +of the series up to and including u<span class="su">n-1</span>. Thus if we write Σu<span class="su">n-1</span> = +C + u<span class="su">n-2</span>, where C is any constant, we shall have</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl">Σu<span class="su">n</span> = Σu<span class="su">n-1</span> + ΔΣu<span class="su">n-1</span> = C + u<span class="su">n-2</span> + u<span class="su">n-1</span>,</td></tr> +<tr><td class="tcl">Σu<span class="su">n+1</span> = C + u<span class="su">n-2</span> + u<span class="su">n-1</span> + u<span class="su">n</span>,</td></tr> +</table> + +<p class="noind">and so on. This is true whatever C may be, so that the knowledge +of ... u<span class="su">n-1</span>, u<span class="su">n</span>, ... gives us no knowledge of the exact value +of Σu<span class="su">n</span>; in other words, C is an arbitrary constant, the value of +which must be supposed to be the same throughout any operations in +which we are concerned with values of Σu corresponding to different +suffixes.</p> + +<p>There is another symbol E, used in conjunction with u to denote +the next term in the series. Thus Eu<span class="su">n</span> means u<span class="su">n+1</span>, so that +Eu<span class="su">n</span> = u<span class="su">n</span> + Δu<span class="su">n</span>.</p> + +<p>10. Corresponding to the advancing-difference notation there is +a <i>receding-difference</i> notation, in which u<span class="su">n+1</span> − u<span class="su">n</span> is regarded as +a difference of u<span class="su">n+1</span>, and may be denoted by Δ′u<span class="su">n+1</span>, and similarly +u<span class="su">n+1</span> − 2u<span class="su">n</span> + u<span class="su">n-1</span> may be denoted by Δ′²u<span class="su">n+1</span>. This notation is only +required for certain special purposes, and the usage is not settled +(§ 19 (ii.)).</p> + +<p>11. The <i>central-difference</i> notation depends on treating +u<span class="su">n+1</span> − 2u<span class="su">n</span> − u<span class="su">n-1</span> as the second <span class="correction" title="amended from dfference">difference</span> of u<span class="su">n</span>, and therefore as +corresponding to the value x<span class="su">n</span>; but there is no settled system of +notation. The following seems to be the most convenient. Since u<span class="su">n</span> is +a function of x<span class="su">n</span>, and the second difference u<span class="su">n+2</span> − 2u<span class="su">n+1</span> + u<span class="su">n</span> is a function +of x<span class="su">n+1</span>, the first difference u<span class="su">n+1</span> − u<span class="su">n</span> must be regarded as a function +of x<span class="su">n+1/2</span>, <i>i.e.</i> of ½(x<span class="su">n</span> + x<span class="su">n+1</span>). We therefore write u<span class="su">n+1</span> − u<span class="su">n</span> = δu<span class="su">n+1/2</span>, +and each difference in the table in § 9 will have the same suffix +as the value of x in the same horizontal line; or, if the difference +is of an odd order, its suffix will be the means of those of the two +nearest values of x. This is shown in the table below.</p> + +<p>In this notation, instead of using the symbol E, we use a symbol μ +to denote the mean of two consecutive values of u, or of two consecutive +differences of the same order, the suffixes being assigned on the +same principle as in the case of the differences. Thus</p> + +<p class="center">μu<span class="su">n+1/2</span> = ½(u<span class="su">n</span> + u<span class="su">n+1</span>, μδu<span class="su">n</span> = ½(δu<span class="su">n-1/2</span> + δu<span class="su">n+1/2</span>, &c.</p> + +<p>If we take the means of the differences of odd order immediately +above and below the horizontal line through any value of x, these +means, with the differences of even order in that line, constitute the +<i>central differences</i> of the corresponding value of u. Thus the table +of central differences is as follows, the values obtained as means +being placed in brackets to distinguish them from the actual +differences:—</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcc allb">x</td> <td class="tcc allb">u</td> <td class="tcc allb">1st Diff.</td> <td class="tcc allb">2nd Diff.</td> <td class="tcc allb">3rd Diff.</td> <td class="tcc allb">4th Diff.</td></tr> + +<tr><td class="tcc lb rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td></tr> +<tr><td class="tcc lb rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td></tr> +<tr><td class="tcc lb rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td></tr> +<tr><td class="tcc lb rb">x<span class="su">n-2</span></td> <td class="tcc rb">u<span class="su">n-2</span></td> <td class="tcc rb">(μδu<span class="su">n-2</span>)</td> <td class="tcc rb">δ²u<span class="su">n-2</span></td> <td class="tcc rb">(μδ³u<span class="su">n-2</span>)</td> <td class="tcc rb">δ<span class="sp">4</span>u<span class="su">n-2</span> ...</td></tr> +<tr><td class="tcc lb rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb">δu<span class="su">n-3/2</span></td> <td class="tcc rb"> </td> <td class="tcc rb">δ³u<span class="su">n-3/2</span></td> <td class="tcc rb"> </td></tr> +<tr><td class="tcc lb rb">x<span class="su">n-1</span></td> <td class="tcc rb">u<span class="su">n-1</span></td> <td class="tcc rb">(μδu<span class="su">n-1</span>)</td> <td class="tcc rb">δ²u<span class="su">n-1</span></td> <td class="tcc rb">(μδ³u<span class="su">n-1</span>)</td> <td class="tcc rb">δ<span class="sp">4</span>u<span class="su">n-1</span> ...</td></tr> +<tr><td class="tcc lb rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb">δu<span class="su">n-1/2</span></td> <td class="tcc rb"> </td> <td class="tcc rb">δ³u<span class="su">n-2</span></td> <td class="tcc rb"> </td></tr> +<tr><td class="tcc lb rb">x<span class="su">n</span></td> <td class="tcc rb">u<span class="su">n</span> </td> <td class="tcc rb">(μδu<span class="su">n</span>) </td> <td class="tcc rb">δ²u<span class="su">n</span> </td> <td class="tcc rb">(μδ³u<span class="su">n</span>) </td> <td class="tcc rb">δ<span class="sp">4</span>u<span class="su">n</span>  ...</td></tr> +<tr><td class="tcc lb rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb">δu<span class="su">n+1/2</span></td> <td class="tcc rb"> </td> <td class="tcc rb">δ³u<span class="su">n+1/2</span></td> <td class="tcc rb"> </td></tr> +<tr><td class="tcc lb rb">x<span class="su">n+1</span></td> <td class="tcc rb">u<span class="su">n+1</span></td> <td class="tcc rb">(μδu<span class="su">n+1</span>)</td> <td class="tcc rb">δ²u<span class="su">n+1</span></td> <td class="tcc rb">(μδ³u<span class="su">n+1</span>)</td> <td class="tcc rb">δ<span class="sp">4</span>u<span class="su">n+1</span> ...</td></tr> +<tr><td class="tcc lb rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb">δu<span class="su">n+3/2</span></td> <td class="tcc rb"> </td> <td class="tcc rb">δ³u<span class="su">n+3/2</span></td> <td class="tcc rb"> </td></tr> +<tr><td class="tcc lb rb">x<span class="su">n+2</span></td> <td class="tcc rb">u<span class="su">n+2</span></td> <td class="tcc rb">(μδu<span class="su">n+2</span>)</td> <td class="tcc rb">δ²u<span class="su">n+2</span></td> <td class="tcc rb">(μδ³u<span class="su">n+2</span>)</td> <td class="tcc rb">δ<span class="sp">4</span>u<span class="su">n+2</span> ...</td></tr> +<tr><td class="tcc lb rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td></tr> +<tr><td class="tcc lb rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td> <td class="tcc rb">·</td></tr> +<tr><td class="tcc lb rb bb">·</td> <td class="tcc rb bb">·</td> <td class="tcc rb bb">·</td> <td class="tcc rb bb">·</td> <td class="tcc rb bb">·</td> <td class="tcc rb bb">·</td></tr> +</table> + +<p>Similarly, by taking the means of consecutive values of u and also +of consecutive differences of even order, we should get a series of +terms and differences central to the intervals x<span class="su">n-2</span> to x<span class="su">n-1</span>, x<span class="su">n-1</span> to +x<span class="su">n</span>, ....</p> + +<p>The terms of the series of which the values of u are the first differences +are denoted by σu, with suffixes on the same principle; the +suffixes being chosen so that δσu<span class="su">n</span> shall be equal to u<span class="su">n</span>. Thus, if</p> + +<p class="center">σu<span class="su">n-3/2</span> = C + u<span class="su">n-2</span>,</p> + +<p class="noind">then</p> + +<p class="center">σu<span class="su">n-1/2</span> = C + u<span class="su">n-2</span> + u<span class="su">n-1</span>, σ<span class="su">n+1/2</span> = C + u<span class="su">n-2</span> + u<span class="su">n-1</span> + u<span class="su">n</span>, &c.,</p> + +<p class="noind">and also</p> + +<p class="center">μσu<span class="su">n-1</span> = C + u<span class="su">n-2</span> + ½u<span class="su">n-1</span>, μσu<span class="su">n</span> = C + u<span class="su">n-2</span> + u<span class="su">n-1</span> + ½u<span class="su">n</span>, &c.,</p> + +<p class="noind">C being an arbitrary constant which must remain the same throughout +any series of operations.</p> + +<p class="center pt2"><i>Operators and Symbolic Methods.</i></p> + +<p>12. There are two further stages in the use of the symbols Δ, Σ, +δ, σ, &c., which are not essential for elementary treatment but +lead to powerful methods of deduction.</p> + +<p>(i.) Instead of treating Δu as a function of x, so that Δu<span class="su">n</span> means +(Δu)<span class="su">n</span>, we may regard Δ as denoting an <i>operation</i> performed on u, +and take Δu<span class="su">n</span> as meaning Δ.u<span class="su">n</span>. This applies to the other symbols +E, δ, &c., whether taken simply or in combination. Thus ΔEu<span class="su">n</span> +means that we first replace u<span class="su">n</span> by u<span class="su">n+1</span>, and then replace this by +u<span class="su">n+2</span> − u<span class="su">n+1</span>.</p> + +<p>(ii.) The operations Δ, E, δ, and μ, 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: <i>e.g.</i> Δ(u<span class="su">n</span> + v<span class="su">n</span>) = Δu<span class="su">n</span> + Δv<span class="su">n</span>, +ΔDu<span class="su">n</span> = DΔu<span class="su">n</span>, &c. Hence the symbols can be separated from the +functions on which the operations are performed, and treated as +if they were algebraical quantities. For instance, we have</p> + +<p class="center">E·u<span class="su">n</span> = u<span class="su">n+1</span> = u<span class="su">n</span> + Δu<span class="su">n</span> = 1·u<span class="su">n</span> + Δ·u<span class="su">n</span>,</p> + +<p class="noind">so that we may write E = 1 + Δ, or Δ = 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<span class="sp">n</span> = (1 + Δ)<span class="sp">n</span>, Δ<span class="sp">n</span> = (E-1)<span class="sp">n</span>, and expand by the binomial +theorem and then operate on u<span class="su">0</span>, we get the general formulae</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">u<span class="su">n</span> = u<span class="su">0</span> + nΔu<span class="su">0</span> +</td> <td>n·n − 1</td> + <td rowspan="2">Δ<span class="sp">2</span>u<span class="su">0</span> + ... + Δ<span class="sp">n</span>u<span class="su">0</span>,</td></tr> +<tr><td class="denom">1·2</td></tr></table> + +<table class="math0" summary="math"> +<tr><td rowspan="2">Δ<span class="sp">n</span>u<span class="su">0</span> = u<span class="su">n</span> − nu<span class="su">n-1</span> +</td> <td>n·n − 1</td> <td rowspan="2">u<span class="su">n-2</span> + ... + (-1)<span class="sp">n</span>u<span class="su">0</span>,</td></tr> +<tr><td class="denom">1·2</td></tr></table> + +<p class="noind">which are identical with the formulae in (ii.) and (i.) of § 3.</p> + +<p>(iii.) What has been said under (ii.) applies, with certain reservations, +to the operations Σ and σ, and to the operation which represents +integration. The latter is sometimes denoted by D<span class="sp">-1</span>; and, since +ΔΣu<span class="su">n</span> = u<span class="su">n</span>, and δσu<span class="su">n</span> = u<span class="su">n</span>, we might similarly replace Σ and σ by +Δ<span class="sp">-1</span> and δ<span class="sp">-1</span>. These symbols can be combined with Δ, E, &c. +according to the ordinary laws of algebra, provided that proper +account is taken of the arbitrary constants introduced by the +operations D<span class="sp">-1</span>, Δ<span class="sp">-1</span>, δ<span class="sp">-1</span>.</p> + +<p class="center pt2"><i>Applications to Algebraical Series.</i></p> + +<p>13. <i>Summation of Series.</i>—If u<span class="su">r</span>, denotes the (r + 1)th term of a +series, and if v<span class="su">r</span> is a function of r such that Δv<span class="su">r</span> = u<span class="su">r</span> for all integral +values of r, then the sum of the terms u<span class="su">m</span>, u<span class="su">m+1</span>, ... u<span class="su">n</span> is +v<span class="su">n+1</span> − v<span class="su">m</span>. Thus the sum of a number of terms of a series may often +be found by inspection, in the same kind of way that an integral +is found.</p> + +<p>14. <i>Rational Integral Functions.</i>—(i.) If u<span class="su">r</span> is a rational integral +function of r of degree p, then Δu<span class="su">r</span>, is a rational integral function of r +of degree p − 1.</p> + +<p>(ii.) A particular case is that of a <i>factorial</i>, <i>i.e.</i> a product of the +form (r + a + 1) (r + a + 2) ... (r + b), each factor exceeding the preceding +factor by 1. We have</p> + +<p class="center">Δ · (r + a + 1) (r + a + 2) ... (r + b) = (b − a)·(r + a + 2) ... (r + b),</p> + +<p class="noind">whence, changing a into a-1,</p> + +<p class="center">Σ(r + a + 1) (r + a + 2) ... (r + b) = <i>const.</i> + (r + a)(r + a + 1) ... (r + b)/(b − a + 1).</p> + +<p class="noind">A similar method can be applied to the series whose (r + 1)th +term is of the form 1/(r + a + 1) (r + a + 2) ... (r + b).</p> + +<p>(iii.) Any rational integral function can be converted into the sum +of a number of factorials; and thus the sum of a series of which such +a function is the general term can be found. For example, it may +be shown in this way that the sum of the pth powers of the first n +natural numbers is a rational integral function of n of degree p + 1, +the coefficient of n<span class="sp">p+1</span> being 1/(p + 1).</p> + +<p>15. <i>Difference-equations.</i>—The summation of the series ... ++ u<span class="su">n+2</span> + u<span class="su">n-1</span> + u<span class="su">n</span> is a solution of the <i>difference-equation</i> Δv<span class="su">n</span> = u<span class="su">n+1</span>, +which may also be written (E − 1)v<span class="su">n</span> = u<span class="su">n+1</span>. This is a simple form +of difference-equation. There are several forms which have been +investigated; a simple form, more general than the above, is the +<i>linear equation</i> with <i>constant coefficients</i>—</p> + +<p class="center">v<span class="su">n+m</span> + a<span class="su">1</span>v<span class="su">n+m-1</span> + a<span class="su">2</span>v<span class="su">n+m-2</span> + ... + a<span class="su">m</span>v<span class="su">n</span> = N,</p> + +<p class="noind">where a<span class="su">1</span>, a<span class="su">2</span>, ... a<span class="su">m</span> are constants, and N is a given function of n. +This may be written</p> + +<p class="center">(E<span class="sp">m</span> + a<span class="su">1</span>E<span class="sp">m-1</span> + ... + a<span class="su">m</span>)v<span class="su">n</span> = N</p> + +<p class="noind">or</p> + +<p class="center">(E − p<span class="su">1</span>)(E − p<span class="su">2</span>) ... (E − p<span class="su">m</span>)v<span class="su">n</span> = N.</p> + +<p class="noind">The solution, if p<span class="su">1</span>, p<span class="su">2</span>, ... p<span class="su">m</span> are all different, is v<span class="su">n</span> = C<span class="su">1</span>p<span class="su">1</span><span class="sp">n</span> + +C<span class="su">2</span>p<span class="su">2</span><span class="sp">n</span> + ... + C<span class="su">m</span>p<span class="su">m</span><span class="sp">n</span> + V<span class="su">n</span>, where C<span class="su">1</span>, C<span class="su">2</span> ... are constants, and +v<span class="su">n</span> = V<span class="su">n</span> is any one solution of the equation. The method of finding +a value for V<span class="su">n</span> depends on the form of N. Certain modifications are +required when two or more of the p’s are equal.</p> + +<p>It should be observed, in all cases of this kind, that, in describing +C<span class="su">1</span>, C<span class="su">2</span> as “constants,” it is meant that the value of any one, as C<span class="su">1</span>, is +the same for all values of n occurring in the series. A “constant” +may, however, be a periodic function of n.</p> + +<p class="center pt2"><i>Applications to Continuous Functions.</i></p> + +<p>16. The cases of greatest practical importance are those in which +u is a continuous function of x. The terms u<span class="su">1</span>, u<span class="su">2</span> ... of the series +then represent the successive values of u corresponding to x = x<span class="su">1</span>, x<span class="su">2</span>.... +The important applications of the theory in these cases are to +(i.) relations between differences and differential coefficients, (ii.) +<span class="pagenum"><a name="page225" id="page225"></a>225</span> +interpolation, or the determination of intermediate values of u, and +(iii.) relations between sums and integrals.</p> + +<p>17. Starting from any pair of values x<span class="su">0</span> and u<span class="su">0</span>, we may suppose +the interval h from x<span class="su">0</span> to x<span class="su">1</span> to be divided into q equal portions. If +we suppose the corresponding values of u to be obtained, and their +differences taken, the successive advancing differences of u<span class="su">0</span> being +denoted by ∂u<span class="su">0</span>, ∂²u<span class="su">0</span> ..., we have (§ 3 (ii.))</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">u<span class="su">1</span> = u<span class="su">0</span> + q∂u<span class="su">0</span> +</td> <td>q·q − 1</td> + <td rowspan="2">∂²u<span class="su">0</span> + ....</td></tr> +<tr><td class="denom">1·2</td></tr></table> + +<p class="noind">When q is made indefinitely great, this (writing ƒ(x) for u) becomes +Taylor’s Theorem (<span class="sc"><a href="#artlinks">Infinitesimal Calculus</a></span>)</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ƒ(x + h) = ƒ(x) + hƒ′(x) +</td> <td>h²</td> + <td rowspan="2">ƒ″(x) + ...,</td></tr> +<tr><td class="denom">1·2</td></tr></table> + +<p class="noind">which, expressed in terms of operators, is</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">E = 1 + hD +</td> <td>h²</td> + <td rowspan="2">D² +</td> <td>h³</td> + <td rowspan="2">D³ + ... = e<span class="sp">hD</span>.</td></tr> +<tr><td class="denom">1·2</td> <td class="denom">1·2·3</td></tr></table> + +<p class="noind">This gives the relation between Δ and D. Also we have</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">u<span class="su">2</span> = u<span class="su">0</span> + 2q∂u<span class="su">0</span> +</td> <td>2q·2q − 1</td> <td rowspan="2">∂²u<span class="su">0</span> + ...</td></tr> +<tr><td class="denom">1·2</td></tr> + +<tr><td rowspan="2">u<span class="su">3</span> = u<span class="su">0</span> + 3q∂u<span class="su">0</span> +</td> <td>3q·3q − 1</td> <td rowspan="2">∂²u<span class="su">0</span> + ...</td></tr> +<tr><td class="denom">1·2</td></tr> + +<tr><td colspan="3">·   ·</td></tr> +<tr><td colspan="3">·   ·</td></tr> +<tr><td colspan="3">·   ·</td></tr> +</table> + +<p class="noind">and, if p is any integer,</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">u<span class="su">p/q</span> = u<span class="su">0</span> + p∂u<span class="su">0</span> +</td> <td>p·p − 1</td> + <td rowspan="2">∂²u<span class="su">0</span> + ....</td></tr> +<tr><td class="denom">1·2</td></tr></table> + +<p class="noind">From these equations u<span class="su">p/q</span> could be expressed in terms of u<span class="su">0</span>, u<span class="su">1</span>, +u<span class="su">2</span>, ...; this is a particular case of interpolation (<i>q.v.</i>).</p> + +<p>18. <i>Differences and Differential Coefficients.</i>—The various formulae +are most quickly obtained by symbolical methods; <i>i.e.</i> by dealing +with the operators Δ, E, D, ... as if they were algebraical +quantities. Thus the relation E = e<span class="sp">hD</span> (§ 17) gives</p> + +<p class="center">hD = log<span class="su">e</span> (1 + Δ) = Δ − ½Δ² + <span class="spp">1</span>⁄<span class="suu">3</span>Δ³ ...</p> + +<p class="noind">or</p> + +<p class="center">h(du/dx)<span class="su">0</span> = Δu<span class="su">0</span> − ½Δ²u<span class="su">0</span> + <span class="spp">1</span>⁄<span class="suu">3</span>Δ³u<span class="su">0</span> ....</p> + +<p>The formulae connecting central differences with differential +coefficients are based on the relations +μ = cosh ½hD = ½(e<span class="sp">1/2hD</span> + e<span class="sp">-1/2hD</span>), +δ = 2 sinh ½hD − e<span class="sp">1/2hD</span> − e<span class="sp">-1/2hD</span>, +and may be grouped as follows:—</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcr">u<span class="su">0</span></td> +<td class="tcl">= u<span class="su">0</span></td> +<td class="tcrm" rowspan="5"><span style="font-size: 8em; font-family: 'Courier New'; color: #a0a0a0;">}</span></td></tr> + +<tr><td class="tcr">μδu<span class="su">0</span></td> +<td class="tcl">= (hD + <span class="spp">1</span>⁄<span class="suu">6</span> h³D³ + <span class="spp">1</span>⁄<span class="suu">120</span> h<span class="sp">5</span>D<span class="sp">5</span> + ...)u<span class="su">0</span></td></tr> + +<tr><td class="tcr">δ²u<span class="su">0</span></td> +<td class="tcl">= (h²D² + <span class="spp">1</span>⁄<span class="suu">12</span> h<span class="sp">4</span>D<span class="sp">4</span> + <span class="spp">1</span>⁄<span class="suu">360</span> h<span class="sp">6</span>D<span class="sp">6</span> + ...)u<span class="su">0</span></td></tr> + +<tr><td class="tcr">μδ³u<span class="su">0</span></td> +<td class="tcl">= (h³D³ + <span class="spp">1</span>⁄<span class="suu">4</span> h<span class="sp">5</span>D<span class="sp">5</span> + ...)u<span class="su">0</span></td></tr> + +<tr><td class="tcr">δ<span class="sp">4</span>u<span class="su">0</span></td> +<td class="tcl">= (h<span class="sp">4</span>D<span class="sp">4</span> + <span class="spp">1</span>⁄<span class="suu">6</span> h<span class="sp">6</span>D<span class="sp">6</span> + ...)u<span class="su">0</span></td></tr> + + +<tr><td> </td> <td class="tcl" colspan="2">·   ·   ·</td></tr> +<tr><td> </td> <td class="tcl" colspan="2">·   ·   ·</td></tr> +<tr><td> </td> <td class="tcl" colspan="2">·   ·   ·</td></tr> + + +<tr><td class="tcr">μu<span class="su">1/2</span></td> +<td class="tcl">= (1 + <span class="spp">1</span>⁄<span class="suu">8</span> h²D² + <span class="spp">1</span>⁄<span class="suu">384</span> h<span class="sp">4</span>D<span class="sp">4</span> + <span class="spp">1</span>⁄<span class="suu">46080</span> h<span class="sp">6</span>D<span class="sp">6</span> + ...)u<span class="su">1/2</span></td> +<td class="tcrm" rowspan="5"><span style="font-size: 8em; font-family: 'Courier New'; color: #a0a0a0;">}</span></td></tr> + +<tr><td class="tcr">δu<span class="su">1/2</span></td> +<td class="tcl">= (hD + <span class="spp">1</span>⁄<span class="suu">24</span> h³D³ + <span class="spp">1</span>⁄<span class="suu">1920</span> h<span class="sp">5</span>D<span class="sp">5</span> + ...)u<span class="su">1/2</span></td></tr> + +<tr><td class="tcr">μδ²u<span class="su">1/2</span></td> +<td class="tcl">= (h²D² + <span class="spp">5</span>⁄<span class="suu">24</span> h<span class="sp">4</span>D<span class="sp">4</span> + <span class="spp">91</span>⁄<span class="suu">5760</span> h<span class="sp">6</span>D<span class="sp">6</span> + ...)u<span class="su">1/2</span></td></tr> + +<tr><td class="tcr">δ³u<span class="su">1/2</span></td> +<td class="tcl">= (h³D³ + <span class="spp">1</span>⁄<span class="suu">8</span> h<span class="sp">5</span>D<span class="sp">5</span> + ...)u<span class="su">1/2</span></td></tr> + +<tr><td class="tcr">μδ<span class="sp">4</span> u<span class="su">1/2</span></td> +<td class="tcl">= (h<span class="sp">4</span>D<span class="sp">4</span> + <span class="spp">7</span>⁄<span class="suu">24</span> h<span class="sp">6</span>D<span class="sp">6</span> + ...)u<span class="su">1/2</span></td></tr> + + +<tr><td> </td> <td class="tcl" colspan="2">·   ·   ·</td></tr> +<tr><td> </td> <td class="tcl" colspan="2">·   ·   ·</td></tr> +<tr><td> </td> <td class="tcl" colspan="2">·   ·   ·</td></tr> + +<tr><td class="tcr">u<span class="su">0</span></td> +<td class="tcl">= u<span class="su">0</span></td> +<td class="tcrm" rowspan="5"><span style="font-size: 8em; font-family: 'Courier New'; color: #a0a0a0;">}</span></td></tr> + +<tr><td class="tcr">hDu<span class="su">0</span></td> +<td class="tcl">= (μδ − <span class="spp">1</span>⁄<span class="suu">6</span> μδ³ + <span class="spp">1</span>⁄<span class="suu">30</span> μδ<span class="sp">5</span> − ...)u<span class="su">0</span></td></tr> + +<tr><td class="tcr">h²D²u<span class="su">0</span></td> +<td class="tcl">= (δ² − <span class="spp">1</span>⁄<span class="suu">12</span> δ<span class="sp">4</span> + <span class="spp">1</span>⁄<span class="suu">90</span> δ<span class="sp">6</span> − ...)u<span class="su">0</span></td></tr> + +<tr><td class="tcr">h³D³u<span class="su">0</span></td> +<td class="tcl">= (μδ³ − <span class="spp">1</span>⁄<span class="suu">4</span> μδ<span class="sp">5</span> + ...)u<span class="su">0</span></td></tr> + +<tr><td class="tcr">h<span class="sp">4</span>D<span class="sp">4</span>u<span class="su">0</span></td> +<td class="tcl">= (δ<span class="sp">4</span> − <span class="spp">1</span>⁄<span class="suu">6</span> δ<span class="sp">6</span> + ...)u<span class="su">0</span></td></tr> + + +<tr><td> </td> <td class="tcl" colspan="2">·   ·   ·</td></tr> +<tr><td> </td> <td class="tcl" colspan="2">·   ·   ·</td></tr> +<tr><td> </td> <td class="tcl" colspan="2">·   ·   ·</td></tr> + + +<tr><td class="tcr">u<span class="su">1/2</span></td> +<td class="tcl">= (μ − <span class="spp">1</span>⁄<span class="suu">8</span> μδ² + <span class="spp">3</span>⁄<span class="suu">128</span> μδ<span class="sp">4</span> − <span class="spp">5</span>⁄<span class="suu">1024</span> μδ<span class="sp">6</span> + ...)u<span class="su">1/2</span></td> +<td class="tcrm" rowspan="5"><span style="font-size: 8em; font-family: 'Courier New'; color: #a0a0a0;">}</span></td></tr> + +<tr><td class="tcr">hDu<span class="su">1/2</span></td> +<td class="tcl">= (δ − <span class="spp">1</span>⁄<span class="suu">24</span> δ³ + <span class="spp">3</span>⁄<span class="suu">640</span> δ<span class="sp">5</span> − ...)u<span class="su">1/2</span></td></tr> + +<tr><td class="tcr">h²D²u<span class="su">1/2</span></td> +<td class="tcl">= (μδ² − <span class="spp">5</span>⁄<span class="suu">24</span> μδ<span class="sp">4</span> + <span class="spp">259</span>⁄<span class="suu">5760</span> μδ<span class="sp">6</span> − ...)u<span class="su">1/2</span></td></tr> + +<tr><td class="tcr">h³D³u<span class="su">1/2</span></td> +<td class="tcl">= (δ³ − <span class="spp">1</span>⁄<span class="suu">8</span> δ<span class="sp">5</span> + ...)u<span class="su">1/2</span></td></tr> + +<tr><td class="tcr">h<span class="sp">4</span>D<span class="sp">4</span> u<span class="su">1/2</span></td> +<td class="tcl">= (μδ<span class="sp">4</span> − <span class="spp">7</span>⁄<span class="suu">24</span> μδ<span class="sp">6</span> + ...)u<span class="su">1/2</span></td></tr> + +<tr><td> </td> <td class="tcl" colspan="2">·   ·   ·</td></tr> +<tr><td> </td> <td class="tcl" colspan="2">·   ·   ·</td></tr> +<tr><td> </td> <td class="tcl" colspan="2">·   ·   ·</td></tr> +</table> + +<p>When u is a rational integral function of x, each of the above series +is a terminating series. In other cases the series will be an infinite +one, and may be divergent; but it may be used for purposes of +approximation up to a certain point, and there will be a “remainder,” +the limits of whose magnitude will be determinate.</p> + +<p>19. <i>Sums and Integrals.</i>—The relation between a sum and an +integral is usually expressed by the <i>Euler-Maclaurin formula</i>. The +principle of this formula is that, if u<span class="su">m</span> and u<span class="su">m+1</span>, are ordinates of a +curve, distant h from one another, then for a first approximation to +the area of the curve between u<span class="su">m</span> and u<span class="su">m+1</span> we have ½h(u<span class="su">m</span> + u<span class="su">m+1</span>), +and the difference between this and the true value of the area can +be expressed as the difference of two expressions, one of which is a +function of x<span class="su">m</span>, and the other is the same function of x<span class="su">m+1</span>. +Denoting these by φ(x<span class="su">m</span>) and φ(x<span class="su">m+1</span>), we have</p> + +<table class="math0" summary="math"> +<tr><td class="vb" rowspan="2"><span class="f200">∫</span></td> <td>x<span class="su">m+1</span></td> + <td rowspan="2">udx = ½h(u<span class="su">m</span> + u<span class="su">m+1</span>) + φ(x<span class="su">m+1</span>) − φ(x<span class="su">m</span>).</td></tr> +<tr><td>x<span class="su">m</span></td></tr></table> + +<p class="noind">Adding a series of similar expressions, we find</p> + +<table class="math0" summary="math"> +<tr><td class="vb" rowspan="2"><span class="f200">∫</span></td> <td>x<span class="su">n</span></td> + <td rowspan="2">udx = h{½u<span class="su">m</span> + u<span class="su">m+1</span> + u<span class="su">m+2</span> + ... + u<span class="su">n-1</span> + ½u<span class="su">n</span>} + φ(x<span class="su">n</span>) − φ(x<span class="su">m</span>).</td></tr> +<tr><td>x<span class="su">m</span></td></tr></table> + +<p>The function φ(x) can be expressed in terms either of differential +coefficients of u or of advancing or central differences; thus there +are three formulae.</p> + +<p>(i.) The Euler-Maclaurin formula, properly so called, (due independently +to Euler and Maclaurin) is</p> + +<table class="math0" summary="math"> +<tr><td class="vb" rowspan="2"><span class="f200">∫</span></td> <td>x<span class="su">n</span></td> + <td rowspan="2">udx = h·μσu<span class="su">n</span> − <span class="spp">1</span>⁄<span class="suu">12</span> h²</td> <td>du<span class="su">n</span></td> + <td rowspan="2">½ + <span class="spp">1</span>⁄<span class="suu">720</span> h<span class="sp">4</span></td> <td>d³u<span class="su">n</span></td> + <td rowspan="2">− <span class="spp">1</span>⁄<span class="suu">30240</span> h<span class="sp">6</span></td> <td>d<span class="sp">5</span>u<span class="su">n</span></td> + <td rowspan="2">+ ... = h·μσu<span class="su">n</span> − </td> <td>B<span class="su">1</span></td> + <td rowspan="2">h<span class="su">2</span></td> <td>du<span class="su">n</span></td> + <td rowspan="2"> + </td> <td>B<span class="su">2</span></td> + <td rowspan="2">h<span class="sp">4</span></td> <td>d<span class="sp">3</span>u<span class="su">n</span></td> + <td rowspan="2">− </td> <td>B<span class="su">3</span></td> + <td rowspan="2">h<span class="sp">6</span></td> <td>d<span class="sp">5</span>u<span class="su">n</span></td> + <td rowspan="2">+ ...</td></tr> +<tr><td> </td> <td class="denom">dx</td> <td class="denom">dx<span class="sp">3</span></td> + <td class="denom">dx<span class="sp">5</span></td> <td class="denom">2!</td> + <td class="denom">dx</td> <td class="denom">4!</td> + <td class="denom">dx<span class="sp">3</span></td> <td class="denom">6!</td> + <td class="denom">dx<span class="sp">5</span></td></tr></table> + +<p class="noind">where B<span class="su">1</span>, B<span class="su">2</span>, B<span class="su">3</span> ... are <i>Bernoulli’s numbers</i>.</p> + +<p>(ii.) If we express differential coefficients in terms of advancing +differences, we get a theorem which is due to Laplace:—</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">1/h </td> <td class="vb" rowspan="2"><span class="f200">∫</span></td> <td>x<span class="su">n</span></td> + <td rowspan="2">udx = μσ(u<span class="su">n</span> − u<span class="su">0</span>) − <span class="spp">1</span>⁄<span class="suu">12</span>(Δu<span class="su">n</span> − + Δu<span class="su">0</span>) + <span class="spp">1</span>⁄<span class="suu">24</span>(Δ²u<span class="su">n</span> − Δ²u<span class="su">0</span>) − + <span class="spp">19</span>⁄<span class="suu">720</span>(Δ³u<span class="su">n</span> − Δ³u<span class="su">0</span>) + <span class="spp">3</span>⁄<span class="suu">160</span>(Δ<span class="sp">4</span>u<span class="su">n</span> − Δ<span class="sp">4</span>u<span class="su">0</span>) − ...</td></tr> +<tr><td>x<span class="su">0</span></td></tr></table> + +<p class="noind">For practical calculations this may more conveniently be written</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">1/h </td> <td class="vb" rowspan="2"><span class="f200">∫</span></td> <td>x<span class="su">n</span></td> +<td rowspan="2">udx = μσ(u<span class="su">n</span> − u<span class="su">0</span>) + <span class="spp">1</span>⁄<span class="suu">12</span>(Δu<span class="su">0</span> − ½Δ²u<span class="su">0</span> + <span class="spp">19</span>⁄<span class="suu">60</span>Δ³u<span class="su">0</span> − ...) + + <span class="spp">1</span>⁄<span class="suu">12</span>(Δ′ u<span class="su">n</span> − ½Δ′ ²u<span class="su">n</span> + <span class="spp">19</span>⁄<span class="suu">60</span>Δ′ ³u<span class="su">n</span> − ...),</td></tr> +<tr><td>x<span class="su">0</span></td></tr></table> + +<p class="noind">where accented differences denote that the values of u are read backwards +from u<span class="su">n</span>; <i>i.e.</i> Δ′u<span class="su">n</span> denotes u<span class="su">n-1</span> − u<span class="su">n</span>, not (as in § 10) u<span class="su">n</span> − u<span class="su">n-1</span>.</p> + +<p>(iii.) Expressed in terms of central differences this becomes</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">1/h </td> <td class="vb" rowspan="2"><span class="f200">∫</span></td> <td>x<span class="su">n</span></td> +<td rowspan="2">udx = μσ(u<span class="su">n</span> − u<span class="su">0</span>) − <span class="spp">1</span>⁄<span class="suu">12</span>μδu<span class="su">n</span> + <span class="spp">11</span>⁄<span class="suu">720</span> μδ³u<span class="su">n</span> − ... + + <span class="spp">1</span>⁄<span class="suu">12</span>μδu<span class="su">0</span> − <span class="spp">11</span>⁄<span class="suu">720</span> μδ³u<span class="su">0</span> + ... = + μ(σ − <span class="spp">1</span>⁄<span class="suu">12</span>δ + <span class="spp">11</span>⁄<span class="suu">720</span>δ³ − <span class="spp">191</span>⁄<span class="suu">60480</span>δ<span class="sp">5</span> + + <span class="spp">2497</span>⁄<span class="suu">3628800</span>δ<span class="sp">7</span> − ...)(u<span class="su">n</span> − u<span class="su">0</span>).</td></tr> +<tr><td>x<span class="su">0</span></td></tr></table> + +<p>(iv.) There are variants of these formulae, due to taking hu<span class="su">m+1/2</span> as +the first approximation to the area of the curve between u<span class="su">m</span> and +u<span class="su">m+1</span>; the formulae involve the sum +u<span class="su">1/2</span> + u<span class="su">3/2</span> + ... + u<span class="su">n-1/2</span> ≡ σ(u<span class="su">n</span> − u<span class="su">0</span>) +(see <span class="sc"><a href="#artlinks">Mensuration</a></span>).</p> + +<p>20. The formulae in the last section can be obtained by symbolical +methods from the relation</p> + +<table class="math0" summary="math"> +<tr><td>1/h </td> <td class="vb"><span class="f200">∫</span></td> + <td>udx = 1/h D<span class="sp">-1</span>u = 1/hD · u.</td></tr></table> + +<p class="noind">Thus for central differences, if we write θ ≡ ½hD, we have μ = cosh θ, +δ = 2 sinh θ, σ = δ<span class="sp">-1</span>, and the result in (iii.) corresponds to the formula</p> + +<p class="center"> +sinh θ = θ cosh θ/(1 + <span class="spp">1</span>⁄<span class="suu">3</span> sinh² θ − <span class="spp">2</span>⁄<span class="suu">3·5</span> sinh<span class="sp">4</span> θ + <span class="spp">2·4</span>⁄<span class="suu">3·5·7</span> sinh<span class="sp">6</span> θ − ...).</p> + +<div class="condensed"> +<p><span class="sc">References.</span>—There is no recent English work on the theory of +finite differences as a whole. G. Boole’s <i>Finite Differences</i> (1st ed., +1860, 2nd ed., edited by J. F. Moulton, 1872) is a comprehensive +treatise, in which symbolical methods are employed very early. +A. A. Markoff’s <i>Differenzenrechnung</i> (German trans., 1896) contains +general formulae. (Both these works ignore central differences.) +<i>Encycl. der math. Wiss.</i> vol. i. pt. 2, pp. 919-935, may also be consulted. +An elementary treatment of the subject will be found in +many text-books, <i>e.g.</i> G. Chrystal’s <i>Algebra</i> (pt. 2, ch. xxxi.). +A. W. Sunderland, <i>Notes on Finite Differences</i> (1885), is intended for +actuarial students. Various central-difference formulae with references +are given in <i>Proc. Lond. Math. Soc.</i> xxxi. pp. 449-488. For +other references see <span class="sc"><a href="#artlinks">Interpolation</a></span>.</p> +</div> +<div class="author">(W. F. Sh.)</div> + + +<hr class="art" /> +<p><span class="bold">DIFFERENTIAL EQUATION,<a name="ar82" id="ar82"></a></span> in mathematics, a relation between +one or more functions and their differential coefficients. +The subject is treated here in two parts: (1) an elementary +introduction dealing with the more commonly recognized types +of differential equations which can be solved by rule; and (2) the +general theory.</p> + +<p class="center pt2"><i>Part I.—Elementary Introduction.</i></p> + +<p>Of equations involving only one independent variable, x (known +as <i>ordinary</i> differential equations), and one dependent variable, y, +and containing only the first differential coefficient dy/dx (and therefore +said to be of the first <i>order</i>), the simplest form is that reducible +to the type</p> + +<p class="center">dy/dx = ƒ(x)/F(y),</p> + +<p class="noind">leading to the result ƒF(y)dy − ƒƒ(x)dx = A, where A is an arbitrary +constant; this result is said to solve the differential equation, the +problem of evaluating the integrals belonging to the integral calculus.</p> + +<p><span class="pagenum"><a name="page226" id="page226"></a>226</span></p> + +<p>Another simple form is</p> + +<p class="center">dy/dx + yP = Q,</p> + +<p class="noind">where P, Q are functions of x only; this is known as the linear equation, +since it contains y and dy/dx only to the first degree. If ƒPdx = u, we clearly have</p> + +<table class="math0" summary="math"> +<tr><td>d</td> <td rowspan="2">(ye<span class="sp">u</span>) = e<span class="sp">u</span></td> <td rowspan="2" class="f200 np">(</td> +<td>dy</td> <td rowspan="2">+ Py</td> <td rowspan="2" class="f200 np">)</td> +<td rowspan="2">= e<span class="sp">u</span>Q,</td></tr> +<tr><td class="denom">dx</td> <td class="denom">dx</td> </tr></table> + +<p class="noind">so that y = e<span class="sp">-u</span>(ƒe<span class="sp">u</span>Qdx + A) solves the equation, and is the only +possible solution, A being an arbitrary constant. The rule for the +solution of the linear equation is thus to multiply the equation by +e<span class="sp">u</span>, where u = ƒPdx.</p> + +<p>A third simple and important form is that denoted by</p> + +<p class="center">y = px + ƒ(p),</p> + +<p class="noind">where p is an abbreviation for dy/dx; this is known as Clairaut’s +form. By differentiation in regard to x it gives</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">p = p + x</td> <td>dp</td> + <td rowspan="2">+ ƒ′(p)</td> <td>dp</td> <td rowspan="2">,</td></tr> +<tr><td class="denom">dx</td> <td class="denom">dx</td></tr></table> + +<p class="noind">where</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ƒ′(p) =</td> <td>d</td> + <td rowspan="2">ƒ(p);</td></tr> +<tr><td class="denom">dp</td></tr></table> + +<p class="noind">thus, either (i.) dp/dx = 0, that is, p is constant on the curve satisfying +the differential equation, which curve is thus any one of the +straight lines y = cx = ƒ(c), where c is an arbitrary constant, or else, +(ii.) x + ƒ′(p) = 0; if this latter hypothesis be taken, and p be eliminated +between x + ƒ′(p) = 0 and y = px + ƒ(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 + ƒ(c).</p> + +<p>In general if a differential equation φ(x, y, dy/dx) = 0 be satisfied +by any one of the curves F(x, y, c) = 0, where c is an arbitrary constant, +it is clear that the envelope of these curves, when existent, must +also satisfy the differential equation; for this equation prescribes +a relation connecting only the co-ordinates x, y and the differential +coefficient dy/dx, and these three quantities are the same at any +point of the envelope for the envelope and for the particular curve +of the family which there touches the envelope. The relation expressing +the equation of the envelope is called a <i>singular</i> solution of +the differential equation, meaning an <i>isolated</i> solution, as not being +one of a family of curves depending upon an arbitrary parameter.</p> + +<p>An extended form of Clairaut’s equation expressed by</p> + +<p class="center">y = xF(p) + ƒ(p)</p> + +<p class="noind">may be similarly solved by first differentiating in regard to p, when +it reduces to a linear equation of which x is the dependent and p the +independent variable; from the integral of this linear equation, and the +original differential equation, the quantity p is then to be eliminated.</p> + +<p>Other types of solvable differential equations of the first order +are (1)</p> + +<p class="center">M dy/dx = N,</p> + +<p class="noind">where M, N are homogeneous polynomials in x and y, of the same +order; by putting v = y/x and eliminating y, the equation becomes +of the first type considered above, in v and x. An equation (aB ≷ bA)</p> + +<p class="center">(ax + by + c)dy/dx = Ax + By + C</p> + +<p class="noind">may be reduced to this rule by first putting x + h, y + k for x and y, +and determining h, k so that ah + bk + c = 0, Ah + Bk + C = 0.</p> + +<p>(2) An equation in which y does not explicitly occur,</p> + +<p class="center">ƒ(x, dy/dx) = 0,</p> + +<p class="noind">may, theoretically, be reduced to the type dy/dx = F(x); similarly +an equation F(y, dy/dx) = 0.</p> + +<p>(3) An equation</p> + +<p class="center">ƒ(dy/dx, x, y) = 0,</p> + +<p class="noind">which is an integral polynomial in dy/dx, may, theoretically, be +solved for dy/dx, as an algebraic equation; to any root dy/dx = F<span class="su">1</span>(x, y) +corresponds, suppose, a solution φ<span class="su">1</span>(x, y, c) = 0, where c is an arbitrary +constant; the product equation φ<span class="su">1</span>(x, y, c)φ<span class="su">2</span>(x, y, c) ... = 0, +consisting of as many factors as there were values of dy/dx, is +effectively as general as if we wrote φ<span class="su">1</span>(x, y, c<span class="su">1</span>)φ<span class="su">2</span>(x, y, c<span class="su">2</span>) ... = 0; +for, to evaluate the first form, we must necessarily consider the +factors separately, and nothing is then gained by the multiple +notation for the various arbitrary constants. The equation +φ<span class="su">1</span>(x, y, c)φ<span class="su">2</span>(x, y, c) ... = 0 is thus the solution of the given differential +equation.</p> + +<p>In all these cases there is, except for cases of singular solutions, one +and only one arbitrary constant in the most general solution of the +differential equation; that this must necessarily be so we may take +as obvious, the differential equation being supposed to arise by +elimination of this constant from the equation expressing its solution +and the equation obtainable from this by differentiation in regard +to x.</p> + +<p>A further type of differential equation of the first order, of the form</p> + +<p class="center">dy/dx = A + By + Cy²</p> + +<p class="noind">in which A, B, C are functions of x, will be briefly considered below +under differential equations of the second order.</p> + +<p>When we pass to ordinary differential equations of the second order, +that is, those expressing a relation between x, y, dy/dx and d²y/dx², +the number of types for which the solution can be found by a known +procedure is very considerably reduced. Consider the general linear +equation</p> + +<table class="math0" summary="math"> +<tr><td>d²y</td> <td rowspan="2">+ P</td> <td>dy</td> <td rowspan="2">+ Qy = R,</td></tr> +<tr><td class="denom">dx²</td> <td class="denom">dx</td></tr></table> + +<p class="noind">where P, Q, R are functions of x only. There is no method always +effective; the main general result for such a linear equation is that +if any particular function of x, say y<span class="su">1</span>, can be discovered, for which</p> + +<table class="math0" summary="math"> +<tr><td>d²y<span class="su">1</span></td> <td rowspan="2">+ P</td> <td>dy<span class="su">1</span></td> <td rowspan="2">+ Qy<span class="su">1</span> = 0,</td></tr> +<tr><td class="denom">dx²</td> <td class="denom">dx</td></tr></table> + +<p class="noind">then the substitution y = y<span class="su">1</span>η in the original equation, with R on +the right side, reduces this to a linear equation of the first order with +the dependent variable dη/dx. In fact, if y = y<span class="su">1</span>η we have</p> + +<table class="math0" summary="math"> +<tr><td>dy</td> + <td rowspan="2">= y<span class="su">1</span></td> <td>dη</td> + <td rowspan="2">+ η</td> <td>dy<span class="su">1</span></td> + <td rowspan="2">and</td> <td>d²y</td> + <td rowspan="2">= y<span class="su">1</span></td> <td>d²η</td> + <td rowspan="2">+ 2</td> <td>dy<span class="su">1</span></td> <td>dη</td> + <td rowspan="2">+ η</td> <td>d²y<span class="su">1</span></td> <td rowspan="2">,</td></tr> +<tr><td class="denom">dx</td> <td class="denom">dx</td> + <td class="denom">dx</td> <td class="denom">dx²</td> + <td class="denom">dx²</td> <td class="denom">dx</td> + <td class="denom">dx</td> <td class="denom">dx²</td></tr></table> + +<p class="noind">and thus</p> + +<table class="math0" summary="math"> +<tr> <td>d²y</td> <td rowspan="2">+ P</td> <td>dy</td> + <td rowspan="2">+ Qy = y<span class="su">1</span></td> <td>d²η</td> + <td rowspan="2">+ <span class="f200">(</span>2</td> <td>dy<span class="su">1</span></td> + <td rowspan="2">+ Py<span class="su">1</span><span class="f200">)</span></td> <td>dη</td> + <td rowspan="2">+ <span class="f200">(</span></td> <td>d²y<span class="su">1</span></td> + <td rowspan="2">+ P</td> <td>dy<span class="su">1</span></td> + <td rowspan="2">+ Qy<span class="su">1</span><span class="f200">)</span>η;</td> <td></td></tr> +<tr><td class="denom">dx²</td> <td class="denom">dx</td> + <td class="denom">dx²</td> <td class="denom">dx</td> + <td class="denom">dx</td> <td class="denom">dx²</td> + <td class="denom">dx</td></tr></table> + +<p class="noind">if then</p> + +<table class="math0" summary="math"> +<tr> <td>d²y<span class="su">1</span></td> <td rowspan="2">+ P</td> <td>dy<span class="su">1</span></td> + <td rowspan="2">+ Qy<span class="su">1</span> = 0,</td></tr> +<tr><td class="denom">dx²</td> <td class="denom">dx</td></tr></table> + +<p class="noind">and z denote dη/dx, the original differential equation becomes</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">y<span class="su">1</span></td> <td>dz</td> +<td rowspan="2">+ <span class="f200">(</span> 2</td> <td>dy<span class="su">1</span></td> +<td rowspan="2">Py<span class="su">1</span><span class="f200">)</span> z = R.</td></tr> +<tr><td class="denom">dx</td> <td class="denom">dx</td></tr></table> + +<p class="noind">From this equation z can be found by the rule given above for +the linear equation of the first order, and will involve one arbitrary +constant; thence y = y<span class="su">1</span> η = y<span class="su">1</span> ∫ zdx + Ay<span class="su">1</span>, +where A is another arbitrary constant, will be the general solution of the original +equation, and, as was to be expected, involves two arbitrary +constants.</p> + +<p>The case of most frequent occurrence is that in which the coefficients +P, Q are constants; we consider this case in some detail. +If θ be a root of the quadratic equation θ² + θP + Q = 0, it can be at +once seen that a particular integral of the differential equation with +zero on the right side is y<span class="su">1</span> = e<span class="sp">θx</span>. Supposing first the roots of the +quadratic equation to be different, and φ to be the other root, so that +φ + θ = -P, the auxiliary differential equation for z, referred to above, +becomes dz/dx + (θ − φ)z = Re<span class="sp">-θx</span> +which leads to ze<span class="sp">(θ-φ)</span> = B + ∫ Re<span class="sp">-θx</span>dx, +where B is an arbitrary constant, and hence to</p> + +<p class="center"> +y = Ae<span class="sp">θ<span class="sp"> x</span></span> + e<span class="sp">θ<span class="sp"> x</span></span> <span class="f150">∫</span> Be<span class="sp">(φ-θ)<span class="sp"> x</span></span> dx + e<span class="sp">θ<span class="sp"> x</span></span> <span class="f150">∫</span> e<span class="sp">(φ-θ)<span class="sp"> x</span></span> <span class="f150">∫</span> Re<span class="sp">-θ<span class="sp"> x</span></span> dxdx,</p> + +<p class="noind">or say to y = Ae<span class="sp">θ<span class="sp"> x</span></span> + Ce<span class="sp">θ<span class="sp"> x</span></span> + U, +where A, C are arbitrary constants and +U is a function of x, not present at all when R = 0. If the quadratic +equation θ² + Pθ + Q = 0 has equal roots, so that 2θ = -P, the +auxiliary equation in z becomes dz/dx = Re<span class="sp">θ<span class="sp"> x</span></span> +giving z = B + ∫ Re<span class="sp">θ<span class="sp"> x</span></span> dx, +where B is an arbitrary constant, and hence</p> + +<p class="center"> +y = (A + Bx)e<span class="sp">θ<span class="sp"> x</span></span> + e<span class="sp">θ<span class="sp"> x</span></span> <span class="f150">∫ ∫</span> Re<span class="sp">-θ<span class="sp"> x</span></span> dxdx,</p> + +<p class="noind">or, say, y = (A + Bx)e<span class="sp">θ<span class="sp"> x</span></span> + U, where A, B are arbitrary constants, and +U is a function of x not present at all when R = 0. The portion +Ae<span class="sp">θ<span class="sp"> x</span></span> + Be<span class="sp">θ<span class="sp"> x</span></span> or (A + Bx)e<span class="sp">θ<span class="sp"> x</span></span> of the solution, which is known as the <i>complementary +function</i>, can clearly be written down at once by inspection +of the given differential equation. The remaining portion U +may, by taking the constants in the complementary function +properly, be replaced by any particular solution whatever of the +differential equation</p> + +<table class="math0" summary="math"> +<tr><td>d²v</td> <td rowspan="2">+ P</td> <td>dy</td> + <td rowspan="2">+ Qy = R;</td></tr> +<tr><td class="denom">dx²</td> <td class="denom">dx</td> </tr></table> + +<p class="noind">for if u be any particular solution, this has a form</p> + +<p class="center">u = A<span class="su">0</span> e<span class="sp">θ<span class="sp"> x</span></span> + B<span class="su">0</span> e<span class="sp">φ<span class="sp"> x</span></span> + U,</p> + +<p class="noind">or a form</p> + +<p class="center">u = (A<span class="su">0</span> + B<span class="su">0</span>x) e<span class="sp">θ<span class="sp"> x</span></span> + U;</p> + +<p class="noind">thus the general solution can be written</p> + +<p class="center"> +(A − A<span class="su">0</span>)e<span class="sp">θ<span class="sp"> x</span></span> + (B − B<span class="su">0</span>)e<span class="sp">θ<span class="sp"> x</span></span> + u, +or {A − A<span class="su">0</span> + (B − B<span class="su">0</span>)x} e<span class="sp">θ<span class="sp"> x</span></span> + u,</p> + +<p class="noind">where A − A<span class="su">0</span>, B − B<span class="su">0</span>, like A, B, are arbitrary constants.</p> + +<p>A similar result holds for a linear differential equation of any order, +say</p> + +<table class="math0" summary="math"> +<tr><td>d<span class="sp">n</span>y</td> <td rowspan="2">+ P<span class="su">1</span></td> + <td>d<span class="sp">n-1</span>y</td> <td rowspan="2">+ ... + P<span class="su">n</span>y = R,</td></tr> +<tr><td class="denom">dx<span class="sp">n</span></td> <td class="denom">dx<span class="sp">n-1</span></td></tr></table> + +<p class="noind">where P<span class="su">1</span>, P<span class="su">2</span>, ... P<span class="su">n</span> are constants, and R is a function of x. If +we form the algebraic equation θ<span class="sp">n</span> + P<span class="su">1</span>θ<span class="sp">n-1</span> + ... + P<span class="su">n</span> = 0, and all the +roots of this equation be different, say they are θ<span class="su">1</span>, θ<span class="su">2</span>, ... θ<span class="su">n</span>, the +general solution of the differential equation is</p> + +<p class="center"> +y = A<span class="su">1</span> e<span class="sp">θ1 x</span> + A<span class="su">2</span> e<span class="sp">θ2 x</span> + ... + A<span class="su">n</span> e<span class="sp">θn x</span> + u,</p> + +<p class="noind">where A<span class="su">1</span>, A<span class="su">2</span>, ... A<span class="su">n</span> are arbitrary constants, and u is any +<span class="pagenum"><a name="page227" id="page227"></a>227</span> +particular solution whatever; but if there be one root θ<span class="su">1</span> repeated +r times, the terms A<span class="su">1</span> e<span class="sp">θ1<span class="sp"> x</span></span> + ... + A<span class="su">r</span> e<span class="sp">θr<span class="sp"> x</span></span> +must be replaced by (A<span class="su">1</span> + A<span class="su">2</span>x + ... + A<span class="su">r</span>x<span class="sp">r-1</span>)e<span class="sp">θ1<span class="sp"> x</span></span> +where A<span class="su">1</span>, ... A<span class="su">n</span> are arbitrary constants; +the remaining terms in the complementary function will +similarly need alteration of form if there be other repeated roots.</p> + +<p>To complete the solution of the differential equation we need some +method of determining a particular integral u; we explain a procedure +which is effective for this purpose in the cases in which R is +a sum of terms of the form e<span class="sp">ax</span>φ(x), where φ(x) is an integral polynomial +in x; this includes cases in which R contains terms of the +form cos bx·φ(x) or sin bx·φ(x). Denote d/dx by D; it is clear that if +u be any function of x, +D(e<span class="sp">ax</span>u) = e<span class="sp">ax</span>Du + ae<span class="sp">ax</span>u, +or say, D(e<span class="sp">ax</span>u) = e<span class="sp">ax</span>(D + a)u; +hence D²(e<span class="sp">ax</span>u), <i>i.e.</i> d²/dx² (e<span class="sp">ax</span>u), being equal to D(e<span class="sp">ax</span>v), +where v = (D + a)u, is equal to e<span class="sp">ax</span>(D + a)v, that is to e<span class="sp">ax</span>(D + a)²u. +In this way we find D<span class="sp">n</span>(e<span class="sp">ax</span>u) = e<span class="sp">ax</span>(D + a)<span class="sp">n</span>u, where n is any positive +integer. Hence if ψ(D) be any polynomial in D with constant coefficients, +ψ(D) (e<span class="sp">ax</span>u) = e<span class="sp">ax</span>ψ(D + a)u. +Next, denoting ∫ udx by D<span class="sp">-1</span>u, +and any solution of the differential equation dz/dx + az = u by z = (d + a)<span class="sp">-1</span>u, +we have D[e<span class="sp">ax</span>(D + a)<span class="sp">-1</span>u] = D(e<span class="sp">ax</span>z) = e<span class="sp">ax</span>(D + a)z = e<span class="sp">ax</span>u, +so that we may write D<span class="sp">-1</span>(e<span class="sp">ax</span>u) = e<span class="sp">ax</span>(D + a)<span class="sp">-1</span>u, +where the meaning is that one value of the left side is equal to one value of the +right side; from this, the expression D<span class="sp">-2</span>(e<span class="sp">ax</span>u), which means +D<span class="sp">-1</span>[D<span class="sp">-1</span>(e<span class="sp">ax</span>u)], +is equal to D<span class="sp">-1</span>(e<span class="sp">ax</span>z) and hence to e<span class="sp">ax</span>(D + a)<span class="sp">-1</span>z, +which we write e<span class="sp">ax</span>(D + a)<span class="sp">-2</span>u; proceeding thus we obtain</p> + +<p class="center">D<span class="sp">-n</span>(e<span class="sp">ax</span>u) = e<span class="sp">ax</span>(D + a)<span class="sp">-n</span>u,</p> + +<p class="noind">where n is any positive integer, and the meaning, as before, is that +one value of the first expression is equal to one value of the second. +More generally, if ψ(D) be any polynomial in D with constant coefficients, +and we agree to denote by [1/ψ(D)]u any solution z of the +differential equation ψ(D)z = u, we have, if v = [1/ψ(D + a)]u, the identity +ψ(D)(e<span class="sp">ax</span>v) = e<span class="sp">ax</span>ψ(D + a)v = e<span class="sp">ax</span>u, +which we write in the form</p> + +<table class="math0" summary="math"> +<tr><td>1</td> <td rowspan="2">(e<span class="sp">ax</span>u) = e<span class="sp">ax</span></td> <td>1</td> + <td rowspan="2">u.</td></tr> +<tr><td class="denom">ψ(D)</td> <td class="denom">ψ(D + a)</td></tr></table> + +<p class="noind">This gives us the first step in the method we are explaining, +namely that a solution of the differential equation +ψ(D)y = e<span class="sp">ax</span>u + e<span class="sp">bx</span>v + ... +where u, v, ... are any functions of x, is any function +denoted by the expression</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">e<span class="sp">ax</span></td> <td>1</td> + <td rowspan="2">u + e<span class="sp">bx</span></td> <td>1</td> + <td rowspan="2">v + ....</td></tr> +<tr><td class="denom">ψ(D + a)</td> <td class="denom">ψ(D + b)</td></tr></table> + +<p>It is now to be shown how to obtain one value of [1/ψ(D + a)]u, +when u is a polynomial in x, namely one solution of the differential equation +ψ(D + a)z = u. Let the highest power of x entering in u be x<span class="sp">m</span>; if t +were a variable quantity, the rational fraction in t, 1/ψ(t + a) by first +writing it as a sum of partial fractions, or otherwise, could be identically +written in the form</p> + +<p class="center"> +K<span class="su">r</span>t<span class="sp">-r</span> + K<span class="su">r-1</span>t<span class="sp">-r+1</span> + ... + K<span class="su">1</span>t<span class="sp">-1</span> + H + H<span class="su">1</span>t + ... + H<span class="su">m</span>t<span class="sp">m</span> + t<span class="sp">m+1</span>φ(t)/ψ(t + a),</p> + +<p class="noind">where φ(t) is a polynomial in t; this shows that there exists an +identity of the form</p> + +<p class="center"> +1 = ψ(t + a)(K<span class="su">r</span>t<span class="sp">−r</span> + ... + K<span class="su">1</span>t<span class="sp">−1</span> + H + H<span class="su">1</span>t + ... + H<span class="su">m</span>t<span class="sp">m</span>) + φ(t)t<span class="sp">m+1</span>,</p> + +<p class="noind">and hence an identity</p> + +<p class="center"> +u = ψ(D + a) [K<span class="su">r</span>D<span class="sp">−r</span> + ... + K<span class="su">1</span>D<span class="sp">−1</span> + H + H<span class="su">1</span>D + ... + H<span class="su">m</span>D<span class="sp">m</span>] u + φ(D) D<span class="sp">m+1</span>u;</p> + +<p class="noind">in this, since u contains no power of x higher than x<span class="sp">m</span>, the second +term on the right may be omitted. We thus reach the conclusion +that a solution of the differential equation ψ(D + a)z = u is given by</p> + +<p class="center"> +z = (K<span class="su">r</span>D<span class="sp">−r</span> + ... + K<span class="su">1</span>D<span class="sp">−1</span> + H + H<span class="su">1</span>D + ... + H<span class="su">m</span>D<span class="sp">m</span>)u,</p> + +<p class="noind">of which the operator on the right is obtained simply by expanding +1/ψ(D + a) in ascending powers of D, as if D were a numerical +quantity, the expansion being carried as far as the highest power of +D which, operating upon u, does not give zero. In this form every +term in z is capable of immediate calculation.</p> + +<p><i>Example.</i>—For the equation</p> + +<table class="math0" summary="math"> +<tr><td>d<span class="sp">4</span>v</td> <td rowspan="2">+ 2</td> <td>d²y</td> + <td rowspan="2">+ y = x³ cos x or (D² + 1)²y = x³ cos x,</td></tr> +<tr><td class="denom">dx<span class="sp">4</span></td> <td class="denom">dx<span class="sp">3</span></td></tr></table> + +<p class="noind">the roots of the associated algebraic equation (θ² + 1)² = 0 are θ = ±i, +each repeated; the complementary function is thus</p> + +<p class="center">(A + Bx)e<span class="sp">ix</span> + (C + Dx)e<span class="sp">−ix</span>,</p> + +<p class="noind">where A, B, C, D are arbitrary constants; this is the same as</p> + +<p class="center">(H + Kx) cos x + (M + Nx) sin x,</p> + +<p class="noind">where H, K, M, N are arbitrary constants. To obtain a particular +integral we must find a value of (1 + D²)−²x³ cos x; +this is the real part of (1 + D²)−² e<span class="sp">ix</span>x³ and hence of e<span class="sp">ix</span> [1 + (D + i)²]−² x³</p> + +<p class="noind">or</p> + +<p class="center">e<span class="sp">ix</span> [2iD(1 + ½iD)]−² x³,</p> + +<p class="noind">or</p> + +<p class="center">−¼e<span class="sp">ix</span> D−² (1 + iD − ¾D² − ½iD³ + <span class="spp">5</span>⁄<span class="suu">16</span>D<span class="sp">4</span> + <span class="spp">3</span>⁄<span class="suu">16</span>iD<span class="sp">5</span> ...)x³,</p> + +<p class="noind">or</p> + +<p class="center">−¼e<span class="sp">ix</span> (<span class="spp">1</span>⁄<span class="suu">20</span>x<span class="sp">5</span> + ¼ix<span class="sp">4</span> − ¾x³ − <span class="spp">3</span>⁄<span class="suu">2</span> ix² + <span class="spp">15</span>⁄<span class="suu">8</span> x + <span class="spp">9</span>⁄<span class="suu">8</span> i);</p> + +<p class="noind">the real part of this is</p> + +<p class="center">−¼ (<span class="spp">1</span>⁄<span class="suu">20</span> x<span class="sp">5</span> − ¾x² + <span class="spp">15</span>⁄<span class="suu">8</span>x) cos x + ¼ (¼x<span class="sp">4</span> − <span class="spp">3</span>⁄<span class="suu">4</span>x² + <span class="spp">9</span>⁄<span class="suu">8</span>) sin x.</p> + +<p class="noind">This expression added to the complementary function found above +gives the complete integral; and no generality is lost by omitting +from the particular integral the terms −<span class="spp">15</span>⁄<span class="suu">32</span>x cos x + <span class="spp">9</span>⁄<span class="suu">32</span> sin x, which +are of the types of terms already occurring in the complementary +function.</p> + +<p>The symbolical method which has been explained has wider applications +than that to which we have, for simplicity of explanation, +restricted it. For example, if ψ(x) be any function of x, and +a<span class="su">1</span>, a<span class="su">2</span>, ... a<span class="su">n</span> be different constants, and [(t + a<span class="su">1</span>) (t + a<span class="su">2</span>) ... (t + a<span class="su">n</span>)]<span class="sp">−1</span> +when expressed in partial fractions be written Σc<span class="su">m</span>(t + a<span class="su">m</span>)<span class="sp">−1</span>, a particular +integral of the differential equation (D + a<span class="su">1</span>)(D + a<span class="su">2</span>) ... +(D + a<span class="su">n</span>)y = ψ(x) is given by</p> + +<p class="center"> +y = Σc<span class="su">m</span>(D + a<span class="su">m</span>)<span class="sp">−1</span> ψ(x) = Σc<span class="su">m</span> (D + a<span class="su">m</span>)<span class="sp">−1</span> e<span class="sp">−a</span>m<span class="sp">x</span>e<span class="sp">a</span>m<span class="sp">x</span> ψ(x) = +Σc<span class="su">m</span>e<span class="sp">−a</span>m<span class="sp">x</span>D<span class="sp">−1</span> <span class="f150">(</span>e<span class="sp">a</span>m<span class="sp">x</span>ψ(x) <span class="f150">)</span> = Σc<span class="su">m</span>e<span class="sp">−a</span>m<span class="sp">x</span> <span class="f150">∫</span> e<span class="sp">a</span>m<span class="sp">x</span>ψ(x)dx.</p> + +<p class="noind">The particular integral is thus expressed as a sum of n integrals. +A linear differential equation of which the left side has the form</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">x<span class="sp">n</span></td> <td>d<span class="sp">n</span>y</td> +<td rowspan="2">+ P<span class="su">1</span>x<span class="sp">n−1</span></td> <td>d<span class="sp">n−1</span>y</td> +<td rowspan="2">+ ... + P<span class="su">n−1</span>x</td> <td>dy</td> +<td rowspan="2">+ P<span class="su">n</span>y,</td></tr> +<tr><td class="denom">dx<span class="sp">n</span></td> <td class="denom">dx<span class="sp">n−1</span></td> <td class="denom">dx</td></tr></table> + +<p class="noind">where P<span class="su">1</span>, ... P<span class="su">n</span> are constants, can be reduced to the case considered +above. Writing x = e<span class="sp">t</span> we have the identity</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">x<span class="sp">m</span></td> <td>d<span class="sp">m</span>u</td> +<td rowspan="2">= θ(θ − 1)(θ − 2) ... (θ − m + 1)u, where θ = d/dt.</td></tr> +<tr><td class="denom">dx<span class="sp">m</span></td></tr></table> + +<p>When the linear differential equation, which we take to be of the +second order, has variable coefficients, though there is no general rule +for obtaining a solution in finite terms, there are some results which +it is of advantage to have in mind. We have seen that if one solution +of the equation obtained by putting the right side zero, say y<span class="su">1</span>, be +known, the equation can be solved. If y<span class="su">2</span> be another solution of</p> + +<table class="math0" summary="math"> +<tr><td>d²y</td> <td rowspan="2">+ P</td> <td>dy</td> +<td rowspan="2">+ Qy = 0,</td></tr> +<tr><td class="denom">dx²</td> <td class="denom">dx</td></tr></table> + +<p class="noind">there being no relation of the form my<span class="su">1</span> + ny<span class="su">2</span> = k, where m, n, k are +constants, it is easy to see that</p> + +<table class="math0" summary="math"> +<tr><td>d</td> <td rowspan="2">(y<span class="su">1</span>′y<span class="su">2</span> − y<span class="su">1</span>y<span class="su">2</span>′) = P(y<span class="su">1</span>′y<span class="su">2</span> − y<span class="su">1</span>y<span class="su">2</span>′),</td></tr> +<tr><td class="denom">dx</td></tr></table> + +<p class="noind">so that we have</p> + +<p class="center">y<span class="su">1</span>′y<span class="su">2</span> − y<span class="su">1</span>y<span class="su">2</span>′ += A exp. <span class="f150">(∫</span> Pdx<span class="f150">)</span>,</p> + +<p class="noind">where A is a suitably chosen constant, and exp. z denotes e<span class="sp">z</span>. In terms +of the two solutions y<span class="su">1</span>, y<span class="su">2</span> of the differential equation having zero on +the right side, the general solution of the equation with R = φ(x) on +the right side can at once be verified to be Ay<span class="su">1</span> + By<span class="su">2</span> + y<span class="su">1</span>u − y<span class="su">2</span>v, +where u, v respectively denote the integrals</p> + +<p class="center"> +u = <span class="f150">∫</span> y<span class="su">2</span>φ(x) (y<span class="su">1</span>′y<span class="su">2</span> − y<span class="su">2</span>′y<span class="su">1</span>)<span class="sp">−1</span>dx, v = <span class="f150">∫</span> y<span class="su">1</span>φ(x) (y<span class="su">1</span>′y<span class="su">2</span> − y<span class="su">2</span>′y<span class="su">1</span>)<span class="sp">−1</span>dx.</p> + +<p>The equation</p> + +<table class="math0" summary="math"> +<tr><td>d²y</td> <td rowspan="2">+ P</td> <td>dy</td> + <td rowspan="2">+ Qy = 0,</td></tr> +<tr><td class="denom">dx²</td> <td class="denom">dx</td></tr></table> + +<p class="noind">by writing y = v exp. (-½ ∫ Pdx), is at once seen to be reduced to +d²v/dx² + Iv = 0, where I = Q − ½dP/dx − ¼P². If η = − 1/v dv/dx, the equation +d²v/dx² + Iv = 0 becomes dη/dx = I + η², a non-linear equation of the first +order.</p> + +<p>More generally the equation</p> + +<table class="math0" summary="math"> +<tr><td>dη</td> <td rowspan="2">= A + Bη + Cη²,</td></tr> +<tr><td class="denom">dx</td></tr></table> + +<p class="noind">where A, B, C are functions of x, is, by the substitution</p> + +<table class="math0" style="border-collapse: separate;" summary="math"> +<tr><td rowspan="2">η = −</td> <td>1</td> <td>dy</td></tr> +<tr><td class="denom">Cy</td> <td class="denom">dx</td></tr></table> + +<p class="noind">reduced to the linear equation</p> + +<table class="math0" style="border-collapse: separate;" summary="math"> +<tr><td>d²y</td> <td rowspan="2">− <span class="f150">(</span>B + </td> <td>1</td> <td>dC</td> + <td rowspan="2"><span class="f150">)</span></td> <td>dy</td> + <td rowspan="2">+ ACy = 0.</td></tr> +<tr><td class="denom">dx²</td> <td class="denom">C</td> + <td class="denom">dx</td> <td class="denom">dx</td></tr></table> + +<p class="noind">The equation</p> + +<table class="math0" summary="math"> +<tr><td>dη</td> <td rowspan="2">= A + Bη + Cη²,</td></tr> +<tr><td class="denom">dx</td></tr></table> + +<p class="noind">known as Riccati’s equation, is transformed into an equation of +the same form by a substitution of the form η = (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 η<span class="su">0</span>, the +<span class="pagenum"><a name="page228" id="page228"></a>228</span> +substitution η = η<span class="su">0</span> − 1/Y enables us at once to obtain the general +solution; for instance, when</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">2B =</td> <td>d</td> + <td rowspan="2">log <span class="f150">(</span></td> <td>A</td> + <td rowspan="2"><span class="f150">)</span>,</td></tr> +<tr><td class="denom">dx</td> <td class="denom">C</td></tr></table> + +<p class="noind">a particular solution is η<span class="su">0</span> = √(-A/C). This is a case of the remark, +often useful in practice, that the linear equation</p> + +<table class="math0" style="border-collapse: separate;" summary="math"> +<tr><td rowspan="2">φ(x)</td> <td>d²y</td> + <td rowspan="2">+ ½</td> <td>dφ</td> <td>dy</td> + <td rowspan="2">+ μy = 0,</td></tr> +<tr><td class="denom">dx²</td> <td class="denom">dx</td> <td class="denom">dx</td></tr></table> + +<p class="noind">where μ is a constant, is reducible to a standard form by taking a new +independent variable z = ∫ dx[φ(x)]<span class="sp">-½</span>.</p> + +<p>We pass to other types of equations of which the solution can be +obtained by rule. We may have cases in which there are two +dependent variables, x and y, and one independent variable t, the +differential coefficients dx/dt, dy/dt being given as functions of x, y +and t. Of such equations a simple case is expressed by the pair</p> + +<table class="math0" summary="math"> +<tr><td>dx</td> <td rowspan="2">= ax + by + c,</td> +<td>dy</td> <td rowspan="2">a′x + b′y + c′,</td></tr> +<tr><td class="denom">dt</td> <td class="denom">dt</td></tr></table> + +<p class="noind">wherein the coefficients a, b, c, a′, b′, c′, are constants. To integrate +these, form with the constant λ the differential coefficient of +z = x + λy, that is dz/dt = (a + λa′)x + (b + λb′)y + c + λc′, the quantity +λ being so chosen that b + λb′ = λ(a + λa′), so that we have +dz/dt = (a + λa′)z + c + λc′; this last equation is at once integrable +in the form z(a + λa′) + c + λc′ = Ae<span class="sp">(a + λa′)t</span>, where A is an arbitrary +constant. In general, the condition b + λb′ = λ(a + λa′) is satisfied by +two different values of λ, say λ<span class="su">1</span>, λ<span class="su">2</span>; the solutions corresponding to +these give the values of x +λ<span class="su">1</span>y and x + λ<span class="su">2</span>y, from which x and y can +be found as functions of t, involving two arbitrary constants. If, +however, the two roots of the quadratic equation for λ are equal, +that is, if (a − b′)² + 4a′b = 0, the method described gives only one +equation, expressing x + λy in terms of t; by means of this equation +y can be eliminated from dx/dt = ax + by + c, leading to an equation +of the form dx/dt = Px + Q + Re<span class="sp">(a + λa′)t</span>, where P, Q, R are constants. +The integration of this gives x, and thence y can be found.</p> + +<p>A similar process is applicable when we have three or more +dependent variables whose differential coefficients in regard to the +single independent variables are given as linear functions of the +dependent variables with constant coefficients.</p> + +<p>Another method of solution of the equations</p> + +<p class="center">dx/dt = ax + by + c, dy/dt = a′x + b′y + c′,</p> + +<p class="noind">consists in differentiating the first equation, thereby obtaining</p> + +<table class="math0" summary="math"> +<tr><td>d²x</td> <td rowspan="2">= a</td> <td>dx</td> +<td rowspan="2">+ b</td> <td>dy</td> <td rowspan="2">;</td></tr> +<tr><td class="denom">dt²</td> <td class="denom">dt</td> <td class="denom">dx</td></tr></table> + +<p class="noind">from the two given equations, by elimination of y, we can express +dy/dt as a linear function of x and dx/dt; we can thus form an +equation of the shape d²x/dt² = P + Qx + Rdx/dt, where P, Q, R are +constants; this can be integrated by methods previously explained, +and the integral, involving two arbitrary constants, gives, +by the equation dx/dt = ax + by + c, the corresponding value of y. +Conversely it should be noticed that any single linear differential +equation</p> + +<table class="math0" summary="math"> +<tr><td>d²x</td> <td rowspan="2">= u + vx + w</td> <td>dx</td> <td rowspan="2">,</td></tr> +<tr><td class="denom">dt²</td> <td class="denom">dt</td></tr></table> + +<p class="noind">where u, v, w are functions of t, by writing y for dx/dt, is equivalent +with the two equations dx/dt = y, dy/dt = u + vx + wy. In fact a +similar reduction is possible for any system of differential equations +with one independent variable.</p> + +<p>Equations occur to be integrated of the form</p> + +<p class="center">Xdx + Ydy + Zdz = 0,</p> + +<p class="noind">where X, Y, Z are functions of x, y, z. We consider only the case in +which there exists an equation φ(x, y, z) = C whose differential</p> + +<table class="math0" summary="math"> +<tr><td>∂φ</td> <td rowspan="2">dx +</td> <td>∂φ</td> +<td rowspan="2">dy +</td> <td>∂φ</td> <td rowspan="2">dz = 0</td></tr> +<tr><td class="denom">∂x</td> <td class="denom">∂y</td> <td class="denom">∂z</td></tr></table> + +<p class="noind">is equivalent with the given differential equation; that is, μ being +a proper function of x, y, z, we assume that there exist equations</p> + +<table class="math0" summary="math"> +<tr><td>∂φ</td> <td rowspan="2">= μX,</td> <td>∂φ</td> +<td rowspan="2">= μY,</td> <td>∂φ</td> +<td rowspan="2">= μZ;</td></tr> +<tr><td class="denom">∂x</td> <td class="denom">∂y</td> <td class="denom">∂z</td></tr></table> + +<p class="noind">these equations require</p> + +<table class="math0" summary="math"> +<tr><td>∂</td> <td rowspan="2">(μY) ≈</td> <td>∂</td> +<td rowspan="2">(μZ), &c.,</td></tr> +<tr><td class="denom">∂z</td> <td class="denom">∂y</td></tr></table> + +<p class="noind">and hence</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">X<span class="f200">(</span></td> <td>∂Z</td> +<td rowspan="2">−</td> <td>∂Y</td> +<td rowspan="2"><span class="f200">)</span> + Y<span class="f200">(</span></td> <td>∂X</td> +<td rowspan="2">−</td> <td>∂Z</td> +<td rowspan="2"><span class="f200">)</span> + Z<span class="f200">(</span></td> <td>∂Y</td> +<td rowspan="2">−</td> <td>∂X</td> <td rowspan="2"><span class="f200">)</span> = 0;</td></tr> +<tr><td class="denom">∂y</td> <td class="denom">∂z</td> + <td class="denom">∂z</td> <td class="denom">∂x</td> + <td class="denom">∂x</td> <td class="denom">∂y</td></tr></table> + +<p class="noind">conversely it can be proved that this is sufficient in order that μ +may exist to render μ(Xdx + Ydy + Zdz) a perfect differential; in +particular it may be satisfied in virtue of the three equations such as</p> + +<table class="math0" summary="math"> +<tr><td>∂Z</td> <td rowspan="2">−</td> <td>∂Y</td> +<td rowspan="2">= 0;</td></tr> +<tr><td class="denom">∂y</td> <td class="denom">∂z</td></tr></table> + +<p class="noind">in which case we may take μ = 1. Assuming the condition in +its general form, take in the given differential equation a plane +section of the surface φ = 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 ψ(x, y, z) = constant, be its +integral, the constant z entering, as a rule, in ψ because it enters in +X and Y. Now differentiate the relation ψ(x, y, z) = ƒ(z), where ƒ +is a function to be determined, so obtaining</p> + +<table class="math0" summary="math"> +<tr><td>∂ψ</td> <td rowspan="2">dx +</td> <td>∂ψ</td> +<td rowspan="2">dy + <span class="f200">(</span></td> <td>∂ψ</td> +<td rowspan="2">−</td> <td>dƒ</td> <td rowspan="2"><span class="f200">)</span> dz = 0;</td></tr> +<tr><td class="denom">∂x </td> <td class="denom">∂y</td> + <td class="denom">∂z</td> <td class="denom">dz</td></tr></table> + +<p class="noind">there exists a function σ of x, y, z such that</p> + +<table class="math0" summary="math"> +<tr> <td>∂ψ</td> <td rowspan="2">= σX,</td> <td>∂ψ</td> +<td rowspan="2">= σY,</td></tr> +<tr><td class="denom">∂x</td> <td class="denom">∂y</td></tr></table> + +<p class="noind">because ψ = constant, is the integral of Xdx + Ydy = 0; we desire to +prove that ƒ can be chosen so that also, in virtue of ψ(x, y, z) = ƒ(z), +we have</p> + +<table class="math0" summary="math"> +<tr><td>∂ψ</td> <td rowspan="2">−</td> <td>dƒ</td> +<td rowspan="2">= σZ, namely</td> <td>dƒ</td> +<td rowspan="2">=</td> <td>∂ψ</td> <td rowspan="2">− σZ;</td></tr> +<tr><td class="denom">∂z</td> <td class="denom">dz</td> + <td class="denom">dz</td> <td class="denom">∂z</td></tr></table> + +<p class="noind">if this can be proved the relation ψ(x, y, z) − ƒ(z) = constant, will be +the integral of the given differential equation. To prove this it is +enough to show that, in virtue of ψ(x, y, z) = ƒ(z), the function +∂ψ/∂x − σZ can be expressed in terms of z only. Now in consequence +of the originally assumed relations,</p> + +<table class="math0" summary="math"> +<tr><td>∂ψ</td> <td rowspan="2">= μX,</td> <td>∂φ</td> +<td rowspan="2">= μY,</td> <td>∂φ</td> <td rowspan="2">= μZ,</td></tr> +<tr><td class="denom">∂x</td> <td class="denom">∂y</td> <td class="denom">∂z</td></tr></table> + +<p class="noind">we have</p> + +<table class="math0" summary="math"> +<tr><td>∂ψ</td> <td rowspan="2"><span class="f200">/</span></td> <td>∂φ</td> +<td rowspan="2">=</td> <td>σ</td> +<td rowspan="2">=</td> <td>∂ψ</td> +<td rowspan="2"><span class="f200">/</span></td> <td>∂φ</td> <td rowspan="2">,</td></tr> +<tr><td class="denom">∂x</td> <td class="denom">∂x</td> + <td class="denom">μ</td> <td class="denom">∂y</td> <td class="denom">∂y</td></tr></table> + +<p class="noind">and hence</p> + +<table class="math0" style="border-collapse: separate;" summary="math"> +<tr><td>∂ψ</td> <td>∂φ</td> <td rowspan="2">−</td> + <td>∂ψ</td> <td>∂φ</td> <td rowspan="2">= 0;</td></tr> +<tr><td class="denom">∂x</td> <td class="denom">∂y</td> + <td class="denom">∂y</td> <td class="denom">∂x</td></tr></table> + +<p class="noind">this shows that, as functions of x and y, ψ is a function of φ (see the +note at the end of part i. of this article, on Jacobian determinants), +so that we may write ψ = F(z, φ), from which</p> + +<table class="math0" style="border-collapse: separate;" summary="math"> +<tr><td>σ</td> <td rowspan="2">=</td> <td>∂F</td> +<td rowspan="2">; then</td> <td>∂ψ</td> +<td rowspan="2">=</td> <td>∂F</td> +<td rowspan="2">+</td> <td>∂F</td> <td>∂φ</td> +<td rowspan="2">=</td> <td>∂F</td> +<td rowspan="2">+</td> <td>σ</td> +<td rowspan="2">· μZ =</td> <td>∂F</td> +<td rowspan="2">+ σZ or</td> <td>∂ψ</td> +<td rowspan="2">− σZ =</td> <td>∂F</td> <td rowspan="2">;</td></tr> +<tr><td class="denom">μ</td> <td class="denom">∂φ</td> +<td class="denom">∂z</td> <td class="denom">∂z</td> +<td class="denom">∂φ</td> <td class="denom">∂z</td> +<td class="denom">∂z</td> <td class="denom">μ</td> +<td class="denom">∂z</td> <td class="denom">∂z</td> <td class="denom">∂z</td></tr></table> + +<p class="noind">in virtue of ψ(x, y, z) = ƒ(z), and ψ = F(z, φ), the function φ can be +written in terms of z only, thus ∂F/∂z can be written in terms of z only, +and what we required to prove is proved.</p> + +<p>Consider lastly a simple type of differential equation containing +<i>two</i> independent variables, say x and y, and one dependent variable +z, namely the equation</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">P</td> <td>∂z</td> +<td rowspan="2">+ Q</td> <td>∂z</td> <td rowspan="2">= R,</td></tr> +<tr><td class="denom">∂x</td> <td class="denom">∂y</td></tr></table> + +<p class="noind">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</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">P</td> <td>∂u</td> +<td rowspan="2">+ Q</td> <td>∂u</td> +<td rowspan="2">+ R</td> <td>∂u</td> +<td rowspan="2">= 0 and P</td> <td>∂v</td> +<td rowspan="2">+ Q</td> <td>∂v</td> +<td rowspan="2">+ R</td> <td>∂v</td> <td rowspan="2">= 0.</td></tr> +<tr><td class="denom">∂x</td> <td class="denom">∂y</td> +<td class="denom">∂z</td> <td class="denom">∂x</td> +<td class="denom">∂y</td> <td class="denom">∂z</td></tr></table> + +<p class="noind">Then if F(x, y, z) = 0 be a relation satisfying the original differential +equations, this relation giving rise to</p> + +<table class="math0" style="border-collapse: separate;" summary="math"> +<tr><td>∂F</td> <td rowspan="2">+</td> <td>∂F</td> <td>∂z</td> +<td rowspan="2">= 0 and</td> <td>∂F</td> +<td rowspan="2">+</td> <td>∂F</td> <td>∂z</td> +<td rowspan="2">= 0, we have P</td> <td>∂F</td> +<td rowspan="2">+ Q</td> <td>∂F</td> +<td rowspan="2">+ R</td> <td>∂F</td> <td rowspan="2">= 0.</td></tr> +<tr><td class="denom">∂x</td> <td class="denom">∂z</td> <td class="denom">∂x</td> +<td class="denom">∂y</td> <td class="denom">∂z</td> <td class="denom">∂y</td> +<td class="denom">∂x</td> <td class="denom">∂y</td> <td class="denom">∂z</td></tr></table> + +<p class="noind">It follows that the determinant of three rows and columns vanishes +whose first row consists of the three quantities ∂F/∂x, ∂F/∂y, ∂F/∂z, +whose second row consists of the three quantities ∂u/∂x, ∂u/∂y, ∂u/∂z, +whose third row consists similarly of the partial derivatives of v. +The vanishing of this so-called Jacobian determinant is known to +imply that F is expressible as a function of u and v, unless these are +themselves functionally related, which is contrary to hypothesis +(see the note below on Jacobian determinants). Conversely, any +relation φ(u, v) = 0 can easily be proved, in virtue of the equations +satisfied by u and v, to lead to</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">P</td> <td>dz</td> +<td rowspan="2">+ Q</td> <td>dz</td> <td rowspan="2">= R.</td></tr> +<tr><td class="denom">dx</td> <td class="denom">dx</td></tr></table> + +<p class="noind">The solution of this partial equation is thus reduced to the solution +of the two ordinary differential equations expressed by +dx/P = dy/Q = dz/R. In regard to this problem one remark may be +made which is often of use in practice: when one equation u = a +has been found to satisfy the differential equations, we may utilize +this to obtain the second equation v = b; for instance, we may, by +means of u = a, eliminate z—when then from the resulting equations +in x and y a relation v = b has been found containing x and y and a, +the substitution a = u will give a relation involving x, y, z.</p> + +<p><i>Note on Jacobian Determinants.</i>—The fact assumed above that the +vanishing of the Jacobian determinant whose elements are the partial +derivatives of three functions F, u, v, of three variables x, y, z, +<span class="pagenum"><a name="page229" id="page229"></a>229</span> +involves that there exists a functional relation connecting the three +functions F, u, v, may be proved somewhat roughly as follows:—</p> + +<p>The corresponding theorem is true for any number of variables. +Consider first the case of two functions p, q, of two variables x, y. +The function p, not being constant, must contain one of the variables, +say x; we can then suppose x expressed in terms of y and the function +p; thus the function q can be expressed in terms of y and the function +p, say q = Q(p, y). This is clear enough in the simplest cases which +arise, when the functions are rational. Hence we have</p> + +<table class="math0" style="border-collapse: separate;" summary="math"> +<tr><td>∂q</td> <td rowspan="2">=</td> <td>∂Q</td> <td>∂p</td> +<td rowspan="2">and</td> <td>∂q</td> +<td rowspan="2">=</td> <td>∂Q</td> <td>∂p</td> +<td rowspan="2">+</td> <td>∂Q</td> <td rowspan="2">;</td></tr> +<tr><td class="denom">∂x</td> <td class="denom">∂p</td> <td class="denom">∂x</td> +<td class="denom">∂y</td> <td class="denom">∂p</td> <td class="denom">∂y</td> +<td class="denom">∂y</td></tr></table> + +<p class="noind">these give</p> + +<table class="math0" summary="math"> +<tr><td>∂p</td> <td>∂q</td> <td rowspan="2">−</td> +<td>∂p</td> <td>∂q</td> <td rowspan="2">=</td> +<td>∂p</td> <td>∂Q</td> <td rowspan="2">;</td></tr> + <tr><td class="ov">∂x</td> <td class="ov">∂y</td> + <td class="ov">∂y</td> <td class="ov">∂x</td> + <td class="ov">∂x</td> <td class="ov">∂y</td></tr></table> + +<p class="noind">by hypothesis ∂p/∂x is not identically zero; therefore if the Jacobian +determinant of p and q in regard to x and y is zero identically, so is +∂Q/∂y, or Q does not contain y, so that q is expressible as a function +of p only. Conversely, such an expression can be seen at once to +make the Jacobian of p and q vanish identically.</p> + +<p>Passing now to the case of three variables, suppose that the +Jacobian determinant of the three functions F, u, v in regard to +x, y, z is identically zero. We prove that if u, v are not themselves +functionally connected, F is expressible as a function of u and v. +Suppose first that the minors of the elements of ∂F/∂x, ∂F/∂y, ∂F/∂z +in the determinant are all identically zero, namely the three determinants +such as</p> + +<table class="math0" summary="math"> +<tr><td>∂u</td> <td>∂v</td> <td rowspan="2">−</td> +<td>∂u</td> <td>∂v</td> <td rowspan="2">;</td></tr> +<tr><td class="ov">∂y</td> <td class="ov">∂z</td> +<td class="ov">∂z</td> <td class="ov">∂y</td></tr></table> + +<p class="noind">then by the case of two variables considered above there exist three +functional relations. ψ<span class="su">1</span>(u, v, x) = 0, ψ<span class="su">2</span>(u, v, y) = 0, ψ<span class="su">3</span>(u, v, z) = 0, of which +the first, for example, follows from the vanishing of</p> + +<table class="math0" summary="math"> +<tr><td>∂u</td> <td>∂v</td> <td rowspan="2">−</td> +<td>∂u</td> <td>∂v</td> <td rowspan="2">.</td></tr> +<tr><td class="ov">∂y</td> <td class="ov">∂z</td> +<td class="ov">∂z</td> <td class="ov">∂y</td></tr></table> + +<p class="noind">We cannot assume that x is absent from ψ<span class="su">1</span>, or y from ψ<span class="su">2</span>, or z from ψ<span class="su">3</span>; +but conversely we cannot simultaneously have x entering in ψ<span class="su">1</span>, and +y in ψ<span class="su">2</span>, and z in ψ<span class="su">3</span>, or else by elimination of u and v from the three +equations ψ<span class="su">1</span> = 0, ψ<span class="su">2</span> = 0, ψ<span class="su">3</span> = 0, we should find a necessary relation +connecting the three independent quantities x, y, z; which is absurd. +Thus when the three minors of ∂F/∂x, ∂F/∂y, ∂F/∂z in the Jacobian +determinant are all zero, there exists a functional relation connecting +u and v only. Suppose no such relation to exist; we can then +suppose, for example, that</p> + +<table class="math0" summary="math"> +<tr><td>∂u</td> <td>∂v</td> <td rowspan="2">−</td> +<td>∂u</td> <td>∂v</td></tr> +<tr><td class="ov">∂y</td> <td class="ov">∂z</td> +<td class="ov">∂z</td> <td class="ov">∂y</td></tr></table> + +<p class="noind">is not zero. Then from the equations u(x, y, z) = u, v(x, y, z) = v we can +express y and z in terms of u, v, and x (the attempt to do this could +only fail by leading to a relation connecting u, v and x, and the +existence of such a relation would involve that the determinant</p> + +<table class="math0" summary="math"> +<tr><td>∂u</td> <td>∂v</td> <td rowspan="2">−</td> +<td>∂u</td> <td>∂v</td></tr> +<tr><td class="ov">∂y</td> <td class="ov">∂z</td> +<td class="ov">∂z</td> <td class="ov">∂y</td></tr></table> + +<p class="noind">was zero), and so write F in the form F(x, y, z) = Φ(u, v, x). We then +have</p> + +<table class="math0" summary="math"> +<tr><td>∂F</td> <td rowspan="2">=</td> <td>∂Φ</td> <td>∂u</td> +<td rowspan="2">+</td> <td>∂Φ</td> <td>∂v</td> +<td rowspan="2">+</td> <td>∂Φ</td> <td rowspan="2">,</td> <td>∂F</td> +<td rowspan="2">=</td> <td>∂Φ</td> <td>∂u</td> <td rowspan="2">+</td> <td>∂Φ</td> <td>∂v</td> +<td rowspan="2">,</td> <td>∂F</td> +<td rowspan="2">=</td> <td>∂Φ</td> <td>∂u</td> <td rowspan="2">+</td> <td>∂Φ</td> <td>∂v</td> +<td rowspan="2">;</td> +</tr> +<tr><td class="ov">∂x</td> <td class="ov">∂u</td> <td class="ov">∂x</td> <td class="ov">∂v</td> +<td class="ov">∂x</td> <td class="ov">∂x</td> <td class="ov">∂y</td> <td class="ov">∂u</td> +<td class="ov">∂y</td> <td class="ov">∂v</td> <td class="ov">∂y</td> <td class="ov">∂z</td> +<td class="ov">∂u</td> <td class="ov">∂z</td> <td class="ov">∂v</td> <td class="ov">∂z</td></tr></table> + +<p class="noind">thereby the Jacobian determinant of F, u, v is reduced to</p> + +<table class="math0" summary="math"> +<tr><td>∂Φ</td> <td rowspan="2"><span class="f150">(</span></td> +<td>∂u</td> <td>∂v</td> <td rowspan="2">−</td> +<td>∂u</td> <td>∂v</td> <td rowspan="2"><span class="f150">)</span>;</td></tr> +<tr><td class="ov">∂x</td> <td class="ov">∂y</td> <td class="ov">∂z</td> +<td class="ov">∂z</td> <td class="ov">∂y</td></tr></table> + +<p class="noind">by hypothesis the second factor of this does not vanish identically; +hence ∂Φ/∂x = 0 identically, and Φ does not contain x; so that F +is expressible in terms of u, v only; as was to be proved.</p> + +<p class="center pt2"><i>Part II.—General Theory.</i></p> + +<p>Differential equations arise in the expression of the relations +between quantities by the elimination of details, either unknown +or regarded as unessential to the formulation of the relations in +question. They give rise, therefore, to the two closely connected +problems of determining what arrangement of details is consistent +with them, and of developing, apart from these details, the general +properties expressed by them. Very roughly, two methods of +study can be distinguished, with the names Transformation-theories, +Function-theories; the former is concerned with the +reduction of the algebraical relations to the fewest and simplest +forms, eventually with the hope of obtaining explicit expressions +of the dependent variables in terms of the independent variables; +the latter is concerned with the determination of the general +descriptive relations among the quantities which are involved by +the differential equations, with as little use of algebraical calculations +as may be possible. Under the former heading we may, +with the assumption of a few theorems belonging to the latter, +arrange the theory of partial differential equations and Pfaff’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.</p> + +<p>When we speak of a function without qualification, it is to be +understood that in the immediate neighbourhood of a particular +set x<span class="su">0</span>, y<span class="su">0</span>, ... of values of the independent variables x, y, ... +of the function, at whatever point of the range of values for +x, y, ... under consideration x<span class="su">0</span>, y<span class="su">0</span>, ... may be chosen, the +function can be expressed as a series of positive integral powers +of the differences x − x<span class="su">0</span>, y − y<span class="su">0</span>, ..., convergent when these are +sufficiently small (see <span class="sc"><a href="#artlinks">Function: Functions of Complex Variables</a></span>). +Without this condition, which we express by saying that +the function is developable about x<span class="su">0</span>, y<span class="su">0</span>, ..., many results +provisionally stated in the transformation theories would be +unmeaning or incorrect. If, then, we have a set of k functions, +ƒ<span class="su">1</span> ... ƒ<span class="su">k</span> of n independent variables x<span class="su">1</span> ... x<span class="su">n</span>, we say that +they are independent when n ≥ 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 dƒ<span class="su">r</span>/dx<span class="su">1</span>, ... dƒ<span class="su">r</span>/dx<span class="su">n</span>; +the justification being in the theorem, which we assume, that if +the determinant involving, for instance, the first k columns be not +zero for x<span class="su">1</span> = xº<span class="su">1</span> ... x<span class="su">n</span> = xº<span class="su">n</span>, and the functions be developable +about this point, then from the equations ƒ<span class="su">1</span> = c<span class="su">1</span>, ... ƒ<span class="su">k</span> = c<span class="su">k</span> we +can express x<span class="su">1</span>, ... x<span class="su">k</span> by convergent power series in the +differences x<span class="su">k+1</span> − xº<span class="su">k+1</span>, ... x<span class="su">n</span> − x<span class="su">nº</span>, and so regard x<span class="su">1</span>, ... x<span class="su">k</span> +as functions of the remaining variables. This we often express by +saying that the equations ƒ<span class="su">1</span> = c<span class="su">1</span>, ... ƒ<span class="su">k</span> = c<span class="su">k</span> can be solved for +x<span class="su">1</span>, ... x<span class="su">k</span>. The explanation is given as a type of explanation +often understood in what follows.</p> + +<p>We may conveniently begin by stating the theorem: If each of +the n functions φ<span class="su">1</span>, ... φ<span class="su">n</span> of the (n + 1) variables x<span class="su">1</span>, ... x<span class="su">n</span>t be developable +<span class="sidenote">Ordinary equations of the first order.</span> +about the values xº<span class="su">1</span>, ... x<span class="su">n</span><span class="sp">0</span>t<span class="sp">0</span>, the n differential +equations of the form dx<span class="su">1</span>/dt = φ<span class="su">1</span>(tx<span class="su">1</span>, ... x<span class="su">n</span>) are satisfied +by convergent power series</p> + +<p class="center">x<span class="su">r</span> = xº<span class="su">r</span> + (t − t<span class="sp">0</span>) A<span class="su">r1</span> + (t − t<span class="su">0</span>)² A<span class="su">r2</span> + ...</p> + +<p class="noind">reducing respectively to xº<span class="su">1</span>, ... xº<span class="su">n</span> when t = t<span class="sp">0</span>; and the +only functions satisfying the equations and reducing respectively to +xº<span class="su">1</span>, ... xº<span class="su">n</span> when t = t<span class="sp">0</span>, are those determined by continuation of these +series. If the result of solving these n equations for xº<span class="su">1</span>, ... xº<span class="su">n</span> be +written in the form ω<span class="su">1</span>(x<span class="su">1</span>, ... x<span class="su">n</span>t) = xº<span class="su">1</span>, ... ω<span class="su">n</span>(x<span class="su">1</span>, ... x<span class="su">n</span>t) = xº<span class="su">n</span>, +<span class="sidenote">Single homogeneous partial equation of the first order.</span> +it is at once evident that the differential equation</p> + +<p class="center">dƒ/dt + φ<span class="su">1</span>dƒ/dx<span class="su">1</span> + ... + φ<span class="su">n</span>dƒ/dx<span class="su">n</span> = 0</p> + +<p class="noind">possesses n integrals, namely, the functions ω<span class="su">1</span>, ... ω<span class="su">n</span>, +which are developable about the values (xº<span class="su">1</span> ... x<span class="su">n</span><span class="sp">0</span>t<span class="sp">0</span>) and +reduce respectively to x<span class="su">1</span>, ... x<span class="su">n</span> when t = t<span class="sp">0</span>. And in fact it +has no other integrals so reducing. Thus this equation +also possesses a unique integral reducing when t = t<span class="sp">0</span> to an arbitrary +function ψ(x<span class="su">1</span>, ... x<span class="su">n</span>), this integral being. ψ(ω<span class="su">1</span>, ... ω<span class="su">n</span>). Conversely +the existence of these <i>principal</i> integrals ω<span class="su">1</span>, ... ω<span class="su">n</span> of the partial +equation establishes the existence of the specified solutions of the +ordinary equations dx<span class="su">i</span>/dt = φ<span class="su">i</span>. The following sketch of the proof of +the existence of these principal integrals for the case n = 2 will show +the character of more general investigations. Put x for x − x<span class="sp">0</span>, &c., +and consider the equation a(xyt) dƒ/dx + b(xyt) dƒ/dy = dƒ/dt, wherein +the functions a, b are developable about x = 0, y = 0, t = 0; say</p> + +<p class="center">a(xyt) = a<span class="su">0</span> + ta<span class="su">1</span> + t²a<span class="su">2</span>/2! + ..., b(xyt) = b<span class="su">0</span> + tb<span class="su">1</span> + t²b<span class="su">2</span>/2! + ...,</p> + +<p class="noind">so that</p> + +<p class="center">ad/dx + bd/dy = δ<span class="su">0</span> + tδ<span class="su">1</span> + ½t²δ<span class="su">2</span> + ...,</p> + +<p class="noind">where δ = a<span class="su">r</span>d/dx + b<span class="su">r</span>d/dy. In order that</p> + +<p class="center">ƒ = p<span class="su">0</span> + tp<span class="su">1</span> + t²p<span class="su">2</span>/2! + ...</p> + +<p><span class="pagenum"><a name="page230" id="page230"></a>230</span></p> + +<p>wherein p<span class="su">0</span>, p<span class="su">1</span> ... are power series in x, y, should satisfy the equation, +it is necessary, as we find by equating like terms, that</p> + +<p class="center">p<span class="su">1</span> = δ<span class="su">0</span>p<span class="su">0</span>, p<span class="su">2</span> = δ<span class="su">0</span>p<span class="su">1</span> + δ<span class="su">1</span>p<span class="su">0</span>, &c.</p> + +<p class="noind">and in general<span class="sidenote">Proof of the existence of integrals.</span></p> + +<p class="center"> +p<span class="su">s+1</span> = δ<span class="su">0</span>p<span class="su">s</span> + s<span class="su">1</span>δ<span class="su">1</span>p<span class="su">s-1</span> + s<span class="su">2</span>δ<span class="su">2</span>p<span class="su">s-2</span> +... + δ<span class="su">s</span>p<span class="su">0</span>,</p> + +<p class="noind">where</p> + +<p class="center">s<span class="su">r</span> = (s!)/(r!) (s − r)!</p> + +<p>Now compare with the given equation another equation</p> + +<p class="center">A(xyt)dF/dx + B(xyt)dF/dy = dF/dt,</p> + +<p class="noind">wherein each coefficient in the expansion of either A or B is real and +positive, and not less than the absolute value of the corresponding +coefficient in the expansion of a or b. In the second equation let us +substitute a series</p> + +<p class="center"> +F = P<span class="su">0</span> + tP<span class="su">1</span> + t²P<span class="su">2</span>/2! + ...,</p> + +<p class="noind">wherein the coefficients in P<span class="su">0</span> are real and positive, and each not less +than the absolute value of the corresponding coefficient in p<span class="su">0</span>; then +putting Δ<span class="su">r</span> = A<span class="su">r</span>d/dx + B<span class="su">r</span>d/dy we obtain necessary equations of the +same form as before, namely,</p> + +<p class="center">P<span class="su">1</span> = Δ<span class="su">0</span>P<span class="su">0</span>, P<span class="su">2</span> = Δ<span class="su">0</span>P<span class="su">1</span> + Δ<span class="su">1</span>P<span class="su">0</span>, ...</p> + +<p class="noind">and in general P<span class="su">s+1</span> = Δ<span class="su">0</span>P<span class="su">s</span>, + s<span class="su">1</span>Δ<span class="su">1</span>P<span class="su">s-1</span> + ... + Δ<span class="su">s</span>P<span class="su">0</span>. These give for +every coefficient in P<span class="su">s+1</span> an integral aggregate with real positive +coefficients of the coefficients in P<span class="su">s</span>, P<span class="su">s-1</span>, ..., P<span class="su">0</span> and the coefficients +in A and B; and they are the same aggregates as would be given by +the previously obtained equations for the corresponding coefficients +in p<span class="su">s+1</span> in terms of the coefficients in p<span class="su">s</span>, p<span class="su">s-1</span>, ..., p<span class="su">0</span> and the coefficients +in a and b. Hence as the coefficients in P<span class="su">0</span> and also in A, B +are real and positive, it follows that the values obtained in succession +for the coefficients in P<span class="su">1</span>, P<span class="su">2</span>, ... are real and positive; and further, +taking account of the fact that the absolute value of a sum of terms +is not greater than the sum of the absolute values of the terms, it +follows, for each value of s, that every coefficient in p<span class="su">s+1</span> is, in absolute +value, not greater than the corresponding coefficient in P<span class="su">s+1</span>. Thus +if the series for F be convergent, the series for ƒ will also be; and we +are thus reduced to (1), specifying functions A, B with real positive +coefficients, each in absolute value not less than the corresponding +coefficient in a, b; (2) proving that the equation</p> + +<p class="center">AdF/dx + BdF/dy = dF/dt</p> + +<p class="noind">possesses an integral P<span class="su">0</span> + tP<span class="su">1</span> + t²P<span class="su">2</span>/2! + ... in which the coefficients +in P<span class="su">0</span> are real and positive, and each not less than the absolute value +of the corresponding coefficient in p<span class="su">0</span>. If a, b be developable for x, y +both in absolute value less than r and for t less in absolute value than +R, and for such values a, b be both less in absolute value than the +real positive constant M, it is not difficult to verify that we may +take A = B = M[1 − (x + y)/r]<span class="sp">-1</span> (1 − t/R)<span class="sp">-1</span>, +and obtain</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">F = r − (r − x − y) <span class="f150">[</span> 1 −</td> <td>4MR</td> +<td rowspan="2"><span class="f150">(</span>1 −</td> <td>x + y</td> +<td rowspan="2"><span class="f150">)</span></td> <td><span class="sp bk1">−2</span></td> +<td rowspan="2">log <span class="f150">(</span> 1 −</td> <td>t</td> +<td rowspan="2"><span class="f150">)</span></td> <td><span class="sp bk1">−1</span></td> +<td rowspan="2"><span class="f150">]</span></td> <td><span class="sp bk1">1/2</span></td> <td rowspan="2">,</td></tr> +<tr><td class="denom">r</td> <td class="denom">r</td> +<td> </td> <td class="denom">R</td> +<td> </td> <td> </td></tr></table> + +<p class="noind">and that this solves the problem when x, y, t are sufficiently small +for the two cases p<span class="su">0</span> = x, p<span class="su">0</span> = y. One obvious application of the +general theorem is to the proof of the existence of an integral of +an ordinary linear differential equation given by the n equations +dy/dx = y<span class="su">1</span>, dy<span class="su">1</span>/dx = y<span class="su">2</span>, ...,</p> + +<p class="center">dy<span class="su">n-1</span>/dx = p − p<span class="su">1</span>y<span class="su">n-1</span> − ... − p<span class="su">n</span>y;</p> + +<p class="noind">but in fact any simultaneous system of ordinary equations is reducible +to a system of the form</p> + +<p class="center">dx<span class="su">i</span>/dt = φ<span class="su">i</span>(tx<span class="su">1</span>, ... x<span class="su">n</span>).</p> + +<p>Suppose we have k homogeneous linear partial equations of the +first order in n independent variables, the general equation being +a<span class="su">σ1</span>dƒ/dx<span class="su">1</span> + ... + a<span class="su">σn</span>dƒ/dx<span class="su">n</span> = 0, +where σ = 1, ... k, and that +<span class="sidenote">Simultaneous linear partial equations.</span> +we desire to know whether the equations have common +solutions, and if so, how many. It is to be understood +that the equations are linearly independent, which implies +that k ≤ n and not every determinant of k rows and columns +is identically zero in the matrix in which the i-th element of the σ-th +row is a<span class="su">σi</span>}(i = 1, ... n, σ = 1, ... k). Denoting the left side of the +σ-th equation by Pσƒ, it is clear that every common solution of the +two equations P<span class="su">σ</span>ƒ = 0, P<span class="su">ρ</span>ƒ = 0, +is also a solution of the equation +P<span class="su">ρ</span>(p<span class="su">σ</span>ƒ), P<span class="su">σ</span>(p<span class="su">ρ</span>ƒ), +We immediately find, however, that this is +also a linear equation, namely, ΣH<span class="su">i</span>dƒ/dx<span class="su">i</span> = 0 +where H<span class="su">i</span> = P<span class="su">ρ</span>a<span class="su">σ</span> − P<span class="su">σ</span>a<span class="su">ρ</span>, +and if it be not already contained among the given equations, or be +linearly deducible from them, it may be added to them, as not introducing +any additional limitation of the possibility of their having +common solutions. Proceeding thus with every pair of the original +equations, and then with every pair of the possibly augmented +system so obtained, and so on continually, we shall arrive at a +system of equations, linearly independent of each other and therefore +not more than n in number, such that the combination, in the way +described, of every pair of them, leads to an equation which is +linearly deducible from them. If the number of this so-called +<i>complete system</i> is n, the equations give dƒ/dx<span class="su">1</span> = 0 ... dƒ/dx<span class="su">n</span> = 0, +leading to the nugatory result ƒ = a constant. Suppose, then, the +number of this system to be r < n; suppose, further, that from the +<span class="sidenote">Complete systems of linear partial equations.</span> +matrix of the coefficients a determinant of r rows and +columns not vanishing identically is that formed by the +coefficients of the differential coefficients of ƒ in regard +to x<span class="su">1</span> ... x<span class="su">r</span>; also that the coefficients are all developable +about the values x<span class="su">1</span> = xº<span class="su">1</span>, ... x<span class="su">n</span>= xº<span class="su">n</span>, and that for these +values the determinant just spoken of is not zero. +Then the main theorem is that the complete system of r equations, +and therefore the originally given set of k equations, +have in common n − r solutions, say ω<span class="su">r+1</span>, ... ω<span class="su">n</span>, which reduce +respectively to x<span class="su">r+1</span>, ... x<span class="su">n</span> when in them for x<span class="su">1</span>, ... x<span class="su">r</span> are respectively +put xº<span class="su">1</span>, ... xº<span class="su">r</span>; so that also the equations have in common a +solution reducing when x<span class="su">1</span> = xº<span class="su">1</span>, ... x<span class="su">r</span> = xº<span class="su">r</span> to an arbitrary function +ψ(x<span class="su">r+1</span>, ... x<span class="su">n</span>) which is developable about xº<span class="su">r+1</span>, ... xº<span class="su">n</span>, namely, +this common solution is ψ(ω<span class="su">r+1</span>, ... ω<span class="su">n</span>). It is seen at once +that this result is a generalization of the theorem for r = 1, and its +proof is conveniently given by induction from that case. It can be +verified without difficulty (1) that if from the r equations of the +complete system we form r independent linear aggregates, with +coefficients not necessarily constants, the new system is also a complete +system; (2) that if in place of the independent variables +x<span class="su">1</span>, ... x<span class="su">n</span> we introduce any other variables which are independent +functions of the former, the new equations also form a complete +system. It is convenient, then, from the complete system of r +equations to form r new equations by solving separately for dƒ/dx<span class="su">1</span>, ..., +dƒ/dx<span class="su">r</span>; suppose the general equation of the new system to be</p> + +<p class="center">Q<span class="su">σ</span>ƒ = dƒ/dx<span class="su">σ</span> + c<span class="su">σjr+1</span>dƒ/dx<span class="su">r+1</span> + ... + c<span class="su">σn</span>dƒ/dx<span class="su">n</span> = 0 (σ = 1, ... r).</p> + +<p class="noind">Then it is easily obvious that the equation Q<span class="su">ρ</span>Q<span class="su">σ</span>ƒ − Q<span class="su">σ</span>Q<span class="su">ρ</span>ƒ = 0 +contains only the differential coefficients of ƒ in regard to x<span class="su">r+1</span> ... x<span class="su">n</span>; as +it is at most a linear function of Q<span class="su">1</span>ƒ, ... Q<span class="su">r</span>ƒ, it must be identically +zero. So reduced the system is called a Jacobian system. Of this +system Q<span class="su">1</span>ƒ=0 has n − 1 principal solutions reducing respectively +<span class="sidenote">Jacobian systems.</span> +to x<span class="su">2</span>, ... x<span class="su">n</span> when</p> + +<p class="center">x<span class="su">1</span> = xº<span class="su">1</span>,</p> + +<p class="noind">and its form shows that of these the first r − 1 are exactly x<span class="su">2</span> ... x<span class="su">r</span>. +Let these n − 1 functions together with x<span class="su">1</span> be introduced as n new +independent variables in all the r equations. Since the first equation +is satisfied by n − 1 of the new independent variables, it will contain +no differential coefficients in regard to them, and will reduce therefore +simply to dƒ/dx<span class="su">1</span> = 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.</p> + +<p>The analogy with the case of one equation is, however, still closer. +With the coefficients c<span class="su">σj</span>, of the equations Q<span class="su">σ</span>ƒ = 0 in transposed +array (σ = 1, ... r, j = r + 1, ... n) we can put down the +(n − r) equations, dx<span class="su">j</span> = c<span class="su">1j</span>dx<span class="su">1</span> + ... + c<span class="su">rj</span>dx<span class="su">r</span>, equivalent to +<span class="sidenote">System of total differential equations.</span> +the r(n − r) equations dx<span class="su">j</span>/dx<span class="su">σ</span> = c<span class="su">σr</span>. That consistent +with them we may be able to regard x<span class="su">r+1</span>, ... x<span class="su">n</span> as +functions of x<span class="su">1</span>, ... x<span class="su">r</span>, these being regarded as independent +variables, it is clearly necessary that when we differentiate c<span class="su">σj</span> in +regard to x<span class="su">ρ</span> on this hypothesis the result should be the same as when +we differentiate c<span class="su">ρj</span>, in regard to x<span class="su">σ</span> on this hypothesis. The differential +coefficient of a function ƒ of x<span class="su">1</span>, ... x<span class="su">n</span> on this hypothesis, in +regard to x<span class="su">ρj</span> is, however,</p> + +<p class="center">dƒ/dx<span class="su">ρ</span> + c<span class="su">ρjr+1</span>dƒ/dx<span class="su">r+1</span> + ... + c<span class="su">ρn</span>dƒ/dx<span class="su">n</span>,</p> + +<p class="noind">namely, is Q<span class="su">ρ</span>ƒ. Thus the consistence of the n − r total equations +requires the conditions Q<span class="su">ρ</span>c<span class="su">σj</span> − Q<span class="su">σ</span>c<span class="su">ρj</span> = 0, which are, however, +verified in virtue of Q<span class="su">ρ</span>(Q<span class="su">σ</span>ƒ) − Q<span class="su">σ</span>(Q<span class="su">ρ</span>ƒ) = 0. And it can in fact be +easily verified that if ω<span class="su">r+1</span>, ... ω<span class="su">n</span> be the principal solutions of the +Jacobian system, Q<span class="su">σ</span>ƒ = 0, reducing respectively to x<span class="su">r+1</span>, ... x<span class="su">n</span> when +x<span class="su">1</span> = xº<span class="su">1</span>, ... x<span class="su">r</span> = xº<span class="su">r</span>, and the equations ω<span class="su">r+1</span> = x<span class="sp">0</span><span class="su">r+1</span>, ... ω<span class="su">n</span> = xº<span class="su">n</span> +be solved for x<span class="su">r+1</span>, ... x<span class="su">n</span> to give x<span class="su">j</span> = ψ<span class="su">j</span>(x<span class="su">1</span>, ... x<span class="su">r</span>, x<span class="sp">0</span><span class="su">r+1</span>, ... xº<span class="su">n</span>), these +values solve the total equations and reduce respectively to x<span class="sp">0</span><span class="su">r+1</span>, ... xº<span class="su">n</span> +when x<span class="su">1</span> = xº<span class="su">1</span> ... x<span class="su">r</span> = xº<span class="su">r</span>. And the total equations have no +other solutions with these initial values. Conversely, the existence +of these solutions of the total equations can be deduced a priori +and the theory of the Jacobian system based upon them. The +theory of such total equations, in general, finds its natural place +under the heading <i>Pfaffian Expressions</i>, below.</p> + +<p>A practical method of reducing the solution of the r equations +of a Jacobian system to that of a single equation in n − r + 1 variables +may be explained in connexion with a geometrical interpretation +which will perhaps be clearer in a particular +<span class="sidenote">Geometrical interpretation and solution.</span> +case, say n = 3, r = 2. There is then only one total +equation, say dz = adz + bdy; if we do not take account +of the condition of integrability, which is in this case +da/dy + bda/dz = db/dx + adb/dz, this equation may be regarded +as defining through an arbitrary point (x<span class="su">0</span>, y<span class="su">0</span>, z<span class="su">0</span>) of three-dimensioned +space (about which a, b are developable) a plane, namely, +z − z<span class="su">0</span> = a<span class="su">0</span>(x − x<span class="su">0</span>) + b<span class="su">0</span>(y − y<span class="su">0</span>), and therefore, through this arbitrary +point ∞<span class="sp">2</span> directions, namely, all those in the plane. If now there be +a surface z = ψ(x, y), satisfying dz = adz + bdy and passing through +(x<span class="su">0</span>, y<span class="su">0</span>, z<span class="su">0</span>), this plane will touch the surface, and the operations of +passing along the surface from (x<span class="su">0</span>, y<span class="su">0</span>, z<span class="su">0</span>) to</p> + +<p class="center">(x<span class="su">0</span> + dx<span class="su">0</span>, y<span class="su">0</span>, z<span class="su">0</span> + dz<span class="su">0</span>)</p> + +<p class="noind">and then to (x<span class="su">0</span> + dx<span class="su">0</span>, y<span class="su">0</span> + dy<span class="su">0</span>, Z<span class="su">0</span> + d<span class="sp">1</span>z<span class="su">0</span>), ought to lead to the same +value of d<span class="sp">1</span>z<span class="su">0</span> as do the operations of passing along the surface from +(x<span class="su">0</span>, y<span class="su">0</span>, z<span class="su">0</span>) to (x<span class="su">0</span>, y<span class="su">0</span> + dy<span class="su">0</span>, z<span class="su">0</span> + δz<span class="su">0</span>), and then to</p> + +<p class="center">(x<span class="su">0</span> + dx<span class="su">0</span>, y<span class="su">0</span> + dy<span class="su">0</span>, z<span class="su">0</span> + δ<span class="sp">1</span>z<span class="su">0</span>),</p> + +<p class="noind">namely, δ<span class="sp">1</span>z<span class="su">0</span> ought to be equal to d<span class="sp">1</span>z<span class="su">0</span>. But we find</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">a<span class="su">0</span>dx<span class="su">0</span> + b<span class="su">0</span>dy<span class="su">0</span> + dx<span class="su">0</span>dy<span class="su">0</span><span class="f150">(</span></td> <td>db</td> +<td rowspan="2">+ a<span class="su">0</span></td> <td>db</td> +<td rowspan="2"><span class="f150">)</span>,</td></tr> +<tr><td class="denom">dx<span class="su">0</span></td> <td class="denom">dz<span class="su">0</span></td></tr></table> + +<p class="noind">and so at once reach the condition of integrability. If now we put +<span class="pagenum"><a name="page231" id="page231"></a>231</span> +x = x<span class="su">0</span> + t, y = y<span class="su">0</span> + mt, and regard m as constant, we shall in fact be +considering the section of the surface by a fixed plane y − y<span class="su">0</span> = m(x − x<span class="su">0</span>); +along this section dz = dt(a + bm); if we then integrate the equation +dx/dt = a + bm, where a, b are expressed as functions of m and t, with +m kept constant, finding the solution which reduces to z<span class="su">0</span> for t = 0, +and in the result again replace m by (y − y<span class="su">0</span>)/(x − x<span class="su">0</span>), we shall have the +surface in question. In the general case the equations</p> + +<p class="center">dx<span class="su">j</span> = c<span class="su">ij</span>dx<span class="su">1</span> + ... c<span class="su">rj</span>dx<span class="su">r</span></p> + +<p class="noind">similarly determine through an arbitrary point xº<span class="su">1</span>, ... xº<span class="su">n</span> +<span class="sidenote">Mayer’s method of integration.</span> +a planar manifold of r dimensions in space of n dimensions, +and when the conditions of integrability are satisfied, +every direction in this manifold through this point is +tangent to the manifold of r dimensions, expressed by +ω<span class="su">r+1</span> = x<span class="sp">0</span><span class="su">r+1</span>, ... ω_ = xº<span class="su">n</span>, +which satisfies the equations and passes through this +point. If we put +x<span class="su">1</span> − xº<span class="su">1</span> = t, x<span class="su">2</span> − xº<span class="su">2</span> = m<span class="su">2</span>t, ... x<span class="su">r</span> − xº<span class="su">r</span> = m<span class="su">r</span>t, +and regard m<span class="su">2</span>, ... m<span class="su">r</span> as fixed, the (n − r) total equations take the form +dx<span class="su">j</span>/dt = c<span class="su">1j</span> + m<span class="su">2</span>c<span class="su">2j</span> + ... + m<span class="su">r</span>c<span class="su">rj</span>, +and their integration is equivalent to that of the single partial equation</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">dƒ/dt + <span class="f150">Σ</span></td> <td class="bk">n</td> +<td rowspan="2">(c<span class="su">1j</span> + m<span class="su">2</span>c<span class="su">2j</span> + ... + m<span class="su">r</span>c<span class="su">rj</span>) dƒ/dx<span class="su">j</span> = 0</td></tr> +<tr><td class="bk">j=r+1</td></tr></table> + +<p class="noind">in the n − r + 1 variables t, x<span class="su">r+1</span>, ... x<span class="su">n</span>. Determining the solutions +Ω<span class="su">r+1</span>, ... Ω<span class="su">n</span> which reduce to respectively x<span class="su">r+1</span>, ... x<span class="su">n</span> when t = 0, and substituting +t = x<span class="su">1</span> − xº<span class="su">1</span>, m<span class="su">2</span> = (x<span class="su">2</span> − xº<span class="su">2</span>)/(x<span class="su">1</span> − xº<span class="su">1</span>), ... m<span class="su">r</span> = (x<span class="su">r</span> − xº<span class="su">r</span>)/(x<span class="su">1</span> − xº<span class="su">1</span>), +we obtain the solutions of the original system of partial equations +previously denoted by ω<span class="su">r+1</span>, ... ω<span class="su">n</span>. It is to be remarked, +however, that the presence of the fixed parameters m<span class="su">2</span>, ... m<span class="su">r</span> in +the single integration may frequently render it more difficult than if +they were assigned numerical quantities.</p> + +<p>We have above considered the integration of an equation</p> + +<p class="center">dz = adz + bdy</p> + +<p class="noind">on the hypothesis that the condition</p> + +<p class="center">da/dy + bda/dz = db/dz + adb/dz.</p> + +<p>It is natural to inquire what relations among x, y, z, if any, +<span class="sidenote">Pfaffian Expressions.</span> +are implied by, or are consistent with, a differential relation +adx + bdy + cdx = 0, when a, b, c are unrestricted functions +of x, y, z. This problem leads to the consideration of the +so-called <i>Pfaffian Expression</i> adx + bdy + cdz. It can be shown (1) if +each of the quantities db/dz − dc/dy, dc/dx − da/dz, da/dy − db/dz, which +we shall denote respectively by u<span class="su">23</span>, u<span class="su">31</span>, u<span class="su">12</span>, be identically zero, the +expression is the differential of a function of x, y, z, equal to dt say; +(2) that if the quantity au<span class="su">23</span> + bu<span class="su">31</span> + cu<span class="su">12</span> is identically zero, the expression +is of the form udt, <i>i.e.</i> it can be made a perfect differential +by multiplication by the factor 1/u; (3) that in general the expression +is of the form dt + u<span class="su">1</span>dt<span class="su">1</span>. Consider the matrix of four +rows and three columns, in which the elements of the first row are +a, b, c, and the elements of the (r + 1)-th row, for r = 1, 2, 3, are the +quantities u<span class="su">r1</span>, u<span class="su">r2</span>, u<span class="su">r3</span>, where u<span class="su">11</span> = u<span class="su">22</span> = u<span class="su">33</span> = 0. Then it is easily +seen that the cases (1), (2), (3) above correspond respectively to the +cases when (1) every determinant of this matrix of two rows and +columns is zero, (2) every determinant of three rows and columns +is zero, (3) when no condition is assumed. This result can be generalized +as follows: if a<span class="su">1</span>, ... a<span class="su">n</span> be any functions of x<span class="su">1</span>, ... x<span class="su">n</span>, the so-called +Pfaffian expression a<span class="su">1</span>dx<span class="su">1</span> + ... + a<span class="su">n</span>dx<span class="su">n</span> can be reduced to one +or other of the two forms</p> + +<p class="center">u<span class="su">1</span>dt<span class="su">1</span> + ... + u<span class="su">k</span>dt<span class="su">k</span>, dt + u<span class="su">1</span>dt<span class="su">1</span> + ... + u<span class="su">k-1</span>dt<span class="su">k-1</span>,</p> + +<p class="noind">wherein t, u<span class="su">1</span> ..., t<span class="su">1</span>, ... are independent functions of x<span class="su">1</span>, ... x<span class="su">n</span>, and k +is such that in these two cases respectively 2k or 2k − 1 is the rank of +a certain matrix of n + 1 rows and n columns, that is, the greatest +number of rows and columns in a non-vanishing determinant of the +matrix; the matrix is that whose first row is constituted by the +quantities a<span class="su">1</span>, ... a<span class="su">n</span>, whose s-th element in the (r + 1)-th row is the +quantity da<span class="su">r</span>/dx<span class="su">s</span> − da<span class="su">s</span>/dx<span class="su">r</span>. The proof of such a reduced form can +be obtained from the two results: (1) If t be any given function +of the 2m independent variables u<span class="su">1</span>, ... u<span class="su">m</span>, t<span class="su">1</span>, ... t<span class="su">m</span>, the expression +dt + u<span class="su">1</span>dt<span class="su">1</span> + ... + u<span class="su">m</span>dt<span class="su">m</span> can be put into the form u′<span class="su">1</span>dt′<span class="su">1</span> + ... + u′<span class="su">m</span>dt′<span class="su">m</span>. +(2) If the quantities u<span class="su">1</span>, ..., u<span class="su">1</span>, t<span class="su">1</span>, ... t<span class="su">m</span> be connected by a relation, +the expression n<span class="su">1</span>dt<span class="su">1</span> + ... + u<span class="su">m</span>dt<span class="su">m</span> can be put into the format dt′ + u′<span class="su">1</span>dt′<span class="su">1</span> + + ... + u′<span class="su">m-1</span>dt′<span class="su">m-1</span>; and if the relation connecting u<span class="su">1</span>, u<span class="su">m</span>, t<span class="su">1</span>, ... t<span class="su">m</span> +be homogeneous in u<span class="su">1</span>, ... u<span class="su">m</span>, then t′ can be taken to be zero. These +two results are deductions from the theory of <i>contact transformations</i> +(see below), and their demonstration requires, beside elementary +algebraical considerations, only the theory of complete systems of +linear homogeneous partial differential equations of the first order. +When the existence of the reduced form of the Pfaffian expression +containing only independent quantities is thus once assured, the +identification of the number k with that defined by the specified +matrix may, with some difficulty, be made <i>a posteriori</i>.</p> + +<p>In all cases of a single Pfaffian equation we are thus led to consider +what is implied by a relation dt − u<span class="su">1</span>dt<span class="su">1</span> − ... − u<span class="su">m</span>dt<span class="su">m</span> = 0, in which +t, u<span class="su">1</span>, ... u<span class="su">m</span>, t<span class="su">1</span> ..., t<span class="su">m</span> are, except for this equation, +independent variables. This is to be satisfied in virtue of +<span class="sidenote">Single linear Pfaffian equation.</span> +one or several relations connecting the variables; these +must involve relations connecting t, t<span class="su">1</span>, ... t<span class="su">m</span> only, and +in one of these at least t must actually enter. We can +then suppose that in one actual system of relations in virtue of which +the Pfaffian equation is satisfied, all the relations connecting t, t<span class="su">1</span> ... +t<span class="su">m</span> only are given by</p> + +<p class="center">t = ψ(t<span class="su">s+1</span> ... t<span class="su">m</span>), t<span class="su">1</span> = ψ<span class="su">1</span>(t<span class="su">s+1</span> ... t<span class="su">m</span>), ... t<span class="su">s</span> = ψ<span class="su">s</span>(t<span class="su">s+1</span> ... t<span class="su">m</span>);</p> + +<p class="noind">so that the equation</p> + +<p class="center">dψ − u<span class="su">1</span>dψ<span class="su">1</span> − ... − u<span class="su">s</span>dψ<span class="su">s</span> − u<span class="su">s+1</span>dt<span class="su">s+1</span> − ... − u<span class="su">m</span>dt<span class="su">m</span> = 0</p> + +<p class="noind">is identically true in regard to u<span class="su">1</span>, ... u<span class="su">m</span>, t<span class="su">s+1</span> ..., t<span class="su">m</span>; equating to +zero the coefficients of the differentials of these variables, we thus +obtain m − s relations of the form</p> + +<p class="center">dψ/dt<span class="su">j</span> − u<span class="su">1</span>dψ<span class="su">1</span>/dt<span class="su">j</span> − ... − u<span class="su">s</span>dψ<span class="su">s</span>/dt<span class="su">j</span> − u<span class="su">j</span> = 0;</p> + +<p class="noind">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 ψ,ψ<span class="su">1</span>, ... ψ<span class="su">s</span>. 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, t<span class="su">1</span>, ... t<span class="su">m</span> only, +and are consistent with the m − s relations connecting u<span class="su">1</span>, ... u<span class="su">m</span>. It +is now evident that, essentially, the integration of a Pfaffian equation</p> + +<p class="center">a<span class="su">1</span>dx<span class="su">1</span> + ... + a<span class="su">n</span>dx<span class="su">n</span> = 0,</p> + +<p class="noind">wherein a<span class="su">1</span>, ... a<span class="su">n</span> are functions of x<span class="su">1</span>, ... x<span class="su">n</span>, is effected by the +processes necessary to bring it to its reduced form, involving only +independent variables. And it is easy to see that if we suppose this +reduction to be carried out in all possible ways, there is no need to +distinguish the classes of integrals corresponding to the various +values of s; for it can be verified without difficulty that by putting +t′ = t − u<span class="su">1</span>t<span class="su">1</span> − ... − u<span class="su">s</span>t<span class="su">s</span>, t′<span class="su">1</span> = u<span class="su">1</span>, ... t′<span class="su">s</span> = u<span class="su">s</span>, u′<span class="su">1</span> = −t<span class="su">1</span>, ..., u′<span class="su">s</span> = −t<span class="su">s</span>, +t′<span class="su">s+1</span> = t<span class="su">s+1</span>, ... t′<span class="su">m</span> = t<span class="su">m</span>, u′<span class="su">s+1</span> = u<span class="su">s+1</span>, ... u′<span class="su">m</span> = u<span class="su">m</span>, +the reduced equation +becomes changed to dt′ − u′<span class="su">1</span>dt′<span class="su">1</span> − ... − u′<span class="su">m</span>dt′<span class="su">m</span> = 0, and the general +relations changed to</p> + +<p class="center">t′ = ψ(t′<span class="su">s+1</span>, ... t′<span class="su">m</span>) − t′<span class="su">1</span>ψ<span class="su">1</span>(t′<span class="su">s+1</span>, ... t′<span class="su">m</span>) − ... − t′<span class="su">s</span>ψ<span class="su">s</span>(t′<span class="su">s+1</span>, ... t′<span class="su">m</span>), = φ,</p> + +<p class="noind">say, together with u′<span class="su">1</span> = dφ/dt′<span class="su">1</span>, ..., u′<span class="su">m</span> = dφ/dt′<span class="su">m</span>, which contain only +one relation connecting the variables t′, t′<span class="su">1</span>, ... t′<span class="su">m</span> only.</p> + +<p>This method for a single Pfaffian equation can, strictly speaking, +be generalized to a simultaneous system of (n − r) Pfaffian equations +dx<span class="su">j</span> = c<span class="su">1j</span>dx<span class="su">1</span> + ... + c<span class="su">rj</span>dx<span class="su">r</span> only in the case already treated, +<span class="sidenote">Simultaneous Pfaffian equations.</span> +when this system is satisfied by regarding x<span class="su">r+1</span>, ... x<span class="su">n</span> as +suitable functions of the independent variables x<span class="su">1</span>, ... x<span class="su">r</span>; +in that case the integral manifolds are of r dimensions. +When these are non-existent, there may be integral manifolds +of higher dimensions; for if</p> + +<p class="center">dφ = φ<span class="su">1</span>dx<span class="su">1</span> + ... + φ<span class="su">r</span>dx<span class="su">r</span> + φ<span class="su">r+1</span>(c<span class="su">1,r+1</span>dx<span class="su">1</span> + ... + c<span class="su">r,r+1</span>dx<span class="su">r</span>) + φ<span class="su">r+2</span>( ) + ...</p> + +<p class="noind">be identically zero, then φσ + cσ,<span class="su">r+1</span>φ<span class="su">r+1</span> + ... + cσ,<span class="su">n</span>φ<span class="su">n</span> ≈ 0, or φ 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 − μ equations, having therefore +n − r − μ independent integrals, the total equations are satisfied over +a manifold of r + μ dimensions (see E. v. Weber, <i>Math. Annal.</i> 1v. +(1901), p. 386).</p> + +<p>It seems desirable to add here certain results, largely of algebraic +character, which naturally arise in connexion with the theory of +contact transformations. For any two functions of the 2n +<span class="sidenote">Contact transformations.</span> +independent variables x<span class="su">1</span>, ... x<span class="su">n</span>, p<span class="su">1</span>, ... p<span class="su">n</span> we denote by (φψ) +the sum of the n terms such as dφdψ/dp<span class="su">i</span>dx<span class="su">i</span> − dψdφ/dp<span class="su">i</span>dx<span class="su">i</span> For two +functions of the (2n + 1) independent variables z, x<span class="su">1</span>, ... x<span class="su">n</span>, p<span class="su">1</span>, ... p<span class="su">n</span> +we denote by φψ the sum of the n terms such as</p> + +<table class="math0" summary="math"> +<tr><td>dφ</td> <td rowspan="2"><span class="f150">(</span></td> <td>dψ</td> +<td rowspan="2">+ p<span class="su">i</span></td> <td>dψ</td> +<td rowspan="2"><span class="f150">)</span> −</td> <td>dψ</td> +<td rowspan="2"><span class="f150">(</span></td> <td>dφ</td> +<td rowspan="2">p<span class="su">i</span></td> <td>dφ</td> <td rowspan="2"><span class="f150">)</span>.</td></tr> +<tr><td class="denom">dp<span class="su">i</span></td> <td class="denom">dx<span class="su">i</span></td> +<td class="denom">dz</td> <td class="denom">dp<span class="su">i</span></td> +<td class="denom">dx<span class="su">i</span></td> <td class="denom">dz</td></tr></table> + +<p class="noind">It can at once be verified that for any three functions [ƒ[φψ]] + [φ[ψƒ]] ++ [psi[ƒφ]] = dƒ/dz [φψ] + dφ/dz [ψƒ] + dψ/dz [ƒφ], +which when ƒ, φ,ψ do not contain z +becomes the identity (ƒ(φψ)) + (phi(ψƒ)) + (ψ(ƒφ)) = 0.Then, if X<span class="su">1</span>, ... X<span class="su">n</span>, +P<span class="su">1</span>, ... P<span class="su">n</span> be such functions Of x<span class="su">1</span>, ... x<span class="su">n</span>, p<span class="su">1</span> ... p<span class="su">n</span> that P<span class="su">1</span>dX<span class="su">1</span> ++ ... + P<span class="su">n</span>dX<span class="su">n</span> is identically equal to p<span class="su">1</span>dx<span class="su">1</span> + ... + p<span class="su">n</span>dx<span class="su">n</span>, it can be +shown by elementary algebra, after equating coefficients of independent +differentials, (1) that the functions X<span class="su">1</span>, ... P<span class="su">n</span> are independent +functions of the 2n variables x<span class="su">1</span>, ... p<span class="su">n</span>, so that the equations +x′<span class="su">i</span> = X<span class="su">i</span>, p′<span class="su">i</span> = P<span class="su">i</span> can be solved for x<span class="su">1</span>, ... x<span class="su">n</span>, p<span class="su">1</span>, ... p<span class="su">n</span>, and represent +therefore a transformation, which we call a homogeneous contact +transformation; (2) that the X<span class="su">1</span>, ... X<span class="su">n</span> are homogeneous functions of +p<span class="su">1</span>, ... p<span class="su">n</span> of zero dimensions, the P<span class="su">1</span>, ... P<span class="su">n</span> are homogeneous functions +of p<span class="su">1</span>, ... p<span class="su">n</span> of dimension one, and the ½n(n − 1) relations (X<span class="su">i</span>X<span class="su">j</span>) = 0 +are verified. So also are the n² relations (P<span class="su">i</span>X<span class="su">i</span> = 1, (P<span class="su">i</span>X<span class="su">j</span>) = 0, +(P<span class="su">i</span>P<span class="su">j</span>) = 0. Conversely, if X<span class="su">1</span>, ... X<span class="su">n</span> be independent functions, each +homogeneous of zero dimension in p<span class="su">1</span>, ... p<span class="su">n</span> satisfying the ½n(n − 1) +relations (X<span class="su">i</span>X<span class="su">j</span>) = 0, then P<span class="su">1</span>, ... P<span class="su">n</span> can be uniquely determined, by +solving linear algebraic equations, such that P<span class="su">1</span>dX<span class="su">1</span> + ... + P<span class="su">n</span>dX<span class="su">n</span> += p<span class="su">1</span>dx<span class="su">1</span> + ... + p<span class="su">n</span>dx<span class="su">n</span>. If now we put n + 1 for n, put z for x<span class="su">n+1</span>, +Z for X<span class="su">n+1</span>, Q<span class="su">i</span> for -P<span class="su">i</span>/P<span class="su">n+1</span>, for i = 1, ... n, put q<span class="su">i</span> for -p<span class="su">i</span>/p<span class="su">n+1</span> and σ +for q<span class="su">n+1</span>/Q<span class="su">n+1</span>, and then finally write P<span class="su">1</span>, ... P<span class="su">n</span>, p<span class="su">1</span>, ... p<span class="su">n</span> for Q<span class="su">1</span>, ... Q<span class="su">n</span>, +q<span class="su">1</span>, ... q<span class="su">n</span>, we obtain the following results: If ZX<span class="su">1</span> ... X<span class="su">n</span>, P<span class="su">1</span>, ... P<span class="su">n</span> +be functions of z, x<span class="su">1</span>, ... x<span class="su">n</span>, p<span class="su">1</span>, ... p<span class="su">n</span>, such that the expression +dZ − P<span class="su">1</span>dX<span class="su">1</span> − ... − P<span class="su">n</span>dX<span class="su">n</span> is identically equal to σ(dz − p<span class="su">1</span>dx<span class="su">1</span> − ... − p<span class="su">n</span>dx<span class="su">n</span>), +and σ not zero, then (1) the functions Z, X<span class="su">1</span>, ... X<span class="su">n</span>, P<span class="su">1</span>, ... P<span class="su">n</span> +are independent functions of z, x<span class="su">1</span>, ... x<span class="su">n</span>, p<span class="su">1</span>, ... p<span class="su">n</span>, so that the +equations z′ = Z, x′<span class="su">i</span> = X<span class="su">i</span>, p′<span class="su">i</span> = P<span class="su">i</span> can be solved for z, x<span class="su">1</span>, ... x<span class="su">n</span>, p<span class="su">1</span>, ... p<span class="su">n</span> +and determine a transformation which we call a (non-homogeneous) +contact transformation; (2) the Z, X<span class="su">1</span>, ... X<span class="su">n</span> verify the ½n(n + 1) +<span class="pagenum"><a name="page232" id="page232"></a>232</span> +identities [ZX<span class="su">i</span>] = 0, [X<span class="su">i</span>X<span class="su">j</span>] = 0. And the further identities</p> + +<p class="center">[P<span class="su">i</span>X<span class="su">i</span>] = σ, [P<span class="su">i</span>X<span class="su">j</span>] = 0, [P<span class="su">i</span>Z] = σP<span class="su">i</span>, [P<span class="su">i</span>P<span class="su">j</span>] = 0,</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">[Zσ] = σ</td> <td>dZ</td> +<td rowspan="2">− σ², [X<span class="su">i</span>σ] = σ</td> <td>dX<span class="su">i</span></td> +<td rowspan="2">, [P<span class="su">i</span>σ] =</td> <td>dP<span class="su">i</span></td></tr> +<tr><td class="denom">dz</td> <td class="denom">dz</td> <td class="denom">dz</td></tr></table> + +<p class="noind">are also verified. Conversely, if Z, x<span class="su">1</span>, ... X<span class="su">n</span> be independent functions +satisfying the identities [ZX<span class="su">i</span>] = 0, [X<span class="su">i</span>X<span class="su">j</span>] = 0, then σ, other than +zero, and P<span class="su">1</span>, ... P<span class="su">n</span> can be uniquely determined, by solution of +algebraic equations, such that</p> + +<p class="center">dZ − P<span class="su">1</span>dX<span class="su">1</span> − ... − P<span class="su">n</span>dX<span class="su">n</span> = σ(dz − p<span class="su">1</span>dx<span class="su">1</span> − ... − p<span class="su">n</span>dx<span class="su">n</span>).</p> + +<p>Finally, there is a particular case of great importance arising when +σ = 1, which gives the results: (1) If U, X<span class="su">1</span>, ... X<span class="su">n</span>, P<span class="su">1</span>, ... P<span class="su">n</span> be +2n + 1 functions of the 2n independent variables x<span class="su">1</span>, ... x<span class="su">n</span>, p<span class="su">1</span>, +... p<span class="su">n</span>, satisfying the identity</p> + +<p class="center">dU + P<span class="su">1</span>dx<span class="su">1</span> + ... + P<span class="su">n</span>dX<span class="su">n</span> = p<span class="su">1</span>dx<span class="su">1</span> + ... + p<span class="su">n</span>dx<span class="su">n</span>,</p> + +<p class="noind">then the 2n functions P<span class="su">1</span>, ... P<span class="su">n</span>, X<span class="su">1</span>, ... X<span class="su">n</span> are independent, +and we have</p> + +<p class="center">(X<span class="su">i</span>X<span class="su">j</span>) = 0, (X<span class="su">i</span>U) = δX<span class="su">i</span>, (P<span class="su">i</span>X<span class="su">i</span>) = 1, (P<span class="su">i</span>X<span class="su">j</span>) = 0, (P<span class="su">i</span>P<span class="su">j</span>) = 0, (P<span class="su">i</span>U) + P<span class="su">i</span> = δP<span class="su">i</span>,</p> + +<p class="noind">where δ denotes the operator p<span class="su">1</span>d/dp<span class="su">1</span> + ... + p<span class="su">n</span>d/dp<span class="su">n</span>; (2) If +X<span class="su">1</span>, ... X<span class="su">n</span> be independent functions of x<span class="su">1</span>, ... x<span class="su">n</span>, p<span class="su">1</span>, ... p<span class="su">n</span>, +such that (X<span class="su">i</span>X<span class="su">j</span>) = 0, then U can be found by a quadrature, such that</p> + +<p class="center">(X<span class="su">i</span>U) = δX<span class="su">i</span>;</p> + +<p class="noind">and when X<span class="su">i</span>, ... X<span class="su">n</span>, U satisfy these ½n(n + 1) conditions, then +P<span class="su">1</span>, ... P<span class="su">n</span> can be found, by solution of linear algebraic equations, to +render true the identity dU + P<span class="su">1</span>dX<span class="su">1</span> + ... + P<span class="su">n</span>dX<span class="su">n</span> = p<span class="su">1</span>dx<span class="su">1</span> + ... + p<span class="su">n</span>dx<span class="su">n</span>; +(3) Functions X<span class="su">1</span>, ... X<span class="su">n</span>, P<span class="su">1</span>, ... P<span class="su">n</span> can be found to satisfy +this differential identity when U is an arbitrary given function of +x<span class="su">1</span>, ... x<span class="su">n</span>, p<span class="su">1</span>, ... p<span class="su">n</span>; but this requires integrations. In order +to see what integrations, it is only necessary to verify the statement +that if U be an arbitrary given function of x<span class="su">1</span>, ... x<span class="su">n</span>, p<span class="su">1</span>, ... p<span class="su">n</span>, +and, for r < n, X<span class="su">1</span>, ... X<span class="su">r</span> be independent functions of these variables, +such that (XσU) = δXσ, (XρXσ) = 0, for ρ, σ = 1 ... r, then +the r + 1 homogeneous linear partial differential equations of the +first order (Uƒ) + δƒ = 0, (Xρƒ) = 0, form a complete system. It will +be seen that the assumptions above made for the reduction of +Pfaffian expressions follow from the results here enunciated for +contact transformations.</p> + +<p>We pass on now to consider the solution of any partial +differential equation of the first order; we attempt to explain +certain ideas relatively to a single equation with any +number of independent variables (in particular, an +<span class="sidenote">Partial differential equation of the first order.</span> +ordinary equation of the first order with one independent +variable) by speaking of a single equation with +two independent variables x, y, and one dependent +variable z. It will be seen that we are naturally led to +consider systems of such simultaneous equations, which we +consider below. The central discovery of the transformation +theory of the solution of an equation F(x, y, z, dz/dx, dz/dy) = 0 +is that its solution can always be reduced to the solution of +partial equations which are <i>linear</i>. For this, however, we must +regard dz/dx, dz/dy, during the process of integration, not as the +differential coefficients of a function z in regard to x and y, but as +variables independent of x, y, z, the too great indefiniteness that +might thus appear to be introduced being provided for in another +way. We notice that if z = ψ(x, y) be a solution of the differential +equation, then dz = dxdψ/dx + dydψ/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 = ψ(x, y) +will necessarily be associated with the equations p = dz/dx, +q = dz/dy, and z will satisfy the equation in its original form. We +have previously seen (under <i>Pfaffian Expressions</i>) that if five +variables x, y, z, p, q, otherwise independent, be subject to +dz − 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</p> + +<p class="center">Z − z = p(X − x) + q(Y − y)</p> + +<p class="noind">passing through it, where X, Y, Z are current co-ordinates, and +call this association a surface-element; and if two consecutive +elements of which the point(x + dx, y + dy, z + dz) of one lies on the +plane of the other, for which, that is, the condition dz = pdx + qdy +is satisfied, be said to be <i>connected,</i> and an infinity of connected +elements following one another continuously be called a <i>connectivity</i>, +then our statement is that a connectivity consists of not +more than ∞² elements, the whole number of elements (x, y, z, p, q) +that are possible being called ∞<span class="sp">5</span>. The solution of an equation +F(x, y, z, dz/dx, dz/dy) = 0 is then to be understood to mean finding +in all possible ways, from the ∞<span class="sp">4</span> elements (x, y, z, p, q) which +satisfy F(x, y, z, p, q) = 0 a set of ∞² elements forming a connectivity; +or, more analytically, finding in all possible ways two +relations G = 0, H = 0 connecting x, y, z, p, q and independent of +F = 0, so that the three relations together may involve</p> + +<p class="center">dz = pdx + qdy.</p> + +<p class="noind">Such a set of three relations may, for example, be of the form +z = ψ(x, y), p = dψ/dx, q = dψ/dy; but it may also, as another +case, involve two relations z = ψ(y), x = ψ<span class="su">1</span>(y) connecting x, y, z, +the third relation being</p> + +<p class="center">ψ′(y) = pψ′<span class="su">1</span>(y) + q,</p> + +<p class="noind">the connectivity consisting in that case, geometrically, of a curve +in space taken with ∞¹ of its tangent planes; or, finally, a +connectivity is constituted by a fixed point and all the planes +passing through that point. This generalized view of the meaning +of a solution of F = 0 is of advantage, moreover, in view of +anomalies otherwise arising from special forms of the equation +<span class="sidenote">Meaning of a solution of the equation.</span> +itself. For instance, we may include the case, sometimes +arising when the equation to be solved is obtained +by transformation from another equation, in which F +does not contain either p or q. Then the equation has +∞² solutions, each consisting of an arbitrary point of the surface +F = 0 and all the ∞² planes passing through this point; it also +has ∞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 ∞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 ∞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 ∞² solutions, each +consisting of one of the curves defined by</p> + +<p class="center">dx/P = dy/Q = dz/R</p> + +<p class="noind">taken with all the tangent planes of this curve; and the same +equation has ∞² solutions, each consisting of the points of a +surface containing ∞¹ of these curves and the tangent planes of +this surface. And for the case of n variables there is similarly +the possibility of n + 1 kinds of solution of an equation +F(x<span class="su">1</span>, ... x<span class="su">n</span>, z, p<span class="su">1</span>, ... p<span class="su">n</span>) = 0; these can, however, by a +simple contact transformation be reduced to one kind, in which +there is only one relation z′ = ψ(x′<span class="su">1</span>, ... x′<span class="su">n</span>) connecting the +new variables x’<span class="su">1</span>, ... x′<span class="su">n</span>, z′ (see under <span class="sc"><a href="#artlinks">Pfaffian Expressions</a></span>); +just as in the case of the solution</p> + +<p class="center">z = ψ(y), x = ψ<span class="su">1</span>(y), ψ′(y) = pψ′<span class="su">1</span>(y) + q</p> + +<p class="noind">of the equation Pp + Qq = R the transformation z’ = z − px, +x′ = p, p′ = −x, y′ = y, q′ = q gives the solution</p> + +<p class="center">z′ = ψ(y′) + x′ψ<span class="su">1</span>(y′), p′ = dz′/dx′, q′ = dz′/dy′</p> + +<p class="noind">of the transformed equation. These explanations take no +account of the possibility of p and q being infinite; this can be +dealt with by writing p = -u/w, q = -v/w, and considering +homogeneous equations in u, v, w, with udx + vdy + wdz = 0 as the +differential relation necessary for a connectivity; in practice we +use the ideas associated with such a procedure more often without +the appropriate notation.</p> + +<p>In utilizing these general notions we shall first consider +the theory of characteristic chains, initiated by Cauchy, which +shows well the nature of the relations implied by the given +differential equation; the alternative ways of carrying +<span class="sidenote">Order of the ideas.</span> +out the necessary integrations are suggested by considering +the method of Jacobi and Mayer, while a good +summary is obtained by the formulation in terms of a Pfaffian +expression.</p> + +<p>Consider a solution of F = 0 expressed by the three independent +equations F = 0, G = 0, H = 0. If it be a solution in which there is +more than one relation connecting x, y, z, let new variables x′, y′, z′, p′, q′ +be introduced, as before explained under <span class="sc"><a href="#artlinks">Pfaffian Expressions</a></span>, +<span class="sidenote">Characteristic chains.</span> +in which z’ is of the form</p> + +<p class="center">z′ = z − p<span class="su">1</span>x<span class="su">1</span> − ... − p<span class="su">s</span>x<span class="su">s</span> (s = 1 or 2),</p> + +<p class="noind">so that the solution becomes of a form z’ = ψ(x′y′), +p′ = dψ/dx′, q′ = dψ/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</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">dx′<span class="f150">/</span></td> <td>dF′</td> +<td rowspan="2">= dy′<span class="f150">/</span></td> <td>dF′</td> +<td rowspan="2">= dz′<span class="f150">/ (</span> p′</td> <td>dF′</td> +<td rowspan="2">+ q′</td> <td>dF′</td> <td rowspan="2"><span class="f150">)</span>.</td></tr> +<tr><td class="denom">dp′</td> <td class="denom">dq′</td> +<td class="denom">dp′</td> <td class="denom">dq′</td></tr></table> + +<p class="noind">Passing in this direction on the surface z′ = ψ(x′, y′) the tangent +<span class="pagenum"><a name="page233" id="page233"></a>233</span> +plane of the surface at this consecutive point is (p′ + dp′, q′ + dq′), +where, since F′(x′, y′, ψ, dψ/dx′, dψ/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,</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">dx′<span class="f150">/</span></td> <td>dF′</td> +<td rowspan="2">= dy′<span class="f150">/</span></td> <td>dF′</td> +<td rowspan="2">= dz′<span class="f150">/ (</span> p′</td> <td>dF′</td> +<td rowspan="2">+ q′</td> <td>dF′</td> +<td rowspan="2"><span class="f150">)</span> = dp′<span class="f150">/ (</span> −</td> <td>dF′</td> +<td rowspan="2">− p′</td> <td>dF′</td> +<td rowspan="2"><span class="f150">)</span> = dq′<span class="f150">/ (</span> −</td> <td>dF′</td> +<td rowspan="2">− q′</td> <td>dF′</td> <td rowspan="2"><span class="f150">)</span></td></tr> +<tr><td class="denom">dp′</td> <td class="denom">dq′</td> +<td class="denom">dp′</td> <td class="denom">dq′</td> +<td class="denom">dx′</td> <td class="denom">dz′</td> +<td class="denom">dy′</td> <td class="denom">dz′</td></tr></table> + +<p class="noind">are satisfied along a connectivity of ∞¹ elements consisting of a curve +on z′ = ψ(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′ + δx′, y′ <span class="correction" title="amended from =">+</span> δy′, z′ + δz′) be a consecutive point of this curve, we +find at once</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">δx′<span class="f150">(</span></td> <td>dF′</td> +<td rowspan="2">+ p′</td> <td>dF′</td> +<td rowspan="2"><span class="f150">)</span> + δy′<span class="f150">(</span></td> <td>dF′</td> +<td rowspan="2">q′</td> <td>dF′</td> <td rowspan="2"><span class="f150">)</span> = 0;</td></tr> +<tr><td class="denom">dx′</td> <td class="denom">dz′</td> +<td class="denom">dy′</td> <td class="denom">dz′</td></tr></table> + +<p class="noind">thus the equations above give δx′dp′ + δy′dq′ = 0, or the tangent line +of the plane curve, is, on the surface z′ = ψ(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′<span class="su">0</span>, y′<span class="su">0</span>, z′<span class="su">0</span>, p′<span class="su">0</span>, q′<span class="su">0</span>, +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 ∞¹ elements, which we call +a <i>characteristic chain</i>; and it is remarkable that when we transform +again to the original variables (x, y, z, p, q), the form of the differential +equations for the chain is unaltered, so that they can be written +down at once from the equation F = 0. Thus we have proved that +the characteristic chain starting from any ordinary element of any +integral of this equation F = 0 consists only of elements belonging +to this integral. For instance, if the equation do not contain p, q, +the characteristic chain, starting from an arbitrary plane through +an arbitrary point of the surface F = 0, consists of a pencil of planes +whose axis is a tangent line of the surface F = 0. Or if F = 0 be of +the form Pp + Qq = R, the chain consists of a curve satisfying +dx/P = dy/Q = dz/R and a single infinity of tangent planes of this +curve, determined by the tangent plane chosen at the initial point. +In all cases there are ∞³ characteristic chains, whose aggregate may +therefore be expected to exhaust the ∞<span class="sp">4</span> elements satisfying F = 0.</p> + + + +<p>Consider, in fact, a single infinity of connected elements each +satisfying F = 0, say a chain connectivity T, consisting of elements +specified by x<span class="su">0</span>, y<span class="su">0</span>, z<span class="su">0</span>, p<span class="su">0</span>, q<span class="su">0</span>, which we suppose expressed as +<span class="sidenote">Complete integral constructed with characteristic chains.</span> +functions of a parameter u, so that</p> + +<p class="center">U<span class="su">0</span> = dz<span class="su">0</span>/du − p<span class="su">0</span>dx<span class="su">0</span>/du − q<span class="su">0</span>dy<span class="su">0</span>/du</p> + +<p class="noind">is everywhere zero on this chain; further, suppose that +each of F, dF/dp, ... , dF/dx + pdF/dz is developable +about each element of this chain T, and that T is <i>not</i> a +characteristic chain. Then consider the aggregate of the +characteristic chains issuing from all the elements of T. +The ∞² elements, consisting of the aggregate of these +characteristic chains, satisfy F = 0, provided the chain +connectivity T consists of elements satisfying F = 0; for each +characteristic chain satisfies dF = 0. It can be shown that these +chains are connected; in other words, that if x, y, z, p, q, be any +element of one of these characteristic chains, not only is</p> + +<p class="center">dz/dt − pdx/dt − qdy/dt = 0,</p> + +<p class="noind">as we know, but also U = dz/du − pdx/du − qdy/du is also zero. For +we have</p> + +<table class="math0" summary="math"> +<tr><td>dU</td> <td rowspan="2">=</td> <td>d</td> +<td rowspan="2"><span class="f150">(</span></td> <td>dz</td> +<td rowspan="2">− p</td> <td>dx</td> +<td rowspan="2">− q</td> <td>dy</td> +<td rowspan="2"><span class="f150">)</span> −</td> <td>d</td> +<td rowspan="2"><span class="f150">(</span></td> <td>dz</td> +<td rowspan="2">− p</td> <td>dx</td> +<td rowspan="2">− q</td> <td>dy</td> <td rowspan="2"><span class="f150">)</span></td></tr> +<tr><td class="denom">dt</td> <td class="denom">dt</td> <td class="denom">du</td> +<td class="denom">du</td> <td class="denom">du</td> <td class="denom">du</td> +<td class="denom">dt</td> <td class="denom">dt</td> <td class="denom">dt</td></tr></table> + +<table class="math0" summary="math"> +<tr><td rowspan="2">=</td> <td>dp</td> <td>dx</td> +<td rowspan="2">−</td> <td>dp</td> <td>dx</td> +<td rowspan="2">+</td> <td>dq</td> <td>dy</td> +<td rowspan="2">−</td> <td>dq</td> <td>dy</td> <td rowspan="2">,</td></tr> +<tr><td class="ov">du</td> <td class="ov">dt</td> <td class="ov">dt</td> +<td class="ov">du</td> <td class="ov">du</td> <td class="ov">dt</td> +<td class="ov">dt</td> <td class="ov">du</td></tr></table> + +<p class="noind">which is equal to</p> + +<table class="math0" summary="math"> +<tr><td>dp</td> <td>dF</td> <td rowspan="2">+</td> <td>dx</td> + <td rowspan="2"><span class="f150">(</span></td> <td>dF</td> + <td rowspan="2">+ p</td> <td>dF</td> <td rowspan="2"><span class="f150">)</span> +</td> + <td>dq</td> <td>dF</td> <td rowspan="2">+</td> <td>dy</td> + <td rowspan="2"><span class="f150">(</span></td> <td>dF</td> + <td rowspan="2">+ q</td> <td>dF</td> <td rowspan="2"><span class="f150">)</span> = −</td> + <td>dF</td> <td rowspan="2">U.</td></tr> +<tr><td class="ov">du</td> <td class="ov">dp</td> <td class="ov">du</td> + <td class="ov">dx</td> <td class="ov">dz</td> <td class="ov">du</td> + <td class="ov">dq</td> <td class="ov">du</td> <td class="ov">dy</td> + <td class="ov">dz</td> <td class="ov">dz</td></tr></table> + +<p class="noind">As dF/dz is a developable function of t, this, giving</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">U = U<span class="su">0</span> exp<span class="f150">(</span> − <span class="f150">∫</span></td> <td class="bk">t</td> <td>dF</td> +<td rowspan="2">dt <span class="f150">)</span>,</td></tr> +<tr><td class="bk">t<span class="su">0</span></td> <td class="ov">dz</td></tr></table> + +<p class="noind">shows that U is everywhere zero. Thus integrals of F = 0 are +obtainable by considering the aggregate of characteristic chains +issuing from arbitrary chain connectivities T satisfying F = 0; and +such connectivities T are, it is seen at once, determinable without +integration. Conversely, as such a chain connectivity T can be taken +out from the elements of any given integral all possible integrals +are obtainable in this way. For instance, an arbitrary curve in +space, given by x<span class="su">0</span> = θ(u), y<span class="su">0</span> = φ(u), z<span class="su">0</span> = ψ(u), determines by the two +equations F(x<span class="su">0</span>, y<span class="su">0</span>, z<span class="su">0</span>, p<span class="su">0</span>, q<span class="su">0</span>) = 0, ψ′(u) = p<span class="su">0</span>θ′(u) + q<span class="su">0</span>φ′(u), such a +chain connectivity T, through which there passes a perfectly +definite integral of the equation F = 0. By taking ∞² initial chain +connectivities T, as for instance by taking the curves x<span class="su">0</span> = θ, y<span class="su">0</span> = φ, +z<span class="su">0</span> = ψ to be the ∞² curves upon an arbitrary surface, we thus obtain +∞² integrals, and so ∞<span class="sp">4</span> elements satisfying F = 0. In general, if +functions G, H, independent of F, be obtained, such that the +equations F = 0, G = b, H = c represent an integral for all values of the +constants b, c, these equations are said to constitute a <i>complete +integral</i>. Then ∞<span class="sp">4</span> elements satisfying F = 0 are known, and in fact +every other form of integral can be obtained without further integrations.</p> + +<p>In the foregoing discussion of the differential equations of a +characteristic chain, the denominators dF/dp, ... may be supposed +to be modified in form by means of F = 0 in any way conducive to +a simple integration. In the immediately following explanation of +ideas, however, we consider indifferently all equations F = constant; +when a function of x, y, z, p, q is said to be zero, it is meant that this +is so identically, not in virtue of F = 0; in other words, we consider +the integration of F = a, where a is an arbitrary constant. In the +theory of linear partial equations we have seen that the integration +<span class="sidenote">Operations necessary for integration of F = a.</span> +of the equations of the characteristic chains, from which, +as has just been seen, that of the equation F = a follows +at once, would be involved in completely integrating +the single linear homogeneous partial differential equation +of the first order [Fƒ] = 0 where the notation is that +explained above under <span class="sc"><a href="#artlinks">Contact Transformations</a></span>. One +obvious integral is ƒ = F. Putting F = a, where a is arbitrary, +and eliminating one of the independent variables, we can reduce +this equation [Fƒ] = 0 to one in four variables; and so on. Calling, then, +the determination of a single integral of a single homogeneous partial +differential equation of the first order in n independent variables, <i>an +operation of order</i> n − 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 +[Fƒ] = 0, which is an operation of order three. Then it follows from +the identity [ƒ[φψ]] + [φ[ψƒ]] + [ψ[ƒφ]] = dƒ/dz [ψφ] + dφ/dz [ψƒ] + dψ/dz [ƒφ] before +remarked, by putting φ = F, ψ = G, and then [Fƒ] = A(ƒ), [Gƒ] = B(ƒ), +that AB(ƒ) − BA(ƒ) = dF/dz B(ƒ) − dG/dz A(ƒ), so that the two linear equations +[Fƒ] = 0, [Gƒ] = 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 ∞² elements satisfying F = a, G = b, H = c, wherein a, b, c are +assigned constants, can then be seen to constitute an integral of F = a. +For the conditions that a characteristic chain of G = b issuing from +an element satisfying F = a, G = b, H = c should consist only of +elements satisfying these three equations are simply [FG] = 0, [GH] = 0. +Thus, starting from an arbitrary element of (F = a, G = b, H = c), we +can single out a connectivity of elements of (F = a, G = b, H = c) +forming a characteristic chain of G = b; then the aggregate of the +characteristic chains of F = a issuing from the elements of this +characteristic chain of G = b will be a connectivity consisting only of +elements of</p> + +<p class="center">(F = a, G = b, H = c),</p> + +<p class="noind">and will therefore constitute an integral of F = a; further, it will +include all elements of (F = a, G = b, H = c). This result follows also +from a theorem given under <i>Contact Transformations</i>, which shows, +moreover, that though the characteristic chains of F = a are not +determined by the three equations F = a, G = b, H = c, no further +integration is now necessary to find them. By this theorem, since +identically [FG] = [GH] = [FH] = 0, we can find, by the solution of +linear algebraic equations only, a non-vanishing function σ and two +functions A, C, such that</p> + +<p class="center">dG − AdF − CdH = σ(dz − pdz − qdy);</p> + +<p class="noind">thus all the elements satisfying F = a, G = b, H = c, satisfy dz = pdx + qdy +and constitute a connectivity, which is therefore an integral of +F = a. While, further, from the associated theorems, F, G, H, A, C +are independent functions and [FC] = 0. Thus C may be taken to +be the remaining integral independent of G, H, of the equation +[Fƒ] = 0, whereby the characteristic chains are entirely determined.</p> + +<p>When we consider the particular equation F = 0, neglecting the +case when neither p nor q enters, and supposing p to enter, we may +express p from F = 0 in terms of x, y, z, q, and then eliminate it from +all other equations. Then instead of the equation [Fƒ] = 0, we +have, if F = 0 give p = ψ(x, y, z, q), the equation</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">Σƒ = − <span class="f150">(</span></td> <td>dƒ</td> +<td rowspan="2">+ ψ</td> <td>dƒ</td> +<td rowspan="2"><span class="f150">)</span> +</td> <td>dψ</td> +<td rowspan="2"><span class="f150">(</span></td> <td>dƒ</td> +<td rowspan="2">+ q</td> <td>dƒ</td> +<td rowspan="2"><span class="f150">)</span> − <span class="f150">(</span></td> <td>dψ</td> +<td rowspan="2">+ q</td> <td>dψ</td> +<td rowspan="2"><span class="f150">)</span></td> <td>dƒ</td> <td rowspan="2">= 0,</td></tr> +<tr><td class="denom">dx</td> <td class="denom">dz</td> <td class="denom">dq</td> +<td class="denom">dy</td> <td class="denom">dz</td> <td class="denom">dy</td> +<td class="denom">dz</td> <td class="denom">dq</td></tr></table> + +<p class="noind">moreover obtainable by omitting the term in dƒ/dp in [p − ψ, ƒ] = 0. +Let x<span class="su">0</span>, y<span class="su">0</span>, z<span class="su">0</span>, q<span class="su">0</span>, be values about which the coefficients in +<span class="sidenote">The single equation F = 0 and Pfaffian formulations.</span> +this equation are developable, and let ζ, η, ω be the +principal solutions reducing respectively to z, y and q +when x = x<span class="su">0</span>. Then the equations p = ψ, ζ = z<span class="su">0</span>, η = y<span class="su">0</span>, ω = q<span class="su">0</span> +represent a characteristic chain issuing from the element +x<span class="su">0</span>, y<span class="su">0</span>, z<span class="su">0</span>, ψ<span class="su">0</span>, q<span class="su">0</span>; we have seen that the aggregate of +such chains issuing from the elements of an arbitrary +chain satisfying</p> + +<p class="center">dz<span class="su">0</span> = p<span class="su">0</span>dx<span class="su">0</span> − q<span class="su">0</span>dy<span class="su">0</span> = 0</p> + +<p class="noind">constitute an integral of the equation p = ψ. Let this arbitrary +<span class="pagenum"><a name="page234" id="page234"></a>234</span> +chain be taken so that x<span class="su">0</span> is constant; then the condition for initial +values is only</p> + +<p class="center">dz<span class="su">0</span> − q<span class="su">0</span>dy<span class="su">0</span> = 0,</p> + +<p class="noind">and the elements of the integral constituted by the characteristic +chains issuing therefrom satisfy</p> + +<p class="center">dζ − ωdη = 0.</p> + +<p class="noind">Hence this equation involves dz − ψdx − qdy = 0, or we have</p> + +<p class="center">dz − ψdx − qdy = σ(dζ − ωdη),</p> + +<p class="noind">where σ is not zero. Conversely, the integration of p = ψ is, essentially, +the problem of writing the expression dz − ψdx − qdy in the form +σ(dζ − ωdη), as must be possible (from what was said under <span class="sc"><a href="#artlinks">Pfaffian +Expressions</a></span>).</p> + +<p>To integrate a system of simultaneous equations of the first +order X<span class="su">1</span> = a<span class="su">1</span>, ... X<span class="su">r</span> = a<span class="su">r</span> in n independent variables x<span class="su">1</span>, ... x<span class="su">n</span> +and one dependent variable z, we write p<span class="su">1</span> for dz/dx<span class="su">1</span>, &c., +<span class="sidenote">System of equations of the first order.</span> +and attempt to find n + 1 − r further functions Z, X<span class="su">r+1</span> +... X<span class="su">n</span>, such that the equations Z = a, X<span class="su">i</span> = a<span class="su">i</span>,(i = 1, ... n) +involve dz − p<span class="su">1</span>dx<span class="su">1</span> − ... − p<span class="su">n</span>dx<span class="su">n</span> = 0. By an argument +already given, the common integral, if existent, must be satisfied +by the equations of the characteristic chains of any one equation +X<span class="su">i</span> = a<span class="su">i</span>; thus each of the expressions [X<span class="su">i</span>X<span class="su">j</span>] must vanish in virtue +of the equations expressing the integral, and we may without loss of +generality assume that each of the corresponding ½r(r − 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 [X<span class="su">1</span>ƒ] = 0, ... [X<span class="su">r</span>ƒ] = 0; or having +obtained one integral of this system other than X<span class="su">1</span>, ... X<span class="su">r</span>, say +X<span class="su">r+1</span>, we may consider the system [X<span class="su">1</span>ƒ] = 0, ... [X<span class="su">r+1</span>ƒ] = 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 p<span class="su">1</span> = φ<span class="su">1</span>, ... p<span class="su">r</span> = φ<span class="su">r</span>, wherein p<span class="su">1</span>, ... p<span class="su">r</span> are not involved in +φ<span class="su">1</span>, ... φ<span class="su">r</span>, if it satisfies the ½r(r − 1) relations [p<span class="su">i</span> − φ<span class="su">i</span>, p<span class="su">j</span> − φ<span class="su">j</span>] = 0, +has a solution z = ψ(x<span class="su">1</span>, ... x<span class="su">n</span>), p<span class="su">1</span> = dψ/dx<span class="su">1</span>, ... p<span class="su">n</span> = dψ/dx<span class="su">n</span>, +reducing to an arbitrary function of x<span class="su">r+1</span>, ... x<span class="su">n</span> only, when x<span class="su">1</span> = xº<span class="su">1</span>, +... x<span class="su">r</span> = xº<span class="su">r</span> under certain conditions as to developability; a +generalization of the theorem for linear equations. The problem of +integration of this system is, as before, to put</p> + +<p class="center">dz − φ<span class="su">1</span>dx<span class="su">1</span> − ... − φ<span class="su">r</span>dx<span class="su">r</span> − p<span class="su">r+1</span>dx<span class="su">r+1</span> − ... − p<span class="su">n</span>dx<span class="su">n</span></p> + +<p class="noind">into the form σ(dζ − ω<span class="su">r+1</span> + dξ<span class="su">r+1</span> − ... − ω<span class="su">n</span>dξ<span class="su">n</span>); and here ζ, ξ<span class="su">r+1</span>, ... ξ<span class="su">n</span>, +ω<span class="su">r+1</span>, ... ω<span class="su">n</span> may be taken, as before, to be principal integrals +of a certain complete system of linear equations; those, namely, +determining the characteristic chains.</p> + +<p>If L be a function of t and of the 2n quantities x<span class="su">1</span>, ... x<span class="su">n</span>, ẋ<span class="su">1</span>, ... +ẋ<span class="su">n</span>, where ẋ<span class="su">i</span>, denotes dx<span class="su">i</span>/dt, &c., and if in the n equations</p> + +<table class="math0" summary="math"> +<tr><td>d</td> <td rowspan="2"><span class="f150">(</span></td> <td>dL</td> +<td rowspan="2"><span class="f150">)</span> =</td> <td>dL</td></tr> +<tr><td class="denom">dt</td> <td class="denom">dx<span class="su">i</span></td> <td class="denom">dx<span class="su">i</span></td></tr></table> + +<p class="noind">we put p<span class="su">i</span> = dL/dẋ<span class="su">i</span>, and so express ẋ<span class="su">i</span>, ... ẋ<span class="su">n</span> in terms of t, x<span class="su">i</span>, ... +x<span class="su">n</span>, p<span class="su">1</span>, ... p<span class="su">n</span>, assuming that the determinant of the quantities +d²L/dx<span class="su">i</span>dẋ<span class="su">j</span> is not zero; if, further, H denote the function of t, x<span class="su">1</span>, ... +x<span class="su">n</span>, p<span class="su">1</span>, ... p<span class="su">n</span>, numerically equal to p<span class="su">1</span>ẋ<span class="su">1</span> + ... + p<span class="su">n</span>ẋ<span class="su">n</span> − L, it is easy +<span class="sidenote">Equations of dynamics.</span> +to prove that dp<span class="su">i</span>/dt = −dH/dx<span class="su">i</span>, dx<span class="su">i</span>/dt = dH/dp<span class="su">i</span>. These +so-called <i>canonical</i> equations form part of those for +the characteristic chains of the single partial equation +dz/dt + H(t, x<span class="su">1</span>, ... x<span class="su">n</span>, dz/dx<span class="su">1</span>, ..., dz/dx<span class="su">n</span>) = 0, to which +then the solution of the original equations for x<span class="su">1</span> ... x<span class="su">n</span> can be +reduced. It may be shown (1) that if z = ψ(t, x<span class="su">1</span>, ... x<span class="su">n</span>, c<span class="su">1</span>, .. c<span class="su">n</span>) + c +be a complete integral of this equation, then p<span class="su">i</span> = dψ/dx<span class="su">i</span>, dψ/dc<span class="su">i</span> = e<span class="su">i</span> +are 2n equations giving the solution of the canonical equations +referred to, where c<span class="su">1</span> ... c<span class="su">n</span> and e<span class="su">1</span>, ... e<span class="su">n</span> are arbitrary constants; +(2) that if x<span class="su">i</span> = X<span class="su">i</span>(t, x<span class="sp">0</span><span class="su">1</span>, ... pº<span class="su">n</span>), p<span class="su">i</span> = P<span class="su">i</span>(t, xº<span class="su">1</span>, ... p<span class="sp">0</span><span class="su">n</span>) be the principal +solutions of the canonical equations for t = t<span class="sp">0</span>, and ω denote the result +of substituting these values in p<span class="su">1</span>dH/dp<span class="su">1</span> + ... + p<span class="su">n</span>dH/dp<span class="su">n</span> − H, and +Ω = ∫<span class="sp1">t</span><span class="su">t0</span> ωdt, where, after integration, Ω is to be expressed as a function +of t, x<span class="su">1</span>, ... x<span class="su">n</span>, xº<span class="su">1</span>, ... xº<span class="su">n</span>, then z = Ω + z<span class="sp">0</span> is a complete integral of +the partial equation.</p> + +<p>A system of differential equations is said to allow a certain +continuous group of transformations (see <span class="sc"><a href="#artlinks">Groups, Theory of</a></span>) +when the introduction for the variables in the differential +equations of the new variables given by the +<span class="sidenote">Application of theory of continuous groups to formal theories.</span> +equations of the group leads, for all values of the +parameters of the group, to the same differential equations +in the new variables. It would be interesting +to verify in examples that this is the case in at least +the majority of the differential equations which are +known to be integrable in finite terms. We give a theorem of +very general application for the case of a simultaneous complete +system of linear partial homogeneous differential equations of the +first order, to the solution of which the various differential equations +discussed have been reduced. It will be enough to consider +whether the given differential equations allow the infinitesimal +transformations of the group.</p> + +<p>It can be shown easily that sufficient conditions in order that a +complete system Π<span class="su">1</span>ƒ = 0 ... Π<span class="su">k</span>ƒ = 0, in n independent variables, +should allow the infinitesimal transformation Pƒ = 0 are expressed +by k equations Π<span class="su">i</span>Pƒ − PΠ<span class="su">i</span>ƒ = λ<span class="su">i1</span>Π<span class="su">1</span>ƒ + ... + λ<span class="su">ik</span>Π<span class="su">k</span>ƒ. Suppose now +a complete system of n − r equations in n variables to allow a +group of r infinitesimal transformations (P<span class="su">1</span>f, ..., P<span class="su">r</span>ƒ) which has +an invariant subgroup of r − 1 parameters (P<span class="su">1</span>ƒ, ..., P<span class="su">r-1</span>ƒ), it +being supposed that the n quantities Π<span class="su">1</span>ƒ, ..., Π<span class="su">n-r</span>ƒ, P<span class="su">1</span>ƒ, ..., +P<span class="su">r</span>ƒ 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 Π<span class="su">i</span>Pσƒ − PσΠ<span class="su">i</span>f is a linear +function of Π<span class="su">1</span>ƒ, ..., Π<span class="su">n-r</span>ƒ and the simultaneous system of independent +equations Π<span class="su">1</span>ƒ = 0, ... Π<span class="su">n-r</span>ƒ = 0, P<span class="su">1</span>ƒ = 0, ... P<span class="su">r-1</span>ƒ = 0 +is therefore a complete system, allowing the infinitesimal transformation +P<span class="su">r</span>ƒ. This complete system of n − 1 equations has therefore +one common solution ω, and P<span class="su">r</span>(ω) is a function of ω. By +choosing ω suitably, we can then make P<span class="su">r</span>(ω) = 1. From this +equation and the n − 1 equations Π<span class="su">i</span>ω = 0, P<span class="su">σω</span> = 0, we can determine +ω by a quadrature only. Hence can be deduced a much more +general result, <i>that if the group of r parameters be integrable, the +complete system can be entirety solved by quadratures</i>; it is only +necessary to introduce the solution found by the first quadrature as +an independent variable, whereby we obtain a complete system of +n − 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′ = ψ(x, y) allow the infinitesimal transformation +ξdƒ/dx + ηdƒ/dy, the integral curves ω(x, y) = y<span class="sp">0</span>, wherein ω(x, y) is +the solution of dƒ/dx + ψ(x, y) dƒ/dy = 0 reducing to y for x = x<span class="sp">0</span>, are +interchanged among themselves by the infinitesimal transformation, +or ω(x, y) can be chosen to make ξd<span class="su">ω</span>/dx + ηd<span class="su">ω</span>/dy = 1; this, with +dω/dx + ψdω/dy = 0, determines ω as the integral of the complete +differential (dy − ψdx)/(η − ψξ). 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 ξdƒ/dx + ηdƒ/dy can by change of variables +(after integration) be brought to the form dƒ/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” = ψ(x, y, y′), equivalent to dy/dx = y<span class="su">1</span>, dy<span class="su">1</span>/dx = ψ(x, y, y<span class="su">1</span>), +if H, H<span class="su">1</span> be the solutions for y and y<span class="su">1</span> chosen to reduce to y<span class="sp">0</span> and +yº<span class="su">1</span> when x = x<span class="sp">0</span>, and the equations H = y, H<span class="su">1</span>= y<span class="su">1</span> be equivalent +to ω = y<span class="sp">0</span>, ω<span class="su">1</span> = yº<span class="su">1</span>, then ω, ω<span class="su">1</span> are the principal solutions of +Πƒ = dƒ/dx + y<span class="su">1</span>dƒ/dy + ψdƒ/dy<span class="su">1</span> = 0. If the original equation allow +an infinitesimal transformation whose first <i>extended</i> form (see +<span class="sc"><a href="#artlinks">Groups</a></span>) is Pƒ = ξdƒ/dx + ηdƒ/dy + η<span class="su">1</span>dƒ/dy<span class="su">1</span>, where η<span class="su">1</span>δt is the increment +of dy/dx when ξδt, ηδt are the increments of x, y, and is to be +expressed in terms of x, y, y<span class="su">1</span>, then each of Pω and Pω<span class="su">1</span> must +be functions of ω and ω<span class="su">1</span>, or the partial differential equation Πƒ +must allow the group Pƒ. 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 Πƒ and the +two infinitesimal transformations are not linearly connected. It +can be shown, from the fact that η<span class="su">1</span> is a quadratic polynomial in y<span class="su">1</span>, +that no differential equation of the second order can allow more +than 8 really independent infinitesimal transformations, and that +every homogeneous linear differential equation of the second order +allows just 8, being in fact reducible to d²y/dx² = 0. Since every +group of more than two parameters has subgroups of two parameters, +a differential equation of the second order allowing a group +of more than two parameters can, as a rule, be solved by quadratures. +By transforming the group we see that if a differential equation of +the second order allows a single infinitesimal transformation, it can +be transformed to the form F(x, dγ/dx, d²γ/dx²); this is not the case +for every differential equation of the second order. (3) For an +ordinary differential equation of the third order, allowing an integrable +group of three parameters whose infinitesimal transformations +are not linearly connected with the partial equation to which the +solution of the given ordinary equation is reducible, the similar +result follows that it can be integrated by quadratures. But if the +group of three parameters be simple, this result must be replaced +by the statement that the integration is reducible to quadratures +and that of a so-called Riccati equation of the first order, of the +form dy/dx = A + By + Cy², where A, B, C are functions of x. (4) Similarly +for the integration by quadratures of an ordinary equation +y<span class="su">n</span> = ψ(x, y, y<span class="su">1</span>, ... y<span class="su">n-1</span>) of any order. Moreover, the group allowed +by the equation may quite well consist of extended contact transformations. +An important application is to the case where the differential +equation is the resolvent equation defining the group of +<span class="pagenum"><a name="page235" id="page235"></a>235</span> +transformations or rationality group of another differential equation +(see below); in particular, when the rationality group of an ordinary +linear differential equation is integrable, the equation can be solved +by quadratures.</p> + +<p>Following the practical and provisional division of theories +of differential equations, to which we alluded at starting, into +transformation theories and function theories, we pass +now to give some account of the latter. These are both +<span class="sidenote">Consideration of function theories of differential equations.</span> +a necessary logical complement of the former, and the +only remaining resource when the expedients of the +former have been exhausted. While in the former +investigations we have dealt only with values of the +independent variables about which the functions are +developable, the leading idea now becomes, as was long ago +remarked by G. Green, the consideration of the neighbourhood of +the values of the variables for which this developable character +ceases. Beginning, as before, with existence theorems applicable +for ordinary values of the variables, we are to consider the cases of +failure of such theorems.</p> + +<p>When in a given set of differential equations the number of +equations is greater than the number of dependent variables, the +equations cannot be expected to have common solutions unless +certain conditions of compatibility, obtainable by equating +different forms of the same differential coefficients deducible from +the equations, are satisfied. We have had examples in systems +of linear equations, and in the case of a set of equations +p<span class="su">1</span> = φ<span class="su">1</span>, ..., p<span class="su">r</span> = φ<span class="su">r</span>. For the case when the number of equations +is the same as that of dependent variables, the following is a +general theorem which should be referred to: Let there be r +equations in r dependent variables z<span class="su">1</span>, ... z<span class="su">r</span> and n independent +<span class="sidenote">A general existence theorem.</span> +variables x<span class="su">1</span>, ... x<span class="su">n</span>; let the differential coefficient of +z<span class="su">σ</span> of highest order which enters be of order h<span class="su">σ</span>, and +suppose d<span class="sp">hσ</span>z<span class="su">σ</span> / dx<span class="su">1</span><span class="sp">hσ</span> to enter, so that the equations can be +written d<span class="sp">hσ</span>z<span class="su">σ</span> / dx<span class="su">1</span><span class="sp">hσ</span> = Φ<span class="su">σ</span>, where in the general differential +coefficient of z<span class="su">ρ</span> which enters in Φ<span class="su">σ</span>, say</p> + +<p class="center">d<span class="sp">k1 + ... + kn</span> z<span class="su">ρ</span> / dx<span class="su">1</span><span class="sp">k1</span> ... dx<span class="su">n</span><span class="sp">kn</span>,</p> + +<p class="noind">we have k<span class="su">1</span> < h<span class="su">ρ</span> and k<span class="su">1</span> + ... + k<span class="su">n</span> ≤ h<span class="su">ρ</span>. Let a<span class="su">1</span>, ... a<span class="su">n</span>, +b<span class="su">1</span>, ... b<span class="su">r</span>, and bρ<span class="su">k1 ... kn</span> be a set of values of</p> + +<p class="center">x<span class="su">1</span>, ... x<span class="su">n</span>, z<span class="su">1</span>, ... z<span class="su">r</span></p> + +<p class="noind">and of the differential coefficients entering in Φ<span class="su">σ</span> about which +all the functions Φ<span class="su">1</span>, ... Φ<span class="su">r</span>, are developable. Corresponding +to each dependent variable z<span class="su">σ</span>, we take now a set of h<span class="su">σ</span> functions of +x<span class="su">2</span>, ... x<span class="su">n</span>, say φ<span class="su">σ</span>, φ<span class="su">σ</span>;<span class="sp">(1)</span>, ... ,φ<span class="su">σ</span><span class="sp">h−1</span> arbitrary save that they must +be developable about a<span class="su">2</span>, a<span class="su">3</span>, ... a<span class="su">n</span>, and such that for these +values of x<span class="su">2</span>, ... x<span class="su">n</span>, the function φ<span class="su">ρ</span> reduces to b<span class="su">ρ</span>, and the +differential coefficient</p> + +<p class="center">d<span class="sp">k2 + ... + kn</span> φ<span class="su">ρ</span><span class="sp">k1</span> / dx<span class="su">2</span><span class="sp">k2</span> ... dx<span class="su">n</span><span class="sp">kn</span></p> + +<p class="noind">reduces to b<span class="sp">ρ</span><span class="su">k1 ... kn</span>. Then the theorem is that there exists +one, and only one, set of functions z<span class="su">1</span>, ... z<span class="su">r</span>, of x<span class="su">2</span>, ... x<span class="su">n</span> +developable about a<span class="su">1</span>, ... a<span class="su">n</span> satisfying the given differential +equations, and such that for x<span class="su">1</span> = a<span class="su">1</span> we have</p> + +<p class="center">z<span class="su">σ</span> = φ<span class="su">σ</span>, dz<span class="su">σ</span> / dx<span class="su">1</span> = φ<span class="su">σ</span><span class="sp">(1)</span>, ... d<span class="sp">hσ−1</span>z<span class="su">σ</span> / d<span class="sp">hσ−1</span>x<span class="su">1</span> = φ<span class="su">σ</span><span class="sp">hσ−1</span>.</p> + +<p class="noind">And, moreover, if the arbitrary functions φ<span class="su">σ</span>, φ<span class="su">σ</span><span class="sp">(1)</span> ... contain a +certain number of arbitrary variables t<span class="su">1</span>, ... t<span class="su">m</span>, and be developable +about the values tº<span class="su">1</span>, ... tº<span class="su">m</span> of these variables, the +solutions z<span class="su">1</span>, ... z<span class="su">r</span> will contain t<span class="su">1</span>, ... t<span class="su">m</span>, and be developable +about tº<span class="su">1</span>, ... tº<span class="su">m</span>.</p> + +<p>The proof of this theorem may be given by showing that if +ordinary power series in x<span class="su">1</span> − a<span class="su">1</span>, ... x<span class="su">n</span> − a<span class="su">n</span>, t<span class="su">1</span> − tº<span class="su">1</span>, ... t<span class="su">m</span> − tº<span class="su">m</span> +be substituted in the equations wherein in z<span class="su">σ</span> the coefficients of +(x<span class="su">1</span> − a<span class="su">1</span>)º, x<span class="su">1</span> − a<span class="su">1</span>, ..., (x<span class="su">1</span> − a<span class="su">1</span>)<span class="sp">hσ−1</span> are the arbitrary functions +φ<span class="su">σ</span>, φ<span class="su">σ</span><span class="sp">(1)</span>, ..., φ<span class="su">σ</span><span class="sp">h−1</span>, 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 <span class="sc"><a href="#artlinks">Function</a></span>), +a function being determined entirely by its development in the +neighbourhood of one set of values of the independent variables, +from which all its other values arise by <i>continuation</i>; it being of +course understood that the coefficients in the differential equations +are to be continued at the same time. But it is to be remarked that +there is no ground for believing, if this method of continuation be +utilized, that the function is single-valued; we may quite well return +to the same values of the independent variables with a different +<span class="sidenote">Singular points of solutions.</span> +value of the function; belonging, as we say, to a different +branch of the function; and there is even no reason for +assuming that the number of branches is finite, or that +different branches have the same singular points and +regions of existence. Moreover, and this is the most difficult consideration +of all, all these circumstances may be dependent upon the +values supposed given to the arbitrary constants of the integral; in +other words, the singular points may be either <i>fixed</i>, being determined +by the differential equations themselves, or they may be +<i>movable</i> with the variation of the arbitrary constants of integration. +Such difficulties arise even in establishing the reversion of an elliptic +integral, in solving the equation</p> + +<p class="center">(dx/ds)² = (x − a<span class="su">1</span>)(x − a<span class="su">2</span>)(x − a<span class="su">3</span>)(x − a<span class="su">4</span>);</p> + +<p class="noind">about an ordinary value the right side is developable; if we put +x − a<span class="su">1</span> = t<span class="su">1</span>², the right side becomes developable about t<span class="su">1</span> = 0; if we +put x = 1/t, the right side of the changed equation is developable +about t = 0; it is quite easy to show that the integral reducing to a +definite value x<span class="su">0</span> for a value s<span class="su">0</span> is obtainable by a series in integral +powers; this, however, must be supplemented by showing that for +no value of s does the value of x become entirely undetermined.</p> + +<p>These remarks will show the place of the theory now to be +sketched of a particular class of ordinary linear homogeneous +<span class="sidenote">Linear differential equations with rational coefficients.</span> +differential equations whose importance arises from +the completeness and generality with which they can +be discussed. We have seen that if in the equations</p> + +<p class="center">dy/dx = y<span class="su">1</span>, dy<span class="su">1</span>/dx = y<span class="su">2</span>, ..., dy<span class="su">n−2</span>/dx = y<span class="su">n−1</span>,<br /> +dy<span class="su">n−1</span>/dx = a<span class="su">n</span>y + a<span class="su">n−1</span>y<span class="su">1</span> + ... + a<span class="su">1</span>y<span class="su">n−1</span>,</p> + +<p class="noind">where a<span class="su">1</span>, a<span class="su">2</span>, ..., a<span class="su">n</span> are now to be taken to be rational +functions of x, the value x = xº be one for which no one of +these rational functions is infinite, and yº, yº<span class="su">1</span>, ..., yº<span class="su">n−1</span> be quite +arbitrary finite values, then the equations are satisfied by</p> + +<p class="center">y = yºu + yº<span class="su">1</span>u<span class="su">1</span> + ... + yº<span class="su">n−1</span>u<span class="su">n−1</span>,</p> + +<p class="noind">where u, u<span class="su">1</span>, ..., u<span class="su">n−1</span> are functions of x, independent of yº, ... +yº<span class="su">n−1</span>, developable about x = xº; this value of y is such that for +x = xº the functions y, y<span class="su">1</span> ... y<span class="su">n−1</span> reduce respectively to yº, yº<span class="su">1</span>, +... yº<span class="su">n−1</span>; it can be proved that the region of existence of these +series extends within a circle centre xº and radius equal to the +distance from xº of the nearest point at which one of a<span class="su">1</span>, ... a<span class="su">n</span> +becomes infinite. Now consider a region enclosing xº and only one +of the places, say Σ, at which one of a<span class="su">1</span>, ... a<span class="su">n</span> becomes infinite. +When x is made to describe a closed curve in this region, including +this point Σ in its interior, it may well happen that the continuations +of the functions u, u<span class="su">1</span>, ..., u<span class="su">n−1</span> give, when we have returned to +the point x, values v, v<span class="su">1</span>, ..., v<span class="su">n−1</span>, so that the integral under consideration +becomes changed to yº + yº<span class="su">1</span>v<span class="su">1</span> + ... + yº<span class="su">n−1</span>v<span class="su">n−1</span>. At +xº let this branch and the corresponding values of y<span class="su">1</span>, ... y<span class="su">n−1</span> be +ηº, ηº<span class="su">1</span>, ... ηº<span class="su">n−1</span>; then, as there is only one series satisfying the +equation and reducing to (ηº, ηº<span class="su">1</span>, ... ηº<span class="su">n−1</span>) for x = xº and the +coefficients in the differential equation are single-valued functions, +we must have ηºu + ηº<span class="su">1</span>u<span class="su">1</span> + ... + ηº<span class="su">n−1</span>u<span class="su">n−1</span> = yºv + yº<span class="su">1</span>v<span class="su">1</span> + ... ++ yº<span class="su">n−1</span>v<span class="su">n−1</span>; as this holds for arbitrary values of yº ... yº<span class="su">n−1</span>, upon +which u, ... u<span class="su">n−1</span> and v, ... v<span class="su">n−1</span> do not depend, it follows that +each of v, ... v<span class="su">n−1</span> is a linear function of u, ... u<span class="su">n−1</span> with constant +coefficients, say v<span class="su">i</span> = A<span class="su">i1</span>u + ... + A<span class="su">in</span>u<span class="su">n−1</span>. Then</p> + +<p class="center">yºv + ... + yº<span class="su">n−1</span>v<span class="su">n−1</span> = (Σ<span class="su">i</span> A<span class="su">i1</span> yº<span class="su">i</span>)u + ... + (Σ<span class="su">i</span> A<span class="su">in</span> yº<span class="su">i</span>) u<span class="su">n−1</span>;</p> + +<p class="noind">this is equal to μ(yºu + ... + yº<span class="su">n−1</span>u<span class="su">n−1</span>) if Σ<span class="su">i</span> A<span class="su">ir</span> yº<span class="su">i</span> = μyº<span class="su">r−1</span>; +eliminating yº ... yº<span class="su">n−1</span> from these linear equations, we have a +determinantal equation of order n for μ; let μ<span class="su">1</span> be one of its roots; +determining the ratios of yº, y<span class="su">1</span>º, ... yº<span class="su">n−1</span> to satisfy the linear +equations, we have thus proved that there exists an integral, +H, of the equation, which when continued round the point Σ and +back to the starting-point, becomes changed to H<span class="su">1</span> = μ<span class="su">1</span>H. Let now +ξ be the value of x at Σ and r<span class="su">1</span> one of the values of (½πi) log μ<span class="su">1</span>; +consider the function (x − ξ)<span class="sp">−r1</span>H; when x makes a circuit round x = ξ, +this becomes changed to</p> + +<p class="center">exp (-2πir<span class="su">1</span>) (x − ξ)<span class="sp">−r1</span> μH,</p> + +<p class="noind">that is, is unchanged; thus we may put H = (x − ξ)<span class="sp">r1</span>φ<span class="su">1</span>, φ<span class="su">1</span> being a +function single-valued for paths in the region considered described +about Σ, and therefore, by Laurent’s Theorem (see <span class="sc"><a href="#artlinks">Function</a></span>), +capable of expression in the annular region about this point by a +series of positive and negative integral powers of x − ξ, which in +general may contain an infinite number of negative powers; there is, +however, no reason to suppose r<span class="su">1</span> to be an integer, or even real. +Thus, if all the roots of the determinantal equation in μ are different, +we obtain n integrals of the forms (x − ξ)<span class="sp">r1</span>φ<span class="su">1</span>, ..., (x − ξ)<span class="sp">rn</span>φ<span class="su">n</span>. +In general we obtain as many integrals of this form as there are +really different roots; and the problem arises to discover, in case a +root be k times repeated, k − 1 equations of as simple a form as +possible to replace the k − 1 equations of the form yº + ... + +yº<span class="su">n−1</span>v<span class="su">n−1</span> = μ(yº + ... + yº<span class="su">n−1</span>u<span class="su">n−1</span>) which would have existed had +the roots been different. The most natural method of obtaining +a suggestion lies probably in remarking that if r<span class="su">2</span> = r<span class="su">1</span> + h, there is an +integral [(x − ξ)<span class="sp">r1 + h</span>φ<span class="su">2</span> − (x − ξ)<span class="sp">r1</span>φ<span class="su">1</span>] / h, where the coefficients in φ<span class="su">2</span> are +<span class="pagenum"><a name="page236" id="page236"></a>236</span> +the same functions of r<span class="su">1</span> + h as are the coefficients in φ<span class="su">1</span> of r<span class="su">1</span>; when +h vanishes, this integral takes the form</p> + +<p class="center">(x − ξ)<span class="sp">r1</span> [dφ<span class="su">1</span>/dr<span class="su">1</span> + φ<span class="su">1</span> log (x − ξ)],</p> + +<p class="noind">or say</p> + +<p class="center">(x − ξ)<span class="sp">r1</span> [φ<span class="su">1</span> + ψ<span class="su">1</span> log (x − ξ)];</p> + +<p class="noind">denoting this by 2πiμ<span class="su">1</span>K, and (x − ξ)<span class="sp">r1</span> φ<span class="su">1</span> by H, a circuit of the point +ξ changes K into</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">K′ =</td> <td>1</td> +<td rowspan="2"><span class="f150">[</span>e<span class="sp">2πir1</span> (x − ξ)<span class="sp">r1</span> ψ<span class="su">1</span> + e<span class="sp">2πir j</span> (x − ξ)<span class="sp">r1</span> φ<span class="su">1</span> (2πi + log(x − ξ) )<span class="f150">]</span> = μ<span class="su">1</span>K + H.</td></tr> +<tr><td class="denom">2πiμ<span class="su">1</span></td></tr></table> + +<p class="noind">A similar artifice suggests itself when three of the roots of the determinantal +equation are the same, and so on. We are thus led to the +result, which is justified by an examination of the algebraic conditions, +that whatever may be the circumstances as to the roots of the +determinantal equation, n integrals exist, breaking up into +batches, the values of the constituents H<span class="su">1</span>, H<span class="su">2</span>, ... of a batch after +circuit about x = ξ being H<span class="su">1</span>′ = μ<span class="su">1</span>H<span class="su">1</span>, H<span class="su">2</span>′ = μ<span class="su">1</span>H<span class="su">2</span> + H<span class="su">1</span>, H<span class="su">3</span>′ = μ<span class="su">1</span>H<span class="su">3</span> + H<span class="su">2</span>, +and so on. And this is found to lead to the forms (x − ξ)<span class="sp">r1</span>φ<span class="su">1</span>, +(x − ξ)<span class="sp">r1</span> [ψ<span class="su">1</span> + φ<span class="su">1</span> log (x − ξ)], +(x − ξ)<span class="sp">r1</span> [χ<span class="su">1</span> + χ<span class="su">2</span> log (x − ξ) + φ<span class="su">1</span>(log(x − ξ) )²], +and so on. Here each of φ<span class="su">1</span>, ψ<span class="su">1</span>, χ<span class="su">1</span>, χ<span class="su">2</span>, ... is a series of positive +and negative integral powers of x − ξ in which the number of negative +powers may be infinite.</p> + +<p>It appears natural enough now to inquire whether, under proper +conditions for the forms of the rational functions a<span class="su">1</span>, ... a<span class="su">n</span>, it may +be possible to ensure that in each of the series φ<span class="su">1</span>, ψ<span class="su">1</span>, [chi]<span class="su">1</span>, ... +the number of negative powers shall be finite. Herein +<span class="sidenote">Regular equations.</span> +lies, in fact, the limitation which experience has shown +to be justified by the completeness of the results obtained. Assuming +n integrals in which in each of φ<span class="su">1</span>, ψ<span class="su">1</span>, χ<span class="su">1</span> ... the number of +negative powers is finite, there is a definite homogeneous linear +differential equation having these integrals; this is found by +forming it to have the form</p> + +<p class="center">y′ <span class="sp">n</span> = (x − ξ)<span class="sp">−1</span> b<span class="su">1</span>y′ <span class="sp">(n−1)</span> + (x − ξ)<span class="sp">−2</span> b<span class="su">2</span>y′ <span class="sp">(n−2)</span> + ... + (x − ξ)<span class="sp">−n</span> b<span class="su">n</span>y,</p> + +<p class="noind">where b<span class="su">1</span>, ... b<span class="su">n</span> are finite for x = ξ. Conversely, assume the +equation to have this form. Then on substituting a series of +the form (x − ξ)<span class="sp">r</span> [1 + A<span class="su">1</span>(x − ξ) + A<span class="su">2</span>(x − ξ)² + ... ] and equating the +coefficients of like powers of x − ξ, 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 − ξ)<span class="sp">r</span> and replacing each of b<span class="su">1</span>, ... b<span class="su">n</span> by their values at +x = ξ; arrange the roots r<span class="su">1</span>, r<span class="su">2</span>, ... of this equation so that the +real part of r<span class="su">i</span> is equal to, or greater than, the real part of r<span class="su">i+1</span>, +and take r equal to r<span class="su">1</span>; it is found that the coefficients A<span class="su">1</span>, A<span class="su">2</span> ... +are uniquely determinate, and that the series converges within a +circle about x = ξ which includes no other of the points at which +the rational functions a<span class="su">1</span> ... a<span class="su">n</span> become infinite. We have thus a +solution H<span class="su">1</span> = (x − ξ)<span class="sp">r1</span>φ<span class="su">1</span> of the differential equation. If we now +substitute in the equation y = H<span class="su">1</span>∫ηdx, it is found to reduce to an +equation of order n − 1 for η of the form</p> + +<p class="center">η′ <span class="sp">(n−1)</span> = (x − ξ)<span class="sp">−1</span> c<span class="su">1</span>η′ <span class="sp">(n−2)</span> + ... + (x − ξ)<span class="sp">(n−1)</span> c<span class="su">n−1</span>η,</p> + +<p class="noind">where c<span class="su">1</span>, ... c<span class="su">n−1</span> are not infinite at x = ξ. To this equation +precisely similar reasoning can then be applied; its index equation +has in fact the roots r<span class="su">2</span> − r<span class="su">1</span> − 1, ..., r<span class="su">n</span> − r<span class="su">1</span> − 1; if r<span class="su">2</span> − r<span class="su">1</span> be zero, +the integral (x − ξ)<span class="sp">−1</span>ψ<span class="su">1</span> of the η equation will give an integral of the +original equation containing log (x − ξ); if r<span class="su">2</span> − r<span class="su">1</span> be an integer, and +therefore a negative integer, the same will be true, unless in ψ<span class="su">1</span> the +term in (x − ξ)<span class="sp">r1 − r2</span> be absent; if neither of these arise, the original +equation will have an integral (x − ξ)<span class="sp">r2</span>φ<span class="su">2</span>. The η equation can now, +by means of the one integral of it belonging to the index r<span class="su">2</span> − r<span class="su">1</span> − 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 <i>regular</i> about x = ξ.</p> + +<p>We may examine in this way the behaviour of the integrals at +all the points at which any one of the rational functions a<span class="su">1</span> ... a<span class="su">n</span> +becomes infinite; in general we must expect that beside +these the value x = ∞ will be a singular point for the +<span class="sidenote">Fuchsian equations.</span> +solutions of the differential equation. To test this we +put x = 1/t throughout, and examine as before at t = 0. For instance, +the ordinary linear equation with constant coefficients has no singular +point for finite values of x; at x = ∞ it has a singular point and is not +regular; or again, Bessel’s equation x²y″ + xy′ + (x² − n²)y = 0 is +regular about x = 0, but not about x = ∞. An equation regular at all +the finite singularities and also at x = ∞ is called a Fuchsian equation. +We proceed to examine particularly the case of an equation of the +second order</p> + +<p class="center">y″ + ay′ + by = 0.</p> + +<p class="noind">Putting x = 1/t, it becomes</p> + +<p class="center">d²y/dt² + (2t<span class="sp">−1</span> − at<span class="sp">−2</span>) dy/dt + bt<span class="sp">−4</span> y = 0,</p> + +<p class="noind">which is not regular about t = 0 unless 2 − at<span class="sp">−1</span> and bt<span class="sp">−2</span>, that is, +unless ax and bx² are finite at x = ∞; which we thus assume; putting +y = t<span class="sp">r</span>(1 + A<span class="su">1</span>t + ... ), we find for the index equation at x = ∞ +the equation r(r − 1) + r(2 − ax)<span class="su">0</span> + (bx²)<span class="su">0</span> = 0. If there be +<span class="sidenote">Equation of the second order.</span> +finite singular points at ξ<span class="su">1</span>, ... ξ<span class="su">m</span>, where we assume +m > 1, the cases m = 0, m = 1 being easily dealt with, and +if φ(x) = (x − ξ<span class="su">1</span>) ... (x − ξ<span class="su">m</span>), we must have a·φ(x) +and b·[φ(x)]² finite for all finite values of x, equal say to the respective +polynomials ψ(x) and θ(x), of which by the conditions at +x = ∞ the highest respective orders possible are m − 1 and 2(m − 1). +The index equation at x = ξ<span class="su">1</span> is r(r − 1) + rψ(ξ<span class="su">1</span>) / φ′ (ξ<span class="su">1</span>) + θ(ξ)<span class="su">1</span> / [φ′(ξ<span class="su">1</span>)]² = 0, +and if α<span class="su">1</span>, β<span class="su">1</span> be its roots, we have α<span class="su">1</span> + β<span class="su">1</span> = 1 − ψ(ξ<span class="su">1</span>) / φ′ (ξ<span class="su">1</span>) and +α<span class="su">1</span>β<span class="su">1</span> = θ(ξ)<span class="su">1</span> / [φ′(ξ<span class="su">1</span>)]². Thus by an elementary theorem of algebra, +the sum Σ(1 − α<span class="su">i</span> − β<span class="su">i</span>) / (x − ξ<span class="su">i</span>), extended to the m finite singular +points, is equal to ψ(x) / φ(x), and the sum Σ(1 − α<span class="su">i</span> − β<span class="su">i</span>) is equal to +the ratio of the coefficients of the highest powers of x in ψ(x) and +φ(x), and therefore equal to 1 + α + β, where α, β are the indices at +x = ∞. Further, if (x, 1)<span class="su">m−2</span> denote the integral part of the quotient +θ(x) / φ(x), we have Σ α<span class="su">i</span>β<span class="su">i</span>φ′ (ξ<span class="su">i</span>) / (x = ξ<span class="su">i</span>) equal to −(x, 1)<span class="su">m−2</span> + θ(x)/φ(x), +and the coefficient of x<span class="sp">m−2</span> in (x, 1)<span class="su">m−2</span> is αβ. Thus the differential +equation has the form</p> + +<p class="center">y″ + y′Σ (1 − α<span class="su">i</span> − β<span class="su">i</span>) / (x − ξ<span class="su">i</span>) + y[(x, 1)<span class="su">m-2</span> + Σ α<span class="su">i</span>β<span class="su">i</span>φ′(ξ<span class="su">i</span>) / (x − ξ<span class="su">i</span>)]/φ(x) = 0.</p> + +<p class="noind">If, however, we make a change in the dependent variable, putting +y = (x − ξ<span class="su">1</span>)<span class="sp">α1</span> ... (x − ξ<span class="su">m</span>)<span class="sp">α mη</span>, it is easy to see that the equation +changes into one having the same singular points about each of +which it is regular, and that the indices at x = ξ<span class="su">i</span> become 0 and β<span class="su">i</span> − α<span class="su">i</span>, +which we shall denote by λ<span class="su">i</span>, for (x − ξ<span class="su">i</span>)<span class="sp">αj</span> can be developed in positive +integral powers of x − ξ<span class="su">i</span> about x = ξ<span class="su">i</span>; by this transformation the +indices at x = ∞ are changed to</p> + +<p class="center">α + α<span class="su">1</span> + ... + α<span class="su">m</span>, β + β<span class="su">1</span> + ... + β<span class="su">m</span></p> + +<p class="noind">which we shall denote by λ, μ. If we suppose this change to have +been introduced, and still denote the independent variable by y, +the equation has the form</p> + +<p class="center">y″ + y′Σ (1 − λ<span class="su">i</span>) / (x − ξ<span class="su">i</span>) + y(x, 1)<span class="su">m−2</span> / φ(x) = 0,</p> + +<p class="noind">while λ + μ + λ<span class="su">1</span> + ... + λ<span class="su">m</span> = m − 1. Conversely, it is easy to verify +that if λμ be the coefficient of x<span class="sp">m−2</span> in (x, 1)<span class="su">m−2</span>, this equation has +the specified singular points and indices whatever be the other +coefficients in (x, 1)<span class="su">m−2</span>.</p> + +<p>Thus we see that (beside the cases m = 0, m = 1) the “Fuchsian +equation” of the second order with <i>two</i> finite singular points is +distinguished by the fact that it has a definite form +when the singular points and the indices are assigned. +<span class="sidenote">Hypergeometric equation.</span> +In that case, putting (x − ξ<span class="su">1</span>) / (x − ξ<span class="su">2</span>) = t / (t − 1), the singular +points are transformed to 0, 1, ∞, and, as is clear, without +change of indices. Still denoting the independent variable by x, +the equation then has the form</p> + +<p class="center">x(1 − x)y″ + y′[1 − λ<span class="su">1</span> − x(1 + λ + μ)] − λμy = 0,</p> + +<p class="noind">which is the ordinary hypergeometric equation. Provided none +of λ<span class="su">1</span>, λ<span class="su">2</span>, λ − μ be zero or integral about x = 0, it has the solutions</p> + +<p class="center">F(λ, μ, 1 − λ<span class="su">1</span>, x), x<span class="sp">λ1</span> F(λ + λ<span class="su">1</span>, μ + λ<span class="su">1</span>, 1 + λ<span class="su">1</span>, x);</p> + +<p class="noind">about x = 1 it has the solutions</p> + +<p class="center">F(λ, μ, 1 − λ<span class="su">2</span>, 1 − x), (1 − x)<span class="sp">λ2</span> F(λ + λ<span class="su">2</span>, μ + λ<span class="su">2</span>, 1 + λ<span class="su">2</span>, 1 − x),</p> + +<p class="noind">where λ + μ + λ<span class="su">1</span> + λ<span class="su">2</span> = 1; about x = ∞ it has the solutions</p> + +<p class="center">x<span class="sp">−λ</span> F(λ, λ + λ<span class="su">1</span>, λ − μ + 1, x<span class="sp">−1</span>), x<span class="sp">−μ</span> F(μ, μ + λ<span class="su">1</span>, μ − λ + 1, x<span class="sp">−1</span>),</p> + +<p class="noind">where F(α, β, γ, x) is the series</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">1 +</td> <td>αβx</td> +<td rowspan="2">+</td> <td>α(α + 1)β(β + 1)x²</td> <td rowspan="2">...,</td></tr> +<tr><td class="denom">γ</td> <td class="denom">1·2·γ(γ + 1)</td></tr></table> + +<p class="noind">which converges when |x| < 1, whatever α, β, γ may be, converges +for all values of x for which |x| = 1 provided the real part of γ − α − β < 0 +algebraically, and converges for all these values except x = 1 provided +the real part of γ − α − β > −1 algebraically.</p> + +<p>In accordance with our general theory, logarithms are to be expected +in the solution when one of λ<span class="su">1</span>, λ<span class="su">2</span>, λ − μ is zero or integral. +Indeed when λ<span class="su">1</span> is a negative integer, not zero, the second solution +about x = 0 would contain vanishing factors in the denominators +of its coefficients; in case λ or μ be one of the positive integers +1, 2, ... (−λ<span class="su">1</span>), vanishing factors occur also in the numerators; +and then, in fact, the second solution about x = 0 becomes x<span class="sp">λ1</span> times +an integral polynomial of degree (−λ<span class="su">1</span>) − λ or of degree (−λ<span class="su">1</span>) − μ. +But when λ<span class="su">1</span> is a negative integer including zero, and neither λ nor μ +is one of the positive integers 1, 2 ... (−λ<span class="su">1</span>), the second solution +about x = 0 involves a term having the factor log x. When λ<span class="su">1</span> is a +positive integer, not zero, the second solution about x = 0 persists as +a solution, in accordance with the order of arrangement of the roots +of the index equation in our theory; the first solution is then +replaced by an integral polynomial of degree -λ or −μ<span class="su">1</span>, when λ or μ +is one of the negative integers 0, −1, −2, ..., 1 − λ<span class="su">1</span>, but otherwise +contains a logarithm. Similarly for the solutions about x = 1 or +x = ∞; it will be seen below how the results are deducible from +those for x = 0.</p> + +<p>Denote now the solutions about x = 0 by u<span class="su">1</span>, u<span class="su">2</span>; those about x = 1 +by v<span class="su">1</span>, v<span class="su">2</span>; and those about x = ∞ by w<span class="su">1</span>, w<span class="su">2</span>; in the region (S<span class="su">0</span>S<span class="su">1</span>) +common to the circles S<span class="su">0</span>, S<span class="su">1</span> of radius 1 whose centres +are the points x = 0, x = 1, all the first four are valid, +<span class="sidenote">March of the Integral.</span> +and there exist equations u<span class="su">1</span> =Av<span class="su">1</span> + Bv<span class="su">2</span>, u<span class="su">2</span> = Cv<span class="su">1</span> + Dv<span class="su">2</span> +where A, B, C, D are constants; in the region (S<span class="su">1</span>S) +lying inside the circle S<span class="su">1</span> and outside the circle S<span class="su">0</span>, those that are +valid are v<span class="su">1</span>, v<span class="su">2</span>, w<span class="su">1</span>, w<span class="su">2</span>, and there exist equations v<span class="su">1</span> = Pw<span class="su">1</span> + Qw<span class="su">2</span>, +v<span class="su">2</span> = Rw<span class="su">1</span> + Tw<span class="su">2</span>, where P, Q, R, T are constants; thus considering +any integral whose expression within the circle S<span class="su">0</span> is au<span class="su">1</span> + bu<span class="su">2</span>, where +a, b are constants, the same integral will be represented within the +circle S<span class="su">1</span> by (aA + bC)v<span class="su">1</span> + (aB + bD)v<span class="su">2</span>, and outside these circles will be +represented by</p> + +<p class="center">[aA + bC)P + (aB + bD)R]w<span class="su">1</span> + [(aA + bC)Q + (aB + bD)T]w<span class="su">2</span>.</p> + +<p class="noind">A single-valued branch of such integral can be obtained by making +a barrier in the plane joining ∞ to 0 and 1 to ∞; for instance, by +excluding the consideration of real negative values of x and of real +<span class="pagenum"><a name="page237" id="page237"></a>237</span> +positive values greater than 1, and defining the phase of x and x − 1 +for real values between 0 and 1 as respectively 0 and π.</p> + +<p>We can form the Fuchsian equation of the second order with +three arbitrary singular points ξ<span class="su">1</span>, ξ<span class="su">2</span>, ξ<span class="su">3</span>, and no singular point +at x = ∞, and with respective indices α<span class="su">1</span>, β<span class="su">1</span>, α<span class="su">2</span>, β<span class="su">2</span>, α<span class="su">3</span>, β<span class="su">3</span> such +that α<span class="su">1</span> + β<span class="su">1</span> + α<span class="su">2</span> + β<span class="su">2</span> + α<span class="su">3</span> + β<span class="su">3</span> = 1. This equation can then be +<span class="sidenote">Transformation of the equation into itself.</span> +transformed into the hypergeometric equation in 24 ways; +for out of ξ<span class="su">1</span>, ξ<span class="su">2</span>, ξ<span class="su">3</span> we can in six ways choose two, say +ξ<span class="su">1</span>, ξ<span class="su">2</span>, which are to be transformed respectively into +0 and 1, by (x − ξ<span class="su">1</span>)/(x − ξ<span class="su">2</span>) = 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 α<span class="su">1</span> or β<span class="su">1</span> at t = 0, +and simultaneously either α<span class="su">2</span> or β<span class="su">2</span> at t = 1. Thus the hypergeometric +equation itself can be transformed into itself in 24 ways, +and from the expression F(λ, μ, 1 − λ<span class="su">1</span>, x) which satisfies it follow 23 +other forms of solution; they involve four series in each of the arguments, +x, x − 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 = ∞; and from the principles by which these +were obtained it is immediately clear that the 24 forms are, in value, +equal in fours.</p> + +<p>The quarter periods K, K′ of Jacobi’s theory of elliptic functions, +of which K = ∫<span class="sp1">π/2</span><span class="su1">0</span> (1 − h sin ²θ)<span class="sp">−½</span>dθ, and K′ is the same function of +1-h, can easily be proved to be the solutions of a hypergeometric +<span class="sidenote">Inversion. Modular functions.</span> +equation of which h is the independent variable. When K, K′ 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 <span class="sc"><a href="#artlinks">Function</a></span>) in terms of single-valued functions. +We may then attempt to investigate, in general, in what cases the +independent variable x of a hypergeometric equation is a single-valued +function of the ratio s of two independent integrals of the equation. +The same inquiry is suggested by the problem of ascertaining in what +cases the hypergeometric series F(α, β, γ, 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 -∞ to 0 and from 1 to +∞, and, supposing logarithms +not to enter about x = 0, choose two quite definite integrals y<span class="su">1</span>, y<span class="su">2</span> of +the equation, say</p> + +<p class="center">y<span class="su">1</span> = F(λ, μ, 1 − λ<span class="su">1</span>, x), +y<span class="su">2</span> = x<span class="sp">λ1</span> F(λ + λ<span class="su">1</span>, μ + λ<span class="su">1</span>, 1 + λ<span class="su">1</span>, x),</p> + +<p class="noind">with the condition that the phase of x is zero when x is real +and between 0 and 1. Then the value of ς = y<span class="su">2</span>/y<span class="su">1</span> is definite for all +values of x in the divided plane, ς being a single-valued monogenic +branch of an analytical function existing and without singularities +all over this region. If, now, the values of ς that so arise be plotted +on to another plane, a value p + iq of σ being represented by a point +(p, q) of this ς-plane, and the value of x from which it arose being +mentally associated with this point of the σ-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 ς-plane can arise twice, or in other words, whether +the region of the ς-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 -∞ 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 σ, 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 σ be similarly plotted on the ς-plane, we shall +again obtain a region which, while not overlapping itself, may quite +possibly overlap the former region. In that case two values of x +would arise for the same value or values of the quotient y<span class="su">2</span>/y<span class="su">1</span>, arising +from two different branches of this quotient. We shall understand +then, by the condition that x is to be a single-valued function of x, +that the region in the ς-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,</p> + +<p class="center">ς = x<span class="sp">λ1</span> F(λ + λ<span class="su">1</span>, μ + λ<span class="su">1</span>, 1 + λ<span class="su">1</span>, x) / F(λ, μ, 1 − λ<span class="su">1</span>, x)</p> + +<p class="noind">is changed to σ = ςe<span class="sp">2πiλ1</span>. Thus the two portions of boundary of the +s-region corresponding to the two sides of the barrier (−∞, 0) meet +(at ς = 0 if the real part of λ<span class="su">1</span> be positive) at an angle 2πL<span class="su">1</span>, where L<span class="su">1</span> +is the absolute value of the real part of λ<span class="su">1</span>; the same is true for the +σ-region representing the branch σ. The condition that the s-region +shall not overlap itself requires, then, L<span class="su">1</span> = 1. But, further, we may +form an infinite number of branches σ = ςe<span class="sp">2πiλ1</span>, σ<span class="su">1</span> = e<span class="sp">2πiλ1</span>, ... +in the same way, and the corresponding regions in the plane upon which +y<span class="su">2</span>/y<span class="su">1</span> is represented will have a common point and each have an +angle 2πL<span class="su">1</span>; if neither overlaps the preceding, it will happen, if L<span class="su">1</span> +is not zero, that at length one is reached overlapping the first, unless +for some positive integer α we have 2παL<span class="su">1</span> = 2π, in other words +L<span class="su">1</span> = 1/α. If this be so, the branch σ<span class="su">α−1</span> = ςe<span class="sp">2πiαλ1</span> will be represented +by a region having the angle at the common point common with the +region for the branch ς; but not altogether coinciding with this last +region unless λ<span class="su">1</span> be real, and therefore = ±1/α; then there is only +a finite number, α, of branches obtainable in this way by crossing +the barrier (−∞, 0). In precisely the same way, if we had begun +by taking the quotient</p> + +<p class="center">ς′ = (x − 1)<span class="sp">λ2</span> F(λ + λ<span class="su">2</span>, μ + λ<span class="su">2</span>, 1 + λ<span class="su">2</span>, 1 − x) / F(λ, μ, 1 − λ<span class="su">2</span>, 1 − x)</p> + +<p class="noind">of the two solutions about x = 1, we should have found that x is not +a single-valued function of ς′ unless λ<span class="su">2</span> is the inverse of an integer, or +is zero; as ς′ is of the form (A<span class="su">σ</span> + B)/(C<span class="su">ς</span> + D), A, B, C, D constants, +the same is true in our case; equally, by considering the integrals +about x = ∞ we find, as a third condition necessary in order that x +may be a single-valued function of ς, that λ − μ must be the inverse +of an integer or be zero. These three differences of the indices, +namely, λ<span class="su">1</span>, λ<span class="su">2</span>, λ − μ, are the quantities which enter in the differential +equation satisfied by x as a function of ς, which is easily found to be</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">−</td> <td>x<span class="su">111</span></td> +<td rowspan="2">+</td> <td>3x²<span class="su">11</span></td> +<td rowspan="2">½(h − h<span class="su">1</span> − h<span class="su">2</span>)x<span class="sp">−1</span>(x − 1)<span class="sp">−1</span> + ½h<span class="su">1</span>x<span class="sp">−2</span> + ½h<span class="su">2</span>(x − 1)<span class="sp">−2</span>,</td></tr> +<tr><td class="denom">x<span class="su">1</span>³</td> <td class="denom">2x<span class="su">1</span><span class="sp">4</span></td></tr></table> + +<p class="noind">where x<span class="su">1</span> = dx/dς, &c.; and h<span class="su">1</span> = 1 − y<span class="su">1</span>², h<span class="su">2</span> = 1 − λ<span class="su">2</span>², h<span class="su">3</span> = 1 − (λ − μ)². Into +the converse question whether the three conditions are sufficient +to ensure (1) that the σ region corresponding to any branch does +not overlap itself, (2) that no two such regions overlap, we have no +space to enter. The second question clearly requires the inquiry +whether the group (that is, the monodromy group) of the differential +equation is properly discontinuous. (See <span class="sc"><a href="#artlinks">Groups, Theory of</a></span>.)</p> + +<p>The foregoing account will give an idea of the nature of the +function theories of differential equations; it appears essential +not to exclude some explanation of a theory intimately related +both to such theories and to transformation theories, which is a +generalization of Galois’s theory of algebraic equations. We deal +only with the application to homogeneous linear differential +equations.</p> + +<p>In general a function of variables x<span class="su">1</span>, x<span class="su">2</span> ... is said to be rational +when it can be formed from them and the integers 1, 2, 3, ... by a +finite number of additions, subtractions, multiplications +and divisions. We generalize this definition. Assume that +<span class="sidenote">Rationality group of a linear equation.</span> +we have assigned a fundamental series of quantities and +functions of x, in which x itself is included, such that all +quantities formed by a finite number of additions, subtractions, +multiplications, divisions <i>and differentiations in regard to x</i>, +of the terms of this series, are themselves members of this series. +Then the quantities of this series, and only these, are called <i>rational</i>. +By a rational function of quantities p, q, r, ... is meant a function +formed from them and any of the fundamental rational quantities +by a finite number of the five fundamental operations. Thus it is a +function which would be called, simply, rational if the fundamental +series were widened by the addition to it of the quantities p, q, r, ... +and those derivable from them by the five fundamental operations. +A rational ordinary differential equation, with x as independent and +y as dependent variable, is then one which equates to zero a rational +function of y, the order k of the differential equation being that of the +highest differential coefficient y<span class="sp">(k)</span> which enters; only such equations +are here discussed. Such an equation P = 0 is called <i>irreducible</i> when, +firstly, being arranged as an integral polynomial in y<span class="sp">(k)</span>, this polynomial +<span class="sidenote">Irreducibility of a rational equation.</span> +is not the product of other polynomials in y<span class="sp">(k)</span> also +of rational form; and, secondly, the equation has no +solution satisfying also a rational equation of lower order. +From this it follows that if an irreducible equation P = 0 +have one solution satisfying another rational equation Q = 0 +of the same or higher order, then all the solutions of P = 0 also satisfy +Q = 0. For from the equation P = 0 we can by differentiation express +y<span class="sp">(k+1)</span>, y<span class="sp">(k+2)</span>, ... in terms of x, y, y<span class="sp">(1)</span>, ... , y<span class="sp">(k)</span>, and so put the +function Q rationally in terms of these quantities only. It is +sufficient, then, to prove the result when the equation Q = 0 is of the +same order as P = 0. Let both the equations be arranged as integral +polynomials in y<span class="sp">(k)</span>; their algebraic eliminant in regard to y<span class="sp">(k)</span> must +then vanish identically, for they are known to have one common +solution not satisfying an equation of lower order; thus the equation +P = 0 involves Q = 0 for all solutions of P = 0.</p> + +<p>Now let y<span class="sp">(n)</span> = a<span class="su">1</span>y<span class="sp">(n−1)</span> + ... + a<span class="su">n</span>y be a given rational homogeneous +linear differential equation; let y<span class="su">1</span>, ... y<span class="su">n</span> be n particular +functions of x, unconnected by any equation with constant coefficients +of the form c<span class="su">1</span>y<span class="su">1</span> + ... + c<span class="su">n</span>y<span class="su">n</span> = 0, all satisfying +<span class="sidenote">The variant function for a linear equation.</span> +the differential equation; let η<span class="su">1</span>, ... η<span class="su">n</span> be linear functions +of y<span class="su">1</span>, ... y<span class="su">n</span>, say η<span class="su">i</span> = A<span class="su">i1</span>y<span class="su">1</span> + ... + A<span class="su">in</span>y<span class="su">n</span>, where the +constant coefficients A<span class="su">ij</span> have a non-vanishing determinant; +write (η) = 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 φ(η), or say φ(A(y)), of η<span class="su">1</span>, ... η, in which the +η² constants A<span class="su">ij</span> shall all be essential, and not reduce effectively to a +fewer number, as they would, for instance, if the y<span class="su">1</span>, ... y<span class="su">n</span> were +connected by a linear equation with constant coefficients. Such a +function is in fact given, if the solutions y<span class="su">1</span>, ... y<span class="su">n</span> be developable +<span class="pagenum"><a name="page238" id="page238"></a>238</span> +in positive integral powers about x = a, by φ(η) = η<span class="su">1</span> + (x − a)<span class="sp">n</span> η<span class="su">2</span> + ... + +(x − a)<span class="sp">(n−1)n</span> η<span class="su">n</span>. Such a function, V, we call a <i>variant</i>.</p> + +<p>Then differentiating V in regard to x, and replacing η<span class="su">i</span><span class="sp">(n)</span> by its +value a<span class="su">1</span>η<span class="sp">(n−1)</span> + ... + a<span class="su">n</span>η, we can arrange dV/dx, and similarly each +of d²/dx² ... d<span class="sp">N</span>V/dx<span class="sp">N</span>, where N = n², as a linear function of +the N quantities η<span class="su">1</span>, ... η<span class="su">n</span>, ... η<span class="su">1</span><span class="sp">(n−1)</span>, ... η<span class="su">n</span><span class="sp">(n−1)</span>, and +<span class="sidenote">The resolvent eqution.</span> +thence by elimination obtain a linear differential equation +for V of order N with rational coefficients. This we +denote by F = 0. Further, each of η<span class="su">1</span> ... η<span class="su">n</span> is expressible +as a linear function of V, dV/dx, ... d<span class="sp">N−1</span>V / dx<span class="sp">N−1</span>, with rational coefficients +not involving any of the n² coefficients A<span class="su">ij</span>, since otherwise +V would satisfy a linear equation of order less than N, which is +impossible, as it involves (linearly) the n² arbitrary coefficients A<span class="su">ij</span>, +which would not enter into the coefficients of the supposed equation. +In particular, y<span class="su">1</span> ,.. y<span class="su">n</span> are expressible rationally as linear functions +of ω, dω/dx, ... d<span class="sp">N−1</span>ω / dx<span class="sp">N−1</span>, where ω is the particular function +φ(y). Any solution W of the equation F = 0 is derivable from +functions ζ<span class="su">1</span>, ... ζ<span class="su">n</span>, which are linear functions of y<span class="su">1</span>, ... y<span class="su">n</span>, just +as V was derived from η<span class="su">1</span>, ... η<span class="su">n</span>; but it does not follow that these +functions ζ<span class="su">i</span>, ... ζ<span class="su">n</span> are obtained from y<span class="su">1</span>, ... y<span class="su">n</span> by a transformation +of the linear group A, B, ... ; for it may happen that the +determinant d(ζ<span class="su">1</span>, ... ζ<span class="su">n</span>) / (dy<span class="su">1</span>, ... y<span class="su">n</span>) is zero. In that case +ζ<span class="su">1</span>, ... ζ<span class="su">n</span> may be called a singular set, and W a singular solution; it +satisfies an equation of lower than the N-th order. But every solution +V, W, ordinary or singular, of the equation F = 0, is expressible +rationally in terms of ω, dω / dx, ... d<span class="sp">N−1</span>ω / dx<span class="sp">N−1</span>; we shall write, +simply, V = r(ω). Consider now the rational irreducible equation +of lowest order, not necessarily a linear equation, which is satisfied +by ω; as y<span class="su">1</span>, ... y<span class="su">n</span> are particular functions, it may quite well +be of order less than N; we call it the <i>resolvent equation</i>, suppose it +of order p, and denote it by γ(v). Upon it the whole theory turns. +In the first place, as γ(v) = 0 is satisfied by the solution ω of F = 0, all +the solutions of γ(v) are solutions F = 0, and are therefore rationally +expressible by ω; any one may then be denoted by r(ω). If this +solution of F = 0 be not singular, it corresponds to a transformation +A of the linear group (A, B, ...), effected upon y<span class="su">1</span>, ... y<span class="su">n</span>. The +coefficients A<span class="su">ij</span> of this transformation follow from the expressions +before mentioned for η<span class="su">1</span> ... η<span class="su">n</span> in terms of V, dV/dx, d²V/dx², ... by +substituting V = r(ω); thus they depend on the p arbitrary parameters +which enter into the general expression for the integral of +the equation γ(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 ω, enables us to define a linear homogeneous +group of transformations of y<span class="su">1</span> ... y<span class="su">n</span> depending on p parameters; +and every operation of this (continuous) group corresponds to a +rational transformation of the solution of the resolvent equation. +This is the group called the <i>rationality group</i>, or the <i>group of transformations</i> +of the original homogeneous linear differential equation.</p> + +<p>The group must not be confounded with a subgroup of itself, +the <i>monodromy group</i> of the equation, often called simply the group +of the equation, which is a set of transformations, not depending +on arbitrary variable parameters, arising for one particular +fundamental set of solutions of the linear equation (see <span class="sc"><a href="#artlinks">Groups, +Theory of</a></span>).</p> + +<p>The importance of the rationality group consists in three propositions. +(1) Any rational function of y<span class="su">1</span>, ... y<span class="su">n</span> which is unaltered in +value by the transformations of the group can be written +in rational form. (2) If any rational function be changed +<span class="sidenote">The fundamental theorem in regard to the rationality group.</span> +in form, becoming a rational function of y<span class="su">1</span>, ... y<span class="su">n</span>, a +transformation of the group applied to its new form will +leave its value unaltered. (3) Any homogeneous linear +transformation leaving unaltered the value of every +rational function of y<span class="su">1</span>, ... y<span class="su">n</span> which has a rational value, +belongs to the group. It follows from these that any +group of linear homogeneous transformations having the +properties (1) (2) is identical with the group in question. It is clear +that with these properties the group must be of the greatest importance +in attempting to discover what functions of x must be regarded as +rational in order that the values of y<span class="su">1</span> ... y<span class="su">n</span> may be expressed. +And this is the problem of solving the equation from another point +of view.</p> + +<div class="condensed"> +<p><span class="sc">Literature.</span>—(α) <i>Formal or Transformation Theories for Equations +of the First Order</i>:—E. Goursat, <i>Leçons sur l’intégration des équations +aux dérivées partielles du premier ordre</i> (Paris, 1891); E. v. +Weber, <i>Vorlesungen über das Pfaff’sche Problem und die Theorie der +partiellen Differentialgleichungen erster Ordnung</i> (Leipzig, 1900); +S. Lie und G. Scheffers, <i>Geometrie der Berührungstransformationen</i>, +Bd. i. (Leipzig, 1896); Forsyth, <i>Theory of Differential Equations, +Part i., Exact Equations and Pfaff’s Problem</i> (Cambridge, 1890); +S. Lie, “Allgemeine Untersuchungen über Differentialgleichungen, die +eine continuirliche endliche Gruppe gestatten” (Memoir), <i>Mathem. +Annal.</i>xxv. (1885), pp. 71-151; S. Lie und G. Scheffers, <i>Vorlesungen +über Differentialgleichungen mit bekannten infinitesimalen Transformationen</i> +(Leipzig, 1891). A very full bibliography is given in the book +of E. v. Weber referred to; those here named are perhaps sufficiently +representative of modern works. Of classical works may be named: +Jacobi, <i>Vorlesungen über Dynamik</i> (von A. Clebsch, Berlin, 1866); +<i>Werke, Supplementband</i>; G Monge, <i>Application de l’analyse à la +géométrie</i> (par M. Liouville, Paris, 1850); J. L. Lagrange, <i>Leçons +sur le calcul des fonctions</i> (Paris, 1806), and <i>Théorie des fonctions +analytiques</i> (Paris, Prairial, an V); G. Boole, <i>A Treatise on Differential +Equations</i> (London, 1859); and <i>Supplementary Volume</i> +(London, 1865); Darboux, <i>Leçons sur la théorie générale des +surfaces</i>, tt. i.-iv. (Paris, 1887-1896); S. Lie, <i>Théorie der transformationsgruppen</i> +ii. (on Contact Transformations) (Leipzig, 1890).</p> + +<p>(β) <i>Quantitative or Function Theories for Linear Equations</i>:—C. +Jordan, <i>Cours d’analyse</i>, t. iii. (Paris, 1896); E. Picard, <i>Traité +d’analyse</i>, tt. ii. and iii. (Paris, 1893, 1896); Fuchs, <i>Various +Memoirs, beginning with that in Crelle’s Journal</i>, Bd. lxvi. p. 121; +Riemann, <i>Werke</i>, 2<span class="sp">r</span> Aufl. (1892); Schlesinger, <i>Handbuch der +Theorie der linearen Differentialgleichungen</i>, Bde. i.-ii. (Leipzig, +1895-1898); Heffter, <i>Einleitung in die Theorie der linearen Differentialgleichungen +mit einer unabhängigen Variablen</i> (Leipzig, 1894); +Klein, <i>Vorlesungen über lineare Differentialgleichungen der zweiten +Ordnung</i> (Autographed, Göttingen, 1894); and <i>Vorlesungen über +die hypergeometrische Function</i> (Autographed, Göttingen, 1894); +Forsyth, <i>Theory of Differential Equations, Linear Equations</i>.</p> + +<p>(γ) <i>Rationality Group (of Linear Differential Equations)</i>:—Picard, +<i>Traité d’Analyse</i>, as above, t. iii.; Vessiot, <i>Annales de +l’École Normale</i>, série III. t. ix. p. 199 (Memoir); S. Lie, +<i>Transformationsgruppen</i>, as above, iii. A connected account is +given in Schlesinger, as above, Bd. ii., erstes Theil.</p> + +<p>(δ) <i>Function Theories of Non-Linear Ordinary Equations</i>:—Painlevé, +<i>Leçons sur la théorie analytique des équations différentielles</i> +(Paris, 1897, Autographed); Forsyth, <i>Theory of Differential Equations, +Part ii., Ordinary Equations not Linear</i> (two volumes, ii. and iii.) +(Cambridge, 1900); Königsberger, <i>Lehrbuch der Theorie der Differentialgleichungen</i> +(Leipzig, 1889); Painlevé, <i>Leçons sur l’intégration +des équations differentielles de la mécanique et applications</i> (Paris, +1895).</p> + +<p>(ε) <i>Formal Theories of Partial Equations of the Second and Higher +Orders</i>:—E. Goursat, <i>Leçons sur l’intégration des équations aux +dérivées partielles du second ordre</i>, tt. i. and ii. (Paris, 1896, 1898); +Forsyth, <i>Treatise on Differential Equations</i> (London, 1889); and +<i>Phil. Trans. Roy. Soc.</i> (A.), vol. cxci. (1898), pp. 1-86.</p> + +<p>(ζ) See also the six extensive articles in the second volume of +the German <i>Encyclopaedia of Mathematics</i>.</p> +</div> +<div class="author">(H. F. Ba.)</div> + + +<hr class="art" /> +<p><span class="bold">DIFFLUGIA<a name="ar83" id="ar83"></a></span> (L. Leclerc), a genus of lobose Rhizopoda, characterized +by a shell formed of sand granules cemented together; +these are swallowed by the animal, and during the process of +bud-fission they pass to the surface of the daughter-bud and +are cemented there. <i>Centropyxis</i> (Steia) and <i>Lecqueureuxia</i> +(Schlumberg) differ only in minor points.</p> + + +<hr class="art" /> +<p><span class="bold">DIFFRACTION OF LIGHT.<a name="ar84" id="ar84"></a></span>—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.</p> + +<p>2. <i>Shadows.</i>—In the infancy of the undulatory theory the +objection most frequently urged against it was the difficulty of +explaining the very existence of shadows. Thanks to Fresnel +and his followers, this department of optics is now precisely the +one in which the theory has gained its greatest triumphs. The +principle employed in these investigations is due to C. Huygens, +and may be thus formulated. If round the origin of waves an +ideal closed surface be drawn, the whole action of the waves in the +region beyond may be regarded as due to the motion continually +propagated across the various elements of this surface. The wave +motion due to any element of the surface is called a <i>secondary</i> +wave, and in estimating the total effect regard must be paid to the +phases as well as the amplitudes of the components. It is usually +convenient to choose as the surface of resolution a <i>wave-front</i>, <i>i.e.</i> +a surface at which the primary vibrations are in one phase. Any +obscurity that may hang over Huygens’s principle is due mainly to +the indefiniteness of thought and expression which we must be +content to put up with if we wish to avoid pledging ourselves as +to the character of the vibrations. In the application to sound, +where we know what we are dealing with, the matter is simple +enough in principle, although mathematical difficulties would +often stand in the way of the calculations we might wish to make. +<span class="pagenum"><a name="page239" id="page239"></a>239</span> +The ideal surface of resolution may be there regarded as a flexible +lamina; and we know that, if by forces locally applied every +element of the lamina be made to move normally to itself exactly +as the air at that place does, the external aerial motion is fully +determined. By the principle of superposition the whole effect +may be found by integration of the partial effects due to each +element of the surface, the other elements remaining at rest.</p> + +<table class="nobctr" style="float: left; width: 220px;" summary="Illustration"> +<tr><td class="figleft1"><img style="width:169px; height:178px" src="images/img239.jpg" alt="" /></td></tr> +<tr><td class="caption sc">Fig. 1.</td></tr></table> + +<p>We will now consider in detail the important case in which uniform +plane waves are resolved at a surface coincident with a wave-front +(OQ). We imagine a wave-front divided +into elementary rings or zones—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 ½λ. There are thus marked out a series +of circles, whose radii x are given by +x² + r² = (r + ½nλ)², or x² = nλr nearly; so that +the rings are at first of nearly equal area. +Now the effect upon P of each element of the +plane is proportional to its area; but it +depends also upon the distance from P, and possibly upon the +inclination of the secondary ray to the direction of vibration and +to the wave-front.</p> + +<p>The latter question can only be treated in connexion with the +dynamical theory (see below, § 11); but under all ordinary circumstances +the result is independent of the precise answer that may be +given. All that it is necessary to assume is that the effects of the +successive zones gradually diminish, whether from the increasing +obliquity of the secondary ray or because (on account of the limitation +of the region of integration) the zones become at last more and +more incomplete. The component vibrations at P due to the +successive zones are thus nearly equal in amplitude and opposite in +phase (the phase of each corresponding to that of the infinitesimal +circle midway between the boundaries), and the series which we have +to sum is one in which the terms are alternately opposite in sign +and, while at first nearly constant in numerical magnitude, gradually +diminish to zero. In such a series each term may be regarded as very +nearly indeed destroyed by the halves of its immediate neighbours, +and thus the sum of the whole series is represented by half the first +term, which stands over uncompensated. The question is thus +reduced to that of finding the effect of the first zone, or central +circle, of which the area is πλr.</p> + +<p>We have seen that the problem before us is independent of the +law of the secondary wave as regards obliquity; but the result of +the integration necessarily involves the law of the intensity and +phase of a secondary wave as a function of r, the distance from the +origin. And we may in fact, as was done by A. Smith (<i>Camb. Math. +Journ.</i>, 1843, 3, p. 46), determine the law of the secondary wave, by +comparing the result of the integration with that obtained by supposing +the primary wave to pass on to P without resolution.</p> + +<p>Now as to the phase of the secondary wave, it might appear +natural to suppose that it starts from any point Q with the phase +of the primary wave, so that on arrival at P, it is retarded by the +amount corresponding to QP. But a little consideration will prove +that in that case the series of secondary waves could not reconstitute +the primary wave. For the aggregate effect of the secondary waves +is the half of that of the first Fresnel zone, and it is the central +element only of that zone for which the distance to be travelled is +equal to r. Let us conceive the zone in question to be divided +into infinitesimal rings of equal area. The effects due to each of +these rings are equal in amplitude and of phase ranging uniformly +over half a complete period. The phase of the resultant is midway +between those of the extreme elements, that is to say, a quarter of +a period behind that due to the element at the centre of the circle. +It is accordingly necessary to suppose that the secondary waves +start with a phase one-quarter of a period in advance of that of the +primary wave at the surface of resolution.</p> + +<p>Further, it is evident that account must be taken of the variation +of phase in estimating the magnitude of the effect at P of the first +zone. The middle element alone contributes without deduction; +the effect of every other must be found by introduction of a resolving +factor, equal to cos θ, if θ represent the difference of phase +between this element and the resultant. Accordingly, the amplitude +of the resultant will be less than if all its components had the same +phase, in the ratio</p> + +<table class="math0" summary="math"> +<tr><td class="np" rowspan="2"><span class="f200">∫</span></td> <td>+½π</td> +<td rowspan="2">cos θdθ : π,</td></tr> +<tr><td>-½π</td></tr></table> + +<p class="noind">or 2 : π. Now 2 area /π = 2λr; so that, in order to reconcile the +amplitude of the primary wave (taken as unity) with the half effect +of the first zone, the amplitude, at distance r, of the secondary wave +emitted from the element of area dS must be taken to be</p> + +<p class="center">dS/λr     (1).</p> + +<p class="noind">By this expression, in conjunction with the quarter-period acceleration +of phase, the law of the secondary wave is determined.</p> + +<p>That the amplitude of the secondary wave should vary as r<span class="sp">-1</span> was +to be expected from considerations respecting energy; but the +occurrence of the factor λ<span class="sp">-1</span>, and the acceleration of phase, have +sometimes been regarded as mysterious. It may be well therefore +to remember that precisely these laws apply to a secondary wave +of sound, which can be investigated upon the strictest mechanical +principles.</p> + +<p>The recomposition of the secondary waves may also be treated +analytically. If the primary wave at O be cos kat, the effect of the +secondary wave proceeding from the element dS at Q is</p> + +<table class="math0" summary="math"> +<tr><td>dS</td> <td rowspan="2">cos k(at − ρ + ¼λ) = −</td> <td>dS</td> +<td rowspan="2">sin k(at − ρ).</td></tr> +<tr><td class="denom">λρ</td> <td class="denom">λρ</td></tr></table> + +<p class="noind">If dS = 2πxdx, we have for the whole effect</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">−</td> <td>2π</td> +<td rowspan="2" class="f200 np">∫</td> <td>∞</td> <td>sin k(at − ρ)x dx</td> +<td rowspan="2">,</td></tr> +<tr><td class="denom">λ</td> <td>0</td> <td class="denom">ρ</td></tr></table> + +<p class="noind">or, since xdx = ρdρ, k = 2π/λ,</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">−k <span class="f150">∫</span></td> <td class="bk">∞</td> +<td rowspan="2"> sin k(at − ρ)dρ = <span class="f150">[</span> −cos k(at − ρ) <span class="f150">]</span></td> <td class="bk">∞</td> <td rowspan="2"></td> +<td rowspan="2">.</td></tr> +<tr><td class="bk">r</td> <td class="bk">r</td></tr></table> + +<p class="noind">In order to obtain the effect of the primary wave, as retarded by +traversing the distance r, viz. cos k(at − r), it is necessary to suppose +that the integrated term vanishes at the upper limit. And it is important +to notice that without some further understanding the +integral is really ambiguous. According to the assumed law of +the secondary wave, the result must actually depend upon the +precise radius of the outer boundary of the region of integration, +supposed to be exactly circular. This case is, however, at most +very special and exceptional. We may usually suppose that a large +number of the outer rings are incomplete, so that the integrated term +at the upper limit may properly be taken to vanish. If a formal +proof be desired, it may be obtained by introducing into the integral +a factor such as e<span class="sp">-hρ</span>, in which h is ultimately made to diminish +without limit.</p> + +<p>When the primary wave is plane, the area of the first Fresnel +zone is πλr, and, since the secondary waves vary as r<span class="sp">-1</span>, the intensity +is independent of r, as of course it should be. If, however, the +primary wave be spherical, and of radius a at the wave-front of +resolution, then we know that at a distance r further on the +amplitude of the primary wave will be diminished in the ratio +a : (r + a). This may be regarded as a consequence of the altered +area of the first Fresnel zone. For, if x be its radius, we have</p> + +<p class="center">{(r + ½λ)² − x²} + √ {a² − x²} = r + a,</p> + +<p class="noind">so that</p> + +<p class="center">x² = λar/(a + r) nearly.</p> + +<p class="noind">Since the distance to be travelled by the secondary waves is still +r, we see how the effect of the first zone, and therefore of the whole +series is proportional to a/(a + r). In like manner may be treated +other cases, such as that of a primary wave-front of unequal principal +curvatures.</p> + +<p>The general explanation of the formation of shadows may also +be conveniently based upon Fresnel’s zones. If the point under +consideration be so far away from the geometrical shadow that a +large number of the earlier zones are complete, then the illumination, +determined sensibly by the first zone, is the same as if there +were no obstruction at all. If, on the other hand, the point be well +immersed in the geometrical shadow, the earlier zones are altogether +missing, and, instead of a series of terms beginning with finite +numerical magnitude and gradually diminishing to zero, we have +now to deal with one of which the terms diminish to zero <i>at both +ends</i>. The sum of such a series is very approximately zero, each term +being neutralized by the halves of its immediate neighbours, which +are of the opposite sign. The question of light or darkness then +depends upon whether the series begins or ends abruptly. With few +exceptions, abruptness can occur only in the presence of the first +term, viz. when the secondary wave of least retardation is unobstructed, +or when a <i>ray</i> passes through the point under consideration. +According to the undulatory theory the light cannot be regarded +strictly as travelling along a ray; but the existence of an unobstructed +ray implies that the system of Fresnel’s zones can be commenced, +and, if a large number of these zones are fully developed and do not +terminate abruptly, the illumination is unaffected by the neighbourhood +of obstacles. Intermediate cases in which a few zones only are +formed belong especially to the province of diffraction.</p> + +<p>An interesting exception to the general rule that full brightness +requires the existence of the first zone occurs when the obstacle +assumes the form of a small circular disk parallel to the plane of +the incident waves. In the earlier half of the 18th century R. Delisle +found that the centre of the circular shadow was occupied by a +bright point of light, but the observation passed into oblivion +until S. D. Poisson brought forward as an objection to Fresnel’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 +<span class="pagenum"><a name="page240" id="page240"></a>240</span> +half of that of the first existing zone, and this is sensibly the same +as if there were no obstruction.</p> + +<p>When light passes through a small circular or annular aperture, +the illumination at any point along the axis depends upon the +precise relation between the aperture and the distance from it at +which the point is taken. If, as in the last paragraph, we imagine +a system of zones to be drawn commencing from the inner circular +boundary of the aperture, the question turns upon the manner in +which the series terminates at the outer boundary. If the aperture +be such as to fit exactly an integral number of zones, the aggregate +effect may be regarded as the half of those due to the first and last +zones. If the number of zones be even, the action of the first and last +zones are antagonistic, and there is complete darkness at the point. +If on the other hand the number of zones be odd, the effects conspire; +and the illumination (proportional to the square of the amplitude) +is four times as great as if there were no obstruction at all.</p> + +<p>The process of augmenting the resultant illumination at a particular +point by stopping some of the secondary rays may be carried +much further (Soret, <i>Pogg. Ann.</i>, 1875, 156, p. 99). By the aid of +photography it is easy to prepare a plate, transparent where the zones +of odd order fall, and opaque where those of even order fall. Such +a plate has the power of a condensing lens, and gives an illumination +out of all proportion to what could be obtained without it. An even +greater effect (fourfold) can be attained by providing that the +stoppage of the light from the alternate zones is replaced by a +phase-reversal without loss of amplitude. R. W. Wood (<i>Phil. Mag.</i>, +1898, 45, p 513) has succeeded in constructing zone plates upon this +principle.</p> + +<p>In such experiments the narrowness of the zones renders necessary +a pretty close approximation to the geometrical conditions. Thus +in the case of the circular disk, equidistant (r) from the source of +light and from the screen upon which the shadow is observed, the +width of the first exterior zone is given by</p> + +<p class="center">dx = λ(2r)/4(2x),</p> + +<p>2x being the diameter of the disk. If 2r = 1000 cm., 2x = 1 cm., +λ = 6 × 10<span class="sp">-5</span> cm., then dx = .0015 cm. Hence, in order that this +zone may be perfectly formed, there should be no error in the circumference +of the order of .001 cm. (It is easy to see that the radius of +the bright spot is of the same order of magnitude.) The experiment +succeeds in a dark room of the length above mentioned, with a +threepenny bit (supported by three threads) as obstacle, the origin +of light being a small needle hole in a plate of tin, through which the +sun’s rays shine horizontally after reflection from an external mirror. +In the absence of a heliostat it is more convenient to obtain a point of +light with the aid of a lens of short focus.</p> + +<p>The amplitude of the light at any point in the axis, when plane +waves are incident perpendicularly upon an annular aperture, is, +as above,</p> + +<p class="center">cos k(at − r<span class="su">1</span>) − cos k(at − r<span class="su">2</span>) = 2 sin kat sin k(r<span class="su">1</span> − r<span class="su">2</span>),</p> + +<p class="noind">r<span class="su">2</span>, r<span class="su">1</span> being the distances of the outer and inner boundaries +from the point in question. It is scarcely necessary to remark +that in all such cases the calculation applies in the first instance +to homogeneous light, and that, in accordance with Fourier’s +theorem, each homogeneous component of a mixture may be treated +separately. When the original light is white, the presence of some +components and the absence of others will +usually give rise to coloured effects, variable +with the precise circumstances of the case.</p> + +<table class="nobctr" style="float: left; width: 220px;" summary="Illustration"> +<tr><td class="figleft1"><img style="width:168px; height:205px" src="images/img240.jpg" alt="" /></td></tr> +<tr><td class="caption sc">Fig. 2.</td></tr></table> + +<p>Although the matter can be fully treated +only upon the basis of a dynamical theory, it +is proper to point out at once that there is an +element of assumption in the application of +Huygens’s principle to the calculation of the +effects produced by opaque screens of limited +extent. Properly applied, the principle could +not fail; but, as may readily be proved in +the case of sonorous waves, it is not in strictness +sufficient to assume the expression for +a secondary wave suitable when the primary +wave is undisturbed, with mere limitation of +the integration to the transparent parts of the screen. But, except +perhaps in the case of very fine gratings, it is probable that the error +thus caused is insignificant; for the incorrect estimation of the +secondary waves will be limited to distances of a few wave-lengths +only from the boundary of opaque and transparent parts.</p> + +<p>3. <i>Fraunhofer’s Diffraction Phenomena.</i>—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.</p> + +<p>If ƒ be the radius of the spherical wave at the place of resolution, +where the vibration is represented by cos kat, then at any point +M (fig. 2) in the recipient screen the vibration due to an element dS +of the wave-front is (§ 2)</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">−</td> <td>dS</td> +<td rowspan="2">sin k(at − ρ),</td></tr> +<tr><td class="denom">λρ</td></tr></table> + +<p class="noind">ρ being the distance between M and the element dS.</p> + +<p>Taking co-ordinates in the plane of the screen with the centre of +the wave as origin, let us represent M by ξ, η, and P (where dS is +situated) by x, y, z.</p> + +<p class="noind">Then</p> + +<p class="center">ρ² = (x − ξ)² + (y − η)² + z², ƒ² = x² + y² + z²;</p> + +<p class="noind">so that</p> + +<p class="center">ρ² = ƒ² − 2xξ − 2yη + ξ² + η².</p> + +<p class="noind">In the applications with which we are concerned, ξ, η are very +small quantities; and we may take</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ρ = ƒ<span class="f150">{</span> 1 −</td> <td>xξ + yη</td> +<td rowspan="2"><span class="f150">}</span>.</td></tr> +<tr><td class="denom">ƒ²</td></tr></table> + +<p class="noind">At the same time dS may be identified with dxdy, and in the denominator +ρ may be treated as constant and equal to ƒ. Thus the +expression for the vibration at M becomes</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">−</td> <td>1</td> +<td rowspan="2"><span class="f150">∫∫</span>sin k <span class="f150">{</span> at − ƒ +</td> <td>xξ + yη</td> +<td rowspan="2"><span class="f150">}</span> dxdy    (1);</td></tr> +<tr><td class="denom">λƒ</td> <td class="denom">ƒ</td></tr></table> + +<p class="noind">and for the intensity, represented by the square of the amplitude,</p> + + +<table class="math0" summary="math"> +<tr><td rowspan="2">I² =</td> <td>1</td> +<td rowspan="2"><span class="f150">[ ∫∫</span> sin k</td> <td>xξ + yη</td> +<td rowspan="2">dxdy <span class="f150">]</span></td> <td>²</td> +<td rowspan="2"> </td></tr> +<tr><td class="denom">벃²</td> <td class="denom">ƒ</td> <td> </td></tr> + +<tr><td rowspan="2">  +</td> <td>1</td> +<td rowspan="2"><span class="f150">[ ∫∫</span> cos k</td> <td>xξ + yη</td> +<td rowspan="2">dxdy <span class="f150">]</span></td> <td>²</td> +<td rowspan="2">     (2).</td></tr> +<tr><td class="denom">벃²</td> <td class="denom">ƒ</td> <td> </td></tr> +</table> + +<p class="noind">This expression for the intensity becomes rigorously applicable when +ƒ is indefinitely great, so that ordinary optical aberration disappears. +The incident waves are thus plane, and are limited to a plane aperture +coincident with a wave-front. The integrals are then properly +functions of the <i>direction</i> in which the light is to be estimated.</p> + +<p>In experiment under ordinary circumstances it makes no difference +whether the collecting lens is in front of or behind the diffracting +aperture. It is usually most convenient to employ a telescope +focused upon the radiant point, and to place the diffracting apertures +immediately in front of the object-glass. What is seen through the +eye-piece in any case is the same as would be depicted upon a screen +in the focal plane.</p> + +<p>Before proceeding to special cases it may be well to call attention +to some general properties of the solution expressed by (2) (see +Bridge, <i>Phil. Mag.</i>, 1858).</p> + +<p>If when the aperture is given, the wave-length (proportional to +k<span class="sp">-1</span>) varies, the composition of the integrals is unaltered, provided +ξ and η are taken universely proportional to λ. A diminution of +λ thus leads to a simple proportional shrinkage of the diffraction +pattern, attended by an augmentation of brilliancy in proportion +to λ<span class="sp">-2</span>.</p> + +<p>If the wave-length remains unchanged, similar effects are produced +by an increase in the scale of the aperture. The linear +dimension of the diffraction pattern is inversely as that of the +aperture, and the brightness at corresponding points is as the +<i>square</i> of the area of aperture.</p> + +<p>If the aperture and wave-length increase in the same proportion, +the size and shape of the diffraction pattern undergo no change.</p> + +<p>We will now apply the integrals (2) to the case of a rectangular +aperture of width a parallel to x and of width b parallel to y. The +limits of integration for x may thus be taken to be −½a and +½a, +and for y to be −½b, +½b. We readily find (with substitution for +k of 2π/λ)</p> + +<table class="math0" summary="math"> +<tr><td colspan="2"> </td> <td rowspan="4">·</td> + <td rowspan="2">sin²</td> <td>πaξ</td> + <td rowspan="4">·</td> <td rowspan="2">sin²</td> + <td>πbη</td> <td rowspan="4">    (3),</td></tr> +<tr><td rowspan="2">I² =</td> <td>a²b²</td> <td class="denom">ƒλ</td> + <td class="denom">ƒλ</td></tr> +<tr><td class="denom">ƒ²λ²</td> + <td class="denom" colspan="2">π²a²ξ²</td> <td class="denom" colspan="2">π²b²η²</td></tr> +<tr><td colspan="2"> </td> + <td class="ov" colspan="2">ƒ²λ²</td> <td class="ov" colspan="2">ƒ²λ²</td></tr> +</table> + +<p class="noind">as representing the distribution of light in the image of a mathematical +point when the aperture is rectangular, as is often the case +in spectroscopes.</p> + +<p>The second and third factors of (3) being each of the form sin²u/u², +we have to examine the character of this function. It vanishes +when u = mπ, m being any whole number other than zero. When +u = 0, it takes the value unity. The maxima occur when</p> + +<p class="center">u = tan u,    (4),</p> + +<p class="noind">and then</p> + +<p class="center">sin²u / u² = cos²u     (5).</p> + +<p class="noind">To calculate the roots of (5) we may assume</p> + +<p class="center">u = (m + ½)π − y = U − y,</p> + +<p><span class="pagenum"><a name="page241" id="page241"></a>241</span></p> + +<p>where y is a positive quantity which is small when u is large. Substituting +this, we find cot y = U − y, whence</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">y =</td> <td>1</td> +<td rowspan="2"><span class="f200">(</span>1 +</td> <td>y</td> +<td rowspan="2">+</td> <td>y-</td> +<td rowspan="2">+ ...<span class="f200">)</span> −</td> <td>y³</td> +<td rowspan="2">−</td> <td>2y<span class="sp">5</span></td> +<td rowspan="2">−</td> <td>17y<span class="sp">7</span></td> <td rowspan="2">.</td></tr> +<tr><td class="denom">U</td> <td class="denom">U</td> <td class="denom">U²</td> + <td class="denom">3</td> <td class="denom">15</td> <td class="denom">315</td></tr></table> + +<p class="noind">This equation is to be solved by successive approximation. It will +readily be found that</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">u = U − y = U − U<span class="sp">−1</span> −</td> <td>2</td> +<td rowspan="2">U<span class="sp">−3</span> −</td> <td>13</td> +<td rowspan="2">U<span class="sp">−5</span> −</td> <td>146</td> <td rowspan="2">U<span class="sp">−7</span> − ...   (6).</td></tr> +<tr><td class="denom">3</td> <td class="denom">15</td> <td class="denom">105</td></tr></table> + +<p>In the first quadrant there is no root after zero, since tan u > 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/π (calculated in another manner by F. M. +Schwerd) are 1.4303, 2.4590, 3.4709, 4.4747, 5.4818, 6.4844, &c.</p> + +<p>Since the maxima occur when u = (m + ½)π nearly, the successive +values are not very different from</p> + +<table class="math0" summary="math"> +<tr><td>4</td> <td rowspan="2">,</td> <td>4</td> +<td rowspan="2">,</td> <td>4</td> +<td rowspan="2">, &c.</td></tr> +<tr><td class="denom">9π²</td> <td class="denom">25π</td> <td class="denom">49π²</td></tr></table> + +<p>The application of these results to (3) shows that the field is +brightest at the centre ξ = 0, η = 0, viz. at the geometrical image +of the radiant point. It is traversed by dark lines whose equations +are</p> + +<p class="center">ξ = mfλ / a, η = mfλ / b.</p> + +<p class="noind">Within the rectangle formed by pairs of consecutive dark lines, +and not far from its centre, the brightness rises to a maximum; +but these subsequent maxima are in all cases much inferior to the +brightness at the centre of the entire pattern (ξ = 0, η = 0).</p> + +<p>By the principle of energy the illumination over the entire focal +plane must be equal to that over the diffracting area; and thus, in +accordance with the suppositions by which (3) was obtained, its +value when integrated from ξ = ∞ to ξ = +∞, and from η = −∞ +to η = +∞ should be equal to ab. This integration, employed +originally by P. Kelland (<i>Edin. Trans.</i> 15, p. 315) to determine the +absolute intensity of a secondary wave, may be at once effected by +means of the known formula</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">∫</span></td> <td class="bk">+∞</td> <td>sin²u</td> +<td rowspan="2">du = <span class="f150">∫</span></td> <td class="bk">+∞</td> <td>sin u</td> +<td rowspan="2">du = π.</td></tr> +<tr><td class="bk">−∞</td> <td class="denom">u²</td> +<td class="bk">−∞</td> <td class="denom">u</td></tr></table> + +<p class="noind">It will be observed that, while the total intensity is proportional to +ab, the intensity at the focal point is proportional to a²b². If the +aperture be increased, not only is the total brightness over the focal +plane increased with it, but there is also a concentration of the +diffraction pattern. The form of (3) shows immediately that, if +a and b be altered, the co-ordinates of any characteristic point in the +pattern vary as a<span class="sp">−1</span> and b<span class="sp">−1</span>.</p> + +<p>The contraction of the diffraction pattern with increase of aperture +is of fundamental importance in connexion with the resolving power +of optical instruments. According to common optics, where images +are absolute, the diffraction pattern is supposed to be infinitely +small, and two radiant points, however near together, form separated +images. This is tantamount to an assumption that λ is infinitely +small. The actual finiteness of λ imposes a limit upon the separating +or resolving power of an optical instrument.</p> + +<p>This indefiniteness of images is sometimes said to be due to +diffraction by the edge of the aperture, and proposals have even been +made for curing it by causing the transition between the interrupted +and transmitted parts of the primary wave to be less abrupt. Such +a view of the matter is altogether misleading. What requires +explanation is not the imperfection of actual images so much as the +possibility of their being as good as we find them.</p> + +<p>At the focal point (ξ = 0, η = 0) all the secondary waves agree in +phase, and the intensity is easily expressed, whatever be the form +of the aperture. From the general formula (2), if A be the <i>area</i> of +aperture,</p> + +<p class="center">I<span class="su">0</span>² = A² / 벃²     (7).</p> + +<p>The formation of a sharp image of the radiant point requires +that the illumination become insignificant when ξ, η 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 ¼λ.</p> + +<p>When the difference of phase amounts to λ, 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 ƒ to be infinite. In the direction +(suppose horizontal) for which η = 0, ξ/ƒ = sin θ, the phases of the +secondary waves range over a complete period when sin θ = λ/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 θ = <span class="spp">3</span>⁄<span class="suu">2</span>λ/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 θ = 0. The phase of the +resultant amplitude is the same as that due to the central secondary +wave, and the discrepancies of phase among the components reduce +the amplitude in the proportion</p> + +<table class="math0" summary="math"> +<tr><td>1</td> <td rowspan="2"><span class="f150">∫</span></td> <td class="bk">+<span class="spp">3</span>⁄<span class="suu">2</span>π</td> +<td rowspan="2">cos φ dφ: 1</td></tr> +<tr><td class="denom">3π</td> <td class="bk">−<span class="spp">3</span>⁄<span class="suu">2</span>π</td></tr></table> + +<p class="noind">or -<span class="spp">2</span>⁄<span class="suu">3</span>π : 1; so that the brightness in this direction is <span class="spp">4</span>⁄<span class="suu">9</span>π² of the +maximum at θ = 0. In like manner we may find the illumination +in any other direction, and it is obvious that it vanishes when sin θ +is any multiple of λ/a.</p> + +<p>The reason of the augmentation of resolving power with aperture +will now be evident. The larger the aperture the smaller are the +angles through which it is necessary to deviate from the principal +direction in order to bring in specified discrepancies of phase—the +more concentrated is the image.</p> + +<p>In many cases the subject of examination is a luminous line of +uniform intensity, the various points of which are to be treated as +independent sources of light. If the image of the line be ξ = 0, +the intensity at any point ξ, η of the diffraction pattern may be +represented by</p> + +<table class="math0" summary="math"> +<tr><td colspan="4"> </td> <td rowspan="2">sin²</td> + <td>πaξ</td> <td rowspan="4">     (8),</td></tr> +<tr><td rowspan="2"><span class="f150">∫</span></td> + <td class="bk">+∞</td> <td rowspan="2">I²dη =</td> <td>a²b</td> <td class="ov">λf</td></tr> +<tr><td class="bk">−∞</td> <td class="ov">λƒ</td> + <td class="denom" colspan="2">π²a²ξ²</td></tr> +<tr><td colspan="4"> </td> <td class="ov" colspan="2">λ²f²</td></tr> +</table> + +<p class="noind">the same law as obtains for a luminous point when horizontal +directions are alone considered. The definition of a fine vertical +line, and consequently the resolving power for contiguous vertical +lines, is thus <i>independent of the vertical aperture of the instrument</i>, +a law of great importance in the theory of the spectroscope.</p> + +<p>The distribution of illumination in the image of a luminous line +is shown by the curve ABC (fig. 3), representing the value of the +function sin²u/u² from u = 0 to u = 2π. The part corresponding to +negative values of u is similar, OA being a line of symmetry.</p> + +<table class="nobctr" style="float: right; width: 230px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:185px; height:181px" src="images/img241.jpg" alt="" /></td></tr> +<tr><td class="caption sc">Fig. 3.</td></tr></table> + +<p>Let us now consider the distribution of brightness in the image +of a double line whose components are of equal strength, and at +such an angular interval that the central line in the image of one +coincides with the first zero of brightness in the image of the other. +In fig. 3 the curve of brightness for one component is ABC, and +for the other OA′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 = π, is such that the phases +of the secondary waves range over a complete +period, <i>i.e.</i> such that the projection of +the horizontal aperture upon this direction +is one wave-length. We conclude that a <i>double line cannot be +fairly resolved unless its components subtend an angle exceeding that +subtended by the wave-length of light at a distance equal to the horizontal +aperture</i>. This rule is convenient on account of its simplicity; and +it is sufficiently accurate in view of the necessary uncertainty as to +what exactly is meant by resolution.</p> + +<p>If the angular interval between the components of a double line +be half as great again as that supposed in the figure, the brightness +midway between is .1802 as against 1.0450 at the central lines of +each image. Such a falling off in the middle must be more than +sufficient for resolution. If the angle subtended by the components +of a double line be twice that subtended by the wave-length at a +distance equal to the horizontal aperture, the central bands are +just clear of one another, and there is a line of absolute blackness +in the middle of the combined images.</p> + +<p>The resolving power of a telescope with circular or rectangular +aperture is easily investigated experimentally. The best object for +examination is a grating of fine wires, about fifty to the inch, backed +by a sodium flame. The object-glass is provided with diaphragms +pierced with round holes or slits. One of these, of width equal, say, +to one-tenth of an inch, is inserted in front of the object-glass, and +the telescope, carefully focused all the while, is drawn gradually back +from the grating until the lines are no longer seen. From a measurement +of the maximum distance the least angle between consecutive +lines consistent with resolution may be deduced, and a comparison +made with the rule stated above.</p> + +<p>Merely to show the dependence of resolving power on aperture it is +not necessary to use a telescope at all. It is sufficient to look at wire +gauze backed by the sky or by a flame, through a piece of blackened +cardboard, pierced by a needle and held close to the eye. By +varying the distance the point is easily found at which resolution +ceases; and the observation is as sharp as with a telescope. The +<span class="pagenum"><a name="page242" id="page242"></a>242</span> +function of the telescope is in fact to allow the use of a wider, and +therefore more easily measurable, aperture. An interesting modification +of the experiment may be made by using light of various +wave-lengths.</p> + +<p>Since the limitation of the width of the central band in the image +of a luminous line depends upon discrepancies of phase among the +secondary waves, and since the discrepancy is greatest for the waves +which come from the edges of the aperture, the question arises +how far the operation of the central parts of the aperture is advantageous. +If we imagine the aperture reduced to two equal +narrow slits bordering its edges, compensation will evidently be +complete when the projection on an oblique direction is equal to +½λ, instead of λ as for the complete aperture. By this procedure +the width of the central band in the diffraction pattern is halved, +and so far an advantage is attained. But, as will be evident, the +bright bands bordering the central band are now not inferior to it +in brightness; in fact, a band similar to the central band is reproduced +an indefinite number of times, so long as there is no sensible +discrepancy of phase in the secondary waves proceeding from the +various parts of the <i>same</i> slit. Under these circumstances the +narrowing of the band is paid for at a ruinous price, and the arrangement +must be condemned altogether.</p> + +<p>A more moderate suppression of the central parts is, however, +sometimes advantageous. Theory and experiment alike prove that +a double line, of which the components are equally strong, is better +resolved when, for example, one-sixth of the horizontal aperture is +blocked off by a central screen; or the rays quite at the centre may +be allowed to pass, while others a little farther removed are blocked +off. Stops, each occupying one-eighth of the width, and with centres +situated at the points of trisection, answer well the required purpose.</p> + +<p>It has already been suggested that the principle of energy requires +that the general expression for I² in (2) when integrated over the +whole of the plane ξ, η should be equal to A, where A is the area of +the aperture. A general analytical verification has been given by +Sir G. G. Stokes (<i>Edin. Trans.</i>, 1853, 20, p. 317). Analytically +expressed—</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">∫∫</span></td> <td class="bk">+∞</td> +<td rowspan="2">I² dξdη = <span class="f150">∫∫</span> dxdy = A     (9).</td></tr> +<tr><td class="bk">−∞</td></tr></table> + +<p>We have seen that I<span class="su">0</span>² (the intensity at the focal point) was equal to +A²/λ²f². If A′ be the area over which the intensity must be I<span class="su">0</span>² in +order to give the actual total intensity in accordance with</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">A′ I<span class="su">0</span>² = <span class="f150">∫∫</span></td> <td class="bk">+∞</td> + <td rowspan="2">I² dξdη,</td></tr> +<tr><td class="bk">−∞</td></tr></table> + +<p class="noind">the relation between A and A′ is AA′ = λ²f². 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.</p> + +<p>4. <i>Theory of Circular Aperture.</i>—We will now consider the +important case where the form of the aperture is circular.</p> + +<p>Writing for brevity</p> + +<p class="center">kξ/f = p, kη/f = q,     (1),</p> + +<p class="noind">we have for the general expression (§ 11) of the intensity</p> + +<p class="center">λ²f²I² = S² + C²     (2),</p> + +<p class="noind">where</p> + +<p class="center">S = <span class="f150">∫∫</span> sin(px + qy)dx dy,     (3),</p> + +<p class="center">C = <span class="f150">∫∫</span> cos(px + qy)dx dy,     (4).</p> + +<p class="noind">When, as in the application to rectangular or circular apertures, +the form is symmetrical with respect to the axes both of x and y, +S = 0, and C reduces to</p> + +<p class="center">C = <span class="f150">∫∫</span> cos px cos qy dx dy,     (5).</p> + +<p class="noind">In the case of the circular aperture the distribution of light is of +course symmetrical with respect to the focal point p = 0, q = 0; and +C is a function of p and q only through √(p² + q²). It is thus +sufficient to determine the intensity along the axis of p. Putting +q = 0, we get</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">C = <span class="f150">∫∫</span> cos px dx dy = 2 <span class="f150">∫</span></td> <td class="bk">+R</td> +<td rowspan="2">cos px √(R² − x²) dx,</td></tr> +<tr><td class="bk">−R</td></tr></table> + +<p class="noind">R being the radius of the aperture. This integral is the Bessel’s +function of order unity, defined by</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">J<span class="su">1</span>(z) =</td> <td>z</td> +<td rowspan="2"><span class="f150">∫</span></td> <td class="bk">π</td> +<td rowspan="2">cos (z cos φ) sin² φ dφ     (6).</td></tr> +<tr><td class="denom">π</td> <td class="bk">0</td></tr></table> + +<p class="noind">Thus, if x = R cos φ,</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">C = π²R</td> <td>2J<span class="su">1</span>(pR)</td> +<td rowspan="2">     (7);</td></tr> +<tr><td class="denom">pR</td></tr></table> + +<p class="noind">and the illumination at distance r from the focal point is</p> + +<table class="math0" summary="math"> +<tr><td colspan="2"> </td> <td rowspan="4">·</td> + <td rowspan="2">4J<span class="su">1</span>² <span class="f150">(</span></td> + <td>2πRr</td> <td rowspan="2"><span class="f150">)</span></td> + <td rowspan="4">     (8).</td></tr> +<tr><td rowspan="2">I² =</td> <td>π²R<span class="sp">4</span></td> <td class="ov">ƒλ</td></tr> +<tr><td class="denom">λ²f²</td> <td class="denom" rowspan="2">   <span class="f150">(</span></td> + <td class="denom">2πRr</td> <td class="denom" rowspan="2"><span class="f150">)</span> ²</td></tr> +<tr><td colspan="2"> </td> <td class="ov">ƒλ</td></tr> +</table> + +<p>The ascending series for J<span class="su">1</span>(z), used by Sir G. B. Airy (<i>Camb. Trans.</i>, +1834) in his original investigation of the diffraction of a circular +object-glass, and readily obtained from (6), is</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">J<span class="su">1</span>(z) =</td> <td>z</td> +<td rowspan="2">−</td> <td>z³</td> +<td rowspan="2">+</td> <td>z<span class="sp">5</span></td> +<td rowspan="2">−</td> <td>z<span class="sp">7</span></td> <td rowspan="2">+ ...     (9).</td></tr> +<tr><td class="denom">2</td> <td class="denom">2²·4</td> +<td class="denom">2²·4²·6</td> <td class="denom">2²·4²·6²·8</td></tr></table> + +<p class="noind">When z is great, we may employ the semi-convergent series</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">J<span class="su">1</span>(z) = <span class="f150">√(</span></td> <td>2</td> +<td rowspan="2"><span class="f150">)</span> sin (z − ¼π) <span class="f150">{</span> 1 +</td> <td>3·5·1</td> +<td rowspan="2"><span class="f150">(</span></td> <td>1</td> +<td rowspan="2"><span class="f150">)</span></td> <td>²</td> +<td rowspan="2">−</td> <td>3·5·7·9·1·3·5</td> +<td rowspan="2"><span class="f150">(</span></td> <td>1</td> +<td rowspan="2"><span class="f150">)</span></td> <td><span class="sp np">4</span></td> +<td rowspan="2">+ ... <span class="f150">}</span></td></tr> +<tr><td class="denom">πz</td> <td class="denom">8·16</td> +<td class="denom">z</td> <td> </td> +<td class="denom">8·16·24·32</td> <td class="denom">z</td> <td> </td></tr></table> + +<table class="math0" summary="math"> +<tr><td rowspan="2">+ <span class="f150">√(</span></td> <td>2</td> +<td rowspan="2"><span class="f150">)</span> cos (z − ¼π) <span class="f150">{</span>3/8 · 1/z −</td> +<td>3·5·7·1·3</td> <td rowspan="2"><span class="f150">(</span></td> <td>1</td> +<td rowspan="2"><span class="f150">)</span></td> <td>³</td> +<td rowspan="2">+</td> <td>3·5·7·9·11·1·3·5·7</td> +<td rowspan="2"><span class="f150">(</span></td> <td>1</td> <td rowspan="2"><span class="f150">)</span></td> +<td><span class="sp">5</span></td> <td rowspan="2">− ... <span class="f150">}</span>     (10).</td></tr> +<tr><td class="denom">πz</td> <td class="denom">8·16·24</td> +<td class="denom">z</td> <td> </td> +<td class="denom">8·16·24·32·40</td> <td class="denom">z</td> <td> </td></tr></table> + +<p class="noind">A table of the values of 2z<span class="sp">-1</span>J<span class="su">1</span>(z) has been given by E. C. J. Lommel +(<i>Schlömilch</i>, 1870, 15, p. 166), to whom is due the first systematic +application of Bessel’s functions to the diffraction integrals.</p> + +<p>The illumination vanishes in correspondence with the roots of the +equation J<span class="su">1</span>(z) = 0. If these be called z<span class="su">1</span> z<span class="su">2</span>, z<span class="su">3</span>, ... the radii of the +dark rings in the diffraction pattern are</p> + +<table class="math0" summary="math"> +<tr><td>ƒλz<span class="su">1</span></td> <td rowspan="2"></td> +<td>ƒλz<span class="su">2</span></td> <td rowspan="2">, ...</td></tr> +<tr><td class="denom">2πR</td> <td class="denom">2πR</td></tr></table> + +<p class="noind">being thus <i>inversely</i> proportional to R.</p> + +<p>The integrations may also be effected by means of polar co-ordinates, +taking first the integration with respect to φ so as to +obtain the result for an infinitely thin annular aperture. Thus, if</p> + +<p class="center">x = ρ cos φ, y = ρ sin φ,</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">C = <span class="f150">∫∫</span> cos px dx dy = <span class="f150">∫</span></td> <td class="bk">R</td> +<td rowspan="2"><span class="f150">∫</span></td> <td class="bk">2π</td> <td rowspan="2">cos (pρ cos θ) ρdρ dθ.</td></tr> +<tr><td class="bk">0</td> <td class="bk">0</td></tr></table> + +<p class="noind">Now by definition</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">J<span class="su">0</span>(z) =</td> <td>2</td> +<td rowspan="2"><span class="f150">∫</span></td> <td class="bk">½π</td> +<td rowspan="2">cos (z cos θ) dθ = 1 −</td> <td>z²</td> +<td rowspan="2">+</td> <td>z<span class="sp">4</span></td> +<td rowspan="2">−</td> <td>z<span class="sp">6</span></td> +<td rowspan="2">+ ...     (11).</td></tr> +<tr><td class="denom">π</td> <td class="bk">0</td> +<td class="denom">2²</td> <td class="denom">2²·4²</td> <td class="denom">2²·4²·6²</td></tr></table> + +<p class="noind">The value of C for an annular aperture of radius r and width dr is +thus</p> + +<p class="center">dC = 2 π J<span class="su">0</span> (pρ) ρ dρ,     (12).</p> + +<p class="noind">For the complete circle,</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">C =</td> <td>2π</td> +<td rowspan="2"><span class="f150">∫</span></td> <td class="bk">pr</td> +<td rowspan="2">J<span class="su">0</span>(z) zdz =</td> <td>2π</td> +<td rowspan="2"><span class="f150">{</span></td> <td>p²R²</td> +<td rowspan="2">−</td> <td>p<span class="sp">4</span>R<span class="sp">4</span></td> +<td rowspan="2">+</td> <td>p<span class="sp">6</span>R<span class="sp">6</span></td> +<td rowspan="2">− ...<span class="f150">}</span>= πR² ·</td> <td>2J<span class="su">1</span>(pR)</td> +<td rowspan="2">as before.</td></tr> +<tr><td class="denom">p²</td> <td class="bk">0</td> +<td class="denom">p²</td> <td class="denom">2</td> +<td class="denom">2²·4²</td> <td class="denom">2²·4²·6²</td> <td class="denom">pR</td></tr></table> + +<p>In these expressions we are to replace p by kξ/ƒ, or rather, since +the diffraction pattern is symmetrical, by kr/ƒ, where r is the distance +of any point in the focal plane from the centre of the system.</p> + +<p>The roots of J<span class="su">0</span>(z) after the first may be found from</p> + +<table class="math0" summary="math"> +<tr><td>z</td> <td rowspan="2">= i − .25 +</td> <td>.050561</td> +<td rowspan="2">−</td> <td>.053041</td> +<td rowspan="2">+</td> <td>.262051</td> <td rowspan="2">     (13),</td></tr> +<tr><td class="denom">π</td> <td class="denom">4i − 1</td> +<td class="denom">(4i − 1)³</td> <td class="denom">(4i − 1)<span class="sp">5</span></td></tr></table> + +<p class="noind">and those of J<span class="su">1</span>(z) from</p> + +<table class="math0" summary="math"> +<tr><td>z</td> <td rowspan="2">= i + .25 −</td> <td>.151982</td> +<td rowspan="2">+</td> <td>.015399</td> +<td rowspan="2">−</td> <td>.245835</td> <td rowspan="2">     (14),</td></tr> +<tr><td class="denom">π</td> <td class="denom">4i + 1</td> +<td class="denom">(4i + 1)³</td> <td class="denom">(4i + 1)<span class="sp">5</span></td></tr></table> + +<p class="noind">formulae derived by Stokes (<i>Camb. Trans.</i>, 1850, vol. ix.) from the +descending series.<a name="fa1g" id="fa1g" href="#ft1g"><span class="sp">1</span></a> The following table gives the actual values:—</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcc allb">i</td> <td class="tcc allb">z/π for J<span class="su">0</span>(z) = 0</td> <td class="tcc allb">z/π for J<span class="su">1</span>(z) = 0</td></tr> +<tr><td class="tcr lb rb">1</td> <td class="tcr rb">7655</td> <td class="tcr rb">1 2197</td></tr> +<tr><td class="tcr lb rb">2</td> <td class="tcr rb">1 7571</td> <td class="tcr rb">2 2330</td></tr> +<tr><td class="tcr lb rb">3</td> <td class="tcr rb">2 7546</td> <td class="tcr rb">3 2383</td></tr> +<tr><td class="tcr lb rb">4</td> <td class="tcr rb">3 7534</td> <td class="tcr rb">4 2411</td></tr> +<tr><td class="tcr lb rb">5</td> <td class="tcr rb">4 7527</td> <td class="tcr rb">5 2428</td></tr> +<tr><td class="tcr lb rb">6</td> <td class="tcr rb">5 7522</td> <td class="tcr rb">6 2439</td></tr> +<tr><td class="tcr lb rb">7</td> <td class="tcr rb">6 7519</td> <td class="tcr rb">7 2448</td></tr> +<tr><td class="tcr lb rb">8</td> <td class="tcr rb">7 7516</td> <td class="tcr rb">8 2454</td></tr> +<tr><td class="tcr lb rb">9</td> <td class="tcr rb">8 7514</td> <td class="tcr rb">9 2459</td></tr> +<tr><td class="tcr lb rb bb">10</td> <td class="tcr rb bb">9 7513</td> <td class="tcr rb bb">10 2463</td></tr> +</table> + +<p>In both cases the image of a mathematical point is thus a +symmetrical ring system. The greatest brightness is at the centre, +where</p> + +<p class="center">dC = 2πρ dρ, C = π R².</p> + +<p class="noind">For a certain distance outwards this remains sensibly unimpaired +and then gradually diminishes to zero, as the secondary waves +become discrepant in phase. The subsequent revivals of brightness +forming the bright rings are necessarily of inferior brilliancy as +compared with the central disk.</p> + +<p>The first dark ring in the diffraction pattern of the complete +circular aperture occurs when</p> + +<p class="center">r/ƒ = 1.2197 × λ/2R     (15).</p> + +<p><span class="pagenum"><a name="page243" id="page243"></a>243</span></p> + +<p class="noind">We may compare this with the corresponding result for a rectangular +aperture of width a,</p> + +<p class="center">ξ/ƒ =λ/a;</p> + +<p class="noind">and it appears that in consequence of the preponderance of the +central parts, the compensation in the case of the circle does not +set in at so small an obliquity as when the circle is replaced by a +rectangular aperture, whose side is equal to the diameter of the +circle.</p> + +<p>Again, if we compare the complete circle with a narrow annular +aperture of the same radius, we see that in the latter case the first +dark ring occurs at a much smaller obliquity, viz.</p> + +<p class="center">r/ƒ = .7655 × λ/2R.</p> + +<p>It has been found by Sir William Herschel and others that the +definition of a telescope is often improved by stopping off a part of +the central area of the object-glass; but the advantage to be obtained +in this way is in no case great, and anything like a reduction of the +aperture to a narrow annulus is attended by a development of the +external luminous rings sufficient to outweigh any improvement +due to the diminished diameter of the central area.<a name="fa2g" id="fa2g" href="#ft2g"><span class="sp">2</span></a></p> + +<p>The maximum brightnesses and the places at which they occur +are easily determined with the aid of certain properties of the +Bessel’s functions. It is known (see <span class="sc"><a href="#artlinks">Spherical Harmonics</a></span>) that</p> + +<p class="center">J<span class="su">0</span>′(z) = −J<span class="su">1</span>(z),     (16);</p> + +<p class="center">J<span class="su">2</span>(z) = (1/z) J<span class="su">1</span>(z) − J<span class="su">1</span>′(z)     (17);</p> + +<p class="center">J<span class="su">0</span>(z) + J<span class="su">2</span>(z) = (2/z) J<span class="su">1</span>(z)     (18).</p> + +<p class="noind">The maxima of C occur when</p> + +<table class="math0" summary="math"> +<tr><td>d</td> <td rowspan="2"><span class="f150">(</span></td> <td>J<span class="su">1</span>(z)</td> +<td rowspan="2"><span class="f150">)</span> =</td> <td>J<span class="su">1</span>′(z)</td> +<td rowspan="2">−</td> <td>J<span class="su">1</span>(z)</td> <td rowspan="2"> = 0;</td></tr> +<tr><td class="denom">dz</td> <td class="denom">z</td> <td class="denom">z</td> <td class="denom">z²</td></tr></table> + +<p class="noind">or by 17 when J<span class="su">2</span>(z) = 0. When z has one of the values thus +determined,</p> + +<table class="math0" summary="math"> +<tr><td>2</td><td rowspan="2">J<span class="su">1</span>(z) = J<span class="su">0</span>(z).</td></tr> +<tr><td class="denom">z</td></tr></table> + +<p class="noind">The accompanying table is given by Lommel, in which the first +column gives the roots of J<span class="su">2</span>(z) = 0, and the second and third columns +the corresponding values of the functions specified. If appears that +the maximum brightness in the first ring is only about <span class="spp">1</span>⁄<span class="suu">57</span> of the +brightness at the centre.</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcc lb tb">z</td> <td class="tcc tb">2z<span class="sp">−1</span>J<span class="su">1</span>(z)</td> <td class="tcc rb tb">4z<span class="sp">−2</span>J<span class="su">1</span>²(z)</td></tr> + +<tr><td class="tcr lb">.000000</td> <td class="tcr">+1.000000</td> <td class="tcr rb">1.000000</td></tr> +<tr><td class="tcr lb">5.135630</td> <td class="tcr">− .132279</td> <td class="tcr rb">.017498</td></tr> +<tr><td class="tcr lb">8.417236</td> <td class="tcr">+ .064482</td> <td class="tcr rb">.004158</td></tr> +<tr><td class="tcr lb">11.619857</td> <td class="tcr">− .040008</td> <td class="tcr rb">.001601</td></tr> +<tr><td class="tcr lb">14.795938</td> <td class="tcr">+ .027919</td> <td class="tcr rb">.000779</td></tr> +<tr><td class="tcr lb bb">17.959820</td> <td class="tcr bb">− .020905</td> <td class="tcr rb bb">.000437</td></tr> +</table> + +<p>We will now investigate the total illumination distributed over +the area of the circle of radius r. We have</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">I² =</td> <td>π²R<span class="sp">4</span></td> +<td rowspan="2">·</td> <td>4J<span class="su">1</span>²(z)</td> <td rowspan="2">     (19),</td></tr> +<tr><td class="denom">벃²</td> <td class="denom">z²</td></tr></table> + +<p class="noind">where</p> + +<p class="center">z = 2πRr/λf     (20).</p> + +<p class="noind">Thus</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">2π <span class="f150">∫</span> I²rdr =</td> <td>λ²f²</td> +<td rowspan="2"><span class="f150">∫</span> I²zdz = πR²· 2 <span class="f150">∫</span> z<span class="sp">−1</span>J<span class="su">1</span>²(z)dz.</td></tr> +<tr><td class="denom">2πR²</td></tr></table> + +<p class="noind">Now by (17), (18)</p> + +<p class="center">z<span class="sp">-1</span>J<span class="su">1</span>(z) = J<span class="su">0</span>(z) − J<span class="su">1</span>′(z);</p> + +<p class="noind">so that</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">z<span class="sp">-1</span>J<span class="su">1</span>²(z) = − ½</td> <td>d</td> +<td rowspan="2">J<span class="su">0</span>² − ½</td> <td>d</td> +<td rowspan="2">J<span class="su">1</span>²(z),</td></tr> +<tr><td class="denom">dz</td> <td class="denom">dz</td></tr></table> + +<p class="noind">and</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">2 <span class="f150">∫</span></td> <td class="bk">z</td> +<td rowspan="2">z<span class="sp">-1</span>J<span class="su">1</span>²(z)dz = 1 − J<span class="su">0</span>²(z) − J<span class="su">1</span>²(z)     (21).</td></tr> +<tr><td class="bk">0</td></tr></table> + +<p class="noind">If r, or z, be infinite, J<span class="su">0</span>(z), J<span class="su">1</span>(z) vanish, and the whole illumination +is expressed by πR², in accordance with the general principle. In +any case the proportion of the whole illumination to be found outside +the circle of radius r is given by</p> + +<p class="center">J<span class="su">0</span>²(z) + J<span class="su">1</span>²(z).</p> + +<p class="noind">For the dark rings J<span class="su">1</span>(z) = 0; so that the fraction of illumination +outside any dark ring is simply J<span class="su">0</span>²(z). Thus for the first, second, +third and fourth dark rings we get respectively .161, .090, .062, .047, +showing that more than <span class="spp">9</span>⁄<span class="suu">10</span>ths of the whole light is concentrated +within the area of the second dark ring (<i>Phil. Mag.</i>, 1881).</p> + +<p>When z is great, the descending series (10) gives</p> + +<table class="math0" summary="math"> +<tr><td>2J<span class="su">1</span>(z)</td> <td rowspan="2">=</td> <td>2</td> +<td rowspan="2"><span class="f150">√(</span></td> <td>2</td> <td rowspan="2"><span class="f150">)</span> sin(z − ¼π)     (22);</td></tr> +<tr><td class="denom">z</td> <td class="denom">z</td> <td class="denom">πz</td></tr></table> + +<p class="noind">so that the places of maxima and minima occur at equal intervals.</p> + +<p>The mean brightness varies as z<span class="sp">-3</span> (or as r<span class="sp">-3</span>), and the integral +found by multiplying it by zdz and integrating between 0 and ∞ +converges.</p> + +<p>It may be instructive to contrast this with the case of an infinitely +narrow annular aperture, where the brightness is proportional to +J<span class="su">0</span>²(z). When z is great,</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">J<span class="su">0</span>(z) = <span class="f150">√(</span></td> <td>2</td> +<td rowspan="2"><span class="f150">)</span> cos(z<span class="sp">-1/4</span>π).</td></tr> +<tr><td class="denom">πz</td></tr></table> + +<p class="noind">The mean brightness varies as z<span class="sp">-1</span>; and the integral +∫<span class="sp1">∞</span><span class="su1">0</span> J<span class="su">0</span>²(z)z dz +is not convergent.</p> + +<p>5. <i>Resolving Power of Telescopes.</i>—The efficiency of a telescope +is of course intimately connected with the size of the disk +by which it represents a mathematical point. In estimating +theoretically the resolving power on a double star we have to +consider the illumination of the field due to the superposition of +the two independent images. If the angular interval between the +components of a double star were equal to twice that expressed +in equation (15) above, the central disks of the diffraction patterns +would be just in contact. Under these conditions there is no +doubt that the star would appear to be fairly resolved, since the +brightness of its external ring system is too small to produce any +material confusion, unless indeed the components are of very +unequal magnitude. The diminution of the star disks with +increasing aperture was observed by Sir William Herschel, and in +1823 Fraunhofer formulated the law of inverse proportionality. +In investigations extending over a long series of years, the +advantage of a large aperture in separating the components of +close double stars was fully examined by W. R. Dawes.</p> + +<p>The resolving power of telescopes was investigated also by +J. B. L. Foucault, who employed a scale of equal bright and dark +alternate parts; it was found to be proportional to the aperture +and independent of the focal length. In telescopes of the best +construction and of moderate aperture the performance is not +sensibly prejudiced by outstanding aberration, and the limit +imposed by the finiteness of the waves of light is practically +reached. M. E. Verdet has compared Foucault’s results with +theory, and has drawn the conclusion that the radius of the +visible part of the image of a luminous point was equal to half the +radius of the first dark ring.</p> + +<p>The application, unaccountably long delayed, of this principle +to the microscope by H. L. F. Helmholtz in 1871 is the foundation +of the important doctrine of the <i>microscopic limit</i>. It is true that +in 1823 Fraunhofer, inspired by his observations upon gratings, +had very nearly hit the mark.<a name="fa3g" id="fa3g" href="#ft3g"><span class="sp">3</span></a> And a little before Helmholtz, +E. Abbe published a somewhat more complete investigation, also +founded upon the phenomena presented by gratings. But +although the argument from gratings is instructive and convenient +in some respects, its use has tended to obscure the essential unity +of the principle of the limit of resolution whether applied to +telescopes or microscopes.</p> + +<table class="nobctr" style="float: right; width: 340px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:288px; height:145px" src="images/img243.jpg" alt="" /></td></tr> +<tr><td class="caption sc">Fig. 4.</td></tr></table> + +<p>In fig. 4, AB represents the axis of an optical instrument (telescope +or microscope), A being a point of the object and B a point +of the image. By the operation of the object-glass LL′ all the rays +issuing from A arrive in the same phase at B. Thus if A be self-luminous, +the illumination is +a maximum at B, where all +the secondary waves agree in +phase. B is in fact the centre +of the diffraction disk which +constitutes the image of A. +At neighbouring points the +illumination is less, in consequence +of the discrepancies of +phase which there enter. In +like manner if we take a neighbouring +point P, also self-luminous, +in the plane of the object, the waves which issue from +it will arrive at B with phases no longer absolutely concordant, +and the discrepancy of phase will increase as the interval AP +<span class="pagenum"><a name="page244" id="page244"></a>244</span> +increases. When the interval is very small the discrepancy, +though mathematically existent, produces no practical effect; and +the illumination at B due to P is as important as that due to A, +the intensities of the two luminous sources being supposed equal. +Under these conditions it is clear that A and P are not separated in +the image. The question is to what amount must the distance AP +be increased in order that the difference of situation may make itself +felt in the image. This is necessarily a question of degree; but it +does not require detailed calculations in order to show that the +discrepancy first becomes conspicuous when the phases corresponding +to the various secondary waves which travel from P to B range over +a complete period. The illumination at B due to P then becomes +comparatively small, indeed for some forms of aperture evanescent. +The extreme discrepancy is that between the waves which travel +through the outermost parts of the object-glass at L and L′; 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 (λ). +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 α, where α is the angular semi-aperture L′AB. In like manner +PL − AL has the same value, so that</p> + +<p class="center">PL − PL′ = 2AP sin α.</p> + +<p>According to the standard adopted, the condition of resolution is +therefore that AP, or ε, should exceed ½λ/sin α. If ε be less than this, +the images overlap too much; while if ε greatly exceed the above +value the images become unnecessarily separated.</p> + +<p>In the above argument the whole space between the object and +the lens is supposed to be occupied by matter of one refractive index, +and λ represents the wave-length <i>in this medium</i> of the kind of light +employed. If the restriction as to uniformity be violated, what we +have ultimately to deal with is the wave-length in the medium +immediately surrounding the object.</p> + +<p>Calling the refractive index μ, we have as the critical value of ε,</p> + +<p class="center">ε = ½λ<span class="su">0</span> /μ sin α,     (1),</p> + +<p class="noind">λ<span class="su">0</span> being the wave-length <i>in vacuo</i>. The denominator μ sin α is the +quantity well known (after Abbe) as the “numerical aperture.”</p> + +<p>The extreme value possible for α is a right angle, so that for the +microscopic limit we have</p> + +<p class="center">ε = ½λ<span class="su">0</span>/μ     (2).</p> + +<p class="noind">The limit can be depressed only by a diminution in λ<span class="su">0</span>, such as +photography makes possible, or by an increase in μ, the refractive +index of the medium in which the object is situated.</p> + +<p>The statement of the law of resolving power has been made in a +form appropriate to the microscope, but it admits also of immediate +application to the telescope. If 2R be the diameter of the object-glass +and D the distance of the object, the angle subtended by AP is +ε/D, and the angular resolving power is given by</p> + +<p class="center">λ/2D sin α = λ/2R     (3).</p> + +<p class="noind">This method of derivation (substantially due to Helmholtz) makes +it obvious that there is no essential difference of principle between +the two cases, although the results are conveniently stated in different +forms. In the case of the telescope we have to deal with a linear +measure of aperture and an angular limit of resolution, whereas in +the case of the microscope the limit of resolution is linear, and it is +expressed in terms of angular aperture.</p> + +<p>It must be understood that the above argument distinctly assumes +that the different parts of the object are self-luminous, or at least +that the light proceeding from the various points is without phase +relations. As has been emphasized by G. J. Stoney, the restriction +is often, perhaps usually, violated in the microscope. A different +treatment is then necessary, and for some of the problems which +arise under this head the method of Abbe is convenient.</p> + +<p>The importance of the general conclusions above formulated, as +imposing a limit upon our powers of direct observation, can hardly +be overestimated; but there has been in some quarters a tendency +to ascribe to it a more precise character than it can bear, or even to +mistake its meaning altogether. A few words of further explanation +may therefore be desirable. The first point to be emphasized is that +nothing whatever is said as to the smallness of a single object that +may be made visible. The eye, unaided or armed with a telescope, +is able to see, as points of light, stars subtending no sensible angle. +The visibility of a star is a question of brightness simply, and has +nothing to do with resolving power. The latter element enters only +when it is a question of recognizing the duplicity of a double star, +or of distinguishing detail upon the surface of a planet. So in the +microscope there is nothing except lack of light to hinder the visibility +of an object however small. But if its dimensions be much +less than the half wave-length, it can only be seen as a whole, and its +parts cannot be distinctly separated, although in cases near the border +line some inference may be possible, founded upon experience of what +appearances are presented in various cases. Interesting observations +upon particles, <i>ultra-microscopic</i> in the above sense, have been +recorded by H. F. W. Siedentopf and R. A. Zsigmondy (<i>Drude’s Ann.</i>, +1903, 10, p. 1).</p> + +<p>In a somewhat similar way a dark linear interruption in a bright +ground may be visible, although its actual width is much inferior +to the half wave-length. In illustration of this fact a simple experiment +may be mentioned. In front of the naked eye was held a piece +of copper foil perforated by a fine needle hole. Observed through +this the structure of some wire gauze just disappeared at a distance +from the eye equal to 17 in., the gauze containing 46 meshes to +the inch. On the other hand, a single wire 0.034 in. in diameter +remained fairly visible up to a distance of 20 ft. The ratio between +the limiting angles subtended by the periodic structure of the gauze +and the diameter of the wire was (.022/.034) × (240/17) = 9.1. For +further information upon this subject reference may be made to +<i>Phil. Mag.</i>, 1896, 42, p. 167; <i>Journ. R. Micr. Soc.</i>, 1903, p. 447.</p> + +<p>6. <i>Coronas or Glories.</i>—The results of the theory of the diffraction +patterns due to circular apertures admit of an interesting +application to <i>coronas</i>, such as are often seen encircling the sun +and moon. They are due to the interposition of small spherules +of water, which act the part of diffracting obstacles. In order to +the formation of a well-defined corona it is essential that the +particles be exclusively, or preponderatingly, of one size.</p> + +<p>If the origin of light be treated as infinitely small, and be seen +in focus, whether with the naked eye or with the aid of a telescope, +the whole of the light in the absence of obstacles would be concentrated +in the immediate neighbourhood of the focus. At other +parts of the field the effect is the same, in accordance with the +principle known as Babinet’s, whether the imaginary screen in front +of the object-glass is generally transparent but studded with a number +of opaque circular disks, or is generally opaque but perforated with +corresponding apertures. Since at these points the resultant due to +the whole aperture is zero, any two portions into which the whole +may be divided must give equal and opposite resultants. Consider +now the light diffracted in a direction many times more oblique than +any with which we should be concerned, were the whole aperture +uninterrupted, and take first the effect of a single small aperture. +The light in the proposed direction is that determined by the size of +the small aperture in accordance with the laws already investigated, +and its phase depends upon the position of the aperture. If we take +a direction such that the light (of given wave-length) from a single +aperture vanishes, the evanescence continues even when the whole +series of apertures is brought into contemplation. Hence, whatever +else may happen, there must be a system of dark rings formed, +the same as from a single small aperture. In directions other than +these it is a more delicate question how the partial effects should be +compounded. If we make the extreme suppositions of an infinitely +small source and absolutely homogeneous light, there is no escape +from the conclusion that the light in a definite direction is arbitrary, +that is, dependent upon the chance distribution of apertures. If, +however, as in practice, the light be heterogeneous, the source of +finite area, the obstacles in motion, and the discrimination of different +directions imperfect, we are concerned merely with the mean brightness +found by varying the arbitrary phase-relations, and this is +obtained by simply multiplying the brightness due to a single +aperture by the number of apertures (n) (see <span class="sc"><a href="#artlinks">Interference of +Light</a></span>, § 4). The diffraction pattern is therefore that due to a single +aperture, merely brightened n times.</p> + +<p>In his experiments upon this subject Fraunhofer employed plates +of glass dusted over with lycopodium, or studded with small metallic +disks of uniform size; and he found that the diameters of the rings +were proportional to the length of the waves and inversely as the +diameter of the disks.</p> + +<p>In another respect the observations of Fraunhofer appear at +first sight to be in disaccord with theory; for his measures of the +diameters of the red rings, visible when white light was employed, +correspond with the law applicable to dark rings, and not to the +different law applicable to the luminous maxima. Verdet has, +however, pointed out that the observation in this form is essentially +different from that in which homogeneous red light is employed, +and that the position of the red rings would correspond to the +<i>absence</i> of blue-green light rather than to the greatest abundance of +red light. Verdet’s own observations, conducted with great care, +fully confirm this view, and exhibit a complete agreement with +theory.</p> + +<p>By measurements of coronas it is possible to infer the size of the +particles to which they are due, an application of considerable +interest in the case of natural coronas—the general rule being the +larger the corona the smaller the water spherules. Young employed +this method not only to determine the diameters of cloud particles +(<i>e.g.</i> <span class="spp">1</span>⁄<span class="suu">1000</span> in.), but also those of fibrous material, for which the +theory is analogous. His instrument was called the <i>eriometer</i> +(see “Chromatics,” vol. iii. of supp. to <i>Ency. Brit.</i>, 1817).</p> + +<p>7. <i>Influence of Aberration. Optical Power of Instruments.</i>—Our +investigations and estimates of resolving power have thus +far proceeded upon the supposition that there are no optical +imperfections, whether of the nature of a regular aberration or +dependent upon irregularities of material and workmanship. In +<span class="pagenum"><a name="page245" id="page245"></a>245</span> +practice there will always be a certain aberration or error of phase, +which we may also regard as the deviation of the actual wave-surface +from its intended position. In general, we may say that +aberration is unimportant when it nowhere (or at any rate over a +relatively small area only) exceeds a small fraction of the wave-length +(λ). 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 ¼λ can have but little effect.</p> + +<p>The only case in which the influence of small aberration upon +the entire image has been calculated (<i>Phil. Mag.</i>, 1879) is that of a +rectangular aperture, traversed by a cylindrical wave with aberration +equal to cx³. The aberration is here unsymmetrical, the wave being +in advance of its proper place in one half of the aperture, but behind +in the other half. No terms in x or x² need be considered. The +first would correspond to a general turning of the beam; and the +second would imply imperfect focusing of the central parts. The +effect of aberration may be considered in two ways. We may +suppose the aperture (a) constant, and inquire into the operation +of an increasing aberration; or we may take a given value of c (<i>i.e.</i> +a given wave-surface) and examine the effect of a varying aperture. +The results in the second case show that an increase of aperture +up to that corresponding to an extreme aberration of half a period +has no ill effect upon the central band (§ 3), but it increases unduly +the intensity of one of the neighbouring lateral bands; and the +practical conclusion is that the best results will be obtained from an +aperture giving an extreme aberration of from a quarter to half a +period, and that with an increased aperture aberration is not so +much a direct cause of deterioration as an obstacle to the attainment +of that improved definition which should accompany the increase +of aperture.</p> + +<p>If, on the other hand, we suppose the aperture given, we find +that aberration begins to be distinctly mischievous when it amounts +to about a quarter period, <i>i.e.</i> when the wave-surface deviates at +each end by a quarter wave-length from the true plane.</p> + +<p>As an application of this result, let us investigate what amount +of temperature disturbance in the tube of a telescope may be expected +to impair definition. According to J. B. Biot and F. J. D. +Arago, the index μ for air at t° C. and at atmospheric pressure is given +by</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">μ − 1 =</td> <td>.00029</td> <td rowspan="2">.</td></tr> +<tr><td class="denom">1 + .0037 t</td></tr></table> + +<p class="noind">If we take 0° C. as standard temperature,</p> + +<p class="center"> δμ = -1.1 × 10<span class="sp">-6</span>.</p> + +<p class="noind">Thus, on the supposition that the irregularity of temperature t +extends through a length l, and produces an acceleration of a quarter +of a wave-length,</p> + +<p class="center">¼λ = 1.1 lt × 10<span class="sp">-6</span>;</p> + +<p class="noind">or, if we take λ = 5.3 × 10<span class="sp">-5</span>,</p> + +<p class="center">lt = 12,</p> + +<p class="noind">the unit of length being the centimetre.</p> + +<p>We may infer that, in the case of a telescope tube 12 cm. long, +a stratum of air heated 1° C. lying along the top of the tube, and +occupying a moderate fraction of the whole volume, would produce +a not insensible effect. If the change of temperature progressed +uniformly from one side to the other, the result would be a lateral +displacement of the image without loss of definition; but in general +both effects would be observable. In longer tubes a similar disturbance +would be caused by a proportionally less difference of +temperature. S. P. Langley has proposed to obviate such ill-effects +by stirring the air included within a telescope tube. It has long been +known that the definition of a carbon bisulphide prism may be much +improved by a vigorous shaking.</p> + +<p>We will now consider the application of the principle to the +formation of images, unassisted by reflection or refraction (<i>Phil. Mag.</i>, +1881). The function of a lens in forming an image is to compensate +by its variable thickness the differences of phase which would otherwise +exist between secondary waves arriving at the focal point from +various parts of the aperture. If we suppose the diameter of the +lens to be given (2R), and its focal length ƒ gradually to increase, the +original differences of phase at the image of an infinitely distant +luminous point diminish without limit. When ƒ attains a certain +value, say ƒ<span class="su">1</span>, the extreme error of phase to be compensated falls +to ¼λ. But, as we have seen, such an error of phase causes no sensible +deterioration in the definition; so that from this point onwards +the lens is useless, as only improving an image already sensibly as +perfect as the aperture admits of. Throughout the operation of +increasing the focal length, the resolving power of the instrument, +which depends only upon the aperture, remains unchanged; and +we thus arrive at the rather startling conclusion that a telescope +of any degree of resolving power might be constructed without an +object-glass, if only there were no limit to the admissible focal length. +This last proviso, however, as we shall see, takes away almost all +practical importance from the proposition.</p> + +<p>To get an idea of the magnitudes of the quantities involved, let us +take the case of an aperture of <span class="spp">1</span>⁄<span class="suu">5</span> in., about that of the pupil of the +eye. The distance ƒ<span class="su">1</span>, which the actual focal length must exceed, is +given by</p> + +<p class="center">√ (ƒ<span class="su">1</span>² + R²) − ƒ<span class="su">1</span> = ¼λ;</p> + +<p class="noind">so that</p> + +<p class="center">ƒ<span class="su">1</span> = 2R²/λ     (1).</p> + +<p class="noind">Thus, if λ = <span class="spp">1</span>⁄<span class="suu">40000</span>, R = <span class="spp">1</span>⁄<span class="suu">10</span>, we find</p> + +<p class="center">ƒ<span class="su">1</span> = 800 inches.</p> + +<p>The image of the sun thrown upon a screen at a distance exceeding +66 ft., through a hole <span class="spp">1</span>⁄<span class="suu">5</span> in. in diameter, is therefore at least as well +defined as that seen direct.</p> + +<p>As the minimum focal length increases with the square of the +aperture, a quite impracticable distance would be required to rival +the resolving power of a modern telescope. Even for an aperture of +4 in., ƒ<span class="su">1</span> would have to be 5 miles.</p> + +<p>A similar argument may be applied to find at what point an +achromatic lens becomes sensibly superior to a single one. The +question is whether, when the adjustment of focus is correct for the +central rays of the spectrum, the error of phase for the most extreme +rays (which it is necessary to consider) amounts to a quarter of a +wave-length. If not, the substitution of an achromatic lens will be +of no advantage. Calculation shows that, if the aperture be <span class="spp">1</span>⁄<span class="suu">5</span> in., +an achromatic lens has no sensible advantage if the focal length +be greater than about 11 in. If we suppose the focal length to be +66 ft., a single lens is practically perfect up to an aperture of 1.7 in.</p> + +<p>Another obvious inference from the necessary imperfection of +optical images is the uselessness of attempting anything like an +absolute destruction of spherical aberration. An admissible error +of phase of ¼λ will correspond to an error of <span class="spp">1</span>⁄<span class="suu">8</span>λ in a reflecting and ½λ +in a (glass) refracting surface, the incidence in both cases being +perpendicular. If we inquire what is the greatest admissible longitudinal +aberration (δƒ) in an object-glass according to the above +rule, we find</p> + +<p class="center">δƒ = λα<span class="sp">-2</span>     (2),</p> + +<p class="noind">α being the angular semi-aperture.</p> + +<p>In the case of a single lens of glass with the most favourable curvatures, +δƒ is about equal to ᲃ, so that α<span class="sp">4</span> must not exceed λ/ƒ. For +a lens of 3 ft. focus this condition is satisfied if the aperture does +not exceed 2 in.</p> + +<p>When parallel rays fall directly upon a spherical mirror the +longitudinal aberration is only about one-eighth as great as for the +most favourably shaped single lens of equal focal length and aperture. +Hence a spherical mirror of 3 ft. focus might have an +aperture of 2½ in., and the image would not suffer materially from +aberration.</p> + +<p>On the same principle we may estimate the least visible displacement +of the eye-piece of a telescope focused upon a distant object, +a question of interest in connexion with range-finders. It appears +(<i>Phil. Mag.</i>, 1885, 20, p. 354) that a displacement δf from the true focus +will not sensibly impair definition, provided</p> + +<p class="center">δƒ < ƒ²λ/R²     (3),</p> + +<p class="noind">2R being the diameter of aperture. The linear accuracy required +is thus a function of the <i>ratio</i> of aperture to focal length. The +formula agrees well with experiment.</p> + +<p>The principle gives an instantaneous solution of the question of +the ultimate optical efficiency in the method of “mirror-reading,” +as commonly practised in various physical observations. A rotation +by which one edge of the mirror advances ¼λ (while the other edge +retreats to a like amount) introduces a phase-discrepancy of a whole +period where before the rotation there was complete agreement. A +rotation of this amount should therefore be easily visible, but the +limits of resolving power are being approached; and the conclusion +is independent of the focal length of the mirror, and of the employment +of a telescope, provided of course that the reflected image is +seen in focus, and that the full width of the mirror is utilized.</p> + +<table class="nobctr" style="float: right; width: 270px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:220px; height:101px" src="images/img245.jpg" alt="" /></td></tr> +<tr><td class="caption sc">Fig. 5.</td></tr></table> + +<p>A comparison with the method of a material pointer, attached to +the parts whose rotation is under observation, and viewed through +a microscope, is of interest. The +limiting efficiency of the microscope +is attained when the angular aperture +amounts to 180°; and it is evident +that a lateral displacement of the point +under observation through ½λ entails +(at the old image) a phase-discrepancy +of a whole period, one extreme ray +being accelerated and the other retarded +by half that amount. We may infer that the limits of +efficiency in the two methods are the same when the length of the +pointer is equal to the width of the mirror.</p> + +<p>We have seen that in perpendicular reflection a surface error not +exceeding <span class="spp">1</span>⁄<span class="suu">8</span>λ may be admissible. In the case of oblique reflection +at an angle φ, the error of retardation due to an elevation BD (fig. 5) +is</p> + +<p class="center"> +QQ′ − QS = BD sec φ(1 − cos SQQ′) = BD sec φ (1 + cos 2φ) = 2BD cos φ;</p> + +<p><span class="pagenum"><a name="page246" id="page246"></a>246</span></p> + +<p class="noind">from which it follows that an error of given magnitude in the figure +of a surface is less important in oblique than in perpendicular +reflection. It must, however, be borne in mind that errors can +sometimes be compensated by altering adjustments. If a surface +intended to be flat is affected with a slight general curvature, a +remedy may be found in an alteration of focus, and the remedy is +the less complete as the reflection is more oblique.</p> + +<p>The formula expressing the optical power of prismatic spectroscopes +may readily be investigated upon the principles of the wave +theory. Let A<span class="su">0</span>B<span class="su">0</span> be a plane wave-surface of the light before it falls +upon the prisms, AB the corresponding wave-surface for a particular +part of the spectrum after the light has passed the prisms, or after it +has passed the eye-piece of the observing telescope. The path of a +ray from the wave-surface A<span class="su">0</span>B<span class="su">0</span> to A or B is determined by the condition +that the optical distance, ∫ μ ds, is a minimum; and, as AB +is by supposition a wave-surface, this optical distance is the same +for both points. Thus</p> + +<p class="center"><span class="f150">∫</span> μ ds (for A) = <span class="f150">∫</span> μ ds (for B)     (4).</p> + +<p class="noind">We have now to consider the behaviour of light belonging to a +neighbouring part of the spectrum. The path of a ray from the +wave-surface A<span class="su">0</span>B<span class="su">0</span> to the point A is changed; but in virtue of the +minimum property the change may be neglected in calculating the +optical distance, as it influences the result by quantities of the second +order only in the changes of refrangibility. Accordingly, the optical +distance from A<span class="su">0</span>B<span class="su">0</span> to A is represented by ∫(μ + δμ)ds, the integration +being along the original path A<span class="su">0</span> ... A; and similarly the optical +distance between A<span class="su">0</span>B<span class="su">0</span> and B is represented by ∫ (μ + δμ)ds, the +integration being along B<span class="su">0</span> ... B. In virtue of (4) the difference +of the optical distances to A and B is</p> + +<p class="center"><span class="f150">∫</span> δμ ds (along B<span class="su">0</span> ... B) − <span class="f150">∫</span> δμ ds (along A<span class="su">0</span> ... A)     (5).</p> + +<p class="noind">The new wave-surface is formed in such a position that the optical +distance is constant; and therefore the <i>dispersion</i>, or the angle +through which the wave-surface is turned by the change of refrangibility, +is found simply by dividing (5) by the distance AB. If, as +in common flint-glass spectroscopes, there is only one dispersing +substance, ∫ δμ ds = δμ·s, where s is simply the thickness traversed +by the ray. If t<span class="su">2</span> and t<span class="su">1</span> be the thicknesses traversed by the extreme +rays, and a denote the width of the emergent beam, the dispersion +θ is given by</p> + +<p class="center">θ = δμ(t<span class="su">2</span> − t<span class="su">1</span>)/a,</p> + +<p class="noind">or, if t<span class="su">1</span> be negligible,</p> + +<p class="center">θ = δμt/a     (6).</p> + +<p class="noind">The condition of resolution of a double line whose components +subtend an angle θ is that θ must exceed λ/a. Hence, in order +that a double line may be resolved whose components have indices +μ and μ + δμ, it is necessary that t should exceed the value given +by the following equation:—</p> + +<p class="center">t = λ/δμ     (7).</p> + +<p>8. <i>Diffraction Gratings.</i>—Under the heading “Colours of +Striated Surfaces,” Thomas Young (<i>Phil. Trans.</i>, 1802) in his +usual summary fashion gave a general explanation of these +colours, including the law of sines, the striations being supposed +to be straight, parallel and equidistant. Later, in his article +“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.</p> + +<p>The theory was further developed by A. J. Fresnel (1815), who +gave a formula equivalent to (5) below. But it is to J. von +Fraunhofer that we owe most of our knowledge upon this subject. +His recent discovery of the “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 <i>reflected</i>. To eliminate the light returned +from the hinder surface of an engraved grating, he covered it with +a black varnish. It then appeared that under certain angles of +incidence parts of the resulting spectra were <i>completely polarized</i>. +These remarkable researches of Fraunhofer, carried out in the +years 1817-1823, are republished in his <i>Collected Writings</i> +(Munich, 1888).</p> + +<p>The principle underlying the action of gratings is identical with +that discussed in § 2, and exemplified in J. L. Soret’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, <i>e.g.</i> 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 +<i>Tracts</i>, to Verdet’s <i>Leçons</i>, or to R. W. Wood’s <i>Physical Optics</i>. If, +however, we assume the theory of a simple rectangular aperture +(§ 3); the results of the ruling can be inferred by elementary methods, +which are perhaps more instructive.</p> + +<p>Apart from the ruling, we know that the image of a mathematical +line will be a series of narrow bands, of which the central one is +by far the brightest. At the middle of this band there is complete +agreement of phase among the secondary waves. The dark lines +which separate the bands are the places at which the phases of the +secondary wave range over an integral number of periods. If now +we suppose the aperture AB to be covered by a great number of +opaque strips or bars of width d, separated by transparent intervals +of width a, the condition of things in the directions just spoken of +is not materially changed. At the central point there is still complete +agreement of phase; but the amplitude is diminished in the ratio of +a : a + d. In another direction, making a small angle with the last, +such that the projection of AB upon it amounts to a few wave-lengths, +it is easy to see that the mode of interference is the same as +if there were no ruling. For example, when the direction is such that +the projection of AB upon it amounts to one wave-length, the +elementary components neutralize one another, because their phases +are distributed symmetrically, though discontinuously, round the +entire period. The only effect of the ruling is to diminish the +amplitude in the ratio a : a + d; and, except for the difference in +illumination, the appearance of a line of light is the same as if the +aperture were perfectly free.</p> + +<p>The lateral (spectral) images occur in such directions that the +projection of the element (a + d) of the grating upon them is an exact +multiple of λ. 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 mnλ, 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)λ. +In considering the relative brightnesses of the different spectra, it +is therefore sufficient to attend merely to the principal directions, +provided that the whole deviation be not so great that its cosine +differs considerably from unity.</p> + +<p>We have now to consider the amplitude due to a single element, +which we may conveniently regard as composed of a transparent +part a bounded by two opaque parts of width ½d. The phase of +the resultant effect is by symmetry that of the component which +comes from the middle of a. The fact that the other components +have phases differing from this by amounts ranging between +± amπ/(a + d) causes the resultant amplitude to be less than for +the central image (where there is complete phase agreement). +<span class="pagenum"><a name="page247" id="page247"></a>247</span> +If B<span class="su">m</span> denote the brightness of the m<span class="sp">th</span> lateral image, and B<span class="su">0</span> that +of the central image, we have</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">B<span class="su">m</span> : B <span class="su">0</span> = <span class="f150">[∫</span></td> <td class="bk">+ amπ/(a+d)</td> +<td rowspan="2">cosx dx ÷</td> <td>2amπ</td> +<td rowspan="2"><span class="f150">]</span></td> <td>²</td> +<td rowspan="2">= <span class="f150">(</span></td> <td>a + d</td> +<td rowspan="2"><span class="f150">)</span></td> <td>²</td> +<td rowspan="2">sin²</td> <td>amπ</td> +<td rowspan="2">     (1).</td></tr> +<tr><td class="bk">− amπ/(a+d)</td> <td class="denom">a + d</td> + <td> </td> <td class="denom">amπ</td> + <td> </td> <td class="denom">a + d</td></tr></table> + +<p class="noind">If B denotes the brightness of the central image when the whole of +the space occupied by the grating is transparent, we have</p> + +<p class="center">B<span class="su">0</span> : B = a² : (a + d)²,</p> + +<p class="noind">and thus</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">B<span class="su">m</span> : B =</td> <td>1</td> +<td rowspan="2">sin²</td> <td>amπ</td> <td rowspan="2">     (2).</td></tr> +<tr><td class="denom">m²π²</td> <td class="denom">a + d</td></tr></table> + +<p>The sine of an angle can never be greater than unity; and consequently +under the most favourable circumstances only 1/m²π² of +the original light can be obtained in the m<span class="sp">th</span> spectrum. We conclude +that, with a grating composed of transparent and opaque +parts, the utmost light obtainable in any one spectrum is in the first, +and there amounts to 1/π², or about <span class="spp">1</span>⁄<span class="suu">10</span>, and that for this purpose +a and d must be equal. When d = a the general formula becomes</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">B<span class="su">m</span> : B =</td> <td>sin² ½mπ</td> +<td rowspan="2">     (3),</td></tr> +<tr><td class="denom">m²π²</td></tr></table> + +<p class="noind">showing that, when m is even, B<span class="su">m</span> vanishes, and that, when m is odd,</p> + +<p class="center">B<span class="su">m</span> : B = 1/m²π².</p> + +<p class="noind">The third spectrum has thus only <span class="spp">1</span>⁄<span class="suu">9</span> of the brilliancy of the first.</p> + +<p>Another particular case of interest is obtained by supposing a +small relatively to (a + d). Unless the spectrum be of very high +order, we have simply</p> + +<p class="center">B<span class="su">m</span> : B = {a/(a + d)}²     (4);</p> + +<p class="noind">so that the brightnesses of all the spectra are the same.</p> + +<p>The light stopped by the opaque parts of the grating, together +with that distributed in the central image and lateral spectra, ought +to make up the brightness that would be found in the central image, +were all the apertures transparent. Thus, if a = d, we should have</p> + +<p class="center">1 = ½ + ¼ + 2/π² (1 + <span class="spp">1</span>⁄<span class="suu">9</span> + <span class="spp">1</span>⁄<span class="suu">25</span> + ...),</p> + +<p class="noind">which is true by a known theorem. In the general case</p> + +<table class="math0" summary="math"> +<tr><td>a</td> <td rowspan="2">= <span class="f150">(</span></td> <td>a</td> + <td rowspan="2"><span class="f150">)</span></td> <td>²</td> + <td rowspan="2">+</td> <td>2</td> + <td rowspan="2"><span class="f150">Σ</span></td> <td class="bk">m=∞</td> <td>1</td> + <td rowspan="2">sin²<span class="f150">(</span></td> <td>mπa</td> + <td rowspan="2"><span class="f150">)</span>,</td></tr> +<tr><td class="denom">a + d</td> <td class="denom">a + d</td> + <td> </td> <td class="denom">π²</td> + <td class="bk">m=1</td> <td class="denom">m²</td> <td class="denom">a + d</td></tr></table> + +<p class="noind">a formula which may be verified by Fourier’s theorem.</p> + +<p>According to a general principle formulated by J. Babinet, the +brightness of a lateral spectrum is not affected by an interchange +of the transparent and opaque parts of the grating. The vibrations +corresponding to the two parts are precisely antagonistic, since if +both were operative the resultant would be zero. So far as the +application to gratings is concerned, the same conclusion may be +derived from (2).</p> + +<table class="nobctr" style="float: left; width: 190px;" summary="Illustration"> +<tr><td class="figleft1"><img style="width:136px; height:178px" src="images/img247a.jpg" alt="" /></td></tr> +<tr><td class="caption sc">Fig. 6.</td></tr></table> + +<p>From the value of B<span class="su">m</span> : B<span class="su">0</span> we see that no lateral spectrum can +surpass the central image in brightness; but this result depends upon +the hypothesis that the ruling acts by opacity, which is generally +very far from being the case in practice. In an engraved glass +grating there is no opaque material present by which light could be +absorbed, and the effect depends upon a difference of retardation in +passing the alternate parts. It is possible to prepare gratings which +give a lateral spectrum brighter than the central image, and the explanation +is easy. For if the alternate parts were equal and alike +transparent, but so constituted as to give a relative retardation of +½λ, it is evident that the central image would be entirely extinguished, +while the first spectrum would be four times as bright as if the +alternate parts were opaque. If it were possible to introduce at +every part of the aperture of the grating an arbitrary retardation, +all the light might be concentrated in any desired spectrum. By +supposing the retardation to vary uniformly and continuously we +fall upon the case of an ordinary prism: but there +is then no diffraction spectrum in the usual sense. +To obtain such it would be necessary that the +retardation should gradually alter by a wave-length +in passing over any element of the grating, +and then fall back to its previous value, thus +springing suddenly over a wave-length (<i>Phil. +Mag.</i>, 1874, 47, p. 193). It is not likely that such +a result will ever be fully attained in practice; but +the case is worth stating, in order to show that +there is no theoretical limit to the concentration +of light of assigned wave-length in one spectrum, +and as illustrating the frequently observed unsymmetrical +character of the spectra on the two sides of the central +image.<a name="fa4g" id="fa4g" href="#ft4g"><span class="sp">4</span></a></p> + +<p>We have hitherto supposed that the light is incident perpendicularly +upon the grating; but the theory is easily extended. If +the incident rays make an angle θ with the normal (fig. 6), and the +diffracted rays make an angle φ (upon the same side), the relative +retardation from each element of width (a + d) to the next is +(a + d) (sinθ + sinφ); and this is the quantity which is to be equated +to mλ. Thus</p> + +<p class="center">sinθ + sinφ = 2sin ½(θ + φ) cos ½ (θ − φ) = mλ/(a + d)     (5).</p> + +<p>The “deviation” is (θ + φ), and is therefore a minimum when +θ = φ, <i>i.e.</i> when the grating is so situated that the angles of incidence +and diffraction are equal.</p> + +<table class="nobctr" style="float: right; width: 220px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:165px; height:102px" src="images/img247b.jpg" alt="" /></td></tr> +<tr><td class="caption sc">Fig. 7.</td></tr></table> + +<p>In the case of a reflection grating the same method applies. If +θ and φ 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 (θ + φ), and is a minimum +when θ = φ, that is, when the diffracted +rays return upon the course of the incident +rays.</p> + +<p>In either case (as also with a prism) the +position of minimum deviation leaves the +width of the beam unaltered, <i>i.e.</i> neither magnifies nor diminishes the +angular width of the object under view.</p> + +<p>From (5) we see that, when the light falls perpendicularly upon +a grating (θ = 0), there is no spectrum formed (the image corresponding +to m = 0 not being counted as a spectrum), if the grating +interval σ or (a + d) is less than λ. 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 λ; and the inference +is plausible, to say the least (<i>Phil. Mag.</i>, 1886). Fraunhofer +should, however, have fixed the microscopic limit at ½λ, as appears +from (5), when we suppose θ = ½π, φ = ½π.</p> + +<table class="nobctr" style="float: right; width: 210px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:163px; height:114px" src="images/img247c.jpg" alt="" /></td></tr> +<tr><td class="caption sc">Fig. 8.</td></tr></table> + +<p>We will now consider the important subject of the resolving +power of gratings, as dependent upon the +number of lines (n) and the order of the spectrum +observed (m). Let BP (fig. 8) be the +direction of the principal maximum (middle +of central band) for the wave-length λ in the +m<span class="sp">th</span> spectrum. Then the relative retardation +of the extreme rays (corresponding to the +edges A, B of the grating) is mnλ. 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)λ. +Suppose now that λ + δλ is the wave-length for which BQ gives the +principal maximum, then</p> + +<p class="center">(mn + 1)λ = mn(λ + δλ);</p> + +<p class="noind">whence</p> + +<p class="center">δλ/λ = 1/mn     (6).</p> + +<p class="noind">According to our former standard, this gives the smallest difference +of wave-lengths in a double line which can be just resolved; and +we conclude that the resolving power of a grating depends only +upon the total number of lines, and upon the order of the spectrum, +without regard to any other considerations. It is here of course +assumed that the n lines are really utilized.</p> + +<p>In the case of the D lines the value of δλ/λ is about 1/1000; so +that to resolve this double line in the first spectrum requires 1000 +lines, in the second spectrum 500, and so on.</p> + +<p>It is especially to be noticed that the resolving power does not +depend directly upon the closeness of the ruling. Let us take the +case of a grating 1 in. broad, and containing 1000 lines, and consider +the effect of interpolating an additional 1000 lines, so as to bisect +the former intervals. There will be destruction by interference of +the first, third and odd spectra generally; while the advantage +gained in the spectra of even order is not in dispersion, nor in +resolving power, but simply in brilliancy, which is increased four +times. If we now suppose half the grating cut away, so as to leave +1000 lines in half an inch, the dispersion will not be altered, while the +brightness and resolving power are halved.</p> + +<p>There is clearly no theoretical limit to the resolving power of +gratings, even in spectra of given order. But it is possible that, +as suggested by Rowland,<a name="fa5g" id="fa5g" href="#ft5g"><span class="sp">5</span></a> the structure of natural spectra may +be too coarse to give opportunity for resolving powers much higher +than those now in use. However this may be, it would always +be possible, with the aid of a grating of given resolving power, to +construct artificially from white light mixtures of slightly different +wave-length whose resolution or otherwise would discriminate +between powers inferior and superior to the given one.<a name="fa6g" id="fa6g" href="#ft6g"><span class="sp">6</span></a></p> + +<p><span class="pagenum"><a name="page248" id="page248"></a>248</span></p> + +<p>If we define as the “dispersion” in a particular part of the +spectrum the ratio of the angular interval dθ to the corresponding +increment of wave-length dλ, 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 +mndλ. Hence, if a be the width of the diffracted beam, and dθ the +angle through which the wave-front is turned,</p> + +<p class="center">adθ = mn dλ,</p> + +<p class="noind">or</p> + +<p class="center">dispersion = mn/a     (7).</p> + +<p>The resolving power and the width of the emergent beam fix +the optical character of the instrument. The latter element must +eventually be decreased until less than the diameter of the pupil +of the eye. Hence a wide beam demands treatment with further +apparatus (usually a telescope) of high magnifying power.</p> + +<p>In the above discussion it has been supposed that the ruling is +accurate, and we have seen that by increase of m a high resolving +power is attainable with a moderate number of lines. But this +procedure (apart from the question of illumination) is open to the +objection that it makes excessive demands upon accuracy. According +to the principle already laid down it can make but little difference +in the principal direction corresponding to the first spectrum, +provided each line lie within a quarter of an interval (a + d) from its +theoretical position. But, to obtain an equally good result in the +m<span class="sp">th</span> spectrum, the error must be less than 1/m of the above amount.<a name="fa7g" id="fa7g" href="#ft7g"><span class="sp">7</span></a></p> + +<p>There are certain errors of a systematic character which demand +special consideration. The spacing is usually effected by means of +a screw, to each revolution of which corresponds a large number +(<i>e.g.</i> one hundred) of lines. In this way it may happen that although +there is almost perfect periodicity with each revolution of the screw +after (say) 100 lines, yet the 100 lines themselves are not equally +spaced. The “ghosts” thus arising were first described by G. H. +Quincke (<i>Pogg. Ann.</i>, 1872, 146, p. 1), and have been elaborately +investigated by C. S. Peirce (<i>Ann. Journ. Math.</i>, 1879, 2, p. 330), both +theoretically and experimentally. The general nature of the effects +to be expected in such a case may be made clear by means of an illustration +already employed for another purpose. Suppose two similar +and accurately ruled transparent gratings to be superposed in such +a manner that the lines are parallel. If the one set of lines exactly +bisect the intervals between the others, the grating interval is +practically halved, and the previously existing spectra of odd order +vanish. But a very slight relative displacement will cause the +apparition of the odd spectra. In this case there is approximate +periodicity in the half interval, but complete periodicity only after +the whole interval. The advantage of approximate bisection lies +in the superior brilliancy of the surviving spectra; but in any case +the compound grating may be considered to be perfect in the +longer interval, and the definition is as good as if the bisection were +accurate.</p> + +<table class="nobctr" summary="Illustration"> +<tr><td class="figcenter" colspan="4"><img style="width:473px; height:87px" src="images/img248a.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 9.</span>—x².</td> <td class="caption"><span class="sc">Fig. 10.</span>—y².</td> +<td class="caption"><span class="sc">Fig. 11.</span>—x³.</td> <td class="caption"><span class="sc">Fig. 12.</span>—xy².</td></tr></table> + +<table class="nobctr" style="float: left; width: 400px;" summary="Illustration"> +<tr><td class="figleft1"><img style="width:347px; height:82px" src="images/img248b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 13.</span>—xy.       <span class="sc">Fig. 14.</span>—x²y.       <span class="sc">Fig. 15.</span>—y³.</td></tr></table> + +<p>The effect of a gradual increase in the interval (fig. 9) as we +pass across the grating has been investigated by M. A. Cornu +(<i>C.R.</i>, 1875, 80, p. 655), who thus explains an anomaly observed by +E. E. N. Mascart. The latter found that certain gratings exercised +a converging power upon the spectra formed upon one side, and a +corresponding diverging power upon the spectra on the other side. +Let us suppose that the light is incident perpendicularly, and that +the grating interval increases from the centre towards that edge +which lies nearest to the spectrum under observation, and decreases +towards the hinder edge. It is evident that the waves from <i>both</i> +halves of the +grating are accelerated +in an +increasing degree, +as we pass from +the centre outwards, +as compared +with the +phase they would possess were the central value of the grating +interval maintained throughout. The irregularity of spacing has +thus the effect of a convex lens, which accelerates the marginal +relatively to the central rays. On the other side the effect is +reversed. This kind of irregularity may clearly be present in a +degree surpassing the usual limits, without loss of definition, when +the telescope is focused so as to secure the best effect.</p> + +<p>It may be worth while to examine further the other variations +from correct ruling which correspond to the various terms expressing +the deviation of the wave-surface from a perfect plane. If x and y +be co-ordinates in the plane of the wave-surface, the axis of y being +parallel to the lines of the grating, and the origin corresponding +to the centre of the beam, we may take as an approximate equation +to the wave-surface</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">z =</td> <td>x²</td> +<td rowspan="2">+ Bxy +</td> <td>y²</td> +<td rowspan="2">+ αx³ + βx²y + γxy² + δy³ + ...     (8);</td></tr> +<tr><td class="denom">2ρ</td> <td class="denom">2ρ′</td></tr></table> + +<p class="noind">and, as we have just seen, the term in x² corresponds to a linear +error in the spacing. In like manner, the term in y² corresponds +to a general <i>curvature</i> of the lines (fig. 10), and does not influence +the definition at the (primary) focus, although it may introduce +astigmatism.<a name="fa8g" id="fa8g" href="#ft8g"><span class="sp">8</span></a> If we suppose that everything is symmetrical on +the two sides of the primary plane y = 0, the coefficients B, β, δ +vanish. In spite of any inequality between ρ and ρ’, the definition +will be good to this order of approximation, provided α and γ vanish. +The former measures the <i>thickness</i> of the primary focal line, and the +latter measures its <i>curvature</i>. The error of ruling giving rise to α is +one in which the intervals increase or decrease in <i>both</i> directions +from the centre outwards (fig. 11), and it may often be compensated +by a slight rotation in azimuth of the object-glass of the observing +telescope. The term in γ corresponds to a <i>variation</i> of curvature +in crossing the grating (fig. 12).</p> + +<p>When the plane zx is not a plane of symmetry, we have to consider +the terms in xy, x²y, and y³. The first of these corresponds to a deviation +from parallelism, causing the interval to alter gradually as we pass +<i>along</i> the lines (fig. 13). The error thus arising may be compensated +by a rotation of the object-glass about one of the diameters y = ± x. +The term in x²y corresponds to a deviation from parallelism in the +same direction on both sides of the central line (fig. 14); and that in +y³ would be caused by a curvature such that there is a point of +inflection at the middle of each line (fig. 15).</p> + +<p>All the errors, except that depending on α, and especially those +depending on γ and δ, can be diminished, without loss of resolving +power, by contracting the <i>vertical</i> aperture. A linear error in the +spacing, and a general curvature of the lines, are eliminated in the +ordinary use of a grating.</p> + +<p>The explanation of the difference of focus upon the two sides as +due to unequal spacing was verified by Cornu upon gratings purposely +constructed with an increasing interval. He has also shown how to +rule a plane surface with lines so disposed that the grating shall of +itself give well-focused spectra.</p> + +<table class="nobctr" style="float: right; width: 230px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:184px; height:190px" src="images/img248c.jpg" alt="" /></td></tr> +<tr><td class="caption1 sc">Fig. 16.</td></tr></table> + +<p>A similar idea appears to have guided H. A. Rowland to his +brilliant invention of concave gratings, by +which spectra can be photographed without +any further optical appliance. In these +instruments the lines are ruled upon a +spherical surface of speculum metal, and +mark the intersections of the surface by a +system of parallel and equidistant planes, +of which the middle member passes through +the centre of the sphere. If we consider for +the present only the primary plane of symmetry, +the figure is reduced to two dimensions. +Let AP (fig. 16) represent the surface +of the grating, O being the centre of the +circle. Then, if Q be any radiant point and +Q’ its image (primary focus) in the spherical mirror AP, we have</p> + +<table class="math0" summary="math"> +<tr><td>1</td> <td rowspan="2">+</td> <td>1</td> +<td rowspan="2">=</td> <td>2cosφ</td> <td rowspan="2">,</td></tr> +<tr><td class="denom">v<span class="su">1</span></td> <td class="denom">u</td> <td class="denom">a</td></tr></table> + +<p class="noind">where v<span class="su">1</span> = AQ′, u = AQ, a = OA, φ = 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 φ, 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.</p> + +<p>This disposition is adopted in Rowland′s instrument; only, in +addition to the central image formed at the angle φ′ = φ, there are +a series of spectra with various values of φ’, 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 (<i>Phil. Mag.</i>, 1883).</p> + +<p>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 = φ, AOP = ω,</p> + +<p class="center">QP² = u² + 4a²sin²½ω − 4au sin ½ω sin (½ω − φ)<br /> += (u + a sin φ sin ω)² − a² sin²φ sin²ω + 4a sin² ½ω(a − u cosφ).</p> + +<p><span class="pagenum"><a name="page249" id="page249"></a>249</span></p> + +<p class="noind">Now as far as ω<span class="sp">4</span></p> + +<p class="center">4 sin² ½ω = sin²ω + ¼sin<span class="sp">4</span>ω,</p> + +<p class="noind">and thus to the same order</p> + +<p class="center">QP² = (u + a sin φ sin ω)²<br /> +− a cos φ(u − a cos φ) sin²ω + ¼ a(a − u cos φ) sin<span class="sp">4</span> ω.</p> + +<p class="noind">pose that Q lies on the circle u = a cos φ, the +middle term vanishes, and we get, correct as far as ω<span class="sp">4</span>,</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">QP = (u + a sin φ sin ω) <span class="f150">√ {</span>1 +</td> <td>a² sin² φ sin<span class="sp">4</span>ω</td> +<td rowspan="2"><span class="f150">}</span>;</td></tr> +<tr><td class="denom">4u</td></tr></table> + +<p class="noind">so that</p> + +<p class="center">QP − u = a sin φ sin ω + <span class="spp">1</span>⁄<span class="suu">8</span>a sin φ tan φ sin<span class="sp">4</span> ω     (9),</p> + +<p class="noind">in which it is to be noticed that the adjustment necessary to secure +the disappearance of sin²ω is sufficient also to destroy the term in +sin³ω.</p> + +<p>A similar expression can be found for Q’P − Q′A; and thus, if +Q′A = v, Q′AO = φ′, where v = a cos φ′, we get</p> + +<p class="center">QP + PQ′ − QA -AQ′ = a sin ω (sin φ − sin φ′)<br /> ++ <span class="spp">1</span>⁄<span class="suu">8</span>a sin<span class="sp">4</span> ω (sin φ tan φ + sin φ′ tan φ′)     (10).</p> + +<p class="noind">If φ′ = φ, the term of the first order vanishes, and the reduction of +the difference of path <i>via</i> P and <i>via</i> A to a term of the fourth order +proves not only that Q and Q′ are conjugate foci, but also that the +foci are exempt from the most important term in the aberration. +In the present application φ′ is not necessarily equal to φ; 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 ± mλ (m integral), and therefore +without influence, provided</p> + +<p class="center">σ (sin φ − sinφ′) = ± mλ     (11),</p> + +<p class="noind">where σ 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.</p> + +<p>In order that a large part of the field of view may be in focus at +once, it is desirable that the locus of the focused spectrum should +be nearly perpendicular to the line of vision. For this purpose +Rowland places the eye-piece at O, so that φ = 0, and then by (11) +the value of φ′ in the m<span class="sp">th</span> spectrum is</p> + +<p class="center">σ sin φ’ = ± mλ     (12).</p> + +<p>If ω now relate to the edge of the grating, on which there are +altogether n lines,</p> + +<p class="center">nσ = 2a sin ω,</p> + +<p class="noind">and the value of the last term in (10) becomes</p> + +<p class="center"><span class="spp">1</span>⁄<span class="suu">16 </span>nσsin³ ω sin φ′ tan φ′,</p> + +<p class="noind">or</p> + +<p class="center"><span class="spp">1</span>⁄<span class="suu">16 </span>mnλ sin³ω tan φ′     (13).</p> + +<p>This expresses the retardation of the extreme relatively to the +central ray, and is to be reckoned positive, whatever may be the +signs of ω, and φ′. If the semi-angular aperture (ω) be <span class="spp">1</span>⁄<span class="suu">100</span>, and +tan φ′ = 1, mn might be as great as four millions before the error of +phase would reach ¼λ. 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 σ slightly variable towards the +edges. Or, retaining σ constant, we might attain compensation by so +polishing the surface as to bring the circumference slightly forward +in comparison with the position it would occupy upon a true sphere.</p> + +<p>It may be remarked that these calculations apply to the rays in +the primary plane only. The image is greatly affected with astigmatism; +but this is of little consequence, if γ 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 γ is in fact small. Its value does not appear +to have been calculated. The other coefficients in (8) vanish in +virtue of the symmetry.</p> + +<p>The mechanical arrangements for maintaining the focus are of +great simplicity. The grating at A and the eye-piece at O are +rigidly attached to a bar AO, whose ends rest on carriages, moving +on rails OQ, AQ at right angles to each other. A tie between the +middle point of the rod OA and Q can be used if thought desirable.</p> + +<p>The absence of chromatic aberration gives a great advantage in +the comparison of overlapping spectra, which Rowland has turned +to excellent account in his determinations of the relative wave-lengths +of lines in the solar spectrum (<i>Phil. Mag.</i>, 1887).</p> + +<p>For absolute determinations of wave-lengths plane gratings are +used. It is found (Bell, <i>Phil. Mag.</i>, 1887) that the angular +measurements present less difficulty than the comparison of the +grating interval with the standard metre. There is also some +uncertainty as to the actual temperature of the grating when in +use. In order to minimize the heating action of the light, it might +be submitted to a preliminary prismatic analysis before it reaches +the slit of the spectrometer, after the manner of Helmholtz.</p> + +<p>In spite of the many improvements introduced by Rowland and +of the care with which his observations were made, recent workers +have come to the conclusion that errors of unexpected amount +have crept into his measurements of wave-lengths, and there is +even a disposition to discard the grating altogether for fundamental +work in favour of the so-called “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.</p> + +<p>Since the time of Fraunhofer many skilled mechanicians have +given their attention to the ruling of gratings. Those of Nobert +were employed by A. J. Ångström in his celebrated researches +upon wave-lengths. L. M. Rutherfurd introduced into common +use the reflection grating, finding that speculum metal was less +trying than glass to the diamond point, upon the permanence of +which so much depends. In Rowland’s dividing engine the +screws were prepared by a special process devised by him, and +the resulting gratings, plane and concave, have supplied the +means for much of the best modern optical work. It would +seem, however, that further improvements are not excluded.</p> + +<p>There are various copying processes by which it is possible +to reproduce an original ruling in more or less perfection. The +earliest is that of Quincke, who coated a glass grating with a +chemical silver deposit, subsequently thickened with copper in +an electrolytic bath. The metallic plate thus produced formed, +when stripped from its support, a reflection grating reproducing +many of the characteristics of the original. It is best to commence +the electrolytic thickening in a silver acetate bath. At +the present time excellent reproductions of Rowland’s speculum +gratings are on the market (Thorp, Ives, Wallace), prepared, after +a suggestion of Sir David Brewster, by coating the original with a +varnish, <i>e.g.</i> of celluloid. Much skill is required to secure that +the film when stripped shall remain undeformed.</p> + +<p>A much easier method, applicable to glass originals, is that +of photographic reproduction by contact printing. In several +papers dating from 1872, Lord Rayleigh (see <i>Collected Papers</i>, +i. 157, 160, 199, 504; iv. 226) has shown that success may +be attained by a variety of processes, including bichromated +gelatin and the old bitumen process, and has investigated the +effect of imperfect approximation during the exposure between the +prepared plate and the original. For many purposes the copies, +containing lines up to 10,000 to the inch, are not inferior. It is +to be desired that transparent gratings should be obtained from +first-class ruling machines. To save the diamond point it might +be possible to use something softer than ordinary glass as the +material of the plate.</p> + +<p>9. <i>Talbot’s Bands.</i>—These very remarkable bands are seen +under certain conditions when a tolerably pure spectrum is regarded +with the naked eye, or with a telescope, <i>half the aperture +being covered by a thin plate</i>, <i>e.g.</i> <i>of glass or mica</i>. The view of the +matter taken by the discoverer (<i>Phil. Mag.</i>, 1837, 10, p. 364) was +that any ray which suffered in traversing the plate a retardation +of an odd number of half wave-lengths would be extinguished, +and that thus the spectrum would be seen interrupted by a +number of dark bars. But this explanation cannot be accepted as +it stands, being open to the same objection as Arago’s theory of +stellar scintillation.<a name="fa9g" id="fa9g" href="#ft9g"><span class="sp">9</span></a> It is as far as possible from being true that +a body emitting homogeneous light would disappear on merely +covering half the aperture of vision with a half-wave plate. +Such a conclusion would be in the face of the principle of energy, +which teaches plainly that the retardation in question leaves +the aggregate brightness unaltered. The actual formation of +<span class="pagenum"><a name="page250" id="page250"></a>250</span> +the bands comes about in a very curious way, as is shown by a +circumstance first observed by Brewster. When the retarding +plate is held on the side towards the red of the spectrum, <i>the bands +are not seen</i>. Even in the contrary case, the thickness of the plate +must not exceed a certain limit, dependent upon the purity of +the spectrum. A satisfactory explanation of these bands was first +given by Airy (<i>Phil. Trans.</i>, 1840, 225; 1841, 1), but we shall here +follow the investigation of Sir G. G. Stokes (<i>Phil. Trans.</i>, 1848, +227), limiting ourselves, however, to the case where the retarded +and unretarded beams are contiguous and of equal width.</p> + +<p>The aperture of the unretarded beam may thus be taken to be +limited by x = -h, x = 0, y = -l, y= +l; and that of the beam retarded +by R to be given by x = 0, x = h, y= -l, y = +l. For the +former (1) § 3 gives</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">−</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span></td> <td class="bk">0</td> +<td rowspan="2"><span class="f150">∫</span></td> <td class="bk">+l</td> +<td rowspan="2">sin k<span class="f150">{</span> at − ƒ +</td> <td>xξ + yη</td> +<td rowspan="2"><span class="f150">}</span>dxdy</td></tr> +<tr><td class="denom">λƒ</td> <td class="bk">−h</td> +<td class="bk">−l</td> <td class="denom">ƒ</td></tr></table> + +<table class="math0" summary="math"> +<tr><td rowspan="2">= −</td> <td>2lh</td> +<td rowspan="2">·</td> <td>ƒ</td> +<td rowspan="2">sin</td> <td>kηl</td> +<td rowspan="2">·</td> <td>2ƒ</td> +<td rowspan="2">sin</td> <td>kξh</td> +<td rowspan="2">· sin k<span class="f150">{</span>at − ƒ −</td> <td>ξh</td> +<td rowspan="2"><span class="f150">}</span>     (1),</td></tr> +<tr><td class="denom">λƒ</td> <td class="denom">kηl</td> +<td class="denom">ƒ</td> <td class="denom">kξh</td> +<td class="denom">2ƒ</td> <td class="denom">2ƒ</td></tr></table> + +<p class="noind">on integration and reduction.</p> + +<p>For the retarded stream the only difference is that we must subtract +R from at, and that the limits of x are 0 and +h. We thus +get for the disturbance at ξ, η, due to this stream</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">−</td> <td>2lh</td> +<td rowspan="2">·</td> <td>ƒ</td> +<td rowspan="2">sin</td> <td>kηl</td> +<td rowspan="2">·</td> <td>2ƒ</td> +<td rowspan="2">sin</td> <td>kξh</td> +<td rowspan="2">· sin k<span class="f150">{</span>at − ƒ − R +</td> <td>ξh</td> +<td rowspan="2"><span class="f150">}</span>     (2)</td></tr> +<tr><td class="denom">λƒ</td> <td class="denom">kηl</td> +<td class="denom">ƒ</td> <td class="denom">kξh</td> +<td class="denom">2ƒ</td> <td class="denom">2ƒ</td></tr></table> + +<p class="noind">If we put for shortness π for the quantity under the last circular +function in (1), the expressions (1), (2) may be put under the forms +u sin τ, v sin (τ − α) respectively; and, if I be the intensity, I will be +measured by the sum of the squares of the coefficients of sin τ and +cos τ in the expression</p> + +<p class="center">u sin τ + v sin (τ − α),</p> + +<p class="noind">so that</p> + +<p class="center">I = u² + v² + 2uv cos α,</p> + +<p class="noind">which becomes on putting for u, v, and α their values, and putting</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">{</span></td> <td>ƒ</td> +<td rowspan="2">sin</td> <td>kηl</td> +<td rowspan="2"><span class="f150">}</span></td> <td>²</td> +<td rowspan="2">= Q     (3),</td></tr> +<tr><td class="denom">kηl</td> <td class="denom">ƒ</td> +<td> </td></tr></table> + +<table class="math0" summary="math"> +<tr><td rowspan="2">I = Q ·</td> <td>4l²</td> +<td rowspan="2">sin²</td> <td>πξh</td> +<td rowspan="2"><span class="f150">{</span>2 + 2 cos<span class="f150">(</span></td> <td>2πR</td> +<td rowspan="2">−</td> <td>2πξh</td> +<td rowspan="2"><span class="f150">)}</span>     (4).</td></tr> +<tr><td class="denom">π²ξ²</td> <td class="denom">λƒ</td> +<td class="denom">λ</td> <td class="denom">λƒ</td></tr></table> + +<p class="noind">If the subject of examination be a luminous line parallel to η, we +shall obtain what we require by integrating (4) with respect to η +from −∞ to +∞. The constant multiplier is of no especial interest +so that we may take as applicable to the image of a line</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">I =</td> <td>2</td> +<td rowspan="2">sin²</td> <td>πξh</td> +<td rowspan="2"><span class="f150">{</span>1 + cos<span class="f150">(</span></td> <td>2πR</td> +<td rowspan="2">−</td> <td>2πξh</td> +<td rowspan="2"><span class="f150">)}</span>     (5).</td></tr> +<tr><td class="denom">ξ²</td> <td class="denom">λƒ</td> +<td class="denom">λ</td> <td class="denom">λƒ</td></tr></table> + +<p class="noind">If R = ½λ, I vanishes at ξ= 0; but the whole illumination, represented +by ∫<span class="sp1">+∞</span><span class="su1">−∞</span> I dξ, is independent of the value of R. If R = 0, +I = (1/ξ²) sin² (2πξh/λƒ), in agreement with § 3, where a has the meaning +here attached to 2h.</p> + +<p>The expression (5) gives the illumination at ξ due to that part +of the complete image whose geometrical focus is at ξ = 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. ξ 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 ξ from −∞ to +∞. To each value of ξ corresponds +a different value of λ, and (in consequence of the dispersing power +of the plate) of R. The variation of λ may, however, be neglected +in the integration, except in 2πR/λ, where a small variation of λ +entails a comparatively large alteration of phase. If we write</p> + +<p class="center">ρ = 2πR/λ     (6),</p> + +<p class="noind">we must regard ρ as a function of ξ, and we may take with sufficient +approximation under any ordinary circumstances</p> + +<p class="center">ρ = ρ′ + <span class="ov">ω</span>ξ     (7),</p> + +<p class="noind">where ρ′ denotes the value of ρ at O, and <span class="ov">ω</span> is a constant, which is +positive when the retarding plate is held at the side on which the +lue of the spectrum <i>is seen</i>. The possibility of dark bands depends +upon <span class="ov">ω</span> being positive. Only in this case can</p> + +<p class="center">cos {ρ′ + (<span class="ov">ω</span> − 2πh/λƒ) ξ}</p> + +<p class="noind">retain the constant value -1 throughout the integration, and then +only when</p> + +<p class="center"><span class="ov">ω</span> = 2πh / λƒ     (8)</p> + +<p class="noind">and</p> + +<p class="center">cos ρ′ = −1     (9).</p> + +<p class="noind">The first of these equations is the condition for the formation of +dark bands, and the second marks their situation, which is the +same as that determined by the imperfect theory.</p> + +<p>The integration can be effected without much difficulty. For +the first term in (5) the evaluation is effected at once by a known +formula. In the second term if we observe that</p> + +<p class="center">cos {ρ′ +(<span class="ov">ω</span> − 2πh/λƒ) ξ} = cos {ρ′ − g<span class="su">1</span>ξ}<br /> += cos ρ′ cos g<span class="su">1</span>ξ + sin ρ′ sin g<span class="su">1</span>ξ,</p> + +<p class="noind">we see that the second part vanishes when integrated, and that +the remaining integral is of the form</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">w = <span class="f150">∫</span></td> <td class="bk">+∞</td> +<td rowspan="2">sin² h<span class="su">1</span>ξ cos g<span class="su">1</span>ξ</td> +<td>dξ</td> <td rowspan="2">,</td></tr> +<tr><td class="bk">−∞</td> <td class="denom">ξ²</td></tr></table> + +<p class="noind">where</p> + +<p class="center">h<span class="su">1</span> = πh/λƒ,   g<span class="su">1</span> = ω − 2πh/λƒ     (10).</p> + +<p class="noind">By differentiation with respect to g<span class="su">1</span> it may be proved that</p> + +<table class="ws" summary="Contents"> + +<tr><td class="tcl">w = 0</td> <td class="tcl">from g<span class="su">1</span> = −∞</td> <td class="tcl">to g<span class="su">1</span> = −2h<span class="su">1</span>,</td></tr> +<tr><td class="tcl">w = ½π(2h<span class="su">1</span> + g<span class="su">1</span>)</td> <td class="tcl">from g<span class="su">1</span> = −2h<span class="su">1</span></td> <td class="tcl">to g<span class="su">1</span> = 0,</td></tr> +<tr><td class="tcl">w = ½π(2h<span class="su">1</span> − g<span class="su">1</span>)</td> <td class="tcl">from g<span class="su">1</span> = 0</td> <td class="tcl">to g<span class="su">1</span> = 2h<span class="su">1</span>,</td></tr> +<tr><td class="tcl">w = 0</td> <td class="tcl">from g<span class="su">1</span> = 2h<span class="su">1</span></td> <td class="tcl">to g<span class="su">1</span> = ∞.</td></tr> +</table> + +<p class="noind">The integrated intensity, I′, or</p> + +<p class="center">2πh<span class="su">1</span> + 2 cos ρw,</p> + +<p class="noind">is thus</p> + +<p class="center">I′ = 2πh<span class="su">1</span>     (11),</p> + +<p class="noind">when g<span class="su">1</span> numerically exceeds 2h<span class="su">1</span>; and, when g<span class="su">1</span> lies between ±2h<span class="su">1</span>,</p> + +<p class="center">I = π{2h<span class="su">1</span> + (2h<span class="su">1</span> − √ g<span class="su">1</span>²) cos ρ′}     (12).</p> + +<p>It appears therefore that there are no bands at all unless ω lies +between 0 and +4h<span class="su">1</span>, and that within these limits the best bands are +formed at the middle of the range when ω = 2h<span class="su">1</span>. The formation +of bands thus requires that the retarding plate be held upon the +side already specified, so that ω be positive; and that the thickness +of the plate (to which ω is proportional) do not exceed a certain +limit, which we may call 2T<span class="su">0</span>. At the best thickness T<span class="su">0</span> the bands +are black, and not otherwise.</p> + +<p>The linear width of the band (e) is the increment of ξ which alters +ρ by 2π, so that</p> + +<p class="center">e = 2π / <span class="ov">ω</span>     (13).</p> + +<p class="noind">With the best thickness</p> + +<p class="center"><span class="ov">ω</span> = 2πh/λƒ     (14),</p> + +<p class="noind">so that in this case</p> + +<p class="center">e = λƒ / h     (15).</p> + +<p class="noind">The bands are thus of the same width as those due to two infinitely +narrow apertures coincident with the central lines of the retarded +and unretarded streams, the subject of examination being itself a +fine luminous line.</p> + +<p>If it be desired to see a given number of bands in the whole or +in any part of the spectrum, the thickness of the retarding plate +is thereby determined, independently of all other considerations. +But in order that the bands may be really visible, and still more in +order that they may be black, another condition must be satisfied. +It is necessary that the aperture of the pupil be accommodated +to the angular extent of the spectrum, or reciprocally. Black +bands will be too fine to be well seen unless the aperture (2h) of +the pupil be somewhat contracted. One-twentieth to one-fiftieth +of an inch is suitable. The aperture and the number of bands being +both fixed, the condition of blackness determines the angular magnitude +of a band and of the spectrum. The use of a grating is very +convenient, for not only are there several spectra in view at the same +time, but the dispersion can be varied continuously by sloping the +grating. The slits may be cut out of tin-plate, and half covered by +mica or “microscopic glass,” held in position by a little cement.</p> + +<p>If a telescope be employed there is a distinction to be observed, +according as the half-covered aperture is between the eye and the +ocular, or in front of the object-glass. In the former case the +function of the telescope is simply to increase the dispersion, and +the formation of the bands is of course independent of the particular +manner in which the dispersion arises. If, however, the +half-covered aperture be in front of the object-glass, the phenomenon +is magnified as a whole, and the desirable relation between +the (unmagnified) dispersion and the aperture is the same as without +the telescope. There appears to be no further advantage in the +use of a telescope than the increased facility of accommodation, +and for this of course a very low power suffices.</p> + +<p>The original investigation of Stokes, here briefly sketched, extends +also to the case where the streams are of unequal width h, k, +and are separated by an interval 2g. In the case of unequal width +the bands cannot be black; but if h = k, the finiteness of 2g does +not preclude the formation of black bands.</p> + +<p>The theory of Talbot’s bands with a half-covered <i>circular</i> aperture +has been considered by H. Struve (<i>St Peters. Trans.</i>, 1883, 31, No. 1).</p> + +<p>The subject of “Talbot’s bands” has been treated in a very +instructive manner by A. Schuster (<i>Phil. Mag.</i>, 1904), whose point +of view offers the great advantage of affording an instantaneous +explanation of the peculiarity noticed by Brewster. A plane +<i>pulse</i>, <i>i.e.</i> a disturbance limited to an infinitely thin slice of the +medium, is supposed to fall upon a parallel grating, which again may +<span class="pagenum"><a name="page251" id="page251"></a>251</span> +be regarded as formed of infinitely thin wires, or infinitely narrow +lines traced upon glass. The secondary pulses diverted by the ruling +fall upon an object-glass as usual, and on arrival at the focus +constitute a procession equally spaced in time, the interval between +consecutive members depending upon the obliquity. If a retarding +plate be now inserted so as to operate upon the pulses which come +from one side of the grating, while leaving the remainder unaffected, +we have to consider what happens at the focal point chosen. A full +discussion would call for the formal application of Fourier’s theorem, +but some conclusions of importance are almost obvious.</p> + +<p>Previously to the introduction of the plate we have an effect +corresponding to wave-lengths closely grouped around the principal +wave-length, viz. σ sin φ, where σ is the grating-interval and φ 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 <i>interfere</i>. The formation of +black bands is thus explained, and it requires that the plate be +introduced upon one particular side, and that the amount of the +retardation be adjusted to a particular value. If the retardation +be too little, the overlapping of the processions is incomplete, so that +besides the procession of half period there are residues of the original +processions of full period. The same thing occurs if the retardation +be too great. If it exceed the double of the value necessary for +black bands, there is again no overlapping and consequently no +interference. If the plate be introduced upon the other side, so as +to retard the procession originally in arrear, there is no overlapping, +whatever may be the amount of retardation. In this way the +principal features of the phenomenon are accounted for, and +Schuster has shown further how to extend the results to spectra +having their origin in prisms instead of gratings.</p> + +<p>10. <i>Diffraction when the Source of Light is not seen in Focus.</i>—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.</p> + +<p>Fresnel commenced his researches with an examination of the +fringes, external and internal, which accompany the shadow of a +narrow opaque strip, such as a wire. As a source of light he used +sunshine passing through a very small hole perforated in a metal +plate, or condensed by a lens of short focus. In the absence of a +heliostat the latter was the more convenient. Following, unknown +to himself, in the footsteps of Young, he deduced the +principle of interference from the circumstance that the darkness +of the interior bands requires the co-operation of light from both +sides of the obstacle. At first, too, he followed Young in the view +that the exterior bands are the result of interference between the +direct light and that reflected from the edge of the obstacle, but +he soon discovered that the character of the edge—<i>e.g.</i> 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.</p> + +<table class="nobctr" style="float: right; width: 230px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:183px; height:125px" src="images/img251.jpg" alt="" /></td></tr> +<tr><td class="caption sc">Fig. 17.</td></tr></table> + +<p>In theoretical investigations these problems are usually treated +as of two dimensions only, everything being referred to the plane +passing through the luminous point and perpendicular to the diffracting +edges, supposed to be straight and parallel. In strictness this +idea is appropriate only when the source is a luminous line, emitting +cylindrical waves, such as might be obtained from a luminous point +with the aid of a cylindrical lens. When, in order to apply Huygens’s +principle, the wave is supposed to be broken up, the phase is the same +at every element of the surface of resolution which lies upon a line +perpendicular to the plane of reference, and +thus the effect of the whole line, or rather +infinitesimal strip, is related in a constant +manner to that of the element which lies +in the plane of reference, and may be +considered to be represented thereby. The +same method of representation is applicable +to spherical waves, issuing from a <i>point</i>, if +the radius of curvature be large; for, although +there is variation of phase along the +length of the infinitesimal strip, the whole effect depends practically +upon that of the central parts where the phase is sensibly constant.<a name="fa10g" id="fa10g" href="#ft10g"><span class="sp">10</span></a></p> + +<p>In fig. 17 APQ is the arc of the circle representative of the wave-front +of resolution, the centre being at O, and the radius QA being +equal to a. B is the point at which the effect is required, distant +a + b from O, so that AB = b, AP = s, PQ = ds.</p> + +<p>Taking as the standard phase that of the secondary wave from +A, we may represent the effect of PQ by</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">cos 2π<span class="f150">(</span></td> <td>t</td> +<td rowspan="2">−</td> <td>δ</td> +<td rowspan="2"><span class="f150">)</span>· ds,</td></tr> +<tr><td class="denom">r</td> <td class="denom">λ</td></tr></table> + +<p class="noind">where δ = BP − AP is the retardation at B of the wave from P +relatively to that from A.</p> + +<p class="noind">Now</p> + +<p class="center">δ = (a + b) s²/2ab     (1),</p> + +<p class="noind">so that, if we write</p> + +<table class="math0" summary="math"> +<tr><td>2πδ</td><td rowspan="2">=</td> <td>π(a + b)s²</td> +<td rowspan="2">=</td> <td>π</td> +<td rowspan="2">v²     (2),</td></tr> +<tr><td class="denom">λ</td> <td class="denom">abλ</td> <td class="denom">2</td></tr></table> + +<p class="noind">the effect at B is</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">{</span></td> <td>abλ</td> +<td rowspan="2"><span class="f150">}</span></td> <td>½</td> +<td rowspan="2"><span class="f150">{</span> cos</td> <td>2πt</td> +<td rowspan="2"><span class="f150">∫</span> cos ½πv²·dv + sin</td> <td>2πt</td> +<td rowspan="2"><span class="f150">∫</span> sin ½πv²·dv <span class="f150">}</span>     (3)</td></tr> +<tr><td class="denom">2(a + b)</td> <td> </td> +<td class="denom">τ</td> <td class="denom">τ</td></tr></table> + +<p class="noind">the limits of integration depending upon the disposition of the +diffracting edges. When a, b, λ 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.</p> + +<p>The intensity I², the quantity with which we are principally +concerned, may thus be expressed</p> + +<p class="center">I²= <span class="f150">{ ∫</span> cos ½πv²·dv<span class="f150">}</span>² + <span class="f150">{ ∫</span> sin ½πv²·dv <span class="f150">}</span>²     (4).</p> + +<p class="noind">These integrals, taken from v = 0, are known as Fresnel’s integrals; +we will denote them by C and S, so that</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">C = <span class="f150">∫</span></td> <td class="bk">v</td> +<td rowspan="2">cos ½πv²·dv,     S = <span class="f150">∫</span></td> <td class="bk">v</td> +<td rowspan="2">sin ½πv²·dv     (5).</td></tr> +<tr><td class="bk">0</td> <td class="bk">0</td></tr></table> + +<p class="noind">When the upper limit is infinity, so that the limits correspond to +the inclusion of half the primary wave, C and S are both equal to +½, by a known formula; and on account of the rapid fluctuation +of sign the parts of the range beyond very moderate values of v +contribute but little to the result.</p> + +<p>Ascending series for C and S were given by K. W. Knockenhauer, +and are readily investigated. Integrating by parts, we find</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">C + iS = <span class="f150">∫</span></td> <td class="bk">v</td> +<td rowspan="2">e</td> <td class="bk">i·½πv²</td> +<td rowspan="2">dv = e</td> <td class="bk">i·½πv²</td> +<td rowspan="2">· v − <span class="spp">1</span>⁄<span class="suu">3</span> iπ <span class="f150">∫</span></td> <td class="bk">v</td> +<td rowspan="2">e</td> <td class="bk">i·½πv²</td> +<td rowspan="2">dv³;</td></tr> +<tr><td class="bk">0</td> <td> </td> +<td> </td> <td class="bk">0</td> <td> </td></tr></table> + +<p class="noind">and, by continuing this process,</p> + +<table class="math0" style="border-collapse: separate;" summary="math"> +<tr><td rowspan="2">C + iS = e</td> <td class="bk">i·½πv²</td> +<td rowspan="2"><span class="f150">{</span> v −</td> <td>iπ</td> +<td rowspan="2">v³ +</td> <td>iπ</td> <td>iπ</td> +<td rowspan="2">v<span class="sp">5</span> −</td> <td>iπ</td> <td>iπ</td> <td>iπ</td> +<td rowspan="2">v<span class="sp">7</span> + ... <span class="f150">}</span>.</td></tr> +<tr><td> </td> <td class="denom">3</td> <td class="denom">3</td> <td class="denom">5</td> +<td class="denom">3</td> <td class="denom">5</td> <td class="denom">7</td></tr></table> + +<p class="noind">By separation of real and imaginary parts,</p> + +<table class="math0" summary="math"> +<tr><td>C = M cos ½πv² − N sin ½πv²</td> +<td rowspan="2"><span class="f200">}</span>     (6),</td></tr> +<tr><td>S = M sin ½πv² − N cos ½πv²</td></tr></table> + +<p class="noind">where</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">M =</td> <td>v</td> +<td rowspan="2">−</td> <td>π²v<span class="sp">5</span></td> +<td rowspan="2">+</td> <td>π<span class="sp">4</span>v<span class="sp">9</span></td> +<td rowspan="2">− ...     (7),</td></tr> +<tr><td class="denom">1</td> <td class="denom">3·5</td> <td class="denom">3·5·7·9</td></tr></table> + +<table class="math0" summary="math"> +<tr><td rowspan="2">N =</td> <td>πv³</td> +<td rowspan="2">−</td> <td>π<span class="sp">3</span>v<span class="sp">7</span></td> +<td rowspan="2">+</td> <td>π<span class="sp">5</span>v<span class="sp">11</span></td> +<td rowspan="2">...     (8).</td></tr> +<tr><td class="denom">1·3</td> <td class="denom">1·3·5·7</td> <td class="denom">1·3·5·7·9·11</td></tr></table> + +<p class="noind">These series are convergent for all values of v, but are practically +useful only when v is small.</p> + +<p>Expressions suitable for discussion when v is large were obtained +<span class="pagenum"><a name="page252" id="page252"></a>252</span> +by L. P. Gilbert (<i>Mem. cour. de l’Acad. de Bruxelles</i>, 31, p. 1). Taking</p> + +<p class="center">½πv² = u     (9),</p> + +<p class="noind">we may write</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">C + iS =</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span></td> <td class="bk">u</td> <td>e<span class="sp">iu</span>du</td> +<td rowspan="2">     (10).</td></tr> +<tr><td class="denom">√(2π)</td> <td class="bk">0</td> +<td class="denom">√u</td></tr></table> + +<p class="noind">Again, by a known formula,</p> + +<table class="math0" summary="math"> +<tr><td>1</td> <td rowspan="2">=</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span></td> <td class="bk">∞</td> <td>e<span class="sp">−ux</span>dx</td> +<td rowspan="2">     (11).</td></tr> +<tr><td class="denom">√ u</td> <td class="denom">√π</td> +<td class="bk">0</td> <td class="denom">√x</td></tr></table> + +<p class="noind">Substituting this in (10), and inverting the order of integration, we +get</p> + +<table class="math0" style="border-collapse: separate;" summary="math"> +<tr><td rowspan="2">C + iS =</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span></td> <td class="bk">∞</td> <td>dx</td> +<td rowspan="2"><span class="f150">∫</span></td> <td class="bk">u</td> +<td rowspan="2">e</td> <td class="bk">u(i − x)</td> +<td rowspan="2">dx =</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span></td> <td class="bk">∞</td> +<td>dx</td> <td>e<span class="sp">u(i − x)</span> − 1</td> +<td rowspan="2">     (12).</td></tr> +<tr><td class="denom">π√2</td> <td class="bk">0</td> +<td class="denom">√x</td> <td class="bk">0</td> +<td> </td> <td class="denom">π√2</td> +<td> </td> <td class="denom">√x</td> <td class="denom">i − x</td></tr></table> + +<p class="noind">Thus, if we take</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">G =</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span></td> <td class="bk">∞</td> <td>e<span class="sp">−ux</span> √x · dx</td> +<td rowspan="2">, H =</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span></td> <td class="bk">∞</td> <td>e<span class="sp">−ux</span> dx</td> +<td rowspan="2">     (13),</td></tr> +<tr><td class="denom">π√2</td> <td class="bk">0</td> +<td class="denom">1 + x²</td> <td class="denom">π√2</td> +<td class="bk">0</td> <td class="denom">√x · (1 + x²)</td></tr></table> + +<p class="center">C = ½ − G cos u + H sin u,   S = ½ − G sin u − H cos u     (14).</p> + +<p class="noind">The constant parts in (14), viz. ½, may be determined by direct +integration of (12), or from the observation that by their constitution +G and H vanish when u = ∞, coupled with the fact that C and +S then assume the value ½.</p> + +<p>Comparing the expressions for C, S in terms of M, N, and in terms +of G, H, we find that</p> + +<p class="center">G = ½ (cos u + sin u) − M,   H = ½ (cos u − sin u) + N         (15),</p> + +<p class="noind">formulae which may be utilized for the calculation of G, H when +u (or v) is small. For example, when u = 0, M = 0, N = 0, and consequently +G = H = ½.</p> + +<p>Descending series of the semi-convergent class, available for +numerical calculation when u is moderately large, can be obtained +from (12) by writing x = uy, and expanding the denominator in +powers of y. The integration of the several terms may then be +effected by the formula</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">∫</span></td> <td class="bk">∞</td> +<td rowspan="2">e<span class="sp">−y</span> y<span class="sp">q−½</span>dy = Γ(q + ½) = (q − ½)(q − <span class="spp">3</span>⁄<span class="suu">2</span>) ... ½ √π;</td></tr> +<tr><td class="bk">0</td></tr></table> + +<p class="noind">and we get in terms of v</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">G =</td> <td>1</td> +<td rowspan="2">−</td> <td>1·3·5</td> +<td rowspan="2">+</td> <td>1·3·5·9</td> +<td rowspan="2">− ...     (16),</td></tr> +<tr><td class="denom">π²v³</td> <td class="denom">π<span class="sp">4</span>v<span class="sp">7</span></td> <td class="denom">π<span class="sp">6</span>v<span class="sp">11</span></td></tr></table> + +<table class="math0" summary="math"> +<tr><td rowspan="2">H =</td> <td>1</td> +<td rowspan="2">−</td> <td>1·3</td> +<td rowspan="2">+</td> <td>1·3·5·7</td> +<td rowspan="2">− ...     (17).</td></tr> +<tr><td class="denom">πv</td> <td class="denom">π³v<span class="sp">5</span></td> <td class="denom">π<span class="sp">5</span>v<span class="sp">9</span></td></tr></table> + +<p class="noind">The corresponding values of C and S were originally derived by +A. L. Cauchy, without the use of Gilbert’s integrals, by direct +integration by parts.</p> + +<p>From the series for G and H just obtained it is easy to verify that</p> + +<table class="math0" summary="math"> +<tr><td>dH</td> <td rowspan="2">= − πvG,     </td> <td>dG</td> +<td rowspan="2">= πvH − 1     (18).</td></tr> +<tr><td class="denom">dv</td> <td class="denom">dv</td></tr></table> + +<p>We now proceed to consider more particularly the distribution of +light upon a screen PBQ near the shadow of a straight edge A. +At a point P within the geometrical shadow of the obstacle, the +half of the wave to the right of C (fig. 18), the nearest point on the +wave-front, is wholly intercepted, and on the left the integration +is to be taken from s = CA to s = ∞. If V be the value of v corresponding +to CA, viz.</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">V = <span class="f150">√{</span></td> <td>2(a + b)</td> +<td rowspan="2"><span class="f150">}</span> · CA,     (19),</td></tr> +<tr><td class="denom">abλ</td></tr></table> + +<p class="noind">we may write</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">I² = <span class="f150">( ∫</span></td> <td class="bk">∞</td> +<td rowspan="2">cos ½πv² · dv <span class="f150">)</span></td> <td>²</td> +<td rowspan="2">+ <span class="f150">( ∫</span></td> <td class="bk">∞</td> +<td rowspan="2">sin ½πv² · dv <span class="f150">)</span></td> <td>²</td> +<td rowspan="2">     (20),</td></tr> +<tr><td class="bk">v</td> <td> </td> +<td class="bk">v</td> <td> </td></tr></table> + +<p class="noind">or, according to our previous notation,</p> + +<p class="center">I²=(½ − C<span class="su">v</span>)² + (½ − S<span class="su">v</span>)² = G² + H²     (21).</p> + +<table class="nobctr" style="float: left; width: 230px;" summary="Illustration"> +<tr><td class="figleft1"><img style="width:183px; height:182px" src="images/img252.jpg" alt="" /></td></tr> +<tr><td class="caption sc">Fig. 18.</td></tr></table> + +<p>Now in the integrals represented by G and H every element +diminishes as V increases from zero. Hence, +as CA increases, viz. as the point P is more +and more deeply immersed in the shadow, +the illumination <i>continuously</i> decreases, and +that without limit. It has long been known +from observation that there are no bands +on the interior side of the shadow of the +edge.</p> + +<p>The law of diminution when V is moderately +large is easily expressed with the aid +of the series (16), (17) for G, H. We have +ultimately G = 0, H = (πV)<span class="sp">−1</span>, so that</p> + +<p class="center">I² = 1/π²V²,</p> + +<p class="noind">or the illumination is inversely as the square +of the distance from the shadow of the edge.</p> + +<p>For a point Q outside the shadow the integration extends over +<i>more</i> than half the primary wave. The intensity may be expressed by</p> + +<p class="center">I² = (½ + C<span class="su">v</span>)² + (½ + S<span class="su">v</span>)²     (22);</p> + +<p class="noind">and the maxima and minima occur when</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">(½ + C<span class="su">v</span>)</td> <td>dC</td> +<td rowspan="2">+ (½ + S<span class="su">v</span>)</td> <td>dS</td> +<td rowspan="2">= 0,</td></tr> +<tr><td class="denom">dV</td> <td class="denom">dV</td></tr></table> + +<p class="noind">whence</p> + +<p class="center">sin ½πV² + cos ½πV² = G     (23).</p> + +<p class="noind">When V = 0, viz. at the edge of the shadow, I² = ½; when V = ∞, +I² = 2, on the scale adopted. The latter is the intensity due to the +uninterrupted wave. The quadrupling of the intensity in passing +outwards from the edge of the shadow is, however, accompanied by +fluctuations giving rise to bright and dark bands. The position +of these bands determined by (23) may be very simply expressed +when V is large, for then sensibly G = 0, and</p> + +<p class="center">½πV² = ¾π + nπ     (24),</p> + +<p class="noind">n being an integer. In terms of δ, we have from (2)</p> + +<p class="center">δ = (<span class="spp">3</span>⁄<span class="suu">8</span> + ½n)λ     (25).</p> + +<p class="noind">The first maximum in fact occurs when δ = <span class="spp">3</span>⁄<span class="suu">8</span>λ − .0046λ, and the +first minimum when δ = <span class="spp">7</span>⁄<span class="suu">8</span>λ − .0016λ, the corrections being readily +obtainable from a table of G by substitution of the approximate +value of V.</p> + +<p>The position of Q corresponding to a given value of V, that is, +to a band of given order, is by (19)</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">BQ =</td> <td>a + b</td> +<td rowspan="2">AD = V <span class="f150">√ {</span></td> <td>bλ(a + b)</td> +<td rowspan="2"><span class="f150">}</span>     (26).</td></tr> +<tr><td class="denom">a</td> <td class="denom">2a</td></tr></table> + +<p class="noind">By means of this expression we may trace the locus of a band of +given order as b varies. With sufficient approximation we may +regard BQ and b as rectangular co-ordinates of Q. Denoting them +by x, y, so that AB is axis of y and a perpendicular through A the +axis of x, and rationalizing (26), we have</p> + +<p class="center">2ax² − V²λy² − V²aλy = 0,</p> + +<p class="noind">which represents a hyperbola with vertices at O and A.</p> + +<p>From (24), (26) we see that the width of the bands is of the order +√{bλ(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 ω be the apparent magnitude of the source seen from A, +ωb should be much smaller than the above quantity, or</p> + +<p class="center">ω < √{λ(a + b)/ab}     (27).</p> + +<p class="noind">If a be very great in relation to b, the condition becomes</p> + +<p class="center">ω < √(λ / b)     (28).</p> + +<p class="noind">so that if b is to be moderately great (1 metre), the apparent magnitude +of the sun must be greatly reduced before it can be used as a +source. The values of V for the maxima and minima of intensity, +and the magnitudes of the latter, were calculated by Fresnel. An +extract from his results is given in the accompanying table.</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcc allb"> </td> <td class="tcc allb">V</td> <td class="tcc allb">I²</td></tr> + +<tr><td class="tcl lb rb">First maximum</td> <td class="tcc rb">1.2172</td> <td class="tcc rb">2.7413</td></tr> +<tr><td class="tcl lb rb">First minimum</td> <td class="tcc rb">1.8726</td> <td class="tcc rb">1.5570</td></tr> +<tr><td class="tcl lb rb">Second maximum</td> <td class="tcc rb">2.3449</td> <td class="tcc rb">2.3990</td></tr> +<tr><td class="tcl lb rb">Second minimum </td> <td class="tcc rb">2.7392</td> <td class="tcc rb">1.6867</td></tr> +<tr><td class="tcl lb rb">Third maximum.</td> <td class="tcc rb">3.0820</td> <td class="tcc rb">2.3022</td></tr> +<tr><td class="tcl lb rb bb">Third minimum</td> <td class="tcc rb bb">3.3913</td> <td class="tcc rb bb">1.7440</td></tr> +</table> + +<p>A very thorough investigation of this and other related questions, +accompanied by fully worked-out tables of the functions concerned, +will be found in a paper by E. Lommel (<i>Abh. bayer. Akad. d. Wiss.</i> +II. CI., 15, Bd., iii. Abth., 1886).</p> + +<p>When the functions C and S have once been calculated, the +discussion of various diffraction problems is much facilitated by +the idea, due to M. A. Cornu (<i>Journ. de Phys.</i>, 1874, 3, p. 1; a similar +suggestion was made independently by G. F. Fitzgerald), of exhibiting +as a curve the relationship between C and S, considered as the +rectangular co-ordinates (x, y) of a point. Such a curve is shown in +fig. 19, where, according to the definition (5) of C, S,</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">x = <span class="f150">∫</span></td> <td class="bk">v</td> +<td rowspan="2">cos ½πv²·dv,   y = <span class="f150">∫</span></td> <td class="bk">v</td> +<td rowspan="2">sin ½πv²·dv     (29).</td></tr> +<tr><td class="bk">0</td> <td class="bk">0</td></tr></table> + +<p class="noind">The origin of co-ordinates O corresponds to v = 0; and the asymptotic +points J, J′, round which the curve revolves in an ever-closing spiral, +correspond to v = ±∞.</p> + +<p>The intrinsic equation, expressing the relation between the arc +σ (measured from O) and the inclination φ of the tangent at any +points to the axis of x, assumes a very simple form. For</p> + +<p class="center">dx = cos ½πv²·dv,   dy = sin ½πv²·dv;</p> + +<p class="noind">so that</p> + +<p class="center">σ = <span class="f150">∫</span> √(dx² + dy²) = v,     (30),</p> + +<p class="center">φ = tan<span class="sp">−1</span>(dy/dx) = ½πv²     (31).</p> + +<p><span class="pagenum"><a name="page253" id="page253"></a>253</span></p> + +<p class="noind">Accordingly,</p> + +<p class="center">φ = ½πσ²     (32);</p> + +<p class="noind">and for the curvature,</p> + +<p class="center">dφ / dσ = πσ     (33).</p> + +<table class="nobctr" style="float: left; width: 360px;" summary="Illustration"> +<tr><td class="figleft1"><img style="width:311px; height:290px" src="images/img253.jpg" alt="" /></td></tr> +<tr><td class="caption sc">Fig. 19.</td></tr></table> + +<p>Cornu remarks that this equation suffices to determine the general +character of the curve. For the osculating circle at any point +includes the whole of the +curve which lies beyond; +and the successive convolutions +envelop one another +without intersection.</p> + +<p>The utility of the curve +depends upon the fact that +the elements of arc represent, +in amplitude and +phase, the component vibrations +due to the corresponding +portions of the +primary wave-front. For +by (30) dσ = 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 2πδ/λ, +or φ. 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 (σ<span class="su">2</span> − σ<span class="su">1</span>).</p> + +<p>In applying the curve in special cases of diffraction to exhibit +the effect at any point P (fig. 18) the centre of the curve O is to be +considered to correspond to that point C of the primary wave-front +which lies nearest to P. The operative part, or parts, of the curve +are of course those which represent the unobstructed portions of +the primary wave.</p> + +<p>Let us reconsider, following Cornu, the diffraction of a screen +unlimited on one side, and on the other terminated by a straight +edge. On the illuminated side, at a distance from the shadow, the +vibration is represented by JJ′. The co-ordinates of J, J′ being +(½, ½), (−½, −½), I² is 2; and the phase is <span class="spp">1</span>⁄<span class="suu">8</span> period in arrear of +that of the element at O. As the point under contemplation is +supposed to approach the shadow, the vibration is represented by the +chord drawn from J to a point on the other half of the curve, which +travels inwards from J′ towards O. The amplitude is thus subject +to fluctuations, which increase as the shadow is approached. At +the point O the intensity is one-quarter of that of the entire wave, +and after this point is passed, that is, when we have entered the +geometrical shadow, the intensity falls off gradually to zero, <i>without +fluctuations</i>. The whole progress of the phenomenon is thus exhibited +to the eye in a very instructive manner.</p> + +<p>We will next suppose that the light is transmitted by a slit, and +inquire what is the effect of varying the width of the slit upon the +illumination at the projection of its centre. Under these circumstances +the arc to be considered is bisected at O, and its length is +proportional to the width of the slit. It is easy to see that the +length of the chord (which passes in all cases through O) increases +to a maximum near the place where the phase-retardation is <span class="spp">3</span>⁄<span class="suu">8</span> of +a period, then diminishes to a minimum when the retardation is +about <span class="spp">7</span>⁄<span class="suu">8</span> of a period, and so on.</p> + +<p>If the slit is of constant width and we require the illumination +at various points on the screen behind it, we must regard the arc +of the curve as of <i>constant length</i>. The intensity is then, as always, +represented by the square of the length of the chord. If the slit +be narrow, so that the arc is short, the intensity is constant over +a wide range, and does not fall off to an important extent until +the discrepancy of the extreme phases reaches about a quarter of a +period.</p> + +<p>We have hitherto supposed that the shadow of a diffracting +obstacle is received upon a diffusing screen, or, which comes to +nearly the same thing, is observed with an eye-piece. If the eye, +provided if necessary with a perforated plate in order to reduce the +aperture, be situated inside the shadow at a place where the illumination +is still sensible, and be focused upon the diffracting edge, the +light which it receives will appear to come from the neighbourhood +of the edge, and will present the effect of a silver lining. This is +doubtless the explanation of a “pretty optical phenomenon, seen +in Switzerland, when the sun rises from behind distant trees standing +on the summit of a mountain.”<a name="fa11g" id="fa11g" href="#ft11g"><span class="sp">11</span></a></p> + +<p>II. <i>Dynamical Theory of Diffraction.</i>—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” (<i>Camb. Phil. Trans.</i>, 1849) on the basis +of the elastic solid theory.</p> + +<p>Let x, y, z be the co-ordinates of any particle of the medium in +its natural state, and χ, η, ζ the displacements of the same particle +at the end of time t, measured in the directions of the three axes +respectively. Then the first of the equations of motion may be put +under the form</p> + +<table class="math0" summary="math"> +<tr><td>d²ξ</td> <td rowspan="2">= b² <span class="f150">(</span></td> <td>d²ξ</td> +<td rowspan="2">+</td> <td>d²ξ</td> +<td rowspan="2">+</td> <td>d²ξ</td> +<td rowspan="2"><span class="f150">)</span> + (a² − b²)</td> <td>d²</td> +<td rowspan="2"><span class="f150">(</span></td> <td>d²ξ</td> +<td rowspan="2">+</td> <td>d²η</td> +<td rowspan="2">+</td> <td>d²ζ</td> +<td rowspan="2"><span class="f150">)</span>,</td></tr> +<tr><td class="denom">dt²</td> <td class="denom">dx²</td> +<td class="denom">dy²</td> <td class="denom">dz²</td> +<td class="denom">dx</td> <td class="denom">dx²</td> +<td class="denom">dy²</td> <td class="denom">dz²</td></tr></table> + +<p class="noind">where a² and b² denote the two arbitrary constants. Put for shortness</p> + +<table class="math0" summary="math"> +<tr><td>d²ξ</td> <td rowspan="2">+</td> <td>d²η</td> +<td rowspan="2">+</td> <td>d²ζ</td> +<td rowspan="2">≈ δ  (1),</td></tr> +<tr><td class="denom">dx²</td> <td class="denom">dy²</td> <td class="denom">dz²</td></tr></table> + +<p class="noind">and represent by Δ²χ the quantity multiplied by b². According to +this notation, the three equations of motion are</p> + +<table class="math0" summary="math"> +<tr><td>d²ξ</td> <td rowspan="2">b²Δ²ξ + (a² − b²)</td> <td>dδ</td> +<td rowspan="6"><span style="font-size: 8em; font-family: 'Courier New'; color: #a0a0a0;">}</span>  (2).</td></tr> +<tr><td class="denom">dt²</td> <td class="denom">dx</td></tr> + +<tr><td>d²η</td> <td rowspan="2">b²Δ²η + (a² − b²)</td> <td>dδ</td></tr> +<tr><td class="denom">dt²</td> <td class="denom">dy</td></tr> + +<tr><td>d²ζ</td> <td rowspan="2">b²Δ²ζ + (a² − b²)</td> <td>dδ</td></tr> +<tr><td class="denom">dt²</td> <td class="denom">dz</td></tr></table> + +<p class="noind">It is to be observed that S denotes the dilatation of volume of the +element situated at (x, y, z). In the limiting case in which the +medium is regarded as absolutely incompressible δ vanishes; but, +in order that equations (2) may preserve their generality, we must +suppose a at the same time to become infinite, and replace a²δ by +a new function of the co-ordinates.</p> + +<p>These equations simplify very much in their application to plane +waves. If the ray be parallel to OX, and the direction of vibration +parallel to OZ, we have ξ = 0, η = 0, while ζ is a function of x and +t only. Equation (1) and the first pair of equations (2) are thus +satisfied identically. The third equation gives</p> + +<table class="math0" summary="math"> +<tr><td>d²ζ</td> <td rowspan="2">= b²</td> <td>d²ζ</td> +<td rowspan="2">  (3),</td></tr> +<tr><td class="denom">dt²</td> <td class="denom">dx²</td></tr></table> + +<p class="noind">of which the solution is</p> + +<p class="center">ζ = ƒ(bt − x)  (4),</p> + +<p class="noind">where ƒ is an arbitrary function.</p> + +<p>The question as to the law of the secondary waves is thus answered +by Stokes. “Let ξ = 0, η = 0, ζ = ƒ(bt − x) be the displacements +corresponding to the incident light; let O<span class="su">1</span> be any point in the plane +P (of the wave-front), dS an element of that plane adjacent to O<span class="su">1</span>, +and consider the disturbance due to that portion only of the incident +disturbance which passes continually across dS. Let O be any point +in the medium situated at a distance from the point O<span class="su">1</span> which is +large in comparison with the length of a wave; let O<span class="su">1</span>O = r, and let +this line make an angle θ with the direction of propagation of the +incident light, or the axis of x, and φ with the direction of vibration, +or axis of z. Then the displacement at O will take place in a direction +perpendicular to O<span class="su">1</span>O, and lying in the plane ZO<span class="su">1</span>O; and, if ζ′ be the +displacement at O, reckoned positive in the direction nearest to +that in which the incident vibrations are reckoned positive,</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ζ′ =</td> <td>dS</td> +<td rowspan="2">(1 + cos θ) sin φ ƒ′(bt − r).</td></tr> +<tr><td class="denom">4πr</td></tr></table> + +<p class="noind">In particular, if</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ƒ(bt − x) = c sin</td> <td>2π</td> +<td rowspan="2">(bt − x)  (5),</td></tr> +<tr><td class="denom">λ</td></tr></table> + +<p class="noind">we shall have</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ζ′ =</td> <td>cdS</td> +<td rowspan="2">(1 + cos θ) sin φcos</td> <td>2π</td> +<td rowspan="2">(bt − r)  (6).”</td></tr> +<tr><td class="denom">2λr</td> <td class="denom">λ</td></tr></table> + +<p class="noind">It is then verified that, after integration with respect to dS, (6) +gives the same disturbance as if the primary wave had been supposed +to pass on unbroken.</p> + +<p>The occurrence of sin φ as a factor in (6) shows that the relative +intensities of the primary light and of that diffracted in the direction +θ 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 φ = 1; but, if the primary vibration be in the plane of +diffraction, sin φ = cos θ. This result was employed by Stokes as +a criterion of the direction of vibration; and his experiments, conducted +with gratings, led him to the conclusion that the vibrations +<span class="pagenum"><a name="page254" id="page254"></a>254</span> +of polarized light are executed in a direction <i>perpendicular</i> to the +plane of polarization.</p> + +<p>The factor (1 + cos θ) shows in what manner the secondary disturbance +depends upon the direction in which it is propagated with +respect to the front of the primary wave.</p> + +<p>If, as suffices for all practical purposes, we limit the application +of the formulae to points in advance of the plane at which the wave +is supposed to be broken up, we may use simpler methods of resolution +than that above considered. It appears indeed that the purely +mathematical question has no definite answer. In illustration of +this the analogous problem for sound may be referred to. Imagine +a flexible lamina to be introduced so as to coincide with the plane +at which resolution is to be effected. The introduction of the lamina +(supposed to be devoid of inertia) will make no difference to the +propagation of plane parallel sonorous waves through the position +which it occupies. At every point the motion of the lamina will be +the same as would have occurred in its absence, the pressure of the +waves impinging from behind being just what is required to generate +the waves in front. Now it is evident that the aerial motion in front +of the lamina is determined by what happens at the lamina without +regard to the cause of the motion there existing. Whether the +necessary forces are due to aerial pressures acting on the rear, or to +forces directly impressed from without, is a matter of indifference. +The conception of the lamina leads immediately to two schemes, +according to which a primary wave may be supposed to be broken +up. In the first of these the element dS, the effect of which is to be +estimated, is supposed to execute its actual motion, while every other +element of the plane lamina is maintained at rest. The resulting +aerial motion in front is readily calculated (see Rayleigh, <i>Theory of +Sound</i>, § 278); it is symmetrical with respect to the origin, <i>i.e.</i> independent +of θ. When the secondary disturbance thus obtained is +integrated with respect to dS over the entire plane of the lamina, the +result is necessarily the same as would have been obtained had the +primary wave been supposed to pass on without resolution, for this +is precisely the motion generated when every element of the lamina +vibrates with a common motion, equal to that attributed to dS. +The only assumption here involved is the evidently legitimate one +that, when two systems of variously distributed motion at the +lamina are superposed, the corresponding motions in front are +superposed also.</p> + +<p>The method of resolution just described is the simplest, but it is +only one of an indefinite number that might be proposed, and which +are all equally legitimate, so long as the question is regarded as a +merely mathematical one, without reference to the physical properties +of actual screens. If, instead of supposing the <i>motion</i> at dS +to be that of the primary wave, and to be zero elsewhere, we suppose +the <i>force</i> operative over the element dS of the lamina to be that +corresponding to the primary wave, and to vanish elsewhere, we +obtain a secondary wave following quite a different law. In this +case the motion in different directions varies as cosθ, vanishing at +right angles to the direction of propagation of the primary wave. +Here again, on integration over the entire lamina, the aggregate +effect of the secondary waves is necessarily the same as that of the +primary.</p> + +<p>In order to apply these ideas to the investigation of the secondary +wave of light, we require the solution of a problem, first treated +by Stokes, viz. the determination of the motion in an infinitely +extended elastic solid due to a locally applied periodic force. If +we suppose that the force impressed upon the element of mass +D dx dy dz is</p> + +<p class="center">DZ dx dy dz,</p> + +<p class="noind">being everywhere parallel to the axis of Z, the only change required +in our equations (1), (2) is the addition of the term Z to the second +member of the third equation (2). In the forced vibration, now +under consideration, Z, and the quantities ξ, η, ζ, δ expressing the +resulting motion, are to be supposed proportional to e<span class="sp">int</span>, where +i = √(-1), and n = 2π/τ, τ being the periodic time. Under these +circumstances the double differentiation with respect to t of any +quantity is equivalent to multiplication by the factor -n², and thus +our equations take the form</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">(b²Δ² + n²)ξ + (a² − b²)</td> <td>dδ</td> +<td rowspan="2">= 0</td> <td rowspan="6"><span style="font-size: 8em; font-family: 'Courier New'; color: #a0a0a0;">}</span>  (7).</td></tr> +<tr><td class="denom">dx</td></tr> + +<tr><td rowspan="2">(b²Δ² + n²)η + (a² − b²)</td> <td>dδ</td> +<td rowspan="2">= 0</td></tr> +<tr><td class="denom">dy</td></tr> + +<tr><td rowspan="2">(b²Δ² + n²)ζ + (a² − b²)</td> <td>dδ</td> +<td rowspan="2">= −Z</td></tr> +<tr><td class="denom">dz</td></tr></table> + +<p class="noind">It will now be convenient to introduce the quantities.<span class="ov">ω</span><span class="su">1</span>, <span class="ov">ω</span><span class="su">2</span>, <span class="ov">ω</span><span class="su">3</span> +which express the <i>rotations</i> of the elements of the medium round axes +parallel to those of co-ordinates, in accordance with the equations</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="ov">ω</span><span class="su">3</span> =</td> <td>dξ</td> +<td rowspan="2">−</td> <td>dη</td> +<td rowspan="2">, <span class="ov">ω</span><span class="su">1</span> =</td> <td>dη</td> +<td rowspan="2">−</td> <td>dζ</td> +<td rowspan="2">, <span class="ov">ω</span><span class="su">2</span></td> <td>dζ</td> +<td rowspan="2">−</td> <td>dξ</td> +<td rowspan="2">  (8).</td></tr> +<tr><td class="denom">dy</td> <td class="denom">dx′</td> +<td class="denom">dz</td> <td class="denom">dy′</td> +<td class="denom">dx</td> <td class="denom">dz′</td></tr></table> + +<p class="noind">In terms of these we obtain from (7), by differentiation and subtraction,</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl">(b²Δ² + n²) <span class="ov">ω</span><span class="su">3</span> = 0</td> <td class="tcl" rowspan="3"><span style="font-size: 4em; font-family: 'Courier New'; color: #a0a0a0;">}</span>  (9).</td></tr> +<tr><td class="tcl">(b²Δ² + n²) <span class="ov">ω</span><span class="su">1</span> = dZ/dy</td></tr> +<tr><td class="tcl">(b²Δ² + n²) <span class="ov">ω</span><span class="su">2</span> = −dZ/dx</td></tr></table> + +<p class="noind">The first of equations (9) gives</p> + +<p class="center"><span class="ov">ω</span><span class="su">3</span> = 0  (10).</p> + +<p class="noind">For <span class="ov">ω</span><span class="su">1</span>, we have</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="ov">ω</span><span class="su">1</span> =</td> <td>1</td> +<td rowspan="2"><span class="f150">∫∫∫</span></td> <td>dZ</td> <td>e<span class="sp">−ikr</span></td> +<td rowspan="2">dx dy dz  (11),</td></tr> +<tr><td class="denom">4πb²</td> <td class="ov">dy</td> <td class="denom">r</td></tr></table> + +<p class="noind">where r is the distance between the element dx dy dz and the point +where <span class="ov">ω</span><span class="su">1</span> is estimated, and</p> + +<p class="center">k = n/b = 2π/λ  (12),</p> + +<p class="noind">λ being the wave-length.</p> + +<p>(This solution may be verified in the same manner as Poisson’s +theorem, in which k = 0.)</p> + +<p>We will now introduce the supposition that the force Z acts +only within a small space of volume T, situated at (x, y, z), and for +simplicity suppose that it is at the origin of co-ordinates that the +rotations are to be estimated. Integrating by parts in (11), we get</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">∫</span></td> <td>e<span class="sp">−ikr</span></td> <td>dZ</td> +<td rowspan="2">dy = <span class="f150">[</span> Z</td> <td>e<span class="sp">−ikr</span></td> +<td rowspan="2"><span class="f150">]</span> − <span class="f150">∫</span> Z</td> <td>d</td> +<td rowspan="2"><span class="f150">(</span></td> <td>e<span class="sp">−ikr</span></td> +<td rowspan="2"><span class="f150">)</span> dy,</td></tr> +<tr><td class="denom">r</td> <td class="ov">dy</td> +<td class="denom">r</td> <td class="denom">dy</td> <td class="denom">r</td></tr></table> + +<p class="noind">in which the integrated terms at the limits vanish, Z being finite +only within the region T. Thus</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="ov">ω</span><span class="su">1</span> =</td> <td>1</td> +<td rowspan="2"><span class="f150">∫∫∫</span> Z</td> <td>d</td> +<td rowspan="2"><span class="f150">(</span></td> <td>e<span class="sp">−ikr</span></td> +<td rowspan="2"><span class="f150">)</span> dx dy dz.</td></tr> +<tr><td class="denom">4πb²</td> <td class="denom">dy</td> <td class="denom">r</td></tr></table> + +<p class="noind">Since the dimensions of T are supposed to be very small in comparison +with λ, the factor d/dy (e<span class="sp">−ikr</span>/r) is sensibly constant; so that, +if Z stand for the mean value of Z over the volume T, we may write</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="ov">ω</span><span class="su">1</span> =</td> <td>TZ</td> +<td rowspan="2">·</td> <td>y</td> +<td rowspan="2">·</td> <td>d</td> +<td rowspan="2"><span class="f150">(</span></td> <td>e<span class="sp">−ikr</span></td> +<td rowspan="2"><span class="f150">)</span>  (13).</td></tr> +<tr><td class="denom">4πb²</td> <td class="denom">r</td> +<td class="denom">dr</td> <td class="denom">r</td></tr></table> + +<p class="noind">In like manner we find</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="ov">ω</span><span class="su">2</span> = −</td> <td>TZ</td> +<td rowspan="2">·</td> <td>x</td> +<td rowspan="2">·</td> <td>d</td> +<td rowspan="2"><span class="f150">(</span></td> <td>e<span class="sp">−ikr</span></td> +<td rowspan="2"><span class="f150">)</span>  (14).</td></tr> +<tr><td class="denom">4πb²</td> <td class="denom">r</td> +<td class="denom">dr</td> <td class="denom">r</td></tr></table> + +<p class="noind">From (10), (13), (14) we see that, as might have been expected, +the rotation at any point is about an axis perpendicular both to +the direction of the force and to the line joining the point to the +source of disturbance. If the resultant rotation be ω, we have</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="ov">ω</span> =</td> <td>TZ</td> +<td rowspan="2">·</td> <td>√(x² + y²)</td> +<td rowspan="2">·</td> <td>d</td> +<td rowspan="2"><span class="f150">(</span></td> <td>e<span class="sp">−ikr</span></td> +<td rowspan="2"><span class="f150">)</span> =</td> <td>TZ sin φ</td> <td>d</td> +<td rowspan="2"><span class="f150">(</span></td> <td>e<span class="sp">−ikr</span></td> +<td rowspan="2"><span class="f150">)</span>,</td></tr> +<tr><td class="denom">4πb²</td> <td class="denom">r</td> +<td class="denom">dr</td> <td class="denom">r</td> +<td class="denom">4πb²</td> <td class="ov">dr</td> +<td class="denom">r</td></tr></table> + +<p class="noind">φ denoting the angle between r and z. In differentiating e<span class="sp">−ikr</span>/r +with respect to r, we may neglect the term divided by r² as altogether +insensible, kr being an exceedingly great quantity at any moderate +distance from the origin of disturbance. Thus</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="ov">ω</span> =</td> <td>−ik · TZ sin φ</td> +<td rowspan="2">·</td> <td>e<span class="sp">−ikr</span></td> +<td rowspan="2">  (15),</td></tr> +<tr><td class="denom">4πb²</td> <td class="denom">r</td></tr></table> + +<p class="noind">which completely determines the rotation at any point. For a disturbing +force of given integral magnitude it is seen to be everywhere +about an axis perpendicular to r and the direction of the force, and +in magnitude dependent only upon the angle (φ) between these two +directions and upon the distance (r).</p> + +<p>The intensity of light is, however, more usually expressed in +terms of the actual displacement in the plane of the wave. This +displacement, which we may denote by ζ′, is in the plane containing +z and r, and perpendicular to the latter. Its connexion with <span class="ov">ω</span>is +expressed by <span class="ov">ω</span> = dζ′/dr; so that</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ζ′ =</td> <td>TZ sin φ</td> +<td rowspan="2">·</td> <td>e′ <span class="sp">(at−kr)</span></td> +<td rowspan="2">  (16),</td></tr> +<tr><td class="denom">4πb²</td> <td class="denom">r</td></tr></table> + +<p class="noind">where the factor e<span class="sp">int</span> is restored.</p> + +<p>Retaining only the real part of (16), we find, as the result of a +local application of force equal to</p> + +<p class="center">DTZ cos nt  (17),</p> + +<p class="noind">the disturbance expressed by</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ζ′ =</td> <td>TZ sin φ</td> +<td rowspan="2">·</td> <td>cos (nt − kr)</td> +<td rowspan="2">  (18).</td></tr> +<tr><td class="denom">4πb²</td> <td class="denom">r</td></tr></table> + +<p>The occurrence of sin φ shows that there is no disturbance +radiated in the direction of the force, a feature which might have +been anticipated from considerations of symmetry.</p> + +<p>We will now apply (18) to the investigation of a law of secondary +disturbance, when a primary wave</p> + +<p class="center">ζ = sin(nt − kx)  (19)</p> + +<p class="noind">is supposed to be broken up in passing the plane x = 0. The first step +is to calculate the force which represents the reaction between the +parts of the medium separated by x = 0. The force operative upon +the positive half is parallel to OZ, and of amount per unit of area +equal to</p> + +<p class="center">−b²D dζ/dx = b²kD cos nt;</p> + +<p class="noind">and to this force acting over the whole of the plane the actual +motion on the positive side may be conceived to be due. The +<span class="pagenum"><a name="page255" id="page255"></a>255</span> +secondary disturbance corresponding to the element dS of the plane +may be supposed to be that caused by a force of the above magnitude +acting over dS and vanishing elsewhere; and it only remains to +examine what the result of such a force would be.</p> + +<p>Now it is evident that the force in question, supposed to act +upon the positive half only of the medium, produces just double of +the effect that would be caused by the same force if the medium +were undivided, and on the latter supposition (being also localized +at a point) it comes under the head already considered. According +to (18), the effect of the force acting at dS parallel to OZ, and of +amount equal to</p> + +<p class="center">2b²kD dS cos nt,</p> + +<p class="noind">will be a disturbance</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ζ′ =</td> <td>dS sin φ</td> +<td rowspan="2">cos (nt − kr)  (20),</td></tr> +<tr><td class="denom">λr</td></tr></table> + +<p class="noind">regard being had to (12). This therefore expresses the secondary +disturbance at a distance r and in a direction making an angle φ +with OZ (the direction of primary vibration) due to the element dS +of the wave-front.</p> + +<p>The proportionality of the secondary disturbance to sin φ is +common to the present law and to that given by Stokes, but here +there is no dependence upon the angle θ between the primary and +secondary rays. The occurrence of the factor λr<span class="sp">−1</span>, and the +necessity of supposing the phase of the secondary wave accelerated +by a quarter of an undulation, were first established by Archibald +Smith, as the result of a comparison between the primary wave, +supposed to pass on without resolution, and the integrated effect +of all the secondary waves (§ 2). The occurrence of factors such +as sin φ, or ½(1 + cos θ), in the expression of the secondary wave +has no influence upon the result of the integration, the effects of +all the elements for which the factors differ appreciably from unity +being destroyed by mutual interference.</p> + +<p>The choice between various methods of resolution, all mathematically +admissible, would be guided by physical considerations +respecting the mode of action of obstacles. Thus, to refer again to +the acoustical analogue in which plane waves are incident upon +a perforated rigid screen, the circumstances of the case are best +represented by the first method of resolution, leading to symmetrical +secondary waves, in which the normal motion is supposed to be zero +over the unperforated parts. Indeed, if the aperture is very small, +this method gives the correct result, save as to a constant factor. In +like manner our present law (20) would apply to the kind of obstruction +that would be caused by an actual physical division of the elastic +medium, extending over the whole of the area supposed to be occupied +by the intercepting screen, but of course not extending to the parts +supposed to be perforated.</p> + +<p>On the electromagnetic theory, the problem of diffraction becomes +definite when the properties of the obstacle are laid down. The +simplest supposition is that the material composing the obstacle +is perfectly conducting, <i>i.e.</i> perfectly reflecting. On this basis +A. J. W. Sommerfeld (<i>Math. Ann.</i>, 1895, 47, p. 317), with great mathematical +skill, has solved the problem of the shadow thrown by a +semi-infinite plane screen. A simplified exposition has been given by +Horace Lamb (<i>Proc. Lond. Math. Soc.</i>, 1906, 4, p. 190). It appears that +Fresnel’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.</p> +<div class="author">(R.)</div> + +<hr class="foot" /> <div class="note"> + +<p><a name="ft1g" id="ft1g" href="#fa1g"><span class="fn">1</span></a> The descending series for J<span class="su">0</span>(z) appears to have been first given +by Sir W. Hamilton in a memoir on “Fluctuating Functions,” +<i>Roy. Irish Trans.</i>, 1840.</p> + +<p><a name="ft2g" id="ft2g" href="#fa2g"><span class="fn">2</span></a> 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.”</p> + +<p><a name="ft3g" id="ft3g" href="#fa3g"><span class="fn">3</span></a> <i>”Man kann daraus schliessen, was moglicher Weise durch Mikroskope +noch zu sehen ist. Ein mikroskopischer Gegenstand z. B, dessen +Durchmesser = (λ) 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”</i> (<i>Gilbert’s Ann.</i> +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.</p> + +<p><a name="ft4g" id="ft4g" href="#fa4g"><span class="fn">4</span></a> 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.]</p> + +<p><a name="ft5g" id="ft5g" href="#fa5g"><span class="fn">5</span></a> Compare also F. F. Lippich, <i>Pogg. Ann.</i> cxxxix. p. 465, 1870; +Rayleigh, <i>Nature</i> (October 2, 1873).</p> + +<p><a name="ft6g" id="ft6g" href="#fa6g"><span class="fn">6</span></a> The power of a grating to construct light of nearly definite wave-length +is well illustrated by Young’s comparison with the production +of a musical note by reflection of a sudden sound from a row of +palings. The objection raised by Herschel (<i>Light</i>, § 703) to this +comparison depends on a misconception.</p> + +<p><a name="ft7g" id="ft7g" href="#fa7g"><span class="fn">7</span></a> It must not be supposed that errors of this order of magnitude are +unobjectionable in all cases. The position of the middle of the bright +band representative of a mathematical line can be fixed with a +spider-line micrometer within a small fraction of the width of the +band, just as the accuracy of astronomical observations far transcends +the separating power of the instrument.</p> + +<p><a name="ft8g" id="ft8g" href="#fa8g"><span class="fn">8</span></a> “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,” +<i>Phil. Mag.</i>, September 1883).</p> + +<p><a name="ft9g" id="ft9g" href="#fa9g"><span class="fn">9</span></a> On account of inequalities in the atmosphere giving a variable +refraction, the light from a star would be irregularly distributed over +a screen. The experiment is easily made on a laboratory scale, with +a small source of light, the rays from which, in their course +towards a rather distant screen, are disturbed by the neighbourhood +of a heated body. At a moment when the eye, or object-glass of a +telescope, occupies a dark position, the star vanishes. A fraction +of a second later the aperture occupies a bright place, and the star +reappears. According to this view the chromatic effects depend +entirely upon atmospheric dispersion.</p> + +<p><a name="ft10g" id="ft10g" href="#fa10g"><span class="fn">10</span></a> In experiment a line of light is sometimes substituted for a point +in order to increase the illumination. The various parts of the line +are here <i>independent</i> sources, and should be treated accordingly. +To assume a cylindrical form of primary wave would be justifiable +only when there is synchronism among the secondary waves issuing +from the various centres.</p> + +<p><a name="ft11g" id="ft11g" href="#fa11g"><span class="fn">11</span></a> H. Necker (<i>Phil. Mag.</i>, November 1832); Fox Talbot (<i>Phil. Mag.</i>, +June 1833). “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.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DIFFUSION<a name="ar85" id="ar85"></a></span> (from the Lat. <i>diffundere; dis-</i>, asunder, and +<i>fundere</i>, to pour out), in general, a spreading out, scattering +or circulation; in physics the term is applied to a special +phenomenon, treated below.</p> + +<p>1. <i>General Description.</i>—When two different substances are +placed in contact with each other they sometimes remain +separate, but in many cases a gradual mixing takes place. In the +case where both the substances are gases the process of mixing +continues until the result is a uniform mixture. In other cases +the proportions in which two different substances can mix +lie between certain fixed limits, but the mixture is distinguished +from a chemical compound by the fact that between these limits +the composition of the mixture is capable of continuous variation, +while in chemical compounds, the proportions of the different +constituents can only have a discrete series of numerical values, +each different ratio representing a different compound. If we +take, for example, air and water in the presence of each other, air +will become dissolved in the water, and water will evaporate into +the air, and the proportions of either constituent absorbed by the +other will vary continuously. But a limit will come when the air +will absorb no more water, and the water will absorb no more air, +and throughout the change a definite surface of separation will +exist between the liquid and the gaseous parts. When no surface +of separation ever exists between two substances they must +necessarily be capable of mixing in all proportions. If they are +not capable of mixing in all proportions a discontinuous change +must occur somewhere between the regions where the substances +are still unmixed, thus giving rise to a surface of separation.</p> + +<p>The phenomena of mixing thus involves the following processes:—(1) +A motion of the substances relative to one another +throughout a definite <i>region</i> of space in which mixing is taking +place. This relative motion is called “diffusion.” (2) The passage +of portions of the mixing substances across the <i>surface</i> of +separation when such a surface exists. These surface actions +are described under various terms such as solution, evaporation, +condensation and so forth. For example, when a soluble salt is +placed in a liquid, the process which occurs at the surface of the +salt is called “solution,” but the salt which enters the liquid by +solution is transported from the surface into the interior of the +liquid by “diffusion.”</p> + +<p>Diffusion may take place in solids, that is, in regions occupied +by matter which continues to exhibit the properties of the solid +state. Thus if two liquids which can mix are separated by a +membrane or partition, the mixing may take place through the +membrane. If a solution of salt is separated from pure water by +a sheet of parchment, part of the salt will pass through the parchment +into the water. If water and glycerin are separated in this +way most of the water will pass into the glycerin and a little +glycerin will pass through in the opposite direction, a property +frequently used by microscopists for the purpose of gradually +transferring minute algae from water into glycerin. A still more +interesting series of examples is afforded by the passage of gases +through partitions of metal, notably the passage of hydrogen +through platinum and palladium at high temperatures. When +the process is considered with reference to a membrane or partition +taken as a whole, the passage of a substance from one side to the +other is commonly known as “osmosis” or “transpiration” +(see <span class="sc"><a href="#artlinks">Solution</a></span>), but what occurs in the material of the membrane +itself is correctly described as diffusion.</p> + +<p>Simple cases of diffusion are easily observed qualitatively. If a +solution of a coloured salt is carefully introduced by a funnel into +the bottom of a jar containing water, the two portions will at first +be fairly well defined, but if the mixture can exist in all proportions, +the surface of separation will gradually disappear; and the +rise of the colour into the upper part and its gradual weakening +in the lower part, may be watched for days, weeks or even longer +intervals. The diffusion of a strong aniline colouring matter into +the interior of gelatine is easily observed, and is commonly seen in +copying apparatus. Diffusion of gases may be shown to exist by +taking glass jars containing vapours of hydrochloric acid and +ammonia, and placing them in communication with the heavier +gas downmost. The precipitation of ammonium chloride shows +that diffusion exists, though the chemical action prevents this +example from forming a typical case of diffusion. Again, when +a film of Canada balsam is enclosed between glass plates, the +disappearance during a few weeks of small air bubbles enclosed +in the balsam can be watched under the microscope.</p> + +<p>In fluid media, whether liquids or gases, the process of mixing +is greatly accelerated by stirring or agitating the fluids, and +liquids which might take years to mix if left to themselves +can thus be mixed in a few seconds. It is necessary to carefully +distinguish the effects of agitation from those of diffusion proper. +By shaking up two liquids which do not mix we split them up +into a large number of different portions, and so greatly increase +the area of the surface of separation, besides decreasing the +thicknesses of the various portions. But even when we produce +the appearance of a uniform turbid mixture, the small portions +remain quite distinct. If however the fluids can really mix, the +final process must in every case depend on diffusion, and all we +do by shaking is to increase the sectional area, and decrease the +thickness of the diffusing portions, thus rendering the completion +of the operation more rapid. If a gas is shaken up in a liquid +the process of absorption of the bubbles is also accelerated by +capillary action, as occurs in an ordinary sparklet bottle. To +state the matter precisely, however finely two fluids have been +<span class="pagenum"><a name="page256" id="page256"></a>256</span> +subdivided by agitation, the molecular constitution of the +different portions remains unchanged. The ultimate process +by which the individual molecules of two different substances +become mixed, producing finally a homogeneous mixture, is in +every case diffusion. In other words, diffusion is that relative +motion of the molecules of two different substances by which the +proportions of the molecules in any region containing a finite +number of molecules are changed.</p> + +<div class="condensed"> +<p>In order, therefore, to make accurate observations of diffusion in +fluids it is necessary to guard against any cause which may set up +currents; and in some cases this is exceedingly difficult. Thus, if +gas is absorbed at the upper surface of a liquid, and if the gaseous +solution is heavier than the pure liquid, currents may be set up, and +a steady state of diffusion may cease to exist. This has been tested +experimentally by C. G. von Hüfner and W. E. Adney. The same +thing may happen when a gas is evolved into a liquid at the surface +of a solid even if no bubbles are formed; thus if pieces of aluminium +are placed in caustic soda, the currents set up by the evolution of +hydrogen are sufficient to set the aluminium pieces in motion, and +it is probable that the motions of the Diatomaceae are similarly +caused by the evolution of oxygen. In some pairs of substances +diffusion may take place more rapidly than in others. Of course the +progress of events in any experiment necessarily depends on various +causes, such as the size of the containing vessels, but it is easy to see +that when experiments with different substances are carried out under +similar conditions, however these “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.</p> +</div> + +<p>2 <i>Quantitative Methods of observing Diffusion.</i>—The simplest +plan of determining the progress of diffusion between two liquids +would be to draw off and examine portions from different strata +at some stage in the process; the disturbance produced would, +however, interfere with the subsequent process of diffusion, and +the observations could not be continued. By placing in the +liquid column hollow glass beads of different average densities, +and observing at what height they remain suspended, it is +possible to trace the variations of density of the liquid column +at different depths, and different times. In this method, which +was originally introduced by Lord Kelvin, difficulties were +caused by the adherence of small air bubbles to the beads.</p> + +<p>In general, optical methods are the most capable of giving +exact results, and the following may be distinguished, (a) <i>By +refraction in a horizontal plane.</i> If the containing vessel is in +the form of a prism, the deviation of a horizontal ray of light in +passing through the prism determines the index of refraction, +and consequently the density of the stratum through which the +ray passes, (b) <i>By refraction in a vertical plane.</i> Owing to the +density varying with the depth, a horizontal ray entering the +liquid also undergoes a small vertical deviation, being bent +downwards towards the layers of greater density. The observation +of this vertical deviation determines not the actual density, +but its rate of variation with the depth, <i>i.e.</i> the “density gradient” +at any point, (c) <i>By the saccharimeter.</i> In the cases of solutions +of sugar, which cause rotation of the plane of polarized light, +the density of the sugar at any depth may be determined by +observing the corresponding angle of rotation, this was done +originally by W. Voigt.</p> + +<p>3. <i>Elementary Definitions of Coefficient of Diffusion.</i>—The +simplest case of diffusion is that of a substance, say a gas, diffusing +in the interior of a homogeneous solid medium, which remains at +rest, when no external forces act on the system. We may regard +it as the result of experience that: (1) if the density of the diffusing +substance is everywhere the same no diffusion takes place, and +(2) if the density of the diffusing substance is different at different +points, diffusion will take place from places of greater to those of +lesser density, and will not cease until the density is everywhere +the same. It follows that the rate of flow of the diffusing substance +at any point in any direction must depend on the density +gradient at that point in that direction, <i>i.e.</i> on the rate at which +the density of the diffusing substance decreases as we move in +that direction. We may define the <i>coefficient of diffusion</i> as the +ratio of the total mass per unit area which flows across any +small section, to the rate of decrease of the density per unit +distance in a direction perpendicular to that section.</p> + +<p>In the case of steady diffusion parallel to the axis of x, if ρ 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 -dρ/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(dρ/dx).</p> + +<p>Further, the rate at which the quantity of substance is increasing +in an element between the distances x and x+dx is equal to the +difference of the rates of flow in and out of the two faces, whence as +in hydrodynamics, we have dρ/dt =-dq/dx.</p> + +<p>It follows that the equation of diffusion in this case assumes the +form</p> + +<table class="math0" summary="math"> +<tr><td>dρ</td> <td rowspan="2">=</td> <td>d</td> +<td rowspan="2"><span class="f150">(</span> K</td> <td>dρ</td> +<td rowspan="2"><span class="f150">)</span>,</td></tr> +<tr><td class="denom">dt</td> <td class="denom">dx</td> <td class="denom">dx</td></tr></table> + +<p class="noind">which is identical with the equations representing conduction of +heat, flow of electricity and other physical phenomena. For motion +in three dimensions we have in like manner</p> + +<table class="math0" summary="math"> +<tr><td>dρ</td> <td rowspan="2">=</td> <td>d</td> +<td rowspan="2"><span class="f150">(</span> K</td> <td>dρ</td> +<td rowspan="2"><span class="f150">)</span> +</td> <td>d</td> +<td rowspan="2"><span class="f150">(</span> K</td> <td>dρ</td> +<td rowspan="2"><span class="f150">)</span> +</td> <td>d</td> +<td rowspan="2"><span class="f150">(</span> K</td> <td>dρ</td> +<td rowspan="2"><span class="f150">)</span>;</td></tr> +<tr><td class="denom">dt</td> <td class="denom">dx </td> +<td class="denom">dx </td> <td class="denom">dy</td> +<td class="denom">dy</td> <td class="denom">dz</td> <td class="denom">dz</td></tr></table> + +<p class="noind">and the corresponding equations in electricity and heat for anisotropic +substances would be available to account for any parallel +phenomena, which may arise, or might be conceived, to exist in +connexion with diffusion through a crystalline solid.</p> + +<p>In the case of a very dilute solution, the coefficient of diffusion +of the dissolved substance can be defined in the same way as +when the diffusion takes place in a solid, because the effects of +diffusion will not have any perceptible influence on the solvent, +and the latter may therefore be regarded as remaining practically +at rest. But in most cases of diffusion between two fluids, both +of the fluids are in motion, and hence there is far greater difficulty +in determining the motion, and even in defining the coefficient of +diffusion. It is important to notice in the first instance, that it +is only the relative motion of the two substances which constitutes +diffusion. Thus when a current of air is blowing, under +ordinary circumstances the changes which take place are purely +mechanical, and do not depend on the separate diffusions of the +oxygen and nitrogen of which the air is mainly composed. It is +only when two gases are flowing with unequal velocity, that +is, when they have a relative motion, that these changes of +relative distribution, which are called diffusion, take place. The +best way out of the difficulty is to investigate the separate motions +of the two fluids, taking account of the mechanical actions +exerted on them, and supposing that the mutual action of the +fluids causes either fluid to resist the relative motion of the other.</p> + +<p>4. <i>The Coefficient of Resistance.</i>—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 ρ<span class="su">1</span>, ρ<span class="su">2</span> are the densities cf the two fluids, +u<span class="su">1</span>, u<span class="su">2</span> their velocities, C the coefficient of resistance, then the +portion of the fluid A contained in a small element of volume v +will experience from the fluid B a resistance Cρ<span class="su">1</span>ρ<span class="su">2</span>v(u<span class="su">1</span> − u<span class="su">2</span>), and +the fluid B contained in the same volume element will experience +from the fluid A an equal and opposite resistance, Cρ<span class="su">1</span>ρ<span class="su">2</span>v(u<span class="su">2</span> − u<span class="su">1</span>).</p> + +<p>This definition implies the following laws of resistance to +diffusion, which must be regarded as based on experience, and +not as self-evident truths: (1) each fluid tends to assume, so far +as diffusion is concerned, the same equüibrium distribution that +it would assume if its motion were unresisted by the presence of +the other fluid. (Of course, the mutual attraction of gravitation +of the two fluids might affect the final distribution, but this is +practically negligible. Leaving such actions as this out of +<span class="pagenum"><a name="page257" id="page257"></a>257</span> +account the following statement is correct.) In a state of +equilibrium, the density of each fluid at any point thus depends +only on the partial pressure of that fluid alone, and is the same +as if the other fluids were absent. It does not depend on the +partial pressures of the other fluids. If this were not the case, +the resistance to diffusion would be analogous to friction, and +would contain terms which were independent of the relative +velocity u<span class="su">2</span> − u<span class="su">1</span>. (2) For slow motions the resistance to diffusion +is (approximately at any rate) proportional to the relative +velocity. (3) The coefficient of resistance C is not necessarily +always constant; it may, for example, and, in general, does, +depend on the temperature.</p> + +<p>If we form the equations of hydrodynamics for the different fluids +occurring in any mixture, taking account of diffusion, but neglecting +viscosity, and using suffixes 1, 2 to denote the separate fluids, these +assume the form given by James Clerk Maxwell (“Diffusion,” in +<i>Ency. Brit.</i>, 9th ed.):—</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ρ<span class="su">1</span></td> <td>Du<span class="su">1</span></td> +<td rowspan="2">+</td> <td>dp<span class="su">1</span></td> +<td rowspan="2">− X<span class="su">1</span>ρ<span class="su">1</span> + C<span class="su">12</span>ρ<span class="su">1</span>ρ<span class="su">2</span>(u<span class="su">1</span> − u<span class="su">2</span>) + &c. = 0,</td></tr> +<tr><td class="denom">Dt</td> <td class="denom">dx</td></tr></table> + +<p class="noind">where</p> + +<table class="math0" summary="math"> +<tr><td>Du<span class="su">1</span></td> <td rowspan="2">=</td> <td>du<span class="su">1</span></td> +<td rowspan="2">+ u<span class="su">1</span></td> <td>du<span class="su">1</span></td> +<td rowspan="2">+ v<span class="su">1</span></td> <td>du<span class="su">1</span></td> +<td rowspan="2">+ w<span class="su">1</span></td> <td>du<span class="su">1</span></td> +<td rowspan="2">,</td></tr> +<tr><td class="denom">Dt</td> <td class="denom">dt</td> +<td class="denom">dx</td> <td class="denom">dy</td> <td class="denom">dz</td></tr></table> + +<p class="noind">and these equations imply that when diffusion and other motions +cease, the fluids satisfy the separate conditions of equilibrium +dp<span class="su">1</span>/dx − X<span class="su">1</span>ρ<span class="su">1</span> = 0. The assumption made in the following account is +that terms such as Du<span class="su">1</span>/Dt may be neglected in the cases considered.</p> + +<p>A further property based on experience is that the motions set +up in a mixture by diffusion are very slow compared with those +set up by mechanical actions, such as differences of pressure. +Thus, if two gases at equal temperature and pressure be allowed +to mix by diffusion, the heavier gas being below the lighter, the +process will take a long time; on the other hand, if two gases, +or parts of the same gas, at different pressures be connected, +equalization of pressure will take place almost immediately. +It follows from this property that the forces required to overcome +the “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.</p> + +<p>5. <i>Slow Diffusion of two Gases. Relation between the Coefficients +of Resistance and of Diffusion.</i>—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.</p> + +<p>The densities of the gases are denoted by ρ<span class="su">1</span>, ρ<span class="su">2</span>, their velocities of +diffusion by u<span class="su">1</span>, u<span class="su">2</span>, and if their partial pressures are p<span class="su">1</span>, p<span class="su">2</span>, we have by +Boyle’s law p<span class="su">1</span> = k<span class="su">1</span>ρ<span class="su">1</span>, p<span class="su">2</span> = k<span class="su">2</span>ρ<span class="su">2</span>, where k<span class="su">1</span>,k<span class="su">2</span> are constants for the two +gases, the temperature being constant. The axis of the cylinder is +taken as the axis of x.</p> + +<p>From the considerations of the preceding section, the effects of +inertia of the diffusing gases may be neglected, and at any instant of +the process either of the gases is to be treated as kept in equilibrium +by its partial pressure and the resistance to diffusion produced by +the other gas. Calling this resistance per unit volume R, and putting +R = Cρ<span class="su">1</span>ρ<span class="su">2</span>(u<span class="su">1</span> − u<span class="su">2</span>), where C is the coefficient of resistance, the equations +of equilibrium give</p> + +<table class="math0" summary="math"> +<tr><td>dp<span class="su">1</span></td> <td rowspan="2">+ Cρ<span class="su">1</span>ρ<span class="su">2</span>(u<span class="su">1</span> − u<span class="su">2</span>) = 0, and</td> <td>dp<span class="su">2</span></td> +<td rowspan="2">+ Cρ<span class="su">1</span>ρ<span class="su">2</span>(u<span class="su">2</span> − u<span class="su">1</span>) = 0  (1)</td></tr> +<tr><td class="denom">dx</td> <td class="denom">dx</td></tr></table> + +<p class="noind">These involve</p> + +<table class="math0" summary="math"> +<tr><td>dp<span class="su">1</span></td> <td rowspan="2">+</td> <td>dp<span class="su">2</span></td> +<td rowspan="2">= 0 or p<span class="su">1</span> + p<span class="su">2</span> = P  (2)</td></tr> +<tr><td class="denom">dx</td> <td class="denom">dx</td></tr></table> + +<p class="noind">where P is the total pressure of the mixture, and is everywhere +constant, consistently with the conditions of mechanical equilibrium.</p> + +<p>Now dp<span class="su">1</span>/dx is the pressure-gradient of the first gas, and is, by +Boyle’s law, equal to k<span class="su">1</span> times the corresponding density-gradient. +Again ρ<span class="su">1</span>u<span class="su">1</span> is the mass of gas flowing across any section per unit +time, and k<span class="su">1</span>ρ<span class="su">1</span>u<span class="su">1</span> or p<span class="su">1</span>u<span class="su">1</span> can be regarded as representing the flux of +partial pressure produced by the motion of the gas. Since the total +pressure is everywhere constant, and the ends of the cylinder are +supposed fixed, the fluxes of partial pressure due to the two gases +are equal and opposite, so that</p> + +<p class="center">p<span class="su">1</span>u<span class="su">1</span> + p<span class="su">2</span>u<span class="su">2</span> = 0 or k<span class="su">1</span>ρ<span class="su">1</span>u<span class="su">1</span> + k<span class="su">2</span>ρ<span class="su">2</span>u<span class="su">2</span> = 0  (3).</p> + +<p class="noind">From (2) (3) we find by elementary algebra</p> + +<p class="center">u<span class="su">1</span>/p<span class="su">2</span> = −u<span class="su">2</span>/p<span class="su">1</span> = (u<span class="su">1</span> − u<span class="su">2</span>)/(p<span class="su">1</span> + p<span class="su">2</span>) = (u<span class="su">1</span> − u<span class="su">2</span>)/P,</p> + +<p class="noind">and therefore</p> + +<p class="center">p<span class="su">2</span>u<span class="su">1</span> = −p<span class="su">2</span>u<span class="su">2</span> = p<span class="su">1</span>p<span class="su">2</span>(u<span class="su">1</span> − u<span class="su">2</span>)/P = k<span class="su">1</span>k<span class="su">2</span>ρ<span class="su">1</span>ρ<span class="su">2</span>(u<span class="su">1</span> − u<span class="su">2</span>)/P</p> + +<p class="noind">Hence equations (1) (2) gives</p> + +<table class="math0" summary="math"> +<tr><td>dp<span class="su">1</span></td> <td rowspan="2">+</td> <td>CP</td> +<td rowspan="2">(p<span class="su">1</span>u<span class="su">1</span>) = 0, and</td> <td>dp<span class="su">2</span></td> +<td rowspan="2">+</td> <td>CP</td> +<td rowspan="2">(p<span class="su">2</span>u<span class="su">2</span>) = 0;</td></tr> +<tr><td class="denom">dx</td> <td class="denom">k<span class="su">1</span>k<span class="su">2</span></td> +<td class="denom">dx</td> <td class="denom">k<span class="su">1</span>k<span class="su">2</span></td></tr></table> + +<p class="noind">whence also substituting p<span class="su">1</span> = k<span class="su">1</span>ρ<span class="su">1</span>, p<span class="su">2</span> = k<span class="su">2</span>ρ<span class="su">2</span>, and by transposing</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ρ<span class="su">1</span>u<span class="su">1</span> = −</td> <td>k<span class="su">1</span>k<span class="su">2</span></td> <td>dρ<span class="su">1</span></td> +<td rowspan="2">, and ρ<span class="su">2</span>u<span class="su">2</span> = −</td> <td>k<span class="su">1</span>k<span class="su">2</span></td> <td>dρ<span class="su">2</span></td> +<td rowspan="2">.</td></tr> +<tr><td class="denom">CP</td> <td class="ov">dx</td> +<td class="denom">CP</td> <td class="ov">dx</td></tr></table> + +<p>We may now define the “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 k<span class="su">1</span>k<span class="su">2</span>/CP. The ratios of the +fluxes of partial pressure to the corresponding pressure-gradients are +also equal to the same coefficient. Calling this coefficient K, we also +observe that the equations of continuity for the two gases are</p> + +<table class="math0" summary="math"> +<tr><td>dρ<span class="su">1</span></td> <td rowspan="2">+</td> <td>d(ρ<span class="su">1</span>u<span class="su">1</span>)</td> +<td rowspan="2">= 0, and</td> <td>dρ<span class="su">2</span></td> +<td rowspan="2">+</td> <td>d(ρ<span class="su">2</span>u<span class="su">2</span>)</td> +<td rowspan="2">= 0,</td></tr> +<tr><td class="denom">dt</td> <td class="denom">dx</td> +<td class="denom">dt</td> <td class="denom">dx</td></tr></table> + +<p class="noind">leading to the equations of diffusion</p> + +<table class="math0" summary="math"> +<tr><td>dρ<span class="su">1</span></td> <td rowspan="2">=</td> <td>d</td> +<td rowspan="2"><span class="f150">(</span>K</td> <td>dρ<span class="su">1</span></td> +<td rowspan="2"><span class="f150">)</span>, and</td> <td>dρ<span class="su">2</span></td> +<td rowspan="2">=</td> <td>d</td> +<td rowspan="2"><span class="f150">(</span>K</td> <td>dρ<span class="su">2</span></td> +<td rowspan="2"><span class="f150">)</span>,</td></tr> +<tr><td class="denom">dt</td> <td class="denom">dx</td> +<td class="denom">dx</td> <td class="denom">dt</td> +<td class="denom">dx</td> <td class="denom">dx</td></tr></table> + +<p class="noind">exactly as in the case of diffusion through a solid.</p> + +<p>If we attempt to treat diffusion in liquids by a similar method, +it is, in the first place, necessary to define the “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 <span class="sc"><a href="#artlinks">Solution</a></span>. For dilute solutions at constant temperature, +the assumption that the osmotic pressure is proportional to the +density, leads to results agreeing fairly closely with experience, +and this fact may be represented by the statement that a substance +occurring in a dilute solution behaves like a perfect gas.</p> + +<p>6. <i>Relation of the Coefficient of Diffusion to the Units of Length +and Time.</i>—We may write the equation defining K in the form</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">−K ×</td> <td>I</td> <td>dρ</td> +<td rowspan="2">.</td></tr> +<tr><td class="ov">ρ</td> <td class="ov">dx</td></tr></table> + +<p>Here −dρ/ρdx 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>i.e.</i> +dimensions L²/T.</p> + +<div class="condensed"> +<p><i>Example 1.</i> Taking K = 0.1423 for carbon dioxide and air (at +temperature 0° C. and pressure 76 cm. of mercury) referred to a +centimetre and a second as units, we may interpret the result as +follows:—Supposing in a mixture of carbon dioxide and air, the +density of the carbon dioxide decreases by, say, 1, 2 or 3% of +itself in a distance of 1 cm., then the corresponding velocities +of the diffusing carbon dioxide will be respectively 0.01, 0.02 and +0.03 times 0.1423, that is, 0.001423, 0.002846 and 0.004269 cm. +per second in the three cases.</p> + +<p><i>Example 2.</i> If we wished to take a foot and a second as our units, +we should have to divide the value of the coefficient of diffusion in +Example 1 by the square of the number of centimetres in 1 ft., that +is, roughly speaking, by 900, giving the new value of K = 0.00016 +roughly.</p> +</div> + +<p>7. <i>Numerical Values of the Coefficient of Diffusion.</i>—The +table on p. 258 gives the values of the coefficient of diffusion of +several of the principal pairs of gases at a pressure of 76 cm. of +mercury, and also of a number of other substances. In the gases +the centimetre and second are taken as fundamental units, in +other cases the centimetre and day.</p> + +<p>8. <i>Irreversible Changes accompanying Diffusion.</i>—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 +<span class="pagenum"><a name="page258" id="page258"></a>258</span> +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.”</p> + +<table class="ws" summary="Contents"> + +<tr><td class="tcc allb">Substances.</td> <td class="tcc allb">Temp.</td> <td class="tcc allb">K.</td> <td class="tcc allb">Author.</td></tr> + +<tr><td class="tcl lb rb">Carbon dioxide and air</td> <td class="tcr rb">0°C.</td> <td class="tcl rb">0.1423 cm²/sec.</td> <td class="tcl rb">J. Loschmidt.</td></tr> +<tr><td class="tcl lb rb"> ”   ”   hydrogen</td> <td class="tcr rb">0°C.</td> <td class="tcl rb">0.5558  ”</td> <td class="tcc rb">”</td></tr> +<tr><td class="tcl lb rb"> ”   ”   oxygen</td> <td class="tcr rb">0°C.</td> <td class="tcl rb">0.1409  ”</td> <td class="tcc rb">”</td></tr> +<tr><td class="tcl lb rb"> ”   ”   carbon monoxide</td> <td class="tcr rb">0°C.</td> <td class="tcl rb">0.1406  ”</td> <td class="tcc rb">”</td></tr> +<tr><td class="tcl lb rb"> ”   ”   marsh gas (methane)</td> <td class="tcr rb">0°C.</td> <td class="tcl rb">0.1586  ”</td> <td class="tcc rb">”</td></tr> +<tr><td class="tcl lb rb"> ”   ”   nitrous oxide</td> <td class="tcr rb">0°C.</td> <td class="tcl rb">0.0983  ”</td> <td class="tcc rb">”</td></tr> +<tr><td class="tcl lb rb">Hydrogen and oxygen</td> <td class="tcr rb">0°C.</td> <td class="tcl rb">0.7214  ”</td> <td class="tcc rb">”</td></tr> +<tr><td class="tcl lb rb"> ”   ” carbon monoxide</td> <td class="tcr rb">0°C.</td> <td class="tcl rb">0.6422  ”</td> <td class="tcc rb">”</td></tr> +<tr><td class="tcl lb rb"> ”   ” sulphur dioxide</td> <td class="tcr rb">0°C.</td> <td class="tcl rb">0.4800  ”</td> <td class="tcc rb">”</td></tr> +<tr><td class="tcl lb rb">Oxygen and carbon monoxide</td> <td class="tcr rb">0°C.</td> <td class="tcl rb">0.1802  ”</td> <td class="tcc rb">”</td></tr> +<tr><td class="tcl lb rb">Water and ammonia</td> <td class="tcr rb">20°C.</td> <td class="tcl rb">1.250   ”</td> <td class="tcl rb">G. Hüfner.</td></tr> +<tr><td class="tcl lb rb"> ”   ”</td> <td class="tcr rb">5°C.</td> <td class="tcl rb">0.822   ”</td> <td class="tcc rb">”</td></tr> +<tr><td class="tcl lb rb"> ”   common salt (density 1.0269)</td> <td class="tcr rb"> </td> <td class="tcl rb">0.355   ”</td> <td class="tcl rb">J. Graham.</td></tr> +<tr><td class="tcl lb rb"> ”     ”   ”</td> <td class="tcr rb">14.33°C.</td> <td class="tcl rb">1.020, 0.996, 0.972, 0.932 cm²/day.</td> <td class="tcl rb"> F. Heimbrodt.</td></tr> +<tr><td class="tcl lb rb"> ”   zinc sulphate (0.312 gm/cm³)</td> <td class="tcr rb"> </td> <td class="tcl rb">0.1162 cm²/day.</td> <td class="tcl rb">W. Seitz.</td></tr> +<tr><td class="tcl lb rb"> ”   zinc sulphate (normal)</td> <td class="tcr rb"> </td> <td class="tcl rb">0.2355  ”</td> <td class="tcc rb">”</td></tr> +<tr><td class="tcl lb rb"> ”   zinc acetate (double normal)</td> <td class="tcr rb"> </td> <td class="tcl rb">0.1195  ”</td> <td class="tcc rb">”</td></tr> +<tr><td class="tcl lb rb"> ”   zinc formate (half normal)</td> <td class="tcr rb"> </td> <td class="tcl rb">0.4654  ”</td> <td class="tcc rb">”</td></tr> +<tr><td class="tcl lb rb"> ”   cadmium sulphate (double normal)</td> <td class="tcc rb">· ·</td> <td class="tcl rb">0.2456  ”</td> <td class="tcc rb">”</td></tr> +<tr><td class="tcl lb rb"> ”   glycerin (<span class="spp">1</span>⁄<span class="suu">8</span>n, ½n, <span class="spp">7</span>⁄<span class="suu">8</span>n, 1.5n)</td> <td class="tcr rb">10.14°C.</td> <td class="tcl rb">0.356, 0.350, 0.342, 0.315 cm²/day.</td> <td class="tcl rb">F. Heimbrodt.</td></tr> +<tr><td class="tcl lb rb"> ”   urea   ”   ”</td> <td class="tcr rb">14.83°C.</td> <td class="tcl rb">0.973, 0.946, 0.926, 0.883 cm²/day.</td> <td class="tcc rb">”</td></tr> +<tr><td class="tcl lb rb"> ”   hydrochloric acid</td> <td class="tcr rb">14.30°C.</td> <td class="tcl rb">2.208, 2.331, 2.480 cm²/day.</td> <td class="tcc rb">”</td></tr> +<tr><td class="tcl lb rb">Gelatin 20% and ammonia</td> <td class="tcr rb">17°C.</td> <td class="tcl rb">127.1 cm²/day.</td> <td class="tcl rb">A. Hagenbach.</td></tr> +<tr><td class="tcl lb rb"> ”  ”  carbon dioxide</td> <td class="tcc rb">· ·</td> <td class="tcl rb">0.845   ”</td> <td class="tcc rb">”</td></tr> +<tr><td class="tcl lb rb"> ”  ”  nitrous oxide</td> <td class="tcc rb">· ·</td> <td class="tcl rb">0.509   ”</td> <td class="tcc rb">”</td></tr> +<tr><td class="tcl lb rb"> ”  ”  oxygen</td> <td class="tcc rb">· ·</td> <td class="tcl rb">0.230   ”</td> <td class="tcc rb">”</td></tr> +<tr><td class="tcl lb rb bb"> ”  ”  hydrogen</td> <td class="tcc rb bb">· ·</td> <td class="tcl rb bb">0.0565  ”</td> <td class="tcc rb bb">”</td></tr> +</table> + +<div class="condensed"> +<p>Let us now assume that we have any syste m such as the gases +above considered, and that it is in the presence of an indefinitely +extended medium which we shall call the “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.</p> + +<p>The amount of unavailable energy associated with any given +transformation is proportional to the absolute temperature of the +auxiliary medium. When divided by that temperature the quotient +is called the change of “entropy” associated with the given change +(see <span class="sc"><a href="#artlinks">Thermodynamics</a></span>). Thus if a body at temperature T receives +a quantity of heat Q, and if T<span class="su">0</span> is the temperature of the auxiliary +medium, the quantity of work which could be obtained from Q by +means of ideal thermodynamic engines would be Q(1 − T<span class="su">0</span>/T), and +the balance, which is QT<span class="su">0</span>/T, would take the form of unavailable +or waste energy given to the medium. The quotient of this, when +divided by T<span class="su">0</span>, is Q/T, and this represents the quantity of entropy +associated with Q units of heat at temperature T.</p> + +<p>Any irreversible change for which a compensating transformation +of energy exists represents, therefore, an increase of unavailable +energy, which is measurable in terms of entropy. The increase of +entropy is independent of the temperature of the auxiliary medium. +It thus affords a measure of the extent to which energy has run +to waste during the change. Moreover, when a body is heated, the +increase of entropy is the factor which determines how much of the +energy imparted to the body is unavailable for conversion into work +under given conditions. In all cases we have</p> + +<table class="math0" summary="math"> +<tr><td>increase of unavailable energy</td> <td rowspan="2">= increase of entropy.</td></tr> +<tr><td class="denom">temperature of auxiliary medium</td></tr></table> + +<p>When diffusion takes place between two gases inside a closed +vessel at uniform pressure and temperature no energy in the form +of heat or work is received from without, and hence the entropy +gained by the gases from without is zero. But the irreversible +processes inside the vessel may involve a gain of entropy, and this +can only be estimated by examining +by what means mixed +gases can be separated, and, in +particular, under what conditions +the process of mixing +and separating the gases could +(theoretically) be made reversible.</p> +</div> + +<p>9. <i>Evidence derived from +Liquefaction of one or both of +the Gases.</i>—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 <i>partial</i> pressure in the other vessel, notwithstanding the fact +that the <i>total</i> pressure in the latter vessel is greater by the +pressure of the air.</p> + +<div class="condensed"> +<p>To separate mixed gases by liquefaction, they must be compressed +and cooled till one separates in the form of a liquid. If no changes are +to take place outside the system, the separate components must be +allowed to expand until the work of expansion is equal to the work +of compression, and the heat given out in compression is reabsorbed +in expansion. The process may be made as nearly reversible as we +like by performing the operations so slowly that the substances +are practically in a state of equilibrium at every stage. This is a +consequence of an important axiom in thermodynamics according +to which “any small change in the neighbourhood of a state of +equilibrium is to a first approximation reversible.”</p> + +<p>Suppose now that at any stage of the compression the partial +pressures of the two gases are p<span class="su">1</span> and p<span class="su">2</span>, and that the volume is +changed from V to V − dV. The work of compression is (p<span class="su">1</span> + p<span class="su">2</span>)dV, +and this work will be restored at the corresponding stage if each +of the separated gases increases in volume from V − dV to V. The +ultimate state of the separated gases will thus be one in which +each gas occupies the volume V originally occupied by the mixture.</p> + +<p>We may now obtain an estimate of the amount of energy rendered +unavailable by diffusion. We suppose two gases occupying volumes +V<span class="su">1</span> and V<span class="su">2</span> at equal pressure p to mix by diffusion, so that the final +volume is V<span class="su">1</span> + V<span class="su">2</span>. Then if before mixing each gas had been allowed +to expand till its volume was V<span class="su">1</span> + V<span class="su">2</span>, work would have been done +in the expansion, and the gases could still have been mixed by a +reversal of the process above described. In the actual diffusion this +work of expansion is lost, and represents energy rendered unavailable +at the temperature at which diffusion takes place. When divided +by that temperature the quotient gives the increase of entropy. +Thus the irreversible processes, and, in particular, the entropy +changes associated with diffusion of two gases at uniform pressure, +are the same as would take place if each of the gases in turn were to +expand by rushing into a vacuum, till it occupied the whole volume +of the mixture. A more rigorous proof involves considerations of +the thermodynamic potentials, following the methods of J. Willard +Gibbs (see <span class="sc"><a href="#artlinks">Energetics</a></span>).</p> + +<p><span class="pagenum"><a name="page259" id="page259"></a>259</span></p> + +<p>Another way in which two or more mixed gases can be separated +is by placing them in the presence of a liquid which can freely absorb +one of the gases, but in which the other gas or gases are insoluble. +Here again it is found by experience that when equilibrium exists +at a given temperature between the dissolved and undissolved +portions of the first gas, the partial pressure of that gas in the +mixture depends on the temperature alone, and is independent of +the partial pressures of the insoluble gases with which it is mixed, +so that the conclusions are the same as before.</p> +</div> + +<p>10. <i>Diffusion through a Membrane or Partition. Theory of the +semi-permeable Membrane.</i>—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.”</p> + +<p>In such cases as can be tested experimentally it has been found +that a gas always tends to pass through a membrane from the side +where its density, and therefore its partial pressure, is greater +to the side where it is less; so that for equilibrium the partial +pressures on the two sides must be equal. This result is unaffected +by the presence of other gases on one or both sides of the +membrane. For example, if different gases at the same pressure +are separated by a partition through which one gas can pass more +rapidly than the other, the diffusion will give rise to a difference of +pressure on the two sides, which is capable of doing mechanical +work in moving the partition. In evidence of this conclusion +Max Planck quotes a test experiment made by him in the Physical +Institute of the university of Munich in 1883, depending on the +fact that platinum foil at white heat is permeable to hydrogen but +impermeable to air, so that if a platinum tube filled with hydrogen +be heated the hydrogen will diffuse out, leaving a vacuum.</p> + +<div class="condensed"> +<p>The details of the experiment may be quoted here:—“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.”</p> +</div> + +<table class="nobctr" style="float: left; width: 370px;" summary="Illustration"> +<tr><td class="figleft1"><img style="width:317px; height:156px" src="images/img259.jpg" alt="" /></td></tr></table> + +<p>In order that diffusion through a membrane may be reversible +so far as a particular gas is concerned, the process must take place +so slowly that equilibrium is set up at every stage (see § 9 above). +In order to separate one +gas from another consistently +with this condition +it is necessary +that no diffusion of the +latter gas should accompany +the process. +The name “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.</p> + +<div class="condensed"> +<p>We suppose the gases contained in a cylindrical tube; P, Q, R, S +are four pistons, of which P and R are joined to one connecting rod, +Q and S to another. P, S are impermeable to both gases; Q is +semi-permeable, allowing the gas A to pass through but not B, similarly +R allows the gas B to pass through but not A. The distance PR +is equal to the distance QS, so that if the rods are pushed towards each +other as far as they will go, P and Q will be in contact, as also R and +S. Imagine the space RQ filled with a mixture of the two gases +under these conditions. Then by slowly drawing the connecting +rods apart until R, Q touch, the gas A will pass into the space PQ, +and B will pass into the space RS, and the gases will finally be completely +separated; similarly, by pushing the connecting rods together, +the two gases will be remixed in the space RQ. By performing the +operations slowly enough we may make the processes as nearly +reversible as we please, so that no available energy is lost in either +change. The gas A being at every instant in equilibrium on the two +sides of the piston Q, its density, and therefore its partial pressure, +is the same on both sides, and the same is true regarding the gas B +on the two sides of R. Also <i>no work is done in moving the pistons</i>, for +the partial pressures of B on the two sides of R balance each other, +consequently, the resultant thrust on R is due to the gas A alone, +and is equal and opposite to its resultant thrust on P, so that the +connecting rods are at every instant in a state of mechanical equilibrium +so far as the pressures of the gases A and B are concerned. We +conclude that in the reversible separation of the gases by this method +at constant temperature without the production or absorption of +mechanical work, the densities and the partial pressures of the two +separated gases are the same as they were in the mixture. These +conclusions are in entire agreement with those of the preceding +section. If this agreement did not exist it would be possible, theoretically, +to obtain perpetual motion from the gases in a way that +would be inconsistent with the second law of thermodynamics.</p> +</div> + +<p>Most physicists admit, as Planck does, that it is impossible to +obtain an ideal semi-permeable substance; indeed such a substance +would necessarily have to possess an infinitely great resistance +to diffusion for such gases as could not penetrate it. But in +an experiment performed under actual conditions the losses of +available energy arising from this cause would be attributable +to the imperfect efficiency of the partitions and not to the gases +themselves; moreover, these losses are, in every case, found to be +completely in accordance with the laws of irreversible thermodynamics. +The reasoning in this article being somewhat condensed +the reader must necessarily be referred to treatises on +thermodynamics for further information on points of detail +connected with the argument. Even when he consults these +treatises he may find some points omitted which have been +examined in full detail at some time or other, but are not sufficiently +often raised to require mention in print.</p> + +<p>II. <i>Kinetic Models of Diffusion.</i>—Imagine in the first instance +that a very large number of red balls are distributed over one half +of a billiard table, and an equal number of white balls over the +other half. If the balls are set in motion with different velocities +in various directions, diffusion will take place, the red balls finding +their way among the white ones, and vice versa; and the +process will be retarded by collisions between the balls. The +simplest model of a perfect gas studied in the kinetic theory of +gases (see <span class="sc"><a href="#artlinks">Molecule</a></span>) differs from the above illustration in that +the bodies representing the molecules move in space instead of in +a plane, and, unlike billiard balls, their motion is unresisted, +and they are perfectly elastic, so that no kinetic energy is lost +either during their free motions, or at a collision.</p> + +<div class="condensed"> +<p>The mathematical analysis connected with the application of the +kinetic theory to diffusion is very long and cumbersome. We shall +therefore confine our attention to regarding a medium formed of +elastic spheres as a mechanical model, by which the most important +features of diffusion can be illustrated. We shall assume the results +of the kinetic theory, according to which:—(1) In a dynamical +model of a perfect gas the mean kinetic energy of translation of the +molecules represents the absolute temperature of the gas. (2) The +pressure at any point is proportional to the product of the number +of molecules in unit volume about that point into the mean square +of the velocity. (The mean square of the velocity is different from +but proportional to the square of the mean velocity, and in the +subsequent arguments either of these two quantities can generally +be taken.) (3) In a gas mixture represented by a mixture of molecules +of unequal masses, the mean kinetic energies of the different +kinds are equal.</p> + +<p>Consider now the problem of diffusion in a region containing two +kinds of molecules A and B of unequal mass. The molecules of A +in the neighbourhood of any point will, by their motion, spread out +in every direction until they come into collision with other molecules +of either kind, and this spreading out from every point of the medium +will give rise to diffusion. If we imagine the velocities of the A +molecules to be equally distributed in all directions, as they would +be in a homogeneous mixture, it is obvious that the process of diffusion +will be greater, <i>ceteris paribus</i>, the greater the velocity of the molecules, +and the greater the length of the free path before a collision +takes place. If we assume consistently with this, that the coefficient +of diffusion of the gas A is proportional to the mean value of +W{a}l{a}, where w{a} is the velocity and l{a} is the length of the path of a +<span class="pagenum"><a name="page260" id="page260"></a>260</span> +molecule of A, this expression for the coefficient of diffusion is of the +right dimensions in length and time. If, moreover, we observe that +when diffusion takes place in a fixed direction, say that of the axis +of x, it depends only on the resolved part of the velocity and length +of path in that direction: this hypothesis readily leads to our taking +the mean value of <span class="spp">1</span>⁄<span class="suu">3</span>w<span class="su">a</span>l<span class="su">a</span> as the coefficient of diffusion for the gas A. +This value was obtained by O. E. Meyer and others.</p> + +<p>Unfortunately, however, it makes the coefficients of diffusion +unequal for the two gases, a result inconsistent with that obtained +above from considerations of the coefficient of resistance, and +leading to the consequence that differences of pressure would be +set up in different parts of the gas. To equalize these differences of +pressure, Meyer assumed that a counter current is set up, this current +being, of course, very slow in practice; and J. Stefan assumed that +the diffusion of one gas was not affected by collisions between molecules +of the <i>same gas</i>. When the molecules are mixed in equal +proportions both hypotheses lead to the value <span class="spp">1</span>⁄<span class="suu">6</span>([w<span class="su">a</span>l<span class="su">a</span>] + [w<span class="su">b</span>l<span class="su">b</span>]), +(square brackets denoting mean values). When one gas preponderates +largely over the other, the phenomena of diffusion are too +difficult of observation to allow of accurate experimental tests +being made. Moreover, in this case no difference exists unless the +molecules are different in size or mass.</p> + +<p>Instead of supposing a velocity of translation added after the +mathematical calculations have been performed, a better plan is to +assume from the outset that the molecules of the two gases have +small velocities of translation in opposite directions, superposed on +the distribution of velocity, which would occur in a medium representing +a gas at rest. When a collision occurs between molecules +of different gases a transference of momentum takes place between +them, and the quantity of momentum so transferred in one second +in a unit of volume gives a dynamical measure of the resistance to +diffusion. It is to be observed that, however small the relative +velocity of the gases A and B, it plays an all-important part in +determining the coefficient of resistance; for without such relative +motion, and with the velocities evenly distributed in all directions, no +transference of momentum could take place. The coefficient of +resistance being found, the motion of each of the two gases may be +discussed separately.</p> +</div> + +<p>One of the most important consequences of the kinetic theory +is that if the volume be kept constant the coefficient of diffusion +varies as the square root of the absolute temperature. To prove +this, we merely have to imagine the velocity of each molecule to +be suddenly increased n fold; the subsequent processes, including +diffusion, will then go on n times as fast; and the temperature +T, being proportional to the kinetic energy, and therefore to the +square of the velocity, will be increased n² fold. Thus K, the +coefficient of diffusion, varies as √T.</p> + +<p>The relation of K to the density when the temperature remains +constant is more difficult to discuss, but it may be sufficient to +notice that if the number of molecules is increased n fold, the +chances of a collision are n times as great, and the distance +traversed between collisions is (not <i>therefore</i> but as the result of +more detailed reasoning) on the average 1/n of what it was before. +Thus the free path, and therefore the coefficient of diffusion, +varies inversely as the density, or directly as the volume. If the +pressure p and temperature T be taken as variables, K varies +inversely as p and directly as √T³.</p> + +<p>Now according to the experiments first made by J. C. Maxwell +and J. Loschmidt, it appeared that with constant density K +was proportional to T more nearly than to √T. The inference is +that in this respect a medium formed of colliding spheres fails to +give a correct mechanical model of gases. It has been found by +L. Boltzmann, Maxwell and others that a system of particles +whose mutual actions vary according to the inverse fifth power of +the distance between them represents more correctly the relation +between the coefficient of diffusion and temperature in actual +gases. Other recent theories of diffusion have been advanced +by M. Thiesen, P. Langevin and W. Sutherland. On the other +hand, J. Thovert finds experimental evidence that the coefficient +of diffusion <i>is</i> proportional to molecular velocity in the cases +examined of non-electrolytes dissolved in water at 18° at 2.5 +grams per litre.</p> + +<div class="condensed"> +<p><span class="sc">Bibliography.</span>—The best introduction to the study of theories +of diffusion is afforded by O. E. Meyer’s Kinetic <i>Theory of Gases</i>, +translated by Robert E. Baynes (London, 1899). The mathematical +portion, though sufficient for ordinary purposes, is mostly of the +simplest possible character. Another useful treatise is R. Ruhlmann’s +<i>Handbuch der mechanischen Wärmetheorie</i> (Brunswick, 1885). For +a shorter sketch the reader may refer to J. C. Maxwell’s <i>Theory of +Heat</i>, chaps, xix. and xxii., or numerous other treatises on physics. +The theory of the semi-permeable membrane is discussed by +M. Planck in his <i>Treatise on Thermodynamics</i>, English translation +by A. Ogg (1903), also in treatises on thermodynamics by W. Voigt +and other writers. For a more detailed study of diffusion in general +the following papers may be consulted:—L. Boltzmann, “Zur +Integration der Diffusionsgleichung,” <i>Sitzung. der k. bayer. Akad math.-phys. +Klasse</i> (May 1894); T. des Coudres, “Diffusionsvorgänge in +einem Zylinder,” <i>Wied. Ann.</i> lv. (1895), p. 213; J. Loschmidt, +“Experimentaluntersuchungen über Diffusion,” <i>Wien. Sitz.</i> lxi., +lxii. (1870); J. Stefan, “Gleichgewicht und ... Diffusion von Gasmengen,” +<i>Wien. Sitz.</i> lxiii., “Dynamische Theorie der Diffusion,” +<i>Wien. Sitz.</i> lxv. (April 1872); M. Toepler, “Gas-diffusion,” <i>Wied. +Ann.</i> lviii. (1896), p. 599; A. Wretschko, “Experimentaluntersuchungen +über die Diffusion von Gasmengen,” <i>Wien. Sitz.</i> lxii. +The mathematical theory of diffusion, according to the kinetic +theory of gases, has been treated by a number of different methods, +and for the study of these the reader may consult L. Boltzmann, +<i>Vorlesungen über Gastheorie</i> (Leipzig, 1896-1898); S. H. Burbury, +<i>Kinetic Theory of Gases</i> (Cambridge, 1899), and papers by L. Boltzmann +in <i>Wien. Sitz.</i> lxxxvi. (1882), lxxxvii. (1883); P. G. Tait, +“Foundations of the Kinetic Theory of Gases,” <i>Trans. R.S.E.</i> +xxxiii., xxxv., xxvi., or <i>Scientific Papers</i>, ii. (Cambridge, 1900). +For recent work reference should be made to the current issues +of <i>Science Abstracts</i> (London), and entries under the heading +“Diffusion” will be found in the general index at the end of each +volume.</p> +</div> +<div class="author">(G. H. Br.)</div> + + +<hr class="art" /> +<p><span class="bold">DIGBY, SIR EVERARD<a name="ar86" id="ar86"></a></span> (1578-1606), English conspirator, son +of Everard Digby of Stoke Dry, Rutland, was born on the 16th +of May 1578. He inherited a large estate at his father’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 (<i>q.v.</i>) sought his +financial support, and he joined eagerly in the conspiracy. His +particular share in the plan was the organization of a rising in the +Midlands; and on the pretence of a hunting party he assembled a +body of gentlemen together at Danchurch in Warwickshire on the +5th of November, who were to take action immediately the news +arrived from London of the successful destruction of the king +and the House of Lords, and to seize the person of the princess +Elizabeth, who was residing in the neighbourhood. The conspirators +arrived late on the evening of the 6th to tell their story +of failure and disaster, and Digby, who possibly might have +escaped the more serious charge of high treason, was persuaded by +Catesby, with a false tale that the king and Salisbury were dead, +to further implicate himself in the plot and join the small band of +conspirators in their hopeless endeavour to raise the country. He +accompanied them, the same day, to Huddington in Worcestershire +and on the 7th to Holbeche in Staffordshire. The following +morning, however, he abandoned his companions, dismissed his +servants except two, who declared “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.</p> + +<p>Digby was a handsome man, of fine presence. Father Gerard +<span class="pagenum"><a name="page261" id="page261"></a>261</span> +extols his skill in sport, his “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 <i>The +Gunpowder Treason</i> by Thomas Barlow, bishop of Lincoln, in +1679. He left two sons, of whom the elder, Sir Kenelm Digby, +was the well-known author and diplomatist.</p> + +<div class="condensed"> +<p>See works on the Gunpowder Plot; Narrative of Father Gerard, +in <i>Condition of the Catholics under James I.</i> by J. Morris (1872), +&c. A life of Digby under the title of <i>A Life of a Conspirator</i>, +by a Romish Recusant (Thomas Longueville), was published in +1895.</p> +</div> +<div class="author">(P. C. Y.)</div> + + +<hr class="art" /> +<p><span class="bold">DIGBY, SIR KENELM<a name="ar87" id="ar87"></a></span> (1603-1665), English author, diplomatist +and naval commander, son of Sir Everard Digby (<i>q.v.</i>), +was born on the 11th of July 1603, and after his father’s execution +in 1606 resided with his mother at Gayhurst, being brought up +apparently as a Roman Catholic. In 1617 he accompanied his +cousin, Sir John Digby, afterwards 1st earl of Bristol, and then +ambassador in Spain, to Madrid. On his return in April 1618 he +entered Gloucester Hall (now Worcester College), Oxford, and +studied under Thomas Allen (1542-1632), the celebrated mathematician, +who was much impressed with his abilities and called +him the <i>Mirandula</i>, <i>i.e.</i> the infant prodigy, of his age.<a name="fa1h" id="fa1h" href="#ft1h"><span class="sp">1</span></a> He left +the university without taking a degree in 1620, and travelled +in France, where, according to his own account, he inspired an +uncontrollable passion in the queen-mother, Marie de’ 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.”<a name="fa2h" id="fa2h" href="#ft2h"><span class="sp">2</span></a> 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 <i>Eupheme</i>, and by other poets of +the day. Digby retired to Gresham College, and exhibited extravagant +grief, maintaining a seclusion for two years. About +this time Digby professed himself a Protestant, but by October +1635, while in France, he had already returned to the Roman +Catholic faith.<a name="fa3h" id="fa3h" href="#ft3h"><span class="sp">3</span></a> In a letter dated the 27th of March 1636 Laud +remonstrates with him, but assures him of the continuance of his +friendship.<a name="fa4h" id="fa4h" href="#ft4h"><span class="sp">4</span></a> In 1638 he published <i>A Conference with a Lady about +choice of a Religion</i>, in which he argues that the Roman Church, +possessing alone the qualifications of universality, unity of +doctrine and uninterrupted apostolic succession, is the only true +church, and that the intrusion of error into it is impossible. The +same subject is treated in letters to George Digby, afterwards +2nd earl of Bristol, dated the 2nd of November 1638 and the 29th +of November 1639, which were published in 1651, as well as in +a further <i>Discourse concerning Infallibility in Religion</i> in 1652. +Returning to England he associated himself with the queen and +her Roman Catholic friends, and joined in the appeal to the +English Romanists for money to support the king’s Scottish +expedition.<a name="fa5h" id="fa5h" href="#ft5h"><span class="sp">5</span></a> In consequence he was summoned to the bar of +the House of Commons on the 27th of January 1641, and the +king was petitioned to remove him with other recusants from his +councils. He left England, and while at Paris killed in a duel a +French lord who had insulted Charles I. in his presence. Louis +XIII. took his part, and furnished him with a military escort into +Flanders. Returning home he was imprisoned, by order of the +House of Commons, early in 1642, successively in the “Three +Tobacco Pipes nigh Charing Cross,” where his delightful conversation +is said to have transformed the prison into “a place of +delight,”<a name="fa6h" id="fa6h" href="#ft6h"><span class="sp">6</span></a> and at Winchester House. He was finally released and +allowed to go to France on the 30th of July 1643, through the +intervention of the queen of France, Anne of Austria, on condition +that he would neither promote nor conceal any plots abroad +against the English government.</p> + +<p>Before leaving England an attempt was made to draw from +him an admission that Laud, with whom he had been intimate, +had desired to be made a cardinal, but Digby denied that the +archbishop had any leanings towards Rome. On the 1st of +November 1643 it was resolved by the Commons to confiscate his +property. He published in London the same year <i>Observations +on the 22nd stanza in the 9th canto of the 2nd book of Spenser’s +“Faërie Queene,”</i> the MS. of which is in the Egerton collection +(British Museum, No. 2725 f. 117 b), and <i>Observations</i> on a +surreptitious and unauthorized edition of the <i>Religio Medici</i>, by +Sir Thomas Browne, from the Roman Catholic point of view, +which drew a severe rebuke from the author. After his arrival +in Paris he published his chief philosophical works, <i>Of Bodies</i> +and <i>Of the Immortality of Man’s Soul</i> (1644), autograph MSS. of +which are in the Bibliothèque Ste Geneviève at Paris, and made +the acquaintance of Descartes. He was appointed by Queen +Henrietta Maria her chancellor, and in the summer of 1645 he was +despatched by her to Rome to obtain assistance. Digby promised +the conversion of Charles and of his chief supporters. At first his +eloquence made a great impression. Pope Innocent X. declared +that he spoke not merely as a Catholic but as an ecclesiastic. +But the absence of any warrant from Charles himself roused +suspicions as to the solidity of his assurances, and he obtained +nothing but a grant of 20,000 crowns. A violent quarrel with the +pope followed, and he returned in 1646, having consented in the +queen’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 +<span class="pagenum"><a name="page262" id="page262"></a>262</span> +Spain, and was engaged by Cromwell, much to the scandal of both +Royalists and Roundheads, in negotiations abroad, of which the +aim was probably to prevent a union between those two foreign +powers. He visited Germany, in 1660 was in Paris, and at the +Restoration returned to England. He was well received in spite +of his former relations with Cromwell, and was confirmed in his +post as Queen Henrietta Maria’s chancellor. In January 1661 +he delivered a lecture, which was published the same month, at +Gresham College, on the vegetation of plants, and became an +original member of the Royal Society in 1663. In January 1664 +he was forbidden to appear at court, the cause assigned being that +he had interposed too far in favour of the 2nd earl of Bristol, +disgraced by the king on account of the charge of high treason +brought by him against Clarendon into the House of Lords. The +rest of his life was spent in the enjoyment of literary and scientific +society at his house in Covent Garden. He died on the 11th of +June 1665. He had five children, of whom two, a son and one +daughter, survived him.</p> + +<p>Digby, though he possessed for the time a considerable knowledge +of natural science, and is said to have been the first to +explain the necessity of oxygen to the existence of plants, bears +no high place in the history of science. He was a firm believer in +astrology and alchemy, and the extraordinary fables which he +circulated on the subject of his discoveries are evidence of anything +rather than of the scientific spirit. In 1656 he made public +a marvellous account of a city in Tripoli, petrified in a few hours, +which he printed in the <i>Mercurius Politicus</i>. Malicious reports +had been current that his wife had been poisoned by one of his +prescriptions, viper wine, taken to preserve her beauty. Evelyn, +who visited him in Paris in 1651, describes him as an “errant +mountebank.” Henry Stubbes characterizes him as “the very +Pliny of our age for lying,” and Lady Fanshawe refers to the same +“infirmity.”<a name="fa7h" id="fa7h" href="#ft7h"><span class="sp">7</span></a> 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 <i>History of +Generation</i> (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 <i>The Closet of the +Eminently Learned Sir Kenelm Digby Knt. Opened</i> (publ. 1677), +are said to make less demand upon the faith of patients, and his +injunction on the subject of the making of tea, to let the water +“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 <i>Institutionum Peripateticorum libri +quinque</i>, purporting to expound Digby’s “peripatetic philosophy,” +but going far beyond Digby’s published treatises. +Digby’s <i>Memoirs</i> are composed in the high-flown fantastic manner +then usual when recounting incidents of love and adventure, +but the style of his more sober works is excellent. In 1632 he +presented to the Bodleian library a collection of 236 MSS., bequeathed +to him by his former tutor Thomas Allen, and described +in <i>Catalogi codicum manuscriptorum bibliothecae Bodleianae</i>, by +W. D. Macray, part ix. Besides the works already mentioned +Digby translated <i>A Treatise of adhering to God written by Albert +the Great, Bishop of Ratisbon</i> (1653); and he was the author of +<i>Private Memoirs</i>, published by Sir N. H. Nicholas from <i>Harleian +MS. 6758</i> with introduction (1827); <i>Journal of the Scanderoon +Voyage in 1628</i>, printed by J. Bruce with preface (Camden +Society, 1868); <i>Poems from Sir Kenelm Digby’s Papers</i>... with +preface and notes (Roxburghe Club, 1877); in the <i>Add. MSS.</i> +34,362 f. 66 is a poem <i>Of the Miserys of Man</i>, probably by Digby; +<i>Choice of Experimental Receipts in Physick and Chirurgery</i> ... +<i>collected by Sir K. Digby</i> (1668), and <i>Chymical Secrets and Rare +Experiments</i> (1683), were published by G. Hartman, who describes +himself as Digby’s steward and laboratory assistant.</p> + +<div class="condensed"> +<p>See the <i>Life of Sir Kenelm Digby by one of his Descendants</i> +(T. Longueville), 1896.</p> +</div> +<div class="author">(P. C. Y.)</div> + +<hr class="foot" /> <div class="note"> + +<p><a name="ft1h" id="ft1h" href="#fa1h"><span class="fn">1</span></a> <i>Letters by Eminent Persons</i> (Aubrey’s Lives), ii. 324.</p> + +<p><a name="ft2h" id="ft2h" href="#fa2h"><span class="fn">2</span></a> <i>Life and Continuation.</i></p> + +<p><a name="ft3h" id="ft3h" href="#fa3h"><span class="fn">3</span></a> Strafford’s <i>Letters</i>, i. 474.</p> + +<p><a name="ft4h" id="ft4h" href="#fa4h"><span class="fn">4</span></a> Laud’s <i>Works</i>, vi. 447.</p> + +<p><a name="ft5h" id="ft5h" href="#fa5h"><span class="fn">5</span></a> <i>Thomason Tracts</i>, Brit. Mus. E 164 (15).</p> + +<p><a name="ft6h" id="ft6h" href="#fa6h"><span class="fn">6</span></a> <i>Archaeologia Cantiana</i>, ii. 190.</p> + +<p><a name="ft7h" id="ft7h" href="#fa7h"><span class="fn">7</span></a> <i>Dict. of Nat. Biog.</i> sub “Digby.” See also Robert Boyle’s +<i>Works</i> (1744), v. 302.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DIGBY, KENELM HENRY<a name="ar88" id="ar88"></a></span> (1800-1880), English writer, +youngest son of William Digby, dean of Clonfert, was born at +Clonfert, Ireland, in 1800. He was educated at Trinity College, +Cambridge, and soon after taking his B.A. degree there in 1819 +became a Roman Catholic. He spent most of his life, which was +mainly devoted to literary pursuits, in London, where he died on +the 22nd of March 1880. Digby’s reputation rests chiefly on his +earliest publication, <i>The Broadstone of Honour, or Rules for the +Gentlemen of England</i> (1822), which contains an exhaustive survey +of medieval customs, full of quotations from varied sources. The +work was subsequently enlarged and issued (1826-1827) in four +volumes entitled: <i>Godefridus</i>, <i>Tancredus</i>, <i>Morus</i> and <i>Orlandus</i> +(numerous re-impressions, the best of which is the edition +brought out by B. Quaritch in five volumes, 1876-1877).</p> + +<div class="condensed"> +<p>Among Digby’s other works are: <i>Mores Catholici, or Ages of +Faith</i> (11 vols., London, 1831-1840); <i>Compitum; or the Meeting of +the Ways at the Catholic Church</i> (7 vols., London, 1848-1854); <i>The +Lovers’ Seat, Kathemérina; or Common Things in relation to Beauty, +Virtue and Faith</i> (2 vols., London, 1856). A complete list is given +in J. Gillow’s <i>Bibliographical Dictionary of English Catholics</i>, ii. +81-83.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DIGENES ACRITAS, BASILIUS,<a name="ar89" id="ar89"></a></span> Byzantine national hero, +probably lived in the 10th century. He is named Digenes (of +double birth) as the son of a Moslem father and a Christian +mother; Acritas (<span class="grk" title="akra">ἄκρα</span>, frontier, boundary), as one of the frontier +guards of the empire, corresponding to the Roman <i>milites +limitanei</i>. The chief duty of these <i>acritae</i> consisted in repelling +Moslem inroads and the raids of the <i>apelatae</i> (cattle-lifters), +brigands who may be compared with the more modern Klephts. +The original Digenes epic is lost, but four poems are extant, in +which the different incidents of the legend have been worked +up by different hands. The first of these consists of about 4000 +lines, written in the so-called “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 <i>Chanson de Roland</i> and the <i>Romance of the Cid</i>, +undoubtedly contains a kernel of fact, although it cannot be +regarded as in any sense an historical record. The scene of action +is laid in Cappadocia and the district of the Euphrates.</p> + +<div class="condensed"> +<p>Editions of the Trebizond MS. by C. Sathas and E. Legrand in +the <i>Collection des monuments pour servir à l’étude de la langue néohellénique</i>, +new series, vi. (1875), and by S. Joannides (Constantinople, +1887). See monographs by A. Luber (Salzburg, 1885) and G. +Wartenberg (Berlin, 1897). Full information will be found in +C. Krumbacher’s <i>Geschichte der byzantinischen Litteratur</i>, p. 827 +(2nd ed., 1897); see also G. Schlumberger, <i>L’Épopée Byzantine à +la fin du dixième siècle</i> (1897).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DIGEST,<a name="ar90" id="ar90"></a></span> a term used generally of any digested or carefully +arranged collection or compendium of written matter, but more +particularly in law of a compilation in condensed form of a body +of law digested in a systematical method; <i>e.g.</i> the Digest (<i>Digesta</i>) +or Pandects (<span class="grk" title="Pandektai">Πάνδεκται</span>) of Justinian, a collection of extracts +from the earlier jurists compiled by order of the emperor +Justinian. The word is also given to the compilations of the +main points (marginal or hand-notes) of decided cases, usually +arranged in alphabetical and subject order, and published under +such titles as “Common Law Digest,” “Annual Digest,” &c.</p> + + +<hr class="art" /> +<p><span class="bold">DIGESTIVE ORGANS<a name="ar91" id="ar91"></a></span> (<span class="sc">Pathology</span>). Several facts of importance +have to be borne in mind for a proper appreciation of +the pathology of the organs concerned in digestive processes (for +the anatomy see <span class="sc"><a href="#artlinks">Alimentary Canal</a></span> and allied articles). In +the first place, more than all other systems, the digestive comprises +greater range of structure and exhibits wider diversity of function +within its domain. Each separate structure and each different +function presents special pathological signs and symptoms. +Again, the duties imposed upon the system have to be performed +<span class="pagenum"><a name="page263" id="page263"></a>263</span> +notwithstanding constant variations in the work set them. The +crude articles of diet offered them vary immensely in nature, bulk +and utility, from which they must elaborate simple food-elements +for absorption, incorporate them after absorption into complex +organic substances properly designed to supply the constant needs +of cellular activity, of growth and repair, and fitly harmonized +to fulfil the many requirements of very divergent processes and +functions. Any form of unphysiological diet, each failure to +cater for the wants of any special tissue engaged in, or of any +processes of, metabolism, carry with them pathological signs. +Perhaps in greater degree than elsewhere are the individual +sections of the digestive system dependent upon, and closely +correlated with, one another. The lungs can only yield oxygen +to the blood when the oxygen is uncombined; no compounds +are of use. The digestive organs have to deal with an enormous +variety of compound bodies, from which to obtain the elements +necessary for protoplasmic upkeep and activity. Morbid lesions +of the respiratory and circulatory systems are frequently capable +of compensation through increased activity elsewhere, and the +symptoms they give rise to follow chiefly along one line; diseases +of the digestive organs are more liable to occasion disorders +elsewhere than to excite compensatory actions. The digestive +system includes every organ, function and process concerned +with the utilization of food-stuffs, from the moment of their +entrance into the mouth, their preparation in the canal, assimilation +with the tissues, their employment therein, up to their +excretion or expulsion in the form of waste. Each portion +resembles a link of a continuous chain; each link depends upon +the integrity of the others, the weakening or breaking of one +straining or making impotent the chain as a whole.</p> + +<p>The mucous membrane lining the alimentary tract is the part +most subject to pathological alterations, and in this connexion +it should be remembered that this membrane differs both in +structure and functions throughout the tract. Chiefly protective +from the mouth to the cardia, it is secretory and absorbent in the +stomach and bowel; while the glandular cells forming part of it +secrete both acid and alkaline fluids, several ferments or mucus. +Over the dorsum of the tongue its modified cells subserve the +sense of taste. Without, connected with it by the submucous +connective tissue, is placed the muscular coat, and externally over +the greater portion of its length the peritoneal serous membrane. +All parts are supplied with blood-vessels, lymph-ducts and +nerves, the last belonging either to local or to central circuits. +Associated with the tract are the salivary glands, the liver and +the pancreas; while, in addition, lymphoid tissue is met with +diffusely scattered throughout the lining membranes in the tonsils, +appendix, solitary glands and Peyer’s patches, and the mesenteric +glands. The functions of the various parts of the system in whose +lesions we are here interested are many in number, and can only +be summarized here. (For the physiology of digestion see +<span class="sc"><a href="#artlinks">Nutrition</a></span>.) Broadly, they maybe given as: (1) Ingestion and +swallowing of food, transmission of it through the tract, and +expulsion of the waste material; (2) secretion of acids and +alkalis for the performance of digestive processes, aided by (3) +elaboration and addition of complex bodies, termed enzymes +or ferments; (4) secretion of mucus; (5) protection of the body +against organismal infection, and against toxic products; (6) +absorption of food elements and reconstitution of them into +complex substances fitted for metabolic application; and (7) +excretion of the waste products of protoplasmic action. These +functions may be altered by disease, singly or in conjunction; it +is rare, however, to find but one affected, while an apparently +identical disturbance of function may often arise from totally +different organic lesions. Another point of importance is seen in +the close interdependence which exists between the secretions of +acid and those of alkaline reaction. The difference in reaction +seems to act <i>mutatis mutandis</i> as a stimulant in each instance.</p> + +<p class="center pt1"><i>General Diseases.</i></p> + +<p>In all sections of the alimentary canal actively engaged in the +digestion of food, a well-marked local engorgement of the blood-vessels +supplying the walls occurs. The hyperaemia abates soon +after completion of the special duties of the individual sections. +<span class="sidenote">Vascular lesions.</span> +This normal condition may be abnormally exaggerated by overstimulation +from irritant poisons introduced into the +canal; from too rich, too copious or indigestible +articles of diet; or from too prolonged an experience +of some unvaried kind of food-stuff, especially if large quantities +of it are necessary for metabolic needs; entering into the first +stage of inflammation, acute hyperaemia. More important, +because productive of less tractable lesions, is passive congestion +of the digestive organs. Whenever the flow of blood into the +right side of the heart is hindered, whether it arise from disease +of the heart itself, or of the lungs, or proceed from obstruction in +some part of the portal system, the damming-back of the venous +circulation speedily produces a more or less pronounced stasis of +the blood in the walls of the alimentary canal and in the associated +abdominal glands. The lack of a sufficiently vigorous flow of +blood is followed by deficient secretion of digestive agents from +the glandular elements involved, by decreased motility of the +muscular coats of the stomach and bowel, and lessened adaptability +throughout for dealing with even slight irregular demands +on their powers. The mucous membrane of the stomach and +bowel, less able to withstand the effects of irritation, even of a +minor character, readily passes into a condition of chronic +catarrh, while it frequently is the seat of small abrasions, +haemorrhagic erosions, which may cause vomiting of blood and +the appearance of blood in the stools. Obstruction to the flow +of blood from the liver leads to dilatation of its blood-vessels, +consequent pressure upon the hepatic cells adjoining them, and +their gradual loss of function, or even atrophy and degeneration. +In addition to the results of such passive congestion exhibited +by the stomach and bowel as noted above, passive congestion +of the liver is often accompanied by varicose enlargement of the +abdominal veins, in particular of those which surround the lower +end of the oesophagus, the lowest part of the rectum and anus. +In the latter position these dilated veins constitute what are +known as haemorrhoids or piles, internal or external as their +site lies within or outside the anal aperture.</p> + +<p>The mucous and serous membranes of the canal and the +glandular elements of the associated organs are the parts most +subject to inflammatory affections. Among the several sections +of the digestive tract itself, the oesophagus and jejunum are +singularly exempt from inflammatory processes; the fauces, +stomach, caecum and appendix, ileum, mouth and duodenum +(including the opening of the common bile-duct), are more +commonly involved. <i>Stomatitis</i>, or inflammation of the mouth, +<span class="sidenote">Inflammatory lesions.</span> +has many predisposing factors, but it has now been +definitely determined that its exciting cause is always +some form of micro-organism. Any condition favouring +oral sepsis, as carious teeth, pyorrhoea alveolaris (a discharge +of pus due to inflamed granulations round carious teeth), +granulations beneath thick crusts of tartar, or an irritating tooth +plate, favours the growth of pyogenic organisms and hence of +stomatitis. Many varieties of this disease have been described, +but all are forms of “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 +<i>Oidium albicans</i>. It is always preceded by a gastro-enteritis and +dry mouth, and if this is not attended to, soon attracts attention +by the little white raised patches surrounded by a dusky red zone +<span class="pagenum"><a name="page264" id="page264"></a>264</span> +scattered on tongue and cheeks. Epidemics have occurred in +hospitals and orphanages. Mouth breathing is the cause of many +ills. As a result of this, the mucous membrane of the tongue, &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 <i>Oidium albicans</i> may spread +from the mouth, and also a diphtheritic inflammation spreads +from the fauces into the oesophagus. A catarrhal oesophagitis +is rarely seen, but the commonest form is traumatic, due to the +swallowing of boiling water, corrosive or irritant substances, &c. +A non-malignant ulceration may result which later leads on to +an oesophageal stricture. The physical changes presented by the +coats of the stomach and the intestine, the subjects of catarrhal +attacks, closely resemble one another, but differ symptomatically. +Acute catarrh of the stomach is associated with intense +hyperaemia of its lining coats, with visible engorgement and +swelling of the mucous membrane, and an excessive secretion of +mucus. The formation of active gastric juice is arrested, digestion +ceases, peristaltic movements are sluggish or absent, unless so +over-stimulated that they act in a direction the reverse of the +normal, and induce expulsion of the gastric contents by vomiting. +The gastric contents, in whatever degree of dilution or concentration +they may have been ingested, when ejected are of porridge-thick +consistency, and often but slightly digested. Such +conditions may succeed a severe alcoholic bout, be caused by +irritant substances taken in by the mouth or arise from fermentative +processes in the stomach contents themselves. Should +the irritating material succeed in passing from the stomach into +the bowel, similar physical signs are present; but as the quickest +path offered for the expulsion of the offending substances from +the body is downwards, peristalsis is increased, the flow of fluid +from the intestinal glands is larger in bulk, though of less potency +as regards its normal actions, than in health, and diarrhoea, with +removal of the irritant, follows. As a general rule, the more +marked the involvement of the large bowel, the severer and more +fluid is the resultant diarrhoea. Inflammation of the stomach +may be due to mechanical injury, thermal or chemical irritants +or invasion by micro-organisms. Also all the symptoms of +gastric catarrh may be brought on by any acute emotion. The +commonest mechanical injury is that due to an excess of food, +especially when following on a fast; poisons act as irritants, and +also the weevils of cheese and the larvae of insects.</p> + +<p>Inflammatory affections of the caecum and its attached +appendix vermiformis are very common, and give rise to several +special symptoms and signs. Acute inflammatory appendicitis +appears to be increasing in frequency, and is associated by many +with the modern deterioration in the teeth. Constipation +certainly predisposes to it, and it appears to be more prevalent +among medical men, commercial travellers, or any engaged in +arduous callings, subjected to irregular meals, fatigue and +exposure. A foreign body is the exciting cause in many cases, +though less commonly so than was formerly imagined. The +inflammation in the appendix varies in intensity from a very +slight catarrhal or simple form to an ulcerative variety, and much +more rarely to the acute fulminating appendicitis in which +necrosis of the appendix with abscess formation occurs. It is +always accompanied by more or less peritonitis, which is protective +in nature, shutting in the inflammatory process. Very +similar symptomatically is the condition termed perityphlitis, +doubtless in former days frequently due to the appendix, an acute +or chronic inflammation of the walls of the caecum often leading +to abscess formation outside the gut, with or without direct +communication with the canal. The colon is subject to three +main forms of inflammation. In simple <i>colitis</i> the mucous +membrane of the colon is intensely injected, bright red in colour, +and secreting a thick mucus, but there is no accompanying +ulceration. It is often found in association with some constitutional +disease, as Bright’s disease, and also with cancer of the +bowel. But when it has no association with other trouble it is +probably bacterial in origin, the <i>Bacillus enteritidis spirogenes</i> +having been isolated in many cases. The motions always contain +large quantities of mucus and more or less blood. A second very +severe form of inflammation of the colon is known as “membranous +colitis,” and this may be either dyspeptic, or secondary to +other diseases. In this trouble membranes are passed <i>per anum</i>, +accompanied by a pain so intense as often to cause fainting. In +severe cases complete tubular casts of the intestine have been +found. Often the motions contain very little faecal matter, but +consist only of membranes, mucus and a little blood. A third +form is that known as “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.</p> + +<p>Chronic catarrhal affections of the stomach are very common, +and often follow upon repeated acute attacks. In them the +connective tissue increases at the expense of the glandular +elements; the mucous membrane becomes thickened and less +active in function. Should the muscular coat be involved, the +elasticity and contractility of the organ suffer; peristaltic movement +is weakened; expulsion of the contents through the pylorus +hindered; and, aggravated by these effects, the condition +becomes worse, atonic dyspepsia in its most pronounced form +results, with or without dilatation. Chronic vascular congestion +may occasion in process of time similar signs and symptoms.</p> + +<p>Duodenal catarrh is constantly associated with jaundice, indeed +is most probably the commonest cause of catarrhal jaundice; often +it is accompanied by catarrh of the common bile-duct. Chronic +inflammation of the small intestine gives rise to less prominent +symptoms than in the stomach. It generally arises from more than +one cause; or rather secondary causes rapidly become as important +as the primary in its incidence. Chronic congestion and prolonged +irritation lead to deficient secretion and sluggish peristalsis; +these effects encourage intestinal putrefaction and auto-intoxication; +and these latter, in turn, increase the local unrest.</p> + +<p>The intestinal mucous membrane, the peritoneum and the +mesenteric glands are the chief sites of tubercular infection in +the digestive organs. Rarely met with in the gullet and +stomach, and comparatively seldom in the mouth and +<span class="sidenote">Infective lesions.</span> +lips, tubercular inflammation of the small intestine +and peritoneum is common. Tubercular enteritis is a frequent +accompaniment of phthisis, but may occur apart from tubercle +of other organs. Children are especially subject to the primary +form. Tubercular peritonitis often is present also. The inflammatory +process readily tends towards ulcer formation, with +haemorrhage and sometimes perforation. If in the large bowel, +the symptoms are usually less acute than those characterizing +tubercular inflammation of the small intestine. The appendix +has been found to be the seat of tubercular processes; in the +rectum they form the general cause of the fistulae and abscesses so +commonly met with here. Tubercular peritonitis may be primary +or secondary, acute or chronic; occasionally very acute cases are +seen running a rapid course; the majority are chronic in type. +<span class="pagenum"><a name="page265" id="page265"></a>265</span> +The tubercles spread over the surface of the serous membrane, +and if small and not very numerous may give rise in chronic +cases to few symptoms; if larger, and especially when they +involve and obstruct the lymph- and blood-vessels, ascites follows. +It is hardly possible that tubercular invasion of the mesenteric +glands can ever occur unaccompanied by peritoneal infection; +but when the infection of the glands constitutes the most prominent +sign, the term <i>tabes mesenterica</i> is sometimes employed. +Here the glands, enlarged, form a doughy mass in the abdomen, +leading to marked protrusion of the abdominal walls, with +wasting elsewhere and diarrhoea.</p> + +<p>The liver is seldom attacked by tubercle, unless in cases of +general miliary tuberculosis. Now and then it contains large +caseous tubercular masses in its substance.</p> + +<p>An important fact with regard to the tubercular processes in +the digestive organs lies in the ready response to treatment shown +by many cases of peritoneal or mesenteric invasion, particularly +in the young.</p> + +<p>The later sequelae of syphilis display a predilection for the +rectum and the liver, usually leading to the development of a +stricture in the former, to a diffuse hepatitis or the formation +of gummata in the second. In inherited syphilis the temporary +teeth usually appear early, are discoloured and soon crumble +away. The permanent teeth may be sound and healthy, but are +often—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.</p> + +<p>The <i>Bacillus typhosus</i> discovered by Eberth is the causal agent +of typhoid fever, and has its chief seat of activity in the small +intestine, more especially in the lower half of the ileum. Attacking +the lymphoid follicles in the mucous membrane, it causes first +inflammatory enlargement, then necrosis and ulceration. The +adjacent portions of the mucous membrane show acute catarrhal +changes. Diarrhoea, of a special “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.</p> + +<p>True Asiatic cholera is due to the comma-bacillus or spirillum +of cholera, which is found in the rice-water evacuations, in the +contents of the intestine after death, and in the mucous membrane +of the intestine just beneath the epithelium. It has not been +found in the blood. It produces an intense irritation of the bowel, +seldom of the stomach, without giving rise locally to any marked +physical change; it causes violent diarrhoea and copious discharges +of “rice-water” stools, consisting largely of serum +swarming with the organism.</p> + +<p>Dysentery gives rise to an inflammation of the large intestine +and sometimes of the lower part of the ileum, resulting in extensive +ulceration and accompanied by faecal discharges of mucus, +muco-pus or blood. In some forms a protozoan, the <i>Amoeba +dysenteriae</i>, is found in the stools—this is the amoebic dysentery; +in other cases a bacillus, <i>Bacillus dysenteriae</i>, is found—the +bacillary dysentery.</p> + +<p>Acute parotitis, or mumps, is an infectious disease of the parotid +glands, chiefly interesting because of the association between it +and the testes in males, inflammation of these glands occasionally +following or replacing the affection of the parotids. The causal +agent is probably organismal, but has as yet escaped detection.</p> + +<p>The relative frequency with which malignant growths occur in +the different organs of the digestive system may be gathered from +the tabular analysis, on p. 266, of 1768 cases recorded in +the books of the Edinburgh Royal Infirmary as having +<span class="sidenote">New growths.</span> +been treated in the medical and surgical wards between +the years 1892 and 1899 inclusive. Of these, 1263, or 71.44%, +were males; 505, or 28.56%, females. (See Table I. p. 266.)</p> + +<p>If the figures there given be classified upon broader lines, the +results are as given in Table II. p. 266, and speak for themselves.</p> + +<p>The digestive organs are peculiarly subject to malignant +disease, a result of the incessant changes from passive to active +conditions, and vice versa, called for by repeated introduction +of food; while the comparative frequency with which different +parts are attacked depends, in part, upon the degree of irritation +or changes of function imposed upon them. Scirrhous, encephaloid +and colloid forms of carcinoma occur. In the stomach +and oesophagus the scirrhous form is most common, the soft +encephaloid form coming next. The most common situation for +cancerous growth in the stomach is the pyloric region. Walsh out +of 1300 cases found 60.8% near the pylorus, 11.4% over the +lesser curvature, and 4.7% more or less over the whole organ. +The small intestine is rarely attacked by cancer; the large +intestine frequently. The rectum, sigmoid flexure, caecum and +colon are affected, and in this order, the cylindrical-celled form +being the most common. Carcinoma of the peritoneum is +generally colloid in character, and is often secondary to growths in +other organs. Cancer of the liver follows cancer of the stomach +and rectum in frequency of occurrence, and is relatively more +common in females than males. Secondary invasion of the liver +is a frequent sequel to gastric cancer. The pancreas occasionally +is the seat of cancerous growth.</p> + +<p>Sarcomata are not so often met with in the digestive organs. +When present, they generally involve the peritoneum or the +mesenteric glands. The liver is sometimes attacked, the stomach +rarely.</p> + +<p>Benign tumours are not of common occurrence in the digestive +organs. Simple growths of the salivary glands, cysts of the +pancreas and polypoid tumours of the rectum are the most +frequent.</p> + +<p>The intestinal canal is the habitat of the majority of animal +parasites found in man. Frequently their presence leads to no +morbid symptoms, local or general; nor are the symptoms, when +they do arise, always characteristic of the presence of +<span class="sidenote">Animal parasites.</span> +parasites alone. Discovery of their bodies, or of their +eggs, in the stools is in most instances the only satisfactory +proof of their presence. The parasites found in the bowel +belong principally to two natural groups, Protozoa and Metazoa. +The great class of the Protozoa furnish amoebae, members of +Sporozoa and Infusoria. The amoebae are almost invariably +found in the large intestine; one species, indeed, is termed <i>Amoeba +coli</i>. The frequently observed relation between attacks of +dysentery and the presence of amoebae in the stools has led to the +proposition that an <i>Amoeba dysenterica</i> exists, causing the disease—a +theory supported by the detection of amoebae in the contents +of dysenteric abscesses of the liver. No symptoms of injury to +health appear to accompany the presence of Sporozoa in the +bowel, while the species of Infusoria found in it, the <i>Cercomonas</i>, +and <i>Trichomonas intestinalis</i>, and the <i>Balantidium coli</i>, may or +may not be guilty of prolonging conditions within the bowel +as have previously set up diarrhoea.</p> + +<p>The Metazoa supply examples of intestinal parasites from the +classes Annuloida and Nematoidea. To the former class belong +the various tapeworms found in the small intestine of man. +They, like other intestinal parasites, are destitute of any power +of active digestion, simply absorbing the nutritious proceeds of +the digestive processes of their hosts. Nematode worms infest +both the small and large intestine; <i>Ascaris lumbricoides</i>, the +common round worm, and the male <i>Oxyuris vermicularis</i> are +found in the small bowel, the adult female <i>Oxyuris vermicularis</i> +and the <i>Tricocephalus dispar</i> in the large.</p> + +<p>The eggs of the <i>Trichina spiralis</i>, when introduced with the +food, develop in the bowel into larval forms which invade the +tissues of the body, to find in the muscles congenial spots wherein +to reach maturity. Similarly, the eggs of the Echinococcus +are hatched in the bowel, and the embryos proceed to take +up their abode in the tissues of the body, developing into cysts +capable of growth into mature worms after their ingestion by +dogs.</p> + +<p><span class="pagenum"><a name="page266" id="page266"></a>266</span></p> + +<p>Numbers of bacterial forms habitually infest the alimentary +canal. Many of them are non-pathogenic; some develop pathogenic +characters only under provocation or when a +suitable environment induces them to act in such a +<span class="sidenote">Vegetable parasites.</span> +manner; others may form the <i>materies morbi</i> of special +lesions, or be casual visitors capable of originating disease if +opportunity occurs. Apart from those organisms associated with +acute infective diseases, disturbances of function and physical +lesions may be the result of abnormal bacterial activity in the +canal; and these disturbances may be both local and general. +Many of the bacteria commonly present produce putrefactive +changes in the contents of the tract by their metabolic processes. +They render the medium they grow in alkaline, produce different +gases and elaborate more or less virulent toxins. Other species +set up an acid fermentation, seldom accompanied by gas or toxin +formation. The products of either class are inimical to the free +growth of members of the other. The species which produce acids +are more resistant to the action of acids. Thus, when the contents +of the stomach possess a normal or excessive proportion of free +hydrochloric acid, a much larger number of putrefactive and +pathogenic organisms in the food are destroyed or inhibited than +of the bacteria of acid fermentation. Diminished gastric acidity +allows of the entry of a greater number of putrefactive (and +pathogenic) types, with, as a consequence, increased facilities +for their growth and activity, and the appearance of intestinal +derangements.</p> + +<p class="center pt2"><span class="sc">Table I.</span></p> + +<table class="ws" summary="Contents"> + +<tr><td class="tccm allb" colspan="2">Males.</td> <td class="tccm allb" colspan="2">Females.</td> <td class="tccm allb" colspan="2">Both Sexes.</td></tr> +<tr><td class="tccm allb">Organ or Tissue in<br />Order of Frequency.</td> <td class="tccm allb">Per-<br />centage</td> +<td class="tccm allb">Organ or Tissue in<br />Order of Frequency.</td> <td class="tccm allb">Per-<br />centage</td> +<td class="tccm allb">Organ or Tissue in<br />Order of Frequency.</td> <td class="tccm allb">Per-<br />centage</td></tr> + +<tr><td class="tcl lb rb"> 1 Stomach</td> <td class="tcr rb">22.56</td> <td class="tcl rb"> 1 Stomach</td> <td class="tcr rb">22.37</td> <td class="tcl rb"> 1 Stomach</td> <td class="tcr rb">22.49</td></tr> +<tr><td class="tcl lb rb"> 2 Lip</td> <td class="tcr rb">12.94</td> <td class="tcl rb"> 2 Rectum</td> <td class="tcr rb">17.24</td> <td class="tcl rb"> 2 Rectum</td> <td class="tcr rb">13.12</td></tr> +<tr><td class="tcl lb rb"> 3 Rectum</td> <td class="tcr rb">11.57</td> <td class="tcl rb"> 3 Liver</td> <td class="tcr rb">15.50</td> <td class="tcl rb"> 3 Liver</td> <td class="tcr rb">10.02</td></tr> +<tr><td class="tcl lb rb"> 4 Tongue</td> <td class="tcr rb">11.36</td> <td class="tcl rb"> 4 Peritoneum</td> <td class="tcr rb">7.86</td> <td class="tcl rb"> 4 Lip</td> <td class="tcr rb">9.89</td></tr> +<tr><td class="tcl lb rb"> 5 Oesophagus</td> <td class="tcr rb">10.90</td> <td class="tcl rb"> 5 Oesophagus</td> <td class="tcr rb">5.33</td> <td class="tcl rb"> 5 Oesophagus</td> <td class="tcr rb">9.29</td></tr> +<tr><td class="tcl lb rb"> 6 Liver</td> <td class="tcr rb">7.80</td> <td class="tcl rb"> 6 Sigmoid</td> <td class="tcr rb">4.53</td> <td class="tcl rb"> 6 Tongue</td> <td class="tcr rb">8.96</td></tr> +<tr><td class="tcl lb rb"> 7 Jaw</td> <td class="tcr rb">6.38</td> <td class="tcl rb"> 7 Pancreas</td> <td class="tcr rb">3.52</td> <td class="tcl rb"> 7 Jaw</td> <td class="tcr rb">5.65</td></tr> +<tr><td class="tcl lb rb"> 8 Mouth</td> <td class="tcr rb">2.88</td> <td class="tcl rb"> 8 Tongue</td> <td class="tcr rb">3.12</td> <td class="tcl rb"> 8 Peritoneum</td> <td class="tcr rb">2.94</td></tr> +<tr><td class="tcl lb rb"> 9 Tonsils</td> <td class="tcr rb">2.09</td> <td class="tcl rb"> 9 Omentum</td> <td class="tcr rb">2.98</td> <td class="tcl rb"> 9 Sigmoid</td> <td class="tcr rb">2.56</td></tr> +<tr><td class="tcl lb rb">10 Sigmoid flexure</td> <td class="tcr rb">1.77</td> <td class="tcl rb">10 Lip</td> <td class="tcr rb">2.57</td> <td class="tcl rb">10 Mouth</td> <td class="tcr rb">2.40</td></tr> +<tr><td class="tcl lb rb">11 Parotid</td> <td class="tcr rb">1.10</td> <td class="tcl rb">11 Jaw</td> <td class="tcr rb">1.97</td> <td class="tcl rb">11 Pancreas</td> <td class="tcr rb">1.80</td></tr> +<tr><td class="tcl lb rb">12 Pancreas</td> <td class="tcc rb">”</td> <td class="tcl rb">12 Colon</td> <td class="tcr rb">1.84</td> <td class="tcl rb">12 Tonsils</td> <td class="tcr rb">1.35</td></tr> +<tr><td class="tcl lb rb">13 Caecum</td> <td class="tcr rb">0.94 </td> <td class="tcl rb">13 Abdomen</td> <td class="tcc rb">”</td> <td class="tcl rb">13 Omentum</td> <td class="tcr rb">1.25</td></tr> +<tr><td class="tcl lb rb">14 Peritoneum</td> <td class="tcc rb">”</td> <td class="tcl rb">14 Intestine</td> <td class="tcr rb">1.56</td> <td class="tcl rb">14 Parotid</td> <td class="tcr rb">1.12</td></tr> +<tr><td class="tcl lb rb">15 Colon</td> <td class="tcr rb">0.89 </td> <td class="tcl rb">15 Caecum</td> <td class="tcr rb">1.37</td> <td class="tcl rb">15 Colon</td> <td class="tcc rb">”</td></tr> +<tr><td class="tcl lb rb">16 Pharynx</td> <td class="tcr rb">0.79</td> <td class="tcl rb">16 Mouth</td> <td class="tcr rb">1.18</td> <td class="tcl rb">16 Caecum</td> <td class="tcr rb">1.08</td></tr> +<tr><td class="tcl lb rb">17 Intestine (site unknown)</td> <td class="tcc rb">”</td> <td class="tcl rb">17 Parotid</td> <td class="tcc rb">”</td> <td class="tcl rb">17 Intestine</td> <td class="tcr rb">1.00</td></tr> +<tr><td class="tcl lb rb">18 Abdomen</td> <td class="tcr rb">0.71</td> <td class="tcl rb">18 Splenic flexure</td> <td class="tcr rb">0.98</td> <td class="tcl rb">18 Abdomen</td> <td class="tcc rb">”</td></tr> +<tr><td class="tcl lb rb">19 Mesentery</td> <td class="tcr rb">0.55</td> <td class="tcl rb">19 Jejunum and ileum</td> <td class="tcr rb">0.78</td> <td class="tcl rb">19 Pharynx</td> <td class="tcr rb">0.62</td></tr> +<tr><td class="tcl lb rb">20 Omentum</td> <td class="tcc rb">”</td> <td class="tcl rb">20 Tonsils</td> <td class="tcr rb">0.68</td> <td class="tcl rb">20 Mesentery</td> <td class="tcr rb">0.52</td></tr> +<tr><td class="tcl lb rb">21 Hepatic flexure</td> <td class="tcr rb">0.39</td> <td class="tcl rb">21 Pharynx</td> <td class="tcr rb">0.40</td> <td class="tcl rb">21 Jejunum and ileum</td> <td class="tcr rb">0.44</td></tr> +<tr><td class="tcl lb rb">22 Submaxillary gland</td> <td class="tcr rb">0.31</td> <td class="tcl rb">22 Hepatic flexure</td> <td class="tcc rb">”</td> <td class="tcl rb">22 Hepatic flexure</td> <td class="tcc rb">”</td></tr> +<tr><td class="tcl lb rb">23 Jejunum and ileum</td> <td class="tcc rb">”</td> <td class="tcl rb">23 Mesentery</td> <td class="tcc rb">”</td> <td class="tcl rb">23 Splenic flexure</td> <td class="tcc rb">”</td></tr> +<tr><td class="tcl lb rb">24 Duodenum</td> <td class="tcr rb">0.23</td> <td class="tcl rb">24 Submaxillary</td> <td class="tcr rb">0.20</td> <td class="tcl rb">24 Submaxillary</td> <td class="tcr rb">0.28</td></tr> +<tr><td class="tcl lb rb bb">25 Splenic flexure</td> <td class="tcr rb bb">0.15</td> <td class="tcl rb bb">25 Duodenum</td> <td class="tcc rb bb">”</td> <td class="tcl rb bb">25 Duodenum</td> <td class="tcr rb bb">0.22</td></tr> +<tr><td class="tcc f90" colspan="6"><i>Note.</i>—The figures where several organs are bracketed apply to each organ separately.</td></tr> +</table> + +<p>In a healthy new-born infant the mouth is free from micro-organisms, +and very few are found in a breast-fed baby, but +<i>Bacillus lactis</i> may be found where the child is bottle fed. +If there is trouble with the first dentition and food is allowed +to collect, staphylococci, streptococci, pneumococci and colon +bacilli may be present. Even in healthy babies <i>Oidium albicans</i> +may be present, and in older children the pseudo-diphtheria +bacillus. From carious teeth may be isolated streptothrix, +leptothrix, spirilla and fusiform bacilli. Under conditions of +health these micro-organisms live in the mouth as saprophytes, +and show no virulence when cultivated +and injected into animals. +The two common pyogenetic organisms, +<i>Staphylococcus albus</i> and +<i>brevis</i>, show no virulence. Also +the pneumococcus, though often +present, must be raised in virulence +before it can produce untoward +results. The foulness of the mouth +is supposed to be due to the colon +bacillus and its allies, but those +obtained from the mouth are innocuous. +Also to enable the <i>Oidium +albicans</i> to attack the mucous membrane +there must be some slight +inflammation or injury. The micro-organisms +found in the stomach +gain access to that organ in the +food or by regurgitation from the +small intestine. Most are relatively +inert, but some have a special fermentative +action on the food (see +<span class="sc"><a href="#artlinks">Nutrition</a></span>). Abelous isolated sixteen +distinct species of organism +from a healthy stomach, including +Sarcinae, <i>B. lactis</i>, <i>pyocyaneus</i>, +<i>subtilis</i>, <i>lactis erythrogenes</i>, <i>amylobacter</i>, +<i>megatherium</i>, and <i>Vibrio +rugula</i>.</p> + +<p>Hare-lip, cleft palate, hernia +and imperforate anus are physical +abnormalities which are interesting to the surgeon rather than to +the pathologist. The oesophagus may be the seat of a diverticulum, +or blind pouch, usually situated in its lower half, which in +<span class="sidenote">Physical abnormalities</span> +most instances is probably partly acquired and partly +congenital; a local weakness succumbing to pressure. +Hypertrophy of the muscular coat of the pyloric region +is an infrequent congenital gastric anomaly in infants, +preventing the passage of food into the bowel, and causing death +in a short time. Incomplete closure +of the vitelline duct results in +the presence of a diverticulum—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.</p> + +<p class="center pt2"><span class="sc">Table II.</span></p> + +<table class="ws" summary="Contents"> +<tr><td class="tccm allb">Males.</td> <td class="tccm allb">Per-<br />centage.</td> <td class="tccm allb">Females.</td> <td class="tccm allb">Per-<br />centage.</td> <td class="tccm allb">Total.</td> <td class="tccm allb">Per-<br />centage.</td></tr> + +<tr><td class="tcl lb rb">1 Mouth and pharynx</td> <td class="tcr rb">37.85</td> <td class="tcl rb">1 Intestines</td> <td class="tcr rb">28.9</td> <td class="tcl rb">1 Oesophagus and stomach</td> <td class="tcr rb">31.78</td></tr> +<tr><td class="tcl lb rb">2 Oesophagus and stomach</td> <td class="tcr rb">33.46</td> <td class="tcl rb">2 Oesophagus and stomach</td> <td class="tcr rb">27.7</td> <td class="tcl rb">2 Mouth and pharynx</td> <td class="tcr rb">30.27</td></tr> +<tr><td class="tcl lb rb">3 Intestines </td> <td class="tcr rb">17.04</td> <td class="tcl rb">3 Liver</td> <td class="tcr rb">15.5</td> <td class="tcl rb">3 Intestines</td> <td class="tcr rb">20.42</td></tr> +<tr><td class="tcl lb rb">4 Liver</td> <td class="tcr rb">7.8 </td> <td class="tcl rb">4 Peritoneum</td> <td class="tcr rb">13.1</td> <td class="tcl rb">4 Liver</td> <td class="tcr rb">10.02</td></tr> +<tr><td class="tcl lb rb">5 Peritoneum</td> <td class="tcr rb">2.75</td> <td class="tcl rb">5 Mouth and pharynx</td> <td class="tcr rb">11.3</td> <td class="tcl rb">5 Peritoneum</td> <td class="tcr rb">5.71</td></tr> +<tr><td class="tcl lb rb bb">6 Pancreas </td> <td class="tcr rb bb">1.1 </td> <td class="tcl rb bb">6 Pancreas</td> <td class="tcr rb bb">3.5</td> <td class="tcl rb bb">6 Pancreas</td> <td class="tcr rb bb">1.80</td></tr> +</table> + +<p>Traction diverticula of the oesophagus +not uncommonly occur as +sequels to suppurative inflammation +of cervical lymphatic glands. +More frequently dilatation of a section is met with, due as a +rule to the presence of a stricture. The stomach often diverges +from the normal in size, shape and position. Normally capable +in the adult of containing from fifty to sixty ounces, either by +reason of organic disease, or as the result of functional disturbance, +its capacity may vary enormously. The writer has seen +post mortem a stomach which held a gallon (160 ounces), and +again one holding only two ounces. Cancer spread over a large +area and cirrhosis of the stomach wall cause diminution in +capacity; pyloric obstruction, weakness of the muscular coat, +and nervous influences are associated with dilatation. A peculiar +distortion of the shape of the stomach follows cicatrization of +<span class="pagenum"><a name="page267" id="page267"></a>267</span> +ulcers of greater or lesser curvature; the gastric cavity becomes +“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.</p> + +<p>Every pathological lesion shown by digestive organs is closely +associated with the state of the nervous system, general or local; +so stoppage of active gastric digestive processes after +profound nervous shock, and occurrence of nervous +<span class="sidenote">Influence of the nervous system.</span> +diarrhoea from the same cause. Gastric dyspepsia +of nervous origin presents most varied and contradictory +symptoms: diminished acidity of the gastric juice, +hyper-acidity, over-production, arrest of secretion, lessened or +increased movements, greater sensitiveness to the presence of +contents, dilatation or spasm. Often the nervous cause can +be traced back farther,—in females, frequently to the pelvic +organs; in both sexes, to the condition of the blood, the brain or +the bowel. Unhealthy conditions related to evacuation of the +bowel-contents commonly induce reflex nervous manifestations of +abnormal character referred to the stomach and liver. Gastric +disturbances similarly react upon the proper conduct of intestinal +functions.</p> + +<p class="center pt2"><i>Local Diseases.</i></p> + +<p><i>The Mouth.</i>—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.</p> + +<p><i>The Stomach.</i>—It were futile to attempt to enumerate all the +protean manifestations of disturbance which proceed from a disordered +stomach. The possible permutations and combinations +of the causes of gastric vagaries almost reach infinity. Idiosyncrasy, +past and present gastric education, penury or plethora, +actual digestive power, motility, bodily requirements and conditions, +environment, mental influences, local or adjacent organic +lesions, and, not least, reflex impressions from other organs, all +contribute to the variance.</p> + +<p>Ulcer of the stomach, however—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.</p> + +<p>Irritation of the sensory nerve-endings in the stomach wall +from the presence of an increased proportion of acid, organic or +mineral, in the stomach contents is accountable for the well known +symptom heartburn. Water-brash is a term applied to +eructation of a colourless, almost tasteless fluid, probably saliva, +which has collected in the lower part of the oesophagus from +failure of the cardiac sphincter of the stomach to relax; reversed +oesophageal peristalsis causing regurgitation. A similar reversed +action serves in merycism, or rumination, occasionally found in +man, to raise part of the food, lately ingested, from the stomach to +the mouth. Vomiting also is aided by reversed peristaltic action, +both of the stomach and the oesophagus, with the help of the +diaphragm and the muscles of the anterior abdominal wall. +Emesis may be caused both by local nervous influence, and +through the central nervous mechanism either reflexly or from +the direct action of substances circulating in the blood. Further, +the causal agent acting on the central nervous apparatus may be +organic or functional, as well as medicinal. Vomiting without +any apparent cause suggests nervous lesions, organic or reflex. +The obstinate vomiting of pregnancy is a case in point. Here the +primary cause proceeds reflexly from the pelvis. In females the +pelvic organs are often the true source of emesis. Haematemesis +accompanies gastric ulcer, cancer, chronic congestion with +haemorrhagic erosion, congestion of the liver, or may follow +violent acts of vomiting. In cases of ulcer the blood is usually +bright and in considerable amount; in cancer, darker, like coffee-grounds; +and in cases of erosion, in smaller quantity and of bright +colour. The reaction of the stomach contents, if the cause be +doubtful, yields valuable aid towards a diagnosis. Of increased +acidity in gastric ulcer, normal in hepatic congestion, it is +diminished in cancer; but as the acid present in cancer is largely +lactic, analysis of the gastric contents must often be a <i>sine qua +non</i>, because hyperacidity from lactic may obscure hypoacidity of +hydrochloric acid.</p> + +<p>Flatulence usually results from fermentative processes in +the stomach and bowel, as the outcome of bacterial activity. A +different form of flatulence is common in neurotic individuals: +in such the gas evolved consists simply in carbonic acid liberated +from the blood, and its evolution is generally characterized by +rapid development and by lack of all fermentative signs.</p> + +<p><i>The Liver.</i>—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 <i>sui generis</i>. Of rare +occurrence, possibly of toxic origin, it is marked by jaundice, at +first of usual type, later becoming most intense; by vomiting; +haemorrhages widely distributed; rapid diminution in the size of +the liver; the appearance of leucin and tyrosin in the urine, with +lessened urea; and in two or three days, death. The liver after +death is soft, of a reddish colour dotted with yellow patches, and +weighs only about a third part of the normal—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 +<span class="pagenum"><a name="page268" id="page268"></a>268</span> +epidemic form. In this the spleen and liver are commonly but +not always swollen, and the liver is often tender on pressure. As +a large proportion of the sufferers from this disease have been +butchers, and the epidemics have occurred in the hot season of +the year, it probably arises from contact with decomposing +animal matter. Hepatic abscess may follow on an attack of +amoebic dysentery, and is produced either by infection through +the portal vein, or by direct infection from the adjacent colon. +In general pyaemia multiple small abscesses may occur in the +liver.</p> + +<p><i>The Gall-Bladder.</i>—The formation of biliary calculi in the gall-bladder +is the chief point of interest here. At least 75% of such +cases occur in women, especially in those who have borne children. +Tight-lacing has been stated to act as an exciting cause, owing to +the consequent retardation of the flow of bile. Gall-stones may +number from one to many thousands. They are largely composed +of cholesterin, combined with small amounts of bile-pigments +and acids, lime and magnesium salts. Their presence +may give rise to no symptoms, or may cause violent biliary colic, +and, if the bile-stream be obstructed, to jaundice. Inflammatory +processes may be initiated in the gall-bladder or the bile-ducts, +catarrhal or suppurative in character.</p> + +<p><i>The Pancreas.</i>—Haemorrhages into the body of the pancreas, +acute and chronic inflammation, calculi, cysts and tumours, +among which cancer is by far the most common, are recognized as +occurring in this organ; the point of greatest interest regarding +them lies in the relations established between pancreatic disease +and diabetes mellitus, affections of the gland frequently being +complicated by, and probably causing, the appearance of sugar in +the urine.</p> + +<p><i>The Small Intestine.</i>—Little remains to be added to the account +of inflammatory lesions in connexion with the small intestine. It +offers but few conditions peculiar to itself, save in typhoid fever, +and the ease with which it contrives to become kinked, or intussuscepted, +producing obstruction, or to take part in hernial +protrusions. The first section, the duodenum, is subject to +development of ulcers very similar to those of the gastric mucous +membrane. For long duodenal ulceration has been regarded as a +complication of extensive burns of the skin, but the relationship +between them has not yet been quite satisfactorily explained. +The condition of colic in the bowel usually arises from overdistension +of some part of the small gut with gas, the frequent +sharp turns of the gut facilitating temporary closure of its lumen +by pressure of the dilated gut near a curve against the part +beyond. In the large bowel accumulations of gas seldom cause +such acute symptoms, having a readier exit.</p> + +<p><i>The Large Intestine.</i>—The colon, especially the ascending +portion, may become immensely dilated, usually after prolonged +constipation and paralysis of the gut; occasionally the condition +is congenital. Straining efforts made in defaecation may often +account for prolapse of the lower end of the rectum through the +anus. Haemorrhage from the bowel is usually a sign of disease +situated in the large intestine: if bright in colour, the source is +probably low down; if dark, from the caecum or from above the +ileo-caecal valve. Blood after a short stay in any section of the +alimentary canal darkens, and eventually becomes almost black +in colour.</p> +<div class="author">(A. L. G.; M. F.*)</div> + + +<hr class="art" /> +<p><span class="bold">DIGGES, WEST<a name="ar92" id="ar92"></a></span> (1720-1786), English actor, made his first stage +appearance in Dublin in 1749 as Jaffier in <i>Venice Preserved</i>; and +both there and in Edinburgh until 1764 he acted in many tragic +rôles with success. He was the original “young Norval” in +Home’s <i>Douglas</i> (1756). His first London appearance was as +Cato in the Haymarket in 1777, and he afterwards played Lear, +Macbeth, Shylock and Wolsey. In 1881 he returned to Dublin +and retired in 1784.</p> + + +<hr class="art" /> +<p><span class="bold">DIGIT<a name="ar93" id="ar93"></a></span> (Lat. <i>digitus</i>, finger), literally a finger or toe, and so used +to mean, from counting on the fingers, a single numeral, or, from +measuring, a finger’s breadth. In astronomy a digit is the twelfth +part of the diameter of the sun or moon; it is used to express the +magnitude of an eclipse.</p> + + +<hr class="art" /> +<p><span class="bold">DIGITALIS.<a name="ar94" id="ar94"></a></span> The leaves of the foxglove (<i>q.v.</i>), gathered from +wild plants when about two-thirds of their flowers are expanded, +deprived usually of the petiole and the thicker part of the midrib, +bitter taste; and to preserve their properties they must be kept +excluded from light in stoppered bottles. They are occasionally +adulterated with the leaves of <i>Inula Conyza</i>, ploughman’s +spikenard, which may be distinguished by their greater roughness, +their less divided margins, and their odour when rubbed; +also with the leaves of <i>Symphytum officinale</i>, comfrey, and of +<i>Verbascum Thapsus</i>, great mullein, which unlike those of the +foxglove have woolly upper and under surfaces. The earliest +known descriptions of the foxglove are those given by Leonhard +Fuchs and Tragus about the middle of the 16th century, but its +virtues were doubtless known to herbalists at a much remoter +period. J. Gerarde, in his <i>Herbal</i> (1597), advocates the use of +foxglove for a variety of complaints; and John Parkinson, in the +<i>Theatrum Botanicum</i>, or <i>Theater of Plants</i> (1640), and later W. +Salmon, in <i>The New London Dispensatory</i>, similarly praised the +remedy. Digitalis was first brought prominently under the +notice of the medical profession by Dr W. Withering, who, in his +<i>Account of the Foxglove</i> (1785), gave details of upwards of 200 +cases chiefly dropsical, in which it was used.</p> + +<p>Digitalis contains four important glucosides, of which three are +cardiac stimulants. The most powerful is <i>digitoxin</i> C<span class="su">34</span>H<span class="su">54</span>O<span class="su">11</span>, +an extremely poisonous and cumulative drug, insoluble in water. +<i>Digitalin</i>, C<span class="su">35</span>H<span class="su">56</span>O<span class="su">14</span>, is crystalline and is also insoluble in water. +<i>Digitalein</i> is amorphous but readily soluble in water. It can +therefore be administered subcutaneously, in doses of about one-hundredth +of a grain. <i>Digitonin</i>, on the other hand, is a cardiac +depressant, and has been found to be identical with saponin, +the chief constituent of senega root. There are numerous preparations, +patent and pharmacopeial, their composition being +extremely varied, so that, unless one has reason to be certain of +any particular preparation, it is almost better to use only the +dried leaves themselves in the form of a powder (dose ½-2 grains). +The pharmacopeial tincture may be given in doses of five to +fifteen minims, and the infusion has the unusually small dose of +two to four drachms—the dose of other infusions being an ounce +or more. The tincture contains a fair proportion of both digitalin +and digitoxin.</p> + +<p>Digitalis leaves have no definite external action. Taken by the +mouth, the drug is apt to cause considerable digestive disturbance, +varying in different cases and sometimes so severe as to cause +serious difficulty. This action is probably due to the digitonin, +which is thus a constituent in every way undesirable. The all-important +property of the drug is its action on the circulation. +Its first action on any of the body-tissues is upon unstriped +muscle, so that the first consequence of its absorption is a contraction +of the arteries and arterioles. No other known drug has +an equally marked action in contracting the arterioles. As the +vaso-motor centre in the medulla oblongata is also stimulated, as +well as the contractions of the heart, there is thus trebly caused a +very great rise in the blood-pressure.</p> + +<p>The clinical influence of digitalis upon the heart is very well +defined. After the taking of a moderate dose the pulse is +markedly slowed. This is due to a very definite influence upon +the different portions of the cardiac cycle. The systole is not +altered in length, but the diastole is very much prolonged, and +since this is the period not only of cardiac rest but also of cardiac +“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 +<span class="pagenum"><a name="page269" id="page269"></a>269</span> +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.</p> + +<p>The action of this drug on the kidney is of importance only +second to its action on the circulation. In small or moderate +doses it is a powerful diuretic. Though Heidenhain asserts that +rise in the renal blood-pressure has not a diuretic action per se, +it seems probable that this influence of the drug is due to a rise +in the general blood-pressure associated with a relatively dilated +condition of the renal vessels. In large doses, on the other hand, +the renal vessels also are constricted and the amount of urine falls. +It is probable that digitalis increases the amount of water rather +than that of the urinary solids. In large doses the action of +digitalis on the circulation causes various cerebral symptoms, +such as seeing all objects blue, and various other disturbances of +the special senses. There appears also to be a specific action of +lowering the reflex excitability of the spinal cord.</p> + +<p>Digitalis is used in therapeutics exclusively for its action on the +circulation. In prescribing this drug it must be remembered that +fully three days elapse before it gets into the system, and thus it +must always be combined with other remedies to tide the patient +over this period. It must never be prescribed in large doses to +begin with, as some patients are quite unable to take it, intractable +vomiting being caused. The three days that must pass before +any clinical effect is obtained renders it useless in an emergency. +A certain consequence of its use is to cause or increase cardiac +hypertrophy—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.</p> + + +<hr class="art" /> +<p><span class="bold">DIGNE,<a name="ar95" id="ar95"></a></span> the chief town of the department of the Basses Alpes, +in S.E. France, 14 m. by a branch line from the main railway +line between Grenoble and Avignon. Pop. (1906), town, 4628; +commune, 7456. The Ville Haute is built on a mountain spur +running down to the left bank of the Bléone river, and is composed +of a labyrinth of narrow winding streets, above which towers the +present cathedral church, dating from the end of the 15th century, +but largely reconstructed in modern times, and the former +bishop’s palace (now the prison). The fine Boulevard Gassendi +separates the Ville Haute from the Ville Basse, which is of modern +date. The old cathedral (Notre Dame du Bourg) is a building of +the 13th century, but is now disused except for funerals: it +stands at the east end of the Ville Basse. The neighbourhood of +Digne is rich in orchards, which have long made the town famous +in France for its preserved fruits and confections. It is the <i>Dinia</i> +of the Romans, and was the capital of the Bodiontii. From the +early 6th century at least it has been an episcopal see, which till +1790 was in the ecclesiastical province of Embrun, but since 1802 +in that of Aix en Provence. The history of Digne in the middle +ages is bound up with that of its bishops, under whom it prospered +greatly. But it suffered much during the religious wars of the +16th and 17th centuries, when it was sacked several times. A +little way off, above the right bank of the Bléone, is Champtercier, +the birthplace of the astronomer Gassendi (1592-1655), whose +name has been given to the principal thoroughfare of the little +town.</p> + +<div class="condensed"> +<p>See F. Guichard, <i>Souvenirs historiques sur la ville de Digne et ses +environs</i> (Digne, 1847).</p> +</div> +<div class="author">(W. A. B. C.)</div> + + +<hr class="art" /> +<p><span class="bold">DIGOIN,<a name="ar96" id="ar96"></a></span> a town of east-central France, in the department of +Saône-et-Loire, on the right bank of the Loire, 55 m. W.N.W. +of Mâcon on the Paris-Lyon railway. Pop. (1906) 5321. It is +situated at the meeting places of the Loire, the Lateral canal of the +Loire and the Canal du Centre, which here crosses the Loire by a +fine aqueduct. The town carries on considerable manufactures of +faience, pottery and porcelain. The port on the Canal du Centre +has considerable traffic in timber, sand, iron, coal and stone.</p> + + +<hr class="art" /> +<p><span class="bold">DIJON,<a name="ar97" id="ar97"></a></span> a town of eastern France, capital of the department of +Côte d’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 <i>enceinte</i>, but on the east side detached +forts, 3 to 4 m. distant from the centre, command all the great +roads, while the hilly ground to the west is protected by Fort +Hauteville to the N.W. and the “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.</p> + +<p>The cathedral of St Bénigne, originally an abbey church, +was built in the latter half of the 13th century on the site of a +Romanesque basilica, of which the crypt remains. The west +front is flanked by two towers and the crossing is surmounted by +a slender timber spire. The plan consists of three naves, short +transepts and a small choir, without ambulatory, terminating in +three apses. In the interior there is a fine organ and a quantity of +statuary, and the vaults contain the remains of Philip the Bold, +duke of Burgundy, and Anne of Burgundy, daughter of John +the Fearless. The site of the abbey buildings is occupied by +the bishop’s palace and an ecclesiastical seminary. The church +of Notre-Dame, typical of the Gothic style of Burgundy, was +erected from 1252 to 1334, and is distinguished for the grace of +its interior and the beauty of the western façade. The portal +consists of three arched openings, above which are two stages of +arcades, open to the light and supported on slender columns. +A row of gargoyles surmounts each storey of the façade, which is +also ornamented by sculptured friezes. A turret to the right of +the portal carries a clock called the Jaquemart, on which the hours +are struck by two figures. The church of St Michel belongs to the +15th century. The west façade, the most remarkable feature of +the church, is, however, of the Renaissance period. The vaulting +of the three portals is of exceptional depth owing to the projection +of the lower storey of the façade. Above this storey rise two +towers of five stages, the fifth stage being formed by an octagonal +cupola. The columns decorating the façade represent all the four +orders. The design of this façade is wrongly attributed to Hugues +<span class="pagenum"><a name="page270" id="page270"></a>270</span> +Sambin (fl. c. 1540), a native of Dijon, and pupil of Leonardo da +Vinci, but the sculpture of the portals, including “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 <i>parlement</i> of Burgundy. Dijon possesses +several houses of the 15th, 16th and 17th centuries, notably the +Maison Richard in the Gothic, and the Hôtel Vogüé in the +Renaissance style. St Bernard, the composer J. P. Rameau and +the sculptor François Rude have statues in the town, of which +they were natives. There are also monuments to those inhabitants +of Dijon who fell in the engagement before the town +in 1870, and to President Carnot and Garibaldi.</p> + +<p>The town is important as the seat of a prefecture, a bishopric, a +court of appeal and a court of assizes, and as centre of an académie +(educational district). There are tribunals of first instance and of +commerce, a board of trade-arbitrators, a chamber of commerce, +an exchange (occupying the former cathedral of St Étienne), and +an important branch of the Bank of France. Its educational +establishments include faculties of law, of science and of letters, a +preparatory school of medicine and pharmacy, a higher school of +commerce, a school of fine art, a conservatoire of music, <i>lycées</i> and +training colleges, and there is a public library with about 100,000 +volumes.</p> + +<p>Dijon is well known for its mustard, and for the black currant +liqueur called <i>cassis de Dijon</i>; its industries include the manufacture +of machinery, automobiles, bicycles, soap, biscuits, +brandy, leather, boots and shoes, candles and hosiery. There +are also flour mills, breweries, important printing works, vinegar +works and, in the vicinity, nursery gardens. The state has a +large tobacco manufactory in the town. Dijon has considerable +trade in cereals and wool, and is the second market for the wines +of Burgundy.</p> + +<p>Under the Romans Dijon (<i>Divonense castrum</i>) was a <i>vicus</i> in +the <i>civitas</i> of Langres. In the 2nd century it was the scene of +the martyrdom of St Benignus (Bénigne, vulg. Berin, Berain), +the apostle of Burgundy. About 274 the emperor Aurelian +surrounded it with ramparts. Gregory of Tours, in the 6th +century, comments on the strength and pleasant situation +of the place, expressing surprise that it does not rank as +a <i>civitas</i>. During the middle ages the fortunes of Dijon +followed those of Burgundy, the dukes of which acquired it +early in the 11th century. The communal privileges, conferred +on the town in 1182 by Hugh III., duke of Burgundy, were +confirmed by Philip Augustus in 1183, and in the 13th century +the dukes took up their residence there. For the decoration of the +palace and other monuments built by them, eminent artists were +gathered from northern France and Flanders, and during this +period the town became one of the great intellectual centres of +France. The union of the duchy with the crown in 1477 deprived +Dijon of the splendour of the ducal court; but to counterbalance +this loss it was made the capital of the province and seat of a +<i>parlement</i>. Its fidelity to the monarchy was tested in 1513, +when the citizens were besieged by 50,000 Swiss and Germans, +and forced to agree to a treaty so disadvantageous that Louis XII. +refused to ratify it. In the wars of religion Dijon sided with the +League, and only opened its gates to Henry IV. in 1595. The +18th century was a brilliant period for the city; it became the +seat of a bishopric, its streets were improved, its commerce +developed, and an academy of science and letters founded; +while its literary salons were hardly less celebrated than those of +Paris. The neighbourhood was the scene of considerable fighting +during the Franco-German War, which was, however, indirectly +of some advantage to the city owing to the impetus given to its +industries by the immigrants from Alsace.</p> + +<div class="condensed"> +<p>See H. Chabeuf, <i>Dijon à travers les âges</i> (Dijon, 1897), and <i>Dijon, +monuments et souvenirs</i> (Dijon, 1894).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DIKE,<a name="ar98" id="ar98"></a></span> or <span class="sc">Dyke</span> (Old Eng. <i>dic</i>, a word which appears in various +forms in many Teutonic languages, cf. Dutch <i>dijk</i>, German <i>Teich</i>, +Danish <i>dige</i>, and in French, derived from Teutonic, <i>digue</i>; it is +the same word as “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 <span class="sc"><a href="#artlinks">Holland</a></span> and <span class="sc"><a href="#artlinks">Fens</a></span>). +In Scotland any wall, fence or even hedge, used as a boundary is +called a dyke. In geology the term is applied to wall-like masses +of rock (sometimes projecting beyond the surrounding surface) +which fill up vertical or highly inclined fissures in the strata.</p> + + +<hr class="art" /> +<p><span class="bold">DIKKA,<a name="ar99" id="ar99"></a></span> a term in Mahommedan architecture for the tribune +raised upon columns, from which the Koran is recited and the +prayers intoned by the Imam of the mosque.</p> + + +<hr class="art" /> +<p><span class="bold">DILAPIDATION<a name="ar100" id="ar100"></a></span> (Lat. for “scattering the stones,” <i>lapides</i>, of a +building), a term meaning in general a falling into decay, but more +particularly used in the plural in English law for (1) the waste +committed by the incumbent of an ecclesiastical living; (2) the +disrepair for which a tenant is usually liable when he has agreed +to give up his premises in good repair (see <span class="sc"><a href="#artlinks">Easement</a></span>; <span class="sc"><a href="#artlinks">Flat</a></span>; +<span class="sc"><a href="#artlinks">Landlord and Tenant</a></span>). By the general law a tenant for +life has no power to cut down timber, destroy buildings, &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 (<i>Ross</i> v. +<i>Adcock</i>, 1868, L.R. 3 C.P. 657) it was said that the court was +acquainted with no precedent or decision extending the liability +of the executors of a deceased incumbent to any species of waste +beyond dilapidation of the house, chancel or other buildings or +fences of the benefice. And it has been held that the mere mismanagement +or miscultivation of the ecclesiastical lands will not +give rise to an action for dilapidations. To place the law relating +to dilapidations on a more satisfactory footing, the Ecclesiastical +Dilapidations Act 1871 was passed. The buildings to which the +act applies are defined to be such houses of residence, chancels, +walls, fences and other buildings and things as the incumbent of +the benefice is by law and custom bound to maintain in repair. +In each diocese a surveyor is appointed by the archdeacons and +rural deans subject to the approval of the bishop; and such +surveyor shall by the direction of the bishop examine the buildings +on the following occasions—viz. (1) when the benefice is +sequestrated; (2) when it is vacant; (3) at the request of the +incumbent or on complaint by the archdeacon, rural dean or +patron. The surveyor specifies the works required, and gives an +<span class="pagenum"><a name="page271" id="page271"></a>271</span> +estimate of their probable cost. In the case of a vacant benefice, +the new incumbent and the old incumbent or his representatives +may lodge objections to the surveyor’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.</p> + + +<hr class="art" /> +<p><span class="bold">DILATATION<a name="ar101" id="ar101"></a></span> (from Lat. <i>dis-</i>, distributive, and <i>latus</i>, wide), a +widening or enlarging; a term used in physiology, &c.</p> + + +<hr class="art" /> +<p><span class="bold">DILATORY<a name="ar102" id="ar102"></a></span> (from Lat. <i>dilatus</i>, from <i>differre</i>, to put off or +delay), delaying, or slow; in law a “dilatory plea” is one +made merely for delaying the suit.</p> + + +<hr class="art" /> +<p><span class="bold">DILEMMA<a name="ar103" id="ar103"></a></span> (Gr. <span class="grk" title="dilêmma">δίλημμα</span>, a double proposition, from <span class="grk" title="di-">δί-</span> and +<span class="grk" title="lambanein">λαμβάνειν</span>), a term used technically in logic, and popularly +in common parlance and rhetoric. (1) The latter use has no +exact definition, but in general it describes a situation wherein +from either of two (or more) possible alternatives an unsatisfactory +conclusion results. The alternatives are called the +“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) <i>Simple Constructive</i>: If A, then C; if B, +then C, but either B or A; therefore C. (b) <i>Simple Destructive</i>: +If A is true, B is true; if A is true, C is true; B and C are not both +true; therefore A is not true. (c) <i>Complex Constructive</i>: If A, +then B; if C, then D; but either A or C; therefore either B or D. +(d) <i>Complex Destructive</i>: If A is true, B is true; if C is true, D is +true; but B and D are not both true; hence A and C are not +both true. The soundness of the dilemmatic argument in general +depends on the alternative possibilities. Unless the alternatives +produced exhaust the possibilities of the case, the conclusion is +invalid. The logical form of the argument makes it especially +valuable in public speaking, before uncritical audiences. It is, in +fact, important rather as a rhetorical subtlety than as a serious +argument.</p> + +<p><i>Dilemmist</i> is also a term used to translate <i>Vaibhashikas</i>, the +name of a Buddhist school of philosophy.</p> + + +<hr class="art" /> +<p><span class="bold">DILETTANTE,<a name="ar104" id="ar104"></a></span> an Italian word for one who delights in the fine +arts, especially in music and painting, so a lover of the fine arts +in general. The Ital. <i>dilettare</i> is from Lat. <i>delectare</i>, to delight. +Properly the word refers to an “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.</p> + + +<hr class="art" /> +<p><span class="bold">DILIGENCE,<a name="ar105" id="ar105"></a></span> in law, the care which a person is bound to +exercise in his relations with others. The possible degrees of +diligence are of course numerous, and the same degree is not +required in all cases. Thus a mere depositary would not be held +bound to the same degree of diligence as a person borrowing an +article for his own use and benefit. Jurists, following the divisions +of the civil law, have concurred in fixing three approximate +standards of diligence—viz. ordinary (<i>diligentia</i>), less than +ordinary (<i>levissima diligentia</i>) and more than ordinary +(<i>exactissima diligentia</i>). Ordinary or common diligence is defined +by Story (<i>On Bailments</i>) as “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” (<i>Essay on Bailments</i>). The highest degree of +diligence would be that which only very prudent persons bestow +on their own concerns; the lowest, that which even careless +persons bestow on their own concerns. The want of these various +degrees of diligence is negligence in corresponding degrees. These +approximations indicate roughly the greater or less severity with +which the law will judge the performance of different classes of +contracts; but English judges have been inclined to repudiate +the distinction as a useless refinement of the jurists. Thus Baron +Rolfe could see no difference between negligence and gross +negligence; it was the same thing with the addition of a vituperative +epithet. See <span class="sc"><a href="#artlinks">Negligence</a></span>.</p> + +<p><i>Diligence</i>, in Scots law, is a general term for the process by +which persons, lands or effects are attached on execution, or in +security for debt.</p> + + +<hr class="art" /> +<p><span class="bold">DILKE, SIR CHARLES WENTWORTH,<a name="ar106" id="ar106"></a></span> Bart. (1810-1869), +English politician, son of Charles Wentworth Dilke, proprietor +and editor of <i>The Athenaeum</i>, was born in London on the 18th +of February 1810, and was educated at Westminster school and +Trinity Hall, Cambridge. He studied law, and in 1834 took his +degree of LL.B., but did not practise. He assisted his father in +his literary work, and was for some years chairman of the council +of the Society of Arts, besides taking a prominent part in the +affairs of the Royal Horticultural Society and other bodies. He +was one of the most zealous promoters of the Great Exhibition +(1851), and a member of the executive committee. At the close +of the exhibition he was honoured by foreign sovereigns, and the +queen offered him knighthood, which, however, he did not accept; +he also declined a large remuneration offered by the royal commission. +In 1853 Dilke was one of the English commissioners at +the New York Industrial Exhibition, and prepared a report on it. +He again declined to receive any money reward for his services. +He was appointed one of the five royal commissioners for the +Great Exhibition of 1862; and soon after the death of the prince +consort he was created a baronet. In 1865 he entered parliament +as member for Wallingford. In 1869 he was sent to Russia as +representative of England at the horticultural exhibition held +at St Petersburg. His health, however, had been for some time +failing, and he died suddenly in that city, on the 10th of May 1869. +A selection from his writings, <i>Papers of a Critic</i> (2 vols., 1875), +contains a biographical sketch by his son.</p> + +<p>His son, <span class="sc">Sir Charles Wentworth Dilke, Bart.</span> (1843-  ), +became a prominent Liberal politician, as M.P. for Chelsea +(1868-1886), under-secretary for foreign affairs (1880-1882), and +president of the local government board (1882-1885); and he +was then marked out as one of the best-informed and ablest of the +advanced Radicals. He was chairman of the royal commission +on the housing of the working classes in 1884-1885. But his +sensational appearance as co-respondent in a divorce case of a +peculiarly unpleasant character in 1885 cast a cloud over his +career. He was defeated in Chelsea in 1886, and did not return +to parliament till 1892, when he was elected for the Forest of +Dean; and though his knowledge of foreign affairs and his +powers as a critic and writer on military and naval questions were +admittedly of the highest order, his official position in public life +could not again be recovered. His military writings are <i>The +British Army</i> (1888); <i>Army Reform</i> (1898) and, with Mr Spenser +Wilkinson, <i>Imperial Defence</i> (1892). On colonial questions he +wrote with equal authority. His <i>Greater Britain</i> (2 vols., 1866-1867) +reached a fourth edition in 1868, and was followed by +<i>Problems of Greater Britain</i> (2 vols., 1890) and <i>The British +Empire</i> (1899). He was twice married, his second wife (née +<span class="pagenum"><a name="page272" id="page272"></a>272</span> +Emilia Frances Strong), the widow of Mark Pattison, being +an accomplished art critic and collector. She died in 1904. The +most important of her books were the studies on <i>French Painters +of the Eighteenth Century</i> (1899) and three subsequent volumes on +the architects and sculptors, furniture and decoration, engravers +and draughtsmen of the same period, the last of which appeared +in 1902. A posthumous volume, <i>The Book of the Spiritual Life</i> +(1905), contains a memoir of her by Sir Charles Dilke.</p> + + +<hr class="art" /> +<table class="nobctr" style="float: right; width: 360px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:310px; height:260px" src="images/img272.jpg" alt="" /></td></tr> +<tr><td class="caption1"> Dill (<i>Anethum</i> or <i>Peucedanum graveolens</i>), +leaf and inflorescence.</td></tr></table> + +<p><span class="bold">DILL<a name="ar107" id="ar107"></a></span> (<i>Anethum</i> or <i>Peucedanum graveolens</i>), a member of the +natural botanical order Umbelliferae, indigenous to the south of +Europe, Egypt and the Cape of Good Hope. It resembles fennel +in appearance. Its root is long and fusiform; the stem is round, +jointed and about a yard high; the leaves have fragrant leaflets; +and the fruits are brown, +oval and concavo-convex. +The plant flowers +from June till August in +England. The seeds are +sown, preferably as soon +as ripe, either broadcast +or in drills between +6 and 12 in. asunder. +The young plants should +be thinned when 3 or 4 +weeks old, so as to be +at distances of about +10 in. A sheltered spot +and dry soil are needed +for the production of the +seed in the climate of England. The leaves of the dill are used in +soups and sauces, and, as well as the umbels, for flavouring +pickles. The seeds are employed for the preparation of dill-water +and oil of dill; they are largely consumed in the manufacture of +gin, and, when ground, are eaten in the East as a condiment. +The British Pharmacopoeia contains the Aqua Anethi or dill-water +(dose 1-2 oz.), and the Oleum Anethi, almost identical in +composition with caraway oil, and given in doses of ½-3 minims. +Dill-water is largely used as a carminative for children, and as a +vehicle for the exhibition of nauseous drugs.</p> + + +<hr class="art" /> +<p><span class="bold">DILLEN<a name="ar108" id="ar108"></a></span> [<span class="sc">Dillenius</span>], <b>JOHANN JAKOB</b> (1684-1747), English +botanist, was born at Darmstadt in 1684, and was educated at the +university of Giessen, where he wrote several botanical papers for +the <i>Ephemerides naturae curiosorum</i>, and printed, in 1719, his +<i>Catalogus plantarum sponte circa Gissam nascentium</i>, illustrated +with figures drawn and engraved by his own hand, and containing +descriptions of many new species. In 1721, at the instance of the +botanist William Sherard (1659-1728), he came to England, and +in 1724 he published a new edition of Ray’s <i>Synopsis stirpium +Britannicarum</i>. In 1732 he published <i>Hortus Elthamensis</i>, a +catalogue of the rare plants growing at Eltham, Kent, in the +collection of Sherard’s younger brother, James (1666-1738), who, +after making a fortune as an apothecary, devoted himself to +gardening and music. For this work Dillen himself executed 324 +plates, and it was described by Linnaeus, who spent a month +with him at Oxford in 1736, and afterwards dedicated his <i>Critica +botanica</i> to him, as “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 <i>Historia muscorum</i> (1741), died at Oxford, of +apoplexy, on the 2nd of April 1747. His manuscripts, books and +collections of dried plants, with many drawings, were bought by +his successor at Oxford, Dr Humphry Sibthorp (1713-1797), and +ultimately passed into the possession of the university.</p> + +<div class="condensed"> +<p>For an account of his collections preserved at Oxford, see <i>The +Dillenian Herbaria</i>, by G. Claridge Druce (Oxford, 1907).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DILLENBURG,<a name="ar109" id="ar109"></a></span> a town of Germany, in the Prussian province of +Hesse-Nassau, delightfully situated in the midst of a well-wooded +country, on the Dill, 25 m. N.W. from. Giessen on the railway to +Troisdorf. Pop. 4500. On an eminence above it lie the ruins +of the castle of Dillenburg, founded by Count Henry the Rich +of Nassau, about the year 1255, and the birthplace of Prince +William of Orange (1533). It has an Evangelical church, with +the vault of the princes of Nassau-Dillenburg, a Roman Catholic +church, a classical school, a teachers’ seminary and a chamber +of commerce. Its industries embrace iron-works, tanneries and +the manufacture of cigars. Owing to its beautiful surroundings +Dillenburg has become a favourite summer resort.</p> + + +<hr class="art" /> +<p><span class="bold">DILLENS, JULIEN<a name="ar110" id="ar110"></a></span> (1849-1904), Belgian sculptor, was born at +Antwerp on the 8th of June 1849, son of a painter. He studied +under Eugène Simonis at the Brussels Academy of Fine Arts. +In 1877 he received the <i>prix de Rome</i> for “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.</p> + + +<hr class="art" /> +<p><span class="bold">DILLINGEN,<a name="ar111" id="ar111"></a></span> a town of Germany, in the kingdom of Bavaria, +on the left bank of the Danube, 25 m. N.E. from Ulm, on the +railway to Ingolstadt. Pop. (1905) 6078. Its principal buildings +are an old palace, formerly the residence of the bishops of +Augsburg and now government offices, a royal gymnasium, a +Latin school with a library of 75,000 volumes, seven churches +(six Roman Catholic), two episcopal seminaries, a Capuchin +monastery, a Franciscan convent and a deaf and dumb asylum. +The university, founded in 1549, was abolished in 1804, being +converted into a lyceum. The inhabitants are engaged in cattle-rearing, +the cultivation of corn, hops and fruit, shipbuilding and +the shipping trade, and the manufacture of cloth, paper and +cutlery. In the vicinity is the Karolinen canal, which cuts off a +bend in the Danube between Lauingen and Dillingen. In 1488 +Dillingen became the residence of the bishops of Augsburg; was +taken by the Swedes in 1632 and 1648, by the Austrians in 1702, +and on the 17th of June 1800 by the French. In 1803 it passed +to Bavaria.</p> + + +<hr class="art" /> +<p><span class="bold">DILLMANN, CHRISTIAN FRIEDRICH AUGUST<a name="ar112" id="ar112"></a></span> (1823-1894), +German orientalist and biblical scholar, the son of a Württemberg +schoolmaster, was born at Illingen on the 25th of April 1823. He +was educated at Tübingen, where he became a pupil and friend of +Heinrich Ewald, and studied under F. C. Baur, though he did not +join the new Tübingen school. For a short time he worked as +pastor at Gersheim, near his native place, but he soon came to +feel that his studies demanded his whole time. He devoted himself +to the study of Ethiopic MSS. in the libraries of Paris, London +and Oxford, and this work caused a revival of Ethiopic study in +the 19th century. In 1847 and 1848 he prepared catalogues of +the Ethiopic MSS. in the British Museum and the Bodleian +library at Oxford. He then set to work upon an edition of the +Ethiopic bible. Returning to Tübingen in 1848, in 1853 he was +appointed professor extraordinarius. Subsequently he became +<span class="pagenum"><a name="page273" id="page273"></a>273</span> +professor of philosophy at Kiel (1854), and of theology at Giessen +(1864) and Berlin (1869). He died on the 4th of July 1894.</p> + +<p>In 1851 he had published the <i>Book of Enoch</i> in Ethiopian +(German, 1853), and at Kiel he completed the first part of the +Ethiopic bible, <i>Octateuchus Aethiopicus</i> (1853-1855). In 1857 +appeared his <i>Grammatik der äthiopischen Sprache</i> (2nd ed. by +C. Bezold, 1899); in 1859 the <i>Book of Jubilees</i>; in 1861 and 1871 +another part of the Ethiopic bible, <i>Libri Regum</i>; in 1865 his +great <i>Lexicon linguae aethiopicae</i>; in 1866 his <i>Chrestomathia +aethiopica</i>. Always a theologian at heart, however, he returned +to theology in 1864. His Giessen lectures were published under +the titles, <i>Ursprung der alttestamentlichen Religion</i> (1865) and +<i>Die Propheten des alten Bundes nach ihrer politischen Wirksamkeit</i> +(1868). In 1869 appeared his <i>Commentar zum Hiob</i> (4th ed. 1891) +which stamped him as one of the foremost Old Testament +exegetes. His renown as a theologian, however, was mainly +founded by the series of commentaries, based on those of August +Wilhelm Knobels’ <i>Genesis</i> (Leipzig, 1875; 6th ed. 1892; Eng. +trans, by W. B. Stevenson, Edinburgh, 1897); <i>Exodus und +Leviticus</i>, 1880, revised edition by V. Ryssel, 1897; <i>Numeri, +Deuteronomium und Josua</i>, with a dissertation on the origin of +the Hexateuch, 1886; <i>Jesaja</i>, 1890 (revised edition by Rudolf +Kittel in 1898). In 1877 he published the <i>Ascension of Isaiah</i> +in Ethiopian and Latin. He was also a contributor to D. +Schenkel’s <i>Bibellexikon</i>, Brockhaus’s <i>Conversationslexikon</i>, and +Herzog’s <i>Realencyklopädie</i>. His lectures on Old Testament +theology, <i>Vorlesungen über Theologie des Allen Testamentes</i>, were +published by Kittel in 1895.</p> + +<div class="condensed"> +<p>See the articles in Herzog-Hauck, <i>Realencyklopädie</i>, and the +<i>Allgemeine deutsche Biographie</i>; F. Lichtenberger, <i>History of +German Theology in the Nineteenth Century</i> (1889); Wolf Baudissin, +<i>A. Dillmann</i> (Leipzig, 1895).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DILLON, ARTHUR RICHARD<a name="ar113" id="ar113"></a></span> (1721-1807), French archbishop, +was the son of Arthur Dillon (1670-1733), an Irish +gentleman who became general in the French service. He was +born at St Germain, entered the priesthood and was successively +curé of Elan near Mezières, vicar-general of Pontoise (1747), +bishop of Evreux (1753) and archbishop of Toulouse (1758), +archbishop of Narbonne in 1763, and in that capacity, president +of the estates of Languedoc. He devoted himself much less to +the spiritual direction of his diocese than to its temporal welfare, +carrying out many works of public utility, bridges, canals, roads, +harbours, &c.; had chairs of chemistry and of physics created at +Montpellier and at Toulouse, and tried to reduce the poverty, +especially in Narbonne. In 1787 and in 1788 he was a member of +the Assembly of Notables called together by Louis XVI., and in +1788 presided over the assembly of the clergy. Having refused +to accept the civil constitution of the clergy, Dillon had to leave +Narbonne in 1790, then to emigrate to Coblenz in 1791. Soon +afterwards he went to London, where he lived until his death in +1807, never accepting the Concordat, which had suppressed his +archiepiscopal see.</p> + +<div class="condensed"> +<p>See L. Audibret, <i>Le Dernier Président des États du Languedoc, Mgr. +Arthur Richard Dillon, archevêque de Narbonne</i> (Bordeaux, 1868); +L. de Lavergne, <i>Les Assemblées provinciales sous Louis XVI</i> +(Paris, 1864).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">DILLON, JOHN<a name="ar114" id="ar114"></a></span> (1851-  ), 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.</p> + + +<hr class="art" /> +<p><span class="bold">DILUVIUM<a name="ar115" id="ar115"></a></span> (Lat. for “deluge,” from <i>diluere</i>, to wash away), +a term in geology for superficial deposits formed by flood-like +operations of water, and so contrasted with alluvium (<i>q.v.</i>) or +alluvial deposits formed by slow and steady aqueous agencies. +The term was formerly given to the “boulder clay” deposits, +supposed to have been caused by the Noachian deluge.</p> + + +<hr class="art" /> +<p><span class="bold">DIME<a name="ar116" id="ar116"></a></span> (from the Lat. <i>decima</i>, a tenth, through the O. Fr. +<i>disme</i>), the tenth part, the tithe paid as church dues, or as tribute +to a temporal power. In this sense it is obsolete, but is found in +Wycliffe’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.”</p> + + +<hr class="art" /> +<p><span class="bold">DIMENSION<a name="ar117" id="ar117"></a></span> (from Lat. <i>dimensio</i>, a measuring), in geometry, a +magnitude measured in a specified direction, <i>i.e.</i> length, breadth +and thickness; thus a line has only length and is said to be of +one dimension, a surface has length and breadth, and has two +dimensions, a solid has length, breadth and thickness, and has +three dimensions. This concept is extended to algebra: since +a line, surface and solid are represented by linear, quadratic and +cubic equations, and are of one, two and three dimensions; a +biquadratic equation has its highest terms of four dimensions, +and, in general, an equation in any number of variables which has +the greatest sum of the indices of any term equal to n is said to +have n dimensions. The “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 <span class="sc"><a href="#artlinks">Units, Dimensions of</a></span>.</p> + + +<hr class="art" /> +<p><span class="bold">DIMITY,<a name="ar118" id="ar118"></a></span> derived from the Gr. <span class="grk" title="dimitos">δίμιτος</span> “double thread,” +through the Ital. <i>dimito</i>, “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.</p> + + +<hr class="art" /> +<p><span class="bold">DINAJPUR,<a name="ar119" id="ar119"></a></span> a town (with a population in 1901 of 13,430) and +district of <span class="correction" title="amended from Britsh">British</span> India, in the Rajshahi division of Eastern +Bengal and Assam. The earthquake of the 12th of June 1897 +caused serious damage to most of the public buildings of the town. +There is a railway station and a government high school. The +district comprises an area of 3946 sq. m. It is traversed in every +direction by a network of channels and water courses. Along the +banks of the Kulik river, the undulating ridges and long lines of +<span class="pagenum"><a name="page274" id="page274"></a>274</span> +mango-trees give the landscape a beauty which is not found elsewhere. +Dinajpur forms part of the rich arable tract lying between +the Ganges and the southern slopes of the Himalayas. Although +essentially a fluvial district, it does not possess any river navigable +throughout the year by boats of 4 tons burden. Rice forms the +staple agricultural product. The climate of the district, although +cooler than that of Calcutta, is very unhealthy, and the people +have a sickly appearance. The worst part of the year is at the +close of the rains in September and October, during which months +few of the natives escape fever. The average maximum temperature +is 92.3° F., and the minimum 74.8°. The average rainfall +is 85.54 in. In 1901 the population was 1,567,080, showing an +increase of 6% in the decade. The district is partly traversed +by the main line of the Eastern Bengal railway and by two branch +lines. Save between 1404 and 1442, when it was the seat of +an independent <i>raj</i>, founded by Raja Ganesh, a Hindu turned +Mussulman, Dinajpur has no separate history. Pillars and +copper-plate inscriptions have yielded numerous records of the +Pal kings who ruled the country from the 9th century onwards, +and the district is famous for many other antiquities, some of +which are connected by legend with an immemorial past (see +<i>Reports, Arch. Survey of India</i>, xv.; <i>Epigraphia Indica</i>, ii.).</p> + + +<hr class="art" /> +<p><span class="bold">DINAN,<a name="ar120" id="ar120"></a></span> a town of north-western France, capital of an +arrondissement in the department of Côtes-du-Nord, 37 m. E. of +St Brieuc on the Western railway. Pop. (1906) 8588. Dinan is +situated on a height on the left bank of the Ranee (here canalized), +some 17 m. above its mouth at St Malo, with which it communicates +by means of small steamers. It is united to the village +of Lanvallay on the right bank of the river by a granite viaduct +130 ft. in height. The town is almost entirely encircled by the +ramparts of the middle ages, strengthened at intervals by towers +and defended on the south by a castle of the late 14th century, +which now serves as prison. Three old gateways are also preserved. +Dinan has two interesting churches; that of St Malo, of +late Gothic architecture, and St Sauveur, in which the Romanesque +and Gothic styles are intermingled. In the latter church a +granite monument contains the heart of Bertrand Du Guesclin, +whose connexion with the town is also commemorated by an +equestrian statue. The quaint winding streets of Dinan are often +bordered by medieval houses. Its picturesqueness attracts large +numbers of visitors and there are many English residents in the +town and its vicinity. About three-quarters of a mile from the +town are the ruins of the château and the Benedictine abbey at +Léhon; near the neighbouring village of St Esprit stands the +large lunatic asylum of Les Bas Foins, founded in 1836; and at +no great distance is the now dismantled château of La Garaye, +which was rendered famous in the 18th century by the philanthropic +devotion of the count and countess whose story is told +in Mrs Norton’s <i>Lady of La Garaye</i>. Dinan is the seat of a subprefect +and has a tribunal of first instance, and a communal +college. There is trade in grain, cider, wax, butter and other +agricultural products. The industries include the manufacture +of leather, farm-implements and canvas.</p> + +<p>The principal event in the history of Dinan, which was a stronghold +of the dukes of Brittany, is the siege by the English under the +duke of Lancaster in 1359, during which Du Guesclin and an +English knight called Thomas of Canterbury engaged in single +combat.</p> + + +<hr class="art" /> +<p><span class="bold">DINANT,<a name="ar121" id="ar121"></a></span> an ancient town on the right bank of the Meuse in +the province of Namur, Belgium, connected by a bridge with the +left bank, on which are the station and the suburb of St Medard. +Pop. (1904) 7674. The name is supposed to be derived from +Diana, and as early as the 7th century it was named as one of the +dependencies of the bishopric of Tongres. In the 10th century it +passed under the titular sway of Liége, and remained the fief of the +prince-bishopric till the French revolution put an end to that +survival of feudalism. In the middle of the 15th century Dinant +reached the height of its prosperity. With a population of +60,000, and 8000 workers in copper, it was one of the most +flourishing cities in Walloon Belgium, until it incurred the wrath +of Charles the Bold. Belief in the strength of its walls and of the +castle that occupied the centre bridge, thus effectually commanding +navigation by the river, engendered arrogance and overconfidence, +and the people of Dinant thought they could defy the +full power of Burgundy. Perhaps they also expected aid from France or Liége. In 1466 Charles, in his father’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.</p> + + +<hr class="art" /> +<p><span class="bold">DINAPUR,<a name="ar122" id="ar122"></a></span> a town and military station of British India, in the +Patna district of Bengal, on the right bank of the Ganges, 12 m. +W. of Patna city by rail. Pop. (1901) 33,699. It is the largest +military cantonment in Bengal, with accommodation for two +batteries of artillery, a European and a native infantry regiment. +In 1857 the sepoy garrison of the place initiated the mutiny of +that year in Patna district, but after a conflict with the European +troops were forced to retire from the town, and subsequently laid +siege to Arrah.</p> + + +<hr class="art" /> +<p><span class="bold">DINARCHUS,<a name="ar123" id="ar123"></a></span> last of the “ten” Attic orators, son of Sostratus +(or, according to Suidas, Socrates), born at Corinth about 361 +<span class="sc">b.c.</span> He settled at Athens early in life, and when not more than +twenty-five was already active as a writer of speeches for the law +courts. As an alien, he was unable to take part in the debates. +He had been the pupil both of Theophrastus and of Demetrius +Phalereus, and had early acquired a certain fluency and versatility +of style. In 324 the Areopagus, after inquiry, reported +that nine men had taken bribes from Harpalus, the fugitive +treasurer of Alexander. Ten public prosecutors were appointed. +Dinarchus wrote, for one or more of these prosecutors, the three +speeches which are still extant—<i>Against Demosthenes</i>, <i>Against +Aristogeiton</i>, <i>Against Philocles</i>. The sympathies of Dinarchus +were in favour of an Athenian oligarchy under Macedonian +control; but it should be remembered that he was not an +Athenian citizen. Aeschines and Demades had no such excuse. +In the Harpalus affair, Demosthenes was doubtless innocent, +and so, probably, were others of the accused. Yet Hypereides, +the most fiery of the patriots, was on the same side as Dinarchus.</p> + +<p>Under the regency of his old master, Demetrius Phalereus, +Dinarchus exercised much political influence. The years 317-307 +were the most prosperous of his life. On the fall of Demetrius +Phalereus and the restoration of the democracy by Demetrius +Poliorcetes, Dinarchus was condemned to death and withdrew +into exile at Chalcis in Euboea. About 292, thanks to his friend +Theophrastus, he was able to return to Attica, and took up his +abode in the country with a former associate, Proxenus. He +afterwards brought an action against Proxenus on the ground +that he had robbed him of some money and plate. Dinarchus +died at Athens about 291.</p> + +<p>According to Suidas, Dinarchus wrote 160 speeches; and Dionysius held +that, out of 85 extant speeches bearing his name, 58 were genuine,—28 +relating to public, 30 to private causes. Although the authenticity of +the three speeches mentioned above is generally admitted, Demetrius of +Magnesia doubted that of the speech <i>Against Demosthenes</i>, while A. +Westermann rejected all three. Dinarchus had little individual style and +imitated by turns Lysias, Hypereides and Demosthenes. He is called by +Hermogenes <span class="grk" title="o critinos demostenes">ὁ κριθινὸς Δημοσθένης</span>, a metaphor taken from barley compared +with wheat, or beer compared with wine,—a Demosthenes whose strength +is rougher, without flavour or sparkle.</p> + +<div class="condensed"> +<p>Editions: (text and exhaustive commentary) E. Mätzner (1842); (text) T. +Thalheim (1887), F. Blass (1888); see L.L. Forman, <i>Index Andocideus, +Lycurgeus, Dinarcheus</i> (1897); and, in general, F. Blass, <i>Attische +Beredsamkeit</i>, iii. There is a valuable treatise on the life and speeches +of Dinarchus by Dionysius of Halicarnassus.</p> +</div> + + +<hr class="art" /> + + + + + + + + + +<pre> + + + + + +End of the Project Gutenberg EBook of Encyclopaedia Britannica, 11th +Edition, Volume 8, Slice 4, by Various + +*** END OF THIS PROJECT GUTENBERG EBOOK ENCYC. BRITANNICA, VOL 8 SL 4 *** + +***** This file should be named 32607-h.htm or 32607-h.zip ***** +This and all associated files of various formats will be found in: + https://www.gutenberg.org/3/2/6/0/32607/ + +Produced by Marius Masi, Don Kretz and the Online +Distributed Proofreading Team at https://www.pgdp.net + + +Updated editions will replace the previous one--the old editions +will be renamed. + +Creating the works from public domain print editions means that no +one owns a United States copyright in these works, so the Foundation +(and you!) can copy and distribute it in the United States without +permission and without paying copyright royalties. Special rules, +set forth in the General Terms of Use part of this license, apply to +copying and distributing Project Gutenberg-tm electronic works to +protect the PROJECT GUTENBERG-tm concept and trademark. Project +Gutenberg is a registered trademark, and may not be used if you +charge for the eBooks, unless you receive specific permission. If you +do not charge anything for copies of this eBook, complying with the +rules is very easy. You may use this eBook for nearly any purpose +such as creation of derivative works, reports, performances and +research. They may be modified and printed and given away--you may do +practically ANYTHING with public domain eBooks. Redistribution is +subject to the trademark license, especially commercial +redistribution. + + + +*** START: FULL LICENSE *** + +THE FULL PROJECT GUTENBERG LICENSE +PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK + +To protect the Project Gutenberg-tm mission of promoting the free +distribution of electronic works, by using or distributing this work +(or any other work associated in any way with the phrase "Project +Gutenberg"), you agree to comply with all the terms of the Full Project +Gutenberg-tm License (available with this file or online at +https://gutenberg.org/license). + + +Section 1. General Terms of Use and Redistributing Project Gutenberg-tm +electronic works + +1.A. By reading or using any part of this Project Gutenberg-tm +electronic work, you indicate that you have read, understand, agree to +and accept all the terms of this license and intellectual property +(trademark/copyright) agreement. If you do not agree to abide by all +the terms of this agreement, you must cease using and return or destroy +all copies of Project Gutenberg-tm electronic works in your possession. +If you paid a fee for obtaining a copy of or access to a Project +Gutenberg-tm electronic work and you do not agree to be bound by the +terms of this agreement, you may obtain a refund from the person or +entity to whom you paid the fee as set forth in paragraph 1.E.8. + +1.B. "Project Gutenberg" is a registered trademark. It may only be +used on or associated in any way with an electronic work by people who +agree to be bound by the terms of this agreement. There are a few +things that you can do with most Project Gutenberg-tm electronic works +even without complying with the full terms of this agreement. See +paragraph 1.C below. There are a lot of things you can do with Project +Gutenberg-tm electronic works if you follow the terms of this agreement +and help preserve free future access to Project Gutenberg-tm electronic +works. See paragraph 1.E below. + +1.C. The Project Gutenberg Literary Archive Foundation ("the Foundation" +or PGLAF), owns a compilation copyright in the collection of Project +Gutenberg-tm electronic works. Nearly all the individual works in the +collection are in the public domain in the United States. If an +individual work is in the public domain in the United States and you are +located in the United States, we do not claim a right to prevent you from +copying, distributing, performing, displaying or creating derivative +works based on the work as long as all references to Project Gutenberg +are removed. Of course, we hope that you will support the Project +Gutenberg-tm mission of promoting free access to electronic works by +freely sharing Project Gutenberg-tm works in compliance with the terms of +this agreement for keeping the Project Gutenberg-tm name associated with +the work. You can easily comply with the terms of this agreement by +keeping this work in the same format with its attached full Project +Gutenberg-tm License when you share it without charge with others. + +1.D. The copyright laws of the place where you are located also govern +what you can do with this work. Copyright laws in most countries are in +a constant state of change. If you are outside the United States, check +the laws of your country in addition to the terms of this agreement +before downloading, copying, displaying, performing, distributing or +creating derivative works based on this work or any other Project +Gutenberg-tm work. The Foundation makes no representations concerning +the copyright status of any work in any country outside the United +States. + +1.E. Unless you have removed all references to Project Gutenberg: + +1.E.1. The following sentence, with active links to, or other immediate +access to, the full Project Gutenberg-tm License must appear prominently +whenever any copy of a Project Gutenberg-tm work (any work on which the +phrase "Project Gutenberg" appears, or with which the phrase "Project +Gutenberg" is associated) is accessed, displayed, performed, viewed, +copied or distributed: + +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 + +1.E.2. If an individual Project Gutenberg-tm electronic work is derived +from the public domain (does not contain a notice indicating that it is +posted with permission of the copyright holder), the work can be copied +and distributed to anyone in the United States without paying any fees +or charges. If you are redistributing or providing access to a work +with the phrase "Project Gutenberg" associated with or appearing on the +work, you must comply either with the requirements of paragraphs 1.E.1 +through 1.E.7 or obtain permission for the use of the work and the +Project Gutenberg-tm trademark as set forth in paragraphs 1.E.8 or +1.E.9. + +1.E.3. If an individual Project Gutenberg-tm electronic work is posted +with the permission of the copyright holder, your use and distribution +must comply with both paragraphs 1.E.1 through 1.E.7 and any additional +terms imposed by the copyright holder. Additional terms will be linked +to the Project Gutenberg-tm License for all works posted with the +permission of the copyright holder found at the beginning of this work. + +1.E.4. Do not unlink or detach or remove the full Project Gutenberg-tm +License terms from this work, or any files containing a part of this +work or any other work associated with Project Gutenberg-tm. + +1.E.5. Do not copy, display, perform, distribute or redistribute this +electronic work, or any part of this electronic work, without +prominently displaying the sentence set forth in paragraph 1.E.1 with +active links or immediate access to the full terms of the Project +Gutenberg-tm License. + +1.E.6. You may convert to and distribute this work in any binary, +compressed, marked up, nonproprietary or proprietary form, including any +word processing or hypertext form. However, if you provide access to or +distribute copies of a Project Gutenberg-tm work in a format other than +"Plain Vanilla ASCII" or other format used in the official version +posted on the official Project Gutenberg-tm web site (www.gutenberg.org), +you must, at no additional cost, fee or expense to the user, provide a +copy, a means of exporting a copy, or a means of obtaining a copy upon +request, of the work in its original "Plain Vanilla ASCII" or other +form. Any alternate format must include the full Project Gutenberg-tm +License as specified in paragraph 1.E.1. + +1.E.7. Do not charge a fee for access to, viewing, displaying, +performing, copying or distributing any Project Gutenberg-tm works +unless you comply with paragraph 1.E.8 or 1.E.9. + +1.E.8. You may charge a reasonable fee for copies of or providing +access to or distributing Project Gutenberg-tm electronic works provided +that + +- You pay a royalty fee of 20% of the gross profits you derive from + the use of Project Gutenberg-tm works calculated using the method + you already use to calculate your applicable taxes. The fee is + owed to the owner of the Project Gutenberg-tm trademark, but he + has agreed to donate royalties under this paragraph to the + Project Gutenberg Literary Archive Foundation. Royalty payments + must be paid within 60 days following each date on which you + prepare (or are legally required to prepare) your periodic tax + returns. Royalty payments should be clearly marked as such and + sent to the Project Gutenberg Literary Archive Foundation at the + address specified in Section 4, "Information about donations to + the Project Gutenberg Literary Archive Foundation." + +- You provide a full refund of any money paid by a user who notifies + you in writing (or by e-mail) within 30 days of receipt that s/he + does not agree to the terms of the full Project Gutenberg-tm + License. You must require such a user to return or + destroy all copies of the works possessed in a physical medium + and discontinue all use of and all access to other copies of + Project Gutenberg-tm works. + +- You provide, in accordance with paragraph 1.F.3, a full refund of any + money paid for a work or a replacement copy, if a defect in the + electronic work is discovered and reported to you within 90 days + of receipt of the work. + +- You comply with all other terms of this agreement for free + distribution of Project Gutenberg-tm works. + +1.E.9. If you wish to charge a fee or distribute a Project Gutenberg-tm +electronic work or group of works on different terms than are set +forth in this agreement, you must obtain permission in writing from +both the Project Gutenberg Literary Archive Foundation and Michael +Hart, the owner of the Project Gutenberg-tm trademark. Contact the +Foundation as set forth in Section 3 below. + +1.F. + +1.F.1. Project Gutenberg volunteers and employees expend considerable +effort to identify, do copyright research on, transcribe and proofread +public domain works in creating the Project Gutenberg-tm +collection. Despite these efforts, Project Gutenberg-tm electronic +works, and the medium on which they may be stored, may contain +"Defects," such as, but not limited to, incomplete, inaccurate or +corrupt data, transcription errors, a copyright or other intellectual +property infringement, a defective or damaged disk or other medium, a +computer virus, or computer codes that damage or cannot be read by +your equipment. + +1.F.2. LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the "Right +of Replacement or Refund" described in paragraph 1.F.3, the Project +Gutenberg Literary Archive Foundation, the owner of the Project +Gutenberg-tm trademark, and any other party distributing a Project +Gutenberg-tm electronic work under this agreement, disclaim all +liability to you for damages, costs and expenses, including legal +fees. YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT +LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSE +PROVIDED IN PARAGRAPH F3. YOU AGREE THAT THE FOUNDATION, THE +TRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE +LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL, PUNITIVE OR +INCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCH +DAMAGE. + +1.F.3. LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover a +defect in this electronic work within 90 days of receiving it, you can +receive a refund of the money (if any) you paid for it by sending a +written explanation to the person you received the work from. If you +received the work on a physical medium, you must return the medium with +your written explanation. The person or entity that provided you with +the defective work may elect to provide a replacement copy in lieu of a +refund. If you received the work electronically, the person or entity +providing it to you may choose to give you a second opportunity to +receive the work electronically in lieu of a refund. If the second copy +is also defective, you may demand a refund in writing without further +opportunities to fix the problem. + +1.F.4. Except for the limited right of replacement or refund set forth +in paragraph 1.F.3, this work is provided to you 'AS-IS' WITH NO OTHER +WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO +WARRANTIES OF MERCHANTIBILITY OR FITNESS FOR ANY PURPOSE. + +1.F.5. Some states do not allow disclaimers of certain implied +warranties or the exclusion or limitation of certain types of damages. +If any disclaimer or limitation set forth in this agreement violates the +law of the state applicable to this agreement, the agreement shall be +interpreted to make the maximum disclaimer or limitation permitted by +the applicable state law. The invalidity or unenforceability of any +provision of this agreement shall not void the remaining provisions. + +1.F.6. INDEMNITY - You agree to indemnify and hold the Foundation, the +trademark owner, any agent or employee of the Foundation, anyone +providing copies of Project Gutenberg-tm electronic works in accordance +with this agreement, and any volunteers associated with the production, +promotion and distribution of Project Gutenberg-tm electronic works, +harmless from all liability, costs and expenses, including legal fees, +that arise directly or indirectly from any of the following which you do +or cause to occur: (a) distribution of this or any Project Gutenberg-tm +work, (b) alteration, modification, or additions or deletions to any +Project Gutenberg-tm work, and (c) any Defect you cause. + + +Section 2. Information about the Mission of Project Gutenberg-tm + +Project Gutenberg-tm is synonymous with the free distribution of +electronic works in formats readable by the widest variety of computers +including obsolete, old, middle-aged and new computers. It exists +because of the efforts of hundreds of volunteers and donations from +people in all walks of life. + +Volunteers and financial support to provide volunteers with the +assistance they need are critical to reaching Project Gutenberg-tm's +goals and ensuring that the Project Gutenberg-tm collection will +remain freely available for generations to come. In 2001, the Project +Gutenberg Literary Archive Foundation was created to provide a secure +and permanent future for Project Gutenberg-tm and future generations. +To learn more about the Project Gutenberg Literary Archive Foundation +and how your efforts and donations can help, see Sections 3 and 4 +and the Foundation web page at https://www.pglaf.org. + + +Section 3. Information about the Project Gutenberg Literary Archive +Foundation + +The Project Gutenberg Literary Archive Foundation is a non profit +501(c)(3) educational corporation organized under the laws of the +state of Mississippi and granted tax exempt status by the Internal +Revenue Service. The Foundation's EIN or federal tax identification +number is 64-6221541. Its 501(c)(3) letter is posted at +https://pglaf.org/fundraising. Contributions to the Project Gutenberg +Literary Archive Foundation are tax deductible to the full extent +permitted by U.S. federal laws and your state's laws. + +The Foundation's principal office is located at 4557 Melan Dr. S. +Fairbanks, AK, 99712., but its volunteers and employees are scattered +throughout numerous locations. Its business office is located at +809 North 1500 West, Salt Lake City, UT 84116, (801) 596-1887, email +business@pglaf.org. Email contact links and up to date contact +information can be found at the Foundation's web site and official +page at https://pglaf.org + +For additional contact information: + Dr. Gregory B. Newby + Chief Executive and Director + gbnewby@pglaf.org + + +Section 4. Information about Donations to the Project Gutenberg +Literary Archive Foundation + +Project Gutenberg-tm depends upon and cannot survive without wide +spread public support and donations to carry out its mission of +increasing the number of public domain and licensed works that can be +freely distributed in machine readable form accessible by the widest +array of equipment including outdated equipment. Many small donations +($1 to $5,000) are particularly important to maintaining tax exempt +status with the IRS. + +The Foundation is committed to complying with the laws regulating +charities and charitable donations in all 50 states of the United +States. Compliance requirements are not uniform and it takes a +considerable effort, much paperwork and many fees to meet and keep up +with these requirements. We do not solicit donations in locations +where we have not received written confirmation of compliance. To +SEND DONATIONS or determine the status of compliance for any +particular state visit https://pglaf.org + +While we cannot and do not solicit contributions from states where we +have not met the solicitation requirements, we know of no prohibition +against accepting unsolicited donations from donors in such states who +approach us with offers to donate. + +International donations are gratefully accepted, but we cannot make +any statements concerning tax treatment of donations received from +outside the United States. U.S. laws alone swamp our small staff. + +Please check the Project Gutenberg Web pages for current donation +methods and addresses. Donations are accepted in a number of other +ways including including checks, online payments and credit card +donations. To donate, please visit: https://pglaf.org/donate + + +Section 5. General Information About Project Gutenberg-tm electronic +works. + +Professor Michael S. Hart was the originator of the Project Gutenberg-tm +concept of a library of electronic works that could be freely shared +with anyone. For thirty years, he produced and distributed Project +Gutenberg-tm eBooks with only a loose network of volunteer support. + + +Project Gutenberg-tm eBooks are often created from several printed +editions, all of which are confirmed as Public Domain in the U.S. +unless a copyright notice is included. Thus, we do not necessarily +keep eBooks in compliance with any particular paper edition. + + +Most people start at our Web site which has the main PG search facility: + + https://www.gutenberg.org + +This Web site includes information about Project Gutenberg-tm, +including how to make donations to the Project Gutenberg Literary +Archive Foundation, how to help produce our new eBooks, and how to +subscribe to our email newsletter to hear about new eBooks. + + +</pre> + +</body> +</html> diff --git a/32607-h/images/img158a.jpg b/32607-h/images/img158a.jpg Binary files differnew file mode 100644 index 0000000..0e3e052 --- /dev/null +++ b/32607-h/images/img158a.jpg diff --git a/32607-h/images/img158b.jpg b/32607-h/images/img158b.jpg Binary files differnew file mode 100644 index 0000000..d301b52 --- /dev/null +++ b/32607-h/images/img158b.jpg diff --git a/32607-h/images/img158c.jpg b/32607-h/images/img158c.jpg Binary files differnew file mode 100644 index 0000000..417f9b5 --- /dev/null +++ b/32607-h/images/img158c.jpg diff --git a/32607-h/images/img158d.jpg b/32607-h/images/img158d.jpg Binary files differnew file mode 100644 index 0000000..11a96a3 --- /dev/null +++ b/32607-h/images/img158d.jpg diff --git a/32607-h/images/img160a1.jpg b/32607-h/images/img160a1.jpg Binary files differnew file mode 100644 index 0000000..5a27934 --- /dev/null +++ b/32607-h/images/img160a1.jpg diff --git a/32607-h/images/img160a2.jpg b/32607-h/images/img160a2.jpg Binary files differnew file mode 100644 index 0000000..a7ebe9a --- /dev/null +++ b/32607-h/images/img160a2.jpg diff --git a/32607-h/images/img160a3.jpg b/32607-h/images/img160a3.jpg Binary files differnew file mode 100644 index 0000000..47c84a1 --- /dev/null +++ b/32607-h/images/img160a3.jpg diff --git a/32607-h/images/img160b1.jpg b/32607-h/images/img160b1.jpg Binary files differnew file mode 100644 index 0000000..472b6f2 --- /dev/null +++ b/32607-h/images/img160b1.jpg diff --git a/32607-h/images/img160b2.jpg b/32607-h/images/img160b2.jpg Binary files differnew file mode 100644 index 0000000..eca9da5 --- /dev/null +++ b/32607-h/images/img160b2.jpg diff --git a/32607-h/images/img161.jpg b/32607-h/images/img161.jpg Binary files differnew file mode 100644 index 0000000..e149007 --- /dev/null +++ b/32607-h/images/img161.jpg diff --git a/32607-h/images/img166a.jpg b/32607-h/images/img166a.jpg Binary files differnew file mode 100644 index 0000000..7112a86 --- /dev/null +++ b/32607-h/images/img166a.jpg diff --git a/32607-h/images/img166b.jpg b/32607-h/images/img166b.jpg Binary files differnew file mode 100644 index 0000000..07d394c --- /dev/null +++ b/32607-h/images/img166b.jpg diff --git a/32607-h/images/img169a.jpg b/32607-h/images/img169a.jpg Binary files differnew file mode 100644 index 0000000..84fdedc --- /dev/null +++ b/32607-h/images/img169a.jpg diff --git a/32607-h/images/img169b.jpg b/32607-h/images/img169b.jpg Binary files differnew file mode 100644 index 0000000..7e8c826 --- /dev/null +++ b/32607-h/images/img169b.jpg diff --git a/32607-h/images/img169c.jpg b/32607-h/images/img169c.jpg Binary files differnew file mode 100644 index 0000000..a66928c --- /dev/null +++ b/32607-h/images/img169c.jpg diff --git a/32607-h/images/img169d.jpg b/32607-h/images/img169d.jpg Binary files differnew file mode 100644 index 0000000..048bae0 --- /dev/null +++ b/32607-h/images/img169d.jpg diff --git a/32607-h/images/img169e.jpg b/32607-h/images/img169e.jpg Binary files differnew file mode 100644 index 0000000..ef62cb8 --- /dev/null +++ b/32607-h/images/img169e.jpg diff --git a/32607-h/images/img170.jpg b/32607-h/images/img170.jpg Binary files differnew file mode 100644 index 0000000..83b01b8 --- /dev/null +++ b/32607-h/images/img170.jpg diff --git a/32607-h/images/img174a.jpg b/32607-h/images/img174a.jpg Binary files differnew file mode 100644 index 0000000..8fad652 --- /dev/null +++ b/32607-h/images/img174a.jpg diff --git a/32607-h/images/img174b.jpg b/32607-h/images/img174b.jpg Binary files differnew file mode 100644 index 0000000..754b7e1 --- /dev/null +++ b/32607-h/images/img174b.jpg diff --git a/32607-h/images/img174c.jpg b/32607-h/images/img174c.jpg Binary files differnew file mode 100644 index 0000000..73452a0 --- /dev/null +++ b/32607-h/images/img174c.jpg diff --git a/32607-h/images/img174d.jpg b/32607-h/images/img174d.jpg Binary files differnew file mode 100644 index 0000000..1379261 --- /dev/null +++ b/32607-h/images/img174d.jpg diff --git a/32607-h/images/img174e.jpg b/32607-h/images/img174e.jpg Binary files differnew file mode 100644 index 0000000..da793c1 --- /dev/null +++ b/32607-h/images/img174e.jpg diff --git a/32607-h/images/img174f.jpg b/32607-h/images/img174f.jpg Binary files differnew file mode 100644 index 0000000..a0ff272 --- /dev/null +++ b/32607-h/images/img174f.jpg diff --git a/32607-h/images/img175.jpg b/32607-h/images/img175.jpg Binary files differnew file mode 100644 index 0000000..98a2fee --- /dev/null +++ b/32607-h/images/img175.jpg diff --git a/32607-h/images/img209.jpg b/32607-h/images/img209.jpg Binary files differnew file mode 100644 index 0000000..a9922e5 --- /dev/null +++ b/32607-h/images/img209.jpg diff --git a/32607-h/images/img223.jpg b/32607-h/images/img223.jpg Binary files differnew file mode 100644 index 0000000..83c2071 --- /dev/null +++ b/32607-h/images/img223.jpg diff --git a/32607-h/images/img239.jpg b/32607-h/images/img239.jpg Binary files differnew file mode 100644 index 0000000..b1d306a --- /dev/null +++ b/32607-h/images/img239.jpg diff --git a/32607-h/images/img240.jpg b/32607-h/images/img240.jpg Binary files differnew file mode 100644 index 0000000..96f2244 --- /dev/null +++ b/32607-h/images/img240.jpg diff --git a/32607-h/images/img241.jpg b/32607-h/images/img241.jpg Binary files differnew file mode 100644 index 0000000..6703d8d --- /dev/null +++ b/32607-h/images/img241.jpg diff --git a/32607-h/images/img243.jpg b/32607-h/images/img243.jpg Binary files differnew file mode 100644 index 0000000..dbce055 --- /dev/null +++ b/32607-h/images/img243.jpg diff --git a/32607-h/images/img245.jpg b/32607-h/images/img245.jpg Binary files differnew file mode 100644 index 0000000..f3b33ec --- /dev/null +++ b/32607-h/images/img245.jpg diff --git a/32607-h/images/img247a.jpg b/32607-h/images/img247a.jpg Binary files differnew file mode 100644 index 0000000..17ce55e --- /dev/null +++ b/32607-h/images/img247a.jpg diff --git a/32607-h/images/img247b.jpg b/32607-h/images/img247b.jpg Binary files differnew file mode 100644 index 0000000..20fa659 --- /dev/null +++ b/32607-h/images/img247b.jpg diff --git a/32607-h/images/img247c.jpg b/32607-h/images/img247c.jpg Binary files differnew file mode 100644 index 0000000..8d43b25 --- /dev/null +++ b/32607-h/images/img247c.jpg diff --git a/32607-h/images/img248a.jpg b/32607-h/images/img248a.jpg Binary files differnew file mode 100644 index 0000000..862da44 --- /dev/null +++ b/32607-h/images/img248a.jpg diff --git a/32607-h/images/img248b.jpg b/32607-h/images/img248b.jpg Binary files differnew file mode 100644 index 0000000..626ddc1 --- /dev/null +++ b/32607-h/images/img248b.jpg diff --git a/32607-h/images/img248c.jpg b/32607-h/images/img248c.jpg Binary files differnew file mode 100644 index 0000000..28e61d7 --- /dev/null +++ b/32607-h/images/img248c.jpg diff --git a/32607-h/images/img251.jpg b/32607-h/images/img251.jpg Binary files differnew file mode 100644 index 0000000..29a1700 --- /dev/null +++ b/32607-h/images/img251.jpg diff --git a/32607-h/images/img252.jpg b/32607-h/images/img252.jpg Binary files differnew file mode 100644 index 0000000..aa4be3a --- /dev/null +++ b/32607-h/images/img252.jpg diff --git a/32607-h/images/img253.jpg b/32607-h/images/img253.jpg Binary files differnew file mode 100644 index 0000000..7c650f7 --- /dev/null +++ b/32607-h/images/img253.jpg diff --git a/32607-h/images/img259.jpg b/32607-h/images/img259.jpg Binary files differnew file mode 100644 index 0000000..a5318d7 --- /dev/null +++ b/32607-h/images/img259.jpg diff --git a/32607-h/images/img272.jpg b/32607-h/images/img272.jpg Binary files differnew file mode 100644 index 0000000..2ff68bf --- /dev/null +++ b/32607-h/images/img272.jpg |
