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-} -i { - font-style: italic; - padding-right: 0.15em; - vertical-align: baseline; -} -.spitc { /*this is a class for sup or sub elements*/ - font-style: italic; -} -.nowrap { - display: inline-block; -} -.xxpn { - font-size: 0.68em; - font-weight: normal; - color: #865; - text-decoration: none; - position: absolute; - right: 0; - line-height: 1.81; /*assumes lh of container is 1.3*/ -} -.hrblk { - margin: 0.7em 50%; - border: thin white solid; -} -.hrblksht { - margin: 0.2em 50%; - border: thin white solid; -} -.dmaths { - page-break-inside: avoid; - page-break-before: auto; -} -.psmprnt2 { - font-size: 0.94em; - line-height: 1.2; - padding: 0.7em 0; -} -.psmprnt3 { - font-size: 0.78em; - line-height: 1.4; - padding: 0.7em 0; -} -.emltr { - font-family: Tahoma, Verdana, Arial, "Lucida Sans Unicode", Impact, sans-serif; - font-style: normal; -} - -/* === handheld === */ -@media screen and (max-width: 600px) { - .xxpn { - position: static; - line-height: inherit; - } - body { - margin-left: 5%; - margin-right: 5%; - } - div, - p { - max-height: none; - } - .dright { - float: right; - } - .dleft, - .pleftfloat { - float: left; - } -} - -</style> -</head> -<body> - -<div style='text-align:center; font-size:1.2em; font-weight:bold'>The Project Gutenberg eBook of On Growth and Form, by D'Arcy Wentworth Thompson</div> -<div style='display:block; margin:1em 0'> -This eBook is for the use of anyone anywhere in the United States and -most other parts of the world 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 <a href="https://www.gutenberg.org">www.gutenberg.org</a>. If you -are not located in the United States, you will have to check the laws of the -country where you are located before using this eBook. -</div> -<div style='display:block; margin-top:1em; margin-bottom:1em; margin-left:2em; text-indent:-2em'>Title: On Growth and Form</div> -<div style='display:block; margin-top:1em; margin-bottom:1em; margin-left:2em; text-indent:-2em'>Author: D'Arcy Wentworth Thompson</div> -<div style='display:block; margin:1em 0'>Release Date: August 4, 2017 [EBook #55264]<br> -[Most recently updated: June 30, 2023]</div> -<div style='display:block; margin:1em 0'>Language: English</div> -<div style='display:block; margin:1em 0'>Character set encoding: UTF-8</div> -<div style='display:block; margin-left:2em; text-indent:-2em'>Produced by: Chris Curnow, RichardW, and the Online -Distributed Proofreading Team at http://www.pgdp.net (This -file was produced from images generously made available -by The Internet Archive)</div> -<div style='margin-top:2em; margin-bottom:4em'>*** START OF THE PROJECT GUTENBERG EBOOK ON GROWTH AND FORM ***</div> - - -<div class="dctr02"> -<img id="coverpage" src="images/cover.jpg" width="600" height="800" alt=""> -</div> - -<div class="dfront"> -<h1 class="h1herein">GROWTH AND FORM</h1></div> - -<div class="dfront"> -<div class="fsz5">CAMBRIDGE UNIVERSITY PRESS</div> - -<div class="fsz5">C. F. CLAY, <span class="smcap">M<b>ANAGER</b></span></div> - -<div class="fsz5">London: FETTER LANE, E.C.</div> - -<div class="fsz5">Edinburgh: 100 PRINCES STREET</div> - -<div id="dcolophon"><img src="images/i003.png" - width="272" height="286" alt=""></div> - -<div class="fsz7">New York: G. P. PUTNAM’S SONS</div> - -<div class="fsz7">Bombay, Calcutta and Madras: MACMILLAN AND Co., <span class="smcap">L<b>TD.</b></span></div> - -<div class="fsz7">Toronto: J. M. DENT AND SONS, <span class="smcap">L<b>TD.</b></span></div> - -<div class="fsz7">Tokyo: THE MARUZEN-KABUSHIKI-KAISHA</div> - -<div class="fsz7 padtopa"><i>All rights reserved</i></div> -</div><!--dfront--> - -<div class="dfront"> -<div class="fsz4">ON</div> -<div class="fsz2">GROWTH AND FORM</div> - -<div class="fsz6 padtopa">BY</div> - -<div class="fsz5">D’ARCY WENTWORTH THOMPSON</div> - -<div class="fsz5 padtopa">Cambridge:</div> -<div class="fsz6">at the University Press</div> -<div class="fsz6">1917</div> -</div><!--dfront--> - -<div class="dfront"> -<p>“The reasonings about the wonderful and intricate operations -of nature are so full of uncertainty, that, as the Wise-man truly -observes, <i>hardly do we guess aright at the things that are upon -earth, and with labour do we find the things that are before us</i>.” -Stephen Hales, <i>Vegetable Staticks</i> (1727), p. 318, 1738.</p></div> - -<div class="chapter"> -<h2 class="h2herein" title="Prefatory Note.">PREFATORY NOTE</h2> - -<p class="pfirst">This -book of mine has little need of preface, for indeed it is -“all preface” from beginning to end. I have written it as -an easy introduction to the study of organic Form, by methods -which are the common-places of physical science, which are by -no means novel in their application to natural history, but which -nevertheless naturalists are little accustomed to employ.</p></div> - -<p>It is not the biologist with an inkling of mathematics, but -the skilled and learned mathematician who must ultimately -deal with such problems as are merely sketched and adumbrated -here. I pretend to no mathematical skill, but I have made what -use I could of what tools I had; I have dealt with simple cases, -and the mathematical methods which I have introduced are of -the easiest and simplest kind. Elementary as they are, my book -has not been written without the help—the indispensable help—of -many friends. Like Mr Pope translating Homer, when I felt -myself deficient I sought assistance! And the experience which -Johnson attributed to Pope has been mine also, that men of -learning did not refuse to help me.</p> - -<p>My debts are many, and I will not try to proclaim them all: -but I beg to record my particular obligations to Professor Claxton -Fidler, Sir George Greenhill, Sir Joseph Larmor, and Professor -A. McKenzie; to a much younger but very helpful friend, -Mr John Marshall, Scholar of Trinity; lastly, and (if I may say -so) most of all, to my colleague Professor William Peddie, whose -advice has made many useful additions to my book and whose -criticism has spared me many a fault and blunder.</p> - -<p>I am under obligations also to the authors and publishers of -many books from which illustrations have been borrowed, and -especially to the following:―</p> - -<p>To the Controller of H.M. Stationery Office, for leave to -reproduce a number of figures, chiefly of Foraminifera and of -Radiolaria, from the Reports of the Challenger Expedition. -<span class="xxpn">{vi}</span></p> - -<p>To the Council of the Royal Society of Edinburgh, and to that -of the Zoological Society of London:—the former for letting me -reprint from their <i>Transactions</i> the greater part of the text and -illustrations of my concluding chapter, the latter for the use of a -number of figures for my chapter on Horns.</p> - -<p>To Professor E. B. Wilson, for his well-known and all but -indispensable figures of the cell (figs. <a href="#fig42" -title="go to Fig. 42">42</a>–<a href="#fig51" title="go to Fig. 51">51</a>, -<a href="#fig53" title="go to Fig. 53">53</a>); to M. A. Prenant, -for other figures (<a href="#fig41" title="go to Fig. 41">41</a>, -<a href="#fig48" title="go to Fig. 48">48</a>) in the same chapter; to Sir Donald -MacAlister and Mr Edwin Arnold for certain figures -(<a href="#fig335" title="go to Fig. 335">335</a>–<a href="#fig337" -title="go to Fig. 337">7</a>), -and to Sir Edward Schäfer and Messrs Longmans for another -(<a href="#fig334" title="go to Fig. 334">334</a>), -illustrating the minute trabecular structure of bone. To Mr -Gerhard Heilmann, of Copenhagen, for his beautiful diagrams -(figs. <a href="#fig388" title="go to Fig. 388">388</a>–<a - href="#fig393" title="go to Fig. 393">93</a>, - <a href="#fig401" title="go to Fig. 401">401</a>, - <a href="#fig402" title="go to Fig. 402">402</a>) included in my last chapter. To Professor -Claxton Fidler and to Messrs Griffin, for letting me use, -with more or less modification or simplification, a number of -illustrations (figs. - <a href="#fig339" title="go to Fig. 339">339</a>–<a href="#fig346" title="go to Fig. 346">346</a>) -from Professor Fidler’s <i>Textbook of -Bridge Construction</i>. To Messrs Blackwood and Sons, for several -cuts (figs. - <a href="#fig127" title="go to Fig. 127">127</a>–<a href="#fig9" title="go to Fig. 9">9</a>, - <a href="#fig131" title="go to Fig. 131">131</a>, - <a href="#fig173" title="go to Fig. 173">173</a>) from Professor Alleyne Nicholson’s -<i>Palaeontology</i>; to Mr Heinemann, for certain figures -(<a href="#fig57" title="go to Fig. 57">57</a>, - <a href="#fig122" title="go to Fig. 122">122</a>, - <a href="#fig123" title="go to Fig. 123">123</a>, -<a href="#fig205" title="go to Fig. 205">205</a>) from Dr Stéphane Leduc’s <i>Mechanism of Life</i>; to Mr A. M. -Worthington and to Messrs Longmans, for figures - (<a href="#fig71" title="go to Fig. 71">71</a>, - <a href="#fig75" title="go to Fig. 75">75</a>) from -<i>A Study of Splashes</i>, and to Mr C. R. Darling and to Messrs E. -and S. Spon for those (fig. - <a href="#fig85" title="go to Fig. 85">85</a>) from Mr Darling’s <i>Liquid Drops -and Globules</i>. To Messrs Macmillan and Co. for two figures -(<a href="#fig304" title="go to Fig. 304">304</a>, - <a href="#fig305" title="go to Fig. 305">305</a>) from Zittel’s <i>Palaeontology</i>, to the Oxford University -Press for a diagram (fig. - <a href="#fig28" title="go to Fig. 28">28</a>) from Mr J. W. Jenkinson’s <i>Experimental -Embryology</i>; and to the Cambridge University Press for -a number of figures from Professor Henry Woods’s <i>Invertebrate -Palaeontology</i>, for one (fig. - <a href="#fig210" title="go to Fig. 210">210</a>) from Dr Willey’s <i>Zoological Results</i>, -and for another (fig. - <a href="#fig321" title="go to Fig. 321">321</a>) from “Thomson and Tait.”</p> - -<div class="dkeeptgth"> -<p>Many more, and by much the greater part of my diagrams, -I owe to the untiring help of Dr Doris L. Mackinnon, D.Sc., and -of Miss Helen Ogilvie, M.A., B.Sc., of this College.</p> - -<p class="psignature">D’ARCY WENTWORTH THOMPSON.</p> - -<p class="fsz7 padtopb"><span class="smcap">U<b>NIVERSITY</b></span> -<span class="smcap">C<b>OLLEGE,</b></span> -<span class="smcap">D<b>UNDEE.</b></span></p> - -<p class="fsz7"><i>December, 1916.</i></p></div> - -<div class="chapter"> - -<h2 class="h2herein" title="Contents.">CONTENTS</h2> -<table> -<tr class="trkeeptgth"> - <th class="fsz8">CHAP.</th> - <th class="fsz8"></th> - <th class="fsz8">PAGE</th></tr> -<tr class="trkeeptgth"> - <td class="tdright">I.</td> - <td class="tdleft"><span class="smcap">I<b>NTRODUCTORY</b></span></td> - <td class="tdright"><a class="aplain" href="#p001" title="go to pg. 1">1</a></td></tr> -<tr class="trkeeptgth"> - <td class="tdright">II.</td> - <td class="tdleft"><span class="smcap">O<b>N</b></span> <span class="smcap">M<b>AGNITUDE</b></span></td> - <td class="tdright"><a class="aplain" href="#p016" title="go to pg. 16">16</a></td></tr> -<tr class="trkeeptgth"> - <td class="tdright">III.</td> - <td class="tdleft"><span class="smcap">T<b>HE</b></span> <span class="smcap">R<b>ATE</b></span> <span class="smmaj">OF</span> <span class="smcap">G<b>ROWTH</b></span></td> - <td class="tdright"><a class="aplain" href="#p050" - title="go to pg. 50">50</a></td></tr> -<tr class="trkeeptgth"> - <td class="tdright">IV.</td> - <td class="tdleft"><span class="smcap">O<b>N</b></span> <span class="smmaj">THE</span> <span class="smcap">I<b>NTERNAL</b></span> <span class="smcap">F<b>ORM</b></span> <span class="smmaj">AND</span> <span class="smcap">S<b>TRUCTURE</b></span> <span class="smmaj">OF</span> <span class="smmaj">THE</span> <span class="smcap">C<b>ELL</b></span></td> - <td class="tdright"><a class="aplain" href="#p156" - title="go to pg. 156">156</a></td></tr> -<tr class="trkeeptgth"> - <td class="tdright">V.</td> - <td class="tdleft"><span class="smcap">T<b>HE</b></span> <span class="smcap">F<b>ORMS</b></span> <span class="smmaj">OF</span> <span class="smcap">C<b>ELLS</b></span></td> - <td class="tdright"><a class="aplain" href="#p201" - title="go to pg. 201">201</a></td></tr> -<tr class="trkeeptgth"> - <td class="tdright">VI.</td> - <td class="tdleft">A <span class="smcap">N<b>OTE</b></span> <span class="smmaj">ON</span> <span class="smcap">A<b>DSORPTION</b></span></td> - <td class="tdright"><a class="aplain" href="#p277" - title="go to pg. 277">277</a></td></tr> -<tr class="trkeeptgth"> - <td class="tdright">VII.</td> - <td class="tdleft"><span class="smcap">T<b>HE</b></span> <span class="smcap">F<b>ORMS</b></span> <span class="smmaj">OF</span> <span class="smcap">T<b>ISSUES,</b></span> <span class="smmaj">OR</span> <span class="smcap">C<b>ELL-AGGREGATES</b></span></td> - <td class="tdright"><a class="aplain" href="#p293" - title="go to pg. 293">293</a></td></tr> -<tr class="trkeeptgth"> - <td class="tdright">VIII.</td> - <td class="tdleft"><span class="smcap">T<b>HE</b></span> <span class="smmaj">SAME</span> (<i>continued</i>)</td> - <td class="tdright"><a class="aplain" href="#p346" - title="go to pg. 346">346</a></td></tr> -<tr class="trkeeptgth"> - <td class="tdright">IX.</td> - <td class="tdleft"><span class="smcap">O<b>N</b></span> - <span class="smcap">C<b>ONCRETIONS,</b></span> - <span class="smcap">S<b>PICULES,</b></span> - <span class="smmaj">AND</span> - <span class="smcap">S<b>PICULAR</b></span> - <span class="smcap">S<b>KELETONS</b></span></td> - <td class="tdright"><a class="aplain" href="#p411" - title="go to pg. 411">411</a></td></tr> -<tr class="trkeeptgth"> - <td class="tdright">X.</td> - <td class="tdleft">A <span class="smcap">P<b>ARENTHETIC</b></span> <span class="smcap">N<b>OTE</b></span> <span class="smmaj">ON</span> <span class="smcap">G<b>EODETICS</b></span></td> - <td class="tdright"><a class="aplain" href="#p488" - title="go to pg. 488">488</a></td></tr> -<tr class="trkeeptgth"> - <td class="tdright">XI.</td> - <td class="tdleft"><span class="smcap">T<b>HE</b></span> <span class="smcap">L<b>OGARITHMIC</b></span> <span class="smcap">S<b>PIRAL</b></span></td> - <td class="tdright"><a class="aplain" href="#p493" - title="go to pg. 493">493</a></td></tr> -<tr class="trkeeptgth"> - <td class="tdright">XII.</td> - <td class="tdleft"><span class="smcap">T<b>HE</b></span> <span class="smcap">S<b>PIRAL</b></span> <span class="smcap">S<b>HELLS</b></span> <span class="smmaj">OF</span> <span class="smmaj">THE</span> <span class="smcap">F<b>ORAMINIFERA</b></span></td> - <td class="tdright"><a class="aplain" href="#p587" - title="go to pg. 587">587</a></td></tr> -<tr class="trkeeptgth"> - <td class="tdright">XIII.</td> - <td class="tdleft"><span class="smcap">T<b>HE</b></span> <span class="smcap">S<b>HAPES</b></span> <span class="smmaj">OF</span> <span class="smcap">H<b>ORNS,</b></span> <span class="smmaj">AND</span> <span class="smmaj">OF</span> <span class="smcap">T<b>EETH</b></span> <span class="smmaj">OR</span> <span class="smcap">T<b>USKS:</b></span> <span class="smmaj">WITH</span> <span class="smmaj">A</span> <span class="smcap">N<b>OTE</b></span> <span class="smmaj">ON</span> <span class="smcap">T<b>ORSION</b></span></td> - <td class="tdright"><a class="aplain" href="#p612" - title="go to pg. 612">612</a></td></tr> -<tr class="trkeeptgth"> - <td class="tdright">XIV.</td> - <td class="tdleft"><span class="smcap">O<b>N</b></span> <span class="smcap">L<b>EAF-ARRANGEMENT,</b></span> <span class="smmaj">OR</span> <span class="smcap">P<b>HYLLOTAXIS</b></span></td> - <td class="tdright"><a class="aplain" href="#p635" - title="go to pg. 635">635</a></td></tr> -<tr class="trkeeptgth"> - <td class="tdright">XV.</td> - <td class="tdleft"><span class="smcap">O<b>N</b></span> <span class="smmaj">THE</span> <span class="smcap">S<b>HAPES</b></span> <span class="smmaj">OF</span> <span class="smcap">E<b>GGS,</b></span> <span class="smmaj">AND</span> <span class="smmaj">OF</span> <span class="smmaj">CERTAIN</span> <span class="smmaj">OTHER</span> <span class="smcap">H<b>OLLOW</b></span> <span class="smcap">S<b>TRUCTURES</b></span></td> - <td class="tdright"><a class="aplain" href="#p652" - title="go to pg. 652">652</a></td></tr> -<tr class="trkeeptgth"> - <td class="tdright">XVI.</td> - <td class="tdleft"><span class="smcap">O<b>N</b></span> <span class="smcap">F<b>ORM</b></span> <span class="smmaj">AND</span> <span class="smcap">M<b>ECHANICAL</b></span> <span class="smcap">E<b>FFICIENCY</b></span></td> - <td class="tdright"><a class="aplain" href="#p670" - title="go to pg. 670">670</a></td></tr> -<tr class="trkeeptgth"> - <td class="tdright">XVII.</td> - <td class="tdleft"><span class="smcap">O<b>N</b></span> <span class="smmaj">THE</span> <span class="smcap">T<b>HEORY</b></span> <span class="smmaj">OF</span> <span class="smcap">T<b>RANSFORMATIONS,</b></span> <span class="smmaj">OR</span> <span class="smmaj">THE</span> <span class="smcap">C<b>OMPARISON</b></span> <span class="smmaj">OF</span> <span class="smcap">R<b>ELATED</b></span> <span class="smcap">F<b>ORMS</b></span></td> - <td class="tdright"><a class="aplain" href="#p719" - title="go to pg. 719">719</a></td></tr> -<tr class="trkeeptgth"> - <td class="tdright"></td> - <td class="tdleft"><span class="smcap">E<b>PILOGUE</b></span></td> - <td class="tdright"><a class="aplain" href="#p778" - title="go to pg. 778">778</a></td></tr> -<tr class="trkeeptgth"> - <td class="tdright"></td> - <td class="tdleft"><span class="smcap">I<b>NDEX</b></span></td> - <td class="tdright"><a class="aplain" href="#p780" - title="go to pg. 780">780</a></td></tr> -</table></div><!--chapter--> - -<h2 class="h2herein" title="List of Illustrations."> - LIST OF ILLUSTRATIONS</h2> -<div class="chapter"> -<table class="fsz6"> -<tr class="trkeeptgth"> - <th class="fsz8">Fig.</th> - <th class="fsz8"></th> - <th class="fsz8">Page</th></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7">  <a class="aplain" href="#fig1" - title="go to Fig. 1">1</a>.</td> - <td class="tdlefthng">Nerve-cells, from larger and smaller animals (Minot, - after Irving Hardesty)</td> - <td class="tdrightloi fsz8">37</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7">  <a class="aplain" href="#fig2" - title="go to Fig. 2">2</a>.</td> - <td class="tdlefthng">Relative magnitudes of some minute organisms (Zsigmondy)</td> - <td class="tdrightloi fsz8">39</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7">  <a class="aplain" href="#fig3" - title="go to Fig. 3">3</a>.</td> - <td class="tdlefthng">Curves of growth in man (Quetelet and Bowditch)</td> - <td class="tdrightloi fsz8">61</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7">  <a class="aplain" href="#fig4" - title="go to Fig. 4">4</a>, 5.</td> - <td class="tdlefthng">Mean annual increments of stature and weight in man (<i>do.</i>)</td> - <td class="tdrightloi fsz8">66, 69</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7">  <a class="aplain" href="#fig6" - title="go to Fig. 6">6</a>.</td> - <td class="tdlefthng">The ratio, throughout life, of female weight to male (<i>do.</i>)</td> - <td class="tdrightloi fsz8">71</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7">  <a class="aplain" href="#fig7" - title="go to Fig. 7">7</a>–9.</td> - <td class="tdlefthng">Curves of growth of child, before and after birth (His and Rüssow)</td> - <td class="tdrightloi fsz8">74–6</td></tr> -<tr> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig10" - title="go to Fig. 10">10</a>.</td> - <td class="tdlefthng">Curve of growth of bamboo (Ostwald, after Kraus)</td> - <td class="tdrightloi fsz8">77</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig11" - title="go to Fig. 11">11</a>.</td> - <td class="tdlefthng">Coefficients of variability in human stature (Boas and Wissler)</td> - <td class="tdrightloi fsz8">80</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig12" - title="go to Fig. 12">12</a>.</td> - <td class="tdlefthng">Growth in weight of mouse (Wolfgang Ostwald)</td> - <td class="tdrightloi fsz8">83</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig13" - title="go to Fig. 13">13</a>.</td> - <td class="tdlefthng"><i>Do.</i> of silkworm (Luciani and Lo Monaco)</td> - <td class="tdrightloi fsz8">84</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig14" - title="go to Fig. 14">14</a>.</td> - <td class="tdlefthng"><i>Do.</i> of tadpole (Ostwald, after Schaper)</td> - <td class="tdrightloi fsz8">85</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig15" - title="go to Fig. 15">15</a>.</td> - <td class="tdlefthng">Larval eels, or <i>Leptocephali</i>, and young elver (Joh. Schmidt)</td> - <td class="tdrightloi fsz8">86</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig16" - title="go to Fig. 16">16</a>.</td> - <td class="tdlefthng">Growth in length of <i>Spirogyra</i> (Hofmeister)</td> - <td class="tdrightloi fsz8">87</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig17" - title="go to Fig. 17">17</a>.</td> - <td class="tdlefthng">Pulsations of growth in <i>Crocus</i> (Bose)</td> - <td class="tdrightloi fsz8">88</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig18" - title="go to Fig. 18">18</a>.</td> - <td class="tdlefthng">Relative growth of brain, heart and body of man (Quetelet)</td> - <td class="tdrightloi fsz8">90</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig19" - title="go to Fig. 19">19</a>.</td> - <td class="tdlefthng">Ratio of stature to span of arms (<i>do.</i>)</td> - <td class="tdrightloi fsz8">94</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig20" - title="go to Fig. 20">20</a>.</td> - <td class="tdlefthng">Rates of growth near the tip of a bean-root (Sachs)</td> - <td class="tdrightloi fsz8">96</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig21" - title="go to Fig. 21">21</a>, 22.</td> - <td class="tdlefthng">The weight-length ratio of the plaice, and its annual periodic changes</td> - <td class="tdrightloi fsz8">99, 100</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig23" - title="go to Fig. 23">23</a>.</td> - <td class="tdlefthng">Variability of tail-forceps in earwigs (Bateson)</td> - <td class="tdrightloi fsz8">104</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig24" - title="go to Fig. 24">24</a>.</td> - <td class="tdlefthng">Variability of body-length in plaice</td> - <td class="tdrightloi fsz8">105</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig25" - title="go to Fig. 25">25</a>.</td> - <td class="tdlefthng">Rate of growth in plants in relation to temperature (Sachs)</td> - <td class="tdrightloi fsz8">109</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig26" - title="go to Fig. 26">26</a>.</td> - <td class="tdlefthng"><i>Do.</i> in maize, observed (Köppen), and calculated curves</td> - <td class="tdrightloi fsz8">112</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig27" - title="go to Fig. 27">27</a>.</td> - <td class="tdlefthng"><i>Do.</i> in roots of peas (Miss I. Leitch)</td> - <td class="tdrightloi fsz8">113</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig28" - title="go to Fig. 28">28</a>, 29.</td> - <td class="tdlefthng">Rate of growth of frog in relation to temperature (Jenkinson, after O. Hertwig), and calculated curves of <i>do.</i></td> - <td class="tdrightloi fsz8">115, 6</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig30" - title="go to Fig. 30">30</a>.</td> - <td class="tdlefthng">Seasonal fluctuation of rate of growth in man (Daffner)</td> - <td class="tdrightloi fsz8">119</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig31" - title="go to Fig. 31">31</a>.</td> - <td class="tdlefthng"><i>Do.</i> in the rate of growth of trees (C. E. Hall)</td> - <td class="tdrightloi fsz8">120</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig32" - title="go to Fig. 32">32</a>.</td> - <td class="tdlefthng">Long-period fluctuation in the rate of growth of Arizona trees (A. E. Douglass)</td> - <td class="tdrightloi fsz8">122</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig33" - title="go to Fig. 33">33</a>, 34.</td> - <td class="tdlefthng">The varying form of brine-shrimps (<i>Artemia</i>), in relation to salinity (Abonyi)</td> - <td class="tdrightloi fsz8">128, 9</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig35" - title="go to Fig. 35">35</a>–39.</td> - <td class="tdlefthng">Curves of regenerative growth in tadpoles’ tails (M. L. Durbin)</td> - <td class="tdrightloi fsz8">140–145</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig40" - title="go to Fig. 40">40</a>.</td> - <td class="tdlefthng">Relation between amount of tail removed, amount restored, and time required for restoration (M. M. Ellis)</td> - <td class="tdrightloi fsz8">148</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig41" - title="go to Fig. 41">41</a>.</td> - <td class="tdlefthng">Caryokinesis in trout’s egg (Prenant, after Prof. P. Bouin)</td> - <td class="tdrightloi fsz8">169</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig42" - title="go to Fig. 42">42</a>–51.</td> - <td class="tdlefthng">Diagrams of mitotic cell-division (Prof. E. B. Wilson)</td> - <td class="tdrightloi fsz8">171–5</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig52" - title="go to Fig. 52">52</a>.</td> - <td class="tdlefthng">Chromosomes in course of splitting and separation (Hatschek and Flemming)</td> - <td class="tdrightloi fsz8">180</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig53" - title="go to Fig. 53">53</a>.</td> - <td class="tdlefthng">Annular chromosomes of mole-cricket (Wilson, after vom Rath)</td> - <td class="tdrightloi fsz8">181</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig54" - title="go to Fig. 54">54</a>–56.</td> - <td class="tdlefthng">Diagrams illustrating a hypothetic field of force in caryokinesis (Prof. W. Peddie)</td> - <td class="tdrightloi fsz8">182–4</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig57" - title="go to Fig. 57">57</a>.</td> - <td class="tdlefthng">An artificial figure of caryokinesis (Leduc)</td> - <td class="tdrightloi fsz8">186</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig58" - title="go to Fig. 58">58</a>.</td> - <td class="tdlefthng">A segmented egg of <i>Cerebratulus</i> (Prenant, after Coe)</td> - <td class="tdrightloi fsz8">189</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig59" - title="go to Fig. 59">59</a>.</td> - <td class="tdlefthng">Diagram of a field of force with two like poles</td> - <td class="tdrightloi fsz8">189</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig60" - title="go to Fig. 60">60</a>.</td> - <td class="tdlefthng">A budding yeast-cell</td> - <td class="tdrightloi fsz8">213</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig61" - title="go to Fig. 61">61</a>.</td> - <td class="tdlefthng">The roulettes of the conic sections</td> - <td class="tdrightloi fsz8">218</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig62" - title="go to Fig. 62">62</a>.</td> - <td class="tdlefthng">Mode of development of an unduloid from a cylindrical tube</td> - <td class="tdrightloi fsz8">220</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig63" - title="go to Fig. 63">63</a>–65.</td> - <td class="tdlefthng">Cylindrical, unduloid, nodoid and catenoid oil-globules (Plateau)</td> - <td class="tdrightloi fsz8">222, 3</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig66" - title="go to Fig. 66">66</a>.</td> - <td class="tdlefthng">Diagram of the nodoid, or elastic curve</td> - <td class="tdrightloi fsz8">224</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig67" - title="go to Fig. 67">67</a>.</td> - <td class="tdlefthng">Diagram of a cylinder capped by the corresponding portion of a sphere</td> - <td class="tdrightloi fsz8">226</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig68" - title="go to Fig. 68">68</a>.</td> - <td class="tdlefthng">A liquid cylinder breaking up into spheres</td> - <td class="tdrightloi fsz8">227</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig69" - title="go to Fig. 69">69</a>.</td> - <td class="tdlefthng">The same phenomenon in a protoplasmic cell of <i>Trianea</i></td> - <td class="tdrightloi fsz8">234</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig70" - title="go to Fig. 70">70</a>.</td> - <td class="tdlefthng">Some phases of a splash (A. M. Worthington)</td> - <td class="tdrightloi fsz8">235</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig71" - title="go to Fig. 71">71</a>.</td> - <td class="tdlefthng">A breaking wave (<i>do.</i>)</td> - <td class="tdrightloi fsz8">236</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig72" - title="go to Fig. 72">72</a>.</td> - <td class="tdlefthng">The calycles of some campanularian zoophytes</td> - <td class="tdrightloi fsz8">237</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig73" - title="go to Fig. 73">73</a>.</td> - <td class="tdlefthng">A flagellate monad, <i>Distigma proteus</i> (Saville Kent)</td> - <td class="tdrightloi fsz8">246</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig74" - title="go to Fig. 74">74</a>.</td> - <td class="tdlefthng"><i>Noctiluca miliaris</i>, diagrammatic</td> - <td class="tdrightloi fsz8">246</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig75" - title="go to Fig. 75">75</a>.</td> - <td class="tdlefthng">Various species of <i>Vorticella</i> (Saville Kent and others)</td> - <td class="tdrightloi fsz8">247</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig76" - title="go to Fig. 76">76</a>.</td> - <td class="tdlefthng">Various species of <i>Salpingoeca</i> (<i>do.</i>)</td> - <td class="tdrightloi fsz8">248</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig77" - title="go to Fig. 77">77</a>.</td> - <td class="tdlefthng">Species of <i>Tintinnus</i>, <i>Dinobryon</i> and <i>Codonella</i> (<i>do.</i>)</td> - <td class="tdrightloi fsz8">248</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig78" - title="go to Fig. 78">78</a>.</td> - <td class="tdlefthng">The tube or cup of <i>Vaginicola</i></td> - <td class="tdrightloi fsz8">248</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig79" - title="go to Fig. 79">79</a>.</td> - <td class="tdlefthng">The same of <i>Folliculina</i></td> - <td class="tdrightloi fsz8">249</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig80" - title="go to Fig. 80">80</a>.</td> - <td class="tdlefthng"><i>Trachelophyllum</i> (Wreszniowski)</td> - <td class="tdrightloi fsz8">249</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig81" - title="go to Fig. 81">81</a>.</td> - <td class="tdlefthng"><i>Trichodina pediculus</i></td> - <td class="tdrightloi fsz8">252</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig82" - title="go to Fig. 82">82</a>.</td> - <td class="tdlefthng"><i>Dinenymplia gracilis</i> (Leidy)</td> - <td class="tdrightloi fsz8">253</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig83" - title="go to Fig. 83">83</a>.</td> - <td class="tdlefthng">A “collar-cell” of <i>Codosiga</i></td> - <td class="tdrightloi fsz8">254</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig84" - title="go to Fig. 84">84</a>.</td> - <td class="tdlefthng">Various species of <i>Lagena</i> (Brady)</td> - <td class="tdrightloi fsz8">256</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig85" - title="go to Fig. 85">85</a>.</td> - <td class="tdlefthng">Hanging drops, to illustrate the unduloid form (C. R. Darling)</td> - <td class="tdrightloi fsz8">257</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig86" - title="go to Fig. 86">86</a>.</td> - <td class="tdlefthng">Diagram of a fluted cylinder</td> - <td class="tdrightloi fsz8">260</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig87" - title="go to Fig. 87">87</a>.</td> - <td class="tdlefthng"><i>Nodosaria scalaris</i> (Brady)</td> - <td class="tdrightloi fsz8">262</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig88" - title="go to Fig. 88">88</a>.</td> - <td class="tdlefthng">Fluted and pleated gonangia of certain Campanularians (Allman)</td> - <td class="tdrightloi fsz8">262</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig89" - title="go to Fig. 89">89</a>.</td> - <td class="tdlefthng">Various species of <i>Nodosaria</i>, <i>Sagrina</i> and <i>Rheophax</i> (Brady)</td> - <td class="tdrightloi fsz8">263</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig90" - title="go to Fig. 90">90</a>.</td> - <td class="tdlefthng"><i>Trypanosoma tineae</i> and <i>Spirochaeta anodontae</i>, to shew undulating membranes (Minchin and Fantham)</td> - <td class="tdrightloi fsz8">266</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig91" - title="go to Fig. 91">91</a>.</td> - <td class="tdlefthng">Some species of <i>Trichomastix</i> and <i>Trichomonas</i> (Kofoid)</td> - <td class="tdrightloi fsz8">267</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig92" - title="go to Fig. 92">92</a>.</td> - <td class="tdlefthng"><i>Herpetomonas</i> assuming the undulatory membrane of a Trypanosome (D. L. Mackinnon)</td> - <td class="tdrightloi fsz8">268</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig93" - title="go to Fig. 93">93</a>.</td> - <td class="tdlefthng">Diagram of a human blood-corpuscle</td> - <td class="tdrightloi fsz8">271</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig94" - title="go to Fig. 94">94</a>.</td> - <td class="tdlefthng">Sperm-cells of decapod crustacea, <i>Inachus</i> and <i>Galathea</i> (Koltzoff)</td> - <td class="tdrightloi fsz8">273</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig95" - title="go to Fig. 95">95</a>.</td> - <td class="tdlefthng">The same, in saline solutions of varying density (<i>do.</i>)</td> - <td class="tdrightloi fsz8">274</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig96" - title="go to Fig. 96">96</a>.</td> - <td class="tdlefthng">A sperm-cell of <i>Dromia</i> (<i>do.</i>)</td> - <td class="tdrightloi fsz8">275</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig97" - title="go to Fig. 97">97</a>.</td> - <td class="tdlefthng">Chondriosomes in cells of kidney and pancreas (Barratt and Mathews)</td> - <td class="tdrightloi fsz8">285</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig98" - title="go to Fig. 98">98</a>.</td> - <td class="tdlefthng">Adsorptive concentration of potassium salts in various plant-cells (Macallum)</td> - <td class="tdrightloi fsz8">290</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"> <a class="aplain" href="#fig99" - title="go to Fig. 99">99</a>–101.</td> - <td class="tdlefthng">Equilibrium of surface-tension in a floating drop</td> - <td class="tdrightloi fsz8">294, 5</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig102" - title="go to Fig. 102">102</a>.</td> - <td class="tdlefthng">Plateau’s “bourrelet” in plant-cells; diagrammatic (Berthold)</td> - <td class="tdrightloi fsz8">298</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig103" - title="go to Fig. 103">103</a>.</td> - <td class="tdlefthng">Parenchyma of maize, shewing the same phenomenon</td> - <td class="tdrightloi fsz8">298</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig104" - title="go to Fig. 104">104</a>, 5.</td> - <td class="tdlefthng">Diagrams of the partition-wall between two soap-bubbles</td> - <td class="tdrightloi fsz8">299, 300</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig106" - title="go to Fig. 106">106</a>.</td> - <td class="tdlefthng">Diagram of a partition in a conical cell</td> - <td class="tdrightloi fsz8">300</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig107" - title="go to Fig. 107">107</a>.</td> - <td class="tdlefthng">Chains of cells in <i>Nostoc</i>, <i>Anabaena</i> and other low algae</td> - <td class="tdrightloi fsz8">300</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig108" - title="go to Fig. 108">108</a>.</td> - <td class="tdlefthng">Diagram of a symmetrically divided soap-bubble</td> - <td class="tdrightloi fsz8">301</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig109" - title="go to Fig. 109">109</a>.</td> - <td class="tdlefthng">Arrangement of partitions in dividing spores of <i>Pellia</i> (Campbell)</td> - <td class="tdrightloi fsz8">302</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig110" - title="go to Fig. 110">110</a>.</td> - <td class="tdlefthng">Cells of <i>Dictyota</i> (Reinke)</td> - <td class="tdrightloi fsz8">303</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig111" - title="go to Fig. 111">111</a>, 2.</td> - <td class="tdlefthng">Terminal and other cells of <i>Chara</i>, and young antheridium of <i>do.</i></td> - <td class="tdrightloi fsz8">303</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig113" - title="go to Fig. 113">113</a>.</td> - <td class="tdlefthng">Diagram of cell-walls and partitions under various conditions of tension</td> - <td class="tdrightloi fsz8">304</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig114" - title="go to Fig. 114">114</a>, 5.</td> - <td class="tdlefthng">The partition-surfaces of three interconnected bubbles</td> - <td class="tdrightloi fsz8">307, 8</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig116" - title="go to Fig. 116">116</a>.</td> - <td class="tdlefthng">Diagram of four interconnected cells or bubbles</td> - <td class="tdrightloi fsz8">309</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig117" - title="go to Fig. 117">117</a>.</td> - <td class="tdlefthng">Various configurations of four cells in a frog’s egg (Rauber)</td> - <td class="tdrightloi fsz8">311</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig118" - title="go to Fig. 118">118</a>.</td> - <td class="tdlefthng">Another diagram of two conjoined soap-bubbles</td> - <td class="tdrightloi fsz8">313</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig119" - title="go to Fig. 119">119</a>.</td> - <td class="tdlefthng">A froth of bubbles, shewing its outer or “epidermal” layer</td> - <td class="tdrightloi fsz8">314</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig120" - title="go to Fig. 120">120</a>.</td> - <td class="tdlefthng">A tetrahedron, or tetrahedral system, shewing its centre of symmetry</td> - <td class="tdrightloi fsz8">317</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig121" - title="go to Fig. 121">121</a>.</td> - <td class="tdlefthng">A group of hexagonal cells (Bonanni)</td> - <td class="tdrightloi fsz8">319</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig122" - title="go to Fig. 122">122</a>, 3.</td> - <td class="tdlefthng">Artificial cellular tissues (Leduc)</td> - <td class="tdrightloi fsz8">320</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig124" - title="go to Fig. 124">124</a>.</td> - <td class="tdlefthng">Epidermis of <i>Girardia</i> (Goebel)</td> - <td class="tdrightloi fsz8">321</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig125" - title="go to Fig. 125">125</a>.</td> - <td class="tdlefthng">Soap-froth, and the same under compression (Rhumbler)</td> - <td class="tdrightloi fsz8">322</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig126" - title="go to Fig. 126">126</a>.</td> - <td class="tdlefthng">Epidermal cells of <i>Elodea canadensis</i> (Berthold)</td> - <td class="tdrightloi fsz8">322</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig127" - title="go to Fig. 127">127</a>.</td> - <td class="tdlefthng"><i>Lithostrotion Martini</i> (Nicholson)</td> - <td class="tdrightloi fsz8">325</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig128" - title="go to Fig. 128">128</a>.</td> - <td class="tdlefthng"><i>Cyathophyllum hexagonum</i> (Nicholson, after Zittel)</td> - <td class="tdrightloi fsz8">325</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig129" - title="go to Fig. 129">129</a>.</td> - <td class="tdlefthng"><i>Arachnophyllum pentagonum</i> (Nicholson)</td> - <td class="tdrightloi fsz8">326</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig130" - title="go to Fig. 130">130</a>.</td> - <td class="tdlefthng"><i>Heliolites</i> (Woods)</td> - <td class="tdrightloi fsz8">326</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig131" - title="go to Fig. 131">131</a>.</td> - <td class="tdlefthng">Confluent septa in <i>Thamnastraea</i> and <i>Comoseris</i> (Nicholson, after Zittel)</td> - <td class="tdrightloi fsz8">327</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig132" - title="go to Fig. 132">132</a>.</td> - <td class="tdlefthng">Geometrical construction of a bee’s cell</td> - <td class="tdrightloi fsz8">330</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig133" - title="go to Fig. 133">133</a>.</td> - <td class="tdlefthng">Stellate cells in the pith of a rush; diagrammatic</td> - <td class="tdrightloi fsz8">335</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig134" - title="go to Fig. 134">134</a>.</td> - <td class="tdlefthng">Diagram of soap-films formed in a cubical wire skeleton (Plateau)</td> - <td class="tdrightloi fsz8">337</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig135" - title="go to Fig. 135">135</a>.</td> - <td class="tdlefthng">Polar furrows in systems of four soap-bubbles (Robert)</td> - <td class="tdrightloi fsz8">341</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig136" - title="go to Fig. 136">136</a>–8.</td> - <td class="tdlefthng">Diagrams illustrating the division of a cube by partitions of minimal area</td> - <td class="tdrightloi fsz8">347–50</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig139" - title="go to Fig. 139">139</a>.</td> - <td class="tdlefthng">Cells from hairs of <i>Sphacelaria</i> (Berthold)</td> - <td class="tdrightloi fsz8">351</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig140" - title="go to Fig. 140">140</a>.</td> - <td class="tdlefthng">The bisection of an isosceles triangle by minimal partitions</td> - <td class="tdrightloi fsz8">353</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig141" - title="go to Fig. 141">141</a>.</td> - <td class="tdlefthng">The similar partitioning of spheroidal and conical cells</td> - <td class="tdrightloi fsz8">353</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig142" - title="go to Fig. 142">142</a>.</td> - <td class="tdlefthng">S-shaped partitions from cells of algae and mosses (Reinke and others)</td> - <td class="tdrightloi fsz8">355</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig143" - title="go to Fig. 143">143</a>.</td> - <td class="tdlefthng">Diagrammatic explanation of the S-shaped partitions</td> - <td class="tdrightloi fsz8">356</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig144" - title="go to Fig. 144">144</a>.</td> - <td class="tdlefthng">Development of <i>Erythrotrichia</i> (Berthold)</td> - <td class="tdrightloi fsz8">359</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig145" - title="go to Fig. 145">145</a>.</td> - <td class="tdlefthng">Periclinal, anticlinal and radial partitioning of a quadrant</td> - <td class="tdrightloi fsz8">359</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig146" - title="go to Fig. 146">146</a>.</td> - <td class="tdlefthng">Construction for the minimal partitioning of a quadrant</td> - <td class="tdrightloi fsz8">361</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig147" - title="go to Fig. 147">147</a>.</td> - <td class="tdlefthng">Another diagram of anticlinal and periclinal partitions</td> - <td class="tdrightloi fsz8">362</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig148" - title="go to Fig. 148">148</a>.</td> - <td class="tdlefthng">Mode of segmentation of an artificially flattened frog’s egg (Roux)</td> - <td class="tdrightloi fsz8">363</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig149" - title="go to Fig. 149">149</a>.</td> - <td class="tdlefthng">The bisection, by minimal partitions, of a prism of small angle</td> - <td class="tdrightloi fsz8">364</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig150" - title="go to Fig. 150">150</a>.</td> - <td class="tdlefthng">Comparative diagram of the various modes of bisection of a prismatic sector</td> - <td class="tdrightloi fsz8">365</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig151" - title="go to Fig. 151">151</a>.</td> - <td class="tdlefthng">Diagram of the further growth of the two halves of a quadrantal cell</td> - <td class="tdrightloi fsz8">367</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig152" - title="go to Fig. 152">152</a>.</td> - <td class="tdlefthng">Diagram of the origin of an epidermic layer of cells</td> - <td class="tdrightloi fsz8">370</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig153" - title="go to Fig. 153">153</a>.</td> - <td class="tdlefthng">A discoidal cell dividing into octants</td> - <td class="tdrightloi fsz8">371</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig154" - title="go to Fig. 154">154</a>.</td> - <td class="tdlefthng">A germinating spore of <i>Riccia</i> (after Campbell), to shew the manner of space-partitioning in the cellular tissue</td> - <td class="tdrightloi fsz8">372</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig155" - title="go to Fig. 155">155</a>, 6.</td> - <td class="tdlefthng">Theoretical arrangement of successive partitions in a discoidal cell</td> - <td class="tdrightloi fsz8">373</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig157" - title="go to Fig. 157">157</a>.</td> - <td class="tdlefthng">Sections of a moss-embryo (Kienitz-Gerloff)</td> - <td class="tdrightloi fsz8">374</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig158" - title="go to Fig. 158">158</a>.</td> - <td class="tdlefthng">Various possible arrangements of partitions in groups of four to eight cells</td> - <td class="tdrightloi fsz8">375</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig159" - title="go to Fig. 159">159</a>.</td> - <td class="tdlefthng">Three modes of partitioning in a system of six cells</td> - <td class="tdrightloi fsz8">376</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig160" - title="go to Fig. 160">160</a>, 1.</td> - <td class="tdlefthng">Segmenting eggs of <i>Trochus</i> (Robert), and of <i>Cynthia</i> (Conklin)</td> - <td class="tdrightloi fsz8">377</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig162" - title="go to Fig. 162">162</a>.</td> - <td class="tdlefthng">Section of the apical cone of <i>Salvinia</i> (Pringsheim)</td> - <td class="tdrightloi fsz8">377</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig163" - title="go to Fig. 163">163</a>, 4.</td> - <td class="tdlefthng">Segmenting eggs of <i>Pyrosoma</i> (Korotneff), and of <i>Echinus</i> (Driesch)</td> - <td class="tdrightloi fsz8">377</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig165" - title="go to Fig. 165">165</a>.</td> - <td class="tdlefthng">Segmenting egg of a cephalopod (Watase)</td> - <td class="tdrightloi fsz8">378</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig166" - title="go to Fig. 166">166</a>, 7.</td> - <td class="tdlefthng">Eggs segmenting under pressure: of <i>Echinus</i> and <i>Nereis</i> (Driesch), and of a frog (Roux)</td> - <td class="tdrightloi fsz8">378</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig168" - title="go to Fig. 168">168</a>.</td> - <td class="tdlefthng">Various arrangements of a group of eight cells on the surface of a frog’s egg (Rauber)</td> - <td class="tdrightloi fsz8">381</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig169" - title="go to Fig. 169">169</a>.</td> - <td class="tdlefthng">Diagram of the partitions and interfacial contacts in a system of eight cells</td> - <td class="tdrightloi fsz8">383</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig170" - title="go to Fig. 170">170</a>.</td> - <td class="tdlefthng">Various modes of aggregation of eight oil-drops (Roux)</td> - <td class="tdrightloi fsz8">384</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig171" - title="go to Fig. 171">171</a>.</td> - <td class="tdlefthng">Forms, or species, of <i>Asterolampra</i> (Greville)</td> - <td class="tdrightloi fsz8">386</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig172" - title="go to Fig. 172">172</a>.</td> - <td class="tdlefthng">Diagrammatic section of an alcyonarian polype</td> - <td class="tdrightloi fsz8">387</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig173" - title="go to Fig. 173">173</a>, 4.</td> - <td class="tdlefthng">Sections of <i>Heterophyllia</i> (Nicholson and Martin Duncan)</td> - <td class="tdrightloi fsz8">388, 9</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig175" - title="go to Fig. 175">175</a>.</td> - <td class="tdlefthng">Diagrammatic section of a ctenophore (<i>Eucharis</i>)</td> - <td class="tdrightloi fsz8">391</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig176" - title="go to Fig. 176">176</a>, 7.</td> - <td class="tdlefthng">Diagrams of the construction of a Pluteus larva</td> - <td class="tdrightloi fsz8">392, 3</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig178" - title="go to Fig. 178">178</a>, 9.</td> - <td class="tdlefthng">Diagrams of the development of stomata, in <i>Sedum</i> and in the hyacinth</td> - <td class="tdrightloi fsz8">394</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig180" - title="go to Fig. 180">180</a>.</td> - <td class="tdlefthng">Various spores and pollen-grains (Berthold and others)</td> - <td class="tdrightloi fsz8">396</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig181" - title="go to Fig. 181">181</a>.</td> - <td class="tdlefthng">Spore of <i>Anthoceros</i> (Campbell)</td> - <td class="tdrightloi fsz8">397</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig182" - title="go to Fig. 182">182</a>, 4, 9.</td> - <td class="tdlefthng">Diagrammatic modes of division of a cell under certain conditions of asymmetry</td> - <td class="tdrightloi fsz8">400–5</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig183" - title="go to Fig. 183">183</a>.</td> - <td class="tdlefthng">Development of the embryo of <i>Sphagnum</i> (Campbell)</td> - <td class="tdrightloi fsz8">402</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig185" - title="go to Fig. 185">185</a>.</td> - <td class="tdlefthng">The gemma of a moss (<i>do.</i>)</td> - <td class="tdrightloi fsz8">403</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig186" - title="go to Fig. 186">186</a>.</td> - <td class="tdlefthng">The antheridium of <i>Riccia</i> (<i>do.</i>)</td> - <td class="tdrightloi fsz8">404</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig187" - title="go to Fig. 187">187</a>.</td> - <td class="tdlefthng">Section of growing shoot of <i>Selaginella</i>, diagrammatic</td> - <td class="tdrightloi fsz8">404</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig188" - title="go to Fig. 188">188</a>.</td> - <td class="tdlefthng">An embryo of <i>Jungermannia</i> (Kienitz-Gerloff)</td> - <td class="tdrightloi fsz8">404</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig190" - title="go to Fig. 190">190</a>.</td> - <td class="tdlefthng">Development of the sporangium of <i>Osmunda</i> (Bower)</td> - <td class="tdrightloi fsz8">406</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig191" - title="go to Fig. 191">191</a>.</td> - <td class="tdlefthng">Embryos of <i>Phascum</i> and of <i>Adiantum</i> (Kienitz-Gerloff)</td> - <td class="tdrightloi fsz8">408</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig192" - title="go to Fig. 192">192</a>.</td> - <td class="tdlefthng">A section of <i>Girardia</i> (Goebel)</td> - <td class="tdrightloi fsz8">408</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig193" - title="go to Fig. 193">193</a>.</td> - <td class="tdlefthng">An antheridium of <i>Pteris</i> (Strasburger)</td> - <td class="tdrightloi fsz8">409</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig194" - title="go to Fig. 194">194</a>.</td> - <td class="tdlefthng">Spicules of <i>Siphonogorgia</i> and <i>Anthogorgia</i> (Studer)</td> - <td class="tdrightloi fsz8">413</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig195" - title="go to Fig. 195">195</a>–7.</td> - <td class="tdlefthng">Calcospherites, deposited in white of egg (Harting)</td> - <td class="tdrightloi fsz8">421, 2</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig198" - title="go to Fig. 198">198</a>.</td> - <td class="tdlefthng">Sections of the shell of <i>Mya</i> (Carpenter)</td> - <td class="tdrightloi fsz8">422</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig199" - title="go to Fig. 199">199</a>.</td> - <td class="tdlefthng">Concretions, or spicules, artificially deposited in cartilage (Harting)</td> - <td class="tdrightloi fsz8">423</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig200" - title="go to Fig. 200">200</a>.</td> - <td class="tdlefthng">Further illustrations of alcyonarian spicules: <i>Eunicea</i> (Studer)</td> - <td class="tdrightloi fsz8">424</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig201" - title="go to Fig. 201">201</a>–3.</td> - <td class="tdlefthng">Associated, aggregated and composite calcospherites (Harting)</td> - <td class="tdrightloi fsz8">425, 6</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig204" - title="go to Fig. 204">204</a>.</td> - <td class="tdlefthng">Harting’s “conostats”</td> - <td class="tdrightloi fsz8">427</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig205" - title="go to Fig. 205">205</a>.</td> - <td class="tdlefthng">Liesegang’s rings (Leduc)</td> - <td class="tdrightloi fsz8">428</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig206" - title="go to Fig. 206">206</a>.</td> - <td class="tdlefthng">Relay-crystals of common salt (Bowman)</td> - <td class="tdrightloi fsz8">429</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig207" - title="go to Fig. 207">207</a>.</td> - <td class="tdlefthng">Wheel-like crystals in a colloid medium (<i>do.</i>)</td> - <td class="tdrightloi fsz8">429</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig208" - title="go to Fig. 208">208</a>.</td> - <td class="tdlefthng">A concentrically striated calcospherite or spherocrystal (Harting)</td> - <td class="tdrightloi fsz8">432</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig209" - title="go to Fig. 209">209</a>.</td> - <td class="tdlefthng">Otoliths of plaice, shewing “age-rings” (Wallace)</td> - <td class="tdrightloi fsz8">432</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig210" - title="go to Fig. 210">210</a>.</td> - <td class="tdlefthng">Spicules, or calcospherites, of <i>Astrosclera</i> (Lister)</td> - <td class="tdrightloi fsz8">436</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig211" - title="go to Fig. 211">211</a>. 2.</td> - <td class="tdlefthng">C- and S-shaped spicules of sponges and holothurians (Sollas and Théel)</td> - <td class="tdrightloi fsz8">442</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig213" - title="go to Fig. 213">213</a>.</td> - <td class="tdlefthng">An amphidisc of <i>Hyalonema</i></td> - <td class="tdrightloi fsz8">442</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig214" - title="go to Fig. 214">214</a>–7.</td> - <td class="tdlefthng">Spicules of calcareous, tetractinellid and hexactinellid sponges, and of various holothurians (Haeckel, Schultze, Sollas and Théel)</td> - <td class="tdrightloi fsz8">445–452</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig218" - title="go to Fig. 218">218</a>.</td> - <td class="tdlefthng">Diagram of a solid body confined by surface-energy to a liquid boundary-film</td> - <td class="tdrightloi fsz8">460</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig219" - title="go to Fig. 219">219</a>.</td> - <td class="tdlefthng"><i>Astrorhiza limicola</i> and <i>arenaria</i> (Brady)</td> - <td class="tdrightloi fsz8">464</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig220" - title="go to Fig. 220">220</a>.</td> - <td class="tdlefthng">A nuclear “<i>reticulum plasmatique</i>” (Carnoy)</td> - <td class="tdrightloi fsz8">468</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig221" - title="go to Fig. 221">221</a>.</td> - <td class="tdlefthng">A spherical radiolarian, <i>Aulonia hexagona</i> (Haeckel)</td> - <td class="tdrightloi fsz8">469</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig222" - title="go to Fig. 222">222</a>.</td> - <td class="tdlefthng"><i>Actinomma arcadophorum</i> (<i>do.</i>)</td> - <td class="tdrightloi fsz8">469</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig223" - title="go to Fig. 223">223</a>.</td> - <td class="tdlefthng"><i>Ethmosphaera conosiphonia</i> (<i>do.</i>)</td> - <td class="tdrightloi fsz8">470</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig224" - title="go to Fig. 224">224</a>.</td> - <td class="tdlefthng">Portions of shells of <i>Cenosphaera favosa</i> and <i>vesparia</i> (<i>do.</i>)</td> - <td class="tdrightloi fsz8">470</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig225" - title="go to Fig. 225">225</a>.</td> - <td class="tdlefthng"><i>Aulastrum triceros</i> (<i>do.</i>)</td> - <td class="tdrightloi fsz8">471</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig226" - title="go to Fig. 226">226</a>.</td> - <td class="tdlefthng">Part of the skeleton of <i>Cannorhaphis</i> (<i>do.</i>)</td> - <td class="tdrightloi fsz8">472</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig227" - title="go to Fig. 227">227</a>.</td> - <td class="tdlefthng">A Nassellarian skeleton, <i>Callimitra carolotae</i> (<i>do.</i>)</td> - <td class="tdrightloi fsz8">472</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig228" - title="go to Fig. 228">228</a>, 9.</td> - <td class="tdlefthng">Portions of <i>Dictyocha stapedia</i> (<i>do.</i>)</td> - <td class="tdrightloi fsz8">474</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig230" - title="go to Fig. 230">230</a>.</td> - <td class="tdlefthng">Diagram to illustrate the conformation of <i>Callimitra</i></td> - <td class="tdrightloi fsz8">476</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig231" - title="go to Fig. 231">231</a>.</td> - <td class="tdlefthng">Skeletons of various radiolarians (Haeckel)</td> - <td class="tdrightloi fsz8">479</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig232" - title="go to Fig. 232">232</a>.</td> - <td class="tdlefthng">Diagrammatic structure of the skeleton of <i>Dorataspis</i> (<i>do.</i>)</td> - <td class="tdrightloi fsz8">481</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig233" - title="go to Fig. 233">233</a>, 4.</td> - <td class="tdlefthng"><i>Phatnaspis cristata</i> (Haeckel), and a diagram of the same</td> - <td class="tdrightloi fsz8">483</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig235" - title="go to Fig. 235">235</a>.</td> - <td class="tdlefthng"><i>Phractaspis prototypus</i> (Haeckel)</td> - <td class="tdrightloi fsz8">484</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig236" - title="go to Fig. 236">236</a>.</td> - <td class="tdlefthng">Annular and spiral thickenings in the walls of plant-cells</td> - <td class="tdrightloi fsz8">488</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig237" - title="go to Fig. 237">237</a>.</td> - <td class="tdlefthng">A radiograph of the shell of <i>Nautilus</i> (Green and Gardiner)</td> - <td class="tdrightloi fsz8">494</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig238" - title="go to Fig. 238">238</a>.</td> - <td class="tdlefthng">A spiral foraminifer, <i>Globigerina</i> (Brady)</td> - <td class="tdrightloi fsz8">495</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig239" - title="go to Fig. 239">239</a>–42.</td> - <td class="tdlefthng">Diagrams to illustrate the development or growth of a logarithmic spiral</td> - <td class="tdrightloi fsz8">407–501</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig243" - title="go to Fig. 243">243</a>.</td> - <td class="tdlefthng">A helicoid and a scorpioid cyme</td> - <td class="tdrightloi fsz8">502</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig244" - title="go to Fig. 244">244</a>.</td> - <td class="tdlefthng">An Archimedean spiral</td> - <td class="tdrightloi fsz8">503</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig245" - title="go to Fig. 245">245</a>–7.</td> - <td class="tdlefthng">More diagrams of the development of a logarithmic spiral</td> - <td class="tdrightloi fsz8">505, 6</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig248" - title="go to Fig. 248">248</a>–57.</td> - <td class="tdlefthng">Various diagrams illustrating the mathematical theory of gnomons</td> - <td class="tdrightloi fsz8">508–13</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig258" - title="go to Fig. 258">258</a>.</td> - <td class="tdlefthng">A shell of <i>Haliotis</i>, to shew how each increment of the shell constitutes a gnomon to the preexisting structure</td> - <td class="tdrightloi fsz8">514</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig259" - title="go to Fig. 259">259</a>, 60.</td> - <td class="tdlefthng">Spiral foraminifera, <i>Pulvinulina</i> and <i>Cristellaria</i>, to illustrate the same principle</td> - <td class="tdrightloi fsz8">514, 5</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig261" - title="go to Fig. 261">261</a>.</td> - <td class="tdlefthng">Another diagram of a logarithmic spiral</td> - <td class="tdrightloi fsz8">517</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig262" - title="go to Fig. 262">262</a>.</td> - <td class="tdlefthng">A diagram of the logarithmic spiral of <i>Nautilus</i> (Moseley)</td> - <td class="tdrightloi fsz8">519</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig263" - title="go to Fig. 263">263</a>, 4.</td> - <td class="tdlefthng">Opercula of <i>Turbo</i> and of <i>Nerita</i> (Moseley)</td> - <td class="tdrightloi fsz8">521, 2</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig265" - title="go to Fig. 265">265</a>.</td> - <td class="tdlefthng">A section of the shell of <i>Melo ethiopicus</i></td> - <td class="tdrightloi fsz8">525</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig266" - title="go to Fig. 266">266</a>.</td> - <td class="tdlefthng">Shells of <i>Harpa</i> and <i>Dolium</i>, to illustrate generating curves and gene</td> - <td class="tdrightloi fsz8">526</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig267" - title="go to Fig. 267">267</a>.</td> - <td class="tdlefthng">D’Orbigny’s Helicometer</td> - <td class="tdrightloi fsz8">529</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig268" - title="go to Fig. 268">268</a>.</td> - <td class="tdlefthng">Section of a nautiloid shell, to shew the “protoconch”</td> - <td class="tdrightloi fsz8">531</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig269" - title="go to Fig. 269">269</a>–73.</td> - <td class="tdlefthng">Diagrams of logarithmic spirals, of various angles</td> - <td class="tdrightloi fsz8">532–5</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig274" - title="go to Fig. 274">274</a>, 6, 7.</td> - <td class="tdlefthng">Constructions for determining the angle of a logarithmic spiral</td> - <td class="tdrightloi fsz8">537, 8</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig275" - title="go to Fig. 275">275</a>.</td> - <td class="tdlefthng">An ammonite, to shew its corrugated surface pattern</td> - <td class="tdrightloi fsz8">537</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig278" - title="go to Fig. 278">278</a>–80.</td> - <td class="tdlefthng">Illustrations of the “angle of retardation”</td> - <td class="tdrightloi fsz8">542–4</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig281" - title="go to Fig. 281">281</a>.</td> - <td class="tdlefthng">A shell of <i>Macroscaphites</i>, to shew change of curvature</td> - <td class="tdrightloi fsz8">550</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig282" - title="go to Fig. 282">282</a>.</td> - <td class="tdlefthng">Construction for determining the length of the coiled spire</td> - <td class="tdrightloi fsz8">551</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig283" - title="go to Fig. 283">283</a>.</td> - <td class="tdlefthng">Section of the shell of <i>Triton corrugatus</i> (Woodward)</td> - <td class="tdrightloi fsz8">554</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig284" - title="go to Fig. 284">284</a>.</td> - <td class="tdlefthng"><i>Lamellaria perspicua</i> and <i>Sigaretus haliotoides</i> (<i>do.</i>)</td> - <td class="tdrightloi fsz8">555</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig285" - title="go to Fig. 285">285</a>, 6.</td> - <td class="tdlefthng">Sections of the shells of <i>Terebra maculata</i> and <i>Trochus niloticus</i></td> - <td class="tdrightloi fsz8">559, 60</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig287" - title="go to Fig. 287">287</a>–9.</td> - <td class="tdlefthng">Diagrams illustrating the lines of growth on a lamellibranch shell</td> - <td class="tdrightloi fsz8">563–5</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig290" - title="go to Fig. 290">290</a>.</td> - <td class="tdlefthng"><i>Caprinella adversa</i> (Woodward)</td> - <td class="tdrightloi fsz8">567</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig291" - title="go to Fig. 291">291</a>.</td> - <td class="tdlefthng">Section of the shell of <i>Productus</i> (Woods)</td> - <td class="tdrightloi fsz8">567</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig292" - title="go to Fig. 292">292</a>.</td> - <td class="tdlefthng">The “skeletal loop” of <i>Terebratula</i> (<i>do.</i>)</td> - <td class="tdrightloi fsz8">568</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig293" - title="go to Fig. 293">293</a>, 4.</td> - <td class="tdlefthng">The spiral arms of <i>Spirifer</i> and of <i>Atrypa</i> (<i>do.</i>)</td> - <td class="tdrightloi fsz8">569</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig295" - title="go to Fig. 295">295</a>–7.</td> - <td class="tdlefthng">Shells of <i>Cleodora</i>, <i>Hyalaea</i> and other pteropods (Boas)</td> - <td class="tdrightloi fsz8">570, 1</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig298" - title="go to Fig. 298">298</a>, 9.</td> - <td class="tdlefthng">Coordinate diagrams of the shell-outline in certain pteropods</td> - <td class="tdrightloi fsz8">572, 3</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig300" - title="go to Fig. 300">300</a>.</td> - <td class="tdlefthng">Development of the shell of <i>Hyalaea tridentata</i> (Tesch)</td> - <td class="tdrightloi fsz8">573</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig301" - title="go to Fig. 301">301</a>.</td> - <td class="tdlefthng">Pteropod shells, of <i>Cleodora</i> and <i>Hyalaea</i>, viewed from the side (Boas)</td> - <td class="tdrightloi fsz8">575</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig302" - title="go to Fig. 302">302</a>, 3.</td> - <td class="tdlefthng">Diagrams of septa in a conical shell</td> - <td class="tdrightloi fsz8">579</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig304" - title="go to Fig. 304">304</a>.</td> - <td class="tdlefthng">A section of <i>Nautilus</i>, shewing the logarithmic spirals of the septa to which the shell-spiral is the evolute</td> - <td class="tdrightloi fsz8">581</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig305" - title="go to Fig. 305">305</a>.</td> - <td class="tdlefthng">Cast of the interior of the shell of <i>Nautilus</i>, to shew the contours of the septa at their junction with the shell-wall</td> - <td class="tdrightloi fsz8">582</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig306" - title="go to Fig. 306">306</a>.</td> - <td class="tdlefthng"><i>Ammonites Sowerbyi</i>, to shew septal outlines (Zittel, after Steinmann and Döderlein)</td> - <td class="tdrightloi fsz8">584</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig307" - title="go to Fig. 307">307</a>.</td> - <td class="tdlefthng">Suture-line of <i>Pinacoceras</i> (Zittel, after Hauer)</td> - <td class="tdrightloi fsz8">584</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig308" - title="go to Fig. 308">308</a>.</td> - <td class="tdlefthng">Shells of <i>Hastigerina</i>, to shew the “mouth” (Brady)</td> - <td class="tdrightloi fsz8">588</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig309" - title="go to Fig. 309">309</a>.</td> - <td class="tdlefthng"><i>Nummulina antiquior</i> (V. von Möller)</td> - <td class="tdrightloi fsz8">591</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig310" - title="go to Fig. 310">310</a>.</td> - <td class="tdlefthng"><i>Cornuspira foliacea</i> and <i>Operculina complanata</i> (Brady)</td> - <td class="tdrightloi fsz8">594</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig311" - title="go to Fig. 311">311</a>.</td> - <td class="tdlefthng"><i>Miliolina pulchella</i> and <i>linnaeana</i> (Brady)</td> - <td class="tdrightloi fsz8">596</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig312" - title="go to Fig. 312">312</a>, 3.</td> - <td class="tdlefthng"><i>Cyclammina cancellata</i> (<i>do.</i>), and diagrammatic figure of the same</td> - <td class="tdrightloi fsz8">596, 7</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig314" - title="go to Fig. 314">314</a>.</td> - <td class="tdlefthng"><i>Orbulina universa</i> (Brady)</td> - <td class="tdrightloi fsz8">598</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig315" - title="go to Fig. 315">315</a>.</td> - <td class="tdlefthng"><i>Cristellaria reniformis</i> (<i>do.</i>)</td> - <td class="tdrightloi fsz8">600</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig316" - title="go to Fig. 316">316</a>.</td> - <td class="tdlefthng"><i>Discorbina bertheloti</i> (<i>do.</i>)</td> - <td class="tdrightloi fsz8">603</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig317" - title="go to Fig. 317">317</a>.</td> - <td class="tdlefthng"><i>Textularia trochus</i> and <i>concava</i> (<i>do.</i>)</td> - <td class="tdrightloi fsz8">604</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig318" - title="go to Fig. 318">318</a>.</td> - <td class="tdlefthng">Diagrammatic figure of a ram’s horns (Sir V. Brooke)</td> - <td class="tdrightloi fsz8">615</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig319" - title="go to Fig. 319">319</a>.</td> - <td class="tdlefthng">Head of an Arabian wild goat (Sclater)</td> - <td class="tdrightloi fsz8">616</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig320" - title="go to Fig. 320">320</a>.</td> - <td class="tdlefthng">Head of <i>Ovis Ammon</i>, shewing St Venant’s curves</td> - <td class="tdrightloi fsz8">621</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig321" - title="go to Fig. 321">321</a>.</td> - <td class="tdlefthng">St Venant’s diagram of a triangular prism under torsion (Thomson and Tait)</td> - <td class="tdrightloi fsz8">623</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig322" - title="go to Fig. 322">322</a>.</td> - <td class="tdlefthng">Diagram of the same phenomenon in a ram’s horn</td> - <td class="tdrightloi fsz8">623</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig323" - title="go to Fig. 323">323</a>.</td> - <td class="tdlefthng">Antlers of a Swedish elk (Lönnberg)</td> - <td class="tdrightloi fsz8">629</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig324" - title="go to Fig. 324">324</a>.</td> - <td class="tdlefthng">Head and antlers of <i>Cervus duvauceli</i> (Lydekker)</td> - <td class="tdrightloi fsz8">630</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig325" - title="go to Fig. 325">325</a>, 6.</td> - <td class="tdlefthng">Diagrams of spiral phyllotaxis (P. G. Tait)</td> - <td class="tdrightloi fsz8">644, 5</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig327" - title="go to Fig. 327">327</a>.</td> - <td class="tdlefthng">Further diagrams of phyllotaxis, to shew how various spiral appearances may arise out of one and the same angular leaf-divergence</td> - <td class="tdrightloi fsz8">648</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig328" - title="go to Fig. 328">328</a>.</td> - <td class="tdlefthng">Diagrammatic outlines of various sea-urchins</td> - <td class="tdrightloi fsz8">664</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig329" - title="go to Fig. 329">329</a>, 30.</td> - <td class="tdlefthng">Diagrams of the angle of branching in blood-vessels (Hess)</td> - <td class="tdrightloi fsz8">667, 8</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig331" - title="go to Fig. 331">331</a>, 2.</td> - <td class="tdlefthng">Diagrams illustrating the flexure of a beam</td> - <td class="tdrightloi fsz8">674, 8</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig333" - title="go to Fig. 333">333</a>.</td> - <td class="tdlefthng">An example of the mode of arrangement of bast-fibres in a plant-stem (Schwendener)</td> - <td class="tdrightloi fsz8">680</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig334" - title="go to Fig. 334">334</a>.</td> - <td class="tdlefthng">Section of the head of a femur, to shew its trabecular structure (Schäfer, after Robinson)</td> - <td class="tdrightloi fsz8">681</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig335" - title="go to Fig. 335">335</a>.</td> - <td class="tdlefthng">Comparative diagrams of a crane-head and the head of a femur (Culmann and H. Meyer)</td> - <td class="tdrightloi fsz8">682</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig336" - title="go to Fig. 336">336</a>.</td> - <td class="tdlefthng">Diagram of stress-lines in the human foot (Sir D. MacAlister, after H. Meyer)</td> - <td class="tdrightloi fsz8">684</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig337" - title="go to Fig. 337">337</a>.</td> - <td class="tdlefthng">Trabecular structure of the <i>os calcis</i> (<i>do.</i>)</td> - <td class="tdrightloi fsz8">685</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig338" - title="go to Fig. 338">338</a>.</td> - <td class="tdlefthng">Diagram of shearing-stress in a loaded pillar</td> - <td class="tdrightloi fsz8">686</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig339" - title="go to Fig. 339">339</a>.</td> - <td class="tdlefthng">Diagrams of tied arch, and bowstring girder (Fidler)</td> - <td class="tdrightloi fsz8">693</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig340" - title="go to Fig. 340">340</a>, 1.</td> - <td class="tdlefthng">Diagrams of a bridge: shewing proposed span, the corresponding stress-diagram and reciprocal plan of construction (<i>do.</i>)</td> - <td class="tdrightloi fsz8">696</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig342" - title="go to Fig. 342">342</a>.</td> - <td class="tdlefthng">A loaded bracket and its reciprocal construction-diagram (Culmann)</td> - <td class="tdrightloi fsz8">697</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig343" - title="go to Fig. 343">343</a>, 4.</td> - <td class="tdlefthng">A cantilever bridge, with its reciprocal diagrams (Fidler)</td> - <td class="tdrightloi fsz8">698</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig345" - title="go to Fig. 345">345</a>.</td> - <td class="tdlefthng">A two-armed cantilever of the Forth Bridge (<i>do.</i>)</td> - <td class="tdrightloi fsz8">700</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig346" - title="go to Fig. 346">346</a>.</td> - <td class="tdlefthng">A two-armed cantilever with load distributed over two pier-heads, as in the quadrupedal skeleton</td> - <td class="tdrightloi fsz8">700</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig347" - title="go to Fig. 347">347</a>–9.</td> - <td class="tdlefthng">Stress-diagrams. or diagrams of bending moments, in the backbones of the horse, of a Dinosaur, and of <i>Titanotherium</i></td> - <td class="tdrightloi fsz8">701–4</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig350" - title="go to Fig. 350">350</a>.</td> - <td class="tdlefthng">The skeleton of <i>Stegosaurus</i></td> - <td class="tdrightloi fsz8">707</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig351" - title="go to Fig. 351">351</a>.</td> - <td class="tdlefthng">Bending-moments in a beam with fixed ends, to illustrate the mechanics of chevron-bones</td> - <td class="tdrightloi fsz8">709</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig352" - title="go to Fig. 352">352</a>, 3.</td> - <td class="tdlefthng">Coordinate diagrams of a circle, and its deformation into an ellipse</td> - <td class="tdrightloi fsz8">729</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig354" - title="go to Fig. 354">354</a>.</td> - <td class="tdlefthng">Comparison, by means of Cartesian coordinates, of the cannon-bones of various ruminant animals</td> - <td class="tdrightloi fsz8">729</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig355" - title="go to Fig. 355">355</a>, 6.</td> - <td class="tdlefthng">Logarithmic coordinates, and the circle of Fig. 352 inscribed therein</td> - <td class="tdrightloi fsz8">729, 31</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig357" - title="go to Fig. 357">357</a>, 8.</td> - <td class="tdlefthng">Diagrams of oblique and radial coordinates</td> - <td class="tdrightloi fsz8">731</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig359" - title="go to Fig. 359">359</a>.</td> - <td class="tdlefthng">Lanceolate, ovate and cordate leaves, compared by the help of radial coordinates</td> - <td class="tdrightloi fsz8">732</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig360" - title="go to Fig. 360">360</a>.</td> - <td class="tdlefthng">A leaf of <i>Begonia daedalea</i></td> - <td class="tdrightloi fsz8">733</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig361" - title="go to Fig. 361">361</a>.</td> - <td class="tdlefthng">A network of logarithmic spiral coordinates</td> - <td class="tdrightloi fsz8">735</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig362" - title="go to Fig. 362">362</a>, 3.</td> - <td class="tdlefthng">Feet of ox, sheep and giraffe, compared by means of Cartesian coordinates</td> - <td class="tdrightloi fsz8">738, 40</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig364" - title="go to Fig. 364">364</a>, 6.</td> - <td class="tdlefthng">“Proportional diagrams” of human physiognomy (Albert Dürer)</td> - <td class="tdrightloi fsz8">740, 2</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig365" - title="go to Fig. 365">365</a>.</td> - <td class="tdlefthng">Median and lateral toes of a tapir, compared by means of rectangular and oblique coordinates</td> - <td class="tdrightloi fsz8">741</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig367" - title="go to Fig. 367">367</a>, 8.</td> - <td class="tdlefthng">A comparison of the copepods <i>Oithona</i> and <i>Sapphirina</i></td> - <td class="tdrightloi fsz8">742</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig369" - title="go to Fig. 369">369</a>.</td> - <td class="tdlefthng">The carapaces of certain crabs, <i>Geryon</i>, <i>Corystes</i> and others, compared by means of rectilinear and curvilinear coordinates</td> - <td class="tdrightloi fsz8">744</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig370" - title="go to Fig. 370">370</a>.</td> - <td class="tdlefthng">A comparison of certain amphipods, <i>Harpinia</i>, <i>Stegocephalus</i> and <i>Hyperia</i></td> - <td class="tdrightloi fsz8">746</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig371" - title="go to Fig. 371">371</a>.</td> - <td class="tdlefthng">The calycles of certain campanularian zoophytes, inscribed in corresponding Cartesian networks</td> - <td class="tdrightloi fsz8">747</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig372" - title="go to Fig. 372">372</a>.</td> - <td class="tdlefthng">The calycles of certain species of <i>Aglaophenia</i>, similarly compared by means of curvilinear coordinates</td> - <td class="tdrightloi fsz8">748</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig373" - title="go to Fig. 373">373</a>, 4.</td> - <td class="tdlefthng">The fishes <i>Argyropelecus</i> and <i>Sternoptyx</i>, compared by means of rectangular and oblique coordinate systems</td> - <td class="tdrightloi fsz8">748</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig375" - title="go to Fig. 375">375</a>, 6.</td> - <td class="tdlefthng"><i>Scarus</i> and <i>Pomacanthus</i>, similarly compared by means of rectangular and coaxial systems</td> - <td class="tdrightloi fsz8">749</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig377" - title="go to Fig. 377">377</a>–80.</td> - <td class="tdlefthng">A comparison of the fishes <i>Polyprion</i>, <i>Pseudopriacanthus</i>, <i>Scorpaena</i> and <i>Antigonia</i></td> - <td class="tdrightloi fsz8">750</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig381" - title="go to Fig. 381">381</a>, 2.</td> - <td class="tdlefthng">A similar comparison of <i>Diodon</i> and <i>Orthagoriscus</i></td> - <td class="tdrightloi fsz8">751</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig383" - title="go to Fig. 383">383</a>.</td> - <td class="tdlefthng">The same of various crocodiles: <i>C. porosus</i>, <i>C. americanus</i> and <i>Notosuchus terrestris</i></td> - <td class="tdrightloi fsz8">753</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig384" - title="go to Fig. 384">384</a>.</td> - <td class="tdlefthng">The pelvic girdles of <i>Stegosaurus</i> and <i>Camptosaurus</i></td> - <td class="tdrightloi fsz8">754</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig385" - title="go to Fig. 385">385</a>, 6.</td> - <td class="tdlefthng">The shoulder-girdles of <i>Cryptocleidus</i> and of <i>Ichthyosaurus</i></td> - <td class="tdrightloi fsz8">755</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig387" - title="go to Fig. 387">387</a>.</td> - <td class="tdlefthng">The skulls of <i>Dimorphodon</i> and of <i>Pteranodon</i></td> - <td class="tdrightloi fsz8">756</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig388" - title="go to Fig. 388">388</a>–92.</td> - <td class="tdlefthng">The pelves of <i>Archaeopteryx</i> and of <i>Apatornis</i> compared, and a method illustrated whereby intermediate configurations may be found by interpolation (G. Heilmann)</td> - <td class="tdrightloi fsz8">757–9</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig393" - title="go to Fig. 393">393</a>.</td> - <td class="tdlefthng">The same pelves, together with three of the intermediate or interpolated forms</td> - <td class="tdrightloi fsz8">760</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig394" - title="go to Fig. 394">394</a>, 5.</td> - <td class="tdlefthng">Comparison of the skulls of two extinct rhinoceroses, <i>Hyrachyus</i> and <i>Aceratherium</i> (Osborn)</td> - <td class="tdrightloi fsz8">761</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig396" - title="go to Fig. 396">396</a>.</td> - <td class="tdlefthng">Occipital views of various extinct rhinoceroses (<i>do.</i>)</td> - <td class="tdrightloi fsz8">762</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig397" - title="go to Fig. 397">397</a>–400.</td> - <td class="tdlefthng">Comparison with each other, and with the skull of <i>Hyrachyus</i>, of the skulls of <i>Titanotherium</i>, tapir, horse and rabbit</td> - <td class="tdrightloi fsz8">763, 4</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig401" - title="go to Fig. 401">401</a>, 2.</td> - <td class="tdlefthng">Coordinate diagrams of the skulls of <i>Eohippus</i> and of <i>Equus</i>, with various actual and hypothetical intermediate types (Heilmann)</td> - <td class="tdrightloi fsz8">765–7</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig403" - title="go to Fig. 403">403</a>.</td> - <td class="tdlefthng">A comparison of various human scapulae (Dwight)</td> - <td class="tdrightloi fsz8">769</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig404" - title="go to Fig. 404">404</a>.</td> - <td class="tdlefthng">A human skull, inscribed in Cartesian coordinates</td> - <td class="tdrightloi fsz8">770</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig405" - title="go to Fig. 405">405</a>.</td> - <td class="tdlefthng">The same coordinates on a new projection, adapted to the skull of the chimpanzee</td> - <td class="tdrightloi fsz8">770</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig406" - title="go to Fig. 406">406</a>.</td> - <td class="tdlefthng">Chimpanzee’s skull, inscribed in the network of Fig. 405</td> - <td class="tdrightloi fsz8">771</td></tr> -<tr class="trkeeptgth"> - <td class="tdlftloi fsz7"><a class="aplain" href="#fig407" - title="go to Fig. 407">407</a>, 8.</td> - <td class="tdlefthng">Corresponding diagrams of a baboon’s skull, and of a dog’s</td> - <td class="tdrightloi fsz8">771, 3</td></tr> -</table></div><!--chapter--> - -<div class="chapter"> -<p class="padtopa0">“Cum formarum naturalium et corporalium esse non consistat nisi in -unione ad materiam, ejusdem agentis esse videtur eas producere cujus -est materiam transmutare. Secundo, quia cum hujusmodi formae non -excedant virtutem et ordinem et facultatem principiorum agentium in -natura, nulla videtur necessitas eorum originem in principia reducere -altiora.” Aquinas, <i>De Pot. Q.</i> iii, a, 11. (Quoted in <i>Brit. Assoc. -Address</i>, <i>Section D</i>, 1911.)</p> - -<p class="padtopb">“...I would that all other natural phenomena might similarly be -deduced from mechanical principles. For many things move me to suspect -that everything depends upon certain forces, in virtue of which the -particles of bodies, through forces not yet understood, are either -impelled together so as to cohere in regular figures, or are repelled -and recede from one another.” Newton, in Preface to the <i>Principia</i>. -(Quoted by Mr W. Spottiswoode, <i>Brit. Assoc. Presidential Address</i>, -1878.)</p> - -<p class="padtopb">“When Science shall have subjected all natural phenomena to the laws -of Theoretical Mechanics, when she shall be able to predict the result -of every combination as unerringly as Hamilton predicted conical -refraction, or Adams revealed to us the existence of Neptune,—that we -cannot say. That day may never come, and it is certainly far in the dim -future. We may not anticipate it, we may not even call it possible. -But none the less are we bound to look to that day, and to labour for -it as the crowning triumph of Science:—when Theoretical Mechanics -shall be recognised as the key to every physical enigma, the chart for -every traveller through the dark Infinite of Nature.” J. H. Jellett, in -<i>Brit. Assoc. Address</i>, <i>Section A</i>, 1874.</p></div><!--chapter--> - -<div class="chapter" id="p001"> -<h2 class="h2herein" title="I. Introductory.">CHAPTER I -<span class="h2ttl"> -INTRODUCTORY</span></h2></div> - -<p>Of the chemistry of his day and generation, Kant declared -that it was “a science, but not science,”—“eine Wissenschaft, -aber nicht Wissenschaft”; for that the criterion of physical -science lay in its relation to mathematics. And a hundred years -later Du Bois Reymond, profound student of the many sciences -on which physiology is based, recalled and reiterated the old -saying, declaring that chemistry would only reach the rank of -science, in the high and strict sense, when it should be found -possible to explain chemical reactions in the light of their causal -relation to the velocities, tensions and conditions of equilibrium -of the component molecules; that, in short, the chemistry of the -future must deal with molecular mechanics, by the methods and -in the strict language of mathematics, as the astronomy of Newton -and Laplace dealt with the stars in their courses. We know how -great a step has been made towards this distant and once hopeless -goal, as Kant defined it, since van’t Hoff laid the firm foundations -of a mathematical chemistry, and earned his proud epitaph, -<i>Physicam chemiae adiunxit</i><a class="afnanch" href="#fn1" id="fnanch1">1</a>.</p> - -<p>We need not wait for the full realisation of Kant’s desire, in -order to apply to the natural sciences the principle which he -urged. Though chemistry fall short of its ultimate goal in mathematical -mechanics, nevertheless physiology is vastly strengthened -and enlarged by making use of the chemistry, as of the physics, -of the age. Little by little it draws nearer to our conception of -a true science, with each branch of physical -science which it <span class="xxpn" id="p002">{2}</span> -brings into relation with itself: with every physical law and every -mathematical theorem which it learns to take into its employ. -Between the physiology of Haller, fine as it was, and that of -Helmholtz, Ludwig, Claude Bernard, there was all the difference -in the world.</p> - -<p>As soon as we adventure on the paths of the physicist, we -learn to <i>weigh</i> and to <i>measure</i>, to deal with time and space and -mass and their related concepts, and to find more and more -our knowledge expressed and our needs satisfied through the -concept of <i>number</i>, as in the dreams and visions of Plato and -Pythagoras; for modern chemistry would have gladdened the -hearts of those great philosophic dreamers.</p> - -<p>But the zoologist or morphologist has been slow, where the -physiologist has long been eager, to invoke the aid of the physical -or mathematical sciences; and the reasons for this difference lie -deep, and in part are rooted in old traditions. The zoologist has -scarce begun to dream of defining, in mathematical language, even -the simpler organic forms. When he finds a simple geometrical -construction, for instance in the honey-comb, he would fain refer -it to psychical instinct or design rather than to the operation of -physical forces; when he sees in snail, or nautilus, or tiny -foraminiferal or radiolarian shell, a close approach to the perfect -sphere or spiral, he is prone, of old habit, to believe that it is -after all something more than a spiral or a sphere, and that in -this “something more” there lies what neither physics nor -mathematics can explain. In short he is deeply reluctant to -compare the living with the dead, or to explain by geometry or -by dynamics the things which have their part in the mystery of -life. Moreover he is little inclined to feel the need of such -explanations or of such extension of his field of thought. He is -not without some justification if he feels that in admiration of -nature’s handiwork he has an horizon open before his eyes as -wide as any man requires. He has the help of many fascinating -theories within the bounds of his own science, which, though -a little lacking in precision, serve the purpose of ordering his -thoughts and of suggesting new objects of enquiry. His art of -classification becomes a ceaseless and an endless search after the -blood-relationships of things living, and the -pedigrees of things <span class="xxpn" id="p003">{3}</span> -dead and gone. The facts of embryology become for him, as -Wolff, von Baer and Fritz Müller proclaimed, a record not only -of the life-history of the individual but of the annals of its race. -The facts of geographical distribution or even of the migration of -birds lead on and on to speculations regarding lost continents, -sunken islands, or bridges across ancient seas. Every nesting -bird, every ant-hill or spider’s web displays its psychological -problems of instinct or intelligence. Above all, in things both -great and small, the naturalist is rightfully impressed, and finally -engrossed, by the peculiar beauty which is manifested in apparent -fitness or “adaptation,”—the flower for the bee, the berry for the -bird.</p> - -<p>Time out of mind, it has been by way of the “final cause,” -by the teleological concept of “end,” of “purpose,” or of “design,” -in one or another of its many forms (for its moods are many), -that men have been chiefly wont to explain the phenomena of -the living world; and it will be so while men have eyes to see -and ears to hear withal. With Galen, as with Aristotle, it was -the physician’s way; with John Ray, as with Aristotle, it was the -naturalist’s way; with Kant, as with Aristotle, it was the philosopher’s -way. It was the old Hebrew way, and has its splendid -setting in the story that God made “every plant of the field before -it was in the earth, and every herb of the field before it grew.” -It is a common way, and a great way; for it brings with it a -glimpse of a great vision, and it lies deep as the love of nature -in the hearts of men.</p> - -<p>Half overshadowing the “efficient” or physical cause, the -argument of the final cause appears in eighteenth century physics, -in the hands of such men as Euler<a class="afnanch" href="#fn2" id="fnanch2">2</a> -and Maupertuis, to whom -Leibniz<a class="afnanch" href="#fn3" id="fnanch3">3</a> -had passed it on. Half overshadowed by the mechanical -concept, it runs through Claude Bernard’s <i>Leçons -sur les <span class="xxpn" id="p004">{4}</span> -phénomènes de la Vie</i><a class="afnanch" href="#fn4" id="fnanch4">4</a>, -and abides in much of modern physiology<a class="afnanch" href="#fn5" id="fnanch5">5</a>. -Inherited from Hegel, it dominated Oken’s <i>Naturphilosophie</i> -and lingered among his later disciples, who were wont to liken -the course of organic evolution not to the straggling branches of -a tree, but to the building of a temple, divinely planned, and the -crowning of it with its polished minarets<a class="afnanch" href="#fn6" id="fnanch6">6</a>.</p> - -<p>It is retained, somewhat crudely, in modern embryology, by -those who see in the early processes of growth a significance -“rather prospective than retrospective,” such that the embryonic -phenomena must be “referred directly to their usefulness in -building the body of the future animal<a class="afnanch" href="#fn7" id="fnanch7">7</a>”:—which -is no more, and -no less, than to say, with Aristotle, that the organism is the τέλος, -or final cause, of its own processes of generation and development. -It is writ large in that Entelechy<a class="afnanch" href="#fn8" id="fnanch8">8</a> -which Driesch rediscovered, -and which he made known to many who had neither learned of it -from Aristotle, nor studied it with Leibniz, nor laughed at it with -Voltaire. And, though it is in a very curious way, we are told that -teleology was “refounded, reformed or rehabilitated<a class="afnanch" href="#fn9" id="fnanch9">9</a>” -by Darwin’s -theory of natural selection, whereby “every variety of form and -colour was urgently and absolutely called upon to produce its -title to existence either as an active useful agent, or as a survival” -of such active usefulness in the past. But in this last, and very -important case, we have reached a “teleology” -without a τέλος, <span class="xxpn" id="p005">{5}</span> -as men like Butler and Janet have been prompt to shew: a teleology -in which the final cause becomes little more, if anything, than the -mere expression or resultant of a process of sifting out of the -good from the bad, or of the better from the worse, in short of -a process of mechanism<a class="afnanch" href="#fn10" id="fnanch10">10</a>. -The apparent manifestations of “purpose” -or adaptation become part of a mechanical philosophy, -according to which “chaque chose finit toujours par s’accommoder -à son milieu<a class="afnanch" href="#fn11" id="fnanch11">11</a>.” -In short, by a road which resembles but is not -the same as Maupertuis’s road, we find our way to the very world -in which we are living, and find that if it be not, it is ever tending -to become, “the best of all possible worlds<a class="afnanch" href="#fn12" id="fnanch12">12</a>.”</p> - -<p>But the use of the teleological principle is but one way, not -the whole or the only way, by which we may seek to learn how -things came to be, and to take their places in the harmonious complexity -of the world. To seek not for ends but for “antecedents” -is the way of the physicist, who finds “causes” in what he has -learned to recognise as fundamental properties, or inseparable -concomitants, or unchanging laws, of matter and of energy. In -Aristotle’s parable, the house is there that men may live in it; -but it is also there because the builders have laid one stone upon -another: and it is as a <i>mechanism</i>, or a mechanical construction, -that the physicist looks upon the world. Like warp and woof, -mechanism and teleology are interwoven together, and we must -not cleave to the one and despise the other; for their union is -“rooted in the very nature of totality<a class="afnanch" href="#fn13" id="fnanch13">13</a>.”</p> - -<p>Nevertheless, when philosophy bids us hearken and obey the -lessons both of mechanical and of teleological interpretation, the -precept is hard to follow: so that oftentimes it has come to pass, -just as in Bacon’s day, that a leaning to the side of the final -cause “hath intercepted the severe and diligent -inquiry of all <span class="xxpn" id="p006">{6}</span> -real and physical causes,” and has brought it about that “the -search of the physical cause hath been neglected and passed in -silence.” So long and so far as “fortuitous variation<a class="afnanch" href="#fn14" id="fnanch14">14</a>” -and the -“survival of the fittest” remain engrained as fundamental and -satisfactory hypotheses in the philosophy of biology, so long will -these “satisfactory and specious causes” tend to stay “severe and -diligent inquiry,” “to the great arrest and prejudice of future -discovery.”</p> - -<p>The difficulties which surround the concept of active or “real” -causation, in Bacon’s sense of the word, difficulties of which -Hume and Locke and Aristotle were little aware, need scarcely -hinder us in our physical enquiry. As students of mathematical -and of empirical physics, we are content to deal with those antecedents, -or concomitants, of our phenomena, without which the -phenomenon does not occur,—with causes, in short, which, <i>aliae -ex aliis aptae et necessitate nexae</i>, are no more, and no less, than -conditions <i>sine quâ non</i>. Our purpose is still adequately fulfilled: -inasmuch as we are still enabled to correlate, and to equate, our -particular phenomena with more and ever more of the physical -phenomena around, and so to weave a web of connection and -interdependence which shall serve our turn, though the metaphysician -withhold from that interdependence the title of causality. -We come in touch with what the schoolmen called a <i>ratio -cognoscendi</i>, though the true <i>ratio efficiendi</i> is still enwrapped in -many mysteries. And so handled, the quest of physical causes -merges with another great Aristotelian theme,—the search for -relations between things apparently disconnected, and for “similitude -in things to common view unlike.” Newton did not shew -the cause of the apple falling, but he shewed a similitude between -the apple and the stars.</p> - -<p>Moreover, the naturalist and the physicist will continue to -speak of “causes,” just as of old, though it may be with some -mental reservations: for, as a French philosopher said, in a -kindred difficulty: “ce sont là des -manières de s’exprimer, <span class="xxpn" id="p007">{7}</span> -et si elles sont interdites il faut renoncer à parler de ces -choses.”</p> - -<p>The search for differences or essential contrasts between the -phenomena of organic and inorganic, of animate and inanimate -things has occupied many mens’ minds, while the search for -community of principles, or essential similitudes, has been followed -by few; and the contrasts are apt to loom too large, great as -they may be. M. Dunan, discussing the “Problème de la Vie<a class="afnanch" href="#fn15" id="fnanch15">15</a>” -in an essay which M. Bergson greatly commends, declares: “Les -lois physico-chimiques sont aveugles et brutales; là où elles -règnent seules, au lieu d’un ordre et d’un concert, il ne peut y -avoir qu’incohérence et chaos.” But the physicist proclaims -aloud that the physical phenomena which meet us by the way -have their manifestations of form, not less beautiful and scarce -less varied than those which move us to admiration among living -things. The waves of the sea, the little ripples on the shore, the -sweeping curve of the sandy bay between its headlands, the -outline of the hills, the shape of the clouds, all these are so many -riddles of form, so many problems of morphology, and all of -them the physicist can more or less easily read and adequately -solve: solving them by reference to their antecedent phenomena, -in the material system of mechanical forces to which they belong, -and to which we interpret them as being due. They have also, -doubtless, their <i>immanent</i> teleological significance; but it is on -another plane of thought from the physicist’s that we contemplate -their intrinsic harmony and perfection, and “see that they are -good.”</p> - -<p>Nor is it otherwise with the material forms of living things. -Cell and tissue, shell and bone, leaf and flower, are so many -portions of matter, and it is in obedience to the laws of physics -that their particles have been moved, -moulded and conformed<a class="afnanch" href="#fn16" id="fnanch16">16</a>. -<span class="xxpn" id="p008">{8}</span> -They are no exception to the rule that Θεὸς ἀεὶ γεωμετρεῖ. Their -problems of form are in the first instance mathematical problems, -and their problems of growth are essentially physical problems; -and the morphologist is, <i>ipso facto</i>, a student of physical science.</p> - -<p>Apart from the physico-chemical problems of modern physiology, -the road of physico-mathematical or dynamical investigation -in morphology has had few to follow it; but the pathway is old. -The way of the old Ionian physicians, of Anaxagoras<a class="afnanch" href="#fn17" id="fnanch17">17</a>, -of -Empedocles and his disciples in the days before Aristotle, lay -just by that highwayside. It was Galileo’s and Borelli’s way. -It was little trodden for long afterwards, but once in a while -Swammerdam and Réaumur looked that way. And of later -years, Moseley and Meyer, Berthold, Errera and Roux have -been among the little band of travellers. We need not wonder -if the way be hard to follow, and if these wayfarers have yet -gathered little. A harvest has been reaped by others, and the -gleaning of the grapes is slow.</p> - -<p>It behoves us always to remember that in physics it has taken -great men to discover simple things. They are very great names -indeed that we couple with the explanation of the path of a stone, -the droop of a chain, the tints of a bubble, the shadows in a cup. -It is but the slightest adumbration of a dynamical morphology -that we can hope to have, until the physicist and the mathematician -shall have made these problems of ours their own, or till a new -Boscovich shall have written for the naturalist the new <i>Theoria -Philosophiae Naturalis</i>.</p> - -<p>How far, even then, mathematics will <i>suffice</i> to describe, and -physics to explain, the fabric of the body no man can foresee. -It may be that all the laws of energy, and all the properties of -matter, and all the chemistry of all the colloids are as powerless -to explain the body as they are impotent to comprehend the -soul. For my part, I think it is not so. Of how it is that the -soul informs the body, physical science teaches me nothing: -consciousness is not explained to my comprehension by all the -nerve-paths and “neurones” of the physiologist; nor do I ask of -physics how goodness shines in one man’s face, and evil betrays -itself in another. But of the construction and -growth and working <span class="xxpn" id="p009">{9}</span> -of the body, as of all that is of the earth earthy, physical science -is, in my humble opinion, our only teacher and guide<a class="afnanch" href="#fn18" id="fnanch18">18</a>.</p> - -<p>Often and often it happens that our physical knowledge is -inadequate to explain the mechanical working of the organism; -the phenomena are superlatively complex, the procedure is -involved and entangled, and the investigation has occupied but -a few short lives of men. When physical science falls short of -explaining the order which reigns throughout these manifold -phenomena,—an order more characteristic in its totality than any -of its phenomena in themselves,—men hasten to invoke a guiding -principle, an entelechy, or call it what you will. But all the while, -so far as I am aware, no physical law, any more than that of -gravity itself, not even among the puzzles of chemical “stereometry,” -or of physiological “surface-action” or “osmosis,” is -known to be <i>transgressed</i> by the bodily mechanism.</p> - -<p>Some physicists declare, as Maxwell did, that atoms or molecules -more complicated by far than the chemist’s hypotheses -demand are requisite to explain the phenomena of life. If what -is implied be an explanation of psychical phenomena, let the -point be granted at once; we may go yet further, and decline, -with Maxwell, to believe that anything of the nature of <i>physical</i> -complexity, however exalted, could ever suffice. Other physicists, -like Auerbach<a class="afnanch" href="#fn19" id="fnanch19">19</a>, -or Larmor<a class="afnanch" href="#fn20" id="fnanch20">20</a>, -or Joly<a class="afnanch" href="#fn21" id="fnanch21">21</a>, -assure us that our laws of -thermodynamics do not suffice, or are “inappropriate,” to explain -the maintenance or (in Joly’s phrase) the -“accelerative absorption” <span class="xxpn" id="p010">{10}</span> -of the bodily energies, and the long battle against the cold and -darkness which is death. With these weighty problems I am not -for the moment concerned. My sole purpose is to correlate with -mathematical statement and physical law certain of the simpler -outward phenomena of organic growth and structure or form: -while all the while regarding, <i>ex hypothesi</i>, for the purposes of -this correlation, the fabric of the organism as a material and -mechanical configuration.</p> - -<p>Physical science and philosophy stand side by side, and one upholds the -other. Without something of the strength of physics, philosophy would -be weak; and without something of philosophy’s wealth, physical science -would be poor. “Rien ne retirera du tissu de la science les fils d’or -que la main du philosophe y a -introduits<a class="afnanch" href="#fn22" id="fnanch22">22</a>.” -But there are fields -where each, for a while at least, must work alone; and where physical -science reaches its limitations, physical science itself must help us -to discover. Meanwhile the appropriate and legitimate postulate of the -physicist, in approaching the physical problems of the body, is that -with these physical phenomena no alien influence interferes. But the -postulate, though it is certainly legitimate, and though it is the -proper and necessary prelude to scientific enquiry, may some day be -proven to be untrue; and its disproof will not be to the physicist’s -confusion, but will come as his reward. In dealing with forms which are -so concomitant with life that they are seemingly controlled by life, it -is in no spirit of arrogant assertiveness that the physicist begins his -argument, after the fashion of a most illustrious exemplar, with the -old formulary of scholastic challenge,—<i>An Vita sit? Dico quod non.</i></p> - -<hr class="hrblk"> - -<p>The terms Form and Growth, which make up the title of this little -book, are to be understood, as I need hardly say, in their relation -to the science of organisms. We want to see how, in some cases at -least, the forms of living things, and of the parts of living things, -can be explained by physical considerations, and to realise that, in -general, no organic forms exist save such as are in conformity with -ordinary physical laws. And while growth is a somewhat vague word for a -complex matter, which may <span class="xxpn" id="p011">{11}</span> -depend on various things, from simple -imbibition of water to the complicated results of the chemistry of -nutrition, it deserves to be studied in relation to form, whether it -proceed by simple increase of size without obvious alteration of form, -or whether it so proceed as to bring about a gradual change of form and -the slow development of a more or less complicated structure.</p> - -<p>In the Newtonian language of elementary physics, force is -recognised by its action in producing or in changing motion, or -in preventing change of motion or in maintaining rest. When we -deal with matter in the concrete, force does not, strictly speaking, -enter into the question, for force, unlike matter, has no independent -objective existence. It is energy in its various forms, known or -unknown, that acts upon matter. But when we abstract our -thoughts from the material to its form, or from the thing moved -to its motions, when we deal with the subjective conceptions of -form, or movement, or the movements that change of form implies, -then force is the appropriate term for our conception of the causes -by which these forms and changes of form are brought about. -When we use the term force, we use it, as the physicist always -does, for the sake of brevity, using a symbol for the magnitude -and direction of an action in reference to the symbol or diagram -of a material thing. It is a term as subjective and symbolic as -form itself, and so is appropriately to be used in connection -therewith.</p> - -<p>The form, then, of any portion of matter, whether it be living -or dead, and the changes of form that are apparent in its movements -and in its growth, may in all cases alike be described as due to -the action of force. In short, the form of an object is a “diagram -of forces,” in this sense, at least, that from it we can judge of or -deduce the forces that are acting or have acted upon it: in this -strict and particular sense, it is a diagram,—in the case of a solid, -of the forces that <i>have</i> been impressed upon it when its conformation -was produced, together with those that enable it to retain its -conformation; in the case of a liquid (or of a gas) of the forces that -are for the moment acting on it to restrain or balance its own -inherent mobility. In an organism, great or small, it is not -merely the nature of the <i>motions</i> of the living substance that we -must interpret in terms of force (according to -kinetics), but also <span class="xxpn" id="p012">{12}</span> -the <i>conformation</i> of the organism itself, whose permanence or -equilibrium is explained by the interaction or balance of forces, -as described in statics.</p> - -<p>If we look at the living cell of an Amoeba or a Spirogyra, we -see a something which exhibits certain active movements, and -a certain fluctuating, or more or less lasting, form; and its form -at a given moment, just like its motions, is to be investigated by -the help of physical methods, and explained by the invocation of -the mathematical conception of force.</p> - -<p>Now the state, including the shape or form, of a portion of -matter, is the resultant of a number of forces, which represent or -symbolise the manifestations of various kinds of energy; and it -is obvious, accordingly, that a great part of physical science must -be understood or taken for granted as the necessary preliminary -to the discussion on which we are engaged. But we may at -least try to indicate, very briefly, the nature of the principal -forces and the principal properties of matter with which our -subject obliges us to deal. Let us imagine, for instance, the case -of a so-called “simple” organism, such as <i>Amoeba</i>; and if our -short list of its physical properties and conditions be helpful -to our further discussion, we need not consider how far it -be complete or adequate from the wider physical point of -view<a class="afnanch" href="#fn23" id="fnanch23">23</a>.</p> - -<p>This portion of matter, then, is kept together by the intermolecular -force of cohesion; in the movements of its particles -relatively to one another, and in its own movements relative to -adjacent matter, it meets with the opposing force of friction. -It is acted on by gravity, and this force tends (though slightly, -owing to the Amoeba’s small mass, and to the small difference -between its density and that of the surrounding fluid), to flatten -it down upon the solid substance on which it may be creeping. -Our Amoeba tends, in the next place, to be deformed by any -pressure from outside, even though slight, which may be applied -to it, and this circumstance shews it to consist of matter in a -fluid, or at least semi-fluid, state: which state is further indicated -when we observe streaming or current motions in its interior. <span class="xxpn" id="p013">{13}</span> -Like other fluid bodies, its surface, whatsoever other substance, -gas, liquid or solid, it be in contact with, and in varying degree -according to the nature of that adjacent substance, is the seat -of molecular force exhibiting itself as a surface-tension, from the -action of which many important consequences follow, which -greatly affect the form of the fluid surface.</p> - -<p>While the protoplasm of the Amoeba reacts to the slightest -pressure, and tends to “flow,” and while we therefore speak of it -as a fluid, it is evidently far less mobile than such a fluid, for -instance, as water, but is rather like treacle in its slow creeping -movements as it changes its shape in response to force. Such -fluids are said to have a high viscosity, and this viscosity obviously -acts in the way of retarding change of form, or in other words -of retarding the effects of any disturbing action of force. When -the viscous fluid is capable of being drawn out into fine threads, -a property in which we know that the material of some Amoebae -differs greatly from that of others, we say that the fluid is also -<i>viscid</i>, or exhibits viscidity. Again, not by virtue of our Amoeba -being liquid, but at the same time in vastly greater measure than if it -were a solid (though far less rapidly than if it were a gas), a process -of molecular diffusion is constantly going on within its substance, -by which its particles interchange their places within the mass, -while surrounding fluids, gases and solids in solution diffuse into -and out of it. In so far as the outer wall of the cell is different -in character from the interior, whether it be a mere pellicle as -in Amoeba or a firm cell-wall as in Protococcus, the diffusion -which takes place <i>through</i> this wall is sometimes distinguished -under the term <i>osmosis</i>.</p> - -<p>Within the cell, chemical forces are at work, and so also in -all probability (to judge by analogy) are electrical forces; and -the organism reacts also to forces from without, that have their -origin in chemical, electrical and thermal influences. The processes -of diffusion and of chemical activity within the cell result, -by the drawing in of water, salts, and food-material with or without -chemical transformation into protoplasm, in growth, and this -complex phenomenon we shall usually, without discussing its -nature and origin, describe and picture as a <i>force</i>. Indeed we -shall manifestly be inclined to use the term growth -in two senses, <span class="xxpn" id="p014">{14}</span> -just indeed as we do in the case of attraction or gravitation, -on the one hand as a <i>process</i>, and on the other hand as a -<i>force</i>.</p> - -<p>In the phenomena of cell-division, in the attractions or repulsions -of the parts of the dividing nucleus and in the “caryokinetic” -figures that appear in connection with it, we seem to see in operation -forces and the effects of forces, that have, to say the least of -it, a close analogy with known physical phenomena; and to this -matter we shall afterwards recur. But though they resemble -known physical phenomena, their nature is still the subject of -much discussion, and neither the forms produced nor the forces -at work can yet be satisfactorily and simply explained. We may -readily admit, then, that besides phenomena which are obviously -physical in their nature, there are actions visible as well as -invisible taking place within living cells which our knowledge -does not permit us to ascribe with certainty to any known physical -force; and it may or may not be that these phenomena will yield -in time to the methods of physical investigation. Whether or -no, it is plain that we have no clear rule or guide as to what is -“vital” and what is not; the whole assemblage of so-called vital -phenomena, or properties of the organism, cannot be clearly -classified into those that are physical in origin and those that are -<i>sui generis</i> and peculiar to living things. All we can do meanwhile -is to analyse, bit by bit, those parts of the whole to which the -ordinary laws of the physical forces more or less obviously and -clearly and indubitably apply.</p> - -<p>Morphology then is not only a study of material things and -of the forms of material things, but has its dynamical aspect, -under which we deal with the interpretation, in terms of force, -of the operations of Energy. And here it is well worth while -to remark that, in dealing with the facts of embryology or the -phenomena of inheritance, the common language of the books -seems to deal too much with the <i>material</i> elements concerned, as -the causes of development, of variation or of hereditary transmission. -Matter as such produces nothing, changes nothing, does -nothing; and however convenient it may afterwards be to abbreviate -our nomenclature and our descriptions, we must most -carefully realise in the outset that the -spermatozoon, the nucleus, <span class="xxpn" id="p015">{15}</span> -the chromosomes or the germ-plasm can never <i>act</i> as matter alone, -but only as seats of energy and as centres of force. And this is but -an adaptation (in the light, or rather in the conventional symbolism, -of modern physical science) of the old saying of the philosopher: -ἀρχὴ γὰρ ἡ φύσις μᾶλλον τῆς ὕλης.</p> - -<div class="chapter" id="p016"> -<h2 class="h2herein" title="II. On Magnitude.">CHAPTER II. -<span class="h2ttl"> -ON MAGNITUDE</span></h2></div> - -<p>To terms of magnitude, and of direction, must we refer all -our conceptions of form. For the form of an object is defined -when we know its magnitude, actual or relative, in various -directions; and growth involves the same conceptions of magnitude -and direction, with this addition, that they are supposed to alter -in time. Before we proceed to the consideration of specific form, -it will be worth our while to consider, for a little while, certain -phenomena of spatial magnitude, or of the extension of a body -in the several dimensions of space<a class="afnanch" href="#fn24" id="fnanch24">24</a>.</p> - -<p>We are taught by elementary mathematics that, in similar -solid figures, the surface increases as the square, and -the volume as the cube, of the linear dimensions. If we -take the simple case of a sphere, with radius <i>r</i>, the -area of its surface is equal to 4π<i>r</i><sup>2</sup> , and its volume to -(<sup>4</sup>⁄<sub>3</sub>)π<i>r</i><sup>3</sup> ; from which it follows that the ratio of volume -to surface, or -<sup class="spitc">V</sup>⁄<sub class="spitc">S</sub> , is -(<sup>1</sup>⁄<sub>3</sub>)<i>r</i>. -In other words, the -greater the radius (or the larger the sphere) the greater -will be its volume, or its mass (if it be uniformly dense -throughout), in comparison with its superficial area. And, -taking <i>L</i> to represent any linear dimension, we may write -the general equations in the form</p> - -<div class="dmaths"> -<div><i>S</i> ∝ <i>L</i><sup>2</sup> , <i>V</i> -∝ <i>L</i><sup>3</sup> ,</div> - -<p class="pcontinue pleftfloat">or</p> - -<div><i>S</i> -= <i>k · L</i><sup>2</sup> , and <i>V</i> -= <i>k′ · L</i><sup>3</sup> ; -<br class="brclrfix"></div> - -<p class="pcontinue pleftfloat">and</p> - -<div><sup class="spitc">V</sup>⁄<sub class="spitc">S</sub> -∝ <i>L</i>.<br class="brclrfix"></div></div><!--dmaths--> - -<p>From these elementary principles a great number of consequences -follow, all more or less interesting, and some of them of -great importance. In the first place, though growth in length -(let <span class="xxpn" id="p017">{17}</span> -us say) and growth in volume (which is usually tantamount to -mass or weight) are parts of one and the same process or phenomenon, -the one attracts our <i>attention</i> by its increase, very much -more than the other. For instance a fish, in doubling its length, -multiplies its weight by no less than eight times; and it all but -doubles its weight in growing from four inches long to five.</p> - -<p>In the second place we see that a knowledge of the correlation -between length and weight in any particular species of animal, -in other words a determination of <i>k</i> in the -formula <i>W</i> -= <i>k · L</i><sup>3</sup> , -enables us at any time to translate the one magnitude into the -other, and (so to speak) to weigh the animal with a measuring-rod; -this however being always subject to the condition that the -animal shall in no way have altered its form, nor its specific -gravity. That its specific gravity or density should materially or -rapidly alter is not very likely; but as long as growth lasts, -changes of form, even though inappreciable to the eye, are likely -to go on. Now weighing is a far easier and far more accurate -operation than measuring; and the measurements which would -reveal slight and otherwise imperceptible changes in the form of -a fish—slight relative differences between length, breadth and -depth, for instance,—would need to be very delicate indeed. But -if we can make fairly accurate determinations of the length, -which is very much the easiest dimension to measure, and then -correlate it with the weight, then the value of <i>k</i>, according to -whether it varies or remains constant, will tell us at once whether -there has or has not been a tendency to gradual alteration in the -general form. To this subject we shall return, when we come to -consider more particularly the rate of growth.</p> - -<p>But a much deeper interest arises out of this changing ratio -of dimensions when we come to consider the inevitable changes -of physical relations with which it is bound up. We are apt, and -even accustomed, to think that magnitude is so purely relative -that differences of magnitude make no other or more essential -difference; that Lilliput and Brobdingnag are all alike, according -as we look at them through one end of the glass or the other. -But this is by no means so; for <i>scale</i> has a very marked effect -upon physical phenomena, and the effect of scale constitutes what -is known as the principle of similitude, or -of dynamical similarity. <span class="xxpn" id="p018">{18}</span></p> - -<p>This effect of scale is simply due to the fact that, of the physical -forces, some act either directly at the surface of a body, or otherwise -in <i>proportion</i> to the area of surface; and others, such as -gravity, act on all particles, internal and external alike, and exert -a force which is proportional to the mass, and so usually to the -volume, of the body.</p> - -<p>The strength of an iron girder obviously varies with the -cross-section of its members, and each cross-section varies as the -square of a linear dimension; but the weight of the whole structure -varies as the cube of its linear dimensions. And it follows at once -that, if we build two bridges geometrically similar, the larger is -the weaker of the two<a class="afnanch" href="#fn25" id="fnanch25">25</a>. -It was elementary engineering experience -such as this that led Herbert Spencer<a class="afnanch" href="#fn26" id="fnanch26">26</a> -to apply the principle of -similitude to biology.</p> - -<p>The same principle had been admirably applied, in a few clear -instances, by Lesage<a class="afnanch" href="#fn27" id="fnanch27">27</a>, -a celebrated eighteenth century physician -of Geneva, in an unfinished and unpublished work<a class="afnanch" href="#fn28" id="fnanch28">28</a>. -Lesage -argued, for instance, that the larger ratio of surface to mass would -lead in a small animal to excessive transpiration, were the skin -as “porous” as our own; and that we may hence account for -the hardened or thickened skins of insects and other small terrestrial -animals. Again, since the weight of a fruit increases as the cube -of its dimensions, while the strength of the stalk increases as the -square, it follows that the stalk should grow out of apparent due -proportion to the fruit; or alternatively, that tall trees should -not bear large fruit on slender branches, and that melons and -pumpkins must lie upon the ground. And again, that in quadrupeds -a large head must be supported on a neck -which is either <span class="xxpn" id="p019">{19}</span> -excessively thick and strong, like a bull’s, or very short like the -neck of an elephant.</p> - -<p>But it was Galileo who, wellnigh 300 years ago, had first laid -down this general principle which we now know by the name of the -principle of similitude; and he did so with the utmost possible -clearness, and with a great wealth of illustration, drawn from -structures living and dead<a class="afnanch" href="#fn29" id="fnanch29">29</a>. -He showed that neither can man -build a house nor can nature construct an animal beyond a certain -size, while retaining the same proportions and employing the -same materials as sufficed in the case of a smaller structure<a class="afnanch" href="#fn30" id="fnanch30">30</a>. -The thing will fall to pieces of its own weight unless we either -change its relative proportions, which will at length cause it to -become clumsy, monstrous and inefficient, or else we must find -a new material, harder and stronger than was used before. Both -processes are familiar to us in nature and in art, and practical -applications, undreamed of by Galileo, meet us at every turn in -this modern age of steel.</p> - -<p>Again, as Galileo was also careful to explain, besides the -questions of pure stress and strain, of the strength of muscles to -lift an increasing weight or of bones to resist its crushing stress, -we have the very important question of <i>bending moments</i>. This -question enters, more or less, into our whole range of problems; -it affects, as we shall afterwards see, or even determines the whole -form of the skeleton, and is very important in such a case as that -of a tall tree<a class="afnanch" href="#fn31" id="fnanch31">31</a>.</p> - -<p>Here we have to determine the point at which the tree will -curve under its own weight, if it be ever so little displaced from -the perpendicular<a class="afnanch" href="#fn32" id="fnanch32">32</a>. -In such an investigation -we have to make <span class="xxpn" id="p020">{20}</span> -some assumptions,—for instance, with regard to the trunk, that -it tapers uniformly, and with regard to the branches that their -sectional area varies according to some definite law, or (as Ruskin -assumed<a class="afnanch" href="#fn33" id="fnanch33">33</a>) -tends to be constant in any horizontal plane; and the -mathematical treatment is apt to be somewhat difficult. But -Greenhill has shewn that (on such assumptions as the above), -a certain British Columbian pine-tree, which yielded the Kew flagstaff -measuring 221 ft. in height with a diameter at the base of -21 inches, could not possibly, by theory, have grown to more -than about 300 ft. It is very curious that Galileo suggested -precisely the same height (<i>dugento braccia alta</i>) as the utmost -limit of the growth of a tree. In general, as Greenhill shews, the -diameter of a homogeneous body must increase as the power 3 ⁄ 2 -of the height, which accounts for the slender proportions of young -trees, compared with the stunted appearance of old and large -ones<a class="afnanch" href="#fn34" id="fnanch34">34</a>. -In short, as Goethe says in <i>Wahrheit und Dichtung</i>, “Es -ist dafür gesorgt dass die Bäume nicht in den Himmel wachsen.” -But Eiffel’s great tree of steel (1000 feet high) is built to a -very different plan; for here the profile of the tower follows the -logarithmic curve, giving <i>equal strength</i> throughout, according -to a principle which we shall have occasion to discuss when we -come to treat of “form and mechanical efficiency” in connection -with the skeletons of animals.</p> - -<p>Among animals, we may see in a general way, without the help -of mathematics or of physics, that exaggerated bulk brings with -it a certain clumsiness, a certain inefficiency, a new element of -risk and hazard, a vague preponderance of disadvantage. The -case was well put by Owen, in a passage which has an interest -of its own as a premonition (somewhat like De Candolle’s) of the -“struggle for existence.” Owen wrote as follows<a class="afnanch" href="#fn35" id="fnanch35">35</a>: -“In proportion -to the bulk of a species is the difficulty of the contest -which, as a living organised whole, the individual -of such species <span class="xxpn" id="p021">{21}</span> -has to maintain against the surrounding agencies that are ever -tending to dissolve the vital bond, and subjugate the living -matter to the ordinary chemical and physical forces. Any -changes, therefore, in such external conditions as a species may -have been originally adapted to exist in, will militate against that -existence in a degree proportionate, perhaps in a geometrical ratio, -to the bulk of the species. If a dry season be greatly prolonged, -the large mammal will suffer from the drought sooner than the -small one; if any alteration of climate affect the quantity of -vegetable food, the bulky Herbivore will first feel the effects of -stinted nourishment.”</p> - -<p>But the principle of Galileo carries us much further and along -more certain lines.</p> - -<p>The tensile strength of a muscle, like that of a rope or of our -girder, varies with its cross-section; and the resistance of a bone -to a crushing stress varies, again like our girder, with its cross-section. -But in a terrestrial animal the weight which tends to -crush its limbs or which its muscles have to move, varies as the -cube of its linear dimensions; and so, to the possible magnitude -of an animal, living under the direct action of gravity, there is -a definite limit set. The elephant, in the dimensions of its limb-bones, -is already shewing signs of a tendency to disproportionate -thickness as compared with the smaller mammals; its movements -are in many ways hampered and its agility diminished: it is -already tending towards the maximal limit of size which the -physical forces permit. But, as Galileo also saw, if the animal -be wholly immersed in water, like the whale, (or if it be partly -so, as was in all probability the case with the giant reptiles of our -secondary rocks), then the weight is counterpoised to the extent -of an equivalent volume of water, and is completely counterpoised -if the density of the animal’s body, with the included air, be -identical (as in a whale it very nearly is) with the water around. -Under these circumstances there is no longer a physical barrier -to the indefinite growth in magnitude of -the animal<a class="afnanch" href="#fn36" id="fnanch36">36</a>. -Indeed, -<span class="xxpn" id="p022">{22}</span> -in the case of the aquatic animal there is, as Spencer pointed out, -a distinct advantage, in that the larger it grows the greater is -its velocity. For its available energy depends on the mass of -its muscles; while its motion through the water is opposed, not -by gravity, but by “skin-friction,” which increases only as the -square of its dimensions; all other things being equal, the bigger -the ship, or the bigger the fish, the faster it tends to go, but only -in the ratio of the square root of the increasing length. For the -mechanical work (<i>W</i>) of which the fish is capable being proportional -to the mass of its muscles, or the cube of its linear -dimensions: and again this work being wholly done in producing -a velocity (<i>V</i>) against a resistance (<i>R</i>) which increases as the -square of the said linear dimensions; we have at once</p> - -<div class="dmaths"> -<div><i>W</i> = <i>l</i><sup>3</sup> ,</div> - -<p class="pcontinue">and also</p> - -<div><i>W</i> -= <i>RV</i><sup>2</sup> -= <i>l</i><sup>2</sup><i>V</i><sup>2</sup> . -<br class="brclrfix"></div> - -<p class="pcontinue">Therefore</p> - -<div><i>l</i><sup>3</sup> -= <i>l</i><sup>2</sup><i>V</i><sup>2</sup> ,  and  <i>V</i> -= √<i>l</i>.</div> - -<p class="pcontinue">This is what is known as Froude’s Law of the -<i>correspondence of speeds</i>.</p> -</div><!--dmaths--> - -<p>But there is often another side to these questions, which makes -them too complicated to answer in a word. For instance, the -work (per stroke) of which two similar engines are capable should -obviously vary as the cubes of their linear dimensions, for it -varies on the one hand with the <i>surface</i> of the piston, and on the -other, with the <i>length</i> of the stroke; so is it likewise in the animal, -where the corresponding variation depends on the cross-section of -the muscle, and on the space through which it contracts. But -in two precisely similar engines, the actual available horse-power -varies as the square of the linear dimensions, and not as the -cube; and this for the obvious reason that the actual energy -developed depends upon the <i>heating-surface</i> of the -boiler<a class="afnanch" href="#fn37" id="fnanch37">37</a>. -So -likewise must there be a similar tendency, among animals, for the -rate of supply of kinetic energy to vary with -the surface of the <span class="xxpn" id="p023">{23}</span> -lung, that is to say (other things being equal) with the <i>square</i> of -the linear dimensions of the animal. We may of course (departing -from the condition of similarity) increase the heating-surface of -the boiler, by means of an internal system of tubes, without -increasing its outward dimensions, and in this very way nature -increases the respiratory surface of a lung by a complex system -of branching tubes and minute air-cells; but nevertheless in -two similar and closely related animals, as also in two steam-engines -of precisely the same make, the law is bound to hold that -the rate of working must tend to vary with the square of the -linear dimensions, according to Froude’s law of <i>steamship comparison</i>. -In the case of a very large ship, built for speed, the -difficulty is got over by increasing the size and number of the -boilers, till the ratio between boiler-room and engine-room is -far beyond what is required in an ordinary small vessel<a class="afnanch" href="#fn38" id="fnanch38">38</a>; -but -though we find lung-space increased among animals where -greater rate of working is required, as in general among birds, -I do not know that it can be shewn to increase, as in the -“over-boilered” ship, with the size of the animal, and in a ratio -which outstrips that of the other bodily dimensions. If it be the -case then, that the working mechanism of the muscles should be -able to exert a force proportionate to the cube of the linear -bodily dimensions, while the respiratory mechanism can only -supply a store of energy at a rate proportional to the square of -the said dimensions, the singular result ought to follow that, in -swimming for instance, the larger fish ought to be able to put on -a spurt of speed far in excess of the smaller one; but the distance -travelled by the year’s end should be very much alike for both -of them. And it should also follow that the -curve of fatigue <span class="xxpn" id="p024">{24}</span> -should be a steeper one, and the staying power should be -less, in the smaller than in the larger individual. This -is the case of long-distance racing, where the big winner -puts on his big spurt at the end. And for an analogous -reason, wise men know that in the ’Varsity boat-race it is -judicious and prudent to bet on the heavier crew.</p> - -<p>Leaving aside the question of the supply of energy, and keeping -to that of the mechanical efficiency of the machine, we may find -endless biological illustrations of the principle of similitude.</p> - -<p>In the case of the flying bird (apart from the initial difficulty of -raising itself into the air, which involves another problem) it may -be shewn that the bigger it gets (all its proportions remaining the -same) the more difficult it is for it to maintain itself aloft in flight. -The argument is as follows:</p> - -<p>In order to keep aloft, the bird must communicate to the air -a downward momentum equivalent to its own weight, and therefore -proportional to <i>the cube of its own linear dimensions</i>. But -the momentum so communicated is proportional to the mass of -air driven downwards, and to the rate at which it is driven: the -mass being proportional to the bird’s wing-area, and also (with -any given slope of wing) to the speed of the bird, and the rate -being again proportional to the bird’s speed; accordingly the -whole momentum varies as the wing-area, i.e. as <i>the square of the -linear dimensions, and also as the square of the speed</i>. Therefore, -in order that the bird may maintain level flight, its speed must -be proportional to <i>the square root of its linear dimensions</i>.</p> - -<p>Now the rate at which the bird, in steady flight, has to -work in order to drive itself forward, is the rate at which -it communicates energy to the air; and this is proportional -to <i>mV</i><sup>2</sup> , i.e. to the mass and to the square of the -velocity of the air displaced. But the mass of air -displaced per second is proportional to the wing-area and -to the speed of the bird’s motion, and therefore to the -power 2½ of the linear dimensions; and the speed at -which it is displaced is proportional to the bird’s speed, -and therefore to the square root of the linear dimensions. -Therefore the energy communicated per second (being -proportional to the mass and to the square of the speed) -is jointly proportional to the power 2½ of the linear -dimensions, as above, and to the -first power thereof: <span class="xxpn" id="p025">{25}</span> -that is to say, it increases in proportion <i>to the power</i> 3½ <i>of the -linear dimensions</i>, and therefore faster than the weight of the -bird increases.</p> - -<div class="dmaths"> -<p>Put in mathematical form, the equations are as follows:</p> - -<p class="pcontinue">(<i>m</i> -= the mass of air thrust downwards; <i>V</i> its velocity, -proportional to that of the bird; <i>M</i> its momentum; <i>l</i> a linear -dimension of the bird; <i>w</i> its weight; <i>W</i> the work done in moving -itself forward.)</p> - -<div><i>M</i> -= <i>w</i> -= <i>l</i><sup>3</sup> .</div> - -<p>But</p> - -<div><i>M</i> -= <i>m V</i>,  and  <i>m</i> -= <i>l</i><sup>2</sup> <i>V</i>.</div> - -<p>Therefore</p> - -<div><i>M</i> -= <i>l</i><sup>2</sup> <i>V</i><sup>2</sup> , -   and</div> - -<div><i>l</i><sup>2</sup> <i>V</i><sup>2</sup> -= <i>l</i><sup>3</sup> , -   or</div> - -<div><i>V</i> = √<i>l</i>.</div> - -<p>But, again,</p> - -<div><i>W</i> -= <i>m V</i><sup>2</sup> -<div class="nowrap pleft dvaligntop">= <i>l</i><sup>2</sup> <i>V × V</i><sup>2</sup> -<br> -= <i>l</i><sup>2</sup> × √<i>l × l</i><br> -= <i>l</i><sup>3½</sup> .</div> -</div></div><!--dmaths--> - -<p>The work requiring to be done, then, varies as the power 3½ of -the bird’s linear dimensions, while the work of which the bird is -capable depends on the mass of its muscles, and therefore varies -as the cube of its linear dimensions<a class="afnanch" href="#fn39" id="fnanch39">39</a>. -The disproportion does not -seem at first sight very great, but it is quite enough to tell. It is -as much as to say that, every time we double the linear dimensions -of the bird, the difficulty of flight is increased in the ratio of -2<sup>3</sup> : 2<sup>3½</sup> , or 8 : 11·3, -or, say, 1 : 1·4. If we take the ostrich to -exceed the sparrow in linear dimensions as 25 : 1, which seems well -within the mark, we have the ratio between 25<sup>3½</sup> and 25<sup>3</sup> , or -between 5<sup>7</sup> : 5<sup>6</sup> ; in other words, flight is just five times more -difficult for the larger than for the smaller bird<a class="afnanch" href="#fn40" id="fnanch40">40</a>.</p> - -<p>The above investigation includes, besides the final result, a -number of others, explicit or implied, which are of not less importance. -Of these the simplest and also -the most important is <span class="xxpn" id="p026">{26}</span> -contained in the equation <i>V</i> -= √<i>l</i>, a result which happens to be -identical with one we had also arrived at in the case of the fish. -In the bird’s case it has a deeper significance than in the other; -because it implies here not merely that the velocity will tend to -increase in a certain ratio with the length, but that it <i>must</i> do so -as an essential and primary condition of the bird’s remaining aloft. -It is accordingly of great practical importance in aeronautics, for -it shews how a provision of increasing speed must accompany every -enlargement of our aeroplanes. If a given machine weighing, say, -500 lbs. be stable at 40 miles an hour, then one geometrically -similar which weighs, say, a couple of tons must have its speed -determined as follows:</p> - -<div class="dmaths"> -<div><i>W</i> : <i>w</i> :: <i>L</i><sup>3</sup> : <i>l</i><sup>3</sup> :: 8 : 1. -</div> - -<p>Therefore</p> - -<div><i>L</i> : <i>l</i> :: 2 : 1. -</div> - -<p>But</p> - -<div><i>V</i><sup>2</sup> : <i>v</i><sup>2</sup> :: <i>L</i> : <i>l</i>. -</div> - -<p>Therefore</p> - -<div><i>V</i> : <i>v</i> :: √2 : 1 -= 1·414 : 1.</div></div><!--dmaths--> - -<p class="pcontinue">That is to say, the larger machine must be capable of a speed -equal to 1·414 × 40, or about 56½ miles per hour.</p> - -<p>It is highly probable, as Lanchester<a class="afnanch" href="#fn41" id="fnanch41">41</a> -remarks, that Lilienthal -met his untimely death not so much from any intrinsic fault in -the design or construction of his machine, but simply because his -engine fell somewhat short of the power required to give the -speed which was necessary for stability. An arrow is a very -imperfectly designed aeroplane, but nevertheless it is evidently -capable, to a certain extent and at a high velocity, of acquiring -“stability” and hence of actual “flight”: the duration and -consequent range of its trajectory, as compared with a bullet of -similar initial velocity, being correspondingly benefited. When -we return to our birds, and again compare the ostrich with the -sparrow, we know little or nothing about the speed in flight of -the latter, but that of the swift is estimated<a class="afnanch" href="#fn42" id="fnanch42">42</a> -to vary from a -minimum of 20 to 50 feet or more per second,—say from 14 to -35 miles per hour. Let us take the same lower limit as not far -from the minimal velocity of the sparrow’s -flight also; and it <span class="xxpn" id="p027">{27}</span> -would follow that the ostrich, of 25 times the sparrow’s linear -dimensions, would be compelled to fly (if it flew at all) with -a <i>minimum</i> velocity of 5 × 14, or 70 miles an hour.</p> - -<p>The same principle of <i>necessary speed</i>, or the indispensable -relation between the dimensions of a flying object and the minimum -velocity at which it is stable, accounts for a great number of -observed phenomena. It tells us why the larger birds have a -marked difficulty in rising from the ground, that is to say, in -acquiring to begin with the horizontal velocity necessary for their -support; and why accordingly, as Mouillard<a class="afnanch" href="#fn43" id="fnanch43">43</a> -and others have -observed, the heavier birds, even those weighing no more than -a pound or two, can be effectively “caged” in a small enclosure -open to the sky. It tells us why very small birds, especially -those as small as humming-birds, and <i>à fortiori</i> the still smaller -insects, are capable of “stationary flight,” a very slight and -scarcely perceptible velocity <i>relatively to the air</i> being sufficient for -their support and stability. And again, since it is in all cases -velocity relative to the air that we are speaking of, we comprehend -the reason why one may always tell which way the wind blows -by watching the direction in which a bird <i>starts</i> to fly.</p> - -<p>It is not improbable that the ostrich has already reached -a magnitude, and we may take it for certain that the moa did -so, at which flight by muscular action, according to the normal -anatomy of a bird, has become physiologically impossible. The -same reasoning applies to the case of man. It would be very -difficult, and probably absolutely impossible, for a bird to fly -were it the bigness of a man. But Borelli, in discussing this -question, laid even greater stress on the obvious fact that a man’s -pectoral muscles are so immensely less in proportion than those -of a bird, that however we may fit ourselves with wings we can -never expect to move them by any power of our own relatively -weaker muscles; so it is that artificial flight only became possible -when an engine was devised whose efficiency was extraordinarily -great in comparison with its weight and size.</p> - -<p>Had Leonardo da Vinci known what Galileo knew, he would -not have spent a great part of his life on vain efforts to make to -himself wings. Borelli had learned the -lesson thoroughly, and <span class="xxpn" id="p028">{28}</span> -in one of his chapters he deals with the proposition, “Est impossible, -ut homines propriis viribus artificiose volare possint<a class="afnanch" href="#fn44" id="fnanch44">44</a>.”</p> - -<p>But just as it is easier to swim than to fly, so is it obvious -that, in a denser atmosphere, the conditions of flight would be -altered, and flight facilitated. We know that in the carboniferous -epoch there lived giant dragon-flies, with wings of a span far -greater than nowadays they ever attain; and the small bodies -and huge extended wings of the fossil pterodactyles would seem -in like manner to be quite abnormal according to our present -standards, and to be beyond the limits of mechanical efficiency -under present conditions. But as Harlé suggests<a class="afnanch" href="#fn45" id="fnanch45">45</a>, -following -upon a suggestion of Arrhenius, we have only to suppose that in -carboniferous and jurassic days the terrestrial atmosphere was -notably denser than it is at present, by reason, for instance, of -its containing a much larger proportion of carbonic acid, and we -have at once a means of reconciling the apparent mechanical -discrepancy.</p> - -<p>Very similar problems, involving in various ways the principle -of dynamical similitude, occur all through the physiology of -locomotion: as, for instance, when we see that a cockchafer can -carry a plate, many times his own weight, upon his back, or that -a flea can jump many inches high.</p> - -<p>Problems of this latter class have been admirably treated both -by Galileo and by Borelli, but many later writers have remained -ignorant of their work. Linnaeus, for instance, remarked that, -if an elephant were as strong in proportion as a stag-beetle, it -would be able to pull up rocks by the root, and to level mountains. -And Kirby and Spence have a well-known passage directed to -shew that such powers as have been conferred upon the insect -have been withheld from the higher animals, for the reason that -had these latter been endued therewith they would have “caused -the early desolation of the world<a class="afnanch" href="#fn46" id="fnanch46">46</a>.” -<span class="xxpn" id="p029">{29}</span></p> - -<p>Such problems as that which is presented by the flea’s jumping -powers, though essentially physiological in their nature, have their -interest for us here: because a steady, progressive diminution of -activity with increasing size would tend to set limits to the possible -growth in magnitude of an animal just as surely as those factors -which tend to break and crush the living fabric under its own -weight. In the case of a leap, we have to do rather with a sudden -impulse than with a continued strain, and this impulse should be -measured in terms of the velocity imparted. The velocity is -proportional to the impulse (<i>x</i>), and inversely proportional to the -mass (<i>M</i>) moved: <i>V</i> -= <i>x ⁄ M</i>. But, according to what we still speak -of as “Borelli’s law,” the impulse (i.e. the work of the impulse) is -proportional to the volume of the muscle by which it is -produced<a class="afnanch" href="#fn47" id="fnanch47">47</a>, -that is to say (in similarly constructed animals) to the mass of the -whole body; for the impulse is proportional on the one hand to -the cross-section of the muscle, and on the other to the distance -through which it contracts. It follows at once from this that the -velocity is constant, whatever be the size of the animals: in -other words, that all animals, provided always that they are -similarly fashioned, with their various levers etc., in like proportion, -ought to jump, not to the same relative, but to the same actual -height<a class="afnanch" href="#fn48" id="fnanch48">48</a>. -According to this, then, the flea is not a better, but -rather a worse jumper than a horse or a man. As a matter of -fact, Borelli is careful to point out that in the act of leaping the -impulse is not actually instantaneous, as in the blow of a hammer, -but takes some little time, during which the levers are being -extended by which the centre of gravity of the animal is being -propelled forwards; and this interval of time will be longer in -the case of the longer levers of the larger animal. To some extent, -then, this principle acts as a corrective to -the more general one, <span class="xxpn" id="p030">{30}</span> -and tends to leave a certain balance of advantage, in regard to -leaping power, on the side of the larger animal<a class="afnanch" href="#fn49" id="fnanch49">49</a>.</p> - -<p>But on the other hand, the question of strength of materials -comes in once more, and the factors of stress and strain and -bending moment make it, so to speak, more and more difficult -for nature to endow the larger animal with the length of lever -with which she has provided the flea or the grasshopper.</p> - -<p>To Kirby and Spence it seemed that “This wonderful strength -of insects is doubtless the result of something peculiar in the -structure and arrangement of their muscles, and principally their -extraordinary power of contraction.” This hypothesis, which is -so easily seen, on physical grounds, to be unnecessary, has been -amply disproved in a series of excellent papers by F. Plateau<a class="afnanch" href="#fn50" id="fnanch50">50</a>.</p> - -<p>A somewhat simple problem is presented to us by the act of -walking. It is obvious that there will be a great economy of -work, if the leg swing at its normal <i>pendulum-rate</i>; and, though -this rate is hard to calculate, owing to the shape and the jointing -of the limb, we may easily convince ourselves, by counting our -steps, that the leg does actually swing, or tend to swing, just as -a pendulum does, at a certain definite rate<a class="afnanch" href="#fn51" id="fnanch51">51</a>. -When we walk -quicker, we cause the leg-pendulum to describe a greater arc, but -we do not appreciably cause it to swing, or vibrate, quicker, until -we shorten the pendulum and begin to run. Now let two individuals, -<i>A</i> and <i>B</i>, walk in a similar fashion, that is to say, with -a similar <i>angle</i> of swing. The <i>arc</i> through which the leg swings, -or the <i>amplitude</i> of each step, will therefore vary as the length -of leg, or say as <i>a ⁄ b</i>; but the time of swing -will vary as the square <span class="xxpn" id="p031">{31}</span> -root of the pendulum-length, or √<i>a</i> ⁄ √<i>b</i>. -Therefore the velocity, -which is measured by amplitude ⁄ time, will also vary as the square-roots -of the length of leg: that is to say, the average velocities of -<i>A</i> and <i>B</i> are in the ratio of √<i>a</i> : √<i>b</i>.</p> - -<p>The smaller man, or smaller animal, is so far at a disadvantage -compared with the larger in speed, but only to the extent of the -ratio between the square roots of their linear dimensions: whereas, -if the rate of movement of the limb were identical, irrespective -of the size of the animal,—if the limbs of the mouse for instance -swung at the same rate as those of the horse,—then, as F. Plateau -said, the mouse would be as slow or slower in its gait than the -tortoise. M. Delisle<a class="afnanch" href="#fn52" id="fnanch52">52</a> -observed a “minute fly” walk three inches -in half-a-second. This was good steady walking. When we -walk five miles an hour we go about 88 inches in a second, or -88 ⁄ 6 -= 14·7 times the pace of M. Delisle’s fly. We should walk -at just about the fly’s pace if our stature were 1 ⁄ (14·7)<sup>2</sup> , or 1 ⁄ 216 -of our present height,—say 72 ⁄ 216 inches, or one-third of an inch -high.</p> - -<p>But the leg comprises a complicated system of levers, by whose -various exercise we shall obtain very different results. For -instance, by being careful to rise upon our instep, we considerably -increase the length or amplitude of our stride, and very considerably -increase our speed accordingly. On the other hand, in running, -we bend and so shorten the leg, in order to accommodate it to -a quicker rate of pendulum-swing<a class="afnanch" href="#fn53" id="fnanch53">53</a>. -In short, the jointed structure -of the leg permits us to use it as the shortest possible pendulum -when it is swinging, and as the longest possible lever when it is -exerting its propulsive force.</p> - -<p>Apart from such modifications as that described in the last -paragraph,—apart, that is to say, from differences in mechanical -construction or in the manner in which the mechanism is used,—we -have now arrived at a curiously simple and uniform result. -For in all the three forms of locomotion -which we have attempted <span class="xxpn" id="p032">{32}</span> -to study, alike in swimming, in flight and in walking, the general -result, attained under very different conditions and arrived at by -very different modes of reasoning, is in every case that the velocity -tends to vary as the square root of the linear dimensions of the -organism.</p> - -<p>From all the foregoing discussion we learn that, as Crookes -once upon a time remarked<a class="afnanch" href="#fn54" id="fnanch54">54</a>, -the form as well as the actions of our -bodies are entirely conditioned (save for certain exceptions in the -case of aquatic animals, nicely balanced with the density of the -surrounding medium) by the strength of gravity upon this globe. -Were the force of gravity to be doubled, our bipedal form would -be a failure, and the majority of terrestrial animals would resemble -short-legged saurians, or else serpents. Birds and insects would -also suffer, though there would be some compensation for them -in the increased density of the air. While on the other hand if -gravity were halved, we should get a lighter, more graceful, more -active type, requiring less energy and less heat, less heart, less -lungs, less blood.</p> - -<p>Throughout the whole field of morphology we may find -examples of a tendency (referable doubtless in each case to some -definite physical cause) for surface to keep pace with volume, -through some alteration of its form. The development of “villi” -on the inner surface of the stomach and intestine (which enlarge -its surface much as we enlarge the effective surface of a bath-towel), -the various valvular folds of the intestinal lining, including -the remarkable “spiral fold” of the shark’s gut, the convolutions -of the brain, whose complexity is evidently correlated (in part -at least) with the magnitude of the animal,—all these and many -more are cases in which a more or less constant ratio tends to be -maintained between mass and surface, which ratio would have -been more and more departed from had it not been for the -alterations of surface-form<a class="afnanch" href="#fn55" id="fnanch55">55</a>. -<span class="xxpn" id="p033">{33}</span></p> - -<p>In the case of very small animals, and of individual cells, the -principle becomes especially important, in consequence of the -molecular forces whose action is strictly limited to the superficial -layer. In the cases just mentioned, action is <i>facilitated</i> by increase -of surface: diffusion, for instance, of nutrient liquids or respiratory -gases is rendered more rapid by the greater area of surface; but -there are other cases in which the ratio of surface to mass may -make an essential change in the whole condition of the system. -We know, for instance, that iron rusts when exposed to moist -air, but that it rusts ever so much faster, and is soon eaten away, -if the iron be first reduced to a heap of small filings; this is a -mere difference of degree. But the spherical surface of the raindrop -and the spherical surface of the ocean (though both happen -to be alike in mathematical form) are two totally different phenomena, -the one due to surface-energy, and the other to that form -of mass-energy which we ascribe to gravity. The contrast is still -more clearly seen in the case of waves: for the little ripple, whose -form and manner of propagation are governed by surface-tension, -is found to travel with a velocity which is inversely as the square -root of its length; while the ordinary big waves, controlled by -gravitation, have a velocity directly proportional to the square -root of their wave-length. In like manner we shall find that the -form of all small organisms is largely independent of gravity, and -largely if not mainly due to the force of surface-tension: either -as the direct result of the continued action of surface tension on -the semi-fluid body, or else as the result of its action at a prior -stage of development, in bringing about a form which subsequent -chemical changes have rendered rigid and lasting. In either case, -we shall find a very great tendency in small organisms to assume -either the spherical form or other simple forms related to ordinary -inanimate surface-tension phenomena; which forms do not recur -in the external morphology of large animals, or if they in part -recur it is for other reasons. <span class="xxpn" id="p034">{34}</span></p> - -<p>Now this is a very important matter, and is a notable illustration -of that principle of similitude which we have already discussed -in regard to several of its manifestations. We are coming easily -to a conclusion which will affect the whole course of our argument -throughout this book, namely that there is an essential difference -in kind between the phenomena of form in the larger and the -smaller organisms. I have called this book a study of <i>Growth -and Form</i>, because in the most familiar illustrations of organic -form, as in our own bodies for example, these two factors are -inseparably associated, and because we are here justified in thinking -of form as the direct resultant and consequence of growth: of -growth, whose varying rate in one direction or another has produced, -by its gradual and unequal increments, the successive -stages of development and the final configuration of the whole -material structure. But it is by no means true that form and -growth are in this direct and simple fashion correlative or complementary -in the case of minute portions of living matter. For in -the smaller organisms, and in the individual cells of the larger, -we have reached an order of magnitude in which the intermolecular -forces strive under favourable conditions with, and at length -altogether outweigh, the force of gravity, and also those other -forces leading to movements of convection which are the prevailing -factors in the larger material aggregate.</p> - -<p>However we shall require to deal more fully with this matter -in our discussion of the rate of growth, and we may leave it meanwhile, -in order to deal with other matters more or less directly -concerned with the magnitude of the cell.</p> - -<p>The living cell is a very complex field of energy, and of energy -of many kinds, surface-energy included. Now the whole surface-energy -of the cell is by no means restricted to its <i>outer</i> surface; -for the cell is a very heterogeneous structure, and all its protoplasmic -alveoli and other visible (as well as invisible) heterogeneities -make up a great system of internal surfaces, at every -part of which one “phase” comes in contact with another “phase,” -and surface-energy is accordingly manifested. But still, the -external surface is a definite portion of the system, with a definite -“phase” of its own, and however little we may know of the distribution -of the total energy of the system, it is at -least plain that <span class="xxpn" id="p035">{35}</span> -the conditions which favour equilibrium will be greatly altered by -the changed ratio of external surface to mass which a change of -magnitude, unaccompanied by change of form, produces in the cell. -In short, however it may be brought about, the phenomenon of -division of the cell will be precisely what is required to keep -approximately constant the ratio between surface and mass, and -to restore the balance between the surface-energy and the other -energies of the system. When a germ-cell, for instance, divides -or “segments” into two, it does not increase in mass; at least if -there be some slight alleged tendency for the egg to increase in -mass or volume during segmentation, it is very slight indeed, -generally imperceptible, and wholly denied by some<a class="afnanch" href="#fn56" id="fnanch56">56</a>. -The -development or growth of the egg from a one-celled stage to -stages of two or many cells, is thus a somewhat peculiar kind -of growth; it is growth which is limited to increase of surface, -unaccompanied by growth in volume or in mass.</p> - -<p>In the case of a soap-bubble, by the way, if it divide into two -bubbles, the volume is actually diminished<a class="afnanch" href="#fn57" id="fnanch57">57</a> -while the surface-area -is greatly increased. This is due to a cause which we shall have -to study later, namely to the increased pressure due to the greater -curvature of the smaller bubbles.</p> - -<p>An immediate and remarkable result of the principles just -described is a tendency on the part of all cells, according to their -kind, to vary but little about a certain mean size, and to have, -in fact, certain absolute limitations of magnitude.</p> - -<p>Sachs<a class="afnanch" href="#fn58" id="fnanch58">58</a> -pointed out, in 1895, that there is a tendency for each -nucleus to be only able to gather around itself a certain definite -amount of protoplasm. Driesch<a class="afnanch" href="#fn59" id="fnanch59">59</a>, -a little later, found that, by -artificial subdivision of the egg, it was possible to rear dwarf -sea-urchin larvae, one-half, one-quarter, or -even one-eighth of their <span class="xxpn" id="p036">{36}</span> -normal size; and that these dwarf bodies were composed of only a -half, a quarter or an eighth of the normal number of cells. Similar -observations have been often repeated and amply confirmed. For -instance, in the development of <i>Crepidula</i> (a little American -“slipper-limpet,” now much at home on our own oyster-beds), -Conklin<a class="afnanch" href="#fn60" id="fnanch60">60</a> -has succeeded in rearing dwarf and giant individuals, -of which the latter may be as much as twenty-five times as big -as the former. But nevertheless, the individual cells, of skin, gut, -liver, muscle, and of all the other tissues, are just the same size -in one as in the other,—in dwarf and in giant<a class="afnanch" href="#fn61" id="fnanch61">61</a>. -Driesch has laid -particular stress upon this principle of a “fixed cell-size.”</p> - -<p>We get an excellent, and more familiar illustration of the same -principle in comparing the large brain-cells or ganglion-cells, both -of the lower and of the higher animals<a class="afnanch" href="#fn62" id="fnanch62">62</a>.</p> - -<div class="dctr01" id="fig1"> -<img src="images/i037.png" width="600" height="274" alt=""> - <div class="dcaption">Fig. 1. Motor ganglion-cells, from - the cervical spinal cord.<br> (From Minot, after Irving - Hardesty.)</div></div> - -<p>In Fig. <a href="#fig1" title="go to Fig. 1">1</a> we have certain identical nerve-cells taken from -various mammals, from the mouse to the elephant, all represented -on the same scale of magnification; and we see at once that they -are all of much the same <i>order</i> of magnitude. The nerve-cell of -the elephant is about twice that of the mouse in linear dimensions, -and therefore about eight times greater in volume, or mass. But -making some allowance for difference of shape, the linear dimensions -of the elephant are to those of the mouse in a ratio certainly -not less than one to fifty; from which it would follow that the -bulk of the larger animal is something like 125,000 times that of -the less. And it also follows, the size of the -nerve-cells being <span class="xxpn" id="p037">{37}</span> -about as eight to one, that, in corresponding parts of the nervous -system of the two animals, there are more than 15,000 times as -many individual cells in one as in the other. In short we may -(with Enriques) lay it down as a general law that among animals, -whether large or small, the ganglion-cells vary in size within -narrow limits; and that, amidst all the great variety of structural -type of ganglion observed in different classes of animals, it is -always found that the smaller species have simpler ganglia than -the larger, that is to say ganglia containing a smaller number -of cellular elements<a class="afnanch" href="#fn63" id="fnanch63">63</a>. -The bearing of such simple facts as this -upon the cell-theory in general is not to be disregarded; and the -warning is especially clear against exaggerated attempts to -correlate physiological processes with the visible mechanism of -associated cells, rather than with the system of energies, or the -field of force, which is associated with them. -For the life of <span class="xxpn" id="p038">{38}</span> -the body is more than the <i>sum</i> of the properties of the cells of -which it is composed: as Goethe said, “Das Lebendige ist zwar -in Elemente zerlegt, aber man kann es aus diesen nicht wieder -zusammenstellen und beleben.”</p> - -<p>Among certain lower and microscopic organisms, such for -instance as the Rotifera, we are still more palpably struck by the -small number of cells which go to constitute a usually complex -organ, such as kidney, stomach, ovary, etc. We can sometimes -number them in a few units, in place of the thousands that make -up such an organ in larger, if not always higher, animals. These -facts constitute one among many arguments which combine to -teach us that, however important and advantageous the subdivision -of organisms into cells may be from the constructional, or from -the dynamical point of view, the phenomenon has less essential -importance in theoretical biology than was once, and is often still, -assigned to it.</p> - -<p>Again, just as Sachs shewed that there was a limit to the amount -of cytoplasm which could gather round a single nucleus, so Boveri -has demonstrated that the nucleus itself has definite limitations -of size, and that, in cell-division after fertilisation, each new -nucleus has the same size as its parent-nucleus<a class="afnanch" href="#fn64" id="fnanch64">64</a>.</p> - -<p>In all these cases, then, there are reasons, partly no doubt -physiological, but in very large part purely physical, which set -limits to the normal magnitude of the organism or of the cell. -But as we have already discussed the existence of absolute and -definite limitations, of a physical kind, to the <i>possible</i> increase in -magnitude of an organism, let us now enquire whether there be -not also a lower limit, below which the very existence of an -organism is impossible, or at least where, under changed conditions, -its very nature must be profoundly modified.</p> - -<p>Among the smallest of known organisms we have, for instance, -<i>Micromonas mesnili</i>, Bonel, a flagellate infusorian, which measures -about ·34 <i>µ</i>, or ·00034 mm., by ·00025 mm.; smaller even than -this we have a pathogenic micrococcus of the rabbit, <i>M. progrediens</i>, -Schröter, the diameter of which is said to be only ·00015 -mm. or ·15 <i>µ</i>, or 1·5 × 10<sup>−5</sup> cm.,—about -equal to the thickness of <span class="xxpn" id="p039">{39}</span> -the thinnest gold-leaf; and as small if not smaller still are a few -bacteria and their spores. But here we have reached, or all but -reached the utmost limits of ordinary microscopic vision; and -there remain still smaller organisms, the so-called “filter-passers,” -which the ultra-microscope reveals, but which are mainly brought -within our ken only by the maladies, such as hydrophobia, foot-and-mouth -disease, or the “mosaic” disease of the tobacco-plant, -to which these invisible micro-organisms give rise<a class="afnanch" href="#fn65" id="fnanch65">65</a>. -Accordingly, -since it is only by the diseases which they occasion that these -tiny bodies are made known to us, we might be tempted to -suppose that innumerable other invisible organisms, smaller and -yet smaller, exist unseen and unrecognised by man.</p> - -<div class="dctr01" id="fig2"> -<img src="images/i039.png" width="600" height="403" alt=""> - <div class="pcaption">Fig. 2. Relative magnitudes of: A, human - blood-corpuscle (7·5 µ in diameter); B, <i>Bacillus anthracis</i> - (4 – 15 µ × 1 µ); C, various - Micrococci (diam. 0·5 – 1 µ, rarely 2 µ); D, - <i>Micromonas progrediens</i>, Schröter (diam. 0·15 µ).</div></div> - -<p>To illustrate some of these small magnitudes I have adapted -the preceding diagram from one given by Zsigmondy<a class="afnanch" href="#fn66" id="fnanch66">66</a>. -Upon -the <span class="xxpn" id="p040">{40}</span> -same scale the minute ultramicroscopic particles of colloid gold -would be represented by the finest dots which we could make -visible to the naked eye upon the paper.</p> - -<p>A bacillus of ordinary, typical size is, say, 1 µ in length. The -length (or height) of a man is about a million and three-quarter -times as great, i.e. 1·75 metres, or 1·75 × 10<sup>6</sup> µ; and the mass of -the man is in the neighbourhood of five million, million, million -(5 × 10<sup>18</sup>) times greater than that of the bacillus. If we ask -whether there may not exist organisms as much less than the -bacillus as the bacillus is less than the dimensions of a man, it -is very easy to see that this is quite impossible, for we are rapidly -approaching a point where the question of molecular dimensions, -and of the ultimate divisibility of matter, begins to call for our -attention, and to obtrude itself as a crucial factor in the case.</p> - -<div class="dmaths"> -<p>Clerk Maxwell dealt with this matter in his article “Atom<a class="afnanch" href="#fn67" id="fnanch67">67</a>,” -and, in somewhat greater detail, Errera discusses the question on -the following lines<a class="afnanch" href="#fn68" id="fnanch68">68</a>. -The weight of a hydrogen molecule is, -according to the physical chemists, somewhere about 8·6 × 2 × 10<sup>−22</sup> -milligrammes; and that of any other element, whose molecular -weight is <i>M</i>, is given by the equation</p> - -<div>(<i>M</i>) -= 8·6 × <i>M</i> × 10<sup>−22</sup> .</div> - -<p class="pcontinue">Accordingly, the weight of the atom of sulphur may be taken as</p> - -<div>8·6 × 32 × 10<sup>−22</sup> mgm. -= 275 × 10<sup>−22</sup> mgm.</div> -</div><!--dmaths--> - -<div class="dmaths"> -<p>The analysis of ordinary bacteria shews them to -consist<a class="afnanch" href="#fn69" id="fnanch69">69</a> -of about 85% of water, and 15% of solids; while the solid -residue of vegetable protoplasm contains about one part in -a thousand of sulphur. We may assume, therefore, that the -living protoplasm contains about</p> - -<div><sup>1</sup>⁄<sub>1000</sub> × <sup>15</sup>⁄<sub>100</sub> -= 15 × 10<sup>−5</sup></div> - -<p class="pcontinue">parts of sulphur, taking the total weight as -= 1.</p></div><!--dmaths--> - -<div class="dmaths"> -<p>But our little micrococcus, of 0·15 µ in diameter, would, if it -were spherical, have a volume of</p> - -<div><sup>π</sup>⁄<sub>6</sub> × 0·15<sup>3</sup> µ -= 18 × 10<sup>−4</sup> cubic microns; -<span class="xxpn" id="p041">{41}</span></div> - -<p class="pcontinue">and therefore (taking its density as equal to that of water), a -weight of</p> - -<div>18 × 10<sup>−4</sup> × 10<sup>−9</sup> -= 18 × 10<sup>−13</sup> mgm.</div> - -<p class="pcontinue">But of this total weight, the sulphur represents only</p> - -<div>18 × 10<sup>−13</sup> × 15 × 10<sup>−5</sup> -= 27 × 10<sup>−17</sup> mgm.</div> - -<p class="pcontinue">And if we divide this by the weight of an atom of sulphur, we have</p> - -<div>(27 × 10<sup>−17</sup>) ÷ (275 × 10<sup>−22</sup>) -= 10,000, or thereby.</div> -</div><!--dmaths--> - -<p class="pcontinue">According to this estimate, then, our little <i>Micrococcus progrediens</i> -should contain only about 10,000 atoms of sulphur, an element -indispensable to its protoplasmic constitution; and it follows that -an organism of one-tenth the diameter of our micrococcus would -only contain 10 sulphur-atoms, and therefore only ten chemical -“molecules” or units of protoplasm!</p> - -<div class="dmaths"> -<p>It may be open to doubt whether the presence of sulphur -be really essential to the constitution of the proteid or -“protoplasmic” molecule; but Errera gives us yet another -illustration of a similar kind, which is free from this -objection or dubiety. The molecule of albumin, as is -generally agreed, can scarcely be less than a thousand -times the size of that of such an element as sulphur: -according to one particular determination<a class="afnanch" -href="#fn70" id="fnanch70">70</a>, serum albumin has -a constitution corresponding to a molecular weight -of 10,166, and even this may be far short of the true -complexity of a typical albuminoid molecule. The weight of -such a molecule is</p> - -<div>8·6 × 10166 × 10<sup>−22</sup> -= 8·7 × 10<sup>−18</sup> mgm.</div> - -<p class="pcontinue">Now the bacteria contain about 14% of albuminoids, these -constituting by far the greater part of the dry residue; and -therefore (from equation (5)), the weight of albumin in our micrococcus -is about</p> - -<div><sup>14</sup>⁄<sub>100</sub> × 18 × 10<sup>−13</sup> -= 2·5 × 10<sup>−13</sup> mgm.</div> - -<p class="pcontinue">If we divide this weight by that which we have arrived at as the -weight of an albumin molecule, we have</p> - -<div>2·5 × 10<sup>−13</sup> ÷ (8·7 × 10<sup>−18</sup>) -= 2·9 × 10<sup>−4</sup> ,</div> - -<p class="pcontinue">in other words, our micrococcus apparently contains something -less than 30,000 molecules of albumin. <span class="xxpn" id="p042">{42}</span></p> -</div><!--dmaths--> - -<p>According to the most recent estimates, the weight of the -hydrogen molecule is somewhat less than that on which Errera -based his calculations, namely about 16 × 10<sup>−22</sup> mgms. and -according to this value, our micrococcus would contain just about -27,000 albumin molecules. In other words, whichever determination -we accept, we see that an organism one-tenth as large as our -micrococcus, in linear dimensions, would only contain some thirty -molecules of albumin; or, in other words, our micrococcus is only -about thirty times as large, in linear dimensions, as a single albumin -molecule<a class="afnanch" href="#fn71" id="fnanch71">71</a>.</p> - -<p>We must doubtless make large allowances for uncertainty in -the assumptions and estimates upon which these calculations are -based; and we must also remember that the data with which the -physicist provides us in regard to molecular magnitudes are, to -a very great extent, <i>maximal</i> values, above which the molecular -magnitude (or rather the sphere of the molecule’s range of motion) -is not likely to lie: but below which there is a greater element of -uncertainty as to its possibly greater minuteness. But nevertheless, -when we shall have made all reasonable allowances for uncertainty -upon the physical side, it will still be clear that the smallest known -bodies which are described as organisms draw nigh towards -molecular magnitudes, and we must recognise that the subdivision -of the organism cannot proceed to an indefinite extent, and in all -probability cannot go very much further than it appears to have -done in these already discovered forms. For, even, after giving -all due regard to the complexity of our unit (that is to say the -albumin-molecule), with all the increased possibilities of interrelation -with its neighbours which this complexity implies, we -cannot but see that physiologically, and comparatively speaking, -we have come down to a very simple thing.</p> - -<p>While such considerations as these, based on the chemical -composition of the organism, teach us that there must be a definite -lower limit to its magnitude, other considerations of a purely -physical kind lead us to the same conclusion. For our discussion -of the principle of similitude has already taught us that, long -before we reach these almost infinitesimal -magnitudes, the <span class="xxpn" id="p043">{43}</span> -diminishing organism will have greatly changed in all its physical -relations, and must at length arrive under conditions which must -surely be incompatible with anything such as we understand by -life, at least in its full and ordinary development and manifestation.</p> - -<p>We are told, for instance, that the powerful force of surface-tension, -or capillarity, begins to act within a range of about -1 ⁄ 500,000 of an inch, or say 0·05 µ. A soap-film, or a film of oil -upon water, may be attenuated to far less magnitudes than this; -the black spots upon a soap-bubble are known, by various concordant -methods of measurement, to be only about 6 × 10<sup>−7</sup> cm., -or about ·006 µ thick, and Lord Rayleigh and M. Devaux<a class="afnanch" href="#fn72" id="fnanch72">72</a> -have -obtained films of oil of ·002 µ, or even ·001 µ in thickness.</p> - -<p>But while it is possible for a fluid film to exist in these almost -molecular dimensions, it is certain that, long before we reach -them, there must arise new conditions of which we have little -knowledge and which it is not easy even to imagine.</p> - -<p>It would seem that, in an organism of ·1 µ in diameter, or even -rather more, there can be no essential distinction between the -interior and the surface layers. No hollow vesicle, I take it, can -exist of these dimensions, or at least, if it be possible for it to do -so, the contained gas or fluid must be under pressures of a formidable -kind<a class="afnanch" href="#fn73" id="fnanch73">73</a>, -and of which we have no knowledge or experience. -Nor, I imagine, can there be any real complexity, or heterogeneity, -of its fluid or semi-fluid contents; there can be no vacuoles within -such a cell, nor any layers defined within its fluid substance, for -something of the nature of a boundary-film is the necessary -condition of the existence of such layers. Moreover, the whole -organism, provided that it be fluid or semi-fluid, can only be -spherical in form. What, then, can we attribute, in the way of -properties, to an organism of a size as small as, or smaller than, -say ·05 µ? It must, in all probability, be a homogeneous, structureless -sphere, composed of a very small number of albuminoid or -other molecules. Its vital properties and functions must be -extraordinarily limited; its specific outward characters, even if we -could see it, must be <i>nil</i>; and its specific properties must be little -more than those of an ion-laden corpuscle, -enabling it to perform <span class="xxpn" id="p044">{44}</span> -this or that chemical reaction, or to produce this or that pathogenic -effect. Even among inorganic, non-living bodies, there -must be a certain grade of minuteness at which the ordinary -properties become modified. For instance, while under ordinary -circumstances crystallisation starts in a solution about a minute -solid fragment or crystal of the salt, Ostwald has shewn that we -may have particles so minute that they fail to serve as a nucleus -for crystallisation,—which is as much as to say that they are too -minute to have the form and properties of a “crystal”; and again, -in his thin oil-films, Lord Rayleigh has noted the striking change -of physical properties which ensues when the film becomes -attenuated to something less than one close-packed layer of -molecules<a class="afnanch" href="#fn74" id="fnanch74">74</a>.</p> - -<p>Thus, as Clerk Maxwell put it, “molecular science sets us face -to face with physiological theories. It forbids the physiologist -from imagining that structural details of infinitely small dimensions -[such as Leibniz assumed, one within another, <i>ad infinitum</i>] -can furnish an explanation of the infinite variety which exists in -the properties and functions of the most minute organisms.” -And for this reason he reprobates, with not undue severity, those -advocates of pangenesis and similar theories of heredity, who -would place “a whole world of wonders within a body so small -and so devoid of visible structure as a germ.” But indeed it -scarcely needed Maxwell’s criticism to shew forth the immense -physical difficulties of Darwin’s theory of Pangenesis: which, -after all, is as old as Democritus, and is no other than that -Promethean <i>particulam undique desectam</i> of which we have read, -and at which we have smiled, in our Horace.</p> - -<p>There are many other ways in which, when we “make a long -excursion into space,” we find our ordinary rules of physical -behaviour entirely upset. A very familiar case, analysed by -Stokes, is that the viscosity of the surrounding medium has a -relatively powerful effect upon bodies below a certain size. -A droplet of water, a thousandth of an inch (25 µ) in diameter, -cannot fall in still air quicker than about an inch and a half per -second; and as its size decreases, its resistance varies as the -diameter, and not (as with larger bodies) as -the surface of the <span class="xxpn" id="p045">{45}</span> -drop. Thus a drop one-tenth of that size (2·5 µ), the size, -apparently, of the drops of water in a light cloud, will fall a -hundred times slower, or say an inch a minute; and one again -a tenth of this diameter (say ·25 µ, or about twice as big, in linear -dimensions, as our micrococcus), will scarcely fall an inch in two -hours. By reason of this principle, not only do the smaller -bacteria fall very slowly through the air, but all minute bodies -meet with great proportionate resistance to their movements in -a fluid. Even such comparatively large organisms as the diatoms -and the foraminifera, laden though they are with a heavy shell -of flint or lime, seem to be poised in the water of the ocean, and -fall in it with exceeding slowness.</p> - -<p>The Brownian movement has also to be reckoned with,—that -remarkable phenomenon studied nearly a century ago (1827) by -Robert Brown, <i>facile princeps botanicorum</i>. It is one more of those -fundamental physical phenomena which the biologists have contributed, -or helped to contribute, to the science of physics.</p> - -<p>The quivering motion, accompanied by rotation, and even by -translation, manifested by the fine granular particles issuing from -a crushed pollen-grain, and which Robert Brown proved to have -no vital significance but to be manifested also by all minute -particles whatsoever, organic and inorganic, was for many years -unexplained. Nearly fifty years after Brown wrote, it was said -to be “due, either directly to some calorical changes continually -taking place in the fluid, or to some obscure chemical action -between the solid particles and the fluid which is indirectly -promoted by heat<a class="afnanch" href="#fn75" id="fnanch75">75</a>.” -Very shortly after these last words were -written, it was ascribed by Wiener to molecular action, and we -now know that it is indeed due to the impact or bombardment of -molecules upon a body so small that these impacts do not for -the moment, as it were, “average out” to approximate equality -on all sides. The movement becomes manifest with particles of -somewhere about 20 µ in diameter, it is admirably displayed by -particles of about 12 µ in diameter, and becomes more marked -the smaller the particles are. The bombardment causes our -particles to behave just like molecules of -uncommon size, and this <span class="xxpn" id="p046">{46}</span> -behaviour is manifested in several ways<a class="afnanch" href="#fn76" id="fnanch76">76</a>. -Firstly, we have the -quivering movement of the particles; secondly, their movement -backwards and forwards, in short, straight, disjointed paths; -thirdly, the particles rotate, and do so the more rapidly the smaller -they are, and by theory, confirmed by observation, it is found -that particles of 1 µ in diameter rotate on an average through -100° per second, while particles of 13 µ in diameter turn through -only 14° per minute. Lastly, the very curious result appears, that -in a layer of fluid the particles are not equally distributed, nor do -they all ever fall, under the influence of gravity, to the bottom. -But just as the molecules of the atmosphere are so distributed, -under the influence of gravity, that the density (and therefore the -number of molecules per unit volume) falls off in geometrical -progression as we ascend to higher and higher layers, so is it with -our particles, even within the narrow limits of the little portion -of fluid under our microscope. It is only in regard to particles -of the simplest form that these phenomena have been theoretically -investigated<a class="afnanch" href="#fn77" id="fnanch77">77</a>, -and we may take it as certain that more complex -particles, such as the twisted body of a Spirillum, would show -other and still more complicated manifestations. It is at least -clear that, just as the early microscopists in the days before Robert -Brown never doubted but that these phenomena were purely -vital, so we also may still be apt to confuse, in certain cases, the -one phenomenon with the other. We cannot, indeed, without the -most careful scrutiny, decide whether the movements of our -minutest organisms are intrinsically “vital” (in the sense of being -beyond a physical mechanism, or working model) or not. For example, -Schaudinn has suggested that the undulating movements of -<i>Spirochaete pallida</i> must be due to the presence of a minute, unseen, -“undulating membrane”; and Doflein says of the same species -that “sie verharrt oft mit eigenthümlich zitternden Bewegungen -zu einem Orte.” Both movements, the -trembling or quivering <span class="xxpn" id="p047">{47}</span> -movement described by Doflein, and the undulating or rotating -movement described by Schaudinn, are just such as may be easily -and naturally interpreted as part and parcel of the Brownian -phenomenon.</p> - -<p>While the Brownian movement may thus simulate in a deceptive -way the active movements of an organism, the reverse statement -also to a certain extent holds good. One sometimes lies awake of -a summer’s morning watching the flies as they dance under the -ceiling. It is a very remarkable dance. The dancers do not -whirl or gyrate, either in company or alone; but they advance -and retire; they seem to jostle and rebound; between the rebounds -they dart hither or thither in short straight snatches of hurried -flight; and turn again sharply in a new rebound at the end of each -little rush. Their motions are wholly “erratic,” independent of -one another, and devoid of common purpose. This is nothing else -than a vastly magnified picture, or simulacrum, of the Brownian -movement; the parallel between the two cases lies in their -complete irregularity, but this in itself implies a close resemblance. -One might see the same thing in a crowded market-place, always -provided that the bustling crowd had no <i>business</i> whatsoever. -In like manner Lucretius, and Epicurus before him, watched the -dust-motes quivering in the beam, and saw in them a mimic -representation, <i>rei simulacrum et imago</i>, of the eternal motions of -the atoms. Again the same phenomenon may be witnessed under -the microscope, in a drop of water swarming with Paramoecia or -suchlike Infusoria; and here the analogy has been put to a numerical -test. Following with a pencil the track of each little swimmer, -and dotting its place every few seconds (to the beat of a metronome), -Karl Przibram found that the mean successive distances from a -common base-line obeyed with great exactitude the “Einstein -formula,” that is to say the particular form of the “law of chance” -which is applicable to the case of the Brownian movement<a class="afnanch" href="#fn78" id="fnanch78">78</a>. -The -phenomenon is (of course) merely analogous, and by no means -identical with the Brownian movement; for the range of motion -of the little active organisms, whether they be gnats or infusoria, -is vastly greater than that of the minute -particles which are <span class="xxpn" id="p048">{48}</span> -passive under bombardment; but nevertheless Przibram is -inclined to think that even his comparatively large infusoria are -small enough for the molecular bombardment to be a stimulus, -though not the actual cause, of their irregular and interrupted -movements.</p> - -<p>There is yet another very remarkable phenomenon which may -come into play in the case of the minutest of organisms; and this -is their relation to the rays of light, as Arrhenius has told us. -On the waves of a beam of light, a very minute particle (<i>in -vacuo</i>) should be actually caught up, and carried along with -an immense velocity; and this “radiant pressure” exercises -its most powerful influence on bodies which (if they be of -spherical form) are just about ·00016 mm., or ·16 µ in diameter. -This is just about the size, as we have seen, of some of -our smallest known protozoa and bacteria, while we have -some reason to believe that others yet unseen, and perhaps -the spores of many, are smaller still. Now we have seen that -such minute particles fall with extreme slowness in air, even at -ordinary atmospheric pressures: our organism measuring ·16 µ -would fall but 83 metres in a year, which is as much as to say -that its weight offers practically no impediment to its transference, -by the slightest current, to the very highest regions of the atmosphere. -Beyond the atmosphere, however, it cannot go, until -some new force enable it to resist the attraction of terrestrial -gravity, which the viscosity of an atmosphere is no longer at -hand to oppose. But it is conceivable that our particle <i>may</i> go -yet farther, and actually break loose from the bonds of earth. -For in the upper regions of the atmosphere, say fifty miles high, -it will come in contact with the rays and flashes of the Northern -Lights, which consist (as Arrhenius maintains) of a fine dust, or -cloud of vapour-drops, laden with a charge of negative electricity, -and projected outwards from the sun. As soon as our particle -acquires a charge of negative electricity it will begin to be repelled -by the similarly laden auroral particles, and the amount of charge -necessary to enable a particle of given size (such as our little -monad of ·16 µ) to resist the attraction of gravity may be calculated, -and is found to be such as the actual conditions can easily supply. -Finally, when once set free from the entanglement -of the earth’s <span class="xxpn" id="p049">{49}</span> -atmosphere, the particle may be propelled by the “radiant -pressure” of light, with a velocity which will carry it.—like -Uriel gliding on a sunbeam,—as far as the orbit of Mars in -twenty days, of Jupiter in eighty days, and as far as the nearest -fixed star in three thousand years! This, and much more, is -Arrhenius’s contribution towards the acceptance of Lord Kelvin’s -hypothesis that life may be, and may have been, disseminated -across the bounds of space, throughout the solar system and the -whole universe!</p> - -<p>It may well be that we need attach no great practical importance -to this bold conception; for even though stellar space be shewn to -be <i>mare liberum</i> to minute material travellers, we may be sure that -those which reach a stellar or even a planetary bourne are infinitely, -or all but infinitely, few. But whether or no, the remote possibilities -of the case serve to illustrate in a very vivid way the profound -differences of physical property and potentiality which are -associated in the scale of magnitude with simple differences of -degree.</p> - -<div class="chapter" id="p050"> -<h2 class="h2herein" title="III. The Rate of Growth.">CHAPTER III -<span class="h2ttl"> -THE RATE OF GROWTH</span></h2></div> - -<p>When we study magnitude by itself, apart, that is to say, -from the gradual changes to which it may be subject, we are -dealing with a something which may be adequately represented -by a number, or by means of a line of definite length; it is what -mathematicians call a <i>scalar</i> phenomenon. When we introduce -the conception of change of magnitude, of magnitude which varies -as we pass from one direction to another in space, or from one -instant to another in time, our phenomenon becomes capable of -representation by means of a line of which we define both the -length and the direction; it is (in this particular aspect) what is -called a <i>vector</i> phenomenon.</p> - -<p>When we deal with magnitude in relation to the dimensions -of space, the vector diagram which we draw plots magnitude in -one direction against magnitude in another,—length against -height, for instance, or against breadth; and the result is simply -what we call a picture or drawing of an object, or (more correctly) -a “plane projection” of the object. In other words, what we -call Form is a <i>ratio of magnitudes</i>, referred to direction in space.</p> - -<p>When in dealing with magnitude we refer its variations to -successive intervals of time (or when, as it is said, we <i>equate</i> it -with time), we are then dealing with the phenomenon of <i>growth</i>; -and it is evident, therefore, that this term growth has wide -meanings. For growth may obviously be positive or negative; -that is to say, a thing may grow larger or smaller, greater or less; -and by extension of the primitive concrete signification of the -word, we easily and legitimately apply it to non-material things, -such as temperature, and say, for instance, that a body “grows” -hot or cold. When in a two-dimensional diagram, we represent -a magnitude (for instance length) in relation to -time (or “plot” <span class="xxpn" id="p051">{51}</span> -length against time, as the phrase is), we get that kind of vector -diagram which is commonly known as a “curve of growth.” We -perceive, accordingly, that the phenomenon which we are now -studying is a <i>velocity</i> (whose “dimensions” are -<sup>Space</sup>⁄<sub>Time</sub> or -<sup class="spitc">L</sup>⁄<sub class="spitc">T</sub>); and -this phenomenon we shall speak of, simply, as a rate of growth.</p> - -<p>In various conventional ways we can convert a two-dimensional -into a three-dimensional diagram. We do so, for example, by -means of the geometrical method of “perspective” when we -represent upon a sheet of paper the length, breadth and depth of -an object in three-dimensional space; but we do it more simply, -as a rule, by means of “contour-lines,” and always when time is -one of the dimensions to be represented. If we superimpose upon -one another (or even set side by side) pictures, or plane projections, -of an organism, drawn at successive intervals of time, we have -such a three-dimensional diagram, which is a partial representation -(limited to two dimensions of <i>space</i>) of the organism’s gradual -change of form, or course of development; and in such a case -our contour-lines may, for the purposes of the embryologist, be -separated by intervals representing a few hours or days, or, for -the purposes of the palaeontologist, by interspaces of unnumbered -and innumerable years<a class="afnanch" href="#fn79" id="fnanch79">79</a>.</p> - -<p>Such a diagram represents in two of its three dimensions form, -and in two, or three, of its dimensions growth; and so we see how -intimately the two conceptions are correlated or inter-related to -one another. In short, it is obvious that the form of an animal -is determined by its specific rate of growth in various directions; -accordingly, the phenomenon of rate of growth deserves to be -studied as a necessary preliminary to the theoretical study of -form, and, mathematically speaking, organic form itself appears -to us as a <i>function of time</i><a class="afnanch" href="#fn80" id="fnanch80">80</a>. -<span class="xxpn" id="p052">{52}</span></p> - -<p>At the same time, we need only consider this part of our -subject somewhat briefly. Though it has an essential bearing on -the problems of morphology, it is in greater degree involved with -physiological problems; and furthermore, the statistical or -numerical aspect of the question is peculiarly adapted for the -mathematical study of variation and correlation. On these -important subjects we shall scarcely touch; for our main purpose -will be sufficiently served if we consider the characteristics of a -rate of growth in a few illustrative cases, and recognise that this -rate of growth is a very important specific property, with its own -characteristic value in this organism or that, in this or that part -of each organism, and in this or that phase of its existence.</p> - -<p>The statement which we have just made that “the form of an -organism is determined by its rate of growth in various directions,” -is one which calls (as we have partly seen in the foregoing chapter) -for further explanation and for some measure of qualification. -Among organic forms we shall have frequent occasion to see that -form is in many cases due to the immediate or direct action of -certain molecular forces, of which surface-tension is that which plays -the greatest part. Now when surface-tension (for instance) causes -a minute semi-fluid organism to assume a spherical form, or gives -the form of a catenary or an elastic curve to a film of protoplasm -in contact with some solid skeletal rod, or when it acts in various -other ways which are productive of definite contours, this is a process -of conformation that, both in appearance and reality, is very -different from the process by which an ordinary plant or animal -<i>grows</i> into its specific form. In both cases, change of form is -brought about by the movement of portions of matter, and in -both cases it is <i>ultimately</i> due to the action of molecular forces; -but in the one case the movements of the particles of matter lie -for the most part <i>within molecular range</i>, while in the other we -have to deal chiefly with the transference of portions of matter -into the system from without, and from one widely distant part -of the organism to another. It is to this latter class of phenomena -that we usually restrict the term growth; and it is in regard to -them that we are in a position to study the <i>rate of action</i> in -different directions, and to see that it is merely on a difference -of velocities that the modification of -form essentially depends. <span class="xxpn" id="p053">{53}</span> -The difference between the two classes of phenomena is somewhat -akin to the difference between the forces which determine the -form of a rain-drop and those which, by the flowing of the waters -and the sculpturing of the solid earth, have brought about the -complex configuration of a river; <i>molecular</i> forces are paramount -in the conformation of the one, and <i>molar</i> forces are dominant -in the other.</p> - -<p>At the same time it is perfectly true that <i>all</i> changes of form, -inasmuch as they necessarily involve changes of actual and relative -magnitude, may, in a sense, be properly looked upon as phenomena -of growth; and it is also true, since the movement of matter must -always involve an element of time<a class="afnanch" href="#fn81" id="fnanch81">81</a>, -that in all cases the rate of -growth is a phenomenon to be considered. Even though the -molecular forces which play their part in modifying the form of -an organism exert an action which is, theoretically, all but -instantaneous, that action is apt to be dragged out to an appreciable -interval of time by reason of viscosity or some other form of -resistance in the material. From the physical or physiological -point of view the rate of action even in such cases may be well -worth studying; for example, a study of the rate of cell-division -in a segmenting egg may teach us something about the work done, -and about the various energies concerned. But in such cases the -action is, as a rule, so homogeneous, and the form finally attained -is so definite and so little dependent on the time taken to effect -it, that the specific rate of change, or rate of growth, does not -enter into the <i>morphological</i> problem.</p> - -<p>To sum up, we may lay down the following general statements. -The form of organisms is a phenomenon to be referred in part -to the direct action of molecular forces, in part to a more complex -and slower process, indirectly resulting from chemical, osmotic -and other forces, by which material is introduced into the organism -and transferred from one part of it to another. It is this latter -complex phenomenon which we usually -speak of as “growth.” <span class="xxpn" id="p054">{54}</span></p> - -<p>Every growing organism, and every part of such a growing -organism, has its own specific rate of growth, referred to a particular -direction. It is the ratio between the rates of growth in various -directions by which we must account for the external forms of -all, save certain very minute, organisms. This ratio between -rates of growth in various directions may sometimes be of a -<i>simple</i> kind, as when it results in the mathematically definable -outline of a shell, or in the smooth curve of the margin of a leaf. -It may sometimes be a very <i>constant</i> one, in which case the -organism, while growing in bulk, suffers little or no perceptible -change in form; but such equilibrium seldom endures for more -than a season, and when the <i>ratio</i> tends to alter, then we have -the phenomenon of morphological “development,” or steady and -persistent change of form.</p> - -<p>This elementary concept of Form, as determined by varying -rates of Growth, was clearly apprehended by the mathematical -mind of Haller,—who had learned his mathematics of the great -John Bernoulli, as the latter in turn had learned his physiology -from the writings of Borelli. Indeed it was this very point, the -apparently unlimited extent to which, in the development of the -chick, inequalities of growth could and did produce changes of -form and changes of anatomical “structure,” that led Haller to -surmise that the process was actually without limits, and that all -development was but an unfolding, or “<i>evolutio</i>,” in which no -part came into being which had not essentially existed before<a class="afnanch" href="#fn82" id="fnanch82">82</a>. -In short the celebrated doctrine of “preformation” implied on the -one hand a clear recognition of what, throughout the later stages -of development, growth can do, by hastening the increase in size -of one part, hindering that of another, changing their relative -magnitudes and positions, and altering their forms; while on the -other hand it betrayed a failure (inevitable in those days) to -recognise the essential difference between these movements of -masses and the molecular processes which -precede and accompany <span class="xxpn" id="p055">{55}</span> -them, and which are characteristic of another order of magnitude.</p> - -<p>By other writers besides Haller the very general, though not -strictly universal connection between form and rate of growth -has been clearly recognised. Such a connection is implicit in -those “proportional diagrams” by which Dürer and some of his -brother artists were wont to illustrate the successive changes of -form, or of relative dimensions, which attend the growth of the -child, to boyhood and to manhood. The same connection was -recognised, more explicitly, by some of the older embryologists, -for instance by Pander<a class="afnanch" href="#fn83" id="fnanch83">83</a>, -and appears, as a survival of the -doctrine of preformation, in his study of the development of -the chick. And long afterwards, the embryological aspect of -the case was emphasised by His, who pointed out, for instance, -that the various foldings of the blastoderm, by which the neural -and amniotic folds were brought into being, were essentially -and obviously the resultant of unequal rates of growth,—of -local accelerations or retardations of growth,—in what to begin -with was an even and uniform layer of embryonic tissue. If -we imagine a flat sheet of paper, parts of which are caused -(as by moisture or evaporation) to expand or to contract, the -plane surface is at once dimpled, or “buckled,” or folded, by -the resultant forces of expansion or contraction: and the various -distortions to which the plane surface of the “germinal disc” is -subject, as His shewed once and for all, are precisely analogous. -An experimental demonstration still more closely comparable to -the actual case of the blastoderm, is obtained by making an -“artificial blastoderm,” of little pills or pellets of dough, which -are caused to grow, with varying velocities, by the addition -of varying quantities of yeast. Here, as Roux is careful to -point out<a class="afnanch" href="#fn84" id="fnanch84">84</a>, -we observe that it is not only the <i>growth</i> of the -individual cells, but the <i>traction</i> exercised through their mutual -interconnections, which brings about the foldings and other distortions -of the entire structure. <span class="xxpn" id="p056">{56}</span></p> - -<p>But this again was clearly present to Haller’s mind, and formed -an essential part of his embryological doctrine. For he has no -sooner treated of <i>incrementum</i>, or <i>celeritas incrementi</i>, than he -proceeds to deal with the contributory and complementary phenomena -of expansion, traction (<i>adtractio</i>)<a class="afnanch" href="#fn85" id="fnanch85">85</a>, -and pressure, and the -more subtle influences which he denominates <i>vis derivationis et -revulsionis</i><a class="afnanch" href="#fn86" id="fnanch86">86</a>: -these latter being the secondary and correlated -effects on growth in one part, brought about, through such -changes as are produced (for instance) in the circulation, by the -growth of another.</p> - -<p>Let us admit that, on the physiological side, Haller’s or His’s -methods of explanation carry us back but a little way; yet even -this little way is something gained. Nevertheless, I can well -remember the harsh criticism, and even contempt, which His’s -doctrine met with, not merely on the ground that it was inadequate, -but because such an explanation was deemed wholly inappropriate, -and was utterly disavowed<a class="afnanch" href="#fn87" id="fnanch87">87</a>. -Hertwig, for instance, asserted that, -in embryology, when we found one embryonic stage preceding -another, the existence of the former was, for the embryologist, -an all-sufficient “causal explanation” of the latter. “We consider -(he says), that we are studying and explaining a causal relation -when we have demonstrated that the gastrula arises by invagination -of a blastosphere, or the neural canal by the infolding of a -cell plate so as to constitute a tube<a class="afnanch" href="#fn88" id="fnanch88">88</a>.” -For Hertwig, therefore, as <span class="xxpn" id="p057">{57}</span> -Roux remarks, the task of investigating a physical mechanism in -embryology,—“der Ziel das Wirken zu erforschen,”—has no -existence at all. For Balfour also, as for Hertwig, the mechanical -or physical aspect of organic development had little or no attraction. -In one notable instance, Balfour himself adduced a physical, or -quasi-physical, explanation of an organic process, when he referred -the various modes of segmentation of an ovum, complete or partial, -equal or unequal and so forth, to the varying amount or the -varying distribution of food yolk in association with the germinal -protoplasm of the egg<a class="afnanch" href="#fn89" id="fnanch89">89</a>. -But in the main, Balfour, like all the -other embryologists of his day, was engrossed by the problems of -phylogeny, and he expressly defined the aims of comparative -embryology (as exemplified in his own textbook) as being “twofold: -(1) to form a basis for Phylogeny. and (2) to form a basis -for Organogeny or the origin and evolution of organs<a class="afnanch" href="#fn90" id="fnanch90">90</a>.”</p> - -<p>It has been the great service of Roux and his fellow-workers -of the school of “Entwickelungsmechanik,” and of many other -students to whose work we shall refer, to try, as His tried<a class="afnanch" href="#fn91" id="fnanch91">91</a> -to -import into embryology, wherever possible, the simpler concepts -of physics, to introduce along with them the method of experiment, -and to refuse to be bound by the narrow limitations which such -teaching as that of Hertwig would of necessity impose on the -work and the thought and on the whole philosophy of the biologist.</p> - -<hr class="hrblk"> - -<p>Before we pass from this general discussion to study some of -the particular phenomena of growth, let me give a single illustration, -from Darwin, of a point of view which is in marked contrast to -Haller’s simple but essentially mathematical conception of Form.</p> - -<p>There is a curious passage in the <i>Origin of Species</i><a class="afnanch" href="#fn92" id="fnanch92">92</a>, -where -Darwin is discussing the leading facts of embryology, and in -particular Von Baer’s “law of embryonic resemblance.” Here -Darwin says “We are so much accustomed to -see a difference in <span class="xxpn" id="p058">{58}</span> -structure between the embryo and the adult, that we are tempted -to look at this difference as in some necessary manner contingent -on growth. <i>But there is no reason why, for instance, the wing of -a bat, or the fin of a porpoise, should not have been sketched out with -all their parts in proper proportion, as soon as any part became -visible.</i>” After pointing out with his habitual care various -exceptions, Darwin proceeds to lay down two general principles, -viz. “that slight variations generally appear at a not very early -period of life,” and secondly, that “at whatever age a variation -first appears in the parent, it tends to reappear at a corresponding -age in the offspring.” He then argues that it is with nature as -with the fancier, who does not care what his pigeons look like -in the embryo, so long as the full-grown bird possesses the desired -qualities; and that the process of selection takes place when -the birds or other animals are nearly grown up,—at least on the -part of the breeder, and presumably in nature as a general rule. -The illustration of these principles is set forth as follows; “Let -us take a group of birds, descended from some ancient form and -modified through natural selection for different habits. Then, -from the many successive variations having supervened in the -several species at a not very early age, and having been inherited -at a corresponding age, the young will still resemble each other -much more closely than do the adults,—just as we have seen -with the breeds of the pigeon .... Whatever influence long-continued -use or disuse may have had in modifying the limbs or other parts -of any species, this will chiefly or solely have affected it when -nearly mature, when it was compelled to use its full powers to -gain its own living; and the effects thus produced will have been -transmitted to the offspring at a corresponding nearly mature -age. Thus the young will not be modified, or will be modified -only in a slight degree, through the effects of the increased use or -disuse of parts.” This whole argument is remarkable, in more -ways than we need try to deal with here; but it is especially -remarkable that Darwin should begin by casting doubt upon the -broad fact that a “difference in structure between the embryo -and the adult” is “in some necessary manner contingent on -growth”; and that he should see no reason why complicated -structures of the adult “should not have -been sketched out <span class="xxpn" id="p059">{59}</span> -with all their parts in proper proportion, as soon as any part -became visible.” It would seem to me that even the most -elementary attention to form in its relation to growth would have -removed most of Darwin’s difficulties in regard to the particular -phenomena which he is here considering. For these phenomena -are phenomena of form, and therefore of relative magnitude; -and the magnitudes in question are attained by growth, proceeding -with certain specific velocities, and lasting for certain long periods -of time. And it is accordingly obvious that in any two related -individuals (whether specifically identical or not) the differences -between them must manifest themselves gradually, and be but -little apparent in the young. It is for the same simple reason -that animals which are of very different sizes when adult, differ -less and less in size (as well as in form) as we trace them backwards -through the foetal stages.</p> - -<hr class="hrblk"> - -<p>Though we study the visible effects of varying rates of growth -throughout wellnigh all the problems of morphology, it is not very -often that we can directly measure the velocities concerned. -But owing to the obvious underlying importance which the -phenomenon has to the morphologist we must make shift to study -it where we can, even though our illustrative cases may seem to -have little immediate bearing on the morphological problem<a class="afnanch" href="#fn93" id="fnanch93">93</a>.</p> - -<p>In a very simple organism, of spherical symmetry, such as the -single spherical cell of Protococcus or of Orbulina, growth is -reduced to its simplest terms, and indeed it becomes so simple -in its outward manifestations that it is no longer of special interest -to the morphologist. The rate of growth is measured by the rate -of change in length of a radius, i.e. <i>V</i> -= (<i>R′</i> -− <i>R</i>) ⁄ <i>T</i>, and from -this we may calculate, as already indicated, the rate of growth in -terms of surface and of volume. The growing body remains of -constant form, owing to the symmetry of the system; because, -that is to say, on the one hand the pressure exerted by the growing -protoplasm is exerted equally in all directions, after the manner -of a hydrostatic pressure, which indeed it actually is: while on -the other hand, the “skin” or surface layer of -the cell is sufficiently <span class="xxpn" id="p060">{60}</span> -homogeneous to exert at every point an approximately uniform -resistance. Under these conditions then, the rate of growth is -uniform in all directions, and does not affect the form of the -organism.</p> - -<p>But in a larger or a more complex organism the study of growth, -and of the rate of growth, presents us with a variety of problems, -and the whole phenomenon becomes a factor of great morphological -importance. We no longer find that it tends to be uniform in -all directions, nor have we any right to expect that it should. -The resistances which it meets with will no longer be uniform. -In one direction but not in others it will be opposed by the -important resistance of gravity; and within the growing system -itself all manner of structural differences will come into play, -setting up unequal resistances to growth by the varying rigidity -or viscosity of the material substance in one direction or another. -At the same time, the actual sources of growth, the chemical and -osmotic forces which lead to the intussusception of new matter, -are not uniformly distributed; one tissue or one organ may well -manifest a tendency to increase while another does not; a series -of bones, their intervening cartilages, and their surrounding -muscles, may all be capable of very different rates of increment. -The differences of form which are the resultants of these differences -in rate of growth are especially manifested during that part of -life when growth itself is rapid: when the organism, as we say, -is undergoing its <i>development</i>. When growth in general has -become slow, the relative differences in rate between different -parts of the organism may still exist, and may be made manifest -by careful observation, but in many, or perhaps in most cases, the -resultant change of form does not strike the eye. Great as are -the differences between the rates of growth in different parts of -an organism, the marvel is that the ratios between them are so -nicely balanced as they actually are, and so capable, accordingly, -of keeping for long periods of time the form of the growing organism -all but unchanged. There is the nicest possible balance of forces -and resistances in every part of the complex body; and when -this normal equilibrium is disturbed, then we get abnormal -growth, in the shape of tumours, exostoses, and malformations -of every kind. <span class="xxpn" id="p061">{61}</span></p> - -<div class="section"> -<h3><i>The rate of growth in Man.</i></h3> - -<p>Man will serve us as well as another organism for our first -illustrations of rate of growth; and we cannot do better than go -for our first data concerning him to Quetelet’s <i>Anthropométrie</i><a class="afnanch" href="#fn94" id="fnanch94">94</a>, -an -epoch-making book for the biologist. For not only is it packed -with information, some of it still unsurpassed, in regard to human -growth and form, but it also merits our highest admiration as the -first great essay in scientific statistics, and the first work in which -organic variation was discussed from the point of view of the -mathematical theory of probabilities.</p></div> - -<div class="dctr01" id="fig3"> -<img src="images/i061.png" width="800" height="514" alt=""> - <div class="pcaption">Fig. 3. Curve of Growth in Man, from - birth to 20 yrs <span class="nowrap">(<img class="iglyph-a" -src="images/iglyph-malesign.png" width="28" height="47" -alt="♂">);</span>) from Quetelet’s Belgian data. The upper -curve of stature from Bowditch’s Boston data.</div></div> - -<p>If the child be some 20 inches, or say 50 cm. tall at birth, and -the man some six feet high, or say 180 cm., at twenty, we may -say that his <i>average</i> rate of growth has been -(180 − 50) ⁄ 20 cm., or -6·5 centimetres per annum. But we know very -well that this is <span class="xxpn" id="p062">{62}</span> -but a very rough preliminary statement, and that the boy grew -quickly during some, and slowly during other, of his twenty years. -It becomes necessary therefore to study the phenomenon of growth -in successive small portions; to study, that is to say, the successive -lengths, or the successive small differences, or increments, of -length (or of weight, etc.), attained in successive short increments -of time. This we do in the first instance in the usual way, by -the “graphic method” of plotting length against time, and so constructing -our “curve of growth.” Our curve of growth, whether -of weight or length (Fig. <a href="#fig3" title="go to Fig. 3">3</a>), has always a certain characteristic -form, or characteristic <i>curvature</i>. This is our immediate proof of -the fact that the <i>rate of growth</i> changes as time goes on; for had -it not been so, had an equal increment of length been added in -each equal interval of time, our “curve” would have appeared -as a straight line. Such as it is, it tells us not only that the rate -of growth tends to alter, but that it alters in a definite and orderly -way; for, subject to various minor interruptions, due to secondary -causes, our curves of growth are, on the whole, “smooth” curves.</p> - -<p>The curve of growth for length or stature in man indicates -a rapid increase at the outset, that is to say during the quick -growth of babyhood; a long period of slower, but still rapid and -almost steady growth in early boyhood; as a rule a marked -quickening soon after the boy is in his teens, when he comes to -“the growing age”; and finally a gradual arrest of growth as the -boy “comes to his full height,” and reaches manhood.</p> - -<p>If we carried the curve further, we should see a very curious thing. -We should see that a man’s full stature endures but for a spell; long -before fifty<a class="afnanch" href="#fn95" id="fnanch95">95</a> -it -has begun to abate, by sixty it is notably lessened, in extreme old -age the old man’s frame is shrunken and it is but a memory that “he -once was tall.” We have already seen, and here we see again, that -growth may have a “negative value.” The phenomenon of negative growth -in old age extends to weight also, and is evidently largely chemical -in origin: the organism can no longer add new material to its fabric -fast enough to keep pace with the wastage of time. Our curve <span -class="xxpn" id="p063">{63}</span> of growth is in fact a diagram -of activity, or “time-energy” diagram<a class="afnanch" href="#fn96" -id="fnanch96">96</a>. -As the organism grows it is absorbing energy -beyond its daily needs, and accumulating it at a rate depicted in -our</p> - -<div class="dtblbox"> -<table class="fsz7 borall"> -<caption class="captioncntr fsz4"> -<i>Stature, weight, and span of outstretched arms.</i><br> -(<i>After Quetelet</i>, <i>pp.</i> 193, 346.)</caption> -<tr> - <th class="borall"></th> - <th class="borall" colspan="2">Stature in metres</th> - <th class="borall"></th> - <th class="borall" colspan="2">Weight in kgm.</th> - <th class="borall"></th> - <th class="borall" rowspan="2">Span of arms, male</th> - <th class="borall" rowspan="2">% ratio of stature to span</th></tr> -<tr> - <th class="borall">Age</th> - <th class="borall">Male</th> - <th class="borall">Female</th> - <th class="borall">% F ⁄ M</th> - <th class="borall">Male</th> - <th class="borall">Female</th> - <th class="borall">% F ⁄ M</th></tr> -<tr> - <td class="tdright">0</td> - <td class="tdright">0·500</td> - <td class="tdright">0·494</td> - <td class="tdright">98·8</td> - <td class="tdright">3·2</td> - <td class="tdright">2·9</td> - <td class="tdright">90·7</td> - <td class="tdright">0·496</td> - <td class="tdright">100·8</td></tr> -<tr> - <td class="tdright">1</td> - <td class="tdright">0·698</td> - <td class="tdright">0·690</td> - <td class="tdright">98·8</td> - <td class="tdright">9·4</td> - <td class="tdright">8·8</td> - <td class="tdright">93·6</td> - <td class="tdright">0·695</td> - <td class="tdright">100·4</td></tr> -<tr> - <td class="tdright">2</td> - <td class="tdright">0·791</td> - <td class="tdright">0·781</td> - <td class="tdright">98·7</td> - <td class="tdright">11·3</td> - <td class="tdright">10·7</td> - <td class="tdright">94·7</td> - <td class="tdright">0·789</td> - <td class="tdright">100·3</td></tr> -<tr> - <td class="tdright">3</td> - <td class="tdright">0·864</td> - <td class="tdright">0·854</td> - <td class="tdright">98·8</td> - <td class="tdright">12·4</td> - <td class="tdright">11·8</td> - <td class="tdright">95·2</td> - <td class="tdright">0·863</td> - <td class="tdright">100·1</td></tr> -<tr> - <td class="tdright">4</td> - <td class="tdright">0·927</td> - <td class="tdright">0·915</td> - <td class="tdright">98·7</td> - <td class="tdright">14·2</td> - <td class="tdright">13·0</td> - <td class="tdright">91·5</td> - <td class="tdright">0·927</td> - <td class="tdright">100·0</td></tr> -<tr> - <td class="tdright">5</td> - <td class="tdright">0·987</td> - <td class="tdright">0·974</td> - <td class="tdright">98·7</td> - <td class="tdright">15·8</td> - <td class="tdright">14·4</td> - <td class="tdright">91·1</td> - <td class="tdright">0·988</td> - <td class="tdright">99·9</td></tr> -<tr> - <td class="tdright">6</td> - <td class="tdright">1·046</td> - <td class="tdright">1·031</td> - <td class="tdright">98·5</td> - <td class="tdright">17·2</td> - <td class="tdright">16·0</td> - <td class="tdright">93·0</td> - <td class="tdright">1·048</td> - <td class="tdright">99·8</td></tr> -<tr> - <td class="tdright">7</td> - <td class="tdright">1·104</td> - <td class="tdright">1·087</td> - <td class="tdright">98·4</td> - <td class="tdright">19·1</td> - <td class="tdright">17·5</td> - <td class="tdright">91·6</td> - <td class="tdright">1·107</td> - <td class="tdright">99·7</td></tr> -<tr> - <td class="tdright">8</td> - <td class="tdright">1·162</td> - <td class="tdright">1·142</td> - <td class="tdright">98·2</td> - <td class="tdright">20·8</td> - <td class="tdright">19·1</td> - <td class="tdright">91·8</td> - <td class="tdright">1·166</td> - <td class="tdright">99·6</td></tr> -<tr> - <td class="tdright">9</td> - <td class="tdright">1·218</td> - <td class="tdright">1·196</td> - <td class="tdright">98·2</td> - <td class="tdright">22·6</td> - <td class="tdright">21·4</td> - <td class="tdright">94·7</td> - <td class="tdright">1·224</td> - <td class="tdright">99·5</td></tr> -<tr> - <td class="tdright">10</td> - <td class="tdright">1·273</td> - <td class="tdright">1·249</td> - <td class="tdright">98·1</td> - <td class="tdright">24·5</td> - <td class="tdright">23·5</td> - <td class="tdright">95·9</td> - <td class="tdright">1·281</td> - <td class="tdright">99·4</td></tr> -<tr> - <td class="tdright">11</td> - <td class="tdright">1·325</td> - <td class="tdright">1·301</td> - <td class="tdright">98·2</td> - <td class="tdright">27·1</td> - <td class="tdright">25·6</td> - <td class="tdright">94·5</td> - <td class="tdright">1·335</td> - <td class="tdright">99·2</td></tr> -<tr> - <td class="tdright">12</td> - <td class="tdright">1·375</td> - <td class="tdright">1·352</td> - <td class="tdright">98·3</td> - <td class="tdright">29·8</td> - <td class="tdright">29·8</td> - <td class="tdright">100·0</td> - <td class="tdright">1·388</td> - <td class="tdright">99·1</td></tr> -<tr> - <td class="tdright">13</td> - <td class="tdright">1·423</td> - <td class="tdright">1·400</td> - <td class="tdright">98·4</td> - <td class="tdright">34·4</td> - <td class="tdright">32·9</td> - <td class="tdright">95·6</td> - <td class="tdright">1·438</td> - <td class="tdright">98·9</td></tr> -<tr> - <td class="tdright">14</td> - <td class="tdright">1·469</td> - <td class="tdright">1·446</td> - <td class="tdright">98·4</td> - <td class="tdright">38·8</td> - <td class="tdright">36·7</td> - <td class="tdright">94·6</td> - <td class="tdright">1·489</td> - <td class="tdright">98·7</td></tr> -<tr> - <td class="tdright">15</td> - <td class="tdright">1·513</td> - <td class="tdright">1·488</td> - <td class="tdright">98·3</td> - <td class="tdright">43·6</td> - <td class="tdright">40·4</td> - <td class="tdright">92·7</td> - <td class="tdright">1·538</td> - <td class="tdright">99·4</td></tr> -<tr> - <td class="tdright">16</td> - <td class="tdright">1·554</td> - <td class="tdright">1·521</td> - <td class="tdright">97·8</td> - <td class="tdright">49·7</td> - <td class="tdright">43·6</td> - <td class="tdright">87·7</td> - <td class="tdright">1·584</td> - <td class="tdright">98·1</td></tr> -<tr> - <td class="tdright">17</td> - <td class="tdright">1·594</td> - <td class="tdright">1·546</td> - <td class="tdright">97·0</td> - <td class="tdright">52·8</td> - <td class="tdright">47·3</td> - <td class="tdright">89·6</td> - <td class="tdright">1·630</td> - <td class="tdright">97·9</td></tr> -<tr> - <td class="tdright">18</td> - <td class="tdright">1·630</td> - <td class="tdright">1·563</td> - <td class="tdright">95·9</td> - <td class="tdright">57·8</td> - <td class="tdright">49·0</td> - <td class="tdright">84·8</td> - <td class="tdright">1·670</td> - <td class="tdright">97·6</td></tr> -<tr> - <td class="tdright">19</td> - <td class="tdright">1·655</td> - <td class="tdright">1·570</td> - <td class="tdright">94·9</td> - <td class="tdright">58·0</td> - <td class="tdright">51·6</td> - <td class="tdright">89·0</td> - <td class="tdright">1·705</td> - <td class="tdright">97·1</td></tr> -<tr> - <td class="tdright">20</td> - <td class="tdright">1·669</td> - <td class="tdright">1·574</td> - <td class="tdright">94·3</td> - <td class="tdright">60·1</td> - <td class="tdright">52·3</td> - <td class="tdright">87·0</td> - <td class="tdright">1·728</td> - <td class="tdright">96·6</td></tr> -<tr> - <td class="tdright">25</td> - <td class="tdright">1·682</td> - <td class="tdright">1·578</td> - <td class="tdright">93·8</td> - <td class="tdright">62·9</td> - <td class="tdright">53·3</td> - <td class="tdright">84·7</td> - <td class="tdright">1·731</td> - <td class="tdright">97·2</td></tr> -<tr> - <td class="tdright">30</td> - <td class="tdright">1·686</td> - <td class="tdright">1·580</td> - <td class="tdright">93·7</td> - <td class="tdright">63·7</td> - <td class="tdright">54·3</td> - <td class="tdright">85·3</td> - <td class="tdright">1·766</td> - <td class="tdright">95·5</td></tr> -<tr> - <td class="tdright">40</td> - <td class="tdright">1·686</td> - <td class="tdright">1·580</td> - <td class="tdright">93·7</td> - <td class="tdright">63·7</td> - <td class="tdright">55·2</td> - <td class="tdright">86·7</td> - <td class="tdright">1·766</td> - <td class="tdright">95·5</td></tr> -<tr> - <td class="tdright">50</td> - <td class="tdright">1·686</td> - <td class="tdright">1·580</td> - <td class="tdright">93·7</td> - <td class="tdright">63·5</td> - <td class="tdright">56·2</td> - <td class="tdright">88·4</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td></tr> -<tr> - <td class="tdright">60</td> - <td class="tdright">1·676</td> - <td class="tdright">1·571</td> - <td class="tdright">93·7</td> - <td class="tdright">61·9</td> - <td class="tdright">54·3</td> - <td class="tdright">87·7</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td></tr> -<tr> - <td class="tdright">70</td> - <td class="tdright">1·660</td> - <td class="tdright">1·556</td> - <td class="tdright">93·7</td> - <td class="tdright">59·5</td> - <td class="tdright">51·5</td> - <td class="tdright">86·5</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td></tr> -<tr> - <td class="tdright">80</td> - <td class="tdright">1·636</td> - <td class="tdright">1·534</td> - <td class="tdright">93·8</td> - <td class="tdright">57·8</td> - <td class="tdright">49·4</td> - <td class="tdright">85·5</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td></tr> -<tr> - <td class="tdright">90</td> - <td class="tdright">1·610</td> - <td class="tdright">1·510</td> - <td class="tdright">93·8</td> - <td class="tdright">57·8</td> - <td class="tdright">49·3</td> - <td class="tdright">85·3</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td></tr> -</table></div><!--dtblbox--> - -<p class="pcontinue">curve; but the time comes when it accumulates no longer, and at -last it is constrained to draw upon its dwindling store. But in part, -the slow decline in stature is an expression of an unequal contest -between our bodily powers and the -unchanging force of gravity, <span class="xxpn" id="p064">{64}</span> -which draws us down when we would fain rise up<a class="afnanch" href="#fn97" id="fnanch97">97</a>. -For against -gravity we fight all our days, in every movement of our limbs, in -every beat of our hearts; it is the indomitable force that defeats -us in the end, that lays us on our deathbed, that lowers us to the -grave<a class="afnanch" href="#fn98" id="fnanch98">98</a>.</p> - -<p>Side by side with the curve which represents growth in length, -or stature, our diagram shows the curve of weight<a class="afnanch" href="#fn99" id="fnanch99">99</a>. -That this -curve is of a very different shape from the former one, is accounted -for in the main (though not wholly) by the fact which we have -already dealt with, that, whatever be the law of increment in a -linear dimension, the law of increase in volume, and therefore in -weight, will be that these latter magnitudes tend to vary as -the cubes of the linear dimensions. This however does not -account for the change of direction, or “point of inflection” -which we observe in the curve of weight at about one or two -years old, nor for certain other differences between our two curves -which the scale of our diagram does not yet make clear. These -differences are due to the fact that the form of the child is altering -with growth, that other linear dimensions are varying somewhat -differently from length or stature, and that consequently the -growth in bulk or weight is following a more complicated law.</p> - -<p>Our curve of growth, whether for weight or length, is a direct -picture of velocity, for it represents, as a connected series, the -successive epochs of time at which successive weights or lengths -are attained. But, as we have already in part seen, a great part -of the interest of our curve lies in the fact that we can see from -it, not only that length (or some other magnitude) is changing, -but that the <i>rate of change</i> of magnitude, or rate of growth, is -itself changing. We have, in short, to study the phenomenon of -<i>acceleration</i>: we have begun by studying a -velocity, or rate of <span class="xxpn" id="p065">{65}</span> -change of magnitude; we must now study an acceleration, or -rate of change of velocity. The rate, or velocity, of growth is -measured by the <i>slope</i> of the curve; where the curve is steep, it -means that growth is rapid, and when growth ceases the curve -appears as a horizontal line. If we can find a means, then, of -representing at successive epochs the corresponding slope, or -steepness, of the curve, we shall have obtained a picture of the -rate of change of velocity, or the acceleration of growth. The -measure of the steepness of a curve is given by the tangent to -the curve, or we may estimate it by taking for equal intervals -of time (strictly speaking, for each infinitesimal interval of time) -the actual increment added during that interval of time: and in -practice this simply amounts to taking the successive <i>differences</i> -between the values of length (or of weight) for the successive -ages which we have begun by studying. If we then plot these -successive <i>differences</i> against time, we obtain a curve each point -upon which represents a velocity, and the whole curve indicates -the rate of change of velocity, and we call it an acceleration-curve. -It contains, in truth, nothing whatsoever that was not implicit -in our former curve; but it makes clear to our eye, and brings -within the reach of further investigation, phenomena that were -hard to see in the other mode of representation.</p> - -<p>The acceleration-curve of height, which we here illustrate, in -Fig. <a href="#fig4" title="go to Fig. 4">4</a>, is very different in form from the curve of growth which -we have just been looking at; and it happens that, in this case, -there is a very marked difference between the curve which we -obtain from Quetelet’s data of growth in height and that which -we may draw from any other series of observations known to me -from British, French, American or German writers. It begins (as -will be seen from our next table) at a very high level, such -as it never afterwards attains; and still stands too high, during -the first three or four years of life, to be represented on the scale -of the accompanying diagram. From these high velocities it falls -away, on the whole, until the age when growth itself ceases, and -when the rate of growth, accordingly, has, for some years together, -the constant value of <i>nil</i>; but the rate of fall, or rate of change of -velocity, is subject to several changes or interruptions. During -the first three or four years of life the fall is -continuous and rapid, <span class="xxpn" id="p066">{66}</span> -but it is somewhat arrested for a while in childhood, from about -five years old to eight. According to Quetelet’s data, there is -another slight interruption in the falling rate between the ages of -about fourteen and sixteen; but in place of this almost insignificant -interruption, the English and other statistics indicate a sudden</p> - -<div class="dctr04" id="fig4"> -<img src="images/i066.png" width="799" height="1056" alt=""> - <div class="dcaption">Fig. 4. Mean annual increments of - stature <span class="nowrap">(<img class="iglyph-a" -src="images/iglyph-malesign.png" width="28" height="47" -alt="♂">),</span> Belgian and American.</div></div> - -<p class="pcontinue">and very marked acceleration of growth beginning at about -twelve years of age, and lasting for three or four years; when -this period of acceleration is over, the rate begins to fall again, -and does so with great rapidity. We do not know how far the -absence of this striking feature in the Belgian curve is due to the -imperfections of Quetelet’s data, or whether it is a real and -significant feature in the small-statured race which he investigated.</p> - -<div class="dtblbox"> -<table class="fsz8 borall"> -<caption class="captioncntr fsz4"> -<i>Annual Increment of Stature (in cm.) from Belgian and -American Statistics.</i></caption> -<tr> - <th class="borall"></th> - <th class="borall" colspan="2">Belgian (Quetelet, p. 344)</th> - <th class="borall" colspan="4">Paris* (Variot et Chaumet, p. 55)</th> - <th class="borall" colspan="3">Toronto† (Boas, p. 1547)</th> - <th class="borall" colspan="4">Worcester‡, Mass. (Boas, p. 1548)</th></tr> -<tr> - <th class="borall" rowspan="2">Age</th> - <th class="borall" rowspan="2">Height (Boys)</th> - <th class="borall" rowspan="2">Ann. increment</th> - <th class="borall" colspan="2">Height</th> - <th class="borall" colspan="2">Increment</th> - <th class="borall" rowspan="2">Height (Boys)</th> - <th class="borall" rowspan="2">Variability of do. (6)</th> - <th class="borall" rowspan="2">Ann. increment</th> - <th class="borall" rowspan="2">Ann. increment (Boys)</th> - <th class="borall" rowspan="2">Variability of do.</th> - <th class="borall" rowspan="2">Ann. increment (Girls)</th> - <th class="borall" rowspan="2">Variability of do.</th></tr> -<tr> - <th class="borall">Boys</th> - <th class="borall">Girls</th> - <th class="borall">Boys</th> - <th class="borall">Girls</th></tr> -<tr> - <td class="tdright">0</td> - <td class="tdright">50·0</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td></tr> -<tr> - <td class="tdright">1</td> - <td class="tdright">69·8</td> - <td class="tdright">19·8</td> - <td class="tdright">74·2</td> - <td class="tdright">73·6</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td></tr> -<tr> - <td class="tdright">2</td> - <td class="tdright">79·1</td> - <td class="tdright">9·3</td> - <td class="tdright">82·7</td> - <td class="tdright">81·8</td> - <td class="tdright">8·5</td> - <td class="tdright">8·2</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td></tr> -<tr> - <td class="tdright">3</td> - <td class="tdright">86·4</td> - <td class="tdright">7·3</td> - <td class="tdright">89·1</td> - <td class="tdright">88·4</td> - <td class="tdright">6·4</td> - <td class="tdright">6·6</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td></tr> -<tr> - <td class="tdright">4</td> - <td class="tdright">92·7</td> - <td class="tdright">6·3</td> - <td class="tdright">96·8</td> - <td class="tdright">95·8</td> - <td class="tdright">7·7</td> - <td class="tdright">7·4</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td></tr> -<tr> - <td class="tdright">5</td> - <td class="tdright">98·7</td> - <td class="tdright">6·0</td> - <td class="tdright">103·3</td> - <td class="tdright">101·9</td> - <td class="tdright">6·5</td> - <td class="tdright">6·1</td> - <td class="tdright">105·90</td> - <td class="tdright">4·40</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td></tr> -<tr> - <td class="tdright">6</td> - <td class="tdright">104·0</td> - <td class="tdright">5·9</td> - <td class="tdright">109·9</td> - <td class="tdright">108·9</td> - <td class="tdright">6·6</td> - <td class="tdright">7·0</td> - <td class="tdright">111·58</td> - <td class="tdright">4·62</td> - <td class="tdright">5·68</td> - <td class="tdright">6·55</td> - <td class="tdright">1·57</td> - <td class="tdright">5·75</td> - <td class="tdright">0·88</td></tr> -<tr> - <td class="tdright">7</td> - <td class="tdright">110·4</td> - <td class="tdright">5·8</td> - <td class="tdright">114·4</td> - <td class="tdright">113·8</td> - <td class="tdright">4·5</td> - <td class="tdright">4·9</td> - <td class="tdright">116·83</td> - <td class="tdright">4·93</td> - <td class="tdright">5·25</td> - <td class="tdright">5·70</td> - <td class="tdright">0·68</td> - <td class="tdright">5·90</td> - <td class="tdright">0·98</td></tr> -<tr> - <td class="tdright">8</td> - <td class="tdright">116·2</td> - <td class="tdright">5·8</td> - <td class="tdright">119·7</td> - <td class="tdright">119·5</td> - <td class="tdright">5·3</td> - <td class="tdright">5·7</td> - <td class="tdright">122·04</td> - <td class="tdright">5·34</td> - <td class="tdright">5·21</td> - <td class="tdright">5·37</td> - <td class="tdright">0·86</td> - <td class="tdright">5·70</td> - <td class="tdright">1·10</td></tr> -<tr> - <td class="tdright">9</td> - <td class="tdright">121·8</td> - <td class="tdright">5·6</td> - <td class="tdright">125·0</td> - <td class="tdright">124·7</td> - <td class="tdright">5·3</td> - <td class="tdright">4·8</td> - <td class="tdright">126·91</td> - <td class="tdright">5·49</td> - <td class="tdright">4·87</td> - <td class="tdright">4·89</td> - <td class="tdright">0·96</td> - <td class="tdright">5·50</td> - <td class="tdright">0·97</td></tr> -<tr> - <td class="tdright">10</td> - <td class="tdright">127·3</td> - <td class="tdright">5·5</td> - <td class="tdright">130·3</td> - <td class="tdright">129·5</td> - <td class="tdright">5·3</td> - <td class="tdright">5·2</td> - <td class="tdright">131·78</td> - <td class="tdright">5·75</td> - <td class="tdright">4·87</td> - <td class="tdright">5·10</td> - <td class="tdright">1·03</td> - <td class="tdright">5·97</td> - <td class="tdright">1·23</td></tr> -<tr> - <td class="tdright">11</td> - <td class="tdright">132·5</td> - <td class="tdright">5·2</td> - <td class="tdright">133·6</td> - <td class="tdright">134·4</td> - <td class="tdright">3·3</td> - <td class="tdright">4·9</td> - <td class="tdright">136·20</td> - <td class="tdright">6·19</td> - <td class="tdright">4·42</td> - <td class="tdright">5·02</td> - <td class="tdright">0·88</td> - <td class="tdright">6·17</td> - <td class="tdright">1·85</td></tr> -<tr> - <td class="tdright">12</td> - <td class="tdright">137·5</td> - <td class="tdright">5·0</td> - <td class="tdright">137·6</td> - <td class="tdright">141·5</td> - <td class="tdright">4·0</td> - <td class="tdright">7·1</td> - <td class="tdright">140·74</td> - <td class="tdright">6·66</td> - <td class="tdright">4·54</td> - <td class="tdright">4·99</td> - <td class="tdright">1·26</td> - <td class="tdright">6·98</td> - <td class="tdright">1·89</td></tr> -<tr> - <td class="tdright">13</td> - <td class="tdright">142·3</td> - <td class="tdright">4·8</td> - <td class="tdright">145·1</td> - <td class="tdright">148·6</td> - <td class="tdright">7·5</td> - <td class="tdright">7·1</td> - <td class="tdright">146·00</td> - <td class="tdright">7·54</td> - <td class="tdright">5·26</td> - <td class="tdright">5·91</td> - <td class="tdright">1·86</td> - <td class="tdright">6·71</td> - <td class="tdright">2·06</td></tr> -<tr> - <td class="tdright">14</td> - <td class="tdright">146·9</td> - <td class="tdright">4·6</td> - <td class="tdright">153·8</td> - <td class="tdright">152·9</td> - <td class="tdright">8·7</td> - <td class="tdright">4·3</td> - <td class="tdright">152·39</td> - <td class="tdright">8·49</td> - <td class="tdright">6·39</td> - <td class="tdright">7·88</td> - <td class="tdright">2·39</td> - <td class="tdright">5·44</td> - <td class="tdright">2·89</td></tr> -<tr> - <td class="tdright">15</td> - <td class="tdright">151·3</td> - <td class="tdright">4·4</td> - <td class="tdright">159·6</td> - <td class="tdright">154·2</td> - <td class="tdright">5·8</td> - <td class="tdright">1·3</td> - <td class="tdright">159·72</td> - <td class="tdright">8·78</td> - <td class="tdright">7·33</td> - <td class="tdright">6·23</td> - <td class="tdright">2·91</td> - <td class="tdright">5·34</td> - <td class="tdright">2·71</td></tr> -<tr> - <td class="tdright">16</td> - <td class="tdright">155·4</td> - <td class="tdright">4·1</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdright">164·90</td> - <td class="tdright">7·73</td> - <td class="tdright">5·18</td> - <td class="tdright">5·64</td> - <td class="tdright">3·46</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td></tr> -<tr> - <td class="tdright">17</td> - <td class="tdright">159·4</td> - <td class="tdright">4·0</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdright">168·91</td> - <td class="tdright">7·22</td> - <td class="tdright">4·01</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td></tr> -<tr> - <td class="tdright">18</td> - <td class="tdright">163·0</td> - <td class="tdright">3·6</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdright">171·07</td> - <td class="tdright">6·74</td> - <td class="tdright">2·16</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td></tr> -<tr> - <td class="tdright">19</td> - <td class="tdright">165·5</td> - <td class="tdright">2·5</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td></tr> -<tr> - <td class="tdright">20</td> - <td class="tdright">167·0</td> - <td class="tdright">1·5</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td></tr> -</table> - -<p class="ptblfn">* Ages from 1–2, 2–3, etc.</p> - -<p class="ptblfn">† The epochs are, in this table, 5·5, 6·5, years, etc.</p> - -<p class="ptblfn">‡ Direct observations on actual, or individualised, -increase of stature from year to year: between the ages of -5–6, 6–7, etc.</p> -</div><!--dtblbox--> - -<p>Even apart from these data of Quetelet’s (which seem to -constitute an extreme case), it is evident that -there are very <span class="xxpn" id="p068">{68}</span> -marked differences between different races, as we shall presently -see there are between the two sexes, in regard to the epochs of -acceleration of growth, in other words, in the “phase” of the -curve.</p> - -<p>It is evident that, if we pleased, we might represent the <i>rate -of change of acceleration</i> on yet another curve, by constructing a -table of “second differences”; this would bring out certain very -interesting phenomena, which here however we must not stay to -discuss.</p> - -<div class="dtblbox"><div class="nowrap"> -<table class="fsz6 borall"> -<caption class="captioncntr fsz5"> -<i>Annual Increment of Weight in Man</i> (<i>kgm.</i>).<br> -(After Quetelet, <i>Anthropométrie</i>, p. 346*.)</caption> -<tr> - <th class="borall"></th> - <th class="borall" colspan="2">Increment</th> - <th>    </th> - <th class="borall"></th> - <th class="borall" colspan="2">Increment</th></tr> -<tr> - <th class="borall">Age</th> - <th class="borall">Male</th> - <th class="borall">Female</th> - <th>    </th> - <th class="borall">Age</th> - <th class="borall">Male</th> - <th class="borall">Female</th></tr> -<tr> - <td class="tdright">0–1 </td> - <td class="tdright">5·9</td> - <td class="tdright">5·6</td> - <td>    </td> - <td class="tdright">12–13</td> - <td class="tdright">4·1</td> - <td class="tdright">3·5</td></tr> -<tr> - <td class="tdright">1–2 </td> - <td class="tdright">2·0</td> - <td class="tdright">2·4</td> - <td>    </td> - <td class="tdright">13–14</td> - <td class="tdright">4·0</td> - <td class="tdright">3·8</td></tr> -<tr> - <td class="tdright">2–3 </td> - <td class="tdright">1·5</td> - <td class="tdright">1·4</td> - <td>    </td> - <td class="tdright">14–15</td> - <td class="tdright">4·1</td> - <td class="tdright">3·7</td></tr> -<tr> - <td class="tdright">3–4 </td> - <td class="tdright">1·5</td> - <td class="tdright">1·5</td> - <td>    </td> - <td class="tdright">15–16</td> - <td class="tdright">4·2</td> - <td class="tdright">3·5</td></tr> -<tr> - <td class="tdright">4–5 </td> - <td class="tdright">1·9</td> - <td class="tdright">1·4</td> - <td>    </td> - <td class="tdright">16–17</td> - <td class="tdright">4·3</td> - <td class="tdright">3·3</td></tr> -<tr> - <td class="tdright">5–6 </td> - <td class="tdright">1·9</td> - <td class="tdright">1·4</td> - <td>    </td> - <td class="tdright">17–18</td> - <td class="tdright">4·2</td> - <td class="tdright">3·0</td></tr> -<tr> - <td class="tdright">6–7 </td> - <td class="tdright">1·9</td> - <td class="tdright">1·1</td> - <td>    </td> - <td class="tdright">18–19</td> - <td class="tdright">3·7</td> - <td class="tdright">2·3</td></tr> -<tr> - <td class="tdright">7–8 </td> - <td class="tdright">1·9</td> - <td class="tdright">1·2</td> - <td>    </td> - <td class="tdright">19–20</td> - <td class="tdright">1·9</td> - <td class="tdright">1·1</td></tr> -<tr> - <td class="tdright">8–9 </td> - <td class="tdright">1·9</td> - <td class="tdright">2·0</td> - <td>    </td> - <td class="tdright">20–21</td> - <td class="tdright">1·7</td> - <td class="tdright">1·1</td></tr> -<tr> - <td class="tdright">9–10</td> - <td class="tdright">1·7</td> - <td class="tdright">2·1</td> - <td>    </td> - <td class="tdright">21–22</td> - <td class="tdright">1·7</td> - <td class="tdright">0·5</td></tr> -<tr> - <td class="tdright">10–11</td> - <td class="tdright">1·8</td> - <td class="tdright">2·4</td> - <td>    </td> - <td class="tdright">22–23</td> - <td class="tdright">1·6</td> - <td class="tdright">0·4</td></tr> -<tr> - <td class="tdright">11–12</td> - <td class="tdright">2·0</td> - <td class="tdright">3·5</td> - <td>    </td> - <td class="tdright">23–24</td> - <td class="tdright">0·9</td> - <td class="tdright">−0·2</td></tr> -<tr> - <td class="tdright">12–13</td> - <td class="tdright">4·1</td> - <td class="tdright">3·5</td> - <td>    </td> - <td class="tdright">24–25</td> - <td class="tdright">0·8</td> - <td class="tdright">−0·2</td></tr> -</table></div> -<p class="ptblfn">* The values given in this table are not in precise accord -with those of the Table on p. <a href="#p063" title="go to pg. 63">63</a>. The latter represent -Quetelet’s results arrived at in 1835; the former are the -means of his determinations in 1835–40.</p> -</div><!--dtblbox--> - -<p>The acceleration-curve for man’s weight (Fig. -<a href="#fig5" title="go to Fig. 5">5</a>), whether we -draw it from Quetelet’s data, or from the British, American and -other statistics of later writers, is on the whole similar to that -which we deduce from the statistics of these latter writers in -regard to height or stature; that is to say, it is not a curve which -continually descends, but it indicates a rate of growth which is -subject to important fluctuations at certain epochs of life. We see -that it begins at a high level, and falls -continuously and rapidly<a class="afnanch" href="#fn100" id="fnanch100">100</a> -<span class="xxpn" id="p069">{69}</span> -during the first two or three years of life. After a slight recovery, -it runs nearly level during boyhood from about five to twelve -years old; it then rapidly rises, in the “growing period” of the -early teens, and slowly and steadily falls from about the age of -sixteen onwards. It does not reach the base-line till the man is -about seven or eight and twenty, for normal increase of weight -continues during the years when the man is “filling out,” long -after growth in height has ceased; but at last, somewhere about -thirty, the velocity reaches zero, and even falls below it, for then -the man usually begins to lose weight a little. The subsequent -slow changes in this acceleration-curve we need not stop to deal -with.</p> - -<div class="dctr01" id="fig5"> -<img src="images/i069.png" width="800" height="557" alt=""> - <div class="dcaption">Fig. 5. Mean annual increments of - weight, in man and woman; from Quetelet’s data.</div></div> - -<p>In the same diagram (Fig. <a href="#fig5" title="go to Fig. 5">5</a>) I have set forth the acceleration-curves -in respect of increment of weight for both man and woman, -according to Quetelet. That growth in boyhood and growth in -girlhood follow a very different course is a matter of common -knowledge; but if we simply plot the ordinary curve of growth, -or velocity-curve, the difference, on the small scale -of our diagrams, <span class="xxpn" id="p070">{70}</span> -is not very apparent. It is admirably brought out, however, in -the acceleration-curves. Here we see that, after infancy, say -from three years old to eight, the velocity in the girl is steady, -just as in the boy, but it stands on a lower level in her case than -in his: the little maid at this age is growing slower than the boy. -But very soon, and while his acceleration-curve is still represented -by a straight line, hers has begun to ascend, and until the girl -is about thirteen or fourteen it continues to ascend rapidly. -After that age, as after sixteen or seventeen in the boy’s case, it -begins to descend. In short, throughout all this period, it is a very -<i>similar</i> curve in the two sexes; but it has its notable differences, -in amplitude and especially in <i>phase</i>. Last of all, we may notice -that while the acceleration-curve falls to a negative value in the -male about or even a little before the age of thirty years, this -does not happen among women. They continue to grow in -weight, though slowly, till very much later in life; until there -comes a final period, in both sexes alike, during which weight, -and height and strength all alike diminish.</p> - -<div class="psmprnt3"> -<p>From certain corrected, or “typical” values, given for -American children by Boas and Wissler (<i>l.c.</i> p. 42), -we obtain the following still clearer comparison of the -annual increments of <i>stature</i> in boys and girls: the -typical stature at the commencement of the period, i.e. -at the age of eleven, being 135·1 cm. and 136·9 cm. for -the boys and girls respectively, and the annual increments -being as follows:</p> - -<div class="dtblbox"><div class="nowrap"> -<table> -<tr> - <td class="tdleft">Age</td> - <td class="tdcntr">12</td> - <td class="tdcntr">13</td> - <td class="tdcntr">14</td> - <td class="tdcntr">15</td> - <td class="tdcntr">16</td> - <td class="tdcntr">17</td> - <td class="tdcntr">18</td> - <td class="tdcntr">19</td> - <td class="tdcntr">20</td></tr> -<tr> - <td class="tdleft">Boys (cm.)</td> - <td class="tdright">4·1</td> - <td class="tdright">6·3</td> - <td class="tdright">8·7</td> - <td class="tdright">7·9</td> - <td class="tdright">5·2</td> - <td class="tdright">3·2</td> - <td class="tdright">1·9</td> - <td class="tdright">0·9</td> - <td class="tdright">0·3</td></tr> -<tr> - <td class="tdleft">Girls (cm.)</td> - <td class="tdright">7·5</td> - <td class="tdright">7·0</td> - <td class="tdright">4·6</td> - <td class="tdright">2·1</td> - <td class="tdright">0·9</td> - <td class="tdright">0·4</td> - <td class="tdright">0·1</td> - <td class="tdright">0·0</td> - <td class="tdright">0·0</td></tr> -<tr> - <td class="tdleft">Difference</td> - <td class="tdright">−3·4</td> - <td class="tdright">−0·7</td> - <td class="tdright">4·1</td> - <td class="tdright">5·8</td> - <td class="tdright">4·3</td> - <td class="tdright">2·8</td> - <td class="tdright">1·8</td> - <td class="tdright">0·9</td> - <td class="tdright">0·3</td></tr> -</table></div></div><!--dtblbox--> -</div><!--psmprnt3--> - -<p>The result of these differences (which are essentially <i>phase</i>-differences) -between the two sexes in regard to the velocity of -growth and to the rate of change of that velocity, is to cause the -<i>ratio</i> between the weights of the two sexes to fluctuate in a somewhat -complicated manner. At birth the baby-girl weighs on the -average nearly 10 per cent. less than the boy. Till about two -years old she tends to gain upon him, but she then loses again -until the age of about five; from five she gains for a few years -somewhat rapidly, and the girl of ten to twelve is only some -3 per cent. less in weight than the boy. The boy in -his teens gains <span class="xxpn" id="p071">{71}</span> -steadily, and the young woman of twenty is nearly 15 per cent. -lighter than the man. This ratio of difference again slowly -diminishes, and between fifty and sixty stands at about 12 per -cent., or not far from the mean for all ages; but once more as -old age advances, the difference tends, though very slowly, to -increase (Fig. <a href="#fig6" title="go to Fig. 6">6</a>).</p> - -<div class="dctr01" id="fig6"> -<img src="images/i071.png" width="799" height="389" alt=""> - <div class="dcaption">Fig. 6. Percentage ratio, throughout life, -of female weight to male; from Quetelet’s data.</div></div> - -<p>While careful observations on the rate of growth in other animals -are somewhat scanty, they tend to show so far as they go that the -general features of the phenomenon are always much the same. Whether -the animal be long-lived, as man or the elephant, or short-lived, -like horse or dog, it passes through the same phases of growth<a -class="afnanch" href="#fn101" id="fnanch101">101</a>. -In all cases -growth begins slowly; it attains a maximum velocity early in its -course, and afterwards slows down (subject to temporary accelerations) -towards a point where growth ceases altogether. But especially in the -cold-blooded animals, such as fishes, the slowing-down period is very -greatly protracted, and the size of the creature would seem never -actually to reach, but only to approach asymptotically, to a maximal -limit.</p> - -<p>The size ultimately attained is a resultant of the rate, and of -<span class="xxpn" id="p072">{72}</span> the duration, of growth. It is -in the main true, as Minot has said, that the rabbit is bigger than the -guinea-pig because he grows the faster; but that man is bigger than the -rabbit because he goes on growing for a longer time.</p> - -<hr class="hrblk"> - -<p>In ordinary physical investigations dealing with velocities, as -for instance with the course of a projectile, we pass at once from -the study of acceleration to that of momentum and so to that of -force; for change of momentum, which is proportional to force, -is the product of the mass of a body into its acceleration or change -of velocity. But we can take no such easy road of kinematical -investigation in this case. The “velocity” of growth is a very -different thing from the “velocity” of the projectile. The forces -at work in our case are not susceptible of direct and easy treatment; -they are too varied in their nature and too indirect in their action -for us to be justified in equating them directly with the mass of -the growing structure.</p> - -<div class="psmprnt3"> -<p>It was apparently from a feeling that the velocity of -growth ought in some way to be equated with the mass of -the growing structure that Minot<a class="afnanch" href="#fn102" id="fnanch102">102</a> -introduced a curious, -and (as it seems to me) an unhappy method of representing -growth, in the form of what he called “percentage-curves”; -a method which has been followed by a number of other -writers and experimenters. Minot’s method was to deal, -not with the actual increments added in successive -periods, such as years or days, but with these increments -represented as <i>percentages</i> of the amount which had been -reached at the end of the former period. For instance, -taking Quetelet’s values for the height in centimetres of -a male infant from birth to four years old, as follows:</p> - -<div class="dtblbox"><div class="nowrap"> -<table> -<tr> - <td class="tdleft">Years</td> - <td class="tdcntr">0</td> - <td class="tdcntr">1</td> - <td class="tdcntr">2</td> - <td class="tdcntr">3</td> - <td class="tdcntr">4</td></tr> -<tr> - <td class="tdleft">cm.</td> - <td class="tdright">50·0</td> - <td class="tdright">69·8</td> - <td class="tdright">79·1</td> - <td class="tdright">86·4</td> - <td class="tdright">92·7</td></tr> -</table> -</div></div><!--dtblbox--> - -<p class="pcontinue">Minot would state the percentage growth in each of the -four annual periods at 39·6, 13·3, 9·6 and 7·3 per cent. -respectively.</p> - -<p>Now when we plot actual length against time, we have a -perfectly definite thing. When we differentiate this -<i>L ⁄ T</i>, we have <i>dL ⁄ dT</i>, which is (of course) velocity; -and from this, by a second differentiation, we obtain <span class="nowrap"> -<i>d</i><sup>2</sup> <i>L ⁄ dT</i><sup>2</sup> ,</span> -that is to say, the acceleration. -<span class="xxpn" id="p073">{73}</span></p> - -<div class="dmaths"> -<p>But when you take percentages of <i>y</i>, you are determining <i>dy ⁄ y</i>, and when -you plot this against <i>dx</i>, you have</p> - -<div>(<i>dy ⁄ y</i>) ⁄ <i>dx</i>, -or <i>dy</i> ⁄ (<i>y</i> · <i>dx</i>), -or (1 ⁄ <i>y</i>) · (<i>dy ⁄ dx</i>),</div> - -<p class="pcontinue">that -is to say, you are multiplying the thing you wish to represent by another -quantity which is itself continually varying; and the result is that you are -dealing with something very much less easily grasped by the mind than the -original factors. Professor Minot is, of course, dealing with a perfectly -legitimate function of <i>x</i> and <i>y</i>; and his method is practically tantamount to -plotting log <i>y</i> against <i>x</i>, that is to say, the logarithm of the increment against -the time. This could only be defended and justified if it led to some simple -result, for instance if it gave us a straight line, or some other simpler curve -than our usual curves of growth. As a matter of fact, it is manifest that it -does nothing of the kind.</p></div><!--dmaths--> -</div><!--psmprnt3--> - -<div class="section"> -<h3><i>Pre-natal and post-natal growth.</i></h3></div> - -<p>In the acceleration-curves which we have shown above -(Figs. <a href="#fig2" title="go to Fig. 2">2</a>, 3), -it will be seen that the curve starts at a considerable interval from -the actual date of birth; for the first two increments which we can as -yet compare with one another are those attained during the first and -second complete years of life. Now we can in many cases “interpolate” -with safety <i>between</i> known points upon a curve, but it is very -much less safe, and is not very often justifiable (at least until -we understand the physical principle involved, and its mathematical -expression), to “extrapolate” beyond the limits of our observations. In -short, we do not yet know whether our curve continued to ascend as we -go backwards to the date of birth, or whether it may not have changed -its direction, and descended, perhaps, to zero-value. In regard to -length, or stature, however, we can obtain the requisite information -from certain tables of Rüssow’s<a class="afnanch" href="#fn103" -id="fnanch103">103</a>, -who gives the stature of the infant month by -month during the first year of its life, as follows:</p> - -<div class="dtblbox"><div class="nowrap"> -<table class="fsz7 borall"> -<tr> - <td class="tdleft">Age in months</td> - <td class="tdcntr borall">0</td> - <td class="tdcntr borall">1</td> - <td class="tdcntr borall">2</td> - <td class="tdcntr borall">3</td> - <td class="tdcntr borall">4</td> - <td class="tdcntr borall">5</td> - <td class="tdcntr borall">6</td> - <td class="tdcntr borall">7</td> - <td class="tdcntr borall">8</td> - <td class="tdcntr borall">9</td> - <td class="tdcntr borall">10</td> - <td class="tdcntr borall">11</td> - <td class="tdcntr borall">12</td></tr> -<tr> - <td class="tdleft">Length in cm.</td> - <td class="tdxl">(50)</td> - <td class="tdxl">54</td> - <td class="tdxl">58</td> - <td class="tdxl">60</td> - <td class="tdxl">62</td> - <td class="tdxl">64</td> - <td class="tdxl">65</td> - <td class="tdxl">66</td> - <td class="tdxl">67·5</td> - <td class="tdxl">68</td> - <td class="tdxl">69</td> - <td class="tdxl">70·5</td> - <td class="tdxl">72</td></tr> -<tr> - <td class="tdleft">[Differences (in cm.)</td> - <td></td> - <td class="tdl">4</td> - <td class="tdl">4</td> - <td class="tdl">2</td> - <td class="tdl">2</td> - <td class="tdl">2</td> - <td class="tdl">1</td> - <td class="tdl">1</td> - <td class="tdl">1·5</td> - <td class="tdl">·5</td> - <td class="tdl">1</td> - <td class="tdl">1·5</td> - <td class="tdl">1·5]</td></tr> -</table></div></div><!--dtblbox--> - -<p>If we multiply these <i>monthly</i> differences, or mean monthly -velocities, by 12, to bring them into a form -comparable with the <span class="xxpn" id="p074">{74}</span> -<i>annual</i> velocities already represented on our acceleration-curves, -we shall see that the one series of observations joins on very well -with the other; and in short we see at once that our acceleration-curve -rises steadily and rapidly as we pass back towards the date -of birth.</p> - -<div class="dctr02" id="fig7"> -<img src="images/i074.png" width="800" height="728" alt=""> - <div class="pcaption">Fig. 7. Curve of growth (in length or -stature) of child, before and after birth. (From His and -Rüssow’s data.)</div></div> - -<p>But birth itself, in the case of a viviparous animal, is but an -unimportant epoch in the history of growth. It is an epoch whose -relative date varies according to the particular animal: the foal -and the lamb are born relatively later, that is to say when development -has advanced much farther, than in the case of man; the -kitten and the puppy are born earlier and therefore more helpless -than we are; and the mouse comes into the world still earlier -and more inchoate, so much so that even the little marsupial is -scarcely more unformed and embryonic. In all these cases alike, -we must, in order to study the curve of growth in its entirety, -take full account of prenatal -or intra-uterine growth. <span class="xxpn" id="p075">{75}</span></p> - -<p>According to His<a class="afnanch" href="#fn104" -id="fnanch104">104</a>, -the following are the mean lengths of the -unborn human embryo, from month to month.</p> - -<div class="dtblbox"><div class="nowrap"> -<table class="fsz7"> -<tr> - <td class="tdleft">Months</td> - <td class="tdcntr">0</td> - <td class="tdcntr">1</td> - <td class="tdcntr">2</td> - <td class="tdcntr">3</td> - <td class="tdcntr">4</td> - <td class="tdcntr">5</td> - <td class="tdcntr">6</td> - <td class="tdcntr">7</td> - <td class="tdcntr">8</td> - <td class="tdcntr">9</td> - <td class="tdcntr">10 (Birth)</td></tr> -<tr> - <td class="tdleft">Length in mm.</td> - <td class="tdcntr">0</td> - <td class="tdcntr">7·5</td> - <td class="tdcntr">40</td> - <td class="tdcntr">84</td> - <td class="tdcntr">162</td> - <td class="tdcntr">275</td> - <td class="tdcntr">352</td> - <td class="tdcntr">402</td> - <td class="tdcntr">443</td> - <td class="tdcntr">472</td> - <td class="tdcntr">490–500</td></tr> -<tr> - <td class="tdleft">Increment per month in mm.</td> - <td class="tdcntr">—</td> - <td class="tdcntr">7·5</td> - <td class="tdcntr">32·5</td> - <td class="tdcntr">44</td> - <td class="tdcntr">78</td> - <td class="tdcntr">113</td> - <td class="tdcntr">77</td> - <td class="tdcntr">50</td> - <td class="tdcntr">41</td> - <td class="tdcntr">29</td> - <td class="tdcntr">18–28</td></tr> -</table></div></div><!--dtblbox--> - -<div class="dctr01" id="fig8"> -<img src="images/i075.png" width="800" height="558" alt=""> - <div class="dcaption">Fig. 8. Mean monthly increments of - length or stature of child (in cms.).</div></div> - -<p>These data link on very well to those of Rüssow, which we -have just considered, and (though His’s measurements for the -pre-natal months are more detailed than are those of Rüssow for -the first year of post-natal life) we may draw a continuous curve of -growth (Fig. <a href="#fig7" title="go to Fig. 7">7</a>) and curve of acceleration of growth (Fig. <a href="#fig8" title="go to Fig. 8">8</a>) for the -combined periods. It will at once be seen that there is a “point -of inflection” somewhere about the fifth month of intra-uterine -life<a class="afnanch" href="#fn105" id="fnanch105">105</a>: -up to that date growth proceeds with -a continually increasing <span class="xxpn" id="p076">{76}</span> -velocity; but after that date, though growth is still rapid, its -velocity tends to fall away. There is a slight break between our -two separate sets of statistics at the date of birth, while this is -the very epoch regarding which we should particularly like to -have precise and continuous information. Undoubtedly there is -a certain slight arrest of growth, or diminution of the rate of -growth, about the epoch of birth: the -sudden change in the <span class="xxpn" id="p077">{77}</span> -method of nutrition has its inevitable effect; but this slight -temporary set-back is immediately followed by a secondary, and -temporary, acceleration.</p> - -<div class="dctr02" id="fig9"> -<img src="images/i076.png" width="728" height="723" alt=""> - <div class="pcaption">Fig. 9. Curve of pre-natal growth - (length or stature) of child; and corresponding curve of mean - monthly increments (mm.).</div></div> - -<p>It is worth our while to draw a separate curve to illustrate on -a larger scale His’s careful data for the ten months of pre-natal -life (Fig. <a href="#fig9" title="go to Fig. 9">9</a>). We see that this curve of growth is a beautifully -regular one, and is nearly symmetrical on either side of that point -of inflection of which we have already spoken; it is a curve for -which we might well hope to find a simple mathematical expression. -The acceleration-curve shown in Fig. <a href="#fig9" title="go to Fig. 9">9</a> together with the pre-natal -curve of growth, is not taken directly from His’s recorded data, -but is derived from the tangents drawn to a smoothed curve, -corresponding as nearly as possible to the actual curve of growth: -the rise to a maximal velocity about the fifth month and the -subsequent gradual fall are now demonstrated even more clearly -than before. In Fig. <a href="#fig10" title="go to Fig. 10">10</a>, which is a curve of growth of the -bamboo<a class="afnanch" href="#fn106" id="fnanch106">106</a>, -we see (so far as it goes) the -same essential features, <span class="xxpn" id="p078">{78}</span> -the slow beginning, the rapid increase of velocity, the point of -inflection, and the subsequent slow negative -acceleration<a class="afnanch" href="#fn107" id="fnanch107">107</a>.</p> - -<div class="dctr03" id="fig10"> -<img src="images/i077.png" width="800" height="701" alt=""> - <div class="pcaption">Fig. 10. Curve of growth of bamboo (from - Ostwald, after Kraus).</div></div> - -<div class="section"> -<h3><i>Variability and Correlation of Growth.</i></h3></div> - -<p>The magnitudes and velocities which we are here dealing with -are, of course, mean values derived from a certain number, sometimes -a large number, of individual cases. But no statistical -account of mean values is complete unless we also take account -of the <i>amount of variability</i> among the individual cases from which -the mean value is drawn. To do this throughout would lead us -into detailed investigations which lie far beyond the scope of this -elementary book; but we may very briefly illustrate the nature -of the process, in connection with the phenomena of growth -which we have just been studying.</p> - -<p>It was in connection with these phenomena, in the case of -man, that Quetelet first conceived the statistical study of variation, -on lines which were afterwards expounded and developed by -Galton, and which have grown, in the hands of Karl Pearson and -others, into the modern science of Biometrics.</p> - -<p>When Quetelet tells us, for instance, that the mean stature -of the ten-year old boy is 1·273 metres, this implies, according to -the law of error, or law of probabilities, that all the individual -measurements of ten-year-old boys group themselves <i>in an orderly -way</i>, that is to say according to a certain definite law, about this -mean value of 1·273. When these individual measurements are -grouped and plotted as a curve, so as to show the number of -individual cases at each individual length, we obtain a characteristic -curve of error or curve of frequency; and the “spread” of this -curve is a measure of the amount of variability in this particular -case. A certain mathematical measure of this “spread,” as -described in works upon statistics, is called the Index of Variability, -or Standard Deviation, and is usually denominated by the letter σ. -It is practically equivalent to a determination of the point upon -the frequency curve where it <i>changes its curvature</i> on either side -of the mean, and where, from being concave towards the middle -line, it spreads out to be convex thereto. -When we divide this <span class="xxpn" id="p079">{79}</span> -value by the mean, we get a figure which is independent of -any particular units, and which is called the Coefficient of Variability. -(It is usually multiplied by 100, to make it of a more -convenient amount; and we may then define this coefficient, <i>C</i>, -as -= (σ ⁄ <i>M</i>) × 100.)</p> - -<p>In regard to the growth of man, Pearson has determined this -coefficient of variability as follows: in male new-born infants, -the coefficient in regard to weight is 15·66, and in regard to -stature, 6·50; in male adults, for weight 10·83, and for stature, 3·66. -The amount of variability tends, therefore, to decrease with -growth or age.</p> - -<p>Similar determinations have been elaborated by Bowditch, by Boas -and Wissler, and by other writers for intermediate ages, especially -from about five years old to eighteen, so covering a great part of -the whole period of growth in man<a class="afnanch" href="#fn108" -id="fnanch108">108</a>.</p> - -<div class="dtblbox"><div class="nowrap"> -<table class="fsz6"> -<caption class="captioncntr"> <i>Coefficient of Variability</i> -(σ ⁄ <i>M</i> × 100) <i>in Man, -at various ages.</i></caption> -<tr> - <td class="tdleft">Age</td> - <td class="tdcntr">5</td> - <td class="tdcntr">6</td> - <td class="tdcntr">7</td> - <td class="tdcntr">8</td> - <td class="tdcntr">9</td></tr> -<tr> - <td class="tdleft">Stature (Bowditch)</td> - <td class="tdright">4·76</td> - <td class="tdright">4·60</td> - <td class="tdright">4·42</td> - <td class="tdright">4·49</td> - <td class="tdright">4·40</td></tr> -<tr> - <td class="tdleft">Stature (Boas and Wissler)</td> - <td class="tdright">4·15</td> - <td class="tdright">4·14</td> - <td class="tdright">4·22</td> - <td class="tdright">4·37</td> - <td class="tdright">4·33</td></tr> -<tr> - <td class="tdleft">Weight (Bowditch)</td> - <td class="tdright">11·56</td> - <td class="tdright">10·28</td> - <td class="tdright">11·08</td> - <td class="tdright">9·92</td> - <td class="tdright">11·04</td></tr> -<tr> - <td class="tdleft">Age</td> - <td class="tdcntr">10</td> - <td class="tdcntr">11</td> - <td class="tdcntr">12</td> - <td class="tdcntr">13</td> - <td class="tdcntr">14</td></tr> -<tr> - <td class="tdleft">Stature (Bowditch)</td> - <td class="tdright">4·55</td> - <td class="tdright">4·70</td> - <td class="tdright">4·90</td> - <td class="tdright">5·47</td> - <td class="tdright">5·79</td></tr> -<tr> - <td class="tdleft">Stature (Boas and Wissler)</td> - <td class="tdright">4·36</td> - <td class="tdright">4·54</td> - <td class="tdright">4·73</td> - <td class="tdright">5·16</td> - <td class="tdright">5·57</td></tr> -<tr> - <td class="tdleft">Weight (Bowditch)</td> - <td class="tdright">11·60</td> - <td class="tdright">11·76</td> - <td class="tdright">13·72</td> - <td class="tdright">13·60</td> - <td class="tdright">16·80</td></tr> -<tr> - <td class="tdleft">Age</td> - <td class="tdcntr">15</td> - <td class="tdcntr">16</td> - <td class="tdcntr">17</td> - <td class="tdcntr">18</td> - <td></td></tr> -<tr> - <td class="tdleft">Stature (Bowditch)</td> - <td class="tdright">5·57</td> - <td class="tdright">4·50</td> - <td class="tdright">4·55</td> - <td class="tdright">3·69</td><td></td></tr> -<tr> - <td class="tdleft">Stature (Boas and Wissler)</td> - <td class="tdright">5·50</td> - <td class="tdright">4·69</td> - <td class="tdright">4·27</td> - <td class="tdright">3·94</td><td></td></tr> -<tr> - <td class="tdleft">Weight (Bowditch)</td> - <td class="tdright">15·32</td> - <td class="tdright">13·28</td> - <td class="tdright">12·96</td> - <td class="tdright">10·40</td><td></td></tr> -</table></div></div><!--dtblbox--> - -<p>The result is very curious indeed. We see, from Fig. <a href="#fig11" title="go to Fig. 11">11</a>, -that the curve of variability is very similar to what we have called -the acceleration-curve (Fig. -<a href="#fig4" title="go to Fig. 4">4</a>): that is to say, it descends when the -rate of growth diminishes, and rises very markedly again when, in -late boyhood, the rate of growth is -temporarily accelerated. We <span class="xxpn" id="p080">{80}</span> -see, in short, that the amount of <i>variability</i> in stature or in weight -is a function of the <i>rate of growth</i> in these magnitudes, though -we are not yet in a position to equate the terms precisely, one with -another.</p> - -<div class="dctr04" id="fig11"> -<img src="images/i080.png" width="600" height="750" alt=""> - <div class="pcaption">Fig. 11. Coefficients of variability of - stature in Man <span class="nowrap">(<img class="iglyph-a" -src="images/iglyph-malesign.png" width="28" height="47" -alt="♂">).</span> from Boas and Wissler’s data.</div></div> - -<div class="psmprnt2"> -<p>If we take not merely the variability of stature or weight at -a given age, but the variability of the actual successive increments -in each yearly period, we see that this latter coefficient of variability -tends to increase steadily, and more and more rapidly, within -the limits of age for which we have information; and this phenomenon -is, in the main, easy of explanation. For a great part of -the difference, in regard to rate of growth, between one individual -and another is a difference of <i>phase</i>,—a difference in the epochs -of acceleration and retardation, and finally in the epoch when -growth comes to an end. And it follows that the variability of -rate will be more and more marked, as we approach and reach -the period when some individuals still continue, and others have -already ceased, to grow. In the -following epitomised table, <span class="xxpn" id="p081">{81}</span> -I have taken Boas’s determinations of variability (σ) (<i>op. cit.</i> -p. 1548), converted them into the corresponding coefficients of -variability (σ ⁄ <i>M</i> × 100), and then -smoothed the resulting numbers.</p> - -<div class="dtblbox"><div class="nowrap"> -<table class="fsz6"> -<caption><i>Coefficients of Variability in Annual Increment of -Stature.</i></caption> -<tr> - <td class="tdleft">Age</td> - <td class="tdcntr">7</td> - <td class="tdcntr">8</td> - <td class="tdcntr">9</td> - <td class="tdcntr">10</td> - <td class="tdcntr">11</td> - <td class="tdcntr">12</td> - <td class="tdcntr">13</td> - <td class="tdcntr">14</td> - <td class="tdcntr">15</td></tr> -<tr> - <td class="tdleft">Boys</td> - <td class="tdright">17·3</td> - <td class="tdright">15·8</td> - <td class="tdright">18·6</td> - <td class="tdright">19·1</td> - <td class="tdright">21·0</td> - <td class="tdright">24·7</td> - <td class="tdright">29·0</td> - <td class="tdright">36·2</td> - <td class="tdright">46·1</td></tr> -<tr> - <td class="tdleft">Girls</td> - <td class="tdright">17·1</td> - <td class="tdright">17·8</td> - <td class="tdright">19·2</td> - <td class="tdright">22·7</td> - <td class="tdright">25·9</td> - <td class="tdright">29·3</td> - <td class="tdright">37·0</td> - <td class="tdright">44·8</td> - <td class="tdcntr">—</td></tr> -</table></div></div><!--dtblbox--> - -<p>The greater variability of annual increment in the girls, as -compared with the boys, is very marked, and is easily explained -by the more rapid rate at which the girls run through the several -phases of the phenomenon.</p> - -</div><!--psmprnt2--> - -<div class="psmprnt3"> -<p>Just as there is a marked difference in “phase” between the growth-curves -of the two sexes, that is to say a difference in the periods when growth -is rapid or the reverse, so also, within each sex, will there be room for similar, -but individual phase-differences. Thus we may have children of accelerated -development, who at a given epoch after birth are both rapidly growing and -already “big for their age”; and others of retarded development who are -comparatively small and have not reached the period of acceleration which, -in greater or less degree, will come to them in turn. In other words, there -must under such circumstances be a strong positive “coefficient of correlation” -between stature and rate of growth, and also between the rate of growth in -one year and the next. But it does not by any means follow that a child who -is precociously big will continue to grow rapidly, and become a man or woman -of exceptional stature. On the contrary, when in the case of the precocious -or “accelerated” children growth has begun to slow down, the backward -ones may still be growing rapidly, and so making up (more or less completely) -to the others. In other words, the period of high positive correlation between -stature and increment will tend to be followed by one of negative correlation. -This interesting and important point, due to Boas and -Wissler<a class="afnanch" href="#fn109" id="fnanch109">109</a>, -is confirmed -by the following table:―</p> - -<div class="dtblbox"><div class="nowrap"> -<table class="fsz7"> -<caption class="fsz4"><i>Correlation -of Stature and Increment in Boys and Girls.</i><br> -(<i>From Boas and Wissler.</i>)</caption> -<tr> - <td class="tdleft">Age</td> - <td class="tdleft"></td> - <td class="tdcntr">6</td> - <td class="tdcntr">7</td> - <td class="tdcntr">8</td> - <td class="tdcntr">9</td> - <td class="tdcntr">10</td> - <td class="tdcntr">11</td> - <td class="tdcntr">12</td> - <td class="tdcntr">13</td> - <td class="tdcntr">14</td> - <td class="tdcntr">15</td></tr> -<tr> - <td class="tdleft">Stature</td> - <td class="tdleft">(B)</td> - <td class="tdright">112·7 </td> - <td class="tdright">115·5 </td> - <td class="tdright">123·2 </td> - <td class="tdright">127·4 </td> - <td class="tdright">133·2 </td> - <td class="tdright">136·8 </td> - <td class="tdright">142·7 </td> - <td class="tdright">147·3 </td> - <td class="tdright">155·9 </td> - <td class="tdright">162·2 </td></tr> -<tr> - <td class="tdright"></td> - <td class="tdleft">(G)</td> - <td class="tdright">111·4 </td> - <td class="tdright">117·7 </td> - <td class="tdright">121·4 </td> - <td class="tdright">127·9 </td> - <td class="tdright">131·8 </td> - <td class="tdright">136·7 </td> - <td class="tdright">144·6 </td> - <td class="tdright">149·7 </td> - <td class="tdright">153·8 </td> - <td class="tdright">157·2 </td></tr> -<tr> - <td class="tdleft">Increment</td> - <td class="tdleft">(B)</td> - <td class="tdright">5·7 </td> - <td class="tdright">5·3 </td> - <td class="tdright">4·9 </td> - <td class="tdright">5·1 </td> - <td class="tdright">5·0 </td> - <td class="tdright">4·7 </td> - <td class="tdright">5·9 </td> - <td class="tdright">7·5 </td> - <td class="tdright">6·2 </td> - <td class="tdright">5·2 </td></tr> -<tr> - <td class="tdright"></td> - <td class="tdleft">(G)</td> - <td class="tdright">5·9 </td> - <td class="tdright">5·5 </td> - <td class="tdright">5·5 </td> - <td class="tdright">5·9 </td> - <td class="tdright">6·2 </td> - <td class="tdright">7·2 </td> - <td class="tdright">6·5 </td> - <td class="tdright">5·4 </td> - <td class="tdright">3·3 </td> - <td class="tdright">1·7 </td></tr> -<tr> - <td class="tdleft">Correlation</td> - <td class="tdleft">(B)</td> - <td class="tdright">·25</td> - <td class="tdright">·11</td> - <td class="tdright">·08</td> - <td class="tdright">·25</td> - <td class="tdright">·18</td> - <td class="tdright">·18</td> - <td class="tdright">·48</td> - <td class="tdright">·29</td> - <td class="tdright">− ·42</td> - <td class="tdright">− ·44</td></tr> -<tr> - <td class="tdright"></td> - <td class="tdleft">(G)</td> - <td class="tdright">·44</td> - <td class="tdright">·14</td> - <td class="tdright">·24</td> - <td class="tdright">·47</td> - <td class="tdright">·18</td> - <td class="tdright">− ·18</td> - <td class="tdright">− ·42</td> - <td class="tdright">− ·39</td> - <td class="tdright">− ·63</td> - <td class="tdright">·11</td></tr> -</table></div></div><!--dtblbox--> - -<span class="xxpn" id="p082">{82}</span> - -<p>A minor, but very curious point brought out by the same investigators -is that, if instead of stature we deal with height in the sitting posture (or, -practically speaking, with length of trunk or back), then the correlations -between this height and its annual increment are throughout negative. In -other words, there would seem to be a general tendency for the long trunks -to grow slowly throughout the whole period under investigation. It is a -well-known anatomical fact that tallness is in the main due not to length of -body but to length of limb.</p> -</div><!--psmprnt2--> - -<p>The whole phenomenon of variability in regard to magnitude -and to rate of increment is in the highest degree suggestive: -inasmuch as it helps further to remind and to impress upon us -that specific rate of growth is the real physiological factor which -we want to get at, of which specific magnitude, dimensions and -form, and all the variations of these, are merely the concrete and -visible resultant. But the problems of variability, though they -are intimately related to the general problem of growth, carry us -very soon beyond our present limitations.</p> - -<div class="section"> -<h3 title="Rate of growth in other organisms."><i>Rate of growth in other -organisms<a class="afnanchlow" href="#fn110" id="fnanch110" title="go to -note 110">*</a>.</i></h3></div> - -<p>Just as the human curve of growth has its slight but well-marked -interruptions, or variations in rate, coinciding with such -epochs as birth and puberty, so is it with other animals, and this -phenomenon is particularly striking in the case of animals which -undergo a regular metamorphosis.</p> - -<p>In the accompanying curve of growth in weight of the mouse -(Fig. <a href="#fig12" title="go to Fig. 12">12</a>), based on W. Ostwald’s observations<a class="afnanch" href="#fn111" id="fnanch111">111</a>, -we see a distinct -slackening of the rate when the mouse is about a fortnight old, -at which period it opens its eyes and very soon afterwards is -weaned. At about six weeks old there is another well-marked -retardation of growth, following on a very rapid period, and -coinciding with the epoch of puberty. <span class="xxpn" id="p083">{83}</span></p> - -<p>Fig. <a href="#fig13" title="go to Fig. 13">13</a> shews the curve of growth of the silkworm<a class="afnanch" href="#fn112" id="fnanch112">112</a>, -during its -whole larval life, up to the time of its entering the chrysalis stage.</p> - -<p>The silkworm moults four times, at intervals of about a week, -the first moult being on the sixth or seventh day after hatching. -A distinct retardation of growth is exhibited on our curve in the -case of the third and fourth moults; while a similar retardation -accompanies the first and second moults also, but the scale of -our diagram does not render it visible. When the worm is about -seven weeks old, a remarkable process of “purgation” takes place, -as a preliminary to entering on the pupal, or chrysalis, stage; -and the great and sudden loss of weight which accompanies this -process is the most marked feature of our curve.</p> - -<div class="dctr02" id="fig12"> -<img src="images/i083.png" width="800" height="667" alt=""> - <div class="dcaption">Fig. 12. Growth in weight of Mouse. - (After W. Ostwald.)</div></div> - -<p>The rate of growth in the tadpole<a class="afnanch" href="#fn113" id="fnanch113">113</a> -(Fig. <a href="#fig14" title="go to Fig. 14">14</a>) is likewise marked -by epochs of retardation, and finally by a sudden and drastic -change. There is a slight diminution in -weight immediately after <span class="xxpn" id="p084">{84}</span> -the little larva frees itself from the egg; there is a retardation of -growth about ten days later, when the external gills disappear; -and finally, the complete metamorphosis, with the loss of the tail, -the growth of the legs and the cessation of branchial respiration, -is accompanied by a loss of weight amounting to wellnigh half -the weight of the full-grown larva. <span class="xxpn" id="p085">{85}</span></p> - -<div class="dctr01" id="fig13"> -<img src="images/i084.png" width="800" height="882" alt=""> - <div class="dcaption">Fig. 13. Growth in weight of Silkworm. - (From Ostwald, after Luciani and Lo Monaco.)</div></div> - -<p>While as a general rule, the better the animals be fed the -quicker they grow and the sooner they metamorphose, Barfürth -has pointed out the curious fact that a short spell of starvation, -just before metamorphosis is due, appears to hasten the change.</p> - -<div class="dctr03" id="fig14"> -<img src="images/i085.png" width="700" height="961" alt=""> - <div class="dcaption">Fig. 14. Growth in weight of Tadpole. (From - Ostwald, after Schaper.)</div></div> - -<p>The negative growth, or actual loss of bulk and weight -which often, and perhaps always, accompanies metamorphosis, -is well shewn in the case of the eel<a class="afnanch" href="#fn114" id="fnanch114">114</a>. -The contrast of -size is great between <span class="xxpn" id="p087">{87}</span> -the flattened, lancet-shaped Leptocephalus larva and the little -black cylindrical, almost thread-like elver, whose magnitude is -less than that of the Leptocephalus in every dimension, even, at -first, in length (Fig. <a href="#fig15" title="go to Fig. 15">15</a>).</p> - -<div class="dctr04" id="fig15"> -<img src="images/i086.jpg" width="692" height="1200" alt=""> - <div class="pcaption">Fig. 15. Development of Eel; from - Leptocephalus larvae to young Elver. (From Ostwald after - Joh. Schmidt.)</div></div> - -<div class="dctr01" id="fig16"> -<img src="images/i087.png" width="800" height="453" alt=""> - <div class="dcaption">Fig. 16. Growth in length of Spirogyra. - (From Ostwald, after Hofmeister.)</div></div> - -<p>From the higher study of the physiology of growth we learn -that such fluctuations as we have described are but special interruptions -in a process which is never actually continuous, but is -perpetually interrupted in a rhythmic manner<a class="afnanch" href="#fn115" id="fnanch115">115</a>. -Hofmeister -shewed, for instance, that the growth of Spirogyra proceeds by -fits and starts, by periods of activity and rest, which alternate -with one another at intervals of so many minutes (Fig. <a href="#fig16" title="go to Fig. 16">16</a>). And -Bose, by very refined methods of experiment, has shewn that -plant-growth really proceeds by tiny and perfectly rhythmical -pulsations recurring at regular intervals of a few seconds of time. -Fig. <a href="#fig17" title="go to Fig. 17">17</a> shews, according to Bose’s observations<a class="afnanch" href="#fn116" id="fnanch116">116</a>, -the growth of -a crocus, under a very high magnification. The stalk grows by -little jerks, each with an amplitude of -about ·002 mm., every <span class="xxpn" id="p088">{88}</span> -twenty seconds or so, and after each little increment there is a -partial recoil.</p> - -<div class="dctr05" id="fig17"> -<img src="images/i088.png" width="450" height="322" alt=""> - <div class="pcaption">Fig. 17. Pulsations of growth in Crocus, in - micro-millimetres. (After Bose.)</div></div> - -<div class="section"> -<h3 title="The rate of growth of various parts or organs."><i>The rate of -growth of various parts or organs<a class="afnanchlow" href="#fn117" -id="fnanch117" title="go to note 117">*</a>.</i></h3></div> - -<p>The differences in regard to rate of growth between various -parts or organs of the body, internal and external, can be -amply illustrated in the case of man, and also, but chiefly -in regard to external form, in some few other creatures<a -class="afnanch" href="#fn118" id="fnanch118">118</a>. It -is obvious that there lies herein an endless field for the -mathematical study of correlation and of variability, but with -this aspect of the case we cannot deal.</p> - -<p>In the accompanying table, I shew, from some of Vierordt’s -data, the <i>relative</i> weights, at various ages, compared with the -weight at birth, of the entire body, of -the brain, heart and liver; <span class="xxpn" id="p089">{89}</span> -and also the percentage relation which each of these organs bears, -at the several ages, to the weight of the whole body.</p> - -<div class="dtblbox"> -<table class="fsz7 borall"> -<caption class="captionblk"><i>Weight -of Various Organs, compared with the Total Weight of -the Human Body (male).</i> (<i>After Vierordt, Anatom. Tabellen, pp. 38, -39.</i>)</caption> -<tr> - <th class="borall"></th> - <th class="borall">Weight of body†</th> - <th class="borall" colspan="4">Relative weights of</th> - <th class="borall" colspan="4">Percentage weights compared with total body-weights</th></tr> -<tr> - <th class="borall">Age</th> - <th class="borall">in kg.</th> - <th class="borall">Body</th> - <th class="borall">Brain</th> - <th class="borall">Heart</th> - <th class="borall">Liver</th> - <th class="borall">Body</th> - <th class="borall">Brain</th> - <th class="borall">Heart</th> - <th class="borall">Liver</th></tr> -<tr> - <td class="tdright">0</td> - <td class="tdright">3·1</td> - <td class="tdright">1   </td> - <td class="tdright">1   </td> - <td class="tdright">1   </td> - <td class="tdright">1   </td> - <td class="tdright">100</td> - <td class="tdright">12·29</td> - <td class="tdright">0·76</td> - <td class="tdright">4·57</td></tr> -<tr> - <td class="tdright">1</td> - <td class="tdright">9·0</td> - <td class="tdright">2·90</td> - <td class="tdright">2·48</td> - <td class="tdright">1·75</td> - <td class="tdright">2·35</td> - <td class="tdright">100</td> - <td class="tdright">10·50</td> - <td class="tdright">0·46</td> - <td class="tdright">3·70</td></tr> -<tr> - <td class="tdright">2</td> - <td class="tdright">11·0</td> - <td class="tdright">3·55</td> - <td class="tdright">2·69</td> - <td class="tdright">2·20</td> - <td class="tdright">3·02</td> - <td class="tdright">100</td> - <td class="tdright">9·32</td> - <td class="tdright">0·47</td> - <td class="tdright">3·89</td></tr> -<tr> - <td class="tdright">3</td> - <td class="tdright">12·5</td> - <td class="tdright">4·03</td> - <td class="tdright">2·91</td> - <td class="tdright">2·75</td> - <td class="tdright">3·42</td> - <td class="tdright">100</td> - <td class="tdright">8·86</td> - <td class="tdright">0·52</td> - <td class="tdright">3·88</td></tr> -<tr> - <td class="tdright">4</td> - <td class="tdright">14·0</td> - <td class="tdright">4·52</td> - <td class="tdright">3·49</td> - <td class="tdright">3·14</td> - <td class="tdright">4·15</td> - <td class="tdright">100</td> - <td class="tdright">9·50</td> - <td class="tdright">0·53</td> - <td class="tdright">4·20</td></tr> -<tr> - <td class="tdright">5</td> - <td class="tdright">15·9</td> - <td class="tdright">5·13</td> - <td class="tdright">3·32</td> - <td class="tdright">3·43</td> - <td class="tdright">3·80</td> - <td class="tdright">100</td> - <td class="tdright">7·94</td> - <td class="tdright">0·51</td> - <td class="tdright">3·39</td></tr> -<tr> - <td class="tdright">6</td> - <td class="tdright">17·8</td> - <td class="tdright">5·74</td> - <td class="tdright">3·57</td> - <td class="tdright">3·60</td> - <td class="tdright">4·34</td> - <td class="tdright">100</td> - <td class="tdright">7·63</td> - <td class="tdright">0·48</td> - <td class="tdright">3·45</td></tr> -<tr> - <td class="tdright">7</td> - <td class="tdright">19·7</td> - <td class="tdright">6·35</td> - <td class="tdright">3·54</td> - <td class="tdright">3·95</td> - <td class="tdright">4·86</td> - <td class="tdright">100</td> - <td class="tdright">6·84</td> - <td class="tdright">0·47</td> - <td class="tdright">3·49</td></tr> -<tr> - <td class="tdright">8</td> - <td class="tdright">21·6</td> - <td class="tdright">6·97</td> - <td class="tdright">3·62</td> - <td class="tdright">4·02</td> - <td class="tdright">4·59</td> - <td class="tdright">100</td> - <td class="tdright">6·38</td> - <td class="tdright">0·44</td> - <td class="tdright">3·01</td></tr> -<tr> - <td class="tdright">9</td> - <td class="tdright">23·5</td> - <td class="tdright">7·58</td> - <td class="tdright">3·74</td> - <td class="tdright">4·59</td> - <td class="tdright">4·95</td> - <td class="tdright">100</td> - <td class="tdright">6·06</td> - <td class="tdright">0·46</td> - <td class="tdright">2·99</td></tr> -<tr> - <td class="tdright">10</td> - <td class="tdright">25·2</td> - <td class="tdright">8·13</td> - <td class="tdright">3·70</td> - <td class="tdright">5·41</td> - <td class="tdright">5·90</td> - <td class="tdright">100</td> - <td class="tdright">5·59</td> - <td class="tdright">0·51</td> - <td class="tdright">3·32</td></tr> -<tr> - <td class="tdright">11</td> - <td class="tdright">27·0</td> - <td class="tdright">8·71</td> - <td class="tdright">3·57</td> - <td class="tdright">5·97</td> - <td class="tdright">6·14</td> - <td class="tdright">100</td> - <td class="tdright">5·04</td> - <td class="tdright">0·52</td> - <td class="tdright">3·22</td></tr> -<tr> - <td class="tdright">12</td> - <td class="tdright">29·0</td> - <td class="tdright">9·35</td> - <td class="tdright">3·78</td> - <td class="tdright">(4·13)</td> - <td class="tdright">6·21</td> - <td class="tdright">100</td> - <td class="tdright">4·88</td> - <td class="tdright">(0·34)</td> - <td class="tdright">3·03</td></tr> -<tr> - <td class="tdright">13</td> - <td class="tdright">33·1</td> - <td class="tdright">10·68</td> - <td class="tdright">3·90</td> - <td class="tdright">6·95</td> - <td class="tdright">7·31</td> - <td class="tdright">100</td> - <td class="tdright">4·49</td> - <td class="tdright">0·50</td> - <td class="tdright">3·13</td></tr> -<tr> - <td class="tdright">14</td> - <td class="tdright">37·1</td> - <td class="tdright">11·97</td> - <td class="tdright">3·38</td> - <td class="tdright">9·16</td> - <td class="tdright">8·39</td> - <td class="tdright">100</td> - <td class="tdright">3·47</td> - <td class="tdright">0·58</td> - <td class="tdright">3·20</td></tr> -<tr> - <td class="tdright">15</td> - <td class="tdright">41·2</td> - <td class="tdright">13·29</td> - <td class="tdright">3·91</td> - <td class="tdright">8·45</td> - <td class="tdright">9·22</td> - <td class="tdright">100</td> - <td class="tdright">3·62</td> - <td class="tdright">0·48</td> - <td class="tdright">3·17</td></tr> -<tr> - <td class="tdright">16</td> - <td class="tdright">45·9</td> - <td class="tdright">14·81</td> - <td class="tdright">3·77</td> - <td class="tdright">9·76</td> - <td class="tdright">9·45</td> - <td class="tdright">100</td> - <td class="tdright">3·16</td> - <td class="tdright">0·51</td> - <td class="tdright">2·95</td></tr> -<tr> - <td class="tdright">17</td> - <td class="tdright">49·7</td> - <td class="tdright">16·03</td> - <td class="tdright">3·70</td> - <td class="tdright">10·63</td> - <td class="tdright">10·46</td> - <td class="tdright">100</td> - <td class="tdright">2·84</td> - <td class="tdright">0·51</td> - <td class="tdright">2·98</td></tr> -<tr> - <td class="tdright">18</td> - <td class="tdright">53·9</td> - <td class="tdright">17·39</td> - <td class="tdright">3·73</td> - <td class="tdright">10·33</td> - <td class="tdright">10·65</td> - <td class="tdright">100</td> - <td class="tdright">2·64</td> - <td class="tdright">0·46</td> - <td class="tdright">2·80</td></tr> -<tr> - <td class="tdright">19</td> - <td class="tdright">57·6</td> - <td class="tdright">18·58</td> - <td class="tdright">3·67</td> - <td class="tdright">11·42</td> - <td class="tdright">11·61</td> - <td class="tdright">100</td> - <td class="tdright">2·43</td> - <td class="tdright">0·51</td> - <td class="tdright">2·86</td></tr> -<tr> - <td class="tdright">20</td> - <td class="tdright">59·5</td> - <td class="tdright">19·19</td> - <td class="tdright">3·79</td> - <td class="tdright">12·94</td> - <td class="tdright">11·01</td> - <td class="tdright">100</td> - <td class="tdright">2·43</td> - <td class="tdright">0·51</td> - <td class="tdright">2·62</td></tr> -<tr> - <td class="tdright">21</td> - <td class="tdright">61·2</td> - <td class="tdright">19·74</td> - <td class="tdright">3·71</td> - <td class="tdright">12·59</td> - <td class="tdright">11·48</td> - <td class="tdright">100</td> - <td class="tdright">2·31</td> - <td class="tdright">0·49</td> - <td class="tdright">2·66</td></tr> -<tr> - <td class="tdright">22</td> - <td class="tdright">62·9</td> - <td class="tdright">20·29</td> - <td class="tdright">3·54</td> - <td class="tdright">13·24</td> - <td class="tdright">11·82</td> - <td class="tdright">100</td> - <td class="tdright">2·14</td> - <td class="tdright">0·50</td> - <td class="tdright">2·66</td></tr> -<tr> - <td class="tdright">23</td> - <td class="tdright">64·5</td> - <td class="tdright">20·81</td> - <td class="tdright">3·66</td> - <td class="tdright">12·42</td> - <td class="tdright">10·79</td> - <td class="tdright">100</td> - <td class="tdright">2·16</td> - <td class="tdright">0·46</td> - <td class="tdright">2·37</td></tr> -<tr> - <td class="tdright">24</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdright">3·74</td> - <td class="tdright">13·09</td> - <td class="tdright">13·04</td> - <td class="tdright">100</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td></tr> -<tr> - <td class="tdright">25</td> - <td class="tdright">66·2</td> - <td class="tdright">21·36</td> - <td class="tdright">3·76</td> - <td class="tdright">12·74</td> - <td class="tdright">12·84</td> - <td class="tdright">100</td> - <td class="tdright">2·16</td> - <td class="tdright">0·46</td> - <td class="tdright">2·75</td></tr> -</table> -<p class="ptblfn">† From Quetelet.</p> -</div><!--dtblbox--> - -<p>From the first portion of the table, it will be seen that none -of these organs by any means keep pace with the body as a whole -in regard to growth in weight; in other words, there must be -some other part of the fabric, doubtless the muscles and the bones, -which increase <i>more</i> rapidly than the average increase of the body. -Heart and liver both grow nearly at the same rate, -and by the <span class="xxpn" id="p090">{90}</span> -age of twenty-five they have multiplied their weight at birth by -about thirteen times, while the weight of the entire body has been -multiplied by about twenty-one; but the weight of the brain has -meanwhile been multiplied only about three and a quarter times. -In the next place, we see the very remarkable phenomenon that -the brain, growing rapidly till the child is about four years old, then -grows more much slowly till about eight or nine years old, and -after that time there is scarcely any further perceptible increase. -These phenomena are diagrammatically illustrated in Fig. <a href="#fig18" title="go to Fig. 18">18</a>.</p> - -<div class="dctr02" id="fig18"> -<img src="images/i090.png" width="750" height="624" alt=""> - <div class="dcaption">Fig. 18. Relative growth in weight (in Man) of - Brain, Heart, and whole Body.</div></div> - -<div class="psmprnt3"> -<p>Many statistics indicate a decrease of brain-weight during adult -life. Boas<a class="afnanch" href="#fn119" id="fnanch119">119</a> -was -inclined to attribute this apparent phenomenon to our statistical -methods, and to hold that it could “hardly be explained in any other -way than by assuming an increased death-rate among men with very large -brains, at an age of about twenty years.” But Raymond Pearl has shewn -that there is evidence of a steady and very gradual decline in the -weight of the brain with advancing age, beginning at or before the -twentieth year, and continuing throughout adult life<a class="afnanch" -href="#fn120" id="fnanch120">120</a>. -<span class="xxpn" -id="p091">{91}</span></p></div><!--psmprnt3--> - -<p>The second part of the table shews the steadily decreasing -weights of the organs in question as compared with the body; -the brain falling from over 12 per cent. at birth to little over -2 per cent. at five and twenty; the heart from ·75 to ·46 per -cent.; and the liver from 4·57 to 2·75 per cent. of the whole -bodily weight.</p> - -<p>It is plain, then, that there is no simple and direct relation, -holding good <i>throughout life</i>, between the size of the body as a -whole and that of the organs we have just discussed; and the -changing ratio of magnitude is especially marked in the case of -the brain, which, as we have just seen, constitutes about one-eighth -of the whole bodily weight at birth, and but one-fiftieth at five -and twenty. The same change of ratio is observed in other -animals, in equal or even greater degree. For instance, Max -Weber<a class="afnanch" href="#fn121" id="fnanch121">121</a> -tells us that in the lion, at five weeks, four months, -eleven months, and lastly when full-grown, the brain-weight -represents the following fractions of the weight of the whole -body, viz. 1 ⁄ 18, 1 ⁄ 80, 1 ⁄ 184, and 1 ⁄ 546. And Kellicott has, in -like manner, shewn that in the dogfish, while some organs (e.g. -rectal gland, pancreas, etc.) increase steadily and very nearly -proportionately to the body as a whole, the brain, and some other -organs also, grow in a diminishing ratio, which is capable of -representation, approximately, by a logarithmic curve<a class="afnanch" href="#fn122" id="fnanch122">122</a>.</p> - -<p>But if we confine ourselves to the adult, then, as Raymond -Pearl has shewn in the case of man, the relation of brain-weight -to age, to stature, or to weight, becomes a comparatively simple -one, and may be sensibly expressed by a straight line, or simple -equation.</p> - -<div class="psmprnt2"> -<div class="dmaths"> -<p>Thus, if <i>W</i> be the brain-weight (in grammes), and <i>A</i> be the -age, or <i>S</i> the stature, of the individual, then (in the case of Swedish -males) the following simple equations suffice to give the required -ratios:</p> - -<div><i>W</i> -= 1487·8 − 1·94 <i>A</i> -= 915·06 + 2·86 <i>S</i>.</div> - -<p class="pcontinue" id="p092">These equations are -applicable to ages between fifteen and eighty; if we take -narrower limits, say between fifteen and fifty, we can get -a closer agreement by using somewhat altered constants. -In the two sexes, and in different races, these empirical -constants will be greatly changed<a class="afnanch" -href="#fn123" id="fnanch123">123</a>. Donaldson has further -shewn that the correlation between brain-weight and -body-weight is very much closer in the rat than in man<a -class="afnanch" href="#fn124" id="fnanch124">124</a>.</p> -</div><!--dmaths--></div><!--psmprnt2--> - -<div class="psmprnt3"> -<p>The falling ratio of weight of brain to body with increase of size or age -finds its parallel in comparative anatomy, in the general law that the larger -the animal the less is the relative weight of the brain.</p> - -<div class="dtblbox"><div class="nowrap"> -<table class="borall"> -<tr> - <th class="borall"></th> - <th class="borall">Weight of<br>entire animal<br>gms.</th> - <th class="borall">Weight<br>of brain<br>gms.</th> - <th class="borall">Ratio</th></tr> -<tr> - <td class="tdleft">Marmoset</td> - <td class="tdright">335</td> - <td class="tdright">12·5</td> - <td class="tdleft">1 : 26</td></tr> -<tr> - <td class="tdleft">Spider monkey</td> - <td class="tdright">1845</td> - <td class="tdright">126  </td> - <td class="tdleft">1 : 15</td></tr> -<tr> - <td class="tdleft">Felis minuta</td> - <td class="tdright">1234</td> - <td class="tdright">23·6</td> - <td class="tdleft">1 : 56</td></tr> -<tr> - <td class="tdleft">F. domestica</td> - <td class="tdright">3300</td> - <td class="tdright">31  </td> - <td class="tdleft">1 : 107</td></tr> -<tr> - <td class="tdleft">Leopard</td> - <td class="tdright">27,700</td> - <td class="tdright">164  </td> - <td class="tdleft">1 : 168</td></tr> -<tr> - <td class="tdleft">Lion</td> - <td class="tdright">119,500</td> - <td class="tdright">219  </td> - <td class="tdleft">1 : 546</td></tr> -<tr> - <td class="tdleft">Elephant</td> - <td class="tdright">3,048,000</td> - <td class="tdright">5430  </td> - <td class="tdleft">1 : 560</td></tr> -<tr> - <td class="tdleft">Whale (Globiocephalus)</td> - <td class="tdright">1,000,000</td> - <td class="tdright">2511  </td> - <td class="tdleft">1 : 400</td></tr> -</table></div></div><!--dtblbox--> - -<p>For much information on this subject, see Dubois, “Abhängigkeit des -Hirngewichtes von der Körpergrösse bei den Säugethieren,” <i>Arch. f. -Anthropol.</i> <span class="smmaj">XXV,</span> 1897. Dubois has attempted, -but I think with very doubtful success, to equate the weight of the -brain with that of the animal. We may do this, in a very simple way, by -representing the weight of the body as a <i>power</i> of that of the brain; -thus, in the above table of the weights of brain and body in four -species of cat, if we call <i>W</i> the weight of the body (in grammes), and -<i>w</i> the weight of the brain, then if in all four cases we express the -ratio by <i>W</i> -= <i>w</i><sup class="spitc">n</sup> , we find that <i>n</i> is almost -constant, and differs little from 2·24 in all four species: the values -being respectively, in the order of the table 2·36, 2·24, 2·18, and -2·17. But this evidently amounts to no more than an empirical rule; -for we can easily see that it depends on the particular scale which we -have used, and that if the weights had been taken, for instance, in -kilogrammes or in milligrammes, the agreement or coincidence would not -have occurred<a class="afnanch" href="#fn125" id="fnanch125">125</a>. -<span class="xxpn" id="p093">{93}</span></p></div><!--psmprnt3--> - -<div class="dtblbox"><div class="nowrap"> -<table class="borall fsz6"> -<caption><i>The Length of the Head in Man at various Ages.</i><br> -(<i>After Quetelet, p. 207.</i>)</caption> -<tr> - <th class="borall" rowspan="2">Age</th> - <th class="borall" colspan="3">Men</th> - <th class="borall" colspan="3">Women</th></tr> -<tr> - <th class="borall">Total height<br>m.</th> - <th class="borall">Head<br>m.</th> - <th class="borall">Ratio</th> - <th class="borall">Height<br>m.</th> - <th class="borall">Head†<br>m.</th> - <th class="borall">Ratio</th></tr> -<tr> - <td class="tdleft"> Birth</td> - <td class="tdright">0·500</td> - <td class="tdright">0·111</td> - <td class="tdright">4·50</td> - <td class="tdright">0·494</td> - <td class="tdright">0·111</td> - <td class="tdright">4·45</td></tr> -<tr> - <td class="tdleft"> 1 year</td> - <td class="tdright">0·698</td> - <td class="tdright">0·154</td> - <td class="tdright">4·53</td> - <td class="tdright">0·690</td> - <td class="tdright">0·154</td> - <td class="tdright">4·48</td></tr> -<tr> - <td class="tdleft"> 2 years</td> - <td class="tdright">0·791</td> - <td class="tdright">0·173</td> - <td class="tdright">4·57</td> - <td class="tdright">0·781</td> - <td class="tdright">0·172</td> - <td class="tdright">4·54</td></tr> -<tr> - <td class="tdleft"> 3 years</td> - <td class="tdright">0·864</td> - <td class="tdright">0·182</td> - <td class="tdright">4·74</td> - <td class="tdright">0·854</td> - <td class="tdright">0·180</td> - <td class="tdright">4·74</td></tr> -<tr> - <td class="tdleft"> 5 years</td> - <td class="tdright">0·987</td> - <td class="tdright">0·192</td> - <td class="tdright">5·14</td> - <td class="tdright">0·974</td> - <td class="tdright">0·188</td> - <td class="tdright">5·18</td></tr> -<tr> - <td class="tdleft">10 years</td> - <td class="tdright">1·273</td> - <td class="tdright">0·205</td> - <td class="tdright">6·21</td> - <td class="tdright">1·249</td> - <td class="tdright">0·201</td> - <td class="tdright">6·21</td></tr> -<tr> - <td class="tdleft">15 years</td> - <td class="tdright">1·513</td> - <td class="tdright">0·215</td> - <td class="tdright">7·04</td> - <td class="tdright">1·488</td> - <td class="tdright">0·213</td> - <td class="tdright">6·99</td></tr> -<tr> - <td class="tdleft">20 years</td> - <td class="tdright">1·669</td> - <td class="tdright">0·227</td> - <td class="tdright">7·35</td> - <td class="tdright">1·574</td> - <td class="tdright">0·220</td> - <td class="tdright">7·15</td></tr> -<tr> - <td class="tdleft">30 years</td> - <td class="tdright">1·686</td> - <td class="tdright">0·228</td> - <td class="tdright">7·39</td> - <td class="tdright">1·580</td> - <td class="tdright">0·221</td> - <td class="tdright">7·15</td></tr> -<tr> - <td class="tdleft">40 years</td> - <td class="tdright">1·686</td> - <td class="tdright">0·228</td> - <td class="tdright">7·39</td> - <td class="tdright">1·580</td> - <td class="tdright">0·221</td> - <td class="tdright">7·15</td></tr> -</table></div> -<p class="ptblfn">† A smooth curve, very similar to this, for the growth in -“auricular height” of the girl’s head, is given by Pearson, -in <i>Biometrika</i>, <span class="smmaj">III,</span> p. 141. 1904.</p> -</div><!--dtblbox--> - -<p>As regards external form, very similar differences exist, which -however we must express in terms not of weight but of length. -Thus the annexed table shews the changing ratios of the vertical -length of the head to the entire stature; and while this ratio -constantly diminishes, it will be seen that the rate of change is -greatest (or the coefficient of acceleration highest) between the -ages of about two and five years.</p> - -<p>In one of Quetelet’s tables (<i>supra</i>, p. 63), he gives measurements -of the total span of the outstretched arms in man, from -year to year, compared with the vertical stature. The two -measurements are so nearly identical in actual magnitude that a -direct comparison by means of curves becomes unsatisfactory; -but I have reduced Quetelet’s data to percentages, and it will be -seen from Fig. <a href="#fig19" title="go to Fig. 19">19</a> that the percentage proportion of span to -height undergoes a remarkable and steady change from birth to -the age of twenty years; the man grows more rapidly in stretch -of arms than he does in height, and the -span which was less than <span class="xxpn" id="p094">{94}</span> -the stature at birth by about 1 per cent. exceeds it at the age of -twenty by about 4 per cent. After the age of twenty, Quetelet’s -data are few and irregular, but it is clear that the span goes on -for a long while increasing in proportion to the stature. How -far the phenomenon is due to actual growth of the arms and -how far to the increasing breadth of the chest is not yet -ascertained.</p> - -<div class="dctr01" id="fig19"> -<img src="images/i094.png" width="800" height="528" alt=""> - <div class="dcaption">Fig. 19. Ratio of stature in Man, to span of - outstretched arms.<br> - (From Quetelet’s data.)</div></div> - -<p>The differences of rate of growth in different parts of the body -are very simply brought out by the following table, which shews -the relative growth of certain parts and organs of a young trout, -at intervals of a few days during the period of most rapid development. -It would not be difficult, from a picture of the little -trout at any one of these stages, to draw its approximate form -at any other, by the help of the numerical data here set -forth<a class="afnanch" href="#fn126" id="fnanch126">126</a>. -<span class="xxpn" id="p095">{95}</span></p> - -<div class="dtblbox"><div class="nowrap"> -<table class="fsz7 borall"> -<caption class="fsz5"><i>Trout (Salmo fario): proportionate growth of various organs.</i><br> -(<i>From Jenkinson’s data.</i>) -</caption> -<tr> - <th class="borall">Days<br>old</th> - <th class="borall">Total<br>length</th> - <th class="borall">Eye</th> - <th class="borall">Head</th> - <th class="borall">1st<br>dorsal</th> - <th class="borall">Ventral<br>fin</th> - <th class="borall">2nd<br>dorsal</th> - <th class="borall">Tail-fin</th> - <th class="borall">Breadth<br>of tail</th></tr> -<tr> - <td class="tdright"> 49</td> - <td class="tdright">100  </td> - <td class="tdright">100  </td> - <td class="tdright">100  </td> - <td class="tdright">100   </td> - <td class="tdright">100   </td> - <td class="tdright">100  </td> - <td class="tdright">100  </td> - <td class="tdright">100  </td></tr> -<tr> - <td class="tdright"> 63</td> - <td class="tdright">129·9</td> - <td class="tdright">129·4</td> - <td class="tdright">148·3</td> - <td class="tdright">148·6 </td> - <td class="tdright">148·5 </td> - <td class="tdright">108·4</td> - <td class="tdright">173·8</td> - <td class="tdright">155·9</td></tr> -<tr> - <td class="tdright"> 77</td> - <td class="tdright">154·9</td> - <td class="tdright">147·3</td> - <td class="tdright">189·2</td> - <td class="tdright">(203·6)</td> - <td class="tdright">(193·6)</td> - <td class="tdright">139·2</td> - <td class="tdright">257·9</td> - <td class="tdright">220·4</td></tr> -<tr> - <td class="tdright"> 92</td> - <td class="tdright">173·4</td> - <td class="tdright">179·4</td> - <td class="tdright">220·0</td> - <td class="tdright">(193·2)</td> - <td class="tdright">(182·1)</td> - <td class="tdright">154·5</td> - <td class="tdright">307·6</td> - <td class="tdright">272·2</td></tr> -<tr> - <td class="tdright">106</td> - <td class="tdright">194·6</td> - <td class="tdright">192·5</td> - <td class="tdright">242·5</td> - <td class="tdright">173·2 </td> - <td class="tdright">165·3 </td> - <td class="tdright">173·4</td> - <td class="tdright">337·3</td> - <td class="tdright">287·7</td></tr> -</table></div></div><!--dtblbox--> - -<p>While it is inequality of growth in <i>different</i> directions that we -can most easily comprehend as a phenomenon leading to gradual change -of outward form, we shall see in another chapter<a class="afnanch" -href="#fn127" id="fnanch127">127</a> -that differences of rate at -different parts of a longitudinal system, though always in the same -direction, also lead to very notable and regular transformations. Of -this phenomenon, the difference in rate of longitudinal growth between -head and body is a simple case, and the difference which accompanies -and results from it in the bodily form of the child and the man is -easy to see. A like phenomenon has been studied in much greater detail -in the case of plants, by Sachs and certain other botanists, after -a method in use by Stephen Hales a hundred and fifty years before<a -class="afnanch" href="#fn128" id="fnanch128">128</a>.</p> - -<p>On the growing root of a bean, ten narrow zones were marked -off, starting from the apex, each zone a millimetre in breadth. -After twenty-four hours’ growth, at a certain constant temperature, -the whole marked portion had grown from 10 mm. to 33 mm. in -length; but the individual zones had grown at very unequal rates, -as shewn in the annexed table<a class="afnanch" href="#fn129" -id="fnanch129">129</a>.</p> - -<div class="dtblbox"><div class="nowrap"> -<table class="borall"> -<tr> - <th class="borall">Zone</th> - <th class="borall">Increment<br>mm.</th> - <th>  </th> - <th class="borall">Zone</th> - <th class="borall">Increment<br>mm.</th></tr> -<tr> - <td class="tdright">Apex</td> - <td class="tdright">1·5</td> - <td class="tdright">  </td> - <td class="tdright">6th</td> - <td class="tdright">1·3</td></tr> -<tr> - <td class="tdright">2nd</td> - <td class="tdright">5·8</td> - <td class="tdright">  </td> - <td class="tdright">7th</td> - <td class="tdright">0·5</td></tr> -<tr> - <td class="tdright">3rd</td> - <td class="tdright">8·2</td> - <td class="tdright">  </td> - <td class="tdright">8th</td> - <td class="tdright">0·3</td></tr> -<tr> - <td class="tdright">4th</td> - <td class="tdright">3·5</td> - <td class="tdright">  </td> - <td class="tdright">9th</td> - <td class="tdright">0·2</td></tr> -<tr> - <td class="tdright">5th</td> - <td class="tdright">1·6</td> - <td class="tdright">  </td> - <td class="tdright">10th</td> - <td class="tdright">0·1</td></tr> -</table></div></div><!--dtblbox--> - -<div><span class="xxpn" id="p096">{96}</span></div> - -<div class="dctr04" id="fig20"> -<img src="images/i096.png" width="600" height="679" alt=""> - <div class="pcaption">Fig. 20. Rate of growth in successive zones - near the tip of the bean-root.</div></div> - -<p>The several values in this table lie very nearly (as we see by -Fig. <a href="#fig20" title="go to Fig. 20">20</a>) in a smooth curve; in other words a definite law, or -principle of continuity, connects the rates of growth at successive -points along the growing axis of the root. Moreover this curve, -in its general features, is singularly like those acceleration-curves -which we have already studied, in which we plotted the rate of -growth against successive intervals of time, as here we have -plotted it against successive spatial intervals of an actual growing -structure. If we suppose for a moment that the velocities of -growth had been transverse to the axis, instead of, as in this case, -longitudinal and parallel with it, it is obvious that these same -velocities would have given us a leaf-shaped structure, of which -our curve in Fig. <a href="#fig20" title="go to Fig. 20">20</a> (if drawn to a suitable scale) would represent -the actual outline on either side of the median axis; or, again, -if growth had been not confined to one plane but symmetrical -about the axis, we should have had a sort -of turnip-shaped root, <span class="xxpn" id="p097">{97}</span> -having the form of a surface of revolution generated by the same -curve. This then is a simple and not unimportant illustration of -the direct and easy passage from velocity to form.</p> - -<div class="psmprnt3"> -<p>A kindred problem occurs when, instead of “zones” artificially marked -out in a stem, we deal with the rates of growth in successive actual -“internodes”; and an interesting variation of this problem occurs when -we consider, not the actual growth of the internodes, but the varying -number of leaves which they successively produce. Where we have whorls -of leaves at each node, as in Equisetum and in many water-weeds, then -the problem presents itself in a simple form, and in one such case, -namely in Ceratophyllum, it has been carefully investigated by Mr -Raymond Pearl<a class="afnanch" href="#fn130" id="fnanch130">130</a>.</p> - -<p>It is found that the mean number of leaves per whorl increases with -each successive whorl; but that the rate of increment diminishes from -whorl to whorl, as we ascend the axis. In other words, the increase in -the number of leaves per whorl follows a logarithmic ratio; and if <i>y</i> -be the mean number of leaves per whorl, and <i>x</i> the successional number -of the whorl from the root or main stem upwards, then</p> - -<div class="maths"> -<i>y</i> -= <i>A</i> + <i>C</i> log(<i>x</i> − <i>a</i>), -</div><!--maths--> - -<p class="pcontinue">where <i>A</i>, <i>C</i>, and <i>a</i> are certain specific -constants, varying with the part of the plant which we happen to be -considering. On the main stem, the rate of change in the number of -leaves per whorl is very slow; when we come to the small twigs, or -“tertiary branches,” it has become rapid, as we see from the following -abbreviated table:</p> - -<div class="dtblbox"><div class="nowrap"> -<table> -<caption><i>Number of leaves per whorl on the tertiary branches of -Ceratophyllum.</i></caption> -<tr> - <td class="tdleft">Position of whorl</td> - <th>1</th> - <th>2</th> - <th>3</th> - <th>4</th> - <th>5</th> - <th>6</th></tr> -<tr> - <td class="tdleft">Mean number of leaves</td> - <td class="tdright">6·55</td> - <td class="tdright">8·07</td> - <td class="tdright">9·00</td> - <td class="tdright">9·20</td> - <td class="tdright">9·75 </td> - <td class="tdright">10·00 </td></tr> -<tr> - <td class="tdleft">Increment</td> - <td class="tdcntr">—</td> - <td class="tdright">1·52</td> - <td class="tdright">·93</td> - <td class="tdright">·20</td> - <td class="tdright">(·55)</td> - <td class="tdright">(·25)</td></tr> -</table></div></div><!--dtblbox--> -</div><!--psmprnt3--> - -<p>We have seen that a slow but definite change of form is a common -accompaniment of increasing age, and is brought about as the simple -and natural result of an altered ratio between the rates of growth in -different dimensions: or rather by the progressive change necessarily -brought about by the difference in their accelerations. There are many -cases however in which the change is all but imperceptible to ordinary -measurement, and many others in which some one dimension is easily -measured, but others are hard to measure with corresponding accuracy. -<span class="xxpn" id="p098">{98}</span> For instance, in any ordinary -fish, such as a plaice or a haddock, the length is not difficult to -measure, but measurements of breadth or depth are very much more -uncertain. In cases such as these, while it remains difficult to define -the precise nature of the change of form, it is easy to shew that -such a change is taking place if we make use of that ratio of length -to weight which we have spoken of in the preceding chapter. Assuming, -as we may fairly do, that weight is directly proportional to bulk or -volume, we may express this relation in the form <i>W ⁄ L</i><sup>3</sup> -= <i>k</i>, where <i>k</i> is a constant, to be determined for each -particular case. (<i>W</i> and <i>L</i> are expressed in grammes and centimetres, -and it is usual to multiply the result by some figure, such as 1000, so -as to give the constant <i>k</i> a value near to unity.)</p> - -<div class="section"> -<div class="dctr03"> -<table class="borall"> -<caption class="captionblk"><i>Plaice caught in a certain area, March, -1907. Variation of k (the weight-length coefficient) with size. (Data -taken from the Department of Agriculture and Fisheries’ Plaice-Report, -vol.</i> <span class="smmaj">I,</span> <i>p.</i> 107, 1908.)</caption> -<tr> - <th class="borall">Size<br>in cm.</th> - <th class="borall">Weight<br>in gm.</th> - <th class="borall"><i>W ⁄ L</i><sup>3</sup><br>× 10,000</th> - <th class="borall"><i>W ⁄ L</i><sup>3</sup><br>(smoothed)</th></tr> -<tr> - <td class="tdright">23</td> - <td class="tdright">113</td> - <td class="tdright">92·8</td> - <td class="tdcntr">—</td></tr> -<tr> - <td class="tdright">24</td> - <td class="tdright">128</td> - <td class="tdright">92·6</td> - <td class="tdright">94·3</td></tr> -<tr> - <td class="tdright">25</td> - <td class="tdright">152</td> - <td class="tdright">97·3</td> - <td class="tdright">96·1</td></tr> -<tr> - <td class="tdright">26</td> - <td class="tdright">173</td> - <td class="tdright">98·4</td> - <td class="tdright">97·9</td></tr> -<tr> - <td class="tdright">27</td> - <td class="tdright">193</td> - <td class="tdright">98·1</td> - <td class="tdright">99·0</td></tr> -<tr> - <td class="tdright">28</td> - <td class="tdright">221</td> - <td class="tdright">100·6</td> - <td class="tdright">100·4</td></tr> -<tr> - <td class="tdright">29</td> - <td class="tdright">250</td> - <td class="tdright">102·5</td> - <td class="tdright">101·2</td></tr> -<tr> - <td class="tdright">30</td> - <td class="tdright">271</td> - <td class="tdright">100·4</td> - <td class="tdright">101·2</td></tr> -<tr> - <td class="tdright">31</td> - <td class="tdright">300</td> - <td class="tdright">100·7</td> - <td class="tdright">100·4</td></tr> -<tr> - <td class="tdright">32</td> - <td class="tdright">328</td> - <td class="tdright">100·1</td> - <td class="tdright">99·8</td></tr> -<tr> - <td class="tdright">33</td> - <td class="tdright">354</td> - <td class="tdright">98·5</td> - <td class="tdright">98·8</td></tr> -<tr> - <td class="tdright">34</td> - <td class="tdright">384</td> - <td class="tdright">97·7</td> - <td class="tdright">98·0</td></tr> -<tr> - <td class="tdright">35</td> - <td class="tdright">419</td> - <td class="tdright">97·7</td> - <td class="tdright">97·6</td></tr> -<tr> - <td class="tdright">36</td> - <td class="tdright">454</td> - <td class="tdright">97·3</td> - <td class="tdright">96·7</td></tr> -<tr> - <td class="tdright">37</td> - <td class="tdright">492</td> - <td class="tdright">95·2</td> - <td class="tdright">96·3</td></tr> -<tr> - <td class="tdright">38</td> - <td class="tdright">529</td> - <td class="tdright">96·4</td> - <td class="tdright">95·6</td></tr> -<tr> - <td class="tdright">39</td> - <td class="tdright">564</td> - <td class="tdright">95·1</td> - <td class="tdright">95·0</td></tr> -<tr> - <td class="tdright">40</td> - <td class="tdright">614</td> - <td class="tdright">95·9</td> - <td class="tdright">95·0</td></tr> -<tr> - <td class="tdright">41</td> - <td class="tdright">647</td> - <td class="tdright">93·9</td> - <td class="tdright">93·8</td></tr> -<tr> - <td class="tdright">42</td> - <td class="tdright">679</td> - <td class="tdright">91·6</td> - <td class="tdright">92·5</td></tr> -<tr> - <td class="tdright">43</td> - <td class="tdright">732</td> - <td class="tdright">92·1</td> - <td class="tdright">92·5</td></tr> -<tr> - <td class="tdright">44</td> - <td class="tdright">800</td> - <td class="tdright">93·9</td> - <td class="tdright">94·0</td></tr> -<tr> - <td class="tdright">45</td> - <td class="tdright">875</td> - <td class="tdright">96·0</td> - <td class="tdcntr">—</td></tr> -</table></div><!--dtblbox--></div> - -<div><span class="xxpn" id="p099">{99}</span></div> - -<p>Now while this <i>k</i> may be spoken of as a “constant,” having -a certain mean value specific to each species of organism, and -depending on the form of the organism, any change to which it -may be subject will be a very delicate index of progressive changes -of form; for we know that our measurements of length are, on -the average, very accurate, and weighing is a still more delicate -method of comparison than any linear measurement.</p> - -<div class="dctr03" id="fig21"> -<img src="images/i099.png" width="600" height="412" alt=""> - <div class="pcaption">Fig. 21. Changes in the weight-length ratio - of Plaice, with increasing size.</div></div> - -<p>Thus, in the case of plaice, when we deal with the mean values -for a large number of specimens, and when we are careful to deal -only with such as are caught in a particular locality and at a particular -time, we see that <i>k</i> is by no means constant, but steadily -increases to a maximum, and afterwards slowly declines with the -increasing size of the fish (Fig. <a href="#fig21" title="go to Fig. 21">21</a>). To begin with, therefore, the -weight is increasing more rapidly than the cube of the length, and -it follows that the length itself is increasing less rapidly than some -other linear dimension; while in later life this condition is reversed. -The maximum is reached when the length of the fish is somewhere -near to 30 cm., and it is tempting to suppose that with this “point -of inflection” there is associated some well-marked epoch in the -fish’s life. As a matter of fact, the size of 30 cm. is approximately -that at which sexual maturity may be said to begin, or is at least -near enough to suggest a close connection between the two -phenomena. The first step towards further -investigation of the <span class="xxpn" id="p100">{100}</span> -apparent coincidence would be to determine the coefficient <i>k</i> of -the two sexes separately, and to discover whether or not the point -of inflection is reached (or sexual maturity is reached) at a smaller -size in the male than in the female plaice; but the material for -this investigation is at present scanty.</p> - -<div class="dctr02" id="fig22"> -<img src="images/i100.png" width="700" height="671" alt=""> - <div class="dcaption">Fig. 22. Periodic annual change - in the weight-length ratio of Plaice.</div></div> - -<p>A still more curious and more unexpected result appears when -we compare the values of <i>k</i> for the same fish at different seasons of -the year<a class="afnanch" href="#fn131" id="fnanch131">131</a>. -When for simplicity’s sake (as in the accompanying -table and Fig. <a href="#fig22" title="go to Fig. 22">22</a>) we restrict ourselves to fish of one particular -size, it is not necessary to determine the value of <i>k</i>, because a -change in the ratio of length to weight is obvious enough; but -when we have small numbers, and various sizes, to deal with, -the determination of <i>k</i> may help us very much. It will be seen, -then, that in the case of plaice the ratio of weight to length -exhibits a regular periodic variation with the -course of the seasons. <span class="xxpn" id="p101">{101}</span></p> - -<div class="dtblbox"><div class="nowrap" id="p101table"> -<table class="borall"> -<caption class="captionblk"><i>Relation of Weight to Length in -Plaice of 55 cm. long, from Month to Month. (Data taken from the -Department of Agriculture and Fisheries Plaice-Report, vol.</i> <span -class="smmaj">II,</span> <i>p.</i> 92, 1909.)</caption> -<tr> - <th class="borall"></th> - <th class="borall">Average<br>weight<br>in<br>grammes</th> - <th class="borall"><i>W ⁄ L</i><sup>3</sup><br>× 100</th> - <th class="borall"><i>W ⁄ L</i><sup>3</sup><br>(smoothed)</th></tr> -<tr> - <td class="tdleft">Jan.</td> - <td class="tdcntr">2039</td> - <td class="tdcntr">1·226</td> - <td class="tdcntr">1·157</td></tr> -<tr> - <td class="tdleft">Feb.</td> - <td class="tdcntr">1735</td> - <td class="tdcntr">1·043</td> - <td class="tdcntr">1·080</td></tr> -<tr> - <td class="tdleft">March</td> - <td class="tdcntr">1616</td> - <td class="tdcntr">0·971</td> - <td class="tdcntr">0·989</td></tr> -<tr> - <td class="tdleft">April</td> - <td class="tdcntr">1585</td> - <td class="tdcntr">0·953</td> - <td class="tdcntr">0·967</td></tr> -<tr> - <td class="tdleft">May</td> - <td class="tdcntr">1624</td> - <td class="tdcntr">0·976</td> - <td class="tdcntr">0·985</td></tr> -<tr> - <td class="tdleft">June</td> - <td class="tdcntr">1707</td> - <td class="tdcntr">1·026</td> - <td class="tdcntr">1·005</td></tr> -<tr> - <td class="tdleft">July</td> - <td class="tdcntr">1686</td> - <td class="tdcntr">1·013</td> - <td class="tdcntr">1·037</td></tr> -<tr> - <td class="tdleft">August</td> - <td class="tdcntr">1783</td> - <td class="tdcntr">1·072</td> - <td class="tdcntr">1·042</td></tr> -<tr> - <td class="tdleft">Sept.</td> - <td class="tdcntr">1733</td> - <td class="tdcntr">1·042</td> - <td class="tdcntr">1·111</td></tr> -<tr> - <td class="tdleft">Oct.</td> - <td class="tdcntr">2029</td> - <td class="tdcntr">1·220</td> - <td class="tdcntr">1·160</td></tr> -<tr> - <td class="tdleft">Nov.</td> - <td class="tdcntr">2026</td> - <td class="tdcntr">1·218</td> - <td class="tdcntr">1·213</td></tr> -<tr> - <td class="tdleft">Dec.</td> - <td class="tdcntr">1998</td> - <td class="tdcntr">1·201</td> - <td class="tdcntr">1·215</td></tr> -</table></div></div><!--dtblbox--> - -<p class="pcontinue">With unchanging length, the weight and therefore the bulk of the -fish falls off from about November to March or April, and again -between May or June and November the bulk and weight are -gradually restored. The explanation is simple, and depends -wholly on the process of spawning, and on the subsequent building -up again of the tissues and the reproductive organs. It follows -that, by this method, without ever seeing a fish spawn, and without -ever dissecting one to see the state of its reproductive system, we -can ascertain its spawning season, and determine the beginning -and end thereof, with great accuracy.</p> - -<hr class="hrblk"> - -<p>As a final illustration of the rate of growth, and of unequal -growth in various directions, I give the following table of data -regarding the ox, extending over the first three years, or nearly -so, of the animal’s life. The observed data are (1) the weight of -the animal, month by month, (2) the length of the back, from the -occiput to the root of the tail, and (3) the height to the withers. -To these data I have added (1) the ratio of length to height, -(2) the coefficient (<i>k</i>) expressing the ratio of weight to the cube of -the length, and (3) a similar coefficient (<i>k′</i>) for the height of the -animal. It will be seen that, while all these ratios tend to alter -continuously, shewing that the animal’s form is steadily altering -as it approaches maturity, the ratio between -length and weight <span class="xxpn" id="p102">{102}</span> -changes comparatively little. The simple ratio between length -and height increases considerably, as indeed we should expect; -for we know that in all Ungulate animals the legs are remarkably</p> - -<div class="dtblboxin10"> -<table class="fsz7 borall"> -<caption class="captionblk fsz5"><i>Relations -between the Weight and certain Linear Dimensions of -the Ox. (Data from Przibram, after Cornevin†.)</i></caption> -<tr> - <th class="borall">Age in<br>months</th> - <th class="borall"><i>W</i>, wt.<br>in kg.</th> - <th class="borall"><i>L</i>,<br>length<br>of back</th> - <th class="borall"><i>H</i>,<br>height</th> - <th class="borall"><i>L ⁄ H</i></th> - <th class="borall"><i>k</i><br>= <i>W ⁄ L</i><sup>3</sup><br>× 10</th> - <th class="borall"><i>k′</i><br>= <i>W ⁄ H</i><sup>3</sup><br>× 10</th></tr> -<tr> - <td class="tdright">0</td> - <td class="tdright">37  </td> - <td class="tdright">·78 </td> - <td class="tdright">·70 </td> - <td class="tdright">1·114</td> - <td class="tdright">·779</td> - <td class="tdright">1·079</td></tr> -<tr> - <td class="tdright">1</td> - <td class="tdright">55·3</td> - <td class="tdright">·94 </td> - <td class="tdright">·77 </td> - <td class="tdright">1·221</td> - <td class="tdright">·665</td> - <td class="tdright">1·210</td></tr> -<tr> - <td class="tdright">2</td> - <td class="tdright">86·3</td> - <td class="tdright">1·09 </td> - <td class="tdright">·85 </td> - <td class="tdright">1·282</td> - <td class="tdright">·666</td> - <td class="tdright">1·406</td></tr> -<tr> - <td class="tdright">3</td> - <td class="tdright">121·3</td> - <td class="tdright">1·207</td> - <td class="tdright">·94 </td> - <td class="tdright">1·284</td> - <td class="tdright">·690</td> - <td class="tdright">1·460</td></tr> -<tr> - <td class="tdright">4</td> - <td class="tdright">150·3</td> - <td class="tdright">1·314</td> - <td class="tdright">·95 </td> - <td class="tdright">1·383</td> - <td class="tdright">·662</td> - <td class="tdright">1·754</td></tr> -<tr> - <td class="tdright">5</td> - <td class="tdright">179·3</td> - <td class="tdright">1·404</td> - <td class="tdright">1·040</td> - <td class="tdright">1·350</td> - <td class="tdright">·649</td> - <td class="tdright">1·600</td></tr> -<tr> - <td class="tdright">6</td> - <td class="tdright">210·3</td> - <td class="tdright">1·484</td> - <td class="tdright">1·087</td> - <td class="tdright">1·365</td> - <td class="tdright">·644</td> - <td class="tdright">1·638</td></tr> -<tr> - <td class="tdright">7</td> - <td class="tdright">247·3</td> - <td class="tdright">1·524</td> - <td class="tdright">1·122</td> - <td class="tdright">1·358</td> - <td class="tdright">·699</td> - <td class="tdright">1·751</td></tr> -<tr> - <td class="tdright">8</td> - <td class="tdright">267·3</td> - <td class="tdright">1·581</td> - <td class="tdright">1·147</td> - <td class="tdright">1·378</td> - <td class="tdright">·677</td> - <td class="tdright">1·791</td></tr> -<tr> - <td class="tdright">9</td> - <td class="tdright">282·8</td> - <td class="tdright">1·621</td> - <td class="tdright">1·162</td> - <td class="tdright">1·395</td> - <td class="tdright">·664</td> - <td class="tdright">1·802</td></tr> -<tr> - <td class="tdright">10</td> - <td class="tdright">303·7</td> - <td class="tdright">1·651</td> - <td class="tdright">1·192</td> - <td class="tdright">1·385</td> - <td class="tdright">·675</td> - <td class="tdright">1·793</td></tr> -<tr> - <td class="tdright">11</td> - <td class="tdright">327·7</td> - <td class="tdright">1·694</td> - <td class="tdright">1·215</td> - <td class="tdright">1·394</td> - <td class="tdright">·674</td> - <td class="tdright">1·794</td></tr> -<tr> - <td class="tdright">12</td> - <td class="tdright">350·7</td> - <td class="tdright">1·740</td> - <td class="tdright">1·238</td> - <td class="tdright">1·405</td> - <td class="tdright">·666</td> - <td class="tdright">1·849</td></tr> -<tr> - <td class="tdright">13</td> - <td class="tdright">374·7</td> - <td class="tdright">1·765</td> - <td class="tdright">1·254</td> - <td class="tdright">1·407</td> - <td class="tdright">·682</td> - <td class="tdright">1·900</td></tr> -<tr> - <td class="tdright">14</td> - <td class="tdright">391·3</td> - <td class="tdright">1·785</td> - <td class="tdright">1·264</td> - <td class="tdright">1·412</td> - <td class="tdright">·688</td> - <td class="tdright">1·938</td></tr> -<tr> - <td class="tdright">15</td> - <td class="tdright">405·9</td> - <td class="tdright">1·804</td> - <td class="tdright">1·270</td> - <td class="tdright">1·420</td> - <td class="tdright">·692</td> - <td class="tdright">1·982</td></tr> -<tr> - <td class="tdright">16</td> - <td class="tdright">417·9</td> - <td class="tdright">1·814</td> - <td class="tdright">1·280</td> - <td class="tdright">1·417</td> - <td class="tdright">·700</td> - <td class="tdright">2·092</td></tr> -<tr> - <td class="tdright">17</td> - <td class="tdright">423·9</td> - <td class="tdright">1·832</td> - <td class="tdright">1·290</td> - <td class="tdright">1·420</td> - <td class="tdright">·689</td> - <td class="tdright">1·974</td></tr> -<tr> - <td class="tdright">18</td> - <td class="tdright">423·9</td> - <td class="tdright">1·859</td> - <td class="tdright">1·297</td> - <td class="tdright">1·433</td> - <td class="tdright">·660</td> - <td class="tdright">1·943</td></tr> -<tr> - <td class="tdright">19</td> - <td class="tdright">427·9</td> - <td class="tdright">1·875</td> - <td class="tdright">1·307</td> - <td class="tdright">1·435</td> - <td class="tdright">·649</td> - <td class="tdright">1·916</td></tr> -<tr> - <td class="tdright">20</td> - <td class="tdright">437·9</td> - <td class="tdright">1·884</td> - <td class="tdright">1·311</td> - <td class="tdright">1·437</td> - <td class="tdright">·655</td> - <td class="tdright">1·944</td></tr> -<tr> - <td class="tdright">21</td> - <td class="tdright">447·9</td> - <td class="tdright">1·893</td> - <td class="tdright">1·321</td> - <td class="tdright">1·433</td> - <td class="tdright">·661</td> - <td class="tdright">1·943</td></tr> -<tr> - <td class="tdright">22</td> - <td class="tdright">464·4</td> - <td class="tdright">1·901</td> - <td class="tdright">1·333</td> - <td class="tdright">1·426</td> - <td class="tdright">·676</td> - <td class="tdright">1·960</td></tr> -<tr> - <td class="tdright">23</td> - <td class="tdright">480·9</td> - <td class="tdright">1·909</td> - <td class="tdright">1·345</td> - <td class="tdright">1·419</td> - <td class="tdright">·691</td> - <td class="tdright">1·977</td></tr> -<tr> - <td class="tdright">24</td> - <td class="tdright">500·9</td> - <td class="tdright">1·914</td> - <td class="tdright">1·352</td> - <td class="tdright">1·416</td> - <td class="tdright">·714</td> - <td class="tdright">2·027</td></tr> -<tr> - <td class="tdright">25</td> - <td class="tdright">520·9</td> - <td class="tdright">1·919</td> - <td class="tdright">1·359</td> - <td class="tdright">1·412</td> - <td class="tdright">·737</td> - <td class="tdright">2·075</td></tr> -<tr> - <td class="tdright">26</td> - <td class="tdright">534·1</td> - <td class="tdright">1·924</td> - <td class="tdright">1·361</td> - <td class="tdright">1·414</td> - <td class="tdright">·750</td> - <td class="tdright">2·119</td></tr> -<tr> - <td class="tdright">27</td> - <td class="tdright">547·3</td> - <td class="tdright">1·929</td> - <td class="tdright">1·363</td> - <td class="tdright">1·415</td> - <td class="tdright">·762</td> - <td class="tdright">2·162</td></tr> -<tr> - <td class="tdright">28</td> - <td class="tdright">554·5</td> - <td class="tdright">1·929</td> - <td class="tdright">1·363</td> - <td class="tdright">1·415</td> - <td class="tdright">·772</td> - <td class="tdright">2·190</td></tr> -<tr> - <td class="tdright">29</td> - <td class="tdright">561·7</td> - <td class="tdright">1·929</td> - <td class="tdright">1·363</td> - <td class="tdright">1·415</td> - <td class="tdright">·782</td> - <td class="tdright">2·218</td></tr> -<tr> - <td class="tdright">30</td> - <td class="tdright">586·2</td> - <td class="tdright">1·949</td> - <td class="tdright">1·383</td> - <td class="tdright">1·409</td> - <td class="tdright">·792</td> - <td class="tdright">2·216</td></tr> -<tr> - <td class="tdright">31</td> - <td class="tdright">610·7</td> - <td class="tdright">1·969</td> - <td class="tdright">1·403</td> - <td class="tdright">1·403</td> - <td class="tdright">·800</td> - <td class="tdright">2·211</td></tr> -<tr> - <td class="tdright">32</td> - <td class="tdright">625·7</td> - <td class="tdright">1·983</td> - <td class="tdright">1·420</td> - <td class="tdright">1·396</td> - <td class="tdright">·803</td> - <td class="tdright">2·186</td></tr> -<tr> - <td class="tdright">33</td> - <td class="tdright">640·7</td> - <td class="tdright">1·997</td> - <td class="tdright">1·437</td> - <td class="tdright">1·390</td> - <td class="tdright">·805</td> - <td class="tdright">2·159</td></tr> -<tr> - <td class="tdright">34</td> - <td class="tdright">655·7</td> - <td class="tdright">2·011</td> - <td class="tdright">1·454</td> - <td class="tdright">1·383</td> - <td class="tdright">·806</td> - <td class="tdright">2·133</td></tr> -</table> - -<p class="ptblfn">† Cornevin, Ch., Études sur la croissance, <i>Arch. de -Physiol. norm. et pathol.</i> (5), <span class="smmaj">IV,</span> p. 477, -1892.</p></div><!--dtblboxin10--> - -<div><span class="xxpn" id="p103">{103}</span></div> - -<p class="pcontinue">long at birth in comparison with other dimensions of the body. -It is somewhat curious, however, that this ratio seems to fall off -a little in the third year of growth, the animal continuing to grow -in height to a marked degree after growth in length has become -very slow. The ratio between height and weight is by much the -most variable of our three ratios; the coefficient <i>W ⁄ H</i><sup>3</sup> steadily -increases, and is more than twice as great at three years old as -it was at birth. This illustrates the important, but obvious fact, -that the coefficient <i>k</i> is most variable in the case of that -dimension which grows most uniformly, that is to say most nearly -in proportion to the general bulk of the animal. In short, the -successive values of <i>k</i>, as determined (at successive epochs) for -one dimension, are a measure of the <i>variability</i> of the others.</p> - -<hr class="hrblk"> - -<p>From the whole of the foregoing discussion we see that a certain -definite rate of growth is a characteristic or specific phenomenon, -deep-seated in the physiology of the organism; and that a very -large part of the specific morphology of the organism depends upon -the fact that there is not only an average, or aggregate, rate of -growth common to the whole, but also a variation of rate in -different parts of the organism, tending towards a specific rate -characteristic of each different part or organ. The smallest change -in the relative magnitudes of these partial or localised velocities -of growth will be soon manifested in more and more striking -differences of form. This is as much as to say that the time-element, -which is implicit in the idea of growth, can never (or -very seldom) be wholly neglected in our consideration of -form<a class="afnanch" href="#fn132" id="fnanch132">132</a>. -It is scarcely necessary to enlarge here upon our statement, for -not only is the truth of it self-evident, but it will find illustration -again and again throughout this book. Nevertheless, let us go -out of our way for a moment to consider it in reference to a -particular case, and to enquire whether it helps to remove any of -the difficulties which that case appears -to present. <span class="xxpn" id="p104">{104}</span></p> - -<div class="dctr03" id="fig23"> -<img src="images/i104.png" width="600" height="446" alt=""> - <div class="pcaption">Fig. 23. Variability of length of - tail-forceps in a sample of Earwigs. (After Bateson, <i>P. Z. - S.</i> 1892, p. 588.)</div></div> - -<p>In a very well-known paper, Bateson shewed that, among a -large number of earwigs, collected in a particular locality, the -males fell into two groups, characterised by large or by small -tail-forceps, with very few instances of intermediate magnitude. -This distribution into two groups, according to magnitude, is -illustrated in the accompanying diagram (Fig. <a href="#fig23" title="go to Fig. 23">23</a>); and the -phenomenon was described, and has been often quoted, as one -of dimorphism, or discontinuous variation. In this diagram the -time-element does not appear; but it is certain, and evident, that -it lies close behind. Suppose we take some organism which is -born not at all times of the year (as man is) but at some one -particular season (for instance a fish), then any random sample -will consist of individuals whose <i>ages</i>, and therefore whose <i>magnitudes</i>, -will form a discontinuous series; and by plotting these -magnitudes on a curve in relation to the number of individuals -of each particular magnitude, we obtain a curve such as that -shewn in Fig. <a href="#fig24" title="go to Fig. 24">24</a>, the first practical use of which is to enable us -to analyse our sample into its constituent “age-groups,” or in -other words to determine approximately the age, or ages of the -fish. And if, instead of measuring the whole length of our fish, -we had confined ourselves to particular parts, such -as head, or <span class="xxpn" id="p105">{105}</span> -tail or fin, we should have obtained discontinuous curves of -distribution, precisely analogous to those for the entire animal. -Now we know that the differences with which Bateson was dealing -were entirely a question of magnitude, and we cannot help seeing -that the discontinuous distributions of magnitude represented by -his earwigs’ tails are just such as are illustrated by the magnitudes -of the older and younger fish; we may indeed go so far as to say -that the curves are precisely comparable, for in both cases we see -a characteristic feature of detail, namely that the “spread” of the -curve is greater in the second wave than in the first, that is to -say (in the case of the fish) in the older as well as larger series. -Over the reason for this phenomenon, which is simple and all but -obvious, we need not pause.</p> - -<div class="dctr03" id="fig24"> -<img src="images/i105.png" width="600" height="444" alt=""> - <div class="dcaption">Fig. 24. Variability of length - of body in a sample of Plaice.</div></div> - -<p>It is evident, then, that in this case of “dimorphism,” the tails -of the one group of earwigs (which Bateson calls the “high males”) -have either grown <i>faster</i>, or have been growing for a longer period -of time, than those of the “low males.” If we could be certain -that the whole random sample of earwigs were of one and the -same age, then we should have to refer the phenomenon of dimorphism -to a physiological phenomenon, simple in kind (however -remarkable and unexpected); viz. that there -were two alternative <span class="xxpn" id="p106">{106}</span> -values, very different from one another, for the mean velocity of -growth, and that the individual earwigs varied around one or -other of these mean values, in each case according to the law of -probabilities. But on the other hand, if we could believe that -the two groups of earwigs were <i>of different ages</i>, then the phenomenon -would be simplicity itself, and there would be no more to -be said about it<a class="afnanch" href="#fn133" id="fnanch133">133</a>.</p> - -<hr class="hrblk"> - -<p>Before we pass from the subject of the relative rate of growth of -different parts or organs, we may take brief note of the fact that -various experiments have been made to determine whether the normal -ratios are maintained under altered circumstances of nutrition, and -especially in the case of partial starvation. For instance, it has been -found possible to keep young rats alive for many weeks on a diet such -as is just sufficient to maintain life without permitting any increase -of weight. The rat of three weeks old weighs about 25 gms., and under a -normal diet should weigh at ten weeks old about 150 gms., in the male, -or 115 gms. in the female; but the underfed rat is still kept at ten -weeks old to the weight of 25 gms. Under normal diet the proportions -of the body change very considerably between the ages of three and ten -weeks. For instance the tail gets relatively longer; and even when the -<i>total</i> growth of the rat is prevented by underfeeding, the <i>form</i> -continues to alter so that this increasing length of the tail is still -manifest<a class="afnanch" href="#fn134" id="fnanch134">134</a>. -<span -class="xxpn" id="p107">{107}</span></p> - -<div class="dtblbox"><div class="nowrap"> -<table class="borall"> -<tr> - <th class="borall" colspan="5"><i>Full-fed Rats.</i></th></tr> -<tr> - <th class="borall">Age in<br>weeks</th> - <th class="borall">Length<br>of body<br>(mm.)</th> - <th class="borall">Length<br>of tail<br>(mm.)</th> - <th class="borall">Total<br>length</th> - <th class="borall">% of<br>tail</th></tr> -<tr> - <td class="tdright">0</td> - <td class="tdright">48·7</td> - <td class="tdright">16·9</td> - <td class="tdright">65·6</td> - <td class="tdright">25·8</td></tr> -<tr> - <td class="tdright">1</td> - <td class="tdright">64·5</td> - <td class="tdright">29·4</td> - <td class="tdright">93·9</td> - <td class="tdright">31·3</td></tr> -<tr> - <td class="tdright">3</td> - <td class="tdright">90·4</td> - <td class="tdright">59·1</td> - <td class="tdright">149·5</td> - <td class="tdright">39·5</td></tr> -<tr> - <td class="tdright">6</td> - <td class="tdright">128·0</td> - <td class="tdright">110·0</td> - <td class="tdright">238·0</td> - <td class="tdright">46·2</td></tr> -<tr> - <td class="tdright">10</td> - <td class="tdright">173·0</td> - <td class="tdright">150·0</td> - <td class="tdright">323·0</td> - <td class="tdright">46·4</td></tr> -<tr> - <th class="borall" colspan="5"><i>Underfed Rats.</i></th></tr> -<tr> - <td class="tdright">6</td> - <td class="tdright">98·0</td> - <td class="tdright">72·3</td> - <td class="tdright">170·3</td> - <td class="tdright">42·5</td></tr> -<tr> - <td class="tdright">10</td> - <td class="tdright">99·6</td> - <td class="tdright">83·9</td> - <td class="tdright">183·5</td> - <td class="tdright">45·7</td></tr> -</table></div></div><!--dtblbox--> - -<p>Again as physiologists have long been aware, there is a marked -difference in the variation of weight of the different organs, -according to whether the animal’s total weight remain constant, -or be caused to diminish by actual starvation; and further striking -differences appear when the diet is not only scanty, but ill-balanced. -But these phenomena of abnormal growth, however interesting -from the physiological view, are of little practical importance to -the morphologist.</p> - -<div class="section"> -<h3><i>The effect of temperature<a class="afnanchlow" href="#fn135" -id="fnanch135" title="go to note 135">*</a>.</i></h3></div> - -<p>The rates of growth which we have hitherto dealt with are based on -special investigations, conducted under particular local conditions. -For instance, Quetelet’s data, so far as we have used them to -illustrate the rate of growth in man, are drawn from his study of the -population of Belgium. But apart from that “fortuitous” individual -variation which we have already considered, it is obvious that the -normal rate of growth will be found to vary, in man and in other -animals, just as the average stature varies, in different localities, -and in different “races.” This phenomenon is a very complex one, and is -doubtless a resultant of many undefined contributory causes; but we at -least gain something in regard to it, when we discover that the rate -of growth is directly affected by temperature, and probably by other -physical <span class="xxpn" id="p108">{108}</span> conditions. Réaumur -was the first to shew, and the observation was repeated by Bonnet<a -class="afnanch" href="#fn136" id="fnanch136">136</a>, -that the rate -of growth or development of the chick was dependent on temperature, -being retarded at temperatures below and somewhat accelerated at -temperatures above the normal temperature of incubation, that is -to say the temperature of the sitting hen. In the case of plants -the fact that growth is greatly affected by temperature is a matter -of familiar knowledge; the subject was first carefully studied by -Alphonse De Candolle, and his results and those of his followers are -discussed in the textbooks of Botany<a class="afnanch" href="#fn137" -id="fnanch137">137</a>.</p> - -<div class="psmprnt3"> -<p>That variation of temperature constitutes only one factor in -determining the rate of growth is admirably illustrated in the case -of the Bamboo. It has been stated (by Lock) that in Ceylon the rate -of growth of the Bamboo is directly proportional to the humidity of -the atmosphere: and again (by Shibata) that in Japan it is directly -proportional to the temperature. The two statements have been -ingeniously and satisfactorily reconciled by Blackman<a class="afnanch" -href="#fn138" id="fnanch138">138</a>, -who suggests that in Ceylon the -temperature-conditions are all that can be desired, but moisture is -apt to be deficient: while in Japan there is rain in abundance but the -average temperature is somewhat too low. So that in the one country it -is the one factor, and in the other country it is the other, which is -<i>essentially</i> variable.</p> -</div><!--psmprnt3--> - -<p>The annexed diagram (Fig. <a href="#fig25" title="go to Fig. 25">25</a>), shewing the growth in length -of the roots of some common plants during an identical period -of forty-eight hours, at temperatures varying from about 14° to -37° C., is a sufficient illustration of the phenomenon. We see that -in all cases there is a certain optimum temperature at which the -rate of growth is a maximum, and we can also see that on either -side of this optimum temperature the acceleration of growth, -positive or negative, with increase of temperature is rapid, while -at a distance from the optimum it is very slow. From the -data given by Sachs and others, we see further that this optimum -temperature is very much the same for all the common plants of -our own climate which have as yet been studied; -in them it is <span class="xxpn" id="p109">{109}</span> -somewhere about 26° C. (or say 77° F.), or about the temperature -of a warm summer’s day; while it is found, very naturally, to be -considerably higher in the case of plants such as the melon or the -maize, which are at home in warmer regions that our own.</p> - -<hr class="hrblk"> - -<div class="dctr03" id="fig25"> -<img src="images/i109.png" width="600" height="569" alt=""> - <div class="pcaption">Fig. 25. Relation of rate of growth to - temperature in certain plants. (From Sachs’s data.)</div></div> - -<p>In a large number of physical phenomena, and in a very marked degree -in all chemical reactions, it is found that rate of action is affected, -and for the most part accelerated, by rise of temperature; and this -effect of temperature tends to follow a definite “exponential” law, -which holds good within a considerable range of temperature, but is -altered or departed from when we pass beyond certain normal limits. The -law, as laid down by van’t Hoff for chemical reactions, is, that for -an interval of <i>n</i> degrees the velocity varies as -<i>x</i><sup><i>n</i></sup> , -<i>x</i> being called the “temperature coefficient”<a class="afnanch" -href="#fn139" id="fnanch139">139</a> -for the reaction in question. -<span class="xxpn" id="p110">{110}</span></p> - -<p>Van’t Hoff’s law, which has become a fundamental principle -of chemical mechanics, is likewise applicable (with certain qualifications) -to the phenomena of vital chemistry; and it follows that, -on very much the same lines, we may speak of the “temperature -coefficient” of growth. At the same time we must remember -that there is a very important difference (though we can scarcely -call it a <i>fundamental</i> one) between the purely physical and the -physiological phenomenon, in that in the former we study (or -seek and profess to study) one thing at a time, while in the latter -we have always to do with various factors which intersect and -interfere; increase in the one case (or change of any kind) tends -to be continuous, in the other case it tends to be brought to arrest. -This is the simple meaning of that <i>Law of Optimum</i>, laid down by -Errera and by Sachs as a general principle of physiology: namely -that <i>every</i> physiological process which varies (like growth itself) -with the amount or intensity of some external influence, does so -according to a law in which progressive increase is followed by -progressive decrease; in other words the function has its <i>optimum</i> -condition, and its curve shews a definite <i>maximum</i>. In the case -of temperature, as Jost puts it, it has on the one hand its accelerating -effect which tends to follow van’t Hoff’s law. But it has also -another and a cumulative effect upon the organism: “Sie schädigt -oder sie ermüdet ihn, und je höher sie steigt, desto rascher macht -sie die Schädigung geltend und desto schneller schreitet sie voran.” -It would seem to be this double effect of temperature in the case -of the organism which gives us our “optimum” curves, which are -the expression, accordingly, not of a primary phenomenon, but -of a more or less complex resultant. Moreover, as Blackman and -others have pointed out, our “optimum” temperature is very -ill-defined until we take account also of the <i>duration</i> of our experiment; -for obviously, a high temperature may lead to a short, -but exhausting, spell of rapid growth, while the slower rate -manifested at a lower temperature may be -the best in the end. <span class="xxpn" id="p111">{111}</span> -The mile and the hundred yards are won by different runners; -and maximum rate of working, and maximum amount of work -done, are two very different things<a class="afnanch" href="#fn140" id="fnanch140">140</a>.</p> - -<hr class="hrblk"> - -<p>In the case of maize, a certain series of experiments shewed that -the growth in length of the roots varied with the temperature as -follows<a class="afnanch" href="#fn141" id="fnanch141">141</a>:</p> - -<div class="dtblbox"><div class="nowrap"> -<table> -<tr> - <th>Temperature<br>°C.</th> - <th>Growth in<br>48 hours<br>mm.</th></tr> -<tr> - <td class="tdright">18·0</td> - <td class="tdright">1·1</td></tr> -<tr> - <td class="tdright">23·5</td> - <td class="tdright">10·8</td></tr> -<tr> - <td class="tdright">26·6</td> - <td class="tdright">29·6</td></tr> -<tr> - <td class="tdright">28·5</td> - <td class="tdright">26·5</td></tr> -<tr> - <td class="tdright">30·2</td> - <td class="tdright">64·6</td></tr> -<tr> - <td class="tdright">33·5</td> - <td class="tdright">69·5</td></tr> -<tr> - <td class="tdright">36·5</td> - <td class="tdright">20·7</td></tr> -</table></div></div><!--dtblbox--> - -<div class="dmaths"> -<p>Let us write our formula in the form</p> - -<div><i>V</i><sub>(<i>t+n</i>)</sub> / <i>V<sub>t</sub></i> -= <i>x</i><sup class="spitc">n</sup> .</div> - -<p>Then choosing two values out of the above experimental -series (say the second and the second-last), we have <i>t</i> -= 23·5, <i>n</i> = 10, and <i>V</i>, <i>V′</i> = 10·8 -and 69·5 respectively.</p> - -<div> -<p class="pleftfloat">Accordingly</p> - -<div>69·5 / 10·8 -= 6·4 -= <i>x</i><sup>10</sup> . -<br class="brclrfix"></div> - -<p class="pleftfloat">Therefore</p> - -<div>(log 6·4) / 10, or ·0806 -= log <i>x</i>. -<br class="brclrfix"></div> - -<p class="pleftfloat">And,</p> - -<div><i>x</i> -= 1·204 (for an interval of 1° C.). -<br class="brclrfix"></div></div> -</div><!--dmaths--> - -<p>This first approximation might be considerably improved by -taking account of all the experimental values, two only of -which we have as yet made use of; but even as it is, we see -by Fig. <a href="#fig26" title="go to Fig. 26">26</a> that it is in very fair accordance with the actual -results of observation, <i>within those particular limits</i> -of temperature to which the experiment is confined. <span -class="xxpn" id="p112">{112}</span></p> - -<p>For an experiment on <i>Lupinus albus</i>, quoted by Asa -Gray<a class="afnanch" href="#fn142" id="fnanch142">142</a>, -I have worked out the corresponding coefficient, but a little -more carefully. Its value I find to be 1·16, or very nearly -identical with that we have just found for the maize; and the -correspondence between the calculated curve and the actual -observations is now a close one.</p> - -<div class="dctr03" id="fig26"> -<img src="images/i112.png" width="600" height="504" alt=""> - <div class="pcaption">Fig. 26. Relation of rate of growth to - temperature in Maize. Observed values (after Köppen), and - calculated curve.</div></div> - -<div class="psmprnt3"> -<p>Since the above paragraphs were written, new data have come to -hand. Miss I. Leitch has made careful observations of the rate -of growth of rootlets of the Pea; and I have attempted a further -analysis of her principal results<a class="afnanch" href="#fn143" -id="fnanch143">143</a>. -In Fig. <a href="#fig27" title="go to Fig. 27">27</a> are shewn the mean rates of -growth (based on about a hundred experiments) at some thirty-four -different temperatures between 0·8° and 29·3°, each experiment lasting -rather less than twenty-four hours. Working out the mean temperature -coefficient for a great many combinations of these values, I obtain -a value of 1·092 per C.°, or 2·41 for an interval of 10°, and a mean -value for the whole series showing a rate of growth of just about 1 mm. -per hour at a temperature of 20°. My curve in Fig. <a href="#fig27" title="go to Fig. 27">27</a> is drawn from -these determinations; and it will be seen that, while it is by no means -exact at the lower temperatures, and will of course fail us altogether -at very high <span class="xxpn" id="p113">{113}</span> temperatures, -yet it serves as a very satisfactory guide to the relations between -rate and temperature within the ordinary limits of healthy growth. Miss -Leitch holds that the curve is <i>not</i> a van’t Hoff curve; and this, in -strict accuracy, we need not dispute. But the phenomenon seems to me to -be one into which the van’t Hoff ratio enters largely, though doubtless -combined with other factors which we cannot at present determine or -eliminate.</p> -</div><!--psmprnt3--> - -<div class="dctr01" id="fig27"> -<img src="images/i113.png" width="700" height="775" alt=""> - <div class="pcaption">Fig. <a href="#fig27" title="go to Fig. 27">27</a>. Relation of rate of growth to - temperature in rootlets of Pea. (From Miss I. Leitch’s - data.)</div></div> - -<p>While the above results conform fairly well to the law of -the temperature coefficient, it is evident that the imbibition -of water plays so large a part in the process of elongation -of the root or stem that the phenomenon is rather a physical -than a chemical one: and on this account, as Blackman has -remarked, the data commonly given for the rate of growth in -plants are apt to be <span class="xxpn" id="p114">{114}</span> -irregular, and sometimes (we might even say) misleading<a -class="afnanch" href="#fn144" id="fnanch144">144</a>. The -fact also, which we have already learned, that the elongation -of a shoot tends to proceed by jerks, rather than smoothly, -is another indication that the phenomenon is not purely -and simply a chemical one. We have abundant illustrations, -however, among animals, in which we may study the temperature -coefficient under circumstances where, though the phenomenon -is always complicated by osmotic factors, true metabolic -growth or chemical combination plays a larger role. Thus Mlle. -Maltaux and Professor Massart<a class="afnanch" href="#fn145" -id="fnanch145">145</a> have studied the rate of division in -a certain flagellate, <i>Chilomonas paramoecium</i>, and found -the process to take 29 minutes at 15° C., 12 at 25°, and -only 5 minutes at 35° C. These velocities are in the ratio -of 1 : 2·4 : 5·76, which ratio -corresponds precisely to a temperature coefficient of 2·4 for -each rise of 10°, or about 1·092 for each degree centigrade.</p> - -<p>By means of this principle we may throw light on the apparently -complicated results of many experiments. For instance, Fig. <a href="#fig28" title="go to Fig. 28">28</a> -is an illustration, which has been often copied, of O. Hertwig’s -work on the effect of temperature on the rate of development of -the tadpole<a class="afnanch" href="#fn146" id="fnanch146">146</a>.</p> - -<p>From inspection of this diagram, we see that the time taken -to attain certain stages of development (denoted by the numbers -III–VII) was as follows, at 20° and at 10° C., respectively.</p> - -<div class="dtblbox"><div class="nowrap"> -<table> -<tr> - <th></th> - <th></th> - <th>At 20°</th> - <th>At 10°</th> - <th></th></tr> -<tr> - <td class="tdleft">Stage</td> - <td class="tdright">III</td> - <td class="tdright">2·0</td> - <td class="tdright">6·5</td> - <td class="tdright">days</td></tr> -<tr> - <td class="tdcntr">″</td> - <td class="tdright">IV</td> - <td class="tdright">2·7</td> - <td class="tdright">8·1</td> - <td class="tdcntr">″</td></tr> -<tr> - <td class="tdcntr">″</td> - <td class="tdright">V</td> - <td class="tdright">3·0</td> - <td class="tdright">10·7</td> - <td class="tdcntr">″</td></tr> -<tr> - <td class="tdcntr">″</td> - <td class="tdright">VI</td> - <td class="tdright">4·0</td> - <td class="tdright">13·5</td> - <td class="tdcntr">″</td></tr> -<tr> - <td class="tdcntr">″</td> - <td class="tdright">VII</td> - <td class="tdright">5·0</td> - <td class="tdright">16·8</td> - <td class="tdcntr">″</td></tr> -<tr> - <td class="tdleft">Total</td> - <td class="tdcntr"></td> - <td class="tdright">16·7</td> - <td class="tdright">55·6</td> - <td class="tdcntr">″</td></tr> -</table></div></div><!--dtblbox--> - -<p>That is to say, the time taken to produce a given result at <span class="xxpn" id="p115">{115}</span> -10° was (on the average) somewhere about 55·6 ⁄ 16·7, or 3·33, -times as long as was required at 20°.</p> - -<div class="dctr02" id="fig28"> -<img src="images/i115.png" width="700" height="912" alt=""> - <div class="pcaption">Fig. 28. Diagram shewing time taken (in - days), at various temperatures (°C.), to reach certain stages - of development in the Frog: viz. I, gastrula; II, medullary - plate; III, closure of medullary folds; IV, tail-bud; V, tail - and gills; VI, tail-fin; VII, operculum beginning; VIII, do. - closing; IX, first appearance of hind-legs. (From Jenkinson, - after O. Hertwig, 1898.)</div></div> - -<div class="dmaths"> -<p>We may then put our equation again in -the simple form, <span class="xxpn" id="p116">{116}</span></p> - -<div><i>x</i><sup>10</sup> -= 3·33.<br class="brclrfix"></div> - -<p class="pleftfloat">Or,</p> - -<div>10 log <i>x</i> -= log 3·33 -= ·52244.<br class="brclrfix"></div> - -<p class="pleftfloat">Therefore</p> - -<div>log <i>x</i> = ·05224,<br class="brclrfix"></div> - -<p class="pcontinue pleftfloat">and</p> - -<div><i>x</i> = 1·128.<br class="brclrfix"></div> -</div><!--dmaths--> - -<p>That is to say, between the intervals of 10° and 20° C., if it -take <i>m</i> days, at a certain given temperature, for a certain stage -of development to be attained, it will take <i>m</i> × 1·128<sup class="spitc">n</sup> days, -when the temperature is <i>n</i> degrees less, for the same stage to -be arrived at.</p> - -<div class="dctr03" id="fig29"> -<img src="images/i116.png" width="600" height="677" alt=""> - <div class="dcaption">Fig. 29. Calculated values, corresponding - to preceding figure.</div></div> - -<p>Fig. <a href="#fig29" title="go to Fig. 29">29</a> is -calculated throughout from this value; and it will be seen -that it is extremely concordant with the original diagram, -as regards all the stages of development and the whole -range of temperatures shewn: in spite of the fact that the -coefficient on which it is based was derived by an easy -method from a very few points in the original curves. <span -class="xxpn" id="p117">{117}</span></p> - -<p>Karl Peter<a class="afnanch" href="#fn147" id="fnanch147">147</a>, -experimenting chiefly on echinoderm eggs, and also making use -of Hertwig’s experiments on young tadpoles, gives the normal -temperature coefficients for intervals of 10° C. (commonly written -<i>Q</i><sub>10</sub>) as follows.</p> - -<div class="dtblbox"><div class="nowrap"> -<table> -<tr> - <td class="tdleft">Sphaerechinus</td> - <td class="tdright">2·15,</td></tr> -<tr> - <td class="tdleft">Echinus</td> - <td class="tdright">2·13,</td></tr> -<tr> - <td class="tdleft">Rana</td> - <td class="tdright">2·86.</td></tr> -</table></div></div><!--dtblbox--> - -<p>These values are not only concordant, but are evidently of the -same order of magnitude as the temperature-coefficient in ordinary -chemical reactions. Peter has also discovered the very interesting -fact that the temperature-coefficient alters with age, usually but -not always becoming smaller as age increases.</p> - -<div class="dtblbox"><div class="nowrap"> -<table> -<tr> - <td class="tdleft">Sphaerechinus;</td> - <td class="tdleft">Segmentation</td> - <td class="tdright"><i>Q</i><sup>10</sup></td> - <td class="tdright">= 2·29,</td></tr> -<tr> - <td class="tdleft"></td> - <td class="tdleft">Later stages</td> - <td class="tdcntr">″</td> - <td class="tdright">= 2·03.</td></tr> -<tr> - <td class="tdleft">Echinus;</td> - <td class="tdleft">Segmentation</td> - <td class="tdcntr">″</td> - <td class="tdright">= 2·30,</td></tr> -<tr> - <td class="tdleft"></td> - <td class="tdleft">Later stages</td> - <td class="tdcntr">″</td> - <td class="tdright">= 2·08.</td></tr> -<tr> - <td class="tdleft">Rana;</td> - <td class="tdleft">Segmentation</td> - <td class="tdcntr">″</td> - <td class="tdright">= 2·23,</td></tr> -<tr> - <td class="tdleft"></td> - <td class="tdleft">Later stages</td> - <td class="tdcntr">″</td> - <td class="tdright">= 3·34.</td></tr> -</table></div></div><!--dtblbox--> - -<p>Furthermore, the temperature coefficient varies with the -temperature, diminishing as the temperature rises,—a rule which -van’t Hoff has shewn to hold in ordinary chemical operations. -Thus, in Rana the temperature coefficient at low temperatures -may be as high as 5·6: which is just another way of saying that -at low temperatures development is exceptionally retarded.</p> - -<hr class="hrblk"> - -<p>In certain fish, such as plaice and haddock, I and others have -found clear evidence that the ascending curve of growth is subject -to seasonal interruptions, the rate during the winter months -being always slower than in the months of summer: it is as though -we superimposed a periodic, annual, sine-curve upon the continuous -curve of growth. And further, as growth itself grows less and less -from year to year, so will the difference between the winter and -the summer rate also grow less and less. -The fluctuation in rate <span class="xxpn" id="p118">{118}</span> -will represent a vibration which is gradually dying out; the amplitude -of the sine-curve will gradually diminish till it disappears; -in short, our phenomenon is simply expressed by what is known -as a “damped sine-curve.” Exactly the same thing occurs in -man, though neither in his case nor in that of the fish have we -sufficient data for its complete illustration.</p> - -<p>We can demonstrate the fact, however, in the case of man by the help -of certain very interesting measurements which have been recorded by -Daffner<a class="afnanch" href="#fn148" id="fnanch148">148</a>, -of the -height of German cadets, measured at half-yearly intervals.</p> - -<div class="dtblbox"> -<table class="fsz7 borall"> -<caption class="fsz5"><i>Growth in height of German military Cadets, in half-yearly -periods.</i> (<i>Daffner.</i>)</caption> -<tr> - <th class="borall" colspan="2"></th> - <th class="borall" colspan="3">Height in cent.</th> - <th class="borall" colspan="3">Increment in cm.</th></tr> -<tr> - <th class="borall">Number observed</th> - <th class="borall">Age</th> - <th class="borall">October</th> - <th class="borall">April</th> - <th class="borall">October</th> - <th class="borall">Winter ½-year</th> - <th class="borall">Summer ½-year</th> - <th class="borall">Year</th></tr> -<tr> - <td class="tdright">12</td> - <td class="tdright">11–12</td> - <td class="tdright">139·4</td> - <td class="tdright">141·0</td> - <td class="tdright">143·3</td> - <td class="tdright">1·6</td> - <td class="tdright">2·3</td> - <td class="tdright">3·9</td></tr> -<tr> - <td class="tdright">80</td> - <td class="tdright">12–13</td> - <td class="tdright">143·0</td> - <td class="tdright">144·5</td> - <td class="tdright">147·4</td> - <td class="tdright">1·5</td> - <td class="tdright">2·9</td> - <td class="tdright">4·4</td></tr> -<tr> - <td class="tdright">146</td> - <td class="tdright">13–14</td> - <td class="tdright">147·5</td> - <td class="tdright">149·5</td> - <td class="tdright">152·5</td> - <td class="tdright">2·0</td> - <td class="tdright">3·0</td> - <td class="tdright">5·0</td></tr> -<tr> - <td class="tdright">162</td> - <td class="tdright">14–15</td> - <td class="tdright">152·2</td> - <td class="tdright">155·0</td> - <td class="tdright">158·5</td> - <td class="tdright">2·5</td> - <td class="tdright">3·5</td> - <td class="tdright">6·0</td></tr> -<tr> - <td class="tdright">162</td> - <td class="tdright">15–16</td> - <td class="tdright">158·5</td> - <td class="tdright">160·8</td> - <td class="tdright">163·8</td> - <td class="tdright">2·3</td> - <td class="tdright">3·0</td> - <td class="tdright">5·3</td></tr> -<tr> - <td class="tdright">150</td> - <td class="tdright">16–17</td> - <td class="tdright">163·5</td> - <td class="tdright">165·4</td> - <td class="tdright">167·7</td> - <td class="tdright">1·9</td> - <td class="tdright">2·3</td> - <td class="tdright">4·2</td></tr> -<tr> - <td class="tdright">82</td> - <td class="tdright">17–18</td> - <td class="tdright">167·7</td> - <td class="tdright">168·9</td> - <td class="tdright">170·4</td> - <td class="tdright">1·2</td> - <td class="tdright">1·5</td> - <td class="tdright">2·7</td></tr> -<tr> - <td class="tdright">22</td> - <td class="tdright">18–19</td> - <td class="tdright">169·8</td> - <td class="tdright">170·6</td> - <td class="tdright">171·5</td> - <td class="tdright">0·8</td> - <td class="tdright">0·9</td> - <td class="tdright">1·7</td></tr> -<tr> - <td class="tdright">6</td> - <td class="tdright">19–20</td> - <td class="tdright">170·7</td> - <td class="tdright">171·1</td> - <td class="tdright">171·5</td> - <td class="tdright">0·4</td> - <td class="tdright">0·4</td> - <td class="tdright">0·8</td></tr> -</table> -</div><!--dtblbox--> - -<p>In the accompanying diagram (Fig. <a href="#fig30" title="go to Fig. 30">30</a>) the half-yearly increments -are set forth, from the above table, and it will be seen that -they form two even and entirely separate series. The curve -joining up each series of points is an acceleration-curve; and the -comparison of the two curves gives a clear view of the relative -rates of growth during winter and summer, and the fluctuation -which these velocities undergo during the years in question. The -dotted line represents, approximately, the acceleration-curve in -its continuous fluctuation of alternate seasonal decrease and -increase.</p> - -<hr class="hrblk"> - -<p>In the case of trees, the seasonal fluctuations of growth<a -class="afnanch" href="#fn149" id="fnanch149">149</a> -admit <span -class="xxpn" id="p119">{119}</span> of easy determination, and it is a -point of considerable interest to compare the phenomenon in evergreen -and in deciduous trees. I happen to have no measurements at hand with -which to make this comparison in the case of our native trees, but -from a paper by Mr Charles E. Hall<a class="afnanch" href="#fn150" -id="fnanch150">150</a> -I have compiled certain mean values for growth -in the climate of Uruguay.</p> - -<div class="dctr01" id="fig30"> -<img src="images/i119.png" width="800" height="394" alt=""> - <div class="dcaption">Fig. 30. Half-yearly increments of growth, - in cadets of various ages. (From Daffner’s data.)</div></div> - -<div class="dtblbox"> -<table class="fsz8 twdth100"> -<caption class="captionblk fsz4"><i>Mean monthly increase in Girth of -Evergreen and Deciduous Trees, at San Jorge, Uruguay.</i> (<i>After -C. E. Hall.</i>) <i>Values expressed as percentages of total annual -increase.</i></caption> -<tr> - <th class="thsnug"> </th> - <th class="thsnug">Jan.</th> - <th class="thsnug">Feb.</th> - <th class="thsnug">Mar.</th> - <th class="thsnug">Apr.</th> - <th class="thsnug">May</th> - <th class="thsnug">June</th> - <th class="thsnug">July</th> - <th class="thsnug">Aug.</th> - <th class="thsnug">Sept.</th> - <th class="thsnug">Oct.</th> - <th class="thsnug">Nov.</th> - <th class="thsnug">Dec.</th></tr> -<tr> - <td class="tdleft borall">Evergreens</td> - <td class="tdsnug"> 9·1</td> - <td class="tdsnug"> 8·8</td> - <td class="tdsnug">8·6</td> - <td class="tdsnug">8·9</td> - <td class="tdsnug">7·7</td> - <td class="tdsnug">5·4</td> - <td class="tdsnug">4·3</td> - <td class="tdsnug">6·0</td> - <td class="tdsnug">9·1</td> - <td class="tdsnug">11·1</td> - <td class="tdsnug">10·8</td> - <td class="tdsnug">10·2</td></tr> -<tr> - <td class="tdleft borall">Deciduous trees</td> - <td class="tdsnug">20·3</td> - <td class="tdsnug">14·6</td> - <td class="tdsnug">9·0</td> - <td class="tdsnug">2·3</td> - <td class="tdsnug">0·8</td> - <td class="tdsnug">0·3</td> - <td class="tdsnug">0·7</td> - <td class="tdsnug">1·3</td> - <td class="tdsnug">3·5</td> - <td class="tdsnug"> 9·9</td> - <td class="tdsnug">16·7</td> - <td class="tdsnug">21·0</td></tr> -</table></div><!--dtblbox--> - -<p>The measurements taken were those of the girth of the tree, -in mm., at three feet from the ground. The evergreens included -species of Pinus, Eucalyptus and Acacia; the deciduous trees -included Quercus, Populus, Robinia and Melia. I have merely -taken mean values for these two groups, and expressed the -monthly values as percentages of the mean annual increase. The -result (as shewn by Fig. <a href="#fig31" title="go to Fig. 31">31</a>) is very much what we might have -expected. The growth of the deciduous trees is completely -arrested in winter-time, and the arrest is -all but complete over <span class="xxpn" id="p120">{120}</span> -a considerable period of time; moreover, during the warm season, -the monthly values are regularly graded (approximately in a -sine-curve) with a clear maximum (in the southern hemisphere) -about the month of December. In the evergreen trees, on the -other hand, the amplitude of the periodic wave is very much -less; there is a notable amount of growth all the year round, -and, while there is a marked diminution in rate during the coldest -months, there is a tendency towards equality over a considerable -part of the warmer season. It is probable that some of the -species examined, and especially the pines, were definitely retarded -in growth, either by a temperature above their optimum, or by -deficiency of moisture, during the hottest period of the year; -with the result that the seasonal curve in our diagram has (as it -were) its region of maximum cut off.</p> - -<div class="dctr02" id="fig31"> -<img src="images/i120.png" width="700" height="604" alt=""> - <div class="pcaption">Fig. 31. Periodic annual fluctuation in rate -of growth of trees (in the southern hemisphere).</div></div> - -<p>In the case of trees, the seasonal periodicity of growth is so well -marked that we are entitled to make use of the phenomenon in a converse -way, and to draw deductions as to variations in <span class="xxpn" -id="p121">{121}</span> climate during past years from the record of -varying rates of growth which the tree, by the thickness of its annual -rings, has preserved for us. Mr. A. E. Douglass, of the University of -Arizona, has made a careful study of this question<a class="afnanch" -href="#fn151" id="fnanch151">151</a>, -and I have received (through -Professor H. H. Turner of Oxford) some measurements of the average -width of the successive annual rings in “yellow pine,” 500 years -old, from Arizona, in which trees the annual rings are very clearly -distinguished. From the year 1391 to 1518, the mean of two trees was -used; from 1519 to 1912, the mean of five; and the means of these, -and sometimes of larger numbers, were found to be very concordant. A -correction was applied by drawing a long, nearly straight line through -the curve for the whole period, which line was assumed to represent -the slowly diminishing mean width of ring accompanying the increase -of size, or age, of the tree; and the actual growth as measured was -equated with this diminishing mean. The figures used give, accordingly, -the ratio of the actual growth in each year to the mean growth -corresponding to the age or magnitude of the tree at that epoch.</p> - -<p>It was at once manifest that the rate of growth so determined -shewed a tendency to fluctuate in a long period of between 100 and -200 years. I then smoothed in groups of 100 (according to Gauss’s -method) the yearly values, so that each number thus found -represented the mean annual increase during a century: that is -to say, the value ascribed to the year 1500 represented the <i>average -annual growth</i> during the whole period between 1450 and 1550, -and so on. These values give us a curve of beautiful and surprising -smoothness, from which we seem compelled to draw the direct -conclusion that the climate of Arizona, during the last 500 years, -has fluctuated with a regular periodicity of almost precisely 150 -years. Here again we should be left in doubt -(so far as these <span class="xxpn" id="p123">{123}</span> -observations go) whether the essential factor be a fluctuation of -temperature or an alternation of moisture and aridity; but the -character of the Arizona climate, and the known facts of recent -years, encourage the belief that the latter is the more direct and -more important factor.</p> - -<div class="dctr01" id="fig32"> -<img src="images/i122.png" width="800" height="355" alt=""> - <div class="pcaption">Fig. 32. Long-period fluctuation in rate of - growth of Arizona trees (smoothed in 100-year periods), - from <span class="smmaj">A.D.</span> 1390–1490 to - <span class="smmaj">A.D.</span> 1810–1910.</div></div> - -<p>It has been often remarked that our common European trees, such -for instance as the elm or the cherry, tend to have larger leaves the -further north we go; but in this case the phenomenon is to be ascribed -rather to the longer hours of daylight than to any difference of -temperature<a class="afnanch" href="#fn152" id="fnanch152">152</a>. -The -point is a physiological one, and consequently of little importance to -us here<a class="afnanch" href="#fn153" id="fnanch153">153</a>; -the -main point for the morphologist is the very simple one that physical -or climatic conditions have greatly influenced the rate of growth. The -case is analogous to the direct influence of temperature in modifying -the colouration of organisms, such as certain butterflies. Now if -temperature affects the rate of growth in strict uniformity, alike -in all directions and in all parts or organs, its direct effect must -be limited to the production of local races or varieties differing -from one another in actual magnitude, as the Siberian goldfinch or -bullfinch, for instance, differ from our own. But if there be even ever -so little of a discriminating action in the enhancement of growth by -temperature, such that it accelerates the growth of one tissue or one -organ more than another, then it is evident that it must at once lead -to an actual difference of racial, or even “specific” form.</p> - -<p>It is not to be doubted that the various factors of climate -have some such discriminating influence. The leaves of our -northern trees may themselves be an instance of it; -and we have, <span class="xxpn" id="p124">{124}</span> -probably, a still better instance of it in the case of Alpine -plants<a class="afnanch" href="#fn154" id="fnanch154">154</a>, -whose general habit is dwarfed, though their floral organs suffer -little or no reduction. The subject, however, has been little -investigated, and great as its theoretic importance would be to -us, we must meanwhile leave it alone.</p> - -<div class="section"> -<h3><i>Osmotic factors in growth.</i></h3></div> - -<p>The curves of growth which we have now been studying -represent phenomena which have at least a two-fold interest, -morphological and physiological. To the morphologist, who -recognises that form is a “function” of growth, the important -facts are mainly these: (1) that the rate of growth is an orderly -phenomenon, with general features common to very various -organisms, while each particular organism has its own characteristic -phenomena, or “specific constants”; (2) that rate of growth -varies with temperature, that is to say with season and with -climate, and with various other physical factors, external and -internal; (3) that it varies in different parts of the body, and -according to various directions or axes; such variations being -definitely correlated with one another, and thus giving rise to -the characteristic proportions, or form, of the organism, and to -the changes in form which it undergoes in the course of its -development. But to the physiologist, the phenomenon suggests -many other important considerations, and throws much light on -the very nature of growth itself, as a manifestation of chemical -and physical energies.</p> - -<p>To be content to shew that a certain rate of growth occurs in -a certain organism under certain conditions, or to speak of the -phenomenon as a “reaction” of the living organism to its environment -or to certain stimuli, would be but an example of that “lack -of particularity<a class="afnanch" href="#fn155" id="fnanch155">155</a>” -in regard to the actual mechanism of physical -cause and effect with which we are apt in biology to be too easily -satisfied. But in the case of rate of -growth we pass somewhat <span class="xxpn" id="p125">{125}</span> -beyond these limitations; for the affinity with certain types of -chemical reaction is plain, and has been recognised by a great -number of physiologists.</p> - -<p>A large part of the phenomenon of growth, both in animals -and still more conspicuously in plants, is associated with “turgor,” -that is to say, is dependent on osmotic conditions; in other words, -the velocity of growth depends in great measure (as we have already -seen, p. <a href="#p113" title="go to pg. 113">113</a>) -on the amount of water taken up into the living -cells, as well as on the actual amount of chemical metabolism -performed by them<a class="afnanch" href="#fn156" id="fnanch156">156</a>. -Of the chemical phenomena which result -in the actual increase of protoplasm we shall speak presently, but -the rôle of water in growth deserves also a passing word, even in -our morphological enquiry.</p> - -<p>It has been shewn by Loeb that in Cerianthus or Tubularia, -for instance, the cells in order to grow must be turgescent; and -this turgescence is only possible so long as the salt water in which -the cells lie does not overstep a certain limit of concentration. The -limit, in the case of Tubularia, is passed when the salt amounts -to about 5·4 per cent. Sea-water contains some 3·0 to 3·5 p.c. -of salts; but it is when the salinity falls much below this normal, -to about 2·2 p.c., that Tubularia exhibits its maximal turgescence, -and maximal growth. A further dilution is said to act as a poison -to the animal. Loeb has also shewn<a class="afnanch" href="#fn157" id="fnanch157">157</a> -that in certain eggs (e.g. -those of the little fish <i>Fundulus</i>) an increasing concentration of -the sea-water (leading to a diminishing “water-content” of the -egg) retards the rate of segmentation and at length renders -segmentation impossible; though nuclear division, by the way, -goes on for some time longer.</p> - -<p>Among many other observations of the same kind, those of -Bialaszewicz<a class="afnanch" href="#fn158" id="fnanch158">158</a>, -on the early growth of the frog, are notable. -He shews that the growth of the embryo while -still <i>within the <span class="xxpn" id="p126">{126}</span> -vitelline membrane</i> depends wholly on the absorption of water; -that whether rate of growth be fast or slow (in accordance with -temperature) the quantity of water absorbed is constant; and -that successive changes of form correspond to definite quantities -of water absorbed. The solid residue, as Davenport has also -shewn, may actually and notably diminish, while the embryo -organism is increasing rapidly in bulk and weight.</p> - -<p>On the other hand, in later stages and especially in the higher -animals, the percentage of water tends to diminish. This has -been shewn by Davenport in the frog, by Potts in the chick, and -particularly by Fehling in the case of man<a class="afnanch" href="#fn159" id="fnanch159">159</a>. -Fehling’s results -are epitomised as follows:</p> - -<div class="dtblbox"> -<table class="fsz7 twdth100"> -<tr> - <td class="tdright">Age in weeks</td> - <td class="tdright">6</td> - <td class="tdright">17</td> - <td class="tdright">22</td> - <td class="tdright">24</td> - <td class="tdright">26</td> - <td class="tdright">30</td> - <td class="tdright">35</td> - <td class="tdright">39</td></tr> -<tr> - <td class="tdright">Percentage of water</td> - <td class="tdright">97·5</td> - <td class="tdright">91·8</td> - <td class="tdright">92·0</td> - <td class="tdright">89·9</td> - <td class="tdright">86·4</td> - <td class="tdright">83·7</td> - <td class="tdright">82·9</td> - <td class="tdright">74·2</td></tr> -</table> - -<p>And the following illustrate Davenport’s results for the frog:</p> - -<table class="fsz7 twdth100"> -<tr> - <td class="tdright">Age in weeks</td> - <td class="tdright">1</td> - <td class="tdright">2</td> - <td class="tdright">5</td> - <td class="tdright">7</td> - <td class="tdright">9</td> - <td class="tdright">14</td> - <td class="tdright">41</td> - <td class="tdright">84</td></tr> -<tr> - <td class="tdright">Percentage of water</td> - <td class="tdright">56·3</td> - <td class="tdright">58·5</td> - <td class="tdright">76·7</td> - <td class="tdright">89·3</td> - <td class="tdright">93·1</td> - <td class="tdright">95·0</td> - <td class="tdright">90·2</td> - <td class="tdright">87·5</td></tr> -</table></div><!--dtblbox--> - -<p>To such phenomena of osmotic balance as the above, or in other -words to the dependence of growth on the uptake of water, Höber<a class="afnanch" href="#fn160" id="fnanch160">160</a> -and also Loeb are inclined to refer the modifications of form -which certain phyllopod crustacea undergo, when the highly -saline waters which they inhabit are further concentrated, or are -abnormally diluted. Their growth, according to Schmankewitsch, -is retarded by increase of concentration, so that the individuals -from the more saline waters appear stunted and dwarfish; and -they become altered or transformed in other ways, which for the -most part suggest “degeneration,” or a failure to attain full and -perfect development<a class="afnanch" href="#fn161" id="fnanch161">161</a>. -Important physiological changes also -ensue. The rate of multiplication is increased, and parthenogenetic -reproduction is encouraged. Male individuals become -plentiful in the less saline waters, and here the -females bring forth <span class="xxpn" id="p127">{127}</span> -their young alive; males disappear altogether in the more concentrated -brines, and then the females lay eggs, which, however, -only begin to develop when the salinity is somewhat reduced.</p> - -<p>The best-known case is the little “brine-shrimp,” <i>Artemia -salina</i>, found, in one form or another, all the world over, and first -discovered more than a century and a half ago in the salt-pans at -Lymington. Among many allied forms, one, <i>A. milhausenii</i>, -inhabits the natron-lakes of Egypt and Arabia, where, under the -name of “loul,” or “Fezzan-worm,” it is eaten by the Arabs<a class="afnanch" href="#fn162" id="fnanch162">162</a>. -This fact is interesting, because it indicates (and investigation -has apparently confirmed) that the tissues of the creature are not -impregnated with salt, as is the medium in which it lives. The -fluids of the body, the <i>milieu interne</i> (as Claude Bernard called -them<a class="afnanch" href="#fn163" id="fnanch163">163</a>), -are no more salt than are those of any ordinary crustacean -or other animal, but contain only some 0·8 per cent. of -NaCl<a class="afnanch" href="#fn164" id="fnanch164">164</a>, -while the <i>milieu externe</i> may contain 10, 20, or more per -cent. of this and other salts; which is as much as to say that -the skin, or body-wall, of the creature acts as a “semi-permeable -membrane,” through which the dissolved salts are not permitted -to diffuse, though water passes through freely: until a statical -equilibrium (doubtless of a complex kind) is at length attained.</p> - -<p>Among the structural changes which result from increased -concentration of the brine (partly during the life-time of the -individual, but more markedly during the short season which -suffices for the development of three or four, or perhaps more, -successive generations), it is found that the tail comes to bear -fewer and fewer bristles, and the tail-fins themselves tend at last -to disappear; these changes corresponding to -what have been <span class="xxpn" id="p128">{128}</span> -described as the specific characters of <i>A. milhausenii</i>, and of a -still more extreme form, <i>A. köppeniana</i>; while on the other -hand, progressive dilution of the water tends to precisely opposite -conditions, resulting in forms which have also been described as -separate species, and even referred to a separate genus, Callaonella, -closely akin to Branchipus (Fig. <a href="#fig33" title="go to Fig. 33">33</a>). <i>Pari passu</i> with these changes, -there is a marked change in the relative lengths of the fore and -hind portions of the body, that is to say, of the “cephalothorax” -and abdomen: the latter growing relatively longer, the salter the -water. In other words, not only is the rate of growth of the whole</p> - -<div class="dctr02" id="fig33"> -<img src="images/i128.png" width="600" height="416" alt=""> - <div class="pcaption">Fig. 33. Brine-shrimps (Artemia), from more - or less saline water. Upper figures shew tail-segment and - tail-fins; lower figures, relative length of cephalothorax - and abdomen. (After Abonyi.)</div></div> - -<p class="pcontinue">animal lessened by the saline concentration, but the specific rates -of growth in the parts of its body are relatively changed. This -latter phenomenon lends itself to numerical statement, and Abonyi -has lately shewn that we may construct a very regular curve, by -plotting the proportionate length of the creature’s abdomen -against the salinity, or density, of the water; and the several -species of Artemia, with all their other correlated specific characters, -are then found to occupy successive, more or less well-defined, and -more or less extended, regions of the curve (Fig. <a href="#fig33" title="go to Fig. 33">33</a>). In short, the -density of the water is so clearly a “function” -of the specific <span class="xxpn" id="p129">{129}</span> -character, that we may briefly define the species <i>Artemia</i> (<i>Callaonella</i>) -<i>Jelskii</i>, for instance, as the Artemia of density 1000–1010 -(NaCl), or the typical <i>A. salina</i>, or <i>principalis</i>, as the Artemia -of density 1018–1025, and so forth. It is a most interesting -fact that these Artemiae, under the protection of their semi-permeable -skins, are capable of living in waters not only of -great density, but of very varied chemical composition. The -natron-lakes, for instance, contain large quantities of magnesium</p> - -<div class="dctr01" id="fig34"> -<img src="images/i129.png" width="800" height="645" alt=""> - <div class="pcaption">Fig. 34. Percentage ratio of length of - abdomen to cephalothorax in brine-shrimps, at various - salinities. (After Abonyi.)</div></div> - -<p class="pcontinue">sulphate; and the Artemiae continue to live equally well in -artificial solutions where this salt, or where calcium chloride, has -largely taken the place of sodium chloride in the more common -habitat. Furthermore, such waters as those of the natron-lakes -are subject to very great changes of chemical composition as -concentration proceeds, owing to the different solubilities of the -constituent salts. It appears that the forms which the Artemiae -assume, and the changes which they undergo, -are identical or <span class="xxpn" id="p130">{130}</span> -indistinguishable, whichever of the above salts happen to exist, -or to predominate, in their saline habitat. At the same time we -still lack (so far as I know) the simple, but crucial experiments -which shall tell us whether, in solutions of different chemical -composition, it is <i>at equal densities</i>, or at “<i>isotonic</i>” concentrations -(that is to say, under conditions where the osmotic pressure, -and consequently the rate of diffusion, is identical), that the -same structural changes are produced, or corresponding phases -of equilibrium attained.</p> - -<p>While Höber and others<a class="afnanch" href="#fn165" id="fnanch165">165</a> -have referred all these phenomena to -osmosis, Abonyi is inclined to believe that the viscosity, or -mechanical resistance, of the fluid also reacts upon the organism; -and other possible modes of operation have been suggested. -But we may take it for certain that the phenomenon as a whole -is not a simple one; and that it includes besides the passive -phenomena of intermolecular diffusion, some other form of activity -which plays the part of a regulatory mechanism<a class="afnanch" href="#fn166" id="fnanch166">166</a>.</p> - -<div class="section"> -<h3><i>Growth and catalytic action.</i></h3></div> - -<p>In ordinary chemical reactions we have to deal (1) with a -specific velocity proper to the particular reaction, (2) with variations -due to temperature and other physical conditions, (3) according -to van’t Hoff’s “Law of Mass,” with variations due to the actual -quantities present of the reacting substances, and (4) in certain -cases, with variations due to the presence of “catalysing agents.” -In the simpler reactions, the law of mass involves a steady, gradual -slowing-down of the process, according to a logarithmic ratio, as -the reaction proceeds and as the initial amount of substance -diminishes; a phenomenon, however, which -need not necessarily <span class="xxpn" id="p131">{131}</span> -occur in the organism, part of whose energies are devoted to the -continual bringing-up of fresh supplies.</p> - -<p>Catalytic action occurs when some substance, often in very -minute quantity, is present, and by its presence produces or -accelerates an action, by opening “a way round,” without -the catalytic agent itself being diminished or used up<a class="afnanch" href="#fn167" id="fnanch167">167</a>. -Here the velocity curve, though quickened, is not necessarily -altered in form, for gradually the law of mass exerts its -effect and the rate of the reaction gradually diminishes. But -in certain cases we have the very remarkable phenomenon that -a body acting as a catalyser is necessarily formed as a product, -or bye-product, of the main reaction, and in such a case as this -the reaction-velocity will tend to be steadily accelerated. Instead -of dwindling away, the reaction will continue with an ever-increasing -velocity: always subject to the reservation that limiting -conditions will in time make themselves felt, such as a failure of -some necessary ingredient, or a development of some substance -which shall antagonise or finally destroy the original reaction. -Such an action as this we have learned, from Ostwald, to describe -as “autocatalysis.” Now we know that certain products of -protoplasmic metabolism, such as the enzymes, are very powerful -catalysers, and we are entitled to speak of an autocatalytic action -on the part of protoplasm itself. This catalytic activity of protoplasm -is a very important phenomenon. As Blackman says, -in the address already quoted, the botanists (or the zoologists) -“call it <i>growth</i>, attribute it to a specific power of protoplasm for -assimilation, and leave it alone as a fundamental phenomenon; -but they are much concerned as to the distribution of new growth -in innumerable specifically distinct forms.” While the chemist, on -the other hand, recognises it as a familiar phenomenon, and refers it -to the same category as his other known examples -of <i>autocatalysis</i>. <span class="xxpn" id="p132">{132}</span></p> - -<p>This very important, and perhaps even fundamental phenomenon -of growth would seem to have been first recognised by -Professor Chodat of Geneva, as we are told by his pupil Monnier<a class="afnanch" href="#fn168" id="fnanch168">168</a>. -“On peut bien, ainsi que M. Chodat l’a proposé, considérer -l’accroissement comme une réaction chimique complexe, dans -laquelle le catalysateur est la cellule vivante, et les corps en -présence sont l’eau, les sels, et l’acide carbonique.”</p> - -<p>Very soon afterwards a similar suggestion was made by Loeb<a class="afnanch" href="#fn169" id="fnanch169">169</a>, -in connection with the synthesis of <i>nuclein</i> or nuclear protoplasm; -for he remarked that, as in an autocatalysed chemical reaction, -the velocity of the synthesis increases during the initial stage of -cell-division in proportion to the amount of nuclear matter already -synthesised. In other words, one of the products of the reaction, -i.e. one of the constituents of the nucleus, accelerates the production -of nuclear from cytoplasmic material.</p> - -<p>The phenomenon of autocatalysis is by no means confined to -living or protoplasmic chemistry, but at the same time it is -characteristically, and apparently constantly, associated therewith. -And it would seem that to it we may ascribe a considerable part -of the difference between the growth of the organism and the -simpler growth of the crystal<a class="afnanch" href="#fn170" id="fnanch170">170</a>: -the fact, for instance, that the cell -can grow in a very low concentration of its nutritive solution, -while the crystal grows only in a supersaturated one; and the -fundamental fact that the nutritive solution need only contain -the more or less raw materials of the complex constituents of the -cell, while the crystal grows only in a solution of its own actual -substance<a class="afnanch" href="#fn171" id="fnanch171">171</a>.</p> - -<p>As F. F. Blackman has pointed out, the multiplication of an -organism, for instance the prodigiously rapid -increase of a bacterium, <span class="xxpn" id="p133">{133}</span> -which tends to double its numbers every few minutes, till (were -it not for limiting factors) its numbers would be all but incalculable -in a day<a class="afnanch" href="#fn172" id="fnanch172">172</a>, -is a simple but most striking illustration of the potentialities -of protoplasmic catalysis; and (apart from the large share -taken by mere “turgescence” or imbibition of water) the same -is true of the growth, or cell-multiplication, of a multicellular -organism in its first stage of rapid acceleration.</p> - -<p>It is not necessary for us to pursue this subject much further, -for it is sufficiently clear that the normal “curve of growth” of -an organism, in all its general features, very closely resembles the -velocity-curve of chemical autocatalysis. We see in it the first -and most typical phase of greater and greater acceleration; this -is followed by a phase in which limiting conditions (whose details -are practically unknown) lead to a falling off of the former -acceleration; and in most cases we come at length to a third phase, -in which retardation of growth is succeeded by actual diminution -of mass. Here we may recognise the influence of processes, or -of products, which have become actually deleterious; their -deleterious influence is staved off for a while, as the organism draws -on its accumulated reserves, but they lead ere long to the stoppage -of all activity, and to the physical phenomenon of death. But -when we have once admitted that the limiting conditions of -growth, which cause a phase of retardation to follow a phase -of acceleration, are very imperfectly known, it is plain that, -<i>ipso facto</i>, we must admit that a resemblance rather than an -identity between this phenomenon and that of chemical autocatalysis -is all that we can safely assert meanwhile. Indeed, as -Enriques has shewn, points of contrast between the two phenomena -are not lacking; for instance, as the chemical reaction draws to -a close, it is by the gradual attainment of chemical equilibrium: -but when organic growth draws to a close, it is by reason of a very -different kind of equilibrium, due in the main to the gradual -differentiation of the organism into parts, -among whose peculiar <span class="xxpn" id="p134">{134}</span> -and specialised functions that of cell-multiplication tends to fall -into abeyance<a class="afnanch" href="#fn173" id="fnanch173">173</a>.</p> - -<p>It would seem to follow, as a natural consequence, from what -has been said, that we could without much difficulty reduce our -curves of growth to logarithmic formulae<a class="afnanch" href="#fn174" id="fnanch174">174</a> -akin to those which -the physical chemist finds applicable to his autocatalytic reactions. -This has been diligently attempted by various writers<a class="afnanch" href="#fn175" id="fnanch175">175</a>; -but the -results, while not destructive of the hypothesis itself, are only -partially successful. The difficulty arises mainly from the fact -that, in the life-history of an organism, we have usually to deal -(as indeed we have seen) with several recurrent periods of relative -acceleration and retardation. It is easy to find a formula which -shall satisfy the conditions during any one of these periodic -phases, but it is very difficult to frame a comprehensive formula -which shall apply to the entire period of growth, or to the whole -duration of life.</p> - -<p>But if it be meanwhile impossible to formulate or to solve in -precise mathematical terms the equation to the growth of an -organism, we have yet gone a very long way towards the solution -of such problems when we have found a “qualitative expression,” -as Poincaré puts it; that is to say, when we have gained a fair -approximate knowledge of the general curve which represents the -unknown function.</p> - -<hr class="hrblk"> - -<p>As soon as we have touched on such matters as the chemical -phenomenon of catalysis, we are on the threshold of a subject -which, if we were able to pursue it, would soon lead us far into -the special domain of physiology; and there it would be necessary -to follow it if we were dealing with growth as a phenomenon in -itself, instead of merely as a help to our study and comprehension -of form. For instance the whole question of <i>diet</i>, of overfeeding -and underfeeding, would present itself for discussion<a class="afnanch" href="#fn176" id="fnanch176">176</a>. -But -without attempting to open up this large -subject, we may say a <span class="xxpn" id="p135">{135}</span> -further passing word upon the essential fact that certain chemical -substances have the power of accelerating or of retarding, or in -some way regulating, growth, and of so influencing directly the -morphological features of the organism.</p> - -<p>Thus lecithin has been shewn by Hatai<a class="afnanch" href="#fn177" id="fnanch177">177</a>, -Danilewsky<a class="afnanch" href="#fn178" id="fnanch178">178</a> -and -others to have a remarkable power of stimulating growth in -various animals; and the so-called “auximones,” which Professor -Bottomley prepares by the action of bacteria upon peat appear -to be, after a somewhat similar fashion, potent accelerators of -the growth of plants. But by much the most interesting cases, -from our point of view, are those where a particular substance -appears to exert a differential effect, stimulating the growth of -one part or organ of the body more than another.</p> - -<p>It has been known for a number of years that a diseased -condition of the pituitary body accompanies the phenomenon -known as “acromegaly,” in which the bones are variously enlarged -or elongated, and which is more or less exemplified in every -skeleton of a “giant”; while on the other hand, disease or extirpation -of the thyroid causes an arrest of skeletal development, and, -if it take place early, the subject remains a dwarf. These, then, -are well-known illustrations of the regulation of function by some -internal glandular secretion, some enzyme or “hormone” (as -Bayliss and Starling call it), or “harmozone,” as Gley calls it in -the particular case where the function regulated is that of growth, -with its consequent influence on form.</p> - -<p>Among other illustrations (which are plentiful) we have, for -instance the growth of the placental decidua, which Loeb has -shewn to be due to a substance given off by the corpus luteum -of the ovary, giving to the uterine tissues an abnormal capacity -for growth, which in turn is called into action by the contact of -the ovum, or even of any foreign body. And various sexual -characters, such as the plumage, comb and spurs of the cock, -are believed in like manner to arise in response to some particular -internal secretion. When the source of such a secretion is removed -by castration, well-known morphological changes take place in -various animals; and when a converse change takes place, the -female acquires, in greater or less degree, -characters which are <span class="xxpn" id="p136">{136}</span> -proper to the male, as in certain extreme cases, known from time -immemorial, when late in life a hen assumes the plumage of the -cock.</p> - -<p>There are some very remarkable experiments by Gudernatsch, -in which he has shewn that by feeding tadpoles (whether of frogs -or toads) on thyroid gland substance, their legs may be made to -grow out at any time, days or weeks before the normal date of -their appearance<a class="afnanch" href="#fn179" id="fnanch179">179</a>. -No other organic food was found to produce -the same effect; but since the thyroid gland is known to contain -iodine<a class="afnanch" href="#fn180" id="fnanch180">180</a>, -Morse experimented with this latter substance, and found -that if the tadpoles were fed with iodised amino-acids the legs -developed precociously, just as when the thyroid gland itself was -used. We may take it, then, as an established fact, whose full -extent and bearings are still awaiting investigation, that there -exist substances both within and without the organism which -have a marvellous power of accelerating growth, and of doing so -in such a way as to affect not only the size but the form or proportions -of the organism.</p> - -<hr class="hrblk"> - -<p>If we once admit, as we are now bound to do, the existence -of such factors as these, which, by their physiological activity -and apart from any direct action of the nervous system, tend -towards the acceleration of growth and consequent modification -of form, we are led into wide fields of speculation by an easy and -a legitimate pathway. Professor Gley carries such speculations -a long, long way: for he says<a class="afnanch" href="#fn181" id="fnanch181">181</a> -that by these chemical influences -“Toute une partie de la construction des êtres parait s’expliquer -d’une façon toute mécanique. La forteresse, si longtemps inaccessible, -du vitalisme est entamée. Car la notion morphogénique -était, suivant le mot de Dastre<a class="afnanch" href="#fn182" id="fnanch182">182</a>, -comme ‘le dernier réduit de la -force vitale.’ ”</p> - -<p>The physiological speculations we need not discuss: but, to -take a single example from morphology, we begin to understand -the possibility, and to comprehend the probable -meaning, of the <span class="xxpn" id="p137">{137}</span> -all but sudden appearance on the earth of such exaggerated and -almost monstrous forms as those of the great secondary reptiles -and the great tertiary mammals<a class="afnanch" href="#fn183" id="fnanch183">183</a>. -We begin to see that it is in -order to account, not for the appearance, but for the disappearance -of such forms as these that natural selection must be invoked. -And we then, I think, draw near to the conclusion that what is -true of these is universally true, and that the great function of -natural selection is not to originate, but to remove: <i>donec ad -interitum genus id natura redegit</i><a class="afnanch" href="#fn184" id="fnanch184">184</a>.</p> - -<p>The world of things living, like the world of things inanimate, -grows of itself, and pursues its ceaseless course of creative evolution. -It has room, wide but not unbounded, for variety of living form -and structure, as these tend towards their seemingly endless, but -yet strictly limited, possibilities of permutation and degree: it -has room for the great and for the small, room for the weak and -for the strong. Environment and circumstance do not always -make a prison, wherein perforce the organism must either live -or die; for the ways of life may be changed, and many a refuge -found, before the sentence of unfitness is pronounced and the -penalty of extermination paid. But there comes a time when -“variation,” in form, dimensions, or other qualities of the organism, -goes farther than is compatible with all the means at hand of -health and welfare for the individual and the stock; when, under -the active and creative stimulus of forces from within and from -without, the active and creative energies of growth pass the -bounds of physical and physiological equilibrium: and so reach -the limits which, as again Lucretius tells us, natural law has set -between what may and what may not be,</p> - -<div class="dpoem"><div class="nowrap"> -<div class="pv0"><span class="spqut">“</span>et quid quaeque queant per foedera naturai</div> -<div class="pv0"><span class="spqutspc">quid</span> porro nequeant.”</div> -</div></div><!--dpoem--> - -<p class="pcontinue">Then, at last, we are entitled to use the customary metaphor, -and to see in natural selection an inexorable force, -whose function <span class="xxpn" id="p138">{138}</span> -is not to create but to destroy,—to weed, to prune, to cut down -and to cast into the fire<a class="afnanch" href="#fn185" id="fnanch185">185</a>.</p> - -<div class="section"> -<h3><i>Regeneration, or growth and repair.</i></h3></div> - -<p>The phenomenon of regeneration, or the restoration of lost or -amputated parts, is a particular case of growth which deserves -separate consideration. As we are all aware, this property is -manifested in a high degree among invertebrates and many cold-blooded -vertebrates, diminishing as we ascend the scale, until at -length, in the warm-blooded animals, it lessens down to no more -than that <i>vis medicatrix</i> which heals a wound. Ever since the -days of Aristotle, and especially since the experiments of Trembley, -Réaumur and Spallanzani in the middle of the eighteenth century, -the physiologist and the psychologist have alike recognised that -the phenomenon is both perplexing and important. The general -phenomenon is amply discussed elsewhere, and we need only -deal with it in its immediate relation to growth<a class="afnanch" href="#fn186" id="fnanch186">186</a>.</p> - -<p>Regeneration, like growth in other cases, proceeds with a -velocity which varies according to a definite law; the rate varies -with the time, and we may study it as velocity and as acceleration.</p> - -<p>Let us take, as an instance, Miss M. L. Durbin’s measurements -of the rate of regeneration of tadpoles’ tails: the rate being here -measured in terms, not of mass, but of length, or longitudinal -increment<a class="afnanch" href="#fn187" id="fnanch187">187</a>.</p> - -<p>From a number of tadpoles, whose average length was 34·2 mm., -their tails being on an average 21·2 mm. long, -about half the tail <span class="xxpn" id="p139">{139}</span> -(11·5 mm.) was cut off, and the amounts regenerated in successive -periods are shewn as follows:</p> - -<div class="dtblbox"> -<table class="fsz7 twdth100"> -<tr> - <td class="tdleft">Days after operation</td> - <td class="tdcntr">3</td> - <td class="tdcntr">7</td> - <td class="tdcntr">10</td> - <td class="tdcntr">14</td> - <td class="tdcntr">18</td> - <td class="tdcntr">24</td> - <td class="tdcntr">30</td></tr> -<tr> - <td class="tdleft">(1) Amount regenerated in mm.</td> - <td class="tdright">1·4 </td> - <td class="tdright">3·4 </td> - <td class="tdright">4·3 </td> - <td class="tdright">5·2 </td> - <td class="tdright">5·5 </td> - <td class="tdright">6·2 </td> - <td class="tdright">6·5 </td></tr> -<tr> - <td class="tdleft">(2) Increment during each period</td> - <td class="tdright">1·4 </td> - <td class="tdright">2·0 </td> - <td class="tdright">0·9 </td> - <td class="tdright">0·9 </td> - <td class="tdright">0·3 </td> - <td class="tdright">0·7 </td> - <td class="tdright">0·3 </td></tr> -<tr> - <td class="tdleft">(3)(?) Rate per day during each period</td> - <td class="tdright">0·46</td> - <td class="tdright">0·50</td> - <td class="tdright">0·30</td> - <td class="tdright">0·25</td> - <td class="tdright">0·07</td> - <td class="tdright">0·12</td> - <td class="tdright">0·05</td></tr> -</table></div><!--dtblbox--> - -<p>The first line of numbers in this table, if plotted as a curve -against the number of days, will give us a very satisfactory view -of the “curve of growth” within the period of the observations: -that is to say, of the successive relations of length to time, or the -<i>velocity</i> of the process. But the third line is not so satisfactory, -and must not be plotted directly as an acceleration curve. For -it is evident that the “rates” here determined do not correspond -to velocities <i>at</i> the dates to which they are referred, but are the -mean velocities over a preceding period; and moreover the periods -over which these means are taken are here of very unequal length. -But we may draw a good deal more information from this experiment, -if we begin by drawing a smooth curve, as nearly as possible -through the points corresponding to the amounts regenerated -(according to the first line of the table); and if we then interpolate -from this smooth curve the actual lengths attained, day by -day, and derive from these, by subtraction, the successive daily -increments, which are the measure of the daily mean <i>velocities</i> -(Table, p. <a href="#p141" title="go to pg. 141">141</a>). (The more accurate and strictly correct method -would be to draw successive tangents to the curve.)</p> - -<p>In our curve of growth (Fig. <a href="#fig35" title="go to Fig. 35">35</a>) we cannot safely interpolate -values for the first three days, that is to say for the dates between -amputation and the first actual measurement of the regenerated -part. What goes on in these three days is very important; but -we know nothing about it, save that our curve descended to zero -somewhere or other within that period. As we have already -learned, we can more or less safely interpolate between known -points, or actual observations; but here we have no known -starting-point. In short, for all that the observations tell us, -and for all that the appearance of the curve can suggest, the -curve of growth may have descended evenly to the base-line, -which it would then have reached about the end -of the second <span class="xxpn" id="p140">{140}</span> -day; or it may have had within the first three days a change of -direction, or “point of inflection,” and may then have sprung -at once from the base-line at zero. That is to say, there may</p> - -<div class="dctr03" id="fig35"> -<img src="images/i140a.png" width="600" height="461" alt=""> - <div class="pcaption">Fig. 35. Curve of regenerative growth in -tadpoles’ tails. (From M. L. Durbin’s data.)</div></div> - -<p class="pcontinue">have been an intervening “latent period,” during -which no growth occurred, between the time of injury and the -first measurement of regenerative growth;</p> - -<div class="dctr03" id="fig36"> -<img src="images/i140b.png" width="600" height="465" alt=""> - <div class="dcaption">Fig. 36. Mean daily increments, -corresponding to Fig. <a href="#fig35" title="go to Fig. 35">35</a>.</div></div> - -<div><span class="xxpn" id="p141">{141}</span></div> - -<p class="pcontinue">or, for all we yet know, -regeneration may have begun at once, but with a velocity much less than -that which it afterwards attained. This apparently trifling difference -would correspond to a very great difference in the nature of the -phenomenon, and would lead to a very striking difference in the curve -which we have next to draw.</p> - -<p>The curve already drawn (Fig. <a href="#fig35" title="go to Fig. 35">35</a>) illustrates, as we have seen, the -relation of length to time, i.e. <i>L ⁄ T</i> -= <i>V</i>. The second (Fig. -<a href="#fig36" title="go to Fig. 36">36</a>) represents the rate of change of velocity; it sets <i>V</i> against -<i>T</i>;</p> - -<div class="dtblboxin10"> -<table class="fsz7"> -<caption><i>The foregoing table, extended by graphic -interpolation.</i></caption> -<tr> - <th>Days</th> - <th>Total<br>increment</th> - <th>Daily<br>increment</th> - <th>Logs<br>of do.</th></tr> -<tr> - <td class="tdrtsht">1</td> - <td class="tdctrsht">—</td> - <td class="tdctrsht">—</td> - <td class="tdctrsht">—</td></tr> -<tr> - <td class="tdrtsht">2</td> - <td class="tdctrsht">—</td> - <td class="tdctrsht">—</td> - <td class="tdctrsht">—</td></tr> -<tr> - <td class="tdrtsht">3</td> - <td class="tdrtsht">1·40</td> - <td class="tdrtsht">·60</td> - <td class="tdrtsht">1·78</td></tr> -<tr> - <td class="tdrtsht">4</td> - <td class="tdrtsht">2·00</td> - <td class="tdrtsht">·52</td> - <td class="tdrtsht">1·72</td></tr> -<tr> - <td class="tdrtsht">5</td> - <td class="tdrtsht">2·52</td> - <td class="tdrtsht">·45</td> - <td class="tdrtsht">1·65</td></tr> -<tr> - <td class="tdrtsht">6</td> - <td class="tdrtsht">2·97</td> - <td class="tdrtsht">·43</td> - <td class="tdrtsht">1·63</td></tr> -<tr> - <td class="tdrtsht">7</td> - <td class="tdrtsht">3·40</td> - <td class="tdrtsht">·32</td> - <td class="tdrtsht">1·51</td></tr> -<tr> - <td class="tdrtsht">8</td> - <td class="tdrtsht">3·72</td> - <td class="tdrtsht">·30</td> - <td class="tdrtsht">1·48</td></tr> -<tr> - <td class="tdrtsht">9</td> - <td class="tdrtsht">4·02</td> - <td class="tdrtsht">·28</td> - <td class="tdrtsht">1·45</td></tr> -<tr> - <td class="tdrtsht">10</td> - <td class="tdrtsht">4·30</td> - <td class="tdrtsht">·22</td> - <td class="tdrtsht">1·34</td></tr> -<tr> - <td class="tdrtsht">11</td> - <td class="tdrtsht">4·52</td> - <td class="tdrtsht">·21</td> - <td class="tdrtsht">1·32</td></tr> -<tr> - <td class="tdrtsht">12</td> - <td class="tdrtsht">4·73</td> - <td class="tdrtsht">·19</td> - <td class="tdrtsht">1·28</td></tr> -<tr> - <td class="tdrtsht">13</td> - <td class="tdrtsht">4·92</td> - <td class="tdrtsht">·18</td> - <td class="tdrtsht">1·26</td></tr> -<tr> - <td class="tdrtsht">14</td> - <td class="tdrtsht">5·10</td> - <td class="tdrtsht">·17</td> - <td class="tdrtsht">1·23</td></tr> -<tr> - <td class="tdrtsht">15</td> - <td class="tdrtsht">5·27</td> - <td class="tdrtsht">·13</td> - <td class="tdrtsht">1·11</td></tr> -<tr> - <td class="tdrtsht">16</td> - <td class="tdrtsht">5·40</td> - <td class="tdrtsht">·14</td> - <td class="tdrtsht">1·15</td></tr> -<tr> - <td class="tdrtsht">17</td> - <td class="tdrtsht">5·54</td> - <td class="tdrtsht">·13</td> - <td class="tdrtsht">1·11</td></tr> -<tr> - <td class="tdrtsht">18</td> - <td class="tdrtsht">5·67</td> - <td class="tdrtsht">·11</td> - <td class="tdrtsht">1·04</td></tr> -<tr> - <td class="tdrtsht">19</td> - <td class="tdrtsht">5·78</td> - <td class="tdrtsht">·10</td> - <td class="tdrtsht">1·00</td></tr> -<tr> - <td class="tdrtsht">20</td> - <td class="tdrtsht">5·88</td> - <td class="tdrtsht">·10</td> - <td class="tdrtsht">1·00</td></tr> -<tr> - <td class="tdrtsht">21</td> - <td class="tdrtsht">5·98</td> - <td class="tdrtsht">·09</td> - <td class="tdrtsht">·95</td></tr> -<tr> - <td class="tdrtsht">22</td> - <td class="tdrtsht">6·07</td> - <td class="tdrtsht">·07</td> - <td class="tdrtsht">·85</td></tr> -<tr> - <td class="tdrtsht">23</td> - <td class="tdrtsht">6·14</td> - <td class="tdrtsht">·07</td> - <td class="tdrtsht">·84</td></tr> -<tr> - <td class="tdrtsht">24</td> - <td class="tdrtsht">6·21</td> - <td class="tdrtsht">·08</td> - <td class="tdrtsht">·90</td></tr> -<tr> - <td class="tdrtsht">25</td> - <td class="tdrtsht">6·29</td> - <td class="tdrtsht">·06</td> - <td class="tdrtsht">·78</td></tr> -<tr> - <td class="tdrtsht">26</td> - <td class="tdrtsht">6·35</td> - <td class="tdrtsht">·06</td> - <td class="tdrtsht">·78</td></tr> -<tr> - <td class="tdrtsht">27</td> - <td class="tdrtsht">6·41</td> - <td class="tdrtsht">·05</td> - <td class="tdrtsht">·70</td></tr> -<tr> - <td class="tdrtsht">28</td> - <td class="tdrtsht">6·46</td> - <td class="tdrtsht">·04</td> - <td class="tdrtsht">·60</td></tr> -<tr> - <td class="tdrtsht">29</td> - <td class="tdrtsht">6·50</td> - <td class="tdrtsht">·03</td> - <td class="tdrtsht">·48</td></tr> -<tr> - <td class="tdrtsht">30</td> - <td class="tdrtsht">6·53</td> - <td class="tdctrsht">—</td> - <td class="tdctrsht">—</td></tr> -</table></div><!--dtblbox--> - -<div><span class="xxpn" id="p142">{142}</span></div> - -<p class="pcontinue">and <i>V ⁄ T</i> or <i>L ⁄ T</i><sup>2</sup> , represents -(as we have learned) the <i>acceleration</i> of growth, this being simply -the “differential coefficient,” the first derivative of the former -curve.</p> - -<div class="dctr02" id="fig37"> -<img src="images/i142.png" width="600" height="477" alt=""> - <div class="dcaption">Fig. 37. Logarithms - of values shewn in Fig. - <a href="#fig36" title="go to Fig. 36">36</a>.</div></div> - -<p>Now, plotting this acceleration curve from the date of the -first measurement made three days after the amputation of the -tail (Fig. <a href="#fig36" title="go to Fig. 36">36</a>), we see that it has no point of inflection, but falls -steadily, only more and more slowly, till at last it comes down -nearly to the base-line. The velocities of growth are continually -diminishing. As regards the missing portion at the beginning of -the curve, we cannot be sure whether it bent round and came down -to zero, or whether, as in our ordinary acceleration curves of growth -from birth onwards, it started from a maximum. The former is, -in this case, obviously the more probable, but we cannot be sure.</p> - -<p>As regards that large portion of the curve which we are -acquainted with, we see that it resembles the curve known as -a rectangular hyperbola, which is the form assumed when two -variables (in this case <i>V</i> and <i>T</i>) vary inversely as one another. -If we take the logarithms of the velocities (as given in the table) -and plot them against time (Fig. <a href="#fig37" title="go to Fig. 37">37</a>), we see that they fall, approximately, -into a straight line; and if this curve be -plotted on the <span class="xxpn" id="p143">{143}</span> -proper scale we shall find that the angle which it makes with the -base is about 25°, of which the tangent is ·46, or in round numbers ½.</p> - -<p>Had the angle been 45° (tan 45° -= 1), the -curve would have been actually a rectangular hyperbola, -with <i>V T</i> -= constant. As it is, we may -assume, provisionally, that it belongs to the same family of -curves, so that <span class="nowrap"> -<i>V</i><sup class="spitc">m</sup> <i>T</i><sup -class="spitc">n</sup> ,</span> or <span class="nowrap"> -<i>V</i><sup class="spitc">m ⁄ n</sup> <i>T</i>,</span> or -<span class="nowrap"> -<i>V T</i><sup class="spitc">n ⁄ m</sup> ,</span> -are all severally constant. In other -words, the velocity varies inversely as some power of the time, -or <i>vice versa</i>. And in this particular case, the equation -<span class="nowrap"><i>V T</i><sup>2</sup></span> -= constant, holds very nearly -true; that is to say the velocity varies, or tends to vary, -inversely as the square of the time. If some general law akin -to this could be established as a general law, or even as a -common rule, it would be of great importance.</p> - -<div class="dctr02" id="fig38"> -<img src="images/i143.png" width="600" height="484" alt=""> - <div class="dcaption">Fig. 38. Rate of regenerative - growth in larger tadpoles.</div></div> - -<p>But though neither in this case nor in any other can the -minute increments of growth during the first few hours, or the -first couple of days, after injury, be directly measured, yet -the most important point is quite capable of solution. What the -foregoing curve leaves us in ignorance of, is simply whether -growth starts at zero, with zero velocity, and works up quickly -to a maximum velocity from which it afterwards gradually -falls away; or whether after a latent period, it begins, -so to speak, in full force. The answer <span class="xxpn" -id="p144">{144}</span> to this question-depends on whether, -in the days following the first actual measurement, we can or -cannot detect a daily <i>increment</i> in velocity, before that -velocity begins its normal course of diminution. Now this -preliminary ascent to a maximum, or point of inflection of -the curve, though not shewn in the above-quoted experiment, -has been often observed: as for instance, in another similar -experiment by the author of the former, the tadpoles being in -this case of larger size (average 49·1 mm.)<a class="afnanch" -href="#fn188" id="fnanch188">188</a>.</p> - -<div class="dtblbox"> -<table class="fsz7 twdth100"> -<tr> - <td class="tdleft">Days</td> - <td class="tdcntr">3</td> - <td class="tdcntr">5</td> - <td class="tdcntr">7</td> - <td class="tdcntr">10</td> - <td class="tdcntr">12</td> - <td class="tdcntr">14</td> - <td class="tdcntr">17</td> - <td class="tdcntr">24</td> - <td class="tdcntr">28</td> - <td class="tdcntr">31</td></tr> -<tr> - <td class="tdleft">Increment</td> - <td class="tdright">0·86</td> - <td class="tdright">2·15</td> - <td class="tdright">3·66</td> - <td class="tdright">5·20</td> - <td class="tdright">5·95</td> - <td class="tdright">6·38</td> - <td class="tdright">7·10</td> - <td class="tdright">7·60</td> - <td class="tdright">8·20</td> - <td class="tdright">8·40</td></tr> -</table></div><!--dtblbox--> - -<p class="pcontinue">Or, by graphic interpolation,</p> - -<div class="dtblboxin10"> -<table class="fsz6"> -<tr> - <th>Days</th> - <th>Total<br>increment</th> - <th>Daily<br>do.</th></tr> -<tr> - <td class="tdright">1</td> - <td class="tdright">·23</td> - <td class="tdright">·23</td></tr> -<tr> - <td class="tdright">2</td> - <td class="tdright">·53</td> - <td class="tdright">·30</td></tr> -<tr> - <td class="tdright">3</td> - <td class="tdright">·86</td> - <td class="tdright">·33</td></tr> -<tr> - <td class="tdright">4</td> - <td class="tdright">1·30</td> - <td class="tdright">·44</td></tr> -<tr> - <td class="tdright">5</td> - <td class="tdright">2·00</td> - <td class="tdright">·70</td></tr> -<tr> - <td class="tdright">6</td> - <td class="tdright">2·78</td> - <td class="tdright">·78</td></tr> -<tr> - <td class="tdright">7</td> - <td class="tdright">3·58</td> - <td class="tdright">·80</td></tr> -<tr> - <td class="tdright">8</td> - <td class="tdright">4·30</td> - <td class="tdright">·72</td></tr> -<tr> - <td class="tdright">9</td> - <td class="tdright">4·90</td> - <td class="tdright">·60</td></tr> -<tr> - <td class="tdright">10</td> - <td class="tdright">5·29</td> - <td class="tdright">·39</td></tr> -<tr> - <td class="tdright">11</td> - <td class="tdright">5·62</td> - <td class="tdright">·33</td></tr> -<tr> - <td class="tdright">12</td> - <td class="tdright">5·90</td> - <td class="tdright">·28</td></tr> -<tr> - <td class="tdright">13</td> - <td class="tdright">6·13</td> - <td class="tdright">·23</td></tr> -<tr> - <td class="tdright">14</td> - <td class="tdright">6·38</td> - <td class="tdright">·25</td></tr> -<tr> - <td class="tdright">15</td> - <td class="tdright">6·61</td> - <td class="tdright">·23</td></tr> -<tr> - <td class="tdright">16</td> - <td class="tdright">6·81</td> - <td class="tdright">·20</td></tr> -<tr> - <td class="tdright">17</td> - <td class="tdright">7·00</td> - <td class="tdright">·19<br>etc.</td></tr> -</table></div><!--dtblbox--> - -<p>The acceleration curve is drawn in Fig. <a href="#fig39" title="go to Fig. 39">39</a>.</p> - -<p>Here we have just what we lacked in the former case, namely -a visible point of inflection in the curve about the seventh day -(Figs. <a href="#fig38" title="go to Fig. 38">38</a>, 39), whose existence is confirmed by successive observations -on the 3rd, 5th, 7th and 10th days, and which justifies to -some extent our extrapolation for the otherwise unknown period -up to and ending with the third day; but even here there is a -short space near the very beginning during which we are not -quite sure of the precise slope of the curve.</p> - -<hr class="hrblk"> - -<p>We have now learned that, according to these experiments, -with which many others are in substantial agreement, the rate of -growth in the regenerative process is as follows. After a very -short latent period, not yet actually proved but whose existence -is highly probable, growth commences with a -velocity which very <span class="xxpn" id="p145">{145}</span> -rapidly increases to a maximum. The curve quickly,—almost -suddenly,—changes its direction, as the velocity begins to fall; -and the rate of fall, that is, the negative acceleration, proceeds -at a slower and slower rate, which rate varies inversely as some -power of the time, and is found in both of the above-quoted -experiments to be very approximately as 1 ⁄ <i>T</i><sup>2</sup> . But it is obvious -that the value which we have found for the latter portion of the -curve (however closely it be conformed to) is only an empirical -value; it has only a temporary usefulness, and must in time give -place to a formula which shall represent the entire phenomenon, -from start to finish.</p> - -<div class="dctr03" id="fig39"> -<img src="images/i145.png" width="600" height="538" alt=""> - <div class="dcaption">Fig. 39. Daily increment, or amount - regenerated, corresponding to Fig. <a href="#fig38" title="go to Fig. 38">38</a>.</div></div> - -<p>While the curve of regenerative growth is apparently different -from the curve of ordinary growth as usually drawn (and while -this apparent difference has been commented on and treated as -valid by certain writers) we are now in a position to see that it -only looks different because we are able to study it, if not from -the beginning, at least very nearly so: while an ordinary curve -of growth, as it is usually presented to us, is one -which dates, not <span class="xxpn" id="p146">{146}</span> -from the beginning of growth, but from the comparatively late, -and unimportant, and even fallacious epoch of birth. A complete -curve of growth, starting from zero, has the same essential characteristics -as the regeneration curve.</p> - -<p>Indeed the more we consider the phenomenon of regeneration, -the more plainly does it shew itself to us as but a particular case -of the general phenomenon of growth<a class="afnanch" href="#fn189" id="fnanch189">189</a>, -following the same lines, -obeying the same laws, and merely started into activity by the -special stimulus, direct or indirect, caused by the infliction of a -wound. Neither more nor less than in other problems of physiology -are we called upon, in the case of regeneration, to indulge in -metaphysical speculation, or to dwell upon the beneficent purpose -which seemingly underlies this process of healing and restoration.</p> - -<hr class="hrblk"> - -<p>It is a very general rule, though apparently not a universal -one, that regeneration tends to fall somewhat short of a <i>complete</i> -restoration of the lost part; a certain percentage only of the lost -tissues is restored. This fact was well known to some of those -old investigators, who, like the Abbé Trembley and like Voltaire, -found a fascination in the study of artificial injury and the regeneration -which followed it. Sir John Graham Dalyell, for instance, -says, in the course of an admirable paragraph on regeneration<a class="afnanch" href="#fn190" id="fnanch190">190</a>: -“The reproductive faculty ... is not confined to one portion, but -may extend over many; and it may ensue even in relation to the -regenerated portion more than once. Nevertheless, the faculty -gradually weakens, so that in general every successive regeneration -is smaller and more imperfect than the organisation preceding it; -and at length it is exhausted.”</p> - -<p>In certain minute animals, such as the Infusoria, in which the -capacity for “regeneration” is so great that the entire animal -may be restored from the merest fragment, it becomes of great -interest to discover whether there be some definite size at which -the fragment ceases to display this power. -This question has <span class="xxpn" id="p147">{147}</span> -been studied by Lillie<a class="afnanch" href="#fn191" id="fnanch191">191</a>, -who found that in Stentor, while still -smaller fragments were capable of surviving for days, the smallest -portions capable of regeneration were of a size equal to a sphere of -about 80 µ in diameter, that is to say of a volume equal to about -one twenty-seventh of the average entire animal. He arrives at -the remarkable conclusion that for this, and for all other species -of animals, there is a “minimal organisation mass,” that is to say -a “minimal mass of definite size consisting of nucleus and cytoplasm -within which the organisation of the species can just find -its latent expression.” And in like manner, Boveri<a class="afnanch" href="#fn192" id="fnanch192">192</a> -has shewn -that the fragment of a sea-urchin’s egg capable of growing up into -a new embryo, and so discharging the complete functions of an -entire and uninjured ovum, reaches its limit at about one-twentieth -of the original egg,—other writers having found a limit at about -one-fourth. These magnitudes, small as they are, represent -objects easily visible under a low power of the microscope, and so -stand in a very different category to the minimal magnitudes in -which life itself can be manifested, and which we have discussed -in chapter II.</p> - -<p>A number of phenomena connected with the linear rate of -regeneration are illustrated and epitomised in the accompanying -diagram (Fig. <a href="#fig40" title="go to Fig. 40">40</a>), which I have constructed from certain data -given by Ellis in a paper on the relation of the amount of tail -<i>regenerated</i> to the amount <i>removed</i>, in Tadpoles. These data are -summarised in the next Table. The tadpoles were all very much -of a size, about 40 mm.; the average length of tail was very near -to 26 mm., or 65 per cent. of the whole body-length; and in four -series of experiments about 10, 20, 40 and 60 per cent. of the tail -were severally removed. The amount regenerated in successive -intervals of three days is shewn in our table. By plotting the -actual amounts regenerated against these three-day intervals of -time, we may interpolate values for the time taken to regenerate -definite percentage amounts, 5 per cent., 10 per -cent., etc. of the <span class="xxpn" id="p148">{148}</span></p> - -<div class="dtblbox"> -<table class="fsz8 borall"> -<caption class="fsz4"><i>The Rate of Regenerative Growth -in Tadpoles’ Tails.</i> (<i>After M. M. Ellis, J. Exp. Zool.</i> <span -class="smmaj">VII,</span> <i>p.</i> 421, 1909.)</caption> - -<tr> - <th class="thsnug" rowspan="2">Series†</th> - <th class="thsnug" rowspan="2">Body length mm.</th> - <th class="thsnug" rowspan="2">Tail length mm.</th> - <th class="thsnug" rowspan="2">Amount removed mm.</th> - <th class="thsnug" rowspan="2">Per cent. of tail removed</th> - <th class="thsnug" colspan="7">% amount regenerated in days</th></tr> -<tr> - <th class="thsnug">3</th> - <th class="thsnug">6</th> - <th class="thsnug">9</th> - <th class="thsnug">12</th> - <th class="thsnug">15</th> - <th class="thsnug">18</th> - <th class="thsnug">32</th></tr> -<tr> - <td class="tdcntr"><i>O</i></td> - <td class="tdsnug">39·575</td> - <td class="tdsnug">25·895</td> - <td class="tdsnug">3·2 </td> - <td class="tdsnug">12·36</td> - <td class="tdsnug">13</td> - <td class="tdsnug">31</td> - <td class="tdsnug">44</td> - <td class="tdsnug">44</td> - <td class="tdsnug">44</td> - <td class="tdsnug">44</td> - <td class="tdsnug">44</td></tr> -<tr> - <td class="tdcntr"><i>P</i></td> - <td class="tdsnug">40·21 </td> - <td class="tdsnug">26·13 </td> - <td class="tdsnug">5·28</td> - <td class="tdsnug">20·20</td> - <td class="tdsnug">10</td> - <td class="tdsnug">29</td> - <td class="tdsnug">40</td> - <td class="tdsnug">44</td> - <td class="tdsnug">44</td> - <td class="tdsnug">44</td> - <td class="tdsnug">44</td></tr> -<tr> - <td class="tdcntr"><i>R</i></td> - <td class="tdsnug">39·86 </td> - <td class="tdsnug">25·70 </td> - <td class="tdsnug">10·4 </td> - <td class="tdsnug">40·50</td> - <td class="tdsnug">6</td> - <td class="tdsnug">20</td> - <td class="tdsnug">31</td> - <td class="tdsnug">40</td> - <td class="tdsnug">48</td> - <td class="tdsnug">48</td> - <td class="tdsnug">48</td></tr> -<tr> - <td class="tdcntr"><i>S</i></td> - <td class="tdsnug">40·34 </td> - <td class="tdsnug">26·11 </td> - <td class="tdsnug">14·8 </td> - <td class="tdsnug">56·7 </td> - <td class="tdsnug">0</td> - <td class="tdsnug">16</td> - <td class="tdsnug">33</td> - <td class="tdsnug">39</td> - <td class="tdsnug">45</td> - <td class="tdsnug">48</td> - <td class="tdsnug">48</td></tr> -</table> -<p class="ptblfn">† Each series gives the mean of 20 experiments.</p> -</div><!--dtblbox--> - -<div class="dctr01" id="fig40"> -<img src="images/i148.png" width="800" height="629" alt=""> - <div class="pcaption">Fig. 40. Relation between the percentage amount of tail - removed, the percentage restored, and the time required for - its restoration. (From M. M. Ellis’s data.)</div></div> - -<p class="pcontinue">amount removed; and my diagram is constructed from the four -sets of values thus obtained, that is to say from the four sets of -experiments which differed from one another in the amount of -tail amputated. To these we have to add the general result of a -fifth series of experiments, which shewed that when as much as -75 per cent. of the tail was cut off, no regeneration took place at -all, but the animal presently died. In our -diagram, then, each <span class="xxpn" id="p149">{149}</span> -curve indicates the time taken to regenerate <i>n</i> per cent. of the -amount removed. All the curves converge towards infinity, when -the amount removed (as shewn by the ordinate) approaches 75 -per cent.; and all of the curves start from zero, for nothing is -regenerated where nothing had been removed. Each curve approximates -in form to a cubic parabola.</p> - -<p>The amount regenerated varies also with the age of the tadpole -and with other factors, such as temperature; in other words, for -any given age, or size, of tadpole and also for various specific -temperatures, a similar diagram might be constructed.</p> - -<hr class="hrblk"> - -<p>The power of reproducing, or regenerating, a lost limb is -particularly well developed in arthropod animals, and is sometimes -accompanied by remarkable modification of the form of -the regenerated limb. A case in point, which has attracted -much attention, occurs in connection with the claws of certain -Crustacea<a class="afnanch" href="#fn193" id="fnanch193">193</a>.</p> - -<p>In many Crustacea we have an asymmetry of the great claws, -one being larger than the other and also more or less different in -form. For instance, in the common lobster, one claw, the larger -of the two, is provided with a few great “crushing” teeth, while -the smaller claw has more numerous teeth, small and serrated. -Though Aristotle thought otherwise, it appears that the crushing-claw -may be on the right or left side, indifferently; whether it -be on one or the other is a problem of “chance.” It is otherwise -in many other Crustacea, where the larger and more powerful -claw is always left or right, as the case may be, according to the -species: where, in other words, the “probability” of the large -or the small claw being left or being right is tantamount to -certainty<a class="afnanch" href="#fn194" id="fnanch194">194</a>.</p> - -<p>The one claw is the larger because it has -grown the faster; <span class="xxpn" id="p150">{150}</span> -it has a higher “coefficient of growth,” and accordingly, as age -advances, the disproportion between the two claws becomes more -and more evident. Moreover, we must assume that the characteristic -form of the claw is a “function” of its magnitude; the -knobbiness is a phenomenon coincident with growth, and we -never, under any circumstances, find the smaller claw with big -crushing teeth and the big claw with little serrated ones. There -are many other somewhat similar cases where size and form are -manifestly correlated, and we have already seen, to some extent, -that the phenomenon of growth is accompanied by certain ratios -of velocity that lead inevitably to changes of form. Meanwhile, -then, we must simply assume that the essential difference between -the two claws is one of magnitude, with which a certain differentiation -of form is inseparably associated.</p> - -<p>If we amputate a claw, or if, as often happens, the crab “casts -it off,” it undergoes a process of regeneration,—it grows anew, -and evidently does so with an accelerated velocity, which acceleration -will cease when equilibrium of the parts is once more attained: -the accelerated velocity being a case in point to illustrate that -<i>vis revulsionis</i> of Haller, to which we have already referred.</p> - -<p>With the help of this principle, Przibram accounts for certain -curious phenomena which accompany the process of regeneration. -As his experiments and those of Morgan shew, if the large or -knobby claw (<i>A</i>) be removed, there are certain cases, e.g. the -common lobster, where it is directly regenerated. In other cases, -e.g. Alpheus<a class="afnanch" href="#fn195" id="fnanch195">195</a>, -the other claw (<i>B</i>) assumes the size and form of that -which was amputated, while the latter regenerates itself in the -form of the other and weaker one; <i>A</i> and <i>B</i> have apparently -changed places. In a third case, as in the crabs, the <i>A</i>-claw regenerates -itself as a small or <i>B</i>-claw, but the <i>B</i>-claw remains for a -time unaltered, though slowly and in the course of repeated moults -it later on assumes the large and heavily toothed <i>A</i>-form.</p> - -<p>Much has been written on this phenomenon, but in essence it -is very simple. It depends upon the respective rates of growth, -upon a ratio between the rate of regeneration and the rate of -growth of the uninjured limb: complicated a -little, however, by <span class="xxpn" id="p151">{151}</span> -the possibility of the uninjured limb growing all the faster for -a time after the animal has been relieved of the other. From the -time of amputation, say of <i>A</i>, <i>A</i> begins to grow from zero, with -a high “regenerative” velocity; while <i>B</i>, starting from a definite -magnitude, continues to increase, with its normal or perhaps -somewhat accelerated velocity. The ratio between the two -velocities of growth will determine whether, by a given time, -<i>A</i> has equalled, outstripped, or still fallen short of the magnitude -of <i>B</i>.</p> - -<p>That this is the gist of the whole problem is confirmed (if -confirmation be necessary) by certain experiments of Wilson’s. -It is known that by section of the nerve to a crab’s claw, its -growth is retarded, and as the general growth of the animal -proceeds the claw comes to appear stunted or dwarfed. Now in -such a case as that of Alpheus, we have seen that the rate of -regenerative growth in an amputated large claw fails to let it -reach or overtake the magnitude of the growing little claw: -which latter, in short, now appears as the big one. But if at the -same time as we amputate the big claw we also sever the nerve -to the lesser one, we so far slow down the latter’s growth that -the other is able to make up to it, and in this case the two claws -continue to grow at approximately equal rates, or in other words -continue of coequal size.</p> - -<hr class="hrblk"> - -<p>The phenomenon of regeneration goes some way towards -helping us to comprehend the phenomenon of “multiplication by -fission,” as it is exemplified at least in its simpler cases in many -worms and worm-like animals. For physical reasons which we -shall have to study in another chapter, there is a natural tendency -for any tube, if it have the properties of a fluid or semi-fluid -substance, to break up into segments after it comes to a certain -length; and nothing can prevent its doing so, except the presence -of some controlling force, such for instance as may be due to the -pressure of some external support, or some superficial thickening -or other intrinsic rigidity of its own substance. If we add to this -natural tendency towards fission of a cylindrical or tubular worm, -the ordinary phenomenon of regeneration, we have all that is -essentially implied in “reproduction by fission.” And -in so far <span class="xxpn" id="p152">{152}</span> -as the process rests upon a physical principle, or natural tendency, -we may account for its occurrence in a great variety of animals, -zoologically dissimilar; and also for its presence here and absence -there, in forms which, though materially different in a physical -sense, are zoologically speaking very closely allied.</p> - -<div class="section"> -<h3><span class="smcap">C<b>ONCLUSION</b></span> -<span class="smmaj">AND</span> -<span class="smcap">S<b>UMMARY.</b></span></h3></div> - -<p>But the phenomena of regeneration, like all the other -phenomena of growth, soon carry us far afield, and we must draw -this brief discussion to a close.</p> - -<p>For the main features which appear to be common to all -curves of growth we may hope to have, some day, a physical -explanation. In particular we should like to know the meaning -of that point of inflection, or abrupt change from an increasing -to a decreasing velocity of growth which all our curves, and -especially our acceleration curves, demonstrate the existence of, -provided only that they include the initial stages of the whole -phenomenon: just as we should also like to have a full physical -or physiological explanation of the gradually diminishing velocity -of growth which follows, and which (though subject to temporary -interruption or abeyance) is on the whole characteristic of growth in -all cases whatsoever. In short, the characteristic form of the curve -of growth in length (or any other linear dimension) is a phenomenon -which we are at present unable to explain, but which presents -us with a definite and attractive problem for future solution. -It would seem evident that the abrupt change in velocity must be -due, either to a change in that pressure outwards from within, -by which the “forces of growth” make themselves manifest, or -to a change in the resistances against which they act, that is to -say the <i>tension</i> of the surface; and this latter force we do not by -any means limit to “surface-tension” proper, but may extend to -the development of a more or less resistant membrane or “skin,” -or even to the resistance of fibres or other histological elements, -binding the boundary layers to the parts within. I take it that -the sudden arrest of velocity is much more likely to be due to a -sudden increase of resistance than to a sudden diminution of -internal energies: in other words, I suspect that it is coincident -with some notable event of histological -differentiation, such as <span class="xxpn" id="p153">{153}</span> -the rapid formation of a comparatively firm skin; and that the -dwindling of velocities, or the negative acceleration, which follows, -is the resultant or composite effect of waning forces of growth on -the one hand, and increasing superficial resistance on the other. -This is as much as to say that growth, while its own energy tends -to increase, leads also, after a while, to the establishment of -resistances which check its own further increase.</p> - -<p>Our knowledge of the whole complex phenomenon of growth -is so scanty that it may seem rash to advance even this tentative -suggestion. But yet there are one or two known facts which -seem to bear upon the question, and to indicate at least the manner -in which a varying resistance to expansion may affect the velocity -of growth. For instance, it has been shewn by Frazee<a class="afnanch" href="#fn196" id="fnanch196">196</a> -that -electrical stimulation of tadpoles, with small current density and -low voltage, increases the rate of regenerative growth. As just -such an electrification would tend to lower the surface-tension, -and accordingly decrease the external resistance, the experiment -would seem to support, in some slight degree, the suggestion -which I have made.</p> - -<div class="psmprnt3"> -<p>Delage<a class="afnanch" href="#fn197" -id="fnanch197">197</a> has lately made use of the principle -of specific rate of growth, in considering the question of -heredity itself. We know that the chromatin of the fertilised -egg comes from the male and female parent alike, in equal or -nearly equal shares; we know that the initial chromatin, so -contributed, multiplies many thousand-fold, to supply the -chromatin for every cell of the offspring’s body; and it has, -therefore, a high “coefficient of growth.” If we admit, with -Van Beneden and others, that the initial contributions of -male and female chromatin continue to be transmitted to the -succeeding generations of cells, we may then conceive these -chromatins to retain each its own coefficient of growth; and if -these differed ever so little, a gradual preponderance of one -or other would make itself felt in time, and might conceivably -explain the preponderating influence of one parent or the other -upon the characters of the offspring. Indeed O. Hertwig is said -(according to Delage’s interpretation) to have actually shewn -that we can artificially modify the rate of growth of one or -other chromatin, and so increase or diminish the influence of -the maternal or paternal heredity. This theory of Delage’s has -its fascination, but it calls for somewhat large assumptions; -and in particular, it seems (like so many other theories -relating to the chromosomes) to rest far too much upon material -elements, rather than on the imponderable dynamic factors of -the cell. <span class="xxpn" id="p154">{154}</span></p> -</div><!--psmprnt3--> - -<p>We may summarise, as follows, the main results of the foregoing -discussion:</p> - -<ul> -<li><p>(1) Except in certain minute organisms and minute parts of -organisms, whose form is due to the direct action of molecular -forces, we may look upon the form of the organism as a “function -of growth,” or a direct expression of a rate of growth which varies -according to its different directions.</p></li> - -<li><p>(2) Rate of growth is subject to definite laws, and the -velocities in different directions tend to maintain a <i>ratio</i> which is -more or less constant for each specific organism; and to this -regularity is due the fact that the form of the organism is in general -regular and constant.</p></li> - -<li><p>(3) Nevertheless, the ratio of velocities in different directions -is not absolutely constant, but tends to alter or fluctuate in a -regular way; and to these progressive changes are due the -changes of form which accompany “development,” and the slower -changes of form which continue perceptibly in after life.</p></li> - -<li><p>(4) The rate of growth is a function of the age of the organism, -it has a maximum somewhat early in life, after which epoch of -maximum it slowly declines.</p></li> - -<li><p>(5) The rate of growth is directly affected by temperature, -and by other physical conditions.</p></li> - -<li><p>(6) It is markedly affected, in the way of acceleration or -retardation, at certain physiological epochs of life, such as birth, -puberty, or metamorphosis.</p></li> - -<li><p>(7) Under certain circumstances, growth may be <i>negative</i>, the -organism growing smaller: and such negative growth is a common -accompaniment of metamorphosis, and a frequent accompaniment -of old age.</p></li> - -<li><p>(8) The phenomenon of regeneration is associated with a large -temporary increase in the rate of growth (or “<i>acceleration</i>” of -growth) of the injured surface; in other respects, regenerative -growth is similar to ordinary growth in all its essential phenomena.</p></li> -</ul> - -<hr class="hrblk"> - -<p>In this discussion of growth, we have left out of account a -vast number of processes, or phenomena, by which, in the physiological -mechanism of the body, growth is effected and controlled. -We have dealt with growth in its relation to -magnitude, and to <span class="xxpn" id="p155">{155}</span> -that relativity of magnitudes which constitutes form; and so we -have studied it as a phenomenon which stands at the beginning -of a morphological, rather than at the end of a physiological -enquiry. Under these restrictions, we have treated it as far as -possible, or in such fashion as our present knowledge permits, on -strictly physical lines.</p> - -<p>In all its aspects, and not least in its relation to form, the -growth of organisms has many analogies, some close and some -perhaps more remote, among inanimate things. As the waves -grow when the winds strive with the other forces which govern -the movements of the surface of the sea, as the heap grows when -we pour corn out of a sack, as the crystal grows when from the -surrounding solution the proper molecules fall into their appropriate -places: so in all these cases, very much as in the organism -itself, is growth accompanied by change of form, and by a development -of definite shapes and contours. And in these cases (as -in all other mechanical phenomena), we are led to equate our -various magnitudes with time, and so to recognise that growth is -essentially a question of rate, or of velocity.</p> - -<p>The differences of form, and changes of form, which are brought -about by varying rates (or “laws”) of growth, are essentially the -same phenomenon whether they be, so to speak, episodes in the -life-history of the individual, or manifest themselves as the normal -and distinctive characteristics of what we call separate species of -the race. From one form, or ratio of magnitude, to another there -is but one straight and direct road of transformation, be the -journey taken fast or slow; and if the transformation take place -at all, it will in all likelihood proceed in the self-same way, whether -it occur within the life-time of an individual or during the long -ancestral history of a race. No small part of what is known as -Wolff’s or von Baer’s law, that the individual organism tends to -pass through the phases characteristic of its ancestors, or that the -life-history of the individual tends to recapitulate the ancestral -history of its race, lies wrapped up in this simple account of the -relation between rate of growth and form.</p> - -<p>But enough of this discussion. Let us leave for a while the -subject of the growth of the organism, and attempt to study the -conformation, within and without, of -the individual cell.</p> - -<div class="chapter" id="p156"><h2 class="h2herein" -title="IV. On the Internal Form and Structure of the -Cell.">CHAPTER IV<span class="h2ttl">ON THE INTERNAL FORM AND -STRUCTURE OF THE CELL</span></h2></div> - -<p>In the early days of the cell-theory, more than seventy years -ago, Goodsir was wont to speak of cells as “centres of growth” -or “centres of nutrition,” and to consider them as essentially -“centres of force.” He looked forward to a time when the forces -connected with the cell should be particularly investigated: when, -that is to say, minute anatomy should be studied in its dynamical -aspect. “When this branch of enquiry,” he says “shall have -been opened up, we shall expect to have a science of organic -forces, having direct relation to anatomy, the science of organic -forms<a class="afnanch" href="#fn198" id="fnanch198">198</a>.” -And likewise, long afterwards, Giard contemplated a -science of <i>morphodynamique</i>,—but still looked upon it as forming -so guarded and hidden a “territoire scientifique, que la plupart -des naturalistes de nos jours ne le verront que comme Moïse vit -la terre promise, seulement de loin et sans pouvoir y entrer<a class="afnanch" href="#fn199" id="fnanch199">199</a>.”</p> - -<p>To the external forms of cells, and to the forces which produce -and modify these forms, we shall pay attention in a later chapter. -But there are forms and configurations of matter within the cell, -which also deserve to be studied with due regard to the forces, -known or unknown, of whose resultant they are the visible -expression.</p> - -<p>In the long interval since Goodsir’s day, the visible structure, -the conformation and configuration, of the cell, has been studied -far more abundantly than the purely dynamic problems that are -associated therewith. The overwhelming progress of microscopic -observation has multiplied our knowledge of cellular and intracellular -structure; and to the multitude of -visible structures it <span class="xxpn" id="p157">{157}</span> -has been often easier to attribute virtues than to ascribe intelligible -functions or modes of action. But here and there nevertheless, -throughout the whole literature of the subject, we find recognition -of the inevitable fact that dynamical problems lie behind the -morphological problems of the cell.</p> - -<p>Bütschli pointed out forty years ago, with emphatic clearness, -the failure of morphological methods, and the need for physical -methods, if we were to penetrate deeper into the essential nature -of the cell<a class="afnanch" href="#fn200" id="fnanch200">200</a>. -And such men as Loeb and Whitman, Driesch and -Roux, and not a few besides, have pursued the same train of -thought and similar methods of enquiry.</p> - -<p>Whitman<a class="afnanch" href="#fn201" id="fnanch201">201</a>, -for instance, puts the case in a nutshell when, in -speaking of the so-called “caryokinetic” phenomena of nuclear -division, he reminds us that the leading idea in the term “<i>caryokinesis</i>” -is <i>motion</i>,—“motion viewed as an exponent of forces -residing in, or acting upon, the nucleus. It regards the nucleus -as a <i>seat of energy, which displays itself in phenomena of motion</i><a class="afnanch" href="#fn202" id="fnanch202">202</a>.”</p> - -<p>In short it would seem evident that, except in relation to a -dynamical investigation, the mere study of cell structure has but -little value of its own. That a given cell, an ovum for instance, -contains this or that visible substance or structure, germinal -vesicle or germinal spot, chromatin or achromatin, chromosomes -or centrosomes, obviously gives no explanation of the <i>activities</i> of -the cell. And in all such hypotheses as that of “pangenesis,” in -all the theories which attribute specific -properties to micellae, <span class="xxpn" id="p158">{158}</span> -idioplasts, ids, or other constituent particles of protoplasm or of -the cell, we are apt to fall into the error of attributing to <i>matter</i> -what is due to <i>energy</i> and is manifested in force: or, more strictly -speaking, of attributing to material particles individually what is -due to the energy of their collocation.</p> - -<p>The tendency is a very natural one, as knowledge of structure -increases, to ascribe particular virtues to the material structures -themselves, and the error is one into which the disciple is likely -to fall, but of which we need not suspect the master-mind. The -dynamical aspect of the case was in all probability kept well in -view by those who, like Goodsir himself, first attacked the problem -of the cell and originated our conceptions of its nature and -functions.</p> - -<p>But if we speak, as Weismann and others speak, of an -“hereditary <i>substance</i>,” a substance which is split off from the -parent-body, and which hands on to the new generation the -characteristics of the old, we can only justify our mode of speech -by the assumption that that particular portion of matter is the -essential vehicle of a particular charge or distribution of energy, -in which is involved the capability of producing motion, or of -doing “work.”</p> - -<p>For, as Newton said, to tell us that a thing “is endowed with -an occult specific quality, by which it acts and produces manifest -effects, is to tell us nothing; but to derive two or three general -principles of motion<a class="afnanch" href="#fn203" id="fnanch203">203</a> -from phenomena would be a very great step -in philosophy, though the causes of these principles were not yet -discovered.” The <i>things</i> which we see in the cell are less important -than the <i>actions</i> which we recognise in the cell; and these latter -we must especially scrutinize, in the hope of discovering how far -they may be attributed to the simple and well-known physical -forces, and how far they be relevant or irrelevant to the phenomena -which we associate with, and deem essential to, the manifestation -of <i>life</i>. It may be that in this way we shall in time draw nigh to -the recognition of a specific and ultimate residuum. <span class="xxpn" id="p159">{159}</span></p> - -<p>And lacking, as we still do lack, direct knowledge of the actual -forces inherent in the cell, we may yet learn something of their -distribution, if not also of their nature, from the outward and -inward configuration of the cell, and from the changes taking -place in this configuration; that is to say from the movements -of matter, the kinetic phenomena, which the forces in action set up.</p> - -<p>The fact that the germ-cell develops into a very complex -structure, is no absolute proof that the cell itself is structurally -a very complicated mechanism: nor yet, though this is somewhat -less obvious, is it sufficient to prove that the forces at work, or -latent, within it are especially numerous and complex. If we blow -into a bowl of soapsuds and raise a great mass of many-hued and -variously shaped bubbles, if we explode a rocket and watch the -regular and beautiful configuration of its falling streamers, if we -consider the wonders of a limestone cavern which a filtering stream -has filled with stalactites, we soon perceive that in all these cases -we have begun with an initial system of very slight complexity, -whose structure in no way foreshadowed the result, and whose -comparatively simple intrinsic forces only play their part by -complex interaction with the equally simple forces of the surrounding -medium. In an earlier age, men sought for the visible embryo, -even for the <i>homunculus</i>, within the reproductive cells; and to -this day, we scrutinize these cells for visible structure, unable to -free ourselves from that old doctrine of “pre-formation<a class="afnanch" href="#fn204" id="fnanch204">204</a>.”</p> - -<p>Moreover, the microscope seemed to substantiate the idea -(which we may trace back to Leibniz<a class="afnanch" href="#fn205" id="fnanch205">205</a> -and to Hobbes<a class="afnanch" href="#fn206" id="fnanch206">206</a>), -that -there is no limit to the mechanical complexity which we may -postulate in an organism, and no limit, therefore, to the hypotheses -which we may rest thereon.</p> - -<p>But no microscopical examination of a stick of sealing-wax, -no study of the material of which it is -composed, can enlighten <span class="xxpn" id="p160">{160}</span> -us as to its electrical manifestations or properties. Matter of -itself has no power to do, to make, or to become: it is in energy -that all these potentialities reside, energy invisibly associated with -the material system, and in interaction with the energies of the -surrounding universe.</p> - -<p>That “function presupposes structure” has been declared an -accepted axiom of biology. Who it was that so formulated the -aphorism I do not know; but as regards the structure of the cell -it harks back to Brücke, with whose demand for a mechanism, -or organisation, within the cell histologists have ever since -been attempting to comply<a class="afnanch" href="#fn207" id="fnanch207">207</a>. -But unless we mean to include -thereby invisible, and merely chemical or molecular, structure, -we come at once on dangerous ground. For we have seen, in -a former chapter, that some minute “organisms” are already -known of such all but infinitesimal magnitudes that everything -which the morphologist is accustomed to conceive as “structure” -has become physically impossible; and moreover recent research -tends generally to reduce, rather than to extend, our conceptions -of the visible structure necessarily inherent in living protoplasm. -The microscopic structure which, in the last resort or in the simplest -cases, it seems to shew, is that of a more or less viscous colloid, -or rather mixture of colloids, and nothing more. Now, as Clerk -Maxwell puts it, in discussing this very problem, “one material -system can differ from another only in the configuration and -motion which it has at a given instant<a class="afnanch" href="#fn208" id="fnanch208">208</a>.” -If we cannot assume -differences in structure, we must assume differences in <i>motion</i>, that -is to say, in <i>energy</i>. And if we cannot do this, then indeed we are -thrown back upon modes of reasoning unauthorised in physical -science, and shall find ourselves constrained to assume, or to -“admit, that the properties of a germ are not those of a purely -material system.” <span class="xxpn" id="p161">{161}</span></p> - -<p>But we are by no means necessarily in this dilemma. For -though we come perilously near to it when we contemplate the -lowest orders of magnitude to which life has been attributed, yet -in the case of the ordinary cell, or ordinary egg or germ which is -going to develop into a complex organism, if we have no reason -to assume or to believe that it comprises an intricate “mechanism,” -we may be quite sure, both on direct and indirect evidence, that, -like the powder in our rocket, it is very heterogeneous in its -structure. It is a mixture of substances of various kinds, more -or less fluid, more or less mobile, influenced in various ways by -chemical, electrical, osmotic, and other forces, and in their -admixture separated by a multitude of surfaces, or boundaries, at -which these, or certain of these forces are made manifest.</p> - -<p>Indeed, such an arrangement as this is already enough to -constitute a “mechanism”; for we must be very careful not to -let our physical or physiological concept of mechanism be narrowed -to an interpretation of the term derived from the delicate and -complicated contrivances of human skill. From the physical -point of view, we understand by a “mechanism” whatsoever -checks or controls, and guides into determinate paths, the workings -of energy; in other words, whatsoever leads in the degradation -of energy to its manifestation in some determinate form of <i>work</i>, -at a stage short of that ultimate degradation which lapses in -uniformly diffused heat. This, as Warburg has well explained, is -the general effect or function of the physiological machine, and in -particular of that part of it which we call “cell-structure<a class="afnanch" href="#fn209" id="fnanch209">209</a>.” -The normal muscle-cell is something which turns energy, derived -from oxidation, into work; it is a mechanism which arrests and -utilises the chemical energy of oxidation in its downward course; -but the same cell when injured or disintegrated, loses its “usefulness,” -and sets free a greatly increased proportion of its energy -in the form of heat.</p> - -<p>But very great and wonderful things are done after this manner -by means of a mechanism (whether natural or artificial) of -extreme simplicity. A pool of water, by -virtue of its surface, <span class="xxpn" id="p162">{162}</span> -is an admirable mechanism for the making of waves; with a lump -of ice in it, it becomes an efficient and self-contained mechanism -for the making of currents. The great cosmic mechanisms are -stupendous in their simplicity; and, in point of fact, every great -or little aggregate of heterogeneous matter (not identical in -“phase”) involves, <i>ipso facto</i>, the essentials of a mechanism. -Even a non-living colloid, from its intrinsic heterogeneity, is in -this sense a mechanism, and one in which energy is manifested -in the movement and ceaseless rearrangement of the constituent -particles. For this reason Graham (if I remember rightly) speaks -somewhere or other of the colloid state as “the dynamic state of -matter”; or in the same philosopher’s phrase (of which Mr -Hardy<a class="afnanch" href="#fn210" id="fnanch210">210</a> -has lately reminded us), it possesses “<i>energia</i><a class="afnanch" href="#fn211" id="fnanch211">211</a>.”</p> - -<p>Let us turn then to consider, briefly and diagrammatically, the -structure of the cell, a fertilised germ-cell or ovum for instance, -not in any vain attempt to correlate this structure with the -structure or properties of the resulting and yet distant organism; -but merely to see how far, by the study of its form and its changing -internal configuration, we may throw light on certain forces which -are for the time being at work within it.</p> - -<p>We may say at once that we can scarcely hope to learn more -of these forces, in the first instance, than a few facts regarding -their direction and magnitude; the nature and specific identity -of the force or forces is a very different matter. This latter -problem is likely to be very difficult of elucidation, for the reason, -among others, that very different forces are often very much alike -in their outward and visible manifestations. So it has come to -pass that we have a multitude of discordant hypotheses as to the -nature of the forces acting within the cell, and producing, in cell -division, the “caryokinetic” figures of which we are about to -speak. One student may, like Rhumbler, choose to account for -them by an hypothesis of mechanical traction, acting on a reticular -web of protoplasm<a class="afnanch" href="#fn212" id="fnanch212">212</a>; -another, like -Leduc, may shew us how in <span class="xxpn" id="p163">{163}</span> -many of their most striking features they may be admirably -simulated by the diffusion of salts in a colloid medium; others -again, like Gallardo<a class="afnanch" href="#fn213" id="fnanch213">213</a> -and Hartog, and Rhumbler (in his earlier -papers)<a class="afnanch" href="#fn214" id="fnanch214">214</a>, -insist on their resemblance to the phenomena of -electricity and magnetism<a class="afnanch" href="#fn215" id="fnanch215">215</a>; -while Hartog believes that the force -in question is only analogous to these, and has a specific identity -of its own<a class="afnanch" href="#fn216" id="fnanch216">216</a>. -All these conflicting views are of secondary importance, -so long as we seek only to account for certain <i>configurations</i> -which reveal the direction, rather than the nature, of a force. -One and the same system of lines of force may appear in a field -of magnetic or of electrical energy, of the osmotic energy of -diffusion, of the gravitational energy of a flowing stream. In short, -we may expect to learn something of the pure or abstract dynamics, -long before we can deal with the special physics of the cell. For -indeed (as Maillard has suggested), just as uniform expansion -about a single centre, to whatsoever physical cause it may be due -will lead to the configuration of a sphere, so will any two centres -or foci of potential (of whatsoever kind) lead to the configurations -with which Faraday made us familiar under the name of “lines -of force<a class="afnanch" href="#fn217" id="fnanch217">217</a>”; -and this is as much as to say -that the phenomenon, <span class="xxpn" id="p164">{164}</span> -though physical in the concrete, is in the abstract purely mathematical, -and in its very essence is neither more nor less than <i>a -property of three-dimensional space</i>.</p> - -<p>But as a matter of fact, in this instance, that is to say in -trying to explain the leading phenomena of the caryokinetic -division of the cell, we shall soon perceive that any explanation -which is based, like Rhumbler’s, on mere mechanical traction, is -obviously inadequate, and we shall find ourselves limited to the -hypothesis of some polarised and polarising force, such as we deal -with, for instance, in the phenomena of magnetism or electricity.</p> - -<p>Let us speak first of the cell itself, as it appears in a state of -rest, and let us proceed afterwards to study the more active -phenomena which accompany its division.</p> - -<hr class="hrblk"> - -<p>Our typical cell is a spherical body; that is to say, the uniform -surface-tension at its boundary is balanced by the outward -resistance of uniform forces within. But at times the surface-tension -may be a fluctuating quantity, as when it produces the -rhythmical contractions or “Ransom’s waves” on the surface of -a trout’s egg; or again, while the egg is in contact with other -bodies, the surface-tension may be locally unequal and variable, -giving rise to an amoeboid figure, as in the egg of Hydra<a class="afnanch" href="#fn218" id="fnanch218">218</a>.</p> - -<p>Within the ovum is a nucleus or germinal vesicle, also spherical, -and consisting as a rule of portions of “chromatin,” aggregated -together within a more fluid drop. The fact has often been -commented upon that, in cells generally, there is no correlation -of <i>form</i> (though there apparently is of <i>size</i>) between the nucleus -and the “cytoplasm,” or main body of the cell. So Whitman<a class="afnanch" href="#fn219" id="fnanch219">219</a> -remarks that “except during the process of division the nucleus -seldom departs from its typical spherical form. It divides and -sub-divides, ever returning to the same round or oval -form .... How different with the cell. It preserves the spherical form as -rarely as the nucleus departs from it. Variation in form marks -the beginning and the end of every important -chapter in its <span class="xxpn" id="p165">{165}</span> -history.” On simple dynamical grounds, the contrast is easily -explained. So long as the fluid substance of the nucleus is qualitatively -different from, and incapable of mixing with, the fluid -or semi-fluid protoplasm which surrounds it, we shall expect it -to be, as it almost always is, of spherical form. For, on the one -hand, it is bounded by a liquid film, whose surface-tension is -uniform; and on the other, it is immersed in a medium which -transmits on all sides a uniform fluid pressure<a class="afnanch" href="#fn220" id="fnanch220">220</a>. -For a similar -reason the contractile vacuole of a Protozoon is spherical in form: -it is just a “drop” of fluid, bounded by a uniform surface-tension -and through whose boundary-film diffusion is taking place. -But here, owning to the small difference between the fluid constituting, -and that surrounding, the drop, the surface-tension equilibrium -is unstable; it is apt to vanish, and the rounded outline -of the drop, like a burst bubble, disappears in a moment<a class="afnanch" href="#fn221" id="fnanch221">221</a>. -The case of the spherical nucleus is closely akin to the spherical -form of the yolk within the bird’s egg<a class="afnanch" href="#fn222" id="fnanch222">222</a>. -But if the substance of -the cell acquire a greater solidity, as for -instance in a muscle <span class="xxpn" id="p166">{166}</span> -cell, or by reason of mucous accumulations in an epithelium cell, -then the laws of fluid pressure no longer apply, the external -pressure on the nucleus tends to become unsymmetrical, and its -shape is modified accordingly. “Amoeboid” movements may be -set up in the nucleus by anything which disturbs the symmetry of -its own surface-tension. And the cases, as in many Rhizopods, -where “nuclear material” is scattered in small portions throughout -the cell instead of being aggregated in a single nucleus, are probably -capable of very simple explanation by supposing that the “phase -difference” (as the chemists say) between the nuclear and the -protoplasmic substance is comparatively slight, and the surface-tension -which tends to keep them separate is correspondingly -small<a class="afnanch" href="#fn223" id="fnanch223">223</a>.</p> - -<p>It has been shewn that ordinary nuclei, isolated in a living -or fresh state, easily flow together; and this fact is enough to -suggest that they are aggregations of a particular substance rather -than bodies deserving the name of particular organs. It is by -reason of the same tendency to confluence or aggregation of -particles that the ordinary nucleus is itself formed, until the -imposition of a new force leads to its disruption.</p> - -<p>Apart from that invisible or ultra-microscopic heterogeneity -which is inseparable from our notion of a “colloid,” there is a -visible heterogeneity of structure within both the nucleus and the -outer protoplasm. The former, for instance, contains a rounded -nucleolus or “germinal spot,” certain conspicuous granules or -strands of the peculiar substance called chromatin, and a coarse -meshwork of a protoplasmic material known as “linin” or achromatin; -the outer protoplasm, or cytoplasm, is generally believed -to consist throughout of a sponge-work, or rather alveolar meshwork, -of more and less fluid substances; and lastly, there are -generally to be detected one or more very minute bodies, usually -in the cytoplasm, sometimes within the nucleus, known as the -centrosome or centrosomes.</p> - -<p>The morphologist is accustomed to speak of -a “polarity” of <span class="xxpn" id="p167">{167}</span> -the cell, meaning thereby a symmetry of visible structure about -a particular axis. For instance, whenever we can recognise in -a cell both a nucleus and a centrosome, we may consider a -line drawn through the two as the morphological axis of polarity; -in an epithelium cell, it is obvious that the cell is morphologically -symmetrical about a median axis passing from its free surface to -its attached base. Again, by an extension of the term “polarity,” -as is customary in dynamics, we may have a “radial” polarity, -between centre and periphery; and lastly, we may have several -apparently independent centres of polarity within the single cell. -Only in cells of quite irregular, or amoeboid form, do we fail to -recognise a definite and symmetrical “polarity.” The <i>morphological</i> -“polarity” is accompanied by, and is but the outward -expression (or part of it) of a true <i>dynamical</i> polarity, or distribution -of forces; and the “lines of force” are rendered visible by concatenation -of particles of matter, such as come under the influence -of the forces in action.</p> - -<p>When the lines of force stream inwards from the periphery -towards a point in the interior of the cell, the particles susceptible -of attraction either crowd towards the surface of the cell, or, when -retarded by friction, are seen forming lines or “fibrillae” which -radiate outwards from the centre and constitute a so-called -“aster.” In the cells of columnar or ciliated epithelium, where -the sides of the cell are symmetrically disposed to their neighbours -but the free and attached surfaces are very diverse from one -another in their external relations, it is these latter surfaces which -constitute the opposite poles; and in accordance with the parallel -lines of force so set up, we very frequently see parallel lines of -granules which have ranged themselves perpendicularly to the -free surface of the cell (cf. fig. <a href="#fig97" title="go to Fig. 97">97</a>).</p> - -<p>A simple manifestation of “polarity” may be well illustrated -by the phenomenon of diffusion, where we may conceive, and may -automatically reproduce, a “field of force,” with its poles and -visible lines of equipotential, very much as in Faraday’s conception -of the field of force of a magnetic system. Thus, in one of Leduc’s -experiments<a class="afnanch" href="#fn224" id="fnanch224">224</a>, -if we spread a layer of salt -solution over a level <span class="xxpn" id="p168">{168}</span> -plate of glass, and let fall into the middle of it a drop of indian -ink, or of blood, we shall find the coloured particles travelling -outwards from the central “pole of concentration” along the lines -of diffusive force, and so mapping out for us a “monopolar field” -of diffusion: and if we set two such drops side by side, their -lines of diffusion will oppose, and repel, one another. Or, instead -of the uniform layer of salt solution, we may place at a little -distance from one another a grain of salt and a drop of blood, -representing two opposite poles: and so obtain a picture of a -“bipolar field” of diffusion. In either case, we obtain results -closely analogous to the “morphological,” but really <i>dynamical</i>, -polarity of the organic cell. But in all probability, the dynamical -polarity, or asymmetry of the cell is a very complicated phenomenon: -for the obvious reason that, in any system, one asymmetry -will tend to beget another. A chemical asymmetry will induce an -inequality of surface-tension, which will lead directly to a modification -of form; the chemical asymmetry may in turn be due to a -process of electrolysis in a polarised electrical field; and again -the chemical heterogeneity may be intensified into a chemical -“polarity,” by the tendency of certain substances to seek a locus -of greater or less surface-energy. We need not attempt to -grapple with a subject so complicated, and leading to so many -problems which lie beyond the sphere of interest of the morphologist. -But yet the morphologist, in his study of the cell, -cannot quite evade these important issues; and we shall return -to them again when we have dealt somewhat with the form of -the cell, and have taken account of some of the simpler phenomena -of surface-tension.</p> - -<hr class="hrblk"> - -<p>We are now ready, and in some measure prepared, to study -the numerous and complex phenomena which usually accompany -the division of the cell, for instance of the fertilised egg.</p> - -<p>Division of the cell is essentially accompanied, and preceded, -by a change from radial or monopolar to a definitely bipolar -polarity.</p> - -<p>In the hitherto quiescent, or apparently quiescent cell, we perceive -certain movements, which correspond precisely to what must -accompany and result from a “polarisation” of -forces within the <span class="xxpn" id="p169">{169}</span> -cell: of forces which, whatever may be their specific nature, at least -are capable of polarisation, and of producing consequent attraction -or repulsion between charged particles of matter. The opposing -forces which were distributed in equilibrium throughout the substance -of the cell become focussed at two “centrosomes,” which -may or may not be already distinguished as visible portions of -matter; in the egg, one of these is always near to, and the other -remote from, the “animal pole” of the egg, which pole is visibly -as well as chemically different from the other, and is the region in -which the more rapid and conspicuous developmental changes will -presently begin. Between the two centrosomes, a spindle-shaped</p> - -<div class="dctr02" id="fig41"> -<img src="images/i169.jpg" width="600" height="283" alt=""> - <div class="pcaption">Fig. 41. Caryokinetic figure in a dividing - cell (or blastomere) of the Trout’s egg. (After Prenant, - from a preparation by Prof. P. Bouin.)</div></div> - -<p class="pcontinue">figure appears, whose striking resemblance to the lines of force -made visible by iron-filings between the poles of a magnet, was at -once recognised by Hermann Fol, when in 1873 he witnessed for -the first time the phenomenon in question. On the farther side -of the centrosomes are seen star-like figures, or “asters,” in which -we can without difficulty recognise the broken lines of force which -run externally to those stronger lines which lie nearer to the polar -axis and which constitute the “spindle.” The lines of force are -rendered visible or “material,” just as in the experiment of the -iron-filings, by the fact that, in the heterogeneous substance of -the cell, certain portions of matter are more “permeable” to the -acting force than the rest, become themselves -polarised after the <span class="xxpn" id="p170">{170}</span> -fashion of a magnetic or “paramagnetic” body, arrange themselves -in an orderly way between the two poles of the field of force, cling -to one another as it were in threads<a class="afnanch" href="#fn225" id="fnanch225">225</a>, -and are only prevented by -the friction of the surrounding medium from approaching and -congregating around the adjacent poles.</p> - -<p>As the field of force strengthens, the more will the lines of force -be drawn in towards the interpolar axis, and the less evident will -be those remoter lines which constitute the terminal, or extrapolar, -asters: a clear space, free from materialised lines of force, may -thus tend to be set up on either side of the spindle, the -so-called “Bütschli space” of the histologists<a class="afnanch" href="#fn226" id="fnanch226">226</a>. -On the other -hand, the lines of force constituting the spindle will be less concentrated -if they find a path of less resistance at the periphery -of the cell: as happens, in our experiment of the iron-filings, when -we encircle the field of force with an iron ring. On this principle, -the differences observed between cells in which the spindle is well -developed and the asters small, and others in which the spindle -is weak and the asters enormously developed, can be easily -explained by variations in the potential of the field, the large, -conspicuous asters being probably correlated with a marked -permeability of the surface of the cell.</p> - -<p>The visible field of force, though often called the “nuclear -spindle,” is formed outside of, but usually near to, the nucleus. -Let us look a little more closely into the structure of this body, -and into the changes which it presently undergoes.</p> - -<p>Within its spherical outline (Fig. <a href="#fig42" title="go to Fig. 42">42</a>), it -contains an “alveolar” <span class="xxpn" id="p171">{171}</span> -meshwork (often described, from its appearance in optical section, -as a “reticulum”), consisting of more solid substances, with more -fluid matter filling up the interalveolar meshes. This phenomenon -is nothing else than what we call in ordinary language, a “froth” -or a “foam.” It is a surface-tension phenomenon, due to the -interacting surface-tensions of two intermixed fluids, not very -different in density, as they strive to separate. Of precisely the -same kind (as Bütschli was the first to shew) are the minute alveolar -networks which are to be discerned in the cytoplasm of the cell<a class="afnanch" href="#fn227" id="fnanch227">227</a>, -and which we now know to be not inherent in the nature of -protoplasm, or of living matter in general, but to be due to various -causes, natural as well as artificial. The microscopic honeycomb -structure of cast metal under various conditions of cooling, even -on a grand scale the columnar structure of basaltic rock, is an -example of the same surface-tension phenomenon. <span class="xxpn" id="p172">{172}</span></p> - -<div class="dctr01" id="fig42"><div id="fig43"> -<img src="images/i171.png" width="800" height="461" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 42.</td> - <td></td> - <td>Fig. 43.</td></tr></table> -</div></div></div><!--dctr01--> - -<div class="psmprnt3"> -<p>But here we touch the brink of a subject so important that we -must not pass it by without a word, and yet so contentious -that we must not enter into its details. The question involved -is simply whether the great mass of recorded observations -and accepted beliefs with regard to the visible structure -of protoplasm and of the cell constitute a fair picture of -the actual <i>living cell</i>, or be based on appearances which -are incident to death itself and to the artificial treatment -which the microscopist is accustomed to apply. The great bulk -of histological work is done by methods which involve the -sudden killing of the cell or organism by strong reagents, -the assumption being that death is so rapid that the visible -phenomena exhibited during life are retained or “fixed” in our -preparations. While this assumption is reasonable and justified -as regards the general outward form of small organisms or -of individual cells, enough has been done of late years to -shew that the case is totally different in the case of the -minute internal networks, granules, etc., which represent the -alleged <i>structure</i> of protoplasm. For, as Hardy puts it, “It -is notorious that the various fixing reagents are coagulants -of organic colloids, and that they produce precipitates which -have a certain figure or structure, ... and -that the figure varies, other things being equal, according -to the reagent used.” So it comes to pass that some writers<a -class="afnanch" href="#fn228" id="fnanch228">228</a> have -altogether denied the existence in the living cell-protoplasm -of a network or alveolar “foam”; others<a class="afnanch" -href="#fn229" id="fnanch229">229</a> have cast doubts on the -main tenets of recent histology regarding nuclear structure; -and Hardy, discussing the structure of certain gland-cells, -declares that “there is no evidence that the structure -discoverable in the cell-substance of these cells after -fixation has any counterpart in the cell when living.” “A large -part of it” he goes on to say “is an artefact. The profound -difference in the minute structure of a secretory cell of a -mucous gland according to the reagent which is used to fix -it would, it seems to me, almost suffice to establish this -statement in the absence of other evidence.”</p> - -<p>Nevertheless, histological study proceeds, especially on -the part of the morphologists, with but little change in -theory or in method, in spite of these and many other -warnings. That certain visible structures, nucleus, vacuoles, -“attraction-spheres” or centrosomes, etc., are actually -present in the living cell, we know for certain; and to this -class belong the great majority of structures (including -the nuclear “spindle” itself) with which we are at present -concerned. That many other alleged structures are artificial -has also been placed beyond a doubt; but where to draw -the dividing line we often do not know<a class="afnanch" -href="#fn230" id="fnanch230">230</a>. <span class="xxpn" -id="p173">{173}</span></p> -</div><!--psmprnt3--> - -<p>The following is a brief epitome of the visible changes -undergone by a typical cell, leading up to the act of -segmentation, and constituting the phenomenon of mitosis or -caryokinetic division. In the egg of a sea-urchin, we see with -almost diagrammatic completeness what is set forth here<a -class="afnanch" href="#fn231" id="fnanch231">231</a>.</p> - -<div class="dctr01" id="fig44"><div id="fig45"> -<img src="images/i173.png" width="800" height="448" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 44.</td> - <td></td> - <td>Fig. 45.</td></tr></table> -</div></div></div><!--dctr01--> - -<ul> -<li><p>1. The chromatin, which to begin with was distributed in -granules on the otherwise achromatic reticulum (Fig. <a href="#fig42" title="go to Fig. 42">42</a>), concentrates -to form a skein or <i>spireme</i>, which may be a continuous -thread from the first (Figs. <a href="#fig43" title="go to Fig. 43">43</a>, 44), or from the first segmented. -In any case it divides transversely sooner or later into a number -of <i>chromosomes</i> (Fig. <a href="#fig45" title="go to Fig. 45">45</a>), which as a rule have the shape of little -rods, straight or curved, often bent into a V, but which may -also be ovoid, or round, or even annular. Certain deeply staining -masses, the nucleoli, which may be present in the resting nucleus, -do not take part in the process of chromosome formation; they -are either cast out of the nucleus and are dissolved in the cytoplasm, -or fade away <i>in situ</i>.</p></li> - -<li><p>2. Meanwhile, the deeply staining granule (here extra-nuclear), -known as the <i>centrosome</i>, has divided in two. The two -resulting granules travel to opposite poles -of the nucleus, and <span class="xxpn" id="p174">{174}</span> -there each becomes surrounded by a system of radiating lines, the -<i>asters</i>; immediately around the centrosome is a clear space, the -<i>centrosphere</i> (Figs. <a href="#fig43" title="go to Fig. 43">43</a>–45). Between the two centrosomes with -their asters stretches a bundle of achromatic fibres, the <i>spindle</i>.</p></li> - -<li><p>3. The surface-film bounding the nucleus has broken down, -the definite nuclear boundaries are lost, and the spindle now -stretches through the nuclear material, in which lie the chromosomes -(Figs. <a href="#fig45" title="go to Fig. 45">45</a>, 46). These chromosomes now arrange themselves -midway between the poles of the spindle, where they form -what is called the <i>equatorial plate</i> (Fig. <a href="#fig47" title="go to Fig. 47">47</a>).</p> - -<div class="dctr01" id="fig46"><div id="fig47"> -<img src="images/i174.png" width="800" height="432" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 46.</td> - <td></td> - <td>Fig. 47.</td></tr></table> -</div></div></div><!--dctr01--></li> - -<li> -<p>4. Each chromosome splits longitudinally into two: usually -at this stage,—but it is to be noticed that the splitting may have -taken place so early as the spireme stage (Fig. -<a href="#fig48" title="go to Fig. 48">48</a>).</p></li> - -<li><p>5. The halves of the split chromosomes now separate from -one another, and travel in opposite directions towards the two -poles (Fig. <a href="#fig49" title="go to Fig. 49">49</a>). As they move, it becomes apparent that the spindle -consists of a median bundle of “fibres,” the central spindle, running -from pole to pole, and a more superficial sheath of “mantle-fibres,” -to which the chromosomes seem to be attached, and by -which they seem to be drawn towards the asters.</p></li> - -<li><p>6. The daughter chromosomes, arranged now in two groups, -become closely crowded in a mass near the centre -of each aster <span class="xxpn" id="p175">{175}</span> -(Fig. <a href="#fig50" title="go to Fig. 50">50</a>). They fuse together and form once more an alveolar reticulum -and may occasionally at this stage form another spireme.</p> - -<div class="dctr01" id="fig48"><div id="fig49"> -<img src="images/i175a.png" width="800" height="437" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 48.</td> - <td></td> - <td>Fig. 49.</td></tr></table> -</div></div></div><!--dctr01--> - -<p class="pcontinue">A boundary or surface wall is now developed round each reconstructed -nuclear mass, and the spindle-fibres disappear (Fig. <a href="#fig51" title="go to Fig. 51">51</a>). -The centrosome remains, as a rule, outside the nucleus.</p> - -<div class="dctr01" id="fig50"><div id="fig51"> -<img src="images/i175b.png" width="800" height="383" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 50.</td> - <td></td> - <td>Fig. 51.</td></tr></table> -</div></div></div><!--dctr01--> -</li> - -<li><p>7. On the central spindle, in the position of the -equatorial plate, there has appeared during the migration of -the chromosomes, a “cell-plate” of deeply staining thickenings -(Figs. <a href="#fig50" title="go to Fig. 50">50</a>, 51). This is more conspicuous in plant-cells. <span -class="xxpn" id="p176">{176}</span></p></li> - -<li><p>8. A constriction has meanwhile appeared in the -cytoplasm, and the cell divides through the equatorial plane. -In plant-cells the line of this division is foreshadowed by -the “cell-plate,” which extends from the spindle across the -entire cell, and splits into two layers, between which appears -the membrane by which the daughter cells are cleft asunder. In -animal cells the cell-plate does not attain such dimensions, -and no cell-wall is formed.</p></li> -</ul> - -<hr class="hrblk"> - -<p>The whole, or very nearly the whole of these nuclear phenomena -may be brought into relation with that polarisation of forces, in -the cell as a whole, whose field is made manifest by the “spindle” -and “asters” of which we have already spoken: certain particular -phenomena, directly attributable to surface-tension and diffusion, -taking place in more or less obvious and inevitable dependence -upon the polar system*.</p> - -<p class="ptblfn">* The reference numbers in the following account -refer to the paragraphs and figures of the preceding -summary of visible nuclear phenomena.</p> - -<p>At the same time, in attempting to explain the phenomena, we -cannot say too clearly, or too often, that all that we are meanwhile -justified in doing is to try to shew that such and such actions lie -<i>within the range</i> of known physical actions and phenomena, or that -known physical phenomena produce effects similar to them. We -want to feel sure that the whole phenomenon is not <i>sui generis</i>, but -is somehow or other capable of being referred to dynamical laws, -and to the general principles of physical science. But when we -speak of some particular force or mode of action, using it as an -illustrative hypothesis, we must stop far short of the implication -that this or that force is necessarily the very one which is actually -at work within the living cell; and certainly we need not attempt -the formidable task of trying to reconcile, or to choose between, -the various hypotheses which have already been enunciated, or -the several assumptions on which they depend.</p> - -<hr class="hrblk"> - -<p>Any region of space within which action is manifested is a -field of force; and a simple example is a bipolar field, in which -the action is symmetrical with reference to the line joining two -points, or poles, and also with reference to the “equatorial” -plane equidistant from both. We have such -a “field of force” in <span class="xxpn" id="p177">{177}</span> -the neighbourhood of the centrosome of the ripe cell or ovum, -when it is about to divide; and by the time the centrosome has -divided, the field is definitely a bipolar one.</p> - -<p>The <i>quality</i> of a medium filling the field of force may be uniform, -or it may vary from point to point. In particular, it may depend -upon the magnitude of the field; and the quality of one medium -may differ from that of another. Such variation of quality, -within one medium, or from one medium to another, is capable -of diagrammatic representation by a variation of the direction or -the strength of the field (other conditions being the same) from the -state manifested in some uniform medium taken as a standard. -The medium is said to be <i>permeable</i> to the force, in greater or less -degree than the standard medium, according as the variation of -the density of the lines of force from the standard case, under -otherwise identical conditions, is in excess or defect. <i>A body -placed in the medium will tend to move towards regions of greater or -less force according as its permeability is greater or less than that of -the surrounding medium</i><a class="afnanch" href="#fn232" id="fnanch232">232</a>. -In the common experiment of placing -iron-filings between the two poles of a magnetic field, the filings -have a very high permeability; and not only do they themselves -become polarised so as to attract one another, but they tend to -be attracted from the weaker to the stronger parts of the field, and -as we have seen, were it not for friction or some other resistance, -they would soon gather together around the nearest pole. But -if we repeat the same experiment with such a metal as bismuth, -which is very little permeable to the magnetic force, then the -conditions are reversed, and the particles, being repelled from the -stronger to the weaker parts of the field, tend to take up their -position as far from the poles as possible. The particles have -become polarised, but in a sense opposite to that of the surrounding, -or adjacent, field.</p> - -<p>Now, in the field of force whose opposite -poles are marked by <span class="xxpn" id="p178">{178}</span> -the centrosomes the nucleus appears to act as a more or less permeable -body, as a body more permeable than the surrounding medium, -that is to say the “cytoplasm” of the cell. It is accordingly -attracted by, and drawn into, the field of force, and tries, as it -were, to set itself between the poles and as far as possible from -both of them. In other words, the centrosome-foci will be -apparently drawn over its surface, until the nucleus as a whole -is involved within the field of force, which is visibly marked out -by the “spindle” (par. 3, Figs. <a href="#fig44" title="go to Fig. 44">44</a>, 45).</p> - -<p>If the field of force be electrical, or act in a fashion analogous -to an electrical field, the charged nucleus will have its surface-tensions -diminished<a class="afnanch" href="#fn233" id="fnanch233">233</a>: -with the double result that the inner -alveolar meshwork will be broken up (par. 1), and that the -spherical boundary of the whole nucleus will disappear (par. 2). -The break-up of the alveoli (by thinning and rupture of their -partition walls) leads to the formation of a net, and the further -break-up of the net may lead to the unravelling of a thread or -“spireme” (Figs. <a href="#fig43" title="go to Fig. 43">43</a>, 44).</p> - -<p>Here there comes into play a fundamental principle which, -in so far as we require to understand it, can be explained in simple -words. The effect (and we might even say the <i>object</i>) of drawing -the more permeable body in between the poles, is to obtain an -“easier path” by which the lines of force may travel; but it is -obvious that a longer route through the more permeable body -may at length be found less advantageous than a shorter route -through the less permeable medium. That is to say, the more -permeable body will only tend to be drawn in to the field of force -until a point is reached where (so to speak) the way <i>round</i> and -the way <i>through</i> are equally advantageous. We should accordingly -expect that (on our hypothesis) there would be found cases in -which the nucleus was wholly, and others in which it was only -partially, and in greater or less degree, drawn in to the field -between the centrosomes. This is precisely what is found to -occur in actual fact. Figs. <a href="#fig44" title="go to Fig. 44">44</a> and 45 represent two so-called -“types,” of a phase which follows that represented in Fig. <a href="#fig43" title="go to Fig. 43">43</a>. -According to the usual descriptions (and in -particular to Professor <span class="xxpn" id="p179">{179}</span> -E. B. Wilson’s<a class="afnanch" href="#fn234" id="fnanch234">234</a>), -we are told that, in such a case as Fig. <a href="#fig44" title="go to Fig. 44">44</a>, the -“primary spindle” disappears and the centrosomes diverge to -opposite poles of the nucleus; such a condition being found in -many plant-cells, and in the cleavage-stages of many eggs. In -Fig. <a href="#fig45" title="go to Fig. 45">45</a>, on the other hand, the primary spindle persists, and -subsequently comes to form the main or “central” spindle; -while at the same time we see the fading away of the nuclear -membrane, the breaking up of the spireme into separate chromosomes, -and an ingrowth into the nuclear area of the “astral rays,”—all -as in Fig. <a href="#fig46" title="go to Fig. 46">46</a>, which represents the next succeeding phase of -Fig. <a href="#fig45" title="go to Fig. 45">45</a>. This condition, of Fig. <a href="#fig46" title="go to Fig. 46">46</a>, occurs in a variety of cases; -it is well seen in the epidermal cells of the salamander, and is -also on the whole characteristic of the mode of formation of the -“polar bodies.” It is clear and obvious that the two “types” -correspond to mere differences of degree, and are such as would -naturally be brought about by differences in the relative permeabilities -of the nuclear mass and of the surrounding cytoplasm, -or even by differences in the magnitude of the former body.</p> - -<p>But now an important change takes place, or rather an -important difference appears; for, whereas the nucleus as a whole -tended to be drawn in to the <i>stronger</i> parts of the field, when it -comes to break up we find, on the contrary, that its contained -spireme-thread or separate chromosomes tend to be repelled to -the <i>weaker</i> parts. Whatever this difference may be due to,—whether, -for instance, to actual differences of permeability, or -possibly to differences in “surface-charge,”—the fact is that the -chromatin substance now <i>behaves</i> after the fashion of a “diamagnetic” -body, and is repelled from the stronger to the weaker -parts of the field. In other words, its particles, lying in the -inter-polar field, tend to travel towards the equatorial plane -thereof (Figs. <a href="#fig47" title="go to Fig. 47">47</a>, 48), and further tend to move outwards towards -the periphery of that plane, towards what the histologist -calls the “mantle-fibres,” or outermost of the lines of force of -which the spindle is made up (par. 5, Fig. <a href="#fig47" title="go to Fig. 47">47</a>). And if this comparatively -non-permeable chromatin substance come to consist of -separate portions, more or less elongated in form, these portions, -or separate “chromosomes,” will adjust -themselves longitudinally, <span class="xxpn" id="p180">{180}</span> -in a peripheral equatorial circle (Figs. <a href="#fig48" title="go to Fig. 48">48</a>, 49). This is precisely -what actually takes place. Moreover, before the breaking up of -the nucleus, long before the chromatin material has broken up -into separate chromosomes, and at the very time when it is being -fashioned into a “spireme,” this body already lies in a polar field, -and must already have a tendency to set itself in the equatorial -plane thereof. But the long, continuous spireme thread is unable, -so long as the nucleus retains its spherical boundary wall, to -adjust itself in a simple equatorial annulus; in striving to do so, -it must tend to coil and “kink” itself, and in so doing (if all this -be so), it must tend to assume the characteristic convolutions of -the “spireme.”</p> - -<div class="dctr01" id="fig52"> -<img src="images/i180.png" width="800" height="283" alt=""> - <div class="dcaption">Fig. 52. Chromosomes, undergoing - splitting and separation.<br>(After Hatschek and Flemming, - diagrammatised.)</div></div> - -<p>After the spireme has broken up into separate chromosomes, -these particles come into a position of temporary, and unstable, -equilibrium near the periphery of the equatorial plane, and -here they tend to place themselves in a symmetrical arrangement -(Fig. <a href="#fig52" title="go to Fig. 52">52</a>). The particles are rounded, linear, sometimes -annular, similar in form and size to one another; and -lying as they do in a fluid, and subject to a symmetrical system -of forces, it is not surprising that they arrange themselves -in a symmetrical manner, the precise arrangement depending -on the form of the particles themselves. This symmetry may -perhaps be due, as has already been suggested, to induced -electrical charges. In discussing Brauer’s observations on the -splitting of the chromatic filament, and the symmetrical arrangement -of the separate granules, in -<i>Ascaris megalocephala</i>, Lillie<a class="afnanch" href="#fn235" id="fnanch235">235</a> -<span class="xxpn" id="p181">{181}</span> -remarks: “This behaviour is strongly suggestive of the division -of a colloidal particle under the influence of its surface electrical -charge, and of the effects of mutual repulsion in keeping the -products of division apart.” It is also probable that surface-tensions -between the particles and the surrounding protoplasm -would bring about an identical result, and would sufficiently -account for the obvious, and at first sight, very curious, symmetry. -We know that if we float a couple of matches in water they tend -to approach one another, till they lie close together, side by side; -and, if we lay upon a smooth wet plate four matches, half broken -across, a precisely similar attraction brings the four matches -together in the form of a symmetrical cross. Whether one of -these, or some other, be the actual explanation of the phenomenon, -it is at least plain that by some physical cause, some mutual and -symmetrical attraction or repulsion of the particles, we must seek</p> - -<div class="dctr02" id="fig53"> -<img src="images/i181.png" width="600" height="254" alt=""> - <div class="pcaption">Fig. 53. Annular chromosomes, formed in the - spermatogenesis of the Mole-cricket. (From Wilson, after - Vom Rath.)</div></div> - -<p class="pcontinue"> -to account for the curious symmetry of these so-called “tetrads.” -The remarkable <i>annular</i> chromosomes, shewn in Fig. -<a href="#fig53" title="go to Fig. 53">53</a>, can also -be easily imitated by means of loops of thread upon a soapy film -when the film within the annulus is broken or its tension reduced.</p> - -<hr class="hrblk"> - -<p>So far as we have now gone, there is no great difficulty in -pointing to simple and familiar phenomena of a field of force -which are similar, or comparable, to the phenomena which we -witness within the cell. But among these latter phenomena -there are others for which it is not so easy to suggest, in accordance -with known laws, a simple mode of physical causation. It is not -at once obvious how, in any simple system -of symmetrical forces, <span class="xxpn" id="p182">{182}</span> -the chromosomes, which had at first been apparently repelled -from the poles towards the equatorial plane, should then be split -asunder, and should presently be attracted in opposite directions, -some to one pole and some to the other. Remembering that it is -not our purpose to <i>assert</i> that some one particular mode of action -is at work, but merely to shew that there do exist physical forces, -or distributions of force, which are capable of producing the -required result, I give the following suggestive hypothesis, which -I owe to my colleague Professor W. Peddie.</p> - -<p>As we have begun by supposing that the nuclear, or chromosomal -matter differs in <i>permeability</i> from the medium, that is to -say the cytoplasm, in which it lies, let us now make the further -assumption that its permeability is variable, and depends upon the -<i>strength of the field</i>.</p> - -<div class="dctr01" id="fig54"> -<img src="images/i182.png" width="800" height="470" alt=""> - <div class="dcaption">Fig. 54.</div></div> - -<p>In Fig. <a href="#fig54" title="go to Fig. 54">54</a>, we have a field of force (representing our cell), -consisting of a homogeneous medium, and including two opposite -poles: lines of force are indicated by full lines, and <i>loci of constant -magnitude of force</i> are shewn by dotted lines.</p> - -<p>Let us now consider a body whose permeability (µ) depends -on the strength of the field <i>F</i>. At two field-strengths, such -as <i>F<sub>a</sub></i>, <i>F<sub>b</sub></i>, let the permeability -of the body be equal to that of the <span class="xxpn" -id="p183">{183}</span> medium, and let the curved line -in Fig. <a href="#fig55" title="go to Fig. 55">55</a> represent generally its permeability at other -field-strengths; and let the outer and inner dotted curves in -Fig. <a href="#fig54" title="go to Fig. 54">54</a> represent respectively the loci of the field-strengths -<i>F<sub>b</sub></i> and <i>F<sub>a</sub></i>. The body if it be placed -in the medium within either branch of the inner curve, -or outside the outer curve, will tend to move into the -neighbourhood of the adjacent pole. If it be placed in the -region intermediate to the two dotted curves, it will tend to -move towards regions of weaker field-strength.</p> - -<div class="dctr05" id="fig55"> -<img src="images/i183.png" width="500" height="381" alt=""> - <div class="dcaption">Fig. 55.</div></div> - -<p>The locus <i>F<sub>b</sub></i> is therefore a locus of stable position, towards -which the body tends to move; the locus <i>F<sub>a</sub></i> is a locus of unstable -position, from which it tends to move. If the body were placed -across <i>F<sub>a</sub></i>, it might be torn asunder into two portions, the split -coinciding with the locus <i>F<sub>a</sub></i>.</p> - -<p>Suppose a number of such bodies to be scattered throughout -the medium. Let at first the regions <i>F<sub>a</sub></i> and <i>F<sub>b</sub></i> be entirely outside -the space where the bodies are situated: and, in making this -supposition we may, if we please, suppose that the loci which we -are calling <i>F<sub>a</sub></i> and <i>F<sub>b</sub></i> are meanwhile situated somewhat farther -from the axis than in our figure, that (for instance) <i>F<sub>a</sub></i> is situated -where we have drawn <i>F<sub>b</sub></i>, and that <i>F<sub>b</sub></i> is still further out. The -bodies then tend towards the poles; but the tendency may be -very small if, in Fig. <a href="#fig55" title="go to Fig. 55">55</a>, the curve and its intersecting straight line -do not diverge very far from one another beyond -<i>F<sub>a</sub></i>; in other <span class="xxpn" id="p184">{184}</span> -words, if, when situated in this region, the permeability of the -bodies is not very much in excess of that of the medium.</p> - -<p>Let the poles now tend to separate farther and farther from -one another, the strength of each pole remaining unaltered; in -other words, let the centrosome-foci recede from one another, as -they actually do, drawing out the spindle-threads between them. -The loci <i>F<sub>a</sub></i>, <i>F<sub>b</sub></i>, will close in to nearer relative distances from the -poles. In doing so, when the locus <i>F<sub>a</sub></i> crosses one of the bodies, -the body may be torn asunder; if the body be of elongated shape, -and be crossed at more points than one, the forces at work will -tend to exaggerate its foldings, and the tendency to rupture is -greatest when <i>F<sub>a</sub></i> is in some median position (Fig. <a href="#fig56" title="go to Fig. 56">56</a>).</p> - -<div class="dctr06" id="fig56"> -<img src="images/i184.png" width="426" height="243" alt=""> - <div class="dcaption">Fig. 56.</div></div> - -<p>When the locus <i>F<sub>a</sub></i> has passed entirely over the body, the body -tends to move towards regions of weaker force; but when, in -turn, the locus <i>F<sub>b</sub></i> has crossed it, then the body again moves towards -regions of stronger force, that is to say, towards the nearest pole. -And, in thus moving towards the pole, it will do so, as appears -actually to be the case in the dividing cell, along the course of -the outer lines of force, the so-called “mantle-fibres” of the -histologist<a class="afnanch" href="#fn236" id="fnanch236">236</a>.</p> - -<p>Such considerations as these give general results, easily open -to modification in detail by a change of any of the arbitrary -postulates which have been made for the sake of simplicity. -Doubtless there are many other assumptions which would more -or less meet the case; for instance, that of -Ida H. Hyde that, <span class="xxpn" id="p185">{185}</span> -during the active phase of the chromatin molecule (during which -it decomposes and sets free nucleic acid) it carries a charge opposite -to that which it bears during its resting, or alkaline phase; and -that it would accordingly move towards different poles under the -influence of a current, wandering with its negative charge in an -alkaline fluid during its acid phase to the anode, and to the kathode -during its alkaline phase. A whole field of speculation is opened -up when we begin to consider the cell not merely as a polarised -electrical field, but also as an electrolytic field, full of wandering -ions. Indeed it is high time we reminded ourselves that we have -perhaps been dealing too much with ordinary physical analogies: -and that our whole field of force within the cell is of an order of -magnitude where these grosser analogies may fail to serve us, -and might even play us false, or lead us astray. But our sole -object meanwhile, as I have said more than once, is to demonstrate, -by such illustrations as these, that, whatever be the actual -and as yet unknown <i>modus operandi</i>, there are physical conditions -and distributions of force which <i>could</i> produce just such phenomena -of movement as we see taking place within the living cell. -This, and no more, is precisely what Descartes is said to have -claimed for his description of the human body as a “mechanism<a class="afnanch" href="#fn237" id="fnanch237">237</a>.”</p> - -<hr class="hrblk"> - -<p>The foregoing account is based on the provisional assumption -that the phenomena of caryokinesis are analogous to, if not identical -with those of a bipolar electrical field; and this comparison, in -my opinion, offers without doubt the best available series of -analogies. But we must on no account omit to mention the -fact that some of Leduc’s diffusion-experiments offer very remarkable -analogies to the diagrammatic phenomena of caryokinesis, as -shewn in the annexed figure<a class="afnanch" href="#fn238" id="fnanch238">238</a>. -Here we have two identical (not -opposite) poles of osmotic concentration, formed by placing a drop -of indian ink in salt water, and then on either side of this central -drop, a hypertonic drop of salt solution more lightly coloured. -On either side the pigment of the central drop has been drawn -towards the focus nearest to it; but in -the middle line, the pigment <span class="xxpn" id="p186">{186}</span> -is drawn in opposite directions by equal forces, and so tends to -remain undisturbed, in the form of an “equatorial plate.”</p> - -<p>Nor should we omit to take account (however briefly and -inadequately) of a novel and elegant hypothesis put forward by -A. B. Lamb. This hypothesis makes use of a theorem of Bjerknes, -to the effect that synchronously vibrating or pulsating bodies in -a liquid field attract or repel one another according as their -oscillations are identical or opposite in phase. Under such -circumstances, true currents, or hydrodynamic lines of force, are -produced, identical in form with the lines of force of a magnetic -field; and other particles floating, though not necessarily pulsating, -in the liquid field, tend to be attracted or repelled by the pulsating -bodies according as they are lighter or heavier than the surrounding -fluid. Moreover (and this is the most remarkable point of all), -the lines of force set up by the <i>oppositely</i> pulsating bodies are the -same as those which are produced by <i>opposite</i> magnetic poles: -though in the former case repulsion, and in the latter case attraction, -takes place between the two poles<a class="afnanch" href="#fn239" id="fnanch239">239</a>.</p> - -<div class="dctr03" id="fig57"> -<img src="images/i186.png" width="600" height="329" alt=""> - <div class="pcaption">Fig. 57. Artificial caryokinesis (after - Leduc), for comparison with Fig. <a href="#fig41" title="go to Fig. 41">41</a>, - p. <a href="#p169" title="go to pg. 169">169</a>.</div></div> - -<hr class="hrblk"> - -<p>But to return to our general discussion.</p> - -<p>While it can scarcely be too often repeated that our enquiry -is not directed towards the solution of -physiological problems, save <span class="xxpn" id="p187">{187}</span> -only in so far as they are inseparable from the problems presented -by the visible configurations of form and structure, and while we -try, as far as possible, to evade the difficult question of what -particular forces are at work when the mere visible forms produced -are such as to leave this an open question, yet in this particular -case we have been drawn into the use of electrical analogies, and -we are bound to justify, if possible, our resort to this particular -mode of physical action. There is an important paper by R. S. Lillie, -on the “Electrical Convection of certain Free Cells and Nuclei<a class="afnanch" href="#fn240" id="fnanch240">240</a>,” -which, while I cannot quote it in direct support of the suggestions -which I have made, yet gives just the evidence we need in order -to shew that electrical forces act upon the constituents of the -cell, and that their action discriminates between the two species -of colloids represented by the cytoplasm and the nuclear chromatin. -And the difference is such that, in the presence of an electrical -current, the cell substance and the nuclei (including sperm-cells) -tend to migrate, the former on the whole with the positive, the -latter with the negative stream: a difference of electrical potential -being thus indicated between the particle and the surrounding -medium, just as in the case of minute suspended particles of various -kinds in various feebly conducing media<a class="afnanch" href="#fn241" id="fnanch241">241</a>. -And the electrical -difference is doubtless greatest, in the case of the cell constituents, -just at the period of mitosis: when the chromatin is invariably -in its most deeply staining, most strongly acid, and therefore, -presumably, in its most electrically -negative phase. In short, <span class="xxpn" id="p188">{188}</span> -Lillie comes easily to the conclusion that “electrical theories of -mitosis are entitled to more careful consideration than they have -hitherto received.”</p> - -<p>Among other investigations, all leading towards the same -general conclusion, namely that differences of electric potential -play a great part in the phenomenon of cell division, I would -mention a very noteworthy paper by Ida H. Hyde<a class="afnanch" href="#fn242" id="fnanch242">242</a>, -in which the -writer shews (among other important observations) that not only -is there a measurable difference of potential between the animal -and vegetative poles of a fertilised egg (<i>Fundulus</i>, toad, turtle, -etc.), but that this difference is not constant, but fluctuates, or -actually reverses its direction, periodically, at epochs coinciding -with successive acts of segmentation or other important phases -in the development of the egg<a class="afnanch" href="#fn243" id="fnanch243">243</a>; -just as other physical rhythms, -for instance in the production of CO<sub>2</sub> , had already been shewn -to do. Hence we shall be by no means surprised to find that the -“materialised” lines of force, which in the earlier stages form the -convergent curves of the spindle, are replaced in the later phases -of caryokinesis by divergent curves, indicating that the two foci, -which are marked out within the field by the divided and reconstituted -nuclei, are now alike in their polarity (Figs. <a href="#fig58" title="go to Fig. 58">58</a>, 59).</p> - -<p>It is certain, to my mind, that these observations of Miss -Hyde’s, and of Lillie’s, taken together with those of many writers -on the behaviour of colloid particles generally in their relation -to an electrical field, have a close bearing upon the physiological -side of our problem, the full discussion of which lies outside our -present field.</p> - -<hr class="hrblk"> - -<p>The break-up of the nucleus, already referred to and ascribed -to a diminution of its surface-tension, is accompanied by certain -diffusion phenomena which are sometimes visible to the eye; and -we are reminded of Lord Kelvin’s view that -diffusion is implicitly <span class="xxpn" id="p189">{189}</span> -associated with surface-tension changes, of which the first step -is a minute puckering of the surface-skin, a sort of interdigitation -with the surrounding medium. For instance, Schewiakoff -has observed in <i>Euglypha</i><a class="afnanch" href="#fn244" id="fnanch244">244</a> -that, just before the break-up -of the nucleus, a system of rays appears, concentred about it, -but having nothing to do with the polar asters: and during the -existence of this striation, the nucleus enlarges very considerably, -evidently by imbibition of fluid from the surrounding protoplasm. -In short, diffusion is at work, hand in hand with, and as it were -in opposition to, the surface-tensions which define the nucleus. -By diffusion, hand in hand with surface-tension, the alveoli of -the nuclear meshwork are formed, enlarged, and finally ruptured: -diffusion sets up the movements which give rise to the appearance -of rays, or striae, around the nucleus: and through increasing -diffusion, and weakening surface-tension, the rounded outline of -the nucleus finally disappears. <span class="xxpn" id="p190">{190}</span></p> - -<div class="dctr01" id="fig58"><div id="fig59"> -<img src="images/i189.png" width="800" height="501" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td><p>Fig. 58. Final stage in the first segmentation of the egg of - Cerebratulus. (From Prenant, after Coe.)<a class="afnanch" - href="#fn245" id="fnanch245">245</a></p></td> - <td></td> - <td><p>Fig. 59. Diagram of field of force with two similar - poles.</p></td></tr></table> -</div></div></div><!--dctr01--> - -<p>As we study these manifold phenomena, in the individual cases -of particular plants and animals, we recognise a close identity of -type, coupled with almost endless variation of specific detail; -and in particular, the order of succession in which certain of the -phenomena occur is variable and irregular. The precise order of -the phenomena, the time of longitudinal and of transverse fission -of the chromatin thread, of the break-up of the nuclear wall, and -so forth, will depend upon various minor contingencies and -“interferences.” And it is worthy of particular note that these -variations, in the order of events and in other subordinate details, -while doubtless attributable to specific physical conditions, would -seem to be without any obvious classificatory value or other -biological significance<a class="afnanch" href="#fn246" id="fnanch246">246</a>.</p> - -<hr class="hrblk"> - -<p>As regards the actual mechanical division of the cell into two -halves, we shall see presently that, in certain cases, such as that -of a long cylindrical filament, surface-tension, and what is known -as the principle of “minimal area,” go a long way to explain the -mechanical process of division; and in all cells whatsoever, the -process of division must somehow be explained as the result of -a conflict between surface-tension and its opposing forces. But -in such a case as our spherical cell, it is not very easy to see what -physical cause is at work to disturb its equilibrium and its integrity.</p> - -<p>The fact that, when actual division of the cell takes place, it -does so at right angles to the polar axis and precisely in the -direction of the equatorial plane, would lead us to suspect that -the new surface formed in the equatorial plane sets up an annular -tension, directed inwards, where it meets the outer surface layer -of the cell itself. But at this point, the problem becomes more -complicated. Before we could hope to comprehend it, we should -have not only to enquire into the potential distribution at the -surface of the cell in relation to that which we have seen to exist -in its interior, but we should probably also have to take account -of the differences of potential which the material arrangements -along the lines of force must themselves -tend to produce. Only <span class="xxpn" id="p191">{191}</span> -thus could we approach a comprehension of the balance of forces -which cohesion, friction, capillarity and electrical distribution -combine to set up.</p> - -<p>The manner in which we regard the phenomenon would seem -to turn, in great measure, upon whether or no we are justified in -assuming that, in the liquid surface-film of a minute spherical cell, -local, and symmetrically localised, differences of surface-tension -are likely to occur. If not, then changes in the conformation of -the cell such as lead immediately to its division must be ascribed -not to local changes in its surface-tension, but rather to direct -changes in internal pressure, or to mechanical forces due to an -induced surface-distribution of electrical potential.</p> - -<p>It has seemed otherwise to many writers, and we have a number -of theories of cell division which are all based directly on inequalities -or asymmetry of surface-tension. For instance, Bütschli -suggested, some forty years ago<a class="afnanch" href="#fn247" id="fnanch247">247</a>, -that cell division is brought -about by an increase of surface-tension in the equatorial region -of the cell. This explanation, however, can scarcely hold; for -it would seem that such an increase of surface-tension in the -equatorial plane would lead to the cell becoming flattened out into -a disc, with a sharply curved equatorial edge, and to a streaming -of material towards the equator. In 1895, Loeb shewed that the -streaming went on from the equator towards the divided nuclei, -and he supposed that the violence of these streaming movements -brought about actual division of the cell: a hypothesis which was -adopted by many other physiologists<a class="afnanch" href="#fn248" id="fnanch248">248</a>. -This streaming movement -would suggest, as Robertson has pointed out, a <i>diminution</i> -of surface-tension in the region of the equator. Now Quincke has -shewn that the formation of soaps at the surface of an oil-droplet -results in a diminution of the surface-tension of the latter; and -that if the saponification be local, that part of the surface tends to -spread. By laying a thread moistened with a dilute solution of -caustic alkali, or even merely smeared with soap, across a drop -of oil, Robertson has further shewn that the drop at once divides -into two: the edges of the drop, that is to say -the ends of the <span class="xxpn" id="p192">{192}</span> -diameter across which the thread lies, recede from the thread, -so forming a notch at each end of the diameter, while violent -streaming motions are set up at the surface, away from the thread -in the direction of the two opposite poles. Robertson<a class="afnanch" href="#fn249" id="fnanch249">249</a> -suggests, -accordingly, that the division of the cell is actually brought about -by a lowering of the equatorial surface-tension, and that this in -turn is due to a chemical action, such as a liberation of cholin, -or of soaps of cholin, through the splitting of lecithin in nuclear -synthesis.</p> - -<p>But purely chemical changes are not of necessity the fundamental -cause of alteration in the surface-tension of the egg, for -the action of electrolytes on surface-tension is now well known -and easily demonstrated. So, according to other views than -those with which we have been dealing, electrical charges are -sufficient in themselves to account for alterations of surface-tension; -while these in turn account for that protoplasmic -streaming which, as so many investigators agree, initiates the -segmentation of the egg<a class="afnanch" href="#fn250" id="fnanch250">250</a>. -A great part of our difficulty arises -from the fact that in such a case as this the various phenomena -are so entangled and apparently concurrent that it is hard to say -which initiates another, and to which this or that secondary -phenomenon may be considered due. Of recent years the phenomenon -of <i>adsorption</i> has been adduced (as we have already briefly -said) in order to account for many of the events and appearances -which are associated with the asymmetry, and lead towards the -division, of the cell. But our short discussion of this phenomenon -may be reserved for another chapter.</p> - -<p>However, we are not directly concerned here with the -phenomena of segmentation or cell division in themselves, except -only in so far as visible changes of form are capable of easy and -obvious correlation with the play of force. The very fact of -“development” indicates that, while it lasts, the equilibrium of -the egg is never complete<a class="afnanch" href="#fn251" id="fnanch251">251</a>. -And we may -simply conclude the <span class="xxpn" id="p193">{193}</span> -matter by saying that, if you have caryokinetic figures developing -inside the cell, that of itself indicates that the dynamic system -and the localised forces arising from it are in continual alteration; -and, consequently, changes in the outward configuration of the -system are bound to take place.</p> - -<hr class="hrblk"> - -<p>As regards the phenomena of fertilisation,—of the union of -the spermatozoon with the “pronucleus” of the egg,—we might -study these also in illustration, up to a certain point, of the -polarised forces which are manifestly at work. But we shall -merely take, as a single illustration, the paths of the male and -female pronuclei, as they travel to their ultimate meeting place.</p> - -<p>The spermatozoon, when within a very short distance of the -egg-cell, is attracted by it. Of the nature of this attractive force -we have no certain knowledge, though we would seem to have -a pregnant hint in Loeb’s discovery that, in the neighbourhood -of other substances, such even as a fragment, or bead, of glass, -the spermatozoon undergoes a similar attraction. But, whatever -the force may be, it is one acting normally to the surface of the -ovum, and accordingly, after entry, the sperm-nucleus points -straight towards the centre of the egg; from the fact that other -spermatozoa, subsequent to the first, fail to effect an entry, we -may safely conclude that an immediate consequence of the entry -of the spermatozoon is an increase in the surface-tension of the -egg<a class="afnanch" href="#fn252" id="fnanch252">252</a>. -Somewhere or other, near or far away, within the egg, lies -its own nuclear body, the so-called female pronucleus, and we -find after a while that this has fused with the head of the spermatozoon -(or male pronucleus), and that the body resulting from -their fusion has come to occupy the centre of the egg. This <i>must</i> -be due (as Whitman pointed out long ago) to a force of attraction -acting between the two bodies, and another force acting upon -one or other or both in the direction of the centre of the cell. -Did we know the magnitude of these several forces, it would be -a very easy task to calculate the precise path which the two -pronuclei would follow, leading to -conjugation and the central <span class="xxpn" id="p194">{194}</span> -position. As we do not know the magnitude, but only the direction, -of these forces we can only make a general statement: (1) the -paths of both moving bodies will lie wholly within a plane triangle -drawn between the two bodies and the centre of the cell; (2) unless -the two bodies happen to lie, to begin with, precisely on a diameter -of the cell, their paths until they meet one another will be curved -paths, the convexity of the curve being towards the straight line -joining the two bodies; (3) the two bodies will meet a little before -they reach the centre; and, having met and fused, will travel -on to reach the centre in a straight line. The actual study and -observation of the path followed is not very easy, owing to the -fact that what we usually see is not the path itself, but only a -<i>projection</i> of the path upon the plane of the microscope; but the -curved path is particularly well seen in the frog’s egg, where the -path of the spermatozoon is marked by a little streak of brown -pigment, and the fact of the meeting of the pronuclei before -reaching the centre has been repeatedly seen by many observers.</p> - -<p>The problem is nothing else than a particular case of the -famous problem of three bodies, which has so occupied the -astronomers; and it is obvious that the foregoing brief description -is very far from including all possible cases. Many of these are -particularly described in the works of Fol, Roux, Whitman and -others<a class="afnanch" href="#fn253" id="fnanch253">253</a>.</p> - -<hr class="hrblk"> - -<p>The intracellular phenomena of which we have now spoken -have assumed immense importance in biological literature and -discussion during the last forty years; but it is open to us to doubt -whether they will be found in the end to possess more than a -remote and secondary biological significance. Most, if not all of -them, would seem to follow immediately and inevitably from very -simple assumptions as to the physical constitution of the cell, and -from an extremely simple distribution of polarised forces within -it. We have already seen that how a thing grows, and what it -grows into, is a dynamic and not a merely material problem; so -far as the material substance is concerned, it -is so only by reason <span class="xxpn" id="p195">{195}</span> -of the chemical, electrical or other forces which are associated -with it. But there is another consideration which would lead us -to suspect that many features in the structure and configuration -of the cell are of very secondary biological importance; and that -is, the great variation to which these phenomena are subject in -similar or closely related organisms, and the apparent impossibility -of correlating them with the peculiarities of the organism as a -whole. “Comparative study has shewn that almost every detail -of the processes (of mitosis) described above is subject to variation -in different forms of cells<a class="afnanch" href="#fn254" id="fnanch254">254</a>.” -A multitude of cells divide to the -accompaniment of caryokinetic phenomena; but others do so -without any visible caryokinesis at all. Sometimes the polarised -field of force is within, sometimes it is adjacent to, and at other -times it lies remote from the nucleus. The distribution of potential -is very often symmetrical and bipolar, as in the case described; -but a less symmetrical distribution often occurs, with the result that -we have, for a time at least, numerous centres of force, instead -of the two main correlated poles: this is the simple explanation -of the numerous stellate figures, or “Strahlungen,” which have -been described in certain eggs, such as those of <i>Chaetopterus</i>. In -one and the same species of worm (<i>Ascaris megalocephala</i>), one -group or two groups of chromosomes may be present. And -remarkably constant, in general, as the number of chromosomes in -any one species undoubtedly is, yet we must not forget that, in -plants and animals alike, the whole range of observed numbers is -but a small one; for (as regards the germ-nuclei) few organisms -have less than six chromosomes, and fewer still have more than -sixteen<a class="afnanch" href="#fn255" id="fnanch255">255</a>. -In closely related animals, such as various species of -Copepods, and even in the same species of worm or insect, the -form of the chromosomes, and their arrangement in relation to -the nuclear spindle, have been found to differ in the various ways -alluded to above. In short, there seem to be strong grounds for -believing that these and many similar phenomena are in no way -specifically related to the particular organism -in which they have <span class="xxpn" id="p196">{196}</span> -been observed, and are not even specially and indisputably connected -with the organism as such. They include such manifestations -of the physical forces, in their various permutations and -combinations, as may also be witnessed, under appropriate -conditions, in non-living things.</p> - -<p>When we attempt to separate our purely morphological or -“purely embryological” studies from physiological and physical -investigations, we tend <i>ipso facto</i> to regard each particular structure -and configuration as an attribute, or a particular “character,” of -this or that particular organism. From this assumption we are -apt to go on to the drawing of new conclusions or the framing of -new theories as to the ancestral history, the classificatory position, -the natural affinities of the several organisms: in fact, to apply -our embryological knowledge mainly, and at times exclusively, to -the study of <i>phylogeny</i>. When we find, as we are not long of -finding, that our phylogenetic hypotheses, as drawn from embryology, -become complex and unwieldy, we are nevertheless -reluctant to admit that the whole method, with its fundamental -postulates, is at fault. And yet nothing short of this would -seem to be the case, in regard to the earlier phases at least of -embryonic development. All the evidence at hand goes, as it -seems to me, to shew that embryological data, prior to and even -long after the epoch of segmentation, are essentially a subject for -physiological and physical investigation and have but the very -slightest link with the problems of systematic or zoological -classification. Comparative embryology has its own facts to -classify, and its own methods and principles of classification. -Thus we may classify eggs according to the presence or absence, -the paucity or abundance, of their associated food-yolk, the -chromosomes according to their form and their number, the -segmentation according to its various “types,” radial, bilateral, -spiral, and so forth. But we have little right to expect, and in -point of fact we shall very seldom and (as it were) only accidentally -find, that these embryological categories coincide with the lines -of “natural” or “phylogenetic” classification which have been -arrived at by the systematic zoologist.</p> - -<hr class="hrblk"> - -<p>The cell, which Goodsir spoke of as a “centre of -force,” is in <span class="xxpn" id="p197">{197}</span> -reality a “sphere of action” of certain more or less localised -forces; and of these, surface-tension is the particular force which -is especially responsible for giving to the cell its outline and its -morphological individuality. The partially segmented differs from -the totally segmented egg, the unicellular Infusorian from the -minute multicellular Turbellarian, in the intensity and the range of -those surface-tensions which in the one case succeed and in the -other fail to form a visible separation between the “cells.” Adam -Sedgwick used to call attention to the fact that very often, even -in eggs that appear to be totally segmented, it is yet impossible -to discover an actual separation or cleavage, through and through -between the cells which on the surface of the egg are so clearly -delimited; so far and no farther have the physical forces effectuated -a visible “cleavage.” The vacuolation of the protoplasm in -<i>Actinophrys</i> or <i>Actinosphaerium</i> is due to localised surface-tensions, -quite irrespective of the multinuclear nature of the latter -organism. In short, the boundary walls due to surface-tension -may be present or may be absent with or without the delimination -of the other specific fields of force which are usually -correlated with these boundaries and with the independent -individuality of the cells. What we may safely admit, however, -is that one effect of these circumscribed fields of force is usually -such a separation or segregation of the protoplasmic constituents, -the more fluid from the less fluid and so forth, as to give a field -where surface-tension may do its work and bring a visible boundary -into being. When the formation of a “surface” is once effected, -its physical condition, or phase, will be bound to differ notably -from that of the interior of the cell, and under appropriate chemical -conditions the formation of an actual cell-wall, cellulose or other, -is easily intelligible. To this subject we shall return again, in -another chapter.</p> - -<p>From the moment that we enter on a dynamical conception -of the cell, we perceive that the old debates were in vain as to -what visible portions of the cell were active or passive, living or -non-living. For the manifestations of force can only be due to -the <i>interaction</i> of the various parts, to the transference of energy -from one to another. Certain properties may be manifested, -certain functions may be carried on, by -the protoplasm apart <span class="xxpn" id="p198">{198}</span> -from the nucleus; but the interaction of the two is necessary, -that other and more important properties or functions may be -manifested. We know, for instance, that portions of an Infusorian -are incapable of regenerating lost parts in the absence of a nucleus, -while nucleated pieces soon regain the specific form of the organism: -and we are told that reproduction by fission cannot be <i>initiated</i>, -though apparently all its later steps can be carried on, independently -of nuclear action. Nor, as Verworn pointed out, can the -nucleus possibly be regarded as the “sole vehicle of inheritance,” -since only in the conjunction of cell and nucleus do we find the -essentials of cell-life. “Kern und Protoplasma sind nur <i>vereint</i> -lebensfähig,” as Nussbaum said. Indeed we may, with E. B. -Wilson, go further, and say that “the terms ‘nucleus’ and ‘cell-body’ -should probably be regarded as only topographical expressions -denoting two differentiated areas in a common structural -basis.”</p> - -<p>Endless discussion has taken place regarding the centrosome, -some holding that it is a specific and essential structure, a permanent -corpuscle derived from a similar pre-existing corpuscle, a -“fertilising element” in the spermatozoon, a special “organ of -cell-division,” a material “dynamic centre” of the cell (as Van -Beneden and Boveri call it); while on the other hand, it is pointed -out that many cells live and multiply without any visible centrosomes, -that a centrosome may disappear and be created anew, -and even that under artificial conditions abnormal chemical -stimuli may lead to the formation of new centrosomes. We may -safely take it that the centrosome, or the “attraction sphere,” -is essentially a “centre of force,” and that this dynamic centre -may or may not be constituted by (but will be very apt to produce) -a concrete and visible concentration of matter.</p> - -<p>It is far from correct to say, as is often done, that the cell-wall, -or cell-membrane, belongs “to the passive products of protoplasm -rather than to the living cell itself”; or to say that in the animal -cell, the cell-wall, because it is “slightly developed,” is relatively -unimportant compared with the important role which it assumes -in plants. On the contrary, it is quite certain that, whether -visibly differentiated into a semi-permeable membrane, or merely -constituted by a liquid film, the surface of the cell is -the seat of <span class="xxpn" id="p199">{199}</span> -important forces, capillary and electrical, which play an essential -part in the dynamics of the cell. Even in the thickened, largely -solidified cellulose wall of the plant-cell, apart from the mechanical -resistances which it affords, the osmotic forces developed in connection -with it are of essential importance.</p> - -<p>But if the cell acts, after this fashion, as a whole, each part -interacting of necessity with the rest, the same is certainly true -of the entire multicellular organism: as Schwann said of old, in -very precise and adequate words, “the whole organism subsists -only by means of the <i>reciprocal action</i> of the single elementary -parts<a class="afnanch" href="#fn256" id="fnanch256">256</a>.”</p> - -<p>As Wilson says again, “the physiological autonomy of the -individual cell falls into the background ... and the apparently -composite character which the multicellular organism may exhibit -is owing to a secondary distribution of its energies among local -centres of action<a class="afnanch" href="#fn257" id="fnanch257">257</a>.”</p> - -<p>It is here that the homology breaks down which is so often -drawn, and overdrawn, between the unicellular organism and the -individual cell of the metazoon<a class="afnanch" href="#fn258" id="fnanch258">258</a>.</p> - -<p>Whitman, Adam Sedgwick<a class="afnanch" href="#fn259" id="fnanch259">259</a>, -and others have lost no -opportunity of warning us against a too literal acceptation -of the cell-theory, against the view that the multicellular -organism is a colony (or, as Haeckel called it (in the case -of the plant), a “republic”) of independent units of life<a class="afnanch" href="#fn260" id="fnanch260">260</a>. -As Goethe said long ago, “Das lebendige -ist zwar in Elemente <span class="xxpn" id="p200">{200}</span> -zerlegt, aber man kann es aus diesen nicht wieder zusammenstellen -und beleben;” the dictum of the <i>Cellularpathologie</i> being just -the opposite, “Jedes Thier erscheint als eine Summe vitaler -Einheiten, von denen <i>jede den vollen Charakter des Lebens an -sich trägt</i>.”</p> - -<p>Hofmeister and Sachs have taught us that in the plant the -growth of the mass, the growth of the organ, is the primary fact, -that “cell formation is a phenomenon very general in organic -life, but still only of secondary significance.” “Comparative -embryology” says Whitman, “reminds us at every turn that the -organism dominates cell-formation, using for the same purpose -one, several, or many cells, massing its material and directing its -movements and shaping its organs, as if cells did not exist<a class="afnanch" href="#fn261" id="fnanch261">261</a>.” -So Rauber declared that, in the whole world of organisms, “das -Ganze liefert die Theile, nicht die Theile das Ganze: letzteres -setzt die Theile zusammen, nicht diese jenes<a class="afnanch" href="#fn262" id="fnanch262">262</a>.” -And on the -botanical side De Bary has summed up the matter in an aphorism, -“Die Pflanze bildet Zellen, nicht die Zelle bildet Pflanzen.”</p> - -<p>Discussed almost wholly from the concrete, or morphological -point of view, the question has for the most part been made to turn -on whether actual protoplasmic continuity can be demonstrated -between one cell and another, whether the organism be an actual -reticulum, or syncytium. But from the dynamical point of view -the question is much simpler. We then deal not with material -continuity, not with little bridges of connecting protoplasm, but -with a continuity of forces, a comprehensive field of force, which -runs through and through the entire organism and is by no means -restricted in its passage to a protoplasmic continuum. And such -a continuous field of force, somehow shaping the whole organism, -independently of the number, magnitude and form of the individual -cells, which enter, like a froth, into its fabric, seems to me certainly -and obviously to exist. As Whitman says, “the fact that physiological -unity is not broken by cell-boundaries is confirmed in so -many ways that it must be accepted as one of the fundamental -truths of biology<a class="afnanch" href="#fn263" id="fnanch263">263</a>.”</p> - -<div class="chapter" id="p201"> -<h2 class="h2herein" title="V. The Forms of Cells">CHAPTER V -<span class="h2ttl"> -THE FORMS OF CELLS</span></h2></div> - -<p>Protoplasm, as we have already said, is a fluid or rather a -semifluid substance, and we need not pause here to attempt to -describe the particular properties of the semifluid, colloid, or -jelly-like substances to which it is allied; we should find it no -easy matter. Nor need we appeal to precise theoretical definitions -of fluidity, lest we come into a debateable land. It is in the most -general sense that protoplasm is “fluid.” As Graham said (of -colloid matter in general), “its softness <i>partakes of fluidity</i>, and -enables the colloid to become a vehicle for liquid diffusion, like -water itself<a class="afnanch" href="#fn264" id="fnanch264">264</a>.” -When we can deal with protoplasm in sufficient -quantity we see it flow; particles move freely through it, air-bubbles -and liquid droplets shew round or spherical within it; -and we shall have much to say about other phenomena manifested -by its own surface, which are those especially characteristic of -liquids. It may encompass and contain solid bodies, and it may -“secrete” within or around itself solid substances; and very -often in the complex living organism these solid substances -formed by the living protoplasm, like shell or nail or horn or -feather, may remain when the protoplasm which formed them -is dead and gone; but the protoplasm itself is fluid or semifluid, -and accordingly permits of free (though not necessarily rapid) -<i>diffusion</i> and easy <i>convection</i> of particles within itself. This simple -fact is of elementary importance in connection with form, and -with what appear at first sight to be common characteristics or -peculiarities of the forms of living things.</p> - -<p>The older naturalists, in discussing the differences between -inorganic and organic bodies, laid stress upon the fact or statement -that the former grow by “agglutination,” -and the latter by <span class="xxpn" id="p202">{202}</span> -what they termed “intussusception.” The contrast is true, -rather, of solid as compared with jelly-like bodies of all kinds, -living or dead, the great majority of which as it so happens, but -by no means all, are of organic origin.</p> - -<p>A crystal “grows” by deposition of new molecules, one by -one and layer by layer, superimposed or aggregated upon the -solid substratum already formed. Each particle would seem to -be influenced, practically speaking, only by the particles in its -immediate neighbourhood, and to be in a state of freedom and -independence from the influence, either direct or indirect, of its -remoter neighbours. As Lord Kelvin and others have explained -the formation and the resulting forms of crystals, so we believe -that each added particle takes up its position in relation to its -immediate neighbours already arranged, generally in the holes and -corners that their arrangement leaves, and in closest contact with -the greatest number<a class="afnanch" href="#fn265" id="fnanch265">265</a>. -And hence we may repeat or imitate this -process of arrangement, with great or apparently even with -precise accuracy (in the case of the simpler crystalline systems), -by piling up spherical pills or grains of shot. In so doing, we must -have regard to the fact that each particle must drop into the -place where it can go most easily, or where no easier place offers. -In more technical language, each particle is free to take up, and -does take up, its position of least potential energy relative to those -already deposited; in other words, for each particle motion is -induced until the energy of the system is so distributed that no -tendency or resultant force remains to move it more. The -application of this principle has been shewn to lead to the production -of <i>planes</i><a class="afnanch" href="#fn266" id="fnanch266">266</a> -(in all cases where by the limitation of material, -surfaces <i>must</i> occur); and where we have planes, straight edges -and solid angles must obviously also occur; -and, if equilibrium is <span class="xxpn" id="p203">{203}</span> -to follow, must occur symmetrically. Our piling up of shot, or -manufacture of mimic crystals, gives us visible demonstration -that the result is actually to obtain, as in the natural crystal, -plane surfaces and sharp angles, symmetrically disposed.</p> - -<p>But the living cell grows in a totally different way, very much -as a piece of glue swells up in water, by “imbibition,” or by interpenetration -into and throughout its entire substance. The semifluid -colloid mass takes up water, partly to combine chemically -with its individual molecules<a class="afnanch" href="#fn267" id="fnanch267">267</a>, -partly by physical diffusion into -the interstices between these molecules, and partly, as it would -seem, in other ways; so that the entire phenomenon is a very -complex and even an obscure one. But, so far as we are concerned, -the net result is a very simple one. For the equilibrium or -tendency to equilibrium of fluid pressure in all parts of its interior -while the process of imbibition is going on, the constant rearrangement -of its fluid mass, the contrast in short with the crystalline -method of growth where each particle comes to rest to move -(relatively to the whole) no more, lead the mass of jelly to swell -up, very much as a bladder into which we blow air, and so, by -a <i>graded</i> and harmonious distribution of forces, to assume everywhere -a rounded and more or less bubble-like external form<a class="afnanch" href="#fn268" id="fnanch268">268</a>. -So, when the same school of older naturalists called attention to -a new distinction or contrast of form between the organic and -inorganic objects, in that the contours of the former tended to -roundness and curvature, and those of the latter to be bounded -by straight lines, planes and sharp angles, we see that this contrast -was not a new and different one, but only another aspect of -their former statement, and an immediate consequence of the -difference between the processes of agglutination and intussusception.</p> - -<p>This common and general contrast between the form of the -crystal on the one hand, and of the colloid or of the organism on -the other, must by no means be pressed -too far. For Lehmann, <span class="xxpn" id="p204">{204}</span> -in his great work on so-called Fluid Crystals<a class="afnanch" href="#fn269" id="fnanch269">269</a>, -to which we shall -afterwards return, has shewn how, under certain circumstances, -surface-tension phenomena may coexist with crystallisation, and -produce a form of minimal potential which is a resultant of both: -the fact being that the bonds maintaining the crystalline arrangement -are now so much looser than in the solid condition that the -tendency to least total surface-area is capable of being satisfied. -Thus the phenomenon of “liquid crystallisation” does not destroy -the distinction between crystalline and colloidal forms, but gives -added unity and continuity to the whole series of phenomena<a class="afnanch" href="#fn270" id="fnanch270">270</a>. -Lehmann has also demonstrated phenomena within the crystal, -known for instance as transcrystallisation, which shew us that we -must not speak unguardedly of the growth of crystals as limited -to deposition upon a surface, and Bütschli has already pointed out -the possible great importance to the biologist of the various -phenomena which Lehmann has described<a class="afnanch" href="#fn271" id="fnanch271">271</a>.</p> - -<p>So far then, as growth goes on, unaffected by pressure or other -external force, the fluidity of protoplasm, its mobility internal -and external, and the manner in which particles move with -comparative freedom from place to place within, all manifestly -tend to the production of swelling, rounded surfaces, and to their -great predominance over plane surfaces in the contour of the -organism. These rounded contours will tend to be preserved, for -a while, in the case of naked protoplasm by its viscosity, and in -the presence of a cell-wall by its very lack of fluidity. In a general -way, the presence of curved boundary surfaces will be especially -obvious in the unicellular organisms, and still more generally in -the <i>external</i> forms of all organisms; and wherever mutual pressure -between adjacent cells, or other adjacent parts, has not come into -play to flatten the rounded surfaces into planes.</p> - -<p>But the rounded contours that are -assumed and exhibited by <span class="xxpn" id="p205">{205}</span> -a piece of hard glue, when we throw it into water and see it expand -as it sucks the water up, are not nearly so regular or so beautiful -as are those which appear when we blow a bubble, or form a -drop, or pour water into a more or less elastic bag. For these -curving contours depend upon the properties of the bag itself, -of the film or membrane that contains the mobile gas, or that -contains or bounds the mobile liquid mass. And hereby, in the -case of the fluid or semifluid mass, we are introduced to the -subject of <i>surface tension</i>: of which indeed we have spoken in -the preceding chapter, but which we must now examine with -greater care.</p> - -<hr class="hrblk"> - -<p>Among the forces which determine the forms of cells, whether -they be solitary or arranged in contact with one another, this -force of surface-tension is certainly of great, and is probably of -paramount importance. But while we shall try to separate out -the phenomena which are directly due to it, we must not forget -that, in each particular case, the actual conformation which we -study may be, and usually is, the more or less complex resultant -of surface tension acting together with gravity, mechanical -pressure, osmosis, or other physical forces.</p> - -<p>Surface tension is that force by which we explain the form of -a drop or of a bubble, of the surfaces external and internal of -a “froth” or collocation of bubbles, and of many other things of -like nature and in like circumstances<a class="afnanch" href="#fn272" id="fnanch272">272</a>. -It is a property of liquids -(in the sense at least with which our subject is concerned), and it -is manifested at or very near the surface, where the liquid comes -into contact with another liquid, a solid or a gas. We note here -that the term <i>surface</i> is to be interpreted in a wide sense; for -wherever we have solid particles imbedded in a fluid, wherever -we have a non-homogeneous fluid or semi-fluid -such as a particle <span class="xxpn" id="p206">{206}</span> -of protoplasm, wherever we have the presence of “impurities,” as -in a mass of molten metal, there we have always to bear in mind -the existence of “surfaces” and of surface tensions, not only -on the exterior of the mass but also throughout its interstices, -wherever like meets unlike.</p> - -<p>Surface tension is due to molecular force, to force that is to -say arising from the action of one molecule upon another, and it -is accordingly exerted throughout a small thickness of material, -comparable to the range of the molecular forces. We imagine -that within the interior of the liquid mass such molecular interactions -negative one another: but that at and near the free -surface, within a layer or film approximately equal to the range -of the molecular force, there must be a lack of such equilibrium -and consequently a manifestation of force.</p> - -<p>The action of the molecular forces has been variously explained. -But one simple explanation (or mode of statement) is that the -molecules of the surface layer (whose thickness is definite and -constant) are being constantly attracted into the interior by those -which are more deeply situated, and that consequently, as -molecules keep quitting the surface for the interior, the bulk of -the latter increases while the surface diminishes; and the process -continues till the surface itself has become a minimum, the <i>surface-shrinkage</i> -exhibiting itself as a <i>surface-tension</i>. This is a sufficient -description of the phenomenon in cases where a portion of liquid -is subject to no other than <i>its own molecular forces</i>, and (since the -sphere has, of all solids, the smallest surface for a given volume) -it accounts for the spherical form of the raindrop, of the grain -of shot, or of the living cell in many simple organisms. It accounts -also, as we shall presently see, for a great number of much more -complicated forms, manifested under less simple conditions.</p> - -<p>Let us here briefly note that surface tension is, in itself, a -comparatively small force, and easily measurable: for instance -that of water is equivalent to but a few grains per linear inch, -or a few grammes per metre. But this small tension, when it -exists in a <i>curved</i> surface of very great curvature, gives rise to a -very great pressure directed towards the centre of curvature. We -can easily calculate this pressure, and so satisfy ourselves that, -when the radius of curvature is of -molecular dimensions, the <span class="xxpn" id="p207">{207}</span> -pressure is of the magnitude of thousands of atmospheres,—a conclusion -which is supported by other physical considerations.</p> - -<p>The contraction of a liquid surface and other phenomena of -surface tension involve the doing of work, and the power to do -work is what we call energy. It is obvious, in such a simple case -as we have just considered, that the whole energy of the system -is diffused throughout its molecules; but of this whole stock of -energy it is only that part which comes into play at or very near -to the surface which normally manifests itself in work, and hence -we may speak (though the term is open to some objections) of -a specific <i>surface energy</i>. The consideration of surface energy, -and of the manner in which its amount is increased and multiplied -by the multiplication of surfaces due to the subdivision of the -organism into cells, is of the highest importance to the physiologist; -and even the morphologist cannot wholly pass it by, if he desires -to study the form of the cell in its relation to the phenomena of -surface tension or “capillarity.” The case has been set forth with -the utmost possible lucidity by Tait and by Clerk Maxwell, on -whose teaching the following paragraphs are based: they having -based their teaching upon that of Gauss,—who rested on Laplace.</p> - -<p>Let <i>E</i> be the whole potential energy of a mass <i>M</i> of liquid; -let <i>e</i><sub>0</sub> be the energy per unit mass of the interior liquid (we may -call it the <i>internal energy</i>); and let <i>e</i> be the energy per unit mass -for a layer of the skin, of surface <i>S</i>, of thickness <i>t</i>, and density -ρ (<i>e</i> being what we call the <i>surface energy</i>). It is obvious that the -total energy consists of the internal <i>plus</i> the surface energy, and -that the former is distributed through the whole mass, minus its -surface layers. That is to say, in mathematical language,</p> - -<div class="dmaths"> -<div><i>E</i> -= (<i>M</i> − <i>S</i> · Σ <i>t</i> ρ) <i>e</i><sub>0</sub> + <i>S</i> · Σ <i>t</i> ρ <i>e</i> . -</div> - -<p class="pcontinue">But this is equivalent to writing:</p> - -<div>= <i>M e</i><sub>0</sub> + <i>S</i> · Σ <i>t</i> ρ(<i>e</i> − <i>e</i><sub>0</sub>) ; -</div></div><!--dmaths--> - -<p class="pcontinue">and this is as much as to say that the total energy of the system -may be taken to consist of two portions, one uniform throughout -the whole mass, and another, which is proportional on the one hand -to the amount of surface, and on the other hand is proportional -to the difference between <i>e</i> and <i>e</i><sub>0</sub> , that is to say to the difference -between the unit values of the internal and -the surface energy. <span class="xxpn" id="p208">{208}</span></p> - -<p>It was Gauss who first shewed after this fashion how, from -the mutual attractions between all the particles, we are led to an -expression which is what we now call the <i>potential energy</i> of the -system; and we know, as a fundamental theorem of dynamics, -that the potential energy of the system tends to a minimum, and -in that minimum finds, as a matter of course, its stable equilibrium.</p> - -<hr class="hrblk"> - -<p>We see in our last equation that the term <span class="nowrap"> -<i>M e</i><sub>0</sub></span> is irreducible, -save by a reduction of the mass itself. But the other term may -be diminished (1) by a reduction in the area of surface, <i>S</i>, or -(2) by a tendency towards equality of <i>e</i> and <i>e</i><sub>0</sub> , that is to say by -a diminution of the specific surface energy, <i>e</i>.</p> - -<p>These then are the two methods by which the energy of the -system will manifest itself in work. The one, which is much the -more important for our purposes, leads always to a diminution of -surface, to the so-called “principle of minimal areas”; the other, -which leads to the lowering (under certain circumstances) of -surface tension, is the basis of the theory of Adsorption, to which -we shall have some occasion to refer as the <i>modus operandi</i> in the -development of a cell-wall, and in a variety of other histological -phenomena. In the technical phraseology of the day, the -“capacity factor” is involved in the one case, and the “intensity -factor” in the other.</p> - -<p>Inasmuch as we are concerned with the form of the cell it is -the former which becomes our main postulate: telling us that -the energy equations of the surface of a cell, or of the free surfaces -of cells partly in contact, or of the partition-surfaces of cells in -contact with one another or with an adjacent solid, all indicate -a minimum of potential energy in the system, by which the system -is brought, <i>ipso facto</i>, into equilibrium. And we shall not fail to -observe, with something more than mere historical interest and -curiosity, how deeply and intrinsically there enter into this whole -class of problems the “principle of least action” of Maupertuis, -the “<i>lineae curvae maximi minimive proprietate gaudentes</i>” of -Euler, by which principles these old natural philosophers explained -correctly a multitude of phenomena, and drew the lines whereon -the foundations of great part of modern physics are well and -truly laid. <span class="xxpn" id="p209">{209}</span></p> - -<p>In all cases where the principle of maxima and minima comes -into play, as it conspicuously does in the systems of liquid films -which are governed by the laws of surface-tension, the figures and -conformations produced are characterised by obvious and remarkable -<i>symmetry</i>. Such symmetry is in a high degree characteristic -of organic forms, and is rarely absent in living things,—save in such -cases as amoeba, where the equilibrium on which symmetry depends -is likewise lacking. And if we ask what physical equilibrium has -to do with formal symmetry and regularity, the reason is not far -to seek; nor can it be put better than in the following words of -Mach’s<a class="afnanch" href="#fn273" id="fnanch273">273</a>. -“In every symmetrical system every deformation that -tends to destroy the symmetry is complemented by an equal and -opposite deformation that tends to restore it. In each deformation -positive and negative work is done. One condition, therefore, -though not an absolutely sufficient one, that a maximum or -minimum of work corresponds to the form of equilibrium, is thus -supplied by symmetry. Regularity is successive symmetry. -There is no reason, therefore, to be astonished that the forms of -equilibrium are often symmetrical and regular.”</p> - -<hr class="hrblk"> - -<p>As we proceed in our enquiry, and especially when we approach -the subject of <i>tissues</i>, or agglomerations of cells, we shall have -from time to time to call in the help of elementary mathematics. -But already, with very little mathematical help, we find ourselves -in a position to deal with some simple examples of organic forms.</p> - -<p>When we melt a stick of sealing-wax in the flame, surface -tension (which was ineffectively present in the solid but finds play -in the now fluid mass), rounds off its sharp edges into curves, so -striving towards a surface of minimal area; and in like manner, -by melting the tip of a thin rod of glass, Leeuwenhoek made the -little spherical beads which served him for a microscope<a class="afnanch" href="#fn274" id="fnanch274">274</a>. -When -any drop of protoplasm, either over all its surface or at some free -end, as at the extremity of the pseudopodium -of an amoeba, is <span class="xxpn" id="p210">{210}</span> -seen likewise to “round itself off,” that is not an effect of “vital -contractility,” but (as Hofmeister shewed so long ago as 1867) -a simple consequence of surface tension; and almost immediately -afterwards Engelmann<a class="afnanch" href="#fn275" id="fnanch275">275</a> -argued on the same lines, that the forces -which cause the contraction of protoplasm in general may “be -just the same as those which tend to make every non-spherical -drop of fluid become spherical!” We are not concerned here with -the many theories and speculations which would connect the -phenomena of surface tension with contractility, muscular movement -or other special <i>physiological</i> functions, but we find ample -room to trace the operation of the same cause in producing, under -conditions of rest and equilibrium, certain definite and inevitable -forms of surface.</p> - -<p>It is however of great importance to observe that the living -cell is one of those cases where the phenomena of surface tension -are by no means limited to the <i>outer</i> surface; for within the -heterogeneous substance of the cell, between the protoplasm and -its nuclear and other contents, and in the alveolar network of the -cytoplasm itself (so far as that “alveolar structure” is actually -present in life), we have a multitude of interior surfaces; and, -especially among plants, we may have a large inner surface of -“interfacial” contact, where the protoplasm contains cavities -or “vacuoles” filled with a different and more fluid material, the -“cell-sap.” Here we have a great field for the development of -surface tension phenomena: and so long ago as 1865, Nägeli and -Schwendener shewed that the streaming currents of plant cells -might be very plausibly explained by this phenomenon. Even -ten years earlier, Weber had remarked upon the resemblance -between these protoplasmic streamings and the streamings to be -observed in certain inanimate drops, for which no cause but -surface tension could be assigned<a class="afnanch" href="#fn276" id="fnanch276">276</a>.</p> - -<p>The case of amoeba, though it is an elementary case, is at the -same time a complicated one. While it remains “amoeboid,” it -is never at rest or in equilibrium; it is always moving, from one -to another of its protean changes of configuration; its surface -tension is constantly varying from point -to point. Where the <span class="xxpn" id="p211">{211}</span> -surface tension is greater, that portion of the surface will contract -into spherical or spheroidal forms; where it is less the surface -will correspondingly extend. While generally speaking the surface -energy has a minimal value, it is not necessarily constant. It may -be diminished by a rise of temperature; it may be altered by -contact with adjacent substances<a class="afnanch" href="#fn277" id="fnanch277">277</a>, -by the transport of constituent -materials from the interior to the surface, or again by actual -chemical and fermentative change. Within the cell, the surface -energies developed about its heterogeneous contents will constantly -vary as these contents are affected by chemical metabolism. As -the colloid materials are broken down and as the particles in -suspension are diminished in size the “free surface energy” -will be increased, but the osmotic energy will be diminished<a class="afnanch" href="#fn278" id="fnanch278">278</a>. -Thus arise the various fluctuations of surface tension and the -various phenomena of amoeboid form and motion, which Bütschli -and others have reproduced or imitated by means of the fine -emulsions which constitute their “artificial amoebae.” A multitude -of experiments shew how extraordinarily delicate is the -adjustment of the surface tension forces, and how sensitive they -are to the least change of temperature or chemical state. Thus, -on a plate which we have warmed at one side, a drop of alcohol -runs towards the warm area, a drop of oil away from it; and a -drop of water on the glass plate exhibits -lively movements when <span class="xxpn" id="p212">{212}</span> -we bring into its neighbourhood a heated wire, or a glass rod -dipped in ether. When we find that a plasmodium of Aethalium, -for instance, creeps towards a damp spot, or towards a warm spot, -or towards substances that happen to be nutritious, and again -creeps away from solutions of sugar or of salt, we seem to be -dealing with phenomena every one of which can be paralleled by -ordinary phenomena of surface tension<a class="afnanch" href="#fn279" id="fnanch279">279</a>. -Even the soap-bubble -itself is imperfectly in equilibrium, for the reason that its film, -like the protoplasm of amoeba or Aethalium, is an excessively -heterogeneous substance. Its surface tensions vary from point -to point, and chemical changes and changes of temperature -increase and magnify the variation. The whole surface of the -bubble is in constant movement as the concentrated portions of -the soapy fluid make their way outwards from the deeper layers; -it thins and it thickens, its colours change, currents are set up in -it, and little bubbles glide over it; it continues in this state of -constant movement, as its parts strive one with another in all -their interactions towards equilibrium<a class="afnanch" href="#fn280" id="fnanch280">280</a>.</p> - -<p>In the case of the naked protoplasmic cell, as the amoeboid -phase is emphatically a phase of freedom and activity, of chemical -and physiological change, so, on the other hand, is the spherical -form indicative of a phase of rest or comparative inactivity. In -the one phase we see unequal surface tensions manifested in the -creeping movements of the amoeboid body, in the rounding off -of the ends of the pseudopodia, in the flowing out of its substance -over a particle of “food,” and in the current-motions in the interior -of its mass; till finally, in the other phase, when internal homogeneity -and equilibrium have been attained -and the potential <span class="xxpn" id="p213">{213}</span> -energy of the system is for the time being at a minimum, the -cell assumes a rounded or spherical form, passing into a state -of “rest,” and (for a reason which we shall presently see) -becoming at the same time “encysted.”</p> - -<div class="dright dwth-h" id="fig60"> -<img src="images/i213.png" width="200" height="216" alt=""> - <div class="dcaption">Fig. 60.</div></div> - -<p>In a budding yeast-cell (Fig. <a href="#fig60" title="go to Fig. 60">60</a>), we see a more definite and -restricted change of surface tension. When a “bud” appears, -whether with or without actual growth by osmosis -or otherwise of the mass, it does so because at a -certain part of the cell-surface the surface tension -has more or less suddenly diminished, and the -area of that portion expands accordingly; but in -turn the surface tension of the expanded area will -make itself felt, and the bud will be rounded off -into a more or less spherical form.</p> - -<p>The yeast-cell with its bud is a simple example of a principle -which we shall find to be very important. Our whole treatment -of cell-form in relation to surface-tension depends on the fact -(which Errera was the first to point out, or to give clear expression -to) that the <i>incipient</i> cell-wall retains with but little impairment -the properties of a liquid film<a class="afnanch" href="#fn281" id="fnanch281">281</a>, -and that the growing cell, in spite -of the membrane by which it has already begun to be surrounded, -behaves very much like a fluid drop. But even the ordinary -yeast-cell shows, by its ovoid and non-spherical form, that it has -acquired its shape under the influence of some force other than -that uniform and symmetrical surface-tension which would be -productive of a sphere; and this or any other asymmetrical form, -once acquired, may be retained by virtue of the solidification and -consequent rigidity of the membranous wall of the cell. Unless -such rigidity ensue, it is plain that such a conformation as that of -the cell with its attached bud could not be long retained, amidst -the constantly varying conditions, as a figure of even partial -equilibrium. But as a matter of fact, the cell in this case is not -in equilibrium at all; it is in <i>process</i> of budding, and is slowly -altering its shape by rounding off the bud. It is plain that over -its surface the surface-energies are unequally distributed, owing -to some heterogeneity of the substance; and to this matter we -shall afterwards return. In like manner the -developing egg <span class="xxpn" id="p214">{214}</span> -through all its successive phases of form is never in complete -equilibrium; but is merely responding to constantly changing -conditions, by phases of partial, transitory, unstable and conditional -equilibrium.</p> - -<p>It is obvious that there are innumerable solitary plant-cells, -and unicellular organisms in general, which, like the yeast-cell, do -not correspond to any of the simple forms that may be generated -under the influence of simple and homogeneous surface-tension; -and in many cases these forms, which we should expect to be -unstable and transitory, have become fixed and stable by reason -of the comparatively sudden or rapid solidification of the envelope. -This is the case, for instance, in many of the more complicated forms -of diatoms or of desmids, where we are dealing, in a less striking -but even more curious way than in the budding yeast-cell, not -with one simple act of formation, but with a complicated result -of successive stages of localised growth, interrupted by phases of -partial consolidation. The original cell has acquired or assumed -a certain form, and then, under altering conditions and new -distributions of energy, has thickened here or weakened there, -and has grown out or tended (as it were) to branch, at particular -points. We can often, or indeed generally, trace in each particular -stage of growth or at each particular temporary growing point, -the laws of surface tension manifesting themselves in what is -for the time being a fluid surface; nay more, even in the adult -and completed structure, we have little difficulty in tracing and -recognising (for instance in the outline of such a desmid as Euastrum) -the rounded lobes that have successively grown or flowed -out from the original rounded and flattened cell. What we see in -a many chambered foraminifer, such as Globigerina or Rotalia, is -just the same thing, save that it is carried out in greater completeness -and perfection. The little organism as a whole is not a figure -of equilibrium or of minimal area; but each new bud or separate -chamber is such a figure, conditioned by the forces of surface -tension, and superposed upon the complex aggregate of similar -bubbles after these latter have become consolidated one by one -into a rigid system.</p> - -<hr class="hrblk"> - -<p>Let us now make some enquiry regarding -the various forms <span class="xxpn" id="p215">{215}</span> -which, under the influence of surface tension, a surface can possibly -assume. In doing so, we are obviously limited to conditions -under which other forces are relatively unimportant, that is to -say where the “surface energy” is a considerable fraction of -the whole energy of the system; and this in general will be -the case when we are dealing with portions of liquid so small -that their dimensions come within what we have called the -molecular range, or, more generally, in which the “specific -surface” is large<a class="afnanch" href="#fn282" id="fnanch282">282</a>: -in other words it will be small or minute -organisms, or the small cellular elements of larger organisms, -whose forms will be governed by surface-tension; while the -general forms of the larger organisms will be due to other and -non-molecular forces. For instance, a large surface of water sets -itself level because here gravity is predominant; but the surface -of water in a narrow tube is manifestly curved, for the reason -that we are here dealing with particles which are mutually within -the range of each other’s molecular forces. The same is the case -with the cell-surfaces and cell-partitions which we are presently -to study, and the effect of gravity will be especially counteracted -and concealed when, as in the case of protoplasm in a watery -fluid, the object is immersed in a liquid of nearly its own specific -gravity.</p> - -<p>We have already learned, as a fundamental law of surface-tension -phenomena, that a liquid film <i>in equilibrium</i> assumes a -form which gives it a minimal area under the conditions to which -it is subject. And these conditions include (1) the form of the -boundary, if such exist, and (2) the pressure, if any, to which the -film is subject; which pressure is closely related to the volume, -of air or of liquid, which the film (if it be a closed one) may have -to contain. In the simplest of cases, when we take up a soap-film -on a plane wire ring, the film is exposed to equal atmospheric -pressure on both sides, and it obviously has its minimal area in -the form of a plane. So long as our wire ring lies in one plane -(however irregular in outline), the film stretched across it will -still be in a plane; but if we bend the ring so that it lies no longer -in a plane, then our film will become curved into a surface which -may be extremely complicated, but is still -the smallest possible <span class="xxpn" id="p216">{216}</span> -surface which can be drawn continuously across the uneven -boundary.</p> - -<p>The question of pressure involves not only external pressures -acting on the film, but also that which the film itself is capable -of exerting. For we have seen that the film is always contracting -to its smallest limits; and when the film is curved, this obviously -leads to a pressure directed inwards,—perpendicular, that is to -say, to the surface of the film. In the case of the soap-bubble, -the uniform contraction of whose surface has led to its spherical -form, this pressure is balanced by the pressure of the air within; -and if an outlet be given for this air, then the bubble contracts -with perceptible force until it stretches across the mouth of the -tube, for instance the mouth of the pipe through which we have -blown the bubble. A precisely similar pressure, directed inwards, -is exercised by the surface layer of a drop of water or a globule -of mercury, or by the surface pellicle on a portion or “drop” of -protoplasm. Only we must always remember that in the soap-bubble, -or the bubble which a glass-blower blows, there is a twofold -pressure as compared with that which the surface-film exercises -on the drop of liquid of which it is a part; for the bubble consists -(unless it be so thin as to consist of a mere layer of molecules<a class="afnanch" href="#fn283" id="fnanch283">283</a>) -of a liquid layer, with a free surface within and another without, -and each of these two surfaces exercises its own independent and -coequal tension, and corresponding pressure<a class="afnanch" href="#fn284" id="fnanch284">284</a>.</p> - -<p>If we stretch a tape upon a flat table, whatever be the tension -of the tape it obviously exercises no pressure upon the table -below. But if we stretch it over a <i>curved</i> surface, a cylinder for -instance, it does exercise a downward pressure; and the more -curved the surface the greater is this pressure, that is to say the -greater is this share of the entire force of tension which is resolved -in the downward direction. In mathematical language, the -pressure (<i>p</i>) varies directly as the tension (<i>T</i>), and inversely as -the radius of curvature (<i>R</i>): that is to say, <i>p</i> -= <i>T ⁄ R</i>, per unit of -surface. <span class="xxpn" id="p217">{217}</span></p> - -<p>If instead of a cylinder, which is curved only in one -direction, we take a case where there are curvatures in two -dimensions (as for instance a sphere), then the effects of -these must be simply added to one another, and the resulting -pressure <i>p</i> is equal to <i>T ⁄ R</i> + <i>T ⁄ R′</i> -or <i>p</i> -= <i>T</i>(1 ⁄ <i>R</i> + 1 ⁄ <i>R′</i>)<a -class="afnanch" href="#fn285" id="fnanch285" title="go to note -285">*</a>.</p> - -<p>And if in addition to the pressure <i>p</i>, which is due to surface -tension, we have to take into account other pressures, <i>p′</i>, <i>p″</i>, etc., -which are due to gravity or other forces, then we may say that -the <i>total pressure</i>, <i>P</i> -= <i>p′</i> + <i>p″</i> + <i>T</i>(1 ⁄ <i>R</i> + 1 ⁄ <i>R′</i>). While in some -cases, for instance in speaking of the shape of a bird’s egg, we -shall have to take account of these extraneous pressures, in the -present part of our subject we shall for the most part be able to -neglect them.</p> - -<p>Our equation is an equation of equilibrium. The resistance -to compression,—the pressure outwards,—of our fluid mass, is a -constant quantity (<i>P</i>); the pressure inwards, <i>T</i>(1 ⁄ <i>R</i> + 1 ⁄ <i>R′</i>), is -also constant; and if (unlike the case of the mobile amoeba) the -surface be homogeneous, so that <i>T</i> is everywhere equal, it follows -that throughout the whole surface 1 ⁄ <i>R</i> + 1 ⁄ <i>R′</i> -= <i>C</i> (a constant).</p> - -<p>Now equilibrium is attained after the surface contraction has -done its utmost, that is to say when it has reduced the surface -to the smallest possible area; and so we arrive, from the physical -side, at the conclusion that a surface such that 1 ⁄ <i>R</i> + 1 ⁄ <i>R′</i> -= <i>C</i>, -in other words a surface which has the same <i>mean curvature</i> at -all points, is equivalent to a surface of minimal area: and to the -same conclusion we may also arrive through purely analytical -mathematics. It is obvious that the plane and the sphere are two -examples of such surfaces, for in both cases the radius of curvature -is everywhere constant, being equal to infinity in the case of the -plane, and to some definite magnitude in the case of the sphere.</p> - -<p>From the fact that we may extend a soap-film across a ring of -wire however fantastically the latter may be bent, we realise that -there is no limit to the number of surfaces of minimal area which -may be constructed or may be imagined; and while some of these -are very complicated indeed, some, for instance a spiral helicoid -screw, are relatively very simple. But if we -limit ourselves to <span class="xxpn" id="p218">{218}</span> -<i>surfaces of revolution</i> (that is to say, to surfaces symmetrical about -an axis), we find, as Plateau was the first to shew, that those which -meet the case are very few in number. They are six in all, -namely the plane, the sphere, the cylinder, the catenoid, the -unduloid, and a curious surface which Plateau called the nodoid.</p> - -<p>These several surfaces are all closely related, and the passage -from one to another is generally easy. Their mathematical interrelation -is expressed by the fact (first shewn by Delaunay<a class="afnanch" href="#fn286" id="fnanch286">286</a>, -in 1841) -that the plane curves by whose rotation they are generated are -themselves generated as “roulettes” of the conic sections.</p> - -<p>Let us imagine a straight line upon which a circle, an ellipse -or other conic section rolls; the focus of the conic section will -describe a line in some relation to the fixed axis, and this line -(or roulette), rotating around the axis, will describe in space one or -other of the six surfaces of revolution with which we are dealing.</p> - -<div class="dctr01" id="fig61"> -<img src="images/i218.png" width="800" height="117" alt=""> - <div class="dcaption">Fig. 61.</div></div> - -<p>If we imagine an ellipse so to roll over a line, either of its foci -will describe a sinuous or wavy line -<span class="nowrap">(Fig. <a href="#fig61" title="go to Fig. 61">61</a><span class="smmaj">B</span>)</span> -at a distance -alternately maximal and minimal from the axis; and this wavy -line, by rotation about the axis, becomes the meridional line of -the surface which we call the <i>unduloid</i>. The more unequal the -two axes are of our ellipse, the more pronounced will be the -sinuosity of the described roulette. If the two axes be equal, -then our ellipse becomes a circle, and the path described by its -rolling centre is a straight line parallel to the axis (A); and -obviously the solid of revolution generated therefrom will be a -<i>cylinder</i>. If one axis of our ellipse vanish, while the other remain -of finite length, then the ellipse is reduced to a straight line, and -its roulette will appear as a succession of semicircles touching one -another upon the axis (C); the solid of revolution will be a series of -equal <i>spheres</i>. If as before one axis of the ellipse vanish, but the -other be infinitely long, then the curve -described by the rotation <span class="xxpn" id="p219">{219}</span> -of this latter will be a circle of infinite radius, i.e. a straight line -infinitely distant from the axis; and the surface of rotation is now -a <i>plane</i>. If we imagine one focus of our ellipse to remain at a -given distance from the axis, but the other to become infinitely -remote, that is tantamount to saying that the ellipse becomes -transformed into a parabola; and by the rolling of this curve -along the axis there is described a catenary (D), whose solid of -revolution is the <i>catenoid</i>.</p> - -<p>Lastly, but this is a little more difficult to imagine, we have -the case of the hyperbola.</p> - -<p>We cannot well imagine the hyperbola rolling upon a fixed -straight line so that its focus shall describe a continuous curve. -But let us suppose that the fixed line is, to begin with, asymptotic -to one branch of the hyperbola, and that the rolling proceed -until the line is now asymptotic to the other branch, that is to -say touching it at an infinite distance; there will then be mathematical -continuity if we recommence rolling with this second -branch, and so in turn with the other, when each has run its -course. We shall see, on reflection, that the line traced by one -and the same focus will be an “elastic curve” describing a succession -of kinks or knots (E), and the solid of revolution described -by this meridional line about the axis is the so-called <i>nodoid</i>.</p> - -<p>The physical transition of one of these surfaces into another -can be experimentally illustrated by means of soap-bubbles, or -better still, after the method of Plateau, by means of a large -globule of oil, supported when necessary by wire rings, within a -fluid of specific gravity equal to its own.</p> - -<p>To prepare a mixture of alcohol and water of a density precisely -equal to that of the oil-globule is a troublesome matter, and a -method devised by Mr C. R. Darling is a great improvement on -Plateau’s<a class="afnanch" href="#fn287" id="fnanch287">287</a>. -Mr Darling uses the oily liquid orthotoluidene, which -does not mix with water, has a beautiful and conspicuous red -colour, and has precisely the same density as water when both -are kept at a temperature of 24° C. We have therefore only to -run the liquid into water at this temperature in order to produce -beautifully spherical drops of any required -size: and by adding <span class="xxpn" id="p220">{220}</span> -a little salt to the lower layers of water, the drop may be made -to float or rest upon the denser liquid.</p> - -<p>We have already seen that the soap-bubble, spherical to begin -with, is transformed into a plane when we relieve its internal -pressure and let the film shrink back upon the orifice of the pipe. -If we blow a small bubble and then catch it up on a second pipe, -so that it stretches between, we may gradually draw the two pipes -apart, with the result that the spheroidal surface will be gradually -flattened in a longitudinal direction, and the bubble will be transformed -into a cylinder. But if we draw the pipes yet farther -apart, the cylinder will narrow in the middle into a sort of hourglass -form, the increasing curvature of its transverse section being -balanced by a gradually increasing <i>negative</i> curvature in the -longitudinal section. The cylinder has, in turn, been converted -into an unduloid. When we hold a portion of a soft glass tube in -the flame, and “draw it out,” we are in the same identical fashion -converting a cylinder into an unduloid -<span class="nowrap">(Fig. <a href="#fig62" title="go to Fig. 62">62</a><span class="smmaj">A</span>);</span> -when on the -other hand we stop the end and blow, we again convert the -cylinder into an unduloid -<span class="nowrap">(<span class="smmaj">B</span>),</span> -but into one which is now positively, -while the former was negatively curved. The two figures are -essentially the same, save that the two halves of the one are -reversed in the other.</p> - -<div class="dctr03" id="fig62"> -<img src="images/i220.png" width="600" height="134" alt=""> - <div class="dcaption">Fig. 62.</div></div> - -<p>That spheres, cylinders and unduloids are of the commonest -occurrence among the forms of small unicellular organism, or of -individual cells in the simpler aggregates, and that in the processes -of growth, reproduction and development transitions are frequent -from one of these forms to another, is obvious to the naturalist, -and we shall deal presently with a few illustrations of these -phenomena.</p> - -<p>But before we go further in this enquiry, it will be necessary -to consider, to some small extent at least, the <i>curvatures</i> of the -six different surfaces, that is to say, to -determine what modification <span class="xxpn" id="p221">{221}</span> -is required, in each case, of the general equation which applies -to them all. We shall find that with this question is closely -connected the question of the <i>pressures</i> exercised by, or impinging -on the film, and also the very important question of -the limitations which, from the nature of the case, exist to -prevent the extension of certain of the figures beyond certain -bounds. The whole subject is mathematical, and we shall only -deal with it in the most elementary way.</p> - -<p>We have seen that, in our general formula, the expression -1 ⁄ <i>R</i> + 1 ⁄ <i>R′</i> -= <i>C</i>, a constant; and that this is, in all cases, the -condition of our surface being one of minimal area. In other -words, it is always true for one and all of the six surfaces which -we have to consider. But the constant <i>C</i> may have any value, -positive, negative, or nil.</p> - -<p>In the case of the plane, where <i>R</i> and <i>R′</i> are both infinite, it -is obvious that 1 ⁄ <i>R</i> + 1 ⁄ <i>R′</i> -= 0. The expression therefore vanishes, -and our dynamical equation of equilibrium becomes <i>P</i> -= <i>p</i>. In -short, we can only have a plane film, or we shall only find a plane -surface in our cell, when on either side thereof we have equal -pressures or no pressure at all. A simple case is the plane partition -between two equal and similar cells, as in a filament of spirogyra.</p> - -<p>In the case of the sphere, the radii are all equal, <i>R</i> -= <i>R′</i>; -they are also positive, and <span class="nowrap"> -<i>T</i> (1 ⁄ <i>R</i> + 1 ⁄ <i>R′</i>),</span> -or <span class="nowrap">2 <i>T ⁄ R</i>,</span> is a positive -quantity, involving a positive pressure <i>P</i>, on the other side of the -equation.</p> - -<p>In the cylinder, one radius of curvature has the finite and -positive value <i>R</i>; but the other is infinite. Our formula becomes -<i>T ⁄ R</i>, to which corresponds a positive pressure <i>P</i>, supplied by the -surface-tension as in the case of the sphere, but evidently of just -half the magnitude developed in the latter case for a given value -of the radius <i>R</i>.</p> - -<div class="dmaths"> -<p>The catenoid has the remarkable property that its -curvature in one direction is precisely equal and opposite -to its curvature in the other, this property holding -good for all points of the surface. That is to say, -<i>R</i> -= −<i>R′</i>; and the expression becomes</p> - -<div>(1 ⁄ <i>R</i> + 1 ⁄ <i>R′</i>) -= (1 ⁄ <i>R</i> − 1 ⁄ <i>R</i>) -= 0;</div> - -<p class="pcontinue">in other words, the surface, as in the case of the -plane, has <i>no <span class="xxpn" id="p222">{222}</span> -curvature</i>, and exercises no pressure. There are no other surfaces, -save these two, which share this remarkable property; and it -follows, as a simple corollary, that we may expect at times to have -the catenoid and the plane coexisting, as parts of one and the -same boundary system; just as, in a cylindrical drop or cell, the -cylinder is capped by portions of spheres, such that the cylindrical -and spherical portions of the wall exert equal positive pressures.</p> -</div><!--dmaths--> - -<p>In the unduloid, unlike the four surfaces which we have just -been considering, it is obvious that the curvatures change from -one point to another. At the middle of one of the swollen -portions, or “beads,” the two curvatures are both positive; the -expression (1 ⁄ <i>R</i> + 1 ⁄ <i>R′</i>) is therefore positive, and it is also finite. -The film, accordingly, exercises a positive tension inwards, which -must be compensated by a finite and positive outward pressure -<i>P</i>. At the middle of one of the narrow necks, between two -adjacent beads, there is obviously, in the transverse direction, -a much stronger curvature than in the former case, and the curvature -which balances it is now a negative one. But the sum of the -two must remain positive, as well as constant; and we therefore -see that the convex or positive curvature must always be greater -than the concave or negative curvature at the same point. This -is plainly the case in our figure of the unduloid.</p> - -<p>The nodoid is, like the unduloid, a continuous curve which -keeps altering its curvature as it alters its distance from the axis; -but in this case the resultant pressure inwards is negative instead -of positive. But this curve is a complicated one, and a full -discussion of it would carry us beyond our scope.</p> - -<div class="dleft dwth-i" id="fig63"> -<img src="images/i222.png" width="136" height="239" alt=""> - <div class="dcaption">Fig. 63.</div></div> - -<p>In one of Plateau’s experiments, a bubble of oil (protected from -gravity by the specific gravity of the surrounding fluid being -identical with its own) is balanced between two -annuli. It may then be brought to assume the form -of Fig. <a href="#fig63" title="go to Fig. 63">63</a>, that is to say the form of a cylinder with -spherical ends; and there is then everywhere, owing -to the convexity of the surface film, a pressure -inwards upon the fluid contents of the bubble. If -the surrounding liquid be ever so little heavier or -lighter than that which constitutes the drop, then -the conditions of equilibrium -will be accordingly <span class="xxpn" id="p223">{223}</span> -modified, and the cylindrical drop will assume the form of an -unduloid (Fig. <a href="#fig64" title="go to Fig. 64">64</a> <span class="smmaj">A,</span> -<span class="nowrap"><span class="smmaj">B</span>),</span> -with its dilated portion below or above,<br class="brclrfix"></p> - -<div class="dright dwth-e" id="fig64"> -<img src="images/i223a.png" width="316" height="269" alt=""> - <div class="dcaption">Fig. 64.</div></div> - -<p class="pcontinue">as the case may be; and our cylinder -may also, of course, be converted into an unduloid either by -elongating it further, or by abstracting a portion of its oil, -until at length rupture ensues and the cylinder breaks up into -two new spherical drops. In all cases alike, the unduloid, like -the original cylinder, will be capped by spherical ends, which -are the sign, and the consequence, of the positive pressure -produced by the curved walls of the unduloid. But if our -initial cylinder, instead of being tall, be a flat or dumpy one -(with certain definite relations of height to breadth), then -new phenomena may be exhibited. For now, if a little oil be -cautiously withdrawn from the mass by help of a small syringe, -the cylinder may be made to flatten down so that its upper and -lower surfaces become plane; which is of itself an indication -that the pressure inwards is now <i>nil</i>. But at the very moment -when the upper and lower surfaces become plane, it will be -found that the sides curve inwards, in the fashion shewn in -Fig. <a href="#fig65" title="go to Fig. 65">65</a><span class="smmaj">B.</span> This figure is a catenoid, -which, as<br class="brclrfix"></p> - -<div class="dctr03" id="fig65"> -<img src="images/i223b.png" width="600" height="168" alt=""> - <div class="dcaption">Fig. 65.</div></div> - -<p class="pcontinue">we have already seen, is, like the -plane itself, a surface exercising no pressure, and which -therefore may coexist with the plane as part of one and the -same system. We may continue to withdraw more oil from our -bubble, drop by drop, and now the upper and lower surfaces -dimple down into concave portions of spheres, as the result of -the <i>negative</i> internal pressure; and thereupon the peripheral -catenoid surface alters its form (perhaps, on this small -scale, imperceptibly), and becomes a portion of a nodoid <span -class="nowrap">(Fig. <a href="#fig65" title="go to Fig. 65">65</a><span class="smmaj">A</span>).</span> -<span class="xxpn" id="p224">{224}</span> It represents, -in fact, that portion of the nodoid, which in Fig. <a href="#fig66" title="go to Fig. 66">66</a> lies -between such points as <span class="smmaj">O,</span> <span -class="smmaj">P.</span> While it is easy to<br class="brclrfix"></p> - -<div class="dleft dwth-e" id="fig66"> -<img src="images/i224.png" width="397" height="203" alt=""> - <div class="dcaption">Fig. 66.</div></div> - -<p class="pcontinue">draw the outline, or meridional -section, of the nodoid (as in -Fig. <a href="#fig66" title="go to Fig. 66">66</a>), it is obvious that the -solid of revolution to be derived -from it, can never be realised in -its entirety: for one part of the -solid figure would cut, or entangle -with, another. All that -we can ever do, accordingly, is to realise isolated portions of the -nodoid.<br class="brclrfix"></p> - -<p>If, in a sequel to the preceding experiment of Plateau’s, we -use solid discs instead of annuli, so as to enable us to exert direct -mechanical pressure upon our globule of oil, we again begin by -adjusting the pressure of these discs so that the oil assumes the -form of a cylinder: our discs, that is to say, are adjusted to -exercise a mechanical pressure equal to what in the former case -was supplied by the surface-tension of the spherical caps or ends -of the bubble. If we now increase the pressure slightly, the -peripheral walls will become convexly curved, exercising a precisely -corresponding pressure. Under these circumstances the -form assumed by the sides of our figure will be that of a portion -of an unduloid. If we increase the pressure between the discs, -the peripheral surface of oil will bulge out more and more, and -will presently constitute a portion of a sphere. But we may -continue the process yet further, and within certain limits we shall -find that the system remains perfectly stable. What is this new -curved surface which has arisen out of the sphere, as the latter -was produced from the unduloid? It is no other than a portion -of a nodoid, that part which in Fig. <a href="#fig66" title="go to Fig. 66">66</a> lies between such limits as -<span class="smmaj">M</span> -and <span class="smmaj">N.</span> But this surface, which is concave in both directions -towards the surface of the oil within, is exerting a pressure upon -the latter, just as did the sphere out of which a moment ago it -was transformed; and we had just stated, in considering the -previous experiment, that the pressure inwards exerted by the -nodoid was a negative one. The explanation of this seeming -discrepancy lies in the simple fact that, if we -follow the outline <span class="xxpn" id="p225">{225}</span> -of our nodoid curve in Fig. <a href="#fig66" title="go to Fig. 66">66</a> from <span class="smmaj">O,</span> <span class="smmaj">P,</span> the surface concerned -in the former case, to <span class="smmaj">M,</span> <span class="smmaj">N,</span> that concerned in the present, we shall -see that in the two experiments the surface of the liquid is not -homologous, but lies on the positive side of the curve in the one -case and on the negative side in the other.</p> - -<hr class="hrblk"> - -<p>Of all the surfaces which we have been describing, the sphere -is the only one which can enclose space; the others can only help -to do so, in combination with one another or with the sphere itself. -Thus we have seen that, in normal equilibrium, the cylindrical -vesicle is closed at either end by a portion of a sphere, and so on. -Moreover the sphere is not only the only one of our figures which -can enclose a finite space; it is also, of all possible figures, that -which encloses the greatest volume with the least area of surface; -it is strictly and absolutely the surface of minimal area, and it -is therefore the form which will be naturally assumed by a unicellular -organism (just as by a raindrop), when it is practically -homogeneous and when, like Orbulina floating in the ocean, its -surroundings are likewise practically homogeneous and symmetrical. -It is only relatively speaking that all the rest are -surfaces <i>minimae areae</i>; they are so, that is to say, under the -given conditions, which involve various forms of pressure or -restraint. Such restraints are imposed, for instance, by the -pipes or annuli with the help of which we draw out our cylindrical -or unduloid oil-globule or soap-bubble; and in the case of the -organic cell, similar restraints are constantly supplied by solidification, -partial or complete, local or general, of the cell-wall.</p> - -<p>Before we pass to biological illustrations of our surface-tension -figures, we have still another preliminary matter to deal with. -We have seen from our description of two of Plateau’s classical -experiments, that at some particular point one type of surface -gives place to another; and again, we know that, when we draw -out our soap-bubble into and then beyond a cylinder, there comes -a certain definite point at which our bubble breaks in two, and -leaves us with two bubbles of which each is a sphere, or a portion -of a sphere. In short there are certain definite limits to the -<i>dimensions</i> of our figures, within which limits equilibrium is -stable but at which it becomes unstable, and -above which it <span class="xxpn" id="p226">{226}</span> -breaks down. Moreover in our composite surfaces, when the -cylinder for instance is capped by two spherical cups or lenticular -discs, there is a well-defined ratio which regulates their respective -curvatures, and therefore their respective dimensions. These two -matters we may deal with together.</p> - -<p>Let us imagine a liquid drop which by appropriate conditions -has been made to assume the form of a cylinder; we have already -seen that its ends will be terminated by portions of spheres. -Since one and the same liquid film covers the sides and ends of -the drop (or since one and the same delicate membrane encloses -the sides and ends of the cell), we assume the surface-tension (<i>T</i>) -to be everywhere identical; and it follows, since the internal -fluid-pressure is also everywhere identical, that the expression -(1 ⁄ <i>R</i> + 1 ⁄ <i>R′</i>) for the cylinder is equal to the corresponding expression, -which we may call (1 ⁄ <i>r</i> + 1 ⁄ <i>r′</i>), in the case of the terminal -spheres. But in the cylinder 1 ⁄ <i>R′</i> -= 0, and in the sphere 1 ⁄ <i>r</i> -= 1 ⁄ <i>r′</i>. -Therefore our relation of equality becomes 1 ⁄ <i>R</i> -= 2 ⁄ <i>r</i>, or <i>r</i> <span class="nowrap"> -= 2 <i>R</i>;</span> -that is to say, the sphere in question has just twice the radius of -the cylinder of which it forms a cap.</p> - -<div class="dleft dwth-e" id="fig67"> -<img src="images/i226.png" width="316" height="396" alt=""> - <div class="dcaption">Fig. 67.</div></div> - -<div class="dmaths"> -<p>And if <i>Ob</i>, the radius of the sphere, be equal to twice the radius -(<i>Oa</i>) of the cylinder, it follows that the angle <i>aOb</i> is an angle of -60°, and <i>bOc</i> is also an angle of 60°; -that is to say, the arc <i>bc</i> is equal to <span class="nowrap"> -(<sup>1</sup>⁄<sub>3</sub>) π.</span> -In other words, the spherical -disc which (under the given conditions) -caps our cylinder, is not a portion -taken at haphazard, but is neither -more nor less than that portion of a -sphere which is subtended by a cone -of 60°. Moreover, it is plain that -the height of the spherical cap, <i>de</i>,</p> - -<div>= <i>Ob</i> − <i>ab</i> -= <i>R</i> (2 − √3) -= 0·27 <i>R</i>,</div> - -<p class="pcontinue">where <i>R</i> is the radius of our cylinder, -or one-half the radius of our spherical -cap: in other words the normal height of the spherical cap over -the end of the cylindrical cell is just a very little more than one-eighth -of the diameter of the cylinder, or of the -radius of the <span class="xxpn" id="p227">{227}</span> -sphere. And these are the proportions which we recognise, under -normal circumstances, in such a case as the cylindrical cell of -Spirogyra where its free end is capped by a portion of a sphere. -<br class="brclrfix"></p></div><!--dmaths--> - -<hr class="hrblk"> - -<p>Among the many important theoretical discoveries which we -owe to Plateau, one to which we have just referred is of peculiar -importance: namely that, with the exception of the sphere and -the plane, the surfaces with which we have been dealing are only -in complete equilibrium within certain dimensional limits, or in -other words, have a certain definite limit of stability; only the plane -and the sphere, or any portions of a sphere, are perfectly stable, -because they are perfectly symmetrical, figures. For experimental -demonstration, the case of the cylinder is the simplest. If we -produce a liquid film having the form of a cylinder, either by</p> - -<div class="dctr04" id="fig68"> -<img src="images/i227.png" width="580" height="250" alt=""> - <div class="dcaption">Fig. 68.</div></div> - -<p class="pcontinue">drawing out a bubble or by supporting between two rings a -globule of oil, the experiment proceeds easily until the length of -the cylinder becomes just about three times as great as its diameter. -But somewhere about this limit the cylinder alters its form; it -begins to narrow at the waist, so passing into an unduloid, and -the deformation progresses quickly until at last our cylinder -breaks in two, and its two halves assume a spherical form. It is -found, by theoretical considerations, that the precise limit of -stability is at the point when the length of the cylinder is exactly -equal to its circumference, that is to say, when <i>L</i> -= 2π<i>R</i>, or when -the ratio of length to diameter is represented by π.</p> - -<p>In the case of the catenoid, Plateau’s experimental procedure -was as follows. To support his globule of oil (in, as usual, a -mixture of alcohol and water of its own specific -gravity), he used <span class="xxpn" id="p228">{228}</span> -a pair of metal rings, which happened to have a diameter of -71 millimetres; and, in a series of experiments, he set these rings -apart at distances of 55, 49, 47, 45, and 43 mm. successively. -In each case he began by bringing his oil-globule into a cylindrical -form, by sucking superfluous oil out of the drop until this result -was attained; and always, for the reason with which we are now -acquainted, the cylindrical sides were associated with spherical -ends to the cylinder. On continuing to withdraw oil in the hope -of converting these spherical ends into planes, he found, naturally, -that the sides of the cylinder drew in to form a concave surface; -but it was by no means easy to get the extremities actually plane: -and unless they were so, thus indicating that the surface-pressure -of the drop was nil, the curvature of the sides could not be that -of a catenoid. For in the first experiment, when the rings were -55 mm. apart, as soon as the convexity of the ends was to a certain -extent diminished, it spontaneously increased again; and the -transverse constriction of the globule correspondingly deepened, -until at a certain point equilibrium set in anew. Indeed, the more -oil he removed, the more convex became the ends, until at last -the increasing transverse constriction led to the breaking of the -oil-globule into two. In the third experiment, when the rings -were 47 mm. apart, it was easy to obtain end-surfaces that were -actually plane, and they remained so even though more oil was -withdrawn, the transverse constriction deepening accordingly. -Only after a considerable amount of oil had been sucked up did -the plane terminal surface become gradually convex, and presently -the narrow waist, narrowing more and more, broke across in the -usual way. Finally in the fifth experiment, where the rings were -still nearer together, it was again possible to bring the ends of the -oil-globule to a plane surface, as in the third and fourth experiments, -and to keep this surface plane in spite of some continued withdrawal -of oil. But very soon the ends became gradually concave, -and the concavity deepened as more and more oil was withdrawn, -until at a certain limit, the whole oil-globule broke up in general -disruption.</p> - -<p>We learn from this that the limiting size of the catenoid was -reached when the distance of the supporting rings was to their -diameter as 47 to 71, or, as nearly as possible, as -two to three; <span class="xxpn" id="p229">{229}</span> -and as a matter of fact it can be shewn that 2 ⁄ 3 is the true -theoretical value. Above this limit of 2 ⁄ 3, the inevitable convexity -of the end-surfaces shows that a positive pressure inwards is being -exerted by the surface film, and this teaches us that the sides of -the figure actually constitute not a catenoid but an unduloid, -whose spontaneous changes tend to a form of greater stability. -Below the 2 ⁄ 3 limit the catenoid surface is essentially unstable, -and the form into which it passes under certain conditions of -disturbance such as that of the excessive withdrawal of oil, is -that of a nodoid -<span class="nowrap">(Fig. <a href="#fig65" title="go to Fig. 65">65</a><span class="smmaj">A</span>).</span></p> - -<p>The unduloid has certain peculiar properties as regards its -limitations of stability. But as to these we need mention two -facts only: (1) that when the unduloid, which we produce with -our soap-bubble or our oil-globule, consists of the figure containing -a complete constriction, it has somewhat wide limits of stability; -but (2) if it contain the swollen portion, then equilibrium is limited -to the condition that the figure consists simply of one complete -unduloid, that is to say that its ends are constituted by the -narrowest portions, and its middle by the widest portion of the -entire curve. The theoretical proof of this latter fact is difficult, -but if we take the proof for granted, the fact will serve to throw -light on what we have learned regarding the stability of the cylinder. -For, when we remember that the meridional section of our unduloid -is generated by the rolling of an ellipse upon a straight line in its -own plane, we shall easily see that the length of the entire unduloid -is equal to the circumference of the generating ellipse. As the -unduloid becomes less and less sinuous in outline, it gradually -approaches, and in time reaches, the form of a cylinder; and -correspondingly, the ellipse which generated it has its foci more -and more approximated until it passes into a circle. The cylinder -of a length equal to the circumference of its generating circle is -therefore precisely homologous to an unduloid whose length is -equal to the circumference of its generating ellipse; and this is -just what we recognise as constituting one complete segment of -the unduloid.</p> - -<hr class="hrblk"> - -<p>While the figures of equilibrium which are at the same time -surfaces of revolution are only six in number, there -is an infinite <span class="xxpn" id="p230">{230}</span> -number of figures of equilibrium, that is to say of surfaces of -constant mean curvature, which are not surfaces of revolution; -and it can be shewn mathematically that any given contour can -be occupied by a finite portion of some one such surface, in stable -equilibrium. The experimental verification of this theorem lies in -the simple fact (already noted) that however we may bend a wire -into a closed curve, plane or not plane, we may always, under -appropriate precautions, fill the entire area with an unbroken -film.</p> - -<p>Of the regular figures of equilibrium, that is to say surfaces -of constant mean curvature, apart from the surfaces of revolution -which we have discussed, the helicoid spiral is the most interesting -to the biologist. This is a helicoid generated by a straight line -perpendicular to an axis, about which it turns at a uniform rate -while at the same time it slides, also uniformly, along this same -axis. At any point in this surface, the curvatures are equal and -of opposite sign, and the sum of the curvatures is accordingly nil. -Among what are called “ruled surfaces” (which we may describe -as surfaces capable of being defined by a system of stretched -strings), the plane and the helicoid are the only two whose mean -curvature is null, while the cylinder is the only one whose curvature -is finite and constant. As this simplest of helicoids corresponds, -in three dimensions, to what in two dimensions is merely a plane -(the latter being generated by the rotation of a straight line about -an axis without the superadded gliding motion which generates -the helicoid), so there are other and much more complicated -helicoids which correspond to the sphere, the unduloid and the -rest of our figures of revolution, the generating planes of these -latter being supposed to wind spirally about an axis. In the case -of the cylinder it is obvious that the resulting figure is indistinguishable -from the cylinder itself. In the case of the unduloid we -obtain a grooved spiral, such as we may meet with in nature (for -instance in Spirochætes, <i>Bodo gracilis</i>, etc.), and which accordingly -it is of interest to us to be able to recognise as a surface of minimal -area or constant curvature.</p> - -<p>The foregoing considerations deal with a small part only -of the theory of surface tension, or of capillarity: with that -part, namely, which relates to the forms of -surface which are <span class="xxpn" id="p231">{231}</span> -capable of subsisting in equilibrium under the action of that force, -either of itself or subject to certain simple constraints. And as -yet we have limited ourselves to the case of a single surface, or -of a single drop or bubble, leaving to another occasion a discussion -of the forms assumed when such drops or vesicles meet and combine -together. In short, what we have said may help us to understand -the form of a <i>cell</i>,—considered, as with certain limitations -we may legitimately consider it, as a liquid drop or liquid vesicle; -the conformation of a <i>tissue</i> or cell-aggregate must be dealt with -in the light of another series of theoretical considerations. In -both cases, we can do no more than touch upon the fringe of a -large and difficult subject. There are many forms capable of -realisation under surface tension, and many of them doubtless to -be recognised among organisms, which we cannot touch upon in -this elementary account. The subject is a very general one; it -is, in its essence, more mathematical than physical; it is part of -the mathematics of surfaces, and only comes into relation with -surface tension, because this physical phenomenon illustrates and -exemplifies, in a concrete way, most of the simple and symmetrical -conditions with which the general mathematical theory is capable -of dealing. And before we pass to illustrate by biological examples -the physical phenomena which we have described, we must be -careful to remember that the physical conditions which we have -hitherto presupposed will never be wholly realised in the organic -cell. Its substance will never be a perfect fluid, and hence -equilibrium will be more or less slowly reached; its surface will -seldom be perfectly homogeneous, and therefore equilibrium will -(in the fluid condition) seldom be perfectly attained; it will very -often, or generally, be the seat of other forces, symmetrical or -unsymmetrical; and all these causes will more or less perturb the -effects of surface tension acting by itself. But we shall find that, -on the whole, these effects of surface tension though modified are -not obliterated nor even masked; and accordingly the phenomena -to which I have devoted the foregoing pages will be found -manifestly recurring and repeating themselves among the phenomena -of the organic cell.</p> - -<hr class="hrblk"> - -<p>In a spider’s web we find exemplified several -of the principles <span class="xxpn" id="p232">{232}</span> -of surface tension which we have now explained. The thread is -formed out of the fluid secretion of a gland, and issues from the -body as a semi-fluid cylinder, that is to say in the form of a surface -of equilibrium, the force of expulsion giving it its elongation and -that of surface tension giving it its circular section. It is prevented, -by almost immediate solidification on exposure to the air, from -breaking up into separate drops or spherules, as it would otherwise -tend to do as soon as the length of the cylinder had passed its -limit of stability. But it is otherwise with the sticky secretion -which, coming from another gland, is simultaneously poured over -the issuing thread when it is to form the spiral portion of the -web. This latter secretion is more fluid than the first, and retains -its fluidity for a very much longer time, finally drying up after -several hours. By capillarity it “wets” the thread, spreading -itself over it in an even film, which film is now itself a cylinder. -But this liquid cylinder has its limit of stability when its length -equals its own circumference, and therefore just at the points so -defined it tends to disrupt into separate segments: or rather, in -the actual case, at points somewhat more distant, owing to the -imperfect fluidity of the viscous film, and still more to the frictional -drag upon it of the inner solid cylinder, or thread, with which it -is in contact. The cylinder disrupts in the usual manner, passing -first into the wavy outline of an unduloid, whose swollen portions -swell more and more till the contracted parts break asunder, and -we arrive at a series of spherical drops or beads, of equal size, -strung at equal intervals along the thread. If we try to spread -varnish over a thin stretched wire, we produce automatically the -same identical result<a class="afnanch" href="#fn288" id="fnanch288">288</a>; -unless our varnish be such as to dry almost -instantaneously, it gathers into beads, and do what we can, we -fail to spread it smooth. It follows that, according to the viscidity -and drying power of the varnish, the process may stop or seem to -stop at any point short of the formation of the perfect spherules; -it is quite possible, therefore, that as our final stage we may only -obtain half-formed beads, or the wavy outline of an unduloid. -The formation of the beads may be facilitated or hastened by -jerking the stretched thread, as the spider -actually does: the <span class="xxpn" id="p233">{233}</span> -effect of the jerk being to disturb and destroy the unstable -equilibrium of the viscid cylinder<a class="afnanch" href="#fn289" id="fnanch289">289</a>. -Another very curious -phenomenon here presents itself.</p> - -<p>In Plateau’s experimental separation of a cylinder of oil into -two spherical portions, it was noticed that, when contact was -nearly broken, that is to say when the narrow neck of the unduloid -had become very thin, the two spherical bullae, instead of absorbing -the fluid out of the narrow neck into themselves as they had done -with the preceding portion, drew out this small remaining part of -the liquid into a thin thread as they completed their spherical -form and consequently receded from one another: the reason being -that, after the thread or “neck” has reached a certain tenuity, -the internal friction of the fluid prevents or retards its rapid exit -from the little thread to the adjacent spherule. It is for the same -reason that we are able to draw a glass rod or tube, which we have -heated in the middle, into a long and uniform cylinder or thread, -by quickly separating the two ends. But in the case of the glass -rod, the long thin intermediate cylinder quickly cools and solidifies, -while in the ordinary separation of a liquid cylinder the corresponding -intermediate cylinder remains liquid; and therefore, like -any other liquid cylinder, it is liable to break up, provided that its -dimensions exceed the normal limit of stability. And its length -is generally such that it breaks at two points, thus leaving two -terminal portions continuous with the spheres and becoming -confluent with these, and one median portion which resolves itself -into a comparatively tiny spherical drop, midway between the -original and larger two. Occasionally, the same process of formation -of a connecting thread repeats itself a second time, between -the small intermediate spherule and the large spheres; and in this -case we obviously obtain two additional spherules, still smaller in -size, and lying one on either side of our first little one. This whole -phenomenon, of equal and regularly interspaced beads, often with -little beads regularly interspaced between the larger ones, and -possibly also even a third series of still smaller beads regularly -intercalated, may be easily observed in a spider’s web, such as -that of <i>Epeira</i>, very often with beautiful -regularity,—which <span class="xxpn" id="p234">{234}</span> -naturally, however, is sometimes interrupted and disturbed owing -to a slight want of homogeneity in the secreted fluid; and the -same phenomenon is repeated on a grosser scale when the web is -bespangled with dew, and every thread bestrung with pearls -innumerable. To the older naturalists, these regularly arranged -and beautifully formed globules on the spider’s web were a cause -of great wonder and admiration. Blackwall, counting some -twenty globules in a tenth of an inch, calculated that a large -garden-spider’s web comprised about 120,000 globules; the net -was spun and finished in about forty minutes, and Blackwall was -evidently filled with astonishment at the skill and quickness with -which the spider manufactured these little beads. And no wonder, -for according to the above estimate they had to be made at the -rate of about 50 per second<a class="afnanch" href="#fn290" id="fnanch290">290</a>.</p> - -<div class="dctr03" id="fig69"> -<img src="images/i234.png" width="600" height="104" alt=""> - <div class="dcaption">Fig. 69. Hair of <i>Trianea</i>, -in glycerine. (After Berthold.)</div></div> - -<p>The little delicate beads which stud the long thin -pseudopodia of a foraminifer, such as <i>Gromia</i>, or which in -like manner appear upon the cylindrical film of protoplasm -which covers the long radiating spicules of <i>Globigerina</i>, -represent an identical phenomenon. Indeed there are many cases, -in which we may study in a protoplasmic filament the whole -process of formation of such beads. If we squeeze out on to -a slide the viscid contents of a mistletoe berry, the long -sticky threads into which the substance runs shew the whole -phenomenon particularly well. Another way to demonstrate it was -noticed many years ago by Hofmeister and afterwards explained -by Berthold. The hairs of certain water-plants, such as -Hydrocharis or Trianea, constitute very long cylindrical cells, -the protoplasm being supported, and maintained in equilibrium -by its contact with the cell-wall. But if we immerse the -filament in some dense fluid, a little sugar-solution for -instance, or dilute glycerine, the cell-sap tends to diffuse -outwards, the protoplasm parts company with its surrounding and -supporting wall, <span class="xxpn" id="p235">{235}</span> and -lies free as a protoplasmic cylinder in the interior of the -cell. Thereupon it immediately shews signs of instability, and -commences to disrupt. It tends to gather into spheres, which -however, as in our illustration, may be prevented by their -narrow quarters from assuming the complete spherical form; -and in between these spheres, we have more or less regularly -alternate ones, of smaller size<a class="afnanch" href="#fn291" -id="fnanch291">291</a>. Similar, but less regular, beads or -droplets may be caused to appear, under stimulation by an -alternating current, in the protoplasmic threads within the -living cells of the hairs of Tradescantia. The explanation -usually given is, that the viscosity of the protoplasm -is reduced, or its fluidity increased; but an increase -of the surface tension would seem a more likely reason<a -class="afnanch" href="#fn292" id="fnanch292">292</a>.</p> - -<hr class="hrblk"> - -<p>We may take note here of a remarkable series of phenomena, -which, though they seem at first sight to be of a very different -order, are closely related to the phenomena which attend and -which bring about the breaking-up of a liquid cylinder or thread.</p> - -<div class="dctr01" id="fig70"> -<img src="images/i235.png" width="800" height="474" alt=""> - <div class="dcaption">Fig. 70. Phases of a Splash. - (From Worthington.)</div></div> - -<p>In some of Mr Worthington’s most beautiful -experiments on <span class="xxpn" id="p236">{236}</span> -splashes, it was found that the fall of a round pebble into water -from a considerable height, caused the rise of a filmy sheet of water -in the form of a cup or cylinder; and the edge of this cylindrical -film tended to be cut up into alternate lobes and notches, and the -prominent lobes or “jets” tended, in more extreme cases, to break -off or to break up into spherical beads (Fig. -<a href="#fig70" title="go to Fig. 70">70</a>)<a class="afnanch" href="#fn293" id="fnanch293">293</a>. -A precisely -similar appearance is seen, on a great scale, in the thin edge of a -breaking wave: when the smooth cylindrical edge, at a given -moment, shoots out an array of tiny jets which break up into -the droplets which constitute “spray” (Fig. <a href="#fig71" title="go to Fig. 71">71</a>, <i>a</i>, <i>b</i>). We -are at once reminded of the beautifully symmetrical notching on -the calycles of many hydroids, which little cups before they became -stiff and rigid had begun their existence as liquid or semi-liquid -films.</p> - -<div class="dctr01" id="fig71"> -<img src="images/i236.png" width="800" height="177" alt=""> - <div class="dcaption">Fig. 71. A breaking wave. (From Worthington.)</div></div> - -<p>The phenomenon is two-fold. In the first place, the edge of -our tubular or crater-like film forms a liquid ring or annulus, -which is closely comparable with the liquid thread or cylinder -which we have just been considering, if only we conceive the thread -to be bent round into the ring. And accordingly, just as the thread -spontaneously segments, first into an unduloid, and then into -separate spherical drops, so likewise will the edge of our annulus -tend to do. This phase of notching, or beading, of the edge of -the film is beautifully seen in many of Worthington’s experiments<a class="afnanch" href="#fn294" id="fnanch294">294</a>. -In the second place, the very fact of the rising of the crater means -that liquid is flowing up from below towards the rim; and the -segmentation of the rim means that channels -of easier flow are <span class="xxpn" id="p237">{237}</span> -created, along which the liquid is led, or is driven, into the protuberances: -and these are thus exaggerated into the jets or arms -which are sometimes so conspicuous at the edge of the crater. -In short, any film or film-like cup, fluid or semi-fluid in its consistency, -will, like the straight liquid cylinder, be unstable: and its -instability will manifest itself (among other ways) in a tendency -to segmentation or notching of the edge; and just such a peripheral -notching is a conspicuous feature of many minute organic cup-like -structures. In the case of the hydroid calycle -(Fig. <a href="#fig72" title="go to Fig. 72">72</a>), we are led -to the conclusion that the two common and conspicuous features -of notching or indentation of the cup, and of constriction or -annulation of the long cylindrical stem, are phenomena of the -same order and are due to surface-tension in both cases alike.</p> - -<div class="dctr01" id="fig72"> -<img src="images/i237.png" width="800" height="349" alt=""> - <div class="pcaption">Fig. 72. Calycles of Campanularian - zoophytes.  (A) <i>C. integra</i>;  (B) <i>C. - groenlandica</i>;  (C) <i>C. bispinosa</i>;  (D) <i>C. - raridentata</i>.</div></div> - -<p>Another phenomenon displayed in the same experiments is the -formation of a rope-like or cord-like thickening of the edge of the -annulus. This is due to the more or less sudden checking at the -rim of the flow of liquid rising from below: and a similar peripheral -thickening is frequently seen, not only in some of our hydroid -cups, but in many Vorticellas (cf. Fig. <a href="#fig75" title="go to Fig. 75">75</a>), and other organic -cup-like conformations. A perusal of Mr Worthington’s book -will soon suggest that these are not the only manifestations of -surface-tension in connection with splashes which present curious -resemblances and analogies to phenomena of organic form.</p> - -<p>The phenomena of an ordinary liquid splash -are so swiftly <span class="xxpn" id="p238">{238}</span> -transitory that their study is only rendered possible by “instantaneous” -photography: but this excessive rapidity is not an -essential part of the phenomenon. For instance, we can repeat -and demonstrate many of the simpler phenomena, in a permanent -or quasi-permanent form, by splashing water on to a surface of -dry sand, or by firing a bullet into a soft metal target. There is -nothing, then, to prevent a slow and lasting manifestation, in -a viscous medium such as a protoplasmic organism, of phenomena -which appear and disappear with prodigious rapidity in a more -mobile liquid. Nor is there anything peculiar in the “splash” -itself; it is simply a convenient method of setting up certain -motions or currents, and producing certain surface-forms, in a -liquid medium,—or even in such an extremely imperfect fluid as -is represented (in another series of experiments) by a bed of sand. -Accordingly, we have a large range of possible conditions under -which the organism might conceivably display configurations -analogous to, or identical with, those which Mr Worthington has -shewn us how to exhibit by one particular experimental method.</p> - -<p>To one who has watched the potter at his wheel, it is plain -that the potter’s thumb, like the glass-blower’s blast of air, -depends for its efficacy upon the physical properties of the -medium on which it operates, which for the time being is essentially -a fluid. The cup and the saucer, like the tube and the bulb, -display (in their simple and primitive forms) beautiful surfaces of -equilibrium as manifested under certain limiting conditions. -They are neither more nor less than glorified “splashes,” formed -slowly, under conditions of restraint which enhance or reveal -their mathematical symmetry. We have seen, and we shall see -again before we are done, that the art of the glass-blower is full -of lessons for the naturalist as also for the physicist: illustrating -as it does the development of a host of mathematical configurations -and organic conformations which depend essentially on the -establishment of a constant and uniform pressure within a <i>closed</i> -elastic shell or fluid envelope. In like manner the potter’s art -illustrates the somewhat obscurer and more complex problems -(scarcely less frequent in biology) of a figure of equilibrium which -is an <i>open</i> surface, or solid, of revolution. It is clear, at the same -time, that the two series of problems are closely -akin; for the <span class="xxpn" id="p239">{239}</span> -glass-blower can make most things that the potter makes, by -cutting off <i>portions</i> of his hollow ware. And besides, when this -fails, and the glass-blower, ceasing to blow, begins to use his rod -to trim the sides or turn the edges of wineglass or of beaker, he -is merely borrowing a trick from the craft of the potter.</p> - -<p>It would be venturesome indeed to extend our comparison -with these liquid surface-tension phenomena from the cup or -calycle of the hydrozoon to the little hydroid polype within: and -yet I feel convinced that there is something to be learned by such -a comparison, though not without much detailed consideration -and mathematical study of the surfaces concerned. The cylindrical -body of the tiny polype, the jet-like row of tentacles, the -beaded annulations which these tentacles exhibit, the web-like -film which sometimes (when they stand a little way apart) conjoins -their bases, the thin annular film of tissue which surrounds the -little organism’s mouth, and the manner in which this annular -“peristome” contracts<a class="afnanch" href="#fn295" id="fnanch295">295</a>, -like a shrinking soap-bubble, to close the -aperture, are every one of them features to which we may find -a singular and striking parallel in the surface-tension phenomena -which Mr Worthington has illustrated and demonstrated in the -case of the splash.</p> - -<p>Here however, we may freely confess that we are for the -present on the uncertain ground of suggestion and conjecture; -and so must we remain, in regard to many other simple and -symmetrical organic forms, until their form and dynamical -stability shall have been investigated by the mathematician: in -other words, until the mathematicians shall have become persuaded -that there is an immense unworked field wherein they may labour, -in the detailed study of organic form.</p> - -<hr class="hrblk"> - -<p>According to Plateau, the viscidity of the liquid, while it -helps to retard the breaking up of the cylinder and so increases -the length of the segments beyond that which theory demands, -has nevertheless less influence in this direction than we might -have expected. On the other hand, any external support or -adhesion, such as contact with a solid body, will be equivalent to -a reduction of surface-tension and so will very -greatly increase the <span class="xxpn" id="p240">{240}</span> -stability of our cylinder. It is for this reason that the mercury -in our thermometer tubes does not as a rule separate into drops, -though it occasionally does so, much to our inconvenience. And -again it is for this reason that the protoplasm in a long and growing -tubular or cylindrical cell does not necessarily divide into separate -cells and internodes, until the length of these far exceeds the -theoretic limits. Of course however and whenever it does so, we -must, without ever excluding the agency of surface tension, -remember that there may be other forces affecting the latter, and -accelerating or retarding that manifestation of surface tension by -which the cell is actually rounded off and divided.</p> - -<p>In most liquids, Plateau asserts that, on the average, the -influence of viscosity is such as to cause the cylinder to segment -when its length is about four times, or at most from four to six -times that of its diameter: instead of a fraction over three times -as, in a perfect fluid, theory would demand. If we take it at -four times, it may then be shewn that the resulting spheres would -have a diameter of about 1·8 times, and their distance apart would -be equal to about 2·2 times the diameter of the original cylinder. -The calculation is not difficult which would shew how these -numbers are altered in the case of a cylinder formed around a solid -core, as in the case of the spider’s web. Plateau has also made -the interesting observation that the <i>time</i> taken in the process of -division of the cylinder is directly proportional to the diameter -of the cylinder, while varying considerably with the nature of the -liquid. This question, of the time occupied in the division of a -cell or filament, in relation to the dimensions of the latter, has not -so far as I know been enquired into by biologists.</p> - -<hr class="hrblk"> - -<p>From the simple fact that the sphere is of all surfaces that -whose surface-area for a given volume is an absolute minimum, -we have already seen it to be plain that it is the one and only -figure of equilibrium which will be assumed under surface-tension -by a drop or vesicle, when no other disturbing factors are present. -One of the most important of these disturbing factors will be -introduced, in the form of complicated tensions and pressures, -when one drop is in contact with another drop and when a system -of intermediate films or partition walls is -developed between them. <span class="xxpn" id="p241">{241}</span> -This subject we shall discuss later, in connection with cell-aggregates -or tissues, and we shall find that further theoretical -considerations are needed as a preliminary to any such enquiry. -Meanwhile let us consider a few cases of the forms of cells, either -solitary, or in such simple aggregates that their individual form is -little disturbed thereby.</p> - -<p>Let us clearly understand that the cases we are about to -consider are those cases where the perfect symmetry of the sphere -is replaced by another symmetry, less complete, such as that of -an ellipsoidal or cylindrical cell. The cases of asymmetrical -deformation or displacement, such as is illustrated in the production -of a bud or the development of a lateral branch, are much simpler. -For here we need only assume a slight and localised variation of -surface-tension, such as may be brought about in various ways -through the heterogeneous chemistry of the cell; to this point -we shall return in our chapter on Adsorption. But the diffused -and graded asymmetry of the system, which brings about for -instance the ellipsoidal shape of a yeast-cell, is another matter.</p> - -<p>If the sphere be the one surface of complete symmetry and -therefore of independent equilibrium, it follows that in every cell -which is otherwise conformed there must be some definite force -to cause its departure from sphericity; and if this cause be the -very simple and obvious one of the resistance offered by a solidified -envelope, such as an egg-shell or firm cell-wall, we must still seek -for the deforming force which was in action to bring about the -given shape, prior to the assumption of rigidity. Such a cause -may be either external to, or may lie within, the cell itself. On -the one hand it may be due to external pressure or to some form -of mechanical restraint: as it is in all our experiments in which -we submit our bubble to the partial restraint of discs or rings or -more complicated cages of wire; and on the other hand it may be -due to intrinsic causes, which must come under the head either of -differences of internal pressure, or of lack of homogeneity or -isotropy in the surface itself<a class="afnanch" href="#fn296" id="fnanch296">296</a>. -<span class="xxpn" id="p242">{242}</span></p> - -<p>Our full formula of equilibrium, or equation to an elastic -surface, is <i>P</i> -= <i>p<sub>e</sub></i> + (<i>T ⁄ R</i> + <i>T′ ⁄ R′</i>), where <i>P</i> is the internal -pressure, <i>p<sub>e</sub></i> any extraneous pressure normal to the surface, <i>R</i>, <i>R′</i> -the radii of curvature at a point, and <i>T</i>, <i>T′</i>, the corresponding -tensions, normal to one another, of the envelope.</p> - -<p>Now in any given form which we are seeking to account for, -<i>R</i>, <i>R′</i> are known quantities; but all the other factors of the equation -are unknown and subject to enquiry. And somehow or other, by -this formula, we must account for the form of any solitary cell -whatsoever (provided always that it be not formed by successive -stages of solidification), the cylindrical cell of Spirogyra, the -ellipsoidal yeast-cell, or (as we shall see in another chapter) the -shape of the egg of any bird. In using this formula hitherto, we -have taken it in a simplified form, that is to say we have made -several limiting assumptions. We have assumed that <i>P</i> was -simply the uniform hydrostatic pressure, equal in all directions, -of a body of liquid; we have assumed that the tension <i>T</i> was -simply due to surface-tension in a homogeneous liquid film, and -was therefore equal in all directions, so that <i>T</i> -= <i>T′</i>; and we have -only dealt with surfaces, or parts of a surface, where extraneous -pressure, <i>p<sub>n</sub></i>, was non-existent. Now in the case of a bird’s egg, -the external pressure <i>p<sub>n</sub></i>, that is to say the pressure exercised by -the walls of the oviduct, will be found to be a very important -factor; but in the case of the yeast-cell or the Spirogyra, wholly -immersed in water, no such external pressure comes into play. -We are accordingly left, in such cases as these last, with two -hypotheses, namely that the departure from a spherical form is due -to inequalities in the internal pressure <i>P</i>, or else to inequalities in -the tension <i>T</i>, that is to say to a difference between <i>T</i> and <i>T′</i>. -In other words, it is theoretically possible that the oval form of -a yeast-cell is due to a greater internal pressure, a greater -“tendency to grow,” in the direction of the longer axis of the -ellipse, or alternatively, that with equal and symmetrical tendencies -to growth there is associated a difference of -external resistance in <span class="xxpn" id="p243">{243}</span> -respect of the tension of the cell-wall. Now the former hypothesis -is not impossible; the protoplasm is far from being a perfect fluid; -it is the seat of various internal forces, sometimes manifestly -polar; and accordingly it is quite possible that the internal -forces, osmotic and other, which lead to an increase of the content -of the cell and are manifested in pressure outwardly directed -upon its wall may be unsymmetrical, and such as to lead to a -deformation of what would otherwise be a simple sphere. But -while this hypothesis is not impossible, it is not very easy of -acceptance. The protoplasm, though not a perfect fluid, has yet -on the whole the properties of a fluid; within the small compass -of the cell there is little room for the development of unsymmetrical -pressures; and, in such a case as Spirogyra, where a large part of -the cavity is filled by a fluid and watery cell-sap, the conditions -are still more obviously those under which a uniform hydrostatic -pressure is to be expected. But in variations of <i>T</i>, that is to say -of the specific surface-tension per unit area, we have an ample -field for all the various deformations with which we shall have to -deal. Our condition now is, that (<i>T ⁄ R</i> + <i>T′ ⁄ R′</i>) -= a constant; but -it no longer follows, though it may still often be the case, that this -will represent a surface of absolute minimal area. As soon as <i>T</i> -and <i>T′</i> become unequal, it is obvious that we are no longer dealing -with a perfectly liquid surface film; but its departure from a -perfect fluidity may be of all degrees, from that of a slight non-isotropic -viscosity to the state of a firm elastic membrane<a class="afnanch" href="#fn297" id="fnanch297">297</a>. -And -it matters little whether this viscosity or semi-rigidity be manifested -in the self-same layer which is still a part of the protoplasm -of the cell, or in a layer which is completely differentiated into a -distinct and separate membrane. As soon as, by secretion or -“adsorption,” the molecular constitution of the surface layer is -altered, it is clearly conceivable that the alteration, or the secondary -chemical changes which follow it, may be such as to produce an -anisotropy, and to render the molecular forces less capable in -one direction than another of exerting that contractile force by -which they are striving to reduce to an -absolute minimum the <span class="xxpn" id="p244">{244}</span> -surface area of the cell. A slight inequality in two opposite -directions will produce the ellipsoid cell, and a very great inequality -will give rise to the cylindrical cell<a class="afnanch" href="#fn298" id="fnanch298">298</a>.</p> - -<p>I take it therefore, that the cylindrical cell of Spirogyra, or -any other cylindrical cell which grows in freedom from any -manifest external restraint, has assumed that particular form -simply by reason of the molecular constitution of its developing -surface-membrane; and that this molecular constitution was -anisotropous, in such a way as to render extension easier in one -direction than another.</p> - -<p>Such a lack of homogeneity or of isotropy, in the cell-wall is -often rendered visible, especially in plant-cells, in various ways, -in the form of concentric lamellae, annular and spiral striations, -and the like.</p> - -<p>But this phenomenon, while it brings about a certain departure -from complete symmetry, is still compatible with, and coexistent -with, many of the phenomena which we have seen to be associated -with surface-tension. The symmetry of tensions still leaves the -cell a solid of revolution, and its surface is still a surface of equilibrium. -The fluid pressure within the cylinder still causes the -film or membrane which caps its ends to be of a spherical form. -And in the young cell, where the surface pellicle is absent or but -little differentiated, as for instance in the oögonium of Achlya, -or in the young zygospore of Spirogyra, we always see the tendency -of the entire structure towards a spherical form reasserting itself: -unless, as in the latter case, it be overcome by direct compression -within the cylindrical mother-cell. Moreover, in those cases -where the adult filament consists of cylindrical cells, we see that -the young, germinating spore, at first spherical, very soon assumes -with growth an elliptical or ovoid form: the direct result of an -incipient anisotropy of its envelope, which when more developed -will convert the ovoid into a cylinder. We may also notice that -a truly cylindrical cell is comparatively rare; for in most cases, -what we call a cylindrical cell shews a distinct bulging of its sides; -it is not truly a cylinder, but a portion of a -spheroid or ellipsoid. <span class="xxpn" id="p245">{245}</span></p> - -<p>Unicellular organisms in general, including the protozoa, the -unicellular cryptogams, the various bacteria, and the free, -isolated cells, spores, ova, etc. of higher organisms, are referable -for the most part to a very small number of typical forms; but -besides a certain number of others which may be so referable, -though obscurely, there are obviously many others in which -either no symmetry is to be recognized, or in which the form is -clearly not one of equilibrium. Among these latter we have -Amoeba itself, and all manner of amoeboid organisms, and also -many curiously shaped cells, such as the Trypanosomes and various -other aberrant Infusoria. We shall return to the consideration of -these; but in the meanwhile it will suffice to say that, as their -surfaces are not equilibrium-surfaces, so neither are the living -cells themselves in any stable equilibrium. On the contrary, they -are in continual flux and movement, each portion of the surface -constantly changing its form, and passing from one phase to -another of an equilibrium which is never stable for more than -a moment. The former class, which rest in stable equilibrium, -must fall (as we have seen) into two classes,—those whose equilibrium -arises from liquid surface-tension alone, and those in -whose conformation some other pressure or restraint has been -superimposed upon ordinary surface-tension.</p> - -<p>To the fact that these little organisms belong to an order of -magnitude in which form is mainly, if not wholly, conditioned and -controlled by molecular forces, is due the limited range of -forms which they actually exhibit. These forms vary according -to varying physical conditions. Sometimes they do so in so regular -and orderly a way that we instinctively explain them merely as -“phases of a life-history,” and leave physical properties and -physical causation alone: but many of their variations of form we -treat as exceptional, abnormal, decadent or morbid, and are apt -to pass these over in neglect, while we give our attention to what -we suppose to be the typical or “characteristic” form or attitude. -In the case of the smallest organisms, the bacteria, micrococci, -and so forth, the range of form is especially limited, owing to their -minuteness, the powerful pressure which their highly curved -surfaces exert, and the comparatively homogeneous nature of their -substance. But within their narrow range -of possible diversity <span class="xxpn" id="p246">{246}</span> -these minute organisms are protean in their changes of form. -A certain species will not only change its shape from stage to -stage of its little “cycle” of life; but it will be remarkably different -in outward form according to the circumstances under which we -find it, or the histological treatment to which we submit it. Hence -the pathological student, commencing the study of bacteriology, -is early warned to pay little heed to differences of <i>form</i>, for purposes -of recognition or specific identification. Whatever grounds we -may have for attributing to these organisms a permanent or stable -specific identity (after the fashion of the higher plants and animals), -we can seldom safely do so on the ground of definite and always -recognisable <i>form</i>: we may</p> - -<div class="dctr01" id="fig73"><div id="fig74"> -<img src="images/i246.png" width="800" height="374" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td class="tdleft">Fig. 73. A flagellate “monad,” <i>Distigma - proteus</i>, Ehr. (After Saville Kent.)</td> - <td></td> - <td>Fig. 74. <i>Noctiluca miliaris.</i></td></tr></table> -</div></div></div><!--dctr01--> - -<p class="pcontinue">often be inclined, in short, to ascribe -to them a physiological (sometimes a “pathogenic”), rather than -a morphological specificity.</p> - -<hr class="hrblk"> - -<p>Among the Infusoria, we have a small number of forms whose -symmetry is distinctly spherical, for instance among the small -flagellate monads; but even these are seldom actually spherical -except when we see them in a non-flagellate and more or less -encysted or “resting” stage. In this condition, it need hardly be -remarked that the spherical form is common and general among -a great variety of unicellular organisms. When our little monad -developes a flagellum, that is in itself an indication of “polarity” -or symmetrical non-homogeneity of the cell; -and accordingly, we <span class="xxpn" id="p247">{247}</span> -usually see signs of an unequal tension of the membrane in the -neighbourhood of the base of the flagellum. Here the tension is -usually less than elsewhere, and the radius of curvature is accordingly -less: in other words that end of the cell is drawn out to a -tapering point (Fig. <a href="#fig73" title="go to Fig. 73">73</a>). But sometimes it is the other way, as -in Noctiluca, where the large flagellum springs from a depression -in the otherwise uniformly rounded cell. In this case the explanation -seems to lie in the many strands of radiating protoplasm -which converge upon this point, and may be supposed to keep it -relatively fixed by their viscosity, while the rest of the cell-surface -is free to expand (Fig. <a href="#fig74" title="go to Fig. 74">74</a>).</p> - -<div class="dctr01" id="fig75"> -<img src="images/i247.png" width="800" height="133" alt=""> - <div class="dcaption">Fig. 75. Various species of Vorticella. - (Mostly after Saville Kent.)</div></div> - -<p>A very large number of Infusoria represent unduloids, or -portions of unduloids, and this type of surface appears and -reappears in a great variety of forms. The cups of the various -species of Vorticella (Fig. <a href="#fig75" title="go to Fig. 75">75</a>) are nothing in the world but a -beautiful series of unduloids, or partial unduloids, in every gradation -from a form that is all but cylindrical to one that is all but -a perfect sphere. These unduloids are not completely symmetrical, -but they are such unduloids as develop themselves when we -suspend an oil-globule between two unequal rings, or blow a -soap-bubble between two unequal pipes; for, just as in these -cases, the surface of our Vorticella bell finds its terminal supports, -on the one hand in its attachment to its narrow stalk, and on the -other in the thickened ring from which spring its circumoral cilia. -And here let me say, that a point or zone from which cilia arise -would seem always to have a peculiar relation to the surrounding -tensions. It usually forms a sharp salient, a prominent point -or ridge, as in our little monads of Fig. <a href="#fig73" title="go to Fig. 73">73</a>; shewing that, -in its formation, the surface tension had here locally diminished. -But if such a ridge or fillet consolidate in the least degree, it -becomes a source of strength, and a <i>point d’appui</i> for the adjacent -film. We shall deal with this point again in -the next chapter. <span class="xxpn" id="p248">{248}</span></p> - -<p>Precisely the same series of unduloid forms may be traced in -even greater variety among various other families or genera of the</p> - -<div class="dctr01" id="fig76"> -<img src="images/i248a.png" width="800" height="321" alt=""> - <div class="dcaption">Fig. 76. Various species of <i>Salpingoeca</i>.</div></div> - -<div class="dctr01" id="fig77"> -<img src="images/i248b.png" width="800" height="257" alt=""> - <div class="dcaption">Fig. 77. Various species of <i>Tintinnus</i>, - <i>Dinobryon</i> and <i>Codonella</i>.<br>(After Saville Kent and - others.)</div></div> - -<p class="pcontinue">Infusoria. Sometimes, as in Vorticella itself, the unduloid is seen -merely in the contour of the soft semifluid body of the living -animal. At other times, as in Salpingoeca, Tintinnus, and many</p> - -<div class="dleft dwth-i" id="fig78"> -<img src="images/i248c.png" width="147" height="277" alt=""> - <div class="dcaption">Fig. 78. <i>Vaginicola.</i></div></div> - -<p class="pcontinue">other genera, we have a distinct -membranous cup, separate from the animal, but -originally secreted by, and moulded upon, its -semifluid living surface. Here we have an excellent -illustration of the contrast between the different -ways in which such a structure may be regarded -and interpreted. The teleological explanation is -that it is developed for the sake of protection, as a -domicile and shelter for the little organism within. -The mechanical explanation of the physicist (seeking -only after the “efficient,” and not the “final” cause), is -that it is <span class="xxpn" id="p249">{249}</span> -present, and has its actual conformation, by reason of certain -chemico-physical conditions: that it was inevitable, under the -given<br class="brclrfix"></p> - -<div class="dright dwth-g" id="fig79"> -<img src="images/i249a.png" width="261" height="283" alt=""> - <div class="dcaption">Fig. 79. <i>Folliculina.</i></div></div> - -<p class="pcontinue">conditions, that certain constituent -substances actually present in the protoplasm -should be aggregated by molecular -forces in its surface layer; that under this -adsorptive process, the conditions continuing -favourable, the particles should -accumulate and concentrate till they -formed (with the help of the surrounding -medium) a pellicle or membrane, thicker -or thinner as the case might be; that this -surface pellicle or membrane was inevitably bound, by molecular -forces, to become a surface of the least<br class="brclrfix"></p> - -<div class="dright dwth-g" id="fig80"> -<img src="images/i249b.png" width="261" height="706" alt=""> - <div class="dcaption">Fig. 80. <i>Trachelophyllum.</i> (After - Wreszniowski.)</div></div> - -<p class="pcontinue">possible area which the circumstances -permitted; that in the present case, the symmetry and “freedom” -of the system permitted, and <i>ipso facto</i> caused, this surface -to be a surface of revolution; and that of the few surfaces of -revolution which, as being also surfaces <i>minimae areae</i>, were -available, the unduloid was manifestly the one permitted, and -<i>ipso facto</i> caused, by the dimensions of the organisms and -other circumstances of the case. And just as the thickness or -thinness of the pellicle was obviously a subordinate matter, a -mere matter of degree, so we also see that the actual outline -of this or that particular unduloid is also a very subordinate -matter, such as physico-chemical variants of a minute kind -would suffice to bring about; for between the various unduloids -which the various species of Vorticella represent, there -is no more real difference than that difference of ratio -or degree which exists between two circles of different -diameter, or two lines of unequal length. <span class="xxpn" -id="p250">{250}</span></p> - -<p>In very many cases (of which Fig. <a href="#fig80" title="go to Fig. 80">80</a> is an example), we have -an unduloid form exhibited, not by a surrounding pellicle or shell, -but by the soft, protoplasmic body of a ciliated organism. In -such cases the form is mobile, and continually changes from one -to another unduloid contour, according to the movements of the -animal. We have here, apparently, to deal with an unstable -equilibrium, and also sometimes with the more complicated -problem of “stream-lines,” as in the difficult problems suggested -by the form of a fish. But this whole class of cases, and of -problems, we can merely take note of in passing, for their treatment -is too hard for us.</p> - -<hr class="hrblk"> - -<p>In considering such series of forms as the various unduloids -which we have just been regarding, we are brought sharply -up (as in the case of our Bacteria or Micrococci) against the biological -concept of organic <i>species</i>. In the intense classificatory -activity of the last hundred years, it has come about that every -form which is apparently characteristic, that is to say which is -capable of being described or portrayed, and capable of being -recognised when met with again, has been recorded as a species,—for -we need not concern ourselves with the occasional discussions, -or individual opinions, as to whether such and such a form deserve -“specific rank,” or be “only a variety.” And this secular labour -is pursued in direct obedience to the precept of the <i>Systema -Naturae</i>,—“<i>ut sic in summa confusione rerum apparenti, summus -conspiciatur Naturae ordo</i>.” In like manner the physicist records, -and is entitled to record, his many hundred “species” of snow-crystals<a class="afnanch" href="#fn299" id="fnanch299">299</a>, -or of crystals of calcium carbonate. But regarding -these latter species, the physicist makes no assumptions: he -records them <i>simpliciter</i>, as specific “forms”; he notes, as best -he can, the circumstances (such as temperature or humidity) -under which they occur, in the hope of elucidating the conditions -determining their formation; but above all, he -does not introduce <span class="xxpn" id="p251">{251}</span> -the element of time, and of succession, or discuss their origin and -affiliation as an <i>historical</i> sequence of events. But in biology, the -term species carries with it many large, though often vague -assumptions. Though the doctrine or concept of the “permanence -of species” is dead and gone, yet a certain definite value, or sort -of quasi-permanency, is still connoted by the term. Thus if a tiny -foraminiferal shell, a Lagena for instance, be found living to-day, -and a shell indistinguishable from it to the eye be found fossil -in the Chalk or some other remote geological formation, the -assumption is deemed legitimate that that species has “survived,” -and has handed down its minute specific character or characters, -from generation to generation, unchanged for untold myriads of -years<a class="afnanch" href="#fn300" id="fnanch300">300</a>. -Or if the ancient forms be like to, rather than identical -with the recent, we still assume an unbroken descent, accompanied -by the hereditary transmission of common characters and progressive -variations. And if two identical forms be discovered at -the ends of the earth, still (with occasional slight reservations on -the score of possible “homoplasy”), we build hypotheses on this -fact of identity, taking it for granted that the two appertain to -a common stock, whose dispersal in space must somehow be -accounted for, its route traced, its epoch determined, and its -causes discussed or discovered. In short, the naturalist admits -no exception to the rule that a “natural classification” can only -be a <i>genealogical</i> one, nor ever doubts that “<i>The fact that we are -able to classify organisms at all in accordance with the structural -characteristics which they present, is due to the fact of their being -related by descent</i><a class="afnanch" href="#fn301" id="fnanch301">301</a>.” -But this great generalisation is apt in my -opinion, to carry us too far. It may be safe and sure and helpful -and illuminating when we apply it to such complex entities,—such -thousand-fold resultants of the combination and permutation -of many variable characters,—as a horse, a lion or an eagle; -but (to my mind) it has a very different look, and a far less firm -foundation, when we attempt to extend it to minute organisms -whose specific characters are few and simple, -whose simplicity <span class="xxpn" id="p252">{252}</span> -becomes much more manifest when we regard it from the point -of view of physical and mathematical description and analysis, -and whose form is referable, or (to say the least of it) is very -largely referable, to the direct and immediate action of a particular -physical force. When we come to deal with the minute skeletons -of the Radiolaria we shall again find ourselves dealing with endless -modifications of form, in which it becomes still more difficult to -discern, or to apply, the guiding principle of affiliation or <i>genealogy</i>.</p> - -<div class="dleft dwth-e" id="fig81"> -<img src="images/i252.png" width="369" height="283" alt=""> - <div class="dcaption">Fig. 81.</div></div> - -<p>Among the more aberrant forms of Infusoria is a little species -known as <i>Trichodina pedicidus</i>, a parasite on the Hydra, or fresh-water -polype (Fig. <a href="#fig81" title="go to Fig. 81">81</a>.) This Trichodina has the form of a more or less -flattened circular disc, with a ring -of cilia around both its upper and -lower margins. The salient ridge -from which these cilia spring may -be taken, as we have already said, -to play the part of a strengthening -“fillet.” The circular base of the -animal is flattened, in contact with -the flattened surface of the Hydra -over which it creeps, and the opposite, -upper surface may be flattened nearly to a plane, or may at -other times appear slightly convex or slightly concave. The sides -of the little organism are contracted, forming a symmetrical -equatorial groove between the upper and lower discs; and, on -account of the minute size of the animal and its constant -movements, we cannot submit the curvature of this concavity to -measurement, nor recognise by the eye its exact contour. But -it is evident that the conditions are precisely similar to those -described on p. <a href="#p223" title="go to pg. 223">223</a>, where we were considering the conditions -of stability of the catenoid. And it is further evident that, when -the upper disc is actually plane, the equatorial groove is strictly -a catenoid surface of revolution; and when on the other hand it -is depressed, then the equatorial groove will tend to assume -the form of a nodoidal surface.<br class="brclrfix"></p> - -<p>Another curious type is the flattened spiral -of <i>Dinenympha</i><a class="afnanch" href="#fn302" id="fnanch302">302</a> -<span class="xxpn" id="p253">{253}</span> -which reminds us of the cylindrical spiral of a Spirillum among -the bacteria. In Dinenympha we have a symmetrical figure, whose -two opposite surfaces each constitute a surface of constant mean -curvature; it is evidently a figure of equilibrium under certain -special conditions of restraint. The cylindrical coil of the -Spirillum, on the other hand, is a surface of constant mean curvature, -and therefore of equilibrium, as truly, and in the same sense, -as the cylinder itself.</p> - -<div class="dctr05" id="fig82"> -<img src="images/i253.png" width="449" height="567" alt=""> - <div class="dcaption">Fig. 82. <i>Dinenympha gracilis</i>, Leidy.</div></div> - -<p>A very curious conformation is that of the vibratile “collar,” -found in Codosiga and the other “Choanoflagellates,” and which -we also meet with in the “collar-cells” which line the interior -cavities of a sponge. Such collar-cells are always very minute, -and the collar is constituted of a very delicate film, which -shews an undulatory or rippling motion. It is a surface of -revolution, and as it maintains itself in equilibrium (though a -somewhat unstable and fluctuating one), it must be, under the -restricted circumstances of its case, a surface of minimal area. -But it is not so easy to see what these special circumstances are; -and it is obvious that the collar, if left to itself, -must at once <span class="xxpn" id="p254">{254}</span> -contract downwards towards its base, and become confluent with</p> - -<div class="dleft dwth-h" id="fig83"> -<img src="images/i254.png" width="201" height="437" alt=""> - <div class="dcaption">Fig. 83.</div></div> - -<p class="pcontinue">the general surface of the cell; for it -has no longitudinal supports and no strengthening ring at -its periphery. But in all these collar-cells, there stands -within the annulus of the collar a large and powerful cilium -or flagellum, in constant movement; and by the action of -this flagellum, and doubtless in part also by the intrinsic -vibrations of the collar itself, there is set up a constant -steady current in the surrounding water, whose direction would -seem to be such that it passes up the outside of the collar, -down its inner side, and out in the middle in the direction of -the flagellum; and there is a distinct eddy, in which foreign -particles tend to be caught, around the peripheral margin of -the collar. When the cell dies, that is to say when motion -ceases, the collar immediately shrivels away and disappears. -It is notable, by the way, that the edge of this little mobile -cup is always smooth, never notched or lobed as in the cases -we have discussed on p. <a href="#p236" title="go to pg. 236">236</a>: this latter condition being -the outcome of a definite instability, marking the close of -a period of equilibrium; while in the vibratile collar of -Codosiga the equilibrium, such as it is, is being constantly -renewed and perpetuated like that of a juggler’s pole, by the -motions of the system. I take it that, somehow, its existence -(in a state of partial equilibrium) is due to the current -motions, and to the traction exerted upon it through the -friction of the stream which is constantly passing by. I think, -in short, that it is formed very much in the same way as the -cup-like ring of streaming ribbons, which we see fluttering -and vibrating in the air-current of a ventilating fan.<br -class="brclrfix"></p> - -<p>It is likely enough, however, that a different and -much better explanation may yet be found; and if we turn -once more to Mr Worthington’s <i>Study of Splashes</i>, we may -find a curious suggestion of analogy in the beautiful -craters encircling a central jet (as the collar of Codosiga -encircles the flagellum), which we see produced in the -later stages of the splash of a pebble<a class="afnanch" -href="#fn303" id="fnanch303">303</a>. <span class="xxpn" -id="p255">{255}</span></p> - -<p>Among the Foraminifera we have an immense variety of forms, -which, in the light of surface tension and of the principle of -minimal area, are capable of explanation and of reduction to a -small number of characteristic types. Many of the Foraminifera -are composite structures, formed by the successive imposition of -cell upon cell, and these we shall deal with later on; let us glance -here at the simpler conformations exhibited by the single chambered -or “monothalamic” genera, and perhaps one or two of the -simplest composites.</p> - -<p>We begin with forms, like Astrorhiza (Fig. <a href="#fig219" title="go to Fig. 219">219</a>, -p. 464), which -are in a high degree irregular, and end with others which manifest a -perfect and mathematical regularity. The broad difference between -these two types is that the former are characterised, like Amoeba, -by a variable surface tension, and consequently by unstable equilibrium; -but the strong contrast between these and the regular forms -is bridged over by various transition-stages, or differences of degree. -Indeed, as in all other Rhizopods, the very fact of the emission of -pseudopodia, which reach their highest development in this group -of animals, is a sign of unstable surface-equilibrium; and we must -therefore consider that those forms which indicate symmetry and -equilibrium in their shells have secreted these during periods when -rest and uniformity of surface conditions alternated with the -phases of pseudopodial activity. The irregular forms are in -almost all cases arenaceous, that is to say they have no solid shells -formed by steady adsorptive secretion, but only a looser covering -of sand grains with which the protoplasmic body has come in -contact and cohered. Sometimes, as in Ramulina, we have a -calcareous shell combined with irregularity of form; but here we -can easily see a partial and as it were a broken regularity, the -regular forms of sphere and cylinder being repeated in various -parts of the ramified mass. When we look more closely at the -arenaceous forms, we find that the same thing is true of them; -they represent, either in whole or part, approximations to the form -of surfaces of equilibrium, spheres, cylinders and so forth. In -Aschemonella we have a precise replica of the calcareous Ramulina; -and in Astrorhiza itself, in the forms distinguished by naturalists -as <i>A. crassatina</i>, what is described as -the “subsegmented interior<a class="afnanch" href="#fn304" id="fnanch304">304</a>” -<span class="xxpn" id="p256">{256}</span> -seems to shew the natural, physical tendency of the long semifluid -cylinder of protoplasm to contract, at its limit of stability, into -unduloid constrictions, as a step towards the breaking up into -separate spheres: the completion of which process is restrained or -prevented by the rigidity and friction of the arenaceous covering.</p> - -<div class="dctr01" id="fig84"> -<img src="images/i256.png" width="800" height="651" alt=""> - <div class="dcaption">Fig. 84. Various species of <i>Lagena</i>. - (After Brady.)</div></div> - -<p>Passing to the typical, calcareous-shelled Foraminifera, we have -the most symmetrical of all possible types in the perfect sphere of -Orbulina; this is a pelagic organism, whose floating habitat places -it in a position of perfect symmetry towards all external forces. -Save for one or two other forms which are also spherical, or -approximately so, like Thurammina, the rest of the monothalamic -calcareous Foraminifera are all comprised by naturalists within -the genus Lagena. This large and varied genus consists of “flask-shaped” -shells, whose surface is simply that of an unduloid, or -more frequently, like that of a flask itself, an unduloid combined -with a portion of a sphere. We do not -know the circumstances <span class="xxpn" id="p257">{257}</span> -under which the shell of Lagena is formed, nor the nature of the -force by which, during its formation, the surface is stretched out -into the unduloid form; but we may be pretty sure that it is -suspended vertically in the sea, that is to say in a position of -symmetry as regards its vertical axis, about which the unduloid -surface of revolution is symmetrically formed. At the same time -we have other types of the same shell in which the form is more -or less flattened; and these are doubtless the cases in which such -symmetry of position was not present, or was replaced by a broader, -lateral contact with the surface pellicle<a class="afnanch" href="#fn305" id="fnanch305">305</a>.</p> - -<div class="dctr01" id="fig85"> -<img src="images/i257.png" width="800" height="321" alt=""> - <div class="dcaption">Fig. 85. (After Darling.)</div></div> - -<p>While Orbulina is a simple spherical drop, Lagena suggests to -our minds a “hanging drop,” drawn out to a long and slender -neck by its own weight, aided by the viscosity of the material. -Indeed the various hanging drops, such as Mr C. R. Darling shews -us, are the most beautiful and perfect unduloids, with spherical -ends, that it is possible to conceive. A suitable liquid, a little -denser than water and incapable of mixing with it (such as -ethyl benzoate), is poured on a surface of -water. It spreads <span class="xxpn" id="p258">{258}</span> -over the surface and gradually forms a hanging drop, approximately -hemispherical; but as more liquid is added the drop -sinks or rather grows downwards, still adhering to the surface -film; and the balance of forces between gravity and surface -tension results in the unduloid contour, as the increasing weight -of the drop tends to stretch it out and finally break it in two. -At the moment of rupture, by the way, a tiny droplet is formed -in the attenuated neck, such as we described in the normal -division of a cylindrical thread (p. <a href="#p233" title="go to pg. 233">233</a>).</p> - -<div class="psmprnt3"> -<p>To pass to a much more highly organised class of animals, -we find the unduloid beautifully exemplified in the little -flask-shaped shells of certain Pteropod mollusca, e.g. -Cuvierina<a class="afnanch" href="#fn306" id="fnanch306">306</a>. -Here again the symmetry of the figure would -at once lead us to suspect that the creature lived in a -position of symmetry to the surrounding forces, as for -instance if it floated in the ocean in an erect position, -that is to say with its long axis coincident with the -direction of gravity; and this we know to be actually the -mode of life of the little Pteropod.</p> -</div><!--psmprnt3--> - -<p>Many species of Lagena are complicated and beautified by a -pattern, and some by the superaddition to the shell of plane -extensions or “wings.” These latter give a secondary, bilateral -symmetry to the little shell, and are strongly suggestive of a -phase or period of growth in which it lay horizontally on the -surface, instead of hanging vertically from the surface-film: in -which, that is to say, it was a floating and not a hanging -drop. The pattern is of two kinds. Sometimes it consists -of a sort of fine reticulation, with rounded or more or -less hexagonal interspaces: in other cases it is produced by a -symmetrical series of ridges or folds, usually longitudinal, on the -body of the flask-shaped cell, but occasionally transversely arranged -upon the narrow neck. The reticulated and folded patterns we -may consider separately. The netted pattern is very similar to the -wrinkled surface of a dried pea, or to the more regular wrinkled -patterns upon many other seeds and even pollen-grains. If a -spherical body after developing a “skin” begin to shrink a little, -and if the skin have so far lost its elasticity as to be unable to -keep pace with the shrinkage of the inner mass, it will tend to -fold or wrinkle; and if the shrinkage be uniform, and the elasticity -and flexibility of the skin be also uniform, -then the amount of <span class="xxpn" id="p259">{259}</span> -folding will be uniformly distributed over the surface. Little -concave depressions will appear, regularly interspaced, and -separated by convex folds. The little concavities being of equal -size (unless the system be otherwise perturbed) each one will tend -to be surrounded by six others; and when the process has reached -its limit, the intermediate boundary-walls, or raised folds, will be -found converted into a regular pattern of hexagons.</p> - -<p>But the analogy of the mechanical wrinkling of the coat of -a seed is but a rough and distant one; for we are evidently dealing -with molecular rather than with mechanical forces. In one of -Darling’s experiments, a little heavy tar-oil is dropped onto a -saucer of water, over which it spreads in a thin film showing -beautiful interference colours after the fashion of those of a soap-bubble. -Presently tiny holes appear in the film, which gradually -increase in size till they form a cellular pattern or honeycomb, -the oil gathering together in the meshes or walls of the cellular -net. Some action of this sort is in all probability at work in a -surface-film of protoplasm covering the shell. As a physical -phenomenon the actions involved are by no means fully understood, -but surface-tension, diffusion and cohesion doubtless play -their respective parts therein<a class="afnanch" href="#fn307" id="fnanch307">307</a>. -The very perfect cellular patterns -obtained by Leduc (to which we shall have occasion to refer in -a subsequent chapter) are diffusion patterns on a larger scale, but -not essentially different.</p> - -<div class="dleft dwth-g" id="fig86"> -<img src="images/i260.png" width="263" height="246" alt=""> - <div class="dcaption">Fig. 86.</div></div> - -<p>The folded or pleated pattern is doubtless to be explained, in -a general way, by the shrinkage of a -surface-film under certain <span class="xxpn" id="p260">{260}</span> -conditions of viscous or frictional restraint. A case which (as it -seems to me) is closely analogous to that of our foraminiferal -shells is described by Quincke<a class="afnanch" href="#fn308" id="fnanch308">308</a>, -who let a film of albumin or of -resin set and harden upon a surface of quicksilver, and found -that the little solid pellicle had been -thrown into a pattern of symmetrical -folds. If the surface thus thrown into -folds be that of a cylinder, or any other -figure with one principal axis of symmetry, -such as an ellipsoid or unduloid, -the direction of the folds will tend to -be related to the axis of symmetry, -and we might expect accordingly to -find regular longitudinal, or regular transverse wrinkling. Now -as a matter of fact we almost invariably find in the Lagena -the former condition: that is to say, in our ellipsoid or unduloid -cell, the puckering takes the form of the vertical fluting on -a column, rather than that of the transverse pleating of an -accordion. And further, there is often a tendency for such -longitudinal flutings to be more or less localised at the end of the -ellipsoid, or in the region where the unduloid merges into its -spherical base. In this latter region we often meet with a regular -series of short longitudinal folds, as we do in the forms of Lagena -denominated <i>L. semistriata</i>. All these various forms of surface -can be imitated, or rather can be precisely reproduced, by the art -of the glass-blower<a class="afnanch" href="#fn309" id="fnanch309">309</a>. -<br class="brclrfix"></p> - -<p>Furthermore, they remind one, in a striking way, of the -regular ribs or flutings in the film or sheath which splashes up to -envelop a smooth ball which has been dropped into a liquid, as -Mr Worthington has so beautifully shewn<a class="afnanch" href="#fn310" id="fnanch310">310</a>. -<span class="xxpn" id="p261">{261}</span></p> - -<div class="dmaths"> -<p>In Mr Worthington’s experiment, there appears to -be something of the nature of a viscous drag in the -surface-pellicle; but whatever be the actual cause of variation -of tension, it is not difficult to see that there must be -in general a tendency towards <i>longitudinal</i> puckering -or “fluting” in the case of a thin-walled cylindrical -or other elongated body, rather than a tendency towards -transverse puckering, or “pleating.” For let us suppose -that some change takes place involving an increase of -surface-tension in some small area of the curved wall, and -leading therefore to an increase of pressure: that is to -say let <i>T</i> become <i>T</i> + <i>t</i>, and <i>P</i> become -<i>P</i> + <i>p</i>. Our new equation of equilibrium, then, -in place of <i>P</i> -= <i>T ⁄ r</i> + <i>T ⁄ r′</i> -becomes</p> - -<div><i>P</i> + <i>p</i> -= (<i>T</i> + <i>t</i>) ⁄ <i>r</i> + (<i>T</i> + <i>t</i>) ⁄ <i>r′</i>, -</div> - -<p class="pcontinue">and by subtraction,</p> - -<div><i>p</i> -= <i>t ⁄ r</i> + <i>t ⁄ r′</i>. -</div> - -<p class="pcontinue">Now if</p> - -<div><i>r</i> < <i>r′</i>,       -<i>t ⁄ r</i> > <i>t ⁄ r′</i>.</div> - -<p class="pcontinue">Therefore, in order to produce the small increment of pressure <i>p</i>, -it is easier to do so by increasing <i>t ⁄ r</i> than <i>t ⁄ r′</i>; that is to say, the -easier way is to alter, or diminish <i>r</i>. And the same will hold good -if the tension and pressure be diminished instead of increased.</p> -</div><!--dmaths--> - -<p>This is as much as to say that, when corrugation or “rippling” -of the walls takes place owing to small changes of surface-tension, -and consequently of pressure, such corrugation is more likely to -take place in the plane of <i>r</i>,—that is to say, <i>in the plane of greatest -curvature</i>. And it follows that in such a figure as an ellipsoid, -wrinkling will be most likely to take place not only in a longitudinal -direction but near the extremities of the figure, that is to say again -in the region of greatest curvature.</p> - -<div class="dctr01" id="fig87"><div id="fig88"> -<img src="images/i262.png" width="800" height="442" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td class="tdleft">Fig. 87. - <i>Nodosaria scalaris</i>, Batsch.</td> - <td></td> - <td class="tdleft">Fig. 88. - Gonangia of Campanularians. (<i>a</i>) <i>C. gracilis</i>; - (<i>b</i>) <i>C. grandis</i>. (After Allman.)</td></tr></table> -</div></div></div><!--dctr01--> - -<p>The longitudinal wrinkling of the flask-shaped bodies of our -Lagenae, and of the more or less cylindrical cells of many other -Foraminifera (Fig. <a href="#fig87" title="go to Fig. 87">87</a>), is in complete accord with the above theoretical -considerations; but nevertheless, we soon find that our result -is not a general one, but is defined by certain limiting conditions, -and is accordingly subject to what are, at first sight, important -exceptions. For instance, when we turn to the narrow neck of -the Lagena we see at once that our theory no -longer holds; for <span class="xxpn" id="p262">{262}</span> -the wrinkling which was invariably longitudinal in the body of -the cell is as invariably transverse in the narrow neck. The reason -for the difference is not far to seek. The conditions in the neck -are very different from those in the expanded portion of the cell: -the main difference being that the thickness of the wall is no longer -insignificant, but is of considerable magnitude as compared with -the diameter, or circumference, of the neck. We must accordingly -take it into account in considering the <i>bending moments</i> at any -point in this region of the shell-wall. And it is at once obvious -that, in any portion of the narrow neck, <i>flexure</i> of a wall in a -transverse direction will be very difficult, while flexure in a -longitudinal direction will be comparatively easy; just as, in the -case of a long narrow strip of iron, we may easily bend it into -folds running transversely to its long axis, but not the other way. -The manner in which our little Lagena-shell tends to fold or wrinkle, -longitudinally in its wider part, and transversely or annularly in -its narrow neck, is thus completely and easily explained.</p> - -<p>An identical phenomenon is apt to occur in the little flask-shaped -gonangia, or reproductive capsules, of some of the hydroid -zoophytes. In the annexed drawings of these gonangia in two -species of Campanularia, we see that in one case -the little vesicle <span class="xxpn" id="p263">{263}</span> -has the flask-shaped or unduloid configuration of a Lagena; and -here the walls of the flask are longitudinally fluted, just after the -manner we have witnessed in the latter genus. But in the other -Campanularian the vesicles are long, narrow and tubular, and here -a transverse folding or pleating takes the place of the longitudinally -fluted pattern. And the very form of the folds or pleats is -enough to suggest that we are not dealing here with a simple -phenomenon of surface-tension, but with a condition in which -surface-tension and <i>stiffness</i> are both present, and play their -parts in the resultant form.</p> - -<div class="dctr01" id="fig89"> -<img src="images/i263.png" width="800" height="354" alt=""> - <div class="pcaption">Fig. 89. Various Foraminifera (after Brady), <i>a</i>, - <i>Nodosaria simplex</i>; <i>b</i>, <i>N. pygmaea</i>; <i>c</i>, <i>N. - costulata</i>; <i>e</i>, <i>N. hispida</i>; <i>f</i>, <i>N. elata</i>; <i>d</i>, - <i>Rheophax</i> (<i>Lituola</i>) <i>distans</i>; <i>g</i>, <i>Sagrina - virgata</i>.</div></div> - -<p>Passing from the solitary flask-shaped cell of Lagena, we have, -in another series of forms, a constricted cylinder, or succession -of unduloids; such as are represented in Fig. <a href="#fig89" title="go to Fig. 89">89</a>, illustrating -certain species of Nodosaria, Rheophax and Sagrina. In some of -these cases, and certainly in that of the arenaceous genus Rheophax, -we have to do with the ordinary phenomenon of a segmenting or -partially segmenting cylinder. But in others, the structure is -not developed out of a continuous protoplasmic cylinder, but as -we can see by examining the interior of the shell, it has been -formed in successive stages, beginning with a simple unduloid -“Lagena,” about whose neck, after its solidification, another drop -of protoplasm accumulated, and in turn assumed the unduloid, -or lagenoid, form. The chains of -interconnected bubbles which <span class="xxpn" id="p264">{264}</span> -Morey and Draper made many years ago of melted resin are a -very similar if not identical phenomenon<a class="afnanch" href="#fn311" id="fnanch311">311</a>.</p> - -<hr class="hrblk"> - -<p>There now remain for our consideration, among the Protozoa, -the great oceanic group of the Radiolaria, and the little group of -their freshwater allies, the Heliozoa. In nearly all these forms we -have this specific chemical difference from the Foraminifera, that -when they secrete, as they generally do secrete, a hard skeleton, -it is composed of silica instead of lime. These organisms and the -various beautiful and highly complicated skeletal fabrics which -they develop give us many interesting illustrations of physical -phenomena, among which the manifestations of surface-tension -are very prominent. But the chief phenomena connected with -their skeletons we shall deal with in another place, under the head -of spicular concretions.</p> - -<p>In a simple and typical Heliozoan, such as the Sun-animalcule, -<i>Actinophrys sol</i>, we have a “drop” of protoplasm, contracted by -its surface tension into a spherical form. Within the heterogeneous -protoplasmic mass are more fluid portions, and at the surface -which separates these from the surrounding protoplasm a similar -surface tension causes them also to assume the form of spherical -“vacuoles,” which in reality are little clear drops within the big -one; unless indeed they become numerous and closely packed, in -which case, instead of isolated spheres or droplets they will -constitute a “froth,” their mutual pressures and tensions giving -rise to regular configurations such as we shall study in the next -chapter. One or more of such clear spaces may be what is called -a “contractile vacuole”: that is to say, a droplet whose surface -tension is in unstable equilibrium and is apt to vanish altogether, -so that the definite outline of the vacuole suddenly disappears<a class="afnanch" href="#fn312" id="fnanch312">312</a>. -Again, within the protoplasm are one or more nuclei, whose own -surface tension (at the surface between the nucleus and the -surrounding protoplasm), has drawn them in -turn into the shape <span class="xxpn" id="p265">{265}</span> -of spheres. Outwards through the protoplasm, and stretching far -beyond the spherical surface of the cell, there run stiff linear -threads of modified or differentiated protoplasm, replaced or -reinforced in some cases by delicate siliceous needles. In either -case we know little or nothing about the forces which lead to their -production, and we do not hide our ignorance when we ascribe -their development to a “radial polarisation” of the cell. In the -case of the protoplasmic filament, we may (if we seek for a -hypothesis), suppose that it is somehow comparable to a viscid -stream, or “liquid vein,” thrust or squirted out from the body of -the cell. But when it is once formed, this long and comparatively -rigid filament is separated by a distinct surface from the neighbouring -protoplasm, that is to say from the more fluid surface-protoplasm -of the cell; and the latter begins to creep up the -filament, just as water would creep up the interior of a glass tube, -or the sides of a glass rod immersed in the liquid. It is the simple -case of a balance between three separate tensions: (1) that between -the filament and the adjacent protoplasm, (2) that between the -filament and the adjacent water, and (3) that between the water -and the protoplasm. Calling these tensions respectively -<i>T<sub>fp</sub></i>, <i>T<sub>fw</sub></i>, -and <i>T<sub>wp</sub></i>, equilibrium will be attained when the angle of contact -between the fluid protoplasm and the filament is such that -cos α -= (<i>T<sub>fw</sub></i> − <i>T<sub>wp</sub></i>) ⁄ <i>T<sub>fp</sub></i>. -It is evident in this case that the angle is -a very small one. The precise form of the curve is somewhat -different from that which, under ordinary circumstances, is assumed -by a liquid which creeps up a solid surface, as water in contact -with air creeps up a surface of glass; the difference being due to -the fact that here, owing to the density of the protoplasm being -practically identical with that of the surrounding medium, the -whole system is practically immune from gravity. Under normal -circumstances the curve is part of the “elastic curve” by which -that surface of revolution is generated which we have called, -after Plateau, the nodoid; but in the present case it is apparently -a catenary. Whatever curve it be, it obviously forms a surface -of revolution around the filament.</p> - -<p>Since the attraction exercised by this surface tension is -symmetrical around the filament, the latter will -be pulled equally <span class="xxpn" id="p266">{266}</span> -in all directions; in other words it will tend to be set normally -to the surface of the sphere, that is to say radiating directly -outwards from the centre. If the distance between two adjacent -filaments be considerable, the curve will simply meet the filament -at the angle α already referred to; but if they be sufficiently near -together, we shall have a continuous catenary curve forming a -hanging loop between one filament and the other. And when this -is so, and the radial filaments are more or less symmetrically -interspaced, we may have a beautiful system of honeycomb-like -depressions over the surface of the organism, each cell of the -honeycomb having a strictly defined geometric configuration.</p> - -<div class="dctr01" id="fig90"> -<img src="images/i266.png" width="800" height="240" - alt=""> <div class="pcaption">Fig. 90. A, <i>Trypanosoma - tineae</i> (after Minchin); B, <i>Spirochaeta anodontae</i> (after - Fantham).</div></div> - -<p>In the simpler Radiolaria, the spherical form of the entire -organism is equally well-marked; and here, as also in the more -complicated Heliozoa (such as Actinosphaerium), the organism is -differentiated into several distinct layers, each boundary surface -tending to be spherical, and so constituting sphere within sphere. -One of these layers at least is close packed with vacuoles, forming -an “alveolar meshwork,” with the configurations of which we shall -attempt in another chapter to correlate the characteristic structure -of certain complex types of skeleton.</p> - -<hr class="hrblk"> - -<p>An exceptional form of cell, but a beautiful manifestation of -surface-tension (or so I take it to be), occurs in Trypanosomes, those -tiny parasites of the blood that are associated with sleeping-sickness -and many other grave or dire maladies. These tiny -organisms consist of elongated solitary cells down one side of which -runs a very delicate frill, or “undulating membrane,” the free -edge of which is seen to be slightly thickened, and -the whole of <span class="xxpn" id="p267">{267}</span> -which undergoes rhythmical and beautiful wavy movements. -When certain Trypanosomes are artificially cultivated (for instance -<i>T. rotatorium</i>, from the blood of the frog), phases of growth are -witnessed in which the organism has no undulating membrane, -but possesses a long cilium or “flagellum,” springing from near -the front end, and exceeding the whole body in length<a class="afnanch" href="#fn313" id="fnanch313">313</a>. -Again, -in <i>T. lewisii</i>, when it reproduces by “multiple fission,” the -products of this division are likewise devoid of an undulating -membrane, but are provided with a long free flagellum<a class="afnanch" href="#fn314" id="fnanch314">314</a>. -It is -a plausible assumption to suppose that, as the flagellum waves -about, it comes to lie near and parallel to the body of the cell, -and that the frill or undulating membrane is formed by the clear, -fluid protoplasm of the surface layer springing up in a film to run -up and along the flagellum, just as a soap-film would be formed in -similar circumstances.</p> - -<div class="dctr04" id="fig91"> -<img src="images/i267.png" width="535" height="556" alt=""> - <div class="pcaption">Fig. 91. A, <i>Trichomonas muris</i>, Hartmann; - B, <i>Trichomastix serpentis</i>, Dobell; C, <i>Trichomonas - angusta</i>, Alexeieff. (After Kofoid.)</div></div> - -<p>This mode of formation of the undulating membrane or frill -appears to be confirmed by the appearances -shewn in Fig. <a href="#fig91" title="go to Fig. 91">91</a>. <span class="xxpn" id="p268">{268}</span> -Here we have three little organisms closely allied to the ordinary -Trypanosomes, of which one, Trichomastix (<i>B</i>), possesses four -flagella, and the other two, Trichomonas, apparently three only: -the two latter possess the frill, which is lacking in the first<a class="afnanch" href="#fn315" id="fnanch315">315</a>. -But -it is impossible to doubt that when the frill is present (as in <i>A</i> and -<i>C</i>), its outer edge is constituted by the apparently missing flagellum -(<i>a</i>), which has become <i>attached</i> to the body of the creature at the -point <i>c</i>, near its posterior end; and all along its course, the superficial -protoplasm has been drawn out into a film, between the -flagellum (<i>a</i>) and the adjacent surface or edge of the body (<i>b</i>).</p> - -<div class="dleft dwth-e" id="fig92"> -<img src="images/i268.png" width="322" height="459" alt=""> - <div class="pcaption">Fig. 92. Herpetomonas assuming the -undulatory membrane of a Trypanosome. (After D. L. -Mackinnon.)</div></div> - -<p>Moreover, this mode of formation has been actually witnessed -and described, though in a somewhat exceptional case. The little -flagellate monad Herpetomonas is normally destitute of an undulating -membrane, but possesses a single long terminal flagellum. -According to Dr D. L. Mackinnon, the cytoplasm in a certain stage -of growth becomes somewhat “sticky,” a phrase which we may -in all probability interpret to mean that its surface tension is -being reduced. For this stickiness is -shewn in two ways. In the first place, -the long body, in the course of its -various bending movements, is apt to -adhere head to tail (so to speak), giving -a rounded or sometimes annular form -to the organism, such as has also been -described in certain species or stages -of Trypanosomes. But again, the -long flagellum, if it get bent backwards -upon the body, tends to adhere -to its surface. “Where the flagellum -was pretty long and active, its efforts -to continue movement under these -abnormal conditions resulted in the -gradual lifting up from the cytoplasm -of the body of a sort of <i>pseudo</i>-undulating -membrane (Fig. <a href="#fig92" title="go to Fig. 92">92</a>). The movements of this structure -were so exactly those of a true undulating -membrane that it was <span class="xxpn" id="p269">{269}</span> -difficult to believe one was not dealing with a small, blunt -trypanosome<a class="afnanch" href="#fn316" id="fnanch316">316</a>.” -This in short is a precise description of the -mode of development which, from theoretical considerations -alone, we should conceive to be the natural if not the only -possible way in which the undulating membrane could come into -existence.</p> - -<p>There is a genus closely allied to Trypanosoma, viz. Trypanoplasma, -which possesses one free flagellum, together with an -undulating membrane; and it resembles the neighbouring genus -Bodo, save that the latter has two flagella and no undulating -membrane. In like manner, Trypanosoma so closely resembles -Herpetomonas that, when individuals ascribed to the former genus -exhibit a free flagellum only, they are said to be in the “Herpetomonas -stage.” In short all through the order, we have pairs -of genera, which are presumed to be separate and distinct, viz. -Trypanosoma-Herpetomonas, Trypanoplasma-Bodo, Trichomastix-Trichomonas, -in which one differs from the other mainly if not -solely in the fact that a free flagellum in the one is replaced by an -undulating membrane in the other. We can scarcely doubt that -the two structures are essentially one and the same.</p> - -<p>The undulating membrane of a Trypanosome, then, according -to our interpretation of it, is a liquid film and must obey the law -of constant mean curvature. It is under curious limitations of -freedom: for by one border it is attached to the comparatively -motionless body, while its free border is constituted by a flagellum -which retains its activity and is being constantly thrown, like the -lash of a whip, into wavy curves. It follows that the membrane, -for every alteration of its longitudinal curvature, must at the same -instant become curved in a direction perpendicular thereto; it -bends, not as a tape bends, but with the accompaniment of beautiful -but tiny waves of double curvature, all tending towards the -establishment of an “equipotential surface”; and its characteristic -undulations are not originated by an active mobility of the -membrane but are due to the molecular tensions which produce -the very same result in a soap-film under similar circumstances.</p> - -<p>In certain Spirochaetes, <i>S. anodontae</i> (Fig. <a href="#fig90" title="go to Fig. 90">90</a>) -and <i>S. balbiani</i> <span class="xxpn" id="p270">{270}</span> -(which we find in oysters), a very similar undulating membrane -exists, but it is coiled in a regular spiral round the body of the cell. -It forms a “screw-surface,” or helicoid, and, though we might -think that nothing could well be more curved, yet its mathematical -properties are such that it constitutes a “ruled surface” whose -“mean curvature” is everywhere <i>nil</i>; and this property (as we -have seen) it shares with the plane, and with the plane alone. -Precisely such a surface, and of exquisite beauty, may be -produced by bending a wire upon itself so that part forms an -axial rod and part a spiral wrapping round the axis, and then -dipping the whole into a soapy solution.</p> - -<p>These undulating and helicoid surfaces are exactly reproduced -among certain forms of spermatozoa. The tail of a spermatozoon -consists normally of an axis surrounded by clearer and more fluid -protoplasm, and the axis sometimes splits up into two or more -slender filaments. To surface tension operating between these -and the surface of the fluid protoplasm (just as in the case of the -flagellum of the Trypanosome), I ascribe the formation of the -undulating membrane which we find, for instance, in the spermatozoa -of the newt or salamander; and of the helicoid membrane, -wrapped in a far closer and more beautiful spiral than that which -we saw in Spirochaeta, which is characteristic of the spermatozoa -of many birds.</p> - -<hr class="hrblk"> - -<p>Before we pass from the subject of the conformation of the -solitary cell we must take some account of certain other exceptional -forms, less easy of explanation, and still less perfectly understood. -Such is the case, for instance, with the red blood-corpuscles of man -and other vertebrates; and among the sperm-cells of the decapod -crustacea we find forms still more aberrant and not less perplexing. -These are among the comparatively few cells or cell-like structures -whose form <i>seems</i> to be incapable of explanation by theories of -surface-tension.</p> - -<p>In all the mammalia (save a very few) the red blood-corpuscles -are flattened circular discs, dimpled in upon their two opposite -sides. This configuration closely resembles that of an india-rubber -ball when we pinch it tightly between finger and thumb; -and we may also compare it with that -experiment of Plateau’s <span class="xxpn" id="p271">{271}</span> -(described on p. <a href="#p223" title="go to pg. 223">223</a>), where a flat cylindrical oil-drop, of certain -relative dimensions, can, by sucking away a little of the contained -oil, be made to assume the form of a biconcave disc, whose periphery -is part of a nodoidal surface. From the relation of the nodoid -to the “elastic curve,” we perceive that these two examples are -closely akin one to the other.</p> - -<div class="dright dwth-e" id="fig93"> -<img src="images/i271.png" width="390" height="165" alt=""> - <div class="dcaption">Fig. 93.</div></div> - -<p>The form of the corpuscle is symmetrical, and its surface is -a surface of revolution; but it -is obviously not a surface of -constant mean curvature, nor of -constant pressure. For we see -at once that, in the sectional -diagram (Fig. <a href="#fig93" title="go to Fig. 93">93</a>), the pressure -inwards due to surface tension -is positive at <i>A</i>, and negative at <i>C</i>; at <i>B</i> there is no -curvature in the plane of the paper, while perpendicular to -it the curvature is negative, and the pressure therefore is also -negative. Accordingly, from the point of view of surface tension -alone, the blood-corpuscle is not a surface of equilibrium; or in -other words, it is not a fluid drop suspended in another liquid. -It is obvious therefore that some other force or forces must be -at work, and the simple effect of mechanical pressure is here -excluded, because the blood-corpuscle exhibits its characteristic -shape while floating freely in the blood. In the lower vertebrates -the blood-corpuscles have the form of a flattened oval disc, with -rather sharp edges and ellipsoidal surfaces, and this again is -manifestly not a surface of equilibrium.</p> - -<p>Two facts are especially noteworthy in connection with the -form of the blood-corpuscle. In the first place, its form is only -maintained, that is to say it is only in equilibrium, in relation to -certain properties of the medium in which it floats. If we add a -little water to the blood, the corpuscle quickly loses its characteristic -shape and becomes a spherical drop, that is to say a true -surface of minimal area and of stable equilibrium. If on the other -hand we add a strong solution of salt, or a little glycerine, the -corpuscle contracts, and its surface becomes puckered and uneven. -In these phenomena it is so far obeying the laws of diffusion and -of surface tension. <span class="xxpn" id="p272">{272}</span></p> - -<p>In the second place, it can be exactly imitated artificially by -means of other colloid substances. Many years ago Norris made the -very interesting observation that in an emulsion of glue the drops -assumed a biconcave form resembling that of the mammalian corpuscles<a class="afnanch" href="#fn317" id="fnanch317">317</a>. -The glue was impure, and doubtless contained lecithin; -and it is possible (as Professor Waymouth Reid tells me) to make -a similar emulsion with cerebrosides and cholesterin oleate, in -which the same conformation of the drops or particles is beautifully -shewn. Now such cholesterin bodies have an important place -among those in which Lehmann and others have shewn and studied -the formation of fluid crystals, that is to say of bodies in which -the forces of crystallisation and the forces of surface tension are -battling with one another<a class="afnanch" href="#fn318" id="fnanch318">318</a>; -and, for want of a better explanation, -we may in the meanwhile suggest that some such cause is at the -bottom of the conformation the explanation of which presents so -many difficulties. But we must not, perhaps, pass from this -subject without adding that the case is a difficult and complex -one from the physiological point of view. For the surface of a -blood-corpuscle consists of a “semi-permeable membrane,” through -which certain substances pass freely and not others (for the most -part anions and not cations), and it may be, accordingly, that we -have in life a continual state of osmotic inequilibrium, of negative -osmotic tension within, to which comparatively simple cause the -imperfect distension of the corpuscle may be also due<a class="afnanch" href="#fn319" id="fnanch319">319</a>. -The whole -phenomenon would be comparatively easy to understand if we -might postulate a stiffer peripheral region to the corpuscle, in the -form for instance of a peripheral elastic ring. Such an annular -thickening or stiffening, like the “collapse-rings” which an engineer -inserts in a boiler, has been actually asserted to exist, but its -presence is not authenticated.</p> - -<p>But it is not at all improbable that we have still much to -learn about the phenomena of osmosis itself, as manifested in the -case of minute bodies such as a blood-corpuscle; and (as Professor -Peddie suggests to me) it is by no means -impossible that <i>curvature</i> <span class="xxpn" id="p273">{273}</span> -of the surface may itself modify the osmotic or perhaps the adsorptive -action. If it should be found that osmotic action tended to -stop, or to reverse, on change of curvature, it would follow that -this phenomenon would give rise to internal currents; and the -change of pressure consequent on these would tend to intensify -the change of curvature when once started<a class="afnanch" href="#fn320" id="fnanch320">320</a>.</p> - -<div class="dctr01" id="fig94"> -<img src="images/i273.png" width="800" height="406" alt=""> - <div class="pcaption">Fig. 94. Sperm-cells of Decapod Crustacea -(after Koltzoff). <i>a</i>, <i>Inachus scorpio</i>; <i>b</i>, <i>Galathea -squamifera</i>; <i>c</i>, <i>do.</i> after maceration, to shew spiral -fibrillae.</div></div> - -<p>The sperm-cells of the Decapod crustacea exhibit various -singular shapes. In the Crayfish they are flattened cells with -stiff curved processes radiating outwards like a St Catherine’s -wheel; in Inachus there are two such circles of stiff processes; -in Galathea we have a still more complex form, with long and -slightly twisted processes. In all these cases, just as in the case -of the blood-corpuscle, the structure alters, and finally loses, its -characteristic form when the nature or constitution (or as we may -assume in particular—the density) of the surrounding medium is -changed.</p> - -<p>Here again, as in the blood-corpuscle, we have to do with a -very important force, which we had not hitherto considered in this -connection,—the force of osmosis, manifested under conditions -similar to those of Pfeffer’s classical experiments on the plant-cell. -The surface of the cell acts as a -“semi-permeable membrane,” <span class="xxpn" id="p274">{274}</span> -permitting the passage of certain dissolved substances (or their -“ions”) and including or excluding others; and thus rendering -manifest and measurable the existence of a definite “osmotic -pressure.” In the case of the sperm-cells of Inachus, certain -quantitative experiments have been performed<a class="afnanch" href="#fn321" id="fnanch321">321</a>. -The sperm-cell -exhibits its characteristic conformation while lying in the serous -fluid of the animal’s body, in ordinary sea-water, or in a 5 per -cent. solution of potassium nitrate; these three fluids being all -“isotonic” with one another. As we alter the concentration of -potassium nitrate, the cell assumes certain definite forms corresponding -to definite concentrations of the salt; and, as a further -and final proof that the phenomenon is entirely physical, it is -found that other salts produce an identical effect when their -concentration is proportionate to their molecular weight, and -whatever identical effect is produced by various salts in their -respective concentrations, a similarly identical effect is produced -when these concentrations are doubled or otherwise proportionately -changed<a class="afnanch" href="#fn322" id="fnanch322">322</a>.</p> - -<div class="dctr01" id="fig95"> -<img src="images/i274.png" width="800" height="207" alt=""> - <div class="pcaption">Fig. 95. Sperm-cells of <i>Inachus</i>, as they - appear in saline solutions of varying density. (After - Koltzoff.)</div></div> - -<div class="section"> -<p>Thus the following table shews the percentage concentrations -of certain salts necessary to bring the cell into the -forms <i>a</i> and <i>c</i> of Fig. <a href="#fig95" title="go to Fig. 95">95</a>; in each case the quantities -are proportional to the molecular weights, and in each -case twice the quantity is necessary to produce the effect -of Fig. <a href="#fig95" title="go to Fig. 95">95</a><i>c</i> compared with that which gives rise to the -all but spherical form of Fig. <a href="#fig95" title="go to Fig. 95">95</a><i>a</i>. <span class="xxpn" -id="p275">{275}</span></p> - -<div class="dtblbox"><div class="nowrap"> -<table class="fsz6"> -<tr> - <th rowspan="2"></th> - <th colspan="2">% concentration<br>of salts - in which the<br>sperm-cell of Inachus<br>assumes the form of</th></tr> -<tr> - <th>fig. <i>a</i></th> - <th>fig. <i>c</i></th></tr> -<tr> - <td class="tdleft">Sodium chloride</td> - <td class="tdright">0·6 </td> - <td class="tdright">1·2</td></tr> -<tr> - <td class="tdleft">Sodium nitrate</td> - <td class="tdright">0·85</td> - <td class="tdright">1·7</td></tr> -<tr> - <td class="tdleft">Potassium nitrate</td> - <td class="tdright">1·0 </td> - <td class="tdright">2·0</td></tr> -<tr> - <td class="tdleft">Acetic acid</td> - <td class="tdright">2·2 </td> - <td class="tdright">4·5</td></tr> -<tr> - <td class="tdleft">Cane sugar</td> - <td class="tdright">5·0 </td> - <td class="tdright">10·0</td></tr> -</table></div></div><!--dtblbox--></div><!--section--> - -<p>If we look then, upon the spherical form of the cell -as its true condition of symmetry and of equilibrium, we -see that what we call its normal appearance is just one of -many intermediate phases of shrinkage, brought about by -the abstraction of fluid from its interior as the result -of an osmotic pressure greater outside than inside the -cell, and where the shrinkage of <i>volume</i> is not kept -pace with by a contraction of the <i>surface-area</i>. In the -case of the blood-corpuscle, the shrinkage is of no great -amount, and the resulting deformation is symmetrical; such -structural inequality as may be necessary to account for -it need be but small. But in the case of the sperm-cells, -we must have, and we actually do find, a somewhat -complicated arrangement of more or less rigid or elastic -structures in the wall of the cell, which like the wire -framework in Plateau’s experiments, restrain and modify -the forces acting on the drop. In one form of Plateau’s -experiments, instead of</p> - -<div class="dright dwth-e" id="fig96"> -<img src="images/i275.png" width="332" height="281" alt=""> - <div class="dcaption">Fig. 96. Sperm-cell of <i>Dromia</i>. -(After Koltzoff.)</div></div> - -<p class="pcontinue">supporting his drop on -rings or frames of wire, he laid upon its surface one or -more elastic coils; and then, on withdrawing oil from -the centre of his globule, he saw its uniform shrinkage -counteracted by the spiral springs, with the result that -the centre of each elastic coil seemed to shoot out into -a prominence. Just such spiral coils are figured (after -Koltzoff) in Fig. <a href="#fig96" title="go to Fig. 96">96</a>; and they may be regarded as precisely -akin to those local thickenings, spiral and other, to -which we have already ascribed the cylindrical form of -the Spirogyra cell. In all probability we must in like -manner attribute the peculiar spiral and other forms, for -instance of many Infusoria, to the <span class="xxpn" -id="p276">{276}</span> presence, among the multitudinous -other differentiations of their protoplasmic substance, -of such more or less elastic fibrillae, which play as it -were the part of a microscopic skeleton<a class="afnanch" -href="#fn323" id="fnanch323">323</a>. -<br class="brclrfix"></p> - -<hr class="hrblk"> - -<p>But these cases which we have just dealt with, lead us to -another consideration. In a semi-permeable membrane, through -which water passes freely in and out, the conditions of a liquid -surface are greatly modified; and, in the ideal or ultimate case, -there is neither surface nor surface tension at all. And this would -lead us somewhat to reconsider our position, and to enquire -whether the true surface tension of a liquid film is actually -responsible for <i>all</i> that we have ascribed to it, or whether certain -of the phenomena which we have assigned to that cause may not -in part be due to the contractility of definite and elastic membranes. -But to investigate this question, in particular cases, is rather for -the physiologist: and the morphologist may go on his way, -paying little heed to what is no doubt a difficulty. In surface -tension we have the production of a film with the properties of an -elastic membrane, and with the special peculiarity that contraction -continues with the same energy however far the process may have -already gone; while the ordinary elastic membrane contracts to -a certain extent, and contracts no more. But within wide limits -the essential phenomena are the same in both cases. Our -fundamental equations apply to both cases alike. And accordingly, -so long as our purpose is <i>morphological</i>, so long as what we -seek to explain is regularity and definiteness of form, it matters -little if we should happen, here or there, to confuse surface tension -with elasticity, the contractile forces manifested at a liquid -surface with those which come into play at the complex internal -surfaces of an elastic solid.</p> - -<div class="chapter" id="p277"> -<h2 class="h2herein" title="VI. A Note on Adsorption.">CHAPTER VI -<span class="h2ttl"> -A NOTE ON ADSORPTION</span></h2></div> - -<p>A very important corollary to, or amplification of the theory -of surface tension is to be found in the modern chemico-physical -doctrine of Adsorption<a class="afnanch" href="#fn324" id="fnanch324">324</a>. -In its full statement this subject soon -becomes complicated, and involves physical conceptions and -mathematical treatment which go beyond our range. But it is -necessary for us to take account of the phenomenon, though it -be in the most elementary way.</p> - -<p>In the brief account of the theory of surface tension with which -our last chapter began, it was pointed out that, in a drop of liquid, -the potential energy of the system could be diminished, and work -manifested accordingly, in two ways. In the first place we saw -that, at our liquid surface, surface tension tends to set up an -equilibrium of form, in which the surface is reduced or contracted -either to the absolute minimum of a sphere, or at any rate to the -least possible area which is permitted by the various circumstances -and conditions; and if the two bodies which comprise our system, -namely the drop of liquid and its surrounding medium, be simple -substances, and the system be uncomplicated by other distributions -of force, then the energy of the system will have done its work -when this equilibrium of form, this minimal area of surface, is -once attained. This phenomenon of the production of a minimal -surface-area we have now seen to be of fundamental importance -in the external morphology of the cell, and especially (so far -as we have yet gone) of the solitary cell -or unicellular organism. <span class="xxpn" id="p278">{278}</span></p> - -<p>But we also saw, according to Gauss’s equation, that the -potential energy of the system will be diminished (and its diminution -will accordingly be manifested in work) if from any cause -the specific surface energy be diminished, that is to say if it be -brought more nearly to an equality with the specific energy of the -molecules in the interior of the liquid mass. This latter is a -phenomenon of great moment in modern physiology, and, while -we need not attempt to deal with it in detail, it has a bearing on -cell-form and cell-structure which we cannot afford to overlook.</p> - -<p>In various ways a diminution of the surface energy may be -brought about. For instance, it is known that every isolated drop -of fluid has, under normal circumstances, a surface-charge of -electricity: in such a way that a positive or negative charge (as -the case may be) is inherent in the surface of the drop, while a -corresponding charge, of contrary sign, is inherent in the -immediately adjacent molecular layer of the surrounding medium. -Now the effect of this distribution, by which all the surface -molecules of our drop are similarly charged, is that by virtue of -this charge they tend to repel one another, and possibly also to -draw other molecules, of opposite charge, from the interior of the -mass; the result being in either case to antagonise or cancel, -more or less, that normal tendency of the surface molecules to -attract one another which is manifested in surface tension. In -other words, an increased electrical charge concentrating at the -surface of a drop tends, whether it be positive or negative, to -<i>lower</i> the surface tension.</p> - -<p>But a still more important case has next to be considered. -Let us suppose that our drop consists no longer of a single chemical -substance, but contains other substances either in suspension or -in solution. Suppose (as a very simple case) that it be a watery -fluid, exposed to air, and containing droplets of oil: we know that -the specific surface tension of oil in contact with air is much less -than that of water, and it follows that, if the watery surface of -our drop be replaced by an oily surface the specific surface energy -of the system will be notably diminished. Now under these -circumstances it is found that (quite apart from gravity, by which -the oil might <i>float</i> to the surface) the oil has a tendency to be -<i>drawn</i> to the surface; and this phenomenon -of molecular attraction <span class="xxpn" id="p279">{279}</span> -or “adsorption” represents the work done, equivalent to the -diminished potential energy of the system<a class="afnanch" href="#fn325" id="fnanch325">325</a>. -In more general -terms, if a liquid (or one or other of two adjacent liquids) be a -chemical mixture, some one constituent in which, if it entered -into or increased in amount in the surface layer, would have the -effect of diminishing its surface tension, then that constituent will -have a tendency to accumulate or concentrate at the surface: the -surface tension may be said, as it were, to exercise an attraction -on this constituent substance, drawing it into the surface layer, -and this tendency will proceed until at a certain “surface concentration” -equilibrium is reached, its opponent being that osmotic -force which tends to keep the substance in uniform solution or -diffusion.</p> - -<p>In the complex mixtures which constitute the protoplasm of -the living cell, this phenomenon of “adsorption” has abundant -play: for many of these constituents, such as oils, soaps, albumens, -etc. possess the required property of diminishing surface tension.</p> - -<p>Moreover, the more a substance has the power of lowering the -surface tension of the liquid in which it happens to be dissolved, -the more will it tend to displace another and less effective substance -from the surface layer. Thus we know that protoplasm always -contains fats or oils, not only in visible drops, but also in the -finest suspension or “colloidal solution.” If under any impulse, -such for instance as might arise from the Brownian movement, -a droplet of oil be brought close to the surface, it is at once drawn -into that surface, and tends to spread itself in a thin layer over -the whole surface of the cell. But a soapy surface (for instance) -would have in contact with the surrounding water a surface tension -even less than that of the film of oil: and consequently, if soap -be present in the water it will in turn be adsorbed, and will tend -to displace the oil from the surface pellicle<a class="afnanch" href="#fn326" id="fnanch326">326</a>. -And this is all as <span class="xxpn" id="p280">{280}</span> -much as to say that the molecules of the dissolved or suspended -substance or substances will so distribute themselves throughout -the drop as to lead towards an equilibrium, for each small unit -of volume, between the superficial and internal energy; or so, in -other words, as to lead towards a reduction to a minimum of the -potential energy of the system. This tendency to concentration -at the surface of any substance within the cell by which the surface -tension tends to be diminished, or <i>vice versa</i>, constitutes, then, -the phenomenon of <i>Adsorption</i>; and the general statement by -which it is defined is known as the Willard-Gibbs, or Gibbs-Thomson -law<a class="afnanch" href="#fn327" id="fnanch327">327</a>.</p> - -<p>Among the many important physical features or concomitants -of this phenomenon, let us take note at present that we need -not conceive of a strictly superficial distribution of the adsorbed -substance, that is to say of its direct association with the surface -layer of molecules such as we imagined in the case of the electrical -charge; but rather of a progressive tendency to concentrate, -more and more, as the surface is nearly approached. Indeed we -may conceive the colloid or gelatinous precipitate in which, in the -case of our protoplasmic cell, the dissolved substance tends often -to be thrown down, to constitute one boundary layer after another, -the general effect being intensified and multiplied by the repeated -addition of these new surfaces.</p> - -<p>Moreover, it is not less important to observe that the process -of adsorption, in the neighbourhood of the surface of a heterogeneous -liquid mass, is a process which <i>takes time</i>; the tendency -to surface concentration is a gradual and progressive one, and will -fluctuate with every minute change in the composition of our -substance and with every change in the area of its surface. In -other words, it involves (in every heterogeneous substance) a -continual instability of equilibrium: and -a constant manifestation <span class="xxpn" id="p281">{281}</span> -of motion, sometimes in the mere invisible transfer of molecules -but often in the production of visible currents of fluid or manifest -alterations in the form or outline of the system.</p> - -<hr class="hrblk"> - -<p>The physiologist, as we have already remarked, takes account -of the general phenomenon of adsorption in many ways: particularly -in connection with various results and consequences of -osmosis, inasmuch as this process is dependent on the presence -of a membrane, or membranes, such as the phenomenon of adsorption -brings into existence. For instance it plays a leading part -in all modern theories of muscular contraction, in which phenomenon -a connection with surface tension was first indicated by -FitzGerald and d’Arsonval nearly forty years ago<a class="afnanch" href="#fn328" id="fnanch328">328</a>. -And, as -W. Ostwald was the first to shew, it gives us an entirely new -conception of the relation of gases (that is to say, of oxygen and -carbon dioxide) to the red corpuscles of the blood<a class="afnanch" href="#fn329" id="fnanch329">329</a>.</p> - -<p>But restricting ourselves, as much as may be, to our morphological -aspect of the case, there are several ways in which adsorption -begins at once to throw light upon our subject.</p> - -<p>In the first place, our preliminary account, such as it is, is -already tantamount to a description of the process of development -of a cell-membrane, or cell-wall. The so-called “secretion” -of this cell-wall is nothing more than a sort of exudation, or -striving towards the surface, of certain constituent molecules or -particles within the cell; and the Gibbs-Thomson law formulates, -in part at least, the conditions under which they do so. The -adsorbed material may range from the almost unrecognisable -pellicle of a blood-corpuscle to the distinctly differentiated -“ectosarc” of a protozoan, and again to the development of a -fully formed cell-wall, as in the cellulose partitions of a vegetable -tissue. In such cases, the dissolved and adsorbable material has -not only the property of lowering the surface -tension, and hence <span class="xxpn" id="p282">{282}</span> -of itself accumulating at the surface, but has also the property -of increasing the viscosity and mechanical rigidity of the material -in which it is dissolved or suspended, and so of constituting -a visible and tangible “membrane<a class="afnanch" href="#fn330" id="fnanch330">330</a>.” -The “zoogloea” around a -group of bacteria is probably a phenomenon of the same order. -In the superficial deposition of inorganic materials we see the -same process abundantly exemplified. Not only do we have the -simple case of the building of a shell or “test” upon the outward -surface of a living cell, as for instance in a Foraminifer, but in a -subsequent chapter, when we come to deal with various spicules -and spicular skeletons such as those of the sponges and of the -Radiolaria, we shall see that it is highly characteristic of the -whole process of spicule-formation for the deposits to be laid -down just in the “interfacial” boundaries between cells or -vacuoles, and that the form of the spicular structures tends in -many cases to be regulated and determined by the arrangement -of these boundaries.</p> - -<div class="psmprnt3"> -<p>In physical chemistry, an important distinction is drawn -between adsorption and <i>pseudo-adsorption</i><a class="afnanch" -href="#fn331" id="fnanch331">331</a>, the former being -a <i>reversible</i>, the latter an irreversible or permanent -phenomenon. That is to say, adsorption, strictly speaking, -implies the surface-concentration of a dissolved substance, -under circumstances which, if they be altered or reversed, -will cause the concentration to diminish or disappear. But -pseudo-adsorption includes cases, doubtless originating -in adsorption proper, where subsequent changes leave the -concentrated substance incapable of re-entering the liquid -system. It is obvious that many (though not all) of our -biological illustrations, for instance the formation of -spicules or of permanent cell-membranes, belong to the class -of so-called pseudo-adsorption phenomena. But the apparent -contrast between the two is in the main a secondary one, and -however important to the chemist is of little consequence to -us. <span class="xxpn" id="p283">{283}</span></p> -</div><!--psmprnt3--> - -<p>While this brief sketch of the theory of membrane-formation -is cursory and inadequate, it is enough to shew that the physical -theory of adsorption tends in part to overturn, in part to simplify -enormously, the older histological descriptions. We can no longer -be content with such statements as that of Strasbürger, that -membrane-formation in general is associated with the “activity -of the kinoplasm,” or that of Harper that a certain spore-membrane -arises directly from the astral rays<a class="afnanch" href="#fn332" id="fnanch332">332</a>. -In short, we have easily -reached the general conclusion that, the formation of a cell-wall -or cell-membrane is a chemico-physical phenomenon, which the -purely objective methods of the biological microscopist do not -suffice to interpret.</p> - -<hr class="hrblk"> - -<p>If the process of adsorption, on which the formation of a -membrane depends, be itself dependent on the power of the -adsorbed substance to lower the surface tension, it is obvious that -adsorption can only take place when the surface tension already -present is greater than zero. It is for this reason that films or -threads of creeping protoplasm shew little tendency, or none, to -cover themselves with an encysting membrane; and that it is -only when, in an altered phase, the protoplasm has developed -a positive surface tension, and has accordingly gathered itself up -into a more or less spherical body, that the tendency to form a -membrane is manifested, and the organism develops its “cyst” -or cell-wall.</p> - -<p>It is found that a rise of temperature greatly reduces the -adsorbability of a substance, and this doubtless comes, either in -part or whole, from the fact that a rise of temperature is itself -a cause of the lowering of surface tension. We may in all probability -ascribe to this fact and to its converse, or at least associate -with it, such phenomena as the encystment of unicellular organisms -at the approach of winter, or the frequent formation of strong -shells or membranous capsules in “winter-eggs.”</p> - -<p>Again, since a film or a froth (which is a system of films) can -only be maintained by virtue of a certain -viscosity or rigidity of <span class="xxpn" id="p284">{284}</span> -the liquid, it may be quickly caused to disappear by the presence -in its neighbourhood of some substance capable of reducing the -surface tension; for this substance, being adsorbed, may displace -from the adsorptive layer a material to which was due the rigidity -of the film. In this way a “bathytonic” substance such as ether -causes most foams to subside, and the pouring oil on troubled -waters not only stills the waves but still more quickly dissipates -the foam of the breakers. The process of breaking up an alveolar -network, such as occurs at a certain stage in the nuclear division -of the cell, may perhaps be ascribed in part to such a cause, as -well as to the direct lowering of surface tension by electrical -agency.</p> - -<p>Our last illustration has led us back to the subject of a previous -chapter, namely to the visible configuration of the interior of the -cell; and in connection with this wide subject there are many -phenomena on which light is apparently thrown by our knowledge -of adsorption, and of which we took little or no account in our -former discussion. One of these phenomena is that visible or -concrete “polarity,” which we have already seen to be in some way -associated with a dynamical polarity of the cell.</p> - -<p>This morphological polarity may be of a very simple kind, as -when, in an epithelial cell, it is manifested by the outward shape -of the elongated or columnar cell itself, by the essential difference -between its free surface and its attached base, or by the presence -in the neighbourhood of the former of mucous or other products -of the cell’s activity. But in a great many cases, this “polarised” -symmetry is supplemented by the presence of various fibrillae, or -of linear arrangements of particles, which in the elongated or -“monopolar” cell run parallel with its axis, and which tend to -a radial arrangement in the more or less rounded or spherical -cell. Of late years especially, an immense importance has been -attached to these various linear or fibrillar arrangements, as they -occur (<i>after staining</i>) in the cell-substance of intestinal epithelium, -of spermatocytes, of ganglion cells, and most abundantly and -most frequently of all in gland cells. Various functions, which -seem somewhat arbitrarily chosen, have been assigned, and many -hard names given to them; for these structures now include your -mitochondria and your chondriokonts (both of -these being varieties <span class="xxpn" id="p285">{285}</span> -of chondriosomes), your Altmann’s granules, your microsomes, -pseudo-chromosomes, epidermal fibrils and basal filaments, your -archeoplasm and ergastoplasm, and probably your idiozomes, -plasmosomes, and many other histological minutiae<a class="afnanch" href="#fn333" id="fnanch333">333</a>.</p> - -<div class="dctr01" id="fig97"> -<img src="images/i285.png" width="800" height="293" alt=""> - <div class="pcaption">Fig. 97. <i>A</i>, <i>B</i>, Chondriosomes - in kidney-cells, prior to and during secretory activity - (after Barratt); <i>C</i>, do. in pancreas of frog (after - Mathews).</div></div> - -<p>The position of these bodies with regard to the other cell-structures -is carefully described. Sometimes they lie in the -neighbourhood of the nucleus itself, that is to say in proximity to -the fluid boundary surface which separates the nucleus from the -cytoplasm; and in this position they often form a somewhat cloudy -sphere which constitutes the <i>Nebenkern</i>. In the majority of cases, -as in the epithelial cells, they form filamentous structures, and rows -of granules, whose main direction is parallel to the axis of the -cell, and which may, in some cases, and in some forms, be conspicuous -at the one end, and in some cases at the other end of -the cell. But I do not find that the histologists attempt to explain, -or to correlate with other phenomena, the tendency of these bodies -to lie parallel with the axis, and perpendicular to the extremities -of the cell; it is merely noted as a peculiarity, or a specific character, -of these particular structures. Extraordinarily complicated and -diverse functions have been ascribed to them. Engelmann’s -“Fibrillenkonus,” which was almost certainly another aspect of -the same phenomenon, was held by him and by cytologists like -Breda and Heidenhain, to be an apparatus -connected in some <span class="xxpn" id="p286">{286}</span> -unexplained way with the mechanism of ciliary movement. -Meves looked upon the chondriosomes as the actual carriers or -transmitters of heredity<a class="afnanch" href="#fn334" id="fnanch334">334</a>. -Altmann invented a new aphorism, -<i>Omne granulum e granulo</i>, as a refinement of Virchow’s <i>omnis -cellula e cellula</i>; and many other histologists, more or less in accord, -accepted the chondriosomes as important entities, <i>sui generis</i>, -intermediate in grade between the cell itself and its ultimate -molecular components. The extreme cytologists of the Munich -school, Popoff, Goldschmidt and others, following Richard Hertwig, -declaring these structures to be identical with “chromidia” (under -which name Hertwig ranked all extra-nuclear chromatin), would -assign them complex functions in maintaining the balance between -nuclear and cytoplasmic material; and the “chromidial hypothesis,” -as every reader of recent cytological literature knows, has -become a very abstruse and complicated thing<a class="afnanch" href="#fn335" id="fnanch335">335</a>. -With the help -of the “binuclearity hypothesis” of Schaudinn and his school, it -has given us the chromidial net, the chromidial apparatus, the -trophochromidia, idiochromidia, gametochromidia, the protogonoplasm, -and many other novel and original conceptions. The -names are apt to vary somewhat in significance from one writer -to another.</p> - -<p>The outstanding fact, as it seems to me, is that physiological -science has been heavily burdened in this matter, with a jargon -of names and a thick cloud of hypotheses; while, from the physical -point of view we are tempted to see but little mystery in the -whole phenomenon, and to ascribe it, in all probability and in -general terms, to the gathering or “clumping” together, under -surface tension, of various constituents of the heterogeneous cell-content, -and to the drawing out of these little clumps along the -axis of the cell towards one or other of its extremities, in relation -to osmotic currents, as these in turn are set up -in direct relation <span class="xxpn" id="p287">{287}</span> -to the phenomena of surface energy and of adsorption<a class="afnanch" href="#fn336" id="fnanch336">336</a>. -And -all this implies that the study of these minute structures, if it -teach us nothing else, at least surely and certainly reveals to us -the presence of a definite “field of force,” and a dynamical polarity -within the cell.</p> - -<hr class="hrblk"> - -<p>Our next and last illustration of the effects of adsorption, -which we owe to the investigations of Professor Macallum, is of -great importance; for it introduces us to a series of phenomena -in regard to which we seem now to stand on firmer ground than -in some of the foregoing cases, though we cannot yet consider that -the whole story has been told. In our last chapter we were -restricted mainly, though not entirely, to a consideration of figures -of equilibrium, such as the sphere, the cylinder or the unduloid; -and we began at once to find ourselves in difficulties when we were -confronted by departures from symmetry, as for instance in the -simple case of the ellipsoidal yeast-cell and the production of its -bud. We found the cylindrical cell of Spirogyra, with its plane -or spherical ends, a comparatively simple matter to understand; -but when this uniform cylinder puts out a lateral outgrowth, in -the act of conjugation, we have a new and very different system -of forces to explain. The analogy of the soap-bubble, or of the -simple liquid drop, was apt to lead us to suppose that the surface -tension was, on the whole, uniform over the surface of our cell; -and that its departures from symmetry of form were therefore -likely to be due to variations in external resistance. But if we -have been inclined to make such an -assumption we must now <span class="xxpn" id="p288">{288}</span> -reconsider it, and be prepared to deal with important localised -variations in the surface tension of the cell. For, as a matter of -fact, the simple case of a perfectly symmetrical drop, with uniform -surface, at which adsorption takes place with similar uniformity, -is probably rare in physics, and rarer still (if it exist at all) in the -fluid or fluid-containing system which we call in biology a cell. -We have mostly to do with cells whose general heterogeneity of -substance leads to qualitative differences of surface, and hence to -varying distributions of surface tension. We must accordingly -investigate the case of a cell which displays some definite and -regular heterogeneity of its liquid surface, just as Amoeba displays -a heterogeneity which is complex, irregular and continually -fluctuating in amount and distribution. Such heterogeneity as -we are speaking of must be essentially chemical, and the preliminary -problem is to devise methods of “microchemical” analysis, -which shall reveal <i>localised</i> accumulations of particular substances -within the narrow limits of a cell, in the hope that, their normal -effect on surface tension being ascertained, we may then correlate -with their presence and distribution the actual indications of -varying surface tension which the form or movement of the cell -displays. In theory the method is all that we could wish, but in -practice we must be content with a very limited application of it; -for the substances which may have such action as we are looking -for, and which are also actual or possible constituents of the cell, -are very numerous, while the means are very seldom at hand to -demonstrate their precise distribution and localisation. But in -one or two cases we have such means, and the most notable is in -connection with the element potassium. As Professor Macallum -has shewn, this element can be revealed, in very minute quantities, -by means of a certain salt, a nitrite of cobalt and sodium<a class="afnanch" href="#fn337" id="fnanch337">337</a>. -This -salt penetrates readily into the tissues and into the interior of the -cell; it combines with potassium to form a sparingly soluble -nitrite of cobalt, sodium and potassium; and this, on subsequent -treatment with ammonium sulphide, is converted into a characteristic -black precipitate of cobaltic sulphide<a class="afnanch" href="#fn338" id="fnanch338">338</a>. -<span class="xxpn" id="p289">{289}</span></p> - -<p>By this means Macallum demonstrated some years ago the -unexpected presence of accumulations of potassium (i.e. of chloride -or other salts of potassium) localised in particular parts of various -cells, both solitary cells and tissue cells; and he arrived at the -conclusion that the localised accumulations in question were -simply evidences of <i>concentration</i> of the dissolved potassium salts, -formed and localised in accordance with the Gibbs-Thomson law. -In other words, these accumulations, occurring as they actually do -in connection with various boundary surfaces, are evidence, when -they appear irregularly distributed over such a surface, of inequalities -in its surface tension<a class="afnanch" href="#fn339" id="fnanch339">339</a>; -and we may safely take it that -our potassium salts, like inorganic substances in general, tend to -<i>raise</i> the surface tension, and will therefore be found concentrating -at a portion of the surface whose tension is weak<a class="afnanch" href="#fn340" id="fnanch340">340</a>.</p> - -<p>In Professor Macallum’s figure (Fig. <a href="#fig98" title="go to Fig. 98">98</a>, 1) of the little green -alga Pleurocarpus, we see that one side of the cell is beginning to -bulge out in a wide convexity. This bulge is, in the first place, -a sign of weakened surface tension on one side of the cell, which as -a whole had hitherto been a symmetrical cylinder; in the second -place, we see that the bulging area corresponds to the position of -a great concentration of the potassic salt; while in the third place, -from the physiological point of view, we call the phenomenon -the first stage in the process of conjugation. In Fig. <a href="#fig98" title="go to Fig. 98">98</a>, 2, of -Mesocarpus (a close ally of Spirogyra), we see the same phenomenon -admirably exemplified in a later stage. From the adjacent cells -distinct outgrowths are being emitted, where the surface tension has -been weakened: just as the glass-blower warms and softens a small -part of his tube to blow out the softened area into a bubble or -diverticulum; and in our Mesocarpus cells (besides a certain -amount of potassium rendered visible over -the boundary which <span class="xxpn" id="p290">{290}</span> -separates the green protoplasm from the cell-sap), there is a very -large accumulation precisely at the point where the tension of the -originally cylindrical cell is weakening to produce the bulge. -But in a still later stage, when the boundary between the two -conjugating cells is lost and the cytoplasm of the two cells becomes -fused together, then the signs of potassic concentration quickly -disappear, the salt becoming generally diffused through the now -symmetrical and spherical “zygospore.”</p> - -<div class="dctr01" id="fig98"> -<img src="images/i290.png" width="800" height="550" alt=""> - <div class="pcaption">Fig. 98. Adsorptive concentration - of potassium salts in (1) cell of <i>Pleurocarpus</i> about - to conjugate; (2) conjugating cells of <i>Mesocarpus</i>; - (3) sprouting spores of <i>Equisetum</i>. (After - Macallum.)</div></div> - -<p>In a spore of Equisetum (Fig. <a href="#fig98" title="go to Fig. 98">98</a>, 3), while it is still a single cell, -no localised concentration of potassium is to be discerned; but as -soon as the spore has divided, by an internal partition, into two -cells, the potassium salt is found to be concentrated in the smaller -one, and especially towards its outer wall, which is marked by a -pronounced convexity. And as this convexity (which corresponds -to one pole of the now asymmetrical, or quasi-ellipsoidal spore) -grows out into the root-hair, the potassium salt accompanies its -growth, and is concentrated under its wall. -The concentration is, <span class="xxpn" id="p291">{291}</span> -accordingly, a concomitant of the diminished surface tension which -is manifested in the altered configuration of the system.</p> - -<p>In the case of ciliate or flagellate cells, there is to be found a -characteristic accumulation of potassium at and near the base of -the cilia. The relation of ciliary movement to surface tension -lies beyond our range, but the fact which we have just mentioned -throws light upon the frequent or general presence of a little -protuberance of the cell-surface just where a flagellum is given -off (cf. p. <a href="#p247" title="go to pg. 247">247</a>), and of a little projecting ridge or fillet at the base -of an isolated row of cilia, such as we find in Vorticella.</p> - -<p>Yet another of Professor Macallum’s demonstrations, though -its interest is mainly physiological, will help us somewhat further -to comprehend what is implied in our phenomenon. In a normal -cell of Spirogyra, a concentration of potassium is revealed along -the whole surface of the spiral coil of chlorophyll-bearing, or -“chromatophoral,” protoplasm, the rest of the cell being wholly -destitute of the former substance: the indication being that, at -this particular boundary, between chromatophore and cell-sap, -the surface tension is small in comparison with any other interfacial -surface within the system.</p> - -<p>Now as Macallum points out, the presence of potassium is -known to be a factor, in connection with the chlorophyll-bearing -protoplasm, in the synthetic production of starch from CO<sub>2</sub> under -the influence of sunlight. But we are left in some doubt as to -the consecutive order of the phenomena. For the lowered surface -tension, indicated by the presence of the potassium, may be -itself a cause of the carbohydrate synthesis; while on the other -hand, this synthesis may be attended by the production of substances -(e.g. formaldehyde) which lower the surface tension, and -so conduce to the concentration of potassium. All we know for -certain is that the several phenomena are associated with one -another, as apparently inseparable parts or inevitable concomitants -of a certain complex action.</p> - -<hr class="hrblk"> - -<p>And now to return, for a moment, to the question of cell-form. -When we assert that the form of a cell (in the absence of mechanical -pressure) is essentially dependent on surface tension, and even when -we make the preliminary assumption that -protoplasm is essentially <span class="xxpn" id="p292">{292}</span> -a fluid, we are resting our belief on a general consensus of evidence, -rather than on compliance with any one crucial definition. The -simple fact is that the agreement of cell-forms with the forms -which physical experiment and mathematical theory assign to -liquids under the influence of surface tension, is so frequently and -often so typically manifested, that we are led, or driven, to accept -the surface tension hypothesis as generally applicable and as -equivalent to a universal law. The occasional difficulties or -apparent exceptions are such as call for further enquiry, but fall -short of throwing doubt upon our hypothesis. Macallum’s -researches introduce a new element of certainty, a “nail in a sure -place,” when they demonstrate that, in certain movements or -changes of form which we should naturally attribute to weakened -surface tension, a chemical concentration which would naturally -accompany such weakening actually takes place. They further -teach us that in the cell a chemical heterogeneity may exist of -a very marked kind, certain substances being accumulated here -and absent there, within the narrow bounds of the system.</p> - -<p>Such localised accumulations can as yet only be demonstrated -in the case of a very few substances, and of a single one in particular; -and these are substances whose presence does not produce, -but whose concentration tends to follow, a weakening of surface -tension. The physical cause of the localised inequalities of surface -tension remains unknown. We may assume, if we please, that it -is due to the prior accumulation, or local production, of chemical -bodies which would have this direct effect; though we are by -no means limited to this hypothesis.</p> - -<p>But in spite of some remaining difficulties and uncertainties, -we have arrived at the conclusion, as regards unicellular organisms, -that not only their general configuration but also <i>their departures -from symmetry</i> may be correlated with the molecular forces -manifested in their fluid -or semi-fluid surfaces.</p> - -<div class="chapter" id="p293"> -<h2 class="h2herein" title="VII. The -Forms of Tissues Or Cell-aggregates.">CHAPTER VII -<span class="h2ttl"> -THE FORMS OF TISSUES OR CELL-AGGREGATES</span></h2></div> - -<p>We now pass from the consideration of the solitary cell to that -of cells in contact with one another,—to what we may call in -the first instance “cell-aggregates,”—through which we shall be led -ultimately to the study of complex tissues. In this part of our -subject, as in the preceding chapters, we shall have to give some -consideration to the effects of various forces; but, as in the case -of the conformation of the solitary cell, we shall probably find, -and we may at least begin by assuming, that the agency of surface -tension is especially manifest and important. The effect of this -surface tension will chiefly manifest itself in the production of -surfaces <i>minimae areae</i>: where, as Plateau was always careful to -point out, we must understand by this expression not an absolute, -but a relative minimum, an area, that is to say, which approximates -to an absolute minimum as nearly as circumstances and the -conditions of the case permit.</p> - -<p>There are certain fundamental principles, or fundamental -equations, besides those which we have already considered, which -we shall need in our enquiry. For instance the case which we -briefly touched upon (on p. <a href="#p265" title="go to pg. 265">265</a>) of the angle of contact between -the protoplasm and the axial filament in a Heliozoan we shall -now find to be but a particular case of a general and elementary -theorem.</p> - -<p>Let us re-state as follows, in terms of <i>Energy</i>, the general -principle which underlies the theory of surface tension or capillarity.</p> - -<p>When a fluid is in contact with another fluid, or with a solid -or a gas, a portion of the energy of the system (that, namely, -which we call surface energy), is proportional to the area of the -surface of contact: it is also proportional to a coefficient which -is specific for each particular pair of substances, and which is -constant for these, save only in so far as it may -be modified by <span class="xxpn" id="p294">{294}</span> -changes of temperature or of electric charge. The condition of -<i>minimum potential energy</i> in the system, which is the condition of -equilibrium, will accordingly be obtained by the utmost possible -diminution in the area of the surfaces in contact. When we have -<i>three</i> bodies in contact, the case becomes a little more complex. -Suppose for instance we have a drop of some fluid, <i>A</i>, floating on -another fluid, <i>B</i>, and exposed to air, <i>C</i>. The whole surface energy -of the system may now be considered as divided into two parts, -one at the surface of the drop, and the other outside of the same; -the latter portion is inherent in the surface <i>BC</i>, between the mass -of fluid <i>B</i> and the superincumbent air, <i>C</i>; but the former portion -consists of two parts, for it is divided between the two surfaces <i>AB</i> -and <i>AC</i>, that namely which separates the drop from the surrounding -fluid and that which separates it from the atmosphere. So far as</p> - -<div class="dctr05" id="fig99"> -<img src="images/i294.png" width="434" height="89" alt=""> - <div class="dcaption">Fig. 99.</div></div> - -<p class="pcontinue">the drop is concerned, then, equilibrium depends on a proper -balance between the energy, per unit area, which is resident in -its own two surfaces, and that which is external thereto: that is -to say, if we call <i>E<sub>bc</sub></i> the energy at the surface between the two -fluids, and so on with the other two pairs of surface energies, the -condition of equilibrium, or of maintenance of the drop, is that</p> - -<div class="dmaths"> -<div><i>E<sub>bc</sub></i> < <i>E<sub>ab</sub></i> + <i>E<sub>ac</sub></i>. -</div> - -<p class="pcontinue">If, on the other hand, the fluid -<i>A</i> happens to be oil and the fluid <i>B</i>, water, then the -energy <i>per unit area</i> of the water-air surface is greater -than that of the oil-air surface and that of the oil-water -surface together; i.e.</p> - -<div><i>E<sub>wa</sub></i> > <i>E<sub>oa</sub></i> + <i>E<sub>ow</sub></i>. -</div> - -<p class="pcontinue">Here there is no equilibrium, and in order to obtain it the water-air -surface must always tend to decrease and the other two interfacial -surfaces to increase; which is as much as to say that the water -tends to become covered by a spreading film of oil, and the water-air -surface to be abolished. <span class="xxpn" id="p295">{295}</span></p> -</div><!--dmaths--> - -<p>The surface energy of which we have here spoken is manifested -in that contractile force, or “tension,” of which we have already -had so much to say<a class="afnanch" href="#fn341" id="fnanch341">341</a>. -In any part of the free water surface, for -instance, one surface particle attracts another surface particle, and -the resultant of these multitudinous attractions is an equilibrium -of tension throughout this particular surface. In the case of our -three bodies in contact with one another, and within a small area -very near to the point of contact, a water particle (for instance) -will be pulled outwards by another water particle; but on the -opposite side, so to speak, there will be no water surface, and no -water particle, to furnish the counterbalancing pull; this counterpull,</p> - -<div class="dctr05" id="fig100"><div id="fig101"> -<img src="images/i295.png" width="419" height="341" alt=""> -<div class="dcaption"><div class="nowrap"> -<table> -<tr> - <td>Fig. 100.</td></tr> -<tr> - <td>Fig. 101.</td></tr></table> -</div></div></div></div><!--dctr01--> - -<p class="pcontinue">which is necessary for equilibrium, must therefore be provided -by the tensions existing in the <i>other two</i> surfaces of contact. In -short, if we could imagine a single particle placed at the very point -of contact, it would be drawn upon by three different forces, -whose directions would lie in the three surface planes, and whose -magnitude would be proportional to the specific tensions characteristic -of the two bodies which in each case combine to form the -“interfacial” surface. Now for three forces acting at a point to -be in equilibrium, they must be capable of representation, in -magnitude and direction, by the three sides of a triangle, taken in -order, in accordance with the elementary theorem of the Triangle -of Forces. So, if we know the form of our floating drop (Fig. <a href="#fig100" title="go to Fig. 100">100</a>), -then by drawing tangents from <i>O</i> (the point -of mutual contact), <span class="xxpn" id="p296">{296}</span> -we determine the three angles of our triangle (Fig. <a href="#fig101" title="go to Fig. 101">101</a>), and we -therefore know the relative magnitudes of the three surface -tensions, which magnitudes are proportional to its sides; and -conversely, if we know the magnitudes, or relative magnitudes, -of the three sides of the triangle, we also know its angles, and these -determine the form of the section of the drop. It is scarcely -necessary to mention that, since all points on the edge of the -drop are under similar conditions, one with another, the form of -the drop, as we look down upon it from above, must be circular, -and the whole drop must be a solid of revolution.</p> - -<hr class="hrblk"> - -<div class="dmaths"> -<p>The principle of the Triangle of Forces is expanded, as follows, -by an old seventeenth-century theorem, called Lami’s Theorem: -“<i>If three forces acting at a point be in equilibrium, each force is -proportional to the sine of the angle contained between the directions -of the other two.</i>” That is to say</p> - -<div><i>P</i> : <i>Q</i> : <i>R</i> -: = sin <i>QOR</i> : sin <i>POR</i> : sin <i>POQ</i>. -<br class="brclrfix"></div> - -<p class="pcontinue pleftfloat">or</p> - -<div><i>P</i> ⁄ sin <i>QOR</i> -= <i>Q</i> ⁄ sin <i>ROP</i> -= <i>R</i> ⁄ sin <i>POQ</i>. -<br class="brclrfix"></div> - -<p class="pcontinue">And from this, in turn, we derive the equivalent formulae, by -which each force is expressed in terms of the other two, and of the -angle between them:</p> - -<div><i>P</i><sup>2</sup> -= <i>Q</i><sup>2</sup> + <i>R</i><sup>2</sup> + 2 <i>QR</i> cos(<i>QOR</i>), etc. -</div></div><!--dmaths--> - -<p>From this and the foregoing, we learn the following important -and useful deductions:</p> - -<ul> -<li><p>(1) The three forces can only be in equilibrium when any one -of them is less than the sum of the other two: for otherwise, the -triangle is impossible. Now in the case of a drop of olive-oil -upon a clean water surface, the relative magnitudes of the three -tensions (at 15° C.) have been determined as follows:</p> - -<div class="dtblbox"><div class="nowrap"> -<table> -<tr> - <td class="tdleft">Water-air surface</td> - <td class="tdright">75</td></tr> -<tr> - <td class="tdleft">Oil-air surface</td> - <td class="tdright">32</td></tr> -<tr> - <td class="tdleft">Oil-water surface</td> - <td class="tdright">21</td></tr> -</table></div></div><!--dtblbox--> - -<p>No triangle having sides of these relative magnitudes is possible; -and no such drop therefore can -remain in equilibrium. <span class="xxpn" id="p297">{297}</span></p></li> - -<li><p>(2) The three surfaces may be all alike: as when a soap-bubble -floats upon soapy water, or when two soap-bubbles are -joined together, on either side of a partition-film. In this case, -the three tensions are all equal, and therefore the three angles -are all equal; that is to say, when three similar liquid surfaces -meet together, they always do so at an angle of 120°. Whether -our two conjoined soap-bubbles be equal or unequal, this is still -the invariable rule; because the specific tension of a particular -surface is unaffected by any changes of magnitude or form.</p></li> - -<li><p>(3) If two only of the surfaces be alike, then two of the -angles will be alike, and the other will be unlike; and this last -will be the difference between 360° and the sum of the other two. -A particular case is when a film is stretched between solid and -parallel walls, like a soap-film within a cylindrical tube. Here, so -long as there is no external pressure applied to either side, so long -as both ends of the tube are open or closed, the angles on either -side of the film will be equal, that is to say the film will set itself -at right angles to the sides.</p> - -<p>Many years ago Sachs laid it down as a principle, which has -become celebrated in botany under the name of Sachs’s Rule, -that one cell-wall always tends to set itself at right angles to another -cell-wall. This rule applies to the case which we have just illustrated; -and such validity as the rule possesses is due to the fact -that among plant-tissues it very frequently happens that one -cell-wall has become solid and rigid before another and later -partition-wall is developed in connection with it.</p></li> - -<li><p>(4) There is another important principle which arises not -out of our equations but out of the general considerations -by which we were led to them. We have seen that, at and -near the point of contact between our several surfaces, -there is a continued balance of forces, carried, so to -speak, across the interval; in other words, there is -<i>physical continuity</i> between one surface and another. It -follows necessarily from this that the surfaces merge one -into another by a continuous curve. Whatever be the form -of our surfaces and whatever the angle between them, this -small intervening surface, approximately spherical, is -always there to bridge over the line of contact<a class="afnanch" href="#fn342" id="fnanch342">342</a>; -and -this little fillet, or “bourrelet,” <span class="xxpn" id="p298">{298}</span> -as Plateau called it, is large enough to be a common and conspicuous -feature in the microscopy of tissues (Fig. <a href="#fig102" title="go to Fig. 102">102</a>). For -instance, the so-called “splitting” of the cell-wall, which is conspicuous -at the angles of the large “parenchymatous” cells in the -succulent tissues of all higher plants (Fig. <a href="#fig103" title="go to Fig. 103">103</a>), is nothing more -than a manifestation of Plateau’s “bourrelet,” or surface of -continuity<a class="afnanch" href="#fn343" id="fnanch343">343</a>.</p></li> -</ul> - -<hr class="hrblk"> - -<p>We may now illustrate some of the foregoing principles, -before we proceed to the more complex cases in which more -bodies than three are in mutual contact. But in doing so, we -must constantly bear in mind the principles set forth in our -chapter on the forms of cells, and especially those relating to the -pressure exercised by a curved film.</p> - -<div class="dctr01" id="fig102"><div id="fig103"> -<img src="images/i298.png" width="800" height="356" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 102. (After Berthold.)</td> - <td></td> - <td>Fig. 103. Parenchyma of Maize.</td></tr></table> -</div></div></div><!--dctr01--> - -<p>Let us look for a moment at the case presented by the partition-wall -in a double soap-bubble. As we have just seen, the three -films in contact (viz. the outer walls of the two bubbles and the -partition-wall between) being all composed -of the same substance <span class="xxpn" id="p299">{299}</span> -and all alike in contact with air, the three surface tensions must -be equal; and the three films must therefore, in all cases, meet -at an angle of 120°. But, unless the two bubbles be of precisely -equal size (and therefore of equal curvature) it is obvious that the -tangents to the spheres will not meet the plane of their circle -of contact at equal angles, and therefore that the partition-wall -must be a <i>curved</i> surface: it is only plane when it divides two -equal and symmetrical cells. It is also obvious, from the symmetry -of the figure, that the centres of the spheres, the centre of -the partition, and the centres of the two spherical surfaces are -all on one and the same straight line.</p> - -<div class="dctr01" id="fig104"> -<img src="images/i299.png" width="800" height="202" alt=""> - <div class="dcaption">Fig. 104.</div></div> - -<p>Now the surfaces of the two bubbles exert a pressure inwards -which is inversely proportional to their radii: that is to say -<i>p</i> : <i>p′</i> :: 1 ⁄ <i>r′</i> : 1 ⁄ <i>r</i>; -and the partition wall must, for equilibrium, -exert a pressure (<i>P</i>) which is equal to the difference between these -two pressures, that is to say, -<i>P</i> -= 1 ⁄ <i>R</i> -= 1 ⁄ <i>r′</i> − 1 ⁄ <i>r</i> -<span class="nowrap"> -= (<i>r</i> − <i>r′</i>) ⁄ <i>r r′</i>.</span> It -follows that the curvature of the partition wall must be just such -a curvature as is capable of exerting this pressure, that is to say, -<i>R</i> <span class="nowrap"> -= <i>r r′</i> ⁄ (<i>r</i> − <i>r′</i>).</span> -The partition wall, then, is always a portion of -a spherical surface, whose radius is equal to the product, divided -by the difference, of the radii of the two vesicles. It follows at -once from this that if the two bubbles be equal, the radius of -curvature of the partition is infinitely great, that is to say the -partition is (as we have already seen) a plane surface.</p> - -<p>The geometrical construction by which we obtain the position -of the centres of the two spheres and also of the partition surface -is very simple, always provided that the surface tensions are -uniform throughout the system. If <i>p</i> be a point of contact -between the two spheres, and <i>cp</i> be a radius of one of them, then -make the angle <i>cpm</i> -= 60°, and mark off on <i>pm</i>, <i>pc′</i> -equal to the <span class="xxpn" id="p300">{300}</span> -radius of the other sphere; in like manner, make the angle -<i>c′pn</i> -= 60°, cutting the line <i>cc′</i> in <i>c″</i>; then <i>c′</i> will be the centre -of the second sphere, and <i>c″</i> that of the spherical partition.</p> - -<div class="dctr01" id="fig105"><div id="fig106"> -<img src="images/i300a.png" width="800" height="240" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 105.</td> - <td></td> - <td>Fig. 106.</td></tr></table> -</div></div></div><!--dctr01--> - -<p>Whether the partition be or be not a plane surface, it is obvious -that its <i>line of junction</i> with the rest of the system lies in a plane, -and is at right angles to the axis of symmetry. The actual -curvature of the partition-wall is easily seen in optical section; -but in surface view, the line of junction is <i>projected</i> as a plane -(Fig. <a href="#fig106" title="go to Fig. 106">106</a>), perpendicular to the axis, and this appearance has -also helped to lend support and authority to “Sachs’s Rule.”</p> - -<hr class="hrblk"> - -<div class="dleft dwth-e" id="fig107"> -<img src="images/i300b.png" width="340" height="393" alt=""> - <div class="pcaption">Fig. 107. Filaments, or chains of - cells, in various lower Algae. - (A) <i>Nostoc</i>; (B) <i>Anabaena</i>; (C) - <i>Rivularia</i>; (D) <i>Oscillatoria</i>.</div></div> - -<p>Many spherical cells, such as -Protococcus, divide into two equal -halves, which are therefore separated -by a plane partition. Among -the other lower Algae, akin to -Protococcus, such as the Nostocs -and Oscillatoriae, in which the -cells are imbedded in a gelatinous -matrix, we find a series of forms -such as are represented in Fig. <a href="#fig107" title="go to Fig. 107">107</a>. -Sometimes the cells are solitary -or disunited; sometimes they run -in pairs or in rows, separated one -from another by flat partitions; -and sometimes the conjoined cells -are approximately hemispherical, but at other times each half -is more than a hemisphere. These -various conditions depend, <span class="xxpn" id="p301">{301}</span> -according to what we have already learned, upon the relative -magnitudes of the tensions at the surface of the cells and at the -boundary between them<a class="afnanch" href="#fn344" id="fnanch344">344</a>. -<br class="brclrfix"></p> - -<div class="dmaths"> -<p>In the typical case of an equally divided cell, such as a double -and co-equal soap-bubble, where the partition-wall and the outer -walls are similar to one another and in contact with similar substances, -we can easily determine the form of the system. For, at -any point of the boundary of the partition-wall, <i>O</i>, the tensions -being equal, the angles <i>QOP</i>, <i>ROP</i>, <i>QOR</i> are all equal, and each -is, therefore, an angle of 120°. But <i>OQ</i>, <i>OR</i> being tangents, the -centres of the two spheres (or circular arcs in the figure) lie on -perpendiculars to them; therefore the radii <i>CO</i>, <i>C′O</i> meet at an</p> - -<div class="dctr05" id="fig108"> -<img src="images/i301.png" width="477" height="352" alt=""> - <div class="dcaption">Fig. 108.</div></div> - -<p class="pcontinue">angle of 60°, and <i>COC′</i> is an -equilateral triangle. That is to say, the centre of -each circle lies on the circumference of the other; the -partition lies midway between the two centres; and the -length (i.e. the diameter) of the partition-wall, <i>PO</i>, -is</p> - -<div>2 sin 60° = 1·732</div> - -<p class="pcontinue">times the radius, or ·866 times the -diameter, of each of the cells. This gives us, then, the -<i>form</i> of an aggregate of two equal cells under uniform -conditions.</p> -</div><!--dmaths--> - -<p>As soon as the tensions become unequal, whether from changes -in their own substance or from differences in the substances with -which they are in contact, then the form alters. -If the tension <span class="xxpn" id="p302">{302}</span> -along the partition, <i>P</i>, diminishes, the partition itself enlarges, -and the angle <i>QOR</i> increases: until, when the tension <i>P</i> is very -small compared to <i>Q</i> or <i>R</i>, the whole figure becomes a circle, and -the partition-wall, dividing it into two hemispheres, stands at -right angles to the outer wall. This is the case when the outer -wall of the cell is practically solid. On the other hand, if <i>P</i> begins -to increase relatively to <i>Q</i> and <i>R</i>, then the partition-wall contracts, -and the two adjacent cells become larger and larger segments of -a sphere, until at length the system becomes divided into two -separate cells.</p> - -<div class="dctr03" id="fig109"> -<img src="images/i302.png" width="661" height="227" alt=""> - <div class="dcaption">Fig. 109. Spore of <i>Pellia</i>. - (After Campbell.)</div></div> - -<p>In the spores of Liverworts (such as <i>Pellia</i>), the first partition-wall -(the equatorial partition in Fig. <a href="#fig109" title="go to Fig. 109">109</a>, <i>a</i>) divides the spore into -two equal halves, and is therefore a plane surface, normal to the -surface of the cell; but the next partitions arise near to either -end of the original spherical or elliptical cell. Each of these latter -partitions will (like the first) tend to set itself normally to the -cell-wall; at least the angles on either side of the partition will -be identical, and their magnitude will depend upon the tension -existing between the cell-wall and the surrounding medium. -They will only be right angles if the cell-wall is already practically -solid, and in all probability (rigidity of the cell-wall not being -quite attained) they will be somewhat greater. In either case -the partition itself will be a portion of a sphere, whose curvature -will now denote a difference of pressures in the two chambers or -cells, which it serves to separate. (The later stages of cell-division, -represented in the figures <i>b</i> and <i>c</i>, we are not yet in a position to -deal with.)</p> - -<p>We have innumerable cases, near the tip of a growing filament, -where in like manner the partition-wall which cuts -off the terminal <span class="xxpn" id="p303">{303}</span> -cell constitutes a spherical lens-shaped surface, set normally to -the adjacent walls. At the tips of the branches of many Florideae, -for instance, we find such a lenticular partition. In <i>Dictyota -dichotoma</i>, as figured by Reinke, we have a succession of such -partitions; and, by the way, in such cases as these, where the -tissues are very transparent, we have often in optical section a -puzzling confusion of lines; one being the optical section of the -curved partition-wall, the other being the straight linear projection -of its outer edge to which we have already referred. In the -conical terminal cell of Chara, we have the same lens-shaped -curve, but a little lower down, where the sides of the shoot are -approximately parallel, we have flat transverse partitions, at the -edges of which, however, we recognise a convexity of the outer -cell-wall and a definite angle of contact, equal on the two sides -of the partition.</p> - -<div class="dctr01" id="fig110"><div id="fig111"> -<img src="images/i303.png" width="800" height="216" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 110. Cells of <i>Dictyota</i>. - (After Reinke.)</td> - <td></td> - <td>Fig. 111. Terminal and other cells - of <i>Chara</i>.</td></tr></table> -</div></div></div><!--dctr01--> - -<div class="dright dwth-h" id="fig112"> -<img src="images/i303-2.png" width="212" height="287" alt=""> - <div class="dcaption">Fig. 112. Young antheridium of - <i>Chara</i>.</div></div> - -<p>In the young antheridia of Chara (Fig. <a href="#fig112" title="go to Fig. 112">112</a>), and in the not -dissimilar case of the sporangium (or conidiophore) of Mucor, we -easily recognise the hemispherical form of the septum which shuts -off the large spherical cell from the cylindrical -filament. Here, in the first phase of development, -we should have to take into consideration -the different pressures exerted by the single -curvature of the cylinder and the double -curvature of its spherical cap (p. <a href="#p221" title="go to pg. 221">221</a>); and -we should find that the partition would have -a somewhat low curvature, with a radius <i>less</i> -than the diameter of the cylinder; which it -would have exactly equalled but for the -additional pressure inwards -which it receives <span class="xxpn" id="p304">{304}</span> -from the curvature of the large surrounding sphere. But as the -latter continues to grow, its curvature decreases, and so likewise -does the inward pressure of its surface; and accordingly the little -convex partition bulges out more and more. -<br class="brclrfix"></p> - -<hr class="hrblk"> - -<p>In order to epitomise the foregoing facts let the annexed -diagrams (Fig. <a href="#fig113" title="go to Fig. 113">113</a>) represent a system of three films, of which -one is a partition-wall between the other two; and let the tensions -at the three surfaces, or the tractions exercised upon a point at -their meeting-place, be proportional to <i>T</i>, <i>T′</i> and <i>t</i>. Let α, β, γ -be, as in the figure, the opposite angles. Then:</p> - -<ul> -<li><p>(1) If <i>T</i> be equal to <i>T′</i>, and <i>t</i> be relatively insignificant, -the angles α, β will be of 90°.</p> - -<div class="dctr01" id="fig113"> -<img src="images/i304.png" width="800" height="198" alt=""> - <div class="dcaption">Fig. 113.</div></div> -</li> - -<li><p>(2) If <i>T</i> -= <i>T′</i>, but be a little greater than <i>t</i>, then <i>t</i> will exert -an appreciable traction, and α, β will be more than 90°, say, for -instance, 100°.</p></li> - -<li><p>(3) If <i>T</i> -= <i>T′</i> -= <i>t</i>, then α, β, γ will all equal 120°.</p></li> -</ul> - -<p>The more complicated cases, when <i>t</i>, <i>T</i> and <i>T′</i> are all unequal, -are already sufficiently explained.</p> - -<hr class="hrblk"> - -<p>The biological facts which the foregoing considerations go a -long way to explain and account for have been the subject of much -argument and discussion, especially on the part of the botanists. -Let me recapitulate, in a very few words, the history of this long -discussion.</p> - -<p>Some fifty years ago, Hofmeister laid it down as a general law -that “The partition-wall stands always perpendicular to what was -previously the principal direction of growth in the cell,”—or, in -other words, perpendicular to the long axis -of the cell<a class="afnanch" href="#fn345" id="fnanch345">345</a>. -Ten <span class="xxpn" id="p305">{305}</span> -years later, Sachs formulated his rule, or principle, of “rectangular -section,” declaring that in all tissues, however complex, the -cell-walls cut one another (at the time of their formation) at right -angles<a class="afnanch" href="#fn346" id="fnanch346">346</a>. -Years before, Schwendener had found, in the final -results of cell-division, a universal system of “orthogonal trajectories<a class="afnanch" href="#fn347" id="fnanch347">347</a>”; -and this idea Sachs further developed, introducing -complicated systems of confocal ellipses and hyperbolæ, and -distinguishing between periclinal walls, whose curves approximate -to the peripheral contours, radial partitions, which cut these at -an angle of 90°, and finally anticlines, which stand at right angles -to the other two.</p> - -<p>Reinke, in 1880, was the first to throw some doubt upon this -explanation. He pointed out various cases where the angle was -not a right angle, but was very definitely an acute one; and -he saw, apparently, in the more common rectangular symmetry -merely what he calls a necessary, but <i>secondary</i>, result of growth<a class="afnanch" href="#fn348" id="fnanch348">348</a>.</p> - -<p>Within the next few years, a number of botanical writers were -content to point out further exceptions to Sachs’s Rule<a class="afnanch" href="#fn349" id="fnanch349">349</a>; -and in -some cases to show that the <i>curvatures</i> of the partition-walls, -especially such cases of lenticular curvature as we have described, -were by no means accounted for by either Hofmeister or Sachs; -while within the same period, Sachs himself, and also Rauber, -attempted to extend the main generalisation to animal tissues<a class="afnanch" href="#fn350" id="fnanch350">350</a>.</p> - -<p>While these writers regarded the form and arrangement of the -cell-walls as a biological phenomenon, with little if any direct -relation to ordinary physical laws, or with but a vague reference -to “mechanical conditions,” the physical side of the case was -soon urged by others, with more or less force and cogency. Indeed -the general resemblance between a cellular -tissue and a “froth” <span class="xxpn" id="p306">{306}</span> -had been pointed out long before, by Melsens, who had made an -“artificial tissue” by blowing into a solution of white of egg<a class="afnanch" href="#fn351" id="fnanch351">351</a>.</p> - -<p>In 1886, Berthold published his <i>Protoplasmamechanik</i>, in which -he definitely adopted the principle of “minimal areas,” and, -following on the lines of Plateau, compared the forms of many -cell-surfaces and the arrangement of their partitions with those -assumed under surface tension by a system of “weightless films.” -But, as Klebs<a class="afnanch" href="#fn352" id="fnanch352">352</a> -points out in reviewing Berthold’s book, Berthold -was careful to stop short of attributing the biological phenomena -to a definite mechanical cause. They remained for him, as they -had done for Sachs, so many “phenomena of growth,” or -“properties of protoplasm.”</p> - -<p>In the same year, but while still apparently unacquainted with -Berthold’s work, Errera<a class="afnanch" href="#fn353" id="fnanch353">353</a> -published a short but very lucid article, -in which he definitely ascribed to the cell-wall (as Hofmeister had -already done) the properties of a semi-liquid film and drew from -this as a logical consequence the deduction that it <i>must</i> assume the -various configurations which the law of minimal areas imposes on -the soap-bubble. So what we may call <i>Errera’s Law</i> is formulated -as follows: A cellular membrane, at the moment of its formation, -tends to assume the form which would be assumed, under the -same conditions, by a liquid film destitute of weight.</p> - -<p>Soon afterwards Chabry, in discussing the embryology of the -Ascidians, indicated many of the points in which the contacts -between cells repeat the surface-tension phenomena of the soap-bubble, -and came to the conclusion that part, at least, of the -embryological phenomena were purely physical<a class="afnanch" href="#fn354" id="fnanch354">354</a>; -and the same -line of investigation and thought were pursued and developed by -Robert, in connection with the embryology of the Mollusca<a class="afnanch" href="#fn355" id="fnanch355">355</a>. -Driesch again, in a series of papers, continued to draw attention -to the presence of capillary phenomena in -the segmenting cells <span class="xxpn" id="p307">{307}</span> -of various embryos, and came to the conclusion that the mode of -segmentation was of little importance as regards the final result<a class="afnanch" href="#fn356" id="fnanch356">356</a>.</p> - -<p>Lastly de Wildeman<a class="afnanch" href="#fn357" id="fnanch357">357</a>, -in a somewhat wider, but also vaguer -generalisation than Errera’s, declared that “The form of the -cellular framework of vegetables, and also of animals, in its -essential features, depends upon the forces of molecular physics.”</p> - -<hr class="hrblk"> - -<p>Let us return to our problem of the arrangement of partition -films. When we have three bubbles in contact, instead of two as -in the case already considered, the phenomenon is strictly analogous -to our former case. The three bubbles will be separated by three -partition surfaces, whose curvature will depend upon the relative</p> - -<div class="dctr01" id="fig114"> -<img src="images/i307.png" width="800" height="389" alt=""> - <div class="dcaption">Fig. 114.</div></div> - -<p class="pcontinue">size -of the spheres, and which will be plane if the latter are all of -the same dimensions; but whether plane or curved, the three -partitions will meet one another at an angle of 120°, in an axial -line. Various pretty geometrical corollaries accompany this arrangement. -For instance, if Fig. <a href="#fig114" title="go to Fig. 114">114</a> represent the three associated -bubbles in a plane drawn through their centres, <i>c</i>, <i>c′</i>, <i>c″</i> (or what -is the same thing, if it represent the base of three bubbles resting -on a plane), then the lines <i>uc</i>, <i>uc″</i>, or <i>sc</i>, <i>sc′</i>, -etc., drawn to the <span class="xxpn" id="p308">{308}</span> -centres from the points of intersection of the circular arcs, will -always enclose an angle of 60°. Again (Fig. <a href="#fig115" title="go to Fig. 115">115</a>), if we make the -angle <i>c″uf</i> equal to 60°, and produce <i>uf</i> to meet <i>cc″</i> in <i>f</i>, <i>f</i> will be -the centre of the circular arc which constitutes the partition <i>Ou</i>; -and further, the three points <i>f</i>, <i>g</i>, <i>h</i>, successively determined in this</p> - -<div class="dctr05" id="fig115"> -<img src="images/i308.png" width="481" height="910" alt=""> - <div class="dcaption">Fig. 115.</div></div> - -<p class="pcontinue">manner, will lie on one and the same straight line. In the case -of coequal bubbles or cells (as in Fig. <a href="#fig114" title="go to Fig. 114">114</a>, B), it is obvious that -the lines joining their centres form an equilateral triangle; and -consequently, that the centre of each circle (or sphere) lies on the -circumference of the other two; it is also obvious that -<i>uf</i> is now <span class="xxpn" id="p309">{309}</span> -parallel to <i>cc″</i>, and accordingly that the centre of curvature of -the partition is now infinitely distant, or (as we have already said), -that the partition itself is plane.</p> - -<p>When we have four bubbles in conjunction, they would seem -to be capable of arrangement in two symmetrical ways: either, -as in Fig. <a href="#fig116" title="go to Fig. 116">116</a> (A), with the four partition-walls meeting at right -angles, or, as in (B), with <i>five</i> partitions meeting, three and three, -at angles of 120°. This latter arrangement is strictly analogous -to the arrangement of three bubbles in Fig. <a href="#fig114" title="go to Fig. 114">114</a>. Now, though -both of these figures, from their symmetry, are apparently figures of -equilibrium, yet, physically, the former turns out to be of unstable</p> - -<div class="dctr01" id="fig116"> -<img src="images/i309.png" width="800" height="482" alt=""> - <div class="dcaption">Fig. 116.</div></div> - -<p class="pcontinue">and the latter of stable equilibrium. -If we try to bring our four bubbles into the form of Fig. -<a href="#fig116" title="go to Fig. 116">116</a>, A, such an arrangement endures only for an instant; -the partitions glide upon each other, a median wall springs -into existence, and the system at once assumes the form -of our second figure (B). This is a direct consequence of -the law of minimal areas: for it can be shewn, by somewhat -difficult mathematics (as was first done by Lamarle), -that, in dividing a closed space into a given number of -chambers by means of partition-walls, the least possible -area of these partition-walls, taken together, can only -be attained when they meet together in groups of three, -at equal angles, that is to say at angles of 120°. <span -class="xxpn" id="p310">{310}</span></p> - -<p>Wherever we have a true cellular complex, an arrangement of -cells in actual physical contact by means of a boundary film, we -find this general principle in force; we must only bear in mind -that, for its perfect recognition, we must be able to view the -object in a plane at right angles to the boundary walls. For -instance, in any ordinary section of a vegetable parenchyma, we -recognise the appearance of a “froth,” precisely resembling that -which we can construct by imprisoning a mass of soap-bubbles in -a narrow vessel with flat sides of glass; in both cases we see the -cell-walls everywhere meeting, by threes, at angles of 120°, irrespective -of the size of the individual cells: whose relative size, on -the other hand, determines the <i>curvature</i> of the partition-walls. -On the surface of a honey-comb we have precisely the same -conjunction, between cell and cell, of three boundary walls, -meeting at 120°. In embryology, when we examine a segmenting -egg, of four (or more) segments, we find in like manner, in the great -majority of cases, if not in all, that the same principle is still -exemplified; the four segments do not meet in a common centre, -but each cell is in contact with two others, and the three, and only -three, common boundary walls meet at the normal angle of 120°. -A so-called <i>polar furrow</i><a class="afnanch" href="#fn358" id="fnanch358">358</a>, -the visible edge of a vertical partition-wall, -joins (or separates) the two triple contacts, precisely as in -Fig. <a href="#fig116" title="go to Fig. 116">116</a>, B.</p> - -<p>In the four-celled stage of the frog’s egg, Rauber (an exceptionally -careful observer) shews us three alternative modes in which -the four cells may be found to be conjoined (Fig. <a href="#fig117" title="go to Fig. 117">117</a>). In (A) we -have the commonest arrangement, which is that which we have -just studied and found to be the simplest theoretical one; that -namely where a straight “polar furrow” intervenes, and where, -at its extremities, the partition-walls are conjoined three by three. -In (B), we have again a polar furrow, which is now seen to be a -portion of the first “segmentation-furrow” (cf. Fig. <a href="#fig155" title="go to Fig. 155">155</a> etc.) by -which the egg was originally divided into two; the four-celled -stage being reached by the appearance of -the transverse furrows <span class="xxpn" id="p311">{311}</span> -and their corresponding partitions. In this case, the polar -furrow is seen to be sinuously curved, and Rauber tells us that -its curvature gradually alters: as a matter of fact, it (or rather -the partition-wall corresponding to it) is gradually setting itself -into a position of equilibrium, that is to say of equiangular contact -with its neighbours, which position of equilibrium is already -attained or nearly so in Fig. <a href="#fig117" title="go to Fig. 117">117</a>, A. In Fig. <a href="#fig117" title="go to Fig. 117">117</a>, C, we have a -very different condition, with which we shall deal in a moment.</p> - -<div class="dctr03" id="fig117"> -<img src="images/i311.png" width="600" height="158" alt=""> - <div class="pcaption">Fig. 117. Various ways in which the four - cells are co-arranged in the four-celled stage of the - frog’s egg. (After Rauber.)</div></div> - -<p>According to the relative magnitude of the bodies in contact, -this “polar furrow” may be longer or shorter, and it may be so -minute as to be not easily discernible; but it is quite certain that -no simple and homogeneous system of fluid films such as we -are dealing with is in equilibrium without its presence. In the -accounts given, however, by embryologists of the segmentation of -the egg, while the polar furrow is depicted in the great majority -of cases, there are others in which it has not been seen and some -in which its absence is definitely asserted<a class="afnanch" href="#fn359" id="fnanch359">359</a>. -The cases where four -cells, lying in one plane, meet <i>in a point</i>, such as were frequently -figured by the older embryologists, are very difficult to verify, -and I have not come across a single clear case in recent literature. -Considering the physical stability of the other arrangement, the -great preponderance of cases in which it is known to occur, the -difficulty of recognising the polar furrow in cases where it is -very small and unless it be specially looked for, and the natural -tendency of the draughtsman to make an all but symmetrical -structure appear wholly so, I am much -inclined to attribute to <span class="xxpn" id="p312">{312}</span> -error or imperfect observation all those cases where the junction-lines -of four cells are represented (after the manner of Fig. <a href="#fig116" title="go to Fig. 116">116</a>, A) -as a simple cross<a class="afnanch" href="#fn360" id="fnanch360">360</a>.</p> - -<p>But while a true four-rayed intersection, or simple cross, is -theoretically impossible (save as a transitory and highly unstable -condition), there is another condition which may closely simulate -it, and which is common enough. There are plenty of representations -of segmenting eggs, in which, instead of the triple -junction and polar furrow, the four cells (and in like manner their -more numerous successors) are represented as <i>rounded off</i>, and -separated from one another by an empty space, or by a little drop -of an extraneous fluid, evidently not directly miscible with the -fluid surfaces of the cells. Such is the case in the obviously -accurate figure which Rauber gives (Fig. <a href="#fig117" title="go to Fig. 117">117</a>, C) of the third mode -of conjunction in the four-celled stage of the frog’s egg. Here -Rauber is most careful to point out that the furrows do not simply -“cross,” or meet in a point, but are separated by a little space, -which he calls the <i>Polgrübchen</i>, and asserts to be constantly present -whensoever the polar furrow, or <i>Brechungslinie</i>, is not to be -discerned. This little interposed space, with its contained drop -of fluid, materially alters the case, and implies a new condition -of theoretical and actual equilibrium. For, on the one hand, we -see that now the four intercellular partitions do not meet <i>one -another at all</i>; but really impinge upon four new and separate -partitions, which constitute interfacial contacts, not between cell -and cell, but between the respective cells and the intercalated -drop. And secondly, the angles at which these four little surfaces -will meet the four cell-partitions, will be determined, in the usual -way, by the balance between the respective tensions of these several -surfaces. In an extreme case (as in some pollen-grains) it may be -found that the cells under the observed circumstances are not truly -in surface contact: that they are so many drops which touch but -do not “wet” one another, and which are merely held together -by the pressure of the surrounding envelope. -But even supposing, <span class="xxpn" id="p313">{313}</span> -as is in all probability the actual case, that they are in actual fluid -contact, the case from the point of view of surface tension presents -no difficulty. In the case of the conjoined soap-bubbles, we were -dealing with <i>similar</i> contacts and with <i>equal</i> surface tensions throughout -the system; but in the system of protoplasmic cells which -constitute the segmenting egg we must make allowance for <i>an inequality</i> -of tensions, between the surfaces where cell meets cell, and -where on the other hand cell-surface is in contact with the surrounding -medium,—in this case generally water or one of the fluids -of the body. Remember that our general condition is that, in our entire</p> - -<div class="dright dwth-d" id="fig118"> -<img src="images/i313.png" width="366" height="266" alt=""> - <div class="dcaption">Fig. 118.</div></div> - -<p class="pcontinue">system, the <i>sum of the surface energies</i> is a minimum; and, -while this is attained by the <i>sum -of the surfaces</i> being a minimum -in the case where the energy is -uniformly distributed, it is not -necessarily so under non-uniform -conditions. In the diagram (Fig. -<a href="#fig118" title="go to Fig. 118">118</a>) if the energy per unit area -be greater along the contact -surface <i>cc′</i>, where cell meets cell, -than along <i>ca</i> or <i>cb</i>, where cell-surface -is in contact with the surrounding medium, these latter -surfaces will tend to increase and the surface of cell-contact -to diminish. In short there will be the usual balance of forces -between the tension along the surface <i>cc′</i>, and the two opposing -tensions along <i>ca</i> and <i>cb</i>. If the former be greater than either -of the other two, the outside angle will be less than 120°; and if -the tension along the surface <i>cc′</i> be as much or more than the -sum of the other two, then the drops will stand in contact only, -save for the possible effect of external pressure, at a point. This is -the explanation, in general terms, of the peculiar conditions -obtaining in Nostoc and its allies (p. <a href="#p300" title="go to pg. 300">300</a>), and it also leads us to -a consideration of the general properties and characters of an -“epidermal” layer.<br class="brclrfix"></p> - -<hr class="hrblk"> - -<p>While the inner cells of the honey-comb are symmetrically -situated, sharing with their neighbours in equally distributed -pressures or tensions, and therefore all tending -with great accuracy <span class="xxpn" id="p314">{314}</span> -to identity of form, the case is obviously different with the cells -at the borders of the system. So it is, in like manner, with our -froth of soap-bubbles. The bubbles, or cells, in the interior of -the mass are all alike in general character, and if they be equal -in size are alike in every respect: their sides are uniformly -flattened<a class="afnanch" href="#fn361" id="fnanch361">361</a>, -and tend to meet at equal angles of 120°. But the -bubbles which constitute the outer layer retain their spherical -surfaces, which however still tend to meet the partition-walls -connected with them at constant angles of 120°. This outer layer -of bubbles, which forms the surface of our froth, constitutes after -a fashion what we should call in botany an “epidermal” layer. -But in our froth of soap-bubbles we have, as a rule, the same kind -of contact (that is to say, contact with <i>air</i>) both within and without -the bubbles; while in our living cell, the outer wall of the epidermal -cell is exposed to air on the one side, but is in contact with the</p> - -<div class="dctr04" id="fig119"> -<img src="images/i314.png" width="483" height="191" alt=""> - <div class="dcaption">Fig. 119.</div></div> - -<p class="pcontinue">protoplasm of the cell on the other: and this involves a difference -of tensions, so that the outer walls and their adjacent partitions -are no longer likely to meet at equal angles of 120°. Moreover, -a chemical change, due for instance to oxidation or possibly also -to adsorption, is very likely to affect the external wall, and may -tend to its consolidation; and this process, as we have seen, is -tantamount to a large increase, and at the same time an -equalisation, of tension in that outer wall, and will lead the -adjacent partitions to impinge upon it at angles more and -more nearly approximating to 90°: the bubble-like, or spherical, -surfaces of the individual cells being more and more flattened -in consequence. Lastly, the chemical changes which affect the -outer walls of the superficial cells may extend, in greater or -less degree, to their inner walls also: with the -result that these <span class="xxpn" id="p315">{315}</span> -cells will tend to become more or less rectangular throughout, and -will cease to dovetail into the interstices of the next subjacent -layer. These then are the general characters which we recognise -in an epidermis; and we perceive that the fundamental character -of an epidermis simply is that it lies on the outside, and that its -main physical characteristics follow, as a matter of course, from -the position which it occupies and from the various consequences -which that situation entails. We have however by no means -exhausted the subject in this short account; for the botanist is -accustomed to draw a sharp distinction between a true epidermis -and what is called epidermal tissue. The latter, which is found in -such a sea-weed as Laminaria and in very many other cryptogamic -plants, consists, as in the hypothetical case we have described, -of a more or less simple and direct modification of the general or -fundamental tissue. But a “true epidermis,” such as we have it -in the higher plants, is something with a long morphological history, -something which has been laid down or differentiated in an early -stage of the plant’s growth, and which afterwards retains its -separate and independent character. We shall see presently that -a physical reason is again at hand to account, under certain -circumstances, for the early partitioning off, from a mass of -embryonic tissue, of an outer layer of cells which from their first -appearance are marked off from the rest by their rectangular and -flattened form.</p> - -<hr class="hrblk"> - -<p>We have hitherto considered our cells, or bubbles, as lying in -a plane of symmetry, and further, we have only considered the -appearance which they present as projected on that plane: in -simpler words, we have been considering their appearance in -surface or in sectional view. But we have further to consider -them as solids, whether they be still grouped in relation to a single -plane (like the four cells in Fig. <a href="#fig116" title="go to Fig. 116">116</a>) or heaped upon one another, -as for instance in a tetrahedral form like four cannon-balls; and in -either case we have to pass from the problems of plane to those of -solid geometry. In short, the further development of our theme -must lead us along two paths of enquiry, which continually -intercross, namely (1) the study of more complex cases of partition -and of contact in a plane, and (2) the whole question -of the surfaces <span class="xxpn" id="p316">{316}</span> -and angles presented by solid figures in symmetrical juxtaposition. -Let us take a simple case of the latter kind, and again afterwards, -so far as possible, let us try to keep the two themes separate.</p> - -<p>Where we have three spheres in contact, as in Fig. <a href="#fig114" title="go to Fig. 114">114</a> or in -either half of Fig. <a href="#fig116" title="go to Fig. 116">116</a>, B, let us consider the point of contact -(<i>O</i>, Fig. <a href="#fig114" title="go to Fig. 114">114</a>) not as a point in the plane section of the diagram, but -as a point where three <i>furrows</i> meet on the surface of the system. -At this point, <i>three cells</i> meet; but it is also obvious that there meet -here <i>six surfaces</i>, namely the outer, spherical walls of the three -bubbles, and the three partition-walls which divide them, two and -two. Also, <i>four</i> lines or <i>edges</i> meet here; viz. the three external arcs -which form the outer boundaries of the partition-walls (and which -correspond to what we commonly call the “furrows” in the segmenting -egg); and as a fourth edge, the “arris” or junction of the -three partitions (perpendicular to the plane of the paper), where -they all three meet together, as we have seen, at equal angles of -120°. Lastly, there meet at the point <i>four solid angles</i>, each -bounded by three surfaces: to wit, within each bubble a solid -angle bounded by two partition-walls and by the surface wall; -and (fourthly) an external solid angle bounded by the outer -surfaces of all three bubbles. Now in the case of the soap-bubbles -(whose surfaces are all in contact with air, both outside and in), -the six films meeting at the point, whether surface films or partition -films, are all similar, with similar tensions. In other words the -tensions, or forces, acting at the point are all similar and symmetrically -arranged, and it at once follows from this that the angles, -solid as well as plane, are all equal. It is also obvious that, as -regards the point of contact, the system will still be symmetrical, -and its symmetry will be quite unchanged, if we add a fourth -bubble in contact with the other three: that is to say, if where -we had merely the outer air before, we now replace it by the air -in the interior of another bubble. The only difference will be that -the pressure exercised by the walls of this fourth bubble will alter -the curvature of the surfaces of the others, so far as it encloses -them; and, if all four bubbles be identical in size, these surfaces -which formerly we called external and which have now come to -be internal partitions, will, like the others, be flattened by equal -and opposite pressure, into planes. We are now -dealing, in short, <span class="xxpn" id="p317">{317}</span> -with six planes, meeting symmetrically in a point, and constituting -there four equal solid angles.</p> - -<div class="dctr05" id="fig120"> -<img src="images/i317.png" width="433" height="450" alt=""> - <div class="dcaption">Fig. 120.</div></div> - -<p>If we make a wire cage, in the form of a regular tetrahedron, -and dip it into soap-solution, then when we withdraw it we see -that to each one of the six edges of the tetrahedron, i.e. to each -one of the six wires which constitute the little cage, a film has -attached itself; and these six films meet internally at a point, and -constitute in every respect the symmetrical figure which we have -just been describing. In short, the system of films we have -hereby automatically produced is precisely the system of partition-walls -which exist in our tetrahedral aggregation of four spherical -bubbles:—precisely the same, that is to say, in the neighbourhood -of the meeting-point, and only differing in that we have made the -wires of our tetrahedron straight, instead of imitating the circular -arcs which actually form the intersections of our bubbles. This -detail we can easily introduce in our wire model if we please.</p> - -<div class="dmaths"> -<p>Let us look for a moment at the geometry of our figure. Let <i>o</i> -(Fig. <a href="#fig120" title="go to Fig. 120">120</a>) be the centre of the tetrahedron, i.e. the centre of symmetry -where our films meet; and let <i>oa</i>, <i>ob</i>, <i>oc</i>, <i>od</i>, be lines drawn to -the four corners of the tetrahedron. Produce <i>ao</i> to meet the base -in <i>p</i>; then <i>apd</i> is a right-angled triangle. It is not difficult to -prove that in such a figure, <i>o</i> (the centre of gravity -of the system) <span class="xxpn" id="p318">{318}</span> -lies just three-quarters of the way between an apex, <i>a</i>, and a point, -<i>p</i>, which is the centre of gravity of the opposite base. Therefore</p> - -<div><i>op</i> -= <i>oa</i> ⁄ 3 -= <i>od</i> ⁄ 3.</div> - -<p class="pcontinue">Therefore</p> - -<div>cos <i>dop</i> -= 1 ⁄ 3    and    cos <i>aod</i> -= − 1 ⁄ 3.</div> -</div><!--dmaths--> - -<p>That is to say, the angle <i>aod</i> is just, as nearly as possible, -109° 28′ 16″. This angle, then, of 109° 28′ 16″, or very nearly -109 degrees and a half, is the angle at which, in this and <i>every -other solid system</i> of liquid films, the edges of the partition-walls -meet one another at a point. It is the fundamental angle in the -solid geometry of our systems, just as 120° was the fundamental -angle of symmetry so long as we considered only the plane projection, -or plane section, of three films meeting in an edge.</p> - -<hr class="hrblk"> - -<p>Out of these two angles, we may construct a great variety of -figures, plane and solid, which become all the more varied and -complex when, by considering the case of unequal as well as equal -cells, we admit curved (e.g. spherical) as well as plane boundary -surfaces. Let us consider some examples and illustrations of -these, beginning with those which we need only consider in reference -to a plane.</p> - -<p>Let us imagine a system of equal cylinders, or equal spheres, -in contact with one another in a plane, and represented in section -by the equal and contiguous circles of Fig. <a href="#fig121" title="go to Fig. 121">121</a>. I borrow my -figure, by the way, from an old Italian naturalist, Bonanni (a -contemporary of Borelli, of Hay and Willoughby and of Martin -Lister), who dealt with this matter in a book chiefly devoted to -molluscan shells<a class="afnanch" href="#fn362" id="fnanch362">362</a>.</p> - -<p>It is obvious, as a simple geometrical fact, that each of these -equal circles is in contact with six surrounding circles. Imagine -now that the whole system comes under some uniform stress. -It may be of uniform surface tension at the boundaries of all the -cells; it may be of pressure caused by uniform growth or expansion -within the cells; or it may be due to some uniformly applied -constricting pressure from without. In all of these cases the <i>points</i> -of contact between the circles in the diagram -will be extended into <span class="xxpn" id="p319">{319}</span> -<i>lines</i> of contact, representing <i>surfaces</i> of contact in the actual -spheres or cylinders; and the equal circles of our diagram will -be converted into regular and equal hexagons. The angles of -these hexagons, at each of which three hexagons meet, are of -course angles of 120°. So far as the form is concerned, so long as -we are concerned only with a morphological result and not with -a physiological process, the result is precisely the same whatever -be the force which brings the bodies together in symmetrical -apposition; it is by no means necessary for us, in the first instance, -even to enquire whether it be surface tension or mechanical -pressure or some other physical force which is the cause, or the -main cause, of the phenomenon.</p> - -<div class="dctr01" id="fig121"> -<img src="images/i319.png" width="800" height="515" alt=""> - <div class="dcaption">Fig. 121. Diagram of hexagonal cells. - (After Bonanni.)</div></div> - -<p>The production by mutual interaction of polyhedral cells, -which, under conditions of perfect symmetry, become regular -hexagons, is very beautifully illustrated by Prof. Bénard’s -“<i>tourbillons cellulaires</i>” (cf. p. 259), and also in some of Leduc’s -diffusion experiments. A weak (5 per cent.) solution of gelatine -is allowed to set on a plate of glass, and little drops of a 5 or -10 per cent. solution of ferrocyanide of potassium are then placed -at regular intervals upon the gelatine. Immediately each little -drop becomes the centre, or pole, of a system -of diffusion currents, <span class="xxpn" id="p320">{320}</span> -and the several systems conflict with and repel one another, so -that presently each little area becomes the seat of a double current -system, from its centre outwards and back again; until at length -the concentration of the field becomes equalised -and the currents <span class="xxpn" id="p321">{321}</span></p> - -<div class="dctr04" id="fig122"> -<img src="images/i320a.png" width="555" height="419" alt=""> - <div class="pcaption">Fig. 122. An “artificial tissue,” formed by -coloured drops of sodium chloride solution diffusing in a -less dense solution of the same salt. (After Leduc.)</div></div> - -<div class="dctr04" id="fig123"> -<img src="images/i320b.png" width="555" height="530" alt=""> - <div class="pcaption">Fig. 123. An artificial cellular tissue, -formed by the diffusion in gelatine of drops of a solution -of potassium ferrocyanide. (After Leduc.)</div></div> - -<p class="pcontinue">cease. After equilibrium is attained, and when the gelatinous -mass is permitted to dry, we have an artificial tissue of more or -less regularly hexagonal “cells,” which simulate in the closest way -an organic parenchyma. And by varying the experiment, in ways -which Leduc describes, we may simulate various forms of tissue, -and produce cells with thick walls or with thin, cells in close -contact or with wide intercellular spaces, cells with plane or with -curved partitions, and so forth.</p> - -<hr class="hrblk"> - -<p>The hexagonal pattern is illustrated among organisms in countless -cases, but those in which the pattern is perfectly regular, by -reason of perfect uniformity of force and perfect equality of the -individual cells, are not so numerous. The hexagonal epithelium-cells -of the pigment layer of the eye, external to the retina, are -a good example. Here we have a single layer of uniform cells, -reposing on the one hand upon a basement membrane, supported</p> - -<div class="dctr04" id="fig124"> -<img src="images/i321.png" width="520" height="152" alt=""> - <div class="dcaption">Fig. 124. Epidermis of <i>Girardia</i>. -(After Goebel.)</div></div> - -<p class="pcontinue">behind by the solid -wall of the sclerotic, and exposed on the other -hand to the uniform fluid pressure of the vitreous humour. The -conditions all point, and lead, to a perfectly symmetrical result: -that is to say, the cells, uniform in size, are flattened out to a -uniform thickness by the fluid pressure acting radially; and their -reaction on each other converts the flattened discs into regular -hexagons. In an ordinary columnar epithelium, such as that of -the intestine, we see again that the columnar cells have been -compressed into hexagonal prisms; but here as a rule the cells -are less uniform in size, small cells are apt to be intercalated -among the larger, and the perfect symmetry is accordingly lost. -The same is true of ordinary vegetable parenchyma; the originally -spherical cells are approximately equal in size, but only approximately; -and there are accordingly all degrees in the regularity and -symmetry of the resulting tissue. But -obviously, wherever we <span class="xxpn" id="p322">{322}</span> -have, in addition to the forces which tend to produce the regular -hexagonal symmetry, some other asymmetrical component arising -from growth or traction, then our regular hexagons will be distorted -in various simple ways. This condition is illustrated in -the accompanying diagram of the epidermis of Girardia; it also -accounts for the more or less pointed or fusiform cells, each still -in contact (as a rule) with six others, which form the epithelial -lining of the blood-vessels: and other similar, or analogous, -instances are very common.</p> - -<div class="dctr03" id="fig125"> -<img src="images/i322a.png" width="595" height="304" alt=""> - <div class="dcaption">Fig. 125. Soap-froth under pressure. - (After Rhumbler.)</div></div> - -<p>In a soap-froth imprisoned between two glass plates, we have -a symmetrical system of cells, which appear in optical section (as -in Fig. <a href="#fig125" title="go to Fig. 125">125</a>, B) as regular hexagons; but if we press the plates a -little closer together, the hexagons become deformed or flattened -(Fig. <a href="#fig125" title="go to Fig. 125">125</a>, A). In this case, however, if -we cease to apply further pressure, the -tension of the films throughout the -system soon adjusts itself again, and in a -short time the system has regained the -former symmetry of Fig. <a href="#fig125" title="go to Fig. 125">125</a>, B.</p> - -<div class="dleft dwth-f" id="fig126"> -<img src="images/i322b.png" width="270" height="328" alt=""> - <div class="dcaption">Fig. 126. From leaf of - <i>Elodea canadensis</i>. (After - Berthold.)</div></div> - -<p>In the growth of an ordinary dicotyledonous -leaf, we once more see reflected in -the form of its epidermal cells the tractions, -irregular but on the whole longitudinal, -which growth has superposed on the tensions -of the partition-walls (Fig. <a href="#fig126" title="go to Fig. 126">126</a>). In -the narrow elongated leaf of a Monocotyledon, -such as a hyacinth, the -elongated, apparently quadrangular <span class="xxpn" id="p323">{323}</span> -cells of the epidermis appear as a necessary consequence of the -simpler laws of growth which gave its simple form to the leaf as -a whole. In this last case, however, as in all the others, the rule -still holds that only three partitions (in surface view) meet in a -point; and at their point of meeting the walls are for a short -distance manifestly curved, so as to permit the junction to take -place at or nearly at the normal angle of 120°.</p> - -<p>Briefly speaking, wherever we have a system of cylinders or -spheres, associated together with sufficient mutual interaction to -bring them into complete surface contact, there, in section or in -surface view, we tend to get a pattern of hexagons.</p> - -<div class="psmprnt3"> -<p>While the formation of an hexagonal pattern on the basis -of ready-formed and symmetrically arranged material units -is a very common, and indeed the general way, it does not -follow that there are not others by which such a pattern can -be obtained. For instance, if we take a little triangular -dish of mercury and set it vibrating (either by help of a -tuning-fork, or by simply tapping on the sides) we shall have -a series of little waves or ripples starting inwards from each -of the three faces; and the intercrossing, or interference of -these three sets of waves produces crests and hollows, and -intermediate points of no disturbance, <i>whose loci are seen</i> -as a beautiful pattern of minute hexagons. It is possible -that the very minute and astonishingly regular pattern of -hexagons which we see, for instance, on the surface of many -diatoms, may be a phenomenon of this order<a class="afnanch" -href="#fn363" id="fnanch363">363</a>. The same may be the -case also in Arcella, where an apparently hexagonal pattern -is found not to consist of simple hexagons, but of “straight -lines in three sets of parallels, the lines of each set making -an angle of sixty degrees with those of the other two sets<a -class="afnanch" href="#fn364" id="fnanch364">364</a>.” We -must also bear in mind, in the case of the minuter forms, the -large possibilities of optical illusion. For instance, in -one of Abbe’s “diffraction-plates,” a pattern of dots, set -at equal interspaces, is reproduced on a very minute scale -by photography; but under certain conditions of microscopic -illumination and focussing, these isolated dots appear as a -pattern of hexagons.</p> - -<hr class="hrblksht"> - -<p>A symmetrical arrangement of hexagons, -such as we have just been studying, suggests various simple -geometrical corollaries, of which the following may perhaps be -a useful one.</p> - -<p>We may sometimes desire to estimate the number of -hexagonal areas or facets in some structure where these -are numerous, such for instance as the <span class="xxpn" -id="p324">{324}</span> cornea of an insect’s eye, or in the -minute pattern of hexagons on many diatoms. An approximate -enumeration is easily made as follows.</p> - -<p>For the area of a hexagon (if we call δ the short -diameter, that namely which bisects two of the opposite -sides) is δ<sup>2</sup> × (√3) ⁄ 2, the area -of a circle being <i>d</i><sup>2</sup> · π ⁄ 4. -Then, if the diameter (<i>d</i>) of a circular area -include <i>n</i> hexagons, the area of that circle equals -(<i>n</i> · δ)<sup>2</sup> × π ⁄ 4. -And, dividing this number by the area of a single -hexagon, we obtain for the number of areas in the -circle, each equal to a hexagonal facet, the expression -<i>n</i><sup>2</sup> × π ⁄ 4 × 2 ⁄ √3 -= 0·907<i>n</i><sup>2</sup> , or -(9 ⁄ 10) · <i>n</i><sup>2</sup> , nearly.</p> - -<p>This calculation deals, not only with the complete facets, -but with the areas of the broken hexagons at the periphery -of the circle. If we neglect these latter, and consider our -whole field as consisting of successive rings of hexagons -about a central one, we may obtain a still simpler rule<a -class="afnanch" href="#fn365" id="fnanch365">365</a>. For -obviously, around our central hexagon there stands a zone of -six, and around these a zone of twelve, and around these a zone -of eighteen, and so on. And the total number, excluding the -central hexagon, is accordingly:</p> - -<div class="dtblbox"><div class="nowrap"> -<table> -<tr> - <td class="tdleft">For one zone</td> - <td class="tdright">6</td> - <td class="tdright">= 2 ×  3</td> - <td class="tdright">= 3 × 1 × 2,</td></tr> -<tr> - <td class="tdleft"> ″  two zones</td> - <td class="tdright">18</td> - <td class="tdright">= 3 ×  6</td> - <td class="tdright">= 3 × 2 × 3,</td></tr> -<tr> - <td class="tdleft"> ″  three zones</td> - <td class="tdright">36</td> - <td class="tdright">= 4 ×  9</td> - <td class="tdright">= 3 × 3 × 4,</td></tr> -<tr> - <td class="tdleft"> ″  four zones</td> - <td class="tdright">60</td> - <td class="tdright">= 5 × 12</td> - <td class="tdright">= 3 × 4 × 5,</td></tr> -<tr> - <td class="tdleft"> ″  five zones</td> - <td class="tdright">90</td> - <td class="tdright">= 6 x 15</td> - <td class="tdright">= 3 × 5 × 6,</td></tr> -</table></div></div><!--dtblbox--> - -<p class="pcontinue">and so forth. If <i>N</i> be the number of zones, and if we add one to the above -numbers for the odd central hexagon, the rule evidently is, that the total -number, <i>H</i>, -= 3<i>N</i>(<i>N</i> + 1) + 1. Thus, if in a preparation of a fly’s cornea, -I can count twenty-five facets in a line from a central one, the total number -in the entire circular field is (3 × 25 × 26) + 1 -= 1951<a class="afnanch" href="#fn366" id="fnanch366">366</a>.</p> -</div><!--psmprnt3--> - -<hr class="hrblk"> - -<p>The same principles which account for the development of -hexagonal symmetry hold true, as a matter of course, not only -of individual <i>cells</i> (in the biological sense), but of any close-packed -bodies of uniform size and originally circular outline; -and the hexagonal pattern is therefore of very common occurrence, -under widely different circumstances. The curious reader may -consult Sir Thomas Browne’s quaint and beautiful account, in the -<i>Garden of Cyrus</i>, of hexagonal (and also of quincuncial) symmetry -in plants and animals, which “doth neatly declare how nature -Geometrizeth, and observeth order in all things.” <span class="xxpn" id="p325">{325}</span></p> - -<p>We have many varied examples of this principle among corals, -wherever the polypes are in close juxtaposition, with neither -empty space nor accumulations of matrix between their adjacent -walls. <i>Favosites gothlandica</i>, for instance, furnishes us with an -excellent example. In the great genus Lithostrotion we have some -species that are “massive” and others that are “fasciculate”; in -other words in some the long cylindrical corallites are in close contact -with one another, and in others they are separate and loosely -bundled (Fig. <a href="#fig127" title="go to Fig. 127">127</a>). Accordingly in the former the corallites are</p> - -<div class="dctr01" id="fig127"><div id="fig128"> -<img src="images/i325.png" width="800" height="541" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 127. <i>Lithostrotion Martini.</i> - (After Nicholson.)</td> - <td></td> - <td>Fig. 128. <i>Cyathophyllum hexagonum.</i> - (From Nicholson, after Zittel.)</td></tr></table> -</div></div></div><!--dctr01--> - -<p class="pcontinue">squeezed into hexagonal prisms, while -in the latter they retain their cylindrical form. Where -the polypes are comparatively few, and so have room to -spread, the mutual pressure ceases to work or only tends to -push them asunder, letting them remain circular in outline -(e.g. Thecosmilia). Where they vary gradually in size, as -for instance in <i>Cyathophyllum hexagonum</i>, they are more -or less hexagonal but are not regular hexagons; and where -there is greater and more irregular variation in size, the -cells will be <i>on the average</i> hexagonal, but some will -have fewer and some more sides than six, as in the annexed -figure of Arachnophyllum (Fig. <a href="#fig129" title="go to Fig. 129">129</a>). <span class="xxpn" -id="p326">{326}</span> Where larger and smaller cells, -corresponding to two different kinds of zooids, are mixed -together, we may get various results. If the larger cells -are numerous enough to be more or less in contact with -one another (e.g. various Monticuliporae) they will be -irregular hexagons, while the smaller cells between them -will be crushed into all manner of irregular angular forms. -If on the other hand the large cells are comparatively few -and are large and strong-walled compared with their smaller -neighbours, then the latter alone will be squeezed into -hexagons, while the larger ones will tend to retain their -circular outline undisturbed (e.g. Heliopora, Heliolites, -etc.).</p> - -<div class="dctr01" id="fig129"><div id="fig130"> -<img src="images/i326.png" width="800" height="313" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 129. <i>Arachnophyllum pentagonum.</i> - (After Nicholson.)</td> - <td></td> - <td>Fig. 130. <i>Heliolites.</i> <br>(After - Woods.)</td></tr></table> -</div></div></div><!--dctr01--> - -<p>When, as happens in certain corals, the peripheral walls or -“thecae” of the individual polypes remain undeveloped but -the radiating septa are formed and calcified, then we obtain new -and beautiful mathematical configurations (Fig. <a href="#fig131" title="go to Fig. 131">131</a>). For the -radiating septa are no longer confined to the circular or hexagonal -bounds of a polypite, but tend to meet and become confluent -with their neighbours on every side; and, tending to assume -positions of equilibrium, or of minimal area, under the restraints -to which they are subject, they fall into congruent curves; and -these correspond, in a striking manner, to the lines of force running, -in a common field of force, between a number of secondary centres. -Similar patterns may be produced in various ways, by the play -of osmotic or magnetic forces; and a particular and very curious -case is to be found in those complicated forms -of nuclear division <span class="xxpn" id="p327">{327}</span> -known as triasters, polyasters, etc., whose relation to a field of -force Hartog has explained<a class="afnanch" href="#fn367" id="fnanch367">367</a>. -It is obvious that, in our corals, -these curving septa are all orthogonal to the non-existent hexagonal -boundaries. As the phenomenon is wholly due to the imperfect -development or non-existence of a thecal wall, it is not surprising -that we find identical configurations among various corals, or -families of corals, not otherwise related to one another; we find -the same or very similar patterns displayed, for instance, in -Synhelia (<i>Oculinidae</i>), in Phillipsastraea (<i>Rugosa</i>), in Thamnastraea -(<i>Fungida</i>), and in many more.</p> - -<hr class="hrblk"> - -<p>The most famous of all hexagonal conformations and perhaps -the most beautiful is that of the bee’s cell. Here we have, as in</p> - -<div class="dctr01" id="fig131"> -<img src="images/i327.png" width="800" height="312" alt=""> - <div class="pcaption">Fig. 131. Surface-views of Corals with - undeveloped thecae and confluent septa. A, <i>Thamnastraea</i>; - B, <i>Comoseris</i>. (From Nicholson, after Zittel.)</div></div> - -<p class="pcontinue">our last examples, a series of equal cylinders, compressed by -symmetrical forces into regular hexagonal prisms. But in this -case we have two rows of such cylinders, set opposite to one -another, end to end; and we have accordingly to consider also -the conformation of their ends. We may suppose our original -cylindrical cells to have spherical ends, which is their normal and -symmetrical mode of termination; and, for closest packing, it is -obvious that the end of any one cylinder will touch, and fit in -between, the ends of three cylinders in the opposite row. It is -just as when we pile round-shot in a heap; -each sphere that we <span class="xxpn" id="p328">{328}</span> -set down fits into its nest between three others, and the four -form a regular tetrahedral arrangement. Just as it was obvious, -then, that by mutual pressure from the six <i>laterally</i> adjacent cells, -any one cell would be squeezed into a hexagonal prism, so is it also -obvious that, by mutual pressure against the three <i>terminal</i> -neighbours, the end of any one cell will be compressed into a solid -trihedral angle whose edges will meet, as in the analogous case -already described of a system of soap-bubbles, at a plane angle -of 109° and so many minutes and seconds. What we have to -comprehend, then, is how the <i>six</i> sides of the cell are to be combined -with its <i>three</i> terminal facets. This is done by bevelling off three -alternate angles of the prism, in a uniform manner, until we have -tapered the prism to a point; and by so doing, we evidently -produce three <i>rhombic</i> surfaces, each of which is double of the -triangle formed by joining the apex to the three untouched angles -of the prism. If we experiment, not with cylinders, but with -spheres, if for instance we pile together a mass of bread-pills (or -pills of plasticine), and then submit the whole to a uniform pressure, -it is obvious that each ball (like the seeds in a pomegranate, as -Kepler said), will be in contact with <i>twelve</i> others,—six in its own -plane, three below and three above, and in compression it will -therefore develop twelve plane surfaces. It will in short repeat, -above and below, the conditions to which the bee’s cell is subject -at one end only; and, since the sphere is symmetrically situated -towards its neighbours on all sides, it follows that the twelve plane -sides to which its surface has been reduced will be all similar, -equal and similarly situated. Moreover, since we have produced -this result by squeezing our original spheres close together, it is -evident that the bodies so formed completely fill space. The -regular solid which fulfils all these conditions is the <i>rhombic -dodecahedron</i>. The bee’s cell, then, is this figure incompletely -formed: it is a hexagonal prism with one open or unfinished end, -and one trihedral apex of a rhombic dodecahedron.</p> - -<p>The geometrical form of the bee’s cell must have attracted the -attention and excited the admiration of mathematicians from time -immemorial. Pappus the Alexandrine has left us (in the introduction -to the Fifth Book of his <i>Collections</i>) an account of its -hexagonal plan, and he drew from its -mathematical symmetry the <span class="xxpn" id="p329">{329}</span> -conclusion that the bees were endowed with reason: “There -being, then, three figures which of themselves can fill up the -space round a point, viz. the triangle, the square and the hexagon, -the bees have wisely selected for their structure that which contains -most angles, suspecting indeed that it could hold more honey than -either of the other two.” Erasmus Bartholinus was apparently -the first to suggest that this hypothesis was not warranted, and -that the hexagonal form was no more than the necessary result -of equal pressures, each bee striving to make its own little circle -as large as possible.</p> - -<p>The investigation of the ends of the cell was a more difficult -matter, and came later, than that of its sides. In general terms -this arrangement was doubtless often studied and described: as -for instance, in the <i>Garden of Cyrus</i>: “And the Combes themselves -so regularly contrived that their mutual intersections -make three Lozenges at the bottom of every Cell; which severally -regarded make three Rows of neat Rhomboidall Figures, connected -at the angles, and so continue three several chains throughout the -whole comb.” But Maraldi<a class="afnanch" href="#fn368" id="fnanch368">368</a> -(Cassini’s nephew) was the first to -measure the terminal solid angle or determine the form of the -rhombs in the pyramidal ending of the cell. He tells us that the -angles of the rhomb are 110° and 70°: “Chaque base d’alvéole -est formée par trois rhombes presque toujours égaux et semblables, -qui, suivant les mesures que nous avons prises, ont les deux angles -obtus chacun de 110 degrés, et par conséquent les deux aigus -chacun de 70°.” He also stated that the angles of the trapeziums -which form the sides of the body of the cell were identical angles, -of 110° and 70°; but in the same paper he speaks of the angles as -being, respectively, 109° 28′ and 70° 32′. Here a singular confusion -at once arose, and has been perpetuated in the books<a class="afnanch" href="#fn369" id="fnanch369">369</a>. -“Unfortunately Réaumur chose to look upon this second determination -of Maraldi’s as being, as well as the first, a direct result -of measurement, whereas it is in reality theoretical. He speaks of -it as Maraldi’s more precise measurement, and this error has been -repeated in spite of its absurdity to the -present day; nobody <span class="xxpn" id="p330">{330}</span> -appears to have thought of the impossibility of measuring such a -thing as the end of a bee’s cell to the nearest minute.” At any -rate, it now occurred to Réaumur (as curiously enough, it had not -done to Maraldi) that, just as the closely packed hexagons gave -the minimal extent of boundary in a plane, so the actual solid -figure, as determined by Maraldi, might be that which, for a given -solid content, gives the minimum of surface: or which, in other -words, would hold the most honey for the least wax. He set this -problem before Koenig, and the geometer confirmed his conjecture, -the result of his calculations agreeing within two minutes (109° 26′ -and 70° 34′) with Maraldi’s determination. But again, Maclaurin<a class="afnanch" href="#fn370" id="fnanch370">370</a> -and Lhuilier<a class="afnanch" href="#fn371" id="fnanch371">371</a>, -by different methods, obtained a result identical -with Maraldi’s; and were able to shew that the discrepancy of -2′ was due</p> - -<div class="dleft dwth-e" id="fig132"> -<img src="images/i330.png" width="350" height="625" alt=""> - <div class="dcaption">Fig. 132.</div></div> - -<p class="pcontinue">to an -error in Koenig’s calculation (of tan θ -= √2),—that -is to say to the imperfection -of his logarithmic tables,—not -(as the books say<a class="afnanch" href="#fn372" id="fnanch372">372</a>) -“to a -mistake on the part of the Bee.” -“Not to a mistake on the part of -Maraldi” is, of course, all that we -are entitled to say.</p> - -<p>The theorem may be proved as -follows:</p> - -<p><i>ABCDEF</i>, <i>abcdef</i>, is a right -prism upon a regular hexagonal base. -The corners <i>BDF</i> are cut off by -planes through the lines <i>AC</i>, <i>CE</i>, -<i>EA</i>, meeting in a point <i>V</i> on the -axis <i>VN</i> of the prism, and intersecting -<i>Bb</i>, <i>Dd</i>, <i>Ff</i>, at <i>X</i>, <i>Y</i>, <i>Z</i>. It is -evident that the volume of the figure -thus formed is the same as that of -the original prism with hexagonal -ends. For, if the axis cut the -hexagon <i>ABCDEF</i> in <i>N</i>, the volumes <i>ACVN</i>, <i>ACBX</i> -are equal. <span class="xxpn" id="p331">{331}</span></p> - -<p>It is required to find the inclination of the faces forming -the trihedral angle at <i>V</i> to the axis, such that the surface -of the figure may be a minimum.</p> - -<p>Let the angle <i>NVX</i>, which is half the solid angle of -the prism, -= θ; the side of the hexagon, as <i>AB</i>, -= <i>a</i>; and the height, as <i>Aa</i>, -= <i>h</i>.<br class="brclrfix"></p> - -<div class="dmaths"> -<p>Then,</p> - -<div><i>AC</i> -= 2<i>a</i> cos 30° -= <i>a</i>√3.</div> - -<p>And <i>VX</i> -= <i>a</i> ⁄ sin θ -(from inspection of the triangle <i>LXB</i>)</p> - -<p>Therefore the area of the rhombus -<i>VAXC</i> -= (<i>a</i><sup>2</sup> √3) ⁄ (2 sin θ).</p> - -<p>And the area of <i>AabX</i> -= (<i>a</i> ⁄ 2)(2<i>h</i> − ½<i>VX</i> cos θ)</p> - -<div>= (<i>a</i> ⁄ 2)(2<i>h</i> − <i>a</i> ⁄ 2 · cot θ). -</div></div><!--dmaths--> - -<div class="dmaths"> -<p>Therefore the total area of the figure</p> - -<div>= hexagon <i>abcdef</i> + 3<i>a</i>(2<i>h</i> − (<i>a</i> ⁄ 2) cot θ) -+ (3<i>a</i><sup>2</sup> √3) ⁄ (2 sin θ). -</div><!--dmaths--></div> - -<div class="dmaths"> -<p>Therefore</p> - -<div><i>d</i>(Area) ⁄ <i>d</i>θ -= (3<i>a</i><sup>2</sup> ⁄ 2)((1 ⁄ sin<sup>2</sup> θ) -− (√3 cos θ) ⁄ (sin<sup>2</sup> θ)).</div> - -<p class="pcontinue">But this expression vanishes, that is to say, <i>d</i>(Area) ⁄ <i>d</i>θ -= 0, -when cos θ -<span class="nowrap">= 1 ⁄ √3,</span> that is when θ -<span class="nowrap">= 54° 44′ 8″</span></p> - -<div>= ½(109° 28′ 16″).</div> -</div><!--dmaths--> - -<p>This then is the condition under which the total area of the -figure has its minimal value.</p> - -<hr class="hrblk"> - -<p>That the beautiful regularity of the bee’s architecture is due -to some automatic play of the physical forces, and that it were -fantastic to assume (with Pappus and Réaumur) that the bee -intentionally seeks for a method of economising wax, is certain, -but the precise manner of this automatic action is not so clear. -When the hive-bee builds a solitary cell, or a small cluster of cells, -as it does for those eggs which are to develop into queens, it makes -but a rude production. The queen-cells are lumps of coarse wax -hollowed out and roughly bitten into shape, bearing the marks of -the bee’s jaws, like the marks of a blunt adze on a rough-hewn log. -Omitting the simplest of all cases, when (as among some humble-bees) -the old cocoons are used to hold honey, the cells built by -the “solitary” wasps and bees are of various kinds. They may -be formed by partitioning off little chambers in -a hollow stem; <span class="xxpn" id="p332">{332}</span> -they may be rounded or oval capsules, often very neatly constructed, -out of mud, or vegetable <i>fibre</i> or little stones, agglutinated -together with a salivary glue; but they shew, except for their -rounded or tubular form, no mathematical symmetry. The social -wasps and many bees build, usually out of vegetable matter -chewed into a paste with saliva, very beautiful nests of “combs”; -and the close-set papery cells which constitute these combs are -just as regularly hexagonal as are the waxen cells of the hive-bee. -But in these cases (or nearly all of them) the cells are in a single -row; their sides are regularly hexagonal, but their ends, from the -want of opponent forces, remain simply spherical. In <i>Melipona -domestica</i> (of which Darwin epitomises Pierre Huber’s description) -“the large waxen honey-cells are nearly spherical, nearly equal in -size, and are aggregated into an irregular mass.” But the spherical -form is only seen on the outside of the mass; for inwardly each -cell is flattened into “two, three or more flat surfaces, according -as the cell adjoins two, three or more other cells. When one cell -rests on three other cells, which from the spheres being nearly -of the same size is very frequently and necessarily the case, the -three flat surfaces are united into a pyramid; and this pyramid, as -Huber has remarked, is manifestly a gross imitation of the three-sided -pyramidal base of the cell of the hive-bee<a class="afnanch" href="#fn373" id="fnanch373">373</a>.” -The question -is, to what particular force are we to ascribe the plane surfaces -and definite angles which define the sides of the cell in all these -cases, and the ends of the cell in cases where one row meets and -opposes another. We have seen that Bartholin suggested, and it -is still commonly believed, that this result is due to simple physical -pressure, each bee enlarging as much as it can the cell which it -is a-building, and nudging its wall outwards till it fills every -intervening gap and presses hard against the similar efforts of -its neighbour in the cell next door<a class="afnanch" href="#fn374" id="fnanch374">374</a>. -But it -is very doubtful <span class="xxpn" id="p333">{333}</span> -whether such physical or mechanical pressure, more or less intermittently -exercised, could produce the all but perfectly smooth, -plane surfaces and the all but perfectly definite and constant -angles which characterise the cell, whether it be constructed of -wax or papery pulp. It seems more likely that we have to do -with a true surface-tension effect; in other words, that the walls -assume their configuration when in a semi-fluid state, while the -papery pulp is still liquid, or while the wax is warm under the -high temperature of the crowded hive<a class="afnanch" href="#fn375" id="fnanch375">375</a>. -Under these circumstances, -the direct efforts of the wasp or bee may be supposed -to be limited to the making of a tubular cell, as thin as the nature -of the material permits, and packing these little cells as close as -possible together. It is then easily conceivable that the symmetrical -tensions of the adjacent films (though somewhat retarded -by viscosity) should suffice to bring the whole system into equilibrium, -that is to say into the precise configuration which the -comb actually presents. In short, the Maraldi pyramids which -terminate the bee’s cell are precisely identical with the facets of -a rhombic dodecahedron, such as we have assumed to constitute -(and which doubtless under certain conditions do constitute) the -surfaces of contact in the interior of a mass of soap-bubbles or -of uniform parenchymatous cells; and there is every reason to -believe that the physical explanation is identical, and not merely -mathematically analogous.</p> - -<p>The remarkable passage in which Buffon discusses the bee’s -cell and the hexagonal configuration in general is of such historical -importance, and tallies so closely with the whole trend of our -enquiry, that I will quote it in full: “Dirai-je encore un mot; -ces cellules des abeilles, tant vantées, tant admirées, me fournissent -une preuve de plus contre l’enthousiasme et l’admiration; cette -figure, toute géométrique et toute régulière qu’elle nous paraît, et -qu’elle est en effet dans la spéculation, n’est ici qu’un résultat -mécanique et assez imparfait qui se trouve -souvent dans la nature, <span class="xxpn" id="p334">{334}</span> -et que l’on remarque même dans les productions les plus brutes; -les cristaux et plusieurs autres pierres, quelques sels, etc., prennent -constamment cette figure dans leur formation. Qu’on observe les -petites écailles de la peau d’une roussette, on verra qu’elles sont -hexagones, parce que chaque écaille croissant en même temps se -fait obstacle, et tend à occuper le plus d’espace qu’il est possible -dans un espace donné: on voit ces mêmes hexagones dans le -second estomac des animaux ruminans, on les trouve dans les -graines, dans leurs capsules, dans certaines fleurs, etc. Qu’on -remplisse un vaisseau de pois, ou plûtot de quelque autre graine -cylindrique, et qu’on le ferme exactement après y avoir versé -autant d’eau que les intervalles qui restent entre ces graines -peuvent en recevoir; qu’on fasse bouillir cette eau, tous ces -cylindres deviendront de colonnes à six pans<a class="afnanch" href="#fn376" id="fnanch376">376</a>. -On y voit clairement -la raison, qui est purement mécanique; chaque graine, dont -la figure est cylindrique, tend par son renflement à occuper le -plus d’espace possible dans un espace donné, elles deviennent donc -toutes nécessairement hexagones par la compression réciproque. -Chaque abeille cherche à occuper de même le plus d’espace possible -dans un espace donné, il est donc nécessaire aussi, puisque le -corps des abeilles est cylindrique, que leurs cellules sont hexagones,—par -la même raison des obstacles réciproques. On donne plus -d’esprit aux mouches dont les ouvrages sont les plus réguliers; -les abeilles sont, dit-on, plus ingénieuses que les guêpes, que les -frélons, etc., qui savent aussi l’architecture, mais dont les constructions -sont plus grossières et plus irrégulières que celles des -abeilles: on ne veut pas voir, ou l’on ne se doute pas que cette -régularité, plus ou moins grande, dépend uniquement du nombre -et de la figure, et nullement de l’intelligence de ces petites bêtes; -plus elles sont nombreuses, plus il y a des forces qui agissent -également et s’opposent de même, plus il y a par conséquent de -contrainte mécanique, de régularité forcée, et de perfection -apparente dans leurs productions<a class="afnanch" href="#fn377" id="fnanch377">377</a>.” -<span class="xxpn" id="p335">{335}</span></p> - -<p>A very beautiful hexagonal symmetry, as seen in section, or -dodecahedral, as viewed in the solid, is presented by the cells -which form the pith of certain rushes (e.g. <i>Juncus effusus</i>), and -somewhat less diagrammatically by those which make the pith -of the banana. These cells are stellate in form, and the tissue -presents in section the appearance of a network of six-rayed -stars (Fig. <a href="#fig133" title="go to Fig. 133">133</a>, <i>c</i>), linked together by the tips of the rays, and -separated by symmetrical, air-filled, intercellular spaces. In thick -sections, the solid twelve-rayed stars may be very beautifully seen -under the binocular microscope.</p> - -<div class="dctr03" id="fig133"> -<img src="images/i335.png" width="596" height="512" alt=""> - <div class="pcaption">Fig. 133. Diagram of development of “stellate cells,” in - pith of <i>Juncus</i>. (The dark, or shaded, areas represent - the cells; the light areas being the gradually enlarging - “intercellular spaces.”)</div></div> - -<p>What has happened here is not difficult to understand. -Imagine, as before, a system of equal spheres all in contact, each -one therefore touching six others in an equatorial plane; and let -the cells be not only in contact, but become attached at the points -of contact. Then instead of each cell expanding, so as to encroach -on and fill up the intercellular spaces, let each cell tend to contract -or shrivel up, by the withdrawal of fluid from -its interior. The <span class="xxpn" id="p336">{336}</span> -result will obviously be that the intercellular spaces will increase; -the six equatorial attachments of each cell (Fig. <a href="#fig133" title="go to Fig. 133">133</a>, <i>a</i>) (or its twelve -attachments in all, to adjacent cells) will remain fixed, and the -portions of cell-wall between these points of attachment will be -withdrawn in a symmetrical fashion (<i>b</i>) towards the centre. As -the final result (<i>c</i>) we shall have a “dodecahedral star” or star-polygon, -which appears in section as a six-rayed figure. It is -obviously necessary that the pith-cells should not only be attached -to one another, but that the outermost layer should be firmly -attached to a boundary wall, so as to preserve the symmetry of -the system. What actually occurs in the rush is tantamount to -this, but not absolutely identical. Here it is not so much the -pith-cells which tend to shrivel within a boundary of constant -size, but rather the boundary wall (that is, the peripheral ring of -woody and other tissues) which continues to expand after the -pith-cells which it encloses have ceased to grow or to multiply. -The twelve points of attachment on the spherical surface of each -little pith-cell are uniformly drawn asunder; but the content, or -volume, of the cell does not increase correspondingly; and the -remaining portions of the surface, accordingly, shrink inwards and -gradually constitute the complicated surface of a twelve-pointed -star, which is still a symmetrical figure and is still also a surface -of minimal area under the new conditions.</p> - -<hr class="hrblk"> - -<p>A few years after the publication of Plateau’s book, Lord -Kelvin shewed, in a short but very beautiful paper<a class="afnanch" href="#fn378" id="fnanch378">378</a>, -that we must -not hastily assume from such arguments as the foregoing, that -a close-packed assemblage of rhombic dodecahedra will be the true -and general solution of the problem of dividing space with a -minimum partitional area, or will be present in a cellular liquid -“foam,” in which it is manifest that the problem is actually and -automatically solved. The general mathematical solution of the -problem (as we have already indicated) is, that every interface or -partition-wall must have constant curvature throughout; that -where such partitions meet in an edge, they must intersect at -angles such that equal forces, in planes -perpendicular to the line <span class="xxpn" id="p337">{337}</span> -of intersection, shall balance; and finally, that no more than three -such interfaces may meet in a line or edge, whence it follows that -the angle of intersection of the film-surfaces must be exactly 120°. -An assemblage of equal and similar rhombic dodecahedra goes far -to meet the case: it completely fills up space; all its surfaces or -interfaces are planes, that is to say, surfaces of constant curvature -throughout; and these surfaces all meet together at angles of 120°. -Nevertheless, the proof that our rhombic dodecahedron (such as -we find exemplified in the bee’s cell) is a surface of minimal area, -is not a comprehensive proof; it is limited to certain conditions, -and practically amounts to no more than this, that of the regular -solids, with all sides plane and similar, this one has the least surface -for its solid content.</p> - -<div class="dright dwth-e" id="fig134"> -<img src="images/i337.png" width="329" height="300" alt=""> - <div class="dcaption">Fig. 134.</div></div> - -<p>The rhombic dodecahedron has six tetrahedral angles, and -eight trihedral angles; and it is obvious, on consideration, that -at each of the former six dodecahedra meet in a point, and that, -where the four tetrahedral facets of each coalesce with their -neighbours, we have twelve plane films, or interfaces, meeting in -a point. In a precisely similar fashion, we may imagine twelve -plane films, drawn inwards from the twelve edges of a cube, to -meet at a point in the centre of the cube. But, as Plateau -discovered<a class="afnanch" href="#fn379" id="fnanch379">379</a>, -when we dip a cubical -wire skeleton into soap-solution and -take it out again, the twelve films -which are thus generated do <i>not</i> -meet in a point, but are grouped -around a small central, plane, quadrilateral -film (Fig. <a href="#fig134" title="go to Fig. 134">134</a>). In other -words, twelve plane films, meeting in -a point, are <i>essentially unstable</i>. If -we blow upon our artificial film-system, -the little quadrilateral alters -its place, setting itself parallel now to one and now to another of -the paired faces of the cube; but we never get rid of it. Moreover, -the size and shape of the quadrilateral, as of all the other films in the -system, are perfectly definite. Of the twelve -films (which we had <span class="xxpn" id="p338">{338}</span> -expected to find all plane and all similar) four are plane isosceles -triangles, and eight are slightly curved quadrilateral figures. The -former have two curved sides, meeting at an angle of 109° 28′, -and their apices coincide with the corners of the central quadrilateral, -whose sides are also curved, and also meet at this identical -angle;—which (as we observe) is likewise an angle which we have -been dealing with in the simpler case of the bee’s cell, and indeed -in all the regular solids of which we have yet treated.</p> - -<p>By completing the assemblage of polyhedra of which -Plateau’s skeleton-cube gives a part, Lord Kelvin -shewed that we should obtain a set of equal and similar -fourteen-sided figures, or “tetrakaidecahedra”; and that -by means of an assemblage of these figures space is -homogeneously partitioned—that is to say, into equal, -similar and similarly situated cells—with an economy -of surface in relation to area even greater than in an -assemblage of rhombic dodecahedra.</p> - -<p>In the most generalised case, the tetrakaidecahedron is bounded -by three pairs of equal and parallel quadrilateral faces, and four -pairs of equal and parallel hexagonal faces, neither the quadrilaterals -nor the hexagons being necessarily plane. In a certain -particular case, the quadrilaterals are plane surfaces, but the -hexagons slightly curved “anticlastic” surfaces; and these latter -have at every point equal and opposite curvatures, and are -surfaces of minimal curvature for a boundary of six curved edges. -The figure has the remarkable property that, like the plane -rhombic dodecahedron, it so partitions space that three faces -meeting in an edge do so everywhere at equal angles of 120° <a class="afnanch" href="#fn380" id="fnanch380">380</a>.</p> - -<p>We may take it as certain that, in a system of <i>perfectly</i> fluid -films, like the interior of a mass of soap-bubbles, where the films -are perfectly free to glide or to rotate over one another, the mass -is actually divided into cells of this -remarkable conformation. <span class="xxpn" id="p339">{339}</span> -And it is quite possible, also, that in the cells of a vegetable -parenchyma, by carefully macerating them apart, the same conformation -may yet be demonstrated under suitable conditions; -that is to say when the whole tissue is highly symmetrical, and the -individual cells are as nearly as possible equal in size. But in an -ordinary microscopic <i>section</i>, it would seem practically impossible -to distinguish the fourteen-sided figure from the twelve-sided. -Moreover, if we have anything whatsoever interposed so as to -prevent our twelve films meeting in a point, and (so to speak) to -take the place of our little central quadrilateral,—if we have, for -instance, a tiny bead or droplet in the centre of our artificial -system, or even a little thickening, or “bourrelet” as Plateau called -it, of the cell-wall, then it is no longer necessary that the -tetrakaidecahedron should be formed. Accordingly, it is very -probably the case that, in the parenchymatous tissue, under the -actual conditions of restraint and of very imperfect fluidity, it is -after all the rhombic dodecahedral configuration which, even under -perfectly symmetrical conditions, is generally assumed.</p> - -<hr class="hrblk"> - -<p>It follows from all that we have said, that the problems -connected with the conformation of cells, and with the manner in -which a given space is partitioned by them, soon become exceedingly -complex. And while this is so even when all our cells are equal -and symmetrically placed, it becomes vastly more so when cells -varying even slightly in size, in hardness, rigidity or other qualities, -are packed together. The mathematics of the case very soon -become too hard for us; but in its essence, the phenomenon -remains the same. We have little reason to doubt, and no just -cause to disbelieve, that the whole configuration, for instance of -an egg in the advanced stages of segmentation, is accurately -determined by simple physical laws, just as much as in the early -stages of two or four cells, during which early stages we are able to -recognise and demonstrate the forces and their resultant effects. -But when mathematical investigation has become too difficult, it -often happens that physical experiment can reproduce for us the -phenomena which Nature exhibits to us, and which we are striving -to comprehend. For instance, in an admirable research, M. Robert -shewed, some years ago, not only that the -early segmentation of <span class="xxpn" id="p340">{340}</span> -the egg of <i>Trochus</i> (a marine univalve mollusc) proceeded in -accordance with the laws of surface tension, but he also succeeded -in imitating by means of soap-bubbles, several stages, one after -another, of the developing egg.</p> - -<div class="dctr04" id="fig135"> -<img src="images/i341.png" width="497" height="715" alt=""> - <div class="pcaption">Fig. 135. Aggregations of four soap-bubbles, to shew - various arrangements of the intermediate partition and - polar furrows. (After Robert.)</div></div> - -<p>M. Robert carried his experiments as far as the stage of -sixteen cells, or bubbles. It is not easy to carry the artificial -system quite so far, but in the earlier stages the experiment is -easy; we have merely to blow our bubbles in a little dish, adding -one to another, and adjusting their sizes to produce a symmetrical -system. One of the simplest and prettiest parts of his investigation -concerned the “polar furrow” of which we have spoken on p. <a href="#p310" title="go to pg. 310">310</a>. -On blowing four little contiguous bubbles he found (as we may -all find with the greatest ease) that they form a symmetrical system, -two in contact with one another by a laminar film, and two, -which are elevated a little above the others, and which are separated -by the length of the aforesaid lamina. The bubbles are thus in -contact three by three, their partition-walls making with one -another equal angles of 120°. The upper and lower edges of the -intermediate lamina (the lower one visible through the transparent -system) constitute the two polar furrows of the embryologist -(Fig. <a href="#fig135" title="go to Fig. 135">135</a>, 1–3). The lamina itself is plane when the system is -symmetrical, but it responds by a corresponding curvature to -the least inequality of the bubbles on either side. In the -experiment, the upper polar furrow is usually a little shorter -than the lower, but parallel to it; that is to say, the lamina -is of trapezoidal form: this lack of perfect symmetry being -due (in the experimental case) to the lower portion of the -bubbles being somewhat drawn asunder by the tension of their -attachments to the sides of the dish (Fig. <a href="#fig135" title="go to Fig. 135">135</a>, 4). A similar -phenomenon is usually found in Trochus, according to Robert, -and many other observers have likewise found the upper furrow -to be shorter than the one below. In the various species of the -genus Crepidula, Conklin asserts that the two furrows are equal -in <i>C. convexa</i>, that the upper one is the shorter in <i>C. fornicata</i>, -and that the upper one all but disappears in <i>C. plana</i>; but we may -well be permitted to doubt, without the evidence of very special -investigations, whether these slight physical differences are -actually characteristic of, and constant in, -particular allied <i>species</i>. <span class="xxpn" id="p341">{341}</span> -Returning to the experimental case, Robert found that by withdrawing -a little air from, and so diminishing the bulk of the two -terminal bubbles (i.e. those at the ends of the intermediate lamina), -the upper polar furrow was caused to elongate, till it became equal -in length to the lower; and by continuing the process it became -the longer in its turn. These two conditions have again been -described by investigators as characteristic of this embryo or that; -for instance in Unio, Lillie has described the two furrows as -gradually altering their respective lengths<a class="afnanch" href="#fn381" id="fnanch381">381</a>; -and Wilson (as Lillie -remarks) had already pointed out that “the reduction of the -apical cross-furrow, as compared with that at -the vegetative pole <span class="xxpn" id="p342">{342}</span> -in molluscs and annelids ‘stands in obvious relation to the different -size of the cells produced at the two poles<a class="afnanch" href="#fn382" id="fnanch382">382</a>.’ ”</p> - -<p>When the two lateral bubbles are gradually reduced in size, -or the two terminal ones enlarged, the upper furrow becomes -shorter and shorter; and at the moment when it is about to -vanish, a new furrow makes its instantaneous appearance in a -direction perpendicular to the old one; but the inferior furrow, -constrained by its attachment to the base, remains unchanged, -and accordingly our two polar furrows, which were formerly -parallel, are now at right angles to one another. Instead of a -single plane quadrilateral partition, we have now two triangular -ones, meeting in the middle of the system by their apices, and -lying in planes at right angles to one another (Fig. <a href="#fig135" title="go to Fig. 135">135</a>, 5–7)<a class="afnanch" href="#fn383" id="fnanch383">383</a>. -Two such polar furrows, equal in length and arranged in a cross, -have again been frequently described by the embryologists. -Robert himself found this condition in Trochus, as an occasional -or exceptional occurrence: it has been described as normal in -Asterina by Ludwig, in Branchipus by Spangenberg, and in -Podocoryne and Hydractinia by Bunting. It is evident that it -represents a state of unstable equilibrium, only to be maintained -under certain conditions of restraint within the system.</p> - -<p>So, by slight and delicate modifications in the relative size of -the cells, we may pass through all the possible arrangements of the -median partition, and of the “furrows” which correspond to its -upper and lower edges; and every one of these arrangements has -been frequently observed in the four-celled stage of various embryos. -As the phases pass one into the other, they are accompanied by -changes in the curvature of the partition, which in like manner -correspond precisely to phenomena which the embryologists have -witnessed and described. And all these configurations belong to -that large class of phenomena whose distribution among embryos, -or among organisms in general, bears no relation to the boundaries -of zoological classification; through -molluscs, worms, <span class="xxpn" id="p343">{343}</span> -coelenterates, vertebrates and what not, we meet with now one and now -another, in a medley which defies classification. They are not -“vital phenomena,” or “functions” of the organism, or special -characteristics of this or that organism, but purely physical -phenomena. The kindred but more complicated phenomena -which correspond to the polar furrow when a larger number of -cells than four are associated together, we shall deal with in the -next chapter.</p> - -<p>Having shewn that the capillary phenomena are patent and -unmistakable during the earlier stages of embryonic development, -but soon become more obscure and incapable of experimental -reproduction in the later stages, when the cells have increased in -number, various writers including Robert himself have been -inclined to argue that the physical phenomena die away, and are -overpowered and cancelled by agencies of a very different order. -Here we pass into a region where direct observation and experiment -are not at hand to guide us, and where a man’s trend of -thought, and way of judging the whole evidence in the case, must -shape his philosophy. We must remember that, even in a froth -of soap-bubbles, we can apply an exact analysis only to the simplest -cases and conditions of the phenomenon; we cannot describe, -but can only imagine, the forces which in such a froth control the -respective sizes, positions and curvatures of the innumerable -bubbles and films of which it consists; but our knowledge is -enough to leave us assured that what we have learned by investigation -of the simplest cases includes the principles which -determine the most complex. In the case of the growing embryo -we know from the beginning that surface tension is only one of -the physical forces at work; and that other forces, including -those displayed within the interior of each living cell, play their -part in the determination of the system. But we have no evidence -whatsoever that at this point, or that point, or at any, the dominion -of the physical forces over the material system gives place to a -new condition where agencies at present unknown to the physicist -impose themselves on the living matter, and become responsible -for the conformation of its material fabric.</p> - -<hr class="hrblk"> - -<p>Before we leave for the present the subject -of the segmenting <span class="xxpn" id="p344">{344}</span> -egg, we must take brief note of two associated problems: viz. -(1) the formation and enlargement of the segmentation cavity, or -central interspace around which the cells tend to group themselves -in a single layer, and (2) the formation of the gastrula, that is to -say (in a typical case) the conversion “by invagination,” of the -one-layered ball into a two-layered cup. Neither problem is free -from difficulty, and all we can do meanwhile is to state them in -general terms, introducing some more or less plausible assumptions.</p> - -<p>The former problem is comparatively easy, as regards the -tendency of a segmentation cavity to <i>enlarge</i>, when once it has -been established. We may then assume that subdivision of the -cells is due to the appearance of a new-formed septum within each -cell, that this septum has a tendency to shrink under surface -tension, and that these changes will be accompanied on the whole -by a diminution of surface energy in the system. This being so, -it may be shewn that the volume of the divided cells must be less -than it was prior to division, or in other words that part of their -contents must exude during the process of segmentation<a class="afnanch" href="#fn384" id="fnanch384">384</a>. -Accordingly, the case where the segmentation cavity enlarges and -the embryo developes into a hollow blastosphere may, under the -circumstances, be simply described as the case where that outflow -or exudation from the cells of the blastoderm is directed on the -whole inwards.</p> - -<p>The physical forces involved in the invagination of the cell-layer -to form the gastrula have been repeatedly discussed<a class="afnanch" href="#fn385" id="fnanch385">385</a>, -but -the true explanation seems as yet to be by no means clear. The -case, however, is probably not a very difficult one, provided that -we may assume a difference of osmotic pressure at the two poles -of the blastosphere, that is to say between the cells which are -being differentiated into outer and inner, into epiblast and hypoblast. -It is plain that a blastosphere, or hollow vesicle bounded -by a layer of vesicles, is under very different physical conditions -from a single, simple vesicle or bubble. The blastosphere has no -effective surface tension of its own, such as to -exert pressure on <span class="xxpn" id="p345">{345}</span> -its contents or bring the whole into a spherical form; nor will local -variations of surface energy be directly capable of affecting the -form of the system. But if the substance of our blastosphere be -sufficiently viscous, then osmotic forces may set up currents -which, reacting on the external fluid pressure, may easily cause -modifications of shape; and the particular case of invagination -itself will not be difficult to account for on this assumption of -non-uniform -exudation and imbibition.</p> - -<div class="chapter" id="p346"> -<h2 class="h2herein" title="VIII. The Forms of Tissues -or Cell-aggregates (continued)">CHAPTER VIII -<span class="h2ttl"> -THE FORMS OF TISSUES OR CELL-AGGREGATES (<i>continued</i>)</span></h2></div> - -<p>The problems which we have been considering, and especially -that of the bee’s cell, belong to a class of “isoperimetrical” -problems, which deal with figures whose surface is a minimum for -a definite content or volume. Such problems soon become -difficult, but we may find many easy examples which lead us -towards the explanation of biological phenomena; and the -particular subject which we shall find most easy of approach is -that of the division, in definite proportions, of some definite -portion of space, by a partition-wall of minimal area. The -theoretical principles so arrived at we shall then attempt to apply, -after the manner of Berthold and Errera, to the actual biological -phenomena of cell-division.</p> - -<p>This investigation we may approach in two ways: by considering, -namely, the partitioning off from some given space or -area of one-half (or some other fraction) of its content; or again, -by dealing simultaneously with the partitions necessary for the -breaking up of a given space into a definite number of compartments.</p> - -<p>If we take, to begin with, the simple case of a cubical cell, it -is obvious that, to divide it into two halves, the smallest possible -partition-wall is one which runs parallel to, and midway between, -two of its opposite sides. If we call <i>a</i> the length of one of the -edges of the cube, then <i>a</i><sup>2</sup> is the area, alike of one of its sides, and -of the partition which we have interposed parallel, or normal, -thereto. But if we now consider the bisected cube, and wish to -divide the one-half of it again, it is obvious that another partition -parallel to the first, so far from being the smallest possible, is -precisely twice the size of a cross-partition -perpendicular to it; <span class="xxpn" id="p347">{347}</span> -for the area of this new partition is <i>a</i> × <i>a</i> ⁄ 2. And again, for a -third bisection, our next partition must be perpendicular to the -other two, and it is obviously a little square, with an area of -(½ <i>a</i>)<sup>2</sup> -= ¼ <i>a</i><sup>2</sup> .</p> - -<p>From this we may draw the simple rule that, for a rectangular -body or parallelopiped to be divided equally by means of a -partition of minimal area, (1) the partition must cut across the -longest axis of the figure; and (2) in the event of successive -bisections, each partition must run at right angles to its immediate -predecessor.</p> - -<div class="dctr04" id="fig136"> -<img src="images/i347.png" width="496" height="509" alt=""> - <div class="dcaption">Fig. 136. (After Berthold.)</div></div> - -<p>We have already spoken of “Sachs’s Rules,” which are an -empirical statement of the method of cell-division in plant-tissues; -and we may now set them forth in full.</p> - -<ul> -<li><p>(1) The cell typically tends to divide into two co-equal parts.</p></li> - -<li><p>(2) Each new plane of division tends to intersect at right -angles the preceding plane of division.</p></li> -</ul> - -<p>The first of these rules is a statement of -physiological fact, not without its exceptions, but so -generally true that it will justify us in limiting our -enquiry, for the most part, to cases of equal subdivision. -That it is by no means universally true for cells -generally is shewn, for instance, by such well-known -cases <span class="xxpn" id="p348">{348}</span> as the -unequal segmentation of the frog’s egg. It is true when -the dividing cell is homogeneous, and under the influence -of symmetrical forces; but it ceases to be true when the -field is no longer dynamically symmetrical, for instance, -when the parts differ in surface tension or internal -pressure. This latter condition, of asymmetry of field, is -frequent in segmenting eggs<a class="afnanch" href="#fn386" -id="fnanch386">386</a>, and is then equivalent to the -principle upon which Balfour laid stress, as leading to -“unequal” or to “partial” segmentation of the egg,—viz. the -unequal or asymmetrical distribution of protoplasm and of -food-yolk.</p> - -<p>The second rule, which also has its exceptions, is true in a -large number of cases; and it owes its validity, as we may judge -from the illustration of the repeatedly bisected cube, solely to the -guiding principle of minimal areas. It is in short subordinate -to, and covers certain cases included under, a much more important -and fundamental rule, due not to Sachs but to Errera; that (3) the -incipient partition-wall of a dividing cell tends to be such that its -area is the least possible by which the given space-content can be -enclosed.</p> - -<hr class="hrblk"> - -<p>Let us return to the case of our cube, and let us suppose that, -instead of bisecting it, we desire to shut off some small portion -only of its volume. It is found in the course of experiments upon -soap-films, that if we try to bring a partition-film too near to one -side of a cubical (or rectangular) space, it becomes unstable; and -is easily shifted to a totally new position, in which it constitutes -a curved cylindrical wall, cutting off one corner of the cube. -It meets the sides of the cube at right angles (for reasons which we -have already considered); and, as we may see -from the symmetry <span class="xxpn" id="p349">{349}</span> -of the case, it constitutes precisely one-quarter of a cylinder. -Our plane transverse partition, wherever it was placed, had always -the same area, viz. <i>a</i><sup>2</sup> ; and it is obvious that a cylindrical wall, -if it cut off a small corner, may be much less than this. We want, -accordingly, to determine what is the particular volume which -might be partitioned off with equal economy of wall-space in one -way as the other, that is to say, what area of cylindrical wall -would be neither more nor less than the area <i>a</i><sup>2</sup> . The calculation -is very easy.</p> - -<p>The <i>surface-area</i> of a cylinder of length <i>a</i> is -2π<i>r</i> · <i>a</i>, and that of our quarter-cylinder -is, therefore, <i>a</i> · π<i>r</i> ⁄ 2; and this being, -by hypothesis, -= <i>a</i><sup>2</sup> , we have <i>a</i> -= π<i>r</i> ⁄ 2, or <i>r</i> -= 2<i>a</i> ⁄ π.</p> - -<p>The <i>volume</i> of a cylinder, of length <i>a</i>, is -<i>a</i>π<i>r</i><sup>2</sup> , and that of our quarter-cylinder -is <i>a</i> · π<i>r</i><sup>2</sup> ⁄ 4, which -(by substituting the value of <i>r</i>) is equal to -<i>a</i><sup>3</sup> ⁄ π.</p> - -<p>Now precisely this same volume is, obviously, shut off by a -transverse partition of area <i>a</i><sup>2</sup> , if the third side -of the rectangular space be equal to <i>a</i> ⁄ π. And this fraction, -if we take <i>a</i> -= 1, is equal to 0·318..., or rather -less than one-third. And, as we have just seen, the radius, -or side, of the corresponding quarter-cylinder will be twice -that fraction, or equal to ·636 times the side of the cubical -cell.</p> - -<div class="dright dwth-e" id="fig137"> -<img src="images/i349.png" width="327" height="324" alt=""> - <div class="dcaption">Fig. 137.</div></div> - -<p>If then, in the process of division of a cubical -cell, it so divide that the two portions be not equal in -volume but that one portion by anything less than about -three-tenths of the whole, or three-sevenths of the other -portion, there will be a tendency for the cell to divide, -not by means of a plane transverse partition, but by means -of a curved, cylindrical wall cutting off one corner of the -original cell; and the part so cut off will be one-quarter -of a cylinder. <br class="brclrfix"></p> - -<p>By a similar calculation we can shew that a <i>spherical</i> -wall, cutting off one solid angle of the cube, and -constituting an octant of a sphere, would likewise be of -less area than a plane partition as soon as the volume to -be enclosed was not greater than about <span class="xxpn" -id="p350">{350}</span> one-quarter of the original cell<a -class="afnanch" href="#fn387" id="fnanch387">387</a>. But -while both the cylindrical wall and the spherical wall -would be of less area than the plane transverse partition -after that limit (of one-quarter volume) was passed, the -cylindrical would still be the better of the two up to a -further limit. It is only when the volume to be partitioned -off <span class="xxpn" id="p351">{351}</span> is no greater -than about 0·15, or somewhere about one-seventh, of the -whole, that the spherical cell-wall in an angle of the -cubical cell, that is to say the octant of a sphere, is -definitely of less area than the quarter-cylinder. In the -accompanying diagram (Fig. <a href="#fig138" title="go to Fig. 138">138</a>) the relative areas of the -three partitions are shewn for all fractions, less than -one-half, of the divided cell.</p> - -<div class="psmprnt3"> - -<div class="dctr04" id="fig138"> -<img src="images/i350.png" width="563" height="844" alt=""> - <div class="dcaption">Fig. 138.</div></div> - -<p>In this figure, we see that the plane transverse partition, whatever fraction -of the cube it cut off, is always of the same dimensions, that is to say is -always equal to <i>a</i><sup>2</sup> , or -= 1. If one-half of the cube have to be cut off, this -plane transverse partition is much the best, for we see by the diagram that a -cylindrical partition cutting off an equal volume would have an area about -25%, and a spherical partition would have an area about 50% greater. -The point <i>A</i> in the diagram corresponds to the point where the cylindrical -partition would begin to have an advantage over the plane, that is to say -(as we have seen) when the fraction to be cut off is about one-third, or ·318 -of the whole. In like manner, at <i>B</i> the spherical octant begins to have an -advantage over the plane; and it is not till we reach the point <i>C</i> that the -spherical octant becomes of less area than the quarter-cylinder.</p> -</div><!--psmprnt3--> - -<div class="dright dwth-f" id="fig139"> -<img src="images/i351.png" width="288" height="238" alt=""> - <div class="dcaption">Fig. 139.</div></div> - -<p>The case we have dealt with is of little practical -importance to the biologist, because the cases in which -a cubical, or rectangular, cell divides unequally, and -unsymmetrically, are apparently few; but we can find, as -Berthold pointed out, a few examples, for instance in the -hairs within the reproductive “conceptacles” of certain -Fuci (Sphacelaria, etc., Fig. <a href="#fig139" title="go to Fig. 139">139</a>), or in the “paraphyses” -of mosses (Fig. <a href="#fig142" title="go to Fig. 142">142</a>). But it is of great theoretical -importance: as serving to introduce us to a large class -of cases, in which the shape and the relative dimensions -of the original cavity lead, according to the principle -of minimal areas, to cell-division in very definite and -sometimes unexpected ways. It is not easy, nor indeed -possible, to give a generalised account of these cases, -for the limiting conditions are somewhat complex, and -the mathematical treatment soon becomes difficult. But -it is easy to comprehend a few simple cases, which of -themselves will carry us a good long way; and which will -go far to convince the student that, in other cases <span -class="xxpn" id="p352">{352}</span> which we cannot fully -master, the same guiding principle is at the root of the -matter. <br class="brclrfix"></p> - -<hr class="hrblk"> - -<p>The bisection of a solid (or the subdivision of its volume in -other definite proportions) soon leads us into a geometry which, -if not necessarily difficult, is apt to be unfamiliar; but in such -problems we can go a long way, and often far enough for our -particular purpose, if we merely consider the plane geometry of -a side or section of our figure. For instance, in the case of the -cube which we have been just considering, and in the case of the -plane and cylindrical partitions by which it has been divided, it -is obvious that, since these two partitions extend symmetrically -from top to bottom of our cube, that we need only consider (so -far as they are concerned) the manner in which they subdivide -the <i>base</i> of the cube. The whole problem of the solid, up to a -certain point, is contained in our plane diagram of Fig. <a href="#fig138" title="go to Fig. 138">138</a>. And -when our particular solid is a solid of revolution, then it is obvious -that a study of its plane of symmetry (that is to say any plane -passing through its axis of rotation) gives us the solution of the -whole problem. The right cone is a case in point, for here the -investigation of its modes of symmetrical subdivision is completely -met by an examination of the isosceles triangle which constitutes -its plane of symmetry.</p> - -<p>The bisection of an isosceles triangle by a line which -shall be the shortest possible is a very easy problem. Let -<i>ABC</i> be such a triangle of which <i>A</i> is the apex; it may be -shewn that, for its shortest line of bisection, we are limited -to three cases: viz. to a vertical line <i>AD</i>, bisecting the -angle at <i>A</i> and the side <i>BC</i>; to a transverse line parallel -to the base <i>BC</i>; or to an oblique line parallel to <i>AB</i> or -to <i>AC</i>. The respective magnitudes, or lengths, of these -partition lines follow at once from the magnitudes of the -angles of our triangle. For we know, to begin with, since the -areas of similar figures vary as the squares of their linear -dimensions, that, in order to bisect the area, a line parallel -to one side of our triangle must always have a length equal -to 1 ⁄ √2 of that side. If then, we take -our base, <i>BC</i>, in all cases of a length -= 2, the -transverse partition drawn parallel to it will always have a -length equal to 2 ⁄ √2, or -= √2. -The vertical <span class="xxpn" id="p353">{353}</span> -partition, <i>AD</i>, since <i>BD</i> -= 1, will always equal -tan β (β being the angle <i>ABC</i>). And the oblique -partition, <i>GH</i>, being equal to <i>AB</i> ⁄ √2 -= 1 ⁄ (√2 cos β). If then we -call our vertical, transverse</p> - -<div class="dctr03" id="fig140"> -<img src="images/i353a.png" width="653" height="297" alt=""> - <div class="dcaption">Fig. 140.</div></div> - -<div class="dmaths"> -<p class="pcontinue">and oblique partitions, <i>V</i>, <i>T</i>, and <i>O</i>, -we have <i>V</i> -= tan β; <i>T</i> -= √2; and <i>O</i> -= 1 ⁄ (√2 cos β), or</p> - -<div><i>V</i> : <i>T</i> : <i>O</i> -= tan β ⁄ √2 : 1 : 1 ⁄ (2 -cos β).</div> - -<p class="pcontinue">And, working out these equations -for various values of β, we very soon see that the -vertical partition (<i>V</i>) is the least of the three until β -= 45°, at which limit <i>V</i> and <i>O</i> are each equal to -1 ⁄ √2 -= ·707; and that again, when -β -= 60°, <i>O</i> and <i>T</i> are each -= 1, after which -<i>T</i> (whose value always -= 1) is the shortest of the -three partitions. And, as we have seen, these results are at -once applicable, not only to the case of the plane triangle, -but also to that of the conical cell.</p> -</div><!--dmaths--> - -<div class="dctr03" id="fig141"> -<img src="images/i353b.png" width="653" height="196" alt=""> - <div class="dcaption">Fig. 141.</div></div> - -<p>In like manner, if we have a spheroidal body, less than -a hemisphere, such for instance as a low, watch-glass shaped -cell (Fig. <a href="#fig141" title="go to Fig. 141">141</a>, <i>a</i>), it is obvious that the smallest possible -partition by which we can divide it into -two equal halves <span class="xxpn" id="p354">{354}</span> -is (as in our flattened disc) a median vertical one. And -likewise, the hemisphere itself can be bisected by no smaller -partition meeting the walls at right angles than that median -one which divides it into two similar quadrants of a sphere. -But if we produce our hemisphere into a more elevated, conical -body, or into a cylinder with spherical cap, it is obvious that there -comes a point where a transverse, horizontal partition will bisect -the figure with less area of partition-wall than a median vertical -one (<i>c</i>). And furthermore, there will be an intermediate region, -a region where height and base have their relative dimensions -nearly equal (as in <i>b</i>), where an oblique partition will be better -than either the vertical or the transverse, though here the analogy -of our triangle does not suffice to give us the precise limiting -values. We need not examine these limitations in detail, but we -must look at the curvatures which accompany the several conditions. -We have seen that a film tends to set itself at equal -angles to the surface which it meets, and therefore, when that -surface is a solid, to meet it (or its tangent if it be a curved surface) -at right angles. Our <i>vertical</i> partition is, therefore, everywhere -normal to the original cell-walls, and constitutes a plane surface.</p> - -<p>But in the taller, conical cell with transverse partition, the -latter still meets the opposite sides of the cell at right angles, and -it follows that it must itself be curved; moreover, since the -tension, and therefore the curvature, of the partition is everywhere -uniform, it follows that its curved surface must be a portion -of a sphere, concave towards the apex of the original, now divided, -cell. In the intermediate case, where we have an oblique partition, -meeting both the base and the curved sides of the mother-cell, -the contact must still be everywhere at right angles: provided -we continue to suppose that the walls of the mother-cell (like those -of our diagrammatic cube) have become practically rigid before -the partition appears, and are therefore not affected and deformed -by the tension of the latter. In such a case, and especially when -the cell is elliptical in cross-section, or is still more complicated -in form, it is evident that the partition, in adapting itself to -circumstances and in maintaining itself as a surface of minimal -area subject to all the conditions of the case, may have to assume -a complex curvature. <span class="xxpn" id="p355">{355}</span></p> - -<div class="dctr01" id="fig142"> -<img src="images/i355.png" width="800" height="473" alt=""> - <div class="pcaption">Fig. 142. - <span class="nowrap"><img class="iglyph-a" - src="images/glyph-s.png" width="32" height="46" alt="S" ->-shaped</span> partitions: <i>A</i>, from <i>Taonia atomaria</i> - (after Reinke); <i>B</i>, from paraphyses of <i>Fucus</i>; - <i>C</i>, from rhizoids of Moss; <i>D</i>, from paraphyses of - <i>Polytrichum</i>.</div></div> - -<p>While in very many cases the partitions (like the walls of the -original cell) will be either plane or spherical, a more complex -curvature will be assumed under a variety of conditions. It will -be apt to occur, for instance, when the mother-cell is irregular in -shape, and one particular case of such asymmetry will be that in -which (as in Fig. <a href="#fig143" title="go to Fig. 143">143</a>) the cell has begun to branch, or give off a -diverticulum, before division takes place. A very complicated -case of a different kind, though not without its analogies to the -cases we are considering, will occur in the partitions of minimal -area which subdivide the spiral tube of a nautilus, as we shall -presently see. And again, whenever we have a marked internal -asymmetry of the cell, leading to irregular and anomalous modes -of division, in which the cell is not necessarily divided into two -equal halves and in which the partition-wall may assume an -oblique position, then apparently anomalous curvatures will tend -to make their appearance<a class="afnanch" href="#fn388" id="fnanch388">388</a>.</p> - -<p>Suppose that a more or less oblong cell have a tendency to -divide by means of an oblique partition (as may happen through -various causes or conditions of asymmetry), such a partition will -still have a tendency to set itself at right -angles to the rigid walls <span class="xxpn" id="p356">{356}</span> -of the mother-cell: and it will at once follow that our oblique -partition, throughout its whole extent, will assume the form of -a complex, saddle-shaped or anticlastic surface.</p> - -<div class="dctr03" id="fig143"> -<img src="images/i356.png" width="600" height="377" alt=""> - <div class="dcaption">Fig. 143. Diagrammatic explanation of - <span class="nowrap"><img class="iglyph-a" - src="images/glyph-s.png" width="32" height="46" alt="S" ->-shaped</span> partition.</div></div> - -<p>Many such cases of partitions with complex or double curvature -exist, but they are not always easy of recognition, nor is the -particular case where they appear in a <i>terminal</i> cell a common -one. We may see them, for instance, in the roots (or rhizoids) -of Mosses, especially at the point of development of a new rootlet -(Fig. <a href="#fig142" title="go to Fig. 142">142</a>, C); and again among Mosses, in the “paraphyses” of -the male prothalli (e.g. in <i>Polytrichum</i>), we find more or less -similar partitions (D). They are frequent also among many Fuci, -as in the hairs or paraphyses of Fucus itself (B). In <i>Taonia -atomaria</i>, as figured in Reinke’s memoir on the Dictyotaceae of -the Gulf of Naples<a class="afnanch" href="#fn389" id="fnanch389">389</a>, -we see, in like manner, <i>oblique</i> partitions, -which on more careful examination are seen to be curves of -double curvature (Fig. <a href="#fig142" title="go to Fig. 142">142</a>, A).</p> - -<p>The physical cause and origin of these -<span class="nowrap"><img class="iglyph-a" -src="images/glyph-s.png" width="32" height="46" alt="S">-shaped</span> -partitions is -somewhat obscure, but we may attempt a tentative explanation. -When we assert a tendency for the cell to divide transversely to -its long axis, we are not only stating empirically that the partition -tends to appear in a small, rather than a large cross-section of the -cell: but we are also implicitly ascribing to the cell a longitudinal -<i>polarity</i> (Fig. <a href="#fig143" title="go to Fig. 143">143</a>, A), and implicitly asserting -that it tends to <span class="xxpn" id="p357">{357}</span> -divide (just as the segmenting egg does), by a partition transverse -to its polar axis. Such a polarity may conceivably be due to -a chemical asymmetry, or anisotropy, such as we have learned -of (from Professor Macallum’s experiments) in our chapter on -Adsorption. Now if the chemical concentration, on which this -anisotropy or polarity (by hypothesis) depends, be unsymmetrical, -one of its poles being as it were deflected to one side, where a little -branch or bud is being (or about to be) given off,—all in precise -accordance with the adsorption phenomena described on p. <a href="#p289" title="go to pg. 289">289</a>,—then -our “polar axis” would necessarily be a curved axis, and the -partition, being constrained (again <i>ex hypothesi</i>) to arise transversely -to the polar axis, would lie obliquely to the <i>apparent</i> axis of the -cell (Fig. <a href="#fig143" title="go to Fig. 143">143</a>, B, C). And if the oblique partition be so situated -that it has to meet the <i>opposite</i> walls (as in C), then, in order to -do so symmetrically (i.e. either perpendicularly, as when the -cell-wall is already solidified, or at least at equal angles on either -side), it is evident that the partition, in its course from one side -of the cell to the other, must necessarily assume a more or less -<span class="nowrap"><img class="iglyph-a" -src="images/glyph-s.png" width="32" height="46" alt="S">-shaped</span> -curvature (Fig. <a href="#fig143" title="go to Fig. 143">143</a>, D).</p> - -<p>As a matter of fact, while we have abundant simple illustrations -of the principles which we have now begun to study, apparent -exceptions to this simplicity, due to an asymmetry of the cell -itself, or of the system of which the single cell is but a part, are -by no means rare. For example, we know that in cambium-cells, -division frequently takes place parallel to the long axis of the -cell, when a partition of much less area would suffice if it were -set cross-ways: and it is only when a considerable disproportion -has been set up between the length and breadth of the cell, that -the balance is in part redressed by the appearance of a transverse -partition. It was owing to such exceptions that Berthold was -led to qualify and even to depreciate the importance of the law -of minimal areas as a factor in cell-division, after he himself had -done so much to demonstrate and elucidate it<a class="afnanch" href="#fn390" id="fnanch390">390</a>. -He was deeply -and rightly impressed by the fact that other forces -besides surface <span class="xxpn" id="p358">{358}</span> -tension, both external and internal to the cell, play their part -in the determination of its partitions, and that the answer to -our problem is not to be given in a word. How fundamentally -important it is, however, in spite of all conflicting tendencies and -apparent exceptions, we shall see better and better as we proceed.</p> - -<hr class="hrblk"> - -<p>But let us leave the exceptions and return to a consideration -of the simpler and more general phenomena. And in so doing, -let us leave the case of the cubical, quadrangular or cylindrical -cell, and examine the case of a spherical cell and of its successive -divisions, or the still simpler case of a circular, discoidal cell.</p> - -<p>When we attempt to investigate mathematically the position -and form of a partition of minimal area, it is plain that we shall -be dealing with comparatively simple cases wherever even one -dimension of the cell is much less than the other two. Where two -dimensions are small compared with the third, as in a thin cylindrical -filament like that of Spirogyra, we have the problem at its -simplest; for it is at once obvious, then, that the partition must -lie transversely to the long axis of the thread. But even where -one dimension only is relatively small, as for instance in a flattened -plate, our problem is so far simplified that we see at once that the -partition cannot be parallel to the extended plane, but must cut -the cell, somehow, at right angles to that plane. In short, the -problem of dividing a much flattened solid becomes identical with -that of dividing a simple <i>surface</i> of the same form.</p> - -<p>There are a number of small Algae, growing in the form of -small flattened discs, consisting (for a time at any rate) of but a -single layer of cells, which, as Berthold shewed, exemplify this -comparatively simple problem; and we shall find presently that -it is also admirably illustrated in the cell-divisions which occur in -the egg of a frog or a sea-urchin, when the egg for the sake of -experiment is flattened out under artificial pressure.</p> - -<div class="dctr02" id="fig144"> -<img src="images/i359a.png" width="704" height="401" alt=""> - <div class="dcaption">Fig. 144. Development of <i>Erythrotrichia</i>. -(After Berthold.)</div></div> - -<p>Fig. <a href="#fig144" title="go to Fig. 144">144</a> (taken from Berthold’s <i>Monograph of the Naples -Bangiaciae</i>) represents younger and older discs of the little alga -<i>Erythrotrichia discigera</i>; and it will be seen that, in all stages save -the first, we have an arrangement of cell-partitions which looks -somewhat complex, but into which we must attempt to throw some -light and order. Starting with the original -single, and flattened, <span class="xxpn" id="p359">{359}</span> -cell, we have no difficulty with the first two cell-divisions; for -we know that no bisecting partitions can possibly be shorter than -the two diameters, which divide the cell into halves and into -quarters. We have only to remember that, for the sum total of -partitions to be a minimum, three only must meet in a point; -and therefore, the four quadrantal walls must shift a little, producing -the usual little median partition, or cross-furrow, instead -of one common, central point of junction. This little intermediate -wall, however, will be very small, and to all intents and purposes</p> - -<div class="dright dwth-f" id="fig145"> -<img src="images/i359b.png" width="351" height="348" alt=""> - <div class="dcaption">Fig. 145.</div></div> - -<p class="pcontinue">we may deal with the case as -though we had now to do with four equal cells, each -one of them a perfect quadrant. And so our problem -is, to find the shortest line which shall divide the -quadrant of a circle into two halves of equal area. A -radial partition (Fig. <a href="#fig145" title="go to Fig. 145">145</a>, <span class="nowrap"><span -class="smmaj">A</span>),</span> starting from the apex of -the quadrant, is at once excluded, for a reason similar -to that just referred to; our choice must lie therefore -between two modes of division such as are illustrated -in Fig. <a href="#fig145" title="go to Fig. 145">145</a>, where the partition is either (as in <span -class="nowrap"><span class="smmaj">B</span>)</span> -<span class="xxpn" id="p360">{360}</span> concentric -with the outer border of the cell, or else (as in <span -class="nowrap"><span class="smmaj">C</span>)</span> cuts -that outer border; in other words, our partition may <span -class="nowrap">(<span class="smmaj">B</span>)</span> -cut <i>both</i> radial walls, or <span class="nowrap">(<span -class="smmaj">C</span>)</span> may cut <i>one</i> radial -wall and the periphery. These are the two methods -of division which Sachs called, respectively, <span -class="nowrap">(<span class="smmaj">B</span>)</span> -<i>periclinal</i>, and <span class="nowrap">(<span -class="smmaj">C</span>)</span> <i>anticlinal</i><a -class="afnanch" href="#fn391" id="fnanch391">391</a>. -We may either treat the walls of the dividing quadrant -as already solidified, or at least as having a tension -compared with which that of the incipient partition film -is inconsiderable. In either case the partition must meet -the cell-wall, on either side, at right angles, and (its -own tension and curvature being everywhere uniform) it must -take the form of a circular arc.<br class="brclrfix"></p> - -<p>Now we find that a flattened cell which is approximately -a quadrant of a circle invariably divides after the manner -of Fig. <a href="#fig145" title="go to Fig. 145">145</a>, <span class="smmaj">C,</span> that is to say, -by an approximately circular, <i>anticlinal</i> wall, such as we -now recognise in the eight-celled stage of Erythrotrichia -(Fig. <a href="#fig144" title="go to Fig. 144">144</a>); let us then consider that Nature has solved our -problem for us, and let us work out the actual geometric -conditions.</p> - -<p>Let the quadrant <i>OAB</i> (in Fig. <a href="#fig146" title="go to Fig. 146">146</a>) be divided into -two parts of equal area, by the circular arc <i>MP</i>. It is -required to determine (1) the position of <i>P</i> upon the -arc of the quadrant, that is to say the angle <i>BOP</i>; (2) -the position of the point <i>M</i> on the side <i>OA</i>; and (3) -the length of the arc <i>MP</i> in terms of a radius of the -quadrant.</p> - -<ul> -<li> -<div class="dmaths"> -<p>(1) Draw <i>OP</i>; also <i>PC</i> a tangent, meeting <i>OA</i> in <i>C</i>; and -<i>PN</i>, perpendicular to <i>OA</i>. Let us call <i>a</i> a radius; and θ the angle -at <i>C</i>, which is obviously equal to <i>OPN</i>, or <i>POB</i>. Then</p> - -<div class="pleft nowrap"> -<i>CP</i> -= <i>a</i> cot θ;    <i>PN</i> -= <i>a</i> cos θ;<br> -<i>NC</i> -= <i>CP</i> cos θ -= <i>a</i> · (cos<sup>2</sup> θ) ⁄ (sin θ). -</div> - -<p>The area of the portion <i>PMN</i></p> - -<div class="pleft nowrap"> -= ½ <i>C P</i><sup>2</sup> θ − ½ <i>PN</i> · <i>NC</i> -<br> -= ½ <i>a</i><sup>2</sup> cot<sup>2</sup> θ -− ½ <i>a</i> cos θ · <i>a</i> cos<sup>2</sup> θ ⁄ sin θ -<br> -= ½ <i>a</i><sup>2</sup>(cot<sup>2</sup> θ − cos<sup>3</sup> θ  ⁄ sin θ). -</div></div><!--dmaths--> - -<div><span class="xxpn" id="p361">{361}</span></div> - -<div class="dmaths"> -<p>And the area of the portion <i>PNA</i></p> - -<div class="pleft nowrap"> -= ½ <i>a</i><sup>2</sup>(π ⁄ 2 − θ) − ½ <i>ON</i> · <i>NP</i> -<br> -= ½ <i>a</i><sup>2</sup>(π ⁄ 2 -− θ) -− ½ <i>a</i> sin θ · <i>a</i> cos θ -<br> -= ½ <i>a</i><sup>2</sup>(π ⁄ 2 − θ − sin θ · cos θ). -</div> - -<p>Therefore the area of the whole portion <i>PMA</i></p> - -<div class="pleft nowrap"> -= <i>a</i><sup>2</sup> ⁄ 2 (π ⁄ 2 − θ + θ cot<sup>2</sup> θ -− cos<sup>3</sup> θ ⁄ sin θ − sin θ · cos θ) -<br> -= <i>a</i><sup>2</sup> ⁄ 2 (π ⁄ 2 − θ + θ cot<sup>2</sup> θ − cot θ), -</div> - -<p class="pcontinue">and also, by hypothesis, -= ½ · area of the quadrant, -= π <i>a</i><sup>2</sup> ⁄ 8.</p> -</div><!--dmaths--> - -<div class="dctr01" id="fig146"> -<img src="images/i361.png" width="800" height="492" alt=""> - <div class="dcaption">Fig. 146.</div></div> - -<div class="dmaths"> -<p>Hence θ is defined by the equation</p> - -<div><i>a</i><sup>2</sup> ⁄ 2 (π ⁄ 2 − θ + θ cot<sup>2</sup> θ − cot θ) -= π <i>a</i><sup>2</sup> ⁄ 8, -<br class="brclrfix"></div> - -<p class="pcontinue pleftfloat">or</p> - -<div> π ⁄ 4 − θ + θ cot<sup>2</sup> θ − cot θ -= 0.<br class="brclrfix"></div> -</div><!--dmaths--> - -<div class="section"> -<p>We may solve this equation by constructing a table (of which -the following is a small portion) for various values of θ.</p> - -<div class="dtblbox"><div class="nowrap"> -<table> -<tr> - <th>θ</th> - <th>π ⁄ 4</th> - <th>− θ</th> - <th>− cot θ</th> - <th>+ θ cot<sup>2</sup> θ</th> - <th>= <i>x</i></th></tr> -<tr> - <td class="tdright">34° 34′</td> - <td class="tdright">·7854</td> - <td class="tdright">− ·6033</td> - <td class="tdright">− 1·4514</td> - <td class="tdright">+ 1·2709</td> - <td class="tdright">=  ·0016</td></tr> -<tr> - <td class="tdright">    35′</td> - <td class="tdright">·7854</td> - <td class="tdright">·6036</td> - <td class="tdright">1·4505</td> - <td class="tdright">1·2700</td> - <td class="tdright">·0013</td></tr> -<tr> - <td class="tdright">    36′</td> - <td class="tdright">·7854</td> - <td class="tdright">·6039</td> - <td class="tdright">1·4496</td> - <td class="tdright">1·2690</td> - <td class="tdright">·0009</td></tr> -<tr> - <td class="tdright">    37′</td> - <td class="tdright">·7854</td> - <td class="tdright">·6042</td> - <td class="tdright">1·4487</td> - <td class="tdright">1·2680</td> - <td class="tdright">·0005</td></tr> -<tr> - <td class="tdright">    38′</td> - <td class="tdright">·7854</td> - <td class="tdright">·6045</td> - <td class="tdright">1·4478</td> - <td class="tdright">1·2671</td> - <td class="tdright">·0002</td></tr> -<tr> - <td class="tdright">    39′</td> - <td class="tdright">·7854</td> - <td class="tdright">·6048</td> - <td class="tdright">1·4469</td> - <td class="tdright">1·2661</td> - <td class="tdright">− ·0002</td></tr> -<tr> - <td class="tdright">    40′</td> - <td class="tdright">·7854</td> - <td class="tdright">·6051</td> - <td class="tdright">1·4460</td> - <td class="tdright">1·2652</td> - <td class="tdright">− ·0005</td></tr> -</table></div></div><!--dtblbox--></div><!--section--> - -<div><span class="xxpn" id="p362">{362}</span></div> - -<p>We see accordingly that the equation is solved (as accurately -as need be) when θ is an angle somewhat over 34° 38′, or say -34° 38½′. That is to say, a quadrant of a circle is bisected by a -circular arc cutting the side and the periphery of the quadrant -at right angles, when the arc is such as to include (90° − 34° 38′), -i.e. 55° 22′ of the quadrantal arc.</p> - -<p>This determination of ours is practically identical with that -which Berthold arrived at by a rough and ready method, without -the use of mathematics. He simply tried various ways of dividing -a quadrant of paper by means of a circular arc, and went on doing -so till he got the weights of his two pieces of paper approximately -equal. The angle, as he thus determined it, was 34·6°, or say -34° 36′.</p></li> - -<li><p>(2) The position of <i>M</i> on the side of -the quadrant <i>OA</i> is given by the equation <i>OM</i> -= <i>a</i> cosec θ − <i>a</i> cot θ; -the value of which expression, for the angle which we have just -discovered, is ·3028. That is to say, the radius (or side) of -the quadrant will be divided by the new partition into two -parts, in the proportions of nearly three to seven.</p></li> - -<li><p>(3) The length of the arc <i>MP</i> is equal to -<i>a</i> θ cot θ; and the -value of this for the given angle is ·8751. This is as much as to -say that the curved partition-wall which we are considering is -shorter than a radial partition in the proportion of 8¾ to 10, or -seven-eights almost exactly.</p></li></ul> - -<div class="dleft dwth-g" id="fig147"> -<img src="images/i362.png" width="226" height="220" alt=""> - <div class="dcaption">Fig. 147.</div></div> - -<p>But we must also compare the length of this curved “anticlinal” -partition-wall (<i>MP</i>) with that of the concentric, -or periclinal, one (<i>RS</i>, Fig. <a href="#fig147" title="go to Fig. 147">147</a>) by -which the quadrant might also be bisected. -The length of this partition is obviously -equal to the arc of the quadrant (i.e. the -peripheral wall of the cell) divided by √2; -or, in terms of the radius, -= π ⁄ 2 √2 -= 1·111. -So that, not only is the anticlinal partition -(such as we actually find in nature) notably the best, but the -periclinal one, when it comes to dividing an entire quadrant, is -very considerably larger even than a radial partition. -<br class="brclrfix"></p> - -<p>The two cells into which our original quadrant is now divided, -while they are equal in volume, are of very -different shapes; the <span class="xxpn" id="p363">{363}</span> -one is a triangle (<i>MAP</i>) with two sides formed of circular arcs, -and the other is a four-sided figure (<i>MOBP</i>), which we may call -approximately oblong. We cannot say as yet how the triangular -portion ought to divide; but it is obvious that the least possible -partition-wall which shall bisect the other must run across the -long axis of the oblong, that is to say periclinally. This, also, is -precisely what tends actually to take place. In the following -diagrams (Fig. <a href="#fig148" title="go to Fig. 148">148</a>) of a frog’s egg dividing under pressure, that -is to say when reduced to the form of a flattened plate, we see, -firstly, the division into four quadrants (by the partitions 1, 2); -secondly, the division of each quadrant by means of an anticlinal -circular arc (3, 3), cutting the peripheral wall of the quadrant -approximately in the</p> - -<div class="dctr01" id="fig148"> -<img src="images/i363.png" width="800" height="298" alt=""> - <div class="dcaption">Fig. 148. Segmentation of frog’s egg, under - artificial compression. (After Roux.)</div></div> - -<p class="pcontinue">proportions of three to seven; and thirdly, -we see that of the eight cells (four triangular and four oblong) -into which the whole egg is now divided, the four which we have -called oblong now proceed to divide by partitions transverse to -their long axes, or roughly parallel to the periphery of the egg.</p> - -<hr class="hrblk"> - -<p>The question how the other, or triangular, portion of the divided -quadrant will next divide leads us to another well-defined problem, -which is only a slight extension, making allowance for the circular -arcs, of that elementary problem of the triangle we have already -considered. We know now that an entire quadrant must divide -(so that its bisecting wall shall have the least possible area) by -means of an anticlinal partition, but how about any smaller -sectors of circles? It is obvious in the case of -a small prismatic <span class="xxpn" id="p364">{364}</span> -sector, such as that shewn in Fig. <a href="#fig149" title="go to Fig. 149">149</a>, that a <i>periclinal</i> partition -is the smallest by which we can possibly bisect the cell; we want, -accordingly, to know the limits below which the periclinal partition -is always the best, and above which the anticlinal arc, as in the -case of the whole quadrant, has the advantage in regard to smallness -of surface area.</p> - -<p>This may be easily determined; for the preceding -investigation is a perfectly general one, and the results hold -good for sectors of any other arc, as well as for the quadrant, -or arc of 90°. That is to say, the length of the partition-wall -<i>MP</i> is always determined by the angle θ, according to our -equation <i>MP</i> -= <i>a</i> θ cot θ; and the angle -θ has a definite relation to α, the angle of arc.</p> - -<div class="dctr05" id="fig149"> -<img src="images/i364.png" width="478" height="252" alt=""> - <div class="dcaption">Fig. 149.</div></div> - -<div class="dmaths"> -<p>Moreover, in the case of the periclinal boundary, <i>RS</i> -(Fig. <a href="#fig147" title="go to Fig. 147">147</a>) -(or <i>ab</i>, Fig. <a href="#fig149" title="go to Fig. -149">149</a>), we know that, if it bisect the cell,</p> - -<div><i>RS</i> -= <i>a</i> · α ⁄ √2.</div> - -<p>Accordingly, the arc <i>RS</i> will be just equal to the arc <i>MP</i> when</p> - -<div>θ cot θ -= α ⁄ √2.<br class="brclrfix"></div> - -<p class="pcontinue pleftfloat">When</p> - -<div>θ cot θ > α ⁄ √2 -    or     -<i>MP</i> > <i>RS</i>, -<br class="brclrfix"></div> - -<p class="pcontinue">then division will take place as in <i>RS</i>.</p> - -<p class="pcontinue pleftfloat">When</p> - -<div>θ cot θ < α ⁄ √2, -    or     -<i>MP</i> < <i>RS</i>, -<br class="brclrfix"></div> - -<p class="pcontinue">then division will take place as in <i>MP</i>.</p> -</div><!--dmaths--> - -<p>In the accompanying diagram (Fig. <a href="#fig150" title="go to Fig. 150">150</a>), I have plotted the -various magnitudes with which we are concerned, in order to -exhibit the several limiting values. Here we see, in the first -place, the curve marked α, which shews on the (left-hand) vertical -scale the various possible magnitudes of that angle -(viz. the angle <span class="xxpn" id="p365">{365}</span> -of arc of the whole sector which we wish to divide), and on the -horizontal scale the corresponding values of θ, or the angle which -determines</p> - -<div class="dctr01" id="fig150"> -<img src="images/i365.png" width="800" height="931" alt=""> - <div class="dcaption">Fig. 150.</div></div> - -<p class="pcontinue">the point on the periphery where it is cut by the -partition-wall, <i>MP</i>. Two limiting cases are to be noticed here: -(1) at 90° (point <i>A</i> in diagram), because we are -at present only <span class="xxpn" id="p366">{366}</span> -dealing with arcs no greater than a quadrant; and (2), the point -(<i>B</i>) where the angle θ comes to equal the angle α, for after that -point the construction becomes impossible, since an anticlinal -bisecting partition-wall would be partly outside the cell. The only -partition which, after the point, can possibly exist, is a periclinal -one. This point, as our diagram shews us, occurs when the angles -(α and θ) are each rather under 52°.</p> - -<p>Next I have plotted, on the same diagram, and in relation to -the same scales of angles, the corresponding lengths of the two -partitions, viz. <i>RS</i> and <i>MP</i>, their lengths being expressed (on -the right-hand side of the diagram) in relation to the radius of -the circle (<i>a</i>), that is to say the side wall, <i>OA</i>, of our cell.</p> - -<p>The limiting values here are (1), <i>C</i>, <i>C′</i>, where the angle of arc -is 90°, and where, as we have already seen, the two partition-walls -have the relative magnitudes of <i>MP</i> : <i>RS</i> -= 0·875 : 1·111; (2) the -point <i>D</i>, where <i>RS</i> equals unity, that is to say where the periclinal -partition has the same length as a radial one; this occurs when -α is rather under 82° (cf. the points <i>D</i>, <i>D′</i>); (3) the point <i>E</i>, where -<i>RS</i> and <i>MP</i> intersect; that is to say the point at which the two -partitions, periclinal and anticlinal, are of the same magnitude; -this is the case, according to our diagram, when the angle of arc -is just over 62½°. We see from this, then, that what we have -called an anticlinal partition, as <i>MP</i>, is only likely to occur in -a triangular or prismatic cell whose angle of arc lies between -90° and 62½°. In all narrower or more tapering cells, the periclinal -partition will be of less area, and will therefore be more and more -likely to occur.</p> - -<p>The case (<i>F</i>) where the angle α is just 60° is of some interest. -Here, owing to the curvature of the peripheral border, and the -consequent fact that the peripheral angles are somewhat greater -than the apical angle α, the periclinal partition has a very slight -and almost imperceptible advantage over the anticlinal, the -relative proportions being about as <i>MP</i> : <i>RS</i> -= 0·73 : 0·72. But if -the equilateral triangle be a plane spherical triangle, i.e. a plane -triangle bounded by circular arcs, then we see that there is no -longer any distinction at all between our two partitions; <i>MP</i> -and <i>RS</i> are now identical.</p> - -<p>On the same diagram, I have inserted the curve -for values of <span class="xxpn" id="p367">{367}</span> -cosec θ − cot θ -= <i>OM</i>, that is to say the distances from the centre, -along the side of the cell, of the starting-point (<i>M</i>) of the anticlinal -partition. The point <i>C″</i> represents its position in the case of -a quadrant, and shews it to be (as we have already said) about -3 ⁄ 10 of the length of the radius from the centre. If, on the other -hand, our cell be an equilateral triangle, then we have to read off -the point on this curve corresponding to α -= 60°, and we find it -at the point <i>F‴</i> (vertically under <i>F</i>), which tells us that the -partition now starts 4·5 ⁄ 10, or nearly halfway, along the radial -wall.</p> - -<hr class="hrblk"> - -<p>The foregoing considerations carry us a long way in our -investigations of many of the simpler forms of cell-division. -Strictly speaking they are limited to the case of flattened cells, -in which we can treat the problem as though we were simply -partitioning a plane surface. But it is obvious that, though they -do not teach us the whole conformation of the partition which -divides a more complicated solid into two halves, yet they do, even -in such a case, enlighten us so far, that they tell us the appearance -presented in one plane of the actual solid. And as this is all that -we see in a microscopic section, it follows that the results we have -arrived at will greatly help us in the interpretation of microscopic -appearances, even in comparatively complex cases of cell-division.</p> - -<div class="dright dwth-d" id="fig151"> -<img src="images/i367.png" width="380" height="394" alt=""> - <div class="dcaption">Fig. 151.</div></div> - -<p>Let us now return to our -quadrant cell (<i>OAPB</i>), which we -have found to be divided into -a triangular and a quadrilateral -portion, as in Fig. <a href="#fig147" title="go to Fig. 147">147</a> or Fig. <a href="#fig151" title="go to Fig. 151">151</a>; -and let us now suppose the whole -system to grow, in a uniform -fashion, as a prelude to further -subdivision. The whole quadrant, -growing uniformly (or with equal -radial increments), will still remain -a quadrant, and it is -obvious, therefore, that for every -new increment of size, more will -be added to the margin of its triangular portion -than to the <span class="xxpn" id="p368">{368}</span> -narrower margin of its quadrilateral portion; and these increments -will be in proportion to the angles of arc, viz. 55° 22′ : 34° 38′, -or as ·96 : ·60, i.e. as 8 : 5. And accordingly, if we may assume -(and the assumption is a very plausible one), that, just as the -quadrant itself divided into two halves after it got to a certain -size, so each of its two halves will reach the same size before -again dividing, it is obvious that the triangular portion will be -doubled in size, and therefore ready to divide, a considerable -time before the quadrilateral part. To work out the problem in -detail would lead us into troublesome mathematics; but if -we simply assume that the increments are proportional to the -increasing radii of the circle, we have the following equations:―</p> - -<p>Let us call the triangular cell <i>T</i>, and the quadrilateral, <i>Q</i> -(Fig. <a href="#fig151" title="go to Fig. 151">151</a>); let the radius, <i>OA</i>, of the original quadrantal cell -= <i>a</i> -= 1; and let the increment which is required to add on a -portion equal to <i>T</i> (such as <i>PP′A′A</i>) be called <i>x</i>, and let that -required, similarly, for the doubling of <i>Q</i> be called <i>x′</i>.</p> - -<div class="dmaths"> -<p>Then we see that the area of the original quadrant</p> - -<div>= <i>T</i> + <i>Q</i> -= ¼ π <i>a</i><sup>2</sup> -= ·7854<i>a</i><sup>2</sup> ,</div> - -<p class="pcontinue">while the area of <i>T</i></p> - -<div>= <i>Q</i> -= ·3927<i>a</i><sup>2</sup> .</div> - -<p>The area of the enlarged sector, <i>p′OA′</i>,</p> - -<div>= (<i>a</i> + <i>x</i>)<sup>2</sup> × (55° 22′) ÷ 2 -= ·4831(<i>a</i> + <i>x</i>)<sup>2</sup> ,</div> - -<p class="pcontinue">and the area <i>OPA</i></p> - -<div>= <i>a</i><sup>2</sup> × (55° 22′) ÷ 2  -= ·4831<i>a</i><sup>2</sup> .</div> - -<p class="pcontinue">Therefore the area of the added portion, <i>T′</i>,</p> - -<div>= ·4831 {(<i>a</i> + <i>x</i>)<sup>2</sup> − <i>a</i><sup>2</sup>}. -</div> - -<p class="pcontinue">And this, by hypothesis,</p> - -<div>= <i>T</i> -= ·3927<i>a</i><sup>2</sup> .</div> - -<p>We get, accordingly, since <i>a</i> -= 1,</p> - -<div><i>x</i><sup>2</sup> + 2<i>x</i> -= ·3927 ⁄ ·4831 -= ·810,</div> - -<p class="pcontinue">and, solving,</p> - -<div><i>x</i> + 1 -= √(1·81) -= 1·345,  or  <i>x</i> -= 0·345.</div> - -<p>Working out <i>x′</i> in the same way, we arrive at the approximate -value, <i>x′</i> + 1 -= 1·517. <span class="xxpn" id="p369">{369}</span></p> -</div><!--dmaths--> - -<p>This is as much as to say that, supposing each cell tends to -divide into two halves when (and not before) its original size is -doubled, then, in our flattened disc, the triangular cell <i>T</i> will tend -to divide when the radius of the disc has increased by about a -third (from 1 to 1·345), but the quadrilateral cell, <i>Q</i>, will not tend -to divide until the linear dimensions of the disc have increased -by about a half (from 1 to 1·517).</p> - -<p>The case here illustrated is of no small general importance. -For it shews us that a uniform and symmetrical growth of the -organism (symmetrical, that is to say, under the limitations of a -plane surface, or plane section) by no means involves a uniform -or symmetrical growth of the individual cells, but may, under -certain conditions, actually lead to inequality among these; and -this inequality may be further emphasised by differences which -arise out of it, in regard to the order of frequency of further -subdivision. This phenomenon (or to be quite candid, this -hypothesis, which is due to Berthold) is entirely independent of -any change or variation in individual surface tensions; and -accordingly it is essentially different from the phenomenon of -unequal segmentation (as studied by Balfour), to which we have -referred on p. <a href="#p348" title="go to pg. 348">348</a>.</p> - -<p>In this fashion, we might go on to consider the manner, and -the order of succession, in which the subsequent cell-divisions -would tend to take place, as governed by the principle of minimal -areas. But the calculations would grow more difficult, or the -results got by simple methods would grow less and less exact. -At the same time, some of these results would be of great interest, -and well worth the trouble of obtaining. For instance, the precise -manner in which our triangular cell, <i>T</i>, would next divide would -be interesting to know, and a general solution of this problem is -certainly troublesome to calculate. But in this particular case -we can see that the width of the triangular cell near <i>P</i> is so -obviously less than that near either of the other two angles, that -a circular arc cutting off that angle is bound to be the shortest -possible bisecting line; and that, in short, our triangular cell -will tend to subdivide, just like the original quadrant, into a -triangular and a quadrilateral portion.</p> - -<p>But the case will be different next time, because -in this new <span class="xxpn" id="p370">{370}</span> -triangle, <i>PRQ</i>, the least width is near the innermost angle, that -at <i>Q</i>; and the bisecting circular arc will therefore be opposite to <i>Q</i>, -or (approximately) parallel to <i>PR</i>. The importance of this fact is -at once evident; for it means to say that there soon comes a -time when, whether by the division of triangles or of quadrilaterals, -we find only quadrilateral cells adjoining the periphery of our -circular disc. In the subsequent division of these quadrilaterals, -the partitions will arise transversely to their long axes, that is to -say, <i>radially</i> (as <i>U</i>, <i>V</i>); and we shall consequently have a superficial -or peripheral layer of quadrilateral cells, with sides approximately -parallel, that is to say what we are accustomed to call <i>an -epidermis</i>. And this epidermis or superficial layer will be in clear -contrast with the more irregularly shaped cells, the products of -triangles and quadrilaterals, which make up the deeper, underlying -layers of tissue.</p> - -<div class="dctr01" id="fig152"> -<img src="images/i370.png" width="800" height="200" alt=""> - <div class="dcaption">Fig. 152.</div></div> - -<p>In following out these theoretic principles and others like to -them, in the actual division of living cells, we must always bear -in mind certain conditions and qualifications. In the first place, -the law of minimal area and the other rules which we have arrived -at are not absolute but relative: they are links, and very important -links, in a chain of physical causation; they are always at work, -but their effects may be overridden and concealed by the operation -of other forces. Secondly, we must remember that, in the great -majority of cases, the cell-system which we have in view is constantly -increasing in magnitude by active growth; and by this -means the form and also the proportions of the cells are continually -liable to alteration, of which phenomenon we have already had -an example. Thirdly, we must carefully remember that, until -our cell-walls become absolutely solid and rigid, they are always -apt to be modified in form owing to the tension -of the adjacent <span class="xxpn" id="p371">{371}</span> -walls; and again, that so long as our partition films are fluid or -semifluid, their points and lines of contact with one another may -shift, like the shifting outlines of a system of soap-bubbles. This -is the physical cause of the movements frequently seen among -segmenting cells, like those to which Rauber called attention in -the segmenting ovum of the frog, and like those more striking -movements or accommodations which give rise to a so-called -“spiral” type of segmentation.</p> - -<hr class="hrblk"> - -<p>Bearing in mind, then, these considerations, let us see what -our flattened disc is likely to look like, after a few successive -divisions</p> - -<div class="dctr05" id="fig153"> -<img src="images/i371.png" width="446" height="406" alt=""> - <div class="pcaption">Fig. 153. Diagram of flattened or discoid - cell dividing into octants: to shew gradual tendency - towards a position of equilibrium.</div></div> - -<p class="pcontinue">into -component cells. In Fig. <a href="#fig153" title="go to Fig. 153">153</a>, <i>a</i>, we have a diagrammatic -representation of our disc, after it has divided into four -quadrants, and each of these in turn into a triangular and a -quadrilateral portion; but as yet, this figure scarcely suggests -to us anything like the normal look of an aggregate of living cells. -But let us go a little further, still limiting ourselves, however, -to the consideration of the eight-celled stage. Wherever one of -our radiating partitions meets the peripheral wall, there will (as -we know) be a mutual tension between the three convergent films, -which will tend to set their edges at equal angles to one another, -angles that is to say of 120°. In consequence of this, the outer -wall of each individual cell will (in this surface view -of our disc) <span class="xxpn" id="p372">{372}</span> -be an arc of a circle of which we can determine the centre by the -method used on p. <a href="#p307" title="go to pg. 307">307</a>; and, furthermore, the narrower cells, -that is to say the quadrilaterals, will have this outer border -somewhat more curved than their broader neighbours. We arrive, -then, at the condition shewn in Fig. <a href="#fig153" title="go to Fig. 153">153</a>, <i>b</i>. Within the cell, -also, wherever wall meets wall, the angle of contact must tend, -in every case, to be an angle of 120°; and in no case may more -than three films (as seen in section) meet in a point (<i>c</i>); and -this condition, of the partitions meeting three by three, and at -co-equal angles, will obviously involve the curvature of some, if -not all, of the partitions (<i>d</i>) which in our preliminary investigation -we treated as plane. To solve this problem in a general way is -no easy matter; but it is a problem which Nature solves in -every case where, as in the case we are considering, eight bubbles, -or eight cells, meet together in a (plane or curved) surface. An -approximate solution has been given in Fig. <a href="#fig153" title="go to Fig. 153">153</a>, <i>d</i>; and it will now -at once be recognised that this figure has vastly more resemblance -to an aggregate of living cells than had the diagram of Fig. <a href="#fig153" title="go to Fig. 153">153</a>, <i>a</i> -with which we began.</p> - -<div class="dleft dwth-k" id="fig154"> -<img src="images/i372.png" width="116" height="126" alt=""> - <div class="dcaption">Fig. 154.</div></div> - -<p>Just as we have constructed in this case a series of purely -diagrammatic or schematic figures, so it will be as a rule possible -to diagrammatise, with but little alteration, the -complicated appearances presented by any ordinary -aggregate of cells. The accompanying little figure -(Fig. <a href="#fig154" title="go to Fig. 154">154</a>), of a germinating spore of a Liverwort -(Riccia), after a drawing of Professor Campbell’s, -scarcely needs further explanation: for it is well-nigh a -typical diagram of the method of space-partitioning which we are -now considering. Let us look again at our figures (on p. <a href="#p359" title="go to pg. 359">359</a>) of the -disc of Erythrotrichia, from Berthold’s <i>Monograph of the Bangiaceae</i> -and redraw the earlier stages in diagrammatic fashion. In the -following series of diagrams the new partitions, or those just about -to form, are in each case outlined; and in the next succeeding -stage they are shewn after settling down into position, and after -exercising their respective tractions on the walls previously laid -down. It is clear, I think, that these four diagrammatic figures -represent all that is shewn in the first five stages drawn by -Berthold from the plant itself; but -the correspondence cannot <span class="xxpn" id="p373">{373}</span> -in this case be precisely accurate, for the simple reason that -Berthold’s figures are taken from different individuals, and are -therefore only approximately consecutive and not strictly continuous. -The last of the six drawings in Fig. <a href="#fig144" title="go to Fig. 144">144</a> is already too</p> - -<div class="dctr01" id="fig155"> -<img src="images/i373a.png" width="800" height="212" alt=""> - <div class="pcaption">Fig. 155. Theoretical arrangement of -successive partitions in a discoid cell; for comparison -with Fig. <a href="#fig144" title="go to Fig. 144">144</a>.</div></div> - -<p class="pcontinue">complicated for diagrammatisation, that is to say it is too complicated -for us to decipher with certainty the precise order of -appearance of the numerous partitions which it contains. But -in Fig. <a href="#fig156" title="go to Fig. 156">156</a> I shew one more diagrammatic figure, of a disc which</p> - -<div class="dctr01" id="fig156"> -<img src="images/i373b.png" width="800" height="490" alt=""> - <div class="pcaption">Fig. 156. Theoretical division of a discoid -cell into sixty-four chambers: no allowance being made for -the mutual tractions of the cell-walls.</div></div> - -<p class="pcontinue">has divided, according to the theoretical plan, into about sixty-four -cells; and making due allowance for the successive changes -which the mutual tensions and tractions of -the partitions must <span class="xxpn" id="p374">{374}</span> -bring about, increasing in complexity with each succeeding stage, -we can see, even at this advanced and complicated stage, a very -considerable resemblance between the actual picture (Fig. <a href="#fig144" title="go to Fig. 144">144</a>) -and the diagram which we have here constructed in obedience to -a few simple rules.</p> - -<p>In like manner, in the annexed figures, representing sections -through a young embryo of a Moss, we have very little difficulty -in discerning the successive stages that must have intervened -between the two stages shewn: so as to lead from the just divided -quadrants (one of which, by the way, has not yet divided in our -figure (<i>a</i>)) to the stage (<i>b</i>) in which a well-marked epidermal -layer surrounds an at first sight irregular agglomeration of -“fundamental” tissue.</p> - -<div class="dctr05" id="fig157"> -<img src="images/i374.png" width="453" height="266" alt=""> - <div class="pcaption">Fig. 157. Sections of embryo of a moss. - (After Kienitz-Gerloff.)</div></div> - -<p>In the last paragraph but one, I have spoken of the difficulty -of so arranging the meeting-places of a number of cells that at -each junction only three cell-walls shall meet in a line, and all -three shall meet it at equal angles of 120°. As a matter of fact, the -problem is soluble in a number of ways; that is to say, when we -have a number of cells, say eight as in the case considered, enclosed -in a common boundary, there are various ways in which their -walls can be made to meet internally, three by three, at equal -angles; and these differences will entail differences also in the -curvature of the walls, and consequently in the shape of the cells. -The question is somewhat complex; it has been dealt with by -Plateau, and treated mathematically by M. Van Rees<a class="afnanch" href="#fn392" id="fnanch392">392</a>.</p> - -<div class="dctr03" id="fig158"> -<img src="images/i375.png" width="609" height="477" alt=""> - <div class="pcaption">Fig. 158. Various possible - arrangements of intermediate partitions, in groups of 4, - 5, 6, 7 or 8 cells.</div></div> - -<p>If within our boundary we have three cells -all meeting <span class="xxpn" id="p375">{375}</span> -internally, they must meet in a point; furthermore, they tend to -do so at equal angles of 120°, and there is an end of the matter. -If we have four cells, then, as we have already seen, the conditions -are satisfied by interposing a little intermediate wall, the two -extremities of which constitute the meeting-points of three cells -each, and the upper edge of which marks the “polar furrow.” -Similarly, in the case of five cells, we require <i>two</i> little intermediate -walls, and two polar furrows; and we soon arrive at the rule that, -for <i>n</i> cells, we require -<i>n</i> − 3 -little longitudinal partitions (and -corresponding polar furrows), connecting the triple junctions of -the cells; and these little walls, like all the rest within the system, -must be inclined to one another at angles of 120°. Where we -have only one such wall (as in the case of four cells), or only two -(as in the case of five cells), there is no room for ambiguity. But -where we have three little connecting-walls, as in the case of six -cells, it is obvious that we can arrange them in three different -ways, as in the annexed Fig. <a href="#fig159" title="go to Fig. 159">159</a>. In the system of seven cells, -the four partitions can be arranged in four ways; and the five -partitions required in the case of eight cells can be arranged in no -less than thirteen different ways, of which Fig. <a href="#fig158" title="go to Fig. 158">158</a> shews some -half-dozen only. It does not follow that, so to -speak, these various <span class="xxpn" id="p376">{376}</span> -arrangements are all equally good; some are known to be much -more stable than others, and some have never yet been realised -in actual experiment.</p> - -<p>The conditions which lead to the presence of any one of them, -in preference to another, are as yet, so far as I am aware, undetermined, -but to this point we shall return.</p> - -<hr class="hrblk"> - -<p>Examples of these various arrangements meet us at every -turn, and not only in cell-aggregates, but in various cases where -non-rigid and semi-fluid partitions (or partitions that were so to -begin with) meet together. And it is a necessary consequence of -this physical phenomenon, and of the limited and very small -number of possible arrangements, that we get similar appearances, -capable of representation by the same diagram, in the most -diverse fields of biology<a class="afnanch" href="#fn393" id="fnanch393">393</a>.</p> - -<div class="dctr01" id="fig159"> -<img src="images/i376.png" width="800" height="241" alt=""> - <div class="dcaption">Fig. 159.</div></div> - -<p>Among the published figures of embryonic stages and other -cell aggregates, we only discern these little intermediate partitions -in cases where the investigator has drawn carefully just what lay -before him, without any preconceived notions as to radial or other -symmetry; but even in other cases we can generally recognise, -without much difficulty, what the actual arrangement was whereby -the cell-walls met together in equilibrium. I have a strong suspicion -that a leaning towards Sachs’s Rule, that one cell-wall tends -to set itself at right angles to another cell-wall (a rule whose strict -limitations, and narrow range of application, -we have already <span class="xxpn" id="p377">{377}</span> -considered) is responsible for many inaccurate or incomplete -representations of the mutual arrangement of aggregated cells.</p> - -<div class="dctr01" id="fig160"><div id="fig161"> -<img src="images/i377a.png" width="800" height="270" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 160. Segmenting egg of <i>Trochus</i>. -(After Robert.)</td> - <td></td> - <td>Fig. 161. Two views of segmenting egg of - <i>Cynthia partita</i>. (After Conklin.)</td></tr></table> -</div></div></div><!--dctr01--> - -<div class="dctr01" id="fig162"> -<img src="images/i377b.png" width="800" height="322" alt=""> - <div class="pcaption">Fig. 162. (<i>a</i>) Section of apical - cone of <i>Salvinia</i>. (After Pringsheim<a class="afnanch" - href="#fn394" id="fnanch394">394</a>.) (<i>b</i>) Diagram of - probable actual arrangement.</div></div> - -<div class="dctr01" id="fig163"><div id="fig164"> -<img src="images/i377c.png" width="800" height="275" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 163. Egg of <i>Pyrosoma</i>. (After - Korotneff).</td> - <td></td> - <td>Fig. 164. Egg of <i>Echinus</i>, segmenting under - pressure. (After Driesch.)</td></tr></table> -</div></div></div><!--dctr01--> - -<p>In the accompanying series of figures -(Figs. <a href="#fig160" title="go to Fig. 160">160</a>–167) I have <span class="xxpn" id="p378">{378}</span> -set forth a few aggregates of eight cells, mostly from drawings of -segmenting eggs. In some cases they shew clearly the manner -in which the cells meet one another, always at angles of 120°, -and always with the help of five intermediate boundary walls -within the eight-celled system; in other cases I have added a -slightly altered drawing, so as to shew, with as -little change as <span class="xxpn" id="p379">{379}</span> -possible, the arrangement of boundaries which probably actually -existed, and gave rise to the appearance which the observer drew. -These drawings may be compared with the various diagrams of -Fig. <a href="#fig158" title="go to Fig. 158">158</a>, in which some seven out of the possible thirteen arrangements -of five intermediate partitions (for a system of eight cells) -have been already set forth.</p> - -<div class="dctr02" id="fig165"> -<img src="images/i378a.png" width="704" height="185" alt=""> - <div class="pcaption">Fig. 165. (<i>a</i>) Part of segmenting - egg of Cephalopod (after Watase); (<i>b</i>) probable actual - arrangement.</div></div> - -<div class="dctr02" id="fig166"> -<img src="images/i378b.png" width="704" height="347" alt=""> - <div class="pcaption">Fig. 166. (<i>a</i>) Egg of <i>Echinus</i>; - (<i>b</i>) do. of <i>Nereis</i>, under pressure. (After - Driesch).</div></div> - -<div class="dctr02" id="fig167"> -<img src="images/i378c.png" width="704" height="327" alt=""> - <div class="pcaption">Fig. 167. (<i>a</i>) Egg of frog, - under pressure (after Roux); (<i>b</i>) probable actual - arrangement.</div></div> - -<p>It will be seen that M. Robert-Tornow’s figure of the segmenting -egg of Trochus (Fig. <a href="#fig160" title="go to Fig. 160">160</a>) clearly shews the cells grouped after the -fashion of Fig. <a href="#fig158" title="go to Fig. 158">158</a>, <i>a</i>. In like manner, Mr Conklin’s figure of the -ascidian egg (<i>Cynthia</i>) shews equally clearly the arrangement <i>g</i>.</p> - -<p>A sea-urchin egg, segmenting under pressure, as figured by -Driesch, scarcely requires any modification of the drawing to -appear as a diagram of the type <i>d</i>. Turning for a moment to a -botanical illustration, we have a figure of Pringsheim’s shewing an -eight-celled stage in the apex of the young cone of Salvinia; it -is in all probability referable, as in my modified diagram, to type -<i>c</i>. Beside it is figured a very different object, a segmenting egg -of the Ascidian <i>Pyrosoma</i>, after Korotneff; it may be that this -also is to be referred to type <i>c</i>, but I think it is more easily referable -to type <i>b</i>. For there is a difference between this diagram and -that of Salvinia, in that here apparently, of the pairs of lateral -cells, the upper and the lower cell are alternately the larger, while -in the diagram of Salvinia the lower lateral cells both appear much -larger than the upper ones; and this difference tallies with the -appearance produced if we fill in the eight cells according to the -type <i>b</i> or the type <i>c</i>. In the segmenting cuttlefish egg, there -is again a slight dubiety as to which type it should be referred to, -but it is in all probability referable, like Driesch’s Echinus egg, -to <i>d</i>. Lastly, I have copied from Roux a curious figure of the -egg of <i>Rana esculenta</i>, viewed from the animal pole, which appears -to me referable, in all probability, to type <i>g</i>. Of type <i>f</i>, in which -the five partitions form a figure with four re-entrant angles, that -is to say a figure representing the five sides of a hexagon, I have -found no examples among segmenting eggs, and that arrangement -in all probability is a very unstable one.</p> - -<hr class="hrblk"> - -<p>It is obvious enough, without more ado, that these phenomena -are in the strictest and completest way common -to both plants <span class="xxpn" id="p380">{380}</span> -and animals. In other words they tally with, and they further -extend, the general and fundamental conclusions laid down by -Schwann, in his <i>Mikroskopische Untersuchungen über die Uebereinstimmung -in der Struktur und dem Wachsthum der Thiere und -Pflanzen</i>.</p> - -<p>But now that we have seen how a certain limited number of -types of eight-celled segmentation (or of arrangements of eight -cell-partitions) appear and reappear, here and there, throughout -the whole world of organisms, there still remains the very important -question, whether <i>in each particular organism</i> the conditions are -such as to lead to one particular arrangement being predominant, -characteristic, or even invariable. In short, is a particular arrangement -of cell-partitions to be looked upon (as the published figures -of the embryologist are apt to suggest) as a <i>specific character</i>, or -at least a constant or normal character, of the particular organism? -The answer to this question is a direct negative, but it is only in -the work of the most careful and accurate observers that we find -it revealed. Rauber (whom we have more than once had occasion -to quote) was one of those embryologists who recorded just what -he saw, without prejudice or preconception; as Boerhaave said -of Swammerdam, <i>quod vidit id asseruit</i>. Now Rauber has put on -record a considerable number of variations in the arrangement of -the first eight cells, which form a discoid surface about the dorsal -(or “animal”) pole of the frog’s egg. In a certain number of -cases these figures are identical with one another in type, identical -(that is to say) save for slight differences in magnitude, relative -proportions, or orientation. But I have selected (Fig. <a href="#fig168" title="go to Fig. 168">168</a>) six -diagrammatic figures, which are all <i>essentially different</i>, and these -diagrams seem to me to bear intrinsic evidence of their accuracy: -the curvatures of the partition-walls, and the angles at which -they meet agree closely with the requirements of theory, and when -they depart from theoretical symmetry they do so only to the -slight extent which we should naturally expect in a material and -imperfectly homogeneous system<a class="afnanch" href="#fn395" id="fnanch395">395</a>. -<span class="xxpn" id="p381">{381}</span></p> - -<p>Of these six illustrations, two are exceptional. In Fig. <a href="#fig168" title="go to Fig. 168">168</a>, 5, -we observe that one of the eight cells is surrounded on all sides -by the other seven. This is a perfectly natural condition, and -represents, like the rest, a phase of partial or conditional equilibrium. -But it is not included in the series we are now considering, -which is restricted to the case of eight cells extending outwards -to a common boundary. The condition shewn in Fig. <a href="#fig168" title="go to Fig. 168">168</a>, 6, is -again peculiar, and is probably rare; but it is included under the -cases considered on p. <a href="#p312" title="go to pg. 312">312</a>, in which the cells are not in complete</p> - -<div class="dctr01" id="fig168"> -<img src="images/i381.png" width="800" height="529" alt=""> - <div class="pcaption">Fig. 168. Various modes of grouping - of eight cells, at the dorsal or epiblastic pole of the - frog’s egg. (After Rauber.)</div></div> - -<p class="pcontinue">fluid contact, but are separated -by little droplets of extraneous matter; it needs no -further comment. But the other four cases are beautiful -diagrams of space-partitioning, similar to those we have -just been considering, but so exquisitely clear that they -need no modification, no “touching-up,” to exhibit their -mathematical regularity. It will easily be recognised -that in Fig. <a href="#fig168" title="go to Fig. 168">168</a>, 1 and 2, we have the arrangements -corresponding to <i>a</i> and <i>d</i> of our diagram (Fig. <a href="#fig158" title="go to Fig. 158">158</a>): -but the other two (i.e. 3 and 4) represent other of the -thirteen possible arrangements, which are not included in -that <span class="xxpn" id="p382">{382}</span> diagram. -It would be a curious and interesting investigation to -ascertain, in a large number of frogs’ eggs, all at -this stage of development, the percentage of cases in -which these various arrangements occur, with a view of -correlating their frequency with the theoretical conditions -(so far as they are known, or can be ascertained) of -relative stability. One thing stands out as very certain -indeed: that the elementary diagram of the frog’s egg -commonly given in text-books of embryology,—in which the -cells are depicted as uniformly symmetrical quadrangular -bodies,—is entirely inaccurate and grossly misleading<a -class="afnanch" href="#fn396" id="fnanch396">396</a>.</p> - -<p>We now begin to realise the remarkable fact, which may even -appear a startling one to the biologist, that all possible groupings -or arrangements whatsoever of eight cells (where all take part in -the <i>surface</i> of the group, none being submerged or wholly enveloped -by the rest) are referable to some one or other of <i>thirteen</i> types or -forms. And that all the thousands and thousands of drawings -which diligent observers have made of such eight-celled structures, -animal or vegetable, anatomical, histological or embryological, are -one and all representations of some one or another of these thirteen -types:—or rather indeed of somewhat less than the whole thirteen, -for there is reason to believe that, out of the total number of -possible groupings, a certain small number are essentially unstable, -and have at best, in the concrete, but a transitory and evanescent -existence.</p> - -<hr class="hrblk"> - -<p>Before we leave this subject, on which a vast deal more might -be said, there are one or two points which we must not omit to -consider. Let us note, in the first place, that the appearance -which our plane diagrams suggest of inequality of the several -cells is apt to be deceptive; for the differences of magnitude -apparent in one plane may well be, and probably generally are, -balanced by equal and opposite differences in another. Secondly, -let us remark that the rule which we are -considering refers only <span class="xxpn" id="p383">{383}</span> -to angles, and to the number, not to the length of the intermediate -partitions; it is to a great extent by variations in the length of these -that the magnitudes of the cells may be equalised, or otherwise -balanced, and the whole system brought into equilibrium. Lastly, -there is a curious point to consider, in regard to the number of -actual contacts, in the various cases, between cell and cell. If we -inspect the diagrams in Fig. <a href="#fig169" title="go to Fig. 169">169</a> (which represent three out of our -thirteen possible arrangements of eight cells) we shall see that, in -the case of type <i>b</i>, two cells are each in contact with two others, -two cells with three others, and four cells each with four other cells. -In type <i>a</i> four cells are each in contact with two, two with four, -and two with five. In type <i>f</i>, two are in contact with two, four -with three, and one with no less than seven. In all cases the</p> - -<div class="dctr01" id="fig169"> -<img src="images/i383.png" width="800" height="280" alt=""> - <div class="dcaption">Fig. 169.</div></div> - -<p class="pcontinue">number of contacts is twenty-six in all; or, in other words, there -are thirteen internal partitions, besides the eight peripheral walls. -For it is easy to see that, in all cases of <i>n</i> cells with a common -external boundary, the number of internal partitions is 2<i>n</i> − 3; -or the number of what we call the internal or interfacial contacts -is 2(2<i>n</i> − 3). But it would appear that the most stable arrangements -are those in which the total number of contacts is most -evenly divided, and the least stable are those in which some one -cell has, as in type <i>f</i>, a predominant number of contacts. In a -well-known series of experiments, Roux has shewn how, by means -of oil-drops, various arrangements, or aggregations, of cells can -be simulated; and in Fig. <a href="#fig170" title="go to Fig. 170">170</a> I shew a number of Roux’s figures, -and have ascribed them to what seem to be their appropriate -“types” among those which we have just -been considering; but <span class="xxpn" id="p384">{384}</span> -it will be observed that in these figures of Roux’s the drops are not -always in complete contact, a little air-bubble often keeping them -apart at their apical junctions, so that we see the configuration -towards which the system is <i>tending</i> rather than that which it has -fully attained<a class="afnanch" href="#fn397" id="fnanch397">397</a>. -The type which we have called <i>f</i> was found by -Roux to be unstable, the large (or apparently large) drop <i>a″</i> -quickly passing into the centre of the system, and here taking up -a position of equilibrium in which, as usual, three cells meet -throughout in a point, at equal angles, and in which, in this case, -all the cells have an equal number of “interfacial” contacts.</p> - -<div class="dctr02" id="fig170"> -<img src="images/i384.png" width="705" height="539" alt=""> - <div class="pcaption">Fig. 170. Aggregations of oil-drops. (After -Roux.) Figs. <a href="#fig4" title="go to Fig. 4">4</a>–6 represent successive changes in a single -system.</div></div> - -<p>We need by no means be surprised to find that, in such -arrangements, the commonest and most stable distributions -are those in which the cell-contacts are distributed as -uniformly as possible between the several cells. We always -expect to find some such tendency to equality in cases -where we have to do with small oscillations on either side -of a symmetrical condition. <span class="xxpn" id="p385">{385}</span></p> - -<p>The rules and principles which we have arrived at from the -point of view of surface tension have a much wider bearing than is -at once suggested by the problems to which we have applied them; -for in this elementary study of the cell-boundaries in a segmenting -egg or tissue we are on the verge of a difficult and important -subject in pure mathematics. It is a subject adumbrated by -Leibniz, studied somewhat more deeply by Euler, and greatly -developed of recent years. It is the <i>Geometria Situs</i> of Gauss, the -<i>Analysis Situs</i> of Riemann, the Theory of Partitions of Cayley, -and of Spatial Complexes of Listing<a class="afnanch" href="#fn398" id="fnanch398">398</a>. -The crucial point for the -biologist to comprehend is, that in a closed surface divided into -a number of faces, the arrangement of all the faces, lines and -points in the system is capable of analysis, and that, when the -number of faces or areas is small, the number of possible arrangements -is small also. This is the simple reason why we meet in -such a case as we have been discussing (viz. the arrangement of -a group or system of eight cells) with the same few types recurring -again and again in all sorts of organisms, plants as well as animals, -and with no relation to the lines of biological classification: and -why, further, we find similar configurations occurring to mark -the symmetry, not of cells merely, but of the parts and organs of -entire animals. The phenomena are not “functions,” or specific -characters, of this or that tissue or organism, but involve general -principles which lie within the province of the mathematician.</p> - -<hr class="hrblk"> - -<p>The theory of space-partitioning, to which the segmentation -of the egg gives us an easy practical introduction, is illustrated in -much more complex ways in other fields of natural history. A -very beautiful but immensely complicated case is furnished by -the “venation” of the wings of insects. Here we have sometimes -(as in the dragon-flies), a general reticulum of small, more or less -hexagonal “cells”: but in most other cases, in flies, bees, butterflies, -etc., we have a moderate number of cells, whose partitions -always impinge upon one another three by three, and whose -arrangement, therefore, includes of necessity a number of small -intermediate partitions, analogous to our -polar furrows. I think <span class="xxpn" id="p386">{386}</span> -that a mathematical study of these, including an investigation of -the “deformation” of the wing (that is to say, of the changes in -shape and changes in the form of its “cells” which it undergoes -during the life of the individual, and from one species to another) -would be of great interest. In very many cases, the entomologist -relies upon this venation, and upon the occurrence of this or that -intermediate vein, for his classification, and therefore for his -hypothetical phylogeny of particular groups; which latter procedure -hardly commends itself to the physicist or the mathematician.</p> - -<div class="dctr01" id="fig171"> -<img src="images/i386.png" width="800" height="282" alt=""> - <div class="pcaption">Fig. 171. (A) <i>Asterolampra marylandica</i>, - Ehr.; (B, C) <i>A. variabilis</i>, Grev. (After Greville.)</div></div> - -<p>Another case, geometrically akin but biologically very -different, is to be found in the little diatoms of the genus Asterolampra, -and their immediate congeners<a class="afnanch" href="#fn399" id="fnanch399">399</a>. -In Asterolampra we -have a little disc, in which we see (as it were) radiating spokes of -one material, alternating with intervals occupied on the flattened -wheel-like disc by another (Fig. <a href="#fig171" title="go to Fig. 171">171</a>). The spokes vary in number, -but the general appearance is in a high degree suggestive of the -Chladni figures produced by the vibration of a circular plate. -The spokes broaden out towards the centre, and interlock by -visible junctions, which obey the rule of triple intersection, and -accordingly exemplify the partition-figures with which we are -dealing. But whereas we have found the particular arrangement -in which one cell is in contact with all the rest to be unstable, -according to Roux’s oil-drop experiments, -and to be conspicuous <span class="xxpn" id="p387">{387}</span> -by its absence from our diagrams of segmenting eggs, here in -Asterolampra, on the other hand, it occurs frequently, and is -indeed the commonest arrangement<a class="afnanch" href="#fn400" id="fnanch400">400</a> -(Fig. <a href="#fig171" title="go to Fig. 171">171</a>, B). In all probability, -we are entitled to consider this marked difference natural -enough. For we may suppose that in Asterolampra (unlike the -case of the segmenting egg) the tendency is to perfect radial -symmetry, all the spokes emanating from a point in the centre: -such a condition would be eminently unstable, and would break -down under the least asymmetry. A very simple, perhaps the -simplest case, would be that one single spoke should differ slightly -from the rest, and should so tend to be drawn in amid the others, -these latter remaining similar and symmetrical among themselves. -Such a configuration would be vastly less unstable than the -original one in which all the boundaries meet in a point; and the -fact that further progress is not made towards other configurations -of still greater stability may be sufficiently accounted for by -viscosity, rapid solidification, or other conditions of restraint. -A perfectly stable condition would of course be obtained if, as in -the case of Roux’s oil-drop (Fig. <a href="#fig170" title="go to Fig. 170">170</a>, 6), one of the cellular spaces -passed into the centre of the system, the other partitions radiating -outwards from its circular wall to the periphery of the whole -system. Precisely such a condition occurs among our diatoms; -but when it does so, it is looked -upon as the mark and characterisation -of the <i>allied genus</i> Arachnoidiscus.</p> - -<hr class="hrblk"> - -<div class="dright dwth-e" id="fig172"> -<img src="images/i387.png" width="342" height="338" alt=""> - <div class="dcaption">Fig. 172. Section of Alcyonarian - polype.</div></div> - -<p>In a diagrammatic section of an Alcyonarian polype (Fig. -<a href="#fig172" title="go to Fig. 172">172</a>), we have eight chambers set, symmetrically, about a -ninth, which constitutes the “stomach.” In this arrangement -there is no difficulty, for it is obvious that, throughout -the system, three boundaries meet (in plane section) in a -point. In many corals we have as <span class="xxpn" id="p388">{388}</span> -simple, or even simpler conditions, for the radiating calcified -partitions either converge upon a central chamber, or fail to -meet it and end freely. But in a few cases, the partitions or -“septa” converge to meet <i>one another</i>, there being no central -chamber on which they may impinge; and here the manner in -which contact is effected becomes complicated, and involves -problems identical with those which we are now studying. -<br class="brclrfix"></p> - -<div class="dleft dwth-f" id="fig173"> -<img src="images/i388.png" width="298" height="265" alt=""> - <div class="dcaption">Fig. 173. <i>Heterophyllia angulata</i>. -(After Nicholson.)</div></div> - -<p>In the great majority of corals we have as simple or -even simpler conditions than those of Alcyonium; for as a -rule the calcified partitions or septa of the coral either -converge upon a central chamber (or central “columella”), -or else fail to meet it and end freely. In the latter -case the problem of space-partitioning does not arise; in -the former, however numerous the septa be, their separate -contacts with the wall of the central chamber comply with -our fundamental rule according to which three lines and no -more meet in a point, and from this simple and symmetrical -arrangement there is little tendency to variation. But -in a few cases, the septal partitions converge to meet -<i>one another</i>, there being no central chamber on which -they may impinge; and here the manner in which contact -is effected becomes complicated, and involves problems -of space-partitioning identical with those which we are -now studying. In the genus Heterophyllia and in a few -allied forms we have such conditions, and students of -the Coelenterata have found them very puzzling. McCoy<a -class="afnanch" href="#fn401" id="fnanch401">401</a>, -their first discoverer, pronounced these corals to be -“totally unlike” any other group, recent or fossil; and -Professor Martin Duncan, writing a memoir on Heterophyllia -and its allies<a class="afnanch" href="#fn402" -id="fnanch402">402</a>, described them as “paradoxical in -their anatomy.” <br class="brclrfix"></p> - -<div class="dctr03" id="fig174"> -<img src="images/i389.png" width="611" height="580" alt=""> - <div class="dcaption">Fig. 174. <i>Heterophyllia</i> sp. - (After Martin Duncan.)</div></div> - -<p>The simplest or youngest Heterophylliae known have six septa -(as in Fig. <a href="#fig174" title="go to Fig. 174">174</a>, <i>a</i>); in the case figured, four of these septa are -conjoined two and two, thus forming the usual triple junctions -together with their intermediate -partition-walls: and in the <span class="xxpn" id="p389">{389}</span> -case of the other two we may fairly assume that their proper -and original arrangement was that of our type 6<i>b</i> (Fig. <a href="#fig158" title="go to Fig. 158">158</a>), -though the central intermediate partition has been crowded out -by partial coalescence. When with increasing age the septa -become more numerous, their arrangement becomes exceedingly -variable; for the simple reason that, from the mathematical -point of view, the number of possible arrangements, of 10, 12 -or more cellular partitions in triple contact, tends to increase -with great rapidity, and there is little to choose between many -of them in regard to symmetry and equilibrium. But while, -mathematically speaking, each particular case among the multitude -of possible cases is an orderly and definite arrangement, -from the purely biological point of view on the other hand no -law or order is recognisable; and so McCoy described the genus -as being characterised by the possession of septa “destitute of any -order of arrangement, but irregularly branching and coalescing in -their passage from the solid external walls towards some indefinite -point near the centre where the few main lamellae irregularly -anastomose.” <span class="xxpn" id="p390">{390}</span></p> - -<p>In the two examples figured (Fig. <a href="#fig174" title="go to Fig. 174">174</a>), both comparatively -simple ones, it will be seen that, of the main chambers, one is in -each case an unsymmetrical one; that is to say, there is one -chamber which is in contact with a greater number of its neighbours -than any other, and which at an earlier stage must have had -contact with them all; this was the case of our type <i>f</i>, in the -eight-celled system (Fig. <a href="#fig158" title="go to Fig. 158">158</a>). Such an asymmetrical chamber -(which may occur in a system of any number of cells greater than -six), constitutes what is known to students of the Coelenterata as -a “fossula”; and we may recognise it not only here, but also in -Zaphrentis and its allies, and in a good many other corals besides. -Moreover certain corals are described as having more than one -fossula: this appearance being naturally produced under certain -of the other asymmetrical variations of normal space-partitioning. -Where a single fossula occurs, we are usually told that it is a -symptom of “bilaterality”; and this is in turn interpreted as -an indication of a higher grade of organisation than is implied -in the purely “radial symmetry” of the commoner types of coral. -The mathematical aspect of the case gives no warrant for this -interpretation.</p> - -<p>Let us carefully notice (lest we run the risk of confusing two -distinct problems) that the space-partitioning of Heterophyllia -by no means agrees with the details of that which we have studied -in (for instance) the case of the developing disc of Erythrotrichia: -the difference simply being that Heterophyllia illustrates the -general case of cell-partitioning as Plateau and Van Rees studied -it, while in Erythrotrichia, and in our other embryological and -histological instances, we have found ourselves justified in making -the additional assumption that each new partition divided a cell -into <i>co-equal parts</i>. No such law holds in Heterophyllia, whose -case is essentially different from the others: inasmuch as the -chambers whose partition we are discussing in the coral are mere -empty spaces (empty save for the mere access of sea-water); while -in our histological and embryological instances, we were speaking -of the division of a cellular unit of living protoplasm. Accordingly, -among other differences, the “transverse” or “periclinal” partitions, -which were bound to appear at regular intervals and in -definite positions, when co-equal bisection was a -feature of the <span class="xxpn" id="p391">{391}</span> -case, are comparatively few and irregular in the earlier stages of -Heterophyllia, though they begin to appear in numbers after the -main, more or less radial, partitions have become numerous, and -when accordingly these radiating partitions come to bound narrow -and almost parallel-sided interspaces; then it is that the transverse -or periclinal partitions begin to come in, and form what the student -of the Coelenterata calls the “dissepiments” of the coral. We -need go no further into the configuration and anatomy of the -corals; but it seems to me beyond a doubt that the whole question -of the complicated arrangement of septa and dissepiments throughout -the group (including the curious vesicular or bubble-like -tissue of the Cyathophyllidae and the general structural plan of -the Tetracoralla,</p> - -<div class="dctr02" id="fig175"> -<img src="images/i391.png" width="736" height="413" alt=""> - <div class="dcaption">Fig. 175. Diagrammatic section of a Ctenophore - (<i>Eucharis</i>).</div></div> - -<p class="pcontinue">such as Streptoplasma and its allies) is well -worth investigation from the physical and mathematical point of -view, after the fashion which is here slightly adumbrated.</p> - -<hr class="hrblk"> - -<p>The method of dividing a circular, or spherical, system into -eight parts, equal as to their areas but unequal in their peripheral -boundaries, is probably of wide biological application; that is to -say, without necessarily supposing it to be rigorously followed, the -typical configuration which it yields seems to recur again and -again, with more or less approximation to precision, and under -widely different circumstances. I am inclined to think, for instance, -that the unequal division of the surface of a -Ctenophore by its <span class="xxpn" id="p392">{392}</span> -meridian-like ciliated bands is a case in point (Fig. <a href="#fig175" title="go to Fig. 175">175</a>). Here, if we -imagine each quadrant to be twice bisected by a curved anticline, -we shall get what is apparently a close approximation to the actual -position of the ciliated bands. The case however is complicated -by the fact that the sectional plan of the organism is never quite -circular, but always more or less elliptical. One point, at least, -is clearly seen in the symmetry of the Ctenophores; and that is -that the radiating canals which pass outwards to correspond in -position with the ciliated bands, have no common centre, but -diverge from one another by repeated bifurcations, in a manner -comparable to the conjunctions of our cell-walls.</p> - -<p>In like manner I am inclined to suggest that the same principle -may help us to understand the apparently complex arrangement -of the skeletal rods of a larval Echinoderm, and the very complex -conformation of the larva which is brought about by the presence -of these long, slender skeletal radii.</p> - -<div class="dctr04" id="fig176"> -<img src="images/i392.png" width="528" height="329" alt=""> - <div class="pcaption">Fig. 176. Diagrammatic arrangement of -partitions, represented by skeletal rods, in larval -Echinoderm (<i>Ophiura</i>).</div></div> - -<p>In Fig. <a href="#fig176" title="go to Fig. 176">176</a> I have divided a circle into its four quadrants, and -have bisected each quadrant by a circular arc (<i>BC</i>), passing from -radius to periphery, as in the foregoing cases of cell-division; and -I have again bisected, in a similar way, the triangular halves of -each quadrant (<i>DD</i>). I have also inserted a small circle in the -middle of the figure, concentric with the large one. If now we -imagine those lines in the figure which I have drawn black to be -replaced by solid rods we shall have at once the frame-work of an -Ophiurid (Pluteus) larva. Let us imagine all these -arms to be <span class="xxpn" id="p393">{393}</span> -bent symmetrically downwards, so that the plane of the paper is -transformed into a spheroidal surface, such as that of a hemisphere, -or that of a tall conical figure with curved sides; let a membrane -be spread, umbrella-like, between the outstretched skeletal rods, -and let its margin loop from rod to rod in curves which are possibly -catenaries, but are more probably portions of an “elastic curve,” -and the outward resemblance to a Pluteus larva is now complete. -By various slight modifications, by altering the relative lengths -of the rods, by modifying their curvature or by replacing the curved -rod by a tangent to itself, we can ring the changes which lead us -from one known type of Pluteus to another. The case of the -Bipinnaria larvae of Echinids is certainly analogous, but it becomes</p> - -<div class="dctr04" id="fig177"> -<img src="images/i393.png" width="527" height="283" alt=""> - <div class="dcaption">Fig. 177. Pluteus-larva of Ophiurid.</div></div> - -<p class="pcontinue">very much more complicated; we have to -do with a more complex partitioning of space, and I confess -that I am not yet able to represent the more complicated -forms in so simple a way.</p> - -<hr class="hrblk"> - -<p>There are a few notable exceptions (besides the various unequally -segmenting eggs) to the general rule that in cell-division -the mother-cell tends to divide into equal halves; and one of these -exceptional cases is to be found in connection with the development -of “stomata” in the leaves of plants. The epidermal cells -by which the leaf is covered may be of various shapes; sometimes, -as in a hyacinth, they are oblong, but more often they have an -irregular shape in which we can recognise, more or less clearly, -a distorted or imperfect hexagon. In the case of the oblong cells, -a transverse partition will be the least possible, whether the cell -be equally or unequally divided, unless (as we -have already seen) <span class="xxpn" id="p394">{394}</span> -the space to be cut off be a very small one, not more than</p> - -<div class="dctr01" id="fig178"> -<img src="images/i394a.png" width="800" height="460" alt=""> - <div class="pcaption">Fig. 178. Diagrammatic development - of Stomata in <i>Sedum</i>. (Cf. fig. in Sachs’s <i>Botany</i>, - 1882, p. 103.)</div></div> - -<p class="pcontinue">about -three-tenths the area of a square based on the <i>short</i> side of the -original rectangular cell. As the portion usually cut off is not -nearly so small as this, we get the form of partition shewn in -Fig. <a href="#fig179" title="go to Fig. 179">179</a>, and the cell so cut off is next bisected by a partition at -right angles to the first; this latter partition splits, and the two -last-formed cells constitute the so-called “guard-cells” of the -stoma. In</p> - -<div class="dleft dwth-d" id="fig179"> -<img src="images/i394b.png" width="384" height="281" alt=""> - <div class="dcaption">Fig. 179. Diagrammatic development of -stomata in Hyacinth.</div></div> - -<p class="pcontinue">other cases, as in Fig. -<a href="#fig178" title="go to Fig. 178">178</a>, there will come a point where the minimal partition -necessary to cut off the required fraction of the -cell-content is no longer a transverse one, but is a -portion of a cylindrical wall (2) cutting off one corner -of the mother-cell. The cell so cut off is now a certain -segment of a circle, with an arc of approximately 120°; -and its next division will be by means of a curved wall -cutting it into a triangular and a quadrangular portion -(3). The triangular portion will continue to divide in -a similar way (4, 5), and at length (for a reason which -is not yet clear) the partition wall <span class="xxpn" -id="p395">{395}</span> between the new-formed cells splits, -and again we have the phenomenon of a “stoma” with its -attendant guard-cells. In Fig. <a href="#fig179" title="go to Fig. 179">179</a> are shewn the successive -stages of division, and the changing curvatures of the -various walls which ensue as each subsequent partition -appears, introducing a new tension into the system.<br -class="brclrfix"></p> - -<p>It is obvious that in the case of the oblong cells of the epidermis -in the hyacinth the stomata will be found arranged in regular rows, -while they will be irregularly distributed over the surface of the -leaf in such a case as we have depicted in Sedum.</p> - -<p>While, as I have said, the mechanical cause of the split which -constitutes the orifice of the stoma is not quite clear, yet there -can be little or no doubt that it, like the rest of the phenomenon, -is related to surface tension. It might well be that it is directly -due to the presence underneath this portion of epidermis of the -hollow air-space which the stoma is apparently developed “for -the purpose” of communicating with; this air-surface on both -sides of the delicate epidermis might well cause such an alteration -of tensions that the two halves of the dividing cell would tend to -part company. In short, if the surface-energy in a cell-air contact -were half or less than half that in a contact between cell and cell, -then it is obvious that our partition would tend to split, and give -us a two-fold surface in contact with air, instead of the original -boundary or interface between one cell and the other. In Professor -Macallum’s experiments, which we have briefly discussed in our -short chapter on Adsorption, it was found that large quantities -of potassium gathered together along the outer walls of the guard-cells -of the stoma, thereby indicating a low surface-tension along -these outer walls. The tendency of the guard-cells to bulge -outwards is so far explained, and it is possible that, under the -existing conditions of restraint, we may have here a force tending, -or helping, to split the two cells asunder. It is clear enough, -however, that the last stage in the development of a stoma, is, -from the physical point of view, not yet properly understood.</p> - -<hr class="hrblk"> - -<p>In all our foregoing examples of the development of a “tissue” -we have seen that the process consists in the <i>successive</i> division -of cells, each act of division being accompanied -by the formation <span class="xxpn" id="p396">{396}</span> -of a boundary-surface, which, whether it become at once a solid -or semi-solid partition or whether it remain semi-fluid, exercises -in all cases an effect on the position and the form of the boundary -which comes into being with the next act of division. In contrast -to this general process stands the phenomenon known as “free -cell-formation,” in which, out of a common mass of protoplasm, -a number of separate cells are <i>simultaneously</i>, or all but simultaneously, -differentiated. In a number of cases it happens that, -to begin with, a number of “mother-cells” are formed simultaneously, -and each of these divides, by two successive</p> - -<div class="dctr01" id="fig180"> -<img src="images/i396.png" width="800" height="432" alt=""> - <div class="pcaption">Fig. 180. Various pollen-grains - and spores (after Berthold, Campbell, Goebel and - others). (1) <i>Epilobium</i>; (2) <i>Passiflora</i>; (3) - <i>Neottia</i>; (4) <i>Periploca graeca</i>; (5) <i>Apocynum</i>; (6) - <i>Erica</i>; (7) Spore of <i>Osmunda</i>; (8) Tetraspore of - <i>Callithamnion</i>.</div></div> - -<p class="pcontinue">divisions, into four “daughter-cells.” -These daughter-cells will tend to group -themselves, just as would four soap-bubbles, into a “tetrad,” the -four cells corresponding to the angles of a regular tetrahedron. -For the system of four bodies is evidently here in perfect symmetry; -the partition-walls and their respective edges meet at equal -angles: three walls everywhere meeting in an edge, and the four -edges converging to a point in the geometrical centre of the -system. This is the typical mode of development of pollen-grains, -common among Monocotyledons and all but universal -among Dicotyledonous plants. By a loosening of the surrounding -tissue and an expansion of the cavity, or -anther-cell, in which <span class="xxpn" id="p397">{397}</span> -they lie, the pollen-grains afterwards fall apart, and their individual -form will depend upon whether or no their walls have</p> - -<div class="dleft dwth-g" id="fig181"> -<img src="images/i397.png" width="226" height="224" alt=""> - <div class="dcaption">Fig. 181. Dividing spore of <i>Anthoceros</i>. -(After Campbell.)</div></div> - -<p class="pcontinue">solidified before this liberation takes place. -For if not, then the separate grains will be -free to assume a spherical form as a consequence -of their own individual and unrestricted -growth; but if they become solid -or rigid prior to the separation of the -tetrad, then they will conserve more or less -completely the plane interfaces and sharp -angles of the elements of the tetrahedron. -The latter is the case, for instance, in -the pollen-grains of Epilobium (Fig. <a href="#fig180" title="go to Fig. 180">180</a>, 1) and in many -others. In the Passion-flower (2) we have an intermediate -condition: where we can still see an indication of the facets -where the grains abutted on one another in the tetrad, but -the plane faces have been swollen by growth into spheroidal or -spherical surfaces. It is obvious that there may easily be cases -where the tetrads of daughter-cells are prevented from assuming -the tetrahedral form: cases, that is to say, where the four cells -are forced and crushed into one plane. The figures given by -Goebel of the development of the pollen of Neottia (3, <i>a</i>–<i>e</i>: all -the figures referring to grains taken from a single anther), illustrate -this to perfection; and it will be seen that, when the four cells -lie in a plane, they conform exactly to our typical diagram of the -first four cells in a segmenting ovum. Occasionally, though the -four cells lie in a plane, the diagram seems to fail us, for the cells -appear to meet in a simple cross (as in 5); but here we soon -perceive that the cells are not in complete interfacial contact, -but are kept apart by a little intervening drop of fluid or bubble -of air. The spores of ferns (7) develop in very much the same -way as pollen-grains; and they also very often retain traces of -the shape which they assumed as members of a tetrahedral figure. -Among the “tetraspores” (8) of the Florideae, or Red Seaweeds, -we have a phenomenon which is in every respect analogous. -<br class="brclrfix"></p> - -<p>Here again it is obvious that, apart from differences in actual -magnitude, and apart from superficial or “accidental” differences -(referable to other physical phenomena) in the -way of colour, <span class="xxpn" id="p398">{398}</span> -texture and minute sculpture or pattern, it comes to pass, through -the laws of surface-tension and the principles of the geometry of -position, that a very small number of diagrammatic figures will -sufficiently represent the outward forms of all the tetraspores, -four-celled pollen-grains, and other four-celled aggregates which -are known or are even capable of existence.</p> - -<hr class="hrblk"> - -<p>We have been dealing hitherto (save for some slight exceptions) -with the partitioning of cells on the assumption that the system -either remains unaltered in size or else that growth has proceeded -uniformly in all directions. But we extend the scope of our -enquiry very greatly when we begin to deal with <i>unequal growth</i>, -with growth, that is to say, which produces a greater extension -along some one axis than another. And here we come close in -touch with that great and still (as I think) insufficiently appreciated -generalisation of Sachs, that the manner in which the cells divide -is <i>the result</i>, and not the cause, of the form of the dividing -structure: that the form of the mass is caused by its growth -as a whole, and is not a resultant of the growth of the -cells individually considered<a class="afnanch" href="#fn403" id="fnanch403">403</a>. -Such asymmetry of growth -may be easily imagined, and may conceivably arise from a -variety of causes. In any individual cell, for instance, it may -arise from molecular asymmetry of the structure of the cell-wall, -giving it greater rigidity in one direction than another, while all -the while the hydrostatic pressure within the cell remains constant -and uniform. In an aggregate of cells, it may very well arise -from a greater chemical, or osmotic, activity in one than another, -leading to a localised increase in the fluid pressure, and to a -corresponding bulge over a certain area of the external surface. -It might conceivably occur as a direct result of the preceding -cell-divisions, when these are such as to produce many peripheral -or concentric walls in one part and few or none in another, with -the obvious result of strengthening the common boundary wall -and resisting the outward pressure of growth in parts where the -former is the case; that is to say, in our -dividing quadrant, if <span class="xxpn" id="p399">{399}</span> -its quadrangular portion subdivide by periclines, and the triangular -portion by oblique anticlines (as we have seen to be the natural -tendency), then we might expect that external growth would be -more manifest over the latter than over the former areas. As -a direct and immediate consequence of this we might expect a -tendency for special outgrowths, or “buds,” to arise from the -triangular rather than from the quadrangular cells; and this -turns out to be not merely a tendency towards which theoretical -considerations point, but a widespread and important factor in the -morphology of the cryptogams. But meanwhile, without enquiring -further into this complicated question, let us simply take -it that, if we start from such a simple case as a round cell which -has divided into two halves, or four quarters (as the case may be), -we shall at once get bilateral symmetry about a main axis, and -other secondary results arising therefrom, as soon as one of the -halves, or one of the quarters, begins to shew a rate of growth in -advance of the others; for the more rapidly growing cell, or the -peripheral wall common to two or more such rapidly growing cells, -will bulge out into an ellipsoid form, and may finally extend -into a cylinder with rounded or ellipsoid end.</p> - -<p>This latter very simple case is illustrated in the development -of a pollen-tube, where the rapidly growing cell develops into the -elongated cylindrical tube, and the slow-growing or quiescent part -remains behind as the so-called “vegetative” cell or cells.</p> - -<p>Just as we have found it easier to study the segmentation of -a circular disc than that of a spherical cell, so let us begin in the -same way, by enquiring into the divisions which will ensue if the -disc tend to grow, or elongate, in some one particular direction, -instead of in radial symmetry. The figures which we shall then -obtain will not only apply to the disc, but will also represent, in -all essential features, a projection or longitudinal section of a solid -body, spherical to begin with, preserving its symmetry as a solid -of revolution, and subject to the same general laws as we have -studied in the disc<a class="afnanch" href="#fn404" id="fnanch404">404</a>. -<span class="xxpn" id="p400">{400}</span></p> - -<ul> -<li><p>(1) Suppose, in the first place, that the axis of growth lies -symmetrically in one of the original quadrantal cells of a segmenting -disc; and let this growing cell elongate with comparative rapidity -before it subdivides. When it does divide, it will necessarily do -so by a transverse partition, concave towards the apex of the -cell: and, as further elongation takes place, the cylindrical -structure which will be developed thereby will tend to be again -and again subdivided by similar concave transverse partitions. -If at any time, through this process of concurrent elongation and -subdivision, the apical cell become equivalent to, or less than, -a hemisphere, it will next divide by means of a longitudinal, or -vertical partition; and similar longitudinal partitions will arise in -the other segments of the cylinder, as soon as it comes about that -their length (in the direction of the axis) is less than their breadth.</p> - -<div class="dctr03" id="fig182"> -<img src="images/i400.png" width="610" height="397" alt=""> - <div class="dcaption">Fig. 182.</div></div> - -<p>But when we think of this structure in the solid, we at once -perceive that each of these flattened segments of the cylinder, -into which our cylinder has divided, is equivalent to a flattened -circular disc; and its further division will accordingly tend to -proceed like any other flattened disc, namely into four quadrants, -and afterwards by anticlines and -periclines in the usual way. <span class="xxpn" id="p401">{401}</span> -A section across the cylinder, then, will tend to shew us precisely -the same arrangements as we have already so fully studied in -connection with the typical division of a circular cell into quadrants, -and of these quadrants into triangular and quadrangular portions, -and so on.</p> - -<p>But there are other possibilities to be considered, in regard to -the mode of division of the elongating quasi-cylindrical portion, as -it gradually develops out of the growing and bulging quadrantal -cell; for the manner in which this latter cell divides will simply -depend upon the form it has assumed before each successive act -of division takes place, that is to say upon the ratio between its -rate of growth and the frequency of its successive divisions. For, -as we have already seen, if the growing cell attain a markedly -oblong or cylindrical form before division ensues, then the partition -will arise transversely to the long axis; if it be but a little more -than a hemisphere, it will divide by an oblique partition; and if -it be less than a hemisphere (as it may come to be after successive -transverse divisions) it will divide by a vertical partition, that is -to say by one coinciding with its axis of growth. An immense -number of permutations and combinations may arise in this way, -and we must confine our illustrations to a small number of cases. -The important thing is not so much to trace out the various -conformations which may arise, but to grasp the fundamental -principle: which is, that the forces which dominate the <i>form</i> of -each cell regulate the manner of its subdivision, that is to say -the form of the new cells into which it subdivides; or in other -words, the form of the growing organism regulates the form and -number of the cells which eventually constitute it. The complex -cell-network is not the cause but the result of the general configuration, -which latter has its essential cause in whatsoever physical -and chemical processes have led to a varying velocity of growth -in one direction as compared with another.</p> - -<div class="dleft dwth-e" id="fig183"> -<img src="images/i402a.png" width="336" height="422" alt=""> - <div class="dcaption">Fig. 183. Development of -<i>Sphagnum</i>. (After Campbell.)</div></div> - -<p>In the annexed figure of an embryo of Sphagnum we see a -mode of development almost precisely corresponding to the -hypothetical case which we have just described,—the case, that -is to say, where one of the four original quadrants of the mother-cell -is the chief agent in future growth and development. We -see at the base of our first figure (<i>a</i>), the -three stationary, or <span class="xxpn" id="p402">{402}</span> -undivided quadrants, one of which has further slowly divided -in the stage <i>b</i>. The active quadrant -has grown quickly into a cylindrical -structure, which inevitably divides, in -the next place, into a series of transverse -partitions; and accordingly, this -mode of development carries with it -the presence of a single “apical cell,” -whose lower wall is a spherical surface -with its convexity downwards. Each -cell of the subdivided cylinder now appears -as a more or less flattened disc, -whose mode of further sub-division -we may prognosticate according to -our former investigation, to which -subject we shall presently return.<br class="brclrfix"></p></li> - -<li> -<div class="dctr03" id="fig184"> -<img src="images/i402b.png" width="608" height="382" alt=""> - <div class="dcaption">Fig. 184.</div></div> - -<p>(2) In the next place, still keeping to the case where only one -of the original quadrant-cells continues to grow and develop, let -us suppose that this growing cell falls to be divided when by -growth it has become just a little greater than a hemisphere; it -will then divide, as in Fig. <a href="#fig184" title="go to Fig. 184">184</a>, 2, by an oblique partition, in the -usual way, whose precise position and inclination to the base will -depend entirely on the configuration of the cell itself, save only, -of course, that we may have also to take into account the possibility -of the division being into two unequal halves. -By our hypothesis, <span class="xxpn" id="p403">{403}</span> -the growth of the whole system is mainly in a vertical direction, -which is as much as to say that the more actively growing protoplasm, -or at least the strongest osmotic force, will be found -near the apex; where indeed there is obviously more external -surface for osmotic action. It will therefore be that one of -the two cells which contains, or constitutes, the apex which -will grow more rapidly than the other, and which therefore will -be the first to divide, and indeed in any case, it will usually be -this one of the two which will tend to divide first, inasmuch -as the triangular and not the quadrangular half is bound to -constitute the apex<a class="afnanch" href="#fn405" id="fnanch405">405</a>. -It is obvious that (unless the act of division -be so long postponed that the cell has become quasi-cylindrical) -it will divide by another oblique partition, starting from, and -running at right angles to, the first. And so division will proceed,</p> - -<div class="dleft dwth-k" id="fig185"> -<img src="images/i403.png" width="121" height="310" alt=""> - <div class="dcaption">Fig. 185. - Gemma of Moss. - (After Campbell.)</div></div> - -<p class="pcontinue">by oblique alternate partitions, each one tending to -be, at first, perpendicular to that on which it is based -and also to the peripheral wall; but all these points of -contact soon tending, by reason of the equal tensions -of the three films or surfaces which meet there, to form -angles of 120°. There will always be, in such a case, -a single apical cell, of a more or less distinctly -triangular form. The annexed figure of the developing -antheridium of a Liverwort (Riccia) is a typical example -of such a case. In Fig. <a href="#fig185" title="go to Fig. 185">185</a> which represents a -“gemma” of a Moss, we see just the same thing; -with this addition, that here the lower of the two -original cells has grown even more quickly than the -other, constituting a long cylindrical stalk, and dividing in accordance -with its shape, by means of transverse septa. -<br class="brclrfix"></p> - -<p>In all such cases as these, the cells whose development we have -studied will in turn tend to subdivide, and the manner in which -they will do so must depend upon their own proportions; and in -all cases, as we have already seen, there will sooner or later be -a tendency to the formation of periclinal walls, cutting off an -“epidermal layer of cells,” as Fig. <a href="#fig186" title="go to Fig. 186">186</a> illustrates very well.</p> - -<div class="dctr02" id="fig186"> -<img src="images/i404a.png" width="702" height="409" alt=""> - <div class="dcaption">Fig. 186. Development of antheridium of - <i>Riccia</i>. (After Campbell.)</div></div> - -<div class="dctr02" id="fig187"><div id="fig188"> -<img src="images/i404b.png" width="702" height="278" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 187. Section of growing shoot of - Selaginella, diagrammatic.</td> - <td></td> - <td>Fig. 188. Embryo of Jungermannia. (After - Kienitz-Gerloff.)</td></tr></table> -</div></div></div><!--dctr01--> - -<p>The method of division by means of oblique partitions is a -common one in the case of ‘growing points’; -for it evidently <span class="xxpn" id="p404">{404}</span> -includes all cases in which the act of cell-division does not lag -far behind that elongation which is determined by the specific rate -of growth. And it is also obvious that, under a common type, -there must here be included a variety of cases which will, at first -sight, present a very different appearance one from another. -For instance, in Fig. <a href="#fig187" title="go to Fig. 187">187</a> which represents a growing shoot of -Selaginella, and somewhat less diagrammatically in the young -embryo of Jungermannia (Fig. <a href="#fig188" title="go to Fig. 188">188</a>), we have the appearance of -an almost straight vertical partition running up in the axis of the -system, and the primary cell-walls are set almost at right angles -to it,—almost transversely, that is to say to the outer walls and -to the long axis of the structure. We -soon recognise, however, <span class="xxpn" id="p405">{405}</span> -that the difference is merely a difference of degree. The more -remote the partitions are, that is to say the greater the velocity -of growth relatively to division, the less abrupt will be the -alternate kinks or curvatures of the portions which lie in the -neighbourhood of the axis, and the more will these portions -appear to constitute a single unbroken wall.</p></li> - -<li> -<div class="dctr01" id="fig189"> -<img src="images/i405.png" width="791" height="238" alt=""> - <div class="dcaption">Fig. 189.</div></div> - -<p>(3) But an appearance nearly, if not quite, indistinguishable -from this may be got in another way, namely, when the original -growing cell is so nearly hemispherical that it is actually divided -by a vertical partition, into two quadrants; and from this vertical -partition, as it elongates, lateral partition-walls will arise on either -side. And by the tensions exercised by these, the vertical partition -will be bent into little portions set at 120° one to another, and the -whole will come to look just like that which, in the former case, -was made up of portions of many successive oblique partitions.</p></li> -</ul> - -<hr class="hrblk"> - -<p>Let us now, in one or two cases, follow out a little further the -stages of cell-division whose beginning we have studied in the last -paragraphs. In the antheridium of Riccia, after the successive -oblique partitions have produced the longitudinal series of cells -shewn in Fig. <a href="#fig186" title="go to Fig. 186">186</a>, it is plain that the next partitions will arise -periclinally, that is to say parallel to the outer wall, which in -this particular case represents the short axis of the oblong cells. -The effect is at once to produce an epidermal layer, whose cells -will tend to subdivide further by means of partitions perpendicular -to the free surface, that is to say crossing the flattened cells by -their shortest diameter. The inner mass, beneath the epidermis, -consists of cells which are still more or less oblong, -or which become <span class="xxpn" id="p406">{406}</span> -definitely so in process of growth; and these again divide, parallel -to their short axes, into squarish cells, which as usual, by the -mutual tension of their walls, become hexagonal, as seen in a plane -section. There is a clear distinction, then, in form as well as in -position, between the outer covering-cells and those which lie -within this envelope; the latter are reduced to a condition which -merely fulfils the mechanical function of a protective coat, while -the former undergo less modification, and give rise to the actively -living, reproductive elements.</p> - -<div class="dctr04" id="fig190"> -<img src="images/i406.png" width="526" height="352" alt=""> - <div class="pcaption">Fig. 190. Development of sporangium of -<i>Osmunda</i>. (After Bower.)</div></div> - -<p>In Fig. <a href="#fig190" title="go to Fig. 190">190</a> is shewn the development of the sporangium of a -fern (Osmunda). We may trace here the common phenomenon -of a series of oblique partitions, built alternately on one another, -and cutting off a conspicuous triangular apical cell. Over the -whole system an epidermal layer has been formed, in the manner -we have described; and in this case it covers the apical cell also, -owing to the fact that it was of such dimensions that, at one stage -of growth, a periclinal partition wall, cutting off its outer end, -was indicated as of less area than an anticlinal one. This periclinal -wall cuts down the apical cell to the proportions, very nearly, -of an equilateral triangle, but the solid form of the cell is obviously -that of a tetrahedron with curved faces; and accordingly, the -least possible partitions by which further subdivision can be -effected will run successively parallel to its four sides (or its three -sides when we confine ourselves to the appearances -as seen in <span class="xxpn" id="p407">{407}</span> -section). The effect, as seen in section, is to cut off on each side -a characteristically flattened cell, oblong as seen in section, still -leaving a triangular (or strictly speaking, a tetrahedral) one in -the centre. The former cells, which constitute no specific structure -or perform no specific physiological function, but which merely -represent certain directions in space towards which the whole -system of partitioning has gradually led, are called by botanists -the “tapetum.” The active growing tetrahedral cell which lies -between them, and from which in a sense every other cell in the -system has been either directly or indirectly segmented off, still -manifests, as it were, its vigour and activity, and now, by -internal subdivision, becomes the mother-cell of the spores.</p> - -<hr class="hrblk"> - -<p>In all these cases, for simplicity’s sake, we have merely considered -the appearances presented in a single, longitudinal, plane -of optical section. But it is not difficult to interpret from these -appearances what would be seen in another plane, for instance -in a transverse section. In our first example, for instance, that -of the developing embryo of Sphagnum (Fig. <a href="#fig183" title="go to Fig. 183">183</a>), we can see that, -at appropriate levels, the cells of the original cylindrical row have -divided into transverse rows of four, and then of eight cells. We -may be sure that the four cells represent, approximately, quadrants -of a cylindrical disc, the four cells, as usual, not meeting in a point, -but intercepted by a small intermediate partition. Again, where -we have a plate of eight cells, we may well imagine that the eight -octants are arranged in what we have found to be the way -naturally resulting from the division of four quadrants, that is to -say into alternately triangular and quadrangular portions; and -this is found by means of sections to be the case. The accompanying -figure is precisely comparable to our previous diagrams of the -arrangement of an aggregate of eight cells in a dividing disc, save -only that, in two cases, the cells have already undergone a further -subdivision.</p> - -<p>It follows in like manner, that in a host of cases we meet with -this characteristic figure, in one or other of its possible, and -strictly limited, variations,—in the cross sections of growing -embryonic structures, just as we have already seen that it appears -in a host of cases where the entire system (or a -portion of its <span class="xxpn" id="p408">{408}</span> -surface) consists of eight cells only. For example, in Fig. <a href="#fig191" title="go to Fig. 191">191</a>,</p> - -<div class="dctr01" id="fig191"> -<img src="images/i408a.png" width="800" height="264" alt=""> - <div class="pcaption">Fig. 191. (A, B,) Sections of younger and -older embryos of <i>Phascum</i>; (C) do. of <i>Adiantum</i>. (After -Kienitz-Gerloff.)</div></div> - -<p class="pcontinue"> -we have it again, in a section of a young embryo of a moss (Phascum), -and in a section of an embryo of a fern (Adiantum). In</p> - -<div class="dleft dwth-d" id="fig192"> -<img src="images/i408b.png" width="385" height="351" alt=""> - <div class="dcaption">Fig. 192. Section through frond of <i>Girardia -sphacelaria</i>. (After Goebel.)</div></div> - -<p class="pcontinue">Fig. <a href="#fig192" title="go to Fig. 192">192</a> shewing a section through a -growing frond of a sea-weed (Girardia) we have a case where -the partitions forming the eight octants have conformed -to the usual type; but instead of the usual division by -periclines of the four quadrangular spaces, these latter -are dividing by means of oblique septa, apparently owing -to the fact that the cell is not dividing into two equal, -but into two unequal portions. In this last figure we -have a peculiar look of stiffness or formality, such that -it appears at first to bear little resemblance to the -rest. The explanation is of the simplest. The mode of -partitioning differs little (except to some slight extent -in the way already mentioned) from the normal type; but in -this case the partition walls are so thick and become so -quickly comparatively solid and rigid, that the secondary -curvatures due to their successive mutual tractions are -here imperceptible. -<br class="brclrfix"></p> - -<p>A curious and beautiful case, apparently aberrant but which -would doubtless be found conforming strictly to -physical laws, if <span class="xxpn" id="p409">{409}</span> -only we clearly understood the actual conditions, is indicated in</p> - -<div class="dright dwth-e" id="fig193"> -<img src="images/i409.png" width="318" height="305" alt=""> - <div class="dcaption">Fig. 193. Development of antheridium of -<i>Pteris</i>. (After Strasbürger.)</div></div> - -<p class="pcontinue"> -the development of the antheridium -of a fern, as described by Strasbürger. -Here the antheridium develops from -a single cell, whose form has grown -to be something more than a hemisphere; -and the first partition, instead -of stretching transversely across the -cell, as we should expect it to do if -the cell were actually spherical, has -as it were sagged down to come in -contact with the base, and so to develop -into an annular partition, running -round the lower margin of the cell. The phenomenon is akin to that -cutting off of the corner of a cubical cell by a spherical partition, -of which we have spoken on p. <a href="#p349" title="go to pg. 349">349</a>, and the annular film is very -easy to reproduce by means of a soap-bubble in the bottom of -a cylindrical dish or beaker. The next partition is a periclinal -one, concentric with the outer surface of the young antheridium; -and this in turn is followed by a concave partition which cuts off -the apex of the original cell: but which becomes connected with -the second, or periclinal partition in precisely the same annular -fashion as the first partition did with the base of the little -antheridium. The result is that, at this stage, we have four -cell-cavities in the little antheridium: (1) a central cavity; -(2) an annular space around the lower margin; (3) a narrow annular -or cylindrical space around the sides of the antheridium; and -(4) a small terminal or apical cell. It is evident that the tendency, -in the next place, will be to subdivide the flattened external cells -by means of anticlinal partitions, and so to convert the whole -structure into a single layer of epidermal cells, surrounding a -central cell within which, in course of time, the antherozoids are -developed. <br class="brclrfix"></p> - -<hr class="hrblk"> - -<p>The foregoing account deals only with a few elementary phenomena, -and may seem to fall far short of an attempt to deal in general -with “the forms of tissues.” But it is the principle involved, -and not its ultimate and very complex results, that -we can alone <span class="xxpn" id="p410">{410}</span> -attempt to grapple with. The stock-in-trade of mathematical -physics, in all the subjects with which that science deals, is for the -most part made up of simple, or simplified, cases of phenomena -which in their actual and concrete manifestations are usually too -complex for mathematical analysis; and when we attempt to -apply its methods to our biological and histological phenomena, -in a preliminary and elementary way, we need not wonder if we -be limited to illustrations which are obviously of a simple kind, -and which cover but a small part of the phenomena with which -the histologist has become familiar. But it is only relatively that -these phenomena to which we have found the method applicable -are to be deemed simple and few. They go already far beyond -the simplest phenomena of all, such as we see in the dividing -Protococcus, and in the first stages, two-celled or four-celled, of -the segmenting egg. They carry us into stages where the cells -are already numerous, and where the whole conformation has -become by no means easy to depict or visualise, without the help -and guidance which the phenomena of surface-tension, the laws -of equilibrium and the principle of minimal areas are at hand -to supply. And so far as we have gone, and so far as we can -discern, we see no sign of the guiding principles failing us, or of -the simple laws ceasing -to hold good.</p> - -<div class="chapter" id="p411"> -<h2 class="h2herein" title="IX. On Concretions, Spicules, - and Spicular Skeletons.">CHAPTER IX - <span class="h2ttl"> - ON CONCRETIONS, SPICULES, AND SPICULAR - SKELETONS</span></h2></div> - -<p>The deposition of inorganic material in the living body, usually -in the form of calcium salts or of silica, is a very common and -wide-spread phenomenon. It begins in simple ways, by the -appearance of small isolated particles, crystalline or non-crystalline, -whose form has little relation or sometimes none to -the structure of the organism; it culminates in the complex -skeletons of the vertebrate animals, in the massive skeletons of -the corals, or in the polished, sculptured and mathematically -regular molluscan shells. Even among many very simple organisms, -such as the Diatoms, the Radiolarians, the Foraminifera, -or the Sponges, the skeleton displays extraordinary variety and -beauty, whether by reason of the intrinsic form of its elementary -constituents or the geometric symmetry with which these are -arranged and interconnected.</p> - -<p>With regard to the form of these various structures (and this -is all that immediately concerns us here), it is plain that we have -to do with two distinct problems, which however, though -theoretically distinct, may merge with one another. For the -form of the spicule or other skeletal element may depend simply -upon its chemical nature, as for instance, to take a simple but -not the only case, when the form is purely crystalline; or the -inorganic solid material may be laid down in conformity with the -shapes assumed by the cells, tissues or organs, and so be, as it -were, moulded to the shape of the living organism; and again, -there may well be intermediate stages in which both phenomena -may be simultaneously recognised, the molecular forces playing -their part in conjunction with, and under the restraint of, the -other forces inherent -in the system. <span class="xxpn" id="p412">{412}</span></p> - -<p>So far as the problem is a purely chemical one, we must deal -with it very briefly indeed; and all the more because special -investigations regarding it have as yet been few, and even the -main facts of the case are very imperfectly known. This at least -is evident, that the whole series of phenomena with which we are -about to deal go deep into the subject of colloid chemistry, and -especially with that branch of the science which deals with the -properties of colloids in connection with capillary or surface -phenomena. It is to the special student of colloid chemistry that -we must ultimately and chiefly look for the elucidation of our -problem<a class="afnanch" href="#fn406" id="fnanch406">406</a>.</p> - -<p>In the first and simplest part of our subject, the essential -problem is the problem of crystallisation in presence of colloids. -In the cells of plants, true crystals are found in comparative -abundance, and they consist, in the great majority of cases, of -calcium oxalate. In the stem and root of the rhubarb, for instance, -in the leaf-stalk of Begonia, and in countless other cases, sometimes -within the cell, sometimes in the substance of the cell-wall, we -find large and well-formed crystals of this salt; their varieties of -form, which are extremely numerous, are simply the crystalline -forms proper to the salt itself, and belong to the two systems, -cubic and monoclinic, in one or other of which, according to -the amount of water of crystallisation, this salt is known to -crystallise. When calcium oxalate crystallises according to the -latter system (as it does when its molecule is combined with two -molecules of water of crystallisation), the microscopic crystals -have the form of fine needles, or “raphides,” such as are very -common in plants; and it has been found that these are artificially -produced when the salt is crystallised out in presence of glucose -or of dextrin<a class="afnanch" href="#fn407" id="fnanch407">407</a>.</p> - -<div class="dctr01" id="fig194"> -<img src="images/i413.png" width="800" height="594" alt=""> - <div class="dcaption">Fig. 194. Alcyonarian spicules: <i>Siphonogorgia</i> -and <i>Anthogorgia</i>. (After Studer.)</div></div> - -<p>Calcium carbonate, on the other hand, when it occurs in plant-cells -(as it does abundantly, for instance in the “cystoliths” of the -Urticaceae and Acanthaceae, and in great -quantities in Melobesia <span class="xxpn" id="p413">{413}</span> -and the other calcareous or “stony” algae), appears in the form -of fine rounded granules, whose inherent crystalline structure -is not outwardly visible, but is only revealed (like that of a -molluscan shell) under polarised light. Among animals, a skeleton -of carbonate of lime occurs under a multitude of forms, of which -we need only mention now a very few of the most conspicuous. -The spicules of the calcareous sponges are triradiate, occasionally -quadriradiate, bodies, with pointed rays, not crystalline in outward -form but with a definitely crystalline internal structure. We shall -return again to these, and find for them what would seem to be -a satisfactory explanation of their form. Among the Alcyonarian -zoophytes we have a great variety of spicules<a class="afnanch" href="#fn408" id="fnanch408">408</a>, -which are sometimes -straight and slender rods, sometimes flattened and more or -less striated plates, and still more often rounded or branched -concretions with rough or knobby surfaces (Figs. <a href="#fig194" title="go to Fig. 194">194</a>, 200). A -third type, presented by several very different things, such as -a pearl, or the ear-bone of a bony fish, consists -of a more or less <span class="xxpn" id="p414">{414}</span> -rounded body, sometimes spherical, sometimes flattened, in which -the calcareous matter is laid down in concentric zones, denser -and clearer layers alternating with one another. In the development -of the molluscan shell and in the calcification of a bird’s -egg or the shell of a crab, for instance, spheroidal bodies with -similar concentric striation make their appearance; but instead of -remaining separate they become crowded together, and as they -coalesce they combine to form a pattern of hexagons. In some -cases, the carbonate of lime on being dissolved away by acid -leaves behind it a certain small amount of organic residue; in -most cases other salts, such as phosphates of lime, ammonia or -magnesia are present in small quantities; and in most cases if -not all the developing spicule or concretion is somehow or other -so associated with living cells that we are apt to take it for granted -that it owes its peculiarities of form to the constructive or plastic -agency of these.</p> - -<p>The appearance of direct association with living cells, however, -is apt to be fallacious; for the actual <i>precipitation</i> takes place, -as a rule, not in actively living, but in dead or at least inactive -tissue<a class="afnanch" href="#fn409" id="fnanch409">409</a>: -that is to say in the “formed material” or matrix which -(as for instance in cartilage) accumulates round the living cells, -in the interspaces between these latter, or at least, as often happens, -in connection with the cell-wall or cell-membrane rather than -within the substance of the protoplasm itself. We need not go -the length of asserting that this is a rule without exception; but, -so far as it goes, it is of great importance and to its consideration -we shall presently return<a class="afnanch" href="#fn410" id="fnanch410">410</a>.</p> - -<p>Cognate with this is the fact that it is known, at least in some -cases, that the organism can go on living and multiplying with -apparently unimpaired health, when stinted or even wholly -deprived of the material of which it is wont to -make its spicules <span class="xxpn" id="p415">{415}</span> -or its shell. Thus, Pouchet and Chabry<a class="afnanch" href="#fn411" id="fnanch411">411</a> -have shown that the -eggs of sea-urchins reared in lime-free water develop in apparent -health, into larvae entirely destitute of the usual skeleton of -calcareous rods, and in which, accordingly, the long arms of the -Pluteus larva, which the rods support and distend, are entirely -suppressed. And again, when Foraminifera are kept for generations -in water from which they gradually exhaust the lime, their -shells grow hyaline and transparent, and seem to consist only of -chitinous material. On the other hand, in the presence of excess -of lime, the shells become much altered, strengthened with various -“ornaments,” and assuming characters described as proper to -other varieties and even species<a class="afnanch" href="#fn412" id="fnanch412">412</a>.</p> - -<p>The crucial experiment, then, is to attempt the formation of -similar structures or forms, apart from the living organism: but, -however feasible the attempt may be in theory, we shall be prepared -from the first to encounter difficulties, and to realise that, though -the actions involved may be wholly within the range of chemistry -and physics, yet the actual conditions of the case may be so -complex, subtle and delicate, that only now and then, and in the -simplest of cases, shall we find ourselves in a position to imitate -them completely and successfully. Such an investigation is only -part of that much wider field of enquiry through which Stephane -Leduc and many other workers<a class="afnanch" href="#fn413" id="fnanch413">413</a> -have sought to produce, by -synthetic means, forms similar to those of living things; but it -is a well-defined and circumscribed part of that wider investigation. -When by chemical or physical experiment we obtain configurations -similar, for instance, to the phenomena of nuclear division, or -conformations similar to a pattern of hexagonal cells, or a group -of vesicles which resemble some particular tissue or cell-aggregate, -we indeed prove what it is the main object of this book to illustrate, -namely, that the physical forces are capable of producing particular -organic forms. But it is by no means always that we can feel -perfectly assured that the physical forces which we deal with in -our experiment are identical with, and not -merely analogous to, <span class="xxpn" id="p416">{416}</span> -the physical forces which, at work in nature, are bringing about -the result which we have succeeded in imitating. In the present -case, however, our enquiry is restricted and apparently simplified; -we are seeking in the first instance to obtain by purely chemical -means a purely chemical result, and there is little room for -ambiguity in our interpretation of the experiment.</p> - -<hr class="hrblk"> - -<p>When we find ourselves investigating the forms assumed by -chemical compounds under the peculiar circumstances of association -with a living body, and when we find these forms to be -characteristic or recognisable, and somehow different from those -which, under other circumstances, the same substance is wont -to assume, an analogy presents itself to our minds, captivating -though perhaps somewhat remote, between this subject of ours -and certain synthetic problems of the organic chemist. There is -doubtless an essential difference, as well as a difference of scale, -between the visible form of a spicule or concretion and the hypothetical -form of an individual molecule; but molecular form is -a very important concept; and the chemist has not only succeeded, -since the days of Wöhler, in synthesising many substances which -are characteristically associated with living matter, but his task -has included the attempt to account for the molecular <i>forms</i> of -certain “asymmetric” substances, glucose, malic acid and many -more, as they occur in nature. These are bodies which, when -artificially synthesised, have no optical activity, but which, as we -actually find them in organisms, turn (when <i>in solution</i>) the plane -of polarised light in one direction or the other; thus dextro-glucose -and laevomalic acid are common products of plant -metabolism; but dextromalic acid and laevo-glucose do not occur -in nature at all. The optical activity of these bodies depends, -as Pasteur shewed more than fifty years ago<a class="afnanch" href="#fn414" id="fnanch414">414</a>, -upon the form, -right-handed or left-handed, of their molecules, which molecular -asymmetry further gives rise to a corresponding right or left-handedness -(or enantiomorphism) in the crystalline aggregates. -It is a distinct problem in organic or -physiological chemistry, <span class="xxpn" id="p417">{417}</span> -and by no means without its interest for the morphologist, to -discover how it is that nature, for each particular substance, -habitually builds up, or at least selects, its molecules in a one-sided -fashion, right-handed or left-handed as the case may be. -It will serve us no better to assert that this phenomenon has its -origin in “fortuity,” than to repeat the Abbé Galiani’s saying, -“<i>les dés de la nature sont pipés.</i>”</p> - -<p>The problem is not so closely related to our immediate subject -that we need discuss it at length; but at the same time it has its -clear relation to the general question of <i>form</i> in relation to vital -phenomena, and moreover it has acquired interest as a theme -of long-continued discussion and new importance from some -comparatively recent discoveries.</p> - -<p>According to Pasteur, there lay in the molecular asymmetry -of the natural bodies and the symmetry of the artificial products, -one of the most deep-seated differences between vital and non-vital -phenomena: he went further, and declared that “this was -perhaps the <i>only</i> well-marked line of demarcation that can at -present [1860] be drawn between the chemistry of dead and of -living matter.” Nearly forty years afterwards the same theme -was pursued and elaborated by Japp in a celebrated lecture<a class="afnanch" href="#fn415" id="fnanch415">415</a>, -and the distinction still has its weight, I believe, in the minds of -many if not most chemists.</p> - -<p>“We arrive at the conclusion,” said Professor Japp, “that the -production of single asymmetric compounds, or their isolation -from the mixture of their enantiomorphs, is, as Pasteur firmly -held, the prerogative of life. Only the living organism, or the -living intelligence with its conception of asymmetry, can produce -this result. Only asymmetry can beget asymmetry.” In these -last words (which, so far as the chemist and the biologist are -concerned, we may acknowledge to be -perfectly true<a class="afnanch" href="#fn416" id="fnanch416">416</a>) -lies the <span class="xxpn" id="p418">{418}</span> -crux of the difficulty; for they at once bid us enquire whether in -nature, external to and antecedent to life, there be not some -asymmetry to which we may refer the further propagation or -“begetting” of the new asymmetries: or whether in default -thereof, we be rigorously confined to the conclusion, from which -Japp “saw no escape,” that “at the moment when life first arose, -a directive force came into play,—a force of precisely the same -character as that which enables the intelligent operator, by the -exercise of his will, to select one crystallised enantiomorph and -reject its asymmetric opposite<a class="afnanch" href="#fn417" id="fnanch417">417</a>.”</p> - -<p>Observe that it is only the first beginnings of chemical -asymmetry that we need to discover; for when asymmetry is once -manifested, it is not disputed that it will continue “to beget -asymmetry.” A plausible suggestion is now at hand, which if it -be confirmed and extended will supply or at least sufficiently -illustrate the kind of explanation which is required<a class="afnanch" href="#fn418" id="fnanch418">418</a>.</p> - -<p>We know in the first place that in cases where ordinary non-polarised -light acts upon a chemical substance, the amount of -chemical action is proportionate to the amount of light absorbed. -We know in the second place<a class="afnanch" href="#fn419" id="fnanch419">419</a>, -in certain cases, that light circularly -polarised is absorbed in different amounts by the right-handed or -left-handed varieties, as the case may be, of an asymmetric -substance. And thirdly, we know that a portion of the light -which comes to us from the sun is already plane-polarised light, -which becomes in part circularly polarised, by reflection (according -to Jamin) at the surface of the sea, and then rotated in a -particular direction under the influence of terrestrial magnetism. -We only require to be assured that the relation between absorption -of light and chemical activity will continue to hold -good in the case of circularly polarised -light; that is to say <span class="xxpn" id="p419">{419}</span> -that the formation of some new substance or other, under the -influence of light so polarised, will proceed asymmetrically in -consonance with the asymmetry of the light itself; or conversely, -that the asymmetrically polarised light will tend to more rapid -decomposition of those molecules by which it is chiefly absorbed. -This latter proof is now said to be furnished by Byk<a class="afnanch" href="#fn420" id="fnanch420">420</a>, -who asserts -that certain tartrates become unsymmetrical under the continued -influence of the asymmetric rays. Here then we seem to have -an example, of a particular kind and in a particular instance, an -example limited but yet crucial (<i>if confirmed</i>), of an asymmetric -force, non-vital in its origin, which might conceivably be the -starting-point of that asymmetry which is characteristic of so -many organic products.</p> - -<p>The mysteries of organic chemistry are great, and the differences -between its processes or reactions as they are carried out in the -organism and in the laboratory are many<a class="afnanch" href="#fn421" id="fnanch421">421</a>. -The actions, catalytic -and other, which go on in the living cell are of extraordinary -complexity. But the contention that they are different in kind -from what we term ordinary chemical operations, or that in the -production of single asymmetric compounds there is actually to -be witnessed, as Pasteur maintained, a “prerogative of life,” -would seem to be no longer safely tenable. And furthermore, it -behoves us to remember that, even though failure continued to -attend all artificial attempts to originate the asymmetric or -optically active compounds which organic nature produces in -abundance, this would only prove that a certain <i>physical force</i>, or -mode of <i>physical action</i>, is at work among living things though -unknown elsewhere. It is a mode of action which we can easily -imagine, though the actual mechanism we cannot set agoing when -we please. And it follows that such a difference between living -matter and dead would carry us but a little way, for it would still -be confined strictly to the physical or mechanical plane.</p> - -<p>Our historic interest in the whole question is -increased by the <span class="xxpn" id="p420">{420}</span> -fact, or the great probability, that “the tenacity with which -Pasteur fought against the doctrine of spontaneous generation was -not unconnected with his belief that chemical compounds of one-sided -symmetry could not arise save under the influence of life<a class="afnanch" href="#fn422" id="fnanch422">422</a>.” -But the question whether spontaneous generation be a fact or not -does not depend upon theoretical considerations; our negative -response is based, and is so far soundly based, on repeated failures -to demonstrate its occurrence. Many a great law of physical -science, not excepting gravitation itself, has no higher claim on -our acceptance.</p> - -<hr class="hrblk"> - -<p>Let us return then, after this digression, to the general subject -of the forms assumed by certain chemical bodies when deposited -or precipitated within the organism, and to the question of how -far these forms may be artificially imitated or theoretically -explained.</p> - -<p>Mr George Rainey, of St Bartholomew’s Hospital (to whom -we have already referred), and Professor P. Harting, of Utrecht, -were the first to deal with this specific problem. Mr Rainey -published, between 1857 and 1861, a series of valuable and -thoughtful papers to shew that shell and bone and certain other -organic structures were formed “by a process of molecular -coalescence, demonstrable in certain artificially-formed products<a class="afnanch" href="#fn423" id="fnanch423">423</a>.” -Professor Harting, after thirty years of experimental work, -published in 1872 a paper, which has become classical, entitled -<i>Recherches de Morphologie Synthétique, sur la production artificielle -de quelques formations calcaires organiques</i>; his aim was to pave -the way for a “morphologie synthétique,” as Wöhler had laid the -foundations of a “chimie synthétique,” by his classical discovery -forty years before. <span class="xxpn" id="p421">{421}</span></p> - -<p>Rainey and Harting used similar methods, and these were -such as many other workers have continued to employ,—partly -with the direct object of explaining the genesis of organic forms -and partly as an integral part of what is now known as Colloid -Chemistry. The whole gist of the method was to bring some soluble -salt of lime, such as the chloride or nitrate, into solution within a -colloid medium, such as gum, gelatine or albumin; and then to -precipitate it out in the form of some insoluble compound, such -as the carbonate or oxalate. Harting found that, when he added -a little sodium or potassium carbonate to a concentrated solution -of calcium chloride in albumin, he got at first a gelatinous mass, -or “colloid precipitate”: which slowly transformed by the</p> - -<div class="dctr02" id="fig195"><div id="fig196"> -<img src="images/i421.png" width="703" height="350" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 195. Calcospherites, or concretions - of calcium carbonate, deposited in white of egg. (After - Harting.)</td> - <td></td> - <td>Fig. 196. A single calcospherite, with - central “nucleus,” and striated, iridescent border. (After - Harting.)</td></tr></table> -</div></div></div><!--dctr01--> - -<p class="pcontinue"> -appearance of tiny microscopic particles, at first motionless, but -afterwards as they grew larger shewing the typical Brownian -movement. So far, very much the same phenomena were witnessed -whether the solution were albuminous or not, and similar -appearances indeed had been witnessed and recorded by Gustav -Rose, so far back as 1837<a class="afnanch" href="#fn424" id="fnanch424">424</a>; -but in the later stages the presence -of albuminoid matter made a great difference. Now, after a few -days, the calcium carbonate was seen to be deposited in the form -of large rounded concretions, with a more or less distinct central -nucleus, and with a surrounding structure at -once radiate and <span class="xxpn" id="p422">{422}</span> -concentric; the presence of concentric zones or lamellae, alternately -dark and clear, was especially characteristic. These -round “calcospherites” shewed a tendency to aggregate together</p> - -<div class="dctr01" id="fig197"> -<img src="images/i422.png" width="800" height="338" alt=""> - <div class="dcaption">Fig. 197. Later stages in the same - experiment.</div></div> - -<p class="pcontinue"> -in layers, and then to assume polyhedral, or often regularly -hexagonal, outlines. In this latter condition they closely resemble</p> - -<div class="dctr01" id="fig198"> -<img src="images/i422b.png" width="800" height="317" alt=""> - <div class="dcaption">Fig. 198, A. Section of shell of Mya; B. - Section of hinge-tooth of do. (After Carpenter.)</div></div> - -<p class="pcontinue"> -the early stages of calcification in a molluscan (Fig. <a href="#fig198" title="go to Fig. 198">198</a>), or still -more in a crustacean shell<a class="afnanch" href="#fn425" id="fnanch425">425</a>; -while in their -isolated condition <span class="xxpn" id="p423">{423}</span> -they very closely resemble the little calcareous bodies in the -tissues of a trematode or a cestode worm, or in the oesophageal -glands of an earthworm<a class="afnanch" href="#fn426" id="fnanch426">426</a>.</p> - -<div class="dctr05" id="fig199"> -<img src="images/i423.png" width="448" height="408" alt=""> - <div class="pcaption">Fig. 199. Large irregular calcareous -concretions, or spicules, deposited in a piece of dead -cartilage, in presence of calcium phosphate. (After -Harting.)</div></div> - -<p>When the albumin was somewhat scanty, or when it was mixed -with gelatine, and especially when a -little phosphate of lime was <span class="xxpn" id="p424">{424}</span> -added to the mixture, the spheroidal globules tended to become -rough, by an outgrowth of spinous or digitiform projections; and -in some cases, but not without the presence of the phosphate, the -result was an irregularly shaped knobby spicule, precisely similar -to those which are characteristic of the Alcyonaria<a class="afnanch" href="#fn427" id="fnanch427">427</a>.</p> - -<div class="psmprnt3"> -<p>The rough spicules of the Alcyonaria are extraordinarily -variable in shape and size, as, looking at them from the -chemist’s or the physicist’s point of view, we should expect -them to be. Partly upon the form of these spicules, and partly -on the general form or mode of branching of the entire colony</p> - -<div class="dctr01" id="fig200"> -<img src="images/i424.png" width="800" height="521" alt=""> - <div class="dcaption">Fig. 200. - Additional illustrations of Alcyonarian spicules: - <i>Eunicea</i>. (After Studer.)</div></div> - -<p class="pcontinue"> -of polypes, a vast number of separate “species” have been -based by systematic zoologists. But it is now admitted that -even in specimens of a single species, from one and the same -locality, the spicules may vary immensely in shape and size: -and Professor Hickson declares (in a paper published while -these sheets are passing through the press) that after many -years of laborious work in striving to determine species of -these animal colonies, he feels “quite convinced that we have -been engaged in a more or less fruitless task<a class="afnanch" -href="#fn428" id="fnanch428">428</a>”.</p> - -<p>The formation of a tooth has very lately been shown -to be a phenomenon of the same order. That is to say, -“calcification in both dentine and enamel <span class="xxpn" -id="p425">{425}</span> is in great part a physical -phenomenon; the actual deposit in both tissues occurs in -the form of calcospherites, and the process in mammalian -tissue is identical in every point with the same process -occurring in lower organisms<a class="afnanch" href="#fn429" -id="fnanch429">429</a>.” The ossification of bone, we may be -sure, is in the same sense and to the same extent a physical -phenomenon.</p> -</div><!--psmprnt3--> - -<p>The typical structure of a calcospherite is no other than that -of a pearl, nor does it differ essentially from that of the otolith -of a mollusc or of a bony fish. (The otoliths, by the way, of the -elasmobranch fishes, like those of reptiles and birds, are not -developed after this fashion, but are true crystals of calc-spar.)</p> - -<p>Throughout these phenomena, the effect of surface-tension is -manifest. It is by surface-tension that ultra-microscopic particles -are brought together in the first floccular precipitate or coagulum;</p> - -<div class="dctr03" id="fig201"><div id="fig202"> -<img src="images/i425.png" width="610" height="252" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 201. A “crust” of close-packed -calcareous concretions, precipitated -at the surface of an albuminous -solution. (After Harting.)</td> - <td></td> - <td>Fig. 202. Aggregated calcospherites. -(After Harting.)</td></tr></table> -</div></div></div><!--dctr01--> - -<p class="pcontinue"> by the same agency, the coarser -particles are in turn agglutinated into visible lumps; -and the form of the calcospherites, whether it be that of -the solitary spheres or that assumed in various stages of -aggregation (e.g. Fig. <a href="#fig202" title="go to Fig. 202">202</a>)<a class="afnanch" href="#fn430" -id="fnanch430">430</a>, is likewise due to the same -agency.</p> - -<p>From the point of view of colloid chemistry the whole phenomenon -is very important and significant; and not the least -significant part is this tendency of the solidified deposits to assume -the form of “spherulites,” and other rounded contours. In the -phraseology of that science, we are dealing with a <i>two-phase</i> -system, which finally consists of solid particles in suspension in -a liquid (the former being styled the <i>disperse -phase</i>, the latter the <span class="xxpn" id="p426">{426}</span> -<i>dispersion medium</i>). In accordance with a rule first recognised -by Ostwald<a class="afnanch" href="#fn431" id="fnanch431">431</a>, -when a substance begins to separate out from a -solution, so making its appearance as a <i>new phase</i>, it always -makes its appearance first as a liquid<a class="afnanch" href="#fn432" id="fnanch432">432</a>. -Here is a case in point. -The minute quantities of material, on their way from a state of -solution to a state of “suspension,” pass through a liquid to a -solid form; and their temporary sojourn in the former leaves its -impress in the rounded contours which surface-tension brought -about while the little aggregate was still labile or fluid: while -coincidently with this surface-tension effect upon the surface, -crystallisation tended to take place throughout the little liquid -mass, or in such portion of it as had not yet consolidated and -crystallised.</p> - -<div class="dctr04" id="fig203"> -<img src="images/i426.png" width="531" height="242" alt=""> - <div class="dcaption">Fig. 203. (After Harting.)</div></div> - -<p>Where we have simple aggregates of two or three calcospherites, -the resulting figure is precisely that of so many contiguous soap-bubbles. -In other cases, composite forms result which are not -so easily explained, but which, if we could only account for them, -would be of very great interest to the biologist. For instance, -when smaller calcospheres seem, as it were, to invade the substance -of a larger one, we get curious conformations which in the closest -possible way resemble the outlines of certain of the Diatoms -(Fig. <a href="#fig203" title="go to Fig. 203">203</a>). Another very curious formation, which Harting calls -a “conostat,” is of frequent occurrence, and in it we see at least -a suggestion of analogy with the configuration which, in a protoplasmic -structure, we have spoken of as a -“collar-cell.” The <span class="xxpn" id="p427">{427}</span> -conostats, which are formed in the surface layer of the solution, -consist of a portion of a spheroidal calcospherite, whose upper -part is continued into a thin spheroidal collar, of somewhat larger -radius than the solid sphere; but the precise manner in which -the collar is formed, possibly around a bubble of gas, possibly -about a vortex-like diffusion-current<a class="afnanch" href="#fn433" id="fnanch433">433</a> -is not obvious.</p> - -<hr class="hrblk"> - -<p>Among these various phenomena, the concentric striation -observed in the calcospherite has acquired a special interest and -importance<a class="afnanch" href="#fn434" id="fnanch434">434</a>. -It is part of a phenomenon now widely known, and -recognised as an important factor in colloid chemistry, under the -name of “Liesegang’s Rings<a class="afnanch" href="#fn435" id="fnanch435">435</a>.”</p> - -<div class="dctr02" id="fig204"> -<img src="images/i427.png" width="704" height="265" alt=""> - <div class="dcaption">Fig. 204. - Conostats. (After Harting.)</div></div> - -<p>If we dissolve, for instance, a little bichromate of potash in -gelatine, pour it on to a glass plate, and after it is set place upon -it a drop of silver nitrate solution, there appears in the course -of a few hours the phenomenon of Liesegang’s rings. At first the -silver forms a central patch of abundant reddish brown chromate -precipitate; but around this, as the silver nitrate diffuses slowly -through the gelatine, the precipitate no longer comes down in -a continuous, uniform layer, but forms a series of zones, beautifully -regular, which alternate with clear interspaces of jelly, and which -stand farther and farther apart, in logarithmic ratio, as they -recede from the centre. For a discussion of -the <i>raison d’être</i> of <span class="xxpn" id="p428">{428}</span> -this phenomenon, still somewhat problematic, the student must -consult the text-books of physical and colloid chemistry<a class="afnanch" href="#fn436" id="fnanch436">436</a>.</p> - -<p>But, speaking very generally, we may say the appearance of -Liesegang’s rings is but a particular and striking case of a more -general phenomenon, namely the influence on crystallisation of -the presence of foreign bodies or “impurities,” represented in this -case by the “gel” or colloid matrix<a class="afnanch" href="#fn437" id="fnanch437">437</a>. -Faraday shewed long ago -that to the presence of slight impurities might be ascribed the -banded structure of ice, of banded quartz or agate, onyx, etc.; -and Quincke and Tomlinson have added to our scanty knowledge -of the same phenomenon<a class="afnanch" href="#fn438" id="fnanch438">438</a>.</p> - -<div class="dctr04" id="fig205"> -<img src="images/i428.png" width="528" height="324" alt=""> - <div class="dcaption">Fig. 205. Liesegang’s Rings. (After Leduc.)</div></div> - -<p>Besides the tendency to rhythmic action, as manifested in -Liesegang’s rings, the association of colloid matter with a crystalloid -in solution may lead to other well-marked effects. These, -according to Professor J. H. Bowman<a class="afnanch" href="#fn439" id="fnanch439">439</a>, -may be grouped somewhat -as follows: (1) total prevention of crystallisation; (2) suppression of -certain of the lines of crystalline growth; (3) extension of the crystal -to abnormal proportions, with a tendency for it to become a compound -crystal; (4) a curving or gyrating of the -crystal or its parts. <span class="xxpn" id="p429">{429}</span></p> - -<div class="dctr02" id="fig206"> -<img src="images/i429a.png" width="726" height="276" alt=""> - <div class="dcaption">Fig. 206. Relay-crystals - of common salt. (After Bowman.)</div></div> - -<p>For instance, it would seem that, if the supply of material to -the growing crystal be not forthcoming in sufficient quantity (as -may well happen in a colloid medium, for lack of convection-currents), -then growth will follow only the strongest lines of -crystallising force, and will be suppressed or partially suppressed -along other axes. The crystal will have a tendency to become -filiform, or “fibrous”; and the raphides of our plant-cells are -a case in point. Again, the long slender crystal so formed, pushing -its way into new material, may initiate a new centre of crystallisation: -we get the phenomenon known as a “relay,” along the</p> - -<div class="dright dwth-d" id="fig207"> -<img src="images/i429b.png" width="397" height="335" alt=""> - <div class="dcaption">Fig. 207. Wheel-like crystals in a -colloid. (After Bowman.)</div></div> - -<p class="pcontinue"> principal lines of force, -and sometimes along subordinate axes as well. This -phenomenon is illustrated in the accompanying figure of -crystallisation in a colloid medium of common salt; and it -may possibly be that we have here an explanation, or part -of an explanation, of the compound siliceous spicules of -the Hexactinellid sponges. Lastly, when the crystallising -force is nearly equalled by the resistance of the viscous -medium, the crystal takes the line of least resistance, -with very various results. One of these results would -seem to be a gyratory course, giving to the crystal a -curious wheel-like shape, as in Fig. <a href="#fig207" title="go to Fig. 207">207</a>; and other -results are the feathery, fern-like <span class="xxpn" -id="p430">{430}</span> or arborescent shapes so frequently -seen in microscopic crystallisation. <br class="brclrfix" -></p> - -<p>To return to Liesegang’s rings, the typical appearance of -concentric rings upon a gelatinous plate may be modified in -various experimental ways. For instance, our gelatinous medium -may be placed in a capillary tube immersed in a solution of the -precipitating salt, and in this case we shall obtain a vertical -succession of bands or zones regularly interspaced: the result being -very closely comparable to the banded pigmentation which we see -in the hair of a rabbit or a rat. In the ordinary plate preparation, -the free surface of the gelatine is under different conditions to the -lower layers and especially to the lowest layer in contact with -the glass; and therefore it often happens that we obtain a double -series of rings, one deep and the other superficial, which by -occasional blending or interlacing, may produce a netted pattern. -In some cases, as when only the inner surface of our capillary -tube is covered with a layer of gelatine, there is a tendency for -the deposit to take place in a continuous spiral line, rather than -in concentric and separate zones. By such means, according to -Küster<a class="afnanch" href="#fn440" id="fnanch440">440</a> -various forms of annular, spiral and reticulated thickenings -in the vascular tissue of plants may be closely imitated; and he -and certain other writers have of late been inclined to carry the -same chemico-physical phenomenon a very long way, in the -explanation of various banded, striped, and other rhythmically -successional types of structure or pigmentation. For example, -the striped pigmentation of the leaves in many plants (such as -<i>Eulalia japonica</i>), the striped or clouded colouring of many -feathers or of a cat’s skin, the patterns of many fishes, such for -instance as the brightly coloured tropical Chaetodonts and the like, -are all regarded by him as so many instances of “diffusion-figures” -closely related to the typical Liesegang phenomenon. Gebhardt -has made a particular study of the same subject in the case of -insects<a class="afnanch" href="#fn441" id="fnanch441">441</a>. -He declares, for instance, that the banded wings of -<i>Papilio podalirius</i> are precisely imitated in Liesegang’s experiments; -that the finer markings on the wings of the Goatmoth -(<i>Cossus ligniperda</i>) shew the double arrangement -of larger and of <span class="xxpn" id="p431">{431}</span> -smaller intermediate rhythms, likewise manifested in certain cases -of the same kind; that the alternate banding of the antennae -(for instance in <i>Sesia spheciformis</i>), a pigmentation not concurrent -with the segmented structure of the antenna, is explicable in the -same way; and that the “ocelli,” for instance of the Emperor -moth, are typical illustrations of the common concentric type. -Darwin’s well-known disquisition<a class="afnanch" href="#fn442" id="fnanch442">442</a> -on the ocellar pattern of the -feathers of the Argus Pheasant, as a result of sexual selection, -will occur to the reader’s mind, in striking contrast to this or -to any other direct physical explanation<a class="afnanch" href="#fn443" id="fnanch443">443</a>. -To turn from the distribution -of pigment to more deeply seated structural characters, -Leduc has shewn how, for instance, the laminar structure of the -cornea or the lens is again, apparently, a similar phenomenon. -In the lens of the fish’s eye, we have a very curious appearance, -the consecutive lamellae being roughened or notched by close-set, -interlocking sinuosities; and precisely the same appearance, save -that it is not quite so regular, is presented in one of Küster’s -figures as the effect of precipitating a little sodium phosphate in -a gelatinous medium. Biedermann has studied, from the same -point of view, the structure and development of the molluscan -shell, the problem which Rainey had first attacked more than -fifty years before<a class="afnanch" href="#fn444" id="fnanch444">444</a>; -and Liesegang himself has applied his results -to the formation of pearls, and to the -development of bone<a class="afnanch" href="#fn445" id="fnanch445">445</a>. -<span class="xxpn" id="p432">{432}</span></p> - -<p>Among all the many cases where this phenomenon of Liesegang’s -comes to the naturalist’s aid in explanation of rhythmic or -zonary configurations in organic forms, it has a special interest -where the presence of concentric zones or rings appears, at -first sight, as a sure and certain sign of periodicity of growth, -depending on the seasons, and capable therefore of serving as -a mark and record of the creature’s age. This is the case, for -instance, with the scales, bones and otoliths of fishes; and a -kindred phenomena in starch-grains has given rise, in like manner, -to the belief that they indicate a diurnal and nocturnal periodicity -of activity and rest<a class="afnanch" href="#fn446" id="fnanch446">446</a>.</p> - -<div class="dleft dwth-j" id="fig208"> -<img src="images/i432a.png" width="144" height="180" alt=""> - <div class="dcaption">Fig. 208.</div></div> - -<p>That this is actually the case in growing starch-grains is -generally believed, on the authority of Meyer<a class="afnanch" href="#fn447" id="fnanch447">447</a>; -but while under -certain circumstances a marked alternation of growing and resting -periods may occur, and may leave its impress on the structure -of the grain, there is now great reason to believe that, apart from -such external influences, the internal phenomena of -diffusion may, just as in the typical Liesegang -experiment, produce the well-known concentric -rings. The spherocrystals of inulin, in like manner, -shew, like the “calcospherites” of Harting (Fig. -<a href="#fig208" title="go to Fig. 208">208</a>), a concentric structure which in all likelihood -has had no causative impulse save from within.<br class="brclrfix"></p> - -<div class="dleft dwth-d" id="fig209"> -<img src="images/i432b.png" width="384" height="263" alt=""> - <div class="dcaption">Fig. 209. Otoliths of Plaice, showing -four zones or “age-rings.” (After -Wallace.)</div></div> - -<p>The striation, or concentric lamellation, of the scales -and otoliths of fishes has been much employed of recent -years as a trustworthy and unmistakeable mark of the fish’s -age. There are difficulties in the way of accepting this -hypothesis, not the least of which is the fact that the -otolith-zones, for instance, are extremely well marked -even in the case of some fishes which spend their lives -in deep water, <span class="xxpn" id="p433">{433}</span> -where the temperature and other physical conditions shew -little or no appreciable fluctuation with the seasons -of the year. There are, on the other hand, phenomena -which seem strongly confirmatory of the hypothesis: for -instance the fact (if it be fully established) that in -such a fish as the cod, zones of growth, <i>identical in -number</i>, are found both on the scales and in the otoliths<a -class="afnanch" href="#fn448" id="fnanch448">448</a>. The -subject has become a much debated one, and this is not the -place for its discussion; but it is at least obvious, with -the Liesegang phenomenon in view, that we have no right -to <i>assume</i> that an appearance of rhythm and periodicity -in structure and growth is necessarily bound up with, and -indubitably brought about by, a periodic recurrence of -particular <i>external</i> conditions.</p> - -<p>But while in the Liesegang phenomenon we have rhythmic -precipitation which depends only on forces intrinsic to the -system, and is independent of any corresponding rhythmic -changes in temperature or other external conditions, we -have not far to seek for instances of chemico-physical -phenomena where rhythmic alternations of appearance or -structure are produced in close relation to periodic -fluctuations of temperature. A well-known instance is that -of the Stassfurt deposits, where the rock-salt alternates -regularly with thin layers of “anhydrite,” or (in another -series of beds) with “polyhalite<a class="afnanch" -href="#fn449" id="fnanch449">449</a>”: and where these -zones are commonly regarded as marking years, and their -alternate bands as having been formed in connection with -the seasons. A discussion, however, of this remarkable and -significant phenomenon, and of how the chemist explains it, -by help of the “phase-rule,” in connection with temperature -conditions, would lead us far beyond our scope<a -class="afnanch" href="#fn450" id="fnanch450">450</a>. <br -class="brclrfix"></p> - -<hr class="hrblk"> - -<p>We now see that the methods by which we attempt to study -the chemical or chemico-physical phenomena which accompany -the development of an inorganic concretion -or spicule within the <span class="xxpn" id="p434">{434}</span> -body of an organism soon introduce us to a multitude of kindred -phenomena, of which our knowledge is still scanty, and which we -must not attempt to discuss at greater length. As regards our -main point, namely the formation of spicules and other elementary -skeletal forms, we have seen that certain of them may be safely -ascribed to simple precipitation or crystallisation of inorganic -materials, in ways more or less modified by the presence of -albuminous or other colloid substances. The effect of these -latter is found to be much greater in the case of some crystallisable -bodies than in others. For instance, Harting, and Rainey also, -found as a rule that calcium oxalate was much less affected by -a colloid medium than was calcium carbonate; it shewed in -their hands no tendency to form rounded concretions or “calcospherites” -in presence of a colloid, but continued to crystallise, -either normally, or with a tendency to form needles or raphides. -It is doubtless for this reason that, as we have seen, <i>crystals</i> of -calcium oxalate are so common in the tissues of plants, while -those of other calcium salts are rare. But true calcospherites, -or spherocrystals, of the oxalate are occasionally found, for -instance in certain Cacti, and Bütschli<a class="afnanch" href="#fn451" id="fnanch451">451</a> -has succeeded in making -them artificially in Harting’s usual way, that is to say by crystallisation -in a colloid medium.</p> - -<p>There link on to these latter observations, and to the statement -already quoted that calcareous deposits are associated with the -dead products rather than with the living cells of the organism, -certain very interesting facts in regard to the <i>solubility</i> of salts -in colloid media, which have been made known to us of late, and -which go far to account for the presence (apart from the form) -of calcareous precipitates within the organism<a class="afnanch" href="#fn452" id="fnanch452">452</a>. -It has been -shewn, in the first place, that the presence of albumin has a notable -effect on the solubility in a watery solution of calcium salts, -increasing the solubility of the phosphate in a marked degree, -and that of the carbonate in still greater -proportion; but the <span class="xxpn" id="p435">{435}</span> -sulphate is only very little more soluble in presence of albumin -than in pure water, and the rarity of its occurrence within the -organism is so far accounted for. On the other hand, the bodies -derived from the breaking down of the albumins, their “catabolic” -products, such as the peptones, etc., dissolve the calcium salts to -a much less degree than albumin itself; and in the case of the -phosphate, its solubility in them is scarcely greater than in water. -The probability is, therefore, that the actual precipitation of the -calcium salts is not due to the direct action of carbonic acid, etc. -on a more soluble salt (as was at one time believed); but to catabolic -changes in the proteids of the organism, which tend to throw -down the salts already formed, which had remained hitherto in -albuminous solution. The very slight solubility of calcium phosphate -under such circumstances accounts for its predominance -in, for instance, mammalian bone<a class="afnanch" href="#fn453" id="fnanch453">453</a>; -and wherever, in short, the -supply of this salt has been available to the organism.</p> - -<p>To sum up, we see that, whether from food or from sea-water, -calcium sulphate will tend to pass but little into solution in the -albuminoid substances of the body: calcium carbonate will enter -more freely, but a considerable part of it will tend to remain in -solution: while calcium phosphate will pass into solution in -considerable amount, but will be almost wholly precipitated -again, as the albumin becomes broken down in the normal process -of metabolism.</p> - -<p>We have still to wait for a similar and equally illuminating -study of the solution and precipitation of <i>silica</i>, in presence of -organic colloids.</p> - -<hr class="hrblk"> - -<p>From the comparatively small group of inorganic formations -which, arising within living organisms, owe their form solely to -precipitation or to crystallisation, that is to say to chemical or other -molecular forces, we shall presently pass to that other and larger -group which appear to be conformed in direct relation to the forms -and the arrangement of the cells or other -protoplasmic elements<a class="afnanch" href="#fn454" id="fnanch454">454</a>. -<span class="xxpn" id="p436">{436}</span> -The two principles of conformation are both illustrated in the -spicular skeletons of the Sponges.</p> - -<div class="dctr03" id="fig210"> -<img src="images/i436.png" width="606" height="523" alt=""> - <div class="pcaption">Fig. 210. Close-packed - calcospherites, or so-called “spicules,” of Astrosclera. - (After Lister.)</div></div> - -<p>In a considerable number, but withal a minority of cases, the -form of the sponge-spicule may be deemed sufficiently explained -on the lines of Harting’s and Rainey’s experiments, that is to say -as the direct result of chemical or physical phenomena associated -with the deposition of lime or of silica in presence of colloids<a class="afnanch" href="#fn455" id="fnanch455">455</a>. -This is the case, for instance, with various small spicules of a -globular or spheroidal form, formed of amorphous silica, concentrically -striated within, and often developing irregular knobs -or tiny tubercles over their surfaces. In the aberrant sponge -<i>Astrosclera</i><a class="afnanch" href="#fn456" id="fnanch456">456</a>, -we have, to begin with, rounded, striated discs or -globules, which in like manner are nothing more -or less than the <span class="xxpn" id="p437">{437}</span> -“calcospherites” of Harting’s experiments; and as these grow -they become closely aggregated together (Fig. <a href="#fig210" title="go to Fig. 210">210</a>), and assume an -angular, polyhedral form, once more in complete accordance with -the results of experiment<a class="afnanch" href="#fn457" id="fnanch457">457</a>. -Again, in many Monaxonid sponges, -we have irregularly shaped, or branched spicules, roughened or -tuberculated by secondary superficial deposits, and reminding one -of the spicules of some Alcyonaria. These also must be looked -upon as the simple result of chemical deposition, the form of the -deposit being somewhat modified in conformity with the surrounding -tissues, just as in the simple experiment the form of the concretionary -precipitate is affected by the heterogeneity, visible or -invisible, of the matrix. Lastly, the simple needles of amorphous -silica, which constitute one of the commonest types of spicule, -call for little in the way of explanation; they are accretions or -deposits about a linear axis, or fine thread of organic material, -just as the ordinary rounded calcospherite is deposited about -some minute point or centre of crystallisation, and as ordinary -crystallisation is often started by a particle of atmospheric dust; -in some cases they also, like the others, are apt to be roughened -by more irregular secondary deposits, which probably, as in -Harting’s experiments, appear in this irregular form when the -supply of material has become relatively scanty.</p> - -<hr class="hrblk"> - -<p>Our few foregoing examples, diverse as they are in look and -kind and ranging from the spicules of Astrosclera or Alcyonium -to the otoliths of a fish, seem all to have their free origin in some -larger or smaller fluid-containing space, or cavity of the body: -pretty much as Harting’s calcospheres made their appearance in -the albuminous content of a dish. But we now come at last to -a much larger class of spicular and skeletal structures, for whose -regular and often complex forms some other explanation than the -intrinsic forces of crystallisation or molecular adhesion is manifestly -necessary. As we enter on this subject, which is certainly -no small or easy one, it may conduce to simplicity, -and to brevity, <span class="xxpn" id="p438">{438}</span> -if we try to make a rough classification, by way of forecast, of -the chief conditions which we are likely to meet with.</p> - -<p>Just as we look upon animals as constituted, some of a vast -number of cells, and others of a single cell or of a very few, and -just as the shape of the former has no longer a visible relation to -the individual shapes of its constituent cells, while in the latter -it is cell-form which dominates or is actually equivalent to the -form of the organism, so shall we find it to be, with more or less -exact analogy, in the case of the skeleton. For example, our own -skeleton consists of bones, in the formation of each of which a -vast number of minute living cellular elements are necessarily -concerned; but the form and even the arrangement of these -bone-forming cells or corpuscles are monotonously simple, and we -cannot find in these a physical explanation of the outward and -visible configuration of the bone. It is as part of a far larger -field of force,—in which we must consider gravity, the action of -various muscles, the compressions, tensions and bending moments -due to variously distributed loads, the whole interaction of a very -complex mechanical system,—that we must explain (if we are to -explain at all) the configuration of a bone.</p> - -<p>In contrast to these massive skeletons, or constituents of a -skeleton, we have other skeletal elements whose whole magnitude, -or whose magnitude in some dimension or another, is commensurate -with the magnitude of a single living cell, or (as comes to very -much the same thing) is comparable to the range of action of the -molecular forces. Such is the case with the ordinary spicules of -a sponge, with the delicate skeleton of a Radiolarian, or with the -denser and robuster shells of the Foraminifera. The effect of -<i>scale</i>, then, of which we had so much to say in our introductory -chapter on Magnitude, is bound to be apparent in the study of -skeletal fabrics, and to lead to essential differences between the -big and the little, the massive and the minute, in regard to their -controlling forces and their resultant forms. And if all this be -so, and if the range of action of the molecular forces be in truth -the important and fundamental thing, then we may somewhat -extend our statement of the case, and include in it not only -association with the living cellular elements of the body, but also -association with any bubbles, drops, vacuoles -or vesicles which <span class="xxpn" id="p439">{439}</span> -may be comprised within the bounds of the organism, and which -are (as their names and characters connote) of the order of -magnitude of which we are speaking.</p> - -<p>Proceeding a little farther in our classification, we may conceive -each little skeletal element to be associated, in one case, with -a single cell or vesicle, and in another with a cluster or “system” -of consociated cells. In either case there are various possibilities. -For instance, the calcified or other skeletal material may tend -to overspread the entire outer surface of the cell or cluster of cells, -and so tend accordingly to assume some configuration comparable -to that of a fluid drop or of an aggregation of drops; this, in brief, -is the gist and essence of our story of the foraminiferal shell. -Another common, but very different condition will arise if, in the -case of the cell-aggregates, the skeletal material tends to accumulate -in the interstices <i>between</i> the cells, in the partition-walls which -separate them, or in the still more restricted distribution indicated -by the <i>lines</i> of junction between these partition-walls. Conditions -such as these will go a very long way to help us in our understanding -of many sponge-spicules and of an immense variety of -radiolarian skeletons. And lastly (for the present), there is a -possible and very interesting case of a skeletal element associated -with the surface of a cell, not so as to cover it like a shell, but -only so as to pursue a course of its own within it, and subject to -the restraints imposed by such confinement to a curved and -limited surface. With this curious condition we shall deal -immediately.</p> - -<p>This preliminary and much simplified classification of skeletal -forms (as is evident enough) does not pretend to completeness. -It leaves out of account some kinds of conformation and configuration -with which we shall attempt to deal, and others which -we must perforce omit. But nevertheless it may help to clear -or to mark our way towards the subjects which this chapter has -to consider, and the conditions by which they are at least partially -defined.</p> - -<hr class="hrblk"> - -<p>Among the several possible, or conceivable, types of microscopic -skeletons let us choose, to begin with, the case of a spicule, more -or less simply linear as far as its <i>intrinsic</i> powers -of growth are <span class="xxpn" id="p440">{440}</span> -concerned, but which owes its now somewhat complicated form -to a restraint imposed by the individual cell to which it is confined, -and within whose bounds it is generated. The conception of a -spicule developed under such conditions we owe to a distinguished -physicist, the late Professor G. F. FitzGerald.</p> - -<p>Many years ago, Sollas pointed out that if a spicule begin to -grow in some particular way, presumably under the control or -constraint imposed by the organism, it continues to grow by -further chemical deposition in the same form or direction even -after it has got beyond the boundaries of the organism or its -cells. This phenomenon is what we see in, and this imperfect -explanation goes so far to account for, the continued growth in -straight lines of the long calcareous spines of Globigerina or -Hastigerina, or the similarly radiating but siliceous spicules of -many Radiolaria. In physical language, if our crystalline -structure has once begun to be laid down in a definite orientation, -further additions tend to accrue in a like regular fashion and in -an identical direction; and this corresponds to the phenomenon -of so-called “orientirte Adsorption,” as described by Lehmann.</p> - -<p>In Globigerina or in Acanthocystis the long needles grow out -freely into the surrounding medium, with nothing to impede their -rectilinear growth and their approximately radiate distribution. -But let us consider some simple cases to illustrate the forms which -a spicule will tend to assume when, striving (as it were) to grow -straight, it comes under the influence of some simple and constant -restraint or compulsion.</p> - -<p>If we take any two points on some curved surface, such as -that of a sphere or an ellipsoid, and imagine a string stretched -between them, we obtain what is known in mathematics as a -“geodetic” curve. It is the shortest line which can be traced -between the two points, upon the surface itself; and the most -familiar of all cases, from which the name is derived, is that curve -upon the earth’s surface which the navigator learns to follow in -the practice of “great-circle sailing.” Where the surface is -spherical, the geodetic is always literally a “great circle,” a circle, -that is to say, whose centre is the centre of the sphere. If instead -of a sphere we be dealing with an ellipsoid, the geodetic becomes -a variable figure, according to the position of -our two points. <span class="xxpn" id="p441">{441}</span> -For obviously, if they lie in a line perpendicular to the long axis -of the ellipsoid, the geodetic which connects them is a circle, also -perpendicular to that axis; and if they lie in a line parallel to -the axis, their geodetic is a portion of that ellipse about which -the whole figure is a solid of revolution. But if our two points -lie, relatively to one another, in any other direction, then their -geodetic is part of a spiral curve in space, winding over the surface -of the ellipsoid.</p> - -<p>To say, as we have done, that the geodetic is the shortest line -between two points upon the surface, is as much as to say that -it is a <i>projection</i> of some particular straight line upon the surface -in question; and it follows that, if any linear body be confined -to that surface, while retaining a tendency to grow by successive -increments always (save only for its confinement to that surface) -in a straight line, the resultant form which it will assume will be -that of a geodetic. In mathematical language, it is a property -of a geodetic that the plane of any two consecutive elements is -a plane perpendicular to that in which the geodetic lies; or, in -simpler words, any two consecutive elements lie in a straight line -<i>in the plane of the surface</i>, and only diverge from a straight line -in space by the actual curvature of the surface to which they are -restrained.</p> - -<p>Let us now imagine a spicule, whose natural tendency is to -grow into a straight linear element, either by reason of its own -molecular anisotropy, or because it is deposited about a thread-like -axis; and let us suppose that it is confined either within a -cell-wall or in adhesion thereto; it at once follows that its line -of growth will be simply a geodetic to the surface of the cell. -And if the cell be an imperfect sphere, or a more or less regular -ellipsoid, the spicule will tend to grow into one or other of three -forms: either a plane curve of circular arc; or, more commonly, -a plane curve which is a portion of an ellipse; or, most commonly -of all, a curve which is a portion of a spiral in space. In the -latter case, the number of turns of the spiral will depend, not only -on the length of the spicule, but on the relative dimensions of -the ellipsoidal cell, as well as upon the angle by which the spicule -is inclined to the ellipsoid axes; but a very common case will -probably be that in which the spicule looks at first -sight to be <span class="xxpn" id="p442">{442}</span> -a plane <span class="nowrap"><em class="emltr">C</em>-shaped</span> -figure, but is discovered, on more careful inspection, -to lie not in one plane but in a more complicated spiral twist.</p> - -<div class="dctr01" id="fig211"> -<img src="images/i442a.png" width="800" height="257" alt=""> - <div class="dcaption">Fig. 211. Sponge and Holothurian spicules.</div></div> - -<p class="pcontinue">This investigation includes a series -of forms which are abundantly represented among actual -sponge-spicules, as illustrated in</p> - -<div class="dleft dwth-i" id="fig212"> -<img src="images/i442b.png" width="175" height="185" alt=""> - <div class="dcaption">Fig. 212.</div></div> - -<p class="pcontinue">Figs. <a href="#fig211" title="go to Fig. 211">211</a> and 212. If the spicule -be not restricted to linear growth, but have a tendency -to expand, or to branch out from a main axis, we shall -obtain a series of more complex figures, all related to -the geodetic system of curves. A very simple case will -arise where the spicule occupies, in the first instance, -the axis of the containing cell, and then, on reaching its -boundary, tends to branch or spread outwards. We shall now -get various figures, in some -of which the spicule will appear as an axis -expanding into a disc or wheel at either -end; and in other cases, the terminal disc -<br class="brclrfix"></p> - -<div class="dright dwth-g" id="fig213"> -<img src="images/i442c.png" width="241" height="347" alt=""> - <div class="dcaption">Fig. 213. An “amphidisc” - of Hyalonema.</div></div> - -<p class="pcontinue"> -will be replaced, or represented, by a series -of rays or spokes, with a reflex curvature, -corresponding to the spherical or ellipsoid -curvature of the surface of the cell. Such -spicules as these are again exceedingly -common among various sponges (Fig. <a href="#fig213" title="go to Fig. 213">213</a>).</p> - -<p>Furthermore, if these mechanical methods -of conformation, and others like to these, -be the true cause of the shapes which the -spicules assume, it is plain that the production -of these spicular shapes is not a specific function of -sponges or of any particular sponge, but that -we should expect <span class="xxpn" id="p443">{443}</span> -the same or very similar phenomena to occur in other organisms, -wherever the conditions of inorganic secretion within closed cells -was very much the same. As a matter of fact, in the group of -Holothuroidea, where the formation of intracellular spicules is a -characteristic feature of the group, all the principal types of -conformation which we have just described can be closely -paralleled. Indeed in many cases, the forms of the Holothurian -spicules are identical and indistinguishable from those of the -sponges<a class="afnanch" href="#fn458" id="fnanch458">458</a>. -But the Holothurian spicules are composed of calcium -carbonate while those which we have just described in the case -of sponges are usually, if not always, siliceous: this being just -another proof of the fact that in such cases the form of the -spicule is not due to its chemical nature or molecular structure, -but to the external forces to which, during its growth, the -spicule is submitted.<br class="brclrfix"></p> - -<hr class="hrblk"> - -<p>So much for that comparatively limited class of sponge-spicules -whose forms seem capable of explanation on the hypothesis -that they are developed within, or under the restraint imposed by, -the surface of a cell or vesicle. Such spicules are usually of small -size, as well as of comparatively simple form; and they are greatly -outstripped in number, in size, and in supposed importance as -guides to zoological classification, by another class of spicules. -This new class includes such as we have supposed to be capable -of explanation on the assumption that they develop in association -(of some sort or another) with the <i>lines of junction</i> of contiguous -cells. They include the triradiate spicules of the calcareous -sponges, the quadriradiate or “tetractinellid” spicules which occur -in the same group, but more characteristically in certain siliceous -sponges known as the Tetractinellidae, and lastly perhaps (though -these last are admittedly somewhat harder to understand) the -six-rayed spicules of the Hexactinellids.</p> - -<p>The spicules of the calcareous sponges are commonly -triradiate, and the three radii are usually inclined to one -another at equal, or nearly equal angles; in certain cases, -two of the three rays are nearly in a straight line, and at -right angles to the <span class="xxpn" id="p444">{444}</span> -third<a class="afnanch" href="#fn459" id="fnanch459">459</a>. -They are seldom in a plane, but are usually inclined to -one another in a solid, trihedral angle, not easy of precise measurement -under the microscope. The three rays are very often -supplemented by a fourth, which is set tetrahedrally, making, that -is to say, coequal angles with the other three. The calcareous -spicule consists mainly of carbonate of lime, in the form of calcite, -with (according to von Ebner) some admixture of soda and -magnesia, of sulphates and of water. According to the same -writer (but the fact, though it would seem easy to test, is still -disputed) there is no organic matter in the spicule, either in the -form of an axial filament or otherwise, and the appearance of -stratification, often simulating the presence of an axial fibre, is -due to “mixed crystallisation” of the various constituents. The -spicule is a true crystal, and therefore its existence and its form -are <i>primarily</i> due to the molecular forces of crystallisation; moreover -it is a single crystal and not a group of crystals, as is at once -seen by its behaviour in polarised light. But its axes are not -crystalline axes, and its form neither agrees with, nor in any way -resembles, any one of the many polymorphic forms in which -calcite is capable of crystallising. It is as though it were carved -out of a solid crystal; it is, in fact, a crystal under restraint, -a crystal growing, as it were, in an artificial mould; and this -mould is constituted by the surrounding cells, or structural -vesicles of the sponge.</p> - -<p>We have already studied in an elementary way, but amply -for our present purpose, the manner in which three or more cells, -or bubbles, tend to meet together under the influence of surface-tension, -and also the outwardly similar phenomena which may be -brought about by a uniform distribution of mechanical pressure. -We have seen that when we confine ourselves to a plane assemblage -of such bodies, we find them meeting one another in threes; that -in a section or plane projection of such an assemblage we see the -partition-walls meeting one another at equal angles of 120°; that -when the bodies are uniform in size, the partitions are straight -lines, which combine to form regular hexagons; -and that when <span class="xxpn" id="p445">{445}</span> -the bodies are unequal in size, the partitions are curved, and -combine to form other and less regular polygons. It is plain, -accordingly, that in any flattened or stratified assemblage of such -cells, a solidified skeletal deposit which originates or accumulates -either between the cells or within the thickness of their mutual -partitions, will tend to take the form of triradiate bodies, whose -rays (in a typical case) will be set at equal angles of 120° (Fig. <a href="#fig214" title="go to Fig. 214">214</a>, <i>F</i>). -And this latter condition of equality will be open to modification -in various ways. It will be</p> - -<div class="dctr03" id="fig214"> -<img src="images/i445.png" width="607" height="637" alt=""> - <div class="dcaption">Fig. 214. Spicules of Grantia and - other calcareous sponges. (After Haeckel.)</div></div> - -<p class="pcontinue"> -modified by any inequality in the -specific tensions of adjacent cells; as a special case, it will be apt -to be greatly modified at the surface of the system, where a spicule -happens to be formed in a plane perpendicular to the cell-layer, -so that one of its three rays lies between two adjacent cells and -the other two are associated with the surface of contact between -the cells and the surrounding medium; in such a case (as in the -cases considered in connection with the forms of -the cells themselves <span class="xxpn" id="p446">{446}</span> -on p. <a href="#p314" title="go to pg. 314">314</a>), we shall tend to obtain a spicule with two equal angles -and one unequal (Fig. <a href="#fig214" title="go to Fig. 214">214</a>, <i>A</i>, <i>C</i>). In the last case, the two outer, -or superficial rays, will tend to be markedly curved. Again, the -equiangular condition will be departed from, and more or less -curvature will be imparted to the rays, wherever the cells of the -system cease to be uniform in size, and when the hexagonal -symmetry of the system is lost accordingly. Lastly, although we -speak of the rays as meeting at certain definite angles, this statement -applies to their <i>axes</i>, rather than to the rays themselves. -For, if the triradiate spicule be developed in the <i>interspace</i> between -three juxtaposed cells, it is obvious that its sides will tend to be -concave, for the interspace between our three contiguous equal -circles is an equilateral, curvilinear triangle; and even if our -spicule be deposited, not in the space between our three cells, -but in the thickness of the intervening wall, then we may recollect -(from p. <a href="#p297" title="go to pg. 297">297</a>) that the several partitions never actually meet at -sharp angles, but the angle of contact is always bridged over by -a small accumulation of material (varying in amount according -to its fluidity) whose boundary takes the form of a circular arc, -and which constitutes the “bourrelet” of Plateau.</p> - -<p>In any sample of the triradiate spicules of Grantia, or in any -series of careful drawings, such as those of Haeckel among others, -we shall find that all these various configurations are precisely -and completely illustrated.</p> - -<p>The tetrahedral, or rather tetractinellid, spicule needs no -explanation in detail (Fig. <a href="#fig214" title="go to Fig. 214">214</a>, <i>D</i>, <i>E</i>). For just as a triradiate -spicule corresponds to the case of three cells in mutual contact, -so does the four-rayed spicule to that of a solid aggregate of four -cells: these latter tending to meet one another in a tetrahedral -system, shewing four edges, at each of which four surfaces meet, -the edges being inclined to one another at equal angles of about -109°. And even in the case of a single layer, or superficial layer, -of cells, if the skeleton originate in connection with all the edges -of mutual contact, we shall, in complete and typical cases, have -a four-rayed spicule, of which one straight limb will correspond -to the line of junction between the three cells, and the other three -limbs (which will then be curved limbs) will correspond to the edges -where two cells meet one another on the surface -of the system. <span class="xxpn" id="p447">{447}</span></p> - -<p>But if such a physical explanation of the forms of our spicules -is to be accepted, we must seek at once for some physical agency -by which we may explain the presence of the solid material just -at the junctions or interfaces of the cells, and for the forces by -which it is confined to, and moulded to the form of, these intercellular -or interfacial contacts. It is to Dreyer that we chiefly -owe the physical or mechanical theory of spicular conformation -which I have just described,—a theory which ultimately rests -on the form assumed, under surface-tension, by an aggregation -of cells or vesicles. But this fundamental point being granted, -we have still several possible alternatives by which to explain the -details of the phenomenon.</p> - -<p>Dreyer, if I understand him aright, was content to assume that -the solid material, secreted or excreted by the organism, accumulated -in the interstices between the cells, and was there subjected -to mechanical pressure or constraint as the cells got more and -more crowded together by their own growth and that of the -system generally. As far as the general form of the spicules goes, -such explanation is not inadequate, though under it we may have -to renounce some of our assumptions as to what takes place at -the outer surface of the system.</p> - -<p>But in all (or most) cases where, but a few years ago, the -concepts of secretion or excretion seemed precise enough, we are -now-a-days inclined to turn to the phenomenon of adsorption as -a further stage towards the elucidation of our facts. Here we -have a case in point. In the tissues of our sponge, wherever two -cells meet, there we have a definite <i>surface</i> of contact, and there -accordingly we have a manifestation of surface-energy; and the -concentration of surface-energy will tend to be a maximum at -the <i>lines</i> or edges whereby the three, or four, such surfaces are -conjoined. Of the micro-chemistry of the sponge-cells our -ignorance is great; but (without venturing on any hypothesis -involving the chemical details of the process) we may safely assert -that there is an inherent probability that certain substances will -tend to be concentrated and ultimately deposited just in these lines -of intercellular contact and conjunction. In other words, adsorptive -concentration, under osmotic pressure, at and in the surface-film -which constitutes the mutual -boundary between contiguous <span class="xxpn" id="p448">{448}</span> -cells, emerges as an alternative (and, as it seems to me, a highly -preferable alternative) to Dreyer’s conception of an accumulation -under mechanical pressure in the vacant spaces left between one -cell and another.</p> - -<p>But a purely chemical, or purely molecular adsorption, is not -the only form of the hypothesis on which we may rely. For -from the purely physical point of view, angles and edges of contact -between adjacent cells will be <i>loci</i> in the field of distribution of -surface-energy, and any material particles whatsoever will tend -to undergo a diminution of freedom on entering one of those -boundary regions. In a very simple case, let us imagine a couple -of soap bubbles in contact with one another. Over the surface -of each bubble there glide in every direction, as usual, a multitude -of tiny bubbles and droplets; but as soon as these find their way -into the groove or re-entrant angle between the two bubbles, -there their freedom of movement is so far restrained, and out of -that groove they have little or no tendency to emerge. A cognate -phenomenon is to be witnessed in microscopic sections of steel or -other metals. Here, amid the “crystalline” structure of the -metal (where in cooling its imperfectly homogeneous material has -developed a cellular structure, shewing (in section) hexagonal or -polygonal contours), we can easily observe, as Professor Peddie -has shewn me, that the little particles of graphite and other -foreign bodies common in the matrix, have tended to aggregate -themselves in the walls and at the angles of the polygonal -cells—this being a direct result of the diminished freedom -which the particles undergo on entering one of these boundary -regions<a class="afnanch" href="#fn460" id="fnanch460">460</a>.</p> - -<p>It is by a combination of these two principles, chemical adsorption -on the one hand, and physical quasi-adsorption or concentration -of grosser particles on the other, that I conceive the substance -of the sponge-spicule to be concentrated and aggregated at the -cell boundaries; and the forms of the triradiate and tetractinellid -spicules are in precise conformity with this hypothesis. A few -general matters, and a few particular cases, -remain to be considered.</p> - -<p>It matters little or not at all, for the phenomenon in -question, <span class="xxpn" id="p449">{449}</span> -what is the histological nature or “grade” of the vesicular structures -on which it depends. In some cases (apart from sponges), they -may be no more than the little alveoli of the intracellular protoplasmic -network, and this would seem to be the case at least in -one known case, that of the protozoan <i>Entosolenia aspera</i>, in which, -within the vesicular protoplasm of the single cell, Möbius has -described tiny spicules in the shape of little tetrahedra with -concave sides. It is probably also the case in the small beginnings -of the Echinoderm spicules, which are likewise intracellular, and -are of similar shape. In the case of our sponges we have many -varying conditions, which we need not attempt to examine in -detail. In some cases there is evidence for believing that the -spicule is formed at the boundaries of true cells or histological -units. But in the case of the larger triradiate or tetractinellid -spicules of the sponge-body, they far surpass in size the actual -“cells”; we find them lying, regularly and symmetrically -arranged, between the “pore-canals” or “ciliated chambers,” -and it is in conformity with the shape and arrangement of these -rounded or spheroidal structures that their shape is assumed.</p> - -<p>Again, it is not necessarily at variance with our hypothesis -to find that, in the adult sponge, the larger spicules may greatly -outgrow the bounds not only of actual cells but also of the -ciliated chambers, and may even appear to project freely from the -surface of the sponge. For we have already seen that the spicule -is capable of growing, without marked change of form, by further -deposition, or crystallisation, of layer upon layer of calcareous -molecules, even in an artificial solution; and we are entitled to -believe that the same process may be carried on in the tissues of -the sponge, without greatly altering the symmetry of the spicule, -long after it has established its characteristic form of a system of -slender trihedral or tetrahedral rays.</p> - -<p>Neither is it of great importance to our hypothesis whether -the rayed spicule necessarily arises as a single structure, or does -so from separate minute centres of aggregation. Minchin has -shewn that, in some cases at least, the latter is the case; the -spicule begins, he tells us, as three tiny rods, separate from one -another, each developed in the interspace between two sister-cells, -which are themselves the results of the division of -one of a <span class="xxpn" id="p450">{450}</span> -little trio of cells; and the little rods meet and fuse together while -still very minute, when the whole spicule is only about 1 ⁄ 200 of a -millimetre long. At this stage, it is interesting to learn that the -spicule is non-crystalline; but the new accretions of calcareous -matter are soon deposited in crystalline form.</p> - -<p>This observation threw considerable difficulties in the way of -former mechanical theories of the conformation of the spicule, and -was quite at variance with Dreyer’s theory, according to which -the spicule was bound to begin from a central nucleus coinciding -with the meeting-place of the three contiguous cells, or rather the -interspace between them. But the difficulty is removed when we -import the concept of adsorption; for by this agency it is natural -enough, or conceivable enough, that the process of deposition -should go on at separate parts of a common system of surfaces; -and if the cells tend to meet one another by their interfaces before -these interfaces extend to the angles and so complete the polygonal -cell, it is again conceivable and natural that the spicule should -first arise in the form of separate and detached limbs or rays.</p> - -<div class="dleft dwth-e" id="fig215"> -<img src="images/i450.png" width="337" height="554" alt=""> - <div class="pcaption">Fig. 215. Spicules of tetractinellid - sponges (after Sollas). <i>a</i>–<i>e</i>, anatriaenes; <i>d</i>–<i>f</i>, - protriaenes.</div></div> - -<p>Among the tetractinellid sponges, whose spicules are -composed of amorphous silica or opal, all or most of the -above-described main types of spicule occur, and, as the -name of the group implies, the four-rayed, tetrahedral -spicules are especially represented. A somewhat frequent -type of spicule is one in which one of the four rays is -greatly developed, and the other three constitute small -prongs diverging at equal angles from the main or axial -ray. In all probability, as Dreyer suggests, we have here -had to do with a group of four vesicles, of which three -were large and co-equal, while a fourth and very much -smaller one lay above and between the other three. In -certain cases where we have likewise one large and three -much smaller <span class="xxpn" id="p451">{451}</span> -rays, the latter are recurved, as in Fig. <a href="#fig215" title="go to Fig. 215">215</a>. This type, -save for the constancy of the number of rays, and the -limitation of the terminal ones to three, and save also -for the more important difference that they occur only -at one and not at both ends of the long axis, is similar -to the type of spicule illustrated in Fig. <a href="#fig213" title="go to Fig. 213">213</a>, which we -have explained as being probably developed within an oval -cell, by whose walls its branches have been conformed to -geodetic curves. But it is much more probable that we have -here to do with a spicule developed in the midst of a group -of three coequal and more or less elongated or cylindrical -cells or vesicles, the long axial ray corresponding to -their common line of contact, and the three short rays -having each lain in the surface furrow between two out of -the three adjacent cells.<br class="brclrfix"></p> - -<div class="dctr01" id="fig216"> -<img src="images/i451.png" width="800" height="398" alt=""> - <div class="dcaption">Fig. 216. Various holothurian spicules. -(After Théel.)</div></div> - -<p>Just as in the case of the little curved or -<span class="nowrap"><img class="iglyph-a" -src="images/glyph-s.png" width="32" height="46" alt="S">-shaped</span> -spicules, -formed apparently within the bounds of a single cell, so also in -the case of the larger tetractinellid and analogous types do we -find among the Holothuroidea the same configurations reproduced -as we have dealt with in the sponges. The holothurian spicules -are a little less neatly formed, a little rougher, than the sponge-spicules; -and certain forms occur among the former group which -do not present themselves among the latter; but for the most -part a community of type is obvious and striking (Fig. <a href="#fig216" title="go to Fig. 216">216</a>).</p> - -<p>A curious and, physically speaking, strictly analogous formation -to the tetrahedral spicules of the sponges is -found in the <span class="xxpn" id="p452">{452}</span> -spores of a certain little group of parasitic protozoa, the Actinomyxidia. -These spores are formed from clusters of six cells, -of which three come to constitute the capsule of the spore; and -this capsule, always triradiate in its symmetry, is in some species -drawn out into long rays, of which one constitutes a straight -central axis, while the others, coming off from it at equal angles, -are recurved in wide circular arcs. The account given of the -development of this structure by its discoverers<a class="afnanch" href="#fn461" id="fnanch461">461</a> -is somewhat -obscure to me, but I think that, on physical grounds, there can -be no doubt whatever that the quadriradiate capsule has been -somehow modelled upon a group of three surrounding cells, its -axis lying between the three, and its three radial arcs occupying -the furrows between adjacent pairs.</p> - -<div class="dctr02" id="fig217"> -<img src="images/i452.png" width="705" height="429" alt=""> - <div class="dcaption">Fig. 217. Spicules of hexactinellid sponges. - (After F. E. Schultze.)</div></div> - -<p>The typically six-rayed siliceous spicules of the hexactinellid -sponges, while they are perhaps the most regular and beautifully -formed spicules to be found within the entire group, have been -found very difficult to explain, and Dreyer has confessed his -complete inability to account for their conformation. But, -though it is doubtless only throwing the difficulty a little further -back, we may so far account for them by considering that the -cells or vesicles by which they are conformed -are not arranged in <span class="xxpn" id="p453">{453}</span> -what is known as “closest packing,” but in linear series; so that in -their arrangement, and by their mutual compression, we tend to -get a pattern, not of hexagons, but of squares: or, looking to -the solid, not of dodecahedra but of cubes or parallelopipeda. -This indeed appears to be the case, not with the individual cells -(in the histological sense), but with the larger units or vesicles -which make up the body of the hexactinellid. And this being -so, the spicules formed between the linear, or cubical series of -vesicles, will have the same tendency towards a “hexactinellid” -shape, corresponding to the angles and adjacent edges of a system -of cubes, as in our former case they had to a triradiate or a -tetractinellid form, when developed in connection with the angles -and edges of a system of hexagons, or a system of dodecahedra.</p> - -<p>Histologically, the case is illustrated by a well-known phenomenon -in embryology. In the segmenting ovum, there is a -tendency for the cells to be budded off in linear series; and so -they often remain, in rows side by side, at least for a considerable -time and during the course of several consecutive cell divisions. -Such an arrangement constitutes what the embryologists call the -“radial type” of segmentation<a class="afnanch" href="#fn462" id="fnanch462">462</a>. -But in what is described as the -“spiral type” of segmentation, it is stated that, as soon as the -first horizontal furrow has divided the cells into an upper and -a lower layer, those of “the upper layer are shifted in respect -to the lower layer, by means of a rotation about the vertical -axis<a class="afnanch" href="#fn463" id="fnanch463">463</a>.” -It is, of course, evident that the whole process is -merely that which is familiar to physicists as “close packing.” -It is a very simple case of what Lord Kelvin used to call -“a problem in tactics.” It is a mere question of the rigidity -of the system, of the freedom of movement on the part of -its constituent cells, whether or at what stage this tendency -to slip into the closest propinquity, or position of minimum -potential, will be found to manifest itself.</p> - -<p>However the hexactinellid spicules be arranged -(and this is <span class="xxpn" id="p454">{454}</span> -not at all easy to determine) in relation to the tissues and chambers -of the sponge, it is at least clear that, whether they be separate -or be fused together (as often happens) in a composite skeleton, -they effect a symmetrical partitioning of space according to the -cubical system, in contrast to that closer packing which is represented -and effected by the tetrahedral system<a class="afnanch" href="#fn464" id="fnanch464">464</a>.</p> - -<hr class="hrblk"> - -<p>This question of the origin and causation of the forms of -sponge-spicules, with which we have now briefly dealt, is all the -more important and all the more interesting because it has been -discussed time and again, from points of view which are characteristic -of very different schools of thought in biology. Haeckel -found in the form of the sponge-spicule a typical illustration of -his theory of “bio-crystallisation”; he considered that these -“biocrystals” represented “something midway—<i>ein Mittelding</i>—between -an inorganic crystal and an organic secretion”; that -there was a “compromise between the crystallising efforts of the -calcium carbonate and the formative activity of the fused cells -of the syncytium”; and that the semi-crystalline secretions of -calcium carbonate “were utilised by natural selection as ‘spicules’ -for building up a skeleton, and afterwards, by the interaction of -adaptation and heredity, became modified in form and differentiated -in a vast variety of ways in the struggle for existence<a class="afnanch" href="#fn465" id="fnanch465">465</a>.” -What Haeckel precisely signified by these words is not clear to me.</p> - -<p>F. E. Schultze, perceiving that identical forms of spicule were -developed whether the material were crystalline or non-crystalline, -abandoned all theories based upon crystallisation; he simply saw -in the form and arrangement of the spicules something which -was “best fitted” for its purpose, that is to say for the support -and strengthening of the porous walls of the sponge, and found -clear evidence of “utility” in the specific structure of these -skeletal elements. <span class="xxpn" id="p455">{455}</span></p> - -<p>Sollas and Dreyer, as we have seen, introduced in various -ways the conception of physical causation,—as indeed Haeckel -himself had done in regard to one particular, when he supposed -the <i>position</i> of the spicules to be due to the constant passage of -the water-currents. Though even here, by the way, if I understand -Haeckel aright, he was thinking not merely of a direct or immediate -physical causation, but of one manifesting itself through -the agency of natural selection<a class="afnanch" href="#fn466" id="fnanch466">466</a>. -Sollas laid stress upon the “path -of least resistance” as determining the direction of growth; -while Dreyer dealt in greater detail with the various tensions -and pressures to which the growing spicule was exposed, amid -the alveolar or vesicular structure which was represented alike -by the chambers of the sponge, by the reticulum of constituent -cells, or by the minute structure of the intracellular protoplasm. -But neither of these writers, so far as I can discover, was inclined -to doubt for a moment the received canon of biology, which sees -in such structures as these the characteristics of true organic -species, and the indications of an hereditary affinity by which -blood-relationship and the succession of evolutionary descent -throughout geologic time can be ultimately deduced.</p> - -<p>Lastly, Minchin, in a well-known paper<a class="afnanch" href="#fn467" id="fnanch467">467</a>, -took sides with -Schultze, and gave reasons for dissenting from such mechanical -theories as those of Sollas and of Dreyer. For example, after -pointing out that all protoplasm contains a number of “granules” -or microsomes, contained in the alveolar framework and lodged -at the nodes of the reticulum, he argued that these also ought to -acquire a form such as the spicules possess, if it were the case that -these latter owed their form to their very similar or identical -position. “If vesicular tension cannot in any other instance cause -the granules at the nodes to assume a tetraxon form, why should -it do so for the sclerites?” In all probability the answer to this -question is not far to seek. If the force which the “mechanical” -hypothesis has in view were simply that -of mechanical <i>pressure</i>, <span class="xxpn" id="p456">{456}</span> -as between solid bodies, then indeed we should expect that any -substances whatsoever, lying between the impinging spheres, -would tend (unless they were infinitely hard) to assume the -quadriradiate or “tetraxon” form; but this conclusion does not -follow at all, in so far as it is to <i>surface-energy</i> that we ascribe the -phenomenon. Here the specific nature of the substances involved -makes all the difference. We cannot argue from one substance -to another; adsorptive attraction shews its effect on one and not -on another; and we have not the least reason to be surprised if -we find that the little granules of protoplasmic material, which -as they lie bathed in the more fluid protoplasm have (presumably, -and as their shape indicates) a strong surface-tension of their -own, behave towards the adjacent vesicles in a very different -fashion to the incipient aggregations of calcareous or siliceous -matter in a colloid medium. “The ontogeny of the spicules,” says -Professor Minchin, “points clearly to their regular form being a -<i>phylogenetic adaptation, which has become fixed and handed on by -heredity, appearing in the ontogeny as a prophetic adaptation</i>.” -And again, “The forms of the spicules are the result of adaptation -to the requirements of the sponge as a whole, produced by <i>the -action of natural selection upon variation in every direction</i>.” It -would scarcely be possible to illustrate more briefly and more -cogently than by these few words (or the similar words of Haeckel -quoted on p. <a href="#p454" title="go to pg. 454">454</a>), the fundamental difference between the -Darwinian conception of the causation and determination of -Form, and that which is characteristic of the physical sciences.</p> - -<hr class="hrblk"> - -<p>If I have dealt comparatively briefly with the inorganic -skeleton of sponges, in spite of the obvious importance of this -part of our subject from the physical or mechanical point of view, -it has been owing to several reasons. In the first place, though -the general trend of the phenomena is clear, it must be at once -admitted that many points are obscure, and could only be discussed -at the cost of a long argument. In the second place, the physical -theory is (as I have shewn) in manifest conflict with the accounts -given by various embryologists of the development of the spicules, -and of the current biological theories which their descriptions -embody; it is beyond our scope to deal -with such descriptions <span class="xxpn" id="p457">{457}</span> -in detail. Lastly, we find ourselves able to illustrate the same -physical principles with greater clearness and greater certitude in -another group of animals, namely the Radiolaria. In our description -of the skeletons occurring within this group we shall by no -means abandon the preliminary classification of microscopic -skeletons which we have laid down; but we shall have occasion -to blend with it the consideration of certain other more or less -correlated phenomena.</p> - -<p>The group of microscopic organisms known as the Radiolaria -is extraordinarily rich in diverse forms, or “species.” I do not -know how many of such species have been described and defined -by naturalists, but some thirty years ago the number was said -to be over four thousand, arranged in more than seven hundred -genera<a class="afnanch" href="#fn468" id="fnanch468">468</a>. -Of late years there has been a tendency to reduce the -number, it being found that some of the earlier species and even -genera are but growth-stages of one and the same form, sometimes -mere fragments or “fission-products” common to several species, -or sometimes forms so similar and so interconnected by intermediate -forms that the naturalist denominates them not “species” -but “varieties.” It has to be admitted, in short, that the conception -of species among the Radiolaria has not hitherto been, -and is not yet, on the same footing as that among most other -groups of animals. But apart from the extraordinary multiplicity -of forms among the Radiolaria, there are certain other features -in this multiplicity which arrest our attention. For instance, -the distribution of species in space is curious and vague; many -species are found all over the world, or at least every here and -there, with no evidence of specific limitations of geographical -habitat; others occur in the neighbourhood of the two poles; -some are confined to warm and others to cold currents of the -ocean. In time also their distribution is not less vague: so much -so that it has been asserted of them that “from the Cambrian -age downwards, the families and even genera appear identical -with those now living.” Lastly, except perhaps in the case of -a few large “colonial forms,” we seldom if ever -find, as is usual <span class="xxpn" id="p458">{458}</span> -in most animals, a local predominance of one particular species. -On the contrary, in a little pinch of deep-sea mud or of some fossil -“Radiolarian earth,” we shall probably find scores, and it may be -even hundreds, of different forms. Moreover, the radiolarian -skeletons are of quite extraordinary delicacy and complexity, in -spite of their minuteness and the comparative simplicity of the -“unicellular” organisms within which they grow; and these -complex conformations have a wonderful and unusual appearance -of geometric regularity. All these <i>general</i> considerations seem -such as to prepare us for the special need of some physical -hypothesis of causation. The little skeletal fabrics remind us of -such objects as snow-crystals (themselves almost endless in their -diversity), rather than of a collection of distinct animals, constructed -in apparent accordance with functional needs, and distributed -in accordance with their fitness for particular situations. -Nevertheless great efforts have been made of recent years to -attach “a biological meaning” to these elaborate structures; -and “to justify the hope that in time the utilitarian character -[of the skeleton] will be more completely recognised<a class="afnanch" href="#fn469" id="fnanch469">469</a>.”</p> - -<p>In the majority of cases, the skeleton of the Radiolaria is -composed, like that of so many sponges, of silica; in one large -family, the Acantharia (and perhaps in some others), it is composed, -in great part at least, of a very unusual constituent, namely -strontium sulphate<a class="afnanch" href="#fn470" id="fnanch470">470</a>. -There is no fundamental or important -morphological character in which the shells formed of these two -constituents differ from one another; and in no case can the -chemical properties of these inorganic materials be said to influence -the form of the complex skeleton or shell, save only in this general -way that, by their rigidity and toughness, they may give rise to -a fabric far more delicate and slender than we find developed -among calcareous organisms.</p> - -<p>A slight exception to this rule is found in the presence of true -crystals, which occur within the central -capsules of certain <span class="xxpn" id="p459">{459}</span> -Radiolaria, for instance the genus Collosphaera<a class="afnanch" href="#fn471" id="fnanch471">471</a>. -Johannes Müller -(whose knowledge and insight never fail to astonish us) remarked -that these were identical in form with crystals of celestine, a -sulphate of strontium and barium; and Bütschli’s discovery of -sulphates of strontium and of barium in kindred forms render it -all but certain that they are actually true crystals of celestine<a class="afnanch" href="#fn472" id="fnanch472">472</a>.</p> - -<p>In its typical form, the Radiolarian body consists of a spherical -mass of protoplasm, around which, and separated from it by some -sort of porous “capsule,” lies a frothy mass, composed of protoplasm -honeycombed into a multitude of alveoli or vacuoles, filled -with a fluid which can scarcely differ much from sea-water<a class="afnanch" href="#fn473" id="fnanch473">473</a>. -According to their surface-tension conditions, these vacuoles may -appear more or less isolated and spherical, or joining together in -a “froth” of polygonal cells; and in the latter, which is the -commoner condition, the cells tend to be of equal size, and the -resulting polygonal meshwork beautifully regular. In many cases, -a large number of such simple individual organisms are associated -together, forming a floating colony, and it is highly probable that -many other forms, with whose scattered skeletons we are alone -acquainted, had in life formed part likewise of a colonial organism.</p> - -<p>In contradistinction to the sponges, in which the skeleton -always begins as a loose mass of isolated spicules, which only in -a few exceptional cases (such as Euplectella and Farrea) fuse into -a continuous network, the characteristic feature of the Radiolarians -lies in the possession of a continuous skeleton, in the form of a -netted mesh or perforated lacework, sometimes however replaced -by and often associated with minute independent spicules. Before -we proceed to treat of the more complex skeletons, we may begin, -then, by dealing with these comparatively simple cases where -either the entire skeleton or a considerable part of it is represented, -not by a continuous fabric, but by a quantity of loose, separate -spicules, or aciculae, which seem, like the -spicules of Alcyonium, <span class="xxpn" id="p460">{460}</span> -to be developed as free and isolated formations or deposits, -precipitated in the colloid matrix, with no relation of form to -the cellular or vesicular boundaries. These simple acicular spicules -occupy a definite position in the organism. Sometimes, as for -instance among the fresh-water Heliozoa (e.g. Raphidiophrys), they -lie on the outer surface of the organism, and not infrequently -(when the spicules are few in number) they tend to collect round -the bases of the pseudopodia, or around the large radiating -spicules, or axial rays, in the cases where these latter are present. -When the spicules are thus localised around some prominent centre, -they tend to take up a position of symmetry in regard to it; instead -of forming a tangled or felted layer, they come to lie side by side, -in a radiating cluster round the focus. In other cases (as for -instance in the well-known Radiolarian <i>Aulacantha scolymantha</i>) -the felted layer of aciculae lies at some depth below the surface, -forming a sphere concentric with the entire spherical organism. -In either case, whether the layer of spicules be deep or be superficial, -it tends to mark a “surface of discontinuity,” a meeting -place between two distinct layers of protoplasm or between the -protoplasm and the water around; and it is obvious that, in either -case, there are manifestations of surface-energy at the boundary, -which cause the spicules to be retained there, and to take up their -position in its plane. The case is somewhat, though not directly, -analogous to that of a cirrus cloud, -which marks the place of a surface -of discontinuity in a stratified atmosphere.</p> - -<div class="dleft dwth-e" id="fig218"> -<img src="images/i460.png" width="336" height="407" alt=""> - <div class="dcaption">Fig. 218.</div></div> - -<p>We have, then, to enquire what are the conditions which -shall, apart from gravity, confine an extraneous body to a -surface-film; and we may do this very simply, by considering -the surface-energy of the entire system. In Fig. <a href="#fig218" title="go to Fig. 218">218</a> we -have two fluids in contact with one another (let us call -them water and protoplasm), and a body (<i>b</i>) which may be -immersed in either, or may be restricted to the boundary -<span class="xxpn" id="p461">{461}</span> between. We have -here three possible “interfacial contacts” each with its own -specific surface-energy, per unit of surface area: namely, -that between our particle and the water (let us call it α), -that between the particle and the protoplasm (β), and that -between water and protoplasm (γ). When the body lies in the -boundary of the two fluids, let us say half in one and half in -the other, the surface-energies concerned are equivalent to -(<i>S</i> ⁄ 2)α + (<i>S</i> ⁄ 2)β; -but we must also remember that, by the presence of the -particle, a small portion (equal to its sectional area <i>s</i>) -of the original contact-surface between water and protoplasm -has been obliterated, and with it a proportionate quantity -of energy, equivalent to <i>s</i>γ, has been set free. When, on -the other hand, the body lies entirely within one or other -fluid, the surface-energies of the system (so far as we are -concerned) are equivalent to <span class="nowrap"> -<i>S</i>α + <i>s</i>γ,</span> or <span class="nowrap"> -<i>S</i>β + <i>s</i>γ,</span> as the case may be. According as α -be less or greater than β, the particle will have a tendency -to remain immersed in the water or in the protoplasm; but if <span class="nowrap"> -(<i>S</i> ⁄ 2)(α + β) − <i>s</i>γ</span> -be less than either <i>S</i>α or <i>S</i>β, then the condition of minimal -potential will be found when the particle lies, as we have -said, in the boundary zone, half in one fluid and half in the -other; and, if we were to attempt a more general solution of -the problem, we should evidently have to deal with possible -conditions of equilibrium under which the necessary balance of -energies would be attained by the particle rising or sinking in -the boundary zone, so as to adjust the relative magnitudes of -the surface-areas concerned. It is obvious that this principle -may, in certain cases, help us to explain the position even -of a <i>radial</i> spicule, which is just a case where the surface -of the solid spicule is distributed between the fluids with a -minimal disturbance, or minimal replacement, of the original -surface of contact between the one fluid and the other.<br class="brclrfix"></p> - -<p>In like manner we may provide for the case (a common and -an important one) where the protoplasm “creeps up” the spicule, -covering it with a delicate film. In Acanthocystis we have -yet another special case, where the radial spicules plunge only -a certain distance into the protoplasm of the cell, being arrested -at a boundary-surface between an inner and an outer layer of -cytoplasm; here we have only to assume that there -is a tension <span class="xxpn" id="p462">{462}</span> -at this surface, between the two layers of protoplasm, sufficient -to balance the tensions which act directly on the spicule<a class="afnanch" href="#fn474" id="fnanch474">474</a>.</p> - -<p>In various Acanthometridae, besides such typical characters -as the radial symmetry, the concentric layers of protoplasm, and -the capillary surfaces in which the outer, vacuolated protoplasm -is festooned upon the projecting radii, we have another curious -feature. On the surface of the protoplasm where it creeps up -the sides of the long radial spicules, we find a number of elongated -bodies, forming in each case one or several little groups, and -lying neatly arranged in parallel bundles. A Russian naturalist, -Schewiakoff, whose views have been accepted in the text-books, -tells us that these are muscular structures, serving to raise or -lower the conical masses of protoplasm about the radial spicules, -which latter serve as so many “tent-poles” or masts, on which -the protoplasmic membranes are hoisted up; and the little -elongated bodies are dignified with various names, such as -“myonemes” or “myophriscs,” in allusion to their supposed -muscular nature<a class="afnanch" href="#fn475" id="fnanch475">475</a>. -This explanation is by no means convincing. -To begin with, we have precisely similar festoons of protoplasm -in a multitude of other cases where the “myonemes” are lacking; -from their minute size (·006–·012 mm.) and the amount of contraction -they are said to be capable of, the myonemes can hardly -be very efficient instruments of traction; and further, for them -to act (as is alleged) for a specific purpose, namely the “hydrostatic -regulation” of the organism giving it power to sink or to swim, -would seem to imply a mechanism of action and of coordination -which is difficult to conceive in these minute and simple organisms. -The fact is (as it seems to me), that the whole method of explanation -is unnecessary. Just as the supposed “hauling up” of the -protoplasmic festoons is at once explained by capillary phenomena, -so also, in all probability, is the position and arrangement of -the little elongated bodies. Whatever the actual nature of these -bodies may be, whether they are truly portions of differentiated -protoplasm, or whether they are foreign bodies or spicular -structures (as bodies occupying a similar position in other cases -undoubtedly are), we can explain their -situation on the surface <span class="xxpn" id="p463">{463}</span> -of the protoplasm, and their arrangement around the radial -spicules, all on the principles of surface-tension<a -class="afnanch" href="#fn476" id="fnanch476">476</a>.</p> - -<p>This last case is not of the simplest; and I do not forget that -my explanation of it, which is wholly theoretical, implies a doubt -of Schewiakoff’s statements, which are founded on direct personal -observation. This I am none too willing to do; but whether it -be justly done in this case or not, I hold that it is in principle -justifiable to look with great suspicion upon a number of kindred -statements where it is obvious that the observer has left out of -account the purely physical aspect of the phenomenon, and all -the opportunities of simple explanation which the consideration -of that aspect might afford.</p> - -<hr class="hrblk"> - -<p>Whether it be wholly applicable to this particular and complex -case or no, our general theorem of the localisation and arrestment -of solid particles in a surface-film is of very great biological -importance; for on it depends the power displayed by many -little naked protoplasmic organisms of covering themselves with -an “agglutinated” shell. Sometimes, as in <i>Difflugia</i>, <i>Astrorhiza</i> -(Fig. <a href="#fig219" title="go to Fig. 219">219</a>) and others, this covering consists of sand-grains picked -up from the surrounding medium, and sometimes, on the other -hand, as in <i>Quadrula</i>, it consists of solid particles which are said -to arise, as inorganic deposits or concretions, within the protoplasm -itself, and which find their way outwards to a position of equilibrium -in the surface-layer; and in both cases, the mutual capillary -attractions between the particles, confined to the boundary-layer -but enjoying a certain measure of freedom therein, tends to the -orderly arrangement of the particles one with another, and even -to the appearance of a regular “pattern” as the result of this -arrangement.</p> - -<div class="dctr02" id="fig219"> -<img src="images/i464.png" width="528" height="700" alt=""> - <div class="pcaption">Fig. 219. Arenaceous Foraminifera; - <i>Astrorhiza limicola</i> and <i>arenaria</i>. (From Brady’s - <i>Challenger Monograph</i>.)</div></div> - -<p>The “picking up” by the protoplasmic organism of a solid -particle with which “to build its house” (for it is hard to avoid -this customary use of anthropomorphic figures of speech, misleading -though they be), is a physical phenomenon kindred to that by which -an Amoeba “swallows” a particle of food. This latter process -has been reproduced or imitated in various -pretty experimental <span class="xxpn" id="p465">{465}</span> -ways. For instance, Rhumbler has shewn that if a thread of -glass be covered with shellac and brought near a drop of -chloroform suspended in water, the drop takes in the spicule, -robs it of its shellac covering, and then passes it out again<a class="afnanch" href="#fn477" id="fnanch477">477</a>. -It is all a question of relative surface-energies, leading to different -degrees of “adhesion” between the chloroform and the glass or -its covering. Thus it is that the Amoeba takes in the diatom, -dissolves off its proteid covering, and casts out the shell.</p> - -<p>Furthermore, as the whole phenomenon depends on a distribution -of surface-energy, the amount of which is specific to certain -particular substances in contact with one another, we have no -difficulty in understanding the <i>selective action</i>, which is very often -a conspicuous feature in the phenomenon<a class="afnanch" href="#fn478" id="fnanch478">478</a>. -Just as some caddis-worms -make their houses of twigs, and others of shells and again -others of stones, so some Rhizopods construct their agglutinated -“test” out of stray sponge-spicules, or frustules of diatoms, or -again of tiny mud particles or of larger grains of sand. In all -these cases, we have apparently to deal -with differences in specific <span class="xxpn" id="p466">{466}</span> -surface-energies, and also doubtless with differences in the total -available amount of surface-energy in relation to gravity or other -extraneous forces. In my early student days, Wyville Thomson -used to tell us that certain deep-sea “Difflugias,” after constructing -a shell out of particles of the black volcanic sand common in parts -of the North Atlantic, finished it off with “a clean white collar” -of little grains of quartz. Even this phenomenon may be accounted -for on surface-tension principles, if we assume that the surface-energy -ratios have tended to change, either with the growth of -the protoplasm or by reason of external variation of temperature -or the like; and we are by no means obliged to attribute the -phenomenon to a manifestation of volition, or taste, or aesthetic -skill, on the part of the microscopic organism. Nor, when certain -Radiolaria tend more than others to attract into their own substance -diatoms and such-like foreign bodies, is it scientifically -correct to speak, as some text-books do, of species “in which -diatom selection has become <i>a regular habit</i>.” To do so is an -exaggerated misuse of anthropomorphic phraseology.</p> - -<p>The formation of an “agglutinated” shell is thus seen to be -a purely physical phenomenon, and indeed a special case of a -more general physical phenomenon which has many other -important consequences in biology. For the shell to assume the -solid and permanent character which it acquires, for instance, in -Difflugia, we have only to make the additional assumption that -some small quantities of a cementing substance are secreted by -the animal, and that this substance flows or creeps by capillary -attraction between all the interstices of the little quartz grains, -and ends by binding them all firmly together. Rhumbler<a class="afnanch" href="#fn479" id="fnanch479">479</a> -has -shewn us how these agglutinated tests, of spicules or of sand-grains, -can be precisely imitated, and how they are formed with -greater or less ease, and greater or less rapidity, according to the -nature of the materials employed, that is to say, according to -the specific surface-tensions which are involved. For instance if -we mix up a little powdered glass with chloroform, and set a drop -of the mixture in water, the glass particles gather neatly round -the surface of the drop so quickly that the eye -cannot follow the <span class="xxpn" id="p467">{467}</span> -operation. If we perform the same experiment with oil and fine sand, -dropped into 70 per cent. alcohol, a still more beautiful artificial -Rhizopod shell is formed, but it takes some three hours to do.</p> - -<p>It is curious that, just at the very time when Rhumbler was -thus demonstrating the purely physical nature of the Difflugian -shell, Verworn was studying the same and kindred organisms -from the older standpoint of an incipient psychology<a class="afnanch" href="#fn480" id="fnanch480">480</a>. -But, as -Rhumbler himself admits, Verworn was very careful not to overestimate -the apparent signs of volition, or selective choice, in the -little organism’s use of the material of its dwelling.</p> - -<hr class="hrblk"> - -<p>This long parenthesis has led us away, for the time being, -from the subject of the Radiolarian skeleton, and to that subject -we must now return. Leaving aside, then, the loose and scattered -spicules, which we have sufficiently discussed, the more perfect -Radiolarian skeletons consist of a continuous and regular structure; -and the siliceous (or other inorganic) material of which this framework -is composed tends to be deposited in one or other of two -ways or in both combined: (1) in the form of long spicular axes, -usually conjoined at, or emanating from, the centre of the protoplasmic -body, and forming a symmetric radial system; (2) in the -form of a crust, developed in various ways, either on the outer -surface of the organism or in relation to the various internal -surfaces which separate its concentric layers or its component -vesicles. Not unfrequently, this superficial skeleton comes to -constitute a spherical shell, or a system of concentric or otherwise -associated spheres.</p> - -<p>We have already learned that a great part of the body -of the Radiolarian, and especially that outer portion to -which Haeckel has given the name of the “calymma,” is -built up of a great mass of “vesicles,” forming a sort of -stiff</p> - -<div class="dleft dwth-d" id="fig220"> -<img src="images/i468.png" width="384" height="496" alt=""> - <div class="dcaption">Fig. 220. “Reticulum plasmatique.” - (After Carnoy.)</div></div> - -<p class="pcontinue"> -froth, and equivalent in the physical sense (though -not necessarily in the biological sense) to “cells,” -inasmuch as the little vesicles have their own well-defined -boundaries, and their own surface phenomena. In short, all -that we have said of cell-surfaces, and cell conformations, -in our discussion of cells and of tissues, will apply -in like manner, and under appropriate conditions, to -these. In certain cases, even in <span class="xxpn" -id="p468">{468}</span> so common and simple a one as the -vacuolated substance of an Actinosphaerium, we may see a -very close resemblance, or formal analogy, to an ordinary -cellular or “parenchymatous” tissue, in the close-packed -arrangement and consequent configuration of these vesicles, -and even at times in a slight membranous hardening -of their walls. Leidy has figured<a class="afnanch" -href="#fn481" id="fnanch481">481</a> some curious little -bodies, like small masses of consolidated froth, which -seem to be nothing else than the dead and empty husks, -or filmy skeletons, of Actinosphaerium. And Carnoy<a -class="afnanch" href="#fn482" id="fnanch482">482</a> has -demonstrated in certain cell-nuclei an all but precisely -similar framework, of extreme delicacy and minuteness, -as the result of partial solidification of interstitial -matter in a close-packed system of alveoli (Fig. <a href="#fig220" title="go to Fig. 220">220</a>).<br -class="brclrfix"></p> - -<p>Let us now suppose that, -in our Radiolarian, the outer -surface of the animal is covered by a layer of froth-like vesicles, -uniform or nearly so in size. We know that their tensions will -tend to conform them into a “honeycomb,” or regular meshwork -of hexagons, and that the free end of each hexagonal prism will -be a little spherical cap. Suppose now that it be at the outer -surface of the protoplasm (that namely which is in contact with -the surrounding sea-water), that the siliceous particles have a -tendency to be secreted or adsorbed; it will at once follow that -they will show a tendency to aggregate in the grooves which -separate the vesicles, and the result will be the development of -a most delicate sphere composed of tiny rods arranged in a regular -hexagonal network (e.g. <i>Aulonia</i>). Such a -conformation is <span class="xxpn" id="p469">{469}</span> -extremely common, and among its many variants may be found -cases in which (e.g. <i>Actinomma</i>), the vesicles have</p> - -<div class="dctr03" id="fig221"> -<img src="images/i469a.png" width="608" height="459" alt=""> - <div class="dcaption">Fig. 221. <i>Aulonia hexagona</i>, Hkl.</div></div> - -<div class="dctr03" id="fig222"> -<img src="images/i469b.png" width="608" height="551" alt=""> - <div class="dcaption">Fig. 222. <i>Actinomma arcadophorum</i>, - Hkl.</div></div> - -<p class="pcontinue">been less regular in size, and -some in which the hexagonal meshwork has been developed -not only on one outer surface, but at successive <span -class="xxpn" id="p470">{470}</span> surfaces, producing a -system of concentric spheres. If the siliceous material -be not limited to the linear junctions of the cells, but -spread over a portion of the outer spherical surfaces or -caps, then we shall have the condition represented in Fig. -<a href="#fig223" title="go to Fig. 223">223</a> (<i>Ethmosphaera</i>), where the shell appears perforated -by circular instead of hexagonal apertures, and the -circular pores are set on slight spheroidal eminences; and, -interconnected with such types as this, we have others in -which the accumulating pellicles of skeletal matter have -extended from the edges into the substance of the boundary -walls</p> - -<div class="dctr01" id="fig223"><div id="fig224"> -<img src="images/i470.png" width="800" height="446" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td class="tdleft">Fig. 223. <i>Ethmosphaera conosiphonia</i>, Hkl.</td> - <td></td> - <td class="tdleft">Fig. 224. Portions of shells of two “species” of - <i>Cenosphaera</i>: upper figure, <i>C. favosa</i>, lower, <i>C. - vesparia</i>, Hkl.</td></tr></table> -</div></div></div><!--dctr01--> - -<p class="pcontinue"> and have so produced a system of -films, normal to the surface of the sphere, constituting -a very perfect honeycomb, as in <i>Cenosphaera favosa</i> -and <i>vesparia</i><a class="afnanch" href="#fn483" -id="fnanch483">483</a>.</p> - -<p>In one or two very simple forms, such as the fresh-water -<i>Clathrulina</i>, just such a spherical perforated shell is produced out -of some organic, acanthin-like substance; and in some examples -of <i>Clathrulina</i> the chitinous lattice-work of the -shell is just as <span class="xxpn" id="p471">{471}</span> -regular and delicate, with the meshes just as beautifully hexagonal, -as in the siliceous shells of the oceanic Radiolaria. This is only -another proof (if proof be needed) that the peculiar conformation -of these little skeletons is not due to the material of which they -are composed, but to the moulding of that material upon an underlying -vesicular structure.</p> - -<div class="dctr04" id="fig225"> -<img src="images/i471.png" width="528" height="483" alt=""> - <div class="dcaption">Fig. 225. - <i>Aulastrum triceros</i>, Hkl.</div></div> - -<p>Let us next suppose that, upon some such lattice-work as has -just been described, another and external layer of cells or vesicles -is developed, and that instead of (or perhaps only in addition to) -a second hexagonal lattice-work, which might develop concentrically -to the first in the boundary-furrows of this new layer of -cells, the siliceous matter now tends to be deposited radially, -or normally to the surface of the sphere, just in the lines where -the external layer of vesicles meet one another, three by three. -The result will be that, when the vesicles themselves are removed, -a series of radiating spicules will be revealed, directed outwards -from each of the angles of the original hexagon; as is seen -in Fig. <a href="#fig225" title="go to Fig. 225">225</a>. And it may further happen that these radiating -skeletal rods are continued at their distal ends into divergent -rays, forming a triple fork, and corresponding -(after a fashion <span class="xxpn" id="p472">{472}</span> -which we have already described as occurring in certain sponge-spicules) -to the three superficial furrows between the adjacent -cells. This last is, as it were, an intermediate stage between the -simple rods and the complete formation of another concentric -sphere of latticed hexagons. Another possible case is when the -large and uniform vesicles of the outer protoplasm are mixed -with, or replaced by, much smaller vesicles, piled on one another -in more or less concentric layers; in this case the radiating</p> - -<div class="dctr01" id="fig226"><div id="fig227"> -<img src="images/i472.png" width="800" height="667" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td class="tdleft">Fig. 226.</td> - <td></td> - <td class="tdleft">Fig. 227. A Nassellarian - skeleton, <i>Callimitra carolotae</i>, Hkl.</td></tr></table> -</div></div></div><!--dctr01--> - -<p class="pcontinue">rods -will no longer be straight, but will be bent into a zig-zag pattern, -with angles in three vertical planes, corresponding to the successive -contacts of the groups of cells around the axis (Fig. <a href="#fig226" title="go to Fig. 226">226</a>).</p> - -<hr class="hrblk"> - -<p>Among a certain group called the Nassellaria, we find geometrical -forms of peculiar simplicity and beauty,—such for instance -as that which I have represented in Fig. <a href="#fig227" title="go to Fig. 227">227</a>. It is obvious at -a glance that this is such a skeleton as may -have been formed <span class="xxpn" id="p473">{473}</span> -(I think we may go so far as to say <i>must</i> have been formed) at -the interfaces of a little tetrahedral group of cells, the four equal -cells of the tetrahedron being in this particular case supplemented -by a little one in the centre of the system. We see, precisely as -in the internal boundary-system of an artificial group of four -soap-bubbles, the plane surfaces of contact, six in number; the -relation to one another of each triple set of interfacial planes, -meeting one another at equal angles of 120°; and finally the -relation of the four lines or edges of triple contact, which tend -(but for the little central vesicle) to meet at co-equal solid angles -in the centre of the system, all as we have described on p. <a href="#p318" title="go to pg. 318">318</a>. -In short, each triple-walled re-entrant angle of the little shell has -essentially the configuration (or a part thereof) of what we have -called a “Maraldi pyramid” in our account of the architecture of -the honeycomb, on p. -<a href="#p329" title="go to pg. 329">329</a><a - class="afnanch" href="#fn484" id="fnanch484">484</a>.</p> - -<p>There are still two or three remarkable or peculiar features in -this all but mathematically perfect shell, and they are in part easy -and in part they seem more difficult of interpretation.</p> - -<p>We notice that the amount of solid matter deposited in the -plane interfacial boundaries is greatly increased at the outer -margin of each boundary wall, where it merges or coincides with -the superficial furrow which separates the free, spherical surfaces -of the bubbles from one another; and we may sometimes find that, -along these edges, the skeleton remains complete and strong, -while it shows signs of imperfect development or of breaking -away over great part of the rest of the interfacial surfaces. In -this there is nothing anomalous, for we have already recognised -that it is at the edges or margins of the interfacial partition-walls -that the manifestation of surface-energy will tend to reach its -maximum. And just as we have seen that, in certain of our -“multicellular” spherical Radiolarians, it is -at the superficial <span class="xxpn" id="p474">{474}</span> -edges or borders of</p> - -<div class="dleft dwth-e" id="fig228"> -<img src="images/i474a.png" width="336" height="279" alt=""> - <div class="dcaption">Fig. 228. An isolated portion of - the skeleton of <i>Dictyocha</i>.</div></div> - -<p class="pcontinue">the partitions, and here only, that -skeletal formation occurs (giving rise to the netted shell -with its hexagonal meshes of Fig. <a href="#fig221" title="go to Fig. 221">221</a>), so also at times, -in the case of such little aggregates of cells or vesicles -as the four-celled system of Callimitra, it may happen -that about the external boundary-<i>lines</i>, and not in the -interior boundary-<i>planes</i>, the whole of the skeletal -matter is aggregated. In Fig. <a href="#fig228" title="go to Fig. 228">228</a> we see a curious little -skeletal structure or complex spicule, whose -conformation is easily accounted for -after this<br class="brclrfix"></p> - -<div class="dctr03" id="fig229"> -<img src="images/i474b.png" width="609" height="241" alt=""> - <div class="dcaption">Fig. 229. <i>Dictyocha stapedia</i>, Hkl.</div></div> - -<p class="pcontinue"> fashion. Little spicules such as this form -isolated portions of the skeleton in the genus <i>Dictyocha</i>, -and occur scattered over the spherical surface of the -organism (Fig. <a href="#fig229" title="go to Fig. 229">229</a>). The more or less basket-shaped spicule -has evidently been developed about a little cluster of -four cells or vesicles, lying in or on the plane of the -surface of the organism, and therefore arranged, not in -the tetrahedral form of Callimitra, but in the manner in -which four contiguous cells lying side by side normally set -themselves, like the four cells of a segmenting egg: that -is to say with an intervening “polar furrow,” whose ends -mark the meeting place, at equal angles, of the cells in -groups of three.</p> - -<p>The little projecting spokes, or spikes, which are set normally -to the main basket-work, seem to be incompleted portions of -a larger basket, or in other words imperfectly formed elements -corresponding to the interfacial contacts in -the surrounding parts <span class="xxpn" id="p475">{475}</span> -of the system. Similar but more complex formations, all explicable -as basket-like frameworks developed around a cluster of cells, are -known in great variety.</p> - -<p>In our Nassellarian itself, and in many other cases where the -plane interfacial boundary-walls are skeletonised, we see that the -siliceous matter is not deposited in an even and continuous layer, -like the waxen walls of a bee’s cell, but constitutes a meshwork -of fine curvilinear threads; and the curves seem to run, on the -whole, isogonally, and to form three main series, one approximately -parallel to, or concentric with, the outer or free edge of -the partition, and the other two related severally to its two edges -of attachment. Sometimes (as may also be seen in our figure), -the system is still further complicated by a fourth series of linear -elements, which tend to run radially from the centre of the system -to the free edge of each partition. As regards the former, their -arrangement is such as would result if deposition or solidification -had proceeded in waves, starting independently from each of the -three boundaries of the little partition-wall; and something of -this kind is doubtless what has happened. We are reminded at -once of the wave-like periodicity of the Liesegang phenomenon. -But apart from this we might conceive of other explanations. -For instance, the liquid film which originally constitutes the -partition must easily be thrown into <i>vibrations</i>, and (like the dust -upon a Chladni’s plate) minute particles of matter in contact with -the film would tend to take up their position in a symmetrical -arrangement, in direct relation to the nodal points or lines of the -vibrating surface<a class="afnanch" href="#fn485" id="fnanch485">485</a>. -Some such explanation as this (to my thinking) -must be invoked to account for the minute and varied and very -beautiful patterns upon many diatoms, the resemblance of which -patterns (in certain of their simpler cases) to the Chladni figures -is sometimes striking and obvious. But the many special problems -which the diatom skeleton suggests I have not attempted -to consider.</p> - -<div class="dctr02" id="fig230"> -<img src="images/i476.png" width="704" height="309" alt=""> - <div class="dcaption">Fig. 230.</div></div> - -<p>The last peculiarity of our Nassellarian lies in an -apparent departure from what we should at first expect in -the way of its <span class="xxpn" id="p476">{476}</span> -external symmetry. Were the system actually composed of four -spherical vesicles in mutual contact, the outer margin of each of -the six interfacial planes would obviously be a circular arc; and -accordingly, at each angle of the tetrahedron, we should expect -to have a depressed, or re-entrant angle, instead of a prominent -cusp. This is all doubtless due to some simple balance of tensions, -whose precise nature and distribution is meanwhile a matter of -conjecture. But it seems as though an extremely simple explanation -would go a long way, and possibly the whole way, to meet -this particular case. In our ordinary plane diagram of three cells, -or soap-bubbles, in contact, we know (and we have just said) -that the tensions of the three partitions draw inwards the outer -walls of the system, till at each point of triple contact (<i>P</i>) we tend -to get a triradiate, equiangular junction. But if we introduce -another bubble into the centre of the system (Fig. <a href="#fig230" title="go to Fig. 230">230</a>), then, as -Plateau shewed, the tensions of its walls and those of the three -partitions by which it is now suspended, again balance one -another, and the central bubble appears (in plane projection) as -a curvilinear, equilateral triangle. We have only got to convert -this plane diagram into that of a tetrahedral solid to obtain <i>almost</i> -precisely the configuration which we are seeking to explain. -Now we observe that, so far as our figure of Callimitra informs -us, this is just the shape of the little bubble which occupies the -centre of the tetrahedral system in that Radiolarian skeleton. -And I conceive, accordingly, that the entire organism was not -limited to the four cells or vesicles (together with -the little central <span class="xxpn" id="p477">{477}</span> -fifth) which we have hitherto been imagining, but there must have -been an outer tetrahedral system, enclosing the cells which fabricated -the skeleton, just as these latter enclosed, and deformed, -the little bubble in the centre of all. We have only to suppose -that this hypothetical tetrahedral series, forming the outer layer or -surface of the whole system, was for some chemico-physical reason -incapable of secreting at its interfacial contacts a skeletal fabric<a class="afnanch" href="#fn486" id="fnanch486">486</a>.</p> - -<p>In this hypothetical case, the edges of the skeletal system would -be circular arcs, meeting one another at an angle of 120°, or, in the -solid pyramid, of 109°: and this latter is <i>very nearly</i> the condition -which our little skeleton actually displays. But we observe in -Fig. <a href="#fig227" title="go to Fig. 227">227</a> that, in the immediate neighbourhood of the tetrahedral -angle, the circular arcs are slightly drawn out into projecting -cusps (cf. Fig. <a href="#fig230" title="go to Fig. 230">230</a>, <i>B</i>). There is -no <span class="nowrap"><img class="iglyph-a" -src="images/glyph-s.png" width="32" height="46" alt="S">-shaped</span> -curvature of the -tetrahedral edges as a whole, but a very slight one, a very slight -change of curvature; close to the apex. This, I conceive, is -nothing more than what, in a material system, we are bound to -have, to represent a “surface of continuity.” It is a phenomenon -precisely analogous to Plateau’s “bourrelet,” which we have -already seen to be a constant feature of all cellular systems, -rounding off the sharp angular contacts by which (in our more -elementary treatment) we expect one film to make its junction -with another<a class="afnanch" href="#fn487" id="fnanch487">487</a>.</p> - -<hr class="hrblk"> - -<p>In the foregoing examples of Radiolaria, the symmetry which -the organism displays would seem to be identical with that -symmetry of forces which is due to the assemblage of surface-tensions -in the whole system; this symmetry being displayed, in -one class of cases, in a complex spherical mass -of froth, and in <span class="xxpn" id="p478">{478}</span> -another class in a simpler aggregate of a few, otherwise isolated, -vesicles. But among the vast number of other known Radiolaria, -there are certain forms (especially among the Phaeodaria and -Acantharia) which display a still more remarkable symmetry, the -origin of which is by no means clear, though surface-tension -doubtless plays a part in its causation. These are cases in which -(as in some of those already described) the skeleton consists -(1) of radiating spicular rods, definite in number and position, -and (2) of interconnecting rods or plates, tangential to the more -or less spherical body of the organism, whose form becomes, -accordingly, that of a geometric, polyhedral solid. It may be -that there is no mathematical difference, save one of degree, -between such a hexagonal polyhedron as we have seen in <i>Aulacantha</i>, -and those which we are about to describe; but the greater -regularity, the numerical symmetry, and the apparent simplicity -of these latter, makes of them a class apart, and suggests problems -which have not been solved nor even investigated.</p> - -<p>The matter is sufficiently illustrated by the -accompanying figures, all drawn from Haeckel’s Monograph of -the Challenger Radiolaria<a class="afnanch" href="#fn488" -id="fnanch488">488</a>. In one of these we see a -regular octahedron, in another a regular, or pentagonal -dodecahedron, in a third a regular icosahedron. In all -cases the figure appears to be perfectly symmetrical, -though neither the triangular facets of the octahedron and -icosahedron, nor the pentagonal facets of the dodecahedron, -are necessarily plane surfaces. In all of these cases, -the radial spicules correspond to the solid angles of the -figure; and they are, accordingly, six in number in the -octahedron, twenty in the dodecahedron, and twelve in the -icosahedron. If we add to these three figures the regular -tetrahedron, which we have had frequent occasion to study, -and the cube (which is represented, at least in outline, -in the skeleton of the hexactinellid sponges), we have -completed the series of the five regular polyhedra known -to geometers, the <i>Platonic bodies</i><a class="afnanch" -href="#fn489" id="fnanch489">489</a> of the older -mathematicians. It is</p> - -<div class="dctr02" id="fig231"> -<img src="images/i479.png" width="528" height="683" alt=""> - <div class="pcaption">Fig. 231. Skeletons of - various Radiolarians, after Haeckel. 1. <i>Circoporus - sexfurcus</i>; 2. <i>C. octahedrus</i>; 3. <i>Circogonia - icosahedra</i>; 4. <i>Circospathis novena</i>; 5. <i>Circorrhegma - dodecahedra</i>.</div></div> - -<p class="pcontinue">at first sight all the more remarkable that -we should here meet <span class="xxpn" id="p480">{480}</span> -with the whole five regular polyhedra, when we remember that, -among the vast variety of crystalline forms known among minerals, -the regular dodecahedron and icosahedron, simple as they are -from the mathematical point of view, never occur. Not only do -these latter never occur in Crystallography, but (as is explained -in text-books of that science) it has been shewn that they cannot -occur, owing to the fact that their indices (or numbers expressing -the relation of the faces to the three primary axes) involve an -irrational quantity: whereas it is a fundamental law of crystallography, -involved in the whole theory of space-partitioning, that -“the indices of any and every face of a crystal are small whole -numbers<a class="afnanch" href="#fn490" id="fnanch490">490</a>.” -At the same time, an imperfect pentagonal dodecahedron, -whose pentagonal sides are non-equilateral, is common -among crystals. If we may safely judge from Haeckel’s figures, -the pentagonal dodecahedron of the Radiolarian is perfectly -regular, and we must presume, accordingly, that it is not brought -about by principles of space-partitioning similar to those which -manifest themselves in the phenomenon of crystallisation. It -will be observed that in all these radiolarian polyhedral shells, -the surface of each external facet is formed of a minute hexagonal -network, whose probable origin, in relation to a vesicular -structure, is such as we have already discussed.</p> - -<p>In certain allied Radiolaria (Fig. <a href="#fig232" title="go to Fig. 232">232</a>), which, like the dodecahedral -form figured in Fig. <a href="#fig231" title="go to Fig. 231">231</a>, 5, have twenty radial spines, these -latter are commonly described as being arranged in a certain very -singular way. It is stated that their arrangement -may be referred <span class="xxpn" id="p481">{481}</span> -to a series of five parallel circles on the sphere, corresponding to the -equator (<i>c</i>), the tropics (<i>b</i>, <i>d</i>) and the polar circles (<i>a</i>, <i>e</i>); and that -beginning with four equidistant spines in the equator, we have -alternating whorls of four, radiating outwards from the sphere in -each of the other parallel zones. This rule was laid down by the -celebrated Johannes Müller, and has ever since been used and -quoted as Müller’s law. The chief point in this alleged arrangement -which strikes us at first sight as very curious, is that there -is said to be no spine at either pole; and when we come to examine -carefully the figure of the organism, we find that the received</p> - -<div class="dctr01" id="fig232"> -<img src="images/i481.png" width="800" height="461" alt=""> - <div class="dcaption">Fig. 232. <i>Dorataspis</i> sp.; - diagrammatic.</div></div> - -<p class="pcontinue">description does not do justice to the facts. We see, in the first -place, from such figures as Figs. <a href="#fig232" title="go to Fig. 232">232</a>, 234, that here, unlike our -former cases, the radial spines issue through the facets (and through -<i>all</i> the facets) of the polyhedron, instead of through its solid angles; -and accordingly, that our twenty spines correspond (not, as before, -to a dodecahedron) but to some sort of an icosahedron. We see -in the next place, that this icosahedron is composed of faces, or -plates, of two different kinds, some hexagonal and some pentagonal; -and when we look closer, we discover that the whole -figure is that of a hexagonal prism, whose twelve solid angles are -replaced by pentagonal facets. Both -hexagons and pentagons <span class="xxpn" id="p482">{482}</span> -appear to be perfectly equilateral, but if we try to construct a -plane-sided polyhedron of this kind, we soon find that it is -impossible; for into the angles between the six equatorial hexagons -those of the six united pentagons will not fit. The figure however -can be easily constructed if we replace the straight edges (or some -of them) by curves, and the plane facets by corresponding, slightly -curved, surfaces. The true symmetry of this figure, then, is -hexagonal, with a polar axis, produced into two polar spicules; -with six equatorial spicules, or rays; and with two sets of six -spicular rays, interposed between the polar axis and the equatorial -rays, and alternating in position with the latter.</p> - -<div class="psmprnt3"> -<p>Müller’s description was emended by Brandt, and what is now known as -“Brandt’s law,” viz. that the symmetry consists of two polar rays, and three -whorls of six each, coincides with the above description so far as the spicular -axes go: save only that Brandt specifically states that the intermediate -whorls stand equidistant between the equator and the poles, i.e. in latitude 45°. -While not far from the truth, this statement is not exact; for according to -the geometry of the figure, the intermediate cycles obviously stand in a slightly -higher latitude, but this latitude I have not attempted to determine; for -the calculation seems to be a little troublesome owing to the curvature of -the sides of the figure, and the enquiring mathematician will perform it more -easily than I. Brandt, if I understand him rightly, did not propose his -“law” as a substitute for Müller’s law, but as a second law applicable to a few -particular cases. I on the other hand can find no case to which Müller’s law -properly applies.</p> -</div><!--psmprnt3--> - -<p>If we construct such a polyhedron, and set it in the position -of Fig. <a href="#fig232" title="go to Fig. 232">232</a>, <i>B</i>, we shall easily see that it is capable of explanation -(though improperly) in accordance with Müller’s law; for the -four equatorial rays of Müller (<i>c</i>) now correspond to the two polar -and to two opposite equatorial facets of our polyhedron: the -four “polar” rays of Müller (<i>a</i> or <i>e</i>) correspond to two adjacent -hexagons and two intermediate pentagons of the figure: and -Müller’s “tropical” rays (<i>b</i> or <i>d</i>) are those which emanate from the -remaining four pentagonal facets, in each half of the figure. In -some cases, such as Haeckel’s <i>Phatnaspis cristata</i> (Fig. <a href="#fig233" title="go to Fig. 233">233</a>), we -have an ellipsoidal body, from which the spines emerge in the -order described, but which is not obviously divided by facets. -In Fig. <a href="#fig234" title="go to Fig. 234">234</a> I have indicated the facets corresponding to the rays, -and dividing the surface in the -usual symmetrical way. <span class="xxpn" id="p483">{483}</span></p> - -<div class="dctr03" id="fig233"> -<img src="images/i483a.png" width="607" height="592" alt=""> - <div class="dcaption">Fig. 233. <i>Phatnaspis cristata</i>, Hkl.</div></div> - -<div class="dctr03" id="fig234"> -<img src="images/i483b.png" width="607" height="589" alt=""> - <div class="dcaption">Fig. 234. The same, diagrammatic.</div></div> - -<div><span class="xxpn" id="p484">{484}</span></div> - -<p>Within any polyhedron we may always inscribe another -polyhedron, whose corners correspond in number to the sides or -facets of the original figure, or (in alternative cases) to a certain -number of these sides; and a similar result is obtained by bevelling -off the corners of the original polyhedron. We may obtain a -precisely similar symmetrical result if (in such a case as these -Radiolarians which we are describing), we imagine the radial -spines to be interconnected by tangential rods, instead of by the -complete facets which we have just been dealing with. In our -complicated polyhedron with its twenty radial spines arranged in -the manner described there are various symmetrical ways in which -we may imagine these interconnecting bars to be arranged. The -most symmetrical of these is one in which the whole surface is -divided into eighteen rhomboidal areas, obtained by systematically -connecting each group of four adjacent radii. This figure has -eighteen faces (<i>F</i>), twenty corners (<i>C</i>), and therefore thirty-six -edges (<i>E</i>), in conformity with Euler’s theorem, <i>F</i> + <i>C</i> -= <i>E</i> + 2.</p> - -<div class="dleft dwth-d" id="fig235"> -<img src="images/i484.png" width="385" height="433" alt=""> - <div class="dcaption">Fig. 235. <i>Phractaspis prototypus</i>, Hkl.</div></div> - -<p class="pcontinue">Another symmetrical arrangement -will divide the surface into fourteen rhombs and eight -triangles. This latter arrangement is obtained by linking -up the radial rods as follows: <i>aaaa</i>, <i>aba</i>, <i>abcb</i>, -<i>bcdc</i>, etc. Here we have again twenty corners, but we -have twenty-two faces; the number of edges, or tangential -spicular bars, will be found, therefore, by the above -formula, to be forty. In Haeckel’s figure of <i>Phractaspis -prototypus</i> we have a spicular skeleton which appears -to be constructed precisely upon this plan, and to be -derivable from the faceted polyhedron precisely after this -manner. <br class="brclrfix"></p> - -<p>In all these latter cases it is the arrangement of the axial -rods, or in other words the “polar symmetry” of the entire -organism, which lies at the root of the matter, and -which, if only <span class="xxpn" id="p485">{485}</span> -we could account for it, would make it comparatively easy to -explain the superficial configuration. But there are no obvious -mechanical forces by which we can so explain this peculiar -polarity. This at least is evident, that it arises in the central -mass of protoplasm, which is the essential living portion of the -organism as distinguished from that frothy peripheral mass whose -structure has helped us to explain so many phenomena of the -superficial or external skeleton. To say that the arrangement -depends upon a specific polarisation of the cell is merely to refer -the problem to other terms, and to set it aside for future solution. -But it is possible that we may learn something about the lines in -which <i>to seek for</i> such a solution by considering the case of -Lehmann’s “fluid crystals,” and the light which they throw upon -the phenomena of molecular aggregation.</p> - -<p>The phenomenon of “fluid crystallisation” is found in a -number of chemical bodies; it is exhibited at a specific temperature -for each substance; and it would seem to be limited to bodies -in which there is a more or less elongated, or “chain-like” arrangement -of the atoms in the molecule. Such bodies, at the appropriate -temperature, tend to aggregate themselves into masses, which are -sometimes spherical drops or globules (the so-called “spherulites”), -and sometimes have the definite form of needle-like or prismatic -crystals. In either case they remain liquid, and are also doubly -refractive, polarising light in brilliant colours. Together with -them are formed ordinary solid crystals, also with characteristic -polarisation, and into such solid crystals all the fluid material -ultimately turns. It is evident that in these liquid crystals, -though the molecules are freely mobile, just as are those of water, -they are yet subject to, or endowed with, a “directive force,” -a force which confers upon them a definite configuration or -“polarity,” the <i>Gestaltungskraft</i> of Lehmann.</p> - -<p>Such an hypothesis as this had been gradually extruded from -the theories of mathematical crystallography<a class="afnanch" href="#fn491" id="fnanch491">491</a>; -and it had come -to be believed that the symmetrical conformation of a homogeneous -crystalline structure was sufficiently explained by the -mere mechanical fitting together of appropriate structural units -along the easiest and simplest lines of “close -packing”: just as <span class="xxpn" id="p486">{486}</span> -a pile of oranges becomes definite, both in outward form and -inward structural arrangement, without the play of any <i>specific</i> -directive force. But while our conceptions of the tactical arrangement -of crystalline molecules remain the same as before, and our -hypotheses of “modes of packing” or of “space-lattices” remain -as useful as ever for the definition and explanation of the -molecular arrangements, an entirely new theoretical conception -is introduced when we find such space-lattices maintained in -what has hitherto been considered the molecular freedom of a -liquid field; and we are constrained, accordingly, to postulate -a specific molecular force, or “Gestaltungskraft” (not unlike -Kepler’s “facultas formatrix”), to account for the phenomenon.</p> - -<p>Now just as some sort of specific “Gestaltungskraft” had -been of old the <i>deus ex machina</i> accounting for all crystalline -phenomena (<i>gnara totius geometriæ, et in ea exercita</i>, as Kepler -said), and as such an hypothesis, after being dethroned and -repudiated, has now fought its way back and has made good its -right to be heard, so it may be also in biology. We begin by an -easy and general assumption of <i>specific properties</i>, by which each -organism assumes its own specific form; we learn later (as it is -the purpose of this book to shew) that throughout the whole -range of organic morphology there are innumerable phenomena of -form which are not peculiar to living things, but which are more -or less simple manifestations of ordinary physical law. But every -now and then we come to certain deep-seated signs of protoplasmic -symmetry or polarisation, which seem to lie beyond the -reach of the ordinary physical forces. It by no means follows -that the forces in question are not essentially physical forces, more -obscure and less familiar to us than the rest; and this would seem -to be the crucial lesson for us to draw from Lehmann’s surprising -and most beautiful discovery. For Lehmann seems actually to -have demonstrated, in non-living, chemical bodies, the existence -of just such a determinant, just such a “Gestaltungskraft,” as -would be of infinite help to us if we might postulate it for the -explanation (for instance) of our Radiolarian’s axial symmetry. -But further than this we cannot go; for such analogy as we seem -to see in the Lehmann phenomenon soon evades us, and refuses -to be pressed home. Not only is it the case, as -we have already <span class="xxpn" id="p487">{487}</span> -seen, that certain of the geometric forms assumed by the symmetrical -Radiolarian shells are just such as the “space-lattice” -theory would seem to be inapplicable to, but it is in other ways -obvious that symmetry of <i>crystallisation</i>, whether liquid or solid, -has no close parallel, but only a series of analogies, in the protoplasmic -symmetry of the living cell.</p> - -<div class="chapter" id="p488"> -<h2 class="h2herein" title="X. A Parenthetic Note on Geodetics.">CHAPTER X -<span class="h2ttl"> -A PARENTHETIC NOTE ON GEODETICS</span></h2></div> - -<p>We have made use in the last chapter of the mathematical -principle of Geodetics (or Geodesics) in order to explain the conformation -of a certain class of sponge-spicules; but the principle -is of much wider application in morphology, and would seem to -deserve attention which it has not yet received.</p> - -<div class="dleft dwth-d" id="fig236"> -<img src="images/i488.png" width="384" height="436" alt=""> - <div class="dcaption">Fig. 236. Annular and spiral thickenings - in the walls of plant-cells.</div></div> - -<p>Defining, meanwhile, our geodetic line (as we have -already done) as the shortest distance between two points -on the surface of a solid of revolution, we find that the -geodetics of the cylinder give us one of the simplest of -cases. Here it is plain that the geodetics are of three -kinds: (1) a series of annuli around the cylinder, that -is to say, a system of circles, in planes parallel to one -another and at right angles to the axis of the cylinder -(Fig. <a href="#fig236" title="go to Fig. 236">236</a>, <i>a</i>); (2) a series of straight lines parallel to -the axis; and (3) a series of spiral curves winding round -the wall of the cylinder (<i>b</i>, <i>c</i>). These three systems -are all of frequent occurrence, and are all illustrated in -the local thickenings of the wall of the cylindrical cells -or vessels of plants. <br class="brclrfix"></p> - -<p>The spiral, or rather helicoid, geodetic is particularly common -in cylindrical structures, and is beautifully shewn for instance in -the spiral coil which stiffens the tracheal tubes of an insect, or -the so-called “tracheides” of a woody -stem. A similar <span class="xxpn" id="p489">{489}</span> -phenomenon is often witnessed in the splitting of a glass tube. If a -crack appear in a thin tube, such as a test-tube, it has a tendency -to be prolonged in its own direction, and the more perfectly -homogeneous and isotropic be the glass the more evenly will the -split tend to follow the straight course in which it began. As -a result, the crack in our test-tube is often seen to continue till -the tube is split into a continuous spiral ribbon.</p> - -<p>In a right cone, the spiral geodetic falls into closer and closer -coils as the diameter of the cone narrows; and a very beautiful -geodetic of this kind is exemplified in the sutural line of a spiral -shell, such as Turritella, or in the striations which run parallel -with the spiral suture. Similarly, in an ellipsoidal surface, we -have a spiral geodetic, whose coils get closer together as we -approach the ends of the long axis of the ellipse; in the splitting -of the integument of an Equisetum-spore, by which are formed -the spiral “elaters” of the spore, we have a case of this kind, -though the spiral is not sufficiently prolonged to shew all its -features in detail.</p> - -<p>We have seen in these various cases, that our original definition -of a geodetic requires to be modified; for it is only subject to -conditions that it is “the shortest distance between two points -on the surface of the solid,” and one of the commonest of these -restricting conditions is that our geodetic may be constrained to -go twice, or many times, round the surface on its way. In short, -we must redefine our geodetic, as a curve drawn upon a surface, -such that, if we take any two <i>adjacent</i> points on the curve, -the curve gives the shortest distance between them. Again, -in the geodetic systems which we meet with in morphology, it -sometimes happens that we have two opposite systems of geodetic -spirals separate and distinct from one another, as in Fig. <a href="#fig236" title="go to Fig. 236">236</a>, <i>c</i>; -and it is also common to find the two systems interfering with -one another, and forming a criss-cross, or reticulated arrangement. -This is a very common source of reticulated patterns.</p> - -<p>Among the ciliated Infusoria, we have in the spiral lines along -which their cilia are arranged a great variety of beautiful geodetic -curves; though it is probable enough that in some complicated -cases these are not simple geodetics, but projections of curves -other than a straight line upon the surface -of the solid. <span class="xxpn" id="p490">{490}</span></p> - -<p>Lastly, a very instructive case is furnished by the arrangement -of the muscular fibres on the surface of a hollow organ, such as -the heart or the stomach. Here we may consider the phenomenon -from the point of view of mechanical efficiency, as well as from -that of purely descriptive or objective anatomy. In fact we have -an <i>a priori</i> right to expect that the muscular fibres covering such -hollow or tubular organs will coincide with geodetic lines, in the -sense in which we are now using the term. For if we imagine a -contractile fibre, or elastic band, to be fixed by its two ends upon -a curved surface, it is obvious that its first effort of contraction -will tend to expend itself in accommodating the band to the -form of the surface, in “stretching it tight,” or in other words -in causing it to assume a direction which is the shortest possible -line <i>upon the surface</i> between the two extremes: and it is only -then that further contraction will have the effect of constricting -the tube and so exercising pressure on its contents. Thus the -muscular fibres, as they wind over the curved surface of an organ, -arrange themselves automatically in geodesic curves: in precisely -the same manner as we also automatically construct complex -systems of geodesics whenever we wind a ball of wool or a spindle -of tow, or when the skilful surgeon bandages a limb. In these -latter cases we see the production of those “figures-of-eight,” to -which, in the case for instance of the heart-muscles, Pettigrew -and other anatomists have ascribed peculiar importance. In the -case of both heart and stomach we must look upon these organs -as developed from a simple cylindrical tube, after the fashion of -the glass-blower, as is further discussed on p. <a href="#p737" title="go to pg. 737">737</a> of this book, -the modification of the simple cylinder consisting of various degrees -of dilatation and of twisting. In the primitive undistorted -cylinder, as in an artery or in the intestine, the muscular fibres -run in geodetic lines, which as a rule are not spiral, but are merely -either annular or longitudinal; these are the ordinary “circular -and longitudinal coats,” which form the normal musculature of -all tubular organs, or of the body-wall of a cylindrical worm<a class="afnanch" href="#fn492" id="fnanch492">492</a>. -If -we consider each muscular fibre as an elastic strand, imbedded in -the elastic membrane which constitutes the wall -of the organ, it <span class="xxpn" id="p491">{491}</span> -is evident that, whatever be the distortion suffered by the entire -organ, the individual fibre will follow the same course, which will -still, in a sense, be a geodetic. But if the distortion be considerable, -as for instance if the tube become bent upon itself, or if at -some point its walls bulge outwards in a diverticulum or pouch, -it is obvious that the old system of geodetics will only mark the -shortest distance between two points more or less approximate to -one another, and that new systems of geodetics will tend to -appear, peculiar to the new surface, and linking up points more -remote from one another. This is evidently the case in the -human stomach. We still have the systems, or their unobliterated -remains, of circular and longitudinal muscles; but we also see -two new systems of fibres, both obviously geodetic (or rather, -when we look more closely, both parts of one and the same -geodetic system), in the form of annuli encircling the pouch or -diverticulum at the cardiac end of the stomach, and of oblique -fibres taking a spiral course from the neighbourhood of the -oesophagus over the sides of the organ.</p> - -<hr class="hrblk"> - -<p>In the heart we have a similar, but more complicated -phenomenon. Its musculature consists, in great part, of the -original simple system of circular and longitudinal muscles -which enveloped the original arterial tubes, which tubes, after -a process of local thickening, expansion, and especially <i>twisting</i>, -came together to constitute the composite, or double, mammalian -heart; and these systems of muscular fibres, geodetic to begin -with, remain geodetic (in the sense in which we are using the -word) after all the twisting to which the primitive cylindrical tube -or tubes have been subjected. That is to say, these fibres still -run their shortest possible course, from start to finish, over the -complicated curved surface of the organ; and it is only because -they do so that their contraction, or longitudinal shortening, is -able to produce its direct effect, as Borelli well understood, in -the contraction or systole of the heart<a class="afnanch" href="#fn493" id="fnanch493">493</a>. -<span class="xxpn" id="p492">{492}</span></p> - -<p>As a parenthetic corollary to the case of the spiral pattern -upon the wall of a cylindrical cell, we may consider for a -moment the spiral line which many small organisms tend to -follow in their path of locomotion<a class="afnanch" href="#fn494" id="fnanch494">494</a>. -The helicoid spiral, traced -around the wall of our cylinder, may be explained as a composition -of two velocities, one a uniform velocity in the direction of the -axis of the cylinder, the other a uniform velocity in a circle -perpendicular to the axis. In a somewhat analogous fashion, the -smaller ciliated organisms, such as the ciliate and flagellate -Infusoria, the Rotifers, the swarm-spores of various Protists, and -so forth, have a tendency to combine a direct with a revolving -path in their ordinary locomotion. The means of locomotion -which they possess in their cilia are at best somewhat primitive -and inefficient; they have no apparent means of steering, or -modifying their direction; and, if their course tended to swerve -ever so little to one side, the result would be to bring them round -and round again in an approximately circular path (such as a man -astray on the prairie is said to follow), with little or no progress -in a definite longitudinal direction. But as a matter of fact, -either through the direct action of their cilia or by reason of a -more or less unsymmetrical form of the body, all these creatures -tend more or less to <i>rotate</i> about their long axis while they swim. -And this axial rotation, just as in the case of a rifle-bullet, causes -their natural swerve, which is always in the same direction as -regards their own bodies, to be in a continually changing direction -as regards space: in short, to make a spiral course around, and -more or less approximate to, a straight axial line.</p> - -<div class="chapter" id="p493"> -<h2 class="h2herein" title="XI. The Logarithmic Spiral.">CHAPTER XI -<span class="h2ttl"> -THE LOGARITHMIC SPIRAL</span></h2></div> - -<p>The very numerous examples of spiral conformation which we -meet with in our studies of organic form are peculiarly adapted -to mathematical methods of investigation. But ere we begin to -study them, we must take care to define our terms, and we had -better also attempt some rough preliminary classification of the -objects with which we shall have to deal.</p> - -<p>In general terms, a Spiral Curve is a line which, starting from -a point of origin, continually diminishes in curvature as it recedes -from that point; or, in other words, whose <i>radius of curvature</i> -continually increases. This definition is wide enough to include -a number of different curves, but on the other hand it excludes -at least one which in popular speech we are apt to confuse with -a true spiral. This latter curve is the simple Screw, or cylindrical -Helix, which curve, as is very evident, neither starts from a definite -origin, nor varies in its curvature as it proceeds. The “spiral” -thickening of a woody plant-cell, the “spiral” thread within an -insect’s tracheal tube, or the “spiral” twist and twine of a climbing -stem are not, mathematically speaking, <i>spirals</i> at all, but <i>screws -or helices</i>. They belong to a distinct, though by no means very -remote, family of curves. Some of these helical forms we have -just now treated of, briefly and parenthetically, under the subject -of Geodetics.</p> - -<p>Of true organic spirals we have no lack<a -class="afnanch" href="#fn495" id="fnanch495">495</a>. -We think at once of the beautiful spiral curves of the -horns of ruminants, and of the still more varied, if not -more beautiful, spirals of molluscan shells. Closely -related spirals may be traced in the arrangement <span -class="xxpn" id="p494">{494}</span> of the florets in -the sunflower; a true spiral, though not, by the way, so -easy of investigation, is presented to us by the outline -of a cordate leaf; and yet again, we can recognise -typical though transitory spirals in the coil of an -elephant’s trunk, in the “circling <span class="xxpn" -id="p495">{495}</span> spires” of a snake, in the coils -of a cuttle-fish’s arm, or of a monkey’s or a chameleon’s -tail.</p> - -<div class="dctr02" id="fig237"> -<img src="images/i494.png" width="528" height="673" alt=""> - <div class="pcaption">Fig. 237. The shell of <i>Nautilus - pompilius</i>, from a radiograph: to shew the logarithmic - spiral of the shell, together with the arrangement of - the internal septa. (From Messrs Green and Gardiner, in - <i>Proc. Malacol. Soc.</i> <span class="smmaj">II,</span> - 1897.)</div></div> - -<p>Among such forms as these, and the many others which we -might easily add to them, it is obvious that we have to do with -things which, though mathematically similar, are biologically -speaking fundamentally different. And not only are they biologically -remote, but they are also physically different, in regard -to the nature of the forces to which they are severally due. For -in the first place, the spiral coil of the elephant’s trunk or of the -chameleon’s tail is, as we have said, but a transitory configuration, -and is plainly the result of certain muscular forces acting upon -a structure of a definite, and normally an essentially different, -form. It is rather a position, or an <i>attitude</i>, than a <i>form</i>, in the -sense in which we have been using this latter term; and, unlike -most of the forms which we have been studying, it has little or no -direct relation to the phenomenon of Growth.</p> - -<div class="dctr03" id="fig238"> -<img src="images/i495.png" width="608" height="257" alt=""> - <div class="dcaption">Fig. 238. A Foraminiferal - shell (Globigerina).</div></div> - -<p>Again, there is a manifest and not unimportant difference -between such a spiral conformation as is built up by the separate -and successive florets in the sunflower, and that which, in the -snail or Nautilus shell, is apparently a single and indivisible unit. -And a similar, if not identical difference is apparent between the -Nautilus shell and the minute shells of the Foraminifera, which -so closely simulate it; inasmuch as the spiral shells of these latter -are essentially composite structures, combined out of successive -and separate chambers, while the molluscan shell, though it may -(as in Nautilus) become secondarily subdivided, has grown as -one continuous tube. It follows from all this -that there cannot <span class="xxpn" id="p496">{496}</span> -possibly be a physical or dynamical, though there may well be -a mathematical <i>Law of Growth</i>, which is common to, and which -defines, the spiral form in the Nautilus, in the Globigerina, in the -ram’s horn, and in the disc of the sunflower.</p> - -<p>Of the spiral forms which we have now mentioned, every one -(with the single exception of the outline of the cordate leaf) is an -example of the remarkable curve known as the Logarithmic Spiral. -But before we enter upon the mathematics of the logarithmic -spiral, let us carefully observe that the whole of the organic forms -in which it is clearly and permanently exhibited, however different -they may be from one another in outward appearance, in nature -and in origin, nevertheless all belong, in a certain sense, to one -particular class of conformations. In the great majority of cases, -when we consider an organism in part or whole, when we look (for -instance) at our own hand or foot, or contemplate an insect or -a worm, we have no reason (or very little) to consider one part -of the existing structure as <i>older</i> than another; through and -through, the newer particles have been merged and commingled, -by intussusception, among the old; the whole outline, such as it -is, is due to forces which for the most part are still at work to -shape it, and which in shaping it have shaped it as a whole. But -the horn, or the snail-shell, is curiously different; for in each of -these, the presently existing structure is, so to speak, partly old -and partly new; it has been conformed by successive and continuous -increments; and each successive stage of growth, starting -from the origin, remains as an integral and unchanging portion -of the still growing structure, and so continues to represent what -at some earlier epoch constituted for the time being the structure -in its entirety.</p> - -<p>In a slightly different, but closely cognate way, the same is -true of the spirally arranged florets of the sunflower. For here -again we are regarding serially arranged portions of a composite -structure, which portions, similar to one another in form, <i>differ -in age</i>; and they differ also in magnitude in a strict ratio according -to their age. Somehow or other, in the logarithmic spiral the -<i>time-element</i> always enters in; and to this important fact, full of -curious biological as well as mathematical significance, we shall -afterwards return. <span class="xxpn" id="p497">{497}</span></p> - -<p>It is, as we have so often seen, an essential part of our whole -problem, to try to understand what distribution of forces is capable -of producing this or that organic form,—to give, in short, a -dynamical expression to our descriptive morphology. Now the -<i>general</i> distribution of forces which lead to the formation of a -spiral (whether logarithmic or other) is very easily understood; -and need not carry us beyond the use of very elementary mathematics.</p> - -<div class="dctr04" id="fig239"> -<img src="images/i497.png" width="532" height="512" alt=""> - <div class="dcaption">Fig. 239.</div></div> - -<p>If we imagine growth to act in a perpendicular direction, as for -example the upward force of growth in a growing stem (<i>OA</i>), then, -in the absence of other forces, elongation will as a matter of course -proceed in an unchanging direction, that is to say the stem will -grow straight upwards. Suppose now that there be some constant -<i>external force</i>, such as the wind, impinging on the growing stem; -and suppose (for simplicity’s sake) that this external force be in a -constant direction (<i>AB</i>) perpendicular to the intrinsic force of growth. -The direction of actual growth will be in the line of the resultant -of the two forces: and, since the external force is (by hypothesis) -constant in direction, while the internal force tends always to act in -the line of actual growth, it is obvious that our growing organism -will tend to be bent into a curve, to which, for -the time being, <span class="xxpn" id="p498">{498}</span> -the actual force of growth will be acting at a tangent. So long -as the two forces continue to act, the curve will approach, but -will never attain, the direction of <i>AB</i>, perpendicular to the original -direction <i>OA</i>. If the external force be constant in amount the -curve will approximate to the form of a hyperbola; and, at any -rate, it is obvious that it will never tend to assume a spiral -form.</p> - -<p>In like manner, if we consider a horizontal beam, fixed at one -end, the imposition of a weight at the other will bend the beam -into a curve, which, as the beam elongates or the weight increases, -will bring the weighted end nearer and nearer to the vertical. -But such a force, constant in direction, will obviously never curve -the beam into a spiral,—a fact so patent and obvious that it would -be superfluous to state it, were it not that some naturalists have -been in the habit of invoking gravity as the force to which may be -attributed the spiral flexure of the shell.</p> - -<p>But if, on the other hand, the deflecting force be <i>inherent</i> in -the growing body, or so connected with it in a system that its -direction (instead of being constant, as in the former case) changes -with the direction of growth, and is perpendicular (or inclined at -some constant angle) to this changing direction of the growing -force, then it is plain that there is no such limit to the deflection -from the normal, but the growing curve will tend to wind round -and round its point of origin. In the typical case of the snail-shell, -such an intrinsic force is manifestly present in the action -of the columellar muscle.</p> - -<p>Many other simple illustrations can be given of a spiral course -being impressed upon what is primarily rectilinear motion, by -any steady deflecting force which the moving body carries, so -to speak, along with it, and which continually gives a lop-sided -tendency to its forward movement. For instance, we have been told -that a man or a horse, travelling over a great prairie, is very apt -to find himself, after a long day’s journey, back again near to his -starting point. Here some small and imperceptible bias, such as -might for instance be caused by one leg being in a minute degree -longer or stronger than the other, has steadily deflected the forward -movement to one side; and has gradually brought the traveller -back, perhaps in a circle to the very point from which -he set out, <span class="xxpn" id="p499">{499}</span> -or else by a spiral curve, somewhere within reach and recognition -of it.</p> - -<p>We come to a similar result when we consider, for instance, -a cylindrical body in which forces of growth are at work tending -to its elongation, but these forces are unsymmetrically distributed. -Let the tendency to elongation along <i>AB</i> be of a magnitude proportional -to <i>BB′</i>, and that along <i>CD</i> be of a magnitude proportional -to <i>DD′</i>; and in each element parallel to <i>AB</i> and <i>CD</i>, let a parallel -force of growth, proportionately intermediate in magnitude, be at -work: and let <i>EFF′</i> be the middle line. Then at any cross-section -<i>BFD</i>, if we deduct the mean force <i>FF′</i>, we have a certain -positive force at <i>B</i>, equal to <i>Bb</i>, and an equal and opposite force -at <i>D</i>, equal to <i>Dd</i>. But <i>AB</i> and <i>CD</i> are not separate</p> - -<div class="dctr04" id="fig240"> -<img src="images/i499.png" width="529" height="317" alt=""> - <div class="dcaption">Fig. 240.</div></div> - -<p class="pcontinue">structures, -but are connected together, either by a solid core, or by the walls -of a tubular shell; and the forces which tend to separate <i>B</i> and -<i>D</i> are opposed, accordingly, by a <i>tension</i> in <i>BD</i>. It follows therefore, -that there will be a resultant force <i>BG</i>, acting in a direction -intermediate between <i>Bb</i> and <i>BD</i>, and also a resultant, <i>DH</i>, -acting at <i>D</i> in an opposite direction; and accordingly, after a -small increment of growth, the growing end of the cylinder will -come to lie, not in the direction <i>BD</i>, but in the direction <i>GH</i>. -The problem is therefore analogous to that of a beam to which -we apply a bending moment; and it is plain that the unequal -force of growth is equivalent to a “<i>couple</i>” which will impart to -our structure a curved form. For, if we regard the part <i>ABDC</i> -as practically rigid, and the part <i>BB′D′D</i> as -pliable, this couple <span class="xxpn" id="p500">{500}</span> -will tend to turn strips such as <i>B′D′</i> about an axis perpendicular -to the plane of the diagram, and passing through an intermediate -point <i>F′</i>. It is plain, also, since all the forces under consideration -are <i>intrinsic to the system</i>, that this tendency will be continuous, -and that as growth proceeds the curving body will assume either -a circular or a spiral form. But the tension which we have here -assumed to exist in the direction <i>BD</i> will obviously disappear if -we suppose a sufficiently rapid rate of growth in that direction. -For if we may regard the mouth of our tubular shell as <i>perfectly -extensible</i> in its own plane, so that it exerts no traction whatsoever -on the sides, then it will be drawn out into more and more elongated -ellipses, forming the more and more oblique orifices of a <i>straight</i> -tube. In other words, in such a structure as we have presupposed, -the existence or</p> - -<div class="dctr05" id="fig241"> -<img src="images/i500.png" width="449" height="287" alt=""> - <div class="dcaption">Fig. 241.</div></div> - -<p class="pcontinue"> -maintenance of a constant ratio between the -rates of extension or growth in the vertical and transverse directions -will lead, in general, to the development of a logarithmic spiral; -the magnitude of that ratio will determine the character (that is -to say, the constant angle) of the spiral; and the spirals so produced -will include, as special or limiting cases, the circle and the -straight line.</p> - -<p>We may dispense with the hypothesis of bending moments, -if we simply presuppose that the increments of growth take -place at a constant angle to the growing surface (as -<i>AB</i>), but more rapidly at <i>A</i> (which we shall call the -“outer edge”) than at <i>B</i>, and that this difference of -velocity maintains a constant ratio. Let us also assume -that the whole structure is rigid, the new accretions -solidifying as soon as they are laid on. For example, <span -class="xxpn" id="p501">{501}</span> let Fig. <a href="#fig242" title="go to Fig. 242">242</a> represent -in section the early growth of a Nautilus-shell, and let -the part <i>ARB</i> represent the earliest stage of all, which -in Nautilus is nearly semicircular. We have to find a law -governing the growth of the shell, such that each edge -shall develop into an equiangular spiral; and this law, -accordingly, must be the same for each edge, namely that at -each instant the direction of growth makes a constant angle -with a line drawn from a fixed point (called the pole of -the spiral) to the point at which growth is taking place. -This growth, we now find, may be considered as effected by -the continuous addition of similar quadrilaterals. Thus, -in Fig. <a href="#fig241" title="go to Fig. 241">241</a>, <i>AEDB</i> is a quadrilateral with <i>AE</i>, <i>DB</i> -parallel, and with the angle <i>EAB</i> of a certain definite</p> - -<div class="dctr04" id="fig242"> -<img src="images/i501.png" width="529" height="405" alt=""> - <div class="dcaption">Fig. 242.</div></div> - -<p class="pcontinue">magnitude, -= γ. Let <i>AB</i> and <i>ED</i> meet, when produced, in <i>C</i>; -and call the angle <i>ACE</i> (or <i>xCy</i>) -= β. Make the angle <i>yCz</i> -= angle -<i>xCy</i>, -= β. Draw <i>EG</i>, so that the angle <i>yEG</i> -= γ, meeting <i>Cz</i> in -<i>G</i>; and draw <i>DF</i> parallel to <i>EG</i>. It is then easy to show that -<i>AEDB</i> and <i>EGFD</i> are similar quadrilaterals. And, when we -consider the quadrilateral <i>AEDB</i> as having infinitesimal sides, -<i>AE</i> and <i>BD</i>, the angle γ tends to α, the constant angle of an equiangular -spiral which passes through the points <i>AEG</i>, and of a -similar spiral which passes through the points <i>BDF</i>; and the point -<i>C</i> is the pole of both of these spirals. In a particular limiting case, -when our quadrilaterals are all equal as well as similar,—which -will be the case when the angle γ (or the angles <i>EAC</i>, -etc.) is a <span class="xxpn" id="p502">{502}</span> -right angle,—the “spiral” curve will be a circular arc, <i>C</i> being the -centre of the circle.</p> - -<div class="psmprnt3"> -<p>Another, and a very simple illustration may be drawn from the -“cymose inflorescences” of the botanists, though the actual -mode of development of some of these structures is open to -dispute, and their nomenclature is involved in extraordinary -historical confusion<a class="afnanch" href="#fn496" -id="fnanch496">496</a>.</p> - -<div class="dleft dwth-f" id="fig243"> -<img src="images/i502.png" width="288" height="346" alt=""> - <div class="dcaption">Fig. 243. <i>A</i>, a helicoid, <i>B</i>, a scorpioid - cyme.</div></div> - -<p>In Fig. <a href="#fig243" title="go to Fig. 243">243</a><i>B</i> (which represents the <i>Cicinnus</i> of Schimper, or -<i>cyme unipare scorpioide</i> of Bravais, as seen in the Borage), -we begin with a primary shoot from which is given off, at a -certain definite angle, a secondary shoot: and from that in -turn, on the same side and at the same angle, another shoot, -and so on. The deflection, or curvature, is continuous and -progressive, for it is caused by no external force but only -by causes intrinsic in the system. And the whole system is -symmetrical: the angles at which the successive shoots are -given off being all equal, and the lengths of the shoots -diminishing <i>in constant ratio</i>. The result is that the -successive shoots, or successive increments of growth, are -tangents to a curve, and this curve is a true logarithmic -spiral. But while, in this simple case, the successive shoots -are depicted as lying <i>in a plane</i>, it may also happen that, -in addition to their successive angular divergence from one -another within that plane, they also tend to diverge by -successive equal angles <i>from</i> that plane of reference; and -by this means, there will be superposed upon the logarithmic -spiral a helicoid twist or screw. And, in the particular case -where this latter angle of divergence is just equal to 180°, -or two right angles, the successive shoots will once more -come to lie in a plane, but they will appear to come off from -one another on <i>alternate</i> sides, as in Fig. <a href="#fig243" title="go to Fig. 243">243</a> <i>A</i>. -This is the <i>Schraubel</i> or <i>Bostryx</i> of Schimper, the <i>cyme -unipare hélicoide</i> of Bravais. The logarithmic spiral is still -latent in it, as in the other; but is concealed from view by -the deformation resulting from the helicoid. The confusion of -nomenclature would seem to have arisen from the fact that many -botanists did not recognise (as the brothers Bravais did) the -mathematical significance of the latter case; but were led, by -the snail-like spiral of the scorpioid cyme, to transfer the -name “helicoid” to it. <br class="brclrfix"></p> -</div><!--psmprnt3--> - -<p>In the study of such curves as these, then, we speak of the -point of origin as the pole (<i>O</i>); a straight line having its extremity -in the pole and revolving about it, is called -the radius vector; <span class="xxpn" id="p503">{503}</span> -and a point (<i>P</i>) which is conceived as travelling along the radius -vector under definite conditions of velocity, will then describe our -spiral curve.</p> - -<p>Of several mathematical curves whose form and development -may be so conceived, the two most important (and the only two -with which we need deal), are those which are known as (1) the -equable spiral, or spiral of Archimedes, and (2) the logarithmic, -or equiangular spiral.</p> - -<div class="dctr05" id="fig244"> -<img src="images/i503.png" width="449" height="421" alt=""> - <div class="dcaption">Fig. 244.</div></div> - -<p>The former may be illustrated by the spiral coil in which a -sailor coils a rope upon the deck; as the rope is of uniform thickness, -so in the whole spiral coil is each whorl of the same breadth -as that which precedes and as that which follows it. Using -its ancient definition, we may define it by saying, that “If a -straight line revolve uniformly about its extremity, a point which -likewise travels uniformly along it will describe the equable -spiral<a class="afnanch" href="#fn497" id="fnanch497">497</a>.” -Or, putting the same thing into our more modern -words, “If, while the radius vector revolve uniformly about the -pole, a point (<i>P</i>) travel with uniform velocity along it, the curve -described will be that called the equable spiral, or spiral of -Archimedes.” <span class="xxpn" id="p504">{504}</span> -<br class="brclrfix"></p> - -<p>It is plain that the spiral of Archimedes may be compared to -a <i>cylinder</i> coiled up. And it is plain also that a radius (<i>r</i> -= <i>OP</i>), -made up of the successive and equal whorls, will increase in -<i>arithmetical</i> progression: and will equal a certain constant -quantity (<i>a</i>) multiplied by the whole number of whorls, or (more -strictly speaking) multiplied by the whole angle (θ) through -which it has revolved: so that <i>r</i> -= <i>a</i>θ.</p> - -<p>But, in contrast to this, in the logarithmic spiral of the Nautilus -or the snail-shell, the whorls gradually increase in breadth, -and do so in a steady and unchanging ratio. Our definition is -as follows: “If, instead of travelling with a <i>uniform</i> velocity, -our point move along the radius vector with <i>a velocity increasing -as its distance from the pole</i>, then the path described is called a -logarithmic spiral.” Each whorl which the radius vector intersects -will be broader than its predecessor in a definite ratio; the -radius vector will increase in length in <i>geometrical</i> progression, -as it sweeps through successive equal angles; and the equation -to the spiral will be <i>r</i> -= <i>a</i><sup>θ</sup> . As the spiral of Archimedes, in our -example of the coiled rope, might be looked upon as a coiled -cylinder, so may the logarithmic spiral, in the case of the shell, -be pictured as a <i>cone</i> coiled upon itself.</p> - -<p>Now it is obvious that if the whorls increase very slowly indeed, -the logarithmic spiral will come to look like a spiral of Archimedes, -with which however it never becomes identical; for it is incorrect -to say, as is sometimes done, that the Archimedean spiral is a -“limiting case” of the logarithmic spiral. The Nummulite is a -case in point. Here we have a large number of whorls, very -narrow, very close together, and apparently of equal breadth, -which give rise to an appearance similar to that of our coiled -rope. And, in a case of this kind, we might actually find that -the whorls <i>were</i> of equal breadth, being produced (as is apparently -the case in the Nummulite) not by any very slow and gradual -growth in thickness of a continuous tube, but by a succession of -similar cells or chambers laid on, round and round, determined as -to their size by constant surface-tension conditions and therefore -of unvarying dimensions. But even in this case we should -have no Archimedean spiral, but only a logarithmic spiral in -which the constant angle -approximated to 90°. <span class="xxpn" id="p505">{505}</span></p> - -<div class="psmprnt3"> -<p>For, in the logarithmic spiral, when α tends to 90°, -the expression <i>r</i> -= <i>a</i><sup>θ cot α</sup> -tends to <i>r</i> -= <i>a</i>(1 + θ cot α); -while the equation to the Archimedean spiral is -<i>r</i> -= <i>b</i>θ. The nummulite must always have a central core, or initial cell, -around which the coil is not only wrapped, <i>but out of which it springs</i>; and -this initial chamber corresponds to our <i>a′</i> in the expression -<i>r</i> -= <i>a′</i> + <i>a</i>θ cot α. The -outer whorls resemble those of an Archimedean spiral, because -of the other term <i>a</i>θ cot α in the same -expression. It follows from this that in all such cases the -whorls must be of excessively small breadth.</p> -</div><!--psmprnt3--> - -<p>There are many other specific properties of the logarithmic -spiral, so interrelated to one another that we may choose pretty -well any one of them as the basis of our definition, and deduce the -others from it either by analytical methods or by the methods of -elementary geometry. For instance, the equation <i>r</i> -= <i>a</i><sup>θ</sup> may be -written in the form log <i>r</i> -= θ log <i>a</i>, or θ -= (log <i>r</i>) ⁄ (log <i>a</i>), -or (since <i>a</i> is -a constant), θ -= <i>k</i> log <i>r</i>. Which is as much as to say that the -vector angles about the pole are proportional to the logarithms -of the successive radii; from which circumstance the name of the -“logarithmic spiral” is derived.</p> - -<div class="dright dwth-f" id="fig245"> -<img src="images/i505.png" width="288" height="367" alt=""> - <div class="dcaption">Fig. 245.</div></div> - -<p>Let us next regard our logarithmic spiral from the -dynamical point of view, as when we consider the forces -concerned in the growth of a material, concrete spiral. -In a growing structure, let the forces of growth exerted -at any point <i>P</i> be a force <i>F</i> acting along the line -joining <i>P</i> to a pole <i>O</i> and a force <i>T</i> acting in a -direction perpendicular to <i>OP</i>; and let the magnitude -of these forces be in the same constant ratio at all -points. It follows that the resultant of the forces <i>F</i> -and <i>T</i> (as <i>PQ</i>) makes a constant angle with the radius -vector. But the constancy of the angle between tangent -and radius vector at any point is a fundamental property -of the logarithmic spiral, and may be shewn to follow -from our definition of the curve: it gives to the curve -its alternative name of <i>equiangular spiral</i>. Hence in -a structure growing under the above conditions the form -of the boundary will be a logarithmic spiral. <span -class="xxpn" id="p506">{506}</span> <br class="brclrfix"></p> - -<div class="dleft dwth-i" id="fig246"> -<img src="images/i506a.png" width="177" height="307" alt=""> - <div class="dcaption">Fig. 246.</div></div> - -<div class="dmaths"> -<p>In such a spiral, radial growth and growth in the direction of -the curve bear a constant ratio to one another. For, if we consider -a consecutive radius vector, <i>OP′</i>, whose increment -as compared with <i>OP</i> is <i>dr</i>, while <i>ds</i> is the small -arc <i>PP′</i>, then</p> - -<div><i>dr ⁄ ds</i> -= cos α -= constant.</div></div><!--dmaths--> - -<p>In the concrete case of the shell, the distribution -of forces will be, originally, a little more complicated -than this, though by resolving the forces in question, -the system may be reduced to this simple form. And -furthermore, the actual distribution of forces will not -always be identical; for example, there is a distinct -difference between the cases (as in the snail) where -a columellar muscle exerts a definite traction in the -direction of the pole, and those (such as Nautilus) -where there is no columellar muscle, and where some other -force must be discovered, or postulated, to account for the -flexure. In the most frequent case, we have, as in Fig. <a -href="#fig247" title="go to Fig. 247">247</a>, three forces -to deal with, acting at a point, <i>p</i> : <i>L</i>, -acting<br class="brclrfix"></p> - -<div class="dleft dwth-c" id="fig247"> -<img src="images/i506b.png" width="433" height="418" alt=""> - <div class="dcaption">Fig. 247.</div></div> - -<p class="pcontinue"> -in the direction of the tangent to the curve, -and representing the force of longitudinal growth; <i>T</i>, -perpendicular to <i>L</i>, and representing the organism’s -tendency to grow in breadth; and <i>P</i>, the traction -exercised, in the direction of the pole, by the columellar -muscle. Let us resolve <i>L</i> and <i>T</i> into components along -<i>P</i> (namely <i>A′</i>, <i>B′</i>), and perpendicular to <i>P</i> (namely -<i>A</i>, <i>B</i>); we have now only two forces to consider, -viz. <i>P</i> − <i>A′</i> − <i>B′</i>, and -<i>A</i> − <i>B</i>. And these two latter we can -again resolve, if we please, so as to deal only with -forces in the direction of <i>P</i> and <i>T</i>. Now, the ratio -of these forces remaining constant, the locus of the -point <i>p</i> is an equiangular spiral. <span class="xxpn" -id="p507">{507}</span></p> - -<p>Furthermore we see how any <i>slight</i> change in any one of the -forces <i>P</i>, <i>T</i>, <i>L</i> will tend to modify the angle α, and produce a slight -departure from the absolute regularity of the logarithmic spiral. -Such slight departures from the absolute simplicity and uniformity -of the theoretic law we shall not be surprised to find, more or less -frequently, in Nature, in the complex system of forces presented -by the living organism. <br class="brclrfix"></p> - -<p>In the growth of a shell, we can conceive no simpler law than -this, namely, that it shall widen and lengthen in the same unvarying -proportions: and this simplest of laws is that which Nature tends -to follow. The shell, like the creature within it, grows in size -<i>but does not change its shape</i>; and the existence of this constant -relativity of growth, or constant similarity of form, is of the essence, -and may be made the basis of a definition, of the logarithmic -spiral.</p> - -<p>Such a definition, though not commonly used by mathematicians, -has been occasionally employed; and it is one from -which the other properties of the curve can be deduced with -great ease and simplicity. In mathematical language it would run -as follows: “Any [plane] curve proceeding from a fixed point -(which is called the pole), and such that the arc intercepted between -this point and any other whatsoever on the curve is always similar -to itself, is called an equiangular, or logarithmic, spiral<a class="afnanch" href="#fn498" id="fnanch498">498</a>.”</p> - -<p>In this definition, we have what is probably the most fundamental -and “intrinsic” property of the curve, namely the property -of continual similarity: and this is indeed the very property by -reason of which it is peculiarly associated with organic growth in -such structures as the horn or the shell, or the scorpioid cyme -which is described on p. <a href="#p502" title="go to pg. 502">502</a>. For it is peculiarly characteristic -of the spiral of a shell, for instance, that (under all normal circumstances) -it does not alter its shape as it grows; each increment is -geometrically similar to its predecessor, and the whole, at any -epoch, is similar to what constituted the whole at another and an -earlier epoch. We feel no surprise when the animal which secretes -the shell, or any other animal whatsoever, -grows by such <span class="xxpn" id="p508">{508}</span> -<i>symmetrical</i> expansion as to preserve its form unchanged; though -even there, as we have already seen, the unchanging form denotes -a nice balance between the rates of growth in various directions, -which is but seldom accurately maintained for long. But the -shell retains its unchanging form in spite of its <i>asymmetrical</i> -growth; it grows at one end only, and so does the horn. And -this remarkable property of increasing by <i>terminal</i> growth, but -nevertheless retaining unchanged the form of the entire figure, is -characteristic of the logarithmic spiral, and of no other mathematical -curve.</p> - -<div class="dctr05" id="fig248"> -<img src="images/i508.png" width="448" height="220" alt=""> - <div class="dcaption">Fig. 248.</div></div> - -<p>We may at once illustrate this curious phenomenon by drawing -the outline of a little Nautilus shell within a big one. We know, -or we may see at once, that they are of precisely the same shape; -so that, if we look at the little shell through a magnifying glass, -it becomes identical with the big one. But we know, on the other -hand, that the little Nautilus shell grows into the big one, not by -uniform growth or magnification in all directions, as is (though -only approximately) the case when the boy grows into the man, -but by growing <i>at one end only</i>.</p> - -<hr class="hrblk"> - -<p>Though of all curves, this property of continued similarity is -found only in the logarithmic spiral, there are very many rectilinear -figures in which it may be observed. For instance, as we may -easily see, it holds good of any right cone; for evidently, in Fig. <a href="#fig248" title="go to Fig. 248">248</a>, -the little inner cone (represented in its triangular section) may -become identical with the larger one either by magnification all -round (as in <i>a</i>), or simply by an increment at one end (as in <i>b</i>); -indeed, in the case of the cone, we have yet a third possibility, -for the same result is attained when it increases all round, save -only at the base, that is to say when the -triangular section increases <span class="xxpn" id="p509">{509}</span> -on two of its sides, as in <i>c</i>. All this is closely associated with the -fact, which we have already noted, that the Nautilus shell is but -a cone rolled up; in other words, the cone is but a particular -variety, or “limiting case,” of the spiral shell.</p> - -<p>This property, which we so easily recognise in the cone, would -seem to have engaged the particular attention of the most ancient -mathematicians even from the days of Pythagoras, and so, with -little doubt, from the more ancient days of that Egyptian school -whence he derived the foundations of his learning<a class="afnanch" href="#fn499" id="fnanch499">499</a>; -and its bearing -on our biological problem of the shell, though apparently indirect, -is yet so close that it deserves our further consideration.</p> - -<div class="dctr02" id="fig249"><div id="fig250"> -<img src="images/i509.png" width="705" height="242" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 249.</td> - <td></td> - <td>Fig. 250.</td></tr></table> -</div></div></div><!--dctr01--> - -<p>If, as in Fig. <a href="#fig249" title="go to Fig. -249">249</a>, we add to two sides of a square a symmetrical -<span class="nowrap"><em class="emltr">L</em>-shaped</span> -portion, similar in shape to what we call a “carpenter’s -square,” the resulting figure is still a square; -and the portion which we have added is called, by -Aristotle (<i>Phys.</i> <span class="smmaj">III,</span> 4), -a “gnomon.” Euclid extends the term to include the case -of any parallelogram<a class="afnanch" href="#fn500" -id="fnanch500">500</a>, whether rectangular or not (Fig. -<a href="#fig250" title="go to Fig. 250">250</a>); and -Hero of Alexandria specifically defines a “gnomon” (as -indeed Aristotle implicitly defines it), as any figure -which, being added to any figure whatsoever, leaves the -resultant figure similar to the original. Included in this -important definition is the case of numbers, considered -geometrically; that is to say, the εἰδητικοὶ ἀριθμοί, which -can be translated into <i>form</i>, by means of rows of dots or -other signs (cf. Arist. <i>Metaph.</i> 1092 b 12), -or in the pattern of a tiled floor: all according to “the -mystical way of <span class="xxpn" id="p510">{510}</span> -Pythagoras, and the secret magick of numbers.” Thus for -example, the odd numbers are “gnomonic numbers,” because</p> - -<div class="dmaths"><div class="nowrap pleft"> -0 + 1 -= 1<sup>2</sup> ,<br> - -1<sup>2</sup> + 3 -= 2<sup>2</sup> ,<br> - -2<sup>2</sup> + 5 -= 3<sup>2</sup> ,<br> - -3<sup>2</sup> + 7 -= 4<sup>2</sup> <i>etc.</i>,</div> -</div><!--dmaths--> - -<p class="pcontinue">which relation we may illustrate -graphically σχηματογραφεῖν by the successive numbers of -dots which keep the annexed figure a perfect square<a -class="afnanch" href="#fn501" id="fnanch501">501</a>: as -follows:</p> - -<div class="dctr09"> -<img src="images/i510a.png" width="256" height="229" alt=""></div> - -<p>There are other gnomonic figures more curious still. For -instance, if we make a rectangle (Fig. <a href="#fig251" title="go to Fig. 251">251</a>) such that the two sides</p> - -<div class="dctr02" id="fig251"><div id="fig252"> -<img src="images/i510b.png" width="704" height="445" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 251.</td> - <td></td> - <td>Fig. 252.</td></tr></table> -</div></div></div><!--dctr01--> - -<p class="pcontinue"> are in the ratio of -1 : √2, it is obvious that, on doubling -it, we obtain a precisely similar figure; for -1 : √2 :: √2 : 2; -and <span class="xxpn" id="p511">{511}</span> each half -of the figure, accordingly, is now a gnomon to the -other. Another elegant example is when we start with -a rectangle (<i>A</i>) whose sides are in the proportion -of 1 : ½(√5 − 1), or, -approximately, 1 : 0·618. The gnomon to this -figure is a square (<i>B</i>) erected on its longer side, and so -on successively (Fig. <a href="#fig252" title="go to Fig. 252">252</a>).</p> - -<div class="dctr02" id="fig253"><div id="fig254"> -<img src="images/i511.png" width="704" height="362" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 253.</td> - <td></td> - <td>Fig. 254.</td></tr></table> -</div></div></div><!--dctr01--> - -<p>In any triangle, as Aristotle tells us, one part is always a -gnomon to the other part. For instance, in the triangle <i>ABC</i> -(Fig. <a href="#fig253" title="go to Fig. 253">253</a>), let us draw <i>CD</i>, so as to make the angle <i>BCD</i> equal to -the angle <i>A</i>. Then the part <i>BCD</i> is a triangle similar to the -whole triangle <i>ABC</i>, and <i>ADC</i> is a gnomon to <i>BCD</i>. A very -elegant case is when the original triangle <i>ABC</i> is an isosceles -triangle having one angle of 36°, and the other two angles, therefore, -each equal to 72° (Fig. <a href="#fig254" title="go to Fig. 254">254</a>). Then, by bisecting one of the -angles of the base, we subdivide the large isosceles triangle into -two isosceles triangles, of which one is similar to the whole figure -and the other is its gnomon<a class="afnanch" href="#fn502" id="fnanch502">502</a>. -There is good reason to believe -that this triangle was especially studied by the Pythagoreans; -for it lies at the root of many interesting geometrical constructions, -such as the regular pentagon, and the mystical “pentalpha,” and -a whole range of other curious figures beloved of the ancient -mathematicians<a class="afnanch" href="#fn503" id="fnanch503">503</a>. -<span class="xxpn" id="p512">{512}</span></p> - -<div class="dleft dwth-f" id="fig255"> -<img src="images/i512a.png" width="287" height="404" alt=""> - <div class="dcaption">Fig. 255.</div></div> - -<p>If we take any one of these figures, for instance the isosceles -triangle which we have just described, -and add to it (or subtract from it) in -succession a series of gnomons, so converting -it into larger and larger (or smaller -and smaller) triangles all similar to the -first, we find that the apices (or other -corresponding points) of all these triangles -have their <i>locus</i> upon a logarithmic spiral: -a result which follows directly from that -alternative definition of the logarithmic -spiral which I have quoted from Whitworth -(p. <a href="#p507" title="go to pg. 507">507</a>).</p> - -<p>Again, we may build up a series of -right-angled triangles, each of which is a -gnomon to the preceding figure; and here again, a logarithmic -spiral is the locus of corresponding points in these successive -triangles. And lastly, whensoever we fill up -space with <br class="brclrfix"></p> - -<div class="dctr03" id="fig256"> -<img src="images/i512.png" width="609" height="612" alt=""> - <div class="dcaption">Fig. 256. Logarithmic spiral derived from - corresponding points in a system of squares.</div></div> - -<p class="pcontinue"> -a <span class="xxpn" id="p513">{513}</span> -collection of either equal or similar figures, similarly situated, -as in Figs. <a href="#fig256" title="go to Fig. 256">256</a>, 257, there we can always discover a series of -inscribed or escribed logarithmic spirals.</p> - -<p>Once more, then, we may modify our definition, and say that: -“Any plane curve proceeding from a fixed point (or pole), and such -that the vectorial area of any sector is always a gnomon to the -whole preceding figure, is called an equiangular, or logarithmic, -spiral.” And we may now introduce this new concept and -nomenclature into our description of the Nautilus shell and -other related organic forms, by saying that: (1) if a growing</p> - -<div class="dctr01" id="fig257"> -<img src="images/i513.png" width="800" height="495" alt=""> - <div class="dcaption">Fig. 257. The same in a system of - hexagons.</div></div> - -<p class="pcontinue"> -structure be built up of successive parts, similar and similarly -situated, we can always trace through corresponding points -a series of logarithmic spirals (Figs. <a href="#fig258" title="go to Fig. 258">258</a>, 259, etc.); (2) it is -characteristic of the growth of the horn, of the shell, and of -all other organic forms in which a logarithmic spiral can be -recognised, that <i>each successive increment of growth is a gnomon -to the entire pre-existing structure</i>. And conversely (3) it follows -obviously, that in the logarithmic spiral outline of the shell -or of the horn we can always inscribe an endless variety of -other gnomonic figures, having no necessary relation, -save as a <span class="xxpn" id="p514">{514}</span> -mathematical accident, to the nature or mode of development -of the actual structure<a class="afnanch" href="#fn504" id="fnanch504">504</a>. -<span class="xxpn" id="p515">{515}</span></p> - -<div class="dctr01" id="fig258"> -<img src="images/i514a.png" width="800" height="453" alt=""> - <div class="pcaption">Fig. 258. A shell of Haliotis, with - two of the many lines of growth, or generating curves, - marked out in black: the areas bounded by these lines of - growth being in all cases “gnomons” to the pre-existing - shell.</div></div> - -<div class="dctr01" id="fig259"> -<img src="images/i514b.png" width="800" height="428" alt=""> - <div class="pcaption">Fig. 259. A spiral foraminifer - (<i>Pulvinulina</i>), to show how each successive chamber - continues the symmetry of, or constitutes a <i>gnomon</i> to, - the rest of the structure.</div></div> - -<p>Of these three propositions, the second is of very great use -and advantage for our easy understanding and simple description -of the molluscan shell, and of a great variety of other structures -whose mode of growth is analogous, and whose mathematical -properties are therefore identical. We see at once that the -successive chambers of a spiral Nautilus (Fig. <a href="#fig237" title="go to Fig. 237">237</a>) or of a straight -Orthoceras (Fig. <a href="#fig300" title="go to Fig. 300">300</a>), each whorl or part of a whorl of a periwinkle -or other gastropod (Fig. <a href="#fig258" title="go to Fig. 258">258</a>), each new increment of the -operculum of a gastropod (Fig. <a href="#fig263" title="go to Fig. 263">263</a>), each additional increment of</p> - -<div class="dright dwth-e" id="fig260"> -<img src="images/i515.png" width="337" height="354" alt=""> - <div class="dcaption">Fig. 260. Another spiral foraminifer, -<i>Cristellaria</i>.</div></div> - -<p class="pcontinue"> -an elephant’s tusk, or each new -chamber of a spiral foraminifer -(Figs. <a href="#fig259" title="go to Fig. 259">259</a> and 260), has its leading -characteristic at once described and -its form so far explained by the -simple statement that it constitutes -a <i>gnomon</i> to the whole previously -existing structure. And herein lies -the explanation of that “time-element” -in the development of -organic spirals of which we have -spoken already, in a preliminary -and empirical way. For it follows -as a simple corollary to this -theorem of gnomons that we must not expect to find the -logarithmic spiral manifested in a structure whose parts are -simultaneously produced, as for instance in the margin of a -leaf, or among the many curves that make the contour of a -fish. But we must rather look for it wherever the organism -retains for us, and still presents to us at a single view, the successive -phases of preceding growth, the successive magnitudes attained, -the successive outlines occupied, as the organism or a part thereof -pursued the even tenour of its growth, year by year and day by -day. And it easily follows from this, that it is in the hard parts -of organisms, and not the soft, fleshy, actively growing parts, -that this spiral is commonly and characteristically found; not -in the fresh mobile tissues whose form is constrained merely by -the active forces of the moment; but in things like shell and tusk, -and horn and claw, where the object is visibly -composed of parts <span class="xxpn" id="p516">{516}</span> -successively, and permanently, laid down. In the main, the -logarithmic spiral is characteristic, not of the living tissues, but -of the dead. And for the same reason, it will always or nearly -always be accompanied, and adorned, by a pattern formed of -“lines of growth,” the lasting record of earlier and successive -stages of form and magnitude. <br class="brclrfix"> -</p> - -<hr class="hrblk"> - -<p>It is evident that the spiral curve of the shell is, in a sense, -a vector diagram of its own growth; for it shews at each instant -of time, the direction, radial and tangential, of growth, and the -unchanging ratio of velocities in these directions. Regarding the -<i>actual</i> velocity of growth in the shell, we know very little (or -practically nothing), by way of experimental measurement; but -if we make a certain simple assumption, then we may go a good -deal further in our description of the logarithmic spiral as it appears -in this concrete case.</p> - -<p>Let us make the assumption that <i>similar</i> increments are added -to the shell in <i>equal</i> times; that is to say, that the amount of -growth in unit time is measured by the areas subtended by equal -angles. Thus, in the outer whorl of a spiral shell a definite area -marked out by ridges, tubercles, etc., has very different linear -dimensions to the corresponding areas of the inner whorl, but the -symmetry of the figure implies that it subtends an equal angle -with these; and it is reasonable to suppose that the successive -regions, marked out in this way by successive natural boundaries -or patterns, are produced in equal intervals of time.</p> - -<p>If this be so, the radii measured from the pole to the boundary -of the shell will in each case be proportional to the velocity of -growth at this point upon the circumference, and at the time when -it corresponded with the outer lip, or region of active growth; -and while the direction of the radius vector corresponds with the -direction of growth in thickness of the animal, so does the tangent -to the curve correspond with the direction, for the time being, of -the animal’s growth in length. The successive radii are a measure -of the acceleration of growth, and the spiral curve of the shell -itself is no other than the <i>hodograph</i> of the growth of the contained -organism. <span class="xxpn" id="p517">{517}</span></p> - -<p>So far as we have now gone, we have studied the elementary -properties of the logarithmic spiral, including its fundamental -property of <i>continued similarity</i>; and we have accordingly learned -that the shell or the horn tends <i>necessarily</i> to assume the form -of this mathematical figure, because in these structures growth -proceeds by successive increments, which are always similar in -form, similarly situated, and of constant relative magnitude one -to another. Our chief objects in enquiring further into the -mathematical properties of the logarithmic spiral will be: (1) to -find means of confirming and verifying the fact that the shell (or -other organic curve) is actually a logarithmic spiral; (2) to learn -how, by the properties of the curve, we may further extend our -knowledge or simplify our descriptions of the shell; and (3) to -understand the factors by which the characteristic form of any -particular logarithmic spiral is determined, and so to comprehend -the nature of the specific or generic characters by which one spiral -shell is found to differ from another.</p> - -<p>Of the elementary properties of the logarithmic spiral, so far as -we have now enumerated them, the following are those which we -may most easily investigate in the concrete case, such as we have -to do with in the molluscan shell: (1) that the polar radii of points -whose vectorial angles are in arithmetical progression, are themselves -in geometrical progression; and (2) that the tangent at any -point of a logarithmic spiral makes a constant -angle (called the <i>angle of the spiral</i>) with the -polar radius vector.</p> - -<div class="dright dwth-h" id="fig261"> -<img src="images/i517.png" width="208" height="415" alt=""> - <div class="dcaption">Fig. 261.</div></div> - -<p>The former of these two propositions may be written -in what is, perhaps, a simpler form, as follows: -radii which form equal angles about the pole of -the logarithmic spiral, are themselves continued -proportionals. That is to say, in Fig. <a href="#fig261" title="go to Fig. 261">261</a>, when -the angle <i>ROQ</i> is equal to the angle <i>QOP</i>, then -<i>OR</i> : <i>OQ</i> :: <i>OQ</i> : <i>OP</i>.</p> - -<p>A particular case of this proposition is when the equal -angles are each angles of 360°: that is to say when in -each case the radius vector makes a complete revolution, -and when, therefore <i>P</i>, <i>Q</i> and <i>R</i> all lie upon the same -radius. <span class="xxpn" id="p518">{518}</span></p> - -<p>It was by observing, with the help of very careful -measurement, this continued proportionality, that Moseley -was enabled to verify his first assumption, based on -the general appearance of the shell, that the shell of -Nautilus was actually a logarithmic spiral, and this -demonstration he was immediately afterwards in a position -to generalise by extending it to all the spiral Ammonitoid -and Gastropod mollusca<a class="afnanch" href="#fn505" -id="fnanch505">505</a>. <br class="brclrfix"></p> - -<p>For, taking a median transverse section of a <i>Nautilus pompilius</i>, -and carefully measuring the successive breadths of the whorls -(from the dark line which marks what was originally the outer -surface, before it was covered up by fresh deposits on the part -of the growing and advancing shell), Moseley found that “the -distance of any two of its whorls measured upon a radius vector -is one-third that of the two next whorls measured upon the same -radius vector<a class="afnanch" href="#fn506" id="fnanch506">506</a>. -Thus (in Fig. <a href="#fig262" title="go to Fig. 262">262</a>), <i>ab</i> is one-third of <i>bc</i>, <i>de</i> of -<i>ef</i>, <i>gh</i> of <i>hi</i>, and <i>kl</i> of <i>lm</i>. The curve is therefore a logarithmic -spiral.”</p> - -<p>The numerical ratio in the case of the Nautilus happens to -be one of unusual simplicity. Let us take, with Moseley, a -somewhat more complicated example.</p> - -<p>From the apex of a large specimen of <i>Turbo duplicatus</i><a class="afnanch" href="#fn507" id="fnanch507">507</a> -a <span class="xxpn" id="p519">{519}</span> -line was drawn across its whorls, and their widths were measured -upon it in succession, beginning with the last but one. The -measurements were, as before, made with a fine pair of compasses -and a diagonal scale. The sight was assisted by a magnifying -glass. In a parallel column to the following admeasurements -are the terms of a geometric progression, whose first term is the -width of the widest whorl measured, and whose common ratio is -1·1804.</p> - -<div class="section"> -<div class="dctr04" id="fig262"> -<img src="images/i519.png" width="529" height="382" alt=""> - <div class="dcaption">Fig. 262.</div></div> - -<div class="dtblbox"><div class="nowrap"> -<table class="fsz7"> -<tr> - <th> - Widths of successive<br> - whorls measured in inches<br> - and parts of an inch</th> - <th> - Terms of a geometrical progression,<br> - whose first term is the width of<br> - the widest whorl, and whose<br> - common ratio is 1·1804</th></tr> -<tr> - <td class="tdcntr">1·31</td> - <td class="tdcntr">1·31   </td></tr> -<tr> - <td class="tdcntr">1·12</td> - <td class="tdcntr">1·1098 </td></tr> -<tr> - <td class="tdcntr"> ·94</td> - <td class="tdcntr"> ·94018</td></tr> -<tr> - <td class="tdcntr"> ·80</td> - <td class="tdcntr"> ·79651</td></tr> -<tr> - <td class="tdcntr"> ·67</td> - <td class="tdcntr"> ·67476</td></tr> -<tr> - <td class="tdcntr"> ·57</td> - <td class="tdcntr"> ·57164</td></tr> -<tr> - <td class="tdcntr"> ·48</td> - <td class="tdcntr"> ·48427</td></tr> -<tr> - <td class="tdcntr"> ·41</td> - <td class="tdcntr"> ·41026</td></tr> -</table></div></div><!--dtblbox--></div><!--section--> - -<p>The close coincidence between the observed and the calculated -figures is very remarkable, and is amply sufficient to justify the -conclusion that we are here dealing with a true logarithmic -spiral.</p> - -<div class="dmaths"> -<p>Nevertheless, in order to verify his conclusion still further, -and to get partially rid of the inaccuracies due -to successive small <span class="xxpn" id="p520">{520}</span> -measurements, Moseley proceeded to investigate the same shell, -measuring not single whorls, but groups of whorls, taken several -at a time: making use of the following property of a geometrical -progression, that “if µ represent the ratio of the sum of every -even number (<i>m</i>) of its terms to the sum of half that number of -terms, then the common ratio (<i>r</i>) of the series is represented by -the formula</p> - -<div><i>r</i> -= (µ − 1)<sup>2 ⁄ <i>m</i></sup> .” -</div></div><!--dmaths--> - -<p>Accordingly, Moseley made the following measurements, -beginning from the second and third whorls respectively:</p> - -<div class="dtblbox"><div class="nowrap"> -<table> -<tr> - <th colspan="2">Width of</th> - <th rowspan="2">Ratio µ</th></tr> -<tr> - <th>Six whorls</th> - <th>Three whorls</th></tr> -<tr> - <td class="tdcntr">5·37</td> - <td class="tdcntr">2·03</td> - <td class="tdcntr">2·645</td></tr> -<tr> - <td class="tdcntr">4·55</td> - <td class="tdcntr">1·72</td> - <td class="tdcntr">2·645</td></tr> -<tr> - <th>Four whorls</th> - <th>Two whorls</th> - <th>Ratio µ</th></tr> -<tr> - <td class="tdcntr">4·15</td> - <td class="tdcntr">1·74</td> - <td class="tdcntr">2·385</td></tr> -<tr> - <td class="tdcntr">3·52</td> - <td class="tdcntr">1·47</td> - <td class="tdcntr">2·394</td></tr> -</table></div></div><!--dtblbox--> - -<div class="dmaths"> -<p>“By the ratios of the two first admeasurements, the formula -gives</p> - -<div><i>r</i> -= (1·645)<sup>1 ⁄ 3</sup> -= 1·1804.</div> - -<p class="pcontinue">By the mean of the ratios deduced from the -second two admeasurements, it gives</p> - -<div><i>r</i> -= (1·389)<sup>1 ⁄ 2</sup> -= 1·1806. -</div></div><!--dmaths--> - -<p>“It is scarcely possible to imagine a more accurate verification -than is deduced from these larger admeasurements, and we may -with safety annex to the species <i>Turbo duplicatus</i> the characteristic -number 1·18.”</p> - -<p>By similar and equally concordant observations, Moseley found -for <i>Turbo phasianus</i> the characteristic ratio, 1·75; and for <i>Buccinum -subulatum</i> that of 1·13.</p> - -<p>From the table referring to <i>Turbo duplicatus</i>, on page <a href="#p519" title="go to pg. 519">519</a>, it -is perhaps worth while to illustrate the logarithmic statement of -the same facts: that is to say, the elementary corollary to the -fact that the successive radii are in geometric progression, that -their logarithms differ from one another by -a constant amount. <span class="xxpn" id="p521">{521}</span></p> - -<div class="dtblbox"><div class="nowrap"> -<table class="borall"> -<caption><i>Turbo duplicatus.</i></caption> -<tr> - <th class="borall">Relative<br>widths of<br>successive<br>whorls</th> - <th class="borall">Logarithms<br>of successive<br>whorls</th> - <th class="borall">Difference<br>of successive<br>logarithms</th></tr> -<tr> - <td class="tdright">131</td> - <td class="tdright">2·11727 </td> - <td class="tdcntr">—</td></tr> -<tr> - <td class="tdright">112</td> - <td class="tdright">2·04922 </td> - <td class="tdright">·06805</td></tr> -<tr> - <td class="tdright">94</td> - <td class="tdright">1·97313 </td> - <td class="tdright">·07609</td></tr> -<tr> - <td class="tdright">80</td> - <td class="tdright">1·90309 </td> - <td class="tdright">·07004</td></tr> -<tr> - <td class="tdright">67</td> - <td class="tdright">1·82607 </td> - <td class="tdright">·07702</td></tr> -<tr> - <td class="tdright">57</td> - <td class="tdright">1·75587 </td> - <td class="tdright">·07020</td></tr> -<tr> - <td class="tdright">48</td> - <td class="tdright">1·68124 </td> - <td class="tdright">·07463</td></tr> -<tr> - <td class="tdright">41</td> - <td class="tdright">1·161278</td> - <td class="tdright">·06846</td></tr> -<tr> - <td class="tdright" colspan="3">Mean difference ·07207</td></tr> -</table></div></div><!--dtblbox--> - -<p class="pcontinue">And ·07207 is the logarithm of 1·1805.</p> - -<div class="dctr04" id="fig263"> -<img src="images/i521.png" width="529" height="442" alt=""> - <div class="dcaption">Fig. 263. Operculum of Turbo.</div></div> - -<p>The logarithmic spiral is not only very beautifully manifested -in the molluscan shell, but also, in certain cases, in the little lid -or “operculum” by which the entrance to the tubular shell is -closed after the animal has withdrawn itself within. In the spiral -shell of <i>Turbo</i>, for instance, the operculum is a thick calcareous -structure, with a beautifully curved outline, which grows by -successive increments applied to one portion of its edge, and shews, -accordingly, a spiral line of growth upon its surface. The successive -increments leave their traces on the surface -of the operculum <span class="xxpn" id="p522">{522}</span> -(Fig. <a href="#fig264" title="go to Fig. 264">264</a>, 1), which traces have the form of curved lines in -Turbo, and of straight lines in (e.g.) Nerita (Fig. <a href="#fig264" title="go to Fig. 264">264</a>, 2); that -is to say, apart from the side constituting the outer edge of the -operculum (which side is always and of necessity curved) the -successive increments constitute curvilinear triangles in the one -case, and rectilinear triangles in the other. The sides of these -triangles are tangents to the spiral line of the operculum, and -may be supposed to generate it by their consecutive intersections.</p> - -<div class="dctr01" id="fig264"> -<img src="images/i522.png" width="800" height="389" alt=""> - <div class="dcaption">Fig. 264. Opercula of (1) Turbo, - (2) Nerita. (After Moseley.)</div></div> - -<p>In a number of such opercula, Moseley measured the breadths -of the successive whorls along a radius vector<a class="afnanch" href="#fn508" id="fnanch508">508</a>, -just in the same -way as he did with the entire shell in the foregoing cases; and -here is one example of his results.</p> - -<div class="dtblbox"><div class="nowrap"> -<table class="fsz7"> -<caption><i>Operculum of Turbo sp.; breadth (in inches) of -successive whorls, measured from the pole.</i></caption> -<tr> - <th>Distance</th> - <th>Ratio</th> - <th>Distance</th> - <th>Ratio</th> - <th>Distance</th> - <th>Ratio</th> - <th>Distance</th> - <th>Ratio</th></tr> -<tr> - <td class="tdrtsht">·24</td> - <td></td> - <td class="tdrtsht">·16</td> - <td></td> - <td class="tdrtsht">·2 </td> - <td></td> - <td class="tdrtsht">·18</td> - <td></td></tr> -<tr> - <td class="tdrtsht"></td> - <td class="tdrtsht">2·28</td> - <td></td> - <td class="tdrtsht">2·31</td> - <td></td> - <td class="tdrtsht">2·30</td> - <td></td> - <td class="tdrtsht">2·30</td></tr> -<tr> - <td class="tdrtsht">·55</td> - <td></td> - <td class="tdrtsht">·37</td> - <td></td> - <td class="tdrtsht">·6 </td> - <td></td> - <td class="tdrtsht">·42</td> - <td></td></tr> -<tr> - <td class="tdrtsht"></td> - <td class="tdrtsht">2·32</td> - <td></td> - <td class="tdrtsht">2·30</td> - <td></td> - <td class="tdrtsht">2·30</td> - <td></td> - <td class="tdrtsht">2·24</td></tr> -<tr> - <td class="tdrtsht">1·28</td> - <td></td> - <td class="tdrtsht">·85</td> - <td></td> - <td class="tdrtsht">1·38</td> - <td></td> - <td class="tdrtsht">·94</td> - <td></td></tr> -</table></div></div><!--dtblbox--> - -<div><span class="xxpn" id="p523">{523}</span></div> - -<p>The ratio is approximately constant, and this spiral also is, -therefore, a logarithmic spiral.</p> - -<p>But here comes in a very beautiful illustration of that property -of the logarithmic spiral which causes its whole shape to remain -unchanged, in spite of its apparently unsymmetrical, or unilateral, -mode of growth. For the mouth of the tubular shell, into which -the operculum has to fit, is growing or widening on all sides: -while the operculum is increasing, not by additions made at the -same time all round its margin, but by additions made only on -one side of it at each successive stage. One edge of the operculum -thus remains unaltered as it is advanced into each new position, -and as it is placed in a newly formed section of the tube, similar -to but greater than the last. Nevertheless, the two apposed -structures, the chamber and its plug, at all times fit one another -to perfection. The mechanical problem (by no means an easy -one), is thus solved: “How to shape a tube of a variable section, -so that a piston driven along it shall, by one side of its margin, -coincide continually with its surface as it advances, provided only -that the piston be made at the same time continually to revolve -in its own plane.”</p> - -<p>As Moseley puts it: “That the same edge which fitted a portion -of the first less section should be capable of adjustment, so as to -fit a portion of the next similar but greater section, supposes -a geometrical provision in the curved form of the chamber of -great apparent complication and difficulty. But God hath -bestowed upon this humble architect the practical skill of a -learned geometrician, and he makes this provision with admirable -precision in that curvature of the logarithmic spiral which he -gives to the section of the shell. This curvature obtaining, he -has only to turn his operculum slightly round in its own plane as -he advances it into each newly formed portion of his chamber, -to adapt one margin of it to a new and larger surface and a different -curvature, leaving the space to be filled up by increasing the -operculum wholly on the other margin.”</p> - -<p>But in many, and indeed more numerous Gastropod mollusca, -the operculum does not grow in this remarkable spiral fashion, -but by the apparently much simpler process of accretion by -concentric rings. This suggests to -us another mathematical <span class="xxpn" id="p524">{524}</span> -feature of the logarithmic spiral. We have already seen that the -logarithmic spiral has a number of “limiting cases,” apparently -very diverse from one another. Thus the right cone is a logarithmic -spiral in which the revolution of the radius vector is infinitely -slow; and, in the same sense, the straight line itself is a limiting -case of the logarithmic spiral. The spiral of Archimedes, though -not a limiting case of the logarithmic spiral, closely resembles -one in which the angle of the spiral is very near to 90°, and the -spiral is coiled around a central core. But if the angle of the -spiral were actually 90°, the radius vector would describe a circle, -identical with the “core” of which we have just spoken; and -accordingly it may be said that the circle is, in this sense, a true -limiting case of the logarithmic spiral. In this sense, then, the -circular concentric operculum, for instance of Turritella or -Littorina, does not represent a breach of continuity, but a “limiting -case” of the spiral operculum of <i>Turbo</i>; the successive “gnomons” -are now not lateral or terminal additions, but complete concentric -rings.</p> - -<hr class="hrblk"> - -<p>Viewed in regard to its own fundamental properties and to -those of its limiting cases, the logarithmic spiral is the simplest -of all known curves; and the rigid uniformity of the simple laws, -or forces, by which it is developed sufficiently account for its -frequent manifestation in the structures built up by the slow and -steady growth of organisms.</p> - -<p>In order to translate into precise terms the whole form and -growth of a spiral shell, we should have to employ a mathematical -notation, considerably more complicated than any that I have -attempted to make use of in this book. But, in the most elementary -language, we may now at least attempt to describe the -general method, and some of the variations, of the mathematical -development of the shell.</p> - -<p>Let us imagine a closed curve in space, whether circular or -elliptical or of some other and more complex specific form, not -necessarily in a plane: such a curve as we see before us when we -consider the mouth, or terminal orifice, of our tubular shell; and -let us imagine some one characteristic point within this closed -curve, such as its centre of gravity. Then, starting -from a fixed <span class="xxpn" id="p525">{525}</span> -origin, let this centre of gravity describe an equiangular spiral in -space, about a fixed axis (namely the axis of the shell), while at -the same time the generating curve grows, with each angular -increment of rotation, in such a way as to preserve the symmetry -of the entire figure, with or without a simultaneous movement -of translation along the axis.</p> - -<div class="dctr06" id="fig265"> -<img src="images/i525.png" width="401" height="603" alt=""> - <div class="dcaption">Fig. 265. <i>Melo ethiopicus</i>, L.</div></div> - -<p>It is plain that the entire resulting shell may now be looked -upon in either of two ways. It is, on the one hand, an <i>ensemble -of similar closed curves</i> spirally arranged in space, gradually increasing -in dimensions, in proportion to the increase of their -vectorial angle from the pole. In other words, we can imagine -our shell cut up into a system of rings, following one another in -continuous spiral succession from that terminal and largest one, -which constitutes the lip of the orifice of the shell. Or, on the -other hand, we may figure to ourselves the whole shell as made -up of an <i>ensemble of spiral lines</i> in space, each -spiral having been <span class="xxpn" id="p526">{526}</span> -traced out by the gradual growth and revolution of a radius -vector from the pole to a given point of the generating curve.</p> - -<p>Both systems of lines, the <i>generating spirals</i> (as these latter -may be called), and the closed <i>generating curves</i> corresponding -to successive margins or lips of the shell, may be easily traced -in a great variety of cases. Thus, for example, in Dolium, -Eburnea, and a host of others, the generating spirals are beautifully -marked out</p> - -<div class="dctr02" id="fig266"> -<img src="images/i526.png" width="704" height="572" alt=""> - <div class="pcaption">Fig. 266. 1, <i>Harpa</i>; 2, <i>Dolium</i>. - The ridges on the shell correspond in (1) to generating - curves, in (2) to generating spirals.</div></div> - -<p class="pcontinue"> -by ridges, tubercles or bands of colour. In Trophon, -Scalaria, and (among countless others) in the Ammonites, it is -the successive generating curves which more conspicuously leave -their impress on the shell. And in not a few cases, as in -Harpa, <i>Dolium perdix</i>, etc., both alike are conspicuous, ridges -and colour-bands intersecting one another in a beautiful isogonal -system. <span class="xxpn" id="p527">{527}</span></p> - -<p>In the complete mathematical formula (such as I have not -ventured to set forth<a class="afnanch" href="#fn509" id="fnanch509">509</a>) -for any given turbinate shell, we should -have, accordingly, to include factors for at least the following -elements: (1) for the specific form of the section of the tube, -which we have called the generating curve; (2) for the specific -rate of growth of this generating curve; (3) for its specific rate -of angular rotation about the pole, perpendicular to the axis; -(4) in turbinate (as opposed to nautiloid) shells, for its rate of -shear, or screw-translation parallel to the axis. There are also -other factors of which we should have to take account, and which -would help to make our whole expression a very complicated one. -We should find, for instance, (5) that in very many cases our -generating curve was not a plane curve, but a sinuous curve in -three dimensions; and we should also have to take account -(6) of the inclination of the plane of this generating curve to the -axis, a factor which will have a very important influence on the -form and appearance of the shell. For instance in Haliotis it is -obvious that the generating curve lies in a plane very oblique to -the axis of the shell. Lastly, we at once perceive that the ratios -which happen to exist between these various factors, the ratio -for instance between the growth-factor and the rate of angular -revolution, will give us endless possibilities of permutation of -form. For instance (7) with a given velocity of vectorial rotation, -a certain rate of growth in the generating curve will give us a -spiral shell of which each successive whorl will just touch its -predecessor and no more; with a slower growth-factor, the whorls -will stand asunder, as in a ram’s horn; with a quicker growth-factor, -each whorl will cut or intersect its predecessor, as in an -Ammonite or the majority of gastropods, and so on (cf. p. <a href="#p541" title="go to pg. 541">541</a>).</p> - -<p>In like manner (8) the ratio between the growth-factor and -the rate of screw-translation parallel to the axis will determine -the apical angle of the resulting conical structure: will give us -the difference, for example, between the sharp, pointed cone of -Turritella, the less acute one of Fusus or -Buccinum, and the <span class="xxpn" id="p528">{528}</span> -obtuse one of Harpa or Dolium. In short it is obvious that <i>all</i> -the differences of form which we observe between one shell and -another are referable to matters of <i>degree</i>, depending, one and all, -upon the relative magnitudes of the various factors in the complex -equation to the curve.</p> - -<hr class="hrblk"> - -<p>The paper in which, nearly eighty years ago, Canon Moseley -thus gave a simple mathematical expression to the spiral forms of -univalve shells, is one of the classics of Natural History. But -other students before him had come very near to recognising -this mathematical simplicity of form and structure. About the -year 1818, Reinecke had suggested that the relative breadths of -the adjacent whorls in an Ammonite formed a constant and -diagnostic character; and Leopold von Buch accepted and -developed the idea<a class="afnanch" href="#fn510" id="fnanch510">510</a>. -But long before, Swammerdam, with a -deeper insight, had grasped the root of the whole matter: for, -taking a few diverse examples, such as Helix and Spirula, he -shewed that they and all other spiral shells whatsoever were -referable to one common type, namely to that of a simple tube, -variously curved according to definite mathematical laws; that -all manner of ornamentation, in the way of spines, tuberosities, -colour-bands and so forth, might be superposed upon them, but -the type was one throughout, and specific differences were of a -geometrical kind. “Omnis enim quae inter eas animadvertitur -differentia ex sola nascitur diversitate gyrationum: quibus si -insuper externa quaedam adjunguntur ornamenta pinnarum, -sinuum, anfractuum, planitierum, eminentiarum, profunditatum, -extensionum, impressionum, circumvolutionum, colorumque: ... tunc -deinceps facile est, quarumcumque Cochlearum figuras -geometricas, curvosque, obliquos atque rectos angulos, ad unicam -omnes speciem redigere: ad oblongum videlicet tubulum, qui -vario modo curvatus, crispatus, extrorsum et introrsum flexus, -ita concrevit<a class="afnanch" href="#fn511" id="fnanch511">511</a>.” -<span class="xxpn" id="p529">{529}</span></p> - -<div class="dright dwth-e" id="fig267"> -<img src="images/i529.png" width="337" height="951" alt=""> - <div class="dcaption">Fig. 267. D’Orbigny’s - Helicometer.</div></div> - -<p>For some years after the appearance of Moseley’s -paper, a number of writers followed in his footsteps, -and attempted, in various ways, to put his conclusions -to practical use. For instance, D’Orbigny devised a -very simple protractor, which he called a Helicometer<a -class="afnanch" href="#fn512" id="fnanch512">512</a>, and -which is represented in Fig. <a href="#fig267" title="go to Fig. 267">267</a>. By means of this little -instrument, the apical angle of the turbinate shell was -immediately read off, and could then be used as a specific -and diagnostic character. By keeping one limb of the -protractor parallel to the side of the cone while the other -was brought into line with the suture between two adjacent -whorls, another specific angle, the “sutural angle,” could -in like manner be recorded. And, by the linear scale upon -the instrument, the relative breadths of the consecutive -whorls, and that of the terminal chamber to the rest of the -shell, might also, though somewhat roughly, be determined. -For instance, in <i>Terebra dimidiata</i>, the apical angle was -found to be 13°, the sutural angle 109°, and so forth.</p> - -<p>It was at once obvious that, in such a shell as is -represented in Fig. <a href="#fig267" title="go to Fig. 267">267</a> the entire outline of the shell -(always excepting that of the immediate neighbourhood of -<span class="xxpn" id="p530">{530}</span> the mouth) could -be restored from a broken fragment. For if we draw our -tangents to the cone, it follows from the symmetry of the -figure that we can continue the projection of the sutural -line, and so mark off the successive whorls, by simply -drawing a series of consecutive parallels, and by then -filling into the quadrilaterals so marked off a series of -curves similar to one another, and to the whorls which are -still intact in the broken shell. <br class="brclrfix"></p> - -<p>But the use of the helicometer soon shewed that it was by no -means universally the case that one and the same right cone was -tangent to all the turbinate whorls; in other words, there was not -always one specific apical angle which held good for the entire -system. In the great majority of cases, it is true, the same -tangent touches all the whorls, and is a straight line. But in -others, as in the large <i>Cerithium nodosum</i>, such a line is slightly -convex to the axis of the shell; and in the short spire of Dolium, -for instance, the convexity is marked, and the apex of the spire -is a distinct cusp. On the other hand, in Pupa and Clausilia, the -common tangent is concave to the axis of the shell.</p> - -<p>So also is it, as we shall presently see, among the Ammonites: -where there are some species in which the ratio of whorl to whorl -remains, to all appearance, perfectly constant; others in which -it gradually, though only slightly increases; and others again in -which it slightly and gradually falls away. It is obvious that, -among the manifold possibilities of growth, such conditions as -these are very easily conceivable. It is much more remarkable -that, among these shells, the relative velocities of growth in various -dimensions should be as constant as it is, than that there should -be an occasional departure from perfect regularity. In such cases -as these latter, the logarithmic law of growth is only approximately -true. The shell is no longer to be represented as a <i>right</i> cone -which has been rolled up, but as a cone which had grown trumpet-shaped, -or conversely whose mouth had narrowed in, and which -in section is a curvilinear instead of a rectilinear triangle. But -all that has happened is that a new factor, usually of small or all -but imperceptible magnitude, has been introduced into the case; -so that the ratio, log <i>r</i> -= θ log α, is no longer constant, but varies -slightly, and in accordance with -some simple law. <span class="xxpn" id="p531">{531}</span></p> - -<p>Some writers, such as Naumann and Grabau, maintained that -the molluscan spiral was no true logarithmic spiral, but differed -from it specifically, and they gave to it the name of <i>Conchospiral</i>. -They pointed out that the logarithmic spiral originates in a -mathematical point, while the molluscan shell starts with a little -embryonic shell, or central chamber (the “protoconch” of the -conchologists), around which the spiral is subsequently wrapped. -It is plain that this undoubted and obvious fact need not -affect the logarithmic law of the shell as a whole; we have -only to add a small constant to our equation, which becomes -<i>r</i> -= <i>m</i> + <i>a</i><sup>θ</sup> .</p> - -<p>There would seem, by the way, to be considerable confusion -in the books with regard to the so-called “protoconch.” In many -cases it is a definite structure, of simple form, representing the -more or less globular embryonic shell before it began to elongate -into its conical or spiral form. But in many cases what is described -as the “protoconch” is merely an empty space in the middle of</p> - -<div class="dright dwth-f" id="fig268"> -<img src="images/i531.png" width="289" height="309" alt=""> - <div class="dcaption">Fig. 268.</div></div> - -<p class="pcontinue">the spiral coil, resulting from -the fact that the actual spiral shell has a definite -magnitude to begin with, and that we cannot follow it down -to its vanishing point in infinity. For instance, in the -accompanying figure, the large space <i>a</i> is styled the -protoconch, but it is the little bulbous or hemispherical -chamber within it, at the end of the spire, which is -the real beginning of the tubular shell. The form and -magnitude of the space <i>a</i> are determined by the “angle of -retardation,” or ratio of rate of growth between the inner -and outer curves of the spiral shell. They are independent -of the shape and size of the embryo, and depend only (as we -shall see better presently) on the direction and relative -rate of growth of the double contour of the shell.<br -class="brclrfix"></p> - -<hr class="hrblk"> - -<div class="section"> -<div class="dleft dwth-h" id="fig269"> -<img src="images/i532.png" width="209" height="406" alt=""> - <div class="dcaption">Fig. 269.</div></div> - -<p>Now that we have dealt, in a very general way, with some of -the more obvious properties of the logarithmic spiral, let us -consider certain of them a little more -particularly, keeping in <span class="xxpn" id="p532">{532}</span> -view as our chief object the investigation (on elementary lines) -of the possible manner and range of variation of the molluscan -shell.</p> - -<div class="dmaths"> -<p>There is yet another equation to the logarithmic spiral, -very commonly employed, and without the -help of which we shall find that we cannot -get far. It is as follows:</p> - -<div><i>r</i> -= ε<sup>θ cot α</sup> .</div> - -<p>This follows directly from the fact that -the angle α (the angle between the radius -vector and the tangent to the curve) is -constant.</p> - -<p>For, then,</p> - -<div>tan α (= tan ϕ) -= <i>r d</i>θ ⁄ <i>dr</i>,</div> - -<p class="pcontinue">therefore</p> - -<div><i>dr ⁄ r</i> -= <i>d</i>θ cot α,</div> - -<p class="pcontinue">and, integrating,</p> - -<div>log <i>r</i> -= θ cot α ,  or</div> - -<div><i>r</i> -= ε<sup>θ cot α</sup> .</div> -</div><!--dmaths--></div><!--section--> - -<hr class="hrblk"> - -<p>As we have seen throughout our preliminary discussion, the -two most important constants (or chief “specific characters,” as -the naturalist would say) in any given logarithmic spiral, are -(1) the magnitude of the angle of the spiral, or “constant angle,” -α, and (2) the rate of increase of the radius vector for any given -angle of revolution, θ. Of this latter, the simplest case is when -θ -= 2π, or 360°; that is to say when we compare the breadths, -along the same radius vector, of two successive whorls. As our -two magnitudes, that of the constant angle, and that of the ratio -of the radii or breadths of whorl, are related to one another, we -may determine either of them by actual measurement and proceed -to calculate the other. -<br class="brclrfix"></p> - -<div class="dmaths"> -<p>In any complete spiral, such as that of Nautilus, it is (as we -have seen) easy to measure any two radii (<i>r</i>), or -the breadths in <span class="xxpn" id="p533">{533}</span> -a radial direction of any two whorls (<i>W</i>). We have then merely -to apply the formula</p> - -<div><i>r</i><sub><i>n</i> + 1</sub> ⁄ <i>r<sub>n</sub></i> -= <i>e</i><sup>θ cot α</sup> ,  or -  <i>W</i><sub><i>n</i> + 1</sub> ⁄ <i>W<sub>n</sub></i> -= <i>e</i><sup>θ cot α</sup> ,</div> - -<p class="pcontinue">which we may simply write <i>r</i> -= <i>e</i><sup>θ cot α</sup> , etc.; since our first radius -or whorl is regarded, for the purpose of comparison, as being equal -to unity.</p> -</div><!--dmaths--> - -<p>Thus, in the diagram, <i>OC ⁄ OE</i> , -or <i>EF ⁄ BD</i> , or -<i>DC ⁄ EF</i> , being in each case radii, or -diameters, at right angles to one another, are all equal to -<i>e</i><sup>(π ⁄ 2) cot α</sup> . -While in like manner, <i>EO ⁄ OF</i> , -<i>EG ⁄ FH</i> , or -<i>GO ⁄ HO</i> , all -equal <i>e</i><sup>π cot α</sup> ; -and <i>BC ⁄ BA</i> , -or <i>CO ⁄ OB</i> -= <i>e</i><sup>2π cot α</sup> .</p> - -<div class="dctr04" id="fig270"> -<img src="images/i533.png" width="528" height="357" alt=""> - <div class="dcaption">Fig. 270.</div></div> - -<p>As soon, then, as we have prepared tables for these values, -the determination of the constant angle α in a particular shell -becomes a very simple matter.</p> - -<p>A complete table would be cumbrous, and it will be sufficient -to deal with the simple case of the ratio between the breadths of -adjacent, or immediately succeeding, whorls.</p> - -<p>Here we have <i>r</i> -= <i>e</i><sup>2π cot α</sup> , or log <i>r</i> -= log <i>e</i> × 2π × cot α , from -which we obtain the following figures<a class="afnanch" href="#fn513" id="fnanch513">513</a>: -<span class="xxpn" id="p534">{534}</span></p> - -<div class="dtblboxin10"> -<table class="fsz6"> -<tr> - <th>Ratio of breadth of<br> - each whorl to the<br> - next preceding<br> - <i>r</i> ⁄ 1</th> - <th>Constant<br> - angle<br> - α</th></tr> -<tr> - <td class="tdright">1·1 </td> - <td class="tdright">89°  8′</td></tr> -<tr> - <td class="tdright">1·25</td> - <td class="tdright">87  58 </td></tr> -<tr> - <td class="tdright">1·5 </td> - <td class="tdright">86  18 </td></tr> -<tr> - <td class="tdright">2·0 </td> - <td class="tdright">83  42 </td></tr> -<tr> - <td class="tdright">2·5 </td> - <td class="tdright">81  42 </td></tr> -<tr> - <td class="tdright">3·0 </td> - <td class="tdright">80   5 </td></tr> -<tr> - <td class="tdright">3·5 </td> - <td class="tdright">78  43 </td></tr> -<tr> - <td class="tdright">4·0 </td> - <td class="tdright">77  34 </td></tr> -<tr> - <td class="tdright">4·5 </td> - <td class="tdright">76  32 </td></tr> -<tr> - <td class="tdright">5·0 </td> - <td class="tdright">75  38 </td></tr> -<tr> - <td class="tdright">10·0 </td> - <td class="tdright">69  53 </td></tr> -<tr> - <td class="tdright">20·0 </td> - <td class="tdright">64  31 </td></tr> -<tr> - <td class="tdright">50·0 </td> - <td class="tdright">58   5 </td></tr> -<tr> - <td class="tdright">100·0 </td> - <td class="tdright">53  46 </td></tr> -<tr> - <td class="tdright">1,000·0 </td> - <td class="tdright">42  17 </td></tr> -<tr> - <td class="tdright">10,000   </td> - <td class="tdright">34  19 </td></tr> -<tr> - <td class="tdright">100,000   </td> - <td class="tdright">28  37 </td></tr> -<tr> - <td class="tdright">1,000,000   </td> - <td class="tdright">24  28 </td></tr> -<tr> - <td class="tdright">10,000,000   </td> - <td class="tdright">21  18 </td></tr> -<tr> - <td class="tdright">100,000,000   </td> - <td class="tdright">18  50 </td></tr> -<tr> - <td class="tdright">1,000,000,000   </td> - <td class="tdright">16  52 </td></tr> -</table></div><!--dtblbox--> - -<p>We learn several interesting things from this short table. -We see, in the first place, that where each whorl is about -three times the breadth of its neighbour and predecessor, -as is the case in Nautilus,</p> - -<div class="dleft dwth-f" id="fig271"> -<img src="images/i534.png" width="256" height="268" alt=""> - <div class="dcaption">Fig. 271.</div></div> - -<p class="pcontinue"> -the constant angle is in the -neighbourhood of 80°; and hence also that, in all the -ordinary Ammonitoid shells, and in all the typically spiral -shells of the Gastropods<a class="afnanch" href="#fn514" -id="fnanch514">514</a>, the constant angle is also a large one, -being very seldom less than 80°, and usually between 80° and -85°. In the next place, we see that with smaller angles the -apparent form of the spiral is greatly altered, and the very -fact of its being a spiral soon ceases to be apparent (Figs. -<a href="#fig271" title="go to Fig. 271">271</a>, 272). Suppose one whorl to be an inch in breadth, then, -if the angle of the spiral were 80°, the <span class="xxpn" -id="p535">{535}</span> next whorl would (as we have just seen) -be about three inches broad; if it were 70°, the next whorl -would be nearly ten inches, and if it were 60°, the next whorl -would be nearly four feet broad. If the angle were 28°, the -next whorl would be a mile and a half in breadth; and if it -were 17°, the next would be some 15,000 miles broad. -<br class="brclrfix"></p> - -<div class="dctr03" id="fig272"> -<img src="images/i535a.png" width="609" height="241" alt=""> - <div class="dcaption">Fig. 272.</div></div> - -<p>In other words, the spiral shells of gentle curvature, or of -small constant angle, such as Dentalium or Nodosaria, are true -logarithmic spirals, just as are those of Nautilus or Rotalia: -from which they differ only in degree, in the magnitude of an -angular constant. But this diminished magnitude of the angle -causes the spiral to dilate with such immense rapidity that, so -to speak, “it never comes round”; and so, in such a shell as -Dentalium, we never see but a small portion of the initial whorl.</p> - -<div class="dctr03" id="fig273"> -<img src="images/i535b.png" width="609" height="341" alt=""> - <div class="dcaption">Fig. 273.</div></div> - -<div class="psmprnt3"> -<p>We might perhaps be inclined to suppose that, in such a shell as Dentalium, -the lack of a visible spiral convolution was only due to our seeing but a small -portion of the curve, at a distance from the pole, and -when, therefore, its <span class="xxpn" id="p536">{536}</span> -curvature had already greatly diminished. That is to say we might suppose -that, however small the angle a, and however rapidly the whorls accordingly -increased, there would nevertheless be a manifest spiral convolution in the -immediate neighbourhood of the pole, as the starting point of the curve. -But it may be shewn that this is not so.</p> - -<div class="dmaths"> -<p>For, taking the formula</p> - -<div><i>r</i> -= <i>a</i>ε<sup>θ cot α</sup> ,</div> - -<p class="pcontinue">this, for any given spiral, is equivalent to -<i>a</i>ε<sup><i>k</i>θ</sup> .</p> - -<p>Therefore</p> - -<div>log(<i>r ⁄ a</i>) -= <i>k</i>θ,<br class="brclrfix"></div> - -<p class="pcontinue pleftfloat">or,</p> - -<div>1 ⁄ <i>k</i> -= θ ⁄ log(<i>r ⁄ a</i>). -<br class="brclrfix"></div> - -<p class="pcontinue">Then, if θ increase by 2π, while <i>r</i> increases to <i>r</i><sub>1</sub> ,</p> - -<div>1 ⁄ <i>k</i> -= (θ + 2π) ⁄ log(<i>r</i><sub>1</sub> ⁄ <i>a</i>), -</div> - -<p class="pcontinue">which leads, by subtraction to</p> - -<div>1 ⁄ <i>k</i> · log(<i>r</i><sub>1</sub> ⁄ <i>r</i>) -= 2π.</div> - -<p>Now, as α tends to 0, <i>k</i> (i.e. cot α) -tends to ∞, and therefore, as <i>k</i> → ∞, -log(<i>r</i><sub>1</sub> ⁄ <i>r</i>) → ∞ -and also -<i>r</i><sub>1</sub> ⁄ <i>r</i> → ∞.</p> -</div><!--dmaths--> - -<p>Therefore if one whorl exists, the radius vector of the -other is infinite; in other words, there is nowhere, even in -the near neighbourhood of the pole, a complete revolution of -the spire. Our spiral shells of small constant angle, such -as Dentalium, may accordingly be considered to represent -sufficiently well the true commencement of their respective -spirals.</p> -</div><!--psmprnt3--> - -<p>Let us return to the problem of how to ascertain, by direct -measurement, the spiral angle of any particular shell. The -method already employed is only applicable to complete spirals, -that is to say to those in which the angle of the spiral is large, -and furthermore it is inapplicable to portions, or broken fragments, -of a shell. In the case of the broken fragment, it is plain that the -determination of the angle is not merely of theoretic interest, -but may be of great practical use to the conchologist as being the -one and only way by which he may restore the outline of the -missing portions. We have a considerable choice of methods, -which have been summarised by, and are partly due to, a very -careful student of the Cephalopoda, the late -Rev. J. F. Blake<a class="afnanch" href="#fn515" id="fnanch515">515</a>. -<span class="xxpn" id="p537">{537}</span></p> - -<ul> -<li> -<div class="dright dwth-h" id="fig274"> -<img src="images/i537a.png" width="209" height="399" alt=""> - <div class="dcaption">Fig. 274.</div></div> - -<p>(1) The following method is useful and easy when we -have a portion of a single whorl, such as to shew both its -inner and its outer edge. A broken whorl of an Ammonite, a -curved shell such as Dentalium, or a horn of similar form -to the latter, will fall under this head. We have merely to -draw a tangent, <i>GEH</i>, to the outer whorl at any point <i>E</i>; -then draw to the inner whorl a tangent parallel to <i>GEH</i>, -touching the curve in some point <i>F</i>. The straight line -joining the points of contact, <i>EF</i>, must evidently pass -through the pole: and, accordingly, the angle <i>GEF</i> is the -angle required. In shells which bear <i>longitudinal</i> striae -or other ornaments, any pair of these will suffice for our -purpose, instead of the actual boundaries of the whorl. -But it is obvious that this method will be apt to fail us -when the angle α is very small; and when, consequently, the -points <i>E</i> and <i>F</i> are very remote. <br class="brclrfix" -></p></li> - -<li> -<div class="dctr01" id="fig275"><div id="fig276"> -<img src="images/i537b.png" width="800" height="322" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 275. An Ammonite, to - shew corrugated surface-pattern.</td> - <td></td> - <td>Fig. 276.</td></tr></table> -</div></div></div><!--dctr01--> - -<p>(2) In shells (or horns) shewing rings, or other <i>transverse</i> -ornamentation, we may take it that these ornaments are set at -a constant angle to the spire, and therefore to the radii. The angle -(θ) between two of them, as <i>AC</i>, <i>BD</i>, is therefore equal to the -angle θ between the polar radii from <i>A</i> and <i>B</i>, or from <i>C</i> and <i>D</i>; -and therefore <i>BD ⁄ AC</i> -= <i>e</i><sup>θ cot α</sup> , which gives us the angle α in terms -of known quantities. <span class="xxpn" id="p538">{538}</span></p></li> - -<li><p>(3) If only the outer edge be available, we have the ordinary -geometrical problem,—given an arc of an equiangular spiral, to -find its pole and spiral angle. The methods we may employ -depend (1) on determining directly the position of the pole, and -(2) on determining the radius of curvature.</p></li> -</ul> - -<div class="dleft dwth-d" id="fig277"> -<img src="images/i538.png" width="384" height="321" alt=""> - <div class="dcaption">Fig. 277.</div></div> - -<p>The first method is theoretically -simple, but difficult in -practice; for it requires great -accuracy in determining the -points. Let <i>AD</i>, <i>DB</i>, be two -tangents drawn to the curve. -Then a circle drawn through the -points <i>ABD</i> will pass through -the pole <i>O</i>; since the angles <i>OAD</i>, -<i>OBE</i> (the supplement of <i>OBD</i>), -are equal. The point <i>O</i> may be -determined by the intersection of two such circles; and the angle -<i>DBO</i> is then the angle, α, required.</p> - -<div class="dmaths"> -<p>Or we may determine, graphically, at two points, the radii -of curvature, <span class="nowrap">ρ<sub>1</sub>ρ<sub>2</sub> .</span> -Then, if <i>s</i> be the length of the arc between them (which may -be determined with fair accuracy by rolling the margin of the -shell along a ruler) <br class="brclrfix"></p> - -<div>cot α -= (ρ<sub>1</sub> − ρ<sub>2</sub>) ⁄ <i>s</i>. -</div></div><!--dmaths--> - -<div class="psmprnt3"> -<div class="dmaths"> -<p>The following method<a class="afnanch" href="#fn516" -id="fnanch516">516</a>, given by Blake, will save actual -determination of the radii of curvature.</p> - -<p>Measure along a tangent to the curve, the distance, -<i>AC</i>, at which a certain small offset, <i>CD</i>, is made by the -curve; and from another point <i>B</i>, measure the distance at -which the curve makes an equal offset. Then, calling the -offset μ; the arc <i>AB</i>, <i>s</i>; and <i>AC</i>, <i>BE</i>, respectively -<i>x</i><sub>1</sub> , <i>x</i><sub>2</sub> , we have</p> - -<div>ρ<sub>1</sub> -= (<i>x</i><sub>1</sub><sup>2</sup> + μ<sup>2</sup>) ⁄ 2μ , approximately, -and</div> - -<div>cot α -= (<i>x</i><sub>2</sub><sup>2</sup> − <i>x</i><sub>1</sub><sup>2</sup>) ⁄ 2μ<i>s</i> . -</div></div><!--dmaths--> -</div><!--psmprnt3--> - -<p>Of all these methods by which the mathematical constants, -or specific characters, of a given spiral shell may be determined, -the only one of which much use has been made is that which -Moseley first employed, namely, the simple -method of determining <span class="xxpn" id="p539">{539}</span> -the relative breadths of the whorl at distances separated by some -convenient vectorial angle (such as 90°, 180°, or 360°).</p> - -<p>Very elaborate measurements of a number of Ammonites have -been made by Naumann<a class="afnanch" href="#fn517" id="fnanch517">517</a>, -by Sandberger<a class="afnanch" href="#fn518" id="fnanch518">518</a>, -and by Grabau<a class="afnanch" href="#fn519" id="fnanch519">519</a>, -among which we may choose a couple of cases for consideration. -In the following table I have taken a portion of Grabau’s determinations -of the breadth of the whorls in <i>Ammonites</i> (<i>Arcestes</i>)</p> - -<div class="dtblbox"><div class="nowrap"> -<table class="borall"> -<caption><i>Ammonites intuslabiatus.</i></caption> -<tr> - <th class="borall">Breadth of<br> - whorls (180°<br> - apart) mm.</th> - <th class="borall">Ratio of breadth of<br> - successive whorls<br> - (360° apart)</th> - <th class="borall">The angle (α)<br> - as calculated</th></tr> -<tr> - <td class="tdright">0·30</td> - <td class="tdcntr">—</td> - <td class="tdright">—  — </td></tr> -<tr> - <td class="tdright">0·30</td> - <td class="tdcntr">1·333</td> - <td class="tdright">87° 23′</td></tr> -<tr> - <td class="tdright">0·40</td> - <td class="tdcntr">1·500</td> - <td class="tdright">86  19 </td></tr> -<tr> - <td class="tdright">0·45</td> - <td class="tdcntr">1·500</td> - <td class="tdright">86  19 </td></tr> -<tr> - <td class="tdright">0·60</td> - <td class="tdcntr">1·444</td> - <td class="tdright">86  39 </td></tr> -<tr> - <td class="tdright">0·65</td> - <td class="tdcntr">1·417</td> - <td class="tdright">86  49 </td></tr> -<tr> - <td class="tdright">0·85</td> - <td class="tdcntr">1·692</td> - <td class="tdright">85  13 </td></tr> -<tr> - <td class="tdright">1·10</td> - <td class="tdcntr">1·588</td> - <td class="tdright">85  47 </td></tr> -<tr> - <td class="tdright">1·35</td> - <td class="tdcntr">1·545</td> - <td class="tdright">86   2 </td></tr> -<tr> - <td class="tdright">1·70</td> - <td class="tdcntr">1·630</td> - <td class="tdright">85  33 </td></tr> -<tr> - <td class="tdright">2·20</td> - <td class="tdcntr">1·441</td> - <td class="tdright">86  40 </td></tr> -<tr> - <td class="tdright">2·45</td> - <td class="tdcntr">1·432</td> - <td class="tdright">86  43 </td></tr> -<tr> - <td class="tdright">3·15</td> - <td class="tdcntr">1·735</td> - <td class="tdright">85   0 </td></tr> -<tr> - <td class="tdright">4·25</td> - <td class="tdcntr">1·683</td> - <td class="tdright">85  16 </td></tr> -<tr> - <td class="tdright">5·30</td> - <td class="tdcntr">1·482</td> - <td class="tdright">86  25 </td></tr> -<tr> - <td class="tdright">6·30</td> - <td class="tdcntr">1·519</td> - <td class="tdright">86  12 </td></tr> -<tr> - <td class="tdright">8·05</td> - <td class="tdcntr">1·635</td> - <td class="tdright">85  32 </td></tr> -<tr> - <td class="tdright">10·30</td> - <td class="tdcntr">1·416</td> - <td class="tdright">86  50 </td></tr> -<tr> - <td class="tdright">11·40</td> - <td class="tdcntr">1·252</td> - <td class="tdright">87  57 </td></tr> -<tr> - <td class="tdright">12·90</td> - <td class="tdcntr">—</td> - <td class="tdright">—  — </td></tr> -<tr> - <td class="tdright" colspan="2">Mean</td> - <td class="tdright">86° 15′</td></tr> -</table></div></div><!--dtblbox--> - -<div><span class="xxpn" id="p540">{540}</span></div> - -<p class="pcontinue"><i>intuslabiatus</i>; these measurements Grabau gives for every 45° of -arc, but I have only set forth one quarter of these measurements, -that is to say, the breadths of successive whorls measured along -one diameter on both sides of the pole. The ratio between -<i>alternate</i> measurements is therefore the same ratio as Moseley -adopted, namely the ratio of breadth between <i>contiguous whorls</i> -along a radius vector. I have then added to these observed -values the corresponding calculated values of the angle α, as -obtained from our usual formula.</p> - -<p>There is considerable irregularity in the ratios derived from -these measurements, but it will be seen that this irregularity only -implies a variation of the angle of the spiral between about 85° -and 87°; and the values fluctuate pretty regularly about the -mean, which is 86° 15′. Considering the difficulty of measuring -the whorls, especially towards the centre, and in particular the -difficulty of determining with precise accuracy the position of the -pole, it is clear that in such a case as this we are scarcely justified -in asserting that the law of the logarithmic spiral is departed from.</p> - -<p>In some cases, however, it is undoubtedly departed from. -Here for instance is another table from Grabau, shewing the -corresponding ratios in an Ammonite of the group of <i>Arcestes -tornatus</i>. In this case we see a distinct tendency of the ratios to</p> - -<div class="dtblboxin10"><div class="nowrap"> -<table class="borall"> -<caption><i>Ammonites tornatus.</i></caption> -<tr> - <th class="borall">Breadth of<br> - whorls (180°<br> - apart) mm.</th> - <th class="borall">Ratio of breadth of<br> - successive whorls<br> - (360° apart)</th> - <th class="borall">The spiral<br> - angle (α) as<br> - calculated</th></tr> -<tr> - <td class="tdright">0·25</td> - <td class="tdcntr">—</td> - <td class="tdright">—  — </td></tr> -<tr> - <td class="tdright">0·30</td> - <td class="tdcntr">1·400</td> - <td class="tdright">86° 56′</td></tr> -<tr> - <td class="tdright">0·35</td> - <td class="tdcntr">1·667</td> - <td class="tdright">85  21 </td></tr> -<tr> - <td class="tdright">0·50</td> - <td class="tdcntr">2·000</td> - <td class="tdright">83  42 </td></tr> -<tr> - <td class="tdright">0·70</td> - <td class="tdcntr">2·000</td> - <td class="tdright">83  42 </td></tr> -<tr> - <td class="tdright">1·00</td> - <td class="tdcntr">2·000</td> - <td class="tdright">83  42 </td></tr> -<tr> - <td class="tdright">1·40</td> - <td class="tdcntr">2·100</td> - <td class="tdright">83  16 </td></tr> -<tr> - <td class="tdright">2·10</td> - <td class="tdcntr">2·179</td> - <td class="tdright">82  56 </td></tr> -<tr> - <td class="tdright">3·05</td> - <td class="tdcntr">2·238</td> - <td class="tdright">82  42 </td></tr> -<tr> - <td class="tdright">4·70</td> - <td class="tdcntr">2·492</td> - <td class="tdright">81  44 </td></tr> -<tr> - <td class="tdright">7·60</td> - <td class="tdcntr">2·574</td> - <td class="tdright">81  27 </td></tr> -<tr> - <td class="tdright">12·10</td> - <td class="tdcntr">2·546</td> - <td class="tdright">81  33 </td></tr> -<tr> - <td class="tdright">19·35</td> - <td class="tdcntr">—</td> - <td class="tdright">—  — </td></tr> -<tr> - <td class="tdright"></td> - <td class="tdright">Mean</td> - <td class="tdright">83° 22′</td></tr> -</table></div></div><!--dtblbox--> - -<div><span class="xxpn" id="p541">{541}</span></div> - -<p class="pcontinue">increase as we pass from the centre of the coil outwards, and -consequently for the values of the angle α to diminish. The case -is precisely comparable to that of a cone with slightly curving -sides: in which, that is to say, there is a slight acceleration -of growth in a transverse as compared with the longitudinal -direction.</p> - -<hr class="hrblk"> - -<p>In a tubular spiral, whether plane or helicoid, the consecutive -whorls may either be (1) isolated and remote from one another; -or (2) they may precisely meet, so that the outer border of one -and the inner border of the next just coincide; or (3) they may -overlap, the vector plane of each outer whorl cutting that of its -immediate predecessor or predecessors.</p> - -<p>Looking, as we have done, upon the spiral shell as being -essentially a cone rolled up, it is plain that, for a given spiral -angle, intersection or non-intersection of the successive whorls -will depend upon <i>the apical angle</i> of the original cone. For the -wider the cone, the more rapidly will its inner border tend to -encroach on the outer border of the preceding whorl.</p> - -<p>But it is also plain that the greater be the apical angle of the -cone, and the broader, consequently, the cone itself be, the greater -difference will there be between the total <i>lengths</i> of its inner and -outer border, under given conditions of flexure. And, since the -inner and outer borders are describing precisely the same spiral -about the pole, it is plain that we may consider the inner border -as being <i>retarded</i> in growth as compared with the outer, and as -being always identical with a smaller and earlier part of the -latter.</p> - -<div class="dmaths"> -<p>If λ be the ratio of growth between the outer and the inner -curve, then, the outer curve being represented by</p> - -<div><i>r</i> -= <i>a e</i><sup>θ cot α</sup> ,</div> - -<p class="pcontinue">the equation to the inner one will be</p> - -<div><i>r′</i> -= <i>a</i>λ<i>e</i><sup>θ cot α</sup> , -   or</div> - -<div><i>r′</i> -= <i>a e</i><sup>(θ − β)cot α</sup> ,</div> - -<p class="pcontinue">and β may then be called the angle of retardation, to which the -inner curve is subject by virtue of its slower -rate of growth. <span class="xxpn" id="p542">{542}</span></p> -</div><!--dmaths--> - -<p>Dispensing with mathematical formulae, the several conditions -may be illustrated as follows:</p> - -<div class="dctr02" id="fig278"> -<img src="images/i542.png" width="706" height="441" alt=""> - <div class="dcaption">Fig. 278.</div></div> - -<p>In the diagrams (Fig. <a href="#fig278" title="go to Fig. 278">278</a>), <span class="nowrap"> -<i>O P</i><sub>1</sub> <i>P</i><sub>2</sub> <i>P</i><sub>3</sub> ,</span> -etc. represents a radius, on which <i>P</i><sub>1</sub> , -<i>P</i><sub>2</sub> , <i>P</i><sub>3</sub> , are -the points attained by the outer border of the tubular -shell after as many entire consecutive revolutions. And -<i>P</i><sub>1</sub>′, <i>P</i><sub>2</sub>′, <i>P</i><sub>3</sub>′, are -the points similarly intersected by the inner border; -<i>OP ⁄ OP′</i> being always = λ, -which is the ratio of growth, or “cutting-down factor.” Then, -obviously, when <span class="nowrap"> -<i>O P</i><sub>1</sub></span> is less than <span class="nowrap"> -<i>O P</i><sub>2</sub>′</span> the whorls will be separated by -an interspace (<i>a</i>); (2) when <span class="nowrap"> -<i>O P</i><sub>1</sub></span> <span class="nowrap"> -= <i>O P</i><sub>2</sub>′</span> they will be in contact -(<i>b</i>), and (3) when <span class="nowrap"> -<i>O P</i><sub>1</sub></span> is greater -than <span class="nowrap"><i>O P</i><sub>2</sub>′</span> there will a greater or -less extent of overlapping, that is to say of concealment of -the surfaces of the earlier by the later whorls (<i>c</i>). And -as a further case (4), it is plain that if λ be very large, -that is to say if <span class="nowrap"> -<i>O P</i><sub>1</sub></span> be greater, -not only than <span class="nowrap"> -<i>O P</i><sub>2</sub>′</span> but also than <span class="nowrap"> -<i>O P</i><sub>3</sub>′,</span> <span class="nowrap"> -<i>O P</i><sub>4</sub>′,</span> -etc., we shall have complete, or all but complete concealment -by the last formed whorl, of the whole of its predecessors. -This latter condition is completely attained in <i>Nautilus -pompilius</i>, and approached, though not quite attained, in <i>N. -umbilicatus</i>; and the difference between these two forms, -or “species,” is constituted accordingly by a difference in -the value of λ. (5) There is also a final case, not easily -distinguishable externally from (4), where <i>P′</i> lies on <span -class="xxpn" id="p543">{543}</span> the opposite side of the -radius vector to <i>P</i>, and is therefore imaginary. This final -condition is exhibited in Argonauta.</p> - -<p>The limiting values of λ are easily ascertained.</p> - -<div class="dright dwth-d" id="fig279"> -<img src="images/i543.png" width="385" height="173" alt=""> - <div class="dcaption">Fig 279.</div></div> - -<div class="dmaths"> -<p>In Fig. <a href="#fig279" title="go to Fig. 279">279</a> we have portions of two successive whorls, whose -corresponding points on the same radius vector (as <i>R</i> and -<i>R′</i>) are, therefore, at a distance apart corresponding to 2π. -Let <i>r</i> and <i>r′</i> refer to the inner, and <i>R</i>, <i>R′</i> to the outer -sides of the two whorls. Then, if we consider</p> - -<div><i>R</i> -= <i>a e</i><sup>θ cot α</sup> ,</div> - -<p class="pcontinue">it follows that <br class="brclrfix"></p> - -<div><i>R′</i> -= <i>a e</i><sup>(θ + 2π)cot α</sup> ,</div> - -<div><i>r</i> -= λ<i>a e</i><sup>θ cot α</sup> -= <i>a e</i><sup>(θ − β)cot α</sup> ,</div> - -<p class="pcontinue">and</p> - -<div><i>r′</i> -= λ<i>a e</i><sup>(θ + 2π)cot α</sup> -= <i>a e</i><sup>(θ + 2π − β)cot α</sup> . -</div> - -<p>Now in the three cases (<i>a</i>, <i>b</i>, <i>c</i>) represented in Fig. <a href="#fig278" title="go to Fig. 278">278</a>, it is -plain that <i>r′</i> -<span class="nowrap"><img class="iglyph-a" -src="images/iglyph-gtheqlth.png" width="33" height="60" -alt="⪌"></span> <i>R</i>, -respectively. That is to say,</p> - -<div>λ<i>a e</i><sup>(θ + 2π)cot α</sup> -<span class="nowrap"><img class="iglyph-a" -src="images/iglyph-gtheqlth.png" width="33" height="60" -alt="⪌"></span> <i>a e</i><sup>θ cot α</sup> , -<br class="brclrfix"></div> - -<p class="pcontinue pleftfloat">and</p> - -<div>λ<i>e</i><sup>2π cot α</sup> -<span class="nowrap"><img class="iglyph-a" -src="images/iglyph-gtheqlth.png" width="33" height="60" -alt="⪌"></span> 1.<br class="brclrfix"></div> -</div><!--dmaths--> - -<p>The case in which λ<i>e</i><sup>2π cot α</sup> -= 1, or −log λ -= 2π cot α log ε, is -the case represented in Fig. <a href="#fig278" title="go to Fig. 278">278</a>, <i>b</i>: that is to say, the particular -case, for each value of α, where the consecutive whorls just -touch, without interspace or overlap. For such cases, then, we -may tabulate the values of λ, as follows:</p> - -<div class="dtblbox"><div class="nowrap"> -<table> -<tr> - <th>Constant angle<br> - α of spiral</th> - <th>Ratio (λ) of rate<br> - of growth of inner border<br> - of tube, as compared with<br> - that of the outer border</th></tr> -<tr> - <td class="tdcntr">89°</td> - <td class="tdcntr">·896 </td></tr> -<tr> - <td class="tdcntr">88 </td> - <td class="tdcntr">·803 </td></tr> -<tr> - <td class="tdcntr">87 </td> - <td class="tdcntr">·720 </td></tr> -<tr> - <td class="tdcntr">86 </td> - <td class="tdcntr">·645 </td></tr> -<tr> - <td class="tdcntr">85 </td> - <td class="tdcntr">·577 </td></tr> -<tr> - <td class="tdcntr">80 </td> - <td class="tdcntr">·330 </td></tr> -<tr> - <td class="tdcntr">75 </td> - <td class="tdcntr">·234 </td></tr> -<tr> - <td class="tdcntr">70 </td> - <td class="tdcntr">·1016</td></tr> -<tr> - <td class="tdcntr">65 </td> - <td class="tdcntr">·0534</td></tr> -</table></div></div><!--dtblbox--> - -<div><span class="xxpn" id="p544">{544}</span></div> - -<p>We see, accordingly, that in plane spirals whose constant -angle lies, say, between 65° and 70°, we can only obtain -contact between consecutive whorls if the rate of growth of -the inner border of the tube be a small fraction,—a tenth or -a twentieth—of that of the outer border. In spirals whose -constant angle is 80°, contact is attained when the respective -rates of growth are, approximately, as 3 to 1; while in spirals -of constant angle from about 85° to 89°, contact is attained -when the rates of growth are in the ratio of from about -3 ⁄ 5 to 9 ⁄ 10.</p> - -<div class="dctr07" id="fig280"> -<img src="images/i544.png" width="353" height="496" alt=""> - <div class="dcaption">Fig. 280.</div></div> - -<p>If on the other hand we have, for any given value of α, a -value of λ greater or less than the value given in the above -table, then we have, respectively, the conditions of separation -or of overlap which are exemplified in Fig. <a href="#fig278" title="go to Fig. 278">278</a>, <i>a</i> and <i>c</i>. -And, just as we have constructed this table of values of λ -for the particular case of simple contact between the whorls, -so we could construct similar tables for various degrees of -separation, or degrees of overlap.</p> - -<p>For instance, a case which admits of simple solution is that -in which the interspace between the whorls is everywhere a mean -proportional between the breadths of the whorls themselves -(Fig. <a href="#fig280" title="go to Fig. 280">280</a>). -<span class="xxpn" id="p545">{545}</span></p> - -<div class="dmaths"> -<p>In this case, let us call <i>OA</i> -= <i>R</i>, <i>OC</i> -= <i>R</i><sub>1</sub> and <i>OB</i> -= <i>r</i>. -We then have</p> - -<div><i>R</i><sub>1</sub> -= <i>OA</i> -= <i>a e</i><sup>θ cot α</sup> ,</div> - -<div><i>R</i><sub>2</sub> -= <i>OC</i> -= <i>a e</i><sup>(θ + 2π) cot α</sup> ,</div> - -<div><i>R</i><sub>1</sub> <i>R</i><sub>2</sub> -= <i>a e</i><sup>2(θ + π) cot α</sup> -= <i>r</i><sup>2</sup> <a class="afnanchlow" href="#fn520" -id="fnanch520" title="go to note 520">†</a>.</div> - -<p class="pcontinue pleftfloat">And</p> - -<div><i>r</i><sup>2</sup> -= (1 ⁄ λ)<sup>2</sup> · ε<sup>2θ cot α</sup> , -<br class="brclrfix"></div> - -<p class="pcontinue">whence, equating,</p> - -<div>1 ⁄ λ -= <i>e</i><sup>π cot α</sup> .</div> -</div><!--dmaths--> - -<p>The corresponding values of λ are as follows:</p> - -<div class="dtblboxin10"> -<table class="fsz7"> -<tr> - <th>Constant angle (α)</th> - <th>Ratio (λ) of rates - of growth of outer and inner border, - such as to produce a spiral with - interspaces between the whorls, the - breadth of which interspaces is a mean - proportional between the breadths of - the whorls themselves</th></tr> -<tr> - <td class="tdcntr">90°</td> - <td class="tdright">1·00  (imaginary)</td> -</tr> -<tr> - <td class="tdcntr">89 </td> - <td class="tdright"> ·95 </td></tr> -<tr> - <td class="tdcntr">88 </td> - <td class="tdright"> ·89 </td></tr> -<tr> - <td class="tdcntr">87 </td> - <td class="tdright"> ·85 </td></tr> -<tr> - <td class="tdcntr">86 </td> - <td class="tdright"> ·81 </td></tr> -<tr> - <td class="tdcntr">85 </td> - <td class="tdright"> ·76 </td></tr> -<tr> - <td class="tdcntr">80 </td> - <td class="tdright"> ·57 </td></tr> -<tr> - <td class="tdcntr">75 </td> - <td class="tdright"> ·43 </td></tr> -<tr> - <td class="tdcntr">70 </td> - <td class="tdright"> ·32 </td></tr> -<tr> - <td class="tdcntr">65 </td> - <td class="tdright"> ·23 </td></tr> -<tr> - <td class="tdcntr">60 </td> - <td class="tdright"> ·18 </td></tr> -<tr> - <td class="tdcntr">55 </td> - <td class="tdright"> ·13 </td></tr> -<tr> - <td class="tdcntr">50 </td> - <td class="tdright"> ·090</td></tr> -<tr> - <td class="tdcntr">45 </td> - <td class="tdright"> ·063</td></tr> -<tr> - <td class="tdcntr">40 </td> - <td class="tdright"> ·042</td></tr> -<tr> - <td class="tdcntr">35 </td> - <td class="tdright"> ·026</td></tr> -<tr> - <td class="tdcntr">30 </td> - <td class="tdright"> ·016</td></tr> -</table></div><!--dtblbox--> - -<div class="dmaths"> -<p>As regards the angle of retardation, β, in the formula</p> - -<div><i>r′</i> -= λ<i>e</i><sup>θ cot α</sup> ,   or   <i>r′</i> -= <i>e</i><sup>(θ − β)cot α</sup> ,</div> - -<p class="pcontinue">and in the case</p> - -<div><i>r′</i> -= <i>e</i><sup>(2π − β)cot α</sup> , -  or   -−log λ -= (2π − β)cot α,</div> - -<div><span class="xxpn" id="p546">{546}</span></div> - -<p class="pcontinue">it is evident that when β -= 2π, that will mean that λ -= 1. In -other words, the outer and inner borders of the tube are identical, -and the tube is constituted by one continuous line.</p> -</div><!--dmaths--> - -<p>When λ is a very small fraction, that is to say when the rates -of growth of the two borders of the tube are very diverse, then -β will tend towards infinity—tend that is to say towards a condition -in which the inner border of the tube never grows at all. -This condition is not infrequently approached in nature. The -nearly parallel-sided cone of Dentalium, or the widely separated -whorls of Lituites, are evidently cases where λ nearly approaches -unity in the one case, and is still large in the other, β being -correspondingly small; while we can easily find cases where β is -very large, and λ is a small fraction, for instance in Haliotis, or -in Gryphaea.</p> - -<p>For the purposes of the morphologist, then, the main result -of this last general investigation is to shew that all the various -types of “open” and “closed” spirals, all the various degrees of -separation or overlap of the successive whorls, are simply the -outward expression of a varying ratio in the <i>rate of growth</i> of the -outer as compared with the inner border of the tubular shell.</p> - -<hr class="hrblk"> - -<p>The foregoing problem of contact, or intersection, of the successive -whorls, is a very simple one in the case of the discoid shell -but a more complex one in the turbinate. For in the discoid shell -contact will evidently take place when the retardation of the -inner as compared with the outer whorl is just 360°, and the -shape of the whorls need not be considered.</p> - -<p>As the angle of retardation diminishes from 360°, the whorls -will stand further and further apart in an open coil; as it increases -beyond 360°, they will more and more overlap; and when the -angle of retardation is infinite, that is to say when the true inner -edge of the whorl does not grow at all, then the shell is said to -be completely involute. Of this latter condition we have a -striking example in Argonauta, and one a little more obscure in -<i>Nautilus pompilius</i>.</p> - -<p>In the turbinate shell, the problem of contact is twofold, for -we have to deal with the possibilities of contact on the <i>same</i> side -of the axis (which is what we have dealt with in -the discoid) and <span class="xxpn" id="p547">{547}</span> -also with the new possibility of contact or intersection on the -<i>opposite</i> side; it is this latter case which will determine the -presence or absence of an <i>umbilicus</i>, and whether, if present, it -will be an open conical space or a twisted cone. It is further -obvious that, in the case of the turbinate, the question of contact -or no contact will depend on the shape of the generating curve; -and if we take the simple case where this generating curve may -be considered as an ellipse, then contact will be found to depend -on the angle which the major axis of this ellipse makes with the -axis of the shell. The question becomes a complicated one, and -the student will find it treated in Blake’s paper already referred to.</p> - -<p>When one whorl overlaps another, so that the generating -curve cuts its predecessor (at a distance of 2π) on the same radius -vector, the locus of intersection will follow a spiral line upon the -shell, which is called the “suture” by conchologists. It is evidently -one of that <i>ensemble</i> of spiral lines in space of which, as we have -seen, the whole shell may be conceived to be constituted; and we -might call it a “contact-spiral,” or “spiral of intersection.” In -discoid shells, such as an Ammonite or a Planorbis, or in <i>Nautilus -umbilicatus</i>, there are obviously two such contact-spirals, one on -each side of the shell, that is to say one on each side of a plane -perpendicular to the axis. In turbinate shells such a condition -is also possible, but is somewhat rare. We have it for instance, -in <i>Solarium perspectivum</i>, where the one contact-spiral is visible -on the exterior of the cone, and the other lies internally, -winding round the open cone of the umbilicus<a class="afnanch" href="#fn521" id="fnanch521">521</a>; -but this second -contact-spiral is usually imaginary, or concealed within the -whorls of the turbinated shell. Again, in Haliotis, one of the -contact-spirals is non-existent, because of the extreme obliquity -of the plane of the generating curve. In <i>Scalaria pretiosa</i> and -in Spirula there is no contact-spiral, because the growth of the -generating curve has been too slow, in comparison with the vector -rotation of its plane. In Argonauta and in Cypraea, there is no -contact-spiral, because the growth of the generating curve has -been too quick. Nor, of course, is there any contact-spiral in -Patella or in Dentalium, because the angle α is too small ever to -give us a complete revolution of the spire. <span class="xxpn" id="p548">{548}</span></p> - -<p>The various forms of straight or spiral shells among the -Cephalopods, which we have seen to be capable of complete -definition by the help of elementary mathematics, have received -a very complicated descriptive nomenclature from the palaeontologists. -For instance, the straight cones are spoken of as -<i>orthoceracones</i> or <i>bactriticones</i>, the loosely coiled forms as <i>gyroceracones</i> -or <i>mimoceracones</i>, the more closely coiled shells, in which -one whorl overlaps the other, as <i>nautilicones</i> or <i>ammoniticones</i>, -and so forth. In such a succession of forms the biologist sees -undoubted and unquestioned evidence of ancestral descent. For -instance we read in Zittel’s <i>Palaeontology</i><a class="afnanch" href="#fn522" id="fnanch522">522</a>: -“The bactriticone -obviously represents the primitive or primary radical of the -Ammonoidea, and the mimoceracone the next or secondary radical -of this order”; while precisely the opposite conclusion was drawn -by Owen, who supposed that the straight chambered shells of -such fossil cephalopods as Orthoceras had been produced by the -gradual unwinding of a coiled nautiloid shell<a class="afnanch" href="#fn523" id="fnanch523">523</a>. -<i>To such phylogenetic -hypotheses the mathematical or dynamical study of the forms of -shells lends no valid support.</i> If we have two shells in which the -constant angle of the spire be respectively 80° and 60°, that fact -in itself does not at all justify an assertion that the one is more -primitive, more ancient, or more “ancestral” than the other. -Nor, if we find a third in which the angle happens to be 70°, -does that fact entitle us to say that this shell is intermediate -between the other two, in time, or in blood relationship, or in -any other sense whatsoever save only the strictly formal and -mathematical one. For it is evident that, though these particular -arithmetical constants manifest themselves in visible and recognisable -differences of form, yet they are not necessarily more -deep-seated or significant than are those which manifest themselves -only in difference of magnitude; and the student of -phylogeny scarcely ventures to draw conclusions as to the relative -antiquity of two allied organisms on the ground that one happens -to be bigger or less, or longer or shorter, than the other. <span class="xxpn" id="p549">{549}</span></p> - -<p>At the same time, while it is obviously unsafe to rest conclusions -upon such features as these, unless they be strongly supported -and corroborated in other ways,—for the simple reason that there -is unlimited room for <i>coincidence</i>, or separate and independent -attainment of this or that magnitude or numerical ratio,—yet on -the other hand it is certain that, in particular cases, the evolution -of a race has actually involved gradual increase or decrease in -some one or more numerical factors, magnitude itself included,—that -is to say increase or decrease in some one or more of the -actual and relative velocities of growth. When we do meet with -a clear and unmistakable series of such progressive magnitudes or -ratios, manifesting themselves in a progressive series of “allied” -forms, then we have the phenomenon of “<i>orthogenesis</i>.” For -orthogenesis is simply that phenomenon of continuous lines or -series of form (and also of functional or physiological capacity), -which was the foundation of the Theory of Evolution, alike to -Lamarck and to Darwin and Wallace; and which we see to exist -whatever be our ideas of the “origin of species,” or of the nature -and origin of “functional adaptations.” And to my mind, the -mathematical (as distinguished from the purely physical) study -of morphology bids fair to help us to recognise this phenomenon -of orthogenesis in many cases where it is not at once patent to -the eye; and also, on the other hand, to warn us, in many other -cases, that even strong and apparently complex resemblances in -form may be capable of arising independently, and may sometimes -signify no more than the equally accidental numerical coincidences -which are manifested in identity of length or weight, or any other -simple magnitudes.</p> - -<hr class="hrblk"> - -<p>I have already referred to the fact that, while in general a -very great and remarkable regularity of form is characteristic of -the molluscan shell, that complete regularity is apt to be departed -from. We have clear cases of such a departure in Pupa, Clausilia, -and various Bulimi, where the enveloping cone of the spire is -not a right cone but a cone whose sides are curved. It is plain -that this condition may arise in two ways: either by a gradual -change in the ratio of growth of the whorls, that is to say in -the logarithmic spiral itself, or by a change in -the velocity of <span class="xxpn" id="p550">{550}</span> -translation along the axis, that is to say in the helicoid which, -in all turbinate shells, is superposed upon the spiral. Very careful -measurements will be necessary to determine to which of these -factors, or in what proportions to each, the phenomenon is due. -But in many Ammonitoidea where the helicoid factor does not -enter into the case, we have a clear illustration of gradual and -marked changes in the spiral angle itself, that is to say of the ratio -of growth corresponding to increase of vectorial angle. We have -seen from some of Naumann’s and Grabau’s measurements that -such a tendency to vary, such an acceleration or retardation, -may be detected even in Ammonites which present nothing -abnormal to the eye. But let us suppose that the spiral angle -increases somewhat rapidly; we shall then get a spiral with -gradually narrowing whorls, and this condition is characteristic</p> - -<div class="dctr03" id="fig281"> -<img src="images/i550.png" width="608" height="240" alt=""> - <div class="pcaption">Fig. 281. An ammonitoid shell - (<i>Macroscaphites</i>) to shew change of curvature.</div></div> - -<p class="pcontinue"> -of Oekotraustes, a subgenus of Ammonites. If on the other hand, -the angle α gradually diminishes, and even falls away to zero, we -shall have the spiral curve opening out, as it does in Scaphites, -Ancyloceras and Lituites, until the spiral coil is replaced by a spiral -curve so gentle as to seem all but straight. Lastly, there are a -few cases, such as <i>Bellerophon expansus</i> and some Goniatites, -where the outer spiral does not perceptibly change, but the whorls -become more “embracing” or the whole shell more involute. -Here it is the angle of retardation, the ratio of growth between -the outer and inner parts of the whorl, which undergoes a gradual -change.</p> - -<hr class="hrblk"> - -<p>In order to understand the relation of a close-coiled shell -to one of its straighter congeners, to compare (for example) -an <span class="xxpn" id="p551">{551}</span> Ammonite with -an Orthoceras, it is necessary to estimate the length of -the right cone which has, so to speak, been coiled up into -the spiral shell. Our problem then is, To find the length -of a plane logarithmic spiral, in terms of the radius and -the constant angle α. In the annexed diagram, if <i>OP</i> be a -radius vector, <i>OQ</i> a line of reference perpendicular to <i>OP</i>, -and <i>PQ</i> a tangent to the curve, <i>PQ</i>, or sec α, is equal -in length to the spiral arc <i>OP</i>. And this is practically -obvious: for <i>PP′ ⁄ PR′</i> -= <i>ds ⁄ dr</i> -= sec α, and -therefore sec α -= <i>s ⁄ r</i>, or the ratio of arc to radius -vector.</p> - -<div class="dctr05" id="fig282"> -<img src="images/i551.png" width="449" height="305" alt=""> - <div class="dcaption">Fig. 282.</div></div> - -<p>Accordingly, the ratio of <i>l</i>, the total length, to <i>r</i>, the radius -vector up to which the total length is to be measured, is expressed -by a simple table of secants; as follows:</p> - -<div class="dtblbox"><div class="nowrap"> -<table> -<tr> - <th>α</th> - <th><i>l ⁄ r</i></th></tr> -<tr> - <td class="tdright">5°    </td> - <td class="tdright">1·004</td></tr> -<tr> - <td class="tdright">10     </td> - <td class="tdright">1·015</td></tr> -<tr> - <td class="tdright">20     </td> - <td class="tdright">1·064</td></tr> -<tr> - <td class="tdright">30     </td> - <td class="tdright">1·165</td></tr> -<tr> - <td class="tdright">40     </td> - <td class="tdright">1·305</td></tr> -<tr> - <td class="tdright">50     </td> - <td class="tdright">1·56 </td></tr> -<tr> - <td class="tdright">60     </td> - <td class="tdright">2·0  </td></tr> -<tr> - <td class="tdright">70     </td> - <td class="tdright">2·9  </td></tr> -<tr> - <td class="tdright">75     </td> - <td class="tdright">3·9  </td></tr> -<tr> - <td class="tdright">80     </td> - <td class="tdright">5·8  </td></tr> -<tr> - <td class="tdright">85     </td> - <td class="tdright">11·5  </td></tr> -<tr> - <td class="tdright">86     </td> - <td class="tdright">14·3  </td></tr> -<tr> - <td class="tdright">87     </td> - <td class="tdright">19·1  </td></tr> -<tr> - <td class="tdright">88     </td> - <td class="tdright">28·7  </td></tr> -<tr> - <td class="tdright">89     </td> - <td class="tdright">57·3  </td></tr> -<tr> - <td class="tdright">89° 10′</td> - <td class="tdright">68·8  </td></tr> -<tr> - <td class="tdright">    20</td> - <td class="tdright">85·9  </td></tr> -<tr> - <td class="tdright">    30</td> - <td class="tdright">114·6  </td></tr> -<tr> - <td class="tdright">    40</td> - <td class="tdright">171·9  </td></tr> -<tr> - <td class="tdright">    50</td> - <td class="tdright">343·8  </td></tr> -<tr> - <td class="tdright">    55</td> - <td class="tdright">687·5  </td></tr> -<tr> - <td class="tdright">    59</td> - <td class="tdright">3437·7  </td></tr> -<tr> - <td class="tdright">90     </td> - <td class="tdcntr">Infinite</td></tr> -</table></div></div><!--dtblbox--> - -<p>Putting the same table inversely, so as to -shew the total <span class="xxpn" id="p552">{552}</span> -length in whole numbers, in terms of the radius, we have as -follows:</p> - -<div class="dtblbox"><div class="nowrap"> -<table> -<tr> - <th> - Total length<br> - (in terms of<br> - the radius)</th> - <th>Constant angle</th></tr> -<tr> - <td class="tdcntr">     2</td> - <td class="tdcntr">60°        </td></tr> -<tr> - <td class="tdcntr">     3</td> - <td class="tdcntr">70  31′    </td></tr> -<tr> - <td class="tdcntr">     4</td> - <td class="tdcntr">75  32     </td></tr> -<tr> - <td class="tdcntr">     5</td> - <td class="tdcntr">78  28     </td></tr> -<tr> - <td class="tdcntr">    10</td> - <td class="tdcntr">84  16     </td></tr> -<tr> - <td class="tdcntr">    20</td> - <td class="tdcntr">87   8     </td></tr> -<tr> - <td class="tdcntr">    30</td> - <td class="tdcntr">88   6     </td></tr> -<tr> - <td class="tdcntr">    40</td> - <td class="tdcntr">88  34     </td></tr> -<tr> - <td class="tdcntr">    50</td> - <td class="tdcntr">88  51     </td></tr> -<tr> - <td class="tdcntr">   100</td> - <td class="tdcntr">89  26     </td></tr> -<tr> - <td class="tdcntr">  1000</td> - <td class="tdcntr">89  56′ 36″</td></tr> -<tr> - <td class="tdcntr">10,000</td> - <td class="tdcntr">89  59  30 </td></tr> -</table></div></div><!--dtblbox--> - -<p>Accordingly, we see that (1), when the constant angle of the -spiral is small, the spiral itself is scarcely distinguishable from -a straight line, and its length is but very little greater than that -of its own radius vector. This remains pretty much the case for -a considerable increase of angle, say from 0° to 20° or more; -(2) for a very considerably greater increase of the constant angle, -say to 50° or more, the shell would only have the appearance of -a gentle curve; (3) the characteristic close coils of the Nautilus -or Ammonite would be typically represented only when the -constant angle lies within a few degrees on either side of about -80°. The coiled up spiral of a Nautilus, with a constant angle -of about 80°, is about six times the length of its radius vector, -or rather more than three times its own diameter; while that of -an Ammonite, with a constant angle of, say, from 85° to 88°, is -from about six to fifteen times as long as its own diameter. And -(4) as we approach an angle of 90° (at which point the spiral -vanishes in a circle), the length of the coil increases with enormous -rapidity. Our spiral would soon assume the appearance of the -close coils of a Nummulite, and the successive increments of -breadth in the successive whorls would become inappreciable to -the eye. The logarithmic spiral of high constant angle would, -as we have already seen, tend to become indistinguishable, without -the most careful measurement, from an Archimedean spiral. -And it is obvious, moreover, that our -ordinary methods of <span class="xxpn" id="p553">{553}</span> -determining the constant angle of the spiral would not in these -cases be accurate enough to enable us to measure the length of -the coil: we should have to devise a new method, based on the -measurement of radii or diameters over a large number of whorls.</p> - -<p>The geometrical form of the shell involves many other beautiful -properties, of great interest to the mathematician, but which it -is not possible to reduce to such simple expressions as we have -been content to use. For instance, we may obtain an equation -which shall express completely the surface of any shell, in terms -of polar or of rectangular coordinates (as has been done by Moseley -and by Blake), or in Hamiltonian vector notation. It is likewise -possible (though of little interest to the naturalist) to determine -the area of a conchoidal surface, or the volume of a conchoidal -solid, and to find the centre of gravity of either surface or -solid<a class="afnanch" href="#fn524" id="fnanch524">524</a>. -And Blake has further shewn, with considerable elaboration, how -we may deal with the symmetrical distortion, due to pressure, -which fossil shells are often found to have undergone, and how -we may reconstitute by calculation their original undistorted -form,—a problem which, were the available methods only a little -easier, would be very helpful to the palaeontologist; for, as -Blake himself has shewn, it is easy to mistake a symmetrically -distorted specimen of (for instance) an Ammonite, for a new and -distinct species of the same genus. But it is evident that to deal -fully with the mathematical problems contained in, or suggested -by, the spiral shell, would require a whole treatise, rather than -a single chapter of this elementary book. Let us then, leaving -mathematics aside, attempt to summarise, and perhaps to extend, -what has been said about the general possibilities of form in this -class of organisms.</p> - -<div class="section"> -<h3><i>The Univalve Shell: a summary.</i></h3></div> - -<p>The surface of any shell, whether discoid or turbinate, may be -imagined to be generated by the revolution about a fixed axis of -a closed curve, which, remaining always geometrically similar to -itself, increases continually its dimensions: and, since the rate of -growth of the generating curve and its velocity of rotation follow -the same law, the curve traced in space by -corresponding points <span class="xxpn" id="p554">{554}</span> -in the generating curve is, in all cases, a logarithmic spiral. In -discoid shells, the generating figure revolves in a plane perpendicular -to the axis, as in Nautilus, the Argonaut and the Ammonite. -In turbinate shells, it slides continually along the axis of revolution, -and the curve in space generated by any given point partakes, -therefore, of the character of a helix, as well as of a logarithmic -spiral; it may be strictly entitled a helico-spiral. Such turbinate -or helico-spiral shells include the snail, the periwinkle and all the -common typical Gastropods.</p> - -<p>The generating figure, as represented by the mouth of the -shell, is sometimes a plane curve, of simple form; in other and -more numerous cases, it becomes more complicated in form and -its boundaries do not lie in one plane: but in such cases as these -we</p> - -<div class="dleft dwth-d" id="fig283"> -<img src="images/i554.png" width="289" height="565" alt=""> - <div class="dcaption">Fig. 283. Section of a spiral, or - turbinate, univalve, <i>Triton corrugatus</i>, Lam. (From - Woodward.)</div></div> - -<p class="pcontinue"> -may replace it by its “trace,” on a -plane at some definite angle to the direction -of growth, for instance by its form as it -appears in a section through the axis of -the helicoid shell. The generating curve -is of very various shapes. It is circular -in Scalaria or Cyclostoma, and in Spirula; -it may be considered as a segment of a -circle in Natica or in Planorbis. It is -approximately triangular in Conus, and -rhomboidal in Solarium or Potamides. It -is very commonly more or less elliptical: -the long axis of the ellipse being parallel -to the axis of the shell in Oliva and Cypraea; -all but perpendicular to it in many Trochi; -and oblique to it in many well-marked -cases, such as Stomatella, Lamellaria, -<i>Sigaretus haliotoides</i> (Fig. <a href="#fig284" title="go to Fig. 284">284</a>) and Haliotis. -In <i>Nautilus pompilius</i> it is approximately -a semi-ellipse, and in <i>N. umbilicatus</i> rather -more than a semi-ellipse, the long axis -lying in both cases perpendicular to the axis -of the shell<a class="afnanch" href="#fn525" id="fnanch525">525</a>. -Its <span class="xxpn" id="p555">{555}</span> -form is seldom open to easy mathematical expression, save when -it is an actual circle or ellipse; but an exception to this rule may -be found in certain Ammonites, forming the group “Cordati,” -where (as Blake points out) the curve is very nearly represented -by a cardioid, whose equation is <i>r</i> -= <i>a</i>(1 + cos θ). -<br class="brclrfix"></p> - -<p>The generating curve may grow slowly or quickly; its growth-factor -is very slow in Dentalium or Turritella, very rapid in Nerita, -or Pileopsis, or Haliotis or the Limpet. It may contain the axis -in its plane, as in Nautilus; it may be parallel to the axis, as in -the majority of Gastropods; or it may be inclined to the axis, as -it is in a very marked degree in Haliotis. In fact, in Haliotis -the generating curve is so oblique to the axis of the shell that -the latter appears to grow by additions to one margin only (cf. -Fig. <a href="#fig258" title="go to Fig. 258">258</a>), as in the case of the opercula of Turbo and Nerita -referred to on p. <a href="#p522" title="go to pg. 522">522</a>; and this is what Moseley supposed it to do.</p> - -<div class="dctr02" id="fig284"> -<img src="images/i555.png" width="705" height="219" alt=""> - <div class="dcaption">Fig. 284. <i>A, Lamellaria - perspicua; B, Sigaretus haliotoides.</i><br>(After - Woodward.)</div></div> - -<p>The general appearance of the entire shell is determined (apart -from the form of its generating curve) by the magnitude of three -angles; and these in turn are determined, as has been sufficiently -explained, by the ratios of certain velocities of growth. These -angles are (1) the constant angle of the logarithmic spiral (α); -(2) in turbinate shells, the enveloping angle of the cone, or (taking -half that angle) the angle (θ) which a tangent to the whorls makes -with the axis of the shell; and (3) an angle called the “angle of -retardation” (β), which expresses the retardation -in growth of <span class="xxpn" id="p556">{556}</span> -the inner as compared with the outer part of each whorl, and -therefore measures the extent to which one whorl overlaps, or the -extent to which it is separated from, another.</p> - -<p>The spiral angle (α) is very small in a limpet, where it is usually -taken as -= 0°; but it is evidently of a significant amount, though -obscured by the shortness of the tubular shell. In Dentalium -it is still small, but sufficient to give the appearance of a regular -curve; it amounts here probably to about 30° to 40°. In Haliotis -it is from about 70° to 75°; in Nautilus about 80°; and it lies -between 80° and 85°, or even more, in the majority of Gastropods.</p> - -<p>The case of Fissurella is curious. Here we have, apparently, -a conical shell with no trace of spiral curvature, or (in other -words) with a spiral angle which approximates to 0°; but in the -minute embryonic shell (as in that of the limpet) a spiral convolution -is distinctly to be seen. It would seem, then, that what we have -to do with here is an unusually large growth-factor in the generating -curve, which causes the shell to dilate into a cone of very wide -angle, the apical portion of which has become lost or absorbed, -and the remaining part of which is too short to show clearly its -intrinsic curvature. In the closely allied Emarginula, there is -likewise a well-marked spiral in the embryo, which however is -still manifested in the curvature of the adult, nearly conical, shell. -In both cases we have to do with a very wide-angled cone, and -with a high retardation-factor for its inner, or posterior, border. -The series is continued, from the apparently simple cone to the -complete spiral, through such forms as Calyptraea.</p> - -<p>The angle α, as we have seen, is not always, nor rigorously, -a constant angle. In some Ammonites it may increase with age, -the whorls becoming closer and closer; in others it may decrease -rapidly, and even fall to zero, the coiled shell then straightening -out, as in Lituites and similar forms. It diminishes somewhat, -also, in many Orthocerata, which are slightly curved in youth, -but straight in age. It tends to increase notably in some common -land-shells, the Pupae and Bulimi; and it decreases in Succinea.</p> - -<p>Directly related to the angle α is the ratio which subsists -between the breadths of successive whorls. The following table -gives a few illustrations of this ratio in particular cases, -in addition to those which we have already studied. <span -class="xxpn" id="p557">{557}</span></p> - -<div class="dtblboxin10"> -<table class="fsz7"> -<caption class="fsz4"><i>Ratio of breadth of - consecutive whorls.</i></caption> -<tr> - <th colspan="2">Pointed Turbinates</th> - <th></th> - <th colspan="2">Obtuse Turbinates and Discoids</th></tr> -<tr> - <td class="tdleft"> <i>Telescopium fuscum</i></td> - <td class="tdright">1·14</td> - <td class="tdright"> </td> - <td class="tdleft"> <i>Conus virgo</i></td> - <td class="tdright">1·25</td></tr> -<tr> - <td class="tdleft"> <i>Acus subulatus</i></td> - <td class="tdright">1·16</td> - <td class="tdright"> </td> - <td class="tdleft"> <i>Conus litteratus</i></td> - <td class="tdright">1·40</td></tr> -<tr> - <td class="tdleft">*<i>Turritella terebellata</i></td> - <td class="tdright">1·18</td> - <td class="tdright"> </td> - <td class="tdleft"> <i>Conus betulina</i></td> - <td class="tdright">1·43</td></tr> -<tr> - <td class="tdleft">*<i>Turritella imbricata</i></td> - <td class="tdright">1·20</td> - <td class="tdright"> </td> - <td class="tdleft">*<i>Helix nemoralis</i></td> - <td class="tdright">1·50</td></tr> -<tr> - <td class="tdleft"> <i>Cerithium palustre</i></td> - <td class="tdright">1·22</td> - <td class="tdright"> </td> - <td class="tdleft">*<i>Solarium perspectivum</i></td> - <td class="tdright">1·50</td></tr> -<tr> - <td class="tdleft"> <i>Turritella duplicata</i></td> - <td class="tdright">1·23</td> - <td class="tdright"> </td> - <td class="tdleft"> <i>Solarium trochleare</i></td> - <td class="tdright">1·62</td></tr> -<tr> - <td class="tdleft"> <i>Melanopsis terebralis</i></td> - <td class="tdright">1·23</td> - <td class="tdright"> </td> - <td class="tdleft"> <i>Solarium magnificum</i></td> - <td class="tdright">1·75</td></tr> -<tr> - <td class="tdleft"> <i>Cerithium nodulosum</i></td> - <td class="tdright">1·24</td> - <td class="tdright"> </td> - <td class="tdleft">*<i>Natica aperta</i></td> - <td class="tdright">2·00</td></tr> -<tr> - <td class="tdleft">*<i>Turritella carinata</i></td> - <td class="tdright">1·25</td> - <td class="tdright"> </td> - <td class="tdleft"> <i>Euomphalus pentangulatus</i></td> - <td class="tdright">2·00</td></tr> -<tr> - <td class="tdleft"> <i>Acus crenulatus</i></td> - <td class="tdright">1·25</td> - <td class="tdright"> </td> - <td class="tdleft"> <i>Planorbis corneas</i></td> - <td class="tdright">2·00</td></tr> -<tr> - <td class="tdleft"> <i>Terebra maculata</i> (Fig. <a href="#fig285" title="go to Fig. 285">285</a>)</td> - <td class="tdright">1·25</td> - <td class="tdright"> </td> - <td class="tdleft"> <i>Solaropsis pellis-serpentis</i></td> - <td class="tdright">2·00</td></tr> -<tr> - <td class="tdleft">*<i>Cerithium lignitarum</i></td> - <td class="tdright">1·26</td> - <td class="tdright"> </td> - <td class="tdleft"> <i>Dolium zonatum</i></td> - <td class="tdright">2·10</td></tr> -<tr> - <td class="tdleft"> <i>Acus dimidiatus</i></td> - <td class="tdright">1·28</td> - <td class="tdright"> </td> - <td class="tdleft">*<i>Natica glaucina</i></td> - <td class="tdright">3·00</td></tr> -<tr> - <td class="tdleft"> <i>Cerithium sulcatum</i></td> - <td class="tdright">1·32</td> - <td class="tdright"> </td> - <td class="tdleft"> <i>Nautilus pompilius</i></td> - <td class="tdright">3·00</td></tr> -<tr> - <td class="tdleft"> <i>Fusus longissimus</i></td> - <td class="tdright">1·34</td> - <td class="tdright"> </td> - <td class="tdleft"> <i>Haliotis excavatus</i></td> - <td class="tdright">4·20</td></tr> -<tr> - <td class="tdleft">*<i>Pleurotomaria conoidea</i></td> - <td class="tdright">1·34</td> - <td class="tdright"> </td> - <td class="tdleft"> <i>Haliotis parvus</i></td> - <td class="tdright">6·00</td></tr> -<tr> - <td class="tdleft"> <i>Trochus niloticus</i> (Fig. <a href="#fig286" title="go to Fig. 286">286</a>)</td> - <td class="tdright">1·41</td> - <td class="tdright"> </td> - <td class="tdleft"> <i>Delphinula atrata</i></td> - <td class="tdright">6·00</td></tr> -<tr> - <td class="tdleft"> <i>Mitra episcopalis</i></td> - <td class="tdright">1·43</td> - <td class="tdright"> </td> - <td class="tdleft"> <i>Haliotis rugoso-plicata</i></td> - <td class="tdright">9·30</td></tr> -<tr> - <td class="tdleft"> <i>Fusus antiquus</i></td> - <td class="tdright">1·50</td> - <td class="tdright"> </td> - <td class="tdleft"> <i>Haliotis viridis</i></td> - <td class="tdright">10·00</td></tr> -<tr> - <td class="tdleft"> <i>Scalaria pretiosa</i></td> - <td class="tdright">1·56</td> - <td class="tdright"> </td> - <td class="tdright"> </td> - <td class="tdright"> </td></tr> -<tr> - <td class="tdleft"> <i>Fusus colosseus</i></td> - <td class="tdright">1·71</td> - <td class="tdright"> </td> - <td class="tdright"> </td> - <td class="tdright"> </td></tr> -<tr> - <td class="tdleft"> <i>Phasianella bulloides</i></td> - <td class="tdright">1·80</td> - <td class="tdright"> </td> - <td class="tdright"> </td> - <td class="tdright"> </td></tr> -<tr> - <td class="tdleft"> <i>Helicostyla polychroa</i></td> - <td class="tdright">2·00</td> - <td class="tdright"> </td> - <td class="tdright"> </td> - <td class="tdright"> </td></tr> -<tr> - <td class="tdright" colspan="5">Those marked * from Naumann; the rest from Macalister<a -class="afnanch" href="#fn526" id="fnanch526">526</a>.</td></tr> -</table> - -</div><!--dtblbox--> - -<div class="dmaths"> -<p>In the case of turbinate shells, we require to take into account -the angle θ, in order to determine the spiral angle α from the -ratio of the breadths of consecutive whorls; for the short table -given on p. <a href="#p534" title="go to pg. 534">534</a> is only applicable to discoid shells, in which -the angle θ is an angle of 90°. Our formula, as mentioned on -p. <a href="#p518" title="go to pg. 518">518</a> now becomes</p> - -<div><i>R</i> -= ε<sup>2π sin θ cot α</sup> . -</div></div><!--dmaths--> - -<p>For this formula I have worked out the following table. -<span class="xxpn" id="p558">{558}</span></p> - -<div class="dtblbox"> -<table class="fsz7 borall"> -<caption><i>Table shewing values of the spiral angle α -corresponding to certain ratios of breadth of successive whorls -of the shell, for various values of the apical semi-angle -θ.</i></caption> -<tr> - <th class="thsnug borall">Ratio<br> - <i>R</i> ⁄ 1</th> - <th class="thsnug borall">θ = 5°</th> - <th class="thsnug borall">10°</th> - <th class="thsnug borall">15°</th> - <th class="thsnug borall">20°</th> - <th class="thsnug borall">30°</th> - <th class="thsnug borall">40°</th> - <th class="thsnug borall">50°</th> - <th class="thsnug borall">60°</th> - <th class="thsnug borall">70°</th> - <th class="thsnug borall">80°</th> - <th class="thsnug borall">90°</th></tr> -<tr> - <td class="tdsnug">  1·1 </td> - <td class="tdsnug">80°  8′</td> - <td class="tdsnug">85°  0′</td> - <td class="tdsnug">86° 44′</td> - <td class="tdsnug">87° 28′</td> - <td class="tdsnug">88° 16′</td> - <td class="tdsnug">88° 39′</td> - <td class="tdsnug">88° 52′</td> - <td class="tdsnug">89°  0′</td> - <td class="tdsnug">89°  4′</td> - <td class="tdsnug">89°  7′</td> - <td class="tdsnug">89°  8′</td></tr> -<tr> - <td class="tdsnug">  1·25</td> - <td class="tdsnug">67  51 </td> - <td class="tdsnug">78  27 </td> - <td class="tdsnug">82  11 </td> - <td class="tdsnug">84   5 </td> - <td class="tdsnug">85  56 </td> - <td class="tdsnug">86  50 </td> - <td class="tdsnug">87  21 </td> - <td class="tdsnug">87  39 </td> - <td class="tdsnug">87  50 </td> - <td class="tdsnug">87  56 </td> - <td class="tdsnug">87  58 </td></tr> -<tr> - <td class="tdsnug">  1·5 </td> - <td class="tdsnug">53  30 </td> - <td class="tdsnug">69  37 </td> - <td class="tdsnug">76   0 </td> - <td class="tdsnug">79  21 </td> - <td class="tdsnug">82  39 </td> - <td class="tdsnug">84  16 </td> - <td class="tdsnug">85  13 </td> - <td class="tdsnug">85  44 </td> - <td class="tdsnug">86   4 </td> - <td class="tdsnug">86  15 </td> - <td class="tdsnug">86  18 </td></tr> -<tr> - <td class="tdsnug">  2·0 </td> - <td class="tdsnug">38  20 </td> - <td class="tdsnug">57  35 </td> - <td class="tdsnug">66  55 </td> - <td class="tdsnug">73  11 </td> - <td class="tdsnug">77  34 </td> - <td class="tdsnug">80  16 </td> - <td class="tdsnug">81  52 </td> - <td class="tdsnug">82  45 </td> - <td class="tdsnug">83  18 </td> - <td class="tdsnug">83  37 </td> - <td class="tdsnug">83  42 </td></tr> -<tr> - <td class="tdsnug">  2·5 </td> - <td class="tdsnug">30  53 </td> - <td class="tdsnug">50   0 </td> - <td class="tdsnug">60  35 </td> - <td class="tdsnug">67   0 </td> - <td class="tdsnug">73  45 </td> - <td class="tdsnug">77  13 </td> - <td class="tdsnug">79  19 </td> - <td class="tdsnug">80  26 </td> - <td class="tdsnug">81  11 </td> - <td class="tdsnug">81  35 </td> - <td class="tdsnug">81  42 </td></tr> -<tr> - <td class="tdsnug">  3·0 </td> - <td class="tdsnug">26  32 </td> - <td class="tdsnug">44  50 </td> - <td class="tdsnug">56   0 </td> - <td class="tdsnug">63   0 </td> - <td class="tdsnug">70  45 </td> - <td class="tdsnug">74  45 </td> - <td class="tdsnug">77  17 </td> - <td class="tdsnug">78  35 </td> - <td class="tdsnug">79  28 </td> - <td class="tdsnug">79  56 </td> - <td class="tdsnug">80   5 </td></tr> -<tr> - <td class="tdsnug">  3·5 </td> - <td class="tdsnug">23  37 </td> - <td class="tdsnug">41   5 </td> - <td class="tdsnug">52  25 </td> - <td class="tdsnug">59  50 </td> - <td class="tdsnug">68  15 </td> - <td class="tdsnug">72  45 </td> - <td class="tdsnug">75  35 </td> - <td class="tdsnug">77   2 </td> - <td class="tdsnug">78   1 </td> - <td class="tdsnug">78  33 </td> - <td class="tdsnug">78  43 </td></tr> -<tr> - <td class="tdsnug">  4·0 </td> - <td class="tdsnug">21  35 </td> - <td class="tdsnug">38  10 </td> - <td class="tdsnug">49  35 </td> - <td class="tdsnug">57  15 </td> - <td class="tdsnug">66  10 </td> - <td class="tdsnug">71   3 </td> - <td class="tdsnug">74   9 </td> - <td class="tdsnug">75  42 </td> - <td class="tdsnug">76  47 </td> - <td class="tdsnug">77  22 </td> - <td class="tdsnug">77  34 </td></tr> -<tr> - <td class="tdsnug">  4·5 </td> - <td class="tdsnug">20   0 </td> - <td class="tdsnug">36   0 </td> - <td class="tdsnug">47  15 </td> - <td class="tdsnug">55   5 </td> - <td class="tdsnug">64  25 </td> - <td class="tdsnug">69  35 </td> - <td class="tdsnug">72  54 </td> - <td class="tdsnug">74  33 </td> - <td class="tdsnug">75  43 </td> - <td class="tdsnug">76  20 </td> - <td class="tdsnug">76  35 </td></tr> -<tr> - <td class="tdsnug">  5·0 </td> - <td class="tdsnug">18  45 </td> - <td class="tdsnug">34  10 </td> - <td class="tdsnug">45  20 </td> - <td class="tdsnug">53  15 </td> - <td class="tdsnug">62  55 </td> - <td class="tdsnug">68  15 </td> - <td class="tdsnug">71  48 </td> - <td class="tdsnug">73  31 </td> - <td class="tdsnug">74  45 </td> - <td class="tdsnug">75  25 </td> - <td class="tdsnug">75  38 </td></tr> -<tr> - <td class="tdsnug"> 10·0 </td> - <td class="tdsnug">13  25 </td> - <td class="tdsnug">25  20 </td> - <td class="tdsnug">35  15 </td> - <td class="tdsnug">43   5 </td> - <td class="tdsnug">53  45 </td> - <td class="tdsnug">60  20 </td> - <td class="tdsnug">64  57 </td> - <td class="tdsnug">67   4 </td> - <td class="tdsnug">68  42 </td> - <td class="tdsnug">69  35 </td> - <td class="tdsnug">69  53 </td></tr> -<tr> - <td class="tdsnug"> 20·0 </td> - <td class="tdsnug">10  25 </td> - <td class="tdsnug">20   0 </td> - <td class="tdsnug">28  30 </td> - <td class="tdsnug">35  45 </td> - <td class="tdsnug">46  25 </td> - <td class="tdsnug">53  25 </td> - <td class="tdsnug">58  52 </td> - <td class="tdsnug">61  10 </td> - <td class="tdsnug">63   6 </td> - <td class="tdsnug">64  10 </td> - <td class="tdsnug">64  31 </td></tr> -<tr> - <td class="tdsnug"> 50·0 </td> - <td class="tdsnug"> 8   0 </td> - <td class="tdsnug">15  35 </td> - <td class="tdsnug">22  35 </td> - <td class="tdsnug">28  50 </td> - <td class="tdsnug">38  45 </td> - <td class="tdsnug">45  55 </td> - <td class="tdsnug">52   1 </td> - <td class="tdsnug">54  18 </td> - <td class="tdsnug">56  28 </td> - <td class="tdsnug">57  42 </td> - <td class="tdsnug">58   6 </td></tr> -<tr> - <td class="tdsnug">100·0 </td> - <td class="tdsnug"> 6  50 </td> - <td class="tdsnug">13  20 </td> - <td class="tdsnug">19  30 </td> - <td class="tdsnug">25   5 </td> - <td class="tdsnug">34  20 </td> - <td class="tdsnug">41  15 </td> - <td class="tdsnug">47  35 </td> - <td class="tdsnug">49  45 </td> - <td class="tdsnug">52   3 </td> - <td class="tdsnug">53  20 </td> - <td class="tdsnug">53  46 </td></tr> -</table></div><!--dtblbox--> - -<div><span class="xxpn" id="p559">{559}</span></div> - -<p>From this table, by interpolation, we may easily fill in the -approximate values of α, as soon as we have determined the -apical angle θ and measured the ratio <i>R</i>; as follows:</p> - -<div class="dtblbox"><div class="nowrap"> -<table class="fsz6"> -<tr> - <th></th> - <th><i>R</i></th> - <th>θ</th> - <th>α</th></tr> -<tr> - <td class="tdleft"><i>Turritella</i> sp.</td> - <td class="tdright">1·12</td> - <td class="tdright">7°</td> - <td class="tdright">81°</td></tr> -<tr> - <td class="tdleft"><i>Cerithium nodulosum</i></td> - <td class="tdright">1·24</td> - <td class="tdright">15 </td> - <td class="tdright">82 </td></tr> -<tr> - <td class="tdleft"><i>Conus virgo</i></td> - <td class="tdright">1·25</td> - <td class="tdright">70 </td> - <td class="tdright">88 </td></tr> -<tr> - <td class="tdleft"><i>Mitra episcopalis</i></td> - <td class="tdright">1·43</td> - <td class="tdright">16 </td> - <td class="tdright">78 </td></tr> -<tr> - <td class="tdleft"><i>Scalaria pretiosa</i></td> - <td class="tdright">1·56</td> - <td class="tdright">26 </td> - <td class="tdright">81 </td></tr> -<tr> - <td class="tdleft"><i>Phasianella bulloides</i></td> - <td class="tdright">1·80</td> - <td class="tdright">26 </td> - <td class="tdright">80 </td></tr> -<tr> - <td class="tdleft"><i>Solarium perspectivum</i></td> - <td class="tdright">1·50</td> - <td class="tdright">53 </td> - <td class="tdright">85 </td></tr> -<tr> - <td class="tdleft"><i>Natica aperta</i></td> - <td class="tdright">2·00</td> - <td class="tdright">70 </td> - <td class="tdright">83 </td></tr> -<tr> - <td class="tdleft"><i>Planorbis corneus</i></td> - <td class="tdright">2·00</td> - <td class="tdright">90 </td> - <td class="tdright">84 </td></tr> -<tr> - <td class="tdleft"><i>Euomphalus pentangulatus</i></td> - <td class="tdright">2·00</td> - <td class="tdright">90 </td> - <td class="tdright">84 </td></tr> -</table></div></div><!--dtblbox--> - -<p>We see from this that shells so different in appearance as -Cerithium, Solarium, Natica and Planorbis differ very little indeed -in the magnitude of the spiral angle α, that is to say in the relative -velocities of radial and tangential growth. It is upon the angle θ</p> - -<div class="dright dwth-e" id="fig285"> -<img src="images/i559.png" width="337" height="569" alt=""> - <div class="dcaption">Fig. 285. <i>Terebra maculata</i>, L.</div></div> - -<p class="pcontinue"> -that the difference in their form -mainly depends: that is to say the -amount of longitudinal shearing, -or displacement parallel to the axis -of the shell.</p> - -<p>The enveloping angle, or rather -semi-angle (θ), of the cone may be -taken as 90° in the discoid shells, -such as Nautilus and Planorbis. It -is still a large angle, of 70° or 75°, -in Conus or in Cymba, somewhat -less in Cassis, Harpa, Dolium or -Natica; it is about 50° to 55° in -the various species of Solarium, -about 35° in the typical Trochi, -such as <i>T. niloticus</i> or <i>T. zizyphinus</i>, -and about 25° or 26° in <i>Scalaria -pretiosa</i> and <i>Phasianella bulloides</i>; it -becomes a very acute angle, of -15°, 10°, or even less, in Eulima, -Turritella or Cerithium. The costly <i>Conus gloria-maris</i>, -one of the <span class="xxpn" id="p560">{560}</span> -great treasures of the conchologist, differs from its congeners in -no important particular save in the somewhat “produced” spire, -that is to say in the comparatively low value of the angle θ. -<br class="brclrfix"></p> - -<p>A variation with advancing age of θ is common, but (as Blake -points out) it is often not to be distinguished or disentangled from -an alteration of α. Whether alone, or combined with a change in -α, we find it in all those many Gastropods whose whorls cannot -all be touched by the same enveloping cone, and whose spire is -accordingly described as <i>concave</i> or <i>convex</i>. The former condition, -as we have it in Cerithium, and in the cusp-like spire of Cassis,</p> - -<div class="dctr03" id="fig286"> -<img src="images/i560.png" width="609" height="473" alt=""> - <div class="dcaption">Fig. 286. <i>Trochus niloticus</i>, - L.</div></div> - -<p class="pcontinue"> -Dolium and some Cones, is much the commoner of the two. -And such tendency to decrease may lead to θ becoming a negative -angle; in which case we have a depressed spire, as in the -Cypraeae.</p> - -<p>When we find a “reversed shell,” a whelk or a snail for instance -whose spire winds to the left instead of to the right, we may -describe it mathematically by the simple statement that the angle -θ has <i>changed sign</i>. In the genus Ampullaria, or Apple-snails, -inhabiting tropical or sub-tropical rivers, we have a remarkable -condition; for in certain “species” the spiral turns to the right, -in others to the left, and in others again we -have a flattened <span class="xxpn" id="p561">{561}</span> -“discoid” shell; and furthermore we have numerous intermediate -stages, on either side, shewing right and left-handed spirals of -varying degrees of acuteness<a class="afnanch" href="#fn527" id="fnanch527">527</a>. -In this case, the angle θ may be -said to vary, within the limits of a genus, from somewhere about -35° to somewhere about 125°.</p> - -<p>The angle of retardation (β) is very small in Dentalium and -Patella; it is very large in Haliotis. It becomes infinite in -Argonauta and in Cypraea. Connected with the angle of retardation -are the various possibilities of contact or separation, in various -degrees, between adjacent whorls in the discoid, and between -both adjacent and opposite whorls in the turbinated shell. But -with these phenomena we have already dealt sufficiently.</p> - -<div class="section"> -<h3><i>Of Bivalve Shells.</i></h3></div> - -<p>Hitherto we have dealt only with univalve shells, and it is in -these that all the mathematical problems connected with the -spiral, or helico-spiral, are best illustrated. But the case of the -bivalve shell, of Lamellibranchs or of Brachiopods, presents no -essential difference, save only that we have here to do with two -conjugate spirals, whose two axes have a definite relation to one -another, and some freedom of rotatory movement relatively to -one another.</p> - -<p>The generating curve is particularly well seen in the bivalve, -where it simply constitutes what we call “the outline of the shell.” -It is for the most part a plane curve, but not always; for there -are forms, such as Hippopus, Tridacna and many Cockles, or -Rhynchonella and Spirifer among the Brachiopods, in which the -edges of the two valves interlock, and others, such as Pholas, -Mya, etc., where in part they fail to meet. In such cases as these -the generating curves are conjugate, having a similar relation, but -of opposite sign, to a median plane of reference. A great variety -of form is exhibited by these generating curves among the bivalves. -In a good many cases the curve is approximately circular, as in -Anomia, Cyclas, Artemis, Isocardia; it is nearly semi-circular in -Argiope. It is approximately elliptical in Orthis and in Anodon; -it may be called semi-elliptical in Spirifer. It is -a nearly rectilinear <span class="xxpn" id="p562">{562}</span> -triangle in Lithocardium, and a curvilinear triangle in Mactra. -Many apparently diverse but more or less related forms may be -shewn to be deformations of a common type, by a simple application -of the mathematical theory of “Transformations,” which we -shall have to study in a later chapter. In such a series as is -furnished, for instance, by Gervillea, Perna, Avicula, Modiola, -Mytilus, etc., a “simple shear” accounts for most, if not all, of -the apparent differences.</p> - -<p>Upon the surface of the bivalve shell we usually see with great -clearness the “lines of growth” which represent the successive -margins of the shell, or in other words the successive positions -assumed during growth by the growing generating curve; and -we have a good illustration, accordingly, of how it is characteristic -of the generating curve that it should constantly increase, while -never altering its geometric similarity.</p> - -<p>Underlying these “lines of growth,” which are so characteristic -of a molluscan shell (and of not a few other organic formations), -there is, then, a “law of growth” which we may attempt to enquire -into and which may be illustrated in various ways. The simplest -cases are those in which we can study the lines of growth on a -more or less flattened shell, such as the one valve of an oyster, -a Pecten or a Tellina, or some such bivalve mollusc. Here around -an origin, the so-called “umbo” of the shell, we have a series of -curves, sometimes nearly circular, sometimes elliptical, and often -asymmetrical; and such curves are obviously not “concentric,” -though we are often apt to call them so, but are always “co-axial.” -This manner of arrangement may be illustrated by various -analogies. We might for instance compare it to a series of waves, -radiating outwards from a point, through a medium which offered -a resistance increasing, with the angle of divergence, according to -some simple law. We may find another, and perhaps a simpler -illustration as follows:</p> - -<div class="dmaths"> -<div class="dright dwth-e" id="fig287"> -<img src="images/i563.png" width="336" height="356" alt=""> - <div class="dcaption">Fig. 287.</div></div> - -<p>In a very simple and beautiful theorem, Galileo shewed -that, if we imagine a number of inclined planes, or -gutters, sloping downwards (in a vertical plane) at various -angles from a common starting-point, and if we imagine a -number of balls rolling each down its own gutter under the -influence of gravity (and without hindrance from friction), -then, at any given instant, the locus of <span class="xxpn" -id="p563">{563}</span> all these moving bodies is a circle -passing through the point of origin. For the acceleration -along any one of the sloping paths, for instance <i>AB</i> (Fig. -<a href="#fig287" title="go to Fig. 287">287</a>), is such -that <br class="brclrfix"></p> - -<div><i>AB</i> -<div class="pleft nowrap dvaligntop"> - = ½ <i>g</i> cos θ · t<sup>2</sup> -<br> -= ½ <i>g</i> · <i>AB ⁄ AC</i> · t<sup>2</sup> . -</div></div> - -<p>Therefore</p> - -<div><i>t</i><sup>2</sup> -= 2 ⁄ <i>g</i> · <i>AC</i>. -</div></div><!--dmaths--> - -<p>That is to say, all the balls reach the circumference -of the circle at the same moment as the ball which drops -vertically from <i>A</i> to <i>C</i>.</p> - -<p>Where, then, as often happens, the generating curve of the -shell is approximately a circle passing through the point of origin, -we may consider the acceleration of growth along various radiants -to be governed by a simple mathematical law, closely akin to -that simple law of acceleration which governs the movements of -a falling body. And, <i>mutatis mutandis</i>, a similar definite law -underlies the cases where the generating curve is continually -elliptical, or where it assumes some more complex, but still regular -and constant form.</p> - -<p>It is easy to extend the proposition to the particular case where -the lines of growth may be considered elliptical. In such a case -we have <i>x</i><sup>2</sup> ⁄ <i>a</i><sup>2</sup> + <i>y</i><sup>2</sup> ⁄ <i>b</i><sup>2</sup> -= 1, where <i>a</i> and <i>b</i> are the major and minor -axes of the ellipse.</p> - -<div class="dmaths"> -<p>Or, changing the origin to the vertex of the figure</p> - -<div><i>x</i><sup>2</sup> ⁄ <i>a</i><sup>2</sup> − 2<i>x ⁄ a</i> + <i>y</i><sup>2</sup> ⁄ <i>b</i><sup>2</sup> -= 0,</div> - -<p class="pcontinue">giving</p> - -<div>(<i>x</i> − <i>a</i>)<sup>2</sup> ⁄ <i>a</i><sup>2</sup> + <i>y</i><sup>2</sup> ⁄ <i>b</i><sup>2</sup> -= 1.</div> - -<p>Then, transferring to polar coordinates, where <i>r</i> · cos θ -= <i>x</i>, -<i>r</i> · sin θ -= <i>y</i>, we have</p> - -<div>(<i>r</i> · cos<sup>2</sup> θ) ⁄ <i>a</i><sup>2</sup> − (2 cos θ) ⁄ <i>a</i> + (<i>r</i> · sin θ) ⁄ <i>b</i><sup>2</sup> -= 0,</div> - -<div><span class="xxpn" id="p564">{564}</span></div> - -<p class="pcontinue">which is equivalent to</p> - -<div><i>r</i> -= 2 <i>a b</i><sup>2</sup> cos θ ⁄ (<i>b</i><sup>2</sup> cos<sup>2</sup> θ + <i>a</i><sup>2</sup> sin<sup>2</sup> θ), -</div> - -<p class="pcontinue">or, eliminating the sine-function,</p> - -<div><i>r</i> -= 2 <i>a b</i><sup>2</sup> cos θ ⁄ ((<i>b</i><sup>2</sup> − <i>a</i><sup>2</sup>) cos<sup>2</sup> θ + <i>a</i><sup>2</sup>). -</div></div><!--dmaths--> - -<p>Obviously, in the case when <i>a</i> -= <i>b</i>, this gives us the circular -system which we have already considered. For other values, or -ratios, of <i>a</i> and <i>b</i>, and for all values of θ, we can easily construct -a table, of which the following is a sample:</p> - -<div class="dtblbox"><div class="nowrap"> -<table class="fsz6 borall"> -<caption><i>Chords of an ellipse, whose major and minor axes (a, b) -are in certain given ratios.</i></caption> -<tr> - <th class="borall">θ</th> - <th class="borall"><i>a ⁄ b</i><br> - = 1 ⁄ 3</th> - <th class="borall">1 ⁄ 2</th> - <th class="borall">2 ⁄ 3</th> - <th class="borall">1 ⁄ 1</th> - <th class="borall">3 ⁄ 2</th> - <th class="borall">2 ⁄ 1</th> - <th class="borall">3 ⁄ 1</th></tr> -<tr> - <td class="tdleft"> 0°</td> - <td class="tdright">1·0  </td> - <td class="tdright">1·0  </td> - <td class="tdright">1·0  </td> - <td class="tdright">1·0  </td> - <td class="tdright">1·0  </td> - <td class="tdright">1·0  </td> - <td class="tdright">1·0  </td></tr> -<tr> - <td class="tdleft">10</td> - <td class="tdright">1·01 </td> - <td class="tdright">1·01 </td> - <td class="tdright">1·002</td> - <td class="tdright">·985</td> - <td class="tdright">·948</td> - <td class="tdright">·902</td> - <td class="tdright">·793</td></tr> -<tr> - <td class="tdleft">20</td> - <td class="tdright">1·05 </td> - <td class="tdright">1·03 </td> - <td class="tdright">1·005</td> - <td class="tdright">·940</td> - <td class="tdright">·820</td> - <td class="tdright">·695</td> - <td class="tdright">·485</td></tr> -<tr> - <td class="tdleft">30</td> - <td class="tdright">1·115</td> - <td class="tdright">1·065</td> - <td class="tdright">1·005</td> - <td class="tdright">·866</td> - <td class="tdright">·666</td> - <td class="tdright">·495</td> - <td class="tdright">·289</td></tr> -<tr> - <td class="tdleft">40</td> - <td class="tdright">1·21 </td> - <td class="tdright">1·11 </td> - <td class="tdright">·995</td> - <td class="tdright">·766</td> - <td class="tdright">·505</td> - <td class="tdright">·342</td> - <td class="tdright">·178</td></tr> -<tr> - <td class="tdleft">50</td> - <td class="tdright">1·34 </td> - <td class="tdright">1·145</td> - <td class="tdright">·952</td> - <td class="tdright">·643</td> - <td class="tdright">·372</td> - <td class="tdright">·232</td> - <td class="tdright">·113</td></tr> -<tr> - <td class="tdleft">60</td> - <td class="tdright">1·50 </td> - <td class="tdright">1·142</td> - <td class="tdright">·857</td> - <td class="tdright">·500</td> - <td class="tdright">·258</td> - <td class="tdright">·152</td> - <td class="tdright">·071</td></tr> -<tr> - <td class="tdleft">70</td> - <td class="tdright">1·59 </td> - <td class="tdright">1·015</td> - <td class="tdright">·670</td> - <td class="tdright">·342</td> - <td class="tdright">·163</td> - <td class="tdright">·092</td> - <td class="tdright">·042</td></tr> -<tr> - <td class="tdleft">80</td> - <td class="tdright">1·235</td> - <td class="tdright">·635</td> - <td class="tdright">·375</td> - <td class="tdright">·174</td> - <td class="tdright">·078</td> - <td class="tdright">·045</td> - <td class="tdright">·020</td></tr> -<tr> - <td class="tdleft">90</td> - <td class="tdright">0·0  </td> - <td class="tdright">0·0  </td> - <td class="tdright">0·0  </td> - <td class="tdright">0·0  </td> - <td class="tdright">0·0  </td> - <td class="tdright">0·0  </td> - <td class="tdright">0·0  </td></tr> -</table></div></div><!--dtblbox--> - -<div class="dleft dwth-h" id="fig288"> -<img src="images/i564.png" width="209" height="391" alt=""> - <div class="dcaption">Fig. 288.</div></div> - -<p>The coaxial ellipses which we then draw, from the -values given in the table, are such as are shewn in -Fig. <a href="#fig288" title="go to Fig. 288">288</a> for the ratio <i>a ⁄ b</i> -= 3 ⁄ 1, and in Fig. <a href="#fig289" title="go to Fig. 289">289</a> for -the ratio <i>a ⁄ b</i> -= 1 ⁄ 2 ; these -are fair approximations to the actual outlines, and to the -actual arrangement of the lines of growth, in such forms as -Solecurtus or Cultellus, and in Tellina or Psammobia. It is not -difficult to introduce a constant into our equation to meet -the case of a shell which is somewhat unsymmetrical on either -side of the median axis. It is a somewhat more troublesome -matter, however, to bring these configurations into relation -with a “law of growth,” as was so easily done in the case of -the circular figure: in other words, to <span class="xxpn" -id="p565">{565}</span> formulate a law of acceleration -according to which points starting from the origin <i>O</i>, and -moving along radial lines, would all lie, at any future epoch, -on an ellipse passing through <i>O</i>; and this calculation we need -not enter into. <br class="brclrfix"></p> - -<div class="dctr01" id="fig289"> -<img src="images/i565.png" width="800" height="407" alt=""> - <div class="dcaption">Fig. 289.</div></div> - -<p>All that we are immediately concerned with is the simple fact -that where a velocity, such as our rate of growth, varies with its -direction,—varies that is to say as a function of the angular -divergence from a certain axis,—then, in a certain simple case, -we get lines of growth laid down as a system of coaxial circles, -and, when the function is a more complex one, as a system of -ellipses or of other more complicated coaxial figures, which figures -may or may not be symmetrical on either side of the axis. Among -our bivalve mollusca we shall find the lines of growth to be -approximately circular in, for instance, Anomia; in Lima (e.g. -<i>L. subauriculata</i>) we have a system of nearly symmetrical ellipses -with the vertical axis about twice the transverse; in <i>Solen pellucidus</i>, -we have again a system of lines of growth which are not far -from being symmetrical ellipses, in which however the transverse -is between three and four times as great as the vertical axis. In -the great majority of cases, we have a similar phenomenon with -the further complication of slight, but occasionally very considerable, -lateral asymmetry.</p> - -<p>In certain little Crustacea (of the genus Estheria) the carapace -takes the form of a bivalve shell, closely simulating -that of a <span class="xxpn" id="p566">{566}</span> -lamellibranchiate mollusc, and bearing lines of growth in all -respects analogous to or even identical with those of the latter. -The explanation is very curious and interesting. In ordinary -Crustacea the carapace, like the rest of the chitinised and calcified -integument, is shed off in successive moults, and is restored again -as a whole. But in Estheria (and one or two other small crustacea) -the moult is incomplete: the old carapace is retained, and the -new, growing up underneath it, adheres to it like a lining, and -projects beyond its edge: so that in course of time the margins -of successive old carapaces appear as “lines of growth” upon the -surface of the shell. In this mode of formation, then (but not -in the usual one), we obtain a structure which “is partly old and -partly new,” and whose successive increments are all similar, -similarly situated, and enlarged in a continued progression. We -have, in short, all the conditions appropriate and necessary for -the development of a logarithmic spiral; and this logarithmic -spiral (though it is one of small angle) gives its own character to -the structure, and causes the little carapace to partake of the -characteristic conformation of the molluscan shell.</p> - -<p>The essential simplicity, as well as the great regularity of the -“curves of growth” which result in the familiar configurations of -our bivalve shells, sufficiently explain, in a general way, the ease -with which they may be imitated, as for instance in the so-called -“artificial shells” which Kappers has produced from the conchoidal -form and lamination of lumps of melted and quickly cooled -paraffin<a class="afnanch" href="#fn528" id="fnanch528">528</a>.</p> - -<div class="psmprnt3"> -<p>In the above account of the mathematical form of the bivalve shell, we -have supposed, for simplicity’s sake, that the pole or origin of the system is -at a point where all the successive curves touch one another. But such an -arrangement is neither theoretically probable, nor is it actually the case; -for it would mean that in a certain direction growth fell, not merely to a -minimum, but to zero. As a matter of fact, the centre of the system (the -“umbo” of the conchologists) lies not at the edge of the system, but very -near to it; in other words, there is a certain amount of growth all round. -But to take account of this condition would involve more troublesome mathematics, -and it is obvious that the foregoing illustrations are a sufficiently near -approximation to the actual case. <span class="xxpn" id="p567">{567}</span></p> -</div><!--psmprnt3--> - -<p>Among the bivalves the spiral angle (α) is very small in the -flattened shells, such as Orthis, Lingula or Anomia. It is larger, -as a rule, in the Lamellibranchs than in the Brachiopods, but in -the latter it is of considerable magnitude among the Pentameri. -Among the Lamellibranchs it is largest in such forms as Isocardia -and Diceras, and in the very curious genus Caprinella; in all of -these last-named genera its magnitude leads to the production of -a spiral shell of several whorls, precisely as in the univalves. The -angle is usually equal, but of opposite sign, in the two valves of -the Lamellibranch, and usually of opposite sign but unequal in -the two valves of the Brachiopod. It is very unequal in many -Ostreidae, and especially in such forms as Gryphaea, or in Caprinella, -which is a kind of exaggerated Gryphaea. Occasionally it -is of the same sign in both valves (that is to say, both valves curve -the same way) as we see sometimes in Anomia, and much better -in Productus or Strophomena.</p> - -<div class="dctr02" id="fig290"><div id="fig291"> -<img src="images/i567.png" width="705" height="471" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 290. <i>Caprinella adversa.</i> - (After Woodward.)</td> - <td></td> - <td>Fig. 291. Section of <i>Productus</i> - (<i>Strophomena</i>) sp. (From - Woods.)</td></tr></table> -</div></div></div><!--dctr01--> - -<p>Owing to the large growth-factor of the generating curve, and -the comparatively small angle of the spiral, the whole shell seldom -assumes a spiral form so conspicuous as to manifest in a typical -way the helical twist or shear which is so -conspicuous in the <span class="xxpn" id="p568">{568}</span> -majority of univalves, or to let us measure or estimate the -magnitude of the apical angle (θ) of the enveloping cone. This -however we can do in forms like Isocardia and Diceras; while in -Caprinella we see that the whorls lie in a plane perpendicular to -the axis, forming a discoidal spire. As in the latter shell, so also -universally among the Brachiopods, there is no lateral asymmetry -in the plane of the generating curve such as to lead to the development -of a helix; but in the majority of the Lamellibranchiata -it is obvious, from the obliquity of the lines of growth, that the -angle θ is significant in amount.</p> - -<hr class="hrblk"> - -<p>The so-called “spiral arms” of Spirifer and many other -Brachiopods are not difficult to explain. They begin as a single -structure, in the form</p> - -<div class="dleft dwth-e" id="fig292"> -<img src="images/i568.png" width="336" height="336" alt=""> - <div class="dcaption">Fig. 292. Skeletal loop of - <i>Terebratula</i>. (From Woods.)</div></div> - -<p class="pcontinue">of a loop of -shelly substance, attached to the -dorsal valve of the shell, in the -neighbourhood of the hinge. This -loop has a curvature of its own, similar -to but not necessarily identical with -that of the valve to which it is -attached; and this curvature will tend -to be developed, by continuous and -symmetrical growth, into a fully -formed logarithmic spiral, so far as -it is permitted to do so under the -constraint of the shell in which it is -contained. In various Terebratulae we see the spiral growth of -the loop, more or less flattened and distorted by the restraining -pressure of the ventral valve. In a number of cases the loop -remains small, but gives off two nearly parallel branches or offshoots, -which continue to grow. And these, starting with just -such a slight curvature as the loop itself possessed, grow on and -on till they may form close-wound spirals, always provided that -the “spiral angle” of the curve is such that the resulting spire -can be freely contained within the cavity of the shell. Owing to -the bilateral symmetry of the whole system, the case will be rare, -and unlikely to occur, in which each separate arm will coil strictly -<i>in a plane</i>, so as to constitute a discoid spiral; -for the original <span class="xxpn" id="p569">{569}</span> -direction of each of the two branches, parallel to the valve (or -nearly so) and outwards from the middle line, will tend to constitute -a curve of double curvature, and so, on further growth, -to develop into a helicoid. This is what actually occurs, in the -great majority of cases. But the curvature may be such that -the helicoid grows outwards from the middle line, or inwards -towards the middle line, a <i>very</i> slight difference in the initial -curvature being sufficient to direct the spire the one way or the -other; the middle course of an undeviating discoid spire will be -rare, from the usual lack of any obvious controlling force to prevent -its deviation. The cases in which the helicoid spires point towards, -or point away from, the middle line are ascribed, in zoological -classification, to particular “families” of Brachiopods, the former -condition defining <br class="brclrfix"></p> - -<div class="dctr02" id="fig293"><div id="fig294"> -<img src="images/i569.png" width="704" height="286" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 293. Spiral arms of <i>Spirifer</i>. (From - Woods.)</td> - <td></td> - <td>Fig. 294. Inwardly directed spiral arms of - <i>Atrypa</i>.</td></tr></table> -</div></div></div><!--dctr01--> - -<p class="pcontinue"> (or helping to define) the Atrypidae and the -latter the Spiriferidae and Athyridae. It is obvious that the -incipient curvature of the arms, and consequently the form and -direction of the spirals, will be influenced by the surrounding -pressures, and these in turn by the general shape of the shell. -We shall expect, accordingly, to find the long outwardly directed -spirals associated with shells which are transversely elongated, as -Spirifer is; while the more rounded Atrypas will tend to the -opposite condition. In a few cases, as in Cyrtina or Reticularia, -where the shell is comparatively narrow but long, and where the -uncoiled basal support of the arms is long also, the spiral coils -into which the latter grow are turned backwards, in the direction -where there is room for them. And in the few cases where the -shell is very considerably flattened, the spirals (if -they find room <span class="xxpn" id="p570">{570}</span> -to grow at all) will be constrained to do so in a discoid or nearly -discoid fashion, and this is actually the case in such flattened -forms as Koninckina or Thecidium.</p> - -<div class="section"> -<h3><i>The Shells of Pteropods.</i></h3></div> - -<p>While mathematically speaking we are entitled to look upon -the bivalve shell of the Lamellibranch as consisting of two distinct -elements, each comparable to the entire shell of the univalve, we -have no biological grounds for such a statement; for the shell -arises from a single embryonic origin, and afterwards becomes split -into portions which constitute the two separate valves. We can -perhaps throw some indirect light upon this phenomenon, and -upon several other phenomena connected with shell-growth, by -a consideration of the simple conical or tubular shells of the -Pteropods. The shells of the latter are in few cases suitable for -simple mathematical investigation, but nevertheless they are of -very considerable interest in connection with our general problem.</p> - -<div class="dleft dwth-f" id="fig295"> -<img src="images/i570.png" width="287" height="465" alt=""> - <div class="dcaption">Fig. 295. Pteropod shells: -(1) <i>Cuvierina columnella</i>; -(2) <i>Cleodora chierchiae</i>; -(3) <i>C. pygmaea</i>. (After -Boas.)</div></div> - -<p class="pcontinue">The morphology of the Pteropods is -by no means well understood, and in speaking of them I -will assume that there are still grounds for believing (in -spite of Boas’ and Pelseneer’s arguments) that they are -directly related to, or may at least be directly compared -with, the Cephalopoda<a class="afnanch" href="#fn529" -id="fnanch529">529</a>.</p> - -<p>The simplest shells among the Pteropods have the form of -a tube, more or less cylindrical (Cuvierina), more often -conical (Creseis, Clio); and this tubular shell (as we have -already had occasion to remark, on p. <a href="#p258" title="go to pg. 258">258</a>), frequently -tends, when it is very small and delicate, to assume the -character of an unduloid. (In such a case it is more than -likely that the tiny shell, or that portion of it which -constitutes the unduloid, has not grown by successive <span -class="xxpn" id="p571">{571}</span> increments or “rings -of growth,” but has developed as a whole.) A thickened -“rib” is often, perhaps generally, present on the dorsal -side of the little conical shell. In a few cases (Limacina, -Peraclis) the tube becomes spirally coiled, in a normal -logarithmic spiral or helico-spiral. <br class="brclrfix" -></p> - -<div class="dctr04" id="fig296"> -<img src="images/i571a.png" width="528" height="338" alt=""> - <div class="pcaption">Fig. 296. Diagrammatic transverse sections, -or outlines of the mouth, in certain Pteropod shells: -A, B, <i>Cleodora australis</i>; C, <i>C. pyramidalis</i>; D, <i>C. -balantium</i>; E, <i>C. cuspidata</i>. (After Boas.)</div></div> - -<div class="dctr01" id="fig297"> -<img src="images/i571b.png" width="800" height="522" alt=""> - <div class="pcaption">Fig. 297. Shells of thecosome Pteropods -(after Boas). (1) <i>Cleodora cuspidata</i>; (2) <i>Hyalaea -trispinosa</i>; (3) <i>H. globulosa</i>; (4) <i>H. uncinata</i>; (5) <i>H. -inflexa</i>.</div></div> - -<p>In certain cases (e.g. Cleodora, Hyalaea) the tube or cone is -curiously modified. In the first place, -its cross-section, originally <span class="xxpn" id="p572">{572}</span> -circular or nearly so, becomes flattened or compressed dorso-ventrally; -and the angle, or rather edge, where dorsal and ventral -walls meet, becomes more and more drawn out into a ridge or -keel. Along the free margin, both of the dorsal and the ventral -portion of the shell, growth proceeds with a regularly varying -velocity, so that these margins, or lips, of the shell become regularly -curved or markedly sinuous. At the same time, growth in a -transverse direction proceeds with an acceleration which manifests -itself in a curvature of the sides, replacing the straight borders of -the original cone. In other words, the cross-section of the cone, -or what we have been calling the generating curve, increases its -dimensions more rapidly than its distance from the pole.</p> - -<div class="dctr03" id="fig298"> -<img src="images/i572.png" width="607" height="389" alt=""> - <div class="dcaption">Fig. 298. <i>Cleodora cuspidata.</i></div></div> - -<p>In the above figures, for instance in that of <i>Cleodora cuspidata</i>, -the markings of the shell which represent the successive edges of -the lip at former stages of growth, furnish us at once with a -“graph” of the varying velocities of growth as measured, radially, -from the apex. We can reveal more clearly the nature of these -variations in the following way which is simply tantamount to -converting our radial into rectangular coordinates. Neglecting -curvature (if any) of the sides and treating the shell (for simplicity’s -sake) as a right cone, we lay off equal angles from the apex <i>O</i>, -along the radii <i>Oa</i>, <i>Ob</i>, etc. If we then plot, as vertical equidistant -ordinates, the magnitudes <i>Oa</i>, <i>Ob</i> ... <i>OY</i>, and again on to -<i>Oa′</i>, we obtain a diagram such as the following -(Fig. <a href="#fig299" title="go to Fig. 299">299</a>); by <span class="xxpn" id="p573">{573}</span> -help of which we not only see more clearly the way in which the -growth-rate varies from point to point, but we also recognise -much better than before, the similar nature of the law which -governs this variation in the different species.</p> - -<div class="dctr04" id="fig299"> -<img src="images/i573a.png" width="527" height="457" alt=""> - <div class="pcaption">Fig. 299. Curves obtained by transforming -radial ordinates, as in Fig. <a href="#fig298" title="go to Fig. 298">298</a>, into vertical equidistant -ordinates. 1, <i>Hyalaea trispinosa</i>; 2, <i>Cleodora -cuspidata</i>.</div></div> - -<p>Furthermore, the young shell having become differentiated into a -dorsal and a ventral part, marked off from one another by a lateral -edge or keel, and the inequality of growth being such as to cause -each portion</p> - -<div class="dctr01" id="fig300"> -<img src="images/i573b.png" width="800" height="329" alt=""> - <div class="pcaption">Fig. 300. Development of the shell of -<i>Hyalaea</i> (<i>Cavolinia</i>) <i>tridentata</i>, Forskal: the earlier -stages being the “<i>Pleuropus longifilis</i>” of Troschel. -(After Tesch.)</div></div> - -<p class="pcontinue"> -to increase most rapidly in the median line, it follows -that the entire shell will appear to have been split into a dorsal -and a ventral plate, both connected with, -and projecting from, <span class="xxpn" id="p574">{574}</span> -what remains of the original undivided cone. Putting the same -thing in other words, we may say that the generating figure, which -lay at first in a plane perpendicular to the axis of the cone, has -now, by unequal growth, been sharply bent or folded, so as to -lie approximately in two planes, parallel to the anterior and -posterior faces of the cone. We have only to imagine the apical -connecting portion to be further reduced, and finally to disappear -or rupture, and we should have a <i>bivalve shell</i> developed out of -the original simple cone.</p> - -<p>In its outer and growing portion, the shell of our Pteropod -now consists of two parts which, though still connected together -at the apex, may be treated as growing practically independently. -The shell is no longer a simple tube, or simple cone, in which -regular inequalities of growth will lead to the development of a -spiral; and this for the simple reason that we have now two -opposite maxima of growth, instead of a maximum on the one side -and a minimum on the other side of our tubular shell. As a matter -of fact, the dorsal and the ventral plate tend to curve in opposite -directions, towards the middle line, the dorsal curving ventrally -and the ventral curving towards the dorsal side.</p> - -<p>In the case of the Lamellibranch or the Brachiopod, it is quite -possible for both valves to grow into more or less pronounced -spirals, for the simple reason that they are <i>hinged</i> upon one another; -and each growing edge, instead of being brought to a standstill -by the growth of its opposite neighbour, is free to move out of -the way, by the rotation about the hinge of the plane in which -it lies.</p> - -<p>But where, as in the Pteropod, there is no such hinge, the -dorsal and ventral halves of the shell (or dorsal and ventral -valves, if we may call them so), if they curved towards one -another (as they do in a cockle), would soon interfere with -one another’s progress, and the development of a pair of -conjugate spirals would become impossible. Nevertheless, there -is obviously, in both dorsal and ventral valve, a <i>tendency</i> to -the development of a spiral curve, that of the ventral valve -being more marked than that of the larger and overlapping -dorsal one, exactly as in the two unequal valves of Terebratula. -In many cases (e.g. <i>Cleodora cuspidata</i>), the dorsal -valve or plate, <span class="xxpn" id="p575">{575}</span> -strengthened and stiffened by its midrib, is nearly straight, while -the curvature of the other is well displayed. But the case will -be materially altered and simplified if growth be arrested or -retarded in either half of the shell. Suppose for instance that -the dorsal valve grew so slowly that after a while, in comparison -with the other, we might speak of it as being absent altogether: -or suppose that it merely became so reduced in relative size as to -form no impediment to the continued growth of the ventral one; -the latter would continue to grow in the direction of its natural -curvature, and would end by forming a complete and coiled -logarithmic spiral. It would be precisely analogous to the spiral -shell of Nautilus, and, in regard to its</p> - -<div class="dctr01" id="fig301"> -<img src="images/i575.png" width="800" height="285" alt=""> - <div class="pcaption">Fig. 301. Pteropod shells, from the side: - (1) <i>Cleodora cuspidata</i>; (2) <i>Hyalaea longirostris</i>; (3) - <i>H. trispinosa</i>. (After Boas.)</div></div> - -<p class="pcontinue">ventral position, concave -towards the dorsal side, it would even deserve to be called directly -homologous with it. Suppose, on the other hand, that the ventral -valve were to be greatly reduced, and even to disappear, the -dorsal valve would then pursue its unopposed growth; and, were -it to be markedly curved, it would come to form a logarithmic -spiral, concave towards the ventral side, as is the case in the shell -of Spirula<a class="afnanch" href="#fn530" id="fnanch530">530</a>. -Were the dorsal valve to be destitute of any marked -curvature (or in other words, to have but a low spiral angle), it -would form a simple plate, as in the shells of Sepia or Loligo. Indeed, -in the shells of these latter, and especially in that of Sepia, -we seem to recognise a manifest resemblance to the dorsal plate of -the Pteropod shell, as we have it (e.g.) in -Cleodora or Hyalaea; <span class="xxpn" id="p576">{576}</span> -the little “rostrum” of Sepia is but the apex of the primitive cone, -and the rounded anterior extremity has grown according to a law -precisely such as that which has produced the curved margin of -the dorsal valve in the Pteropod. The ventral portion of the -original cone is nearly, but not wholly, wanting. It is represented -by the so-called posterior wall of the “siphuncular space.” In -many decapod cuttle-fishes also (e.g. Todarodes, Illex, etc.) we -still see at the posterior end of the “pen,” a vestige of the primitive -cone, whose dorsal margin only has continued to grow; and the -same phenomenon, on an exaggerated scale, is represented in the -Belemnites.</p> - -<p>It is not at all impossible that we may explain on the same -lines the development of the curious “operculum” of the Ammonites. -This consists of a single horny plate (<i>Anaptychus</i>), or of -a thicker, more calcified plate divided into two symmetrical -halves (<i>Aptychi</i>), often found inside the terminal chamber of the -Ammonite, and occasionally to be seen lying <i>in situ</i>, as an -operculum which partially closes the mouth of the shell; this -structure is known to exist even in connection with the early -embryonic shell. In form the Anaptychus, or the pair of conjoined -Aptychi, shew an upper and a lower border, the latter -strongly convex, the former sometimes slightly concave, sometimes -slightly convex, and usually shewing a median projection or -slightly developed rostrum. From this “rostral” border the -curves of growth start, and course round parallel to, finally -constituting, the convex border. It is this convex border which -fits into the free margin of the mouth of the Ammonite’s shell, -while the other is applied to and overlaps the preceding whorl of -the spire. Now this relationship is precisely what we should -expect, were we to imagine as our starting-point a shell similar -to that of Hyalaea, in which however the dorsal part of the split -cone had become separate from the ventral half, had remained -flat, and had grown comparatively slowly, while at the same time -it kept slipping forward over the growing and coiling spire into -which the ventral half of the original shell develops<a class="afnanch" href="#fn531" id="fnanch531">531</a>. -In short, -I think there is reason to believe, or at least -to suspect, that we <span class="xxpn" id="p577">{577}</span> -have in the shell and Aptychus of the Ammonites, two portions -of a once united structure; of which other Cephalopods retain -not both parts but only one or other, one as the ventrally -situated shell of Nautilus, the other as the dorsally placed shell -for example of Sepia or of Spirula.</p> - -<p>In the case of the bivalve shells of the Lamellibranchs or of -the Brachiopods, we have to deal with a phenomenon precisely -analogous to the split and flattened cone of our Pteropods, save -only that the primitive cone has been split into two portions, not -incompletely as in the Pteropod (Hyalaea), but completely, so -as to form two separate valves. Though somewhat greater -freedom is given to growth now that the two valves are separate -and hinged, yet still the two valves oppose and hamper one -another, so that in the longitudinal direction each is capable of -only a moderate curvature. This curvature, as we have seen, is -recognisable as a logarithmic spiral, but only now and then does -the growth of the spiral continue so far as to develop successive -coils: as it does in a few symmetrical forms such as <i>Isocardia cor</i>; -and as it does still more conspicuously in a few others, such as -Gryphaea and Caprinella, where one of the two valves is stunted, -and the growth of the other is (relatively speaking) unopposed.</p> - -<div class="section"> -<h3><i>Of Septa.</i></h3></div> - -<p>Before we leave the subject of the molluscan shell, we have -still another problem to deal with, in regard to the form and -arrangement of the septa which divide up the tubular shell into -chambers, in the Nautilus, the Ammonite and their allies (Fig. -<a href="#fig304" title="go to Fig. 304">304</a>, etc.).</p> - -<p>The existence of septa in a Nautiloid shell may probably be -accounted for as follows. We have seen that it is a property of -a cone that, while growing by increments at one end only, it -conserves its original shape: therefore the animal within, which -(though growing by a different law) also conserves its shape, will -continue to fill the shell if it actually fills it to begin with: as -does a snail or other Gastropod. But suppose that our mollusc -fills a part only of a conical shell (as it does in the case of Nautilus); -then, unless it alter its shape, it must move upward as it grows in -the growing cone, until it come to occupy a space -similar in form <span class="xxpn" id="p578">{578}</span> -to that which it occupied before: just, indeed, as a little ball -drops far down into the cone, but a big one must stay farther up. -Then, when the animal after a period of growth has moved farther -up in the shell, the mantle-surface continues its normal secretory -activity, and that portion which had been in contact with the -former septum secretes a septum anew. In short, at any given -epoch, the creature is not secreting a tube and a septum by -separate operations, but is secreting a shelly case about its rounded -body, of which case one part appears to us as the continuation -of the tube, and the other part, merging with it by indistinguishable -boundaries, appears to us as the septum<a class="afnanch" href="#fn532" id="fnanch532">532</a>.</p> - -<p>The various forms assumed by the septa in spiral shells<a class="afnanch" href="#fn533" id="fnanch533">533</a> -present us with a number of problems of great beauty, simple in -their essence, but whose full investigation would soon lead us -into mathematics of a very high order.</p> - -<p>We do not know in great detail how these septa are laid down; -but the essential facts are clear<a class="afnanch" href="#fn534" id="fnanch534">534</a>. -The septum begins as a very -thin cuticular membrane (composed apparently of a substance -called conchyolin), which is secreted by the skin, or mantle-surface, -of the animal; and upon this membrane nacreous matter -is gradually laid down on the mantle-side (that is to say between -the animal’s body and the cuticular membrane which has been -thrown off from it), so that the membrane remains as a thin pellicle -over the <i>hinder</i> surface of the septum, and so that, to begin with, -the membranous septum is moulded on the flexible and elastic -surface of the animal, within which the fluids of the body must -exercise a uniform, or nearly uniform pressure.</p> - -<div class="dmaths"> -<p>Let us think, then, of the septa as they would appear -in their uncalcified condition, formed of, or at least -superposed upon, an <span class="xxpn" id="p579">{579}</span> -elastic membrane. They must then follow the general law, -applicable to all elastic membranes under uniform pressure, that -the tension varies inversely as the radius of curvature; and we -come back once more to our old equation of Laplace, that</p> - -<div><i>P</i> -= <i>T</i>(1 ⁄ <i>r</i> + 1 ⁄ <i>r′</i>). -</div></div><!--dmaths--> - -<div class="dright dwth-g" id="fig302"> -<img src="images/i579a.png" width="240" height="437" alt=""> - <div class="dcaption">Fig. 302.</div></div> - -<p>Moreover, since the cavity below the septum is -practically closed, and is filled either with air or with -water, <i>P</i> will be constant over the whole area of the -septum. And further, we must assume, at least to begin -with, that the membrane constituting the incipient septum -is homogeneous or isotropic.</p> - -<p>Let us take first the case of a straight cone, of -circular section, more or less like an Orthoceras; and -let us suppose that the septum is attached to the shell -in a plane perpendicular to its axis. The septum itself -must then obviously be spherical. Moreover the extent of -the spherical surface is constant, and easily determined. -For obviously, in Fig. <a href="#fig302" title="go to -Fig. 302">302</a>, the angle <i>LCL′</i> equals the supplement -of the angle (<i>LOL′</i>) of the cone; that is to say, the -circle of contact subtends an angle at the centre of the -spherical surface, which is constant, and which is equal to -π − 2θ. The case is not excluded where, owing -to an asymmetry of tensions, the septum meets the side -walls of the cone at other than a right angle, -<br class="brclrfix"></p> - -<div class="dright dwth-h" id="fig303"> -<img src="images/i579b.png" width="208" height="379" alt=""> - <div class="dcaption">Fig. 303.</div></div> - -<p class="pcontinue">as in Fig. <a href="#fig303" title="go -to Fig. 303">303</a>; and here, while the septa still -remain portions of spheres, the geometrical construction -for the position of their centres is equally easy.</p> - -<p>If, on the other hand, the attachment of the -septum to the inner walls of the cone be in a -plane oblique to the axis, then it is evident that -the outline of the septum will be an ellipse, and -its surface an <span class="xxpn" id="p580">{580}</span> -ellipsoid. If the attachment of the septum be not in one -plane, but form a sinuous line of contact with the cone, then -the septum will be a saddle-shaped surface, of great complexity -and beauty. In all cases, provided only that the membrane be -isotropic, the form assumed will be precisely that of a soap-bubble -under similar conditions of attachment: that is to say, it will be -(with the usual limitations or conditions) a surface of minimal -area.</p> - -<div class="dmaths"> -<p>If our cone be no longer straight, but curved, then the septa -will be symmetrically deformed in consequence. A beautiful and -interesting case is afforded us by Nautilus itself. Here the -outline of the septum, referred to a plane, is approximately -bounded by two elliptic curves, similar and similarly situated, -whose areas are to one another in a definite ratio, namely as -<br class="brclrfix"></p> - -<div><i>A</i><sub>1</sub> ⁄ <i>A</i><sub>2</sub> -= (<i>r</i><sub>1</sub> <i>r′</i><sub>1</sub>) ⁄ (<i>r</i><sub>2</sub> <i>r′</i><sub>2</sub>) -= ε<sup>−4π cot α</sup> ,</div> - -<p class="pcontinue">and a similar ratio exists in Ammonites -and all other close-whorled spirals, in which however we -cannot always make the simple assumption of elliptical form. -In a median section of Nautilus, we see each septum forming a -tangent to the inner and to the outer wall, just as it did in a -section of the straight Orthoceras; but the curvatures in the -neighbourhood of these two points of contact are not identical, -for they now vary inversely as the radii, drawn from the pole -of the spiral shell. The contour of the septum in this median -plane is a spiral curve identical with the original logarithmic -spiral. Of this it is the “invert,” and the fact that the -original curve and its invert are both identical is one of -the most beautiful properties of the logarithmic spiral<a -class="afnanch" href="#fn535" id="fnanch535">535</a>.</p> -</div><!--dmaths--> - -<p>But while the outline of the septum in median section is simple -and easy to determine, the curved surface of the septum in its -entirety is a very complicated matter, even in Nautilus which is -one of the simplest of actual cases. For, in the first place, since -the form of the septum, as seen in median section, is that of a -logarithmic spiral, and as therefore its curvature is constantly -altering, it follows that, in successive -<i>transverse</i> sections, the <span class="xxpn" id="p581">{581}</span> -curvature is also constantly altering. But in the case of Nautilus, -there are other aspects of the phenomenon, which we can illustrate, -but only in part, in the following simple manner. Let us imagine</p> - -<div class="dctr03" id="fig304"> -<img src="images/i581.png" width="608" height="787" alt=""> - <div class="pcaption">Fig. 304. Section of <i>Nautilus</i>, shewing -the contour of the septa in the median plane: the septa -being (in this plane) logarithmic spirals, of which the -shell-spiral is the evolute.</div></div> - -<p class="pcontinue"> -a pack of cards, in which we have cut out of each card a similar -concave arc of a logarithmic spiral, such as we actually see in the -median section of the septum of a Nautilus. Then, while we hold -the cards together, foursquare, in the ordinary -position of the <span class="xxpn" id="p582">{582}</span> -pack, we have a simple “ruled” surface, which in any longitudinal -section has the form of a logarithmic spiral but in any transverse -section is a straight horizontal line. If we shear or slide the -cards upon one another, thrusting the middle cards of the pack -forward in advance of the others, till the one end of the pack is -a convex, and the other a concave, ellipse, the cut edges which -combine to represent our septum will now form a curved surface</p> - -<div class="dctr02" id="fig305"> -<img src="images/i582.png" width="705" height="662" alt=""> - <div class="pcaption">Fig. 305. Cast of the interior of -<i>Nautilus</i>: to shew the contours of the septa at their -junction with the shell-wall.</div></div> - -<p class="pcontinue">of much greater complexity; and this -is part, but not by any means all, of the deformation -produced as a direct consequence of the form in Nautilus -of the section of the tube within which the septum has -to lie. And the complex curvature of the surface will -be manifested in a sinuous outline of the edge, or line -of attachment of the septum to the tube, and will vary -according to the configuration of the latter. In the -case of Nautilus, it is easy to shew empirically (though -not perhaps easy to demonstrate <span class="xxpn" -id="p583">{583}</span> mathematically) that the sinuous or -saddle-shaped form of the “suture” (or line of attachment -of the septum to the tube) is such as can be precisely -accounted for in this manner. It is also easy to see that, -when the section of the tube (or “generating curve”) is -more complicated in form, when it is flattened, grooved, -or otherwise ornamented, the curvature of the septum -and the outline of its sutural attachment will become -very complicated indeed<a class="afnanch" href="#fn536" -id="fnanch536">536</a>; but it will be comparatively simple -in the case of the first few sutures of the young shell, -laid down before any overlapping of whorls has taken place, -and this comparative simplicity of the first-formed sutures -is a marked feature among Ammonites<a class="afnanch" -href="#fn537" id="fnanch537">537</a>.</p> - -<p>We have other sources of complication, besides those which -are at once introduced by the sectional form of the tube. For -instance, the siphuncle, or little inner tube which perforates the -septa, exercises a certain amount of tension, sometimes evidently -considerable, upon the latter; so that we can no longer consider -each septum as an isotropic surface, under uniform pressure; and -there may be other structural modifications, or inequalities, in -that portion of the animal’s body with which the septum is in -contact, and by which it is conformed. It is hardly likely, for -all these reasons, that we shall ever attain to a full and particular -explanation of the septal surfaces and their sutural outlines -throughout the whole range of Cephalopod shells; but in general -terms, the problem is probably not beyond the reach of mathematical -analysis. The problem might be approached experimentally, -after the manner of Plateau’s experiments, -by bending <span class="xxpn" id="p584">{584}</span> -a wire into the complicated form of the suture-line, and studying -the form of the liquid film which constitutes the corresponding -surface <i>minimae areae</i>.</p> - -<div class="dctr04" id="fig306"> -<img src="images/i584a.png" width="528" height="545" alt=""> - <div class="pcaption">Fig. 306. <i>Ammonites</i> (<i>Sonninia</i>) - <i>Sowerbyi</i>. (From Zittel, after Steinmann and - Döderlein.)</div></div> - -<p>In certain Ammonites the septal outline is further complicated -in another way. Superposed upon the usual sinuous outline, with -its “lobes” and “saddles,” we have here a minutely ramified, or -arborescent outline,</p> - -<div class="dctr01" id="fig307"> -<img src="images/i584b.png" width="800" height="210" alt=""> - <div class="dcaption">Fig. 307. Suture-line of a Triassic Ammonite -(<i>Pinacoceras</i>). (From Zittel, after Hauer.)</div></div> - -<p class="pcontinue"> -in which all the branches terminate in wavy, -more or less circular arcs,—looking just like the ‘landscape -marble’ from the Bristol Rhaetic. We have no difficulty in -recognising in this a surface-tension phenomenon. The figures -are precisely such as we can imitate (for instance) -by pouring a <span class="xxpn" id="p585">{585}</span> -few drops of milk upon a greasy plate, or of oil upon an alkaline -solution.</p> - -<p>We have very far from exhausted, we have perhaps little -more than begun, the study of the logarithmic spiral and the -associated curves which find exemplification in the multitudinous -diversities of molluscan shells. But, with a closing word or two, -we must now bring this chapter to an end.</p> - -<p>In the spiral shell we have a problem, or a phenomenon, of -growth, immensely simplified by the fact that each successive -increment is irrevocably fixed in regard to magnitude and position, -instead, of remaining in a state of flux and sharing in the further -changes which the organism undergoes. In such a structure, then, -we have certain primary phenomena of growth manifested in their -original simplicity, undisturbed by secondary and conflicting -phenomena. What actually <i>grows</i> is merely the lip of an orifice, -where there is produced a ring of solid material, whose form we -have treated of under the name of the generating curve; and -this generating curve grows in magnitude without alteration of -its form. Besides its increase in areal magnitude, the growing -curve has certain strictly limited degrees of freedom, which define -its motions in space: that is to say, it has a vector motion at -right angles to the axis of the shell; and it has a sliding motion -along that axis. And, though we may know nothing whatsoever -about the actual velocities of any of these motions, we do know -that they are so correlated together that their <i>relative</i> velocities -remain constant, and accordingly the form and symmetry of the -whole system remain in general unchanged.</p> - -<p>But there is a vast range of possibilities in regard to every -one of these factors: the generating curve may be of various -forms, and even when of simple form, such as an ellipse, its axes -may be set at various angles to the system; the plane also in -which it lies may vary, almost indefinitely, in its angle relatively -to that of any plane of reference in the system; and in the several -velocities of growth, of rotation and of translation, and therefore -in the ratios between all these, we have again a vast range of -possibilities. We have then a certain definite type, or group of -forms, mathematically isomorphous, but presenting infinite diversities -of outward appearance: which -diversities, as Swammerdam <span class="xxpn" id="p586">{586}</span> -said, <i>ex sola nascuntur diversitate gyrationum</i>; and which accordingly -are seen to have their origin in differences of rate, or of -magnitude, and so to be, essentially, neither more nor less than -<i>differences of degree</i>.</p> - -<p>In nature, we find these forms presenting themselves with -but little relation to the character of the creature by which they -are produced. Spiral forms of certain particular kinds are common -to Gastropods and to Cephalopods, and to diverse families of -each; while outside the class of molluscs altogether, among the -Foraminifera and among the worms (as in Spirorbis, Spirographis, -and in the Dentalium-like shell of Ditrupa), we again meet with -similar and corresponding forms.</p> - -<p>Again, we find the same forms, or forms which (save for external -ornament) are mathematically identical, repeating themselves in -all periods of the world’s geological history; and, irrespective of -climate or local conditions, we see them mixed up, one with -another, in the depths and on the shores of every sea. It is hard -indeed (to my mind) to see where Natural Selection necessarily -enters in, or to admit that it has had any share whatsoever in the -production of these varied conformations. Unless indeed we use -the term Natural Selection in a sense so wide as to deprive it of -any purely biological significance; and so recognise as a sort of -natural selection whatsoever nexus of causes suffices to differentiate -between the likely and the unlikely, the scarce and the -frequent, the easy and the hard: and leads accordingly, under -the peculiar conditions, limitations and restraints which we call -“ordinary circumstances,” one type of crystal, one form of cloud, -one chemical compound, to be of frequent occurrence and another -to be rare.</p> - -<div class="chapter" id="p587"> -<h2 class="h2herein" -title="XII. The Spiral Shells of the Foraminifera.">CHAPTER -XII <span class="h2ttl"> THE SPIRAL SHELLS OF THE -FORAMINIFERA</span></h2></div> - -<p>We have already dealt in a few simple cases with the shells of -the Foraminifera<a class="afnanch" href="#fn538" id="fnanch538">538</a>; -and we have seen that wherever the shell is -but a single unit or single chamber, its form may be explained -in general by the laws of surface tension: the assumption being -that the little mass of protoplasm which makes the simple shell -behaves as a <i>fluid drop</i>, the form of which is perpetuated when -the protoplasm acquires its solid covering. Thus the spherical -Orbulinae and the flask-shaped Lagenae represent drops in -equilibrium, under various conditions of freedom or constraint; -while the irregular, amoeboid body of Astrorhiza is a manifestation -not of equilibrium, but of a varying and fluctuating distribution -of surface energy. When the foraminiferal shell becomes multilocular, -the same general principles continue to hold; the growing -protoplasm increases drop by drop, and each successive drop has -its particular phenomena of surface energy, manifested at its fluid -surface, and tending to confer upon it a certain place in the system -and a certain shape of its own.</p> - -<p>It is characteristic and even diagnostic of this particular -group of Protozoa (1) that development proceeds by a well-marked -alternation of rest and of activity—of activity during which the -protoplasm increases, and of rest during which the shell is formed; -(2) that the shell is formed at the outer surface of the protoplasmic -organism, and tends to constitute a continuous or all but continuous -covering; and it follows (3) from these two factors taken together -that each successive increment is added on outside of and distinct -from its predecessors, that the successive -parts or chambers of <span class="xxpn" id="p588">{588}</span> -the shell are of different and successive ages, that one part of the -shell is always relatively new, and the rest old in various grades -of seniority.</p> - -<p>The forms which we set together in the sister-group of Radiolaria -are very differently characterised. Here the cells or vesicles -of which each little composite organism is made up are but little -separated, and in no way walled off, from one another; the hard -skeletal matter tends to be deposited in the form of isolated -spicules or of little connected rods or plates, at the angles, the -edges or the interfaces of the vesicles; the cells or vesicles form -a coordinated and cotemporaneous rather than a successive series. -In a word, the whole quasi-fluid protoplasmic body may be -likened to a little mass of froth or foam: that is to say, to an -aggregation of simultaneously formed drops or bubbles, whose -physical properties and geometrical relations are very different -from those of a system of drops or bubbles which are formed one -after another, each solidifying before the next is formed.</p> - -<div class="dctr01" id="fig308"> -<img src="images/i588.png" width="800" height="387" alt=""> - <div class="dcaption">Fig. 308. <i>Hastigerina</i> sp.; - to shew the “mouth.”</div></div> - -<p>With the actual origin or mode of development of the foraminiferal -shell we are now but little concerned. The main factor -is the adsorption, and subsequent precipitation at the surface of -the organism, of calcium carbonate,—the shell so formed being -interrupted by pores or by some larger interspace or “mouth” -(Fig. <a href="#fig308" title="go to Fig. 308">308</a>), which interruptions we may doubtless interpret as -being due to unequal distributions of surface -energy. In many <span class="xxpn" id="p589">{589}</span> -cases the fluid protoplasm “picks up” sand-grains and other -foreign particles, after a fashion which we have already described -(p. <a href="#p463" title="go to pg. 463">463</a>); and it cements these together with more or less of -calcareous material. The calcareous shell is a crystalline structure, -and the micro-crystals of calcium carbonate are so set that their -little prisms radiate outwards in each chamber through the thickness -of the wall:—which symmetry is subject to corresponding -modification when the spherical chambers are more or less symmetrically -deformed<a class="afnanch" href="#fn539" id="fnanch539">539</a>.</p> - -<p>In various ways the rounded, drop-like shells of the Foraminifera, -both simple and compound, have been artificially -imitated. Thus, if small globules of mercury be immersed in -water in which a little chromic acid is allowed to dissolve, as the -little beads of quicksilver become slowly covered with a crystalline -coat of mercuric chromate they assume various forms reminiscent -of the monothalamic Foraminifera. The mercuric chromate has -a higher atomic volume than the mercury which it replaces, and -therefore the fluid contents of the drop are under pressure, which -increases with the thickness of the pellicle; hence at some weak -spot in the latter the contents will presently burst forth, so forming -a mouth to the little shell. Sometimes a long thread is formed, -just as in <i>Rhabdammina linearis</i>; and sometimes unduloid -swellings make their appearance on such a thread, just as in -<i>R. discreta</i>. And again, by appropriate modifications of the -experimental conditions, it is possible (as Rhumbler has shewn) -to build up a chambered shell<a class="afnanch" href="#fn540" id="fnanch540">540</a>.</p> - -<p>In a few forms, such as Globigerina and its close allies, the -shell is beset during life with excessively long and delicate -calcareous spines or needles. It is only in oceanic forms that -these are present, because only when poised -in water can such <span class="xxpn" id="p590">{590}</span> -delicate structures endure; in dead shells, such as we are much -more familiar with, every trace of them is broken and rubbed -away. The growth of these long needles is explained (as we have -already briefly mentioned, on p. <a href="#p440" title="go to pg. 440">440</a>) by the phenomenon which -Lehmann calls <i>orientirte Adsorption</i>—the tendency for a crystalline -structure to grow by accretion, not necessarily in the outward form -of a “crystal,” but continuing in any direction or orientation -which has once been impressed upon it: in this case the spicular -growth is simply in direct continuation of the radial symmetry -of the micro-crystalline elements of the shell-wall. Over the -surface of the shell the radiating spicules tend to occur in a -hexagonal pattern, symmetrically grouped around the pores which -perforate the shell. Rhumbler has suggested that this arrangement -is due to diffusion-currents, forming little eddies about the -base of the pseudopodia issuing from the pores: the idea being -borrowed from Bénard, to whom is due the discovery of this type -or order of vortices<a class="afnanch" href="#fn541" id="fnanch541">541</a>. -In one of Bénard’s experiments a thin -layer of paraffin is strewn with particles of graphite, then warmed -to melting, whereupon each little solid granule becomes the centre -of a vortex; by the interaction of these vortices the particles tend -to be repelled to equal distances from one another, and in the -end they are found to be arranged in a hexagonal pattern<a class="afnanch" href="#fn542" id="fnanch542">542</a>. -The -analogy is plain between this experiment and those diffusion -experiments by which Leduc produces his beautiful hexagonal -systems of artificial cells, with which we have dealt in a previous -chapter (p. <a href="#p320" title="go to pg. 320">320</a>).</p> - -<p>But let us come back to the shell itself, and consider particularly -its spiral form. That the shell in the Foraminifera should -tend towards a spiral form need not surprise us; for we have -learned that one of the fundamental conditions of the production -of a concrete spiral is just precisely what we have here, namely -the gradual development of a structure by means of successive -increments superadded to its exterior, which then form part, -successively, of a permanent and rigid -structure. This condition <span class="xxpn" id="p591">{591}</span> -is obviously forthcoming in the foraminiferal, but not at all in -the radiolarian, shell. Our second fundamental condition of the -production of a logarithmic spiral is that each successive increment -shall be so posited and so conformed that its addition to the -system leaves the form of the whole system unchanged. We -have now to enquire into this latter condition; and to determine -whether the successive increments, or successive chambers, of the -foraminiferal shell actually constitute <i>gnomons</i> to the entire -structure.</p> - -<p>It is obvious enough that the spiral shells of the Foraminifera -closely resemble true logarithmic spirals. Indeed so precisely do -the minute shells of many Foraminifera repeat or simulate the -spiral shells of Nautilus and its allies that to the naturalists of the -early nineteenth century they were known as the <i>Céphalopodes -microscopiques</i><a class="afnanch" href="#fn543" id="fnanch543">543</a>, -until Dujardin shewed that their little bodies -comprised no complex anatomy of organs, but consisted merely -of that slime-like organic matter which he taught us to call -“sarcode,” and which we learned afterwards from Schwann to -speak of as “protoplasm.”</p> - -<div class="dctr03" id="fig309"> -<img src="images/i591.png" width="604" height="388" alt=""> - <div class="dcaption">Fig. 309. <i>Nummulina antiquior</i>, R. and V. -(After V. von Möller.)</div></div> - -<p>One striking difference, however, is apparent between the -shell of Nautilus and the little nautiloid or rotaline -shells of the Foraminifera: namely that the septa in these -latter, and in all other <span class="xxpn" id="p592">{592}</span> -chambered Foraminifera, are convex outwards (Fig. <a href="#fig308" title="go to Fig. 308">308</a>), whereas -they are concave outwards in Nautilus (Fig. <a href="#fig304" title="go to Fig. 304">304</a>) and in the rest -of the chambered molluscan shells. The reason is perfectly -simple. In both cases the curvature of the septum was determined -before it became rigid, and at a time when it had the -properties either of a fluid film or an elastic membrane. In both -cases the actual curvature is determined by the tensions of the -membrane and the pressures to which it was exposed. Now it -is obvious that the extrinsic pressure which the tension of the -membrane has to withstand is on opposite sides in the two cases. -In Nautilus, the pressure to be resisted is that produced by the -growing body of the animal, lying to the <i>outer side</i> of the septum, -in the outer, wider portion of the tubular shell. In the Foraminifer -the septum at the time of its formation was no septum at all; -it was but a portion of the convex surface of a drop-that portion -namely which afterwards became overlapped and enclosed by the -succeeding drop; and the curvature of the septum is concave -towards the pressure to be resisted, which latter is <i>inside</i> the -septum, being simply the hydrostatic pressure of the fluid contents -of the drop. The one septum is, speaking generally, the reverse -of the other; the organism, so to speak, is outside the one and -inside the other; and in both cases alike, the septum tends to -assume the form of a surface of minimal area, as permitted, or as -defined, by all the circumstances of the case.</p> - -<p>The logarithmic spiral is easily recognisable in typical cases<a class="afnanch" href="#fn544" id="fnanch544">544</a> -(and especially where the spire makes more than one visible -revolution about the pole), by its fundamental property of continued -similarity: that is to say, by reason of the fact that the -big many-chambered shell is of just the same shape as the smaller -and younger shell—which phenomenon is apparent and even -obvious in the nautiloid Foraminifera, as in Nautilus itself: but -nevertheless the nature of the curve must be verified by careful -measurement, just as Moseley determined -or verified it in his <span class="xxpn" id="p593">{593}</span> -original study of nautilus (cf. p. <a href="#p518" title="go to pg. 518">518</a>). This has accordingly been -done, by various writers: and in the first instance by Valerian -von Möller, in an elaborate study of Fusulina—a palaeozoic genus -whose little shells have built up vast tracts of carboniferous -limestone over great part of European Russia<a class="afnanch" href="#fn545" id="fnanch545">545</a>.</p> - -<p>In this genus a growing surface of protoplasm may be conceived -as wrapping round and round a small initial chamber, in -such a way as to produce a fusiform or ellipsoidal shell—a transverse -section of which reveals the close-wound spiral coil. The -following are examples of measurements of the successive whorls -in a couple of species of this genus.</p> - -<div class="dtblbox"><div class="nowrap"> -<table class="fsz7 borall"> -<tr> - <th rowspan="2"></th> - <th class="borall" colspan="2"><i>F. cylindrica</i>, Fischer</th> - <th class="borall" colspan="2"><i>F. Böcki</i>, v. Möller</th></tr> -<tr> - <th class="borall" colspan="4">Breadth (in millimetres).</th></tr> -<tr> - <th class="borall">Whorl</th> - <th class="borall">Observed</th> - <th class="borall">Calculated</th> - <th class="borall">Observed</th> - <th class="borall">Calculated</th></tr> -<tr> - <td class="tdcntr">I</td> - <td class="tdcntr">·132</td> - <td class="tdcntr">—</td> - <td class="tdcntr">·079</td> - <td class="tdcntr">—</td></tr> -<tr> - <td class="tdcntr">II</td> - <td class="tdcntr">·195</td> - <td class="tdcntr">·198</td> - <td class="tdcntr">·120</td> - <td class="tdcntr">·119</td></tr> -<tr> - <td class="tdcntr">III</td> - <td class="tdcntr">·300</td> - <td class="tdcntr">·297</td> - <td class="tdcntr">·180</td> - <td class="tdcntr">·179</td></tr> -<tr> - <td class="tdcntr">IV</td> - <td class="tdcntr">·449</td> - <td class="tdcntr">·445</td> - <td class="tdcntr">·264</td> - <td class="tdcntr">·267</td></tr> -<tr> - <td class="tdcntr">V</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">·396</td> - <td class="tdcntr">·401</td></tr> -</table></div></div><!--dtblbox--> - -<p>In both cases the successive whorls are very nearly in the -ratio of 1 : 1·5; and on this ratio the calculated values are -based.</p> - -<p>Here is another of von Möller’s series of measurements of -<i>F. cylindrica</i>, the measurements being those of opposite whorls—that -is to say of whorls 180° apart:</p> - -<div class="dtblbox"><div class="nowrap"> -<table class="fsz7"> -<tr> - <td class="tdleft">Breadth in mm.</td> - <td class="tdcntr">·096</td> - <td class="tdcntr">·117</td> - <td class="tdcntr">·144</td> - <td class="tdcntr">·176</td> - <td class="tdcntr">·216</td> - <td class="tdcntr">·264</td> - <td class="tdcntr">·323</td> - <td class="tdcntr">·395</td></tr> -<tr> - <td class="tdleft">Log. of mm.</td> - <td class="tdcntr">·982</td> - <td class="tdcntr">·068</td> - <td class="tdcntr">·158</td> - <td class="tdcntr">·246</td> - <td class="tdcntr">·334</td> - <td class="tdcntr">·422</td> - <td class="tdcntr">·509</td> - <td class="tdcntr">·597</td></tr> -<tr> - <td class="tdleft">Diff. of logs.</td> - <td class="tdcntr">—</td> - <td class="tdcntr">·086</td> - <td class="tdcntr">·090</td> - <td class="tdcntr">·088</td> - <td class="tdcntr">·088</td> - <td class="tdcntr">·088</td> - <td class="tdcntr">·087</td> - <td class="tdcntr">·088</td></tr> -</table></div></div><!--dtblbox--> - -<p>The mean logarithmic difference is here ·088, -= log 1·225; or -the mean difference of alternate logs (corresponding to a vector -angle of 2π, i.e. to consecutive measurements along the <i>same</i> -radius) is ·176, -= log 1·5, the same value as before. And this -ratio of 1·5 between the breadths of successive whorls corresponds -(as we see by our table on p. <a href="#p534" title="go to pg. 534">534</a>) to a constant -angle of about <span class="xxpn" id="p594">{594}</span> -86°, or just such a spiral as we commonly meet with in the -Ammonites<a class="afnanch" href="#fn546" id="fnanch546">546</a> -(cf. p. <a href="#p539" title="go to pg. 539">539</a>).</p> - -<div class="dctr01" id="fig310"> -<img src="images/i594.png" width="799" height="493" alt=""> - <div class="dcaption">Fig. 310. A, <i>Cornuspira foliacea</i>, Phil.; B, -<i>Operculina complanata</i>, Defr.</div></div> - -<p>In Fusulina, and in some few other Foraminifera (cf. Fig. -<a href="#fig310" title="go to Fig. 310">310</a>, <span class="nowrap"><span class="smmaj">A</span>),</span> -the spire seems to wind evenly on, with little or no -external sign of the successive periods of growth, or successive -chambers of the shell. The septa which mark off the chambers, -and correspond to retardations or cessations in the periodicity of -growth, are still to be found in sections of the shell of Fusulina; -but they are somewhat irregular and comparatively inconspicuous; -the measurements we have just spoken of are taken without -reference to the segments or chambers, but only with reference -to the whorls, or in other words with direct reference to the -vectorial angle.</p> - -<p>The linear dimensions of successive chambers have been -<span class="xxpn" id="p595">{595}</span> measured in a -number of cases. Van Iterson<a class="afnanch" href="#fn547" -id="fnanch547">547</a> has done so in various Miliolinidae, -with such results as the following:</p> - -<div class="dtblboxin10"> -<table class="fsz7"> -<caption><i>Triloculina rotunda</i>, d’Orb.</caption> -<tr> - <td class="tdleft">No. of chamber</td> - <td class="tdcntr">1</td> - <td class="tdcntr">2</td> - <td class="tdcntr">3</td> - <td class="tdcntr">4</td> - <td class="tdcntr">5</td> - <td class="tdcntr">6</td> - <td class="tdcntr">7</td> - <td class="tdcntr">8</td> - <td class="tdcntr">9</td> - <td class="tdcntr">10</td></tr> -<tr> - <td class="tdleft">Breadth of chamber in <i>µ</i></td> - <td class="tdcntr">—</td> - <td class="tdcntr">34</td> - <td class="tdcntr">45</td> - <td class="tdcntr">61</td> - <td class="tdcntr">84</td> - <td class="tdcntr">114</td> - <td class="tdcntr">142</td> - <td class="tdcntr">182</td> - <td class="tdcntr">246</td> - <td class="tdcntr">319</td></tr> -<tr> - <td class="tdleft">Breadth of chamber in <i>µ</i>, calculated</td> - <td class="tdcntr">—</td> - <td class="tdcntr">34</td> - <td class="tdcntr">45</td> - <td class="tdcntr">60</td> - <td class="tdcntr">79</td> - <td class="tdcntr">105</td> - <td class="tdcntr">140</td> - <td class="tdcntr">187</td> - <td class="tdcntr">243</td> - <td class="tdcntr">319</td></tr> -</table></div><!--dtblbox--> - -<p>Here the mean ratio of breadth of consecutive chambers may -be taken as 1·323 (that is to say, the eighth root of 319 ⁄ 34); and -the calculated values, as given above, are based on this determination.</p> - -<p>Again, Rhumbler has measured the linear dimensions of a -number of rotaline forms, for instance <i>Pulvinulina menardi</i> -(Fig. <a href="#fig259" title="go to Fig. 259">259</a>): in which common species he finds the mean linear -ratio of consecutive chambers to be about 1·187. In both cases, -and especially in the latter, the ratio is not strictly constant from -chamber to chamber, but is subject to a small secondary fluctuation<a class="afnanch" href="#fn548" id="fnanch548">548</a>.</p> - -<div class="dctr03" id="fig311"> -<img src="images/i596a.png" width="608" height="626" alt=""> - <div class="dcaption">Fig. 311. 1, 2, <i>Miliolina pulchella</i>, -d’Orb.; 3–5, <i>M. linnaeana</i>, d’Orb. (After Brady.)</div></div> - -<p>When the linear dimensions of successive chambers are in -continued proportion, then, in order that the whole shell may -constitute a logarithmic spiral, it is necessary that the several -chambers should subtend equal angles of revolution at the pole. -In the case of the Miliolidae this is obviously the case (Fig. <a href="#fig311" title="go to Fig. 311">311</a>); -for in this family the chambers lie in two rows (Biloculina), or -three rows (Triloculina), or in some other small number of series: -so that the angles subtended by them are large, simple fractions -of the circular arc, such as 180° or 120°. In many of the nautiloid -forms, such as Cyclammina (Fig. <a href="#fig312" title="go to Fig. 312">312</a>), the angles</p> - -<div class="dctr03" id="fig312"> -<img src="images/i596b.png" width="608" height="468" alt=""> - <div class="dcaption">Fig. 312. <i>Cyclammina cancellata</i>, - Brady.</div></div> - -<p class="pcontinue">subtended, -though of less magnitude, are still remarkably -constant, as we <span class="xxpn" id="p597">{597}</span> -may see by Fig. <a href="#fig313" title="go to Fig. 313">313</a>; where the angle subtended by each chamber -is made equal to 20°, and this diagrammatic figure is not perceptibly -different from the other. In some cases the subtended -angle is less constant; and in these it would be necessary to equate -the several linear dimensions with the corresponding vector angles, -according to our equation <i>r</i> -= <i>e</i><sup>θ cot α</sup> . It is probable that, by so -taking account of variations of θ, such variations of <i>r</i> as (according -to Rhumbler’s measurements) Pulvinulina and other genera -appear to shew, would be found to diminish or even to disappear.</p> - -<div class="dctr03" id="fig313"> -<img src="images/i597.png" width="609" height="546" alt=""> - <div class="dcaption">Fig. 313. <i>Cyclammina</i> - sp. (Diagrammatic.)</div></div> - -<div class="dmaths"> -<p>The law of increase by which each chamber bears a constant -ratio of magnitude to the next may be looked upon as -a simple consequence of the structural uniformity or -homogeneity of the organism; we have merely to suppose -(as this uniformity would naturally lead us to do) that -the rate of increase is at each instant proportional to -the whole existing mass. For if <i>V</i><sub>0</sub> , <i>V</i><sub>1</sub> etc., -be the volumes of the successive chambers, let <i>V</i><sub>1</sub> -bear a constant proportion to <i>V</i><sub>0</sub> , so that <i>V</i><sub>1</sub> -<span class="nowrap"> -= <i>q V</i><sub>0</sub> ,</span> -and let <i>V</i><sub>2</sub> bear the same proportion to -the whole pre-existing volume: then</p> - -<div><i>V</i><sub>2</sub> -= <i>q</i>(<i>V</i><sub>0</sub> + <i>V</i><sub>1</sub>) -= <i>q</i>(<i>V</i><sub>0</sub> + <i>q V</i><sub>0</sub>) -</div> - -<div>= <i>q V</i><sub>0</sub>(1 + <i>q</i>) -    and     -<i>V</i><sub>2</sub> ⁄ <i>V</i><sub>1</sub> -= 1 + <i>q</i>. -</div></div><!--dmaths--> - -<div><span class="xxpn" id="p598">{598}</span></div> - -<p>This ratio of 1 ⁄ (1 + <i>q</i>) is easily shewn to be the constant ratio -running through the whole series, from chamber to chamber; -and if this ratio of volumes be constant, so also are the ratios -of corresponding surfaces, and of corresponding linear dimensions, -provided always that the successive increments, or successive -chambers, are similar in form.</p> - -<p>We have still to discuss the similarity of form and the symmetry -of position which characterise the successive chambers, and which, -together with the law of continued proportionality of size, are the -distinctive characters and the indispensable conditions of a series -of “gnomons.”</p> - -<div class="dctr05" id="fig314"> -<img src="images/i598.png" width="448" height="350" alt=""> - <div class="dcaption">Fig. 314. <i>Orbulina - universa</i>, d’Orb.</div></div> - -<p>The minute size of the foraminiferal shell or at least of each -successive increment thereof, taken in connection with the fluid -or semi-fluid nature of the protoplasmic substance, is enough to -suggest that the molecular forces, and especially the force of -surface-tension, must exercise a controlling influence over the form -of the whole structure; and this suggestion, or belief, is already -implied in our statement that each successive increment of growing -protoplasm constitutes a separate <i>drop</i>. These “drops,” partially -concealed by their successors, but still shewing in part their -rounded outlines, are easily recognisable in the various foraminiferal -shells which are illustrated in this chapter.</p> - -<p>The accompanying figure represents, to begin with, the spherical -shell characteristic of the common, floating, oceanic Orbulina. -In the specimen illustrated, a second chamber, -superadded to the <span class="xxpn" id="p599">{599}</span> -first, has arisen as a drop of protoplasm which exuded through the -pores of the first chamber, accumulated on its surface, and spread -over the latter till it came to rest in a position of equilibrium. -We may take it that this position of equilibrium is determined, -at least in the first instance, by the “law of the constant angle,” -which holds, or tends to hold, in all cases where the free surface -of a given liquid is in contact with a given solid, in presence of -another liquid or a gas. The corresponding equations are precisely -the same as those which we have used in discussing the -form of a drop (on p. <a href="#p294" title="go to pg. 294">294</a>); though some slight modification must -be made in our definitions, inasmuch as the consideration of -surface-<i>tension</i> is no longer appropriate at the solid surfaces, and -the concept of surface-<i>energy</i> must take its place. Be that as it -may, it is enough for us to observe that, in such a case as ours, -when a given fluid (namely protoplasm) is in surface contact with -a solid (viz. a calcareous shell), in presence of another fluid (sea-water), -then the angle of contact, or angle by which the common -surface (or interface) of the two liquids abuts against the solid wall, -tends to be constant: and that being so, the drop will have a -certain definite form, depending (<i>inter alia</i>) on the form of the -surface with which it is in contact. After a period of rest, during -which the surface of our second drop becomes rigid by calcification, -a new period of growth will recur and a new drop of protoplasm -be accumulated. Circumstances remaining the same, this new -drop will meet the solid surface of the shell at the same angle as -did the former one; and, the other forces at work on the system -remaining the same, the form of the whole drop, or chamber, will -be the same as before.</p> - -<p>According to Rhumbler, this “law of the constant angle” is -the fundamental principle in the mechanical conformation of the -foraminiferal shell, and provides for the symmetry of form as -well as of position in each succeeding drop of protoplasm: which -form and position, once acquired, become rigid and fixed with the -onset of calcification. But Rhumbler’s explanation brings with -it its own difficulties. It is by no means easy of verification, for -on the very complicated curved surfaces of the shell it seems to -me extraordinarily difficult to measure, or even to recognise, the -actual angle of contact: of which angle of contact, -by the way, <span class="xxpn" id="p600">{600}</span> -but little is known, save only in the particular case where one of -the three bodies is air, as when a surface of water is exposed to -air and in contact with glass. It is easy moreover to see that in -many of our Foraminifera the angle of contact, though it may be -constant in homologous positions from chamber to chamber, is -by no means constant at all points along the boundary of each -chamber. In Cristellaria, for instance (Fig. <a href="#fig315" title="go to Fig. 315">315</a>), it would seem -to be (and Rhumbler</p> - -<div class="dctr06" id="fig315"> -<img src="images/i600.png" width="400" height="400" alt=""> - <div class="dcaption">Fig. 315. <i>Cristellaria - reniformis</i>, d’Orb.</div></div> - -<p class="pcontinue">asserts that it actually is) about 90° on the -outer side and only about 50° on the inner side of each septal -partition; in Pulvinulina (Fig. <a href="#fig259" title="go to Fig. 259">259</a>), according to Rhumbler, the -angles adjacent to the mouth are of 90°, and the opposite angles -are of 60°, in each chamber. For these and other similar discrepancies -Rhumbler would account by simply invoking the heterogeneity -of the protoplasmic drop: that is to say, by assuming that -the protoplasm has a different composition and different properties -(including a very different distribution of surface-energy), at -points near to and remote from the mouth of the shell. Whether -the differences in angle of contact be as great as Rhumbler takes -them to be, whether marked heterogeneities of the protoplasm -occur, and whether these be enough to account for the differences -of angle, I cannot tell. But it seems to me that we had better -rest content with a general statement, and that Rhumbler has -taken too precise and narrow a view. -<span class="xxpn" id="p601">{601}</span></p> - -<p>In the molecular growth of a crystal, although we must of -necessity assume that each molecule settles down in a position of -minimum potential energy, we find it very hard indeed to explain -precisely, even in simple cases and after all the labours of modern -crystallographers, why this or that position is actually a place of -minimum potential. In the case of our little Foraminifer (just -as in the case of the crystal), let us then be content to assert that -each drop or bead of protoplasm takes up a position of minimum -potential energy, in relation to all the circumstances of the case; -and let us not attempt, in the present state of our knowledge, to -define that position of minimum potential by reference to angle -of contact or any other particular condition of equilibrium. In -most cases the whole exposed surface, on some portion of which -the drop must come to rest, is an extremely complicated one, and -the forces involved constitute a system which, in its entirety, is -more complicated still; but from the symmetry of the case and -the continuity of the whole phenomenon, we are entitled to believe -that the conditions are just the same, or very nearly the same, -time after time, from one chamber to another: as the one chamber -is conformed so will the next tend to be, and as the one is situated -relatively to the system so will its successor tend to be situated in -turn. The physical law of minimum potential (including also the -law of minimal area) is all that we need in order to explain, <i>in -general terms</i>, the continued similarity of one chamber to another; -and the physiological law of growth, by which a continued proportionality -of size tends to run through the series of successive -chambers, impresses upon this series of similar increments the -form of a logarithmic spiral.</p> - -<p>In each particular case the nature of the logarithmic spiral, -as defined by its constant angle, will be chiefly determined by -the rate of growth; that is to say by the particular ratio in which -each new chamber exceeds its predecessor in magnitude. But -shells having the same constant angle (α) may still differ from one -another in many ways—in the general form and relative position -of the chambers, in their extent of overlap, and hence in the actual -contour and appearance of the shell; and these variations must -correspond to particular distributions of energy within the system, -which is governed as a whole by the law -of minimum potential. <span class="xxpn" id="p602">{602}</span></p> - -<p>Our problem, then, becomes reduced to that of investigating -the possible configurations which may be derived from the successive -symmetrical apposition of similar bodies whose magnitudes -are in continued proportion; and it is obvious, mathematically -speaking, that the various possible arrangements all come under -the head of the logarithmic spiral, together with the limiting cases -which it includes. Since the difference between one such form -and another depends upon the numerical value of certain -coefficients of magnitude, it is plain that any one must tend to -pass into any other by small and continuous gradations; in -other words, that a <i>classification</i> of these forms must (like any -classification whatsoever of logarithmic spirals or of any other -mathematical curves), be theoretic or “artificial.” But we may -easily make such an artificial classification, and shall probably -find it to agree, more or less, with the usual methods of classification -recognised by biological students of the Foraminifera.</p> - -<p>Firstly we have the typically spiral shells, which occur in -great variety, and which (for our present purpose) we need hardly -describe further. We may merely notice how in certain cases, -for instance Globigerina, the individual chambers are little removed -from spheres; in other words, the area of contact between the -adjacent chambers is small. In such forms as Cyclammina and -Pulvinulina, on the other hand, each chamber is greatly overlapped -by its successor, and the spherical form of each is lost in -a marked asymmetry. Furthermore, in Globigerina and some -others we have a tendency to the development of a helicoid spiral -in space, as in so many of our univalve molluscan shells. The -mathematical problem of how a shell should grow, under the -assumptions which we have made, would probably find its most -general statement in such a case as that of Globigerina, where the -whole organism lives and grows freely poised in a medium whose -density is little different from its own.</p> - -<p>The majority of spiral forms, on the other hand, are plane -or discoid spirals, and we may take it that in these cases some -force has exercised a controlling influence, so as to keep all the -chambers in a plane. This is especially the case in forms like -Rotalia or Discorbina (Fig. <a href="#fig316" title="go to Fig. 316">316</a>), where the organism lives attached -to a rock or a frond of sea-weed; for here (just as in -the case of <span class="xxpn" id="p603">{603}</span> -the coiled tubes which little worms such as Serpula and Spirorbis -make, under similar conditions) the spiral disc is itself asymmetrical, -its whorls being markedly flattened on their attached surfaces.</p> - -<div class="dctr06" id="fig316"> -<img src="images/i603.png" width="400" height="950" alt=""> - <div class="dcaption">Fig. 316. <i>Discorbina - bertheloti</i>, d’Orb.</div></div> - -<p>We may also conceive, among other conditions, the -very curious case in which the protoplasm may entirely -overspread the surface of the shell without reaching a -position of equilibrium; in which case a new shell will -be formed <i>enclosing</i> the old one, <span class="xxpn" -id="p604">{604}</span> whether the old one be in the form -of a single, solitary chamber, or have already attained to -the form of a chambered or spiral shell. This is precisely -what often happens in the case of Orbulina, when within -the spherical shell we find a small, but perfectly formed, -spiral “Globigerina<a class="afnanch" href="#fn549" -id="fnanch549">549</a>.”</p> - -<p>The various Miliolidae (Fig. <a href="#fig311" title="go to Fig. 311">311</a>), only differ from the typical -spiral, or rotaline forms, in the large angle subtended by each -chamber, and the consequent abruptness of their inclination to -each other. In these cases the <i>outward</i> appearance of a spiral -tends to be lost; and it behoves us to recollect, all the more, -that our spiral curve is not necessarily identical with the <i>outline</i> -of the shell, but is always a line drawn through corresponding -points in the successive chambers of the latter.</p> - -<div class="dctr02" id="fig317"> -<img src="images/i604.png" width="705" height="365" alt=""> - <div class="dcaption">Fig. 317. A, <i>Tertularia trochus</i>, d’Orb. B, -<i>T. concava</i>, Karrer.</div></div> - -<p>We reach a limiting case of the logarithmic spiral -when the chambers are arranged in a straight line; and -the eye will tend to associate with this limiting case -the much more numerous forms in which the spiral angle -is small, and the shell only exhibits a gentle curve with -no succession of enveloping whorls. This constitutes the -Nodosarian type (Fig. <a href="#fig87" title="go to Fig. 87">87</a>, p. <a href="#p262" title="go to pg. 262">262</a>); and here again, we must -postulate some force which has tended to keep the chambers -in a rectilinear series: such for instance as gravity, -acting on a system of “hanging drops.” <span class="xxpn" -id="p605">{605}</span></p> - -<p>In Textularia and its allies (Fig. <a href="#fig317" title="go to Fig. 317">317</a>), we have a precise -parallel to the helicoid cyme of the botanists (cf. p. <a href="#p502" title="go to pg. 502">502</a>): that -is to say we have a screw translation, perpendicular to the plane -of the underlying logarithmic spiral. In other words, in tracing -a genetic spiral through the whole succession of chambers, we do -so by a continuous vector rotation, through successive angles of -180° (or 120° in some cases), while the pole moves along an axis -perpendicular to the original plane of the spiral.</p> - -<p>Another type is furnished by the “cyclic” shells of the -Orbitolitidae, where small and numerous chambers tend to be -added on round and round the system, so building up a circular -flattened disc. This again we perceive to be, mathematically, a -limiting case of the logarithmic spiral, where the spiral has become -a circle and the constant angle is now an angle of 90°.</p> - -<p>Lastly there are a certain number of Foraminifera in which, -without more ado, we may simply say that the arrangement of -the chambers is irregular, neither the law of constant ratio of -magnitude nor that of constant form being obeyed. The chambers -are heaped pell-mell upon one another, and such forms are known -to naturalists as the Acervularidae.</p> - -<p>While in these last we have an extreme lack of regularity, we -must not exaggerate the regularity or constancy which the more -ordinary forms display. We may think it hard to believe that -the simple causes, or simple laws, which we have described should -operate, and operate again and again, in millions of individuals to -produce the same delicate and complex conformations. But we -are taking a good deal for granted if we assert that they do so, -and in particular we are assuming, with very little proof, the -“constancy of species” in this group of animals. Just as Verworn -has shewn that the typical <i>Amoeba proteus</i>, when a trace of alkali -is added to the water in which it lives, tends, by alteration of -surface tensions, to protrude the more delicate pseudopodia -characteristic of <i>A. radiosa</i>,—and again when the water is rendered -a little more alkaline, to turn apparently into the so-called <i>A. -limax</i>,—so it is evident that a very slight modification in the -surface-energies concerned, might tend to turn one so-called -species into another among the Foraminifera. To what extent -this process actually occurs, we -do not know. <span class="xxpn" id="p606">{606}</span></p> - -<p>But that this, or something of the kind, does actually occur -we can scarcely doubt. For example in the genus Peneroplis, the -first portion of the shell consists of a series of chambers arranged -in a spiral or nautiloid series; but as age advances the spiral is -apt to be modified in various ways<a class="afnanch" href="#fn550" id="fnanch550">550</a>. -Sometimes the successive -chambers grow rapidly broader, the whole shell becoming fan-shaped. -Sometimes the chambers become narrower, till they no -longer enfold the earlier chambers but only come in contact each -with its immediate predecessor: the result being that the shell -straightens out, and (taking into account the earlier spiral portion) -may be described as crozier-shaped. Between these extremes of -shape, and in regard to other variations of thickness or thinness, -roughness or smoothness, and so on, there are innumerable -gradations passing one into another and intermixed without regard -to geographical distribution:—“wherever Peneroplides abound -this wide variation exists, and nothing can be more easy than to -pick out a number of striking specimens and give to each a distinctive -name, but <i>in no other way can they be divided into</i> -‘<i>species.</i>’<a class="afnanch" href="#fn551" id="fnanch551">551</a>” -Some writers have wondered at the peculiar -variability of this particular shell<a class="afnanch" href="#fn552" id="fnanch552">552</a>; -but for all we know of the -life-history of the Foraminifera, it may well be that a great -number of the other forms which we distinguish as separate species -and even genera are no more than temporary manifestations of -the same variability<a class="afnanch" href="#fn553" id="fnanch553">553</a>. -<span class="xxpn" id="p607">{607}</span></p> - -<div class="section"> -<h3><i>Conclusion.</i></h3></div> - -<p>If we can comprehend and interpret on some such lines as -these the form and mode of growth of the foraminiferal shell, we -may also begin to understand two striking features of the group, -namely, on the one hand the large number of diverse types or -families which exist and the large number of species and varieties -within each, and on the other the persistence of forms which in -many cases seem to have undergone little change or none at all -from the Cretaceous or even from earlier periods to the present -day. In few other groups, perhaps only among the Radiolaria, -do we seem to possess so nearly complete a picture of all possible -transitions between form and form, and of the whole branching -system of the evolutionary tree: as though little or nothing of it -had ever perished, and the whole web of life, past and present, -were as complete as ever. It leads one to imagine that these -shells have grown according to laws so simple, so much in harmony -with their material, with their environment, and with all the -forces internal and external to which they are exposed, that none -is better than another and none fitter or less fit to survive. It -invites one also to contemplate the possibility of the lines of -possible variation being here so narrow and determinate that -identical forms may have come independently into being again -and again.</p> - -<p>While we can trace in the most complete and beautiful manner -the passage of one form into another among these little shells, -and ascribe them all at last (if we please) to a series which starts -with the simple sphere of Orbulina or with the amoeboid body of -Astrorhiza, the question stares us in the face whether this be an -“evolution” which we have any right to correlate with historic -<i>time</i>. The mathematician can trace one conic section into -another, and “evolve” for example, through innumerable graded -ellipses, the circle from the straight line: which tracing of continuous -steps is a true “evolution,” though time has no part -therein. It was after this fashion that Hegel, and for that matter -Aristotle himself, was an evolutionist—to -whom evolution was <span class="xxpn" id="p608">{608}</span> -a mental concept, involving order and continuity in thought, but -not an actual sequence of events in time. Such a conception of -evolution is not easy for the modern biologist to grasp, and harder -still to appreciate. And so it is that even those who, like Dreyer<a class="afnanch" href="#fn554" id="fnanch554">554</a> -and like Rhumbler, study the foraminiferal shell as a physical -system, who recognise that its whole plan and mode of growth is -closely akin to the phenomena exhibited by fluid drops under -particular conditions, and who explain the conformation of the -shell by help of the same physical principles and mathematical -laws—yet all the while abate no jot or tittle of the ordinary -postulates of modern biology, nor doubt the validity and universal -applicability of the concepts of Darwinian evolution. For these -writers the <i>biogenetisches Grundgesetz</i> remains impregnable. The -Foraminifera remain for them a great family tree, whose actual -pedigree is traceable to the remotest ages; in which historical -evolution has coincided with progressive change; and in which -structural fitness for a particular function (or functions) has -exercised its selective action and ensured “the survival of the -fittest.” By successive stages of historic evolution we are supposed -to pass from the irregular Astrorhiza to a Rhabdammina with its -more concentrated disc; to the forms of the same genus which -consist of but a single tube with central chamber; to those where -this chamber is more and more distinctly segmented; so to the -typical many-chambered Nodosariae; and from these, by another -definite advance and later evolution to the spiral Trochamminae. -After this fashion, throughout the whole varied series of the -Foraminifera, Dreyer and Rhumbler (following Neumayr) recognise -so many successions of related forms, one passing into another, -and standing towards it in a definite relationship of ancestry or -descent. Each evolution of form, from simpler to more complex, -is deemed to have been attended by an advantage to the -organism, an enhancement of its chances of survival or perpetuation; -hence the historically older forms are, on the whole, -structurally the simpler; or conversely the simpler forms, such -as the simple sphere, were the first to come into being in primeval -seas; and finally, the gradual development -and increasing <span class="xxpn" id="p609">{609}</span> -complication of the individual within its own lifetime is held to -be at least a partial recapitulation of the unknown history of -its race and dynasty<a class="afnanch" href="#fn555" id="fnanch555">555</a>.</p> - -<p>We encounter many difficulties when we try to extend such -concepts as these to the Foraminifera. We are led for instance -to assert, as Rhumbler does, that the increasing complexity of the -shell, and of the manner in which one chamber is fitted on another, -makes for advantage; and the particular advantage on which -Rhumbler rests his argument is <i>strength</i>. Increase of strength, <i>die -Festigkeitssteigerung</i>, is according to him the guiding principle in -foraminiferal evolution, and marks the historic stages of their -development in geologic time. But in days gone by I used to -see the beach of a little Connemara bay bestrewn with millions -upon millions of foraminiferal shells, simple Lagenae, less simple -Nodosariae, more complex Rotaliae: all drifted by wave and -gentle current from their sea-cradle to their sandy grave: all -lying bleached and dead: one more delicate than another, but all -(or vast multitudes of them) perfect and unbroken. And so I -am not inclined to believe that niceties of form affect the case -very much: nor in general that foraminiferal life involves a -struggle for existence wherein breakage is a constant danger to -be averted, and increased strength an advantage to be ensured<a class="afnanch" href="#fn556" id="fnanch556">556</a>.</p> - -<p>In the course of the same argument Rhumbler remarks that -Foraminifera are absent from the coarse sands and gravels<a class="afnanch" href="#fn557" id="fnanch557">557</a>, -as -Williamson indeed had observed many years ago: -so averting, or <span class="xxpn" id="p610">{610}</span> -at least escaping, the dangers of concussion. But this is after -all a very simple matter of mechanical analysis. The coarseness -or fineness of the sediment on the sea-bottom is a measure of the -current: where the current is strong the larger stones are washed -clean, where there is perfect stillness the finest mud settles down; -and the light, fragile shells of the Foraminifera find their appropriate -place, like every other graded sediment, in this spontaneous -order of lixiviation.</p> - -<p>The theorem of Organic Evolution is one thing; the problem -of deciphering the lines of evolution, the order of phylogeny, the -degrees of relationship and consanguinity, is quite another. Among -the higher organisms we arrive at conclusions regarding these -things by weighing much circumstantial evidence, by dealing with -the resultant of many variations, and by considering the probability -or improbability of many coincidences of cause and effect; but -even then our conclusions are at best uncertain, our judgments -are continually open to revision and subject to appeal, and all -the proof and confirmation we can ever have is that which comes -from the direct, but fragmentary evidence of palaeontology<a class="afnanch" href="#fn558" id="fnanch558">558</a>.</p> - -<p>But in so far as forms can be shewn to depend on the play of -physical forces, and the variations of form to be directly due to -simple quantitative variations in these, just so far are we thrown -back on our guard before the biological conception of consanguinity, -and compelled to revise the vague canons which connect -classification with phylogeny.</p> - -<p>The physicist explains in terms of the properties of matter, -and classifies according to a mathematical analysis, all the drops -and forms of drops and associations of drops, all the kinds of -froth and foam, which he may discover among inanimate things; -and his task ends there. But when such forms, such conformations -and configurations, occur among <i>living</i> things, then at once the -biologist introduces his concepts of heredity, of historical evolution, -of succession in time, of recapitulation of remote ancestry in -individual growth, of common origin (unless contradicted by -direct evidence) of similar forms remotely separated by geographic -space or geologic time, of fitness -for a function, of <span class="xxpn" id="p611">{611}</span> -adaptation to an environment, of higher and lower, of “better” -and “worse.” This is the fundamental difference between the -“explanations” of the physicist and those of the biologist.</p> - -<p>In the order of physical and mathematical complexity there is -no question of the sequence of historic time. The forces that -bring about the sphere, the cylinder or the ellipsoid are the same -yesterday and to-morrow. A snow-crystal is the same to-day as -when the first snows fell. The physical forces which mould the -forms of Orbulina, of Astrorhiza, of Lagena or of Nodosaria to-day -were still the same, and for aught we have reason to believe the -physical conditions under which they worked were not appreciably -different, in that yesterday which we call the Cretaceous epoch; -or, for aught we know, throughout all that duration of time which -is marked, but not measured, by the geological record.</p> - -<p>In a word, the minuteness of our organism brings its conformation -as a whole within the range of the molecular forces; the -laws of its growth and form appear to lie on simple lines; what -Bergson calls<a class="afnanch" href="#fn559" id="fnanch559">559</a> -the “ideal kinship” is plain and certain, but the -“material affiliation” is problematic and obscure; and, in the -end and upshot, it seems to me by no means certain that the -biologist’s usual mode of reasoning is appropriate to the case, or -that the concept of continuous historical evolution must necessarily, -or may safely and legitimately, be employed.</p> - -<div class="chapter" id="p612"> -<h2 class="h2herein" -title="XIII. The Shapes of Horns, and of Teeth Or Tusks: -With a Note on Torsion.">CHAPTER XIII <span class="h2ttl"> -THE SHAPES OF HORNS, AND OF TEETH OR TUSKS: WITH A NOTE ON -TORSION</span></h2></div> - -<p>We have had so much to say on the subject of shell-spirals -that we must deal briefly with the analogous problems which are -presented by the horns of sheep, goats, antelopes and other -horned quadrupeds; and all the more, because these horn-spirals -are on the whole less symmetrical, less easy of measurement than -those of the shell, and in other ways also are less easy of investigation. -Let us dispense altogether in this case with mathematics; -and be content with a very simple account of the configuration -of a horn.</p> - -<p>There are three types of horn which deserve separate consideration: -firstly, the horn of the rhinoceros; secondly the -horns of the sheep, the goat, the ox or the antelope, that is to say, -of the so-called hollow-horned ruminants; and thirdly, the solid -bony horns, or “antlers,” which are characteristic of the deer.</p> - -<p>The horn of the rhinoceros presents no difficulty. It is -physiologically equivalent to a mass of consolidated hairs, and, -like ordinary hair, it consists of non-living or “formed” material, -continually added to by the living tissues at its base. In section, -that is to say in the form of its “generating curve,” the horn is -approximately elliptical, with the long axis fore-and-aft, or, in -some species, nearly circular. Its longitudinal growth proceeds -with a maximum velocity anteriorly, and a minimum posteriorly; -and the ratio of these velocities being constant, the horn curves -into the form of a logarithmic spiral in the manner that we have -already studied. The spiral is of small angle, but in the longer-horned -species, such as the great white rhinoceros (Ceratorhinus), -the spiral form is distinctly to be recognised. -As the horn <span class="xxpn" id="p613">{613}</span> -occupies a median position on the head,—a position, that is to say, -of symmetry in respect to the field of force on either side,—there -is no tendency towards a lateral twist, and the horn accordingly -develops as a <i>plane</i> logarithmic spiral. When two horns coexist, -the hinder one is much the smaller of the two: which is as much -as to say that the force, or rate, of growth diminishes as we pass -backwards, just as it does within the limits of the single horn. -And accordingly, while both horns have <i>essentially</i> the same -shape, the spiral curvature is less manifest in the second one, -simply by reason of its comparative shortness.</p> - -<p>The paired horns of the ordinary hollow-horned ruminants, -such as the sheep or the goat, grow under conditions which are -in some respects similar, but which differ in other and important -respects from the conditions under which the horn grows in the -rhinoceros. As regards its structure, the entire horn now consists -of a bony core with a covering of skin; the inner, or dermal, -layer of the latter is richly supplied with nutrient blood-vessels, -while the outer layer, or epidermis, develops the fibrous or -chitinous material, chemically and morphologically akin to a -mass of cemented or consolidated hairs, which constitutes the -“sheath” of the horn. A zone of active growth at the base of -the horn keeps adding to this sheath, ring by ring, and the specific -form of this annular zone is, accordingly, that of the “generating -curve” of the horn. Each horn no longer lies, as it does in the -rhinoceros, in the plane of symmetry of the animal of which it -forms a part; and the limited field of force concerned in the -genesis and growth of the horn is bound, accordingly, to be more -or less laterally asymmetrical. But the two horns are in symmetry -one with another; they form “conjugate” spirals, one -being the “mirror-image” of the other. Just as in the hairy coat -of the animal each hair, on either side of the median “parting,” -tends to have a certain definite direction of its own axis, inclined -away from the median axial plane of the whole system, so is it -both with the bony core of the horn and with the consolidated -mass of hairs or hair-like substance which constitutes its sheath; -the primary axis of the horn is more or less inclined to, and may -even be nearly perpendicular to, the axial plane of the animal.</p> - -<p>The growth of the horny sheath is not continuous, -but more or <span class="xxpn" id="p614">{614}</span> -less definitely periodic: sometimes, as in the sheep, this periodicity -is particularly well-marked, and causes the horny sheath to be -composed of a series of all but separate rings, which are supposed -to be formed year by year, and so to record the age of the animal<a class="afnanch" href="#fn560" id="fnanch560">560</a>.</p> - -<p>Just as we sought for the true generating curve in the orifice, -or “lip,” of the molluscan shell, so we might be apt to assume -that in the spiral horn the generating curve corresponded to the -lip or margin of one of the horny rings or annuli. This annular -margin, or boundary of the ring, is usually a sinuous curve, not -lying in a plane, but such as would form the boundary of an -anticlastic surface of great complexity: to the meaning and origin -of which phenomenon we shall return presently. But, as we have -already seen in the case of the molluscan shell, the complexities -of the lip itself, or of the corresponding lines of growth upon the -shell, need not concern us in our study of the development of the -spiral: inasmuch as we may substitute for these actual boundary -lines, their “trace,” or projection on a plane perpendicular to the -axis—in other words the simple outline of a transverse section -of the whorl. In the horn, this transverse section is often circular -or nearly so, as in the oxen and many antelopes: it now and then -becomes of somewhat complicated polygonal outline, as in a -highland ram; but in many antelopes, and in most of the sheep, -the outline is that of an isosceles, or sometimes nearly equilateral -triangle, a form which is typically displayed, for instance, in -<i>Ovis Ammon</i>. The horn in this latter case is a trihedral prism, -whose three faces are, (1) an upper, or frontal face, in continuation -of the plane of the frontal bone; (2) an outer, or orbital, starting -from the upper margin of the orbit; and (3) an inner, or “nuchal,” -abutting on the parietal bone<a class="afnanch" href="#fn561" id="fnanch561">561</a>. -Along these three faces, and -their corresponding angles or edges, we can trace in the fibrous -substance of the horn a series of homologous -spirals, such as we <span class="xxpn" id="p615">{615}</span> -have called in a preceding chapter the “<i>ensemble</i> of generating -spirals” which constitute the surface.</p> - -<p>In some few cases, of which the male musk ox is one of the -most notable, the horn is not developed in a continuous spiral -curve. It changes its shape as growth proceeds; and this, as -we have seen, is enough to show that it does not constitute a -logarithmic spiral. The reason is that the bony exostoses, or -horn-cores, about which the horny sheath is shaped and moulded, -neither grow continuously nor even remain of constant size after -attaining their full growth. But as the horns grow heavy the -bony core is bent downwards by their weight, and so guides</p> - -<div class="dctr01" id="fig318"> -<img src="images/i615.png" width="800" height="385" alt=""> - <div class="pcaption">Fig. 318. Diagram of Ram’s horns. - (After Sir Vincent Brooke, from <i>P.Z.S.</i>) <i>a</i>, frontal; - <i>b</i>, orbital; <i>c</i>, nuchal surface.</div></div> - -<p class="pcontinue"> -the growth of the horn in a new direction. Moreover as age advances, -the horn-core is further weakened and to a great extent absorbed: -and the horny sheath or horn proper, deprived of its support, -continues to grow, but in a flattened curve very different from -its original spiral<a class="afnanch" href="#fn562" id="fnanch562">562</a>. -The chamois is a somewhat analogous case. -Here the terminal, or oldest, part of the horn is curved; it tends -to assume a spiral form, though from its comparative shortness -it seems merely to be bent into a hook. But later on, the bony -core within, as it grows and strengthens, stiffens the horn, and -guides it into a straighter course or form. -The same phenomenon <span class="xxpn" id="p616">{616}</span> -of change of curvature, manifesting itself at the time when, or -the place where, the horn is freed from the support of the internal -core, is seen in a good many other antelopes (such as the hartebeest) -and in many buffaloes; and the cases where it is most manifest -appear to be those where the bony core is relatively short, or -relatively weak.</p> - -<div class="dctr03" id="fig319"> -<img src="images/i616.png" width="608" height="708" alt=""> - <div class="dcaption">Fig. 319. Head of Arabian - Wild Goat, <i>Capra sinaitica</i>. (After Sclater, from - <i>P.Z.S.</i>)</div></div> - -<p>But in the great majority of horns, we have no difficulty in -recognising a continuous logarithmic spiral, nor in referring it, as -before, to an unequal rate of growth (parallel to the axis) on two -opposite sides of the horn, the inequality maintaining a constant -ratio as long as growth proceeds. In certain antelopes, such as -the gemsbok, the spiral angle is very small, or in other words -the horn is very nearly straight; in other species of the same -genus Oryx, such as the Beisa antelope and the -Leucoryx, a gentle <span class="xxpn" id="p617">{617}</span> -curve (not unlike though generally less than that of a Dentalium -shell) is evident; and the spiral angle, according to the few -measurements I have made, is found to measure from about -20° to nearly 40°. In some of the large wild goats, such as the -Scinde wild goat, we have a beautiful logarithmic spiral, with a -constant angle of rather less than 70°; and we may easily arrange -a series of forms, such for example as the Siberian ibex, the -moufflon, <i>Ovis Ammon</i>, etc., and ending with the long-horned -Highland ram: in which, as we pass from one to another, we -recognise precisely homologous spirals, with an increasing angular -constant, the spiral angle being, for instance, about 75° or rather -less in <i>Ovis Ammon</i>, and in the Highland ram a very little more. -We have already seen that in the neighbourhood of 70° or 80° -a small change of angle makes a marked difference in the appearance -of the spire; and we know also that the actual length of the -horn makes a very striking difference, for the spiral becomes -especially conspicuous to the eye when a horn or shell is long -enough to shew several whorls, or at least a considerable part of -one entire whorl.</p> - -<p>Even in the simplest cases, such as the wild goats, the spiral -is never (strictly speaking) a plane or discoid spiral: but in -greater or less degree there is always superposed upon the plane -logarithmic spiral a helical spiral in space. Sometimes the latter -is scarcely apparent, for the helical curvature is comparatively -small, and the horn (though long, as in the said wild goats) is not -nearly long enough to shew a complete convolution: at other -times, as in the ram, and still better in many antelopes, such as -the koodoo, the helicoid or corkscrew curve of the horn is its -most characteristic feature.</p> - -<p>Accordingly we may study, as in the molluscan shell, the -helicoid component of the spire—in other words the variation in -what we have called (on p. <a href="#p555" title="go to pg. 555">555</a>) the angle <i>θ</i>. This factor it is -which, more than the constant angle of the logarithmic spiral, -imparts a characteristic appearance to the various species of -sheep, for instance to the various closely allied species of Asiatic -wild sheep, or Argali. In all of these the constant angle of the -logarithmic spiral is very much the same, but the shearing component -differs greatly. And thus the long drawn -out horns of <span class="xxpn" id="p618">{618}</span> -<i>Ovis Poli</i>, four feet or more from tip to tip, differ conspicuously -from those of <i>Ovis Ammon</i> or <i>O. hodgsoni</i>, in which a very similar -logarithmic spiral is wound (as it were) round a much blunter cone.</p> - -<hr class="hrblk"> - -<p>The ram’s horn then, like the snail’s shell, is a curve of double -curvature, in which one component has imposed upon the structure -a plane logarithmic spiral, and the other has produced a continuous -displacement, or “shear,” proportionate in magnitude to, and -perpendicular or otherwise inclined in direction to, the axis of -the former spiral curvature. The result is precisely analogous to -that which we have studied in the snail and other spiral univalves; -but while the form, and therefore the resultant forces, are similar, -the original distribution of force is not the same: for we have not -here, as we had in the snail-shell, a “columellar” muscle, to -introduce the component acting in the direction of the axis. We -have, it is true, the central bony core, which in part performs an -analogous function; but the main phenomenon here is apparently -a complex distribution of rates of growth, perpendicular to the -plane of the generating curve.</p> - -<p>Let us continue to dispense with mathematics, for the mathematical -treatment of a curve of double curvature is never very -simple, and let us deal with the matter by experiment. We have -seen that the generating curve, or transverse section, of a typical -ram’s horn is triangular in form. Measuring (along the curve of -the horn) the length of the three edges of the trihedral structure -in a specimen of <i>Ovis Ammon</i>, and calling them respectively the -outer, inner, and hinder edges (from their position at the base of -the horn, relatively to the skull), I find the outer edge to measure -80 cm., the inner 74 cm., and the posterior 45 cm.; let us say -that, roughly, they are in the ratio of 9 : 8 : 5. Then, if we make -a number of little cardboard triangles, equip each with three little -legs (I make them of cork), whose relative lengths are as 9 : 8 : 5, -and pile them up and stick them all together, we straightway -build up a curve of double curvature precisely analogous to the -ram’s horn: except only that, in this first approximation, we have -not allowed for the gradual increment (or decrement) of the -triangular surfaces, that is to say, for the <i>tapering</i> of the horn -due to the growth in its own plane of -the generating curve. <span class="xxpn" id="p619">{619}</span></p> - -<p>In this case then, and in most other trihedral or three-sided -horns, one of the three components, or three unequal velocities of -growth, is of relatively small magnitude, but the other two are -nearly equal one to the other. It would involve but little change -for these latter to become precisely equal; and again but little to -turn the balance of inequality the other way. But the immediate -consequence of this altered ratio of growth would be that the -horn would appear to wind the other way, as it does in the -antelopes, and also in certain goats, e.g. the markhor, <i>Capra -falconeri</i>.</p> - -<div class="psmprnt3"> -<p>For these two opposite directions of twist Dr Wherry has introduced a -convenient nomenclature. When the horn winds so that we follow it from -base to apex in the direction of the hands of a watch, it is customary to call -it a “left-handed” spiral. Such a spiral we have in the horn on the left-hand -side of a ram’s head. Accordingly, Dr Wherry calls the condition <i>homonymous</i>, -where, as in the sheep, a right-handed spiral is on the right side of the head, -and a left-handed spiral on the left side; while he calls the opposite condition -<i>heteronymous</i>, as we have it in the antelopes, where the right-handed twist -is on the left side of the head, and the left-handed twist on the right-hand side. -Among the goats, we may have either condition. Thus the domestic and -most of the wild goats agree with the sheep; but in the markhor the twisted -horns are heteronymous, as in the antelopes. The difference, as we have -seen, is easily explained; and (very much as in the case of our opposite spirals -in the apple-snail, referred to on p. <a href="#p560" title="go to pg. 560">560</a>), it has no very deep importance.</p> -</div><!--psmprnt3--> - -<p>Summarised then, in a very few words, the argument by which -we account for the spiral conformation of the horn is as follows: -The horn elongates by dint of continual growth within a narrow -zone, or annulus, at its base. If the rate of growth be identical -on all sides of this zone, the horn will grow straight; if it be -greater on one side than on the other, the horn will become curved: -and it probably <i>will</i> be greater on one side than on the other, -because each single horn occupies an unsymmetrical field with -reference to the plane of symmetry of the animal. If the maximal -and minimal velocities of growth be precisely at opposite sides -of the zone of growth, the resultant spiral will be a plane spiral; -but if they be not precisely or diametrically opposite, then the -spiral will be a spiral in space, with a winding or helical component; -and it is by no means likely that the maximum and -minimum <i>will</i> occur at precisely opposite ends of -a diameter, for <span class="xxpn" id="p620">{620}</span> -no such plane of symmetry is manifested in the field of force to -which the growing annulus corresponds or appertains.</p> - -<p>Now we must carefully remember that the rates of growth of -which we are here speaking are the net rates of longitudinal -increment, in which increment the activity of the living cells in -the zone of growth at the base of the horn is only one (though it -is the fundamental) factor. In other words, if the horny sheath -were continually being added to with equal rapidity all round its -zone of active growth, but at the same time had its elongation -more retarded on one side than the other (prior to its complete -solidification) by varying degrees of adhesion or membranous -attachment to the bone core within, then the net result would be -a spiral curve precisely such as would have arisen from initial -inequalities in the rate of growth itself. It seems highly probable -that this is a very important factor, and sometimes even the -chief factor in the case. The same phenomenon of attachment to -the bony core, and the consequent friction or retardation with -which the sheath slides over its surface, will lead to various -subsidiary phenomena: among others to the presence of transverse -folds or corrugations upon the horn, and to their unequal distribution -upon its several faces or edges. And while it is perfectly true -that nearly all the characters of the horn can be accounted -for by unequal velocities of longitudinal growth upon its different -sides, it is also plain that the actual field of force is a very complicated -one indeed. For example, we can easily see that (at least -in the great majority of cases) the direction of growth of the -horny fibres of the sheath is by no means parallel to the axis of -the core within; accordingly these fibres will tend to wind in a -system of helicoid curves around the core, and not only this -helicoid twist but any other tendency to spiral curvature on the -part of the sheath will tend to be opposed or modified by the -resistance of the core within. But on the other hand living bone -is a very plastic structure, and yields easily though slowly to any -forces tending to its deformation; and so, to a considerable -extent, the bony core itself will tend to be modelled by the curvature -which the growing sheath assumes, and the final result will -be determined by an equilibrium between these two systems of -forces. <span class="xxpn" id="p621">{621}</span></p> - -<p>While it is not very safe, perhaps, to lay down any general -rule as to what horns are more, and what are less spirally curved, -I think it may be said that, on the whole, the thicker the horn, -the greater is its spiral curvature. It is the slender horns, of such -forms as the Beisa antelope, which are gently curved, and it is -the robust horns of goats or of sheep in which the curvature is -more pronounced. Other things being the same, this is what we -should expect to find; for it is where the transverse section of -the horn is large that we may expect to find the more marked -differences in the intensity of the field of force, whether of active -growth or of retardation, on opposite sides or in different sectors -thereof.</p> - -<div class="dctr03" id="fig320"> -<img src="images/i621.png" width="608" height="367" alt=""> - <div class="dcaption">Fig. 320. Head of <i>Ovis Ammon</i>, shewing St -Venant’s curves.</div></div> - -<p>But there is yet another and a very remarkable phenomenon -which we may discern in the growth of a horn, when it takes the -form of a curve of double curvature, namely, an effect of torsional -strain; and this it is which gives rise to the sinuous “lines of -growth,” or sinuous boundaries of the separate horny rings, of -which we have already spoken. It is not at first sight obvious -that a mechanical strain of torsion is necessarily involved in the -growth of the horn. In our experimental illustration (p. <a href="#p618" title="go to pg. 618">618</a>), we -built up a twisted coil of separate elements, and no torsional -strain attended the development of the system. So would it -be if the horny sheath grew by successive annular increments, -free save for their relation to one another, and having no attachment -to the solid core within. But as a matter of -fact there is <span class="xxpn" id="p622">{622}</span> -such an attachment, by subcutaneous connective tissue, to the -bony core; and accordingly a torsional strain will be set up in -the growing horny sheath, again provided that the forces of growth -therein be directed more or less obliquely to the axis of the core; -for a “couple” is thus introduced, giving rise to a strain which -the sheath would not experience were it free (so to speak) to slip -along, impelled only by the pressure of its own growth from below. -And furthermore, the successive small increments of the growing -horn (that is to say, of the horny sheath) are not instantaneously -converted from living to solid and rigid substance; but there is -an intermediate stage, probably long-continued, during which -the new-formed horny substance in the neighbourhood of the zone -of active growth is still plastic and capable of deformation.</p> - -<p>Now we know, from the celebrated experiments of St Venant<a class="afnanch" href="#fn563" id="fnanch563">563</a>, -that in the torsion of an elastic body, other than a cylinder of -circular section, a very remarkable state of strain is introduced. -If the body be thus cylindrical (whether solid or hollow), then a -twist leaves each circular section unchanged, in dimensions and -in figure. But in all other cases, such as an elliptic rod or a -prism of any particular sectional form, forces are introduced which -act parallel to the axis of the structure, and which warp each -section into a complex anticlastic surface. Thus in the case of a -triangular and equilateral prism, such as is shewn in section in -Fig. <a href="#fig321" title="go to Fig. 321">321</a>, if the part of the rod represented in the section be twisted -by a force acting in the direction of the arrow, then the originally -plane section will be warped as indicated in the diagram:—where -the full contour-lines represent elevation above, and the dotted -lines represent depression below, the original level. On the -external surface of the prism, then, contour-lines which were -originally parallel and horizontal, will be found warped into sinuous -curves, such that, on each of the three faces, the curve will be -convex upwards on one half, and concave upwards on the other -half of the face. The ram’s horn, and still better that of <i>Ovis -Ammon</i>, is comparable to such a prism, save that in section it -is not quite equilateral, and that its three faces are not plane. -The warping is therefore not precisely identical -on the three faces <span class="xxpn" id="p623">{623}</span> -of the horn; but, in the general distribution of the curves, it is -in complete accordance with theory. Similar anticlastic curves -are well seen in many antelopes; but they are conspicuous by -their absence in the <i>cylindrical</i> horns of oxen.</p> - -<p>The better to illustrate this phenomenon, the nature of which -is indeed obvious enough from a superficial examination of the -horn, I made a plaster cast of one of the horny rings in a horn of -<i>Ovis Ammon</i>, so as to get an accurate pattern of its sinuous edge: -and then, filling the mould up with wet clay, I modelled an anticlastic -surface, such as to correspond as nearly as possible with -the sinuous outline<a class="afnanch" href="#fn564" id="fnanch564">564</a>. -Finally, after making a plaster cast of this -sectional surface, I drew its contour-lines (as shewn in Fig. <a href="#fig322" title="go to Fig. 322">322</a>), -with the help of a simple form of spherometer. It will be seen -that in great part this diagram is precisely</p> - -<div class="dctr01" id="fig321"><div id="fig322"> -<img src="images/i623.png" width="800" height="369" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 321.</td> - <td></td> - <td>Fig. 322.</td></tr></table> -</div></div></div><!--dctr01--> - -<p class="pcontinue">similar to St Venant’s -diagram of the cross-section of a twisted triangular prism; and -this is especially the case in the neighbourhood of the sharp angle -of our prismatic section. That in parts the diagram is somewhat -asymmetrical is not to be wondered at: and (apart from inaccuracies -due to the somewhat rough means by which it was made) -this asymmetry can be sufficiently accounted for by anisotropy -of the material, by inequalities in thickness of different parts of -the horny sheath, and especially (I think) by unequal distributions -of rigidity due to the presence of the smaller -corrugations of the <span class="xxpn" id="p624">{624}</span> -horn. It is apparently on account of these minor corrugations -that, in such horns as the Highland ram’s, where they are strongly -marked, the main St Venant effect is not nearly so well shewn as -in the smoother horns such as those of <i>O. Ammon</i> and its immediate -congeners<a class="afnanch" href="#fn565" id="fnanch565">565</a>.</p> - -<div class="section"> -<h3><i>A further Note upon Torsion.</i></h3></div> - -<p>The phenomenon of torsion, to which we have been thus -introduced, opens up many wide questions in connection with -form. Some of the associated phenomena are admirably illustrated -in the case of climbing plants; but we can only deal with these -still more briefly and parenthetically.</p> - -<p>The subject of climbing plants has been elaborately dealt -with not only in Darwin’s books<a class="afnanch" href="#fn566" id="fnanch566">566</a>, -but also by a very large number -of earlier and later writers. In “twining” plants, which constitute -the greater number of “climbers,” the essential phenomenon is a -tendency of the growing shoot to revolve about a vertical axis—a -tendency long ago discussed and investigated by such writers -as Palm, H. von Mohl and Dutrochet<a class="afnanch" href="#fn567" id="fnanch567">567</a>. -This tendency to -revolution—“circumvolution,” -as Darwin calls it, “revolving nutation,” -as Sachs puts it—is very closely comparable to the process by which -an antelope’s horn (such as the koodoo’s) grows into its spiral -or rather helicoid form; and it is simply due, in like manner, to -inequalities in the rate of growth on different sides of the growing -stem. There is only this difference between the two cases, that -in the antelope’s horn the zone of active growth is confined to -the base of the horn, while in the climbing stem the same -phenomenon is at work throughout the whole length of the growing -structure. This growth is in the main due to “turgescence,” -that is to the extension, or elongation, of ready-formed cells -through the imbibition of water; it is a phenomenon due to -osmotic pressure. The particular stimuli to which these movements -(that is to say, these inequalities of growth) -have been <span class="xxpn" id="p625">{625}</span> -ascribed, such as contact (thigmotaxis), exposure to light -(heliotropism), and so forth, need not be discussed here<a class="afnanch" href="#fn568" id="fnanch568">568</a>.</p> - -<p>A simple stem growing upright in the dark, or in uniformly -diffused light, would be in a position of equilibrium to a field of -force radially symmetrical about its vertical axis. But this -complete radial symmetry will not often occur; and the radial -anomalies may be such as arise intrinsically from structural -peculiarities in the stem itself, or externally to it by reason of -unequal illumination or through various other localised forces. -The essential fact, so far as we are concerned, is that in twining -plants we have a very marked tendency to inequalities in longitudinal -growth on different aspects of the stem—a tendency which -is but an exaggerated manifestation of one which is more or less -present, under certain conditions, in all plants whatsoever. Just -as in the case of the ruminants’ horns so we find here, that this -inequality may be, so to speak, positive or negative, the maximum -lying to the one side or the other of the twining stem; and so it -comes to pass that some climbers twine to the one side and some -to the other: the hop and the honeysuckle following the sun, -and the field-convolvulus twining in the reverse direction; there -are also some, like the woody nightshade (<i>Solanum Dulcamara</i>) -which twine indifferently either way.</p> - -<p>Together with this circumnutatory movement, there is very -generally to be seen an actual <i>torsion</i> of the twining stem—a -twist, that is to say, about its own axis; and Mohl made the -curious observation, confirmed by Darwin, that when a stem -twines around a smooth cylindrical stick the torsion does not take -place, save “only in that degree which follows as a mechanical -necessity from the spiral winding”: but that stems which had -climbed around a rough stick were all more or less, and generally -much, twisted. Here Darwin did not refrain from introducing -that teleological argument which pervades his whole train of -reasoning: “The stem,” he says, “probably gains rigidity by -being twisted (on the same principle that a much -twisted rope <span class="xxpn" id="p626">{626}</span> -is stiffer than a slackly twisted one), and is thus indirectly -benefited so as to be able to pass over inequalities in its spiral -ascent, and to carry its own weight when allowed to revolve -freely.” The mechanical explanation would appear to be very -simple, and such as to render the teleological hypothesis unnecessary. -In the case of the roughened support, there is a -temporary adhesion or “clinging” between it and the growing -stem which twines around it; and a system of forces is thus set -up, producing a “couple,” just as it was in the case of the ram’s -or antelope’s horn through direct adhesion of the bony core to -the surrounding sheath. The twist is the direct result of this -couple, and it disappears when the support is so smooth that no -such force comes to be exerted.</p> - -<p>Another important class of climbers includes the so-called -“leaf-climbers.” In these, some portion of the leaf, generally the -petiole, sometimes (as in the fumitory) the elongated midrib, -curls round a support; and a phenomenon of like nature occurs -in many, though not all, of the so-called “tendril-bearers.” -Except that a different part of the plant, leaf or tendril instead of -stem, is concerned in the twining process, the phenomenon here -is strictly analogous to our former case; but in the resulting -helix there is, as a rule, this obvious difference, that, while the -twining stem, for instance of the hop, makes a slow revolution -about its support, the typical leaf-climber makes a close, firm -coil: the axis of the latter is nearly perpendicular and parallel -to the axis of its support, while in the twining stem the angle -between the two axes is comparatively small. Mathematically -speaking, the difference merely amounts to this, that the component -in the direction of the vertical axis is large in the one -case, and the corresponding component is small, if not absent, -in the other; in other words, we have in the climbing stem a -considerable vertical component, due to its own tendency to grow -in height, while this longitudinal or vertical extension of the -whole system is not apparent, or little apparent, in the other -cases. But from the fact that the twining stem tends to run -obliquely to its support, and the coiling petiole of the leaf-climber -tends to run transversely to the axis of its support, there -immediately follows this marked difference, -that the phenomenon <span class="xxpn" id="p627">{627}</span> -of <i>torsion</i>, so manifest in the former case, will be absent in the -latter.</p> - -<hr class="hrblk"> - -<p>There is one other phenomenon which meets us in the twining -and twisted stem, and which is doubtless illustrated also, though -not so well, in the antelope’s horn; it is a phenomenon which -forms the subject of a second chapter of St Venant’s researches on -the effects of torsional strain in elastic bodies. We have already -seen how one effect of torsion, in for instance a prism, is to -produce strains parallel to the axis, elevating parts and depressing -other parts of each transverse section. But in addition to this, -the same torsion has the effect of materially altering the form of -the section itself, as we may easily see by twisting a square or -oblong piece of india-rubber. If we start with a cylinder, such as -a round piece of catapult india-rubber, and twist it on its own -long axis, we have already seen that it suffers no other distortion; -it still remains a cylinder, that is to say, it is still in section everywhere -circular. But if it be of any other shape than cylindrical -the case is quite different, for now the sectional shape tends to -alter under the strain of torsion. Thus, if our rod be elliptical -in section to begin with, it will, under torsion, become a more -elongated ellipse; if it be square, its angles will become more -prominent, and its sides will curve inwards, till at length the -square assumes the appearance of a four-pointed star, with -rounded angles. Furthermore, looking at the results of this -process of modification, we find experimentally that the resultant -figures are more easily twisted, less resistant to torsion, than -were those from which we evolved them; and this is a very -curious physical or mathematical fact. So a cylinder, which is -especially resistant to torsion, is very easily bent or flexed; while -projecting ribs or angles, such as an engineer makes in a bar or -pillar of iron for the purpose of greatly increasing its strength in -the way of resistance to <i>bending</i>, actually make it much weaker -than before (for the same amount of metal per unit length) in the -way of resistance to <i>torsion</i>.</p> - -<p>In the hop itself, and in a very considerable number of other -twining and twisting stems, the ribbed or channelled form of the -stem is a conspicuous feature. We may safely take -it, (1) that <span class="xxpn" id="p628">{628}</span> -such stems are especially susceptible of torsion; and (2) that the -effect of torsion will be to intensify any such peculiarities of -sectional outline which they may possess, though not to initiate -them in an originally cylindrical structure. In the leaf-climbers -the case does not present itself, for there, as we have seen, torsion -itself is not, or is very slightly, manifested. There are very -distinct traces of the phenomenon in the horns of certain antelopes, -but the reason why it is not a more conspicuous feature of the -antelope’s horn or of the ram’s is apparently a very simple one: -namely, that the presence of the bony core within tends to check -that deformation which is perpendicular, while it permits that -which is parallel, to the axis of the horn.</p> - -<div class="section"> -<h3><i>Of Deer’s Antlers.</i></h3></div> - -<p>But let us return to our subject of the shapes of horns, -and consider briefly our last class of these structures, -namely the bony antlers of the various species of elk and -deer<a class="afnanch" href="#fn569" id="fnanch569">569</a>. -The problems which these present to us are very different from -those which we have had to do with in the antelope or the -sheep.</p> - -<p>With regard to its structure, it is plain that the bony antler -corresponds, upon the whole, to the bony core of the antelope’s -horn; while in place of the hard horny sheath of the latter, we -have the soft “velvet,” which every season covers the new growing -antler, and protects the large nutrient blood-vessels by help of -which the antler grows<a class="afnanch" href="#fn570" id="fnanch570">570</a>. -The main difference lies in the fact -that, in the one case, the bony core, imprisoned within its sheath, -is rendered incapable of branching and incapable also of lateral -expansion, and the whole horn is only permitted to grow in length, -while retaining a sectional contour that is identical with (or but -little altered from) that which it possesses -at its growing base: <span class="xxpn" id="p629">{629}</span> -but in the antler, on the other hand, no such restraint is imposed, -and the living, growing fabric of bone may expand into a broad -flattened plate over which the blood-vessels run. In the immediate -neighbourhood of the main blood-vessels growth will be most -active; in the interspaces between, it may wholly fail: with the -result that we may have great notches cut out of the flattened -plate, or may at length find it reduced to the form of a simple -branching structure. The main point, as it seems to me, is that -the “horn” is essentially an <i>axial rod</i>, while the “antler” is</p> - -<div class="dctr01" id="fig323"> -<img src="images/i629.png" width="800" height="477" alt=""> - <div class="dcaption">Fig. 323. Antlers of Swedish Elk. -(After Lönnberg, from <i>P.Z.S.</i>)</div></div> - -<p class="pcontinue"> essentially an outspread <i>surface</i><a -class="afnanch" href="#fn571" id="fnanch571">571</a>. In -other words, I believe that the whole configuration of an -antler is more easily understood by conceiving it as a -plate or a surface, more and more notched and scolloped -till but a slender skeleton may remain, than to look upon -it the other way, namely as an axial stem (or beam) giving -<span class="xxpn" id="p630">{630}</span> off branches (or -tines), the interspaces between which latter may sometimes -be filled up to form a continuous plate.</p> - -<p>In a sense it matters very little whether we regard the broad -plate-like antlers of the elk or the slender branching antlers of the -stag as the more primitive type; for we are not concerned here -with the question of hypothetical phylogeny. And even from the -mathematical point of view it makes little or no difference whether -we describe the plate as constituted by the interconnection of -the branches, or the branches derived by a</p> - -<div class="dctr03" id="fig324"> -<img src="images/i630.png" width="608" height="587" alt=""> - <div class="dcaption">Fig. 324. Head and antlers of a Stag -(<i>Cervus Duvauceli</i>). (After Lydekker, from <i>P.Z.S.</i>)</div></div> - -<p class="pcontinue"> -process of notching -or incision from the plate. The important point for us is to -recognise that (save for occasional slight irregularities) the -branching system in the one <i>conforms</i> essentially to the curved -plate or surface which we see plainly in the other. In short the -arrangement of the branches is more or less comparable to that -of the veins in a leaf, or to that of the blood-vessels as they course -over the curved surface of an organ. It is a process of ramification, -not, like that of a tree, in various planes, -but strictly limited <span class="xxpn" id="p631">{631}</span> -to a single surface. And just as the veins within a leaf are not -necessarily confined (as they happen to be in most ordinary -leaves) to a <i>plane</i> surface, but, as in the petal of a tulip or the -capsule of a poppy, may have to run their course within a curved -surface, so does the analogy of the leaf lead us directly to the -mode of branching which is characteristic of the antler. The -surface to which the branches of the antler tend to be confined -is a more or less spheroidal, or occasionally an ellipsoidal one; -and furthermore, when we inspect any well-developed pair of -antlers, such as those of a red deer, a sambur or a wapiti, we have -no difficulty in seeing that the two antlers make up between them -<i>a single surface</i>, and constitute a symmetrical figure, each half -being the mirror-image of the other.</p> - -<p>To put the case in another way, a pair of antlers (apart from -occasional slight irregularities) tends to constitute a figure such -that we could conceive an elastic sheet stretched over or round -the entire system, so as to form one continuous and even surface; -and not only would the surface curvature be on the whole smooth -and even, but the boundary of the surface would also tend to be -an even curve: that is to say the tips of all the tines would -approximately have their locus in a continuous curve.</p> - -<p>It follows from this that if we want to make a simple model of -a set of antlers, we shall be very greatly helped by taking some -appropriate spheroidal surface as our groundwork or scaffolding. -The best form of surface is a matter for trial and investigation in -each particular case; but even in a sphere, by selecting appropriate -areas thereof, we can obtain sufficient varieties of surface to meet -all ordinary cases. With merely a bit of sculptor’s clay or plasticine, -we should be put hard to it to model the horns of a wapiti -or a reindeer: but if we start with an orange (or a round florence -flask) and lay our little tapered rolls of plasticine upon it, in simple -natural curves, it is surprising to see how quickly and successfully -we can imitate one type of antler after another. In doing so, -we shall be struck by the fact that our model may vary in its -mode of branching within very considerable limits, and yet look -perfectly natural. For the same wide range of variation is characteristic -of the natural antlers themselves. As Sir V. Brooke says -(<i>op. cit.</i> p. 892), “No two antlers are ever exactly -alike; and the <span class="xxpn" id="p632">{632}</span> -variation to which the antlers are subject is so great that in the -absence of a large series they would be held to be indicative of -several distinct species<a class="afnanch" href="#fn572" id="fnanch572">572</a>.” -But all these many variations lie -within a limited range, for they are all subject to our general -rule that the entire structure is essentially confined to a single -curved surface.</p> - -<p>It is plain that in the curvatures both of the beam and of its -tines, in the angles by which these latter meet the beam, and in -the contours of the entire system, there are involved many elegant -mathematical problems with which we cannot at present attempt -to deal. Nor must we attempt meanwhile to enquire into the -physical meaning or origin of these phenomena, for as yet the clue -seems to be lacking and we should only heap one hypothesis upon -another. That there is a complete contrast of mathematical -properties between the horn and the antler is the main lesson with -which, in the meantime, we must rest content.</p> - -<div class="section"> -<h3><i>Of Teeth, and of Beak and Claw.</i></h3></div> - -<p>In a fashion similar to that manifested in the shell or the -horn, we find the logarithmic spiral to be implicit in a great many -other organic structures where the phenomena of growth proceed -in a similar way: that is to say, where about an axis there is some -asymmetry leading to unequal rates of longitudinal growth, and -where the structure is of such a kind that each new increment is -added on as a permanent and unchanging part of the entire -conformation. Nail and claw, beak and tooth, all come under -this category. The logarithmic spiral <i>always</i> tends to manifest -itself in such structures as these, though it usually only attracts -our attention in elongated structures, where (that is to say) the -radius vector has described a considerable angle. When the -canary-bird’s claws grow long from lack of use, or when the -incisor tooth of a rabbit or a rat grows long by reason of an injury -to the opponent tooth against which it was wont to bite, we know -that the tooth or claw tends to grow into a spiral curve, and we -speak of it as a malformation. But there has been no fundamental -change of form, save only an abnormal -increase in length; <span class="xxpn" id="p633">{633}</span> -the elongated tooth or claw has the selfsame curvature that it had -when it was short, but the spiral curvature becomes more and more -manifest the longer it grows. A curious analogous case is that -of the New Zealand huia bird, in which the beak of the female -is described as being comparatively short and straight, while that -of the male is long and curved; it is easy to see that there is a -slight curvature also in the beak of the female, and that the beak -of the male shows nothing but the same curve produced. In the -case of the more curved beaks, such as those of an eagle or a parrot, -we may, if we please, determine the constant angle of the logarithmic -spiral, just as we have done in the case of the Nautilus -shell; and here again, as the bird grows older or the beak longer, -the spiral nature of the curve becomes more and more apparent, -as in the hooked beak of an old eagle, or as in the great beak of -some large parrot such as a hyacinthine macaw.</p> - -<p>Let us glance at one or two instances to illustrate the spiral -curvature of teeth.</p> - -<p>A dentist knows that every tooth has a curvature of its own, -and that in pulling the tooth he must follow the direction of the -curve; but in an ordinary tooth this curvature is scarcely visible, -and is least so when the diameter of the tooth is large compared -with its length.</p> - -<p>In the simply formed, more or less conical teeth, such as are -those of the dolphin, and in the more or less similarly shaped canines -and incisors of mammals in general, the curvature of the tooth -is particularly well seen. We see it in the little teeth of a hedgehog, -and in the canines of a dog or a cat it is very obvious indeed. -When the great canine of the carnivore becomes still further -enlarged or elongated, as in Machairodus, it grows into the -strongly curved sabre-tooth of that great extinct tiger. In rodents, -it is the incisors which undergo a great elongation; their rate of -growth differs, though but slightly, on the two sides, anterior and -posterior, of the axis, and by summation of these slight differences -in the rapid growth of the tooth an unmistakeable logarithmic -spiral is gradually built up. We see it admirably in the beaver, -or in the great ground-rat, Geomys. The elephant is a similar -case, save that the tooth, or tusk, remains, owing to comparative -lack of wear, in a more perfect condition. In the rodent (save -only in those abnormal cases mentioned on the -last page) the <span class="xxpn" id="p634">{634}</span> -anterior, first-formed, part of the tooth wears away as fast as it -is added to from behind; and in the grown animal, all those -portions of the tooth near to the pole of the logarithmic spiral -have long disappeared. In the elephant, on the other hand, we -see, practically speaking, the whole unworn tooth, from point to -root; and its actual tip nearly coincides with the pole of the -spiral. If we assume (as with no great inaccuracy we may do) -that the tip actually coincides with the pole, then we may very -easily construct the continuous spiral of which the existing tusk -constitutes a part; and by so doing, we see the short, gently -curved tusk of our ordinary elephant growing gradually into the -spiral tusk of the mammoth. No doubt, just as in the case of -our molluscan shells, we have a tendency to variation, both -individual and specific, in the constant angle of the spiral; some -elephants, and some species of elephant, undoubtedly have a -higher spiral angle than others. But in most cases, the angle -would seem to be such that a spiral configuration would become -very manifest indeed if only the tusk pursued its steady growth, -unchanged otherwise in form, till it attained the dimensions -which we meet with in the mammoth. In a species such as -<i>Mastodon angustidens</i>, or <i>M. arvernensis</i>, the specific angle is -low and the tusk comparatively straight; but the American -mastodons and the existing species of elephant have tusks which -do not differ appreciably, except in size, from the great spiral -tusks of the mammoth, though from their comparative shortness -the spiral is little developed and only appears to the eye as a -gentle curve. Wherever the tooth is very long indeed, as in the -mammoth or the beaver, the effect of some slight and all but -inevitable lateral asymmetry in the rate of growth begins to shew -itself: in other words, the spiral is seen to lie not absolutely in -a plane, but to be a curve of double curvature, like a twisted -horn. We see this condition very well in the huge canine tusks -of the Babirussa; it is a conspicuous feature in the mammoth, -and it is more or less perceptible in any large tusk of the ordinary -elephants.</p> - -<p>The form of a molar tooth, which is essentially a branching or -budding system, and in which such longitudinal growth as gives -rise to a spiral curve is but little manifest, constitutes an entirely -different problem with which I shall not at present -attempt to deal.</p> - -<div class="chapter" id="p635"> -<h2 class="h2herein" -title="XIV. On Leaf-arrangement, Or Phyllotaxis.">CHAPTER -XIV <span class="h2ttl"> ON LEAF-ARRANGEMENT, OR -PHYLLOTAXIS</span></h2></div> - -<p>The beautiful configurations produced by the orderly arrangement -of leaves or florets on a stem have long been an object of -admiration and curiosity. Leonardo da Vinci would seem, as Sir -Theodore Cook tells us, to have been the first to record his thoughts -upon this subject; but the old Greek and Egyptian geometers -are not likely to have left unstudied or unobserved the spiral -traces of the leaves upon a palm-stem, or the spiral curves of the -petals of a lotus or the florets in a sunflower.</p> - -<p>The spiral leaf-order has been regarded by many learned -botanists as involving a fundamental law of growth, of the deepest -and most far-reaching importance; while others, such as Sachs, -have looked upon the whole doctrine of “phyllotaxis” as “a sort -of geometrical or arithmetical playing with ideas,” and “the -spiral theory as a mode of view gratuitously introduced into the -plant.” Sachs even goes so far as to declare this doctrine “in -direct opposition to scientific investigation, and based upon the -idealistic direction of the Naturphilosophie,”—the mystical biology -of Oken and his school.</p> - -<p>The essential facts of the case are not difficult to understand; -but the theories built upon them are so varied, so conflicting, and -sometimes so obscure, that we must not attempt to submit them -to detailed analysis and criticism. There are two chief ways by -which we may approach the question, according to whether we -regard, as the more fundamental and typical, one or other of the -two chief modes in which the phenomenon presents itself. That -is to say, we may hold that the phenomenon is displayed in its -essential simplicity by the corkscrew spirals, or helices, which -mark the position of the leaves upon a cylindrical stem -or on an <span class="xxpn" id="p636">{636}</span> -elongated fir-cone; or, on the other hand, we may be more -attracted by, and regard as of greater importance, the logarithmic -spirals which we trace in the curving rows of florets in the discoidal -inflorescence of a sunflower. Whether one way or the other be -the better, or even whether one be not positively correct and the -other radically wrong, has been vehemently debated. In my -judgment they are, both mathematically and biologically, to be -regarded as inseparable and correlative phenomena.</p> - -<p>The helical arrangement (as in the fir-cone) was carefully -studied in the middle of the eighteenth century by the celebrated -Bonnet, with the help of Calandrini, the mathematician. Memoirs -published about 1835, by Schimper and Braun, greatly amplified -Bonnet’s investigations, and introduced a nomenclature which -still holds its own in botanical textbooks. Naumann and the -brothers Bravais are among those who continued the investigation -in the years immediately following, and Hofmeister, in 1868, gave -an admirable account and summary of the work of these and -many other writers<a class="afnanch" href="#fn573" id="fnanch573">573</a>.</p> - -<p>Starting from some given level and proceeding upwards, let -us mark the position of some one leaf (<i>A</i>) upon a cylindrical stem. -Another, and a younger leaf (<i>B</i>) will be found standing at a certain -distance <i>around</i> the stem, and a certain distance -<i>along</i> the stem, <span class="xxpn" id="p637">{637}</span> -from the first. The former distance may be expressed as a -fractional “divergence” (such as two-fifths of the circumference -of the stem) as the botanists describe it, or by an “angle of -azimuth” (such as ϕ -= 144°) as the mathematician would be more -likely to state it. The position of <i>B</i> relatively to <i>A</i> must be -determined, not only by this angle ϕ, in the horizontal plane, but -also by an angle (θ) in the vertical plane; for the height of <i>B</i> above -the level of <i>A</i>, in comparison with the diameter of the cylinder, -will obviously make a great difference in the appearance of the -whole system, in short the position of each leaf must be expressed -by <i>F</i>(ϕ · sin θ). But this matter botanical students have not -concerned themselves with; in other words, their studies have -been limited (or mainly limited) to the relation of the leaves to -one another in <i>azimuth</i>.</p> - -<p>Whatever relation we have found between <i>A</i> and <i>B</i>, let -precisely the same relation subsist between <i>B</i> and <i>C</i>: and so on. -Let the growth of the system, that is to say, be continuous and -uniform; it is then evident that we have the elementary conditions -for the development of a simple cylindrical helix; and this -“primary helix” or “genetic spiral” we can now trace, winding -round and round the stem, through <i>A</i>, <i>B</i>, <i>C</i>, etc. But if we can -trace such a helix through <i>A</i>, <i>B</i>, <i>C</i>, it follows from the symmetry -of the system, that we have only to join <i>A</i> to some other leaf to -trace another spiral helix, such, for instance, as <i>A</i>, <i>C</i>, <i>E</i>, etc.; -parallel to which will run another and similar one, namely in this -case <i>B</i>, <i>D</i>, <i>F</i>, etc. And these spirals will run in the opposite -direction to the spiral <i>ABC</i>.</p> - -<p>In short, the existence of one helical arrangement of points -implies and involves the existence of another and then another -helical pattern, just as, in the pattern of a wall-paper, our eye -travels from one linear series to another.</p> - -<p>A modification of the helical system will be introduced when, -instead of the leaves appearing, or standing, in singular succession, -we get two or more appearing simultaneously upon the same level. -If there be two such, then we shall have two generating spirals -precisely equivalent to one another; and we may call them -<i>A</i>, <i>B</i>, <i>C</i>, etc., and <i>A′</i>, <i>B′</i>, <i>C′</i>, and so on. These are the cases -which we call “whorled” leaves, or in the -simplest case, where <span class="xxpn" id="p638">{638}</span> -the whorl consists of two opposite leaves only, we call them -decussate.</p> - -<hr class="hrblk"> - -<p>Among the phenomena of phyllotaxis, two points in particular -have been found difficult of explanation, and have aroused discussion. -These are (1), the presence of the logarithmic spirals -such as we have already spoken of in the sunflower; and (2) the -fact that, as regards the number of the helical or spiral rows, -certain numerical coincidences are apt to recur again and again, -to the exclusion of others, and so to become characteristic features -of the phenomenon.</p> - -<p>The first of these appears to me to present no difficulty. It -is a mere matter of strictly mathematical “deformation.” The -stem which we have begun to speak of as a cylinder is not strictly -so, inasmuch as it tapers off towards its summit. The curve -which winds evenly around this stem is, accordingly, not a true -helix, for that term is confined to the curve which winds evenly -around the <i>cylinder</i>: it is a curve in space which (like the spiral -curve we have studied in our turbinate shells) partakes of the -characters of a helix and of a logarithmic spiral, and which is in -fact a logarithmic spiral with its pole drawn out of its original -plane by a force acting in the direction of the axis. If we imagine -a tapering cylinder, or cone, projected, by vertical projection, on -a plane, it becomes a circular disc; and a helix described about -the cone necessarily becomes in the disc a logarithmic spiral -described about a focus which corresponds to the apex of our cone. -In like manner we may project an identical spiral in space upon -such surfaces as (for instance) a portion of a sphere or of an ellipsoid; -and in all these cases we preserve the spiral configuration, which -is the more clearly brought into view the more we reduce the -vertical component by which it was accompanied. The converse -is, of course, equally true, and equally obvious, namely that any -logarithmic spiral traced upon a circular disc or spheroidal surface -will be transformed into a corresponding spiral helix when the -plane or spheroidal disc is extended into an elongated cone -approximating to a cylinder. This mathematical conception is -translated, in botany, into actual fact. The fir-cone may be -looked upon as a cylindrical axis contracted at -both ends, until <span class="xxpn" id="p639">{639}</span> -it becomes approximately an ellipsoidal solid of revolution, -generated about the long axis of the ellipse; and the semi-ellipsoidal -capitulum of the teasel, the more or less hemispherical one -of the thistle, and the flattened but still convex one of the sunflower, -are all beautiful and successive deformations of what is -typically a long, conical, and all but cylindrical stem. On the -other hand, every stem as it grows out into its long cylindrical -shape is but a deformation of the little spheroidal or ellipsoidal -surface, or cone, which was its forerunner in the bud.</p> - -<p>This identity of the helical spirals around the stem with spirals -projected on a plane was clearly recognised by Hofmeister, who -was accustomed to represent his diagrams of leaf-arrangement -either in one way or the other, though not in a strictly geometrical -projection<a class="afnanch" href="#fn574" id="fnanch574">574</a>.</p> - -<hr class="hrblk"> - -<p>According to Mr A. H. Church<a class="afnanch" href="#fn575" id="fnanch575">575</a>, -who has dealt very carefully -and elaborately with the whole question of phyllotaxis, the -logarithmic spirals such as we see in the disc of the sunflower have -a far greater importance and a far deeper meaning than this brief -treatment of mine would accord to them: and Sir Theodore Cook, -in his book on the <i>Curves of Life</i>, has adopted and has helped to -expound and popularise Mr Church’s investigations.</p> - -<p>Mr Church, regarding the problem as one of “uniform growth,” -easily arrives at the conclusion that, <i>if</i> this growth can be conceived -as taking place symmetrically about a central point or “pole,” -the uniform growth would then manifest itself in logarithmic -spirals, including of course the limiting cases of the circle and -straight line. With this statement I have little fault to find; it -is in essence identical with much that I have said in a previous -chapter. But other statements of Mr Church’s, and many theories -woven about them by Sir T. Cook and himself, I am less able to -follow. Mr Church tells us that the essential phenomenon in the -sunflower disc is a series of orthogonally intersecting logarithmic -spirals. Unless I wholly misapprehend Mr Church’s meaning, I -should say that this is very far from -essential. The spirals <span class="xxpn" id="p640">{640}</span> -intersect isogonally, but orthogonal intersection would be only -one particular case, and in all probability a very infrequent one, -in the intersection of logarithmic spirals developed about a -common pole. Again on the analogy of the hydrodynamic lines -of force in certain vortex movements, and of similar lines of -force in certain magnetic phenomena, Mr Church proceeds to -argue that the energies of life follow lines comparable to those of -electric energy, and that the logarithmic spirals of the sunflower -are, so to speak, lines of equipotential<a class="afnanch" href="#fn576" id="fnanch576">576</a>. -And Sir T. Cook -remarks that this “theory, if correct, would be fundamental for -all forms of growth, though it would be more easily observed in -plant construction than in animals.” The parallel I am not able -to follow.</p> - -<p>Mr Church sees in phyllotaxis an organic mystery, a something -for which we are unable to suggest any precise cause: a phenomenon -which is to be referred, somehow, to waves of growth emanating -from a centre, but on the other hand not to be explained by the -division of an apical cell, or any other histological factor. As -Sir T. Cook puts it, “at the growing point of a plant where the -new members are being formed, there is simply <i>nothing to see</i>.”</p> - -<p>But it is impossible to deal satisfactorily, in brief space, either -with Mr Church’s theories, or my own objections to them<a class="afnanch" href="#fn577" id="fnanch577">577</a>. -Let -it suffice to say that I, for my part, see no subtle mystery in the -matter, other than what lies in the steady production of similar -growing parts, similarly situated, at similar successive intervals -of time. If such be the case, then we are -bound to have in <span class="xxpn" id="p641">{641}</span> -consequence a series of symmetrical patterns, whose nature will -depend upon the form of the entire surface. If the surface be -that of a cylinder we shall have a system, or systems, of spiral -helices: if it be a plane, with an infinitely distant focus, such as -we obtain by “unwrapping” our cylindrical surface, we shall -have straight lines; if it be a plane containing the focus within -itself, or if it be any other symmetrical surface containing the -focus, then we shall have a system of logarithmic spirals. The -appearance of these spirals is sometimes spoken of as a “subjective” -phenomenon, but the description is inaccurate: it is a purely -mathematical phenomenon, an inseparable secondary result of -other arrangements which we, for the time being, regard as primary. -When the bricklayer builds a factory chimney, he lays his bricks -in a certain steady, orderly way, with no thought of the spiral -patterns to which this orderly sequence inevitably leads, and which -spiral patterns are by no means “subjective.” The designer of -a wall-paper not only has no intention of producing a pattern -of criss-cross lines, but on the contrary he does his best to avoid -them; nevertheless, so long as his design is a symmetrical one, -the criss-cross intersections inevitably come.</p> - -<p>Let us, however, leave this discussion, and return to the facts -of the case.</p> - -<hr class="hrblk"> - -<p>Our second question, which relates to the numerical coincidences -so familiar to all students of phyllotaxis, is not to be set and -answered in a word.</p> - -<p>Let us, for simplicity’s sake, avoid consideration of simultaneous -or whorled leaf origins, and consider only the more frequent -cases where a single “genetic spiral” can be traced throughout -the entire system.</p> - -<p>It is seldom that this primary, genetic spiral catches the eye, -for the leaves which immediately succeed one another in this -genetic order are usually far apart on the circumference of the -stem, and it is only in close-packed arrangements that the eye -readily apprehends the continuous series. Accordingly in such -a case as a fir-cone, for instance, it is certain of the secondary -spirals or “parastichies” which catch the eye; and among -fir-cones, we can easily count these, and we find -them to be <span class="xxpn" id="p642">{642}</span> -on the whole very constant in number, according to the -species.</p> - -<p>Thus in many cones, such as those of the Norway spruce, we -can trace five rows of scales winding steeply up the cone in one -direction, and three rows winding less steeply the other way; in -certain other species, such as the common larch, the normal -number is eight rows in the one direction and five in the other; -while in the American larch we have again three in the one direction -and five in the other. It not seldom happens that two arrangements -grade into one another on different parts of one and the -same cone. Among other cases in which such spiral series are -readily visible we have, for instance, the crowded leaves of the -stone-crops and mesembryanthemums, and (as we have said) the -crowded florets of the composites. Among these we may find -plenty of examples in which the numbers of the serial rows are -similar to those of the fir-cones; but in some cases, as in the daisy -and others of the smaller composites, we shall be able to trace -thirteen rows in one direction and twenty-one in the other, or -perhaps twenty-one and thirty-four; while in a great big sunflower -we may find (in one and the same species) thirty-four and fifty-five, -fifty-five and eighty-nine, or even as many as eighty-nine and -one hundred and forty-four. On the other hand, in an ordinary -“pentamerous” flower, such as a ranunculus, we may be able to -trace, in the arrangement of its sepals, petals and stamens, shorter -spiral series, three in one direction and two in the other. It will -be at once observed that these arrangements manifest themselves -in connection with very different things, in the orderly interspacing -of single leaves and of entire florets, and among all kinds of leaf-like -structures, foliage-leaves, bracts, cone-scales, and the various -parts or members of the flower. Again we must be careful to -note that, while the above numerical characters are by much the -most common, so much so as to be deemed “normal,” many -other combinations are known to occur.</p> - -<p>The arrangement, as we have seen, is apt to vary when the -entire structure varies greatly in size, as in the disc of the sunflower. -It is also subject to less regular variation within one and -the same species, as can always be discovered when we examine -a sufficiently large sample of fir-cones. For instance, -out of 505 <span class="xxpn" id="p643">{643}</span> -cones of the Norway spruce, Beal<a class="afnanch" href="#fn578" id="fnanch578">578</a> -found 92 per cent. in which -the spirals were in five and eight rows; in 6 per cent. the rows -were four and seven, and in 4 per cent. they were four and six. -In each case they were nearly equally divided as regards direction; -for instance of the 467 cones shewing the five-eight arrangement, -the five-series ran in right-handed spirals in 224 cases, and in -left-handed spirals in 243.</p> - -<p>Omitting the “abnormal” cases, such as we have seen to occur -in a small percentage of our cones of the spruce, the arrangements -which we have just mentioned may be set forth as follows, (the -fractional number used being simply an abbreviated symbol for -the number of associated helices or parastichies which we can -count running in the opposite directions): -2 ⁄ 3, - 3 ⁄ 5, - 5 ⁄ 8, - 8 ⁄ 13, - 13 ⁄ 21, - 21 ⁄ 34, - 34 ⁄ 55, - 55 ⁄ 89, - 89 ⁄ 144. Now these numbers form a -very interesting series, which happens to have a number of curious -mathematical properties<a class="afnanch" href="#fn579" id="fnanch579">579</a>. -We see, for instance, that the denominator -of each fraction is the numerator of the next; and further, -that each successive numerator, or denominator, is the sum of -the preceding two. Our immediate problem, then, is to determine, -if possible, how these numerical coincidences come about, and -why these particular numbers should be so -commonly met with <span class="xxpn" id="p644">{644}</span> -as to be considered “normal” and characteristic features of the -general phenomenon of phyllotaxis. The following account is -based on a short paper by Professor P. G. Tait<a class="afnanch" href="#fn580" id="fnanch580">580</a>.</p> - -<div class="dleft dwth-d" id="fig325"> -<img src="images/i644.png" width="384" height="255" alt=""> - <div class="dcaption">Fig. 325.</div></div> - -<p>Of the two following diagrams, Fig. <a href="#fig325" title="go to Fig. 325">325</a> represents the -general case, and Fig. <a href="#fig326" title="go to Fig. 326">326</a> a particular one, for the sake -of possibly greater simplicity. Both diagrams represent a -portion of a branch, or fir-cone, regarded as cylindrical, -and unwrapped to form a plane surface. <i>A</i>, <i>a</i>, at the two -ends of the base-line, represent the same initial leaf or -scale: <i>O</i> is a leaf which can be reached from <i>A</i> by <i>m</i> -steps in a right-hand spiral (developed into the straight -line <i>AO</i>), and by <i>n</i> steps from <i>a</i> in a left-handed -spiral <i>aO</i>. Now it is obvious in our fir-cone, that we can -include <i>all</i> the scales upon the cone by taking so many -spirals in the one direction, and again include them all -by so many in the other. Accordingly, in our diagrammatic -construction, the spirals <i>AO</i> and <i>aO</i> <i>must</i>, and always -<i>can</i>, be so taken that <i>m</i> spirals parallel to <i>aO</i>, and -<i>n</i> spirals parallel to <i>AO</i>, shall separately include all -the leaves upon the stem or cone. <br class="brclrfix" -></p> - -<p>If <i>m</i> and <i>n</i> have a common factor, <i>l</i>, it can easily be shewn that -the arrangement is composite, and that there are <i>l</i> fundamental, -or genetic spirals, and <i>l</i> leaves (including <i>A</i>) which are situated -exactly on the line <i>Aa</i>. That is to say, we have here a <i>whorled</i> -arrangement, which we have agreed to leave unconsidered in -favour of the simpler case. We restrict ourselves, accordingly, -to the cases where there is but one genetic spiral, and when -<i>therefore</i> <i>m</i> and <i>n</i> are prime to one another.</p> - -<p>Our fundamental, or genetic, spiral, as we have seen, is that -which passes from <i>A</i> (or <i>a</i>) to the leaf which is situated nearest to -the base-line <i>Aa</i>. The fundamental spiral will thus be right-handed -(<i>A</i>, <i>P</i>, etc.) if <i>P</i>, which is nearer to <i>A</i> than to <i>a</i>, be this -leaf—left-handed if it be <i>p</i>. That is to say, we make it a convention -that we shall always, for our fundamental -spiral, run <span class="xxpn" id="p645">{645}</span> -round the system, from one leaf to the next, <i>by the shortest -way</i>.</p> - -<p>Now it is obvious, from the symmetry of the figure (as further -shewn in Fig. <a href="#fig326" title="go to Fig. 326">326</a>), that, besides the spirals running along <i>AO</i> and -<i>aO</i>, we have a series running <i>from the steps on</i> <i>aO</i> to the steps on -<i>AO</i>. In other words we can find a leaf (<i>S</i>) upon <i>AO</i>, which, like -the leaf <i>O</i>, is reached directly by a spiral series from <i>A</i> and from -<i>a</i>, such that <i>aS</i> includes <i>n</i> steps, and <i>AS</i> (being part of the old -spiral line <i>AO</i>) now includes -<span class="nowrap"><i>m−n</i></span></p> - -<div class="dctr03" id="fig326"> -<img src="images/i645.png" width="608" height="678" alt=""> - <div class="dcaption">Fig. 326.</div></div> - -<p class="pcontinue"> -steps. And, since <i>m</i> and <i>n</i> -are prime to one another (for otherwise the system would have -been a composite or whorled one), it is evident that we can -continue this process of convergence until we come down to a -<span class="nowrap">1, 1</span> -arrangement, that is to say to a leaf which is reached by a -single step, in opposite directions from <i>A</i> and from <i>a</i>, which leaf -is therefore the first leaf, next to <i>A</i>, of the fundamental or -generating spiral. <span class="xxpn" id="p646">{646}</span></p> - -<p>If our original lines along <i>AO</i> and <i>aO</i> contain, -for instance, 13 and 8 steps respectively (i.e. <i>m</i> -<span class="nowrap">= 13,</span> <i>n</i> <span -class="nowrap">= 8),</span> then our next series, -observable in the same cone, will be 8 and <span -class="nowrap">(13 − 8)</span> or 5; the next 5 -and <span class="nowrap">(8 − 5)</span> or 3; -the next <span class="nowrap">3, 2;</span> and the next <span -class="nowrap">2, 1;</span> leading to the ultimate condition -of <span class="nowrap">1, 1.</span> These are the very series -which we have found to be common, or normal; and so far as our -investigation has yet gone, it has proved to us that, if one -of these exists, it entails, <i>ipso facto</i>, the presence of the -rest.</p> - -<p>In following down our series, according to the above construction, -we have seen that at every step we have changed -direction, the longer and the shorter sides of our triangle changing -places every time. Let us stop for a moment, when we come to -the 1, 2 series, or <i>AT</i>, <i>aT</i> of Fig. <a href="#fig326" title="go to Fig. 326">326</a>. It is obvious that there is -nothing to prevent us making a new 1, 3 series if we please, by -continuing the generating spiral through three leaves, and connecting -the leaf so reached directly with our initial one. But in -the case represented in Fig. <a href="#fig326" title="go to Fig. 326">326</a>, it is obvious that these two -series (<i>A</i>, 1, 2, 3, etc., and <i>a</i>, 3, 6, etc.) will be running in the same -direction; i.e. they will both be right-handed, or both left-handed -spirals. The simple meaning of this is that the third leaf of the -generating spiral was distant from our initial leaf by <i>more than the -circumference</i> of the cylindrical stem; in other words, that there -were more than two, but <i>less than three</i> leaves in a single turn of -the fundamental spiral.</p> - -<p>Less than two there can obviously never be. When there are -exactly two, we have the simplest of all possible arrangements, -namely that in which the leaves are placed alternately on -opposite sides of the stem. When there are more than two, but -less than three, we have the elementary condition for the -production of the series which we have been considering, namely -<span class="nowrap">1, 2;</span> <span class="nowrap">2, -3;</span> <span class="nowrap">3, 5,</span> etc. To put -the latter part of this argument in more precise language, -let us say that: If, in our descending series, we come to -steps 1 and <i>t</i>, where <i>t</i> is determined by the condition -that 1 and <i>t</i> + 1 would give spirals both -right-handed, or both left-handed; it follows that there are -less than <i>t</i> + 1 leaves in a single turn of -the fundamental spiral. And, determined in this manner, it is -found in the great majority of cases, in fir-cones and a host -of other examples of phyllotaxis, that <i>t</i> = 2. In other -words, in the <span class="xxpn" id="p647">{647}</span> great -majority of cases, we have what corresponds to an arrangement -next in order of simplicity to the simplest case of all: next, -that is to say, to the arrangement which consists of opposite -and alternate leaves.</p> - -<p>“These simple considerations,” as Tait says, “explain -completely the so-called mysterious appearance of terms of the -recurring series 1, 2, 3, 5, 8, 13, etc.<a class="afnanch" -href="#fn581" id="fnanch581">581</a> The other natural series, -usually but misleadingly represented by convergents to an -infinitely extended continuous fraction, are easily explained, -as above, by taking <i>t</i> = 3, 4, 5, etc., etc.” Many -examples of these latter series have been given by Dickson<a -class="afnanch" href="#fn582" id="fnanch582">582</a> and other -writers.</p> - -<hr class="hrblk"> - -<p>We have now learned, among other elementary facts, that -wherever any one system of helical spirals is present, certain -others invariably and of necessity accompany it, and are definitely -related to it. In any diagram, such as Fig. <a href="#fig326" title="go to Fig. 326">326</a>, in which we -represent our leaf-arrangement by means of uniform and regularly -interspaced dots, we can draw one series of spirals after another, -and one as easily as another. But in our fir-cone, for instance, -one particular series, or rather two conjugate series, are always -conspicuous, while the others are sought and found with comparative -difficulty.</p> - -<p>The phenomenon is illustrated by Fig. <a href="#fig327" title="go to Fig. 327">327</a>, <i>a</i>–<i>d</i>. The ground-plan -of all these diagrams is identically the same. The generating -spiral in each case represents a divergence of 3 ⁄ 8, or 135° of -azimuth; and the points succeed one another at the same successional -distances parallel to the axis. The rectangular outlines, -which correspond to the exposed surface of the leaves or cone-scales, -are of equal area, and of equal number. Nevertheless -the appearances presented by these diagrams are very different; -for in one the eye catches a 5 ⁄ 8 arrangement, in another a 3 ⁄ 5; -and so on, down to an arrangement of 1 ⁄ 1. The mathematical -side of this very curious phenomenon I have not attempted to -investigate. But it is quite obvious that, -in a system within <span class="xxpn" id="p648">{648}</span> -which various spirals are implicitly contained, the conspicuousness -of one set or another does not depend upon angular divergence. -It depends on the</p> - -<div class="dctr04" id="fig327"> -<img src="images/i648.png" width="528" height="917" alt=""> - <div class="dcaption">Fig. 327.</div></div> - -<p class="pcontinue"> -relative proportions in length and breadth of -the leaves themselves; or, more strictly speaking, on the ratio of -the diagonals of the rhomboidal figure by which each leaf-area is -circumscribed. When, as in the fir-cone, the scales by mutual -compression conform to these rhomboidal outlines, their inclined -edges at once guide the eye in the direction of some one particular -spiral; and we shall not fail to notice that in such -cases the usual <span class="xxpn" id="p649">{649}</span> -result is to give us arrangements corresponding to the middle -diagrams in Fig. <a href="#fig327" title="go to Fig. 327">327</a>, which are the configurations in which the -quadrilateral outlines approach most nearly to a rectangular -form, and give us accordingly the least possible ratio (under the -given conditions) of sectional boundary-wall to surface area.</p> - -<p>The manner in which one system of spirals may be caused -to slide, so to speak, into another, has been ingeniously -demonstrated by Schwendener on a mechanical model, -consisting essentially of a framework which can be opened -or closed to correspond with one after another of the -above series of diagrams<a class="afnanch" href="#fn583" -id="fnanch583">583</a>.</p> - -<p>The determination of the precise angle of divergence -of two consecutive leaves of the generating spiral does -not enter into the above general investigation (though -Tait gives, in the same paper, a method by which it may -be easily determined); and the very fact that it does -not so enter shews it to be essentially unimportant. The -determination of so-called “orthostichies,” or precisely -vertical successions of leaves, is also unimportant. We -have no means, other than observation, of determining that -one leaf is vertically above another, and spiral series -such as we have been dealing with will appear, whether -such orthostichies exist, whether they be near or remote, -or whether the angle of divergence be such that no precise -vertical superposition ever occurs. And lastly, the fact -that the successional numbers, expressed as fractions, -1 ⁄ 2, 2 ⁄ 3, -3 ⁄ 5, represent a convergent series, -whose final term is equal to 0·61803..., the <i>sectio aurea</i> -or “golden mean” of unity, is seen to be a mathematical -coincidence, devoid of biological significance; it is -but a particular case of Lagrange’s theorem that the -roots of every numerical equation of the second degree -can be expressed by a periodic continued fraction. The -same number has a multitude of curious arithmetical -properties. It is the final term of all similar series to -that with which we have been dealing, such for instance -as 1 ⁄ 3, 3 ⁄ 4, -4 ⁄ 7, etc., or -1 ⁄ 4, 4 ⁄ 5, -5 ⁄ 9, etc. It is a number beloved -of the circle-squarer, and of all those who seek to -find, and then to penetrate, the secrets of the Great -Pyramid. It is deep-set in Pythagorean as well as in -Euclidean geometry. It enters (as the chord of an angle -of 36°), <span class="xxpn" id="p650">{650}</span> into -the thrice-isosceles triangle of which we have spoken on -p. <a href="#p511" title="go to pg. 511">511</a>; it is a number which becomes (by the addition of -unity) its own reciprocal; its properties never end. To -Kepler (as Naber tells us) it was a symbol of Creation, -or Generation. Its recent application to biology and -art-criticism by Sir Theodore Cook and others is not new. -Naber’s book, already quoted, is full of it. Zeising, -in 1854, found in it the key to all morphology, and the -same writer, later on<a class="afnanch" href="#fn584" -id="fnanch584">584</a>, declared it to dominate both -architecture and music. But indeed, to use Sir Thomas -Browne’s words (though it was of another number that -he spoke): “To enlarge this contemplation into all the -mysteries and secrets accommodable unto this number, were -inexcusable Pythagorisme.”</p> - -<p>If this number has any serious claim at all to enter into -the biological question of phyllotaxis, this must depend on the -fact, first emphasized by Chauncey Wright<a class="afnanch" -href="#fn585" id="fnanch585">585</a>, that, if the successive -leaves of the fundamental spiral be placed at the particular -azimuth which divides the circle in this “sectio aurea,” -then no two leaves will ever be superposed; and thus we are -said to have “the most thorough and rapid distribution of -the leaves round the stem, each new or higher leaf falling -over the angular space between the two older ones which are -nearest in direction, so as to divide it in the same ratio -(<i>K</i>), in which the first two or any two successive ones divide -the circumference. Now 5 ⁄ 8 and all -successive fractions differ inappreciably from <i>K</i>.” To this -view there are many simple objections. In the first place, -even 5 ⁄ 8, or ·625, is but a moderately -close approximation to the “golden mean”; in the second place -the arrangements by which a better approximation is got, such -as 8 ⁄ 13, 13 ⁄ 21, -and the very close approximations such as -34 ⁄ 55, 55 ⁄ 89, -89 ⁄ 144, etc., are comparatively -rare, while the much less close approximations of -3 ⁄ 5 or 2 ⁄ 3, -or even 1 ⁄ 2, are extremely common. -Again, the general type of argument such as that which -asserts that the plant is “aiming at” something which we may -call an “ideal angle” is one that cannot commend itself to -a plain student of physical science: nor is the hypothesis -rendered more acceptably when Sir T. Cook qualifies it by -telling us that “all that a plant can do <span class="xxpn" -id="p651">{651}</span> is to vary, to make blind shots at -constructions, or to ‘mutate’ as it is now termed; and the -most suitable of these constructions will in the long run be -isolated by the action of Natural Selection.” Finally, and this -is the most concrete objection of all, the supposed isolation -of the leaves, or their most complete “distribution to the -action of the surrounding atmosphere” is manifestly very little -affected by any conditions which are confined to the angle of -azimuth. If we could imagine a case in which all the leaves of -the stem, or all the scales of a fir-cone, were crushed down to -one and the same level, into a simple ring or whorl of leaves, -then indeed they would have their most equable distribution -under the condition of the “ideal angle,” that is to say of -the “golden mean.” But if it be (so to speak) Nature’s object -to set them further apart than they actually are, to give them -freer exposure to the air than they actually have, then it is -surely manifest that the simple way to do so is to elongate the -axis, and to set the leaves further apart, lengthways on the -stem. This has at once a far more potent effect than any nice -manipulation of the “angle of divergence.” For it is obvious that in -<span class="nowrap"><i>F</i>(ϕ · sin θ)</span> -we have a greater range of variation by altering θ than by -altering ϕ. We come then, without more ado, to the conclusion -that the “Fibonacci series,” and its supposed usefulness, and -the hypothesis of its introduction into plant-structure through -natural selection, are all matters which deserve no place in -the plain study of botanical phenomena. As Sachs shrewdly -recognised years ago, all such speculations as these hark back -to a school of mystical idealism.</p> - -<div class="chapter" id="p652"> -<h2 class="h2herein" title="XV. On the Shapes of Eggs, and of Certain Other - Hollow Structures.">CHAPTER XV - <span class="h2ttl">ON THE SHAPES OF EGGS, AND OF CERTAIN OTHER - HOLLOW STRUCTURES</span></h2></div> - -<p>The eggs of birds and all other hard-shelled eggs, such as those -of the tortoise and the crocodile, are simple solids of revolution; -but they differ greatly in form, according to the configuration of -the plane curve by the revolution of which the egg is, in a mathematical -sense, generated. Some few eggs, such as those of the -owl, the penguin, or the tortoise, are spherical or very nearly so; a -few more, such as the grebe’s, the cormorant’s or the pelican’s, are -approximately ellipsoidal, with symmetrical or nearly symmetrical -ends, and somewhat similar are the so-called “cylindrical” eggs -of the megapodes and the sand-grouse; the great majority, like -the hen’s egg, are ovoid, a little blunter at one end than the other; -and some, by an exaggeration of this lack of antero-posterior -symmetry, are blunt at one end but characteristically pointed at -the other, as is the case with the eggs of the guillemot and puffin, -the sandpiper, plover and curlew. It is an obvious but by no -means negligible fact that the egg, while often pointed, is never -flattened or discoidal; it is a prolate, but never an oblate, spheroid.</p> - -<p>The careful study and collection of birds’ eggs would seem to -have begun with the Count de Marsigli<a class="afnanch" href="#fn586" id="fnanch586">586</a>, -the same celebrated -naturalist who first studied the “flowers” of the coral, and who -wrote the <i>Histoire physique de la mer</i>; and the specific form, as -well as the colour and other attributes of the egg have been -again and again discussed, and not least by the many dilettanti -naturalists of the eighteenth century who soon followed in -Marsigli’s footsteps<a class="afnanch" href="#fn587" id="fnanch587">587</a>. -<span class="xxpn" id="p653">{653}</span></p> - -<p>We need do no more than mention Aristotle’s belief, doubtless -old in his time, that the more pointed egg produces the male -chicken, and the blunter egg the hen; though this theory survived -into modern times<a class="afnanch" href="#fn588" id="fnanch588">588</a> -and perhaps still lingers on. Several naturalists, -such as Günther (1772) and Bühle (1818), have taken the -trouble to disprove it by experiment. A more modern and more -generally accepted explanation has been that the form of the egg -is in direct relation to that of the bird which has to be hatched -within—a view that would seem to have been first set forth by -Naumann and Bühle, in their great treatise on eggs<a class="afnanch" href="#fn589" id="fnanch589">589</a>, -and adopted -by Des Murs<a class="afnanch" href="#fn590" id="fnanch590">590</a> -and many other well-known writers.</p> - -<p>In a treatise by de Lafresnaye<a class="afnanch" href="#fn591" id="fnanch591">591</a>, -an elaborate comparison is -made between the skeleton and the egg of the various birds, to -shew, for instance, how those birds with a deep-keeled sternum -laid rounded eggs, which alone could accommodate the form of the -young. According to this view, that “Nature had foreseen<a class="afnanch" href="#fn592" id="fnanch592">592</a>” -the form adapted to and necessary for the growing embryo, it -was easy to correlate the owl with its spherical egg, the diver -with its elliptical one, and in like manner the round egg of the -tortoise and the elongated one of the crocodile with the shape of -the creatures which had afterwards to be hatched therein. A few -writers, such as Thienemann<a class="afnanch" href="#fn593" id="fnanch593">593</a>, -looked at the same facts the other -way, and asserted that the form of the egg was determined by -that of the bird by which it was laid, and in whose body it had -been conformed.</p> - -<p>In more recent times, other theories, based upon the principles -of Natural Selection, have been current and very generally accepted, -to account for these diversities of form. The pointed, conical -egg of the guillemot is generally supposed -to be an adaptation, <span class="xxpn" id="p654">{654}</span> -advantageous to the species in the circumstances under which -the egg is laid; the pointed egg is less apt than a spherical one to -roll off the narrow ledge of rock on which this bird is said to lay -its solitary egg, and the more pointed the egg, so much the fitter -and likelier is it to survive. The fact that the plover or the -sandpiper, breeding in very different situations, lay eggs that are -also conical, elicits another explanation, to the effect that here -the conical form permits the many large eggs to be packed closely -under the mother bird<a class="afnanch" href="#fn594" id="fnanch594">594</a>. -Whatever truth there be in these apparent -adaptations to existing circumstances, it is only by a very hasty -logic that we can accept them as a <i>vera causa</i>, or adequate -explanation of the facts; and it is obvious that, in the bird’s egg, -we have an admirable case for the direct investigation of the -mechanical or physical significance of its form<a class="afnanch" href="#fn595" id="fnanch595">595</a>.</p> - -<p>Of all the many naturalists of the eighteenth and nineteenth -centuries who wrote on the subject of eggs, one alone (so far as -I am aware) ascribed the form of the egg to direct mechanical -causes. Günther<a class="afnanch" href="#fn596" id="fnanch596">596</a>, -in 1772, declared that the more or less rounded -or pointed form of the egg is a mechanical consequence of the -pressure of the oviduct at a time when the shell is yet unformed -or unsolidified; and that accordingly, to explain the round egg of -the owl or the kingfisher, we have only to admit that the oviduct -of these birds is somewhat larger than that of most others, or -less subject to violent contractions. This statement contains, in -essence, the whole story of the mechanical conformation of the egg.</p> - -<p>Let us consider, very briefly, the conditions to which the egg -is subject in its passage down the oviduct<a class="afnanch" href="#fn597" id="fnanch597">597</a>.</p> - -<ul> -<li><p>(1) The “egg,” as it enters the oviduct, consists of the yolk -only, enclosed in its vitelline membrane. As it passes down the -first portion of the oviduct, the white is -gradually superadded, <span class="xxpn" id="p655">{655}</span> -and becomes in turn surrounded by the “shell-membrane.” -About this latter the shell is secreted, rapidly and at a late period; -the egg having meanwhile passed on into a wider portion of the -oviducal tube, called (by loose analogy, as Owen says) the “uterus.” -Here the egg assumes its permanent form, here it becomes rigid, -and it is to this portion of the “oviduct” that our argument -principally refers.</p></li> - -<li><p>(2) Both the yolk and the entire egg tend to fill completely -their respective membranes, and, whether this be due to growth -or imbibition on the part of the contents or to contraction on the -part of the surrounding membranes, the resulting tendency is for -both yolk and egg to be, in the first instance, spherical, unless -otherwise distorted by external pressure.</p></li> - -<li><p>(3) The egg is subject to pressure within the oviduct, which -is an elastic, muscular tube, along the walls of which pass peristaltic -waves of contraction. These muscular contractions may -be described as the contraction of successive annuli of muscle, -giving annular (or radial) pressure to successive portions of the -egg; they drive the egg forward against the frictional resistance -of the tube, while tending at the same time to distort its form. -While nothing is known, so far as I am aware, of the muscular -physiology of the oviduct, it is well known in the case of the -intestine that the presence of an obstruction leads to the development -of violent contractions in its rear, which waves of contraction -die away, and are scarcely if at all propagated in advance of the -obstruction.</p></li> - -<li><p>(4) It is known by observation that a hen’s egg is always -laid blunt end foremost.</p></li> - -<li><p>(5) It can be shown, at least as a very common rule, that -those eggs which are most unsymmetrical, or most tapered off -posteriorly, are also eggs of a large size relatively to the parent -bird. The guillemot is a notable case in point, and so also are -the curlews, sandpipers, phaleropes and terns. We may accordingly -presume that the more pointed eggs are those that are large -relatively to the tube or oviduct through which they have to pass, -or, in other words, are those which are subject to the greatest -pressure while being forced along. So general is this relation -that we may go still further, and presume -with great plausibility <span class="xxpn" id="p656">{656}</span> -in the few exceptional cases (of which the apteryx is the most -conspicuous) where the egg is relatively large though not markedly -unsymmetrical, that in these cases the oviduct itself is in all -probability large (as Günther had suggested) in proportion to the -size of the bird. In the case of the common fowl we can trace a -direct relation between the size and shape of the egg, for the first -eggs laid by a young pullet are usually smaller, and at the same -time are much more nearly spherical than the later ones; and, -moreover, some breeds of fowls lay proportionately smaller eggs -than others, and on the whole the former eggs tend to be rounder -than the latter<a class="afnanch" href="#fn598" id="fnanch598">598</a>.</p></li> -</ul> - -<hr class="hrblk"> - -<p>We may now proceed to inquire more particularly how the form -of the egg is controlled by the pressures to which it is subjected.</p> - -<p>The egg, just prior to the formation of the shell, is, as we have -seen, a fluid body, tending to a spherical shape and <i>enclosed within -a membrane</i>.</p> - -<p>Our problem, then, is: Given a practically incompressible -fluid, contained in a deformable capsule, which is either (<i>a</i>) entirely -inextensible, or (<i>b</i>) slightly extensible, and which is placed in a -long elastic tube the walls of which are radially contractile, to -determine the shape under pressure.</p> - -<p>If the capsule be spherical, inextensible, and completely filled -with the fluid, absolutely no deformation can take place. The -few eggs that are actually or approximately spherical, such as -those of the tortoise or the owl, may thus be alternatively explained -as cases where little or no deforming pressure has been applied -prior to the solidification of the shell, or else as cases where the -capsule was so little capable of extension and so completely filled -as to preclude the possibility of deformation.</p> - -<p>If the capsule be not spherical, but be inextensible, then -deformation can take place under the external -radial compression, <span class="xxpn" id="p657">{657}</span> -only provided that the pressure tends to make the shape more -nearly spherical, and then only on the further supposition that -the capsule is also not entirely filled as the deformation proceeds. -In other words, an incompressible fluid contained in an inextensible -envelope cannot be deformed without puckering of the -envelope taking place.</p> - -<p>Let us next assume, as the conditions by which this result -may be avoided, (<i>a</i>) that the envelope is to some extent extensible, -or (<i>b</i>) that the whole structure grows under relatively fixed -conditions. The two suppositions are practically identical with -one another in effect. It is obvious that, on the presumption -that the envelope is only moderately extensible, the whole structure -can only be distorted to a moderate degree away from the spherical -or spheroidal form.</p> - -<p>At all points the shape is determined by the law of the -distribution of <i>radial pressure within the given region of the tube</i>, -surface friction helping to maintain the egg in position. If -the egg be under pressure from the oviduct, but without any -marked component either in a forward or backward direction, -the egg will be compressed in the middle, and will tend more or -less to the form of a cylinder with spherical ends. The eggs of -the grebe, cormorant, or crocodile may be supposed to receive -their shape in such circumstances.</p> - -<p>When the egg is subject to the peristaltic contraction of the -oviduct during its formation, then from the nature and direction -of motion of the peristaltic wave the pressure will be greatest -somewhere behind the middle of the egg; in other words, the tube -is converted for the time being into a more conical form, and the -simple result follows that the anterior end of the egg becomes the -broader and the posterior end the narrower.</p> - -<p>With a given shape and size of body, equilibrium in the tube -may be maintained under greater radial pressure towards one end -than towards the other. For example, a cylinder having conical -ends, of semi-angles θ and θ′ respectively, remains in equilibrium, -apart from friction, if <i>p</i>cos<sup>2</sup> θ -= <i>p′</i>cos<sup>2</sup> θ′, so that at the more -tapered end where θ is small <i>p</i> is small. Therefore the whole -structure might assume such a configuration, or grow under such -conditions, finally becoming rigid by solidification -of the envelope. <span class="xxpn" id="p658">{658}</span> -According to the preceding paragraph, we must assume some -initial distribution of pressure, some squeeze applied to the -posterior part of the egg, in order to give it its tapering form. But, -that form once acquired, the egg may remain in equilibrium both -as regards form and position within the tube, even after that -excess of pressure on the posterior part is relieved. Moreover, -the above equation shews that a normal pressure no greater and -(within certain limits) actually less acting upon the posterior part -than on the anterior part of the egg after the shell is formed will -be sufficient to communicate to it a forward motion. This is an -important consideration, for it shews that the ordinary form of -an egg, and even the conical form of an extreme case such as the -guillemot’s, is directly favourable to the movement of the egg -within the oviduct, blunt end foremost.</p> - -<p>The mathematical statement of the whole case is as follows: -In our egg, consisting of an extensible membrane filled with an -incompressible fluid and under external pressure, the equation of -the envelope is <i>p<sub>n</sub></i> + <i>T</i>(1 ⁄ <i>r</i> + 1 ⁄ <i>r′</i>) -= <i>P</i>, where <i>p<sub>n</sub></i> is the normal -component of external pressure at a point where <i>r</i> and <i>r′</i> are the -radii of curvature, <i>T</i> is the tension of the envelope, and <i>P</i> the -internal fluid pressure. This is simply the equation of an elastic -surface where <i>T</i> represents the coefficient of elasticity; in other -words, a flexible elastic shell has the same mathematical properties -as our fluid, membrane-covered egg. And this is the identical -equation which we have already had so frequent occasion to employ -in our discussion of the forms of cells; save only that in these -latter we had chiefly to study the tension <i>T</i> (i.e. the surface-tension -of the semi-fluid cell) and had little or nothing to do with the -factor of external pressure (<i>p<sub>n</sub></i>), which in the case of the egg becomes -of chief importance.</p> - -<p>The above equation is the <i>equation of equilibrium</i>, so that it -must be assumed either that the whole body is at rest or that its -motion while under pressure is not such as to affect the result. -Tangential forces, which have been neglected, could modify the -form by alteration of <i>T</i>. In our case we must, and may very -reasonably, assume that any movement of the egg down the -oviduct during the period when its form is being impressed upon -it is very slow, being possibly balanced by the -advance of the <span class="xxpn" id="p659">{659}</span> -peristaltic wave which causes the movement, as well as by -friction.</p> - -<p>The quantity <i>T</i> is the tension of the enclosing capsule—the -surrounding membrane. If <i>T</i> be constant or symmetrical about -the axis of the body, the body is symmetrical. But the abnormal -eggs that a hen sometimes lays, cylindrical, annulated, or quite -irregular, are due to local weakening of the membrane, in other -words, to asymmetry of <i>T</i>. Not only asymmetry of <i>T</i>, but also -asymmetry of <i>p<sub>n</sub></i>, will render the body subject to deformation, -and this factor, the unknown but regularly varying, largely -radial, pressure applied by successive annuli of the oviduct, is the -essential cause of the form, and variations of form, of the egg. -In fact, in so far as the postulates correspond near enough to -actualities, the above equation is the equation of <i>all eggs</i> in the -universe. At least this is so if we generalise it in the form -<i>p<sub>n</sub></i> + <i>T ⁄ r</i> + <i>T′ ⁄ r′</i> -= <i>P</i> in recognition of a possible difference between -the principal tensions.</p> - -<p>In the case of the spherical egg it is obvious that <i>p<sub>n</sub></i> is everywhere -equal. The simplest case is where <i>p<sub>n</sub></i> -= 0, in other words, -where the egg is so small as practically to escape deforming -pressure from the tube. But we may also conceive the tube to -be so thin-walled and extensible as to press with practically -equal force upon all parts of the contained sphere. If while our -egg be in process of conformation the envelope be free at any -part from external pressure (that is to say, if <i>p<sub>n</sub></i> -= 0), then it is -obvious that that part (if of circular section) will be a portion of -a sphere. This is not unlikely to be the case actually or approximately -at one or both poles of the egg, and is evidently the case -over a considerable portion of the anterior end of the plover’s -egg.</p> - -<p>In the case of the conical egg with spherical ends, as is more -or less the case in the plover’s and the guillemot’s, then at either -end of the egg <i>r</i> and <i>r′</i> are identical, and they are greater at the -blunt anterior end than at the other. If we may assume that <i>p<sub>n</sub></i> -vanishes at the poles of the egg, then it is plain that <i>T</i> varies in -the neighbourhood of these poles, and, further, that the tension -<i>T</i> is greatest at and near the small end of the egg. It is here, -in short, that the egg is most likely to be -irregularly distorted or <span class="xxpn" id="p660">{660}</span> -even to burst, and it is here that we most commonly find irregularities -of shape in abnormal eggs.</p> - -<p>If one portion of the envelope were to become practically stiff -before <i>p</i> ceases to vary, that would be tantamount to a sudden -variation of <i>T</i>, and would introduce asymmetry by the imposition -of a boundary condition in addition to the above equation.</p> - -<p>Within the egg lies the yolk, and the yolk is invariably spherical -or very nearly so, whatever be the form of the entire egg. The -reason is simple, and lies in the fact that the yolk is itself enclosed -in another membrane, between which and the outer membrane -lies a fluid the presence of which makes <i>p<sub>n</sub></i> for the inner membrane -practically constant. The smallness of friction is indicated by -the well-known fact that the “germinal spot” on the surface of -the yolk is always found uppermost, however we may place and -wherever we may open the egg; that is to say, the yolk easily -rotates within the egg, bringing its lighter pole uppermost. So, -owing to this lack of friction in the outer fluid, or white, whatever -shear is produced within the egg will not be easily transmitted -to the yolk, and, moreover, owing to the same fluidity, the yolk -will easily recover its normal sphericity after the egg-shell is -formed and the unequal pressure relieved.</p> - -<p>These, then, are the general principles involved in, and illustrated -by, the configuration of an egg; and they take us as far -as we can safely go without actual quantitative determinations, -in each particular case, of the forces concerned.</p> - -<hr class="hrblk"> - -<p>In certain cases among the invertebrates, we again find -instances of hard-shelled eggs which have obviously been -moulded by the oviduct, or so-called “ootype,” in which they -have lain: and not merely in such a way as to shew the effects -of peristaltic pressure upon a uniform elastic envelope, but so -as to impress upon the egg the more or less irregular form -of the cavity, within which it had been for a time contained -and compressed. After this fashion Dr -Looss<a class="afnanch" href="#fn599" id="fnanch599">599</a> -of Cairo has <span class="xxpn" id="p661">{661}</span> -explained the curious form of the egg in <i>Bilharzia</i> (<i>Schistosoma</i>) -<i>haematobium</i>, a formidable parasitic worm to which is due a disease -wide-spread in Africa and Arabia, and an especial scourge of the -Mecca pilgrims. The egg in this worm is provided at one end -with a little spine, which now and then is found to be placed not -terminally but laterally or ventrally, and which when so placed -has been looked upon as the mark of a supposed new species, -<i>S. Mansoni</i>. As Looss has now shewn, the little spine must be -explained as having been moulded within a little funnel-shaped -expansion of the uterus, just where it communicates with the -common duct leading from the ovary and yolk-gland; by the -accumulation of eggs in the ootype, the one last formed is crowded -into a sideways position, and then, where the side-wall of the egg -bulges in the funnel-shaped orifice of the duct, a little lateral -“spine” is formed. In another species, <i>S. japonicum</i>, the egg is -described as bulging into a so-called “calotte,” or bubble-like -convexity at the end opposite to the spine. This, I think, may, -with very little doubt, be ascribed to hardening of the egg-shell -having taken place just at the period when partial relief from -pressure was being experienced by the egg in the neighbourhood -of the dilated orifice of the oviduct.</p> - -<p>This case of Bilharzia is not, from our present point of view, a -very important one, but nevertheless it is interesting. It ascribes -to a mechanical cause a curious peculiarity of form; it shews, by -reference to this mechanical principle, that two conditions which -were very different to the systematic naturalist’s eye, were really -only two simple mechanical modifications of the same thing; -and it destroys the chief evidence for the existence of a supposed -new species of worm, a continued belief in which, among worms -of such great pathogenic importance, might lead to gravely -erroneous pathological deductions.</p> - -<div class="section"> -<h3><i>On the Form of Sea-urchins</i></h3></div> - -<p>As a corollary to the problem of the bird’s egg, we may consider -for a moment the forms assumed by the shells of the sea-urchins. -These latter are commonly divided into two classes, the Regular -and the Irregular Echinids. The regular -sea-urchins, save in <span class="xxpn" id="p662">{662}</span> -slight details which do not affect our problem, have a complete -radial symmetry. The axis of the animal’s body is vertical, -with mouth below and the intestinal outlet above; and around -this axis the shell is built as a symmetrical system. It follows -that in horizontal section the shell is everywhere circular, and we -shall have only to consider its form as seen in vertical section or -projection. The irregular urchins (very inaccurately so-called) -have the anal extremity of the body removed from its central, -dorsal situation; and it follows that they have now a single plane -of symmetry, about which the organism, shell and all, is bilaterally -symmetrical. We need not concern ourselves in detail with the -shapes of their shells, which may be very simply interpreted, by -the help of radial co-ordinates, as deformations of the circular or -“regular” type.</p> - -<p>The sea-urchin shell consists of a membrane, stiffened into -rigidity by calcareous deposits, which constitute a beautiful -skeleton of separate, neatly fitting “ossicles.” The rigidity of -the shell is more apparent than real, for the entire structure is, -in a sluggish way, plastic; inasmuch as each little ossicle is -capable of growth, and the entire shell grows by increments to -each and all of these multitudinous elements, whose individual -growth involves a certain amount of freedom to move relatively -to one another; in a few cases the ossicles are so little developed -that the whole shell appears soft and flexible. The viscera of the -animal occupy but a small part of the space within the shell, the -cavity being mainly filled by a large quantity of watery fluid, -whose density must be very near to that of the external sea-water.</p> - -<p>Apart from the fact that the sea-urchin continues to grow, it -is plain that we have here the same general conditions as in the -egg-shell, and that the form of the sea-urchin is subject to a similar -equilibrium of forces. But there is this important difference, that -an external muscular pressure (such as the oviduct administers -during the consolidation of egg-shell), is now lacking. In its -place we have the steady continuous influence of gravity, and -there is yet another force which in all probability we require to -take into consideration.</p> - -<p>While the sea-urchin is alive, an immense number of delicate -“tube-feet,” with suckers at their tips, pass -through minute pores <span class="xxpn" id="p663">{663}</span> -in the shell, and, like so many long cables, moor the animal to -the ground. They constitute a symmetrical system of forces, -with one resultant downwards, in the direction of gravity, and -another outwards in a radial direction; and if we look upon the -shell as originally spherical, both will tend to depress the sphere -into a flattened cake. We need not consider the radial component, -but may treat the case as that of a spherical shell symmetrically -depressed under the influence of gravity. This is precisely the -condition which we have to deal with in a drop of liquid lying on -a plate; the form of which is determined by its own uniform -surface-tension, plus gravity, acting against the uniform internal -hydrostatic pressure. Simple as this system is, the full mathematical -investigation of the form of a drop is not easy, and we -can scarcely hope that the systematic study of the Echinodermata -will ever be conducted by methods based on Laplace’s differential -equation<a class="afnanch" href="#fn600" id="fnanch600">600</a>; -but we have no difficulty in seeing that the various -forms represented in a series of sea-urchin shells are no other than -those which we may easily and perfectly imitate in drops.</p> - -<p>In the case of the drop of water (or of any other particular -liquid) the specific surface-tension is always constant, and the -pressure varies inversely as the radius of curvature; therefore -the smaller the drop the more nearly is it able to conserve the -spherical form, and the larger the drop the more does it become -flattened under gravity. We can represent the phenomenon by -using india-rubber balls filled with water, of different sizes; the -little ones will remain very nearly spherical, but the larger will -fall down “of their own weight,” into the form of more and more -flattened cakes; and we see the same thing when we let drops of -heavy oil (such as the orthotoluidene spoken of on p. <a href="#p219" title="go to pg. 219">219</a>), fall -through a tall column of water, the little ones remaining round, -and the big ones getting more and more flattened as they sink. -In the case of the sea-urchin, the same series of forms may be -assumed to occur, irrespective of size, through variations in <i>T</i>, -the specific tension, or “strength,” of the enveloping shell. -Accordingly we may study, entirely from this point of view, -such a series as the following (Fig. <a href="#fig328" title="go to Fig. 328">328</a>). In a very few cases, -such as the fossil Palaeechinus, we have an -approximately spherical <span class="xxpn" id="p664">{664}</span> -shell, that is to say a shell so strong that the influence of gravity -becomes negligible as a cause of deformation. The ordinary -species of Echinus begin to display a pronounced depression, and -this reaches its maximum in such soft-shelled flexible forms as -Phormosoma. On the general question I took the opportunity -of consulting Mr C. R. Darling, who is an acknowledged expert -in drops, and he at once agreed with me that such forms as are -represented in Fig. <a href="#fig328" title="go to Fig. 328">328</a> are no other than diagrammatic illustrations</p> - -<div class="dctr01" id="fig328"> -<img src="images/i664.png" width="800" height="556" alt=""> - <div class="pcaption">Fig. 328. Diagrammatic vertical - outlines of various Sea-urchins: A, Palaeechinus; B, - <i>Echinus acutus</i>; C, Cidaris; D, D′ Coelopleurus; E, E′ - Genicopatagus; F, <i>Phormosoma luculenter</i>; G, P. <i>tenuis</i>; - H, Asthenosoma; I, Urechinus.</div></div> - -<p class="pcontinue">of various kinds of drops, “most of which can easily be reproduced -in outline by the aid of liquids of approximately equal density to -water, although some of them are fugitive.” He found a difficulty -in the case of the outline which represents Asthenosoma, but the -reason for the anomaly is obvious; the flexible shell has flattened -down until it has come in contact with the hard skeleton of the -jaws, or “Aristotle’s lantern,” within, and the curvature of the -outline is accordingly disturbed. The elevated, conical shells -such as those of Urechinus and Coelopleurus evidently call for -some further explanation; for there is here some -cause at work <span class="xxpn" id="p665">{665}</span> -to elevate, rather than to depress the shell. Mr Darling tells me -that these forms “are nearly identical in shape with globules I -have frequently obtained, in which, on standing, bubbles of gas -rose to the summit and pressed the skin upwards, without being -able to escape.” The same condition may be at work in the -sea-urchin; but a similar tendency would also be manifested by -the presence in the upper part of the shell of any accumulation -of substance lighter than water, such as is actually present in the -masses of fatty, oily eggs.</p> - -<div class="section"> -<h3><i>On the Form and Branching of Blood-vessels</i></h3></div> - -<p>Passing to what may seem a very different subject, we may -investigate a number of interesting points in connection with the -form and structure of the blood-vessels, on the same principle -and by help of the same equations as those we have used, for -instance, in studying the egg-shell.</p> - -<div class="dmaths"> -<p>We know that the fluid pressure (<i>P</i>) within the vessel -is balanced by (1) the tension (<i>T</i>) of the wall, divided -by the radius of curvature, and (2) the external pressure -(<i>p<sub>n</sub></i>), normal to the wall: according to our -formula</p> - -<div><i>P</i> -= <i>p<sub>n</sub></i> + <i>T</i>(1 ⁄ <i>r</i> + 1 ⁄ <i>r′</i>). -</div> - -<p class="pcontinue">If we neglect the external pressure, that is to say any support -which may be given to the vessel by the surrounding tissues, and -if we deal only with a cylindrical vein or artery, this formula -becomes simplified to the form <i>P</i> -= <i>T ⁄ R</i>. That is to say, under -constant pressure, the tension varies as the radius. But the -tension, per unit area of the vessel, depends upon the thickness -of the wall, that is to say on the amount of membranous and -especially of muscular tissue of which it is composed.</p> -</div><!--dmaths--> - -<p>Therefore, so long as the pressure is constant, the thickness -of the wall should vary as the radius, or as the diameter, of the -blood-vessel. But it is not the case that the pressure is constant, -for it gradually falls off, by loss through friction, as we pass from -the large arteries to the small; and accordingly we find that while, -for a time, the cross-sections of the larger and smaller vessels are -symmetrical figures, with the wall-thickness proportional to the -size of the tube, this proportion is gradually lost, -and the walls <span class="xxpn" id="p666">{666}</span> -of the small arteries, and still more of the capillaries, become -exceedingly thin, and more so than in strict proportion to the -narrowing of the tube.</p> - -<hr class="hrblk"> - -<p>In the case of the heart we have, within each of its cavities, a -pressure which, at any given moment, is constant over the whole -wall-area, but the thickness of the wall varies very considerably. -For instance, in the left ventricle, the apex is by much the thinnest -portion, as it is also that with the greatest curvature. We may -assume, therefore (or at least suspect), that the formula, -<i>t</i>(1 ⁄ <i>r</i> + 1 ⁄ <i>r′</i>) -= <i>C</i>, holds good; that is to say, that the thickness (<i>t</i>) -of the wall varies inversely as the mean curvature. This may be -tested experimentally, by dilating a heart with alcohol under a -known pressure, and then measuring the thickness of the walls -in various parts after the whole organ has become hardened. -By this means it is found that, for each of the cavities, the law -holds good with great accuracy<a class="afnanch" href="#fn601" id="fnanch601">601</a>. -Moreover, if we begin by -dilating the right ventricle and then dilate the left in like manner, -until the whole heart is equally and symmetrically dilated, we -find (1) that we have had to use a pressure in the left ventricle -from six to seven times as great as in the right ventricle, and -(2) that the thickness of the walls is just in the same proportion<a class="afnanch" href="#fn602" id="fnanch602">602</a>.</p> - -<hr class="hrblk"> - -<p>A great many other problems of a mechanical or hydrodynamical -kind arise in connection with the blood-vessels<a class="afnanch" href="#fn603" id="fnanch603">603</a>, -and -while these are chiefly interesting to the physiologist they have -also their interest for the morphologist in so far as they bear upon -structure and form. As an example of -such mechanical problems <span class="xxpn" id="p667">{667}</span> -we may take the conditions which determine or help to determine -the manner of branching of an artery, or the angle at which its -branches are given off; for, as John Hunter said<a class="afnanch" href="#fn604" id="fnanch604">604</a>, -“To keep up a -circulation sufficient for the part, and no more, Nature has varied -the angle of the origin of the arteries accordingly.” The general -principle is that the form and arrangement of the blood-vessels is -such that the circulation proceeds with a minimum of effort, and -with a minimum of wall-surface, the latter condition leading to a -minimum of friction and being therefore included in the first. -What, then, should be the angle of branching, such that there -shall be the least possible loss of energy in the course of the -circulation? In order to solve this problem in any particular -case we should obviously require to know (1) how the loss of -energy depends upon the distance travelled, and (2) how the loss -of energy varies with the diameter of the vessel. The loss of -energy is evidently greater in a narrow tube than in a wide one, -and greater, obviously, in a long journey than a short. If the</p> - -<div class="dright dwth-f" id="fig329"> -<img src="images/i667.png" width="288" height="312" alt=""> - <div class="dcaption">Fig. 329.</div></div> - -<p class="pcontinue">large artery, <i>AB</i>, give off a comparatively -narrow branch leading to <i>P</i> (such as <i>CP</i>, -or <i>DP</i>), the route <i>ACP</i> is evidently -shorter than <i>ADP</i>, but on the other -hand, by the latter path, the blood has -tarried longer in the wide vessel <i>AB</i>, -and has had a shorter course in the -narrow branch. The relative advantage -of the two paths will depend on the loss -of energy in the portion <i>CD</i>, as compared -with that in the alternative portion -<i>CD′</i>, the latter being short and narrow, the former long and wide. -If we ask, then, which factor is the more important, length or -width, we may safely take it that the question is one of degree: -and that the factor of width will become much the more important -wherever the artery and its branch are markedly unequal in size. -In other words, it would seem that for small branches a large -angle of bifurcation, and for large branches a small one, is always -the better. Roux has laid down certain rules in regard to the -branching of arteries, which correspond -with the general <span class="xxpn" id="p668">{668}</span> -conclusions which we have just arrived at. The most important of -these are as follows: (1) If an artery bifurcate into two equal -branches, these branches come off at equal angles to the main -stem. (2) If one of the two branches be smaller than the other, -then the main branch, or continuation of the original artery, -makes with the latter a smaller angle than does the smaller or -“lateral” branch. And (3) all branches which are so small that -they scarcely seem to weaken or diminish the main stem come off -from it at a large angle, from about 70° to 90°. -<br class="brclrfix"></p> - -<div class="dleft dwth-e" id="fig330"> -<img src="images/i668.png" width="336" height="534" alt=""> - <div class="dcaption">Fig. 330.</div></div> - -<p>We may follow Hess in a further investigation of this -phenomenon. Let <i>AB</i> be an artery, from which a branch has -to be given off so as to reach <i>P</i>, and let <i>ACP</i>, <i>ADP</i>, -etc., be alternative courses which the branch may follow: -<i>CD</i>, <i>DE</i>, etc., in the diagram, being equal distances -(= <i>l</i>) along <i>AB</i>. Let us call the angles <i>PCD</i>, -<i>PCE</i>, <i>x</i><sub>1</sub> , <i>x</i><sub>2</sub> , -etc.: and the distances <i>CD′</i>, <i>DE′</i>, by which each -branch exceeds the next in length, we shall call -<i>l</i><sub>1</sub> , <i>l</i><sub>2</sub> , etc. Now -it is evident that, of the courses shewn, <i>ACP</i> is the -shortest which the blood can take, but it is also that by -which its transit through the narrow branch is the longest. -We may reduce its transit through the narrow branch more -and more, till we come to <i>CGP</i>, or rather to a point where -the branch comes off at right angles to the main stem; -but in so doing we very considerably increase the whole -distance travelled. We may take it that there will be -some intermediate point which will strike the balance of -advantage. <br class="brclrfix"></p> - -<p>Now it is easy to shew that if, in Fig. <a href="#fig330" title="go to Fig. 330">330</a>, the route <i>ADP</i> and -<i>AEP</i> (two contiguous routes) be equally favourable, then any -other route on either side of these, such as <i>ACP</i> or <i>AFP</i>, must -be less favourable than either. Let <i>ADP</i> and <i>AEP</i>, then, be -equally favourable; that is to say, let the loss of energy which -the blood suffers in its passage along these two -routes be equal. <span class="xxpn" id="p669">{669}</span> -Then, if we make the distance <i>DE</i> very small, the angles <i>x</i><sub>2</sub> and -<i>x</i><sub>3</sub> are nearly equal, and may be so treated. And again, if <i>DE</i> -be very small, then <i>DE′E</i> becomes a right angle, and <i>l</i><sub>2</sub> (or -<i>DE′</i>) -= <i>l</i> cos <i>x</i><sub>2</sub> .</p> - -<p>But if <i>L</i> be the loss of energy per unit distance in -the wide tube <i>AB</i>, and <i>L′</i> be the corresponding loss -of energy in the narrow tube <i>DP</i>, etc., then <i>lL</i> -<span class="nowrap"> -= <i>l</i><sub>2</sub> <i>L′</i>,</span> because, as we have assumed, the loss of -energy on the route <i>DP</i> is equal to that on the whole -route <i>DEP</i>. Therefore <i>lL</i> -= <i>lL′</i> cos <i>x</i><sub>2</sub> , and -cos <i>x</i><sub>2</sub> -= <i>L ⁄ L′</i>. That is to say, the most favourable -angle of branching will be such that the cosine of the -angle is equal to the ratio of the loss of energy which the -blood undergoes, per unit of length, in the main vessel, as -compared with that which it undergoes in the branch.</p> - -<p>While these statements are so far true, and while they -undoubtedly cover a great number of observed facts, yet it is -plain that, as in all such cases, we must regard them not as a -complete explanation, but as <i>factors</i> in a complicated phenomenon: -not forgetting that (as the most learned of all students of the -heart and arteries, Dr Thomas Young, said in his Croonian -lecture<a class="afnanch" href="#fn605" id="fnanch605">605</a>) -all such questions as these, and all matters connected -with the muscular and elastic powers of the blood-vessels, -“belong to the most refined departments of hydraulics.” Some -other explanation must be sought in order to account for a -phenomenon which particularly impressed John Hunter’s mind, -namely the gradually altering angle at which the successive intercostal -arteries are given off from the thoracic aorta: the special -interest of this case arising from the regularity and symmetry of -the series, for “there is not another set of arteries in the body -whose origins are so much the same, whose offices are so much -the same, whose distances from their origin to the place of use, -and whose uses [? sizes]<a class="afnanch" href="#fn606" id="fnanch606">606</a> -are so much -the same.”</p> - -<div class="chapter" id="p670"> -<h2 class="h2herein" title="XVI. On Form and Mechanical -Efficiency.">CHAPTER XVI - <span class="h2ttl">ON FORM AND - MECHANICAL EFFICIENCY</span></h2></div> - -<p>There is a certain large class of morphological problems of -which we have not yet spoken, and of which we shall be able to -say but little. Nevertheless they are so important, so full of -deep theoretical significance, and are so bound up with the general -question of form and of its determination as a result of growth, -that an essay on growth and form is bound to take account of -them, however imperfectly and briefly. The phenomena which -I have in mind are just those many cases where <i>adaptation</i>, in the -strictest sense, is obviously present, in the clearly demonstrable -form of mechanical fitness for the exercise of some particular -function or action which has become inseparable from the life -and well-being of the organism.</p> - -<p>When we discuss certain so-called “adaptations” to outward -circumstance, in the way of form, colour and so forth, we are often -apt to use illustrations convincing enough to certain minds but -unsatisfying to others—in other words, incapable of demonstration. -With regard to colouration, for instance, it is by colours -“cryptic,” “warning,” “signalling,” “mimetic,” and so -on<a class="afnanch" href="#fn607" id="fnanch607">607</a>, -that we prosaically expound, and slavishly profess to justify, the -vast Aristotelian synthesis that Nature makes all things with a -purpose and “does nothing in vain.” Only for a moment let us -glance at some few instances by which the modern teleologist -accounts for this or that manifestation of colour, and is led on -and on to beliefs and doctrines to which it becomes more and more -difficult to subscribe. <span class="xxpn" id="p671">{671}</span></p> - -<p>Some dangerous and malignant animals are said (in sober -earnest) to wear a perpetual war-paint, in order to “remind their -enemies that they had better leave them -alone<a class="afnanch" href="#fn608" id="fnanch608">608</a>.” -The wasp and -the hornet, in gallant black and gold, are terrible as an army -with banners; and the Gila Monster (the poison-lizard of the -Arizona desert) is splashed with scarlet—its dread and black -complexion stained with heraldry more dismal. But the wasp-like -livery of the noisy, idle hover-flies and drone-flies is but -stage armour, and in their tinsel suits the little counterfeit cowardly -knaves mimic the fighting crew.</p> - -<p>The jewelled splendour of the peacock and the humming-bird, -and the less effulgent glory of the lyre-bird and the Argus pheasant, -are ascribed to the unquestioned prevalence of vanity in the one -sex and wantonness in the other<a class="afnanch" href="#fn609" id="fnanch609">609</a>.</p> - -<p>The zebra is striped that it may graze unnoticed on the plain, -the tiger that it may lurk undiscovered in the jungle; the banded -Chaetodont and Pomacentrid fishes are further bedizened to the -hues of the coral-reefs in which they dwell<a class="afnanch" href="#fn610" id="fnanch610">610</a>. -The tawny lion is -yellow as the desert sand; but the leopard wears its dappled hide -to blend, as it crouches on the branch, with the sun-flecks peeping -through the leaves.</p> - -<p>The ptarmigan and the snowy owl, the arctic fox and the polar -bear, are white among the snows; but go he north or go he south, -the raven (like the jackdaw) is boldly and impudently black.</p> - -<p>The rabbit has his white scut, and sundry antelopes their -piebald flanks, that one timorous fugitive may hie after another, -spying the warning signal. The primeval -terrier or collie-dog <span class="xxpn" id="p672">{672}</span> -had brown spots over his eyes that he might seem awake when he -was sleeping<a class="afnanch" href="#fn611" id="fnanch611">611</a>: -so that an enemy might let the sleeping dog lie, -for the singular reason that he imagined him to be awake. And -a flock of flamingos, wearing on rosy breast and crimson wings -a garment of invisibility, fades away into the sky at dawn or -sunset like a cloud incarnadine<a class="afnanch" href="#fn612" id="fnanch612">612</a>.</p> - -<p>To buttress the theory of natural selection the same -instances of “adaptation” (and many more) are used, which in -an earlier but not distant age testified to the wisdom of the -Creator and revealed to simple piety the high purpose of God. -In the words of a certain learned theologian<a class="afnanch" -href="#fn613" id="fnanch613">613</a>, “The free use of -final causes to explain what seems obscure was temptingly -easy .... Hence the finalist was often the man -who made a liberal use of the <i>ignava ratio</i>, or lazy argument: -when you failed to explain a thing by the ordinary process of -causality, you could “explain” it by reference to some purpose -of nature or of its Creator. This method lent itself with -dangerous facility to the well-meant endeavours of the older -theologians to expound and emphasise the beneficence of the -divine purpose.” <i>Mutatis mutandis</i>, the passage carries its -plain message to the naturalist.</p> - -<p>The fate of such arguments or illustrations is always the -same. They attract and captivate for awhile; they go to the -building of a creed, which contemporary orthodoxy defends under -its severest penalties: but the time comes when they lose their -fascination, they somehow cease to satisfy and to convince, -their foundations are discovered to be insecure, and in the end -no man troubles to controvert them.</p> - -<p>But of a very different order from all such “adaptations” as -these, are those very perfect adaptations of form which, for -instance, fit a fish for swimming or a bird for -flight. Here we are <span class="xxpn" id="p673">{673}</span> -far above the region of mere hypothesis, for we have to deal with -questions of mechanical efficiency where statical and dynamical -considerations can be applied and established in detail. The -naval architect learns a great part of his lesson from the investigation -of the stream-lines of a fish; and the mathematical study -of the stream-lines of a bird, and of the principles underlying the -areas and curvatures of its wings and tail, has helped to lay the -very foundations of the modern science of aeronautics. When, -after attempting to comprehend the exquisite adaptation of the -swallow or the albatross to the navigation of the air, we try to -pass beyond the empirical study and contemplation of such -perfection of mechanical fitness, and to ask how such fitness came -to be, then indeed we may be excused if we stand wrapt in wonderment, -and if our minds be occupied and even satisfied with the -conception of a final cause. And yet all the while, with no loss -of wonderment nor lack of reverence, do we find ourselves constrained -to believe that somehow or other, in dynamical principles -and natural law, there lie hidden the steps and stages of physical -causation by which the material structure was so shapen to its -ends<a class="afnanch" href="#fn614" id="fnanch614">614</a>.</p> - -<p>But the problems associated with these phenomena are -difficult at every stage, even long before we approach to the -unsolved secrets of causation; and for my part I readily confess -that I lack the requisite knowledge for even an elementary -discussion of the form of a fish or of a bird. But in the form of -a bone we have a problem of the same kind and order, so far -simplified and particularised that we may to some extent deal -with it, and may possibly even find, in our partial comprehension -of it, a partial clue to the principles of causation underlying this -whole class of problems.</p> - -<hr class="hrblk"> - -<p>Before we speak of the form of a bone, let us say a word about, -the mechanical properties of the material of which -it is built<a class="afnanch" href="#fn615" id="fnanch615">615</a>, -in <span class="xxpn" id="p674">{674}</span> -relation to the strength it has to manifest or the forces it has to -resist: understanding always that we mean thereby the properties -of fresh or living bone, with all its organic as well as inorganic -constituents, for dead, dry bone is a very different thing. In all -the structures raised by the engineer, in beams, pillars and girders -of every kind, provision has to be made, somehow or other, for -strength of two kinds, strength to resist compression or crushing, -and strength to resist tension or pulling asunder. The evenly -loaded column is designed with a view to supporting a downward -pressure, the wire-rope, like the tendon of a muscle, is adapted -only to resist a tensile stress; but in many or most cases the two -functions are very closely inter-related and combined. The case -of a loaded beam is a familiar one; though, by the way, we are -now told that it is by no means so simple as it looks, and indeed -that “the stresses and strains in this log of timber are so complex -that the problem has not yet been solved in a manner that reasonably -accords with the known strength of the beam as found by -actual experiment<a class="afnanch" href="#fn616" id="fnanch616">616</a>.” -However, be that as it may, we know,</p> - -<div class="dleft dwth-e" id="fig331"> -<img src="images/i674.png" width="336" height="216" alt=""> - <div class="dcaption">Fig. 331.</div></div> - -<p class="pcontinue">roughly, that when the beam is -loaded in the middle and supported at both ends, it -tends to be bent into an arc, in which condition its -lower fibres are being stretched, or are undergoing a -tensile stress, while its upper fibres are undergoing -compression. It follows that in some intermediate layer -there is a “neutral zone,” where the fibres of the wood -are subject to no stress of either kind. In like manner, -a vertical pillar if unevenly loaded (as, for instance, -the shaft of our thigh-bone normally is) will tend to -bend, and so to endure compression on its concave, and -tensile stress upon its convex side. In many cases it -is the business of the engineer to separate out, as far -as possible, the pressure-lines from the tension-lines, -in order to use separate modes of construction, or even -different materials for each. In a <span class="xxpn" -id="p675">{675}</span> suspension-bridge, for instance, -a great part of the fabric is subject to tensile strain -only, and is built throughout of ropes or wires; but -the massive piers at either end of the bridge carry the -weight of the whole structure and of its load, and endure -all the “compression-strains” which are inherent in the -system. Very much the same is the case in that wonderful -arrangement of struts and ties which constitute, or -complete, the skeleton of an animal. The “skeleton,” as we -see it in a Museum, is a poor and even a misleading picture -of mechanical efficiency<a class="afnanch" href="#fn617" -id="fnanch617">617</a>. From the engineer’s point of view, -it is a diagram showing all the compression-lines, but by -no means all of the tension-lines of the construction; it -shews all the struts, but few of the ties, and perhaps we -might even say <i>none</i> of the principal ones; it falls all -to pieces unless we clamp it together, as best we can, in -a more or less clumsy and immobilised way. But in life, -that fabric of struts is surrounded and interwoven with a -complicated system of ties: ligament and membrane, muscle -and tendon, run between bone and bone; and the beauty and -strength of the mechanical construction lie not in one part -or in another, but in the complete fabric which all the -parts, soft and hard, rigid and flexible, tension-bearing -and pressure-bearing, make up together<a class="afnanch" -href="#fn618" id="fnanch618">618</a>. <br class="brclrfix" -></p> - -<p>However much we may find a tendency, whether in nature or -art, to separate these two constituent factors of tension and -compression, we cannot do so completely; and accordingly the -engineer seeks for a material which shall, as nearly as possible, -offer equal resistance to both kinds of strain. In the following -table—I borrow it from Sir Donald MacAlister—we see approximately -the relative breaking (or tearing) limit and crushing limit -in a few substances. <span class="xxpn" id="p676">{676}</span></p> - -<div class="dtblbox"><div class="nowrap"> -<table class="fsz6"> -<caption><i>Average Strength of Materials (in kg. per -sq. mm.).</i></caption> -<tr> - <th></th> - <th>Tensile<br>strength</th> - <th>Crushing<br>strength</th></tr> -<tr> - <td class="tdleft">Steel</td> - <td class="tdleft">100</td> - <td class="tdleft">145</td></tr> -<tr> - <td class="tdleft">Wrought Iron</td> - <td class="tdleft"> 40</td> - <td class="tdleft"> 20</td></tr> -<tr> - <td class="tdleft">Cast Iron</td> - <td class="tdleft"> 12</td> - <td class="tdleft"> 72</td></tr> -<tr> - <td class="tdleft">Wood</td> - <td class="tdleft">  4</td> - <td class="tdleft">  2</td></tr> -<tr> - <td class="tdleft">Bone</td> - <td class="tdleft">  9–12</td> - <td class="tdleft"> 13–16</td></tr> -</table></div></div><!--dtblbox--> - -<p>At first sight, bone seems weak indeed; but it has the great -and unusual advantage that it is very nearly as good for a tie -as for a strut, nearly as strong to withstand rupture, or tearing -asunder, as to resist crushing. We see that wrought-iron is only -half as strong to withstand the former as the latter; while in -cast-iron there is a still greater discrepancy the other way, for it -makes a good strut but a very bad tie indeed. Cast-steel is not -only actually stronger than any of these, but it also possesses, -like bone, the two kinds of strength in no very great relative -disproportion.</p> - -<p>When the engineer constructs an iron or steel girder, to -take the place of the primitive wooden beam, we know that he -takes advantage of the elementary principle we have spoken of, -and saves weight and economises material by leaving out as -far as possible all the middle portion, all the parts in the -neighbourhood of the “neutral zone”; and in so doing he reduces -his girder to an upper and lower “flange,” connected together -by a “web,” the whole resembling, in cross-section, an <span class="nowrap"><img class="iglyph-a" -src="images/i676-glyph-i.png" width="25" height="60" alt="I"></span> -or an <span class="nowrap"><img class="iglyph-a" -src="images/i676-glyph-ibeam.png" width="45" height="60" alt="⌶">.</span></p> - -<p>But it is obvious that, if the strains in the two -flanges are to be equal as well as opposite, and if the -material be such as cast-iron or wrought-iron, one or -other flange must be made much thicker than the other -in order that it may be equally strong; and if at times -the two flanges have, as it were, to change places, or -play each other’s parts, then there must be introduced a -margin of safety by making both flanges thick enough to -meet that kind of stress in regard to which the material -happens to be weakest. There is great economy, then, in any -material which is, as nearly as possible, equally strong -in both ways; and so we see that, from the engineer’s -or contractor’s point of view, bone is a very good and -suitable material for purposes of construction. <span -class="xxpn" id="p677">{677}</span></p> - -<p>The <span class="nowrap"><img class="iglyph-a" -src="images/i676-glyph-i.png" width="25" height="60" alt="I"></span> -or the <em class="embold">H</em>-girder or rail is designed to resist bending in one -particular direction, but if, as in a tall pillar, it be necessary to -resist bending in all directions alike, it is obvious that the tubular -or cylindrical construction best meets the case; for it is plain -that this hollow tubular pillar is but the -<span class="nowrap"><img class="iglyph-a" -src="images/i676-glyph-i.png" width="25" height="60" -alt="I">-girder</span> turned round -every way, in a “solid of revolution,” so that on any two opposite -sides compression and tension are equally met and resisted, and -there is now no need for any substance at all in the way of web -or “filling” within the hollow core of the tube. And it is not only -in the supporting pillar that such a construction is useful; it is -appropriate in every case where <i>stiffness</i> is required, where bending -has to be resisted. The long bone of a bird’s wing has little or -no weight to carry, but it has to withstand powerful bending -moments; and in the arm-bone of a long-winged bird, such as -an albatross, we see the tubular construction manifested in its -perfection, the bony substance being reduced to a thin, perfectly -cylindrical, and almost empty shell. The quill of the bird’s -feather, the hollow shaft of a reed, the thin tube of the wheat-straw -bearing its heavy burden in the ear, are all illustrations -which Galileo used in his account of this mechanical principle<a class="afnanch" href="#fn619" id="fnanch619">619</a>.</p> - -<p>Two points, both of considerable importance, present themselves -here, and we may deal with them before we go further. In the -first place, it is not difficult to see that, in our bending beam, the -strain is greatest at its middle; if we press our walking-stick hard -against the ground, it will tend to snap midway. Hence, if our -cylindrical column be exposed to strong bending stresses, it will -be prudent and economical to make its walls thickest in the middle -and thinning off gradually towards the ends; and if we look at -a longitudinal section of a thigh-bone, we shall see that this is -just what nature has done. The thickness of the walls is nothing -less than a diagram, or “graph,” of the “bending-moments” -from one point to another along the length of the bone.</p> - -<div class="dleft dwth-f" id="fig332"> -<img src="images/i678.png" width="288" height="183" alt=""> - <div class="dcaption">Fig. 332.</div></div> - -<p>The second point requires a little more explanation. If -we <span class="xxpn" id="p678">{678}</span> imagine our -loaded beam to be supported at one end only (for instance, -by being built into a wall), so as to form what is called -a “bracket” or “cantilever,” then we can see, without -much difficulty, that the lines of stress in the beam -run somewhat as in the accompanying diagram. Immediately -under the load, the “compression-lines” tend to run -vertically downward; but where the bracket is fastened to -the wall, there is pressure directed horizontally against -the wall in the lower part of the surface of attachment; -and the vertical beginning and the horizontal end of -these pressure-lines must be continued into one another -in the form of some even mathematical curve—which, as it -happens, is part of a parabola. The tension-lines are -identical in form with the compression-lines, of which they -constitute the “mirror-image”; and where the two systems -intercross, they do so at right angles, or “orthogonally” -to one another. Such systems of stress-lines as these we -shall deal with again; but let us take note here of the -important, though well-nigh obvious fact, that while in -the beam they both unite to carry the load, yet it is -always possible to weaken one set of lines at the expense -of the other, and in some cases to do altogether away with -one set or the other. For example, when we replace our -end-supported beam by a curved bracket, bent upwards or -downwards as the case may be, we have evidently cut away -in the one case the greater part of the tension-lines, and -in the other the greater part of the compression-lines. -And if instead of bridging a stream with our beam of -wood we bridge it with a rope, it is evident that this -new construction contains all the tension-lines, but -none of the compression-lines of the old. The biological -interest connected with this principle lies chiefly in -the mechanical construction of the rush or the straw, or -any other typically cylindrical stem. The material of -which the stalk is constructed is very weak to withstand -compression, but parts of it have a very great tensile -strength. Schwendener, who was both botanist and engineer, -has elaborately investigated the factor of strength in -the cylindrical stem, which Galileo was the first to call -attention to. <span class="xxpn" id="p679">{679}</span> -Schwendener<a class="afnanch" href="#fn620" -id="fnanch620">620</a> shewed that the strength was -concentrated in the little bundles of “bast-tissue” but -that these bast-fibres had a tensile strength per square -mm. of section, up to the limit of elasticity, not less -than that of steel-wire of such quality as was in use in -his day. <br class="brclrfix"></p> - -<p>For instance, we see in the following table the load which -various fibres, and various wires, were found capable of sustaining, -not up to the breaking-point, but up to the “elastic limit,” or -point beyond which complete recovery to the original length took -place no longer after release of the load.</p> - -<div class="dtblbox"><div class="nowrap"> -<table class="fsz7"> -<tr> - <th></th> - <th> - Stress, or load in gms.<br> - per sq. mm., at<br> - Limit of Elasticity</th> - <th> - Strain, or amount<br> - of stretching,<br> - per mille</th></tr> -<tr> - <td class="tdleft"><i>Secale cereale</i></td> - <td class="tdcntr">15–20</td> - <td class="tdcntr"> 4·4 </td></tr> -<tr> - <td class="tdleft"><i>Lilium auratum</i></td> - <td class="tdcntr">19  </td> - <td class="tdcntr"> 7·6 </td></tr> -<tr> - <td class="tdleft"><i>Phormium tenax</i></td> - <td class="tdcntr">20  </td> - <td class="tdcntr">13·0 </td></tr> -<tr> - <td class="tdleft"><i>Papyrus antiquorum</i></td> - <td class="tdcntr">20  </td> - <td class="tdcntr">15·2 </td></tr> -<tr> - <td class="tdleft"><i>Molinia coerulea</i></td> - <td class="tdcntr">22  </td> - <td class="tdcntr">11·0 </td></tr> -<tr> - <td class="tdleft"><i>Pincenectia recurvata</i></td> - <td class="tdcntr">25  </td> - <td class="tdcntr">14·5 </td></tr> -<tr> - <td class="tdleft">Copper wire</td> - <td class="tdcntr">12·1</td> - <td class="tdcntr"> 1·0 </td></tr> -<tr> - <td class="tdleft">Brass wire</td> - <td class="tdcntr">13·3</td> - <td class="tdcntr"> 1·35</td></tr> -<tr> - <td class="tdleft">Iron wire</td> - <td class="tdcntr">21·9</td> - <td class="tdcntr"> 1·0 </td></tr> -<tr> - <td class="tdleft">Steel wire</td> - <td class="tdcntr">24·6</td> - <td class="tdcntr"> 1·2 </td></tr> -</table></div></div><!--dtblbox--> - -<p>In other respects, it is true, the plant-fibres were inferior to -the wires; for the former broke asunder very soon after the -limit of elasticity was passed, while the iron-wire could stand, -before snapping, three times the load which was measured by its -limit of elasticity: in the language of a modern engineer, the -bast-fibres had a low “yield-point,” little above the elastic limit. -But nevertheless, within certain limits, plant-fibre and wire were -just as good and strong one as the other. And then Schwendener -proceeds to shew, in many beautiful diagrams, the various ways -in which these strands of strong tensile tissue are arranged in -various cases: sometimes, in the simpler cases, forming numerous -small bundles arranged in a peripheral ring, not quite at the -periphery, for a certain amount of space has to be left for living -and active tissue; sometimes in a sparser ring -of larger and <span class="xxpn" id="p680">{680}</span> -stronger bundles; sometimes with these bundles further strengthened -by radial balks or ridges; sometimes with all the fibres set</p> - -<div class="dleft dwth-d" id="fig333"> -<img src="images/i680.png" width="336" height="306" alt=""> - <div class="dcaption">Fig. 333.</div></div> - -<p class="pcontinue">close together in a continuous hollow -cylinder. In the case figured in Fig. <a href="#fig333" title="go to Fig. 333">333</a> Schwendener -calculated that the resistance to bending was at least -twenty-five times as great as it would have been had the -six main bundles been brought close together in a solid -core. In many cases the centre of the stem is altogether -empty; in all other cases it is filled with soft tissue, -suitable for the ascent of sap or other functions, but -never such as to confer mechanical rigidity. In a tall -conical stem, such as that of a palm-tree, we can see -not only these principles in the construction of the -cylindrical trunk, but we can observe, towards the apex, -the bundles of fibre curving over and intercrossing -orthogonally with one another, exactly after the fashion of -our stress-lines in Fig. <a href="#fig332" title="go to Fig. 332">332</a>; but of course, in this case, -we are still dealing with tensile members, the opposite -bundles taking on in turn, as the tree sways, the alternate -function of resisting tensile strain<a class="afnanch" -href="#fn621" id="fnanch621">621</a>. <br class="brclrfix" -></p> - -<hr class="hrblk"> - -<p>Let us now come, at last, to the mechanical structure of bone, -of which we find a well-known and classical illustration in the -various bones of the human leg. In the case of the tibia, the bone -is somewhat widened out above, and its hollow shaft is capped -by an almost flattened roof, on which the weight of the body -directly rest. It is obvious that, under these circumstances, the -engineer would find it necessary to devise means for supporting -this flat roof, and for distributing the vertical pressures which -impinge upon it to the cylindrical walls of -the shaft. <span class="xxpn" id="p681">{681}</span></p> - -<p>In the case of the bird’s wing-bone, the hollow of the bone is -practically empty, as we have already said, being filled only with -air save for a thin layer of living tissue immediately within the -cylinder of bone; but in our own bones, and all weight-carrying -bones in general, the hollow space is filled with marrow, blood-vessels -and other tissues; and among these living tissues lies a -fine lattice-work of little interlaced “trabeculae” of bone, forming</p> - -<div class="dctr01" id="fig334"> -<img src="images/i681.png" width="608" height="486" alt=""> - <div class="dcaption">Fig. 334. Head of the human femur in -section. (After Schäfer, from a photo by Prof. A. -Robinson.)</div></div> - -<p class="pcontinue"> -the so-called “cancellous tissue.” The older anatomists were -content to describe this cancellous tissue as a sort of “spongy -network,” or irregular honeycomb, until, some fifty years ago, a -remarkable discovery was made regarding it. It was found by -Hermann Meyer (and afterwards shewn in greater detail by -Julius Wolff and others) that the trabeculae, as seen in a longitudinal -section of a long bone, were arranged in a very definite -and orderly way; in the femur, they spread -in beautiful curving <span class="xxpn" id="p682">{682}</span> -lines from the head to the tubular shaft of the bone, and these -bundles of lines were crossed by others, with so nice a regularity -of arrangement that each intercrossing was as nearly as -possible an orthogonal one: that is to say, the one set of fibres -crossed the other everywhere at right angles. A great engineer, -Professor Culmann of Zürich (to whom, by the way, we owe the -whole modern method of “graphic statics”), happened to see -some of Meyer’s drawings and preparations, and he recognised -in a moment that in the arrangement of the trabeculae we had</p> - -<div class="dctr01" id="fig335"> -<img src="images/i682.png" width="800" height="616" alt=""> - <div class="dcaption">Fig. 335. Crane-head and femur. (After -Culmann and H. Meyer.)</div></div> - -<p class="pcontinue">nothing more nor less than a diagram of the lines of stress, or -directions of compression and tension, in the loaded structure: -in short, that nature was strengthening the bone in precisely the -manner and direction in which strength was needed. In the -accompanying diagram of a crane-head, by Culmann, we recognise -a slight modification (caused entirely by the curved shape of the -structure) of the still simpler lines of tension and compression -which we have already seen in our end-supported beam as -represented in Fig. <a href="#fig332" title="go to Fig. 332">332</a>. In the shaft of the -crane, the concave <span class="xxpn" id="p683">{683}</span> -or inner side, overhung by the loaded head, is the “compression-member”; -the outer side is the “tension-member”; and the -pressure-lines, starting from the loaded surface, gather themselves -together, always in the direction of the resultant pressure, till -they form a close bundle running down the compressed side -of the shaft: while the tension-lines, running upwards along the -opposite side of the shaft, spread out through the head, orthogonally -to, and linking together, the system of compression-lines. -The head of the femur (Fig. <a href="#fig335" title="go to Fig. 335">335</a>) is a little more complicated in -form and a little less symmetrical than Culmann’s diagrammatic -crane, from which it chiefly differs in the fact that the load is -divided into two parts, that namely which is borne by the head -of the bone, and that smaller portion which rests upon the great -trochanter; but this merely amounts to saying that a <i>notch</i> has -been cut out of the curved upper surface of the structure, and we -have no difficulty in seeing that the anatomical arrangement of -the trabeculae follows precisely the mechanical distribution of -compressive and tensile stress or, in other words, accords perfectly -with the theoretical stress-diagram of the crane. The lines of -stress are bundled close together along the sides of the shaft, and -lost or concealed there in the substance of the solid wall of bone; -but in and near the head of the bone, a peripheral shell of bone -does not suffice to contain them, and they spread out through the -central mass in the actual concrete form of -bony trabeculae<a class="afnanch" href="#fn622" id="fnanch622">622</a>. -<span class="xxpn" id="p684">{684}</span></p> - -<p><i>Mutatis mutandis</i>, the same phenomenon may be traced in any -other bone which carries weight and is liable to flexure; and in -the <i>os calcis</i> and the tibia, and more or less in all the bones of the -lower limb, the arrangement is found to be very simple and -clear.</p> - -<div class="dctr03" id="fig336"> -<img src="images/i684.png" width="608" height="386" alt=""> - <div class="pcaption">Fig. 336. Diagram of stress-lines in the -human foot. (From Sir D. MacAlister, after H. Meyer.)</div></div> - -<p>Thus, in the <i>os calcis</i>, the weight resting on the head of the -bone has to be transmitted partly through the backward-projecting -heel to the ground, and partly forwards through its articulation -with the cuboid bone, to the arch of the foot. We thus have, -very much as in a triangular roof-tree, two compression-members, -sloping apart from one another; and these have to be bound -together by a “tie” or tension-member, corresponding to the -third, horizontal member of the truss.</p> - -<hr class="hrblk"> - -<p>So far, dealing wholly with the stresses and strains due to -tension and compression, we have altogether omitted to speak -of a third very important factor in the engineer’s calculations, -namely what is known as “shearing stress.” A shearing force is -one which produces “angular distortion” in a figure, or (what -comes to the same thing) which tends to -cause its particles to <span class="xxpn" id="p685">{685}</span> -slide over one another. A shearing stress is a somewhat complicated -thing, and we must try to illustrate it (however -imperfectly) in the simplest possible way. If we build up a pillar, -for instance, of a pile of flat horizontal slates, or of a pack of -cards, a vertical load placed upon it will produce compression, but -will have no tendency to cause one card to slide, or shear, upon -another; and in like manner, if we make up a cable of parallel -wires and, letting it hang vertically, load it evenly with a weight, -again the tensile stress produced has no tendency to cause one -wire to slip or shear upon another. But the case would have</p> - -<div class="dctr02" id="fig337"> -<img src="images/i685.png" width="704" height="457" alt=""> - <div class="dcaption">Fig. 337. Trabecular structure of the os -calcis. (From MacAlister.)</div></div> - -<p class="pcontinue"> -been very different if we had built up our pillar of cards or slates -lying obliquely to the lines of pressure, for then at once there -would have been a tendency for the elements of the pile to slip -and slide asunder, and to produce what the geologists call “a -fault” in the structure.</p> - -<div class="psmprnt3"> -<p>Somewhat more generally, if <i>AB</i> be a bar, or pillar, of cross-section <i>a</i> -under a direct load <i>P</i>, giving a stress per unit area -= <i>p</i>, then the whole -pressure <i>P</i> -= <i>pa</i>. Let <i>CD</i> be an oblique section, inclined at an angle θ to the -cross-section; the pressure on <i>CD</i> will evidently be -= <i>pa</i> cos θ. But at any -point <i>O</i> in <i>CD</i>, the pressure <i>P</i> may be resolved into the force <i>Q</i> acting along -<i>CD</i>, and <i>N</i> perpendicular to it: where <i>N</i> -= <i>P</i> cos θ, and -<i>Q</i> -= <i>P</i> sin θ -= <i>pa</i> sin θ. -The whole force <i>Q</i> upon <i>CD</i> -= <i>q</i> · area of <i>CD</i>, which -is -= <i>q</i> · <i>a</i> ⁄ (cos θ). -<span class="xxpn" id="p686">{686}</span> -Therefore <i>qa</i> ⁄ (cos θ) -= <i>pa</i> sin θ, therefore -<i>q</i> -= <i>p</i> sin θ cos θ, -= ½ <i>p</i> sin 2θ. -Therefore when sin 2θ -= 1, that is, when θ -= 45°, <i>q</i> is a maximum, and -= <i>p</i> ⁄ 2; and when sin 2θ -= 0, that is when θ -= 0° -or 90°, then <i>q</i> vanishes altogether.</p> -</div><!--psmprnt3--> - -<div class="dleft dwth-f" id="fig338"> -<img src="images/i686.png" width="288" height="378" alt=""> - <div class="dcaption">Fig. 338.</div></div> - -<p>This is as much as to say, that a -shearing stress vanishes altogether along -the lines of maximum compression or -tension; it has a definite value in all -other positions, and a maximum value -when it is inclined at 45° to either, or -half-way between the two. This may be -further illustrated in various simple ways. -When we submit a cubical block of iron -to compression in the testing machine, it -does not tend to give way by crumbling -all to pieces; but as a rule it disrupts by shearing, and along -some plane approximately at 45° to the axis of compression. -Again, in the beam which we have already considered under a -bending moment, we know that if we substitute for it a pack of -cards, they will be strongly sheared on one another; and the -shearing stress is greatest in the “neutral zone,” where neither -tension nor compression is manifested: that is to say in the line -which cuts at equal angles of 45° the orthogonally intersecting -lines of pressure and tension. <br class="brclrfix"></p> - -<p>In short we see that, while shearing <i>stresses</i> can by no means -be got rid of, the danger of rupture or breaking-down under -shearing stress is completely got rid of when we arrange the -materials of our construction wholly along the pressure-lines and -tension-lines of the system; for <i>along these lines</i> there is no shear.</p> - -<p>To apply these principles to the growth and development of -our bone, we have only to imagine a little trabecula (or group of -trabeculae) being secreted and laid down fortuitously in any -direction within the substance of the bone. If it lie in the -direction of one of the pressure-lines, for instance, it will be in -a position of comparative equilibrium, or minimal disturbance; -but if it be inclined obliquely to the pressure-lines, the shearing -force will at once tend to act upon it and move it away. This -is neither more nor less than what happens when -we comb our <span class="xxpn" id="p687">{687}</span> -hair, or card a lock of wool: filaments lying in the direction of -the comb’s path remain where they were; but the others, under -the influence of an oblique component of pressure, are sheared -out of their places till they too come into coincidence with the -lines of force. So straws show how the wind blows—or rather -how it has been blowing. For every straw that lies askew to the -wind’s path tends to be sheared into it; but as soon as it has -come to lie the way of the wind it tends to be disturbed no -more, save (of course) by a violence such as to hurl it bodily -away.</p> - -<p>In the biological aspect of the case, we must always remember -that our bone is not only a living, but a highly plastic -structure; the little trabeculae are constantly being formed and -deformed, demolished and formed anew. Here, for once, it is -safe to say that “heredity” need not and cannot be invoked to -account for the configuration and arrangement of the trabeculae: -for we can see them, at any time of life, in the making, under the -direct action and control of the forces to which the system is -exposed. If a bone be broken and so repaired that its parts lie -somewhat out of their former place, so that the pressure-and -tension-lines have now a new distribution, before many weeks are -over the trabecular system will be found to have been entirely -remodelled, so as to fall into line with the new system of forces. -And as Wolff pointed out, this process of reconstruction extends -a long way off from the seat of injury, and so cannot be looked -upon as a mere accident of the physiological process of healing -and repair; for instance, it may happen that, after a fracture of -the <i>shaft</i> of a long bone, the trabecular meshwork is wholly altered -and reconstructed within the distant <i>extremities</i> of the bone. -Moreover, in cases of transplantation of bone, for example when -a diseased metacarpal is repaired by means of a portion taken -from the lower end of the ulna, with astonishing quickness the -plastic capabilities of the bony tissue are so manifested that -neither in outward form nor inward structure can the old portion -be distinguished from the new.</p> - -<p>Herein then lies, so far as we can discern it, a great part at -least of the physical causation of what at first sight strikes us as -a purely functional adaptation: as a phenomenon, -in other words, <span class="xxpn" id="p688">{688}</span> -whose physical cause is as obscure as its final cause or end is, -apparently, manifest.</p> - -<hr class="hrblk"> - -<p>Partly associated with the same phenomenon, and partly to -be looked upon (meanwhile at least) as a fact apart, is the very -important physiological truth that a condition of <i>strain</i>, the -result of a <i>stress</i>, is a direct stimulus to growth itself. This indeed -is no less than one of the cardinal facts of theoretical biology. -The soles of our boots wear thin, but the soles of our feet grow -thick, the more we walk upon them: for it would seem that the -living cells are “stimulated” by pressure, or by what we call -“exercise,” to increase and multiply. The surgeon knows, when -he bandages a broken limb, that his bandage is doing something -more than merely keeping the parts together: and that the even, -constant pressure which he skilfully applies is a direct encouragement -of growth and an active agent in the process of repair. In the -classical experiments of Sédillot<a class="afnanch" href="#fn623" id="fnanch623">623</a>, -the greater part of the shaft of the -tibia was excised in some young puppies, leaving the whole weight -of the body to rest upon the fibula. The latter bone is normally -about one-fifth or sixth of the diameter of the tibia; but under -the new conditions, and under the “stimulus” of the increased -load, it grew till it was as thick or even thicker than the normal -bulk of the larger bone. Among plant tissues this phenomenon -is very apparent, and in a somewhat remarkable way; for a strain -caused by a constant or increasing weight (such as that in the -stalk of a pear while the pear is growing and ripening) produces -a very marked increase of <i>strength</i> without any necessary increase -of bulk, but rather by some histological, or molecular, alteration -of the tissues. Hegler, and also Pfeffer, have investigated this -subject, by loading the young shoot of a plant nearly to its -breaking point, and then redetermining the breaking-strength -after a few days. Some young shoots of the sunflower were found -to break with a strain of 160 gms.; but when loaded with 150 gms., -and retested after two days, they were able to support 250 gms.; -and being again loaded with something short of this, by next day -they sustained 300 gms., and a few days -later even 400 gms. <span class="xxpn" id="p689">{689}</span></p> - -<p>Such experiments have been amply confirmed, but so far as -I am aware, we do not know much more about the matter: we -do not know, for instance, how far the change is accompanied by -increase in number of the bast-fibres, through transformation of -other tissues; or how far it is due to increase in size of these -fibres; or whether it be not simply due to strengthening of the -original fibres by some molecular change. But I should be much -inclined to suspect that the latter had a good deal to do with the -phenomenon. We know nowadays that a railway axle, or any -other piece of steel, is weakened by a constant succession of -frequently interrupted strains; it is said to be “fatigued,” and -its strength is restored by a period of rest. The converse effect -of continued strain in a uniform direction may be illustrated by -a homely example. The confectioner takes a mass of boiled -sugar or treacle (in a particular molecular condition determined -by the temperature to which it has been exposed), and draws the -soft sticky mass out into a rope; and then, folding it up lengthways, -he repeats the process again and again. At first the rope is pulled -out of the ductile mass without difficulty; but as the work goes -on it gets harder to do, until all the man’s force is used to stretch -the rope. Here we have the phenomenon of increasing strength, -following mechanically on a rearrangement of molecules, as the -original isotropic condition is transmuted more and more into -molecular asymmetry or anisotropy; and the rope apparently -“adapts itself” to the increased strain which it is called on to bear, -all after a fashion which at least suggests a parallel to the increasing -strength of the stretched and weighted fibre in the plant. For -increase of strength by rearrangement of the particles we have -already a rough illustration in our lock of wool or hank of tow. -The piece of tow will carry but little weight while its fibres are -tangled and awry: but as soon as we have carded it out, and -brought all its long fibres parallel and side by side, we may at once -make of it a strong and useful cord.</p> - -<p>In some such ways as these, then, it would seem that we may -co-ordinate, or hope to co-ordinate, the phenomenon of growth -with certain of the beautiful structural phenomena which present -themselves to our eyes as “provisions,” or mechanical adaptations, -for the display of strength where strength -is most required. <span class="xxpn" id="p690">{690}</span> -That is to say, the origin, or causation, of the phenomenon would -seem to lie, partly in the tendency of growth to be accelerated -under strain: and partly in the automatic effect of shearing -strain, by which it tends to displace parts which grow obliquely -to the direct lines of tension and of pressure, while leaving those -in place which happen to lie parallel or perpendicular to those -lines: an automatic effect which we can probably trace as working -on all scales of magnitude, and as accounting therefore for the -rearrangement of minute particles in the metal or the fibre, as -well as for the bringing into line of the fibres themselves within -the plant, or of the little trabeculae within the bone.</p> - -<hr class="hrblk"> - -<p>But we may now attempt to pass from the study of the -individual bone to the much wider and not less beautiful problems -of mechanical construction which are presented to us by the -skeleton as a whole. Certain problems of this class are by no -means neglected by writers on anatomy, and many have been -handed down from Borelli, and even from older writers. For -instance, it is an old tradition of anatomical teaching to point -out in the human body examples of the three orders of levers<a class="afnanch" href="#fn624" id="fnanch624">624</a>; -again, the principle that the limb-bones tend to be shortened in -order to support the weight of a very heavy animal is well understood -by comparative anatomists, in accordance with Euler’s law, -that the weight which a column liable to flexure is capable of -supporting varies inversely as the square of its length; and again, -the statical equilibrium of the body, in relation for instance to -the erect posture of man, has long been a favourite theme of the -philosophical anatomist. But the general method, based upon -that of graphic statics, to which we have been introduced in our -study of a bone, has not, so far as I know, been applied to the -general fabric of the skeleton. Yet it is plain -that each bone plays <span class="xxpn" id="p691">{691}</span> -a part in relation to the whole body, analogous to that which a -little trabecula, or a little group of trabeculae, plays within the -bone itself: that is to say, in the normal distribution of forces -in the body, the bones tend to follow the lines of stress, and -especially the pressure-lines. To demonstrate this in a comprehensive -way would doubtless be difficult; for we should be dealing -with a framework of very great complexity, and should have to -take account of a great variety of conditions<a class="afnanch" href="#fn625" id="fnanch625">625</a>. -This framework -is complicated as we see it in the skeleton, where (as we have said) -it is only, or chiefly, the <i>struts</i> of the whole fabric which are -represented; but to understand the mechanical structure in -detail, we should have to follow out the still more complex -arrangement of the <i>ties</i>, as represented by the muscles and -ligaments, and we should also require much detailed information -as to the weights of the various parts and as to the other forces -concerned. Without these latter data we can only treat the -question in a preliminary and imperfect way. But, to take once -again a small and simplified part of a big problem, let us think -of a quadruped (for instance, a horse) in a standing posture, and -see whether the methods and terminology of the engineer may not -help us, as they did in regard to the minute structure of the single -bone.</p> - -<p>Standing four-square upon its forelegs and hindlegs, with the -weight of the body suspended between, the quadruped at once -suggests to us the analogy of a bridge, carried by its two piers. -And if it occurs to us, as naturalists, that we never look at a -standing quadruped without contemplating a bridge, so, conversely, -a similar idea has occurred to the engineer; for Professor -Fidler, in this <i>Treatise on Bridge-Construction</i>, deals with the chief -descriptive part of his subject under the heading of “The Comparative -Anatomy of Bridges.” The designation is most just, for -in studying the various types of bridges we are studying a series -of well-planned <i>skeletons</i><a class="afnanch" href="#fn626" id="fnanch626">626</a>; -and (at the cost of -a little pedantry) <span class="xxpn" id="p692">{692}</span> -we might go even further, and study (after the fashion of the -anatomist) the “osteology” and “desmology” of the structure, -that is to say the bones which are represented by “struts,” and -the ligaments, etc., which are represented by “ties.” Furthermore -after the methods of the comparative anatomist, we may -classify the families, genera and species of bridges according to -their distinctive mechanical features, which correspond to certain -definite conditions and functions.</p> - -<p>In more ways than one, the quadrupedal bridge is a remarkable -one; and perhaps its most remarkable peculiarity is that it is a -jointed and flexible bridge, remaining in equilibrium under -considerable and sometimes great modifications of its curvature, -such as we see, for instance, when a cat humps or flattens her -back. The fact that <i>flexibility</i> is an essential feature in the -quadrupedal bridge, while it is the last thing which an engineer -desires and the first which he seeks to provide against, will impose -certain important limiting conditions upon the design of the -skeletal fabric; but to this matter we shall afterwards return. -Let us begin by considering the quadruped at rest, when he stands -upright and motionless upon his feet, and when his legs exercise -no function save only to carry the weight of the whole body. So -far as that function is concerned, we might now perhaps compare -the horse’s legs with the tall and slender piers of some railway -bridge; but it is obvious that these jointed legs are ill-adapted -to receive the <i>horizontal thrust</i> of any <i>arch</i> that may be placed -atop of them. Hence it follows that the curved backbone of the -horse, which appears to cross like an arch the span between his -shoulders and his flanks, cannot be regarded -as an <i>arch</i>, in the <span class="xxpn" id="p693">{693}</span> -engineer’s sense of the word. It resembles an arch in <i>form</i>, but -not in <i>function</i>, for it cannot act as an arch unless it be held back -at each end (as every arch is held back) by <i>abutments</i> capable of -resisting the horizontal thrust; and these necessary abutments -are not present in the structure. But in various ways the -engineer can modify his superstructure so as to supply the place -of these <i>external</i> reactions, which in the simple arch are obviously -indispensable. Thus, for example, we may begin by inserting a -straight steel tie, <i>AB</i> (Fig. <a href="#fig339" title="go to Fig. 339">339</a>), uniting the ends of the curved rib -<i>AaB</i>; and this tie will supply the place of the external reactions, -converting the structure into a “tied arch,” such as we may see -in the roofs of many railway-stations. Or we may go on to fill -in the space between arch and tie by a “web-system,” converting -it into what the engineer describes as a “parabolic bowstring -girder” (Fig. <a href="#fig339" title="go to Fig. 339">339</a><i>b</i>). In either case, the structure becomes an</p> - -<div class="dctr02" id="fig339"> -<img src="images/i693.png" width="704" height="259" alt=""> - <div class="dcaption">Fig. 339.</div></div> - -<p class="pcontinue">independent “detached girder,” supported at each end but not -otherwise fixed, and consisting essentially of an upper compression-member, -<i>AaB</i>, and a lower tension-member, <i>AB</i>. But again, in -the skeleton of the quadruped, <i>the necessary tie</i>, <i>AB</i>, <i>is not to be -found</i>; and it follows that these comparatively simple types of -bridge do not correspond to, nor do they help us to understand, -the type of bridge which nature has designed in the skeleton of -the quadruped. Nevertheless if we try to look, as an engineer -would look, at the actual design of the animal skeleton and the -actual distribution of its load, we find that, the one is most admirably -adapted to the other, according to the strict principles of -engineering construction. The structure is not an arch, nor a -tied arch, nor a bowstring girder: but it is -strictly and beautifully <span class="xxpn" id="p694">{694}</span> -comparable to the main girder of a double-armed cantilever -bridge.</p> - -<p>Obviously, in our quadrupedal bridge, the superstructure does -not terminate (as it did in our former diagram) at the two points -of support, but it extends beyond them at each end, carrying the -head at one end and the tail at the other, upon a pair of projecting -arms or “cantilevers” (Fig. <a href="#fig346" title="go to Fig. 346">346</a>).</p> - -<p>In a typical cantilever bridge, such as the Forth Bridge -(Fig. <a href="#fig345" title="go to Fig. 345">345</a>), a certain simplification is introduced. For each pier -carries, in this case, its own double-armed cantilever, linked by -a short connecting girder to the next, but so jointed to it that no -weight is transmitted from one cantilever to another. The bridge -in short is <i>cut</i> into separate sections, practically independent of -one another; at the joints a certain amount of bending is not -precluded, but shearing strain is evaded; and each pier carries -only its own load. By this arrangement the engineer finds that -design and construction are alike simplified and facilitated. In -the case of the horse, it is obvious that the two piers of the bridge, -that is to say the fore-legs and the hind-legs, do not bear (as they -do in the Forth Bridge) separate and independent loads, but the -whole system forms a continuous structure. In this case, the -calculation of the loads will be a little more difficult and the -corresponding design of the structure a little more complicated. -We shall accordingly simplify our problem very considerably if, -to begin with, we look upon the quadrupedal skeleton as constituted -of two separate systems, that is to say of two balanced -cantilevers, one supported on the fore-legs and the other on the -hind; and we may deal afterwards with the fact that these two -cantilevers are not independent, but are bound up in one common -field of force and plan of construction.</p> - -<p>In the horse it is plain that the two cantilever systems into -which we may thus analyse the quadrupedal bridge are unequal -in magnitude and importance. The fore-part of the animal is -much bulkier than its hind quarters, and the fact that the fore-legs -carry, as they so evidently do, a greater weight than the hind-legs -has long been known and is easily proved; we have only to walk -a horse onto a weigh-bridge, weigh first his fore-legs and then his -hind-legs, to discover that what we may call his -front half weighs <span class="xxpn" id="p695">{695}</span> -a good deal more than what is carried on his hind feet, say about -three-fifths of the whole weight of the animal.</p> - -<p>The great (or anterior) cantilever then, in the horse, is constituted -by the heavy head and still heavier neck on one side of -the pier which is represented by the fore-legs, and by the dorsal -vertebrae carrying a large part of the weight of the trunk upon -the other side; and this weight is so balanced over the fore-legs -that the cantilever, while “anchored” to the other parts of the -structure, transmits but little of its weight to the hind-legs, and -the amount so transmitted will vary with the position of the -head and with the position of any artificial load<a class="afnanch" href="#fn627" id="fnanch627">627</a>. -Under certain -conditions, as when the head is thrust well forward, it is evident -that the hind-legs will be actually relieved of a portion of the -comparatively small load which is their normal share.</p> - -<p>Our problem now is to discover, in a rough and approximate -way, some of the structural details which the balanced load upon -the double cantilever will impress upon the fabric.</p> - -<hr class="hrblk"> - -<p>Working by the methods of graphic statics, the engineer’s -task is, in theory, one of great simplicity. He begins by drawing -in outline the structure which he desires to erect; he calculates -the stresses and bending-moments necessitated by the dimensions -and load on the structure; he draws a new diagram representing -these forces, and he designs and builds his fabric on the lines of this -statical diagram. He does, in short, precisely what we have seen -<i>nature</i> doing in the case of the bone. For if we had begun, as -it were, by blocking out the femur roughly, and considering its -position and dimensions, its means of support and the load which -it has to bear, we could have proceeded at once to draw the system -of stress-lines which must occupy the field of force: and to -precisely these stress-lines has nature kept in the building of the -bone, down to the minute arrangement of its trabeculae.</p> - -<p>The essential function of a bridge is to stretch across a certain -span, and carry a certain definite load; and -this being so, the <span class="xxpn" id="p696">{696}</span> -chief problem in the designing of a bridge is to provide due -resistance to the “bending-moments” which result from the load. -These bending-moments will vary from point to point along the -girder, and taking the simplest case of a uniform load supported -at both ends, they will be represented by points on a parabola. -If the girder be of uniform depth, that is to say if its two flanges,</p> - -<div class="dctr04" id="fig340"> -<img src="images/i696a.png" width="528" height="295" alt=""> - <div class="pcaption">Fig. 340. A, Span of - proposed bridge. B, Stress diagram, or diagram of - bending-moments<a class="afnanch" href="#fn628" - id="fnanch628">628</a>.</div></div> - -<p class="pcontinue">respectively under tension and compression, be parallel to one -another, then the stress upon these flanges will vary as the bending-moments, -and will accordingly be very severe in the middle and -will dwindle towards the ends. But if we make the <i>depth</i> of the -girder everywhere proportional to the bending-moments, that is</p> - -<div class="dctr04" id="fig341"> -<img src="images/i696b.png" width="528" height="247" alt=""> - <div class="dcaption">Fig. 341. The bridge constructed, as a - parabolic girder.</div></div> - -<p class="pcontinue">to say if we copy in the girder the outlines of the bending-moment -diagram, then our design will automatically meet the circumstances -of the case, for the horizontal stress in each flange will -now be uniform throughout the length of the -girder. In short, in <span class="xxpn" id="p697">{697}</span> -Professor Fidler’s words, “Every diagram of moments represents -the outline of a framed structure which will carry the given load -with a uniform horizontal stress in the principal members.”</p> - -<div class="dright dwth-d" id="fig342"> -<img src="images/i697.png" width="384" height="230" alt=""> - <div class="dcaption">Fig. 342.</div></div> - -<p>In the following diagrams (Fig. <a href="#fig342" title="go to Fig. 342">342</a>, <i>a</i>, <i>b</i>) (which are taken -from the original ones of Culmann), -we see at once that the -loaded beam or bracket (<i>a</i>) has -a “danger-point” close to its -fixed base, that is to say at the -point remotest from its load. -But in the parabolic bracket -(<i>b</i>) there is no danger-point at -all, for the dimensions of the -structure are made to increase <i>pari passu</i> with the bending-moments: -stress and resistance vary together. Again in Fig. <a href="#fig340" title="go to Fig. 340">340</a>, -we have a simple span (A), with its stress diagram (B); and in -Fig. <a href="#fig341" title="go to Fig. 341">341</a> we have the corresponding parabolic girder, whose -stresses are now uniform throughout. In fact we see that, by a -process of conversion, the stress diagram in each case becomes -the structural diagram in the other<a class="afnanch" href="#fn629" id="fnanch629">629</a>. -Now all this is but the -modern rendering of one of Galileo’s most famous propositions. -In the Dialogue which we have already quoted more than once<a class="afnanch" href="#fn630" id="fnanch630">630</a>, -Sagredo says “It would be a fine thing if one could discover the -proper shape to give a solid in order to make it equally resistant -at every point, in which case a load placed at the middle would -not produce fracture more easily than if placed at any other -point.” And Galileo (in the person of Salviati) first puts the -problem into its more general form; and then shews us how, by -giving a parabolic outline to our beam, we have its simple and -comprehensive solution.<br class="brclrfix"></p> - -<p>In the case of our cantilever bridge, we shew -the primitive girder <span class="xxpn" id="p698">{698}</span> -in Fig. <a href="#fig343" title="go to Fig. 343">343</a>, A, with its bending-moment diagram (B); and it is -evident that, if we turn this diagram upside down, it will still be -illustrative, just as before, of the bending-moments from point -to point: for as yet it is merely a diagram, or graph, of relative -magnitudes.</p> - -<p>To either of these two stress diagrams, direct or inverted, we -may fit the design of the construction, as in Figs. <a href="#fig343" title="go to Fig. 343">343</a>, C and 344.</p> - -<div class="dctr01" id="fig343"> -<img src="images/i698a.png" width="800" height="450" alt=""> - <div class="dcaption">Fig. 343.</div></div> - -<div class="dctr01" id="fig344"> -<img src="images/i698b.png" width="800" height="169" alt=""> - <div class="dcaption">Fig. 344.</div></div> - -<p>Now in different animals the amount and distribution of the -load differs so greatly that we can expect no single diagram, -drawn from the comparative anatomy of bridges, to apply equally -well to all the cases met with in the comparative anatomy of -quadrupeds; but nevertheless we have already gained an insight -into the general principles of “structural design” in the quadrupedal -bridge.</p> - -<p>In our last diagram the upper member of the -cantilever is under <span class="xxpn" id="p699">{699}</span> -tension; it is represented in the quadruped by the <i>ligamentum -nuchae</i> on the one side of the cantilever, and by the supraspinous -ligaments of the dorsal vertebrae on the other. The compression -member is similarly represented, on both sides of the cantilever, -by the vertebral column, or rather by the <i>bodies</i> of the vertebrae; -while the web, or “filling,” of the girders, that is to say the upright -or sloping members which extend from one flange to the other, is -represented on the one hand by the spines of the vertebrae, and -on the other hand, by the oblique interspinous ligaments and -muscles. The high spines over the quadruped’s withers are no -other than the high struts which rise over the supporting piers -in the parabolic girder, and correspond to the position of the -maximal bending-moments. The fact that these tall vertebrae -of the withers usually slope backwards, sometimes steeply, in -a quadruped, is easily and obviously explained<a class="afnanch" href="#fn631" id="fnanch631">631</a>. -For each -vertebra tends to act as a “hinged lever,” and its spine, acted -on by the tensions transmitted by the ligaments on either side, -takes up its position as the diagonal of the parallelogram of -forces to which it is exposed.</p> - -<p>It happens that in these comparatively simple types of -cantilever bridge the whole of the parabolic curvature is transferred -to one or other of the principal members, either the -tension-member or the compression-member as the case may be. -But it is of course equally permissible to have both members -curved, in opposite directions. This, though not exactly the case -in the Forth Bridge, is approximately so; for here the main -compression-member is curved or arched, and the main tension-member -slopes downwards on either side from its maximal height -above the piers. In short, the Forth Bridge is a nearer approach -than either of the other cantilever bridges -which we have <span class="xxpn" id="p700">{700}</span> -illustrated to the plan of the quadrupedal skeleton; for the main -compression-member almost exactly recalls the form of the vertebral -column, while the main tension-member, though not so -closely similar to the supraspinous and nuchal ligaments, corresponds -to the plan of these in a somewhat simplified form.</p> - -<div class="dctr02" id="fig345"> -<img src="images/i700a.png" width="704" height="220" alt=""> - <div class="pcaption">Fig. 345. A two-armed cantilever of the -Forth Bridge. Thick lines, compression-members (bones); -thin lines, tension-members (ligaments).</div></div> - -<p>We may now pass without difficulty from the two-armed -cantilever supported on a single pier, as it is in each separate -section of the Forth Bridge, or as we have imagined it to be in -the forequarters of a horse, to the condition which actually exists -in that quadruped, where a two-armed cantilever has its load -distributed over two separate piers. This is not precisely what -an engineer calls a “continuous” girder, for that term is applied -to a girder which, as a continuous structure, crosses two or more -spans, while here there is only one. But nevertheless, this girder</p> - -<div class="dctr02" id="fig346"> -<img src="images/i700b.png" width="704" height="270" alt=""> - <div class="dcaption">Fig. 346.</div></div> - -<p class="pcontinue">is <i>effectively</i> continuous from the head to the tip of the tail; and -at each point of support (<i>A</i> and <i>B</i>) it is subjected to the negative -bending-moment due to the overhanging load on each of the -projecting cantilever arms <i>AH</i> and <i>BT</i>. The diagram of bending-moments -will (according to the ordinary -conventions) lie below <span class="xxpn" id="p701">{701}</span> -the base line (because the moments are negative), and must take -some such form as that shown in the diagram: for the girder -must suffer its greatest bending stress not at the centre, but at -the two points of support <i>A</i> and <i>B</i>, where the moments are -measured by the vertical ordinates. It is plain that this figure -only differs from a representation of <i>two</i> independent two-armed -cantilevers in the fact that there is no point midway in the span -where the bending-moment vanishes, but only a region between -the two piers in which its magnitude tends to diminish.</p> - -<p>The diagram effects a graphic summation of the positive and -negative moments, but its form may assume various modifications -according to the method of graphic summation which we may -choose to adopt; and it is obvious also that the form of the -diagram may assume many modifications of detail according to -the actual distribution of the load. In all cases the essential -points to be observed are these: firstly that the girder which is</p> - -<div class="dctr04" id="fig347"> -<img src="images/i701.png" width="528" height="124" alt=""> - <div class="dcaption">Fig. 347. Stress-diagram of horse’s backbone.</div></div> - -<p class="pcontinue"> -to resist the bending-moments induced by the load must possess -its two principal members—an upper tension-member or tie, -represented by ligament, and a lower compression-member -represented by bone: these members being united by a web -represented by the vertebral spines with their interspinous ligaments, -and being placed one above the other in the order named -because the moments are negative; secondly we observe that the -depth of the web, or distance apart of the principal members,—that -is to say the height of the vertebral spines,—must be proportional -to the bending-moment at each point along the length -of the girder.</p> - -<p>In the case of an animal carrying two-thirds of his weight -upon his fore-legs and only one-third upon his hind-legs, the -bending-moment diagram will be unsymmetrical, after the fashion -of Fig. <a href="#fig347" title="go to Fig. 347">347</a>, the vertical ordinate at <i>A</i> being thrice the height of -that at <i>B</i>. <span class="xxpn" id="p702">{702}</span></p> - -<p>On the other hand the Dinosaur, with his light head and -enormous tail would give us a moment-diagram with the opposite -kind of asymmetry, the greatest bending stress being now found -over the haunches, at <i>B</i> (Fig. <a href="#fig348" title="go to Fig. 348">348</a>). A glance at the skeleton of -<i>Diplodocus Carnegii</i> will shew us the high vertebral spines over -the loins, in precise correspondence with the requirements of this -diagram: just as in the horse, under the opposite conditions of -load, the highest vertebral spines are those of the withers, that -is to say those of the posterior cervical and anterior dorsal -vertebrae.</p> - -<div class="dctr02" id="fig348"> -<img src="images/i702.png" width="704" height="143" alt=""> - <div class="dcaption">Fig. 348. Stress-diagram of backbone - of Dinosaur.</div></div> - -<p>We have now not only dealt with the general resemblance, -both in structure and in function, of the quadrupedal backbone -with its associated ligaments to a double-armed cantilever girder, -but we have begun to see how the characters of the vertebral -system must differ in different quadrupeds, according to the -conditions imposed by the varying distribution of the load: and -in particular how the height of the vertebral spines which constitute -the web will be in a definite relation, as regards magnitude -and position, to the bending-moments induced thereby. We -should require much detailed information as to the actual weights -of the several parts of the body before we could follow out -quantitatively the mechanical efficiency of each type of skeleton; -but in an approximate way what we have already learnt will -enable us to trace many interesting correspondences between -structure and function in this particular part of comparative -anatomy. We must, however, be careful to note that the great -cantilever system is not of necessity constituted by the vertical -column and its ligaments alone, but that the pelvis, firmly united -as it is to the sacral vertebrae, and stretching backwards far -beyond the acetabulum, becomes an intrinsic part of the system; -and helping (as it does) to carry the load of -the abdominal viscera, <span class="xxpn" id="p703">{703}</span> -constitutes a great portion of the posterior cantilever arm, or -even its chief portion in cases where the size and weight of the -tail are insignificant, as is the case in the majority of terrestrial -mammals.</p> - -<p>We may also note here, that just as a bridge is often a -“combined” or composite structure, exhibiting a combination of -principles in its construction, so in the quadruped we have, as -it were, another girder supported by the same piers to carry the -viscera; and consisting of an inverted parabolic girder, whose -compression-member is again constituted by the backbone, its -tension-member by the line of the sternum and the abdominal -muscles, while the ribs and intercostal muscles play the part of -the web or filling.</p> - -<p>A very few instances must suffice to illustrate the chief -variations in the load, and therefore in the bending-moment -diagram, and therefore also in the plan of construction, of various -quadrupeds. But let us begin by setting forth, in a few cases, -the actual weights which are borne by the fore-limbs and the -hind-limbs, in our quadrupedal bridge<a class="afnanch" href="#fn632" id="fnanch632">632</a>.</p> - -<div class="dtblboxin10"> -<table class="fsz7 borall"> -<tr> - <th class="borall" rowspan="2"></th> - <th class="borall" colspan="2">Gross. weight.</th> - <th class="borall">On Fore-feet.</th> - <th class="borall">On Hind-feet.</th> - <th class="borall" rowspan="2">% on Fore-feet.</th> - <th class="borall" rowspan="2">% on Hind-feet.</th></tr> -<tr> - <th class="borall">ton</th> - <th class="borall">cwts.</th> - <th class="borall">cwts.</th> - <th class="borall">cwts.</th></tr> -<tr> - <td class="tdleft">Camel (Bactrian)</td> - <td class="tdcntr">—</td> - <td class="tdcntr">14·25</td> - <td class="tdcntr"> 9·25</td> - <td class="tdcntr"> 4·5  </td> - <td class="tdcntr">67·3</td> - <td class="tdcntr">32·7</td></tr> -<tr> - <td class="tdleft">Llama</td> - <td class="tdcntr">—</td> - <td class="tdcntr"> 2·75</td> - <td class="tdcntr"> 1·75</td> - <td class="tdcntr">  ·875</td> - <td class="tdcntr">66·7</td> - <td class="tdcntr">33·3</td></tr> -<tr> - <td class="tdleft">Elephant (Indian)</td> - <td class="tdcntr">1</td> - <td class="tdcntr">15·75</td> - <td class="tdcntr">20·5 </td> - <td class="tdcntr">14·75 </td> - <td class="tdcntr">58·2</td> - <td class="tdcntr">41·8</td></tr> -<tr> - <td class="tdleft">Horse</td> - <td class="tdcntr">—</td> - <td class="tdcntr"> 8·25</td> - <td class="tdcntr"> 4·75</td> - <td class="tdcntr"> 3·5  </td> - <td class="tdcntr">57·6</td> - <td class="tdcntr">42·4</td></tr> -<tr> - <td class="tdleft">Horse (large Clydesdale)</td> - <td class="tdcntr">—</td> - <td class="tdcntr">15·5 </td> - <td class="tdcntr"> 8·5 </td> - <td class="tdcntr"> 7·0  </td> - <td class="tdcntr">54·8</td> - <td class="tdcntr">45·2</td></tr> -</table></div><!--dtblbox--> - -<p>It will be observed that in all these animals the load upon the -fore-feet preponderates considerably over that upon the hind, the -preponderance being rather greater in the elephant than in the -horse, and markedly greater in the camel and the llama than in -the other two. But while these weights are helpful and suggestive, -it is obvious that they do not go nearly far enough to -give us a full insight into the constructional diagram to which -the animals are conformed. For such a -purpose we should <span class="xxpn" id="p704">{704}</span> -require to weigh the total load, not in two portions, but in many; -and we should also have to take close account of the general form -of the animal, of the relation between that form and the distribution -of the load, and of the actual directions of each bone and -ligament by which the forces of compression and tension were -transmitted. All this lies beyond us for the present; but nevertheless -we may consider, very briefly, the principal cases involved -in our enquiry, of which the above animals form only a partial -and preliminary illustration.</p> - -<ul><li> -<p>(1) Wherever we have a heavily loaded anterior cantilever -arm, that is to say whenever the head and neck represent a -considerable fraction of the whole weight of the body, we tend -to have large bending-moments over the fore-legs, and correspondingly -high spines over the vertebrae of the withers. This</p> - -<div class="dctr02" id="fig349"> -<img src="images/i704.png" width="704" height="208" alt=""> - <div class="dcaption">Fig. 349. Stress-diagram of - Titanotherium.</div></div> - -<p class="pcontinue"> -is the case in the great majority of four-footed, terrestrial animals, -the chief exceptions being found in animals with comparatively -small heads but large and heavy tails, such as the anteaters or -the Dinosaurian reptiles, and also (very naturally) in animals -such as the crocodile, where the “bridge” can scarcely be said -to be developed, for the long heavy body sags down to rest upon -the ground. The case is sufficiently exemplified by the horse, -and still more notably by the stag, the ox, or the pig. It is -illustrated in the accompanying diagram of the conditions in the -great extinct Titanotherium.</p></li> - -<li><p>(2) In the elephant and the camel we have similar conditions, -but slightly modified. In both cases, and especially in the latter, -the weight on the fore-quarters is relatively large; and in both -cases the bending-moments are all the larger, by reason of the -length and forward extension of the camel’s neck, -and the forward <span class="xxpn" id="p705">{705}</span> -position of the heavy tusks of the elephant. In both cases the -dorsal spines are large, but they do not strike us as exceptionally -so; but in both cases, and especially in the elephant, they slope -backwards in a marked degree. Each spine, as already explained, -must in all cases assume the position of the diagonal in the -parallelogram of forces defined by the tensions acting on it at -its extremity; for it constitutes a “hinged lever,” by which the -bending-moments on either side are automatically balanced; and -it is plain that the more the spine slopes backwards the more it -indicates a relatively large strain thrown upon the great ligament -of the neck, and a relief of strain upon the more directly acting, -but weaker, ligaments of the back and loins. In both cases, the -bending-moments would seem to be more evenly distributed over -the region of the back than, for instance, in the stag, with its -light hind-quarters and heavy load of antlers: and in both cases -the high “girder” is considerably prolonged, by an extension of -the tall spines backwards in the direction of the loins. When -we come to such a case as the mammoth, with its immensely -heavy and immensely elongated tusks, we perceive at once that -the bending-moments over the fore-legs are now very severe; -and we see also that the dorsal spines in this region are much -more conspicuously elevated than in the ordinary elephant.</p></li> - -<li><p>(3) In the case of the giraffe we have, without doubt, a very -heavy load upon the fore-legs, though no weighings are at hand -to define the ratio; but as far as possible this disproportionate -load would seem to be relieved, by help of a downward as well -as backward thrust, through the sloping back, to the unusually -low hind-quarters. The dorsal spines of the vertebrae are very -high and strong, and the whole girder-system very perfectly -formed. The elevated, rather than protruding position of the -head lessens the anterior bending-moment as far as possible; but -it leads to a strong compressional stress transmitted almost -directly downwards through the neck: in correlation with which -we observe that the bodies of the cervical vertebrae are exceptionally -large and strong and steadily increase in size and strength -from the head downwards.</p></li> - -<li><p>(4) In the kangaroo, the fore-limbs are entirely relieved of -their load, and accordingly the tall spines over -the withers, which <span class="xxpn" id="p706">{706}</span> -were so conspicuous in all heavy-headed <i>quadrupeds</i>, have now -completely vanished. The creature has become bipedal, and body -and tail form the extremities of <i>a single</i> balanced cantilever, -whose maximal bending-moments are marked by strong, high -lumbar and sacral vertebrae, and by iliac bones of peculiar form -and exceptional strength.</p> - -<p>Precisely the same condition is illustrated in the Iguanodon, -and better still by reason of the great bulk of the creature, and of -the heavy load which falls to be supported by the great cantilever -and by the hind-legs which form its piers. The long and heavy -body and neck require a balance-weight (as in the kangaroo) in -the form of a long heavy tail. And the double-armed cantilever, -so constituted, shews a beautiful parabolic curvature in the graded -heights of the whole series of vertebral spines, which rise to a -maximum over the haunches and die away slowly towards the -neck and the tip of the tail.</p></li> - -<li><p>(5) In the case of some of the great American fossil reptiles, -such as Diplodocus, it has always been a more or less disputed -question whether or not they assumed, like Iguanodon, an erect, -bipedal attitude. In all of these we see an elongated pelvis, and, -in still more marked degree, we see elevated spinous processes of -the vertebrae over the hind-limbs; in all of them we have a long -heavy tail, and in most of them we have a marked reduction in -size and weight both of the fore-limb and of the head itself. The -great size of these animals is not of itself a proof against the erect -attitude; because it might well have been accompanied by an -aquatic or partially submerged habitat, and the crushing stress of -the creature’s huge bulk proportionately relieved. But we must -consider each such case in the whole light of its own evidence; -and it is easy to see that, just as the quadrupedal mammal may -carry the greater part but not all of its weight upon its fore-limbs, -so a heavy-tailed reptile may carry the greater part upon its hind-limbs, -without this process going so far as to relieve its fore-limbs -of all weight whatsoever. This would seem to be the case in such -a form as Diplodocus, and also in Stegosaurus, whose restoration -by Marsh is doubtless substantially correct<a class="afnanch" href="#fn633" id="fnanch633">633</a>. -The fore-limbs, <span class="xxpn" id="p707">{707}</span> -though comparatively small, are obviously fashioned for support, -but the weight which they have to carry is far less than that -which the hind-limbs bear. The head is small and the neck -short, while on the other hand the hind-quarters and the tail are -big and massive. The backbone bends into a great, double-armed -cantilever, culminating over the pelvis and the hind-limbs, and -here furnished with its highest and strongest spines to separate -the tension-member from the compression-member of the girder. -The fore-legs form a secondary supporting pier to this great -cantilever, the greater part of whose weight is poised upon the -hind-limbs alone.</p> - -<div class="dctr02" id="fig350"> -<img src="images/i707.png" width="704" height="338" alt=""> - <div class="dcaption">Fig. 350. Diagram of Stegosaurus.</div></div> -</li> - -<li><p>(6) In a bird, such as an ostrich or a common fowl, the -bipedal habit necessitates the balancing of the load upon a single -double-armed cantilever-girder, just as in the Iguanodon and the -kangaroo, but the construction is effected in a somewhat different -way. The great heavy tail has entirely disappeared; but, though -from the skeleton alone it would seem that nearly all the bulk of -the animal lay in front of the hind-limbs, yet in the living bird -we can easily perceive that the great weight of the abdominal -organs lies suspended <i>behind</i> the socket for the thigh-bone, and -so hangs from the posterior lever-arm of the cantilever, balancing -the head and neck and thorax whose combined -weight hangs from <span class="xxpn" id="p708">{708}</span> -the anterior arm. The great cantilever girder appears, accordingly, -balanced over the hind-legs. It is now constituted in part by -the posterior dorsal or lumbar vertebrae, all traces of special -elevation having disappeared from the anterior dorsals; but the -greater part of the girder is made up of the great iliac bones, -placed side by side, and gripping firmly the sacral vertebrae, often -almost to the extinction of these latter. In the form of these -iliac bones, the arched curvature of their upper border, in their -elongation fore-and-aft to overhang both ways their supporting -pier, and in the coincidence of their greatest height with the -median line of support over the centre of gravity, we recognise -all the characteristic properties of the typical balanced -cantilever<a class="afnanch" href="#fn634" id="fnanch634">634</a>.</p></li> - -<li><p>(7) We find a highly important corollary in the case of -aquatic animals. For here the effect of gravity is neutralised; -we have neither piers nor cantilevers; and we find accordingly -in all aquatic mammals of whatsoever group—whales, seals or -sea-cows—that the high arched vertebral spines over the withers, -or corresponding structures over the hind-limbs, have both -entirely disappeared.</p></li> -</ul> - -<p>Just as the cantilever girder tended to become obsolete in the -aquatic mammal so does it tend to weaken and disappear in the -aquatic bird. There is a very marked contrast between the high-arched -strongly-built pelvis in the ostrich or the hen, and the -long, thin, comparatively straight and weakly bone which represents -it in a diver, a grebe or a penguin.</p> - -<p>But in the aquatic mammal, such as a whale or a dolphin (and -not less so in the aquatic bird), <i>stiffness</i> must be ensured in order -to enable the muscles to act against the resistance of the water -in the act of swimming; and accordingly nature must provide -against bending-moments irrespective of gravity. In the dolphin, -at any rate as regards its tail end, the conditions will be not very -different from those of a column or beam with fixed ends, in -which, under deflexion, there will be two points of contrary -flexure, as at <i>C</i>, <i>D</i>, in Fig. <a href="#fig351" title="go to Fig. 351">351</a>. <span class="xxpn" id="p709">{709}</span></p> - -<div class="dright dwth-e" id="fig351"> -<img src="images/i709.png" width="384" height="109" alt=""> - <div class="dcaption">Fig. 351.</div></div> - -<p>Here, between <i>C</i> and <i>D</i> we have a varying bending-moment, -represented by a continuous curve with its maximal elevation -midway between the points of inflexion. And correspondingly, -in our dolphin, we have a continuous -series of high dorsal -spines, rising to a maximum -about the middle of the animal’s -body, and falling to nil at some -distance from the end of the tail. It is their business (as -usual) to keep the tension-member, represented by the strong -supraspinous ligaments, wide apart from the compression-member, -which is as usual represented by the backbone itself. But in -our diagram we see that on the further side of <i>C</i> and <i>D</i> we -have a <i>negative</i> curve of bending-moments, or bending-moments -in a contrary direction. Without inquiring how these stresses -are precisely met towards the dolphin’s head (where the coalesced -cervical vertebrae suggest themselves as a partial explanation), -we see at once that towards the tail they are met by the strong -series of chevron-bones, which in the caudal region, where tall -<i>dorsal</i> spines are no longer needed, take their place <i>below</i> the -vertebrae, in precise correspondence with the bending-moment -diagram. In many cases other than these aquatic ones, when -we have to deal with animals with long and heavy tails (like the -Iguanodon and the kangaroo of which we have already spoken), -we are apt to meet with similar, though usually shorter chevron-bones; -and in all these cases we may see without difficulty that -a negative bending-moment is there to be resisted. <br class="brclrfix"></p> - -<p>In the dolphin we may find a good illustration of the fact -that not only is it necessary to provide for rigidity in the vertical -direction, but also in the horizontal, where a tendency to bending -must be resisted on either side. This function is effected in part -by the ribs with their associated muscles, but they extend but a -little way and their efficacy for this purpose can be but small. -We have, however, behind the region of the ribs and on either side -of the backbone a strong series of elongated and flattened transverse -processes, forming a web for the support of a tension-member -in the usual form of ligament, and so playing a part precisely -analogous to that performed by the dorsal spines -in the same <span class="xxpn" id="p710">{710}</span> -animal. In an ordinary fish, such as a cod or a haddock, we see -precisely the same thing: the backbone is stiffened by the indispensable -help of its <i>three series</i> of ligament-connected processes, -the dorsal and the two transverse series. And here we see (as -we see partly also among the whales), that these three series of -processes, or struts, tend to be arranged well-nigh at equal angles, -of 120°, with one another, giving the greatest and most uniform -strength of which such a system is capable. On the other hand, -in a flat fish, such as a plaice, where from the natural mode of -progression it is necessary that the backbone should be flexible -in one direction while stiffened in another, we find the whole -outline of the fish comparable to that of a double bowstring -girder, the compression-member being (as usual) the backbone, -the tension-member on either side being constituted by the interspinous -ligaments and muscles, while the web or filling is very -beautifully represented by the long and evenly graded spines, -which spring symmetrically from opposite sides of each individual -vertebra.</p> - -<hr class="hrblk"> - -<p>The main result at which we have now arrived, in regard to -the construction of the vertebral column and its associated parts, -is that we may look upon it as a certain type of <i>girder</i>, whose depth, -as we cannot help seeing, is everywhere very nearly proportional -to the height of the corresponding ordinate in the diagram of -moments: just as it is in the girder of a cantilever bridge as -designed by a modern engineer. In short, after the nineteenth -or twentieth century engineer has done his best in framing the -design of a big cantilever, he may find that some of his best ideas -bad, so to speak, been anticipated ages ago in the fabric of the -great saurians and the larger mammals.</p> - -<p>But it is possible that the modern engineer might be disposed -to criticise the skeleton girder at two or three points; and in -particular he might think the girder, as we see it for instance in -Diplodocus or Stegosaurus, not deep enough for carrying the -animal’s enormous weight of some twenty tons. If we adopt a -much greater depth (or ratio of depth to length) as in the modern -cantilever, we shall greatly increase the <i>strength</i> of the structure; -but at the same time we should greatly increase -its <i>rigidity</i>, and <span class="xxpn" id="p711">{711}</span> -this is precisely what, in the circumstances of the case, it would -seem that nature is bound to avoid. We need not suppose that -the great saurian was by any means active and limber; but a -certain amount of activity and flexibility he was bound to have, -and in a thousand ways he would find the need of a backbone -that should be <i>flexible</i> as well as <i>strong</i>. Now this opens up a -new aspect of the matter and is the beginning of a long, long story, -for in every direction this double requirement of strength and -flexibility imposes new conditions upon our design. To represent -all the correlated quantities we should have to construct not only -a diagram of moments but also a diagram of elastic deflexion and -its so-called “curvature”; and the engineer would want to know -something more about the <i>material</i> of the ligamentous tension-member—its -modulus of elasticity in direct tension, its elastic -limit, and its safe working stress.</p> - -<p>In various ways our structural problem is beset by “limiting -conditions.” Not only must rigidity be associated with flexibility, -but also stability must be ensured in various positions and -attitudes; and the primary function of support or weight-carrying -must be combined with the provision of <i>points d’appui</i> for the -muscles concerned in locomotion. We cannot hope to arrive at -a numerical or quantitative solution of this complicate problem, -but we have found it possible to trace it out in part towards a -qualitative solution. And speaking broadly we may certainly -say that in each case the problem has been solved by nature -herself, very much as she solves the difficult problems of minimal -areas in a system of soap-bubbles; so that each animal is fitted -with a backbone adapted to his own individual needs, or (in -other words) corresponding exactly to the mean resultant of the -stresses to which as a mechanical system it is exposed.</p> - -<hr class="hrblk"> - -<p>Throughout this short discussion of the principles of construction, -limited to one part of the skeleton, we see the same -general principles at work which we recognise in the plan and -construction of an individual bone. That is to say, we see a -tendency for material to be laid down just in the lines of <i>stress</i>, -and so as to evade thereby the distortions and disruptions due to -<i>shear</i>. In these phenomena there lies a definite -law of growth, <span class="xxpn" id="p712">{712}</span> -whatever its ultimate expression or explanation may come to be. -Let us not press either argument or hypothesis too far: but be -content to see that skeletal form, as brought about by growth, -is to a very large extent determined by mechanical considerations, -and tends to manifest itself as a diagram, or reflected image, of -mechanical stress. If we fail, owing to the immense complexity -of the case, to unravel all the mathematical principles involved -in the construction of the skeleton, we yet gain something, and -not a little, by applying this method to the familiar objects of our -anatomical study: <i>obvia conspicimus, nubem pellente mathesi</i><a class="afnanch" href="#fn635" id="fnanch635">635</a>.</p> - -<p>Before we leave this subject of mechanical adaptation, let us -dwell once more for a moment upon the considerations which -arise from our conception of a field of force, or field of stress, in -which tension and compression (for instance) are inevitably -combined, and are met by the materials naturally fitted to resist -them. It has been remarked over and over again how harmoniously -the whole organism hangs together, and how throughout -its fabric one part is related and fitted to another in strictly -functional correlation. But this conception, though never denied, -is sometimes apt to be forgotten in the course of that process of -more and more minute analysis by which, for simplicity’s sake, -we seek to unravel the intricacies of a complex organism.</p> - -<p>We tend, as we analyse a thing into its parts or into its -properties, to magnify these, to exaggerate their apparent -independence, and to hide from ourselves (at least for a time) the -essential integrity and individuality of the composite whole. We -divide the body into its organs, the skeleton into its bones, as -in very much the same fashion we make a subjective analysis of -the mind, according to the teachings of psychology, into component -factors: but we know very well that judgment and knowledge, -courage or gentleness, love or fear, have no separate existence, -but are somehow mere manifestations, or imaginary co-efficients, -of a most complex integral. And likewise, as biologists, we may -go so far as to say that even the bones themselves are only in a -limited and even a deceptive sense, separate and individual -things. The skeleton begins as a <i>continuum</i>, and a <i>continuum</i> it -remains all life long. The things that link -bone with bone, <span class="xxpn" id="p713">{713}</span> -cartilage, ligaments, membranes, are fashioned out of the same -primordial tissue, and come into being <i>pari passu</i>, with the bones -themselves. The entire fabric has its soft parts and its hard, its -rigid and its flexible parts; but until we disrupt and dismember -its bony, gristly and fibrous parts, one from another, it exists -simply as a “skeleton,” as one integral and individual whole.</p> - -<p>A bridge was once upon a time a loose heap of pillars and rods -and rivets of steel. But the identity of these is lost, just as if -they were fused into a solid mass, when once the bridge is built; -their separate functions are only to be recognised and analysed -in so far as we can analyse the stresses, the tensions and the -pressures, which affect this part of the structure or that; and -these forces are not themselves separate entities, but are the -resultants of an analysis of the whole field of force. Moreover -when the bridge is broken it is no longer a bridge, and all its -strength is gone. So is it precisely with the skeleton. In it is -reflected a field of force: and keeping pace, as it were, in action -and interaction with this field of force, the whole skeleton and -every part thereof, down to the minute intrinsic structure of the -bones themselves, is related in form and in position to the lines -of force, to the resistances it has to encounter; for by one of -the mysteries of biology, resistance begets resistance, and where -pressure falls there growth springs up in strength to meet it. -And, pursuing the same train of thought, we see that all this is -true not of the skeleton alone but of the whole fabric of the body. -Muscle and bone, for instance, are inseparably associated and -connected; they are moulded one with another; they come into -being together, and act and react together<a class="afnanch" href="#fn636" id="fnanch636">636</a>. -We may study -them apart, but it is as a concession to our weakness and to the -narrow outlook of our minds. We see, dimly perhaps, but yet -with all the assurance of conviction, that between muscle and -bone there can be no change in the one but it is correlated with -changes in the other; that through and through they are linked -in indissoluble association; that they are -only separate entities <span class="xxpn" id="p714">{714}</span> -in this limited and subordinate sense, that they are <i>parts</i> of a -whole which, when it loses its composite integrity, ceases to -exist.</p> - -<p>The biologist, as well as the philosopher, learns to recognise -that the whole is not merely the sum of its parts. It is this, and -much more than this. For it is not a bundle of parts but an -organisation of parts, of parts in their mutual arrangement, -fitting one with another, in what Aristotle calls “a single and -indivisible principle of unity”; and this is no merely metaphysical -conception, but is in biology the fundamental truth which lies at -the basis of Geoffroy’s (or Goethe’s) law of “compensation,” or -“balancement of growth.”</p> - -<p>Nevertheless Darwin found no difficulty in believing that -“natural selection will tend in the long run to reduce <i>any part</i> -of the organisation, as soon as, through changed habits, it becomes -superfluous: without by any means causing some other part to -be largely developed in a corresponding degree. And conversely, -that natural selection may perfectly well succeed in largely developing -an organ without requiring as a necessary compensation -the reduction of some adjoining part<a class="afnanch" href="#fn637" id="fnanch637">637</a>.” -This view has been -developed into a doctrine of the “independence of single characters” -(not to be confused with the germinal “unit characters” -of Mendelism), especially by the palaeontologists. Thus Osborn -asserts a “principle of hereditary correlation,” combined with a -“principle of <i>hereditary separability</i> whereby the body is a colony, -a mosaic, of single individual and separable characters<a class="afnanch" href="#fn638" id="fnanch638">638</a>.” -I cannot think that there is more than a small element of truth -in this doctrine. As Kant said, “die Ursache der Art der Existenz -bei jedem Theile eines lebenden Körpers <i>ist im Ganzen enthalten</i>.” -And, according to the trend or aspect of our thought, we may -look upon the co-ordinated parts, now as related and fitted <i>to the -end or function</i> of the whole, and now as related to or resulting -<i>from the physical causes</i> inherent in the entire system of forces -to which the whole has been exposed, and under whose influence -it has come into being<a class="afnanch" href="#fn639" id="fnanch639">639</a>. -<span class="xxpn" id="p715">{715}</span></p> - -<p>It would seem to me that the mechanical principles and -phenomena which we have dealt with in this chapter are of no small -importance to the morphologist, all the more when he is inclined -to direct his study of the skeleton exclusively to the problem of -phylogeny; and especially when, according to the methods of -modern comparative morphology, he is apt to take the skeleton -to pieces, and to draw from the comparison of a series of scapulae, -humeri, or individual vertebrae, conclusions as to the descent -and relationship of the animals to which they belong.</p> - -<p>It would, I dare say, be a gross exaggeration to see in every -bone nothing more than a resultant of immediate and direct -physical or mechanical conditions; for to do so would be to deny -the existence, in this connection, of a principle of heredity. And -though I have tried throughout this book to lay emphasis on the -direct action of causes other than heredity, in short to circumscribe -the employment of the latter as a working hypothesis in -morphology, there can still be no question whatsoever but that -heredity is a vastly important as well as a mysterious thing; it -is one of the great factors in biology, however we may attempt to -figure to ourselves, or howsoever we may fail even to imagine, -its underlying physical explanation. But I maintain that it is -no less an exaggeration if we tend to neglect these direct physical -and mechanical modes of causation altogether, and to see in the -characters of a bone merely the results of variation and of heredity, -and to trust, in consequence, to those characters as a sure and -certain and unquestioned guide to affinity and phylogeny. -Comparative anatomy has its physiological side, which filled -men’s minds in John Hunter’s day, and in Owen’s -day; it has its <span class="xxpn" id="p716">{716}</span> -classificatory and phylogenetic aspect, which has all but filled -men’s minds during the last couple of generations; and we can -lose sight of neither aspect without risk of error and misconception.</p> - -<p>It is certain that the question of phylogeny, always difficult, -becomes especially so in cases where a great change of physical -or mechanical conditions has come about, and where accordingly -the physical and physiological factors in connection with change -of form are bound to be large. To discuss these questions at -length would be to enter on a discussion of Lamarck’s philosophy -of biology, and of many other things besides. But let us take -one single illustration.</p> - -<p>The affinities of the whales constitute, as will be readily -admitted, a very hard problem in phylogenetic classification. -We know now that the extinct Zeuglodons are related to the -old Creodont carnivores, and thereby (though distantly) to the -seals; and it is supposed, but it is by no means so certain, that -in turn they are to be considered as representing, or as allied to, -the ancestors of the modern toothed whales<a class="afnanch" href="#fn640" id="fnanch640">640</a>. -The proof of any -such a contention becomes, to my mind, extraordinarily difficult -and complicated; and the arguments commonly used in such cases -may be said (in Bacon’s phrase) to allure, rather than to extort -assent. Though the Zeuglodonts were aquatic animals, we do not -know, and we have no right to suppose or to assume, that they -swam after the fashion of a whale (any more than the seal does), -that they dived like a whale, and leaped like a whale. But the fact -that the whale does these things, and the way in which he does -them, is reflected in many parts of his skeleton—perhaps more -or less in all: so much so that the lines of stress which these -actions impose are the very plan and working-diagram of great -part of his structure. That the Zeuglodon has a scapula like that -of a whale is to my mind no necessary argument that he is akin -by blood-relationship to a whale: that his dorsal vertebrae are -very different from a whale’s is no conclusive -argument that <span class="xxpn" id="p717">{717}</span> -such blood-relationship is lacking. The former fact goes a long -way to prove that he used his flippers very much as a whale does; -the latter goes still farther to prove that his general movements -and equilibrium in the water were totally different. The whale -may be descended from the Carnivora, or might for that matter, -as an older school of naturalists believed, be descended from the -Ungulates; but whether or no, we need not expect to find in him -the scapula, the pelvis or the vertebral column of the lion or of -the cow, for it would be physically impossible that he could live -the life he does with any one of them. In short, when we hope to -find the missing links between a whale and his terrestrial ancestors, -it must be not by means of conclusions drawn from a scapula, an -axis, or even from a tooth, but by the discovery of forms so intermediate -in their general structure as to indicate an organisation -and, <i>ipso facto</i>, a mode of life, intermediate between the terrestrial -and the Cetacean form. There is no valid syllogism to the effect -that <i>A</i> has a flat curved scapula like a seal’s, and <i>B</i> has a flat, -curved scapula like a seal’s: and therefore <i>A</i> and <i>B</i> are related -to the seals and to each other; it is merely a flagrant case of an -“undistributed middle.” But there is validity in an argument -that <i>B</i> shews in its general structure, extending over this bone -and that bone, resemblances both to <i>A</i> and to the seals: and that -therefore he may be presumed to be related to both, in his -hereditary habits of life and in actual kinship by blood. It is -cognate to this argument that (as every palaeontologist knows) -we find clues to affinity more easily, that is to say with less -confusion and perplexity, in certain structures than in others. -The deep-seated rhythms of growth which, as I venture to -think, are the chief basis of morphological heredity, bring about -similarities of form, which endure in the absence of conflicting -forces; but a new system of forces, introduced by altered environment -and habits, impinging on those particular parts of the fabric -which lie within this particular field of force, will assuredly not -be long of manifesting itself in notable and inevitable modifications -of form. And if this be really so, it will further imply that -modifications of form will tend to manifest themselves, not so -much in small and <i>isolated</i> phenomena, in this part of the fabric -or in that, in a scapula for instance or a humerus: -but rather in <span class="xxpn" id="p718">{718}</span> -some slow, <i>general</i>, and more or less uniform or graded modification, -spread over a number of correlated parts, and at times extending -over the whole, or over great portions, of the body. Whether -any such general tendency to widespread and correlated transformation -exists, we shall attempt to discuss in the following -chapter.</p> - -<div class="chapter" id="p719"> - <h2 class="h2herein" - title="XVII. On the Theory of Transformations, Or - the Comparison of Related Forms.">CHAPTER XVII <span - class="h2ttl">ON THE THEORY OF TRANSFORMATIONS, OR THE - COMPARISON OF RELATED FORMS<a class="afnanchlow" href="#fn641" - id="fnanch641" title="go to note 641">*</a></span></h2></div> - -<p>In the foregoing chapters of this book we have attempted to -study the inter-relations of growth and form, and the part which -certain of the physical forces play in this complex interaction; -and, as part of the same enquiry, we have tried in comparatively -simple cases to use mathematical methods and mathematical -terminology in order to describe and define the forms of organisms. -We have learned in so doing that our own study of organic form, -which we call by Goethe’s name of Morphology, is but a portion -of that wider Science of Form which deals with the forms assumed -by matter under all aspects and conditions, and, in a still wider -sense, with forms which are theoretically imaginable.</p> - -<p>The study of form may be descriptive merely, or it may -become analytical. We begin by describing the shape of an object -in the simple words of common speech: we end by defining it -in the precise language of mathematics; and the one method -tends to follow the other in strict scientific order and historical -continuity. Thus, for instance, the form of the earth, of a raindrop -or a rainbow, the shape of the hanging chain, or the path of a stone -thrown up into the air, may all be described, however inadequately, -in common words; but when we have learned to comprehend -and to define the sphere, the catenary, or the parabola, we have -made a wonderful and perhaps a manifold advance. The mathematical -definition of a “form” has a quality of precision which -was quite lacking in our earlier stage of mere description; it is -expressed in few words, or in still briefer -symbols, and these <span class="xxpn" id="p720">{720}</span> -words or symbols are so pregnant with meaning that thought -itself is economised; we are brought by means of it in touch with -Galileo’s aphorism (as old as Plato, as old as Pythagoras, as old -perhaps as the wisdom of the Egyptians), that “the Book of -Nature is written in characters of Geometry.”</p> - -<p>Next, we soon reach through mathematical analysis to mathematical -synthesis; we discover homologies or identities which -were not obvious before, and which our descriptions obscured -rather than revealed: as for instance, when we learn that, however -we hold our chain, or however we fire our bullet, the contour -of the one or the path of the other is always mathematically -homologous. Lastly, and this is the greatest gain of all, we pass -quickly and easily from the mathematical conception of form in -its statical aspect to form in its dynamical relations: we pass from -the conception of form to an understanding of the forces which -gave rise to it; and in the representation of form and in the -comparison of kindred forms, we see in the one case a diagram -of forces in equilibrium, and in the other case we discern the -magnitude and the direction of the forces which have sufficed to -convert the one form into the other. Here, since a change of -material form is only effected by the movement of matter, we have -once again the support of the schoolman’s and the philosopher’s -axiom, “<i>Ignorato motu, ignoratur Natura</i>.”</p> - -<hr class="hrblk"> - -<p>In the morphology of living things the use of mathematical -methods and symbols has made slow progress; and there are -various reasons for this failure to employ a method whose -advantages are so obvious in the investigation of other physical -forms. To begin with, there would seem to be a psychological -reason lying in the fact that the student of living things is by -nature and training an observer of concrete objects and phenomena, -and the habit of mind which he possesses and cultivates is alien -to that of the theoretical mathematician. But this is by no -means the only reason; for in the kindred subject of mineralogy, -for instance, crystals were still treated in the days of Linnaeus -as wholly within the province of the naturalist, and were described -by him after the simple methods in use for animals and plants: -but as soon as Haüy showed the application -of mathematics to <span class="xxpn" id="p721">{721}</span> -the description and classification of crystals, his methods were -immediately adopted and a new science came into being.</p> - -<p>A large part of the neglect and suspicion of mathematical -methods in organic morphology is due (as we have partly seen in -our opening chapter) to an ingrained and deep-seated belief that -even when we seem to discern a regular mathematical figure in -an organism, the sphere, the hexagon, or the spiral which we so -recognise merely resembles, but is never entirely explained by, -its mathematical analogue; in short, that the details in which the -figure differs from its mathematical prototype are more important -and more interesting than the features in which it agrees, and -even that the peculiar aesthetic pleasure with which we regard -a living thing is somehow bound up with the departure from -mathematical regularity which it manifests as a peculiar attribute -of life. This view seems to me to involve a misapprehension. -There is no such essential difference between these phenomena of -organic form and those which are manifested in portions of -inanimate matter<a class="afnanch" href="#fn642" id="fnanch642">642</a>. -No chain hangs in a perfect catenary and no -raindrop is a perfect sphere: and this for the simple reason that -forces and resistances other than the main one are inevitably at -work. The same is true of organic form, but it is for the mathematician -to unravel the conflicting forces which are at work -together. And this process of investigation may lead us on step -by step to new phenomena, as it has done in physics, where -sometimes a knowledge of form leads us to the interpretation of -forces, and at other times a knowledge of the forces at work -guides us towards a better insight into form. I would illustrate -this by the case of the earth itself. After the fundamental advance -had been made which taught us that the world was round, Newton -showed that the forces at work upon it must lead to its being -imperfectly spherical, and in the course of time its oblate spheroidal -shape was actually verified. But now, in turn, it has been shown -that its form is still more complicated, and the next step will be -to seek for the forces that have deformed the -oblate spheroid. <span class="xxpn" id="p722">{722}</span></p> - -<p>The organic forms which we can define, more or less precisely, -in mathematical terms, and afterwards proceed to explain and -to account for in terms of force, are of many kinds, as we have -seen; but nevertheless they are few in number compared with -Nature’s all but infinite variety. The reason for this is not far -to seek. The living organism represents, or occupies, a field of -force which is never simple, and which as a rule is of immense -complexity. And just as in the very simplest of actual cases we -meet with a departure from such symmetry as could only exist -under conditions of <i>ideal</i> simplicity, so do we pass quickly to -cases where the interference of numerous, though still perhaps very -simple, causes leads to a resultant which lies far beyond our powers -of analysis. Nor must we forget that the biologist is much more -exacting in his requirements, as regards form, than the physicist; -for the latter is usually content with either an ideal or a general -description of form, while the student of living things must needs -be specific. The physicist or mathematician can give us perfectly -satisfying expressions for the form of a wave, or even of a heap of -sand; but we never ask him to define the form of any particular -wave of the sea, nor the actual form of any mountain-peak or -hill<a class="afnanch" href="#fn643" id="fnanch643">643</a>. -<span class="xxpn" id="p723">{723}</span></p> - -<p>For various reasons, then, there are a vast multitude of organic -forms which we are unable to account for, or to define, in mathematical -terms; and this is not seldom the case even in forms which -are apparently of great simplicity and regularity. The curved -outline of a leaf, for instance, is such a case; its ovate, lanceolate, -or cordate shape is apparently very simple, but the difficulty of -finding for it a mathematical expression is very great indeed. -To define the complicated outline of a fish, for instance, or of a -vertebrate skull, we never even seek for a mathematical formula.</p> - -<p>But in a very large part of morphology, our essential task lies -in the comparison of related forms rather than in the precise -definition of each; and the <i>deformation</i> of a complicated figure -may be a phenomenon easy of comprehension, though the figure -itself have to be left unanalysed and undefined. This process -of comparison, of recognising in one form a definite permutation -or <i>deformation</i> of another, apart altogether from a precise and -adequate understanding of the original “type” or standard of -comparison, lies within the immediate province of mathematics, -and finds its solution in the elementary use of a certain method -of the mathematician. This method is the Method of Co-ordinates, -on which is based the Theory of Transformations.</p> - -<p>I imagine that when Descartes conceived the method of -co-ordinates, as a generalisation from the proportional diagrams -of the artist and the architect, and long before the immense -possibilities of this analysis could be foreseen, he had in mind a -very simple purpose; it was perhaps no more than to find a way -of translating the <i>form</i> of a curve into <i>numbers</i> and into <i>words</i>. -This is precisely what we do, by the method of co-ordinates, -every time we study a statistical curve; and conversely, we -translate numbers into form whenever we “plot a curve” to -illustrate a table of mortality, a rate of growth, or the daily -variation of temperature or barometric pressure. In precisely -the same way it is possible to inscribe in a net of rectangular -co-ordinates the outline, for instance, of a fish, -and so to translate <span class="xxpn" id="p724">{724}</span> -it into a table of numbers, from which again we may at pleasure -reconstruct the curve.</p> - -<p>But it is the next step in the employment of co-ordinates -which is of special interest and use to the morphologist; and this -step consists in the alteration, or “transformation,” of our system -of co-ordinates and in the study of the corresponding transformation -of the curve or figure inscribed in the co-ordinate -network.</p> - -<p>Let us inscribe in a system of Cartesian co-ordinates the outline -of an organism, however complicated, or a part thereof: such as -a fish, a crab, or a mammalian skull. We may now treat this -complicated figure, in general terms, as a function of <i>x</i>, <i>y</i>. If we -submit our rectangular system to “deformation,” on simple and -recognised lines, altering, for instance, the direction of the axes, -the ratio of <i>x ⁄ y</i>, or substituting -for <i>x</i> and <i>y</i> some more complicated -expressions, then we shall obtain a new system of co-ordinates, -whose deformation from the original type the inscribed figure -will precisely follow. In other words, we obtain a new figure, -which represents the old figure <i>under strain</i>, and is a function of -the new co-ordinates in precisely the same way as the old figure -was of the original co-ordinates <i>x</i> and <i>y</i>.</p> - -<p>The problem is closely akin to that of the cartographer who -transfers identical data to one projection or another; and whose -object is to secure (if it be possible) a complete correspondence, -<i>in each small unit of area</i>, between the one representation and the -other. The morphologist will not seek to draw his organic forms -in a new and artificial projection; but, in the converse aspect of -the problem, he will inquire whether two different but more or -less obviously related forms can be so analysed and interpreted -that each may be shown to be a transformed representation of -the other. This once demonstrated, it will be a comparatively -easy task (in all probability) to postulate the direction and -magnitude of the force capable of effecting the required transformation. -Again, if such a simple alteration of the system of -forces can be proved adequate to meet the case, we may find -ourselves able to dispense with many widely current and more -complicated hypotheses of biological causation. For it is a -maxim in physics that an effect ought not to -be ascribed to <span class="xxpn" id="p725">{725}</span> -the joint operation of many causes if few are adequate to the -production of it: <i>Frustra fit per plura, quod fieri potest per -pauciora.</i></p> - -<hr class="hrblk"> - -<p>It is evident that by the combined action of appropriate -forces any material form can be transformed into any other: -just as out of a “shapeless” mass of clay the potter or the sculptor -models his artistic product; or just as we attribute to Nature -herself the power to effect the gradual and successive transformation -of the simplest into the most complex organism. In -like manner it is possible, at least theoretically, to cause the outline -of any closed curve to appear as a projection of any other whatsoever. -But we need not let these theoretical considerations -deter us from our method of comparison of <i>related</i> forms. We -shall strictly limit ourselves to cases where the transformation -necessary to effect a comparison shall be of a simple kind, and -where the transformed, as well as the original, co-ordinates shall -constitute an harmonious and more or less symmetrical system. -We should fall into deserved and inevitable confusion if, whether -by the mathematical or any other method, we attempted to -compare organisms separated far apart in Nature and in zoological -classification. We are limited, not by the nature of our method, -but by the whole nature of the case, to the comparison of -organisms such as are manifestly related to one another and belong -to the same zoological class.</p> - -<p>Our inquiry lies, in short, just within the limits which Aristotle -himself laid down when, in defining a “genus,” he showed that -(apart from those superficial characters, such as colour, which he -called “accidents”) the essential differences between one “species” -and another are merely differences of proportion, of relative -magnitude, or (as he phrased it) of “excess and defect.” “Save -only for a difference in the way of excess or defect, the parts are -identical in the case of such animals as are of one and the same -genus; and by ‘genus’ I mean, for instance, Bird or Fish.” -And again: “Within the limits of the same genus, as a general -rule, most of the parts exhibit differences ... in the way of multitude -or fewness, magnitude or parvitude, in short, in the way of excess -or defect. For ‘the more’ and ‘the less’ may -be represented as <span class="xxpn" id="p726">{726}</span> -‘excess’ and ‘defect<a class="afnanch" href="#fn644" id="fnanch644">644</a>.’ ” -It is precisely this difference of relative -magnitudes, this Aristotelian “excess and defect” in the case of -form, which our co-ordinate method is especially adapted to -analyse, and to reveal and demonstrate as the main cause of what -(again in the Aristotelian sense) we term “specific” differences.</p> - -<p>The applicability of our method to particular cases will depend -upon, or be further limited by, certain practical considerations -or qualifications. Of these the chief, and indeed the essential, -condition is, that the form of the entire structure under investigation -should be found to vary in a more or less uniform manner, -after the fashion of an approximately homogeneous and isotropic -body. But an imperfect isotropy, provided always that some -“principle of continuity” run through its variations, will not -seriously interfere with our method; it will only cause our transformed -co-ordinates to be somewhat less regular and harmonious -than are those, for instance, by which the physicist depicts the -motions of a perfect fluid or a theoretic field of force in a uniform -medium.</p> - -<p>Again, it is essential that our structure vary in its entirety, -or at least that “independent variants” should be relatively few. -That independent variations occur, that localised centres of -diminished or exaggerated growth will now and then be found, -is not only probable but manifest; and they may even be so -pronounced as to appear to constitute new formations altogether. -Such independent variants as these Aristotle himself clearly -recognised: “It happens further that some have parts that others -have not; for instance, some [birds] have spurs and others not, -some have crests, or combs, and others not; but, as a general -rule, most parts and those that go to make up the bulk of the body -are either identical with one another, or differ from one another -in the way of contrast and of excess and defect. For ‘the more’ -and ‘the less’ may be represented as ‘excess’ or ‘defect.’ ”</p> - -<p>If, in the evolution of a fish, for instance, it be the case that -its several and constituent parts—head, body, and tail, or this -fin and that fin—represent so many independent variants, then -our co-ordinate system will at once become too complex to be -intelligible; we shall be making not one -comparison but several <span class="xxpn" id="p727">{727}</span> -separate comparisons, and our general method will be found -inapplicable. Now precisely this independent variability of parts -and organs—here, there, and everywhere within the organism—would -appear to be implicit in our ordinary accepted notions -regarding variation; and, unless I am greatly mistaken, it is -precisely on such a conception of the easy, frequent, and normal -independent variability of parts that our conception of the process -of natural selection is fundamentally based. For the morphologist, -when comparing one organism with another, describes the -differences between them point by point, and “character” by -“character<a class="afnanch" href="#fn645" id="fnanch645">645</a>.” -If he is from time to time constrained to admit -the existence of “correlation” between characters (as a hundred -years ago Cuvier first showed the way), yet all the while he -recognises this fact of correlation somewhat vaguely, as a phenomenon -due to causes which, except in rare instances, he can hardly -hope to trace; and he falls readily into the habit of thinking and -talking of evolution as though it had proceeded on the lines of his -own descriptions, point by point, and character by character<a class="afnanch" href="#fn646" id="fnanch646">646</a>.</p> - -<p>But if, on the other hand, diverse and dissimilar fishes can be -referred as a whole to identical functions of very different co-ordinate -systems, this fact will of itself constitute a proof that -variation has proceeded on definite and orderly lines, that a -comprehensive “law of growth” has pervaded the whole structure -in its integrity, and that some more or less simple and recognisable -system of forces has been at work. It will not only show -how real and deep-seated is the phenomenon of “correlation,” -in regard to form, but it will also demonstrate the fact that -a correlation which had seemed too complex -for analysis or <span class="xxpn" id="p728">{728}</span> -comprehension is, in many cases, capable of very simple graphic -expression. This, after many trials, I believe to be in general -the case, bearing always in mind that the occurrence of independent -or localised variations must often be considered.</p> - -<div class="psmprnt3"> -<p>We are dealing in this chapter with the forms of related -organisms, in order to shew that the differences between -them are as a general rule simple and symmetrical, and just -such as might have been brought about by a slight and simple -change in the system of forces to which the living and -growing organism was exposed. Mathematically speaking, the -phenomenon is identical with one met with by the geologist, -when he finds a bed of fossils squeezed flat or otherwise -symmetrically deformed by the pressures to which they, and -the strata which contain them, have been subjected. In the -first step towards fossilisation, when the body of a fish -or shellfish is silted over and buried, we may take it that -the wet sand or mud exercises, approximately, a hydrostatic -pressure—that is to say a pressure which is uniform in all -directions, and by which the form of the buried object will -not be appreciably changed. As the strata consolidate and -accumulate, the fossil organisms which they contain will tend -to be flattened by the vast superincumbent load, just as -the stratum which contains them will also be compressed and -will have its molecular arrangement more or less modified<a -class="afnanch" href="#fn647" id="fnanch647">647</a>. But the -deformation due to direct vertical pressure in a horizontal -stratum is not nearly so striking as are the deformations -produced by the oblique or shearing stresses to which inclined -and folded strata have been exposed, and by which their various -“dislocations” have been brought about. And especially in -mountain regions, where these dislocations are especially -numerous and complicated, the contained fossils are apt to be -so curiously and yet so symmetrically deformed (usually by a -simple shear) that they may easily be interpreted as so many -distinct and separate “species<a class="afnanch" href="#fn648" -id="fnanch648">648</a>.” A great number of described species, -and here and there a new genus (as the genus Ellipsolithes -for an obliquely deformed Goniatite or Nautilus) are said to -rest on no other foundation<a class="afnanch" href="#fn649" -id="fnanch649">649</a>.</p> -</div><!--psmprnt3--> - -<hr class="hrblk"> - -<p>If we begin by drawing a net of rectangular equidistant -co-ordinates (about the axes <i>x</i> and <i>y</i>), we may alter or -<i>deform</i> this <span class="xxpn" id="p729">{729}</span> -network in various ways, several of which are very simple -indeed. Thus (1) we may alter the dimensions of our system, -extending it along one or other axis, and so converting each -little square into a corresponding and directly proportionate -oblong (Fig. <a href="#fig353" title="go to Fig. 353">353</a>). It follows that any figure which we may -have inscribed in the</p> - -<div class="dctr01" id="fig352"><div id="fig353"> -<img src="images/i729a.png" width="784" height="394" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 352.</td> - <td></td> - <td>Fig. 353.</td></tr></table> -</div></div></div><!--dctr01--> - -<div class="dctr01" id="fig354"><div id="fig355"> -<img src="images/i729b.png" width="784" height="421" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 354.</td> - <td></td> - <td>Fig. 355.</td></tr></table> -</div></div></div><!--dctr01--> - -<p class="pcontinue"> -original net, and which we transfer -to the new, will thereby be <i>deformed</i> in strict proportion -to the deformation of the entire configuration, being still -defined by corresponding points in the network and being -throughout in conformity with the original figure. For -instance, a circle inscribed in the original “Cartesian” -net will now, after extension in the <i>y</i>-direction, be -found elongated <span class="xxpn" id="p730">{730}</span> -into an ellipse. In elementary mathematical language, -for the original <i>x</i> and <i>y</i> we have substituted -<i>x</i><sub>1</sub> and <i>cy</i><sub>1</sub> , -and the equation to our original circle, -<i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> -= <i>a</i><sup>2</sup> , becomes that of the ellipse, -<i>x</i><sub>1</sub><sup>2</sup> + <i>c</i><sup>2</sup> <i>y</i><sub>1</sub><sup>2</sup> -= <i>a</i><sup>2</sup> .</p> - -<p>If I draw the cannon-bone of an ox (Fig. <a href="#fig354" title="go to Fig. 354">354</a>, A), for instance, -within a system of rectangular co-ordinates, and then transfer -the same drawing, point for point, to a system in which for the -<i>x</i> of the original diagram we substitute <i>x′</i> -= 2<i>x</i> ⁄ 3, we obtain a -drawing (B) which is a very close approximation to the cannon-bone -of the sheep. In other words, the main (and perhaps -the only) difference between the two bones is simply that that of -the sheep is elongated, along the vertical axis, as compared with -that of the ox in the relation of 3 ⁄ 2. And similarly, the long -slender cannon-bone of the giraffe (C) is referable to the same -identical type, subject to a reduction of breadth, or increase -of length, corresponding to <i>x″</i> -= <i>x</i> ⁄ 3.</p> - -<p>(2) The second type is that where extension is not equal or -uniform at all distances from the origin: but grows greater -or less, as, for instance, when we stretch a <i>tapering</i> -elastic band. In such cases, as I have represented it in -Fig. <a href="#fig355" title="go to Fig. 355">355</a>, the ordinate increases logarithmically, and for -<i>y</i> we substitute ε<sup class="spitc">y</sup> . -It is obvious that this logarithmic extension may involve -both abscissae and ordinates, <i>x</i> becoming ε<sup -class="spitc">x</sup> , while <i>y</i> becomes ε<sup -class="spitc">y</sup> . The circle in our original -figure is now deformed into some such shape as that of Fig. -<a href="#fig356" title="go to Fig. 356">356</a>. This method of deformation is a common one, and will often -be of use to us in our comparison of organic forms.</p> - -<p>(3) Our third type is the “simple shear,” where the -rectangular co-ordinates become “oblique,” their axes -being inclined to one another at a certain angle ω. Our -original rectangle now becomes such a figure as that of -Fig. <a href="#fig357" title="go to Fig. 357">357</a>. The system may now be described in terms of the -oblique axes <i>X</i>, <i>Y</i>; or may be directly referred to new -rectangular co-ordinates ξ, η by the simple transposition -<i>x</i> = ξ − η cot ω, <i>y</i> -= η cosec ω.</p> - -<p>(4) Yet another important class of deformations may be -represented by the use of radial co-ordinates, in which one set of -lines are represented as radiating from a point or “focus,” while -the other set are transformed into circular arcs cutting the radii -orthogonally. These radial co-ordinates -are especially applicable <span class="xxpn" id="p731">{731}</span> -to cases where there exists (either within or without the figure) -some part which is supposed to suffer no deformation; a simple -illustration is afforded by the diagrams which illustrate the -flexure of a beam (Fig. <a href="#fig358" title="go to Fig. 358">358</a>). In biology these co-ordinates will</p> - -<div class="dctr06" id="fig356"> -<img src="images/i731a.png" width="400" height="317" alt=""> - <div class="dcaption">Fig. 356.</div></div> - -<div class="dctr03" id="fig357"> -<img src="images/i731b.png" width="608" height="209" alt=""> - <div class="dcaption">Fig. 357.</div></div> - -<div class="dctr06" id="fig358"> -<img src="images/i731c.png" width="401" height="255" alt=""> - <div class="dcaption">Fig. 358.</div></div> - -<p class="pcontinue"> -be especially applicable in cases where the growing structure -includes a “node,” or point where growth is absent or at a -minimum; and about which node the rate of growth may be -assumed to increase symmetrically. Precisely such a case is -furnished us in a leaf of an ordinary dicotyledon. The -leaf of a <span class="xxpn" id="p732">{732}</span> -typical monocotyledon—such as a grass or a hyacinth, for instance—grows -continuously from its base, and exhibits no node or “point -of arrest.” Its sides taper off gradually from its broad base to -its slender tip, according to some law of decrement specific to -the plant; and any alteration in the relative velocities of longitudinal -and transverse growth will merely make the leaf a little -broader or narrower, and will effect no other conspicuous alteration -in its contour. But if there once come into existence a node, or -“locus of no growth,” about which we may assume the growth—which -in the hyacinth leaf was longitudinal and transverse—to -take place radially and transversely to the radii, then we shall</p> - -<div class="dctr02" id="fig359"> -<img src="images/i732.png" width="704" height="485" alt=""> - <div class="dcaption">Fig. 359.</div></div> - -<p class="pcontinue">at once see, in the first place, that the sloping and slightly curved -sides of the hyacinth leaf suffer a transformation into what we -consider a more typical and “leaf-like” shape, the sides of the -figure broadening out to a zone of maximum breadth and then -drawing inwards to the pointed apex. If we now alter the ratio -between the radial and tangential velocities of growth—in other -words, if we increase the angles between corresponding radii—we -pass successively through the various configurations which -the botanist describes as the lanceolate, the ovate, and finally -the cordate leaf. These successive changes may to some extent, -and in appropriate cases, be traced as the -individual leaf grows <span class="xxpn" id="p733">{733}</span> -to maturity; but as a much more general rule, the balance -of forces, the ratio between radial and tangential velocities of -growth, remains so nicely and constantly balanced that the leaf -increases in size without conspicuous modification of form. It is -rather what we may call a long-period variation, a tendency for -the relative velocities to alter from one generation to another, -whose result is brought into view by this method of illustration.</p> - -<p>There are various corollaries to this method of describing the -form of a leaf which may be here alluded to, for we shall not return -again to the subject of radial co-ordinates. For instance, the -so-called unsymmetrical leaf<a class="afnanch" href="#fn650" id="fnanch650">650</a> -of a begonia, in which one side of -the leaf may be merely ovate while the other has a cordate outline,</p> - -<div class="dright dwth-e" id="fig360"> -<img src="images/i733.png" width="384" height="545" alt=""> - <div class="dcaption">Fig. 360. <i>Begonia daedalea.</i></div></div> - -<p class="pcontinue"> -is seen to be really a case of -<i>unequal</i>, and not truly asymmetrical, -growth on either side -of the midrib. There is nothing -more mysterious in its conformation -than, for instance, in that -of a forked twig in which one -limb of the fork has grown -longer than the other. The case -of the begonia leaf is of sufficient -interest to deserve illustration, -and in Fig. <a href="#fig360" title="go to Fig. 360">360</a> I have outlined -a leaf of the large <i>Begonia daedalea</i>. -On the smaller left-hand -side of the leaf I have taken at -random three points, <i>a</i>, <i>b</i>, <i>c</i>, and -have measured the angles, <i>AOa</i>, -etc., which the radii from the -hilus of the leaf to these points make with the median axis. On -the other side of the leaf I have marked the points <i>a′</i>, <i>b′</i>, <i>c′</i>, such -that the radii drawn to this margin of the leaf are equal to the -former, <i>Oa′</i> to <i>Oa</i>, etc. Now if the two sides -of the leaf are <span class="xxpn" id="p734">{734}</span> -mathematically similar to one another, it is obvious that the -respective angles should be in continued proportion, i.e. as <i>AOa</i> -is to <i>AOa′</i>, so should <i>AOb</i> be to <i>AOb′</i>. This proves to be very -nearly the case. For I have measured the three angles on one -side, and one on the other, and have then compared, as follows, -the calculated with the observed values of the other two: -<br class="brclrfix"></p> - -<div class="dtblbox"><div class="nowrap"> -<table class="fsz7"> -<tr> - <th></th> - <th><i>AOa</i></th> - <th><i>AOb</i></th> - <th><i>AOc</i></th> - <th><i>AOa′</i></th> - <th><i>AOb′</i></th> - <th><i>AOc′</i></th></tr> -<tr> - <td class="tdleft">Observed values</td> - <td class="tdcntr">12°</td> - <td class="tdcntr">28.5°</td> - <td class="tdcntr">88°</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">157°</td></tr> -<tr> - <td class="tdleft">Calculated values</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">21.5°</td> - <td class="tdcntr">51.1°</td> - <td class="tdcntr">—</td></tr> -<tr> - <td class="tdleft">Observed values</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">—</td> - <td class="tdcntr">20   </td> - <td class="tdcntr">52   </td> - <td class="tdcntr">—</td></tr> -</table></div></div><!--dtblbox--> - -<p>The agreement is very close, and what discrepancy there is -may be amply accounted for, firstly, by the slight irregularity -of the sinuous margin of the leaf; and secondly, by the fact that -the true axis or midrib of the leaf is not straight but slightly -curved, and therefore that it is curvilinear and not rectilinear -triangles which we ought to have measured. When we understand -these few points regarding the peripheral curvature of the -leaf, it is easy to see that its principal veins approximate closely -to a beautiful system of isogonal co-ordinates. It is also obvious -that we can easily pass, by a process of shearing, from those cases -where the principal veins start from the base of the leaf to those, -as in most dicotyledons, where they arise successively from the -midrib.</p> - -<p>It may sometimes happen that the node, or “point of arrest,” -is at the upper instead of the lower end of the leaf-blade; and -occasionally there may be a node at both ends. In the former case, -as we have it in the daisy, the form of the leaf will be, as it were, -inverted, the broad, more or less heart-shaped, outline appearing -at the upper end, while below the leaf tapers gradually downwards -to an ill-defined base. In the latter case, as in <i>Dionaea</i>, we obtain -a leaf equally expanded, and similarly ovate or cordate, at both -ends. We may notice, lastly, that the shape of a solid fruit, -such as an apple or a cherry, is a solid of revolution, developed -from similar curves and to be explained on the same principle. -In the cherry we have a “point of arrest” at the base of the berry, -where it joins its peduncle, and about this point the fruit (in -imaginary section) swells out into a cordate outline; -while in the <span class="xxpn" id="p735">{735}</span> -apple we have two such well-marked points of arrest, above and -below, and about both of them the same conformation tends to -arise. The bean and the human kidney owe their “reniform” -shape to precisely the same phenomenon, namely, to the existence -of a node or “hilus,” about which the forces of growth are radially -and symmetrically arranged.</p> - -<hr class="hrblk"> - -<p>Most of the transformations which we have hitherto considered -(other than that of the simple shear) are particular cases of a -general transformation, obtainable by the method of conjugate -functions and equivalent to the projection of the original figure -on a new plane. Appropriate transformations, on these general -lines, provide for the cases of a coaxial system where the -Cartesian co-ordinates are replaced by coaxial circles, or a confocal -system in which they are replaced by confocal ellipses and -hyperbolas.</p> - -<p>Yet another curious and important transformation, belonging -to the same class, is that by which a system of straight lines -becomes transformed into a conformal system of logarithmic -spirals: the straight line <i>Y</i>−<i>AX</i> -= <i>c</i> corresponding to the -logarithmic spiral θ − <i>A</i> log <i>r</i> -= <i>c</i> (Fig. <a href="#fig361" title="go to Fig. 361">361</a>). This beautiful and</p> - -<div class="dright dwth-e" id="fig361"> -<img src="images/i735.png" width="336" height="328" alt=""> - <div class="dcaption">Fig. 361.</div></div> - -<p class="pcontinue"> -simple transformation lets us at once -convert, for instance, the straight -conical shell of the Pteropod or the -<i>Orthoceras</i> into the logarithmic spiral -of the Nautiloid; it involves a mathematical -symbolism which is but a -slight extension of that which we -have employed in our elementary -treatment of the logarithmic spiral.</p> - -<p>These various systems of co-ordinates, -which we have now briefly -considered, are sometimes called “isothermal -co-ordinates,” from the fact that, when employed in -this particular branch of physics, they perfectly represent the -phenomena of the conduction of heat, the contour lines of equal -temperature appearing, under appropriate conditions, as the -orthogonal lines of the co-ordinate system. And -it follows that <span class="xxpn" id="p736">{736}</span> -the “law of growth” which our biological analysis by means of -orthogonal co-ordinate systems presupposes, or at least foreshadows, -is one according to which the organism grows or -develops along <i>stream lines</i>, which may be defined by a suitable -mathematical transformation. <br class="brclrfix"></p> - -<p>When the system becomes no longer orthogonal, as in many -of the following illustrations—for instance, that of <i>Orthagoriscus</i> -(Fig. <a href="#fig382" title="go to Fig. 382">382</a>),—then the transformation is no longer within the reach -of comparatively simple mathematical analysis. Such departure -from the typical symmetry of a “stream-line” system is, in the -first instance, sufficiently accounted for by the simple fact that -the developing organism is very far from being homogeneous and -isotropic, or, in other words, does not behave like a perfect fluid. -But though under such circumstances our co-ordinate systems -may be no longer capable of strict mathematical analysis, they -will still indicate <i>graphically</i> the relation of the new co-ordinate -system to the old, and conversely will furnish us with some -guidance as to the “law of growth,” or play of forces, by which -the transformation has been effected.</p> - -<hr class="hrblk"> - -<p>Before we pass from this brief discussion of transformations in -general, let us glance at one or two cases in which the forces applied -are more or less intelligible, but the resulting transformations are, -from the mathematical point of view, exceedingly complicated.</p> - -<p>The “marbled papers” of the bookbinder are a beautiful -illustration of visible “stream lines.” On a dishful of a sort of -semi-liquid gum the workman dusts a few simple lines or patches -of colouring matter; and then, by passing a comb through the -liquid, he draws the colour-bands into the streaks, waves, and -spirals which constitute the marbled pattern, and which he then -transfers to sheets of paper laid down upon the gum. By some -such system of shears, by the effect of unequal traction or unequal -growth in various directions and superposed on an originally -simple pattern, we may account for the not dissimilar marbled -patterns which we recognise, for instance, on a large serpent’s -skin. But it must be remarked, in the case of the marbled paper, -that though the method of application of the forces is simple, -yet in the aggregate the system of forces set up -by the many <span class="xxpn" id="p737">{737}</span> -teeth of the comb is exceedingly complex, and its complexity is -revealed in the complicated “diagram of forces” which constitutes -the pattern.</p> - -<p>To take another and still more instructive illustration. To -turn one circle (or sphere) into two circles would be, from the point -of view of the mathematician, an extraordinarily difficult transformation; -but, physically speaking, its achievement may be -extremely simple. The little round gourd grows naturally, by -its symmetrical forces of expansive growth, into a big, round, or -somewhat oval pumpkin or melon. But the Moorish husbandman -ties a rag round its middle, and the same forces of growth, unaltered -save for the presence of this trammel, now expand the globular -structure into two superposed and connected globes. And -again, by varying the position of the encircling band, or by -applying several such ligatures instead of one, a great variety of -artificial forms of “gourd” may be, and actually are, produced. -It is clear, I think, that we may account for many ordinary -biological processes of development or transformation of form by -the existence of trammels or lines of constraint, which limit and -determine the action of the expansive forces of growth that would -otherwise be uniform and symmetrical. This case has a close -parallel in the operations of the glassblower, to which we have -already, more than once, referred in passing<a class="afnanch" href="#fn651" id="fnanch651">651</a>. -The glassblower -starts his operations with a <i>tube</i>, which he first closes at one end -so as to form a hollow vesicle, within which his blast of air exercises -a uniform pressure on all sides; but the spherical conformation -which this uniform expansive force would naturally tend to -produce is modified into all kinds of forms by the trammels or -resistances set up as the workman lets one part or another of his -bubble be unequally heated or cooled. It was Oliver Wendell -Holmes who first shewed this curious parallel between the -operations of the glassblower and those of Nature, when she starts, -as she so often does, with a simple tube<a class="afnanch" href="#fn652" id="fnanch652">652</a>. -The alimentary canal, <span class="xxpn" id="p738">{738}</span> -the arterial system including the heart, the central nervous -system of the vertebrate, including the brain itself, all begin as -simple tubular structures. And with them Nature does just -what the glassblower does, and, we might even say, no more -than he. For she can expand the tube here and narrow it there; -thicken its walls or thin them; blow off a lateral offshoot or -caecal diverticulum; bend the tube, or twist and coil it; and -infold or crimp its walls as, so to speak, she pleases. Such a form -as that of the human stomach is easily explained when it is -regarded from this point of view; it is simply an ill-blown bubble, -a bubble that has been rendered lopsided by a trammel or restraint -along one side, such as to prevent its symmetrical expansion—such -a trammel as is produced if the glassblower lets one side of -his bubble get cold, and such as is actually present in the stomach -itself in the form of a muscular band.</p> - -<hr class="hrblk"> - -<p>We may now proceed to consider and illustrate a few permutations -or transformations of organic form, out of the vast -multitude which are equally open to this method of inquiry.</p> - -<div class="dleft dwth-e" id="fig362"> -<img src="images/i738.png" width="336" height="381" alt=""> - <div class="dcaption">Fig. 362.</div></div> - -<p>We have already compared in a preliminary fashion the -metacarpal or cannon-bone of the ox, the sheep, and the giraffe -(Fig. <a href="#fig354" title="go to Fig. 354">354</a>); and we have seen that the essential difference in form -between these three bones is a matter -of relative length and breadth, such -that, if we reduce the figures to an -identical standard of length (or identical -values of <i>y</i>), the breadth (or value of -<i>x</i>) will be approximately two-thirds -that of the ox in the case of the sheep -and one-third that of the ox in the -case of the giraffe. We may easily, -for the sake of closer comparison, -determine these ratios more accurately, -for instance, if it be our purpose to -compare the different racial varieties -within the limits of a single species. -And in such cases, by the way, as when we compare with one -another various breeds or races of cattle or of -horses, the ratios <span class="xxpn" id="p739">{739}</span> -of length and breadth in this particular bone are extremely -significant<a class="afnanch" href="#fn653" id="fnanch653">653</a>. -<br class="brclrfix"></p> - -<p>If, instead of limiting ourselves to the cannon-bone, we inscribe -the entire foot of our several Ungulates in a co-ordinate system, -the same ratios of <i>x</i> that served us for the cannon-bones still give -us a first approximation to the required comparison; but even -in the case of such closely allied forms as the ox and the sheep -there is evidently something wanting in the comparison. The -reason is that the relative elongation of the several parts, or -individual bones, has not proceeded equally or proportionately -in all cases; in other words, that the equations for <i>x</i> will not -suffice without some simultaneous modification of the values of -<i>y</i> (Fig. <a href="#fig362" title="go to Fig. 362">362</a>). In such a case it may be found possible to satisfy -the varying values of <i>y</i> by some logarithmic or other formula; -but, even if that be possible, it will probably be somewhat difficult -of discovery or verification in such a case as the present, owing -to the fact that we have too few well-marked points of correspondence -between the one object and the other, and that especially -along the shaft of such long bones as the cannon-bone of the ox, -the deer, the llama, or the giraffe there is a complete lack of easily -recognisable corresponding points. In such a case a brief tabular -statement of apparently corresponding values of <i>y</i>, or of those -obviously corresponding values which coincide with the boundaries -of the several bones of the foot, will, as in the following example, -enable us to dispense with a fresh equation.</p> - -<div class="dtblbox"><div class="nowrap"> -<table class="fsz6"> -<tr> - <th colspan="2"> </th> - <th><i>a</i></th> - <th><i>b</i></th> - <th><i>c</i></th> - <th><i>d</i></th></tr> -<tr> - <td class="tdleft"><i>y</i> (Ox)</td> - <td class="tdcntr">0</td> - <td class="tdcntr">18</td> - <td class="tdcntr">27</td> - <td class="tdcntr">42</td> - <td class="tdcntr">100</td></tr> -<tr> - <td class="tdleft"><i>y′</i> (Sheep)</td> - <td class="tdcntr">0</td> - <td class="tdcntr">10</td> - <td class="tdcntr">19</td> - <td class="tdcntr">36</td> - <td class="tdcntr">100</td></tr> -<tr> - <td class="tdleft"><i>y″</i> (Giraffe)</td> - <td class="tdcntr">0</td> - <td class="tdcntr"> 5</td> - <td class="tdcntr">10</td> - <td class="tdcntr">24</td> - <td class="tdcntr">100</td></tr> -</table></div></div><!--dtblbox--> - -<p class="pcontinue">This summary of values of <i>y′</i>, coupled with the -equations for the <span class="xxpn" id="p740">{740}</span> -value of <i>x</i>, will enable us, from any drawing of the ox’s foot, to -construct a figure of that of the sheep or of the giraffe with -remarkable accuracy.</p> - -<div class="dleft dwth-d" id="fig363"> -<img src="images/i740a.png" width="391" height="377" alt=""> - <div class="dcaption">Fig. 363.</div></div> - -<p>That underlying the varying amounts of extension -to which the parts or segments of the limb have been -subject there is a law, or principle of continuity, may -be discerned from such a diagram as the above (Fig. <a href="#fig363" title="go to Fig. 363">363</a>), -where the values of <i>y</i> in the case of the ox are plotted -as a straight line, and the corresponding values for the -sheep (extracted from the above table) are seen to form a -more or less regular and even curve. This simple graphic -result implies the existence of a comparatively simple -equation between <i>y</i> and <i>y′</i>.</p> - -<p>An elementary application of the principle of -co-ordinates to the study of proportion, as we have -here used it to illustrate the varying proportions of a -bone, was in common use in the sixteenth and seventeenth -centuries by artists in their study of the human form. -The method is probably much more ancient, and may -even be classical<a class="afnanch" href="#fn654" -id="fnanch654">654</a>; <br class="brclrfix"></p> - -<div class="dctr02" id="fig364"> -<img src="images/i740b.png" width="704" height="229" alt=""> - <div class="dcaption">Fig. 364. (After Albert Dürer.)</div></div> - -<p class="pcontinue"> -it is fully described and put in practice by -Albert Dürer in his <i>Geometry</i>, and especially in his <i>Treatise on -Proportion</i><a class="afnanch" href="#fn655" id="fnanch655">655</a>. -In this latter work, the -manner in which the <span class="xxpn" id="p741">{741}</span> -human figure, features, and facial expression are all transformed -and modified by slight variations in the relative magnitude of -the parts is admirably and copiously illustrated (Fig. <a href="#fig364" title="go to Fig. 364">364</a>).</p> - -<p>In a tapir’s foot there is a striking difference, and yet at the -same time there is an obvious underlying resemblance, between -the middle toe and either of its unsymmetrical lateral neighbours. -Let us take the median terminal phalanx and inscribe its outline -in a net of rectangular equidistant co-ordinates (Fig. <a href="#fig365" title="go to Fig. 365">365</a>, <i>a</i>). Let -us then make a similar network about axes which are no longer -at right angles, but inclined to one another at an angle of about -50° (<i>b</i>). If into this new network we fill in, point for point, -an outline precisely corresponding to our original drawing of the -middle toe, we shall find that we have already represented the -main features of the adjacent lateral one. We shall, however, -perceive</p> - -<div class="dctr01" id="fig365"> -<img src="images/i741.png" width="800" height="242" alt=""> - <div class="dcaption">Fig. 365.</div></div> - -<p class="pcontinue"> that our new diagram looks a little -too bulky on one side, the inner side, of the lateral -toe. If now we substitute for our equidistant ordinates, -ordinates which get gradually closer and closer together -as we pass towards the median side of the toe, then we -shall obtain a diagram which differs in no essential -respect from an actual outline copy of the lateral toe -(<i>c</i>). In short, the difference between the outline -of the middle toe of the tapir and the next lateral -toe may be almost completely expressed by saying that -if the one be represented by rectangular equidistant -co-ordinates, the other will be represented by oblique -co-ordinates, whose axes make an angle of 50°, and in -which the abscissal interspaces decrease in a certain -logarithmic ratio. We treated our original complex curve -or projection of the tapir’s toe as a function of the -form <em class="embold"><i>F</i></em>(<i>x</i>, <i>y</i>) -= 0. The figure of the tapir’s lateral -<span class="xxpn" id="p742">{742}</span> toe -is a precisely identical function of the form -<em class="embold"><i>F</i></em>(<i>e</i><sup -class="spitc">x</sup>, <i>y</i><sub>1</sub>) -= 0, where <i>x</i><sub>1</sub> , <i>y</i><sub>1</sub> -are oblique co-ordinate axes inclined to one another at an -angle of 50°.</p> - -<div class="dctr04" id="fig366"> -<img src="images/i742a.png" width="528" height="237" alt=""> - <div class="dcaption">Fig. 366. (After Albert Dürer.)</div></div> - -<p>Dürer was acquainted with these oblique co-ordinates -also, and I have copied two illustrative figures -from his book<a class="afnanch" href="#fn656" -id="fnanch656">656</a>.</p> - -<hr class="hrblk"> - -<p>In Fig. <a href="#fig367" title="go to Fig. 367">367</a> I have sketched the common -Copepod <i>Oithona nana</i>, <span class="xxpn" id="p743">{743}</span> -and have inscribed it in a rectangular net, with abscissae three-fifths -the</p> - -<div class="dctr01" id="fig367"><div id="fig368"> -<img src="images/i742b.png" width="800" height="386" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 367. <i>Oithona nana.</i></td> - <td></td> - <td>Fig. 368. <i>Sapphirina.</i></td></tr></table> -</div></div></div><!--dctr01--> - -<p class="pcontinue"> -length of the ordinates. Side by side (Fig. <a href="#fig368" title="go to Fig. 368">368</a>) is drawn -a very different Copepod, of the genus <i>Sapphirina</i>; and about -it is drawn a network such that each co-ordinate passes (as nearly -as possible) through points corresponding to those of the former -figure. It will be seen that two differences are apparent. (1) The -values of <i>y</i> in Fig. <a href="#fig368" title="go to Fig. 368">368</a> are large in the upper part of the figure, and -diminish rapidly towards its base. (2) The values of <i>x</i> are very -large in the neighbourhood of the origin, but diminish rapidly as -we pass towards either side, away from the median vertical axis; -and it is probable that they do so according to a definite, but -somewhat complicated, ratio. If, instead of seeking for an -actual equation, we simply tabulate our values of <i>x</i> and <i>y</i> in the -second figure as compared with the first (just as we did in comparing -the feet of the Ungulates), we get the dimensions of a net -in which, by simply projecting the figure of <i>Oithona</i>, we obtain -that of <i>Sapphirina</i> without further trouble, e.g.:</p> - -<div class="dtblbox"><div class="nowrap"> -<table class="fsz6"> -<tr> - <td class="tdleft"><i>x</i> (<i>Oithona</i>)</td> - <td class="tdcntr">0</td> - <td class="tdcntr">3</td> - <td class="tdcntr"> 6</td> - <td class="tdcntr"> 9</td> - <td class="tdcntr">12</td> - <td class="tdcntr">15</td> - <td class="tdcntr">—</td></tr> -<tr> - <td class="tdleft"><i>x′</i> (<i>Sapphirina</i>)</td> - <td class="tdcntr">0</td> - <td class="tdcntr">8</td> - <td class="tdcntr">10</td> - <td class="tdcntr">12</td> - <td class="tdcntr">13</td> - <td class="tdcntr">14</td> - <td class="tdcntr">—</td></tr> -<tr> - <td class="tdleft"><i>y</i> (<i>Oithona</i>)</td> - <td class="tdcntr">0</td> - <td class="tdcntr">5</td> - <td class="tdcntr">10</td> - <td class="tdcntr">15</td> - <td class="tdcntr">20</td> - <td class="tdcntr">25</td> - <td class="tdcntr">30</td></tr> -<tr> - <td class="tdleft"><i>y′</i> (<i>Sapphirina</i>)</td> - <td class="tdcntr">0</td> - <td class="tdcntr">2</td> - <td class="tdcntr"> 7</td> - <td class="tdcntr"> 3</td> - <td class="tdcntr">23</td> - <td class="tdcntr">32</td> - <td class="tdcntr">40</td></tr> -</table></div></div><!--dtblbox--> - -<p>In this manner, with a single model or type to copy from, we -may record in very brief space the data requisite for the production -of approximate outlines of a great number of forms. For instance -the difference, at first sight immense, between the attenuated -body of a <i>Caprella</i> and the thick-set body of a <i>Cyamus</i> is obviously -little, and is probably nothing, more than a difference of relative -magnitudes, capable of tabulation by numbers and of complete -expression by means of rectilinear co-ordinates.</p> - -<p>The Crustacea afford innumerable instances of more complex -deformations. Thus we may compare various higher Crustacea -with one another, even in the case of such dissimilar forms as a -lobster and a crab. It is obvious that the whole body of the -former is elongated as compared with the latter, and that the -crab is relatively broad in the region of the carapace, while it -tapers off rapidly towards its attenuated and abbreviated tail. -In a general way, the elongated rectangular -system of co-ordinates <span class="xxpn" id="p744">{744}</span> -in which we may inscribe the outline of the lobster becomes a -shortened triangle in the case of the crab. In a little more detail -we may compare the outline of the carapace in various crabs one -with another: and the comparison will be found easy and significant, -even, in many cases, down to minute details, such as the -number and situation of the marginal spines, though these are in -other cases subject to independent variability.</p> - -<div class="dctr01" id="fig369"> -<img src="images/i744.png" width="800" height="882" alt=""> - <div class="pcaption">Fig. 369. Carapaces of various - crabs. 1, <i>Geryon</i>; 2, <i>Corystes</i>; 3, <i>Scyramathia</i>; 4, - <i>Paralomis</i>; 5, <i>Lupa</i>; 6, <i>Chorinus</i>.</div></div> - -<p>If we choose, to begin with, such a crab as <i>Geryon</i> (Fig. <a href="#fig369" title="go to Fig. 369">369</a>, 1), -and inscribe it in our equidistant rectangular co-ordinates, we shall -see that we pass easily to forms more elongated -in a transverse <span class="xxpn" id="p745">{745}</span> -direction, such as <i>Matuta</i> or <i>Lupa</i> (5), and conversely, by -transverse compression, to such a form as <i>Corystes</i> (2). In -certain other cases the carapace conforms to a triangular diagram, -more or less curvilinear, as in Fig. <a href="#fig4" title="go to Fig. 4">4</a>, which represents -the genus <i>Paralomis</i>. Here we can easily see that the posterior -border is transversely elongated as compared with that of <i>Geryon</i>, -while at the same time the anterior part is longitudinally extended -as compared with the posterior. A system of slightly curved and -converging ordinates, with orthogonal and logarithmically interspaced -abscissal lines, as shown in the figure, appears to satisfy -the conditions.</p> - -<p>In an interesting series of cases, such as the genus <i>Chorinus</i>, -or <i>Scyramathia</i>, and in the spider-crabs generally, we appear to -have just the converse of this. While the carapace of these crabs -presents a somewhat triangular form, which seems at first sight -more or less similar to those just described, we soon see that the -actual posterior border is now narrow instead of broad, the -broadest part of the carapace corresponding precisely, not to -that which is broadest in <i>Paralomis</i>, but to that which was broadest -in <i>Geryon</i>; while the most striking difference from the latter lies -in an antero-posterior lengthening of the forepart of the carapace, -culminating in a great elongation of the frontal region, with its -two spines or “horns.” The curved ordinates here converge -posteriorly and diverge widely in front (Figs. <a href="#fig3" title="go to Fig. 3">3</a> and 6), while -the decremental interspacing of the abscissae is very marked -indeed.</p> - -<p>We put our method to a severer test when we attempt to sketch -an entire and complicated animal than when we simply compare -corresponding parts such as the carapaces of various Malacostraca, -or related bones as in the case of the tapir’s toes. Nevertheless, -up to a certain point, the method stands the test very well. In -other words, one particular mode and direction of variation is -often (or even usually) so prominent and so paramount throughout -the entire organism, that one comprehensive system of co-ordinates -suffices to give a fair picture of the actual phenomenon. To take -another illustration from the Crustacea, I have drawn roughly in -Fig. <a href="#fig370" title="go to Fig. 370">370</a>, 1 a little amphipod of the family Phoxocephalidae -(<i>Harpinia</i> sp.). Deforming the co-ordinates of the -figure into the <span class="xxpn" id="p746">{746}</span> -curved orthogonal system in Fig. <a href="#fig2" title="go to Fig. 2">2</a>, we at once obtain a very fair -representation of an allied genus, belonging to a different family -of amphipods, namely <i>Stegocephalus</i>. As we proceed further from -our type our co-ordinates will require greater deformation, and -the resultant figure will usually be somewhat less accurate. In -Fig. <a href="#fig3" title="go to Fig. 3">3</a> I show a network, to which, if we transfer our diagram of -<i>Harpinia</i> or of</p> - -<div class="dctr05" id="fig370"> -<img src="images/i746.png" width="448" height="665" alt=""> - <div class="pcaption">Fig 370. 1. <i>Harpinia plumosa</i> - Kr. 2. <i>Stegocephalus inflatus</i> Kr. 3. <i>Hyperia - galba</i>.</div></div> - -<p class="pcontinue"> -<i>Stegocephalus</i>, we shall obtain a tolerable representation -of the aberrant genus <i>Hyperia</i>, with its narrow abdomen, -its reduced pleural lappets, its great eyes, and its inflated head.</p> - -<hr class="hrblk"> - -<p>The hydroid zoophytes constitute a “polymorphic” group, -within which a vast number of species have already been distinguished; -and the labours of the systematic naturalist are -constantly adding to the number. The specific distinctions are -for the most part based, not upon -characters directly presented <span class="xxpn" id="p747">{747}</span> -by the living animal, but upon the form, size and arrangement -of the little cups, or “calycles,” secreted and inhabited by the -little individual polypes which compose the compound organism. -The variations, which are apparently infinite, of these conformations -are easily seen to be a question of relative magnitudes, and -are capable of complete expression, sometimes by very simple, -sometimes by somewhat more complex, co-ordinate networks.</p> - -<div class="dctr02" id="fig371"> -<img src="images/i747.png" width="704" height="372" alt=""> - <div class="pcaption">Fig. 371. <i>a</i>, <i>Campanularia - macroscyphus</i>, Allm.; <i>b</i>, <i>Gonothyraea hyalina</i>, Hincks; - <i>c</i>, <i>Clytia Johnstoni</i>, Alder.</div></div> - -<p>For instance, the varying shapes of the simple wineglass-shaped -cups of the Campanularidae are at once sufficiently -represented and compared by means of simple Cartesian co-ordinates -(Fig. <a href="#fig371" title="go to Fig. 371">371</a>). In the two allied families of Plumulariidae and -Aglaopheniidae the calycles are set unilaterally upon a jointed -stem, and small cup-like structures (holding rudimentary polypes) -are associated with the large calycles in definite number and -position. These small calyculi are variable in number, but in the -great majority of cases they accompany the large calycle in -groups of three—two standing by its upper border, and one, -which is especially variable in form and magnitude, lying at its -base. The stem is liable to flexure and, in a high degree, to -extension or compression; and these variations extend, often on -an exaggerated scale, to the related calycles. As a result we find -that we can draw various systems of curved or sinuous co-ordinates, -which express, all but completely, the configuration -of the various <span class="xxpn" id="p748">{748}</span> -hydroids which we inscribe therein (Fig. <a href="#fig372" title="go to Fig. 372">372</a>). The comparative -smoothness or denticulation of the margin of the calycle, and the -number of its denticles, constitutes an independent variation, and</p> - -<div class="dctr01" id="fig372"> -<img src="images/i748a.png" width="800" height="337" alt=""> - <div class="pcaption">Fig. 372. <i>a</i>, <i>Cladocarpus - crenatus</i>, F.; <i>b</i>, <i>Aglaophenia pluma</i>, L.; <i>c</i>, <i>A. - rhynchocarpa</i>, A.; <i>d</i>, <i>A cornuta</i>, K.; <i>e</i>, <i>A. - ramulosa</i>, K.</div></div> - -<p class="pcontinue"> -requires separate description; we have already seen (p. <a href="#p236" title="go to pg. 236">236</a>) that -this denticulation is in all probability due to a particular physical -cause.</p> - -<hr class="hrblk"> - -<p>Among the fishes we discover a great variety of deformations, -some of them of a very simple kind, while others are more striking -and more unexpected. A comparatively simple case, involving a -simple shear,</p> - -<div class="dctr01" id="fig373"><div id="fig374"> -<img src="images/i748b.png" width="800" height="292" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 373. <i>Argyropelecus Olfersi.</i></td> - <td></td> - <td>Fig. 374. <i>Sternoptyx diaphana.</i></td></tr></table> -</div></div></div><!--dctr01--> - -<p class="pcontinue"> -is illustrated by Figs. <a href="#fig373" title="go to -Fig. 373">373</a> and <a href="#fig374" title="go to Fig. -374">374</a>. Fig. <a href="#fig373" title="go to Fig. -373">373</a> represents, within Cartesian co-ordinates, -a certain little oceanic fish known as <i>Argyropelecus -Olfersi</i>. Fig. <a href="#fig374" title="go to Fig. -374">374</a> represents precisely the same outline, -transferred to a system of oblique co-ordinates whose -<span class="xxpn" id="p749">{749}</span> axes are -inclined at an angle of 70°; but this is now (as far as -can be seen on the scale of the drawing) a very good -figure of an allied fish, assigned to a different genus, -under the name of <i>Sternoptyx diaphana</i>. The deformation -illustrated by this case of <i>Argyropelecus</i> is precisely -analogous to the simplest and commonest kind of deformation -to which fossils are subject (as we have seen on p. <a -href="#p553" title="go to pg. 553">553</a>) as the result -of shearing-stresses in the solid rock.</p> - -<div class="dctr01" id="fig375"><div id="fig376"> -<img src="images/i749.png" width="800" height="276" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 375. <i>Scarus</i> sp.</td> - <td></td> - <td>Fig. 376. <i>Pomacanthus.</i></td></tr></table> -</div></div></div><!--dctr01--> - -<p>Fig. <a href="#fig375" title="go to Fig. 375">375</a> is an outline diagram of a typical Scaroid fish. Let us -deform its rectilinear co-ordinates into a system of (approximately) -coaxial circles, as in Fig. <a href="#fig376" title="go to Fig. 376">376</a>, and then filling into the new system, -space by space and point by point, our former diagram of <i>Scarus</i>, -we obtain a very good outline of an allied fish, belonging to a -neighbouring family, of the genus <i>Pomacanthus</i>. This case is all -the more interesting, because upon the body of our <i>Pomacanthus</i> -there are striking colour bands, which correspond in direction -very closely to the lines of our new curved ordinates. In like -manner, the still more bizarre outlines of other fishes of the same -family of Chaetodonts will be found to correspond to very slight -modifications of similar co-ordinates; in other words, to small -variations in the values of the constants of the coaxial curves.</p> - -<p>In Figs. <a href="#fig377" title="go to Fig. 377">377</a>–380 I have represented another series of Acanthopterygian -fishes, not very distantly related to the foregoing. If -we start this series with the figure of <i>Polyprion</i>, in Fig. <a href="#fig377" title="go to Fig. 377">377</a>, we see -that the outlines of <i>Pseudopriacanthus</i> (Fig. <a href="#fig378" title="go to Fig. 378">378</a>) and of <i>Sebastes</i> or -<i>Scorpaena</i> (Fig. <a href="#fig379" title="go to Fig. 379">379</a>) are easily derived by substituting a system -of triangular, or radial, co-ordinates for the -rectangular ones in <span class="xxpn" id="p750">{750}</span> -which we had inscribed <i>Polyprion</i>. The very curious fish <i>Antigonia -capros</i>, an oceanic relative of our own “boar-fish,” conforms -closely to the peculiar deformation represented in Fig. <a href="#fig380" title="go to Fig. 380">380</a>.</p> - -<div class="dctr01" id="fig377"><div id="fig378"> -<img src="images/i750a.png" width="800" height="361" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 377. <i>Polyprion.</i></td> - <td></td> - <td>Fig. 378. <i>Pseudopriacanthus altus.</i></td></tr></table> -</div></div></div><!--dctr01--> - -<div class="dctr01" id="fig379"><div id="fig380"> -<img src="images/i750b.png" width="800" height="400" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 379. <i>Scorpaena</i> sp.</td> - <td></td> - <td>Fig. 380. <i>Antigonia capros.</i></td></tr></table> -</div></div></div><!--dctr01--> - -<p>Fig. <a href="#fig381" title="go to Fig. 381">381</a> is a common, typical <i>Diodon</i> or porcupine-fish, and in -Fig. <a href="#fig382" title="go to Fig. 382">382</a> I have deformed its vertical co-ordinates into a system -of concentric circles, and its horizontal co-ordinates into a system -of curves which, approximately and provisionally, are made to -resemble a system of hyperbolas<a class="afnanch" href="#fn657" id="fnanch657">657</a>. -The -old outline, transferred <span class="xxpn" id="p751">{751}</span> -in its integrity to the new network, appears as a manifest -representation of the closely allied, but very different looking, -sunfish, <i>Orthagoriscus mola</i>. This is a particularly instructive -case of deformation or transformation. It is true that, in a -mathematical sense, it is not a perfectly satisfactory or perfectly -regular deformation, for the system is no longer isogonal; but</p> - -<div class="dctr01" id="fig381"><div id="fig382"> -<img src="images/i751.png" width="800" height="763" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 381. <i>Diodon.</i></td> - <td></td> - <td>Fig. 382. <i>Orthagoriscus.</i></td></tr></table> -</div></div></div><!--dctr01--> - -<p class="pcontinue"> -nevertheless, it is symmetrical to the eye, and obviously approaches -to an isogonal system under certain conditions of friction or -constraint. And as such it accounts, by one single integral -transformation, for all the apparently separate and distinct -external differences between the two fishes. It leaves the parts -near to the origin of the system, the whole region of the head, -the opercular orifice and the pectoral -fin, practically unchanged <span class="xxpn" id="p752">{752}</span> -in form, size and position; and it shews a greater and greater -apparent modification of size and form as we pass from the origin -towards the periphery of the system.</p> - -<p>In a word, it is sufficient to account for the new and striking -contour in all its essential details, of rounded body, exaggerated -dorsal and ventral fins, and truncated tail. In like manner, and -using precisely the same co-ordinate networks, it appears to me -possible to shew the relations, almost bone for bone, of the skeletons -of the two fishes; in other words, to reconstruct the skeleton of -the one from our knowledge of the skeleton of the other, under -the guidance of the same correspondence as is indicated in their -external configuration.</p> - -<hr class="hrblk"> - -<p>The family of the crocodiles has had a special interest for the -evolutionist ever since Huxley pointed out that, in a degree only -second to the horse and its ancestors, it furnishes us with a close -and almost unbroken series of transitional forms, running down -in continuous succession from one geological formation to another. -I should be inclined to transpose this general statement into other -terms, and to say that the Crocodilia constitute a case in which, -with unusually little complication from the presence of independent -variants, the trend of one particular mode of transformation is -visibly manifested. If we exclude meanwhile from our comparison -a few of the oldest of the crocodiles, such as <i>Belodon</i>, which differ -more fundamentally from the rest, we shall find a long series of -genera in which we can refer not only the changing contours of -the skull, but even the shape and size of the many constituent -bones and their intervening spaces or “vacuities,” to one and the -same simple system of transformed co-ordinates. The manner -in which the skulls of various Crocodilians differ from one another -may be sufficiently illustrated by three or four examples.</p> - -<div class="dctr02" id="fig383"> -<img src="images/i753.png" width="704" height="407" alt=""> - <div class="dcaption">Fig. 383. A, <i>Crocodilus porosus</i>. - B, <i>C. americanus</i>. C, <i>Notosuchus terrestris</i>.</div></div> - -<p>Let us take one of the typical modern crocodiles as our standard -of form, e.g. <i>C. porosus</i>, and inscribe it, as in Fig. <a href="#fig383" title="go to Fig. 383">383</a>, <i>a</i>, in the -usual Cartesian co-ordinates. By deforming the rectangular network -into a triangular system, with the apex of the triangle a -little way in front of the snout, as in <i>b</i>, we pass to such a form as -<i>C. americanus</i>. By an exaggeration of the same process we at -once get an approximation to the form of one -of the sharp-snouted, <span class="xxpn" id="p753">{753}</span> -or longirostrine, crocodiles, such as the genus <i>Tomistoma</i>; and, -in the species figured, the oblique position of the orbits, the arched -contour of the occipital border, and certain other characters suggest -a certain amount of curvature, such as I have represented in the -diagram (Fig. <a href="#fig383" title="go to Fig. 383">383</a>, <i>b</i>), on the part of the horizontal co-ordinates. -In the still more elongated skull of such a form as the Indian -Gavial, the whole skull has undergone a great longitudinal -extension, or, in other words, the ratio of <i>x ⁄ y</i> is greatly diminished; -and this extension is not uniform, but is at a maximum in the -region of the nasal and maxillary bones. This especially elongated -region is at the same time narrowed in an exceptional degree, and -its excessive narrowing is represented by a curvature, convex -towards the median axis, on the part of the vertical ordinates. -Let us take as a last illustration one of the Mesozoic crocodiles, -the little <i>Notosuchus</i>, from the Cretaceous formation. This little -crocodile is very different from our type in the proportions of its -skull. The region of the snout, in front of and including the frontal -bones, is greatly shortened; from constituting fully two-thirds of -the whole length of the skull in <i>Crocodilus</i>, it now constitutes less -than half, or, say, three-sevenths of the whole; and the whole -skull, and especially its posterior part, is curiously compact, -broad, and squat. The orbit is unusually large. If in the diagram -of this skull we select a number of -points obviously corresponding <span class="xxpn" id="p754">{754}</span> -to points where our rectangular co-ordinates intersect particular -bones or other recognisable features in our typical crocodile, we -shall easily discover that the lines joining these points in <i>Notosuchus</i> -fall into such a co-ordinate network as that which is -represented in Fig. <a href="#fig383" title="go to Fig. 383">383</a>, <i>c</i>. To all intents and purposes, then, this -not very complex system, representing one harmonious “deformation,” -accounts for <i>all</i> the differences between the two figures, -and is sufficient to enable one at any time to reconstruct a detailed -drawing, bone for bone, of the skull of <i>Notosuchus</i> from the model -furnished by the common crocodile.</p> - -<div class="dctr05" id="fig384"> -<img src="images/i754.png" width="448" height="421" alt=""> - <div class="dcaption">Fig. 384. Pelvis of (A) <i>Stegosaurus</i>; - (B) <i>Camptosaurus</i>.</div></div> - -<p>The many diverse forms of Dinosaurian reptiles, all of which -manifest a strong family likeness underlying much superficial -diversity, furnish us with plentiful material for comparison by -the method of transformations. As an instance, I have figured -the pelvic bones of <i>Stegosaurus</i> and of <i>Camptosaurus</i> (Fig. <a href="#fig384" title="go to Fig. 384">384</a>, -<i>a</i>, <i>b</i>) to show that, when the former is taken as our Cartesian -type, a slight curvature and an approximately logarithmic -extension of the <i>x</i>-axis brings us easily to the configuration of -the other. In the original specimen of <i>Camptosaurus</i> described -by Marsh<a class="afnanch" href="#fn658" id="fnanch658">658</a>, -the anterior portion of the iliac bone is missing; and -in Marsh’s restoration this part of the bone is drawn as though -it came somewhat abruptly to a sharp point. -In my figure I <span class="xxpn" id="p755">{755}</span> -have completed this missing part of the bone in harmony with the -general co-ordinate network which is suggested by our comparison -of the two entire pelves; and I venture to think that the result -is more natural in appearance, and more likely to be correct than -was Marsh’s conjectural restoration. It would seem, in fact, -that there is an obvious field for the employment of the method -of co-ordinates in this task of reproducing missing portions of a -structure to the proper scale and in harmony with related types. -To this subject we shall presently return.</p> - -<div class="dctr04" id="fig385"> -<img src="images/i755a.png" width="528" height="303" alt=""> - <div class="dcaption">Fig. 385. Shoulder-girdle of - <i>Cryptocleidus</i>. <i>a</i>, young; <i>b</i>, adult.</div></div> - -<p>In Fig. <a href="#fig385" title="go to Fig. 385">385</a>, <i>a</i>, <i>b</i>, I have drawn the shoulder-girdle of <i>Cryptocleidus</i>, -a Plesiosaurian reptile, half-grown in the one case and -full-grown in the other. The change of form during growth in -this region of the body is very considerable, and its nature is well -brought out by the two co-ordinate systems. In Fig. <a href="#fig386" title="go to Fig. 386">386</a> I have -drawn the shoulder-girdle of an</p> - -<div class="dctr04" id="fig386"> -<img src="images/i755b.png" width="528" height="215" alt=""> - <div class="dcaption">Fig. 386. Shoulder-girdle of - <i>Ichthyosaurus</i>.</div></div> - -<p class="pcontinue"> -Ichthyosaur, referring it to -<i>Cryptocleidus</i> as a standard of comparison. The interclavicle, -which is present in <i>Ichthyosaurus</i>, is minute and hidden in <i>Cryptocleidus</i>; -but the numerous other differences -between the two <span class="xxpn" id="p756">{756}</span> -forms, chief among which is the great elongation in <i>Ichthyosaurus</i> -of the two clavicles, are all seen by our diagrams to be part and -parcel of one general and systematic deformation.</p> - -<p>Before we leave the group of reptiles we may glance at the -very strangely modified skull of <i>Pteranodon</i>, one of the extinct -flying reptiles, or Pterosauria. In this very curious skull the -region of the jaws, or beak, is greatly elongated and pointed; the -occipital bone is drawn out into an enormous backwardly-directed -crest; the posterior part of the lower jaw is similarly produced -backwards; the orbit is small; and the</p> - -<div class="dctr04" id="fig387"> -<img src="images/i756.png" width="528" height="401" alt=""> - <div class="dcaption">Fig. 387. <i>a</i>, Skull of <i>Dimorphodon</i>. <i>b</i>, -Skull of <i>Pteranodon</i>.</div></div> - -<p class="pcontinue"> -quadrate bone is strongly -inclined downwards and forwards. The whole skull has a configuration -which stands, apparently, in the strongest possible -contrast to that of a more normal Ornithosaurian such as -<i>Dimorphodon</i>. But if we inscribe the latter in Cartesian coordinates -(Fig. <a href="#fig387" title="go to Fig. 387">387</a>, <i>a</i>), and refer our <i>Pteranodon</i> to a system of -oblique co-ordinates (<i>b</i>), in which the two co-ordinate systems of -parallel lines become each a pencil of diverging rays, we make -manifest a correspondence which extends uniformly throughout -all parts of these very different-looking skulls.</p> - -<hr class="hrblk"> - -<p>We have dealt so far, and for the most part we shall continue -to deal, with our co-ordinate method as a means of comparing one -known structure with another. But it is obvious, as -I have said, <span class="xxpn" id="p757">{757}</span> -that it may also be employed for drawing hypothetical structures, -on the assumption that they have varied from a known form in -some definite way. And this process may be especially useful, -and will be most obviously legitimate, when we apply it to the -particular case of representing intermediate stages between two -forms which are actually known to exist, in other words, of reconstructing -the transitional stages through which the course</p> - -<div class="dctr06" id="fig388"> -<img src="images/i757a.png" width="800" height="723" alt=""> - <div class="dcaption">Fig. 388. Pelvis of - <i>Archaeopteryx</i>.</div></div> - -<div class="dctr03" id="fig389"> -<img src="images/i757b.png" width="800" height="350" alt=""> - <div class="dcaption">Fig. 389. Pelvis of <i>Apatornis</i>.</div></div> - -<p class="pcontinue">of -evolution must have successively travelled if it has brought about -the change from some ancestral type to its presumed descendant. -Some little time ago I sent to my friend, Mr Gerhard Heilmann -of Copenhagen, a few of my own rough co-ordinate diagrams, including -some in which the pelves of certain ancient and primitive -birds were compared one with another. Mr Heilmann, who is -both a skilled draughtsman and an able morphologist, returned -me a set of diagrams which are a vast improvement -on my own, <span class="xxpn" id="p758">{758}</span> -and which are reproduced in Figs. <a href="#fig388" title="go to Fig. 388">388</a>–393. Here we have, as -extreme cases, the pelvis of <i>Archaeopteryx</i>, the most ancient of -known birds, and that of <i>Apatornis</i>, one of the fossil “toothed”</p> - -<div class="dctr01" id="fig390"> -<img src="images/i758a.png" width="800" height="533" alt=""> - <div class="dcaption">Fig. 390. The co-ordinate systems of Figs. -<a href="#fig388" title="go to Fig. 388">388</a> and 389, with three intermediate systems interpolated.</div></div> - -<div class="dctr05" id="fig391"> -<img src="images/i758b.png" width="448" height="338" alt=""> - <div class="pcaption">Fig. 391. The first intermediate co-ordinate -network, with its corresponding inscribed pelvis.</div></div> - -<p class="pcontinue">birds from the North American Cretaceous formations—a bird -shewing some resemblance to the modern terns. The pelvis of -<i>Archaeopteryx</i> is taken as our type, and -referred accordingly to <span class="xxpn" id="p759">{759}</span> -Cartesian co-ordinates (Fig. <a href="#fig388" title="go to Fig. 388">388</a>); while the corresponding coordinates -of the very different pelvis of <i>Apatornis</i> are represented -in Fig. <a href="#fig389" title="go to Fig. 389">389</a>. In Fig. <a href="#fig390" title="go to Fig. 390">390</a> the outlines of these two co-ordinate -systems are superposed upon one another, and those of three -intermediate and equidistant co-ordinate systems are interpolated -between them. From each of these latter systems, so determined -by direct interpolation, a complete co-ordinate diagram is drawn, -and the corresponding outline of a pelvis is found from each of</p> - -<div class="dctr04" id="fig392"> -<img src="images/i759.png" width="800" height="873" alt=""> - <div class="pcaption">Fig. 392. The second and - third intermediate co-ordinate networks, with their - corresponding inscribed pelves.</div></div> - -<p class="pcontinue"> -these systems of co-ordinates, as in Figs. <a href="#fig391" title="go to Fig. 391">391</a>, 392. Finally, in -Fig. <a href="#fig393" title="go to Fig. 393">393</a> the complete series is represented, beginning with the -known pelvis of <i>Archaeopteryx</i>, and leading up by our three intermediate -hypothetical types to the known pelvis of <i>Apatornis</i>.</p> - -<hr class="hrblk"> - -<p>Among mammalian skulls I will take two illustrations only, -one drawn from a comparison of the human skull with that of -the higher apes, and another from the -group of Perissodactyle <span class="xxpn" id="p760">{760}</span> -Ungulates, the group which includes the rhinoceros, the tapir, -and the horse.</p> - -<div class="dctr06" id="fig393"> -<img src="images/i760.png" width="415" height="794" alt=""> - <div class="pcaption">Fig. 393. The pelves of - <i>Archaeopteryx</i> and of <i>Apatornis</i>, with three - transitional types interpolated between them.</div></div> - -<p>Let us begin by choosing as our type the skull of <i>Hyrachyus -agrarius</i>, Cope, from the Middle Eocene of North America, as -figured by Osborn in his Monograph of the Extinct Rhinoceroses<a class="afnanch" href="#fn659" id="fnanch659">659</a> -(Fig. <a href="#fig394" title="go to Fig. 394">394</a>).</p> - -<p>The many other forms of primitive rhinoceros described in -the monograph differ from <i>Hyrachyus</i> in various details—in the -characters of the teeth, sometimes in the number of the toes, and -so forth; and they also differ very considerably -in the general <span class="xxpn" id="p761">{761}</span> -appearance of the skull. But these differences in the conformation -of the skull, conspicuous as they are at first sight, will be found -easy to bring under the conception of a simple and homogeneous -transformation, such as would result from the application of some -not very complicated stress. For instance, the corresponding</p> - -<div class="dctr03" id="fig394"> -<img src="images/i761a.png" width="608" height="364" alt=""> - <div class="dcaption">Fig. 394. Skull of <i>Hyrachyus - agrarius</i>. (After Osborn.)</div></div> - -<div class="dctr03" id="fig395"> -<img src="images/i761b.png" width="608" height="414" alt=""> - <div class="dcaption">Fig. 395. Skull of <i>Aceratherium - tridactylum</i>. (After Osborn.)</div></div> - -<p class="pcontinue"> -co-ordinates of <i>Aceratherium tridactylum</i>, as shown in Fig. <a href="#fig395" title="go to Fig. 395">395</a>, -indicate that the essential difference between this skull and the -former one may be summed up by saying that the long axis of the -skull of <i>Aceratherium</i> has undergone a slight double curvature, -while the upper parts of the skull have at the -same time been <span class="xxpn" id="p762">{762}</span> -subject to a vertical expansion, or to growth in somewhat greater -proportion than the lower parts. Precisely the same changes, -on a somewhat greater scale, give us the skull of an existing -rhinoceros.</p> - -<p>Among the species of <i>Aceratherium</i>, the posterior, or occipital, -view of the skull presents specific differences which are perhaps -more conspicuous than those furnished by the side view; and -these differences are very strikingly brought out by the series of -conformal transformations</p> - -<div class="dctr04" id="fig396"> -<img src="images/i762.png" width="528" height="498" alt=""> - <div class="pcaption">Fig. 396. Occipital view of the - skulls of various extinct rhinoceroses (<i>Aceratherium</i> - spp.). (After Osborn.)</div></div> - -<p class="pcontinue"> -which I have represented in Fig. <a href="#fig396" title="go to Fig. 396">396</a>. -In this case it will perhaps be noticed that the correspondence -is not always quite accurate in small details. It could easily -have been made much more accurate by giving a slightly sinuous -curvature to certain of the co-ordinates. But as they stand, -the correspondence indicated is very close, and the simplicity of -the figures illustrates all the better the general character of the -transformation.</p> - -<p>By similar and not more violent changes we pass easily to such -allied forms as the Titanotheres (Fig. <a href="#fig397" title="go to Fig. 397">397</a>); and the well-known -series of species of <i>Titanotherium</i>, by which -Professor Osborn has <span class="xxpn" id="p763">{763}</span> -illustrated the evolution of this genus, constitutes a simple and -suitable case for the application of our method.</p> - -<p>But our method enables us to pass over greater gaps than these, -and to discern the general, and to a very large extent even the -detailed, resemblances between the skull of the rhinoceros and -those of the tapir or the horse. From the Cartesian co-ordinates -in which we have begun by inscribing the skull of a primitive -rhinoceros, we pass to the tapir’s skull (Fig. <a href="#fig398" title="go to Fig. 398">398</a>), firstly, by converting -the rectangular into a triangular network, by which we -represent the depression of the anterior and the progressively -increasing elevation of the posterior part of the skull; and -secondly, by giving to the vertical ordinates a curvature such as -to bring about a certain longitudinal compression, or condensation, -in the forepart of the skull, especially in the nasal and orbital -regions.</p> - -<div class="dctr01" id="fig397"><div id="fig398"> -<img src="images/i763.png" width="800" height="303" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 397. <i>Titanotherium robustum</i>.</td> - <td></td> - <td>Fig. 398. Tapir’s skull.</td></tr></table> -</div></div></div><!--dctr01--> - -<p>The conformation of the horse’s skull departs from that of our -primitive Perissodactyle (that is to say our early type of rhinoceros, -<i>Hyrachyus</i>) in a direction that is nearly the opposite of that taken -by <i>Titanotherium</i> and by the recent species of rhinoceros. For -we perceive, by Fig. <a href="#fig399" title="go to Fig. 399">399</a>, that the horizontal co-ordinates, which -in these latter cases became transformed into curves with the -concavity upwards, are curved, in the case of the horse, in the -opposite direction. And the vertical ordinates, which are also -curved, somewhat in the same fashion as in the tapir, are very -nearly equidistant, instead of being, as in that animal, crowded -together anteriorly. Ordinates and abscissae -form an oblique <span class="xxpn" id="p764">{764}</span> -system, as is shown in the figure. In this case I have attempted -to produce the network beyond the region which is actually -required to include the diagram of the horse’s skull, in order to -show better the form of the general transformation, with a part -only of which we have actually to deal.</p> - -<div class="dctr04" id="fig399"> -<img src="images/i764a.png" width="528" height="287" alt=""> - <div class="dcaption">Fig. 399. Horse’s skull.</div></div> - -<div class="dctr04" id="fig400"> -<img src="images/i764b.png" width="528" height="308" alt=""> - <div class="dcaption">Fig. 400. Rabbit’s skull.</div></div> - -<p>It is at first sight not a little surprising to find that we can pass, -by a cognate and even simpler transformation, from our Perissodactyle -skulls to that of the rabbit; but the fact that we can -easily do so is a simple illustration of the undoubted affinity -which exists between the Rodentia, especially the family of the -Leporidae, and the more primitive Ungulates. For my part, I -would go further; for I think there is strong reason to believe -that the Perissodactyles are more closely related to the Leporidae -than the former are to the other Ungulates, or than the Leporidae -are to the rest of the Rodentia. Be that as it may, it is obvious -from Fig. <a href="#fig400" title="go to Fig. 400">400</a> that the rabbit’s skull conforms to -a system of <span class="xxpn" id="p765">{765}</span> -co-ordinates corresponding to the Cartesian co-ordinates in which -we have inscribed the skull of <i>Hyrachyus</i>, with the difference, -firstly, that the horizontal ordinates of the latter are transformed -into equidistant curved lines, approximately arcs of circles, with -their concavity directed downwards; and secondly, that the -vertical ordinates are transformed into a pencil of rays approximately -orthogonal to the circular arcs. In short, the configuration -of the rabbit’s skull is derived from that of our primitive rhinoceros -by the unexpectedly simple process of submitting the latter to a</p> - -<div class="dctr01" id="fig401"> -<img src="images/i765.png" width="800" height="487" alt=""> - <div class="pcaption">Fig. 401. <i>A</i>, outline diagram - of the Cartesian co-ordinates of the skull of - <i>Hyracotherium</i> or <i>Eohippus</i>, as shewn in Fig. - <a href="#fig402" title="go to Fig. 402">402</a>, - A. <i>H</i>, outline of the corresponding projection of the - horse’s skull. <i>B</i>–<i>G</i>, intermediate, or interpolated, - outlines.</div></div> - -<p class="pcontinue">strong and uniform flexure in the downward direction (cf. Fig. -<a href="#fig358" title="go to Fig. 358">358</a>, -p. <a href="#p731" title="go to pg. 731">731</a>). In the case of the rabbit the configuration of the -individual bones does not conform quite so well to the general -transformation as it does when we are comparing the several -Perissodactyles one with another; and the chief departures -from conformity will be found in the size of the orbit and in the -outline of the immediately surrounding bones. The simple fact -is that the relatively enormous eye of the rabbit constitutes an -independent variation, which cannot be brought into the general -and fundamental transformation, but must</p> - -<div class="dctr01" id="fig402"> -<img src="images/i766.png" width="800" height="474" alt=""> -<img src="images/i766b.png" width="800" height="385" alt=""> -<img src="images/i766c.png" width="800" height="357" alt=""> -<img src="images/i767a.png" width="800" height="263" alt=""> -<img src="images/i767b.png" width="800" height="467" alt=""> -<img src="images/i767c.png" width="800" height="179" alt=""> - <div class="pcaption">Fig. 402. <i>A</i>, skull of - <i>Hyracotherium</i>, from the Eocene, after W. B. Scott; - <i>H</i>, skull of horse, represented as a co-ordinate - transformation of that of <i>Hyracotherium</i>, and to the - same scale of magnitude; <i>B</i>–<i>G</i>, various artificial or - imaginary types, reconstructed as intermediate stages - between <i>A</i> and <i>H</i>; <i>M</i>, skull of <i>Mesohippus</i>, from the - Oligocene, after Scott, for comparison with <i>C</i>; <i>P</i>, - skull of <i>Protohippus</i>, from the Miocene, after Cope, - for comparison with <i>E</i>; <i>Pp</i>, lower jaw of <i>Protohippus - placidus</i> (after Matthew and Gidley), for comparison with - <i>F</i>; <i>Mi</i>, <i>Miohippus</i> (after Osborn), <i>Pa</i>, <i>Parahippus</i> - (after Peterson), shewing resemblance, but less perfect - agreement, with <i>C</i> and <i>D</i>.</div></div> - -<p class="pcontinue"> -be dealt with <span class="xxpn" id="p768">{768}</span> -separately. The enlargement of the eye, like the modification in -form and number of the teeth, is a separate phenomenon, which -supplements but in no way contradicts our general comparison of -the skulls taken in their entirety.</p> - -<hr class="hrblk"> - -<p>Before we leave the Perissodactyla and their allies, let us look -a little more closely into the case of the horse and its immediate -relations or ancestors, doing so with the help of a set of diagrams -which I again owe to Mr Gerard Heilmann<a class="afnanch" href="#fn660" id="fnanch660">660</a>. -Here we start afresh, -with the skull (Fig. <a href="#fig402" title="go to Fig. 402">402</a>, <i>A</i>) of <i>Hyracotherium</i> (or <i>Eohippus</i>), -inscribed in a simple Cartesian network. At the other end of the -series (<i>H</i>) is a skull of Equus, in its own corresponding network; -and the intermediate stages (<i>B</i>–<i>G</i>) are all drawn by direct and -simple interpolation, as in Mr Heilmann’s former series of drawings -of <i>Archaeopteryx</i> and <i>Apatornis</i>. In this present case, the relative -magnitudes are shewn, as well as the forms, of the several skulls. -Alongside of these reconstructed diagrams, are set figures of -certain extinct “horses” (Equidae or Palaeotheriidae), and in -two cases, viz. <i>Mesohippus</i> and <i>Protohippus</i> (<i>M</i>, <i>P</i>), it will be -seen that the actual fossil skull coincides in the most perfect -fashion with one of the hypothetical forms or stages which our -method shews to be implicitly involved in the transition from -<i>Hyracotherium</i> to <i>Equus</i>. In a third case, that of <i>Parahippus</i> -(<i>Pa</i>), the correspondence (as Mr Heilmann points out) is by no -means exact. The outline of this skull comes nearest to that of -the hypothetical transition stage <i>D</i>, but the “fit” is now a bad -one; for the skull of <i>Parahippus</i> is evidently a longer, straighter -and narrower skull, and differs in other minor characters besides. -In short, though some writers have placed <i>Parahippus</i> in the -direct line of descent between <i>Equus</i> and <i>Eohippus</i>, we see at -once that there is no place for it there, and that it must, accordingly, -represent a somewhat divergent branch or offshoot of the -Equidae<a class="afnanch" href="#fn661" id="fnanch661">661</a>. -It may be noticed, especially in the -case of <i>Protohippus</i> <span class="xxpn" id="p769">{769}</span> -(<i>P</i>), that the configuration of the angle of the jaw does not tally -quite so accurately with that of our hypothetical diagrams as do -other parts of the skull. As a matter of fact, this region is -somewhat variable, in different species of a genus, and even in -different individuals of the same species; in the small figure (<i>Pp</i>) -of <i>Protohippus placidus</i> the correspondence is more exact.</p> - -<p>In considering this series of figures we cannot but be -struck, not only with the regularity of the succession -of “transformations,” but also with the slight and -inconsiderable differences which separate the known -and recorded stages, and even the two extremes of the -whole series. These differences are no greater (save in -regard to actual magnitude) than those between one human -skull and another, at least if we take into account the -older or remoter races; and they are again no greater, -but if anything less, than the range of variation, -racial and individual, in certain other human bones, for -instance the scapula<a class="afnanch" href="#fn662" -id="fnanch662">662</a>.</p> - -<div class="dctr01" id="fig403"> -<img src="images/i769.png" width="800" height="300" alt=""> - <div class="pcaption">Fig. 403. Human scapulae (after - Dwight). <i>A</i>, Caucasian; <i>B</i>, Negro; <i>C</i>, North American - Indian (from Kentucky Mountains).</div></div> - -<p>The variability of this latter bone is great, -but it is neither <span class="xxpn" id="p770">{770}</span> -surprising nor peculiar; for it is linked with all the considerations -of mechanical efficiency and functional modification which we -dealt with in our last chapter. The scapula occupies, as it were, -a focus in a very important field of force; and the lines of force -converging on it will be very greatly modified by the varying -development of the muscles over a large area of the body and of -the uses to which they are habitually put.</p> - -<div class="dctr05" id="fig404"> -<img src="images/i770a.png" width="432" height="330" alt=""> - <div class="dcaption">Fig. 404. Human skull.</div></div> - -<div class="dctr05" id="fig405"> -<img src="images/i770b.png" width="432" height="273" alt=""> - <div class="pcaption">Fig. 405. Co-ordinates of chimpanzee’s -skull, as a projection of the Cartesian co-ordinates of -Fig. <a href="#fig404" title="go to Fig. 404">404</a>.</div></div> - -<p>Let us now inscribe in our Cartesian co-ordinates the outline -of a human skull (Fig. <a href="#fig404" title="go to Fig. 404">404</a>), for the purpose of comparing it with -the skulls of some of the higher apes. We know beforehand that -the main differences between the human and the simian types -depend upon the enlargement or expansion of the brain and -braincase in man, and the relative diminution or enfeeblement of -his jaws. Together with these changes, the “facial angle” -increases from an oblique angle to nearly a right -angle in man, <span class="xxpn" id="p771">{771}</span> -and the configuration of every constituent bone of the face and -skull undergoes an alteration. We do not know to begin with, -and we are not shewn by the ordinary methods of comparison, -how far these various changes form part of one harmonious and -congruent transformation, or whether we are to look, for instance, -upon the changes undergone by the frontal, the occipital, the -maxillary, and the mandibular regions as a congeries of separate -modifications or independent variants. But as soon as we have -marked out a number of points in the gorilla’s or chimpanzee’s -skull, corresponding with those which our co-ordinate network -intersected in the human skull, we find that these corresponding -points may be at once linked up by smoothly curved lines of -intersection, which form a new system of co-ordinates and constitute -a simple “projection” of our human skull. The network</p> - -<div class="dctr01" id="fig406"><div id="fig407"> -<img src="images/i771.png" width="800" height="262" alt=""> -<div class="dcaption"> -<table class="twdth100"> -<tr> - <td>Fig. 406. Skull of chimpanzee.</td> - <td></td> - <td>Fig. 407. Skull of baboon.</td></tr></table> -</div></div></div><!--dctr01--> - -<p class="pcontinue"> -represented in Fig. <a href="#fig405" title="go to Fig. 405">405</a> constitutes such a projection of the human -skull on what we may call, figuratively speaking, the “plane” of -the chimpanzee; and the full diagram in Fig. <a href="#fig406" title="go to Fig. 406">406</a> demonstrates -the correspondence. In Fig. <a href="#fig407" title="go to Fig. 407">407</a> I have shewn the similar deformation -in the case of a baboon, and it is obvious that the -transformation is of precisely the same order, and differs only in -an increased intensity or degree of deformation.</p> - -<p>In both dimensions, as we pass from above downwards and -from behind forwards, the corresponding areas of the network -are seen to increase in a gradual and approximately logarithmic -order in the lower as compared with the higher type of skull; -and, in short, it becomes at once manifest that the modifications -of jaws, braincase, and the regions between are all portions of one -continuous and integral process. It is of course easy -to draw the <span class="xxpn" id="p772">{772}</span> -inverse diagrams, by which the Cartesian co-ordinates of the ape -are transformed into curvilinear and non-equidistant co-ordinates -in man.</p> - -<p>From this comparison of the gorilla’s or chimpanzee’s with -the human skull we realise that an inherent weakness underlies -the anthropologist’s method of comparing skulls by reference to -a small number of axes. The most important of these are the -“facial” and “basicranial” axes, which include between them the -“facial angle.” But it is, in the first place, evident that these -axes are merely the principal axes of a system of co-ordinates, -and that their restricted and isolated use neglects all that can be -learned from the filling in of the rest of the co-ordinate network. -And, in the second place, the “facial axis,” for instance, as -ordinarily used in the anthropological comparison of one human -skull with another, or of the human skull with the gorilla’s, is -in all cases treated as a straight line; but our investigation has -shewn that rectilinear axes only meet the case in the simplest -and most closely related transformations; and that, for instance, -in the anthropoid skull no rectilinear axis is homologous with a -rectilinear axis in a man’s skull, but what is a straight line in the -one has become a certain definite curve in the other.</p> - -<p>Mr Heilmann tells me that he has tried, but without success, -to obtain a transitional series between the human skull and some -prehuman, anthropoid type, which series (as in the case of the -Equidae) should be found to contain other known types in direct -linear sequence. It appears impossible, however, to obtain such a -series, or to pass by successive and continuous gradations through -such forms as Mesopithecus, Pithecanthropus, <i>Homo neanderthalensis</i>, -and the lower or higher races of modern man. The -failure is not the fault of our method. It merely indicates that -no one straight line of descent, or of consecutive transformation, -exists; but on the contrary, that among human and anthropoid -types, recent and extinct, we have to do with a complex problem -of divergent, rather than of continuous, variation. And in like -manner, easy as it is to correlate the baboon’s and chimpanzee’s -skulls severally with that of man, and easy as it is to see that the -chimpanzee’s skull is much nearer to the human type than is the -baboon’s, it is also not difficult to perceive that the -series is not, <span class="xxpn" id="p773">{773}</span> -strictly speaking, continuous, and that neither of our two apes -lies <i>precisely</i> on the same direct line or sequence of deformation -by which we may hypothetically connect the other with -man.</p> - -<p>As a final illustration I have drawn the outline of a dog’s -skull (Fig. <a href="#fig408" title="go to Fig. 408">408</a>), and inscribed it in a network comparable with -the Cartesian network of the human skull in Fig. <a href="#fig404" title="go to Fig. 404">404</a>. Here we -attempt to bridge over a wider gulf than we have crossed in any -of our former comparisons. But, nevertheless, it is obvious that -our method still holds good, in spite of the fact that there are -various specific differences, such as the open or closed orbit, etc., -which have to be separately described and accounted for. We -see that the chief essential differences in plan between the dog’s -skull and the man’s lie in the fact that, relatively speaking, the</p> - -<div class="dctr04" id="fig408"> -<img src="images/i773.png" width="528" height="249" alt=""> - <div class="dcaption">Fig. 408. Skull of dog, compared with the -human skull of Fig. <a href="#fig404" title="go to Fig. 404">404</a>.</div></div> - -<p class="pcontinue">former tapers away in front, a -triangular taking the place of a rectangular conformation; -secondly, that, coincident with the tapering off, there -is a progressive elongation, or pulling out, of the whole -forepart of the skull; and lastly, as a minor difference, -that the straight vertical ordinates of the human skull -become curved, with their convexity directed forwards, in -the dog. While the net result is that in the dog, just as -in the chimpanzee, the brain-pan is smaller and the jaws -are larger than in man, it is now conspicuously evident -that the co-ordinate network of the ape is by no means -intermediate between those which fit the other two. The -mode of deformation is on different lines; and, while it -may be correct to say that the chimpanzee and the baboon -are more brute-like, it would be by no means accurate -to assert that they are more dog-like, than man. <span -class="xxpn" id="p774">{774}</span></p> - -<p>In this brief account of co-ordinate transformations and of -their morphological utility I have dealt with plane co-ordinates -only, and have made no mention of the less elementary subject -of co-ordinates in three-dimensional space. In theory there is -no difficulty whatsoever in such an extension of our method; it -is just as easy to refer the form of our fish or of our skull to the -rectangular co-ordinates <i>x</i>, <i>y</i>, <i>z</i>, or to the polar co-ordinates -ξ, η, ζ, as it is to refer their plane projections to the two axes to -which our investigation has been confined. And that it would -be advantageous to do so goes without saying; for it is the shape -of the solid object, not that of the mere drawing of the object, -that we want to understand; and already we have found some -of our easy problems in solid geometry leading us (as in the case -of the form of the bivalve and even of the univalve shell) quickly -in the direction of co-ordinate analysis and the theory of conformal -transformations. But this extended theme I have not attempted -to pursue, and it must be left to other times, and to other hands. -Nevertheless, let us glance for a moment at the sort of simple -cases, the simplest possible cases, with which such an investigation -might begin; and we have found our plane co-ordinate systems -so easily and effectively applicable to certain fishes that we may -seek among them for our first and tentative introduction to the -three-dimensional field.</p> - -<p>It is obvious enough that the same method of description and -analysis which we have applied to one plane, we may apply to -another: drawing by observation, and by a process of trial and -error, our various cross-sections and the co-ordinate systems -which seem best to correspond. But the new and important -problem which now emerges is to <i>correlate</i> the deformation or -transformation which we discover in one plane with that which -we have observed in another: and at length, perhaps, after -grasping the general principles of such correlation, to forecast -approximately what is likely to take place in the other two planes -of reference when we are acquainted with one, that is to say, to -determine the values along one axis in terms of the other two.</p> - -<p>Let us imagine a common “round” fish, and a common “flat” -fish, such as a haddock and a plaice. These two fishes are not as -nicely adapted for comparison by means of -plane co-ordinates as <span class="xxpn" id="p775">{775}</span> -some which we have studied, owing to the presence of essentially -unimportant, but yet conspicuous differences in the position of -the eyes, or in the number of the fins,—that is to say in the manner -in which the continuous dorsal fin of the plaice appears in the -haddock to be cut or scolloped into a number of separate fins. -But speaking broadly, and apart from such minor differences as -these, it is manifest that the chief factor in the case (so far as we -at present see) is simply the broadening out of the plaice’s body, -as compared with the haddock’s, in the dorso-ventral direction, -that is to say, along the <i>y</i> axis; in other words, the ratio <i>x ⁄ y</i> -is much less, (and indeed little more than half as great), in the -haddock than in the plaice. But we also recognise at once that -while the plaice (as compared with the haddock) is expanded in -one direction, it is also flattened, or thinned out, in the other: -<i>y</i> increases, but <i>z</i> diminishes, relatively to <i>x</i>. And furthermore, -we soon see that this is a common or even a general phenomenon. -The high, expanded body in our Antigonia or in our sun-fish is -at the same time flattened or <i>compressed</i> from side to side, in -comparison with the related fishes which we have chosen as -standards of reference or comparison; and conversely, such a -fish as the skate, while it is expanded from side to side in comparison -with a shark or dogfish, is at the same time flattened or -<i>depressed</i> in its vertical section. We proceed then, to enquire -whether there be any simple relation of <i>magnitude</i> discernible -between these twin factors of expansion and compression; and -the very fact that the two dimensions tend to vary <i>inversely</i> -already assures us that, in the general process of deformation, the -<i>volume</i> is less affected than are the <i>linear dimensions</i>. Some years -ago, when I was studying the length-weight co-efficient in fishes -(of which we have already spoken in Chap. III, p. <a href="#p098" title="go to pg. 98">98</a>), that is to -say the coefficient <i>k</i> in the formula <i>W</i> -= <i>kL</i><sup>3</sup> , or <i>k</i> -= <i>W ⁄ L</i><sup>3</sup> , I -was not a little surprised to find that <i>k</i> was all but identical in -two such different looking fishes as our haddock and our plaice: -thus indicating that these two fishes, little as they resemble one -another externally (though they belong to two closely related -families), have approximately the same <i>volume</i> when they are -equal in <i>length</i>; or, in other words, that the extent to which the -plaice’s body has become expanded or broadened -is <i>just about <span class="xxpn" id="p776">{776}</span> -compensated for</i> by the extent to which it has also got flattened -or thinned. In short, if we could permit ourselves to conceive -of a haddock being directly transformed into a plaice, a very -large part of the change would be simply accounted for by supposing -the former fish to be “rolled out,” as a baker rolls a piece of dough. -This is, as it were, an extreme case of the <i>balancement des organes</i>, -or “compensation of parts.”</p> - -<p>Simple Cartesian co-ordinates will not suffice very well to -compare the haddock with the plaice, for the deformation undergone -by the former in comparison with the latter is more on the -lines of that by which we have compared our Antigonia with our -Polyprion; that is to say, the expansion is greater towards the -middle of the fish’s length, and dwindles away towards either -end. But again simplifying our illustration to the utmost, and -being content with a rough comparison, we may assert that, -when haddock and plaice are brought to the same standard of -length, we can inscribe them both (approximately) in rectangular -co-ordinate networks, such that <i>Y</i> in the plaice is about twice -as great as <i>y</i> in the haddock. But if the volumes of the two -fishes be equal, this is as much as to say that <i>xyz</i> in the one case -(or rather the summation of all these values) is equal to <i>XYZ</i> -in the other; and therefore (since <i>X</i> -= <i>x</i>, and <i>Y</i> -= 2<i>y</i>), it follows -that <i>Z</i> -= <i>z</i> ⁄ 2. When we have drawn our vertical transverse -section of the haddock (or projected that fish in the <i>yz</i> plane), we -have reason accordingly to anticipate that we can draw a similar -projection (or section) of the plaice by simply doubling the <i>y</i>’s -and halving the <i>z</i>’s: and, very approximately, this turns out to -be the case. The plaice is (in round numbers) just about twice -as broad and also just about half as thick as the haddock; and -therefore the ratio of breadth to thickness (or <i>y</i> to <i>z</i>) is just about -four times as great in the one case as in the other.</p> - -<p>It is true that this simple, or simplified, illustration carries us -but a very little way, and only half prepares us for much greater -complications. For instance, we have no right or reason to presume -that the equality of weights, or volumes, is a common, -much less a general rule. And again, in all cases of more complex -deformation, such as that by which we have compared Diodon -with the sunfish, we must be prepared for -very much more <span class="xxpn" id="p777">{777}</span> -recondite methods of comparison and analysis, leading doubtless to -very much more complicated results. In this last case, of Diodon -and the sunfish, we have seen that the vertical <i>expansion</i> of the -latter as compared with the former fish, increases rapidly as we -go backwards towards the tail; but we can by no means say that -the lateral <i>compression</i> increases in like proportion. If anything, -it would seem that the said expansion and compression tend to -vary inversely; for the Diodon is very thick in front and greatly -thinned away behind, while the flattened sunfish is more nearly -of the same thickness all the way along. Interesting as the whole -subject is we must meanwhile leave it alone; recognising, however, -that if the difficulties of description and representation could be -overcome, it is by means of such co-ordinates in space that we -should at last obtain an adequate and satisfying picture of the -processes of deformation and of the directions of growth<a class="afnanch" href="#fn663" id="fnanch663">663</a>.</p> - -<div class="chapter" id="p778"> -<h2 class="h2herein" title="Epilogue.">EPILOGUE.</h2></div> - -<p>In the beginning of this book I said that its scope and treatment -were of so prefatory a kind that of other preface it had no -need; and now, for the same reason, with no formal and elaborate -conclusion do I bring it to a close. The fact that I set little store -by certain postulates (often deemed to be fundamental) of our -present-day biology the reader will have discovered and I have -not endeavoured to conceal. But it is not for the sake of polemical -argument that I have written, and the doctrines which I do not -subscribe to I have only spoken of by the way. My task is finished -if I have been able to shew that a certain mathematical aspect of -morphology, to which as yet the morphologist gives little heed, is -interwoven with his problems, complementary to his descriptive -task, and helpful, nay essential, to his proper study and comprehension -of Form. <i>Hic artem remumque repono.</i></p> - -<p>And while I have sought to shew the naturalist how a few -mathematical concepts and dynamical principles may help and -guide him, I have tried to shew the mathematician a field for his -labour,—a field which few have entered and no man has explored. -Here may be found homely problems, such as often tax the -highest skill of the mathematician, and reward his ingenuity all -the more for their trivial associations and outward semblance of -simplicity.</p> - -<p>That I am no skilled mathematician I have had little need to -confess, but something of the use and beauty of mathematics I -think I am able to understand. I know that in the study of -material things, number, order and position are the threefold clue -to exact knowledge; that these three, in the mathematician’s -hands, furnish the “first outlines for a sketch of the Universe”; -that by square and circle we are helped, like Emile Verhaeren’s -carpenter, to conceive “Les lois indubitables et fécondes Qui sont -la règle et la clarté du monde.”</p> - -<p>For the harmony of the world is made manifest in Form and -Number, and the heart and soul and all the -poetry of Natural <span class="xxpn" id="p779">{779}</span> -Philosophy are embodied in the concept of mathematical beauty. -A greater than Verhaeren had this in mind when he told of “the -golden compasses, prepared In God’s eternal store.” A greater -than Milton had magnified the theme and glorified Him “who -sitteth upon the circle of the earth,” saying: He measureth the -waters in the hollow of his hand, he meteth out the heavens with -his span, he comprehendeth the dust of the earth in a measure.</p> - -<p>Moreover the perfection of mathematical beauty is such (as -Maclaurin learned of the bee), that whatsoever is most beautiful -and regular is also found to be most useful and excellent.</p> - -<p>The living and the dead, things animate and inanimate, we -dwellers in the world and this world wherein we -dwell,—πάντα γα μὰν τὰ γιγνωσκόμενα,—are -bound alike by physical and -mathematical law. “Conterminous with space and coeval with -time is the kingdom of Mathematics; within this range her -dominion is supreme; otherwise than according to her order -nothing can exist, and nothing takes place in contradiction to her -laws.” So said, some forty years ago, a certain mathematician; -and Philolaus the Pythagorean had said much the same.</p> - -<p>But with no less love and insight has the science of Form and -Number been appraised in our own day and generation by a very -great Naturalist indeed:—by that old man eloquent, that wise -student and pupil of the ant and the bee, who died but yesterday, -and who in his all but saecular life tasted of the firstfruits of -immortality; who curiously conjoined the wisdom of antiquity -with the learning of to-day; whose Provençal verse seems set to -Dorian music; in whose plainest words is a sound as of bees’ -industrious murmur; and who, being of the same blood and -marrow with Plato and Pythagoras, saw in Number “la clef de la -voûte,” and found in it “le comment et le -pourquoi des choses.”</p> - -<h2 class="h2herein">NOTES:</h2> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch1" id="fn1">1</a> -These sayings of Kant and of Du Bois, and others like -to them, have been the text of many discourses: see, for instance, -Stallo’s <i>Concepts</i>, p. 21, 1882; Höber, <i>Biol. Centralbl.</i> -<span class="smmaj">XIX,</span> p. 284, 1890, etc. Cf. also Jellett, <i>Rep. Brit. Ass.</i> -1874, p. 1.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch2" id="fn2">2</a> -“Quum enim mundi universi fabrica sit perfectissima, atque -a Creatore sapientissimo absoluta, nihil omnino in mundo contingit -in quo non maximi minimive ratio quaepiam eluceat; quamobrem dubium -prorsus est nullum quin omnes mundi effectus ex causis finalibus, ope -methodi maximorum et minimorum, aeque feliciter determinari queant -atque ex ipsis causis efficientibus.” <i>Methodus inveniendi</i>, etc. 1744 -(<i>cit.</i> Mach, <i>Science of Mechanics</i>, 1902, p. 455).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch3" id="fn3">3</a> -Cf. Opp. (ed. Erdmann), p. 106, “Bien loin d’exclure les -causes finales..., c’est de là qu’il faut tout déduire en Physique.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch4" id="fn4">4</a> -Cf. p. 162. “La force vitale dirige des phénomènes qu’elle -ne produit pas: les agents physiques produisent des phénomènes qu’ils -ne dirigent pas.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch5" id="fn5">5</a> -It is now and then conceded with reluctance. Thus -Enriques, a learned and philosophic naturalist, writing “della economia -di sostanza nelle osse cave” (<i>Arch. f. Entw. Mech.</i> <span class="smmaj">XX,</span> -1906), says “una certa impronta di teleologismo quà e là è rimasta, mio -malgrado, in questo scritto.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch6" id="fn6">6</a> -Cf. Cleland, On Terminal Forms of Life, <i>J. Anat. -and Phys.</i> <span class="smmaj">XVIII,</span> 1884.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch7" id="fn7">7</a> -Conklin, Embryology of Crepidula, <i>Journ. of Morphol.</i> -<span class="smmaj">XIII,</span> p. 203, 1897; Lillie, F. R., Adaptation in Cleavage, -<i>Woods Holl Biol. Lectures</i>, pp. 43–67, 1899.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch8" id="fn8">8</a> -I am inclined to trace back Driesch’s teaching of -Entelechy to no less a person than Melanchthon. When Bacon (<i>de -Augm.</i> <span class="smmaj">IV,</span> 3) states with disapproval that the soul “has -been regarded rather as a function than as a substance,” R. L. Ellis -points out that he is referring to Melanchthon’s exposition of the -Aristotelian doctrine. For Melanchthon, whose view of the peripatetic -philosophy had long great influence in the Protestant Universities, -affirmed that, according to the true view of Aristotle’s opinion, the -soul is not a substance, but an ἑντελέχεια, or <i>function</i>. He -defined it as δύναμις <i>quaedam ciens actiones</i>—a description -all but identical with that of Claude Bernard’s “<i>force vitale</i>.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch9" id="fn9">9</a> -Ray Lankester, <i>Encycl. Brit.</i> (9th ed.), art. “Zoology,” -p. 806, 1888.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch10" id="fn10">10</a> -Alfred Russel Wallace, especially in his later years, -relied upon a direct but somewhat crude teleology. Cf. his <i>World of -Life, a Manifestation of Creative Power, Directive Mind and Ultimate -Purpose</i>, 1910.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch11" id="fn11">11</a> -Janet, <i>Les Causes Finales</i>, 1876, p. 350.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch12" id="fn12">12</a> -The phrase is Leibniz’s, in his <i>Théodicée</i>.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch13" id="fn13">13</a> -Cf. (<i>int. al.</i>) Bosanquet, The Meaning of Teleology, -<i>Proc. Brit. Acad.</i> 1905–6, pp. 235–245. Cf. also Leibniz (<i>Discours -de Métaphysique; Lettres inédites, ed.</i> de Careil, 1857, p. 354; -<i>cit.</i> Janet, p. 643), “L’un et l’autre est bon, l’un et l’autre peut -être utile ... et les auteurs qui suivent ces routes différentes ne -devraient point se maltraiter: <i>et seq.</i>”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch14" id="fn14">14</a> -The reader will understand that I speak, not of the -“severe and diligent inquiry” of variation or of “fortuity,” but merely -of the easy assumption that these phenomena are a sufficient basis -on which to rest, with the all-powerful help of natural selection, a -theory of definite and progressive evolution.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch15" id="fn15">15</a> -<i>Revue Philosophique.</i> <span class="smmaj">XXXIII,</span> 1892.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch16" id="fn16">16</a> -This general principle was clearly grasped by Dr George -Rainey (a learned physician of St Bartholomew’s) many years ago, and -expressed in such words as the following: “......it is illogical to -suppose that in the case of vital organisms a distinct force exists -to produce results perfectly within the reach of physical agencies, -especially as in many instances no end could be attained were that the -case, but that of opposing one force by another capable of effecting -exactly the same purpose.” (On Artificial Calculi, <i>Q.J.M.S.</i> (<i>Trans. -Microsc. Soc.</i>), <span class="smmaj">VI,</span> p. 49, 1858.) Cf. also Helmholtz, -<i>infra cit.</i>, p. 9.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch17" id="fn17">17</a> -Whereby he incurred the reproach -of Socrates, -in the <i>Phaedo</i>.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch18" id="fn18">18</a> -In a famous lecture (Conservation of Forces -applied to Organic Nature, <i>Proc. Roy. Instit.</i>, April -12, 1861), Helmholtz laid it down, as “the fundamental -principle of physiology,” that “There may be other agents -acting in the living body than those agents which act in -the inorganic world; but those forces, as far as they cause -chemical and mechanical influence in the body, must be -<i>quite of the same character</i> as inorganic forces: in this -at least, that their effects must be ruled by necessity, -and must always be the same when acting in the same -conditions; and so there cannot exist any arbitrary choice -in the direction of their actions.” It would follow from -this, that, like the other “physical” forces, they must be -subject to mathematical analysis and deduction. Cf. also -Dr T. Young’s Croonian Lecture On the Heart and Arteries, -<i>Phil. Trans.</i> 1809, p. 1; <i>Coll. Works</i>, <span class="smmaj">I,</span> 511.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch19" id="fn19">19</a> -<i>Ektropismus, oder die physikalische Theorie -des Lebens</i>, Leipzig, 1910.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch20" id="fn20">20</a> -Wilde Lecture, <i>Nature</i>, March 12, 1908; -<i>ibid.</i> Sept. 6, 1900, p. 485; <i>Aether and Matter</i>, p. 288. -Cf. also Lord Kelvin, <i>Fortnightly Review</i>, 1892, p. 313.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch21" id="fn21">21</a> -Joly, The Abundance of Life, <i>Proc. Roy. -Dublin Soc.</i> <span class="smmaj">VII,</span> 1890; and in <i>Scientific -Essays</i>, etc. 1915, p. 60 <i>et seq.</i></p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch22" id="fn22">22</a> -Papillon, <i>Histoire de la philosophie moderne</i>, -<span class="smmaj">I,</span> p. 300.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch23" id="fn23">23</a> -With the special and important properties -of <i>colloidal</i> matter we are, for the time being, not -concerned.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch24" id="fn24">24</a> -Cf. Hans Przibram, <i>Anwendung elementarer Mathematik auf Biologische -Probleme</i> (in Roux’s <i>Vorträge</i>, Heft -<span class="nowrap"><span class="smmaj">III</span>),</span> -Leipzig, 1908, p. 10.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch25" id="fn25">25</a> -The subject is treated from an engineering -point of view by Prof. James Thomson, Comparisons of -Similar Structures as to Elasticity, Strength, and -Stability, <i>Trans. Inst. Engineers, Scotland</i>, 1876 -(<i>Collected Papers</i>, 1912, pp. 361–372), and by Prof. A. -Barr, <i>ibid.</i> 1899; see also Rayleigh, <i>Nature</i>, April 22, -1915.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch26" id="fn26">26</a> -Cf. Spencer, The Form of the Earth, etc., -<i>Phil. Mag.</i> <span class="smmaj">XXX,</span> pp. 194–6, 1847; also -<i>Principles of Biology</i>, pt. <span class="smmaj">II,</span> ch. <span class="smmaj">I,</span> -1864 (p. 123, etc.).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch27" id="fn27">27</a> -George Louis Lesage (1724–1803), well known -as the author of one of the few attempts to explain -gravitation. (Cf. Leray, <i>Constitution de la Matière</i>, -1869; Kelvin, <i>Proc. R. S. E.</i> <span class="smmaj">VII,</span> p. 577, 1872, -etc.; Clerk Maxwell, <i>Phil. Trans.</i> vol. 157, p. 50, 1867; -art. “Atom,” <i>Encycl. Brit.</i> 1875, p. 46.)</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch28" id="fn28">28</a> -Cf. Pierre Prévost, <i>Notices de la vie et des -écrits de Lesage</i>, 1805; quoted by Janet, <i>Causes Finales</i>, -app. <span class="smmaj">III.</span></p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch29" id="fn29">29</a> -Discorsi e Dimostrazioni matematiche, intorno à due nuove scienze, -attenenti alla Mecanica, ed ai Movimenti Locali: appresso gli Elzevirii, <span class="smmaj">MDCXXXVIII.</span> -<i>Opere</i>, ed. Favaro, <span class="smmaj">VIII,</span> p. 169 seq. Transl. by Henry Crew and A. de Salvio, -1914, p. 130, etc. See <i>Nature</i>, June 17, 1915.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch30" id="fn30">30</a> -So Werner remarked that Michael Angelo and Bramanti could not have built -of gypsum at Paris on the scale they built of travertin in Rome.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch31" id="fn31">31</a> Sir -G. Greenhill, Determination of the greatest height to which -a Tree of given proportions can grow, <i>Cambr. Phil. Soc. -Pr.</i> <span class="smmaj">IV,</span> p. 65, 1881, and Chree, -<i>ibid.</i> <span class="smmaj">VII,</span> 1892. Cf. Poynting -and Thomson’s <i>Properties of Matter</i>, 1907, p 99.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch32" id="fn32">32</a> -In like manner the wheat-straw bends over -under the weight of the loaded ear, and the tip of the -cat’s tail bends over when held upright,—not because -they “possess flexibility,” but because they outstrip the -dimensions within which stable equilibrium is possible in -a vertical position. The kitten’s tail, on the other hand, -stands up spiky and straight.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch33" id="fn33">33</a> -<i>Modern Painters.</i></p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch34" id="fn34">34</a> -The stem of the giant bamboo may attain a -height of 60 metres, while not more than about 40 cm. in -diameter near its base, which dimensions are not very -far short of the theoretical limits (A. J. Ewart, <i>Phil. -Trans.</i> vol. 198, p. 71, 1906).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch35" id="fn35">35</a> -<i>Trans. Zool. Soc.</i> <span class="smmaj">IV,</span> 1850, p. 27.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch36" id="fn36">36</a> -It would seem to be a common if not a general -rule that marine organisms, zoophytes, molluscs, etc., tend -to be larger than the corresponding and closely related -forms living in fresh water. While the phenomenon may have -various causes, it has been attributed (among others) -to the simple fact that the forces of growth are less -antagonised by gravity in the denser medium (cf. Houssay, -<i>La Forme et la Vie</i>, 1900, p. 815). The effect of -gravity on outward <i>form</i> is illustrated, for instance, by -the contrast between the uniformly upward branching of a -sea-weed and the drooping curves of a shrub or tree.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch37" id="fn37">37</a> -The analogy is not a very strict one. We -are not taking account, for instance, of a proportionate -increase in thickness of the boiler-plates.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch38" id="fn38">38</a> -Let <i>L</i> be the length, <i>S</i> the (wetted) -surface, <i>T</i> the tonnage, <i>D</i> the displacement (or volume) -of a ship; and let it cross the Atlantic at a speed <i>V</i>. -Then, in comparing two ships, similarly constructed but of -different magnitudes, we know that <i>L</i> -= <i>V</i><sup>2</sup> , <i>S</i> -= <i>L</i><sup>2</sup> -= <i>V</i><sup>4</sup> , <i>D</i> -= <i>T</i> -= <i>L</i><sup>3</sup> -= <i>V</i><sup>6</sup> ; also <i>R</i> (resistance) -= <i>S</i> · <i>V</i><sup>2</sup> -= <i>V</i><sup>6</sup> ; <i>H</i> (horse-power) -= <i>R</i> · <i>V</i> -= <i>V</i><sup>7</sup> ; -and the coal (<i>C</i>) necessary for the voyage -= <i>H ⁄ V</i> -= <i>V</i><sup>6</sup> . That is to say, in ordinary engineering language, to -increase the speed across the Atlantic by 1 per cent. the -ship’s length must be increased 2 per cent., her tonnage -or displacement 6 per cent., her coal-consumpt also 6 per -cent., her horse-power, and therefore her boiler-capacity, -7 per cent. Her bunkers, accordingly, keep pace with the -enlargement of the ship, but her boilers tend to increase -out of proportion to the space available.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch39" id="fn39">39</a> -This is the result arrived at by Helmholtz, -Ueber ein Theorem geometrisch ähnliche Bewegungen flüssiger -Körper betreffend, nebst Anwendung auf das Problem -Luftballons zu lenken, <i>Monatsber. Akad. Berlin</i>, 1873, -pp. 501–14. It was criticised and challenged (somewhat -rashly) by K. Müllenhof, Die Grösse der Flugflächen, etc., -<i>Pflüger’s Archiv</i>, <span class="smmaj">XXXV,</span> p. 407, <span class="smmaj">XXXVI,</span> -p. 548, 1885.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch40" id="fn40">40</a> -Cf. also Chabrier, Vol des Insectes, <i>Mém. -Mus. Hist. Nat. Paris</i>, <span class="smmaj">VI</span>–<span class="smmaj">VIII,</span> -1820–22.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch41" id="fn41">41</a> -<i>Aerial Flight</i>, vol. <span class="smmaj">II</span> -(<i>Aerodonetics</i>), 1908, p. 150.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch42" id="fn42">42</a> -By Lanchester, <i>op. cit.</i> p. 131.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch43" id="fn43">43</a> -Cf. <i>L’empire de l’air; ornithologie appliquée -à l’aviation</i>. 1881.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch44" id="fn44">44</a> -<i>De Motu Animalium</i>, I, prop. cciv, ed. 1685, -p. 243.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch45" id="fn45">45</a> -Harlé, On Atmospheric Pressure in past -Geological Ages, <i>Bull. Geol. Soc. Fr.</i> <span class="smmaj">XI,</span> pp. -118–121; or <i>Cosmos</i>, p. 30, July 8, 1911.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch46" id="fn46">46</a> -<i>Introduction to Entomology</i>, 1826, -<span class="smmaj">II,</span> p. 190. K. and S., like many less learned -authors, are fond of popular illustrations of the “wonders -of Nature,” to the neglect of dynamical principles. They -suggest, for instance, that if the white ant were as big as -a man, its tunnels would be “magnificent cylinders of more -than three hundred feet in diameter”; and that if a -certain noisy Brazilian insect were as big as a man, its -voice would be heard all the world over: “so that Stentor -becomes a mute when compared with these insects!” It is an -easy consequence of anthropomorphism, and hence a common -characteristic of fairy-tales, to neglect the principle of -dynamical, while dwelling on the aspect of geometrical, -similarity.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch47" id="fn47">47</a> -I.e. the available energy of muscle, in -ft.-lbs. per lb. of muscle, is the same for all animals: a -postulate which requires considerable qualification when we -are comparing very different <i>kinds</i> of muscle, such as the -insect’s and the mammal’s.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch48" id="fn48">48</a> -Prop. clxxvii. Animalia minora et minus -ponderosa majores saltus efficiunt respectu sui corporis, -si caetera fuerint paria.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch49" id="fn49">49</a> -See also (<i>int. al.</i>), John Bernoulli, <i>de -Motu Musculorum</i>, Basil., 1694; Chabry, Mécanisme du Saut, -<i>J. de l’Anat. et de la Physiol.</i> <span class="smmaj">XIX,</span> 1883; Sur -la longueur des membres des animaux sauteurs, <i>ibid.</i> -<span class="smmaj">XXI,</span> p. 356, 1885; Le Hello, De l’action des -organes locomoteurs, etc., <i>ibid.</i> <span class="smmaj">XXIX,</span> p. 65–93, -1893, etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch50" id="fn50">50</a> -Recherches sur la force absolue des muscles -des Invertébrés, <i>Bull. Acad. E. de Belgique</i> (3), -<span class="smmaj">VI,</span> <span class="smmaj">VII,</span> 1883–84; see also <i>ibid.</i> (2), -<span class="smmaj">XX,</span> 1865, <span class="smmaj">XXII,</span> 1866; <i>Ann. Mag. N. H.</i> -<span class="smmaj">XVII,</span> p. 139, 1866, <span class="smmaj">XIX,</span> p. 95, 1867. -The subject was also well treated by Straus-Dürckheim, in -his <i>Considérations générales sur l’anatomie comparée des -animaux articulés</i>, 1828.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch51" id="fn51">51</a> -The fact that the limb tends to swing in -pendulum-time was first observed by the brothers Weber -(<i>Mechanik der menschl. Gehwerkzeuge</i>, Göttingen, 1836). -Some later writers have criticised the statement (e.g. -Fischer, Die Kinematik des Beinschwingens etc., <i>Abh. math. -phys. Kl. k. Sächs. Ges.</i> <span class="smmaj">XXV</span>–<span class="smmaj">XXVIII,</span> -1899–1903), but for all that, with proper qualifications, -it remains substantially true.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch52" id="fn52">52</a> -Quoted in Mr John Bishop’s interesting article -in Todd’s <i>Cyclopaedia</i>, <span class="smmaj">III,</span> p. 443.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch53" id="fn53">53</a> -There is probably also another factor involved -here: for in bending, and therefore shortening, the leg we -bring its centre of gravity nearer to the pivot, that is to -say, to the joint, and so the muscle tends to move it the -more quickly.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch54" id="fn54">54</a> -<i>Proc. Psychical Soc.</i> <span class="smmaj">XII,</span> pp. -338–355, 1897.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch55" id="fn55">55</a> -For various calculations of the increase of -surface due to histological and anatomical subdivision, -see E. Babak, Ueber die Oberflächenentwickelung bei -Organismen, <i>Biol. Centralbl.</i> <span class="smmaj">XXX,</span> pp. 225–239, -257–267, 1910. In connection with the physical theory -of surface-energy, Wolfgang Ostwald has introduced the -conception of <i>specific surface</i>, that is to say the ratio -of surface to volume, or <i>S ⁄ V</i>. In a cube, <i>V</i> -= <i>l</i><sup>3</sup> , -and <i>S</i> -= 6<i>l</i><sup>2</sup> ; therefore <i>S ⁄ V</i> -= 6 ⁄ <i>l</i>. Therefore if -the side <i>l</i> measure 6 -cm., the ratio <i>S ⁄ V</i> -= 1, and such a cube may be taken -as our standard, or unit of specific surface. A human -blood-corpuscle has, accordingly, a specific surface of -somewhere about 14,000 or 15,000. It is found in physical -chemistry that surface energy becomes an important factor -when the specific surface reaches a value of 10,000 or -thereby.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch56" id="fn56">56</a> -Though the entire egg is not increasing in -mass, this is not to say that its living protoplasm is not -increasing all the while at the expense of the reserve -material.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch57" id="fn57">57</a> -Cf. Tait, <i>Proc. R.S.E.</i> <span class="smmaj">V,</span> 1866, and -<span class="smmaj">VI,</span> 1868.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch58" id="fn58">58</a> -<i>Physiolog. Notizen</i> (9), p. 425, 1895. Cf. -Strasbürger, Ueber die Wirkungssphäre der Kerne und die -Zellgrösse, <i>Histolog. Beitr.</i> (5), pp. 95–129, 1893; J. -J. Gerassimow, Ueber die Grösse des Zellkernes, <i>Beih. -Bot. Centralbl.</i> <span class="smmaj">XVIII,</span> 1905; also G. Levi and -T. Terni, Le variazioni dell’ indice plasmatico-nucleare -durante l’intercinesi, <i>Arch. Ital. di Anat.</i> <span class="smmaj">X,</span> -p. 545, 1911.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch59" id="fn59">59</a> -<i>Arch. f. Entw. Mech.</i> <span class="smmaj">IV,</span> 1898, pp. -75, 247.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch60" id="fn60">60</a> -Conklin, E. G., Cell-size and nuclear-size, -<i>J. Exp. Zool.</i> <span class="smmaj">XII.</span> pp. 1–98, 1912.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch61" id="fn61">61</a> -Thus the fibres of the crystalline lens are of -the same size in large and small dogs; Rabl, <i>Z. f. w. Z.</i> -<span class="smmaj">LXVII,</span> 1899. Cf. (<i>int. al.</i>) Pearson, On the Size -of the Blood-corpuscles in Rana, <i>Biometrika</i>, <span class="smmaj">VI,</span> -p. 403, 1909. Dr Thomas Young caught sight of the -phenomenon, early in last century: “The solid particles of -the blood do not by any means vary in magnitude in the same -ratio with the bulk of the animal,” <i>Natural Philosophy</i>, -ed. 1845, p. 466; and Leeuwenhoek and Stephen Hales were -aware of it a hundred years before. But in this case, -though the blood-corpuscles show no relation of magnitude -to the size of the animal, they do seem to have some -relation to its activity. At least the corpuscles in the -sluggish Amphibia are much the largest known to us, while -the smallest are found among the deer and other agile and -speedy mammals. (Cf. Gulliver, <i>P.Z.S.</i> 1875, p. 474, etc.) -This apparent correlation may have its bearing on modern -views of the surface-condensation or adsorption of oxygen -in the blood-corpuscles, a process which would be greatly -facilitated and intensified by the increase of surface due -to their minuteness.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch62" id="fn62">62</a> -Cf. P. Enriques, La forma come funzione della -grandezza: Ricerche sui gangli nervosi degli Invertebrati, -<i>Arch. f. Entw. Mech.</i> <span class="smmaj">XXV,</span> p. 655, 1907–8.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch63" id="fn63">63</a> -While the difference in cell-volume is vastly -less than that between the volumes, and very much less -also than that between the surfaces, of the respective -animals, yet there <i>is</i> a certain difference; and this it -has been attempted to correlate with the need for each -cell in the many-celled ganglion of the larger animal to -possess a more complex “exchange-system” of branches, for -intercommunication with its more numerous neighbours. -Another explanation is based on the fact that, while -such cells as continue to divide throughout life tend to -uniformity of size in all mammals, those which do not do -so, and in particular the ganglion cells, continue to -grow, and their size becomes, therefore, a function of the -duration of life. Cf. G. Levi, Studii sulla grandezza delle -cellule, <i>Arch. Ital. di Anat. e di Embryolog.</i> <span class="smmaj">V,</span> -p. 291, 1906.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch64" id="fn64">64</a> -Boveri. <i>Zellen-studien, V. Ueber die Abhängigkeit -der Kerngrösse und Zellenzahl -der Seeigellarven von der Chromosomenzahl der Ausgangszellen.</i> Jena, 1905.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch65" id="fn65">65</a> -Recent important researches suggest that such -ultra-minute “filter-passers” are the true cause of certain -acute maladies commonly ascribed to the presence of much -larger organisms; cf. Hort, Lakin and Benians, The true -infective Agent in Cerebrospinal Fever, etc., <i>J. Roy. Army -Med. Corps</i>, Feb. 1910.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch66" id="fn66">66</a> -<i>Zur Erkenntniss der Kolloide</i>, 1905, p. 122; -where there will be found an interesting discussion of -various molecular and other minute magnitudes.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch67" id="fn67">67</a> -<i>Encyclopaedia Britannica</i>, 9th edit., vol. -<span class="smmaj">III,</span> p. 42, 1875.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch68" id="fn68">68</a> -Sur la limite de petitesse des organismes, -<i>Bull. Soc. R. des Sc. méd. et nat. de Bruxelles</i>, Jan. -1903; <i>Rec. d’œuvres</i> (<i>Physiol. générale</i>), p. 325.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch69" id="fn69">69</a> -Cf. A. Fischer, <i>Vorlesungen über Bakterien</i>, -1897, p. 50.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch70" id="fn70">70</a> -F. Hofmeister, quoted in Cohnheim’s <i>Chemie der -Eiweisskörper</i>, 1900, p. 18.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch71" id="fn71">71</a> -McKendrick arrived at a still lower estimate, -of about 1250 proteid molecules in the minutest organisms. -<i>Brit. Ass. Rep.</i> 1901, p. 808.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch72" id="fn72">72</a> -Cf. Perrin, <i>Les Atomes</i>, 1914, p. 74.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch73" id="fn73">73</a> -Cf. Tait, On Compression of Air in small Bubbles, -<i>Proc. R. S. E.</i> <span class="smmaj">V,</span> 1865.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch74" id="fn74">74</a> -<i>Phil. Mag.</i> <span class="smmaj">XLVIII,</span> 1899; <i>Collected -Papers</i>, <span class="smmaj">IV,</span> p. 430.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch75" id="fn75">75</a> -Carpenter, <i>The Microscope</i>, edit. 1862, p. -185.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch76" id="fn76">76</a> -The modern literature on the Brownian Movement -is very large, owing to the value which the phenomenon is -shewn to have in determining the size of the atom. For a -fuller, but still elementary account, see J. Cox, <i>Beyond -the Atom</i>, 1913, pp. 118–128; and see, further, Perrin, -<i>Les Atomes</i>, pp. 119–189.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch77" id="fn77">77</a> -Cf. R. Gans, Wie fallen Stäbe und Scheiben in -einer reibenden Flüssigkeit? <i>Münchener Bericht</i>, 1911, -p. 191; K. Przibram, Ueber die Brown’sche Bewegung nicht -kugelförmiger Teilchen, <i>Wiener Ber.</i> 1912, p. 2339.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch78" id="fn78">78</a> -Ueber die ungeordnete Bewegung niederer -Thiere, <i>Pflüger’s Archiv</i>, <span class="smmaj">CLIII,</span> p. 401, 1913.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch79" id="fn79">79</a> -Sometimes we find one and the same diagram -suffice, whether the intervals of time be great or small; -and we then invoke “Wolff’s Law,” and assert that the -life-history of the individual repeats, or recapitulates, -the history of the race.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch80" id="fn80">80</a> -Our subject is one of Bacon’s “Instances of -the Course,” or studies wherein we “measure Nature by -periods of Time.” In Bacon’s <i>Catalogue of Particular -Histories</i>, one of the odd hundred histories or -investigations which he foreshadowed is precisely that -which we are engaged on, viz. a “History of the Growth and -Increase of the Body, in the whole and in its parts.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch81" id="fn81">81</a> -Cf. Aristotle, <i>Phys.</i> vi, 5, 235 <i>a</i> 11, -ὲπεὶ γὰρ ἅπασα κίνησις ἐν χρόνῳ, κτλ. Bacon -emphasised, in like manner, the fact that “all motion or -natural action is performed in time: some more quickly, -some more slowly, but all in periods determined and fixed -in the nature of things. Even those actions which seem to -be performed suddenly, and (as we say) in the twinkling -of an eye, are found to admit of degree in respect of -duration.” <i>Nov. Org.</i> <span class="smmaj">XLVI.</span></p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch82" id="fn82">82</a> -Cf. (e.g.) <i>Elem. Physiol.</i> ed. 1766, -<span class="smmaj">VIII,</span> p. 114, “Ducimur autem ad evolutionem -potissimum, quando a perfecto animale retrorsum -progredimur, et incrementorum atque mutationum seriem -relegimus. Ita inveniemus perfectum illud animal fuisse -imperfectius, alterius figurae et fabricae, et denique rude -et informe: et tamen idem semper animal sub iis diversis -phasibus fuisse, quae absque ullo saltu perpetuos parvosque -per gradus cohaereant.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch83" id="fn83">83</a> -<i>Beiträge zur Entwickelungsgeschichte des -Hühnchens im Ei</i>, p. 40, 1817. Roux ascribes the same views -also to Von Baer and to R. H. Lotze (<i>Allg. Physiologie</i>, -p. 353, 1851).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch84" id="fn84">84</a> -Roux, <i>Die Entwickelungsmechanik</i>, p. 99, -1905.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch85" id="fn85">85</a> -<i>Op. cit.</i> p. 302, “Magnum hoc naturae -instrumentum, etiam in corpore animato evolvendo potenter -operatur; etc.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch86" id="fn86">86</a> -<i>Ibid.</i> p. 306. “Subtiliora ista, et -aliquantum hypothesi mista, tamen magnum mihi videntur -speciem veri habere.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch87" id="fn87">87</a> -Cf. His, On the Principles of Animal -Morphology, <i>Proc. R. S. E.</i> <span class="smmaj">XV,</span> 1888, p. 294: -“My own attempts to introduce some elementary mechanical -or physiological conceptions into embryology have not -generally been agreed to by morphologists. To one it seemed -ridiculous to speak of the elasticity of the germinal -layers; another thought that, by such considerations, we -‘put the cart before the horse’: and one more recent author -states, that we have better things to do in embryology -than to discuss tensions of germinal layers and similar -questions, since all explanations must of necessity be of -a phylogenetic nature. This opposition to the application -of the fundamental principles of science to embryological -questions would scarcely be intelligible had it not a -dogmatic background. No other explanation of living forms -is allowed than heredity, and any which is founded on -another basis must be rejected ....... To think that heredity -will build organic beings without mechanical means is a -piece of unscientific mysticism.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch88" id="fn88">88</a> -Hertwig, O., <i>Zeit und Streitfragen der -Biologie</i>, <span class="smmaj">II.</span> 1897.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch89" id="fn89">89</a> -Cf. Roux, <i>Gesammelte Abhandlungen</i>, -<span class="smmaj">II,</span> p. 31, 1895.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch90" id="fn90">90</a> -<i>Treatise on Comparative Embryology</i>, -<span class="smmaj">I,</span> p. 4, 1881.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch91" id="fn91">91</a> -Cf. Fick, <i>Anal. Anzeiger</i>, <span class="smmaj">XXV,</span> p. -190, 1904.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch92" id="fn92">92</a> -1st ed. p. 444; 6th ed. p. 390. The student -should not fail to consult the passage in question; -for there is always a risk of misunderstanding or -misinterpretation when one attempts to epitomise Darwin’s -carefully condensed arguments.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch93" id="fn93">93</a> -“In omni rerum naturalium historia utile est -<i>mensuras definiri et numeros</i>,” Haller, <i>Elem. Physiol.</i> -<span class="smmaj">II,</span> p. 258, 1760. Cf. Hales, <i>Vegetable Staticks</i>, -Introduction.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch94" id="fn94">94</a> -Brussels, 1871. Cf. the same author’s -<i>Physique sociale</i>, 1835, and <i>Lettres sur la théorie des -probabilités</i>, 1846. See also, for the general subject, -Boyd, R., Tables of weights of the Human Body, etc. <i>Phil. -Trans.</i> vol. <span class="smmaj">CLI,</span> 1861; Roberts, C., <i>Manual of -Anthropometry</i>, 1878; Daffner, F., <i>Das Wachsthum des -Menschen</i> (2nd ed.), 1902, etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch95" id="fn95">95</a> -Dr Johnson was not far wrong in saying that -“life declines from thirty-five”; though the Autocrat -of the Breakfast-table, like Cicero, declares that “the -furnace is in full blast for ten years longer.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch96" id="fn96">96</a> -Joly, <i>The Abundance of Life</i>, 1915 (1890), p. 86.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch97" id="fn97">97</a> -“<i>Lou pes, mèstre de tout</i> [Le poids, maître -de tout], <i>mèstre sènso vergougno, Que te tirasso en bas de -sa brutalo pougno</i>,” J. H. Fabre, <i>Oubreto prouvençalo</i>, p. -61.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch98" id="fn98">98</a> -The continuity of the phenomenon of growth, -and the natural passage from the phase of increase to that -of decrease or decay, are admirably discussed by Enriques, -in “La morte,” <i>Riv. di Scienza</i>, 1907, and in “Wachsthum -und seine analytische Darstellung,” <i>Biol. Centralbl.</i> -June, 1909. Haller (<i>Elem</i>. <span class="smmaj">VII,</span> p. 68) recognised -<i>decrementum</i> as a phase of growth, not less important -(theoretically) than <i>incrementum</i>: “<i>tristis, sed copiosa, -haec est materies</i>.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch99" id="fn99">99</a> -Cf. (<i>int. al.</i>), Friedenthal, H., Das -Wachstum des Körpergewichtes ... in verschiedenen -Lebensältern, <i>Zeit. f. allg. Physiol.</i> <span class="smmaj">IX,</span> pp. -487–514, 1909.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch100" id="fn100">100</a> -As Haller observed it to do in the chick -(<i>Elem.</i> <span class="smmaj">VIII,</span> p. 294): “Hoc iterum incrementum -miro ordine ita distribuitur, ut in principio incubationis -maximum est: inde perpetuo minuatur.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch101" id="fn101">101</a> -There is a famous passage in Lucretius -(v. 883) where he compares the course of life, -or rate of growth, in the horse and his boyish master: -<i>Principio circum tribus actis impiger annis Floret equus, -puer hautquaquam</i>, etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch102" id="fn102">102</a> -Minot, C. S., Senescence and Rejuvenation, -<i>Journ. of Physiol.</i> <span class="smmaj">XII,</span> pp. 97–153, 1891; The -Problem of Age, Growth and Death, <i>Pop. Science Monthly</i> -(June–Dec.), 1907.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch103" id="fn103">103</a> -Quoted in Vierordt’s <i>Anatomische ... Daten -und Tabellen</i>, 1906. p. 13.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch104" id="fn104">104</a> -<i>Unsere Körperform</i>, Leipzig, 1874.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch105" id="fn105">105</a> -No such point of inflection appears in -the curve of weight according to C. M. Jackson’s data -(On the Prenatal Growth of the Human Body, etc., <i>Amer. -Journ. of Anat.</i> <span class="smmaj">IX,</span> 1009, pp. 126, 156), nor -in those quoted by him from Ahlfeld, Fehling and others. -But it is plain that the very rapid increase of the -monthly weights, approximately in the ratio of the cubes -of the corresponding lengths, would tend to conceal any -such breach of continuity, unless it happened to be very -marked indeed. Moreover in the case of Jackson’s data (and -probably also in the others) the actual age of the embryos -was not determined, but was estimated from their lengths. -The following is Jackson’s estimate of average weights at -intervals of a lunar month:</p> - -<div class="dtblbox"><div class="nowrap"> -<table class="fsz6"> -<tr> - <td class="tdright">Months</td> - <td class="tdcntr">0</td> - <td class="tdcntr">1</td> - <td class="tdcntr">2</td> - <td class="tdcntr">3</td> - <td class="tdcntr">4</td> - <td class="tdcntr">5</td> - <td class="tdcntr">6</td> - <td class="tdcntr">7</td> - <td class="tdcntr">8</td> - <td class="tdcntr">9</td> - <td class="tdcntr">10</td></tr> -<tr> - <td class="tdright">Wt in gms.</td> - <td class="tdcntr">·0</td> - <td class="tdcntr">·04</td> - <td class="tdcntr">3</td> - <td class="tdcntr">36</td> - <td class="tdcntr">120</td> - <td class="tdcntr">330</td> - <td class="tdcntr">600</td> - <td class="tdcntr">1000</td> - <td class="tdcntr">1500</td> - <td class="tdcntr">2200</td> - <td class="tdcntr">3200</td></tr> -</table></div></div><!--dtblbox--></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch106" id="fn106">106</a> -G. Kraus (after Wallich-Martius), <i>Ann. du -Jardin bot. de Buitenzorg</i>, <span class="smmaj">XII,</span> 1, 1894, p. 210. -Cf. W. Ostwald, <i>Zeitliche Eigenschaften</i>, etc. p. 56.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch107" id="fn107">107</a> -Cf. Chodat, R., et Monnier, A., Sur la courbe -de croissance des végétaux, <i>Bull. Herb. Boissier</i> (2), -<span class="smmaj">V,</span> pp. 615, 616, 1905.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch108" id="fn108">108</a> -Cf. Fr. Boas, Growth of Toronto Children, -<i>Rep. of U.S. Comm. of Education</i>, 1896–7, pp. 1541–1599, -1898; Boas and Clark Wissler, Statistics of Growth, -<i>Education Rep.</i> 1904, pp. 25–132, 1906; H. P. Bowditch, -<i>Rep. Mass. State Board of Health</i>, 1877; K. Pearson, On -the Magnitude of certain coefficients of Correlation in -Man, <i>Pr. R. S.</i> <span class="smmaj">LXVI,</span> 1900.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch109" id="fn109">109</a> -<i>l.c.</i> p. 42, and other papers there quoted.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch110" id="fn110">110</a> -See, for an admirable résumé of facts, -Wolfgang Ostwald, <i>Ueber die Zeitliche Eigenschaften der -Entwickelungsvorgänge</i> (71 pp.), Leipzig, 1908 (Roux’s -<i>Vorträge</i>, Heft <span class="nowrap"><span class="smmaj">V</span>):</span> -to which work I am much -indebted. A long list of observations on the growth-rate -of various animals is also given by H. Przibram, <i>Exp. -Zoologie</i>, 1913, pt. <span class="smmaj">IV</span> (<i>Vitalität</i>), pp. -85–87.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch111" id="fn111">111</a> -Cf. St Loup, Vitesse de croissance chez les -Souris, <i>Bull. Soc. Zool. Fr.</i> <span class="smmaj">XVIII,</span> 242, 1893; -Robertson, <i>Arch. f. Entwickelungsmech.</i> <span class="smmaj">XXV,</span> p. -587, 1908; Donaldson. <i>Boas Memorial Volume</i>, New -York, 1906.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch112" id="fn112">112</a> -Luciani e Lo Monaco, <i>Arch. Ital. de -Biologie</i>, <span class="smmaj">XXVII,</span> p. 340, 1897.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch113" id="fn113">113</a> -Schaper, <i>Arch. f. Entwickelungsmech.</i> -<span class="smmaj">XIV,</span> p. 356, 1902. Cf. Barfurth, Versuche über -die Verwandlung der Froschlarven, <i>Arch. f. mikr. Anat.</i> -<span class="smmaj">XXIX,</span> 1887.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch114" id="fn114">114</a> -Joh. Schmidt, Contributions to the -Life-history of the Eel, <i>Rapports du Conseil Intern. pour -l’exploration de la Mer</i>, vol. <span class="smmaj">V,</span> pp. 137–274, -Copenhague, 1906.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch115" id="fn115">115</a> -That the metamorphoses of an insect are -but phases in a process of growth, was firstly clearly -recognised by Swammerdam, <i>Biblia Naturae</i>, 1737, pp. 6, -579 etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch116" id="fn116">116</a> -From Bose, J. C., <i>Plant Response</i>, London, -1906, p. 417.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch117" id="fn117">117</a> -This phenomenon, of <i>incrementum inequale</i>, as -opposed to <i>incrementum in universum</i>, was most carefully -studied by Haller: “Incrementum inequale multis modis fit, -ut aliae partes corporis aliis celerius increscant. Diximus -hepar minus fieri, majorem pulmonem, minimum thymum, etc.” -(<i>Elem.</i> <span class="smmaj">VIII</span> (2), p. 34).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch118" id="fn118">118</a> -See (<i>inter alia</i>) Fischel, A., Variabilität -und Wachsthum des embryonalen Körpers, <i>Morphol. Jahrb.</i> -<span class="smmaj">XXIV,</span> pp. 369–404, 1896. Oppel, <i>Vergleichung -des Entwickelungsgrades der Organe zu verschiedenen -Entwickelungszeiten bei Wirbelthieren</i>, Jena, 1891. Faucon, -A., <i>Pesées et Mensurations fœtales à différents âges de -la grossesse</i>. (Thèse.) Paris, 1897. Loisel, G., Croissance -comparée en poids et en longueur des fœtus mâle et -femelle dans l’espèce humaine, <i>C. R. Soc. de Biologie</i>, -Paris, 1903. Jackson, C. M., Pre-natal growth of the human -body and the relative growth of the various organs and -parts, <i>Am. J. of Anat.</i> <span class="smmaj">IX,</span> 1909; Post-natal -growth and variability of the body and of the various -organs in the albino rat, <i>ibid.</i> <span class="smmaj">XV,</span> 1913.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch119" id="fn119">119</a> -<i>l.c.</i> p. 1542.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch120" id="fn120">120</a> -Variation and Correlation in Brain-weight, -<i>Biometrika</i>, <span class="smmaj">IV,</span> pp. 13–104, 1905.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch121" id="fn121">121</a> -<i>Die Säugethiere</i>, p. 117.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch122" id="fn122">122</a> -<i>Amer. J. of Anatomy</i>, <span class="smmaj">VIII,</span> -pp. 319–353, 1908. Donaldson (<i>Journ. Comp. Neur. and -Psychol.</i> <span class="smmaj">XVIII,</span> pp. 345–392, 1908) also gives -a logarithmic formula for brain-weight (<i>y</i>) as compared -with body-weight (<i>x</i>), which in the case of the white rat -is <i>y</i> -= ·554 − ·569 log(<i>x</i> − 8·7), and the agreement is -very close. But the formula is admittedly empirical and -as Raymond Pearl says (<i>Amer. Nat.</i> 1909, p. 303), “no -ulterior biological significance is to be attached to it.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch123" id="fn123">123</a> -<i>Biometrika</i>, <span class="smmaj">IV,</span> pp. 13–104, 1904.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch124" id="fn124">124</a> -Donaldson, H. H., A Comparison of the White -Rat with Man in respect to the Growth of the entire Body, -<i>Boas Memorial Vol.</i>, New York, 1906, pp. 5–26.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch125" id="fn125">125</a> -Besides many papers quoted by Dubois on the -growth and weight of the brain, and numerous papers in -<i>Biometrika</i>, see also the following: Ziehen, Th., <i>Das -Gehirn: Massverhältnisse</i>, in Bardeleben’s <i>Handb. der -Anat. des Menschen</i>, <span class="smmaj">IV,</span> pp. 353–386, 1899. -Spitzka, E. A., Brain-weight of Animals with special -reference to the Weight of the Brain in the Macaque Monkey, -<i>J. Comp. Neurol.</i> <span class="smmaj">XIII,</span> pp. 9–17, 1903. Warneke, P., -Mitteilung neuer Gehirn und Körpergewichtsbestimmungen -bei Säugern, nebst Zusammenstellung der gesammten bisher -beobachteten absoluten und relativen Gehirngewichte bei -den verschiedenen Species, <i>J. f. Psychol. u. Neurol.</i> -<span class="smmaj">XIII,</span> pp. 355–403, 1909. Donaldson, H. H., On the -regular seasonal Changes in the relative Weight of the -Central Nervous System of the Leopard Frog, <i>Journ. of -Morph.</i> <span class="smmaj">XXII,</span> pp. 663–694, 1911.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch126" id="fn126">126</a> -Cf. Jenkinson, Growth, Variability and Correlation in Young Trout, -<i>Biometrika</i>, <span class="smmaj">VIII,</span> pp. 444–455, 1912.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch127" id="fn127">127</a> -Cf. chap. xvii, p. 739.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch128" id="fn128">128</a> -“ ...I marked in the same manner as the Vine, -young Honeysuckle shoots, etc....; and I found in them -all a gradual scale of unequal extensions, those parts -extending most which were tenderest,” <i>Vegetable Staticks</i>, -Exp. cxxiii.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch129" id="fn129">129</a> -From Sachs, <i>Textbook of Botany</i>, 1882, p. -820.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch130" id="fn130">130</a> -Variation and Differentiation in -Ceratophyllum, <i>Carnegie Inst. Publications</i>, No. 58, -Washington, 1907.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch131" id="fn131">131</a> -Cf. Lämmel, Ueber periodische Variationen in -Organismen, <i>Biol. Centralbl.</i> <span class="smmaj">XXII,</span> pp. 368–376, -1903.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch132" id="fn132">132</a> -Herein lies the easy answer to a contention -frequently raised by Bergson, and to which he ascribes -great importance, that “a mere variation of size is one -thing, and a change of form is another.” Thus he considers -“a change in the form of leaves” to constitute “a profound -morphological difference.” <i>Creative Evolution</i>, p. 71.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch133" id="fn133">133</a> -I do not say that the assumption that these -two groups of earwigs were of different ages is altogether -an easy one; for of course, even in an insect whose -metamorphosis is so simple as the earwig’s, consisting only -in the acquisition of wings or wing-cases, we usually take -it for granted that growth proceeds no more after the final -stage, or “adult form” is attained, and further that this -adult form is attained at an approximately constant age, -and constant magnitude. But even if we are not permitted -to think that the earwig may have grown, or moulted, -after once the elytra were produced, it seems to me far -from impossible, and far from unlikely, that prior to the -appearance of the elytra one more stage of growth, or one -more moult took place in some cases than in others: for the -number of moults is known to be variable in many species of -Orthoptera. Unfortunately Bateson tells us nothing about -the sizes or total lengths of his earwigs; but his figures -suggest that it was bigger earwigs that had the longer -tails; and that the rate of growth of the tails had had -a certain definite ratio to that of the bodies, but not -necessarily a simple ratio of equality.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch134" id="fn134">134</a> -Jackson, C. M., <i>J. of Exp. Zool.</i> -<span class="smmaj">XIX,</span> 1915, p. 99; cf. also Hans Aron, Unters. über -die Beeinflüssung der Wachstum durch die Ernährung, <i>Berl. -klin. Wochenbl.</i> <span class="smmaj">LI,</span> pp. 972–977, 1913, etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch135" id="fn135">135</a> -The temperature limitations of life, and to -some extent of growth, are summarised for a large number of -species by Davenport, <i>Exper. Morphology</i>, cc. viii, xviii, -and by Hans Przibram, <i>Exp. Zoologie</i>, <span class="smmaj">IV,</span> c. v.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch136" id="fn136">136</a> -Réaumur: <i>L’art de faire éclore et élever en -toute saison des oiseaux domestiques, foit par le moyen de -la chaleur du fumier</i>, Paris, 1749.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch137" id="fn137">137</a> -Cf. (<i>int. al.</i>) de Vries, H., Matériaux pour -la connaissance de l’influence de la température sur les -plantes, <i>Arch. Néerl.</i> <span class="smmaj">V,</span> 385–401, 1870. Köppen, -Wärme und Pflanzenwachstum, <i>Bull. Soc. Imp. Nat. Moscou.</i> -<span class="smmaj">XLIII,</span> pp. 41–110, 1870.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch138" id="fn138">138</a> -Blackman, F. F., <i>Ann. of Botany</i>, -<span class="smmaj">XIX,</span> p. 281, 1905.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch139" id="fn139">139</a> -For various instances of a “temperature -coefficient” in physiological processes, see Kanitz, -<i>Zeitschr. f. Elektrochemie</i>, 1907, p. 707; <i>Biol. -Centralbl.</i> <span class="smmaj">XXVII,</span> p. 11, 1907; Hertzog, R. O., -Temperatureinfluss auf die Entwicklungsgeschwindigkeit der -Organismen, <i>Zeitschr. f. Elektrochemie</i>, <span class="smmaj">XI,</span> p -820, 1905; Krogh, -Quantitative Relation between Temperature and -Standard Metabolism, <i>Int. Zeitschr. f. physik.-chem. -Biologie</i>, <span class="smmaj">I,</span> p. 491, 1914; Pütter, A., Ueber -Temperaturkoefficienten, <i>Zeitschr. f. allgem. Physiol.</i> -<span class="smmaj">XVI,</span> p. 574, 1914. Also Cohen, <i>Physical Chemistry -for Physicians and Biologists</i> (English edition), 1903; -Pike, F. H., and Scott. E. L., The Regulation of the -Physico-chemical Condition of the Organism, <i>American -Naturalist</i>, Jan. 1915, and various papers quoted therein.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch140" id="fn140">140</a> -Cf. Errera, L., <i>L’Optimum</i>, 1896 (<i>Rec. -d’Oeuvres, Physiol. générale</i>, pp. 338–368, 1910); -Sachs, <i>Physiologie d. Pflanzen</i>, 1882, p. 233; Pfeffer, -<i>Pflanzenphysiologie</i>, ii, p. 78, 1904; and cf. Jost, -Ueber die Reactionsgeschwindigkeit im Organismus, <i>Biol. -Centralbl.</i> <span class="smmaj">XXVI,</span> pp. 225–244, 1906.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch141" id="fn141">141</a> -After Köppen, <i>Bull. Soc. Nat. Moscou</i>, -<span class="smmaj">XLIII,</span> pp. 41–110, 1871.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch142" id="fn142">142</a> -<i>Botany</i>, p. 387.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch143" id="fn143">143</a> -Leitch, I., Some Experiments on the Influence -of Temperature on the Rate of Growth in <i>Pisum sativum, -Ann. of Botany</i>, <span class="smmaj">XXX,</span> pp. 25–46, 1916. (Cf. -especially Table III, p. 45.)</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch144" id="fn144">144</a> -Blackman, F. F., Presidential Address in -Botany, <i>Brit. Ass.</i> Dublin, 1908.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch145" id="fn145">145</a> -<i>Rec. de l’Inst. Bot. de Bruxelles</i>, -<span class="smmaj">VI,</span> 1906.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch146" id="fn146">146</a> -Hertwig, O., Einfluss der Temperatur auf -die Entwicklung von <i>Rana fusca</i> und <i>R. esculenta</i>, -<i>Arch. f. mikrosk. Anat.</i> <span class="smmaj">LI,</span> p. 319, 1898. -Cf. also Bialaszewicz, K., Beiträge z. Kenntniss d. -Wachsthumsvorgänge bei Amphibienembryonen, <i>Bull. Acad. -Sci. de Cracovie</i>, p. 783, 1908; Abstr. in <i>Arch. f. -Entwicklungsmech.</i> <span class="smmaj">XXVIII,</span> p. 160, 1909.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch147" id="fn147">147</a> -Der Grad der Beschleunigung tierischer -Entwickelung durch erhöhte Temperatur, <i>A. f. Entw.</i> Mech. -<span class="smmaj">XX.</span> p. 130, 1905. More recently, Bialaszewicz has -determined the coefficient for the rate of segmentation in -Rana as being 2·4 per 10° C.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch148" id="fn148">148</a> -<i>Das Wachstum des Menschen</i>, p. 329, 1902.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch149" id="fn149">149</a> -The <i>diurnal</i> periodicity is beautifully -shewn in the case of the Hop by Joh. Schmidt (<i>C. R. -du Laboratoire de Carlsberg</i>, <span class="smmaj">X,</span> pp. 235–248, -Copenhague, 1913).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch150" id="fn150">150</a> -<i>Trans. Botan. Soc. Edinburgh</i>, -<span class="smmaj">XVIII,</span> 1891, p. 456.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch151" id="fn151">151</a> -I had not received, when this was written, -Mr Douglass’s paper, On a method of estimating Rainfall -by the Growth of Trees, <i>Bull. Amer. Geograph. Soc.</i> -<span class="smmaj">XLVI,</span> pp. 321–335, 1914. Mr Douglass does not fail -to notice the long period here described; but he lays more -stress on the occurrence of shorter cycles (of 11, 21 and -33 years), well known to meteorologists. Mr Douglass is -inclined (and I think rightly) to correlate the variations -in growth directly with fluctuations in rainfall, that is -to say with alternate periods of moisture and aridity; but -he points out that the temperature curves (and also the -sunspot curves) are markedly similar.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch152" id="fn152">152</a> -It may well be that the effect is not due to -light after all; but to increased absorption of heat by the -soil, as a result of the long hours of exposure to the sun.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch153" id="fn153">153</a> -On growth in relation to light, see Davenport, -<i>Exp. Morphology</i>, <span class="smmaj">II,</span> ch. xvii. In some cases (as -in the roots of Peas), exposure to light seems to have no -effect on growth; in other cases, as in diatoms (according -to Whipple’s experiments, quoted by Davenport, <span class="smmaj">II,</span> -p. 423), the effect of light on growth or multiplication -is well-marked, measurable, and apparently capable of -expression by a logarithmic formula. The discrepancy would -seem to arise from the fact that, while light-energy always -tends to be absorbed by the chlorophyll of the plant, -converted into chemical energy, and stored in the shape -of starch or other reserve materials, the actual rate of -growth depends on the rate at which these reserves are -drawn on: and this is another matter, in which light-energy -is no longer directly concerned.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch154" id="fn154">154</a> -Cf. for instance, Nägeli’s classical account -of the effect of change of habitat on Alpine and other -plants: <i>Sitzungsber. Baier. Akad. Wiss.</i> 1865, pp. -228–284.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch155" id="fn155">155</a> -Cf. Blackman, F. F., Presidential Address -in Botany, <i>Brit. Ass.</i> Dublin, 1908. The fact was first -enunciated by Baudrimont and St Ange, Recherches sur le -développement du fœtus, <i>Mém. Acad. Sci.</i> <span class="smmaj">XI,</span> p. -469, 1851.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch156" id="fn156">156</a> -Cf. Loeb, <i>Untersuchungen zur physiol. -Morphologie der Thiere</i>, 1892; also Experiments on -Cleavage, <i>J. of Morph.</i> <span class="smmaj">VII,</span> p. 253, 1892; -Zusammenstellung der Ergebnisse einiger Arbeiten über die -Dynamik des thierischen Wachsthum, <i>Arch. f. Entw. Mech.</i> -<span class="smmaj">XV,</span> 1902–3, p. 669; Davenport, On the Rôle of -Water in Growth, <i>Boston Soc. N. H.</i> 1897; Ida H. Hyde, -<i>Am. J. of Physiol.</i> <span class="smmaj">XII,</span> 1905, p. 241, etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch157" id="fn157">157</a> -<i>Pflüger’s Archiv</i>, <span class="smmaj">LV,</span> 1893.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch158" id="fn158">158</a> -Beiträge zur Kenntniss der Wachstumsvorgänge -bei Amphibienembryonen, <i>Bull. Acad. Sci. de Cracovie</i>, -1908, p. 783; cf. <i>Arch. f. Entw. Mech.</i> <span class="smmaj">XXVIII,</span> -p. 160, 1909; <span class="smmaj">XXXIV,</span> p. 489, 1912.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch159" id="fn159">159</a> -Fehling, H., <i>Arch. für Gynaekologie</i>, <span class="smmaj">XI,</span> 1877; cf. -Morgan, <i>Experimental Zoology</i>, p. 240, 1907.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch160" id="fn160">160</a> -Höber, R., Bedeutung der Theorie der -Lösungen für Physiologie und Medizin, <i>Biol. Centralbl.</i> -<span class="smmaj">XIX,</span> 1899; cf. pp. 272–274.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch161" id="fn161">161</a> -Schmankewitsch has made other interesting -observations on change of size and form, after some -generations, in relation to change of density; e.g. in the -flagellate infusorian <i>Anisonema acinus</i>, Bütschli (<i>Z. f. -w. Z.</i> <span class="smmaj">XXIX,</span> p. 429, 1877).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch162" id="fn162">162</a> -These “Fezzan-worms,” when first described, -were supposed to be “insects’ eggs”; cf. Humboldt, -<i>Personal Narrative</i>, <span class="smmaj">VI,</span> i, 8, note; Kirby and -Spence, Letter <span class="smmaj">X.</span></p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch163" id="fn163">163</a> -Cf. <i>Introd. à l’étude de la médecine -expérimentale</i>, 1885, p. 110.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch164" id="fn164">164</a> -Cf. Abonyi, <i>Z. f. w. Z.</i> <span class="smmaj">CXIV,</span> p. -134, 1915. But Frédéricq has shewn that the amount of NaCl -in the blood of Crustacea (<i>Carcinus moenas</i>) varies, and -all but corresponds, with the density -of the water in which the creature has been kept (<i>Arch. -de Zool. Exp. et Gén.</i> (2), <span class="smmaj">III,</span> p. xxxv, 1885); -and other results of Frédéricq’s, -and various data given or quoted by Bottazzi (Osmotischer -Druck und elektrische Leitungsfähigkeit der Flüssigkeiten -der Organismen, in Asher-Spiro’s <i>Ergebn. d. Physiologie</i>, -<span class="smmaj">VII,</span> pp. 160–402, 1908) suggest that the case of -the brine-shrimps must be looked upon as an extreme or -exceptional one.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch165" id="fn165">165</a> -Cf. Schmankewitsch, <i>Z. f. w. Zool.</i> -<span class="smmaj">XXV,</span> 1875, <span class="smmaj">XXIX,</span> 1877, etc.; transl. -in appendix to Packard’s <i>Monogr. of N. American -Phyllopoda</i>, 1883, pp. 466–514; Daday de Deés, <i>Ann. -Sci. Nat.</i> (<i>Zool.</i>), (9), <span class="smmaj">XI,</span> 1910; Samter und -Heymons, <i>Abh. d. K. pr. Akad. Wiss.</i> 1902; Bateson, -<i>Mat. for the Study of Variation</i>, 1894, pp. 96–101; -Anikin, <i>Mitth. Kais. Univ. Tomsk</i>, -<span class="nowrap"><span class="smmaj">XIV</span>:</span> <i>Zool. -Centralbl.</i> <span class="smmaj">VI,</span> pp. 756–760, 1908; Abonyi, <i>Z. -f. w. Z.</i> <span class="smmaj">CXIV,</span> pp. 96–168, 1915 (with copious -bibliography), etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch166" id="fn166">166</a> -According to the empirical canon of -physiology, that (as Frédéricq expresses it) “L’être -vivant est agencé de telle manière que chaque influence -perturbatrice provoque d’elle-même la mise en activité de -l’appareil compensateur qui doit neutraliser et réparer le -dommage.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch167" id="fn167">167</a> -Such phenomena come precisely under the head -of what Bacon called <i>Instances of Magic</i>: “By which I -mean those wherein the material or efficient cause is -scanty and small as compared with the work or effect -produced; so that even when they are common, they seem like -miracles, some at first sight, others even after attentive -consideration. These magical effects are brought about in -three ways ... [of which one is] by excitation or invitation -in another body, as in the magnet which excites numberless -needles without losing any of its virtue, <i>or in yeast and -such-like</i>.” <i>Nov. Org.</i>, cap. li.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch168" id="fn168">168</a> -Monnier, A., Les matières minérales, et la -loi d’accroissement des Végétaux, <i>Publ. de l’Inst. de -Bot. de l’Univ. de Genève</i> (7), <span class="smmaj">III,</span> 1905. Cf. -Robertson, On the Normal Rate of Growth of an Individual, -and its Biochemical Significance, <i>Arch. f. Entw. Mech.</i> -<span class="smmaj">XXV,</span> pp. 581–614, <span class="smmaj">XXVI,</span> pp. 108–118, -1908; Wolfgang Ostwald, <i>Die zeitlichen Eigenschaften der -Entwickelungsvorgänge</i>, 1908; Hatai, S., Interpretation of -Growth-curves from a Dynamical Standpoint, <i>Anat. Record</i>, -<span class="smmaj">V,</span> p. 373, 1911.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch169" id="fn169">169</a> -<i>Biochem. Zeitschr.</i> <span class="smmaj">II,</span> 1906, p. 34.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch170" id="fn170">170</a> -Even a crystal may be said, in a sense, -to display “autocatalysis”: for the bigger its surface -becomes, the more rapidly does the mass go on increasing.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch171" id="fn171">171</a> -Cf. Loeb, The Stimulation of Growth, -<i>Science</i>, May 14, 1915.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch172" id="fn172">172</a> -<i>B. coli-communis</i>, according to Buchner, -tends to double in 22 minutes; in 24 hours, therefore, a -single individual would be multiplied by something like -10<sup>28</sup> ; <i>Sitzungsber. München. Ges. Morphol. u. Physiol.</i> -<span class="smmaj">III,</span> pp. 65–71, 1888. Cf. Marshall Ward, Biology -of <i>Bacillus ramosus</i>, etc. <i>Pr. R. S.</i> <span class="smmaj">LVIII,</span> -265–468, 1895. The comparatively large infusorian -Stylonichia, according to Maupas, would multiply in a month -by 10<sup>43</sup> .</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch173" id="fn173">173</a> -Cf. Enriques, Wachsthum und seine analytisehe -Darstellung, <i>Biol. Centralbl.</i> 1909, p. 337.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch174" id="fn174">174</a> -Cf. (<i>int. al.</i>) Mellor, <i>Chemical Statics and -Dynamics</i>, 1904, p. 291.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch175" id="fn175">175</a> -Cf. Robertson, <i>l.c.</i></p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch176" id="fn176">176</a> -See, for a brief resumé of this subject, -Morgan’s <i>Experimental Zoology</i>, chap. xvi.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch177" id="fn177">177</a> -<i>Amer. J. of Physiol.</i>, <span class="smmaj">X,</span> 1904.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch178" id="fn178">178</a> -<i>C.R.</i> <span class="smmaj">CXXI,</span> <span class="smmaj">CXXII,</span> 1895–96.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch179" id="fn179">179</a> -Cf. Loeb, <i>Science</i>, May 14, 1915.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch180" id="fn180">180</a> -Cf. Baumann u. Roos, Vorkommen von Iod im -Thierkörper, <i>Zeitschr. für Physiol. Chem.</i> <span class="smmaj">XXI,</span> -<span class="smmaj">XXII,</span> 1895, 6.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch181" id="fn181">181</a> -Le Néo-Vitalisme, <i>Rev. Scientifique</i>, Mars -1911, p. 22 (of reprint).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch182" id="fn182">182</a> -<i>La vie et la mort</i>, p. 43, 1902.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch183" id="fn183">183</a> -Cf. Dendy, <i>Evolutionary Biology</i>, 1912, p. -408; <i>Brit. Ass. Report</i> (Portsmouth), 1911, p. 278.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch184" id="fn184">184</a> -Lucret. v, 877. “Lucretius nowhere seems to -recognise the possibility of improvement or change of -species by ‘natural selection’; the animals remain as they -were at the first, except that the weaker and more useless -kinds have been crushed out. Hence he stands in marked -contrast with modern evolutionists.” Kelsey’s note, <i>ad -loc.</i></p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch185" id="fn185">185</a> -Even after we have so narrowed the scope -and sphere of natural selection, it is still hard to -understand; for the causes of <i>extinction</i> are often -wellnigh as hard to comprehend as are those of the <i>origin</i> -of species. If we assert (as has been lightly done) that -Smilodon perished owing to its gigantic tusks, that -Teleosaurus was handicapped by its exaggerated snout, -or Stegosaurus weighed down by its intolerable load of -armour, we may be reminded of other kindred forms to -show that similar conditions did not necessarily lead to -extermination, or that rapid extinction ensued apart from -any such visible or apparent disadvantages. Cf. Lucas, F. -A., On Momentum in Variation, <i>Amer. Nat.</i> xli, p. 46, -1907.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch186" id="fn186">186</a> -See Professor T. H. Morgan’s <i>Regeneration</i> -(316 pp.), 1901 for a full account and copious -bibliography. The early experiments on regeneration, -by Vallisneri, Réaumur, Bonnet, Trembley, Baster, and -others, are epitomised by Haller, <i>Elem. Physiologiae</i>, -<span class="smmaj">VIII,</span> p. 156 <i>seq.</i></p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch187" id="fn187">187</a> -<i>Journ. Experim. Zool.</i> <span class="smmaj">VII,</span> p. 397, -1909.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch188" id="fn188">188</a> -<i>Op. cit.</i> p. 406, Exp. <span class="smmaj">IV.</span></p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch189" id="fn189">189</a> -The experiments of Loeb on the growth of -Tubularia in various saline solutions, referred to on p. -125, might as well or better have been referred to under -the heading of regeneration, as they were performed on cut -pieces of the zoophyte. (Cf. Morgan, <i>op. cit.</i> p. 35.)</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch190" id="fn190">190</a> -<i>Powers of the Creator</i>, <span class="smmaj">I,</span> p. 7, -1851. See also <i>Rare and Remarkable Animals</i>, <span class="smmaj">II,</span> -pp. 17–19, 90, 1847.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch191" id="fn191">191</a> -Lillie, F. R., The smallest Parts of -Stentor capable of Regeneration, <i>Journ. of Morphology</i>, -<span class="smmaj">XII,</span> p. 239, 1897.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch192" id="fn192">192</a> -Boveri, Entwicklungsfähigkeit kernloser -Seeigeleier, etc., <i>Arch. f. Entw. Mech.</i> <span class="smmaj">II,</span> -1895. See also Morgan, Studies of the partial larvae of -Sphaerechinus, <i>ibid.</i> 1895; J. Loeb, On the Limits of -Divisibility of Living Matter, <i>Biol. Lectures</i>, 1894, etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch193" id="fn193">193</a> -Cf. Przibram, H., Scheerenumkehr bei -dekapoden Crustaceen, <i>Arch. f. Entw. Mech.</i> <span class="smmaj">XIX,</span> -181–247, 1905; <span class="smmaj">XXV,</span> 266–344, 1907. Emmel, -<i>ibid.</i> <span class="smmaj">XXII,</span> 542, 1906; Regeneration of lost -parts in Lobster, <i>Rep. Comm. Inland Fisheries, Rhode -Island</i>, <span class="smmaj">XXXV,</span> <span class="smmaj">XXXVI,</span> 1905–6; <i>Science</i> -(n.s.), <span class="smmaj">XXVI,</span> 83–87, 1907. Zeleny, Compensatory -Regulation, <i>J. Exp. Zool.</i> <span class="smmaj">II,</span> 1–102, 347–369, -1905; etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch194" id="fn194">194</a> -Lobsters are occasionally found with two -symmetrical claws: which are then usually serrated, -sometimes (but very rarely) both blunt-toothed. Cf. Calman, -<i>P.Z.S.</i> 1906, pp. 633, 634, and <i>reff.</i></p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch195" id="fn195">195</a> -Wilson, E. B., Reversal of Symmetry in <i>Alpheus heterochelis</i>, <i>Biol. Bull.</i> <span class="smmaj">IV,</span> -p. 197, 1903.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch196" id="fn196">196</a> -<i>J. Exp. Zool.</i> <span class="smmaj">VII,</span> p. 457, 1909.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch197" id="fn197">197</a> -<i>Biologica</i>, <span class="smmaj">III,</span> p. 161, June. 1913.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch198" id="fn198">198</a> -<i>Anatomical and Pathological Observations</i>, p. -3, 1845; <i>Anatomical Memoirs</i>, <span class="smmaj">II,</span> p. 392, 1868.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch199" id="fn199">199</a> -Giard, A., L’œuf et les débuts de l’évolution, -<i>Bull. Sci. du Nord de la Fr.</i> <span class="smmaj">VIII,</span> pp. 252–258, -1876.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch200" id="fn200">200</a> -<i>Entwickelungsvorgänge der Eizelle</i>, 1876; -<i>Investigations on Microscopic Foams and Protoplasm</i>, p. 1, -1894.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch201" id="fn201">201</a> -<i>Journ. of Morphology</i>, <span class="smmaj">I,</span> p. 229, -1887.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch202" id="fn202">202</a> -While it has been very common to look upon -the phenomena of mitosis as sufficiently explained by -the results <i>towards which</i> they seem to lead, we may -find here and there a strong protest against this mode of -interpretation. The following is a case in point: “On a -tenté d’établir dans la mitose dite primitive plusieurs -catégories, plusieurs types de mitose. On a choisi le plus -souvent comme base de ces systèmes des concepts abstraits -et téléologiques: répartition plus ou moins exacte de -la chromatine entre les deux noyaux-fils suivant qu’il -y a ou non des chromosomes (<i>Dangeard</i>), distribution -particulière et signification dualiste des substances -nucléaires (substance kinétique et substance générative ou -héréditaire, <i>Hartmann et ses élèves</i>), etc. Pour moi tous -ces essais sont à rejeter catégoriquement à cause de leur -caractère finaliste; de plus, ils sont construits sur des -concepts non démontrés, et qui parfois représentent des -généralisations absolument erronées.” A. Alexeieff, <i>Archiv -für Protistenkunde</i>, <span class="smmaj">XIX,</span> p. 344, 1913.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch203" id="fn203">203</a> -This is the old philosophic axiom writ large: -<i>Ignorato motu, ignoratur natura</i>; which again is but -an adaptation of Aristotle’s phrase, ἡ ἀρχὴ τῆς κινήσεως, -as equivalent to the “Efficient Cause.” -FitzGerald holds that “all explanation consists in a -description of underlying motions”; <i>Scientific Writings</i>, -1902, p. 385.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch204" id="fn204">204</a> -As when Nägeli concluded that the organism -is, in a certain sense, “vorgebildet”; <i>Beitr. zur wiss. -Botanik</i>, <span class="smmaj">II,</span> 1860. Cf. E. B. Wilson, <i>The Cell, -etc.</i>, p. 302.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch205" id="fn205">205</a> -“La matière arrangée par une sagesse divine -doit être essentiellement organisée partout ... il y -a machine dans les parties de la machine Naturelle à -l’infini.” <i>Sur le principe de la Vie</i>, p. 431 (Erdmann). -This is the very converse of the doctrine of the Atomists, -who could not conceive a condition “<i>ubi dimidiae partis -pars semper habebit Dimidiam partem, nec res praefiniet -ulla</i>.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch206" id="fn206">206</a> -Cf. an interesting passage from the <i>Elements</i> -(<span class="smmaj">I,</span> p. 445, Molesworth’s edit.), quoted by Owen, -<i>Hunterian Lectures on the Invertebrates</i>, 2nd ed. pp. 40, -41, 1855.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch207" id="fn207">207</a> -“Wir müssen deshalb den lebenden Zellen, -abgesehen von der Molekularstructur der organischen -Verbindungen welche sie enthält, noch eine andere und in -anderer Weise complicirte Structur zuschreiben, und diese -es ist welche wir mit dem Namen <i>Organisation</i> bezeichnen,” -Brücke, Die Elementarorganismen, <i>Wiener Sitzungsber.</i> -<span class="smmaj">XLIV,</span> 1861, p. 386; quoted by Wilson, <i>The -Cell</i>, etc. p. 289. Cf. also Hardy, <i>Journ. of Physiol.</i> -<span class="smmaj">XXIV,</span> 1899, p. 159.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch208" id="fn208">208</a> -Precisely as in the Lucretian <i>concursus</i>, -<i>motus</i>, <i>ordo</i>, <i>positura</i>, <i>figurae</i>, whereby bodies -<i>mutato ordine mutant naturam</i>.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch209" id="fn209">209</a> -Otto Warburg, Beiträge zur Physiologie der -Zelle, insbesondere über die Oxidationsgeschwindigkeit in -Zellen; in Asher-Spiro’s <i>Ergebnisse der Physiologie</i>, -<span class="smmaj">XIV,</span> pp. 253–337, 1914 (see p. 315). (Cf. Bayliss, -<i>General Physiology</i>, 1915, p. 590).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch210" id="fn210">210</a> -Hardy, W. B., On some Problems of Living -Matter (Guthrie Lecture), <i>Tr. Physical Soc. London</i>, -xxviii, p. 99–118, 1916.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch211" id="fn211">211</a> -As a matter of fact both phrases occur, side -by side, in Graham’s classical paper on “Liquid Diffusion -applied to Analysis,” <i>Phil. Trans.</i> <span class="smmaj">CLI,</span> p. 184, -1861; <i>Chem. and Phys. Researches</i> (ed. Angus Smith), 1876, -p. 554.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch212" id="fn212">212</a> -L. Rhumbler, Mechanische Erklärung der -Aehnlichkeit zwischen Magnetischen Kraftliniensystemen und -Zelltheilungsfiguren, <i>Arch. f. Entw. Mech.</i> <span class="smmaj">XV,</span> -p. 482, 1903.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch213" id="fn213">213</a> -Gallardo, A., Essai d’interpretation des -figures caryocinétiques, <i>Anales del Museo de Buenos-Aires</i> -(2), <span class="smmaj">II,</span> 1896; La division de la cellule, -phenomène bipolaire de caractère electro-colloidal, <i>Arch. -f. Entw. Mech.</i> <span class="smmaj">XXVIII,</span> 1909, etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch214" id="fn214">214</a> -<i>Arch. f. Entw. Mech.</i> <span class="smmaj">III,</span> -<span class="smmaj">IV,</span> 1896–97.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch215" id="fn215">215</a> -On various theories of the mechanism of -mitosis, see (e.g.) Wilson, <i>The Cell in Development</i>, -etc., pp. 100–114; Meves, <i>Zelltheilung</i>, in Merkel u. -Bonnet’s <i>Ergebnisse der Anatomie</i>, etc., <span class="smmaj">VII,</span> -<span class="smmaj">VIII,</span> 1897–8; Ida H. Hyde, <i>Amer. Journ. of -Physiol.</i> <span class="smmaj">XII,</span> pp. 241–275, 1905; and especially -Prenant, A., Theories et interprétations physiques de la -mitose, <i>J. de l’Anat. et Physiol.</i> <span class="smmaj">XLVI,</span> pp. -511–578, 1910.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch216" id="fn216">216</a> -Hartog, M., Une force nouvelle: le -mitokinétisme, <i>C.R.</i> 11 Juli, 1910; Mitokinetism in the -Mitotic Spindle and in the Polyasters, <i>Arch. f. Entw. -Mech.</i> <span class="smmaj">XXVII,</span> pp. 141–145, 1909; cf. <i>ibid.</i> -<span class="smmaj">XL,</span> pp. 33–64, 1914. Cf. also Hartog’s papers in -<i>Proc. R. S.</i> (B), <span class="smmaj">LXXVI,</span> 1905; <i>Science Progress</i> -(n. s.), <span class="smmaj">I,</span> 1907; <i>Riv. di Scienza</i>, <span class="smmaj">II,</span> -1908; <i>C. R. Assoc. fr. pour l’Avancem. des Sc.</i> 1914, etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch217" id="fn217">217</a> -The configurations, as obtained by the -usual experimental methods, were of course known long -before Faraday’s day, and constituted the “convergent -and divergent magnetic curves” of eighteenth century -mathematicians. As Leslie said, in 1821, they were -“regarded with wonder by a certain class of dreaming -philosophers, who did not hesitate to consider them as -the actual traces of an invisible fluid, perpetually -circulating between the poles of the magnet.” Faraday’s -great advance was to interpret them as indications of -<i>stress in a medium</i>,—of tension or attraction along -the lines, and of repulsion transverse to the lines, of the -diagram.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch218" id="fn218">218</a> -Cf. also the curious phenomenon in a dividing -egg described as “spinning” by Mrs G. F. Andrews, <i>J. of -Morph.</i> <span class="smmaj">XII,</span> pp. 367–389, 1897.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch219" id="fn219">219</a> -Whitman, <i>J. of Morph.</i> <span class="smmaj">II,</span> p. 40, -1889.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch220" id="fn220">220</a> -“Souvent il n’y a qu’une séparation <i>physique</i> -entre le cytoplasme et le suc nucléaire, comme entre deux -liquides immiscibles, etc.;” Alexeieff, Sur la mitose dite -“primitive,” <i>Arch. f. Protistenk.</i> <span class="smmaj">XXIX,</span> p. 357, -1913.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch221" id="fn221">221</a> -The appearance of “vacuolation” is a result -of endosmosis or the diffusion of a less dense fluid -into the denser plasma of the cell. <i>Caeteris paribus</i>, -it is less apparent in marine organisms than in those of -freshwater, and in many or most marine Ciliates and even -Rhizopods a contractile vacuole has not been observed -(Bütschli, in Bronn’s <i>Protozoa</i>, p. 1414); it is also -absent, and probably for the same reason, in parasitic -Protozoa, such as the Gregarines and the Entamoebae. -Rossbach shewed that the contractile vacuole of ordinary -freshwater Ciliates was very greatly diminished in a 5 per -cent. solution of NaCl, and all but disappeared in a 1 -per cent. solution of sugar (<i>Arb. z. z. Inst. Würzburg</i>, -1872, cf. Massart, <i>Arch. de Biol.</i> <span class="smmaj">LX,</span> p. 515, -1889). <i>Actinophrys sol</i>, when gradually acclimatised to -sea-water, loses its vacuoles, and <i>vice versa</i> (Gruber, -<i>Biol. Centralbl.</i> <span class="smmaj">IX,</span> p. 22, 1889); and the -same is true of Amoeba (Zuelzer, <i>Arch. f. Entw. Mech.</i> -1910, p. 632). The gradual enlargement of the contractile -vacuole is precisely analogous to the change of size of -a bubble until the gases on either side of the film are -equally diffused, as described long ago by Draper (<i>Phil. -Mag.</i> (n. s.), <span class="smmaj">XI,</span> p. 559, 1837). Rhumbler has -shewn that contractile or pulsating vacuoles may be well -imitated in chloroform-drops, suspended in water in -which various substances are dissolved (<i>Arch. f. Entw. -Mech.</i> <span class="smmaj">VII,</span> 1898, p. 103). The pressure within -the contractile vacuole, always greater than without, -diminishes with its size, being inversely proportional -to its radius; and when it lies near the surface of the -cell, as in a Heliozoon, it bursts as soon as it reaches -a thinness which its viscosity or molecular cohesion no -longer permits it to maintain.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch222" id="fn222">222</a> -Cf. p. 660.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch223" id="fn223">223</a> -The elongated or curved “macronucleus” of -an Infusorian is to be looked upon as a single mass of -chromatin, rather than as an aggregation of particles in -a fluid drop, as in the case described. It has a shape of -its own, in which ordinary surface-tension plays a very -subordinate part.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch224" id="fn224">224</a> -<i>Théorie physico-chimique de la Vie</i>, p. 73, -1910; <i>Mechanism of Life</i>, p. 56, 1911.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch225" id="fn225">225</a> -Whence the name “mitosis” (Greek μίτος, -a thread), applied first by Flemming to the whole -phenomenon. Kollmann (<i>Biol. Centralbl.</i> <span class="smmaj">II,</span> -p. 107, 1882) called it <i>divisio per fila</i>, or <i>divisio -laqueis implicata</i>. Many of the earlier students, such as -Van Beneden (Rech. sur la maturation de l’œuf, <i>Arch. de -Biol.</i> <span class="smmaj">IV,</span> 1883), and Hermann (Zur Lehre v. d. -Entstehung d. karyokinetischen Spindel, <i>Arch. f. mikrosk. -Anat.</i> <span class="smmaj">XXXVII,</span> 1891) thought they recognised -actual muscular threads, drawing the nuclear material -asunder towards the respective foci or poles; and some -such view was long maintained by other writers, Boveri, -Heidenhain, Flemming, R. Hertwig, and many more. In fact, -the existence of contractile threads, or the ascription -to the spindle rather than to the poles or centrosomes of -the active forces concerned in nuclear division, formed -the main tenet of all those who declined to go beyond the -“contractile properties of protoplasm” for an explanation -of the phenomenon. (Cf. also J. W. Jenkinson, <i>Q. J. M. S.</i> -<span class="smmaj">XLVIII,</span> p. 471, 1904.)</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch226" id="fn226">226</a> -Cf. Bütschli, O., Ueber die künstliche Nachahmung der karyokinetischen -Figur, <i>Verh. Med. Nat. Ver. Heidelberg</i>, <span class="smmaj">V,</span> pp. 28–41 (1892), 1897.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch227" id="fn227">227</a> -Arrhenius, in describing a typical colloid precipitate, does so in terms that -are very closely applicable to the ordinary microscopic appearance of the protoplasm -of the cell. The precipitate consists, he says, “en un réseau d’une substance -solide contenant peu d’eau, dans les mailles duquel est inclus un fluide contenant -un peu de colloide dans beaucoup d’eau ... Evidemment cette structure se forme -à cause de la petite différence de poids spécifique des deux phases, et de la consistance -gluante des particules séparées, qui s’attachent en forme de réseau.” <i>Rev. -Scientifique</i>, Feb. 1911.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch228" id="fn228">228</a> -F. Schwartz, in Cohn’s <i>Beitr. z. Biologie der -Pflanzen</i>, <span class="smmaj">V,</span> p. 1, 1887.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch229" id="fn229">229</a> -Fischer, <i>Anat. Anzeiger</i>, <span class="smmaj">IX,</span> p. -678, 1894, <span class="smmaj">X,</span> p. 769, 1895.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch230" id="fn230">230</a> -See, in particular, W. B. Hardy, On the -structure of Cell Protoplasm, <i>Journ. of Physiol.</i> -<span class="smmaj">XXIV,</span> pp. 158–207, 1889; also Höber, -<i>Physikalische Chemie der Zelle und der Gewebe</i>, 1902. -Cf. (<i>int. al.</i>) Flemming, <i>Zellsubstanz, Kern und -Zelltheilung</i> 1882, p. 51, etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch231" id="fn231">231</a> -My description and diagrams (Figs 42–51) are -based on those of Professor E. B. Wilson.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch232" id="fn232">232</a> -If the word <i>permeability</i> be deemed too -directly suggestive of the phenomena of <i>magnetism</i> we may -replace it by the more general term of <i>specific inductive -capacity</i>. This would cover the particular case, which is -by no means an improbable one, of our phenomena being due -to a “surface charge” borne by the nucleus itself and also -by the chromosomes: this surface charge being in turn the -result of a difference in inductive capacity between the -body or particle and its surrounding medium. (Cf. footnote, -p. 187.)</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch233" id="fn233">233</a> -On the effect of electrical influences in -altering the surface-tensions of the colloid particles, see -Bredig, <i>Anorganische Fermente</i>, pp. 15, 16, 1901.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch234" id="fn234">234</a> -<i>The Cell</i>, etc. p. 66.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch235" id="fn235">235</a> -Lillie, R. S., <i>Amer. J. of Physiol.</i> -<span class="smmaj">VIII,</span> p. 282, 1903.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch236" id="fn236">236</a> -We have not taken account in the above -paragraphs of the obvious fact that the supposed -symmetrical field of force is distorted by the presence -in it of the more or less permeable bodies; nor is it -necessary for us to do so, for to that distorted field the -above argument continues to apply, word for word.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch237" id="fn237">237</a> -M. Foster, <i>Lectures on the History of -Physiology</i>, 1901, p. 62.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch238" id="fn238">238</a> -<i>Op. cit.</i> pp. 110 and 91.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch239" id="fn239">239</a> -Lamb, A. B., A new Explanation of -the Mechanism of Mitosis, <i>Journ. Exp. Zool.</i> -<span class="smmaj">V,</span> pp. 27–33, 1908.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch240" id="fn240">240</a> -<i>Amer. J. of Physiol.</i> <span class="smmaj">VIII,</span> -pp. 273–283, 1903 (<i>vide supra</i>, p. 181); cf. <i>ibid.</i> -<span class="smmaj">XV,</span> pp. 46–84, 1905. Cf. also <i>Biological -Bulletin</i>, <span class="smmaj">IV,</span> p. 175. 1903.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch241" id="fn241">241</a> -In like manner Hardy has shewn that colloid -particles migrate with the negative stream if the reaction -of the surrounding fluid be alkaline, and <i>vice versa</i>. -The whole subject is much wider than these brief allusions -suggest, and is essentially part of Quincke’s theory of -Electrical Diffusion or Endosmosis: according to which the -particles and the fluid in which they float (or the fluid -and the capillary walls through which it flows) each carry -a charge, there being a discontinuity of potential at the -surface of contact, and hence a field of force leading to -powerful tangential or shearing stresses, communicating -to the particles a velocity which varies with the density -per unit area of the surface charge. See W. B. Hardy’s -paper on Coagulation by Electricity, <i>Journ. of Physiol.</i> -<span class="smmaj">XXIV,</span> p. 288–304, 1899, also Hardy and H. W. -Harvey, Surface Electric Charges of Living Cells, <i>Proc. -R. S.</i> <span class="smmaj">LXXXIV</span> (B), pp. 217–226, 1911, and papers -quoted therein. Cf. also E. N. Harvey’s observations on the -convection of unicellular organisms in an electric field -(Studies on the Permeability of Cells, <i>Journ. of Exper. -Zool.</i> <span class="smmaj">X,</span> pp. 508–556, 1911).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch242" id="fn242">242</a> -On Differences in Electrical Potential in -Developing Eggs, <i>Amer. Journ. of Physiol.</i> <span class="smmaj">XII,</span> -pp. 241–275, 1905. This paper contains an excellent summary -of various physical theories of the segmentation of the -cell.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch243" id="fn243">243</a> -Gray has recently demonstrated a temporary -increase of electrical conductivity in sea-urchin eggs -during the process of fertilisation (The Electrical -Conductivity of fertilised and unfertilised Eggs, <i>Journ. -Mar. Biol. Assoc.</i> <span class="smmaj">X,</span> pp. 50–59, 1913).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch244" id="fn244">244</a> -Schewiakoff, Ueber die karyokinetische -Kerntheilung der <i>Euglypha alveolata, Morph. Jahrb.</i> -<span class="smmaj">XIII,</span> pp. 193–258, 1888 (see p. 216).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch245" id="fn245">245</a> -Coe, W. R., Maturation and Fertilization -of the Egg of Cerebratulus, <i>Zool. Jahrbücher</i> (<i>Anat. -Abth.</i>), <span class="smmaj">XII,</span> pp. 425–476, 1899.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch246" id="fn246">246</a> -Thus, for example, Farmer and Digby (On -Dimensions of Chromosomes considered in relation to -Phylogeny, <i>Phil. Trans.</i> (B), <span class="smmaj">CCV,</span> -pp. 1–23, 1914) have been at pains to shew, in confutation -of Meek (<i>ibid.</i> <span class="smmaj">CCIII,</span> pp. 1–74, 1912), that the -width of the chromosomes cannot be correlated with the -order of phylogeny.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch247" id="fn247">247</a> -Cf. also <i>Arch. f. Entw. Mech.</i> <span class="smmaj">X,</span> p. -52, 1900.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch248" id="fn248">248</a> -Cf. Loeb, <i>Am. J. of Physiol.</i> <span class="smmaj">VI,</span> -p. 32, 1902; Erlanger, <i>Biol. Centralbl.</i> -<span class="smmaj">XVII,</span> pp. 152, 339, 1897; Conklin, <i>Biol. -Lectures</i>, <i>Woods Holl</i>, p. 69, etc. 1898–9.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch249" id="fn249">249</a> -Robertson, T. B., Note on the Chemical -Mechanics of Cell Division, <i>Arch. f. Entw. Mech.</i> -<span class="smmaj">XXVII,</span> p. 29, 1909, <span class="smmaj">XXXV,</span> p. 692. 1913. -Cf. R. S. Lillie, <i>J. Exp. Zool.</i> <span class="smmaj">XXI,</span> pp. -369–402, 1916.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch250" id="fn250">250</a> -Cf. D’Arsonval, <i>Arch. de Physiol.</i> p. 460, -1889; Ida H. Hyde, <i>op. cit.</i> p. 242.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch251" id="fn251">251</a> -Cf. Plateau’s remarks (<i>Statique des -liquides</i>, <span class="smmaj">II,</span> p. 154) on the <i>tendency</i> towards -equilibrium, rather than actual equilibrium, in many of his -systems of soap-films.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch252" id="fn252">252</a> -But under artificial conditions, “polyspermy” -may take place, e.g. under the action of dilute poisons, -or of an abnormally high temperature, these being all, -doubtless, conditions under which the surface-tension is -diminished.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch253" id="fn253">253</a> -Fol, H., <i>Recherches sur la fécondation</i>, -1879. Roux, W., Beiträge zur Entwickelungsmechanik des -Embryo, <i>Arch. f. Mikr. Anat.</i> <span class="smmaj">XIX,</span> 1887. Whitman, -C. O., Oökinesis, <i>Journ. of Morph.</i> <span class="smmaj">I,</span> 1887.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch254" id="fn254">254</a> -Wilson. <i>The Cell</i>, p. 77.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch255" id="fn255">255</a> -Eight and twelve are by much the commonest -numbers, six and sixteen coming next in order. If we may -judge by the list given by E. B. Wilson (<i>The Cell</i>, p. -206), over 80 % of the observed cases lie between 6 and 16, -and nearly 60 % between 8 and 12.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch256" id="fn256">256</a> -<i>Theory of Cells</i>, p. 191.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch257" id="fn257">257</a> -<i>The Cell in Development</i>, etc. p. 59; cf. pp. -388, 413.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch258" id="fn258">258</a> -E.g. Brücke, <i>Elementarorganismen</i>, p. 387: -“Wir müssen in der Zelle einen kleinen Thierleib sehen, und -dürfen die Analogien, welche zwischen ihr und den kleinsten -Thierformen existiren, niemals aus den Augen lassen.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch259" id="fn259">259</a> -Whitman, C. O., The Inadequacy of the -Cell-theory, <i>Journ. of Morphol.</i> <span class="smmaj">VIII,</span> pp. -639–658, 1893; Sedgwick, A., On the Inadequacy of the -Cellular Theory of Development, <i>Q.J.M.S.</i> <span class="smmaj">XXXVII,</span> -pp. 87–101, 1895, <span class="smmaj">XXXVIII,</span> pp. 331–337, 1896. -Cf. Bourne, G. C., A Criticism of the Cell-theory; -being an answer to Mr Sedgwick’s article, etc., <i>ibid.</i> -<span class="smmaj">XXXVIII,</span> pp. 137–174, 1896.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch260" id="fn260">260</a> -Cf. Hertwig, O., <i>Die Zelle und die -Gewebe</i>, 1893, p. 1; “Die Zellen, in welche der Anatom -die pflanzlichen und thierischen Organismen zerlegt, -sind die Träger der Lebensfunktionen; sie sind, wie -Virchow sich ausgedrückt hat, die ‘Lebenseinheiten.’ -Von diesem Gesichtspunkt aus betrachtet, erscheint der -Gesammtlebensprocess eines zusammengesetzten Organismus -nichts Anderes zu sein als das höchst verwickelte -Resultat der einzelnen Lebensprocesse seiner zahlreichen, -verschieden functionirenden Zellen.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch261" id="fn261">261</a> -<i>Journ. of Morph.</i> <span class="smmaj">VIII,</span> p. 653, -1893.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch262" id="fn262">262</a> -Neue Grundlegungen zur Kenntniss der Zelle, -<i>Morph. Jahrb.</i> <span class="smmaj">VIII,</span> pp. 272, 313, 333, 1883.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch263" id="fn263">263</a> -<i>Journ. of Morph.</i> <span class="smmaj">II,</span> p. 49, 1889.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch264" id="fn264">264</a> -<i>Phil. Trans.</i> <span class="smmaj">CLI,</span> p. 183, 1861; -<i>Researches</i>, ed. Angus Smith, 1877, p. 553.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch265" id="fn265">265</a> -Cf. Kelvin, On the Molecular Tactics of a -Crystal, <i>The Boyle Lecture</i>, Oxford, 1893, <i>Baltimore -Lectures</i>, 1904, pp. 612–642. Here Kelvin was mainly -following Bravais’s (and Frankenheim’s) theory of -“space-lattices,” but he had been largely anticipated by -the crystallographers. For an account of the development -of the subject in modern crystallography, by Sohncke, von -Fedorow, Schönfliess, Barlow and others, see Tutton’s -<i>Crystallography</i>, chap. ix, pp. 118–134, 1911.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch266" id="fn266">266</a> -In a homogeneous crystalline arrangement, -<i>symmetry</i> compels a locus of one property to be a plane or -set of planes; the locus in this case being that of least -surface potential energy.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch267" id="fn267">267</a> -This is what Graham called the <i>water -of gelatination</i>, on the analogy of <i>water of -crystallisation</i>; <i>Chem. and Phys. Researches</i>, p. 597.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch268" id="fn268">268</a> -Here, in a non-crystalline or random -arrangement of particles, symmetry ensures that the -potential energy shall be the same per unit area of all -surfaces; and it follows from geometrical considerations -that the total surface energy will be least if the surface -be spherical.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch269" id="fn269">269</a> -Lehmann, O., <i>Flüssige Krystalle, sowie -Plasticität von Krystallen im allgemeinen</i>, etc., 264 -pp. 39 pll., Leipsig, 1904. For a -semi-popular, illustrated account, see Tutton’s <i>Crystals</i> -(Int. Sci. Series), 1911.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch270" id="fn270">270</a> -As Graham said of an allied phenomenon (the -so-called blood-crystals of Funke), it “illustrates the -maxim that in nature there are no abrupt transitions, and -that distinctions of class are never absolute.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch271" id="fn271">271</a> -Cf. Przibram, H., Kristall-analogien -zur Entwickelungsmechanik der Organismen, <i>Arch. f. -Entw. Mech.</i> <span class="smmaj">XXII,</span> p. 207, 1906 (with copious -bibliography); Lehmann, Scheinbar lebende Kristalle und -Myelinformen, <i>ibid.</i> <span class="smmaj">XXVI,</span> p. 483, 1908.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch272" id="fn272">272</a> -The idea of a “surface-tension” in liquids -was first enunciated by Segner, <i>De figuris superficierum -fluidarum</i>, in <i>Comment. Soc. Roy. Göttingen</i>, 1751, -p. 301. Hooke, in the <i>Micrographia</i> (1665, Obs. -<span class="smmaj">VIII,</span> etc.), had called attention to the globular -or spherical form of the little morsels of steel struck -off by a flint, and had shewn how to make a powder of such -spherical grains, by heating fine filings to melting point. -“This Phaenomenon” he said “proceeds from a propriety which -belongs to all kinds of fluid Bodies more or less, and -is caused by the Incongruity of the Ambient and included -Fluid, which so acts and modulates each other, that -they acquire, as neer as is possible, a <i>spherical</i> or -<i>globular</i> form....”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch273" id="fn273">273</a> -<i>Science of Mechanics</i>, 1902, p. 395; see -also Mach’s article Ueber die physikalische Bedeutung der -Gesetze der Symmetrie, <i>Lotos</i>, <span class="smmaj">XXI,</span> pp. 139–147, -1871.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch274" id="fn274">274</a> -Similarly, Sir David Brewster and others made -powerful lenses by simply dropping small drops of Canada -balsam, castor oil, or other strongly refractive liquids, -on to a glass plate: <i>On New Philosophical Instruments</i> -(Description of a new Fluid Microscope), Edinburgh, 1813, -p. 413.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch275" id="fn275">275</a> -Beiträge z. Physiologie d. Protoplasma, -<i>Pflüger’s Archiv</i>, <span class="smmaj">II,</span> p. 307, 1869.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch276" id="fn276">276</a> -<i>Poggend. Annalen</i>, <span class="smmaj">XCIV,</span> pp. -447–459, 1855. Cf. Strethill Wright, <i>Phil. Mag.</i> Feb. -1860.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch277" id="fn277">277</a> -Haycraft and Carlier pointed out (<i>Proc. -R.S.E.</i> <span class="smmaj">XV,</span> pp. 220–224, 1888) that the amoeboid -movements of a white blood-corpuscle are only manifested -when the corpuscle is in contact with some solid substance: -while floating freely in the plasma or serum of the -blood, these corpuscles are spherical, that is to say -they are at rest and in equilibrium. The same fact has -recently been recorded anew by Ledingham (On Phagocytosis -from an adsorptive point of view, <i>Journ. of Hygiene</i>, -<span class="smmaj">XII,</span> p. 324, 1912). On the emission of pseudopodia -as brought about by changes in surface tension, see also -(<i>int. al.</i>) Jensen, Ueber den Geotropismus niederer -Organismen, <i>Pflüger’s Archiv</i>, <span class="smmaj">LIII,</span> 1893. Jensen -remarks that in Orbitolites, the pseudopodia issuing -through the pores of the shell first float freely, then -as they grow longer bend over till they touch the ground, -whereupon they begin to display amoeboid and streaming -motions. Verworn indicates (<i>Allg. Physiol.</i> 1895, p. 429), -and Davenport says (<i>Experim. Morphology</i>, <span class="smmaj">II,</span> p. -376) that “this persistent clinging to the substratum is -a ‘thigmotropic’ reaction, and one which belongs clearly -to the category of ‘response.’ ” (Cf. Pütter, Thigmotaxis -bei Protisten, <i>A. f. Physiol.</i> 1900, Suppl. p. 247.) But -it is not clear to my mind that to account for this simple -phenomenon we need invoke other factors than gravity and -surface-action.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch278" id="fn278">278</a> -Cf. Pauli, <i>Allgemeine physikalische Chemie -d. Zellen u. Gewebe</i>, in Asher-Spiro’s <i>Ergebnisse der -Physiologie</i>, 1912; Przibram, <i>Vitalität</i>, 1913, p. 6.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch279" id="fn279">279</a> -The surface-tension theory of protoplasmic -movement has been denied by many. Cf. (e.g.), Jennings, -H. S., Contributions to the Study of the Behaviour of -the Lower Organisms, <i>Carnegie Inst.</i> 1904, pp. 130–230; -Dellinger, O. P., Locomotion of Amoebae, etc. <i>Journ. Exp. -Zool.</i> <span class="smmaj">III,</span> pp. 337–357, 1906; also various papers -by Max Heidenhain, in <i>Anatom. Hefte</i> (Merkel und Bonnet), -etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch280" id="fn280">280</a> -These various movements of a liquid surface, -and other still more striking movements such as those of -a piece of camphor floating on water, were at one time -ascribed by certain physicists to a peculiar force, <i>sui -generis</i>, the <i>force épipolique</i> of Dutrochet: until van -der Mensbrugghe shewed that differences of surface tension -were enough to account for this whole series of phenomena -(Sur la tension superficielle des liquides considérée -au point de vue de certains mouvements observés à leur -surface, <i>Mém. Cour. Acad. de Belgique</i>, <span class="smmaj">XXXIV,</span> -1869; cf. Plateau, p. 283).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch281" id="fn281">281</a> -Cf. <i>infra</i>, p. 306.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch282" id="fn282">282</a> -Cf. p. 32.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch283" id="fn283">283</a> -Or, more strictly speaking, unless its -thickness be less than twice the range of the molecular -forces.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch284" id="fn284">284</a> -It follows that the tension, depending only on -the surface-conditions, is independent of the thickness of -the film.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch285" id="fn285">285</a> -This simple but immensely important formula -is due to Laplace (<i>Mécanique Céleste</i>, Bk. x. suppl. -<i>Théorie de l’action capillaire</i>, 1806).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch286" id="fn286">286</a> -Sur la surface de révolution dont la -courbure moyenne est constante, <i>Journ. de M. Liouville</i>, -<span class="smmaj">VI,</span> p. 309, 1841.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch287" id="fn287">287</a> -See <i>Liquid Drops and Globules</i>, 1914, p. 11. -Robert Boyle used turpentine in much the same way. For -other methods see Plateau, <i>op. cit.</i> p. 154.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch288" id="fn288">288</a> -Felix Plateau recommends the use of a weighted -thread, or plumb-line, drawn up out of a jar of water or -oil; <i>Phil. Mag.</i> <span class="smmaj">XXXIV,</span> p. 246, 1867.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch289" id="fn289">289</a> -Cf. Boys, C. V., On Quartz Fibres, <i>Nature</i>, -July 11, 1889; Warburton, C., The Spinning Apparatus of -Geometric Spiders, <i>Q.J.M.S.</i> <span class="smmaj">XXXI,</span> pp. 29–39, -1890.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch290" id="fn290">290</a> -J. Blackwall, <i>Spiders of Great Britain</i> (Ray -Society), 1859, p. 10; <i>Trans. Linn. Soc.</i> <span class="smmaj">XVI,</span> p. -477, 1833.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch291" id="fn291">291</a> -The intermediate spherules appear, with great -regularity and beauty, whenever a liquid jet breaks up into -drops; see the instantaneous photographs in Poynting and -Thomson’s <i>Properties of Matter</i>, pp. 151, 152, (ed. 1907).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch292" id="fn292">292</a> -Kühne, <i>Untersuchungen über das Protoplasma</i>, -1864, p. 75, etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch293" id="fn293">293</a> -<i>A Study of Splashes</i>, 1908, p. 38, etc.; -Segmentation of a Liquid Annulus, <i>Proc. Roy. Soc.</i> -<span class="smmaj">XXX,</span> pp. 49–60, 1880.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch294" id="fn294">294</a> -Cf. <i>ibid.</i> pp. 17, 77. The same phenomenon -is beautifully and continuously evident when a strong jet -of water from a tap impinges on a curved surface and then -shoots off it.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch295" id="fn295">295</a> -See a <i>Study of Splashes</i>, p. 54.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch296" id="fn296">296</a> -A case which we have not specially considered, -but which may be found to deserve consideration in -biology, is that of a cell or drop suspended in a liquid -of <i>varying</i> density, for instance in the upper layers of -a fluid (e.g. sea-water) at whose surface condensation is -going on, so as to produce a steady density-gradient. In -this case the normally spherical drop will be flattened -into an oval form, with its -maximum surface-curvature lying at the level where the -densities of the drop and the surrounding liquid are just -equal. The sectional outline of the drop has been shewn to -be not a true oval or ellipse, but a somewhat complicated -quartic curve. (Rice, <i>Phil. Mag.</i> Jan. 1915.)</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch297" id="fn297">297</a> -Indeed any non-isotropic <i>stiffness</i>, even -though <i>T</i> remained uniform, would simulate, and be -indistinguishable from, a condition of non-stiffness and -non-isotropic <i>T</i>.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch298" id="fn298">298</a> -A non-symmetry of <i>T</i> and <i>T′</i> might -also be capable of explanation as a result of “liquid -crystallisation.” This hypothesis is referred to, in -connection with the blood-corpuscles, on p. 272.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch299" id="fn299">299</a> -The case of the snow-crystals is a -particularly interesting one; for their “distribution” is -in some ways analogous to what we find, for instance, among -our microscopic skeletons of Radiolarians. That is to say, -we may one day meet with myriads of some one particular -form or species only, and another day with myriads of -another; while at another time and place we may find -species intermingled in inexhaustible variety. (Cf. e.g. -J. Glaisher, <i>Ill. London News</i>, Feb. 17, 1855; <i>Q.J.M.S.</i> -<span class="smmaj">III,</span> pp. 179–185, 1855).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch300" id="fn300">300</a> -Cf. Bergson, <i>Creative Evolution</i>, p. 107: -“Certain Foraminifera have not varied since the Silurian -epoch. Unmoved witnesses of the innumerable revolutions -that have upheaved our planet, the Lingulae are today what -they were at the remotest times of the palaeozoic era.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch301" id="fn301">301</a> -Ray Lankester, <i>A.M.N.H.</i> (4), <span class="smmaj">XI,</span> p. -321, 1873.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch302" id="fn302">302</a> -Leidy, Parasites of the Termites, <i>J. Nat. -Sci., Philadelphia</i>, <span class="smmaj">VIII,</span> pp. 425–447, 1874–81; -cf. Saville Kent’s <i>Infusoria</i>, <span class="smmaj">II,</span> p. 551.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch303" id="fn303">303</a> -<i>Op. cit.</i> p. 79.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch304" id="fn304">304</a> -Brady, <i>Challenger Monograph</i>, pl. <span class="smmaj">XX,</span> p. 233.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch305" id="fn305">305</a> -That the Foraminifera not only can but do hang -from the surface of the water is confirmed by the following -apt quotation which I owe to Mr E. Heron-Allen: “Quand on -place, comme il a été dit, le dépôt provenant du lavage -des fucus dans un flacon que l’on remplit de nouvelle -eau, on voit au bout d’une heure environ les animaux -[<i>Gromia dujardinii</i>] se mettre en mouvement et commencer -à grimper. Six heures après ils tapissent l’extérieur du -flacon, de sorte que les plus élevés sont à trente-six ou -quarante-deux millimetres du fond; le lendemain beaucoup -d’entre eux, <i>après avoir atteint le niveau du liquide, -ont continué à ramper à sa surface, en se laissant pendre -au-dessous</i> comme certains mollusques gastéropodes.” -(Dujardin, F., Observations nouvelles sur les prétendus -céphalopodes microscopiques, <i>Ann. des Sci. Nat.</i> (2), -<span class="smmaj">III,</span> p. 312, 1835.)</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch306" id="fn306">306</a> -Cf. Boas, <i>Spolia Atlantica</i>, 1886, pl. 6.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch307" id="fn307">307</a> -This cellular pattern would seem to be related -to the “cohesion figures” described by Tomlinson in -various surface-films (<i>Phil. Mag.</i> 1861 to 1870); to the -“tesselated structure” in liquids described by Professor -James Thomson in 1882 (<i>Collected Papers</i>, p. 136); and to -the <i>tourbillons cellulaires</i> of Prof. H. Bénard (<i>Ann. -de Chimie</i> (7), <span class="smmaj">XXIII,</span> pp. 62–144, 1901, (8), -<span class="smmaj">XXIV,</span> pp. 563–566, 1911), <i>Rev. génér. des Sci.</i> -<span class="smmaj">XI,</span> p. 1268, 1900; cf. also E. H. Weber. -(<i>Poggend. Ann.</i> -<span class="smmaj">XCIV,</span> p. 452, 1855, etc.). The phenomenon is of -great interest and various appearances have been referred -to it, in biology, geology, metallurgy and even astronomy: -for the flocculent clouds in the solar photosphere shew -an analogous configuration. (See letters by Kerr Grant, -Larmor, Wager and others, in <i>Nature</i>, April 16 to June -11, 1914.) In many instances, marked by strict symmetry -or regularity, it is very possible that the interference -of waves or ripples may play its part in the phenomenon. -But in the majority of cases, it is fairly certain that -localised centres of action, or of diminished tension, are -present, such as might be provided by dust-particles in the -case of Darling’s experiment (cf. <i>infra</i>, p. 590).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch308" id="fn308">308</a> -Ueber physikalischen Eigenschaften dünner, -fester Lamellen, <i>S.B. Berlin. Akad.</i> 1888, pp. 789, 790.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch309" id="fn309">309</a> -Certain palaeontologists (e.g. Haeusler and -Spandel) have maintained that in each family or genus the -plain smooth-shelled forms are the primitive and ancient -ones, and that the ribbed and otherwise ornamented shells -make their appearance at later dates in the course of -a definite evolution (cf. Rhumbler, <i>Foraminiferen der -Plankton-Expedition</i>, 1911, i, p. 21). If this were true it -would be of fundamental importance: but this book of mine -would not deserve to be written.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch310" id="fn310">310</a> -<i>A Study of Splashes</i>, p. 116.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch311" id="fn311">311</a> -See <i>Silliman’s Journal</i>, <span class="smmaj">II,</span> p. 179, -1820; and cf. Plateau, <i>op. cit.</i> <span class="smmaj">II,</span> pp. 134, -461.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch312" id="fn312">312</a> -The presence or absence of the contractile -vacuole or vacuoles is one of the chief distinctions, -in systematic zoology, between the Heliozoa and the -Radiolaria. As we have seen on p. 165 (footnote), it -is probably no more than a physical consequence of the -different conditions of existence in fresh water and in -salt.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch313" id="fn313">313</a> -Cf. Doflein, <i>Lehrbuch der Protozoenkunde</i>, -1911, p. 422.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch314" id="fn314">314</a> -Cf. Minchin, <i>Introduction to the Study of the -Protozoa</i>, 1914 p. 293, Fig. 127.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch315" id="fn315">315</a> -Cf. C. A. Kofoid and Olive Swezy, On -Trichomonad Flagellates, etc. <i>Pr. Amer. Acad. of Arts and -Sci.</i> <span class="smmaj">LI,</span> pp. 289–378, 1915.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch316" id="fn316">316</a> -D. L. Mackinnon, Herpetomonads from the -Alimentary Tract of certain Dungflies, <i>Parasitology</i>, -<span class="smmaj">III,</span> p. 268, 1910.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch317" id="fn317">317</a> -<i>Proc. Roy. Soc.</i> <span class="smmaj">XII,</span> pp. 251–257, -1862–3.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch318" id="fn318">318</a> -Cf. (<i>int. al.</i>) Lehmann, Ueber scheinbar -lebende Kristalle und Myelinformen, <i>Arch. f. Entw. -Mech.</i> <span class="smmaj">XXVI,</span> p. 483, 1908; <i>Ann. d. Physik</i>, -<span class="smmaj">XLIV,</span> p. 969, 1914.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch319" id="fn319">319</a> -Cf. B. Moore and H. C. Roaf, On the Osmotic -Equilibrium of the Red Blood Corpuscle, <i>Biochem. Journal</i>, -<span class="smmaj">III,</span> p. 55, 1908.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch320" id="fn320">320</a> -For an attempt to explain the form of a -blood-corpuscle by surface-tension alone, see Rice, <i>Phil. -Mag.</i> Nov. 1914; but cf. Shorter, <i>ibid.</i> Jan. 1915.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch321" id="fn321">321</a> -Koltzoff, N. K., Studien über die Gestalt -der Zelle, <i>Arch. f. mikrosk. Anat.</i> <span class="smmaj">LXVII,</span> pp. -364–571, 1905; <i>Biol. Centralbl.</i> <span class="smmaj">XXIII,</span> pp. -680–696, 1903, <span class="smmaj">XXVI,</span> pp. 854–863, 1906; <i>Arch. f. -Zellforschung</i>, <span class="smmaj">II,</span> pp. 1–65, 1908, <span class="smmaj">VII,</span> -pp. 344–423, 1911; <i>Anat. Anzeiger</i>, <span class="smmaj">XLI,</span> pp. -183–206, 1912.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch322" id="fn322">322</a> -Cf. <i>supra</i>, p. 129.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch323" id="fn323">323</a> -As Bethe points out (Zellgestalt, Plateausche -Flüssigkeitstigur und Neurofibrille, <i>Anat. Anz.</i> -<span class="smmaj">XL.</span> p. 209, 1911), the spiral fibres of which -Koltzoff speaks must lie <i>in the surface</i>, and not within -the substance, of the cell whose conformation is affected -by them.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch324" id="fn324">324</a> -See for a further but still elementary -account, Michaelis, <i>Dynamics of Surfaces</i>, 1914, -p. 22 <i>seq.</i>; Macallum, <i>Oberflächenspannung und -Lebenserscheinungen</i>, in Asher-Spiro’s <i>Ergebnisse der -Physiologie</i>, <span class="smmaj">XI,</span> pp. 598–658, 1911; see also W. -W. Taylor’s <i>Chemistry of Colloids</i>, 1915, p. 221 <i>seq.</i>, -Wolfgang Ostwald, <i>Grundriss der Kolloidchemie</i>, 1909, -and other text-books of physical chemistry; and Bayliss’s -<i>Principles of General Physiology</i>, pp. 54–73, 1915.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch325" id="fn325">325</a> -The first instance of what we now call an -adsorptive phenomenon was observed in soap-bubbles. -Leidenfrost, in 1756, was aware that the outer layer of -the bubble was covered by an “oily” layer. A hundred years -later Dupré shewed that in a soap-solution the soap tends -to concentrate at the surface, so that the surface-tension -of a very weak solution is very little different from that -of a strong one (<i>Théorie mécanique de la chaleur</i>, 1869, -p. 376; cf. Plateau, <span class="smmaj">II,</span> p. 100).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch326" id="fn326">326</a> -This identical phenomenon was the basis of -Quincke’s theory of amoeboid movement (Ueber periodische -Ausbreitung von Flüssigkeitsoberflächen, etc., <i>SB. Berlin. -Akad.</i> 1888, pp. 791–806; cf. <i>Pflüger’s Archiv</i>, 1879, p. -136).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch327" id="fn327">327</a> -J. Willard Gibbs, Equilibrium of Heterogeneous -Substances, <i>Tr. Conn. Acad.</i> <span class="smmaj">III,</span> pp. 380–400, -1876, also in <i>Collected Papers</i>, <span class="smmaj">I,</span> pp. 185–218, -London, 1906; J. J. Thomson, <i>Applications of Dynamics -to Physics and Chemistry</i>, 1888 (Surface tension of -solutions), p. 190. See also (<i>int. al.</i>) the various -papers by C. M. Lewis, <i>Phil. Mag.</i> (6), <span class="smmaj">XV,</span> p. -499, 1908, <span class="smmaj">XVII,</span> p. 466, 1909, <i>Zeitschr. f. -physik. Chemie</i>, <span class="smmaj">LXX,</span> p. 129, 1910; Milner, <i>Phil. -Mag.</i> (6), <span class="smmaj">XIII,</span> p. 96, 1907, etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch328" id="fn328">328</a> -G. F. FitzGerald, On the Theory of Muscular -Contraction, <i>Brit. Ass. Rep.</i> 1878; also in <i>Scientific -Writings</i>, ed. Larmor, 1902, pp. 34, 75. A. d’Arsonval, -Relations entre l’électricité animale et la tension -superficielle, <i>C. R.</i> <span class="smmaj">CVI,</span> p. 1740. 1888; cf. A. -Imbert, Le mécanisme de la contraction musculaire, déduit -de la considération des forces de tension superficielle, -<i>Arch. de Phys.</i> (5), <span class="smmaj">IX,</span> pp. 289–301, 1897.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch329" id="fn329">329</a> -Ueber die Natur der Bindung der Gase im -Blut und in seinen Bestandtheilen, <i>Kolloid. Zeitschr.</i> -<span class="smmaj">II,</span> pp. 264–272, 294–301, 1908; cf. Loewy, -Dissociationsspannung des Oxyhaemoglobin im Blut, <i>Arch. f. -Anat. und Physiol.</i> 1904, p. 231.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch330" id="fn330">330</a> -We may trace the first steps in the study -of this phenomenon to Melsens, who found that thin -films of white of egg become firm and insoluble (Sur -les modifications apportées à l’albumine ... par l’action -purement mécanique, <i>C. R. Acad. Sci.</i> <span class="smmaj">XXXIII,</span> p. -247, 1851); and Harting made similar observations about -the same time. Ramsden has investigated the same subject, -and also the more general phenomenon of the formation -of albuminoid and fatty membranes by adsorption: cf. -Koagulierung der Eiweisskörper auf mechanischer Wege, -<i>Arch. f. Anat. u. Phys.</i> (<i>Phys. Abth.</i>) 1894, p. 517; -Abscheidung fester Körper in Oberflächenschichten <i>Z. -f. phys. Chem.</i> <span class="smmaj">XLVII,</span> p. 341, 1902; <i>Proc. R. -S.</i> <span class="smmaj">LXXII,</span> p. 156, 1904. For a general review -of the whole subject see H. Zangger, Ueber Membranen -und Membranfunktionen, in Asher-Spiro’s <i>Ergebnisse der -Physiologie</i>, <span class="smmaj">VII,</span> pp. 99–160, 1908.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch331" id="fn331">331</a> -Cf. Taylor, <i>Chemistry of Colloids</i>, p. 252.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch332" id="fn332">332</a> -Strasbürger, Ueber Cytoplasmastrukturen, etc. -<i>Jahrb. f. wiss. Bot.</i> <span class="smmaj">XXX,</span> 1897; R. A. Harper, -Kerntheilung und freie Zellbildung im Ascus, <i>ibid.</i>; cf. -Wilson, <i>The Cell in Development, etc.</i> pp. 53–55.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch333" id="fn333">333</a> -Cf. A. Gurwitsch, <i>Morphologie und Biologie -der Zelle</i>, 1904, pp. 169–185; Meves, Die Chondriosomen -als Träger erblicher Anlagen, <i>Arch. f. mikrosk. Anat.</i> -1908, p. 72; J. O. W. Barratt, Changes in Chondriosomes, -etc. <i>Q.J.M.S.</i> <span class="smmaj">LVIII,</span> pp. 553–566, 1913, etc.; A. -Mathews, Changes in Structure of the Pancreas Cell, etc., -<i>J. of Morph.</i> <span class="smmaj">XV</span> (Suppl.), pp. 171–222, 1899.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch334" id="fn334">334</a> -The question whether chromosomes, -chondriosomes or chromidia be the true vehicles or -transmitters of “heredity” is not without its analogy -to the older problem of whether the pineal gland or the -pituitary body were the actual seat and domicile of the -soul.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch335" id="fn335">335</a> -Cf. C. C. Dobell, Chromidia and the -Binuclearity Hypotheses; a review and a criticism, -<i>Q.J.M.S.</i> <span class="smmaj">LIII,</span> 279–326, 1909; Prenant, A., Les -Mitochondries et l’Ergastoplasme, <i>Journ. de l’Anat. et de -la Physiol.</i> <span class="smmaj">XLVI,</span> pp. 217–285, 1910 (both with -copious bibliography).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch336" id="fn336">336</a> -Traube in particular has maintained that -in differences of surface-tension we have the origin of -the active force productive of osmotic currents, and -that herein we find an explanation, or an approach to an -explanation, of many phenomena which were formerly deemed -peculiarly “vital” in their character. “Die Differenz der -Oberflächenspannungen oder der Oberflächendruck eine Kraft -darstellt, welche als treibende Kraft der Osmose, an die -Stelle des nicht mit dem Oberflächendruck identischen -osmotischen Druckes, zu setzen ist, etc.” (Oberflächendruck -und seine Bedeutung im Organismus, <i>Pflüger’s Archiv</i>, -<span class="smmaj">CV,</span> p. 559, 1904.) Cf. also Hardy (<i>Pr. Phys. -Soc.</i> <span class="smmaj">XXVIII,</span> p. 116, 1916), “If the surface film -of a colloid membrane separating two masses of fluid were -to change in such a way as to lower the potential of the -water in it, water would enter the region from both sides -at once. But if the change of state were to be propagated -as a wave of change, starting at one face and dying out at -the other face, water would be carried along from one side -of the membrane to the other. A succession of such waves -would maintain a flow of fluid.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch337" id="fn337">337</a> -On the Distribution of Potassium in animal and -vegetable Cells; <i>Journ. of Physiol.</i> <span class="smmaj">XXXII,</span> p. -95, 1905.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch338" id="fn338">338</a> -The reader will recognise that there is -a fundamental difference, and contrast, between such -experiments as these of Professor Macallum’s and the -ordinary staining processes of the histologist. The -latter are (as a general rule) purely empirical, while -the former endeavour to reveal the true microchemistry of -the cell. “On peut dire que la microchimie n’est encore -qu’à la période d’essai, et que l’avenir de l’histologie -et spécialement de la cytologie est tout entier dans -la microchimie” (Prenant, A., Méthodes et résultats de -la Microchimie, <i>Journ. de l’Anat. et de la Physiol.</i> -<span class="smmaj">XLVI,</span> pp. 343–404, 1910).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch339" id="fn339">339</a> -Cf. Macallum, Presidential Address, Section I, -<i>Brit. Ass. Rep.</i> (Sheffield), 1910, p. 744.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch340" id="fn340">340</a> -In accordance with a simple <i>corollary</i> to the -Gibbs-Thomson law.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch341" id="fn341">341</a> -It can easily be proved (by equating the -increase of energy stored in an increased surface to the -work done in increasing that surface), that the tension -measured per unit breadth, <i>T<sub>ab</sub></i>, is equal to the -energy per unit area, <i>E<sub>ab</sub></i>.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch342" id="fn342">342</a> -The presence of this little liquid -“bourrelet,” drawn from the material of which the -partition-walls themselves are composed, is obviously -tending to a reduction of the internal surface-area. And it -may be that it is as well, or better, accounted for on this -ground than on Plateau’s assumption that it represents a -“surface of continuity.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch343" id="fn343">343</a> -A similar “bourrelet” is admirably seen at the -line of junction between a floating bubble and the liquid -on which it floats; in which case it constitutes a “masse -annulaire,” whose mathematical properties and relation -to the form of the <i>nearly</i> hemispherical bubble, have -been investigated by van der Mensbrugghe (cf. Plateau, -<i>op. cit.</i>, p. 386). The form of the superficial vacuoles -in Actinophrys or Actinosphaerium involves an identical -problem.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch344" id="fn344">344</a> -In an actual calculation we must of course -always take account of the tensions on <i>both sides</i> of each -film or membrane.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch345" id="fn345">345</a> -Hofmeister, <i>Pringsheim’s Jahrb.</i> -<span class="smmaj">III,</span> p. 272, 1863; <i>Hdb. d. physiol. Bot.</i> -<span class="smmaj">I,</span> 1867, p. 129.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch346" id="fn346">346</a> -Sachs, Ueber die Anordnung der Zellen in -jüngsten Pflanzentheilen, <i>Verh. phys. med. Ges. Würzburg</i>, -<span class="smmaj">XI,</span> pp. 219–242, 1877; Ueber Zellenanordnung und -Wachsthum, <i>ibid.</i> <span class="smmaj">XII,</span> 1878; Ueber die durch -Wachsthum bedingte Verschiebung kleinster Theilchen in -trajectorischen Curven, <i>Monatsber. k. Akad. Wiss. Berlin</i>, -1880; <i>Physiology of Plants</i>, chap. xxvii, pp. 431–459, -Oxford, 1887.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch347" id="fn347">347</a> -Schwendener, Ueber den Bau und das Wachsthum -des Flechtenthallus, <i>Naturf. Ges. Zürich</i>, Febr. 1860, pp. -272–296.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch348" id="fn348">348</a> -Reinke, <i>Lehrbuch der Botanik</i>, 1880, p. 519.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch349" id="fn349">349</a> -Cf. Leitgeb, <i>Unters. über die Lebermoose</i>, -<span class="smmaj">II,</span> p. 4, Graz, 1881.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch350" id="fn350">350</a> -Rauber, Neue Grundlegungen zur Kenntniss der -Zelle, <i>Morph. Jahrb.</i> <span class="smmaj">VIII,</span> pp. 279, 334, 1882.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch351" id="fn351">351</a> -<i>C. R. Acad. Sc.</i> <span class="smmaj">XXXIII,</span> p. 247, -1851; <i>Ann. de chimie et de phys.</i> (3), <span class="smmaj">XXXIII,</span> p. -170, 1851; <i>Bull. R. Acad. Belg.</i> <span class="smmaj">XXIV,</span> p. 531, -1857.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch352" id="fn352">352</a> -Klebs, <i>Biolog. Centralbl.</i> <span class="smmaj">VII,</span> pp. -193–201, 1887.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch353" id="fn353">353</a> -L. Errera, Sur une condition fondamentale -d’équilibre des cellules vivantes, <i>C. R.</i>, <span class="smmaj">CIII,</span> -p. 822, 1886; <i>Bull. Soc. Belge de Microscopie</i>, -<span class="smmaj">XIII,</span> Oct. 1886; <i>Recueil d’œuvres</i> (<i>Physiologie -générale</i>), 1910, pp. 201–205.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch354" id="fn354">354</a> -L. Chabry, Embryologie des Ascidiens, <i>J. -Anat. et Physiol.</i> <span class="smmaj">XXIII,</span> p. 266, 1887.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch355" id="fn355">355</a> -Robert, Embryologie des Troques, <i>Arch. de -Zool. exp. et gén.</i> (3), <span class="smmaj">X,</span> 1892.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch356" id="fn356">356</a> -“Dass der Furchungsmodus etwas für -das Zukünftige unwesentliches ist,” <i>Z. f. w. Z.</i> -<span class="smmaj">LV,</span> 1893, p. 37. With this -statement compare, or contrast, that of Conklin, quoted on -p. 4; cf. also pp. 157, 348 (footnotes).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch357" id="fn357">357</a> -de Wildeman, Etudes sur l’attache des cloisons -cellulaires, <i>Mém. Couronn. de l’Acad. R. de Belgique</i>, -<span class="smmaj">LIII,</span> 84 pp., 1893–4.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch358" id="fn358">358</a> -It was so termed by Conklin in 1897, in his -paper on Crepidula (<i>J. of Morph.</i> <span class="smmaj">XIII,</span> 1897). It -is the <i>Querfurche</i> of Rabl (<i>Morph. Jahrb.</i> <span class="smmaj">V,</span> -1879); the <i>Polarfurche</i> of O. Hertwig (<i>Jen. Zeitschr.</i> -<span class="smmaj">XIV,</span> 1880); the <i>Brechungslinie</i> of Rauber (Neue -Grundlage zur K. der Zelle, <i>M. Jb.</i> <span class="smmaj">VIII,</span> 1882). -It is carefully discussed by Robert, Dév. des Troques, -<i>Arch. de Zool. Exp. et Gén.</i> (3), <span class="smmaj">X,</span> 1892, p. 307 -seq.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch359" id="fn359">359</a> -Thus Wilson (<i>J. of Morph.</i> <span class="smmaj">VIII,</span> -1895) declared that in Amphioxus the polar furrow was -occasionally absent, and Driesch took occasion to criticise -and to throw doubt upon the statement (<i>Arch. f. Entw. -Mech.</i> <span class="smmaj">I,</span> 1895, p. 418).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch360" id="fn360">360</a> -Precisely the same remark was made long -ago by Driesch: “Das so oft sehematisch gezeichnete -Vierzellenstadium mit zwei sich in zwei Punkten -scheidende Medianen kann man wohl getrost aus der Reihe -des Existierenden streichen,” <i>Entw. mech. Studien, Z. -f. w. Z.</i> <span class="smmaj">LIII,</span> p. 166, 1892. Cf. also his -<i>Math. mechanische Bedeutung morphologischer Probleme der -Biologie</i>, Jena, 59 pp. 1891.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch361" id="fn361">361</a> -Compare, however, p. 299.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch362" id="fn362">362</a> -<i>Ricreatione dell’ occhio e della mente, nell’ -Osservatione delle Chiocciole</i>, Roma, 1681.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch363" id="fn363">363</a> -Cf. some of J. H. Vincent’s photographs of -ripples, in <i>Phil. Mag.</i> 1897–1899; or those of F. R. -Watson, in <i>Phys. Review</i>, 1897, 1901, 1916. The appearance -will depend on the rate of the wave, and in turn on the -surface-tension; with a low tension one would probably see -only a moving “jabble.” FitzGerald thought diatom-patterns -might be due to electromagnetic vibrations (<i>Works</i>, p. -503, 1902).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch364" id="fn364">364</a> -Cushman, J. A. and Henderson, W. P., <i>Amer. -Nat.</i> <span class="smmaj">XL,</span> pp. 797–802, 1906.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch365" id="fn365">365</a> -This does not merely neglect the <i>broken</i> -ones but <i>all</i> whose centres lie between this circle and a -hexagon inscribed in it.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch366" id="fn366">366</a> -For more detailed calculations see a paper by -“H.M.” [? H. Munro], in <i>Q. J. M. S.</i> <span class="smmaj">VI,</span> p. 83, -1858.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch367" id="fn367">367</a> -Cf. Hartog, The Dual Force of the Dividing -Cell, <i>Science Progress</i> (n.s.), <span class="smmaj">I,</span> Oct. 1907, and -other papers. Also Baltzer, <i>Ueber mehrpolige Mitosen bei -Seeigeleiern</i>, Inaug. Diss. 1908.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch368" id="fn368">368</a> -Observations sur les Abeilles, <i>Mém. Acad. Sc. -Paris</i>, 1712, p. 299.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch369" id="fn369">369</a> -As explained by Leslie Ellis, in his essay -“On the Form of Bees’ Cells,” in <i>Mathematical and other -Writings</i>, 1853, p. 353; cf. O. Terquem, <i>Nouv. Ann. Math.</i> -1856, p. 178.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch370" id="fn370">370</a> -<i>Phil. Trans.</i> <span class="smmaj">XLII,</span> 1743, pp. -565–571.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch371" id="fn371">371</a> -<i>Mém. de l’Acad. de Berlin</i>, 1781.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch372" id="fn372">372</a> -Cf. Gregory, <i>Examples</i>, p. 106, Wood’s <i>Homes -without Hands</i>, 1865, p. 428, Mach, <i>Science of Mechanics</i>, -1902, p. 453, etc., etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch373" id="fn373">373</a> -<i>Origin of Species</i>, ch. <span class="smmaj">VIII</span> (6th -ed., p. 221). The cells of various bees, humble-bees -and social wasps have been described and mathematically -investigated by K. Müllenhoff, <i>Pflüger’s Archiv</i> -<span class="smmaj">XXXII,</span> p. 589, 1883; but his many interesting -results are too complex to epitomise. For figures of -various nests and combs see (e.g.) von Büttel-Reepen, -<i>Biol. Centralbl.</i> <span class="smmaj">XXXIII,</span> pp. 4, 89, 129, 183, -1903.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch374" id="fn374">374</a> -Darwin had a somewhat similar idea, though -he allowed more play to the bee’s instinct or conscious -intention. Thus, when he noticed certain half-completed -cell-walls to be concave on one side and convex on the -other, but to become perfectly flat when restored for -a short time to the hive, he says: “It was absolutely -impossible, from the extreme thinness of the little plate, -that they could have effected this by gnawing away the -convex side; and I suspect that the bees in such cases -stand on opposite sides and push and bend the ductile and -warm wax (which as I have tried is easily done) into its -proper intermediate plane, and thus flatten it.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch375" id="fn375">375</a> -Since writing the above, I see that Müllenhoff -gives the same explanation, and declares that the waxen -wall is actually a <i>Flüssigkeitshäutchen</i>, or liquid film.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch376" id="fn376">376</a> -Bonnet criticised Buffon’s explanation, on the -ground that his description was incomplete; for Buffon took -no account of the Maraldi pyramids.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch377" id="fn377">377</a> -Buffon, <i>Histoire Naturelle</i>, <span class="smmaj">IV,</span> -p. 99. Among many other papers on the Bee’s cell, see -Barclay, <i>Mem. Wernerian Soc.</i> <span class="smmaj">II,</span> p. 259 (1812), -1818; Sharpe, <i>Phil. Mag.</i> <span class="smmaj">IV,</span> 1828, pp. 19–21; -L. Lalanne, <i>Ann. Sci. Nat.</i> (2) Zool. <span class="smmaj">XIII,</span> -pp. 358–374, 1840; Haughton, <i>Ann. Mag. Nat. Hist.</i> (3), -<span class="smmaj">XI,</span> pp. 415–429, 1863; A. R. Wallace, <i>ibid.</i> -<span class="smmaj">XII,</span> p. 303, 1863; Jeffries Wyman. <i>Pr. Amer. -Acad. of Arts and Sc.</i> <span class="smmaj">VII,</span> pp. 68–83, 1868; -Chauncey Wright, <i>ibid.</i> <span class="smmaj">IV,</span> p. 432, 1860.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch378" id="fn378">378</a> -Sir W. Thomson, On the Division of Space with -Minimum Partitional Area, <i>Phil. Mag.</i> (5), <span class="smmaj">XXIV,</span> -pp. 503–514, Dec. 1887; cf. <i>Baltimore Lectures</i>, 1904, p. -615.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch379" id="fn379">379</a> -Also discovered independently by Sir David -Brewster, <i>Trans. R.S.E.</i> <span class="smmaj">XXIV,</span> p. 505, 1867, -<span class="smmaj">XXV,</span> p. 115, 1869.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch380" id="fn380">380</a> -Von Fedorow had already described (in Russian) -the same figure, under the name of cubo-octahedron, or -hepta-parallelohedron, limited however to the case where -all the faces are plane. This figure, together with the -cube, the hexagonal prism, the rhombic dodecahedron and the -“elongated dodecahedron,” constituted the five plane-faced, -parallel-sided figures by which space is capable of being -completely filled and symmetrically partitioned; the -series so forming the foundation of Von Fedorow’s theory -of crystalline structure. The elongated dodecahedron is, -essentially, the figure of the bee’s cell.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch381" id="fn381">381</a> -F. R. Lillie, Embryology of the Unionidae, -<i>Journ. of Morphology</i>, <span class="smmaj">X,</span> p. 12, 1895.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch382" id="fn382">382</a> -E. B. Wilson, The Cell-lineage of Nereis, -<i>Journ. of Morphology</i>, <span class="smmaj">VI,</span> p. 452, 1892.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch383" id="fn383">383</a> -It is highly probable, and we may reasonably -assume, that the two little triangles do not actually meet -at an apical <i>point</i>, but merge into one another by a -twist, or minute surface of complex curvature, so as not to -contravene the normal conditions of equilibrium.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch384" id="fn384">384</a> -Professor Peddie has given me this interesting -and important result, but the mathematical reasoning is too -lengthy to be set forth here.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch385" id="fn385">385</a> -Cf. Rhumbler, <i>Arch. f. Entw. Mech.</i> -<span class="smmaj">XIV,</span> p. 401, 1902; Assheton, <i>ibid.</i> -<span class="smmaj">XXXI,</span> pp. 46–78, 1910.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch386" id="fn386">386</a> -M. Robert (<i>l. c.</i> p. 305) has compiled a -long list of cases among the molluscs and the worms, where -the initial segmentation of the egg proceeds by equal or -unequal division. The two cases are about equally numerous. -But like many other writers, he would ascribe this equality -or inequality rather to a provision for the future than to -a direct effect of immediate physical causation: “Il semble -assez probable, comme on l’a dit souvent, que la plus -grande taille d’un blastomère est liée à l’importance et -au développement précoce des parties du corps qui doivent -en naître: il y aurait là une sorte de reflet des stades -postérieures du développement sur les premières phénomènes, -ce que M. Ray Lankester appelle <i>precocious segregation</i>. -Il faut avouer pourtant qu’on est parfois assez embarrassé -pour assigner une cause à pareilles différences.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch387" id="fn387">387</a> -The principle is well illustrated in an -experiment of Sir David Brewster’s (<i>Trans. R.S.E.</i> -<span class="smmaj">XXV,</span> p. 111, 1869). A soap-film is drawn over the -rim of a wine-glass, and then covered by a watch-glass. -The film is inclined or shaken till it becomes attached -to the glass covering, and it then immediately changes -place, leaving its transverse position to take up that of a -spherical segment extending from one side of the wine-glass -to its cover, and so enclosing the same volume of air as -formerly but with a great economy of surface, precisely -as in the case of our spherical partition cutting off one -corner of a cube.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch388" id="fn388">388</a> -Cf. Wildeman, <i>Attache des Cloisons</i>, etc., pls. 1, 2.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch389" id="fn389">389</a> -<i>Nova Acta K. Leop. Akad.</i> <span class="smmaj">XI,</span> 1, pl. <span class="smmaj">IV.</span></p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch390" id="fn390">390</a> -Cf. <i>Protoplasmamechanik</i>, p. 229: “Insofern -liegen also die Verhältnisse hier wesentlich anders als -bei der Zertheilung hohler Körperformen durch flüssige -Lamellen. Wenn die Membran bei der Zelltheilung die von dem -Prinzip der kleinsten Flächen geforderte Lage und Krümmung -annimmt, so werden wir den Grund dafür in andrer Weise -abzuleiten haben.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch391" id="fn391">391</a> -There is, I think, some ambiguity or -disagreement among botanists as to the use of this latter -term: the sense in which I am using it, viz. for any -partition which meets the outer or peripheral wall at right -angles (the strictly <i>radial</i> partition being for the -present excluded), is, however, clear.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch392" id="fn392">392</a> -<i>Cit.</i> Plateau, <i>Statique des Liquides</i>, i, p. 358.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch393" id="fn393">393</a> -Even in a Protozoon (<i>Euglena viridis</i>), -when kept alive under artificial compression, Ryder found -a process of cell-division to occur which he compares -to the segmenting blastoderm of a fish’s egg, and which -corresponds in its essential features with that here -described. <i>Contrib. Zool. Lab. Univ. Pennsylvania</i>, -<span class="smmaj">I,</span> pp. 37–50, 1893.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch394" id="fn394">394</a> -This, like many similar figures, is manifestly -drawn under the influence of Sachs’s theoretical views, or -assumptions, regarding orthogonal trajectories, coaxial -circles, confocal ellipses, etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch395" id="fn395">395</a> -Such preconceptions as Rauber entertained -were all in a direction likely to lead him away from such -phenomena as he has faithfully depicted. Rauber had no idea -whatsoever of the principles by which we are guided in -this discussion, nor does he introduce at all the analogy -of surface-tension, or any other purely physical concept. -But he was deeply under the influence of Sachs’s rule of -rectangular intersection; and he was accordingly -disposed to look upon the configuration represented above -in Fig. <a href="#fig168" title="go to Fig. 168">168</a>, 6, as the most typical or most primitive.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch396" id="fn396">396</a> -Cf. Rauber, Neue Grundlage z. K. der Zelle, -<i>Morph. Jahrb.</i> <span class="smmaj">VIII,</span> 1883, pp. 273, 274:</p> - -<p>“Ich betone noch, dass unter meinen Figuren diejenige -gar nicht enthalten ist, welche zum Typus der -Batrachierfurchung gehörig am meisten bekannt -ist .... Es haben so ausgezeichnete Beobachter sie als vorhanden -beschrieben, dass es mir nicht einfallen kann, sie -überhaupt nicht anzuerkennen.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch397" id="fn397">397</a> -Roux’s experiments were performed with drops -of paraffin suspended in dilute alcohol, to which a little -calcium acetate was added to form a soapy pellicle over the -drops and prevent them from reuniting with one another.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch398" id="fn398">398</a> -Cf. (e.g.) Clerk Maxwell, On Reciprocal -Figures, etc., <i>Trans. R. S. E.</i> <span class="smmaj">XXVI,</span> p. 9, 1870.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch399" id="fn399">399</a> -See Greville, K. R., Monograph of the Genus -Asterolampra, <i>Q.J.M.S.</i> <span class="smmaj">VIII,</span> (Trans.), pp. -102–124, 1860; cf. <span class="smmaj">IBID.</span> (n.s.), <span class="smmaj">II,</span> pp. -41–55, 1862.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch400" id="fn400">400</a> -The same is true of the insect’s wing; but in -this case I do not hazard a conjectural explanation.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch401" id="fn401">401</a> -<i>Ann. Mag. N. H.</i> (2), <span class="smmaj">III,</span> p. 126, -1849.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch402" id="fn402">402</a> -<i>Phil. Trans.</i> <span class="smmaj">CLVII,</span> pp. 643–656, -1867.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch403" id="fn403">403</a> -Sachs, <i>Pflanzenphysiologie</i> (<i>Vorlesung</i> -<span class="nowrap"><span class="smmaj">XXIV</span>),</span> 1882; cf. Rauber, Neue Grundlage zur -Kenntniss der Zelle, <i>Morphol. Jahrb.</i> <span class="smmaj">VIII,</span> p. -303 <i>seq.</i>, 1883; E. B. Wilson, Cell-lineage of Nereis, -<i>Journ. of Morphology</i>, <span class="smmaj">VI,</span> p. 448, 1892, etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch404" id="fn404">404</a> -In the following account I follow closely on -the lines laid down by Berthold; <i>Protoplasmamechanik</i>, -cap. vii. Many botanical phenomena identical and similar to -those here dealt with, are elaborately discussed by Sachs -in his <i>Physiology of Plants</i> (chap. xxvii, pp. 431–459, -Oxford, 1887); and in his earlier papers, Ueber die -Anordnung der Zellen in jüngsten Pflanzentheilen, and Ueber -Zellenanordnung und Wachsthum (<i>Arb. d. botan. Inst. Würzburg</i>, -1878, 1879). But Sachs’s treatment differs entirely from -that which I adopt and advocate here: his explanations -being based on his “law” of rectangular succession, and -involving complicated systems of confocal conics, with -their orthogonally intersecting ellipses and hyperbolas.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch405" id="fn405">405</a> -Cf. p. 369.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch406" id="fn406">406</a> -There is much information regarding the -chemical composition and mineralogical structure of shells -and other organic products in H. C. Sorby’s Presidential -Address to the Geological Society (<i>Proc. Geol. Soc.</i> 1879, -pp. 56–93); but Sorby failed to recognise that association -with “organic” matter, or with colloid matter whether -living or dead, introduced a new series of purely physical -phenomena.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch407" id="fn407">407</a> -Vesque, <i>Ann. des Sc. Nat.</i> (<i>Bot.</i>) (5), -<span class="smmaj">XIX,</span> p. 310, 1874.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch408" id="fn408">408</a> -Cf. Kölliker, <i>Icones Histiologicae</i>, -1864, pp. 119, etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch409" id="fn409">409</a> -In an interesting paper by Irvine and Sims -Woodhead on the “Secretion of Carbonate of Lime by Animals” -(<i>Proc. R. S. E.</i> <span class="smmaj">XVI,</span> 1889, p. -351) it is asserted that “lime salts, of whatever form, are -deposited <i>only</i> in vitally inactive tissue.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch410" id="fn410">410</a> -The tube of Teredo shews no trace of organic -matter, but consists of irregular prismatic crystals: the -whole structure “being identical with that of small veins -of calcite, such as are seen in thin sections of rocks” -(Sorby, <i>Proc. Geol. Soc.</i> 1879, p. 58). This, then, would -seem to be a somewhat exceptional case of a shell laid -down completely outside of the animal’s external layer of -organic or colloid substance.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch411" id="fn411">411</a> -<i>C. R. Soc. Biol. Paris</i> (9), <span class="smmaj">I,</span> pp. -17–20, 1889; <i>C. R. Ac. Sc.</i> <span class="smmaj">CVIII,</span> pp. 196–8, -1889.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch412" id="fn412">412</a> -Cf. Heron-Allen, <i>Phil. Trans.</i> (B), vol. -<span class="smmaj">CCVI,</span> p. 262, 1915.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch413" id="fn413">413</a> -See Leduc, <i>Mechanism of Life</i> (1911), ch. -<span class="smmaj">X,</span> for copious references to other works on the -artificial production of “organic” forms.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch414" id="fn414">414</a> -Lectures on the Molecular Asymmetry of Natural -Organic Compounds, <i>Chemical Soc. of Paris</i>, 1860, and -also in Ostwald’s <i>Klassiker d. ex. Wiss.</i> No. 28, and -in <i>Alembic Club Reprints</i>, No. 14, Edinburgh, 1897; cf. -Richardson, G. M., <i>Foundations of Stereochemistry</i>, N. Y. -1901.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch415" id="fn415">415</a> -Japp, Stereometry and Vitalism, <i>Brit. Ass. -Rep.</i> (Bristol), p. 813, 1898; cf. also a voluminous -discussion in <i>Nature</i>, 1898–9.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch416" id="fn416">416</a> -They represent the general theorem of which -particular cases are found, for instance, in the asymmetry -of the ferments (or <i>enzymes</i>) which act upon asymmetrical -bodies, the one fitting the other, according to Emil -Fischer’s well-known phrase, as lock and key. Cf. his -Bedeutung der Stereochemie für die Physiologie, <i>Z. f. -physiol. Chemie</i>, <span class="smmaj">V,</span> p. 60, 1899, and various -papers in the <i>Ber. d. d. chem. Ges.</i> from 1894.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch417" id="fn417">417</a> -In accordance with Emil Fischer’s conception -of “asymmetric synthesis,” it is now held to be more likely -that the process is synthetic than analytic: more likely, -that is to say, that the plant builds up from the first -one asymmetric body to the exclusion of the other, than -that it “selects” or “picks out” (as Japp supposed) the -right-handed or the left-handed molecules from an original, -optically inactive, mixture of the two; cf. A. McKenzie, -Studies in Asymmetric Synthesis, <i>Journ. Chem. Soc.</i> -(Trans.), <span class="smmaj">LXXXV,</span> p. 1249, 1904.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch418" id="fn418">418</a> -See for a fuller discussion, Hans Przibram, -<i>Vitalität</i>, 1913, Kap. iv, Stoffwechsel (Assimilation -und Katalyse).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch419" id="fn419">419</a> -Cf. Cotton, <i>Ann. de Chim. et de Phys.</i> (7), -<span class="smmaj">VIII,</span> pp. 347–432 (cf. p. 373), 1896.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch420" id="fn420">420</a> -Byk, A., Zur Frage der Spaltbarkeit von -Razemverbindungen durch Zirkularpolarisiertes Licht, ein -Beitrag zur primären Entstehung optisch-activer Substanzen, -<i>Zeitsch. f. physikal. Chemie</i>, <span class="smmaj">XLIX,</span> p. 641, -1904. It must be admitted that further positive evidence on -these lines is still awanting.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch421" id="fn421">421</a> -Cf. (<i>int. al.</i>) Emil Fischer, <i>Untersuchungen -über Aminosäuren, Proteine</i>, etc. Berlin, 1906.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch422" id="fn422">422</a> -Japp, <i>l. c.</i> p. 828.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch423" id="fn423">423</a> -Rainey, G., On the Elementary Formation of -the Skeletons of Animals, and other Hard Structures formed -in connection with Living Tissue, <i>Brit. For. Med. Ch. -Rev.</i> <span class="smmaj">XX,</span> pp. 451–476, 1857; published separately -with additions, 8vo. London, 1858. For other papers by -Rainey on kindred subjects see <i>Q. J. M. S.</i> <span class="smmaj">VI</span> -(<i>Tr. Microsc. Soc.</i>), pp. 41–50, 1858, <span class="smmaj">VII,</span> pp. -212–225, 1859, <span class="smmaj">VIII,</span> pp. 1–10, 1860, -<span class="smmaj">I</span> -(n. s.), pp. 23–32, 1861. Cf. also Ord, W. M., On -Molecular Coalescence, and on the influence exercised by -Colloids upon the Forms of Inorganic Matter, <i>Q. J. M. S.</i> -<span class="smmaj">XII,</span> pp. 219–239, 1872; and also the early but -still interesting observations of Mr Charles Hatchett, -Chemical Experiments on Zoophytes; with some observations -on the component parts of Membrane, <i>Phil. Trans.</i> 1800. -pp. 327–402.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch424" id="fn424">424</a> -Cf. Quincke, Ueber unsichtbare -Flüssigkeitsschichten, <i>Ann. der Physik</i>, 1902.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch425" id="fn425">425</a> -See for instance other excellent illustrations -in Carpenter’s article “Shell,” in Todd’s <i>Cyclopædia</i>, -vol. <span class="smmaj">IV.</span> pp. 550–571, 1847–49. According to -Carpenter, the shells of the mollusca (and also of -the crustacea) are “essentially composed of <i>cells</i>, -consolidated by a deposit of carbonate of lime in their -interior.” That is to say, Carpenter supposed that the -spherulites, or calcospherites of Harting, were, to -begin with, just so many living protoplasmic cells. Soon -afterwards however, -Huxley pointed out that the mode of formation, -while at first sight “irresistibly suggesting a cellular -structure, ... is in reality nothing of the kind,” but “is -simply the result of the concretionary manner in which the -calcareous matter is deposited”; <i>ibid.</i> art. “Tegumentary -Organs,” vol. <span class="smmaj">V,</span> p. 487, 1859. Quekett (<i>Lectures -on Histology</i>, vol. <span class="smmaj">II,</span> p. 393, 1854, and <i>Q. -J. M. S.</i> <span class="smmaj">XI,</span> pp. 95–104, 1863) supported -Carpenter; but Williamson (Histological Features in the -Shells of the Crustacea, <i>Q. J. M. S.</i> <span class="smmaj">VIII,</span> -pp. 35–47, 1860) amply confirmed Huxley’s view, which -in the end Carpenter himself adopted (<i>The Microscope</i>, -1862, p. 604). A like controversy arose later in regard -to corals. Mrs Gordon (M. M. Ogilvie) asserted that the -coral was built up “of successive layers of calcified -cells, which hang together at first by their cell-walls, -and ultimately, as crystalline changes continue, form the -individual laminae of the skeletal structures” (<i>Phil. -Trans.</i> <span class="smmaj">CLXXXVII,</span> p. 102, 1896): whereas v. -Koch had figured the coral as formed out of a mass of -“Kalkconcremente” or “crystalline spheroids,” laid down -outside the ectoderm, and precisely similar both in their -early rounded and later polygonal stages (though von Koch -was not aware of the fact) to the calcospherites of Harting -(Entw. d. Kalkskelettes von Asteroides, <i>Mitth. Zool. -St. Neapel</i>, <span class="smmaj">III,</span> pp. 284–290, pl. <span class="smmaj">XX,</span> -1882). Lastly Duerden shewed that external to, and -apparently secreted by the ectoderm lies a homogeneous -organic matrix or membrane, “in which the minute calcareous -crystals forming the skeleton are laid down” (The Coral -<i>Siderastraea radians</i>, etc., <i>Carnegie Inst. Washington</i>, -1904, p. 34). Cf. also M. M. Ogilvie-Gordon, <i>Q. J. M. S.</i> -<span class="smmaj">XLIX,</span> p. 203, 1905, etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch426" id="fn426">426</a> -Cf. Claparède, <i>Z. f. w. Z.</i> <span class="smmaj">XIX,</span> p. -604, 1869.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch427" id="fn427">427</a> -Spicules extremely like those of the -Alcyonaria occur also in a few sponges; cf. (e.g.), Vaughan -Jennings, <i>Journ. Linn. Soc.</i> <span class="smmaj">XXIII,</span> p. 531, pl. -13, fig. 8, 1891.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch428" id="fn428">428</a> -<i>Mem. Manchester Lit. and Phil. Soc.</i> -<span class="smmaj">LX,</span> p. 11, 1916.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch429" id="fn429">429</a> -Mummery, J. H., On Calcification in Enamel and -Dentine, <i>Phil. Trans.</i> <span class="smmaj">CCV</span> (B), pp. 95–111, 1914.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch430" id="fn430">430</a> -The artificial concretion represented in -Fig. <a href="#fig202" title="go to Fig. 202">202</a> is identical in appearance with the concretions -found in the kidney of Nautilus, as figured by Willey -(<i>Zoological Results</i>, p. lxxvi, Fig. 2, 1902).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch431" id="fn431">431</a> -Cf. Taylor’s <i>Chemistry of Colloids</i>, p. 18, -etc., 1915.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch432" id="fn432">432</a> -This rule, undreamed of by Errera, supports -and justifies the cardinal assumption (of which we have -had so much to say in discussing the forms of cells and -tissues) that the <i>incipient</i> cell-wall behaves as, and -indeed actually is, a liquid film (cf. p. 306).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch433" id="fn433">433</a> -Cf. p. 254.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch434" id="fn434">434</a> -Cf. Harting, <i>op. cit.</i>, pp. 22, 50: “J’avais -cru d’abord que ces couches concentriques étaient produites -par l’alternance de la chaleur ou de la lumière, pendant le -jour et la nuit. Mais l’expérience, expressément instituée -pour examiner cette question, y a répondu négativement.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch435" id="fn435">435</a> -Liesegang, R. E., <i>Ueber die Schichtungen bei -Diffusionen</i>, Leipzig, 1907, and other earlier papers.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch436" id="fn436">436</a> -Cf. Taylor’s <i>Chemistry of Colloids</i>, pp. -146–148, 1915.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch437" id="fn437">437</a> -Cf. S. C. Bradford, The Liesegang -Phenomenon and Concretionary Structure in -Rocks, <i>Nature</i>, <span class="smmaj">XCVII,</span> p. 80, 1916; cf. <i>Sci. -Progress</i>, <span class="smmaj">X,</span> p. 369, 1916.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch438" id="fn438">438</a> -Cf. Faraday, On Ice of Irregular Fusibility, -<i>Phil. Trans.</i>, 1858, p. 228; <i>Researches in Chemistry, -etc.</i>, 1859, p. 374; Tyndall, <i>Forms of Water</i>, p. 178, -1872; Tomlinson, C., On some effects of small Quantities -of Foreign Matter on Crystallisation, <i>Phil. Mag.</i> (5) -<span class="smmaj">XXXI,</span> p. 393, 1891, and other papers.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch439" id="fn439">439</a> -A Study in Crystallisation, <i>J. of Soc. of -Chem. Industry</i>, <span class="smmaj">XXV,</span> p. 143, 1906.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch440" id="fn440">440</a> -<i>Ueber Zonenbildung in kolloidalen Medien</i>, -Jena, 1913.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch441" id="fn441">441</a> -<i>Verh. d. d. Zool. Gesellsch.</i> p. 179, 1912.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch442" id="fn442">442</a> -<i>Descent of Man</i>, <span class="smmaj">II,</span> pp. 132–153, -1871.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch443" id="fn443">443</a> -As a matter of fact, the phenomena associated -with the development of an “ocellus” are or may be of great -complexity, inasmuch as they involve not only a graded -distribution of pigment, but also, in “optical” coloration, -a symmetrical distribution of structure or form. The -subject therefore deserves very careful discussion, such as -Bateson gives to it (<i>Variation</i>, chap. xii). This, by the -way, is one of the very rare cases in which Bateson appears -inclined to suggest a purely physical explanation of an -organic phenomenon: “The suggestion is strong that the -whole series of rings (in <i>Morpho</i>) may have been formed -by some one central disturbance, somewhat as a series of -concentric waves may be formed by the splash of a stone -thrown into a pool, etc.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch444" id="fn444">444</a> -Cf. also Sir D. Brewster, On optical -properties of Mother of Pearl, <i>Phil. Trans.</i> 1814, p. 397.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch445" id="fn445">445</a> -Biedermann, W., Ueber die Bedeutung von -Kristallisationsprozessen der Skelette wirbelloser Thiere, -namentlich der Molluskenschalen, <i>Z. f. allg. Physiol.</i> -<span class="smmaj">I,</span> p. 154, 1902; Ueber Bau und Entstehung -der Molluskenschale, <i>Jen. Zeitschr.</i> <span class="smmaj">XXXVI,</span> -pp. 1–164, 1902. Cf. also Steinmann, Ueber Schale und -Kalksteinbildungen, <i>Ber. Naturf. Ges. Freiburg i. Br</i> -<span class="smmaj">IV,</span> 1889; Liesegang, <i>Naturw. Wochenschr.</i> p. 641, -1910.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch446" id="fn446">446</a> -Cf. Bütschli, Ueber die Herstellung -künstlicher Stärkekörner oder von Sphärokrystallen der -Stärke, <i>Verh. nat. med. Ver. Heidelberg</i>, <span class="smmaj">V,</span> pp. -457–472, 1896.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch447" id="fn447">447</a> -<i>Untersuchungen über die Stärkekörner</i>, Jena, -1905.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch448" id="fn448">448</a> -Cf. Winge, <i>Meddel. fra Komm. for -Havundersögelse</i> (<i>Fiskeri</i>), <span class="smmaj">IV,</span> p. 20, -Copenhagen, 1915.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch449" id="fn449">449</a> -The anhydrite is sulphate of lime (CaSO<sub>4</sub>); -the polyhalite is a triple sulphate of lime, magnesia and potash -<span class="nowrap"> -(2 CaSO<sub>4</sub> . MgSO<sub>4</sub> . K<sub>2</sub>SO<sub>4</sub></span> - + 2 H<sub>2</sub>O).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch450" id="fn450">450</a> -Cf. van’t Hoff, <i>Physical Chemistry in the -Service of the Sciences</i>, p. 99 seq. Chicago, 1903.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch451" id="fn451">451</a> -Sphärocrystalle von Kalkoxalat bei Kakteen, -<i>Ber. d. d. Bot. Gesellsch.</i> p. 178, 1885.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch452" id="fn452">452</a> -Pauli, W. u. Samec, M., -Ueber Löslichkeitsbeeinflüssung -von Elektrolyten durch -Eiweisskörper, <i>Biochem. Zeitschr.</i> <span class="smmaj">XVII,</span> p. 235, -1910. Some of these results were known much earlier; cf. -Fokker in <i>Pflüger’s Archiv</i>, <span class="smmaj">VII,</span> p. 274, 1873; -also Irvine and Sims Woodhead, <i>op. cit.</i> p. 347.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch453" id="fn453">453</a> -Which, in 1000 parts of ash, contains about -840 parts of phosphate and 76 parts of calcium carbonate.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch454" id="fn454">454</a> -Cf. Dreyer, Fr., Die Principien der -Gerüstbildung bei Rhizopoden, Spongien und Echinodermen, -<i>Jen. Zeitschr.</i> <span class="smmaj">XXVI,</span> pp. 204–468, 1892.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch455" id="fn455">455</a> -In an anomalous and very remarkable Australian -sponge, just described by Professor Dendy (<i>Nature</i>, May -18, 1916, p. 253) under the name of <i>Collosclerophora</i>, -the spicules are “gelatinous,” consisting of a gel of -colloid silica with a high percentage of water. It is not -stated whether an organic colloid is present together -with the silica. These gelatinous spicules arise as -exudations on the outer surface of cells, and come to lie -in intercellular spaces or vesicles.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch456" id="fn456">456</a> -Lister, in Willey’s <i>Zoological Results</i>, pt -<span class="smmaj">IV,</span> p. 459, 1900.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch457" id="fn457">457</a> -The peculiar spicules of Astrosclera are -now said to consist of spherules, or calcospherites, -of aragonite, spores of a certain red seaweed forming -the nuclei, or starting-points, of the concretions (R. -Kirkpatrick, <i>Proc. R. S.</i> <span class="smmaj">LXXXIV</span> (B), p. 579, -1911).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch458" id="fn458">458</a> -See for instance the plates in Théel’s -Monograph of the Challenger Holothuroidea; also Sollas’s -Tetractinellida, p. lxi.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch459" id="fn459">459</a> -For very numerous illustrations of the -triradiate and quadriradiate spicules of the calcareous -sponges, see (<i>int. al.</i>), papers by Dendy (<i>Q. J. M. S.</i> -<span class="smmaj">XXXV,</span> 1893), Minchin (<i>P. Z. S.</i> 1904), Jenkin -(<i>P. Z. S.</i> 1908), etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch460" id="fn460">460</a> -Cf. again Bénard’s <i>Tourbillons cellulaires</i>, -<i>Ann. de Chimie</i>, 1901, p. 84.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch461" id="fn461">461</a> -Léger, Stolc and others, in Doflein’s -<i>Lehrbuch d. Protozoenkunde</i>, 1911, p. 912.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch462" id="fn462">462</a> -See, for instance, the figures of the -segmenting egg of Synapta (after Selenka), in Korschelt and -Heider’s <i>Vergleichende Entwicklungsgeschichte</i> (Allgem. -Th., 3<sup>te</sup> Lief.), p. 19, 1909. On the spiral type of -segmentation as a secondary derivative, due to mechanical -causes, of the “radial” type of segmentation, see E. B. -Wilson, Cell-lineage of Nereis, <i>Journ. of Morphology</i>, -<span class="smmaj">VI,</span> p. 450, 1892.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch463" id="fn463">463</a> -Korschelt and Heider, p. 16.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch464" id="fn464">464</a> -<i>Chall. Rep. Hexactinellida</i>, pls. xvi, liii, -lxxvi, lxxxviii.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch465" id="fn465">465</a> -“Hierbei nahm der kohlensaure Kalk eine -halb-krystallinische Beschaffenheit an, und gestaltete sich -unter Aufnahme von Krystallwasser und in Verbindung mit -einer geringen Quantität von organischer Substanz zu jenen -individuellen, festen Körpern, welche durch die natürliche -Züchtung als <i>Spicula</i> zur Skeletbildung benützt, und -späterhin durch die Wechselwirkung von Anpassung und -Vererbung im Kampfe ums Dasein auf das Vielfältigste -umgebildet und differenziert wurden.” <i>Die Kalkschwämme</i>, -<span class="smmaj">I,</span> p. 377, 1872; cf. also pp. 482, 483.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch466" id="fn466">466</a> -<i>Op. cit.</i> p. 483. “Die geordnete, oft so -sehr regelmässige und zierliche Zusammensetzung des -Skeletsystems ist zum grössten Theile unmittelbares Product -der Wasserströmung; die characteristische Lagerung der -Spicula ist von der constanten Richtung des Wasserstroms -hervorgebracht; zum kleinsten Theile ist sie die Folge von -Anpassungen an untergeordnete äussere Existenzbedingungen.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch467" id="fn467">467</a> -Materials for a Monograph of the Ascones, <i>Q. -J. M. S.</i> <span class="smmaj">XL.</span> pp. 469–587, 1898.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch468" id="fn468">468</a> -Haeckel, in his <i>Challenger Monograph</i>, p. -clxxxviii (1887) estimated the number of known forms at -4314 species, included in 739 genera. Of these, 3508 -species were described for the first time in that work.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch469" id="fn469">469</a> -Cf. Gamble, <i>Radiolaria</i> (Lankester’s -<i>Treatise on Zoology</i>), vol. <span class="smmaj">I,</span> p. 131, 1909. Cf. -also papers by V. Häcker, in <i>Jen. Zeitschr.</i> -<span class="smmaj">XXXIX,</span> p. 581, 1905, <i>Z. f. wiss. Zool.</i> -<span class="smmaj">LXXXIII,</span> p. 336, 1905, <i>Arch. f. Protistenkunde</i>, -<span class="smmaj">IX,</span> p. 139, 1907, etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch470" id="fn470">470</a> -Bütschli, Ueber die chemische Natur der -Skeletsubstanz der Acantharia, <i>Zool. Anz.</i> <span class="smmaj">XXX,</span> -p. 784, 1906.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch471" id="fn471">471</a> -For figures of these crystals see Brandt, <i>F. -u. Fl. d. Golfes von Neapel</i>, <span class="smmaj">XIII,</span> <i>Radiolaria</i>, -1885, pl. v. Cf. J. Müller, Ueber die Thalassicollen, etc. -<i>Abh. K. Akad. Wiss. Berlin</i>, 1858.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch472" id="fn472">472</a> -Celestine, or celestite, is SrSO<sub>4</sub> with some -BaO replacing SrO.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch473" id="fn473">473</a> -With the colloid chemists, we may adopt (as -Rhumbler has done) the terms <i>spumoid</i> or <i>emulsoid</i> -to denote an agglomeration of fluid-filled vesicles, -restricting the name <i>froth</i> to such vesicles when filled -with air or some other gas.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch474" id="fn474">474</a> -Cf. Koltzoff, Zur Frage der Zellgestalt, -<i>Anat. Anzeiger</i>, <span class="smmaj">XLI,</span> p. 190, 1912.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch475" id="fn475">475</a> -<i>Mém. de l’Acad. des Sci., St. -Pétersbourg</i>, <span class="smmaj">XII,</span> Nr. 10, 1902.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch476" id="fn476">476</a> -The manner in which the minute spicules of -Raphidiophrys arrange themselves round the -bases of the pseudopodial rays is a similar phenomenon.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch477" id="fn477">477</a> -Rhumbler, Physikalische Analyse von -Lebenserscheinungen der Zelle, <i>Arch. f. Entw. Mech.</i> -<span class="smmaj">VII,</span> p. 103, 1898.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch478" id="fn478">478</a> -The whole phenomenon is described by -biologists as a “surprising exhibition of constructive -and selective activity,” and is ascribed, in varying -phraseology, to intelligence, skill, purpose, psychical -activity, or “microscopic mentality”: that is to say, -to Galen’s τεχνικὴ φύσις, or “artistic -creativeness” (cf. Brock’s <i>Galen</i>, 1916, p. xxix). -Cf. Carpenter, <i>Mental Physiology</i>, 1874, p. 41; -Norman, Architectural achievements of Little Masons, -etc., <i>Ann. Mag. Nat. Hist.</i> (5), <span class="smmaj">I,</span> p. 284, -1878; Heron-Allen, Contributions ... to the Study of -the Foraminifera, <i>Phil. Trans.</i> (B), <span class="smmaj">CCVI,</span> -pp. 227–279, 1915; Theory and Phenomena of Purpose and -Intelligence exhibited by the Protozoa, as illustrated -by selection and behaviour in the Foraminifera, <i>Journ. -R. Microscop. Soc.</i> pp. 547–557, 1915; <i>ibid.</i>, pp. -137–140, 1916. Prof. J. A. Thomson (<i>New Statesman</i>, -Oct. 23, 1915) describes a certain little foraminifer, -whose protoplasmic body is overlaid by a crust of -sponge-spicules, as “a psycho-physical individuality -whose experiments in self-expression include a masterly -treatment of sponge-spicules, and illustrate that organic -skill which came before the dawn of Art.” Sir Ray Lankester -finds it “not difficult to conceive of the existence of a -mechanism in the protoplasm of the Protozoa which selects -and rejects building-material, and determines the shapes of -the structures built, comparable to that mechanism which -is assumed to exist in the nervous system of insects and -other animals which ‘automatically’ go through wonderfully -elaborate series of complicated actions.” And he agrees -with “Darwin and others [who] have attributed the building -up of these inherited mechanisms to the age-long action of -Natural Selection, and the survival of those individuals -possessing qualities or ‘tricks’ of life-saving value,” <i>J. -R. Microsc. Soc.</i> April, 1916, p. 136.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch479" id="fn479">479</a> -Rhumbler, <i>Das Protoplasma als physikalisches -System</i>, Jena, p. 591, 1914; also in <i>Arch. f. -Entwickelungsmech.</i> <span class="smmaj">VII,</span> pp. 279–335, 1898.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch480" id="fn480">480</a> -Verworn, <i>Psycho-physiologische -Protisten-Studien</i>, Jena, 1889 (219 pp.).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch481" id="fn481">481</a> -Leidy, J., <i>Fresh-water Rhizopods of N. -America</i>, 1879, p. 262, pl. xli, figs. 11, 12.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch482" id="fn482">482</a> -Carnoy, <i>Biologie Cellulaire</i>, p. 244, fig. -108; cf. Dreyer, <i>op. cit.</i> 1892, fig. 185.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch483" id="fn483">483</a> -In all these latter cases we recognise a -relation to, or extension of, the principle of Plateau’s -<i>bourrelet</i>, or van der Mensbrugghe’s <i>masse annulaire</i>, of -which we have already spoken (p. 297).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch484" id="fn484">484</a> -Apart from the fact that the apex of each -pyramid is interrupted, or truncated, by the presence of -the little central cell, it is also possible that the solid -angles are not precisely equivalent to those of Maraldi’s -pyramids, owing to the fact that there is a certain amount -of distortion, or axial asymmetry, in the Nassellarian -system. In other words (to judge from Haeckel’s figures), -the tetrahedral symmetry in Nassellaria is not absolutely -regular, but has a main axis about which three of the -trihedral pyramids are symmetrical, the fourth having its -solid angle somewhat diminished.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch485" id="fn485">485</a> -Cf. Faraday’s beautiful experiments, On the -Moving Groups of Particles found on Vibrating Elastic -Surfaces, etc., <i>Phil. Trans.</i> 1831, p. 299; <i>Researches in -Chem. and Phys.</i> 1859, pp. 314–358.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch486" id="fn486">486</a> -We need not go so far as to suppose that -the external layer of cells wholly lacked the power of -secreting a skeleton. In many of the Nassellariae figured -by Haeckel (for there are many variant forms or species -besides that represented here), the skeleton of the -partition-walls is very slightly and scantily developed. In -such a case, if we imagine its few and scanty strands to be -broken away, the central tetrahedral figure would be set -free, and would have all the appearance of a complete and -independent structure.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch487" id="fn487">487</a> -The “bourrelet” is not only, as Plateau -expresses it, a “surface of continuity,” but we -also recognise that it tends (so far as material is -available for its production) to further lessen the free -surface-area. On its relation to vapour-pressure and to the -stability of foam, see FitzGerald’s interesting note in -<i>Nature</i>, Feb. 1, 1894 (<i>Works</i>, p. 309).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch488" id="fn488">488</a> -Of the many thousand figures in the hundred -and forty plates of this beautifully illustrated book, -there is scarcely one which does not depict, now patently, -now in pregnant suggestion, some subtle and elegant -geometrical configuration.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch489" id="fn489">489</a> -They were known (of course) long before Plato: -Πλάτων δὲ καὶ ἐν τούτοις πυθαγορίζει.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch490" id="fn490">490</a> -If the equation of any plane face of a crystal -be written in the form <span class="nowrap"> -<i>h x</i> + <i>k y</i> + <i>l z</i></span> -= 1, then -<i>h</i>, <i>k</i>, <i>l</i> are the indices of which we are speaking. -They are the reciprocals of the parameters, or reciprocals -of the distances from the origin at which the plane meets -the several axes. In the case of the regular or pentagonal -dodecahedron these indices are 2, 1 + √5, 0. Kepler -described as follows, briefly but adequately, the common -characteristics of the dodecahedron and icosahedron: “Duo -sunt corpora regularia, dodecaedron et icosaedron, quorum -illud quinquangulis figuratur expresse, hoc triangulis -quidem sed in quinquanguli formam coaptatis. Utriusque -horum corporum ipsiusque adeo quinquanguli <i>structura -perfici non potest sine proportione illa, quam hodierni -geometrae divinam appellant</i>” (<i>De nive sexangula</i> (1611), -Opera, ed. Frisch, <span class="smmaj">VII,</span> p. 723). Here Kepler -was dealing, somewhat after the manner of Sir Thomas -Browne, with the mysteries of the quincunx, and also of -the hexagon; and was seeking for an explanation of the -mysterious or even mystical beauty of the 5-petalled or -3-petalled flower,—<i>pulchritudinis aut proprietatis -figurae, quae animam harum plantarum characterisavit</i>.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch491" id="fn491">491</a> -Cf. Tutton, <i>Crystallography</i>, p. 932, 1911.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch492" id="fn492">492</a> -However, we can often recognise, in a small -artery for instance, that the so-called “circular” fibres -tend to take a slightly oblique, or spiral, course.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch493" id="fn493">493</a> -The spiral fibres, or a large portion of -them, constitute what Searle called “the rope of the -heart” (Todd’s <i>Cyclopaedia</i>, <span class="smmaj">II,</span> p. 621, 1836). -The “twisted sinews of the heart” were known to early -anatomists, and have been frequently and elaborately -studied: for instance, by Gerdy (<i>Bull. Fac. Med. Paris</i>, -1820, pp. 40–148), and by Pettigrew (<i>Phil. Trans.</i> -1864), and of late by J. B. Macallum (<i>Johns Hopkins -Hospital Report</i>, <span class="smmaj">IX,</span> 1900) and by Franklin P. -Mall (<i>Amer. J. of Anat.</i> <span class="smmaj">XI,</span> 1911).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch494" id="fn494">494</a> -Cf. Bütschli, “Protozoa,” in Bronn’s -<i>Thierreich</i>, <span class="smmaj">II,</span> p. 848, <span class="smmaj">III,</span> p. 1785, -etc., 1883–87; Jennings, <i>Amer. Nat.</i> <span class="smmaj">XXXV,</span> p. -369, 1901; Pütter, Thigmotaxie bei Protisten, <i>Arch. f. -Anat. u. Phys.</i> (<i>Phys. Abth. Suppl.</i>), pp. 243–302, 1900.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch495" id="fn495">495</a> -A great number of spiral forms, both organic -and artificial, are described and beautifully illustrated -in Sir T. A. Cook’s <i>Curves of Life</i>, 1914, and <i>Spirals in -Nature and Art</i>, 1903.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch496" id="fn496">496</a> -Cf. Vines, The History of the Scorpioid Cyme, -<i>Journ. of Botany</i> (n.s.), <span class="smmaj">X,</span> pp. 3–9, 1881.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch497" id="fn497">497</a> -Leslie’s <i>Geometry of Curved Lines</i>, p. 417, -1821. This is practically identical with Archimedes’ own -definition (ed. Torelli, p. 219); cf. Cantor, <i>Geschichte -der Mathematik</i>, <span class="smmaj">I,</span> p. 262, 1880.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch498" id="fn498">498</a> -See an interesting paper by Whitworth, W. -A., “The Equiangular Spiral, its chief properties proved -geometrically,” in the <i>Messenger of Mathematics</i> (1), -<span class="smmaj">I,</span> p. 5, 1862.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch499" id="fn499">499</a> -I am well aware that the debt of Greek -science to Egypt and the East is vigorously denied by many -scholars, some of whom go so far as to believe that the -Egyptians never had any science, save only some “rough -rules of thumb for measuring fields and pyramids” (Burnet’s -<i>Greek Philosophy</i>, 1914, p. 5).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch500" id="fn500">500</a> -Euclid (<span class="smmaj">II,</span> def. 2).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch501" id="fn501">501</a> -Cf. Treutlein, <i>Z. f. Math. u. Phys.</i> (<i>Hist. -litt. Abth.</i>), <span class="smmaj">XXVIII,</span> p. 209, 1883.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch502" id="fn502">502</a> -This is the so-called -<i>Dreifachgleichschenkelige Dreieck</i>; cf. Naber, <i>op. infra -cit.</i> The ratio 1 : 0·618 is again not hard to find in this -construction.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch503" id="fn503">503</a> -See, on the mathematical history of the -Gnomon, Heath’s <i>Euclid</i>, <span class="smmaj">I,</span> <i>passim</i>, 1908; -Zeuthen, <i>Theorème de Pythagore</i>, Genève, 1904; also a -curious and interesting book, <i>Das Theorem des Pythagoras</i>, -by Dr. H. A. Naber, Haarlem, 1908.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch504" id="fn504">504</a> -For many beautiful geometrical constructions -based on the molluscan shell, see Colman, S. and Coan, C. -A., <i>Nature’s Harmonic Unity</i> (ch. ix, Conchology), New -York, 1912.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch505" id="fn505">505</a> -The Rev. H. Moseley, On the Geometrical Forms -of Turbinated and Discoid Shells, <i>Phil. Trans.</i> pp. -351–370. 1838.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch506" id="fn506">506</a> -It will be observed that here Moseley, -speaking as a mathematician and considering the <i>linear</i> -spiral, speaks of <i>whorls</i> when he means the linear -boundaries, or lines traced by the revolving radius -vector; while the conchologist usually applies the term -<i>whorl</i> to the whole space between the two boundaries. As -conchologists, therefore, we call the <i>breadth of a whorl</i> -what Moseley looked upon as the <i>distance between two -consecutive whorls</i>. But this latter nomenclature Moseley -himself often uses.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch507" id="fn507">507</a> -In the case of Turbo, and all other -“turbinate” shells, we are dealing not with a plane -logarithmic spiral, as in Nautilus, but with a “gauche” -spiral, such that the radius vector no longer revolves -in a plane perpendicular to the axis of the system, but -is inclined to that axis at some constant angle (θ). The -figure still preserves its continued similarity, and may -with strict accuracy be called a logarithmic spiral in -space. It is evident that its envelope will be a right -circular cone; and indeed it is commonly spoken of as -a logarithmic spiral <i>wrapped upon a cone</i>, its pole -coinciding with the apex of the cone. It follows that the -distances of successive whorls of the spiral measured -on the same straight line passing through the apex of -the cone, are in geometrical progression, and conversely -just as in the former case. But the ratio between any two -consecutive interspaces (i.e. <span class="nowrap"> -<i>R</i><sub>3</sub> − <i>R</i><sub>2</sub> ⁄ <i>R</i><sub>2</sub> − <i>R</i><sub>1</sub>)</span> -is now equal to <span class="nowrap"> -ε<sup>2π sin θ cot α</sup> ,</span> -θ being the -semi-angle of the enveloping cone. (Cf. Moseley, <i>Phil. -Mag.</i> <span class="smmaj">XXI,</span> p. 300, 1842.)</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch508" id="fn508">508</a> -As the successive increments evidently constitute -similar figures, similarly related to the pole (<i>P</i>), it -follows that their linear dimensions are to one another -as the radii vectores drawn to similar points in them: -for instance as -<span class="nowrap"><i>P P</i><sub>1</sub> ,</span> -<span class="nowrap"><i>P P</i><sub>2</sub> ,</span> -which (in Fig. <a href="#fig264" title="go to Fig. 264">264</a>, 1) are -radii vectores drawn to the points where they meet the common -boundary.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch509" id="fn509">509</a> -The equation to the surface of a turbinate -shell is discussed by Moseley (<i>Phil. Trans.</i> tom. cit. -p. 370), both in terms of polar coordinates and of the -rectangular coordinates <i>x</i>, <i>y</i>, <i>z</i>. A more elegant -representation can be given in vector notation, by the -method of quaternions.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch510" id="fn510">510</a> -J. C. M. Reinecke, <i>Maris protogaei Nautilos, -etc.</i>, Coburg, 1818. Leopold von Buch, Ueber die Ammoniten -in den älteren Gebirgsschichten, <i>Abh. Berlin. Akad., Phys. -Kl.</i> pp. 135–158, 1830; <i>Ann. Sc. Nat.</i> <span class="smmaj">XXVIII,</span> -pp. 5–43, 1833; cf. Elie de Beaumont, Sur l’enroulement des -Ammonites, <i>Soc. Philom., Pr. verb.</i> pp. 45–48, 1841.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch511" id="fn511">511</a> -<i>Biblia Naturae sive Historia Insectorum</i>, -Leydae, 1737, p. 152.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch512" id="fn512">512</a> -Alcide D’Orbigny, <i>Bull. de la soc. géol. -Fr.</i> <span class="smmaj">XIII,</span> p. 200, 1842; <i>Cours élém. de -Paléontologie</i>, <span class="smmaj">II,</span> p. 5, 1851. A somewhat -similar instrument was described by Boubée. in <i>Bull. soc. -géol.</i> <span class="smmaj">I,</span> p. 232, 1831. Naumann’s Conchyliometer -(<i>Poggend. Ann.</i> <span class="smmaj">LIV,</span> p. 544, 1845) was an -application of the screw-micrometer; it was provided also -with a rotating stage, for angular measurement. It was -adapted for the Study of a discoid or ammonitoid shell, -while D’Orbigny’s instrument was meant for the study of a -turbinate shell.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch513" id="fn513">513</a> -It is obvious that the ratios of opposite -whorls, or of radii 180° apart, are represented by the -square roots of these values; and the ratios of whorls or -radii 90° apart, by the square roots of these again.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch514" id="fn514">514</a> -For the correction to be applied in the case -of the helicoid, or “turbinate” shells, see p. 557.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch515" id="fn515">515</a> -On the Measurement of the Curves formed -by Cephalopods and other Mollusks. <i>Phil. Mag.</i> (5), -<span class="smmaj">VI,</span> pp. 241–263, 1878.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch516" id="fn516">516</a> -For an example of this method, see Blake, <i>l.c.</i> p. 251.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch517" id="fn517">517</a> -Naumann, C. F., Ueber die Spiralen von -Conchylien, <i>Abh. k. sächs</i>. Ges. pp. 153–196, 1846; -Ueber die cyclocentrische Conchospirale u. über das -Windungsgesetz von <i>Planorbis corneus</i>, <i>ibid.</i> <span class="smmaj">I,</span> -pp. 171–195, 1849; Spirale von Nautilus u. <i>Ammonites -galeatus</i>, <i>Ber. k. sächs. Ges.</i> <span class="smmaj">II,</span> p. 26, 1848; -Spirale von <i>Amm. Ramsaueri</i>, <i>ibid.</i> <span class="smmaj">XVI,</span> p. 21, -1864; see also <i>Poggendorff’s Annalen</i>, <span class="smmaj">L,</span> p. 223, -1840; <span class="smmaj">LI,</span> p. 245, 1841; <span class="smmaj">LIV,</span> -p. 541, 1845, etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch518" id="fn518">518</a> -Sandberger, G., Spiralen des <i>Ammonites -Amaltheus</i>, <i>A. Gaytani</i>, und <i>Goniatites intumescens</i>, -<i>Zeitschr. d. d. Geol. Gesellsch.</i> <span class="smmaj">X,</span> pp. 446–449, -1858.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch519" id="fn519">519</a> -Grabau, A. H., <i>Ueber die Naumannsche -Conchospirale</i>, etc. Inauguraldiss. Leipzig, 1872; <i>Die -Spiralen von Conchylien</i>, etc. Programm, Nr. 502, Leipzig, -1882.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch520" id="fn520">520</a> -It has been pointed out to me that it does not -follow at once and obviously that, because the interspace -<i>AB</i> is a mean proportional between the breadths of the -adjacent whorls, therefore the whole distance <i>OB</i> is -a mean proportional between <i>OA</i> and <i>OC</i>. This is a -corollary which requires to be proved; but the proof is -easy.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch521" id="fn521">521</a> -A beautiful construction: <i>stupendum Naturae -artificium</i>, Linnaeus.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch522" id="fn522">522</a> -English edition, p. 537, 1900. The chapter is -revised by Prof. Alpheus Hyatt, to whom the nomenclature -is largely due. For a more copious terminology, see Hyatt, -<i>Phylogeny of an Acquired Characteristic</i>, p. 422 <i>seq.</i>, -1894.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch523" id="fn523">523</a> -This latter conclusion is adopted by Willey, -<i>Zoological Results</i>, p. 747, 1902.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch524" id="fn524">524</a> -See Moseley, <i>op. cit.</i> pp. 361 <i>seq.</i></p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch525" id="fn525">525</a> -In Nautilus, the “hood” has somewhat different -dimensions in the two sexes, and these differences -are impressed upon the shell, that is to say upon its -“generating curve.” The latter constitutes a somewhat -broader ellipse in the -male than in the female. But this difference is not to be -detected in the young; in other words, the form of the -generating curve perceptibly alters with advancing age. -Somewhat similar differences in the shells of Ammonites -were long ago suspected, by D’Orbigny, to be due to sexual -differences. (Cf. Willey, <i>Natural Science</i>, <span class="smmaj">VI,</span> -p. 411, 1895; <i>Zoological Results</i>, p. 742, 1902.)</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch526" id="fn526">526</a> -Macalister, Alex., Observations on the Mode of Growth of Discoid and -Turbinated Shells, <i>P. R. S.</i> <span class="smmaj">XVIII,</span> pp. 529–532, 1870.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch527" id="fn527">527</a> -See figures in Arnold Lang’s <i>Comparative -Anatomy</i> (English translation), <span class="smmaj">II,</span> p. 161, 1902.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch528" id="fn528">528</a> -Kappers, C. U. A., Die Bildung künstlicher -Molluskenschalen, <i>Zeitschr. f. allg. Physiol.</i> -<span class="smmaj">VII,</span> p. 166, 1908.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch529" id="fn529">529</a> -We need not assume a <i>close</i> relationship, nor -indeed any more than such a one as permits us to compare -the shell of a Nautilus with that of a Gastropod.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch530" id="fn530">530</a> -Cf. Owen, “These shells [Nautilus and -Ammonites] are revolutely spiral or coiled over the back of -the animal, not involute like Spirula”: <i>Palaeontology</i>, -1861, p. 97; cf. <i>Mem. on the Pearly Nautilus</i>, 1832; also -<i>P.Z.S.</i> 1878, p. 955.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch531" id="fn531">531</a> -The case of Terebratula or of Gryphaea would -be closely analogous, if the smaller valve were less -closely connected and co-articulated with the larger.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch532" id="fn532">532</a> -“It has been suggested, and I think in -some quarters adopted as a dogma, that the formation of -successive septa [in Nautilus] is correlated with the -recurrence of reproductive periods. This is not the case, -since, according to my observations, propagation only takes -place after the last septum is formed;” Willey, <i>Zoological -Results</i>, p. 746, 1902.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch533" id="fn533">533</a> -Cf. Woodward, Henry, On the Structure of -Camerated Shells, <i>Pop. Sci. Rev.</i> <span class="smmaj">XI,</span> pp. -113–120, 1872.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch534" id="fn534">534</a> -See Willey, Contributions to the Natural -History of the Pearly Nautilus, <i>Zoological Results</i>, etc. -p. 749, 1902. Cf. also Bather, Shell-growth in Cephalopoda, -<i>Ann. Mag. N. H.</i> (6), <span class="smmaj">I,</span> pp 298–310, 1888; -<i>ibid.</i> pp. 421–427, and other papers by Blake, Riefstahl, -etc. quoted therein.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch535" id="fn535">535</a> -It was this that led James Bernoulli, -in imitation of Archimedes, to have the logarithmic -spiral graven on his tomb, with the pious motto, <i>Eadem -mutata resurgam</i>. On Goodsir’s grave the same symbol is -reinscribed.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch536" id="fn536">536</a> -The “lobes” and “saddles” which arise in this -manner, and on whose arrangement the modern classification -of the nautiloid and ammonitoid shells largely depends, -were first recognised and named by Leopold von Buch, <i>Ann. -Sci. Nat.</i> <span class="smmaj">XXVII,</span> <span class="smmaj">XXVIII,</span> 1829.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch537" id="fn537">537</a> -Blake has remarked upon the fact (<i>op. cit.</i> -p. 248) that in some Cyrtocerata we may have a curved shell -in which the ornaments approximately run at a constant -angular distance from the pole, while the septa approximate -to a radial direction; and that “thus one law of growth is -illustrated by the inside, and another by the outside.” In -this there is nothing at which we need wonder. It is merely -a case where the generating curve is set very obliquely -to the axis of the shell; but where the septa, which have -no necessary relation to the <i>mouth</i> of the shell, take -their places, as usual, at a certain definite angle to the -<i>walls</i> of the tube. This relation of the septa to the -walls of the tube arises after the tube itself is fully -formed, and the obliquity of growth of the open end of the -tube has no relation to the matter.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch538" id="fn538">538</a> -Cf. pp. 255, 463, etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch539" id="fn539">539</a> -In a few cases, according to Awerinzew and -Rhumbler, where the chambers are added on in concentric -series, as in Orbitolites, we have the crystalline -structure arranged radially in the radial walls but -tangentially in the concentric ones: whereby we tend -to obtain, on a minute scale, a system of orthogonal -trajectories, comparable to that which we shall presently -study in connection with the structure of bone. Cf. S. -Awerinzew, Kalkschale der Rhizopoden, <i>Z. f. w. Z.</i> -<span class="smmaj">LXXIV,</span> pp. 478–490, 1903.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch540" id="fn540">540</a> -Rhumbler, L., Die Doppelschalen von -Orbitolites und anderer Foraminiferen, etc., <i>Arch. f. -Protistenkunde</i>, <span class="smmaj">I,</span> pp. 193–296, 1902; and other -papers. Also <i>Die Foraminiferen der Planktonexpedition</i>, -<span class="smmaj">I,</span> 1911, pp. 50–56.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch541" id="fn541">541</a> -Bénard, H, Les tourbillons cellulaires, <i>Ann. -de Chimie</i> (8), <span class="smmaj">XXIV,</span> 1901. Cf. also the -pattern of cilia on an Infusorian, as figured by Bütschli -in Bronn’s <i>Protozoa</i>, <span class="smmaj">III,</span> p. 1281, 1887.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch542" id="fn542">542</a> -A similar hexagonal pattern is obtained by the -mutual repulsion of floating magnets in Mr R. W. Wood’s -experiments, <i>Phil. Mag.</i> <span class="smmaj">XLVI,</span> pp. 162–164, -1898.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch543" id="fn543">543</a> -Cf. D’Orbigny, Alc., Tableau méthodique -de la classe des Céphalopodes, <i>Ann. des Sci. Nat.</i> -(1), <span class="smmaj">VII,</span> pp. 245–315, 1826; Dujardin. Félix, -Observations nouvelles sur les prétendus Céphalopodes -microscopiques, <i>ibid.</i> (2), <span class="smmaj">III,</span> pp. 108, 109, -312–315, 1835; Recherches sur les organismes inférieurs, -<i>ibid.</i> <span class="smmaj">IV,</span> pp. 343–377, 1835, etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch544" id="fn544">544</a> -It is obvious that the actual <i>outline</i> of a -foraminiferal, just as of a molluscan shell, may depart -widely from a logarithmic spiral. When we say here, for -short, that the shell <i>is</i> a logarithmic spiral, we merely -mean that it is essentially related to one: that it can -be inscribed in such a spiral, or that corresponding -points (such, for instance, as the centres of gravity of -successive chambers, or the extremities of successive -septa) wall always be found to lie upon such a spiral.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch545" id="fn545">545</a> -von Möller, V., Die spiral-gewundenen -Foraminifera des russischen Kohlenkalks, <i>Mém. de l’Acad. -Imp. Sci., St Pétersbourg</i> (7), <span class="smmaj">XXV,</span> 1878.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch546" id="fn546">546</a> -As von Möller is careful to explain, -Naumann’s formula for the “cyclocentric conchospiral” is -appropriate to this and other spiral Foraminifera, since -we have in all these cases a central or initial chamber, -approximately spherical, about which the logarithmic spiral -is coiled (cf. Fig. <a href="#fig309" title="go to Fig. 309">309</a>). In species where the central -chamber is especially large, Naumann’s formula is all the -more advantageous. But it is plain that it is only required -when we are dealing with diameters, or with radii; so long -as we are merely comparing the breadths of <i>successive -whorls</i>, the two formulae come to the same thing.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch547" id="fn547">547</a> -Van Iterson, G., <i>Mathem. u. mikrosk.-anat. -Studien über Blattstellungen, nebst Betrachtungen über den -Schalenbau der Miliolinen</i>, 331 pp., Jena, 1907.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch548" id="fn548">548</a> -Hans Przibram asserts that the linear ratio -of successive chambers tends in many Foraminifera to -approximate to 1·26, which -= ∛2; in other words, -that the volumes of successive chambers tend to double. -This Przibram would bring into relation with another law, -viz. that insects and other arthropods tend to moult, or -to metamorphose, just when they double their weights, -or increase their linear dimensions in the ratio of -1 : ∛2. (Die Kammerprogression der Foraminiferen als -Parallele zur Häutungsprogression der Mantiden, <i>Arch. f. -Entw. Mech.</i> <span class="smmaj">XXXIV</span> p. 680, 1813.) Neither rule -seems to me to be well grounded.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch549" id="fn549">549</a> -Cf. Schacko, G., Ueber Globigerina-Einschluss -bei Orbulina, <i>Wiegmann’s Archiv</i>, <span class="smmaj">XLIX,</span> p. 428, -1883; Brady, <i>Chall. Rep.</i>, p. 607, 1884.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch550" id="fn550">550</a> -Cf. Brady, H. B., <i>Challenger Rep.</i>, -<i>Foraminifera</i>, 1884, p. 203, pl. <span class="smmaj">XIII.</span></p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch551" id="fn551">551</a> -Brady, <i>op. cit.</i>, p. 206; Batsch, one of -the earliest writers on Foraminifera, had already noticed -that this whole series of ear-shaped and crozier-shaped -shells was filled in by gradational forms; <i>Conchylien des -Seesandes</i>, 1791, p. 4, pl. <span class="smmaj">VI,</span> fig. 15<i>a</i>–<i>f</i>. See -also, in particular, Dreyer, <i>Peneroplis</i>; <i>eine Studie zur -biologischen Morphologie und zur Speciesfrage</i>, Leipzig, -1898; also Eimer und Fickert, Artbildung und Verwandschaft -bei den Foraminiferen, <i>Tübinger zool. Arbeiten</i>, -<span class="smmaj">III,</span> p. 35, 1899.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch552" id="fn552">552</a> -Doflein, <i>Protozoenkunde</i>, 1911, p. 263; -“Was diese Art veranlässt in dieser Weise gelegentlich zu -varüren, ist vorläufig noch ganz räthselhaft.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch553" id="fn553">553</a> -In the case of Globigerina, some fourteen -species (out of a very much larger number of described -forms) were allowed by Brady (in 1884) to be distinct; -and this list has been, I believe, rather added to than -diminished. But these so-called species depend for the -most part on slight differences of degree, differences in -the angle of the spiral, in the ratio of magnitude of the -segments, or in their area of contact one with another. -Moreover with the exception of one or two “dwarf” forms, -said to be limited to Arctic and Antarctic waters, there is -no principle of geographical distribution to be discerned -amongst them. A species found fossil -in New Britain turns up in the North Atlantic: a species described from the West -Indies is rediscovered at the ice-barrier of the Antarctic.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch554" id="fn554">554</a> -Dreyer, F., Principien der Gerüstbildung bei -Rhizopoden, etc., <i>Jen. Zeitschr.</i> <span class="smmaj">XXVI,</span> pp. -204–468, 1892.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch555" id="fn555">555</a> -A difficulty arises in the case of forms -(like Peneroplis) where the young shell appears to be -more complex than the old, the first formed portion being -closely coiled while the later additions become straight -and simple: “die biformen Arten verhalten sich, kurz -gesagt. gerade umgekehrt als man nach dem biogenetischen -Grundgesetz erwarten sollte,” Rhumbler, <i>op. cit.</i>, p. 33 -etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch556" id="fn556">556</a> -“Das Festigkeitsprinzip als Movens der -Weiterentwicklung ist zu interessant und für die -Aufstellung meines Systems zu wichtig um die Frage -unerörtert zu lassen, warum diese Bevorzügung der -Festigkeit stattgefunden hat. Meiner Ansicht nach lautet -die Antwort auf diese Frage einfach, weil die Foraminiferen -meistens unter Verhältnissen leben, die ihre Schalen in -hohem Grade der Gefahr des Zerbrechens aussetzen; es muss -also eine fortwahrende Auslese des Festeren stattfinden,” -Rhumbler, <i>op. cit.</i>, p. 22.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch557" id="fn557">557</a> -“Die Foraminiferen kiesige oder grobsandige -Gebiete des Meeresbodens <i>nicht lieben</i>, u.s.w.”: where the -last two words have no particular meaning, save only that -(as M. Aurelius says) “of things that use to be, we say -commonly that they love to be.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch558" id="fn558">558</a> -In regard to the Foraminifera, “die -Palaeontologie lässt uns leider an Anfang der -Stammesgeschichte fast gänzlich im Stiche,” Rhumbler, <i>op. -cit.</i>, p. 14.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch559" id="fn559">559</a> -The evolutionist theory, as Bergson puts -it, “consists above all in establishing relations of -ideal kinship, and in maintaining that wherever there -is this relation of, so to speak, <i>logical</i> affiliation -between forms, <i>there is also a relation of chronological -succession between the species in which these forms are -materialised</i>”: <i>Creative Evolution</i>, 1911, p. 26. Cf. -<i>supra</i>, p. 251.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch560" id="fn560">560</a> -In the case of the ram’s horn, the assumption -that the rings are annual is probably justified. In -cattle they are much less conspicuous, but are sometimes -well-marked in the cow; and in Sweden they are then called -“calf-rings,” from a belief that they record the number -of offspring. That is to say, the growth of the horn -is supposed to be retarded during gestation, and to be -accelerated after parturition, when superfluous nourishment -seeks a new outlet. (Cf. Lönnberg, <i>P.Z.S.</i>, p. 689, 1900.)</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch561" id="fn561">561</a> -Cf. Sir V. Brooke, On the Large Sheep of the -Thian Shan, <i>P.Z.S.</i>, p. 511, 1875.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch562" id="fn562">562</a> -Cf. Lönnberg, E., On the Structure of the Musk -Ox, <i>P.Z.S.</i>, pp. 686–718, 1900.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch563" id="fn563">563</a> -St Venant, De la torsion des prismes, avec des -considérations sur leur flexion, etc., <i>Mém. des Savants -Étrangers</i>, Paris, <span class="smmaj">XIV,</span> pp. 233–560, 1856.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch564" id="fn564">564</a> -This is not difficult to do, with considerable -accuracy, if the clay be kept well wetted, or semi-fluid, -and the smoothing be done with a large wet brush.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch565" id="fn565">565</a> -The curves are well shewn in most of Sir V. -Brooke’s figures of the various species of Argali, in the -paper quoted on p. 614.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch566" id="fn566">566</a> -<i>Climbing Plants</i>, 1865 (2nd edit. 1875); -<i>Power of Movement in Plants</i>, 1880.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch567" id="fn567">567</a> -Palm, <i>Ueber das Winden der Pflanzen</i>, -1827; von Mohl, <i>Bau und Winden der Ranken</i>, etc., 1827; -Dutrochet, Mouvements révolutifs spontanés, <i>C.R.</i> 1843, -etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch568" id="fn568">568</a> -Cf. (e.g.) Lepeschkin, Zur Kenntnis des -Mechanismus der Variationsbewegungen, <i>Ber. d. d. Bot. -Gesellsch.</i> <span class="smmaj">XXVI</span> A, pp. 724–735, 1908; also A. -Tröndle, Der Einfluss des Lichtes auf die Permeabilität -des Plasmahaut, <i>Jahrb. wiss. Bot.</i> <span class="smmaj">XLVIII,</span> pp. -171–282, 1910.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch569" id="fn569">569</a> -For an elaborate study of antlers, see Rörig, -A., <i>Arch. f. Entw. Mech.</i> <span class="smmaj">X,</span> pp. 525–644, 1900, -<span class="smmaj">XI,</span> pp. 65–148, 225–309, 1901; Hoffmann, C., -<i>Zur Morphologie der rezenten Hirschen</i>, 75 pp., 23 pls., -1901: also Sir Victor Brooke, On the Classification of the -Cervidae, <i>P.Z.S.</i>, pp. 883–928, 1878. For a discussion -of the development of horns and antlers, see Gadow, H., -<i>P.Z.S.</i>, pp. 206–222, 1902, and works quoted therein.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch570" id="fn570">570</a> -Cf. Rhumbler, L., Ueber die Abhängigkeit des -Geweihwachstums der Hirsche, speziell des Edelhirsches, -vom Verlauf der Blutgefässe im Kolbengeweih, <i>Zeitschr. f. -Forst. und Jagdwesen</i>, 1911, pp. 295–314.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch571" id="fn571">571</a> -The fact that in one very small deer, the -little South American Coassus, the antler is reduced -to a simple short spike, does not preclude the general -distinction which I have drawn. In Coassus we have the -beginnings of an antler, which has not yet manifested its -tendency to expand; and in the many allied species of the -American genus Cariacus, we find the expansion manifested -in various simple modes of ramification or bifurcation. -(Cf. Sir V. Brooke, Classification of the Cervidae, p. -897.)</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch572" id="fn572">572</a> -Cf. also the immense range of variation -in elks’ horns, as described by Lönnberg, <i>P.Z.S.</i> -<span class="smmaj">II,</span> pp. 352–360, 1902.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch573" id="fn573">573</a> -Besides papers referred to below, and many -others quoted in Sach’s <i>Botany</i> and elsewhere, the -following are important: Braun, Alex., Vergl. Untersuchung -über die Ordnung der Schuppen an den Tannenzapfen, etc., -<i>Verh. Car. Leop. Akad.</i> <span class="smmaj">XV,</span> pp. 199–401, 1831; -Dr C. Schimper’s Vorträge über die Möglichkeit eines -wissenschaftlichen Verständnisses der Blattstellung, etc., -<i>Flora</i>, <span class="smmaj">XVIII,</span> pp. 145–191, 737–756, 1835; -Schimper, C. F., Geometrische Anordnung der um eine Axe -peripherische Blattgebilde, <i>Verhandl. Schweiz. Ges.</i>, -pp. 113–117, 1836; Bravais, L. and A., Essai sur la -disposition des feuilles curvisériées, <i>Ann. Sci. Nat.</i> -(2), <span class="smmaj">VII,</span> pp. 42–110, 1837; Sur la disposition -symmétrique des inflorescences, <i>ibid.</i>, pp. 193–221, -291–348, <span class="smmaj">VIII,</span> pp. 11–42, 1838; Sur la disposition -générale des feuilles rectisériées, <i>ibid.</i> <span class="smmaj">XII,</span> -pp. 5–41, 65–77, 1839; Zeising, <i>Normalverhältniss der -chemischen und morphologischen Proportionen</i>, Leipzig, -1856; Naumann, C. F., Ueber den Quincunx als Gesetz der -Blattstellung bei Sigillaria, etc., <i>Neues Jahrb. f. -Miner.</i> 1842, pp. 410–417; Lestiboudois, T., <i>Phyllotaxie -anatomique</i>, Paris, 1848; Henslow, G., <i>Phyllotaxis</i>, -London, 1871; Wiesner, Bemerkungen über rationale und -irrationale Divergenzen, <i>Flora</i>, <span class="smmaj">LVIII,</span> pp. -113–115, 139–143, 1875; Airy, H., On Leaf Arrangement, -<i>Proc. R. S.</i> <span class="smmaj">XXI,</span> p. 176, 1873; Schwendener, -S., <i>Mechanische Theorie der Blattstellungen</i>, Leipzig, -1878; Delpino, F., <i>Causa meccanica della filotassi -quincunciale</i>, Genova, 1880; de Candolle, C., <i>Étude de -Phyllotaxie</i>, Genève, 1881.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch574" id="fn574">574</a> -<i>Allgemeine Morphologie der Gewächse</i>, p. 442, -etc. 1868.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch575" id="fn575">575</a> -<i>Relation of Phyllotaxis to Mechanical Laws</i>, -Oxford, 1901–1903; cf. <i>Ann. of Botany</i>, <span class="smmaj">XV,</span> p. -481, 1901.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch576" id="fn576">576</a> -“The proposition is that the genetic spiral -is a logarithmic spiral, homologous with the line of -current-flow in a spiral vortex; and that in such a system -the action of orthogonal forces will be mapped out by -other orthogonally intersecting logarithmic spirals—the -‘parastichies’ ”; Church, <i>op. cit.</i> <span class="smmaj">I,</span> p. 42.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch577" id="fn577">577</a> -Mr Church’s whole theory, if it be not based -upon, is interwoven with, Sachs’s theory of the orthogonal -intersection of cell-walls, and the elaborate theories of -the symmetry of a growing point or apical cell which are -connected therewith. According to Mr Church, “the law of -the orthogonal intersection of cell-walls at a growing apex -may be taken as generally accepted” (p. 32); but I have -taken a very different view of Sachs’s law, in the eighth -chapter of the present book. With regard to his own and -Sachs’s hypotheses, Mr Church makes the following curious -remark (p. 42): “Nor are the hypotheses here put forward -more imaginative than that of the paraboloid apex of Sachs -which remains incapable of proof, or his construction -for the apical cell of Pteris which does not satisfy the -evidence of his own drawings.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch578" id="fn578">578</a> -<i>Amer. Naturalist</i>, <span class="smmaj">VII,</span> p. 449, -1873.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch579" id="fn579">579</a> -This celebrated series, which appears in the continued -fraction <span class="nowrap"><img id="glyph-fn579" -src="images/fn579.jpg" width="133" height="145" alt=" -(1 ⁄ 1) + (1 ⁄ (1 + ))" -></span> etc. and is closely connected with the <i>Sectio -aurea</i> or Golden Mean, is commonly called the Fibonacci -series, after a very learned twelfth century arithmetician -(known also as Leonardo of Pisa), who has some claims -to be considered the introducer of Arabic numerals into -christian Europe. It is called Lami’s series by some, -after Father Bernard Lami, a contemporary of Newton’s, -and one of the co-discoverers of the parallelogram of -forces. It was well-known to Kepler, who, in his paper -<i>De nive sexangula</i> (cf. <i>supra</i>, p. 480), discussed -it in connection with the form of the dodecahedron and -icosahedron, and with the ternary or quinary symmetry of -the flower. (Cf. Ludwig, F., Kepler über das Vorkommen -der Fibonaccireihe im Pflanzenreich, <i>Bot. Centralbl.</i> -<span class="smmaj">LXVIII,</span> p. 7, 1896). Professor -William Allman, Professor of Botany in Dublin (father of -the historian of Greek geometry), speculating on the same -facts, put forward the curious suggestion that the cellular -tissue of the dicotyledons, or exogens, would be found to -consist of dodecahedra. and that of the monocotyledons or -endogens of icosahedra (<i>On the mathematical connexion -between the parts of Vegetables</i>: abstract of a Memoir -read before the Royal Society in the year 1811 (privately -printed, <i>n.d.</i>). Cf. De Candolle, <i>Organogénie végétale</i>, -<span class="smmaj">I,</span> p. 534).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch580" id="fn580">580</a> -<i>Proc. Roy. Soc. Edin.</i> <span class="smmaj">VII,</span> p. 391, 1872.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch581" id="fn581">581</a> -The necessary existence of these recurring -spirals is also proved, in a somewhat different way, by -Leslie Ellis, On the Theory of Vegetable Spirals, in -<i>Mathematical and other Writings</i>, 1853, pp. 358–372.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch582" id="fn582">582</a> -<i>Proc. Roy. Soc. Edin.</i> <span class="smmaj">VII,</span> p. 397, -1872; <i>Trans. Roy. Soc. Edin.</i> <span class="smmaj">XXVI,</span> p. 505, -1870–71.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch583" id="fn583">583</a> -A common form of pail-shaped waste-paper -basket, with wide rhomboidal meshes of cane, is well-nigh -as good a model as is required.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch584" id="fn584">584</a> -<i>Deutsche Vierteljahrsschrift</i>, p. 261, 1868.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch585" id="fn585">585</a> -<i>Memoirs of Amer. Acad.</i> <span class="smmaj">IX,</span> p. 389.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch586" id="fn586">586</a> -<i>De avibus circa aquas Danubii vagantibus -et de ipsarum Nidis</i> (Vol. <span class="smmaj">V</span> of the <i>Danubius -Pannonico-mysicus</i>), Hagae Com., 1726.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch587" id="fn587">587</a> -Sir Thomas Browne had a collection of eggs at -Norwich, according to Evelyn, in 1671.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch588" id="fn588">588</a> -Cf. Lapierre, in Buffon’s <i>Histoire -Naturelle</i>, ed. Sonnini, 1800.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch589" id="fn589">589</a> -<i>Eier der Vögel Deutschlands</i>, 1818–28 (<i>cit.</i> -des Murs, p. 36).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch590" id="fn590">590</a> -<i>Traité d’Oologie</i>, 1860.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch591" id="fn591">591</a> -Lafresnaye, F. de, Comparaison des œufs -des Oiseaux avec leurs squelettes, comme seul moven de -reconnaître la cause de leurs différentes formes, <i>Rev. -Zool.</i>, 1845, pp. 180–187, 239–244.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch592" id="fn592">592</a> -Cf. Des Murs, p. 67: “Elle devait encore -penser au moment où ce germe aurait besoin de l’espace -nécessaire à son accroissement, à ce moment où ... il devra -remplir exactement l’intervalle circonscrit par sa fragile -prison, etc.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch593" id="fn593">593</a> -Thienemann, F. A. L., <i>Syst. Darstellung der -Fortpflanzung der Vögel Europas</i>. Leipzig, 1825–38.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch594" id="fn594">594</a> -Cf. Newton’s <i>Dictionary of Birds</i>, 1893, p. -191; Szielasko, Gestalt der Vogeleier, <i>J. f. Ornith.</i> -<span class="smmaj">LIII,</span> pp. 273–297, 1905.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch595" id="fn595">595</a> -Jacob Steiner suggested a Cartesian oval, <span class="nowrap"> -<i>r</i> + <i>m r′</i></span> -= <i>c</i>, as a general formula for all eggs (cf. -Fechner, <i>Ber. sächs. Ges.</i>, 1849, p. 57); but this formula -(which fails in such a case as the guillemot), is purely -empirical, and has no mechanical foundation.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch596" id="fn596">596</a> -Günther, F. C., <i>Sammlung von Nestern und -Eyern verschiedener Vögel</i>, Nürnb. 1772. Cf. also Raymond -Pearl, Morphogenetic Activity of the Oviduct, <i>J. Exp. -Zool.</i> <span class="smmaj">VI,</span> pp. 339–359, 1909.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch597" id="fn597">597</a> -The following account is in part reprinted -from <i>Nature</i>, June 4, 1908.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch598" id="fn598">598</a> -In so far as our explanation involves a -shaping or moulding of the egg by the uterus or “oviduct” -(an agency supplemented by the proper tensions of the egg), -it is curious to note that this is very much the same as -that old view of Telesius regarding the formation of the -embryo (<i>De rerum natura</i>, <span class="smmaj">VI,</span> cc. 4 and 10), which -he had inherited from Galen, and of which Bacon speaks -(<i>Nov. Org.</i> cap. 50; cf. Ellis’s note). Bacon expressly -remarks that “Telesius should have been able to shew the -like formation in the shells of eggs.” This old theory of -embryonic modelling survives only in our usage of the term -“matrix” for a “mould.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch599" id="fn599">599</a> -<i>Journal of Tropical Medicine</i>, 15th June, -1911. I leave this paragraph as it was written, though -it is now once more asserted that the terminal and -lateral-spined eggs belong to separate and distinct species -of Bilharzia (Leiper, <i>Brit. Med. Journ.</i>, 18th March, -1916, p. 411).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch600" id="fn600">600</a> -Cf. Bashforth and Adams, <i>Theoretical Forms of -Drops, etc.</i>, Cambridge, 1883.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch601" id="fn601">601</a> -Woods, R. H., On a Physical Theorem applied to -tense Membranes, <i>Journ. of Anat. and Phys.</i> <span class="smmaj">XXVI,</span> -pp. 362–371, 1892. A similar investigation of the tensions -in the uterine wall, and of the varying thickness of -its muscles, was attempted by Haughton in his <i>Animal -Mechanics</i>, pp. 151–158, 1873.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch602" id="fn602">602</a> -This corresponds with a determination of the -normal pressures (in systole) by Krohl, as being in the -ratio of 1 : 6·8.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch603" id="fn603">603</a> -Cf. Schwalbe, G., Ueber Wechselbeziehungen -und ihr Einfluss auf die Gestaltung des Arteriensystem, -<i>Jen. Zeitschr.</i> <span class="smmaj">XII,</span> p. 267, 1878, Roux, Ueber -die Verzweigungen der Blutgefässen des Menschen, <i>ibid.</i> -<span class="smmaj">XII,</span> p. 205, 1878; Ueber die Bedeutung der -Ablenkung des Arterienstämmen bei der Astaufgabe, <i>ibid.</i> -<span class="smmaj">XIII,</span> p. 301, 1879; Hess, Walter, Eine mechanisch -bedingte Gesetzmässigkeit im Bau des Blutgefässsystems, -<i>A. f. Entw. Mech.</i> <span class="smmaj">XVI,</span> p. 632, 1903; Thoma, -R., <i>Ueber die Histogenese und Histomechanik des -Blutgefässsystems</i>, 1893.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch604" id="fn604">604</a> -<i>Essays</i>, etc., edited by Owen, -<span class="smmaj">I,</span> p. 134, 1861.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch605" id="fn605">605</a> -On the Functions of the Heart and Arteries, -<i>Phil. Trans.</i> 1809, pp. 1–31, cf. 1808, pp. 164–186; -<i>Collected Works</i>, <span class="smmaj">I,</span> pp. 511–534, 1855. The -same lesson is conveyed by all such work as that of -Volkmann, E. H. Weber and Poiseuille. Cf. Stephen Hales’ -<i>Statical Essays</i>, <span class="smmaj">II,</span> <i>Introduction</i>: “Especially -considering that they [i.e. animal Bodies] are in a manner -framed of one continued Maze of innumerable Canals, in -which Fluids are incessantly circulating, some with great -Force and Rapidity, others with very different Degrees of -rebated Velocity: Hence, <i>etc.</i>”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch606" id="fn606">606</a> -“Sizes” is Owen’s editorial emendation, which -seems amply justified.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch607" id="fn607">607</a> -For a more elaborate classification, into -colours cryptic, procryptic, anticryptic, apatetic, -epigamic, sematic, episematic, aposematic, etc., see -Poulton’s <i>Colours of Animals</i> (Int. Scientific Series, -<span class="nowrap"><span class="smmaj">LXVIII</span>),</span> -1890; cf. also Meldola, R., Variable -Protective Colouring in Insects, <i>P.Z.S.</i> 1873, pp. -153–162, etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch608" id="fn608">608</a> -Dendy, <i>Evolutionary Biology</i>, p. 336, 1912.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch609" id="fn609">609</a> -Delight in beauty is one of the pleasures of -the imagination; there is no limit to its indulgence, and -no end to the results which we may ascribe to its exercise. -But as for the particular “standard of beauty” which the -bird (for instance) admires and selects (as Darwin says -in the <i>Origin</i>, p. 70, edit. 1884), we are very much in -the dark, and we run the risk of arguing in a circle: for -wellnigh all we can safely say is what Addison says (in the -412th <i>Spectator</i>)—that each different species “is most -affected with the beauties of its own kind .... Hinc merula -in nigro se oblectat nigra marito; ... hinc noctua tetram -Canitiem alarum et glaucos miratur ocellos.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch610" id="fn610">610</a> -Cf. Bridge, T. W., <i>Cambridge Natural History</i> -(Fishes), <span class="smmaj">VII,</span> p. 173, 1904; also Frisch, K. v., -Ueber farbige Anpassung bei Fische, <i>Zool. Jahrb.</i> (<i>Abt. -Allg. Zool.</i>), <span class="smmaj">XXXII,</span> pp. 171–230, 1914.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch611" id="fn611">611</a> -<i>Nature</i>, <span class="smmaj">L,</span> p. 572; <span class="smmaj">LI,</span> pp. -33, 57, 533, 1894–95.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch612" id="fn612">612</a> -They are “wonderfully fitted for ‘vanishment’ -against the flushed, rich-coloured skies of early morning -and evening .... their chief feeding-times”; and “look like a -real sunset or dawn, repeated on the opposite side of the -heavens,—either east or west as the case may be”: Thayer, -<i>Concealing-coloration in the Animal Kingdom</i>, New York, -1909, pp. 154–155. This hypothesis, like the rest, is not -free from difficulty. Twilight is apt to be short in the -homes of the flamingo: and moreover, Mr Abel Chapman, who -watched them on the Guadalquivir, tells us that they <i>feed -by day</i>.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch613" id="fn613">613</a> -Principal Galloway, <i>Philosophy of Religion</i>, -p. 344, 1914.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch614" id="fn614">614</a> -Cf. Professor Flint, in his Preface to -Affleck’s translation of Janet’s <i>Causes finales</i>: “We are, -no doubt, still a long way from a mechanical theory of -organic growth, but it may be said to be the <i>quaesitum</i> of -modern science, and no one can say that it is a chimaera.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch615" id="fn615">615</a> -Cf. Sir Donald MacAlister, How a Bone is -Built, <i>Engl. Ill. Mag.</i> 1884.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch616" id="fn616">616</a> -Professor Claxton Fidler, <i>On Bridge -Construction</i>, p. 22 (4th ed.), 1909; cf. (<i>int. al.</i>) -Love’s <i>Elasticity</i>, p. 20 (<i>Historical Introduction</i>), 2nd -ed., 1906.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch617" id="fn617">617</a> -In preparing or “macerating” a skeleton, the -naturalist nowadays carries on the process till nothing -is left but the whitened bones. But the old anatomists, -whose object was not the study of “comparative” morphology -but the wider theme of comparative physiology, were wont -to macerate by easy stages; and in many of their most -instructive preparations, the ligaments were intentionally -left in connection with the bones, and as part of the -“skeleton.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch618" id="fn618">618</a> -In a few anatomical diagrams, for instance in -some of the drawings in Schmaltz’s <i>Atlas der Anatomie des -Pferdes</i>, we may see the system of “ties” diagrammatically -inserted in the figure of the skeleton. Cf. Gregory, On the -principles of Quadrupedal Locomotion, <i>Ann. N. Y. Acad. of -Sciences</i>, <span class="smmaj">XXII,</span> p. 289, 1912.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch619" id="fn619">619</a> -Galileo, <i>Dialogues concerning Two New -Sciences</i> (1638), Crew and Salvio’s translation, New York, -1914, p. 150; <i>Opere</i>, ed. Favaro, <span class="smmaj">VIII,</span> p. -186. Cf. Borelli, <i>De Motu Animalium</i>, <span class="smmaj">I,</span> prop. -<span class="smmaj">CLXXX,</span> 1685. Cf. also Camper, P., La structure des -os dans les oiseaux, <i>Opp.</i> <span class="smmaj">III,</span> p. 459, ed. 1803; -Rauber, A., Galileo über Knochenformen, <i>Morphol. Jahrb.</i> -<span class="smmaj">VII,</span> pp. 327, 328, 1881; Paolo Enriques, Della -economia di sostanza nelle osse cave, <i>Arch. f. Ent. Mech.</i> -<span class="smmaj">XX,</span> pp. 427–465, 1906.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch620" id="fn620">620</a> -<i>Das mechanische Prinzip. im -anatomischen Bau der Monocotylen</i>, Leipzig, 1874.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch621" id="fn621">621</a> -For further botanical illustrations, see -(<i>int. al.</i>) Hegler, Einfluss der Zugkraften auf die -Festigkeit und die Ausbildung mechanischer Gewebe in -Pflanzen, <i>SB. sächs. Ges. d. Wiss.</i> p. 638, 1891; Kny, -L., Einfluss von Zug und Druck auf die Richtung der -Scheidewande in sich teilenden Pflanzenzellen, <i>Ber. d. -bot. Gesellsch.</i> <span class="smmaj">XIV,</span> 1896; Sachs, Mechanomorphose -und Phylogenie, <i>Flora</i>, <span class="smmaj">LXXVIII,</span> 1894; cf. -also Pflüger, Einwirkung der Schwerkraft, etc., über die -Richtung der Zelltheilung, <i>Archiv</i>, <span class="smmaj">XXXIV,</span> 1884.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch622" id="fn622">622</a> -Among other works on the mechanical -construction of bone see: Bourgery, <i>Traité de l’anatomie</i> -(<i>I. Ostéologie</i>), 1832 (with admirable illustrations -of trabecular structure); Fick, L., <i>Die Ursachen -der Knochenformen</i>, Göttingen, 1857; Meyer, H., Die -Architektur der Spongiosa, <i>Archiv f. Anat. und Physiol.</i> -<span class="smmaj">XLVII,</span> pp. 615–628, 1867; <i>Statik u. Mechanik des -menschlichen Knochengerüstes</i>, Leipzig, 1873; Wolff, J., -Die innere Architektur der Knochen, <i>Arch. f. Anat, und -Phys.</i> <span class="smmaj">L,</span> 1870; <i>Das Gesetz der Transformation -bei Knochen</i>, 1892; von Ebner, V., Der feinere Bau der -Knochensubstanz, <i>Wiener Bericht</i>, <span class="smmaj">LXXII,</span> -1875; Rauber, Anton, <i>Elastizität und Festigkeit der -Knochen</i>, Leipzig, 1876; O. Meserer, <i>Elast, u. Festigk. -d. menschlichen Knochen</i>, Stuttgart, 1880; MacAlister, -Sir Donald, How a Bone is Built, <i>English Illustr. -Mag.</i> pp. 640–649, 1884; Rasumowsky, Architektonik -des Fussskelets, <i>Int. Monatsschr. f. -Anat.</i> p. 197, 1889; Zschokke, <i>Weitere Unters. über das -Verhältniss der Knochenbildung zur Statik und Mechanik -des Vertebratenskelets</i>, Zürich, 1892; Roux, W., <i>Ges. -Abhandlungen über Entwicklungsmechanik der Organismen, Bd. -I, Funktionelle Anpassung</i>, Leipzig, 1895; Triepel, H., -Die Stossfestigkeit der Knochen, <i>Arch. f. Anat. u. Phys.</i> -1900; Gebhardt, Funktionell wichtige Anordnungsweisen der -feineren und gröberen Bauelemente des Wirbelthierknochens, -etc., <i>Arch. f. Entw. Mech.</i> 1900–1910; Kirchner. A., -Architektur der Metatarsalien, <i>A. f. E. M.</i> <span class="smmaj">XXIV,</span> 1907; -Triepel, Herm., Die trajectorielle Structuren (in -<i>Einf. in die Physikalische Anatomie</i>, 1908); Dixon, A. -F., Architecture of the Cancellous Tissue forming the -Upper End of the Femur, <i>Journ. of Anat. and Phys.</i> (3) -<span class="smmaj">XLIV,</span> pp. 223–230, 1910.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch623" id="fn623">623</a> -Sédillot, De l’influence des fonctions sur la -structure et la forme des organes; <i>C. R.</i> <span class="smmaj">LIX,</span> p. -539, 1864; cf. <span class="smmaj">LX,</span> p. 97, 1865, <span class="smmaj">LXVIII.</span> -p. 1444. 1869.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch624" id="fn624">624</a> -E.g. (1) the head, nodding backwards and -forwards on a fulcrum, represented by the atlas vertebra, -lying between the weight and the power; (2) the foot, -raising on tip-toe the weight of the body against the -fulcrum of the ground, where the weight is between the -fulcrum and the power, the latter being represented by the -<i>tendo Achillis</i>; (3) the arm, lifting a weight in the -hand, with the power (i.e. the biceps muscle) between the -fulcrum and the weight. (The second case, by the way, has -been much disputed; cf. Haycraft in Schäfer’s <i>Textbook of -Physiology</i>, p. 251, 1900.)</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch625" id="fn625">625</a> -Our problem is analogous to Dr Thomas Young’s -problem of the best disposition of the timbers in a wooden -ship (<i>Phil. Trans.</i> 1814, p. 303). He was not long of -finding that the forces which may act upon the fabric are -very numerous and very variable, and that the best mode of -resisting them, or best structural arrangement for ultimate -strength, becomes an immensely complicated problem.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch626" id="fn626">626</a> -In like manner, Clerk Maxwell could not help -employing the term “skeleton” in defining the mathematical -conception of a “frame,” constituted by points and their -interconnecting lines: in studying the equilibrium of -which, we consider its different points as mutually acting -on each other with forces whose directions are those of the -lines joining each pair of points. Hence (says Maxwell), -“in order to exhibit the mechanical action of the frame in -the most elementary manner, we may draw it as a <i>skeleton</i>, -in which the different points are joined by straight lines, -and we may indicate by numbers attached to these lines the -tensions or compressions in the corresponding pieces of -the frame” (<i>Trans. R. S. E.</i> <span class="smmaj">XXVI,</span> p. 1, 1870). -It follows that the diagram so constructed represents a -“diagram of forces,” in this limited sense that it is -geometrical as regards the position and direction of the -forces, but arithmetical as regards their magnitude. It is -to just such a diagram that the animal’s skeleton tends to -approximate.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch627" id="fn627">627</a> -When the jockey crouches over the neck -of his race-horse, and when Tod Sloan introduced the -“American seat,” the object in both cases is to relieve the -hind-legs of weight, and so leave them free for the work of -propulsion. Nevertheless, we must not exaggerate the share -taken by the hind-limbs in this latter duty; cf. Stillman, -<i>The Horse in Motion</i>, p. 69, 1882.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch628" id="fn628">628</a> -This and the following diagrams are borrowed -and adapted from Professor Fidler’s <i>Bridge Construction</i>.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch629" id="fn629">629</a> -The method of constructing <i>reciprocal -diagrams</i>, in which one should represent the outlines of -a frame, and the other the system of forces necessary to -keep it in equilibrium, was first indicated in Culmann’s -<i>Graphische Statik</i>; it was greatly developed soon -afterwards by Macquorn Rankine (<i>Phil. Mag.</i> Feb. 1864, -and <i>Applied Mechanics</i>, passim), to whom is mainly due -the general application of the principle to engineering -practice.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch630" id="fn630">630</a> -<i>Dialogues concerning Two New Sciences</i> -(1638): Crew and Salvio’s translation, p. 140 <i>seq.</i></p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch631" id="fn631">631</a> -The form and direction of the vertebral -spines have been frequently and elaborately described; cf. -(e.g.) Gottlieb, H., Die Anticlinie der Wirbelsäule der -Säugethiere, <i>Morphol. Jahrb.</i> <span class="smmaj">LXIX,</span> pp. 179–220, -1915, and many works quoted therein. According to Morita, -Ueber die Ursachen der Richtung und Gestalt der thoracalen -Dornfortsätze der Säugethierwirbelsäule (<i>ibi cit.</i> p. -201), various changes take place in the direction or -inclination of these processes in rabbits, after section of -the interspinous ligaments and muscles. These changes seem -to be very much what we should expect, on simple mechanical -grounds. See also Fischer, O., <i>Theoretische Grundlagen für -eine Mechanik der lebenden Körper</i>, Leipzig, pp. 3, 372, -1906.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch632" id="fn632">632</a> -I owe the first four of these determinations -to the kindness of Dr Chalmers Mitchell, who had them made -for me at the Zoological Society’s Gardens; while the great -Clydesdale carthorse was weighed for me by a friend in -Dundee.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch633" id="fn633">633</a> -This pose of Diplodocus, and of other -Sauropodous reptiles, has been much discussed. Cf. (<i>int. -al.</i>) Abel, O., <i>Abh. k. k. zool. bot. Ges. Wien</i>, -<span class="smmaj">V.</span> 1909–10 (60 pp.); Tornier, <i>SB. Ges. Naturf. -Fr. Berlin</i>, pp. 193–209, 1909; Hay, O. P., <i>Amer. Nat.</i> -Oct. 1908; <i>Tr. Wash. Acad. Sci.</i> <span class="smmaj">XLII,</span> pp. 1–25, -1910; Holland, <i>Amer. Nat.</i> May, 1910, pp. -259–283; Matthew, <i>ibid.</i> pp. 547–560; Gilmore, C. W. -(<i>Restoration of Stegosaurus</i>). <i>Pr. U.S. Nat. Museum</i>, -1915.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch634" id="fn634">634</a> -The form of the cantilever is much less -typical in the small flying birds, where the strength of -the pelvic region is insured in another way, with which we -need not here stop to deal.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch635" id="fn635">635</a> -The motto was Macquorn Rankine’s.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch636" id="fn636">636</a> -John Hunter was seldom wrong; but I cannot -believe that he was right when he said (<i>Scientific Works</i>, -ed. Owen, <span class="smmaj">I,</span> p. 371), “The bones, in a mechanical -view, appear to be the first that are to be considered. -We can study their shape, connexions, number, uses, etc., -<i>without considering any other part of the body</i>.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch637" id="fn637">637</a> -<i>Origin of Species</i>, 6th ed. p. 118.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch638" id="fn638">638</a> -<i>Amer. Naturalist</i>, April, 1915, p. 198, etc. -Cf. <i>infra</i>, p. 727.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch639" id="fn639">639</a> -Driesch sees in “Entelechy” that something -which differentiates the whole from the sum of its parts -in the case of the organism: “The organism, we know, is -a system the single constituents of which are inorganic -in themselves; only the whole constituted by them in -their typical order or arrangement owes its specificity -to ‘Entelechy’ ” (<i>Gifford Lectures</i>, p. 229, 1908): and -I think it could be shewn that many other philosophers -have said precisely the same thing. So far as the argument -goes, I fail to see how <i>this</i> Entelechy is shewn to -be peculiarly or specifically related to the <i>living</i> -organism. The conception that the whole is <i>always</i> -something very different from its parts is a very ancient -doctrine. The reader will perhaps remember how, in another -vein, the theme is treated by Martinus Scriblerus: “In -every Jack there is a <i>meat-roasting</i> Quality, which -neither resides in the fly, nor in the weight, nor in any -particular wheel of the Jack, but is the result of the -whole composition; etc., etc.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch640" id="fn640">640</a> -“There can be no doubt that Fraas is correct -in regarding this type (<i>Procetus</i>) as an annectant form -between the Zeuglodonts and the Creodonta, but, although -the origin of the Zeuglodonts is thus made clear, it still -seems to be by no means so certain as that author believes, -that they may not themselves be the ancestral forms of the -Odontoceti”; Andrews, <i>Tertiary Vertebrata of the Fayum</i>, -1906, p. 235.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch641" id="fn641">641</a> -Reprinted, with some changes and additions, -from a paper in the <i>Trans. Roy. Soc. Edin.</i> <span class="smmaj">L,</span> -pp. 857–95, 1915.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch642" id="fn642">642</a> -M. Bergson repudiates, with peculiar -confidence, the application of mathematics to biology. -Cf. <i>Creative Evolution</i>, p. 21, “Calculation touches, at -most, certain phenomena of organic destruction. Organic -creation, on the contrary, the evolutionary phenomena which -properly constitute life, we cannot in any way subject to a -mathematical treatment.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch643" id="fn643">643</a> -In this there lies a certain justification -for a saying of Minot’s, of the greater part of which, -nevertheless, I am heartily inclined to disapprove. “We -biologists,” he says, “cannot deplore too frequently or too -emphatically the great mathematical delusion by which men -often of great if limited ability have been misled into -becoming advocates of an erroneous conception of accuracy. -The delusion is that no science is accurate until its -results can be expressed mathematically. The error comes -from the assumption that mathematics can express complex -relations. Unfortunately mathematics have a very limited -scope, and are based upon a few extremely rudimentary -experiences, which we make as very little children and -of which no adult has any recollection. The fact that -from this basis men of genius have evolved wonderful -methods of dealing with numerical relations should not -blind us to another fact, namely, that the observational -basis of mathematics is, psychologically speaking, very -minute compared with the observational basis of even a -single minor branch of biology .... While therefore here -and there the mathematical methods may aid us, <i>we need -a kind and degree of accuracy of which mathematics is -absolutely incapable</i> .... With human minds constituted as -they actually are, we cannot anticipate that there will -ever be a mathematical expression for any organ or even a -single cell, although formulae will continue to be useful -for dealing now and then with isolated details...” (<i>op. -cit.</i>, p. 19, 1911). It were easy to discuss and criticise -these sweeping assertions, which perhaps had their origin -and parentage in an <i>obiter dictum</i> of Huxley’s, to the -effect that “Mathematics is that study which knows nothing -of observation, nothing of experiment, -nothing of induction, nothing of causation” (<i>cit.</i> Cajori, -<i>Hist of Elem. Mathematics</i>, p. 283). But Gauss called -mathematics “a science of the eye”; and Sylvester assures -us that “most, if not all, of the great ideas of modern -mathematics have had their origin in observation” (<i>Brit. -Ass. Address</i>, 1869, and <i>Laws of Verse</i>, p. 120, 1870).</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch644" id="fn644">644</a> -<i>Historia Animalium</i> <span class="smmaj">I,</span> 1.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch645" id="fn645">645</a> -Cf. <i>supra</i>, p. 714.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch646" id="fn646">646</a> -Cf. Osborn, H. F., On the Origin of Single -Characters, as observed in fossil and living Animals and -Plants, <i>Amer. Nat.</i> <span class="smmaj">XLIX,</span> pp. 193–239, 1915 (and -other papers); <i>ibid.</i> p. 194, “Each individual is composed -of a vast number of somewhat similar new or old characters, -each character has its independent and separate history, -each character is in a certain stage of evolution, each -character is correlated with the other characters of the -individual .... The real problem has always been that of the -origin and development of characters. Since the <i>Origin of -Species</i> appeared, the terms variation and variability have -always referred to single characters; if a species is said -to be variable, we mean that a considerable number of the -single characters or groups of characters of which it is -composed are variable,” etc.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch647" id="fn647">647</a> -Cf. Sorby, <i>Quart. Journ. Geol. Soc.</i> -(<i>Proc.</i>), 1879, p. 88.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch648" id="fn648">648</a> -Cf. D’Orbigny, Alc., <i>Cours élém. de -Paléontologie</i>, etc., <span class="smmaj">I,</span> pp. 144–148, 1849; -see also Sharpe, Daniel, On Slaty Cleavage, <i>Q.J.G.S.</i> -<span class="smmaj">III,</span> p. 74, 1847.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch649" id="fn649">649</a> -Thus <i>Ammonites erugatus</i>, when compressed, -has been described as <i>A. planorbis</i>: cf. Blake, J. F., -<i>Phil. Mag.</i> (5), <span class="smmaj">VI,</span> p. 260, 1878. Wettstein has -shewn that several species of the fish-genus <i>Lepidopus</i> -have been based on specimens artificially deformed -in various ways: Ueber die Fischfauna des Tertiären -Glarnerschiefers, <i>Abh. Schw. Palaeont. Gesellsch.</i> -<span class="smmaj">XIII,</span> 1886 (see especially pp. 23–38, pl. -<span class="nowrap"><span class="smmaj">I</span>).</span> The whole subject, interesting as it is, has -been little studied: both Blake and Wettstein deal with it -mathematically.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch650" id="fn650">650</a> -Cf. Sir Thomas Browne, in <i>The Garden of -Cyrus</i>: “But why ofttimes one side of the leaf is unequall -unto the other, as in Hazell and Oaks, why on either side -the master vein the lesser and derivative channels stand -not directly opposite, nor at equall angles, respectively -unto the adverse side, but those of one side do often -exceed the other, as the Wallnut and many more, deserves -another enquiry.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch651" id="fn651">651</a> -Where gourds are common, the glass-blower is -still apt to take them for a prototype, as the prehistoric -potter also did. For instance, a tall, annulated Florence -oil-flask is an exact but no longer a conscious imitation -of a gourd which has been converted into a bottle in the -manner described.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch652" id="fn652">652</a> -Cf. <i>Elsie Venner</i>, chap. ii.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch653" id="fn653">653</a> -This significance is particularly remarkable -in connection with the development of speed, for the -metacarpal region is the seat of very important leverage -in the propulsion of the body. In the Museum of the Royal -College of Surgeons in Edinburgh, there stand side by -side the skeleton of an immense carthorse (celebrated for -having drawn all the stones of the Bell Rock Lighthouse -to the shore), and a beautiful skeleton of a racehorse, -which (though the fact is disputed) there is good reason to -believe is the actual skeleton of Eclipse. When I was a boy -my grandfather used to point out to me that the cannon-bone -of the little racer is not only relatively, but actually, -longer than that of the great Clydesdale.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch654" id="fn654">654</a> -Cf. Vitruvius, <span class="smmaj">III,</span> 1.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch655" id="fn655">655</a> -<i>Les quatres livres d’Albert Dürer de la -proportion des parties et pourtraicts des corps humains</i>, -Arnheim, 1613, folio (and earlier editions). Cf. also -Lavater, <i>Essays on Physiognomy</i>, <span class="smmaj">III,</span> p. 271, -1799.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch656" id="fn656">656</a> -It was these very drawings of Dürer’s that -gave to Peter Camper his notion of the “facial angle.” -Camper’s method of comparison was the very same as ours, -save that he only drew the axes, without filling in the -network, of his coordinate system; he saw clearly the -essential fact, that the skull <i>varies as a whole</i>, -and that the “facial angle” is the index to a general -deformation. “The great object was to shew that natural -differences might be reduced to rules, of which the -direction of the facial line forms the <i>norma</i> or canon; -and that these directions and inclinations are always -accompanied by correspondent form, size and position of -the other parts of the cranium,” etc.; from Dr T. Cogan’s -preface to Camper’s work <i>On the Connexion between the -Science of Anatomy and the Arts of Drawing, Painting and -Sculpture</i> (1768?), quoted in Dr R. Hamilton’s Memoir of -Camper, in <i>Lives of Eminent Naturalists</i> (<i>Nat. Libr.</i>), -Edin. 1840.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch657" id="fn657">657</a> -The co-ordinate system of Fig. <a href="#fig382" title="go to Fig. 382">382</a> is somewhat -different from that which I drew and published in my former -paper. It is not unlikely that further investigation will -further simplify the comparison, and shew it to involve a -still more symmetrical system.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch658" id="fn658">658</a> -<i>Dinosaurs of North America</i>, pl. <span class="smmaj">LXXXI,</span> etc. 1896.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch659" id="fn659">659</a> -<i>Mem. Amer. Mus. of Nat. Hist.</i> <span class="smmaj">I,</span> -<span class="smmaj">III,</span> 1898.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch660" id="fn660">660</a> -These and also other coordinate diagrams will -be found in Mr G. Heilmann’s book <i>Fuglenes Afstamning</i>, -398 pp., Copenhagen, 1916; see especially pp. 368–380.</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch661" id="fn661">661</a> -Cf. W. B. Scott (<i>Amer. Journ. of Science</i>, -<span class="smmaj">XLVIII,</span> pp. 335–374, 1894), “We find that any -mammalian series at all complete, such as that of the -horses, is remarkably continuous, and that the progress -of discovery is steadily filling up what few gaps remain. -So closely do successive stages follow upon one another -that it is sometimes extremely difficult to arrange them -all in order, and to distinguish clearly those members -which belong in the main line of descent, and those which -represent incipient branches. Some phylogenies actually -suffer from an embarrassment of riches.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch662" id="fn662">662</a> -Cf. Dwight, T., The Range of Variation of the -Human Scapula, <i>Amer. Nat.</i> <span class="smmaj">XXI,</span> pp. 627–638, -1887. Cf. also Turner, <i>Challenger Rep.</i> <span class="smmaj">XLVII,</span> -on Human Skeletons, p. 86, 1886: “I gather both from my -own measurements, and those of other observers, that the -range of variation in the relative length and breadth of -the scapula is very considerable in the same race, so that -it needs a large number of bones to enable one to obtain an -accurate idea of the mean of the race.”</p></div> - -<div class="dftnt"> -<p class="pcontinue"> -<a class="afnlabel" href="#fnanch663" id="fn663">663</a> -There is a paper on the mathematical study -of organic forms and organic processes by the learned -and celebrated Gustav Theodor Fechner, which I have only -lately read, but which would have been of no little use -and help to our argument had I known it before. (Ueber -die mathematische Behandlung organischer Gestalten und -Processe, <i>Berichte d. k. sächs. Gesellsch.</i>, <i>Math.-phys. -Cl.</i>, Leipzig, 1849, pp. 50–64.) Fechner’s treatment is -more purely mathematical and less physical in its scope and -bearing than ours, and his paper is but a short one; but -the conclusions to which he is led differ little from our -own. Let me quote a single sentence which, together with -its context, runs precisely on the lines of the discussion -with which this chapter of ours began. “So ist also die -mathematische Bestimmbarkeit im Gebiete des Organischen -ganz eben so gut vorhanden als in dem des Unorganischen, -und in letzterem eben solchen oder äquivalenten -Beschränkungen unterworfen als in ersterem; und nur sofern -die unorganischen Formen und das unorganische Geschehen -sich einer einfacheren Gesetzlichkeit mehr nähern als die -organischen, kann die Approximation im unorganischen Gebiet -leichter und weiter getrieben werden als im organischen. -Dies wäre der ganze, sonach rein relative, Unterschied.” -Here in a nutshell, in words written some seventy years -ago, is the gist of the whole matter.</p> - -<p>An interesting little book of Schiaparelli’s (which I ought -to have known long ago)—<i>Forme organiche naturali e forme -geometriche pure</i>, Milano, Hoepli, 1898—has likewise come -into my hands too late for discussion.</p></div> - -<div class="chapter" id="p780"> -<h2 class="h2herein" title="Index.">INDEX.</h2></div> - -<ul class="ulindx"> <li class="liindx">Abbe’s diffraction -plates, <a class="aindx" href="#p323" title="go to pg. -323">323</a></li> - -<li class="liindx">Abel, O., <a class="aindx" href="#p706" -title="go to pg. 706">706</a></li> - -<li class="liindx">Abonyi, A., <a class="aindx" -href="#p127" title="go to pg. 127">127</a></li> - -<li class="liindx">Acantharia, spicules of, <a -class="aindx" href="#p458" title="go to pg. -458">458</a></li> - -<li class="liindx">Acanthometridae, <a class="aindx" -href="#p462" title="go to pg. 462">462</a></li> - -<li class="liindx">Acceleration, <a class="aindx" -href="#p064" title="go to pg. 64">64</a></li> - -<li class="liindx">Aceratherium, <a class="aindx" -href="#p761" title="go to pg. 761">761</a></li> - -<li class="liindx">Achlya, <a class="aindx" href="#p244" -title="go to pg. 244">244</a></li> - -<li class="liindx">Acromegaly, <a class="aindx" -href="#p135" title="go to pg. 135">135</a></li> - -<li class="liindx">Actinomma, <a class="aindx" href="#p469" -title="go to pg. 469">469</a></li> - -<li class="liindx">Actinomyxidia, <a class="aindx" -href="#p452" title="go to pg. 452">452</a></li> - -<li class="liindx">Actinophrys, <a class="aindx" -href="#p165" title="go to pg. 165">165</a>, <a -class="aindx" href="#p197" title="go to pg. 197">197</a>, -<a class="aindx" href="#p264" title="go to pg. -264">264</a>, <a class="aindx" href="#p298" title="go to -pg. 298">298</a></li> - -<li class="liindx">Actinosphaerium, <a class="aindx" -href="#p197" title="go to pg. 197">197</a>, <a -class="aindx" href="#p266" title="go to pg. 266">266</a>, -<a class="aindx" href="#p298" title="go to pg. -298">298</a>, <a class="aindx" href="#p468" title="go to -pg. 468">468</a></li> - -<li class="liindx">Adams, J. C., <a class="aindx" -href="#p663" title="go to pg. 663">663</a></li> - -<li class="liindx">Adaptation, <a class="aindx" -href="#p670" title="go to pg. 670">670</a></li> - -<li class="liindx">Addison, Joseph, <a class="aindx" -href="#p671" title="go to pg. 671">671</a></li> - -<li class="liindx">Adiantum, <a class="aindx" href="#p408" -title="go to pg. 408">408</a></li> - -<li class="liindx">Adsorption, <a class="aindx" -href="#p192" title="go to pg. 192">192</a>, <a -class="aindx" href="#p208" title="go to pg. 208">208</a>, -<a class="aindx" href="#p241" title="go to pg. -241">241</a>, <a class="aindx" href="#p277" title="go to -pg. 277">277</a>, <a class="aindx" href="#p357" title="go -to pg. 357">357</a>; <ul> <li class="liindx">orientirte, -<a class="aindx" href="#p440" title="go to pg. -440">440</a>, <a class="aindx" href="#p590" title="go -to pg. 590">590</a>;</li> <li class="liindx">pseudo, -<a class="aindx" href="#p282" title="go to pg. -282">282</a></li> </ul></li> - -<li class="liindx">Agglutination, <a class="aindx" -href="#p201" title="go to pg. 201">201</a></li> - -<li class="liindx">Aglaophenia, <a class="aindx" -href="#p748" title="go to pg. 748">748</a></li> - -<li class="liindx">Airy, H., <a class="aindx" href="#p636" -title="go to pg. 636">636</a></li> - -<li class="liindx">Albumin molecule, <a class="aindx" -href="#p041" title="go to pg. 41">41</a></li> - -<li class="liindx">Alcyonaria, <a class="aindx" -href="#p387" title="go to pg. 387">387</a>, <a -class="aindx" href="#p413" title="go to pg. 413">413</a>, -<a class="aindx" href="#p424" title="go to pg. -424">424</a>, <a class="aindx" href="#p459" title="go to -pg. 459">459</a></li> - -<li class="liindx">Alexeieff, A., <a class="aindx" -href="#p157" title="go to pg. 157">157</a>, <a -class="aindx" href="#p165" title="go to pg. -165">165</a></li> - -<li class="liindx">Allmann, W., <a class="aindx" -href="#p643" title="go to pg. 643">643</a></li> - -<li class="liindx">Alpheus, claws of, <a class="aindx" -href="#p150" title="go to pg. 150">150</a></li> - -<li class="liindx">Alpine plants, <a class="aindx" -href="#p124" title="go to pg. 124">124</a></li> - -<li class="liindx">Altmann’s granules, <a class="aindx" -href="#p285" title="go to pg. 285">285</a></li> - -<li class="liindx">Alveolar meshwork, <a class="aindx" -href="#p170" title="go to pg. 170">170</a></li> - -<li class="liindx">Ammonites, <a class="aindx" href="#p526" -title="go to pg. 526">526</a>, <a class="aindx" -href="#p530" title="go to pg. 530">530</a>, <a -class="aindx" href="#p537" title="go to pg. 537">537</a>, -<a class="aindx" href="#p539" title="go to pg. -539">539</a>, <a class="aindx" href="#p550" title="go to -pg. 550">550</a>, <a class="aindx" href="#p552" title="go -to pg. 552">552</a>, <a class="aindx" href="#p576" -title="go to pg. 576">576</a>, <a class="aindx" -href="#p583" title="go to pg. 583">583</a>, <a -class="aindx" href="#p584" title="go to pg. 584">584</a>, -<a class="aindx" href="#p728" title="go to pg. -728">728</a></li> - -<li class="liindx">Amoeba, <a class="aindx" href="#p012" -title="go to pg. 12">12</a>, <a class="aindx" href="#p165" -title="go to pg. 165">165</a>, <a class="aindx" -href="#p209" title="go to pg. 209">209</a>, <a -class="aindx" href="#p212" title="go to pg. 212">212</a>, -<a class="aindx" href="#p245" title="go to pg. -245">245</a>, <a class="aindx" href="#p255" title="go to -pg. 255">255</a>, <a class="aindx" href="#p288" title="go -to pg. 288">288</a>, <a class="aindx" href="#p463" -title="go to pg. 463">463</a>, <a class="aindx" -href="#p605" title="go to pg. 605">605</a></li> - -<li class="liindx">Amphidiscs, <a class="aindx" -href="#p440" title="go to pg. 440">440</a></li> - -<li class="liindx">Amphioxus, <a class="aindx" href="#p311" -title="go to pg. 311">311</a></li> - -<li class="liindx">Ampullaria, <a class="aindx" -href="#p560" title="go to pg. 560">560</a></li> - -<li class="liindx">Anabaena, <a class="aindx" href="#p300" -title="go to pg. 300">300</a></li> - -<li class="liindx">Anaxagoras, <a class="aindx" -href="#p008" title="go to pg. 8">8</a></li> - -<li class="liindx">Ancyloceras, <a class="aindx" -href="#p550" title="go to pg. 550">550</a></li> - -<li class="liindx">Andrews, G. F., <a class="aindx" -href="#p164" title="go to pg. 164">164</a>; <ul> <li -class="liindx">C. W., <a class="aindx" href="#p716" -title="go to pg. 716">716</a></li></ul></li> - -<li class="liindx">Anhydrite, <a class="aindx" href="#p433" -title="go to pg. 433">433</a></li> - -<li class="liindx">Anikin, W. P., <a class="aindx" -href="#p130" title="go to pg. 130">130</a></li> - -<li class="liindx">Anisonema, <a class="aindx" href="#p126" -title="go to pg. 126">126</a></li> - -<li class="liindx">Anisotropy, <a class="aindx" -href="#p241" title="go to pg. 241">241</a>, <a -class="aindx" href="#p357" title="go to pg. -357">357</a></li> - -<li class="liindx">Anomia, <a class="aindx" href="#p565" -title="go to pg. 565">565</a>, <a class="aindx" -href="#p567" title="go to pg. 567">567</a></li> - -<li class="liindx">Antelopes, horns of, <a class="aindx" -href="#p614" title="go to pg. 614">614</a>, <a -class="aindx" href="#p671" title="go to pg. -671">671</a></li> - -<li class="liindx">Antheridia, <a class="aindx" -href="#p303" title="go to pg. 303">303</a>, <a -class="aindx" href="#p403" title="go to pg. 403">403</a>, -<a class="aindx" href="#p405" title="go to pg. -405">405</a>, <a class="aindx" href="#p409" title="go to -pg. 409">409</a></li> - -<li class="liindx">Anthoceros, spore of, <a class="aindx" -href="#p397" title="go to pg. 397">397</a></li> - -<li class="liindx">Anthogorgia, spicules of, -<a class="aindx" href="#p413" title="go to pg. -413">413</a></li> - -<li class="liindx">Anthropometry, <a class="aindx" -href="#p051" title="go to pg. 51">51</a></li> - -<li class="liindx">Anticline, <a class="aindx" href="#p360" -title="go to pg. 360">360</a></li> - -<li class="liindx">Antigonia, <a class="aindx" href="#p750" -title="go to pg. 750">750</a>, <a class="aindx" -href="#p775" title="go to pg. 775">775</a></li> - -<li class="liindx">Antlers, <a class="aindx" href="#p628" -title="go to pg. 628">628</a></li> - -<li class="liindx">Apatornis, <a class="aindx" href="#p757" -title="go to pg. 757">757</a></li> - -<li class="liindx">Apocynum, pollen of, <a class="aindx" -href="#p396" title="go to pg. 396">396</a></li> - -<li class="liindx">Aptychus, <a class="aindx" href="#p576" -title="go to pg. 576">576</a></li> - -<li class="liindx">Arachnoidiscus, <a class="aindx" -href="#p387" title="go to pg. 387">387</a></li> - -<li class="liindx">Arachnophyllum, <a class="aindx" -href="#p325" title="go to pg. 325">325</a></li> - -<li class="liindx">Arcella, <a class="aindx" href="#p323" -title="go to pg. 323">323</a></li> - -<li class="liindx">Arcestes, <a class="aindx" href="#p539" -title="go to pg. 539">539</a>, <a class="aindx" -href="#p540" title="go to pg. 540">540</a></li> - -<li class="liindx">Archaeopteryx, <a class="aindx" -href="#p757" title="go to pg. 757">757</a></li> - -<li class="liindx">Archimedes, <a class="aindx" -href="#p580" title="go to pg. 580">580</a>; <ul> -<li class="liindx">spiral of, <a class="aindx" -href="#p503" title="go to pg. 503">503</a>, <a -class="aindx" href="#p524" title="go to pg. 524">524</a>, -<a class="aindx" href="#p552" title="go to pg. -552">552</a></li> </ul></li> - -<li class="liindx">Argali, horns of, <a class="aindx" -href="#p617" title="go to pg. 617">617</a></li> - -<li class="liindx">Argiope, <a class="aindx" href="#p561" -title="go to pg. 561">561</a></li> - -<li class="liindx">Argonauta, <a class="aindx" href="#p546" -title="go to pg. 546">546</a>, <a class="aindx" -href="#p561" title="go to pg. 561">561</a></li> - -<li class="liindx">Argus pheasant, <a class="aindx" -href="#p431" title="go to pg. 431">431</a>, <a -class="aindx" href="#p631" title="go to pg. -631">631</a></li> - -<li class="liindx">Argyropelecus, <a class="aindx" -href="#p748" title="go to pg. 748">748</a></li> - -<li class="liindx">Aristotle, <a class="aindx" href="#p003" -title="go to pg. 3">3</a>, <a class="aindx" href="#p004" -title="go to pg. 4">4</a>, <a class="aindx" href="#p005" -title="go to pg. 5">5</a>, <a class="aindx" href="#p008" -title="go to pg. 8">8</a>, <a class="aindx" href="#p015" -title="go to pg. 15">15</a>, <a class="aindx" href="#p138" -title="go to pg. 138">138</a>, <a class="aindx" -href="#p149" title="go to pg. 149">149</a>, <a -class="aindx" href="#p158" title="go to pg. 158">158</a>, -<a class="aindx" href="#p509" title="go to pg. -509">509</a>, <a class="aindx" href="#p653" title="go to -pg. 653">653</a>, <a class="aindx" href="#p714" title="go -to pg. 714">714</a>, <a class="aindx" href="#p725" -title="go to pg. 725">725</a>, <a class="aindx" -href="#p726" title="go to pg. 726">726</a></li> - -<li class="liindx">Arizona trees, <a class="aindx" -href="#p121" title="go to pg. 121">121</a></li> - -<li class="liindx">Arrhenius, Sv., <a class="aindx" -href="#p028" title="go to pg. 28">28</a>, <a class="aindx" -href="#p048" title="go to pg. 48">48</a>, <a class="aindx" -href="#p171" title="go to pg. 171">171</a></li> - -<li class="liindx">Artemia, <a class="aindx" href="#p127" -title="go to pg. 127">127</a></li> - -<li class="liindx">Artemis, <a class="aindx" href="#p561" -title="go to pg. 561">561</a></li> - -<li class="liindx">Ascaris megalocephala, <a -class="aindx" href="#p180" title="go to pg. 180">180</a>, -<a class="aindx" href="#p195" title="go to pg. -195">195</a></li> - -<li class="liindx">Aschemonella, <a class="aindx" -href="#p255" title="go to pg. 255">255</a></li> - -<li class="liindx">Assheton, R., <a class="aindx" -href="#p344" title="go to pg. 344">344</a></li> - -<li class="liindx">Asterina, <a class="aindx" href="#p342" -title="go to pg. 342">342</a></li> - -<li class="liindx">Asteroides, <a class="aindx" -href="#p423" title="go to pg. 423">423</a></li> - -<li class="liindx">Asterolampra, <a class="aindx" -href="#p386" title="go to pg. 386">386</a></li> - -<li class="liindx">Asters, <a class="aindx" href="#p167" -title="go to pg. 167">167</a>, <a class="aindx" -href="#p174" title="go to pg. 174">174</a></li> - -<li class="liindx">Asthenosoma, <a class="aindx" -href="#p664" title="go to pg. 664">664</a></li> - -<li class="liindx">Astrorhiza, <a class="aindx" -href="#p255" title="go to pg. 255">255</a>, <a -class="aindx" href="#p463" title="go to pg. 463">463</a>, -<a class="aindx" href="#p587" title="go to pg. -587">587</a>, <a class="aindx" href="#p607" title="go to -pg. 607">607</a></li> - -<li class="liindx">Astrosclera, <a class="aindx" -href="#p436" title="go to pg. 436">436</a></li> - -<li class="liindx">Asymmetric substances, <a class="aindx" -href="#p416" title="go to pg. 416">416</a></li> - -<li class="liindx">Asymmetry, <a class="aindx" href="#p241" -title="go to pg. 241">241</a></li> - -<li class="liindx">Atrypa, <a class="aindx" href="#p569" -title="go to pg. 569">569</a></li> - -<li class="liindx">Auerbach, F., <a class="aindx" -href="#p009" title="go to pg. 9">9</a></li> - -<li class="liindx">Aulacantha, <a class="aindx" -href="#p460" title="go to pg. 460">460</a></li> - -<li class="liindx">Aulastrum, <a class="aindx" href="#p471" -title="go to pg. 471">471</a></li> - -<li class="liindx">Aulonia, <a class="aindx" href="#p468" -title="go to pg. 468">468</a></li> - -<li class="liindx">Auricular height, <a class="aindx" -href="#p093" title="go to pg. 93">93</a></li> - -<li class="liindx">Autocatalysis, <a class="aindx" -href="#p131" title="go to pg. 131">131</a></li> - -<li class="liindx">Auximones, <a class="aindx" href="#p135" -title="go to pg. 135">135</a></li> - -<li class="liindx">Awerinzew, S., <a class="aindx" -href="#p589" title="go to pg. 589">589</a></li></ul> - -<ul class="ulindx"> <li class="liindx">Babak, E., <a -class="aindx" href="#p032" title="go to pg. 32">32</a></li> - -<li class="liindx">Babirussa, teeth of, <a class="aindx" -href="#p634" title="go to pg. 634">634</a></li> - -<li class="liindx">Baboon, skull of, <a class="aindx" -href="#p771" title="go to pg. 771">771</a></li> - -<li class="liindx">Bacillus, <a class="aindx" href="#p039" -title="go to pg. 39">39</a>; <ul> <li class="liindx">B. -ramosus, <a class="aindx" href="#p133" title="go to pg. -133">133</a></li> </ul></li> - -<li class="liindx">Bacon, Lord, <a class="aindx" -href="#p004" title="go to pg. 4">4</a>, <a class="aindx" -href="#p005" title="go to pg. 5">5</a>, <a class="aindx" -href="#p051" title="go to pg. 51">51</a>, <a class="aindx" -href="#p053" title="go to pg. 53">53</a>, <a class="aindx" -href="#p131" title="go to pg. 131">131</a>, <a -class="aindx" href="#p656" title="go to pg. 656">656</a>, -<a class="aindx" href="#p716" title="go to pg. -716">716</a></li> - -<li class="liindx">Bacteria, <a class="aindx" href="#p245" -title="go to pg. 245">245</a>, <a class="aindx" -href="#p250" title="go to pg. 250">250</a></li> - -<li class="liindx">Baer, K. E., von, <a class="aindx" -href="#p003" title="go to pg. 3">3</a>, <a class="aindx" -href="#p055" title="go to pg. 55">55</a>, <a class="aindx" -href="#p057" title="go to pg. 57">57</a>, <a class="aindx" -href="#p155" title="go to pg. 155">155</a></li> - -<li class="liindx">Balancement, <a class="aindx" -href="#p714" title="go to pg. 714">714</a>, <a -class="aindx" href="#p776" title="go to pg. -776">776</a></li> - -<li class="liindx">Balfour, F. M., <a class="aindx" -href="#p057" title="go to pg. 57">57</a>, <a class="aindx" -href="#p348" title="go to pg. 348">348</a></li> - -<li class="liindx">Baltzer, Fr., <a class="aindx" -href="#p327" title="go to pg. 327">327</a></li> - -<li class="liindx">Bamboo, growth of, <a class="aindx" -href="#p077" title="go to pg. 77">77</a></li> - -<li class="liindx">Barclay, J., <a class="aindx" -href="#p334" title="go to pg. 334">334</a></li> - -<li class="liindx">Barfurth, D., <a class="aindx" -href="#p085" title="go to pg. 85">85</a></li> - -<li class="liindx">Barlow, W., <a class="aindx" -href="#p202" title="go to pg. 202">202</a></li> - -<li class="liindx">Barratt, J. O. W., <a class="aindx" -href="#p285" title="go to pg. 285">285</a></li> - -<li class="liindx">Bartholinus, E., <a class="aindx" -href="#p329" title="go to pg. 329">329</a></li> - -<li class="liindx">Bashforth, Fr., <a class="aindx" -href="#p663" title="go to pg. 663">663</a></li> - -<li class="liindx">Bast-fibres, strength of, -<a class="aindx" href="#p679" title="go to pg. -679">679</a></li> - -<li class="liindx">Baster, Job, <a class="aindx" -href="#p138" title="go to pg. 138">138</a></li> - -<li class="liindx">Bateson, W., <a class="aindx" -href="#p104" title="go to pg. 104">104</a>, <a -class="aindx" href="#p431" title="go to pg. -431">431</a></li> - -<li class="liindx">Bather, F. A., <a class="aindx" -href="#p578" title="go to pg. 578">578</a></li> - -<li class="liindx">Batsch, A. J. G. K., <a class="aindx" -href="#p606" title="go to pg. 606">606</a></li> - -<li class="liindx">Baudrimont, A., and St Ange, -<a class="aindx" href="#p124" title="go to pg. -124">124</a></li> - -<li class="liindx">Baumann and Roos, <a class="aindx" -href="#p136" title="go to pg. 136">136</a></li> - -<li class="liindx">Bayliss, W. M., <a class="aindx" -href="#p135" title="go to pg. 135">135</a>, <a -class="aindx" href="#p277" title="go to pg. -277">277</a></li> - -<li class="liindx">Beads or globules, <a class="aindx" -href="#p234" title="go to pg. 234">234</a></li> - -<li class="liindx">Beak, shape of, <a class="aindx" -href="#p632" title="go to pg. 632">632</a></li> - -<li class="liindx">Beal, W. J., <a class="aindx" -href="#p643" title="go to pg. 643">643</a></li> - -<li class="liindx">Beam, loaded, <a class="aindx" -href="#p674" title="go to pg. 674">674</a></li> - -<li class="liindx">Bee’s cell, <a class="aindx" -href="#p327" title="go to pg. 327">327</a>, <a -class="aindx" href="#p779" title="go to pg. -779">779</a></li> - -<li class="liindx">Begonia, <a class="aindx" href="#p412" -title="go to pg. 412">412</a>, <a class="aindx" -href="#p733" title="go to pg. 733">733</a></li> - -<li class="liindx">Beisa antelope, horns of, <a -class="aindx" href="#p616" title="go to pg. 616">616</a>, -<a class="aindx" href="#p621" title="go to pg. -621">621</a></li> - -<li class="liindx">Bellerophon, <a class="aindx" -href="#p550" title="go to pg. 550">550</a></li> - -<li class="liindx">Bénard, H., <a class="aindx" -href="#p259" title="go to pg. 259">259</a>, <a -class="aindx" href="#p319" title="go to pg. 319">319</a>, -<a class="aindx" href="#p448" title="go to pg. -448">448</a>, <a class="aindx" href="#p590" title="go to -pg. 590">590</a></li> - -<li class="liindx">Bending moments, <a class="aindx" -href="#p019" title="go to pg. 19">19</a>, <a -class="aindx" href="#p677" title="go to pg. 677">677</a>, -<a class="aindx" href="#p696" title="go to pg. -696">696</a></li> - -<li class="liindx">Beneden, Ed. van, <a class="aindx" -href="#p153" title="go to pg. 153">153</a>, <a -class="aindx" href="#p170" title="go to pg. 170">170</a>, -<a class="aindx" href="#p198" title="go to pg. -198">198</a></li> - -<li class="liindx">Bergson, H., <a class="aindx" -href="#p007" title="go to pg. 7">7</a>, <a class="aindx" -href="#p103" title="go to pg. 103">103</a>, <a -class="aindx" href="#p251" title="go to pg. 251">251</a>, -<a class="aindx" href="#p611" title="go to pg. -611">611</a>, <a class="aindx" href="#p721" title="go to -pg. 721">721</a></li> - -<li class="liindx">Bernard, Claude, <a class="aindx" -href="#p002" title="go to pg. 2">2</a>, <a class="aindx" -href="#p013" title="go to pg. 13">13</a>, <a class="aindx" -href="#p127" title="go to pg. 127">127</a></li> - -<li class="liindx">Bernoulli, James, <a class="aindx" -href="#p580" title="go to pg. 580">580</a>; <ul> <li -class="liindx">John, <a class="aindx" href="#p030" -title="go to pg. 30">30</a>, <a class="aindx" href="#p054" -title="go to pg. 54">54</a></li> </ul></li> - -<li class="liindx">Berthold, G., <a class="aindx" -href="#p008" title="go to pg. 8">8</a>, <a class="aindx" -href="#p234" title="go to pg. 234">234</a>, <a -class="aindx" href="#p298" title="go to pg. 298">298</a>, -<a class="aindx" href="#p306" title="go to pg. -306">306</a>, <a class="aindx" href="#p322" title="go to -pg. 322">322</a>, <a class="aindx" href="#p346" title="go -to pg. 346">346</a>, <a class="aindx" href="#p351" -title="go to pg. 351">351</a>, <a class="aindx" -href="#p357" title="go to pg. 357">357</a>, <a -class="aindx" href="#p358" title="go to pg. 358">358</a>, -<a class="aindx" href="#p372" title="go to pg. -372">372</a>, <a class="aindx" href="#p399" title="go to -pg. 399">399</a></li> - -<li class="liindx">Bethe, A., <a class="aindx" href="#p276" -title="go to pg. 276">276</a></li> - -<li class="liindx">Bialaszewicz, K., <a class="aindx" -href="#p114" title="go to pg. 114">114</a>, <a -class="aindx" href="#p125" title="go to pg. -125">125</a></li> - -<li class="liindx">Biedermann, W., <a class="aindx" -href="#p431" title="go to pg. 431">431</a></li> - -<li class="liindx">Bilharzia, egg of, <a class="aindx" -href="#p656" title="go to pg. 656">656</a></li> - -<li class="liindx">Binuclearity, <a class="aindx" -href="#p286" title="go to pg. 286">286</a></li> - -<li class="liindx">Biocrystallisation, <a class="aindx" -href="#p454" title="go to pg. 454">454</a></li> - -<li class="liindx">Biogenetisches Grundgesetz, -<a class="aindx" href="#p608" title="go to pg. -608">608</a></li> - -<li class="liindx">Biometrics, <a class="aindx" -href="#p078" title="go to pg. 78">78</a></li> - -<li class="liindx">Bird, flight of, <a class="aindx" -href="#p024" title="go to pg. 24">24</a>; <ul> <li -class="liindx">form of, <a class="aindx" href="#p673" -title="go to pg. 673">673</a></li> </ul></li> - -<li class="liindx">Bisection of solids, <a class="aindx" -href="#p352" title="go to pg. 352">352</a>, etc.</li> - -<li class="liindx">Bishop, John <a class="aindx" -href="#p031" title="go to pg. 31">31</a></li> - -<li class="liindx">Bivalve shells, <a class="aindx" -href="#p561" title="go to pg. 561">561</a></li> - -<li class="liindx">Bjerknes, V. <a class="aindx" -href="#p186" title="go to pg. 186">186</a></li> - -<li class="liindx">Blackman, F. F. <a class="aindx" -href="#p108" title="go to pg. 108">108</a>, <a -class="aindx" href="#p110" title="go to pg. 110">110</a>, -<a class="aindx" href="#p114" title="go to pg. -114">114</a>, <a class="aindx" href="#p124" title="go to -pg. 124">124</a>, <a class="aindx" href="#p131" title="go -to pg. 131">131</a>, <a class="aindx" href="#p132" -title="go to pg. 132">132</a></li> - -<li class="liindx">Blackwall, J. <a class="aindx" -href="#p234" title="go to pg. 234">234</a></li> - -<li class="liindx">Blake, J. F. <a class="aindx" -href="#p536" title="go to pg. 536">536</a>, <a -class="aindx" href="#p547" title="go to pg. 547">547</a>, -<a class="aindx" href="#p553" title="go to pg. -553">553</a>, <a class="aindx" href="#p578" title="go to -pg. 578">578</a>, <a class="aindx" href="#p583" title="go -to pg. 583">583</a>, <a class="aindx" href="#p728" -title="go to pg. 728">728</a></li> - -<li class="liindx">Blastosphere, <a class="aindx" -href="#p056" title="go to pg. 56">56</a>, <a class="aindx" -href="#p344" title="go to pg. 344">344</a></li> - -<li class="liindx">Blood-corpuscles, form of, <a -class="aindx" href="#p270" title="go to pg. 270">270</a>; -<ul> <li class="liindx">size of, <a class="aindx" -href="#p036" title="go to pg. 36">36</a></li> </ul></li> - -<li class="liindx">Blood-vessels, <a class="aindx" -href="#p665" title="go to pg. 665">665</a></li> - -<li class="liindx">Boas, Fr., <a class="aindx" href="#p079" -title="go to pg. 79">79</a></li> - -<li class="liindx">Bodo, <a class="aindx" href="#p230" -title="go to pg. 230">230</a>, <a class="aindx" -href="#p269" title="go to pg. 269">269</a></li> - -<li class="liindx">Boerhaave, Hermann, <a class="aindx" -href="#p380" title="go to pg. 380">380</a></li> - -<li class="liindx">Bonanni, F., <a class="aindx" -href="#p318" title="go to pg. 318">318</a></li> - -<li class="liindx">Bone, <a class="aindx" href="#p425" -title="go to pg. 425">425</a>, <a class="aindx" -href="#p435" title="go to pg. 435">435</a>; <ul> -<li class="liindx">repair of, <a class="aindx" -href="#p687" title="go to pg. 687">687</a>;</li> <li -class="liindx">structure of, <a class="aindx" href="#p673" -title="go to pg. 673">673</a>, <a class="aindx" -href="#p680" title="go to pg. 680">680</a></li> </ul></li> - -<li class="liindx">Bonnet, Ch., <a class="aindx" -href="#p108" title="go to pg. 108">108</a>, <a -class="aindx" href="#p138" title="go to pg. 138">138</a>, -<a class="aindx" href="#p334" title="go to pg. -334">334</a>, <a class="aindx" href="#p635" title="go to -pg. 635">635</a></li> - -<li class="liindx">Borelli, J. A., <a class="aindx" -href="#p008" title="go to pg. 8">8</a>, <a class="aindx" -href="#p027" title="go to pg. 27">27</a>, <a class="aindx" -href="#p029" title="go to pg. 29">29</a>, <a class="aindx" -href="#p318" title="go to pg. 318">318</a>, <a -class="aindx" href="#p677" title="go to pg. 677">677</a>, -<a class="aindx" href="#p690" title="go to pg. -690">690</a></li> - -<li class="liindx">Bosanquet, B., <a class="aindx" -href="#p005" title="go to pg. 5">5</a></li> - -<li class="liindx">Boscovich, Father R. J., S.J., <a -class="aindx" href="#p008" title="go to pg. 8">8</a></li> - -<li class="liindx">Bose, J. C., <a class="aindx" -href="#p087" title="go to pg. 87">87</a></li> - -<li class="liindx">Bostryx, <a class="aindx" href="#p502" -title="go to pg. 502">502</a></li> - -<li class="liindx">Bottazzi, F., <a class="aindx" -href="#p127" title="go to pg. 127">127</a></li> - -<li class="liindx">Bottomley, J. T., <a class="aindx" -href="#p135" title="go to pg. 135">135</a></li> - -<li class="liindx">Boubée, N., <a class="aindx" -href="#p529" title="go to pg. 529">529</a></li> - -<li class="liindx">Bourgery, J. M., <a class="aindx" -href="#p683" title="go to pg. 683">683</a></li> - -<li class="liindx">Bourne, G. C., <a class="aindx" -href="#p199" title="go to pg. 199">199</a></li> - -<li class="liindx">Bourrelet, Plateau’s, <a class="aindx" -href="#p297" title="go to pg. 297">297</a>, <a -class="aindx" href="#p339" title="go to pg. 339">339</a>, -<a class="aindx" href="#p446" title="go to pg. -446">446</a>, <a class="aindx" href="#p470" title="go to -pg. 470">470</a>, <a class="aindx" href="#p477" title="go -to pg. 477">477</a></li> - -<li class="liindx">Boveri, Th., <a class="aindx" -href="#p038" title="go to pg. 38">38</a>, <a class="aindx" -href="#p147" title="go to pg. 147">147</a>, <a -class="aindx" href="#p170" title="go to pg. 170">170</a>, -<a class="aindx" href="#p198" title="go to pg. -198">198</a></li> - -<li class="liindx">Bowditch, H. P., <a class="aindx" -href="#p061" title="go to pg. 61">61</a>, <a class="aindx" -href="#p079" title="go to pg. 79">79</a></li> - -<li class="liindx">Bower, F. O., -<a class="aindx" href="#p406" title="go to pg. 406">406</a></li> - -<li class="liindx">Bowman, J. H., <a class="aindx" -href="#p428" title="go to pg. 428">428</a></li> - -<li class="liindx">Boyd, R., <a class="aindx" href="#p061" -title="go to pg. 61">61</a></li> - -<li class="liindx">Boys, C. V., <a class="aindx" -href="#p233" title="go to pg. 233">233</a></li> - -<li class="liindx">Brachiopods, <a class="aindx" -href="#p561" title="go to pg. 561">561</a>, <a -class="aindx" href="#p568" title="go to pg. 568">568</a>, -<a class="aindx" href="#p577" title="go to pg. -577">577</a></li> - -<li class="liindx">Bradford, S. C., <a class="aindx" -href="#p428" title="go to pg. 428">428</a></li> - -<li class="liindx">Brady, H. B., <a class="aindx" -href="#p255" title="go to pg. 255">255</a>, <a -class="aindx" href="#p606" title="go to pg. -606">606</a></li> - -<li class="liindx">Brain, growth of, <a class="aindx" -href="#p089" title="go to pg. 89">89</a>; <ul> <li -class="liindx">weight of, <a class="aindx" href="#p090" -title="go to pg. 90">90</a></li> </ul></li> - -<li class="liindx">Branchipus, <a class="aindx" -href="#p128" title="go to pg. 128">128</a>, <a -class="aindx" href="#p342" title="go to pg. -342">342</a></li> - -<li class="liindx">Brandt, K., <a class="aindx" -href="#p459" title="go to pg. 459">459</a>, <a -class="aindx" href="#p482" title="go to pg. -482">482</a></li> - -<li class="liindx">Brauer, A., <a class="aindx" -href="#p180" title="go to pg. 180">180</a></li> - -<li class="liindx">Braun, A., <a class="aindx" href="#p636" -title="go to pg. 636">636</a></li> - -<li class="liindx">Bravais, L. and A., <a class="aindx" -href="#p202" title="go to pg. 202">202</a>, <a -class="aindx" href="#p502" title="go to pg. 502">502</a>, -<a class="aindx" href="#p636" title="go to pg. -636">636</a></li> - -<li class="liindx">Bredig, G., <a class="aindx" -href="#p178" title="go to pg. 178">178</a></li> - -<li class="liindx">Brewster, Sir D., <a class="aindx" -href="#p209" title="go to pg. 209">209</a>, <a -class="aindx" href="#p337" title="go to pg. 337">337</a>, -<a class="aindx" href="#p350" title="go to pg. -350">350</a>, <a class="aindx" href="#p431" title="go to -pg. 431">431</a></li> - -<li class="liindx">Bridge, T. W., <a class="aindx" -href="#p671" title="go to pg. 671">671</a></li> - -<li class="liindx">Bridge construction, <a class="aindx" -href="#p018" title="go to pg. 18">18</a>, <a class="aindx" -href="#p691" title="go to pg. 691">691</a></li> - -<li class="liindx">Brine shrimps, <a class="aindx" -href="#p127" title="go to pg. 127">127</a></li> - -<li class="liindx">Brooke, Sir V., <a class="aindx" -href="#p614" title="go to pg. 614">614</a>, <a -class="aindx" href="#p624" title="go to pg. 624">624</a>, -<a class="aindx" href="#p628" title="go to pg. -628">628</a>, <a class="aindx" href="#p631" title="go to -pg. 631">631</a></li> - -<li class="liindx">Browne, Sir T., <a class="aindx" -href="#p324" title="go to pg. 324">324</a>, <a -class="aindx" href="#p329" title="go to pg. 329">329</a>, -<a class="aindx" href="#p480" title="go to pg. -480">480</a>, <a class="aindx" href="#p650" title="go to -pg. 650">650</a>, <a class="aindx" href="#p652" title="go -to pg. 652">652</a>, <a class="aindx" href="#p733" -title="go to pg. 733">733</a></li> - -<li class="liindx">Brownian movement, <a class="aindx" -href="#p045" title="go to pg. 45">45</a>, <a -class="aindx" href="#p279" title="go to pg. 279">279</a>, -<a class="aindx" href="#p421" title="go to pg. -421">421</a></li> - -<li class="liindx">Brücke, C., <a class="aindx" -href="#p160" title="go to pg. 160">160</a>, <a -class="aindx" href="#p199" title="go to pg. -199">199</a></li> - -<li class="liindx">Buccinum, <a class="aindx" href="#p520" -title="go to pg. 520">520</a>, <a class="aindx" -href="#p527" title="go to pg. 527">527</a></li> - -<li class="liindx">Buch, Leopold von, <a class="aindx" -href="#p528" title="go to pg. 528">528</a>, <a -class="aindx" href="#p583" title="go to pg. -583">583</a></li> - -<li class="liindx">Buchner, Hans, <a class="aindx" -href="#p133" title="go to pg. 133">133</a></li> - -<li class="liindx">Budding, <a class="aindx" href="#p213" -title="go to pg. 213">213</a>, <a class="aindx" -href="#p399" title="go to pg. 399">399</a></li> - -<li class="liindx">Buffon, on the bee’s cell, -<a class="aindx" href="#p333" title="go to pg. -333">333</a></li> - -<li class="liindx">Bühle, C. A., <a class="aindx" -href="#p653" title="go to pg. 653">653</a></li> - -<li class="liindx">Bulimus, <a class="aindx" href="#p549" -title="go to pg. 549">549</a>, <a class="aindx" -href="#p556" title="go to pg. 556">556</a></li> - -<li class="liindx">Burnet, J., <a class="aindx" -href="#p509" title="go to pg. 509">509</a></li> - -<li class="liindx">Bütschli, O., <a class="aindx" -href="#p165" title="go to pg. 165">165</a>, <a -class="aindx" href="#p170" title="go to pg. 170">170</a>, -<a class="aindx" href="#p171" title="go to pg. -171">171</a>, <a class="aindx" href="#p204" title="go -to pg. 204">204</a>, <a class="aindx" href="#p432" -title="go to pg. 432">432</a>, <a class="aindx" -href="#p434" title="go to pg. 434">434</a>, <a -class="aindx" href="#p458" title="go to pg. 458">458</a>, -<a class="aindx" href="#p492" title="go to pg. -492">492</a></li> - -<li class="liindx">Büttel-Reepen, H. von, <a class="aindx" -href="#p332" title="go to pg. 332">332</a></li> - -<li class="liindx">Byk, A., <a class="aindx" href="#p419" -title="go to pg. 419">419</a></li></ul> - -<ul class="ulindx"> <li class="liindx">Cactus, -sphaerocrystals, in <a class="aindx" href="#p434" title="go -to pg. 434">434</a></li> - -<li class="liindx">Cadets, growth of German, -<a class="aindx" href="#p119" title="go to pg. -119">119</a></li> - -<li class="liindx">Calandrini, G. L., <a class="aindx" -href="#p636" title="go to pg. 636">636</a></li> - -<li class="liindx">Calcospherites, <a class="aindx" -href="#p421" title="go to pg. 421">421</a>, <a -class="aindx" href="#p434" title="go to pg. -434">434</a></li> - -<li class="liindx">Callimitra, <a class="aindx" -href="#p472" title="go to pg. 472">472</a></li> - -<li class="liindx">Callithamnion, spore of, <a -class="aindx" href="#p396" title="go to pg. -396">396</a></li> - -<li class="liindx">Calman, T. W., <a class="aindx" -href="#p149" title="go to pg. 149">149</a></li> - -<li class="liindx">Calyptraea, <a class="aindx" -href="#p556" title="go to pg. 556">556</a></li> - -<li class="liindx">Camel, <a class="aindx" href="#p703" -title="go to pg. 703">703</a>, <a class="aindx" -href="#p704" title="go to pg. 704">704</a></li> - -<li class="liindx">Campanularia, <a class="aindx" -href="#p237" title="go to pg. 237">237</a>, <a -class="aindx" href="#p262" title="go to pg. 262">262</a>, -<a class="aindx" href="#p747" title="go to pg. -747">747</a></li> - -<li class="liindx">Campbell, D. H., <a class="aindx" -href="#p302" title="go to pg. 302">302</a>, <a -class="aindx" href="#p397" title="go to pg. 397">397</a>, -<a class="aindx" href="#p402" title="go to pg. -402">402</a></li> - -<li class="liindx">Camper, P., <a class="aindx" -href="#p742" title="go to pg. 742">742</a></li> - -<li class="liindx">Camptosaurus, <a class="aindx" -href="#p754" title="go to pg. 754">754</a></li> - -<li class="liindx">Cannon bone, <a class="aindx" -href="#p730" title="go to pg. 730">730</a></li> - -<li class="liindx">Cantilever, <a class="aindx" -href="#p678" title="go to pg. 678">678</a>, <a -class="aindx" href="#p694" title="go to pg. -694">694</a></li> - -<li class="liindx">Cantor, Moritz, <a class="aindx" -href="#p503" title="go to pg. 503">503</a></li> - -<li class="liindx">Caprella, <a class="aindx" href="#p743" -title="go to pg. 743">743</a></li> - -<li class="liindx">Caprinella, <a class="aindx" -href="#p567" title="go to pg. 567">567</a>, <a -class="aindx" href="#p577" title="go to pg. -577">577</a></li> - -<li class="liindx">Carapace of crabs, <a class="aindx" -href="#p744" title="go to pg. 744">744</a></li> - -<li class="liindx">Cardium, <a class="aindx" href="#p561" -title="go to pg. 561">561</a></li> - -<li class="liindx">Cariacus, <a class="aindx" href="#p629" -title="go to pg. 629">629</a></li> - -<li class="liindx">Carlier, E. W., <a class="aindx" -href="#p211" title="go to pg. 211">211</a></li> - -<li class="liindx">Carnoy, J. B., <a class="aindx" -href="#p468" title="go to pg. 468">468</a></li> - -<li class="liindx">Carpenter, W. B., <a class="aindx" -href="#p045" title="go to pg. 45">45</a>, <a -class="aindx" href="#p422" title="go to pg. 422">422</a>, -<a class="aindx" href="#p465" title="go to pg. -465">465</a></li> - -<li class="liindx">Caryokinesis, <a class="aindx" -href="#p014" title="go to pg. 14">14</a>, <a class="aindx" -href="#p157" title="go to pg. 157">157</a>, etc.</li> - -<li class="liindx">Cassini, D., <a class="aindx" -href="#p329" title="go to pg. 329">329</a></li> - -<li class="liindx">Cassis, <a class="aindx" href="#p559" -title="go to pg. 559">559</a></li> - -<li class="liindx">Catabolic products, <a class="aindx" -href="#p435" title="go to pg. 435">435</a></li> - -<li class="liindx">Catalytic action, <a class="aindx" -href="#p130" title="go to pg. 130">130</a></li> - -<li class="liindx">Catenoid, <a class="aindx" href="#p218" -title="go to pg. 218">218</a>, <a class="aindx" -href="#p223" title="go to pg. 223">223</a>, <a -class="aindx" href="#p227" title="go to pg. 227">227</a>, -<a class="aindx" href="#p252" title="go to pg. -252">252</a></li> - -<li class="liindx">Causation, <a class="aindx" href="#p006" -title="go to pg. 6">6</a></li> - -<li class="liindx">Cavolinia, <a class="aindx" href="#p573" -title="go to pg. 573">573</a></li> - -<li class="liindx">Cayley, A., <a class="aindx" -href="#p385" title="go to pg. 385">385</a></li> - -<li class="liindx">Celestite, <a class="aindx" href="#p459" -title="go to pg. 459">459</a></li> - -<li class="liindx">Cell-theory, <a class="aindx" -href="#p197" title="go to pg. 197">197</a>, <a -class="aindx" href="#p199" title="go to pg. -199">199</a></li> - -<li class="liindx">Cells, forms of, <a class="aindx" -href="#p201" title="go to pg. 201">201</a>; <ul> <li -class="liindx">sizes of, <a class="aindx" href="#p035" -title="go to pg. 35">35</a></li> </ul></li> - -<li class="liindx">Cellular pathology, <a class="aindx" -href="#p200" title="go to pg. 200">200</a>; <ul> <li -class="liindx">tissue, artificial, <a class="aindx" -href="#p320" title="go to pg. 320">320</a></li> </ul></li> - -<li class="liindx">Cenosphaera, <a class="aindx" -href="#p470" title="go to pg. 470">470</a></li> - -<li class="liindx">Centres of force, <a class="aindx" -href="#p156" title="go to pg. 156">156</a>, <a -class="aindx" href="#p196" title="go to pg. -196">196</a></li> - -<li class="liindx">Centrosome, <a class="aindx" -href="#p167" title="go to pg. 167">167</a>, <a -class="aindx" href="#p168" title="go to pg. 168">168</a>, -<a class="aindx" href="#p173" title="go to pg. -173">173</a></li> - -<li class="liindx">Cephalopods, <a class="aindx" -href="#p548" title="go to pg. 548">548</a>, etc.; <ul> -<li class="liindx">eggs of, <a class="aindx" href="#p378" -title="go to pg. 378">378</a></li> </ul></li> - -<li class="liindx">Ceratophyllum, growth of, <a -class="aindx" href="#p097" title="go to pg. 97">97</a></li> - -<li class="liindx">Ceratorhinus, <a class="aindx" -href="#p612" title="go to pg. 612">612</a></li> - -<li class="liindx">Cerebratulus, egg of, <a class="aindx" -href="#p189" title="go to pg. 189">189</a></li> - -<li class="liindx">Cerianthus, <a class="aindx" -href="#p125" title="go to pg. 125">125</a></li> - -<li class="liindx">Cerithium, <a class="aindx" -href="#p530" title="go to pg. 530">530</a>, <a -class="aindx" href="#p557" title="go to pg. 557">557</a>, -<a class="aindx" href="#p559" title="go to pg. -559">559</a></li> - -<li class="liindx">Chabrier, J., <a class="aindx" -href="#p025" title="go to pg. 25">25</a></li> - -<li class="liindx">Chabry, L., <a class="aindx" -href="#p030" title="go to pg. 30">30</a>, <a -class="aindx" href="#p306" title="go to pg. 306">306</a>, -<a class="aindx" href="#p415" title="go to pg. -415">415</a></li> - -<li class="liindx">Chaetodont fishes, <a class="aindx" -href="#p671" title="go to pg. 671">671</a>, <a -class="aindx" href="#p749" title="go to pg. -749">749</a></li> - -<li class="liindx">Chaetopterus, egg of, <a class="aindx" -href="#p195" title="go to pg. 195">195</a></li> - -<li class="liindx">Chamois, horns of, <a class="aindx" -href="#p615" title="go to pg. 615">615</a></li> - -<li class="liindx">Chapman, Abel, <a class="aindx" -href="#p672" title="go to pg. 672">672</a></li> - -<li class="liindx">Chara, <a class="aindx" href="#p303" -title="go to pg. 303">303</a></li> - -<li class="liindx">Characters, biological, <a -class="aindx" href="#p196" title="go to pg. 196">196</a>, -<a class="aindx" href="#p727" title="go to pg. -727">727</a></li> - -<li class="liindx">Chevron bones, <a class="aindx" -href="#p709" title="go to pg. 709">709</a></li> - -<li class="liindx">Chick, hatching of, <a class="aindx" -href="#p108" title="go to pg. 108">108</a></li> - -<li class="liindx">Chilomonas, <a class="aindx" -href="#p114" title="go to pg. 114">114</a></li> - -<li class="liindx">Chladni figures, <a class="aindx" -href="#p386" title="go to pg. 386">386</a>, <a -class="aindx" href="#p475" title="go to pg. -475">475</a></li> - -<li class="liindx">Chlorophyll, <a class="aindx" -href="#p291" title="go to pg. 291">291</a></li> - -<li class="liindx">Choanoflagellates, <a class="aindx" -href="#p253" title="go to pg. 253">253</a></li> - -<li class="liindx">Chodat, R., <a class="aindx" -href="#p078" title="go to pg. 78">78</a>, <a class="aindx" -href="#p132" title="go to pg. 132">132</a></li> - -<li class="liindx">Cholesterin, <a class="aindx" -href="#p272" title="go to pg. 272">272</a></li> - -<li class="liindx">Chondriosomes, <a class="aindx" -href="#p285" title="go to pg. 285">285</a></li> - -<li class="liindx">Chorinus, <a class="aindx" href="#p744" -title="go to pg. 744">744</a></li> - -<li class="liindx">Chree, C., <a class="aindx" href="#p019" -title="go to pg. 19">19</a></li> - -<li class="liindx">Chromatin, <a class="aindx" href="#p153" -title="go to pg. 153">153</a></li> - -<li class="liindx">Chromidia, <a class="aindx" href="#p286" -title="go to pg. 286">286</a></li> - -<li class="liindx">Chromosomes, <a class="aindx" -href="#p157" title="go to pg. 157">157</a>, <a -class="aindx" href="#p173" title="go to pg. 173">173</a>, -<a class="aindx" href="#p179" title="go to pg. -179">179</a>, <a class="aindx" href="#p181" title="go to -pg. 181">181</a>, <a class="aindx" href="#p190" title="go -to pg. 190">190</a>, <a class="aindx" href="#p195" -title="go to pg. 195">195</a></li> - -<li class="liindx">Church, A. H., <a class="aindx" -href="#p639" title="go to pg. 639">639</a></li> - -<li class="liindx">Cicero, <a class="aindx" href="#p062" -title="go to pg. 62">62</a></li> - -<li class="liindx">Cicinnus, <a class="aindx" href="#p502" -title="go to pg. 502">502</a></li> - -<li class="liindx">Cidaris, <a class="aindx" href="#p664" -title="go to pg. 664">664</a></li> - -<li class="liindx">Circogonia, <a class="aindx" -href="#fig231" title="go to Fig. 231">479</a></li> - -<li class="liindx">Cladocarpus, <a class="aindx" -href="#p748" title="go to pg. 748">748</a></li> - -<li class="liindx">Claparède, E. R, <a class="aindx" -href="#p423" title="go to pg. 423">423</a></li> - -<li class="liindx">Clathrulina, <a class="aindx" -href="#p470" title="go to pg. 470">470</a></li> - -<li class="liindx">Clausilia, <a class="aindx" href="#p520" -title="go to pg. 520">520</a>, <a class="aindx" -href="#p549" title="go to pg. 549">549</a></li> - -<li class="liindx">Claws, <a class="aindx" href="#p149" -title="go to pg. 149">149</a>, <a class="aindx" -href="#p632" title="go to pg. 632">632</a></li> - -<li class="liindx">Cleland, John, <a class="aindx" -href="#p004" title="go to pg. 4">4</a></li> - -<li class="liindx">Cleodora, <a class="aindx" href="#p570" -title="go to pg. 570">570</a>–<a class="aindx" href="#p575" -title="go to pg. 575">575</a></li> - -<li class="liindx">Climate and growth, <a class="aindx" -href="#p121" title="go to pg. 121">121</a></li> - -<li class="liindx">Clio, <a class="aindx" href="#p570" -title="go to pg. 570">570</a></li> - -<li class="liindx">Close packing, <a class="aindx" -href="#p453" title="go to pg. 453">453</a></li> - -<li class="liindx">Clytia, <a class="aindx" href="#p747" -title="go to pg. 747">747</a></li> - -<li class="liindx">Coan, C. A., <a class="aindx" -href="#p514" title="go to pg. 514">514</a></li> - -<li class="liindx">Coassus, <a class="aindx" href="#p629" -title="go to pg. 629">629</a></li> - -<li class="liindx">Cod, otoliths of, <a class="aindx" -href="#p432" title="go to pg. 432">432</a>; <ul> <li -class="liindx">skeleton of, <a class="aindx" href="#p710" -title="go to pg. 710">710</a></li> </ul></li> - -<li class="liindx">Codonella, <a class="aindx" href="#p248" -title="go to pg. 248">248</a></li> - -<li class="liindx">Codosiga, <a class="aindx" href="#p253" -title="go to pg. 253">253</a></li> - -<li class="liindx">Coe, W. R., <a class="aindx" -href="#p189" title="go to pg. 189">189</a></li> - -<li class="liindx">Coefficient of growth, <a class="aindx" -href="#p153" title="go to pg. 153">153</a>; <ul> -<li class="liindx">of temperature, <a class="aindx" -href="#p109" title="go to pg. 109">109</a></li> </ul></li> - -<li class="liindx">Coelopleurus, <a class="aindx" -href="#p664" title="go to pg. 664">664</a></li> - -<li class="liindx">Cogan, Dr T., <a class="aindx" -href="#p742" title="go to pg. 742">742</a></li> - -<li class="liindx">Cohen, A., <a class="aindx" href="#p110" -title="go to pg. 110">110</a></li> - -<li class="liindx">Cohesion figures, <a class="aindx" -href="#p259" title="go to pg. 259">259</a></li> - -<li class="liindx">Collar-cells, <a class="aindx" -href="#p253" title="go to pg. 253">253</a></li> - -<li class="liindx">Colloids, <a class="aindx" href="#p162" -title="go to pg. 162">162</a>, <a class="aindx" -href="#p178" title="go to pg. 178">178</a>, <a -class="aindx" href="#p201" title="go to pg. 201">201</a>, -<a class="aindx" href="#p279" title="go to pg. -279">279</a>, <a class="aindx" href="#p412" title="go to -pg. 412">412</a>, <a class="aindx" href="#p421" title="go -to pg. 421">421</a>, etc.</li> - -<li class="liindx">Collosclerophora, <a class="aindx" -href="#p436" title="go to pg. 436">436</a></li> - -<li class="liindx">Collosphaera, <a class="aindx" -href="#p459" title="go to pg. 459">459</a></li> - -<li class="liindx">Colman, S., <a class="aindx" -href="#p514" title="go to pg. 514">514</a></li> - -<li class="liindx">Comoseris, <a class="aindx" href="#p327" -title="go to pg. 327">327</a></li> - -<li class="liindx">Compensation, law of, <a class="aindx" -href="#p714" title="go to pg. 714">714</a>, <a -class="aindx" href="#p776" title="go to pg. -776">776</a></li> - -<li class="liindx">Conchospiral, <a class="aindx" -href="#p531" title="go to pg. 531">531</a>, <a -class="aindx" href="#p539" title="go to pg. 539">539</a>, -<a class="aindx" href="#p594" title="go to pg. -594">594</a></li> - -<li class="liindx">Conchyliometer, <a class="aindx" -href="#p529" title="go to pg. 529">529</a></li> - -<li class="liindx">Concretions, <a class="aindx" -href="#p410" title="go to pg. 410">410</a>, etc.</li> - -<li class="liindx">Conjugate curves, <a class="aindx" -href="#p561" title="go to pg. 561">561</a>, <a -class="aindx" href="#p613" title="go to pg. -613">613</a></li> - -<li class="liindx">Conklin, E. G., <a class="aindx" -href="#p036" title="go to pg. 36">36</a>, <a class="aindx" -href="#p191" title="go to pg. 191">191</a>, <a -class="aindx" href="#p310" title="go to pg. 310">310</a>, -<a class="aindx" href="#p340" title="go to pg. -340">340</a>, <a class="aindx" href="#p377" title="go to -pg. 377">377</a></li> - -<li class="liindx">Conostats, <a class="aindx" href="#p427" -title="go to pg. 427">427</a></li> - -<li class="liindx">Continuous girder, <a class="aindx" -href="#p700" title="go to pg. 700">700</a></li> - -<li class="liindx">Contractile vacuole, <a class="aindx" -href="#p165" title="go to pg. 165">165</a>, <a -class="aindx" href="#p264" title="go to pg. -264">264</a></li> - -<li class="liindx">Conus, <a class="aindx" href="#p557" -title="go to pg. 557">557</a>, <a class="aindx" -href="#p559" title="go to pg. 559">559</a>, <a -class="aindx" href="#p560" title="go to pg. -560">560</a></li> - -<li class="liindx">Cook, Sir T. A., <a class="aindx" -href="#p493" title="go to pg. 493">493</a>, <a -class="aindx" href="#p635" title="go to pg. 635">635</a>, -<a class="aindx" href="#p639" title="go to pg. -639">639</a>, <a class="aindx" href="#p650" title="go to -pg. 650">650</a></li> - -<li class="liindx">Co-ordinates, <a class="aindx" -href="#p723" title="go to pg. 723">723</a></li> - -<li class="liindx">Corals, <a class="aindx" href="#p325" -title="go to pg. 325">325</a>, <a class="aindx" -href="#p388" title="go to pg. 388">388</a>, <a -class="aindx" href="#p423" title="go to pg. -423">423</a></li> - -<li class="liindx">Cornevin, Ch., <a class="aindx" -href="#p102" title="go to pg. 102">102</a></li> - -<li class="liindx">Cornuspira, <a class="aindx" -href="#p594" title="go to pg. 594">594</a></li> - -<li class="liindx">Correlation, <a class="aindx" -href="#p078" title="go to pg. 78">78</a>, <a class="aindx" -href="#p727" title="go to pg. 727">727</a></li> - -<li class="liindx">Corystes, <a class="aindx" href="#p744" -title="go to pg. 744">744</a></li> - -<li class="liindx">Cotton, A., <a class="aindx" -href="#p418" title="go to pg. 418">418</a></li> - -<li class="liindx">Cox, J., <a class="aindx" href="#p046" -title="go to pg. 46">46</a></li> - -<li class="liindx">Crane-head, <a class="aindx" -href="#p682" title="go to pg. 682">682</a></li> - -<li class="liindx">Crayfish, sperm-cells of, -<a class="aindx" href="#p273" title="go to pg. -273">273</a></li> - -<li class="liindx">Creodonta, <a class="aindx" href="#p716" -title="go to pg. 716">716</a></li> - -<li class="liindx">Crepidula, <a class="aindx" href="#p036" -title="go to pg. 36">36</a>, <a class="aindx" href="#p310" -title="go to pg. 310">310</a>, <a class="aindx" -href="#p340" title="go to pg. 340">340</a></li> - -<li class="liindx">Creseis, <a class="aindx" href="#p570" -title="go to pg. 570">570</a></li> - -<li class="liindx">Cristellaria, <a class="aindx" -href="#p515" title="go to pg. 515">515</a>, <a -class="aindx" href="#p600" title="go to pg. -600">600</a></li> - -<li class="liindx">Crocodile, <a class="aindx" href="#p704" -title="go to pg. 704">704</a>, <a class="aindx" -href="#p752" title="go to pg. 752">752</a></li> - -<li class="liindx">Crocus, growth of, <a class="aindx" -href="#p088" title="go to pg. 88">88</a></li> - -<li class="liindx">Crookes, Sir W., <a class="aindx" -href="#p032" title="go to pg. 32">32</a></li> - -<li class="liindx">Cryptocleidus, <a class="aindx" -href="#p755" title="go to pg. 755">755</a></li> - -<li class="liindx">Crystals, <a class="aindx" href="#p202" -title="go to pg. 202">202</a>, <a class="aindx" -href="#p250" title="go to pg. 250">250</a>, <a -class="aindx" href="#p429" title="go to pg. 429">429</a>, -<a class="aindx" href="#p444" title="go to pg. -444">444</a>, <a class="aindx" href="#p480" title="go to -pg. 480">480</a>, <a class="aindx" href="#p601" title="go -to pg. 601">601</a></li> - -<li class="liindx">Ctenophora, <a class="aindx" -href="#p391" title="go to pg. 391">391</a></li> - -<li class="liindx">Cube, partition of, <a class="aindx" -href="#p346" title="go to pg. 346">346</a></li> - -<li class="liindx">Cucumis, growth of, <a class="aindx" -href="#p109" title="go to pg. 109">109</a></li> - -<li class="liindx">Culmann, Professor C., <a -class="aindx" href="#p682" title="go to pg. 682">682</a>, -<a class="aindx" href="#p697" title="go to pg. -697">697</a></li> - -<li class="liindx">Cultellus, <a class="aindx" href="#p564" -title="go to pg. 564">564</a></li> - -<li class="liindx">Curlew, eggs of, <a class="aindx" -href="#p652" title="go to pg. 652">652</a></li> - -<li class="liindx">Cushman, J. A., <a class="aindx" -href="#p323" title="go to pg. 323">323</a></li> - -<li class="liindx">Cuvier, <a class="aindx" href="#p727" -title="go to pg. 727">727</a></li> - -<li class="liindx">Cuvierina, <a class="aindx" href="#p258" -title="go to pg. 258">258</a>, <a class="aindx" -href="#p570" title="go to pg. 570">570</a></li> - -<li class="liindx">Cyamus, <a class="aindx" href="#p743" -title="go to pg. 743">743</a></li> - -<li class="liindx">Cyathophyllum, <a class="aindx" -href="#p325" title="go to pg. 325">325</a>, <a -class="aindx" href="#p391" title="go to pg. -391">391</a></li> - -<li class="liindx">Cyclammina, <a class="aindx" -href="#p595" title="go to pg. 595">595</a>, <a -class="aindx" href="#fig312" title="go to Fig. 312">596</a>, -<a class="aindx" href="#p602" title="go to pg. -602">602</a></li> - -<li class="liindx">Cyclas, <a class="aindx" href="#p561" -title="go to pg. 561">561</a></li> - -<li class="liindx">Cyclostoma, <a class="aindx" -href="#p554" title="go to pg. 554">554</a></li> - -<li class="liindx">Cylinder, <a class="aindx" -href="#p218" title="go to pg. 218">218</a>, <a -class="aindx" href="#p227" title="go to pg. 227">227</a>, -<a class="aindx" href="#p377" title="go to pg. -377">377</a></li> - -<li class="liindx">Cymba, <a class="aindx" href="#p559" -title="go to pg. 559">559</a></li> - -<li class="liindx">Cyme, <a class="aindx" href="#p502" -title="go to pg. 502">502</a></li> - -<li class="liindx">Cypraea, <a class="aindx" href="#p547" -title="go to pg. 547">547</a>, <a class="aindx" -href="#p554" title="go to pg. 554">554</a>, <a -class="aindx" href="#p560" title="go to pg. 560">560</a>, -<a class="aindx" href="#p561" title="go to pg. -561">561</a></li> - -<li class="liindx">Cyrtina, <a class="aindx" href="#p569" -title="go to pg. 569">569</a></li> - -<li class="liindx">Cyrtocerata, <a class="aindx" -href="#p583" title="go to pg. 583">583</a></li> - -<li class="liindx">Cystoliths, <a class="aindx" -href="#p412" title="go to pg. 412">412</a></li> </ul> - -<ul class="ulindx"> <li class="liindx">Daday de Dees, -E. v., <a class="aindx" href="#p130" title="go to pg. -130">130</a></li> - -<li class="liindx">Daffner, Fr., <a class="aindx" -href="#p061" title="go to pg. 61">61</a>, <a class="aindx" -href="#p118" title="go to pg. 118">118</a></li> - -<li class="liindx">Dalyell, Sir John G., <a class="aindx" -href="#p146" title="go to pg. 146">146</a></li> - -<li class="liindx">Danilewsky, B., <a class="aindx" -href="#p135" title="go to pg. 135">135</a></li> - -<li class="liindx">Darling, C. R., <a class="aindx" -href="#p219" title="go to pg. 219">219</a>, <a -class="aindx" href="#p257" title="go to pg. 257">257</a>, -<a class="aindx" href="#p664" title="go to pg. -664">664</a></li> - -<li class="liindx">D’Arsonval, A., <a class="aindx" -href="#p192" title="go to pg. 192">192</a>, <a -class="aindx" href="#p281" title="go to pg. -281">281</a></li> - -<li class="liindx">Darwin, C., <a class="aindx" -href="#p004" title="go to pg. 4">4</a>, <a class="aindx" -href="#p044" title="go to pg. 44">44</a>, <a class="aindx" -href="#p057" title="go to pg. 57">57</a>, <a class="aindx" -href="#p332" title="go to pg. 332">332</a>, <a -class="aindx" href="#p431" title="go to pg. 431">431</a>, -<a class="aindx" href="#p465" title="go to pg. -465">465</a>, <a class="aindx" href="#p549" title="go to -pg. 549">549</a>, <a class="aindx" href="#p624" title="go -to pg. 624">624</a>, <a class="aindx" href="#p671" -title="go to pg. 671">671</a>, <a class="aindx" -href="#p714" title="go to pg. 714">714</a></li> - -<li class="liindx">Dastre, A., <a class="aindx" -href="#p136" title="go to pg. 136">136</a></li> - -<li class="liindx">Davenport, C. B., <a class="aindx" -href="#p107" title="go to pg. 107">107</a>, <a -class="aindx" href="#p123" title="go to pg. 123">123</a>, -<a class="aindx" href="#p125" title="go to pg. -125">125</a>, <a class="aindx" href="#p126" title="go to -pg. 126">126</a>, <a class="aindx" href="#p211" title="go -to pg. 211">211</a></li> - -<li class="liindx">De Candolle, A., <a class="aindx" -href="#p108" title="go to pg. 108">108</a>, <a -class="aindx" href="#p643" title="go to pg. 643">643</a>; -<ul> <li class="liindx">A. P., <a class="aindx" -href="#p020" title="go to pg. 20">20</a>;</li> <li -class="liindx">C., <a class="aindx" href="#p636" title="go -to pg. 636">636</a></li> </ul></li> - -<li class="liindx">Decapod Crustacea, sperm-cells -of, <a class="aindx" href="#p273" title="go to pg. -273">273</a></li> - -<li class="liindx">Deer, antlers of, <a class="aindx" -href="#p628" title="go to pg. 628">628</a></li> - -<li class="liindx">Deformation, <a class="aindx" -href="#p638" title="go to pg. 638">638</a>, <a -class="aindx" href="#p728" title="go to pg. 728">728</a>, -etc.</li> - -<li class="liindx">Degree, differences of, <a -class="aindx" href="#p586" title="go to pg. 586">586</a>, -<a class="aindx" href="#p725" title="go to pg. -725">725</a></li> - -<li class="liindx">Delage, Yves, <a class="aindx" -href="#p153" title="go to pg. 153">153</a></li> - -<li class="liindx">Delaunay, C. E., <a class="aindx" -href="#p218" title="go to pg. 218">218</a></li> - -<li class="liindx">Delisle, <a class="aindx" href="#p031" -title="go to pg. 31">31</a></li> - -<li class="liindx">Dellinger, O. P., <a class="aindx" -href="#p212" title="go to pg. 212">212</a></li> - -<li class="liindx">Delphinula, <a class="aindx" -href="#p557" title="go to pg. 557">557</a></li> - -<li class="liindx">Delpino, F., <a class="aindx" -href="#p636" title="go to pg. 636">636</a></li> - -<li class="liindx">Democritus, <a class="aindx" -href="#p044" title="go to pg. 44">44</a></li> - -<li class="liindx">Dendy, A., <a class="aindx" href="#p137" -title="go to pg. 137">137</a>, <a class="aindx" -href="#p436" title="go to pg. 436">436</a>, <a -class="aindx" href="#p440" title="go to pg. 440">440</a>, -<a class="aindx" href="#p671" title="go to pg. -671">671</a></li> - -<li class="liindx">Dentalium, <a class="aindx" href="#p535" -title="go to pg. 535">535</a>, <a class="aindx" -href="#p537" title="go to pg. 537">537</a>, <a -class="aindx" href="#p546" title="go to pg. 546">546</a>, -<a class="aindx" href="#p555" title="go to pg. -555">555</a>, <a class="aindx" href="#p556" title="go to -pg. 556">556</a>, <a class="aindx" href="#p561" title="go -to pg. 561">561</a></li> - -<li class="liindx">Dentine, <a class="aindx" href="#p425" -title="go to pg. 425">425</a></li> - -<li class="liindx">Descartes, R., <a class="aindx" -href="#p185" title="go to pg. 185">185</a>, <a -class="aindx" href="#p723" title="go to pg. -723">723</a></li> - -<li class="liindx">Des Murs, O., <a class="aindx" -href="#p653" title="go to pg. 653">653</a></li> - -<li class="liindx">Devaux, H., <a class="aindx" -href="#p043" title="go to pg. 43">43</a></li> - -<li class="liindx">De Vries, H., <a class="aindx" -href="#p108" title="go to pg. 108">108</a></li> - -<li class="liindx">Diatoms, <a class="aindx" -href="#p214" title="go to pg. 214">214</a>, <a -class="aindx" href="#p386" title="go to pg. 386">386</a>, -<a class="aindx" href="#p426" title="go to pg. -426">426</a></li> - -<li class="liindx">Diceras, <a class="aindx" href="#p567" -title="go to pg. 567">567</a></li> - -<li class="liindx">Dickson, Alex., <a class="aindx" -href="#p647" title="go to pg. 647">647</a></li> - -<li class="liindx">Dictyota, <a class="aindx" -href="#p303" title="go to pg. 303">303</a>, <a -class="aindx" href="#p356" title="go to pg. 356">356</a>, -<a class="aindx" href="#p474" title="go to pg. -474">474</a></li> - -<li class="liindx">Diet and growth, <a class="aindx" -href="#p134" title="go to pg. 134">134</a></li> - -<li class="liindx">Difflugia, <a class="aindx" href="#p463" -title="go to pg. 463">463</a>, <a class="aindx" -href="#p466" title="go to pg. 466">466</a></li> - -<li class="liindx">Diffusion figures, <a class="aindx" -href="#p259" title="go to pg. 259">259</a>, <a -class="aindx" href="#p430" title="go to pg. -430">430</a></li> - -<li class="liindx">Dimorphism of earwigs, <a class="aindx" -href="#p105" title="go to pg. 105">105</a></li> - -<li class="liindx">Dimorphodon, <a class="aindx" -href="#p756" title="go to pg. 756">756</a></li> - -<li class="liindx">Dinenympha, <a class="aindx" -href="#p252" title="go to pg. 252">252</a></li> - -<li class="liindx">Dinobryon, <a class="aindx" href="#p248" -title="go to pg. 248">248</a></li> - -<li class="liindx">Dinosaurs, <a class="aindx" -href="#p702" title="go to pg. 702">702</a>, <a -class="aindx" href="#p704" title="go to pg. 704">704</a>, -<a class="aindx" href="#p754" title="go to pg. -754">754</a></li> - -<li class="liindx">Diodon, <a class="aindx" href="#p751" -title="go to pg. 751">751</a>, <a class="aindx" -href="#p777" title="go to pg. 777">777</a></li> - -<li class="liindx">Dionaea, <a class="aindx" href="#p734" -title="go to pg. 734">734</a></li> - -<li class="liindx">Diplodocus, <a class="aindx" -href="#p702" title="go to pg. 702">702</a>, <a -class="aindx" href="#p706" title="go to pg. 706">706</a>, -<a class="aindx" href="#p710" title="go to pg. -710">710</a></li> - -<li class="liindx">Disc, segmentation of a, <a -class="aindx" href="#p367" title="go to pg. -367">367</a></li> - -<li class="liindx">Discorbina, <a class="aindx" -href="#p602" title="go to pg. 602">602</a></li> - -<li class="liindx">Distigma, <a class="aindx" href="#p246" -title="go to pg. 246">246</a></li> - -<li class="liindx">Distribution, geographical, <a -class="aindx" href="#p457" title="go to pg. 457">457</a>, -<a class="aindx" href="#p606" title="go to pg. -606">606</a></li> - -<li class="liindx">Ditrupa, <a class="aindx" href="#p586" -title="go to pg. 586">586</a></li> - -<li class="liindx">Dixon, A. F., <a class="aindx" -href="#p684" title="go to pg. 684">684</a></li> - -<li class="liindx">Dobell, C. C., <a class="aindx" -href="#p286" title="go to pg. 286">286</a></li> - -<li class="liindx">Dodecahedron, <a class="aindx" -href="#p336" title="go to pg. 336">336</a>, <a -class="aindx" href="#p478" title="go to pg. 478">478</a>, -etc.</li> - -<li class="liindx">Doflein, F. J., <a class="aindx" -href="#p046" title="go to pg. 46">46</a>, <a -class="aindx" href="#p267" title="go to pg. 267">267</a>, -<a class="aindx" href="#p606" title="go to pg. -606">606</a></li> - -<li class="liindx">Dog’s skull, <a class="aindx" -href="#p773" title="go to pg. 773">773</a></li> - -<li class="liindx">Dolium, <a class="aindx" href="#p526" -title="go to pg. 526">526</a>, <a class="aindx" -href="#p528" title="go to pg. 528">528</a>, <a -class="aindx" href="#p530" title="go to pg. 530">530</a>, -<a class="aindx" href="#p557" title="go to pg. -557">557</a>, <a class="aindx" href="#p559" title="go to -pg. 559">559</a>, <a class="aindx" href="#p560" title="go -to pg. 560">560</a></li> - -<li class="liindx">Dolphin, skeleton of, <a class="aindx" -href="#p709" title="go to pg. 709">709</a></li> - -<li class="liindx">Donaldson, H. H., <a class="aindx" -href="#p082" title="go to pg. 82">82</a>, <a class="aindx" -href="#p093" title="go to pg. 93">93</a></li> - -<li class="liindx">Dorataspis, <a class="aindx" -href="#p481" title="go to pg. 481">481</a></li> - -<li class="liindx">D’Orbigny, Alc., <a class="aindx" -href="#p529" title="go to pg. 529">529</a>, <a -class="aindx" href="#p555" title="go to pg. 555">555</a>, -<a class="aindx" href="#p591" title="go to pg. -591">591</a>, <a class="aindx" href="#p728" title="go to -pg. 728">728</a></li> - -<li class="liindx">Douglass, A. E., <a class="aindx" -href="#p121" title="go to pg. 121">121</a></li> - -<li class="liindx">Draper, J. W., <a class="aindx" -href="#p165" title="go to pg. 165">165</a>, <a -class="aindx" href="#p264" title="go to pg. -264">264</a></li> - -<li class="liindx">Dreyer, F. R., <a class="aindx" -href="#p435" title="go to pg. 435">435</a>, <a -class="aindx" href="#p447" title="go to pg. 447">447</a>, -<a class="aindx" href="#p455" title="go to pg. -455">455</a>, <a class="aindx" href="#p468" title="go to -pg. 468">468</a>, <a class="aindx" href="#p606" title="go -to pg. 606">606</a>, <a class="aindx" href="#p608" -title="go to pg. 608">608</a></li> - -<li class="liindx">Driesch, H., <a class="aindx" -href="#p004" title="go to pg. 4">4</a>, <a class="aindx" -href="#p035" title="go to pg. 35">35</a>, <a class="aindx" -href="#p157" title="go to pg. 157">157</a>, <a -class="aindx" href="#p306" title="go to pg. 306">306</a>, -<a class="aindx" href="#p310" title="go to pg. -310">310</a>, <a class="aindx" href="#p312" title="go to -pg. 312">312</a>, <a class="aindx" href="#p377" title="go -to pg. 377">377</a>, <a class="aindx" href="#p378" -title="go to pg. 378">378</a>, <a class="aindx" -href="#p714" title="go to pg. 714">714</a></li> - -<li class="liindx">Dromia, <a class="aindx" href="#p275" -title="go to pg. 275">275</a></li> - -<li class="liindx">Drops, <a class="aindx" href="#p044" -title="go to pg. 44">44</a>, <a class="aindx" href="#p257" -title="go to pg. 257">257</a>, <a class="aindx" -href="#p587" title="go to pg. 587">587</a></li> - -<li class="liindx">Du Bois-Reymond, Emil, <a class="aindx" -href="#p001" title="go to pg. 1">1</a>, <a class="aindx" -href="#p092" title="go to pg. 92">92</a></li> - -<li class="liindx">Duerden, J. E., <a class="aindx" -href="#p423" title="go to pg. 423">423</a></li> - -<li class="liindx">Dufour, Louis, <a class="aindx" -href="#p219" title="go to pg. 219">219</a></li> - -<li class="liindx">Dujardin, F., <a class="aindx" -href="#p257" title="go to pg. 257">257</a>, <a -class="aindx" href="#p591" title="go to pg. -591">591</a></li> - -<li class="liindx">Dunan, <a class="aindx" href="#p007" -title="go to pg. 7">7</a></li> - -<li class="liindx">Duncan, P. Martin, <a class="aindx" -href="#p388" title="go to pg. 388">388</a></li> - -<li class="liindx">Dupré, Athanase, <a class="aindx" -href="#p279" title="go to pg. 279">279</a></li> - -<li class="liindx">Durbin, Marion L., <a class="aindx" -href="#p138" title="go to pg. 138">138</a></li> - -<li class="liindx">Dürer, A., <a class="aindx" href="#p055" -title="go to pg. 55">55</a>, <a class="aindx" href="#p740" -title="go to pg. 740">740</a>, <a class="aindx" -href="#p742" title="go to pg. 742">742</a></li> - -<li class="liindx">Dutrochet, R. J. H., <a class="aindx" -href="#p212" title="go to pg. 212">212</a>, <a -class="aindx" href="#p624" title="go to pg. -624">624</a></li> - -<li class="liindx">Dwight, T., <a class="aindx" -href="#p769" title="go to pg. 769">769</a></li> - -<li class="liindx">Dynamical similarity, <a class="aindx" -href="#p017" title="go to pg. 17">17</a></li> </ul> - -<ul class="ulindx"> <li class="liindx">Earthworm, -calcospheres in, <a class="aindx" href="#p423" title="go to -pg. 423">423</a></li> - -<li class="liindx">Earwigs, dimorphism in, <a class="aindx" -href="#p104" title="go to pg. 104">104</a></li> - -<li class="liindx">Ebner, V. von, <a class="aindx" -href="#p444" title="go to pg. 444">444</a>, <a -class="aindx" href="#p683" title="go to pg. -683">683</a></li> - -<li class="liindx">Echinoderms, larval, <a class="aindx" -href="#p392" title="go to pg. 392">392</a>; <ul> <li -class="liindx">spicules of, <a class="aindx" href="#p449" -title="go to pg. 449">449</a></li> </ul></li> - -<li class="liindx">Echinus, <a class="aindx" -href="#p377" title="go to pg. 377">377</a>, <a -class="aindx" href="#p378" title="go to pg. 378">378</a>, -<a class="aindx" href="#p664" title="go to pg. -664">664</a></li> - -<li class="liindx">Eclipse, skeleton of, <a class="aindx" -href="#p739" title="go to pg. 739">739</a></li> - -<li class="liindx">Ectosarc, <a class="aindx" href="#p281" -title="go to pg. 281">281</a></li> - -<li class="liindx">Eel, growth of, <a class="aindx" -href="#p085" title="go to pg. 85">85</a></li> - -<li class="liindx">Efficiency, mechanical, <a class="aindx" -href="#p670" title="go to pg. 670">670</a></li> - -<li class="liindx">Efficient cause, <a class="aindx" -href="#p006" title="go to pg. 6">6</a>, <a class="aindx" -href="#p158" title="go to pg. 158">158</a>, <a -class="aindx" href="#p248" title="go to pg. -248">248</a></li> - -<li class="liindx">Eggs of birds, <a class="aindx" -href="#p652" title="go to pg. 652">652</a></li> - -<li class="liindx">Eiffel tower, <a class="aindx" -href="#p020" title="go to pg. 20">20</a></li> - -<li class="liindx">Eight cells, grouping of, <a -class="aindx" href="#p381" title="go to pg. 381">381</a>, -etc.</li> - -<li class="liindx">Eimer, Th., <a class="aindx" -href="#p606" title="go to pg. 606">606</a></li> - -<li class="liindx">Einstein formula, <a class="aindx" -href="#p047" title="go to pg. 47">47</a></li> - -<li class="liindx">Elastic curve, <a class="aindx" -href="#p219" title="go to pg. 219">219</a>, <a -class="aindx" href="#p265" title="go to pg. 265">265</a>, -<a class="aindx" href="#p271" title="go to pg. -271">271</a></li> - -<li class="liindx">Elaters, <a class="aindx" href="#p489" -title="go to pg. 489">489</a></li> - -<li class="liindx">Electrical convection, <a class="aindx" -href="#p187" title="go to pg. 187">187</a>; <ul> <li -class="liindx">stimulation of growth, <a class="aindx" -href="#p153" title="go to pg. 153">153</a></li> </ul></li> - -<li class="liindx">Elephant, <a class="aindx" href="#p021" -title="go to pg. 21">21</a>, <a class="aindx" -href="#p633" title="go to pg. 633">633</a>, <a -class="aindx" href="#p703" title="go to pg. 703">703</a>, -<a class="aindx" href="#p704" title="go to pg. -704">704</a></li> - -<li class="liindx">Elk, antlers of, <a class="aindx" -href="#p629" title="go to pg. 629">629</a>, <a -class="aindx" href="#p632" title="go to pg. -632">632</a></li> - -<li class="liindx">Ellipsolithes, <a class="aindx" -href="#p728" title="go to pg. 728">728</a></li> - -<li class="liindx">Ellis, R. Leslie, <a class="aindx" -href="#p004" title="go to pg. 4">4</a>, <a class="aindx" -href="#p329" title="go to pg. 329">329</a>, <a -class="aindx" href="#p647" title="go to pg. 647">647</a>; -<ul> <li class="liindx">M. M., <a class="aindx" -href="#p147" title="go to pg. 147">147</a>, <a -class="aindx" href="#p656" title="go to pg. -656">656</a></li> </ul></li> - -<li class="liindx">Elodea, <a class="aindx" href="#p322" -title="go to pg. 322">322</a></li> - -<li class="liindx">Emarginula, <a class="aindx" -href="#p556" title="go to pg. 556">556</a></li> - -<li class="liindx">Emmel, V. E., <a class="aindx" -href="#p149" title="go to pg. 149">149</a></li> - -<li class="liindx">Empedocles, <a class="aindx" -href="#p008" title="go to pg. 8">8</a></li> - -<li class="liindx">Emperor Moth, <a class="aindx" -href="#p431" title="go to pg. 431">431</a></li> - -<li class="liindx">Encystment, <a class="aindx" -href="#p213" title="go to pg. 213">213</a>, <a -class="aindx" href="#p283" title="go to pg. -283">283</a></li> - -<li class="liindx">Engelmann, T. W., <a class="aindx" -href="#p210" title="go to pg. 210">210</a>, <a -class="aindx" href="#p285" title="go to pg. -285">285</a></li> - -<li class="liindx">Enriques, P., <a class="aindx" -href="#p004" title="go to pg. 4">4</a>, <a class="aindx" -href="#p036" title="go to pg. 36">36</a>, <a class="aindx" -href="#p064" title="go to pg. 64">64</a>, <a class="aindx" -href="#p133" title="go to pg. 133">133</a>, <a -class="aindx" href="#p134" title="go to pg. 134">134</a>, -<a class="aindx" href="#p677" title="go to pg. -677">677</a></li> - -<li class="liindx">Entelechy, <a class="aindx" href="#p004" -title="go to pg. 4">4</a>, <a class="aindx" href="#p714" -title="go to pg. 714">714</a></li> - -<li class="liindx">Entosolenia, <a class="aindx" -href="#p449" title="go to pg. 449">449</a></li> - -<li class="liindx">Enzymes, <a class="aindx" href="#p135" -title="go to pg. 135">135</a></li> - -<li class="liindx">Epeira, <a class="aindx" href="#p233" -title="go to pg. 233">233</a></li> - -<li class="liindx">Epicurus, <a class="aindx" href="#p047" -title="go to pg. 47">47</a></li> - -<li class="liindx">Epidermis, <a class="aindx" href="#p314" -title="go to pg. 314">314</a>, <a class="aindx" -href="#p370" title="go to pg. 370">370</a></li> - -<li class="liindx">Epilobium, pollen of, <a class="aindx" -href="#p396" title="go to pg. 396">396</a></li> - -<li class="liindx">Epipolic force, <a class="aindx" -href="#p212" title="go to pg. 212">212</a></li> - -<li class="liindx">Equatorial plate, <a class="aindx" -href="#p174" title="go to pg. 174">174</a></li> - -<li class="liindx">Equiangular spiral, <a class="aindx" -href="#p050" title="go to pg. 50">50</a>, <a class="aindx" -href="#p505" title="go to pg. 505">505</a></li> - -<li class="liindx">Equilibrium, figures of, <a -class="aindx" href="#p227" title="go to pg. -227">227</a></li> - -<li class="liindx">Equipotential lines, <a class="aindx" -href="#p640" title="go to pg. 640">640</a></li> - -<li class="liindx">Equisetum, spores of, <a class="aindx" -href="#p290" title="go to pg. 290">290</a>, <a -class="aindx" href="#p489" title="go to pg. -489">489</a></li> - -<li class="liindx">Errera, Leo, <a class="aindx" -href="#p008" title="go to pg. 8">8</a>, <a class="aindx" -href="#p040" title="go to pg. 40">40</a>, <a class="aindx" -href="#p110" title="go to pg. 110">110</a>, <a -class="aindx" href="#p111" title="go to pg. 111">111</a>, -<a class="aindx" href="#p213" title="go to pg. -213">213</a>, <a class="aindx" href="#p306" title="go to -pg. 306">306</a>, <a class="aindx" href="#p346" title="go -to pg. 346">346</a>, <a class="aindx" href="#p348" -title="go to pg. 348">348</a>, <a class="aindx" -href="#p426" title="go to pg. 426">426</a></li> - -<li class="liindx">Erythrotrichia, <a class="aindx" -href="#p358" title="go to pg. 358">358</a>, <a -class="aindx" href="#p372" title="go to pg. 372">372</a>, -<a class="aindx" href="#p390" title="go to pg. -390">390</a></li> - -<li class="liindx">Ethmosphaera, <a class="aindx" -href="#p470" title="go to pg. 470">470</a></li> - -<li class="liindx">Euastrum, <a class="aindx" href="#p214" -title="go to pg. 214">214</a></li> - -<li class="liindx">Eucharis, <a class="aindx" href="#p391" -title="go to pg. 391">391</a></li> - -<li class="liindx">Euclid, <a class="aindx" href="#p509" -title="go to pg. 509">509</a></li> - -<li class="liindx">Euglena, <a class="aindx" href="#p376" -title="go to pg. 376">376</a></li> - -<li class="liindx">Euglypha, <a class="aindx" href="#p189" -title="go to pg. 189">189</a></li> - -<li class="liindx">Euler, L., <a class="aindx" href="#p003" -title="go to pg. 3">3</a>, <a class="aindx" href="#p208" -title="go to pg. 208">208</a>, <a class="aindx" -href="#p385" title="go to pg. 385">385</a>, <a -class="aindx" href="#p484" title="go to pg. 484">484</a>, -<a class="aindx" href="#p690" title="go to pg. -690">690</a></li> - -<li class="liindx">Eulima, <a class="aindx" href="#p559" -title="go to pg. 559">559</a></li> - -<li class="liindx">Eunicea, spicules of, <a class="aindx" -href="#p424" title="go to pg. 424">424</a></li> - -<li class="liindx">Euomphalus, <a class="aindx" -href="#p557" title="go to pg. 557">557</a>, <a -class="aindx" href="#p559" title="go to pg. -559">559</a></li> - -<li class="liindx">Evelyn, John, <a class="aindx" -href="#p652" title="go to pg. 652">652</a></li> - -<li class="liindx">Evolution, <a class="aindx" href="#p549" -title="go to pg. 549">549</a>, <a class="aindx" -href="#p610" title="go to pg. 610">610</a>, etc.</li> - -<li class="liindx">Ewart, A. J., <a class="aindx" -href="#p020" title="go to pg. 20">20</a></li> </ul> - -<ul class="ulindx"> <li class="liindx">Fabre, J. H., <a -class="aindx" href="#p064" title="go to pg. 64">64</a>, -<a class="aindx" href="#p779" title="go to pg. -779">779</a></li> - -<li class="liindx">Facial angle, <a class="aindx" -href="#p742" title="go to pg. 742">742</a>, <a -class="aindx" href="#p770" title="go to pg. 770">770</a>, -<a class="aindx" href="#p772" title="go to pg. -772">772</a></li> - -<li class="liindx">Faraday, M., <a class="aindx" -href="#p163" title="go to pg. 163">163</a>, <a -class="aindx" href="#p167" title="go to pg. 167">167</a>, -<a class="aindx" href="#p428" title="go to pg. -428">428</a>, <a class="aindx" href="#p475" title="go to -pg. 475">475</a></li> - -<li class="liindx">Farmer, J. B. and Digby, <a -class="aindx" href="#p190" title="go to pg. -190">190</a></li> - -<li class="liindx">Fatigue, molecular, <a class="aindx" -href="#p689" title="go to pg. 689">689</a></li> - -<li class="liindx">Faucon, A., <a class="aindx" -href="#p088" title="go to pg. 88">88</a></li> - -<li class="liindx">Favosites, <a class="aindx" href="#p325" -title="go to pg. 325">325</a></li> - -<li class="liindx">Fechner, G. T., <a class="aindx" -href="#p654" title="go to pg. 654">654</a>, <a -class="aindx" href="#p777" title="go to pg. -777">777</a></li> - -<li class="liindx">Fedorow, E. S. von, <a class="aindx" -href="#p338" title="go to pg. 338">338</a></li> - -<li class="liindx">Fehling, H., <a class="aindx" -href="#p076" title="go to pg. 76">76</a>, <a class="aindx" -href="#p126" title="go to pg. 126">126</a></li> - -<li class="liindx">Ferns, spores of, <a class="aindx" -href="#p396" title="go to pg. 396">396</a></li> - -<li class="liindx">Fertilisation, <a class="aindx" -href="#p193" title="go to pg. 193">193</a></li> - -<li class="liindx">Fezzan-worms, <a class="aindx" -href="#p127" title="go to pg. 127">127</a></li> - -<li class="liindx">Fibonacci, <a class="aindx" href="#p643" -title="go to pg. 643">643</a></li> - -<li class="liindx">Fibrillenkonus, <a class="aindx" -href="#p285" title="go to pg. 285">285</a></li> - -<li class="liindx">Fick, R., <a class="aindx" href="#p057" -title="go to pg. 57">57</a>, <a class="aindx" href="#p683" -title="go to pg. 683">683</a></li> - -<li class="liindx">Fickert, C., <a class="aindx" -href="#p606" title="go to pg. 606">606</a></li> - -<li class="liindx">Fidler, Prof. T. Claxton, <a -class="aindx" href="#p691" title="go to pg. 691">691</a>, -<a class="aindx" href="#p674" title="go to pg. -674">674</a>, <a class="aindx" href="#p696" title="go to -pg. 696">696</a></li> - -<li class="liindx">Films, liquid, <a class="aindx" -href="#p215" title="go to pg. 215">215</a>, <a -class="aindx" href="#p217" title="go to pg. 217">217</a>, -<a class="aindx" href="#p426" title="go to pg. -426">426</a></li> - -<li class="liindx">Filter-passers, <a class="aindx" -href="#p039" title="go to pg. 39">39</a></li> - -<li class="liindx">Final cause, <a class="aindx" -href="#p003" title="go to pg. 3">3</a>, <a class="aindx" -href="#p248" title="go to pg. 248">248</a>, <a -class="aindx" href="#p714" title="go to pg. -714">714</a></li> - -<li class="liindx">Fir-cone, <a class="aindx" href="#p635" -title="go to pg. 635">635</a>, <a class="aindx" -href="#p647" title="go to pg. 647">647</a></li> - -<li class="liindx">Fischel, Alfred, <a class="aindx" -href="#p088" title="go to pg. 88">88</a></li> - -<li class="liindx">Fischer, Alfred, <a class="aindx" -href="#p040" title="go to pg. 40">40</a>, <a class="aindx" -href="#p172" title="go to pg. 172">172</a>; <ul> <li -class="liindx">Emil, <a class="aindx" href="#p417" -title="go to pg. 417">417</a>, <a class="aindx" -href="#p418" title="go to pg. 418">418</a>;</li> <li -class="liindx">Otto, <a class="aindx" href="#p030" -title="go to pg. 30">30</a>, <a class="aindx" href="#p699" -title="go to pg. 699">699</a></li> </ul></li> - -<li class="liindx">Fishes, forms of, <a class="aindx" -href="#p748" title="go to pg. 748">748</a></li> - -<li class="liindx">Fission, multiplication by, -<a class="aindx" href="#p151" title="go to pg. -151">151</a></li> - -<li class="liindx">Fissurella, <a class="aindx" -href="#p556" title="go to pg. 556">556</a></li> - -<li class="liindx">FitzGerald, G. F., <a class="aindx" -href="#p158" title="go to pg. 158">158</a>, <a -class="aindx" href="#p281" title="go to pg. 281">281</a>, -<a class="aindx" href="#p323" title="go to pg. -323">323</a>, <a class="aindx" href="#p440" title="go to -pg. 440">440</a>, <a class="aindx" href="#p477" title="go -to pg. 477">477</a></li> - -<li class="liindx">Flagellum, <a class="aindx" -href="#p246" title="go to pg. 246">246</a>, <a -class="aindx" href="#p267" title="go to pg. 267">267</a>, -<a class="aindx" href="#p291" title="go to pg. -291">291</a></li> - -<li class="liindx">Flemming, W., <a class="aindx" -href="#p170" title="go to pg. 170">170</a>, <a -class="aindx" href="#p172" title="go to pg. 172">172</a>, -<a class="aindx" href="#p180" title="go to pg. -180">180</a></li> - -<li class="liindx">Flight, <a class="aindx" href="#p024" -title="go to pg. 24">24</a></li> - -<li class="liindx">Flint, Professor, <a class="aindx" -href="#p673" title="go to pg. 673">673</a></li> - -<li class="liindx">Fluid crystals, <a class="aindx" -href="#p204" title="go to pg. 204">204</a>, <a -class="aindx" href="#p272" title="go to pg. 272">272</a>, -<a class="aindx" href="#p485" title="go to pg. -485">485</a></li> - -<li class="liindx">Fluted pattern, <a class="aindx" -href="#p260" title="go to pg. 260">260</a></li> - -<li class="liindx">Fly’s cornea, <a class="aindx" -href="#p324" title="go to pg. 324">324</a></li> - -<li class="liindx">Fol, Hermann, <a class="aindx" -href="#p168" title="go to pg. 168">168</a>, <a -class="aindx" href="#p194" title="go to pg. -194">194</a></li> - -<li class="liindx">Folliculina, <a class="aindx" -href="#p249" title="go to pg. 249">249</a></li> - -<li class="liindx">Foraminifera, <a class="aindx" -href="#p214" title="go to pg. 214">214</a>, <a -class="aindx" href="#p255" title="go to pg. 255">255</a>, -<a class="aindx" href="#p415" title="go to pg. -415">415</a>, <a class="aindx" href="#p495" title="go to -pg. 495">495</a>, <a class="aindx" href="#p515" title="go -to pg. 515">515</a></li> - -<li class="liindx">Forth Bridge, <a class="aindx" -href="#p694" title="go to pg. 694">694</a>, <a -class="aindx" href="#p699" title="go to pg. 699">699</a>, -<a class="aindx" href="#p700" title="go to pg. -700">700</a></li> - -<li class="liindx">Fossula, <a class="aindx" href="#p390" -title="go to pg. 390">390</a></li> - -<li class="liindx">Foster, M., <a class="aindx" -href="#p185" title="go to pg. 185">185</a></li> - -<li class="liindx">Fraas, E., <a class="aindx" href="#p716" -title="go to pg. 716">716</a></li> - -<li class="liindx">Frankenheim, M. L., <a class="aindx" -href="#p202" title="go to pg. 202">202</a></li> - -<li class="liindx">Frazee, O. E., <a class="aindx" -href="#p153" title="go to pg. 153">153</a></li> - -<li class="liindx">Frédéricq, L., <a class="aindx" -href="#p127" title="go to pg. 127">127</a>, <a -class="aindx" href="#p130" title="go to pg. -130">130</a></li> - -<li class="liindx">Free cell formation, <a class="aindx" -href="#p396" title="go to pg. 396">396</a></li> - -<li class="liindx">Friedenthal, H., <a class="aindx" -href="#p064" title="go to pg. 64">64</a></li> - -<li class="liindx">Frisch, K. von, <a class="aindx" -href="#p671" title="go to pg. 671">671</a></li> - -<li class="liindx">Frog, egg of, <a class="aindx" -href="#p310" title="go to pg. 310">310</a>, <a -class="aindx" href="#p363" title="go to pg. 363">363</a>, -<a class="aindx" href="#p378" title="go to pg. -378">378</a>, <a class="aindx" href="#p382" title="go -to pg. 382">382</a>; <ul> <li class="liindx">growth -of, <a class="aindx" href="#p093" title="go to pg. -93">93</a>, <a class="aindx" href="#p126" title="go to pg. -126">126</a></li> </ul></li> - -<li class="liindx">Froth or foam, <a class="aindx" -href="#p171" title="go to pg. 171">171</a>, <a -class="aindx" href="#p205" title="go to pg. 205">205</a>, -<a class="aindx" href="#p305" title="go to pg. -305">305</a>, <a class="aindx" href="#p314" title="go to -pg. 314">314</a>, <a class="aindx" href="#p322" title="go -to pg. 322">322</a>, <a class="aindx" href="#p343" -title="go to pg. 343">343</a></li> - -<li class="liindx">Froude, W., <a class="aindx" -href="#p022" title="go to pg. 22">22</a></li> - -<li class="liindx">Fucus, <a class="aindx" href="#p355" -title="go to pg. 355">355</a></li> - -<li class="liindx">Fundulus, <a class="aindx" href="#p125" -title="go to pg. 125">125</a></li> - -<li class="liindx">Fusulina, <a class="aindx" href="#p593" -title="go to pg. 593">593</a>, <a class="aindx" -href="#p594" title="go to pg. 594">594</a></li> - -<li class="liindx">Fusus, <a class="aindx" href="#p527" -title="go to pg. 527">527</a>, <a class="aindx" -href="#p557" title="go to pg. 557">557</a></li> </ul> - -<ul class="ulindx"> <li class="liindx">Gadow, H. -F., <a class="aindx" href="#p628" title="go to pg. -628">628</a></li> - -<li class="liindx">Galathea, <a class="aindx" href="#p273" -title="go to pg. 273">273</a></li> - -<li class="liindx">Galen, <a class="aindx" href="#p003" -title="go to pg. 3">3</a>, <a class="aindx" href="#p465" -title="go to pg. 465">465</a>, <a class="aindx" -href="#p656" title="go to pg. 656">656</a></li> - -<li class="liindx">Galileo, <a class="aindx" href="#p008" -title="go to pg. 8">8</a>, <a class="aindx" href="#p019" -title="go to pg. 19">19</a>, <a class="aindx" href="#p028" -title="go to pg. 28">28</a>, <a class="aindx" -href="#p562" title="go to pg. 562">562</a>, <a -class="aindx" href="#p677" title="go to pg. 677">677</a>, -<a class="aindx" href="#p720" title="go to pg. -720">720</a></li> - -<li class="liindx">Gallardo, A., <a class="aindx" -href="#p163" title="go to pg. 163">163</a></li> - -<li class="liindx">Galloway, Principal, <a class="aindx" -href="#p672" title="go to pg. 672">672</a></li> - -<li class="liindx">Gamble, F. A., <a class="aindx" -href="#p458" title="go to pg. 458">458</a></li> - -<li class="liindx">Ganglion-cells, size of, <a -class="aindx" href="#p037" title="go to pg. 37">37</a></li> - -<li class="liindx">Gans, R., <a class="aindx" href="#p046" -title="go to pg. 46">46</a></li> - -<li class="liindx">Garden of Cyrus, <a class="aindx" -href="#p324" title="go to pg. 324">324</a>, <a -class="aindx" href="#p329" title="go to pg. -329">329</a></li> - -<li class="liindx">Gastrula, <a class="aindx" href="#p344" -title="go to pg. 344">344</a></li> - -<li class="liindx">Gauss, K. F., <a class="aindx" -href="#p207" title="go to pg. 207">207</a>, <a -class="aindx" href="#p278" title="go to pg. 278">278</a>, -<a class="aindx" href="#p723" title="go to pg. -723">723</a></li> - -<li class="liindx">Gebhardt, W., <a class="aindx" -href="#p430" title="go to pg. 430">430</a>, <a -class="aindx" href="#p683" title="go to pg. -683">683</a></li> - -<li class="liindx">Gelatination, water of, <a class="aindx" -href="#p203" title="go to pg. 203">203</a></li> - -<li class="liindx">Generating curves and spirals, <a -class="aindx" href="#p526" title="go to pg. 526">526</a>, -<a class="aindx" href="#p561" title="go to pg. -561">561</a>, <a class="aindx" href="#p615" title="go to -pg. 615">615</a>, <a class="aindx" href="#p637" title="go -to pg. 637">637</a>, <a class="aindx" href="#p641" -title="go to pg. 641">641</a></li> - -<li class="liindx">Geodetics, <a class="aindx" href="#p440" -title="go to pg. 440">440</a>, <a class="aindx" -href="#p488" title="go to pg. 488">488</a></li> - -<li class="liindx">Geoffroy St Hilaire, Et. de, -<a class="aindx" href="#p714" title="go to pg. -714">714</a></li> - -<li class="liindx">Geotropism, <a class="aindx" -href="#p211" title="go to pg. 211">211</a></li> - -<li class="liindx">Gerassimow, J. J., <a class="aindx" -href="#p035" title="go to pg. 35">35</a></li> - -<li class="liindx">Gerdy, P. N., <a class="aindx" -href="#p491" title="go to pg. 491">491</a></li> - -<li class="liindx">Geryon, <a class="aindx" href="#p744" -title="go to pg. 744">744</a></li> - -<li class="liindx">Gestaltungskraft, <a class="aindx" -href="#p485" title="go to pg. 485">485</a></li> - -<li class="liindx">Giard, A., <a class="aindx" href="#p156" -title="go to pg. 156">156</a></li> - -<li class="liindx">Gilmore, C. W., <a class="aindx" -href="#p707" title="go to pg. 707">707</a></li> - -<li class="liindx">Giraffe, <a class="aindx" -href="#p705" title="go to pg. 705">705</a>, <a -class="aindx" href="#p730" title="go to pg. 730">730</a>, -<a class="aindx" href="#p738" title="go to pg. -738">738</a></li> - -<li class="liindx">Girardia, <a class="aindx" href="#p321" -title="go to pg. 321">321</a>, <a class="aindx" -href="#p408" title="go to pg. 408">408</a></li> - -<li class="liindx">Glaisher, J., <a class="aindx" -href="#p250" title="go to pg. 250">250</a></li> - -<li class="liindx">Glassblowing, <a class="aindx" -href="#p238" title="go to pg. 238">238</a>, <a -class="aindx" href="#p737" title="go to pg. -737">737</a></li> - -<li class="liindx">Gley, E., <a class="aindx" href="#p135" -title="go to pg. 135">135</a>, <a class="aindx" -href="#p136" title="go to pg. 136">136</a></li> - -<li class="liindx">Globigerina, <a class="aindx" -href="#p214" title="go to pg. 214">214</a>, <a -class="aindx" href="#p234" title="go to pg. 234">234</a>, -<a class="aindx" href="#p440" title="go to pg. -440">440</a>, <a class="aindx" href="#p495" title="go -to pg. 495">495</a>, <a class="aindx" href="#p589" -title="go to pg. 589">589</a>, <a class="aindx" -href="#p602" title="go to pg. 602">602</a>, <a -class="aindx" href="#p604" title="go to pg. 604">604</a>, -<a class="aindx" href="#p606" title="go to pg. -606">606</a></li> - -<li class="liindx">Gnomon, <a class="aindx" href="#p509" -title="go to pg. 509">509</a>, <a class="aindx" -href="#p515" title="go to pg. 515">515</a>, <a -class="aindx" href="#p591" title="go to pg. -591">591</a></li> - -<li class="liindx">Goat, horns of, <a class="aindx" -href="#p613" title="go to pg. 613">613</a></li> - -<li class="liindx">Goat moth, wings of, <a class="aindx" -href="#p430" title="go to pg. 430">430</a></li> - -<li class="liindx">Goebel, K., <a class="aindx" -href="#p321" title="go to pg. 321">321</a>, <a -class="aindx" href="#p397" title="go to pg. 397">397</a>, -<a class="aindx" href="#p408" title="go to pg. -408">408</a></li> - -<li class="liindx">Goethe, <a class="aindx" href="#p020" -title="go to pg. 20">20</a>, <a class="aindx" href="#p038" -title="go to pg. 38">38</a>, <a class="aindx" -href="#p199" title="go to pg. 199">199</a>, <a -class="aindx" href="#p714" title="go to pg. 714">714</a>, -<a class="aindx" href="#p719" title="go to pg. -719">719</a></li> - -<li class="liindx">Golden Mean, <a class="aindx" -href="#p511" title="go to pg. 511">511</a>, <a -class="aindx" href="#p643" title="go to pg. 643">643</a>, -<a class="aindx" href="#p649" title="go to pg. -649">649</a></li> - -<li class="liindx">Goldschmidt, R., <a class="aindx" -href="#p286" title="go to pg. 286">286</a></li> - -<li class="liindx">Goniatites, <a class="aindx" -href="#p550" title="go to pg. 550">550</a>, <a -class="aindx" href="#p728" title="go to pg. -728">728</a></li> - -<li class="liindx">Gonothyraea, <a class="aindx" -href="#p747" title="go to pg. 747">747</a></li> - -<li class="liindx">Goodsir, John, <a class="aindx" -href="#p156" title="go to pg. 156">156</a>, <a -class="aindx" href="#p196" title="go to pg. 196">196</a>, -<a class="aindx" href="#p580" title="go to pg. -580">580</a></li> - -<li class="liindx">Gottlieb, H., <a class="aindx" -href="#p699" title="go to pg. 699">699</a></li> - -<li class="liindx">Gourd, form of, <a class="aindx" -href="#p737" title="go to pg. 737">737</a></li> - -<li class="liindx">Grabau, A. H., <a class="aindx" -href="#p531" title="go to pg. 531">531</a>, <a -class="aindx" href="#p539" title="go to pg. 539">539</a>, -<a class="aindx" href="#p550" title="go to pg. -550">550</a></li> - -<li class="liindx">Graham, Thomas, <a class="aindx" -href="#p162" title="go to pg. 162">162</a>, <a -class="aindx" href="#p201" title="go to pg. 201">201</a>, -<a class="aindx" href="#p203" title="go to pg. -203">203</a></li> - -<li class="liindx">Grant, Kerr, <a class="aindx" -href="#p259" title="go to pg. 259">259</a></li> - -<li class="liindx">Grantia, <a class="aindx" href="#p445" -title="go to pg. 445">445</a></li> - -<li class="liindx">Graphic statics, <a class="aindx" -href="#p682" title="go to pg. 682">682</a></li> - -<li class="liindx">Gravitation, <a class="aindx" -href="#p012" title="go to pg. 12">12</a>, <a class="aindx" -href="#p032" title="go to pg. 32">32</a></li> - -<li class="liindx">Gray, J., <a class="aindx" href="#p188" -title="go to pg. 188">188</a></li> - -<li class="liindx">Greenhill, Sir A. G., <a class="aindx" -href="#p019" title="go to pg. 19">19</a></li> - -<li class="liindx">Gregory, D. F., <a class="aindx" -href="#p330" title="go to pg. 330">330</a>, <a -class="aindx" href="#p675" title="go to pg. -675">675</a></li> - -<li class="liindx">Greville, R. K., <a class="aindx" -href="#p386" title="go to pg. 386">386</a></li> - -<li class="liindx">Gromia, <a class="aindx" href="#p234" -title="go to pg. 234">234</a>, <a class="aindx" -href="#p257" title="go to pg. 257">257</a></li> - -<li class="liindx">Gruber, A., <a class="aindx" -href="#p165" title="go to pg. 165">165</a></li> - -<li class="liindx">Gryphaea, <a class="aindx" -href="#p546" title="go to pg. 546">546</a>, <a -class="aindx" href="#p576" title="go to pg. 576">576</a>, -<a class="aindx" href="#p577" title="go to pg. -577">577</a></li> - -<li class="liindx">Guard-cells, <a class="aindx" -href="#p394" title="go to pg. 394">394</a></li> - -<li class="liindx">Gudernatsch, J. F., <a class="aindx" -href="#p136" title="go to pg. 136">136</a></li> - -<li class="liindx">Guillemot, egg of, <a class="aindx" -href="#p652" title="go to pg. 652">652</a></li> - -<li class="liindx">Gulliver, G., <a class="aindx" -href="#p036" title="go to pg. 36">36</a></li> - -<li class="liindx">Günther, F. C., <a class="aindx" -href="#p633" title="go to pg. 633">633</a>, <a -class="aindx" href="#p654" title="go to pg. -654">654</a></li> - -<li class="liindx">Gurwitsch, A., <a class="aindx" -href="#p285" title="go to pg. 285">285</a></li> </ul> - -<ul class="ulindx"> <li class="liindx">Häcker, V., -<a class="aindx" href="#p458" title="go to pg. -458">458</a></li> - -<li class="liindx">Haddock, <a class="aindx" href="#p774" -title="go to pg. 774">774</a></li> - -<li class="liindx">Haeckel, E., <a class="aindx" -href="#p199" title="go to pg. 199">199</a>, <a -class="aindx" href="#p445" title="go to pg. 445">445</a>, -<a class="aindx" href="#p454" title="go to pg. -454">454</a>, <a class="aindx" href="#p455" title="go -to pg. 455">455</a>, <a class="aindx" href="#p457" -title="go to pg. 457">457</a>, <a class="aindx" -href="#p467" title="go to pg. 467">467</a>, <a -class="aindx" href="#p480" title="go to pg. 480">480</a>, -<a class="aindx" href="#p481" title="go to pg. -481">481</a></li> - -<li class="liindx">Hair, pigmentation of, <a class="aindx" -href="#p430" title="go to pg. 430">430</a></li> - -<li class="liindx">Hales, Stephen, <a class="aindx" -href="#p036" title="go to pg. 36">36</a>, <a class="aindx" -href="#p059" title="go to pg. 59">59</a>, <a class="aindx" -href="#p095" title="go to pg. 95">95</a>, <a class="aindx" -href="#p669" title="go to pg. 669">669</a></li> - -<li class="liindx">Haliotis, <a class="aindx" href="#p514" -title="go to pg. 514">514</a>, <a class="aindx" -href="#p527" title="go to pg. 527">527</a>, <a -class="aindx" href="#p546" title="go to pg. 546">546</a>, -<a class="aindx" href="#p547" title="go to pg. -547">547</a>, <a class="aindx" href="#p554" title="go to -pg. 554">554</a>, <a class="aindx" href="#p555" title="go -to pg. 555">555</a>, <a class="aindx" href="#p557" -title="go to pg. 557">557</a>, <a class="aindx" -href="#p561" title="go to pg. 561">561</a></li> - -<li class="liindx">Hall, C. E., <a class="aindx" -href="#p119" title="go to pg. 119">119</a></li> - -<li class="liindx">Haller, A. von, <a class="aindx" -href="#p002" title="go to pg. 2">2</a>, <a class="aindx" -href="#p054" title="go to pg. 54">54</a>, <a class="aindx" -href="#p056" title="go to pg. 56">56</a>, <a class="aindx" -href="#p059" title="go to pg. 59">59</a>, <a class="aindx" -href="#p064" title="go to pg. 64">64</a>, <a class="aindx" -href="#p068" title="go to pg. 68">68</a></li> - -<li class="liindx">Hardesty, Irving, <a class="aindx" -href="#p037" title="go to pg. 37">37</a></li> - -<li class="liindx">Hardy, W. B., <a class="aindx" -href="#p160" title="go to pg. 160">160</a>, <a -class="aindx" href="#p162" title="go to pg. 162">162</a>, -<a class="aindx" href="#p172" title="go to pg. -172">172</a>, <a class="aindx" href="#p187" title="go to -pg. 187">187</a>, <a class="aindx" href="#p287" title="go -to pg. 287">287</a></li> - -<li class="liindx">Harlé, N., <a class="aindx" href="#p028" -title="go to pg. 28">28</a></li> - -<li class="liindx">Harmozones, <a class="aindx" -href="#p135" title="go to pg. 135">135</a></li> - -<li class="liindx">Harpa, <a class="aindx" href="#p526" -title="go to pg. 526">526</a>, <a class="aindx" -href="#p528" title="go to pg. 528">528</a>, <a -class="aindx" href="#p559" title="go to pg. -559">559</a></li> - -<li class="liindx">Harper, R. A., <a class="aindx" -href="#p283" title="go to pg. 283">283</a></li> - -<li class="liindx">Harpinia, <a class="aindx" href="#p746" -title="go to pg. 746">746</a></li> - -<li class="liindx">Harting, P., <a class="aindx" -href="#p282" title="go to pg. 282">282</a>, <a -class="aindx" href="#p420" title="go to pg. 420">420</a>, -<a class="aindx" href="#p426" title="go to pg. -426">426</a>, <a class="aindx" href="#p434" title="go to -pg. 434">434</a></li> - -<li class="liindx">Hartog, M., <a class="aindx" -href="#p163" title="go to pg. 163">163</a>, <a -class="aindx" href="#p327" title="go to pg. -327">327</a></li> - -<li class="liindx">Harvey, E. N. and H. W., <a -class="aindx" href="#p187" title="go to pg. -187">187</a></li> - -<li class="liindx">Hatai, S., <a class="aindx" href="#p132" -title="go to pg. 132">132</a>, <a class="aindx" -href="#p135" title="go to pg. 135">135</a></li> - -<li class="liindx">Hatchett, C., <a class="aindx" -href="#p420" title="go to pg. 420">420</a></li> - -<li class="liindx">Hatschek, B., <a class="aindx" -href="#p180" title="go to pg. 180">180</a></li> - -<li class="liindx">Haughton, Rev. S., <a class="aindx" -href="#p334" title="go to pg. 334">334</a>, <a -class="aindx" href="#p666" title="go to pg. -666">666</a></li> - -<li class="liindx">Haüy, R. J., <a class="aindx" -href="#p720" title="go to pg. 720">720</a></li> - -<li class="liindx">Hay, O. P., <a class="aindx" -href="#p707" title="go to pg. 707">707</a></li> - -<li class="liindx">Haycraft, J. B., <a class="aindx" -href="#p211" title="go to pg. 211">211</a>, <a -class="aindx" href="#p690" title="go to pg. -690">690</a></li> - -<li class="liindx">Head, length of, <a class="aindx" -href="#p093" title="go to pg. 93">93</a></li> - -<li class="liindx">Heart, growth of, <a class="aindx" -href="#p089" title="go to pg. 89">89</a>; <ul> <li -class="liindx">muscles of, <a class="aindx" href="#p490" -title="go to pg. 490">490</a></li> </ul></li> - -<li class="liindx">Heath, Sir T., <a class="aindx" -href="#p511" title="go to pg. 511">511</a></li> - -<li class="liindx">Hegel, G. W. F., <a class="aindx" -href="#p004" title="go to pg. 4">4</a></li> - -<li class="liindx">Hegler, <a class="aindx" href="#p680" -title="go to pg. 680">680</a>, <a class="aindx" -href="#p688" title="go to pg. 688">688</a></li> - -<li class="liindx">Heidenhain, M., <a class="aindx" -href="#p170" title="go to pg. 170">170</a>, <a -class="aindx" href="#p212" title="go to pg. -212">212</a></li> - -<li class="liindx">Heilmann, Gerhard, <a class="aindx" -href="#p757" title="go to pg. 757">757</a>, <a -class="aindx" href="#p768" title="go to pg. 768">768</a>, -<a class="aindx" href="#p772" title="go to pg. -772">772</a></li> - -<li class="liindx">Helicoid, <a class="aindx" -href="#p230" title="go to pg. 230">230</a>; <ul> <li -class="liindx">cyme, <a class="aindx" href="#p502" -title="go to pg. 502">502</a>, <a class="aindx" -href="#p605" title="go to pg. 605">605</a></li> </ul></li> - -<li class="liindx">Helicometer, <a class="aindx" -href="#p529" title="go to pg. 529">529</a></li> - -<li class="liindx">Helicostyla, <a class="aindx" -href="#p557" title="go to pg. 557">557</a></li> - -<li class="liindx">Heliolites, <a class="aindx" -href="#p326" title="go to pg. 326">326</a></li> - -<li class="liindx">Heliozoa, <a class="aindx" href="#p264" -title="go to pg. 264">264</a>, <a class="aindx" -href="#p460" title="go to pg. 460">460</a></li> - -<li class="liindx">Helix, <a class="aindx" href="#p528" -title="go to pg. 528">528</a>, <a class="aindx" -href="#p557" title="go to pg. 557">557</a></li> - -<li class="liindx">Helmholtz, H. von, <a class="aindx" -href="#p002" title="go to pg. 2">2</a>, <a class="aindx" -href="#p009" title="go to pg. 9">9</a>, <a class="aindx" -href="#p025" title="go to pg. 25">25</a></li> - -<li class="liindx">Henderson, W. P., <a class="aindx" -href="#p323" title="go to pg. 323">323</a></li> - -<li class="liindx">Henslow, G., <a class="aindx" -href="#p636" title="go to pg. 636">636</a></li> - -<li class="liindx">Heredity, <a class="aindx" -href="#p158" title="go to pg. 158">158</a>, <a -class="aindx" href="#p286" title="go to pg. 286">286</a>, -<a class="aindx" href="#p715" title="go to pg. -715">715</a></li> - -<li class="liindx">Hermann, F., <a class="aindx" -href="#p170" title="go to pg. 170">170</a></li> - -<li class="liindx">Hero of Alexandria, <a class="aindx" -href="#p509" title="go to pg. 509">509</a></li> - -<li class="liindx">Heron-Allen, E., <a class="aindx" -href="#p257" title="go to pg. 257">257</a>, <a -class="aindx" href="#p415" title="go to pg. 415">415</a>, -<a class="aindx" href="#p465" title="go to pg. -465">465</a></li> - -<li class="liindx">Herpetomonas, <a class="aindx" -href="#p268" title="go to pg. 268">268</a></li> - -<li class="liindx">Hertwig, O., <a class="aindx" -href="#p056" title="go to pg. 56">56</a>, <a class="aindx" -href="#p114" title="go to pg. 114">114</a>, <a -class="aindx" href="#p153" title="go to pg. 153">153</a>, -<a class="aindx" href="#p199" title="go to pg. -199">199</a>, <a class="aindx" href="#p310" title="go -to pg. 310">310</a>; <ul> <li class="liindx">R., <a -class="aindx" href="#p170" title="go to pg. 170">170</a>, -<a class="aindx" href="#p285" title="go to pg. -285">285</a></li></ul></li> - -<li class="liindx">Hertzog, R. O., <a class="aindx" -href="#p109" title="go to pg. 109">109</a></li> - -<li class="liindx">Hess, W., <a class="aindx" href="#p666" -title="go to pg. 666">666</a>, <a class="aindx" -href="#p668" title="go to pg. 668">668</a></li> - -<li class="liindx">Heteronymous horns, <a class="aindx" -href="#p619" title="go to pg. 619">619</a></li> - -<li class="liindx">Heterophyllia, <a class="aindx" -href="#p388" title="go to pg. 388">388</a></li> - -<li class="liindx">Hexactinellids, <a class="aindx" -href="#p429" title="go to pg. 429">429</a>, <a -class="aindx" href="#p452" title="go to pg. 452">452</a>, -<a class="aindx" href="#p453" title="go to pg. -453">453</a></li> - -<li class="liindx">Hexagonal symmetry, <a class="aindx" -href="#p319" title="go to pg. 319">319</a>, <a -class="aindx" href="#p323" title="go to pg. 323">323</a>, -<a class="aindx" href="#p471" title="go to pg. -471">471</a>, <a class="aindx" href="#p513" title="go to -pg. 513">513</a></li> - -<li class="liindx">Hickson, S. J., <a class="aindx" -href="#p424" title="go to pg. 424">424</a></li> - -<li class="liindx">Hippopus, <a class="aindx" href="#p561" -title="go to pg. 561">561</a></li> - -<li class="liindx">His, W., <a class="aindx" href="#p055" -title="go to pg. 55">55</a>, <a class="aindx" href="#p056" -title="go to pg. 56">56</a>, <a class="aindx" href="#p074" -title="go to pg. 74">74</a>, <a class="aindx" href="#p075" -title="go to pg. 75">75</a></li> - -<li class="liindx">Hobbes, Thomas, <a class="aindx" -href="#p159" title="go to pg. 159">159</a></li> - -<li class="liindx">Höber, R., <a class="aindx" -href="#p001" title="go to pg. 1">1</a>, <a class="aindx" -href="#p126" title="go to pg. 126">126</a>, <a -class="aindx" href="#p130" title="go to pg. 130">130</a>, -<a class="aindx" href="#p172" title="go to pg. -172">172</a></li> - -<li class="liindx">Hodograph, <a class="aindx" href="#p516" -title="go to pg. 516">516</a></li> - -<li class="liindx">Hoffmann, C., <a class="aindx" -href="#p628" title="go to pg. 628">628</a></li> - -<li class="liindx">Hofmeister, F., <a class="aindx" -href="#p041" title="go to pg. 41">41</a>; W., <a -class="aindx" href="#p087" title="go to pg. 87">87</a>, <a -class="aindx" href="#p210" title="go to pg. 210">210</a>, -<a class="aindx" href="#p234" title="go to pg. -234">234</a>, <a class="aindx" href="#p304" title="go to -pg. 304">304</a>, <a class="aindx" href="#p306" title="go -to pg. 306">306</a>, <a class="aindx" href="#p636" -title="go to pg. 636">636</a>, <a class="aindx" -href="#p639" title="go to pg. 639">639</a></li> - -<li class="liindx">Holland, W. J., <a class="aindx" -href="#p707" title="go to pg. 707">707</a></li> - -<li class="liindx">Holmes, O. W., <a class="aindx" -href="#p062" title="go to pg. 62">62</a>, <a class="aindx" -href="#p737" title="go to pg. 737">737</a></li> - -<li class="liindx">Holothuroid spicules, <a class="aindx" -href="#p440" title="go to pg. 440">440</a>, <a -class="aindx" href="#p451" title="go to pg. -451">451</a></li> - -<li class="liindx">Homonymous horns, <a class="aindx" -href="#p619" title="go to pg. 619">619</a></li> - -<li class="liindx">Homoplasy, <a class="aindx" href="#p251" -title="go to pg. 251">251</a></li> - -<li class="liindx">Hooke, Robert, <a class="aindx" -href="#p205" title="go to pg. 205">205</a></li> - -<li class="liindx">Hop, growth of, <a class="aindx" -href="#p118" title="go to pg. 118">118</a>; <ul> <li -class="liindx">stem of, <a class="aindx" href="#p627" -title="go to pg. 627">627</a></li> </ul></li> - -<li class="liindx">Horace, <a class="aindx" href="#p044" -title="go to pg. 44">44</a></li> - -<li class="liindx">Hormones, <a class="aindx" href="#p135" -title="go to pg. 135">135</a></li> - -<li class="liindx">Horns, <a class="aindx" href="#p612" -title="go to pg. 612">612</a></li> - -<li class="liindx">Horse, <a class="aindx" href="#p694" -title="go to pg. 694">694</a>, <a class="aindx" -href="#p701" title="go to pg. 701">701</a>, <a -class="aindx" href="#p703" title="go to pg. 703">703</a>, -<a class="aindx" href="#p764" title="go to pg. -764">764</a></li> - -<li class="liindx">Houssay, F., <a class="aindx" -href="#p021" title="go to pg. 21">21</a></li> - -<li class="liindx">Huber, P., <a class="aindx" href="#p332" -title="go to pg. 332">332</a></li> - -<li class="liindx">Huia bird, <a class="aindx" href="#p633" -title="go to pg. 633">633</a></li> - -<li class="liindx">Humboldt, A. von, <a class="aindx" -href="#p127" title="go to pg. 127">127</a></li> - -<li class="liindx">Hume, David, <a class="aindx" -href="#p006" title="go to pg. 6">6</a></li> - -<li class="liindx">Hunter, John, <a class="aindx" -href="#p667" title="go to pg. 667">667</a>, <a -class="aindx" href="#p669" title="go to pg. 669">669</a>, -<a class="aindx" href="#p713" title="go to pg. -713">713</a>, <a class="aindx" href="#p715" title="go to -pg. 715">715</a></li> - -<li class="liindx">Huxley, T. H., <a class="aindx" -href="#p423" title="go to pg. 423">423</a>, <a -class="aindx" href="#p722" title="go to pg. 722">722</a>, -<a class="aindx" href="#p752" title="go to pg. -752">752</a></li> - -<li class="liindx">Hyacinth, <a class="aindx" href="#p322" -title="go to pg. 322">322</a>, <a class="aindx" -href="#p394" title="go to pg. 394">394</a></li> - -<li class="liindx">Hyalaea, <a class="aindx" href="#p571" -title="go to pg. 571">571</a>–<a class="aindx" href="#p577" -title="go to pg. 577">577</a></li> - -<li class="liindx">Hyalonema, <a class="aindx" href="#p442" -title="go to pg. 442">442</a></li> - -<li class="liindx">Hyatt, A., <a class="aindx" href="#p548" -title="go to pg. 548">548</a></li> - -<li class="liindx">Hyde, Ida H., <a class="aindx" -href="#p125" title="go to pg. 125">125</a>, <a -class="aindx" href="#p163" title="go to pg. 163">163</a>, -<a class="aindx" href="#p184" title="go to pg. -184">184</a>, <a class="aindx" href="#p188" title="go to -pg. 188">188</a></li> - -<li class="liindx">Hydra, <a class="aindx" href="#p252" -title="go to pg. 252">252</a>; <ul> <li class="liindx">egg -of, <a class="aindx" href="#p164" title="go to pg. -164">164</a></li> </ul></li> - -<li class="liindx">Hydractinia, <a class="aindx" -href="#p342" title="go to pg. 342">342</a></li> - -<li class="liindx">Hydraulics, <a class="aindx" -href="#p669" title="go to pg. 669">669</a></li> - -<li class="liindx">Hydrocharis, <a class="aindx" -href="#p234" title="go to pg. 234">234</a></li> - -<li class="liindx">Hyperia, <a class="aindx" href="#p746" -title="go to pg. 746">746</a></li> - -<li class="liindx">Hyrachyus, <a class="aindx" href="#p760" -title="go to pg. 760">760</a>, <a class="aindx" -href="#p765" title="go to pg. 765">765</a></li> - -<li class="liindx">Hyracotherium, <a class="aindx" -href="#fig402" title="go to Fig. 402">766</a>, <a -class="aindx" href="#p768" title="go to pg. -768">768</a></li> </ul> - -<ul class="ulindx"> <li class="liindx">Ibex, -<a class="aindx" href="#p617" title="go to pg. -617">617</a></li> - -<li class="liindx">Ice, structure of, <a class="aindx" -href="#p428" title="go to pg. 428">428</a></li> - -<li class="liindx">Ichthyosaurus, <a class="aindx" -href="#p755" title="go to pg. 755">755</a></li> - -<li class="liindx">Icosahedron, <a class="aindx" -href="#p478" title="go to pg. 478">478</a></li> - -<li class="liindx">Iguanodon, <a class="aindx" href="#p706" -title="go to pg. 706">706</a>, <a class="aindx" -href="#p708" title="go to pg. 708">708</a></li> - -<li class="liindx">Inachus, sperm-cells of, <a -class="aindx" href="#p273" title="go to pg. -273">273</a></li> - -<li class="liindx">Infusoria, <a class="aindx" href="#p246" -title="go to pg. 246">246</a>, <a class="aindx" -href="#p489" title="go to pg. 489">489</a></li> - -<li class="liindx">Intussusception, <a class="aindx" -href="#p202" title="go to pg. 202">202</a></li> - -<li class="liindx">Inulin, <a class="aindx" href="#p432" -title="go to pg. 432">432</a></li> - -<li class="liindx">Invagination, <a class="aindx" -href="#p056" title="go to pg. 56">56</a>, <a class="aindx" -href="#p344" title="go to pg. 344">344</a></li> - -<li class="liindx">Iodine, <a class="aindx" href="#p136" -title="go to pg. 136">136</a></li> - -<li class="liindx">Irvine, Robert, <a class="aindx" -href="#p414" title="go to pg. 414">414</a>, <a -class="aindx" href="#p434" title="go to pg. -434">434</a></li> - -<li class="liindx">Isocardia, <a class="aindx" href="#p561" -title="go to pg. 561">561</a>, <a class="aindx" -href="#p577" title="go to pg. 577">577</a></li> - -<li class="liindx">Isoperimetrical problems, <a -class="aindx" href="#p208" title="go to pg. 208">208</a>, -<a class="aindx" href="#p346" title="go to pg. -346">346</a></li> - -<li class="liindx">Isotonic solutions, <a class="aindx" -href="#p130" title="go to pg. 130">130</a>, <a -class="aindx" href="#p274" title="go to pg. -274">274</a></li> - -<li class="liindx">Iterson, G. van, <a class="aindx" -href="#p595" title="go to pg. 595">595</a></li> </ul> - -<ul class="ulindx"> <li class="liindx">Jackson, C. -M., <a class="aindx" href="#p075" title="go to pg. -75">75</a>, <a class="aindx" href="#p088" title="go to pg. -88">88</a>, <a class="aindx" href="#p106" title="go to pg. -106">106</a></li> - -<li class="liindx">Jamin, J. C., <a class="aindx" -href="#p418" title="go to pg. 418">418</a></li> - -<li class="liindx">Janet, Paul, <a class="aindx" -href="#p005" title="go to pg. 5">5</a>, <a class="aindx" -href="#p018" title="go to pg. 18">18</a>, <a class="aindx" -href="#p673" title="go to pg. 673">673</a></li> - -<li class="liindx">Japp, F. R., <a class="aindx" -href="#p417" title="go to pg. 417">417</a></li> - -<li class="liindx">Jellett, J. H., <a class="aindx" -href="#p001" title="go to pg. 1">1</a></li> - -<li class="liindx">Jenkin, C. F., <a class="aindx" -href="#p444" title="go to pg. 444">444</a></li> - -<li class="liindx">Jenkinson, J. W., <a class="aindx" -href="#p094" title="go to pg. 94">94</a>, <a -class="aindx" href="#p114" title="go to pg. 114">114</a>, -<a class="aindx" href="#p170" title="go to pg. -170">170</a></li> - -<li class="liindx">Jennings, H. S., <a class="aindx" -href="#p212" title="go to pg. 212">212</a>, <a -class="aindx" href="#p492" title="go to pg. 492">492</a>; -<ul> <li class="liindx">Vaughan, <a class="aindx" -href="#p424" title="go to pg. 424">424</a></li> </ul></li> - -<li class="liindx">Jensen, P., <a class="aindx" -href="#p211" title="go to pg. 211">211</a></li> - -<li class="liindx">Johnson, Dr S., <a class="aindx" -href="#p062" title="go to pg. 62">62</a></li> - -<li class="liindx">Joly, John, <a class="aindx" -href="#p009" title="go to pg. 9">9</a>, <a class="aindx" -href="#p063" title="go to pg. 63">63</a></li> - -<li class="liindx">Jost, L., <a class="aindx" href="#p110" -title="go to pg. 110">110</a>, <a class="aindx" -href="#p111" title="go to pg. 111">111</a></li> - -<li class="liindx">Juncus, pith of, <a class="aindx" -href="#p335" title="go to pg. 335">335</a></li> - -<li class="liindx">Jungermannia, <a class="aindx" -href="#p404" title="go to pg. 404">404</a></li> </ul> - -<ul class="ulindx"> <li class="liindx">Kangaroo, <a -class="aindx" href="#p705" title="go to pg. 705">705</a>, -<a class="aindx" href="#p706" title="go to pg. -706">706</a>, <a class="aindx" href="#p709" title="go to -pg. 709">709</a></li> - -<li class="liindx">Kanitz, Al., <a class="aindx" -href="#p109" title="go to pg. 109">109</a></li> - -<li class="liindx">Kant, Immanuel, <a class="aindx" -href="#p001" title="go to pg. 1">1</a>, <a class="aindx" -href="#p003" title="go to pg. 3">3</a>, <a class="aindx" -href="#p714" title="go to pg. 714">714</a></li> - -<li class="liindx">Kappers, C. U. A., <a class="aindx" -href="#p566" title="go to pg. 566">566</a></li> - -<li class="liindx">Kellicott, W. E., <a class="aindx" -href="#p091" title="go to pg. 91">91</a></li> - -<li class="liindx">Kelvin, Lord, <a class="aindx" -href="#p009" title="go to pg. 9">9</a>, <a class="aindx" -href="#p049" title="go to pg. 49">49</a>, <a class="aindx" -href="#p188" title="go to pg. 188">188</a>, <a -class="aindx" href="#p202" title="go to pg. 202">202</a>, -<a class="aindx" href="#p336" title="go to pg. -336">336</a>, <a class="aindx" href="#p453" title="go to -pg. 453">453</a></li> - -<li class="liindx">Kepler, <a class="aindx" href="#p328" -title="go to pg. 328">328</a>, <a class="aindx" -href="#p480" title="go to pg. 480">480</a>, <a -class="aindx" href="#p486" title="go to pg. 486">486</a>, -<a class="aindx" href="#p643" title="go to pg. -643">643</a>, <a class="aindx" href="#p650" title="go to -pg. 650">650</a></li> - -<li class="liindx">Kienitz-Gerloff, F., <a class="aindx" -href="#p404" title="go to pg. 404">404</a>, <a -class="aindx" href="#p408" title="go to pg. -408">408</a></li> - -<li class="liindx">Kirby and Spence, <a class="aindx" -href="#p028" title="go to pg. 28">28</a>, <a class="aindx" -href="#p030" title="go to pg. 30">30</a>, <a class="aindx" -href="#p127" title="go to pg. 127">127</a></li> - -<li class="liindx">Kirchner, A., <a class="aindx" -href="#p683" title="go to pg. 683">683</a></li> - -<li class="liindx">Kirkpatrick, R., <a class="aindx" -href="#p437" title="go to pg. 437">437</a></li> - -<li class="liindx">Klebs, G., <a class="aindx" href="#p306" -title="go to pg. 306">306</a></li> - -<li class="liindx">Kny, L., <a class="aindx" href="#p680" -title="go to pg. 680">680</a></li> - -<li class="liindx">Koch, G. von, <a class="aindx" -href="#p423" title="go to pg. 423">423</a></li> - -<li class="liindx">Koenig, Samuel, <a class="aindx" -href="#p330" title="go to pg. 330">330</a></li> - -<li class="liindx">Kofoid, C. A., <a class="aindx" -href="#p268" title="go to pg. 268">268</a></li> - -<li class="liindx">Kölliker, A. von, <a class="aindx" -href="#p413" title="go to pg. 413">413</a></li> - -<li class="liindx">Kollmann, M., <a class="aindx" -href="#p170" title="go to pg. 170">170</a></li> - -<li class="liindx">Koltzoff, N. K., <a class="aindx" -href="#p273" title="go to pg. 273">273</a>, <a -class="aindx" href="#p462" title="go to pg. -462">462</a></li> - -<li class="liindx">Koninckina, <a class="aindx" -href="#p570" title="go to pg. 570">570</a></li> - -<li class="liindx">Koodoo, horns of, <a class="aindx" -href="#p624" title="go to pg. 624">624</a></li> - -<li class="liindx">Köppen, Wladimir, <a class="aindx" -href="#p111" title="go to pg. 111">111</a></li> - -<li class="liindx">Korotneff, A., <a class="aindx" -href="#p377" title="go to pg. 377">377</a></li> - -<li class="liindx">Kraus, G., <a class="aindx" href="#p077" -title="go to pg. 77">77</a></li> - -<li class="liindx">Krogh, A., <a class="aindx" href="#p109" -title="go to pg. 109">109</a></li> - -<li class="liindx">Krohl, <a class="aindx" href="#p666" -title="go to pg. 666">666</a></li> - -<li class="liindx">Kühne, W., <a class="aindx" href="#p235" -title="go to pg. 235">235</a></li> - -<li class="liindx">Küster, E., <a class="aindx" -href="#p430" title="go to pg. 430">430</a></li> </ul> - -<ul class="ulindx"> <li class="liindx">Lafresnaye, F. -de, <a class="aindx" href="#p653" title="go to pg. -653">653</a></li> - -<li class="liindx">Lagena, <a class="aindx" href="#p251" -title="go to pg. 251">251</a>, <a class="aindx" -href="#p256" title="go to pg. 256">256</a>, <a -class="aindx" href="#p260" title="go to pg. 260">260</a>, -<a class="aindx" href="#p587" title="go to pg. -587">587</a></li> - -<li class="liindx">Lagrange, J. L., <a class="aindx" -href="#p649" title="go to pg. 649">649</a></li> - -<li class="liindx">Lalanne, L., <a class="aindx" -href="#p334" title="go to pg. 334">334</a></li> - -<li class="liindx">Lamarck, J. B. de, <a class="aindx" -href="#p549" title="go to pg. 549">549</a>, <a -class="aindx" href="#p716" title="go to pg. -716">716</a></li> - -<li class="liindx">Lamb, A. B., <a class="aindx" -href="#p186" title="go to pg. 186">186</a></li> - -<li class="liindx">Lamellaria, <a class="aindx" -href="#p554" title="go to pg. 554">554</a></li> - -<li class="liindx">Lamellibranchs, <a class="aindx" -href="#p561" title="go to pg. 561">561</a></li> - -<li class="liindx">Lami, B., <a class="aindx" href="#p296" -title="go to pg. 296">296</a>, <a class="aindx" -href="#p643" title="go to pg. 643">643</a></li> - -<li class="liindx">Laminaria, <a class="aindx" href="#p315" -title="go to pg. 315">315</a></li> - -<li class="liindx">Lammel, R., <a class="aindx" -href="#p100" title="go to pg. 100">100</a></li> - -<li class="liindx">Lanchester, F. W., <a class="aindx" -href="#p026" title="go to pg. 26">26</a></li> - -<li class="liindx">Lang, Arnold, <a class="aindx" -href="#p561" title="go to pg. 561">561</a></li> - -<li class="liindx">Lankester, Sir E. Ray, <a class="aindx" -href="#p004" title="go to pg. 4">4</a>, <a class="aindx" -href="#p251" title="go to pg. 251">251</a>, <a -class="aindx" href="#p348" title="go to pg. 348">348</a>, -<a class="aindx" href="#p465" title="go to pg. -465">465</a></li> - -<li class="liindx">Laplace, P. S. de, <a class="aindx" -href="#p001" title="go to pg. 1">1</a>, <a class="aindx" -href="#p207" title="go to pg. 207">207</a>, <a -class="aindx" href="#p217" title="go to pg. -217">217</a></li> - -<li class="liindx">Larmor, Sir J., <a class="aindx" -href="#p009" title="go to pg. 9">9</a>, <a class="aindx" -href="#p259" title="go to pg. 259">259</a></li> - -<li class="liindx">Lavater, J. C., <a class="aindx" -href="#p740" title="go to pg. 740">740</a></li> - -<li class="liindx">Law, Borelli’s, <a class="aindx" -href="#p029" title="go to pg. 29">29</a>; <ul> -<li class="liindx">Brandt’s, <a class="aindx" -href="#p482" title="go to pg. 482">482</a>;</li> <li -class="liindx">of Constant Angle, <a class="aindx" -href="#p599" title="go to pg. 599">599</a>;</li> <li -class="liindx">Errera’s, <a class="aindx" href="#p213" -title="go to pg. 213">213</a>, <a class="aindx" -href="#p306" title="go to pg. 306">306</a>;</li> -<li class="liindx">Froude’s, <a class="aindx" -href="#p022" title="go to pg. 22">22</a>;</li> <li -class="liindx">Lamarle’s, <a class="aindx" href="#p309" -title="go to pg. 309">309</a>;</li> <li class="liindx">of -Mass, <a class="aindx" href="#p130" title="go to pg. -130">130</a>;</li> <li class="liindx">Maupertuis’s, -<a class="aindx" href="#p208" title="go to pg. -208">208</a>;</li> <li class="liindx">Müller’s, -<a class="aindx" href="#p481" title="go to pg. -481">481</a>;</li> <li class="liindx">of Optimum, -<a class="aindx" href="#p110" title="go to pg. -110">110</a>;</li> <li class="liindx">van’t Hoff’s, -<a class="aindx" href="#p109" title="go to pg. -109">109</a>;</li> <li class="liindx">Willard-Gibbs’, -<a class="aindx" href="#p280" title="go to pg. -280">280</a>;</li> <li class="liindx">Wolff’s, <a -class="aindx" href="#p003" title="go to pg. 3">3</a>, <a -class="aindx" href="#p051" title="go to pg. 51">51</a>, -<a class="aindx" href="#p155" title="go to pg. -155">155</a></li> </ul></li> - -<li class="liindx">Leaping, <a class="aindx" href="#p029" -title="go to pg. 29">29</a></li> - -<li class="liindx">Leaves, arrangement of, <a class="aindx" -href="#p635" title="go to pg. 635">635</a>; <ul> <li -class="liindx">form of, <a class="aindx" href="#p731" -title="go to pg. 731">731</a></li> </ul></li> - -<li class="liindx">Ledingham, J. C. G., <a class="aindx" -href="#p211" title="go to pg. 211">211</a></li> - -<li class="liindx">Leduc, Stéphane, <a class="aindx" -href="#p162" title="go to pg. 162">162</a>, <a -class="aindx" href="#p167" title="go to pg. 167">167</a>, -<a class="aindx" href="#p185" title="go to pg. -185">185</a>, <a class="aindx" href="#p219" title="go to -pg. 219">219</a>, <a class="aindx" href="#p259" title="go -to pg. 259">259</a>, <a class="aindx" href="#p415" -title="go to pg. 415">415</a>, <a class="aindx" -href="#p428" title="go to pg. 428">428</a>, <a -class="aindx" href="#p431" title="go to pg. 431">431</a>, -<a class="aindx" href="#p590" title="go to pg. -590">590</a></li> - -<li class="liindx">Leeuwenhoek, A. van, <a class="aindx" -href="#p036" title="go to pg. 36">36</a>, <a class="aindx" -href="#p209" title="go to pg. 209">209</a></li> - -<li class="liindx">Leger, L., <a class="aindx" href="#p452" -title="go to pg. 452">452</a></li> - -<li class="liindx">Le Hello, P., <a class="aindx" -href="#p030" title="go to pg. 30">30</a></li> - -<li class="liindx">Lehmann, O., <a class="aindx" -href="#p203" title="go to pg. 203">203</a>, <a -class="aindx" href="#p272" title="go to pg. 272">272</a>, -<a class="aindx" href="#p440" title="go to pg. -440">440</a>, <a class="aindx" href="#p485" title="go to -pg. 485">485</a>, <a class="aindx" href="#p590" title="go -to pg. 590">590</a></li> - -<li class="liindx">Leibniz, G. W. von, <a class="aindx" -href="#p003" title="go to pg. 3">3</a>, <a class="aindx" -href="#p005" title="go to pg. 5">5</a>, <a class="aindx" -href="#p159" title="go to pg. 159">159</a>, <a -class="aindx" href="#p385" title="go to pg. -385">385</a></li> - -<li class="liindx">Leidenfrost, J. G., <a class="aindx" -href="#p279" title="go to pg. 279">279</a></li> - -<li class="liindx">Leidy, J., <a class="aindx" href="#p252" -title="go to pg. 252">252</a>, <a class="aindx" -href="#p468" title="go to pg. 468">468</a></li> - -<li class="liindx">Leiper, R. T., <a class="aindx" -href="#p660" title="go to pg. 660">660</a></li> - -<li class="liindx">Leitch, I., <a class="aindx" -href="#p112" title="go to pg. 112">112</a></li> - -<li class="liindx">Leitgeb, H., <a class="aindx" -href="#p305" title="go to pg. 305">305</a></li> - -<li class="liindx">Length-weight coefficient, <a -class="aindx" href="#p098" title="go to pg. 98">98</a>–<a -class="aindx" href="#p103" title="go to pg. 103">103</a>, -<a class="aindx" href="#p775" title="go to pg. -775">775</a></li> - -<li class="liindx">Leonardo da Vinci, <a class="aindx" -href="#p027" title="go to pg. 27">27</a>, <a class="aindx" -href="#p635" title="go to pg. 635">635</a>; <ul> <li -class="liindx">of Pisa, <a class="aindx" href="#p643" -title="go to pg. 643">643</a></li> </ul></li> - -<li class="liindx">Lepeschkin, <a class="aindx" -href="#p625" title="go to pg. 625">625</a></li> - -<li class="liindx">Leptocephalus, <a class="aindx" -href="#p087" title="go to pg. 87">87</a></li> - -<li class="liindx">Leray, Ad., <a class="aindx" -href="#p018" title="go to pg. 18">18</a></li> - -<li class="liindx">Lesage, G. L., <a class="aindx" -href="#p018" title="go to pg. 18">18</a></li> - -<li class="liindx">Leslie, Sir John, <a class="aindx" -href="#p163" title="go to pg. 163">163</a>, <a -class="aindx" href="#p503" title="go to pg. -503">503</a></li> - -<li class="liindx">Lestiboudois, T., <a class="aindx" -href="#p636" title="go to pg. 636">636</a></li> - -<li class="liindx">Leucocytes, <a class="aindx" -href="#p211" title="go to pg. 211">211</a></li> - -<li class="liindx">Levers, Orders of, <a class="aindx" -href="#p690" title="go to pg. 690">690</a></li> - -<li class="liindx">Levi, G., <a class="aindx" href="#p035" -title="go to pg. 35">35</a>, <a class="aindx" href="#p037" -title="go to pg. 37">37</a></li> - -<li class="liindx">Lewis, C. M., <a class="aindx" -href="#p280" title="go to pg. 280">280</a></li> - -<li class="liindx">Lhuilier, S. A. J., <a class="aindx" -href="#p330" title="go to pg. 330">330</a></li> - -<li class="liindx">Liesegang’s rings, <a class="aindx" -href="#p427" title="go to pg. 427">427</a>, <a -class="aindx" href="#p475" title="go to pg. -475">475</a></li> - -<li class="liindx">Light, pressure of, <a class="aindx" -href="#p048" title="go to pg. 48">48</a></li> - -<li class="liindx">Lillie, F. R., <a class="aindx" -href="#p004" title="go to pg. 4">4</a>, <a class="aindx" -href="#p147" title="go to pg. 147">147</a>, <a -class="aindx" href="#p341" title="go to pg. 341">341</a>; -<ul> <li class="liindx">R. S., <a class="aindx" -href="#p180" title="go to pg. 180">180</a>, <a -class="aindx" href="#p187" title="go to pg. 187">187</a>, -<a class="aindx" href="#p192" title="go to pg. -192">192</a></li> </ul></li> - -<li class="liindx">Lima, <a class="aindx" href="#p565" -title="go to pg. 565">565</a></li> - -<li class="liindx">Limacina, <a class="aindx" href="#p571" -title="go to pg. 571">571</a></li> - -<li class="liindx">Lines of force, <a class="aindx" -href="#p163" title="go to pg. 163">163</a>; <ul> <li -class="liindx">of growth, <a class="aindx" href="#p562" -title="go to pg. 562">562</a></li> </ul></li> - -<li class="liindx">Lingula, <a class="aindx" href="#p251" -title="go to pg. 251">251</a>, <a class="aindx" -href="#p567" title="go to pg. 567">567</a></li> - -<li class="liindx">Linnaeus, <a class="aindx" href="#p028" -title="go to pg. 28">28</a>, <a class="aindx" -href="#p250" title="go to pg. 250">250</a>, <a -class="aindx" href="#p547" title="go to pg. 547">547</a>, -<a class="aindx" href="#p720" title="go to pg. -720">720</a></li> - -<li class="liindx">Lion, brain of, <a class="aindx" -href="#p091" title="go to pg. 91">91</a></li> - -<li class="liindx">Liquid veins, <a class="aindx" -href="#p265" title="go to pg. 265">265</a></li> - -<li class="liindx">Lister, Martin, <a class="aindx" -href="#p318" title="go to pg. 318">318</a>; <ul> <li -class="liindx">J. J., <a class="aindx" href="#p436" -title="go to pg. 436">436</a></li> </ul></li> - -<li class="liindx">Listing, J. B., <a class="aindx" -href="#p385" title="go to pg. 385">385</a></li> - -<li class="liindx">Lithostrotion, <a class="aindx" -href="#p325" title="go to pg. 325">325</a></li> - -<li class="liindx">Littorina, <a class="aindx" href="#p524" -title="go to pg. 524">524</a></li> - -<li class="liindx">Lituites, <a class="aindx" href="#p546" -title="go to pg. 546">546</a>, <a class="aindx" -href="#p550" title="go to pg. 550">550</a></li> - -<li class="liindx">Llama, <a class="aindx" href="#p703" -title="go to pg. 703">703</a></li> - -<li class="liindx">Lobsters’ claws, <a class="aindx" -href="#p149" title="go to pg. 149">149</a></li> - -<li class="liindx">Locke, John, <a class="aindx" -href="#p006" title="go to pg. 6">6</a></li> - -<li class="liindx">Loeb, J., <a class="aindx" href="#p125" -title="go to pg. 125">125</a>, <a class="aindx" -href="#p132" title="go to pg. 132">132</a>, <a -class="aindx" href="#p135" title="go to pg. 135">135</a>, -<a class="aindx" href="#p136" title="go to pg. -136">136</a>, <a class="aindx" href="#p147" title="go to -pg. 147">147</a>, <a class="aindx" href="#p157" title="go -to pg. 157">157</a>, <a class="aindx" href="#p191" -title="go to pg. 191">191</a>, <a class="aindx" -href="#p193" title="go to pg. 193">193</a></li> - -<li class="liindx">Loewy, A., <a class="aindx" href="#p281" -title="go to pg. 281">281</a></li> - -<li class="liindx">Logarithmic spiral, <a class="aindx" -href="#p493" title="go to pg. 493">493</a>, etc.</li> - -<li class="liindx">Loisel, G., <a class="aindx" -href="#p088" title="go to pg. 88">88</a></li> - -<li class="liindx">Loligo, shell of, <a class="aindx" -href="#p575" title="go to pg. 575">575</a></li> - -<li class="liindx">Lo Monaco, <a class="aindx" href="#p083" -title="go to pg. 83">83</a></li> - -<li class="liindx">Lönnberg, E., <a class="aindx" -href="#p614" title="go to pg. 614">614</a>, <a -class="aindx" href="#p632" title="go to pg. -632">632</a></li> - -<li class="liindx">Looss, A., <a class="aindx" href="#p660" -title="go to pg. 660">660</a></li> - -<li class="liindx">Lotze, R. H., <a class="aindx" -href="#p055" title="go to pg. 55">55</a></li> - -<li class="liindx">Love, A. E. H., <a class="aindx" -href="#p674" title="go to pg. 674">674</a></li> - -<li class="liindx">Lucas, F. A., <a class="aindx" -href="#p138" title="go to pg. 138">138</a></li> - -<li class="liindx">Luciani, L., <a class="aindx" -href="#p083" title="go to pg. 83">83</a></li> - -<li class="liindx">Lucretius, <a class="aindx" href="#p047" -title="go to pg. 47">47</a>, <a class="aindx" href="#p071" -title="go to pg. 71">71</a>, <a class="aindx" href="#p137" -title="go to pg. 137">137</a>, <a class="aindx" -href="#p160" title="go to pg. 160">160</a></li> - -<li class="liindx">Ludwig, Carl, <a class="aindx" -href="#p002" title="go to pg. 2">2</a>; <ul> <li -class="liindx">F., <a class="aindx" href="#p643" -title="go to pg. 643">643</a>;</li> <li class="liindx">H. -J., <a class="aindx" href="#p342" title="go to pg. -342">342</a></li> </ul></li> - -<li class="liindx">Lupa, <a class="aindx" href="#p744" -title="go to pg. 744">744</a></li> - -<li class="liindx">Lupinus, growth of, <a class="aindx" -href="#p109" title="go to pg. 109">109</a>, <a -class="aindx" href="#p112" title="go to pg. -112">112</a></li> </ul> - -<ul class="ulindx"> <li class="liindx">Macalister, -A., <a class="aindx" href="#p557" title="go to pg. -557">557</a></li> - -<li class="liindx">MacAlister, Sir D., <a class="aindx" -href="#p673" title="go to pg. 673">673</a>, <a -class="aindx" href="#p683" title="go to pg. -683">683</a></li> - -<li class="liindx">Macallum, A. B., <a class="aindx" -href="#p277" title="go to pg. 277">277</a>, <a -class="aindx" href="#p287" title="go to pg. 287">287</a>, -<a class="aindx" href="#p357" title="go to pg. -357">357</a>, <a class="aindx" href="#p395" title="go -to pg. 395">395</a>; <ul> <li class="liindx">J. -B., <a class="aindx" href="#p492" title="go to pg. -492">492</a></li> </ul></li> - -<li class="liindx">McCoy, F., <a class="aindx" href="#p388" -title="go to pg. 388">388</a></li> - -<li class="liindx">Mach, Ernst, <a class="aindx" -href="#p209" title="go to pg. 209">209</a>, <a -class="aindx" href="#p330" title="go to pg. -330">330</a></li> - -<li class="liindx">Machaerodus, teeth of, <a class="aindx" -href="#p633" title="go to pg. 633">633</a></li> - -<li class="liindx">McKendrick, J. G., <a class="aindx" -href="#p042" title="go to pg. 42">42</a></li> - -<li class="liindx">McKenzie, A., <a class="aindx" -href="#p418" title="go to pg. 418">418</a></li> - -<li class="liindx">Mackinnon, D. L., <a class="aindx" -href="#p268" title="go to pg. 268">268</a></li> - -<li class="liindx">Maclaurin, Colin, <a class="aindx" -href="#p330" title="go to pg. 330">330</a>, <a -class="aindx" href="#p779" title="go to pg. -779">779</a></li> - -<li class="liindx">Macroscaphites, <a class="aindx" -href="#p550" title="go to pg. 550">550</a></li> - -<li class="liindx">Mactra, <a class="aindx" href="#p562" -title="go to pg. 562">562</a></li> - -<li class="liindx">Magnitude, <a class="aindx" href="#p016" -title="go to pg. 16">16</a></li> - -<li class="liindx">Maillard, L., <a class="aindx" -href="#p163" title="go to pg. 163">163</a></li> - -<li class="liindx">Maize, growth of, <a class="aindx" -href="#p109" title="go to pg. 109">109</a>, <a -class="aindx" href="#p111" title="go to pg. 111">111</a>, -<a class="aindx" href="#p298" title="go to pg. -298">298</a></li> - -<li class="liindx">Mall, F. P., <a class="aindx" -href="#p492" title="go to pg. 492">492</a></li> - -<li class="liindx">Maltaux, Mlle, <a class="aindx" -href="#p114" title="go to pg. 114">114</a></li> - -<li class="liindx">Mammoth, <a class="aindx" href="#p634" -title="go to pg. 634">634</a>, <a class="aindx" -href="#p705" title="go to pg. 705">705</a></li> - -<li class="liindx">Man, growth of, <a class="aindx" -href="#p061" title="go to pg. 61">61</a>; <ul> <li -class="liindx">skull of, <a class="aindx" href="#p770" -title="go to pg. 770">770</a></li> </ul></li> - -<li class="liindx">Maraldi, J. P., <a class="aindx" -href="#p329" title="go to pg. 329">329</a>, <a -class="aindx" href="#p473" title="go to pg. -473">473</a></li> - -<li class="liindx">Marbled papers, <a class="aindx" -href="#p736" title="go to pg. 736">736</a></li> - -<li class="liindx">Marcus Aurelius, <a class="aindx" -href="#p609" title="go to pg. 609">609</a></li> - -<li class="liindx">Markhor, horns of, <a class="aindx" -href="#p619" title="go to pg. 619">619</a></li> - -<li class="liindx">Marsh, O. C., <a class="aindx" -href="#p706" title="go to pg. 706">706</a>, <a -class="aindx" href="#p754" title="go to pg. -754">754</a></li> - -<li class="liindx">Marsigli, Comte L. F. de, -<a class="aindx" href="#p652" title="go to pg. -652">652</a></li> - -<li class="liindx">Massart, J., <a class="aindx" -href="#p114" title="go to pg. 114">114</a></li> - -<li class="liindx">Mastodon, <a class="aindx" href="#p634" -title="go to pg. 634">634</a></li> - -<li class="liindx">Mathematics, <a class="aindx" -href="#p719" title="go to pg. 719">719</a>, <a -class="aindx" href="#p778" title="go to pg. 778">778</a>, -etc.</li> - -<li class="liindx">Mathews, A., <a class="aindx" -href="#p285" title="go to pg. 285">285</a></li> - -<li class="liindx">Matrix, <a class="aindx" href="#p656" -title="go to pg. 656">656</a></li> - -<li class="liindx">Matter and energy, <a class="aindx" -href="#p011" title="go to pg. 11">11</a></li> - -<li class="liindx">Matthew, W. D., <a class="aindx" -href="#p707" title="go to pg. 707">707</a></li> - -<li class="liindx">Matuta, <a class="aindx" href="#p744" -title="go to pg. 744">744</a></li> - -<li class="liindx">Maupas, M., <a class="aindx" -href="#p133" title="go to pg. 133">133</a></li> - -<li class="liindx">Maupertuis, <a class="aindx" -href="#p003" title="go to pg. 3">3</a>, <a class="aindx" -href="#p005" title="go to pg. 5">5</a>, <a class="aindx" -href="#p208" title="go to pg. 208">208</a></li> - -<li class="liindx">Maxwell, J. Clerk, <a class="aindx" -href="#p009" title="go to pg. 9">9</a>, <a class="aindx" -href="#p018" title="go to pg. 18">18</a>, <a class="aindx" -href="#p040" title="go to pg. 40">40</a>, <a class="aindx" -href="#p044" title="go to pg. 44">44</a>, <a class="aindx" -href="#p160" title="go to pg. 160">160</a>, <a -class="aindx" href="#p207" title="go to pg. 207">207</a>, -<a class="aindx" href="#p385" title="go to pg. -385">385</a>, <a class="aindx" href="#p691" title="go to -pg. 691">691</a></li> - -<li class="liindx">Mechanical efficiency, <a class="aindx" -href="#p670" title="go to pg. 670">670</a></li> - -<li class="liindx">Mechanism, <a class="aindx" href="#p005" -title="go to pg. 5">5</a>, <a class="aindx" href="#p161" -title="go to pg. 161">161</a>, <a class="aindx" -href="#p185" title="go to pg. 185">185</a>, etc.</li> - -<li class="liindx">Meek, C. F. U., <a class="aindx" -href="#p190" title="go to pg. 190">190</a></li> - -<li class="liindx">Melanchthon, <a class="aindx" -href="#p004" title="go to pg. 4">4</a></li> - -<li class="liindx">Melanopsis, <a class="aindx" -href="#p557" title="go to pg. 557">557</a></li> - -<li class="liindx">Meldola, R., <a class="aindx" -href="#p670" title="go to pg. 670">670</a></li> - -<li class="liindx">Melipona, <a class="aindx" href="#p332" -title="go to pg. 332">332</a></li> - -<li class="liindx">Mellor, J. W., <a class="aindx" -href="#p134" title="go to pg. 134">134</a></li> - -<li class="liindx">Melo, <a class="aindx" href="#p525" -title="go to pg. 525">525</a></li> - -<li class="liindx">Melobesia, <a class="aindx" href="#p412" -title="go to pg. 412">412</a></li> - -<li class="liindx">Melsens, L. H. F., <a class="aindx" -href="#p282" title="go to pg. 282">282</a></li> - -<li class="liindx">Membrane-formation, <a class="aindx" -href="#p281" title="go to pg. 281">281</a></li> - -<li class="liindx">Mensbrugghe, G. van der, <a -class="aindx" href="#p212" title="go to pg. 212">212</a>, -<a class="aindx" href="#p298" title="go to pg. -298">298</a>, <a class="aindx" href="#p470" title="go to -pg. 470">470</a></li> - -<li class="liindx">Meserer, O., <a class="aindx" -href="#p683" title="go to pg. 683">683</a></li> - -<li class="liindx">Mesocarpus, <a class="aindx" -href="#p289" title="go to pg. 289">289</a></li> - -<li class="liindx">Mesohippus, <a class="aindx" -href="#fig402" title="go to fig. 402">766</a></li> - -<li class="liindx">Metamorphosis, <a class="aindx" -href="#p082" title="go to pg. 82">82</a></li> - -<li class="liindx">Meves, F., <a class="aindx" href="#p163" -title="go to pg. 163">163</a>, <a class="aindx" -href="#p285" title="go to pg. 285">285</a></li> - -<li class="liindx">Meyer, Arthur, <a class="aindx" -href="#p432" title="go to pg. 432">432</a>; <ul> <li -class="liindx">G. H., <a class="aindx" href="#p008" -title="go to pg. 8">8</a>, <a class="aindx" href="#p682" -title="go to pg. 682">682</a>, <a class="aindx" -href="#p683" title="go to pg. 683">683</a></li> </ul></li> - -<li class="liindx">Micellae, <a class="aindx" href="#p157" -title="go to pg. 157">157</a></li> - -<li class="liindx">Michaelis, L., <a class="aindx" -href="#p277" title="go to pg. 277">277</a></li> - -<li class="liindx">Microchemistry, <a class="aindx" -href="#p288" title="go to pg. 288">288</a></li> - -<li class="liindx">Micrococci, <a class="aindx" -href="#p039" title="go to pg. 39">39</a>, <a -class="aindx" href="#p245" title="go to pg. 245">245</a>, -<a class="aindx" href="#p250" title="go to pg. -250">250</a></li> - -<li class="liindx">Micromonas, <a class="aindx" -href="#p038" title="go to pg. 38">38</a></li> - -<li class="liindx">Miliolidae, <a class="aindx" -href="#p595" title="go to pg. 595">595</a>, <a -class="aindx" href="#p604" title="go to pg. -604">604</a></li> - -<li class="liindx">Milner, R. S., <a class="aindx" -href="#p280" title="go to pg. 280">280</a></li> - -<li class="liindx">Milton, John, <a class="aindx" -href="#p779" title="go to pg. 779">779</a></li> - -<li class="liindx">Mimicry, <a class="aindx" href="#p671" -title="go to pg. 671">671</a></li> - -<li class="liindx">Minchin, E. A., <a class="aindx" -href="#p267" title="go to pg. 267">267</a>, <a -class="aindx" href="#p444" title="go to pg. 444">444</a>, -<a class="aindx" href="#p449" title="go to pg. -449">449</a>, <a class="aindx" href="#p455" title="go to -pg. 455">455</a></li> - -<li class="liindx">Minimal areas, <a class="aindx" -href="#p208" title="go to pg. 208">208</a>, <a -class="aindx" href="#p215" title="go to pg. 215">215</a>, -<a class="aindx" href="#p225" title="go to pg. -225">225</a>, <a class="aindx" href="#p293" title="go to -pg. 293">293</a>, <a class="aindx" href="#p306" title="go -to pg. 306">306</a>, <a class="aindx" href="#p336" -title="go to pg. 336">336</a>, <a class="aindx" -href="#p349" title="go to pg. 349">349</a></li> - -<li class="liindx">Minot, C. S., <a class="aindx" -href="#p037" title="go to pg. 37">37</a>, <a class="aindx" -href="#p072" title="go to pg. 72">72</a>, <a class="aindx" -href="#p722" title="go to pg. 722">722</a></li> - -<li class="liindx">Miohippus, <a class="aindx" href="#fig402" -title="go to Fig. 402">767</a></li> - -<li class="liindx">Mitchell, P. Chalmers, <a class="aindx" -href="#p703" title="go to pg. 703">703</a></li> - -<li class="liindx">Mitosis, <a class="aindx" href="#p170" -title="go to pg. 170">170</a></li> - -<li class="liindx">Mitra, <a class="aindx" href="#p557" -title="go to pg. 557">557</a>, <a class="aindx" -href="#p559" title="go to pg. 559">559</a></li> - -<li class="liindx">Möbius, K., <a class="aindx" -href="#p449" title="go to pg. 449">449</a></li> - -<li class="liindx">Modiola, <a class="aindx" href="#p562" -title="go to pg. 562">562</a></li> - -<li class="liindx">Mohl, H. von, <a class="aindx" -href="#p624" title="go to pg. 624">624</a></li> - -<li class="liindx">Molar and molecular forces, <a -class="aindx" href="#p053" title="go to pg. 53">53</a></li> - -<li class="liindx">Mole-cricket, chromosomes of, -<a class="aindx" href="#p181" title="go to pg. -181">181</a></li> - -<li class="liindx">Molecular asymmetry, <a class="aindx" -href="#p416" title="go to pg. 416">416</a></li> - -<li class="liindx">Molecules, <a class="aindx" href="#p041" -title="go to pg. 41">41</a></li> - -<li class="liindx">Möller, V. von, <a class="aindx" -href="#p593" title="go to pg. 593">593</a></li> - -<li class="liindx">Monnier, A., <a class="aindx" -href="#p078" title="go to pg. 78">78</a>, <a class="aindx" -href="#p132" title="go to pg. 132">132</a></li> - -<li class="liindx">Monticulipora, <a class="aindx" -href="#p326" title="go to pg. 326">326</a></li> - -<li class="liindx">Moore, B., <a class="aindx" href="#p272" -title="go to pg. 272">272</a></li> - -<li class="liindx">Morey, S., <a class="aindx" href="#p264" -title="go to pg. 264">264</a></li> - -<li class="liindx">Morgan, T. H., <a class="aindx" -href="#p126" title="go to pg. 126">126</a>, <a -class="aindx" href="#p134" title="go to pg. 134">134</a>, -<a class="aindx" href="#p138" title="go to pg. -138">138</a>, <a class="aindx" href="#p147" title="go to -pg. 147">147</a></li> - -<li class="liindx">Morita, <a class="aindx" href="#p699" -title="go to pg. 699">699</a></li> - -<li class="liindx">Morphodynamique, <a class="aindx" -href="#p156" title="go to pg. 156">156</a></li> - -<li class="liindx">Morphologie synthétique, <a -class="aindx" href="#p420" title="go to pg. -420">420</a></li> - -<li class="liindx">Morphology, <a class="aindx" -href="#p719" title="go to pg. 719">719</a>, etc.</li> - -<li class="liindx">Morse, Max, <a class="aindx" -href="#p136" title="go to pg. 136">136</a></li> - -<li class="liindx">Moseley, H., <a class="aindx" -href="#p008" title="go to pg. 8">8</a>, <a class="aindx" -href="#p518" title="go to pg. 518">518</a>, <a -class="aindx" href="#p521" title="go to pg. 521">521</a>, -<a class="aindx" href="#p538" title="go to pg. -538">538</a>, <a class="aindx" href="#p553" title="go to -pg. 553">553</a>, <a class="aindx" href="#p555" title="go -to pg. 555">555</a>, <a class="aindx" href="#p592" -title="go to pg. 592">592</a></li> - -<li class="liindx">Moss, embryo of, <a class="aindx" -href="#p374" title="go to pg. 374">374</a>; <ul> -<li class="liindx">gemma of, <a class="aindx" -href="#p403" title="go to pg. 403">403</a>;</li> <li -class="liindx">rhizoids of, <a class="aindx" href="#p356" -title="go to pg. 356">356</a></li> </ul></li> - -<li class="liindx">Mouillard, L. P., <a class="aindx" -href="#p027" title="go to pg. 27">27</a></li> - -<li class="liindx">Mouse, growth of, <a class="aindx" -href="#p082" title="go to pg. 82">82</a></li> - -<li class="liindx">Mucor, sporangium of, <a class="aindx" -href="#p303" title="go to pg. 303">303</a></li> - -<li class="liindx">Müllenhof, K. von, <a class="aindx" -href="#p025" title="go to pg. 25">25</a>, <a class="aindx" -href="#p332" title="go to pg. 332">332</a></li> - -<li class="liindx">Müller, Fritz, <a class="aindx" -href="#p003" title="go to pg. 3">3</a>; <ul> <li -class="liindx">Johannes, <a class="aindx" href="#p459" -title="go to pg. 459">459</a>, <a class="aindx" -href="#p481" title="go to pg. 481">481</a></li> </ul></li> - -<li class="liindx">Mummery, J. H., <a class="aindx" -href="#p425" title="go to pg. 425">425</a></li> - -<li class="liindx">Munro, H., <a class="aindx" href="#p323" -title="go to pg. 323">323</a></li> - -<li class="liindx">Musk-ox, horns of, <a class="aindx" -href="#p615" title="go to pg. 615">615</a></li> - -<li class="liindx">Mya, <a class="aindx" href="#p422" -title="go to pg. 422">422</a>, <a class="aindx" -href="#p561" title="go to pg. 561">561</a></li> - -<li class="liindx">Myonemes, <a class="aindx" href="#p562" -title="go to pg. 562">562</a></li> </ul> - -<ul class="ulindx"> <li class="liindx">Naber, H. A., <a -class="aindx" href="#p511" title="go to pg. 511">511</a>, -<a class="aindx" href="#p650" title="go to pg. -650">650</a></li> - -<li class="liindx">Nägeli, C., <a class="aindx" -href="#p124" title="go to pg. 124">124</a>, <a -class="aindx" href="#p159" title="go to pg. 159">159</a>, -<a class="aindx" href="#p210" title="go to pg. -210">210</a></li> - -<li class="liindx">Nassellaria, <a class="aindx" -href="#p472" title="go to pg. 472">472</a></li> - -<li class="liindx">Natica, <a class="aindx" href="#p554" -title="go to pg. 554">554</a>, <a class="aindx" -href="#p557" title="go to pg. 557">557</a>, <a -class="aindx" href="#p559" title="go to pg. -559">559</a></li> - -<li class="liindx">Natural selection, <a class="aindx" -href="#p004" title="go to pg. 4">4</a>, <a class="aindx" -href="#p058" title="go to pg. 58">58</a>, <a class="aindx" -href="#p137" title="go to pg. 137">137</a>, <a -class="aindx" href="#p456" title="go to pg. 456">456</a>, -<a class="aindx" href="#p586" title="go to pg. -586">586</a>, <a class="aindx" href="#p609" title="go to -pg. 609">609</a>, <a class="aindx" href="#p651" title="go -to pg. 651">651</a>, <a class="aindx" href="#p653" -title="go to pg. 653">653</a></li> - -<li class="liindx">Naumann, C. F., <a class="aindx" -href="#p529" title="go to pg. 529">529</a>, <a -class="aindx" href="#p531" title="go to pg. 531">531</a>, -<a class="aindx" href="#p539" title="go to pg. -539">539</a>, <a class="aindx" href="#p550" title="go to -pg. 550">550</a>, <a class="aindx" href="#p577" title="go -to pg. 577">577</a>, <a class="aindx" href="#p594" -title="go to pg. 594">594</a>, <a class="aindx" -href="#p636" title="go to pg. 636">636</a>; <ul> <li -class="liindx">J. F., <a class="aindx" href="#p653" -title="go to pg. 653">653</a></li> </ul></li> - -<li class="liindx">Nautilus, <a class="aindx" href="#p355" -title="go to pg. 355">355</a>, <a class="aindx" -href="#p494" title="go to pg. 494">494</a>, <a -class="aindx" href="#p501" title="go to pg. 501">501</a>, -<a class="aindx" href="#p515" title="go to pg. -515">515</a>, <a class="aindx" href="#p518" title="go to -pg. 518">518</a>, <a class="aindx" href="#p532" title="go -to pg. 532">532</a>, <a class="aindx" href="#p535" -title="go to pg. 535">535</a>, <a class="aindx" -href="#p546" title="go to pg. 546">546</a>, <a -class="aindx" href="#p552" title="go to pg. 552">552</a>, -<a class="aindx" href="#p557" title="go to pg. -557">557</a>, <a class="aindx" href="#p575" title="go to -pg. 575">575</a>, <a class="aindx" href="#p577" title="go -to pg. 577">577</a>, <a class="aindx" href="#p580" -title="go to pg. 580">580</a>, <a class="aindx" -href="#p592" title="go to pg. 592">592</a>, <a -class="aindx" href="#p633" title="go to pg. 633">633</a>; -<ul> <li class="liindx">hood of, <a class="aindx" -href="#p554" title="go to pg. 554">554</a>;</li> <li -class="liindx">kidney of, <a class="aindx" href="#p425" -title="go to pg. 425">425</a>;</li> <li class="liindx">N. -umbilicatus, <a class="aindx" href="#p542" title="go to pg. -542">542</a>, <a class="aindx" href="#p547" title="go to -pg. 547">547</a>, <a class="aindx" href="#p554" title="go -to pg. 554">554</a></li> </ul></li> - -<li class="liindx">Nebenkern, <a class="aindx" href="#p285" -title="go to pg. 285">285</a></li> - -<li class="liindx">Neottia, pollen of, <a class="aindx" -href="#p396" title="go to pg. 396">396</a></li> - -<li class="liindx">Nereis, egg of, <a class="aindx" -href="#p342" title="go to pg. 342">342</a>, <a -class="aindx" href="#p378" title="go to pg. 378">378</a>, -<a class="aindx" href="#p453" title="go to pg. -453">453</a></li> - -<li class="liindx">Nerita, <a class="aindx" href="#p522" -title="go to pg. 522">522</a>, <a class="aindx" -href="#p555" title="go to pg. 555">555</a></li> - -<li class="liindx">Neumayr, M., <a class="aindx" -href="#p608" title="go to pg. 608">608</a></li> - -<li class="liindx">Neutral zone, <a class="aindx" -href="#p674" title="go to pg. 674">674</a>, <a -class="aindx" href="#p676" title="go to pg. 676">676</a>, -<a class="aindx" href="#p686" title="go to pg. -686">686</a></li> - -<li class="liindx">Newton, <a class="aindx" href="#p001" -title="go to pg. 1">1</a>, <a class="aindx" href="#p006" -title="go to pg. 6">6</a>, <a class="aindx" href="#p158" -title="go to pg. 158">158</a>, <a class="aindx" -href="#p643" title="go to pg. 643">643</a>, <a -class="aindx" href="#p721" title="go to pg. -721">721</a></li> - -<li class="liindx">Nicholson, H. A., <a class="aindx" -href="#p325" title="go to pg. 325">325</a>, <a -class="aindx" href="#p327" title="go to pg. -327">327</a></li> - -<li class="liindx">Noctiluca, <a class="aindx" href="#p246" -title="go to pg. 246">246</a></li> - -<li class="liindx">Nodoid, <a class="aindx" href="#p218" -title="go to pg. 218">218</a>, <a class="aindx" -href="#p223" title="go to pg. 223">223</a></li> - -<li class="liindx">Nodosaria, <a class="aindx" -href="#p262" title="go to pg. 262">262</a>, <a -class="aindx" href="#p535" title="go to pg. 535">535</a>, -<a class="aindx" href="#p604" title="go to pg. -604">604</a></li> - -<li class="liindx">Norman, A. M., <a class="aindx" -href="#p465" title="go to pg. 465">465</a></li> - -<li class="liindx">Norris, Richard, <a class="aindx" -href="#p272" title="go to pg. 272">272</a></li> - -<li class="liindx">Nostoc, <a class="aindx" href="#p300" -title="go to pg. 300">300</a>, <a class="aindx" -href="#p313" title="go to pg. 313">313</a></li> - -<li class="liindx">Notosuchus, <a class="aindx" -href="#p753" title="go to pg. 753">753</a></li> - -<li class="liindx">Nuclear spindle, <a class="aindx" -href="#p170" title="go to pg. 170">170</a>; <ul> <li -class="liindx">structure, <a class="aindx" href="#p166" -title="go to pg. 166">166</a></li> </ul></li> - -<li class="liindx">Nummulites, <a class="aindx" -href="#p504" title="go to pg. 504">504</a>, <a -class="aindx" href="#p552" title="go to pg. 552">552</a>, -<a class="aindx" href="#p591" title="go to pg. -591">591</a></li> - -<li class="liindx">Nussbaum, M., <a class="aindx" -href="#p198" title="go to pg. 198">198</a></li> </ul> - -<ul class="ulindx"> <li class="liindx">Oekotraustes, -<a class="aindx" href="#p550" title="go to pg. -550">550</a></li> - -<li class="liindx">Ogilvie-Gordon, M. M., <a class="aindx" -href="#p423" title="go to pg. 423">423</a></li> - -<li class="liindx">Oil-globules, Plateau’s, <a -class="aindx" href="#p219" title="go to pg. -219">219</a></li> - -<li class="liindx">Oithona, <a class="aindx" href="#p742" -title="go to pg. 742">742</a></li> - -<li class="liindx">Oken, L., <a class="aindx" href="#p004" -title="go to pg. 4">4</a>, <a class="aindx" href="#p635" -title="go to pg. 635">635</a></li> - -<li class="liindx">Oliva, <a class="aindx" href="#p554" -title="go to pg. 554">554</a></li> - -<li class="liindx">Ootype, <a class="aindx" href="#p660" -title="go to pg. 660">660</a></li> - -<li class="liindx">Operculina, <a class="aindx" -href="#p594" title="go to pg. 594">594</a></li> - -<li class="liindx">Operculum of gastropods, <a -class="aindx" href="#p521" title="go to pg. -521">521</a></li> - -<li class="liindx">Oppel, A., <a class="aindx" href="#p088" -title="go to pg. 88">88</a></li> - -<li class="liindx">Optimum temperature, <a class="aindx" -href="#p110" title="go to pg. 110">110</a></li> - -<li class="liindx">Orbitolites, <a class="aindx" -href="#p605" title="go to pg. 605">605</a></li> - -<li class="liindx">Orbulina, <a class="aindx" href="#p059" -title="go to pg. 59">59</a>, <a class="aindx" href="#p225" -title="go to pg. 225">225</a>, <a class="aindx" -href="#p257" title="go to pg. 257">257</a>, <a -class="aindx" href="#p587" title="go to pg. 587">587</a>, -<a class="aindx" href="#p598" title="go to pg. -598">598</a>, <a class="aindx" href="#p604" title="go to -pg. 604">604</a>, <a class="aindx" href="#p607" title="go -to pg. 607">607</a></li> - -<li class="liindx">Organs, growth of, <a class="aindx" -href="#p088" title="go to pg. 88">88</a></li> - -<li class="liindx">Orthagoriscus, <a class="aindx" -href="#p751" title="go to pg. 751">751</a>, <a -class="aindx" href="#p775" title="go to pg. 775">775</a>, -<a class="aindx" href="#p777" title="go to pg. -777">777</a></li> - -<li class="liindx">Orthis, <a class="aindx" href="#p561" -title="go to pg. 561">561</a>, <a class="aindx" -href="#p567" title="go to pg. 567">567</a></li> - -<li class="liindx">Orthoceras, <a class="aindx" -href="#p515" title="go to pg. 515">515</a>, <a -class="aindx" href="#p548" title="go to pg. 548">548</a>, -<a class="aindx" href="#p551" title="go to pg. -551">551</a>, <a class="aindx" href="#p556" title="go to -pg. 556">556</a>, <a class="aindx" href="#p579" title="go -to pg. 579">579</a>, <a class="aindx" href="#p735" -title="go to pg. 735">735</a></li> - -<li class="liindx">Orthogenesis, <a class="aindx" -href="#p549" title="go to pg. 549">549</a></li> - -<li class="liindx">Orthogonal trajectories, <a -class="aindx" href="#p305" title="go to pg. 305">305</a>, -<a class="aindx" href="#p377" title="go to pg. -377">377</a>, <a class="aindx" href="#p400" title="go to -pg. 400">400</a>, <a class="aindx" href="#p640" title="go -to pg. 640">640</a>, <a class="aindx" href="#p678" -title="go to pg. 678">678</a></li> - -<li class="liindx">Orthostichies, <a class="aindx" -href="#p649" title="go to pg. 649">649</a></li> - -<li class="liindx">Orthotoluidene, <a class="aindx" -href="#p219" title="go to pg. 219">219</a></li> - -<li class="liindx">Oryx, horns of, <a class="aindx" -href="#p616" title="go to pg. 616">616</a></li> - -<li class="liindx">Osborn, H. F., <a class="aindx" -href="#p714" title="go to pg. 714">714</a>, <a -class="aindx" href="#p727" title="go to pg. 727">727</a>, -<a class="aindx" href="#p760" title="go to pg. -760">760</a></li> - -<li class="liindx">Oscillatoria, <a class="aindx" -href="#p300" title="go to pg. 300">300</a></li> - -<li class="liindx">Osmosis, <a class="aindx" href="#p124" -title="go to pg. 124">124</a>, <a class="aindx" -href="#p287" title="go to pg. 287">287</a>, etc.</li> - -<li class="liindx">Osmunda, <a class="aindx" href="#p396" -title="go to pg. 396">396</a>, <a class="aindx" -href="#p406" title="go to pg. 406">406</a></li> - -<li class="liindx">Ostrea, <a class="aindx" href="#p562" -title="go to pg. 562">562</a></li> - -<li class="liindx">Ostrich, <a class="aindx" href="#p025" -title="go to pg. 25">25</a>, <a class="aindx" href="#p707" -title="go to pg. 707">707</a>, <a class="aindx" -href="#p708" title="go to pg. 708">708</a></li> - -<li class="liindx">Ostwald, Wilhelm, <a class="aindx" -href="#p044" title="go to pg. 44">44</a>, <a class="aindx" -href="#p131" title="go to pg. 131">131</a>, <a -class="aindx" href="#p426" title="go to pg. 426">426</a>; -<ul> <li class="liindx">Wolfgang, <a class="aindx" -href="#p032" title="go to pg. 32">32</a>, <a class="aindx" -href="#p077" title="go to pg. 77">77</a>, <a class="aindx" -href="#p082" title="go to pg. 82">82</a>, <a class="aindx" -href="#p132" title="go to pg. 132">132</a>, <a -class="aindx" href="#p277" title="go to pg. 277">277</a>, -<a class="aindx" href="#p281" title="go to pg. -281">281</a></li> </ul></li> - -<li class="liindx">Otoliths, <a class="aindx" href="#p425" -title="go to pg. 425">425</a>, <a class="aindx" -href="#p432" title="go to pg. 432">432</a></li> - -<li class="liindx">Ovis Ammon, <a class="aindx" -href="#p614" title="go to pg. 614">614</a></li> - -<li class="liindx">Owen, Sir R., <a class="aindx" -href="#p020" title="go to pg. 20">20</a>, <a class="aindx" -href="#p575" title="go to pg. 575">575</a>, <a -class="aindx" href="#p654" title="go to pg. 654">654</a>, -<a class="aindx" href="#p669" title="go to pg. -669">669</a>, <a class="aindx" href="#p715" title="go to -pg. 715">715</a></li> - -<li class="liindx">Ox, cannon-bone of, <a class="aindx" -href="#p730" title="go to pg. 730">730</a>, <a -class="aindx" href="#p738" title="go to pg. 738">738</a>; -<ul> <li class="liindx">growth of, <a class="aindx" -href="#p102" title="go to pg. 102">102</a></li> </ul></li> - -<li class="liindx">Oxalate, calcium, <a class="aindx" -href="#p412" title="go to pg. 412">412</a>, <a -class="aindx" href="#p434" title="go to pg. -434">434</a></li> </ul> - -<ul class="ulindx"> <li class="liindx">Palaeechinus, -<a class="aindx" href="#p663" title="go to pg. -663">663</a></li> - -<li class="liindx">Palm, <a class="aindx" href="#p624" -title="go to pg. 624">624</a></li> - -<li class="liindx">Pander, C. H., <a class="aindx" -href="#p055" title="go to pg. 55">55</a></li> - -<li class="liindx">Pangenesis, <a class="aindx" -href="#p044" title="go to pg. 44">44</a>, <a class="aindx" -href="#p157" title="go to pg. 157">157</a></li> - -<li class="liindx">Papillon, Fernand, <a class="aindx" -href="#p010" title="go to pg. 10">10</a></li> - -<li class="liindx">Pappus of Alexandria, <a class="aindx" -href="#p328" title="go to pg. 328">328</a></li> - -<li class="liindx">Parabolic girder, <a class="aindx" -href="#p693" title="go to pg. 693">693</a>, <a -class="aindx" href="#p696" title="go to pg. -696">696</a></li> - -<li class="liindx">Parahippus, <a class="aindx" -href="#fig402" title="go to fig. 402">767</a></li> - -<li class="liindx">Paralomis, <a class="aindx" href="#p744" -title="go to pg. 744">744</a></li> - -<li class="liindx">Paraphyses of mosses, <a class="aindx" -href="#p351" title="go to pg. 351">351</a></li> - -<li class="liindx">Parastichies, <a class="aindx" -href="#p640" title="go to pg. 640">640</a>, <a -class="aindx" href="#p641" title="go to pg. -641">641</a></li> - -<li class="liindx">Passiflora, pollen of, <a class="aindx" -href="#p396" title="go to pg. 396">396</a></li> - -<li class="liindx">Pasteur, L., <a class="aindx" -href="#p416" title="go to pg. 416">416</a></li> - -<li class="liindx">Patella, <a class="aindx" href="#p561" -title="go to pg. 561">561</a></li> - -<li class="liindx">Pauli, W., <a class="aindx" href="#p211" -title="go to pg. 211">211</a>, <a class="aindx" -href="#p434" title="go to pg. 434">434</a></li> - -<li class="liindx">Pearl, Raymond, <a class="aindx" -href="#p090" title="go to pg. 90">90</a>, <a class="aindx" -href="#p097" title="go to pg. 97">97</a>, <a class="aindx" -href="#p654" title="go to pg. 654">654</a></li> - -<li class="liindx">Pearls, <a class="aindx" href="#p425" -title="go to pg. 425">425</a>, <a class="aindx" -href="#p431" title="go to pg. 431">431</a></li> - -<li class="liindx">Pearson, Karl, <a class="aindx" -href="#p036" title="go to pg. 36">36</a>, <a class="aindx" -href="#p078" title="go to pg. 78">78</a></li> - -<li class="liindx">Peas, growth of, <a class="aindx" -href="#p112" title="go to pg. 112">112</a></li> - -<li class="liindx">Pecten, <a class="aindx" href="#p562" -title="go to pg. 562">562</a></li> - -<li class="liindx">Peddie, W., <a class="aindx" -href="#p182" title="go to pg. 182">182</a>, <a -class="aindx" href="#p272" title="go to pg. 272">272</a>, -<a class="aindx" href="#p344" title="go to pg. -344">344</a>, <a class="aindx" href="#p448" title="go to -pg. 448">448</a></li> - -<li class="liindx">Pellia, spore of, <a class="aindx" -href="#p302" title="go to pg. 302">302</a></li> - -<li class="liindx">Pelseneer, P., <a class="aindx" -href="#p570" title="go to pg. 570">570</a></li> - -<li class="liindx">Pendulum, <a class="aindx" href="#p030" -title="go to pg. 30">30</a></li> - -<li class="liindx">Peneroplis, <a class="aindx" -href="#p606" title="go to pg. 606">606</a></li> - -<li class="liindx">Percentage-curves, Minot’s, <a -class="aindx" href="#p072" title="go to pg. 72">72</a></li> - -<li class="liindx">Pericline, <a class="aindx" href="#p360" -title="go to pg. 360">360</a></li> - -<li class="liindx">Periploca, pollen of, <a class="aindx" -href="#p396" title="go to pg. 396">396</a></li> - -<li class="liindx">Peristome, <a class="aindx" href="#p239" -title="go to pg. 239">239</a></li> - -<li class="liindx">Permeability, magnetic, <a -class="aindx" href="#p177" title="go to pg. 177">177</a>, -<a class="aindx" href="#p182" title="go to pg. -182">182</a></li> - -<li class="liindx">Perrin, J., <a class="aindx" -href="#p043" title="go to pg. 43">43</a>, <a class="aindx" -href="#p046" title="go to pg. 46">46</a></li> - -<li class="liindx">Peter, Karl, <a class="aindx" -href="#p117" title="go to pg. 117">117</a></li> - -<li class="liindx">Pettigrew, J. B., <a class="aindx" -href="#p490" title="go to pg. 490">490</a></li> - -<li class="liindx">Pfeffer, W., <a class="aindx" -href="#p111" title="go to pg. 111">111</a>, <a -class="aindx" href="#p273" title="go to pg. 273">273</a>, -<a class="aindx" href="#p688" title="go to pg. -688">688</a></li> - -<li class="liindx">Pflüger, E., <a class="aindx" -href="#p680" title="go to pg. 680">680</a></li> - -<li class="liindx">Phagocytosis, <a class="aindx" -href="#p211" title="go to pg. 211">211</a></li> - -<li class="liindx">Phascum, <a class="aindx" href="#p408" -title="go to pg. 408">408</a></li> - -<li class="liindx">Phase of curve, <a class="aindx" -href="#p068" title="go to pg. 68">68</a>, <a class="aindx" -href="#p081" title="go to pg. 81">81</a>, etc.</li> - -<li class="liindx">Phasianella, <a class="aindx" -href="#p557" title="go to pg. 557">557</a>, <a -class="aindx" href="#p559" title="go to pg. -559">559</a></li> - -<li class="liindx">Phatnaspis, <a class="aindx" -href="#p482" title="go to pg. 482">482</a></li> - -<li class="liindx">Phillipsastraea, <a class="aindx" -href="#p327" title="go to pg. 327">327</a></li> - -<li class="liindx">Philolaus, <a class="aindx" href="#p779" -title="go to pg. 779">779</a></li> - -<li class="liindx">Pholas, <a class="aindx" href="#p561" -title="go to pg. 561">561</a></li> - -<li class="liindx">Phormosoma, <a class="aindx" -href="#p664" title="go to pg. 664">664</a></li> - -<li class="liindx">Phractaspis, <a class="aindx" -href="#p484" title="go to pg. 484">484</a></li> - -<li class="liindx">Phyllotaxis, <a class="aindx" -href="#p635" title="go to pg. 635">635</a></li> - -<li class="liindx">Phylogeny, <a class="aindx" href="#p196" -title="go to pg. 196">196</a>, <a class="aindx" -href="#p251" title="go to pg. 251">251</a>, <a -class="aindx" href="#p548" title="go to pg. 548">548</a>, -<a class="aindx" href="#p716" title="go to pg. -716">716</a></li> - -<li class="liindx">Pike, F. H., <a class="aindx" -href="#p110" title="go to pg. 110">110</a></li> - -<li class="liindx">Pileopsis, <a class="aindx" href="#p555" -title="go to pg. 555">555</a></li> - -<li class="liindx">Pinacoceras, <a class="aindx" -href="#p584" title="go to pg. 584">584</a></li> - -<li class="liindx">Pithecanthropus, <a class="aindx" -href="#p772" title="go to pg. 772">772</a></li> - -<li class="liindx">Pith of rush, <a class="aindx" -href="#p335" title="go to pg. 335">335</a></li> - -<li class="liindx">Plaice, <a class="aindx" href="#p098" -title="go to pg. 98">98</a>, <a class="aindx" href="#p105" -title="go to pg. 105">105</a>, <a class="aindx" -href="#p117" title="go to pg. 117">117</a>, <a -class="aindx" href="#p432" title="go to pg. 432">432</a>, -<a class="aindx" href="#p710" title="go to pg. -710">710</a>, <a class="aindx" href="#p774" title="go to -pg. 774">774</a></li> - -<li class="liindx">Planorbis, <a class="aindx" href="#p539" -title="go to pg. 539">539</a>, <a class="aindx" -href="#p547" title="go to pg. 547">547</a>, <a -class="aindx" href="#p554" title="go to pg. 554">554</a>, -<a class="aindx" href="#p557" title="go to pg. -557">557</a>, <a class="aindx" href="#p559" title="go to -pg. 559">559</a></li> - -<li class="liindx">Plateau, F., <a class="aindx" -href="#p030" title="go to pg. 30">30</a>, <a class="aindx" -href="#p232" title="go to pg. 232">232</a>; <ul> <li -class="liindx">J. A. F., <a class="aindx" href="#p192" -title="go to pg. 192">192</a>, <a class="aindx" -href="#p212" title="go to pg. 212">212</a>, <a -class="aindx" href="#p218" title="go to pg. 218">218</a>, -<a class="aindx" href="#p239" title="go to pg. -239">239</a>, <a class="aindx" href="#p275" title="go to -pg. 275">275</a>, <a class="aindx" href="#p297" title="go -to pg. 297">297</a>, <a class="aindx" href="#p374" -title="go to pg. 374">374</a>, <a class="aindx" -href="#p477" title="go to pg. 477">477</a></li> </ul></li> - -<li class="liindx">Plato, <a class="aindx" href="#p002" -title="go to pg. 2">2</a>, <a class="aindx" href="#p478" -title="go to pg. 478">478</a>, <a class="aindx" -href="#p720" title="go to pg. 720">720</a>; <ul> <li -class="liindx">Platonic bodies, <a class="aindx" -href="#p478" title="go to pg. 478">478</a></li> </ul></li> - -<li class="liindx">Plesiosaurs, <a class="aindx" -href="#p755" title="go to pg. 755">755</a></li> - -<li class="liindx">Pleurocarpus, <a class="aindx" -href="#p289" title="go to pg. 289">289</a></li> - -<li class="liindx">Pleuropus, <a class="aindx" href="#p573" -title="go to pg. 573">573</a></li> - -<li class="liindx">Pleurotomaria, <a class="aindx" -href="#p557" title="go to pg. 557">557</a></li> - -<li class="liindx">Plumulariidae, <a class="aindx" -href="#p747" title="go to pg. 747">747</a></li> - -<li class="liindx">Pluteus larva, <a class="aindx" -href="#p392" title="go to pg. 392">392</a>, <a -class="aindx" href="#p415" title="go to pg. -415">415</a></li> - -<li class="liindx">Podocoryne, <a class="aindx" -href="#p342" title="go to pg. 342">342</a></li> - -<li class="liindx">Poincaré, H., <a class="aindx" -href="#p134" title="go to pg. 134">134</a></li> - -<li class="liindx">Poiseuille, J. L. M., <a class="aindx" -href="#p669" title="go to pg. 669">669</a></li> - -<li class="liindx">Polar bodies, <a class="aindx" -href="#p179" title="go to pg. 179">179</a>; <ul> <li -class="liindx">furrow, <a class="aindx" href="#p310" -title="go to pg. 310">310</a>, <a class="aindx" -href="#p340" title="go to pg. 340">340</a></li> </ul></li> - -<li class="liindx">Polarised light, <a class="aindx" -href="#p418" title="go to pg. 418">418</a></li> - -<li class="liindx">Polarity, morphological, <a -class="aindx" href="#p166" title="go to pg. 166">166</a>, -<a class="aindx" href="#p168" title="go to pg. -168">168</a>, <a class="aindx" href="#p246" title="go to -pg. 246">246</a>, <a class="aindx" href="#p295" title="go -to pg. 295">295</a>, <a class="aindx" href="#p284" -title="go to pg. 284">284</a></li> - -<li class="liindx">Pollen, <a class="aindx" href="#p396" -title="go to pg. 396">396</a>, <a class="aindx" -href="#p399" title="go to pg. 399">399</a></li> - -<li class="liindx">Polyhalite, <a class="aindx" -href="#p433" title="go to pg. 433">433</a></li> - -<li class="liindx">Polyprion, <a class="aindx" href="#p749" -title="go to pg. 749">749</a>, <a class="aindx" -href="#p776" title="go to pg. 776">776</a></li> - -<li class="liindx">Polyspermy, <a class="aindx" -href="#p193" title="go to pg. 193">193</a></li> - -<li class="liindx">Polytrichum, <a class="aindx" -href="#p355" title="go to pg. 355">355</a></li> - -<li class="liindx">Pomacanthus, <a class="aindx" -href="#p749" title="go to pg. 749">749</a></li> - -<li class="liindx">Popoff, M., <a class="aindx" -href="#p286" title="go to pg. 286">286</a></li> - -<li class="liindx">Potamides, <a class="aindx" href="#p554" -title="go to pg. 554">554</a></li> - -<li class="liindx">Potassium, in living cells, -<a class="aindx" href="#p288" title="go to pg. -288">288</a></li> - -<li class="liindx">Potential energy, <a class="aindx" -href="#p208" title="go to pg. 208">208</a>, <a -class="aindx" href="#p294" title="go to pg. 294">294</a>, -<a class="aindx" href="#p601" title="go to pg. -601">601</a>, etc.</li> - -<li class="liindx">Potter’s wheel, <a class="aindx" -href="#p238" title="go to pg. 238">238</a></li> - -<li class="liindx">Potts, R., <a class="aindx" href="#p126" -title="go to pg. 126">126</a></li> - -<li class="liindx">Pouchet, G., <a class="aindx" -href="#p415" title="go to pg. 415">415</a></li> - -<li class="liindx">Poulton, E. B., <a class="aindx" -href="#p670" title="go to pg. 670">670</a></li> - -<li class="liindx">Poynting, J. H., <a class="aindx" -href="#p235" title="go to pg. 235">235</a></li> - -<li class="liindx">Precocious segregation, <a class="aindx" -href="#p348" title="go to pg. 348">348</a></li> - -<li class="liindx">Preformation, <a class="aindx" -href="#p054" title="go to pg. 54">54</a>, <a class="aindx" -href="#p159" title="go to pg. 159">159</a></li> - -<li class="liindx">Prenant, A., <a class="aindx" -href="#p163" title="go to pg. 163">163</a>, <a -class="aindx" href="#p104" title="go to pg. 104">104</a>, -<a class="aindx" href="#p189" title="go to pg. -189">189</a>, <a class="aindx" href="#p286" title="go to -pg. 286">286</a>, <a class="aindx" href="#p289" title="go -to pg. 289">289</a></li> - -<li class="liindx">Prévost, Pierre, <a class="aindx" -href="#p018" title="go to pg. 18">18</a></li> - -<li class="liindx">Pringsheim, N., <a class="aindx" -href="#p377" title="go to pg. 377">377</a></li> - -<li class="liindx">Probabilities, theory of, <a -class="aindx" href="#p061" title="go to pg. 61">61</a></li> - -<li class="liindx">Productus, <a class="aindx" href="#p567" -title="go to pg. 567">567</a></li> - -<li class="liindx">Protective colouration, <a class="aindx" -href="#p671" title="go to pg. 671">671</a></li> - -<li class="liindx">Protococcus, <a class="aindx" -href="#p059" title="go to pg. 59">59</a>, <a -class="aindx" href="#p300" title="go to pg. 300">300</a>, -<a class="aindx" href="#p410" title="go to pg. -410">410</a></li> - -<li class="liindx">Protoconch, <a class="aindx" -href="#p531" title="go to pg. 531">531</a></li> - -<li class="liindx">Protohippus, <a class="aindx" -href="#fig402" title="go to Fig. 402">767</a></li> - -<li class="liindx">Protoplasm, structure of, -<a class="aindx" href="#p172" title="go to pg. -172">172</a></li> - -<li class="liindx">Przibram, Hans, <a class="aindx" -href="#p016" title="go to pg. 16">16</a>, <a class="aindx" -href="#p082" title="go to pg. 82">82</a>, <a class="aindx" -href="#p107" title="go to pg. 107">107</a>, <a -class="aindx" href="#p149" title="go to pg. 149">149</a>, -<a class="aindx" href="#p204" title="go to pg. -204">204</a>, <a class="aindx" href="#p211" title="go -to pg. 211">211</a>, <a class="aindx" href="#p418" -title="go to pg. 418">418</a>, <a class="aindx" -href="#p595" title="go to pg. 595">595</a>; <ul> <li -class="liindx">Karl, <a class="aindx" href="#p046" -title="go to pg. 46">46</a></li> </ul></li> - -<li class="liindx">Psammobia, <a class="aindx" href="#p564" -title="go to pg. 564">564</a></li> - -<li class="liindx">Pseuopriacauthus, <a class="aindx" -href="#p749" title="go to pg. 749">749</a></li> - -<li class="liindx">Pteranodon, <a class="aindx" -href="#p756" title="go to pg. 756">756</a></li> - -<li class="liindx">Pteris, antheridia of, <a class="aindx" -href="#p409" title="go to pg. 409">409</a></li> - -<li class="liindx">Pteropods of, <a class="aindx" -href="#p258" title="go to pg. 258">258</a>, <a -class="aindx" href="#p570" title="go to pg. -570">570</a></li> - -<li class="liindx">Pulvinulina, <a class="aindx" -href="#p514" title="go to pg. 514">514</a>, <a -class="aindx" href="#p595" title="go to pg. 595">595</a>, -<a class="aindx" href="#p600" title="go to pg. -600">600</a>, <a class="aindx" href="#p602" title="go to -pg. 602">602</a></li> - -<li class="liindx">Pupa, <a class="aindx" href="#p530" -title="go to pg. 530">530</a>, <a class="aindx" -href="#p549" title="go to pg. 549">549</a>, <a -class="aindx" href="#p556" title="go to pg. -556">556</a></li> - -<li class="liindx">Pütter, A., <a class="aindx" -href="#p110" title="go to pg. 110">110</a>, <a -class="aindx" href="#p211" title="go to pg. 211">211</a>, -<a class="aindx" href="#p492" title="go to pg. -492">492</a></li> - -<li class="liindx">Pyrosoma, egg of, <a class="aindx" -href="#p377" title="go to pg. 377">377</a></li> - -<li class="liindx">Pythagoras, <a class="aindx" -href="#p002" title="go to pg. 2">2</a>, <a class="aindx" -href="#p509" title="go to pg. 509">509</a>, <a -class="aindx" href="#p651" title="go to pg. 651">651</a>, -<a class="aindx" href="#p720" title="go to pg. -720">720</a>, <a class="aindx" href="#p779" title="go to -pg. 779">779</a></li> </ul> - -<ul class="ulindx"> <li class="liindx">Quadrant, bisection -of, <a class="aindx" href="#p359" title="go to pg. -359">359</a></li> - -<li class="liindx">Quekett, J. T., <a class="aindx" -href="#p423" title="go to pg. 423">423</a></li> - -<li class="liindx">Quetelet, A., <a class="aindx" -href="#p061" title="go to pg. 61">61</a>, <a class="aindx" -href="#p078" title="go to pg. 78">78</a>, <a class="aindx" -href="#p093" title="go to pg. 93">93</a></li> - -<li class="liindx">Quincke, G. H., <a class="aindx" -href="#p187" title="go to pg. 187">187</a>, <a -class="aindx" href="#p191" title="go to pg. 191">191</a>, -<a class="aindx" href="#p279" title="go to pg. -279">279</a>, <a class="aindx" href="#p421" title="go to -pg. 421">421</a></li> </ul> - -<ul class="ulindx"> <li class="liindx">Rabbit, skull -of, <a class="aindx" href="#p764" title="go to pg. -764">764</a></li> - -<li class="liindx">Rabl, K., <a class="aindx" href="#p036" -title="go to pg. 36">36</a>, <a class="aindx" href="#p310" -title="go to pg. 310">310</a></li> - -<li class="liindx">Radial co-ordinates, <a class="aindx" -href="#p730" title="go to pg. 730">730</a></li> - -<li class="liindx">Radiolaria, <a class="aindx" -href="#p252" title="go to pg. 252">252</a>, <a -class="aindx" href="#p264" title="go to pg. 264">264</a>, -<a class="aindx" href="#p457" title="go to pg. -457">457</a>, <a class="aindx" href="#p467" title="go to -pg. 467">467</a>, <a class="aindx" href="#p588" title="go -to pg. 588">588</a>, <a class="aindx" href="#p607" -title="go to pg. 607">607</a></li> - -<li class="liindx">Rainey, George, <a class="aindx" -href="#p007" title="go to pg. 7">7</a>, <a class="aindx" -href="#p420" title="go to pg. 420">420</a>, <a -class="aindx" href="#p431" title="go to pg. 431">431</a>, -<a class="aindx" href="#p434" title="go to pg. -434">434</a></li> - -<li class="liindx">Rainfall and growth, <a class="aindx" -href="#p121" title="go to pg. 121">121</a></li> - -<li class="liindx">Ram, horns of, <a class="aindx" -href="#p613" title="go to pg. 613">613</a>–<a class="aindx" -href="#p624" title="go to pg. 624">624</a></li> - -<li class="liindx">Ramsden, W., <a class="aindx" -href="#p282" title="go to pg. 282">282</a></li> - -<li class="liindx">Ramulina, <a class="aindx" href="#p255" -title="go to pg. 255">255</a></li> - -<li class="liindx">Rankine, W. J. Macquorn, <a -class="aindx" href="#p697" title="go to pg. 697">697</a>, -<a class="aindx" href="#p712" title="go to pg. -712">712</a></li> - -<li class="liindx">Ransom’s waves, <a class="aindx" -href="#p164" title="go to pg. 164">164</a></li> - -<li class="liindx">Raphides, <a class="aindx" -href="#p412" title="go to pg. 412">412</a>, <a -class="aindx" href="#p429" title="go to pg. 429">429</a>, -<a class="aindx" href="#p434" title="go to pg. -434">434</a></li> - -<li class="liindx">Raphidiophrys, <a class="aindx" -href="#p460" title="go to pg. 460">460</a>, <a -class="aindx" href="#p463" title="go to pg. -463">463</a></li> - -<li class="liindx">Rasumowsky, <a class="aindx" -href="#p683" title="go to pg. 683">683</a></li> - -<li class="liindx">Rat, growth of, <a class="aindx" -href="#p106" title="go to pg. 106">106</a></li> - -<li class="liindx">Rath, O. vom, <a class="aindx" -href="#p181" title="go to pg. 181">181</a></li> - -<li class="liindx">Rauber, A., <a class="aindx" -href="#p200" title="go to pg. 200">200</a>, <a -class="aindx" href="#p305" title="go to pg. 305">305</a>, -<a class="aindx" href="#p310" title="go to pg. -310">310</a>, <a class="aindx" href="#p380" title="go -to pg. 380">380</a>, <a class="aindx" href="#p382" -title="go to pg. 382">382</a>, <a class="aindx" -href="#p398" title="go to pg. 398">398</a>, <a -class="aindx" href="#p677" title="go to pg. 677">677</a>, -<a class="aindx" href="#p683" title="go to pg. -683">683</a></li> - -<li class="liindx">Ray, John, <a class="aindx" href="#p003" -title="go to pg. 3">3</a></li> - -<li class="liindx">Rayleigh, Lord, <a class="aindx" -href="#p043" title="go to pg. 43">43</a>, <a class="aindx" -href="#p044" title="go to pg. 44">44</a></li> - -<li class="liindx">Réaumur, R. A. de, <a class="aindx" -href="#p008" title="go to pg. 8">8</a>, <a class="aindx" -href="#p108" title="go to pg. 108">108</a>, <a -class="aindx" href="#p329" title="go to pg. -329">329</a></li> - -<li class="liindx">Reciprocal diagrams, <a class="aindx" -href="#p697" title="go to pg. 697">697</a></li> - -<li class="liindx">Rees, R. van, <a class="aindx" -href="#p374" title="go to pg. 374">374</a></li> - -<li class="liindx">Regeneration, <a class="aindx" -href="#p138" title="go to pg. 138">138</a></li> - -<li class="liindx">Reid, E. Waymouth, <a class="aindx" -href="#p272" title="go to pg. 272">272</a></li> - -<li class="liindx">Reinecke, J. C. M., <a class="aindx" -href="#p528" title="go to pg. 528">528</a></li> - -<li class="liindx">Reinke, J., <a class="aindx" -href="#p303" title="go to pg. 303">303</a>, <a -class="aindx" href="#p305" title="go to pg. 305">305</a>, -<a class="aindx" href="#p355" title="go to pg. -355">355</a>, <a class="aindx" href="#p356" title="go to -pg. 356">356</a></li> - -<li class="liindx">Reniform shape, <a class="aindx" -href="#p735" title="go to pg. 735">735</a></li> - -<li class="liindx">Reticularia, <a class="aindx" -href="#p569" title="go to pg. 569">569</a></li> - -<li class="liindx">Reticulated patterns, <a class="aindx" -href="#p258" title="go to pg. 258">258</a></li> - -<li class="liindx">Réticulum plasmatique, <a class="aindx" -href="#p468" title="go to pg. 468">468</a></li> - -<li class="liindx">Rhabdammina, <a class="aindx" -href="#p589" title="go to pg. 589">589</a></li> - -<li class="liindx">Rheophax, <a class="aindx" href="#p263" -title="go to pg. 263">263</a></li> - -<li class="liindx">Rhinoceros, <a class="aindx" -href="#p612" title="go to pg. 612">612</a>, <a -class="aindx" href="#p760" title="go to pg. -760">760</a></li> - -<li class="liindx">Rhumbler, L., <a class="aindx" -href="#p162" title="go to pg. 162">162</a>, <a -class="aindx" href="#p165" title="go to pg. 165">165</a>, -<a class="aindx" href="#p260" title="go to pg. -260">260</a>, <a class="aindx" href="#p322" title="go to -pg. 322">322</a>, <a class="aindx" href="#p344" title="go -to pg. 344">344</a>, <a class="aindx" href="#p465" -title="go to pg. 465">465</a>, <a class="aindx" -href="#p466" title="go to pg. 466">466</a>, <a -class="aindx" href="#p589" title="go to pg. 589">589</a>, -<a class="aindx" href="#p590" title="go to pg. -590">590</a>, <a class="aindx" href="#p595" title="go to -pg. 595">595</a>, <a class="aindx" href="#p599" title="go -to pg. 599">599</a>, <a class="aindx" href="#p608" -title="go to pg. 608">608</a>, <a class="aindx" -href="#p628" title="go to pg. 628">628</a></li> - -<li class="liindx">Rhynchonella, <a class="aindx" -href="#p561" title="go to pg. 561">561</a></li> - -<li class="liindx">Riccia, <a class="aindx" href="#p372" -title="go to pg. 372">372</a>, <a class="aindx" -href="#p403" title="go to pg. 403">403</a>, <a -class="aindx" href="#p405" title="go to pg. -405">405</a></li> - -<li class="liindx">Rice, J., <a class="aindx" href="#p242" -title="go to pg. 242">242</a>, <a class="aindx" -href="#p273" title="go to pg. 273">273</a></li> - -<li class="liindx">Richardson, G. M., <a class="aindx" -href="#p416" title="go to pg. 416">416</a></li> - -<li class="liindx">Riefstahl, E., <a class="aindx" -href="#p578" title="go to pg. 578">578</a></li> - -<li class="liindx">Riemann, B., <a class="aindx" -href="#p385" title="go to pg. 385">385</a></li> - -<li class="liindx">Ripples, <a class="aindx" href="#p033" -title="go to pg. 33">33</a>, <a class="aindx" href="#p261" -title="go to pg. 261">261</a>, <a class="aindx" -href="#p323" title="go to pg. 323">323</a></li> - -<li class="liindx">Rivularia, <a class="aindx" href="#p300" -title="go to pg. 300">300</a></li> - -<li class="liindx">Roaf, H. C., <a class="aindx" -href="#p272" title="go to pg. 272">272</a></li> - -<li class="liindx">Robert, A., <a class="aindx" -href="#p306" title="go to pg. 306">306</a>, <a -class="aindx" href="#p339" title="go to pg. 339">339</a>, -<a class="aindx" href="#p348" title="go to pg. -348">348</a>, <a class="aindx" href="#p377" title="go to -pg. 377">377</a></li> - -<li class="liindx">Roberts, C., <a class="aindx" -href="#p061" title="go to pg. 61">61</a></li> - -<li class="liindx">Robertson, T. B., <a class="aindx" -href="#p082" title="go to pg. 82">82</a>, <a class="aindx" -href="#p132" title="go to pg. 132">132</a>, <a -class="aindx" href="#p191" title="go to pg. 191">191</a>, -<a class="aindx" href="#p192" title="go to pg. -192">192</a></li> - -<li class="liindx">Robinson, A., <a class="aindx" -href="#p681" title="go to pg. 681">681</a></li> - -<li class="liindx">Rörig, A., <a class="aindx" href="#p628" -title="go to pg. 628">628</a></li> - -<li class="liindx">Rose, Gustav, <a class="aindx" -href="#p421" title="go to pg. 421">421</a></li> - -<li class="liindx">Rossbach, M. J., <a class="aindx" -href="#p165" title="go to pg. 165">165</a></li> - -<li class="liindx">Rotalia, <a class="aindx" -href="#p214" title="go to pg. 214">214</a>, <a -class="aindx" href="#p535" title="go to pg. 535">535</a>, -<a class="aindx" href="#p602" title="go to pg. -602">602</a></li> - -<li class="liindx">Rotifera, cells of, <a class="aindx" -href="#p038" title="go to pg. 38">38</a></li> - -<li class="liindx">Roulettes, <a class="aindx" href="#p218" -title="go to pg. 218">218</a></li> - -<li class="liindx">Roux, W., <a class="aindx" href="#p008" -title="go to pg. 8">8</a>, <a class="aindx" href="#p055" -title="go to pg. 55">55</a>, <a class="aindx" href="#p057" -title="go to pg. 57">57</a>, <a class="aindx" href="#p157" -title="go to pg. 157">157</a>, <a class="aindx" -href="#p194" title="go to pg. 194">194</a>, <a -class="aindx" href="#p378" title="go to pg. 378">378</a>, -<a class="aindx" href="#p383" title="go to pg. -383">383</a>, <a class="aindx" href="#p666" title="go to -pg. 666">666</a>, <a class="aindx" href="#p683" title="go -to pg. 683">683</a></li> - -<li class="liindx">Ruled surfaces, <a class="aindx" -href="#p230" title="go to pg. 230">230</a>, <a -class="aindx" href="#p270" title="go to pg. 270">270</a>, -<a class="aindx" href="#p582" title="go to pg. -582">582</a></li> - -<li class="liindx">Ruskin, John, <a class="aindx" -href="#p020" title="go to pg. 20">20</a></li> - -<li class="liindx">Russow, ——, <a class="aindx" -href="#p073" title="go to pg. 73">73</a>, <a class="aindx" -href="#p075" title="go to pg. 75">75</a></li> - -<li class="liindx">Ryder, J. A., <a class="aindx" -href="#p376" title="go to pg. 376">376</a></li> </ul> - -<ul class="ulindx"> <li class="liindx">Sachs, J., <a -class="aindx" href="#p035" title="go to pg. 35">35</a>, <a -class="aindx" href="#p038" title="go to pg. 38">38</a>, <a -class="aindx" href="#p095" title="go to pg. 95">95</a>, <a -class="aindx" href="#p108" title="go to pg. 108">108</a>, -<a class="aindx" href="#p110" title="go to pg. -110">110</a>, <a class="aindx" href="#p111" title="go to -pg. 111">111</a>, <a class="aindx" href="#p200" title="go -to pg. 200">200</a>, <a class="aindx" href="#p360" -title="go to pg. 360">360</a>, <a class="aindx" -href="#p398" title="go to pg. 398">398</a>, <a -class="aindx" href="#p399" title="go to pg. 399">399</a>, -<a class="aindx" href="#p624" title="go to pg. -624">624</a>, <a class="aindx" href="#p635" title="go to -pg. 635">635</a>, <a class="aindx" href="#p640" title="go -to pg. 640">640</a>, <a class="aindx" href="#p651" -title="go to pg. 651">651</a>, <a class="aindx" -href="#p680" title="go to pg. 680">680</a></li> - -<li class="liindx">Sachs’s rule, <a class="aindx" -href="#p297" title="go to pg. 297">297</a>, <a -class="aindx" href="#p300" title="go to pg. 300">300</a>, -<a class="aindx" href="#p305" title="go to pg. -305">305</a>, <a class="aindx" href="#p347" title="go to -pg. 347">347</a>, <a class="aindx" href="#p376" title="go -to pg. 376">376</a></li> - -<li class="liindx">Saddles, of ammonites, <a class="aindx" -href="#p583" title="go to pg. 583">583</a></li> - -<li class="liindx">Sagrina, <a class="aindx" href="#p263" -title="go to pg. 263">263</a></li> - -<li class="liindx">St Venant, Barré de, <a class="aindx" -href="#p621" title="go to pg. 621">621</a>, <a -class="aindx" href="#p627" title="go to pg. -627">627</a></li> - -<li class="liindx">Salamander, sperm-cells of, -<a class="aindx" href="#p179" title="go to pg. -179">179</a></li> - -<li class="liindx">Salpingoeca, <a class="aindx" -href="#p248" title="go to pg. 248">248</a></li> - -<li class="liindx">Salt, crystals of, <a class="aindx" -href="#p429" title="go to pg. 429">429</a></li> - -<li class="liindx">Salvinia, <a class="aindx" href="#p377" -title="go to pg. 377">377</a></li> - -<li class="liindx">Samec, M., <a class="aindx" href="#p434" -title="go to pg. 434">434</a></li> - -<li class="liindx">Samter, M. and Heymons, <a class="aindx" -href="#p130" title="go to pg. 130">130</a></li> - -<li class="liindx">Sandberger, G., <a class="aindx" -href="#p539" title="go to pg. 539">539</a></li> - -<li class="liindx">Sapphirina, <a class="aindx" -href="#p742" title="go to pg. 742">742</a></li> - -<li class="liindx">Saville Kent, W., <a class="aindx" -href="#p246" title="go to pg. 246">246</a>, <a -class="aindx" href="#p247" title="go to pg. 247">247</a>, -<a class="aindx" href="#p248" title="go to pg. -248">248</a></li> - -<li class="liindx">Scalaria, <a class="aindx" href="#p526" -title="go to pg. 526">526</a>, <a class="aindx" -href="#p547" title="go to pg. 547">547</a>, <a -class="aindx" href="#p554" title="go to pg. 554">554</a>, -<a class="aindx" href="#p557" title="go to pg. -557">557</a>, <a class="aindx" href="#p559" title="go to -pg. 559">559</a></li> - -<li class="liindx">Scale, effect of, <a class="aindx" -href="#p017" title="go to pg. 17">17</a>, <a class="aindx" -href="#p438" title="go to pg. 438">438</a></li> - -<li class="liindx">Scaphites, <a class="aindx" href="#p550" -title="go to pg. 550">550</a></li> - -<li class="liindx">Scapula, human, <a class="aindx" -href="#p769" title="go to pg. 769">769</a></li> - -<li class="liindx">Scarus, <a class="aindx" href="#p749" -title="go to pg. 749">749</a></li> - -<li class="liindx">Schacko, G., <a class="aindx" -href="#p604" title="go to pg. 604">604</a></li> - -<li class="liindx">Schaper, A. A., <a class="aindx" -href="#p083" title="go to pg. 83">83</a></li> - -<li class="liindx">Schaudinn, F., <a class="aindx" -href="#p046" title="go to pg. 46">46</a>, <a class="aindx" -href="#p286" title="go to pg. 286">286</a></li> - -<li class="liindx">Scheerenumkehr, <a class="aindx" -href="#p149" title="go to pg. 149">149</a></li> - -<li class="liindx">Schewiakoff, W., <a class="aindx" -href="#p189" title="go to pg. 189">189</a>, <a -class="aindx" href="#p462" title="go to pg. -462">462</a></li> - -<li class="liindx">Schimper, C. F., <a class="aindx" -href="#p502" title="go to pg. 502">502</a>, <a -class="aindx" href="#p636" title="go to pg. -636">636</a></li> - -<li class="liindx">Schmaltz, A., <a class="aindx" -href="#p675" title="go to pg. 675">675</a></li> - -<li class="liindx">Schmankewitsch, W., <a class="aindx" -href="#p130" title="go to pg. 130">130</a></li> - -<li class="liindx">Schmidt, Johann, <a class="aindx" -href="#p085" title="go to pg. 85">85</a>, <a class="aindx" -href="#p087" title="go to pg. 87">87</a>, <a class="aindx" -href="#p118" title="go to pg. 118">118</a></li> - -<li class="liindx">Schönflies, A., <a class="aindx" -href="#p202" title="go to pg. 202">202</a></li> - -<li class="liindx">Schultze, F. E., <a class="aindx" -href="#p452" title="go to pg. 452">452</a>, <a -class="aindx" href="#p454" title="go to pg. -454">454</a></li> - -<li class="liindx">Schwalbe, G., <a class="aindx" -href="#p666" title="go to pg. 666">666</a></li> - -<li class="liindx">Schwann, Theodor, <a class="aindx" -href="#p199" title="go to pg. 199">199</a>, <a -class="aindx" href="#p380" title="go to pg. 380">380</a>, -<a class="aindx" href="#p591" title="go to pg. -591">591</a></li> - -<li class="liindx">Schwartz, Fr., <a class="aindx" -href="#p172" title="go to pg. 172">172</a></li> - -<li class="liindx">Schwendener, S., <a class="aindx" -href="#p210" title="go to pg. 210">210</a>, <a -class="aindx" href="#p305" title="go to pg. 305">305</a>, -<a class="aindx" href="#p636" title="go to pg. -636">636</a>, <a class="aindx" href="#p678" title="go to -pg. 678">678</a></li> - -<li class="liindx">Scorpaena, <a class="aindx" href="#p749" -title="go to pg. 749">749</a></li> - -<li class="liindx">Scorpioid cyme, <a class="aindx" -href="#p502" title="go to pg. 502">502</a></li> - -<li class="liindx">Scott, E. L., <a class="aindx" -href="#p110" title="go to pg. 110">110</a>; <ul> <li -class="liindx">W. B., <a class="aindx" href="#p768" -title="go to pg. 768">768</a></li> </ul></li> - -<li class="liindx">Scyromathia, <a class="aindx" -href="#p744" title="go to pg. 744">744</a></li> - -<li class="liindx">Searle, H., <a class="aindx" -href="#p491" title="go to pg. 491">491</a></li> - -<li class="liindx">Sea urchins, <a class="aindx" -href="#p661" title="go to pg. 661">661</a>; <ul> -<li class="liindx">egg of, <a class="aindx" -href="#p173" title="go to pg. 173">173</a>;</li> <li -class="liindx">growth of, <a class="aindx" href="#p117" -title="go to pg. 117">117</a>, <a class="aindx" -href="#p147" title="go to pg. 147">147</a></li> </ul></li> - -<li class="liindx">Sebastes, <a class="aindx" href="#p749" -title="go to pg. 749">749</a></li> - -<li class="liindx">Sectio aurea, <a class="aindx" -href="#p511" title="go to pg. 511">511</a>, <a -class="aindx" href="#p643" title="go to pg. 643">643</a>, -<a class="aindx" href="#p649" title="go to pg. -649">649</a></li> - -<li class="liindx">Sedgwick, A., <a class="aindx" -href="#p197" title="go to pg. 197">197</a>, <a -class="aindx" href="#p199" title="go to pg. -199">199</a></li> - -<li class="liindx">Sédillot, Charles E., <a class="aindx" -href="#p688" title="go to pg. 688">688</a></li> - -<li class="liindx">Segmentation of egg, <a class="aindx" -href="#p057" title="go to pg. 57">57</a>, <a class="aindx" -href="#p310" title="go to pg. 310">310</a>, <a -class="aindx" href="#p344" title="go to pg. 344">344</a>, -<a class="aindx" href="#p382" title="go to pg. -382">382</a>, etc.; <ul> <li class="liindx">spiral -do., <a class="aindx" href="#p371" title="go to pg. -371">371</a>, <a class="aindx" href="#p453" title="go to -pg. 453">453</a></li> </ul></li> - -<li class="liindx">Segner, J. A. von, <a class="aindx" -href="#p205" title="go to pg. 205">205</a></li> - -<li class="liindx">Selaginella, <a class="aindx" -href="#p404" title="go to pg. 404">404</a></li> - -<li class="liindx">Semi-permeable membranes, -<a class="aindx" href="#p272" title="go to pg. -272">272</a></li> - -<li class="liindx">Sepia, <a class="aindx" href="#p575" -title="go to pg. 575">575</a>, <a class="aindx" -href="#p577" title="go to pg. 577">577</a></li> - -<li class="liindx">Septa, <a class="aindx" href="#p577" -title="go to pg. 577">577</a>, <a class="aindx" -href="#p592" title="go to pg. 592">592</a></li> - -<li class="liindx">Serpula, <a class="aindx" href="#p603" -title="go to pg. 603">603</a></li> - -<li class="liindx">Sexual characters, <a class="aindx" -href="#p135" title="go to pg. 135">135</a></li> - -<li class="liindx">Sharpe, D., <a class="aindx" -href="#p728" title="go to pg. 728">728</a></li> - -<li class="liindx">Shearing stress, <a class="aindx" -href="#p684" title="go to pg. 684">684</a>, <a -class="aindx" href="#p730" title="go to pg. 730">730</a>, -etc.</li> - -<li class="liindx">Sheep, <a class="aindx" href="#p613" -title="go to pg. 613">613</a>, <a class="aindx" -href="#p730" title="go to pg. 730">730</a>, <a -class="aindx" href="#p738" title="go to pg. -738">738</a></li> - -<li class="liindx">Shell, formation of, <a class="aindx" -href="#p422" title="go to pg. 422">422</a></li> - -<li class="liindx">Sigaretus, <a class="aindx" href="#p554" -title="go to pg. 554">554</a></li> - -<li class="liindx">Silkworm, growth of, <a class="aindx" -href="#p083" title="go to pg. 83">83</a></li> - -<li class="liindx">Similitude, principle of, <a -class="aindx" href="#p017" title="go to pg. 17">17</a></li> - -<li class="liindx">Sims Woodhead, G., <a class="aindx" -href="#p414" title="go to pg. 414">414</a>, <a -class="aindx" href="#p434" title="go to pg. -434">434</a></li> - -<li class="liindx">Siphonogorgia, <a class="aindx" -href="#p413" title="go to pg. 413">413</a></li> - -<li class="liindx">Skeleton, <a class="aindx" href="#p019" -title="go to pg. 19">19</a>, <a class="aindx" href="#p438" -title="go to pg. 438">438</a>, <a class="aindx" -href="#p675" title="go to pg. 675">675</a>, <a -class="aindx" href="#p691" title="go to pg. 691">691</a>, -etc.</li> - -<li class="liindx">Snow crystals, <a class="aindx" -href="#p250" title="go to pg. 250">250</a>, <a -class="aindx" href="#p480" title="go to pg. 480">480</a>, -<a class="aindx" href="#p611" title="go to pg. -611">611</a></li> - -<li class="liindx">Soap-bubbles, <a class="aindx" -href="#p043" title="go to pg. 43">43</a>, <a class="aindx" -href="#p219" title="go to pg. 219">219</a>, <a -class="aindx" href="#p299" title="go to pg. 299">299</a>, -<a class="aindx" href="#p307" title="go to pg. -307">307</a>, etc.</li> - -<li class="liindx">Socrates, <a class="aindx" href="#p008" -title="go to pg. 8">8</a></li> - -<li class="liindx">Sohncke, L. A., <a class="aindx" -href="#p202" title="go to pg. 202">202</a></li> - -<li class="liindx">Solanum, <a class="aindx" href="#p625" -title="go to pg. 625">625</a></li> - -<li class="liindx">Solarium, <a class="aindx" href="#p547" -title="go to pg. 547">547</a>, <a class="aindx" -href="#p554" title="go to pg. 554">554</a>, <a -class="aindx" href="#p557" title="go to pg. 557">557</a>, -<a class="aindx" href="#p559" title="go to pg. -559">559</a></li> - -<li class="liindx">Solecurtus, <a class="aindx" -href="#p564" title="go to pg. 564">564</a></li> - -<li class="liindx">Solen, <a class="aindx" href="#p565" -title="go to pg. 565">565</a></li> - -<li class="liindx">Sollas, W. J., <a class="aindx" -href="#p440" title="go to pg. 440">440</a>, <a -class="aindx" href="#p450" title="go to pg. 450">450</a>, -<a class="aindx" href="#p455" title="go to pg. -455">455</a></li> - -<li class="liindx">Solubility of salts, <a class="aindx" -href="#p434" title="go to pg. 434">434</a></li> - -<li class="liindx">Sorby, H. C., <a class="aindx" -href="#p412" title="go to pg. 412">412</a>, <a -class="aindx" href="#p414" title="go to pg. 414">414</a>, -<a class="aindx" href="#p728" title="go to pg. -728">728</a></li> - -<li class="liindx">Spallanzani, L., <a class="aindx" -href="#p138" title="go to pg. 138">138</a></li> - -<li class="liindx">Span of arms, <a class="aindx" -href="#p063" title="go to pg. 63">63</a>, <a class="aindx" -href="#p093" title="go to pg. 93">93</a></li> - -<li class="liindx">Spangenberg, Fr., <a class="aindx" -href="#p342" title="go to pg. 342">342</a></li> - -<li class="liindx">Specific characters, <a class="aindx" -href="#p246" title="go to pg. 246">246</a>, <a -class="aindx" href="#p380" title="go to pg. -380">380</a>; <ul> <li class="liindx">inductive -capacity, <a class="aindx" href="#p177" title="go to -pg. 177">177</a>;</li> <li class="liindx">surface, <a -class="aindx" href="#p032" title="go to pg. 32">32</a>, -<a class="aindx" href="#p215" title="go to pg. -215">215</a></li> </ul></li> - -<li class="liindx">Spencer, Herbert, <a class="aindx" -href="#p018" title="go to pg. 18">18</a>, <a class="aindx" -href="#p022" title="go to pg. 22">22</a></li> - -<li class="liindx">Spermatozoon, path of, <a class="aindx" -href="#p193" title="go to pg. 193">193</a></li> - -<li class="liindx">Sperm-cells of Crustacea, -<a class="aindx" href="#p273" title="go to pg. -273">273</a></li> - -<li class="liindx">Sphacelaria, <a class="aindx" -href="#p351" title="go to pg. 351">351</a></li> - -<li class="liindx">Sphaerechinus, <a class="aindx" -href="#p117" title="go to pg. 117">117</a>, <a -class="aindx" href="#p147" title="go to pg. -147">147</a></li> - -<li class="liindx">Sphagnum, <a class="aindx" href="#p402" -title="go to pg. 402">402</a>, <a class="aindx" -href="#p407" title="go to pg. 407">407</a></li> - -<li class="liindx">Sphere, <a class="aindx" href="#p218" -title="go to pg. 218">218</a>, <a class="aindx" -href="#p225" title="go to pg. 225">225</a></li> - -<li class="liindx">Spherocrystals, <a class="aindx" -href="#p434" title="go to pg. 434">434</a></li> - -<li class="liindx">Spherulites, <a class="aindx" -href="#p422" title="go to pg. 422">422</a></li> - -<li class="liindx">Spicules, <a class="aindx" href="#p282" -title="go to pg. 282">282</a>, <a class="aindx" -href="#p411" title="go to pg. 411">411</a>, etc.</li> - -<li class="liindx">Spider’s web, <a class="aindx" -href="#p231" title="go to pg. 231">231</a></li> - -<li class="liindx">Spindle, nuclear, <a class="aindx" -href="#p169" title="go to pg. 169">169</a>, <a -class="aindx" href="#p174" title="go to pg. -174">174</a></li> - -<li class="liindx">Spinning of protoplasm, <a class="aindx" -href="#p164" title="go to pg. 164">164</a></li> - -<li class="liindx">Spiral, geodetic, <a class="aindx" -href="#p488" title="go to pg. 488">488</a>; <ul> -<li class="liindx">logarithmic, <a class="aindx" -href="#p493" title="go to pg. 493">493</a>, etc.;</li> <li -class="liindx">segmentation, <a class="aindx" href="#p371" -title="go to pg. 371">371</a>, <a class="aindx" -href="#p453" title="go to pg. 453">453</a></li> </ul></li> - -<li class="liindx">Spireme, <a class="aindx" href="#p173" -title="go to pg. 173">173</a>, <a class="aindx" -href="#p180" title="go to pg. 180">180</a></li> - -<li class="liindx">Spirifer, <a class="aindx" href="#p561" -title="go to pg. 561">561</a>, <a class="aindx" -href="#p568" title="go to pg. 568">568</a></li> - -<li class="liindx">Spirillum, <a class="aindx" href="#p046" -title="go to pg. 46">46</a>, <a class="aindx" href="#p253" -title="go to pg. 253">253</a></li> - -<li class="liindx">Spirochaetes, <a class="aindx" -href="#p046" title="go to pg. 46">46</a>, <a -class="aindx" href="#p230" title="go to pg. 230">230</a>, -<a class="aindx" href="#p266" title="go to pg. -266">266</a></li> - -<li class="liindx">Spirographis, <a class="aindx" -href="#p586" title="go to pg. 586">586</a></li> - -<li class="liindx">Spirogyra, <a class="aindx" href="#p012" -title="go to pg. 12">12</a>, <a class="aindx" href="#p221" -title="go to pg. 221">221</a>, <a class="aindx" -href="#p227" title="go to pg. 227">227</a>, <a -class="aindx" href="#p242" title="go to pg. 242">242</a>, -<a class="aindx" href="#p244" title="go to pg. -244">244</a>, <a class="aindx" href="#p275" title="go to -pg. 275">275</a>, <a class="aindx" href="#p287" title="go -to pg. 287">287</a>, <a class="aindx" href="#p289" -title="go to pg. 289">289</a></li> - -<li class="liindx">Spirorbis, <a class="aindx" href="#p586" -title="go to pg. 586">586</a>, <a class="aindx" -href="#p603" title="go to pg. 603">603</a></li> - -<li class="liindx">Spirula, <a class="aindx" href="#p528" -title="go to pg. 528">528</a>, <a class="aindx" -href="#p547" title="go to pg. 547">547</a>, <a -class="aindx" href="#p554" title="go to pg. 554">554</a>, -<a class="aindx" href="#p575" title="go to pg. -575">575</a>, <a class="aindx" href="#p577" title="go to -pg. 577">577</a></li> - -<li class="liindx">Spitzka, E. A., <a class="aindx" -href="#p092" title="go to pg. 92">92</a></li> - -<li class="liindx">Splashes, <a class="aindx" href="#p235" -title="go to pg. 235">235</a>, <a class="aindx" -href="#p236" title="go to pg. 236">236</a>, <a -class="aindx" href="#p254" title="go to pg. 254">254</a>, -<a class="aindx" href="#p260" title="go to pg. -260">260</a></li> - -<li class="liindx">Sponge-spicules, <a class="aindx" -href="#p436" title="go to pg. 436">436</a>, <a -class="aindx" href="#p440" title="go to pg. -440">440</a></li> - -<li class="liindx">Spontaneous generation, <a class="aindx" -href="#p420" title="go to pg. 420">420</a></li> - -<li class="liindx">Sporangium, <a class="aindx" -href="#p406" title="go to pg. 406">406</a></li> - -<li class="liindx">Spottiswoode, W., <a class="aindx" -href="#p779" title="go to pg. 779">779</a></li> - -<li class="liindx">Spray, <a class="aindx" href="#p236" -title="go to pg. 236">236</a></li> - -<li class="liindx">Stallo, J. B., <a class="aindx" -href="#p001" title="go to pg. 1">1</a></li> - -<li class="liindx">Standard deviation, <a class="aindx" -href="#p078" title="go to pg. 78">78</a></li> - -<li class="liindx">Starch, <a class="aindx" href="#p432" -title="go to pg. 432">432</a></li> - -<li class="liindx">Starling, E. H., <a class="aindx" -href="#p135" title="go to pg. 135">135</a></li> - -<li class="liindx">Stassfurt salt, <a class="aindx" -href="#p433" title="go to pg. 433">433</a></li> - -<li class="liindx">Stegocephalus, <a class="aindx" -href="#p746" title="go to pg. 746">746</a></li> - -<li class="liindx">Stegosaurus, <a class="aindx" -href="#p706" title="go to pg. 706">706</a>, <a -class="aindx" href="#p707" title="go to pg. 707">707</a>, -<a class="aindx" href="#p710" title="go to pg. -710">710</a>, <a class="aindx" href="#p754" title="go to -pg. 754">754</a></li> - -<li class="liindx">Steiner, Jacob, <a class="aindx" -href="#p654" title="go to pg. 654">654</a></li> - -<li class="liindx">Steinmann, G., <a class="aindx" -href="#p431" title="go to pg. 431">431</a></li> - -<li class="liindx">Stellate cells, <a class="aindx" -href="#p335" title="go to pg. 335">335</a></li> - -<li class="liindx">Stentor, <a class="aindx" href="#p147" -title="go to pg. 147">147</a></li> - -<li class="liindx">Stereometry, <a class="aindx" -href="#p417" title="go to pg. 417">417</a></li> - -<li class="liindx">Sternoptyx, <a class="aindx" -href="#p748" title="go to pg. 748">748</a></li> - -<li class="liindx">Stillmann, J. D. B., <a class="aindx" -href="#p695" title="go to pg. 695">695</a></li> - -<li class="liindx">St Loup, R., <a class="aindx" -href="#p082" title="go to pg. 82">82</a></li> - -<li class="liindx">Stokes, Sir G. G., <a class="aindx" -href="#p044" title="go to pg. 44">44</a></li> - -<li class="liindx">Stolc, Ant., <a class="aindx" -href="#p452" title="go to pg. 452">452</a></li> - -<li class="liindx">Stomach, muscles of, <a class="aindx" -href="#p490" title="go to pg. 490">490</a></li> - -<li class="liindx">Stomata, <a class="aindx" href="#p393" -title="go to pg. 393">393</a></li> - -<li class="liindx">Stomatella, <a class="aindx" -href="#p554" title="go to pg. 554">554</a></li> - -<li class="liindx">Strasbürger, E., <a class="aindx" -href="#p035" title="go to pg. 35">35</a>, <a -class="aindx" href="#p283" title="go to pg. 283">283</a>, -<a class="aindx" href="#p409" title="go to pg. -409">409</a></li> - -<li class="liindx">Straus-Dürckheim, H. E., <a -class="aindx" href="#p030" title="go to pg. 30">30</a></li> - -<li class="liindx">Stream-lines, <a class="aindx" -href="#p250" title="go to pg. 250">250</a>, <a -class="aindx" href="#p673" title="go to pg. 673">673</a>, -<a class="aindx" href="#p736" title="go to pg. -736">736</a></li> - -<li class="liindx">Strength of materials, <a -class="aindx" href="#p676" title="go to pg. 676">676</a>, -<a class="aindx" href="#p679" title="go to pg. -679">679</a></li> - -<li class="liindx">Streptoplasma, <a class="aindx" -href="#p391" title="go to pg. 391">391</a></li> - -<li class="liindx">Strophomena, <a class="aindx" -href="#p567" title="go to pg. 567">567</a></li> - -<li class="liindx">Studer, T., <a class="aindx" -href="#p413" title="go to pg. 413">413</a></li> - -<li class="liindx">Stylonichia, <a class="aindx" -href="#p133" title="go to pg. 133">133</a></li> - -<li class="liindx">Succinea, <a class="aindx" href="#p556" -title="go to pg. 556">556</a></li> - -<li class="liindx">Sunflower, <a class="aindx" href="#p494" -title="go to pg. 494">494</a>, <a class="aindx" -href="#p635" title="go to pg. 635">635</a>, <a -class="aindx" href="#p639" title="go to pg. 639">639</a>, -<a class="aindx" href="#p688" title="go to pg. -688">688</a></li> - -<li class="liindx">Surface energy, <a class="aindx" -href="#p032" title="go to pg. 32">32</a>, <a class="aindx" -href="#p034" title="go to pg. 34">34</a>, <a class="aindx" -href="#p191" title="go to pg. 191">191</a>, <a -class="aindx" href="#p207" title="go to pg. 207">207</a>, -<a class="aindx" href="#p278" title="go to pg. -278">278</a>, <a class="aindx" href="#p293" title="go to -pg. 293">293</a>, <a class="aindx" href="#p460" title="go -to pg. 460">460</a>, <a class="aindx" href="#p599" -title="go to pg. 599">599</a></li> - -<li class="liindx">Survival of species, <a class="aindx" -href="#p251" title="go to pg. 251">251</a></li> - -<li class="liindx">Sutures of cephalopods, <a class="aindx" -href="#p583" title="go to pg. 583">583</a></li> - -<li class="liindx">Swammerdam, J., <a class="aindx" -href="#p008" title="go to pg. 8">8</a>, <a class="aindx" -href="#p087" title="go to pg. 87">87</a>, <a class="aindx" -href="#p380" title="go to pg. 380">380</a>, <a -class="aindx" href="#p528" title="go to pg. 528">528</a>, -<a class="aindx" href="#p585" title="go to pg. -585">585</a></li> - -<li class="liindx">Swezy, Olive, <a class="aindx" -href="#p268" title="go to pg. 268">268</a></li> - -<li class="liindx">Sylvester, J. J., <a class="aindx" -href="#p723" title="go to pg. 723">723</a></li> - -<li class="liindx">Symmetry, meaning of, <a class="aindx" -href="#p209" title="go to pg. 209">209</a></li> - -<li class="liindx">Synapta, egg of, <a class="aindx" -href="#p453" title="go to pg. 453">453</a></li> - -<li class="liindx">Syncytium, <a class="aindx" href="#p200" -title="go to pg. 200">200</a></li> - -<li class="liindx">Synhelia, <a class="aindx" href="#p327" -title="go to pg. 327">327</a></li> - -<li class="liindx">Szielasko, A., <a class="aindx" -href="#p654" title="go to pg. 654">654</a></li> </ul> - -<ul class="ulindx"> <li class="liindx">Tadpole, growth -of, <a class="aindx" href="#p083" title="go to pg. -83">83</a>, <a class="aindx" href="#p114" title="go to pg. -114">114</a>, <a class="aindx" href="#p138" title="go to -pg. 138">138</a>, <a class="aindx" href="#p153" title="go -to pg. 153">153</a></li> - -<li class="liindx">Tait, P. G., <a class="aindx" -href="#p035" title="go to pg. 35">35</a>, <a class="aindx" -href="#p043" title="go to pg. 43">43</a>, <a -class="aindx" href="#p207" title="go to pg. 207">207</a>, -<a class="aindx" href="#p644" title="go to pg. -644">644</a></li> - -<li class="liindx">Taonia, <a class="aindx" href="#p355" -title="go to pg. 355">355</a>, <a class="aindx" -href="#p356" title="go to pg. 356">356</a></li> - -<li class="liindx">Tapetum, <a class="aindx" href="#p407" -title="go to pg. 407">407</a></li> - -<li class="liindx">Tapir, <a class="aindx" href="#p741" -title="go to pg. 741">741</a>, <a class="aindx" -href="#p763" title="go to pg. 763">763</a></li> - -<li class="liindx">Taylor, W. W., <a class="aindx" -href="#p277" title="go to pg. 277">277</a>, <a -class="aindx" href="#p282" title="go to pg. 282">282</a>, -<a class="aindx" href="#p426" title="go to pg. -426">426</a>, <a class="aindx" href="#p428" title="go to -pg. 428">428</a></li> - -<li class="liindx">Teeth, <a class="aindx" href="#p424" -title="go to pg. 424">424</a>, <a class="aindx" -href="#p612" title="go to pg. 612">612</a>, <a -class="aindx" href="#p632" title="go to pg. -632">632</a></li> - -<li class="liindx">Telescopium, <a class="aindx" -href="#p557" title="go to pg. 557">557</a></li> - -<li class="liindx">Telesius, Bernardinus, <a class="aindx" -href="#p656" title="go to pg. 656">656</a></li> - -<li class="liindx">Tellina, <a class="aindx" href="#p562" -title="go to pg. 562">562</a></li> - -<li class="liindx">Temperature coefficient, <a -class="aindx" href="#p109" title="go to pg. -109">109</a></li> - -<li class="liindx">Terebra, <a class="aindx" -href="#p529" title="go to pg. 529">529</a>, <a -class="aindx" href="#p557" title="go to pg. 557">557</a>, -<a class="aindx" href="#p559" title="go to pg. -559">559</a></li> - -<li class="liindx">Terebratula, <a class="aindx" -href="#p568" title="go to pg. 568">568</a>, <a -class="aindx" href="#p574" title="go to pg. 574">574</a>, -<a class="aindx" href="#p576" title="go to pg. -576">576</a></li> - -<li class="liindx">Teredo, <a class="aindx" href="#p414" -title="go to pg. 414">414</a></li> - -<li class="liindx">Terni, T., <a class="aindx" href="#p035" -title="go to pg. 35">35</a></li> - -<li class="liindx">Terquem, O., <a class="aindx" -href="#p329" title="go to pg. 329">329</a></li> - -<li class="liindx">Tesch, J. J., <a class="aindx" -href="#p573" title="go to pg. 573">573</a></li> - -<li class="liindx">Tetractinellida, <a class="aindx" -href="#p443" title="go to pg. 443">443</a>, <a -class="aindx" href="#p450" title="go to pg. -450">450</a></li> - -<li class="liindx">Tetrahedral symmetry, <a class="aindx" -href="#p315" title="go to pg. 315">315</a>, <a -class="aindx" href="#p396" title="go to pg. 396">396</a>, -<a class="aindx" href="#p476" title="go to pg. -476">476</a></li> - -<li class="liindx">Tetrakaidecahedron, <a class="aindx" -href="#p337" title="go to pg. 337">337</a></li> - -<li class="liindx">Tetraspores, <a class="aindx" -href="#p396" title="go to pg. 396">396</a></li> - -<li class="liindx">Textularia, <a class="aindx" -href="#p604" title="go to pg. 604">604</a></li> - -<li class="liindx">Thamnastraea, <a class="aindx" -href="#p327" title="go to pg. 327">327</a></li> - -<li class="liindx">Thayer, J. E., <a class="aindx" -href="#p672" title="go to pg. 672">672</a></li> - -<li class="liindx">Thecidium, <a class="aindx" href="#p570" -title="go to pg. 570">570</a></li> - -<li class="liindx">Thecosmilia, <a class="aindx" -href="#p325" title="go to pg. 325">325</a></li> - -<li class="liindx">Théel, H., <a class="aindx" href="#p451" -title="go to pg. 451">451</a></li> - -<li class="liindx">Thienemann, F. A. L., <a class="aindx" -href="#p653" title="go to pg. 653">653</a></li> - -<li class="liindx">Thistle, capitulum of, <a class="aindx" -href="#p639" title="go to pg. 639">639</a></li> - -<li class="liindx">Thoma, R., <a class="aindx" href="#p666" -title="go to pg. 666">666</a></li> - -<li class="liindx">Thomson, James, <a class="aindx" -href="#p018" title="go to pg. 18">18</a>, <a class="aindx" -href="#p259" title="go to pg. 259">259</a>; <ul> <li -class="liindx">J. A., <a class="aindx" href="#p465" -title="go to pg. 465">465</a>;</li> <li class="liindx">J. -J., <a class="aindx" href="#p235" title="go to pg. -235">235</a>, <a class="aindx" href="#p280" title="go -to pg. 280">280</a>;</li> <li class="liindx">Wyville, -<a class="aindx" href="#p466" title="go to pg. -466">466</a></li> </ul></li> - -<li class="liindx">Thurammina, <a class="aindx" -href="#p256" title="go to pg. 256">256</a></li> - -<li class="liindx">Thyroid gland, <a class="aindx" -href="#p136" title="go to pg. 136">136</a></li> - -<li class="liindx">Time-element, <a class="aindx" -href="#p051" title="go to pg. 51">51</a>, <a class="aindx" -href="#p496" title="go to pg. 496">496</a>, etc.; <ul> -<li class="liindx">time-energy diagram, <a class="aindx" -href="#p063" title="go to pg. 63">63</a></li> </ul></li> - -<li class="liindx">Tintinnus, <a class="aindx" href="#p248" -title="go to pg. 248">248</a></li> - -<li class="liindx">Tissues, forms of, <a class="aindx" -href="#p293" title="go to pg. 293">293</a></li> - -<li class="liindx">Titanotherium, <a class="aindx" -href="#p704" title="go to pg. 704">704</a>, <a -class="aindx" href="#p762" title="go to pg. -762">762</a></li> - -<li class="liindx">Tomistoma, <a class="aindx" href="#p753" -title="go to pg. 753">753</a></li> - -<li class="liindx">Tomlinson, C., <a class="aindx" -href="#p259" title="go to pg. 259">259</a>, <a -class="aindx" href="#p428" title="go to pg. -428">428</a></li> - -<li class="liindx">Tornier, G., <a class="aindx" -href="#p707" title="go to pg. 707">707</a></li> - -<li class="liindx">Torsion, <a class="aindx" href="#p621" -title="go to pg. 621">621</a>, <a class="aindx" -href="#p624" title="go to pg. 624">624</a></li> - -<li class="liindx">Trachelophyllum, <a class="aindx" -href="#p249" title="go to pg. 249">249</a></li> - -<li class="liindx">Transformations, theory of, <a -class="aindx" href="#p562" title="go to pg. 562">562</a>, -<a class="aindx" href="#p719" title="go to pg. -719">719</a></li> - -<li class="liindx">Traube, M., <a class="aindx" -href="#p287" title="go to pg. 287">287</a></li> - -<li class="liindx">Trees, growth of, <a class="aindx" -href="#p119" title="go to pg. 119">119</a>; <ul> <li -class="liindx">height of, <a class="aindx" href="#p019" -title="go to pg. 19">19</a></li> </ul></li> - -<li class="liindx">Trembley, Abraham, <a class="aindx" -href="#p138" title="go to pg. 138">138</a>, <a -class="aindx" href="#p146" title="go to pg. -146">146</a></li> - -<li class="liindx">Treutlein, P., <a class="aindx" -href="#p510" title="go to pg. 510">510</a></li> - -<li class="liindx">Trianea, hairs of, <a class="aindx" -href="#p234" title="go to pg. 234">234</a></li> - -<li class="liindx">Triangle, properties of, <a -class="aindx" href="#p508" title="go to pg. 508">508</a>; -<ul> <li class="liindx">of forces, <a class="aindx" -href="#p295" title="go to pg. 295">295</a></li> </ul></li> - -<li class="liindx">Triasters, <a class="aindx" href="#p327" -title="go to pg. 327">327</a></li> - -<li class="liindx">Trichodina, <a class="aindx" -href="#p252" title="go to pg. 252">252</a></li> - -<li class="liindx">Trichomastix, <a class="aindx" -href="#p267" title="go to pg. 267">267</a></li> - -<li class="liindx">Triepel, H., <a class="aindx" -href="#p683" title="go to pg. 683">683</a>, <a -class="aindx" href="#p684" title="go to pg. -684">684</a></li> - -<li class="liindx">Triloculina, <a class="aindx" -href="#p595" title="go to pg. 595">595</a></li> - -<li class="liindx">Triton, <a class="aindx" href="#p554" -title="go to pg. 554">554</a></li> - -<li class="liindx">Trochus, <a class="aindx" href="#p377" -title="go to pg. 377">377</a>, <a class="aindx" -href="#p557" title="go to pg. 557">557</a>, <a -class="aindx" href="#p560" title="go to pg. 560">560</a>; -<ul> <li class="liindx">embryology of, <a class="aindx" -href="#p340" title="go to pg. 340">340</a></li> </ul></li> - -<li class="liindx">Tröndle, A., <a class="aindx" -href="#p625" title="go to pg. 625">625</a></li> - -<li class="liindx">Trophon, <a class="aindx" href="#p526" -title="go to pg. 526">526</a></li> - -<li class="liindx">Trout, growth of, <a class="aindx" -href="#p094" title="go to pg. 94">94</a></li> - -<li class="liindx">Trypanosomes, <a class="aindx" -href="#p245" title="go to pg. 245">245</a>, <a -class="aindx" href="#p266" title="go to pg. 266">266</a>, -<a class="aindx" href="#p269" title="go to pg. -269">269</a></li> - -<li class="liindx">Tubularia, <a class="aindx" -href="#p125" title="go to pg. 125">125</a>, <a -class="aindx" href="#p126" title="go to pg. 126">126</a>, -<a class="aindx" href="#p146" title="go to pg. -146">146</a></li> - -<li class="liindx">Turbinate shells, <a class="aindx" -href="#p534" title="go to pg. 534">534</a></li> - -<li class="liindx">Turbo, <a class="aindx" href="#p518" -title="go to pg. 518">518</a>, <a class="aindx" -href="#p555" title="go to pg. 555">555</a></li> - -<li class="liindx">Turgor, <a class="aindx" href="#p125" -title="go to pg. 125">125</a></li> - -<li class="liindx">Turner, Sir W., <a class="aindx" -href="#p769" title="go to pg. 769">769</a></li> - -<li class="liindx">Turritella, <a class="aindx" -href="#p489" title="go to pg. 489">489</a>, <a -class="aindx" href="#p524" title="go to pg. 524">524</a>, -<a class="aindx" href="#p527" title="go to pg. -527">527</a>, <a class="aindx" href="#p555" title="go to -pg. 555">555</a>, <a class="aindx" href="#p557" title="go -to pg. 557">557</a>, <a class="aindx" href="#p559" -title="go to pg. 559">559</a></li> - -<li class="liindx">Tusks, <a class="aindx" href="#p515" -title="go to pg. 515">515</a>, <a class="aindx" -href="#p612" title="go to pg. 612">612</a></li> - -<li class="liindx">Tutton, A. E. H., <a class="aindx" -href="#p202" title="go to pg. 202">202</a></li> - -<li class="liindx">Twining plants, <a class="aindx" -href="#p624" title="go to pg. 624">624</a></li> - -<li class="liindx">Tyndall, John, <a class="aindx" -href="#p428" title="go to pg. 428">428</a></li> </ul> - -<ul class="ulindx"> <li class="liindx">Umbilicus of -shell, <a class="aindx" href="#p547" title="go to pg. -547">547</a></li> - -<li class="liindx">Underfeeding, effect of, <a -class="aindx" href="#p106" title="go to pg. -106">106</a></li> - -<li class="liindx">Undulatory membrane, <a class="aindx" -href="#p266" title="go to pg. 266">266</a></li> - -<li class="liindx">Unduloid, <a class="aindx" href="#p218" -title="go to pg. 218">218</a>, <a class="aindx" -href="#p222" title="go to pg. 222">222</a>, <a -class="aindx" href="#p229" title="go to pg. 229">229</a>, -<a class="aindx" href="#p246" title="go to pg. -246">246</a>, <a class="aindx" href="#p256" title="go to -pg. 256">256</a></li> - -<li class="liindx">Unio, <a class="aindx" href="#p341" -title="go to pg. 341">341</a></li> - -<li class="liindx">Univalve shells, <a class="aindx" -href="#p553" title="go to pg. 553">553</a></li> - -<li class="liindx">Urechinus, <a class="aindx" href="#p664" -title="go to pg. 664">664</a></li> </ul> - -<ul class="ulindx"> <li class="liindx">Vaginicola, -<a class="aindx" href="#p248" title="go to pg. -248">248</a></li> - -<li class="liindx">Vallisneri, Ant., <a class="aindx" -href="#p138" title="go to pg. 138">138</a></li> - -<li class="liindx">Van Iterson, G., <a class="aindx" -href="#p595" title="go to pg. 595">595</a></li> - -<li class="liindx">Van Rees, R., <a class="aindx" -href="#p374" title="go to pg. 374">374</a></li> - -<li class="liindx">Van’t Hoff, J. H., <a class="aindx" -href="#p001" title="go to pg. 1">1</a>, <a class="aindx" -href="#p110" title="go to pg. 110">110</a>, <a -class="aindx" href="#p433" title="go to pg. -433">433</a></li> - -<li class="liindx">Variability, <a class="aindx" -href="#p078" title="go to pg. 78">78</a>, <a class="aindx" -href="#p103" title="go to pg. 103">103</a></li> - -<li class="liindx">Venation of wings, <a class="aindx" -href="#p385" title="go to pg. 385">385</a></li> - -<li class="liindx">Verhaeren, Emile, <a class="aindx" -href="#p778" title="go to pg. 778">778</a></li> - -<li class="liindx">Verworn, M., <a class="aindx" -href="#p198" title="go to pg. 198">198</a>, <a -class="aindx" href="#p211" title="go to pg. 211">211</a>, -<a class="aindx" href="#p467" title="go to pg. -467">467</a>, <a class="aindx" href="#p605" title="go to -pg. 605">605</a></li> - -<li class="liindx">Vesque, J., <a class="aindx" -href="#p412" title="go to pg. 412">412</a></li> - -<li class="liindx">Vierordt, K., <a class="aindx" -href="#p073" title="go to pg. 73">73</a></li> - -<li class="liindx">Villi, <a class="aindx" href="#p032" -title="go to pg. 32">32</a></li> - -<li class="liindx">Vincent, J. H., <a class="aindx" -href="#p323" title="go to pg. 323">323</a></li> - -<li class="liindx">Vines, S. H., <a class="aindx" -href="#p502" title="go to pg. 502">502</a></li> - -<li class="liindx">Virchow, R., <a class="aindx" -href="#p200" title="go to pg. 200">200</a>, <a -class="aindx" href="#p286" title="go to pg. -286">286</a></li> - -<li class="liindx">Vital phenomena, <a class="aindx" -href="#p014" title="go to pg. 14">14</a>, <a class="aindx" -href="#p417" title="go to pg. 417">417</a>, etc.</li> - -<li class="liindx">Vitruvius, <a class="aindx" href="#p740" -title="go to pg. 740">740</a></li> - -<li class="liindx">Volkmann, A. W., <a class="aindx" -href="#p669" title="go to pg. 669">669</a></li> - -<li class="liindx">Voltaire, <a class="aindx" href="#p004" -title="go to pg. 4">4</a>, <a class="aindx" href="#p146" -title="go to pg. 146">146</a></li> - -<li class="liindx">Vorticella, <a class="aindx" -href="#p237" title="go to pg. 237">237</a>, <a -class="aindx" href="#p246" title="go to pg. 246">246</a>, -<a class="aindx" href="#p291" title="go to pg. -291">291</a></li> </ul> - -<ul class="ulindx"> <li class="liindx">Wager, H. W. -T., <a class="aindx" href="#p259" title="go to pg. -259">259</a></li> - -<li class="liindx">Walking, <a class="aindx" href="#p030" -title="go to pg. 30">30</a></li> - -<li class="liindx">Wallace, A. R., <a class="aindx" -href="#p005" title="go to pg. 5">5</a>, <a class="aindx" -href="#p432" title="go to pg. 432">432</a>, <a -class="aindx" href="#p549" title="go to pg. -549">549</a></li> - -<li class="liindx">Wallich-Martius, <a class="aindx" -href="#p077" title="go to pg. 77">77</a></li> - -<li class="liindx">Warburg, O., <a class="aindx" -href="#p161" title="go to pg. 161">161</a></li> - -<li class="liindx">Warburton, C., <a class="aindx" -href="#p233" title="go to pg. 233">233</a></li> - -<li class="liindx">Ward, H. Marshall, <a class="aindx" -href="#p133" title="go to pg. 133">133</a></li> - -<li class="liindx">Warnecke, P., <a class="aindx" -href="#p093" title="go to pg. 93">93</a></li> - -<li class="liindx">Watase, S., <a class="aindx" -href="#p378" title="go to pg. 378">378</a></li> - -<li class="liindx">Water, in growth, <a class="aindx" -href="#p125" title="go to pg. 125">125</a></li> - -<li class="liindx">Watson, F. R., <a class="aindx" -href="#p323" title="go to pg. 323">323</a></li> - -<li class="liindx">Weber, E. H., <a class="aindx" -href="#p210" title="go to pg. 210">210</a>, <a -class="aindx" href="#p259" title="go to pg. 259">259</a>, -<a class="aindx" href="#p669" title="go to pg. -669">669</a>; <ul> <li class="liindx">E. H. and W. -E., <a class="aindx" href="#p030" title="go to pg. -30">30</a>;</li> <li class="liindx">Max, <a class="aindx" -href="#p091" title="go to pg. 91">91</a></li> </ul></li> - -<li class="liindx">Weight, curve of, <a class="aindx" -href="#p064" title="go to pg. 64">64</a>, etc.</li> - -<li class="liindx">Weismann, A., <a class="aindx" -href="#p158" title="go to pg. 158">158</a></li> - -<li class="liindx">Werner, A. G., <a class="aindx" -href="#p019" title="go to pg. 19">19</a></li> - -<li class="liindx">Wettstein, R. von, <a class="aindx" -href="#p728" title="go to pg. 728">728</a></li> - -<li class="liindx">Whale, affinities, <a class="aindx" -href="#p716" title="go to pg. 716">716</a>; -<ul> <li class="liindx">size, <a class="aindx" -href="#p021" title="go to pg. 21">21</a>;</li> <li -class="liindx">structure, <a class="aindx" href="#p708" -title="go to pg. 708">708</a></li> </ul></li> - -<li class="liindx">Whipple, I. L., <a class="aindx" -href="#p123" title="go to pg. 123">123</a></li> - -<li class="liindx">Whitman, C. O., <a class="aindx" -href="#p157" title="go to pg. 157">157</a>, <a -class="aindx" href="#p164" title="go to pg. 164">164</a>, -<a class="aindx" href="#p193" title="go to pg. -193">193</a>, <a class="aindx" href="#p194" title="go to -pg. 194">194</a>, <a class="aindx" href="#p199" title="go -to pg. 199">199</a>, <a class="aindx" href="#p200" -title="go to pg. 200">200</a></li> - -<li class="liindx">Whitworth, W. A., <a class="aindx" -href="#p506" title="go to pg. 506">506</a>, <a -class="aindx" href="#p512" title="go to pg. -512">512</a></li> - -<li class="liindx">Wiener, A. F., <a class="aindx" -href="#p045" title="go to pg. 45">45</a></li> - -<li class="liindx">Wildeman, E. de, <a class="aindx" -href="#p307" title="go to pg. 307">307</a>, <a -class="aindx" href="#p355" title="go to pg. -355">355</a></li> - -<li class="liindx">Willey, A., <a class="aindx" -href="#p425" title="go to pg. 425">425</a>, <a -class="aindx" href="#p548" title="go to pg. 548">548</a>, -<a class="aindx" href="#p555" title="go to pg. -555">555</a>, <a class="aindx" href="#p578" title="go to -pg. 578">578</a></li> - -<li class="liindx">Williamson, W. C., <a class="aindx" -href="#p423" title="go to pg. 423">423</a>, <a -class="aindx" href="#p609" title="go to pg. -609">609</a></li> - -<li class="liindx">Willughby, Fr., <a class="aindx" -href="#p318" title="go to pg. 318">318</a></li> - -<li class="liindx">Wilson, E. B., <a class="aindx" -href="#p150" title="go to pg. 150">150</a>, <a -class="aindx" href="#p163" title="go to pg. 163">163</a>, -<a class="aindx" href="#p173" title="go to pg. -173">173</a>, <a class="aindx" href="#p195" title="go to -pg. 195">195</a>, <a class="aindx" href="#p199" title="go -to pg. 199">199</a>, <a class="aindx" href="#p311" -title="go to pg. 311">311</a>, <a class="aindx" -href="#p341" title="go to pg. 341">341</a>, <a -class="aindx" href="#p342" title="go to pg. 342">342</a>, -<a class="aindx" href="#p398" title="go to pg. -398">398</a>, <a class="aindx" href="#p453" title="go to -pg. 453">453</a></li> - -<li class="liindx">Winge, O., <a class="aindx" href="#p433" -title="go to pg. 433">433</a></li> - -<li class="liindx">Winter eggs, <a class="aindx" -href="#p283" title="go to pg. 283">283</a></li> - -<li class="liindx">Wissler, Clark, <a class="aindx" -href="#p079" title="go to pg. 79">79</a></li> - -<li class="liindx">Wissner, J., <a class="aindx" -href="#p636" title="go to pg. 636">636</a></li> - -<li class="liindx">Wöhler, Fr., <a class="aindx" -href="#p416" title="go to pg. 416">416</a>, <a -class="aindx" href="#p420" title="go to pg. -420">420</a></li> - -<li class="liindx">Wolff, J., <a class="aindx" href="#p683" -title="go to pg. 683">683</a>; <ul> <li class="liindx">J. -C. F., <a class="aindx" href="#p003" title="go to pg. -3">3</a>, <a class="aindx" href="#p051" title="go to pg. -51">51</a>, <a class="aindx" href="#p155" title="go to pg. -155">155</a></li> </ul></li> - -<li class="liindx">Wood, R. W., <a class="aindx" -href="#p590" title="go to pg. 590">590</a></li> - -<li class="liindx">Woods, R. H., <a class="aindx" -href="#p666" title="go to pg. 666">666</a></li> - -<li class="liindx">Woodward, H., <a class="aindx" -href="#p578" title="go to pg. 578">578</a>; <ul> <li -class="liindx">S. P., <a class="aindx" href="#p554" -title="go to pg. 554">554</a>, <a class="aindx" -href="#p567" title="go to pg. 567">567</a></li> </ul></li> - -<li class="liindx">Worthington, A. M., <a class="aindx" -href="#p235" title="go to pg. 235">235</a>, <a -class="aindx" href="#p254" title="go to pg. -254">254</a></li> - -<li class="liindx">Wreszneowski, A., <a class="aindx" -href="#p249" title="go to pg. 249">249</a></li> - -<li class="liindx">Wright, Chauncey, <a class="aindx" -href="#p335" title="go to pg. 335">335</a></li> - -<li class="liindx">Wright, T. Strethill, <a class="aindx" -href="#p210" title="go to pg. 210">210</a></li> - -<li class="liindx">Wyman, Jeffrey, <a class="aindx" -href="#p335" title="go to pg. 335">335</a></li> </ul> - -<ul class="ulindx"> <li class="liindx">Yeast cell, <a -class="aindx" href="#p213" title="go to pg. 213">213</a>, -<a class="aindx" href="#p242" title="go to pg. -242">242</a></li> - -<li class="liindx">Yield-point, <a class="aindx" -href="#p679" title="go to pg. 679">679</a></li> - -<li class="liindx">Yolk of egg, <a class="aindx" -href="#p165" title="go to pg. 165">165</a>, <a -class="aindx" href="#p660" title="go to pg. -660">660</a></li> - -<li class="liindx">Young, Thomas, <a class="aindx" -href="#p009" title="go to pg. 9">9</a>, <a class="aindx" -href="#p036" title="go to pg. 36">36</a>, <a -class="aindx" href="#p669" title="go to pg. 669">669</a>, -<a class="aindx" href="#p691" title="go to pg. -691">691</a></li> </ul> - -<ul class="ulindx"> <li class="liindx">Zangger, -H., <a class="aindx" href="#p282" title="go to pg. -282">282</a></li> - -<li class="liindx">Zeising, A., <a class="aindx" -href="#p636" title="go to pg. 636">636</a>, <a -class="aindx" href="#p650" title="go to pg. -650">650</a></li> - -<li class="liindx">Zeleny, C., <a class="aindx" -href="#p149" title="go to pg. 149">149</a></li> - -<li class="liindx">Zeuglodon, <a class="aindx" href="#p716" -title="go to pg. 716">716</a></li> - -<li class="liindx">Zeuthen, H. G., <a class="aindx" -href="#p511" title="go to pg. 511">511</a></li> - -<li class="liindx">Ziehen, Ch., <a class="aindx" -href="#p092" title="go to pg. 92">92</a></li> - -<li class="liindx">Zittel, K. A. von, <a class="aindx" -href="#p325" title="go to pg. 325">325</a>, <a -class="aindx" href="#p327" title="go to pg. 327">327</a>, -<a class="aindx" href="#p548" title="go to pg. -548">548</a>, <a class="aindx" href="#p584" title="go to -pg. 584">584</a></li> - -<li class="liindx">Zoogloea, <a class="aindx" href="#p282" -title="go to pg. 282">282</a></li> - -<li class="liindx">Zschokke, F., <a class="aindx" -href="#p683" title="go to pg. 683">683</a></li> - -<li class="liindx">Zsigmondy, <a class="aindx" href="#p039" -title="go to pg. 39">39</a></li> - -<li class="liindx">Zuelzer, M., <a class="aindx" -href="#p165" title="go to pg. 165">165</a></li> -</ul><!--ulindx--> - -<div class="fsz9 padtopa">CAMBRIDGE: PRINTED BY J. B. PEACE, M.A., AT -THE UNIVERSITY PRESS</div> - -<div class="chapter" id="p809"> -<h2 class="h2herein" title="Selection from the General -Catalogue of Books Published by the Cambridge University -Press."><span class="h2ttl"> -SELECTION FROM THE GENERAL CATALOGUE</span> -<span class="h2ttl">OF BOOKS PUBLISHED BY</span> -<span class="h2ttl">THE CAMBRIDGE UNIVERSITY PRESS</span></h2></div> - -<ul id="ullnh1_1"> -<li class="liindx"><em class="embold">Growth -in Length: Embryological Essays.</em> By <span -class="smcap">R<b>ICHARD</b></span> <span -class="smcap">A<b>SSHETON</b>,</span> M.A., Sc.D., F.R.S. -<span class="fsz7">With 42 illustrations. Demy 8vo. 2s 6d -net.</span></li> - -<li class="liindx"><em class="embold">Experimental -Zoology.</em> By <span class="smcap">H<b>ANS</b></span> <span -class="smcap">P<b>RZIBRAM</b>,</span> Ph.D. Part I. Embryogeny, -an account of the laws governing the development of the animal -egg as ascertained by experiment. <span class="fsz7">With 16 plates. Royal 8vo. 7s -6d net.</span></li> - -<li class="liindx"><em class="embold">Zoology. -An Elementary Text-Book.</em> By A. E. <span -class="smcap">S<b>HIPLEY</b>,</span> Sc.D., F.R.S., and E. W. -<span class="smcap">M<b>ACBRIDE</b>,</span> D.Sc., F.R.S. <span class="fsz7">Third -edition, enlarged and re-written. With 360 illustrations. Demy -8vo. 12s 6d net. Cambridge Zoological Series.</span></li> - -<li class="liindx"><em class="embold">The Natural -History of some Common Animals.</em> By <span -class="smcap">O<b>SWALD</b></span> H. <span -class="smcap">L<b>ATTER</b>,</span> M.A. <span class="fsz7">With 54 illustrations. -Crown 8vo. 5s net. Cambridge Biological Series.</span></li> - -<li class="liindx"><em class="embold">The Origin and -Influence of the Thoroughbred Horse.</em> By W. <span -class="smcap">R<b>IDGEWAY</b>,</span> Sc.D., Litt.D., F.B.A., -Disney Professor of Archæology and Fellow of Gonville and -Caius College. <span class="fsz7">With 143 illustrations. Demy 8vo. 12s 6d net. -Cambridge Biological Series.</span></li> - -<li class="liindx"><em class="embold">The -Vertebrate Skeleton.</em> By S. H. <span -class="smcap">R<b>EYNOLDS</b>,</span> M.A., Professor of -Geology in the University of Bristol. <span class="fsz7">Second edition. Demy 8vo. -15s net. Cambridge Zoological Series.</span></li> - -<li class="liindx"><em class="embold">Outlines of -Vertebrate Palæontology for Students of Zoology.</em> -By <span class="smcap">A<b>RTHUR</b></span> -<span class="smcap">S<b>MITH</b></span> <span -class="smcap">W<b>OODWARD</b>,</span> M.A., F.R.S. <span class="fsz7">With -228 illustrations. Demy 8vo. 14s net. Cambridge Biological -Series.</span></li> - -<li class="liindx"><em -class="embold">Palæontology—Invertebrate.</em> -By <span class="smcap">H<b>ENRY</b></span> <span -class="smcap">W<b>OODS</b>,</span> M.A., F.G.S. <span class="fsz7">Fourth edition. -With 151 illustrations. Crown 8vo. 6s net. Cambridge Biological -Series.</span></li> - -<li class="liindx"><em class="embold">Morphology and -Anthropology.</em> A Handbook for Students. By W. L. H. <span -class="smcap">D<b>UCKWORTH</b>,</span> M.A., M.D., Sc.D. <span class="fsz7">Second -edition. Volume I. With 208 illustrations. Demy 8vo. 10s 6d -net.</span></li> - -<li class="liindx"><em class="embold">Studies -from the Morphological Laboratory.</em> Edited -by <span class="smcap">A<b>DAM</b></span> <span -class="smcap">S<b>EDGWICK</b>,</span> M.A., F.R.S. -<span class="fsz7"> Royal 8vo. -Vol. II, Part I. 10s net. Vol. II, Part II. 7s 6d net. Vol. -III, Parts I and II. 7s 6d net each. Vol. IV, Part I. 12s 6d -net. Vol. IV, Part II. 10s net. Vol. IV, Part III. 5s net. Vol. -V, Part I. 7s 6d net. Vol. V, Part II. 5s net. Vol. VI. 15s -net.</span></li> - -<li class="liindx"><em class="embold">The Determination of -Sex.</em> By L. <span class="smcap">D<b>ONCASTER</b>,</span> -Sc.D., Fellow of King’s College, Cambridge. <span class="fsz7">With 23 plates. -Demy 8vo. 7s 6d net.</span></li> - -<li class="liindx"><em class="embold">Conditions of Life in the -Sea.</em> A short account of Quantitative Marine Biological -Research. By <span class="smcap">J<b>AMES</b></span> <span -class="smcap">J<b>OHNSTONE</b>,</span> Fisheries Laboratory, -University of Liverpool. <span class="fsz7">With chart and 31 illustrations. Demy -8vo. 9s net.</span></li> - -<li class="liindx"><em class="embold">Mendel’s Principles of -Heredity.</em> By W. <span class="smcap">B<b>ATESON</b>,</span> -M.A., F.R.S., V.M.H., Director of the John Innes Horticultural -Institution. <span class="fsz7"> -Third impression with additions. With 3 portraits, -6 coloured plates, and 38 figures. Royal 8vo. 12s net.</span></li> - -<li class="liindx"><em class="embold">The Elements of -Botany.</em> By Sir <span class="smcap">F<b>RANCIS</b></span> -<span class="smcap">D<b>ARWIN</b>,</span> Sc.D., M.B., -F.R.S., Fellow of Christ’s College. <span class="fsz7">Second edition. With 94 -illustrations. Crown 8vo. 4s 6d net. Cambridge Biological -Series.</span></li> - -<li class="liindx"><em class="embold">Practical Physiology of -Plants.</em> By Sir <span class="smcap">F<b>RANCIS</b></span> -<span class="smcap">D<b>ARWIN</b>,</span> Sc.D., F.R.S., -and E. <span class="smcap">H<b>AMILTON</b></span> <span -class="smcap">A<b>CTON</b>,</span> M.A. <span class="fsz7">Third edition. With -45 illustrations. Crown 8vo. 4s 6d net. Cambridge Biological -Series.</span></li> - -<li class="liindx"><em class="embold">Botany.</em> -A Text-Book for Senior Students. By D. <span -class="smcap">T<b>HODAY</b>,</span> <span class="fsz7"> -M.A., Lecturer in -Physiological Botany and Assistant Director of the Botanical -Laboratories in the University of Manchester.With 205 figures. -Large crown 8vo. 5s 6d net.</span></li> - -<li class="liindx"><em class="embold">Algæ.</em> Volume -I, Myxophyceæ, Peridinieæ, Bacillarieæ, Chlorophyceæ, -<span class="fsz7">together with a brief summary of the Occurrence and -Distribution of Freshwater Algæ.</span> By G. S. <span -class="smcap">W<b>EST</b>,</span> <span class="fsz7"> -M.A., D.Sc., A.R.C.S., -F.L.S., Mason Professor of Botany in the University of -Birmingham. With 271 illustrations. Large royal 8vo. 25s net. -Cambridge Botanical Handbooks.</span></li> - -<li class="liindx"><em class="embold">The Philosophy of -Biology.</em> By <span class="smcap">J<b>AMES</b></span> <span -class="smcap">J<b>OHNSTONE</b>,</span> D.Sc. <span class="fsz7">Demy 8vo. 9s -net.</span></li> - -<li class="liindx"><em class="embold">Cambridge Manuals -of Science and Literature.</em> General Editors: P. <span -class="smcap">G<b>ILES</b>,</span> Litt.D., and A. C. <span -class="smcap">S<b>EWARD</b>,</span> M.A., F.R.S. Royal -16mo. <span class="fsz7">Cloth, 1s 3d net each. Leather, 2s 6d net each.</span> -<ul> -<li class="liindx"> <em class="embold">The Coming of -Evolution.</em> By <span class="smcap">J<b>OHN</b></span> W. -<span class="smcap">J<b>UDD</b>,</span> C.B., LL.D., F.R.S. -<span class="fsz7">With 4 plates.</span></li> - -<li class="liindx"><em class="embold">Heredity in -the light of recent research.</em> By L. <span -class="smcap">D<b>ONCASTER</b>,</span> Sc.D.<span class="fsz7"> With 12 -figures.</span></li> - -<li class="liindx"><em class="embold">Prehistoric Man.</em> By -W. L. H. <span class="smcap">D<b>UCKWORTH</b>,</span> M.A., -M.D., Sc.D. <span class="fsz7">With 2 tables and 28 figures.</span></li> - -<li class="liindx"><em class="embold">Primitive Animals.</em> -By G. <span class="smcap">S<b>MITH</b>,</span> M.A. <span class="fsz7">With 26 -figures.</span></li> - -<li class="liindx"><em class="embold">The -Life-Story of Insects.</em> By Prof. G. H. <span -class="smcap">C<b>ARPENTER</b>.</span> <span class="fsz7">With 24 -illustrations.</span></li> - -<li class="liindx"><em class="embold">Earthworms and their -Allies.</em> By <span class="smcap">F<b>RANK</b></span> E. -<span class="smcap">B<b>EDDARD</b>,</span> M.A. (Oxon.), -F.R.S., F.R.S.E. <span class="fsz7">With 13 figures.</span></li> - -<li class="liindx"><em class="embold">Spiders.</em> -By <span class="smcap">C<b>ECIL</b></span> <span -class="smcap">W<b>ARBURTON</b>,</span> M.A. <span class="fsz7">With 13 -figures.</span></li> - -<li class="liindx"><em class="embold">Bees and Wasps.</em> By -O. H. <span class="smcap">L<b>ATTER</b>,</span> M.A., F.E.S. -<span class="fsz7">With 21 illustrations.</span></li> - -<li class="liindx"><em class="embold">The Flea.</em> By -H. <span class="smcap">R<b>USSELL</b>.</span> <span class="fsz7">With 9 -illustrations.</span></li> - -<li class="liindx"><em class="embold">Life in the -Sea.</em> By <span class="smcap">J<b>AMES</b></span> <span -class="smcap">J<b>OHNSTONE</b>,</span> B.Sc. <span class="fsz7">With frontispiece, -4 figures and 5 tailpieces.</span></li> - -<li class="liindx"><em class="embold">Plant-Animals.</em> By -F. W. <span class="smcap">K<b>EEBLE</b>,</span> Sc.D. <span class="fsz7">With 23 -figures.</span></li> - -<li class="liindx"><em class="embold">Plant-Life on Land -considered in some of its biological aspects.</em> By F. O. -<span class="smcap">B<b>OWER</b>,</span> Sc.D., F.R.S. <span class="fsz7">With 27 -figures.</span></li> - -<li class="liindx"><em class="embold">Links with -the Past in the Plant World.</em> By A. C. <span -class="smcap">S<b>EWARD</b>,</span> M.A., F.R.S. <span class="fsz7">With -frontispiece and 20 figures.</span></li></ul></li></ul> - -<div class="fsz4 padtopa">Cambridge University Press</div> -<div>Fetter Lane, London: C. F. Clay, Manager</div> - -<div class="section"> -<div class="transnote">TRANSCRIBER’S NOTE - -<p>Original spelling and grammar have been generally retained, -with some exceptions noted below. Original printed page numbers -are shown like this: {52}. Footnotes have been renumbered 1–663 -and relocated to the end of the book, ahead of the Index. The -transcriber produced the cover image and hereby assigns it to -the public domain. Original page images are available from -archive.org—search for<br> “ongrowthform1917thom”.</p> - -<p>Some tables and illustrations have been moved from their -original locations within paragraphs of text to nearby -locations between paragraphs. This includes, for example, -the full-page table printed on page 67, which page number -is removed from these ebook editions. Some other tables -and illustrations have been left where they originally -lay, in the middle of paragraphs of text.</p> - -<p>Some, but not all ditto -marks, including “do.”, have been -eliminated. Enlarged curly brackets { } turned -horizontal, used as graphic devices to -combine information in two or more columns of a table, have -been eliminated. Enlarged curly brackets used as -graphic devices to suggest combination of information over two -or more lines of text, have been eliminated. For example, on -page <a href="#p075" title="go to pg. 75">75</a>, in the last column of the table, two lengths, 490 and -500, were printed, the latter under the former, with a large -right curly bracket combining them. The transcriber has changed -that construction to “490–500”, taking the original to mean a -range.</p> - -<ul> -<li><p class="pfirst"> -Page <a href="#p106" title="go to pg. 106">106</a>. -Changed “it we could believe” to “if we could believe”.</p></li> - -<li><p class="pfirst"> -Page <a href="#p107" title="go to pg. 107">107</a>. -Changed “(m.)” to “(mm.)”, in column 3 of the table.</p></li> - -<li><p class="pfirst"> -Page <a href="#p117" title="go to pg. 117">117</a>. -Both “<i>Q</i><sub>10</sub>” and -<span class="nowrap">“<i>Q</i><sup>10</sup> ”</span> -appear on the page as originally printed.</p></li> - -<li><p class="pfirst"> -Page <a href="#fn317" title="go to footnote 317">272n</a>. -Changed “<i>Proc. R y. Soc.</i> -<span class="nowrap"><span class="smmaj">XII</span>”</span> to “<i>Proc. -Roy. Soc.</i> <span class="smmaj">XII</span>”.</p></li> - -<li><p class="pfirst"> -Page <a href="#p368" title="go to pg. 368">368</a>. -Perhaps, the original “The area of the enlarged -sector, <span class="nowrap"><i>p′OA′</i> ”</span> -should read “The area of the enlarged sector, <span -class="nowrap"><i>P′OA′</i> ”.</span></p></li> - -<li><p class="pfirst"> -Page <a href="#fn437" title="go to footnote 437">428</a>n. -Changed “Phenonemon” to “Phenomenon”.</p></li> - -<li><p class="pfirst"> -Page <a href="#fn476" title="go to footnote 476">463n</a>. -Changed “Raphidophrys” to “Raphidiophrys”.</p></li> - -<li><p class="pfirst"> -Page <a href="#p543" title="go to pg. 543">543</a>. -The Unicode character [⪌ u+2a8c greater-than -above double-line equal above less-than] is pretty rare, -and may not display properly in most fonts. An image is -used instead of the Unicode in all but the simple text -edition.</p></li> - -<li><p class="pfirst"> -Page <a href="#p676" title="go to pg. 676">676</a>. -The Unicode character [⌶ u+2336 APL functional -symbol i-beam] is also unusual. An image is substituted in -all but the simple text edition.</p></li> - -<li><p class="pfirst"> -Page <a href="#p748" title="go to pg. 748">748</a>. -Changed “Fig. 474” to “Fig. 374”.</p></li> - -<li><p class="pfirst"> -Page <a href="#p768" title="go to pg. 768">768</a>. -Changed “in the case of <i>Pro ohippus</i>” to “in -the case of <i>Protohippus</i>”. ¶ There were three footnotes -on this page, but only two footnote anchors. The second -footnote, missing an anchor, said “† Cf. <i>Zittel, Grundzüge -d. Palaeontologie</i>, p. 463, 1911.”</p></li> -</ul> - - -</div></div> - - -<div style='display:block; margin-top:4em'>*** END OF THE PROJECT GUTENBERG EBOOK ON GROWTH AND FORM ***</div> -<div style='text-align:left'> - -<div style='display:block; margin:1em 0'> -Updated editions will replace the previous one—the old editions will -be renamed. -</div> - -<div style='display:block; margin:1em 0'> -Creating the works from print editions not protected by U.S. copyright -law 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™ electronic works to protect the PROJECT GUTENBERG™ -concept and trademark. 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