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diff --git a/old/55264-0.txt b/old/55264-0.txt deleted file mode 100644 index 6cec7e2..0000000 --- a/old/55264-0.txt +++ /dev/null @@ -1,32506 +0,0 @@ -Project Gutenberg's On Growth and Form, by D'Arcy Wentworth Thompson - -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 -www.gutenberg.org. If you are not located in the United States, you'll have -to check the laws of the country where you are located before using this ebook. - -Title: On Growth and Form - -Author: D'Arcy Wentworth Thompson - -Release Date: August 4, 2017 [EBook #55264] - -Language: English - -Character set encoding: UTF-8 - -*** START OF THIS PROJECT GUTENBERG EBOOK ON GROWTH AND FORM *** - - - - -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) - - - - - - - - - -GROWTH AND FORM - - - - - CAMBRIDGE UNIVERSITY PRESS - - C. F. CLAY, MANAGER - - London: FETTER LANE, E.C. - - Edinburgh: 100 PRINCES STREET - - [Illustration] - - New York: G. P. PUTNAM’S SONS - - Bombay, Calcutta and Madras: MACMILLAN AND Co., LTD. - - Toronto: J. M. DENT AND SONS, LTD. - - Tokyo: THE MARUZEN-KABUSHIKI-KAISHA - - - _All rights reserved_ - - - - - ON GROWTH AND FORM - - BY - - D’ARCY WENTWORTH THOMPSON - - - Cambridge: - at the University Press - 1917 - - - - -“The reasonings about the wonderful and intricate operations of nature -are so full of uncertainty, that, as the Wise-man truly observes, -_hardly do we guess aright at the things that are upon earth, and with -labour do we find the things that are before us_.” Stephen Hales, -_Vegetable Staticks_ (1727), p. 318, 1738. - - - - -PREFATORY NOTE - - -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. - -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. - -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. - -I am under obligations also to the authors and publishers of many books -from which illustrations have been borrowed, and especially to the -following:― - -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. {vi} - -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 _Transactions_ 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. - -To Professor E. B. Wilson, for his well-known and all but indispensable -figures of the cell (figs. 42–51, 53); to M. A. Prenant, for other -figures (41, 48) in the same chapter; to Sir Donald MacAlister and Mr -Edwin Arnold for certain figures (335–7), and to Sir Edward Schäfer -and Messrs Longmans for another (334), illustrating the minute -trabecular structure of bone. To Mr Gerhard Heilmann, of Copenhagen, -for his beautiful diagrams (figs. 388–93, 401, 402) 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. 339–346) from Professor Fidler’s -_Textbook of Bridge Construction_. To Messrs Blackwood and Sons, for -several cuts (figs. 127–9, 131, 173) from Professor Alleyne Nicholson’s -_Palaeontology_; to Mr Heinemann, for certain figures (57, 122, -123, 205) from Dr Stéphane Leduc’s _Mechanism of Life_; to Mr A. M. -Worthington and to Messrs Longmans, for figures (71, 75) from _A Study -of Splashes_, and to Mr C. R. Darling and to Messrs E. and S. Spon -for those (fig. 85) from Mr Darling’s _Liquid Drops and Globules_. -To Messrs Macmillan and Co. for two figures (304, 305) from Zittel’s -_Palaeontology_, to the Oxford University Press for a diagram (fig. -28) from Mr J. W. Jenkinson’s _Experimental Embryology_; and to the -Cambridge University Press for a number of figures from Professor -Henry Woods’s _Invertebrate Palaeontology_, for one (fig. 210) from Dr -Willey’s _Zoological Results_, and for another (fig. 321) from “Thomson -and Tait.” - -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. - - D’ARCY WENTWORTH THOMPSON. - - UNIVERSITY COLLEGE, DUNDEE. - - _December, 1916._ - - - - -CONTENTS - - - CHAP. PAGE - I. INTRODUCTORY 1 - - II. ON MAGNITUDE 16 - - III. THE RATE OF GROWTH 50 - - IV. ON THE INTERNAL FORM AND STRUCTURE OF THE CELL 156 - - V. THE FORMS OF CELLS 201 - - VI. A NOTE ON ADSORPTION 277 - - VII. THE FORMS OF TISSUES, OR CELL-AGGREGATES 293 - - VIII. THE SAME (_continued_) 346 - - IX. ON CONCRETIONS, SPICULES, AND SPICULAR SKELETONS 411 - - X. A PARENTHETIC NOTE ON GEODETICS 488 - - XI. THE LOGARITHMIC SPIRAL 493 - - XII. THE SPIRAL SHELLS OF THE FORAMINIFERA 587 - - XIII. THE SHAPES OF HORNS, AND OF TEETH OR TUSKS: WITH - A NOTE ON TORSION 612 - - XIV. ON LEAF-ARRANGEMENT, OR PHYLLOTAXIS 635 - - XV. ON THE SHAPES OF EGGS, AND OF CERTAIN OTHER HOLLOW - STRUCTURES 652 - - XVI. ON FORM AND MECHANICAL EFFICIENCY 670 - - XVII. ON THE THEORY OF TRANSFORMATIONS, OR THE COMPARISON - OF RELATED FORMS 719 - - EPILOGUE 778 - - INDEX 780 - - - - -LIST OF ILLUSTRATIONS - - - 1. Nerve-cells, from larger and smaller animals (Minot, after Irving - Hardesty) . . . 37 - - 2. Relative magnitudes of some minute organisms (Zsigmondy) . . . 39 - - 3. Curves of growth in man (Quetelet and Bowditch) . . . 61 - - 4, 5. Mean annual increments of stature and weight in man (_do._) - . . . 66, 69 - - 6. The ratio, throughout life, of female weight to male (_do._) - . . . 71 - - 7–9. Curves of growth of child, before and after birth (His and - Rüssow) . . . 74–6 - - 10. Curve of growth of bamboo (Ostwald, after Kraus) . . . 77 - - 11. Coefficients of variability in human stature (Boas and Wissler) - . . . 80 - - 12. Growth in weight of mouse (Wolfgang Ostwald) . . . 83 - - 13. _Do._ of silkworm (Luciani and Lo Monaco) . . . 84 - - 14. _Do._ of tadpole (Ostwald, after Schaper) . . . 85 - - 15. Larval eels, or _Leptocephali_, and young elver (Joh. Schmidt) - . . . 86 - - 16. Growth in length of _Spirogyra_ (Hofmeister) . . . 87 - - 17. Pulsations of growth in _Crocus_ (Bose) . . . 88 - - 18. Relative growth of brain, heart and body of man (Quetelet) . . . 90 - - 19. Ratio of stature to span of arms (_do._) . . . 94 - - 20. Rates of growth near the tip of a bean-root (Sachs) . . . 96 - - 21, 22. The weight-length ratio of the plaice, and its annual periodic - changes . . . 99, 100 - - 23. Variability of tail-forceps in earwigs (Bateson) . . . 104 - - 24. Variability of body-length in plaice . . . 105 - - 25. Rate of growth in plants in relation to temperature (Sachs) - . . . 109 - - 26. _Do._ in maize, observed (Köppen), and calculated curves . . . 112 - - 27. _Do._ in roots of peas (Miss I. Leitch) . . . 113 - - 28, 29. Rate of growth of frog in relation to temperature (Jenkinson, - after O. Hertwig), and calculated curves of _do._ . . . 115, 6 - - 30. Seasonal fluctuation of rate of growth in man (Daffner) . . . 119 - - 31. _Do._ in the rate of growth of trees (C. E. Hall) . . . 120 - - 32. Long-period fluctuation in the rate of growth of Arizona trees (A. - E. Douglass) . . . 122 - - 33, 34. The varying form of brine-shrimps (_Artemia_), in relation to - salinity (Abonyi) . . . 128, 9 - - 35–39. Curves of regenerative growth in tadpoles’ tails (M. L. Durbin) - . . . 140–145 - - 40. Relation between amount of tail removed, amount restored, and time - required for restoration (M. M. Ellis) . . . 148 - - 41. Caryokinesis in trout’s egg (Prenant, after Prof. P. Bouin) - . . . 169 - - 42–51. Diagrams of mitotic cell-division (Prof. E. B. Wilson) - . . . 171–5 - - 52. Chromosomes in course of splitting and separation (Hatschek and - Flemming) . . . 180 - - 53. Annular chromosomes of mole-cricket (Wilson, after vom Rath) - . . . 181 - - 54–56. Diagrams illustrating a hypothetic field of force in - caryokinesis (Prof. W. Peddie) . . . 182–4 - - 57. An artificial figure of caryokinesis (Leduc) . . . 186 - - 58. A segmented egg of _Cerebratulus_ (Prenant, after Coe) . . . 189 - - 59. Diagram of a field of force with two like poles . . . 189 - - 60. A budding yeast-cell . . . 213 - - 61. The roulettes of the conic sections . . . 218 - - 62. Mode of development of an unduloid from a cylindrical tube - . . . 220 - - 63–65. Cylindrical, unduloid, nodoid and catenoid oil-globules - (Plateau) . . . 222, 3 - - 66. Diagram of the nodoid, or elastic curve . . . 224 - - 67. Diagram of a cylinder capped by the corresponding portion of a - sphere . . . 226 - - 68. A liquid cylinder breaking up into spheres . . . 227 - - 69. The same phenomenon in a protoplasmic cell of _Trianea_ . . . 234 - - 70. Some phases of a splash (A. M. Worthington) . . . 235 - - 71. A breaking wave (_do._) . . . 236 - - 72. The calycles of some campanularian zoophytes . . . 237 - - 73. A flagellate monad, _Distigma proteus_ (Saville Kent) . . . 246 - - 74. _Noctiluca miliaris_, diagrammatic . . . 246 - - 75. Various species of _Vorticella_ (Saville Kent and others) . . . 247 - - 76. Various species of _Salpingoeca_ (_do._) . . . 248 - - 77. Species of _Tintinnus_, _Dinobryon_ and _Codonella_ (_do._) - . . . 248 - - 78. The tube or cup of _Vaginicola_ . . . 248 - - 79. The same of _Folliculina_ . . . 249 - - 80. _Trachelophyllum_ (Wreszniowski) . . . 249 - - 81. _Trichodina pediculus_ . . . 252 - - 82. _Dinenymplia gracilis_ (Leidy) . . . 253 - - 83. A “collar-cell” of _Codosiga_ . . . 254 - - 84. Various species of _Lagena_ (Brady) . . . 256 - - 85. Hanging drops, to illustrate the unduloid form (C. R. Darling) - . . . 257 - - 86. Diagram of a fluted cylinder . . . 260 - - 87. _Nodosaria scalaris_ (Brady) . . . 262 - - 88. Fluted and pleated gonangia of certain Campanularians (Allman) - . . . 262 - - 89. Various species of _Nodosaria_, _Sagrina_ and _Rheophax_ (Brady) - . . . 263 - - 90. _Trypanosoma tineae_ and _Spirochaeta anodontae_, to shew - undulating membranes (Minchin and Fantham) . . . 266 - - 91. Some species of _Trichomastix_ and _Trichomonas_ (Kofoid) . . . 267 - - 92. _Herpetomonas_ assuming the undulatory membrane of a Trypanosome - (D. L. Mackinnon) . . . 268 - - 93. Diagram of a human blood-corpuscle . . . 271 - - 94. Sperm-cells of decapod crustacea, _Inachus_ and _Galathea_ - (Koltzoff) . . . 273 - - 95. The same, in saline solutions of varying density (_do._) . . . 274 - - 96. A sperm-cell of _Dromia_ (_do._) . . . 275 - - 97. Chondriosomes in cells of kidney and pancreas (Barratt and - Mathews) . . . 285 - - 98. Adsorptive concentration of potassium salts in various plant-cells - (Macallum) . . . 290 - - 99–101. Equilibrium of surface-tension in a floating drop . . . 294, 5 - - 102. Plateau’s “bourrelet” in plant-cells; diagrammatic (Berthold) - . . . 298 - - 103. Parenchyma of maize, shewing the same phenomenon . . . 298 - - 104, 5. Diagrams of the partition-wall between two soap-bubbles - . . . 299, 300 - - 106. Diagram of a partition in a conical cell . . . 300 - - 107. Chains of cells in _Nostoc_, _Anabaena_ and other low algae - . . . 300 - - 108. Diagram of a symmetrically divided soap-bubble . . . 301 - - 109. Arrangement of partitions in dividing spores of _Pellia_ - (Campbell) . . . 302 - - 110. Cells of _Dictyota_ (Reinke) . . . 303 - - 111, 2. Terminal and other cells of _Chara_, and young antheridium of - _do._ . . . 303 - - 113. Diagram of cell-walls and partitions under various conditions of - tension . . . 304 - - 114, 5. The partition-surfaces of three interconnected bubbles - . . . 307, 8 - - 116. Diagram of four interconnected cells or bubbles . . . 309 - - 117. Various configurations of four cells in a frog’s egg (Rauber) - . . . 311 - - 118. Another diagram of two conjoined soap-bubbles . . . 313 - - 119. A froth of bubbles, shewing its outer or “epidermal” layer - . . . 314 - - 120. A tetrahedron, or tetrahedral system, shewing its centre of - symmetry . . . 317 - - 121. A group of hexagonal cells (Bonanni) . . . 319 - - 122, 3. Artificial cellular tissues (Leduc) . . . 320 - - 124. Epidermis of _Girardia_ (Goebel) . . . 321 - - 125. Soap-froth, and the same under compression (Rhumbler) . . . 322 - - 126. Epidermal cells of _Elodea canadensis_ (Berthold) . . . 322 - - 127. _Lithostrotion Martini_ (Nicholson) . . . 325 - - 128. _Cyathophyllum hexagonum_ (Nicholson, after Zittel) . . . 325 - - 129. _Arachnophyllum pentagonum_ (Nicholson) . . . 326 - - 130. _Heliolites_ (Woods) . . . 326 - - 131. Confluent septa in _Thamnastraea_ and _Comoseris_ (Nicholson, - after Zittel) . . . 327 - - 132. Geometrical construction of a bee’s cell . . . 330 - - 133. Stellate cells in the pith of a rush; diagrammatic . . . 335 - - 134. Diagram of soap-films formed in a cubical wire skeleton (Plateau) - . . . 337 - - 135. Polar furrows in systems of four soap-bubbles (Robert) . . . 341 - - 136–8. Diagrams illustrating the division of a cube by partitions of - minimal area . . . 347–50 - - 139. Cells from hairs of _Sphacelaria_ (Berthold) . . . 351 - - 140. The bisection of an isosceles triangle by minimal partitions - . . . 353 - - 141. The similar partitioning of spheroidal and conical cells . . . 353 - - 142. S-shaped partitions from cells of algae and mosses (Reinke and - others) . . . 355 - - 143. Diagrammatic explanation of the S-shaped partitions . . . 356 - - 144. Development of _Erythrotrichia_ (Berthold) . . . 359 - - 145. Periclinal, anticlinal and radial partitioning of a quadrant - . . . 359 - - 146. Construction for the minimal partitioning of a quadrant . . . 361 - - 147. Another diagram of anticlinal and periclinal partitions . . . 362 - - 148. Mode of segmentation of an artificially flattened frog’s egg - (Roux) . . . 363 - - 149. The bisection, by minimal partitions, of a prism of small angle - . . . 364 - - 150. Comparative diagram of the various modes of bisection of a - prismatic sector . . . 365 - - 151. Diagram of the further growth of the two halves of a quadrantal - cell . . . 367 - - 152. Diagram of the origin of an epidermic layer of cells . . . 370 - - 153. A discoidal cell dividing into octants . . . 371 - - 154. A germinating spore of _Riccia_ (after Campbell), to shew the - manner of space-partitioning in the cellular tissue . . . 372 - - 155, 6. Theoretical arrangement of successive partitions in a - discoidal cell . . . 373 - - 157. Sections of a moss-embryo (Kienitz-Gerloff) . . . 374 - - 158. Various possible arrangements of partitions in groups of four to - eight cells . . . 375 - - 159. Three modes of partitioning in a system of six cells . . . 376 - - 160, 1. Segmenting eggs of _Trochus_ (Robert), and of _Cynthia_ - (Conklin) . . . 377 - - 162. Section of the apical cone of _Salvinia_ (Pringsheim) . . . 377 - - 163, 4. Segmenting eggs of _Pyrosoma_ (Korotneff), and of _Echinus_ - (Driesch) . . . 377 - - 165. Segmenting egg of a cephalopod (Watase) . . . 378 - - 166, 7. Eggs segmenting under pressure: of _Echinus_ and _Nereis_ - (Driesch), and of a frog (Roux) . . . 378 - - 168. Various arrangements of a group of eight cells on the surface of - a frog’s egg (Rauber) . . . 381 - - 169. Diagram of the partitions and interfacial contacts in a system of - eight cells . . . 383 - - 170. Various modes of aggregation of eight oil-drops (Roux) . . . 384 - - 171. Forms, or species, of _Asterolampra_ (Greville) . . . 386 - - 172. Diagrammatic section of an alcyonarian polype . . . 387 - - 173, 4. Sections of _Heterophyllia_ (Nicholson and Martin Duncan) - . . . 388, 9 - - 175. Diagrammatic section of a ctenophore (_Eucharis_) . . . 391 - - 176, 7. Diagrams of the construction of a Pluteus larva . . . 392, 3 - - 178, 9. Diagrams of the development of stomata, in _Sedum_ and in the - hyacinth . . . 394 - - 180. Various spores and pollen-grains (Berthold and others) . . . 396 - - 181. Spore of _Anthoceros_ (Campbell) . . . 397 - - 182, 4, 9. Diagrammatic modes of division of a cell under certain - conditions of asymmetry . . . 400–5 - - 183. Development of the embryo of _Sphagnum_ (Campbell) . . . 402 - - 185. The gemma of a moss (_do._) . . . 403 - - 186. The antheridium of _Riccia_ (_do._) . . . 404 - - 187. Section of growing shoot of _Selaginella_, diagrammatic . . . 404 - - 188. An embryo of _Jungermannia_ (Kienitz-Gerloff) . . . 404 - - 190. Development of the sporangium of _Osmunda_ (Bower) . . . 406 - - 191. Embryos of _Phascum_ and of _Adiantum_ (Kienitz-Gerloff) . . . 408 - - 192. A section of _Girardia_ (Goebel) . . . 408 - - 193. An antheridium of _Pteris_ (Strasburger) . . . 409 - - 194. Spicules of _Siphonogorgia_ and _Anthogorgia_ (Studer) . . . 413 - - 195–7. Calcospherites, deposited in white of egg (Harting) . . . 421, 2 - - 198. Sections of the shell of _Mya_ (Carpenter) . . . 422 - - 199. Concretions, or spicules, artificially deposited in cartilage - (Harting) . . . 423 - - 200. Further illustrations of alcyonarian spicules: _Eunicea_ (Studer) - . . . 424 - - 201–3. Associated, aggregated and composite calcospherites (Harting) - . . . 425, 6 - - 204. Harting’s “conostats” . . . 427 - - 205. Liesegang’s rings (Leduc) . . . 428 - - 206. Relay-crystals of common salt (Bowman) . . . 429 - - 207. Wheel-like crystals in a colloid medium (_do._) . . . 429 - - 208. A concentrically striated calcospherite or spherocrystal - (Harting) . . . 432 - - 209. Otoliths of plaice, shewing “age-rings” (Wallace) . . . 432 - - 210. Spicules, or calcospherites, of _Astrosclera_ (Lister) . . . 436 - - 211. 2. C- and S-shaped spicules of sponges and holothurians (Sollas - and Théel) . . . 442 - - 213. An amphidisc of _Hyalonema_ . . . 442 - - 214–7. Spicules of calcareous, tetractinellid and hexactinellid - sponges, and of various holothurians (Haeckel, Schultze, Sollas and - Théel) . . . 445–452 - - 218. Diagram of a solid body confined by surface-energy to a liquid - boundary-film . . . 460 - - 219. _Astrorhiza limicola_ and _arenaria_ (Brady) . . . 464 - - 220. A nuclear “_reticulum plasmatique_” (Carnoy) . . . 468 - - 221. A spherical radiolarian, _Aulonia hexagona_ (Haeckel) . . . 469 - - 222. _Actinomma arcadophorum_ (_do._) . . . 469 - - 223. _Ethmosphaera conosiphonia_ (_do._) . . . 470 - - 224. Portions of shells of _Cenosphaera favosa_ and _vesparia_ (_do._) - . . . 470 - - 225. _Aulastrum triceros_ (_do._) . . . 471 - - 226. Part of the skeleton of _Cannorhaphis_ (_do._) . . . 472 - - 227. A Nassellarian skeleton, _Callimitra carolotae_ (_do._) . . . 472 - - 228, 9. Portions of _Dictyocha stapedia_ (_do._) . . . 474 - - 230. Diagram to illustrate the conformation of _Callimitra_ . . . 476 - - 231. Skeletons of various radiolarians (Haeckel) . . . 479 - - 232. Diagrammatic structure of the skeleton of _Dorataspis_ (_do._) - . . . 481 - - 233, 4. _Phatnaspis cristata_ (Haeckel), and a diagram of the same - . . . 483 - - 235. _Phractaspis prototypus_ (Haeckel) . . . 484 - - 236. Annular and spiral thickenings in the walls of plant-cells - . . . 488 - - 237. A radiograph of the shell of _Nautilus_ (Green and Gardiner) - . . . 494 - - 238. A spiral foraminifer, _Globigerina_ (Brady) . . . 495 - - 239–42. Diagrams to illustrate the development or growth of a - logarithmic spiral . . . 407–501 - - 243. A helicoid and a scorpioid cyme . . . 502 - - 244. An Archimedean spiral . . . 503 - - 245–7. More diagrams of the development of a logarithmic spiral - . . . 505, 6 - - 248–57. Various diagrams illustrating the mathematical theory of - gnomons . . . 508–13 - - 258. A shell of _Haliotis_, to shew how each increment of the shell - constitutes a gnomon to the preexisting structure . . . 514 - - 259, 60. Spiral foraminifera, _Pulvinulina_ and _Cristellaria_, to - illustrate the same principle . . . 514, 5 - - 261. Another diagram of a logarithmic spiral . . . 517 - - 262. A diagram of the logarithmic spiral of _Nautilus_ (Moseley) - . . . 519 - - 263, 4. Opercula of _Turbo_ and of _Nerita_ (Moseley) . . . 521, 2 - - 265. A section of the shell of _Melo ethiopicus_ . . . 525 - - 266. Shells of _Harpa_ and _Dolium_, to illustrate generating curves - and gene . . . 526 - - 267. D’Orbigny’s Helicometer . . . 529 - - 268. Section of a nautiloid shell, to shew the “protoconch” . . . 531 - - 269–73. Diagrams of logarithmic spirals, of various angles . . . 532–5 - - 274, 6, 7. Constructions for determining the angle of a logarithmic - spiral . . . 537, 8 - - 275. An ammonite, to shew its corrugated surface pattern . . . 537 - - 278–80. Illustrations of the “angle of retardation” . . . 542–4 - - 281. A shell of _Macroscaphites_, to shew change of curvature . . . 550 - - 282. Construction for determining the length of the coiled spire - . . . 551 - - 283. Section of the shell of _Triton corrugatus_ (Woodward) . . . 554 - - 284. _Lamellaria perspicua_ and _Sigaretus haliotoides_ (_do._) - . . . 555 - - 285, 6. Sections of the shells of _Terebra maculata_ and _Trochus - niloticus_ . . . 559, 60 - - 287–9. Diagrams illustrating the lines of growth on a lamellibranch - shell . . . 563–5 - - 290. _Caprinella adversa_ (Woodward) . . . 567 - - 291. Section of the shell of _Productus_ (Woods) . . . 567 - - 292. The “skeletal loop” of _Terebratula_ (_do._) . . . 568 - - 293, 4. The spiral arms of _Spirifer_ and of _Atrypa_ (_do._) . . . 569 - - 295–7. Shells of _Cleodora_, _Hyalaea_ and other pteropods (Boas) - . . . 570, 1 - - 298, 9. Coordinate diagrams of the shell-outline in certain pteropods - . . . 572, 3 - - 300. Development of the shell of _Hyalaea tridentata_ (Tesch) . . . 573 - - 301. Pteropod shells, of _Cleodora_ and _Hyalaea_, viewed from the - side (Boas) . . . 575 - - 302, 3. Diagrams of septa in a conical shell . . . 579 - - 304. A section of _Nautilus_, shewing the logarithmic spirals of the - septa to which the shell-spiral is the evolute . . . 581 - - 305. Cast of the interior of the shell of _Nautilus_, to shew the - contours of the septa at their junction with the shell-wall . . . 582 - - 306. _Ammonites Sowerbyi_, to shew septal outlines (Zittel, after - Steinmann and Döderlein) . . . 584 - - 307. Suture-line of _Pinacoceras_ (Zittel, after Hauer) . . . 584 - - 308. Shells of _Hastigerina_, to shew the “mouth” (Brady) . . . 588 - - 309. _Nummulina antiquior_ (V. von Möller) . . . 591 - - 310. _Cornuspira foliacea_ and _Operculina complanata_ (Brady) - . . . 594 - - 311. _Miliolina pulchella_ and _linnaeana_ (Brady) . . . 596 - - 312, 3. _Cyclammina cancellata_ (_do._), and diagrammatic figure of - the same . . . 596, 7 - - 314. _Orbulina universa_ (Brady) . . . 598 - - 315. _Cristellaria reniformis_ (_do._) . . . 600 - - 316. _Discorbina bertheloti_ (_do._) . . . 603 - - 317. _Textularia trochus_ and _concava_ (_do._) . . . 604 - - 318. Diagrammatic figure of a ram’s horns (Sir V. Brooke) . . . 615 - - 319. Head of an Arabian wild goat (Sclater) . . . 616 - - 320. Head of _Ovis Ammon_, shewing St Venant’s curves . . . 621 - - 321. St Venant’s diagram of a triangular prism under torsion (Thomson - and Tait) . . . 623 - - 322. Diagram of the same phenomenon in a ram’s horn . . . 623 - - 323. Antlers of a Swedish elk (Lönnberg) . . . 629 - - 324. Head and antlers of _Cervus duvauceli_ (Lydekker) . . . 630 - - 325, 6. Diagrams of spiral phyllotaxis (P. G. Tait) . . . 644, 5 - - 327. Further diagrams of phyllotaxis, to shew how various spiral - appearances may arise out of one and the same angular leaf-divergence - . . . 648 - - 328. Diagrammatic outlines of various sea-urchins . . . 664 - - 329, 30. Diagrams of the angle of branching in blood-vessels (Hess) - . . . 667, 8 - - 331, 2. Diagrams illustrating the flexure of a beam . . . 674, 8 - - 333. An example of the mode of arrangement of bast-fibres in a - plant-stem (Schwendener) . . . 680 - - 334. Section of the head of a femur, to shew its trabecular structure - (Schäfer, after Robinson) . . . 681 - - 335. Comparative diagrams of a crane-head and the head of a femur - (Culmann and H. Meyer) . . . 682 - - 336. Diagram of stress-lines in the human foot (Sir D. MacAlister, - after H. Meyer) . . . 684 - - 337. Trabecular structure of the _os calcis_ (_do._) . . . 685 - - 338. Diagram of shearing-stress in a loaded pillar . . . 686 - - 339. Diagrams of tied arch, and bowstring girder (Fidler) . . . 693 - - 340, 1. Diagrams of a bridge: shewing proposed span, the corresponding - stress-diagram and reciprocal plan of construction (_do._) . . . 696 - - 342. A loaded bracket and its reciprocal construction-diagram - (Culmann) . . . 697 - - 343, 4. A cantilever bridge, with its reciprocal diagrams (Fidler) - . . . 698 - - 345. A two-armed cantilever of the Forth Bridge (_do._) . . . 700 - - 346. A two-armed cantilever with load distributed over two pier-heads, - as in the quadrupedal skeleton . . . 700 - - 347–9. Stress-diagrams. or diagrams of bending moments, in the - backbones of the horse, of a Dinosaur, and of _Titanotherium_ - . . . 701–4 - - 350. The skeleton of _Stegosaurus_ . . . 707 - - 351. Bending-moments in a beam with fixed ends, to illustrate the - mechanics of chevron-bones . . . 709 - - 352, 3. Coordinate diagrams of a circle, and its deformation into an - ellipse . . . 729 - - 354. Comparison, by means of Cartesian coordinates, of the - cannon-bones of various ruminant animals . . . 729 - - 355, 6. Logarithmic coordinates, and the circle of Fig. 352 inscribed - therein . . . 729, 31 - - 357, 8. Diagrams of oblique and radial coordinates . . . 731 - - 359. Lanceolate, ovate and cordate leaves, compared by the help of - radial coordinates . . . 732 - - 360. A leaf of _Begonia daedalea_ . . . 733 - - 361. A network of logarithmic spiral coordinates . . . 735 - - 362, 3. Feet of ox, sheep and giraffe, compared by means of Cartesian - coordinates . . . 738, 40 - - 364, 6. “Proportional diagrams” of human physiognomy (Albert Dürer) - . . . 740, 2 - - 365. Median and lateral toes of a tapir, compared by means of - rectangular and oblique coordinates . . . 741 - - 367, 8. A comparison of the copepods _Oithona_ and _Sapphirina_ - . . . 742 - - 369. The carapaces of certain crabs, _Geryon_, _Corystes_ and others, - compared by means of rectilinear and curvilinear coordinates . . . 744 - - 370. A comparison of certain amphipods, _Harpinia_, _Stegocephalus_ - and _Hyperia_ . . . 746 - - 371. The calycles of certain campanularian zoophytes, inscribed in - corresponding Cartesian networks . . . 747 - - 372. The calycles of certain species of _Aglaophenia_, similarly - compared by means of curvilinear coordinates . . . 748 - - 373, 4. The fishes _Argyropelecus_ and _Sternoptyx_, compared by means - of rectangular and oblique coordinate systems . . . 748 - - 375, 6. _Scarus_ and _Pomacanthus_, similarly compared by means of - rectangular and coaxial systems . . . 749 - - 377–80. A comparison of the fishes _Polyprion_, _Pseudopriacanthus_, - _Scorpaena_ and _Antigonia_ . . . 750 - - 381, 2. A similar comparison of _Diodon_ and _Orthagoriscus_ . . . 751 - - 383. The same of various crocodiles: _C. porosus_, _C. americanus_ and - _Notosuchus terrestris_ . . . 753 - - 384. The pelvic girdles of _Stegosaurus_ and _Camptosaurus_ . . . 754 - - 385, 6. The shoulder-girdles of _Cryptocleidus_ and of _Ichthyosaurus_ - . . . 755 - - 387. The skulls of _Dimorphodon_ and of _Pteranodon_ . . . 756 - - 388–92. The pelves of _Archaeopteryx_ and of _Apatornis_ compared, and - a method illustrated whereby intermediate configurations may be found - by interpolation (G. Heilmann) . . . 757–9 - - 393. The same pelves, together with three of the intermediate or - interpolated forms . . . 760 - - 394, 5. Comparison of the skulls of two extinct rhinoceroses, - _Hyrachyus_ and _Aceratherium_ (Osborn) . . . 761 - - 396. Occipital views of various extinct rhinoceroses (_do._) . . . 762 - - 397–400. Comparison with each other, and with the skull of - _Hyrachyus_, of the skulls of _Titanotherium_, tapir, horse and rabbit - . . . 763, 4 - - 401, 2. Coordinate diagrams of the skulls of _Eohippus_ and of - _Equus_, with various actual and hypothetical intermediate types - (Heilmann) . . . 765–7 - - 403. A comparison of various human scapulae (Dwight) . . . 769 - - 404. A human skull, inscribed in Cartesian coordinates . . . 770 - - 405. The same coordinates on a new projection, adapted to the skull of - the chimpanzee . . . 770 - - 406. Chimpanzee’s skull, inscribed in the network of Fig. 405 . . . 771 - - 407, 8. Corresponding diagrams of a baboon’s skull, and of a dog’s - . . . 771, 3 - - - - -“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, _De Pot. Q._ iii, a, 11. (Quoted in _Brit. Assoc. -Address_, _Section D_, 1911.) - -“...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 _Principia_. -(Quoted by Mr W. Spottiswoode, _Brit. Assoc. Presidential Address_, -1878.) - -“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 -_Brit. Assoc. Address_, _Section A_, 1874. - -{1} - - - - -CHAPTER I - -INTRODUCTORY - - -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, _Physicam chemiae -adiunxit_[1]. - -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 {2} 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. - -As soon as we adventure on the paths of the physicist, we learn to -_weigh_ and to _measure_, 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 _number_, as -in the dreams and visions of Plato and Pythagoras; for modern chemistry -would have gladdened the hearts of those great philosophic dreamers. - -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 {3} 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. - -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. - -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[2] and Maupertuis, to whom Leibniz[3] had passed it -on. Half overshadowed by the mechanical concept, it runs through Claude -Bernard’s _Leçons sur les {4} phénomènes de la Vie_[4], and abides in -much of modern physiology[5]. Inherited from Hegel, it dominated Oken’s -_Naturphilosophie_ 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[6]. - -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[7]”:—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[8] 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[9]” 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, {5} 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[10]. 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[11].” -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[12].” - -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 _mechanism_, 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[13].” - -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 {6} 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[14]” 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.” - -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, _aliae ex aliis aptae et necessitate nexae_, are no -more, and no less, than conditions _sine quâ non_. 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 _ratio cognoscendi_, though -the true _ratio efficiendi_ 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. - -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, {7} et si elles -sont interdites il faut renoncer à parler de ces choses.” - -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[15]” 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 _immanent_ 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.” - -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[16]. {8} 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, -_ipso facto_, a student of physical science. - -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[17], 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. - -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 _Theoria Philosophiae Naturalis_. - -How far, even then, mathematics will _suffice_ 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 {9} -of the body, as of all that is of the earth earthy, physical science -is, in my humble opinion, our only teacher and guide[18]. - -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 -_transgressed_ by the bodily mechanism. - -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 _physical_ complexity, however exalted, could ever -suffice. Other physicists, like Auerbach[19], or Larmor[20], or -Joly[21], 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” {10} 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, _ex hypothesi_, for the purposes of this -correlation, the fabric of the organism as a material and mechanical -configuration. - -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[22].” 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,—_An Vita sit? Dico quod non._ - -―――――――――― - -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 {11} 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. - -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. - -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 -_have_ 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 _motions_ -of the living substance that we must interpret in terms of force -(according to kinetics), but also {12} the _conformation_ of the -organism itself, whose permanence or equilibrium is explained by the -interaction or balance of forces, as described in statics. - -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. - -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 _Amoeba_; 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[23]. - -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. {13} -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. - -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 _viscid_, 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 _through_ this wall is sometimes -distinguished under the term _osmosis_. - -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 _force_. Indeed we shall manifestly be inclined to use -the term growth in two senses, {14} just indeed as we do in the case -of attraction or gravitation, on the one hand as a _process_, and on -the other hand as a _force_. - -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 _sui generis_ -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. - -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 _material_ -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, -{15} the chromosomes or the germ-plasm can never _act_ 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. - -{16} - - - - -CHAPTER II - -ON MAGNITUDE - - -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[24]. - -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 -_r_, the area of its surface is equal to 4π_r_^2, and its volume to -(4/3)π_r_^3; from which it follows that the ratio of volume to surface, -or _V_/_S_, is (1/3)_r_. 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 _L_ to represent any linear dimension, we may write -the general equations in the form - - _S_ ∝ _L_^2, _V_ ∝ _L_^3, - or _S_ = _k_ ⋅ _L_^2, and _V_ = _k′_ ⋅ _L_^3; - and _V_/_S_ ∝ _L_. - -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 {17} 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 _attention_ 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. - -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 _k_ in the formula _W_ = _k_ ⋅ _L_^3, 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 _k_, 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. - -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 _scale_ -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. {18} - -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 _proportion_ 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. - -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[25]. -It was elementary engineering experience such as this that led Herbert -Spencer[26] to apply the principle of similitude to biology. - -The same principle had been admirably applied, in a few clear -instances, by Lesage[27], a celebrated eighteenth century physician -of Geneva, in an unfinished and unpublished work[28]. 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 {19} excessively thick and strong, like a bull’s, or -very short like the neck of an elephant. - -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[29]. 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[30]. 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. - -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 _bending moments_. 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[31]. - -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[32]. In such an investigation we have to make {20} 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[33]) 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 (_dugento braccia alta_) 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[34]. In short, as -Goethe says in _Wahrheit und Dichtung_, “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 _equal -strength_ 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. - -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[35]: “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 {21} 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.” - -But the principle of Galileo carries us much further and along more -certain lines. - -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[36]. Indeed, {22} 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 (_W_) 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 (_V_) against a resistance (_R_) which increases as the square -of the said linear dimensions; we have at once - - _W_ = _l_^3, - - and also _W_ = _R_ _V_^2 = _l_^2 _V_^2. - - Therefore _l_^3 = _l_^2 _V_^2, and _V_ = √_l_. - -This is what is known as Froude’s Law of the _correspondence of -speeds_. - -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 _surface_ of the piston, and on the other, with the _length_ -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 _heating-surface_ of -the boiler[37]. 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 {23} lung, that is to say (other things being equal) -with the _square_ 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 _steamship comparison_. 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[38]; 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 {24} 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. - -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. - -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: - -In order to keep aloft, the bird must communicate to the air a downward -momentum equivalent to its own weight, and therefore proportional -to _the cube of its own linear dimensions_. 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 _the -square of the linear dimensions, and also as the square of the speed_. -Therefore, in order that the bird may maintain level flight, its speed -must be proportional to _the square root of its linear dimensions_. - -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 _m_ _V_^2, 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: {25} that is to say, it increases in -proportion _to the power_ 3½ _of the linear dimensions_, and therefore -faster than the weight of the bird increases. - -Put in mathematical form, the equations are as follows: - -(_m_ = the mass of air thrust downwards; _V_ its velocity, proportional -to that of the bird; _M_ its momentum; _l_ a linear dimension of the -bird; _w_ its weight; _W_ the work done in moving itself forward.) - - _M_ = _w_ = _l_^3. - - But _M_ = _m_ _V_, and _m_ = _l_^2 _V_. - - Therefore _M_ = _l_^2 _V_^2, - and _l_^2 _V_^2 = _l_^3, - or _V_ = √_l_. - - But, again, _W_ = _m_ _V_^2 = _l_^2 _V_ × _V_^2 - = _l_^2 × √_l_ × _l_ - = _l_^{3½}. - -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[39]. 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 2^3 : 2^{3½}, -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^{3½} and 25^3, or between 5^7 : 5^6; -in other words, flight is just five times more difficult for the larger -than for the smaller bird[40]. - -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 {26} contained in -the equation _V_ = √_l_, 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 _must_ 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: - - _W_ : _w_ :: _L_^3 : _l_^3 :: 8 : 1. - - Therefore _L_ : _l_ :: 2 : 1. - - But _V_^2 : _v_^2 :: _L_ : _l_. - - Therefore _V_ : _v_ :: √2 : 1 = 1·414 : 1. - -That is to say, the larger machine must be capable of a speed equal -to 1·414 × 40, or about 56½ miles per hour. - -It is highly probable, as Lanchester[41] 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[42] 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 {27} 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 _minimum_ velocity of 5 × 14, or 70 miles an hour. - -The same principle of _necessary speed_, 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[43] 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 _à fortiori_ the still -smaller insects, are capable of “stationary flight,” a very slight and -scarcely perceptible velocity _relatively to the air_ 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 _starts_ to fly. - -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. - -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 {28} in one of his -chapters he deals with the proposition, “Est impossible, ut homines -propriis viribus artificiose volare possint[44].” - -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[45], -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. - -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. - -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[46].” {29} - -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 (_x_), -and inversely proportional to the mass (_M_) moved: _V_ = _x_/_M_. 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[47], 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[48]. 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, {30} and tends to leave a certain -balance of advantage, in regard to leaping power, on the side of the -larger animal[49]. - -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. - -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[50]. - -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 _pendulum-rate_; 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[51]. 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, _A_ and _B_, walk in a similar fashion, that is -to say, with a similar _angle_ of swing. The _arc_ through which the -leg swings, or the _amplitude_ of each step, will therefore vary as the -length of leg, or say as _a_/_b_; but the time of swing will vary as -the square {31} root of the pendulum-length, or √_a_/√_b_. 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 _A_ and _B_ are in the ratio of √_a_ : √_b_. - -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[52] -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)^2, or 1/216 of our present height,—say 72/216 inches, or -one-third of an inch high. - -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[53]. 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. - -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 {32} 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. - -From all the foregoing discussion we learn that, as Crookes once upon -a time remarked[54], 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. - -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[55]. {33} - -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 _facilitated_ 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. {34} - -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 _Growth and Form_, 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. - -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. - -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 _outer_ 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 {35} 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[56]. 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. - -In the case of a soap-bubble, by the way, if it divide into two -bubbles, the volume is actually diminished[57] 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. - -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. - -Sachs[58] 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[59], 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 -{36} 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 _Crepidula_ (a little American -“slipper-limpet,” now much at home on our own oyster-beds), Conklin[60] -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[61]. Driesch has laid particular stress -upon this principle of a “fixed cell-size.” - -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[62]. - -[Illustration: Fig. 1. Motor ganglion-cells, from the cervical spinal -cord. (From Minot, after Irving Hardesty.)] - -In Fig. 1 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 _order_ 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 {37} 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[63]. 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 {38} the body is more than -the _sum_ 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.” - -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. - -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[64]. - -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 _possible_ 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. - -Among the smallest of known organisms we have, for instance, -_Micromonas mesnili_, Bonel, a flagellate infusorian, which measures -about ·34 _µ_, or ·00034 mm., by ·00025 mm.; smaller even than this -we have a pathogenic micrococcus of the rabbit, _M. progrediens_, -Schröter, the diameter of which is said to be only ·00015 mm. or -·15 _µ_, or 1·5 × 10^{−5} cm.,—about equal to the thickness of {39} -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[65]. 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. - -[Illustration: Fig. 2. Relative magnitudes of: A, human blood-corpuscle -(7·5 µ in diameter); B, _Bacillus anthracis_ (4 – 15 µ × 1 µ); C, -various Micrococci (diam. 0·5 – 1 µ, rarely 2 µ); D, _Micromonas -progrediens_, Schröter (diam. 0·15 µ).] - -To illustrate some of these small magnitudes I have adapted the -preceding diagram from one given by Zsigmondy[66]. Upon the {40} 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. - -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 × 10^6 µ; and the mass of the man is -in the neighbourhood of five million, million, million (5 × 10^{18}) -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. - -Clerk Maxwell dealt with this matter in his article “Atom[67],” and, in -somewhat greater detail, Errera discusses the question on the following -lines[68]. The weight of a hydrogen molecule is, according to the -physical chemists, somewhere about 8·6 × 2 × 10^{−22} milligrammes; and -that of any other element, whose molecular weight is _M_, is given by -the equation - - (_M_) = 8·6 × _M_ × 10^{−22}. - -Accordingly, the weight of the atom of sulphur may be taken as - - 8·6 × 32 × 10^{−22} mgm. = 275 × 10^{−22} mgm. - -The analysis of ordinary bacteria shews them to consist[69] 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 - - 1/1000 × 15/100 = 15 × 10^{−5} - -parts of sulphur, taking the total weight as = 1. - -But our little micrococcus, of 0·15 µ in diameter, would, if it were -spherical, have a volume of - - π/6 × 0·15^3 µ = 18 × 10^{−4} cubic microns; {41} - -and therefore (taking its density as equal to that of water), a weight -of - - 18 × 10^{−4} × 10^{−9} = 18 × 10^{−13} mgm. - -But of this total weight, the sulphur represents only - - 18 × 10^{−13} × 15 × 10^{−5} = 27 × 10^{−17} mgm. - -And if we divide this by the weight of an atom of sulphur, we have - - (27 × 10^{−17}) ÷ (275 × 10^{−22}) = 10,000, or thereby. - -According to this estimate, then, our little _Micrococcus -progrediens_ 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! - -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[70], 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 - - 8·6 × 10166 × 10^{−22} = 8·7 × 10^{−18} mgm. - -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 - - 14/100 × 18 × 10^{−13} = 2·5 × 10^{−13} mgm. - -If we divide this weight by that which we have arrived at as the -weight of an albumin molecule, we have - - (2·5 × 10^{−13}) ÷ (8·7 × 10^{−18}) = 2·9 × 10^{−4}, - -in other words, our micrococcus apparently contains something less -than 30,000 molecules of albumin. {42} - -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^{−22} 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[71]. - -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, -_maximal_ 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. - -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 {43} 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. - -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^{−7} cm., or about ·006 µ thick, and Lord Rayleigh -and M. Devaux[72] have obtained films of oil of ·002 µ, or even ·001 µ -in thickness. - -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. - -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[73], 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 _nil_; and its specific properties must -be little more than those of an ion-laden corpuscle, enabling it to -perform {44} 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[74]. - -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, _ad infinitum_] 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 _particulam -undique desectam_ of which we have read, and at which we have smiled, -in our Horace. - -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 {45} 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. - -The Brownian movement has also to be reckoned with,—that remarkable -phenomenon studied nearly a century ago (1827) by Robert Brown, -_facile princeps botanicorum_. It is one more of those fundamental -physical phenomena which the biologists have contributed, or helped to -contribute, to the science of physics. - -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[75].” 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 {46} behaviour is manifested in several -ways[76]. 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[77], -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 -_Spirochaete pallida_ 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 {47} 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. - -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 _business_ whatsoever. In like manner -Lucretius, and Epicurus before him, watched the dust-motes quivering -in the beam, and saw in them a mimic representation, _rei simulacrum -et imago_, 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[78]. -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 {48} 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. - -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 (_in vacuo_) 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 _may_ 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 {49} 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! - -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 -_mare liberum_ 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. - -{50} - - - - -CHAPTER III - -THE RATE OF GROWTH - - -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 _scalar_ -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 _vector_ phenomenon. - -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 _ratio of magnitudes_, referred to -direction in space. - -When in dealing with magnitude we refer its variations to successive -intervals of time (or when, as it is said, we _equate_ it with time), -we are then dealing with the phenomenon of _growth_; 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” {51} 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 _velocity_ (whose “dimensions” are Space/Time or _L_/_T_); and this -phenomenon we shall speak of, simply, as a rate of growth. - -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 _space_) 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[79]. - -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 _function of -time_[80]. {52} - -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. - -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 _grows_ 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 _ultimately_ due to the action of molecular forces; -but in the one case the movements of the particles of matter lie for -the most part _within molecular range_, 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 _rate of action_ in different directions, -and to see that it is merely on a difference of velocities that the -modification of form essentially depends. {53} 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; _molecular_ forces -are paramount in the conformation of the one, and _molar_ forces are -dominant in the other. - -At the same time it is perfectly true that _all_ 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[81], 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 _morphological_ problem. - -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.” {54} - -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 _simple_ 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 _constant_ 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 _ratio_ tends to alter, then we have -the phenomenon of morphological “development,” or steady and persistent -change of form. - -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 “_evolutio_,” in -which no part came into being which had not essentially existed -before[82]. 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 {55} them, and which -are characteristic of another order of magnitude. - -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[83], 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[84], we observe that it -is not only the _growth_ of the individual cells, but the _traction_ -exercised through their mutual interconnections, which brings about the -foldings and other distortions of the entire structure. {56} - -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 _incrementum_, or _celeritas incrementi_, than he proceeds -to deal with the contributory and complementary phenomena of expansion, -traction (_adtractio_)[85], and pressure, and the more subtle -influences which he denominates _vis derivationis et revulsionis_[86]: -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. - -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[87]. -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[88].” For Hertwig, -therefore, as {57} 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[89]. 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[90].” - -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[91] 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. - -―――――――――― - -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. - -There is a curious passage in the _Origin of Species_[92], 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 {58} structure between the -embryo and the adult, that we are tempted to look at this difference -as in some necessary manner contingent on growth. _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._” 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 -{59} 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. - -―――――――――― - -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[93]. - -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. _V_ = (_R′_ − _R_)/_T_, 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 {60} 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. - -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 _development_. -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. {61} - - -_The rate of growth in Man._ - -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 _Anthropométrie_[94], 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. - -[Illustration: Fig. 3. Curve of Growth in Man, from birth to 20 yrs -(♂); from Quetelet’s Belgian data. The upper curve of stature from -Bowditch’s Boston data.] - -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 _average_ rate of growth has been (180 − 50)/20 cm., or 6·5 -centimetres per annum. But we know very well that this is {62} 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. 3), has -always a certain characteristic form, or characteristic _curvature_. -This is our immediate proof of the fact that the _rate of growth_ -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. - -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. - -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[95] 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 {63} of growth is in fact a diagram of activity, or -“time-energy” diagram[96]. As the organism grows it is absorbing energy -beyond its daily needs, and accumulating it at a rate depicted in our - - _Stature, weight, and span of outstretched arms._ - - (_After Quetelet_, _pp._ 193, 346.) - - Stature in metres Weight in kgm. Span of % ratio - arms, of stature - Age Male Female % F/M Male Female % F/M male to span - 0 0·500 0·494 98·8 3·2 2·9 90·7 0·496 100·8 - 1 0·698 0·690 98·8 9·4 8·8 93·6 0·695 100·4 - 2 0·791 0·781 98·7 11·3 10·7 94·7 0·789 100·3 - 3 0·864 0·854 98·8 12·4 11·8 95·2 0·863 100·1 - 4 0·927 0·915 98·7 14·2 13·0 91·5 0·927 100·0 - 5 0·987 0·974 98·7 15·8 14·4 91·1 0·988 99·9 - 6 1·046 1·031 98·5 17·2 16·0 93·0 1·048 99·8 - 7 1·104 1·087 98·4 19·1 17·5 91·6 1·107 99·7 - 8 1·162 1·142 98·2 20·8 19·1 91·8 1·166 99·6 - 9 1·218 1·196 98·2 22·6 21·4 94·7 1·224 99·5 - 10 1·273 1·249 98·1 24·5 23·5 95·9 1·281 99·4 - 11 1·325 1·301 98·2 27·1 25·6 94·5 1·335 99·2 - 12 1·375 1·352 98·3 29·8 29·8 100·0 1·388 99·1 - 13 1·423 1·400 98·4 34·4 32·9 95·6 1·438 98·9 - 14 1·469 1·446 98·4 38·8 36·7 94·6 1·489 98·7 - 15 1·513 1·488 98·3 43·6 40·4 92·7 1·538 99·4 - 16 1·554 1·521 97·8 49·7 43·6 87·7 1·584 98·1 - 17 1·594 1·546 97·0 52·8 47·3 89·6 1·630 97·9 - 18 1·630 1·563 95·9 57·8 49·0 84·8 1·670 97·6 - 19 1·655 1·570 94·9 58·0 51·6 89·0 1·705 97·1 - 20 1·669 1·574 94·3 60·1 52·3 87·0 1·728 96·6 - 25 1·682 1·578 93·8 62·9 53·3 84·7 1·731 97·2 - 30 1·686 1·580 93·7 63·7 54·3 85·3 1·766 95·5 - 40 1·686 1·580 93·7 63·7 55·2 86·7 1·766 95·5 - 50 1·686 1·580 93·7 63·5 56·2 88·4 — — - 60 1·676 1·571 93·7 61·9 54·3 87·7 — — - 70 1·660 1·556 93·7 59·5 51·5 86·5 — — - 80 1·636 1·534 93·8 57·8 49·4 85·5 — — - 90 1·610 1·510 93·8 57·8 49·3 85·3 — — - -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, {64} which -draws us down when we would fain rise up[97]. 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[98]. - -Side by side with the curve which represents growth in length, or -stature, our diagram shows the curve of weight[99]. 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. - -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 _rate of -change_ of magnitude, or rate of growth, is itself changing. We have, -in short, to study the phenomenon of _acceleration_: we have begun by -studying a velocity, or rate of {65} 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 _slope_ 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 _differences_ between the values of length (or -of weight) for the successive ages which we have begun by studying. If -we then plot these successive _differences_ 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. - -The acceleration-curve of height, which we here illustrate, in Fig. 4, -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 _nil_; 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, {66} 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 - -[Illustration: Fig. 4. Mean annual increments of stature (♂), Belgian -and American.] - -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. - - _Annual Increment of Stature (in cm.) from Belgian and American - Statistics._ - - D: Belgian (Quetelet, p. 344) - E: Paris* (Variot et Chaumet, p. 55) - F: Toronto† (Boas, p. 1547) - G: Worcester‡, Mass. (Boas, p. 1548) - H: Ann. increment - I: Increment - J: Boys - K: Girls - V: Variability of do. - - ────[D]─── ───────[E]─────── ──────[F]────── ────────[G]─────── - Height Height [I] Height [V] [H] [H] - Age (Boys) [H] [J] [K] [J] [K] (Boys) (6) [H] ([J]) [V] ([K]) [V] - 0 50·0 — — — — — — — — — — — — - 1 69·8 19·8 74·2 73·6 — — — — — — — — — - 2 79·1 9·3 82·7 81·8 8·5 8·2 — — — — — — — - 3 86·4 7·3 89·1 88·4 6·4 6·6 — — — — — — — - 4 92·7 6·3 96·8 95·8 7·7 7·4 — — — — — — — - 5 98·7 6·0 103·3 101·9 6·5 6·1 105·90 4·40 — — — — — - 6 104·0 5·9 109·9 108·9 6·6 7·0 111·58 4·62 5·68 6·55 1·57 5·75 0·88 - 7 110·4 5·8 114·4 113·8 4·5 4·9 116·83 4·93 5·25 5·70 0·68 5·90 0·98 - 8 116·2 5·8 119·7 119·5 5·3 5·7 122·04 5·34 5·21 5·37 0·86 5·70 1·10 - 9 121·8 5·6 125·0 124·7 5·3 4·8 126·91 5·49 4·87 4·89 0·96 5·50 0·97 - 10 127·3 5·5 130·3 129·5 5·3 5·2 131·78 5·75 4·87 5·10 1·03 5·97 1·23 - 11 132·5 5·2 133·6 134·4 3·3 4·9 136·20 6·19 4·42 5·02 0·88 6·17 1·85 - 12 137·5 5·0 137·6 141·5 4·0 7·1 140·74 6·66 4·54 4·99 1·26 6·98 1·89 - 13 142·3 4·8 145·1 148·6 7·5 7·1 146·00 7·54 5·26 5·91 1·86 6·71 2·06 - 14 146·9 4·6 153·8 152·9 8·7 4·3 152·39 8·49 6·39 7·88 2·39 5·44 2·89 - 15 151·3 4·4 159·6 154·2 5·8 1·3 159·72 8·78 7·33 6·23 2·91 5·34 2·71 - 16 155·4 4·1 — — — — 164·90 7·73 5·18 5·64 3·46 — — - 17 159·4 4·0 — — — — 168·91 7·22 4·01 — — — — - 18 163·0 3·6 — — — — 171·07 6·74 2·16 — — — — - 19 165·5 2·5 — — — — — — — — — — — - 20 167·0 1·5 — — — — — — — — — — — - - * Ages from 1–2, 2–3, etc. - - † The epochs are, in this table, 5·5, 6·5, years, etc. - - ‡ Direct observations on actual, or individualised, - increase of stature from year to year: between the ages of - 5–6, 6–7, etc. - -Even apart from these data of Quetelet’s (which seem to constitute -an extreme case), it is evident that there are very {68} 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. - -It is evident that, if we pleased, we might represent the _rate of -change of acceleration_ 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. - - _Annual Increment of Weight in Man_ (_kgm._). - - (After Quetelet, _Anthropométrie_, p. 346*.) - - Increment - Age Male Female - 0–1 5·9 5·6 - 1–2 2·0 2·4 - 2–3 1·5 1·4 - 3–4 1·5 1·5 - 4–5 1·9 1·4 - 5–6 1·9 1·4 - 6–7 1·9 1·1 - 7–8 1·9 1·2 - 8–9 1·9 2·0 - 9–10 1·7 2·1 - 10–11 1·8 2·4 - 11–12 2·0 3·5 - 12–13 4·1 3·5 - 13–14 4·0 3·8 - 14–15 4·1 3·7 - 15–16 4·2 3·5 - 16–17 4·3 3·3 - 17–18 4·2 3·0 - 18–19 3·7 2·3 - 19–20 1·9 1·1 - 20–21 1·7 1·1 - 21–22 1·7 0·5 - 22–23 1·6 0·4 - 23–24 0·9 −0·2 - 24–25 0·8 −0·2 - - * The values given in this table are not in precise accord - with those of the Table on p. 63. The latter represent - Quetelet’s results arrived at in 1835; the former are the - means of his determinations in 1835–40. - -The acceleration-curve for man’s weight (Fig. 5), 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[100] {69} 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. - -[Illustration: Fig. 5. Mean annual increments of weight, in man and -woman; from Quetelet’s data.] - -In the same diagram (Fig. 5) 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, {70} 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 -_similar_ curve in the two sexes; but it has its notable differences, -in amplitude and especially in _phase_. 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. - - From certain corrected, or “typical” values, given for American - children by Boas and Wissler (_l.c._ p. 42), we obtain the following - still clearer comparison of the annual increments of _stature_ 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: - - Age 12 13 14 15 16 17 18 19 20 - Boys (cm.) 4·1 6·3 8·7 7·9 5·2 3·2 1·9 0·9 0·3 - Girls (cm.) 7·5 7·0 4·6 2·1 0·9 0·4 0·1 0·0 0·0 - Difference −3·4 −0·7 4·1 5·8 4·3 2·8 1·8 0·9 0·3 - -The result of these differences (which are essentially -_phase_-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 -_ratio_ 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 {71} 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. 6). - -[Illustration: Fig. 6. Percentage ratio, throughout life, of female -weight to male; from Quetelet’s data.] - -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[101]. 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. - -The size ultimately attained is a resultant of the rate, and of {72} -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. - -―――――――――― - -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. - - 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[102] 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 _percentages_ 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: - - Years 0 1 2 3 4 - cm. 50·0 69·8 79·1 86·4 92·7 - - 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. - - Now when we plot actual length against time, we have a perfectly - definite thing. When we differentiate this _L_/_T_, we have - _dL_/_dT_, which is (of course) velocity; and from this, by a second - differentiation, we obtain _d_^2 _L_/_dT_^2, that is to say, the - acceleration. {73} - - But when you take percentages of _y_, you are determining _dy_/_y_, - and when you plot this against _dx_, you have - - (_dy_/_y_)/_dx_, or _dy_/(_y_ ⋅ _dx_), or (1/_y_) ⋅ (_dy_/_dx_), - - 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 _x_ and _y_; - and his method is practically tantamount to plotting log _y_ against - _x_, 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. - - -_Pre-natal and post-natal growth._ - -In the acceleration-curves which we have shown above (Figs. 2, 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 _between_ 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[103], who gives the stature of the infant month by month -during the first year of its life, as follows: - - Age in months 0 1 2 3 4 5 6 7 8 9 10 11 12 - Length in cm. (50) 54 58 60 62 64 65 66 67·5 68 69 70·5 72 - [Differences (in cm.) 4 4 2 2 2 1 1 1·5 ·5 1 1·5 1·5] - -If we multiply these _monthly_ differences, or mean monthly velocities, -by 12, to bring them into a form comparable with the {74} _annual_ -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. - -[Illustration: Fig. 7. Curve of growth (in length or stature) of child, -before and after birth. (From His and Rüssow’s data.)] - -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. {75} - -According to His[104], the following are the mean lengths of the unborn -human embryo, from month to month. - - Months 0 1 2 3 4 5 6 7 8 9 10 - (Birth) - Length in mm. 0 7·5 40 84 162 275 352 402 443 472 490–500 - Increment per - month in mm. — 7·5 32·5 44 78 113 77 50 41 29 18–28 - -[Illustration: Fig. 8. Mean monthly increments of length or stature of -child (in cms.).] - -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. 7) and curve of -acceleration of growth (Fig. 8) 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[105]: up to that date growth -proceeds with a continually increasing {76} 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 {77} method of nutrition has its inevitable effect; but this -slight temporary set-back is immediately followed by a secondary, and -temporary, acceleration. - -[Illustration: Fig. 9. Curve of pre-natal growth (length or stature) of -child; and corresponding curve of mean monthly increments (mm.).] - -[Illustration: Fig. 10. Curve of growth of bamboo (from Ostwald, after -Kraus).] - -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. 9). 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. 9 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. 10, which is -a curve of growth of the bamboo[106], we see (so far as it goes) the -same essential features, {78} the slow beginning, the rapid increase -of velocity, the point of inflection, and the subsequent slow negative -acceleration[107]. - - -_Variability and Correlation of Growth._ - -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 _amount of variability_ -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. - -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. - -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 _in an orderly way_, 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 _changes its curvature_ -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 {79} 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, _C_, as -= (σ/_M_) × 100.) - -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. - -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[108]. - - _Coefficient of Variability (σ/_M_ × 100) in Man, at various ages._ - - Age 5 6 7 8 9 - Stature (Bowditch) 4·76 4·60 4·42 4·49 4·40 - Stature (Boas and Wissler) 4·15 4·14 4·22 4·37 4·33 - Weight (Bowditch) 11·56 10·28 11·08 9·92 11·04 - - Age 10 11 12 13 14 - Stature (Bowditch) 4·55 4·70 4·90 5·47 5·79 - Stature (Boas and Wissler) 4·36 4·54 4·73 5·16 5·57 - Weight (Bowditch) 11·60 11·76 13·72 13·60 16·80 - - Age 15 16 17 18 - Stature (Bowditch) 5·57 4·50 4·55 3·69 - Stature (Boas and Wissler) 5·50 4·69 4·27 3·94 - Weight (Bowditch) 15·32 13·28 12·96 10·40 - -The result is very curious indeed. We see, from Fig. 11, that the -curve of variability is very similar to what we have called the -acceleration-curve (Fig. 4): 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 {80} see, -in short, that the amount of _variability_ in stature or in weight is a -function of the _rate of growth_ in these magnitudes, though we are not -yet in a position to equate the terms precisely, one with another. - -[Illustration: Fig. 11. Coefficients of variability of stature in Man -(♂). from Boas and Wissler’s data.] - - 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 _phase_,—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, {81} I have taken Boas’s determinations - of variability (σ) (_op. cit._ p. 1548), converted them into the - corresponding coefficients of variability ((σ/_M_) × 100), and then - smoothed the resulting numbers. - - _Coefficients of Variability in Annual Increment of Stature._ - - Age 7 8 9 10 11 12 13 14 15 - Boys 17·3 15·8 18·6 19·1 21·0 24·7 29·0 36·2 46·1 - Girls 17·1 17·8 19·2 22·7 25·9 29·3 37·0 44·8 — - - 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. - - 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[109], is - confirmed by the following table:― - - _Correlation of Stature and Increment in Boys and Girls._ - - (_From Boas and Wissler._) - - Age 6 7 8 9 10 11 12 13 14 15 - Stature (B) 112·7 115·5 123·2 127·4 133·2 136·8 142·7 147·3 155·9 162·2 - (G) 111·4 117·7 121·4 127·9 131·8 136·7 144·6 149·7 153·8 157·2 - Increment (B) 5·7 5·3 4·9 5·1 5·0 4·7 5·9 7·5 6·2 5·2 - (G) 5·9 5·5 5·5 5·9 6·2 7·2 6·5 5·4 3·3 1·7 - Correlation (B) ·25 ·11 ·08 ·25 ·18 ·18 ·48 ·29 −·42 −·44 - (G) ·44 ·14 ·24 ·47 ·18 −·18 −·42 −·39 −·63 ·11 - -{82} - -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. - -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. - - -_Rate of growth in other organisms[110]._ - -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. - -In the accompanying curve of growth in weight of the mouse (Fig. 12), -based on W. Ostwald’s observations[111], 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. {83} - -Fig. 13 shews the curve of growth of the silkworm[112], during its -whole larval life, up to the time of its entering the chrysalis stage. - -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. - -[Illustration: Fig. 12. Growth in weight of Mouse. (After W. Ostwald.)] - -The rate of growth in the tadpole[113] (Fig. 14) is likewise marked -by epochs of retardation, and finally by a sudden and drastic change. -There is a slight diminution in weight immediately after {84} 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. {85} - -[Illustration: Fig. 13. Growth in weight of Silkworm. (From Ostwald, -after Luciani and Lo Monaco.)] - -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. - -[Illustration: Fig. 14. Growth in weight of Tadpole. (From Ostwald, -after Schaper.)] - -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[114]. The contrast of size is great between {87} 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. 15). - -[Illustration: Fig. 15. Development of Eel; from Leptocephalus larvae -to young Elver. (From Ostwald after Joh. Schmidt.)] - -[Illustration: Fig. 16. Growth in length of Spirogyra. (From Ostwald, -after Hofmeister.)] - -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[115]. 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. 16). 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. 17 shews, according to Bose’s observations[116], -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 -{88} twenty seconds or so, and after each little increment there is a -partial recoil. - -[Illustration: Fig. 17. Pulsations of growth in Crocus, in -micro-millimetres. (After Bose.)] - - -_The rate of growth of various parts or organs[117]._ - -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[118]. 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. - -In the accompanying table, I shew, from some of Vierordt’s data, the -_relative_ weights, at various ages, compared with the weight at birth, -of the entire body, of the brain, heart and liver; {89} and also the -percentage relation which each of these organs bears, at the several -ages, to the weight of the whole body. - - _Weight of Various Organs, compared with the Total Weight of the Human - Body (male)._ (_After Vierordt, Anatom. Tabellen, pp. 38, 39._) - - Percentage weights compared - Weight Relative weights of with total body-weights - of body† ───────────────────────── ─────────────────────────── - Age in kg. Body Brain Heart Liver Body Brain Heart Liver - 0 3·1 1 1 1 1 100 12·29 0·76 4·57 - 1 9·0 2·90 2·48 1·75 2·35 100 10·50 0·46 3·70 - 2 11·0 3·55 2·69 2·20 3·02 100 9·32 0·47 3·89 - 3 12·5 4·03 2·91 2·75 3·42 100 8·86 0·52 3·88 - 4 14·0 4·52 3·49 3·14 4·15 100 9·50 0·53 4·20 - 5 15·9 5·13 3·32 3·43 3·80 100 7·94 0·51 3·39 - 6 17·8 5·74 3·57 3·60 4·34 100 7·63 0·48 3·45 - 7 19·7 6·35 3·54 3·95 4·86 100 6·84 0·47 3·49 - 8 21·6 6·97 3·62 4·02 4·59 100 6·38 0·44 3·01 - 9 23·5 7·58 3·74 4·59 4·95 100 6·06 0·46 2·99 - 10 25·2 8·13 3·70 5·41 5·90 100 5·59 0·51 3·32 - 11 27·0 8·71 3·57 5·97 6·14 100 5·04 0·52 3·22 - 12 29·0 9·35 3·78 (4·13) 6·21 100 4·88 (0·34) 3·03 - 13 33·1 10·68 3·90 6·95 7·31 100 4·49 0·50 3·13 - 14 37·1 11·97 3·38 9·16 8·39 100 3·47 0·58 3·20 - 15 41·2 13·29 3·91 8·45 9·22 100 3·62 0·48 3·17 - 16 45·9 14·81 3·77 9·76 9·45 100 3·16 0·51 2·95 - 17 49·7 16·03 3·70 10·63 10·46 100 2·84 0·51 2·98 - 18 53·9 17·39 3·73 10·33 10·65 100 2·64 0·46 2·80 - 19 57·6 18·58 3·67 11·42 11·61 100 2·43 0·51 2·86 - 20 59·5 19·19 3·79 12·94 11·01 100 2·43 0·51 2·62 - 21 61·2 19·74 3·71 12·59 11·48 100 2·31 0·49 2·66 - 22 62·9 20·29 3·54 13·24 11·82 100 2·14 0·50 2·66 - 23 64·5 20·81 3·66 12·42 10·79 100 2·16 0·46 2·37 - 24 — — 3·74 13·09 13·04 100 — — — - 25 66·2 21·36 3·76 12·74 12·84 100 2·16 0·46 2·75 - - † From Quetelet. - -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 _more_ -rapidly than the average increase of the body. Heart and liver both -grow nearly at the same rate, and by the {90} 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. 18. - -[Illustration: Fig. 18. Relative growth in weight (in Man) of Brain, -Heart, and whole Body.] - - Many statistics indicate a decrease of brain-weight during adult life. - Boas[119] 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[120]. {91} - -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. - -It is plain, then, that there is no simple and direct relation, holding -good _throughout life_, 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[121] 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[122]. - -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. - - Thus, if _W_ be the brain-weight (in grammes), and _A_ be the age, - or _S_ the stature, of the individual, then (in the case of Swedish - males) the following simple equations suffice to give the required - ratios: - - _W_ = 1487·8 − 1·94 _A_ = 915·06 + 2·86 _S_. - -{ 92} - - 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[123]. Donaldson has further shewn that the correlation - between brain-weight and body-weight is very much closer in the rat - than in man[124]. - - 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. - - Weight of Weight of - entire animal brain - gms. gms. Ratio - Marmoset 335 12·5 1 : 26 - Spider monkey 1845 126 1 : 15 - Felis minuta 1234 23·6 1 : 56 - F. domestica 3300 31 1 : 107 - Leopard 27,700 164 1 : 168 - Lion 119,500 219 1 : 546 - Elephant 3,048,000 5430 1 : 560 - Whale (Globiocephalus) 1,000,000 2511 1 : 400 - - For much information on this subject, see Dubois, “Abhängigkeit des - Hirngewichtes von der Körpergrösse bei den Säugethieren,” _Arch. f. - Anthropol._ XXV, 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 _power_ 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 _W_ the weight of the body (in grammes), and _w_ the weight - of the brain, then if in all four cases we express the ratio by _W_ - = _w_^{_n_}, we find that _n_ 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[125]. {93} - - _The Length of the Head in Man at various Ages._ - - (_After Quetelet, p. 207._) - - Men Women - ────────────────────────── ────────────────────── - Age Total height Head Ratio Height Head† Ratio - m. m. m. m. - Birth 0·500 0·111 4·50 0·494 0·111 4·45 - 1 year 0·698 0·154 4·53 0·690 0·154 4·48 - 2 years 0·791 0·173 4·57 0·781 0·172 4·54 - 3 years 0·864 0·182 4·74 0·854 0·180 4·74 - 5 years 0·987 0·192 5·14 0·974 0·188 5·18 - 10 years 1·273 0·205 6·21 1·249 0·201 6·21 - 15 years 1·513 0·215 7·04 1·488 0·213 6·99 - 20 years 1·669 0·227 7·35 1·574 0·220 7·15 - 30 years 1·686 0·228 7·39 1·580 0·221 7·15 - 40 years 1·686 0·228 7·39 1·580 0·221 7·15 - - † A smooth curve, very similar to this, for the growth in - “auricular height” of the girl’s head, is given by Pearson, - in _Biometrika_, III, p. 141. 1904. - -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. - -In one of Quetelet’s tables (_supra_, 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. 19 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 {94} 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. - -[Illustration: Fig. 19. Ratio of stature in Man, to span of -outstretched arms. - -(From Quetelet’s data.)] - -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[126]. {95} - - _Trout (Salmo fario): proportionate growth of various organs._ - - (_From Jenkinson’s data._) - - Days Total 1st Ventral 2nd Breadth - old length Eye Head dorsal fin dorsal Tail-fin of tail - 49 100 100 100 100 100 100 100 100 - 63 129·9 129·4 148·3 148·6 148·5 108·4 173·8 155·9 - 77 154·9 147·3 189·2 (203·6) (193·6) 139·2 257·9 220·4 - 92 173·4 179·4 220·0 (193·2) (182·1) 154·5 307·6 272·2 - 106 194·6 192·5 242·5 173·2 165·3 173·4 337·3 287·7 - -While it is inequality of growth in _different_ directions that we can -most easily comprehend as a phenomenon leading to gradual change of -outward form, we shall see in another chapter[127] 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[128]. - -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[129]. - - Zone Increment - mm. - Apex 1·5 - 2nd 5·8 - 3rd 8·2 - 4th 3·5 - 5th 1·6 - 6th 1·3 - 7th 0·5 - 8th 0·3 - 9th 0·2 - 10th 0·1 - -{96} - -[Illustration: Fig. 20. Rate of growth in successive zones near the tip -of the bean-root.] - -The several values in this table lie very nearly (as we see by Fig. -20) 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. 20 (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, {97} -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. - - 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[130]. - - 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 - _y_ be the mean number of leaves per whorl, and _x_ the successional - number of the whorl from the root or main stem upwards, then - - _y_ = _A_ + _C_ log(_x_ − _a_), - - where _A_, _C_, and _a_ 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: - - _Number of leaves per whorl on the tertiary branches of Ceratophyllum._ - - Position of whorl 1 2 3 4 5 6 - Mean number of leaves 6·55 8·07 9·00 9·20 9·75 10·00 - Increment — 1·52 ·93 ·20 (·55) (·25) - -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. {98} 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 _W_/_L_^3 = _k_, where _k_ is a constant, to be determined -for each particular case. (_W_ and _L_ 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 _k_ a value near to unity.) - - _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, _p._ 107, 1908.) - - Size in cm. Weight in gm. _W_/_L_^3 × 10,000 _W_/_L_^3 (smoothed) - 23 113 92·8 — - 24 128 92·6 94·3 - 25 152 97·3 96·1 - 26 173 98·4 97·9 - 27 193 98·1 99·0 - 28 221 100·6 100·4 - 29 250 102·5 101·2 - 30 271 100·4 101·2 - 31 300 100·7 100·4 - 32 328 100·1 99·8 - 33 354 98·5 98·8 - 34 384 97·7 98·0 - 35 419 97·7 97·6 - 36 454 97·3 96·7 - 37 492 95·2 96·3 - 38 529 96·4 95·6 - 39 564 95·1 95·0 - 40 614 95·9 95·0 - 41 647 93·9 93·8 - 42 679 91·6 92·5 - 43 732 92·1 92·5 - 44 800 93·9 94·0 - 45 875 96·0 — - -{99} - -Now while this _k_ 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. - -[Illustration: Fig. 21. Changes in the weight-length ratio of Plaice, -with increasing size.] - -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 _k_ is by no means constant, but steadily increases to a -maximum, and afterwards slowly declines with the increasing size of the -fish (Fig. 21). 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 {100} apparent coincidence would be to determine -the coefficient _k_ 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. - -[Illustration: Fig. 22. Periodic annual change in the weight-length -ratio of Plaice.] - -A still more curious and more unexpected result appears when we compare -the values of _k_ for the same fish at different seasons of the -year[131]. When for simplicity’s sake (as in the accompanying table and -Fig. 22) we restrict ourselves to fish of one particular size, it is -not necessary to determine the value of _k_, 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 _k_ 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. {101} - - _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._ II, _p._ 92, 1909.) - - Average weight - in grammes _W_/_L_^3 × 100 _W_/_L_^3 (smoothed) - Jan. 2039 1·226 1·157 - Feb. 1735 1·043 1·080 - March 1616 0·971 0·989 - April 1585 0·953 0·967 - May 1624 0·976 0·985 - June 1707 1·026 1·005 - July 1686 1·013 1·037 - August 1783 1·072 1·042 - Sept. 1733 1·042 1·111 - Oct. 2029 1·220 1·160 - Nov. 2026 1·218 1·213 - Dec. 1998 1·201 1·215 - -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. - -―――――――――― - -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 (_k_) expressing -the ratio of weight to the cube of the length, and (3) a similar -coefficient (_k′_) 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 {102} 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 - - _Relations between the Weight and certain Linear Dimensions of the Ox. - (Data from Przibram, after Cornevin†.)_ - - _k_ = _k′_ = - Age in _W_, wt. _L_, length _H_, _W_/_L_^3 _W_/_H_^3 - months in kg. of back height _L_/_H_ × 10 × 10 - 0 37 ·78 ·70 1·114 ·779 1·079 - 1 55·3 ·94 ·77 1·221 ·665 1·210 - 2 86·3 1·09 ·85 1·282 ·666 1·406 - 3 121·3 1·207 ·94 1·284 ·690 1·460 - 4 150·3 1·314 ·95 1·383 ·662 1·754 - 5 179·3 1·404 1·040 1·350 ·649 1·600 - 6 210·3 1·484 1·087 1·365 ·644 1·638 - 7 247·3 1·524 1·122 1·358 ·699 1·751 - 8 267·3 1·581 1·147 1·378 ·677 1·791 - 9 282·8 1·621 1·162 1·395 ·664 1·802 - 10 303·7 1·651 1·192 1·385 ·675 1·793 - 11 327·7 1·694 1·215 1·394 ·674 1·794 - 12 350·7 1·740 1·238 1·405 ·666 1·849 - 13 374·7 1·765 1·254 1·407 ·682 1·900 - 14 391·3 1·785 1·264 1·412 ·688 1·938 - 15 405·9 1·804 1·270 1·420 ·692 1·982 - 16 417·9 1·814 1·280 1·417 ·700 2·092 - 17 423·9 1·832 1·290 1·420 ·689 1·974 - 18 423·9 1·859 1·297 1·433 ·660 1·943 - 19 427·9 1·875 1·307 1·435 ·649 1·916 - 20 437·9 1·884 1·311 1·437 ·655 1·944 - 21 447·9 1·893 1·321 1·433 ·661 1·943 - 22 464·4 1·901 1·333 1·426 ·676 1·960 - 23 480·9 1·909 1·345 1·419 ·691 1·977 - 24 500·9 1·914 1·352 1·416 ·714 2·027 - 25 520·9 1·919 1·359 1·412 ·737 2·075 - 26 534·1 1·924 1·361 1·414 ·750 2·119 - 27 547·3 1·929 1·363 1·415 ·762 2·162 - 28 554·5 1·929 1·363 1·415 ·772 2·190 - 29 561·7 1·929 1·363 1·415 ·782 2·218 - 30 586·2 1·949 1·383 1·409 ·792 2·216 - 31 610·7 1·969 1·403 1·403 ·800 2·211 - 32 625·7 1·983 1·420 1·396 ·803 2·186 - 33 640·7 1·997 1·437 1·390 ·805 2·159 - 34 655·7 2·011 1·454 1·383 ·806 2·133 - - † Cornevin, Ch., Études sur la croissance, _Arch. de - Physiol. norm. et pathol._ (5), IV, p. 477, 1892. - -{103} - -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 _W_/_H_^3 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 _k_ 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 _k_, as determined (at successive -epochs) for one dimension, are a measure of the _variability_ of the -others. - -―――――――――― - -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[132]. 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. -{104} - -[Illustration: Fig. 23. Variability of length of tail-forceps in a -sample of Earwigs. (After Bateson, _P. Z. S._ 1892, p. 588.)] - -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. 23); 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 _ages_, and therefore whose _magnitudes_, 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. 24, 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 {105} 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. - -[Illustration: Fig. 24. Variability of length of body in a sample of -Plaice.] - -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 _faster_, 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 {106} 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 _of different ages_, -then the phenomenon would be simplicity itself, and there would be no -more to be said about it[133]. - -―――――――――― - -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 -_total_ growth of the rat is prevented by underfeeding, the _form_ -continues to alter so that this increasing length of the tail is still -manifest[134]. {107} - -_Full-fed Rats._ - - Age in Length of Length of Total - weeks body (mm.) tail (mm.) length % of tail - 0 48·7 16·9 65·6 25·8 - 1 64·5 29·4 93·9 31·3 - 3 90·4 59·1 149·5 39·5 - 6 128·0 110·0 238·0 46·2 - 10 173·0 150·0 323·0 46·4 - - _Underfed Rats._ - - 6 98·0 72·3 170·3 42·5 - 10 99·6 83·9 183·5 45·7 - -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. - - -_The effect of temperature[135]._ - -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 {108} conditions. Réaumur was the first to shew, and -the observation was repeated by Bonnet[136], 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[137]. - - 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[138], 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 _essentially_ variable. - -The annexed diagram (Fig. 25), 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 {109} 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. - -―――――――――― - -[Illustration: Fig. 25. Relation of rate of growth to temperature in -certain plants. (From Sachs’s data.)] - -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 _n_ degrees the velocity varies as _x_^{_n_}, _x_ being -called the “temperature coefficient”[139] for the reaction in question. -{110} - -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 _fundamental_ -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 _Law of -Optimum_, laid down by Errera and by Sachs as a general principle of -physiology: namely that _every_ 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 _optimum_ condition, and its curve shews a definite _maximum_. 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 _duration_ 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. {111} 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[140]. - -―――――――――― - -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[141]: - - Temperature Growth in 48 hours - °C. mm. - 18·0 1·1 - 23·5 10·8 - 26·6 29·6 - 28·5 26·5 - 30·2 64·6 - 33·5 69·5 - 36·5 20·7 - -Let us write our formula in the form - - _V__{(_t_+_n_)}/_V__{_t_} = _x_^{_n_}. - -Then choosing two values out of the above experimental series (say the -second and the second-last), we have _t_ = 23·5, _n_ = 10, and _V_, -_V′_ = 10·8 and 69·5 respectively. - -Accordingly 69·5/10·8 = 6·4 = _x_^{10}. - -Therefore (log 6·4)/10, or ·0806 = log _x_. - -And, _x_ = 1·204 (for an interval of 1° C.). - -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. 26 that it is in -very fair accordance with the actual results of observation, _within -those particular limits_ of temperature to which the experiment is -confined. {112} - -For an experiment on _Lupinus albus_, quoted by Asa Gray[142], 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. - -[Illustration: Fig. 26. Relation of rate of growth to temperature in -Maize. Observed values (after Köppen), and calculated curve.] - - 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[143]. In Fig. 27 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. 27 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 {113} 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 _not_ 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. - -[Illustration: Fig. 27. Relation of rate of growth to temperature in -rootlets of Pea. (From Miss I. Leitch’s data.)] - -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 {114} irregular, -and sometimes (we might even say) misleading[144]. 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[145] have studied the rate of division in a -certain flagellate, _Chilomonas paramoecium_, 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. - -By means of this principle we may throw light on the apparently -complicated results of many experiments. For instance, Fig. 28 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[146]. - -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. - - At 20° At 10° - Stage III 2·0 6·5 days - Stage IV 2·7 8·1 days - Stage V 3·0 10·7 days - Stage VI 4·0 13·5 days - Stage VII 5·0 16·8 days - Total 16·7 55·6 days - -That is to say, the time taken to produce a given result at {115} 10° -was (on the average) somewhere about 55·6/16·7, or 3·33, times as long -as was required at 20°. - -[Illustration: 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.)] - -We may then put our equation again in the simple form, {116} - - _x_^{10} = 3·33. - - Or, 10 log _x_ = log 3·33 = ·52244. - - Therefore log _x_ = ·05224, - - and _x_ = 1·128. - -That is to say, between the intervals of 10° and 20° C., if it take -_m_ days, at a certain given temperature, for a certain stage of -development to be attained, it will take _m_ × 1·128^{_n_} days, when -the temperature is _n_ degrees less, for the same stage to be arrived -at. - -[Illustration: Fig. 29. Calculated values, corresponding to preceding -figure.] - -Fig. 29 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. {117} - -Karl Peter[147], 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 -_Q__{10}) as follows. - - Sphaerechinus 2·15, - Echinus 2·13, - Rana 2·86. - -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. - - Sphaerechinus; Segmentation _Q_^{10} = 2·29, - Later stages _Q_^{10} = 2·03. - Echinus; Segmentation _Q_^{10} = 2·30, - Later stages _Q_^{10} = 2·08. - Rana; Segmentation _Q_^{10} = 2·23, - Later stages _Q_^{10} = 3·34. - -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. - -―――――――――― - -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 {118} 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. - -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[148], of the height of German cadets, measured at half-yearly -intervals. - - _Growth in height of German military Cadets, in half-yearly periods._ - (_Daffner._) - - Height in cent. Increment in cm. - Number ─────────────────────── Winter Summer - observed Age October April October ½-year ½-year Year - 12 11–12 139·4 141·0 143·3 1·6 2·3 3·9 - 80 12–13 143·0 144·5 147·4 1·5 2·9 4·4 - 146 13–14 147·5 149·5 152·5 2·0 3·0 5·0 - 162 14–15 152·2 155·0 158·5 2·5 3·5 6·0 - 162 15–16 158·5 160·8 163·8 2·3 3·0 5·3 - 150 16–17 163·5 165·4 167·7 1·9 2·3 4·2 - 82 17–18 167·7 168·9 170·4 1·2 1·5 2·7 - 22 18–19 169·8 170·6 171·5 0·8 0·9 1·7 - 6 19–20 170·7 171·1 171·5 0·4 0·4 0·8 - -In the accompanying diagram (Fig. 30) 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. - -―――――――――― - -In the case of trees, the seasonal fluctuations of growth[149] admit -{119} 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[150] I have compiled certain mean values for growth in -the climate of Uruguay. - -[Illustration: Fig. 30. Half-yearly increments of growth, in cadets of -various ages. (From Daffner’s data.)] - - _Mean monthly increase in Girth of Evergreen and Deciduous Trees, - at San Jorge, Uruguay._ (_After C. E. Hall._) _Values expressed as - percentages of total annual increase._ - - Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. - Evergreens 9·1 8·8 8·6 8·9 7·7 5·4 4·3 6·0 9·1 11·1 10·8 10·2 - - Deciduous - trees 20·3 14·6 9·0 2·3 0·8 0·3 0·7 1·3 3·5 9·9 16·7 21·0 - -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. 31) 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 {120} -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. - -[Illustration: Fig. 31. Periodic annual fluctuation in rate of growth -of trees (in the southern hemisphere).] - -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 {121} 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[151], 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. - -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 _average annual growth_ 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 {123} 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. - -[Illustration: Fig. 32. Long-period fluctuation in rate of growth of -Arizona trees (smoothed in 100-year periods), from A.D. 1390–1490 to -A.D. 1810–1910.] - -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[152]. The point is a physiological one, and consequently of -little importance to us here[153]; 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. - -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, {124} probably, a -still better instance of it in the case of Alpine plants[154], 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. - - -_Osmotic factors in growth._ - -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. - -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[155]” -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 {125} beyond these limitations; for -the affinity with certain types of chemical reaction is plain, and has -been recognised by a great number of physiologists. - -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. 113) -on the amount of water taken up into the living cells, as well as on -the actual amount of chemical metabolism performed by them[156]. 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. - -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[157] that in certain eggs (e.g. those of the -little fish _Fundulus_) 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. - -Among many other observations of the same kind, those of -Bialaszewicz[158], on the early growth of the frog, are notable. He -shews that the growth of the embryo while still _within the {126} -vitelline membrane_ 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. - -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[159]. Fehling’s results are epitomised as -follows: - - Age in weeks 6 17 22 24 26 30 35 39 - Percentage of water 97·5 91·8 92·0 89·9 86·4 83·7 82·9 74·2 - - And the following illustrate Davenport’s results for the frog: - - Age in weeks 1 2 5 7 9 14 41 84 - Percentage of water 56·3 58·5 76·7 89·3 93·1 95·0 90·2 87·5 - -To such phenomena of osmotic balance as the above, or in other words to -the dependence of growth on the uptake of water, Höber[160] 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[161]. 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 {127} 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. - -The best-known case is the little “brine-shrimp,” _Artemia salina_, -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, _A. milhausenii_, inhabits the natron-lakes of -Egypt and Arabia, where, under the name of “loul,” or “Fezzan-worm,” -it is eaten by the Arabs[162]. 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 _milieu interne_ (as -Claude Bernard called them[163]), are no more salt than are those of -any ordinary crustacean or other animal, but contain only some 0·8 per -cent. of NaCl[164], while the _milieu externe_ 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. - -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 {128} described as the specific characters of _A. -milhausenii_, and of a still more extreme form, _A. köppeniana_; -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. 33). _Pari passu_ -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 - -[Illustration: 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.)] - -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. 33). In short, the density of the water is -so clearly a “function” of the specific {129} character, that we may -briefly define the species _Artemia_ (_Callaonella_) _Jelskii_, for -instance, as the Artemia of density 1000–1010 (NaCl), or the typical -_A. salina_, or _principalis_, 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 - -[Illustration: Fig. 34. Percentage ratio of length of abdomen to -cephalothorax in brine-shrimps, at various salinities. (After Abonyi.)] - -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 {130} 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 _at equal densities_, or -at “_isotonic_” 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. - -While Höber and others[165] 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[166]. - - -_Growth and catalytic action._ - -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 {131} occur in the organism, part -of whose energies are devoted to the continual bringing-up of fresh -supplies. - -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[167]. 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 _growth_, 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 _autocatalysis_. {132} - -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[168]. “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.” - -Very soon afterwards a similar suggestion was made by Loeb[169], in -connection with the synthesis of _nuclein_ 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. - -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[170]: 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[171]. - -As F. F. Blackman has pointed out, the multiplication of an organism, -for instance the prodigiously rapid increase of a bacterium, {133} -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[172], 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. - -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, _ipso -facto_, 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 {134} and specialised functions that of cell-multiplication -tends to fall into abeyance[173]. - -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[174] akin to those which the physical -chemist finds applicable to his autocatalytic reactions. This has -been diligently attempted by various writers[175]; 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. - -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. - -―――――――――― - -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 _diet_, of overfeeding and underfeeding, would present -itself for discussion[176]. But without attempting to open up this -large subject, we may say a {135} 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. - -Thus lecithin has been shewn by Hatai[177], Danilewsky[178] 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. - -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. - -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 {136} 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. - -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[179]. No -other organic food was found to produce the same effect; but since the -thyroid gland is known to contain iodine[180], 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. - -―――――――――― - -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[181] 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[182], comme ‘le dernier -réduit de la force vitale.’ ” - -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 {137} 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[183]. 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: _donec ad interitum genus id natura redegit_[184]. - -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, - - “et quid quaeque queant per foedera naturai - quid porro nequeant.” - -Then, at last, we are entitled to use the customary metaphor, and to -see in natural selection an inexorable force, whose function {138} is -not to create but to destroy,—to weed, to prune, to cut down and to -cast into the fire[185]. - - -_Regeneration, or growth and repair._ - -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 _vis -medicatrix_ 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[186]. - -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. - -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[187]. - -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 {139} -(11·5 mm.) was cut off, and the amounts regenerated in successive -periods are shewn as follows: - - Days after operation 3 7 10 14 18 24 30 - (1) Amount regenerated in mm. 1·4 3·4 4·3 5·2 5·5 6·2 6·5 - (2) Increment during each period 1·4 2·0 0·9 0·9 0·3 0·7 0·3 - (3) (?) Rate per day during - each period 0·46 0·50 0·30 0·25 0·07 0·12 0·05 - -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 _velocity_ 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 _at_ 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 _velocities_ -(Table, p. 141). (The more accurate and strictly correct method would -be to draw successive tangents to the curve.) - -In our curve of growth (Fig. 35) 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 -{140} 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 - -[Illustration: Fig. 35. Curve of regenerative growth in tadpoles’ -tails. (From M. L. Durbin’s data.)] - -have been an intervening “latent period,” during which no growth - -[Illustration: Fig. 36. Mean daily increments, corresponding to Fig. -35.] - -{141} - -occurred, between the time of injury and the first measurement -of regenerative growth; 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. - -The curve already drawn (Fig. 35) illustrates, as we have seen, the -relation of length to time, i.e. _L_/_T_ = _V_. The second (Fig. 36) -represents the rate of change of velocity; it sets _V_ against _T_; - - _The foregoing table, extended by graphic interpolation._ - - Total Daily - Days increment increment Logs of do. - 1 — - — — - 2 — - — — - 3 1·40 - ·60 1·78 - 4 2·00 - ·52 1·72 - 5 2·52 - ·45 1·65 - 6 2·97 - ·43 1·63 - 7 3·40 - ·32 1·51 - 8 3·72 - ·30 1·48 - 9 4·02 - ·28 1·45 - 10 4·30 - ·22 1·34 - 11 4·52 - ·21 1·32 - 12 4·73 - ·19 1·28 - 13 4·92 - ·18 1·26 - 14 5·10 - ·17 1·23 - 15 5·27 - ·13 1·11 - 16 5·40 - ·14 1·15 - 17 5·54 - ·13 1·11 - 18 5·67 - ·11 1·04 - 19 5·78 - ·10 1·00 - 20 5·88 - ·10 1·00 - 21 5·98 - ·09 ·95 - 22 6·07 - ·07 ·85 - 23 6·14 - ·07 ·84 - 24 6·21 - ·08 ·90 - 25 6·29 - ·06 ·78 - 26 6·35 - ·06 ·78 - 27 6·41 - ·05 ·70 - 28 6·46 - ·04 ·60 - 29 6·50 - ·03 ·48 - 30 6·53 - -{142} - -and _V_/_T_ or _L_/_T_^2, represents (as we have learned) the -_acceleration_ of growth, this being simply the “differential -coefficient,” the first derivative of the former curve. - -[Illustration: Fig. 37. Logarithms of values shewn in Fig. 36.] - -Now, plotting this acceleration curve from the date of the first -measurement made three days after the amputation of the tail (Fig. -36), 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. - -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 -_V_ and _T_) vary inversely as one another. If we take the logarithms -of the velocities (as given in the table) and plot them against time -(Fig. 37), we see that they fall, approximately, into a straight line; -and if this curve be plotted on the {143} 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. - -Had the angle been 45° (tan 45° = 1), the curve would have been -actually a rectangular hyperbola, with _V_ _T_ = constant. As it is, -we may assume, provisionally, that it belongs to the same family -of curves, so that _V_^{_m_} _T_^{_n_}, or _V_^{_m_/_n_} _T_, or -_V_ _T_^{_n_/_m_}, are all severally constant. In other words, the -velocity varies inversely as some power of the time, or _vice versa_. -And in this particular case, the equation _V_ _T_^2 = 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. - -[Illustration: Fig. 38. Rate of regenerative growth in larger tadpoles.] - -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 {144} to this -question-depends on whether, in the days following the first actual -measurement, we can or cannot detect a daily _increment_ 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.)[188]. - - Days 3 5 7 10 12 14 17 24 28 31 - Increment 0·86 2·15 3·66 5·20 5·95 6·38 7·10 7·60 8·20 8·40 - -Or, by graphic interpolation, - - Total Daily - Days increment do. - 1 ·23 ·23 - 2 ·53 ·30 - 3 ·86 ·33 - 4 1·30 ·44 - 5 2·00 ·70 - 6 2·78 ·78 - 7 3·58 ·80 - 8 4·30 ·72 - 9 4·90 ·60 - 10 5·29 ·39 - 11 5·62 ·33 - 12 5·90 ·28 - 13 6·13 ·23 - 14 6·38 ·25 - 15 6·61 ·23 - 16 6·81 ·20 - 17 7·00 ·19 etc. - -The acceleration curve is drawn in Fig. 39. - -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. 38, -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. - -―――――――――― - -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 {145} 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/_T_^2. 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. - -[Illustration: Fig. 39. Daily increment, or amount regenerated, -corresponding to Fig. 38.] - -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 {146} 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. - -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[189], 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. - -―――――――――― - -It is a very general rule, though apparently not a universal one, that -regeneration tends to fall somewhat short of a _complete_ 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[190]: “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.” - -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 {147} been studied by -Lillie[191], 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[192] 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. - -A number of phenomena connected with the linear rate of regeneration -are illustrated and epitomised in the accompanying diagram (Fig. 40), -which I have constructed from certain data given by Ellis in a paper -on the relation of the amount of tail _regenerated_ to the amount -_removed_, 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 -{148} - -[Illustration: 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.)] - - _The Rate of Regenerative Growth in Tadpoles’ Tails._ - - (_After M. M. Ellis, J. Exp. Zool._ VII, _p._ 421, 1909.) - - Body Tail Amount Per cent. % amount regenerated in days - length length removed of tail ──────────────────────────── - Series† mm. mm. mm. removed 3 6 9 12 15 18 32 - _O_ 39·575 25·895 3·2 12·36 13 31 44 44 44 44 44 - _P_ 40·21 26·13 5·28 20·20 10 29 40 44 44 44 44 - _R_ 39·86 25·70 10·4 40·50 6 20 31 40 48 48 48 - _S_ 40·34 26·11 14·8 56·7 0 16 33 39 45 48 48 - - † Each series gives the mean of 20 experiments. - -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 {149} curve indicates the -time taken to regenerate _n_ 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. - -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. - -―――――――――― - -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[193]. - -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[194]. - -The one claw is the larger because it has grown the faster; {150} 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. - -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 _vis -revulsionis_ of Haller, to which we have already referred. - -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 (_A_) -be removed, there are certain cases, e.g. the common lobster, where it -is directly regenerated. In other cases, e.g. Alpheus[195], the other -claw (_B_) assumes the size and form of that which was amputated, while -the latter regenerates itself in the form of the other and weaker one; -_A_ and _B_ have apparently changed places. In a third case, as in the -crabs, the _A_-claw regenerates itself as a small or _B_-claw, but the -_B_-claw remains for a time unaltered, though slowly and in the course -of repeated moults it later on assumes the large and heavily toothed -_A_-form. - -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 {151} 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 _A_, _A_ begins to grow from zero, with a high “regenerative” -velocity; while _B_, 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, _A_ has equalled, outstripped, or still fallen short of the -magnitude of _B_. - -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. - -―――――――――― - -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 {152} 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. - - -CONCLUSION AND SUMMARY. - -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. - -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 _tension_ 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 {153} 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. - -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[196] 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. - - Delage[197] 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. {154} - -We may summarise, as follows, the main results of the foregoing -discussion: - -(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. - -(2) Rate of growth is subject to definite laws, and the velocities in -different directions tend to maintain a _ratio_ 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. - -(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. - -(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. - -(5) The rate of growth is directly affected by temperature, and by -other physical conditions. - -(6) It is markedly affected, in the way of acceleration or retardation, -at certain physiological epochs of life, such as birth, puberty, or -metamorphosis. - -(7) Under certain circumstances, growth may be _negative_, the organism -growing smaller: and such negative growth is a common accompaniment of -metamorphosis, and a frequent accompaniment of old age. - -(8) The phenomenon of regeneration is associated with a large temporary -increase in the rate of growth (or “_acceleration_” of growth) of the -injured surface; in other respects, regenerative growth is similar to -ordinary growth in all its essential phenomena. - -―――――――――― - -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 {155} 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. - -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. - -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. - -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. - -{156} - - - - -CHAPTER IV - -ON THE INTERNAL FORM AND STRUCTURE OF THE CELL - - -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[198].” And likewise, long afterwards, Giard -contemplated a science of _morphodynamique_,—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[199].” - -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. - -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 {157} 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. - -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[200]. -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. - -Whitman[201], 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 “_caryokinesis_” is -_motion_,—“motion viewed as an exponent of forces residing in, or -acting upon, the nucleus. It regards the nucleus as a _seat of energy, -which displays itself in phenomena of motion_[202].” - -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 _activities_ of the cell. And in all such -hypotheses as that of “pangenesis,” in all the theories which attribute -specific properties to micellae, {158} idioplasts, ids, or other -constituent particles of protoplasm or of the cell, we are apt to fall -into the error of attributing to _matter_ what is due to _energy_ and -is manifested in force: or, more strictly speaking, of attributing to -material particles individually what is due to the energy of their -collocation. - -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. - -But if we speak, as Weismann and others speak, of an “hereditary -_substance_,” 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.” - -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[203] from phenomena would be a very great step in philosophy, -though the causes of these principles were not yet discovered.” The -_things_ which we see in the cell are less important than the _actions_ -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 _life_. It may be that in this -way we shall in time draw nigh to the recognition of a specific and -ultimate residuum. {159} - -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. - -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 _homunculus_, 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[204].” - -Moreover, the microscope seemed to substantiate the idea (which we may -trace back to Leibniz[205] and to Hobbes[206]), 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. - -But no microscopical examination of a stick of sealing-wax, no study -of the material of which it is composed, can enlighten {160} 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. - -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[207]. 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[208].” If we cannot assume differences in structure, we must -assume differences in _motion_, that is to say, in _energy_. 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.” {161} - -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. - -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 _work_, 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[209].” -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. - -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, {162} 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, _ipso facto_, 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[210] has lately reminded -us), it possesses “_energia_[211].” - -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. - -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[212]; another, like Leduc, may shew -us how in {163} many of their most striking features they may be -admirably simulated by the diffusion of salts in a colloid medium; -others again, like Gallardo[213] and Hartog, and Rhumbler (in his -earlier papers)[214], insist on their resemblance to the phenomena of -electricity and magnetism[215]; while Hartog believes that the force -in question is only analogous to these, and has a specific identity of -its own[216]. All these conflicting views are of secondary importance, -so long as we seek only to account for certain _configurations_ 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[217]”; and this is as much -as to say that the phenomenon, {164} though physical in the concrete, -is in the abstract purely mathematical, and in its very essence is -neither more nor less than _a property of three-dimensional space_. - -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. - -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. - -―――――――――― - -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[218]. - -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 _form_ (though there -apparently is of _size_) between the nucleus and the “cytoplasm,” or -main body of the cell. So Whitman[219] 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 {165} 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[220]. 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[221]. The case of the spherical nucleus is closely akin -to the spherical form of the yolk within the bird’s egg[222]. But if -the substance of the cell acquire a greater solidity, as for instance -in a muscle {166} 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[223]. - -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. - -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. - -The morphologist is accustomed to speak of a “polarity” of {167} -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 _morphological_ “polarity” is accompanied by, and is but the -outward expression (or part of it) of a true _dynamical_ 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. - -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. -97). - -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[224], -if we spread a layer of salt solution over a level {168} 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 _dynamical_, 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. - -―――――――――― - -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. - -Division of the cell is essentially accompanied, and preceded, by a -change from radial or monopolar to a definitely bipolar polarity. - -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 {169} 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 - -[Illustration: Fig. 41. Caryokinetic figure in a dividing cell (or -blastomere) of the Trout’s egg. (After Prenant, from a preparation by -Prof. P. Bouin.)] - -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 {170} 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[225], -and are only prevented by the friction of the surrounding medium from -approaching and congregating around the adjacent poles. - -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[226]. 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. - -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. - -Within its spherical outline (Fig. 42), it contains an “alveolar” -{171} 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[227], 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. {172} - -[Illustration: Fig. 42.] - -[Illustration: Fig. 43.] - - 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 _living cell_, 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 _structure_ 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[228] have altogether denied the existence in the living - cell-protoplasm of a network or alveolar “foam”; others[229] 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.” - - 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[230]. {173} - -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[231]. - -[Illustration: Fig. 44.] - -[Illustration: Fig. 45.] - -1. The chromatin, which to begin with was distributed in granules on -the otherwise achromatic reticulum (Fig. 42), concentrates to form a -skein or _spireme_, which may be a continuous thread from the first -(Figs. 43, 44), or from the first segmented. In any case it divides -transversely sooner or later into a number of _chromosomes_ (Fig. 45), -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 _in situ_. - -2. Meanwhile, the deeply staining granule (here extra-nuclear), known -as the _centrosome_, has divided in two. The two resulting granules -travel to opposite poles of the nucleus, and {174} there each becomes -surrounded by a system of radiating lines, the _asters_; immediately -around the centrosome is a clear space, the _centrosphere_ (Figs. -43–45). Between the two centrosomes with their asters stretches a -bundle of achromatic fibres, the _spindle_. - -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. 45, 46). These -chromosomes now arrange themselves midway between the poles of the -spindle, where they form what is called the _equatorial plate_ (Fig. -47). - -[Illustration: Fig. 46.] - -[Illustration: Fig. 47.] - -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. 48). - -5. The halves of the split chromosomes now separate from one another, -and travel in opposite directions towards the two poles (Fig. 49). 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. - -6. The daughter chromosomes, arranged now in two groups, become closely -crowded in a mass near the centre of each aster {175} (Fig. 50). -They fuse together and form once more an alveolar reticulum and may -occasionally at this stage form another spireme. - -[Illustration: Fig. 48.] - -[Illustration: Fig. 49.] - -A boundary or surface wall is now developed round each reconstructed -nuclear mass, and the spindle-fibres disappear (Fig. 51). The -centrosome remains, as a rule, outside the nucleus. - -[Illustration: Fig. 50.] - -[Illustration: Fig. 51.] - -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. 50, 51). This is -more conspicuous in plant-cells. {176} - -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. - -―――――――――― - -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†. - -† The reference numbers in the following account refer to the -paragraphs and figures of the preceding summary of visible nuclear -phenomena. - -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 _within -the range_ 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 _sui generis_, 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. - -―――――――――― - -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 {177} 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. - -The _quality_ 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 _permeable_ 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. _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_[232]. 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. - -Now, in the field of force whose opposite poles are marked by {178} -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. 44, 45). - -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[233]: 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. 43, 44). - -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 _object_) 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 _round_ and the way _through_ 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. 44 and 45 represent two so-called “types,” -of a phase which follows that represented in Fig. 43. According to -the usual descriptions (and in particular to Professor {179} E. B. -Wilson’s[234]), we are told that, in such a case as Fig. 44, 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. 45, 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. 46, which represents the next succeeding phase -of Fig. 45. This condition, of Fig. 46, 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. - -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 _stronger_ 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 _weaker_ 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 -_behaves_ 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. 47, 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. 47). 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, {180} -in a peripheral equatorial circle (Figs. 48, 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.” - -[Illustration: Fig. 52. Chromosomes, undergoing splitting and -separation. (After Hatschek and Flemming, diagrammatised.)] - -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. 52). 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 -_Ascaris megalocephala_, Lillie[235] {181} 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 to account for -the curious symmetry of these so-called “tetrads.” The remarkable -_annular_ chromosomes, shewn in Fig. 53, 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. - -[Illustration: Fig. 53. Annular chromosomes, formed in the -spermatogenesis of the Mole-cricket. (From Wilson, after Vom Rath.)] - -―――――――――― - -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, {182} 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 _assert_ 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. - -As we have begun by supposing that the nuclear, or chromosomal -matter differs in _permeability_ 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 _strength of -the field_. - -[Illustration: Fig. 54.] - -In Fig. 54, 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 _loci of constant -magnitude of force_ are shewn by dotted lines. - -Let us now consider a body whose permeability (µ) depends on the -strength of the field _F_. At two field-strengths, such as _F_{a}_, -_F_{b}_, let the permeability of the body be equal to that of the -{183} medium, and let the curved line in Fig. 55 represent generally -its permeability at other field-strengths; and let the outer and -inner dotted curves in Fig. 54 represent respectively the loci of the -field-strengths _F_{b}_ and _F_{a}_. 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. - -[Illustration: Fig. 55.] - -The locus _F_{b}_ is therefore a locus of stable position, towards -which the body tends to move; the locus _F_{a}_ is a locus of unstable -position, from which it tends to move. If the body were placed across -_F_{a}_, it might be torn asunder into two portions, the split -coinciding with the locus _F_{a}_. - -Suppose a number of such bodies to be scattered throughout the medium. -Let at first the regions _F_{a}_ and _F_{b}_ 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 _F_{a}_ -and _F_{b}_ are meanwhile situated somewhat farther from the axis than -in our figure, that (for instance) _F_{a}_ is situated where we have -drawn _F_{b}_, and that _F_{b}_ is still further out. The bodies then -tend towards the poles; but the tendency may be very small if, in Fig. -55, the curve and its intersecting straight line do not diverge very -far from one another beyond _F_{a}_; in other {184} words, if, when -situated in this region, the permeability of the bodies is not very -much in excess of that of the medium. - -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 _F_{a}_, -_F_{b}_, will close in to nearer relative distances from the poles. In -doing so, when the locus _F_{a}_ 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 _F_{a}_ is in -some median position (Fig. 56). - -[Illustration: Fig. 56.] - -When the locus _F_{a}_ has passed entirely over the body, the body -tends to move towards regions of weaker force; but when, in turn, the -locus _F_{b}_ 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[236]. - -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, {185} 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 _modus operandi_, there are physical -conditions and distributions of force which _could_ 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[237].” - -―――――――――― - -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[238]. 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 {186} is drawn in opposite directions by -equal forces, and so tends to remain undisturbed, in the form of an -“equatorial plate.” - -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 -_oppositely_ pulsating bodies are the same as those which are produced -by _opposite_ magnetic poles: though in the former case repulsion, and -in the latter case attraction, takes place between the two poles[239]. - -[Illustration: Fig. 57. Artificial caryokinesis (after Leduc), for -comparison with Fig. 41, p. 169.] - -―――――――――― - -But to return to our general discussion. - -While it can scarcely be too often repeated that our enquiry is not -directed towards the solution of physiological problems, save {187} -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[240],” 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[241]. 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, {188} Lillie comes easily to the conclusion that “electrical -theories of mitosis are entitled to more careful consideration than -they have hitherto received.” - -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[242], 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 (_Fundulus_, 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[243]; just as other physical rhythms, for instance in the -production of CO_{2}, 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. 58, 59). - -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. - -―――――――――― - -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 {189} -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 -_Euglypha_[244] 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. {190} - -[Illustration: Fig. 58. Final stage in the first segmentation of the -egg of Cerebratulus. (From Prenant, after Coe.)[245]] - -[Illustration: Fig. 59. Diagram of field of force with two similar -poles.] - -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[246]. - -―――――――――― - -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. - -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 {191} thus could we approach a comprehension of the -balance of forces which cohesion, friction, capillarity and electrical -distribution combine to set up. - -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. - -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[247], 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[248]. -This streaming movement would suggest, as Robertson has pointed out, -a _diminution_ 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 {192} 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[249] 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. - -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[250]. 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 -_adsorption_ 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. - -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[251]. -And we may simply conclude the {193} 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. - -―――――――――― - -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. - -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[252]. 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 _must_ 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 {194} 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 -_projection_ 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. - -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[253]. - -―――――――――― - -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 {195} 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[254].” 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 _Chaetopterus_. In one and the same species of worm (_Ascaris -megalocephala_), 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[255]. 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 {196} 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. - -When we attempt to separate our purely morphological or “purely -embryological” studies from physiological and physical investigations, -we tend _ipso facto_ 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 _phylogeny_. -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. - -―――――――――― - -The cell, which Goodsir spoke of as a “centre of force,” is in {197} -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 _Actinophrys_ or _Actinosphaerium_ -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. - -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 _interaction_ -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 {198} 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 _initiated_, 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 -_vereint_ 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.” - -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. - -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 {199} 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. - -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 _reciprocal action_ of the single elementary parts[256].” - -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[257].” - -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[258]. - -Whitman, Adam Sedgwick[259], 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[260]. As Goethe said long ago, “Das -lebendige ist zwar in Elemente {200} zerlegt, aber man kann es aus -diesen nicht wieder zusammenstellen und beleben;” the dictum of the -_Cellularpathologie_ being just the opposite, “Jedes Thier erscheint -als eine Summe vitaler Einheiten, von denen _jede den vollen Charakter -des Lebens an sich trägt_.” - -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[261].” 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[262].” -And on the botanical side De Bary has summed up the matter in an -aphorism, “Die Pflanze bildet Zellen, nicht die Zelle bildet Pflanzen.” - -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[263].” - -{201} - - - - -CHAPTER V - -THE FORMS OF CELLS - - -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 _partakes of fluidity_, and enables the colloid to become -a vehicle for liquid diffusion, like water itself[264].” 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) _diffusion_ and easy _convection_ -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. - -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 {202} 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. - -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[265]. 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 _planes_[266] (in all cases where -by the limitation of material, surfaces _must_ occur); and where we -have planes, straight edges and solid angles must obviously also occur; -and, if equilibrium is {203} 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. - -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[267], 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 _graded_ and harmonious distribution of forces, -to assume everywhere a rounded and more or less bubble-like external -form[268]. 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. - -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, {204} in his great work on -so-called Fluid Crystals[269], 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[270]. 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[271]. - -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 _external_ 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. - -But the rounded contours that are assumed and exhibited by {205} 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 _surface tension_: of which indeed we have spoken in -the preceding chapter, but which we must now examine with greater care. - -―――――――――― - -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. - -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[272]. 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 _surface_ 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 {206} 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. - -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. - -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 _surface-shrinkage_ exhibiting itself as a -_surface-tension_. This is a sufficient description of the phenomenon -in cases where a portion of liquid is subject to no other than _its -own molecular forces_, 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. - -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 _curved_ -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 {207} pressure is of the magnitude -of thousands of atmospheres,—a conclusion which is supported by other -physical considerations. - -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 _surface energy_. -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. - -Let _E_ be the whole potential energy of a mass _M_ of liquid; let -_e__{0} be the energy per unit mass of the interior liquid (we may call -it the _internal energy_); and let _e_ be the energy per unit mass for -a layer of the skin, of surface _S_, of thickness _t_, and density -ρ (_e_ being what we call the _surface energy_). It is obvious that -the total energy consists of the internal _plus_ the surface energy, -and that the former is distributed through the whole mass, minus its -surface layers. That is to say, in mathematical language, - - _E_ = (_M_ − _S_ ⋅ Σ _t_ ρ) _e__{0} + _S_ ⋅ Σ _t_ ρ _e_. - -But this is equivalent to writing: - - = _M_ _e__{0} + _S_ ⋅ Σ _t_ ρ(_e_ − _e__{0}); - -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 _e_ and _e__{0}, that is to say to the difference between the -unit values of the internal and the surface energy. {208} - -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 _potential energy_ 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. - -―――――――――― - -We see in our last equation that the term _M_ _e__{0} 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, _S_, or (2) by -a tendency towards equality of _e_ and _e__{0}, that is to say by a -diminution of the specific surface energy, _e_. - -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 _modus operandi_ 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. - -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, -_ipso facto_, 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 “_lineae curvae maximi -minimive proprietate gaudentes_” 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. {209} - -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 _symmetry_. 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[273]. “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.” - -―――――――――― - -As we proceed in our enquiry, and especially when we approach the -subject of _tissues_, 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. - -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[274]. 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 {210} 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[275] 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 _physiological_ 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. - -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 _outer_ 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[276]. - -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 {211} 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[277], 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[278]. 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 {212} 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[279]. 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[280]. - -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 {213} 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.” - -[Illustration: Fig. 60.] - -In a budding yeast-cell (Fig. 60), 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. - -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 -_incipient_ cell-wall retains with but little impairment the properties -of a liquid film[281], 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 _process_ 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 {214} 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. - -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. - -―――――――――― - -Let us now make some enquiry regarding the various forms {215} 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[282]: 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. - -We have already learned, as a fundamental law of surface-tension -phenomena, that a liquid film _in equilibrium_ 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 {216} surface which can be drawn continuously across -the uneven boundary. - -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[283]) -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[284]. - -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 _curved_ 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 (_p_) varies directly -as the tension (_T_), and inversely as the radius of curvature (_R_): -that is to say, _p_ = _T_/_R_, per unit of surface. {217} - -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 _p_ is equal to -_T_/_R_ + _T_/_R′_ or _p_ = _T_(1/_R_ + 1/_R′_)[285]. - -And if in addition to the pressure _p_, which is due to surface -tension, we have to take into account other pressures, _p′_, _p″_, -etc., which are due to gravity or other forces, then we may say that -the _total pressure_, _P_ = _p′_ + _p″_ + _T_(1/_R_ + 1/_R′_). 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. - -Our equation is an equation of equilibrium. The resistance to -compression,—the pressure outwards,—of our fluid mass, is a constant -quantity (_P_); the pressure inwards, _T_(1/_R_ + 1/_R′_), is also -constant; and if (unlike the case of the mobile amoeba) the surface -be homogeneous, so that _T_ is everywhere equal, it follows that -throughout the whole surface 1/_R_ + 1/_R′_ = _C_ (a constant). - -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/_R_ + 1/_R′_ = _C_, in other -words a surface which has the same _mean curvature_ 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. - -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 {218} _surfaces -of revolution_ (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. - -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[286], in 1841) that -the plane curves by whose rotation they are generated are themselves -generated as “roulettes” of the conic sections. - -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. - -[Illustration: Fig. 61.] - -If we imagine an ellipse so to roll over a line, either of its -foci will describe a sinuous or wavy line (Fig. 61B) 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 _unduloid_. 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 _cylinder_. 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 _spheres_. If as before -one axis of the ellipse vanish, but the other be infinitely long, then -the curve described by the rotation {219} 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 _plane_. 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 _catenoid_. - -Lastly, but this is a little more difficult to imagine, we have the -case of the hyperbola. - -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 _nodoid_. - -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. - -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[287]. -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 {220} a little salt to the lower layers of water, the -drop may be made to float or rest upon the denser liquid. - -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 _negative_ 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 -(Fig. 62A); when on the other hand we stop the end and blow, we again -convert the cylinder into an unduloid (B), 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. - -[Illustration: Fig. 62.] - -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. - -But before we go further in this enquiry, it will be necessary to -consider, to some small extent at least, the _curvatures_ of the six -different surfaces, that is to say, to determine what modification -{221} 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 _pressures_ 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. - -We have seen that, in our general formula, the expression -1/_R_ + 1/_R′_ = _C_, 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 _C_ may have any value, positive, negative, -or nil. - -In the case of the plane, where _R_ and _R′_ are both infinite, it is -obvious that 1/_R_ + 1/_R′_ = 0. The expression therefore vanishes, -and our dynamical equation of equilibrium becomes _P_ = _p_. 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. - -In the case of the sphere, the radii are all equal, _R_ = _R′_; they -are also positive, and _T_ (1/_R_ + 1/_R′_), or 2 _T_/_R_, is a -positive quantity, involving a positive pressure _P_, on the other side -of the equation. - -In the cylinder, one radius of curvature has the finite and positive -value _R_; but the other is infinite. Our formula becomes _T_/_R_, -to which corresponds a positive pressure _P_, 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 _R_. - -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, _R_ = −_R′_; and the expression becomes - - (1/_R_ + 1/_R′_) = (1/_R_ − 1/_R_) = 0; - -in other words, the surface, as in the case of the plane, has _no -{222} curvature_, 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. - -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/_R_ + 1/_R′_) -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 _P_. 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. - -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. - -[Illustration: Fig. 63.] - -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. 63, 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 {223} modified, and the -cylindrical drop will assume the form of an unduloid (Fig. 64 A, B), -with its dilated portion below or above, - -[Illustration: Fig. 64.] - -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 -_nil_. 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. 65B. This figure is a catenoid, which, as - -[Illustration: Fig. 65.] - -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 _negative_ -internal pressure; and thereupon the peripheral catenoid surface alters -its form (perhaps, on this small scale, imperceptibly), and becomes a -portion of a nodoid (Fig. 65A). {224} It represents, in fact, that -portion of the nodoid, which in Fig. 66 lies between such points as O, -P. While it is easy to - -[Illustration: Fig. 66.] - -draw the outline, or meridional section, of the nodoid (as in Fig. -66), 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. - -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. 66 lies between such limits as M and N. 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 {225} of our nodoid curve in Fig. 66 from O, P, the surface -concerned in the former case, to M, N, 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. - -―――――――――― - -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 _minimae areae_; 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. - -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 _dimensions_ of our figures, within which limits -equilibrium is stable but at which it becomes unstable, and above which -it {226} 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. - -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 (_T_) to be everywhere -identical; and it follows, since the internal fluid-pressure is also -everywhere identical, that the expression (1/_R_ + 1/_R′_) for the -cylinder is equal to the corresponding expression, which we may call -(1/_r_ + 1/_r′_), in the case of the terminal spheres. But in the -cylinder 1/_R′_ = 0, and in the sphere 1/_r_ = 1/_r′_. Therefore our -relation of equality becomes 1/_R_ = 2/_r_, or _r_ = 2 _R_; that is to -say, the sphere in question has just twice the radius of the cylinder -of which it forms a cap. - -[Illustration: Fig. 67.] - -And if _Ob_, the radius of the sphere, be equal to twice the radius -(_Oa_) of the cylinder, it follows that the angle _aOb_ is an angle of -60°, and _bOc_ is also an angle of 60°; that is to say, the arc _bc_ -is equal to (1/3) π. 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, _de_, - - = _Ob_ − _ab_ = _R_ (2 − √3) = 0·27 _R_, - -where _R_ 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 {227} 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. - -―――――――――― - -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 - -[Illustration: Fig. 68.] - -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 _L_ = 2π_R_, or when the ratio of length to diameter is -represented by π. - -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 {228} -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. - -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; {229} 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 (Fig. 65A). - -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. - -―――――――――― - -While the figures of equilibrium which are at the same time surfaces -of revolution are only six in number, there is an infinite {230} -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. - -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, _Bodo gracilis_, etc.), and which accordingly -it is of interest to us to be able to recognise as a surface of minimal -area or constant curvature. - -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 {231} 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 -_cell_,—considered, as with certain limitations we may legitimately -consider it, as a liquid drop or liquid vesicle; the conformation of a -_tissue_ 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. - -―――――――――― - -In a spider’s web we find exemplified several of the principles {232} -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[288]; 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 {233} effect of the jerk -being to disturb and destroy the unstable equilibrium of the viscid -cylinder[289]. Another very curious phenomenon here presents itself. - -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 _Epeira_, very often with -beautiful regularity,—which {234} 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[290]. - -[Illustration: Fig. 69. Hair of _Trianea_, in glycerine. (After -Berthold.)] - -The little delicate beads which stud the long thin pseudopodia of a -foraminifer, such as _Gromia_, or which in like manner appear upon the -cylindrical film of protoplasm which covers the long radiating spicules -of _Globigerina_, 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, -{235} 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[291]. -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[292]. - -[Illustration: Fig. 70. Phases of a Splash. (From Worthington.)] - -―――――――――― - -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. - -In some of Mr Worthington’s most beautiful experiments on {236} -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. 70)[293]. 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. -71, _a_, _b_). 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. - -[Illustration: Fig. 71. A breaking wave. (From Worthington.)] - -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[294]. 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 {237} 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. 72), 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. - -[Illustration: Fig. 72. Calycles of Campanularian zoophytes. (A) -_C. integra_; (B) _C. groenlandica_; (C) _C. bispinosa_; (D) _C. -raridentata_.] - -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. 75), 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. - -The phenomena of an ordinary liquid splash are so swiftly {238} -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. - -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 _closed_ -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 -_open_ surface, or solid, of revolution. It is clear, at the same -time, that the two series of problems are closely akin; for the {239} -glass-blower can make most things that the potter makes, by cutting -off _portions_ 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. - -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[295], 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. - -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. - -―――――――――― - -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 {240} 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. - -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 _time_ 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. - -―――――――――― - -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. {241} 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. - -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. - -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[296]. {242} - -Our full formula of equilibrium, or equation to an elastic surface, -is _P_ = _p_{e}_ + (_T_/_R_ + _T′_/_R′_), where _P_ is the internal -pressure, _p_{e}_ any extraneous pressure normal to the surface, -_R_, _R′_ the radii of curvature at a point, and _T_, _T′_, the -corresponding tensions, normal to one another, of the envelope. - -Now in any given form which we are seeking to account for, _R_, _R′_ -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 _P_ was simply the uniform hydrostatic pressure, equal in -all directions, of a body of liquid; we have assumed that the tension -_T_ was simply due to surface-tension in a homogeneous liquid film, -and was therefore equal in all directions, so that _T_ = _T′_; and we -have only dealt with surfaces, or parts of a surface, where extraneous -pressure, _p_{n}_, was non-existent. Now in the case of a bird’s egg, -the external pressure _p_{n}_, 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 _P_, or else to inequalities in the tension _T_, -that is to say to a difference between _T_ and _T′_. 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 {243} 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 _T_, 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 (_T_/_R_ + _T′_/_R′_) = 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 _T_ and -_T′_ 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[297]. 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 {244} 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[298]. - -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. - -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. - -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. {245} - -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. - -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 -{246} 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 _form_, 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 -_form_: we may often be inclined, in short, to ascribe to them a -physiological (sometimes a “pathogenic”), rather than a morphological -specificity. - -[Illustration: Fig. 73. A flagellate “monad,” _Distigma proteus_, Ehr. -(After Saville Kent.)] - -[Illustration: Fig. 74. _Noctiluca miliaris._] - -―――――――――― - -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 {247} 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. 73). 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. -74). - -[Illustration: Fig. 75. Various species of Vorticella. (Mostly after -Saville Kent.)] - -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. -75) 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. 73; 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 _point d’appui_ for the adjacent film. We shall deal with this point -again in the next chapter. {248} - -[Illustration: Fig. 76. Various species of _Salpingoeca_.] - -[Illustration: Fig. 77. Various species of _Tintinnus_, _Dinobryon_ and -_Codonella_. (After Saville Kent and others.)] - -[Illustration: Fig. 78. _Vaginicola._] - -[Illustration: Fig. 79. _Folliculina._] - -[Illustration: Fig. 80. _Trachelophyllum._ (After Wreszniowski.)] - -Precisely the same series of unduloid forms may be traced in even -greater variety among various other families or genera of the -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 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 -{249} present, and has its actual conformation, by reason of certain -chemico-physical conditions: that it was inevitable, under the given -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 possible area which the circumstances -permitted; that in the present case, the symmetry and “freedom” of -the system permitted, and _ipso facto_ caused, this surface to be a -surface of revolution; and that of the few surfaces of revolution -which, as being also surfaces _minimae areae_, were available, the -unduloid was manifestly the one permitted, and _ipso facto_ 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. -{250} - -In very many cases (of which Fig. 80 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. - -―――――――――― - -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 _species_. 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 _Systema Naturae_,—“_ut sic in summa confusione rerum apparenti, -summus conspiciatur Naturae ordo_.” In like manner the physicist -records, and is entitled to record, his many hundred “species” of -snow-crystals[299], or of crystals of calcium carbonate. But regarding -these latter species, the physicist makes no assumptions: he records -them _simpliciter_, 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 {251} the element of time, and of -succession, or discuss their origin and affiliation as an _historical_ -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[300]. 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 _genealogical_ one, nor ever doubts that -“_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_[301].” 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 {252} -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 _genealogy_. - -[Illustration: Fig. 81.] - -Among the more aberrant forms of Infusoria is a little species known -as _Trichodina pedicidus_, a parasite on the Hydra, or fresh-water -polype (Fig. 81.) 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. 223, 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. - -Another curious type is the flattened spiral of _Dinenympha_[302] -{253} 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. - -[Illustration: Fig. 82. _Dinenympha gracilis_, Leidy.] - -[Illustration: Fig. 83.] - -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 {254} contract downwards towards its base, -and become confluent with 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. 236: 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. - -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 _Study of Splashes_, 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[303]. {255} - -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. - -We begin with forms, like Astrorhiza (Fig. 219, 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 _A. crassatina_, what is -described as the “subsegmented interior[304]” {256} 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. - -[Illustration: Fig. 84. Various species of _Lagena_. (After Brady.)] - -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 {257} 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[305]. - -[Illustration: Fig. 85. (After Darling.)] - -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 {258} 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. 233). - - 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[306]. 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. - -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 {259} 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. - -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[307]. 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. - -[Illustration: Fig. 86.] - -The folded or pleated pattern is doubtless to be explained, in a -general way, by the shrinkage of a surface-film under certain {260} -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[308], 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 _L. semistriata_. All these various forms of surface can -be imitated, or rather can be precisely reproduced, by the art of the -glass-blower[309]. - -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[310]. {261} - -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 _longitudinal_ 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 -_T_ become _T_ + _t_, and _P_ become _P_ + _p_. Our new equation of -equilibrium, then, in place of _P_ = _T_/_r_ + _T_/_r′_ becomes - - _P_ + _p_ = (_T_ + _t_)/_r_ + (_T_ + _t_)/_r′_, - - and by subtraction, - - _p_ = _t_/_r_ + _t_/_r′_. - - Now if _r_ < _r′_, _t_/_r_ > _t_/_r′_. - -Therefore, in order to produce the small increment of pressure _p_, -it is easier to do so by increasing _t_/_r_ than _t_/_r′_; that is -to say, the easier way is to alter, or diminish _r_. And the same -will hold good if the tension and pressure be diminished instead of -increased. - -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 _r_,—that is to say, _in the plane of greatest -curvature_. 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. - -[Illustration: Fig. 87. _Nodosaria scalaris_, Batsch.] - -[Illustration: Fig. 88. Gonangia of Campanularians. (_a_) _C. -gracilis_; (_b_) _C. grandis_. (After Allman.)] - -The longitudinal wrinkling of the flask-shaped bodies of our Lagenae, -and of the more or less cylindrical cells of many other Foraminifera -(Fig. 87), 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 {262} 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 _bending moments_ at any point in this region of the shell-wall. -And it is at once obvious that, in any portion of the narrow neck, -_flexure_ 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. - -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 {263} 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 _stiffness_ are both present, -and play their parts in the resultant form. - -[Illustration: Fig. 89. Various Foraminifera (after Brady), _a_, -_Nodosaria simplex_; _b_, _N. pygmaea_; _c_, _N. costulata_; _e_, _N. -hispida_; _f_, _N. elata_; _d_, _Rheophax_ (_Lituola_) _distans_; _g_, -_Sagrina virgata_.] - -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. 89, 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 {264} Morey and Draper made many years ago of melted -resin are a very similar if not identical phenomenon[311]. - -―――――――――― - -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. - -In a simple and typical Heliozoan, such as the Sun-animalcule, -_Actinophrys sol_, 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[312]. 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 {265} 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 _T__{_fp_}, -_T__{_fw_}, and _T__{_wp_}, equilibrium will be attained when the angle -of contact between the fluid protoplasm and the filament is such that -cos α = (_T__{_fw_} − _T__{_wp_})/_T__{_fp_}. 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. - -Since the attraction exercised by this surface tension is symmetrical -around the filament, the latter will be pulled equally {266} 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. - -[Illustration: Fig. 90. A, _Trypanosoma tineae_ (after Minchin); B, -_Spirochaeta anodontae_ (after Fantham).] - -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. - -―――――――――― - -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 {267} which undergoes -rhythmical and beautiful wavy movements. When certain Trypanosomes are -artificially cultivated (for instance _T. rotatorium_, 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[313]. Again, in _T. lewisii_, 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[314]. -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. - -[Illustration: Fig. 91. A, _Trichomonas muris_, Hartmann; B, -_Trichomastix serpentis_, Dobell; C, _Trichomonas angusta_, Alexeieff. -(After Kofoid.)] - -This mode of formation of the undulating membrane or frill appears to -be confirmed by the appearances shewn in Fig. 91. {268} Here we have -three little organisms closely allied to the ordinary Trypanosomes, of -which one, Trichomastix (_B_), possesses four flagella, and the other -two, Trichomonas, apparently three only: the two latter possess the -frill, which is lacking in the first[315]. But it is impossible to -doubt that when the frill is present (as in _A_ and _C_), its outer -edge is constituted by the apparently missing flagellum (_a_), which -has become _attached_ to the body of the creature at the point _c_, -near its posterior end; and all along its course, the superficial -protoplasm has been drawn out into a film, between the flagellum (_a_) -and the adjacent surface or edge of the body (_b_). - -[Illustration: Fig. 92. Herpetomonas assuming the undulatory membrane -of a Trypanosome. (After D. L. Mackinnon.)] - -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 _pseudo_-undulating membrane (Fig. 92). The movements -of this structure were so exactly those of a true undulating membrane -that it was {269} difficult to believe one was not dealing with a -small, blunt trypanosome[316].” 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. - -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. - -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. - -In certain Spirochaetes, _S. anodontae_ (Fig. 90) and _S. balbiani_ -{270} (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 _nil_; 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. - -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. - -―――――――――― - -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 _seems_ to -be incapable of explanation by theories of surface-tension. - -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 {271} (described on p. 223), -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. - -[Illustration: Fig. 93.] - -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. 93), the pressure inwards due to surface -tension is positive at _A_, and negative at _C_; at _B_ 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. - -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. {272} - -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[317]. 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[318]; 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[319]. 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. - -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 _curvature_ {273} 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[320]. - -[Illustration: Fig. 94. Sperm-cells of Decapod Crustacea (after -Koltzoff). _a_, _Inachus scorpio_; _b_, _Galathea squamifera_; _c_, -_do._ after maceration, to shew spiral fibrillae.] - -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. - -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,” {274} -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[321]. 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[322]. - -[Illustration: Fig. 95. Sperm-cells of _Inachus_, as they appear in -saline solutions of varying density. (After Koltzoff.)] - -Thus the following table shews the percentage concentrations of certain -salts necessary to bring the cell into the forms _a_ and _c_ of Fig. -95; 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. 95_c_ compared with that which gives rise to the all -but spherical form of Fig. 95_a_. {275} - - % concentration of salts - in which the sperm-cell - of Inachus assumes the form of - ───────────────────── - fig. _a_ fig. _c_ - - Sodium chloride 0·6 1·2 - Sodium nitrate 0·85 1·7 - Potassium nitrate 1·0 2·0 - Acetic acid 2·2 4·5 - Cane sugar 5·0 10·0 - -[Illustration: Fig. 96. Sperm-cell of _Dromia_. (After Koltzoff.)] - -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 _volume_ is not kept pace with by a -contraction of the _surface-area_. 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 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. 96; 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 {276} 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[323]. - -―――――――――― - -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 _all_ 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 _morphological_, 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. - -{277} - - - - -CHAPTER VI - -A NOTE ON ADSORPTION - - -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[324]. 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. - -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. {278} - -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. - -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 _lower_ the surface -tension. - -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 _float_ -to the surface) the oil has a tendency to be _drawn_ to the surface; -and this phenomenon of molecular attraction {279} or “adsorption” -represents the work done, equivalent to the diminished potential energy -of the system[325]. 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. - -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. - -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[326]. And this is all as {280} 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 _vice versa_, -constitutes, then, the phenomenon of _Adsorption_; and the general -statement by which it is defined is known as the Willard-Gibbs, or -Gibbs-Thomson law[327]. - -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. - -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 _takes time_; 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 {281} 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. - -―――――――――― - -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[328]. 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[329]. - -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. - -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 {282} 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[330].” 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. - - In physical chemistry, an important distinction is drawn between - adsorption and _pseudo-adsorption_[331], the former being a - _reversible_, 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. {283} - -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[332]. 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. - -―――――――――― - -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. - -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.” - -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 {284} -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. - -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. - -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 (_after staining_) 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 {285} 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[333]. - -[Illustration: Fig. 97. _A_, _B_, Chondriosomes in kidney-cells, prior -to and during secretory activity (after Barratt); _C_, do. in pancreas -of frog (after Mathews).] - -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 -_Nebenkern_. 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 {286} unexplained -way with the mechanism of ciliary movement. Meves looked upon the -chondriosomes as the actual carriers or transmitters of heredity[334]. -Altmann invented a new aphorism, _Omne granulum e granulo_, as a -refinement of Virchow’s _omnis cellula e cellula_; and many other -histologists, more or less in accord, accepted the chondriosomes as -important entities, _sui generis_, 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[335]. 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. - -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 {287} to the phenomena of surface energy and of -adsorption[336]. 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. - -―――――――――― - -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 {288} 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 _localised_ 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[337]. 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[338]. {289} - -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 _concentration_ 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[339]; and we may safely take it that our potassium salts, like -inorganic substances in general, tend to _raise_ the surface tension, -and will therefore be found concentrating at a portion of the surface -whose tension is weak[340]. - -In Professor Macallum’s figure (Fig. 98, 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. 98, 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 -{290} 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.” - -[Illustration: Fig. 98. Adsorptive concentration of potassium salts in -(1) cell of _Pleurocarpus_ about to conjugate; (2) conjugating cells of -_Mesocarpus_; (3) sprouting spores of _Equisetum_. (After Macallum.)] - -In a spore of Equisetum (Fig. 98, 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, {291} accordingly, -a concomitant of the diminished surface tension which is manifested in -the altered configuration of the system. - -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. 247), and of -a little projecting ridge or fillet at the base of an isolated row of -cilia, such as we find in Vorticella. - -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. - -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_{2} 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. - -―――――――――― - -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 -{292} 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. - -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. - -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 _their departures from -symmetry_ may be correlated with the molecular forces manifested in -their fluid or semi-fluid surfaces. - -{293} - - - - -CHAPTER VII - -THE FORMS OF TISSUES OR CELL-AGGREGATES - - -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 _minimae areae_: 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. - -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. 265) 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. - -Let us re-state as follows, in terms of _Energy_, the general principle -which underlies the theory of surface tension or capillarity. - -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 {294} changes of temperature -or of electric charge. The condition of _minimum potential energy_ 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 _three_ bodies in contact, the case becomes -a little more complex. Suppose for instance we have a drop of some -fluid, _A_, floating on another fluid, _B_, and exposed to air, _C_. -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 _BC_, -between the mass of fluid _B_ and the superincumbent air, _C_; but the -former portion consists of two parts, for it is divided between the two -surfaces _AB_ and _AC_, that namely which separates the drop from the -surrounding fluid and that which separates it from the atmosphere. So -far as - -[Illustration: Fig. 99.] - -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 _E__{_bc_} 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 - - _E__{_bc_} < _E__{_ab_} + _E__{_ac_}. - -If, on the other hand, the fluid _A_ happens to be oil and the fluid -_B_, water, then the energy _per unit area_ of the water-air surface -is greater than that of the oil-air surface and that of the oil-water -surface together; i.e. - - _E__{_wa_} > _E__{_oa_} + _E__{_ow_}. - -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. {295} - -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[341]. 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, - -[Illustration: Fig. 100.] - -[Illustration: Fig. 101.] - -which is necessary for equilibrium, must therefore be provided by -the tensions existing in the _other two_ 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. 100), then by drawing tangents from _O_ -(the point of mutual contact), {296} we determine the three angles of -our triangle (Fig. 101), 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. - -―――――――――― - -The principle of the Triangle of Forces is expanded, as follows, by -an old seventeenth-century theorem, called Lami’s Theorem: “_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._” That is to say - - _P_ : _Q_ : _R_ : = sin _QOR_ : sin _POR_ : sin _POQ_. - - or _P_/sin _QOR_ = _Q_/sin _ROP_ = _R_/sin _POQ_. - -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_^2 = _Q_^2 + _R_^2 + 2_Q_ _R_ cos(_QOR_), etc. - -From this and the foregoing, we learn the following important and -useful deductions: - -(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: - - Water-air surface 75 - Oil-air surface 32 - Oil-water surface 21 - -No triangle having sides of these relative magnitudes is possible; and -no such drop therefore can remain in equilibrium. {297} - -(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. - -(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. - -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. - -(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 -_physical continuity_ 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[342]; -and this little fillet, or “bourrelet,” {298} as Plateau called -it, is large enough to be a common and conspicuous feature in the -microscopy of tissues (Fig. 102). 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. 103), is nothing more than a manifestation of Plateau’s -“bourrelet,” or surface of continuity[343]. - -―――――――――― - -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. - -[Illustration: Fig. 102. (After Berthold.)] - -[Illustration: Fig. 103. Parenchyma of Maize.] - -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 {299} 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 _curved_ 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. - -[Illustration: Fig. 104.] - -Now the surfaces of the two bubbles exert a pressure inwards -which is inversely proportional to their radii: that is to say -_p_ : _p′_ :: 1/_r′_ : 1/_r_; and the partition wall must, -for equilibrium, exert a pressure (_P_) which is equal to the -difference between these two pressures, that is to say, _P_ = 1/_R_ -= 1/_r′_ − 1/_r_ = (_r_ − _r′_)/_r_ _r′_. 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, _R_ = _r_ _r′_/(_r_ − _r′_). -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. - -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 _p_ be a point of contact between the two -spheres, and _cp_ be a radius of one of them, then make the angle _cpm_ -= 60°, and mark off on _pm_, _pc′_ equal to the {300} radius of the -other sphere; in like manner, make the angle _c′pn_ = 60°, cutting the -line _cc′_ in _c″_; then _c′_ will be the centre of the second sphere, -and _c″_ that of the spherical partition. - -[Illustration: Fig. 105.] - -[Illustration: Fig. 106.] - -Whether the partition be or be not a plane surface, it is obvious that -its _line of junction_ 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 _projected_ as a plane (Fig. 106), -perpendicular to the axis, and this appearance has also helped to lend -support and authority to “Sachs’s Rule.” - -―――――――――― - -[Illustration: Fig. 107. Filaments, or chains of cells, in various -lower Algae. (A) _Nostoc_; (B) _Anabaena_; (C) _Rivularia_; (D) -_Oscillatoria_.] - -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. 107. -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, {301} 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[344]. - -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, _O_, the tensions being equal, the -angles _QOP_, _ROP_, _QOR_ are all equal, and each is, therefore, an -angle of 120°. But _OQ_, _OR_ being tangents, the centres of the two -spheres (or circular arcs in the figure) lie on perpendiculars to them; -therefore the radii _CO_, _C′O_ meet at an - -[Illustration: Fig. 108.] - -angle of 60°, and _COC′_ 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, _PO_, is - - 2 sin 60° = 1·732 - -times the radius, or ·866 times the diameter, of each of the cells. -This gives us, then, the _form_ of an aggregate of two equal cells -under uniform conditions. - -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 {302} along -the partition, _P_, diminishes, the partition itself enlarges, and -the angle _QOR_ increases: until, when the tension _P_ is very small -compared to _Q_ or _R_, 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 _P_ begins to increase -relatively to _Q_ and _R_, 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. - -[Illustration: Fig. 109. Spore of _Pellia_. (After Campbell.)] - -In the spores of Liverworts (such as _Pellia_), the first -partition-wall (the equatorial partition in Fig. 109, _a_) 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 _b_ and -_c_, we are not yet in a position to deal with.) - -We have innumerable cases, near the tip of a growing filament, where -in like manner the partition-wall which cuts off the terminal {303} -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 _Dictyota dichotoma_, -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. - -[Illustration: Fig. 110. Cells of _Dictyota_. (After Reinke.)] - -[Illustration: Fig. 111. Terminal and other cells of _Chara_.] - -[Illustration: Fig. 112. Young antheridium of _Chara_.] - -In the young antheridia of Chara (Fig. 112), 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. 221); and we should find -that the partition would have a somewhat low curvature, with a radius -_less_ than the diameter of the cylinder; which it would have exactly -equalled but for the additional pressure inwards which it receives -{304} 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. - -―――――――――― - -In order to epitomise the foregoing facts let the annexed diagrams -(Fig. 113) 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 _T_, _T′_ and _t_. Let α, β, γ be, as -in the figure, the opposite angles. Then: - -(1) If _T_ be equal to _T′_, and _t_ be relatively insignificant, the -angles α, β will be of 90°. - -[Illustration: Fig. 113.] - -(2) If _T_ = _T′_, but be a little greater than _t_, then _t_ will -exert an appreciable traction, and α, β will be more than 90°, say, for -instance, 100°. - -(3) If _T_ = _T′_ = _t_, then α, β, γ will all equal 120°. - -The more complicated cases, when _t_, _T_ and _T′_ are all unequal, are -already sufficiently explained. - -―――――――――― - -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. - -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[345]. Ten {305} 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[346]. Years before, Schwendener had found, in the final results -of cell-division, a universal system of “orthogonal trajectories[347]”; -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. - -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 _secondary_, result of growth[348]. - -Within the next few years, a number of botanical writers were content -to point out further exceptions to Sachs’s Rule[349]; and in some cases -to show that the _curvatures_ 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[350]. - -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” {306} had been -pointed out long before, by Melsens, who had made an “artificial -tissue” by blowing into a solution of white of egg[351]. - -In 1886, Berthold published his _Protoplasmamechanik_, 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[352] 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.” - -In the same year, but while still apparently unacquainted with -Berthold’s work, Errera[353] 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 _must_ assume -the various configurations which the law of minimal areas imposes on -the soap-bubble. So what we may call _Errera’s Law_ 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. - -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[354]; and the same line of investigation and thought -were pursued and developed by Robert, in connection with the embryology -of the Mollusca[355]. Driesch again, in a series of papers, continued -to draw attention to the presence of capillary phenomena in the -segmenting cells {307} of various embryos, and came to the conclusion -that the mode of segmentation was of little importance as regards the -final result[356]. - -Lastly de Wildeman[357], 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.” - -―――――――――― - -[Illustration: Fig. 114.] - -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 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. 114 -represent the three associated bubbles in a plane drawn through their -centres, _c_, _c′_, _c″_ (or what is the same thing, if it represent -the base of three bubbles resting on a plane), then the lines _uc_, -_uc″_, or _sc_, _sc′_, etc., drawn to the {308} centres from the -points of intersection of the circular arcs, will always enclose an -angle of 60°. Again (Fig. 115), if we make the angle _c″uf_ equal to -60°, and produce _uf_ to meet _cc″_ in _f_, _f_ will be the centre of -the circular arc which constitutes the partition _Ou_; and further, the -three points _f_, _g_, _h_, successively determined in this - -[Illustration: Fig. 115.] - -manner, will lie on one and the same straight line. In the case -of coequal bubbles or cells (as in Fig. 114, 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 _uf_ is now -{309} parallel to _cc″_, 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. - -[Illustration: Fig. 116.] - -When we have four bubbles in conjunction, they would seem to be capable -of arrangement in two symmetrical ways: either, as in Fig. 116 (A), -with the four partition-walls meeting at right angles, or, as in (B), -with _five_ partitions meeting, three and three, at angles of 120°. -This latter arrangement is strictly analogous to the arrangement of -three bubbles in Fig. 114. Now, though both of these figures, from -their symmetry, are apparently figures of equilibrium, yet, physically, -the former turns out to be of unstable and the latter of stable -equilibrium. If we try to bring our four bubbles into the form of Fig. -116, 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°. {310} - -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 -_curvature_ 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 _polar furrow_[358], the visible edge of a vertical -partition-wall, joins (or separates) the two triple contacts, precisely -as in Fig. 116, B. - -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. 117). 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. 155 etc.) by which the egg was -originally divided into two; the four-celled stage being reached by -the appearance of the transverse furrows {311} 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. 117, A. In Fig. -117, C, we have a very different condition, with which we shall deal in -a moment. - -[Illustration: Fig. 117. Various ways in which the four cells are -co-arranged in the four-celled stage of the frog’s egg. (After Rauber.)] - -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[359]. The cases where four cells, lying in one plane, meet _in -a point_, 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 {312} error or imperfect observation all those cases where the -junction-lines of four cells are represented (after the manner of Fig. -116, A) as a simple cross[360]. - -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 _rounded off_, 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. 117, 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 _Polgrübchen_, and asserts to be -constantly present whensoever the polar furrow, or _Brechungslinie_, 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 _one another -at all_; 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, {313} 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 _similar_ contacts and -with _equal_ surface tensions throughout the system; but in the system -of protoplasmic cells which constitute the segmenting egg we must make -allowance for _an inequality_ 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 - -[Illustration: Fig. 118.] - -entire system, the _sum of the surface energies_ is a minimum; and, -while this is attained by the _sum of the surfaces_ 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. 118) -if the energy per unit area be greater along the contact surface _cc′_, -where cell meets cell, than along _ca_ or _cb_, 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 _cc′_, and the two opposing tensions along _ca_ and _cb_. -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 -_cc′_ 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. 300), -and it also leads us to a consideration of the general properties and -characters of an “epidermal” layer. - -―――――――――― - -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 {314} 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[361], 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 _air_) 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 - -[Illustration: Fig. 119.] - -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 {315} 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. - -―――――――――― - -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. 116) 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 {316} -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. - -Where we have three spheres in contact, as in Fig. 114 or in either -half of Fig. 116, B, let us consider the point of contact (_O_, Fig. -114) not as a point in the plane section of the diagram, but as a point -where three _furrows_ meet on the surface of the system. At this point, -_three cells_ meet; but it is also obvious that there meet here _six -surfaces_, namely the outer, spherical walls of the three bubbles, -and the three partition-walls which divide them, two and two. Also, -_four_ lines or _edges_ 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 _four solid angles_, 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, {317} with six planes, meeting -symmetrically in a point, and constituting there four equal solid -angles. - -[Illustration: Fig. 120.] - -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. - -Let us look for a moment at the geometry of our figure. Let _o_ (Fig. -120) be the centre of the tetrahedron, i.e. the centre of symmetry -where our films meet; and let _oa_, _ob_, _oc_, _od_, be lines drawn -to the four corners of the tetrahedron. Produce _ao_ to meet the base -in _p_; then _apd_ is a right-angled triangle. It is not difficult to -prove that in such a figure, _o_ (the centre of gravity of the system) -{318} lies just three-quarters of the way between an apex, _a_, and -a point, _p_, which is the centre of gravity of the opposite base. -Therefore - - _op_ = _oa_/3 = _od_/3. - - Therefore cos _dop_ = 1/3 and cos _aod_ = − 1/3. - -That is to say, the angle _aod_ 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 _every other -solid system_ 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. - -―――――――――― - -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. - -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. 121. 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[362]. - -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 _points_ of contact between the -circles in the diagram will be extended into {319} _lines_ of contact, -representing _surfaces_ 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. - -[Illustration: Fig. 121. Diagram of hexagonal cells. (After Bonanni.)] - -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 “_tourbillons cellulaires_” -(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, {320} 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 {321} - -[Illustration: 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.)] - -[Illustration: Fig. 123. An artificial cellular tissue, formed by the -diffusion in gelatine of drops of a solution of potassium ferrocyanide. -(After Leduc.)] - -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. - -―――――――――― - -[Illustration: Fig. 124. Epidermis of _Girardia_. (After Goebel.)] - -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 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 {322} -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. - -[Illustration: Fig. 125. Soap-froth under pressure. (After Rhumbler.)] - -In a soap-froth imprisoned between two glass plates, we have a -symmetrical system of cells, which appear in optical section (as in -Fig. 125, B) as regular hexagons; but if we press the plates a little -closer together, the hexagons become deformed or flattened (Fig. 125, -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. -125, B. - -[Illustration: Fig. 126. From leaf of _Elodea canadensis_. (After -Berthold.)] - -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. 126). In the narrow elongated -leaf of a Monocotyledon, such as a hyacinth, the elongated, apparently -quadrangular {323} 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°. - -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. - - 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, _whose loci are seen_ 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[363]. 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[364].” 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. - - ―――――――――― - - 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. - - 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 {324} cornea of an insect’s eye, or in the minute pattern of - hexagons on many diatoms. An approximate enumeration is easily made as - follows. - - For the area of a hexagon (if we call δ the short diameter, that - namely which bisects two of the opposite sides) is δ^2 × (√3)/2, - the area of a circle being _d_^2 ⋅ π/4. Then, if the diameter (_d_) - of a circular area include _n_ hexagons, the area of that circle - equals (_n_ ⋅ δ)^2 × 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 _n_^2 × π/4 × 2/(√3) - = 0·907_n_^2, or (9/10) ⋅ _n_^2, nearly. - - 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[365]. 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: - - For one zone 6 = 2 × 3 = 3 × 1 × 2, - For two zones 18 = 3 × 6 = 3 × 2 × 3, - For three zones 36 = 4 × 9 = 3 × 3 × 4, - For four zones 60 = 5 × 12 = 3 × 4 × 5, - For five zones 90 = 6 x 15 = 3 × 5 × 6, - - and so forth. If _N_ 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, _H_, = 3_N_(_N_ + 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[366]. - -―――――――――― - -The same principles which account for the development of hexagonal -symmetry hold true, as a matter of course, not only of individual -_cells_ (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 _Garden of Cyrus_, of hexagonal -(and also of quincuncial) symmetry in plants and animals, which “doth -neatly declare how nature Geometrizeth, and observeth order in all -things.” {325} - -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. _Favosites -gothlandica_, 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. 127). Accordingly in -the former the corallites are - -[Illustration: Fig. 127. _Lithostrotion Martini._ (After Nicholson.)] - -[Illustration: Fig. 128. _Cyathophyllum hexagonum._ (From Nicholson, -after Zittel.)] - -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 _Cyathophyllum hexagonum_, 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 _on the average_ hexagonal, but -some will have fewer and some more sides than six, as in the annexed -figure of Arachnophyllum (Fig. 129). {326} 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.). - -[Illustration: Fig. 129. _Arachnophyllum pentagonum._ (After -Nicholson.)] - -[Illustration: Fig. 130. _Heliolites._ (After Woods.)] - -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. 131). 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 {327} known as triasters, polyasters, etc., whose -relation to a field of force Hartog has explained[367]. 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 (_Oculinidae_), in Phillipsastraea (_Rugosa_), in Thamnastraea -(_Fungida_), and in many more. - -[Illustration: Fig. 131. Surface-views of Corals with undeveloped -thecae and confluent septa. A, _Thamnastraea_; B, _Comoseris_. (From -Nicholson, after Zittel.)] - -―――――――――― - -The most famous of all hexagonal conformations and perhaps the most -beautiful is that of the bee’s cell. Here we have, as in 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 {328} -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 _laterally_ adjacent cells, any one cell -would be squeezed into a hexagonal prism, so is it also obvious that, -by mutual pressure against the three _terminal_ 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 _six_ sides of -the cell are to be combined with its _three_ 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 _rhombic_ 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 _twelve_ 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 _rhombic dodecahedron_. 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. - -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 _Collections_) an account of its hexagonal plan, -and he drew from its mathematical symmetry the {329} 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. - -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 _Garden of Cyrus_: “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[368] -(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[369]. “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 {330} 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[370] and -Lhuilier[371], by different methods, obtained a result identical with -Maraldi’s; and were able to shew that the discrepancy of 2′ was due 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[372]) “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. - -[Illustration: Fig. 132.] - -The theorem may be proved as follows: - -_ABCDEF_, _abcdef_, is a right prism upon a regular hexagonal base. The -corners _BDF_ are cut off by planes through the lines _AC_, _CE_, _EA_, -meeting in a point _V_ on the axis _VN_ of the prism, and intersecting -_Bb_, _Dd_, _Ff_, at _X_, _Y_, _Z_. 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 _ABCDEF_ in _N_, the -volumes _ACVN_, _ACBX_ are equal. {331} - -It is required to find the inclination of the faces forming the -trihedral angle at _V_ to the axis, such that the surface of the figure -may be a minimum. - -Let the angle _NVX_, which is half the solid angle of the prism, = θ; -the side of the hexagon, as _AB_, = _a_; and the height, as _Aa_, = _h_. - - Then, _AC_ = 2_a_ cos 30° = _a_√3. - - And _VX_ = _a_/sin θ (from inspection of the triangle _LXB_) - - Therefore the area of the rhombus _VAXC_ = (_a_^2 √3)/(2 sin θ). - - And the area of _AabX_ = (_a_/2)(2_h_ − ½_VX_ cos θ) - - = (_a_/2)(2_h_ − _a_/2 ⋅ cot θ). - - Therefore the total area of the figure - - = hexagon _abcdef_ + 3_a_(2_h_ − (_a_/2) cot θ) - + 3((_a_^2 √3)/(2 sin θ)). - - Therefore _d_(Area)/_d_θ = (3_a_^2/2)((1/sin^2 θ) - − (√3 cos θ)/(sin^2 θ)). - -But this expression vanishes, that is to say, _d_(Area)/_d_θ = 0, -when cos θ = 1/√3, that is when θ = 54° 44′ 8″ = ½(109° 28′ 16″). - -This then is the condition under which the total area of the figure has -its minimal value. - -―――――――――― - -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; {332} -they may be rounded or oval capsules, often very neatly constructed, -out of mud, or vegetable _fibre_ 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 _Melipona domestica_ (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[373].” 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[374]. -But it is very doubtful {333} 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[375]. 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. - -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, {334} 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[376]. 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[377].” {335} - -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. _Juncus effusus_), 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. 133, _c_), 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. - -[Illustration: Fig. 133. Diagram of development of “stellate cells,” in -pith of _Juncus_. (The dark, or shaded, areas represent the cells; the -light areas being the gradually enlarging “intercellular spaces.”)] - -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 {336} result will -obviously be that the intercellular spaces will increase; the six -equatorial attachments of each cell (Fig. 133, _a_) (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 (_b_) towards the centre. As the -final result (_c_) 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. - -―――――――――― - -A few years after the publication of Plateau’s book, Lord Kelvin -shewed, in a short but very beautiful paper[378], 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 {337} 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. - -[Illustration: Fig. 134.] - -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[379], when we dip a cubical wire -skeleton into soap-solution and take it out again, the twelve films -which are thus generated do _not_ meet in a point, but are grouped -around a small central, plane, quadrilateral film (Fig. 134). In -other words, twelve plane films, meeting in a point, are _essentially -unstable_. 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 {338} 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. - -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. - -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° [380]. - -We may take it as certain that, in a system of _perfectly_ 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. {339} -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 -_section_, 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. - -―――――――――― - -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 {340} the -egg of _Trochus_ (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. - -[Illustration: Fig. 135. Aggregations of four soap-bubbles, to shew -various arrangements of the intermediate partition and polar furrows. -(After Robert.)] - -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. 310. 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. -135, 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. 135, 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 _C. convexa_, that -the upper one is the shorter in _C. fornicata_, and that the upper -one all but disappears in _C. plana_; 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 _species_. {341} 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[381]; 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 {342} in molluscs and annelids ‘stands in obvious relation to the -different size of the cells produced at the two poles[382].’ ” - -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. 135, 5–7)[383]. 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. - -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, {343} 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. - -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. - -―――――――――― - -Before we leave for the present the subject of the segmenting {344} -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. - -The former problem is comparatively easy, as regards the tendency of a -segmentation cavity to _enlarge_, 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[384]. 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. - -The physical forces involved in the invagination of the cell-layer to -form the gastrula have been repeatedly discussed[385], 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 {345} 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. - -{346} - - - - -CHAPTER VIII - -THE FORMS OF TISSUES OR CELL-AGGREGATES (_continued_) - - -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. - -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. - -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 _a_ the length of one of the edges of -the cube, then _a_^2 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; {347} for the area of this new -partition is _a_ × _a_/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_a_)^2 = ¼(_a_^2). - -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. - -[Illustration: Fig. 136. (After Berthold.)] - -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. - -(1) The cell typically tends to divide into two co-equal parts. - -(2) Each new plane of division tends to intersect at right angles the -preceding plane of division. - -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 -{348} 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[386], 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. - -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. - -―――――――――― - -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 {349} 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. _a_^2; 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 _a_^2. -The calculation is very easy. - -The _surface-area_ of a cylinder of length _a_ is 2π_r_ ⋅ _a_, and that -of our quarter-cylinder is, therefore, _a_ ⋅ π_r_/2; and this being, by -hypothesis, = _a_^2, we have _a_ = π_r_/2, or _r_ = 2_a_/π. - -The _volume_ of a cylinder, of length _a_, is _a_π_r_^2, and that of -our quarter-cylinder is (_a_ ⋅ π_r_^2)/4, which (by substituting the -value of _r_) is equal to (_a_^3)/π. - -Now precisely this same volume is, obviously, shut off by a transverse -partition of area _a_^2, if the third side of the rectangular space -be equal to _a_/π. And this fraction, if we take _a_ = 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. - -[Illustration: Fig. 137.] - -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. - -By a similar calculation we can shew that a _spherical_ 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 {350} -one-quarter of the original cell[387]. 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 {351} -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. 138) the -relative areas of the three partitions are shewn for all fractions, -less than one-half, of the divided cell. - -[Illustration: Fig. 138.] - - 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 _a_^2, 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 _A_ 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 _B_ the spherical octant begins - to have an advantage over the plane; and it is not till we reach the - point _C_ that the spherical octant becomes of less area than the - quarter-cylinder. - -[Illustration: Fig. 139.] - -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. 139), or in the “paraphyses” of mosses -(Fig. 142). 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 {352} which we cannot fully -master, the same guiding principle is at the root of the matter. - -―――――――――― - -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 _base_ of the cube. The whole problem of the -solid, up to a certain point, is contained in our plane diagram of -Fig. 138. 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. - -The bisection of an isosceles triangle by a line which shall be the -shortest possible is a very easy problem. Let _ABC_ be such a triangle -of which _A_ 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 -_AD_, bisecting the angle at _A_ and the side _BC_; to a transverse -line parallel to the base _BC_; or to an oblique line parallel to _AB_ -or to _AC_. 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, _BC_, 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 {353} -partition, _AD_, since _BD_ = 1, will always equal tan β (β being the -angle _ABC_). And the oblique partition, _GH_, being equal to _AB_/√2 -= 1/(√2 cos β). If then we call our vertical, transverse - -[Illustration: Fig. 140.] - -and oblique partitions, _V_, _T_, and _O_, we have _V_ = tan β; _T_ -= √2; and _O_ = 1/(√2 cos β), or - - _V_ : _T_ : _O_ = tan β/√2 : 1 : 1/(2 cos β). - -And, working out these equations for various values of β, we very -soon see that the vertical partition (_V_) is the least of the three -until β = 45°, at which limit _V_ and _O_ are each equal to 1/√2 -= ·707; and that again, when β = 60°, _O_ and _T_ are each = 1, after -which _T_ (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. - -[Illustration: Fig. 141.] - -In like manner, if we have a spheroidal body, less than a hemisphere, -such for instance as a low, watch-glass shaped cell (Fig. 141, _a_), -it is obvious that the smallest possible partition by which we can -divide it into two equal halves {354} 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 (_c_). And furthermore, there will be an intermediate region, a -region where height and base have their relative dimensions nearly -equal (as in _b_), 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 _vertical_ partition -is, therefore, everywhere normal to the original cell-walls, and -constitutes a plane surface. - -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. {355} - -[Illustration: Fig. 142. S-shaped partitions: _A_, from _Taonia -atomaria_ (after Reinke); _B_, from paraphyses of _Fucus_; _C_, from -rhizoids of Moss; _D_, from paraphyses of _Polytrichum_.] - -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. -143) 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[388]. - -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 {356} 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. - -[Illustration: Fig. 143. Diagrammatic explanation of S-shaped -partition.] - -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 _terminal_ 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. 142, C); and again -among Mosses, in the “paraphyses” of the male prothalli (e.g. in -_Polytrichum_), 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 _Taonia atomaria_, as figured in Reinke’s memoir on -the Dictyotaceae of the Gulf of Naples[389], we see, in like manner, -_oblique_ partitions, which on more careful examination are seen to be -curves of double curvature (Fig. 142, A). - -The physical cause and origin of these S-shaped 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 _polarity_ -(Fig. 143, A), and implicitly asserting that it tends to {357} -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. 289,—then our “polar axis” would -necessarily be a curved axis, and the partition, being constrained -(again _ex hypothesi_) to arise transversely to the polar axis, would -lie obliquely to the _apparent_ axis of the cell (Fig. 143, B, C). -And if the oblique partition be so situated that it has to meet the -_opposite_ 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 S-shaped curvature (Fig. 143, D). - -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[390]. He was -deeply and rightly impressed by the fact that other forces besides -surface {358} 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. - -―――――――――― - -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. - -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 _surface_ of the same -form. - -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. - -[Illustration: Fig. 144. Development of _Erythrotrichia_. (After -Berthold.)] - -[Illustration: Fig. 145.] - -Fig. 144 (taken from Berthold’s _Monograph of the Naples Bangiaciae_) -represents younger and older discs of the little alga _Erythrotrichia -discigera_; 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, {359} 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 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. 145, A), 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. 145, where the partition is either (as in B) {360} -concentric with the outer border of the cell, or else (as in C) cuts -that outer border; in other words, our partition may (B) cut _both_ -radial walls, or (C) may cut _one_ radial wall and the periphery. These -are the two methods of division which Sachs called, respectively, (B) -_periclinal_, and (C) _anticlinal_[391]. 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. - -Now we find that a flattened cell which is approximately a quadrant of -a circle invariably divides after the manner of Fig. 145, C, that is to -say, by an approximately circular, _anticlinal_ wall, such as we now -recognise in the eight-celled stage of Erythrotrichia (Fig. 144); let -us then consider that Nature has solved our problem for us, and let us -work out the actual geometric conditions. - -Let the quadrant _OAB_ (in Fig. 146) be divided into two parts of equal -area, by the circular arc _MP_. It is required to determine (1) the -position of _P_ upon the arc of the quadrant, that is to say the angle -_BOP_; (2) the position of the point _M_ on the side _OA_; and (3) the -length of the arc _MP_ in terms of a radius of the quadrant. - -(1) Draw _OP_; also _PC_ a tangent, meeting _OA_ in _C_; and _PN_, -perpendicular to _OA_. Let us call _a_ a radius; and θ the angle at -_C_, which is obviously equal to _OPN_, or _POB_. Then - - _CP_ = _a_ cot θ; _PN_ = _a_ cos θ; _NC_ = _CP_ cos θ = _a_ ⋅ (cos^2 θ)/(sin θ). - -The area of the portion _PMN_ - - = ½_C_(_P_^2)θ − ½_PN_ ⋅ _NC_ - - = ½_a_^2(cot^2 θ) − ½_a_(cos θ) ⋅ _a_(cos^2 θ)/(sin θ) - - = ½_a_^2(cot^2 θ − (cos^3 θ)/(sin θ)). - -{361} - -And the area of the portion _PNA_ - - = ½_a_^2(π/2 − θ) − ½_ON_ ⋅ _NP_ - - = ½_a_^2(π/2 − θ) − ½_a_(sin θ) ⋅ _a_(cos θ) - - = ½_a_^2(π/2 − θ − sin θ ⋅ cos θ). - -Therefore the area of the whole portion _PMA_ - - = (_a_^2)/2 (π/2 − θ + θ cot^2 θ − (cos^3 θ)/sin θ − sin θ ⋅ cos θ) - - = (_a_^2)/2 (π/2 − θ + θ cot^2 θ − cot θ), - -and also, by hypothesis, = ½ ⋅ area of the quadrant, = (π_a_^2)/8. - -[Illustration: Fig. 146.] - -Hence θ is defined by the equation - - _a_^2/2 (π/2 − θ + θ cot^2 θ − cot θ) = (π_a_^2)/8, - - or π/4 − θ + θ cot^2 θ − cot θ = 0. - -We may solve this equation by constructing a table (of which the -following is a small portion) for various values of θ. - - θ π/4 − θ − cot θ + θ cot^2 θ = _x_ - 34° 34′ ·7854 − ·6033 − 1·4514 + 1·2709 = ·0016 - 35′ ·7854 ·6036 1·4505 1·2700 ·0013 - 36′ ·7854 ·6039 1·4496 1·2690 ·0009 - 37′ ·7854 ·6042 1·4487 1·2680 ·0005 - 38′ ·7854 ·6045 1·4478 1·2671 ·0002 - 39′ ·7854 ·6048 1·4469 1·2661 −·0002 - 40′ ·7854 ·6051 1·4460 1·2652 −·0005 - -{362} - -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. - -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′. - -(2) The position of _M_ on the side of the quadrant _OA_ is given -by the equation _OM_ = _a_ cosec θ − _a_ 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. - -(3) The length of the arc _MP_ is equal to _a_ 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. - -But we must also compare the length of this curved “anticlinal” -partition-wall (_MP_) with that of the concentric, or periclinal, one -(_RS_, Fig. 147) 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. - -[Illustration: Fig. 147.] - -The two cells into which our original quadrant is now divided, while -they are equal in volume, are of very different shapes; the {363} one -is a triangle (_MAP_) with two sides formed of circular arcs, and the -other is a four-sided figure (_MOBP_), 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. 148) -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 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. - -[Illustration: Fig. 148. Segmentation of frog’s egg, under artificial -compression. (After Roux.)] - -―――――――――― - -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 {364} sector, such as -that shewn in Fig. 149, that a _periclinal_ 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. - -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 _MP_ is always determined by the angle -θ, according to our equation _MP_ = _a_θ cot θ; and the angle θ has a -definite relation to α, the angle of arc. - -[Illustration: Fig. 149.] - -Moreover, in the case of the periclinal boundary, _RS_ (Fig. 147) (or -_ab_, Fig. 149), we know that, if it bisect the cell, - - _RS_ = _a_ ⋅ α/√2. - -Accordingly, the arc _RS_ will be just equal to the arc _MP_ when - - θ cot θ = α/√2. - - When θ cot θ > α/√2 or _MP_ > _RS_, - - then division will take place as in _RS_. - - When θ cot θ < α/√2, or _MP_ < _RS_, - - then division will take place as in _MP_. - -In the accompanying diagram (Fig. 150), 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 {365} 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 - -[Illustration: Fig. 150.] - -determines the point on the periphery where it is cut by the -partition-wall, _MP_. Two limiting cases are to be noticed here: (1) -at 90° (point _A_ in diagram), because we are at present only {366} -dealing with arcs no greater than a quadrant; and (2), the point (_B_) -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°. - -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. -_RS_ and _MP_, their lengths being expressed (on the right-hand side of -the diagram) in relation to the radius of the circle (_a_), that is to -say the side wall, _OA_, of our cell. - -The limiting values here are (1), _C_, _C′_, where the angle of arc -is 90°, and where, as we have already seen, the two partition-walls -have the relative magnitudes of _MP_ : _RS_ = 0·875 : 1·111; (2) the -point _D_, where _RS_ 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 _D_, _D′_); (3) the point _E_, where -_RS_ and _MP_ 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 _MP_, 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. - -The case (_F_) 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 _MP_ : _RS_ = 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; _MP_ and _RS_ are now identical. - -On the same diagram, I have inserted the curve for values of {367} -cosec θ − cot θ = _OM_, that is to say the distances from the centre, -along the side of the cell, of the starting-point (_M_) of the -anticlinal partition. The point _C″_ 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 _F‴_ (vertically under _F_), which tells us that the partition -now starts 4·5/10, or nearly halfway, along the radial wall. - -―――――――――― - -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. - -[Illustration: Fig. 151.] - -Let us now return to our quadrant cell (_OAPB_), which we have found -to be divided into a triangular and a quadrilateral portion, as in -Fig. 147 or Fig. 151; 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 {368} 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:― - -Let us call the triangular cell _T_, and the quadrilateral, _Q_ (Fig. -151); let the radius, _OA_, of the original quadrantal cell = _a_ = 1; -and let the increment which is required to add on a portion equal to -_T_ (such as _PP′A′A_) be called _x_, and let that required, similarly, -for the doubling of _Q_ be called _x′_. - -Then we see that the area of the original quadrant - - = _T_ + _Q_ = ¼π_a_^2 = ·7854_a_^2, - - while the area of _T_ = _Q_ = ·3927_a_^2. - -The area of the enlarged sector, _p′OA′_, - - = (_a_ + _x_)^2 × (55° 22′) ÷ 2 = ·4831(_a_ + _x_)^2, - - and the area _OPA_ - - = _a_^2 × (55° 22′) ÷ 2 = ·4831_a_^2. - - Therefore the area of the added portion, _T′_, - - = ·4831 ((_a_ + _x_)^2 − _a_^2). - - And this, by hypothesis, - - = _T_ = ·3927_a_^2. - -We get, accordingly, since _a_ = 1, - - _x_^2 + 2_x_ = ·3927/·4831 = ·810, - - and, solving, - - _x_ + 1 = √1·81 = 1·345, or _x_ = 0·345. - -Working out _x′_ in the same way, we arrive at the approximate value, -_x′_ + 1 = 1·517. {369} - -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 _T_ 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, _Q_, will not tend to divide -until the linear dimensions of the disc have increased by about a half -(from 1 to 1·517). - -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. 348. - -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, _T_, 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 _P_ 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. - -But the case will be different next time, because in this new {370} -triangle, _PRQ_, the least width is near the innermost angle, that at -_Q_; and the bisecting circular arc will therefore be opposite to _Q_, -or (approximately) parallel to _PR_. 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, _radially_ (as -_U_, _V_); 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 _an epidermis_. 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. - -[Illustration: Fig. 152.] - -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 -{371} 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. - -―――――――――― - -[Illustration: Fig. 153. Diagram of flattened or discoid cell -dividing into octants: to shew gradual tendency towards a position of -equilibrium.] - -Bearing in mind, then, these considerations, let us see what our -flattened disc is likely to look like, after a few successive divisions -into component cells. In Fig. 153, _a_, 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) {372} be an arc of a circle of which we can determine the -centre by the method used on p. 307; 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. 153, _b_. 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 (_c_); 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 (_d_) 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. 153, _d_; 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. 153, _a_ -with which we began. - -[Illustration: Fig. 154.] - -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. 154), 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. 359) of the disc of Erythrotrichia, from -Berthold’s _Monograph of the Bangiaceae_ 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 {373} 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. 144 is already too - -[Illustration: Fig. 155. Theoretical arrangement of successive -partitions in a discoid cell; for comparison with Fig. 144.] - -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. -156 I shew one more diagrammatic figure, of a disc which - -[Illustration: Fig. 156. Theoretical division of a discoid cell into -sixty-four chambers: no allowance being made for the mutual tractions -of the cell-walls.] - -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 {374} 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. 144) and the diagram which -we have here constructed in obedience to a few simple rules. - -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 (_a_)) to the stage (_b_) -in which a well-marked epidermal layer surrounds an at first sight -irregular agglomeration of “fundamental” tissue. - -[Illustration: Fig. 157. Sections of embryo of a moss. (After -Kienitz-Gerloff.)] - -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[392]. - -[Illustration: Fig. 158. Various possible arrangements of intermediate -partitions, in groups of 4, 5, 6, 7 or 8 cells.] - -If within our boundary we have three cells all meeting {375} -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 _two_ little intermediate walls, and two -polar furrows; and we soon arrive at the rule that, for _n_ cells, -we require _n_ − 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. 159. -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. 158 shews some half-dozen only. It does not follow that, so to -speak, these various {376} 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. - -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. - -―――――――――― - -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[393]. - -[Illustration: Fig. 159.] - -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 {377} considered) is -responsible for many inaccurate or incomplete representations of the -mutual arrangement of aggregated cells. - -[Illustration: Fig. 160. Segmenting egg of _Trochus_. (After Robert.)] - -[Illustration: Fig. 161. Two views of segmenting egg of _Cynthia -partita_. (After Conklin.)] - -[Illustration: Fig. 162. (_a_) Section of apical cone of _Salvinia_. -(After Pringsheim[394].) (_b_) Diagram of probable actual arrangement.] - -[Illustration: Fig. 163. Egg of _Pyrosoma_. (After Korotneff).] - -[Illustration: Fig. 164. Egg of _Echinus_, segmenting under pressure. -(After Driesch.)] - -In the accompanying series of figures (Figs. 160–167) I have {378} -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 {379} 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. 158, 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. - -[Illustration: Fig. 165. (_a_) Part of segmenting egg of Cephalopod -(after Watase); (_b_) probable actual arrangement.] - -[Illustration: Fig. 166. (_a_) Egg of _Echinus_; (_b_) do. of _Nereis_, -under pressure. (After Driesch).] - -[Illustration: Fig. 167. (_a_) Egg of frog, under pressure (after -Roux); (_b_) probable actual arrangement.] - -It will be seen that M. Robert-Tornow’s figure of the segmenting egg of -Trochus (Fig. 160) clearly shews the cells grouped after the fashion of -Fig. 158, _a_. In like manner, Mr Conklin’s figure of the ascidian egg -(_Cynthia_) shews equally clearly the arrangement _g_. - -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 _d_. 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 _c_. Beside -it is figured a very different object, a segmenting egg of the Ascidian -_Pyrosoma_, after Korotneff; it may be that this also is to be referred -to type _c_, but I think it is more easily referable to type _b_. 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 _b_ or the type _c_. 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 _d_. Lastly, I have copied from Roux a -curious figure of the egg of _Rana esculenta_, viewed from the animal -pole, which appears to me referable, in all probability, to type _g_. -Of type _f_, 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. - -―――――――――― - -It is obvious enough, without more ado, that these phenomena are in -the strictest and completest way common to both plants {380} and -animals. In other words they tally with, and they further extend, -the general and fundamental conclusions laid down by Schwann, in -his _Mikroskopische Untersuchungen über die Uebereinstimmung in der -Struktur und dem Wachsthum der Thiere und Pflanzen_. - -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 -_in each particular organism_ 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 _specific character_, 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, _quod vidit id asseruit_. 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. 168) six -diagrammatic figures, which are all _essentially different_, 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[395]. {381} - -[Illustration: Fig. 168. Various modes of grouping of eight cells, at -the dorsal or epiblastic pole of the frog’s egg. (After Rauber.)] - -Of these six illustrations, two are exceptional. In Fig. 168, 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. 168, 6, is again peculiar, and is probably -rare; but it is included under the cases considered on p. 312, in -which the cells are not in complete 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. 168, 1 and 2, we have the arrangements corresponding to _a_ -and _d_ of our diagram (Fig. 158): but the other two (i.e. 3 and 4) -represent other of the thirteen possible arrangements, which are not -included in that {382} 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[396]. - -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 -_surface_ of the group, none being submerged or wholly enveloped by -the rest) are referable to some one or other of _thirteen_ 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. - -―――――――――― - -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 {383} 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. 169 (which represent three out of our -thirteen possible arrangements of eight cells) we shall see that, in -the case of type _b_, two cells are each in contact with two others, -two cells with three others, and four cells each with four other cells. -In type _a_ four cells are each in contact with two, two with four, -and two with five. In type _f_, two are in contact with two, four with -three, and one with no less than seven. In all cases the - -[Illustration: Fig. 169.] - -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 _n_ cells with a common -external boundary, the number of internal partitions is 2_n_ − 3; or -the number of what we call the internal or interfacial contacts is -2(2_n_ − 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 _f_, 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. -170 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 {384} 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 _tending_ rather -than that which it has fully attained[397]. The type which we have -called _f_ was found by Roux to be unstable, the large (or apparently -large) drop _a″_ 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. - -[Illustration: Fig. 170. Aggregations of oil-drops. (After Roux.) Figs. -4–6 represent successive changes in a single system.] - -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. {385} - -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 -_Geometria Situs_ of Gauss, the _Analysis Situs_ of Riemann, the Theory -of Partitions of Cayley, and of Spatial Complexes of Listing[398]. 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. - -―――――――――― - -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 -{386} 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. - -[Illustration: Fig. 171. (A) _Asterolampra marylandica_, Ehr.; (B, C) -_A. variabilis_, Grev. (After Greville.)] - -Another case, geometrically akin but biologically very different, is -to be found in the little diatoms of the genus Asterolampra, and their -immediate congeners[399]. 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. 171). 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 {387} 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[400] (Fig. 171, 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. 170, 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 _allied genus_ -Arachnoidiscus. - -―――――――――― - -[Illustration: Fig. 172. Section of Alcyonarian polype.] - -In a diagrammatic section of an Alcyonarian polype (Fig. 172), 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 {388} 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 _one -another_, 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. - -[Illustration: Fig. 173. _Heterophyllia angulata_. (After Nicholson.)] - -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 _one another_, 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[401], 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[402], described them as “paradoxical in -their anatomy.” - -[Illustration: Fig. 174. _Heterophyllia_ sp. (After Martin Duncan.)] - -The simplest or youngest Heterophylliae known have six septa (as in -Fig. 174, _a_); 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 {389} case of the other -two we may fairly assume that their proper and original arrangement -was that of our type 6_b_ (Fig. 158), 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.” {390} - -In the two examples figured (Fig. 174), 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 _f_, in the eight-celled system (Fig. 158). 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. - -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 _co-equal -parts_. 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 {391} 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, -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. - -[Illustration: Fig. 175. Diagrammatic section of a Ctenophore -(_Eucharis_).] - -―――――――――― - -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 {392} meridian-like -ciliated bands is a case in point (Fig. 175). 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. - -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. - -[Illustration: Fig. 176. Diagrammatic arrangement of partitions, -represented by skeletal rods, in larval Echinoderm (_Ophiura_).] - -In Fig. 176 I have divided a circle into its four quadrants, and have -bisected each quadrant by a circular arc (_BC_), 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 -(_DD_). 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 {393} 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 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. - -[Illustration: Fig. 177. Pluteus-larva of Ophiurid.] - -―――――――――― - -[Illustration: Fig. 178. Diagrammatic development of Stomata in -_Sedum_. (Cf. fig. in Sachs’s _Botany_, 1882, p. 103.)] - -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) {394} the space to be cut -off be a very small one, not more than about three-tenths the area of -a square based on the _short_ 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. 179, 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 - -[Illustration: Fig. 179. Diagrammatic development of stomata in -Hyacinth.] - -stoma. In other cases, as in Fig. 178, 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 -{395} between the new-formed cells splits, and again we have the -phenomenon of a “stoma” with its attendant guard-cells. In Fig. 179 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. - -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. - -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. - -―――――――――― - -[Illustration: Fig. 180. Various pollen-grains and spores (after -Berthold, Campbell, Goebel and others). (1) _Epilobium_; (2) -_Passiflora_; (3) _Neottia_; (4) _Periploca graeca_; (5) _Apocynum_; -(6) _Erica_; (7) Spore of _Osmunda_; (8) Tetraspore of _Callithamnion_.] - -[Illustration: Fig. 181. Dividing spore of _Anthoceros_. (After -Campbell.)] - -In all our foregoing examples of the development of a “tissue” we -have seen that the process consists in the _successive_ division of -cells, each act of division being accompanied by the formation {396} -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 _simultaneously_, 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 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 {397} they lie, -the pollen-grains afterwards fall apart, and their individual form -will depend upon whether or no their walls have 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. 180, 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, _a_–_e_: 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. - -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, {398} -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. - -―――――――――― - -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 _unequal growth_, 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 _the result_, 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[403]. 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 {399} -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. - -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. - -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[404]. {400} - -(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. - -[Illustration: Fig. 182.] - -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. {401} 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. - -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 _form_ 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. - -[Illustration: Fig. 183. Development of _Sphagnum_. (After Campbell.)] - -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 -(_a_), the three stationary, or {402} undivided quadrants, one of -which has further slowly divided in the stage _b_. 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. - -[Illustration: Fig. 184.] - -(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. 184, 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, {403} 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[405]. 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, - -[Illustration: Fig. 185. Gemma of Moss. (After Campbell.)] - -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. 185 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. - -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. -186 illustrates very well. - -[Illustration: Fig. 186. Development of antheridium of _Riccia_. (After -Campbell.)] - -[Illustration: Fig. 187. Section of growing shoot of Selaginella, -diagrammatic.] - -[Illustration: Fig. 188. Embryo of Jungermannia. (After -Kienitz-Gerloff.)] - -The method of division by means of oblique partitions is a common one -in the case of ‘growing points’; for it evidently {404} 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. 187 which represents -a growing shoot of Selaginella, and somewhat less diagrammatically in -the young embryo of Jungermannia (Fig. 188), 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, {405} 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. - -[Illustration: Fig. 189.] - -(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. - -―――――――――― - -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. -186, 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 {406} 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. - -[Illustration: Fig. 190. Development of sporangium of _Osmunda_. (After -Bower.)] - -In Fig. 190 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 {407} 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. - -―――――――――― - -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. 183), 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. - -[Illustration: Fig. 191. (A, B,) Sections of younger and older embryos -of _Phascum_; (C) do. of _Adiantum_. (After Kienitz-Gerloff.)] - -[Illustration: Fig. 192. Section through frond of _Girardia -sphacelaria_. (After Goebel.)] - -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 {408} surface) -consists of eight cells only. For example, in Fig. 191, 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 Fig. 192 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. - -[Illustration: Fig. 193. Development of antheridium of _Pteris_. (After -Strasbürger.)] - -A curious and beautiful case, apparently aberrant but which would -doubtless be found conforming strictly to physical laws, if {409} -only we clearly understood the actual conditions, is indicated -in 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. 349, 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. - -―――――――――― - -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 {410} -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. - -{411} - - - - -CHAPTER IX - -ON CONCRETIONS, SPICULES, AND SPICULAR SKELETONS - - -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. - -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. {412} - -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[406]. - -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[407]. - -[Illustration: Fig. 194. Alcyonarian spicules: _Siphonogorgia_ and -_Anthogorgia_. (After Studer.)] - -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 {413} 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[408], -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. 194, 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 {414} 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. - -The appearance of direct association with living cells, however, is -apt to be fallacious; for the actual _precipitation_ takes place, -as a rule, not in actively living, but in dead or at least inactive -tissue[409]: 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[410]. - -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 {415} or -its shell. Thus, Pouchet and Chabry[411] 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[412]. - -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[413] 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, {416} 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. - -―――――――――― - -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 -_forms_ 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 _in solution_) 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[414], 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, {417} 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, -“_les dés de la nature sont pipés._” - -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 _form_ 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. - -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 _only_ -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[415], and the distinction still has its weight, I -believe, in the minds of many if not most chemists. - -“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[416]) lies the {418} 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[417].” - -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[418]. - -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[419], 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 {419} -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[420], 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 -(_if confirmed_), 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. - -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[421]. 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 _physical -force_, or mode of _physical action_, 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. - -Our historic interest in the whole question is increased by the -{420} 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[422].” 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. - -―――――――――― - -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. - -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[423].” Professor Harting, after thirty years of experimental -work, published in 1872 a paper, which has become classical, entitled -_Recherches de Morphologie Synthétique, sur la production artificielle -de quelques formations calcaires organiques_; 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. {421} - -[Illustration: Fig. 195. Calcospherites, or concretions of calcium -carbonate, deposited in white of egg. (After Harting.)] - -[Illustration: Fig. 196. A single calcospherite, with central -“nucleus,” and striated, iridescent border. (After Harting.)] - -[Illustration: Fig. 197. Later stages in the same experiment.] - -[Illustration: Fig. 198, A. Section of shell of Mya; B. Section of -hinge-tooth of do. (After Carpenter.)] - -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 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[424]; -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 {422} concentric; the presence of concentric zones or -lamellae, alternately dark and clear, was especially characteristic. -These round “calcospherites” shewed a tendency to aggregate together in -layers, and then to assume polyhedral, or often regularly hexagonal, -outlines. In this latter condition they closely resemble the early -stages of calcification in a molluscan (Fig. 198), or still more in a -crustacean shell[425]; while in their isolated condition {423} 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[426]. - -[Illustration: Fig. 199. Large irregular calcareous concretions, or -spicules, deposited in a piece of dead cartilage, in presence of -calcium phosphate. (After Harting.)] - -When the albumin was somewhat scanty, or when it was mixed with -gelatine, and especially when a little phosphate of lime was {424} -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[427]. - -[Illustration: Fig. 200. Additional illustrations of Alcyonarian -spicules: _Eunicea_. (After Studer.)] - - 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 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[428]”. - - 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 {425} 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[429].” The ossification of bone, - we may be sure, is in the same sense and to the same extent a physical - phenomenon. - -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.) - -[Illustration: Fig. 201. A “crust” of close-packed calcareous -concretions, precipitated at the surface of an albuminous solution. -(After Harting.)] - -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; 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. 202)[430], is likewise due to the same agency. - -[Illustration: Fig. 202. Aggregated calcospherites. (After Harting.)] - -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 _two-phase_ system, which finally -consists of solid particles in suspension in a liquid (the former -being styled the _disperse phase_, the latter the {426} _dispersion -medium_). In accordance with a rule first recognised by Ostwald[431], -when a substance begins to separate out from a solution, so making -its appearance as a _new phase_, it always makes its appearance first -as a liquid[432]. 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. - -[Illustration: Fig. 203. (After Harting.)] - -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. 203). 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 {427} 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[433] is not -obvious. - -―――――――――― - -Among these various phenomena, the concentric striation observed in -the calcospherite has acquired a special interest and importance[434]. -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[435].” - -[Illustration: Fig. 204. Conostats. (After Harting.)] - -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 _raison d’être_ of {428} this phenomenon, -still somewhat problematic, the student must consult the text-books of -physical and colloid chemistry[436]. - -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[437]. 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[438]. - -[Illustration: Fig. 205. Liesegang’s Rings. (After Leduc.)] - -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[439], 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. {429} - -[Illustration: Fig. 206. Relay-crystals of common salt. (After -Bowman.)] - -[Illustration: Fig. 207. Wheel-like crystals in a colloid. (After -Bowman.)] - -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 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. 207; and other results are the -feathery, fern-like {430} or arborescent shapes so frequently seen in -microscopic crystallisation. - -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[440] -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 _Eulalia japonica_), 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[441]. He declares, for instance, that the banded -wings of _Papilio podalirius_ are precisely imitated in Liesegang’s -experiments; that the finer markings on the wings of the Goatmoth -(_Cossus ligniperda_) shew the double arrangement of larger and of -{431} smaller intermediate rhythms, likewise manifested in certain -cases of the same kind; that the alternate banding of the antennae (for -instance in _Sesia spheciformis_), 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[442] 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[443]. 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[444]; and Liesegang himself has applied his results to the -formation of pearls, and to the development of bone[445]. {432} - -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[446]. - -[Illustration: Fig. 208.] - -That this is actually the case in growing starch-grains is generally -believed, on the authority of Meyer[447]; 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. 208), a concentric structure which in -all likelihood has had no causative impulse save from within. - -[Illustration: Fig. 209. Otoliths of Plaice, showing four zones or -“age-rings.” (After Wallace.)] - -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, {433} -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, _identical in number_, -are found both on the scales and in the otoliths[448]. 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 _assume_ 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 -_external_ conditions. - -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[449]”: 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[450]. - -―――――――――― - -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 {434} 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, _crystals_ 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[451] has succeeded -in making them artificially in Harting’s usual way, that is to say by -crystallisation in a colloid medium. - -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 _solubility_ 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[452]. 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 {435} 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[453]; and wherever, in short, the supply -of this salt has been available to the organism. - -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. - -We have still to wait for a similar and equally illuminating study of -the solution and precipitation of _silica_, in presence of organic -colloids. - -―――――――――― - -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[454]. {436} -The two principles of conformation are both illustrated in the -spicular skeletons of the Sponges. - -[Illustration: Fig. 210. Close-packed calcospherites, or so-called -“spicules,” of Astrosclera. (After Lister.)] - -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[455]. 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 _Astrosclera_[456], we have, -to begin with, rounded, striated discs or globules, which in like -manner are nothing more or less than the {437} “calcospherites” of -Harting’s experiments; and as these grow they become closely aggregated -together (Fig. 210), and assume an angular, polyhedral form, once more -in complete accordance with the results of experiment[457]. 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. - -―――――――――― - -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, {438} if we try to make a rough classification, by way of -forecast, of the chief conditions which we are likely to meet with. - -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. - -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 _scale_, 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 -{439} 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. - -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 _between_ the cells, in the partition-walls which separate -them, or in the still more restricted distribution indicated by the -_lines_ 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. - -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. - -―――――――――― - -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 _intrinsic_ powers of growth are -{440} 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. - -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. - -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. - -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. {441} 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. - -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 _projection_ 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 _in the plane of the -surface_, and only diverge from a straight line in space by the actual -curvature of the surface to which they are restrained. - -[Illustration: Fig. 211. Sponge and Holothurian spicules.] - -[Illustration: Fig. 212.] - -[Illustration: Fig. 213. An “amphidisc” of Hyalonema.] - -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 {442} a plane C-shaped figure, but is discovered, -on more careful inspection, to lie not in one plane but in a more -complicated spiral twist. This investigation includes a series of -forms which are abundantly represented among actual sponge-spicules, -as illustrated in Figs. 211 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 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. 213). - -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 {443} 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[458]. 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. - -―――――――――― - -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 _lines of junction_ 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. - -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 {444} third[459]. 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 -_primarily_ 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. - -[Illustration: Fig. 214. Spicules of Grantia and other calcareous -sponges. (After Haeckel.)] - -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 {445} 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. 214, _F_). And this latter condition of equality will be open to -modification in various ways. It will be 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 -{446} on p. 314), we shall tend to obtain a spicule with two equal -angles and one unequal (Fig. 214, _A_, _C_). 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 _axes_, -rather than to the rays themselves. For, if the triradiate spicule be -developed in the _interspace_ 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. 297) 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. - -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. - -The tetrahedral, or rather tetractinellid, spicule needs no explanation -in detail (Fig. 214, _D_, _E_). 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. {447} - -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. - -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. - -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 _surface_ 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 _lines_ 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 {448} 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. - -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 _loci_ 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[460]. - -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. - -It matters little or not at all, for the phenomenon in question, {449} -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 _Entosolenia aspera_, 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. - -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. - -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 {450} 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. - -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. - -[Illustration: Fig. 215. Spicules of tetractinellid sponges (after -Sollas). _a_–_e_, anatriaenes; _d_–_f_, protriaenes.] - -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 -{451} rays, the latter are recurved, as in Fig. 215. 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. 213, 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. - -[Illustration: Fig. 216. Various holothurian spicules. (After Théel.)] - -Just as in the case of the little curved or S-shaped 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. 216). - -A curious and, physically speaking, strictly analogous formation to -the tetrahedral spicules of the sponges is found in the {452} 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[461] 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. - -[Illustration: Fig. 217. Spicules of hexactinellid sponges. (After F. -E. Schultze.)] - -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 {453} 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. - -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[462]. 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[463].” 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. - -However the hexactinellid spicules be arranged (and this is {454} 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[464]. - -―――――――――― - -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—_ein Mittelding_—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[465].” What Haeckel precisely signified by these -words is not clear to me. - -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. {455} - -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 _position_ 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[466]. -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. - -Lastly, Minchin, in a well-known paper[467], 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 -_pressure_, {456} 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 _surface-energy_ 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 _phylogenetic adaptation, which has become fixed -and handed on by heredity, appearing in the ontogeny as a prophetic -adaptation_.” And again, “The forms of the spicules are the result of -adaptation to the requirements of the sponge as a whole, produced by -_the action of natural selection upon variation in every direction_.” -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. 454), the fundamental difference between the Darwinian -conception of the causation and determination of Form, and that which -is characteristic of the physical sciences. - -―――――――――― - -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 {457} 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. - -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[468]. 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 -{458} 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 -_general_ 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[469].” - -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[470]. 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. - -A slight exception to this rule is found in the presence of true -crystals, which occur within the central capsules of certain {459} -Radiolaria, for instance the genus Collosphaera[471]. 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[472]. - -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[473]. 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. - -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, {460} 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 _Aulacantha scolymantha_) 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. - -[Illustration: Fig. 218.] - -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. 218 we have two fluids in contact with one another -(let us call them water and protoplasm), and a body (_b_) which may -be immersed in either, or may be restricted to the boundary {461} -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 (_S_/2)α + (_S_/2)β; but we must also remember that, by -the presence of the particle, a small portion (equal to its sectional -area _s_) of the original contact-surface between water and protoplasm -has been obliterated, and with it a proportionate quantity of energy, -equivalent to _s_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 _S_α + _s_γ, -or _S_β + _s_γ, 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 (_S_/2)(α + β) − _s_γ be less than -either _S_α or _S_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 _radial_ -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. - -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 {462} at this surface, between the two layers -of protoplasm, sufficient to balance the tensions which act directly on -the spicule[474]. - -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[475]. 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 {463} of the -protoplasm, and their arrangement around the radial spicules, all on -the principles of surface-tension[476]. - -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. - -―――――――――― - -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 _Difflugia_, _Astrorhiza_ (Fig. 219) and others, -this covering consists of sand-grains picked up from the surrounding -medium, and sometimes, on the other hand, as in _Quadrula_, 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. - -[Illustration: Fig. 219. Arenaceous Foraminifera; _Astrorhiza limicola_ -and _arenaria_. (From Brady’s _Challenger Monograph_.)] - -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 {465} 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[477]. 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. - -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 _selective action_, which is very often a -conspicuous feature in the phenomenon[478]. 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 {466} -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 _a regular habit_.” To do so is an -exaggerated misuse of anthropomorphic phraseology. - -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[479] 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 {467} 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. - -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[480]. 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. - -―――――――――― - -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. - -[Illustration: Fig. 220. “Reticulum plasmatique.” (After Carnoy.)] - -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 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 {468} 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[481] -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[482] 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. 220). - -[Illustration: Fig. 221. _Aulonia hexagona_, Hkl.] - -[Illustration: Fig. 222. _Actinomma arcadophorum_, Hkl.] - -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. _Aulonia_). Such a conformation is {469} extremely -common, and among its many variants may be found cases in which (e.g. -_Actinomma_), the vesicles have been less regular in size, and some -in which the hexagonal meshwork has been developed not only on one -outer surface, but at successive {470} 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. 223 (_Ethmosphaera_), 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 and have so produced a system of films, normal to -the surface of the sphere, constituting a very perfect honeycomb, as in -_Cenosphaera favosa_ and _vesparia_[483]. - -[Illustration: Fig. 223. _Ethmosphaera conosiphonia_, Hkl.] - -[Illustration: Fig. 224. Portions of shells of two “species” of -_Cenosphaera_: upper figure, _C. favosa_, lower, _C. vesparia_, Hkl.] - -In one or two very simple forms, such as the fresh-water _Clathrulina_, -just such a spherical perforated shell is produced out of some organic, -acanthin-like substance; and in some examples of _Clathrulina_ the -chitinous lattice-work of the shell is just as {471} 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. - -[Illustration: Fig. 225. _Aulastrum triceros_, Hkl.] - -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. 225. 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 {472} 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 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. 226). - -[Illustration: Fig. 226.] - -[Illustration: Fig. 227. A Nassellarian skeleton, _Callimitra -carolotae_, Hkl.] - -―――――――――― - -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. 227. It is obvious at a glance that -this is such a skeleton as may have been formed {473} (I think we -may go so far as to say _must_ 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. 318. 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. 329[484]. - -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. - -[Illustration: Fig. 228. An isolated portion of the skeleton of -_Dictyocha_.] - -[Illustration: Fig. 229. _Dictyocha stapedia_, Hkl.] - -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 {474} edges or borders of the partitions, and here only, -that skeletal formation occurs (giving rise to the netted shell with -its hexagonal meshes of Fig. 221), 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-_lines_, -and not in the interior boundary-_planes_, the whole of the skeletal -matter is aggregated. In Fig. 228 we see a curious little skeletal -structure or complex spicule, whose conformation is easily accounted -for after this fashion. Little spicules such as this form isolated -portions of the skeleton in the genus _Dictyocha_, and occur scattered -over the spherical surface of the organism (Fig. 229). 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. - -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 {475} 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. - -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 _vibrations_, 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[485]. 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. - -[Illustration: Fig. 230.] - -The last peculiarity of our Nassellarian lies in an apparent departure -from what we should at first expect in the way of its {476} 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 (_P_) we tend to get a triradiate, equiangular junction. But -if we introduce another bubble into the centre of the system (Fig. -230), 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 _almost_ -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 {477} 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[486]. - -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 _very nearly_ the condition -which our little skeleton actually displays. But we observe in Fig. -227 that, in the immediate neighbourhood of the tetrahedral angle, the -circular arcs are slightly drawn out into projecting cusps (cf. Fig. -230, _B_). There is no S-shaped 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[487]. - -―――――――――― - -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 {478} 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 _Aulacantha_, 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. - -[Illustration: Fig. 231. Skeletons of various Radiolarians, after -Haeckel. 1. _Circoporus sexfurcus_; 2. _C. octahedrus_; 3. _Circogonia -icosahedra_; 4. _Circospathis novena_; 5. _Circorrhegma dodecahedra_.] - -The matter is sufficiently illustrated by the accompanying figures, -all drawn from Haeckel’s Monograph of the Challenger Radiolaria[488]. -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 _Platonic bodies_[489] of the older mathematicians. It -is at first sight all the more remarkable that we should here meet -{480} 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[490].” 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. - -[Illustration: Fig. 232. _Dorataspis_ sp.; diagrammatic.] - -In certain allied Radiolaria (Fig. 232), which, like the dodecahedral -form figured in Fig. 231, 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 {481} to a -series of five parallel circles on the sphere, corresponding to the -equator (_c_), the tropics (_b_, _d_) and the polar circles (_a_, _e_); -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 description does not -do justice to the facts. We see, in the first place, from such figures -as Figs. 232, 234, that here, unlike our former cases, the radial -spines issue through the facets (and through _all_ 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 {482} 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. - - 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. - -If we construct such a polyhedron, and set it in the position of -Fig. 232, _B_, 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 (_c_) now correspond to the two polar and -to two opposite equatorial facets of our polyhedron: the four “polar” -rays of Müller (_a_ or _e_) correspond to two adjacent hexagons and -two intermediate pentagons of the figure: and Müller’s “tropical” -rays (_b_ or _d_) are those which emanate from the remaining four -pentagonal facets, in each half of the figure. In some cases, such as -Haeckel’s _Phatnaspis cristata_ (Fig. 233), we have an ellipsoidal -body, from which the spines emerge in the order described, but which -is not obviously divided by facets. In Fig. 234 I have indicated the -facets corresponding to the rays, and dividing the surface in the usual -symmetrical way. {483} - -[Illustration: Fig. 233. _Phatnaspis cristata_, Hkl.] - -[Illustration: Fig. 234. The same, diagrammatic.] - -{484} - -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 (_F_), twenty corners (_C_), and therefore thirty-six edges -(_E_), in conformity with Euler’s theorem, _F_ + _C_ = _E_ + 2. - -[Illustration: Fig. 235. _Phractaspis prototypus_, Hkl.] - -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: _aaaa_, _aba_, _abcb_, _bcdc_, -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 -_Phractaspis prototypus_ 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. - -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 {485} 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 _to seek for_ such a solution by considering the case -of Lehmann’s “fluid crystals,” and the light which they throw upon the -phenomena of molecular aggregation. - -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 _Gestaltungskraft_ of Lehmann. - -Such an hypothesis as this had been gradually extruded from the -theories of mathematical crystallography[491]; 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 {486} a pile of oranges becomes -definite, both in outward form and inward structural arrangement, -without the play of any _specific_ 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. - -Now just as some sort of specific “Gestaltungskraft” had been of old -the _deus ex machina_ accounting for all crystalline phenomena (_gnara -totius geometriæ, et in ea exercita_, 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 _specific -properties_, 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 {487} -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 _crystallisation_, whether liquid or solid, has no close -parallel, but only a series of analogies, in the protoplasmic symmetry -of the living cell. - -{488} - - - - -CHAPTER X - -A PARENTHETIC NOTE ON GEODETICS - - -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. - -[Illustration: Fig. 236. Annular and spiral thickenings in the walls of -plant-cells.] - -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. 236, _a_); (2) a series of -straight lines parallel to the axis; and (3) a series of spiral curves -winding round the wall of the cylinder (_b_, _c_). 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. - -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 {489} 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. - -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. - -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 _adjacent_ -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. 236, _c_; -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. - -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. {490} - -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 _a priori_ 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 _upon the surface_ 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. 737 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[492]. If we -consider each muscular fibre as an elastic strand, imbedded in the -elastic membrane which constitutes the wall of the organ, it {491} 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. - -―――――――――― - -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 _twisting_, 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[493]. {492} - -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[494]. 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 _rotate_ 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. - -{493} - - - - -CHAPTER XI - -THE LOGARITHMIC SPIRAL - - -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. - -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 _radius of curvature_ 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, _spirals_ at all, but _screws or helices_. 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. - -[Illustration: Fig. 237. The shell of _Nautilus pompilius_, 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 _Proc. Malacol. Soc._ II, 1897.)] - -Of true organic spirals we have no lack[495]. 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 {494} 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 {495} spires” of -a snake, in the coils of a cuttle-fish’s arm, or of a monkey’s or a -chameleon’s tail. - -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 _attitude_, -than a _form_, 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. - -[Illustration: Fig. 238. A Foraminiferal shell (Globigerina).] - -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 {496} possibly be a physical or dynamical, though -there may well be a mathematical _Law of Growth_, 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. - -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 _older_ 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. - -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, _differ in age_; and they -differ also in magnitude in a strict ratio according to their age. -Somehow or other, in the logarithmic spiral the _time-element_ always -enters in; and to this important fact, full of curious biological as -well as mathematical significance, we shall afterwards return. {497} - -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 _general_ -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. - -[Illustration: Fig. 239.] - -If we imagine growth to act in a perpendicular direction, as for -example the upward force of growth in a growing stem (_OA_), 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 _external -force_, such as the wind, impinging on the growing stem; and suppose -(for simplicity’s sake) that this external force be in a constant -direction (_AB_) 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, {498} 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 _AB_, perpendicular to the original direction _OA_. 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. - -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. - -But if, on the other hand, the deflecting force be _inherent_ 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. - -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, {499} or else by a spiral curve, somewhere within -reach and recognition of it. - -[Illustration: Fig. 240.] - -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 _AB_ be of a magnitude proportional to -_BB′_, and that along _CD_ be of a magnitude proportional to _DD′_; -and in each element parallel to _AB_ and _CD_, let a parallel force -of growth, proportionately intermediate in magnitude, be at work: and -let _EFF′_ be the middle line. Then at any cross-section _BFD_, if -we deduct the mean force _FF′_, we have a certain positive force at -_B_, equal to _Bb_, and an equal and opposite force at _D_, equal to -_Dd_. But _AB_ and _CD_ are not separate 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 _B_ and _D_ are opposed, -accordingly, by a _tension_ in _BD_. It follows therefore, that there -will be a resultant force _BG_, acting in a direction intermediate -between _Bb_ and _BD_, and also a resultant, _DH_, acting at _D_ 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 -_BD_, but in the direction _GH_. 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 “_couple_” which -will impart to our structure a curved form. For, if we regard the part -_ABDC_ as practically rigid, and the part _BB′D′D_ as pliable, this -couple {500} will tend to turn strips such as _B′D′_ about an axis -perpendicular to the plane of the diagram, and passing through an -intermediate point _F′_. It is plain, also, since all the forces under -consideration are _intrinsic to the system_, 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 _BD_ 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 _perfectly extensible_ -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 _straight_ tube. In -other words, in such a structure as we have presupposed, the existence -or 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. - -[Illustration: Fig. 241.] - -[Illustration: Fig. 242.] - -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 _AB_), but more rapidly at _A_ (which we -shall call the “outer edge”) than at _B_, 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, {501} let Fig. 242 represent in -section the early growth of a Nautilus-shell, and let the part _ARB_ -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. 241, _AEDB_ is a quadrilateral with -_AE_, _DB_ parallel, and with the angle _EAB_ of a certain definite -magnitude, = γ. Let _AB_ and _ED_ meet, when produced, in _C_; and -call the angle _ACE_ (or _xCy_) = β. Make the angle _yCz_ = angle -_xCy_, = β. Draw _EG_, so that the angle _yEG_ = γ, meeting _Cz_ in -_G_; and draw _DF_ parallel to _EG_. It is then easy to show that -_AEDB_ and _EGFD_ are similar quadrilaterals. And, when we consider the -quadrilateral _AEDB_ as having infinitesimal sides, _AE_ and _BD_, the -angle γ tends to α, the constant angle of an equiangular spiral which -passes through the points _AEG_, and of a similar spiral which passes -through the points _BDF_; and the point _C_ 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 _EAC_, etc.) is a {502} right angle,—the “spiral” -curve will be a circular arc, _C_ being the centre of the circle. - - 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[496]. - - [Illustration: Fig. 243. _A_, a helicoid, _B_, a scorpioid cyme.] - - In Fig. 243_B_ (which represents the _Cicinnus_ of Schimper, or _cyme - unipare scorpioide_ 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 _in constant ratio_. 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 _in a plane_, 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 _from_ 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 _alternate_ sides, as in Fig. 243 _A_. - This is the _Schraubel_ or _Bostryx_ of Schimper, the _cyme unipare - hélicoide_ 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. - -In the study of such curves as these, then, we speak of the point of -origin as the pole (_O_); a straight line having its extremity in the -pole and revolving about it, is called the radius vector; {503} and a -point (_P_) which is conceived as travelling along the radius vector -under definite conditions of velocity, will then describe our spiral -curve. - -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. - -[Illustration: Fig. 244.] - -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[497].” Or, putting the same thing into -our more modern words, “If, while the radius vector revolve uniformly -about the pole, a point (_P_) travel with uniform velocity along it, -the curve described will be that called the equable spiral, or spiral -of Archimedes.” {504} - -It is plain that the spiral of Archimedes may be compared to a -_cylinder_ coiled up. And it is plain also that a radius (_r_ -= _OP_), made up of the successive and equal whorls, will increase in -_arithmetical_ progression: and will equal a certain constant quantity -(_a_) multiplied by the whole number of whorls, or (more strictly -speaking) multiplied by the whole angle (θ) through which it has -revolved: so that _r_ = _a_θ. - -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 _uniform_ velocity, our point move along -the radius vector with _a velocity increasing as its distance from -the pole_, 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 _geometrical_ progression, as it sweeps through successive -equal angles; and the equation to the spiral will be _r_ = _a_^θ. 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 _cone_ coiled upon itself. - -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 _were_ 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°. {505} - - For, in the logarithmic spiral, when α tends to 90°, the expression - _r_ = _a_^{θ cot α} tends to _r_ = _a_(1 + θ cot α); while the - equation to the Archimedean spiral is _r_ = _b_θ. The nummulite must - always have a central core, or initial cell, around which the coil - is not only wrapped, _but out of which it springs_; and this initial - chamber corresponds to our _a′_ in the expression _r_ = _a′_ + _a_θ - cot α. The outer whorls resemble those of an Archimedean spiral, - because of the other term _a_θ cot α in the same expression. It - follows from this that in all such cases the whorls must be of - excessively small breadth. - -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 _r_ = _a_^θ may be written in the -form log _r_ = θ log _a_, or θ = (log _r_)/(log _a_), or (since _a_ -is a constant), θ = _k_ log _r_. 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. - -[Illustration: Fig. 245.] - -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 _P_ be a force _F_ acting along the -line joining _P_ to a pole _O_ and a force _T_ acting in a direction -perpendicular to _OP_; and let the magnitude of these forces be in -the same constant ratio at all points. It follows that the resultant -of the forces _F_ and _T_ (as _PQ_) 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 _equiangular spiral_. -Hence in a structure growing under the above conditions the form of the -boundary will be a logarithmic spiral. {506} - -[Illustration: Fig. 246.] - -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, _OP′_, whose increment as compared with _OP_ -is _dr_, while _ds_ is the small arc _PP′_, then - - _dr_/_ds_ = cos α = constant. - -[Illustration: Fig. 247.] - -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. 247, three forces to deal with, -acting at a point, _p_: _L_, acting in the direction of the tangent -to the curve, and representing the force of longitudinal growth; _T_, -perpendicular to _L_, and representing the organism’s tendency to -grow in breadth; and _P_, the traction exercised, in the direction -of the pole, by the columellar muscle. Let us resolve _L_ and _T_ -into components along _P_ (namely _A′_, _B′_), and perpendicular to -_P_ (namely _A_, _B_); we have now only two forces to consider, viz. -_P_ − _A′_ − _B′_, and _A_ − _B_. And these two latter we can again -resolve, if we please, so as to deal only with forces in the direction -of _P_ and _T_. Now, the ratio of these forces remaining constant, the -locus of the point _p_ is an equiangular spiral. {507} - -Furthermore we see how any _slight_ change in any one of the forces -_P_, _T_, _L_ 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. - -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 _but does -not change its shape_; 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. - -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[498].” - -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. 502. 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 {508} _symmetrical_ 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 -_asymmetrical_ growth; it grows at one end only, and so does the horn. -And this remarkable property of increasing by _terminal_ growth, but -nevertheless retaining unchanged the form of the entire figure, is -characteristic of the logarithmic spiral, and of no other mathematical -curve. - -[Illustration: Fig. 248.] - -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 _at one end only_. - -―――――――――― - -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. 248, the little -inner cone (represented in its triangular section) may become identical -with the larger one either by magnification all round (as in _a_), or -simply by an increment at one end (as in _b_); 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 {509} on two of its sides, -as in _c_. 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. - -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[499]; and its bearing on our -biological problem of the shell, though apparently indirect, is yet so -close that it deserves our further consideration. - -[Illustration: Fig. 249.] - -[Illustration: Fig. 250.] - -If, as in Fig. 249, we add to two sides of a square a symmetrical -L-shaped 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 (_Phys._ III, 4), a “gnomon.” -Euclid extends the term to include the case of any parallelogram[500], -whether rectangular or not (Fig. 250); 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 -_form_, by means of rows of dots or other signs (cf. Arist. _Metaph._ -1092 b 12), or in the pattern of a tiled floor: all according to “the -mystical way of {510} Pythagoras, and the secret magick of numbers.” -Thus for example, the odd numbers are “gnomonic numbers,” because - - 0 + 1 = 1^2, - - 1^2 + 3 = 2^2, - - 2^2 + 5 = 3^2, - - 3^2 + 7 = 4^2 _etc._, - -which relation we may illustrate graphically σχηματογραφεῖν by the -successive numbers of dots which keep the annexed figure a perfect -square[501]: as follows: - - · · · · · · - · · · · · · - · · · · · · - · · · · · · - · · · · · · - · · · · · · - -[Illustration: Fig. 251.] - -[Illustration: Fig. 252.] - -There are other gnomonic figures more curious still. For instance, if -we make a rectangle (Fig. 251) such that the two sides 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 {511} each half of the -figure, accordingly, is now a gnomon to the other. Another elegant -example is when we start with a rectangle (_A_) whose sides are in the -proportion of 1 : ½(√5 − 1), or, approximately, 1 : 0·618. The gnomon -to this figure is a square (_B_) erected on its longer side, and so on -successively (Fig. 252). - -[Illustration: Fig. 253.] - -[Illustration: Fig. 254.] - -In any triangle, as Aristotle tells us, one part is always a gnomon to -the other part. For instance, in the triangle _ABC_ (Fig. 253), let us -draw _CD_, so as to make the angle _BCD_ equal to the angle _A_. Then -the part _BCD_ is a triangle similar to the whole triangle _ABC_, and -_ADC_ is a gnomon to _BCD_. A very elegant case is when the original -triangle _ABC_ is an isosceles triangle having one angle of 36°, and -the other two angles, therefore, each equal to 72° (Fig. 254). 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[502]. 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[503]. {512} - -[Illustration: Fig. 255.] - -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 _locus_ upon a logarithmic spiral: a result which -follows directly from that alternative definition of the logarithmic -spiral which I have quoted from Whitworth (p. 507). - -[Illustration: Fig. 256. Logarithmic spiral derived from corresponding -points in a system of squares.] - -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 -a {513} collection of either equal or similar figures, similarly -situated, as in Figs. 256, 257, there we can always discover a series -of inscribed or escribed logarithmic spirals. - -[Illustration: Fig. 257. The same in a system of hexagons.] - -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 structure be built up of successive parts, similar -and similarly situated, we can always trace through corresponding -points a series of logarithmic spirals (Figs. 258, 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 _each successive increment of growth is a gnomon to the entire -pre-existing structure_. 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 {514} mathematical accident, to the -nature or mode of development of the actual structure[504]. {515} - -[Illustration: 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.] - -[Illustration: Fig. 259. A spiral foraminifer (_Pulvinulina_), to show -how each successive chamber continues the symmetry of, or constitutes a -_gnomon_ to, the rest of the structure.] - -[Illustration: Fig. 260. Another spiral foraminifer, _Cristellaria_.] - -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. 237) or of a straight Orthoceras (Fig. 300), -each whorl or part of a whorl of a periwinkle or other gastropod -(Fig. 258), each new increment of the operculum of a gastropod (Fig. -263), each additional increment of an elephant’s tusk, or each new -chamber of a spiral foraminifer (Figs. 259 and 260), has its leading -characteristic at once described and its form so far explained by -the simple statement that it constitutes a _gnomon_ 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 {516} 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. - -―――――――――― - -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 _actual_ -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. - -Let us make the assumption that _similar_ increments are added to the -shell in _equal_ 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. - -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 -_hodograph_ of the growth of the contained organism. {517} - -So far as we have now gone, we have studied the elementary properties -of the logarithmic spiral, including its fundamental property of -_continued similarity_; and we have accordingly learned that the shell -or the horn tends _necessarily_ 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. - -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 _angle of the spiral_) with -the polar radius vector. - -[Illustration: Fig. 261.] - -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. 261, when the angle _ROQ_ is -equal to the angle _QOP_, then _OR_ : _OQ_ :: _OQ_ : _OP_. - -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 _P_, _Q_ and _R_ all -lie upon the same radius. {518} - -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[505]. - -For, taking a median transverse section of a _Nautilus pompilius_, 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[506]. Thus (in Fig. 262), -_ab_ is one-third of _bc_, _de_ of _ef_, _gh_ of _hi_, and _kl_ of -_lm_. The curve is therefore a logarithmic spiral.” - -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. - -From the apex of a large specimen of _Turbo duplicatus_[507] a {519} -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. - -[Illustration: Fig. 262.] - - Widths of successive Terms of a geometrical progression, - whorls measured in inches whose first term is the width of - and parts of an inch the widest whorl, and whose - common ratio is 1·1804 - - 1·31 1·31 - 1·12 1·1098 - ·94 ·94018 - ·80 ·79651 - ·67 ·67476 - ·57 ·57164 - ·48 ·48427 - ·41 ·41026 - -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. - -Nevertheless, in order to verify his conclusion still further, and -to get partially rid of the inaccuracies due to successive small -{520} 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 (_m_) of its terms to the sum of half that number of terms, then -the common ratio (_r_) of the series is represented by the formula - - _r_ = (µ − 1)^{2/_m_} .” - -Accordingly, Moseley made the following measurements, beginning from -the second and third whorls respectively: - - Width of - ──────────────────────── - Six whorls Three whorls Ratio µ - - 5·37 2·03 2·645 - 4·55 1·72 2·645 - - Four whorls Two whorls - - 4·15 1·74 2·385 - 3·52 1·47 2·394 - -“By the ratios of the two first admeasurements, the formula gives - - _r_ = (1·645)^{1/3} = 1·1804. - -By the mean of the ratios deduced from the second two admeasurements, -it gives - - _r_ = (1·389)^½ = 1·1806. - -“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 _Turbo duplicatus_ the characteristic number 1·18.” - -By similar and equally concordant observations, Moseley found for -_Turbo phasianus_ the characteristic ratio, 1·75; and for _Buccinum -subulatum_ that of 1·13. - -From the table referring to _Turbo duplicatus_, on page 519, 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. {521} - - -_Turbo duplicatus._ - - Relative widths of Logarithms of Difference of - successive whorls successive whorls successive logarithms - 131 2·11727 — - 112 2·04922 ·06805 - 94 1·97313 ·07609 - 80 1·90309 ·07004 - 67 1·82607 ·07702 - 57 1·75587 ·07020 - 48 1·68124 ·07463 - 41 1·161278 ·06846 - Mean difference ·07207 - -And ·07207 is the logarithm of 1·1805. - -[Illustration: Fig. 263. Operculum of Turbo.] - -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 -_Turbo_, 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 {522} (Fig. 264, 1), which -traces have the form of curved lines in Turbo, and of straight lines -in (e.g.) Nerita (Fig. 264, 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. - -[Illustration: Fig. 264. Opercula of (1) Turbo, (2) Nerita. (After -Moseley.)] - -In a number of such opercula, Moseley measured the breadths of the -successive whorls along a radius vector[508], just in the same way as -he did with the entire shell in the foregoing cases; and here is one -example of his results. - - _Operculum of Turbo sp.; breadth (in inches) of successive whorls, - measured from the pole._ - - Distance Ratio Distance Ratio Distance Ratio Distance Ratio - ·24 ·16 ·2 ·18 - 2·28 2·31 2·30 2·30 - ·55 ·37 ·6 ·42 - 2·32 2·30 2·30 2·24 - 1·28 ·85 1·38 ·94 - -{523} - -The ratio is approximately constant, and this spiral also is, -therefore, a logarithmic spiral. - -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.” - -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.” - -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 {524} 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 _Turbo_; the successive “gnomons” are now not lateral or terminal -additions, but complete concentric rings. - -―――――――――― - -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. - -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. - -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 {525} 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. - -[Illustration: Fig. 265. _Melo ethiopicus_, L.] - -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 _ensemble of similar -closed curves_ 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 _ensemble of spiral lines_ in space, -each spiral having been {526} traced out by the gradual growth and -revolution of a radius vector from the pole to a given point of the -generating curve. - -Both systems of lines, the _generating spirals_ (as these latter may be -called), and the closed _generating curves_ 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 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, _Dolium perdix_, etc., both alike are conspicuous, ridges -and colour-bands intersecting one another in a beautiful isogonal -system. {527} - -[Illustration: Fig. 266. 1, _Harpa_; 2, _Dolium_. The ridges on the -shell correspond in (1) to generating curves, in (2) to generating -spirals.] - -In the complete mathematical formula (such as I have not ventured -to set forth[509]) 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. 541). - -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 {528} obtuse one of Harpa or -Dolium. In short it is obvious that _all_ the differences of form which -we observe between one shell and another are referable to matters of -_degree_, depending, one and all, upon the relative magnitudes of the -various factors in the complex equation to the curve. - -―――――――――― - -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[510]. 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[511].” {529} - -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[512], and which -is represented in Fig. 267. 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 _Terebra dimidiata_, the apical angle was -found to be 13°, the sutural angle 109°, and so forth. - -[Illustration: Fig. 267. D’Orbigny’s Helicometer.] - -It was at once obvious that, in such a shell as is represented in Fig. -267 the entire outline of the shell (always excepting that of the -immediate neighbourhood of {530} 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. - -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 -_Cerithium nodosum_, 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. - -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 _right_ 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 _r_ = θ log α, is no longer constant, but -varies slightly, and in accordance with some simple law. {531} - -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 _Conchospiral_. 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 _r_ = _m_ + _a_^θ. - -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 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 _a_ -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 _a_ 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. - -[Illustration: Fig. 268.] - -―――――――――― - -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 {532} view as our chief -object the investigation (on elementary lines) of the possible manner -and range of variation of the molluscan shell. - -[Illustration: Fig. 269.] - -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: - - _r_ = ε^{θ cot α}. - -This follows directly from the fact that the angle α (the angle between -the radius vector and the tangent to the curve) is constant. - -For, then, - - tan α (= tan ϕ) = _r_ _d_θ/_dr_, - - therefore _dr_/_r_ = _d_θ cot α, - - and, integrating, - - log _r_ = θ cot α, - - or _r_ = ε^{θ cot α}. - -―――――――――― - -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. - -In any complete spiral, such as that of Nautilus, it is (as we have -seen) easy to measure any two radii (_r_), or the breadths in {533} a -radial direction of any two whorls (_W_). We have then merely to apply -the formula - - _r__{_n_ + 1}/_r__{_n_} = _e_^{θ cot α}, or _W__{_n_ + 1}/_W__{_n_} - = _e_^{θ cot α}, - -which we may simply write _r_ = _e_^{θ cot α}, etc.; since our first -radius or whorl is regarded, for the purpose of comparison, as being -equal to unity. - -Thus, in the diagram, _OC_/_OE_, or _EF_/_BD_, or _DC_/_EF_, being in -each case radii, or diameters, at right angles to one another, are all -equal to _e_^{π/2 cot α}. While in like manner, _EO_/_OF_, _EG_/_FH_, -or _GO_/_HO_, all equal _e_^{π cot α}; and _BC_/_BA_, or _CO_/_OB_ -= _e_^{2π cot α}. - -[Illustration: Fig. 270.] - -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. - -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. - -Here we have _r_ = _e_^{2π cot α}, or log _r_ = log _e_ × 2π × cot α, -from which we obtain the following figures[513]: {534} - - Ratio of breadth of each - whorl to the next preceding Constant angle - _r_/1 α - 1·1 89° 8′ - 1·25 87 58 - 1·5 86 18 - 2·0 83 42 - 2·5 81 42 - 3·0 80 5 - 3·5 78 43 - 4·0 77 34 - 4·5 76 32 - 5·0 75 38 - 10·0 69 53 - 20·0 64 31 - 50·0 58 5 - 100·0 53 46 - 1,000·0 42 17 - 10,000 34 19 - 100,000 28 37 - 1,000,000 24 28 - 10,000,000 21 18 - 100,000,000 18 50 - 1,000,000,000 16 52 - -[Illustration: Fig. 271.] - -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, -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[514], 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. 271, 272). Suppose one whorl to be an -inch in breadth, then, if the angle of the spiral were 80°, the {535} -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. - -[Illustration: Fig. 272.] - -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. - -[Illustration: Fig. 273.] - - 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 {536} 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. - - For, taking the formula _r_ = _a_ε^{θ cot α}, - - this, for any given spiral, is equivalent to _a_ε^{_k_θ}. - - Therefore log(_r_/_a_) = _k_θ, - - or, 1/_k_ = θ/log(_r_/_a_). - - Then, if θ increase by 2π, while _r_ increases to _r__{1}, - - 1/_k_ = (θ + 2π)/log(_r__{1}/_a_), - - which leads, by subtraction to - - 1/_k_ ⋅ log(_r__{1}/_r_) = 2π. - - Now, as α tends to 0, _k_ (i.e. cot α) tends to ∞, and therefore, as - _k_ → ∞, log(_r__{1}/_r_) → ∞ and also _r__{1}/_r_ → ∞. - - 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. - -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[515]. {537} - -(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, _GEH_, to the outer whorl at any -point _E_; then draw to the inner whorl a tangent parallel to _GEH_, -touching the curve in some point _F_. The straight line joining the -points of contact, _EF_, must evidently pass through the pole: and, -accordingly, the angle _GEF_ is the angle required. In shells which -bear _longitudinal_ 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 _E_ and _F_ -are very remote. - -[Illustration: Fig. 274.] - -(2) In shells (or horns) shewing rings, or other _transverse_ -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 _AC_, _BD_, is therefore equal to the angle -θ between the polar radii from _A_ and _B_, or from _C_ and _D_; and -therefore _BD_/_AC_ = _e_^{θ cot α}, which gives us the angle α in -terms of known quantities. {538} - -[Illustration: Fig. 275. An Ammonite, to shew corrugated -surface-pattern.] - -[Illustration: Fig. 276.] - -(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. - -[Illustration: Fig. 277.] - -The first method is theoretically simple, but difficult in practice; -for it requires great accuracy in determining the points. Let _AD_, -_DB_, be two tangents drawn to the curve. Then a circle drawn through -the points _ABD_ will pass through the pole _O_; since the angles -_OAD_, _OBE_ (the supplement of _OBD_), are equal. The point _O_ may be -determined by the intersection of two such circles; and the angle _DBO_ -is then the angle, α, required. - -Or we may determine, graphically, at two points, the radii of -curvature, ρ_{1} ρ_{2}. Then, if _s_ 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) - - cot α = (ρ_{1} − ρ_{2})/_s_. - - The following method[516], given by Blake, will save actual - determination of the radii of curvature. - - Measure along a tangent to the curve, the distance, _AC_, at which a - certain small offset, _CD_, is made by the curve; and from another - point _B_, measure the distance at which the curve makes an equal - offset. Then, calling the offset μ; the arc _AB_, _s_; and _AC_, _BE_, - respectively _x__{1}, _x__{2}, we have - - ρ_{1} = (_x__{1}^2 + μ^2)/2μ, approximately, - - and cot α = (_x__{2}^2 − _x__{1}^2)/2μ_s_. - -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 {539} the relative breadths -of the whorl at distances separated by some convenient vectorial angle -(such as 90°, 180°, or 360°). - -Very elaborate measurements of a number of Ammonites have been made by -Naumann[517], by Sandberger[518], and by Grabau[519], 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 _Ammonites_ (_Arcestes_) - - _Ammonites intuslabiatus._ - - Ratio of breadth of - Breadth of whorls successive whorls The angle (α) - (180° apart) (360° apart) as calculated - - 0·30 mm. — — — - 0·30 1·333 87° 23′ - 0·40 1·500 86 19 - 0·45 1·500 86 19 - 0·60 1·444 86 39 - 0·65 1·417 86 49 - 0·85 1·692 85 13 - 1·10 1·588 85 47 - 1·35 1·545 86 2 - 1·70 1·630 85 33 - 2·20 1·441 86 40 - 2·45 1·432 86 43 - 3·15 1·735 85 0 - 4·25 1·683 85 16 - 5·30 1·482 86 25 - 6·30 1·519 86 12 - 8·05 1·635 85 32 - 10·30 1·416 86 50 - 11·40 1·252 87 57 - 12·90 — — — - ────── - Mean 86° 15′ - -{540} - -_intuslabiatus_; 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 _alternate_ -measurements is therefore the same ratio as Moseley adopted, namely the -ratio of breadth between _contiguous whorls_ 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. - -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. - -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 _Arcestes tornatus_. In this case we see -a distinct tendency of the ratios to - - _Ammonites tornatus._ - - Ratio of breadth of - Breadth of whorls successive whorls The spiral angle - (180° apart) (360° apart) (α) as calculated - 0·25 mm. — — — - 0·30 1·400 86° 56′ - 0·35 1·667 85 21 - 0·50 2·000 83 42 - 0·70 2·000 83 42 - 1·00 2·000 83 42 - 1·40 2·100 83 16 - 2·10 2·179 82 56 - 3·05 2·238 82 42 - 4·70 2·492 81 44 - 7·60 2·574 81 27 - 12·10 2·546 81 33 - 19·35 — — — - ────── - Mean 83° 22′ - -{541} - -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. - -―――――――――― - -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. - -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 _the apical angle_ 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. - -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 _lengths_ 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 -_retarded_ in growth as compared with the outer, and as being always -identical with a smaller and earlier part of the latter. - -If λ be the ratio of growth between the outer and the inner curve, -then, the outer curve being represented by - - _r_ = _a_ _e_^{θ cot α}, - -the equation to the inner one will be - - _r′_ = _a_λ_e_^{θ cot α}, - - or _r′_ = _a_ _e_^{(θ − β)cot α}, - -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. {542} - -Dispensing with mathematical formulae, the several conditions may be -illustrated as follows: - -[Illustration: Fig. 278.] - -In the diagrams (Fig. 278), _O_ _P__{1} _P__{2} _P__{3}, etc. -represents a radius, on which _P__{1}, _P__{2}, _P__{3}, are the points -attained by the outer border of the tubular shell after as many entire -consecutive revolutions. And _P__{1}′, _P__{2}′, _P__{3}′, are the -points similarly intersected by the inner border; _OP_/_OP′_ being -always = λ, which is the ratio of growth, or “cutting-down factor.” -Then, obviously, when _O_ _P__{1} is less than _O_ _P__{2}′ the -whorls will be separated by an interspace (_a_); (2) when _O_ _P__{1} -= _O_ _P__{2}′ they will be in contact (_b_), and (3) when _O_ _P__{1} -is greater than _O_ _P__{2}′ 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 (_c_). And as a further case (4), -it is plain that if λ be very large, that is to say if _O_ _P__{1} -be greater, not only than _O_ _P__{2}′ but also than _O_ _P__{3}′, -_O_ _P__{4}′, 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 _Nautilus pompilius_, -and approached, though not quite attained, in _N. umbilicatus_; 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 _P′_ -lies on {543} the opposite side of the radius vector to _P_, and is -therefore imaginary. This final condition is exhibited in Argonauta. - -The limiting values of λ are easily ascertained. - -[Illustration: Fig 279] - -In Fig. 279 we have portions of two successive whorls, whose -corresponding points on the same radius vector (as _R_ and _R′_) are, -therefore, at a distance apart corresponding to 2π. Let _r_ and _r′_ -refer to the inner, and _R_, _R′_ to the outer sides of the two whorls. -Then, if we consider - - _R_ = _a_ _e_^{θ cot α}, - - it follows that _R′_ = _a_ _e_^{(θ + 2π)cot α}, - - _r_ = λ_a_ _e_^{θ cot α} = _a_ _e_^{(θ − β)cot α}, - - and _r′_ = λ_a_ _e_^{(θ + 2π)cot α} = _a_ _e_^{(θ + 2π − β)cot α}. - -Now in the three cases (_a_, _b_, _c_) represented in Fig. 278, it is -plain that _r′_ ⪌ _R_, respectively. That is to say, - - λ_a_ _e_^{(θ + 2π)cot α} ⪌ _a_ _e_^{θ cot α}, - - and λ_e_^{2π cot α} ⪌ 1. - -The case in which λ_e_^{2π cot α} = 1, or −log λ = 2π cot α log ε, is -the case represented in Fig. 278, _b_: 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: - - Constant angle Ratio (λ) of rate of growth of inner border of tube, - α of spiral as compared with that of the outer border - - 89° ·896 - 88 ·803 - 87 ·720 - 86 ·645 - 85 ·577 - 80 ·330 - 75 ·234 - 70 ·1016 - 65 ·0534 - -{544} - -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. - -[Illustration: Fig. 280.] - -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. 278, _a_ and _c_. 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. - -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. 280). {545} - -In this case, let us call _OA_ = _R_, _OC_ = _R__{1} and _OB_ = _r_. We -then have - - _R__{1} = _OA_ = _a_ _e_^{θ cot α}, - - _R__{2} = _OC_ = _a_ _e_^{(θ + 2π) cot α}, - - _R__{1} _R__{2} = _a_ _e_^{2(θ + π) cot α} = _r_^2 [520]. - - And _r_^2 = (1/λ)^2 ⋅ ε^{2θ cot α}, - - whence, equating, 1/λ = _e_^{π cot α}. - -The corresponding values of λ are as follows: - - 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 - Constant angle (α) breadths of the whorls themselves - - 90° 1·00 (imaginary) - 89 ·95 - 88 ·89 - 87 ·85 - 86 ·81 - 85 ·76 - 80 ·57 - 75 ·43 - 70 ·32 - 65 ·23 - 60 ·18 - 55 ·13 - 50 ·090 - 45 ·063 - 40 ·042 - 35 ·026 - 30 ·016 - -As regards the angle of retardation, β, in the formula - - _r′_ = λ_e_^{θ cot α}, or _r′_ = _e_^{(θ − β)cot α}, - -and in the case - - _r′_ = _e_^{(2π − β)cot α}, or −log λ = (2π − β)cot α, - -{546} - -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. - -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. - -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 _rate of growth_ of the outer as compared with -the inner border of the tubular shell. - -―――――――――― - -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. - -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 _Nautilus pompilius_. - -In the turbinate shell, the problem of contact is twofold, for we have -to deal with the possibilities of contact on the _same_ side of the -axis (which is what we have dealt with in the discoid) and {547} also -with the new possibility of contact or intersection on the _opposite_ -side; it is this latter case which will determine the presence or -absence of an _umbilicus_, 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. - -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 -_ensemble_ 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 _Nautilus umbilicatus_, -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 _Solarium perspectivum_, -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[521]; 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 _Scalaria -pretiosa_ 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. {548} - -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 _orthoceracones_ or _bactriticones_, the loosely -coiled forms as _gyroceracones_ or _mimoceracones_, the more closely -coiled shells, in which one whorl overlaps the other, as _nautilicones_ -or _ammoniticones_, 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 _Palaeontology_[522]: “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[523]. _To such phylogenetic hypotheses -the mathematical or dynamical study of the forms of shells lends no -valid support._ 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. {549} - -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 _coincidence_, 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 “_orthogenesis_.” 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. - -―――――――――― - -[Illustration: Fig. 281. An ammonitoid shell (_Macroscaphites_) to shew -change of curvature.] - -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 {550} 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 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 _Bellerophon expansus_ 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. - -―――――――――― - -In order to understand the relation of a close-coiled shell to one of -its straighter congeners, to compare (for example) an {551} 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 _OP_ be a radius vector, _OQ_ a line of reference perpendicular to -_OP_, and _PQ_ a tangent to the curve, _PQ_, or sec α, is equal in -length to the spiral arc _OP_. And this is practically obvious: for -_PP′_/_PR′_ = _ds_/_dr_ = sec α, and therefore sec α = _s_/_r_, or the -ratio of arc to radius vector. - -[Illustration: Fig. 282.] - -Accordingly, the ratio of _l_, the total length, to _r_, the radius -vector up to which the total length is to be measured, is expressed by -a simple table of secants; as follows: - - α _l_/_r_ - - 5° 1·004 - 10 1·015 - 20 1·064 - 30 1·165 - 40 1·305 - 50 1·56 - 60 2·0 - 70 2·9 - 75 3·9 - 80 5·8 - 85 11·5 - 86 14·3 - 87 19·1 - 88 28·7 - 89 57·3 - 89° 10′ 68·8 - 20 85·9 - 30 114·6 - 40 171·9 - 50 343·8 - 55 687·5 - 59 3437·7 - 90 Infinite - -Putting the same table inversely, so as to shew the total {552} length -in whole numbers, in terms of the radius, we have as follows: - - Total length (in terms - of the radius) Constant angle - - 2 60° - 3 70 31′ - 4 75 32 - 5 78 28 - 10 84 16 - 20 87 8 - 30 88 6 - 40 88 34 - 50 88 51 - 100 89 26 - 1000 89 56′ 36″ - 10,000 89 59 30 - -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 {553} 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. - -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[524]. 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. - - -_The Univalve Shell: a summary._ - -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 {554} 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. - -[Illustration: Fig. 283. Section of a spiral, or turbinate, univalve, -_Triton corrugatus_, Lam. (From Woodward.)] - -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 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, _Sigaretus -haliotoides_ (Fig. 284) and Haliotis. In _Nautilus pompilius_ it is -approximately a semi-ellipse, and in _N. umbilicatus_ rather more -than a semi-ellipse, the long axis lying in both cases perpendicular -to the axis of the shell[525]. Its {555} 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 _r_ -= _a_(1 + cos θ). - -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. 258), as in the case -of the opercula of Turbo and Nerita referred to on p. 522; and this is -what Moseley supposed it to do. - -[Illustration: Fig. 284. _A, Lamellaria perspicua; B, Sigaretus -haliotoides._ - -(After Woodward.)] - -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 {556} 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. - -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. - -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. - -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. - -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. {557} - - _Ratio of breadth of consecutive whorls._ - - Pointed Turbinates - - _Telescopium fuscum_ 1·14 - _Acus subulatus_ 1·16 - *_Turritella terebellata_ 1·18 - *_Turritella imbricata_ 1·20 - _Cerithium palustre_ 1·22 - _Turritella duplicata_ 1·23 - _Melanopsis terebralis_ 1·23 - _Cerithium nodulosum_ 1·24 - *_Turritella carinata_ 1·25 - _Acus crenulatus_ 1·25 - _Terebra maculata_ (Fig. 285) 1·25 - *_Cerithium lignitarum_ 1·26 - _Acus dimidiatus_ 1·28 - _Cerithium sulcatum_ 1·32 - _Fusus longissimus_ 1·34 - *_Pleurotomaria conoidea_ 1·34 - _Trochus niloticus_ (Fig. 286) 1·41 - _Mitra episcopalis_ 1·43 - _Fusus antiquus_ 1·50 - _Scalaria pretiosa_ 1·56 - _Fusus colosseus_ 1·71 - _Phasianella bulloides_ 1·80 - _Helicostyla polychroa_ 2·00 - - Obtuse Turbinates and Discoids - - _Conus virgo_ 1·25 - _Conus litteratus_ 1·40 - _Conus betulina_ 1·43 - *_Helix nemoralis_ 1·50 - *_Solarium perspectivum_ 1·50 - _Solarium trochleare_ 1·62 - _Solarium magnificum_ 1·75 - *_Natica aperta_ 2·00 - _Euomphalus pentangulatus_ 2·00 - _Planorbis corneas_ 2·00 - _Solaropsis pellis-serpentis_ 2·00 - _Dolium zonatum_ 2·10 - *_Natica glaucina_ 3·00 - _Nautilus pompilius_ 3·00 - _Haliotis excavatus_ 4·20 - _Haliotis parvus_ 6·00 - _Delphinula atrata_ 6·00 - _Haliotis rugoso-plicata_ 9·30 - _Haliotis viridis_ 10·00 - - Those marked * from Naumann; the rest from Macalister[526]. - -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. 534 is -only applicable to discoid shells, in which the angle θ is an angle of -90°. Our formula, as mentioned on p. 518 now becomes - - _R_ = ε^{2π sin θ cot α}. - -For this formula I have worked out the following table. {558} - - _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._ - - Ratio - _R_/1 θ = 5° 10° 15° 20° 30° 40° 50° 60° 70° 80° 90° - - 1·1 80° 8′ 85° 0′ 86° 44′ 87° 28′ 88° 16′ 88° 39′ 88° 52′ 89° 0′ 89° 4′ 89° 7′ 89° 8′ - 1·25 67 51 78 27 82 11 84 5 85 56 86 50 87 21 87 39 87 50 87 56 87 58 - 1·5 53 30 69 37 76 0 79 21 82 39 84 16 85 13 85 44 86 4 86 15 86 18 - 2·0 38 20 57 35 66 55 73 11 77 34 80 16 81 52 82 45 83 18 83 37 83 42 - 2·5 30 53 50 0 60 35 67 0 73 45 77 13 79 19 80 26 81 11 81 35 81 42 - 3·0 26 32 44 50 56 0 63 0 70 45 74 45 77 17 78 35 79 28 79 56 80 5 - 3·5 23 37 41 5 52 25 59 50 68 15 72 45 75 35 77 2 78 1 78 33 78 43 - 4·0 21 35 38 10 49 35 57 15 66 10 71 3 74 9 75 42 76 47 77 22 77 34 - 4·5 20 0 36 0 47 15 55 5 64 25 69 35 72 54 74 33 75 43 76 20 76 35 - 5·0 18 45 34 10 45 20 53 15 62 55 68 15 71 48 73 31 74 45 75 25 75 38 - 10·0 13 25 25 20 35 15 43 5 53 45 60 20 64 57 67 4 68 42 69 35 69 53 - 20·0 10 25 20 0 28 30 35 45 46 25 53 25 58 52 61 10 63 6 64 10 64 31 - 50·0 8 0 15 35 22 35 28 50 38 45 45 55 52 1 54 18 56 28 57 42 58 6 - 100·0 6 50 13 20 19 30 25 5 34 20 41 15 47 35 49 45 52 3 53 20 53 46 - -{559} - -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 _R_; as follows: - - _R_ θ α - - _Turritella_ sp. 1·12 7° 81° - _Cerithium nodulosum_ 1·24 15 82 - _Conus virgo_ 1·25 70 88 - _Mitra episcopalis_ 1·43 16 78 - _Scalaria pretiosa_ 1·56 26 81 - _Phasianella bulloides_ 1·80 26 80 - _Solarium perspectivum_ 1·50 53 85 - _Natica aperta_ 2·00 70 83 - _Planorbis corneus_ 2·00 90 84 - _Euomphalus pentangulatus_ 2·00 90 84 - -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 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. - -[Illustration: Fig. 285. _Terebra maculata_, L.] - -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 -_T. niloticus_ or _T. zizyphinus_, and about 25° or 26° in _Scalaria -pretiosa_ and _Phasianella bulloides_; it becomes a very acute angle, -of 15°, 10°, or even less, in Eulima, Turritella or Cerithium. The -costly _Conus gloria-maris_, one of the {560} 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. - -[Illustration: Fig. 286. _Trochus niloticus_, L.] - -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 _concave_ or _convex_. The former condition, as we have it in -Cerithium, and in the cusp-like spire of Cassis, 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. - -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 _changed -sign_. 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 {561} “discoid” shell; -and furthermore we have numerous intermediate stages, on either -side, shewing right and left-handed spirals of varying degrees of -acuteness[527]. 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°. - -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. - - -_Of Bivalve Shells._ - -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. - -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 -{562} 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. - -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. - -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: - -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 {563} 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 _AB_ (Fig. 287), is such that - - _AB_ = ½_g_ cos θ ⋅ t^2 - = ½_g_ ⋅ _AB_/_AC_ ⋅ t^2. - - Therefore - - _t_^2 = 2/_g_ ⋅ _AC_. - -[Illustration: Fig. 287.] - -That is to say, all the balls reach the circumference of the circle at -the same moment as the ball which drops vertically from _A_ to _C_. - -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, -_mutatis mutandis_, 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. - -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 -_x_^2/_a_^2 + _y_^2/_b_^2 = 1, where _a_ and _b_ are the major and -minor axes of the ellipse. - -Or, changing the origin to the vertex of the figure - - _x_^2/_a_^2 − 2_x_/_a_ + _y_^2/_b_^2 = 0, - - giving (_x_ − _a_)^2/_a_^2 + _y_^2/_b_^2 = 1. - -Then, transferring to polar coordinates, where _r_ ⋅ cos θ = _x_, -_r_ ⋅ sin θ = _y_, we have - - (_r_ ⋅ cos^2 θ)/_a_^2 − (2 cos θ)/_a_ + (_r_ ⋅ sin θ)/_b_^2 = 0, - -{564} - - which is equivalent to - - _r_ = (2_a_ _b_^2 cos θ)/((_b_^2 cos^2 θ) + (_a_^2 sin^2 θ)), - - or, eliminating the sine-function, - - _r_ = (2_a_ _b_^2 cos θ)/((_b_^2 − _a_^2) cos^2 θ + _a_^2). - -Obviously, in the case when _a_ = _b_, this gives us the circular -system which we have already considered. For other values, or ratios, -of _a_ and _b_, and for all values of θ, we can easily construct a -table, of which the following is a sample: - - _Chords of an ellipse, whose major and minor axes (a, b) are in - certain given ratios._ - - θ _a_/_b_ = 1/3 1/2 2/3 1/1 3/2 2/1 3/1 - 0° 1·0 1·0 1·0 1·0 1·0 1·0 1·0 - 10 1·01 1·01 1·002 ·985 ·948 ·902 ·793 - 20 1·05 1·03 1·005 ·940 ·820 ·695 ·485 - 30 1·115 1·065 1·005 ·866 ·666 ·495 ·289 - 40 1·21 1·11 ·995 ·766 ·505 ·342 ·178 - 50 1·34 1·145 ·952 ·643 ·372 ·232 ·113 - 60 1·50 1·142 ·857 ·500 ·258 ·152 ·071 - 70 1·59 1·015 ·670 ·342 ·163 ·092 ·042 - 80 1·235 ·635 ·375 ·174 ·078 ·045 ·020 - 90 0·0 0·0 0·0 0·0 0·0 0·0 0·0 - -[Illustration: Fig. 288.] - -The coaxial ellipses which we then draw, from the values given in -the table, are such as are shewn in Fig. 288 for the ratio _a_/_b_ -= 3/1, and in Fig. 289 for the ratio _a_/_b_ = ½; 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 {565} formulate a law of -acceleration according to which points starting from the origin _O_, -and moving along radial lines, would all lie, at any future epoch, on -an ellipse passing through _O_; and this calculation we need not enter -into. - -[Illustration: Fig. 289.] - -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. _L. subauriculata_) we have a system -of nearly symmetrical ellipses with the vertical axis about twice the -transverse; in _Solen pellucidus_, 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. - -In certain little Crustacea (of the genus Estheria) the carapace -takes the form of a bivalve shell, closely simulating that of a -{566} 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. - -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[528]. - - 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. {567} - -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. - -[Illustration: Fig. 290. _Caprinella adversa._ (After Woodward.)] - -[Illustration: Fig. 291. Section of _Productus_ (_Strophomena_) sp. -(From Woods.)] - -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 {568} -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. - -―――――――――― - -[Illustration: Fig. 292. Skeletal loop of _Terebratula_. (From Woods.)] - -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 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 _in a plane_, so as to constitute a discoid spiral; for -the original {569} 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 _very_ 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 - -[Illustration: Fig. 293. Spiral arms of _Spirifer_. (From Woods.)] - -[Illustration: Fig. 294. Inwardly directed spiral arms of _Atrypa_.] - -condition defining (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 {570} 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. - - -_The Shells of Pteropods._ - -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. - -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[529]. - -[Illustration: Fig. 295. Pteropod shells: (1) _Cuvierina columnella_; -(2) _Cleodora chierchiae_; (3) _C. pygmaea_. (After Boas.)] - -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. 258), 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 {571} 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. - -[Illustration: Fig. 296. Diagrammatic transverse sections, or outlines -of the mouth, in certain Pteropod shells: A, B, _Cleodora australis_; -C, _C. pyramidalis_; D, _C. balantium_; E, _C. cuspidata_. (After -Boas.)] - -[Illustration: Fig. 297. Shells of thecosome Pteropods (after Boas). -(1) _Cleodora cuspidata_; (2) _Hyalaea trispinosa_; (3) _H. globulosa_; -(4) _H. uncinata_; (5) _H. inflexa_.] - -In certain cases (e.g. Cleodora, Hyalaea) the tube or cone is -curiously modified. In the first place, its cross-section, originally -{572} 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. - -[Illustration: Fig. 298. _Cleodora cuspidata._] - -In the above figures, for instance in that of _Cleodora cuspidata_, -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 _O_, along the radii _Oa_, _Ob_, etc. If -we then plot, as vertical equidistant ordinates, the magnitudes _Oa_, -_Ob_ ... _OY_, and again on to _Oa′_, we obtain a diagram such as the -following (Fig. 299); by {573} 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. - -[Illustration: Fig. 299. Curves obtained by transforming radial -ordinates, as in Fig. 298, into vertical equidistant ordinates. 1, -_Hyalaea trispinosa_; 2, _Cleodora cuspidata_.] - -[Illustration: Fig. 300. Development of the shell of _Hyalaea_ -(_Cavolinia_) _tridentata_, Forskal: the earlier stages being the -“_Pleuropus longifilis_” of Troschel. (After Tesch.)] - -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 -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, {574} 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 _bivalve shell_ -developed out of the original simple cone. - -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. - -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 _hinged_ 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. - -[Illustration: Fig. 301. Pteropod shells, from the side: (1) _Cleodora -cuspidata_; (2) _Hyalaea longirostris_; (3) _H. trispinosa_. (After -Boas.)] - -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 -_tendency_ 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. _Cleodora cuspidata_), the dorsal valve or plate, {575} -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 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[530]. 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; {576} 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. - -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 (_Anaptychus_), or of a thicker, more -calcified plate divided into two symmetrical halves (_Aptychi_), often -found inside the terminal chamber of the Ammonite, and occasionally -to be seen lying _in situ_, 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[531]. -In short, I think there is reason to believe, or at least to suspect, -that we {577} 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. - -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 _Isocardia cor_; 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. - - -_Of Septa._ - -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. 304, etc.). - -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 {578} 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[532]. - -The various forms assumed by the septa in spiral shells[533] 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. - -We do not know in great detail how these septa are laid down; but -the essential facts are clear[534]. 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 _hinder_ 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. - -Let us think, then, of the septa as they would appear in their -uncalcified condition, formed of, or at least superposed upon, an {579} -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_ = _T_(1/_r_ + 1/_r′_). - -Moreover, since the cavity below the septum is practically closed, and -is filled either with air or with water, _P_ 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. - -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. 302, the angle _LCL′_ equals the supplement of the angle (_LOL′_) -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, as in Fig. 303; and here, while the septa still remain -portions of spheres, the geometrical construction for the position of -their centres is equally easy. - -[Illustration: Fig. 302.] - -[Illustration: Fig. 303.] - -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 {580} -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. - -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 - - _A__{1}/_A__{2} = (_r__{1} _r′__{1})/(_r__{2} _r′__{2}) = ε^{−4π cot α}, - -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[535]. - -[Illustration: Fig. 304. Section of _Nautilus_, 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.] - -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 _transverse_ sections, the {581} 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 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 {582} 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 - -[Illustration: Fig. 305. Cast of the interior of _Nautilus_: to shew -the contours of the septa at their junction with the shell-wall.] - -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 {583} 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[536]; 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[537]. - -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 {584} a wire into -the complicated form of the suture-line, and studying the form of the -liquid film which constitutes the corresponding surface _minimae areae_. - -[Illustration: Fig. 306. _Ammonites_ (_Sonninia_) _Sowerbyi_. (From -Zittel, after Steinmann and Döderlein.)] - -[Illustration: Fig. 307. Suture-line of a Triassic Ammonite -(_Pinacoceras_). (From Zittel, after Hauer.)] - -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, 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 {585} few drops of milk upon a -greasy plate, or of oil upon an alkaline solution. - -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. - -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 -_grows_ 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 _relative_ velocities remain constant, -and accordingly the form and symmetry of the whole system remain in -general unchanged. - -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 {586} said, _ex sola nascuntur diversitate gyrationum_; 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 -_differences of degree_. - -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. - -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. - -{587} - - - - -CHAPTER XII - -THE SPIRAL SHELLS OF THE FORAMINIFERA - - -We have already dealt in a few simple cases with the shells of the -Foraminifera[538]; 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 _fluid -drop_, 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. - -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 {588} 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. - -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. - -[Illustration: Fig. 308. _Hastigerina_ sp.; to shew the “mouth.”] - -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. 308), which -interruptions we may doubtless interpret as being due to unequal -distributions of surface energy. In many {589} cases the fluid -protoplasm “picks up” sand-grains and other foreign particles, after -a fashion which we have already described (p. 463); 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[539]. - -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 _Rhabdammina linearis_; and sometimes unduloid -swellings make their appearance on such a thread, just as in _R. -discreta_. And again, by appropriate modifications of the experimental -conditions, it is possible (as Rhumbler has shewn) to build up a -chambered shell[540]. - -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 {590} 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. 440) by the -phenomenon which Lehmann calls _orientirte Adsorption_—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[541]. 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[542]. 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. 320). - -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 {591} 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 _gnomons_ to the entire structure. - -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 _Céphalopodes microscopiques_[543], -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.” - -[Illustration: Fig. 309. _Nummulina antiquior_, R. and V. (After V. von -Möller.)] - -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 -{592} chambered Foraminifera, are convex outwards (Fig. 308), whereas -they are concave outwards in Nautilus (Fig. 304) 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 _outer side_ -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 _inside_ 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. - -The logarithmic spiral is easily recognisable in typical cases[544] -(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 {593} original study of nautilus (cf. p. 518). 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[545]. - -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. - - _F. cylindrica_, Fischer _F. Böcki_, v. Möller - Breadth (in millimetres). - Whorl Observed Calculated Observed Calculated - I ·132 — ·079 — - II ·195 ·198 ·120 ·119 - III ·300 ·297 ·180 ·179 - IV ·449 ·445 ·264 ·267 - V — — ·396 ·401 - -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. - -Here is another of von Möller’s series of measurements of _F. -cylindrica_, the measurements being those of opposite whorls—that is to -say of whorls 180° apart: - - Breadth in mm. ·096 ·117 ·144 ·176 ·216 ·264 ·323 ·395 - Log. of mm. ·982 ·068 ·158 ·246 ·334 ·422 ·509 ·597 - Diff. of logs. — ·086 ·090 ·088 ·088 ·088 ·087 ·088 - -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 _same_ 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. -534) to a constant angle of about {594} 86°, or just such a spiral as -we commonly meet with in the Ammonites[546] (cf. p. 539). - -In Fusulina, and in some few other Foraminifera (cf. Fig. 310, A), 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. - -[Illustration: Fig. 310. A, _Cornuspira foliacea_, Phil.; B, -_Operculina complanata_, Defr.] - -The linear dimensions of successive chambers have been {595} measured -in a number of cases. Van Iterson[547] has done so in various -Miliolinidae, with such results as the following: - - _Triloculina rotunda_, d’Orb. - - No. of chamber 1 2 3 4 5 6 7 8 9 10 - Breadth of chamber in _µ_ — 34 45 61 84 114 142 182 246 319 - Breadth of chamber in _µ_, - calculated — 34 45 60 79 105 140 187 243 319 - -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. - -Again, Rhumbler has measured the linear dimensions of a number of -rotaline forms, for instance _Pulvinulina menardi_ (Fig. 259): 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[548]. - -[Illustration: Fig. 311. 1, 2, _Miliolina pulchella_, d’Orb.; 3–5, _M. -linnaeana_, d’Orb. (After Brady.)] - -[Illustration: Fig. 312. _Cyclammina cancellata_, Brady.] - -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. 311); 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. 312), -the angles subtended, though of less magnitude, are still remarkably -constant, as we {597} may see by Fig. 313; 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 _r_ = _e_^{θ cot α}. It is probable that, -by so taking account of variations of θ, such variations of _r_ as -(according to Rhumbler’s measurements) Pulvinulina and other genera -appear to shew, would be found to diminish or even to disappear. - -[Illustration: Fig. 313. _Cyclammina_ sp. (Diagrammatic.)] - -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 _V__{0}, _V__{1} etc., be the volumes of the successive -chambers, let _V__{1} bear a constant proportion to _V__{0}, so that -_V__{1} = _q_ _V__{0}, and let _V__{2} bear the same proportion to the -whole pre-existing volume: then - - _V__{2} = _q_(_V__{0} + _V__{1}) = _q_(_V__{0} + _q_ _V__{0}) - = _q_ _V__{0}(1 + _q_) and _V__{2}/_V__{1} = 1 + _q_. - -{598} - -This ratio of 1/(1 + _q_) 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. - -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.” - -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 _drop_. 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. - -[Illustration: Fig. 314. _Orbulina universa_, d’Orb.] - -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 {599} 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. 294); though -some slight modification must be made in our definitions, inasmuch as -the consideration of surface-_tension_ is no longer appropriate at -the solid surfaces, and the concept of surface-_energy_ 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 (_inter alia_) 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. - -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, {600} 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. 315), -it would seem to be (and Rhumbler 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. 259), 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. - -[Illustration: Fig. 315. _Cristellaria reniformis_, d’Orb.] - -{601} - -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, _in general terms_, 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. - -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. {602} - -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 _classification_ 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. - -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. - -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. 316), where the organism lives attached to a rock or a frond of -sea-weed; for here (just as in the case of {603} 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. - -[Illustration: Fig. 316. _Discorbina bertheloti_, d’Orb.] - -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 _enclosing_ the old one, {604} 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[549].” - -The various Miliolidae (Fig. 311), 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 _outward_ 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 _outline_ of the shell, but is always a -line drawn through corresponding points in the successive chambers of -the latter. - -[Illustration: Fig. 317. A, _Tertularia trochus_, d’Orb. B, _T. -concava_, Karrer.] - -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. 87, p. 262); 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.” {605} - -In Textularia and its allies (Fig. 317), we have a precise parallel -to the helicoid cyme of the botanists (cf. p. 502): 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. - -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°. - -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. - -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 _Amoeba proteus_, -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 _A. radiosa_,—and again when the water is -rendered a little more alkaline, to turn apparently into the so-called -_A. limax_,—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. {606} - -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[550]. 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 _in no other way can they be divided into_ ‘_species._’[551]” -Some writers have wondered at the peculiar variability of this -particular shell[552]; 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[553]. {607} - - -_Conclusion._ - -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. - -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 _time_. 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 {608} 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[554] 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 _biogenetisches -Grundgesetz_ 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 {609} 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[555]. - -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 -_strength_. Increase of strength, _die Festigkeitssteigerung_, 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[556]. - -In the course of the same argument Rhumbler remarks that Foraminifera -are absent from the coarse sands and gravels[557], as Williamson indeed -had observed many years ago: so averting, or {610} 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. - -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[558]. - -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. - -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 _living_ 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 {611} 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. - -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. - -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[559] 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. - -{612} - - - - -CHAPTER XIII - -THE SHAPES OF HORNS, AND OF TEETH OR TUSKS: WITH A NOTE ON TORSION - - -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. - -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. - -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 {613} 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 _plane_ 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 _essentially_ the same -shape, the spiral curvature is less manifest in the second one, simply -by reason of its comparative shortness. - -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. - -The growth of the horny sheath is not continuous, but more or {614} -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[560]. - -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 _Ovis -Ammon_. 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[561]. 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 {615} have called in -a preceding chapter the “_ensemble_ of generating spirals” which -constitute the surface. - -[Illustration: Fig. 318. Diagram of Ram’s horns. (After Sir Vincent -Brooke, from _P.Z.S._) _a_, frontal; _b_, orbital; _c_, nuchal surface.] - -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 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[562]. 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 {616} 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. - -[Illustration: Fig. 319. Head of Arabian Wild Goat, _Capra sinaitica_. -(After Sclater, from _P.Z.S._)] - -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 {617} 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, _Ovis Ammon_, 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 _Ovis Ammon_, 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. - -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. - -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. 555) the angle _θ_. 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 {618} _Ovis Poli_, four feet or more from tip to -tip, differ conspicuously from those of _Ovis Ammon_ or _O. hodgsoni_, -in which a very similar logarithmic spiral is wound (as it were) round -a much blunter cone. - -―――――――――― - -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. - -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 _Ovis Ammon_, 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 _tapering_ of the horn due -to the growth in its own plane of the generating curve. {619} - -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, _Capra falconeri_. - - 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 _homonymous_, 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 _heteronymous_, - 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. - 560), it has no very deep importance. - -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 _will_ -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 _will_ occur at precisely opposite ends of a diameter, for -{620} no such plane of symmetry is manifested in the field of force to -which the growing annulus corresponds or appertains. - -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. {621} - -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. - -[Illustration: Fig. 320. Head of _Ovis Ammon_, shewing St Venant’s -curves.] - -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. 618), 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 {622} 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. - -Now we know, from the celebrated experiments of St Venant[563], 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. 321, 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 _Ovis Ammon_, 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 {623} 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 _cylindrical_ horns of -oxen. - -[Illustration: Fig. 321.] - -[Illustration: Fig. 322.] - -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 _Ovis Ammon_, 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[564]. -Finally, after making a plaster cast of this sectional surface, I drew -its contour-lines (as shewn in Fig. 322), with the help of a simple -form of spherometer. It will be seen that in great part this diagram -is precisely 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 {624} 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 _O. Ammon_ and its -immediate congeners[565]. - - -_A further Note upon Torsion._ - -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. - -The subject of climbing plants has been elaborately dealt with not -only in Darwin’s books[566], 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[567]. 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 {625} ascribed, such as contact -(thigmotaxis), exposure to light (heliotropism), and so forth, need not -be discussed here[568]. - -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 (_Solanum Dulcamara_) which twine -indifferently either way. - -Together with this circumnutatory movement, there is very generally -to be seen an actual _torsion_ 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 {626} 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. - -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 {627} of _torsion_, so manifest in the former -case, will be absent in the latter. - -―――――――――― - -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 _bending_, actually make it much -weaker than before (for the same amount of metal per unit length) in -the way of resistance to _torsion_. - -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 {628} 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. - - -_Of Deer’s Antlers._ - -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[569]. 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. - -[Illustration: Fig. 323. Antlers of Swedish Elk. (After Lönnberg, from -_P.Z.S._)] - -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[570]. 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: {629} 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 _axial rod_, while the -“antler” is essentially an outspread _surface_[571]. 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 -{630} off branches (or tines), the interspaces between which latter -may sometimes be filled up to form a continuous plate. - -[Illustration: Fig. 324. Head and antlers of a Stag (_Cervus -Duvauceli_). (After Lydekker, from _P.Z.S._)] - -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 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 -_conforms_ 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 {631} 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 _plane_ 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 _a single surface_, and constitute a -symmetrical figure, each half being the mirror-image of the other. - -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. - -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 (_op. cit._ p. 892), “No two antlers are ever -exactly alike; and the {632} 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[572].” 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. - -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. - - -_Of Teeth, and of Beak and Claw._ - -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 _always_ 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; {633} 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. - -Let us glance at one or two instances to illustrate the spiral -curvature of teeth. - -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. - -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 {634} 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 _Mastodon angustidens_, or _M. -arvernensis_, 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. - -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. - -{635} - - - - -CHAPTER XIV - -ON LEAF-ARRANGEMENT, OR PHYLLOTAXIS - - -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. - -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. - -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 {636} 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. - -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[573]. - -Starting from some given level and proceeding upwards, let us mark -the position of some one leaf (_A_) upon a cylindrical stem. Another, -and a younger leaf (_B_) will be found standing at a certain distance -_around_ the stem, and a certain distance _along_ the stem, {637} -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 _B_ relatively to _A_ 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 _B_ above the level of _A_, 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 _F_(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 _azimuth_. - -Whatever relation we have found between _A_ and _B_, let precisely the -same relation subsist between _B_ and _C_: 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 _A_, _B_, -_C_, etc. But if we can trace such a helix through _A_, _B_, _C_, it -follows from the symmetry of the system, that we have only to join _A_ -to some other leaf to trace another spiral helix, such, for instance, -as _A_, _C_, _E_, etc.; parallel to which will run another and similar -one, namely in this case _B_, _D_, _F_, etc. And these spirals will run -in the opposite direction to the spiral _ABC_. - -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. - -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 _A_, _B_, _C_, etc., -and _A′_, _B′_, _C′_, and so on. These are the cases which we call -“whorled” leaves, or in the simplest case, where {638} the whorl -consists of two opposite leaves only, we call them decussate. - -―――――――――― - -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. - -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 _cylinder_: 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 {639} 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. - -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[574]. - -―――――――――― - -According to Mr A. H. Church[575], 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 _Curves -of Life_, has adopted and has helped to expound and popularise Mr -Church’s investigations. - -Mr Church, regarding the problem as one of “uniform growth,” easily -arrives at the conclusion that, _if_ 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 {640} 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[576]. 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. - -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 _nothing to see_.” - -But it is impossible to deal satisfactorily, in brief space, either -with Mr Church’s theories, or my own objections to them[577]. 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 {641} -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. - -Let us, however, leave this discussion, and return to the facts of the -case. - -―――――――――― - -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. - -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. - -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 {642} on the whole very constant in number, -according to the species. - -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. - -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 {643} cones of -the Norway spruce, Beal[578] 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. - -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[579]. 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 {644} 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[580]. - -[Illustration: Fig. 325.] - -Of the two following diagrams, Fig. 325 represents the general case, -and Fig. 326 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. _A_, -_a_, at the two ends of the base-line, represent the same initial leaf -or scale: _O_ is a leaf which can be reached from _A_ by _m_ steps in a -right-hand spiral (developed into the straight line _AO_), and by _n_ -steps from _a_ in a left-handed spiral _aO_. Now it is obvious in our -fir-cone, that we can include _all_ 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 _AO_ and _aO_ _must_, and always _can_, be so taken that -_m_ spirals parallel to _aO_, and _n_ spirals parallel to _AO_, shall -separately include all the leaves upon the stem or cone. - -If _m_ and _n_ have a common factor, _l_, it can easily be shewn that -the arrangement is composite, and that there are _l_ fundamental, or -genetic spirals, and _l_ leaves (including _A_) which are situated -exactly on the line _Aa_. That is to say, we have here a _whorled_ -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 _therefore_ _m_ and _n_ -are prime to one another. - -Our fundamental, or genetic, spiral, as we have seen, is that which -passes from _A_ (or _a_) to the leaf which is situated nearest to -the base-line _Aa_. The fundamental spiral will thus be right-handed -(_A_, _P_, etc.) if _P_, which is nearer to _A_ than to _a_, be this -leaf—left-handed if it be _p_. That is to say, we make it a convention -that we shall always, for our fundamental spiral, run {645} round the -system, from one leaf to the next, _by the shortest way_. - -[Illustration: Fig. 326.] - -Now it is obvious, from the symmetry of the figure (as further shewn -in Fig. 326), that, besides the spirals running along _AO_ and _aO_, -we have a series running _from the steps on_ _aO_ to the steps on -_AO_. In other words we can find a leaf (_S_) upon _AO_, which, like -the leaf _O_, is reached directly by a spiral series from _A_ and from -_a_, such that _aS_ includes _n_ steps, and _AS_ (being part of the -old spiral line _AO_) now includes _m_ − _n_ steps. And, since _m_ and -_n_ 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 1, 1 arrangement, that -is to say to a leaf which is reached by a single step, in opposite -directions from _A_ and from _a_, which leaf is therefore the first -leaf, next to _A_, of the fundamental or generating spiral. {646} - -If our original lines along _AO_ and _aO_ contain, for instance, 13 and -8 steps respectively (i.e. _m_ = 13, _n_ = 8), then our next series, -observable in the same cone, will be 8 and (13 − 8) or 5; the next 5 -and (8 − 5) or 3; the next 3, 2; and the next 2, 1; leading to the -ultimate condition of 1, 1. 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, -_ipso facto_, the presence of the rest. - -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 _AT_, _aT_ -of Fig. 326. 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. 326, it is obvious -that these two series (_A_, 1, 2, 3, etc., and _a_, 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 -_more than the circumference_ of the cylindrical stem; in other words, -that there were more than two, but _less than three_ leaves in a single -turn of the fundamental spiral. - -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 1, 2; 2, 3; 3, 5, 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 _t_, where _t_ is -determined by the condition that 1 and _t_ + 1 would give spirals both -right-handed, or both left-handed; it follows that there are less -than _t_ + 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 _t_ -= 2. In other words, in the {647} 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. - -“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.[581] The other natural series, usually but -misleadingly represented by convergents to an infinitely extended -continuous fraction, are easily explained, as above, by taking _t_ = 3, -4, 5, etc., etc.” Many examples of these latter series have been given -by Dickson[582] and other writers. - -―――――――――― - -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. 326, 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. - -[Illustration: Fig. 327.] - -The phenomenon is illustrated by Fig. 327, _a_–_d_. 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 {648} -which various spirals are implicitly contained, the conspicuousness -of one set or another does not depend upon angular divergence. It -depends on the 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 {649} result is to give us -arrangements corresponding to the middle diagrams in Fig. 327, 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. - -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[583]. - -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 _sectio aurea_ 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°), {650} into the thrice-isosceles triangle of -which we have spoken on p. 511; 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[584], 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.” - -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[585], 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 (_K_), in which the first two or any two successive -ones divide the circumference. Now 5/8 and all successive fractions -differ inappreciably from _K_.” 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 {651} 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 _F_(ϕ ⋅ sin θ) 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. - -{652} - - - - -CHAPTER XV - -ON THE SHAPES OF EGGS, AND OF CERTAIN OTHER HOLLOW STRUCTURES - - -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. - -The careful study and collection of birds’ eggs would seem to have -begun with the Count de Marsigli[586], the same celebrated naturalist -who first studied the “flowers” of the coral, and who wrote the -_Histoire physique de la mer_; 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[587]. {653} - -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[588] 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[589], and adopted by Des Murs[590] and many other well-known -writers. - -In a treatise by de Lafresnaye[591], 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[592]” 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[593], 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. - -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, {654} 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[594]. -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 _vera causa_, 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[595]. - -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[596], -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. - -Let us consider, very briefly, the conditions to which the egg is -subject in its passage down the oviduct[597]. - -(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, {655} 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. - -(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. - -(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. - -(4) It is known by observation that a hen’s egg is always laid blunt -end foremost. - -(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 {656} 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[598]. - -―――――――――― - -We may now proceed to inquire more particularly how the form of the egg -is controlled by the pressures to which it is subjected. - -The egg, just prior to the formation of the shell, is, as we have seen, -a fluid body, tending to a spherical shape and _enclosed within a -membrane_. - -Our problem, then, is: Given a practically incompressible fluid, -contained in a deformable capsule, which is either (_a_) entirely -inextensible, or (_b_) slightly extensible, and which is placed in -a long elastic tube the walls of which are radially contractile, to -determine the shape under pressure. - -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. - -If the capsule be not spherical, but be inextensible, then deformation -can take place under the external radial compression, {657} 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. - -Let us next assume, as the conditions by which this result may be -avoided, (_a_) that the envelope is to some extent extensible, or (_b_) -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. - -At all points the shape is determined by the law of the distribution of -_radial pressure within the given region of the tube_, 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. - -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. - -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 -_p_ cos^2 θ = _p′_ cos^2 θ′, so that at the more tapered end where θ -is small _p_ is small. Therefore the whole structure might assume such -a configuration, or grow under such conditions, finally becoming rigid -by solidification of the envelope. {658} 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. - -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 -_p_{n}_ + _T_(1/_r_ + 1/_r′_) = _P_, where _p_{n}_ is the normal -component of external pressure at a point where _r_ and _r′_ are the -radii of curvature, _T_ is the tension of the envelope, and _P_ the -internal fluid pressure. This is simply the equation of an elastic -surface where _T_ 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 _T_ (i.e. the surface-tension of the -semi-fluid cell) and had little or nothing to do with the factor of -external pressure (_p_{n}_), which in the case of the egg becomes of -chief importance. - -The above equation is the _equation of equilibrium_, 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 _T_. -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 {659} peristaltic wave which causes the movement, as well as by -friction. - -The quantity _T_ is the tension of the enclosing capsule—the -surrounding membrane. If _T_ 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 _T_. Not only asymmetry of _T_, but also asymmetry of _p_{n}_, -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 _all eggs_ in the universe. At least this is so if -we generalise it in the form _p_{n}_ + _T_/_r_ + _T′_/_r′_ = _P_ in -recognition of a possible difference between the principal tensions. - -In the case of the spherical egg it is obvious that _p_{n}_ is -everywhere equal. The simplest case is where _p_{n}_ = 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 _p_{n}_ = 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. - -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 _r_ and _r′_ are identical, and they are greater at the -blunt anterior end than at the other. If we may assume that _p_{n}_ -vanishes at the poles of the egg, then it is plain that _T_ varies in -the neighbourhood of these poles, and, further, that the tension _T_ 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 {660} even -to burst, and it is here that we most commonly find irregularities of -shape in abnormal eggs. - -If one portion of the envelope were to become practically stiff before -_p_ ceases to vary, that would be tantamount to a sudden variation of -_T_, and would introduce asymmetry by the imposition of a boundary -condition in addition to the above equation. - -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 _p_{n}_ 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. - -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. - -―――――――――― - -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[599] of -Cairo has {661} explained the curious form of the egg in _Bilharzia_ -(_Schistosoma_) _haematobium_, 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, _S. Mansoni_. -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, _S. japonicum_, -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. - -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. - - -_On the Form of Sea-urchins_ - -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 {662} 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. - -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. - -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. - -While the sea-urchin is alive, an immense number of delicate -“tube-feet,” with suckers at their tips, pass through minute pores -{663} 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[600]; 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. - -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. 219), -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 _T_, 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. 328). In a very few cases, such as the fossil Palaeechinus, -we have an approximately spherical {664} 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. 328 are no other than diagrammatic -illustrations - -[Illustration: Fig. 328. Diagrammatic vertical outlines of various -Sea-urchins: A, Palaeechinus; B, _Echinus acutus_; C, Cidaris; D, D′ -Coelopleurus; E, E′ Genicopatagus; F, _Phormosoma luculenter_; G, P. -_tenuis_; H, Asthenosoma; I, Urechinus.] - -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 {665} 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. - - -_On the Form and Branching of Blood-vessels_ - -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. - -We know that the fluid pressure (_P_) within the vessel is balanced by -(1) the tension (_T_) of the wall, divided by the radius of curvature, -and (2) the external pressure (_p_{n}_), normal to the wall: according -to our formula - - _P_ = _p_{n}_ + _T_(1/_r_ + 1/_r′_). - -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 _P_ = _T_/_R_. 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. - -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 -{666} 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. - -―――――――――― - -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, _t_(1/_r_ + 1/_r′_) -= _C_, holds good; that is to say, that the thickness (_t_) 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[601]. 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[602]. - -―――――――――― - -A great many other problems of a mechanical or hydrodynamical kind -arise in connection with the blood-vessels[603], 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 {667} 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[604], “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 - -[Illustration: Fig. 329.] - -large artery, _AB_, give off a comparatively narrow branch leading -to _P_ (such as _CP_, or _DP_), the route _ACP_ is evidently shorter -than _ADP_, but on the other hand, by the latter path, the blood has -tarried longer in the wide vessel _AB_, 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 _CD_, as compared with that -in the alternative portion _CD′_, 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 {668} 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°. - -[Illustration: Fig. 330.] - -We may follow Hess in a further investigation of this phenomenon. Let -_AB_ be an artery, from which a branch has to be given off so as to -reach _P_, and let _ACP_, _ADP_, etc., be alternative courses which -the branch may follow: _CD_, _DE_, etc., in the diagram, being equal -distances (= _l_) along _AB_. Let us call the angles _PCD_, _PCE_, -_x__{1}, _x__{2}, etc.: and the distances _CD′_, _DE′_, by which each -branch exceeds the next in length, we shall call _l__{1}, _l__{2}, etc. -Now it is evident that, of the courses shewn, _ACP_ 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 _CGP_, 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. - -Now it is easy to shew that if, in Fig. 330, the route _ADP_ and _AEP_ -(two contiguous routes) be equally favourable, then any other route on -either side of these, such as _ACP_ or _AFP_, must be less favourable -than either. Let _ADP_ and _AEP_, 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. {669} Then, if we make the distance -_DE_ very small, the angles _x__{2} and _x__{3} are nearly equal, -and may be so treated. And again, if _DE_ be very small, then _DE′E_ -becomes a right angle, and _l__{2} (or _DE′_) = _l_ cos _x__{2}. - -But if _L_ be the loss of energy per unit distance in the wide tube -_AB_, and _L′_ be the corresponding loss of energy in the narrow tube -_DP_, etc., then _lL_ = _l__{2} _L′_, because, as we have assumed, the -loss of energy on the route _DP_ is equal to that on the whole route -_DEP_. Therefore _lL_ = _lL′_ cos _x__{2}, and cos _x__{2} = _L_/_L′_. -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. - -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 _factors_ 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[605]) 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][606] are so -much the same.” - -{670} - - - - -CHAPTER XVI - -ON FORM AND MECHANICAL EFFICIENCY - - -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 _adaptation_, 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. - -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[607], 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. {671} - -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[608].” 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. - -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[609]. - -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[610]. 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. - -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. - -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 {672} had brown -spots over his eyes that he might seem awake when he was sleeping[611]: -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[612]. - -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[613], “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 _ignava ratio_, 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.” _Mutatis mutandis_, the passage -carries its plain message to the naturalist. - -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. - -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 {673} 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[614]. - -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. - -―――――――――― - -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[615], in -{674} 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[616].” However, be that as it may, we know, - -[Illustration: Fig. 331.] - -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 {675} 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[617]. 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 _none_ 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[618]. - -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. {676} - - _Average Strength of Materials (in kg. per sq. mm.)._ - - Tensile Crushing - strength strength - Steel 100 145 - Wrought Iron 40 20 - Cast Iron 12 72 - Wood 4 2 - Bone 9–12 13–16 - -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. - -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 I or an -⌶. - -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. -{677} - -The I or the H-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 I-girder 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 _stiffness_ 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[619]. - -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. - -[Illustration: Fig. 332.] - -The second point requires a little more explanation. If we {678} -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. {679} Schwendener[620] -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. - -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. - - Stress, or load in gms. Strain, or amount - per sq. mm., at of stretching, - Limit of Elasticity per mille - _Secale cereale_ 15–20 4·4 - _Lilium auratum_ 19 7·6 - _Phormium tenax_ 20 13·0 - _Papyrus antiquorum_ 20 15·2 - _Molinia coerulea_ 22 11·0 - _Pincenectia recurvata_ 25 14·5 - Copper wire 12·1 1·0 - Brass wire 13·3 1·35 - Iron wire 21·9 1·0 - Steel wire 24·6 1·2 - -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 {680} stronger bundles; sometimes with these bundles -further strengthened by radial balks or ridges; sometimes with all the -fibres set - -[Illustration: Fig. 333.] - -close together in a continuous hollow cylinder. In the case figured -in Fig. 333 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. 332; 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[621]. - -―――――――――― - -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. {681} - -[Illustration: Fig. 334. Head of the human femur in section. (After -Schäfer, from a photo by Prof. A. Robinson.)] - -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 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 {682} 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 - -[Illustration: Fig. 335. Crane-head and femur. (After Culmann and H. -Meyer.)] - -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. 332. In the shaft of the -crane, the concave {683} 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. -335) 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 -_notch_ 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[622]. {684} - -_Mutatis mutandis_, the same phenomenon may be traced in any other bone -which carries weight and is liable to flexure; and in the _os calcis_ -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. - -[Illustration: Fig. 336. Diagram of stress-lines in the human foot. -(From Sir D. MacAlister, after H. Meyer.)] - -Thus, in the _os calcis_, 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. - -―――――――――― - -[Illustration: Fig. 337. Trabecular structure of the os calcis. (From -MacAlister.)] - -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 {685} 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 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. - - Somewhat more generally, if _AB_ be a bar, or pillar, of cross-section - _a_ under a direct load _P_, giving a stress per unit area = _p_, - then the whole pressure _P_ = _pa_. Let _CD_ be an oblique section, - inclined at an angle θ to the cross-section; the pressure on _CD_ - will evidently be = _pa_ cos θ. But at any point _O_ in _CD_, - the pressure _P_ may be resolved into the force _Q_ acting along - _CD_, and _N_ perpendicular to it: where _N_ = _P_ cos θ, and _Q_ - = _P_ sin θ = _pa_ sin θ. The whole force _Q_ upon _CD_ = _q_ ⋅ area - of _CD_, which is = _q_ ⋅ _a_/(cos θ). {686} Therefore _qa_/(cos θ) - = _pa_ sin θ, therefore _q_ = _p_ sin θ cos θ, = ½_p_ sin 2θ. - Therefore when sin 2θ = 1, that is, when θ = 45°, _q_ is a maximum, - and = _p_/2; and when sin 2θ = 0, that is when θ = 0° or 90°, then _q_ - vanishes altogether. - -[Illustration: Fig. 338.] - -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. - -In short we see that, while shearing _stresses_ 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 -_along these lines_ there is no shear. - -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 {687} 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. - -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 _shaft_ of a -long bone, the trabecular meshwork is wholly altered and reconstructed -within the distant _extremities_ 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. - -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, {688} whose -physical cause is as obscure as its final cause or end is, apparently, -manifest. - -―――――――――― - -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 _strain_, the result of a -_stress_, 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[623], 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 -_strength_ 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. {689} - -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. - -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. {690} 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. - -―――――――――― - -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[624]; 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 {691} 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[625]. This framework -is complicated as we see it in the skeleton, where (as we have said) -it is only, or chiefly, the _struts_ 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 -_ties_, 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. - -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 _Treatise on -Bridge-Construction_, 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 _skeletons_[626]; and (at the -cost of a little pedantry) {692} 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. - -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 -_flexibility_ 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 _horizontal thrust_ of any _arch_ 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 _arch_, in the -{693} engineer’s sense of the word. It resembles an arch in _form_, -but not in _function_, for it cannot act as an arch unless it be held -back at each end (as every arch is held back) by _abutments_ 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 _external_ -reactions, which in the simple arch are obviously indispensable. -Thus, for example, we may begin by inserting a straight steel tie, -_AB_ (Fig. 339), uniting the ends of the curved rib _AaB_; 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. 339_b_). In either case, the -structure becomes an - -[Illustration: Fig. 339.] - -independent “detached girder,” supported at each end but -not otherwise fixed, and consisting essentially of an upper -compression-member, _AaB_, and a lower tension-member, _AB_. But again, -in the skeleton of the quadruped, _the necessary tie_, _AB_, _is not -to be found_; 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 {694} comparable to the main girder -of a double-armed cantilever bridge. - -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. 346). - -In a typical cantilever bridge, such as the Forth Bridge (Fig. 345), 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 _cut_ 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. - -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 {695} a good deal more than what is -carried on his hind feet, say about three-fifths of the whole weight of -the animal. - -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[627]. 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. - -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. - -―――――――――― - -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 _nature_ 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. - -The essential function of a bridge is to stretch across a certain span, -and carry a certain definite load; and this being so, the {696} 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, - -[Illustration: Fig. 340. A, Span of proposed bridge. B, Stress diagram, -or diagram of bending-moments[628].] - -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 _depth_ of the girder -everywhere proportional to the bending-moments, that is - -[Illustration: Fig. 341. The bridge constructed, as a parabolic girder.] - -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 {697} 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.” - -[Illustration: Fig. 342.] - -In the following diagrams (Fig. 342, _a_, _b_) (which are taken from -the original ones of Culmann), we see at once that the loaded beam or -bracket (_a_) 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 -(_b_) there is no danger-point at all, for the dimensions of the -structure are made to increase _pari passu_ with the bending-moments: -stress and resistance vary together. Again in Fig. 340, we have a -simple span (A), with its stress diagram (B); and in Fig. 341 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[629]. -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[630], 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. - -In the case of our cantilever bridge, we shew the primitive girder -{698} in Fig. 343, 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. - -To either of these two stress diagrams, direct or inverted, we may fit -the design of the construction, as in Figs. 343, C and 344. - -[Illustration: Fig. 343.] - -[Illustration: Fig. 344.] - -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. - -In our last diagram the upper member of the cantilever is under {699} -tension; it is represented in the quadruped by the _ligamentum nuchae_ -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 _bodies_ 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[631]. 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. - -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 -{700} 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. - -[Illustration: Fig. 345. A two-armed cantilever of the Forth Bridge. -Thick lines, compression-members (bones); thin lines, tension-members -(ligaments).] - -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 - -[Illustration: Fig. 346.] - -is _effectively_ continuous from the head to the tip of the tail; and -at each point of support (_A_ and _B_) it is subjected to the negative -bending-moment due to the overhanging load on each of the projecting -cantilever arms _AH_ and _BT_. The diagram of bending-moments will -(according to the ordinary conventions) lie below {701} 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 _A_ -and _B_, where the moments are measured by the vertical ordinates. It -is plain that this figure only differs from a representation of _two_ -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. - -[Illustration: Fig. 347. Stress-diagram of horse’s backbone.] - -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 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. - -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. 347, the -vertical ordinate at _A_ being thrice the height of that at _B_. {702} - -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 _B_ -(Fig. 348). A glance at the skeleton of _Diplodocus Carnegii_ 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. - -[Illustration: Fig. 348. Stress-diagram of backbone of Dinosaur.] - -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, {703} 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. - -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. - -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[632]. - - On On - Fore- Hind- % on % on - Gross weight. feet feet Fore- Hind- - ton cwts. cwts. cwts. feet. feet. - Camel (Bactrian) — 14·25 9·25 4·5 67·3 32·7 - Llama — 2·75 1·75 ·875 66·7 33·3 - Elephant (Indian) 1 15·75 20·5 14·75 58·2 41·8 - Horse — 8·25 4·75 3·5 57·6 42·4 - Horse (large Clydesdale) — 15·5 8·5 7·0 54·8 45·2 - -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 {704} 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. - -[Illustration: Fig. 349. Stress-diagram of Titanotherium.] - -(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 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. - -(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 {705} 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. - -(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. - -(4) In the kangaroo, the fore-limbs are entirely relieved of their -load, and accordingly the tall spines over the withers, which {706} -were so conspicuous in all heavy-headed _quadrupeds_, have now -completely vanished. The creature has become bipedal, and body and tail -form the extremities of _a single_ 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. - -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. - -(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[633]. The -fore-limbs, {707} 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. - -[Illustration: Fig. 350. Diagram of Stegosaurus.] - -(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 _behind_ 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 {708} 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[634]. - -(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. - -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. - -But in the aquatic mammal, such as a whale or a dolphin (and not less -so in the aquatic bird), _stiffness_ 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 _C_, _D_, in Fig. 351. {709} - -[Illustration: Fig. 351.] - -Here, between _C_ and _D_ 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 _C_ and _D_ we have a _negative_ 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 -_dorsal_ spines are no longer needed, take their place _below_ 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. - -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 {710} 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 -_three series_ 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. - -―――――――――― - -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 _girder_, 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. - -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 _strength_ of the structure; but at the same time we should greatly -increase its _rigidity_, and {711} 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 _flexible_ as well as _strong_. 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 _material_ -of the ligamentous tension-member—its modulus of elasticity in direct -tension, its elastic limit, and its safe working stress. - -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 _points d’appui_ 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. - -―――――――――― - -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 _stress_, and so as to evade thereby -the distortions and disruptions due to _shear_. In these phenomena -there lies a definite law of growth, {712} 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: _obvia conspicimus, nubem pellente -mathesi_[635]. - -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. - -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 _continuum_, and a _continuum_ it remains all life -long. The things that link bone with bone, {713} cartilage, ligaments, -membranes, are fashioned out of the same primordial tissue, and come -into being _pari passu_, 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. - -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[636]. 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 {714} -in this limited and subordinate sense, that they are _parts_ of a -whole which, when it loses its composite integrity, ceases to exist. - -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.” - -Nevertheless Darwin found no difficulty in believing that “natural -selection will tend in the long run to reduce _any part_ 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[637].” 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 _hereditary separability_ -whereby the body is a colony, a mosaic, of single individual and -separable characters[638].” 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 _ist im Ganzen -enthalten_.” And, according to the trend or aspect of our thought, we -may look upon the co-ordinated parts, now as related and fitted _to the -end or function_ of the whole, and now as related to or resulting _from -the physical causes_ inherent in the entire system of forces to which -the whole has been exposed, and under whose influence it has come into -being[639]. {715} - -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. - -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 {716} 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. - -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. - -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[640]. 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 {717} 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, _ipso facto_, a mode of -life, intermediate between the terrestrial and the Cetacean form. There -is no valid syllogism to the effect that _A_ has a flat curved scapula -like a seal’s, and _B_ has a flat, curved scapula like a seal’s: and -therefore _A_ and _B_ 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 _B_ shews in its general structure, -extending over this bone and that bone, resemblances both to _A_ 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 _isolated_ phenomena, in this -part of the fabric or in that, in a scapula for instance or a humerus: -but rather in {718} some slow, _general_, 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. - -{719} - - - - -CHAPTER XVII - -ON THE THEORY OF TRANSFORMATIONS, OR THE COMPARISON OF RELATED -FORMS[641] - - -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. - -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 {720} -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.” - -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, “_Ignorato motu, ignoratur -Natura_.” - -―――――――――― - -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 {721} the description and -classification of crystals, his methods were immediately adopted and a -new science came into being. - -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[642]. 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. {722} - -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 _ideal_ 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[643]. {723} - -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. - -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 _deformation_ 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 _deformation_ 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. - -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 _form_ of a curve into -_numbers_ and into _words_. 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 {724} it into a -table of numbers, from which again we may at pleasure reconstruct the -curve. - -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. - -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 _x_, _y_. If we submit -our rectangular system to “deformation,” on simple and recognised -lines, altering, for instance, the direction of the axes, the ratio -of _x_/_y_, or substituting for _x_ and _y_ 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 _under strain_, and is a function of the new co-ordinates -in precisely the same way as the old figure was of the original -co-ordinates _x_ and _y_. - -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, _in each small -unit of area_, 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 {725} the joint operation of -many causes if few are adequate to the production of it: _Frustra fit -per plura, quod fieri potest per pauciora._ - -―――――――――― - -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 -_related_ 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. - -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 {726} ‘excess’ and ‘defect[644].’ ” 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. - -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. - -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.’ ” - -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 {727} 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[645].” 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[646]. - -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 {728} 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. - - 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[647]. 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[648].” - 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[649]. - -―――――――――― - -[Illustration: Fig. 352.] - -[Illustration: Fig. 353.] - -[Illustration: Fig. 354.] - -[Illustration: Fig. 355.] - -If we begin by drawing a net of rectangular equidistant co-ordinates -(about the axes _x_ and _y_), we may alter or _deform_ this {729} -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. 353). It follows -that any figure which we may have inscribed in the original net, and -which we transfer to the new, will thereby be _deformed_ 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 _y_-direction, be found elongated {730} into an ellipse. In -elementary mathematical language, for the original _x_ and _y_ we -have substituted _x__{1} and _c_ _y__{1}, and the equation to our -original circle, _x_^2 + _y_^2 = _a_^2, becomes that of the ellipse, -_x__{1}^2 + _c_^2 _y__{1}^2 = _a_^2. - -If I draw the cannon-bone of an ox (Fig. 354, 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 _x_ of the -original diagram we substitute _x′_ = 2_x_/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 _x″_ = _x_/3. - -(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 _tapering_ elastic band. In such -cases, as I have represented it in Fig. 355, the ordinate increases -logarithmically, and for _y_ we substitute ε^{_y_}. It is obvious that -this logarithmic extension may involve both abscissae and ordinates, -_x_ becoming ε^{_x_}, while _y_ becomes ε^{_y_}. The circle in our -original figure is now deformed into some such shape as that of Fig. -356. This method of deformation is a common one, and will often be of -use to us in our comparison of organic forms. - -(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. 357. The system may now be described in terms of the -oblique axes _X_, _Y_; or may be directly referred to new rectangular -co-ordinates ξ, η by the simple transposition _x_ = ξ − η cot ω, _y_ -= η cosec ω. - -[Illustration: Fig. 356.] - -[Illustration: Fig. 357.] - -[Illustration: Fig. 358.] - -(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 {731} 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. -358). In biology these co-ordinates will 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 {732} 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 - -[Illustration: Fig. 359.] - -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 {733} 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. - -[Illustration: Fig. 360. _Begonia daedalea._] - -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[650] of a begonia, in which one side of the leaf -may be merely ovate while the other has a cordate outline, is seen to -be really a case of _unequal_, 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. 360 I have outlined a leaf of the large _Begonia daedalea_. On the -smaller left-hand side of the leaf I have taken at random three points, -_a_, _b_, _c_, and have measured the angles, _AOa_, 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 _a′_, -_b′_, _c′_, such that the radii drawn to this margin of the leaf are -equal to the former, _Oa′_ to _Oa_, etc. Now if the two sides of the -leaf are {734} mathematically similar to one another, it is obvious -that the respective angles should be in continued proportion, i.e. -as _AOa_ is to _AOa′_, so should _AOb_ be to _AOb′_. 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: - - _AOa_ _AOb_ _AOc_ _AOa′_ _AOb′_ _AOc′_ - Observed values 12° 28.5° 88° — — 157° - Calculated values — — — 21.5° 51.1° — - Observed values — — — 20 52 — - -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. - -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 _Dionaea_, 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 {735} 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. - -―――――――――― - -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. - -[Illustration: Fig. 361.] - -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 _Y_ − _AX_ = _c_ corresponding to the logarithmic -spiral θ − _A_ log _r_ = _c_ (Fig. 361). This beautiful and simple -transformation lets us at once convert, for instance, the straight -conical shell of the Pteropod or the _Orthoceras_ 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. - -These various systems of coordinates, 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 {736} 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 _stream lines_, which may be defined by a suitable mathematical -transformation. - -When the system becomes no longer orthogonal, as in many of the -following illustrations—for instance, that of _Orthagoriscus_ (Fig. -382),—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 _graphically_ -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. - -―――――――――― - -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. - -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 {737} teeth of the -comb is exceedingly complex, and its complexity is revealed in the -complicated “diagram of forces” which constitutes the pattern. - -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[651]. The glassblower starts his operations with a _tube_, -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[652]. The alimentary canal, {738} 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. - -―――――――――― - -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. - -[Illustration: Fig. 362.] - -We have already compared in a preliminary fashion the metacarpal or -cannon-bone of the ox, the sheep, and the giraffe (Fig. 354); 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 _y_), the breadth (or value of _x_) 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 {739} of length and breadth in this -particular bone are extremely significant[653]. - -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 _x_ 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 _x_ will not suffice without some simultaneous -modification of the values of _y_ (Fig. 362). In such a case it may be -found possible to satisfy the varying values of _y_ 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 _y_, 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. - - _a_ _b_ _c_ _d_ - _y_ (Ox) 0 18 27 42 100 - _y′_ (Sheep) 0 10 19 36 100 - _y″_ (Giraffe) 0 5 10 24 100 - -This summary of values of _y′_, coupled with the equations for the -{740} value of _x_, 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. - -[Illustration: Fig. 363.] - -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. -363), where the values of _y_ 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 _y_ and _y′_. - -[Illustration: Fig. 364. (After Albert Dürer.)] - -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[654]; -it is fully described and put in practice by Albert Dürer in his -_Geometry_, and especially in his _Treatise on Proportion_[655]. -In this latter work, the manner in which the {741} 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. 364). - -[Illustration: Fig. 365.] - -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. 365, _a_). 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° (_b_). 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 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 (_c_). -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 _F_ (_x_, _y_) = 0. The figure of the tapir’s lateral {742} toe -is a precisely identical function of the form _F_ (_e_^{_x_}, _y__{1}) -= 0, where _x__{1}, _y__{1} are oblique co-ordinate axes inclined to -one another at an angle of 50°. - -[Illustration: Fig. 366. (After Albert Dürer.)] - -Dürer was acquainted with these oblique co-ordinates also, and I have -copied two illustrative figures from his book[656]. - -―――――――――― - -[Illustration: Fig. 367. _Oithona nana._] - -[Illustration: Fig. 368. _Sapphirina._] - -In Fig. 367 I have sketched the common Copepod _Oithona nana_, -{743} and have inscribed it in a rectangular net, with abscissae -three-fifths the length of the ordinates. Side by side (Fig. 368) is -drawn a very different Copepod, of the genus _Sapphirina_; 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 -_y_ in Fig. 368 are large in the upper part of the figure, and diminish -rapidly towards its base. (2) The values of _x_ 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 _x_ and _y_ 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 _Oithona_, we obtain that of _Sapphirina_ without further -trouble, e.g.: - - _x_ (_Oithona_) 0 3 6 9 12 15 — - _x′_ (_Sapphirina_) 0 8 10 12 13 14 — - - _y_ (_Oithona_) 0 5 10 15 20 25 30 - _y′_ (_Sapphirina_) 0 2 7 3 23 32 40 - -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 -_Caprella_ and the thick-set body of a _Cyamus_ 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. - -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 {744} 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. - -[Illustration: Fig. 369. Carapaces of various crabs. 1, _Geryon_; 2, -_Corystes_; 3, _Scyramathia_; 4, _Paralomis_; 5, _Lupa_; 6, _Chorinus_.] - -If we choose, to begin with, such a crab as _Geryon_ (Fig. 369, 1), -and inscribe it in our equidistant rectangular co-ordinates, we shall -see that we pass easily to forms more elongated in a transverse -{745} direction, such as _Matuta_ or _Lupa_ (5), and conversely, by -transverse compression, to such a form as _Corystes_ (2). In certain -other cases the carapace conforms to a triangular diagram, more or less -curvilinear, as in Fig. 4, which represents the genus _Paralomis_. Here -we can easily see that the posterior border is transversely elongated -as compared with that of _Geryon_, 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. - -In an interesting series of cases, such as the genus _Chorinus_, -or _Scyramathia_, 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 _Paralomis_, but to that which was broadest in _Geryon_; 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. 3 and 6), while the decremental interspacing of the abscissae is -very marked indeed. - -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. 370, 1 a little amphipod of the -family Phoxocephalidae (_Harpinia_ sp.). Deforming the co-ordinates -of the figure into the {746} curved orthogonal system in Fig. 2, -we at once obtain a very fair representation of an allied genus, -belonging to a different family of amphipods, namely _Stegocephalus_. -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. 3 I show a network, to which, if we transfer -our diagram of _Harpinia_ or of _Stegocephalus_, we shall obtain a -tolerable representation of the aberrant genus _Hyperia_, with its -narrow abdomen, its reduced pleural lappets, its great eyes, and its -inflated head. - -[Illustration: Fig 370. 1. _Harpinia plumosa_ Kr. 2. _Stegocephalus -inflatus_ Kr. 3. _Hyperia galba_.] - -―――――――――― - -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 {747} 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. - -[Illustration: Fig. 371. _a_, _Campanularia macroscyphus_, Allm.; _b_, -_Gonothyraea hyalina_, Hincks; _c_, _Clytia Johnstoni_, Alder.] - -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. 371). 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 {748} hydroids -which we inscribe therein (Fig. 372). The comparative smoothness or -denticulation of the margin of the calycle, and the number of its -denticles, constitutes an independent variation, and requires separate -description; we have already seen (p. 236) that this denticulation is -in all probability due to a particular physical cause. - -[Illustration: Fig. 372. _a_, _Cladocarpus crenatus_, F.; _b_, -_Aglaophenia pluma_, L.; _c_, _A. rhynchocarpa_, A.; _d_, _A cornuta_, -K.; _e_, _A. ramulosa_, K.] - -―――――――――― - -[Illustration: Fig. 373. _Argyropelecus Olfersi._] - -[Illustration: Fig. 374. _Sternoptyx diaphana._] - -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, is -illustrated by Figs. 373 and 374. Fig. 373 represents, within Cartesian -co-ordinates, a certain little oceanic fish known as _Argyropelecus -Olfersi_. Fig. 474 represents precisely the same outline, transferred -to a system of oblique co-ordinates whose {749} 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 _Sternoptyx diaphana_. The -deformation illustrated by this case of _Argyropelecus_ is precisely -analogous to the simplest and commonest kind of deformation to which -fossils are subject (as we have seen on p. 553) as the result of -shearing-stresses in the solid rock. - -[Illustration: Fig. 375. _Scarus_ sp.] - -[Illustration: Fig. 376. _Pomacanthus._] - -Fig. 375 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. 376, and then filling into the new system, -space by space and point by point, our former diagram of _Scarus_, -we obtain a very good outline of an allied fish, belonging to a -neighbouring family, of the genus _Pomacanthus_. This case is all the -more interesting, because upon the body of our _Pomacanthus_ 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. - -In Figs. 377–380 I have represented another series of Acanthopterygian -fishes, not very distantly related to the foregoing. If we start -this series with the figure of _Polyprion_, in Fig. 377, we see that -the outlines of _Pseudopriacanthus_ (Fig. 378) and of _Sebastes_ or -_Scorpaena_ (Fig. 379) are easily derived by substituting a system of -triangular, or radial, co-ordinates for the rectangular ones in {750} -which we had inscribed _Polyprion_. The very curious fish _Antigonia -capros_, an oceanic relative of our own “boar-fish,” conforms closely -to the peculiar deformation represented in Fig. 380. - -[Illustration: Fig. 377. _Polyprion._] - -[Illustration: Fig. 378. _Pseudopriacanthus altus._] - -[Illustration: Fig. 379. _Scorpaena_ sp.] - -[Illustration: Fig. 380. _Antigonia capros._] - -[Illustration: Fig. 381. _Diodon._] - -[Illustration: Fig. 382. _Orthagoriscus._] - -Fig. 381 is a common, typical _Diodon_ or porcupine-fish, and in -Fig. 382 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[657]. The old outline, transferred {751} in its -integrity to the new network, appears as a manifest representation of -the closely allied, but very different looking, sunfish, _Orthagoriscus -mola_. 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 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 {752} 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. - -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. - -―――――――――― - -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 _Belodon_, 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. - -[Illustration: Fig. 383. A, _Crocodilus porosus_. B, _C. americanus_. -C, _Notosuchus terrestris_.] - -Let us take one of the typical modern crocodiles as our standard of -form, e.g. _C. porosus_, and inscribe it, as in Fig. 383, _a_, 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 _b_, we pass to such a form as _C. -americanus_. By an exaggeration of the same process we at once get -an approximation to the form of one of the sharp-snouted, {753} or -longirostrine, crocodiles, such as the genus _Tomistoma_; 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. 383, _b_), 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 _x_/_y_ 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 _Notosuchus_, 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 _Crocodilus_, 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 {754} 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 _Notosuchus_ fall into such a co-ordinate -network as that which is represented in Fig. 383, _c_. To all intents -and purposes, then, this not very complex system, representing -one harmonious “deformation,” accounts for _all_ 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 -_Notosuchus_ from the model furnished by the common crocodile. - -[Illustration: Fig. 384. Pelvis of (A) _Stegosaurus_; (B) -_Camptosaurus_.] - -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 -_Stegosaurus_ and of _Camptosaurus_ (Fig. 384, _a_, _b_) to show that, -when the former is taken as our Cartesian type, a slight curvature -and an approximately logarithmic extension of the _x_-axis brings us -easily to the configuration of the other. In the original specimen of -_Camptosaurus_ described by Marsh[658], 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 {755} 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. - -[Illustration: Fig. 385. Shoulder-girdle of _Cryptocleidus_. _a_, -young; _b_, adult.] - -[Illustration: Fig. 386. Shoulder-girdle of _Ichthyosaurus_.] - -In Fig. 385, _a_, _b_, I have drawn the shoulder-girdle of -_Cryptocleidus_, 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. 386 I have drawn -the shoulder-girdle of an Ichthyosaur, referring it to _Cryptocleidus_ -as a standard of comparison. The interclavicle, which is present in -_Ichthyosaurus_, is minute and hidden in _Cryptocleidus_; but the -numerous other differences between the two {756} forms, chief among -which is the great elongation in _Ichthyosaurus_ of the two clavicles, -are all seen by our diagrams to be part and parcel of one general and -systematic deformation. - -[Illustration: Fig. 387. _a_, Skull of _Dimorphodon_. _b_, Skull of -_Pteranodon_.] - -Before we leave the group of reptiles we may glance at the very -strangely modified skull of _Pteranodon_, 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 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 _Dimorphodon_. But if we inscribe the latter in -Cartesian coordinates (Fig. 387, _a_), and refer our _Pteranodon_ to -a system of oblique co-ordinates (_b_), 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. - -―――――――――― - -[Illustration: Fig. 388. Pelvis of _Archaeopteryx_.] - -[Illustration: Fig. 389. Pelvis of _Apatornis_.] - -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, {757} -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 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, {758} and which -are reproduced in Figs. 388–393. Here we have, as extreme cases, the -pelvis of _Archaeopteryx_, the most ancient of known birds, and that of -_Apatornis_, one of the fossil “toothed” - -[Illustration: Fig. 390. The co-ordinate systems of Figs. 388 and 389, -with three intermediate systems interpolated.] - -[Illustration: Fig. 391. The first intermediate co-ordinate network, -with its corresponding inscribed pelvis.] - -birds from the North American Cretaceous formations—a bird shewing -some resemblance to the modern terns. The pelvis of _Archaeopteryx_ -is taken as our type, and referred accordingly to {759} Cartesian -co-ordinates (Fig. 388); while the corresponding coordinates of the -very different pelvis of _Apatornis_ are represented in Fig. 389. In -Fig. 390 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 these systems of co-ordinates, as in Figs. 391, -392. Finally, in Fig. 393 the complete series is represented, beginning -with the known pelvis of _Archaeopteryx_, and leading up by our three -intermediate hypothetical types to the known pelvis of _Apatornis_. - -[Illustration: Fig. 392. The second and third intermediate co-ordinate -networks, with their corresponding inscribed pelves.] - -―――――――――― - -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 {760} Ungulates, the group -which includes the rhinoceros, the tapir, and the horse. - -[Illustration: Fig. 393. The pelves of _Archaeopteryx_ and of -_Apatornis_, with three transitional types interpolated between them.] - -Let us begin by choosing as our type the skull of _Hyrachyus agrarius_, -Cope, from the Middle Eocene of North America, as figured by Osborn in -his Monograph of the Extinct Rhinoceroses[659] (Fig. 394). - -[Illustration: Fig. 394. Skull of _Hyrachyus agrarius_. (After -Osborn.)] - -[Illustration: Fig. 395. Skull of _Aceratherium tridactylum_. (After -Osborn.)] - -The many other forms of primitive rhinoceros described in the monograph -differ from _Hyrachyus_ 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 {761} 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 co-ordinates of _Aceratherium tridactylum_, -as shown in Fig. 395, 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 _Aceratherium_ has undergone a slight double -curvature, while the upper parts of the skull have at the same time -been {762} 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. - -[Illustration: Fig. 396. Occipital view of the skulls of various -extinct rhinoceroses (_Aceratherium_ spp.). (After Osborn.)] - -Among the species of _Aceratherium_, 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 which I have represented in Fig. 396. 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. - -By similar and not more violent changes we pass easily to such allied -forms as the Titanotheres (Fig. 397); and the well-known series of -species of _Titanotherium_, by which Professor Osborn has {763} -illustrated the evolution of this genus, constitutes a simple and -suitable case for the application of our method. - -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. 398), 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. - -[Illustration: Fig. 397. _Titanotherium robustum_.] - -[Illustration: Fig. 398. Tapir’s skull.] - -The conformation of the horse’s skull departs from that of our -primitive Perissodactyle (that is to say our early type of rhinoceros, -_Hyrachyus_) in a direction that is nearly the opposite of that taken -by _Titanotherium_ and by the recent species of rhinoceros. For we -perceive, by Fig. 399, 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 {764} 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. - -[Illustration: Fig. 399. Horse’s skull.] - -[Illustration: Fig. 400. Rabbit’s skull.] - -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. 400 that the rabbit’s skull conforms -to a system of {765} co-ordinates corresponding to the Cartesian -co-ordinates in which we have inscribed the skull of _Hyrachyus_, -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 - -[Illustration: Fig. 401. _A_, outline diagram of the Cartesian -co-ordinates of the skull of _Hyracotherium_ or _Eohippus_, as shewn -in Fig. 402, A. _H_, outline of the corresponding projection of the -horse’s skull. _B_–_G_, intermediate, or interpolated, outlines.] - -strong and uniform flexure in the downward direction (cf. Fig. 358, -p. 731). 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 be dealt with -{768} 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. - -[Illustration: 〈two printed pages〉 - -Fig. 402. _A_, skull of _Hyracotherium_, from the Eocene, after W. B. -Scott; _H_, skull of horse, represented as a co-ordinate transformation -of that of _Hyracotherium_, and to the same scale of magnitude; -_B_–_G_, various artificial or imaginary types, reconstructed as -intermediate stages between _A_ and _H_; _M_, skull of _Mesohippus_, -from the Oligocene, after Scott, for comparison with _C_; _P_, skull of -_Protohippus_, from the Miocene, after Cope, for comparison with _E_; -_Pp_, lower jaw of _Protohippus placidus_ (after Matthew and Gidley), -for comparison with _F_; _Mi_, _Miohippus_ (after Osborn), _Pa_, -_Parahippus_ (after Peterson), shewing resemblance, but less perfect -agreement, with _C_ and _D_.] - -―――――――――― - -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[660]. Here we start afresh, -with the skull (Fig. 402, _A_) of _Hyracotherium_ (or _Eohippus_), -inscribed in a simple Cartesian network. At the other end of the -series (_H_) is a skull of Equus, in its own corresponding network; -and the intermediate stages (_B_–_G_) are all drawn by direct and -simple interpolation, as in Mr Heilmann’s former series of drawings of -_Archaeopteryx_ and _Apatornis_. 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. _Mesohippus_ and _Protohippus_ (_M_, _P_), 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 _Hyracotherium_ to _Equus_. -In a third case, that of _Parahippus_ (_Pa_), 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 -_D_, but the “fit” is now a bad one; for the skull of _Parahippus_ -is evidently a longer, straighter and narrower skull, and differs in -other minor characters besides. In short, though some writers have -placed _Parahippus_ in the direct line of descent between _Equus_ and -_Eohippus_, 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[661]. It may be noticed, especially in the -case of _Protohippus_ {769} (_P_), 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 (_Pp_) of _Protohippus placidus_ the correspondence is -more exact. - -[Illustration: Fig. 403. Human scapulae (after Dwight). _A_, Caucasian; -_B_, Negro; _C_, North American Indian (from Kentucky Mountains).] - -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[662]. - -The variability of this latter bone is great, but it is neither {770} -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. - -[Illustration: Fig. 404. Human skull.] - -[Illustration: Fig. 405. Co-ordinates of chimpanzee’s skull, as a -projection of the Cartesian co-ordinates of Fig. 404.] - -Let us now inscribe in our Cartesian co-ordinates the outline of a -human skull (Fig. 404), 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, {771} 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 represented in Fig. 405 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. 406 demonstrates the -correspondence. In Fig. 407 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. - -[Illustration: Fig. 406. Skull of chimpanzee.] - -[Illustration: Fig. 407. Skull of baboon.] - -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 {772} inverse diagrams, -by which the Cartesian co-ordinates of the ape are transformed into -curvilinear and non-equidistant co-ordinates in man. - -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. - -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, _Homo neanderthalensis_, 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, {773} strictly speaking, continuous, and that neither of -our two apes lies _precisely_ on the same direct line or sequence of -deformation by which we may hypothetically connect the other with man. - -[Illustration: Fig. 408. Skull of dog, compared with the human skull of -Fig. 404.] - -As a final illustration I have drawn the outline of a dog’s skull (Fig. -408), and inscribed it in a network comparable with the Cartesian -network of the human skull in Fig. 404. 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 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. -{774} - -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 _x_, -_y_, _z_, 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. - -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 -_correlate_ 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. - -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 {775} 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 _y_ axis; -in other words, the ratio _x_/_y_ 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: _y_ increases, but _z_ diminishes, relatively to -_x_. 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 _compressed_ 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 _depressed_ -in its vertical section. We proceed then, to enquire whether there -be any simple relation of _magnitude_ discernible between these twin -factors of expansion and compression; and the very fact that the two -dimensions tend to vary _inversely_ already assures us that, in the -general process of deformation, the _volume_ is less affected than -are the _linear dimensions_. Some years ago, when I was studying the -length-weight co-efficient in fishes (of which we have already spoken -in Chap. III, p. 98), that is to say the coefficient _k_ in the formula -_W_ = _k_ _L_^3, or _k_ = _W_/_L_^3, I was not a little surprised to -find that _k_ 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 -_volume_ when they are equal in _length_; or, in other words, that the -extent to which the plaice’s body has become expanded or broadened is -_just about {776} compensated for_ 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 _balancement des organes_, or -“compensation of parts.” - -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 _Y_ in the plaice is about -twice as great as _y_ in the haddock. But if the volumes of the two -fishes be equal, this is as much as to say that _xyz_ in the one case -(or rather the summation of all these values) is equal to _XYZ_ in the -other; and therefore (since _X_ = _x_, and _Y_ = 2_y_), it follows -that _Z_ = _z_/2. When we have drawn our vertical transverse section -of the haddock (or projected that fish in the _yz_ plane), we have -reason accordingly to anticipate that we can draw a similar projection -(or section) of the plaice by simply doubling the _y_’s and halving -the _z_’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 _y_ to _z_) is just about four times as great in the -one case as in the other. - -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 {777} 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 _expansion_ 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 _compression_ 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[663]. - -{778} - - - - -EPILOGUE. - - -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. _Hic artem remumque repono._ - -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. - -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.” - -For the harmony of the world is made manifest in Form and Number, and -the heart and soul and all the poetry of Natural {779} 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. - -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. - -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. - -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.” - - - - -NOTES: - -[1] 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 -_Concepts_, p. 21, 1882; Höber, _Biol. Centralbl._ XIX, p. 284, 1890, -etc. Cf. also Jellett, _Rep. Brit. Ass._ 1874, p. 1. - -[2] “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.” _Methodus inveniendi_, etc. 1744 (_cit._ -Mach, _Science of Mechanics_, 1902, p. 455). - -[3] Cf. Opp. (ed. Erdmann), p. 106, “Bien loin d’exclure les causes -finales..., c’est de là qu’il faut tout déduire en Physique.” - -[4] 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.” - -[5] It is now and then conceded with reluctance. Thus Enriques, a -learned and philosophic naturalist, writing “della economia di sostanza -nelle osse cave” (_Arch. f. Entw. Mech._ XX, 1906), says “una certa -impronta di teleologismo quà e là è rimasta, mio malgrado, in questo -scritto.” - -[6] Cf. Cleland, On Terminal Forms of Life, _J. Anat. and Phys._ XVIII, -1884. - -[7] Conklin, Embryology of Crepidula, _Journ. of Morphol._ XIII, p. -203, 1897; Lillie, F. R., Adaptation in Cleavage, _Woods Holl Biol. -Lectures_, pp. 43–67, 1899. - -[8] I am inclined to trace back Driesch’s teaching of Entelechy to no -less a person than Melanchthon. When Bacon (_de Augm._ IV, 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 _function_. He defined it as δύναμις _quaedam ciens actiones_—a -description all but identical with that of Claude Bernard’s “_force -vitale_.” - -[9] Ray Lankester, _Encycl. Brit._ (9th ed.), art. “Zoology,” p. 806, -1888. - -[10] Alfred Russel Wallace, especially in his later years, relied -upon a direct but somewhat crude teleology. Cf. his _World of Life, a -Manifestation of Creative Power, Directive Mind and Ultimate Purpose_, -1910. - -[11] Janet, _Les Causes Finales_, 1876, p. 350. - -[12] The phrase is Leibniz’s, in his _Théodicée_. - -[13] Cf. (_int. al._) Bosanquet, The Meaning of Teleology, _Proc. -Brit. Acad._ 1905–6, pp. 235–245. Cf. also Leibniz (_Discours de -Métaphysique; Lettres inédites, ed._ de Careil, 1857, p. 354; _cit._ -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: _et seq._” - -[14] 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. - -[15] _Revue Philosophique._ XXXIII, 1892. - -[16] 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, _Q.J.M.S._ (_Trans. Microsc. Soc._), -VI, p. 49, 1858.) Cf. also Helmholtz, _infra cit._, p. 9. - -[17] Whereby he incurred the reproach of Socrates, in the _Phaedo_. - -[18] In a famous lecture (Conservation of Forces applied to Organic -Nature, _Proc. Roy. Instit._, 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 _quite of the same character_ -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, _Phil. Trans._ 1809, p. 1; _Coll. Works_, I, 511. - -[19] _Ektropismus, oder die physikalische Theorie des Lebens_, Leipzig, -1910. - -[20] Wilde Lecture, _Nature_, March 12, 1908; _ibid._ Sept. 6, 1900, p. -485; _Aether and Matter_, p. 288. Cf. also Lord Kelvin, _Fortnightly -Review_, 1892, p. 313. - -[21] Joly, The Abundance of Life, _Proc. Roy. Dublin Soc._ VII, 1890; -and in _Scientific Essays_, etc. 1915, p. 60 _et seq._ - -[22] Papillon, _Histoire de la philosophie moderne_, I, p. 300. - -[23] With the special and important properties of _colloidal_ matter we -are, for the time being, not concerned. - -[24] Cf. Hans Przibram, _Anwendung elementarer Mathematik auf -Biologische Probleme_ (in Roux’s _Vorträge_, Heft III), Leipzig, 1908, -p. 10. - -[25] The subject is treated from an engineering point of view by Prof. -James Thomson, Comparisons of Similar Structures as to Elasticity, -Strength, and Stability, _Trans. Inst. Engineers, Scotland_, 1876 -(_Collected Papers_, 1912, pp. 361–372), and by Prof. A. Barr, _ibid._ -1899; see also Rayleigh, _Nature_, April 22, 1915. - -[26] Cf. Spencer, The Form of the Earth, etc., _Phil. Mag._ XXX, pp. -194–6, 1847; also _Principles of Biology_, pt. II, ch. I, 1864 (p. 123, -etc.). - -[27] George Louis Lesage (1724–1803), well known as the author of one -of the few attempts to explain gravitation. (Cf. Leray, _Constitution -de la Matière_, 1869; Kelvin, _Proc. R. S. E._ VII, p. 577, 1872, etc.; -Clerk Maxwell, _Phil. Trans._ vol. 157, p. 50, 1867; art. “Atom,” -_Encycl. Brit._ 1875, p. 46.) - -[28] Cf. Pierre Prévost, _Notices de la vie et des écrits de Lesage_, -1805; quoted by Janet, _Causes Finales_, app. III. - -[29] Discorsi e Dimostrazioni matematiche, intorno à due nuove -scienze, attenenti alla Mecanica, ed ai Movimenti Locali: appresso gli -Elzevirii, MDCXXXVIII. _Opere_, ed. Favaro, VIII, p. 169 seq. Transl. -by Henry Crew and A. de Salvio, 1914, p. 130, etc. See _Nature_, June -17, 1915. - -[30] 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. - -[31] Sir G. Greenhill, Determination of the greatest height to which -a Tree of given proportions can grow, _Cambr. Phil. Soc. Pr._ IV, p. -65, 1881, and Chree, _ibid._ VII, 1892. Cf. Poynting and Thomson’s -_Properties of Matter_, 1907, p 99. - -[32] 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. - -[33] _Modern Painters._ - -[34] 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, _Phil. Trans._ vol. 198, p. 71, 1906). - -[35] _Trans. Zool. Soc._ IV, 1850, p. 27. - -[36] 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, _La Forme -et la Vie_, 1900, p. 815). The effect of gravity on outward _form_ 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. - -[37] 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. - -[38] Let _L_ be the length, _S_ the (wetted) surface, _T_ the tonnage, -_D_ the displacement (or volume) of a ship; and let it cross the -Atlantic at a speed _V_. Then, in comparing two ships, similarly -constructed but of different magnitudes, we know that _L_ = _V_^2, -_S_ = _L_^2 = _V_^4, _D_ = _T_ = _L_^3 = _V_^6; also _R_ (resistance) -= _S_ ⋅ _V_^2 = _V_^6; _H_ (horse-power) = _R_ ⋅ _V_ = _V_^7; and the -coal (_C_) necessary for the voyage = _H_/_V_ = _V_^6. 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. - -[39] 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, _Monatsber. Akad. -Berlin_, 1873, pp. 501–14. It was criticised and challenged (somewhat -rashly) by K. Müllenhof, Die Grösse der Flugflächen, etc., _Pflüger’s -Archiv_, XXXV, p. 407, XXXVI, p. 548, 1885. - -[40] Cf. also Chabrier, Vol des Insectes, _Mém. Mus. Hist. Nat. Paris_, -VI–VIII, 1820–22. - -[41] _Aerial Flight_, vol. II (_Aerodonetics_), 1908, p. 150. - -[42] By Lanchester, _op. cit._ p. 131. - -[43] Cf. _L’empire de l’air; ornithologie appliquée à l’aviation_. 1881. - -[44] _De Motu Animalium_, I, prop. cciv, ed. 1685, p. 243. - -[45] Harlé, On Atmospheric Pressure in past Geological Ages, _Bull. -Geol. Soc. Fr._ XI, pp. 118–121; or _Cosmos_, p. 30, July 8, 1911. - -[46] _Introduction to Entomology_, 1826, II, 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. - -[47] 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 _kinds_ -of muscle, such as the insect’s and the mammal’s. - -[48] Prop. clxxvii. Animalia minora et minus ponderosa majores saltus -efficiunt respectu sui corporis, si caetera fuerint paria. - -[49] See also (_int. al._), John Bernoulli, _de Motu Musculorum_, -Basil., 1694; Chabry, Mécanisme du Saut, _J. de l’Anat. et de -la Physiol._ XIX, 1883; Sur la longueur des membres des animaux -sauteurs, _ibid._ XXI, p. 356, 1885; Le Hello, De l’action des organes -locomoteurs, etc., _ibid._ XXIX, p. 65–93, 1893, etc. - -[50] Recherches sur la force absolue des muscles des Invertébrés, -_Bull. Acad. E. de Belgique_ (3), VI, VII, 1883–84; see also _ibid._ -(2), XX, 1865, XXII, 1866; _Ann. Mag. N. H._ XVII, p. 139, 1866, XIX, -p. 95, 1867. The subject was also well treated by Straus-Dürckheim, -in his _Considérations générales sur l’anatomie comparée des animaux -articulés_, 1828. - -[51] The fact that the limb tends to swing in pendulum-time was first -observed by the brothers Weber (_Mechanik der menschl. Gehwerkzeuge_, -Göttingen, 1836). Some later writers have criticised the statement -(e.g. Fischer, Die Kinematik des Beinschwingens etc., _Abh. math. phys. -Kl. k. Sächs. Ges._ XXV–XXVIII, 1899–1903), but for all that, with -proper qualifications, it remains substantially true. - -[52] Quoted in Mr John Bishop’s interesting article in Todd’s -_Cyclopaedia_, III, p. 443. - -[53] 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. - -[54] _Proc. Psychical Soc._ XII, pp. 338–355, 1897. - -[55] For various calculations of the increase of surface due to -histological and anatomical subdivision, see E. Babak, Ueber die -Oberflächenentwickelung bei Organismen, _Biol. Centralbl._ XXX, pp. -225–239, 257–267, 1910. In connection with the physical theory of -surface-energy, Wolfgang Ostwald has introduced the conception of -_specific surface_, that is to say the ratio of surface to volume, or -_S_/_V_. In a cube, _V_ = _l_^3, and _S_ = 6_l_^2; therefore _S_/_V_ -= 6/_l_. Therefore if the side _l_ measure 6 cm., the ratio _S_/_V_ -= 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. - -[56] 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. - -[57] Cf. Tait, _Proc. R.S.E._ V, 1866, and VI, 1868. - -[58] _Physiolog. Notizen_ (9), p. 425, 1895. Cf. Strasbürger, Ueber die -Wirkungssphäre der Kerne und die Zellgrösse, _Histolog. Beitr._ (5), -pp. 95–129, 1893; J. J. Gerassimow, Ueber die Grösse des Zellkernes, -_Beih. Bot. Centralbl._ XVIII, 1905; also G. Levi and T. Terni, Le -variazioni dell’ indice plasmatico-nucleare durante l’intercinesi, -_Arch. Ital. di Anat._ X, p. 545, 1911. - -[59] _Arch. f. Entw. Mech._ IV, 1898, pp. 75, 247. - -[60] Conklin, E. G., Cell-size and nuclear-size, _J. Exp. Zool._ XII. -pp. 1–98, 1912. - -[61] Thus the fibres of the crystalline lens are of the same size in -large and small dogs; Rabl, _Z. f. w. Z._ LXVII, 1899. Cf. (_int. al._) -Pearson, On the Size of the Blood-corpuscles in Rana, _Biometrika_, -VI, 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,” _Natural Philosophy_, 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, _P.Z.S._ 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. - -[62] Cf. P. Enriques, La forma come funzione della grandezza: Ricerche -sui gangli nervosi degli Invertebrati, _Arch. f. Entw. Mech._ XXV, p. -655, 1907–8. - -[63] 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 _is_ 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, _Arch. Ital. -di Anat. e di Embryolog._ V, p. 291, 1906. - -[64] Boveri. _Zellen-studien, V. Ueber die Abhängigkeit der Kerngrösse -und Zellenzahl der Seeigellarven von der Chromosomenzahl der -Ausgangszellen._ Jena, 1905. - -[65] 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., _J. -Roy. Army Med. Corps_, Feb. 1910. - -[66] _Zur Erkenntniss der Kolloide_, 1905, p. 122; where there will be -found an interesting discussion of various molecular and other minute -magnitudes. - -[67] _Encyclopaedia Britannica_, 9th edit., vol. III, p. 42, 1875. - -[68] Sur la limite de petitesse des organismes, _Bull. Soc. R. des -Sc. méd. et nat. de Bruxelles_, Jan. 1903; _Rec. d’œuvres_ (_Physiol. -générale_), p. 325. - -[69] Cf. A. Fischer, _Vorlesungen über Bakterien_, 1897, p. 50. - -[70] F. Hofmeister, quoted in Cohnheim’s _Chemie der Eiweisskörper_, -1900, p. 18. - -[71] McKendrick arrived at a still lower estimate, of about 1250 -proteid molecules in the minutest organisms. _Brit. Ass. Rep._ 1901, p. -808. - -[72] Cf. Perrin, _Les Atomes_, 1914, p. 74. - -[73] Cf. Tait, On Compression of Air in small Bubbles, _Proc. R. S. E._ -V, 1865. - -[74] _Phil. Mag._ XLVIII, 1899; _Collected Papers_, IV, p. 430. - -[75] Carpenter, _The Microscope_, edit. 1862, p. 185. - -[76] 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, _Beyond the Atom_, 1913, pp. 118–128; and see, further, Perrin, -_Les Atomes_, pp. 119–189. - -[77] Cf. R. Gans, Wie fallen Stäbe und Scheiben in einer reibenden -Flüssigkeit? _Münchener Bericht_, 1911, p. 191; K. Przibram, Ueber die -Brown’sche Bewegung nicht kugelförmiger Teilchen, _Wiener Ber._ 1912, -p. 2339. - -[78] Ueber die ungeordnete Bewegung niederer Thiere, _Pflüger’s -Archiv_, CLIII, p. 401, 1913. - -[79] 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. - -[80] Our subject is one of Bacon’s “Instances of the Course,” or -studies wherein we “measure Nature by periods of Time.” In Bacon’s -_Catalogue of Particular Histories_, 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.” - -[81] Cf. Aristotle, _Phys._ vi, 5, 235 _a_ 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.” _Nov. Org._ XLVI. - -[82] Cf. (e.g.) _Elem. Physiol._ ed. 1766, VIII, 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.” - -[83] _Beiträge zur Entwickelungsgeschichte des Hühnchens im Ei_, p. 40, -1817. Roux ascribes the same views also to Von Baer and to R. H. Lotze -(_Allg. Physiologie_, p. 353, 1851). - -[84] Roux, _Die Entwickelungsmechanik_, p. 99, 1905. - -[85] _Op. cit._ p. 302, “Magnum hoc naturae instrumentum, etiam in -corpore animato evolvendo potenter operatur; etc.” - -[86] _Ibid._ p. 306. “Subtiliora ista, et aliquantum hypothesi mista, -tamen magnum mihi videntur speciem veri habere.” - -[87] Cf. His, On the Principles of Animal Morphology, _Proc. R. S. -E._ XV, 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.” - -[88] Hertwig, O., _Zeit und Streitfragen der Biologie_, II. 1897. - -[89] Cf. Roux, _Gesammelte Abhandlungen_, II, p. 31, 1895. - -[90] _Treatise on Comparative Embryology_, I, p. 4, 1881. - -[91] Cf. Fick, _Anal. Anzeiger_, XXV, p. 190, 1904. - -[92] 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. - -[93] “In omni rerum naturalium historia utile est _mensuras definiri -et numeros_,” Haller, _Elem. Physiol._ II, p. 258, 1760. Cf. Hales, -_Vegetable Staticks_, Introduction. - -[94] Brussels, 1871. Cf. the same author’s _Physique sociale_, 1835, -and _Lettres sur la théorie des probabilités_, 1846. See also, for the -general subject, Boyd, R., Tables of weights of the Human Body, etc. -_Phil. Trans._ vol. CLI, 1861; Roberts, C., _Manual of Anthropometry_, -1878; Daffner, F., _Das Wachsthum des Menschen_ (2nd ed.), 1902, etc. - -[95] 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.” - -[96] Joly, _The Abundance of Life_, 1915 (1890), p. 86. - -[97] “_Lou pes, mèstre de tout_ [Le poids, maître de tout], _mèstre -sènso vergougno, Que te tirasso en bas de sa brutalo pougno_,” J. H. -Fabre, _Oubreto prouvençalo_, p. 61. - -[98] 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,” _Riv. di Scienza_, -1907, and in “Wachsthum und seine analytische Darstellung,” _Biol. -Centralbl._ June, 1909. Haller (_Elem_. VII, p. 68) recognised -_decrementum_ as a phase of growth, not less important (theoretically) -than _incrementum_: “_tristis, sed copiosa, haec est materies_.” - -[99] Cf. (_int. al._), Friedenthal, H., Das Wachstum des -Körpergewichtes ... in verschiedenen Lebensältern, _Zeit. f. allg. -Physiol._ IX, pp. 487–514, 1909. - -[100] As Haller observed it to do in the chick (_Elem._ VIII, p. 294): -“Hoc iterum incrementum miro ordine ita distribuitur, ut in principio -incubationis maximum est: inde perpetuo minuatur.” - -[101] 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: _Principio circum tribus actis impiger annis Floret equus, puer -hautquaquam_, etc. - -[102] Minot, C. S., Senescence and Rejuvenation, _Journ. of Physiol._ -XII, pp. 97–153, 1891; The Problem of Age, Growth and Death, _Pop. -Science Monthly_ (June–Dec.), 1907. - -[103] Quoted in Vierordt’s _Anatomische ... Daten und Tabellen_, 1906. -p. 13. - -[104] _Unsere Körperform_, Leipzig, 1874. - -[105] 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., _Amer. Journ. of Anat._ IX, 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: - - Months 0 1 2 3 4 5 6 7 8 9 10 - Wt in gms. ·0 ·04 3 36 120 330 600 1000 1500 2200 3200 - -[106] G. Kraus (after Wallich-Martius), _Ann. du Jardin bot. de -Buitenzorg_, XII, 1, 1894, p. 210. Cf. W. Ostwald, _Zeitliche -Eigenschaften_, etc. p. 56. - -[107] Cf. Chodat, R., et Monnier, A., Sur la courbe de croissance des -végétaux, _Bull. Herb. Boissier_ (2), V, pp. 615, 616, 1905. - -[108] Cf. Fr. Boas, Growth of Toronto Children, _Rep. of U.S. Comm. -of Education_, 1896–7, pp. 1541–1599, 1898; Boas and Clark Wissler, -Statistics of Growth, _Education Rep._ 1904, pp. 25–132, 1906; H. P. -Bowditch, _Rep. Mass. State Board of Health_, 1877; K. Pearson, On the -Magnitude of certain coefficients of Correlation in Man, _Pr. R. S._ -LXVI, 1900. - -[109] _l.c._ p. 42, and other papers there quoted. - -[110] See, for an admirable résumé of facts, Wolfgang Ostwald, _Ueber -die Zeitliche Eigenschaften der Entwickelungsvorgänge_ (71 pp.), -Leipzig, 1908 (Roux’s _Vorträge_, Heft V): 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, _Exp. Zoologie_, 1913, pt. IV -(_Vitalität_), pp. 85–87. - -[111] Cf. St Loup, Vitesse de croissance chez les Souris, _Bull. Soc. -Zool. Fr._ XVIII, 242, 1893; Robertson, _Arch. f. Entwickelungsmech._ -XXV, p. 587, 1908; Donaldson. _Boas Memorial Volume_, New York, 1906. - -[112] Luciani e Lo Monaco, _Arch. Ital. de Biologie_, XXVII, p. 340, -1897. - -[113] Schaper, _Arch. f. Entwickelungsmech._ XIV, p. 356, 1902. Cf. -Barfurth, Versuche über die Verwandlung der Froschlarven, _Arch. f. -mikr. Anat._ XXIX, 1887. - -[114] Joh. Schmidt, Contributions to the Life-history of the Eel, -_Rapports du Conseil Intern. pour l’exploration de la Mer_, vol. V, pp. -137–274, Copenhague, 1906. - -[115] That the metamorphoses of an insect are but phases in a process -of growth, was firstly clearly recognised by Swammerdam, _Biblia -Naturae_, 1737, pp. 6, 579 etc. - -[116] From Bose, J. C., _Plant Response_, London, 1906, p. 417. - -[117] This phenomenon, of _incrementum inequale_, as opposed to -_incrementum in universum_, 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.” (_Elem._ VIII (2), p. 34). - -[118] See (_inter alia_) Fischel, A., Variabilität und Wachsthum des -embryonalen Körpers, _Morphol. Jahrb._ XXIV, pp. 369–404, 1896. Oppel, -_Vergleichung des Entwickelungsgrades der Organe zu verschiedenen -Entwickelungszeiten bei Wirbelthieren_, Jena, 1891. Faucon, A., _Pesées -et Mensurations fœtales à différents âges de la grossesse_. (Thèse.) -Paris, 1897. Loisel, G., Croissance comparée en poids et en longueur -des fœtus mâle et femelle dans l’espèce humaine, _C. R. Soc. de -Biologie_, Paris, 1903. Jackson, C. M., Pre-natal growth of the human -body and the relative growth of the various organs and parts, _Am. J. -of Anat._ IX, 1909; Post-natal growth and variability of the body and -of the various organs in the albino rat, _ibid._ XV, 1913. - -[119] _l.c._ p. 1542. - -[120] Variation and Correlation in Brain-weight, _Biometrika_, IV, pp. -13–104, 1905. - -[121] _Die Säugethiere_, p. 117. - -[122] _Amer. J. of Anatomy_, VIII, pp. 319–353, 1908. Donaldson -(_Journ. Comp. Neur. and Psychol._ XVIII, pp. 345–392, 1908) also -gives a logarithmic formula for brain-weight (_y_) as compared -with body-weight (_x_), which in the case of the white rat is _y_ -= ·554 − ·569 log(_x_ − 8·7), and the agreement is very close. But the -formula is admittedly empirical and as Raymond Pearl says (_Amer. Nat._ -1909, p. 303), “no ulterior biological significance is to be attached -to it.” - -[123] _Biometrika_, IV, pp. 13–104, 1904. - -[124] Donaldson, H. H., A Comparison of the White Rat with Man in -respect to the Growth of the entire Body, _Boas Memorial Vol._, New -York, 1906, pp. 5–26. - -[125] Besides many papers quoted by Dubois on the growth and weight of -the brain, and numerous papers in _Biometrika_, see also the following: -Ziehen, Th., _Das Gehirn: Massverhältnisse_, in Bardeleben’s _Handb. -der Anat. des Menschen_, IV, pp. 353–386, 1899. Spitzka, E. A., -Brain-weight of Animals with special reference to the Weight of the -Brain in the Macaque Monkey, _J. Comp. Neurol._ XIII, 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, -_J. f. Psychol. u. Neurol._ XIII, 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, _Journ. of Morph._ XXII, pp. -663–694, 1911. - -[126] Cf. Jenkinson, Growth, Variability and Correlation in Young -Trout, _Biometrika_, VIII, pp. 444–455, 1912. - -[127] Cf. chap. xvii, p. 739. - -[128] “ ...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,” -_Vegetable Staticks_, Exp. cxxiii. - -[129] From Sachs, _Textbook of Botany_, 1882, p. 820. - -[130] Variation and Differentiation in Ceratophyllum, _Carnegie Inst. -Publications_, No. 58, Washington, 1907. - -[131] Cf. Lämmel, Ueber periodische Variationen in Organismen, _Biol. -Centralbl._ XXII, pp. 368–376, 1903. - -[132] 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.” _Creative Evolution_, p. 71. - -[133] 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. - -[134] Jackson, C. M., _J. of Exp. Zool._ XIX, 1915, p. 99; cf. also -Hans Aron, Unters. über die Beeinflüssung der Wachstum durch die -Ernährung, _Berl. klin. Wochenbl._ LI, pp. 972–977, 1913, etc. - -[135] The temperature limitations of life, and to some extent of -growth, are summarised for a large number of species by Davenport, -_Exper. Morphology_, cc. viii, xviii, and by Hans Przibram, _Exp. -Zoologie_, IV, c. v. - -[136] Réaumur: _L’art de faire éclore et élever en toute saison des -oiseaux domestiques, foit par le moyen de la chaleur du fumier_, Paris, -1749. - -[137] Cf. (_int. al._) de Vries, H., Matériaux pour la connaissance -de l’influence de la température sur les plantes, _Arch. Néerl._ V, -385–401, 1870. Köppen, Wärme und Pflanzenwachstum, _Bull. Soc. Imp. -Nat. Moscou._ XLIII, pp. 41–110, 1870. - -[138] Blackman, F. F., _Ann. of Botany_, XIX, p. 281, 1905. - -[139] For various instances of a “temperature coefficient” in -physiological processes, see Kanitz, _Zeitschr. f. Elektrochemie_, -1907, p. 707; _Biol. Centralbl._ XXVII, p. 11, 1907; Hertzog, R. -O., Temperatureinfluss auf die Entwicklungsgeschwindigkeit der -Organismen, _Zeitschr. f. Elektrochemie_, XI, p 820, 1905; Krogh, -Quantitative Relation between Temperature and Standard Metabolism, -_Int. Zeitschr. f. physik.-chem. Biologie_, I, p. 491, 1914; Pütter, -A., Ueber Temperaturkoefficienten, _Zeitschr. f. allgem. Physiol._ -XVI, p. 574, 1914. Also Cohen, _Physical Chemistry for Physicians and -Biologists_ (English edition), 1903; Pike, F. H., and Scott. E. L., The -Regulation of the Physico-chemical Condition of the Organism, _American -Naturalist_, Jan. 1915, and various papers quoted therein. - -[140] Cf. Errera, L., _L’Optimum_, 1896 (_Rec. d’Oeuvres, Physiol. -générale_, pp. 338–368, 1910); Sachs, _Physiologie d. Pflanzen_, 1882, -p. 233; Pfeffer, _Pflanzenphysiologie_, ii, p. 78, 1904; and cf. Jost, -Ueber die Reactionsgeschwindigkeit im Organismus, _Biol. Centralbl._ -XXVI, pp. 225–244, 1906. - -[141] After Köppen, _Bull. Soc. Nat. Moscou_, XLIII, pp. 41–110, 1871. - -[142] _Botany_, p. 387. - -[143] Leitch, I., Some Experiments on the Influence of Temperature on -the Rate of Growth in _Pisum sativum, Ann. of Botany_, XXX, pp. 25–46, -1916. (Cf. especially Table III, p. 45.) - -[144] Blackman, F. F., Presidential Address in Botany, _Brit. Ass._ -Dublin, 1908. - -[145] _Rec. de l’Inst. Bot. de Bruxelles_, VI, 1906. - -[146] Hertwig, O., Einfluss der Temperatur auf die Entwicklung von -_Rana fusca_ und _R. esculenta_, _Arch. f. mikrosk. Anat._ LI, p. -319, 1898. Cf. also Bialaszewicz, K., Beiträge z. Kenntniss d. -Wachsthumsvorgänge bei Amphibienembryonen, _Bull. Acad. Sci. de -Cracovie_, p. 783, 1908; Abstr. in _Arch. f. Entwicklungsmech._ XXVIII, -p. 160, 1909. - -[147] Der Grad der Beschleunigung tierischer Entwickelung durch -erhöhte Temperatur, _A. f. Entw._ Mech. XX. p. 130, 1905. More -recently, Bialaszewicz has determined the coefficient for the rate of -segmentation in Rana as being 2·4 per 10° C. - -[148] _Das Wachstum des Menschen_, p. 329, 1902. - -[149] The _diurnal_ periodicity is beautifully shewn in the case of -the Hop by Joh. Schmidt (_C. R. du Laboratoire de Carlsberg_, X, pp. -235–248, Copenhague, 1913). - -[150] _Trans. Botan. Soc. Edinburgh_, XVIII, 1891, p. 456. - -[151] I had not received, when this was written, Mr Douglass’s paper, -On a method of estimating Rainfall by the Growth of Trees, _Bull. -Amer. Geograph. Soc._ XLVI, 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. - -[152] 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. - -[153] On growth in relation to light, see Davenport, _Exp. Morphology_, -II, 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, II, 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. - -[154] Cf. for instance, Nägeli’s classical account of the effect of -change of habitat on Alpine and other plants: _Sitzungsber. Baier. -Akad. Wiss._ 1865, pp. 228–284. - -[155] Cf. Blackman, F. F., Presidential Address in Botany, _Brit. Ass._ -Dublin, 1908. The fact was first enunciated by Baudrimont and St Ange, -Recherches sur le développement du fœtus, _Mém. Acad. Sci._ XI, p. 469, -1851. - -[156] Cf. Loeb, _Untersuchungen zur physiol. Morphologie der Thiere_, -1892; also Experiments on Cleavage, _J. of Morph._ VII, p. 253, 1892; -Zusammenstellung der Ergebnisse einiger Arbeiten über die Dynamik des -thierischen Wachsthum, _Arch. f. Entw. Mech._ XV, 1902–3, p. 669; -Davenport, On the Rôle of Water in Growth, _Boston Soc. N. H._ 1897; -Ida H. Hyde, _Am. J. of Physiol._ XII, 1905, p. 241, etc. - -[157] _Pflüger’s Archiv_, LV, 1893. - -[158] Beiträge zur Kenntniss der Wachstumsvorgänge bei -Amphibienembryonen, _Bull. Acad. Sci. de Cracovie_, 1908, p. 783; cf. -_Arch. f. Entw. Mech._ XXVIII, p. 160, 1909; XXXIV, p. 489, 1912. - -[159] Fehling, H., _Arch. für Gynaekologie_, XI, 1877; cf. Morgan, -_Experimental Zoology_, p. 240, 1907. - -[160] Höber, R., Bedeutung der Theorie der Lösungen für Physiologie und -Medizin, _Biol. Centralbl._ XIX, 1899; cf. pp. 272–274. - -[161] 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 _Anisonema acinus_, Bütschli -(_Z. f. w. Z._ XXIX, p. 429, 1877). - -[162] These “Fezzan-worms,” when first described, were supposed to be -“insects’ eggs”; cf. Humboldt, _Personal Narrative_, VI, i, 8, note; -Kirby and Spence, Letter X. - -[163] Cf. _Introd. à l’étude de la médecine expérimentale_, 1885, p. -110. - -[164] Cf. Abonyi, _Z. f. w. Z._ CXIV, p. 134, 1915. But Frédéricq has -shewn that the amount of NaCl in the blood of Crustacea (_Carcinus -moenas_) varies, and all but corresponds, with the density of the water -in which the creature has been kept (_Arch. de Zool. Exp. et Gén._ (2), -III, 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 -_Ergebn. d. Physiologie_, VII, pp. 160–402, 1908) suggest that the case -of the brine-shrimps must be looked upon as an extreme or exceptional -one. - -[165] Cf. Schmankewitsch, _Z. f. w. Zool._ XXV, 1875, XXIX, 1877, etc.; -transl. in appendix to Packard’s _Monogr. of N. American Phyllopoda_, -1883, pp. 466–514; Daday de Deés, _Ann. Sci. Nat._ (_Zool._), (9), -XI, 1910; Samter und Heymons, _Abh. d. K. pr. Akad. Wiss._ 1902; -Bateson, _Mat. for the Study of Variation_, 1894, pp. 96–101; Anikin, -_Mitth. Kais. Univ. Tomsk_, XIV: _Zool. Centralbl._ VI, pp. 756–760, -1908; Abonyi, _Z. f. w. Z._ CXIV, pp. 96–168, 1915 (with copious -bibliography), etc. - -[166] 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.” - -[167] Such phenomena come precisely under the head of what Bacon called -_Instances of Magic_: “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, _or in -yeast and such-like_.” _Nov. Org._, cap. li. - -[168] Monnier, A., Les matières minérales, et la loi d’accroissement -des Végétaux, _Publ. de l’Inst. de Bot. de l’Univ. de Genève_ -(7), III, 1905. Cf. Robertson, On the Normal Rate of Growth of an -Individual, and its Biochemical Significance, _Arch. f. Entw. Mech._ -XXV, pp. 581–614, XXVI, pp. 108–118, 1908; Wolfgang Ostwald, _Die -zeitlichen Eigenschaften der Entwickelungsvorgänge_, 1908; Hatai, S., -Interpretation of Growth-curves from a Dynamical Standpoint, _Anat. -Record_, V, p. 373, 1911. - -[169] _Biochem. Zeitschr._ II, 1906, p. 34. - -[170] 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. - -[171] Cf. Loeb, The Stimulation of Growth, _Science_, May 14, 1915. - -[172] _B. coli-communis_, according to Buchner, tends to double in -22 minutes; in 24 hours, therefore, a single individual would be -multiplied by something like 10^{28}; _Sitzungsber. München. Ges. -Morphol. u. Physiol._ III, pp. 65–71, 1888. Cf. Marshall Ward, Biology -of _Bacillus ramosus_, etc. _Pr. R. S._ LVIII, 265–468, 1895. The -comparatively large infusorian Stylonichia, according to Maupas, would -multiply in a month by 10^{43}. - -[173] Cf. Enriques, Wachsthum und seine analytisehe Darstellung, _Biol. -Centralbl._ 1909, p. 337. - -[174] Cf. (_int. al._) Mellor, _Chemical Statics and Dynamics_, 1904, -p. 291. - -[175] Cf. Robertson, _l.c._ - -[176] See, for a brief resumé of this subject, Morgan’s _Experimental -Zoology_, chap. xvi. - -[177] _Amer. J. of Physiol._, X, 1904. - -[178] _C.R._ CXXI, CXXII, 1895–96. - -[179] Cf. Loeb, _Science_, May 14, 1915. - -[180] Cf. Baumann u. Roos, Vorkommen von Iod im Thierkörper, _Zeitschr. -für Physiol. Chem._ XXI, XXII, 1895, 6. - -[181] Le Néo-Vitalisme, _Rev. Scientifique_, Mars 1911, p. 22 (of -reprint). - -[182] _La vie et la mort_, p. 43, 1902. - -[183] Cf. Dendy, _Evolutionary Biology_, 1912, p. 408; _Brit. Ass. -Report_ (Portsmouth), 1911, p. 278. - -[184] 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, _ad loc._ - -[185] Even after we have so narrowed the scope and sphere of natural -selection, it is still hard to understand; for the causes of -_extinction_ are often wellnigh as hard to comprehend as are those of -the _origin_ 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, _Amer. Nat._ xli, p. 46, 1907. - -[186] See Professor T. H. Morgan’s _Regeneration_ (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, _Elem. Physiologiae_, VIII, p. 156 -_seq._ - -[187] _Journ. Experim. Zool._ VII, p. 397, 1909. - -[188] _Op. cit._ p. 406, Exp. IV. - -[189] 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, _op. cit._ p. 35.) - -[190] _Powers of the Creator_, I, p. 7, 1851. See also _Rare and -Remarkable Animals_, II, pp. 17–19, 90, 1847. - -[191] Lillie, F. R., The smallest Parts of Stentor capable of -Regeneration, _Journ. of Morphology_, XII, p. 239, 1897. - -[192] Boveri, Entwicklungsfähigkeit kernloser Seeigeleier, etc., _Arch. -f. Entw. Mech._ II, 1895. See also Morgan, Studies of the partial -larvae of Sphaerechinus, _ibid._ 1895; J. Loeb, On the Limits of -Divisibility of Living Matter, _Biol. Lectures_, 1894, etc. - -[193] Cf. Przibram, H., Scheerenumkehr bei dekapoden Crustaceen, _Arch. -f. Entw. Mech._ XIX, 181–247, 1905; XXV, 266–344, 1907. Emmel, _ibid._ -XXII, 542, 1906; Regeneration of lost parts in Lobster, _Rep. Comm. -Inland Fisheries, Rhode Island_, XXXV, XXXVI, 1905–6; _Science_ (n.s.), -XXVI, 83–87, 1907. Zeleny, Compensatory Regulation, _J. Exp. Zool._ II, -1–102, 347–369, 1905; etc. - -[194] Lobsters are occasionally found with two symmetrical claws: -which are then usually serrated, sometimes (but very rarely) both -blunt-toothed. Cf. Calman, _P.Z.S._ 1906, pp. 633, 634, and _reff._ - -[195] Wilson, E. B., Reversal of Symmetry in _Alpheus heterochelis_, -_Biol. Bull._ IV, p. 197, 1903. - -[196] _J. Exp. Zool._ VII, p. 457, 1909. - -[197] _Biologica_, III, p. 161, June. 1913. - -[198] _Anatomical and Pathological Observations_, p. 3, 1845; -_Anatomical Memoirs_, II, p. 392, 1868. - -[199] Giard, A., L’œuf et les débuts de l’évolution, _Bull. Sci. du -Nord de la Fr._ VIII, pp. 252–258, 1876. - -[200] _Entwickelungsvorgänge der Eizelle_, 1876; _Investigations on -Microscopic Foams and Protoplasm_, p. 1, 1894. - -[201] _Journ. of Morphology_, I, p. 229, 1887. - -[202] While it has been very common to look upon the phenomena of -mitosis as sufficiently explained by the results _towards which_ 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 (_Dangeard_), distribution particulière et -signification dualiste des substances nucléaires (substance kinétique -et substance générative ou héréditaire, _Hartmann et ses élèves_), 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, _Archiv für Protistenkunde_, XIX, p. 344, 1913. - -[203] This is the old philosophic axiom writ large: _Ignorato motu, -ignoratur natura_; 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”; _Scientific Writings_, 1902, p. 385. - -[204] As when Nägeli concluded that the organism is, in a certain -sense, “vorgebildet”; _Beitr. zur wiss. Botanik_, II, 1860. Cf. E. B. -Wilson, _The Cell, etc._, p. 302. - -[205] “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.” _Sur le principe de la Vie_, -p. 431 (Erdmann). This is the very converse of the doctrine of the -Atomists, who could not conceive a condition “_ubi dimidiae partis pars -semper habebit Dimidiam partem, nec res praefiniet ulla_.” - -[206] Cf. an interesting passage from the _Elements_ (I, p. 445, -Molesworth’s edit.), quoted by Owen, _Hunterian Lectures on the -Invertebrates_, 2nd ed. pp. 40, 41, 1855. - -[207] “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 _Organisation_ bezeichnen,” -Brücke, Die Elementarorganismen, _Wiener Sitzungsber._ XLIV, 1861, p. -386; quoted by Wilson, _The Cell_, etc. p. 289. Cf. also Hardy, _Journ. -of Physiol._ XXIV, 1899, p. 159. - -[208] Precisely as in the Lucretian _concursus_, _motus_, _ordo_, -_positura_, _figurae_, whereby bodies _mutato ordine mutant naturam_. - -[209] Otto Warburg, Beiträge zur Physiologie der Zelle, insbesondere -über die Oxidationsgeschwindigkeit in Zellen; in Asher-Spiro’s -_Ergebnisse der Physiologie_, XIV, pp. 253–337, 1914 (see p. 315). (Cf. -Bayliss, _General Physiology_, 1915, p. 590). - -[210] Hardy, W. B., On some Problems of Living Matter (Guthrie -Lecture), _Tr. Physical Soc. London_, xxviii, p. 99–118, 1916. - -[211] As a matter of fact both phrases occur, side by side, in Graham’s -classical paper on “Liquid Diffusion applied to Analysis,” _Phil. -Trans._ CLI, p. 184, 1861; _Chem. and Phys. Researches_ (ed. Angus -Smith), 1876, p. 554. - -[212] L. Rhumbler, Mechanische Erklärung der Aehnlichkeit zwischen -Magnetischen Kraftliniensystemen und Zelltheilungsfiguren, _Arch. f. -Entw. Mech._ XV, p. 482, 1903. - -[213] Gallardo, A., Essai d’interpretation des figures caryocinétiques, -_Anales del Museo de Buenos-Aires_ (2), II, 1896; La division de la -cellule, phenomène bipolaire de caractère electro-colloidal, _Arch. f. -Entw. Mech._ XXVIII, 1909, etc. - -[214] _Arch. f. Entw. Mech._ III, IV, 1896–97. - -[215] On various theories of the mechanism of mitosis, see (e.g.) -Wilson, _The Cell in Development_, etc., pp. 100–114; Meves, -_Zelltheilung_, in Merkel u. Bonnet’s _Ergebnisse der Anatomie_, etc., -VII, VIII, 1897–8; Ida H. Hyde, _Amer. Journ. of Physiol._ XII, pp. -241–275, 1905; and especially Prenant, A., Theories et interprétations -physiques de la mitose, _J. de l’Anat. et Physiol._ XLVI, pp. 511–578, -1910. - -[216] Hartog, M., Une force nouvelle: le mitokinétisme, _C.R._ 11 Juli, -1910; Mitokinetism in the Mitotic Spindle and in the Polyasters, _Arch. -f. Entw. Mech._ XXVII, pp. 141–145, 1909; cf. _ibid._ XL, pp. 33–64, -1914. Cf. also Hartog’s papers in _Proc. R. S._ (B), LXXVI, 1905; -_Science Progress_ (n. s.), I, 1907; _Riv. di Scienza_, II, 1908; _C. -R. Assoc. fr. pour l’Avancem. des Sc._ 1914, etc. - -[217] 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 _stress in a medium_,—of tension or attraction along the lines, and -of repulsion transverse to the lines, of the diagram. - -[218] Cf. also the curious phenomenon in a dividing egg described as -“spinning” by Mrs G. F. Andrews, _J. of Morph._ XII, pp. 367–389, 1897. - -[219] Whitman, _J. of Morph._ II, p. 40, 1889. - -[220] “Souvent il n’y a qu’une séparation _physique_ entre le -cytoplasme et le suc nucléaire, comme entre deux liquides immiscibles, -etc.;” Alexeieff, Sur la mitose dite “primitive,” _Arch. f. -Protistenk._ XXIX, p. 357, 1913. - -[221] The appearance of “vacuolation” is a result of endosmosis or the -diffusion of a less dense fluid into the denser plasma of the cell. -_Caeteris paribus_, 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 _Protozoa_, 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 (_Arb. z. z. Inst. Würzburg_, 1872, cf. Massart, _Arch. de Biol._ -LX, p. 515, 1889). _Actinophrys sol_, when gradually acclimatised -to sea-water, loses its vacuoles, and _vice versa_ (Gruber, _Biol. -Centralbl._ IX, p. 22, 1889); and the same is true of Amoeba (Zuelzer, -_Arch. f. Entw. Mech._ 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 (_Phil. Mag._ (n. s.), XI, 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 (_Arch. f. Entw. Mech._ VII, 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. - -[222] Cf. p. 660. - -[223] 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. - -[224] _Théorie physico-chimique de la Vie_, p. 73, 1910; _Mechanism of -Life_, p. 56, 1911. - -[225] Whence the name “mitosis” (Greek μίτος, a thread), applied first -by Flemming to the whole phenomenon. Kollmann (_Biol. Centralbl._ -II, p. 107, 1882) called it _divisio per fila_, or _divisio laqueis -implicata_. Many of the earlier students, such as Van Beneden (Rech. -sur la maturation de l’œuf, _Arch. de Biol._ IV, 1883), and Hermann -(Zur Lehre v. d. Entstehung d. karyokinetischen Spindel, _Arch. f. -mikrosk. Anat._ XXXVII, 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, _Q. J. M. S._ XLVIII, p. -471, 1904.) - -[226] Cf. Bütschli, O., Ueber die künstliche Nachahmung der -karyokinetischen Figur, _Verh. Med. Nat. Ver. Heidelberg_, V, pp. 28–41 -(1892), 1897. - -[227] 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.” _Rev. Scientifique_, Feb. 1911. - -[228] F. Schwartz, in Cohn’s _Beitr. z. Biologie der Pflanzen_, V, p. -1, 1887. - -[229] Fischer, _Anat. Anzeiger_, IX, p. 678, 1894, X, p. 769, 1895. - -[230] See, in particular, W. B. Hardy, On the structure of Cell -Protoplasm, _Journ. of Physiol._ XXIV, pp. 158–207, 1889; also Höber, -_Physikalische Chemie der Zelle und der Gewebe_, 1902. Cf. (_int. al._) -Flemming, _Zellsubstanz, Kern und Zelltheilung_ 1882, p. 51, etc. - -[231] My description and diagrams (Figs 42–51) are based on those of -Professor E. B. Wilson. - -[232] If the word _permeability_ be deemed too directly suggestive of -the phenomena of _magnetism_ we may replace it by the more general -term of _specific inductive capacity_. 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.) - -[233] On the effect of electrical influences in altering the -surface-tensions of the colloid particles, see Bredig, _Anorganische -Fermente_, pp. 15, 16, 1901. - -[234] _The Cell_, etc. p. 66. - -[235] Lillie, R. S., _Amer. J. of Physiol._ VIII, p. 282, 1903. - -[236] 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. - -[237] M. Foster, _Lectures on the History of Physiology_, 1901, p. 62. - -[238] _Op. cit._ pp. 110 and 91. - -[239] Lamb, A. B., A new Explanation of the Mechanism of Mitosis, -_Journ. Exp. Zool._ V, pp. 27–33, 1908. - -[240] _Amer. J. of Physiol._ VIII, pp. 273–283, 1903 (_vide supra_, p. -181); cf. _ibid._ XV, pp. 46–84, 1905. Cf. also _Biological Bulletin_, -IV, p. 175. 1903. - -[241] In like manner Hardy has shewn that colloid particles migrate -with the negative stream if the reaction of the surrounding fluid -be alkaline, and _vice versa_. 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, _Journ. of Physiol._ XXIV, p. 288–304, -1899, also Hardy and H. W. Harvey, Surface Electric Charges of Living -Cells, _Proc. R. S._ LXXXIV (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, _Journ. of Exper. Zool._ X, pp. 508–556, 1911). - -[242] On Differences in Electrical Potential in Developing Eggs, _Amer. -Journ. of Physiol._ XII, pp. 241–275, 1905. This paper contains an -excellent summary of various physical theories of the segmentation of -the cell. - -[243] 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, -_Journ. Mar. Biol. Assoc._ X, pp. 50–59, 1913). - -[244] Schewiakoff, Ueber die karyokinetische Kerntheilung der _Euglypha -alveolata, Morph. Jahrb._ XIII, pp. 193–258, 1888 (see p. 216). - -[245] Coe, W. R., Maturation and Fertilization of the Egg of -Cerebratulus, _Zool. Jahrbücher_ (_Anat. Abth._), XII, pp. 425–476, -1899. - -[246] Thus, for example, Farmer and Digby (On Dimensions of Chromosomes -considered in relation to Phylogeny, _Phil. Trans._ (B), CCV, pp. 1–23, -1914) have been at pains to shew, in confutation of Meek (_ibid._ -CCIII, pp. 1–74, 1912), that the width of the chromosomes cannot be -correlated with the order of phylogeny. - -[247] Cf. also _Arch. f. Entw. Mech._ X, p. 52, 1900. - -[248] Cf. Loeb, _Am. J. of Physiol._ VI, p. 32, 1902; Erlanger, _Biol. -Centralbl._ XVII, pp. 152, 339, 1897; Conklin, _Biol. Lectures_, _Woods -Holl_, p. 69, etc. 1898–9. - -[249] Robertson, T. B., Note on the Chemical Mechanics of Cell -Division, _Arch. f. Entw. Mech._ XXVII, p. 29, 1909, XXXV, p. 692. -1913. Cf. R. S. Lillie, _J. Exp. Zool._ XXI, pp. 369–402, 1916. - -[250] Cf. D’Arsonval, _Arch. de Physiol._ p. 460, 1889; Ida H. Hyde, -_op. cit._ p. 242. - -[251] Cf. Plateau’s remarks (_Statique des liquides_, II, p. 154) on -the _tendency_ towards equilibrium, rather than actual equilibrium, in -many of his systems of soap-films. - -[252] 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. - -[253] Fol, H., _Recherches sur la fécondation_, 1879. Roux, W., -Beiträge zur Entwickelungsmechanik des Embryo, _Arch. f. Mikr. Anat._ -XIX, 1887. Whitman, C. O., Oökinesis, _Journ. of Morph._ I, 1887. - -[254] Wilson. _The Cell_, p. 77. - -[255] 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 (_The Cell_, p. 206), over 80 % of the observed cases lie -between 6 and 16, and nearly 60 % between 8 and 12. - -[256] _Theory of Cells_, p. 191. - -[257] _The Cell in Development_, etc. p. 59; cf. pp. 388, 413. - -[258] E.g. Brücke, _Elementarorganismen_, 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.” - -[259] Whitman, C. O., The Inadequacy of the Cell-theory, _Journ. of -Morphol._ VIII, pp. 639–658, 1893; Sedgwick, A., On the Inadequacy of -the Cellular Theory of Development, _Q.J.M.S._ XXXVII, pp. 87–101, -1895, XXXVIII, pp. 331–337, 1896. Cf. Bourne, G. C., A Criticism of the -Cell-theory; being an answer to Mr Sedgwick’s article, etc., _ibid._ -XXXVIII, pp. 137–174, 1896. - -[260] Cf. Hertwig, O., _Die Zelle und die Gewebe_, 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.” - -[261] _Journ. of Morph._ VIII, p. 653, 1893. - -[262] Neue Grundlegungen zur Kenntniss der Zelle, _Morph. Jahrb._ VIII, -pp. 272, 313, 333, 1883. - -[263] _Journ. of Morph._ II, p. 49, 1889. - -[264] _Phil. Trans._ CLI, p. 183, 1861; _Researches_, ed. Angus Smith, -1877, p. 553. - -[265] Cf. Kelvin, On the Molecular Tactics of a Crystal, _The Boyle -Lecture_, Oxford, 1893, _Baltimore Lectures_, 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 _Crystallography_, chap. ix, pp. 118–134, 1911. - -[266] In a homogeneous crystalline arrangement, _symmetry_ 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. - -[267] This is what Graham called the _water of gelatination_, on the -analogy of _water of crystallisation_; _Chem. and Phys. Researches_, p. -597. - -[268] 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. - -[269] Lehmann, O., _Flüssige Krystalle, sowie Plasticität von -Krystallen im allgemeinen_, etc., 264 pp. 39 pll., Leipsig, 1904. For a -semi-popular, illustrated account, see Tutton’s _Crystals_ (Int. Sci. -Series), 1911. - -[270] 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.” - -[271] Cf. Przibram, H., Kristall-analogien zur Entwickelungsmechanik -der Organismen, _Arch. f. Entw. Mech._ XXII, p. 207, 1906 (with copious -bibliography); Lehmann, Scheinbar lebende Kristalle und Myelinformen, -_ibid._ XXVI, p. 483, 1908. - -[272] The idea of a “surface-tension” in liquids was first enunciated -by Segner, _De figuris superficierum fluidarum_, in _Comment. Soc. Roy. -Göttingen_, 1751, p. 301. Hooke, in the _Micrographia_ (1665, Obs. -VIII, 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 -_spherical_ or _globular_ form....” - -[273] _Science of Mechanics_, 1902, p. 395; see also Mach’s article -Ueber die physikalische Bedeutung der Gesetze der Symmetrie, _Lotos_, -XXI, pp. 139–147, 1871. - -[274] 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: _On New Philosophical -Instruments_ (Description of a new Fluid Microscope), Edinburgh, 1813, -p. 413. - -[275] Beiträge z. Physiologie d. Protoplasma, _Pflüger’s Archiv_, II, -p. 307, 1869. - -[276] _Poggend. Annalen_, XCIV, pp. 447–459, 1855. Cf. Strethill -Wright, _Phil. Mag._ Feb. 1860. - -[277] Haycraft and Carlier pointed out (_Proc. R.S.E._ XV, 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, _Journ. of Hygiene_, -XII, p. 324, 1912). On the emission of pseudopodia as brought about -by changes in surface tension, see also (_int. al._) Jensen, Ueber -den Geotropismus niederer Organismen, _Pflüger’s Archiv_, LIII, 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 (_Allg. Physiol._ -1895, p. 429), and Davenport says (_Experim. Morphology_, II, 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, _A. f. Physiol._ -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. - -[278] Cf. Pauli, _Allgemeine physikalische Chemie d. Zellen u. Gewebe_, -in Asher-Spiro’s _Ergebnisse der Physiologie_, 1912; Przibram, -_Vitalität_, 1913, p. 6. - -[279] 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, _Carnegie Inst._ 1904, pp. -130–230; Dellinger, O. P., Locomotion of Amoebae, etc. _Journ. Exp. -Zool._ III, pp. 337–357, 1906; also various papers by Max Heidenhain, -in _Anatom. Hefte_ (Merkel und Bonnet), etc. - -[280] 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, _sui generis_, the _force épipolique_ 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, _Mém. Cour. Acad. de Belgique_, -XXXIV, 1869; cf. Plateau, p. 283). - -[281] Cf. _infra_, p. 306. - -[282] Cf. p. 32. - -[283] Or, more strictly speaking, unless its thickness be less than -twice the range of the molecular forces. - -[284] It follows that the tension, depending only on the -surface-conditions, is independent of the thickness of the film. - -[285] This simple but immensely important formula is due to Laplace -(_Mécanique Céleste_, Bk. x. suppl. _Théorie de l’action capillaire_, -1806). - -[286] Sur la surface de révolution dont la courbure moyenne est -constante, _Journ. de M. Liouville_, VI, p. 309, 1841. - -[287] See _Liquid Drops and Globules_, 1914, p. 11. Robert Boyle used -turpentine in much the same way. For other methods see Plateau, _op. -cit._ p. 154. - -[288] Felix Plateau recommends the use of a weighted thread, or -plumb-line, drawn up out of a jar of water or oil; _Phil. Mag._ XXXIV, -p. 246, 1867. - -[289] Cf. Boys, C. V., On Quartz Fibres, _Nature_, July 11, 1889; -Warburton, C., The Spinning Apparatus of Geometric Spiders, _Q.J.M.S._ -XXXI, pp. 29–39, 1890. - -[290] J. Blackwall, _Spiders of Great Britain_ (Ray Society), 1859, p. -10; _Trans. Linn. Soc._ XVI, p. 477, 1833. - -[291] 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 _Properties of -Matter_, pp. 151, 152, (ed. 1907). - -[292] Kühne, _Untersuchungen über das Protoplasma_, 1864, p. 75, etc. - -[293] _A Study of Splashes_, 1908, p. 38, etc.; Segmentation of a -Liquid Annulus, _Proc. Roy. Soc._ XXX, pp. 49–60, 1880. - -[294] Cf. _ibid._ 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. - -[295] See a _Study of Splashes_, p. 54. - -[296] 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 _varying_ 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, _Phil. Mag._ Jan. 1915.) - -[297] Indeed any non-isotropic _stiffness_, even though _T_ remained -uniform, would simulate, and be indistinguishable from, a condition of -non-stiffness and non-isotropic _T_. - -[298] A non-symmetry of _T_ and _T′_ 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. - -[299] 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, _Ill. London News_, Feb. 17, 1855; -_Q.J.M.S._ III, pp. 179–185, 1855). - -[300] Cf. Bergson, _Creative Evolution_, 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.” - -[301] Ray Lankester, _A.M.N.H._ (4), XI, p. 321, 1873. - -[302] Leidy, Parasites of the Termites, _J. Nat. Sci., Philadelphia_, -VIII, pp. 425–447, 1874–81; cf. Saville Kent’s _Infusoria_, II, p. 551. - -[303] _Op. cit._ p. 79. - -[304] Brady, _Challenger Monograph_, pl. XX, p. 233. - -[305] 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 [_Gromia -dujardinii_] 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, _après avoir atteint le niveau du liquide, ont -continué à ramper à sa surface, en se laissant pendre au-dessous_ -comme certains mollusques gastéropodes.” (Dujardin, F., Observations -nouvelles sur les prétendus céphalopodes microscopiques, _Ann. des Sci. -Nat._ (2), III, p. 312, 1835.) - -[306] Cf. Boas, _Spolia Atlantica_, 1886, pl. 6. - -[307] This cellular pattern would seem to be related to the “cohesion -figures” described by Tomlinson in various surface-films (_Phil. Mag._ -1861 to 1870); to the “tesselated structure” in liquids described by -Professor James Thomson in 1882 (_Collected Papers_, p. 136); and to -the _tourbillons cellulaires_ of Prof. H. Bénard (_Ann. de Chimie_ -(7), XXIII, pp. 62–144, 1901, (8), XXIV, pp. 563–566, 1911), _Rev. -génér. des Sci._ XI, p. 1268, 1900; cf. also E. H. Weber. (_Poggend. -Ann._ XCIV, 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 _Nature_, 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. _infra_, p. 590). - -[308] Ueber physikalischen Eigenschaften dünner, fester Lamellen, _S.B. -Berlin. Akad._ 1888, pp. 789, 790. - -[309] 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, _Foraminiferen der -Plankton-Expedition_, 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. - -[310] _A Study of Splashes_, p. 116. - -[311] See _Silliman’s Journal_, II, p. 179, 1820; and cf. Plateau, _op. -cit._ II, pp. 134, 461. - -[312] 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. - -[313] Cf. Doflein, _Lehrbuch der Protozoenkunde_, 1911, p. 422. - -[314] Cf. Minchin, _Introduction to the Study of the Protozoa_, 1914 p. -293, Fig. 127. - -[315] Cf. C. A. Kofoid and Olive Swezy, On Trichomonad Flagellates, -etc. _Pr. Amer. Acad. of Arts and Sci._ LI, pp. 289–378, 1915. - -[316] D. L. Mackinnon, Herpetomonads from the Alimentary Tract of -certain Dungflies, _Parasitology_, III, p. 268, 1910. - -[317] _Proc. Roy. Soc._ XII, pp. 251–257, 1862–3. - -[318] Cf. (_int. al._) Lehmann, Ueber scheinbar lebende Kristalle und -Myelinformen, _Arch. f. Entw. Mech._ XXVI, p. 483, 1908; _Ann. d. -Physik_, XLIV, p. 969, 1914. - -[319] Cf. B. Moore and H. C. Roaf, On the Osmotic Equilibrium of the -Red Blood Corpuscle, _Biochem. Journal_, III, p. 55, 1908. - -[320] For an attempt to explain the form of a blood-corpuscle by -surface-tension alone, see Rice, _Phil. Mag._ Nov. 1914; but cf. -Shorter, _ibid._ Jan. 1915. - -[321] Koltzoff, N. K., Studien über die Gestalt der Zelle, _Arch. f. -mikrosk. Anat._ LXVII, pp. 364–571, 1905; _Biol. Centralbl._ XXIII, -pp. 680–696, 1903, XXVI, pp. 854–863, 1906; _Arch. f. Zellforschung_, -II, pp. 1–65, 1908, VII, pp. 344–423, 1911; _Anat. Anzeiger_, XLI, pp. -183–206, 1912. - -[322] Cf. _supra_, p. 129. - -[323] As Bethe points out (Zellgestalt, Plateausche Flüssigkeitstigur -und Neurofibrille, _Anat. Anz._ XL. p. 209, 1911), the spiral fibres -of which Koltzoff speaks must lie _in the surface_, and not within the -substance, of the cell whose conformation is affected by them. - -[324] See for a further but still elementary account, Michaelis, -_Dynamics of Surfaces_, 1914, p. 22 _seq._; Macallum, -_Oberflächenspannung und Lebenserscheinungen_, in Asher-Spiro’s -_Ergebnisse der Physiologie_, XI, pp. 598–658, 1911; see also W. -W. Taylor’s _Chemistry of Colloids_, 1915, p. 221 _seq._, Wolfgang -Ostwald, _Grundriss der Kolloidchemie_, 1909, and other text-books of -physical chemistry; and Bayliss’s _Principles of General Physiology_, -pp. 54–73, 1915. - -[325] 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 (_Théorie -mécanique de la chaleur_, 1869, p. 376; cf. Plateau, II, p. 100). - -[326] This identical phenomenon was the basis of Quincke’s -theory of amoeboid movement (Ueber periodische Ausbreitung von -Flüssigkeitsoberflächen, etc., _SB. Berlin. Akad._ 1888, pp. 791–806; -cf. _Pflüger’s Archiv_, 1879, p. 136). - -[327] J. Willard Gibbs, Equilibrium of Heterogeneous Substances, _Tr. -Conn. Acad._ III, pp. 380–400, 1876, also in _Collected Papers_, I, -pp. 185–218, London, 1906; J. J. Thomson, _Applications of Dynamics -to Physics and Chemistry_, 1888 (Surface tension of solutions), p. -190. See also (_int. al._) the various papers by C. M. Lewis, _Phil. -Mag._ (6), XV, p. 499, 1908, XVII, p. 466, 1909, _Zeitschr. f. physik. -Chemie_, LXX, p. 129, 1910; Milner, _Phil. Mag._ (6), XIII, p. 96, -1907, etc. - -[328] G. F. FitzGerald, On the Theory of Muscular Contraction, _Brit. -Ass. Rep._ 1878; also in _Scientific Writings_, ed. Larmor, 1902, pp. -34, 75. A. d’Arsonval, Relations entre l’électricité animale et la -tension superficielle, _C. R._ CVI, p. 1740. 1888; cf. A. Imbert, Le -mécanisme de la contraction musculaire, déduit de la considération des -forces de tension superficielle, _Arch. de Phys._ (5), IX, pp. 289–301, -1897. - -[329] Ueber die Natur der Bindung der Gase im Blut und in seinen -Bestandtheilen, _Kolloid. Zeitschr._ II, pp. 264–272, 294–301, 1908; -cf. Loewy, Dissociationsspannung des Oxyhaemoglobin im Blut, _Arch. f. -Anat. und Physiol._ 1904, p. 231. - -[330] 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, _C. R. Acad. Sci._ XXXIII, 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, _Arch. f. Anat. -u. Phys._ (_Phys. Abth._) 1894, p. 517; Abscheidung fester Körper in -Oberflächenschichten _Z. f. phys. Chem._ XLVII, p. 341, 1902; _Proc. -R. S._ LXXII, p. 156, 1904. For a general review of the whole subject -see H. Zangger, Ueber Membranen und Membranfunktionen, in Asher-Spiro’s -_Ergebnisse der Physiologie_, VII, pp. 99–160, 1908. - -[331] Cf. Taylor, _Chemistry of Colloids_, p. 252. - -[332] Strasbürger, Ueber Cytoplasmastrukturen, etc. _Jahrb. f. wiss. -Bot._ XXX, 1897; R. A. Harper, Kerntheilung und freie Zellbildung im -Ascus, _ibid._; cf. Wilson, _The Cell in Development, etc._ pp. 53–55. - -[333] Cf. A. Gurwitsch, _Morphologie und Biologie der Zelle_, 1904, -pp. 169–185; Meves, Die Chondriosomen als Träger erblicher Anlagen, -_Arch. f. mikrosk. Anat._ 1908, p. 72; J. O. W. Barratt, Changes in -Chondriosomes, etc. _Q.J.M.S._ LVIII, pp. 553–566, 1913, etc.; A. -Mathews, Changes in Structure of the Pancreas Cell, etc., _J. of -Morph._ XV (Suppl.), pp. 171–222, 1899. - -[334] 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. - -[335] Cf. C. C. Dobell, Chromidia and the Binuclearity Hypotheses; a -review and a criticism, _Q.J.M.S._ LIII, 279–326, 1909; Prenant, A., -Les Mitochondries et l’Ergastoplasme, _Journ. de l’Anat. et de la -Physiol._ XLVI, pp. 217–285, 1910 (both with copious bibliography). - -[336] 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, _Pflüger’s -Archiv_, CV, p. 559, 1904.) Cf. also Hardy (_Pr. Phys. Soc._ XXVIII, 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.” - -[337] On the Distribution of Potassium in animal and vegetable Cells; -_Journ. of Physiol._ XXXII, p. 95, 1905. - -[338] 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, -_Journ. de l’Anat. et de la Physiol._ XLVI, pp. 343–404, 1910). - -[339] Cf. Macallum, Presidential Address, Section I, _Brit. Ass. Rep._ -(Sheffield), 1910, p. 744. - -[340] In accordance with a simple _corollary_ to the Gibbs-Thomson law. - -[341] 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, _T__{_ab_}, is -equal to the energy per unit area, _E__{_ab_}. - -[342] 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.” - -[343] 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 _nearly_ hemispherical bubble, have -been investigated by van der Mensbrugghe (cf. Plateau, _op. cit._, -p. 386). The form of the superficial vacuoles in Actinophrys or -Actinosphaerium involves an identical problem. - -[344] In an actual calculation we must of course always take account of -the tensions on _both sides_ of each film or membrane. - -[345] Hofmeister, _Pringsheim’s Jahrb._ III, p. 272, 1863; _Hdb. d. -physiol. Bot._ I, 1867, p. 129. - -[346] Sachs, Ueber die Anordnung der Zellen in jüngsten -Pflanzentheilen, _Verh. phys. med. Ges. Würzburg_, XI, pp. 219–242, -1877; Ueber Zellenanordnung und Wachsthum, _ibid._ XII, 1878; Ueber -die durch Wachsthum bedingte Verschiebung kleinster Theilchen in -trajectorischen Curven, _Monatsber. k. Akad. Wiss. Berlin_, 1880; -_Physiology of Plants_, chap. xxvii, pp. 431–459, Oxford, 1887. - -[347] Schwendener, Ueber den Bau und das Wachsthum des Flechtenthallus, -_Naturf. Ges. Zürich_, Febr. 1860, pp. 272–296. - -[348] Reinke, _Lehrbuch der Botanik_, 1880, p. 519. - -[349] Cf. Leitgeb, _Unters. über die Lebermoose_, II, p. 4, Graz, 1881. - -[350] Rauber, Neue Grundlegungen zur Kenntniss der Zelle, _Morph. -Jahrb._ VIII, pp. 279, 334, 1882. - -[351] _C. R. Acad. Sc._ XXXIII, p. 247, 1851; _Ann. de chimie et de -phys._ (3), XXXIII, p. 170, 1851; _Bull. R. Acad. Belg._ XXIV, p. 531, -1857. - -[352] Klebs, _Biolog. Centralbl._ VII, pp. 193–201, 1887. - -[353] L. Errera, Sur une condition fondamentale d’équilibre des -cellules vivantes, _C. R._, CIII, p. 822, 1886; _Bull. Soc. Belge -de Microscopie_, XIII, Oct. 1886; _Recueil d’œuvres_ (_Physiologie -générale_), 1910, pp. 201–205. - -[354] L. Chabry, Embryologie des Ascidiens, _J. Anat. et Physiol._ -XXIII, p. 266, 1887. - -[355] Robert, Embryologie des Troques, _Arch. de Zool. exp. et gén._ -(3), X, 1892. - -[356] “Dass der Furchungsmodus etwas für das Zukünftige unwesentliches -ist,” _Z. f. w. Z._ LV, 1893, p. 37. With this statement compare, -or contrast, that of Conklin, quoted on p. 4; cf. also pp. 157, 348 -(footnotes). - -[357] de Wildeman, Etudes sur l’attache des cloisons cellulaires, _Mém. -Couronn. de l’Acad. R. de Belgique_, LIII, 84 pp., 1893–4. - -[358] It was so termed by Conklin in 1897, in his paper on Crepidula -(_J. of Morph._ XIII, 1897). It is the _Querfurche_ of Rabl (_Morph. -Jahrb._ V, 1879); the _Polarfurche_ of O. Hertwig (_Jen. Zeitschr._ -XIV, 1880); the _Brechungslinie_ of Rauber (Neue Grundlage zur K. der -Zelle, _M. Jb._ VIII, 1882). It is carefully discussed by Robert, Dév. -des Troques, _Arch. de Zool. Exp. et Gén._ (3), X, 1892, p. 307 seq. - -[359] Thus Wilson (_J. of Morph._ VIII, 1895) declared that in -Amphioxus the polar furrow was occasionally absent, and Driesch took -occasion to criticise and to throw doubt upon the statement (_Arch. f. -Entw. Mech._ I, 1895, p. 418). - -[360] 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,” _Entw. mech. Studien, Z. f. w. Z._ LIII, p. -166, 1892. Cf. also his _Math. mechanische Bedeutung morphologischer -Probleme der Biologie_, Jena, 59 pp. 1891. - -[361] Compare, however, p. 299. - -[362] _Ricreatione dell’ occhio e della mente, nell’ Osservatione delle -Chiocciole_, Roma, 1681. - -[363] Cf. some of J. H. Vincent’s photographs of ripples, in _Phil. -Mag._ 1897–1899; or those of F. R. Watson, in _Phys. Review_, 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 (_Works_, p. 503, 1902). - -[364] Cushman, J. A. and Henderson, W. P., _Amer. Nat._ XL, pp. -797–802, 1906. - -[365] This does not merely neglect the _broken_ ones but _all_ whose -centres lie between this circle and a hexagon inscribed in it. - -[366] For more detailed calculations see a paper by “H.M.” [? H. -Munro], in _Q. J. M. S._ VI, p. 83, 1858. - -[367] Cf. Hartog, The Dual Force of the Dividing Cell, _Science -Progress_ (n.s.), I, Oct. 1907, and other papers. Also Baltzer, _Ueber -mehrpolige Mitosen bei Seeigeleiern_, Inaug. Diss. 1908. - -[368] Observations sur les Abeilles, _Mém. Acad. Sc. Paris_, 1712, p. -299. - -[369] As explained by Leslie Ellis, in his essay “On the Form of Bees’ -Cells,” in _Mathematical and other Writings_, 1853, p. 353; cf. O. -Terquem, _Nouv. Ann. Math._ 1856, p. 178. - -[370] _Phil. Trans._ XLII, 1743, pp. 565–571. - -[371] _Mém. de l’Acad. de Berlin_, 1781. - -[372] Cf. Gregory, _Examples_, p. 106, Wood’s _Homes without Hands_, -1865, p. 428, Mach, _Science of Mechanics_, 1902, p. 453, etc., etc. - -[373] _Origin of Species_, ch. VIII (6th ed., p. 221). The cells of -various bees, humble-bees and social wasps have been described and -mathematically investigated by K. Müllenhoff, _Pflüger’s Archiv_ -XXXII, 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, _Biol. Centralbl._ XXXIII, pp. 4, 89, 129, 183, 1903. - -[374] 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.” - -[375] Since writing the above, I see that Müllenhoff gives the -same explanation, and declares that the waxen wall is actually a -_Flüssigkeitshäutchen_, or liquid film. - -[376] Bonnet criticised Buffon’s explanation, on the ground that his -description was incomplete; for Buffon took no account of the Maraldi -pyramids. - -[377] Buffon, _Histoire Naturelle_, IV, p. 99. Among many other papers -on the Bee’s cell, see Barclay, _Mem. Wernerian Soc._ II, p. 259 -(1812), 1818; Sharpe, _Phil. Mag._ IV, 1828, pp. 19–21; L. Lalanne, -_Ann. Sci. Nat._ (2) Zool. XIII, pp. 358–374, 1840; Haughton, _Ann. -Mag. Nat. Hist._ (3), XI, pp. 415–429, 1863; A. R. Wallace, _ibid._ -XII, p. 303, 1863; Jeffries Wyman. _Pr. Amer. Acad. of Arts and Sc._ -VII, pp. 68–83, 1868; Chauncey Wright, _ibid._ IV, p. 432, 1860. - -[378] Sir W. Thomson, On the Division of Space with Minimum Partitional -Area, _Phil. Mag._ (5), XXIV, pp. 503–514, Dec. 1887; cf. _Baltimore -Lectures_, 1904, p. 615. - -[379] Also discovered independently by Sir David Brewster, _Trans. -R.S.E._ XXIV, p. 505, 1867, XXV, p. 115, 1869. - -[380] 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. - -[381] F. R. Lillie, Embryology of the Unionidae, _Journ. of -Morphology_, X, p. 12, 1895. - -[382] E. B. Wilson, The Cell-lineage of Nereis, _Journ. of Morphology_, -VI, p. 452, 1892. - -[383] It is highly probable, and we may reasonably assume, that the two -little triangles do not actually meet at an apical _point_, but merge -into one another by a twist, or minute surface of complex curvature, so -as not to contravene the normal conditions of equilibrium. - -[384] Professor Peddie has given me this interesting and important -result, but the mathematical reasoning is too lengthy to be set forth -here. - -[385] Cf. Rhumbler, _Arch. f. Entw. Mech._ XIV, p. 401, 1902; Assheton, -_ibid._ XXXI, pp. 46–78, 1910. - -[386] M. Robert (_l. c._ 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 _precocious segregation_. -Il faut avouer pourtant qu’on est parfois assez embarrassé pour -assigner une cause à pareilles différences.” - -[387] The principle is well illustrated in an experiment of Sir David -Brewster’s (_Trans. R.S.E._ XXV, 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. - -[388] Cf. Wildeman, _Attache des Cloisons_, etc., pls. 1, 2. - -[389] _Nova Acta K. Leop. Akad._ XI, 1, pl. IV. - -[390] Cf. _Protoplasmamechanik_, 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.” - -[391] 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 _radial_ partition being for the present -excluded), is, however, clear. - -[392] _Cit._ Plateau, _Statique des Liquides_, i, p. 358. - -[393] Even in a Protozoon (_Euglena viridis_), 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. -_Contrib. Zool. Lab. Univ. Pennsylvania_, I, pp. 37–50, 1893. - -[394] 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. - -[395] 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. 168, 6, as the most typical or most primitive. - -[396] Cf. Rauber, Neue Grundlage z. K. der Zelle, _Morph. Jahrb._ VIII, -1883, pp. 273, 274: - -“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.” - -[397] 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. - -[398] Cf. (e.g.) Clerk Maxwell, On Reciprocal Figures, etc., _Trans. R. -S. E._ XXVI, p. 9, 1870. - -[399] See Greville, K. R., Monograph of the Genus Asterolampra, -_Q.J.M.S._ VIII, (Trans.), pp. 102–124, 1860; cf. IBID. (n.s.), II, pp. -41–55, 1862. - -[400] The same is true of the insect’s wing; but in this case I do not -hazard a conjectural explanation. - -[401] _Ann. Mag. N. H._ (2), III, p. 126, 1849. - -[402] _Phil. Trans._ CLVII, pp. 643–656, 1867. - -[403] Sachs, _Pflanzenphysiologie_ (_Vorlesung_ XXIV), 1882; cf. -Rauber, Neue Grundlage zur Kenntniss der Zelle, _Morphol. Jahrb._ VIII, -p. 303 _seq._, 1883; E. B. Wilson, Cell-lineage of Nereis, _Journ. of -Morphology_, VI, p. 448, 1892, etc. - -[404] In the following account I follow closely on the lines laid down -by Berthold; _Protoplasmamechanik_, cap. vii. Many botanical phenomena -identical and similar to those here dealt with, are elaborately -discussed by Sachs in his _Physiology of Plants_ (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 (_Arb. d. botan. Inst. Würzburg_, 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. - -[405] Cf. p. 369. - -[406] 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 (_Proc. Geol. -Soc._ 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. - -[407] Vesque, _Ann. des Sc. Nat._ (_Bot._) (5), XIX, p. 310, 1874. - -[408] Cf. Kölliker, _Icones Histiologicae_, 1864, pp. 119, etc. - -[409] In an interesting paper by Irvine and Sims Woodhead on the -“Secretion of Carbonate of Lime by Animals” (_Proc. R. S. E._ XVI, -1889, p. 351) it is asserted that “lime salts, of whatever form, are -deposited _only_ in vitally inactive tissue.” - -[410] 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, _Proc. Geol. Soc._ 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. - -[411] _C. R. Soc. Biol. Paris_ (9), I, pp. 17–20, 1889; _C. R. Ac. Sc._ -CVIII, pp. 196–8, 1889. - -[412] Cf. Heron-Allen, _Phil. Trans._ (B), vol. CCVI, p. 262, 1915. - -[413] See Leduc, _Mechanism of Life_ (1911), ch. X, for copious -references to other works on the artificial production of “organic” -forms. - -[414] Lectures on the Molecular Asymmetry of Natural Organic Compounds, -_Chemical Soc. of Paris_, 1860, and also in Ostwald’s _Klassiker d. ex. -Wiss._ No. 28, and in _Alembic Club Reprints_, No. 14, Edinburgh, 1897; -cf. Richardson, G. M., _Foundations of Stereochemistry_, N. Y. 1901. - -[415] Japp, Stereometry and Vitalism, _Brit. Ass. Rep._ (Bristol), p. -813, 1898; cf. also a voluminous discussion in _Nature_, 1898–9. - -[416] They represent the general theorem of which particular cases are -found, for instance, in the asymmetry of the ferments (or _enzymes_) -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, _Z. f. physiol. -Chemie_, V, p. 60, 1899, and various papers in the _Ber. d. d. chem. -Ges._ from 1894. - -[417] 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, _Journ. Chem. Soc._ (Trans.), LXXXV, p. 1249, 1904. - -[418] See for a fuller discussion, Hans Przibram, _Vitalität_, 1913, -Kap. iv, Stoffwechsel (Assimilation und Katalyse). - -[419] Cf. Cotton, _Ann. de Chim. et de Phys._ (7), VIII, pp. 347–432 -(cf. p. 373), 1896. - -[420] Byk, A., Zur Frage der Spaltbarkeit von Razemverbindungen durch -Zirkularpolarisiertes Licht, ein Beitrag zur primären Entstehung -optisch-activer Substanzen, _Zeitsch. f. physikal. Chemie_, XLIX, p. -641, 1904. It must be admitted that further positive evidence on these -lines is still awanting. - -[421] Cf. (_int. al._) Emil Fischer, _Untersuchungen über Aminosäuren, -Proteine_, etc. Berlin, 1906. - -[422] Japp, _l. c._ p. 828. - -[423] Rainey, G., On the Elementary Formation of the Skeletons of -Animals, and other Hard Structures formed in connection with Living -Tissue, _Brit. For. Med. Ch. Rev._ XX, pp. 451–476, 1857; published -separately with additions, 8vo. London, 1858. For other papers by -Rainey on kindred subjects see _Q. J. M. S._ VI (_Tr. Microsc. Soc._), -pp. 41–50, 1858, VII, pp. 212–225, 1859, VIII, pp. 1–10, 1860, I (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, _Q. J. M. S._ XII, 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, _Phil. Trans._ 1800. pp. 327–402. - -[424] Cf. Quincke, Ueber unsichtbare Flüssigkeitsschichten, _Ann. der -Physik_, 1902. - -[425] See for instance other excellent illustrations in Carpenter’s -article “Shell,” in Todd’s _Cyclopædia_, vol. IV. pp. 550–571, -1847–49. According to Carpenter, the shells of the mollusca (and also -of the crustacea) are “essentially composed of _cells_, 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”; _ibid._ art. “Tegumentary Organs,” vol. V, p. 487, 1859. -Quekett (_Lectures on Histology_, vol. II, p. 393, 1854, and _Q. J. -M. S._ XI, pp. 95–104, 1863) supported Carpenter; but Williamson -(Histological Features in the Shells of the Crustacea, _Q. J. M. S._ -VIII, pp. 35–47, 1860) amply confirmed Huxley’s view, which in the -end Carpenter himself adopted (_The Microscope_, 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” (_Phil. Trans._ CLXXXVII, 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, -_Mitth. Zool. St. Neapel_, III, pp. 284–290, pl. XX, 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 _Siderastraea radians_, etc., _Carnegie Inst. Washington_, 1904, -p. 34). Cf. also M. M. Ogilvie-Gordon, _Q. J. M. S._ XLIX, p. 203, -1905, etc. - -[426] Cf. Claparède, _Z. f. w. Z._ XIX, p. 604, 1869. - -[427] Spicules extremely like those of the Alcyonaria occur also in a -few sponges; cf. (e.g.), Vaughan Jennings, _Journ. Linn. Soc._ XXIII, -p. 531, pl. 13, fig. 8, 1891. - -[428] _Mem. Manchester Lit. and Phil. Soc._ LX, p. 11, 1916. - -[429] Mummery, J. H., On Calcification in Enamel and Dentine, _Phil. -Trans._ CCV (B), pp. 95–111, 1914. - -[430] The artificial concretion represented in Fig. 202 is identical -in appearance with the concretions found in the kidney of Nautilus, as -figured by Willey (_Zoological Results_, p. lxxvi, Fig. 2, 1902). - -[431] Cf. Taylor’s _Chemistry of Colloids_, p. 18, etc., 1915. - -[432] 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 _incipient_ cell-wall behaves -as, and indeed actually is, a liquid film (cf. p. 306). - -[433] Cf. p. 254. - -[434] Cf. Harting, _op. cit._, 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.” - -[435] Liesegang, R. E., _Ueber die Schichtungen bei Diffusionen_, -Leipzig, 1907, and other earlier papers. - -[436] Cf. Taylor’s _Chemistry of Colloids_, pp. 146–148, 1915. - -[437] Cf. S. C. Bradford, The Liesegang Phenomenon and Concretionary -Structure in Rocks, _Nature_, XCVII, p. 80, 1916; cf. _Sci. Progress_, -X, p. 369, 1916. - -[438] Cf. Faraday, On Ice of Irregular Fusibility, _Phil. Trans._, -1858, p. 228; _Researches in Chemistry, etc._, 1859, p. 374; Tyndall, -_Forms of Water_, p. 178, 1872; Tomlinson, C., On some effects of small -Quantities of Foreign Matter on Crystallisation, _Phil. Mag._ (5) XXXI, -p. 393, 1891, and other papers. - -[439] A Study in Crystallisation, _J. of Soc. of Chem. Industry_, XXV, -p. 143, 1906. - -[440] _Ueber Zonenbildung in kolloidalen Medien_, Jena, 1913. - -[441] _Verh. d. d. Zool. Gesellsch._ p. 179, 1912. - -[442] _Descent of Man_, II, pp. 132–153, 1871. - -[443] 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 (_Variation_, 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 _Morpho_) 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.” - -[444] Cf. also Sir D. Brewster, On optical properties of Mother of -Pearl, _Phil. Trans._ 1814, p. 397. - -[445] Biedermann, W., Ueber die Bedeutung von Kristallisationsprozessen -der Skelette wirbelloser Thiere, namentlich der Molluskenschalen, -_Z. f. allg. Physiol._ I, p. 154, 1902; Ueber Bau und Entstehung der -Molluskenschale, _Jen. Zeitschr._ XXXVI, pp. 1–164, 1902. Cf. also -Steinmann, Ueber Schale und Kalksteinbildungen, _Ber. Naturf. Ges. -Freiburg i. Br_ IV, 1889; Liesegang, _Naturw. Wochenschr._ p. 641, 1910. - -[446] Cf. Bütschli, Ueber die Herstellung künstlicher Stärkekörner oder -von Sphärokrystallen der Stärke, _Verh. nat. med. Ver. Heidelberg_, V, -pp. 457–472, 1896. - -[447] _Untersuchungen über die Stärkekörner_, Jena, 1905. - -[448] Cf. Winge, _Meddel. fra Komm. for Havundersögelse_ (_Fiskeri_), -IV, p. 20, Copenhagen, 1915. - -[449] The anhydrite is sulphate of lime (CaSO_{4}); the polyhalite is -a triple sulphate of lime, magnesia and potash (2 CaSO_{4}. MgSO_{4}. -K_{2}SO_{4} + 2 H_{2}O). - -[450] Cf. van’t Hoff, _Physical Chemistry in the Service of the -Sciences_, p. 99 seq. Chicago, 1903. - -[451] Sphärocrystalle von Kalkoxalat bei Kakteen, _Ber. d. d. Bot. -Gesellsch._ p. 178, 1885. - -[452] Pauli, W. u. Samec, M., Ueber Löslichkeitsbeeinflüssung von -Elektrolyten durch Eiweisskörper, _Biochem. Zeitschr._ XVII, p. 235, -1910. Some of these results were known much earlier; cf. Fokker in -_Pflüger’s Archiv_, VII, p. 274, 1873; also Irvine and Sims Woodhead, -_op. cit._ p. 347. - -[453] Which, in 1000 parts of ash, contains about 840 parts of -phosphate and 76 parts of calcium carbonate. - -[454] Cf. Dreyer, Fr., Die Principien der Gerüstbildung bei Rhizopoden, -Spongien und Echinodermen, _Jen. Zeitschr._ XXVI, pp. 204–468, 1892. - -[455] In an anomalous and very remarkable Australian sponge, just -described by Professor Dendy (_Nature_, May 18, 1916, p. 253) under the -name of _Collosclerophora_, 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. - -[456] Lister, in Willey’s _Zoological Results_, pt IV, p. 459, 1900. - -[457] 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, _Proc. R. S._ LXXXIV (B), p. 579, 1911). - -[458] See for instance the plates in Théel’s Monograph of the -Challenger Holothuroidea; also Sollas’s Tetractinellida, p. lxi. - -[459] For very numerous illustrations of the triradiate and -quadriradiate spicules of the calcareous sponges, see (_int. al._), -papers by Dendy (_Q. J. M. S._ XXXV, 1893), Minchin (_P. Z. S._ 1904), -Jenkin (_P. Z. S._ 1908), etc. - -[460] Cf. again Bénard’s _Tourbillons cellulaires_, _Ann. de Chimie_, -1901, p. 84. - -[461] Léger, Stolc and others, in Doflein’s _Lehrbuch d. -Protozoenkunde_, 1911, p. 912. - -[462] See, for instance, the figures of the segmenting egg of -Synapta (after Selenka), in Korschelt and Heider’s _Vergleichende -Entwicklungsgeschichte_ (Allgem. Th., 3^{te} 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, _Journ. of Morphology_, VI, p. 450, -1892. - -[463] Korschelt and Heider, p. 16. - -[464] _Chall. Rep. Hexactinellida_, pls. xvi, liii, lxxvi, lxxxviii. - -[465] “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 _Spicula_ zur Skeletbildung benützt, -und späterhin durch die Wechselwirkung von Anpassung und Vererbung im -Kampfe ums Dasein auf das Vielfältigste umgebildet und differenziert -wurden.” _Die Kalkschwämme_, I, p. 377, 1872; cf. also pp. 482, 483. - -[466] _Op. cit._ 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.” - -[467] Materials for a Monograph of the Ascones, _Q. J. M. S._ XL. pp. -469–587, 1898. - -[468] Haeckel, in his _Challenger Monograph_, 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. - -[469] Cf. Gamble, _Radiolaria_ (Lankester’s _Treatise on Zoology_), -vol. I, p. 131, 1909. Cf. also papers by V. Häcker, in _Jen. Zeitschr._ -XXXIX, p. 581, 1905, _Z. f. wiss. Zool._ LXXXIII, p. 336, 1905, _Arch. -f. Protistenkunde_, IX, p. 139, 1907, etc. - -[470] Bütschli, Ueber die chemische Natur der Skeletsubstanz der -Acantharia, _Zool. Anz._ XXX, p. 784, 1906. - -[471] For figures of these crystals see Brandt, _F. u. Fl. d. Golfes -von Neapel_, XIII, _Radiolaria_, 1885, pl. v. Cf. J. Müller, Ueber die -Thalassicollen, etc. _Abh. K. Akad. Wiss. Berlin_, 1858. - -[472] Celestine, or celestite, is SrSO_{4} with some BaO replacing SrO. - -[473] With the colloid chemists, we may adopt (as Rhumbler has done) -the terms _spumoid_ or _emulsoid_ to denote an agglomeration of -fluid-filled vesicles, restricting the name _froth_ to such vesicles -when filled with air or some other gas. - -[474] Cf. Koltzoff, Zur Frage der Zellgestalt, _Anat. Anzeiger_, XLI, -p. 190, 1912. - -[475] _Mém. de l’Acad. des Sci., St. Pétersbourg_, XII, Nr. 10, 1902. - -[476] The manner in which the minute spicules of Raphidiophrys arrange -themselves round the bases of the pseudopodial rays is a similar -phenomenon. - -[477] Rhumbler, Physikalische Analyse von Lebenserscheinungen der -Zelle, _Arch. f. Entw. Mech._ VII, p. 103, 1898. - -[478] 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 _Galen_, 1916, -p. xxix). Cf. Carpenter, _Mental Physiology_, 1874, p. 41; Norman, -Architectural achievements of Little Masons, etc., _Ann. Mag. Nat. -Hist._ (5), I, p. 284, 1878; Heron-Allen, Contributions ... to the -Study of the Foraminifera, _Phil. Trans._ (B), CCVI, pp. 227–279, -1915; Theory and Phenomena of Purpose and Intelligence exhibited -by the Protozoa, as illustrated by selection and behaviour in the -Foraminifera, _Journ. R. Microscop. Soc._ pp. 547–557, 1915; _ibid._, -pp. 137–140, 1916. Prof. J. A. Thomson (_New Statesman_, 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,” _J. -R. Microsc. Soc._ April, 1916, p. 136. - -[479] Rhumbler, _Das Protoplasma als physikalisches System_, Jena, p. -591, 1914; also in _Arch. f. Entwickelungsmech._ VII, pp. 279–335, 1898. - -[480] Verworn, _Psycho-physiologische Protisten-Studien_, Jena, 1889 -(219 pp.). - -[481] Leidy, J., _Fresh-water Rhizopods of N. America_, 1879, p. 262, -pl. xli, figs. 11, 12. - -[482] Carnoy, _Biologie Cellulaire_, p. 244, fig. 108; cf. Dreyer, _op. -cit._ 1892, fig. 185. - -[483] In all these latter cases we recognise a relation to, or -extension of, the principle of Plateau’s _bourrelet_, or van der -Mensbrugghe’s _masse annulaire_, of which we have already spoken (p. -297). - -[484] 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. - -[485] Cf. Faraday’s beautiful experiments, On the Moving Groups of -Particles found on Vibrating Elastic Surfaces, etc., _Phil. Trans._ -1831, p. 299; _Researches in Chem. and Phys._ 1859, pp. 314–358. - -[486] 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. - -[487] 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 _Nature_, Feb. 1, 1894 -(_Works_, p. 309). - -[488] 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. - -[489] They were known (of course) long before Plato: Πλάτων δὲ καὶ ἐν -τούτοις πυθαγορίζει. - -[490] If the equation of any plane face of a crystal be written in -the form _h_ _x_ + _k_ _y_ + _l_ _z_ = 1, then _h_, _k_, _l_ 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 -_structura perfici non potest sine proportione illa, quam hodierni -geometrae divinam appellant_” (_De nive sexangula_ (1611), Opera, -ed. Frisch, VII, 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,—_pulchritudinis aut proprietatis figurae, quae animam harum -plantarum characterisavit_. - -[491] Cf. Tutton, _Crystallography_, p. 932, 1911. - -[492] 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. - -[493] The spiral fibres, or a large portion of them, constitute what -Searle called “the rope of the heart” (Todd’s _Cyclopaedia_, II, 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 (_Bull. Fac. Med. Paris_, 1820, pp. 40–148), and by -Pettigrew (_Phil. Trans._ 1864), and of late by J. B. Macallum (_Johns -Hopkins Hospital Report_, IX, 1900) and by Franklin P. Mall (_Amer. J. -of Anat._ XI, 1911). - -[494] Cf. Bütschli, “Protozoa,” in Bronn’s _Thierreich_, II, p. 848, -III, p. 1785, etc., 1883–87; Jennings, _Amer. Nat._ XXXV, p. 369, 1901; -Pütter, Thigmotaxie bei Protisten, _Arch. f. Anat. u. Phys._ (_Phys. -Abth. Suppl._), pp. 243–302, 1900. - -[495] A great number of spiral forms, both organic and artificial, are -described and beautifully illustrated in Sir T. A. Cook’s _Curves of -Life_, 1914, and _Spirals in Nature and Art_, 1903. - -[496] Cf. Vines, The History of the Scorpioid Cyme, _Journ. of Botany_ -(n.s.), X, pp. 3–9, 1881. - -[497] Leslie’s _Geometry of Curved Lines_, p. 417, 1821. This is -practically identical with Archimedes’ own definition (ed. Torelli, p. -219); cf. Cantor, _Geschichte der Mathematik_, I, p. 262, 1880. - -[498] See an interesting paper by Whitworth, W. A., “The Equiangular -Spiral, its chief properties proved geometrically,” in the _Messenger -of Mathematics_ (1), I, p. 5, 1862. - -[499] 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 -_Greek Philosophy_, 1914, p. 5). - -[500] Euclid (II, def. 2). - -[501] Cf. Treutlein, _Z. f. Math. u. Phys._ (_Hist. litt. Abth._), -XXVIII, p. 209, 1883. - -[502] This is the so-called _Dreifachgleichschenkelige Dreieck_; cf. -Naber, _op. infra cit._ The ratio 1 : 0·618 is again not hard to find -in this construction. - -[503] See, on the mathematical history of the Gnomon, Heath’s _Euclid_, -I, _passim_, 1908; Zeuthen, _Theorème de Pythagore_, Genève, 1904; also -a curious and interesting book, _Das Theorem des Pythagoras_, by Dr. H. -A. Naber, Haarlem, 1908. - -[504] For many beautiful geometrical constructions based on the -molluscan shell, see Colman, S. and Coan, C. A., _Nature’s Harmonic -Unity_ (ch. ix, Conchology), New York, 1912. - -[505] The Rev. H. Moseley, On the Geometrical Forms of Turbinated and -Discoid Shells, _Phil. Trans._ pp. 351–370. 1838. - -[506] It will be observed that here Moseley, speaking as a -mathematician and considering the _linear_ spiral, speaks of _whorls_ -when he means the linear boundaries, or lines traced by the revolving -radius vector; while the conchologist usually applies the term _whorl_ -to the whole space between the two boundaries. As conchologists, -therefore, we call the _breadth of a whorl_ what Moseley looked upon -as the _distance between two consecutive whorls_. But this latter -nomenclature Moseley himself often uses. - -[507] 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 _wrapped upon a cone_, 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. _R__{3} − _R__{2}/_R__{2} − _R__{1}) is now equal to -ε^{2π sin θ cot α}, θ being the semi-angle of the enveloping cone. (Cf. -Moseley, _Phil. Mag._ XXI, p. 300, 1842.) - -[508] As the successive increments evidently constitute similar -figures, similarly related to the pole (_P_), it follows that their -linear dimensions are to one another as the radii vectores drawn to -similar points in them: for instance as _P_ _P__{1}, _P_ _P__{2}, which -(in Fig. 264, 1) are radii vectores drawn to the points where they meet -the common boundary. - -[509] The equation to the surface of a turbinate shell is discussed -by Moseley (_Phil. Trans._ tom. cit. p. 370), both in terms of polar -coordinates and of the rectangular coordinates _x_, _y_, _z_. A more -elegant representation can be given in vector notation, by the method -of quaternions. - -[510] J. C. M. Reinecke, _Maris protogaei Nautilos, etc._, Coburg, -1818. Leopold von Buch, Ueber die Ammoniten in den älteren -Gebirgsschichten, _Abh. Berlin. Akad., Phys. Kl._ pp. 135–158, 1830; -_Ann. Sc. Nat._ XXVIII, pp. 5–43, 1833; cf. Elie de Beaumont, Sur -l’enroulement des Ammonites, _Soc. Philom., Pr. verb._ pp. 45–48, 1841. - -[511] _Biblia Naturae sive Historia Insectorum_, Leydae, 1737, p. 152. - -[512] Alcide D’Orbigny, _Bull. de la soc. géol. Fr._ XIII, p. 200, -1842; _Cours élém. de Paléontologie_, II, p. 5, 1851. A somewhat -similar instrument was described by Boubée. in _Bull. soc. géol._ I, -p. 232, 1831. Naumann’s Conchyliometer (_Poggend. Ann._ LIV, 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. - -[513] 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. - -[514] For the correction to be applied in the case of the helicoid, or -“turbinate” shells, see p. 557. - -[515] On the Measurement of the Curves formed by Cephalopods and other -Mollusks. _Phil. Mag._ (5), VI, pp. 241–263, 1878. - -[516] For an example of this method, see Blake, _l.c._ p. 251. - -[517] Naumann, C. F., Ueber die Spiralen von Conchylien, _Abh. k. -sächs_. Ges. pp. 153–196, 1846; Ueber die cyclocentrische Conchospirale -u. über das Windungsgesetz von _Planorbis corneus_, _ibid._ I, pp. -171–195, 1849; Spirale von Nautilus u. _Ammonites galeatus_, _Ber. k. -sächs. Ges._ II, p. 26, 1848; Spirale von _Amm. Ramsaueri_, _ibid._ -XVI, p. 21, 1864; see also _Poggendorff’s Annalen_, L, p. 223, 1840; -LI, p. 245, 1841; LIV, p. 541, 1845, etc. - -[518] Sandberger, G., Spiralen des _Ammonites Amaltheus_, _A. Gaytani_, -und _Goniatites intumescens_, _Zeitschr. d. d. Geol. Gesellsch._ X, pp. -446–449, 1858. - -[519] Grabau, A. H., _Ueber die Naumannsche Conchospirale_, etc. -Inauguraldiss. Leipzig, 1872; _Die Spiralen von Conchylien_, etc. -Programm, Nr. 502, Leipzig, 1882. - -[520] It has been pointed out to me that it does not follow at once -and obviously that, because the interspace _AB_ is a mean proportional -between the breadths of the adjacent whorls, therefore the whole -distance _OB_ is a mean proportional between _OA_ and _OC_. This is a -corollary which requires to be proved; but the proof is easy. - -[521] A beautiful construction: _stupendum Naturae artificium_, -Linnaeus. - -[522] 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, _Phylogeny of an Acquired -Characteristic_, p. 422 _seq._, 1894. - -[523] This latter conclusion is adopted by Willey, _Zoological -Results_, p. 747, 1902. - -[524] See Moseley, _op. cit._ pp. 361 _seq._ - -[525] 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, _Natural -Science_, VI, p. 411, 1895; _Zoological Results_, p. 742, 1902.) - -[526] Macalister, Alex., Observations on the Mode of Growth of Discoid -and Turbinated Shells, _P. R. S._ XVIII, pp. 529–532, 1870. - -[527] See figures in Arnold Lang’s _Comparative Anatomy_ (English -translation), II, p. 161, 1902. - -[528] Kappers, C. U. A., Die Bildung künstlicher Molluskenschalen, -_Zeitschr. f. allg. Physiol._ VII, p. 166, 1908. - -[529] We need not assume a _close_ relationship, nor indeed any more -than such a one as permits us to compare the shell of a Nautilus with -that of a Gastropod. - -[530] Cf. Owen, “These shells [Nautilus and Ammonites] are revolutely -spiral or coiled over the back of the animal, not involute like -Spirula”: _Palaeontology_, 1861, p. 97; cf. _Mem. on the Pearly -Nautilus_, 1832; also _P.Z.S._ 1878, p. 955. - -[531] 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. - -[532] “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, _Zoological Results_, p. 746, -1902. - -[533] Cf. Woodward, Henry, On the Structure of Camerated Shells, _Pop. -Sci. Rev._ XI, pp. 113–120, 1872. - -[534] See Willey, Contributions to the Natural History of the Pearly -Nautilus, _Zoological Results_, etc. p. 749, 1902. Cf. also Bather, -Shell-growth in Cephalopoda, _Ann. Mag. N. H._ (6), I, pp 298–310, -1888; _ibid._ pp. 421–427, and other papers by Blake, Riefstahl, etc. -quoted therein. - -[535] It was this that led James Bernoulli, in imitation of Archimedes, -to have the logarithmic spiral graven on his tomb, with the pious -motto, _Eadem mutata resurgam_. On Goodsir’s grave the same symbol is -reinscribed. - -[536] 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, _Ann. Sci. Nat._ XXVII, XXVIII, 1829. - -[537] Blake has remarked upon the fact (_op. cit._ 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 _mouth_ of the shell, take their places, as usual, at a certain -definite angle to the _walls_ 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. - -[538] Cf. pp. 255, 463, etc. - -[539] 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, _Z. f. w. Z._ LXXIV, -pp. 478–490, 1903. - -[540] Rhumbler, L., Die Doppelschalen von Orbitolites und anderer -Foraminiferen, etc., _Arch. f. Protistenkunde_, I, pp. 193–296, 1902; -and other papers. Also _Die Foraminiferen der Planktonexpedition_, I, -1911, pp. 50–56. - -[541] Bénard, H, Les tourbillons cellulaires, _Ann. de Chimie_ (8), -XXIV, 1901. Cf. also the pattern of cilia on an Infusorian, as figured -by Bütschli in Bronn’s _Protozoa_, III, p. 1281, 1887. - -[542] A similar hexagonal pattern is obtained by the mutual repulsion -of floating magnets in Mr R. W. Wood’s experiments, _Phil. Mag._ XLVI, -pp. 162–164, 1898. - -[543] Cf. D’Orbigny, Alc., Tableau méthodique de la classe des -Céphalopodes, _Ann. des Sci. Nat._ (1), VII, pp. 245–315, 1826; -Dujardin. Félix, Observations nouvelles sur les prétendus Céphalopodes -microscopiques, _ibid._ (2), III, pp. 108, 109, 312–315, 1835; -Recherches sur les organismes inférieurs, _ibid._ IV, pp. 343–377, -1835, etc. - -[544] It is obvious that the actual _outline_ 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 _is_ 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. - -[545] von Möller, V., Die spiral-gewundenen Foraminifera des russischen -Kohlenkalks, _Mém. de l’Acad. Imp. Sci., St Pétersbourg_ (7), XXV, 1878. - -[546] 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. 309). 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 _successive whorls_, the two formulae come to the same -thing. - -[547] Van Iterson, G., _Mathem. u. mikrosk.-anat. Studien über -Blattstellungen, nebst Betrachtungen über den Schalenbau der -Miliolinen_, 331 pp., Jena, 1907. - -[548] 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, _Arch. f. Entw. Mech._ -XXXIV p. 680, 1813.) Neither rule seems to me to be well grounded. - -[549] Cf. Schacko, G., Ueber Globigerina-Einschluss bei Orbulina, -_Wiegmann’s Archiv_, XLIX, p. 428, 1883; Brady, _Chall. Rep._, p. 607, -1884. - -[550] Cf. Brady, H. B., _Challenger Rep._, _Foraminifera_, 1884, p. -203, pl. XIII. - -[551] Brady, _op. cit._, 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; _Conchylien des Seesandes_, 1791, p. 4, pl. VI, fig. 15_a_–_f_. -See also, in particular, Dreyer, _Peneroplis_; _eine Studie zur -biologischen Morphologie und zur Speciesfrage_, Leipzig, 1898; also -Eimer und Fickert, Artbildung und Verwandschaft bei den Foraminiferen, -_Tübinger zool. Arbeiten_, III, p. 35, 1899. - -[552] Doflein, _Protozoenkunde_, 1911, p. 263; “Was diese Art -veranlässt in dieser Weise gelegentlich zu varüren, ist vorläufig noch -ganz räthselhaft.” - -[553] 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. - -[554] Dreyer, F., Principien der Gerüstbildung bei Rhizopoden, etc., -_Jen. Zeitschr._ XXVI, pp. 204–468, 1892. - -[555] 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, _op. cit._, p. 33 etc. - -[556] “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, _op. cit._, p. 22. - -[557] “Die Foraminiferen kiesige oder grobsandige Gebiete des -Meeresbodens _nicht lieben_, 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.” - -[558] In regard to the Foraminifera, “die Palaeontologie lässt uns -leider an Anfang der Stammesgeschichte fast gänzlich im Stiche,” -Rhumbler, _op. cit._, p. 14. - -[559] 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, _logical_ affiliation -between forms, _there is also a relation of chronological succession -between the species in which these forms are materialised_”: _Creative -Evolution_, 1911, p. 26. Cf. _supra_, p. 251. - -[560] 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, -_P.Z.S._, p. 689, 1900.) - -[561] Cf. Sir V. Brooke, On the Large Sheep of the Thian Shan, -_P.Z.S._, p. 511, 1875. - -[562] Cf. Lönnberg, E., On the Structure of the Musk Ox, _P.Z.S._, pp. -686–718, 1900. - -[563] St Venant, De la torsion des prismes, avec des considérations -sur leur flexion, etc., _Mém. des Savants Étrangers_, Paris, XIV, pp. -233–560, 1856. - -[564] 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. - -[565] 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. - -[566] _Climbing Plants_, 1865 (2nd edit. 1875); _Power of Movement in -Plants_, 1880. - -[567] Palm, _Ueber das Winden der Pflanzen_, 1827; von Mohl, _Bau -und Winden der Ranken_, etc., 1827; Dutrochet, Mouvements révolutifs -spontanés, _C.R._ 1843, etc. - -[568] Cf. (e.g.) Lepeschkin, Zur Kenntnis des Mechanismus der -Variationsbewegungen, _Ber. d. d. Bot. Gesellsch._ XXVI A, pp. 724–735, -1908; also A. Tröndle, Der Einfluss des Lichtes auf die Permeabilität -des Plasmahaut, _Jahrb. wiss. Bot._ XLVIII, pp. 171–282, 1910. - -[569] For an elaborate study of antlers, see Rörig, A., _Arch. f. -Entw. Mech._ X, pp. 525–644, 1900, XI, pp. 65–148, 225–309, 1901; -Hoffmann, C., _Zur Morphologie der rezenten Hirschen_, 75 pp., 23 pls., -1901: also Sir Victor Brooke, On the Classification of the Cervidae, -_P.Z.S._, pp. 883–928, 1878. For a discussion of the development of -horns and antlers, see Gadow, H., _P.Z.S._, pp. 206–222, 1902, and -works quoted therein. - -[570] Cf. Rhumbler, L., Ueber die Abhängigkeit des Geweihwachstums der -Hirsche, speziell des Edelhirsches, vom Verlauf der Blutgefässe im -Kolbengeweih, _Zeitschr. f. Forst. und Jagdwesen_, 1911, pp. 295–314. - -[571] 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.) - -[572] Cf. also the immense range of variation in elks’ horns, as -described by Lönnberg, _P.Z.S._ II, pp. 352–360, 1902. - -[573] Besides papers referred to below, and many others quoted in -Sach’s _Botany_ and elsewhere, the following are important: Braun, -Alex., Vergl. Untersuchung über die Ordnung der Schuppen an den -Tannenzapfen, etc., _Verh. Car. Leop. Akad._ XV, pp. 199–401, 1831; Dr -C. Schimper’s Vorträge über die Möglichkeit eines wissenschaftlichen -Verständnisses der Blattstellung, etc., _Flora_, XVIII, pp. 145–191, -737–756, 1835; Schimper, C. F., Geometrische Anordnung der um eine Axe -peripherische Blattgebilde, _Verhandl. Schweiz. Ges._, pp. 113–117, -1836; Bravais, L. and A., Essai sur la disposition des feuilles -curvisériées, _Ann. Sci. Nat._ (2), VII, pp. 42–110, 1837; Sur la -disposition symmétrique des inflorescences, _ibid._, pp. 193–221, -291–348, VIII, pp. 11–42, 1838; Sur la disposition générale des -feuilles rectisériées, _ibid._ XII, pp. 5–41, 65–77, 1839; Zeising, -_Normalverhältniss der chemischen und morphologischen Proportionen_, -Leipzig, 1856; Naumann, C. F., Ueber den Quincunx als Gesetz der -Blattstellung bei Sigillaria, etc., _Neues Jahrb. f. Miner._ 1842, -pp. 410–417; Lestiboudois, T., _Phyllotaxie anatomique_, Paris, 1848; -Henslow, G., _Phyllotaxis_, London, 1871; Wiesner, Bemerkungen über -rationale und irrationale Divergenzen, _Flora_, LVIII, pp. 113–115, -139–143, 1875; Airy, H., On Leaf Arrangement, _Proc. R. S._ XXI, p. -176, 1873; Schwendener, S., _Mechanische Theorie der Blattstellungen_, -Leipzig, 1878; Delpino, F., _Causa meccanica della filotassi -quincunciale_, Genova, 1880; de Candolle, C., _Étude de Phyllotaxie_, -Genève, 1881. - -[574] _Allgemeine Morphologie der Gewächse_, p. 442, etc. 1868. - -[575] _Relation of Phyllotaxis to Mechanical Laws_, Oxford, 1901–1903; -cf. _Ann. of Botany_, XV, p. 481, 1901. - -[576] “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, _op. cit._ I, p. 42. - -[577] 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.” - -[578] _Amer. Naturalist_, VII, p. 449, 1873. - -[579] This celebrated series, which appears in the continued fraction -(1/1) + (1/(1 + )) etc. and is closely connected with the _Sectio -aurea_ 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 _De nive sexangula_ (cf. _supra_, 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, _Bot. Centralbl._ LXVIII, 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 (_On the mathematical -connexion between the parts of Vegetables_: abstract of a Memoir read -before the Royal Society in the year 1811 (privately printed, _n.d._). -Cf. De Candolle, _Organogénie végétale_, I, p. 534). - -[580] _Proc. Roy. Soc. Edin._ VII, p. 391, 1872. - -[581] 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 _Mathematical and other Writings_, 1853, pp. -358–372. - -[582] _Proc. Roy. Soc. Edin._ VII, p. 397, 1872; _Trans. Roy. Soc. -Edin._ XXVI, p. 505, 1870–71. - -[583] 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. - -[584] _Deutsche Vierteljahrsschrift_, p. 261, 1868. - -[585] _Memoirs of Amer. Acad._ IX, p. 389. - -[586] _De avibus circa aquas Danubii vagantibus et de ipsarum Nidis_ -(Vol. V of the _Danubius Pannonico-mysicus_), Hagae Com., 1726. - -[587] Sir Thomas Browne had a collection of eggs at Norwich, according -to Evelyn, in 1671. - -[588] Cf. Lapierre, in Buffon’s _Histoire Naturelle_, ed. Sonnini, 1800. - -[589] _Eier der Vögel Deutschlands_, 1818–28 (_cit._ des Murs, p. 36). - -[590] _Traité d’Oologie_, 1860. - -[591] Lafresnaye, F. de, Comparaison des œufs des Oiseaux avec -leurs squelettes, comme seul moven de reconnaître la cause de leurs -différentes formes, _Rev. Zool._, 1845, pp. 180–187, 239–244. - -[592] 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.” - -[593] Thienemann, F. A. L., _Syst. Darstellung der Fortpflanzung der -Vögel Europas_. Leipzig, 1825–38. - -[594] Cf. Newton’s _Dictionary of Birds_, 1893, p. 191; Szielasko, -Gestalt der Vogeleier, _J. f. Ornith._ LIII, pp. 273–297, 1905. - -[595] Jacob Steiner suggested a Cartesian oval, _r_ + _mr′_ = _c_, as a -general formula for all eggs (cf. Fechner, _Ber. sächs. Ges._, 1849, p. -57); but this formula (which fails in such a case as the guillemot), is -purely empirical, and has no mechanical foundation. - -[596] Günther, F. C., _Sammlung von Nestern und Eyern verschiedener -Vögel_, Nürnb. 1772. Cf. also Raymond Pearl, Morphogenetic Activity of -the Oviduct, _J. Exp. Zool._ VI, pp. 339–359, 1909. - -[597] The following account is in part reprinted from _Nature_, June 4, -1908. - -[598] 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 (_De rerum natura_, VI, cc. 4 and 10), which he had -inherited from Galen, and of which Bacon speaks (_Nov. Org._ 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.” - -[599] _Journal of Tropical Medicine_, 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, _Brit. Med. Journ._, 18th March, 1916, p. -411). - -[600] Cf. Bashforth and Adams, _Theoretical Forms of Drops, etc._, -Cambridge, 1883. - -[601] Woods, R. H., On a Physical Theorem applied to tense Membranes, -_Journ. of Anat. and Phys._ XXVI, 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 _Animal -Mechanics_, pp. 151–158, 1873. - -[602] This corresponds with a determination of the normal pressures (in -systole) by Krohl, as being in the ratio of 1 : 6·8. - -[603] Cf. Schwalbe, G., Ueber Wechselbeziehungen und ihr Einfluss -auf die Gestaltung des Arteriensystem, _Jen. Zeitschr._ XII, p. 267, -1878, Roux, Ueber die Verzweigungen der Blutgefässen des Menschen, -_ibid._ XII, p. 205, 1878; Ueber die Bedeutung der Ablenkung des -Arterienstämmen bei der Astaufgabe, _ibid._ XIII, p. 301, 1879; -Hess, Walter, Eine mechanisch bedingte Gesetzmässigkeit im Bau des -Blutgefässsystems, _A. f. Entw. Mech._ XVI, p. 632, 1903; Thoma, R., -_Ueber die Histogenese und Histomechanik des Blutgefässsystems_, 1893. - -[604] _Essays_, etc., edited by Owen, I, p. 134, 1861. - -[605] On the Functions of the Heart and Arteries, _Phil. Trans._ 1809, -pp. 1–31, cf. 1808, pp. 164–186; _Collected Works_, I, 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’ _Statical Essays_, II, -_Introduction_: “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, _etc._” - -[606] “Sizes” is Owen’s editorial emendation, which seems amply -justified. - -[607] For a more elaborate classification, into colours cryptic, -procryptic, anticryptic, apatetic, epigamic, sematic, episematic, -aposematic, etc., see Poulton’s _Colours of Animals_ (Int. Scientific -Series, LXVIII), 1890; cf. also Meldola, R., Variable Protective -Colouring in Insects, _P.Z.S._ 1873, pp. 153–162, etc. - -[608] Dendy, _Evolutionary Biology_, p. 336, 1912. - -[609] 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 _Origin_, 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 _Spectator_)—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.” - -[610] Cf. Bridge, T. W., _Cambridge Natural History_ (Fishes), VII, -p. 173, 1904; also Frisch, K. v., Ueber farbige Anpassung bei Fische, -_Zool. Jahrb._ (_Abt. Allg. Zool._), XXXII, pp. 171–230, 1914. - -[611] _Nature_, L, p. 572; LI, pp. 33, 57, 533, 1894–95. - -[612] 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, _Concealing-coloration in the Animal Kingdom_, 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 _feed by day_. - -[613] Principal Galloway, _Philosophy of Religion_, p. 344, 1914. - -[614] Cf. Professor Flint, in his Preface to Affleck’s translation -of Janet’s _Causes finales_: “We are, no doubt, still a long way -from a mechanical theory of organic growth, but it may be said to be -the _quaesitum_ of modern science, and no one can say that it is a -chimaera.” - -[615] Cf. Sir Donald MacAlister, How a Bone is Built, _Engl. Ill. Mag._ -1884. - -[616] Professor Claxton Fidler, _On Bridge Construction_, p. 22 (4th -ed.), 1909; cf. (_int. al._) Love’s _Elasticity_, p. 20 (_Historical -Introduction_), 2nd ed., 1906. - -[617] 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.” - -[618] In a few anatomical diagrams, for instance in some of the -drawings in Schmaltz’s _Atlas der Anatomie des Pferdes_, we may see -the system of “ties” diagrammatically inserted in the figure of the -skeleton. Cf. Gregory, On the principles of Quadrupedal Locomotion, -_Ann. N. Y. Acad. of Sciences_, XXII, p. 289, 1912. - -[619] Galileo, _Dialogues concerning Two New Sciences_ (1638), Crew -and Salvio’s translation, New York, 1914, p. 150; _Opere_, ed. Favaro, -VIII, p. 186. Cf. Borelli, _De Motu Animalium_, I, prop. CLXXX, 1685. -Cf. also Camper, P., La structure des os dans les oiseaux, _Opp._ III, -p. 459, ed. 1803; Rauber, A., Galileo über Knochenformen, _Morphol. -Jahrb._ VII, pp. 327, 328, 1881; Paolo Enriques, Della economia di -sostanza nelle osse cave, _Arch. f. Ent. Mech._ XX, pp. 427–465, 1906. - -[620] _Das mechanische Prinzip. im anatomischen Bau der Monocotylen_, -Leipzig, 1874. - -[621] For further botanical illustrations, see (_int. al._) Hegler, -Einfluss der Zugkraften auf die Festigkeit und die Ausbildung -mechanischer Gewebe in Pflanzen, _SB. sächs. Ges. d. Wiss._ p. -638, 1891; Kny, L., Einfluss von Zug und Druck auf die Richtung -der Scheidewande in sich teilenden Pflanzenzellen, _Ber. d. bot. -Gesellsch._ XIV, 1896; Sachs, Mechanomorphose und Phylogenie, _Flora_, -LXXVIII, 1894; cf. also Pflüger, Einwirkung der Schwerkraft, etc., über -die Richtung der Zelltheilung, _Archiv_, XXXIV, 1884. - -[622] Among other works on the mechanical construction of bone -see: Bourgery, _Traité de l’anatomie_ (_I. Ostéologie_), 1832 -(with admirable illustrations of trabecular structure); Fick, L., -_Die Ursachen der Knochenformen_, Göttingen, 1857; Meyer, H., Die -Architektur der Spongiosa, _Archiv f. Anat. und Physiol._ XLVII, pp. -615–628, 1867; _Statik u. Mechanik des menschlichen Knochengerüstes_, -Leipzig, 1873; Wolff, J., Die innere Architektur der Knochen, _Arch. -f. Anat, und Phys._ L, 1870; _Das Gesetz der Transformation bei -Knochen_, 1892; von Ebner, V., Der feinere Bau der Knochensubstanz, -_Wiener Bericht_, LXXII, 1875; Rauber, Anton, _Elastizität und -Festigkeit der Knochen_, Leipzig, 1876; O. Meserer, _Elast, u. -Festigk. d. menschlichen Knochen_, Stuttgart, 1880; MacAlister, Sir -Donald, How a Bone is Built, _English Illustr. Mag._ pp. 640–649, -1884; Rasumowsky, Architektonik des Fussskelets, _Int. Monatsschr. f. -Anat._ p. 197, 1889; Zschokke, _Weitere Unters. über das Verhältniss -der Knochenbildung zur Statik und Mechanik des Vertebratenskelets_, -Zürich, 1892; Roux, W., _Ges. Abhandlungen über Entwicklungsmechanik -der Organismen, Bd. I, Funktionelle Anpassung_, Leipzig, 1895; Triepel, -H., Die Stossfestigkeit der Knochen, _Arch. f. Anat. u. Phys._ 1900; -Gebhardt, Funktionell wichtige Anordnungsweisen der feineren und -gröberen Bauelemente des Wirbelthierknochens, etc., _Arch. f. Entw. -Mech._ 1900–1910; Kirchner. A., Architektur der Metatarsalien, _A. -f. E. M._ XXIV, 1907; Triepel, Herm., Die trajectorielle Structuren -(in _Einf. in die Physikalische Anatomie_, 1908); Dixon, A. F., -Architecture of the Cancellous Tissue forming the Upper End of the -Femur, _Journ. of Anat. and Phys._ (3) XLIV, pp. 223–230, 1910. - -[623] Sédillot, De l’influence des fonctions sur la structure et la -forme des organes; _C. R._ LIX, p. 539, 1864; cf. LX, p. 97, 1865, -LXVIII. p. 1444. 1869. - -[624] 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 _tendo Achillis_; (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 _Textbook of -Physiology_, p. 251, 1900.) - -[625] Our problem is analogous to Dr Thomas Young’s problem of the best -disposition of the timbers in a wooden ship (_Phil. Trans._ 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. - -[626] 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 _skeleton_, 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” (_Trans. R. S. E._ XXVI, 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. - -[627] 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, _The Horse -in Motion_, p. 69, 1882. - -[628] This and the following diagrams are borrowed and adapted from -Professor Fidler’s _Bridge Construction_. - -[629] The method of constructing _reciprocal diagrams_, 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 _Graphische Statik_; it was greatly developed soon afterwards -by Macquorn Rankine (_Phil. Mag._ Feb. 1864, and _Applied Mechanics_, -passim), to whom is mainly due the general application of the principle -to engineering practice. - -[630] _Dialogues concerning Two New Sciences_ (1638): Crew and Salvio’s -translation, p. 140 _seq._ - -[631] 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, _Morphol. Jahrb._ LXIX, -pp. 179–220, 1915, and many works quoted therein. According to -Morita, Ueber die Ursachen der Richtung und Gestalt der thoracalen -Dornfortsätze der Säugethierwirbelsäule (_ibi cit._ 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., _Theoretische Grundlagen für -eine Mechanik der lebenden Körper_, Leipzig, pp. 3, 372, 1906. - -[632] 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. - -[633] This pose of Diplodocus, and of other Sauropodous reptiles, has -been much discussed. Cf. (_int. al._) Abel, O., _Abh. k. k. zool. -bot. Ges. Wien_, V. 1909–10 (60 pp.); Tornier, _SB. Ges. Naturf. -Fr. Berlin_, pp. 193–209, 1909; Hay, O. P., _Amer. Nat._ Oct. 1908; -_Tr. Wash. Acad. Sci._ XLII, pp. 1–25, 1910; Holland, _Amer. Nat._ -May, 1910, pp. 259–283; Matthew, _ibid._ pp. 547–560; Gilmore, C. W. -(_Restoration of Stegosaurus_). _Pr. U.S. Nat. Museum_, 1915. - -[634] 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. - -[635] The motto was Macquorn Rankine’s. - -[636] John Hunter was seldom wrong; but I cannot believe that he was -right when he said (_Scientific Works_, ed. Owen, I, 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., -_without considering any other part of the body_.” - -[637] _Origin of Species_, 6th ed. p. 118. - -[638] _Amer. Naturalist_, April, 1915, p. 198, etc. Cf. _infra_, p. 727. - -[639] 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’ ” -(_Gifford Lectures_, 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 _this_ Entelechy is shewn to -be peculiarly or specifically related to the _living_ organism. The -conception that the whole is _always_ 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 _meat-roasting_ 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.” - -[640] “There can be no doubt that Fraas is correct in regarding this -type (_Procetus_) 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, _Tertiary Vertebrata of the Fayum_, 1906, p. 235. - -[641] Reprinted, with some changes and additions, from a paper in the -_Trans. Roy. Soc. Edin._ L, pp. 857–95, 1915. - -[642] M. Bergson repudiates, with peculiar confidence, the -application of mathematics to biology. Cf. _Creative Evolution_, -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.” - -[643] 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, _we need a kind and degree of accuracy of which -mathematics is absolutely incapable_ .... 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...” (_op. cit._, p. 19, 1911). It were easy to discuss -and criticise these sweeping assertions, which perhaps had their -origin and parentage in an _obiter dictum_ 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” -(_cit._ Cajori, _Hist of Elem. Mathematics_, 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” (_Brit. Ass. Address_, 1869, and _Laws of -Verse_, p. 120, 1870). - -[644] _Historia Animalium_ I, 1. - -[645] Cf. _supra_, p. 714. - -[646] Cf. Osborn, H. F., On the Origin of Single Characters, as -observed in fossil and living Animals and Plants, _Amer. Nat._ XLIX, -pp. 193–239, 1915 (and other papers); _ibid._ 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 _Origin of Species_ 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. - -[647] Cf. Sorby, _Quart. Journ. Geol. Soc._ (_Proc._), 1879, p. 88. - -[648] Cf. D’Orbigny, Alc., _Cours élém. de Paléontologie_, etc., I, pp. -144–148, 1849; see also Sharpe, Daniel, On Slaty Cleavage, _Q.J.G.S._ -III, p. 74, 1847. - -[649] Thus _Ammonites erugatus_, when compressed, has been described as -_A. planorbis_: cf. Blake, J. F., _Phil. Mag._ (5), VI, p. 260, 1878. -Wettstein has shewn that several species of the fish-genus _Lepidopus_ -have been based on specimens artificially deformed in various ways: -Ueber die Fischfauna des Tertiären Glarnerschiefers, _Abh. Schw. -Palaeont. Gesellsch._ XIII, 1886 (see especially pp. 23–38, pl. I). -The whole subject, interesting as it is, has been little studied: both -Blake and Wettstein deal with it mathematically. - -[650] Cf. Sir Thomas Browne, in _The Garden of Cyrus_: “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.” - -[651] 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. - -[652] Cf. _Elsie Venner_, chap. ii. - -[653] 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. - -[654] Cf. Vitruvius, III, 1. - -[655] _Les quatres livres d’Albert Dürer de la proportion des parties -et pourtraicts des corps humains_, Arnheim, 1613, folio (and earlier -editions). Cf. also Lavater, _Essays on Physiognomy_, III, p. 271, 1799. - -[656] 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 _varies as a whole_, 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 _norma_ 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 _On the Connexion between the -Science of Anatomy and the Arts of Drawing, Painting and Sculpture_ -(1768?), quoted in Dr R. Hamilton’s Memoir of Camper, in _Lives of -Eminent Naturalists_ (_Nat. Libr._), Edin. 1840. - -[657] The co-ordinate system of Fig. 382 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. - -[658] _Dinosaurs of North America_, pl. LXXXI, etc. 1896. - -[659] _Mem. Amer. Mus. of Nat. Hist._ I, III, 1898. - -[660] These and also other coordinate diagrams will be found in Mr G. -Heilmann’s book _Fuglenes Afstamning_, 398 pp., Copenhagen, 1916; see -especially pp. 368–380. - -[661] Cf. W. B. Scott (_Amer. Journ. of Science_, XLVIII, 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.” - -[662] Cf. Dwight, T., The Range of Variation of the Human Scapula, -_Amer. Nat._ XXI, pp. 627–638, 1887. Cf. also Turner, _Challenger Rep._ -XLVII, 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.” - -[663] 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, -_Berichte d. k. sächs. Gesellsch._, _Math.-phys. Cl._, 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. - -An interesting little book of Schiaparelli’s (which I ought to have -known long ago)—_Forme organiche naturali e forme geometriche pure_, -Milano, Hoepli, 1898—has likewise come into my hands too late for -discussion. - -{780} - - - - -INDEX. - - - Abbe’s diffraction plates, 323 - - Abel, O., 706 - - Abonyi, A., 127 - - Acantharia, spicules of, 458 - - Acanthometridae, 462 - - Acceleration, 64 - - Aceratherium, 761 - - Achlya, 244 - - Acromegaly, 135 - - Actinomma, 469 - - Actinomyxidia, 452 - - Actinophrys, 165, 197, 264, 298 - - Actinosphaerium, 197, 266, 298, 468 - - Adams, J. C., 663 - - Adaptation, 670 - - Addison, Joseph, 671 - - Adiantum, 408 - - Adsorption, 192, 208, 241, 277, 357; - orientirte, 440, 590; - pseudo, 282 - - Agglutination, 201 - - Aglaophenia, 748 - - Airy, H., 636 - - Albumin molecule, 41 - - Alcyonaria, 387, 413, 424, 459 - - Alexeieff, A., 157, 165 - - Allmann, W., 643 - - Alpheus, claws of, 150 - - Alpine plants, 124 - - Altmann’s granules, 285 - - Alveolar meshwork, 170 - - Ammonites, 526, 530, 537, 539, 550, 552, 576, 583, 584, 728 - - Amoeba, 12, 165, 209, 212, 245, 255, 288, 463, 605 - - Amphidiscs, 440 - - Amphioxus, 311 - - Ampullaria, 560 - - Anabaena, 300 - - Anaxagoras, 8 - - Ancyloceras, 550 - - Andrews, G. F., 164; - C. W., 716 - - Anhydrite, 433 - - Anikin, W. P., 130 - - Anisonema, 126 - - Anisotropy, 241, 357 - - Anomia, 565, 567 - - Antelopes, horns of, 614, 671 - - Antheridia, 303, 403, 405, 409 - - Anthoceros, spore of, 397 - - Anthogorgia, spicules of, 413 - - Anthropometry, 51 - - Anticline, 360 - - Antigonia, 750, 775 - - Antlers, 628 - - Apatornis, 757 - - Apocynum, pollen of, 396 - - Aptychus, 576 - - Arachnoidiscus, 387 - - Arachnophyllum, 325 - - Arcella, 323 - - Arcestes, 539, 540 - - Archaeopteryx, 757 - - Archimedes, 580; - spiral of, 503, 524, 552 - - Argali, horns of, 617 - - Argiope, 561 - - Argonauta, 546, 561 - - Argus pheasant, 431, 631 - - Argyropelecus, 748 - - Aristotle, 3, 4, 5, 8, 15, 138, 149, 158, 509, 653, 714, 725, 726 - - Arizona trees, 121 - - Arrhenius, Sv., 28, 48, 171 - - Artemia, 127 - - Artemis, 561 - - Ascaris megalocephala, 180, 195 - - Aschemonella, 255 - - Assheton, R., 344 - - Asterina, 342 - - Asteroides, 423 - - Asterolampra, 386 - - Asters, 167, 174 - - Asthenosoma, 664 - - Astrorhiza, 255, 463, 587, 607 - - Astrosclera, 436 - - Asymmetric substances, 416 - - Asymmetry, 241 - - Atrypa, 569 - - Auerbach, F., 9 - - Aulacantha, 460 - - Aulastrum, 471 - - Aulonia, 468 - - Auricular height, 93 - - Autocatalysis, 131 - - Auximones, 135 - - Awerinzew, S., 589 - - Babak, E., 32 - - Babirussa, teeth of, 634 - - Baboon, skull of, 771 - - Bacillus, 39; - B. ramosus, 133 - - Bacon, Lord, 4, 5, 51, 53, 131, 656, 716 - - Bacteria, 245, 250 - - Baer, K. E., von, 3, 55, 57, 155 - - Balancement, 714, 776 - - Balfour, F. M., 57, 348 - - Baltzer, Fr., 327 - - Bamboo, growth of, 77 - - Barclay, J., 334 - - Barfurth, D., 85 - - Barlow, W., 202 - - Barratt, J. O. W., 285 - - Bartholinus, E., 329 - - Bashforth, Fr., 663 - - Bast-fibres, strength of, 679 - - Baster, Job, 138 - - Bateson, W., 104, 431 - - Bather, F. A., 578 - - Batsch, A. J. G. K., 606 - - Baudrimont, A., and St Ange, 124 - - Baumann and Roos, 136 - - Bayliss, W. M., 135, 277 - - Beads or globules, 234 - - Beak, shape of, 632 - - Beal, W. J., 643 - - Beam, loaded, 674 - - Bee’s cell, 327, 779 - - Begonia, 412, 733 - - Beisa antelope, horns of, 616, 621 - - Bellerophon, 550 - - Bénard, H., 259, 319, 448, 590 - - Bending moments, 19, 677, 696 - - Beneden, Ed. van, 153, 170, 198 - - Bergson, H., 7, 103, 251, 611, 721 - - Bernard, Claude, 2, 13, 127 - - Bernoulli, James, 580; - John, 30, 54 - - Berthold, G., 8, 234, 298, 306, 322, 346, 351, 357, 358, 372, 399 - - Bethe, A., 276 - - Bialaszewicz, K., 114, 125 - - Biedermann, W., 431 - - Bilharzia, egg of, 656 - - Binuclearity, 286 - - Biocrystallisation, 454 - - Biogenetisches Grundgesetz, 608 - - Biometrics, 78 - - Bird, flight of, 24; - form of, 673 - - Bisection of solids, 352, etc. - - Bishop, John 31 - - Bivalve shells, 561 - - Bjerknes, V. 186 - - Blackman, F. F. 108, 110, 114, 124, 131, 132 - - Blackwall, J. 234 - - Blake, J. F. 536, 547, 553, 578, 583, 728 - - Blastosphere, 56, 344 - - Blood-corpuscles, form of, 270; - size of, 36 - - Blood-vessels, 665 - - Boas, Fr., 79 - - Bodo, 230, 269 - - Boerhaave, Hermann, 380 - - Bonanni, F., 318 - - Bone, 425, 435; - repair of, 687; - structure of, 673, 680 - - Bonnet, Ch., 108, 138, 334, 635 - - Borelli, J. A., 8, 27, 29, 318, 677, 690 - - Bosanquet, B., 5 - - Boscovich, Father R. J., S.J., 8 - - Bose, J. C., 87 - - Bostryx, 502 - - Bottazzi, F., 127 - - Bottomley, J. T., 135 - - Boubée, N., 529 - - Bourgery, J. M., 683 - - Bourne, G. C., 199 - - Bourrelet, Plateau’s, 297, 339, 446, 470, 477 - - Boveri, Th., 38, 147, 170, 198 - - Bowditch, H. P., 61, 79 - - Bower, F. O., 406 - - Bowman, J. H., 428 - - Boyd, R., 61 - - Boys, C. V., 233 - - Brachiopods, 561, 568, 577 - - Bradford, S. C., 428 - - Brady, H. B., 255, 606 - - Brain, growth of, 89; - weight of, 90 - - Branchipus, 128, 342 - - Brandt, K., 459, 482 - - Brauer, A., 180 - - Braun, A., 636 - - Bravais, L. and A., 202, 502, 636 - - Bredig, G., 178 - - Brewster, Sir D., 209, 337, 350, 431 - - Bridge, T. W., 671 - - Bridge construction, 18, 691 - - Brine shrimps, 127 - - Brooke, Sir V., 614, 624, 628, 631 - - Browne, Sir T., 324, 329, 480, 650, 652, 733 - - Brownian movement, 45, 279, 421 - - Brücke, C., 160, 199 - - Buccinum, 520, 527 - - Buch, Leopold von, 528, 583 - - Buchner, Hans, 133 - - Budding, 213, 399 - - Buffon, on the bee’s cell, 333 - - Bühle, C. A., 653 - - Bulimus, 549, 556 - - Burnet, J., 509 - - Bütschli, O., 165, 170, 171, 204, 432, 434, 458, 492 - - Büttel-Reepen, H. von, 332 - - Byk, A., 419 - - Cactus, sphaerocrystals, in 434 - - Cadets, growth of German, 119 - - Calandrini, G. L., 636 - - Calcospherites, 421, 434 - - Callimitra, 472 - - Callithamnion, spore of, 396 - - Calman, T. W., 149 - - Calyptraea, 556 - - Camel, 703, 704 - - Campanularia, 237, 262, 747 - - Campbell, D. H., 302, 397, 402 - - Camper, P., 742 - - Camptosaurus, 754 - - Cannon bone, 730 - - Cantilever, 678, 694 - - Cantor, Moritz, 503 - - Caprella, 743 - - Caprinella, 567, 577 - - Carapace of crabs, 744 - - Cardium, 561 - - Cariacus, 629 - - Carlier, E. W., 211 - - Carnoy, J. B., 468 - - Carpenter, W. B., 45, 422, 465 - - Caryokinesis, 14, 157, etc. - - Cassini, D., 329 - - Cassis, 559 - - Catabolic products, 435 - - Catalytic action, 130 - - Catenoid, 218, 223, 227, 252 - - Causation, 6 - - Cavolinia, 573 - - Cayley, A., 385 - - Celestite, 459 - - Cell-theory, 197, 199 - - Cells, forms of, 201; - sizes of, 35 - - Cellular pathology, 200; - tissue, artificial, 320 - - Cenosphaera, 470 - - Centres of force, 156, 196 - - Centrosome, 167, 168, 173 - - Cephalopods, 548, etc.; - eggs of, 378 - - Ceratophyllum, growth of, 97 - - Ceratorhinus, 612 - - Cerebratulus, egg of, 189 - - Cerianthus, 125 - - Cerithium, 530, 557, 559 - - Chabrier, J., 25 - - Chabry, L., 30, 306, 415 - - Chaetodont fishes, 671, 749 - - Chaetopterus, egg of, 195 - - Chamois, horns of, 615 - - Chapman, Abel, 672 - - Chara, 303 - - Characters, biological, 196, 727 - - Chevron bones, 709 - - Chick, hatching of, 108 - - Chilomonas, 114 - - Chladni figures, 386, 475 - - Chlorophyll, 291 - - Choanoflagellates, 253 - - Chodat, R., 78, 132 - - Cholesterin, 272 - - Chondriosomes, 285 - - Chorinus, 744 - - Chree, C., 19 - - Chromatin, 153 - - Chromidia, 286 - - Chromosomes, 157, 173, 179, 181, 190, 195 - - Church, A. H., 639 - - Cicero, 62 - - Cicinnus, 502 - - Cidaris, 664 - - Circogonia, 479 - - Cladocarpus, 748 - - Claparède, E. R, 423 - - Clathrulina, 470 - - Clausilia, 520, 549 - - Claws, 149, 632 - - Cleland, John, 4 - - Cleodora, 570–575 - - Climate and growth, 121 - - Clio, 570 - - Close packing, 453 - - Clytia, 747 - - Coan, C. A., 514 - - Coassus, 629 - - Cod, otoliths of, 432; - skeleton of, 710 - - Codonella, 248 - - Codosiga, 253 - - Coe, W. R., 189 - - Coefficient of growth, 153; - of temperature, 109 - - Coelopleurus, 664 - - Cogan, Dr T., 742 - - Cohen, A., 110 - - Cohesion figures, 259 - - Collar-cells, 253 - - Colloids, 162, 178, 201, 279, 412, 421, etc. - - Collosclerophora, 436 - - Collosphaera, 459 - - Colman, S., 514 - - Comoseris, 327 - - Compensation, law of, 714, 776 - - Conchospiral, 531, 539, 594 - - Conchyliometer, 529 - - Concretions, 410, etc. - - Conjugate curves, 561, 613 - - Conklin, E. G., 36, 191, 310, 340, 377 - - Conostats, 427 - - Continuous girder, 700 - - Contractile vacuole, 165, 264 - - Conus, 557, 559, 560 - - Cook, Sir T. A., 493, 635, 639, 650 - - Co-ordinates, 723 - - Corals, 325, 388, 423 - - Cornevin, Ch., 102 - - Cornuspira, 594 - - Correlation, 78, 727 - - Corystes, 744 - - Cotton, A., 418 - - Cox, J., 46 - - Crane-head, 682 - - Crayfish, sperm-cells of, 273 - - Creodonta, 716 - - Crepidula, 36, 310, 340 - - Creseis, 570 - - Cristellaria, 515, 600 - - Crocodile, 704, 752 - - Crocus, growth of, 88 - - Crookes, Sir W., 32 - - Cryptocleidus, 755 - - Crystals, 202, 250, 429, 444, 480, 601 - - Ctenophora, 391 - - Cube, partition of, 346 - - Cucumis, growth of, 109 - - Culmann, Professor C., 682, 697 - - Cultellus, 564 - - Curlew, eggs of, 652 - - Cushman, J. A., 323 - - Cuvier, 727 - - Cuvierina, 258, 570 - - Cyamus, 743 - - Cyathophyllum, 325, 391 - - Cyclammina, 595, 596, 602 - - Cyclas, 561 - - Cyclostoma, 554 - - Cylinder, 218, 227, 377 - - Cymba, 559 - - Cyme, 502 - - Cypraea, 547, 554, 560, 561 - - Cyrtina, 569 - - Cyrtocerata, 583 - - Cystoliths, 412 - - Daday de Dees, E. v., 130 - - Daffner, Fr., 61, 118 - - Dalyell, Sir John G., 146 - - Danilewsky, B., 135 - - Darling, C. R., 219, 257, 664 - - D’Arsonval, A., 192, 281 - - Darwin, C., 4, 44, 57, 332, 431, 465, 549, 624, 671, 714 - - Dastre, A., 136 - - Davenport, C. B., 107, 123, 125, 126, 211 - - De Candolle, A., 108, 643; - A. P., 20; - C., 636 - - Decapod Crustacea, sperm-cells of, 273 - - Deer, antlers of, 628 - - Deformation, 638, 728, etc. - - Degree, differences of, 586, 725 - - Delage, Yves, 153 - - Delaunay, C. E., 218 - - Delisle, 31 - - Dellinger, O. P., 212 - - Delphinula, 557 - - Delpino, F., 636 - - Democritus, 44 - - Dendy, A., 137, 436, 440, 671 - - Dentalium, 535, 537, 546, 555, 556, 561 - - Dentine, 425 - - Descartes, R., 185, 723 - - Des Murs, O., 653 - - Devaux, H., 43 - - De Vries, H., 108 - - Diatoms, 214, 386, 426 - - Diceras, 567 - - Dickson, Alex., 647 - - Dictyota, 303, 356, 474 - - Diet and growth, 134 - - Difflugia, 463, 466 - - Diffusion figures, 259, 430 - - Dimorphism of earwigs, 105 - - Dimorphodon, 756 - - Dinenympha, 252 - - Dinobryon, 248 - - Dinosaurs, 702, 704, 754 - - Diodon, 751, 777 - - Dionaea, 734 - - Diplodocus, 702, 706, 710 - - Disc, segmentation of a, 367 - - Discorbina, 602 - - Distigma, 246 - - Distribution, geographical, 457, 606 - - Ditrupa, 586 - - Dixon, A. F., 684 - - Dobell, C. C., 286 - - Dodecahedron, 336, 478, etc. - - Doflein, F. J., 46, 267, 606 - - Dog’s skull, 773 - - Dolium, 526, 528, 530, 557, 559, 560 - - Dolphin, skeleton of, 709 - - Donaldson, H. H., 82, 93 - - Dorataspis, 481 - - D’Orbigny, Alc., 529, 555, 591, 728 - - Douglass, A. E., 121 - - Draper, J. W., 165, 264 - - Dreyer, F. R., 435, 447, 455, 468, 606, 608 - - Driesch, H., 4, 35, 157, 306, 310, 312, 377, 378, 714 - - Dromia, 275 - - Drops, 44, 257, 587 - - Du Bois-Reymond, Emil, 1, 92 - - Duerden, J. E., 423 - - Dufour, Louis, 219 - - Dujardin, F., 257, 591 - - Dunan, 7 - - Duncan, P. Martin, 388 - - Dupré, Athanase, 279 - - Durbin, Marion L., 138 - - Dürer, A., 55, 740, 742 - - Dutrochet, R. J. H., 212, 624 - - Dwight, T., 769 - - Dynamical similarity, 17 - - Earthworm, calcospheres in, 423 - - Earwigs, dimorphism in, 104 - - Ebner, V. von, 444, 683 - - Echinoderms, larval, 392; - spicules of, 449 - - Echinus, 377, 378, 664 - - Eclipse, skeleton of, 739 - - Ectosarc, 281 - - Eel, growth of, 85 - - Efficiency, mechanical, 670 - - Efficient cause, 6, 158, 248 - - Eggs of birds, 652 - - Eiffel tower, 20 - - Eight cells, grouping of, 381, etc. - - Eimer, Th., 606 - - Einstein formula, 47 - - Elastic curve, 219, 265, 271 - - Elaters, 489 - - Electrical convection, 187; - stimulation of growth, 153 - - Elephant, 21, 633, 703, 704 - - Elk, antlers of, 629, 632 - - Ellipsolithes, 728 - - Ellis, R. Leslie, 4, 329, 647; - M. M., 147, 656 - - Elodea, 322 - - Emarginula, 556 - - Emmel, V. E., 149 - - Empedocles, 8 - - Emperor Moth, 431 - - Encystment, 213, 283 - - Engelmann, T. W., 210, 285 - - Enriques, P., 4, 36, 64, 133, 134, 677 - - Entelechy, 4, 714 - - Entosolenia, 449 - - Enzymes, 135 - - Epeira, 233 - - Epicurus, 47 - - Epidermis, 314, 370 - - Epilobium, pollen of, 396 - - Epipolic force, 212 - - Equatorial plate, 174 - - Equiangular spiral, 50, 505 - - Equilibrium, figures of, 227 - - Equipotential lines, 640 - - Equisetum, spores of, 290, 489 - - Errera, Leo, 8, 40, 110, 111, 213, 306, 346, 348, 426 - - Erythrotrichia, 358, 372, 390 - - Ethmosphaera, 470 - - Euastrum, 214 - - Eucharis, 391 - - Euclid, 509 - - Euglena, 376 - - Euglypha, 189 - - Euler, L., 3, 208, 385, 484, 690 - - Eulima, 559 - - Eunicea, spicules of, 424 - - Euomphalus, 557, 559 - - Evelyn, John, 652 - - Evolution, 549, 610, etc. - - Ewart, A. J., 20 - - Fabre, J. H., 64, 779 - - Facial angle, 742, 770, 772 - - Faraday, M., 163, 167, 428, 475 - - Farmer, J. B. and Digby, 190 - - Fatigue, molecular, 689 - - Faucon, A., 88 - - Favosites, 325 - - Fechner, G. T., 654, 777 - - Fedorow, E. S. von, 338 - - Fehling, H., 76, 126 - - Ferns, spores of, 396 - - Fertilisation, 193 - - Fezzan-worms, 127 - - Fibonacci, 643 - - Fibrillenkonus, 285 - - Fick, R., 57, 683 - - Fickert, C., 606 - - Fidler, Prof. T. Claxton, 691, 674, 696 - - Films, liquid, 215, 217, 426 - - Filter-passers, 39 - - Final cause, 3, 248, 714 - - Fir-cone, 635, 647 - - Fischel, Alfred, 88 - - Fischer, Alfred, 40, 172; - Emil, 417, 418; - Otto, 30, 699 - - Fishes, forms of, 748 - - Fission, multiplication by, 151 - - Fissurella, 556 - - FitzGerald, G. F., 158, 281, 323, 440, 477 - - Flagellum, 246, 267, 291 - - Flemming, W., 170, 172, 180 - - Flight, 24 - - Flint, Professor, 673 - - Fluid crystals, 204, 272, 485 - - Fluted pattern, 260 - - Fly’s cornea, 324 - - Fol, Hermann, 168, 194 - - Folliculina, 249 - - Foraminifera, 214, 255, 415, 495, 515 - - Forth Bridge, 694, 699, 700 - - Fossula, 390 - - Foster, M., 185 - - Fraas, E., 716 - - Frankenheim, M. L., 202 - - Frazee, O. E., 153 - - Frédéricq, L., 127, 130 - - Free cell formation, 396 - - Friedenthal, H., 64 - - Frisch, K. von, 671 - - Frog, egg of, 310, 363, 378, 382; - growth of, 93, 126 - - Froth or foam, 171, 205, 305, 314, 322, 343 - - Froude, W., 22 - - Fucus, 355 - - Fundulus, 125 - - Fusulina, 593, 594 - - Fusus, 527, 557 - - Gadow, H. F., 628 - - Galathea, 273 - - Galen, 3, 465, 656 - - Galileo, 8, 19, 28, 562, 677, 720 - - Gallardo, A., 163 - - Galloway, Principal, 672 - - Gamble, F. A., 458 - - Ganglion-cells, size of, 37 - - Gans, R., 46 - - Garden of Cyrus, 324, 329 - - Gastrula, 344 - - Gauss, K. F., 207, 278, 723 - - Gebhardt, W., 430, 683 - - Gelatination, water of, 203 - - Generating curves and spirals, 526, 561, 615, 637, 641 - - Geodetics, 440, 488 - - Geoffroy St Hilaire, Et. de, 714 - - Geotropism, 211 - - Gerassimow, J. J., 35 - - Gerdy, P. N., 491 - - Geryon, 744 - - Gestaltungskraft, 485 - - Giard, A., 156 - - Gilmore, C. W., 707 - - Giraffe, 705, 730, 738 - - Girardia, 321, 408 - - Glaisher, J., 250 - - Glassblowing, 238, 737 - - Gley, E., 135, 136 - - Globigerina, 214, 234, 440, 495, 589, 602, 604, 606 - - Gnomon, 509, 515, 591 - - Goat, horns of, 613 - - Goat moth, wings of, 430 - - Goebel, K., 321, 397, 408 - - Goethe, 20, 38, 199, 714, 719 - - Golden Mean, 511, 643, 649 - - Goldschmidt, R., 286 - - Goniatites, 550, 728 - - Gonothyraea, 747 - - Goodsir, John, 156, 196, 580 - - Gottlieb, H., 699 - - Gourd, form of, 737 - - Grabau, A. H., 531, 539, 550 - - Graham, Thomas, 162, 201, 203 - - Grant, Kerr, 259 - - Grantia, 445 - - Graphic statics, 682 - - Gravitation, 12, 32 - - Gray, J., 188 - - Greenhill, Sir A. G., 19 - - Gregory, D. F., 330, 675 - - Greville, R. K., 386 - - Gromia, 234, 257 - - Gruber, A., 165 - - Gryphaea, 546, 576, 577 - - Guard-cells, 394 - - Gudernatsch, J. F., 136 - - Guillemot, egg of, 652 - - Gulliver, G., 36 - - Günther, F. C., 633, 654 - - Gurwitsch, A., 285 - - Häcker, V., 458 - - Haddock, 774 - - Haeckel, E., 199, 445, 454, 455, 457, 467, 480, 481 - - Hair, pigmentation of, 430 - - Hales, Stephen, 36, 59, 95, 669 - - Haliotis, 514, 527, 546, 547, 554, 555, 557, 561 - - Hall, C. E., 119 - - Haller, A. von, 2, 54, 56, 59, 64, 68 - - Hardesty, Irving, 37 - - Hardy, W. B., 160, 162, 172, 187, 287 - - Harlé, N., 28 - - Harmozones, 135 - - Harpa, 526, 528, 559 - - Harper, R. A., 283 - - Harpinia, 746 - - Harting, P., 282, 420, 426, 434 - - Hartog, M., 163, 327 - - Harvey, E. N. and H. W., 187 - - Hatai, S., 132, 135 - - Hatchett, C., 420 - - Hatschek, B., 180 - - Haughton, Rev. S., 334, 666 - - Haüy, R. J., 720 - - Hay, O. P., 707 - - Haycraft, J. B., 211, 690 - - Head, length of, 93 - - Heart, growth of, 89; - muscles of, 490 - - Heath, Sir T., 511 - - Hegel, G. W. F., 4 - - Hegler, 680, 688 - - Heidenhain, M., 170, 212 - - Heilmann, Gerhard, 757, 768, 772 - - Helicoid, 230; - cyme, 502, 605 - - Helicometer, 529 - - Helicostyla, 557 - - Heliolites, 326 - - Heliozoa, 264, 460 - - Helix, 528, 557 - - Helmholtz, H. von, 2, 9, 25 - - Henderson, W. P., 323 - - Henslow, G., 636 - - Heredity, 158, 286, 715 - - Hermann, F., 170 - - Hero of Alexandria, 509 - - Heron-Allen, E., 257, 415, 465 - - Herpetomonas, 268 - - Hertwig, O., 56, 114, 153, 199, 310; - R., 170, 285 - - Hertzog, R. O., 109 - - Hess, W., 666, 668 - - Heteronymous horns, 619 - - Heterophyllia, 388 - - Hexactinellids, 429, 452, 453 - - Hexagonal symmetry, 319, 323, 471, 513 - - Hickson, S. J., 424 - - Hippopus, 561 - - His, W., 55, 56, 74, 75 - - Hobbes, Thomas, 159 - - Höber, R., 1, 126, 130, 172 - - Hodograph, 516 - - Hoffmann, C., 628 - - Hofmeister, F., 41; W., 87, 210, 234, 304, 306, 636, 639 - - Holland, W. J., 707 - - Holmes, O. W., 62, 737 - - Holothuroid spicules, 440, 451 - - Homonymous horns, 619 - - Homoplasy, 251 - - Hooke, Robert, 205 - - Hop, growth of, 118; - stem of, 627 - - Horace, 44 - - Hormones, 135 - - Horns, 612 - - Horse, 694, 701, 703, 764 - - Houssay, F., 21 - - Huber, P., 332 - - Huia bird, 633 - - Humboldt, A. von, 127 - - Hume, David, 6 - - Hunter, John, 667, 669, 713, 715 - - Huxley, T. H., 423, 722, 752 - - Hyacinth, 322, 394 - - Hyalaea, 571–577 - - Hyalonema, 442 - - Hyatt, A., 548 - - Hyde, Ida H., 125, 163, 184, 188 - - Hydra, 252; - egg of, 164 - - Hydractinia, 342 - - Hydraulics, 669 - - Hydrocharis, 234 - - Hyperia, 746 - - Hyrachyus, 760, 765 - - Hyracotherium, 766, 768 - - Ibex, 617 - - Ice, structure of, 428 - - Ichthyosaurus, 755 - - Icosahedron, 478 - - Iguanodon, 706, 708 - - Inachus, sperm-cells of, 273 - - Infusoria, 246, 489 - - Intussusception, 202 - - Inulin, 432 - - Invagination, 56, 344 - - Iodine, 136 - - Irvine, Robert, 414, 434 - - Isocardia, 561, 577 - - Isoperimetrical problems, 208, 346 - - Isotonic solutions, 130, 274 - - Iterson, G. van, 595 - - Jackson, C. M., 75, 88, 106 - - Jamin, J. C., 418 - - Janet, Paul, 5, 18, 673 - - Japp, F. R., 417 - - Jellett, J. H., 1 - - Jenkin, C. F., 444 - - Jenkinson, J. W., 94, 114, 170 - - Jennings, H. S., 212, 492; - Vaughan, 424 - - Jensen, P., 211 - - Johnson, Dr S., 62 - - Joly, John, 9, 63 - - Jost, L., 110, 111 - - Juncus, pith of, 335 - - Jungermannia, 404 - - Kangaroo, 705, 706, 709 - - Kanitz, Al., 109 - - Kant, Immanuel, 1, 3, 714 - - Kappers, C. U. A., 566 - - Kellicott, W. E., 91 - - Kelvin, Lord, 9, 49, 188, 202, 336, 453 - - Kepler, 328, 480, 486, 643, 650 - - Kienitz-Gerloff, F., 404, 408 - - Kirby and Spence, 28, 30, 127 - - Kirchner, A., 683 - - Kirkpatrick, R., 437 - - Klebs, G., 306 - - Kny, L., 680 - - Koch, G. von, 423 - - Koenig, Samuel, 330 - - Kofoid, C. A., 268 - - Kölliker, A. von, 413 - - Kollmann, M., 170 - - Koltzoff, N. K., 273, 462 - - Koninckina, 570 - - Koodoo, horns of, 624 - - Köppen, Wladimir, 111 - - Korotneff, A., 377 - - Kraus, G., 77 - - Krogh, A., 109 - - Krohl, 666 - - Kühne, W., 235 - - Küster, E., 430 - - Lafresnaye, F. de, 653 - - Lagena, 251, 256, 260, 587 - - Lagrange, J. L., 649 - - Lalanne, L., 334 - - Lamarck, J. B. de, 549, 716 - - Lamb, A. B., 186 - - Lamellaria, 554 - - Lamellibranchs, 561 - - Lami, B., 296, 643 - - Laminaria, 315 - - Lammel, R., 100 - - Lanchester, F. W., 26 - - Lang, Arnold, 561 - - Lankester, Sir E. Ray, 4, 251, 348, 465 - - Laplace, P. S. de, 1, 207, 217 - - Larmor, Sir J., 9, 259 - - Lavater, J. C., 740 - - Law, Borelli’s, 29; - Brandt’s, 482; - of Constant Angle, 599; - Errera’s, 213, 306; - Froude’s, 22; - Lamarle’s, 309; - of Mass, 130; - Maupertuis’s, 208; - Müller’s, 481; - of Optimum, 110; - van’t Hoff’s, 109; - Willard-Gibbs’, 280; - Wolff’s, 3, 51, 155 - - Leaping, 29 - - Leaves, arrangement of, 635; - form of, 731 - - Ledingham, J. C. G., 211 - - Leduc, Stéphane, 162, 167, 185, 219, 259, 415, 428, 431, 590 - - Leeuwenhoek, A. van, 36, 209 - - Leger, L., 452 - - Le Hello, P., 30 - - Lehmann, O., 203, 272, 440, 485, 590 - - Leibniz, G. W. von, 3, 5, 159, 385 - - Leidenfrost, J. G., 279 - - Leidy, J., 252, 468 - - Leiper, R. T., 660 - - Leitch, I., 112 - - Leitgeb, H., 305 - - Length-weight coefficient, 98–103, 775 - - Leonardo da Vinci, 27, 635; - of Pisa, 643 - - Lepeschkin, 625 - - Leptocephalus, 87 - - Leray, Ad., 18 - - Lesage, G. L., 18 - - Leslie, Sir John, 163, 503 - - Lestiboudois, T., 636 - - Leucocytes, 211 - - Levers, Orders of, 690 - - Levi, G., 35, 37 - - Lewis, C. M., 280 - - Lhuilier, S. A. J., 330 - - Liesegang’s rings, 427, 475 - - Light, pressure of, 48 - - Lillie, F. R., 4, 147, 341; - R. S., 180, 187, 192 - - Lima, 565 - - Limacina, 571 - - Lines of force, 163; - of growth, 562 - - Lingula, 251, 567 - - Linnaeus, 28, 250, 547, 720 - - Lion, brain of, 91 - - Liquid veins, 265 - - Lister, Martin, 318; - J. J., 436 - - Listing, J. B., 385 - - Lithostrotion, 325 - - Littorina, 524 - - Lituites, 546, 550 - - Llama, 703 - - Lobsters’ claws, 149 - - Locke, John, 6 - - Loeb, J., 125, 132, 135, 136, 147, 157, 191, 193 - - Loewy, A., 281 - - Logarithmic spiral, 493, etc. - - Loisel, G., 88 - - Loligo, shell of, 575 - - Lo Monaco, 83 - - Lönnberg, E., 614, 632 - - Looss, A., 660 - - Lotze, R. H., 55 - - Love, A. E. H., 674 - - Lucas, F. A., 138 - - Luciani, L., 83 - - Lucretius, 47, 71, 137, 160 - - Ludwig, Carl, 2; - F., 643; - H. J., 342 - - Lupa, 744 - - Lupinus, growth of, 109, 112 - - Macalister, A., 557 - - MacAlister, Sir D., 673, 683 - - Macallum, A. B., 277, 287, 357, 395; - J. B., 492 - - McCoy, F., 388 - - Mach, Ernst, 209, 330 - - Machaerodus, teeth of, 633 - - McKendrick, J. G., 42 - - McKenzie, A., 418 - - Mackinnon, D. L., 268 - - Maclaurin, Colin, 330, 779 - - Macroscaphites, 550 - - Mactra, 562 - - Magnitude, 16 - - Maillard, L., 163 - - Maize, growth of, 109, 111, 298 - - Mall, F. P., 492 - - Maltaux, Mlle, 114 - - Mammoth, 634, 705 - - Man, growth of, 61; - skull of, 770 - - Maraldi, J. P., 329, 473 - - Marbled papers, 736 - - Marcus Aurelius, 609 - - Markhor, horns of, 619 - - Marsh, O. C., 706, 754 - - Marsigli, Comte L. F. de, 652 - - Massart, J., 114 - - Mastodon, 634 - - Mathematics, 719, 778, etc. - - Mathews, A., 285 - - Matrix, 656 - - Matter and energy, 11 - - Matthew, W. D., 707 - - Matuta, 744 - - Maupas, M., 133 - - Maupertuis, 3, 5, 208 - - Maxwell, J. Clerk, 9, 18, 40, 44, 160, 207, 385, 691 - - Mechanical efficiency, 670 - - Mechanism, 5, 161, 185, etc. - - Meek, C. F. U., 190 - - Melanchthon, 4 - - Melanopsis, 557 - - Meldola, R., 670 - - Melipona, 332 - - Mellor, J. W., 134 - - Melo, 525 - - Melobesia, 412 - - Melsens, L. H. F., 282 - - Membrane-formation, 281 - - Mensbrugghe, G. van der, 212, 298, 470 - - Meserer, O., 683 - - Mesocarpus, 289 - - Mesohippus, 766 - - Metamorphosis, 82 - - Meves, F., 163, 285 - - Meyer, Arthur, 432; - G. H., 8, 682, 683 - - Micellae, 157 - - Michaelis, L., 277 - - Microchemistry, 288 - - Micrococci, 39, 245, 250 - - Micromonas, 38 - - Miliolidae, 595, 604 - - Milner, R. S., 280 - - Milton, John, 779 - - Mimicry, 671 - - Minchin, E. A., 267, 444, 449, 455 - - Minimal areas, 208, 215, 225, 293, 306, 336, 349 - - Minot, C. S., 37, 72, 722 - - Miohippus, 767 - - Mitchell, P. Chalmers, 703 - - Mitosis, 170 - - Mitra, 557, 559 - - Möbius, K., 449 - - Modiola, 562 - - Mohl, H. von, 624 - - Molar and molecular forces, 53 - - Mole-cricket, chromosomes of, 181 - - Molecular asymmetry, 416 - - Molecules, 41 - - Möller, V. von, 593 - - Monnier, A., 78, 132 - - Monticulipora, 326 - - Moore, B., 272 - - Morey, S., 264 - - Morgan, T. H., 126, 134, 138, 147 - - Morita, 699 - - Morphodynamique, 156 - - Morphologie synthétique, 420 - - Morphology, 719, etc. - - Morse, Max, 136 - - Moseley, H., 8, 518, 521, 538, 553, 555, 592 - - Moss, embryo of, 374; - gemma of, 403; - rhizoids of, 356 - - Mouillard, L. P., 27 - - Mouse, growth of, 82 - - Mucor, sporangium of, 303 - - Müllenhof, K. von, 25, 332 - - Müller, Fritz, 3; - Johannes, 459, 481 - - Mummery, J. H., 425 - - Munro, H., 323 - - Musk-ox, horns of, 615 - - Mya, 422, 561 - - Myonemes, 562 - - Naber, H. A., 511, 650 - - Nägeli, C., 124, 159, 210 - - Nassellaria, 472 - - Natica, 554, 557, 559 - - Natural selection, 4, 58, 137, 456, 586, 609, 651, 653 - - Naumann, C. F., 529, 531, 539, 550, 577, 594, 636; - J. F., 653 - - Nautilus, 355, 494, 501, 515, 518, 532, 535, 546, 552, 557, 575, 577, - 580, 592, 633; - hood of, 554; - kidney of, 425; - N. umbilicatus, 542, 547, 554 - - Nebenkern, 285 - - Neottia, pollen of, 396 - - Nereis, egg of, 342, 378, 453 - - Nerita, 522, 555 - - Neumayr, M., 608 - - Neutral zone, 674, 676, 686 - - Newton, 1, 6, 158, 643, 721 - - Nicholson, H. A., 325, 327 - - Noctiluca, 246 - - Nodoid, 218, 223 - - Nodosaria, 262, 535, 604 - - Norman, A. M., 465 - - Norris, Richard, 272 - - Nostoc, 300, 313 - - Notosuchus, 753 - - Nuclear spindle, 170; - structure, 166 - - Nummulites, 504, 552, 591 - - Nussbaum, M., 198 - - Oekotraustes, 550 - - Ogilvie-Gordon, M. M., 423 - - Oil-globules, Plateau’s, 219 - - Oithona, 742 - - Oken, L., 4, 635 - - Oliva, 554 - - Ootype, 660 - - Operculina, 594 - - Operculum of gastropods, 521 - - Oppel, A., 88 - - Optimum temperature, 110 - - Orbitolites, 605 - - Orbulina, 59, 225, 257, 587, 598, 604, 607 - - Organs, growth of, 88 - - Orthagoriscus, 751, 775, 777 - - Orthis, 561, 567 - - Orthoceras, 515, 548, 551, 556, 579, 735 - - Orthogenesis, 549 - - Orthogonal trajectories, 305, 377, 400, 640, 678 - - Orthostichies, 649 - - Orthotoluidene, 219 - - Oryx, horns of, 616 - - Osborn, H. F., 714, 727, 760 - - Oscillatoria, 300 - - Osmosis, 124, 287, etc. - - Osmunda, 396, 406 - - Ostrea, 562 - - Ostrich, 25, 707, 708 - - Ostwald, Wilhelm, 44, 131, 426; - Wolfgang, 32, 77, 82, 132, 277, 281 - - Otoliths, 425, 432 - - Ovis Ammon, 614 - - Owen, Sir R., 20, 575, 654, 669, 715 - - Ox, cannon-bone of, 730, 738; - growth of, 102 - - Oxalate, calcium, 412, 434 - - Palaeechinus, 663 - - Palm, 624 - - Pander, C. H., 55 - - Pangenesis, 44, 157 - - Papillon, Fernand, 10 - - Pappus of Alexandria, 328 - - Parabolic girder, 693, 696 - - Parahippus, 767 - - Paralomis, 744 - - Paraphyses of mosses, 351 - - Parastichies, 640, 641 - - Passiflora, pollen of, 396 - - Pasteur, L., 416 - - Patella, 561 - - Pauli, W., 211, 434 - - Pearl, Raymond, 90, 97, 654 - - Pearls, 425, 431 - - Pearson, Karl, 36, 78 - - Peas, growth of, 112 - - Pecten, 562 - - Peddie, W., 182, 272, 344, 448 - - Pellia, spore of, 302 - - Pelseneer, P., 570 - - Pendulum, 30 - - Peneroplis, 606 - - Percentage-curves, Minot’s, 72 - - Pericline, 360 - - Periploca, pollen of, 396 - - Peristome, 239 - - Permeability, magnetic, 177, 182 - - Perrin, J., 43, 46 - - Peter, Karl, 117 - - Pettigrew, J. B., 490 - - Pfeffer, W., 111, 273, 688 - - Pflüger, E., 680 - - Phagocytosis, 211 - - Phascum, 408 - - Phase of curve, 68, 81, etc. - - Phasianella, 557, 559 - - Phatnaspis, 482 - - Phillipsastraea, 327 - - Philolaus, 779 - - Pholas, 561 - - Phormosoma, 664 - - Phractaspis, 484 - - Phyllotaxis, 635 - - Phylogeny, 196, 251, 548, 716 - - Pike, F. H., 110 - - Pileopsis, 555 - - Pinacoceras, 584 - - Pithecanthropus, 772 - - Pith of rush, 335 - - Plaice, 98, 105, 117, 432, 710, 774 - - Planorbis, 539, 547, 554, 557, 559 - - Plateau, F., 30, 232; - J. A. F., 192, 212, 218, 239, 275, 297, 374, 477 - - Plato, 2, 478, 720; - Platonic bodies, 478 - - Plesiosaurs, 755 - - Pleurocarpus, 289 - - Pleuropus, 573 - - Pleurotomaria, 557 - - Plumulariidae, 747 - - Pluteus larva, 392, 415 - - Podocoryne, 342 - - Poincaré, H., 134 - - Poiseuille, J. L. M., 669 - - Polar bodies, 179; - furrow, 310, 340 - - Polarised light, 418 - - Polarity, morphological, 166, 168, 246, 295, 284 - - Pollen, 396, 399 - - Polyhalite, 433 - - Polyprion, 749, 776 - - Polyspermy, 193 - - Polytrichum, 355 - - Pomacanthus, 749 - - Popoff, M., 286 - - Potamides, 554 - - Potassium, in living cells, 288 - - Potential energy, 208, 294, 601, etc. - - Potter’s wheel, 238 - - Potts, R., 126 - - Pouchet, G., 415 - - Poulton, E. B., 670 - - Poynting, J. H., 235 - - Precocious segregation, 348 - - Preformation, 54, 159 - - Prenant, A., 163, 104, 189, 286, 289 - - Prévost, Pierre, 18 - - Pringsheim, N., 377 - - Probabilities, theory of, 61 - - Productus, 567 - - Protective colouration, 671 - - Protococcus, 59, 300, 410 - - Protoconch, 531 - - Protohippus, 767 - - Protoplasm, structure of, 172 - - Przibram, Hans, 16, 82, 107, 149, 204, 211, 418, 595; - Karl, 46 - - Psammobia, 564 - - Pseuopriacauthus, 749 - - Pteranodon, 756 - - Pteris, antheridia of, 409 - - Pteropods of, 258, 570 - - Pulvinulina, 514, 595, 600, 602 - - Pupa, 530, 549, 556 - - Pütter, A., 110, 211, 492 - - Pyrosoma, egg of, 377 - - Pythagoras, 2, 509, 651, 720, 779 - - Quadrant, bisection of, 359 - - Quekett, J. T., 423 - - Quetelet, A., 61, 78, 93 - - Quincke, G. H., 187, 191, 279, 421 - - Rabbit, skull of, 764 - - Rabl, K., 36, 310 - - Radial co-ordinates, 730 - - Radiolaria, 252, 264, 457, 467, 588, 607 - - Rainey, George, 7, 420, 431, 434 - - Rainfall and growth, 121 - - Ram, horns of, 613–624 - - Ramsden, W., 282 - - Ramulina, 255 - - Rankine, W. J. Macquorn, 697, 712 - - Ransom’s waves, 164 - - Raphides, 412, 429, 434 - - Raphidiophrys, 460, 463 - - Rasumowsky, 683 - - Rat, growth of, 106 - - Rath, O. vom, 181 - - Rauber, A., 200, 305, 310, 380, 382, 398, 677, 683 - - Ray, John, 3 - - Rayleigh, Lord, 43, 44 - - Réaumur, R. A. de, 8, 108, 329 - - Reciprocal diagrams, 697 - - Rees, R. van, 374 - - Regeneration, 138 - - Reid, E. Waymouth, 272 - - Reinecke, J. C. M., 528 - - Reinke, J., 303, 305, 355, 356 - - Reniform shape, 735 - - Reticularia, 569 - - Reticulated patterns, 258 - - Réticulum plasmatique, 468 - - Rhabdammina, 589 - - Rheophax, 263 - - Rhinoceros, 612, 760 - - Rhumbler, L., 162, 165, 260, 322, 344, 465, 466, 589, 590, 595, 599, - 608, 628 - - Rhynchonella, 561 - - Riccia, 372, 403, 405 - - Rice, J., 242, 273 - - Richardson, G. M., 416 - - Riefstahl, E., 578 - - Riemann, B., 385 - - Ripples, 33, 261, 323 - - Rivularia, 300 - - Roaf, H. C., 272 - - Robert, A., 306, 339, 348, 377 - - Roberts, C., 61 - - Robertson, T. B., 82, 132, 191, 192 - - Robinson, A., 681 - - Rörig, A., 628 - - Rose, Gustav, 421 - - Rossbach, M. J., 165 - - Rotalia, 214, 535, 602 - - Rotifera, cells of, 38 - - Roulettes, 218 - - Roux, W., 8, 55, 57, 157, 194, 378, 383, 666, 683 - - Ruled surfaces, 230, 270, 582 - - Ruskin, John, 20 - - Russow, ——, 73, 75 - - Ryder, J. A., 376 - - Sachs, J., 35, 38, 95, 108, 110, 111, 200, 360, 398, 399, 624, 635, - 640, 651, 680 - - Sachs’s rule, 297, 300, 305, 347, 376 - - Saddles, of ammonites, 583 - - Sagrina, 263 - - St Venant, Barré de, 621, 627 - - Salamander, sperm-cells of, 179 - - Salpingoeca, 248 - - Salt, crystals of, 429 - - Salvinia, 377 - - Samec, M., 434 - - Samter, M. and Heymons, 130 - - Sandberger, G., 539 - - Sapphirina, 742 - - Saville Kent, W., 246, 247, 248 - - Scalaria, 526, 547, 554, 557, 559 - - Scale, effect of, 17, 438 - - Scaphites, 550 - - Scapula, human, 769 - - Scarus, 749 - - Schacko, G., 604 - - Schaper, A. A., 83 - - Schaudinn, F., 46, 286 - - Scheerenumkehr, 149 - - Schewiakoff, W., 189, 462 - - Schimper, C. F., 502, 636 - - Schmaltz, A., 675 - - Schmankewitsch, W., 130 - - Schmidt, Johann, 85, 87, 118 - - Schönflies, A., 202 - - Schultze, F. E., 452, 454 - - Schwalbe, G., 666 - - Schwann, Theodor, 199, 380, 591 - - Schwartz, Fr., 172 - - Schwendener, S., 210, 305, 636, 678 - - Scorpaena, 749 - - Scorpioid cyme, 502 - - Scott, E. L., 110; - W. B., 768 - - Scyromathia, 744 - - Searle, H., 491 - - Sea urchins, 661; - egg of, 173; - growth of, 117, 147 - - Sebastes, 749 - - Sectio aurea, 511, 643, 649 - - Sedgwick, A., 197, 199 - - Sédillot, Charles E., 688 - - Segmentation of egg, 57, 310, 344, 382, etc.; - spiral do., 371, 453 - - Segner, J. A. von, 205 - - Selaginella, 404 - - Semi-permeable membranes, 272 - - Sepia, 575, 577 - - Septa, 577, 592 - - Serpula, 603 - - Sexual characters, 135 - - Sharpe, D., 728 - - Shearing stress, 684, 730, etc. - - Sheep, 613, 730, 738 - - Shell, formation of, 422 - - Sigaretus, 554 - - Silkworm, growth of, 83 - - Similitude, principle of, 17 - - Sims Woodhead, G., 414, 434 - - Siphonogorgia, 413 - - Skeleton, 19, 438, 675, 691, etc. - - Snow crystals, 250, 480, 611 - - Soap-bubbles, 43, 219, 299, 307, etc. - - Socrates, 8 - - Sohncke, L. A., 202 - - Solanum, 625 - - Solarium, 547, 554, 557, 559 - - Solecurtus, 564 - - Solen, 565 - - Sollas, W. J., 440, 450, 455 - - Solubility of salts, 434 - - Sorby, H. C., 412, 414, 728 - - Spallanzani, L., 138 - - Span of arms, 63, 93 - - Spangenberg, Fr., 342 - - Specific characters, 246, 380; - inductive capacity, 177; - surface, 32, 215 - - Spencer, Herbert, 18, 22 - - Spermatozoon, path of, 193 - - Sperm-cells of Crustacea, 273 - - Sphacelaria, 351 - - Sphaerechinus, 117, 147 - - Sphagnum, 402, 407 - - Sphere, 218, 225 - - Spherocrystals, 434 - - Spherulites, 422 - - Spicules, 282, 411, etc. - - Spider’s web, 231 - - Spindle, nuclear, 169, 174 - - Spinning of protoplasm, 164 - - Spiral, geodetic, 488; - logarithmic, 493, etc.; - segmentation, 371, 453 - - Spireme, 173, 180 - - Spirifer, 561, 568 - - Spirillum, 46, 253 - - Spirochaetes, 46, 230, 266 - - Spirographis, 586 - - Spirogyra, 12, 221, 227, 242, 244, 275, 287, 289 - - Spirorbis, 586, 603 - - Spirula, 528, 547, 554, 575, 577 - - Spitzka, E. A., 92 - - Splashes, 235, 236, 254, 260 - - Sponge-spicules, 436, 440 - - Spontaneous generation, 420 - - Sporangium, 406 - - Spottiswoode, W., 779 - - Spray, 236 - - Stallo, J. B., 1 - - Standard deviation, 78 - - Starch, 432 - - Starling, E. H., 135 - - Stassfurt salt, 433 - - Stegocephalus, 746 - - Stegosaurus, 706, 707, 710, 754 - - Steiner, Jacob, 654 - - Steinmann, G., 431 - - Stellate cells, 335 - - Stentor, 147 - - Stereometry, 417 - - Sternoptyx, 748 - - Stillmann, J. D. B., 695 - - St Loup, R., 82 - - Stokes, Sir G. G., 44 - - Stolc, Ant., 452 - - Stomach, muscles of, 490 - - Stomata, 393 - - Stomatella, 554 - - Strasbürger, E., 35, 283, 409 - - Straus-Dürckheim, H. E., 30 - - Stream-lines, 250, 673, 736 - - Strength of materials, 676, 679 - - Streptoplasma, 391 - - Strophomena, 567 - - Studer, T., 413 - - Stylonichia, 133 - - Succinea, 556 - - Sunflower, 494, 635, 639, 688 - - Surface energy, 32, 34, 191, 207, 278, 293, 460, 599 - - Survival of species, 251 - - Sutures of cephalopods, 583 - - Swammerdam, J., 8, 87, 380, 528, 585 - - Swezy, Olive, 268 - - Sylvester, J. J., 723 - - Symmetry, meaning of, 209 - - Synapta, egg of, 453 - - Syncytium, 200 - - Synhelia, 327 - - Szielasko, A., 654 - - Tadpole, growth of, 83, 114, 138, 153 - - Tait, P. G., 35, 43, 207, 644 - - Taonia, 355, 356 - - Tapetum, 407 - - Tapir, 741, 763 - - Taylor, W. W., 277, 282, 426, 428 - - Teeth, 424, 612, 632 - - Telescopium, 557 - - Telesius, Bernardinus, 656 - - Tellina, 562 - - Temperature coefficient, 109 - - Terebra, 529, 557, 559 - - Terebratula, 568, 574, 576 - - Teredo, 414 - - Terni, T., 35 - - Terquem, O., 329 - - Tesch, J. J., 573 - - Tetractinellida, 443, 450 - - Tetrahedral symmetry, 315, 396, 476 - - Tetrakaidecahedron, 337 - - Tetraspores, 396 - - Textularia, 604 - - Thamnastraea, 327 - - Thayer, J. E., 672 - - Thecidium, 570 - - Thecosmilia, 325 - - Théel, H., 451 - - Thienemann, F. A. L., 653 - - Thistle, capitulum of, 639 - - Thoma, R., 666 - - Thomson, James, 18, 259; - J. A., 465; - J. J., 235, 280; - Wyville, 466 - - Thurammina, 256 - - Thyroid gland, 136 - - Time-element, 51, 496, etc.; - time-energy diagram, 63 - - Tintinnus, 248 - - Tissues, forms of, 293 - - Titanotherium, 704, 762 - - Tomistoma, 753 - - Tomlinson, C., 259, 428 - - Tornier, G., 707 - - Torsion, 621, 624 - - Trachelophyllum, 249 - - Transformations, theory of, 562, 719 - - Traube, M., 287 - - Trees, growth of, 119; - height of, 19 - - Trembley, Abraham, 138, 146 - - Treutlein, P., 510 - - Trianea, hairs of, 234 - - Triangle, properties of, 508; - of forces, 295 - - Triasters, 327 - - Trichodina, 252 - - Trichomastix, 267 - - Triepel, H., 683, 684 - - Triloculina, 595 - - Triton, 554 - - Trochus, 377, 557, 560; - embryology of, 340 - - Tröndle, A., 625 - - Trophon, 526 - - Trout, growth of, 94 - - Trypanosomes, 245, 266, 269 - - Tubularia, 125, 126, 146 - - Turbinate shells, 534 - - Turbo, 518, 555 - - Turgor, 125 - - Turner, Sir W., 769 - - Turritella, 489, 524, 527, 555, 557, 559 - - Tusks, 515, 612 - - Tutton, A. E. H., 202 - - Twining plants, 624 - - Tyndall, John, 428 - - Umbilicus of shell, 547 - - Underfeeding, effect of, 106 - - Undulatory membrane, 266 - - Unduloid, 218, 222, 229, 246, 256 - - Unio, 341 - - Univalve shells, 553 - - Urechinus, 664 - - Vaginicola, 248 - - Vallisneri, Ant., 138 - - Van Iterson, G., 595 - - Van Rees, R., 374 - - Van’t Hoff, J. H., 1, 110, 433 - - Variability, 78, 103 - - Venation of wings, 385 - - Verhaeren, Emile, 778 - - Verworn, M., 198, 211, 467, 605 - - Vesque, J., 412 - - Vierordt, K., 73 - - Villi, 32 - - Vincent, J. H., 323 - - Vines, S. H., 502 - - Virchow, R., 200, 286 - - Vital phenomena, 14, 417, etc. - - Vitruvius, 740 - - Volkmann, A. W., 669 - - Voltaire, 4, 146 - - Vorticella, 237, 246, 291 - - Wager, H. W. T., 259 - - Walking, 30 - - Wallace, A. R., 5, 432, 549 - - Wallich-Martius, 77 - - Warburg, O., 161 - - Warburton, C., 233 - - Ward, H. Marshall, 133 - - Warnecke, P., 93 - - Watase, S., 378 - - Water, in growth, 125 - - Watson, F. R., 323 - - Weber, E. H., 210, 259, 669; - E. H. and W. E., 30; - Max, 91 - - Weight, curve of, 64, etc. - - Weismann, A., 158 - - Werner, A. G., 19 - - Wettstein, R. von, 728 - - Whale, affinities, 716; - size, 21; - structure, 708 - - Whipple, I. L., 123 - - Whitman, C. O., 157, 164, 193, 194, 199, 200 - - Whitworth, W. A., 506, 512 - - Wiener, A. F., 45 - - Wildeman, E. de, 307, 355 - - Willey, A., 425, 548, 555, 578 - - Williamson, W. C., 423, 609 - - Willughby, Fr., 318 - - Wilson, E. B., 150, 163, 173, 195, 199, 311, 341, 342, 398, 453 - - Winge, O., 433 - - Winter eggs, 283 - - Wissler, Clark, 79 - - Wissner, J., 636 - - Wöhler, Fr., 416, 420 - - Wolff, J., 683; - J. C. F., 3, 51, 155 - - Wood, R. W., 590 - - Woods, R. H., 666 - - Woodward, H., 578; - S. P., 554, 567 - - Worthington, A. M., 235, 254 - - Wreszneowski, A., 249 - - Wright, Chauncey, 335 - - Wright, T. Strethill, 210 - - Wyman, Jeffrey, 335 - - Yeast cell, 213, 242 - - Yield-point, 679 - - Yolk of egg, 165, 660 - - Young, Thomas, 9, 36, 669, 691 - - Zangger, H., 282 - - Zeising, A., 636, 650 - - Zeleny, C., 149 - - Zeuglodon, 716 - - Zeuthen, H. G., 511 - - Ziehen, Ch., 92 - - Zittel, K. A. von, 325, 327, 548, 584 - - Zoogloea, 282 - - Zschokke, F., 683 - - Zsigmondy, 39 - - Zuelzer, M., 165 - -CAMBRIDGE: PRINTED BY J. B. PEACE, M.A., AT THE UNIVERSITY PRESS - -{809} - - - - -SELECTION FROM THE GENERAL CATALOGUE OF BOOKS PUBLISHED BY - -THE CAMBRIDGE UNIVERSITY PRESS - - GROWTH IN LENGTH: EMBRYOLOGICAL ESSAYS. By RICHARD ASSHETON, M.A., - Sc.D., F.R.S. With 42 illustrations. Demy 8vo. 2s 6d net. - - EXPERIMENTAL ZOOLOGY. By HANS PRZIBRAM, Ph.D. Part I. Embryogeny, an - account of the laws governing the development of the animal egg as - ascertained by experiment. With 16 plates. Royal 8vo. 7s 6d net. - - ZOOLOGY. AN ELEMENTARY TEXT-BOOK. By A. E. SHIPLEY, Sc.D., F.R.S., and - E. W. 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