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The estate of 
Herbert W, Rand 
January 9, 19^4 






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arovonto: J. M. DENT AND SONS, Ltd. 


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2. A similar comparison of Diodon and Orthagoriscus . . . 

The same of various crocodiles: C. 2^o^osus, C. americanus and 
Notosuchus terrestris ......... 

The pelvic girdles of Stegosaurus and Camptosaurus 
6. The shoulder-girdles of Cryptocleidus and of Ichthyosaurus . 

The skulls of Dimorphodon and of Pteranodon .... 

The pelves of Archaeopteryx and of Apatornis compared, and a 
method illustrated whereby intermediate configurations may be 
found by interpolation (G. Heilmann) .... 

393. The same pelves, together with three of the intermediate or inter 

polated forms ......... 

394, 5. Comparison of the skulls of two extinct rhinoceroses, Hyrachyii 

and Aceratherium (Osbom) ...... 

396. Occipital views of various extinct rhinoceroses (do.) 

397—400. Comparison with each other, and with the skull of Hyrachyus, of 

the skulls of Titanotherium, tapir, horse and rabbit 
401, 2. Coordinate diagrams of the skulls of Eohippus and of Equus, with 

various actual and hypothetical intermediate types (Heilmann) 

403. A comparison of various human scapulae (Dwight) 

404. A human skull, inscribed in Cartesian coordinates 

405. The same coordinates on a new projection, adapted to the skull of 

the chimpanzee . . . . . . 

406. Chimpanzee's skull, inscribed in the network of Fig. 405 

407. 8. Corresponding diagrams of a baboon's skull, and of a dog's 












"Cum formarum naturalium et corporalium esse non consistat iiisi 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." x'^quinas, 
De Pat. Q. ui, 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 
everythmg depends upon certain forces, in wtue 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 ]VIr W. Spottiswoode, 
Brit. Assoc. Presidential Address, 1878.) 

"When Science shall have subjected all natmal 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 futiu-e. 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. 






l^l LIBRARY 1^ 



Cambridge : 

at the University Press 


■'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 ivith labour do we find the things that are before us." 
Stephen Hales, Vegetable Staticks (1727), p. 318, 1738. 


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 httle 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 
Eadiolaria, from the Reports of the Challenger Expedition. 


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 
MacAhster and Mr Edwin Arnold for certain figures (335 — 7), 
and to Sir Edward Schafer 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 Pro- 
fessor Claxton Fidler and to Messrs Grifl&n, 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 Stephane 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. Darhng and to Messrs E. 
and S. Spon for those (fig. 85) from Mr Darhng'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 Experi- 
mental Embryology; and to the Cambridge University Press for 
a number of figures from Professor Henry Woods's Invertebrate 
Palaeontology , for oife (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. 


University College, Dundee. 
December, 1916. 

1 1 I / !" 



I. Introductory \ 

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 




1. Nerve-cells, from larger and smaller animals (Minot, after Irving 

Hardesty) .......... 

2. Relative magnitudes of some minute organisms (Zsigmondy) . 

3. Curves of growth in man (Quetelet and Bowditch) 

4. 5. Mean annual increments of stature and weight in man (do.) 
6. The ratio, throughout life, of female weight to male (do.) 

7—9. Curves of growi;h of child, before and after birth (His and Riissow 

10. Curve of growth of bamboo (Ostwald, after Kraus) 

11. Coefficients of variability in human stature (Boas and Wisslerl 

12. Growth in weight of mouse (Wolfgang Ostwald) 

13. /)•-). cf silkworm fLuciani and Lo Monaco) 
]i. Do. of tadpole (Ostwald, after Schapcr) . 

15. Larval eels, or Leptocephali, aiid young elver (Joh. 

16. Growth in length of Spirogyra (Hofmeister) 

17. Pulsations of growth in Crocus (Bose) 

18. Relative growi;h of brain, heart and body of man 

19. Ratio of stature to span of arms (do.) 

20. Rates of growth near the tip of a bean-root (Sach 

21. 22. The weight-length ratio of the plaice, and its 

changes ....... 

23. Variability of tail-forceps in earwigs (Bateson) 

24. Variability of body-length in plaice 

25. Rate of growth in plants in relation to temperature (Sachs) 

26. Do. in maize, observed (Koppen), and calculated curves . 

27. Do. in roots of peas (Miss I. Leitch) .... 

28. 29. 






66, 69 




annual periodic 


99, 100 


Rate of growth of frog in relation to temperature (Jenkinson 
after O. Hertwig), and calculated curves of do. 

30. Seasonal fluctuation of rate of growth in man (Daffner) 

31. Do. in the rate of growth of trees (C. E. Hall) 

32. Long-period fluctuation in the rate of growth of Arizona trees 

(A. E. Douglass) 122 

33. 34. The varying form of brine-shrimps (Arte7nia), in relation to 

salinity (Abonyi) ......... 128,9 

35-39. Curves of regenerative growth in tadpoles' tails (M. L. Durbin) 140-145 

40. Relation between amoimt of tail removed, amount restored, and 

time required foi 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 

Flemmmg) 180 



of a spher 


53. Annular chromosomes of mole-cricket (Wilson, after vom Rath) . 

54-56. Diagrams illustrating a hypothetic field of force in caryokinesis 

(Prof. W. Peddie) 

57. An artificial figure of caryokinesis (Leduc) 

58. A segmented egg of Cerebratidus (Prenant, after Coe) 

59. Diagram of a field of force with two like poles 

60. A budding yeast-cell ...... 

61. The roulettes of the conic sections .... 

62. Mode of development of an unduloid from a cylindrical tube 
63-65. Cylindrical, unduloid, nodoid and catenoid oil-globules (Plateau) 

Diagram of the nodoid, or elastic curve 

Diagram of a cylinder capped by the corresponding portion 

A liquid cylinder breaking up into spheres 

The same phenomenon in a protoplasmic cell of Trianea 

Some phases of a splash (A. M. Worthington) 

A breaking wave (do.) ...... 

The calycles of some campanularian zoophytes 

A flagellate monad, Disiigma proteiis (Saville Kent) 

Nocliluca miliaris, diagrammatic .... 

Various species of Vorticella (Saville Kent and others) 

Various species of Salpingoeca (do.) 

Species of Tintinnus, Dinobryon and Codonella (do.) 

The tube or cup of Vaginicola ... 

The same of Folliculina ...... 

Trachelophyllum (Wveszniowski) .... 

Trichodina j)e,diculus . . ... 

Dinenympha gracilis (Leidy) ..... 

A "collar-cell" of Codosiga ..... 

Various species of Lagtna (Brady) . 

Hanging drops, to illustrate the unduloid form (C. R. Darling) 

Diagram of a fluted cylinder .... 

Nodosaria scalar i.? (Brady) .... 

Fluted and pleated gonangia of certain Campanularians (AUman) 
Various species of Nodosaria, Sagrina and Rheophax (Brady) . 
Trypanosoma tineae and Spirochaeta anodonfae, to shew undulatin 

membranes (Minchin and Fantham) .... 

Some species of Tricliomastix and Trichowonas (Kofoid) 
Herpetomonas assuming the undulatory membrane of a Trypanosome 

(D. L. Mackinnon) 

Diagram of a human blood-corpuscle 

Sperm-cells of decapod crustacea, Inachus and Galathea (Koltzoff) 

The same, in saline solutions of varying density (do.) 

A sperm-cell of Dromia (do.) .....•• 

Chondriosomes in cells of kidney and pancreas (Barratt and Mathews 

Adsorptive concentration of potassium salts in various plant-cell 

(Macallum) ......••• 

99-101. Equilibrium of surface-tension in a floating drop 

102. Plateau's "bourrelet" in plant-cells; diagrammatic (Berthold) 

103. Parenchyma of maize, shewing the same phenomenon . 







222, 3 









104, 5. Diagrams of the partition -wall between two soap-biibbles 

106. Diagram of a partition in a conical cell 

107. Chains of cells in Nostoc, Anabaena and other low algae 

108. Diagram of a symmetrically divided soap-bubble ... 

109. Arrangement of partitions in dividing spores of Pellia (Campbell) 

110. Cells of Dictyota (Reinke) 

111. 2. Terminal and other cells of Chara, and young antheridium of do 

113. Diagram of cell-walls and partitions under various conditions of 

tension ......... 

114, 5. The partition-sui'faces of three interconnected bubbles 

116. Diagram of four interconnected cells or bubbles 

117. Varioiis configurations of four cells in a frog's egg (Rauber) 

1 18. Another diagram of two conjoined soap-bubbles 

119. A froth of bubbles, shewing its outer or "epidermal" layer 

120. A tetrahedron, or tetrahedral system, shewing its centre of symmetry 

121. A group of hexagonal cells (Bonanni) .... 

122. 3. Artificial cellular tissues (Leduc) ..... 

124. Epidermis of Oirardia (Goebel) ..... 

125. Soap-fioth, and the same under compression (Rhumbler) 

126. Epidermal cells of Elodea canadensis (Berthnld) 

127. Lithostrotion Martini (Nicholson) ..... 

128. Cyathopkylbtm hexagomim (Nicholson, after Zittel) . 

129. Arachnophyllum pentagonum (Nicholson) .... 

130. Hdiolites (Woods) 

131. Confluent septa in Thamnastraea and Comoseris (Nicholson, after 

Zittel) ... 

132. Geometrical construction of a bee's cell .... 

133. Stellate cells in the pith of a rush ; diagrammatic 

134. Diagram of soap-films formed in a cubical wire skeleton (Plateau) 

135. Polar furrows in systems of four soap-bubbles (Robert) 
136-8. Diagrams illustrating the division of a cube by partitions of minimal 

area ........... 

139. Cells from hairs of Sphacelaria (Berthold) .... 

140. The bisection of an isosceles triangle by minimal partitions . \ 

141. The similar partitioning of spheroidal and conical cells . 

142. S-shaped partitions from cells of algae and mosses (Reinke and others) 

143. Diagrammatic explanation of the S-shaped partitions 

144. Development of Erythrotrichia (Berthold) 

145. Periclinal, anticlinal and radial partitioning of a quadrant 

146. Construction for the minimal partitioning of a quadrant 

147. Another diagram of anticlinal and perichnal partitions . 

148. Mode of segmentation of an artificially flattened frog's egg 

(Roux) ......... 

149. The bisection, by minimal partitions, of a prism of small angle 

150. Comparative diagram of the various modes of bisection of a prismatic 

sector ......... 

151. Diagram of the further growth of the two halves of a quadrantal cell 

152. Diagram of the origin of an epidermic layer of cells 

153. A discoidal cell dividing into octants .... 


299, 300 









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 coll 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 Pyrosom.a (Korotneff), and of Echinus (l)riesch) 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. Develoj^ment of the sporangium of Osimmda (Bower) . . ■ 406 

191. Embryos of Phascum and of Adiantum (Kienitz-GerlolT) . . 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 My a (Carpenter) 422 

199. Concretions, or spicules, artificially deposited in cartilage (Hartmg) 423 

200. Further illustrations of alcyonarian spicules: Eunicea (Studer) . 424 
201-3. Associated, aggregated and composite calcospherites (Harting) . 425,6 

204. Harting's "conostats" "^27 

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 

Theel) 442 

An amphidisc of Hyalonema 442 

Spicules of calcareous, tetractinellid and hexactinellid sponges, and 

of various holothurians (Haeckel, Schultze, Sollas and Theel) 445-452 
Diagram of a solid body confined by surface-energy to a liquid 

boundary-film 460 

Astrorhiza limicola and arenaria (Brady) ..... 464 

A nuclear ^'reticulmn plasmatigue'^ (Carnoy) ..... 468 

A spherical radiolarian, Aulonia hexagona (Haeckel) '. . . 469 

Actinomma arcadophorum (do.) ....... 469 

Ethmosphaera conosipJionia (do.) ....... 470 

Portions of shells of Cenosphaera favosa and vesparia (do.) . . 470 
Aulasfrum triceros (do.) . . . . . . . , .471 

Part of the skeleton of CaniiorhapJ'is (do.) 472 

A Nassellarian skeleton, Callimitra. carolotac (do.) .... 472 

228, 9. Portions of Dictyocha stapedia (do.) 474 

230. Diagram to illustrate the conformation of Callimitra . . . 476 

Skeletons of various radiolarians (Haeckel) 479 

Diagrammatic structure of the skeleton of Doratas2ns (do.) . . 481 

4. Phatnaspis cristata (Haeckel), and a diagram of the same . . 483 

Phractasjris prototypus (Haeckel) ....... 484 

Annular and spiral thickenings in the walls of plant-cells . . 488 

A radiograph of the shell of Nautilus (Green and Gardiner) . . 494 

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 Ncrita (Moseley) .... 521, 2 

265. A section of the shell of Melo ethiopicus ..... 525 

266. SheUs of Harpa and Dolium, to illustrate generating curves and 

generating spirals ......... 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 varioiis 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 Macroscaphitps, 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 halioioides (do.) . . . 555 

285. 6. Sections of the shells of Terebra maculata and Trochus nilolicvs 550. (iO 
287-9. Diagrams illustrating the lines of growth on a lamellibranch shell 563-5 

290. CaprineUa adversa (Woodward) ....... 507 

291. Section of the shell of Productus (Woods) ..... 567 

292. The "skeletal loop" of Terehratula (do.) 508 

293. 4. The spiral arms of Spirifer and of Atrypa (do.) .... 5(i9 
295-7. Shells of Cleodora, Hyalam 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 570 

304. A section of Nautikis, 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 Soinerhyi, to shew septal outlines (Zittel, after Stcinmann 

and Doderlein) 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 Moller) ...... 591 

310. Cornuspira foliacea and Operculina complanata (Brady) . . . 594 

311. Miliolina pulchella and liiuiaeana (Brady) ..... 596 

312. 3. Cyclammiria cancellata (do.), and diagrammatic figure of the same 596, 7 

314. Orhulina universa (Brady) ........ 508 

315. Cristellaria reniformis (do.) ........ 600 

316. Discorhina hertheloti (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 (Solater) 616 

320. Head of Ouis 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 (Lonnberg) 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 appear- 

ances 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 

(Schafer, after Robinson) ....... 681 



335 Comparative diagrams of a crane-head and the liead 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 05 cakis (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 

eUipse 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 Diirer) 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 carapanularian zoophytes, inscribed in corre- 
sponding 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 rect- 
angular and coaxial systems ....... 749 

377-80. A comparison of the fishes Polyprion, Pseudopriacanthus, Scorpaena 

and Antigonia .......... 750 



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 equihbrium 
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*. 

We need not wait for the full reahsation 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 mathe- 
matical 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 

* 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 ; Hober, 
Biol Cetdmlbl. xix, p. 284, 1890, etc. Cf. also Jellett, Rep. Brit. Ass. 1874, p. 1. 

T. n. 1 


brings into relation wdth 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 he 
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 beheve 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 ^uch 
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, whieh, 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 


dead and gone. The facts of embryology become for him, as 
Wolff, von Baer and Fritz Miiller 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 intelhgence. 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 

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 philo- 
sopher'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* and Maupertuis, to whom 
Leibniz t had passed it on. Half overshadowed by the me- 
chanical concept, it runs through Claude Bernard's LsQons siir les 

* " 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 
onines mundi effectus ex causis finalibus, ope methodi maximorum et minimorum, 
aeque feliciter determinari queant atque ex ipsis causis efficientibus." Methodtis 
inveniendi, etc. 1744 (cit. Mach, Science of Mechanics, 1902, p. 455). 

t Cf. 0pp. (ed. Erdmann), p. 106, "Bien loin d'exclure les causes finales..., 
c'est de la qu'il faut tout deduire en Physique." 



phenomenes de la Vie*, and abides in much of modern physio- 
logy f . Inherited from Hegel, it dominated Oken's Natur'philosophie 
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 J. 

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§" : — which is no more, and 
no less, than to say, with Aristotle, that the organism is the reXo^, 
or final cause, of its own processes of generation and development. 
It is writ large in that Entelechy|| 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^" 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 reXo'i., 

* Cf. p. 162. "La force vitale dirige des phenomenes qu'elle ne produit pAs: 
les agents physiques produisent des phenomenes qu'ils ne dirigent pas." 

t It is now and then conceded with reluctance. Thus Enriques, a learned 
and philosophic naturalist, writing "deUa economia di sostanza nelle osse cave" 
{Arch.f. Entw. Mech. xx, 1906), says "una certa impronta di teleologismo qua e la 
e rimasta, mio malgrado, in questo scritto." 

% Cf. Cleland, On Terminal Forms of Life, J. Anat. and Phys. xvni, 1884. 

§ Conklin, Embryology of Crepidula, Journ. of Marphol. xm, p. 203, 1897; 
Lillie, F. R., Adaptation in Cleavage, Woods Holl Biol. Lectures, pp. 43-67, 1899. 

II 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. 
ElUs points out that ho 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 ivTeXexe^o., or 
function. He defined it as dvva/jLis qunedam ciens actiones — a description all but 
identical with that of Claude Bernard's ""force vitale.''' 

•1 Ray Lankester, Encijd. Brit. (9th ed.), art. "Zoology," p. 806, 1889. 


as men like Butler and Janet have been pronipt 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 iprocess of mechanism*. The apparent manifestations of "pur- 
pose" or adaptation become part of a mechanical philosophy, 
according to which "chaque chose finit toujours par s'accommoder 
a son miheuf." 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 {." 

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 com- 
plexity 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 §." 

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 

* 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. 

t Janet, Les Causes Finales, 1876, p. 350. 

t The phrase is Leibniz's, in his Theodicee. 

§ Cf. (int. al.) Bosanquet, The Meaning of Teleology, Proc. Brit. Acad. 
1905-6, pp. 235-245. Cf. also Leibniz (Discours de Metaphysiqtte; Lettres inedites, 
€d. de Careil, 1857, p. 354; cit. Janet, p. 643), "L'un et Tautre est bon, I'un et 
I'autre pent etre utile... et les auteurs qui suivent ces routes differentes ne devraient 
point se maltraiter: et seq." 


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*" 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 

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 ante- 
cedents, or concomitants, of our phenomena, without which the 
phenomenon does not occur, — with causes, in short, which, aliae 
ex aliis aftae et necessitate nexae, are no more, and no less, than 
conditions sine qua 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 meta- 
physician 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 " simili- 
tude 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 naturahst 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 la des manieres de s'exprimer, 

* 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. 


et si elles sont interdites il faut renoncer a parler de ces 

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 simiUtudes, has been followed 
by few; and the contrasts are apt to loom too large, great as 
they may be. M. Dunan, discussing ^the "Probleme de la Vie*" 
in an essay which M. Bergson greatly commends, declares: "Les 
lois physico-chimiques sont aveugles et brutales ; la oil elles 
regnent seules, au lieu d'un ordre et d'un concert, il ne pent y 
avoir qu'incoherence 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 Kttle ripples on the shore, the 
sweeping curve of the sandy bay between its headlands, the 
outUne 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 

Nor is it otherwise with the material forms of hving 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 f. 

* Revue Philosophique. xxxiii, 1892. 

t This general prineiple 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 withm the reach of physical 
agencies, especially as hi 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. Infm rit.. p. 9. 


They are no exception to the rule that 0eo<? aei yew/jLerpel. 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 physio- 
logy, 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*, of 
Empedocles and his disciples in the days before Aristotle, lay 
just by that highwayside. It was Gahleo's and BorelU's way. 
It was little trodden for long afterwards, but once in a while 
Swammerdam and Reaumur looked that way. And of latCT 
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 httle. 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 
* Whereby he incurred the reproach of Socrates, in the Phaedo. 


of the body, as of all that is of the earth earthy, physical science 
is, in my humble opinion, our only teacher and guide*. 

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 "stereo- 
metry," 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 mole- 
cules 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 Auerbachf, or LarmorJ, or Joly§, 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" 

* In a famous lecture (Conservation of Forces applied to Organic Nature, 
Proc. Roy. lustit., 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 anatysis and deduction. Cf. also Dr T. Young's Croonian 
Lecture On the Heart and Arteries. Phil. Trans. 1809, p. 1: Coll. Works, i, .511. 

f Ektropismus, oder die physikalische Theorie des Lebens, Leipzig, 1910. 

J 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. 

§ Joly, The Abundance of Life, Proc. Roy. Dublin Soc. vii, 1890; and in 
/Scientific Essays, etc. 1915, p. 60 et seq. 


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*." 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 ahen 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 hfe, it is in no spirit of arrogant assertive- 
ness 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 reahse 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 

* Papillon, Histoire de la philosophie moderne, i, p. 300. 


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 

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 ahke 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 Hquid (or of a gas) of the forces that 
^re for the moment acting on it to restrain or balance its own 
inherent mobihty. In an organism, great or small, it is not 
merely the nature of the ynotions of the living substance that we 
must interpret in terms of force (according to kinetics), but also 


the conformation of the organism itself, whose permanence or 
equihbrium 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 

This portion of matter, then, is kept together by the inter- 
molecular 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 sohd substance on which it may be creeping. 
Our Amoeba tends, in the next place, to be deformed by any 
pressure from outside, even though sUght, which may be apphed 
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. 

* With the special and important properties of colloidal matter we are, for 
the time being, not concerned. 


Like other fluid bodies, its surface, whatsoever other substance, 
gas, Hquid or sohd, 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 afEect 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 sohds 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 pro- 
cesses 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. 


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 

In the phenomena of cell-division, in the attractions or repul- 
sions of the parts of the dividing nucleus and in the " caryokinetic " 
figures that appear in connection with it, we seem to see in opera- 
tion 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 wdth 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 trans- 
mission. Matter as such produces nothing, changes nothing, does 
nothing ; and however convenient it may afterwards be to abbre- 
viate our nomenclature and our descriptions, we must most 
carefully reahse in the outset that the spermatozoon, the nucleus, 


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 con- 
ventional symbolism, of modern physical science) of the old 
saying of the philosopher: apxv l^P V 4>^o-i^ /uLaWov r?}? vXrjq. 



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*. 

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 477/^, 
and its volume to 47rf^; from which it follows that the ratio of 


volume to surface, or V/S, is ^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 dimen- 
sion, we may write the general equations in the form 

SocL^ F oc L\ 

or S^k.L^ and V = k'.L^; 

and (S^ X L. 

From these elementary principles a great number of conse- 
quences follow, all more or less interesting, and some of them of 
great importance. In the first place, though growth in length (let 

* Cf. Hans Przibrain, Anwendung elementarer Mathematik auf Biologische 
Probleme (in Roux's Vortrdge, Heft ra), Leipzig, 1908, p. 10. 


us say) and growth in volume (which is usually tantamount to 
mass or weight) are parts of one and the same process or pheno- 
menon, the one attracts our attention by its increase, very much 
more than the other. For instance a fish, in doubhng its length, 
multipHes 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^, 
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 hkely; 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 shght and otherwise imperceptible changes in the form of 
a fish — shght relative difiierences between length, breadth and 
depth, for instance, — would need to be very dehcate 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 Lilhput and Brobdingnag are all ahke, 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. 


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 other- 
wise in proportion to the area of surface; and others, such as 
gravity, act on all particles, internal and external ahke, 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 *. It was elementary engineering experience 
such as this that led Herbert Spencer f to apply the principle of 
simihtude to biology. 

The same principle had been admirably applied, in a few clear 
instances, by LesageJ, a celebrated eighteenth century physician 
of Geneva, in an unfinished and unpublished work§. Lesage 
argued, for mstance, 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 quad- 
rupeds a large head must be supported on a neck which is either 

* The subject is treated from an engineering j^oint of view by Prof. James 
Thomson, Comparisons of Similar Structures as to Elasticity, Strength, and 
StabUity, Trans. Inst. Engineers, Scotland, 1876 {Collected Papers, 1912, pp. 361- 
372), and by Prof. A. Barr, ibid. 1899; see also Rayleigh, Nattire, April 22, 1915. 

t Cf. Spencer, The Form of the Earth, etc., Phil. Mag. xxx, pp. 194-6, 
1847; also Principles of Biology, pt. n, ch. i, 1864 (p. 123, etc.). 

% George Louis Lesage (1724-1803), well known as the author of one of the few 
attempts to explam gravitation. (Cf. Leray, Constitution de la Matiere, 1869; 
Kelvin, Proc. R. S. E. vn, p. 577, 1872, etc. ; Clerk Maxwell, Phil. Trans, vol. 157, 
p. 50, 1867; art. "Atom," Emycl. Brit. 1875, p. 46.) 

§ Cf. Pierre Prevost, Noticts de la vie et des ecrits de Lesage, 1805; quoted by 
Janet, Causes Finales, app. in. 


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 siniihtude ; and he did so with the utmost possible 
clearness, and with a great wealth of illustration, drawn from 
structures hving and dead*. 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 sufiiced in the case of a smaller structure f. 
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 famihar to us in nature and in art, and practical 
appUcations, undreamed of by Gahleo, meet us at every turn in 
this modern age of steel. 

Again, as Gahleo 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 afiects, 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 J. 

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 §. In such an investigation we have to make 

* Discorsi e Dimostrazioni majematiche, intorno a due nuove scienze. 
attenenti alia Mecanica, ed ai Movimenti Local! : appresso gli Elzevirii, mdcxxxviii. 
Opere, ed. Favaro, vni, p. 169 seq. Transl. by Henry Crew and A. de Salvio, 
1914, p. 130, etc. See Nature, June 17, 191.5. 

t 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. 

X 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. Pojoiting and Thomson's Properties of Matter, 1907, p 99. 

§ 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 withm which stable 



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*) 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 flag- 
staff measuring 22f 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|. In short, as Goethe says in Wahrheit und Dichtung, "Es 
ist dafiir gesorgt dass die Baume nicht in den Himmel wachsen." 
But Eiffel's great tree of steel (1000 feet high) is built to a 
very differeiit 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 inefiiciency, 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 $: "In pro- 
portion to the bulk of a species is the difficulty of the contest 
which, as a living organised whole, the individual of such species 

equilibrium is possible in a vertical position. The kitten's tail, on the other hand, 
stands up spiky and straight. 

* Modern Painters. 

t 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). 

X Trans. Zool. Soc. iv, 1850, p. 27. 


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*. Indeed, 

* 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 


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 pro- 
portional to the mass of its muscles, or the cube of its linear 
dimensions : and again this work being wholly done in producing 
a velocity (F) against a resistance {R) which increases as the 
square of the said linear dimensions ; we have at once 

W = l\ 

and also W = RV'- = l^VK 

Therefore P = IW\ and V = y/l. 

This is what is known as Fronde's Law of the correspondence of 

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*. 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 

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. 

* 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. 


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 steamshij) com- 
parison. 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 * ; 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 

* Let L be the length, S the (wetted) surface, T the tonnage, D the displacement 
(oi" volume) of a ship; and let it cross the Atlantic at a speed V. Then, in com- 
paring two ships, similarly constructed but of different magnitudes, we know that 
L = V", S = L~ = V*, D = T=L^ = V«; also R (resistance) = S . F- = F« ; H (horse- 
power) = R .V ^V ; and the coal (C) necessary for the voyage = HjV = F*. 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. 


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 

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 simihtude. 

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 there- 
fore proportional to tlie 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 ffight, has to work 
in order to drive itself forward, is the rate at which it commmiicates 
energy to the air; and this is proportional to mV^, 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 2| of the hnear dimensions ; and the speed at which it 
is displaced is proportional to the bird's speed, and therefore to 
the square root of the hnear 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 2| of 
the linear dimensions, as above, and to the first power thereof : 


that is to say, it increases in proportion to the power 3| 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 ; F its velocity, 
proportional to that of the bird; M its momentum; I a Hnear 
dimension of the bird ; w its weight ; W the work done in moving 
itself forward.) 

M = w = l^. 

But M^mV, and 7n = PV. 

Therefore M ^ l^ V^ 

and ?2 y2 _ jz^ 

or 7 = V?- 

But, again, W = mV^ ^PV x V^ 

= PxVl^l 

The work requiring to be done, then, varies as the power 3| 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 hnear dimensions*. 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 hnear dimensions 
of the bird, the difficulty of flight is increased in the ratio of 
2^ : 2^-, 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^* and 25^, or 
between 5'^: 5^; in other words, flight is just five times more 
difficult for the larger than for the smaller birdf. 

The above investigation includes, besides the final result, a 
number of others, explicit or implied, which are of not less im- 
portance. Of these the simplest and also the most important is 

* This is the result ari'ived at by Helmholtz, Ueber ein Theorem geometrisch 
ahnliche Bewegungen fliissiger Korper 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. Miillenhof, Die Grosse 
der Flugflachen, etc., Pfluger''s Archiv, xxxv, p. 407, xxxvi, p. 548, 1885. 

t Cf. also Chabrier, Vol des Insectes, 3Iem. 3Ius. Hist. Nat. Paris, vi-viii, 


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 -.wi-.L^: P :: 8 : 1. 

Therefore L:l::2:l. 

But 72 : ^2 :: Z : I. 

Therefore V : v.: V2 : 1 = 1-414 : 1. 

That is to say, the larger machine must be capable of a speed 
equal to 1-414 x 40, or about 56| miles per hour. 

It is highly probable, as Lanchester* 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 f 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 

* Aerial Flight, vol. 11 (Aerodonetics), 1908, p. 150. 
f By Lanchester, op. cit. p. 131. 


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 x 14, or 70 miles an hour. 

The same principle of necessary S'peed, 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* 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 a fortiori the still smaller 
insects, are capable of "stationary flight," a very sHght 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 GaHleo 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 
* Cf. U empire de Vair ; oniitkologie appliquee a Vaviation. 1881. 


in one of his chapters he deals with the proposition, "Est im- 
possibile, ut homines propriis viribus artificiose volare possint*." 

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 facihtated. We know that in the carboniferous 
epoch there hved 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 Harle suggests f, 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 

Very similar problems, involving in various ways the principle 
of dynamical simihtude, 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 Gahleo 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 J." 

* De Motu Animalium, I, prop, cciv, ed. 1685, p. 243. 

f Harle, On Atmospheric Pressure in past Geological Ages, Bull. Geol. Soc. 
Fr. XI, pp; 118-121; or Cosmos, p. 30, July 8, 1911. 

X 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 


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 *, 
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 f. 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, 

hundred feet in diameter"; and that if a certain noisy Brazilian insect were as 
big as a man, its voice would be heard aU 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. 

* 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. 

t Prop, clxxvii. Animaha minora et minus ponderosa majores saltus efficiunt 
respectu sui corporis, si caetera fuerint paria. 


and tends to leave a certain balance of advantage, in regard to 
leaping power, on the side of the larger animal*. 

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 pecuhar 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 "f. 

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 s'wing at its normal 'pendnlum-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 J. 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 indi- 
viduals, 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 a? ajh ; but the time of swing will vary as the square 

* See also (int. al.), John Bernoulli, de Motu Musculorum, Basil., 1694; 
Chabrj', Mecanisme du Saut, J, de VAnat. et de la Physiol, xix, 1883; Sur la 
longueur des membres des animaux sauteurs, ibid, xxi, p. 356, 1885; Le Hello, 
De Taction des organes locomoteurs, etc., ibid, xxix, p. 65-93, 1893, etc. 

t Recherches sur la force absolue des muscles des Invertebres, Bull. Acad. B. 
de Belgique (3), vi, \ti, 1883-84: see also ibid. (2), xx, 1865, xxii, 1866; A^ut. 
Mag. ]<:. H. x\ti, p. 139, 1866, xix, p. 95, 1867. The subject was also well treated 
by Straus-Diirckheim, in his Considerations generales sur Vanatomie comparee des 
animaux articules, 1828. 

X The fact that the limb tends to swing in pendulum-time was first observed 
by the brothers Weber [Mechanik der menschl. Gehicerkzeuxje, Gottingen, 1836). 
Some later writers have criticised the statement (e.g. Fischer, Die Kiuematik des 
Beinschwingens etc., Abh. math. phys. Kl. k. Sachs. Ges. xxv-xxvm, 1899-1903), 
but for all that, with proper qualifications, it remains substantially true. 


root of the peiidulum-length, or -^/ajy/'b. Therefore the velocity, 

, . , . 1 , amplitude .,, , , 

which IS measured by . , will also vary as the square- 
time ■ ^ 

roots of the length of leg : that is to say, the average velocities of 

A and B are in the ratio of y'a : y/h. 

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* 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. Dehsle's fly. We should walk 
at just about the fly's pace if our stature were 1/(14-7)^, or 1/216 
of our present height, — say 72/216 inches, or one-third of an inch 

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 "j". 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 

* Quoted in Mr John Bishop's interesting article in Todd's Cyclopaedia, ni, 
p. 443. 

t There is probably also another factor involved here : for in bending, and there- 
fore 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. 


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 

From all the foregoing discussion we learn that, as Crookes 
once upon a time remarked*, 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 "vilh" 
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 f. 

* Proc. Psychical Soc. xn, pp. 338-355, 1897. 

f For various calculations of the increase of surface due to histological and 
anatomical subdivision, see E. Babak, Ueber die Oberflachenentwickelung bei 
Organismen, Biol. Centralbl. xxs, 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 SjV. 
In a cube, V=P, and S = 61^; therefore SjV — Q/l. Therefore if the side I measure 


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 rain- 
drop and the spherical surface of the ocean (though both happen 
to be aUke in mathematical form) are two totally different pheno- 
mena, 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 httle 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. 

6 cm., the ratio SjV = 1, and such a cube may be taken as our standard, or unit 
of specific surface. A human blood-corpuscle has, accordingly, a si^ecific 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. 

T. G. 3 


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 pro- 
duced, 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 comple- 
mentary 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 prevaihng 
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 mean- 
while, in order to deal with other matters more or less directly 
concerned with the magnitude of the cell. 

The hving 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 proto- 
plasmic alveoh and other visible (as well as in\asible) hetero- 
geneities 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 httle we may know of the distri- 
bution of the total energy of the system, it is at least plain that 


the conditions which favour equihbrium 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. 
Li short, however it may be brought aboi^t, 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 shght 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*. 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 f, 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 J pointed out, in 1895, that there is a tendency for each 
nucleus to be only able to gather around itself a certain definite 
amount of protoplasm. Driesch§, a 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 

* Though the entire egg is not increasing in mass, this is not to say that its 
living protoplasm is not increasing all the whUe at the expense of tbe reserve 

t Of. Tait, Proc. R.S.E. v, 1866, and vi, 1868. 

X Physiolog. Notizen (9), p. 425, 1895. Cf. Strasbiirger, Ueber die Wirkungs- 
sphare der Kerne und die Zellgrosse, Histolog. Beitr. (5), pp. 95-129, 1893; 
J. J. Gerassimow, Ueber die Grosse des Zellkernes, Beih. Bot. Centralbl. xvm, 
1905 ; also G. Levi and T. Terni, Le variazioni dell' indice plasmatico-nucleare 
durante 1' intercinesi. Arch. Ital. di Anat. x, p. 545, 1911. 

§ Arch. f. Entw. Mech. iv, 1898, pp. 75, 247. 



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 Crefidula (a httle American 
" shpper-hmpet," now much at home on our own oyster-beds), 
Conkhn* 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 f. 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 ganghon-cells, both 
of the lower and of the higher animalsf . 

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 Unear dimensions, 
and therefore about eight times greater in volume, or mass. But 
making some allow^ance for difference of shape, the Unear dimen- 
sions 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 

* Conklin, E. G., Cell-size and nuclear-size, J. Exp. Zool. xii. pp. 1-98, 1912. 

t Thus the fibres of the crystalline lens are of the same size in large and small 
dogs ; Eabl, Z. f. w. Z. Lxvn, 1899. Cf. {i7it. al.) Pearson, On the Size of the Blood- 
corpuscles in Rana, Biometrilca, 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 coipuscles in the sluggish Amphibia are much 
the largest known to us, whQe 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 modem 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. 

% Cf. P. Enriques, La forma come funzione della grandezza: Ricerche sui 
gangli nervosi degli Invertebrati, Arch. f. Entw. Mech. xxv, p. 655, 1907-8. 


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 *. The bearing of such simple facts as this 
upon the cell-theory in general is not to be disregarded ; and the 



Fig. 1. Motor ganglion-cells, from the cervical spinal cord. 
(From Minot, after Irving Hardesty.) 

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 

* 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 diiJerenee ; 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. Iktl. di Anat. e di 
Emhryolog. V, p. 291, 1906. 


the body is more than the svm. 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 fertihsation, each new 
nucleus has the same size as its parent-nucleus*. 

In all these cases, then, there are reasons, partly no doubt 
physiological, but in very large part purely physical, which set 
hmits 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 -M fi, or •00034 mm., by •00025 mm.; smaller even than 
this we have a pathogenic micrococcus of the rabbit, M. pro- 
grediens, Schrdter, the diameter of which is said to be only -00015 
mm. or -IS/x, or TS x 10"^ cm., — about equal to the thickness of 

* Boveri, Zellen-studien, V. Ueber die Ahhdngirjkeit der Kerncjrosse und Zellen- 
zahl der Seeigellarven von der Chromosomenzakl der Ausgangszellert . Jena, 1905. 




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*. Accordingly, 



Fig. 2. Relative magnitudes of: A, human blood-corpuscle (7-5^1 in diameter); 
B, Bacillus anthracis {4: — 15 fx x I /j.) ; C. various Micrococci (diam. 0-5 — l/n, 
rarely 2^i); D, Micromonas progrediens, Schroter (diam. 0-15/x). 

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. 

To illustrate some of these small magnitudes I have adapted 
the preceding diagram from one given by Zsigmondyf . Upon the 

* 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. 1916. 

■(• Zur Erkenntniss cler Kolloide, 1905, p. 122; where there wiU be found an 
interesting discussion of various molecular and other minute magnitudes. 


same scale the minute iiltramicroscopic 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 /a in length. The 
length (or height) of a man is about a milhon and three-quarter 
times as great, i.e. 1-75 metres, or 1-75 x 10® /x; and the mass of 
the man is in the neighbourhood of five million, million, million 
(5 X 10^^) 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*,' 
and, in somewhat greater detail, Errera discusses the question on 
the following lines f. The weight of a hydrogen molecule is, 
according to the physical chemists, somewhere about 8*6 x 2 x 10-^^ 
milligrammes ; and that of any other element, whose molecular 
weight is M, is given by the equation 

(M) = 8-6 X ilf x 10-22. 
Accordingly, the weight of the atom of sulphur may be taken as 
8-6 X 32 X 10-22 jj^gni. = 275 x IO-22 mgm. 

The analysis of ordinary bacteria shews them to consist ± 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 

Tooo ^ Too — -L'J A iU 

parts of sulphur, taking the total weight as = 1. 

But our little micrococcus, of 0-15 /x in diameter, would, if it 

were spherical, have a volume of 


7, X 0-15^ u., = 18 X 10-^ cubic microns: 

* Encyclopaedia Britannica, 9th edit., vol. ill, p. 42, 1875. v 

t Sur la limite de petitesse des organismes, Bull. Soc. R. des Sc med. et nat. 
de Bruxelhs, Jan. 1903 ; Rec. d'oeuvres (Physiol, generale), p. 325. 
{ Cf. A. Fischer, Vorlesungen fiber Bakterien, 1897, p. 50. 


and therefore (taking its density as equal to that of water), a 
weight of 

18 X 10-4 X 10-9 == 18 X 10-13 mgm. 

But of this total weight, the sulphur represents only 

18 X 10-13 X 15 X 10-5 = 27 X lO-i^ mgm. 

And if we divide this by the weight of an atom of sulphur, we have 

27 X 10-17 ^ 275 X 10-22 _ io,000, or thereby. 

According to this estimate, then, our little Micrococcus frogrediens 
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*, 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 X 10166 X 10-22 _ 8-7 x lO-i^ 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 micro- 
coccus is about 

^ij^ X 18 X 10-13 _ 2-5 X 10-13 i^gii^_ 

If we divide this weight by that which we have arrived at as the 
weight of an albumin molecule, we have 

2-5 X 10-13 ^ 8-7 X 10-18 = 2-9 x 10*, 

in other w^ords, our micrococcus apparently contains something 
less than 30,000 molecules of albumin. 

* F. Hofmeister, quoted in Cohnheim's Chemie der Eiweisskorper, 1900, p. 18. 


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 x 10"^^ mgms. and 
according to this value, our micrococcus would contain just about 
27,000 albumin molecules. In other words, whichever determina- 
tion we accept, we see that an organism one-tenth as large as our 
micrococcus, in hnear dimensions, would only contain some thirty 
molecules of albumin ; or, in other words, our micrococcus is only 
about thirty times as large, in hnear dimensions, as a single albumin 
molecule *. 

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, tnaximal values, above which the molecular 
magnitude (or rather the sphere of the molecule's range of motion) 
is not Ukely to he : 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 inter- 
relation with its neighbours which this complexity impHes, 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 hmit to its magnitude, other considerations of a purely 
physical kind lead us to the same conclusion. For our discussion 
of the principle of simihtude has already taught us that, long 
before we reach these almost infinitesimal magnitudes, the 

* McKendrick arrived at a still lower estimate, of about 1250 proteid molecules 
in the minutest organisms. Brif. Ass. Rep. 1901, p. 808. 


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 ^u,. 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 con- 
cordant methods of measurement, to be only about 6 x 10-'^ cm.^ 
or about -006 /x thick, and Lord Rayleigh and M. Devaux* have 
obtained films of oil of -002 /x, or even -001 fi 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 Uttle 
knowledge and which it is not easy even to imagine. 

It would seem that, in an organism of -1 /x 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 formid- 
able kindf, 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 [x ? It must, in all probability, be a homogeneous, structure- 
less 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 httle 
more than those of an ion-laden corpuscle, enabhng it to perform 

* Cf. Perrin, Les Atomes, 1914, p. 74. 

t Cf. Tait, On Compression of Air in small 'Bubbles, Proc. B. S. E. V, 1865. 


this or that chemical reaction, or to produce this or that patho- 
genic effect. Even among inorganic, non-Hving bodies, there 
must be a certain grade of minuteness at which the ordinary 
properties become modified. For instance, while under ordinary 
circumstances crystaUisation 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 

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 ivfiiiitiim] 
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 tmdique desectani 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 /x) 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 

, * Phil. Mag. XLvni, 1*899 ; Collected Papers, iv. p. 430. 


drop. Thus a drop one-tenth of that size (2-5 ^u.), the size, 
apparently, of the drops of water in a Ught cloud, will fall a 
hundred times slower, or say an inch a minute; and one again 
a tenth of this diameter (say -25 jj,, 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 
Kobert Brown, facile frincefs botanicorum. It is one more of those 
fundamental physical phenomena which the biologists have con- 
tributed, 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 soUd particles and the fluid w^hich is indirectly 
promoted by heat*." 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 jj. in diameter, it is admirably displayed by 
particles of about 12/^ in diameter, and becomes more marked 
the smaller the particles are. The bombardment causes our 
particles to behave just Hke molecules of uncommon size, and this 

* Carpenter, The Microscope, erlit. 1862, p. 185. 


behaviour is manifested in several ways*. 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 /a in diameter rotate on an average through 
100° per second, while particles of 13 /a 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!, 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 compUcated 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 ex- 
ample, Schaudinn has suggested that the undulating movements of 
Sfirochaete 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 eigenthiimlich zitternden Bewegungen 
zu einem Orte." Both movements, the trembhng or quivering 

* 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 fuUer, but still elementary account, see J. Cox, Beyond the Atom, 1913, 
pp. 118-128; and see, further, Perrin, Les Atomes, pp. 119-189. 

•j- Cf. R. Gans, Wie fallen Stabe mid Scheiben m euier reibenden Fliissigkeit ? 
Munchener Bericht, 1911, p. 191; K. Przibram, Ueber die Brown'sche Bewegung 
nicht kugelformiger Teilchen, Wiener Ber. 1912, p. 2339. 


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 

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 simulacrmn ef 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*. 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 

* Ueber die ungeordnete Bewegung niederer Thiere, Pfliiger^s Archiv, CLin, 
p. 401, 1913. 


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 

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 /x 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 /j, 
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 atmo- 
sphere. 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 ?nay ga 
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 /x) 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 


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 potentiahty which are 
associated in the scale of magnitude with simple differences of 



When we study magnitude by itself, apart, that is to say, 
from the gradual changes to which it may be subject, we are 
deaUng Avith a something which may be adequately represented 
by a number, or by means of a hne 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 deahng 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 groivth ; 
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" 


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 ^t or ^ J ; 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 
(hmited 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*. 

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 iriter-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 foym itself appears 
to us as a function of titnei. 

* 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. 

f 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." 



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 sohd skeletal rod, or when it acts in various 
other ways which are productive of definite contours, this is a pro- 
cess of conformation that, both in appearance and reahty, is very 
difierent 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. 


The difference between the two classes ^i 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 sohd 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*, 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 httle dependent on the time taken to effect 
it, that the specific rate of change, or rate of growth, does not 
enter into the moriihological 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." 

* Cf. Aristotle, Phys. vi, 5, 235 a 11. eVet yap airaaa Kivrjcm iv xP^''Vy '''''^• 
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. 


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, w^hile 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, w^as clearly apprehended by the mathematical 
mind of Haller, — who had learned his mathematics of the great 
John BernoulH, 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 *. 
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 

* Cf. (e.g.) Elem. Physiol, ed. 1766, viii, p. 114, '•Ducimur autem ad evolu- 
tionem potissimum, quando a perfecto animale retrorsum progredimur, et incre- 
mentorum atque nautationum 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 uUo 
saltu perpetuos parvosque per gradus cohaereant." 


them, and which are characteristic of another order of magni- 

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 Diirer 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*, 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 httle 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 outt, 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 dis- 
tortions of the entire structure. 

* Beitrdge zur Entwickelungsgeschichte des Hiihnchens im Ei, p. 40, 1817. Roux 
ascribes the same views also to Von Baer and to R. H. Lotze (Allg. Physiologie, 
p. 353, 1851). 

t Roux, Die Entwickdungsmeclianik, p. 99, 1905. 


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 pheno- 
mena of expansion, traction {adfractio)*, and pressure, and the 
more subtle influences which he denominates vis derivationis et 
revulsio7iis'\ : 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 j. 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 invagina- 
tion of a blastosphere, or the neural canal by the infolding of a 
cell plate so as to constitute a tube §." For Hertwig, therefore, as 

* Op. cif. p. 302, " Magnum hoc naturae instrumentum, etiam in corpore 
animato evolvendo potenter operatur; etc." 

t Ibid. p. 306. "Subtiliora ista, et aliquantum hypotliesi mista, tamen magnum 
mihi videntur speciem veri habere." 

{ Cf. His, On the Principles of Animal Morphology, Proc. JR. 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 explana- 
tion of living forms is allowed than heredity, and any which is founded on another 

basis must be rejected To think that hei'edity will build organic beings 

without mechanical means is a piece of unscientific mysticism." 

§ Hertwig, 0., Zeit und Streitfragen der Biologie, ii, 1897 


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 httle 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*. But in the main, Balfour, hke all the 
other embryologists of his day, was engrossed by the problems of 
phylogeny, and he expressly defined the aims of comparative 
embryology (as exemphfied in his own textbook) as being "two- 
fold: (1) to form a basis for Phylogeny. and (2) to form a basis 
for Organogeny or the origin and evolution of organsf." 

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 J, 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^, 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 

* Cf. Roux, Gesammelte Ahhandlungen, ii, p. 31, 1895. 

t Treatise on Comparative Embryology, i, p. 4, 1881. 

% Cf. Fick, Anal. Anzeiger, xxv, p. 190, 1904. 

§ 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 mis- 
interpretation when one attempts to epitomise Darwin's carefully condensed 


structure between the embryo and the adult, that we are tempted 
to look at this dift'erence as in some necessary manner contingent 
on growth. But there is no reason why, for instance, the wing of 
a hat, or the Jin of a porfoise, 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 shght variations generally appear at a not very early 
period of hfe," 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 hke 
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 hmbs 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 shght 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 


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 dift'erent sizes when adult, differ 
less and less in size (as well as in form) as we trace them back- 
wards 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*. 

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 

* '"In omni rerum naturalium historia utile est mensuras definiri et numeros," 
Haller, Elem. Physiol, ii, p. 258, 1760. Cf. Hales, Vegetable Staficks, Introduction. 


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 

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 miiform 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 itfe develojoment . 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. 




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 Anthro'pometrie* , 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. 

Fig. 3. Curve of Growth in Man, from birth to 20 yrs (3) ; 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 

* Brussels, 1871. Cf. the same author's Physique sociale, 1835, and Lettres 
sur la theorie des probabilites, 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 Aiithropometry, 1878; Daffner, F., Das Wachsthum des Menschen 
(2nd ed.), 1902, etc. 


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 con- 
structing 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* 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 

* Dr Johnson was not far wrong in saying that "life decHnes from thirty-five" ; 
though the Autocrat of the Breakfast-table, like Cicero, declares that "the furnace 
is in full blast for ten years longer." 


of growth is in fact a diagram of activity, or "time-energy" 
diagram*. 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 Qnetelet, pp- 193, 346.) 

Stature in metres Weight in kgm. Span of % ratio 

f ^ ^ , '^ -, arms, of stature 

Age Male Female % F/M Male Female % F/M male to span 

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, 

* Joly, The Abundance of Life, 1915 (1890), p. 86. 


which draws us down when we would fain rise up*. 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 t- 

Side by side with the curve which repiesents growth in length, 
or stature, our diagram shows the curve of weight J. 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 
Unear 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 comphcated law. 

Our curve of growth, whether for weight or length, is a direct 
picturs 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 
itseK changing. We have, in short, to study the phenomenon of 
acceleration: we have begun by studying a velocity, or rate of 

* " Lou pes, mestre de tout [Le poids, maitre de tout], mestre senso vergougnOi 
Que te tirasso en bus de sa brutalo pougno," J. H. Fabre, Oubreto prouvenQcdo, p. 61. 

f 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," Biv. di Scienza, 1907, and in " Wachsthum und seine 
analytische Darstellung," Biol. Ceniralbl. June, 1909. Haller (Elem. vn, p. 68) 
recognised decrementum as a phase of growth, not less important (theoretically) 
than incrementum: "tristis, sed copiosa, haec est materies." 

J Cf. (int. al.), Friedenthal, H., Das Wachstum des 
verschiedenen Lebensaltem, Zeit. f. allg. Physiol, ix, pp. 487-514, 1909. 


change of magnitude ; we must now study an acceleration, or 
rate of change of velocity. The rate, or velocity, of growth is 
measured by th? 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, 

T. G. 5 




but it is somewhat arrested for a wkile in childhood, from about 
five years old to eight. According to Quetelet's data, there is 
another shght interruption in the falhng rate between the ages of 
about fourteen and sixteen ; but in place of this almost insignificant 
interruption, the Enghsh and other statistics indicate a sudden 






< 20 







; \ ;' 


\ \ 


1 '" 

1 Bowditch 


v., X 










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. 
Even apart from these data of Quetelet's (which seem to 
constitute an extreme case), it is evident that there are very 




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<N ^ 


03 i" 



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 

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 

Annual Increment of Weight in Man (kgm.). 
(After Quetelet, Anihropometrie, p. 346*.) 

Increment Increment 














































































- 0-2 

12-13 4-1 3-5 24-25 0-8 - 02 

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 f 

* 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 iii 1835; the 
former are the means of his determinations in 1835-40. 

f 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." 




during the first two or three years of hfe. After a shght 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 

S 3 



Fig. 5. Mean annual increments of weight, in man and woman; 
from Quetelet's data. 

the man usually begins to lose weight a little. The subsequent 
slow changes in this acceleration-curve we need not stop to deal 

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 THE RATE OF C4R0WTH [ch. 

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 Uttle maid at this age is growing slower than the boy. 
But very soon, and while his acceleration-curve is still represented 
by a straight hne, 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 tw^o sexes ; but it has its notable differences, 
in ampUtude and especially in fliase. Last of all, we may notice 
that while the acceleration-curve falls to a negative value in the 
male about or even a httle 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 hfe ; until there 
comes a final period, in both sexes alike, during which weight, 
and height and strength all ahke diminish. 

From certain corrected, or "typical" values, given for American children 
by Boas and Wissler {I.e. p. 42), we obtain the following still clearer comparison 
of the armual 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 gu'ls respectively, and the annual incx'ements 
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 -07 4-1 58 43 2-8 18 09 0-3 

The result of these differences (which are essentially phase- 
difierences) 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 some- 
what 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 


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). 

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 

80 90 

Fig. 6. Percentage ratio, throughout life, of female weight to male; 
from Quetelet's data. 

same phases of growth*. In all cases growth begins slowly; it 
attains a maximum velocity early in its course, and afterwards 
slows down (subject to temporary accelerations) tow^ards 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 

* 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 
trihus actis imfiger annis Floret equus, puer hautquaquam, etc. 


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* intro- 
duced 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: 












Jlinot 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 LjT , we have dL/dT , which is (of course) 
velocity; and from this, by a second differentiation, we obtain d^L/dT-, that 
is to say, the acceleration. 

* Minot, C. S., Senescence and Rejuvenation, Journ. of Physiol, xn, pp. 97— 
153, 1891; The Problem of Age, Growth and Death, Poj). Science Monthly 
(June-Dec), 1907. 


But when you take percentages of y, you are determining dyly, and when 
you plot this against dx, you have 

dijly dy 1 dy 

dx y . ax y ax 

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 fost-natal groivtli. 

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 Riissow's*, who gives the stature of the infant 
month by month during the first year of its life, as follows : 

Age in months 12345 67 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 )nonthlij differences, or mean monthly 
velocities, by 12, to bring them into a form comparable with the 

* Quoted in Vierordt's Anatomische...Daten und Tabellen, 1906. p. 13. 




anmial 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. 

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 








2 4 

6 8 10 12 14 16 18 20 22 


Fig. 7. Curve of growth (in length or stature) of child, before and after 
birth. (From His and Riissow's data.) 

and the lamb are born relatively later, that is to say when develop- 
ment has advanced much farther, than in the case of man ; the 
kitten and the puppy are born earher and therefore more helpless 
than we are; and the mouse comes into the world still earher 
and more inchoate, so much so that even the little marsupial is 
scarcely more unformed and embryonic. In all these cases ahke, 
we must, in order to study the curve of growth in its entirety, 
take full account of prenatal or intra-uterine growth. 


According to His*, the following are the mean lengths of the 
unborn human embryo, from month to month. 

Months 01 2345 6 7 89 10 

Length in mm. 7-5 40 84 162 275 352 402 443 472 490) 


Increment per — 75 325 44 78 113 77 .50 41 29 18 1 

month in mm. 28 I 

These data link on very well to those of Riissow, which we 
have just considered, and (though His's measurements for the 

2 4 6 

Fig. 8. Mean monthly increments of length or stature of child (in cms.) 

8 10 12 14 16 18 20 22 


pre-natal months are more detailed than are those of Riissow 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 f : up to that date growth proceeds with a continually increasing 

* Unsere KorjKrforrn, Leipzig, 1874. 

t No such pomt of inflection appears in the curve of weight according to 
C. M. Jackson's data (On the Prenatal Growth of the Human Body, etc., Arner. 
Journ. of Anat. ix. 1909, jip. 126 156), nor in those quoted by him from Ahlfeld, 




velocity; but after that date, though growth is still rapid, its 
velocity tends to fall away. There is a shght 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 shght arrest of growth, or diminution of the rate of 
growth, about the epoch of birth : the sudden change in the 


y Length 


• / < 

/ ^j 

/ "^ 

/ "^ 

/ '*- 

/ ° 


/ ^ 

/ '^ 

/ o 


' .'^ 





,''' ~^"^v 



/ '^ 

■~^^^^ Acceleration 


" t^^ 

^-xC' ' 



1 1 

1 1 . "" 






2 4 6 8 10 


Fig. 9. Curve of pre-natal growth (length or stature) of child; and 
correspondmg curve of mean monthly increments (mm.). 

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 limar month : 

Months 0123456 7 8 9 10 

Wt in gms. -0 -04 3 36 120 330 600 1000 1500 2200 3200 


method of nutrition has its inevitable effect; but this shght 
temporary set-back is immediately followed by a secondary, and 
temporary, acceleration. 

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 

20 22 24 26 28 30 

Fig. 10. Curve of growth of bamboo (from Ostwald, after Kraus). 


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*, we see (so far as it goes) the same essential features, 

* G. Kraus (after Wallich-Martius), Ann. du Jardin hot. dc Buitenzonj, xii, 1, 
1894, p. 210. Cf. W. Ostwald, Zeitliche Eiyenschaften, etc. p. 56. 


the slow beginning, the rapid increase of velocity, the point of 
inflection, and the subsequent slow negative acceleration *. 

Variability and Correlation of Growth. 

The magnitudes and velocities which we are here deahng with 
are, of course, mean values derived from a certain number, some- 
times 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 he 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 hnes 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 probabihties, that all the individual 
measurements of ten-year-old boys group themselves iti an orderly 
ivay, 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 variabihty in this particular 
case. A certain mathematical measure of this "spread," as 
described in works upon statistics, is called the Index of Variabihty, 
or Standard Deviation, and is usually denominated by the letter cr. 
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 

* Cf. Chodat, R., et Monnier. A., Sur la courbe de croissance des vegetaux. 
Bull. Herb. Boissier (2), v, pp. 615, 616, 1905. 


value by the mean, we get a figure which is independent of 
any particular units, and which is called the Coefl&cient of Varia- 
bility. (It is usually multiphed by 100, to make it of a more 
convenient amount ; and we may then define this coefficient, C, 
as = a/M X 100.) 

In regard to the growth of man, Pearson has determined this 
coefficient of variabihty 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 variabihty tends, therefore, to decrease with 
growth or age. 

Similar determinations have been elaborated by Bowditqh, 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*. 

Coefficient of Variability (ujM x 100) in Man, at various ages. 

Age 5 6 ■? 8 9 

Stature (Bowditch) ... 4-76 4-60 4-42 4-49 4-40 

„ (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 

„ (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 

„ (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 variabihty 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 

* Cf. Fr. Boas, Growth of Toronto Children, Bep. of U.S. Comm. of Education, 
1896-7, pp. 1541-1599, 1898; Boas and Clark Wissler, Statistics of Growth, 
Education Re}]. 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. 




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 witli 

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 



Fig. 11. 

Coefficients of variability of stature in Man {^). from Boas 
and Wissler's data. 

the limits of age for w^hich we have information ; and this pheno- 
menon 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, 


I have taken Boas's determinations of variability (cr) {op. cit. 
p. 1548), converted them into the corresponding coefficients of 
variability {a[M x 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 18-6 191 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*, is confirmed 
by the following table : — 

Correlation of Stature and Increment in Boys and Girls. 
{From Boas and Wissler.) 


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 


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 50 4-7 5-9 7-5 6-2 5-2 


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 


-44 -14 -24 -47 -18 -18 --42 --39 -63 -11 

I.e. p. 42, and other papers there quoted. 


A minor, but very curious point brought out by the same investigators 
is that, if instead of stature we deal with lieight 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 variabiUty in regard to magnitude 
and to rate of increment is in the highest degree : 
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 variabihty, though they 
are intimately related to the general problem of growth, carry us 
very soon beyond our present limitations. 

Rate of grow ih in other organisms'^. 

Just as the human curve of growth has its shght 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 f, 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 ( nother well-marked 
retardation of growth, following on a very rapid period, and 
coinciding with the epoch of puberty. 

* See, for an admirable resume of facts, Wolfgang Ostwald, Ueber die ZeitUche 
Eigenschaften der Entwickelungsvorgdnge (71 pp.), Leip/.ig, 1908 (Roux's Vortrdge, 
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 ( Vitalitdt), pp. 85-87. 

f Cf . St Loup, Vitesse de croissanoe chez les Souris, Bull. Soc. Zool. Fr. xvm, 
242, 1893; Robertson, Arch. f. Entwickelung-smech. xxv, p. 587, 1908; Donaldson. 
Boas Memorial Volume, New York, 1906. 




Fig. 13 shews the curve of growth of the silkworm*, during its 
whole larv 1 hfe, 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, 

Fig. 12. Growth in weight of Mouse. (After W. Ostwald.) 

as a preliminary to entering on the pupal, or chrysaHs, stage; 
and the great and sudden loss of weight which accompanies this 
process is the most marked feature of our curve. 

The rate of growth in the tadpole | (Fig. 14) is hkewise marked 
by epochs of retardation, and finally by a sudden and drastic 
change. There is a slight diminution in weight immediately after 

* Lupiani e Lo Monaco, Arch. Ital. de Biologie, xxvn, p. 340, 1897. 
t Schaper, Arch. f. Entwickdmigsmech. xiv, p. 35fi, 1902. Cf. Barfiirth, Ver- 
suche iiber die Verwandlung der Froschlarven, Arch. f. mikr. Anat. xxix, 1887. 





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, 





















IV / 




III / 


' 1 ; 



Fig. 13 Growth in weight of Silkworm. (From Ostwald, after Luciani 
and Lo Monaco.) 

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. 




While as a general rule, the better the animals be fed the 
quicker they grow and the sooner they metamorphose, Barfiirth 
has pointed out the curious fact that a short spell of starvation, 
just before metamorphosis is due, appears to hasten the change. 






/ "^ 
/ ^ 

/ ^ 

/ o 

/ 5 

/ ^ 




- 1 





- 1 





60 70 80 90 

10 20 30 40 50 

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*. The contrast of size is great between 

* Joh. Schmidt, Contributions to the Life-history of the Eel, Rapports du Conseil 
Intern, pour V exploration de la Mer, vol. v, pp. 137-274, Copenhague, 1906. 

Fig, 15. Development of Eel; from Leptocephalus larvae to youno 
Elver. (From Ostwakl after Joh. Schmidt.) 


the flattened, lancet-shaped Leptocephalus larva and the httle 
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). 

From the higher study of the physiology of growth we learn 
that such fluctuations as we have described are but special inter- 
ruptions in a process which is never actually continuous, but is 
perpetually interrupted in a rhythmic manner*. 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 

20 40 60 80 100 120 140 160 180 200 220 240 260 


Fig. 16. Growth in length of Spirogyra. (From Ostwald, after Hofmeister.) 

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 observationsf, the growth of 
a crocus, under a very high magnification. The stalk grows by 
little jerks, each with an ampHtude of about -002 mm., every 

* That the metamorphoses of an insect are but phases in a process of 
growth, was firstly clearly recognised by Swammerdam, Bihlia Naturae, 1737, 
pp. 6, 579 etc 

t From Bose, J. C, Plant Eesponse, London, 1906, p. 417. 




twenty seconds or so, and after each little increment there is a 
partial recoil. 

Fig. 17. Pulsations of growth in Crocus, in micro-millimetres. 
(After Bose.) 

The rate of growth of various farts or organs*. 

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 creaturesf. It is obvious that 
there hes herein an endless field for the mathematical study of 
correlation and of variabihty, 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 hver ; 

* This phenomenon, of incrementum inequale, as opposed to incrementum in 
iiniversum, was most carefully studied by HaUer: "Incrementum inequale multis 
modis fit, ut aliae partes corporis aliis celerius increscant. Diximus hepar minus 
fieri, majorem pulmonem, minimum thymum. etc." (Elem. \ui (2), p. 34). 

f See [inter alia) Fischel, A., Variabilitat und Wachsthum des embryonalen 
Korpers, Morphol. Jahrb. xxiv, pp. 369-404, 18fi6. Oppel, VergUichung des 
Entwickelungsgrades der Organe zu verschiedenen Entwickelungszeiten bei Wirbel- 
thieren, Jena, 1891. Faucon, A., Pesees ef Mensurations fcetales n differents ages 
de la grossesse. (These.) Paris, 1897. Loisel, G., Croissance comparee en poids 
et en longueur des foetus male et femelle dans I'espece 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. 


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, Anatorn. Tabellen, 
pp. 38, 39.) 


of body'' 

in kg. 


weights 1 



Percentage weights compai-ed 
with total body-weights 






















































































































































37- 1 
























































































































= 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 




age of twenty-five they have multiphed their weight at birth by 
about thirteen times, while the weight of the entire body has been 
multiphed by about twenty-one ; but the weight of the brain has 
meanwhile been multiphed 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. 

5 10 15 20 ^ears 25 

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* 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 con- 
tinuing throughout adult lifef. 

* I.e. p. 1542. 

t Variation and Correlation in Brain-weight, Biometrika, iv, pp. 13-104, 1905. 


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 hver 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* tells us that in the Hon, 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 KelHcott has, in 
hke 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 f. 

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 rnay be sensibly expressed by a straight line, or simple 

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^ = 915-06 -f 2-86 iS. 

* Die Sdugethiere, p. 117. 

t Amer. J. of Anatomy, vm, pp. 319-353, 1908. Donaldson {Journ. Camp. 
Neur. and Psychol, xvm, 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 J, = -554 -^ -569 log (a;— 8-7), and the agreement is very close. But the 
formula i& admittedly empinca and as Raymond Pearl says (Amer. Nat. 1909, 
p. 303), " no ulterior biological significance is to be attached to it." 




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* . Donaldson has further shewn that the correla- 
tion between brain-weight and body-weight is very much closer 
in the rat than in man"]". 

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 









Spider monkey 




FeUs minuta . . . 




F. domestica ... 
















Whale (Globiocephak 

is) 1,000,000 



For much information on this subject, see Dubois, " Abhangigkeit des 
Hirngewichtes von der Korpergrosse bei denSsiUgethieren,''' 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", 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 J. 

* Biometrika, iv, pp. 13-104, 1904. 

t 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. 

t 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 : Massverhaltnisse, in Bardeleben's Handh. 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. 



The Length of the Head in Man at various Ages. 
{After Quetelet, f. 207.) 

Men Women 



Total height 

















1 year 







2 years 







3 „ 







5 „ 







10 „ 







15 „ 







20 „ 







30 „ 







40 „ 







* 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 measure- 
ments 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 

xm, pp. &-17, 1903. Warneke, P., Mitteilimg neuer Gehirn imd Korperge- 
wichtsbestimmungen bei Saugern, Zusammenstellung der gesammten bisher 
beobachteten absoluten imd relativen Gehirngewichte bei den verschiedenen 
Species, J. f. Psychol, u. Neurol, xm, 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. 




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 ■ 

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 develop- 
ment. 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 

* Cf. Jenkinson, Growth, Variability and Correlation in Young Trout, 
Biometrika, viii. pp. 444-455, 1912. 


Trout {Scdnio fario) : pwportionate groivth of various organs. 
{From Jenkinson's data.) 















of tail 





































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* 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 f. 

On the growing root of a bean, ten narrow zones were marked 
off, starting from the apex, each zone a milhmetre in breadth. 
After twenty-four hours' growth, at a certain constant tempera- 
ture, 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 tablet. 



























* Cf. chap, xvii, p. 739. 

t " ...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 Staiicks, Exp. cxxili. 

t From Sachs, Textbook of Botany, 1882, p. 820. 




The several values in this table he 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 hke 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 




II 1 ! 





/ \ 



/ \ 



- / \ 



- / \ . 



- / \ 



/ ^ 


/ 1 1 1 1 

1 1 ■ ' ^Tr 

123456789 10 


Fig. 20. Rate of growth in successive zones near the tip of the bean-root. 

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 outhne 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, 


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*. 

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 pro- 
gressive 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. 

* Variation and Differentiation in Ceratophyllum, Carneqie Inst. Publica- 
tions, No. 58, Washington, 1907. 


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 WJL^ = 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 Fislieries' Plaice- Report, 
vol. I, f. 107, 1908.) 

Size in cm. Weight in gm. WjL^ x 10,000 WjL^ (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 98-1 990 

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 — 




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 dehcate 
method of comparison than any linear measurement. 

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 par- 
ticular time, we see that k is by no means constant, but steadily 
increases to a maximum, and afterwards slowly declines with the 

23 25 27 29 31 33 35 37 39 41 43cms. 
Fig. 21. Changes in the weight-length ratio of Plaice, with increasing size. 

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 fife 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 tov^ards further investigation of the 





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. 

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*. When for simphcity's sake (as in the accompanying 
table and Fig. 22) we restrict ourselves to fish of one particular 


22. Periodic annual change in the weight-length ratio of Plaice. 

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. 

* Cf. Lammel, Ueber periodische Variationen in Organismen, Biol. Centralhl. 
xxn, pp. 368-376, 1903. 


RelMion 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^ X 100 

W/L^ (smoothed) 

















































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 




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 Przihrayn, after Cornevin*.) 

Age in 

If, wt. 
in kg. 

L, length 
of back 

H, height 



/■' = W/H 










•94 . 








































7 . 



































































































































































































2- 133 

* Cornevin, Ch., fitudes sur la croissance, Arch, de Physiol, norm, et pathol. 
(5), IV, p. 477, 1892. 


long ; t 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 jH^ 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*. 
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. 

* Herein lies the easy answer to a contention fiequently raised by Bergson, 
and to which he asciibes 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. 




' 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 

T3 100 

Length of tail-forceps, in mm. 

Fig. 23. Variability of length of tail-forceps in a sample of Earwigs. 
(After Bateson, P. Z. S. 1892, p. 588.) 

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 magni- 
tudes, 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 




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 

20 25 30 

Length of fish, in cm. 

Fig. 24. Variability of length of body in a sample of Plaice. 

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. 

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 di- 
morphism to a physiological phenomenon, simple in kind (however 
remarkable and unexpected) ; viz. that there were two alternative 


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, it we could beheve that 
the two groups of earwigs were of different ages, then the pheno- 
menon would be simphcity itself, and there would be no more to 
be said about it*. 

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 |. 

* 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 proceed? no more 
after the final stage, or "adult form" is attained, and further that this adult form 
is attained at an approximate^ 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. 

■j- Jackson, C. M., J. of Exp. Zool. xix, 1915, p. 99; cf. also Hans Aron, Unters. 


Full-fed Rats. 

Age in 

Length of Length of 



body (mm.) tail (m.) 


Oq of tail 

48-7 16-9 




64-5 29-4 




90-4 59-1 




128-0 110-0 




1730 150-0 

Underfed Rats 




98-0 72-3 




99-6 83-9 



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 interestmg 
from the physiological view, are of little practical importance to 
the morphologist. 

The effect of temperature*. 

The rates of growth which we have hitherto dealt with arc 
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 con- 
sidered, 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 

iiber die Beeinfliissung der Wachstum durch die Ernahrung, Berl. klin. Wochenhl. 
LI, pp. 972-977, 1913, etc. 

* The temperature limitations of life, and to some extent of growth, are summar- 
ised for a large number of species by Davenport, ^x^er. Mor2')}iology , cc. viii, xviii, 
and by Hans Przibram, Exp. Zoologie, iv, c. v. 


conditions. Reaumur was the first to shew, and the observation 
was repeated by Bonnet*, that the rate of growth or development 
of the chick was dependent on tem.perature, 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 f. 

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 
BlackmanJ, 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 sufiicient 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 

* Reaumur : Uart de faire colore et elever en toufe saison des oiseaux domestiques, 
foil ixir le moyeii de la chaleur du fumier, Paris, 1749. 

f Cf. (int. al.) de Vries, H., Materiaux pour la connaissance de rinfluence de 
la temperature sur les plantes, Arch. Neerl. v, 385-401, 1870. Koppen, Warme 
und Pflanzenwachstum, Bull. Soc. Imp. Nat. Moscou. XLiii, pp. 41-110, 1870. 

X Blackman, F. F., Ann. of Botany, xix, p. 281, 1905. 




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. 

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 ; 

14°16 18 20 22 24 26 28' 30 32 34 36 38 40° 


Fig. 25. Relation of rate of growth to temperature in certain plants. 
(From Sachs's data.) 

and this effect of temperature tends to follow a definite "ex- 
ponential" law, which holds good within a considerable range of 
temperature, but is altered or departed from when we pass beyond 
certain normal hmits. 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'^, x being called the "temperature coefficient"* for the 
reaction in question. 

* For various instances of a "temperature coefficient" in physiological pro- 
cesses, see Kanitz, Zeitschr. f. Elektrochemie, 1907, p. 707; Biol. Centralhl. xxvii, 
p. 11, 1907; Hertzog, R. 0., Temperatureinfluss auf die Entwicklungsgesch- 
windigkeit der Organismen, Zeitschr. f. Elektrochemie, xi, p 820, 1905; Krogh, 


Van't Hoffs law, which has become a fundamental principle 
of chemical mechanics, is likewise applicable (with certain qualifica- 
tions) 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 Optimmn, 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 accelerat- 
ing effect which tends to follow van't Hoff's law. But it has also 
another and a cumulative eft'ect upon the organism : " Sie schadigt 
oder sie ermiidet ihn, und je hoher sie steigt, desto rascher macht 
sie die Schadigung 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 experi- 
ment ; 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. 

Quantitative Relation between Temperature and Standard Metabolism, l7it. 
Zeifschr. J. physik.-chem. Biologie, i, p. 491, 1914; Piitter, A., Ueber Temperatur- 
koefficienten, Zeiischr. f. allgern. Phi/siol. 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 
Orsranism, American Naturalist, Jan. 1915, and various papers quoted therein. 


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*. 

In the case of maize, a certain series of experiments shewed that 
the growth in length of the roots varied with the temperature as 
followsl : 


Growth in 48 hours 















Let us write our formula 






n) _ 

■X'' . 


Then choosing two values out of the above experimental series 
(say the second and the second-last), we have / = 23-5, n = 10, 
and F, F' = 10-8 and 69-5 respectively. 

Accordingly YrTo = 6-4 = x^^. 

Therefore ^f^ ' ^^ '^^^^ = ^°g ^• 

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. 

* Cf. Errera, L., UOptimmn, 1896 {Rec. d'Oeuvres, Physiol, generate, pp. 338-368, 
1910); Sachs, Physiologie d. Pflanzen, 1882, p. 233; Pfeffer, Pftunzenphysiologie, 
ii, p. 78, 1904; and cf. Jost, Ueber die Reactionsgeschwindigkeit im Organismus, 
Biol. Centralbl. xxvi, pp. 225-244, 1906. 

t After Koppen, Bull. Hoc. Nat. Moscou, xliii, pp. 41-110, 1871. 




For an experiment on Lupinus albus, quoted by Asa Gray*, 
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. 

18 20 22 

24 26 28 


Fig. 26. 

Relation of rate of growth to temijerature in Maize, 
values (after Koppen). and calculated curve. 



Since the above paragraphs were wi'itten, 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 resultsy. 
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 hotir 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 

* Botany, p. 387. 

t 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. especiallj' 
Table III, p. 45.) 




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. 


12° 16° 20° 

















































28° 32' 

4 2 

2 p 

1-0 ^ 

•8 S 



Fig. 27. Relation of rate of growth to temperature in rootlets 
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 


irregular, and sometimes (we might even say) misleading *. 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 compli- 
cated by osmotic factors, true metabolic growth or chemical 
combination plays a larger role. Thus Mile. Maltaux and Professor 
Massartf have studied the rate of division in a certain flagellate, 
Cliilomonas faramoecium, 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 
comphcated results of many experiments. For instance. Fig. 28 
is an illustration, which has been often copied, of 0. Hertwig's 
work on the effect of temperature on the rate of development of 
the tadpolef , 

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 


6-5 days 

„ IV 


8-1 „ 

„ V 


10-7 „ 

„ VI 


13-5 „ 

„ VII 


16-8 „ 



55-6 „ 

That is to say, the time taken to produce a given result at 

* Blackman, F. F., Presidential Address in Botany, Brit. Ass. Dublin, 1908. 

f Rec. de Vljist. Bot. de Bruxelles, vi, 1906. 

X Hertwig, 0., Einfluss der Temperatur auf die Entwicklung von Bana fusca 
und R. esculenia, Arch. f. mikrosk. Anat. li, p. 319, 1898. Cf. also Bialaszewicz, 
K., Beitrage z. Kenntniss d. Wachsthumsvorgange bei Amphibienembryonen, 
Bull. Acad. Sci. de Cracovie, p. 783, 1908 ; Abstr. in Arch. f. Entwicklung smech. 
xxvm, p. 160, 1909. 




10° was (on the average) somewhere about 55-6/16-7, or 3-33, 
times as long as was required at 20°. 


1 — 







-V - 


















































































































































■ — ■ 

TemrLpXa,lu%e CentigriacLe 

2#-° ZJ" ii" H' iO" 19' 18° It' 16' l{>' IV 13'' n" II' 10° 9° a" '^ 6° 

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 0. Hertwig, 1898.) 

We may then put our equation again in the simple form, 
xi" - 333. 








10 log X = log 3-33 - -52244. 
log X - -05224, 
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 x 1-128" days, 
when the temperature is n degrees less, for the same stage to 
be arrived at. 




15° 10° 


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. 


Karl Peter*, 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^q) 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^" = 2-29, 

Later stages ,, 2-03. 

Echinus; Segmentation ,, 2*30, 

Later stages ,, 2-08. 

Rana; Segmentation ,, 2-23, 

Later stages ,, 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 

* Der Grad der Beschleunigung tierischer Entwickelung durch erhcihte 
Temperatur, A. f. Entiv. 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. 


will represent a vibration which is gradually dying out ; the ampli- 
tude 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 *, of the height of German cadets, 
measured at half-yearly intervals. 

Growth in height of German military Cadets, in half-yearly 
periods. {Dajfner.) 

Increment in cm. 



IgllU lit V.C 




















































































In the accompanying diagram (Fig. 30) the half-yearly incre- 
ments 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 Hne represents, approximately, the acceleration-curve in 
its continuous fluctuation of alternate seasonal decrease and 

In the case of trees, the seasonal fluctuations of growthl admit 

* Das Wachstum des Menschen, p. 329, 1902. 

f The diurnal periodicity is beautifully shewn in the case of the Hop by Joh. 
Schmidt (C E. du Laboratoire de Garlsberg, x, pp. 235-248, Copenhague, 1913). 




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* I have compiled certain mean values for 
growth in the climate of Uruguay. 

19 20 


Fig. 30. Half-yearly increments of growth, in cadets of various ages. 
(From Daffner's data.) 

Mean monthly increase in Girth of Evergreen a?id 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 

trees ... 20-3 14-6 90 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 iVcacia ; the deciduous trees 
included Quercus, Populus, Robinia and Meha. 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. Th^ growth of the deciduous trees is completely 
arrested in winter-time, and the arrest is all but complete over 

* Trans. Botan. Soc. Edinburgh, xvm, 1891, p. 456. 




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 equahty over a considerable 

Fig. 31. Periodic annual fluctuation in rate of growth of trees (in the 
southern hemisphere). 

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. 

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 


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*, and I have 
received (through Professor H. H. Turner of Oxford) some measure- 
ments 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 

* I had not received, when this was written, Mr Douglass's paper, On a method 
of estimating Rainfall by the Growth of Trees, Bull. Arner. 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 inchned (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. 

*i I— t 

O o 

b ;z; 


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. 

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*. The point is a physiological one, 
and consequently of little importance to us here f ; 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 dift'ering 
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, 

* 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. 

t On growth in relation to light, see Davenport, Exp. Morphology, ii, ch. xvii. 
In some cases (as in the roots of Peas), exposure to hght seems to have no effect 
on growth; in other cases, as in diatoms (according to Whipple's experiments, 
quoted by Davenport, n, p. 423), the effect of light on growth or multipHcation 
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. 


probably, a still better instance of it in the case of Alpine plants *, 
whose general habit is dwarfed, though their floral organs suffer 
little or no reduction. The subject, however, has been httle 
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 character- 
istic 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 environ- 
ment or to certain stimuli, would be but an example of that " lack 
of particularity!" 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 

* Cf. for instance, Nageli's classical account of the effect of change of habitat 
on Alpine and other plants : Sitzungsber. Baier. Akad. Wiss. 1865, pp. 228—284. 

j Cf. Blackman, F. F., Presidential Address m Botany, Brit. Ass. Dublin, 1908. 
The fact was first enunciated by Baudrimont and St Ange, Recherches sur le 
developpement du foetus, Mem. Acad. Set. xi, p. 469, 1851. 


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*. Of the chemical phenomena which result 
in the actual increase of protoplasm we shall speak presently, but 
the role 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 w^hich 
the cells he does not overstep a certain Umit 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 sahnity 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 f 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 t , on the early growth of the frog, are notable. 
He shews that the growth of the embryo while still within the 

* Cf. Loeb, Untcr.tuchungen zur physiol. Morphologie der Thiere, 1892; also 
Experiments on Cleavage, J. of Morph. vn, p. 253, 1892; Zusammenstellung der 
Ergebnisse einiger Arbeiten iiber die Dynamik des thierischen Wachsthum, Arch, 
f. Entw. Mech. xv, 1902-3, p. 669 : Davenport, On the Role of Water in Growth, 
Bosto7i Soc. N. H. 1897; Ida H. Hyde, Am. J. of Physiol, xn, 1905, p. 241, etc. 

t Pfldger's Archiv, lv, 1893. 

J Beitrage zur Kenntniss der Wachstumsvorgange bei Amphibienembryonen, 
B^dl. Acad. Sci. de Cracovie, 1908, p. 783 ; cf. Arch. f. Entw. Mech. xxvm, p. 160, 
1909; xxxiv, p. 489, 1912. 


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 Fehhng in the case of man*. Fehhng'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 









Percentage of water 









To such phenomena of osmotic balance as the above, or in other 
words to the dependence of growth on the uptake of water, Hober f 
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 J. Important physiological changes also 
ensue. The rate of multiplication is increased, and partheno- 
genetic reproduction is encouraged. Male individuals become 
plentiful in the less sahne waters, and here the females bring forth 

* Febling, H., Arch, fiir Gynaekologie, xi. 1877; cf. Morgan, Experimental 
Zoology, p. 240, 1907. 

t Hober, R., Bedeutung der Theorie der Losungen fiir Physiologie und 
Medizin, Biol. Centralhl. xix, 1899; cf. pp. 272-274. 

t 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, Biitschli {Z. /. w. Z. xxix, p. 429, 1877). 


their young alive ; males disappear altogether in the more con- 
centrated 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. ynilhausenii, 
inhabits the natron-lakes of Egypt and Arabia, where, under the 
name of "loul," or "Fezzan-worm," it is eaten by the Arabs*. 
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 t), are no more salt than are those of any ordinary crus- 
tacean or other animal, but contain only some 0-8 per cent, of 
NaClt, 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 

* 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. 

t Cf. Introd. a VeUide de la medecine experimentile, 1885, p. 110. 

t Cf. Abonyi, Z. f. w. Z. cxiv, p. 134, 1915. But Fredericq has shewn that 
the amount of NaCl in the blood of Crustacea (Carcinus moenas) varies, -and 
all but corr s onds, with the density of the water in which the creature 
has been kept {Arch, de Zool. Exp. et Gen. (2), in, p. xxxv, 1885); and 
other results of Fredericq's, and various data given or quoted by Bottazzi 
(Osmotischer Druck und elektrische Leitungsfahigkeit der Fliissigkeiten 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 




described as the specific characters of A. milhausenii, and of a 
still more extreme form, A. kop'peniana ; 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 

^ w w 


















Artemia s sir. 


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 sahne 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 




character, that we may briefly define the species Artemia {Callao- 
nella) 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 




^ 120 


1000 1020 1040 1060 

Density of water 



Fig. 34. Percentage ratio of lengtli of abdomen to ceplialothorax 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 

T. G. 9 


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 Hober and others* 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 
wliich plays the part of a regulatory mechanismf . 

Growth and catalytic action. 

In ordinary chemical reactions we have to deal (1) with a 
specific velocity proper to the particular reaction, (2) with varia- 
tions 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 

* Cf. Sehmankewitsch, 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 Dees, Ann. Sci. Nat. (Zool.), (9), xi, 1910; Samter imd 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. Ce7itralM. vi, pp. 756-760 
1908; Abonyi, Z.f. w. Z. cxiv, pp. 96-168, 1915 (with copious bibliograpliy), etc 

t According to the empirical canon of physiology, that (as Fredericq expresses 
it) "L'etre vivant est agence de telle maniere que chaque influence pertui-batrice 
provoque d'elle-meme la mise en activite de I'appareil compensateur qui doit 
neutrahser et reparer le dommage." 


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*. 
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 dwind ing 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 pro- 
toplasm is a very important phenomenon. As Blackman says, 
in the address already quoted, the botanists (or the zoologists) 
"call it groivth, 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. 

* 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 iight, others 
even after attentive consideration. These magical effects are brought about in 
three ways... [of which one isj 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. 



This very important, and perhaps even fundamental pheno- 
menon of growth would seem to have been first recognised by 
Professor Chodat of Geneva, as we are told by his pupil Monnier *. 
"On peut bien, ainsi que M. Chodat I'a propose, considerer 
I'accroissement comme une reaction chimique complexe, dans 
laquelle le catalysateur est la cellule vivante, et les corps en 
presence sont Teau, les sels, et I'acide carbonique." 

Very soon afterwards a similar suggestion was made by Loebt, 
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 pro- 
duction 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 J : 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 § . 

As F. F. Blackman has pointed out, the multiplication of an 
organism, for instance the prodigiously rapid increase of a bacterium, 

* Monnier, A., Les matieres minerales, et la loi d'accroissement des Vegetaux, 
Publ. de Vlnst. de Bot. de VUniv. de, Geneve (7), in, 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 Eigenschafien der Eniwickelungsvonjdnge, 1908; Hatai, S., Interpreta- 
tion of Growth-curves from a Dynamical Standpoint, Anat. Record, v, p. 373, 

t Biochem. Zeitschr. n. 1906, p. 34. 

J 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. 

§ Cf. Loeb, The Stimulation of Growth, Science, May 14, 1915. 


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*, is a simple but most striking illustration of the potenti- 
alities 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 awhile, 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 auto- 
catalysis 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 

* B. coli-comynunis, according to Buchner, tends to double in 22 minutes; in 
24 hours, therefore, a single individual would be multiplied by something like 
10-8; Sitzung.<<ber. Munchen. Ges. MorphoJ. u. Physiol, in, 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*^. 


and specialised functions that of cell-multiplication tends to fall 
into abeyance*. 

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 | akin to those which 
the physical chemist finds apphcable to his autocatalytic reactions. 
This has been diligently attempted by various writersj ; 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 fife. 

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 Poincare 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 §. But 
without attempting to open up this large subject, we may say a 

* Cf. Enriques, Wachsthum und seine analytisohe Darstellung, Biol. Centralbl. 
1909, p. 337. 

f Cf. (int. al.) Mellor, Chemical Statics and Dynamics, 1904, p. 291. 

t Cf. Robertson, I.e. 

§ See, for a brief resume of this subject, Morgan's Experimental Zoology, 
chap. xvi. 


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*, Danilewskyf 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 w^hich is more or less exemplified in every 
skeleton of a '"giant" ; while on the other hand, disease or extirpa- 
tion 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 
Bayhss 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 

* Amer. J. of Physiol, x, 1904. f C'.J?. cxxi, cxxii, 1895-96. 


proper to the male, as in certain extreme cases, known from time 
immemorial, when late in life a hen assumes the plumage of the 

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*. No other organic food was found to produce 
the same effect ; but since the thyroid gland is known to contain 
iodine f , 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 pro- 
portions 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 J that by these chemical influences 
"Toute une partie de la construction des etres parait s'expliquer 
d'une fa^on toute mecanique. La forteresse, si longtemps inacces- 
sible, du vitalisme est entamee. Car la notion morphogenique 
etait, suivant le mot de Dastre§, comme 'le dernier reduit 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 

* Cf. Loeb, Science, May 14, 191.5, 

f Cf. Baumann u. Roos, Vorkommen von lod im Thierkorper, Zeitschr. fur 
Physiol. Chem. xxi, xxii, 1895, 6. 

% Le Neo- Vitalisme, Rev. Scientifique, Mars 1911, p. 22 (of reprint). 
§ La vie et la mort, p. 43, 1902. 


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*. 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 : donee ad 
inter itum genus id natura redegitf. 

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 

* Cf. Dendy, Evolutionary Biology, 1912, p. 408; Brit. Ass. Report (Portsmouth), 
1911, p. 278. 

I 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. 


is not to create but to destroy, — to weed, to prune, to cut down 
and to cast into the fire*. 

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 niedicatrix which heals a wound. Ever since the 
days of Aristotle, and especially since the experiments of Trembley, 
Reaumur and Spallanzani in the middle of the eighteenth century, 
the physiologist and the psychologist have ahke 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 f. 

Regeneration, hke 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 t. 

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 

* 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. xh, p. 46, 1907. 

t See Professor T. H. Morgan's Regeneration (316 pp.), 1901 for a full account 
and copious bibliography. The early experiments on regeneration, by Vallisneri, 
Reaumur, Bonnet, Trembley, Baster, and others, are epitomised by HaUer, Elem. 
Physiologiae, vm, p. 156 seq. 

J Journ. Experim. Zool. vii, p. ,397, 1909. 























(11-5 mm.) was cut o&, and the amounts regenerated in successive 
periods are shewn as follows : 

Days after operation 

(1) Amount regenerated in mm. 

(2) Increment during each period 
(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 experi- 
ment, 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-hne, 
which it would then have reached about the end of the second 




day ; or it may have had within the first three days a change of 
direction, or "point of inflection," and may then have sprung 

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 


Fig. 35. Curve of regenerative gro\\i;h in tadpoles' tails. (From 
M. L. Durbin's data.) 

at once from the base-hne at zero. That is to say, there may 
have been an intervening "latent period," during which no growth 

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 


Fig. 3H. Mean daily increments, corresponding to Fig. 3.5. 


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. LjT = F. The second (Fig. 36) 
represents the rate of change of velocity; it sets F against T ; 


table, extended by graphic 


Total Daily 


increment inci 

ement Logs of do. 




















































































































and V/T or L/T- , represents (as we have learned) the acceleration of 
growth, this being simply the "differential coejficient," the first 
derivative of the former curve. 

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 dowii 












X,- - 




i . 1 ; 1 1 r 1 1 1 1 1 • 

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 


Fig. 37. Logarithms of values shewn in Fig. 36. 

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 
A''ariables (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, approxi- 
mately, into a straight line; and if this curve be plotted on the 


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 |. 
Had the angle been 45° (tan 45° = 1), the curve would have 
been actually a rectangular hyperbola, with VT = constant. As 
it is, we may assume, provisionally, that it belongs to the same 
family of curves, so that F'^T'S or V^'^'T, or VT"'"^; 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 VT^ = constant, holds very nearly true ; that is to say 
the velocity varies, or tends to vary, inversely as the square of 

16 1i 

Fig. 38. Rate of regenerative growth in larger tadpoles. 

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 

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 




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 diminu- 
tion. Now this preUminary 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 bein^ 
in this case of larger size (average 49-1 mm.)*. 




7 10 



17 24 

28 31 



) 2-15 

3-66 5^20 



7-10 7-60 

8-20 8-40 

Or, by gra 

,phic interpolation, 






















































•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 existeiice is confirmed by successive observa- 
tions 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 

* Op. cit. p. 406, Exp. IV. 


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'^. 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 

2 4 6 8 
Fig. 39. Daily increment, or amount regenerated, corresponding to Fig. 38. 

place to a formula which shall represent the entire phenomenon, 
from start to finish. 

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 

T. G. 10 


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 charac- 
teristics 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*, 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 Abbe Trembley and like Voltaire, 
found a fascination in the study of artificial injury and the regenera- 
tion which followed it. Sir John Graham Dalyell, for instance, 
says, in the course of an admirable paragraph on regeneration I : 
"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 

* The experiments of Loeb on the growth of Tubulaiia in various saline 
solutions, referred to on p. 125, might as well or better have been referred to under 
the headmg of regeneration, as they were performed on cut pieces of the 7,oophji;e. 
(Cf. Morgan, op. cii. p. 35.) 

"f Powers of the Creator, i, p. 7, 1851. See also Rare and Remarkable Animals, 
II, pp. 17-19, 90, 1847. 


been studied by Lillie*, 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 jj, 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 cyto- 
plasm within which the organisation of the species can just find 
its latent expression." And in like manner, Boverif 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 

* Lillie, F. R., The smallest Parts of Stentor capable of Regeneration, 
Journ. of Morphology, xii, p. 23.9, 1897. 

t Boveri, Entwicklungsfahigkeit 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, 



The Rate of Regenerative Growth in Tadpoles' Tails. 
{After M. M. Ellis, J. Exp. Zool. vii, p. 421, 1909.) 







Per cent. 

of tail 


amount regenerated 

in days 




9 12 


18 32 







44 44 


44 44 








40 44 


44 44 








31 40 


48 48 







33 39 


48 48 

* Each series gives the mean of 20 experiments. 


Fig. 40. Relation between the percentage amomit of tail removed, the percentage 
restored, and the time required for its restoration. (From M. M. Ellis's 

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 


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 ap- 
proximates 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 hmb is 
particularly well developed in arthropod animals, and is some- 
times 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 

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 

The one claw is the larger because it has grown the faster; 

* 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. n, 1-102, 347-369, 1905; etc. 

t Lobsters are occasionally found with two symmetrical claws : which are then 
usually serrated, sometimes (but very rai-ely) both blunt-toothed. Cf. Caiman, 
P.Z.S. 1906, pp. 633, 634, and reff. 


it has a higher "coefficieDt of growth," and accordingly, as age 
advances, the disproportion between the two claws becomes more 
and more evident. Moreover, we must assume that the character- 
istic 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 th^ 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 magnitudej with which a certain difEerentia- 
tion 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 accelera- 
tion 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*, 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 yl-claw re- 
generates itself as a small or 5-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 ^-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 

* Wilson, E. B., Reversal of Symmetry in Alpheus heterochelis, Biol. Bull, rv, 
p. 197, 1903. 


the possibility of the uninjured Hmb growing all the faster for 
a time after the animal has been relieved of the other. From the 
time of amputation, say of ^, ^ 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 i5. 

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 


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 hkely 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 


the rapid formation of a comparatively firm skin; and that the 
dwindUng 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 afiect the velocity 
of growth. For instance, it has been shewn by Frazee* 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. 

Delagef 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 (Kke 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. 

* J. Exp. Zool. VII, p. 457, 1909. 
f Biologica, ni, p. 161, June,. 1913. 


We may summarise, as follows, the main results of the fore- 
going 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 grbwth, we have left out of account a 
vast number of processes, or phenomena, by which, in the physio- 
logical mechanism of the body, growth is effected and controlled. 
We have dealt with growth in its relation to magnitude, and to 


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 appro- 
priate places : so in all these cases, very much as in the organism 
itself, is growth accompanied by change of form, and by a develop- 
ment 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 hkelihood 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. 



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*." And likewise, long afterwards, Giard contemplated a 
science of mor'phodynamique, — 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 Moise vit 
la terre promise, seulement de loin et sans pouvoir y entrerf." 

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 

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 intra- 
cellular structure ; and to the multitude of visible structures it 

* Anatomical and Pathological Observations, p. 3, 1845; Anatomical Memoirs, 
n, p. 392, 18G8. 

t Giard. A., L'oeuf et les debuts de revolution, Bull. Sci. du Nord de la Fr. 
VIII, pp. 252-258, 1876. 


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. 

Blitschli 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*. 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!, 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 " caryo- 
kinesis'' 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 J." 

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, 

* Entwickelungsvorgdnge der Eizelle, 1876; Investigations on Microscopic Foams 
and Protoplasm, p. 1, 1894. 

t Journ. of Morphology, i, p. 229, 1887. 

J 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 iind 
here and there a strong protest against this mode of interpretation. The following 
is a case in point: "On a tente d'etablir dans la mitose dite primitive plusieurs 
categories, plusieurs types de mitose. On a choisi le plus souvent comme base 
de ces systemes des concepts abstraits et teleologiques : repartition plus ou moins 
exacte de la chromatine entre les deux noyaux-fils suivant qu'il y a ou non des 
chromosomes (Da.ngeard), distribution particuliere et signification dualiste des 
substances nucleaires (substance kmetique et substance generative ou hereditaire, 
Hartmann et ses eleves), etc. Pour moi tous ces essais sont a rejeter categorique- 
ment a cause de leur caractere finaliste ; de plus, ils sont construits sur des concepts 
non demontres, et qui parfois representent des generalisations absolument erronees." 
A. AlexeiefE, Archivfitr Protistenkunde, xix, p. 344, 1913. 


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 

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* 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. 

* This is the old philosophic axiom writ large: Ignorato motu, ignoratur 
natura ; which again is but an adaptation of Aristotle's phrase, 17 dpxv rrjs Kivr]<T€us. 
as equivalent to the "Efficient Cause." FitzGerald holds that ''all explanation 
consists in a desciiption of underlying motions"; Scientific Writings, 1902, p. 385. 


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 comphcated 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 beautifiil configuration of its falUng 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 surround- 
ing 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*." 

Moreover, the microscope seemed to substantiate the idea 
(which we may trace back to Leibniz | and to HobbesJ), that 
there is no limit to the mechanical complexity which we may 
postulate in an organism, and no limit, therefore, to the hypo- 
theses 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 

* As when Nageli concluded that the organism is, in a certain sense, "vorge- 
bildet" ; Beitr. zur iviss. Botanik, II, 1860. Cf. E. B. Wilson, The Cell, etc., p. 302. 

t "La matiere arrangee par une sagesse divine doit etre essentieUement organisee y a machine dans les parties de la machine Naturelle a I'infini." Sur le 
qirincipe de la Vie, p. 431 (Erdmann). This is the very converse of the doctrine 
of the Atomists, who could not conceive a condition "w6i dimidiae jjartis pars 
■semper habebit Dimidiam partem, nee res praefiniet ulla.''' 

1 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. 


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 Briicke, with whose demand for a mechanism, 
or organisation, within the cell histologists have ever since 
been attempting to comply*. 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 t-" If we cannot assume 
differences in structure, we must assume differences in niotion, 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." 

* "Wir miissen deshalb den lebenden Zellen, abgesehen von der Molekular- 
structur der organischen Verbindungen welche sie enthalt, noch eine andere und 
in anderer Weise complicirte Structur zuschreiben, und diese es ist welche wir 
mit dem Namen Organisation bezeiclmen," Briicke^ 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. 

f Precisely as in the Lucretian concursus, motus, ordo, positura,figurae, whereby 
bodies miitato ordine mutant naturam. 


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 ivorh, 
at a stage short of that ultimate degradation which lapses in 
uniformly diffused heat. This, as Warburg has well explained, is 
the general eflect or function of the physiological machine, and in 
particular of that part of it which we call "cell-structure*." 
The normal muscle-cell is something which turns energy, derived 
from oxidation, into work; it is a mechanism which arrests and 
utiUses the chemical energy of oxidation in its downward course ; 
but the same cell when injured or disintegrated, loses its "use- 
fulness," 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, 

* Otto Warburg, Beitrage zur Physiologie der Zelle, insbesondere iiber 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). 

T. G. li 


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 mechanisna. 
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* has lately reminded us), it possesses "ewer^rmf." 

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 J ; another, like Leduc, may shew us how in 

* Hardy, W. B., On some Problems of Living Matter (Guthrie Lecture), 
Tr. Physical Soc. London, xxviii, p. 99-118, 1916. 

I As a matter of fact both phrases occur, side by side, in Graham's classical 
paper on "Liquid Diffusion apphed to Analysis," Phil. Trans, cli, p. 184, 1861; 
Chem. and Phys. Researches (ed. Angus Smith), 1876, p. 554. 

X L. Rhumbler, Mechanische Erklarung der Aehnhchkeit zwischen Magne- 


many of their most striking features they may be admirably 
simulated by the diffusion of salts in a colloid medium; others 
again, like Gallardo* and Hartog, and Rhumbler (in his earher 
papers) f, insist on their resemblance to the phenomena of 
electricity and magnetism J ; while Hartog beUeves that the force 
in question is only analogous to these, and has a specific identity 
of its own §. All these conflicting views are of secondary import- 
ance, 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 Hues 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||"; and this is as much as to say that the phenomenon, 

tischen Kraftliniensystemen unci Zelltheilungsfiguren, Arch. f. Entw. Mech. xv, 
p. 482, 1903. 

* Gallardo, A., Essai d'interijretation des figures caryocinetiques, Anales del 
Miiseo de Buenos-Aires (2), ii, 1896; La division de la cellule, phenoraene bipolaire 
de caractere electro-colloidal. Arch. f. Entw. Mech. xxviii, 1909, etc. 

t Arch. f. Entw. Mech. m, iv, 1896-97. 

J 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 Anatomic, etc., vii, viii, 1897-8; Ida H. Hyde, Amer. Journ. 
of Physiol. XII, pp. 241-275, 1905; and especially Prenant, A., Theories et inter- 
pretations physiques de la mitose, J. de VAnat. et Physiol, xlvi, pp. 511-578, 1910. 

§ Hartog, M., Une force nouvelle: le mitokinetisme, C.R. 11 Juli, 1910; 
Mitokinetism in the Mitotic Spindle and in the Polyasters, Arch. f. Entw. Mech. 
xxvn, 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. pour VAvancem. des Sc. 1914, etc. 

II 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 LesUe 
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 



though physical in the concrete, is in the abstract purely mathe- 
matical, 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*. 

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 "f 

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 

tension or attraction along the lines, and of repulsion transverse to the hnes, of tlie 

* Cf. also the curious phenomenon in a dividing egg described as "spinning" 
by Mrs G. F. Andrews, J. of Morph. xn, pp. 367-389, 1897. 

t Whitman, J. of Morph. n, p. 40, 1889. 



history." On simple dynamical gromids, the contrast is easily 
explained. So long as the fluid substance of the nucleus is quah- 
tatively 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 *. 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, owing to the small difference between the fluid constitut- 
ing, and that surrounding, the drop, the surface-tension equi- 
librium is unstable ; it is apt to vanish, and the rounded outline 
of the drop, like a burst bubble, disappears in a moment t- 
The case of the spherical nucleus is closely akin to the spherical 
form of the yolk within the bird's egg J. But if the substance of 
the cell acquire a greater solidity, as for instance in a muscle 

* "Souvent il n'y a qu'une separation physique entre le cytoplasme et le sue 
nucleaire, comme entre deux liquides immiscibles, etc.;" Alexeieff, Sur la mitosc 
dite ''primitive," Arch. f. Protistenk. xxix, p. 357, 1913. 

f The aj^pearance of " vacuolation " is a result of endosmosis or the diffusion 
of a less dense fluid into the denser plasma of the cell. Caeferis 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 
(Biitschli, 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. Wurzburg, 1872, of. Massart, Arch, de Biol, lx, 
p. 515, 1889). Actmophrys 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 enlarge- 
ment 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. 

t Cf, p. 660. 


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 
difEerence" (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 

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 pecuHar substance called chromatin, and a coarse 
meshwork of a protoplasmic material known as "Hnin" or achro- 
matin; the outer protoplasm, or cytoplasm, is generally believed 
to consist throughout of a sponger work, or rather alveolar mesh- 
work, 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 

* 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. 


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 epithehum 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 morpho- 
logical "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 "hues of force" are rendered visible by con- 
catenation 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 w^e 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*, if we spread a layer of salt solution over a level 

* Theorie physico-chimique de la Vie, p. 73, 1910; Mechanism of Life, p. 56, 


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 travelUng 
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 probabiUty, the dynamical 
polarity, or asymmetry of the cell is a very complicated phenome- 
non : for the obvious reason that, in any system, one asymmetry 
^^'i^ tend to beget another. A chemical asymmetry will induce an 
inequahty of surface-tension, which will lead directly to a modifi- 
cation 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 morph- 
ologist. 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 pheno- 
mena 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 

In the hitherto quiescent, or apparently quiescent cell, we per- 
ceive certain movements, which correspond precisely to what must 
accompany and result from a "polarisation" of forces within the 


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 sub- 
stance 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 

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-fil ngs, 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 


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*, 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 hnes 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 "Biitschli space" of the histologistsf. On the other 
hand, the lines of force constituting the spindle will be less con- 
centrated 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 httle more closely into the structure of this body, 
and into the changes which it presently undergoes. 

Within its spherical outhne (Fig. 42), it contains an "alveolar" 

* Whence the name "mitosis" (Greek /j-ltos, 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 earher students, such as 
Van Beneden (Rech. sur la maturation de I'oeuf, 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 dechned 
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.) 

f Cf. Biitschli, 0., Ueber die kiinsthche Nachahmung der karj'okinetischen 
.Figur, Verh. Med. Nat. Ver. Heidelberg, v, pp. 28-41 (1892), 1897. 




mesh work (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 Biitschli was the first to shew) are the minute alveolar 
networks which are to be discerned in the cytoplasm of the cell*, 
and which we now know to be not inherent in the nature of 

Fig. 42. 

Fig. 43. 

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. 

* Arrhenius, in describing a typical colloid precipitate, does so in terms that 
are very closely applicable to the ordinary microscopic appearance of the protojjlasm 
of the cell. The precipitate consists, he says, "en un reseau d'mie substance 
solide contenant peu d'eau, dans les mailles duquel est inclus un fluide contenant 
un coUoide dans beaucoup d'eau. ..Evidemment cette structure se forme 
a cause de la petite difference de poids specifique des deux phases, et de la con- 
sistance gluante des particules separees. qui s'attachent en forme de reseau." Rev. 
Scientifique, Feb. 1911. 


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 diu'ing 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* have altogether denied the 
existence in the living cell-protoplasm of a network or alveolar "foam"; 
others t 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 

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 sti'uctures, 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 j. 

* F. Schwartz, in Cohn's Beifr. z. Biologie der Pflanzen, v, p. 1, 1887. 

t Fischer, Anat. Anzeiger, ix, p. 678, 1894, x, p. 769, 1895. 

J See, in particular, W. B. Hardy, On the structure of Cell Protoplasm, Journ. 
of Physiol. XXIV, pp. 158-207, 1889; also Hober, Physikalische Chemie der Zelle 
uiid der Gewebe. 1902. Cf. (int. al.) Flemming, Zellsubstanz, Kern und Zelltheilung 
1882, p. 51, etc. 




The following is a brief epitome of the visible changes undergone 
by a typical cell, leading up to the act of segmentation, and con- 
stituting the phenomenon of mitosis or caryokinetic division. In 
the egg of a sea-urchin, we see with almost diagrammatic com- 
pleteness what is set forth here*. 

1. The chromatin, which to begin with was distributed in 
granules on the otherwise achromatic reticulum (Fig. 42), concen- 
trates to form a skein or sfireme, 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 

Fig. 44. 

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 cyto- 
plasm, 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 

* My description and diagrams (Figs 42 — 51) are based on those of Professor 
E. B. WUson. 




there each becomes surrounded by a system of radiating Unes, 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 he the chromo- 
somes (Figs. 45, 46). These chromosomes now arrange them- 
selves midway between the poles of the spindle, where they form 
what is called the equatorial plate (Fig. 47). 

4. Each chromosome splits longitudinally into two : usually 

equatorial plate 

spindle fibres 

Fig. 46. 

Fig. 47. 

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 




(Fig. 50). They fuse together and form once more an alveolar reti- 
culum and may occasionally at this stage form another spireme. 

central spindie 


cplit chrcmosomc 2 

Fig. 48. 

Fis. 49. 

A boundary or surface wall is now developed round each recon- 
structed nuclear mass, and the spindle-fibres disappear (Fig. 51). 
The centrosome remains, as a rule, outside the nucleus. 

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. 


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*. 

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 imphcation 
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 

* The reference numbers in the following account refer to the paragraphs and 
figures of the preceding summary of visible nuclear phenomena. 


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 toivards regions of greater or 
less force according as its 'permeability is greater or less than that oj 
the surrounding medium*. In the common experiment of placing 
iron-fiHngs between the two poles of a magnetic field, the fihngs 
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 surround- 
ing, or adjacent, field. 

Now, in the field of force whose opposite poles are marked by 

* 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.) 

T. G. 12 


the centrosonies the nucleus appears to act as a more or less perme- 
able 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 * : 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 alveoh (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 

* On the effect of electrical influences in altering the surface-tensions of the 
colloid particles, see Bredig, Anorganische Fermente, pp. 15, 16, 1901. 


E. B. Wilson's*), 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 chromo- 
somes, 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 per- 
meabilities 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 permeabihty, or 
possibly to differences in "surface-charge," — the fact is that the 
chromatin substance now behaves after the fashion of a "dia- 
magnetic" 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 com- 
paratively 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, 

* The Cell, etc. p. 66. 





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." 

After the spireme has broken up into separate chromosomes, 
these particles come into a position of temporary, and unstable, 

Fig. 52. Chromosomes, undergoing splitting and separation. 
(After Hatschek and FJemming, diagrammatised.) 

equilibrium near the periphery of the equatorial plane, and 
here they tend to place themselves in a symmetrical arrange- 
ment (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 arrange- 
ment of the separate granules, in Ascaris megalocephala, Lillie* 

* Lillie, R. S., Amer. J. of Physiol, viii, p. 282, 1903. 




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 svmmetrical 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 

-Fig. 53. Annular chromosomes, formed in the spermatogenesis of 
the Mole-cricket, (From Wilson, after Vom Rath.) 

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. 

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, 




the chromosomes, which had at first been apparently repelled 
from the poles towards the equatorial plane, should then be spht 
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 chromo- 
somal matter differs in permeability from the medium, that is to 


Fig. 54. 

say the cytoplasm, in which it Ues, let us now make the further 
assumption that its permeabihty is variable, and depends upon the 
strength of the field. 

In Fig. 54, we have a field of force (representing our cell), 
consisting of a homogeneous medium, and including two opposite 
poles : fines 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 (/x) depends 
on the strength of the field F. At two field-strengths, such as 
Fa, Ff,, let the permeability of the body be equal to that of the 




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^ and F^- 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 

The locus 2^6 is therefore a locus of stable position, towards 
which the body tends to move ; the locus Fg, is a locus of unstable 
position, from which it tends to move. If the body were placed 


Fig. 55. 

across F^, it might be torn asunder into two portions, the split 
coinciding with the locus F^. 

Suppose a number of such bodies to be scattered throughout 
the medium. Let at first the regions F^ and F^ 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 calhng F^ and F^ are meanwhile situated somewhat farther 
from the axis than in our figure, that (for instance) F^ is situated 
where we have drawn Fj,, and that Fj, 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 fine 
do not diverge very far from one another beyond Fa', in other 


words, if, when situated in this region, the permeabihty of the 
bodies is not very much in excess of that of the medium. 

Let the poles now tend to separate farth-er 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^, F^,, will close in to nearer relative distances from the 
poles. In doing so, when the locus F„ 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„ is in some median position (Fig. 56). 

When the locus F^ has passed entirely over the body, the body 
tends to move towards regions of weaker force ; but when, in 


Fig. 5n. 

turn, the locus F^, 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 hues of force, the so-called "mantle-fibres" of the 

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, 

* 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. 


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 
alkahne fluid during its acid phase to the anode, and to the kathode 
during its alkahne 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 deahng 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 demon- 
strate, by such illustrations as these, that, whatever be the actual 
and as yet unknown modiis 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 *." 

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 remark- 
able analogies to the diagrammatic phenomena of caryokinesis, as 
shewn in the annexed figure f. 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 

* M. Foster, Lectures on the History of Physiology, 1901, p. 62. 
t Op. cit. pp. 110 and 91. 




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 hquid 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 hues of force of a magnetic 
field ; and other particles floating, though not necessarily pulsating, 
in the hquid field, tend to be attracted or repelled by the pulsating 

Fig. 57. Artificial caryokinesis (after Leduc), for comparison 
with Fig. 41, p. 169. 

bodies according as they are lighter or heavier than the surrounding 
fluid. Moreover (and this is the most remarkable point of all), 
the fines 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 attrac- 
tion, takes place between the two poles*. 

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 

* Lamb, A. B., A new Explanation of the Mechanism of Mitosis, Journ. Exp. 
Zool. y, pp. 27-33, 1908. 


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*," 
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 conducting media "j". 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, 

* Amer. J. of Physiol, via, pp. 273-283, 1903 [vide, supra, p. 181); cf. ibid. 
XV, pp. 46-84, 1905. Cf. also Biological Btilleti^i, iv, p. 175, 1903. 

f In Uke manner Hardy has shewn that colloid particles migrate with the 
negative stream if the reaction of the surrounding fluid be alkahne, 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. xxn% 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 PermeabiHty of Cells, Journ. of Exper. Zool. x, pp. 508-556, 1911). 


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*, 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 eggt; just as other physical rhythms, 
for instance in the production of CO2, had already been shewm 
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 recon- 
stituted 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 

* On Differences in Electrical Potential in Developing Eggs, Atner. Journ. of 
Physiol, xn, pp. 241-275, 1905. This paper contains an excellent summary of 
various physical theories of the segmentation of the cell. 

t Gray has recently demonstrated a temporary increase of electrical con- 
ductivity in sea-urchin eggs during the process of fertihsation (The Electrical 
Conductivity of fertilised and unfertiUsed Eggs, Journ. Mar. Biol. Assoc, x, pp. 
50-59, 1913). 




associated with surface-tension changes, of which the first step 
is a minute puckering of the surface-skin, a sort of interdigi- 
tation with the surrounding medium. For instance, Schewia- 
kofE has observed in Euglypha* 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. 

Fig. 58. Final stage in tfte first 
segmentation of the egg of Cerebra- 
tulus. (From Prenant, after Coe.)| 

Fig. 59. Diagram of field of force 
with two similar poie.?. 

By diffusion, hand in hand with surface-tension, the alveoli of 
the nuclear meshw^ork 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. 

* Schewiakoff, Ueber die karyokinetische Kerntheilung der Euglypha alveolata, 
Morph. Jahrb. xm, pp. 193-258. 1888 (see p. 216). 

f Coe, W. R., Maturation and Fertihzation of the Egg of Cerebratuhis, Zool. 
Jahrbiicher {Anat. Abth.), xii, pp. 425-476, 1899. 


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 w^ithout any obvious classificatory value or other 
biological significance*. 

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 cyHndrical 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 siee 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 

* Thus, for example, Farmer and Digby (On Dimensions of Chromosomes 
considered in relation to Phylogeny, Phil. Trails. (B), ccv, pp. 1-23, 1914) have 
been at pains to shew, in confutation of Meek (ibid, ccni, pp. 1-74, 1912), that the 
width of the chromosomes cannot be correlated with the order of phylogeny. 


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 hquid surface-film of a minute spherical cell, 
local, and symmetrically locahsed, 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 in- 
equalities or asymmetry of surface-tension. For instance, Biitschli 
suggested, some forty years ago*, 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 f. This streaming move- 
ment 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 tetids 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 

* Cf. also Arch. f. Entw. Mech. x, p. 52, 1900. 

t Cf. Loeb, Am. J. of Physiol, vi, p. 432, 1902. ; Erlanger, Biol. Centralbl. 
xvn, pp. 152, 339, 1897 ; Conklin, Biol. Lectures, Woods Holl, p. 69, etc. 1898-9. 


diameter across which the thread Hes, 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* 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 choUn, 
or of soaps of cholin, through the splitting of lecithin in nuclear 

But purely chemical changes are not of necessity the funda- 
mental cause of alteration in the surface-tension of the egg, for 
the action of electrolytes on surface-tension is now well known 
and easily demonstrated. So, according to other views than 
those with which we have been dealing, electrical charges are 
sufficient in themselves to account for alterations of surface- 
tension ; while these in turn account for that protoplasmic 
streaming which, as so many investigators agree, initiates the 
segmentation of the egg|. A 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 pheno- 
menon of adsorptio7L 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. 

Hov/ever, 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 equihbrium of 
the egg is never complete J. And we may simply conclude the 

* Robertson, T. B., Note on the Chemical Mechanics of Cell Division, Arclu 
/. Entw. Mech. xxvn, p. 29, 1909, xxxv, p. 692. 1913. Cf. R. S. Lilhe, J. Exp. 
Zool. XXI, pp. 369—402, 1916. 

t Cf. D'Arsonval, Arch, de Physiol, p. 460, 1889; Ida H. Hyde, op.cit. p. 242. 

I Cf. Plateau's remarks {Statique des liquides, n, p. 154) on the tendency towarda 
equilibrium, rather than actual equilibrium, in many of his systems of soap-films^ 


matter by saying that, if you have caryokinetic figures developing 
inside the cell, that of itself indicates that the dynamic system 
and the locahsed 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 fertihsation, — 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*. Somewhere or other, near or far away, within the egg, hes 
its own nuclear body, the so-called female pronucleus, and we 
find after a while that this has fused with the head of the sperma- 
tozoon (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 

* 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. 

T. G. 13 


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 he, 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 hne 
joining the two bodies ; (3) the two bodies will meet a httle before 
they reach the centre ; and, having met and fused, will travel 
on to reach the centre in a straight hne. 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 httle 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 

The intracellular phenomena of which we have now spoken 
have assumed immense importance in biological hterature 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 

* Fol, H., Recherches sur la fecondation, 1879. Roux, W., Beitrage zur 
Entwickelungsmechanik des Embryo, Arch. f. Mikr. Anat. xix, 1887. Whitman, 
C. O., Ookinesis, Journ. of Morph. i, 1887. 


of the chemical, electrical or other forces which are associated 
with it. But there is another consideration which would lead us 
to suspect that many features in the structure and configuration 
of the cell are of very secondary biological importance ; and that 
is, the great variation to which these phenomena are subject in 
similar or closely related organisms, and the apparent impossibility 
of correlating them with the peculiarities of the organism as a 
whole. "Comparative study has shewn that almost every detail 
of the processes (of mitosis) described above is subject to variation 
in different forms of cells*." A 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 Hes 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 Cliaetofterus. In 
one and the same species of worm {Ascaris megalocefhala), 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 t- 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 
beheving that these and many similar phenomena are in no way 
specifically related to the particular organism in which they have 

* Wilson. The Cell. p. 77. 

t 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. 



been observed, and are not even specially and indisputably con- 
nected with the organism as such. They include such manifesta- 
tions 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 phytogeny. When we find, as we are not long of 
finding, that our phylogenetic hypotheses, as drawn from em- 
bryology, 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 


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 effect- 
uated 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 dehmi- 
nation 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 


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, indepen- 
dently 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-Hfe. "Kern und Protoplasma sind nur vereint 
lebensfahig," 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 expres- 
sions denoting two difEerentiated areas in a common structural 

Endless discussion has taken place regarding the centrosome, 
some holding that it is a specific and essential structure, a per- 
manent corpuscle derived from a similar pre-existing corpuscle, a 
"fertihsing 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 centro- 
somes, that a centrosome inaj 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 "shghtly 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 hquid film, the surface of the cell is the seat of 


important forces, capillary and electrical, which play an essential 
part in the dynamics of the cell. Even in the thickened, largely 
soHdified cellulose wall of the plant-cell, apart from the mechanical 
resistances which it affords, the osmotic forces developed in con- 
nection 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 

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 actionf. ' 

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 metazoonj. 

Whitman, Adam Sedgwick §, 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||. 
As Goethe said long ago, "Das lebendige ist zwar in Elemente 

* Theory of Cells, p. 191. 

f The Cell in Development, etc. p. 59; cf. pp. 388, 413.. 

J E.g. Briicke; Elementarorganismen, p. 387: "Wir miissen in der Zelle einen 
kleinen Thierleib sehen, und diirfen die Analogien, welche zwischen ihr und den 
kleinsten Thierformen existiren, niemals aus den Augen lassen." 

§ Whitman, C. 0., The Inadequacy of the Cell-theory, Journ. of Morphol. 
vm, 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, xxxvm, pp. 137-174, 1896. 

II Cf. Hertwig, 0., Die Zelle und die Gewebe, 1893, p. 1; "Die Zellen, in welche 
der Anatom die pflanzlichen und thierischen Organismen zerlegt, sind die Trager 
der Lebensfunktionen ; sie sind, wie Virchow sich ausgedriickt hat, die 'Lebensein- 
heiten.' Von diesem Gesichtspunkt aus betrachtet, erscheint der Gesammtlebens- 
process eines zusammengesetzten Organismus nichts Anderes zu sein als das hochst 
verwickelte Resultat der einzelnen Lebensprocesse seiner zahbeichen, verschieden 
functionirenden Zellen." 


zerlegt, aber man kann es aus diesen nicht wieder zusammenstellen 
und beleben;" the dictum of the Cellular fathologie being just 
the opposite, "Jedes Thier erscheint als eine Summe vitaler 
Einheiten, von denen jede den vollen CharaJcter des Lebens an 
sich tragi." 

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*." 
So Rauber declared that, in the whole world of organisms, "das 
Ganze lief ert die Theile, nicht die Theile das Ganze : letzteres 
setzt die Theile zusammen, nicht diese jenes|." 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, Kke a froth, into its fabric, seems to me certainly 
and obviously to exist. As Whitman says, "the fact that physio- 
logical 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 biokgyj." 

* Journ. of Morph. viii. p. 653, 1893. 

•f Neue Grundlegungen zur Kenntniss der Zelle, Morph. Jahrb. vin, pp. 272, 
3] 3, 333, 1883. 

I Journ. of Morph. ii, p. 49, 1889. 



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 partaJces of fluidity, and 
enables the colloid to become a vehicle for liquid diffusion, like 
water itself*." When we can deal with protoplasm in sufficient 
quantity we see it flow ; particles move freely through it, air- 
bubbles and hquid 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 
pecuharities of the forms of living things. 

The older naturalists, in discussing the differences between 
inorganic and organic bodies, laid stress upon the fact or state- 
ment that the former grow by "agglutination," and the latter by 

* Phil. Trans, cli, p. 183, 1861; Researches, ed. Angus Smith, 1877, p. 553. 


what they termed "intussusception." The contrast is true, 
rather, of solid as compared with jelly-Hke 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 beheve 
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*. 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. Li 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 produc- 
tion of planes -f (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 

* 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 (tod 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, Schonfiiess, Barlow 
and others, see Tutton's Crystallography, chap, ix, pp. 118-134, 1911. 

f 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. 


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 dift'erent way, very much 
as a piece of glue swells up in water, by "imbibition," or by inter- 
penetration into and throughout its entire substance. The semi- 
fluid colloid mass takes up water, partly to combine chemically 
with its individual molecules*, 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 con- 
cerned, 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 rearrange- 
ment 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 every- 
where a rounded and more or less bubble-like external formf. 
So, when the same school of older naturahsts 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 intussus- 

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, 

* This is what Graham called the water of gelatination, on the analogy of xvater 
of crystallisation ; Chem. and Phys. Researches, p. 597. 

f 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. 


in his great work on so-called Fluid Crystals*, 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 arrange- 
ment 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 f. 
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 Biitschli has already pointed out 
the possible great importance to the biologist of the various 
phenomena which Lehmann has described t. 

So far then, as growth goes on, unafltected 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 

* Lehmann, 0., Flussige Krysfalle, soivie Plasticitdt von Krystallen im allge- 
meinen, etc., 264 pp. 39 plL, Leipsig, 1904. For a semi-popular, illustrated account, 
see Tutton's Crystals (Int. Soi. Series), 1911. 

t 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." 

X Cf. Przibram, H., Kristall-analogien zur Entwickelungsmechanik der Organ- 
ismen, Arch. f. Entw. Mech. xxii,p. 207, 1906 (with copious bibliography); Lehmann, 
Scheinbar lebende Kristalle und MyeUnformen, ibid. xxAa, p. 483, 1908. 


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*. 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 Ave have solid particles imbedded in a fluid, wherever 
we have a non-homogeneous fluid or semi-fluid such as a particle 

* The idea of a "surface-tension" in liquids was first enunciated by Segner, 
De figuris superficierum fluidarum, in Comment. Soc. Boy. Gottiiigen, 1751, p. 301. 
Hooke, in the Micrographia (1665, Obs. viii, etc.), had called attention to the 
globular or spherical form of the httle 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...." 


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 unhke. 

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 inter- 
actions 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 t|;ie phenomenon in cases where a portion of liquid 
is subject to no other than its oivn 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 


pressure is of the magnitude of thousands of atmospheres, — a con- 
clusion 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 multiphed 
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 Cq 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 
p {e being what we call the surface energy). It is obvious that the 
total energy consists of the internal flus 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 . ^tp) e^ + S . lltpe. 

But this is equivalent to writing : 

= Mco + S .i:tp{e - Co) ; 

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^, that is to say to the difference 
between the unit values of the internal and the surface energy. 


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 fotential 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 Me^ 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 Cq, 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 froprietate gaiidentes" 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. 


111 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 remark- 
able 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 putbetter than in the following words of 
Mach's*. "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 "j". 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 

* Science of Mechanics, 1902, p. 395 ; see also Mach's article Ueber die physika- 
lische Bedeutung der Gesetze der Symnietrie, Lotos, xxi, pp. 139-147, 1871. 

t 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. 

T. G. 14 

210 ' THE FORMS OF CELLS [ch. 

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* 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 move- 
ment 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 ovter 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, Nageli 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 f. 

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 

* Beitrage z. Physiologie d. Protoplasma, Pfluger''s Archiv, n, p. 307, 1869. 
t Poggend. Annalen, xciv, pp. 447-459, 1855. Cf. Strethill Wright, Phil. 
Mag. Feb. 1860. 


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*, 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 f. 
Thus arise the various fluctuations of surface tension and the 
various phenomena of amoeboid form and motion, which Biitschli 
and others have reproduced or imitated by means of the fine 
emulsions which constitute their "artificial amoebae." A multi- 
tude 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 

* Haycraft and Carlier pointed out {Proc. E.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 soUd 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 equihbrium. The same fact has recently been recorded anew 
by Ledingham (On Phagocytosis from an adsorptive point of view, Journ. of Hygiene, 
xu, 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, Pflilger's Archiv, Liii, 1893. Jensen remarks that in OrbitoUtes, the 
pseudopodia issuing through the pores of the shell first float freely, then as they 
grow longer bend over tiU they touch the ground, whereupon they begin to display 
amoeboid and streaming motions. Verworn indicates (Allg. Physiol. 189.5, p. 429), 
and Davenport says {Experim. Morphology, ii, p. 376) that "this persistent cUnging 
to the substratum is a ' thigmotropic ' reaction, and one which belongs clearly to 
the category of 'response.'" (Cf. Piitter, 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. 

t Cf. Pauli, Allgemeine physikalische Chemie d. Zellen u. Gewebe, in Asher-Spiro's 
Ergebnisse der Physiologic, 1912; Przibram, Vitalitdt, 1913, p. 6. 



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*. 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 f. 

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 o£E 
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 homo- 
geneity and equilibrium have been attained and the potential 

* 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; Delhnger, 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. 

"]" 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 e'pipolique 
of Dutrochet : imtil van der Mensbrugghe shewed that differences of surface tension 
were enough to account for this whole series of phenomena (Sur la tension super- 
ficielle des hquides consideree au point de vue de certains mouvements observes 
a leur surface, Mem. Cour. Acad, de Belgique, xxxiv, 1869; cf. Plateau, p. 283). 


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." 

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 of? p- gQ^ 
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 *, 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 ^he 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 

* Cf. infra, p. 306. 


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 con- 
ditional 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 Euas- 
trum) 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 complete- 
ness 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 


wliich, 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*: 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 

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 

* Cf. p. 32. 


surface which can be drawn continuously across the uneven 

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*) 
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 |. 

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 (/j) varies directly as the tension {T), and inversely as 
the radius of curvature {R) : that is to say, p = T/R, per unit of 

* Or, more strictly speaking, unless its thickness be less than twice the range 
of the molecular forces. 

I It follows that the tension, depending only on the surface-conditions, is 
independent of the thickness of the film. 


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 riiust be simply 
added to one another, and the resulting pressure j) is equal to 
T/R + T/R' or p^TiljR+ l/R') *. 

And if in addition to the pressure p, which is due to surface 
tension, we have to take into account other pressures, 7?', j)", etc., 
which are due to gravity or other forces, then we may say that 
the total 'pressure, P = p' + p" + T {l/R + l/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 {l/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 l/R + l/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 l/R + l/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 

* This simple but immensely important formula is due to Laplace {Mecanique 
Celeste, Bk x. suppl. Theorie de Vaction capillaire, 1806). 


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 inter- 
relation is expressed by the fact (first shewn by Delaunay*, 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 elHpse 
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. 

If we imagine an ellipse so to roll over a line, either of its foci 
will describe a sinuous or wavy line (Fig. 61 b) 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 elhpse, the more pronounced will be the 
sinuosity of the described roulette. If the two axes be equal, 
then our elhpse becomes a circle, and the path described by its 
rolhng 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 elhpse vanish, but the 
other be infinitely long, then the curve described by the rotation 

* Sur la surface de revolution dont la courbure moyenne est constante, Journ. 
de M. LiouvilU, vi, p. 309, 1841. 


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 difl&cult 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 mathe- 
matical 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 suc- 
cession 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 *. 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 

* 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. 


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 w^U be trans- 
formed into a cylinder. But if we draw the pipes yet farther 
apart, the cylinder will narrow in the middle into a sort of hour- 
glass 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 

Fig. 62. 

converting a cylinder into an unduloid (Fig. 62 a) ; 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. 

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 

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 


is required, in each case, of the general equation which appHes 
to them all. We shall find that with this question is closely 
connected the question of the pressures exercised by, or im- 
pinging 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 2T/R, is a positive 
quantity, involving a positive pressure P, on the other side of the 

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 


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 {IjR + 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 curva- 
ture 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. 

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 filjn, 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 equihbrium will be accordingly 




modified, and the cylindrical drop will assume the form of an 

miduloid (Fig. 64 a, b), with its dilated portion below or above, 

as the case may be ; and our cyUnder 

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, hke 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 fiat 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. 65 b. This figure is a catenoid, which, as 


Fig. 65. 

we have already seen, is, hke 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. 65 a). 


It represents, in fact, that portion of the nodoid, which 

in Fig. 66 hes between such points as o, p. While it is easy to 

draw the outhne, or meridional 

section, of the nodoid (as in 

Fig. 66), it is obvious that the 

sohd of revolution to be derived 

from it, can never be reahsed in 

its entirety : for one part of the 

solid figure would cut, or en- 
Fie. 66. . 

tangle with, another. All that 

we can ever do, accordingly, is to realise isolated portions of the 

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 cyhnder: our discs, that is to say, are adjusted to 
exercise a mechanical pressure equal to what in the former case 
was supphed by the surface-tension of the spherical caps or ends 
of the bubble. If we now increase the pressure shghtly, the 
peripheral walls will become convexly curved, exercising a pre- 
cisely 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 


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 hquid is not 
homologous, but hes 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 equiUbrium, 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 uni- 
cellular 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 sym- 
metrical. 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 suppUed by solidifica- 
tion, 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 

T. G. 15 




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 + l/R') for the cylinder is equal to the corresponding expres- 
sion, 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 = 2R; 
that is to say, the sphere in question has just twice the radius of 
the cylinder of which it forms a cap. 

And if Ob, the radius of the sphere, be equal to twice the radius 
(Ort) 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 be is equal to 
1 77". 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, 

Fig. 67. 

^Ob-ab = R{2-^3)^ 0-27R, 

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 




sphere. And these are the proportions which we recognise, ander 
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 

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 = IttR, or when 
the ratio of length to diameter is represented by tt. 

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 

15—2 . 


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 with- 
drawal 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, th'e whole oil-globule broke up in general 

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; 


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. 65 a). 

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 difiicult, 
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 equihbrium which are at the same time 
surfaces of revolution are only six in number, there is an infinite 


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 

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 indistinguish- 
able 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 Spirochsetes, 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 


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 com- 
bine together. In short, what we have said may help us to under- 
stand 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 pheno- 
mena of the organic cell. 

In a spider's web we find exemplified several of the principles 


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 cyhnder had passed its 
limit of stabihty. 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 * ; 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 

* 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. 


effect of the jerk being to disturb and destroy the unstable 
equiHbriiim of the viscid cyhnder*. Another very curious 
phenomenon here presents itself. 

In Plateau's experimental separation of a cyhnder 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 samfe 
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 corre- 
sponding 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 w^hich resolves itself 
into a comparatively tiny spherical drop, midway between the 
original and larger two. Occasionally, the same process of forma- 
tion 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 

* 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. 


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*. 

The little delicate beads which stud the long thin pseudopodia 
of a foraminifer, such as Gromia, or which in like manner appear 

Fig. 69. Hair of Tnanea, in gtycerine. (After Berthold.) 

upon the cylindrical film of protoplasm which covers the long 
radiating spicules of 6'^o6i^e>ma, represent an identical phenomenon. 
Indeed there are many cases, in which we may study in a proto- 
plasmic 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 cylin- 
drical 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 proto- 
plasm parts company with its surrounding and supporting wall, 

* J. Blackwall, Spiders of Great Britain (Ray Society), 1859, p. 10; Trans. 
Linn. Soc. xvi, p. 477, 1833. 




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*. Similar, but less regular, beads or droplets may be caused to 
appear, under stimulation by an alternating current, in the proto- 
plasmic threads within the living cells of the hairs of Tradescantia. 
The explanation usually given is, that the viscosity of the proto- 

Fig. 70. Phases of a Splash. (From Worthington. ) 

plasm is reduced, or its fluidity increased ; but an increase of the 
surface tension would seem a more likely reason f. 

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 

* The intermediate spherules appear, with great regularity and beauty, whenever 
a hquid jet breaks up into drops; see the instantaneous photographs in Poynting 
and Thomson's Proiierties of Matter, pp. 151, 152, (ed. 1907). 

■]■ Kiihne, Untersiwhungen iiber das Protoplasma, 1864, p. 75, etc. 




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)*. 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 

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'}". 
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 

* A Study of Splashes, 1908, p. 38, etc. ; Segmentation of a Liquid Annulus, 
Proc. Roy. Soc. xxx, pp. 49-60, 1880. 

f 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. 




created, along which the Hquid is led, or is driven, into the pro- 
tuberances: 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 consis- 
tency, 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. 

Fig. 72. Calycles of Campanularian zoophytes. (A) C. Integra; 
(B) C. groenlandica ; (C) C. bispinosa; (D) C. raridentafa. 

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 peri|)heral 
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 


transitory that their study is only rendered possible by "instan- 
taneous" 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 
niotions 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 configura- 
tions 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 ofen surface, or solid, of revolution. It is clear, at the same 
time, that the two series of problems are closely akin; for the 


glass-blower can make most things that the potter makes, by 
cutting oft' '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 cylin- 
drical 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*, 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 cyhnder 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 
* See a Study of Splashes, p. 54. 


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 tjie 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. 


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 elhpsoidal or cyUndrical 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 sUght and locaHsed 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 elhpsoidal 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*. 

* 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 

T. G, 16 


Our full formula of equilibrium, or equation to an elastic 
surface, is P = 'p^+ {TIE + T'/R'), where P is the internal 
pressure, fg 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 Ave 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, 2^„, was non-existent. Now in the case of a bird's egg, 
the external pressure 2>nj 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 

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.) 


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 w^atery 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 [TjR + 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*. And 
it matters little whether this viscosity or semi-rigidity be mani- 
fested 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 

* Indeed any non-isotropic stiffness, even though T remained uniform, would 
simulate, and be indistinguishable from, a condition of non-stiffness arid non- 
isotropic T. 



surface area of the cell. A slight inequality in two opposite 
directions will produce the ellipsoid cell, and a very great in- 
equality will give rise to the cyHndrical cell*. 

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 equi- 
librium. 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 oogonium 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 cyHnder. 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. 

* 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. 


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 equihbrium. 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 equi- 
librium 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 httle organisms belong to an order of 
magnitude in which form is mainly, if not wholly, conditioned and 
controlled by molecular forces, is due the hmited 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 




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 Uttle " 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 oiform, 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 for7n : we may often be inclined, in short, to ascribe 

Fiw. 73. A flagellate "monad," Distigma 
proteus, Ehr. (After Saville Kent.) 

Fig, 74. Noctiluca miliaris. 

to them a physiological (sometimes a "pathogenic"), rather than 
a morphological specificity. 

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 


usually see signs of an unequal tension of the membrane in the 
neighbourhood of the base of the fiagellum. Here the tension is 
usually less than elsewhere, and the radius of curvature is accord- 
ingly 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 fiagellum springs from a depression 
in the otherwise uniformly rounded cell. In this case the explan- 
ation 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). 

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 

Fig. 75. Various species of Vorticella. (Mostly after Saville Kent.) 

beautiful series of unduloids, or partial unduloids, in every grada- 
tion 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 w^e 
suspend an oil-globule between two unequal rings, or blow a 
soap-bubble betw^een 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 pecuKar 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 'jwint (Vapfui for the adjacent 
film. We shall deal with this point again in the next chapter. 




Precisely the same series of unduloid forms may be traced in 
even greater variety among various other families or genera of the 

Fig. 76. Various species of Salpingoeca. 


Fig. 77. Various species of Tintinnus, Dinobryon and Codonella. (After 
Saville Kent and others.) 

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, Tin- 
tinnus, 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 

Fig. 78. 





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 proto- 
plasm should be aggregated by molecular 
forces in its surface layer ; that under this 
adsorptive process, the conditions con- 
tinuing 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 
?ninimae areae, were available, the undu- 
loid 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. 

Fig. 80. Trachelophyllum. 
(After Wreszniowski.) 


In very many cases (of which Fig. 80 is an example), we have 
an unduloid form exhibited, not by a surrounding pelhcle 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 comphcated 
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 treat- 
ment 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 bio- 
logical 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 reriim apparenti, siimmvs 
conspiciatur Naturae ordo." In hke manner the physicist records, 
and is entitled to record, his many hundred "species" of snow- 
crystals*, 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 

* 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 inter- 
mingled in inexhaustible variety. (Cf. e.g. J. Glaisher, III. London News, Feb. 17, 
1855; Q.J. M.S. iii, pp. 179-185, 1855). 


the element of time, and of succession, or discuss their origin and 
affihation 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 n\yriads of 
vears*. 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 pro- 
gressive 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 ivhich they fresent, is due to the fact of their being 
related by descenti" But this great generahsation 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 

* Cf. Bergson, Creative Evolution, p. 107: "Certain Foraminifera have not 
varied since the Sihirian epoch. Unmoved witnesses of the innumerable revolu- 
tions that have upheaved our planet, the Lingulae are today what they were at 
the remotest times of the palaeozoic era." 

t Ray Lankester, A.M.N.H. (4), xi, p. .321, 1873. 


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 di£6-cult to 
discern, or to apply, the guiding principle of affiliation oi genealogy. 
Among the more aberrant forms of Infusoria is a little species 
known as Trichodina fediculus, a parasite on the Hydra, or fresh- 
water polype (Fig. 8L ) 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 oppo- 
site, 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^ 

* Leidy, Parasites of the Termites, J. Nat. Sci., Philadelphia, vm, pp. 425- 
447, 1874—81; cf. Saville Kent's Infusoria, ii, p. 551. 




which reminds us of the cylindrical spiral of a Spirillum among 
the bacteria. In Dinenympha we have a symmetrical figm-e, 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 curva- 
ture, and therefore of equilibrium, as truly, and in the same sense, 
as the cylinder itself. 


Fig. 82. Dinenympha gracilis, Leidy. 

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 ripphng 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 


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 sur- 
rounding 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 
Fig. 83.^ 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*. 

* Op. cit. p. 79. 



Among the Foraminifera we have an immense variety of forms, 
Avhich, in the hght 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 cham- 
bered 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 equi- 
librium ; 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 sohd 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 Ave 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 w^e 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* " 
* Brady, Challenger Monograph, pi. xx, p. 233. 




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. 
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 

Fig. 84. Various species of Lagena. (After Brady.) 

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 



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*. 

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. 

Fig. 8.5. (After Darling. 

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 

* 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 ete dit, le depot provenant du lavage des 
fucus dans un flacon que Ton remplit de nouveUe eau, on voit au bout d'une heure 
environ les animaux [Gramid dujardinii] se mettre en mouvement et commencer 
a grimper. Six heures apres ils tapissent I'exterieur du flacon, de sorte que les plus 
eleves sont a trenjte-six ou quarante-deux millimetres du fond; le lendemain 
beaucoup d'entre eux, apres avoir atteint le niveau du liquide, ont continue a ramper 
'a sa surface, en se laissani pendre au-dessous comme certains mollusques gastero- 
podes." (Dujardin, F., Observations nouvelles sur les pretendus cephalopodes 
microscopiques, Ann. des Sci. Nat. (2), iii, p. 312, 1835.) 

T. n. 17 


over the surface and gradually forms a hanging drop, approxi- 
mately hemispherical ; but as more liquid is added the drop 
sinks or rattier 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*. 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 

Many species of Lagena are complicated and beautified by a 
pattern, and some by the superadd! tion 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 
* Cf. Boas, Spolia Atlantica, 1886, pi. 6. 


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 hmit, 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 
Darlmg'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 under- 
stood, but surface-tension, diffusion and cohesion doubtless play 
their respective part? therein*. 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. 

The folded or pleated pattern is doubtless to be explained, in 
a general way, by the shrinkage of a surface-film under certain 

* 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 hquids described by Professor James Thomson in 
1882 [Collected Papers, p. 136); and to the tourbiUo7is cellulaves of Prof. H. Benarcl 
(Ann. de Chimie (7), xxni, pp. 62-144, 1901, (8), xxiv, pp. 563-566, 1911), 
Rev. geae'r. des Sci. xi, p. 1268, 1900; cf. also E. H. Weber. Prggend. 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 Cirant, 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 
locahsed 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). 



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*, who let a film of albumin or of 
resin set and harden upon a surface of quicksilver, and found 
that the little solid pelhcle 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 sym- 
metry, 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. semistrinta. All these various forms of surface 
can be imitated, or rather can be precisely reproduced, by the art 
of the glass-blower f. 

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 W( r hington has so beauiifully shewn J. 

* Ueber piiysikalischen Eigenschaften dunner, fester Lamellen, S.B. Berlin. 
Akad. 1888, pp. 789, 790. 

t 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, Foramini/eren 
der Pla:'kton-Exi:)edition, 1911, i, p. 21). If this were true it would be of funda- 
mental importance : but this book of mine would not deserve to be written. 

X A Study of Splashes, p. 116. 


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 + f. 
Our new equation of equilibrium, then, in place of P = Tjr + T/r' 

P^ T+tT+t 

P + P = \ 7— 

and by subtraction, 

]) = t/r + t/r'. 

Now if r < r', t/r > t/r'. 

Therefore, in order to produce the small increment of pressure f, 
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. 

The longitudinal wrinkhng 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 theo- 
retical 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 




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 ynoments at any 
point in this region of the shell-wall. And it is at once obvious 
that, in any portion of the narrow neck, fl,exnre of a wall in a 

Fig. 87. Nodosaria scalaris, 

Fig. 88. Gonangia of Campanularians. 
(a) C. gracilis; (b) C. grandis. 
(After Allman.) 

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 




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. 

Passing from the solitary flask-shaped cell of Lagena, we have, 
in another series of forms, a constricted cylinder, or succession 

a h c d e f g 

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. 

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 


Morey and Draper made many years ago of melted resin are a 
very similar if not identical phenomenon*. 

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 js 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 "j". 
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 

* See SiUi7na7i''s Journal, n, p. 179, 1820; and cf. Plateau, op. cit. ii, pp. 134, 

t 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 con- 
^sequence of the different conditions of existence in fresh water and in salt. 


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 sihceous 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 neigh- 
bouring 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 Tf^.,, Tf^, 
and T^,p, equilibrium will be attained when the angle of contact 
between the fluid protoplasm and the filament is such that 

T — T 

cos a = ^^-, -^ . 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 




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 a 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. 

In the simpler Eadiolaria, 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 

Fig. 90. A, Trypanosoma tineae (after Minchin); B, Spirochaeta anodontae 
(after Fantham). 

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 mesh work," 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 




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*. 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 "j". It is 

Fig. 91. A, Trichomonas muris, Hartmann; B, Trichomastix serpentis, Dobell; 
C, Trichomonas angusta, Alexeieff. (After Kofoid.) 

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. 

This mode of formation of the undulating membrane or frill 
appears to be confirmed by the appearances shewn in Fig. 91. 

* Cf. Doflein, Lehrbuch der Protozoenkmide, 1911, p. 422. 

t Cf. Minchin, Introduction to the Study of the Protozoa, 1914 p. 293, Fig. 127. 




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*. 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 super- 
ficial protoplasm has been drawn out into a film, between the 
flagellum {a) and the adjacent surface or edge of the body (6). 

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 undulat- 
ing 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 back- 
wards 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 fseudo- 
undulating membrane (Fig. 92). The movements of this structure 
were so exactly those of a true undulating membrane that it was 

* Cf. C. A. Kofoid and Olive Swezy, On Trichomonad Flagellates, etc. Pr. 
Amer. Acad, of Arts and Sci. li, pp. 289-378, 1915. 

Fig. 92. Herpetomonas assuming 
the undulatory membrane of' a 
Trypanosome. (After D. L. 
Mackinnon. ) 


difficult to believe one was not dealing with a small, blunt 
trypanosome*.'' 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 

There is a genus closely allied to Trypanosoma, viz. Trypano- 
plasma, which possesses one free flagellum, together with an 
undulating membrane ; and it resembles the neighbouring genus 
Bodo, save that the latter has two fiagella 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 "Her- 
petomonas 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 apd 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. halhiani 

* D. L. Mackinnon, Herpetomonads from the Alimentary Tract of certain 
Dungtlies, Parasitoloyy, ni, p. 268, 1910. 


(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 
fiagellum of the Trypanosome), I ascribe the formation of the 
undulating membrane which we find, for instance, in the spermato- 
zoa 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 seenis to be incapable of explanation by theories of 

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 


(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. 

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 ig 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 character- 
istic 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. 


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 cor- 
puscles*. 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 f ; 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 J. The whole 
phenomenon would be comparatively easy to understand if we 
might postulate a stifEer 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 m^ute bodies such as a blood-corpuscle ; and (as Professor 
Peddie suggests to me) it is by no means impossible that curvature 

* Prnc B y. Soc. xii, pp. 251-257, 1862-3. 

t Cf. (int. al.) Lehmann, Ueber scheinbar lebende KristaUe und Myelinformen, 
Arch. f. Enho. Mech. xxvi, p. 483, 1908; Ann. d. Physik, xliv, p 969, 1914. 

f Cf. B. Moore and H. C. Roaf, On the Osmotic Equilibrium of the Red 
Blood Corpuscle, BiocJiem. Journal, in, p. 55, 1908. 




of the surface may itself modify the osmotic or perhaps the adsorp- 
tive 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*. 

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 stif? processes ; 
in Galathea we have a still more complex form, with long and 

Fig. 94. Sperm-cells of Decapod Crustacea (after Koltzoff). a, Inachus scorpio; 
b, Galathea squamifera ; c, do. after maceration, to shew spiral fibrillae. 

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 
assum.e in particular — the density) of the surrounding medium is 

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," 

* 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. 

T. G. 18 




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." Li the case of the sperm-cells of Inachus, certain 
quantitative experiments have been performed*. 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 corre- 
sponding 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 


Pig. 95. Sperm-cells of Inachus, as they appear in saline solutions of 
varying density. (After Koltzoff.) 

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 f. 

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. 

* Koltzoff, N. K. , Studien iiber die Gestalt der Zelle, Arch. f. mihrosk. Anaf. 
Lxvn, pp. 364-571, 1905; Biol. Centralbl. xxm, 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. 

t Cf. supra, p. 129. 


% concentration of salts in wkicli 

the sperm-cell of Inachus 

assumes the form of 

%. a 

fig. c 

Sodium chloride 



Sodium nitrate 



Potassium nitrate 



Acetic acid 



Cane sugar 



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 with- 
drawing 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 KoltzofE) 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 probabihty we 
must in like manner attribute the 

peculiar spiral and other forms, for -p- nc o ' n j; r> 

^ ^ ' i< ig. 96. bperm-cell of Dromia. 

instance of many Infusoria, to the (After KoltzofE.) 


276 THE FORMS OF CELLS [ch. v 

presence, among the multitudinous other dift'erentiations of their 
protoplasmic substance, of such more or less elastic fibrillae, 
which play as it were the part of a microscopic skeleton*. 

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 gor^e ; 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 accord- 
ingly, so long as our purpose is mofphological, 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. 

* As Bethe points out (Zellgestalt, Plateausche Fliissigkeitsfigur unci Neuro- 
fibrille, 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. 



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*. 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. 

* See for a further but still elementary account, Michaelis, Dynamics of Surfaces, 
1914, p. 22 seq.; Macallum, Oberfldchenspcmnttng und Lebenserscheinungen, in 
Asher-Spiro's Ergebnisse der Physiologic, 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 Bayhss's 
Principles of General Physiology, pp. 54-73, 1915. 


But we also saw, according to Gauss's equation, that the 
potential energy of the system will be diminished (and its diminu- 
tion 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 


or "adsorption" represents the work done, equivalent to the 
diminished potential energy of the system*. 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 con- 
centration" equilibrium is reached, its opponent being that osmotic 
force which tends to keep the substance in uniform solution or 

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 Uquid 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 f. And this is all as 

* 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 Dupre 
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 httle different from that 
of a strong one (Theorie me'canique de la chaleur, 1869, p. 376; cf. Plateau, ii, 
p. 100). 

t This identical phenomenon was the basis of Quincke's theory of amoeboid 


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*. 

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 hetero- 
geneous 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 

movement (Ueber periodische Ausbreitung von Fliissigkeitsoberflachen, etc., SB. 
Berlin. Akad. 1888, pp. 791-806; of. Pfluger's Archiv, 1879, p. 136). 

* J. WiUard Gibbs, Equilibrium of Heterogeneous Substances, Tr. Conn. Acad. 
Ill, 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, ZeUschr. f. physik. Chemie, 
Lxx, p. 129, 1910; Milner, Phil. Mag. (6), xiii, p. 96, 1907, etc. 


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 outUne of the system. 

The physiologist, as we have already remarked, takes account 
of the general phenomenon of adsorption in many ways : particu- 
larly 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 adsorp- 
tion brings into existence. For instance it plays a leading part 
in all modern theories of muscular contraction, in which phenome- 
non a connection with surface tension was first indicated by 
E'itzGerald and d'Arsonval nearly forty years ago*. 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 f. 

But restricting ourselves, as much as may be, to our morpho- 
logical 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 prehminary account, such as it is, is 
already tantamount to a description of the process of develop- 
ment 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 
pelUcle 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 

* G. F. FitzGerald, On the Theory of Muscular Contraction, Brit. Ass. RejJ. 
1878; also in Scientific Writings, ed. Larmor, 1902, pp. 34. 75. A. d'Arsonval, 
Relations entre I'electricite animale et la tension superficielle, C. B. cvi, p. 1740, 
1888; cf. A Imbert, Le mecanisme de la contraction musculaire, deduit de la con- 
sideration des forces de tension superficielle, Arch, de Pliys. (5), ix, pp. 289-301 , 1897. 

t Ueber die Natur der Bindung der G^se im Blut und in seinen Bestandtheilen, 
Kolloid. Zeitschr. ii, pp. 264-272, 294-301, 1908; cf. Loewy, Dissociationsspan- 
nung des Oxy haemoglobin ini Blut, Arch. f. Anat. und Physiol. 1904, p. 231. 


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*." 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 " interf acial " 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'f, the former being a reversible, the latter an irreversible 
or permanent phenomenon. That is to say, adsorption, strictly speaking, 
impUes 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 Uttle 
consequence to us. 

* 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 modi- 
fications apportees a ralbumine...par Taction purement mecanique, 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. KoaguHerung der Eiweisskorper auf mechanischer Wege, Arch. f. Anat. u. Phys. 
{Phys. Ablh.) 1894, p. 517; Abscheidung fester Korper in Oberflachenschichten 
Z.f. phys. Chem. XLVn, p. 341, 1902; Proc. R. S. lxxii, p. 156, 1904. For a general 
review of the whole subject see H. Zangger, Ueber Membranen und Membranfunk- 
tionen, in Asher-Spiro's Ergebnisse der Physiologic, vii, pp. 99-160, 1908. 

t Cf. Taylor, Chemittry of Colloids, p. 252, 


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 Strasbiirger, 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*. 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 pro- 
bability 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 

* Strasbiirger, Ueber Cytoplasmastrukturen, etc. Jahrb. f. luiss. Bot. xxx> 
1897 ; R. A. Harper, Kerntheilung und freie Zellbildung im Ascus, ibid. ; of. 
Wilson, The Cell in Devdo-pment, etc. pp. 53-55. 


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 

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 




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*. 

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 


A B C . 

Fig. 97. A, B, Cliondriosomes in kidney-cells, prior to and during secretory 
activity (after Barratt); C, do. in pancreas of frog (after Mathews). 

cell, and which may, in some cases, and in some forms, be con- 
spicuous 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 
* Cf. A. Gurwitsch, Morphologic und Biologie der Zelle, 1904, pp. 169-185; 
Meves, Die Chondriosomen als Trager 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. 


unexplained way with the mechanism of cihary movement. 
Meves looked upon the chondriosomes as the actual carriers or 
transmitters of heredity*. 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, Goldachmidt and others, following Kichard 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 hypo- 
thesis," as every reader of recent cytological literature knows, has 
become a very abstruse and complicated thing f. 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, gametochroniidia, the protogono- 
plasm, 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 a'nd 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 

* 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. 

f Of. C. C. Dobell, Chromidia and the Binuclearity Hypotheses ; a review and 
a criticism, Q.J. M.S. una, 279-326, 1909; Prenant, A., Les Mitochondries et 
I'Ergastoplasme, Journ. de VAnat. et de la Physiol. XLVi, pp. 217-285, 1910 (both 
with copious bibUography). 


to the phenomena of surface energy and of adsorption*. 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 ivas, 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 

* 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 Oberflachenspannungen oder der Oberflachendruck eine Kraft darsteUt, welche 
als treibende Kraft der Osmose, an die Stelle des nicht mit dem Oberflachendruck 
identischen osmotischen Druckes, zu setzen ist, etc." (Oberflachendi-uck und 
•seine Bedeutung im Organismus, Pfliiger's Archiv, cv, p. 559, 1904.) Cf. also 
Hardy {Pr. Phys. Soc. xxvm, 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." 


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 w^hich is complex, irregular and continually 
fluctuating in amount and distribution. Such heterogeneity as 
we are speaking of must be essentially chemical, and the prelimin- 
ary 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 ^r 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*. 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 character- 
istic black precipitate of cobaltic sulphide f. 

* On the Distribution of Potassium in animal and vegetable Cells; Journ. of 
Physiol. XXXII, p. 95, 1905. 

f The reader will recognise that there is a fundamental difference, and contrast. 


By this means Macalluni 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 in- 
equalities in its surface tension* ; 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 weakf. 

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 ill 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 cyhnder ; 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 

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 pent dire que la microchimie n'est encore qu'a la periode d'essai, et 
que I'avenir de I'histologie et specialement de la cytologie est tout entier dans la 
microchimie" (Prenant, A., Methodes et resultats de la Microchimie, Jourti. de 
VAnat. et de la Physiol, xlvi, pp. 343-404, 1910). 

* Cf. Macallum, Presidential Address, Section I, Brit. Ass. Rep. (Sheffield), 
1910, p. 744. 

t In accordance with a simple corollary to the Gibbs-Thomson law. 

T. G. 19 




separates the green protoplasm from the cell-sap), there is a very 
large accumulation precisely at the point where the tension of the 
originally cyUndrical 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." 


Fig. 98. Adsorptive concentration of potassium salts in (1) cell of Pleurocarpus 
about to conjugate; (2) conjugating cells of Mesocarpus; (3) sprouting spores 
oi 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. 


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 
hes beyond our range, but the fact which we have just mentioned 
throws hght upon the frequent or general presence of a little 
protuberance of the cell-surface just where a flagellum is given 
of£ (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 COg 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 sub- 
stances (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 ine\'itable 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 



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 
eqmvaient 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," wnen tliey 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 par- 
ticular ; 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. 



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 efEects 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 mmimae 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 approxi- 
mates 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 

Let us re-state as follows, in terms of Energij, 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 


changes of temperature or of electric charge. The condition of 
minimum potential energy in the system, which is the condition of 
equihbrium, 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 

the drop is concerned, then, equihbrium 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 Ejjf. 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 

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 > E + E 

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. 


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*. 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 counter- 

Fig. 100. 

Fig. 101. 

pull, 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 charac- 
teristic 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 (the point of mutual contact), 

* It can easily be proved (by equating the increase of energy stored in an 
increased surface to the work done \.\ increasing that surface), that the tension 
measured per unit breadth, T„,,, is equal to the energy per unit area, ^n,,. 


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 he in equilihrium, 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: = sm QOR : sin FOR : sin POQ, 

P Q _ R^ 

®^ sin QOR ~ sin ROP ~ 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 : 

P2 = ^2 + 2^2 + 2QR 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. 


(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 ahke, 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 illus- 
trated; 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 iiito 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* ; and this little fillet, or " bourrelet," 

* The presence of this httle liquid '"bourrelet," drawn from tlie material of which 




as Plateau called it, is large enough to be a common and con- 
spicuous feature in the microscopy of tissues (Fig. 102). For 
instance, the so-called "splitting" of the cell- wall, which is con- 
spicuous 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 

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 purved film. 

Fig. 102. (After Berthold.) 

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 

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 

* 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, oj). cit., p. 386). The form of the superficial vacuoles in Actinophrys or 
Actinosphaerium involves an identical problem. 




and all alike iu 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 sym- 
metry 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 fine. 

Now the surfaces of the two bubbles exert a pressure inwards 
which is inversely proportional to their radii : that is to say 
]) : J)' : : l/r' : 1/r ; and the partition wall must, for equilibrium, 
exert a pressure (P) which is equal to the difference between these 

Fig. 104. 

two pressures, that is to say, P = l/R = l/r' — l/r = (r — r')/rr'. 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 = rr'/{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 ^ 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 




radius of the other sphere ; in Uke manner, make the angle 
c'jyn = 60°, cutting the Hne cc' in c" ; then c' will be the centre 
of the second sphere, and c" that of the spherical partition. 

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. 

Fig. 105. 

Fig. 106. 

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." 


ooooooo ; 


Many spRerical cells, such as 
Protococcus, divide into two equal 
halves, which are therefore separ- 
ated 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, 

c QGoxECDaio: 

Fig. 107. Filaments, or chains of 
cells, in various lower Algae. 
(A) NoHoc; (B) Anabaena; (C) 
Rivularia; (D) Oscillatorin. 




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*. 

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 sub- 
stances, we can easily determine the form of the system. For, at 
any point of the boundary of the partition-wall, 0. 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, CO meet at an 

Fig. 108. 

angle of 60°, and COC is an equilateral triangle. That is to say, 
the centre of each circle hes 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 

* In an actual calculation we must of course always take account of the tensions 
on both sides of each film or membrane. 


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. 

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 

Fig. 109. Spore of Pellia. (After Campbell.) 

end of the original spherical or elliptical cell. Each of these latter 
partitions \vill (like the first) tend to set itself normally to the 
cell-wall; at least the angles on either side of the partition w^ll 
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 h 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 ofE the terminal 




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. 

Fig. 110. Cells of Dictyota. 
(After Reinke.) 

Fig. HI. 

Terminal and other cells 
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 develop- 
ment, 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 Fig- 112. Young 

, ,. . , . , . , . . antheridium of 

adaitionai pressure inwards which it receives Chara. 




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 a, j3, y 
be, as in the figure, the opposite angles. Then : 

(1) If T be equal to T', and t be relatively insignificant, 
the angles a, j8 will be of 90°. 

Fig. 113. 

(2) If T = T', but be a little greater than t, then t will exert 
an appreciable traction, and a, j8 will be more than 90°, say, for 
instance, 100°. 

(3) If T- T' = Athena, ,^,y 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 

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*. Ten 

* Hofmeister, Pringsheini's Jahrb. in, p. 272, 1863; Hdb. d. 
1867, p. 129. 

Bot. I, 


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 
andes*. Years before, Schwendener had found, in the final 
results of cell-division, a universal system of "orthogonal tra- 
jectoriest"; and this idea Sachs further developed, introducing 
complicated systems of confocal ellipses and hyperbolae, and 
distinguishing between periclinal walls, whose curves approximate 
to the peripheral contours, radial partitions, which cut these at 
an angle of 90°, and finally antichnes, which stand at right angles 
bo 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 J. 

Within the next few years, a number of botanical writers were 
content to point out further exceptions to Sachs's Ilule§; 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 ||. 

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"' 

* Sachs, Ueber die Anordnung der Zellen in jiingsten Pflanzentheilen, Verh. 
pkys. med. Ges. Wurzburg, xi, pp. 219-242, 1877 ; Uteber Zellenanordnung und 
Wachsthiim, 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. 

t Schwendener, Ueber den Bau und das Wachsthum des Flechtenthallus, 
Naturf. Ges. Zurich, Febr. 1860, pp. 272-296. 

f Reinke, Lehrbuch der Botanik, 1880, p. 519. 

§ Cf. Leitgeb, Uriters. ilber die Lebermoose, ii, p. 4, Graz, 1881. 

I! Rauber, Neue Grundlegungen zur Kenntniss der Zelle, Mar ph. Jahrb. vnr, 
pp. 279, 334, 1882. 

T. G. 20 


had been pointed out long before, by Melsens, who had made an 
"artificial tissue" by blowing into a solution of white of egg*. 

In 1886, Berthold pubUshed 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| 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, ErreraJ 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 Errerd'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 §; and the same 
line of investigation and thought were pursued and developed by 
Robert, in connection with the embryology of the Mollusca||. 
Driesch again, in a series of papers, continued to draw attention 
to the presence of capillary phenomena in the segmenting cells 

* C. R. Acad. 8c. xxxiii, p. 247, 1851; Ann. de chimie et de phys. (3), xxxm, 
p. 170, 1851; Bull. R. Acad. Belg. xxiv, p. 531, 1857. 

t Klebs, Biolog. Centralbl. vn, pp. 193-201, 1^87. 

{ L. Errera, Sur une condition fondamentale d'equUibre des cellules vivantes, 
C. R., cni, p. 822, 1886; Bull. Soc. Beige de Microscopie, xm, Oct. 1886; Recueil 
fTcBUvres (Physiologie generale), 1910, pp. 201-205. 

§ L. Chabry, Embryologie des Ascidiens, J. Anat. et Physiol, xxm, p. 266, 1887. 

II Robert, Embryologie des Troques, Arch, de Zool. exp. et gen. (3), X, 1892. 




of various embryos, and came to the conclusion that the mode of 
segmentation was of httle importance as regards the final result*. 
Lastly de Wildemanf, in a somewhat wider, but also vaguer 
generalisation than Errera's, declared that "The form of the 
celhilar framework of vegetables, and also of animals, in its 
essential features, depends upon the forces of molecular physics." 

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 

Fisf. Hi. 

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 ar- 
rangement. 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 Unes uc, uc" , or sc, sc' , etc., drawn to the 

* "Dass der Furchungsmodus etwas fiir das Zukiinftige unwesentliches ist," 
Z. f. w. Z. Lv, 1893, p. 37. With this statement compare, or contrast, that of 
Conkhn, quoted on p. 4; of. also pp. 157, 348 (footnotes). 

t de Wildeman, Etudes sur I'attache des cloisons cellulaires, Mem. Couronn. 
de VAcad. R. de Belgique, liii, 84 pp., 1893-4. 





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 /,/ will be 
the centre of the circular arc which constitutes the partition Ou ; 
and further, the three points/,^, h, successively determined in this 


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 




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. 

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^ve 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 

Fig. 116. 

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), tbat, 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°. 


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°, irre- 
spective 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*, 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 exception- 
ally 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" (of. 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 

* It was so termed by Conklin in 1897, in his paper on Crepidula (J. of Morph. 
xm, 1897). It is the Querfurche of Rabl {Morph. Jahrb. v, 1879); the Polarfurche 
of 0. Hertwig {Jen. Zeitschr. xrv, 1880); the Brechungslinie of Rauber (Neue 
Grundlage zur K. der ZeUe, M. Jb. vin, 1882). It is carefully discussed by Robert, 
Dev. des Troques, Arch, de Zool. Exp. et Gen. (3), x, 1892, p. 307 seq. 


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 equihbrium 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. 
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 deahng 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 

A B C 

Fig. 117. Various ways in which the four cells are co-arranged in 
the four-celled stage of the frog's egg. (After Rauber.) 

of cases, there are others in which it has not been seen and some 
in which its absence is definitely asserted*. 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 

* Thus Wilson (J. of Morph. vni, 1895) declared that in Amphioxus the polar 
furrow was occasionally absent, and Driesch took occasion to criticise and to throw 
doubt upon the statement {ArcJi. f. Entw. Mech. i, 1895, j). 418). 


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* . 

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 repre- 
sentations of segmenting eggs, in which, instead of the triple 
junction and polar furrow, the four cells (and in Hke manner their 
more numerous successors) are represented as rounded off, and 
separated from one another by an empty space, or by a httle 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 Ilauber 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 Polgrubchen, 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 equihbrium. For, on the one hand, we 
see that now the four intercellular partitions do not meet ove 
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, 

* Precisely the same remark was made long ago by Driesch: "Das so oft 
schematisch gezeichnete VierzeUenstadium 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. Lni, p. 106, 1892. Cf. also his Math, mechanische 
Bedeutung morphologischer Probleme der Biologie, Jena, 59 pp. 1891. 


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 through- 
out the system; but in the system of protoplasmic cells which 
constitute the segmenting egg we must make allowance for an in- 
equality of tensions, between the surfaces where cell meets cell, and 
where on the other hand cell-surface is in contact with the sur- 
rounding medium, — in this case generally water or one of the fluids 
of the body. Remember that our general condition is that, in our 
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 ch, 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 ch. 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 




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 soaprbubbles. 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*, 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 

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 

* Compare, however, p. 299. 


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 


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 
(0, 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,/oMr 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 seg- 
menting 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 symmet- 
rically 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. 


with six planes, meeting symmetrically in a point, and constituting 
there four equal solid angles. 

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 sym- 
metry where our films meet ; and let oa, oh, oc, od, be hues drawn to 
the four corners of the tetrahedron. Produce ao to meet the base 
in f, then a'pd is a right-angled triangle. It is not difficult to 
prove that in such a figure, o (the centre of gravity of the system) 


lies just three-quarters of the way between an apex, a, and a point, 
J), which is the centre of gravity of the opposite base. Therefore 

op = oa/3 = od/3. 

Therefore cos dop = ^ , and cos aod = — ^ . 

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 
sohd geometry of our systems, just as 120° was the fundamental 
angle of symmetry so long as we considered only the plane pro- 
jection, 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 BorelH, of Ray and Willoughby and of Martin 
Lister), who dealt with this matter in a book chiefly devoted to 
molluscan shells*. 

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 

* Ricreatione delV occhio e delta mente, nelV Osservatione delle Chiocciole, Roma, 




lines of contact, representing surfaces of contact in the actual 
spheres or cyUnders ; 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 

Fig. 121. Diagram of hexagonal cells. (After Bonanni.) 

pressure or some other physical force which is the cause, or the 
main cause, of the phenomenon. 

The production by mutual interaction of polyhedral cells, 
which, under conditions of perfect symmetry, become regular 
hexagons, is very beautifully illustrated by Prof. Benard's 
" tourhillons 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 «f 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. 




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 

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.) 

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. 

The hexagonal pattern is illustrated among organisms in count- 
less 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 

Fig. 124. Epidermis of Girardia. (After Goebel.) 

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 approxi- 
mately ; and there are accordingly all degrees in the regularity and 
symmetry of the resulting tissue. But obviously, wherever we 
T. G. - 21 




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 dis- 
torted 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. 

In a soap-froth imprisoned between two glass plates, we have 
a symmetrical system of cells, which appear in optical section (as 

Fig. 125. Soap-froth under pressure. (After Rhumbler.) 

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. 

In the growth of an ordinary dicoty- 
ledonous 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 ten- 
sions of the partition-walls (Fig. 126). In 
the narrow elongated leaf of a Monocoty- 
ledon, such as a hyacinth, the elongated, apparently quadrangular 

Fig. 126. From leaf of 
Elodea canadensis. (After 


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 procUices 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*. The same maybe the case also inArcella, 
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 f." 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 

* 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 ( Wori.s, p. 503, 1902). 

t Cushman, J. A. and Henderson, W. P., Amer. Nat. xl, pp. 797-802, 1906. 

21 2 


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 8 the short diameter, that namely 
which bisects two of the opposite sides) is 8^ x ^3/2, the area of a circle 
being d'^ . 7r/4. Then, if the diameter (d) of a circular area include n hexagons, 
the area of that circle equals {n . 8)^ x 7r/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^ x 7r/4 x 2/s^B — 0-907»^, or 
9/10 . 71^, 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*. 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: 

^'or one zone 


= 2x3 

= 3x1x2, 

,, two zones 


= 3x6 

= 3x2x3, 

,, three zones 


= 4x9 

= 3x3x4, 

,, four zones 


= 5 X 12 

= 3x4x5, 

,, five zones 


= 6 X 15 

= 3x5x6, 

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, = 3N {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 x 25 x 26) + 1 = 1951t- 

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." 

* This does not merely neglect the broken ones but all whose centres He between 
this circle and a hexagon inscribed in it. 

f For more detailed calculations see a paper by "H.M." [? H. Munro], in 
Q. J. M. S. VI, p. 83, 1858. 




'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 con- 
tact with one another, and in others they are separate and loosely 
bundled (Fig. 127). Accordingly in the former the corallites are 

Fig. 127. Lithostrotion Martini. 
(After Nicholson.) 

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. Thecosmiha). 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). 




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.). 

When, as happens in certain corals,, the peripheral walls or 

Fig. 129. Amchnophylhnii pentagoyium. 
(After Nicholson.) 

"thecae" of the individual polypes remain undevelope'd 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 


known as triasters, poly asters, etc., whose relation to a field of 
force Hartog has explained*. 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 Thamnas- 
traea {Fungida), and in many more. 

The most famous of all hexagonal conformations and perhaps 
the most beautiful is that of the bee's cell. Here we have, as in 

Fig. 131. Surface-views of Corals with undeveloped thecae and confluent septa. 
A, Thamnastraea ; B, Comoseris. (From Nicholson, after Zittel.) 

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 

* 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 Seeigel- 
eiern, Inaug. Diss. 1908. 


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 intro- 
duction to the Fifth Book of his Collections) an account of its 
hexagonal plan, and he drew from its mathematical symmetry the 


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 Barthohnus 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 them- 
selves so regularly contrived that their mutual intersections 
make three Lozenges at the bottom of every Cell ; which severally 
regarded make three Rows of neat Bhomboidall Figures, connected 
at the angles, and so continue three several chains throughout the 
whole comb." But Maraldi* (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'alveole 
est formee par trois rhombes presque toujours egaux et semblables, 
qui, suivant les mesures que nous avons prises, ont les deux angles 
obtus chacun de 110 degres, et par consequent 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 con- 
fusion at once arose, and has been perpetuated in the books f- 
"Unfortunately Reaumur chose to look upon this second deter- 
mination 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 

* Observations sur les Abeilles, Mem. Acad. Sc. Paris, 1712, p. 209. 

t As explained by Leslie EUis, in his essay "On the Form of Bees' Cells," 
in Mathematical and other Writings, 1853, p. 353; cf. 0. Terquem, Nouv. Ann. 
Math. 1856, p. 178. 




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 Reaumur (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* 
and LhuiUert, 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 6 = \/2), 
y — that is to say to the imper- 

fection of his logarithmic tables, — 
not (as the books say J) "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. 

The theorem may be proved as 
follows : 

ABCDEF, abcdef, is a right 
prism upon a regular hexagonal base. 
The corners BDF are cut ofE by 
planes through the lines AC, CE, 
EA, meeting in a point V on the 
axis VN of the prism, and intersect- 
ing Bh, 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 
Fig. 132. & r & 

ends. For, if the axis cut the 

hexagon ABCDEF in N, the volumes ACVN, ACBX are equal. 

* Phil. Trans, xlu, 1743, pp. 5(55-571. t ^^em. de VAcad. de Beilin, 1781. 

J Cf. Gregory, Examples, p. 106, Wood's Homes without Hands, 1865, p. 428, 
Mach, Science of Mechanics, 1902, p. 453, etc., etc. 










\ "^ 





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, 
= 6 ; the side of the hexagon, as AB, = a ; and the height, as 
Aa, = h. 

Then, AC = 2a cos 30° = a^/3. 

And VX = a/sin 6 (from inspection of the triangle LXB) 

Therefore the area of the rhombus VAXC = a^\/3/2 sin 6. 

And the area of AabX = a/2 {2h - |FZ cos d) 
= a/2 {2Ji-al2.cotd). 

Therefore the total area of the figure 

= hexagon abcdef + 3a [ 2h — ^ cot ^ ) + 3 „ . 

Therefore ^ (^""^^^ - ^""^ f ^-^ a/S^os^N 
inereiore — ^^ - ^ l^sin^ ^ ' sin^ ^ / ' 

But this expression vanishes, that is to say, d (Area)/(^^ = 0, 
when cos 6 - 1/^3, that is when d = 54° 44' 8" 

= 1 (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 Reaumur) 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 "soHtary" wasps and bees are of various kinds. They may 
be formed by partitioning off httle chambers in a hollow stem; 


they may be rounded or oval capsules, often very neatly con- 
structed, out of mud, or vegetable^6re 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*." 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 BarthoKn suggested, and it 
is still commonly beheved, 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|. But it is very doubtful 

* Origin of Species, ch. vni (6th ed., p. 221). The cells of various bees, 
humble-bees and social wasps have been described and mathematically investigated 
by K. Miillenhoijf, Pfliiger'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 Biittel-Reepen, Biol. Centralbl. xxxiii, pp. 4, 89, 129, 183, 1903. 

t 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 im- 


whether such physical or mechanical pressure, more or less inter- 
mittently 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 efi'ect ; 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*. Under these circum- 
stances, 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 sym- 
metrical tensions of the adjacent films (though somewhat retarded 
by viscosity) should suffice to bring the whole system into equi- 
librium, 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 Bufton 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 vantees, tant admirees, me fournissent 
une preuve de plus contre I'enthousiasme et I'admiration ; cette 
figure, toute geometrique et toute reguliere qu'elle nous parait, et 
qu'elle est en efEet dans la speculation, n'est ici qu'un resultat 
mecanique et assez imparfait qui se trouve souvent dans la nature, 

possible, from the extreme thinness of the Uttle 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." 
* Since writing the above, I see that MiillenhofE gives the same explanation, 
and declares that the waxen wall is actually a Flussigkeitshdutchen, or liquid film. 


et que Ton remarque meme 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 ecailles de la peau d'une roussette, on verra qu'elles sont 
hexagones, parce que chaque ecaille croissant en meme temps se 
fait obstacle, et tend a occuper le plus d'espace qu'il est possible 
dans un espace donne : on voit ces memes hexagones dans le 
second estomac des animaux ruminans, on les trouve dans les 
graines, dans leurs capsules, dans certaines fieurs, etc. Qu'on 
remplisse un vaisseau de pois, ou plutot de quelque autre graine 
cylindrique, et qu'on le ferme exactement apres y avoir verse 
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 a six pans*. On y voit claire- 
ment la raison, qui est purement mecanique ; chaque graine, dont 
la figure est cylindrique, tend par son renfiement a occuper le 
plus d'espace possible dans un espace donne, elles deviennent done 
toutes necessairement hexagones par la compression reciproque. 
Chaque abeille cherche a occuper de meme le plus d'espace possible 
dans un espace donne, il est done necessaire aussi, puisque le 
corps des abeilles est cylindrique, que leurs cellules sont hexagones, 
— par la meme raison des obstacles reciproques. On donne plus 
d'esprit aux mouches dont les ouvrages sont les plus reguliers; 
les abeilles sont, dit-on, plus ingenieuses que les guepes, que les 
frelons, etc., qui savent aussi I'architecture, mais dont les con- 
structions sont plus grossieres et plus irregulieres que celles des 
abeilles : on ne veut pas voir, ou Ton ne se doute pas que cette 
regularite, plus ou moins grande, depend uniquement du nombre 
et de la figure, et nullement de I'intelligence de ces petites betes ; 
plus elles sont nombreuses, plus il y a des forces qui agissent 
egalement et s'opposent de meme, plus il y a par consequent de 
contrainte mecanique, de regularite forcee, et de perfection 
apparente dans leurs productions f." 

* Bonnet criticised Buffon's explanation, on the ground that his description 
was incomplete ; for Buffon took no account of the Maraldi pyramids. 

t Buffon, Histoire Naturelle, rv, p. 99. Among many other papers on the 
Bee's cell, see Barclay, Mem. Wernerian Soc. ir, p. 25!t (1812), 1818; Sharpe, Phil. 
Mag. IV, 1828, pp. 19-21; L. Lalanne, Ann. Sci. Nat. (2) Zool. xm, pp. 358-374, 
1840; Haughton, Ann. Mag. Nat Hist. (3), xi, pp. 415-429, 1863; A. R. Wallace, 




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 ejfusus), 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. 


: c 

Fig. 133. Diagram of development of "stellate cells," in pith of Juncuf>. 
(The dark, or shaded, areas represent the cells; the Ught 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 

ibid. XII, p. 303, 1863; Jeffries Wyman. Pr. Anier. Acad, of Arts and 8c. vir, pp. 
68-83, 1868; Chauncey Wright, ibid, iv, p. 432, 1860. 


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 *, 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 

* Sir W. Thomson, On the Division of Space with Minimum Partitional Area, 
Phil. Mag. (5), xxrv, pp. 503-.514, Dec. 1887; cf. Baltimore Lectures, 1904, p. 615. 




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. 

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 dis- 
covered*, 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, quadri- 
lateral 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 

* Also discovered independently by Sir David Brewster, Trans. R.S.E. xxiv, 
p. 505, 1867, XXV, p. 115, 1869. 

T. G. 22 

Fig. 134. 


expected to find all plane and all similar) four afe 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 quadri- 
lateral, 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 " tetra- 
kaidecahedra " ; 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 quadri- 
laterals nor the hexagons being necessarily plane. In a certain 
particular case, the quadrilaterals are plane surfaces, but the 
hexagons shghtly 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°*. 

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. 

* Von Fedorow had already described (in Russian) the same figure, under the 
name of cubo-octahedron, or hejita-parallelohedron, Umited 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 founda- 
tion of Von Fedorow's theory of crystaUine structure. The elongated dodecahedron 
is, essentiaUj^ the figure of the bee's cell. 


And it is quite possible, also, that in the cells of a vegetable 
parenchyma, by carefully macerating them apart, the same con- 
formation 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 quahties, 
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 disbeheve, that the whole configuration, for instancef 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 difiicult, 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 



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. 

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 t^ie "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 
at|B.chments 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. 




Returning to the experimental case, Robert found that by with- 
drawing a Uttle 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 

Fig. 135. Aggregations of four soap-bubbles, to shew various arrangements of 
the intermediate partition and polar furrows. (After Robert.) 

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* ; 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 

* F. R. Lillie, Embryology of the Unionidae, Journ. of Morphology, x, p. 12, 


in molluscs and annelids ' stands in obvious relation to the different 
size of the cells produced at the two poles*.'" 

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) f. 
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 sUght 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, coelenter- 

* E. B. Wilson, The Cell-lineage of Nereis, Journ. of Morphology, vi, p. 452, 

t 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 equihbrium. 


ates, 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 experi- 
ment 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 in- 
vestigation 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 


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*. 
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 f, 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 hypo- 
blast. 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 

* Professor Peddie has given me this interesting and important result, but the 
mathematical reasoning is too lengthy to be set forth here. 

t Cf. Rhumbler, Arch. f. Entw. Mech. xrv, p. 401, 1902; Assheton, ibid, xxxi, 
pp. 46-78, 1910. 


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. 



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 con- 
sidering, namely, the partitioning ofi 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 compart- 

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^ 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; 

CH. VIIl] 



for the area of this new partition is a x 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 

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 


/\ ^^y% 








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 hmiting 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 


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 dynami- 
cally symmetrical, for instance, when the parts difier in surface 
tension or internal pressure. This latter condition, of asymmetry 
of field, is frequent in segmenting eggs*, 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- 

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 

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 

* M. Robert {I. 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 equaUty or inequahty rather to a provision 
for the future than to a direct effect of immediate physical causation : " li semble 
assez probable, comm-3 on I'a dit souvent, que la plus grande taille d'un blastomere 
est liee a I'importance et au developpement precoce des parties du corps qui doivent 
en naitre : il y aurait la une sorte de reflet des stades posterieures du developpement 
sur les premieres phenomenes, ce que M. Ray Lankester appelle precocious segrega- 
tion. II faut avouer pourtant qu'on est parfois assez embarrasse pour assignor une 
cause a pareilles differences." 




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^; and it is obvious that a cylindrical wall, 
if it cut ofE 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^. The calculation 
is very easy. 

The surface-area of a cylinder of length a is 277r . a, and that 
of our quarter-cylinder is, therefore, a . 7rr/2 ; and this being, by 
hypothesis, = a'^, we have a = 7Tr/2, or r = 2a/7r. 

The volume of a cylinder, of length a, is a-Trr^, and that of our 
quarter-cylinder is a . -n-r^ji, which (by substituting the value of r) 
is equal to a^/Tr. 

Now precisely this same volume is, obviously, shut off by a 
transverse partition of area a^, if the third side of the rectangular 
space be equal to a/ir. 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. 

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 

Fig. 137. 




one-quarter of the original cell*. But while both the cyhndrical 
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 cyhndrical would still be the better of the two up to 
a further limit. It is only when the volume to be partitioned ofE 

* The principle is well illustrated in an experiment of Sir David Brewster's 
{Trans. R.8.E. xxv, p. Ill, 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 tiU 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. 


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. 

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^, 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 cyhndrical 
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-thiixl, 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. 

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 


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 cyHndrical 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 sohd 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 l/v2 
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 = y/'I. The vertical 




partition, AD, since BD = 1, will always equal tan^ (/3 being 
the angle ABC). And the oblique partition, GH, being equal to 
AB/\/'2=^ l/-V2co& ^. If then we call our vertical, transverse 

and oblique partitions, V, T, and 0, we have F = tan j8 ; 
T = V2 ; and = 1/a/2 cos j8, or 

V -.T :0 = tan^/V2 : 1 : 1/2 cos ^. 

And, working out these equations for various values of ^, we 
very soon see that the vertical partition (F) is the least of the 
three until j8 = 45°, at which limit F and are each equal to 
1/V2 = -707 ; and that again, when ^ = 60°, 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 appHcable, not only to the case of the plane triangle, but 
also to that of the conical cell. 

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 

T. G. 23 


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 obhque 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 con- 
ditions. 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 every- 
where 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 elhptical in cross-section, or is still more compHcated 
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. 


While in very many cases the partitions (hke 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 brahch, or give ofE 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 

C D 

Fig. 142. S-shaped partitions: A, from Taonia atomaria (after Reinke); B, from 
paraphyses of Fucus; C, from rhizoids of Mossj D, from paraphyses of 

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*. 

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 

* Cf. Wildeman, Attache des Clo sons, etc., pis. 1, 2. 





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. 

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 hew 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 



Fig. 143. Diagrammatic explanation of S-shaped partition. 

atomaria, as figured in Reinke's memoir on the Dictyotaceae of 
the Gulf of Naples*, 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 

* Nova Acta K. Leop. Akad. si, 1, pi. iv. 


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 asvmmetry, 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 hyjpoihesi) to arise transversely 
to the polar axis, would lie obliquely to the apparent axis of the 
cell (Fig. 143, B, C). And if the obhque 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 sohdified, 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*. He was deeply 
and rightly impressed by the fact that other forces besides surface 

* Cf. Protoplasmamechanik, p. 229: "Insofern liegen also die Verhaltnisse hier 
wesentlich anders als bei der Zertheiluno; hohler Korperformen durch fliissige 
Lamellen. Wenn die Membran bei der Zelltheilung die von dem Prinzip der 
kleinsten Flachen geforderte Lage und Kriimmung annimmt, so werden wir den 
Grund dafiir in andrer Weise abzuleiten haben." 


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 cylin- 
drical filament Uke 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. 

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. 




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 

Fig. 144. Development of Erythrotrichia. (After Berthold. 

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 httle, pro- 
ducing the usual httle median partition, or cross-furrow, instead 
of one common, central point of junction. This little inter- 
mediate 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 ec[ual 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) 

Fig. 14.5. 


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, respec- 
tively, (b) periclinal, and (c) anticlinal*. 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 6 the angle 
at C, which is obviously equal to OPN, or POB. Then 

CP = a cot d; PN = a cos 6; NC = CP cos 6 = a . cos^ ^/sin d. 

The area of the portion PMN 

= \CP'- e - IPN . NC 

= \a^ cot^ 6 — la cos d . a cos^ 6 /sin 6 

= \a^ (cot2 d - cos3 djsm 6). 

* There is, I tliink, 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. 




And the area of the portion PNA 

= 1^2 (77/2 -6)- ION . NP 
= ^a^ {ttI2 — 6) — |a sin ^ . a cos 6 
= |a2 (77/2 - d-&me . cos 9). 
Therefore the area of the whole portion PMA 
= a2/2 (7r/2 -6+6 cot^ 6 - cos^ dj&m 6 - sin 6 . cos 6) 
= a''/2 (7r/2 -6+6 cot^ 6 - cot 6), 
and also, by hypothesis, = f . area of the quadrant, = Tra^/S. 

Fig. 14(3. 

Hence 6 is defined by the equation 

a^/2 (77/2 - ^ + cot2 6 - cot 0) = 77a2/8, 
or TTJi- 6 + 6 cot^ ^ - cot 6* = 0. 

We may solve this equation by constructing a table (of which 
the following is a small portion) for various values of 6. 





+ d cot2 e 

= x 

34° 34' 

•7854 - 

•6033 - 

- 1-4514 

+ 1-2709 - 






























1-2661 - 

- -0002 






- -0005 


We see accordingly that the equation is solved (as accurately 
as need be) when 6 is an angle somewhat over 34° 38'^ or say 
34° 38|'. 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 6 — a cot 6 ; 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 ad cot 6 ; 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 8f to 10, or 
seven-eights almost exactly. 

But we must also compare the length of this curved " antichnal " 
partition- wall {MP) with that of the con- 
centric, 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 v'2 ; 
or, in terms of the radius, = 7r/2V2 = 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. 

The two cells into which our original quadrant is now divided, 
while they are equal in volume, are of very different shapes ; the 




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 anti- 
clinal circular arc (3, 3), cutting the peripheral wall of the quadrant 
approximately in the proportions of three to seven ; and thirdly. 

Fig. 148. Segmentation of frog's egg, under artificial compression. 
(After Roux.) 

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. 

The question how the other, or triangular, portion of the divided 
qudarant 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 antichnal partition, but how about any smaller 
sectors of circles? It is obvious in the case of a small prismatic 


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 perichnal 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 small- 
ness 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 6, according to our equation MP = ad cot 6 ; and 
the angle 6 has a definite relation to a, the angle of arc. 


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 . a/V2. 
Accordingly, the arc RS will be just equal to the arc MP when 

d cot 6 = a/V2. 

When ^ cot ^ > a/V2, or MP > RS, 

then division will take place as in RS. 

When 6 cot 6 < a/V2, 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 a, which shews on the (left-hand) vertical 
scale the various possible magnitudes of that angle (viz. the angle 




of arc of the whole sector which we wish to di\dde), and on the 
horizontal scale the corresponding values of 6, or the angle which 

Angle (6) determining the intersection of the partition -wall with the outer border 

of the cell. 






1-4 ^ 






1-0 .-s 

- .8 S 







60° 70° 80° 







1 1 






^ 80 




C3 O 









g 60 










"^ Kf? 



.2 50 




\ .<>*/ 



« 40 



2 3d 







U 2d 






~-^ \ ^>^ 




X,,^ \ 

' 1 

1 1 1 

1 1 1- "" 



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 


dealing with arcs no greater than a quadrant; and (2), the point 
{B) where the angle 6 comes to equal the angle a, 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 
(a and 6) 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 
a is rather under 82° (cf. the points Z), D'); (3) the point E, where 
RS and MP intersect ; that is to say the point at which the two 
partitions, periclinal and anticHnal, are of the same magnitude; 
this is the case, according to our diagram, when the angle of arc 
is just over 62|°. We see from this, then, that what we have 
called an anticlinal partition, as MP, is only hkely to occur in 
a triangular or prismatic cell whose angle of arc lies between 
90° and 62|°. 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 a 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 a, the perichnal partition has a very shght 
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 




cosec 6 — cot 6 = 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 a = 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 

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 compUcated 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. 

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 re- 
main a quadrant, and it is 
obvious, therefore, that for every -^ j^j 

new increment of size, more will 
be added to the margin of its triangular portion than to the 


narrower margin of its quadrilateral portion; and these incre- 
ments 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 = l7ra2 = .7854a^ 
while the area of T ^Q= •S927a^. 

The area of the enlarged sector, p'OA', 

= {a + xY X (55^ 22') -^ 2 = -4831 (a + xf, 
and the area OP A 

= a2 X (55° 22') -^ 2 = •4831a2. 

Therefore the area of the added portion, T', 

= -4831{(a + a;)2-a2}. 

And this, by hypothesis, 

= T = •3927a2. 

We get, accordingly, since a = 1, 

a:2 + 2a; - •3927/-4831 = -810, 
and, solving, 

a; + 1 = VbSi = 1-345, or a; - 0-345. 

Working out x' in the same way, we arrive at the approximate 
value, a;' + 1 = 1-517. 


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 diSicult, 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 

T. G. 24 




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 super- 
ficial or peripheral layer of quadrilateral cells, with sides approxi- 
mately 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. 

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 con- 
stantly 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 




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, hke the shifting outhnes of a system of soap-bubbles. This 
is the physical cause of the movements frequently seen among 
segmenting cells, like those to which Kauber 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. 

Bearing in mind, then, these considerations, let us see what 
our flattened disc is likely to look Hke, after a few successive 

Fig. 153. Diagram of flattened or discoid cell dividing into octants : to shew 
gradual tendency towards a position of equilibrium. 

divisions into component cells. In Fig. 153, a, we have a diagram- 
matic 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 hke the normal look of an aggregate of living cells. 
But let us go a httle 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) 



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. 

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 outhned ; and in the next succeeding 
stage they are shewn after setthng 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 




in this case be precisely accurate, for the simple reason that 
Berthold's figures are taken from different individuals, and aie 
therefore only approximately consecutive and not strictly con- 
tinuous. The last of the six drawings in Fig. 144 is already too 

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 com- 
plicated 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 

3 . 






/ / iT 







3 1 

Fig. 156. Theoretical division of a discoid cell into sixty-four chambers: no 
allowance being made for the mutual tractions of the ceU-walfe. 

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 


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. 

a b 

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 Kees*. 

If within our boundary we have three cells all meeting 

* Cit. Plateau, Statiquedes Liquides, i, p. 358. 




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 w — 3 little longitudinal partitions (and 
corresponding polar furrows), connecting the triple junctions of 

8 CeJ/s \Tvpe 

Fig. 158. Various possible arrangements of intermediate partitions, in 
groups of 4, 5, 6, 7 or 8 cells. 

the cells ; and these little walls, like all the rest within the system, 
must be inchned 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 




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, un- 
determined, 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*. 

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 sus- 
picion 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 

* Even in a Protozoon (Euglena viridis), when kept alive under artificial com- 
pression, 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. Confrib. Zool. Lab. Univ. Pennsylvania, i, 
jDp. 37-50, 1893. 




considered) is responsible for many inaccurate or incomplete 
representations of the mutual arrangement of aggregated cells. 
In the accompanying series of figures (Figs. 160-167) I have 

Fig. 160. Segmenting egg 
of Trochus. (After Robert. ) 

Fig. 161. Two views of segmenting egg of Cynthia 
partita. (After Conklin.) 

Fig. 162. (a) Section of apical cone of Salvinia. (After Pringslieim*.) 
(b) Diagram of probable actual arrangement. 

163. Egg of Pyrosoma. 
(After Korotneff). 

Fig. 164. Egg of Echinufi, segmenting 
under pressure. (After Driesch.) 

* Tliis, like many similar figures, is manifestly drawn under the influence of 
Sachs's theoretical views, or assumptions, regarding orthogonal trajectories, coaxial 
circles, confocal eUipses, etc. 




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°, 

Fig. 165. (a) Part of segmenting egg of Cephalopod (after Watase); 
(6) probable actual arrangement. 

Fig. 166. (a) Egg of Echinus; (b) do. of Nereis, under pressure. (After 

Fig. 167. (a) Egg of frog, under pressure (after Roux); (6) probable 
actual arrangement. 

and always with the help of five intermediate boundary walls 
within the eight-cell«d system; in other cases I have added a 
slightly altered drawing, so as to shew, with as little change as 


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 arrange- 
ments of five intermediate partitions (for a system of eight cells) 
have been already set forth. 

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 6 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 probabiHty, to type g. Of type/, 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 arrange- 
ment 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 commoii to both plants 


and animals. In other words they tally with, and they further 
extend, the general and fundamental conclusions laid down by 
Schwann, in his MikrosJiopische Untersuchungen iiher die Ueherein- 
stimmung in der Struktur und deni Wachsthuni der Thiere und 

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 arrange- 
ment of cell-partitions to be looked upon (as the pubUshed 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*. 

* 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 aU the analogy of surface-tension, or any other purely 
physical concept. But he was deeply under the influence of Sachs's rule of rect- 




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 equili- 
brium. 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 

4 — 5 

Fig. 168. Various modes of grouping of eight cells, at the dorsal or 
epiblastic pole of the frog's egg. (After Rauber.) 

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 

angular intersection ; and he was accordingly disposed to look upon the configura- 
tion represented above in Fig. 168, 6, as the most typical or most primitive. 


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*. 

We now begin to realise the remarkable fact, which paay 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 

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 

* Cf. Rauber,NeueGrundlagez.K. der Zelle, Morph. Jahrb. vin, 1883, pp. 273. 
274 : 

"Ich betone noch, dass unter meinen Figuren diejenige gar nicht enthalten ist, 
welche zum Typus der Batrachierfurchung gehorig am meisten bekannt ist....Es 
haben so ausgezeichnete Beobachter sie als vorhanden beschrieben, dass es mir 
nicht einf alien kann, sie iiberhaupt nicht anzuerkennen." 




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/, two are in contact with two, four 
with three, and one with no less than seven. In all cases the 

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 2n — 3 ; 
or the number of what we call the internal or interfacial contacts 
is 2 (2n — 3). But it would appear that the most stable arrange- 
ments 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 /, 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 




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*. The type which we have called/ 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 " interf acial " contacts. 

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 arrange- 
ments, 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. 

* 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. 


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*. 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 arrange- 
ments 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 
w^hy, 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 comphcated 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, butter- 
flies, 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 

* Cf. (e.g.) Clerk Maxwell, On Reciprocal Figures, etc., Trans. R. 8. E. xxvr, 
p. 9, 1870. 

T. G. 25 




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 hfe 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 pro- 
cedure hardly commends itself to the physicist or the mathe- 

Another case, geometrically akin but biologically very 
different, is to be found in the httle diatoms of the genus Astero- 
lampra, and their immediate congeners*. In Asterolampra we 

A B c 

Pig. 171. (A) Asterolampra inarylandica, Ehr. ; (B, C) A. variabilis, Grev. 
(After Greville.) 

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 

* See Greville, K. R., Monograph of the Genus Asterolampra, Q.J. M.S. vin, 
(Trans.), pp. 102-124, 1860; of. ibid, (n.s.), ii, pp. 41-55, 1862. 


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* (Fig. 171, B). In all proba- 
bility, 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 characterisa- 
tion of the allied genus Arachnoid- 

In a diagrammatic section of 
an Alcyonarian polype (Fig. 172), 
we have eight chambers set, sym- 
metrically, about a ninth, which 
constitutes the "stomach." In this 

arrangement there is no difficulty, 

... Fig. 172. Section of Alcyonarian 

tor it IS obvious that, throughout polype. 

the system, three boundaries meet 

(in plane section) in a point. In many corals we have as 

* The same is true of the insect's wing; but in this case I do not hazard a 
conjectural explanation. 



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 comphcated, and involves 
problems identical with those which we are now studying. 

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 
Fig. 173. Heterophyllia angu- chamber comply with our fundamental 
lata. (After Nicholson.) j,^|g 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 comphcated, and 
involves problems of space-partitioning identical with those which 
we are now studying. In the genus Heterophylha and in a few 
alUed forms we have such conditions, and students of the Coelen- 
terata have found them very puzzling. McCoy*, 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 f, described them 
as "paradoxical in their anatomy." 

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 

* Ann. Mag. N. H. (2), m, p. 126, 1849. 
t Phil. Trans. CLvn, pp. 643-656, 1867. 




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 

Fig. 174. Heterophyllia sp. (After Martin Duncan.) 

of them in regard to symmetry and equilibrium. But while, 
mathematically speaking, each particular case among the multi- 
tude 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 


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 /, 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 

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 Heterophylha 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 Heterophylha, 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" parti- 
tions, which w^ere bound to appear at regular intervals and in 
definite positions, when co-equal bisection was a feature of the 




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 through- 
out the group (including the curious vesicular or bubble-like 
tissue of the Cyathophyllidae and the general structural plan of 

Diagrammatic section of a Ctenophore {Eucharis). 

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. 

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 




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 cihated bands. The case however is comphcated 
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 hke manner I am inclined to suggest that the same principle 
may help us to understand the apparently complex arrangement 

Fig. 176. Diagrammatic arrangement of partitions, represented by skeletal 
rods, in larval Echinoderm (Ophiura). 

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. 

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 Unes 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 




bent symmetrically downwards, so that the plane of the paper i? 
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-hke, 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 be- 
comes very much more complicated ; we have to do with a more 

Fig. 177. Pluteus -larva of Ophiurid. 

complex partitioning of space, and I confess that I am not yet 
able to represent the more compUcated forms in so simple a way. 

There are a few notable exceptions (besides the various un- 
equally 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 develop- 
ment 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 




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. 178. Diagrammatic development of Stomata in Sedum. (Cf. fig. in 
Sachs's Botany, 1882, p. 103.) 

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 
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 approxi- 
mately 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), 


Fig. 179. Diagrammatic development ^nd at length (for a reason which 
of stomata in Hyacinth. is not yet clear) the partition wall 


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 

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. 

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 




of a boundary-surface, which, whether it become at once a solid 
or semi-soHd partition or whether it remain semi-fluid, exercises 
in all cases an efiect 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 simul- 
taneously, differentiated. In a number of cases it happens that, 
to begin with, a number of "mother-cells" are formed simul- 
taneously, and each of these divides, by two successive divisions. 

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 

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 


they lie, the pollen-grains afterwards fall apart, and their in- 
dividual form will depend upon whether or no their walls have 
solidified before this hberation takes place. 
For if not, then the separate grains will be 
free to assume a spherical form as a con- 
sequence of their own individual and un- 
restricted 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 Fig. 18 1. Dividing spore 
angles of the elements of the tetrahedron. gi^pbdlT'' ^^^*''' 
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 w^e 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 httle 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. 


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 
generahsation 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*. 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 locahsed 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 

* Sa,chs,Pflanzenphysiologie{Vorlesiingxxiv),'lSS2; cf. Rauber, NeueGrundlage 
zui- Kenntniss der Zelle, Morphol. Jahrb. vin, p. 303 seq., 1883; E. B. Wilson, 
Cell-lineage of Nereis, Joiirn. of Morphology, vi, p. 448, 1892, etc. 


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 en- 
quiring further into this compUcated 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*. 

* In the following account I follow closely on the lines laid down by Berthold ; 
Protojtlasmamechanik, 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 earher papers, Ueber 
die Anordnung der Zellen in jiingsten Pflanzentheilen, and Ueber Zellenanordnung 




(1) Suppose, in the first place, that the axis of growth hes 
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 cyUndrical 
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 

Fig. 182. 

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. 
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. 

und Wachsthum (Arb. d. botan. Inst. Wicrzburg, 1878, 1879). But Sachs's treat- 
ment differs entirely from that which I adopt and advocate here : his explanations 
being based on his "law" of rectangular succession, and involving compUcated 
systems of confocal conies, with their orthogonally intersecting elhpses and hyper- 


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-cyhndrical 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 obhque 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 configura- 
tion, 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. 

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 
T f- 26 




undivided quadrants, one of which has further slowly divided 

in the stage 6. The active quadrant 
has grown quickly into a cylindrical 
structure, which inevitably divides, in 
the next place, into a series of trans- 
verse 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 ap- 
pears 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. 
(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 

Fig. 183. 

Development of 
(After Campbell.) 

Fig. 184. 

will then divide, as in Fig. 184, 2, by an oblique partition, in the 
usual way, whose precise position and incHnation 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 possibiUty 
of the division being into two unequal halves. By our hypothesis. 


the growth of the whole system is mainly in a vertical direction, 
which is as much as to say that the more actively growing proto- 
plasm, 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*. 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, 
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 Fia;. 185 which represents a 

. . Fio-. 185. 

"gemma" of a Moss, we see just the same thing; Gemma of 
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 ac- 
cordance 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. 

The method of division by means of oblique partitions is a 
common one in the case of ' growing points ' ; for it evidently 

* Cf. p. 369. 





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. 

Fig. 186. Development of antheridium of Riccia. (After Campbell.) 

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 

Fig. 187. Section of growing shoot 
of Selaginella, diagrammatic. 

Fig. 188. Embryo of Jungermannia. 
(After Kienitz-Gerloff.) 

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. 




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. 

(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 

Fig, 189. 

whole will come to look just like that which, in the former case, 
was made up of portions of many successive obHque partitions. 

Let us now, in one or two cases, follow out a httle 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 


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. 

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. 

Fig. 190. Development of sporangium of Osmunda. (After Bower.) 

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 perichnal partition wall, cutting off its outer end, 
was indicated as of less area than an antichnal one. This periclinal 
wall cuts down the apical cell to the proportions, very nearly, 
of an equilateral triangle, but the sohd 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 


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 con- 
sidered 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 cyhndrical 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 accompany- 
ing 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 

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 


surface) consists of eight cells only. For example, in Fig. 191, 

Fig. 191. (A, B,) Sections of younger and older embryos of Phascum; 
(C) do. of Adiantmn. (After Kienitz-Gerloff. ) 

we have it again, in a section of a young embryo of a moss (Phas- 
cum), 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 

A curious and beautiful case, apparently aberrant but which 
would doubtless be found conforming strictly to physical laws, if 

Fig. 192. Section through frond 
of Girardia sphacehria. (After 


only we clearly understood the actual conditions, is indicated in 
the development of the antheridium 
of a fern, as described by Strasbiirger. 
Here the antheridium develops from 
a single cell, whose form has grown 
to be something more than a hemi- 
sphere ; 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 Kg. 193. Development of anthe- 

contact with the base, and so to develop ridium of Pteris. (After 

. Strasbiirger.) 

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 perichnal 
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 

The foregoing account deals only with a few elementary pheno- 
mena, 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 THE FORMS OF TISSUES ETC. [ch. viii 

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 prehminary 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. 



The deposition of inorganic material in the Hving body, usually 
in the form of calcium salts or of sihca, 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 organ- 
isms, 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. 


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 

In the first and simplest part of our subject, the essential 
problem is the problem of crystalhsation 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 crystalHne 
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 crystalhsation, 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 crystalhsation), 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 f. 

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 

* There is much information regarding the chemical composition and minera- 
logical 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. 

t Vesque, Ann. des Sc. Nat. (Bot.) (5), xix, p. 310, 1874. 




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 

Fig. 194. Alcyonarian spicules: Siphonogorgia and Anthogorgia. (After Studer.) 

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*, which are some- 
times 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 

* Cf. Kolliker, Icones Histiologicae, 1864, pp. 119, etc. 


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 develop- 
ment 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 Uving 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* : 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 f. 

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 w^hich it is wont to make its spicules 

* In an interesting paper by Irvine and 8ims Woodhead on the "Secretion of 
Carbonate of Lime by Animals" (Proc. E. S. E. xvi, 1889, p. 351) it is asserted 
that "lime salts, of whatever form, are deposited only in vitally inactive tissue." 

t 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 sub- 


or its shell. Thus, Pouchet and Chabry* 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 genera- 
tions 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 hme, the shells become much altered, strengthened with various 
"ornaments," and assuming characters described as proper to 
other varieties and even species f. 

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 Ave 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 J 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, 

* C. R. Soc. Biol. Paris (9), i, pp. 17-20, 1889; C. i?. Ac. 8c. cvm, pp. 196-8, 


t Cf. Heron-Allen, Phil. Trans. (B), vol. ccvi, p. 262, 1915 

J Sec Leduc, Mechanism of Life (1911), oh. x, for copious references to other 

works on the artificial production of "organic" forms. 


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 associa- 
tion with a living body, and when we find these forms to be 
characteristic or recognisable, and fomehow 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 hypo- 
thetical form of an individual molecule; but molecular form is 
a very important concept ; and the chemist has not only succeeded, 
since the days of Wohler, 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, mahc 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 laevomahc 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*, 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, 

* 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; of. Richardson, G. M., 
Foundations of Stereochemistry, N. Y. 1901. 


and by no means without its interest for the morphoJogist, 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 Abbe Galiani's saying, 
"Zes des de la nature sont pipes." 

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 w^ell-marked line of demarcation that can at 
present [1860] be drawn between the chemistry of dead and of 
hving matter." Nearly forty years afterwards the same theme 
was pursued and elaborated by Japp in a celebrated lecture*, 
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 truef) lies the 

* Japp, Stereometrj^ and Vitalism, Brit. Ass. Rep. (Bristol), p. 813, 1898; 
cf. also a voluminous discussion in Nature, 1898-9. 

■f 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 fiir die 
Physiologie, Z. f. physiol. Chemie, v, p. 60, 1899, and various papers in the Ber. 
d. d. chem. Ges. from 1894. 

T fi 27 


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*." 

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 f. 

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 J, 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 hght 
which comes to us from the sun is already plane-polarised hght, 
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 ab- 
sorption of light and chemical activity will continue to hold 
good in the case of circularly polarised light; that is to say 

* 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 sa}% 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 sup- 
posed) 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. 

f See for a fuller discussion, Hans Przibram, Vitalitdt, 1913, Kap. iv, Stoff- 
wechsel (Assimilation und Katalyse). 

X Cf. Cotton, ^Jiw. de Chim. et de Phijs. (7), viii, pp. 347-432 (cf. p. 373), 1896. 


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*, 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 f. 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 hving 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 hving 
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 

* Byk, A., Zur Frage der Spaltbarkeit von Razemverbindungen durch Zirkulai- 
polarisiertes Licht, ein Beitrag zur primaren Entstehung optisch-activer Substanzen, 
Zeitsch.f. j)hysikal. Chemie. xlix, p. 641, 1904. It must be admitted that further 
positive evidence on these Unes is still awanting. 

t Cf. (int. al.) Emil Fischer, U niersuchungen ilber Aminosduren, Proteine, etc. 
Berlin, 1906. 



fact, or the great probability, that "the tenacity with which 
Pasteur fought against the doctrine of spontaneous generation was 
not unconnected with his behef that chemical compounds of one- 
sided symmetry could not arise save under the influence of life*." 
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 \N^thin the organism, and to the question of how 
far these forms may be artificially imitated or theoretically 

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 A\ith this specific problem. Mr Rainey 
pubHshed, 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 f." 
Professor Harting, after thirty years of experimental work, 
pubHshed in 1872 a paper, which has become classical, entitled 
Recherches de MorpJwlogie Synthetique, sur la prodtiction artificielle 
de quelqties formations calcaires brganiques ; his aim was to pave 
the way for a "morphologic synthetique," as Wohler had laid the 
foundations of a "chimie synthetique," by his classical discovery 
forty years before. 

* Japp, I. c. p. 828. 

f 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. 4.51— i76, 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, vn, pp. 212-225, 1859, \Tn, pp. 1-10, 1860. 
I (n. s.), pp. 23-32, 1801. 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 observa- 
tions on the component parts of Membrane, PhiJ. Trans. 1800. pp. 327-402. 


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 

Fig. 195. Calcospherites, or concretions 
of calcium carbonate, deposited in 
white of egg. (After Harting.) 

Fig. 196. A single calco- 
spherite, witli central 
"nucleus," and striated, 
iridescent border. (After 

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 wit- 
nessed whether the solution were albuminous or not, and similar 
appearances indeed had been witnessed and recorded by Gustav 
Rose, so far back as 1837 * ; 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 
* Cf. Quincke, Ueber unsichtbare Fliissigkeitsscliichten, Amt. der Physilc, 1902. 




concentric ; the presence of concentric zones or lamellae, alter- 
nately dark and clear, was especially characteristic. These 
round " calcospherites " shewed a tendency to aggregate together 

Fig. 197. Later stages in the same experiment. 

in layers, and then to assume polyhedral, or often regularly 
hexagonal, outlines. In this latter condition they closely resemble 

Fig. 198, A. Section of shell of Mya; B. Section of hinge- 
tooth of do. (After Carpenter.) 

the early stages of calcification in a molluscan (Fig. 198), or still 
more in a crustacean shell * ; while in their isolated condition 

* See for instance other excellent illustrations in Carpenter's article "Shell," in 
Todd's Cyclopaedia, vol. iv. pp. 556-571, 1847-49. According to Carpenter, the 
shells of the moUusca (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. 


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*. 

When the albumin was somewhat scanty, or when it was mixed 
with gelatine, and especially when a little phosphate of lime was 

Fig. 199. Large irregular calcareous concretions, or spicules, deposited in a piece 
of dead cartilage, in presence of calcium phosphate, (After Harting.) 

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 (Lecttires on 
Histology, vol. n, 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 Microsccpe, 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, in, pp. 284-290, pi. xx. 1882). Lastly Duerden 
shewed that external to, and apparently secreted by the ectoderm lies a homo- 
geneous organic matrix or membrane, "in which the minute calcareous crystals 
forming the skeleton are laid down" (The Coral Sidernstraea radians, etc.. Carnegie Washington, 1904, p. 34). Cf. also M. M. Ogilvie-Gordon, Q. J. 31. S. xlix. 
p. 203. 1905, etc. 

* Cf. Claparede. Z. f. w. Z. xix, p. 604. ISfiO. 


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*. 

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 

Fig. 200. Additional illustrations of Alcyonarian spicules : Eunicea. (After 


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 locahty, 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 con- 
vinced that we have been engaged in a more or less fruitless taskf". 

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 

* .Spicules extremely like those of the Alcyonaria occur also in a few sponges ; 
of. (e.g.), Vaughan Jennings, Journ. Linn.. Soc. xxni, p. 531, pi. 13, fig. 8, 1891. 
t Mem. Manchester Lit. and Phil. Soc. lx, p. 11, 1916. 


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 occiu-ring in lower organisms*." 
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 otohths, 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.) 

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 ; 



Fig. 201. A '"crust" of close-packed 

calcareous concretions, precipitated -r-i- , —. . 11 

at the surface of an albuminous Fig. 202. Aggregated calco- 
solution. (After Harting.) sphentes. (After Harting.) 

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) f, is likewise due to the same agency. 
From the point of view of colloid chemistry the whole phe- 
nomenon 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 ttvo-phase 
system, which finally consists of sohd particles in suspension in 
a liquid (the former being styled the dis'perse phase, the latter the 

* Mummery, J. H., On Calcification in Enamel and Dentine. Phil. Trans, ccv 
(B), pp. 95-lli, 1914. 

t 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. Ixxvi, Fig. 2, 1902). 




dispersion medium). In accordance with a rule first recognised 
by Ostwald*, 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 |. 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 

(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 proto- 
plasmic structure, we have spoken of as a "collar-cell." The 

* Of. Taylor's Chemistry of Colloids, p. 18, etc., 191.5. 

•j- This rule, undreamed of by Errera, supports and justifies the cardinal 
assumption (of which we have had so much to «ay 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). 




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 difiusion-current* is not obvious. 

Among these various phenomena, the concentric stria tion 
observed in the calcospherite has acquired a special interest and 
importance f. 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 J." 

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 th'e 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'etre of 

* Cf. p. 254. 

f Cf. Harting, op. cit., pp. 22, 50: "J'avais cm d'abord qne ces couches 
concentriques etaient produites par I'alternance de la chaleur ou de la lumiere, 
pendant le jour et la nuit. Mais I'experience, expressement instituee pour 
examiner cette question, y a repondu negativement." 

X Liesegang, R. E., Ueber die Schichtungen bei Diffusionen, Leipzig, 1907, and 
other earlier papers. 


this phenomenon, still somewhat problematic, the student must 
consult the text-books of physical and colloid chemistry*. 

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 |. 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 {. 

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 crystal- 
loid in solution may lead to other well-marked effects. These, 
according to Professor J. H. Bowman §, 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 com- 
pound crystal ; (4) a curving or gyrating of the crystal or its parts. 

* Cf. Taylor's Chemistry of Colloids, pp. 146-148, 1915. 

t Cf. S. C. Bradford, The Liesegang Phenonemon and Concretionary Structure 
in Rocks. Nature, xcvii, p. 80. 1916; cf. Sci. Progress, x, p. 369, 1916. 

X 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. 

§ A Study in Crystallisation, /. of Soc. of Chem. Industry, xxv, p. 143, 1906. 




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 
fihform, 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 crystallisa- 
tion: we get the phenomenon known as a "relay," along the 

Fig. 20H. Relay-crystals of common salt. (After Bowman.) 

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, w^hen 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 

207. Wheel-like crystals in a 
colloid. (After Bowman.) 


or arborescent shapes so frequently seen in microscopic crys- 

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 
Kiister * 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 incUned 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 japonina), 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 t- He declares, for instance, that the banded wings of 
Papilio podalirius are precisely imitated in Liesegang's experi- 
ments ; that the finer markings on the wings of the Goatmoth 
{Cossus Ugni/perda) shew the double arrangement of larger and of 

* Ue.ber Zonenbildung in kolloidalen Mediev, Jena, 1913. 
t Vcrh. d. d. Zool. Gesellsch. p. 179, 1912. 


smaller intermediate rhythms, likewise manifested in certain cases 
of the same kind ; that the alternate banding of the antennae 
(for instance in Sesia spheciforniis), 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* 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! . To turn from the dis- 
tribution 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 K lister's 
figures as the effect of precipitating a little sodium phosphate in 
a gelatinous medium. Biedermann has studied, from the sanje 
point of view, the structure and development of the molluscan 
shell, the problem which Rainey had first attacked more than 
fifty years before J ; and Liesegang himself has applied his results 
to the formation of pearls, and to the development of bone§. 

* Descent of Man, ii, pp. 132-153, 1871. 

t 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 symmetrica] 
distribution of structure or form. The subject therefore deserves very careful 
disciission, 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." 

% Cf. also Sir D. Brewster, On optical properties of Mother of Pearl, Phil. Ti-ans. 
1814, p. 397. 

§ Biedermann. W., Ueber die Bedeutung von KristaUisationsprozessen der 
Skelette wirbelloser Thiere, namentlich der MoUuskenschalen, Z. /. allg. Physiol. 
I, p. 154, 1902; Ueber Bau und Entstehung der MoUuskenschale, Jen. Zeitschr. 
xxxvi, pp. 1-164, 1902. Cf. also Steinmann, Ueber Schale mid KalkstembUdungen, 
Ber. Naturf. Ges. Freiburg i. Br iv, 1889; Liesegang, Naturw. Wochenschr. p. 641, 




Among all the many cases where this phenomenon of Liese- 
gang'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 otohths of fishes ; and a 
kindred phenomena in starch-grains has given rise, in like manner, 
to the behef that they indicate a diurnal and nocturnal periodicity 
of activity and rest*. 

That this is actually the case in growing starch-grains is 
generally believed, on the authority of Meyer f ; 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 hkehhood 
has had no causative impulse save from within, 
striation, or concentric lamellation, of the scales and 

otoliths of fishes has been much 
employed of recent years as a 
trustworthy and uninistakeable 
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. 

Fig. 208. 


Fig. 209. Otoliths of Plaice, showing 
four zones or "age-rings." (After 
Wallace. ) 

* Cf. Biitschli, Ueber die Herstellung kiinstlicher Starkekorner oder von 
Spharokrystallen der Starke, Verh. Ver. Heidelberg, \, pp. 457-472, 1896. 
f Untersuchungen ilber die Starkekorner, Jena, 1905. 


where the temperature and other physical conditions shew httle 
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 estabhshed) that in such a fish as the cod, zones of 
growth, identical in mimher, are found both on the scales and 
in the otoliths*. 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 strvicture 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 " poly halite f " : 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 J. 

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 

* Gf. Winge, Me.ddel. fra Komm. for Havundersogelse (Fiskeri), iv, p. 20, Copen- 
hagen, 1915. 

.f The anhydrite is sulphate of lime (CaS04) ; the polyhalite is a triple sulphate 
of lime, magnesia and potash (2CaS04 . MgS04 . K2SO44- 2H2O). 

t Cf. van't Hoff, Physical Chemistry in the. Service of the Sciences, p. 99 seq. 
Chicago, 1903. 

T. a. 28 


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 crystalhsation 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 "calco- 
spherites" in presence of a colloid, but continued to crystalhse, 
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 Biitschli* has succeeded in making 
them artificially in Harting's usual way, that is to say by crystal- 
lisation 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 mthin the organism f. 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 

* Spharocrystalle von Kalkoxalat bei Kakteen, Ber. d. d. Bot. Gesellsch. 
p. 178, 1885. 

f Pauli, W. u. Samec, M., Ueber Loslichkeitsbeeinfliissung von Elektrolyten 
durch Eiweisskorper, Biochem. Zeitschr. xvii, p. 235, 1910. Some of these results 
were known much earUer; cf. Fokker in Pfluger's Archiv, vn, p. 274, 1873; also 
Irvine and Sims Woodhead, op. cit. p. 347. 


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 probabihty 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 cata- 
bolic 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 phos- 
phate under such circumstances accounts for its predominance 
in, for instance, mammalian bone*; 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 mil 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 |. 

* Which, ill 1000 parts of ash, contains about 840 parts of phosphate and 
76 parts of calcium carbonate. 

T Cf. Dreyer, Fr., Die Principien der Geriistbildung bei Rhizopoden, Spongien 
nnd Echinodermen, Jen. Zeitschr. xxvi, pp. 204-468, 1892. 





The two principles of conformation are both illustrated in the 
spicular skeletons of the Sponges. 

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*. 
This is the case, for instance, with various small spicules of a 
globular or spheroidal form, formed of amorphous silica, con- 


Fig. 210. Close-packed calcospherites, or so-called "spicules," 
of Astrosclera. (After Lister.) 

centrically striated within, and often developing irregular knobs 
or tiny tubercles over their surfaces. In the aberrant sponge 
Astrosclera'l , we have, to begin with, rounded, striated discs or 
globules, which in like manner are nothing more or less than the 

* In an anomalous and very remarkable Australian sponge, just described by 
Professor Dendy (Nature, May 18, 1916, p. 253) under the name of Collosclerojyhora, 
the spicules are "gelatinous," consisting of a gel of coUoid 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. 

•j- Lister, in Willey's Zoological Results, pt iv, p. 459, 1900. 


'■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*. 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 surround- 
ing tissues, just as in the simple experiment the form of the con- 
cretionary precipitate is affected by the heterogeneity, visible or 
invisible, of the matrix. Lastly, the simple needles of amorphous 
silica, w^hich constitute one of the commonest types of spicule, 
call for little in the w^ay 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 
crystalhsation 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 irt 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 mani- 
festly necessary. As we enter on this subject, which is certainly 
no small or easy one, it may conduce to simplicity, and to brevity. 

* 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. -u9, 1911. 


if we try to make a rough classification, by way of forecast, of 
the chief conditions which we are Ukely 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 raijge 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 


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 under- 
standing 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 

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 con- 
figuration 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 

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 


concerned, but which owes its now somewhat comphcated 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 crystalUne 
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. 


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 

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- 
hke 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 
elhpsoid, 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 




a plane C-shaped figure, but is discovered, on more careful inspec- 
tion, to lie not in one plane but in a more complicated spiral twist. 

Fig. 212. 

Fig. 211. Sponge and Holothurian spicules. 

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 Unear growth, but have a tendency to ex- 
pand, 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 pro- 
duction of these spicular shapes is not a specific function of 
sponges or of any particular sponge, but that we should expect 

Fig. 213. All " ampliidisc " 
of Hyalonema. 


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*. 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 tri- 
radiate, and the three radii are usually inchned 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 

* See for instance the plates in Theel's Monograph of the Challenger Holo- 
thuroidea; also Sollas's Tetractinellida, p. Ixi. 


third*. They are seldom in a plane, but are usually inclined to 
one another in a solid, trihedral angle, not easy of precise measure- 
ment 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 ; more- 
over 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 crj^stal ; 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 siirrounding cells, or structural 
vesicles of the sponge. 

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 

* For very numerous illustrations of the triradiate and quadriradiate spicules 
of the calcareous sponges, see {int. al.), papers by Dendy [Q. J. M. 8. xxxv, 1893), 
Minchin [P. Z. S. 1904), Jenkin (P. Z. S. 1908), etc. 




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 ] 20° (Fig. 214. F). 
And this latter condition of equality will be open to modification 

Fig. 214. Spicules of Grantia and other calcareous sponges. 
(After Haeckel.) 

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 hes 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 


on p. 314)^ we shall tend to obtain a spicule with two equal angles 
and one unequal (Fig. 214, .4, 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 state- 
ment 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, curvihnear 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 inchned 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. 


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 inter- 
cellular 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, accumu- 
lated 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 inchned 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 probabihty that certain substances will 
tend to be concentrated and ultimately deposited just in these lines 
of intercellular contact and conjunction. In other words, adsorp- 
tive concentration, under osmotic pressure, at and in the surface- 
film which constitutes the mutual boundary between contiguous 


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 adsorptioD, 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 httle 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 

It is by a combination of these two principles, chemical adsorp- 
tion 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 con- 

It matters little or not at all, for the phenomenon in question, 

* Cf. again Benard's Tourhillons ceUulaires, Ann. de Chimie, 1901, p. S-l. 


what is the histological nature or " grade " of the vesicular structures 
on which it depends. In soi;ne cases (apart from sponges), they 
may be no more than the little alveoli of the intracellular proto- 
plasmic 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, Mobius has 
described tiny spicules in the shape of httle 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 "cihated 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 th6 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 

T. G. 29 




little trio of cells ; and the little rods meet and fuse together while 
still very minute, when the whole spicule is only about ^^ 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. 

Among the tetractinellid sponges, 
whose spicules are composed of amor- 
phous 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 
V J t, equal angles from the main or axial 

V^ I ^ j ray. In all probability, as Dreyer 

^■^ ^ suggests, we have here had to do with 

■ ^ M a group of four vesicles, of which 

I I three were large and co-equal, while a 

" fourth and very much smaller one lay 

Pig. 215. Spicules of tetracti- above and between the other three, 
nellid sponges (after Sollas). x . • ^ i i-i 

, . J f In certain cases where we nave nke- 

a-e, anatnaenes; a-j, pro- 

triaenes. wise one large and three much smaller 




rays, the latter are recurved, as in Fig. 215. This type, save for 
the constancy of the number of rays, and the Umitation 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 con- 
formed 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 



Fig. 21(i. Various holothurian spicules. (After Theel.) 

line of contact, and the three short rays having each lain in the 
surface furrow between two out of the three adjacent cells. 

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 forma- 
tion to the tetrahedral spicules of the sponges is found in the 





spores of a certain little group of parasitic protozoa^ the Actino- 
myxidia. 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* 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. 

Pig. 217. Spicules of hexactinellicl sponges. (After F. E. Schultze.) 

The typically six-rayed siHceous 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 difl&cult to explain, and Dreyer has confessed his 
complete inability to account for their conformation. But, 
though it is doubtless only throwing the difl&culty 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 

* Leger, Stole and others, in Doflein's Lehrbuch d. Protozoenkunde, 1911, 
p. 912. 


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 pheno- 
menon 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*. 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 
axisf." 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 

* Se^, for instance, the figures of the segmenting egg of Synapta (after Selenka), 
in Korschelt and Heider's Vergleichende Entwicklutigsgescliichte (AUgem. Th., 3*<^ 
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, 
CeU-Uneage of Nereis, Journ. of Morphology, vi, p. 450, 1892. • 

•f Korschelt and Heider, p. 16. 


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 repre- 
sented and effected by the tetrahedral system*. 

This question of the origin and causation of the forms of 
sponge-spicules, with which we have noAv 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 charac- 
teristic 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-crystalhsation"' ; 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 utihsed by natural selection as ' spicules ' 
for building up a skeleton, and afterwards, by the interaction of 
adaptation and heredity, became modified in form and differen- 
tiated in a vast variety of ways in the struggle for existence f." 
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 "utiUty" in the specific structure of these 
skeletal elements. 

* Cliall. Rej). Heractinellida, pis. xvi, liii, Ixxvi, Ixxxviii. 

f "Hierbei nahm der kolilensaure Kalk eine halb-krystallinische Beschaffen- 
heit an, und gestaltete sich unter Aufnahme von Krystallwasser luid in Verbindung 
mit einer geringen Quantitat von organischer Substanz zu jenen individuellen, 
festen Korpern, welche durch die natiirliche Ziichtung als Spicula zur Skeletbildung 
beniitzt, und spaterhin durch die Wecliselwirkung von Anpassung und Vererbung 
im Kampfe urns Dasein auf das Vielfaltigste umgebildet und differenziert wurden." 
Die Kalkschwdmme, i, p. 377, 1872 ; cf. also pp. 482, 483. 


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 under- 
stand Haeckel aright, he was thinking not merely of a direct or im- 
mediate physical causation , but of one manifesting itself through 
the agency of natural selection *. 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 f, 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