On Growth and Form

few Bork: G. P, PUTNAM’S SONS Bombay, Caleutta anv fHladras: MACMILLAN AND Co,, Lrp, Toronto: J. M. DENT AND SONS, Ltp, Tokyo: THE MARUZEN-KABUSHIKI-KAISHA 381, 2. A similar comparison of Diodon and Orthagoriscus . 383. The same of various crocodiles: C. porosus, C. americanus Pond Notosuchus terrestris . : : : : 2 os 384. The pelvic girdles of Stegosaurus aad ehiipreninus 754 385, 6. The shoulder-girdles of Cryptocleidus and of jist Gace ‘ 55 387. The skulls of Dimorphodon and of Pteranodon ‘ 756 388-92. The pelves of Archaeopteryx and of Apatornis opnnpenea fad a method illustrated whereby intermediate configurations may be

found by interpolation (G. Heilmann) 5 . 157-9 393. The same pelves, together with three of the inte maedines or infor polated forms . é ow 394, 5. Comparison of the ails of we ete Thinoecroree: Op yrachyus and Aceratheriwm (Osborn) : : : aol 396. Occipital views of various extinct MiGnocerosay (ie) : b OS 397-400. Comparison with each other, and with the skull of Hurachyus, of the skulls of Tvtanotheriwm, tapir, horse and rabbit : . 763, 4 401, 2. Coordinate diagrams of the skulls of Hohippus and of Hquus, with various actual and hypothetical intermediate types (Heilmann) . 765-7

403. A comparison of various human scapulae (Dwight) ; : 09 404, A human skull, inscribed in Cartesian coordinates ‘ ; sO 405. The same coordinates on a new Pay adapted to the skull of the chimpanzee . : : ee - 406. Chimpanzee’s skull, inscribed in ie poirot ce Fig. 405 : tale “Cum formarum naturalium et corporalium esse non consistat nisi in unione ad materiam, ejusdem agentis esse videtur eas producere cujus est materiam transmutare. Secundo, quia cum hujusmodi formae non excedant virtutem et ordinem et facultatem principiorum agentium in natura, nulla videtur necessitas eorum originem in principia reducere altiora.” Aquinas, De Pot. Q. iii, a, 11. (Quoted in Brit. Assoc. Address, Section D, 1911.)

“..1 would that all other natural phenomena might similarly be deduced from mechanical principles. For many things move me to suspect that everything depends upon certain forces, in virtue of which the particles of bodies, through forces not yet understood, are either impelled together so as to cohere in regular figures, or are repelled and recede from one another.” Newton, in Preface to the Principia. (Quoted by Mr W. Spottiswoode, Brit. Assoc. Presidential Address, 1878.)

‘““When Science shall have subjected all natural phenomena to the laws of Theoretical Mechanics, when she shall be able to predict the result of every combination as unerringly as Hamilton predicted conical refraction, or Adams revealed to us the existence of Neptune,—that we cannot say. That day may never come, and it is certainly far in the dim future. We may not anticipate it, we may not even call it possible. But none the less are we bound to look to that day, and to labour for it as the crowning triumph of Science :—when Theoretical Mechanics shall be recognised as the key to every physical enigma, the chart for every traveller through the dark Infinite of Nature.” J. H. Jellett, in Brit. Assoc. Address, Section A, 1874.

“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 wpon earth, and with labour do we find the things that are before us.” Stephen Hales, Vegetable Staticks (1727), p. 318, 1738. HIS book of mine has little need of preface, for indeed it is an easy introduction to the study of organic Form, by methods which are the common-places of physical science, which are by

no means novel in their application to natural history, but which nevertheless naturalists are little accustomed to employ. It is not the biologist with an inkling of mathematics, but the skilled and learned mathematician who must ultimately deal with such problems as are merely sketched and adumbrated here. I pretend to no mathematical skill, but I have made what use I could of what tools I had; I have dealt with simple cases, and the mathematical methods which I have introduced are of the easiest and simplest kind. Elementary as they are, my book has not been written without the help—the indispensable help— of many friends. Like Mr Pope translating Homer, when I felt myself deficient I sought assistance! And the experience which Johnson attributed to Pope has been mine also, that men of learning did not refuse to help me.

My debts are many, and I will not try to proclaim them all: but I beg to record my particular obligations to Professor Claxton Fidler, Sir George Greenhill, Sir Joseph Larmor, and Professor A. McKenzie; to a much younger but very helpful friend, Mr John Marshall, Scholar of Trinity; lastly, and (if I may say - so) most of all, to my colleague Professor William Peddie, whose advice has made many useful additions to my book and whose criticism has spared me many a fault and blunder.

I am under obligations also to the authors and publishers of many books from which illustrations have been borrowed, and especially to the following :— To the Controller of H.M. Stationery Office, for leave to reproduce a number of figures, chiefly of Foraminifera and of Radiolaria, from the Reports of the Challenger Expedition. 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 EK. B. Wilson, for his well-known and all but indispensable figures of the cell (figs. 42—51, 53); to M. A. Prenant, for other figures (41, 48) in the same chapter; to Sir Donald MacAlister and Mr Edwin Arnold for certain figures (335—7), and to Sir Edward 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 Professor Claxton Fidler and to Messrs Griffin, for letting me use, with more or less modification or simplification, a number of illustrations (figs. 339—346) from Professor Fidler’s Textbook of Bridge Construction. To Messrs Blackwood and Sons, for several cuts (figs. 127-9, 131, 173) from Professor Alleyne Nicholson’s Palaeontology; to Mr Heinemann, for certain figures (57, 122, 123, 205) from Dr Stéphane Leduc’s Mechanism of Infe; to Mr A. M. Worthington and to Messrs Longmans, for figures (71, 75) from A Study of Splashes, and to Mr C. R. Darling and to Messrs EH. and 8. Spon for those (fig. 85) from Mr Darling’s Liquid Drops and Globules. To Messrs Macmillan and Co. for two figures (304, 305) from Zittel’s Palaeontology; to the Oxford University Press for a diagram (fig. 28) from Mr J. W. Jenkinson’s Hxperi- _mental Embryology; and to the Cambridge University Press for a number of figures from Professor Henry Woods’s Invertebrate Palaeontology, for orte (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. On THE INTERNAL FoRM AND STRUCTURE OF THE CELL THE Forms oF CELLS On THE THEORY OF TRANSFORMATIONS, OR THE COMPARISON oF RELATED FoRMS 1. Nerve-cells, from larger and smaller animals (Minot, after Irving 2. Relative magnitudes of some nantes organisms ‘(Gearon : ‘ 39 4, 5. Mean eas increments of stature and weight in man (do.) . 66, 69 6. The ratio, throughout life. of female weight to male (do.) . - 71 7-9. Curves of growth of child, before and after birth (His and Riissow) 74-6

10. Curve of growth of bamboo (Ostwald. after Kraus) : 2 urs 11. Coefficients cf variability in human stature (Boas and Wiesler) : 80 12. Growth in weight cf mouse (Wolfgang Ostwald) . : 4 2 83 13. Dea, of silkwerm (Luciani and Lo Monaco) : : 4 , : 84 14. Do. cf tadpole (Ostwald, after Schaper) . : 3 85 15. Larval eels, or Leptocephali, arid young elver (Joh. Schmidt) ee TSO 16. Growth in length of Spirogyra (Hofmeister) . : ; ; 2 87 17. Pulsations of growth in Crecus (Bose) . : 88 18. Relative growth of brain, heart and body of man ( Gidetelct) : 90 19. Ratio of stature to span of arms (do.) . : : 5 : 94 20. Rates of growth near the tip of a bean-root (Sachs) F : : 96 21, 22. The weight-length ratio of the plaice, and its annual periodic

changes : ; : : § : 99, 100 23. Variability of tailpee in earwigs (eatoan) ‘ ; : => Oz: 24. Variability of body-length in plaice - F eo EOE 25. Rate of growth in plants in relation to iamrpcestne (Sachs) i? GS 26. Do. in maize, observed (K6ppen), and calculated curves . : ~ tee 27. Do. in roots of peas (Miss I. Leitch) : : 113 28, 29. Rate of growth of frog in relation to ieueetonc (iekicaaree after O. Hertwig), and calculated curves of do. : : - 115,6 30. Seasonal fluctuation of rate of growth in man (Dafiner) P EEG 31. Do. in the rate of growth of trees (C. E. Hall) . : : - 220 32. Long-period fluctuation in the rate of growth of Arizona trees

(A. E. Douglass) : : : : : ee oe 33, 34. The varying form of oe pneeiee NAveonial: in relation to 35-39. Curves of regenerative eens in Gaiseles ‘ails (M. L. Durbis) 140-145 40. Relation between amount of tail removed, amount restored, and 53. Annular chromosomes of mole-cricket (Wilson, after vom Rath) . 181 54-56. Diagrams illustrating a hypothetic field of force in caryokinesis (Prof. W. Peddie) : i : & F . 182-4 57. An artificial figure of enor fede} ‘ : , i, 86 58. A segmented egg of Cerebratulus (Prenant, after Coe) ; : =» 189 59. Diagram of a field of force with two like poles. : : . 189 60. A budding yeast-cell ASS AEWA en eH Fale a Wieeek tie, Ce 61. The roulettes of the conic sections . ; Billie 62. Mode of development of an unduloid from a amuncal fae rl)

91. Some species of Trichomastix and Trichomonas (Kofoid) 5 40 LOI 92. Herpetomonas assuming the undulatory membrane of a Trypanosome (D. L. Mackinnon) . ‘ ; 5 3 ; : . 268 93. Diagram of a human bloodporuncle : 271 97. Chondriosomes in cells of kidney and pancreas (Barat one Mathews) 285 98. Adsorptive concentration of eae aes salts in various plant-cells 103. Parenchyma of maize, shewing the same phenomenon . . »298 106. Diagram of a partition in a conical cell : 300 107. Chains of cells in Nostoc, Anabaena and other low aleae 300 108. Diagram of a symmetrically divided soap-bubble - aol 109. Arrangement of partitions in dividing spores of Pellia (Camppell. 302 110. Cells of Dictyota (Reinke) 303 111, 2. Terminal and other cells of Chari, Rade reane ntheciditird of te 303 113. Diagram of cell-walls and partitions under various conditions of tension 304 114, 5. The partition- \ saettned of fee dorconmiented Bubbles 307,8 116. Diagram of four interconnected cells or bubbles 309 117. Various configurations of four cells in a frog’s egg (Rauber) . 311 118. Another diagram of two conjoined soap-bubbles ‘ 313 119. A froth of bubbles, shewing its outer or “epidermal” layer . 314 120. A tetrahedron, or tetrahedral system, shewing its centre of symmetry 317 121. A group of bexagonal cells (Bonanni) 319 122, 3. Artificial cellular tissues (Leduc) . 320 124. Epidermis of Girardia (Goebel) 321 125. Soap-froth, and the same under compression (Bhasnblen) 322 126. Epidermal cells of Elodea canadensis (Berthold) 322 127. Lithostrotion Martini (Nicholson) : 325 128. Cyathophyllum hexagonum (Nicholson, ter Zittel) : 325 129. Arachnophyllum pentagonum (Nicholson) . 326 130. Heliolites (Woods) ; : . 326 131. Confluent septa in hanna aen al Gomosmis (Nicholson, after Zittel) 327 132. Geometrical construction 3 a hes? s cell : 330 133. Stellate cells in the pith of a rush; diagrammatic 335 134. Diagram of soap-films formed in a cubical wire skeleton (Plateau) 337 135. Polar furrows in systems of four soap-bubbles (Robert) 341 136-8. _ Diagrams illustrating the division of a cube by partitions of minimal area. : : : - 347-50 139. Cells from hairs of Sihiabeloria (Berthold) 351 140. The bisection of an isosceles triangle by minimal patties Hit ene S52: 141.

The similar partitioning of spheroidal and conical cells . 353 142. S-shaped partitions from cells of algae and mosses (Reinke and others) 355 143. Diagrammatic explanation of the S-shaped partitions 356 144. Development of Hrythrotrichia (Berthold) 359 145. Periclinal, anticlinal and radial partitioning of a quadieee 359 146. Construction for the minimal partitioning of a quadrant 361 147. Another diagram of anticlinal and periclinal partitions . 362 148. Mode of segmentation of an artificially flattened frog’s egg (Roux) : 363 149. The bisection, by eee ears aE a prism a al snale : 364 150. Comparative diagram of the various modes of bisection of a prismatic sector 365 151. Diagram of the further sows ge ete) two hale of a quareeael cell» 367 152. Diagram of the origin of an epidermic layer of cells 370 153. <A discoidal cell dividing into octants 371

154. A germinating spore of Riccia (after Campbell), to shew the manner of space-partitioning in the cellular tissue ‘ 155, 6. Theoretical arrangement of successive partitions in a aR ioidsl cell 373 158. Various possible arrangements of De ae in groups of fone toe a cellsie. : 5; : .+) 315 159. Three modes of pecans in a eat of six alls ; eke 160, 1. Segmenting eggs of T'rochus (Robert), and of Cynthia (C ‘onlin F eone 162. Section of the apical cone of Salvinia (Pringsheim) ; 377 163,4. Segmenting eggs of Pyrosoma (Korotneff), and of Echinus @omese h) 377 165. Segmenting egg of a cephalopod (Watase) ; ‘ a Bilis:

166, 7. Eggs segmenting under pressure: of Echinus and Wer els (Driesch), and of a frog (Roux) ; 378 168. Various arrangements of a group of Soles € ei on he pratace of a ae s egg (Rauber) ; : 381 208. A concentrically striated calcospherite or spherocrystal (Hantittg) . 432 214-7. Spicules of calcareous, ioirechinellid and horace cli eae aad of various holothurians (Haeckel, Schultze, Sollas and Théel) 445-452 218. Diagram of a solid body confined by surface-energy to a liquid

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, co fies = ine same principle . : 3 : ; : . 514,5 261. Another diagram of a eevee ea : Seay 262. A diagram of the logarithmic spiral of Riau (ivigecleys Z eas Og 263, 4. Opercula of Turbo and of Nerita (Moseley) : ‘ : - 821,.2 265. <A section of the shell of Melo ethiopicus ; : 525 266. Shells of Harpa and Doliwm, to illustrate generating curves aad

282. Construction for determining the length of the coiled spire 285, 6. Sections of the shells of Terebra maculata and Trochus nilonie Us 287-9. Diagrams illustrating the lines of growth on a lamellibranch shell | 295-7. Shells of Cleodora, Hyalaea and other pteropods (Boas) 298, 9. Coordinate diagrams of the shell-outline in certain pteropods 300. Development of the shell of Hyalaea tridentata (Tesch) . 301. Pteropod shells, of Cleodora and Hyalaea, viewed from the side ( Boas)

304. <A section of Nautilus, shewing the logarithmic spirals of the eats 40 which the shell-spiral is the evolute 305. Cast of the interior of the shell of Nautilus, to ee the eonigine ee the septa at their junction with the shell-wall : 306. Ammonites Sowerbyi, to shew septal outlines (Zittel, after Saini and Déderlein) ; 312, 3. Cyclammina cancellata (do.), and diagrammatic figure af the, same 321. St Venant’s diagram of a triangular prism under torsion (fiomecn

327. Further diagrams of phyllotaxis, to shew how various spiral papeat: ances may arise out of one and the same angular leaf-divergence 329, 30. Diagrams of the angle of branching in bloodeect (Hess) 333. An example of the mode of arrangement of bast-fibres in a late -stem (Schwendener) 334. Section of the head of a Four ta: chew its imapeculas pecheture (Schafer, after Robinson) 335 Comparative diagrams of a crane-head and the head of a femur ~~

(Culmann and H. Meyer) . 3 682 336. Diagram of stress-lines in the human foot (Sir D. MacAlister dea stress-diagram and reciprocal plan of construction (do.) . 2.696 342. A loaded bracket and its reciprocal construction-diagram (Culmann). 697 343, 4. A cantilever bridge, with its reciprocal diagrams (Fidler) . ous 345. <A two-armed cantilever of the Forth Bridge (do.) . ; coe Oe 346. A two-armed cantilever with load distributed over two piereae

as in the quadrupedal skeleton : 700 347-9. Stress-diagrams, or diagrams of bending Toman in fe backbones of the horse, of a Dinosaur, and of Tvtanotherium . : . 701-4 350. The skeleton of Stegosaurus : A ila eeaOd 351. Bending-moments in a beam Site feed ends, to illustrate the mechanics of chevron-bones : oo 8 MU) 352, 3. Coordinate diagrams of a circle, and its dekdrmiation inte an ellipse : 729 354. Comparison, by means oe Orton eae. of the cannonbones

of various ruminant animals. : a 720) 355, 6. Logarithmic coordinates, and the circle of Fig. 352 yea fee 729, 31 357, 8. Diagrams of oblique and radial coordinates . : : ») iit 359. Lanceolate, ovate and cordate leaves, compared by the fap of radial coordinates : ; : 5 2 : : : at hee 360. <A leaf of Begonia daedalea : : F : 4 ‘ of hos 361. A network of logarithmic spiral ponmeinaeee ‘ : 5 a a 362, 3. Feet of ox, sheep and giraffe, compared by means of Cartesian

364, 6. “ Proportional dineearae: of ON Si enore ( 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 Gitians ad ein ; 742 369. The carapaces of certain crabs, Geryon, Corystes and others, Soe by means of rectilinear and curvilinear coordinates : 744 370. A comparison of certain amphipods, Harpinia, ae oad Hyperia : 746 371 The calycles of aor pineaon sn Sues tiecnibed in corresponding Cartesian networks . y 747 372. The calycles of certain species of Aglaophenia, rade compared By means of curvilinear coordinates . 748. 373, 4. The fishes Argyropelecus and Sternoptyz, Garapared by means of rectangular and oblique coordinate systems . 748 375, 6. Scarus and Pomacanthus, similarly compared by means of reat angular and coaxial systems . 749

377-80. Acomparison of the fishes Polyprion, Pecul op eenanihin Sionipeeae and Antigonia . < 5 2 : : : : : 5 7a0 Of the chemistry of his day and generation, Kant declared that it was ‘“‘a science, but not science,’—‘“eine Wissenschaft, aber nicht Wissenschaft”; for that the criterion of physical science lay in its relation to mathematics. And a hundred years later Du Bois Reymond, profound student of the many sciences on which physiology is based, recalled and reiterated the old saying, declaring that chemistry would only reach the rank of science, in the high and strict sense, when it should be found possible to explain chemical reactions in the light of their causal relation to the velocities, tensions and conditions of equilibrium of the component molecules; that, in short, the chemistry of the future must deal with molecular mechanics, by the methods and in the strict language of mathematics, as the astronomy of Newton and Laplace dealt with the stars in their courses. We know how great a step has been made towards this distant and once hopeless goal, as Kant defined it, since van’t Hoff laid the firm foundations of a mathematical chemistry, and earned his proud epitaph, Physicam chemiae adiunxit*.

We need not wait for the full realisation of Kant’s desire, in order to apply to the natural sciences the principle which he urged. Though chemistry fall short of its ultimate goal in mathematical mechanics, nevertheless physiology is vastly strengthened and enlarged by making use of the chemistry, as of the physics, of the age. Little by little it draws nearer to our conception of a true science, with each branch of physical science which it * These sayings of Kant and of Du Bois, and others like to them, have been

brings into relation with itself: with every physical law and every mathematical theorem which it learns to take into its employ. Between the physiology of Haller, fine as it was, and that of Helmholtz, Ludwig, Claude Bernard, there was all the difference in the world. As soon as we adventure on the paths of the physicist, we learn to weigh and to measure, to deal with time and space and mass and their related concepts, and to find more and more our knowledge expressed and our needs satisfied through the concept of number, as in the dreams and visions of Plato and Pythagoras; for modern chemistry would have gladdened the hearts of those great philosophic dreamers.

But the zoologist or morphologist has been slow, where the physiologist has long been eager, to invoke the aid of the physical or mathematical sciences; and the reasons for this difference lie deep, and in part are rooted in old traditions. The zoologist has scarce begun to dream:of defining, in mathematical language, even the simpler organic forms. When he finds a simple geometrical construction, for instance in the honey-comb, he would fain refer it to psychical instinct or design rather than to the operation of physical forces; when he sees in snail, or nautilus, or tiny foraminiferal or radiolarian shell, a close approach to the perfect sphere or spiral, he is prone, of old habit, to beheve that it is after all something more than a spiral or a sphere, and that in this ““something more” there hes what neither physics nor mathematics can explain. In short he is deeply reluctant to compare the living with the dead, or to explain by geometry or by dynamics the things which have their part in the mystery of life. Moreover he is little inclined to feel the need of such explanations or of such extension of his field of thought. He is not without some justification if he feels that in admiration of nature’s handiwork he has an horizon open before his eyes as wide as any man requires. He has the help of many fascinating theories within the bounds of his own science, 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