GROWTH AND FORM Ah instantdnedus photograph of a 'splash' of milk. From Harold E. ' '"*' KdgertbTiTirlassachusetts Institute of Technology See p. 390 ON 94 GROWTH AND FORM ^^ BY D'ARCY WENTWORTH THOMPSON A new edition MARINE BIOLOGICAL LABORATORY LIBRARY WOODS HOLE, MASS. W. H. 0. 1. CAMBRIDGE: AT THE UNIVERSIT NEW YORK: THE MACMILLAN CC Y PRESS )MPANY 1945 "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 jhat are upon earth, and with labour do we find the things that are before us.'' Stephen Hales, Vegetable Staticks (1727), p. 318, 1738. "Ever since I have been enquiring into the works of Nature I have always loved and admired the Simplicity of her Ways." Dr George Martine (a pupil of Boorhaave's), in Medical Essays and Observations, Edinburgh, 1747. PREFATORY NOTE THIS book of mine has little need of preface, for indeed it is "all preface" from beginning to end. I have written it as an easy introduction to the study of organic Form, by methods which are the common-places of physical science, which are by no means novel in their application to natural history, but which nevertheless naturalists are little accustomed to employ. It is not the biologist with an inkling of mathematics, but the skilled and learned mathematician who must ultimately deal with such problems as are 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 mathe- matical 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. I wrote this book in wartime, and its revision has employed me during another war. It gave me solace and occupation, when service was debarred me by my years. Few are left of the friends who helped me write it, but I do not forget the debt I owe them all. Let me add another to these kindly names, that of Dr G. T. Bennett, of Emmanuel College, Cambridge; he has never wearied of collaboration with me, and his criticisms have been an education to receive. D. W. T. 1916-1941. American edition published August, 1942 Reprinted January, 1943 Reprinted May, 1944 Reprinted May, 1945 PRINTED IN THE UNITED STATES OF AMERICA CONTENTS CHAP. PAGE I. Introductory 1 II. On Magnitude 22 III. The Rate of Growth 78 IV. On the Internal Form and Structure of the Cell . . 286 V. The Forms of Cells 346 VI. A Note on Adsorption 444 VII. The Forms of Tissues, or Cell-aggregates . . . 465 VIII. The same (continued) . . 566 IX. On Concretions, Spicules, and Spicular Skeletons . . 645 X. A Parenthetic Note on Geodetics 741 XI. The Equiangular Spiral 748 XII. The Spiral Shells of the Foraminifera .... 850 XIII. The Shapes of Horns, and of Teeth or Tusks: with a Note on Torsion 874 XIV. On Leaf-arrangement, or Phyllotaxis .... 912 XV. On the Shapes of Eggs, and of certain other Hollow Structures 934, XVI. On Form and Mechanical Efficiency .... 958 XVII. On the Theory of Transformations, or the Comparison OF Related Forms 1026 Epilogue 1093 Index 1095 Plates A Splash of Milk Frontispiece The Latter Phase of a Splash . . " . . . . facing page 390 "The mathematicians are well acquainted with the difference between pure science, which has to do only with ideas, and the application of its laws to the use of life, in which they are con- strained to submit to the imperfections of matter and the influence of accident." Dr Johnson, in the fourteenth Rambler, May 5, 1750. "Natural History. . .is either the beginning or the end of physical science." Sir John Herschel, in The Study of Natvml Philosophy, p. 221, 1831. "I believe the day must come when the biologist will — without being a mathematician — not hesitate to use mathematical analysis when he requires it." Karl Pearsonf in Nature, January 17, 1901. CHAPTER I INTRODUCTORY Of the chemistry of his day and generation, Kant declared that it was a science, but not Science — eine Wissenschaft, aber nicht Wissen- schaft — for that the criterion of true science lay in its relation to mathematics*. This was an old story : for Roger Bacon had called mathematics porta et clavis scientiarum, and Leonardo da Vinci had said much the samef. Once again, a hundred years after Kant, Du Bois Reymond, profound student of the many sciences on which physiology is based, recalled the old saying, and declared that chemistry would only reach the rank of science, in the high and strict sense, when it should be found possible to explain che^dcal reactions in the light of their causal relations to the velocities, tensions and conditions of equihbrium of the constituent 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 Newtoii and Laplace dealt with the stars in their courses. We know how great a step was made towards this distant goal as Kant defined it, when van't Hoif laid the firm foundations of a mathematical chemistry, and earned his proud epitaph — Physicam chemiae adiunxitX- We need not wait for the full reahsation of Kant's desire, to apply to the natural sciences the principle which he laid down. Though chemistry fall short of its ultimate goal in mathematical mechanics §, nevertheless physiology is vastly strengthened and enlarged by * " Ich behaupte nur dass in jeder besonderen Naturlehre nur so viel eigentliche Wissenschaft angetroffen konne als darin Mathematik anzutreffen ist" : Gesammelte Schriften, iv, p. 470. t "Nessuna humana investigazione si puo dimandare vera scienzia s'essa non passa per le matematiche dimostrazione." X Cf. also Crum Brown, On an application of Mathematics to Chemistry, Trans. R.S.E. XXIV, pp. 691-700, 1867. § Ultimate, for, as Francis Bacon tells us: Mathesis philosophiam naturalem terminare debet, non generare aut procreare. 2 INTRODUCTORY <^i,J^^H making use of the chemistry, and of the physics, of the age. Littl by httle it draws nearer to our conception of a true science with each branch of physical science which it brings into relation with itself: with every physical law and mathematical theorem which it learns to take into its employ*. Between the physiology of Haller, fine as it was, and that of Liebig, Helmholtz, Ludwig, Claude Bernard, there was all the difference in the worldf. 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 philo- sophic dreamers. Dreams apart, numerical precision is the very soul of science, and its attainment affords the best, perhaps, the only criterion of the truth of theories and the correctness of experi- mentsj:. So said Sir John Herschel, a hundred years ago; and Kant had said that it was Nature herself, and not the mathematician, who brings mathematics into natural philosophy. 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 are partly rooted in old tradition and partly in the diverse minds and temperaments of men. To treat the living body as a mechanism was repugnant, and seemed even ludicrous, to Pascal §; and Goethe, lover of nature as he was, ruled mathematics out of place in natural history. Even now the zoologist has scarce begun to dream of defining in mathematical language even the simplest organic forms. When he meets with a simple geometrical * "Sine profunda Mechanices Scientia nil veri vos intellecturos, nil boni pro- laturos aliis": Boerhaave, De usu ratiocinii Mechanici in Medicina, 1713. t It is well within my own memor how Thomson and Tait, and Klein and Sylvester had to lay stress on the mathematical aspect, and urge the mathematical study, of physical science itself! X Dr Johnson says that "to count is a modern practice, the ancient method was to guess"; but Seneca was alive to the difference — "magnum esse solem philosophus probabit, quantus sit mathematicus." § Cf. Pensees, xxix, "II faut dire, en gros, cela se fait par figure et mouvement, car cela est yrai. Mais de dire quels, et composer la machine, cela est ridicule, car cela est inutile, et incertain, et penible." I] OF ADAPTATION AND FITNESS 3 construction, for instance in the honeycomb, he would fain refer it to psychical instinct, or to skill and ingenuity, rather than to the operation of physical forces or mathematical laws; when he sees in snail, or nautilus, or tiny foraminiferal or radiolarian shell a close approach to sphere or spiral, he is prone of old habit to believe that after all it is something more than a spiral or a sphere, and that in this "something more" there lies what neither mathematics nor physics can explain. In short, he is deeply reluctant to compare the living with the dead, or to explain by geometry or by mechanics thp things which have their part in the mystery of life. Moreover he IS httle inclined to feel the need of such explanations, or of such extension of his field of thought. He is not without some justifi- cation if he feels that in admiration of nature's handiwork he has an horizon open before his eyes as wide as any man requires. He has the help of many fascinating theories within the bounds of his own science, which, though a little lacking in precision, serve the purpose of ordering his thoughts and of suggesting new objects of enquiry. His art of classification becomes an endless search after the blood-relationships of things living and the pedigrees of things dead and gone. The facts of embryology record for him (as Wolff, von Baer and Fritz Miiller proclaimed) not only the life-history of the individual but the ancient 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 problenis of instinct or intel- ligence. Above all, in things both great and small, the naturajist is rightfully impressed and finally engrossed by the peculiar beauty which is manifested in apparent fitness or "adaptation" — the flower for the bee, the berry for the bird. Some lofty concepts, like space and number, involve truths remote from the category of causation; and here we must be content, as Aristotle says, if the mere facts be known*. But natural history deals with ephemeral and accidental, not eternal nor universal * ovK diraiTrjTeop 5 ov8i ttju airiav ofxoiojs, dW LKavou ^v tlctl rb on deLxdrjvai /caXcDj Eth. Nic. 1098a, 33. Teleologist as he was at heart, Aristotle realised that mathematics was on another plane to teleology: rds 5^ fxad-qfiariKas oOdeva iroieladai \6you wepi dyaduv xat KaKQv. Met. 996a, 35. 4 INTRODUCTORY [ch. things ; their causes and effects thru^ themselves on our curiosity, and become the ultimate relations to which our contemplation extends*. 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 of its many forms (for its moods are many), that men have been chiefly wont to explain the phenomena of th« Hving world ; and it will be so while men have eyes to see and ears to hear withal. With Galen, as with Aristotlef, it was the physician's way; with John Ray J, as with Aristotle, it was the naturahst's way; with Kant, as with Aristotle, it was the philosopher's way. It was the old Hebrew way, and has its splendid setting in the story that God made "every plant of the field before it was in the earth, and every herb of the field before it grew." It is a common way, and a great way; for it brings with it a glimpse of a great vision, and it hes deep as the love of nature in the hearts of men. The argument of the final cause is conspicuous in eighteenth- century physics, half overshadowing the "efficient" or physical cause in the hands of such men as Euler§, or Fermat or Maupertuis, to whom Leibniz 1 1 had passed it on. Half overshadowed by the mechanical concept, it runs through Claude Bernard's Legons sur les phenomenes de la Vie^, and abides in much of modern physiology**. * "All reasonings concerning matters of fact seem to be founded on the relation of Cause and Effect. By means of that relation alone we go beyond the evidence of our memory and senses": David Hume, On the Operations of the Understanding. t E.g. "In the works of Nature purpose, not accident, is the main thing": to yap fir) TvxofTWs, d\\' eveKOL rtros, ev tols tt]S (pixreajs ^pyoLs ecrl ko-I jxaXicra. PA, 645a, 24. X E.g. "Quaeri fortasse a nonnullis potest, Quis Papilionum usus? Respondeo, ad ornatum Universi, et ut hominibus spectaculo sint." Joh. Rail, Hist. Insedorum, p. 109. § "Quum enim Mundi universi fabrica sit perfectissima, atque a Creators sapientissimo absoluta, nihil omnino in Mundo contingit in quo non maximi minimive ratio quaepiam eluceat; quamobrem dubium prorsus est nullum quin omnes Mundi effectus ex causis finalibus, ope Methodi maximorum et minimorum, aeque feliciter determinari queant atque ex ipsis causis efficientibus." Methodus inveniendi, etc., 1744, p. 24o {cit. Mach, Science of Mechanics, 1902, p. 455). !| Cf. Opera (ed. Erdmann), p. 106, "Bien loin d'exclure les causes finales... c'est de la qu'il faut tout deduire en Physique": in sharp contrast to Descartes's teaching, " NuUas unquam res naturales a fine, quem Deus aut Natura in iis faciendis sib jproposuit, desumemus, etc." Princip. i, 28. Tj 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." ** It is now and then conceded with reluctance. Thus Paolo Enriques, a learned and philosophic naturalist, writing '"dell' economia di sostanza nelle osse cave" I] OF THE FINAL CAUSE 5 Inherited from Hegel, it dominated Oken's Naturphilosophie and lingered among his later disciples, who were wont to liken the course of organic evolution not to the straggling branches of a tree, but to the building of a temple, divinely planned, and the crowning of it with its pohshed minarets*. It is retained, somewhat crudely, in modern embryology, by those who see in th^ early processes of growth a significance ''rather prospective than retrospective," such that the enlbryonic phenomena must "be referred directly to their usefulness in building up the body of the future animalf": — which is no more, and no less, than to say, with Aristotle, that the organism is the reAo?, or final cause, of its own processes of generation and development. It is writ large in that Entelechyij: 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 Rabelais and Voltaire. And, though it is in a very curious way, we are told that teleology was "refounded, reformed and rehabihtated " by Darwin's concept of the origin of species §; for, just as the older naturalists held (as Addison || puts it) that "the make of every kind of animal is different from that of every other kind; and ye. ti^ere is not the least turn in the muscles, or twist in the fibres of any one, which does not render them more proper for that particular animal's way of life than any other cut or texture of them would have been " : so, by the theory of natural selection, "every variety of form and colour was urgently and absolutely called upon to produce its title {Arch. f. Entw. Mech. xx, 1906), says "una certa impronta di teleologismo qua e la e rimasta, mio malgrado, in questo scritto." * Cf. John Cleland, On terminal forms of life, Journ. Anat. and Physiol. XVIII, 1884. t Conklin, Embryology of Crepidula, Journ. of Morphol. xin, p. 203, 1897; cf. F. R. Lillie, Adaptation in cleavage. Wood's Hole Biol. Lectures, 1899, pp. 43-67. X 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," Leslie Ellis points out that he is referring to Melanchthon's exposition of the Aristotelian doctrine. For Melanchthon, whose view of the peripatetic philosophy had great and lasting influence in the Protestant Universities, affirmed that, according to the true view of Aristotle's opinion, the soul is not a substance but an ivreX^xeia, or function. He defined it as Sufa/xis quaedam ciens actiones — a description all but identical with that of Claude Bernard's '''force vitale.'' § Ray Lankester, art. Zoology, Encycl. Brit. (9th edit.), 1888, p. 806. !l Spectator, No. 120. 6 INTRODUCTORY [ch. 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 rdXos, as men like Butler and Janet have been prompt to shew, an "adapta- tion" without "design," a teleology in which the final cause becomes little more, if anything, than the mere expression or resultant of a sifting out of the good from the bad, or of the better from the worse, in short of a process of mechanism. The apparent manifestations of purpose or adaptation become part of a mechanical philosophy, "une forme methodologique de connaissance*," according to which "la Nature agit tou jours par les moyens les plus simplest," and "chaque chose finit toujours par s'accommoder a son miUeu," as in the Epicurean creed or aphorism that ^atnTe finds a use for every- thing J. 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 complexity of the world. To seek not for end^ 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. It is as a mechanism, or a mechanical construction, that the physicist looks upon the world; and Democritus, first of physicists and one of the greatest of the Greeks, chose to refer all natural phenomena to mechanism and set the final cause aside. * So Newton, in the Preface to the Principia: "Natura enim simplex est, et rerura causis superfluis non -luxuriat"; "Nature is pleased with simplicity, and affects not the pomp of superfluous causes." Modern physics finds the perfection of mathematical beauty in what Newton called the perfection of simplicity. t Janet, Les Causes Finales, 1876, p. 350. X "Nil ideo quoniam natumst in corpore ut uti Possemus sed quod natumst id procreat usum." Lucret. iv, 834. § The phrase is Leibniz's, in his Theodicee: and harks back to Aristotle — If one way be better than another, that you may be sure is Nature's way; Nic. Eth. 10996, 23 et al. I] OF EFFICIENT AND FINAL CAUSES 7 Still, all the whiLe, like warp and woof, mechanism and teleology are interwoven together, and we must not cleave to the one nor despise the other; for their union is rooted in the very nature of totality. We may grow shy or weary of looking to a final cause for an explanation of our phenomena ; but after we have accounted for these on the plainest principles of mechanical causation it may be useful and appropriate to see how the final cause would tally wuth the other, and lead towards the same conclusion*. Maupertuis had Uttle hking for the final cause, and shewed some sympathy with Descartes in his repugnance to its appHcation to physical science. But he found at last, taking the final and the efficient causes one with another, that "I'harmonie de ces deux attributs est si parfaite que sans doute tous les effets de la Nature se pourroient deduire de chacun pris separement. Une Mecanique aveugle et necessaire suit les dessins de I'lntelHgence la plus eclairee et la plus hbref." Boyle also, the Father of Chemistry, wrote, in his latter years, a Disquisition about the Final Causes of Natural Things: Wherein it is Inquired Whether, And {if at all) With what Cautions, a Naturalist should admit Themi He found "that all consideration of final cause is not to be banished from- Natural Philosophy..."; but on the other hand ''that the naturahst who would deserve that name must not let the search and knowledge of final causes make him neglect the in- dustrious indagation of efficients J." In our own day the philosopher neither minimises nor unduly magnifies the mechanical aspect of the Cosmos; nor need the naturahst either exaggerate • or be- little the mechanical phenomena which are profoundly associated with Life, and inseparable from our understanding of Growth and Form. ^ * "S'il est dangereux de se servir des causes finales a priori pour trouver les lois des phenomenes, 11 est peut-etre utile et il est au moins curieux de faire voir com- ment le principe des causes finales s'accorde avec les lois des phenomenes, pourvu qu'on commence par determiner ces lois d'apres les principes de mecanique clairs et incontestables." (D'Aiembert, Art. Causes finales, Encyclopedie, ii, p. 789, 1751.) I SeeJiis essay on the '''Accord des differentes lois de la Nature." X Cf. also Leibniz {Discours de la Metaphysique: Lettres inedites, ed. de Careil, 1857, p. 354), "L'un et I'autre est bon, I'un et I'autre peut etre utile... et les auteurs qui suivent ces deux routes differentes ne devraient pas se maltraiter." Or again in the Monadologie, "Les ames agissent selon les causes finales. ... Les corps agissent selon les lois des causes efficientes ou des mouveraents. Et les deux regnes, celui des causes efficientes et des causes finales sont harmonieux entre eux." 8 INTRODUCTORY [ch. 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 enquiry of all 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 hypo- theses in the philosophy of biology, so long will these "satisfactory and specious causes" tend to stay "severe and diligent enquiry. . . to the great arrest and prejudice of future discovery." Long before the great Lord Keeper wrote these words, Roger Bacon had shewn how easy it is, and how vain, to survey the operations of Nature and idly refer her wondrous works to chance or accident, or to the immediate interposition of Godf. The difiiculties which surround the concept of ultimate or "real" caugation, in Bacon's or Newton's sense of the word, the in- superable difficulty of giving any just and tenable account of the relation of cause and effect from the empirical point of view, need scarcely hinder us in our physical enquiry. As students of mathe- matical and experimental physics we are content to deal with those antecedents, or concomitants, of our phenomena without which the phenomenon does not occur — with causes, in short, which, aliae ex aliis aptaetet necessitate' nexae, are no more, and no less, than con- ditions 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 more of the physical phenomena around, and so to weave a web of connection and interdependence which shall serve our turn, though the metaphysician withhold from that interdependence the title of causality J. We come in touch * The reader will understand that I speak, not of the "severe and diligent enquiry" 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. f Op. tert. (ed. Brewer, p. 99). "Ideo mirabiles actiones naturae, quae tota die fiunt in nobis et in rebus coram oculis nostris, non percipimus; sed aestimamus eas fieri vel per specialem operationem divinam. . .vel a casu et fortuna," t Cf. Fourier's phrase, in his Theorie de la Chaleur, with which Thomson and Tait prefaced their Treatise an Natural Philosophy: "Les causes primordiales ne I] OF ULTIMATE CAUSATION 9 with what the schoohnen called a ratio cognoscendi, though the true ratio efflciendi 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 dis- connected, and for "similitude in things to common view unlike*." Newton did not shew the cause of the apple falling, but he shewed a simihtude ("the more to increase our wonder, with an apple") between the apple and the starsf. By doing so he turned old facts into new knowledge ; and was well content if he could bring diverse phenomena under "two or three Principles of Motion" even "though the Causes of these Principles were not yet discovered". Moreover, the naturalist and the physicist will continue to speak of "causes", just as of old, though it may be with some mental reservations : for, as a French philosopher said in a kindred difficulty : "ce sont la des manieres de s'exprimer, et si elles sont interdites il faut renoncer a parler de ces choses." The search for differences or fundamental contrasts between the phenomena of organic and inorganic, of animate and inanimate, things,, has occupied many men's minds, while the search for com- munity of principles or essential simihtudes has been pursued by few; and the contrasts are apt to loom too large, great though they may be. M. Dunan, discussing the Probleme de la Viel, in an essay which M. Bergson greatly commends, declares that "les lois physico- chimiques sont aveugles et brutales ; la ou 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 forms not less beautiful and scarce less varied than those which move us to admira- nous sont point connues; mais elles sont assujetties a des lois simples et eonstantes, que Ton peut decouvrir par I'observation, et dont I'etude est I'objet de la philosophie naturelle." * "Plurimum amo analogias, fidelissimos meos magistros, omnium Naturae arcanorum conscios," said Kepler; and Perrin speaks with admiration, in Les Atonies, of men like Galileo and Carnot, who "possessed the power of perceiving analogies to an extraordinary degree." Hure^e declared, and Mill said much the same thing, that all reasoning whatsoever depends on resemblance or analogy, and the power to recognise it. Comparative anatomy (as Vicq d'Azyr first called it), or comparative physics (to use a phrase of Mach's), are particular instances of a sustained search for analogy or similitude. t As for Newton's apple, see De Morgan, in Notes and Queries (2), vi, p. 169, 1858. I Revue Philosophique, xxxiii, 1892. 10 INTRODUCTORY [ch. tion among living things. The waves of the sea, the little ripples on the shore, the sweeping curve of the sandy bay between the headlands, the outline of the hills, the shape of the clouds, all these are so many riddles of form, so many problems of morphology, and all of them the physicist can more or less easily read and adequately solve: solving them by reference to their antecedent phenomena, in the material system of mechanical forces to which they belong, and to which we interpret them as being due. They have also, doubtless, their immanent teleological significance; but it is on another plane of thought from the physicist's that we contemplate their intrinsic harmony* and perfection, and "see that they are good." Nor is it otherwise with the material forms of living things. Cell and tissue, shell and bone, leaf and flower, are so many portions of matter, and it is in obedience to the laws of physics that their particles have been moved, moulded and conformedf. They are no exception to the rule that Qeos aet yeajjjLerpeL. Their problems of form are in the first instance mathematical problems, their problems of growth are essentially physical problems, and the morphologist is, ipso facto, a student of physical science. He may learn from that comprehensive science, as the physiologists have not failed to do, the point of view from which her problems are approached, the quantitative methods by which they are attacked, and the whole- some restraints under which all her work is done. He may come to realise that there is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real * What I understand by "holism" is what the Greeks called apfiovia. This is something exhibited not only by a lyre in tune, but by all the handiwork of craftsmen, and by all that is " put together" by art or nature. It is the " composite- ness of any composite whole"; and, like the cognate terms KpSicns or (rvvdeais, implies a balance or attunement. Cf. John Tate, in Class. Review, Feb. 1939. t This general principle was clearly grasped by Mr George Rainey many years ago, and expressed in such words as the following: "It is illogical to suppose that in the case of vital organisms a distinct force exists to produce results perfectly within the reach of physical agencies, especially as in many instances no end could be attained were that the case, but that of opposing one force by another capable of effecting exactly the same purpose." (On artificial calculi, Q.J. M.S. {Trans. Microsc. Soc), vr, p. 49, 18.58.) Cf. also Helmholtz, infra cit. p. 9. (Mr George Rainey, a man of learning and originality, was demonstrator of anatomy at St Thomas's; he followed that modest calling to a great age, and is remembered by a few old pupils with peculiar affection.) I] OF EVOLUTION AND ENTROPY 11 world*. He may even find a certain analogy between the slow, reluctant extension of physical laws to vital phenomena and the slow triumphant demonstration by Tycho Brahe, Copernicus, GaHleo and Newton (all in opposition to the Aristotelian cosmogony), that the heavens are formed of like substance with the earth, and that the movements of both are subject to the selfsame laws. Organic evolution has its physical analogue in the universal law that the world tends, in all its parts and particles, to pass from certain less probable to certain more probable configurations or states. This is the second law of thermodynamics. It has been called the law of evolution of the world'f ; and we call it, after Clausius, the Principle of Entropy, which is a literal translation of Evolution into Greek. The introduction of mathematical concepts into natural' science has seemed to many men no mere stumbling-block, but a very parting of the ways. Bichat was a man of genius, who did immense service to philosophical anatomy, but, like Pascal, he utterly refused to bring physics or mathematics into biology: " On calcule le retour d'un comete, les resistances d'un fluide parcourant un canal inerte, la Vitesse d'un projectile, etc.; mais calculer avec BorelU la force d'un muscle, avec Keil la vitesse du sang, avec Jurine, Lavoisier et d'autres la quantite d'air entrant dans le poumon, c'est batir sur un sable mouvant un edifice sohde par lui-meme, mais qui tombe bientot faute de base assureej." Comte went further still, and said that every attempt to introduce mathematics into chemistry must be deemed profoundly irrational, and contrary to the whole spirit of the science §. But the great makers of modern science have all gone the other way. Von Baer, using a bold metaphor, thought that it might become possible " die bildenden Krafte des thierischen Korpers . auf die allgemeinen Krafte oder Lebenserscheinun^en des Weltganzes zuriickzufiihrenll." Thomas Young shewed, as BorelU had done, how physics may subserve anatomy; he learned from the heart and a,rteries that " the mechanical motions which take place in an animal's body are regulated by the same general laws as the motions of * So said Lobatchevsky. t Cf. Chwolson, Lehrbuch, iii, p. 499, 1905; J. Perrin, Traitd de chimie physique, I, p. 142, 1903; and Lotka's Elements of Physical Biology, 1925, p. 26. t La Vie et la Mort, p. 81. § Philosophie Positive, Bk. rv. ]| Ueber Entwicklung der Thiere: Beobachtungen und Reflexionen, i, p. 22, 1828. 12 INTRODUCTORY [ch. inanimate bodies*." And Theodore Schwann said plainly, a hun- dred years ago, "Ich wiederhole iibrigens dass, wenn hier von einer physikahschen Erklarung der organischen Erscheinungen die Rede ist, darunter nicht nothwendig eine Erklarung durch die bekannten physikalischen Krafte. . .zu verstehen ist, sondern iiberhaupt eine Erklarung durch Krafte, die nach strengen Gesetzen der blinden Nothwendigkeit wie die physikalischen Krafte wirken, mogen diese Krafte auch in der anorganischen Natur auftreten oder nicht f." Helmholtz, in a famous and influential lecture, and surely with these very words of Schwann's in mind, laid it down as the funda- mental principle of physiology that "there may be other agents acting in the hving body than those agents which act in the inorganic world ; but these forces, so 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 eff'ects must be ruled by necessity, and must always be the same when acting under the same conditions ; and so there cannot exist any arbitrary choice in the direction of their actions." It follows further that, like the other "physical" forces, they must be subject to mathematical analysis and deduction J. So much for the physico-chemical problems of physiology. Apart from these, the road of physico-mathematical or dynamical investi- gation in morphology has found 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 highway side. It was Galileo's and Borelli's way; and Harvey's way, when he discovered the circulation of the blood ||. It was little trodden for long afterwards, but once in a while Swammerdam and Reaumur passed thereby. And of later years Moseley and Meyer, Berthold, Errera and Roux have been among * Croonian Lecture on the heart and arteries, Phil. Trans. 1809, p. 1; Collected Works, I, p. 511. t M ikroskopische Untersuchungen, 1839, p. 226. J The conservation of forces applied to organic nature, Proc. Royal Inst. April 12, 1861. § Whereby he incurred the reproach of Socrates, in the Phaedo. See Clerk Maxwell on Anaxagoras as a Physicist, in Phil. Mag. (4), xlvi, pp. 453-460, 1873. II Cf. Harvey's preface to his Exercitationes de Generatione Animalium, 1651: "Quoniam igitur in Generatione animalium (ut etiam in caeteris rebus omnibus de quibus aliquid scire cupimus), inquisitio omnis a caussis petenda est, praesertim a materiali et efficiente: visum est mihi" etc. I] OF NATURAL PHILOSOPHY 13 the little band of travellers. We need not wonder if the way be hard to follow, and if these wayfarers have yet gathered little. A harvest has been reaped by others, and the gleaning of the grapes is slow. It behoves us always to remember that in physics it has taken great men to discover simple things. They are very great names indeed which 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 shghtest 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 naturahst the new TheariaPhilosophiaeNaturalis. 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; and that living matter influences and is influenced by mind is a mystery without a clu'e. 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 of the body, as of all else 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 in- adequate 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 no physical law, any more than gravity itself, not even among the puzzles of stereo- chemistry or of physiological surface-action and osmosis, is known to be transgressed by the bodily mechanism. 14 INTRODUCTORY [ch. Some physicists declare, as Maxwell did, that atoms or molecules more compHcated by far than the chemist's hypotheses demand, are requisite to explain the phenomena of life. If what is impHed be an explanation of psychical phenomena, let the point be granted at once; we may go yet further and decHne, with Maxwell, to believe that anything of the nature of physical complexity, however exalted, could ever suffice. Other physicists, like Auerbach*, or Larmorj, or Joly J, 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, of the bodily energies, the retardation of entropy, 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 state- ment and physical law certain of the simpler outward phenomena of organic growth and structure or form, while all the while regarding the fabric of the organism, ex hypothesi, as a material and mechanical configuration. This is my purpose here. But I would not for the world be thought to beheve that this is the only story which Life and her Children have tp tell. One does not come by studying living things for a lifetime to suppose that physics and chemistry can account for them all§. 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 * Ektropismus, oder die physikalische Theorie des Lebens, Leipzig, 1810. t Wilde Lecture, Nature, March 12, 1908; ibid. Sept. 6, 1900; Aether and Matter, p. 288. Cf. also Kelvin, Fortnightly Review, 1892, p. 313. X The abundance of life, Proc. Roy. Dublin Soc. vii, 1890; Scientific Essays, 1915, p. 60 seq. § That mechanism has its share in the scheme of nature no philosopher has denied. Aristotle (or whosoever wrote the De Mundo) goes so far as to assert that in the most mechanical operations of nature we behold some of the divinest attributes of God. II J. H, Fr. Papillon, Histoire de la jyhilosophie moderne dans ses rapports avec le developpement des sciences de la nature, i, p. 300, 1870. I] OF LIFE ITSELF 15 legitimate postulate of the physicist, in approaching the physical problems of the living 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 life, it is in no spirit of arrogant assertiveness if the physicist begins his argument, after the fashion of a most illustrious exemplar, with the old formula of scholastic challenge: An Vita sit? Dico quod non. The terms Growth and Form, which make up the title of this book, are to be understood, as I need hardly say, in their relation to the study of organisms. We want to see how, in some cases at least, the forms of living things, and of the parts of living things, can be explained by physical considerations, and to realise that in general no organic forms exist save such as are in conformity with physical and mathematical laws. And while growth is a somewhat vague word for a very complex matter, which may 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, * It is neither unnecessary nor superfluous to explain that physics is passing through an empirical phase into a phase of pure mathematical reasoning. But when we use physics to interpret and elucidate our biology, it is the old-fashioned empirical physics which we endeavour, and are alone able, to apply. 16 INTRODUCTORY [ch. or movement, or the movements that change of form impHes, 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 used appropriately in connection therewith. The form, then, of any portion of matter, whether it be living or dead, and the changes of form which are apparent in its movements and in its growth, may in all cases alike be described as due to the action of force. In short, the form of an object is a "diagram of forces," in this sense, at least, that from it we can judge of or deduce the forces that are acting or have acted upon it: in this strict and particular sense, it is a diagram — in the case of a sohd, of the forces which have been impressed upon it when its conformation was produced, together with those which enable it to retain its conformation; in the case of a Hquid (or of a gas) of the forces which are for the moment acting on it to restrain or balance its own inherent mobility. In an organism, great or small, it is not merely the nature of the motions of the hving substance which we must interpret in terms of force (according to kinetics), but also the conformation of the organism itself, whose permanence or equilibrium is explained by the interaction or balance of forces, as described in statics. If we look at the hving 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 under- stood 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 I] OF MATTER AND ENERGY 17 principal properties of matter with which our subject obhges us to deal. Let us imagine, for instance, the case of a so-called "simple" organism, such as Amoeba; and if our short list of its physical •properties and conditions be helpful to our further discussion, we need not consider how far it be complete or adequate from the wider physical point of view*. 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 — without the help of which its creeping movements could not be performed. It is acted on by gravity, and this force tends (though slightly, owing to the Amoeba's small mass, and to the small difference between its density and that of the surrounding fluid) to flatten it down upon the solid substance on which it may be creeping. Our Amoeba tends, in the next place, to be deformed by any pressure from outside, even though slight, which may be applied to it, and this circumstance shews it to consist of matter in a fluid, or at least semi-fluid, state: which state is further indicated when we observe streaming or current motions in its interior. Like other fluid bodies, its surfacef, whatsoever other substance — gas, hquid or solid — it be in contact with, and in varying degree according to the nature of that adjacent substance, is the seat of molecular force exhibiting itself as a surface-tension, from the action of which many important consequences follow, greatly affecting the form of the fluid surface. While the protoplasmj of the Amoeba reacts to the shghtest pressure, and tends to "flow," and while we therefore speak of it * With the special and impprtant properties of colloidal matter we are, for the time being, not concerned. t Whether an animal cell has a membrane, or only a pellicle or zona limitans, was once deemed of great importance, and played a big part in the early contro- versies between the cell-theory of Schwann and the protoplasma-theory of Max Schultze and others, Dujardin came near the truth when he said, somewhat naively, "en niant la presence d'un tegument propre, je ne pretends pas du tout nier i'existence d'une surface." % The word protoplasm is used here in its most general sense, as vaguely as when Huxley spoke of it as the "physical basis of life." Its many changes and shades of meaning in early years are discussed by Van Bambeke in the Bull. Sac. Beige de Microscopie, xxn, pp. 1-16, 1896. 18 INTRODUCTORY [ch. as a fluid*, it is evidently far less mobile than such a fluid (for instance) as water, but is rather Uke 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 resisting change of form, or in other words of retarding the efl'ects 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 some Amoebae differ greatly from 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 sohd (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 soUds 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 pelhcle as in Amoeba or a firm cell-wall as in Protococcus, the diffusion which takes place throtigh this wall is sometimes distinguished under the term osmosis. Within the cell, chemical forces are at work, and so also in all probabihty (to judge by analogy) are electrical forces; and the organism reacts also to forces from without, that have their origin in chemical, electrical and thermal influences. The processes of diffusion and of chemical activity within the cell result, by the drawing in of water, salts, and food-material with or without chemical transformation into protoplasm, in growth, and this com- plex phenomenon we shall usually, without discussing its nature and origin, describe and picture as a force. Indeed we shall manifestly be incHned 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 as a force. In the phenomena of cell-division, in the attractions or repulsions of the parts of the dividing nucleus, and in the " caryokinetic " figures which appear in connection with it, we seem to see in operation forces and the effects of forces which have, to say the * One of the first statements which Dujardin made about protoplasm (or, as he called it, sarcode) was that it was not a fluid; and he relied greatly on this fact to shew that it was a living, or an organised, structure. I] OF VITAL PHENOMENA 19 least of it, a close analogy with known physical phenomena : and to this matter we shall presently return. But though they resemble known physical phenomena, their nature is still the subject of much dubiety and 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 they do or no, it is plain that we have no clear rule or guidance 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. But even the ordinary laws of the physical forces are by no means simple and plain. In the winding up of a clock (so Kelvin once said), and in the properties of matter which it involves, there is enough and more than enough of mystery for our limited under- standing: "a watchspring is much farther beyond our understanding than a gaseous nebula." We learn and learn, but never know all, about the smallest, humblest thing. So said St Bonaventure : " Si per multos annos viveres, adhuc naturam unius festucae seu muscae seu minimae creaturae de mundo ad plenum cognoscere non valeres*." There is a certain fascination in such ignorance ; and we learn (like the Abbe Galiani) without discouragement that Science is "plutot destine a etudier qu'a connaitre, a chercher qu'a trouver la verite." Morphology 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 term^ of force, of the operations of Energyf . And here it is well worth while to remark that, in deahng * Op. V, p. 541 ; cit. E. Gilson. t This is a great theme. Boltzmann, writing in 1886 on the second law of thermodynamics, declared that available energy was the main object at stake in the struggle for existence and the evolution of the world. Cf. Lotka, The energetics of evolution, Proc. Nat. Acad. Sci. 1922, p. 147. 20 INTRODUCTORY [ch. with the facts of embryology or the phenomena of inheritance, the common language of the, books seems to deal too much with the material elements concerned, as the causes of development, of variation or of hereditary transmission. Matter as such produces nothing, changes nothing, does nothing; and however convenient it may afterwards be to abbreviate our nomenclature and our descriptions, we must most carefully realise in the outset that the spermatozoon, the nucleus, the chromosomes or the germ-plasma can never act as matter alone, but only as seats of energy and as centres of force. And this is but an adaptation (in the light, or rather in the conventional symboHsm, of modern science) of the old saying of the philosopher : apx^ yo.p r) voLs fxdXXov rrjs vXrjg. Since this book was written, some five and twenty years ago, certain great physico-mathematical concepts have greatly changed. Newtonian mechanics and Newtonian concepts of space and time are found unsuitable, even untenable or invahd, for the all but infinitely great and the all but infinitely small. The very idea of physical causation is said to be illusory, and the physics of the atom and the electron, and of the quantum theory, are to be elucidated by the laws of probability rather than by the concept of causation and its effects. But the orders of magnitude, whether of space or time, within which these new concepts become useful, or hold true, lie far away. We distinguish, and can never help distinguishing, between the things which are of our own scale and order, to which our minds are accustomed and our senses attuned, and those remote phenomena which ordinary standards fail to measure, in regions where (as Robert Louis Stevenson said) there is no habitable city for the mind of man. It is no wonder if new methods, new laws, new words, new modes of thought are needed when we make bold to contemplate a Universe within which all Newton's is but a speck. But the world of the Hving, wide as it may be, is bounded by a famihar horizon within which our thoughts and senses are at home, our scales of time and magnitude suffice, and the Natural Philosophy of Newton and Gahleo rests secure. We start, like Aristotle, with our own stock-in-trade of know- ledge: dpKTeov OLTTO Tcov rjfjuv yvajpLjjLojv. And only when we are I] OF NEWTONIAN PHYSICS 21 steeped to the marrow (as Henri Poincare once said) in the old laws, and in no danger of forgetting them, may we be allowed to learn how they have their remote but subtle limitations, and cease afar off to be more than approximately true *. Kant's axiom of causahty, that it is denknotwendig — indispensable for thought — remains true however physical science may change. His later aphorism, that all changes take place subject to the law which links cause and effect together — "alle Veranderungen geschehen nach dem Gesetz der Verkniipfung von Ursache und Wirkung" — is still an axiom a priori, independent of experience: for experience itself depends upon its truth t- * So Max Planck himself says somewhere: "In my opinion the teaching of mechanics will still have to begin with Newtonian force, just as optics begins in the sensation of colour and thermodynamics with the sensation of warmth, despite th^ fact that a more precise basis is substituted later on." t "Weil er [der Grundsatz das Kausalverhaltnisses] selbst der grund der Moglich- keit einer solchen Erfahrung ist": Kritik d. reinen Vernunft, ed. Odicke, 1889, p. 221. Cf. also G. W. Kellner, Die Kausalitat in der Physik, Ztschr.f. Physik, lx,iv, pp. 568- 580. 1930. CHAPTER II ON MAGNITUDE To terms of magnitude, and of direction, must we refer all our conceptions of Form. For the form of an object is defined when we know its magnitude, actual or relative, in various directions; and Growth involves the same concepts of magnitude and direction, related to the further concept, or "dimension," of Time. Before we proceed to the consideration of specific form, it will be well to consider certain general phenomena of spatial magnitude, or of the extension of a body in the several dimensions of space. We are taught by elementary mathematics — and by Archimedes himself — that in similar 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 4:7rr^, and its volume to ^ttt^ ; from which it follows that the ratio of its volume to surface, or V/S, is Jr. That is to say, VfS varies as r; or, in other words, the larger the sphere by so much the greater will be its volume (or its mass, if it be uniformly dense throughout) in comparison with its superficial area. And, taking L to represent any linear dimension, we may write the general equations in the form Soz L\ F oc L^ or iS = kL\ and V = k'L\ where k, k', are "factors of proportion," V V k and ^ cc L, or — = j-, L = KL. o ok So, in Lilliput, "His Majesty's Ministers, finding that Gulhver's stature exceeded theirs in the proportion of twelve to one, concluded from the similarity of their bodies that his must contain at least 1728 [or 12^] of theirs, and must needs be rationed accordingly*." * Likewise Gulliver had a whole Lilliputian hogshead for his half-pint of wine: in the due proportion of 1728 half-pints, or 108 gallons, equal to one pipe or CH. II] OF DIMENSIONS 23 From these elementary principles a great many consequences follow, all more or less interesting, and some of them of great importance. In the first place, though growth in length (let us say) and growth in volume (which is usually tantamount to mass or weight) are parts of one and the same process or phenomenon, the one attracts our attention by its increase very much more than the other. For instance a fish, in doubhng its length, multiphes its weight 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 an understanding of the correla- tion 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 likely; but as long as growth lasts changes of form, even though inappreciable to the eye, are apt and hkely to occur. Now weighing is a far easier and far more accurate operation than measuring; and the measurements which would reveal slight and otherwise imperceptible changes in the form of a fish — slight relative differences between length, breadth and depth, for instance — would need to be very dehcate indeed. But if we can make fairly accurate determinations of the length, which is much the easiest linear dimension to measure, and correlate it with the weight, then the value of k, whether it varies or remains constant, will tell us at once whether there has or has not been a tendency to alteration in the general form, or, in other words, a difference in the rates of growth in different directions. To this subject we shall return, when we come to consider more particularly the phenomenon of rate of growth. double-hogshead. But Gilbert White of Selborne could not see what was plain to the Lilliputians; for finding that a certain little long-legged bird, the stilt, weighed 4J oz. and had legs 8 in. long, he thought that a flamingo, weighing 4 lbs., should have legs 10 ft. long, to be in the same proportion as the stilt's. But it is obvious to us that, as the weights of the two birds are as 1 : 15, so the legs (or other linear dimensions) should be as the cube-roots of these numbers, or nearly as 1 : 2^. And on this scale the flamingo's legs should be, as they actually are, about 20 in. long. 24 ON MAGNITUDE [ch. We are accustomed to think of magnitude as a purely relative matter. We call a thing big or little with reference to what it is wont to be, as when we speak of a small elephant or a large rat ; and we are apt accordingly to suppose that size makes no other or more essential difference, and that Lilliput and Brobdingnag* are all alike, according as we look at them through one end of the glass or the other. Gulliver himself declared, in Brobdingnag, that "undoubtedly philosophers are in the right when they tell us that nothing is great and little otherwise than by comparison": and Oliver Heaviside used to say, in like manner, that there is no absolute scale of size in the Universe, for it is boundless towards the great and also boundless towards the small. It is of the very essence of the Newtonian philosophy that we should be able to extend our concepts and deductions from the one extreme of magni- tude to the other; and Sir John Herschel said that "the student must lay his account to finding the distinction of great and little altogether annihilated in nature." All this is true of number, and of relative magnitude. The Universe has its endless gamut of great and small, of near and far, of many and few. Nevertheless, in physical science the scale of absolute magnitude becomes a very real and important thing; and a new and deeper interest arises out of the changing ratio of dimensions when we come to consider the inevitable changes of physical rela- tions with which it is bound up. The effect of scale depends not on a thing in itself, but in relation to its whole environment or milieu ; it is in conformity with the thing's "place in Nature," its field of action and reaction in the Universe. Everywhere Nature works true to scale, and everything has its proper size accordingly. Men and trees, birds and fishes, stars and star-systems, have their appropriate dimensions, and their more or less narrow range of absolute magnitudes. The scale of human observation and ex- perience lies within the narrow bounds of inches, feet or miles, all measured in terms drawn from our own selves or our own doings. Scales which include light-years, parsecs. Angstrom units, or atomic * Swift paid close attention to the arithmetic of magnitude, but none to its physical aspect. See De Morgan, on Lilliput, in N. and Q. (2), vi, pp. 123-125, 1858. On relative magnitude see also Berkeley, in his Essay towards a New Theory of Visio7i, 1709. II] THE EFFECT OF SCALE 25 and sub-atomic magnitudes, belong to other orders of things and other principles of cognition. A common effect of scale is due to the fact that, of the physical forces, some act either directly at the surface of a body, or otherwise in proportion to its surface or area; while others, and above all gravity, act on all particles, internal and external alike, and exert a force which is proportional to the mass, and so usually to the volume of the, body. A simple case is that of two similar weights hung by two similar wires. The forces exerted by the weights are proportional to their masses, and these to their volumes, and so to the cubes of the several Hnear dimensions, including the diameters of the wires. But the areas of cross-section of the wires are as the squares of the said linear dimensions; therefore the stresses in the wires 'per unit area are not identical, but increase in the ratio of the linear dimen- sions, and the larger the structure the more severe the strain becomes : Force l^ A^ ^ r^ ^ ^' and the less the wires are capable of supporting it. In short, it often happens that of the forces in action in a system some vary as one power and some as another, of the masses, distances or other magnitudes involved; the "dimensions" remain the same in our equations of equilibrium, but the relative values alter with the scale. This is known as the "Principle of Similitude," or of dynamical similarity, and it and its consequences are of great importance. In a handful of matter cohesion, capillarity, chemical affinity, electric charge are all potent; across the solar system gravitation* rules supreme; in the mysterious region of the nebulae, it may haply be that gravitation grows negligible again. To come back to homelier things, 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 dimen- * In the early days of the theory of gravitation, it was deemed especially remarkable that the action of gravity "is proportional to the quantity of solid matter in bodies, and not to their surfaces as is usual in mechanical causes; this power, therefore, seems to surpass mere mechanism" (Colin Maclaurin, on Sir Isaac Newton's Philosophical Discoveries, iv, 9). 26 ON MAGNITUDE [ch. sions. It follows at once that, if we build two bridges geometrically similar, the larger is the weaker of the two*, and is so in the ratio of their linear dimensions. It was elementary engineering experience such as this that led Herbert Spencer to apply the principle of simihtude to biologyf. But here, before w^e go further, let us take careful note that increased weakness is no necessary concomitant of increasing size. There are exceptions to the rule, in those exceptional cases where we have to deal only with forces which vary merely with the area on which they impinge. - If in a big and a httle ship two similar masts carry two similar sails, the two sails will be similarly strained, and equally stressed at homologous places, and alike suitable for resisting the force of the same wind. Two similar umbrellas, however differing in size, will serve ahke in the same weather; and the expanse (though not the leverage) of a bird's wing may be enlarged with little alteration. The principle of similitude had been admirably apphed in a few clear instances by Lesage J, a celebrated eighteenth-century physician, in an unfinished and unpublished work. Lesage argued, for example, that the larger ratio of surface to mass in a small animal would lead to excessive transpiration, were the skin as "porous" as our own; and that we may thus account for the hardened or thickened skins of insects and many other small terrestrial animals. Again, since the weight of a fruit increases as the cube of its linear dimensions, while the strength of the stalk increases as the square, it follows that the stalk must needs grow out of apparent due proportion to the fruit: or, alternatively, that tall trees should not bear large * The subject is treated from the engineer's point of view by Prof. James Thomson, Comparison of similar structures as to elasticity, strength and stability, Coll. Papers, 1912, pp. 361-372, and Trans. Inst. Engineers, Scotland, 1876; also by Prof. A. Barr, ibid. 1899. See also Rayleigh, Nature, April 22, 1915; Sir G. Greenhill, On mechanical similitude, Math. Gaz. March 1916, Coll. Works, vi, p. 300. For a mathematical account, sec (e.g.) P. VV. Bridgeman, Dimensional Analysis (2nd ed.), 1931, or F. W. Lanchester, The Theory of Dimensions, 1936. t Herbert Spencer, The form of the earth, etc., Phil. Mag. xxx, pp. 194-6, 1847; also Principles of Biology, pt. ii, p. 123 seq., 1864. I See Pierre Prevost, Notices de la vie et des ecrits de Lesage, 1805. George Louis Lesage, born at Geneva in 1724, devoted sixty-three years of a life of eighty to a mechanical theory of gravitation; see W. Thomson (Lord Kelvin), On the ultramundane corpuscles of Lesage, Proc. E.S.E. vii, pp. 577-589, 1872; Phil. Mag. XLV, pp. 321-345, 1873; and Clerk Maxwell, art. "Atom," Encyd. Brit. (9), p.' 46. II] THE PRINCIPLE OF SIMILITUDE 27 fruit on slender branches, and that melons and pumpkins must lie upon the ground. And yet again, that in quadrupeds a large head must be supported on a neck which is either excessively thick and strong like a bull's, or very short like an elephant's*. But it was Gahleo who, wellnigh three hundred years ago, had first laid down this general principle of simiUtude; and he did so with the utmost possible clearness, and with a great wealth of illustra- tion drawn from structures living and deadf. He said that if we tried building ships, palaces or temples of enormous size, yards, beams and bolts would cease to hold together; nor can Nature grow a tree nor construct an animal beyond a certain size, while retaining the proportions and employing the materials which suffice in the case of a smaller structure J. 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 new material, harder and stronger than was used before. Both processes are famihar to us in Nature and in art, and practical apphcations, undreamed of by Gahleo, meet us at every turn in this modern age of cement and steel §. Again, as Galileo was also careful to explain, besides the questions of pure stress and strain, of the strength of muscles to hft an increasing weight or of bones to resist its crushing stress, we have the important question of bending ynornents. This enters, more or less, into our whole range of problems ; it aifects the whole form of the skeleton, and sets a limit to the height of a tall tree||. * Cf. W. Walton, On the debility of large animals and trees, Quart. Journ. of Math. IX, pp. 179-184, 1868; also L. J. Henderson, On volume in Biology, Proc. Amer. Acad. Sci. ii, pp. 654-658, 1916; etc. t Discorsi e Dimostrazioni matematiche, intorno a due nuove scienze attenenti alia Mecanica ed ai Muovimenti Locali: appresso gli Elzevirii, 1638; Opere, ed. Favaro, viir, p. 169 seq. Transl. by Henry Crew and A. de Salvio, 1914, p. 130. X So Werner remarked that Michael Angelo and Bramanti could not have built of gypsum at Paris on the scale they built of travertin at Rome. § The Chrysler and Empire State Buildings, the latter 1048 ft. high to the foot of its 200 ft. "mooring mast," are the last word, at present, in this brobdingnagian architecture. II It was Euler and Lagrange who first shewed (about 1776-1778) that a column of a certain height would merely be compressed, but one of a greater height would be bent by its own weight. See Euler, De altitudine columnarum etc.. Acta Acad. Sci. Imp. Petropol. 1778, pp. 163-193; G, Greenhill, Determination of the greatest height to which a tree of given proportions can grow, Cambr. Phil. Soc. Proc. rv, p. 65, 1881, and Chree, ibid, vu, 1892. 28 ON MAGNITUDE [ch. We learn in elementary mechanics the simple case of two similar beams, supported at both ends and carrying no other weight than their own. Within the limits of their elasticity they tend to be deflected, or to sag downwards, in proportion to the squares of their linear dimensions ; if a match-stick be two inches long and a similar beam six feet (or 36 times as long), the latter will sag under its own weight thirteen hundred times as much as the other. To counteract this tendency, as the size of an animal increases, the limbs tend to become thicker and shorter and the whole skeleton bulkier and heavier; bones make up some 8 per cent, of the body of mouse or wren, 13 or 14 per cent, of goose or dog, and 17 or 18 per cent, of the body of a man. Elephant and hippopotamus have grown clumsy as well as big, and the elk is of necessity less graceful than the gazelle. It is of high interest, on the other hand, to observe how little the skeletal proportions differ in a httle porpoise and a great whale, even in the limbs and hmb-bones ; for the whole influence of gravity has become neghgible, or nearly so, in both of these. In ifhe problem of the tall tree we have to determine the point at which the tree will begin to bend under its own weight if it be ever so little displaced from the perpendicular*. In such an investigation we have to make certain assumptions — for instance that the trunk tapers uniformly, and that the sectional area of the branches 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 shewed, on such assumptions as the above, that a certain British Columbian pine-tree, of which the Kew flag-staff, which is 221 ft. high and 21 inches in diameter at the base, was made, could not possibly, by theory, have grown to more than about 300 ft. It is very curious that Galileo had suggested precisely the same height (ducento braccie alta) as the utmost limit of the altitude of a tree. In general, as Greenhill shewed, the diameter of a tall homogeneous body must increase as the power 3/2 of its height, which accounts for the slender proportions of young trees compared with the squat * In like manner the wheat-straw bends over under the weight of the loaded ear, and the cat's tail bends over when held erect — not because, they "possess flexibility," but because they outstrip the dimensions within which stable equi- librium is possible in a vertical position. The kitten's tail, on the other hand, stands up spiky and straight. II] OF THE HEIGHT OF A TREE 29 or stunted appearance of old and large ones*. In short, as Goethe says in Dicktung und Wahrheit, "Es ist dafiir gesorgt dass die Baume nicht in den Himmel wachsen." But the tapering pine-tree is but a special case of a wider problem. The oak does not grow so tall as the pine-tree, but it carries a heavier load, and its boll, broad-based upon its spreading roots, shews a different contour. Smeaton took it for the pattern of his Hghthouse, and Eiffel built his great tree of steel, a thousand feet high, to a similar but a stricter plan. Here the profile of tower or tree follows, or tends to follow, a logarithmic curve, giving equal strength throughout, according to a principle which we shall have occasion to discuss later on, when we come to treat of form and mechanical efficiency in the skeletons of animals. In the tree, moreover, anchoring roots form powerful wind-struts, and are most de- veloped opposite to the direction of the prevailing winds; for the lifetime of a tree is affected by the frequency of storms, and its strength is related to the wind-pressure which it must needs with- standf. Among animals we see, without the help of mathematics or of physics, how small birds and beasts are quick and agile, how slower and sedater movements come with larger size, and how exaggerated bulk brings with it a certain clumsiness, a certain inefficiency, an element of risk and hazard, a preponderance of disadvantage. The case was well put by Owen, in a passage which has an interest of its own as a premonition, somewhat Hke De. Candolle's, of the "struggle for existence." Owen wrote as follows J: " In proportion to the bulk of a species is the difficulty of the contest which, as a living organised whole, the individual of each species has to maintain against the surrounding agencies that are ever tending to dissolve the vital bond, and subjugate the Hving matter to the ordinary chemical and physical forces. Any changes, therefore, in such external conditions as a species may have been original|y adapted * 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 fall not far short of the theoretical limits; A. J. Ewart, Phil. Trans, cxcviii, p. 71, 1906. t Cf. {int. al.) T. Fetch, On buttress tree-roots, Ann. R. Bot. Garden, Peradenyia, XI, pp. 277-285, 1930. Also au interesting paper by James Macdonald, on The form of coniferous trees. Forestry, vi, 1 and 2, 1931/2. X Trans. Zool. Soc. iv, p. 27, 1850. 30 ON MAGNITUDE [ch. 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 be the first to feel the effects of stinted nourishment." But the principle of GaHleo carries us further and along more certain lines. The strength of a muscle, like that of a rope or girder, varies with its cross-section; and the resistance of a bone to a crushing stress varies, again hke 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 hnear 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 Hmit of size which the physical forces permit*. The spindleshanks of gnat or daddy-long-legs have their own factor of safety, conditional on the creature's exiguous bulk and weight; for after their own fashion even these small creatures tend towards an inevitable limitation of their natural size. But, as Gahleo also saw, if the animal be wholly immersed in water like the whale, or if it be partly so, as was probably the case with the giant reptiles of the mesozoic age, 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 a whale's very nearly is) with that of the water around^. Under these circumstances there is no longer the same physical barrier to the indefinite growth of the animal. Indeed, in the case of the aquatic animal, there is, as Herbert Spencer pointed out, * Cf. A. Rauber, Galileo iiber Knochenformen, Morphol. Jahrb. vii, p. 327, 1882. t Cf. W. S. Wall, A New Sperm Whale etc., Sydney, 1851, p. 64: "As for the immense size of Cetacea, it evidently proceeds from their buoyancy in the medium in which they live, and their being enabled thus to counteract the force of gravity." II] OF THE SPEED OF A SHIP 31 a distinct advantage, in that the larger it grows the greater is its speed. 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 the linear dimensions*: whence, 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 velocity (F) which the fish attains depends on the work (W) it can do and the resistance (R) it must overcome. Now we^ have seen that the dimensions of W are l^, and of R are l'^ ; and by elementary mechanics WocRV^ or F^oc^. 73 Therefore V^oCr-=l, and F oc Vl. This is what is known as Fronde's Law, of the correspondence of speeds — a simple and most elegant instance of "dimensional theory!." 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 vary as the cubes of their linear dimensions, for it varies on the one hand with the area of the piston, and on the other with the length of the stroke; so is it likewise in the animal, where the corresponding ratio depends on the cross-section of the muscle, and on the distance through which it contracts. But in two similar engines, the available horse-power varies as the square of the linear dimensions, and not as the cube ; and this for the reason that the actual energy developed depends on the heating-surface of the boiler J. So likewise must * We are neglecting "drag" or "head-resistance," which, increasing as the cube of the speed, is a formidable obstacle to an unstreamlined body. But the perfect streamlining of whale or fish or bird lets the surrounding air or water behave like a perfect fluid, gives rise to no "surface of discontinuity," and the creature passes through it without recoil or turbulence. Froude reckoned skin-friction, or surface- resistance, as equal to that of a plane as long as the vessel's water-line, and of area equal to that of the wetted surface of the vessel. t Though, as Lanchester says, the great designer "was not hampered by a knowledge of the theory of dimensions." X The analogy is not a very strict or complete one. We are not taking account, for instance, of the thickness of the boiler-plates. 32 ON MAGNITUDE [ch. there be a similar tendency among animals for the rate of supply o^ kinetic energy to vary with the surface of the lung, that is to say (other things being equal) with the square of the linear dimensions of the animal ; which means that, caeteris paribus, the small animal is stronger (having more power per unit weight) than a large one. 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 the same make, the law is bound to hold that the rate of working tends to vary with the square of the linear dimensions, according to Froude's law of steamship comparison. In the case of a very large ship, built for speed, the difficulty is got over by increasing the size and number of the boilers, till the ratio between boiler-room and engine-room is far beyond what is required in an ordinary small vessel*; 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, * Let L be the length, S the (wetted) surface, T the tonnage, D the displacement (or volume) of a ship; and let it cross the Atlantic at a speed V. Then, in com- paring two ships, similarly constructed but of different magnitudes, we know that L=V\ S=L^ = V\ D = T = L^=V^; also B (resistance) =/Sf. F^^ F«; H (horse- power) = i2 . F = F' ; and the coal (C) necessary for the voy a,ge— HjV = V^. 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- consumption 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. Suppose a steamer 400 ft. long, of 2000 tons, 2000 H.P., and a speed of 14 knots. The corresponding vessel of 800 ft. long should develop a speed of 20 knots (I : 2 :: 14^ : 20^), her tonnage would be 16,000, her H.p. 25,000 or thereby. Such a vessel would probably be driven by four propellers instead of one, each carrying 8000 h.p. See (int. al.) W. J. Millar, On the most economical speed to drive a steamer, Proc. Edin. Math. Soc. vii, pp. 27-29, 1889; Sir James R. Napier, On the most profitable speed for a fully laden cargo steamer for a given voyage, Proc. Phil. Soc, Glasgow, vi, pp. 33-38, 1865. II] OF FROUDE'S LAW 33 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 ahke for both of them. And it should also follow that the curve of fatigue is a steeper one, and the staying power less, in the smaller than ixx the larger individual. This is the c^ase in long-distance racing, where neither draws far ahead until the big winner puts on his big spurt at the end; on which is based an aphorism of the turf, that "a good big 'un is better than a good httle 'un." For an analogous reason wise men know that in the 'Varsity boat-race it is prudent and judicious to bet on the heavier crew. Consider again the dynamical problem of the movements of the body and the hmbs. The work done (W) in moving a Hmb, whose weight is p, over a distance s, is measured hy ps; p varies as the cube of the hnear dimensions, and s, in ordinary locomotion, varies as the linear dimensions, that is to say as the length of Hmb : Wocpsocl^ xl^lK But the work done is limited by the power available, and this varies as the mass of the muscles, or as l^; and under this hmitation neither p nor s increase as they would otherwise tend to do. The limbs grow shorter, relatively, as the animal grows bigger; and spiders, daddy-long-legs and such-hke long-limbed creatures attain no great size. Let us consider more closely the actual energies of the body. A hundred years ago, in Strasburg, a physiologist and a mathema- tician were studying the temperature of warm-blooded animals*. The heat lost must, they said, be proportional to the surface of the animal : and the gain must be equal to the loss, since the temperature of the body keeps constant. It would seem, therefore, that the heat lost by radiation and that gained by oxidation vary both alike, as the surface-area, or the square of the Hnear dimensions, of the animal. But this result is paradoxical; for whereas the heat lost * MM. Rameaux et Sarrus, Bull. Acad. R. de Me'decine, in, pp. 1094-1100, 1838-39. 34 ON MAGNITUDE [ch. may well vary as the surface-area, that produced by oxidation ought rather to vary as the bulk of the animal: one should vary as the square and the other as the cube of the Hnear dimensions. Therefore the ratio of loss to gain, hke that of surface to volume, ought to increase as the size of the creature diminishes. Another physiologist, Carl Bergmann*, took the case a step further. It was he, by the way, whp first said that the real distinction was not between warm-blooded and cold-blooded animals, but between those of constant and those of variable temperature: and who coined the terms homoeothermic and poecilothermic which we use today. He was driven to the conclusion that the smaller animal does produce more heat (per unit of mass) than the large one, in order to keep pace with surface-loss; and that this extra heat-production means more energy spent, more food consumed, more work donef. Sim- plified as it thus was, the problem still perplexed the physiologists for years after. The tissues of one mammal are much like those of another. We can hardly imagine the muscles of a small mammal to produce more heat {caeteris paribus) than those of a large ; and we begin to wonder whether it be not nervous excitation, rather than quahty of muscular tissue, which determines the rate of oxidation and the output of heat. It is evident in certain cases, and may be a general rule, that the smaller animals have the bigger brains; "plus I'animal est petit," says M. Charles Richet, "plus il a des echanges chimiques actifs, et plus son cerveau est volumineuxj." That the smaller animal needs more food is. certain and obvious. The amount of food and oxygen consumed by a small flying insect is enormous; and bees and flies and hawkmoths and humming- * Carl Bergmann, Verhaltnisse der Warmeokonomie der Tiere zu ihrer Grosse, Gottinger Studien, i, pp. 594-708, 1847 — a very original paper. t The metabolic activity of sundry mammals, per 24 hours, has been estimated as follows : Weight (kilo.) Calories per kilo. Guinea-pig 0-7 223 Rabbit 2 58 Man 70 33 Horse 600 22 Elephant 4000 13 Whale 150000 circa 1-7 J Ch. Richet, Recherches de calorimetrie. Arch, de Physiologie (3), vi, pp. 237-291, 450-497, 1885. Cf. also an interesting historical account by M. Elie le Breton, Sur la notion de "masse protoplasmique active": i. Probl^mes poses par la signification de la loi des surfaces, ibid. 1906, p. 606. II] OF BERGMANN'S LAW 35 birds live on nectar, the richest and most concentrated of foods*. Man consumes a fiftieth part of his own weight of food daily, but a mouse will eat half its own weight in a day; its rate of Hving is faster, it breeds faster, and old age comes to it much sooner than to man. A warm-blooded animal much smaller than a mouse becomes an impossibihty; it could neither obtain nor yet digest the food required to maintain its constant temperature, and hence no mammals and no birds are as small as the smallest frogs or fishes. The disadvantage of small size is all the greater when loss of heat is accelerated by conduction as in the Arctic, or by convection as in the sea. The far north is a home of large birds but not of small ; bears but not mice five through an Arctic winter; the 'least of the dolphins Hve in warm waters, and there are no small mammals in the sea. This principle is sometimes spoken of as Bergmann's Law. The whole subject of the conservation of heat and the maintenance of an all but constant temperature in warm-blooded animals interests the physicist and the physiologist ahke. It drew Kelvin's attention many years agof, and led him to shew, in a curious paper, how larger bodies are kept warm by clothing while smaller are only cooled the more. If a current be passed through a thin wire, of which part is covered and part is bare, the thin bare part may glow with heat, while convection-currents streaming round the covered part cool it off and leave it in darkness. The hairy coat of very small animals is apt to look thin and meagre, but it may serve them better than a shaggier covering. 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. All through the physiology of locomotion we meet with it in various ways : as, for instance, when we see a cockchafer carry a plate many times its own weight upon its back, or a flea jump many inches high. ''A dog," says Gahleo, "could probably carry two or three such dogs upon his back; but I believe that a horse could not carry even one of his own size." ♦ Cf. R. A. Davies and G. Fraenkel, The oxygen- consumption of flies during flight, Jl. Exp. Biol. XVII, pp. 402-407, 1940. t W. Thomson, On the efficiency of clothing for maintaining temperature, Nature, xxix, p. 567, 1884. 36 ON MAGNITUDE [ch. Such problems were admirably treated by Galileo and Borelli, but many writers remained ignorant of their work. Linnaeus remarked that if an elephant were as strong in proportion as a stag-beetle, it would be able to pull up rocks and 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*." Such problems as that presented by the flea's jumping powersf , 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 Hmits 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 = xjM. But, according to what we still speak of as "Borelh'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 th^ 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 * Introduction to Entomology, ii, p. 190, 1826. Kirby and Spence, like many less learned authors, are fond of popular illustrations of the "wonders of Nature," to the neglect of dynamical principles. They suggest that if a white ant were as big as a man, its tunnels would be "magnificent cylinders of more than three hundred feet in diameter"; and that if a certain noisy Brazilian insect were as big as a man, its voice would be heard all the world over, "so that Stentor becomes a mute when compared with these insects!" It is an easy consequence of anthropomorphism, and hence a common characteristic of fairy-tales, to neglect the dynamical and dwell on the geometrical aspect of similarity. •f The flea is a very clever jumper; he jumps backwards, is stream-lined ac- cordingly, and alights on his two long hind-legs. Cf. G. I. Watson, in Nature, 21 May 1938. X That is to say, 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 come to compare very different kinds of muscle, such as the insect's and the mammal's. II] OF BORELLI'S LAW 37 contracts. It follows from this that the velpcity is constant, what- ever be the size of the animal. Putting it still more simply, the work done in leaping is propor- tional to the mass and to the height to which it is raised, W oc mH. But the muscular power available for this work is proportional to the mass of muscle, or (in similarly constructed animals) to the mass of the animal, W oc m. It follows that H is, or tends to be, a constant. In other words, all animals, provided always that they are similarly fashioned, with their various levers in Hke proportion, ought to jump not to the same relative but to the same actual height*. The grasshopper seems to be as well planned for jumping as the flea, and the actual heights to which they jump are much of a muchness; but the flea's jump is about 200 times its own height, the grasshopper's at most 20-30 times; and neither flea nor grass- hopper is a better but rather a worse jumper than a horse or a manf . As a matter of fact, Borelli is careful to point out that in the act of leaping the impulse is not actually instantaneous, like the blow of a hammer, but takes some httle time, during which the levers are being extended by which 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, and tends to leave a certain balance of advantage in regard to leaping power on the side of the larger animal J. 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 more and more difficult for nature to endow the larger animal with the length of lever with which she has provided the grasshopper or the flea. To Kirby and Spence it seemed that "This wonderful strength of insects is doubtless the result of something peculiar in the structure and arrangement of their muscles, and principally their extraordinary * Borelli, Prop. CLxxvn. Animalia minora et minus ponderosa majores saltus efficiunt respectu sui corporis, si caetera fuerint paria. t The high jump is nowadays a highly skilled performance. For the jumper contrives that his centre of gravity goes under the bar, while his body, bit by bit, goes over it. J See also {int. al.), John Bernoulli, De Motu Musculorum, Basil., 1694; Chabry, 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, pp. 65-93, 1893; etc. 38 ON MAGNITUDE [ch. 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 Felix Plateau*. From the impulse of the preceding case we may pass to the momentum created (or destroyed) under similar circumstances by a given force acting for a given time : mv = Ft. We know that . m oc P, and t = Ifv, so that lH = Fllv, or v^ = F/l\ But whatsoever force be available, the animal may only exert so much of it as is in proportion to the strength of his own limbs, that is to say to the cross-section of bone, sinew and muscle ; and all of these cross-sections are proportional to P, the square of the linear dimensions. The maximal force, -f'max, which the animal dare exert is proportional, then, to l^; therefore Fraa^JP = coustant. And the maximal speed which the animal can safely reach, namely F'max = ^max/^, IS also constaut, or independent {ceteris paribus) of the dimensions of the animal. A spurt or effort may be well within the capacity of the animal but far beyond the margin of safety, as trainer and athlete well know. This margin is a narrow one, whether for athlete or racehorse; both run a constant risk of overstrain, under which they may "pull" a muscle, lacerate a tendon, or even "break down" a bonet. It is fortunate for their safety that animals do not jump to heights pro- portional to their own. For conceive an animal (of mass m) to jump to a certain altitude, such that it reaches the ground with a velocity v; then if c be the crushing strain at any point of the sectional area (A) of the limbs, the limiting condition is that mv = cA. If the animal vary in magnitude without change in the height to which it jumps (or in the velocity with which it descends), then m P , coc 2 oc ^"2, orZ. The crushing strain varies directly with the linear dimensions of the animal ; and this, a dynamical case, is identical with the usual statical limitalfion of magnitude. * Recherches sur la force absolue des muscles des Inverlebres, Bull. Acad. R, de Belgique (3), vi, vii, 1^83-84: see also ibid. (2), xx, 1865; xxii, 1866; Ann. Mag. N.H. xvii, p. 139, 1866; xix, p. 95, 1867. Cf. M. Radau, Sur la force musculaire des insectes, Revue des deux Mondes, lxiv, p. 770, 1866. The subject had been well treated by Straus-Diirckheim, in his Considerations generates sur Vanatomie comparee des animauz articuUs, 1828. t Cf. The dynamics of sprint -running, by A. V. Hill and others, Proc. R.S. (B), cii, pp. 29-42, 1927; or Muscular Movement in Man, by A. V. Hill, New York, 1927, ch. VI, p. 41. II] OF LOCOMOTION 39 But if the animal, with increasing size or stature, jump to a correspondingly increasing height, the case becomes much more serious. For the final velocity of descent varies as the square root of the altitude reached, and therefore as the square root of tte linear dimensions of the animal. And since, as before, 13 cccmvcc j^ Vy c oc 75 . V ;, or c oc Z». If a creature's jump were in proportion to its height, the crushing strains would so increase that its dimensions would be limited thereby in a much higher degree than was indicated by statical considerations. An animal may grow to a size where lit is unstable dynamically, though still on the safe side statically — a size where it moves with difficulty though it rests secure. It is by reason of dynamical rather than of statical relations that an elephant is of graver deportment than a. mouse. An apparently simple problem, much less simple than it looks, lies in the act of walking, where there will evidently be great economy of work if the leg swing with the help of gravity, that is to say, at a peTidulum-rate. The conical shape and jointing of the limb, the time spent with the foot upon the ground, these and other mechanical differences complicate the case, and make the rate hard to define or calculate. Nevertheless, we may convince ourselves by counting our steps, that the leg does actually tend to swing, as a pendulum does, at a certain definite rate*. So on the same principle, but to the slower beat of a longer pendulum, the scythe swings smoothly in the mower's hands. To walk quicker, we "step out"; we cause the leg-pendulum to describe a greater arc, but it does not swing or vibrate faster until we shorten the pendulum and begin to run. Now let two similar individuals, A and B, walk in a similar fashion, that is to say with a similar an^le of swing (Fig. 1). The arc through which the leg swings, or the amplitude of each step, will then vary as the length of leg (say as a/6), and so as the height or other linear dimension (/) of the manf. But the time of swing varies inversely as'the square * The assertion that the hmb tends to swing in pendulum-time was first made by the brothers Weber {Mechanik der menschl. Gehwerkzeuge, Gottingen, 1836). Some later writers have criticised the statement (e.g. Fischer, Die Kinematik dea Beinschwingens etc., AhJi. math. phys. Kl. k. Sachs. Ges. xxv-xxvin, 1899-1903), but for all that, with proper and large qualifications, it remains substantially true. t So the stride of a Brobdingnagian was 10 yards long, or just twelve times the 2 ft. 6 in., which make the average stride or half-pace of a man. 40 ON MAGNITUDE [CH. root of the pendulum-length, or Va/Vb. Therefore the velocity, which is measured by amphtude/time, or a/b x Vb/Va, will also vary as the square root of the hnear dimen- sions; which is Froude's law over again. The smaller man, or smaller animal, goes slower than the larger, but only in the ratio of the square roots of their linear dimensions ; whereas, if the limbs moved alike, irrespective of the size of the animal — if the limbs of the mouse swung no faster than those of the horse — then the mouse would be as slow in its gait or slower than the tortoise. M. DeHsle* saw a 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)2, or 1/216 of our present height — say 72/216 inches, or one-third of an inch high. Let us note in passing that the number of legs does not matter, any more than the number of wheels to a coach; the centipede runs none the faster for all his hundred legs. But the leg comprises a complicated system of levers, by whose various exercise we obtain very different results. For instance, by being careful to rise upon our instep we increase the length or amplitude of our stride, and improve our speed very materially; and it is curious to see how Nature lengthens this metatarsal joint, or instep-lever, in horse f and hare and greyhound, in ostrich and in kangaroo, and in every speedy animal. Furthermore, 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 * Quoted in Mr John Bishop's interesting article in Todd's Cyclopaedia, iii, p. 443. t The "cannon-bones" are not only relatively longer but may even be actually longer in a little racehorse than a great carthorse. t There is probably another factor involved here: for in bending and thus 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. After all, II] OF RATE OF WALKING 41 of the leg permits us to use it as the shortest possible lever while it is swinging, and as the longest possible lever when it is exerting its propulsive force. The bird's case is of peculiar interest. In running, walking or swimming, we consider the speed which an animal can attain, and the increase of speed which increasing size permits of. But in flight there is a certain necessary speed— a speed (relative to the air) which the bird must attain in order to maintain itself aloft, and which nfiust increase as its size increases. It is highly probable, as Lanchester remarks, that Lilienthal met his untimely death (in August 1896) 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 necessary for its stability. Twenty-five years ago, when this book was written, the bird, or the aeroplane, was thought of as a machine whose sloping wings, held at a given angle and driven horizontally forward, deflect the air downwards and derive support from the upward reaction. In other words, the bird was supposed to communicate to a mass of air a downward momentum equivalent (in unit time) to its own weight, and to do so by direct and continuous impact. The down- ward momentum is then proportional to the mass of air thrust downwards, and to the rate at which it is so thrust or driven: the mass being proportional to the wing-area and to the speed of the bird, and the rate being again proportional to the flying speed; so that the momentum varies as the square of the bird's linear dimen- sions and also as the square of its speed. But in order to balance its weight, this momentum must also be proportional to the cube of the bird's linear dimensions; therefore the bird's necessary speed, such as enables it to maintain level flight, must be pro- portional to the square root of its linear dimensions, and the whole work done must be proportional to the power 3 J of the said linear dimensions. The case stands, so far, as follows : m, the mass of air deflected downwards; M, the momentum so communicated; W, the work done — all in unit time; w, the weight, and F, the velocity of the we know that the pendulum theory is not the whole story, but only an important first approximation to a complex phenomenon. 42 ON MAGNITUDE [ch. bird; I, a linear dimension, the form of the bird being supposed constant. j^j _ ^^ ^ ^3^ but M = mV , and m = W , Therefore M - ^272 _ ^3^ and therefore F = VI and Tf = MF = R The gist of the matter is, or seems to be, that the work which can he done varies with the available weight of muscle, that is to say, with the mass of the bird ; but the work which has to be done varies with mass and distance; so the larger the bird grows, the greater the disadvantage under which all its work is done*. The dispropor- tion does not seem very great at first sight, but it is quite enough to tell. It is as much as to say that, every time we double the linear dimensions of the bird, the difficulty of flight, > or the work which must needs be done in order to fly, is increased in the ratio of 2^ to 2^*, or 1 : V2, or say 1:14. If we take the ostrich to exceed the sparrow in linear dimensions as 25 : 1, which seems well within the mark, the ratio would be that between 25^* and 25^, or between 5' and 5^; in other words, flight would be five times more difficult for the larger than for the smaller bird. But this whole explanation is doubly inadequate. For one thing, it takes no account of gliding flight, in which energy is drawn from the wind, and neither muscular power nor engine power are em- ployed; and we see that the larger birds, vulture, albatross or solan-goose, depend on gliding more and more. Secondly, the old simple account of the impact of the wing upon the air, and the manner in which a downward momentum is communicated and support obtained, is now known to be both inadequate and erroneous. For the science of flight, or aerodynamics, has grown out of the older science of hydrodynamics; both deal with the special properties of a fluid, whether water or air; and in our case, to be content to think of the air as a body of mass m, to which a velocity v is imparted, is to neglect all its fluid properties. How the * This is the result arrived 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. 501-514. It was criticised and challenged (somewhat rashly) by K. Miillenhof, Die Grosse der Flug- flachen etc., PfiUger's Archiv, xxxv, p. 407; xxxvi, p. 548, 1885. II] OF FLIGHT 43 fish or the dolphin swims, and how the bird flies, are up to a certain point analogous problems ; and stream-lining plays an essential part in both. But the bird is much heavier than the air, and the fish has much the same density as the water, so that the problem of keeping afloat or aloft is negligible in the one, and all-important in the other. Furthermore, the one fluid is highly compressible, and the other (to all intents and purposes) incompressible ; and it is this very difference which the bird, or the aeroplane, takes special advantage of, and which helps, or even enables, it to fly. It remains as true as ever that a bird, in order to counteract gravity, must cause air to move downward and obtains an upward reaction thereby. But the air displaced downward beneath the wing accounts for a small and varying part, perhaps a third perhaps a good deal less, of the whole force derived; and the rest is generated above the wing, in a less simple way. For, as the air streams past the slightly sloping wing, as smoothly as the stream-lined form and polished surface permit, it swirls round the front or "leading" edge*, and then streams swiftly over the upper surface of the wing; while it passes comparatively slowly, checked by the opposing slope of the wing, across the lower side. And this is as much as to say that it tends to be compressed below and rarefied above; in other words, that a partial vacuum is formed above the wing and follows it wherever it goes, so long as the stream-lining of the wing and its angle of incidence are suitable, and so long as the bird travels fast enough through the air. The bird's weight is exerting a downward force upon the air, in one way just as in the other; and we can imagine a barometer delicate enough to shew and measure it as the bird flies overhead. But to calculate that force we should have to consider a multitude of component elements; we should have to deal with the stream- lined tubes of flow above and below, and the eddies round the fore- edge of the wing and elsewhere; and the calculation which was too simple before now becomes insuperably difficult. But the principle of necessary speed remains as true as ever. The bigger the bird * The arched form, or "dipping front edge" of the wing, and its use in causing a vacuum above, were first recognised by Mr H. F. Phillips, who put the idea into a patent in 1884. The facts were discovered independently, and soon afterwards, both by Lilienthal and Lanchester. 44 ON MAGNITUDE [ch. becomes, the more swiftly must the air stream over the wing to give rise to the rarefaction or negative pressure which is more and more required; and the harder must it be to fly, so long as work has to be done by the muscles of the bird. The general principle is the same as before, though the quantitative relation does not work out as easily as it did. As a matter of fact, there is probably little difference in the end; and in aeronautics, the "total resultant force" which the bird employs for its support is said, e^npirically, to vary as the square of the air-speed: which is then a result analogous to Froude's law, and is just what we arrived at before in the simpler and less accurate setting of the case. But a comparison between the larger and the smaller bird, like all other comparisons, applies only so long as the other factors in the case remain the same ; and these vary so much in the complicated action of flight that it is hard indeed to compare one bird with another. For not only is the bird continually changing the incidence of its wing, but it alters the lie of every single important feather; and all the ways and means of flight vary so enormously, in big wings and small, and Nature exhibits so many refinements and " improvements" in the mechanism required, that a comparison based on size alone becomes imaginary, and is little worth the making. The above considerations are of great practical importance in aeronautics, for they shew how a provision of increasing speed must ac- company every enlargement of our aeroplanes. Speaking generally, the necessary or minimal speed of an aeroplane varies as the square root of its Unear dimensions; if (ceteris paribus) we make it four times as long, it must, in order to remain aloft, fly twice as fast as before*. If a given machine weighing, say, 500 lb. be stable at 40 miles an hour, then a geometrically similar one which weighs, say, a couple of tons has its speed determined as follows : W:w::L^:l^::S:l. Therefore L:l::2:l. But V^:v^::L:l Therefore V:v::V2:l = 1-414 : 1. * G. H. Bryan, Stability in Aviation, 1911; F. W. Lanchester, Aerodynamics, 1909; cf. (int. al.) George Greenhill, The Dynamics of Mechanical Flight, 1912; F. W. Headley, The Flight of Birds, and recent works. II] OF NECESSARY SPEED 45 That is to say, the larger machine must be capable of a speed of 40 X 1-414, or about 56|, miles per hour. An arrow is a somewhat rudimentary flying-machine; but it is capable, to a certain extent and at a high velocity, of acquiring "stability," and hence of actual flight after the fashion of an aero- plane; the duration and consequent range of its trajectory are vastly superior to those of a bullet of the same initial velocity. Coming back to our birds, and again comparing the ostrich with the sparrow, we find we know little or nothing about the actual speed of the latter ; but the minimal speed of the swift is estimated at 100 ft. per second, or even more — say 70 miles an hour. We shall be on the safe side, and perhaps not far wrong, to take 20 miles an hour as the sparrow's minimal speed; and it would then follow that the ostrich, of 25 times the sparrow's linear dimensions, would have to fly (if it flew at all) with a minimum velocity of 5 x 20, or 100 miles an hour*. The same principle of necessary speed, or the inevitable relation between the dimensions of a flying object and the minimum velocity at which its flight is stable, accounts for a considerable number of * Birds have an ordinary and a forced speed. Meinertzhagen puts the ordinary flight of the swift at 68 m.p.h., which talhes with the old estimate of Athanasius Kircher (Physiologia, ed. 1680, p. 65) of 100 ft. per second for the swallow. Abel Chapman {Retrospect, 1928, ch. xiv) puts the gliding or swooping flight of the swift at over 150 m.p.h., and that of the griffon vulture at 180 m.p.h.; but these skilled fliers doubtless far exceed the necessary minimal speeds which we are speaking of. An airman flying at 70 m.p.h. has seen a golden eagle fly past him easily; but even this speed is exceptional. Several observers agree in giving 50 m.p.h. for grouse and woodcock, and 30 m.p.h. for starling, chaffinch, quail and crow. A migrating flock of lapwing travelled at 41 m.p.h., ten or twelve miles more than the usual speed of the single bird. Lanchester, on theoretical considerations, estimates the speed of the herring gull at 26 m.p.h., and of the albatross at about 34 miles. A tern, a very skilful flier, was seen to fly as slowly as 15 m.p.h. A hornet or a large dragonfly may reach 14 or 18 m.p.h.; but for most insects 2-4 metres per sec, say 4-9 m.p.h., is a common speed (cf, A. Magnan, Vol. des Insectes, 1834, p. 72). The larger diptera are very swift, but their speed is much exaggerated. A deerfly (Cephenomyia) has been said to fly at 400 yards per second, or say 800 m.p.h., an impossible velocity (Irving Langmuir, Science, March 11, 1938). It would mean a pressure on the fly's head of half an atmosphere, probably enough to crush the fly; to maintain it would take half a horsepower; and this would need a food- consumption of 1^ times the fly's weight per second\ 25 m.p.h. is a more reasonable estimate. The naturalist should not forget, though it does not touch our present argument, that the aeroplane is built to the pattern of a beetle rather than of a bird; for the elytra 'are not wings but planes. Cf. int. ah, P. Amans, Geometrie. . .des ailes rigides, C.R. Assoc. Frang. pour Vavancem. des Sc. 1901. 46 ON MAGNITUDE [ch, 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 effectually caged in small enclosures open to the sky. It explains why, as Mr Abel Chapman says, "all ponderous birds, wild swans and geese, great bustard and caper- cailzie, even blackcock, fly faster than they appear to do," while "light-built types with a big wing-areaf, such as herons and harriers, possess no turn of speed at all." For the fact is that the heavy birds must fly quickly, or not at all. 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 slight and scarcely perceptible velocity relatively to the air being sufficient for their support and stabihty. And again, since it is in all these cases velocity relatively to the air which 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. The wing of a bird or insect, like the tail of a fish or the blade of an oar, gives rise at each impulsion to a swirl or vortex, which tends (so to speak) to chng to it and travel along with it;' and the resistance which wing or oar encounter comes much more from these vortices than from the viscosity of the fluid. J We learn as a corollary to this, that vortices form only at the edge of oar or wing — it is only the length and not the breadth of these which matters. A long narrow oar outpaces a broad one, and the efficiency of the long, narrow wing of albatross, swift or hawkmoth is so far accounted for. From the length of the wing we can calculate approximately its rate of swing, and more conjecturally the dimensions of each vortex, and finally the resistance or Ufting power of the stroke; and the result shews once again the advantages of the small-scale * Mouillard, L' empire de Vair; essai d'ornithologie appliqu4e a Vaviationf 1881; transl. in Annual Report of the Smithsonian Institution^ 1892. t On wing-area in relation to weight of bird see Lendenfeld in Naturw. Wochenschr. Nov. 1904, transl. in Smithsonian Inst. Rep. 1904; also E. H. Hankin, Animal Flight, 1913; etc. X Cf. V. Bjerknes, Hydrodynamique physique, n, p. 293, 1934. II] OF MODES OF FLIGHT 47 mechanism, and the disadvantage under which the larger machine or larger creature hes. Length Speed of Radius Force of Specific Weight of wing Beats wing-tip of wing-beat j orce, gm. m. per sec (From V. m./s. Bjerknes) vortex* gra. FjW Stork 3500 0-91 2 5-7 1-5 1480 2:5 Gull 1000 0-60 3 5-7 1-0 640 2:3 Pigeon 350 0-30 6 5-7 0-5 160 1:2 Sparrow 30 Oil 13 4-5 0-18 13 2:5 Bee 0-07 0-01 200 6-3 002 0-2 3i:l Fly 001 0-007 190 4-2 001 004 4:1 * Conjectural. A bird may exert a force at each stroke of its wing equal to one-half, let us say for safety one-quarter, of its own weight, more or less ; but a bee or a fly does twice or thrice the equivalent of its own weight, at a low estimate. If stork, gull or pigeon can thus carry only one-fifth, one-third, one- quarter of their weight by the beating of their wings, it follows that all the rest must be borne by sailing-flight between the wing-beats. But an insect's wings lift it easily and with something to spare; hence saihng-flight, and with it the whole principle of necessary speed, does not concern the lesser insects, nor the smallest birds, at all; for a humming-bird can "stand still" in the air, like a hover-fly, and dart backwards as well as forwards, if it please. There is a little group of Fairy-flies (Mymaridae), far below the size of any small famiUar insects; their eggs are laid and larvae reared within the tiny eggs of larger insects ; their bodies may be no more than J mm. long, and their outspread wings 2 mm. from tip to tip (Fig. 2). It is a pecuharity of some of these that their Httle wings are made of a few hairs or bristles, instead of the continuous membrane of a wing. How these act on the minute quantity of air involved we can only conjecture. It would seem that that small quantity reacts as a viscous fluid to the beat of the wing ; but there are doubtless other unobserved anomahes in the mechanism and the mode of flight of these pigmy creaturesf . The ostrich has apparently reached a magnitude, and the moa certainly did so, at which flight by muscular action, according to t It is obvious that in a still smaller order of magnitude tfie Brownian movement would suffice to make^ flight impossible. 48 ON MAGNITUDE [CH. the normal anatomy of a bird, becomes physiologically impossible. The same reasoning applies to the case of man. It would be very difficult, and probably absolutely impossible, for a bird to flap its way through the air were it of the bigness of a man; but Borelli, in discussing the matter, laid even greater stress on the fact that a man's pectoral muscles are so much less in proportion than those of a bird, that however we might fit ourselves out with wings, we could never expect to flap them by any power of our own weak muscles. Borelli had learned this lesson thoroughly, and in one of his chapters he deals with the proposition : Est impossibile ut homines jyropriis viribus artificiose volare possinV^, But gliding flight, where a ' b 2. Fairy-flies (Mymaridae) : after F. Enock. x 20. wind-force and gravitational energy take the place of muscular power, is another story, and its limitations are of another kind. Nature has many modes and mechanisms of flight, in birds of one kind and another, in bats and beetles, butterflies, dragonflies and what not ; and gliding seems to be the common way of birds, and the flapping flight {remigio alarum) of sparrow and of crow to be the exception rather than the rule. But it were truer to say that gliding and soaring, by which energy is captured from the wind, are modes of flight little needed by the small birds, but more and more essential to the large. BorelH had proved so convincingly that we could never hope to fly propriis viribus, that all through the eighteenth century men tried no more to fly at all. It was in trying to glide that the pioneers of aviation, Cayley, Wenham and Mouillard, * Giovanni Alfonso Borelli, De Motu Animalium, i, Prop, cciv, p. 243, edit. 1685. The part on The Flight of Birds is issued by the Royal Aeronautical Society as No. 6 of its Aeronautical Classics. II] OF GLIDING FLIGHT 49 Langley, Lilienthal and the Wrights — all careful students of birds — renewed the attempt*; and only after the Wrights had learned to ghde did they seek to add power to their glider. Flight, as the Wrights declared, is a matter of practice and of skill, and skill in gliding has now reached a point which more than justifies all Leonardo da Vinci's attempts to fly. Birds shew infinite skill and instinctive knowledge in the use they make of the horizontal accelera- tion of the wind, and the advantage they take of ascending currents in the air. Over the hot sands of the Sahara, where every here and there hot air is going up and cooler coming down, birds keep as best they can to the one, or ghde quickly through the other; so we may watch a big dragonfly planing slowly down a few feet above the heated soil, and only every five minutes or so regaining height with a vigorous stroke of his wings. The albatross uses the upward current on the lee-side of a great ocean- wave ; so, on a lesser scale, does the flying- fish; and the seagull flies in curves, taking every advantage of the varying wind- velocities at different levels over the sea. An Indian vulture flaps his way up for a few laborious yards, then catching an upward current soars in easy spirals to 2000 feet ; here he may stay, effortless, all day long, and come down at sunset. Nor is the modern sail-plane much less efiicient than a soaring bird ; for a skilful pilot in the tropics should be able to roam all day long at willf . A bird's sensitiveness to air-pressure is indicated in other ways besides. Heavy birds, hke duck and partridge, fly low and ap- parently take advantage of air-pressure reflected from the ground. Water-hen and dipper follow the windings of the stream as they fly up or down; a bee-hne would give them a shorter course, but not so smooth a journey. Some small birds — wagtails, woodpeckers and a few others — fly, so to speak, by leaps and bounds ; they fly briskly * Sir George Cayley (1774-1857), father of British aeronautics, was the first to perceive the capabilities of rigid planes, and to experiment on gliding flight. He anticipated all the essential principles of the modern aeroplane, and his first paper "On Aerial Navigation" appeared in Nicholson's Journal for November 1809. F. H. Wenham (1824-1908) studied the flight of birds and estimated the necessary proportion of surface to weight and speed; he held that "the whole secret of success in flight depends upon a proper concave form of the supporting surface." See his paper "On Aerial Locomotion" in the Report of the Aeronautical Society 1866. t Sir Gilbert Walker, in Nature, Oct. 2, 1937. 50 ON MAGNITUDE [ch. for a few moments, then close their wings and shoot along*. The flying-fishes do much the same, save that they keep their wings outspread. The best of them "taxi" along with only their tails in the water, the tail vibrating with great rapidity, and the speed attained lasts the fish on its long glide through the airf. Flying may have begun, as in Man's case it did, with short spells of gUding flight, helped by gravity, and far short of sustained or continuous locomotion. The short wings and long tail of Archae- opteryx would be efficient as a slow-speed ghder; and we may still see a Touraco glide down from his perch looking not much unlike Archaeopteryx in the proportions of his wings and tail. The small bodies, scanty muscles and narrow but vastly elongated wings of a Pterodactyl go far beyond the hmits of mechanical efficiency for ordinary flapping flight; but for ghding they approach perfection J. Sooner or later Nature does everything which is physically possible ; and to glide with skill and safety through the air is a possibihty which she did not overlook. Apart from all differences in the action of the limbs — apart 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 to study, alike in swimming and in walking, and even in the more, complex problem of flight, the general result, obtained under very different conditions and arrived at by different modes of reasoning, shews in every case that speed tends to vary as the square root of the linear dimensions of the animal. While the rate of progress tends to increase slowly with increasing size (according to Froude's law), and the rhythm or pendulum-rate of the limbs to increase rapidly with decreasing size (according to Galileo's law), some such increase of velocity with decreasing * Why large birds cannot do the same is discussed by Lanchester, op. cit. Appendix iv., t Cf. Carl L. Hubbs, On the flight of. . .the Cypselurinae, and remarks on the evolution of the flight of fishes. Papers of the Michigan Acad, of Sci. xvii, pp. 575- 611, 1933. See also E. H. Hankin, P.Z.S. 1920, pp. 467-474; and C. M. Breeder, On the structural specialisation of flying fishes from the standpoint of aero- dynamics, Copeia, 1930, pp. 114-121. X The old conjecture that their flight was helped or rendered possible by a denser atmosphere than ours is thus no longer called for. II] OF THE FORCE OF GRAVITY 51 magnitude is true of all the rhythmic actions of the body, though for reasons not always easy to explain. The elephant's heart beats slower than ours*, the dog's quicker; the rabbit's goes pit-a-pat; the mouse's and the sparrow's are too quick to count. But the very "rate of Hving" (measured by the 0 consumed and COg produced) slows down as size increases; and a rat lives so much faster than a man that the years of its life are three, instead of threescore and ten. From all the foregoing discussion we learn that, as Crookes once upon a time remarked I, the forms as well as the actions of our bodies are entirely conditioned (save for certain exceptions in the case of aquatic animals) by the strength of gravity upon this globe; or, as Sir Charles Bell had put it some sixty years before, the very animals which move upon the surface of the earth are proportioned to its magnitude. 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 suffer Ukewise, though with some com- pensation in the increased density of the air. On the other hand, if gravity were halved, we should get a lighter, slenderer, more active type, needing less energy, less heat, less heart, less lungs, less blood. Gravity not only controls the actions but also influences the forms of all save the least of organisms. The tree under its burden of leaves or fruit has changed its every curve and outline since its boughs were bare, and a mantle of snow will alter its configuration again. Sagging wrinkles, hanging breasts and many another sign of age are part of gravitation's slow relentless handiwork. There are other physical factors besides gravity which help to limit the size to which an animal may grow and to define the con- ditions under which it may Hve. The small insects skating on a pool have their movements controlled and their freedom hmited by the surface-tension between water and air, and the measure of that tension determines the magnitude which they may attain. A man coming wet from his bath carries a few ounces of water, and is perhaps 1 per cent, heavier than before; but a wet fly weighs twice as much as a dry one, and becomes a helpless thing. * A small * Say 28 to 30 beats to the minute. t Proc. Psychical Soc. xn, p. 338-355, 1^97. 52 ON MAGNITUDE [ch. insect finds itself imprisoned in a drop of water, and a fly with two feet in one drop finds it hard to extricate them. The mechanical construction of insect or crustacean is highly efficient up to a certain size, but even crab and lobster never exceed certain moderate dimensions, perfect within these narrow bounds as their construction seems to be. Their body lies within a hollow shell, the stresses within which increase much faster than the mere scale of size ; every hollow structure, every dome or cyhnder, grows weaker as it grows larger, and a tin canister is easy to make but a great boiler is a complicated affair. The boiler has to be strengthened by "stiffening rings" or ridges, and so has the lobster's shell; but there is a limit even to this method of counteracting the weakening effect of size. An ordinary girder-bridge may be made»efficient up to a span of 200 feet or so ; but it is physically incapable of spanning the Firth of Forth. The great Japanese spider-crab, Macrocheira, has a span of some 12 feet across; but Nature meets the difficulty and solves the problem by keeping the body small, and building up the long and slender legs out of short lengths of narrow tubes. A hollow shell is admirable for small animals, but Nature does not and cannot make use of it for the large. In the case of insects, other causes help to keep them of small dimensions. In their peculiar respiratory system blood does not carry oxygen to the tissues, but innumerable fine tubules or tracheae lead air into the interstices of the body. If we imagine them growing even to the size of crab or lobster, a vast complication of tracheal tubules would be necessary, within which friction would increase and diffusion be retarded, and which would soon be an inefficient and inappropriate mechanism. The vibration of vocal chords and auditory drums has this in common with the pendulum-hke motion of a hmb that its rate also tends to vary inversely as the square root of the linear dimen- sions. We know by common experience of fiddle, drum or organ, that pitch rises, or the frequency of vibration increases, as the dimensions of pipe or membrane or string diminish; and in like manner we expect to hear a bass note from the great beasts and a piping treble from the small. The rate of vibration {N) of a stretched string depends on its tension and its density; these beins^ equal, it varies inversely as its own length and as its diameter, i^or similar II] OF EYES AND EARS 53 strings, N oc 1//^, and for a circular membrane, of radius r and thickness e, N oc ll(r^ Ve). But the deHcate drums or tympana of various animals seem to vary much less in thickness than in diameter, and we may be content to write, once more, N oc l/r^. Suppose one animal to be fifty times less than another, vocal chords and all: the one's voice will be pitched 2500 times as many beats, or some ten or eleven octaves, above the other's; and the same comparison, or the same contrast, will apply to the tympanic membranes by which the vibrations are received. But our own perception of musical notes only reaches to 4000 vibrations per second, or thereby; a squeaking mouse or bat is heard by few, and to vibrations of 10,000 per second we are all of us stone-deaf. Structure apart, mere size is enough to give the lesser birds and beasts a music quite different to our own: the humming-bird, for aught we know, may be singing all day long. A minute insect may utter and receive vibrations of prodigious rapidity; even its little wings may beat hundreds of times a second*. Far more things happen to it in a second than to us ; a thousandth part of a second is no longer neghgible, and time itself seems to run a different course to ours. The eye and its retinal elements have ranges of magnitude and Hmitations of magnitude of their own. A big dog's eye is hardly bigger than a little dog's; a squirrel's is much larger, propor- tionately, than an elephant's; and a robin's is but little less than a pigeon's or a crow's. For the rods and cones do not vary with the size of the animal, but have their dimensions optically limited by the interference-patterns of the waves of light, which set bounds to the production of clear retinal images. True, the larger animal may want a larger field of view ; but this makeg little difference, for but a small area of the retina is ever needed or used. The eye, in short, can never be very small and need never be very big; it has its own conditions and limitations apart from the size of the animal. But the insect's eye tells another story. If a fly had an eye like ours, the pupil would be so small that diffraction would render a clear image impossible. The only alternative is to unite a number * The wing-beats are said to be as follows: dragonfly 28 per sec, bee 190, housefly 330; cf. Erhard, Verh. d. d. zool. Gesellsch. 1913, p. 206. 54 ON MAGNITUDE [ch. of small and optically isolated simple eyes into a compound eye, and in the insect Nature adopts this alternative possibiHty*. Our range of vision is limited to a bare octave of "luminous" waves, which is a considerable part of the whole range of Hght-heat rays emitted by the sun; the sun's rays extend into the ultra-violet for another half-octave or more, but the rays to which our eyes are sensitive are just those which pass with the least absorption through a watery medium. Some ancient vertebrate may have learned to see in an ocean which let a certain part of the sun's whole radiation through, which part is our part still ; or perhaps the watery media of the eye itself account sufficiently for the selective filtration. In either case, the dimensions of the retinal elements are so closely related to the wave-lengths of light (or to their interference patterns) that we have good reason to look upon the retina as perfect of its kind, within the hmits which the properties of hght itself impose; and this perfection is further illustrated by the fact that a few light-quanta, perhaps a single one, suffice to produce a sensation "j". The hard eyes of insects are sensitive over a wider range. The bee has two visual optima, one coincident with our own, the other and principal one high up in the ultra-violet J. And with the latt^er the bee is able to see that ultra-violet which is so well reflected by many flowers that flower-photographs have been taken through a filter which passes these but transmits no other rays§. When we talk of hght, and of magnitudes whose order is that of a wave-length of hght, the subtle phenomenon of colour is near at hand. The hues of hving things are due to sundry causes; where they come from chemical pigmentation they are outside our theme, but oftentimes there is no pigment at all, save perhaps as a screen or background, and the tints are those proper to a scale of wave- lengths or range of magnitude. In birds these "optical colours" are of two chief kinds. One kind include certain vivid blues, the * Cf. C. J. van der Horst, The optics of the insect eye, Acta Zoolog. 1933, p. 108. t Cf. Niels Bohr, in Nature, April 1, 1933, p. 457. Also J. Joly, Proc. R.S. (B), xcn, p. 222, 1921. f L. M. Bertholf, Reactions of the honey-bee to light, Journ. of Agric. Res. XLm, p. 379; xliv, p. 763, 1931. § A. Kuhn, Ueber den Farbensinn der Bienen, Ztschr. d. vergl. Physiol, v, pp. 762-800, 1927; cf. F. K. Richtmeyer, Reflection of ultra-violet by flowers, J num. Optical Soc. Amer. vii, pp. 151-168, 1923; etc. II] OF LIGHT AND COLOUR 55 blue of a blue jay, an Indian roller or a macaw; to the other belong the iridescent hues of mother-of-pearl, of the humming-bird, the peacock and the dove: for the dove's grey breast shews many colours yet contains but one — colores inesse plures nee esse plus uno, as Cicero said. The jay's blue feather shews a layer of enamel-like cells beneath a thin horny cuticle, and the cell-walls are spongy with innumerable tiny air-filled pores. These are about 0-3 /it in diameter, in some birds even a little less, and so are not far from the hmits of microscopic vision. A deeper layer carries dark-brown pigment, but there is no blue pigment at all; if the feather be dipped in a fluid of refractive index equal to its own, the blue utterly disappears, to reappear when the feather dries. This blue is like the colour of the sky; it is ''Tyndall's blue," such as is displayed by turbid media, cloudy with dust-motes or tiny bubbles of a size comparable to the wave-lengths of the blue end of the spectrum. The longer waves of red or yellow pass through, the shorter violet rays are reflected or scattered; the intensity of the blue depends on the size and concentration of the particles, while the dark pigment- screen enhances the effect. Rainbow hues are more subtle and more complicated ; but in the peacock and the humming-bird we know for certain* that the colours are those of Newton's rings, and are produced by thin plates or films covering the barbules of the feather. The colours are such as are shewn by films about J [x thick, more or less ; they change towards the blue end of the spectrum as thei hght falls more and more obliquely; or towards the red end if you soak the feather and cause the thin plates to swell. The barbules of the peacock's feather are broad and flat, smooth and shiny, and their cuticular layer sphts into three very thin transparent films, hardly more than 1 jjL thick, all three together. The gorgeous tints of the humming- birds have had their places in Newton's scale defined, and the changes which they exhibit at varying incidence have been predicted * Rayleigh, Phil. Mag. (6), xxxvii, p. 98, 1919. For a review of the whole subject, and a discussion of its many difficulties, see H. Onslow, On a periodic structure in many insect scales, etc., Phil. Trans. (B), ccxi, pp. 1-74, 1921; also C. W. Mason, Journ. Physic. Chemistry, xxvii, xxx, xxxi, 1923-25-27; F. Suffert, Zeitschr. f. Morph. u. Oekol. d. Tiere, i, pp. 171-306, 1924 (scales of butterflies); also B. Reusch and Th. Elsasser in Journ. f. Ornithologie, lxxiti, 1925; etc. 56 ON MAGNITUDE [ch. and explained. The thickness of each film lies on the very limit of microscopic vision, and the least change or irregularity in this minute dimension would throw the whole display of colour out of gear. No phenomenon of organic magnitude is more striking than this constancy of size; none more remarkable than that these fine lamellae should have their tenuity so sharply defined, so uniform in feather after feather, so identical in all the individuals of a species, so constant from one generation to another. A simpler phenomenon, and one which is visible throughout the whole field of morphology, is the tendency (referable doubtless in each case to some definite physical cause) for mere bodily surface to keep pace with volume, through some alteration of its form. The development of villi on the lining of the intestine (which increase 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 valve" of the shark's gut, the lobulation of the kidney in large animals*, the vast increase of respiratory surface in the air-sacs and alveoli of the lung, the development of gills in the larger Crustacea and worms though the general surface of the body suffices for respiration in the smaller species — 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 with increasing size, had it not been for such alteration of surface-form f. A leafy wood, a grassy sward, a piece of sponge, a reef of coral, are all instances of a hke pheno- menon. In fact, a deal of evolution is involved in keeping due balance between surface and mass as growth goes on. In the case of very small animals, and of individual cells, the principle becomes especially important, in consequence of the molecular forces whose resultant action is 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 * Cf. R. Anthony, C.R. clxix, p. 1174, 1919, etc. Cf. also A. Putter, Studien uber physiologische Ahnlichkeit, Pfluger's Archiv, clxviii, pp. 209-246, 1917. t For various calculations of the increase of surface due to histological and anatomical subdivision, see E. Babak, Ueber die Oberflachenentwickelung bei Organismen, Biol. Centralbl. xxx, pp. 225-239, 257-267, 1910. II] OF SURFACE AND VOLUME 57 there are other cases in which the ratio of surface to mass may change the whole condition of the system. Iron rusts when exposed to moist air, but 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 ahke in mathematical form) are two totally different phenomena, the one due to surface-energy, and the other to that form of mass- energy which we ascribe to gravity. The contrast is still more clearly seen in the case of waves: for the little ripple, whose form and manner of propagation are governed by surface-tension, is found to travel with a velocity which is inversely as the square root of its length; while the ordinary big waves, controlled by gravitation, have a velocity directly proportional to the square root of their wave-length. In hke manner we shall find that the form of all very small organisms is 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 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. Now this is a very important matter, and is a notable illustration of that principle of simihtude which we have already discussed in regard to several of its manifestations. We are coming to a con- clusion 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 Fonn, because in the most familiar illustrations of organic form, as in our own bodies for example, these two factors are inseparably asso- ciated, and because we are here justified in thinking of form as the direct resultant and consequence of growth: of growth, whose varying rate in one direction or another has produced, by its gradual and unequal increments, the successive stages of development and 58 ON MAGNITUDE [ch. the final configuration of the whole material structure. But it is by no means true that form and growth are in this direct and simple fashion correlative or complementary in the case of minute portions of living matter. For in the smaller organisms, and in the indi- vidual cells of the larger, we have reached an order of magnitude in which the intermolecular forces strive under favourable conditions with, and at length altogether outweigh, the force of gravity, and also those other forces leading to movements of convection which are the prevailing factors in the larger material aggregate. However, we shall require to deal more fully with this matter in our discussion of the rate of growth, and we may leave it 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, of which surface-energy is not the least. Now the whole surface-energy of the cell is by no means restricted to its outer surface; for the cell is a very heterogeneous structure, and all its protoplasmic alveoli and other visible (as well as invisible) 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 manifested accordingly. But still, the external surface is a definite portion of the system, with a definite "phase" of its own, and however little we may know of the distribution of the total energy of the system, it is at least plain that the conditions which favour equihbrium will be greatly altered by the changed ratio of external surface to mass which a mere change of magnitude produces in the cell. In short, the phenomenon of division of the growing cell, however it be brought about, will be precisely what is wanted to keep fairly constant the ratio between surface and mass, and to retain or restore the balance between surface-energy and the other forces of the system*. But when a germ-cell 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 * Certain cells of the cucumber were found to divide when they had grown to a volume half as large again as that of the "resting cells." Thus the volumes of resting, dividing and daughter cells were as 1:1-5: 0-75; and their surfaces, being as the power 2/3 of these figures, were, roughly, as 1:1-3: 0-8. The ratio of SjV was then as 1 : 0-9 : 1-1, or much nearer equality. Cf. F. T. Lewis, Anat. Record, xlvii, pp. 59-99, 1930. II] OF THE SIZE OF DROPS 59 mass or volume during segmentation it is very slight indeed, generally imperceptible, and wholly denied by some*. The growth or development of the egg from a one-celled stage to stages of two or many cells is thus a somewhat pecuhar kind of growth; it is growth limited to change of form and increase of surface, unaccom- panied 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, while the surface-area is greatly increased! ; the diminution being 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. The diameter of a large parenchymatous cell is perhaps tenfold that of a httle one; but the tallest phanerogams are ten thousand times the height of the least. In short, Nature has her materials of predeterminate dimensions, and keeps to the same bricks whether she build a great house or a small. Even ordinary drops tend towards a certain fixed size, which size is a function of the surface-tension, and may be used (as Quincke used it) as a measure thereof. In a shower of rain the principle is curiously illustrated, as Wilding Roller and V. Bjerknes tell us. The drops are of graded sizes, each twice as big as another, beginning with the minute and uniform droplets of an impalpable mist. They rotate as they fall, and if two rotate in contrary directions they draw together and presently coalesce; but this only happens when two drops are faUing side by side, and since the rate of fall depends on the size it always is a pair of coequal drops which so meet, approach and join together. A supreme instance of constancy or quasi-constancy of size, remote from but yet analogous to the size-hmitation of a rain-drop or a cell, is the fact that the stars of heaven (however else one diifereth from another), and even the nebulae themselves, are all wellnigh co-equal in mass. Gravity draws matter together, condensing it into a world * Though the entire egg is not increasing in mass, that is not to say that its living protoplasm is not increasing all the while at the expense of the reserve material, t Cf. P. G. Tait, Proc. E.S.E. v, 1866 and vi, 1868. 60 ON MAGNITUDE [ch. or into a star; but ethereal pressure is an opponent force leading to disruption, negligible on the small scale but potent on the large. High up in the scale of magnitude, from about 10^^ to 10^^ grams of matter, these two great cosmic forces balance one another; and all the magnitudes of all the stars he within or hard by these narrow limits. In the hving cell, Sachs pointed out (in 1895) that there is a tendency for each nucleus to gather around itself a certain definite amount of protoplasm*. Drieschf, a httle later, found it possible, by artificial subdivision of the egg, to rear dwarf sea-urchin larvae, one-half, one-quarter or even one-eighth of their usual size; which dwarf larvae were composed of only a half, a quarter or an eighth of the normal number of cells. These observations have been often repeated and amply confirmed : and Loeb found the sea-urchin eggs capable of reduction to a certain size, but no further. In the development of Crepidula (an American "shpper-Hmpet," now much at home on our oyster-beds), ConklinJ has succeeded in rearing dwarf and giant individuals, of which the latter may be five-and-twenty times as big as the former. But the individual cells, of skin, gut, liver, muscle and other tissues, are just the same size in one as in the other, in dwarf and in giant §. In like manner * Physiologische Notizen (9), p. 425, 1895. Cf. Amelung, Flora, 1893; Stras- biirger, Ueber die Wirkungssphare der Kerne und die Zellgrosse, Histol. Beitr. (5), pp. 95-129, 1893; R. Hertwig, Ueber Korrelation von Zell- und Kerngrosse (Kernplasraarelation), Biol. Centralbl. xvm, pp. 49-62, 108-119, 1903; G. Levi and T. Terni, Le variazioni dell' indice plasmatico-nucleare durante 1' intercinesi, Arch. Hal. di Anat. x, p. 545, 1911; also E. le Breton and G. Schaeffer, Variations hiuchimiqiies du rapport nucleo-plasmatique, Strasburg, 1923. t Arch.f. Entw. Mech. iv, 1898, pp. 75, 247. X E. G. Conklin, Cell-size and nuclear size, Journ. Exp. Zool. xii, pp. 1-98, 1912; Body-size and cell-size, Journ. of Morphol. xxiii, pp. 159-188, 1912. Cf. M. Popoff, Ueber die Zellgrosse, Arch.f. Zellforschung, iii, 1909. § Thus the fibres of the crystalline lens are of the same size in large and small dogs, Rabl, Z.f. w. Z. lxvii, 1899. Cf. {int. al.) Pearson, On the size of the blood- corpuscles in Rana, Biometrika, vi, p. 403, 1909. Dr Thomas Young caught sight of the phenomenon early in last century: "The solid particles of the blood do not by any means vary in magnitude in the same ratio with the bulk of the animal," Natural Philosophy, ed. 1845, p. 466; and Leeuwenhoek and Stephen Hales were aware of it nearly two hundred years before. Leeuwenhoek indeed had a very good idea of the size of a human blood-corpuscle, and was in the habit of using its diameter — about 1/3000 of an inch — as a standard of comparison. But though the blood-corpuscles shew no relation of magnitude to the size of the animal, they are related without doubt to its activity; for the corpuscles in the OF THE SIZE OF CELLS 61 the leaf-cells are found to be of the same size in an ordinary water- lily, in the great Victoria regia, and in the still hiiger leaf, nearly 3 metres long, of Euryale ferox in Japan*. Driesch has laid par- ticular stress upon this principle of a "fixed cell-size," which has, however, its own limitations and exceptions. Among these excep- tions, or apparent exceptions, are the giant frond-like cell of a Caulerpa or the great undivided plasmodium of a Myxomycete. The flattening of the one and the branching of the other serve (or help) to increase the ratio of surface to content, the nuclei tend to multiply, and streaming currents keep the interior and exterior of the mass in touch with one another. j^ Rabbit Man Dog Fig. 3. Motor ganglion-cells, from the cervical spinal cord. From Minot, after Irving Hardesty. We get a good and even a famiUar illustration of the principle of size-hmitation in comparing the brain-cells or ganghon-cells, whether of the lower or of the higher animals f . In Fig. 3 we shew certain identical nerve-cells from various mammals, from mouse to elephant, all drawn to the same scale of magnification ; and we see that they are all of much the same order of magnitude. The nerve- cell of the elephant is about twice that of the mouse in linear sluggish Amphibia are much the largest known to us, while the smallest are found among the deer and other agile and speedy animals (cf. Gulliver, P.Z.S. 1875, p. 474, etc.). This correlation is explained by the surface condensation or adsorption of oxygen in the blood-corpuscles, a process greatly facilitated and intensified by the increase of surface due to their minuteness. * Okada and Yomosuke, in Sci. Rep. Tohoku Univ. iii, pp. 271-278, 1928. t Cf. P. Enriques, La forma eome funzione della grandezza : Ricerche sui gangli nervosi degli invertebrati, Arch. f. Entw. Mech. xxv, p. 655, 1907-8. 62 ON MAGNITUDE [ch. dimensions, and therefore about eight times greater in volume or in mass. But making due allowance for difference of shape, the linear dimensions of the elephant are to those of the mouse as not less than one to fifty ; and the bulk of the larger animal is something like 125,000 times that of the less. It follows, if the size of the nerve-cells are as eight to one, that, in corresponding parts of the nervous system, there are more than 15,000 times as many individual cells in one animal as in the other. In short we may (with Enriques) lay it down as a general law that among animals, large or small, the gangUon-cells vary in size within narrow limits; and that, amidst all the great variety of structure observed in the nervous system of different classes of animals, it is always found that the smaller species have simpler gangha than the larger, that is to say ganglia containing a smaller number of cellular elements*. The bepiing of such facts as this upon the cell-theory in general is not to be dis- regarded; and the warning is especially clear against exaggerated attempts to correlate physiological processes with the visible mechanism of associated cells, rather than with the system of energies, or the field of force, which is associated with them. For the life of the body is more than the swm 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 microscopic organisms such as the Rotifera (which have the least average size and the narrowest range of size of all the Metazoa), we are still more palpably struck by the small number of cells which go to constitute a usually complex organ, such as kidney, stomach or ovary ; we can sometimes number them in a few * While the difference in cell-volume is vastly less than that between the volumes, and very much less also than that between the surfaces, of the respective animals, yet there is a certain difference ; and this it has been attempted to correlate with the need for each cell in the many-celled ganglion of the larger animal to possess a more complex "exchange-system" of branches, for intercommunication with its more numerous neighbours. Another explanation is based on the fact that, while such cells as continue to divide throughout life tend to uniformity of size in all mammals, those which do not do so, and in particular the ganglion cells, continue to grow, and their size becomes, therefore, a function of the duration of life. Cf. G. Levi, Studii sulla grandezza delle cellule. Arch. Jtal. di Anat. e di Embrioloq. v, p. 291, 1906; cf. also A. Berezowski, Studien liber die Zellgrosse, Arch. f. Zellforsch. v, pp. 375-384, 1910. II] OF THE LEAST OF ORGANISMS 63 units, in place of the many thousands which make up such an organ in larger, if not alwayc higher, animals. We have already spoken of the Fairy-flies, a few score of which would hardly weigh down one of the larger rotifers, and a hundred thousand would weigh less than one honey-bee. Their form is complex and their httle bodies exquisitely beautiful; but I feel sure that their cells are few, and their organs of great histological simplicity. These considerations help, I think, to shew that, however important and advantageous the subdivision of the tissues into cells may be from the construc- tional, or from the dynamic, point of view, the phenomenon has less fundamental importance than was once, and is often still, assigned to it. Just as Sachs shewed there was a hmit to the amount of cytoplasm which could gather round a nucleus, so Boveri has demonstrated that the nucleus itself has its own hmitations of size, and that, in cell-division after fertilisation, each new nucleus has the same size as its parent nucleus*; we may nowadays transfer the statement to the chromosomes. It may be that a bacterium lacks a nucleus for the simple reason that it is too small to hold one, and that the same is true of such small plants as the Cyanophyceae, or blue-green algae. Even a chromatophore with its "pyrenoids" seems to be impossible below a certain sizej. Always then, there are reasons, partly physiological but in large part purely physical, which define or regulate the magnitude of the organism or the cell. And as we have already found definite Hmitations to the 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 becomes impossible. * Boveri, Zellen.studien, V: Ueber die Abhangigkeit der Kerngrosse und Zellenzahl von der Chromosomenzahl der Ausgangszellen. Jena, 1905. Cf. also {int. al.) H. Voss, Kerngrossenverhaltnisse in der Leber etc., Ztschr.f. Zellforschung, VII, pp. 187-200, 1928. t The size of the nucleus may be affected, even determined, by the number of chromosomes it contains. There are giant race* of Oenothera, Primula and Solanum whose cell-nuclei contain twice the normal number of chromosomes, and a dwarf race of a little freshwater crustacean, Cyclops, has half the usual number. The cytoplasm in turn varies with the amount of nuclear matter, the whole cell is unusually large or unusually small; and in these exceptional cases we see a direct relation between the size of the organism and the size of the cell. Cf. {int. al.) R. P. Gregory, Proc. Camb. Phil Soc. xv, pp. 239-246, 1909; F. Keeble, Journ. of Genetics, ii, pp. 163-188, 1912. • 64 ON MAGNITUDE [ch. A bacillus of ordinary size is, say, 1 /x in length. The length (or height) of a man is about a million and three-quarter times as great, i.e. 1-75 metres, or 1-75 x 10^ /x; and the mass of the man is in the neighbourhood of 5 x 10^® (five million, million, milHon) 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 man, it is easy to reply 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, obtrudes itself as a crucial factor in the case. Clerk Maxwell dealt with this matter seventy years ago, in his celebrated article Atom"^. KoUi (or Colley), a Russian chemist, declared in 1893 that the head of a spermatozoon could hold no more than a few protein molecules ; and Errera, ten years later, discussed the same topic with great ingenuity f. But it needs no elaborate calculation to convince us that the smaller bacteria or micrococci nearly approach the smallest magnitudes which we can conceive to have an organised structure. A few small bacteria are the smallest of visible organisms, and a minute species associated with influenza, B. pneumosinter, is said to be the least of them all. Its size is of the order of 0-1 /x, or rather less; and here we are in close touch with the utmost limits of microscopic vision, for the wave-lengths of visible light run only from about 400 to 700 m/x. The largest of the bacteria, B. megatherium, larger than the well-known B. anthracis of splenic fever, has much the same proportion to the least as an elephant to a guinea-pig {. Size of body is no mere accident. Man, respiring as he does, cannot be as small as an insect, nor vice versa; only now and then, as in the Goliath beetle, do the sizes of mouse and beetle meet and overlap. The descending scale of mammals stops short at a weight of about 5 grams, that of beetles at a length of about half a milli- metre, and every group of animals has its upper and its lower limitations of size. So, not far from the lower limit of our vision, does the long series of bacteria come to an end. There remain still smaller particles which the ultra-microscope in part reveals; and * Encyclopaedia Britannica, 9th edition, 1875. f Leo Errera, Siir la limite de la petitesse des organismes, BvlL Soc. Roy. des Sc. me'd. et nat. de Bruxelles, 1903; Recueil d'osnvres {Physiologie gen^rale), p. 325. I Cf. A. E. Boycott, The transition from live to dead, Proc. R. Soc. of Medicine, XXII {Pathology), pp. 55-69, 1928. II] OF MOLECULAR MAGNITUDES 65 here or hereabouts are said to come the so-called viruses or "filter- passers," brought within our ken by the maladies, such as hydro- phobia, or foot-and-mouth disease, or the mosaic diseases of tobacco and potato, to which they give rise. These minute particles, of the order of one-tenth the diameter of our smallest bacteria, have no diffusible contents, no included water — whereby they differ from every living thing. They appear to be inert colloidal (or even crystalloid) aggregates of a nucleo-protein, of perhaps ten times the diameter of an ordinary protein-molecule, and not much larger than the giant molecules of haemoglobin or haemocyanin *. Bejerinck called such a virus a contagium vivum; "infective nucleo-protein" is a newer name. We have stepped down, by a single step, from Hving to non-Hving things, from bacterial dimen- sions to the molecular magnitudes of protein chemistry. And we begin to suspect that the virus-diseases are not due to an "organism, capable of physiological reproduction and multiphcation, but to a mere specific chemical substance, capable of catalysing pre-existing materials and thereby producing more and more molecules hke itself. The spread of the virus in a plant would then be a mere autocatalysis, not involving the transport of matter, but only a progressive change of state in substances already there f." But, after all, a simple tabulation is all we need to shew how nearly the least of organisms approach to molecular magnitudes. The same table will suffice to shew how each main group of animals has its mean and characteristic size, and a range on either side, sometimes greater and sometimes less. Our table of magnitudes is no mere catalogue of isolated facts, but goes deep into the relation between the creature and its world. A certain range, and a- narrow one, contains mouse and elephant, and all whose business it is to walk and run ; this is our own world, * Cf. Svedberg, Journ. Am. Chem. Soc. XLviir, p. 30, 1926. According to the Foot-and-Mouth Disease Research Committee {oth Report, 1937), the foot-and- mouth virus has a diameter, determined by graded filters, of 8-12m/i; while Kenneth Smith and W. D. MacClement {Proc. R.S. (B), cxxv, p. 296, 1938) calculate for certain others a diameter of no more than 4m^, or less than a molecule of haemocyanin. t H. H. Dixon, Croonian lecture on the transport of substances in plants, Proc. R.S. (B), vol. cxxv, pp. 22, 23, 1938. 66 ON MAGNITUDE [CH. with whose dimensions our hves, our hmbs, our senses are in tune. The great whales grow out of this range by throwing the burden of their bulk upon the waiters; the dinosaurs wallowed in the swamp, and the hippopotamus, the sea-elephant and Steller's great sea-cow pass or passed their hves in the rivers or the sea. The things which Linear dimensions of organisms, and other objects cm. (10,000 km.) 10' A quadrant of the earth's circumference (1000 km.) 10« 105 10* Orkney to Land's End Mount Everest (km.) 103 102 Giant trees: Sequoia Large whale Basking shark 101 Elephant; ostrich; man (metre) 10" 10-1 Dog; rat; eagle Small birds and mammals; large insects (cm.) 10-2 Small insects; minute fish (mm.) 10-3 Minute insects 10-* Protozoa; pollen -grains - Cells 10-5 Large bacteria; human blood-corpuscles (micron, /z,) io-« 10-' Minute bacteria Limit of microscopic vision io-« Viruses, or filter-passers r. n j *• i Giant albuminoids, casein, etc.f ^^^^^^ P^^^^^^^^ Starch-molecule (m/ii) io-» Water-molecule (Angstrom unit) 10-10 fly are smaller than the things which walk and run ; the flying birds are never as large as the larger mammals, the lesser birds and mammals are much of a muchness, but insects come down a step in the scale and more. The lessening influence of gravity facilitates flight, but makes it less easy to walk and run; first claws, then hooks and suckers and glandular hairs help to secure a foothold, II] OF SCALES OF MAGNITUDE 67 until to creep upon wall or ceiling becomes as easy as to walk upon the ground. Fishes, by evading gravity, increase their range of magnitude both above and below that of terrestrial animals. Smaller than all these, passing out of our range of vision and going down to the least dimensions of hving things, are protozoa, rotifers, spores, pollen-grains* and bacteria. All save the largest of these float rather than swim; they are buoyed up by air or water, and fall (as Stokes's law explains) with exceeding slowness. There is a certain narrow range of magnitudes where (as we have partly said) gravity and surface tension become comparable forces, nicely balanced with one another. Here a population of small plants and animals not only dwell in the surface waters but are bound to the surface film itself — the whirligig beetles and pond- skaters, the larvae of gnat and mosquito, the duckweeds (Lemna), the tiny Wolffia, and Azolla; even in mid-ocean, one small insect (Halobates) retains this singular habitat. It would be a long story to tell the various ways in which surface-tension is thus taken full advantage of. Gravitation not only Hmits the magnitude but controls the form of things. With the help of gravity the quadruped has its back and its belly, and its limbs upon the ground ; its freedom of motion in a plane perpendicular to gravitational force ; its sense of fore-and-aft, its head and tail, its bilateral symmetry. Gravitation influences both our bodies and our minds. We owe to it our sense of the vertical, our knowledge of up-and-down; our conception of the horizontal plane on which we stand, and our discovery of two axes therein, related to the vertical as to one another; it was gravity which taught us to think of three-dimensional space. Our archi- tecture is controlled by gravity, but gravity has less influence over the architecture of the bee ; a bee might be excused, might even be commended, if it referred space to four dimensions instead of three ! t The plant has its root and its stem", but about this vertical or * Pollen-grains, like protozoa, have a considerable range of magnitude. The largest, such as those of the pumpkin, are about 200/x in diameter; these have to be carried by insects, for they are above the level of Stokes's law, and no longer float upon the air. The smallest pollen-grains, such as those of the forget-me-not, are about 4^ /x in diameter (Wodehouse). f Corresponding, that is to say, to the four axes which, meeting in a pfoint, make co-equal angles (the so-called tetrahedral angles) one with another, as do the basal angles of the honeycomb. (See below, chap, vu.) 68 ON MAGNITUDE [ch. gravitational axis its radiate symmetry remains, undisturbed by directional polarity, save for the sun. Among animals, radiate symmetry is confined to creatures of no great size ; and some form or degree of spherical symmetry becomes the rule in the small world of the protozoon — unless gravity resume its sway through the added burden of a shell. The creatures which swim, walk or run, fly, creep or float are, so to speak, inhabitants and natural proprietors of as many distinct and all but separate worlds. Humming-bird and hawkmoth may, once in a way, be co-tenants of the same world; but for the most part the mammal, the bird, the fish, the insect and the small life of the sea, not only have their zoological distinctions, but each has a physical universe of its own. The world of bacteria is yet another world again, and so is the world of colloids ; but through these small Lilliputs we pass outside the range of hving things. What we call mechanical principles apply to the magnitudes among which we are at home; but lesser worlds are governed by other and appropriate physical laws, of capillarity, adsorption and electric charge. There are other worlds at the far other end of the scale, in the uttermost depths of space, whose vast magnitudes lie within a narrow range. When the globular star-clusters are plotted on a curve, apparent diameter against estimated distance, the curve is a fair approximation to a rectangular hyperbola; which means that, to the same rough approximation, the actual diameter is identical in them all*. It is a remarkable thing, worth pausing to reflect on, that we can pass so easily and in a dozen fines from molecular magnitudes f to the dimensions of a Sequoia or a whale. Addition and subtraction, the old arithmetic of the Egyptians, are not powerful enough for such an operation; but the story of the grains of wheat upon the chessboard shewed the way, and Archimedes and Napier elaborated * See Harlow Shapley and A. B. Sayer, The angular diameters of globular clusters, Proc. Nat. Acad, of Sci. xxi, pp. 593-597, 1935. The same is approxi- mately true of the spiral nebulae also. t We may call (after Siedentopf and Zsigmondi) the smallest visible particles microns, such for instance as small bacteria, or the fine particles of gum-mastich in suspension, measuring 0-5 to 1-0/x; sub-microns are those revealed by the ultra- microscope, such as particles of colloid gold (2-15m/Lt), or starch-moleculea (5m/x); amicrons, under Im^, are not perceptible by either method. A water-molecule measures, probably, about 0-1 m/i. II] OF THIN FILMS 69 the arithmetic of multipHcation. So passing up and down by easy- steps, as Archimedes did when he numbered the sands of the sea, we compare the magnitudes of the great beasts and the small, of che atoms of which they are made, and of the world in which they dwell*. While considerations based on the chemical composition of the organism have taught 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 similitude has already taught us that long before we reach these all but infinitesimal magnitudes the dwindling organism will have experienced great changes in all its physical relations, and must at length arrive at conditions surely incompatible with life, or what we understand as life, in its 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 />t. A soap film, or a film of oil on water, may be attenuated to far less magnitudes than this; the black spots on a soap bubble are known, by various concordant methods of measure- ment, to be only about 6x 10~' cm., or about 6m/x thick, and Lord Rayleigh and M. Devaux have obtained films of oil of 2mjLt, or even 1 m/x in thickness. But while it is possible for a fluid film to exist of these molecular dimensions, it is certain that long before we reach these magnitudes there arise conditions of which we have little knowledge, and which it is not easy to imagine. A bacillus lives in a world, or on the borders of a world, far other than our own, and preconceptions drawn from our experience are not valid there. Even among inorganic, non-living bodies, there comes a certain grade of minuteness at which the ordinary properties become modified. For instance, while under ordinary circumstances crystal- lisation starts in a solution about a minute sohd fragment or crystal * Observe that, following a common custom, we have only used a logarithmic scale for the round numbers representing powers of ten, leaving the interspaces between these to be filled up, if at all, by ordinary numbers. There is nothing to prevent us from using fractional indices, if we please, throughout, and calling a blood-corpuscle, for instance, 10~^"^ cm. in diameter, a man lO^'^^ cm. high, or Sibbald's Rorqual lOi"*^ metres long. This method, implicit in that of Napier of Merchiston, was first set forth by Wallis, in his Arithmetica infinitorum. 70 ON MAGNITUDE [ch. of the salt, Ostwald has shewn that we may have particles so minute that they fail to serve as a nucleus for crystalUsation — which is as much as to say that they are too small to have the form and pro- perties of a "crystal." And again, in his thin oil-films, Lord Rayleigh noted the striking change of physical properties which ensues when the film becomes attenuated to one, or something less than one, close-packed layer of molecules, and when, in short, it no longer has the properties of matter in mass. These attenuated films are now known to be "monomolecular," the long-chain molecules of the fatty acids standing close-packed, like the cells of a honeycomb, and the film being just as thick as the molecules are long. A recent determination makes the several molecules of oleic, palmitic and stearic acids measure 10-4, 14-1 and 15- 1 cm. in length, and in breadth 7-4, 6-0 and 5-5 cm., all by 10~^: in good agreement with Lord Rayleigh and Devaux's lowest estimates (F. J. Hill, Phil. Mag. 1929, pp. 940-946). But it has since been shewn that in aliphatic substances the long-chain molecules are not erect, but inclined to the plane of the film ; that the zig-zag constitution of the molecules permits them to interlock, so giving the film increased stability ; and that the interlock may be by means of a first or second zig-zag, the measured area of the film corresponding precisely to these two dimorphic arrangements. (Cf. C. G. Lyons and E. K. Rideal, Proc. R.S. (A), cxxvin, pp. 468-473, 1930.) The film may be lifted on to a polished surface of metal, or even on a sheet of paper, and one monomolecular layer so added to another; even the complex protein molecule can be unfolded to form a film one amino- acid molecule thick. The whole subject of monomolecular layers, the nature of the film, whether condensed, expanded or gaseous, its astonishing sensitive- ness to the least impurities, apd the manner of spreading of the one liquid over the other, has become of great interest and importance through the work of Irving Langmuir, Devaux, N. K. Adam and others, and throws new light on the whole subject of molecular magnitudes*. The surface-tension of a drop (as Laplace" conceived it) is the cumulative effect, the statistical average, of countless molecular attractions, but we are now entering on dimensions where the molecules are fewf. The free surface-energy of a body begins to vary with the radius, when that radius is of an order comparable to inter-molecular distances; and the whole expression for such energy tends to vanish away when the radius of the drop or particle is less than O-Olfx, or lOm^it. The quahties and properties of our * Cf. (int. al.) Adam, Physics and Chemistry of Surfaces, 1930; Irving Langmuir, Proc. R.S. (A), CLXX, 1939. t See a very interesting paper by Fred Vies, Introduction a la physique bac- terienne, Revue Sclent. 11 juin 1921. Cf. also N. Rashevsky, Zur Theorie d. spontanen Teilung von mikroskopischen Tropfen, Ztschr.f. Physik, xlvi, p. 578, 1928. II] OF MINUTE MAGNITUDES 71 particle suffer an abrupt change here; what then can we attribute, in the way of properties, to a corpuscle or organism as small or smaller than, say, 0-05 or 0-03 /x? It must, in all probability, be a homogeneous structureless body, composed of a very small number of albumenoid or other molecules. Its vital properties and functions must be extremely limited; its specific outward characters, even if we could see it, must be nil; its osmotic pressure and exchanges must be anomalous, and under molecular bombardment they may be rudely disturbed; its properties can be Httle more than those of an ion-laden corpuscle, enabling it to perform this or that specific chemical reaction, to effect this or that disturbing influence, or produce this or that pathogenic effect. Had it sensation, its ex- periences would be strange indeed; for if it could feel, it would regard a fall in temperature as a movement of the molecules around, and if it could see it would be surrounded with light of many shifting colours, like a room filled with rainbows. The dimensions of a cilium are of such an order that its substance is mostly, if not all, under the pecuHar conditions of a surface-layer, and surface-energy is bound to play a leading part in cihary action. A cilium or flagellum is (as it seems to me) a portion of matter in a state sui generis, with properties of its own, just as the film and the jet have theirs. And just as Savart and Plateau have told us about jets and films, so will the physicist some day explain the properties of the cilium and flagellum. It is certain that we shall never understand these remarkable structures so long as we magnify them to another scale, and forget that new and pecuhar physical properties are associated with the scale to which they belong*. As Clerk Maxwell put it, "molecular science sets us face to face with physiological theories. It forbids the physiologist to imagine that structural details of infinitely small dimensions (such as Leibniz assumed, one within another, ad infinitum) can furnish an explana- tion of the infinite variety which exists in the properties and functions of the most minute organisms." And for this reason Maxwell reprobates, with not undue severity, those advocates of pangenesis * The cilia on the gills of bivalve molluscs are of exceptional size, measuring from say 20 to 120^ long. They are thin triangular plates, rather than filaments; they are from 4 to lO/x broad at the base, but less than 1/x thick. Cf. D. Atkins. Q.J. M.S., 1938, and other papers. 72 ON MAGNITUDE [ch. 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 particula undique desecta 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 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 ju,) in diameter, cannot fall in still air quicker than about an inch and a half per second; as its size decreases, its resistance varies as the radius, not (as with larger bodies) as the surface; and its "critical" or terminal velocity varies as the square of the radius, or as the surface of the drop. A minute drop in a misty cloud may be one-tenth that size, and will fall a hundred times slower, say an inch a minute; and one again a tenth of this diameter (say 0-25 />t, or about twice as big as a small micro- coccus) will scarcely fall an inch in two hours*. Not only do dust-particles, spores f and bacteria fall, by reason of this principle, very slowly through the air, but all minute bodies meet with great proportionate resistance to their movements through a fluid. In salt water they have the added influence of a larger coefficient of friction than in fresh J ; and 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 waters of the ocean, and fall with exceeding slowness. * The resistance depends on the radius of the particle, the viscosity, and the rate of fall ( V) ; the eifective weight by which this resistance is to be overcome depends on gravity, on the density of the particle compared with that of the medium, and on the mass, which varies as r^. Resistance =A;rF, and effective weight = )fcV; when these two equal one another we have the critical or terminal velocity, and Vccr^. t A. H. R. BuUer found the spores of a fungus (CoUybia), measuring 5x3/i, to fall at the rate of half a millimetre per second, or rather more than an inch a minute; Studies on Fungi, 1909. X Cf. W. Krause, Biol. Centralbl. i, p. 578, 1881; Fliigel, Meteorol Ztschr. 1881, p. 321. II] OF STOKES'S LAW 73 When we talk of one thing touching another, there may yet be a distance between, not only measurable but even large compared with the magnitudes we have been considering. Two polished plates of glass or steel resting on one another are still about 4/x apart — the average size of the smallest dust; and when all dust- particles are sedulously excluded, the one plate sinks slowly down to within O-S/jl of the other, an apparent separation to be accounted for by minute irregularities of the polished surfaces*. The Brownian movement has also to be reckoned with — that remarkable phenomenon studied more than a century ago by Robert Brown f, Humboldt's /ac?7e princeps botanicorum, and discoverer of the nucleus of the cell J. It is the chief of those fundamental phenomena which the biologists have contributed, or helped to contribute, to the science of physics. The quivering motion, accompanied by rotation and even by translation, manifested by the fine granular particle issuing from a crushed pollen-grain, and which Brown proved to have no vital significance but to be manifested by all minute particles whatsoever, was for many years unexplained. Thirty years and more after Brown wrote, it was said to be "due, either directly to some calorical changes continually taking place in the fluid, or to some obscure chemical action between the solid particles and the fluid which is indirectly promoted by heat§." Soon after these words were * Cf. Hardy and Nottage, Proc. R.S. (A), cxxviii, p. 209, 1928; Baston and Bowden, ibid, cxxxiv, p. 404, 1931. t A Brief Description of Microscopical Observations. . .on the Particles contained in the Pollen of Plants; and on the General Existence of Active Molecules in Organic and Inorganic Bodies, London, 1828. See also Edinb. Netv Philosoph. Journ. v, p. 358, 1828; Edinb. Journ. of Science, i, p. 314, 1829; Ann. Sc. Nat. xiv, pp. 341- 362, 1828; etc. The Brownian movement was hailed by some as supporting Leibniz's theory of Monads, a theory once so deeply rooted and so widely believed that even under Schwann's cell-theory Johannes Miiller and Henle spoke of the cells as "organische Monaden"; cf. Emit du Bois Reymond, Leibnizische Gedanken in der neueren Naturwissenschaft, Monatsber. d. k. Akad. Wiss., Berlin, 1870. J The "nucleus" was first seen in the epidermis of Orchids; but "this areola, or nucleus of the cell as perhaps it might be termed, is not confined to the epidermis," etc. See his paper on Fecundation in Orchideae and Asclepiadae, Trans. Linn. Soc. xvi, 1829-33, also Proc. Linn. Soc. March 30, 1832. § Carpenter, The Microscope, edit. 1862, p. 185. 74 ON MAGNITUDE [ch. written it was ascribed by Christian Wiener * to molecular move- ments within the fluid, and was hailed as visible proof of the atomistic (or molecular) constitution of the same. 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 average out, for the moment, to approximate equality on all sides f. The movement becomes manifest with particles of somewhere about 20 /x, and is better displayed by those of about 10 /x, and especially well by certain colloid suspensions or emulsions whose particles are just below 1/Lt in diameter {. The bombardment causes our particles to behave just hke molecules of unusual size, and this 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 IjjL in diameter rotate on an average through 100° a second, while particles of 13/x turn through only 14° a minute. Lastly, the very curious result appears, that in a layer of fluid the particles are not evenly distributed, nor do they ever fall under the influence of gravity to the bottom. For here gravity and the Brownian move- ment are rival powers, striving for equilibrium; just as gravity is opposed in the atmosphere by the proper motion of the gaseous molecules. And just as equihbrium is attained in the atmosphere when the molecules are so distributed that the density (and therefore the number of molecules per unit volume) falls oif in geometrical * In Poggendorffs Annalen, cxviii, pp. 79-94, 1863. For an account of this remarkable man, see Naturmssensehaften, xv, 1927; cf. also Sigraund Exner, Ueber Brown's Molecularbewegung, Sitzungsher. kk. Akad. Wien, lvi, p. 116, 1867. t Perrin, Les preuves de la realite moleculaire, Ann. de Physique, xvii, p. 549, 1905; XIX, p. 571, 1906. The actual molecular collisions are unimaginably frequent; we see only the residual fluctuations. J Wiener was struck by the fact that the phenomenon becomes conspicuous just when the size of the particles becomes comparable to that of a wave-length of light. § For a full, but still elementary, account, see J. Perrin, Les Atomes; cf. also Th. Svedberg, Die Existenz der Molekiile, 1912; R. A. Millikan, The Electron, 1917, etc. The modern literature of the Brownian movement (by Einstein, Perrin, de Broglie, Smoluchowski and Millikan) is very large, chiefly owing to the value which the phenomenon is shewn to have in determining the size of the atom or the charge on an electron, and of giving, as Ostwald said, experimental proof of the atomic theory. II] OF THE BROWNIAN MOVEMENT 75 progression as we ascend to higher and higher layers, so is it with our particles within the narrow Umits 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 shew other and still more comphcated mani- festations. It is at least clear that, just as the early microscopists in the days before Robert Brown never doubted but that these phenomena were purely vital, so we also may still be apt to confuse, in certain cases, the one phenomenon with the other. We cannot, indeed, without the most careful scrutiny, decide whether the movements of our minutest organisms are intrinsically "vital" (in the sense of being beyond a physical mechanism, or working model) or not. For example, Schaudinn has suggested that the undulating movements of Spirochaete pallida must be due to the presence of a minute, unseen, "undulating membrane"; and Doflein says of the same species that "sie verharrt oft mit eigenthiimlich zitternden Bewegungen zu einem Orte." Both movements, the trembling or quivering movement described by Doflein, and the undulating or rotating movement described by Schaudinn, are just such as may be easily and naturally interpreted as part and parcel of the Brownian phenomenon. While the Brownian movement may thus simulate in a deceptive way the active movements of an organism, the reverse statement also to a certain extent holds good. One sometimes Hes 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 rushf. * Cf. R. Gans, Wie fallen Stabe und Scheiben in einer reibenden Fliissigkeit? Miinchener Bericht, 1911, p. 191; K. Przibram, Ueber die Brown'sche Bewegung nicht kugelformiger Teilchen, Wiener Bericht, 1912, p. 2339; 1913, pp. 1895-1912. t As Clerk Maxwell put it to the British Association at Bradford in 1873, "We cannot do better than observe a swarm of bees, where every individual bee is flying furiously, first in one direction and then in another, while the swarm as a whole is either at rest or sails slowly through the air." 76 ON MAGNITUDE [ch. Their motions are erratic, independent of one another, and devoid of common purpose*. This is nothing else than a vastly magnified picture, or simulacrum, of the Brownian move- ment; 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 busthng crowd had no business whatsoever. In like manner Lucretius, and Epicurus before him, watched the dust-motes quivering in the beam, and saw in them a mimic representation, rei simulacrum et imago, of the eternal motions of the atoms. Again the same phenomenon may be witnessed under the microscope, in a drop of water swarming with Paramoecia or such-Hke 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-hne obeyed with great exactitude the "Einstein formula," that is to say the particular form of the "law of chance" which is apphcable to the case of the Brownian movement f. 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 passive under bombardment ; nevertheless Przibram is inclined to think that even his compara- tively large infusoria are small enough for the molecular bombard- ment to be a stimulus, even though not the actual cause, of their irregular and interrupted movements {. * Nevertheless there may be a certain amount of bias or direction in these seemingly random divagations: cf. J. Brownlee, Proc. R.S.E. xxxi, p. 262, 1910-11; F. H. Edgeworth, Metron, i, p. 75, 1920; Lotka, Elem. of Physical Biology, 1925, p. 344. t That is to say, the mean square of the displacements of a particle, in any direction, is proportional to the interval of time. Cf. K. Przibram, Ueber die ungeordnete Bewegung niederer Tiere, Pfliigefs Archiv, cliii, pp. 401-405, 1913; Arch. f. Entw. Mech. xliii, pp. 20-27, 1917. X All that is actually proven is that "pure chance" has governed the movements of the little organism. Przibram has made the analogous observation that infusoria, when not too crowded together, spread or diffuse through an aperture from one vessel to another at a rate very closely comparable to the ordinary laws of molecular diffusion. II] OF THE EFFECTS OF SCALE 77 George Johnstone Stoney, the remarkable man to whom we owe the name and concept of the electron, went further than this; for he supposed that molecular bombardment might be the source of the life-energy of the bacteria. He conceived the swifter moving molecules to dive deep into the minute body of the organism, and this in turn to be able to make use of these importations of energy*. We draw near the end of this discussion. We found, to begin with, that "scale" had a marked eifect on physical phenomena, and that increase or diminution of magnitude migl^t mean a complete change of statical or dynamical equiUbrium. In the end we begin to see that there are discontinuities in the scale, defining phases in which different forces predominate and different conditions prevail. Life has a range of magnitude narrow indeed compared to that with which physical science deals ; but it is wide enough to include three such discrepant conditions as those in which a man, an insect and a bacillus have their being and play their several roles. Man is ruled by gravitation, and rests on mother earth. A water-beetle finds the surface of a pool a matter of Ufe and death, a perilous entanglement or an indispensable support. In a third world, where the bacillus Hves, gravitation is forgotten, and the viscosity of the Hquid, the resistance defined by Stokes's law, the molecular shocks of the Brownian movement, doubtless also the electric charges of the ionised medium, make up the physical environment and have their potent and immediate influence on the organism. The predominant factors are no longer those of our scale ; we have come to the edge of a world of which we have no experience, and where all our preconceptions must be recast. * Phil. Mag. April 1890. CHAPTER III, THE RATE OF GROWTH When we study magnitude by itself, apart from the gradual changes to which it may be subject, we are deahng with 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 point 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, our diagram plots magnitude in one direction against magnitude in another — length against height, for instance, or against breadth ; and the result is what we call a picture or outhne, 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 growth; and it is evident that this term growth has wide meanings. For growth may be positive or negative, a thing may grow larger or smaller, greater or less; and by extension of the concrete signification of the word we easily arid 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 known as a "curve of growth." We see that the pheno- menon which we are studying is a velocity (whose "dimensions" are space/time, or L/T), and this phenomenon we shall speak of, simply, as a rate of growth. In various conventional wa-ys we convert a two-dimensional into * In Aristotelian logic. Form is a quality. None the less, it is related to qiuirUity't and we jfind the Schoolmen speaking of it as qualitas circa quantitatem. CH. Ill] OF CHANGE OF MAGNITUDE 79 a three-dimensional diagram. We do so, for example, when, by means of the geometrical method of "perspective," we represent upon a sheet of paper the length, breadth and depth of an object in three-dimensional space, but we do it better by means of contour- Hnes or "isopleths." By contour-lines superposed upon a map of a country, we shew its hills and valleys; and by contour-Unes we may shew temperature, rainfall, population, language, or any other *' third dimension" related to the two dimensions of the map. Time is always impHcit, in so far as each map refers to its own date or epoch; but Time as a dimension can only be substituted for one of the three dimensions already there. Thus we may superpose upon our map the successive outlines of the coast from remote antiquity, or of any single isotherm or isobar from day to day. And if in hke manner we superpose on one another, or even set side by side, the outhnes of a growing organism — for instance of a young leaf and an old, we have a three-dimensional diagram which is a partial representation (limited to two dimensions of space) of the organism's gradual change of form, or course of development; in such a case our contours may, for the purposes of the embryologist, be separated by time-intervals of a few hours or days, or, for 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 we see how intimately the two concepts are correlated or interrelated to one another. In short it is obvious that the form of an organism is determined by its rate of growth in various directions ; hence rate of growth deserves to be studied as a necessary preliminary to the theoretical study of form, and organic form itself is found, mathematically speaking, to be di function of time'\. * Sometimes we find one and the same diagram suffice, whether the time-intervals be great or small; and we then invoke " Wolff's law" (or Kielmeyer's), and assert that the life-history of the individual repeats, or recapitulates, the history of the race. This "recapitulation theory" was alt-important in nineteenth -century embryology, but was criticised by Adam Sedgwick {Q.J. M.S. xxxvi, p. 38, 1894) and many later authors; cf. J. Needham, Chemical Embryology, 1931, pp. 1629-1647. t Our subject is one of Bacon's "Instances of the Course" or studies wherein we "measure Nature by periods of Time." In Bacpji's Catalogue of Particular Histories, one of the odd hundred histories or investigations which, he foreshadows 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." 80 THE RATE OF GROWTH [ch. At the same time, we need only consider this large 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; also, the statistical or numerical aspect of the question is peculiarly adapted to the mathematical study of variation and correlation. These important subjects we must not neglect; but our main purpose will be 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 for further explanation and for some measure of quahfication. Among organic forms we shall have many an occasion to see that form may be due in simple cases to the direct action of certain molecular forces, among which surface-tension plays a leading part. Now when surface-tension causes (for instance) a minute semifluid organism to assume a spherical form, or gives to a film of protoplasm the form of a catenary or of an elastic curve, or when it acts in various other ways productive of definite contours — just as it does in the making of a drop, a splash or a jet — this is a process of con- formation very diiferent from that 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 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; it is in regard to them that we are in a position to study the rate of action in different directions and at different times, and to realise that it is on such differences of rate that form and its modifications essentially and ultimately depend. Ill] OF RATE OF ACTION 81 The difference between the two classes of phenomena is akin to the difference between the forces which determine the form of a raindrop and those which, by the flowing of the waters and the sculpturing of the solid earth, have brought about the configuration of a river or a hill ; molecular forces are paramount in the one, and wolar forces are dominant in the other. At the same time, it is true that all changes of form, inasmuch as they necessarily involve changes of actual and relative magnitude, may in a sense be 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 may be well worth studying even in such cases as these; for example, a study of the rate of cell-division in a segmenting egg may teach us something about the w^ork done, and the various energies concerned. But in such cases the action is, as a rule, so homogeneous, and the form finally attained is so definite and so little dependent on the time taken to effect it, that the specific rate of change, or rate of growth, does not enter into the morphological problem. We are deahng with Form in a very concrete way. To Aristotle it was a metaphysical concept; to us it is a quasi-mechanical effect on Matter of the operation of chemico-physical forces f. To * Cf. Aristotle, Phys. VI, 5, 235a, 11, eirel yap awaaa Kiurjffis ev XP^'^V* kt\.; he had already told us that natural science deals with magnitude, with motion and with time: ^iOOC (N (M ' I lo »n lo t:- (N * CO lO -^ GS| l> lO »C op t^ (N t^^ I I 4* -^ -^ »0 >0 »0 « ;D t> 00 CO I> t^ CO I I 05<©QpC>0S00(NI>O'^l>0S0ii^ I I lO -^ «0 !;• Op (N OS F^K pH CO CO I I I CO CO l> CO t^ •"^ A -^ li -^ t^ l> -^ fH I I I I I I lO T*< t- I 1 I obcoi^- io ■«* r- »o CO »c CO CO CO CO o lo i> CO I I I CO CO ■^ O lO lO CO -^ t^ 00 if^ I 1 I Cp00-^0pO5O>Q0lCI>»O-^»CCOOSCSI I 1 CO A ob o F^ CO CO di -^ 05 rh f^ 00 c 00 00 OS O O r-H i-l(M CO CO CO -^ »0 lO § ^ 3 f^ •if '^ (Ni>oscot>qs Qd50sdscocoo4c^cib(NiO* .2P^ d5OSOSCOC0OS' 00 OS OS © rt f-^ (N (N CO CO T*< rj^ lO 00 U5 fH t^ ' CO CO I tJt, ©^00OS©^(MCO'^lOcpt'00OS© ^ I— ll-Hl-Hl— ll-Hl— l|t(r-ll— 1l— (M Ill] OF CURVES OF GROWTH 95 length or stature, and in short that infant, boy and man are not similar figures*. The change of form seems sHght and gradual, but behind it he other and more complex things. Th6 changing ratio between height and weight imphes changes in the child's metabolism^ in the income and expenditure of the body. The infant stores up fat, and the active child "runs it off" again; at four years old or five, bodily metaboUsm and increase of weight are at a minimum; but a fresh start is made, a new "nutritional period" sets in, and the small schoolboy grows stout and sJ3rong|. Our curve of growth shews at successive epochs of time the height or weight which has been reached by then; it plots changing magnitude (y) against advancing time (x). It is essentially a cumulative or summation curve; it sums up or "integrates" all the successive magnitudes which have been added in all the foregoing intervals of time. Where the curve is steep it means that growth was rapid, and when growth ceases the curve becomes a horizontal line. It follows that, by measuring the slope or steepness of our curve of growth at successive epochs, we shall obtain a picture of the successive velocities or growth-rates. The steepness of a curve is measured by its "gradient J," or we may roughly estimate it by taking for equal intervals of time (strictly speaking, for each infinitesimal interval of time) the incre- ment added during that interval; and this amounts in practice to taking the differences between the values given for 'the successive epochs, or ages, which we have begun by studying. Plotting these successive differences against time, we obtain a curve each point on which represents a certain rate at a certain time; and while the former curve shewed a continuous succession of varying magnitudes, this shews a succession of varying velocities. The mathematician calls it a curve of first differences ; we may call it a curve of annual (or other) increments ; but we shall not go wrong if we call it a curve of the rate (or rates) of growth, or still more simply, a velocity-curve. * According to Quetelet's data, man's stature is multiplied by 3-4 and his weight by 20-3, between birth and the age of twenty-one. But the cube of 3-4 is nearly 40; so the weight at birth should be multiplied forty times by the age of twenty-one, if infant, boy and man were similar figures. t Cf. T. W. Adams and E. P. Poulton, Heat production in man, Ouy's Hospital Reports (4), xvn, 1937, and works quoted therein. I That is, by its trigonometrical tangent, referred to the base-line. 96 THE RATE OF GROWTH [CH. We have now obtained two different but closely related curves, based on the selfsame facts or observations, and illustrating them in different ways. One is the inverse of the other ; one is the integral and one the differential of the other; and each makes clear to the eye phenomena which are imphcit, but are less conspicuous, in the other. We are using mathematical terms to describe or designate them; but these "curves of growth" are more comphcated than the curves with which mathematicians are wont to deal. In our study of growth we may well hope to find curves simpler than these; 20» 3 5 7 9 II 13 15 17 19 21 Age in years Fig. 5. Annual increments of growth in man. From Quetelet's Belgian data. but in the successive annual increments of a boy's growth (as Fig. 5 exhibits them) we are deahng with no one continuous operation (such as a mathematical formula might define), but with a succession of events, changing as times and circumstances change. Our curve of increments, or of first differences, for man's stature (Fig. 5) is based, perforce, on annual measurements, and growth alters quickly enough at certain ages to make annual intervals unduly long; nevertheless our curve shews several important things. It suffices to shew, for length or stature, that the growth-rate in early infancy is such as is never afterwards re-attained. From this high early velocity the rate on the whole falls away, until growth itself Ill] OF HUMAN STATURE 97 comes to an end * ; but it does so subject to certain important changes and interruptions, which are much the same whether we draw them from Quetelet's Belgian data, or from the British, American and other statistics of later writers. The curve falls fast and steadily during the first couple of years of the child's hfe (a). It runs nearly level during early boyhood, from four or five years old to nine or ten (6). Then, after a brief but unmistakable period of depression! during which growth slows down still more (c), the boy enters on Annual increments of stature and of weight in man {After Quetelet; see Table, p. 90) Stature A (cm.) Weight Male (kgm.) Age Male Female Fema 0- 1 20 21 6-8 6-4 1- 2 10 9 2-0 1-9 2- 3 6 7 1-2 11 3- 4 7 6 1-9 1-7 4- 5 6 6 1-6 1-3 5- 6 6 6 1-3 1-2 6- 7 6 7 2-1 1-7 7- 8 6 4 21 1-4 8- 9 6 6 1-8 2-6 9-10 5 5 2-0 1-8 10-11 5 3 1-7 2-0 11-12 3 5 3-2 4-3 12-13 4 6 4-3 4-1 13-14 9 6 5-2 3-5 14-15 7 2 5-9 3-2 15-16 5 3 7-0 2-1 16-17 6 4 40 4-6 17-18 3 2 3-9 40 18-19 1 1 21 1-4 19-20 0 0 1-7 — his teens and begins to "grow out of his clothes"; it is his "growing age", and comes to its height when he is about thirteen or fourteen years old (d). The lad goes on growing in stature for some years more, but the rate begins to fall off (e), and soon does so with great rapidity. The corresponding curve of increments in weight is not very different from that for stature, but such differences as there are * As Haller observed it to do in the chick f "Hoc iterum incrementum miro ordine distribuitur, ut in principio incubationis maximum est; inde perpetuo minuatur" {Elementa Pkysiologiae, viii, p. 294). Or as Bichat says, "II y a surabondance de vie dans I'enfant" {Sur la Vie et la Mort, p. 1). t This depression, or slowing down before puberty, seems to be a universal phenomenon, common to all races of men. It is a curious thing that Quetelet's "adjusted figures" (which I used in my first edition) all but smooth out of recognition this characteristic feature of his own observations. 98 THE RATE OF GROWTH [ch. between them are significant enough. There is some tendency for growth in weight to fall off or fluctuate at four or five years old, before the small boy goes to school ; but there is, or should be, httle retardation of weight when growth in height slows down before he enters on his teens*. The healthy lad puts on weight again more and more rapidly, for some httle while after gjowth in stature has slowed down; and normal increase of weight goes on, more slowly, while the man is "fiUing out," long after growth in stature has come to an end. But somewhere about thirty he begins losing weight a httle ; and such subsequent slow changes as men commonly undergo we need not stop to deal with. The differences in stature and build between one race and another are in hke manner a question of growth-rate in the main. Let us take a single instance, and compare the annual increments of growth in Chinese and Enghsh boys. The curves are much the same in form, but differ in amplitude and phase. The Enghsh boy is growing faster all the while; but the minimal rate and the maximal rate come later by a year or more than in the Chinese curve t (Fig. 6). Quetelet was not the first to study man's growth and stature, nor was he the first student of social statistics and "demography." The foundations of modern vital statistics had been laid by Graunt and Petty in the seventeenth century J; the economists developed the subject during the eighteenth §, and parts of it were studied ♦ That the annual increments of weight in boys are nearly constant, and the curve of growth nearly a straight line at this age, especially from about 8 to 11, has been repeatedly noticed. Cf. Elderton, Glasgow School-children, Biometrika, X, p. 283, 1914-15; Fessard and others, Croissance des Ecoliers Pariaiens, 1934, p. 13. But careful measurements of American children, by Katherine Simmons and T. Wingate Todd, shew steadily increasing increments from four years old tiU puberty {Growth, n, pp. 93-133, 1938). t For copious bibliography, see J. Needham, op. cit., also Gaston Backmah, Das Wachstum der Korperlange des Menschen, K. Sv.Vetensk. Akad. Hdlgr. (3), xiv, 1934. X Cf. John Graunt's Natural and Political Observations. . .upon the Bills of Mortality, London, 1662; The Economic Writings of Sir William Petty, ed. by C. H. Hull, 2 vols,, Cambridge, Mass., 1927. Concerning Graimt and Petty — two of the original Fellows of the Royal Society — see {int. al.) H. Westergaard, History of Statistics, 1932, and L. Hogben (and others). Political Arithmetic, 1938. § Besides the many works of the economists, cf. J. G. Roederer, Sermo de pondere et longitudine recens-natorum. Comment. Soe. Beg. Sci. Oottingae, m, 1753; J. F. G. Dietz, De temporum in graviditate et partu aestimatione, Diss., Gottingen, 1757. Ill] OF HUMAN STATURE 99 7 1 1 1 1 1 1 1 1 1 6 *x N^ - 5 \ y^ \ \ \ \ a 4 ^^ '> / \ \ a v r 6 ^ Chinese V 2 \ 1 0 1 f, 1 1 1 1 1 1 1 7 k 9/,o 12/, 3 •6/l7 I7/|8 Age ig. 6. Annual increments of stature. From Roberts (English) and Appleton's (Chinese) data. •Date 1760*61 ^62'63'64'65 W67^68*6970'7r72*73'74V5*76 1777 |6'0 0 1 2 3 4 5 6 7 8 9 10 n 12 13 14 15 16 17 18 Age in years Fig. 7. Curve of growth of a French boy of the eighteenth century. From Scammon, after BufFon. 100 THE RATE OF GROWTH CH. eagerly in the early nineteenth, when the exhaustion of the armies of France and the evils of factory labour in England drew attention to the stature and physique of man and to the difference between the healthy and the stunted child*. A friend of BufPon's, the Count Philibert Gueneau de Montbeillard, kept careful measurements of his own son; and Buifon pubhshed these in 1777, in a supplementary volume of the Histoire Naturelle'f. The child was born in April 1759; it was measured every six months 1 - \ 1 1 I 1 1 1 1 ' ' . 1 . . 20 A — _ \ - - d - - \" - - A - 10 1 1 1 1 b . 1 1 A 1 1 1 1 1 1 10 20 Fig. 8. Annual increments of stature of the said French boy. for seventeen years, and the record gives a curve of great interest and beauty (Fig. 7). There are two ways of studying such a phenomenon — the statistical method based on large numbers, and the careful study of the individual case; the curve of growth of this one French child is to all intents and purposes identical, save that the boy was throughout a trifle taller, with the mean curve yielded by a recent study of forty-four thousand Uttle Parisians {. In young Montbeillard's case the "curve of first differences," or of the successive annual increments of stature (Fig. 8), is clear and beautiful. It shews (a) the rapid, but rapidly diminishing, rate of * Cf. M. Hargenvilliers, Recherches . . . sur . . .le recrutement de Varmie en France, 1817; J. W. Cowell, Measurements of children in Manchester and Stockport, Factory Reports, i; and works referred to by Quetelet. t See Richard E. Scammon, The first seriatim study of human growth, Amer. Journ. of Physical Anthropology, x, pp. 329-336, 1927. J MM. Variot et Chaumet,- Tables de croissance, dressees. . .d'apr^s les mensura- tions de 44,000 enfants parisiens, Bull, et M4m. Soc. d'Anthropologie, iii, p. 55, 1906. Ill] OF GROWTH IN MAN AND WOMAN 101 growth in infancy ; (6) the steady growth in early boyhood ; (c) the period of retardation which precedes, and (d) the rapid growth which accompanies, puberty. Buifon, with his usual wisdom, adds some remarks of his own, which include two notable discoveries. He had observed that a man's stature is measurably diminished by fatigue, and the loss soon made up for in repose; long afterwards Quetelet said, to the same effect, "le lit est favorable a la croissance, et le matin un homme est un peu plus grand que le soir." Buff on asked whether growth varied with the seasons, and Montbeillard's data gave him his reply. Growth was quicker from April to October than during the rest of the year: shewing that "la chaleur, qui agit generalement sur le developpement de tous les etres organisees, influe considerablement sur I'accroissement du corps humain." Between five years old and ten, the child grew seven inches during the five summers, but during the five winters only four; there was a Hke difference again, though not so great, while the boy was growing quickly in his teens; but there were no seasonal differences at all from birth to five years old, when the child was doubtless sheltered from both heat and cold*. On rate of growth in man and ivoman That growth follows a different course in boyhood and in girlhood is a matter of common knowledge; but differences in the curves "of growth are not very apparent on the scale of our diagrams. They are better seen in the annual increments, or first differences; and we may further simplify the comparison by representing the girl's weight or stature as a percentage of the boy's. Taking weight to begin with (Fig. 9), the girl's growth-rate is steady in childhood, from two or three to six or seven years old, * Growth-rates based on the continuqus study of a single individual are rare; we depend mostly on average measurements of many individuals grouped according to their average age. That this is a sound method we take for granted, but we may lose by it as well as gain. (See above, p. 92.) The chief epochs of growth, the chief singularities of the curve, will come out much the same in the individual and in the average curve. But if the individual curves be skew, averaging them will tend to smooth the skewnesa away; and, more curiously, if they be all more or less diverse, though all symmetrical, a certain skewness will tend to develop in the composite or average curve. Cf. Margaret Merrill, The relationship of individual to average growth, Human Biology, iii. pp. 37-70, 1931. 102 THE RATE OF GROWTH [CH. J \ I L J \ L 0 2 4 6 8 10 12 14 16 18 19 Age Fig. 9. Annual increase in weight of Belgian boys and girls. From Quetelet's data. (Smoothed curves.) 10 15 Age Fig. 10. Percentage ratio of female weight and stature to male. From Quetelet's Belgian data. 23 Ill] OF GROWTH IN MAN AND WOMAN 103 just as is the boy's; but her curve stands on a lower level, for the little maid is putting on less weight than the boy (b). Later on, her rate accelerates (c) sooner than does his, but it neyer rises quite so high {d). After a first maximum at eleven or twelve her rate of growth slows down a Httle, then rises to a second maximum when 112 110 108 106 104 102 100 98 96 94 92 90 Matle weight / \y / 1. \ L J I L 1 3 5 7 9 11 13 Fig. i: Relative weight of American boys and girls. From Simmons and Todd's data. she is sixteen or seventeen, after the boy's phase of quickest growth is over and done. This second spurt of growth, this increase of vigour and of weight in the girl of seventeen or eighteen, Quetelet's figures indicate and common observation confirms. Last of all, while men stop adding to their weight about the age of thirty or before, this does not happen to women. They increase in weight, though slowly, till much later on: until there comes a final phase, in both sexes ahke, when weight and height and strength decline together. 104 THE RATE OF GROWTH [CH. 8 1 te OOOOOOOOOOO^^'-^OO I I I I II I oS :3 .(MiCOOOtO-HQO(NC>OlO pq i« »o lb lb -^ lb »« ITS »b »« lb lb lb -^ -^ t^ibeC'^ (Li £ t;-t;-00t;-QOt;-qiNQp'N©-^q5t;-iOCO eO©00O5'*00«OOS©;Ot^'N».OC<500 o O) M 00 iM © — l-- © CO t^ © -H t^ so 't IC LT pq < ?ot*ooo5©^iMeO'^»c«ot^oo Ill OF GROWTH IN CHILDHOOD 105 Stature and weight of American children (Ohio) (From Katherine Simmons and T. Wingate Todd's data) Stature (cm.) % ratio Weight (lbs.) Boys Girls Age Boys Girls % rati 3 months 61-3 59-3 96-7 14-4 13-1 91-1 1 year 761 . 74-2 97-5 23-9 21-9 91-8 2 „ 87-4 86-2 98-6 29-1 27-5 94-7 3 „ 96-2 95-5 99-3 33-5 32-5 96-9 4 „ 103-9 103-2 99-3 38-4 37-1 96-7 o „ 110-9 110-3 99-4 43-2 42-3 98-1 6 „ 117-2 117-4 1001 48-5 48-6 100-0 7 „ 123-9 123-2 99-4 54-7 54-0 98-8 8 „ 130-1 129-3 99-4 62-2 61-5 98-7 9 „ 136-0 135-7 99-7 69-5 70-9 102-0 10 „ 141-4 140-8 99-6 78-5 77-6 98-8 11 » 146-5 147-8 100-7 86-5 87-0 100-6 12 „ 151-1 155-3 102-8 92-7 102-7 110-7 13 „ 156-7 159-9 102-0 102-8 114-6 111-4 Mean of observed incremeyits of stature and weight of American children Increment of stature (mm.) Increment of weight (lbs.) '' % r /o Age Boys Girls ratio Boys Girls ratio 3 m . - 1 yr. 150-4 150-1 99-8 9-32 8-07 93-8 1 yr. - 2 ", 123-9 132-0 106-5 4-97 5-56 1120 2 , - 3 , 88-1 90-0 102-2 4-01 4-18 104-3 3 , - 4 , 73-9 79-1 106-9 4-13 4-46 108-0 4 , - 5 , 69-4 72-2 104-0 4-60 4-55 99-8 5 , - 6 , 67-0 68-0 101-5 4-51 5-08 112-8 6 , - 7 , 64-1 62-6 97-6 5-57 5-40 96-9 7 , - 8 , 61-2 57-8 94-4 6-70 6-65 99-4 8 , —9 , 55-7 60-1 108-0 6-64 7-38 1111 9 , -10 , 54-9 57-7 105-1 7-92 8-12 104-8 10 , -11 , 51-9 61-3 118-2 8-81 9-58 108-7 11 , -12 , 53-2 66-9 125-6 9-54 11-98 1331 12 , -13 , 61-0 55-1 89-0 10-90 10-29 94-7 These differences between the two sexes, which are essentially phase-differences, cause the ratio between their weights to fluctuate in a somewhat comphcated way (Figs. 10, 11). At birth the baby girl's weight is about nine-tenths of the boy's. She gains on him for a year or two, then falls behind again ; from seven or eight onwards she gains rapidly, and the girl of twelve or thirteen is very httle lighter than the boy; indeed in certain American statistics she is by a good deal the heavier of the two. In their teens the boy gains 106 THE RATE OF GROWTH [CH. steadily, and the lad of sixteen is some 15 per cent, heavier than the lass. The disparity tends to diminish for a while, when the maid of seventeen has her second spurt of growth ; but it increases again, though slowly, until at five-and-twenty the young woman is no more than four-fifths the weight of the man. During middle life she gains on him, and at sixty the difference stands at some 12 1 1 \ r J i \ \ \ \ L 0 2 4 6 8 10 12 14 16 18 Age Fig. 12. Annual increments of stature, in boys and girls. From Quetelet's data. (Smoothed curves.) per cent., not far from the mean for all ages; but the old woman shrinks and dwindles, and the difference tends to increase again. The rate of increase of stature, like stature itself, differs notably in the two sexes, and the differences, as in the case of weight, are mostly a question of phase (Fig. 12). The httle girl is adding rather more to her stature than the boy at four years old*, but she grows * This early spurt of growth in the girl is shewn in Enghsh, French and American observations, but not in Quetelet's. Ill OF GROWTH IN CHILDHOOD 107 slower than he does for a few years thereafter (6). At ten years old the girl's growth-rate begins to rise (c), a full year before the boy's; at twelve or thirteen the rate is much ahke for both, but it has reached its maximum for the girl. The boys' rate goes on rising, and at fourteen or fifteen they are growing twice as fast as the girls. So much for the annual increments, as a rough measure of the rates of growth. In actual stature the baby girl is some 2 or 3 per cent, below the boy at birth; she makes up the difference, and there is 7 9 Age Fig. 13. Ratio of female stature to male. II 13 15 From Simmons and Todd's data. From R. M. Fleming's data. good evidence to shew that she is by a very little the taller for a while, at about five years old or six. At twelve or thirteen she is very generally the taller of the two, and we call it her "gawky age" (Fig. 13). Man and woman differ in length of life, just 3,s they do in weight and stature. More baby boys are born than girls by nearly 5 peF cent. The numbers draw towards equality in their teens; after 108 THE RATE OF GROWTH ' [ch. twenty the women begin to outnumber the men, and at eighty-five there are twice as many women as men left in the world*. Men have pondered over the likeness and the unhkeness between the short lifetimes and the long; and some take it to be fallacious to measure all alike by the common timepiece of the sun. Life, they say, has a varying time-scale of its own ; and by this modulus the sparrow hves as long as the eagle and the day-fly as the manf. The time-scale of the living has in each case so strange a property of logarithmic decrement that our days and years are long in childhood, but an old man's minutes hasten to their end. On pre-natal and post-natal growth The rates of growth which we have so far studied are based on annual increments, or "first differences" between yearly determina- tions of magnitude. The first increment indicates the mean rate of growth during the first year of the infant's life, or (on a further assumption) the mean rate at the mean epoch of six months old; there is a gap between that epoch and the epoch of birth, of which we have learned nothing; we do not yet know whether the very high rate shewn within the first year goes on rising, or tends to fall, as the date of birth is approached. We are accustomed to inter- polate freely, and on the whole safely, between known points on a curve: "si timide que Ton soit, il faut bien que Ton interpole," says Henri Poincare; but it is much less safe and seldom justifiable (at least until we understand the physical principle involved and its mathematical expression) to "extrapolate" beyond the limits of our observations. We must look for more detailed observations, and we may learn much to begin with from certain old tables of Russow's J, who gives * Cf. F. E. A. Crew's Presidential Address to Section D of the British Association, 1937. + Cf. Gaston Backman, Die organische Zeit, Lunds Universitets Arsskrift, xxxv, Nr. 7, 1939. J Quoted in Vierordt's Anatomische . . . Daten und Tabellen, 1906, p. 13. See also, among many others, Camerer's data, in Pfaundler and Schlossman's Hdb. d. Kinderheilkunde, i, pp. 49, 424, 1908; Variot, op. cit.; for pre-natal growth, R. E. Scammon and L. A. Calkins, Growth in the Foetal Period, Minneapolis, 1929. Also, on this and many other matters, E. Faure-Fremiet, La cinetique du developpement, Paris, 1925; and, not least, J. Needham, Chemical EinJbryology, 1931. in] OF GROWTH IN CHILDHOOD 109 SIM cc op 9 ss "^ <^' lO© ■^ t- © 1 (M t^ t^ ©do ^^ C! -^ is X 9 1 1 QO 1 t- t- S CO © t^ O CC 1 op <^ lO fh lO CO c^i -^ V 9 10 00 -^ (N (N 1 ©00 CO l> © lO 2!2 1 15 © CO t^ 00 t^ r^ F^ CO CO ?o © ©© lO O-HiO 00 o ^ lO -^ '^ lb © iO © CO 99 9^ ©op 9^ t*9 © -§> ^ © lO-* f-H '.% 00 i> © I> I> -^6 lO lO —I (M ^ cc c -71 © »o l> © rH i-H ■^ -^ f^ CO © 1^ w ©© ©© r-H C^l OGO ^ '* 4^ j3 s" • (k'Zxn eo a t>00 © >o © © I> C i"H f-H r~* Oi ^'T*? 1 s i §5 '^ ©©© 1 T"? Jt^ Tt* 00 00 © ©00 1 ^ c^©© Tl* © © lO lO © 3| CO £ e 1 C3 0 1 t, 99-* ^ ^^© 1 1 CO 9 op ^ 9 © «6 © I> §1 i i »S2I 1 'S ■Ki 1 0) ©00 CO © ii d a ^ I 1 a 0 s r* 9 ss CO© ©CO CO \o £ i -2S§ 1 " CO) 0 lO coco coco ^ -* SB » ^ " 2 1 A 1 o 00 CO ?9 eo© r^ 1 © 1 © ©di CO CO 1 1 1 1 bC f^ CO "t TJH 0 ©© -fs r-H 1-^ ^ 1 1 1 1 0 2 4 6 8 10 months Fig. 18. Curve of a child's pre-natal growth, in length or stature; and corre- sponding curve of mean monthly increments (mm.). (Smoothed curves.) Ill] OF GROWTH IN INFANCY 115 the same curves meet us again and again in the growth of other organisms. The pre-natal growth of the guinea-pig is just the same*. We have the same essential features, the same S-shaped curve, in the growth by weight of an ear of maize (Fig. 19), or the growth in length of the root of a bean (Fig. 20); in both we see the same slow beginning, the rate rapidly increasing to a maximum, and the subsequent slowing down or "negative acceleration "f." One phase passes into another; so far as these curves go, they exhibit growth as a continuous process, with its beginning, its middle and its end— a continuity which Sachs recognised some seventy years ago, and spoke of as the "grand period of growth J." But these simple curves relate to simple instances, to the infant sheltered in the womb, or to plant-growth in the sunny season of the year. They mark a favourable episode, rather than relate the course of a lifetime. A curve of growth to run all life long is only simple in the simplest of organisms, and is usually a very complex affair. Growth in length of Vallisneria^, and root ofbean\\ and weight of ntaize^ VaUisneria A Vicia Zea — -\ Hours Inches Days Mm. Days Gm. 6 0-3 0 1-0 6 1 16 1-7 1 2-8 18 4 42 12-6 2 6-5 30 9 54 15-4 3 240 39 17 65 161 4 40-5 46 26 77 16-7 5 57-5 53 42 88 17-1 6 72-0 60 62 7 79-0 74 71 8 790 93 74 It would seem to be a natural rule, that those offspring which are most highly organised at birth are those which are born largest * See R. L. Draper, Anat. Record, xviii,"p. 369, 1920; cf. Needham, op. cil., p. 1672. t *^f. R. Chodat et A. Monnier; Sur la courbe de croissance chez les vegetaux. Bull. Herbier Boissier (2), v, p. 615, 1905. X Arbeiten a. d. bot. Instit. Wiirzburg, i, p. 569, 1872. § A. Bennett, Trans. Linn. Soc. (2), i (Bot.), p. 133, 1880. II Sachs, I.e. ^ Stefanowska, op. cit. ; G. Backman, Ergebn. d. Physiologie, xxiii, p. 925, 193j 116 THE RATE OF GROWTH [CH. 20 40 60 80 100 Days Fig. 19. Growth in weight of maize. From Gustav Backman, after Stefanowska. 0 5 8 Age in days Fig. 20. Growth in height of a beanstalk. From Sachs's data. Ill] OF GROWTH IN INFANCY 117 relatively to their parents' size. But another rule comes in, which is perhaps less to be expected, that the offspring are born smaller the larger the species to which they belong. Here we shew, roughly, the relative weights of the new-born animal and its mother*: Bear 1:600 Sheep 1:14 Lion 160 Ox 13 Hippopotamus 45 Horse 12 Dog 45-50 Rabbit 40 Cat 25 Mouse , 10-25 Man 22 Guinea-pig 7 These differences at birth are for the most part made up quickly; in other words, there are great differences in the rate of growth during early post-natal life. Two Uourcubs, studied by M. Anthony, grew as follows: Male Female Feb. 23 (born) — — 28 2-0 kilos 1-7 kilos Mar. 8 30 2-6 15 3-8 3-3 22 4-6 4-0 30 5-3 4-6 Apr. 5 61 5-2 12 7-0 6-0 19 8-0 7-0 Thus the lion-cub doubles its weight in the first month, and wellnigh doubles it again in the second; but the newborn child takes fully five months to double its weight, and nearly two years to do so again. The size finally attained is a resultant of the rate and of the duration of growth; and one or other of these may be the more important, in this case or in that. It is on the whole true, as Minot said, that the rabbit is bigger than the guinea-pig because he grows faster, but man is bigger than the rabbit because he goes on growing for a longer time. A bantam and a barn-door fowl differ in their rate of growth, which in either case is definite and specific. Bantams have been bred to match almost every variety of fowl ; and large size or small, quick growth or slow, is inherited or transmitted as a Mendehan * Data from Variot, after Anthony, 118 OF THE RATE OF GROWTH [ch. character in every cross between a bantam and a larger breed. The bantam is not produced by selecting smaller and smaller specimens of a larger breed, as an older school might have supposed ; but always by first crossing with bantam blood, so introducing the "character" of smallness or retarded growth, and then segregating the desired types among the dwarfish offspring. In fact. Darwinian selection plays a small and unimportant part in the process*. From the whole of the foregoing discussion we see that rate of growth is a specific phenomenon, deep-seated in the nature of the organism; wolf and dog, horse and ass, nay man and woman, grow at different rates under the same circumstances, and pass at different epochs through like phases of development. Much the same might be said of mental or intellectual growth ; the girl's mind is more precocious than the boy's, and its development is sooner arrested than the man's f. On variability, and on the curve of frequency or of error The magnitudes which we are dealing with in .this chapter — heights and weights and rates of change — are (with few exceptions) mean values derived from a large number of individual cases. We deal with what (to borrow a word from atomic physics) we may call an ensemble; we employ the equaUsing powei^ of averages, invoke the "law of large numbers J," and claim to obtain results thereby which are more trustworthy than observation itself §. But in ascertaining a mean value we must also take account of the amount of variability, or departure from the mean, among the cases from which the mean value is derived. This leads on far beyond our scope, but we must spare it a passing word ; it was this identical phenomenon, in the case of Man, which suggested to Quetelet the * Cf. Raymond Pearl, The selection problem, Amer. Naturalist, 1917, p. 82; R. C. Punnett and P. G. Bailey, Journ. of Genetics, iv, pp. 23-39, 1914. t Cf. E. Devaux, L'p,llure du developpement dans les deux sexes, Revue gindr. des Sci. 1926, p. 598. X S. D. Poisson, following James Bernoulli's Ars Conjectandi (op. posth. 1713), was the discoverer, or inventor, of the law of large numbers. "Les chos^s de toute nature sont soumises a une loi universelle qu'on pent appeler la loi des grands nombres" {Recherches, 1837, pp. 7-12). § See p. 137, footnote. Ill OF VARIABILITY 119 statistical study of Variation, led Francis Galton to enquire into the laws of Natural Inheritance, and served Karl Pearson as the foundation of his science of Biometrics. When Quetelet tells us that the mean stature of a ten-year-old boy is 1-275 metres, this is found to imply, not only that the measurements of all his ten-year-old boys group themselves about this mean value of 1-275 metres, but that they do so in an orderly way, many departing httle from that mean value, and fewer and fewer departing more and more. In fact, when all the measure- ments are grouped and plotted, so as to shew the number of instances (y) ^ each gradation of size (x), we obtain a characteristic -2cr —cr Fig. 21. The normal curve of frequency, or of error. a, -a, the "standard deviation". configuration, mathematically definable, called the curve of frequency, or oi error (Fig. 21). This is a very remarkable fact. That a "curve of stature" should agree closely with the "normal curve of error" amazed Galton, and (as he said) formed the mainstay of his long and fruitful enquiry into natural inheritance*. The curve is a thing apart, sui generis. It depicts no course of events, it is no time or vector diagram. It merely deals with the variabihty, and variation, of magnitudes; and by magnitudes " we mean anything which can be counted or measured, a regiment of men, a basket of * Stature itself, in a homogeneous population, is a good instance of a normal frequency distribution, save only that the spread or range of variation is unusually low; for one-half of the population of England differs by no more than an inch and a half from the average of them all. Variation is said to be greater among the negroid than among the white races, and it is certainly very great from one race to another: e.g. from the Dinkas of the White Nile with a mean height of 1-8 m. to the Congo pygmies averaging 1-35, or say 5 ft. 11 in. and 4 ft. 6 in. respectively. 120 THE RATE OF GROWTH [ch. niits, the florets of a daisy, the stripes of a zebra, the- nearness of shots to the bull's eye*. It thereby illustrates one of the most far-reaching, some say one of the most fundamental, of nature's laws. We find the curve of error manifesting itself in the departures from a mean value, which seems itself to be merely accidental — as, for instance, the mean height or weight of ten-year-old English boys; but we find it no less well displayed when a certain definite or normal number is indicated by the nature of the case. For instance the Medusae, or jelly-fishes, have a "radiate symmetry" of eight nodes and internodes. But even so, the number eight is subject to variation, and the instances of more or less graup them- selves in a Gaussian curve. Number of " tentaculocysts'' in Medusae {Ephyra and Aurelia) [Data from E. T. Browne, QJ.M.S. xxxvii, p. 245, 1895) 5 6 7 8 9 10 11 12 13 14 15 Ephyra (1893) — „ (1894) 1 4 6 8 34 278 883 22 75 18 61 12 35 14 17 3 3 1 — Aurelia (1894) — 2 18 296 33 16 18 7 — — 1 Percentage numbers: Ephyra — M 0-5 2-2 30 77-4 79-0 61 6-7 5-0 5-4 3-3 31 3-9 1-4 0-8 0-2 — — Aurelia — 0-5 4-7 77-2 8-6 41 2-6 1-8 — — — Mean — 0-7 3-3 77-9 7-1 4-8 3-0 2-4 0-3 — — The curve of error is a "bell-shaped curve," a courbe en cloche. It rises to a maximum, falls away on either side, has neither beginning nor end. It is (normally) symmetrical, for lack of cause to make it otherwise; it falls off faster and then slower the farther it departs from the mean or middle line; it has a "point of inflexion," of necessity on either side, where it changes its curvature and from being concave to the middle line spreads out to become convex * "I know of scarcely anything (says Galton) so apt to impress the imagination as the wonderful form of cosmic order expressed by the Law of Frequency of Error. ... It reigns with serenity and in complete self-effacement amidst the wildest confusion" {Natural Inheritance, p. 62). Observe that Galton calls it the "law of frequency o/ error," which is indeed its older and proper name. Cf. (int. al.) P. G. Tait, Trans R.S.E. xxiv, pp. 139-145, 1867. i Ill] OF THE CURVE OF ERROR 121 thereto. If we pour a bushel of corn out of a sack, the outhne or profile of the heap resembles such a curve ; and wellnigh every hill and mountain in the world is analogous (even though remotely) to that heap of corn *. Causes beyond our ken have cooperated to place and allocate each grain or pebble; and we call the result a "random distribution," and attribute it to fortuity, or chance. Galton devised a very beautiful experiment, in which a slopmg tray is beset with pins, and sand or millet-seed poured in at the top. Every falhng grain has its course deflected again and again; the final distribution is emphatically a random one, and the curve of error builds itself up before our eyes. The curve as defined by Gauss, princeps mathematicorum — who in turn was building on Laplace f — is at once empirical and theoretical J ; and Lippmann is said to have remarked to Poincare : *'Les experimentateurs s'imaginent que c'est un theoreme de mathematique, et les mathematiciens d'etre un fait experimental ! " It is theoretical in so far as its equation is based on certam hypo- thetical considerations: viz. (1) that the arithmetic mean of a number of variants is their best or likeliest average, an axiom which is obviously true in simple cases — but not necessarily in all; (2) that "fortuity" implies the absence of any predominant, decisive or overwhelming cause, and connotes rather the coexistence and joint effect of small, undefined but independent causes, many or few: * If we pour the corn out carefully through a small hole above, the heap becomes a cone, with sides sloping at an "angle of repose"; and the cone of Fujiyama is an exquisite illustration of the same thing. But in these two instances one predominant cause outweighs all the rest, and the distribution is no longer a random one. t The Gaussian curve of error is really the "second curve of error" of Laplace. Laplace's first curve of error (which has uses of its own) consists of two exponential curves, joining in a sharp peak at the median value. Cf. W. J. Luyten, Proc. Nat. Acad. Sci. xvm, pp. 360-365, 1932. I The Gaussian equation to the normal frequency distribution or "curve of error" need not concern us further, but let us state it once for all: J, _(xa-x)* ^ V27T where Xf^ is the abscissa which gives the maximum ordinate, and where the maximum ordinate, y^ = 1/^/(27t). Thus the log of the ordinate is a quadratic function of the abscissa ; and a simple property, fundamental to the curve, is that for equally spaced ordinates (starting anywhere) the square of any ordinate divided by the product of its neighbours gives a scalar quantity which is constant all along (G.T.B.). 122 THE RATE OF GROWTH [ch. producing their several variations, deviations or errors; and potent in their combinations, permutations and interferences*. We begin to see why bodily dimensions lend, or submit, them- selves to this masterful law. Stature is no single, simple thing; it is compounded of bones, cartilages and other elements, variable each in its own way, some lengthening as others shorten, each playing its little part, hke a single pin in Galton's toy, towards a ''fortuitous" resultant. "The beautiful regularity in the statures of a population (says Galton) whenever they are statistically marshalled in the order of their heights, is due to the number of variable and quasi-independent elements of which stature is the sum." In a bagful of pennies fresh from the Mint each coin is made by the single stroke of an identical die, and no ordinary weights and measures suffice to differentiate them; but in a bagful of old-fashioned hand-made nails a slow succession of repeated operations has drawn the rod and cut the lengths and hammered out head, shaft and point of every single nail — and a curve of error depicts the differences between them. The law of error was formulated by Gauss for the sake of the astronomers, who aimed at the highest possible accuracy, and strove so to interpret their observations as to eliminate or minimise their inevitable personal and instrumental errors. It had its roots also in the luck of the gaming-table, and in the discovery by eighteenth-century mathematicians that "chance might be defined in terms of mathematical precision, or mathematical 'law'." It was Quetelet who, beginning as astronomer and meteorologist, applied the "law of frequency of error" for the first time to biological statistics, with which in name and origin it had nothing whatsoever to do. The intrinsic significance of the theory of probabihties and the law of error is hard to understand. It is sometimes said that to forecast the future is the main purpose of statistical study, and expectation, or expectancy, is a common theme. But all the theory * "The curve of error would seem to carry the great lesson that the ultimate differences between individuals are* simple and few ; that they depend on collisions and arrangements, on permutations and combinations, on groupings and inter- ferences, of elementary qualities which are limited iyi variety and finite in extent'' (J. M. Keynes). A connection between this law and Mendelian inheritance is discussed by John Brownlee, P.R.S.E. xxxi, p. 251, 1910. Ill] bF THE LAW OF ERROR 123 in the vorld enables us to foretell no single unknown thing, not even the turn of a card or the fall of a die. The theory of probabiUties is a development of the theory of combinations, and only deals with what occurs, or has occurred, in the long run, among large numbers and many permutations thereof. Large numbers simplify many things; a million men are easier to understand than one man out of a million. As David Hume* said: "What depends on a few persons is in a great measure to be ascribed to chance, or to secret and unknown causes; what arises from a great many may often be accounted for by determinate and known causes." Physics is, or has become, a comparatively simple science, just because its laws are based on the statistical averages of innumerable molecular or primordial elements. In that invisible world we are sometimes told that "chance" reigns, and "uncertainty" is the rule; but such phrases as mere chance, or at random, have no meaning at all except with reference to the knowledge of the observer, and a thing is a "pure matter of chance" when it depends on laws which we do not know, or are not considering f. Ever since its inception the merits and significance of the theory of probabiUties have been variously estimated. Some say it touches the very foundations of know- ledge} ; and others remind us that "avec les chiffres on pent tout demontrer." It is beyond doubt, it is a matter of common ex- perience, that probability plays its part as a guide to reasoning. It extends, so to speak, the theory of the syllogism, and has been called the "logic of uncertain inference "§. In measuring a group of natural objects, our measurements are uncertain on the one hand and the objects variable on the other; and our first care is to measure in such a way, and to such a scale, that our own errors are small compared with the natural variations. Then, having made our careful measurements of. a group, we want to know more of the distribution of the several magnitudes, and * Essay xiv. t So Leslie Ellis and G. B. Airy, in correspondence with Sir J. D. Forbes; see his Life, p. 480. X Cf. Hans Reichenbach, Les fondements logiques du calcul des probabilit^s, Annales de Vinst. Poincare, vii, pp. 267, 1937. § Cf. J. M. Keynes, A Treatise on Probability, 1921; and A. C. Aitken's Statistical Mathematics, 1939. 124 THE RATE OF GROWTH [ch. especially to know two important things. We want a mean value, as a substitute for the true value* if there be such a thing; let us use the arithmetic mean to begin with. About this mean the ob- served values are grouped like a target hit by skilful or unskilful shots ; we want some measure of their inaccuracy, some measure of their spread, or scatter, or dispersion, and there are more ways than one of measuring and of representing this. We do it visibly and graphically every time we draw the curve (or polygon) of frequency ; but we want a means of description or tabulation, in words or in numbers. We find it, according to statistical mathematics, in the so-called index of variability, or standard deviation (o), which merely means the average deviation from the meanf. But we must take some precautions in determining this average; for in the nature of things these deviations err both by excess and defect, they are partly positive and partly negative, and their mean value is the mean of the variants themselves. Their squares, however, are all positive, and the mean of these takes account of the magnitude of each deviation with no risk of cancelling out the positive and negative terms: but the "dimension " of this average of the squares is wrong. The square root of this average of squares restores the correct dimension, and the result is the useful index of variability, or of deviation, which is called o-J. This standard deviation divides the area under the normal curve nearly into equal halves, and nearly coincides with the point of inflexion on either side; it is the simplest algebraic measure of dispersion, as the mean is the simplest arithmetical measure of position. When we divide this value by the mean, we get a figure * It is not always obvious what the "errors" are, nor what it is that they depart or deviate from. We are apt to think of the arithmetic mean, and to leave it at that. But were we to try to ascertain the ratio of circumference to diameter by measuring pennies or cartwheels, our "errors" would be found grouped round a mean value which no simple arithmetic could define. t a, the standard deviation, was chosen for its convenience in mathematical calculation and formulation. It has no special biological significance; and a simpler index, the "inter-quartile distance," has its advantages for the non- mathematician, as we shall see presently. X That is to say : Square the deviation-from-the-mean of each class or ordinate (^); multiply each by the number of instances (or "variates") in ^that class (/); divide by the total number (N) ; and take the square-root of the whole : a^= ~ "^ *' . I Ill] THE COEFFICIENT OF VARIABILITY 125 which is independent of any particular units, and which is called the coefficient of variability''^ . Karl Pearson, measuring the amount of variability in the weight and height of man, found this coefficient to run as follows : In male new-born infants, for weight 15-6, and for stature 6-5; in male adults, for weight 10-8, and for stature 3-6. Here the amount of variability is thrice as great for weight as for stature among grown men, and about 2 J times as great in infancy f. The same curious fact is well brought out in some careful measurements of shell-fish, as follows; Variability of youdg Clams (Mactra sp.)X Average size A Coefficient of variability Age (years) Number in sample 1 2 41 20 1 2 41 20 Length (cm.) Height Thickness 3-2 6-3 2-3 4-7 1-3 2-8 15-3 6-3 140 6-7 9-6 8-3 Weight (gm.) 6-4 59-8 35-4 18-5 The phenomenon is purely mathematical. Weight varies as the product of length, height and depth, or (as we have so often seen) as the cube of any one of these dimensions in the case of similar figures. It is then a mathematical, rather than a biological fact that, for small deviations, the variability of the whole tends to be equal to the sum of that of the three constituent dimensions. For if weight, w, varies as height x, breadth y, and depth z, we may write w = c.xyz. „„ ,.„ ,. ,. dw dx dy dz Whence, ditterentiatmg, = h -^ H . w X y z We see that among the shell-fish there is much more variability in the younger than in the older brood. This may be due to * It is usually multiplied by 100, to make it of a handier amount; and we may then define this coefficient, C, AS — ajM x 100. t Cf. Fr. Boas, Growth of Toronto children,- Rep. of U.S. Cornm. of Education, 1896-7, 1898, pp. 1541-1599; Boas and Clark Wissler, Statist cs of growth, Education Rep. 1904, 1906, pp. 25-132; H. P. Bowditch Rep. Mass. State Board of Health, 1877; K. Pearson, On the magnitude of certain coefficients of correlation in man, Proc. R.S. lxvi, 1900; S. Nagai, Korperkonstitution der Japaner, from Brugsch-Levy, Biologie d. Person, ii, p. 445, 1928; R. M. Fleming, A study of growth and development. Medical Research Council, Special Report, No. 190, 1933. J From F. W. Weymouth, California Fish Bulletin, No. 7, 1923. 126 THE RATE OF GROWTH [ch. inequality of age; for in a population only a few weeks old, a few days sooner or later in the date of birth would make more diiference than later on. But a more important matter, to be seen in man- kind (Fig. 22), is that variability of stature runs j)ari passu, or nearly so, with the rate of growth, or curve of annual increments (cf. Fig. 12). The curve of variability descends when the growth- rate slackens, and rises high when in late boyhood growth is speeded up. In short, the amount of variability in stature or in weight is correlated with, or is a function of, the rate of growth in these magnitudes. Judging from the evidence at hand, we may say that variabihty reaches its height in man about the age of thirteen or fourteen, rather earlier in the girls than in the boys, and rather earher in the case of stature than of weight. The difference in this respect between the boys and the girls is now on one side, now on the other. In infancy variabihty is greater in the girls; the boys shew it the more at five or six years old; about ten years old the girls have it again. From twelve to sixteen the boys are much the more variable, but by seventeen the balance has swung the other way (Fig. 23). Coefficient of variability (ojM x 100) in man, at various ages Age ... o 6 7 8 9 10 11 12 13 14 15 16 17 18 ytature I British (Fleming): Boys •51 5-4 50 5-3 5-4 5-6 5-7 5-6 5-8 5-8 5-8 5-0 4-3 30 Girls 5-2/ 5-2 50 5-5 5-4 5-6 5-8 5-7 5-6 4-7 4-2 3-9 3-7 3-8 American 4-8 4-6 4-4 4-5 4-4 4-6 4-7 4-9 5-5 5-8 5-6 5-5 4-6 3-7 (Bowditch) Japanese (Nagai): Boys 40 — 4-3 — 41 — 40 50 50 4-2 3-2 — — Girls — 4-3 — 41 — 4-5 — 4-5 4-6 3-6 31 30 — — Mean — 4-7 — 4-7 — 4-9 — ^ 5-0 5-3 50 4-6 41 — ■ — Weight American 11-6 10-3 Ill 9-9 110 1-6 1-8 13-7 3-6 6-8 15-3 13-3 130 10-4 Japanese : Boys 10-3 121 — 0-8 — 7-0 51 70 13-8 10-9 — — Girls — 10-2 — 11-2 — 21 — 150 5-6 3-4 11-4 11-5 — — Mean — 10-3 — 111 — 11-5 — 11-9 14-8 15-7 13-5 11-9 — -- Ill] THE COEFFICIENT OF VARIABILITY 127 61 1 1 1 1 1 r 1 ^ y r^ American.-^ / -^ / Japanese^ I I I I I \ L 5 6 7 8 9 10 II 12 13 14 15 16 17 II Age Fig. 22. Variability in stature (boys). After Fleming, Bowditch and Nagai. + z [— + 1 t - B ntish / / / / // // ^ \ \ \ \ \ \ \ \ \ \ \ / / / / \ -1 / 1 Japanese 1 1 3 18 . 10 15 Age in years Fig. 23. Coefficient of variability in stature: excess or defect of this coefficient in the boy over the girl. Data from R. M. Fleming, and from Nagai. 128 THE RATE OF GROWTH [ch. The amount of variability is bound to differ from one race or nationality to another, and we find big differences between the Americans and the Japanese, both in magnitude and phase (Fig. 22). If we take not merely the variability of stature or -weight at a given age, but the variability of the yearly increments, we find that this latter variabihty tends to increase steadily, and more and more rapidly, within the ages for which we have information; and this phenomenon is, in the main, easy of explanation. For a great part of the difference between one individual and another in regard to rate of growth is a mere difference of phase — a difference in the epochs of acceleration and retardation, and finally a difference as to the epoch when growth comes to an end ; it follows that variabihty 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 the standard deviation (ct), converted them into the corresponding coefficients of variabihty {olM x 100), and then smoothed the resulting numbers: Coefficients of variability in annual increments of stature Age ... 7 8 9 10 11 12 13 14 15 Boys Girls 17-3 171 15-8 17-8 18-6 19-2 191 22-7 21-0 25-9 24-7 29-3 29-0 370 36-2 44-8 461 The greater variabihty in the girls is very marked f, and is explained (in part at least) by the jnore rapid rate at which the girls run through the several phases of their growth (Fig. 24). To say that children of a given age vary in the rate at which they are growing would seem to be a more fundamental statement than that they vary in the size to which they have grown. 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 epochs 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 * Op. cit. p. lo48. I That women are on the whole more variable than men was argued by Karl Pearson in one of his earlier essays: The Chances of Death and other Studies, 1897. in THE COEFFICIENT OF VARIABILITY 129 epoch after birth are growing rapidly and are 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 "coefl&cient of correlation" between stature and rate of growth, and also between § 20 7 8 9 10 II 12 13 14 15 Age in years Fig. 24. Coefficients of variability, in annual increments of stature. After Boas. 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 back- * Some first attempts at analysis seem to shew that the size of the embryo at birth, or of the seed at germination, has more, influence than we were wont to suppose on the ultimate size of plant or animal. See (e.g.) Eric Ashby, Heterosis and the inheritance of acquired characters, Proc. R.S. (B), No. 833, pp. 431-441, 1937; and papers quoted therein. 130 THE RATE OF GROWTH [ch. ward ones may still be growing rapidly, and so making up (more or less completely) on 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) Age 6 7 8 9 10 11 12 13 14 15 Stature (B) 112-7 115-5 123-2 127-4 133-2 136-8 142-7 147-3 155-9 162-2 (G) 111-4 117-7 121-4 127-9 131-8 136-7 144-6 149-7 153-8 157-2 Increment (B) 5-7 5-3 4-9 5-1 5-0 4-7 5-9 7-5 6-2 5-2 (G) 5-9 5-5 5-5 5-9 6-2 7-2 6-5 5-4 3-3 1-7 Correlation (B) 0-25 0-11 0-08 0-25 018 0-18 0-48 0-29 -0-42 -0-44 (G) 0-44 0-14 0-24 0-47 018 -0-18 -0-42 -0-39 -0-63 0-11 A minor but very curious point brought out by the same investigators is that, if instead of stature we deal with height in the sitting posture (or, practically speaking, with length of trunk or back), then the correlations between this height and its annual increment are throughout negative. In other words, there would seem to be a general tendency for the long trunks to grow slowly throughout the whole period under investigation. It is a well- known anatomical fact that tallness is in the main due not to length of body but to length of limb. Since growth in height and growth in weight have each, their own velocities, and these fluctuate, and even the amount of their variabihty alters with age, it follows that the correlation between height and weight must not only also vary but must tend to fluctuate in a somewhat complicated way. The fact is, this corre- lation passes through alternate maxima and minima, chief among which are a maximum at about fourteen years of age and a minimum about twenty-one. Other intercorrelations, such as those between height or weight and chest-measurement, shew their periodic variations in like manner; and it is about the time of puberty * I.e. p. 42, and other papers there quoted. Cf. also T. B. Robertson, Criteria of Normality in the Growth of Children, Sydney, 1922. Ill] THE CURVE OF ERROR 131 that correlation tends to be closest, or a norm to be most nearly approached*. The whole subject of variabiHty, both of magnitude and rate of increment, is highly suggestive and instructive: inasmuch as it helps further to impress upon us that growth and specific rate of growth are the main physiological factors, of which specific mag- nitude, dimensions and form are the concrete and visible resultant. Nor may we forget for a moment that growth-rate, and growth itself, are both of them very complex things. The increase of the active tissues, the building of the skeleton and the laying up of fat and other stores, all these and more enter into the complex phenomenon of growth. In the first instance we may treat these many factors as though they were all one. But the breeder and the geneticist will soon want to deal with them apart; and the mathematician will scarce look for a simple expression where so many factors are involved. But the problems of variability, though they are intimately related to the general problem of growth, carry us very soon beyond our hmitations. The curve of error To return to the curve of error. The normal curve is a symmetrical one. Its middle point, or median ordinate, marks the arithmetic mean of all the measurements ; it is also the mode, or class to which the largest number of individual instances belong. Mean, median and mode are three diiferent sorts of average; but they are one and the same in the normal curve. It is easy to produce a related curve which is not symmetrical, and in which mean, median and mode are no longer the same. The heap of corn will be lop-sided or "skew" if the wind be blowing while the grain is falling: in other words, if some prevailing cause disturb the quasi-equihbrium of fortuity ; and there are other ways, some simple, some more subtle, by^ which asymmetry may be impressed upon our curve. The Gaussian curve is only one of many similar bell-shaped curves ; and the binomial coefficients, the numerical coefficients of (a + by, yield a curve so Hke it that we may treat them as the same. The * Cf. Joseph Bergson, Growth -changes in physical correlation, Human Biology, I, p. 4, 1930. 132 THE RATE OF GROWTH [ch. Gaussian curve extends, in theory, to infinity at either end; and this infinite extension, or asymptotism, has its biological significance. We know that this or that athletic record is lowered, slowly but continually, as the years go by. This is due in part, doubtless, to increasing skill and improved technique ; but quite apart from these the record would slowly fall as more and more races are run, owing to the indefinite extension of the Gaussian curve*. On the other hand, while the Gaussian curve extends in theory to infinity, the fact that variation is always limited and that extreme v3,riations are infinitely rare is one of the chief lessons of the law of frequency. If, in a population of 100,000 men, 170 cm. be the mean height and 6 cm. the standard deviation, only 11 per cenl^., or say 130 men, will exceed 188 cm., only 10 men will be over 191 cm., and only one over 193 cm., or 13 J per cent, above the average. The chance is negligible of a single one being found over 210 cm., or 7 ft. high, or 24 per cent, above the average. Yet, widely as the law holds good, it is hardly safe to count it as a universal law. Old Parr at 150 years old, or the giant Chang at more than eight feet high, are not so much extreme instances of a law of probabihty, as exceptional cases due to some peculiar cause or influence coming inf. In a somewhat analogous way, one or two species in a group grow far beyond the average size ; the Atlas moth, the Gohath beetle, the ostrich and the elephant, are far-off outhers from the groups to which they belong. A reason is not easy to. find. It looks as though variations came at last to be in proportion to the size attained, and so to go on by compound interest or geometrical progression. There may be nothing surprising in this ; nevertheless, it is in contradistinction to that summation of small fortuitous differences which lies at the root of the law of error. If size vary in proportion to the magnitude of the variant individuals, not only * This is true up to a certain extent, but would become a mathematical fiction later on. There will be physical limitations (as there are in quantum mechanics) both to record-breaking, and to the measurement of minute extensions of the record. t We may indeed treat old Parr's case on the ordinary lines of actuarial probability, but it is "without much actuarial importance." The chance of his record being broken by a modern centenarian is reckoned at (5)^°, by Major Greenwood and J. C. Irwin, writing on Senility, in Human Biology, xi, pp. 1-23, 1939. Ill] THE CURVE OF ERROR 133 will the frequency curve be obviously skew, but the geometric mean^ not the arithmetic, becomes the most probable value*. Now the logarithm of the geometric mean of a series of numbers is the arithmetic mean of their logarithms; and it follows that in such cases the logarithms of the variants, and not the variants them- selves, will tend to obey the Gaussian law and follow the normal curve of frequency f. The Gaussian curve, and the standard deviation associated with it, were (as we have seen) invented by a mathematician for the use 21 33 36 24 27 30 Length in mm. Fig. 25 A. Curve of frequency of a population of minnows. 39 of an astronomer, and their use in biology has its difficulties and disadvantages. We may do much in a simpler way. Choosing a random example, I take a catch of minnows, measured in 3 mm. groups, as follows (Fig. 25 A): Size (mm.) 13-15 16^18 19-21 22-24 25-27 28^30 31-33 34-36 37-39 Number 1 22 52 67 114 257 177 41 2 * See especially J. C. Kapteyn, Skew frequency curves in biology and statistics, Rec. des Trav. Botan. N4erland., Groningen, xin, pp. 105-158, 1916. Also Axel M. Hemmingsen, Statistical analysis of the differences in body-size of related species, Danske Vidensk. Selsk. Medd. xcvm, pp. 125-160, 1934. t This often holds good. Wealth breeds wealth, hence the distribution of wealth follows a skew curve; but logarithmically this curve becomes a normal one. Weber's law, in physiology, is a well-known instance; on the thresholds of sensations, effects are produced proportional to the magnitudes of those thresholds, and the logs of the thresholds, and not the thresholds themselves, are normaUy distributed. 134 THE RATE OF GROWTH [CH. Let us sum the same figures up, so as to show the whole number above or below the respective sizes. Size (mm.) 15 18 21 24 27 30 33 36 39 Number below 1 23 75 142 256 513 690 731 733 Percentage — 31 10-2 19-4 34-9 70-0 941 99-6 100 Our first set of figures, the actual measurements, would give us the '*courbe en cloche," in the formrof an unsymmetrical (or "skew") Gaussian curve: one, that is to say, with a long sloping talus on Extreme Decile Quartile Median Quartile Decile Extreme Fig. 25 B. 18 33 100% 90 75 .50 36 39 21 24 27 30 Length in mm. 'Curve of distribution" of a population of minnows. one side of the hilj. The other gives us an "S-shaped curve,'' ap- parently hmited, but really asymptotic at both ends (Fig. 25 B) ; and this S-shaped curve is so easy to work with that we may at once divide it into two halves (so finding the ''median" value), or into quarters and tenths (giving the "quartiles" and "deciles"), or as we please. In short, after drawing the curVe to a larger scale, we shall find that we can safely read it to thirds of a miUimetre, and so draw from it the following somewhat rough but very useful tabular epitome of our population of minnows, from which the curve can be recon- structed at aiiy time: mm. 13 210 25-3 28-6 30-6 32-3 Extreme First decile Lower quartile Median Upper quartile Last decile Extreme 39 Ill] OF Multimodal curves 135 This S-shaped "summation-curve" is what Francis Galton called a curve of distribution, and he "Uked it the better the more he used it." The spread or "scatter" is conveniently and immediately estimated by the distance between the two quartiles ; and it happens that this very nearly coincides with the standard deviation of the normal curve. 40 50 60 70 Micrometer-scale units 80 90 100 Fig. 26. A plankton-sample of fish-eggs: North of Scotland, February 1905. (Only eggs without oil-globule are counted here.) A. Dab and Flounder. B, Gadus Esmarckii and G. luscus. C, Cod and Haddock. D, Plaice. There are biological questions for which we want all the accuracy which biometric science can give; but there are many others on which such refinements are thrown away. Mathematically speaking, we cannot integrate the Gaussian curve,, save by using an infinite series; but to all intents and purposes we are doing so, graphically and very easily, in the illustration we have just shewn. In any case, whatever may be the precise character of each, we begin to see how our two simplest curves of growth, the bell-shaped and the S-shaped curve, form a reciprocal pair, the integral and the differential of one another "^ — hke the distance travelled * It is of considerable historical interest to know that this practical method of summation was first used by Edward Wright, in a Table of Latitudes published in his Certain Errors in Navigation corrected, 1599, as a means of virtually integrating sec X. (On this, and on Wright's claim to be the inventor of logarithms, see Florian Cajori, in Napier Memorial Volume, 1915, pp. 94-99.) 136 THE RATE OF GROWTH [ch. and the velocit}^ of a moving body. If ^ = e"^^ be the ordinate of the one, z = le-^'^dx is that of the other. There is one more kind of frequency-curve which we must take passing note of. We begin by. thinking of our curve, whether symmetrical or skew, as the outcome of a single homogeneous group. But if we happen to have two distinct but intermingled groups to deal with, differing by ever so little in kind, age, place or circumstance — leaves of both oak and beech, heights of both men and women — this heterogeneity will tend to manifest itself in two separate cusps, or modes, on the common curve: which is then indeed two curves rolled into one, each keeping something of its own individuality. For example, the floating eggs of the food-fishes are much alike, but differ appreciably in size. A random gathering, netted at the surface of the sea, will yield on measurement a multi- modal curve, each cusp of which is recognisable, more or less certainly, as belonging to a particular kind of fish (Fig. 26). A further note upon curves A statistical "curve", such as Quetelet seems to have been the first to use*, is a device whose peculiar and varied beauty we are apt, through famiharity, to disregard. The curve of frequency which we have been studying depicts (as a rule) the distribution of mag- nitudes in a material system (a population, for instance) at a certain epoch of time ; it represents a given state, and we may call /it a diagram of configuration "f. But we oftener use our curves to compare successive states, or changes of magnitude, as one configuration gives place to another; and such a curve may be called a diagram of displacement. An imaginary point moves in imaginary space, the dimensions of which represent those of the phenomenon in question, dimensions which we may further define and measure by a system of "coordinates"; the movements of our point through its figurative space are thus analogous to, and illus- trative of, the events which constitute the phenomenon. Time is often represented, and measured, on one of the coordinate axes, and our diagram of "displacement" then becomes a diagram of velocity. * In his Theorie des probabilites, 1846. t See Clerk Maxwell's article "Diagrams," in the Encyclopaedia Britannica, 9th edition. Ill] OF STATISTICAL CURVES 137 This simple method (said Kelvin) of shewing to the eye the law of variation, however complicated, of an independent variable, is one of the most beautiful results of mathematics*. We make and use our curves in various ways. We set down on the coordinate network of our chart the points givQ^i by a series of observations, and connect them up into a continuous series as we chart the voyage of a ship from her positions day by day ; we may "smooth" the line, if we so desire. Sometimes we find our points so crowded, or otherwise so dispersed and distributed, that a line can be drawn not from one to another but among them all — a method first used by Sir John Herschel f, when he studied the orbits of the double stars. His dehcate observations were affected by errors, at first sight without rhyme or reason, but a curve drawn where the points lay thickest embodied the common lesson of them all; any one pair of observations would have sufiiced, whether better or worse, for the calculation of an orbit, but Herschel's dot-diagram obtained "from the whole assemblage of observations taken together, and regarded as a single set of data, a single result in whose favour they all conspire." It put us in possession, said Herschel, of something truer than the observations themselves % ; and Whewell remarked that it enabled us to obtain laws of Nature not only from good but from very imperfect observations §. These are some advantages of the use of "curves," which have made them essential to research and discovery. It is often helpful and sometimes necessary to smooth our curves, * Kelvin, Nature, xxix, p. 440, 1884. t Mem. Astron. Soc. v, p. 171, 1830; Nautical Almanack, 1835, p. 495; etc. X Here a certain distinction may be observed. We take the average height of a regiment, because the men actually vary about a mean. But in estimating the place of a star, or the height of Mont Blanc, we average results which only differ by I)ersonal or instrumental error. It is this latter process of averaging which leads, in Herschel's phrase, to results more trustworthy than observation itself. Laplace had made a similar remark long before {Oeuvres,yii, Theorie des probabilites) : that we may ascertain the very small effect of a constant cause, by means of a long series of observations the errors of which exceed the effect itself. He instances the small deviation to the eastward which the rotation of the earth imposes on a falling body. In like manner the mean level of the sea may be determined to the second decimal of an inch by observations of high and low water taken roughly to the nearest inch, provided these are faithfxilly carried out at every tide, for say a hundred years. Cf. my paper on Mean Sea Level, in Scottish Fishery Board's Sci. Report for 1915. § Novum Organum Renovatum (3rd ed.), 1858, p. 20. 138 THE RATE OF GROWTH [ch. whether at free hand or by help of m^athematical rules; it is one way of getting rid of non-essentials — and to do so has been called the very key-note of mathematics*. A simple rule, first used by Gauss, is to replace each point by a mean between it and its two or more neighbours, and so to take a "floating" or "running average." In so doing we trade once more on the "principle of continuity"; and recognise that in a series of observations each one is related to another, and is part of the contributory evidence on which our knowledge of all the rest depends. But all the while we feel that Gaussian smoothing gives us a practical or descriptive result, rather than a mathematical one. Some curves are more elegant than others. We may have to rest content with points in which no. order is apparent, as when we plot the daily rainfall for a month or two; for this phenomenon is one whose regularity only becomes apparent over long periods, when average values lead at last to "statistical uniformity." But the most irregular of curves may be instructive if it coincide with another not less irregular : as when the curve of a nation's birth-rate, in its ups and downs, follows or seems to follow the price of wheat or the spots upon the sun. It seldom happens, outside of the exact sciences, that we com- prehend the mathematical aspect of a phenomenon enough to define (by formulae and constants) the curve which illustrates it. But, failing such thorough comprehension, we can at least speak of the trend of our curves and put into words the character and the course of the phenomena they indicate. We see how this curve or that indicates a uniform velocity, a tendency towards acceleration or retardation, a periodic or non-periodic fluctuation, a start from or an approach to a limit. When the curve becomes, or approximates to, a mathematical one, the types are few to which it is Kkely to belong f. A straight line, a parabola, or hyperbola, an exponential or a logarithmic curve (like x'=ay^), a sine-curve or sinusoid, damped or no, suffice for a wide range of phenomena; we merely modify our scale, and change the names of our coordinates. * Cf. W. H. Young, The mathematic method and its limitations, Atti del Congresso dei Matematici, Bologna, 192/8, i, p. 203. t Hence the engineer usually begins, for his first tentative construction, by drawing one of the familiar curves, catenary, parabola, arc of a circle, or curve of, sines. Ill] OF STATISTICAL CURVES 139 The curves we mostly use, other than the Gaussian curve, are time-diagrams. Each has a beginning and an end; and one and the same curve may illustrate the life of a man, the economic history of a kingdom, the schedule of a train between one st^^tion and another. What it then shews is a velocity, an acceleration, and a subsequent negative acceleration or retardation. It depicts a "mechanism" at work, and helps us to see analogous ' mechanisms in different fields; for Nature rings her many changes on a few simple themes. The same expressions serve for different orders of phenomena. The swing of a pendulum, the flow of a current, the attraction of a magnet, the shock of a blow, have their analogues in a fluctuation of trade, a wave of prosperity, a blow to credit, a tide in the affairs of men. The same exponential curve may illustrate a rate of cooHng, a loss of electric charge, the chemical action of a ferment or a catalyst. The S-shaped population-curve or "logistic curve" of Verhulst (to which we are soon coming) is the hysteresis-curve by which Ewing represented self-induction in a magnetic field ; it is akin to the path of a falhng body under the influence of friction; and Lotka has drawn a curve of the growing mileage of American railways, and found it to be a typical logistic curve. A few bars of music plotted in wave-lengths of the notes might be mistaken for a tidal record. The periodicity of a wave, the acceleration of gravity, retardation by friction, the role of inertia, the explosive action of a spark or an electric contact — these are some of the modes of action or "forms of mechanism" which recur in Hmited number, but in endless shapes and circumstances*. The way in which one curve fits many phenomena is characteristic of mathematics itself, which does not deal with the specific or individual case, but generalises all the while, and is fond (as Henri Poincare said) of giving the same name to different things. Our curves, as we have said, are mostly time-diagrams, and represent a change in time from one magnitude to another; they are diagrams of displacement, in Maxwell's phrase. We may consider four different cases, not equally simple mathematically, but all * See an admirable little book by Michael Petrovich, Les micanismes communs aux phenomenes disparates, Paris, 1921. 140 THE RATE OF GROWTH [ch. capable of explanation, up to a certain point, without mathe- matics. (1) If in our coordinate diagram we have merely to pass from one isolated jpoint to another, a straight line joining the two points is the shortest — and the hkeliest way. (2) To rise and fall alternately, going to and fro from maximum to minimum, a zig-zag rectilinear path would still be, geometrically, the shortest way; but it would be sharply discontinuous at every turn, it would run counter to the "principle of continuity," it is not likely to be nature's way. A wavy course, with no more change of curvature than is absolutely necessary, is the path which nature follows. We call it a simple harmonic motion, and the simplest of The Sine -curve The S- shaped curve The bell-shaped curve Fig. 27. Simple curves, representing a change from one magnitude to another. all such wavy curves we call a sine-curve. If there be but one maximum and one minimum, which our variant alternates between, the vector pathway may be translated into jpolar coordinates; the vector does- what the hands of the clock do, and a circle takes the place of the sine-curve. (3) To pass from a zero-line to a maximum once for all is a very different thing ; for now minimum and maximum are both of them continuous states, and the principle of continuity will cause our vector-variant to leave the one gradually, and arrive gradually at the other. The problem is how to go uphill from one level road to another, with the least possible interruption or discontinuity. The path follows an S-shaped course; it has an inflection midway; and the first phase and the last are represented by horizontal asymptotes. This is an important curve, and a common one. It so far resembles Ill] OF CURVES IN GENERAL 141 an "elastic curve" (though it is not mathematically identical with it) that it may be roughly simulated by a watchspring, lying between two parallel straight lines and touching both of them. It has its kinetic analogue in the motion of a pendulum, which starts from rest and comes to rest again, after passing midway through its maximal velocity. It indicates a balance between production and waste, between growth and decay: an approach on either side to a state of rest and equilibrium. It shows the speed of a train between two stations ; it illustrates the growth of a simple organism, or even of a population of men. A certain simple and symmetrical case is called the Verhulst- Pearl curve, or the logistic curve. (4) Lastly, in order to leave a certain minimum, or zero-hne, and return to it again, the simplest way will be by a curve asymptotic to the base-line at both ends — or rather in both directions; it will be a bell-shaped curve, having a maximum midway, and of necessity a point of inflection on either side ; it is akin to, and under certain precise conditions it becomes, the curve of error or Gaussian curve. Besides the ordinary curve of growth, which is a summation- curve, and the curve of growth-rates, which is its derivative, there are yet others which we may employ. One of these was introduced by Minot*, from 'a feeling that the rate of growth, or the amount of increment, ought in some way to be equated with the growing structure. Minot's method is to deal, not with the actual increments added in successive periods, but with these successive increments represented as percentages of the amount already reached. For instance, taking Quetelet's values for the height (in centimetres) of a male infant, we have as follows: Years 0 1 2 3 4 era. 500 69-8 791 86-4 92-7 But Minot would state the percentage-growth in each of these four annual periods at 39-6, 13-3, 9-2 and 7-3 per cent, respectively: Years 0 1 2 3 4 Height (cm.) 50-0 69-8 79-1 86-4 92-7 Increments (cm.) — 19-8 9-3 7-3 6-3 (per cent.) ~ 39-6 13-3 9-2 7-3 * C. S. Minot, On certain phenomena of growing old, Proc. Amer. Assoc, xxxix, 1890, 21 pp.; Senescence and rejuvenation, Journ. Physiol, xii, pp. 97-153, 1891; etc. Criticised by S. Brody and J. Needham, ojp, cit. pp. 401 seq. 142 THE RATE OF GROWTH [ch. Now, in our first curve of growth we plotted length against time, a very simple thing to do. When we differentiate L with respect to T, we have dL/dT, which is rate or velocity, again a very simple thing ; and from this, by a second differentiation, we obtain, if necessary, d^L/dT^, that is to say, the acceleration. But when you take percentages of y, you are determining dyjy, and w^hen you plot this against dx, you have ^, or -^, or -.^. dx ' y.dx' y' dx' That is to say, you are multiplying the thing whose variations you are studying by another quantity which is itself continually varying; and are dealing with something more complex than the original factors*. Minot's method deals with a perfectly legitimate function of x and y, and is tantamount to plotting log y against x, that is to say, the logarithm of the increment against the time. This would be all to the good if it led to some simple result, a straight line for instance ; but it is seldom if ever, as it seems to me, that it does anything of the kind. It has also been pointed out as a grave fault in his method that, whereas growth is a continuous process, Minot chooses an arbitrary time-interval as his basis of comparison, and uses the same interval in all stages of development. There is little use in comparing the percentage increase fer week of a week- old chick, with that of the same bird at six months old or at six years. The growth of a population After dealing with Man's growth and stature, Quetelet turned to the analogous problem of the growth of a populatfon — all the more analogous in our eyes since we know man himself to be a "statistical unit," an assemblage of organs, a population of cells. He had read * Schmalhausen, among others, uses the same measure of rate of growth, in the form log F-logF dvl^ ^~ k{t-t) ~^ dt v' Arch. f. Entw. Mech. cxm, pp. 462-519, 1928. m] MALTHUS ON POPULATION 143 Malthus's Essay on Population* in a French translation, and was impressed like all the world by the importance of the theme. He saw that poverty and misery ensue when a population outgrows its means of support, and believed that multiphcation is checked both . by lack of food and fear of poverty. He knew that there were, and must be, obstacles of one kind or another to the unrestricted increase of a population; and he knew the more subtle fact that a population, after growing to a certain height, oscillates about an unstable level of equilibrium f. Malthus had said that a population grows by geometrical pro- gression (as 1, 2, 4, 8) while its means of subsistence tend rather to grow by arithmetical (as 1, 2, 3, 4) — that one adds up while the other multiphesj. A geometrical progression is a natural and a * T. R. Malthus, An Essay on the Principle of Population, as it affects the Future Improvement of Society, etc., 1798 (6th ed. 1826; transl. by P. and G. Prevost, Geneva, 1830, 1845). Among the books to which Malthus was most indebted was A Dissertation on the Numbers of Mankind in ancient and modern Times, published anonymously in Edinburgh in 1753, but known to be by Robert Wallace and read by him some years before to the Philosophical Society at Edinburgh. In this remarkable work the writer says (after the manner of Malthus) that mankind naturally increase by successive doubling, and tend to do so thrice in a hundred years. He explains, on the other hand, that "mankind do not actually propagate according to the rule in our tables, or any other constant rule; ^et tables of this nature are not entirely useless, but may serve to shew, how much the increase of mankind is prevented by the various causes which confine their number within such narrow limits." Malthus was also indebted to David Hume's Political Discourse, Of the Populousness of ancient Nations, 1752, a work criticised by Wallace. See also McCulloch's notes to Adam Smith's Wealth of Nations, 1828. ■f" That the nearest approach to equilibrium in a population is long- continued ebb and flow, a mean level and a tide, was known to Herbert Spencer, and was stated mathematically long afterwards by Vito Volterra. See also Spencer's First Principles, ch. ^22, sect. 173: "Every species of plant or animal is perpetually undergoing a rhythmical variation in number — now from abundance of food and absence of enemies rising above its average, and then by a consequent scarcity of food and abundance of enemies being depressed below its average.. . .Amid these oscillations produced by their conflict, lies that average number of the species at which its expansive tendency ie in equilibrium with surrounding repressive tendencies." Cf. A. J. Lotka, Analytical note on certain rhythmic relations in organic systems, Proc. Nat. Acad. Sci. vi, pp. ^10-415, 1920; but cf. also his Elements of Physical Biology, 1915, p. 90. An analogy, and perhaps a close one, may be found on the Bourse or money market. X That a population will soon oiitrun its means of subsistence was a natural assumption in Malthus's day, and in his own thickly populated land. The danger may be postponed and the assumption apparently falsified, as by an Argentine cattle-ranch /or prairie wheat-farm — but only so long as we enjoy world-wide freedom of import and exchange. 144 THE RATE OF GROWTH [ch. common thing, and, apart from the free growth of a population or an organism, we find it in many biological phenomena. An epidemic dechnes, or tends to decline, at a rate corresponding to a geometrical progression; the mortahty from zymotic diseases declines in geo- metrical progression among children from one to ten years old; and the chances of death increase in geometrical progression after a certain time of Hfe for us all*. But in the ascending scale, the story of the horseshoe nails tells us how formidable a thing successive multipHcation becomes f. Enghsh law forbids the protracted accumulation of compound interest; and hkewise Nature deals after her own fashion with the case, and provides her automatic remedies. A fungus is growing on an oaktree — it sheds more spores in a night than the tree drops acorns in a hundred years. A certain bacillus grows up and multiphes by two in two hoTlrs' time; its descendants, did they all survive, would number four thousand in a day, as a man's might in three hundred years. A codfish lays a million eggs and more — all in order that one pair may survive to take their parents' places in the world. On the other hand, the humming-birds lay only two eggs, the auks and guillemots only one; yet the former are multitudinous in their haunts, and some say that the Arctic auks and auklets outnumber all other birds in the world. Linnaeus { shewed that an annual plant would have a miUion offspring in twenty years, if only two seeds grew up to maturity in a year. But multiply as they will, these vast populations have their limits. They reach the end of their tether, the pace slows down, and at last they increase no more. Their world is fully peopled, whether it be an island with its swarms of humming-birds, a test-tube with its myriads of yeast-cells, or a continent with its miUions of mankind. Growth, whether of a population or an individual, draws to its natural end; and Quetelet compares it, by a bold metaphor, to the motion of a body in a resistant medium. A typical population grows slowly from an asymptotic minimum; it multiphes quickly; * According to the Law of Gompertz ; cf. John Brownlee, in Proc. R.S.E. xxxi, pp. 627-634, 1"911. t Herbert Spencer, A theory of population deduced from the general law of animal fertility, Westminster Review, April 1852. X In his essay De Tellure, 1740. Ill] VERHULST'S LAW 145 it draws slowly to an ill-defined and asymptotic maximum. The two ends of the population-curve define, in a general way, the whole curve between; for so beginning and so ending the curve must pass through a point of inflection, it must be an S-shaped curve. It is just such a curve as we have seen imder simple conditions of growth in an individual organism. This general and all but obvious trend of a population-curve has been recognised, with more or less precision, by many writers. It is imphcit in Quetelet's own words, as follows: "Quand une population pent se developper hbrement et sans obstacles, elle croit selon une progression geometrique; si le developpement a heu au miheu d'obstacles de toute espece qui tendent a Farreter, et qui agissent d'une maniere uniforme, c'est a dire si I'etat sociale ne change point, la population n'augmente pas d'une maniere indefinie, mais elle tend de plus en plus a devenir stationnaire*.'' P. F. Verhulst, a mathematical colleague of Quetelet's, was interested in the same things, and tried to give a mathematical shape to the same general conclusions; that is to say, he looked for a "fonction retardatrice" which should turn the Malthusian curve of geometrical progression into the S-shaped, or as he called it, the logistic curve, which should thus constitute the true "law of population," and thereby indicate (among other things) the hmit above which the population^ was not likely to grow f . Verhulst soon saw that he could only solve his problem in a prehminary and tentative way; ''la hi de la population nous est inconnue, parcequ'on ignore la nature de la fonction qui sert de mesure aux obstacles qui s'opposent a la multiphcation indefini de I'espece humaine." The materials at hand were almost unbeUevably scanty and poor. The French statistics were taken from documents "qui ont ete reconnus entierement fictifs"; in England the growth * Physique Sociale, i, p. 27, 1835. But Quetelet's brief account is somewhat ambiguous, and he had in mind a body falling through a resistant medium — which suggests a limiting velocity, or limiting annual increment, rather than a terminal value. See Sir G. Udny Yule, The growth of population, Journ. R. Statist. Soc. Lxxxvm, p. 42, 1925. t P. F. Verhulst, Notice sur la loi que la population suit dans son accroissement. Correspondence math. etc. public par M. A. Quetelet, x, pp. 113-121, 1838; Rech. math, sur la loi etc., Nozw. Mem. de VAcad. R. de Bruxelles, xviii, 38 pp., 1845; deuxieme Mem., ihid. xx, 32 pp., 1847. The term logistic curve had already been used by Edward Wright; see antea, p. 135. footnote. 146 THE RATE OF GROWTH [ch. of the population was estimated by the number of births, and the births by the baptisms in the Church of England, "de maniere que les enfants des dissidents ne sont point portes sur les registres officiels." A law of population, or "loi d'affaibhssement" became a mere matter of conjecture, and the simplest hypothesis seemed to Verhulst to be, to regard "cet affaibhssement comme proportionnel a I'accroissement de la population, depuis le moment ou la difficulty de trouver de bonnes terres a commence a se faire sentir*." Verhulst was making two assumptions. The first, which is beyond question, is that the rate of increase cannot be, and indeed is not, a constant; and the second is that the rate must somehow depend on (or be some function of) the population for the time being. A third assumption, again beyond question, is that the simplest possible function is a linear function. He suggested as the simplest possible case that, once the rate begins to fall (or once the struggle for existence sets in), it will fall the more as the population continues to grow; we shall have a growth-factor and a retardation-factor in proportion to one another. He was making early use of a simple differential equation such as Vito Volterra and others now employ freely in the general study of natural selec.tionf. The point wher^ a struggle for existence first sets in, and where ipso facto the rate of increase begins to diminish, is called by Verhulst the normal level of the population ; he chooses it for the origin of his curve, which is so defined as to be symmetrical on either side of this origin. Thus Verhulst's law, and his logistic curve, owe their form and their precision and all their power to forecast the future to certain hypothetical assumptions; and the tentative solution arrived at is one "sous le point de vue mathematiquej." ♦ Op. cit. p. 8. t Besides many well-known papers by Volterra, see V. A, Kostitzin, Biologie mathematique, Paris, 1937. Cf. also, for the so-called "Malaria equations," Ronald Ross, Prevention of Malaria, 2nd ed. 1911, p. 679; Martini, Zur Epidemiologie d. Malaria, Hamburg, 1921; W. R. Thompson, C.R. clxxiv, p. 1443, 1922, C. N. Watson, Nature, cxi, p. 88, 1923. J Verhulst goes on to say that " une longue serie d 'observations, non interrompues par de grandes catastrophes sociales ou des revolutions du globe, fera probablement decouvrir la fonction retardatrice dont il vient d'etre fait mention." Verhulst simplified his problem to the utmost, but it is more complicated today than ever; he thought it impossible that a country should draw its bread and meat from overseas: "lors meme qu'une partie considerable de la population pourrait etre Ill] THE LOGISTIC CURVE 147 The mathematics of the Verhulst-Pearl curve need hardly concern us; they are fully dealt with in Raymond Pearl's, Lotka's and other books. Verhulst starts, as Malthus does, with a population growing in geometrical progression, and so giving a logarithmic curve: dp He then assumes, as his "loi d'affaibUssement," a coefficient of retardation {n) which increases as the population increases: dp 2 Integrating, p = ^^_^_^_. If the point of inflection be taken as the origin, k = 0; and again for < = 00, jt? = — = L. We may write accordingly: 1 + e-^ Malthus had reckoned on a population doubling itself, if unchecked by want or "accident," every twenty-five years*; but fifty years after, Verhulst shewed that this "grande vitesse d'accroissement" was no longer to be found in France or Belgium or other of the older countries!, but wa^ still being reaUsed in the United States (Fig. 28). All over Europe, "le rapport de I'exces annuel des nais- sances sur les deces, a la population qui I'a fourni, va sans cesse en s'affaiblissant ; de maniere que Faccroissement annuel, dont la valeur absolue augmente continuellement lorsqu'il y a progression geo- metrique, parait suivre une progression tout au plus arithmetique." nourrie de bles etrangers, jamais un gouvernement sage ne consentira a faire dependre I'existence de milliers de citoyens du bon vouloir des souverains etrangers." On this and other problems in the growth of a human population, see L. Hogben's Genetic Problems, etc., 1937, chap. vii. See^also [int. al.) Warren S. Thompson and P. K. Whelpton, Population Trends in the United States, 1933; F. Lorimer and F. Osborn, Dynamics of Population, 1934, etc. * An estimate based, like the rest of Malthus's arithmetic, on very slender evidence. t In Quetelet's time the European countries, far from doubling in twenty-five years, were estimated to do so in from sixty years (Norway) to four hundred years (France); see M. Haushofer, Lehrbuch der Statiatik, 1882. 148 THE RATE OF GROWTH [CH. The "celebrated aphorism" of Malthus was thus, and to this extent, confirmed*. In the United States, the Malthusian estimate of unrestricted increase continued to be reahsed for a hundred years after Malthus wrote; for the 3-93 millions of the U.S. census of 1790 were doubled three times over in the census of 1860, and four times over in that of 1890. A capital which doubles in twenty-five years has grown at 2-85 per cent, per annum, compound interest; the U.S. population did rather more, for it grew at fully 3 per cent, for fifty of those hundred years f. 1790 1800 1850 1900 1930 Fig. 28. Population of the United States, 1790-1930. The population of the whole world and of every continent has increased during modem times, and the increase is large though the rate is low. The rate of increase has been put at about half-a- per-cent per annum for the last three hundred years — a shade more in Europe and a shade less in the rest of the world { : * Op. cit. 1845, p. 7. t Verhulst foretold forty millions as the "extreme limit" of the population of France, and 6^ millions as that of Belgium. The latter estimate he increased to 8 millions later on. The actual populations of France and Belgium at the present time are a little more than the ultimate limit which Verhulst foretold. + From A. M. Carr-Saunders' World Population, 1936, p. 30. in] RAYMOND PEARL 149 An estimate of the population of the world (After W. F. Willcox) Mean rate of 1650 1750 1800 1850 1900 increase Europe 100 140 187 266 401 0-52 % per annum World total 545 728 906 1171 1608 0-49% „ Verhulst was before his time, and his work was neglected and presently forgotten. Only some twenty years ago, Raymond Pearl and L. J. Reed of Baltimore, studying the U.S. population as Verhulst had done, approached the subject in the same way, and came to an identical result; then, soon afterwards (about 1924), Raymond Pearl came across Verhulst's papers, and drew attention to what we now speak of as the Verhulst-Pearl law. Pearl and Reed saw, as Verhulst had dope, that a "law of population" which should cover all the ups and downs of human affairs was not to be found; and yet the general form which such a law must take was plain to see. There must be a limit to the population of a region, great or small; and the curve of growth must sooner or later "turn over," approach the limit, and resolve itself into an S-shaped curve. The rate of growth (or annual increment) will depend (1) on the population at the time, and (2) on "the still unutiUsed reserves of population-support existing" in the available land. Here we have, to all intents and purposes, the growth-factor and retardation-factor of Verhulst, and they lead to the same formula, or the same differential equation, as his*. A hundred years have passed since Verhulst dealt with the first U.S. census returns, and found them verifying the Malthusian expectation of a doubhng every twenty-five years. That "grande vitesse d'accroissement " continued through five decennia; but it ceased some seventy years ago, and a retarding influence has been manifest through all these seventy years (Fig. 29). It is more recently, only after the census of 1910, that the curve seemed to be * Raymond Pearl and L. J. Reed, on the Rate of growth of the population of the U.S. since 1790, and its mathematical representation, Proc. Nat. Acad. Sci. VI, pp. 275-288, 1920; ibid, vin, pp. 365-368, 1922; Metron, m, 1923. In the first edition of Pearl's Medical Biometry and Statistics, 1923 (2nd ed. 1930), Verhulst is not mentioned. See also his Studies in Human Biology, Baltimore, 1924, Natural History of Population, 1939, and other works. 150 THE RATE OF GROWTH [CH. 1800 1850 1900 1940 Fig. 29. Decennial increments of the population of the United States. * The Civil War. * * The "slump". zuu — 1 — 1 -1 — r- 1 1 1 1 1 ! I^H^^ - 1 ' • _ y y - / / ~ • 150 — / _ / _ 1 / / ¥ ^ - t / § - . ^ / 1 '^^ / f 1 - 0 - ^ - - P - 50 - - """i 1 1 1 1 1 1 1 1 1 1 1 1 1 r 1 1 1 1 1800 1850 1900 1950 2000 Fig. 30. Conjectural population of the United States, according to the Verhulst- Pearl Law. Ill] POPULATION OF THE UNITED STATES 151 finding its turning-point, or point of inflection; and only now, since 1940, can we say with full confidence that it has done so. A hundred years ago the conditions were still relatively' simple, but they are far from simple now. Immigration was only beginning to be an important factor; but immigrants made a quarter of the whole increase of the population of the United States during eighty of these hundred years*. Wars and financial crises have made their mark upon the curve; manners and customs, means and standards of living, have changed prodigiously. But the S-shaped curve makes its appearance through all of these, and the Verhulst-Pearl formula meets the case with surprising accuracy. Population of the United States I In ten years Calculated No. of A Total ■\ Increase by multi- by logistic immigrants increase of plication Population curve landed population Percentage in 25 Year xlOOO (Udny Yule) xlOOO xlOOO increase years 1790 3,929 3,929 — — — — 1800 5,308 5,336 — 1,379 351 — 1810 7,240 7,223 — 1,932 36-4 1820 9,638 9,757 250 2,398 331 208 1830 12,866 13,109 228 3,228 33-5 205 1840 17,069 17,506 538 4.203 32-7 2-02 1850 23,192 23,192 1,427 6,123 35-9 206 1860 31,443 30,418 2,748 8,251 35-6 210 1870 38,558 39,372 2,123 7,115 32-6 1-91 1880 50,156 50,177 2,741 11,598 301 1-83 1890 62.948 62,769 5,249 12,792 25-5 1-80 1900 75,995 76,870 3,694 13,047 20-7 1-71 1910 91,972 91,972 8,201 15,977 210 1-63 1920 105,711 — 6,347 13,739 14-9 1-52 1930 122,975 — — 17,264 161 1-46 1940 131,669 — — 8,694 71 1-33 A colony of yeast or of bacteria is a population in its simplest terms, and Verhulst's law was rediscovered in the growth of a bacterial colony some years before Raymond Pearl found it in a population of men, by Colonel M'Kendrick and Dr Kesava Pai, who put their case very simply indeed f. The bacillus grows by geometrical * Without counting the children born to those immigrants after landing, and before the next census return. t A. G. M'Kendrick and M. K. Pai, The rate of multiplication of micro-organisms : a mathematical study, Proc. R.S.E. xxxi, pp. 649-655, 1911. (The period of generation in B. coli, answering to Malthus's twenty-five years for men, was found to be 22^ minutes.) Cf. also Myer Coplans, Journ. of Pathol, and Bacleriol. xiv, p. 1, 1910 and H. G. Thornton, Ana. of Applied Biology, 1922, p. 265. 152 THE RATE OF GROWTH [ch. progression so long as nutriment is enough and to spare; that is to say, the rate of growth is proportional to the number present: dt ^ But in a test-tube colony the supply of nourishment is hmited, and the rate of multiphcation is bound to fall off. If a be the original concentration of food-stuff, it will have dwindled by time t to (a — y). The rate of growth will now be % = by{<^-y), which means that the rate of increase iS proportional to the number of organisms present, and to the concentration of the food-supply. It is Verhulst's case in a nutshell; the differential equation so indicated leads to an S-shaped curve which further experiment confirms; and Sach's "grand period of growth" is seen to accom- phsh itself*. The growth of yeast is studied in the everyday routine of a brewery. But the brewer is concerned only with the phase of unrestricted growth, and the rules of compound interest are all he needs, to find its rate or test its constancy. A population of 1360 yeast-cells grew to 3,550,000 in 35 hours: it had multiplied -2610 times. Accordingly, 1^8^51^^^^ = 0-098 = log 1-254. That is to say, the population had increased at the rate of 25-4 per cent, per hour, during the 35 hours. The time (^2) required to double the population is easily found : log 2 0-301 ^ ^^ , * The sigmoid curve illustrates a theorem which, obvious as it may seem, is of no small philosophical importance, to wit, that a body starting from rest must, in order to attain a certain velocity, pass through all intermediate velocities on its way. Galileo discusses this theorem, and attributes it to Plato: "Platone avendo per avventura avuto concetto non potere alcun mobil passare daUa quiete ad alcun determinate grado di velocita se non col passare per tutti gli altri gradi di velocita minori, etc."; Discorsi e dimostrazioni, ed. 1638, p. 254. Ill] A POPULATION OF YEAST 153 The duplication-period thus determined is known to brewers as the generation-time. Much care is taken to ensure the maximal growth. If the yeast sink to the bottom of the vat only its upper layers enjoy unstinted nutriment; a potent retardation-factor sets in, and the exponential phase of the growth-curve degenerates into a premature horizontal asymptote. Moreover, both the yeast and the bacteria differ in this respect from the typical (or perhaps only simpHfied) case of man, that they not only begin to suffer want as soon as there comes to be a deficiency of any one essential constituent of their food*, but they also produce things which are injurious to their own growth and in time fatal to their existence. Growth stops long before the food-supply is exhausted ; for it does so as soon as a certain balance is reached, depending on the kind or quahty of the yeast, between the alcohol and the sugar in the cellf. If we use the compound-interest law at all, we had better think of Nature's interest as being paid, not once a year nor once an hour as our elementary treatment of the yeast-population assumed, but continuously; and then we learn (in elementary algebra) that in time t, at rate r, a sum P increases to Pe'^\ or P< ^ ^o^^*. Applying this to the growth of our sample of 1360 yeast cells, we have PtIPo = 2610, log 2610 = 3-417, which, multipHed by the modulus 2-303 = 7-868. Dividing by n = 35, the number of hours, 7-868/35 = 0-225 = r. The rate, that is to say, is 22-5 per cent, per hour, continuous compound interest. It becomes a well-defined physiological constant, and we may call it, with V. H. Blackman, an index of efficiency. Our former result, for interest at hourly intervals, was 25-4 per * According to Liebig's "law of the minimum." t T. Carlson, Geschwindigkeit und Grosse'der Hefevermehrung, Biochem. Ztschr. Lvn, pp. 313-334, 1913; A. Slator, Journ. Chem. Soc. cxix, pp. 128-142, 1906; Biochem. Journ. vii, p. 198, 1913; 0. W. Richards, Ann. of Botany, xlh, pp. 271-283, 1928; Alf Klem, Hvalradets Skrifter, nr. 7, pp. 55-91, Oslo, 1933; Per Ottestad, ibid. pp. 30-54. For optimum conditions of temperature, nutriment, pB., etc. see Oscar W. Richards, Analysis of growth as illustrated by yeast. Cold Spring Harbour Symposia, ii, pp. 157-166, 1934. 154 THE RATE OF GROWTH [CH. cent. ; there is no great difference between such short intervals and actual continuity, but there is a deal of difference between continuous payment and payment (say) once a year*. Certain sunflowers (Helianihus) were found to grow as follows, in thirty-seven days: Compound interest rate (%) Weight (gm.) Continuous Discontinuous Giant sunflower Dwarf sunflower Seedling 0033 0035 Plant 17-33 14-81 Per day Per wk. Per day Per wk. 170 119 18-5 228% 16-4 114 17-7 214% When the yeast population is allowed to run its course, it yields a simple S-shaped curve ; and the curve of first differences derived rTi*-2*5 3 4 .■) 6 7 8 9 Estimated length at 1 year old 140 13-2 12-7 I2-0 12- 1 11-8 11-9 11-8 fact and observation that the rings alter in breadth as the fish goes on growing f ; that the oldest or innermost rings grow steadily narrower, while the outermost hardly change or even widen a little ; that- the relative breadths of successive rings alter accordingly; and it follows that when we try to trace the growth of a herring through its lifetime from its scales when it is old, the result is more or less misleading, and the values for the earher years are apt to be much too small. The whole subject is very difficult, as we might well expect it to be; and, I am only concerned to shew some small part of its difficulty J. While careful observations on the rate of growth of the higher animals are scanty, they shew so fax as they go that the general features of the phenomenon are much the same. Whether the animal be long-lived, as man or elephant, or short-lived hke horse § * From T. Emrys Watkin, The Drift Herring of the S.E. of Ireland, Rapports du Conseil pour V Exploration de la Mer, lxxxiv, p. 85, 1933. t Cf. {int. al.) Rosa M. Lee, Methods of age and growth determination in fishes by means of scales, Fishery Investigations, Dept. of Agr. and Fisheries, 1^20. X The copious literature of the subject is epitomised, so far, by Michael Graham, in Fishery lyivestigations (2), xi, No. 3, 1928. § There is a famous passage in Lucretius (v, 883) where he compares the course of life, or rate of growth, in the horse and his boyish master: Principio circum tribus actis impiger annis Floret equus, puer hautquaquam, etc. 184 THE RATE OF GROWTH [ch. or dog, it passes through the same phases of growth ; and, to quote Dr Johnson again, "whatsoever is formed for long duration arrives slowly to its maturity*." In all cases growth begins slowly; it attains a maximum velocity somewhat early in its course, and afterwards slows down (subject to temporary accelerations) towards a point where growth ceases altogether. But in cold-blooded animals, as fish or tortoises, the slowing down is greatly protracted, and the size of the creature would seem never to reach, but only to approach asymptotically, to a maximal limit. This, after all, is an important difference. Among certain still lower animals growth ceases early but Hfe goes on, and draws (apparently) to no predetermined end. So sea-anemones have been kept in captivity for sixty or even eighty years, have fed, flourished and borne offspring all the while, but have shewn no growth at all. The rate of growth of various parts or organs f That the several parts and organs of the body, within and without, have their own rates of growth can be amply demonstrated in the case of man, and illustrated also, but chiefly in regard to external form, in other animals. There lies herein an endless field for the study of correlation and of variability J. In the accompanying table I show, from some of Vierordt's data, the relative weights at various ages, compared with the weight at birth, of the entire body, and of brain, heart and liver; also the changing relation which each of these organs consequently bears, as time goes on, to the weight of the whole body (Fig. 47) §. * All of which is tantamount to a mere change of scale of the time-curve. f This phenomenon, of incrementum inequale, as opposed to incrementum in universum, was most carefully studied by Haller: "Incrementum inequale multis modis fit, ut aliae partes corporis aliis celerius increscant. Diximus hepar minus fieri, majorem pulmonem, minimum thymum, etc." (Elem. viii (2), p. 34.) X See {int. al.) A. Fischel, Variabilitat und Wachsthum des embryonalen Korpers, Morphol. Jahrb. xxiv, pp. 369-404, 1896; Oppel, Vergleickung des Entwickelungsgrades der Organe zu verschiedenen Entwickelungszeiten hei Wirhel- thieren, Jena, 1891; C. M. Jackson, Pre-natal growth of the human body and the relative growth of the various organs and parts, Amer. Journ. of Anat. ix, 1909; and of the albino rat, ibid, xv, 1913; L. A. Calkins, Growth of the human body in the foetal period. Rep. Amer. Assoc. Anat. 1921. For still more detailed measure- ments, see A. Arnold, Korperuntersuchungen an 1656 Leipziger Studenten, Ztschr. f. Konstitutionslehre, xv, pp. 43-113, 1929. § From Vierordt's Anatomische Tabellen, pp. 38, 39, much abbreviated. Ill] OF PARTS OR ORGANS 185 Weight of various organs, compared with the total weight of the human body' (male). (From Vierordfs Anatomische Tahellen) Percentage increase Percentage of body-wt. Wt. r -A. , , . ^ , A s Age (kgm.) Body Brain Heart Liver Brain Heart Live 0 31 10 10 10 10 12-3 0-76 4-6 1 9-0 2-9 2-5 1-8 2-4 10-5 0-46 3-7 2 110 3-6 2-7 2-2 30 9-3 0-47 3-9 3 125 40 2-9 2-8 3-4 8-9 0-52 3-9 4 140 4-5 3-5 31 4-2 9-5 0-53 4-2 5 15-9 51 3-3 3-9 3-8 7-9 0-51 3-4 6 17-8 5-7 3-6 3-6 4-3 7-6 0-48 3-5 7 19-7 6-4 3-5 3-9 4-9 6-8 0-47 3-5 8 21-6 70 3-6 40 4-6 6-4 0-44 30 9 23-5 7-6 3-7 4-6 50 61 0-46 30 10 25-2 8-1 3-7 5-4 5-9 5-6 0-51 3-3 11 27-0 8-7 3-6 6-0 61 50 0-52 3-2 12 290 9-4 3-8 (4.1) 6-2 4-9 (0-34) 30 13 331 10-7 3-9 7-0 7-3 45 0-50 31 14 371 120 3-4 9-2 8-4 3-5 0-58 3-2 15 41-2 13-3 3-9 8-5 9-2 3-6 0-48 32 16 45-9 14-8 3-8 9-8 9-5 3-2 0-51 30 17 49-7 160 3-7 10-6 10-5 2-8 0-51 30 18 53-9 17-4 3-7 10-3 10-7 2-6 0-46 2-8 19 57-6 18-6 3-7 11-4 11-6 2-4 0-51 2-9 20 59-5 19-2 3-8 12-9 110 2-4 0-51 2-6 21 61-2 19-7 3-7 12-5 11-5 2-3 0-49 2-7 22 62-9 20-3 3o 13-2 11-8 2-2 0-50 2-7 23 64-5 20-8 3-6 12-4 10-8 2-2 0-46 2-4 24 3-7 131 13-0 — — — 25 66-2 21-4 3-8 12-7 12-8 2-2 0-46 2-8 Fig. 47. 10 Age in years Relative growth in weight of brain, heart and body of man. From Quetelet's data (smoothed curves). 186 THE RATE OF GROWTH [ch. We see that neither brain, heart nor Hver keeps pace by any means with the growing weight of the whole; there must then be other parts of the fabric, probably the muscles and the bones, which increase more rapidly than the general average. Heart and liver grow nearly at the same rate, the liver keeping a little ahead to begin with, and the heart making up on it in the end; by the age of twenty-five both have multiplied their original weight at birth about thirteen times, but the body as a whole has multiplied by twenty-one. In contrast to these the brain has only multiplied its weight about three and three-quarter times, and shews but little increase since the child was four or five, and hardly any since it was eight years old. Man and the gorilla are born with brains much of a size; but the gorilla's brain stops growing very soon indeed, while the child's has four years of steady increase. The child's brain grows quicker than the gorilla's, but the great ape's body grows much quicker than the child's; at four years old the young gorilla has reached about 80 per cent, of his bodily stature, and the child's brain has reached about 80 per cent, of its full size. Even during foetal life, as well as afterwards, the relative weight of the brain keeps on declining. It is about 18 per cent, of the body- weight in the third month, 16 per cent, in the fourth, 14 per cent, in the fifth; and the ratio falls slowly till it comes to about 12 per cent, at birth, say 10 per cent, a year afterwards, and little more than 2 per cent, at twenty*. Many statistics indicate a further decrease of brain-weight, actual as well as relative. The fact has been doubted and denied ; but Raymond Pearl has shewn evidence of a slow decline continuing throughout adult life f. The latter part of the table shews the decreasing weights of the organs compared with the body as a whole: brain, which was 12 per cent, of the body- weight at birth, falling to 2 per cent, at five-and-twenty ; heart from 0-76 to 0-46 per cent.; liver from 4-6 to 2-78 per cent. The thyroid gland (as we know it in the rat) grows for a few weeks, and then diminishes during all the rest of the creature's Hfetime; even during the brief period of its own growth it is growing slower than the body as a whole. It is plain, then, that there is no simple and direct relation, holding *■ Cf. J. Ariens Kappers, Proc. K. Akad. Wetensch., Amsterdam, xxxix, No. 7, 1936. t R. Pearl, Variation and correlation in brain-weight, Biometrika, iv, pp. 13-104, 1905. Ill] OF PARTS OR ORGANS 187 good throughout life, between the size of the body and its organs; and the ratio of magnitude tends to change not only as the individual grows, but also with change of bodily size from one individual, one race, one species to another. In giant and pigmy breeds of rabbits, the organs have by no means the same ratio to the body- weight ; but if we choose individuals of the same weight, then the ratios tend to be identical, irrespective of breed*. The larger breeds of dogs are for the most part lighter and slenderer than the small, and the organs change their proportions with their size. The spleen keeps pace with the weight of the body ; but the liver, hke the brain, becomes relatively less. It falls from about 6 per cent, of the body-weight in little dogs to rather over 2 per cent, in a great hound f. The changing ratio with increasing magnitude is especially marked in the case of the brain, which constitutes (as we have just seen) an eighth of the body- weight at birth, and but one-fiftieth at twenty-five. This faUing ratio finds its parallel in comparative anatomy, in the general law that the larger the animal the smaller (relatively) is the brain J. A falhng ratio of brain- weight during life is seen in other animals. Max Weber § tells us that in the lion, at five weeks, four months, eleven months and lastly when full-grown, the brain represents the following fractions of the weight of the body : viz. i/18, 1/80, 1/184 and 1/546. And Kellicott has shewn that in the dogfish, while certain organs, e.g. pancreas and rectal gland, grow pari passu with the body, the brain grows in a diminishing ratio, to be represented (roughly) by a logarithmic curve ||. In the grown man, Raymond Pearl has shewn brain-weight to increase with the stature of the indix'idual and to decrease with his age, both in a straight-hne ratio, or linear regression, as the * R. C. Robb, Hereditary size-limitation in the rabbit, Journ. Exp. Biol, vi, 1929. t Cf. H. Vorsteher, Einfiuss d. Gesamtgrosse auf die Zusammensetzung des Kbrpers; Diss., Leipzig, 1923. X Oliver Goldsmith argues in his Animated Nature as follows, regarding the un- likelihood of dwarfs or giants: "Had man been born a dwarf, he could not have been a reasonable creature; for to that end, he must have a jolt head, and then he would not have body and blood enough to supply his brain with spirits; or if he had a small head, proportionable to his body, there would not be brain enough for conducting life. But it is still worse with giants, etc." § Die Sdugethiere, p. 117. II Amer. Journ. of Anatomy, viii, pp. 319-353, 1908. 188 THE RATE OF GROWTH [ch. statisticians call it. Thus the following wholly empirical equations give the required ratios in the case of Swedish males : Brain-weight (gms.) = 1487-8 — 1-94 x age, or = 915-06 + 2-86 X stature. In the two sexes, and in different races, these empirical constants will be greatly changed*; and Donaldson has further shewn that correlation between brain-weight and body-weight is much closer in the rat than in manf. Weight of • entire Weight of animal brain Ratios (gm.) (gm.) J, In r W w w:W y^iv:-^W M>»»=Fr Marmoset 335 12-5 1:26 1:20 w = 2-30 Spider monkey 1,845 126 15 M 1-56 Felis minuta 1,234 23-6 52 1-2 2-25 F. domestica 3,300 31 107 2-4 2-36 Leopard 27,700 164 168 12 200 Lion 119,500 219 546 1-3 217 Dik-dik 4,575 37 124 2-7 2-30 Steinbok 8,600 49-5 173 2-9 2-32 Impala 37,900 148-5 255 2-75 211 Wildebeest 212,200 443 479 2-8 201 Zebra 255,000 541 472 2-7 1-98 »> 297,000 555 536 2-8 200 Rhinoceros 765,000 655 1170 3-6 209 Elephant 3,048,000 5,430 560 20 1-74 Whale (Globiocephalus) 1,000,000 2,511 400 20 1-77 Mean 2-23 206 Brandt, a very philosophical anatomist, argued some seventy years ago that the brain, being essentially a hollow structure, a surface rather than a mass, ought to be equated with the surface rather than the mass of the animal. This we may do by taking the square-root of the brain-weight and the cube-root of the body- weight; and while the ratios so obtained do not point to equality, they do tend to constancy, especially if we hmit our comparison to similar or related animals. Or we may vary the method, and ask (as Dubois has done) to what power the brain-weight must be raised * Biometrika, iv, pp. 13-105, 1904. t H. H. Donaldson, A comparison of the white rat with man, etc., Boas Memorial Volume, New York, 1906, pp. 5-26. Ill] OF PARTS OR ORGANS 189 to equal the body-weight; and here again we find the same tendency towards uniformity*. The converse to the unequal growth of organs is found in their unequal loss of weight under starvation. Chossat found, in a well-known experiment, that a starved pigeon had lost 93 per cent, of its fat, about 70 per cent, of hver and spleen, 40 per cent, of its muscles, and only 2 per cent, of brain and nervous tissues f. The salmon spends many' weeks in the river before spawning, without taking food. The muscles waste enormously, but the reproductive bodies continue to grow. As the internal organs of the body grow at different rates, so that their ratios one to another alter as time goes on, so is it with those hnear dimensions whose inconstant ratios constitute the changing form and proportions of the body. In one of Quetelet's tables he shews the span of the outstretched arms from year to year, com- pared with the vertical stature. It happens that height and span are so nearly co-equal in man that direct comparison means little ; but the ratio of span to height (Fig. 48) undergoes a significant and remarkable change. The man grows faster in stretch of arms than he does in height, and span which was less at birth than stature b}'' about I per cent, exceeds it by about 4 per cent, at the age of twenty. Quetelet's data are few for later years, but it is clear enough that span goes on increasing in proportion to stature. How far this is due to actual growth of the arms and how far to increasing breadth of the chest is another story, and is not yet ascertained. * Cf. A. Brandt, Sur le rapport du poids du cerveau a celui du corps chez diflFerents aniraaux, Bull, de la Soc. Imp. des naturalistes de Moscou, XL, p. 525, 1867; J. Baillanger, De I'etendu de la surface du cerveau, Ann. Med. Psychol. XVII, p. 1, 1853; Th. van Bischoff, Das Hirngewicht des Menschen, Bonn, 1880 (170 pp.), cf. Biol. Centralhl. i, pp. 531-541, 1881; E. Dubois, On the relation between the quantity of brain and the size of the body, Proc. K. Akad. Wetensch., Amsterdam, xvi, 1913. Also, Th. Ziehen,^ Maszverhaltnisse des Gehirns, in Bardeleben's Handh. d. Anatomie des Menschen; P. Warneke, Gehirn u. Korper- gewichtsbestimmungen bei Saugern, Journ. f. Psychol, u. Neurol, xiii, pp. 355-403, 1909; B. Klatt, Studien zum Domestikationsproblem, Bibliotheca genetica, ii, 1921; etc. The case of the heart is somewhat analogous ; see Parrot, Zooi. Ja^r&. (System.), vii, 1894; Piatt, in Biol. Centralhl. xxxix, p. 406, 1919. t C. Chossat, Recherches sur I'inanition, Mem. Acad, des Sci., Paris, 1843, p. 438. 190 THE RATE OF GROWTH [CH. The growth-rates of head and body differ still more; for the height of the head is no more than doubled, but stature is trebled, Height of the head in man at various ages * (After Quetelet, p. 207, abbreviated) Men Women Stature Head Stature Head Age m. ra. Ratio m. m. Rat Birth 0-50 Oil 4-5 0-49 Oil 4-4 1 year 0-70 015 4-5 0-69 015 4-5 2 years 0-79 017 4-6 0-78 0-17 4-5 3 „ 0-86 0-18 4-7 0-85 0-18 4-7 5 „ 0-99 019 51 0-97 019 51 10 „ 1-27 0-21 6-2 1-25 0-20 6-2 20 „ 1-51 0-22 70 1-49 0-21 7 0 25 ,. 1-67 0-23 7-3 1-57 0-22 71 30 „ 1-69 0-23 7-4 1-58 0-22 71 40 „ 1-69 0-23 7-4 1-58 0-22 71 Fig. 48. Ratio of stature in man, to span of outstretched arms. From Quetelet's data. between infancy and manhood. Diirer studied and illustrated this remarkable phenomenon, and the difference which accompanies and * A smooth curve, very similar to this, is given by Karl Pearson for the growth in "auricular height" of the girl's head, in Biometrika, ni, p. 141, 1904. Ill] OF PARTS OR ORGANS 191 results from it in the bodily form of the child and the man is easy to see. The following table shews the relative sizes of certain parts and organs of a young trout during its most rapid development; and so illustrates in a simple way the varying growth-rates in different parts of the body*. It would not be difficult, from a picture of the little trout at any one of these stages, to draw its approximate form at any other by the help of the numerical data here set forth. In like manner a herring's head and tail grow longer, the parts betw^een grow relatively less, and the fins change their places a httle; the same changes take place with their specific differences in related fishes, and herring, sprat and pilchard owe their specific characters to their rates of growth or modes of increment f. Trout (Salmo fario) ; proportionate growth of various organs (From Jenkinson's data) Days Total 1st Ventral 2nd Tail Breadth old length Eye Head dorsal fin dorsal fin • of tail 40 100 100 100 100 100 100 100 100 63 130 129 148 149 149 108 174 156 77 155 147 189 (204) (194) 139 258 220 92 173 179 220 (193) (182) 155 308 272 106 195 193 243 173 165 173 337 288 Sachs studied the same phenomenon in plants, after a method in use by Stephen Hales a hundred and fifty years before. On the growing root of a bean ten narrow zones were marked off, starting from the apex, each zone a millimetre long. After twenty-four hours' growth (at a given temperature) the whole ten zones had grown from 10 to 33 mm., but the several zones had grown very unequally, as shewn in the annexed table J (p. 192): * Cf. J. W. Jenkinson, Growth, variability "and correlation in young trout, Bio- metrika, vin, pp. 444-466. 1912. I Cf. E. Ford, On the transition from larval to adolescent herring, Journ. Mar. Biol. Assoc. XVI, p. 723; xviii, p. 977, 1930-31. 8o also in larval eels, tail and body grow at different rates, which rates differ in different species; cf. Johannes Schmidt, Meddel. Kommlss. Havsiindersok. 1916; L. Bertin, Bull. Zool. France, 1926, p. 327. I From Sachs's Textbook of Botany, 1882, p. 820. 192 THE RATE OF GROWTH [CH. Graded growth of bean-root Increment Increment Zone mm. Zone mm. Apex 1-5 6th 1-3 2nd 5-8 7th 0-5 3rd 8-2 8th 0-3 4th 3-5 9th 0-2 5th 1-6 10th 01 "... I marked in the same manner as the Vine, young Honeysuckle shoots, etc. . . . ; and I found in them all a gradual scale of unequal extensions, those parts extending most which were tenderest," Vegetable Staticks, Exp. cxxiii. The lengths attained by the successive zones He very nearly on a smooth curve or gradient; for a certain law, or principle of continuity, connects and governs the growth-rates along the growing axis. This curve has its family likeness to those differential 10 2 3 4 5 6 7 8 Zones Fig. 49. Rate of growth of bean-root, in successive zones of 1 mm. each, beginning at the tip. curves which we have already studied, in which rate of growth was plotted against time, as here it is plotted against successive spatial intervals of a growing structure ; and its general features are those of a curve, a skew curve, of error. Had the several growth-rates been transverse to the axis, instead of being longitudinal and parallel to it, they would have given us a leaf-shaped structure, of which our curve would represent the outline on either side; or again, if growth had been symmetrical about the axis, it might have given us a turnip-shaped soHd of revolution. There is always an easy passage from growth to form. Ill] CONCERNING GRADIENTS 193 A like problem occurs when we deal with rates of growth in successive natural internodes; and we may then pass from the actual growth of the internodes to the varying number of leaves which they successively produce. Where we have whorls of leaves at each node, as in Equisetum or in many water-weeds, then the problem is simphfied; and one such case has been studied by Ra3rmond Pearl*. In Ceratophyllum the mean number of leaves increases with each successive whorl, but the rate of increase diminishes from whorl to whorl as we ascend. On the main stem the rate of change is very slow; but in the small twigs, or tertiary branches, it becomes rapid, as we see from the following abbreviated table : Number of leaves per whorl on the tertiary branches of Ceratophyllum Order of whorl ... 1 2 3 4 5 6 Mean no. of leaves Smoothed no. 6-55 6-5 8-07 8-0 1 9-00 90 9-20 9-5 9-75 9-8 10-00 100 Raymond Pearl gives a logarithmic formula to fit the case; but the main point is that the numbers form a graded series, and can be plotted as a simple curve. In short, a large part of the morphology of the organism depends on the fact that there is not only an average, or aggregate, rate of growth common to the whole, but also a gradation of rate from one part to another, tending towards a specific rate characteristic of each part or organ. The least change in the ratio, one to another, of these partial or locahsed rates of growth will soon be manifested in more and more striking differences of form; and this is as much as to say that the time-element, which is imphcit in the idea oigrowthy can never (or very seldom) be wholly neglected in our consideration of form I . A flowering spray of Montbretia or lily-of-the-valley exemplifies a growth-gradient, after a simple fashion of its own. Along the * On variation and differentiation in Ceratophylly,m, Carnegie Inst. Publications, No. 58, 1907; see p. 87. t Herein lies the easy answer to a contention raised by Bergson, an(J to which he ascribes much 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). 194 THE RATE OF GROWTH [ch. stalk the growth-rate falls away ; the florets are of descending age, from flower tO' bud: their graded differences of age lead to an exquisite gradation. of size and form; the time-interval between one and another, or the "space-time relation" between them all, gives a peculiar quality — ^we may call it phase-beauty — to the whole. A clump of reeds or rushes shews this same phase-beauty, and so do the waves on a cornfield or on the sea. A jet of water is not much, but a fountain becomes a beautiful thing, and the play of many fountains is an enchantment at Versailles. On the weight-length coefficient, or ponderal itidex So much for the visible changes of form which accompany advancing age, and are brought about by a diversity of rates of growth at successive points or in different directions. But it often happens that an animal's change of form may be so gradual as to pass unnoticed, and even careful measurement of such small changes becomes difficult and uncertain. Sometimes one dimension is easily determined, but others are hard to measure with the same accuracy. The length of a fish is easily measured ; but the breadth and depth of plaice or haddock are vaguer and more uncertain. We may then make use of that ratio of weight to length which we spoke of in the last chapter: viz. that W oc L^, or W ^ kL^, or W/L^ = k, where k, the "ponderal index," is a constant to be determined for each particular case*. We speak of this /; as a "constant," with a mean value specific to each species of animal and dependent on the bodily proportions or form of that animal; yet inasmuch as the animal is continually apt to change its bodily proportions during life, k also is continually subject to change, and is indeed a very delicate index of such * This relation, and how important it is, were clearly recognised by Herbert Spencer in his Recent Discussions in Science, etc., 1871. The formula has been X y/w often, and often independently, employed: first perhaps in the form — y- x 100, by R. Livi, L'indice ponderale, o rapporto tra la statura e il peso, Atti Soc. Romamt Antropologica, v, 1897. Values of k for man and many animals are given by H. Przibram, in Form und Formel, 1922. On its use as an index to the condition or habit of body of an individual, see von Rhode, in Abderhalden's Arbeitsmethoden, IX, 4. The constant k might be called, more strictly, ki, leaving kf, and k,i for the similar constants to be deri-ved from the breadth and depth of the fish. Ill THE PONDERAL INDEX 195 progressive changes : delicate — because our measurements of length are very accurate on the whole, and weighing is a still more dehcate method of comparison. Thus, in the case of plaice, when we deal with mean values for large numbers and with samples so far "homogeneous" that they are taken at one place and time, we find that k is by no means constant, but varies, and varies in an orderly way, with increasing size of the fish. The phenomenon is unexpectedly complex, much more so than I was aware of when I first wrote this book. Fig. 50 120 — .2 o 110 § 100 ^ U 90 ' 1 T — :—t r -j— T- 1 I 1 ' _ December ^ ^ — — ~-y March ^ ^'^"^^■"•..•^ — , , 1 1 1 1 1 1 1 1 1 1 \ , 1 1 1 1 1 1 1 22 25 40 30 35 Length (cm.) Fig. 50. Changes in the weight-length coefficient of plaice with increasing size; from March and December samples. shews the weight-length coefficient, or ponderal index, in two large samples, one taken in the month of March, the other in December. In the latter sample k increases steadily as the plaice grow from about 25 to 40 cm. long; weight, that is to say, increases more rapidly than the cube of the length, and it follows that length itself is increasing less rapidly than some other linear dimension. In other words, the plaice grow thicker, or bulkier, with length and age. The other sample, taken in the month of March, is curiously different; for now k rises to a maximum when the fish are some- where about 30 cm. long, and then decHnes slowly with further increase in size of the fish; and k itself is less in March than in December, the discrepancy being slight in the small fish and great in the large. , The "point of inflection" at 30 cm. or thereby marks 196 THE RATE "OF GROWTH [ch. an epoch in the fish's life; it is about the size when sexual maturity begins, or at least near enough to suggest a connection between the two phenomena*. A step towards further investigation would be to determine k for the two sexes separately, and to see whether or no the point of inflection occurs, as maturity is known to be reached, at a smaller size in the male. This d'Ancona has done, not for the plaice but for the shad {Alosa finta). He finds that the males are the first to reach maturity, first to shew a retardation of the rate of growth, first to reach a maximal value of th^ ponderal index, and in all probability the first to diet • Again we may enquire whether, or how, k varies with the time of year; and this torrelation leads to a striking result J. For the ponderal index fluctuates periodically with the seasons, falling steeply to a minimum in March or April, and rising slowly to an annual maximum in December (Fig. 51) §. The main and obvious explanation hes in the process of spawning, the rapid loss of weight thereby, and the slow subsequent rebuilding* of the reproductive tissues; whence it follows that, without ever seeing the fish spawn, and without ever dissecting one to see the state of its reproductive system, we may by this statistical method ascertain its spawning season, and determine the beginning and end thereof with con- siderable accuracy. But all the while a similar fluctuation, of much less amphtude, is to be found in voung plaice before the spawning age; whence we learn that the* fluctuation is not only due to shedding and replacement of spawn, but in part also to seasonal changes in appetite and general condition. Returning to our former instance, we now see that the March and December samples of plaice, which shewed such discrepant variations of the ponderal index with increasing size, happen to ♦ The carp shews still more striking changes than does the plaice in the weight - length coefl&cient: in other words, still greater changes in bodily shape with advancing age and increasing size; cf. P. H. Stnithers, The Champlain Watershed, Albany, New York, 1930. ■j- U. d'Ancona, II problema dell' accrescimento dei pesci, etc., Mem. R. Acad, dei Lincei (6), n, pp. 497-540, 1928. J Cf. Lammel, Ueber periodische Variationen in Organismen, Biol. Centralbl. xxn, pp. 368-376, 1903. § When we restrict ourselves, for simphcity'a sake, to fish of one particular size, we need not determine the values of k, for changes in weight are obvious enough; but when we have small numbers and various sizes to deal with, the determination of k helps very much. Ill THE PONDERAL INDEX 197 coincide with the beginning and end of the spawning season; the fish were full of spawn in December, but spent and lean in March. The weight-length ratio was, of necessity, higher at the former season; and the faUing-off in condition, and in bulk, which the March sample indicates, is more and more pronounced in the larger and therefore more heavily spawn-laden fish. JFMAMJJASONDJ Fig. 51. Periodic annual change in the weight-length ratio of plaice. Periodi-c relation of weight to length in plaice of 55 cm. long Average weight WJL^ W/L^ (smoothed) decigrams Jan. 204 1-23 116 Feb. 174 104 108 March 162 0-97 0-99 April 159 0-95 0-97 May 162 0-98 0-98 June 171 103 101 July 169 101 104 August 178 1-07 104 Sept. 173 104 Ml Oct. 203 1-22 116 Nov. 203 1-22 1-21 Dec. 200 1-20 1-22 Mean 180 1-08 198 THE RATE OF GROWTH [ch. Plaice caught in a certain area, March 1907 and December 1905. Variation of k, the weight-length coefficient, with size March sample December sample A r Do. Do. cm. gm. WjL^ smoothed gm. W/L^ smoothed 23 113 0-93 24 128 0-93 0-94 — ' — 25 152 0-97 0-96 26 178 0-96 0-98 177 101 27 193 0-98 0-99 209 106 1-06 28 221 101 1-00 241 MO 1-08 29 250 102 101 264 108 109 30 271 1-00 101 294 109 109 31 300 101 100 325 109 110 32 328 100 100 366 112 112 33 354 0-99 0-99 410 114 113 34 384 0-98 0-98 449 M4 115 35 419 0-98 0-98 501 117 117 36 454 0-97 0-97 556 119 117 37 492 0-95 0-96 589 116 118 38 529 0-96 0-96 652 119 119 39 564 0-95 0-95 719 1-21 1-22 40 614 096 0-95 809 1-26 — 41 647 0-94 0-94 — 42 679 0-92 0-93 — — — 43 732 0-92 0-93 — • — — H 800 0-94 0-94 — — — 4l5 875 0-96 — — — — These weights and measurements of plaice are taken from the Department of Agriculture and Fisheries' Plaice-Report, i, pp. 65, 107, 1908; ii, p. 92, 1909. Japanese goldfish* are exposed to a much wider range of tem- perature than our plaice are called on to endure ; they hibernate in winter and feed greedily in the heat of summer. Their weight is low in winter but rises in early spring, it falls as low as ever at the height of the spawning season in the month of May; so for one weight-length fluctuation which the plaice has, the goldfish has a twofold cycle in the year. The index reaches its second and higher maximum in August, and falls thereafter till the end of the year. That it should begin to fall so soon, and fall so quickly, merely means that late autumn is a time of groipth ; the fish are not losing weight, but growing longer f. * Cf. Kichiro Sasaki, Tohoku Soi. Reports (4), i, pp. 239-260, 1926. t Much has been written on the weight-length index in fishes. See (int. al.) A. Meek, The growth of flatfish, Northumberland Sea Fisheries Ctee, 1905, p. 58; W, J. Crozier, Correlations of weight, length, etc, in the weakfish, Cynoscion hi] the PONDERAL index 199 It is the rule in fishes and other cold-blooded vertebrates that growth is asymptotic and size indeterminate, while in the warm- blooded growth comes, sooner or later, to an end. But the characteristic form is established earlier in the former case, and changes less, save for the minor fluctuations we have spoken of. In the higher animals, such as ourselves, the whole course of life is attended by constant alteration and modification of form; and Fig. 52. The ponderal index, or weight-length coefficient, in man. From Quetelet's data. we may use our weight-length formula, or ponderal index, to illus- trate (for instance) the changing relation between height and weight in boyhood, of which we spoke before (Fig. 52). regalis, Bull. U.S. Bureau of Fisheries, xxxni, pp. 141-147, 1913; Selig Hecht, Form and growth in fishes, Journ. of Morphology, xxvii, pp. 379-400, 1916; J. Johnstone (Plaice), Trans. Liverpool Biolog. Soc. xxv, pp. 186-224, 1911; J. J. Tesch (Eel), Journ. du Conseil, iii, 1927; Frances N. Clark (Sardine), Calif. Fish. Bulletin, No. 19, 1928 (with full bibliography). For a discussion on statistical lines, apart from any assumptions such as the "law of the cubes," see G. Buncker, Korrelation zwischen Lange u. Gewicht, etc., Wissensch. Meerestmtersuch. Helgoland, XV, pp. 1-26, 1923. THE RATE OF GROWTH [CH. The weight-length coefficient, or ponderal index, k, in young Belgians (From Quetelefs figures) (years) WJL^ Age (years) WjL^ 0 2-55 10 1-25 1 2-92 11 118 2 2-34 12 1-23 3 2-08 13 1-29 4 1-87 14 1-23 5 1-72 15 1-23 6 1-56 16 1-28 7 1-48 20 1-30 8 1-39 25 1-36 9 1-29 The infant is plump and chubby/ and the ponderal index is at its highest at a year old. As the boy grows, it is in stature that he does so most of all; his ponderal index falls continually, till the growing years are over, and the lad "fills out" and grows stouter again. During prenatal Ufe the index varied httle, and less than we might suppose : Relation between length and weight of the human foetus (From Scammon's data) Length Weight cm. gm. 7-7 13 12-3 41 17-3 115 223 239 27-2 405 32-3 750 37-2 1163 42-2 1758 46-9 2389 51-7 3205 2-9 2-2 2-2 2-2 io 2-2 2-3 2-3 2-3 2-3 As a further illustration of the rate of growth, and of unequal growth in various directions, we have figures for the ox, extending over the first three years of the animal's life, and giving (1) the weight of the animal, month by month, (2) the length of the back, from occiput to tail, and (3) the height to the withers. To these I have added (4) the ratio of length to height, (5) the weight-length coefficient, k, and (6) a similar coefficient, or index-number, k' , for Ill] THE PONDERAL INDEX 201 the height, of the animal. All these ratios change as time goes on. The ratio of length to height increases, at first considerably, for the legs seem disproportionately long at birth in the ox, as in other Relations between the weight and certain linear dimensions of the ox {Data from i Jo^nevin*y abbreviated) Length Age Weight of back Height lonths kgm. m. m. L/H k = WjL^ k' = Wli 0 37 0-78 0-70 Ml 0-78 1-08 1 55 0-94 0-77 1-22 0-66 1-21 2 86 1-09 0-85 1-28 0-67 1-41 3 121 1-21 0-94 1-28 0-69 1-46 4 150 1-31 0-95 1-38 0-66 1-75 5 179 1-40 104 1-35 0-65 1-60 6 210 1-48 109 1-36 0-64 1-64 7 247 1-52 112 1-36 0-70 1-75 8 267 1-58 115 1-38 0-68 1-79 9 283 1-62 116 1-39 0-66 1-80 10 304 1-65 119 1-39 0-68 1-79 11 328 1-69 1-22 1-39 0-67 1-79 12 351 1-74 1-24 1-40 0-67 1-85 13 375 1-77 1-25 1-41 0-68 1-90 14 391 1-79 1-26 1-41 0-69 1-94 15 406 1-80 1-27 1-42 0-69 1-98 16 418 1-81 1-28 1-42 0-70 2-09 17 424 1-83 1-29 1-42 0-69 1-97 18 424 1-86 1-30 1-43 0-66 1-94 19 428 1-88 1-31 1-44 0-65 1-92 20 438 1-88 1-31 1-44 0-66 1-94 21 448 1-89 1-32 1-43 0-66 1-94 22 464 1-90 1-33 1-43 0-68 1-96 23 481 1-91 1-35 1-42 0-69 1-98 24 501 1-91 1-35 1-42 .0-71 203 25 521 1-92 1-36 1-41 0-74 2-08 26 534 1-92 1-36 1-41 0-75 212 27 547 1-93 1-36 1-41 0-76 2-16 28 555 1-93 1-36 1-41 0-77 219 29 562 1-93 1-36 1-41 0-78 2-22 30 586 1-95 1-38 1-41 0-79 2-22 31 611 1-97 1-40 1-40 0-80 2-21 32 626 1-98 1-42 1-40 0-80 219 33 641 200 1-44 1-39 0-81 216 34 656 201 1-45 1-38 0-81 213 ungulate animals; but this ratio reaches its maximum and falls off a little during the third year : so indicating that the beast is growing more in height than length, at a time when growth in both * Ch. Cornevin, ^^tudes sur la croissanee, Arch, de Physiol, norm, et pathol. (5), iv, p. 477, 1892. Cf. also R. Gartner, Ueber das W'achstum d. Tiere, Landxvirtsch. Jahresher. lvii, p. 707, 1922. 202 THE RATE OF GROWTH [CH. dimensions is nearly over*. The ratio W/H^ increases steadily, and at three years old is double what it was at birth. It is the most variable of the' three ratios; and it so illustrates the some- what obvious but not unimportant fact that k varies most for the dimension which varies least, or grows most uniformly; in other words, that the values of k, as determined at successive epochs for any one dimension, are a measure of the variability of the other two. The same ponderal index serves as an index of "build," or bodily proportion; and its mean values have been determined for various ages and f6r many races of mankind. Within one and the ie>u 1 • r- • y y- 150 y y X« 140 my X' y 130 ^ 1 1 1 1 , 1 1 1 \ 60 70 65 Height (inches) Fig. 53. Ratio of height to weight in man. From Goringe's data. same race it varies with stature ; for tall men, and boys too, are apt to be slender and lean, and short ones to be thickset and strong. And so much does the weight-length ratio change with build or stature that, in the following table of mean heights and weights of men between five and six feet high, it will be seen that weight, instead of varying as the cube of the height, is (within the hmits shewn) in nearly simple linear relation to it (Fig. 53) f. * As a matter of fact, the data shew that the animal grows under 7 per cent, in length, but over 11 per cent, in height, between the twentieth and the thirtieth month of its age. t Had the weights varied as the cube of the height, the tallest men should have weighed close on 200 lb., instead of 160 lb. Ill] OF WHALES AND ELEPHANTS 203 Ratio of height to weight in man * No. of Height Weight instances in. lb. WIH Wjm 59 60-5 125 207 5-62 118 61-5 129 213 5-55- 220 62-5 133 213 • 5-45 285 63-5 136 214 5-30 327 64-5 139 215 519 386 65-5 143 218 5-09 346 66-5 146 2-20 4-97 289 67-5 153 2-27 4-96 220 68-5 151 2-20 4-71 116 69-5 156 224 4-64 58 70-5 160 2-27 4-57 The same index may be used as a measure of the condition, even of the quaUty, of an animal; three Burmese elephants had the following heights, weights, and reputations!: Height Weight WjH^ A B C 7 ft. lOi in. 8 1 7 5 7,511 lb. 7,216 4,756 1-54 1-36 115 A famous elephant A good elephant A weak, poor elephant But a great African elephant, 10 ft. 10 in. high, weighed 14,640 Ib.J : whence the weight-height coefficient was no more than 1-15. That is to say, the African elephant is considerably taller than the Indian, and the weight-height ratio is correspondingly less. Lastly, by means of the same index we may judge, to a first rough approximation, the weight of a large animal such as a whale, where weighing is out of the question. Sigurd Rusting has given us many measurements, and many foetal weights, from the Antarctic whale-fishery: among which, choosing at random, we find that a certain foetus of the blue whale, or Sibbald's rorqual, measured 4 ft. 6 in. long, and weighed 23 kilos, or say 46 lb. A whale of the same kind, 45 ft. long, should then weigh 46 x 10^ lb., or about 23 tons; and one of 90 ft., 23 x 2^ tons, or over 180 tons. Again in seven young unborn whales, measuring from 39 to 54 inches and weighing from 10 to 23 kilos, the mean value of the index was found * Data from Sir C. Goringe, The English Convict, H.M. Stationery Office, 1913. See also J. A, Harris and others. The, Measurement of Man, Minnesota, 1930, p. 41. t Data from A. J. Milroy, On the management of elephants, Shillong, 1921. % D. P. Quireng, in Qrcrwth, iii, p. 9, 1939. 204 THE RATE OF GROWTH [ch. to be 15-2, in gramme-inches. From this we calculate the weight of the great rorqual, as follows : 1 5*2 X SOO^ At 25 ft., or 300 inches, W = -— — = 4,100,000 g. lUu = 4,100 kg. = 4 tons, nearly. At 50 ft., TF = 4 X 23 tons = 32 tons. 100 ft. = 32 X 23 tons = 256 tons. 106 tons (the largest known) W = 305 tons, nearly. The two independent estimates are in close agreement. Of surface and volume While the weight-length relation is of especial importance, and is wellnigh fundamental to the understanding of growth and form and magnitude, the corresponding relation of surface-area to weight or volume has in certain cases an interest of its own. At the surface of an animal heat is lost, evaporation takes place, and oxygen may be taken in, all in due proportion as near as may be to the bulk of the animal; and again the bird's wing is a surface, the area of which must be in due proportion to the size of the bird. In hollow organs, such as heart or stomach, area is the important thing rather than weight or mass; and we have seen how the brain, an organ not obviously but essentially and developmentally hollow, tends to shew its due proportions when reckoned as a surface in comparison with the creature's mass. Surface cannot keep pace with increasing volume in bodies of similar form; wing-area does not and cannot long keep pace with the bird's increasing bulk and weight, and this is enough of itself to set limits to the size of the flying bird. It is the ratio between square-root-of-surface and cube-root-of-volume which should, in theory, remain constant; but as a matter of fact this ratio varies (up to a certain extent) with the circumstances, and in the case of the bird's wing with varying modes and capabilities of _flight. The owl, with his silent, effortless flight, capable of short swift spurts of attack, has the largest spread of wings of all ; the kite outstrips the other hawks in spread of wing, in soaring, and perhaps in speed. Ill] OF^SURFACE AND VOLUME 205 Stork and seagull have a great expanse of wing; but other skilled and speedy fliers have long narrow wings rather than large ones. The peregrine has less wing-area than the goshawk or the kestrel; the swift and the swallow have less than the lark. Mean ratio, VS/^^^W, between wing-area and weight of birds {From Mouillard's data) Ratio Owls 1 1 species 2-2 Hawks 7 Gulls 1 Waders 3 Petrels 2 Plovers 3 Passeres 4 1-3 Ducks 2 1-2 To measure the length of an animal is easy, to weigh it is easier still, but to estimate its surface-area is another thing. Hence we know but little of the surface-weight ratios of animals, and what we know is apt to be uncertain and discrepant. Nevertheless, such data as we possess average down to mean values which are more uniform than we might expect*. Mean ratio, VS/VW, in various animals [cm. gm. units) Ape 11-8 I Sheep (shorn) 8 Man 11 1 Snake 12-5 Dog 10-11 1 Frog 10-6 Cat, horse 10 Birds 10 Rabbit 9-75 1 Tortoise 10 Cow, pig, rat 9 A further note on unequal growth, or heterogony An organism is so complex a thing, and growth so complex a phenomenon, that for growth to be so uniform and constant in all the parts as to keep the whole shape unchanged would indeed be an unlikely and an unusual circumstance. Rates ''^ary, propor- tions change, and the whole configuration alters accordingly. In so humble a creature as a medusoid,. manubrium and disc grow at different rates, and certain sectors of the disc faster than others, as when the little Ephyra-lfnya. "develops" into the great Aurdia- jellyfish. Many fishes grow from youth to age with no visible, * From Fr. G. Benedict, Oberflachenbestimmung verschiedener Tiergattungen, Ergebnisae d. Pkysiologiey xxxvi, pp. 300-346, 1934 (with copious bibliography). 206 THE RATE OF GROWTH [ch. hardly a measurable, change of form*; but the shapes and looks of man and woman go on changing long after the growing age is over, even all their lives long. A centipede has its many pairs of legs alike, to all intents and purposes; they begin alike and grow uniformly. But a lobster has his great claws a,nd his small, his lesser legs, his swimmerets and the broad flaps of his tail ; all these begin ahke, and diverse rates of growth make up the difference between them. Moreover, we may sometimes watch a single Umb growing to an unusual size, perhaps in one sex and not in the other, perhaps on one side and not on the other side of the body: such are the "horns," or mandibles, of the stag-beetle, only conspicuous in the male, and the great unsymmetrical claws of the lobster, or of that extreme case the httle fiddler-crab {Uca pugnax). For such well-marked cases of differential growth-ratio between one part and another, JuHan Huxley has introduced the term heterogonyf. Of the fiddler-crabs some four hundred males were weighed, in twenty-five graded samples all nearly of a size, and the weights of the great claw and of the rest of the body recorded separately. To begin with the great claw was about 8 per cent., and at the end about 38 per cent., of the total weight of the unmutilated body. In the female the claw weighs about 8 per cent, of the whole from beginning to end; and this contrast marks the disproportionate, or heterogonic, rate of growth in the male. We know nothing about the actual rate of growth of either body or claw, we cannot plot either against time; but we know the relative proportions, or relative rates of growth of the two parts of the animal, and this is all that matters meanwhile. In Fig. 54, we have set off the successive weights of the body as abscissae, up to 700 mgm., or about one-third of its weight in the adult animal; and the ordinates represent the corresponding weights of the claw. We see that the ratio between the two magnitudes follows a curve, apparently an exponential curve; it does in fact (as Huxley has shewn) follow a compound * Cf, S. Hecht, Form and growth in fishes, Journ. Morphology, xxvii, pp. 379- 400, 1916; F. S. and D. W. Haramett, Proportional length-growth of garfish (Lepidosteus), Growth, in, pp. 197-209, 1939. t See Problems of Relative Growth, 1932, and many papers quoted therein. The term, as Huxley, tells us, had been used by Pezard; but it Jiad been used, in another sense, by Rolleston long before to mean an alternation of generations, or production of offspring dissimilar to the parent. ^73 )([M,j^ J III] OF UNEQUAL GROWTH 207 interest law, which (calhng y and x the weights of the claw and of the rest of the body) may be expressed by the usual formula for compound interest, y = bx^, or log y = \ogb + k log x; and the coefficients (6 and Jc) work out in the case of the fiddler-crab, to begin with, at 2/ = 0-0073 xi-«2. . 300 100 200 300 400 500 600 700 Weight of body (mgm.) Eig. 54. Relative weights of body and claw in the fiddler- crab { Uca 'pugnax). But after a certain age, or certain size, th^e coefficients no longer hold, and new coefficients have to be found. Whether or no, the formula is mathematical rather than biological; there is a lack of either biological or physical significance in a growth-rate which happens to stand, during part of an animaFs fife, at 62 per cent, compound interest. Julian Huxley holds, and many hoki with him, that the exponential or logarithmic formula, or the compound-interest .law, is of general application to cases of differential growth-rates. I do not find it to be so : any more than we have found organ, organism or population to increase by compound interest or geometrical progression, save 208 THE feATE OF GROWTH [ch. under exceptional circumstances and in transient phase. Undoubtedly many of Huxley's instances shew increase by compound interest, during a phase of rapid and unstinted growth; but I find many others following a simple-interest rather than a compound-interest law. Relative weights of claw and body in fiddler-crabs (Uca pugnax). (Data abbreviated from Huxley, Problems of Relative Growth, p. 12) rt. of body less claw Wt. of Ratio Wt. of Wt. of Ratio (mgm.) claw % body claw % 58 5 8-6 618 243 39-3 80 9 11-2 743 319 42-9 109 14 12-8 872 418 47-9 156 25 160 983 461 46-9 200 38 190 1080 537 49-7 238 53 22-3 1166 594 50-9 270 59 21-9 1212 617 50-9 300 78 260 1299 670 51-6 355 105 29-7 1363 699 51-3 420 135 321 1449 773 53-7 470 165 351 1808 1009 55-8 536 196 36-6 2233 1380 61-7 In the common stag-beetle {Lv^anus cervus) we have the following measurements of mandible and elytron or wing-case: which two organs make up the bulk of, and may for our purpose be held as constituting, the "total length" of the beetle. Here a simple equation meets the case; in other words, the length of elytron or of mandible plotted against total length gives what is to all intents and purposes a straight hne, indicating a simple-interest rather than a compound-interest rate of increase. Measurements of iS stag-beetles (Lucanus cervus)* (mm.) Number of specimens 1 4 5 10 5 7 11 5 Length, total (x) 310 38-7 40-5 42-6 450 46-9 49-2 53-6 Length of elytron (y) 250 30-9 31-5 32-6 33-8 351 36-4 39-2 ( „ calculated) (y') 26-9 30-8 31-7 32-8 340 350 36-2 38-5 Length of mandible (z) 60 7-8 90 100 11-2 11-9 12-8 14-4 ( „ calculated) (z') 5-9 7-7 9-3 101 110 11-7 12-6 14-2 * Data, from Julian Huxley, after W. Bateson and H. H. Brindley, in P.Z.S. 1892, pp. 585-594. Ill] OF UNEQUAL GROWTH OR HETEROGONY 209 From the observed data we may solve, by the method of least squares, the simple equations y = a -{- bx, z= c + dx, or in other words, find the equations of the straight lines in closest agreement with the observed data. The solutions are as follows*: y - 11-02 + 0-512^, and z^- 5-64 + 0-368x, the two coefficients 0-368 and 0-512 signifying the difference between the rates *of increase of the two organs. The number of samples is not very large, and some deviation is to be expected; nevertheless, the calculated straight lines come close to the observed values. 40 10 20 Fig. 30 40 50 Total length (mm.) 55. Relative growth of body and mandible in reindeer-beetle {Cydommatus tarandus). The reindeer-beetle {Cydommatus tarandus), belonging to the same family, shews much the same thing. The mandible grows in approximately linear ratio to the body, save that it tends to be at first a little above, and later on a little below, this Unear ratio (Fig. 55). Measurements o/ Cydommatus tarandus f (mm.) Length of mandible (y) 3-9 10-7 ^ 141 19-9 24-0 30-7 34-5 Total length (x) 20-4 331 38-4 47-3 54-2 66-1 74-0 Total length calculated: x = l7y + U-l 20-3 31-9 37-7 47-5 54*5 65-9 72-4 * As determined for me by Dr A. C. Aitken, F.R.S. t Data, much abbreviated, from Huxley, after E. Dudieh, Archivf. Naturgesch. (A), 1923. 22-0 48-3 58-0 73-5 891 102-0 1120 420 65-3 74-5 85-5 99-3 112-6 1200 421 65-2 73-7 87-4 971 112-5 121-2 210 THE RATE OF GROWTH [ch. The facial and cranial parts of a dog's skull tend to grow at different rates (Fig. 56) ; and changes in the ratio between the two go a long way to explain the differences in shape between one dog's skull and another's, between the greyhound's and the pug's. But using Huxley's own data (after Becher) for the sheepdog, I find the ratio between the facial and cranial portions of the skull to be, once again, a simple linear one. Measurements of skull of sheep-dog (30 specimens) * {mm.) Mean length of facial region (y) Mean length of cranial region- (x) Calculated values for cranial region: x = 22-7 +0-88y And now, returning to the fiddler-crab, we find that after the crab has reached a certain size and the first phase of rapid growth is over, claw and body grow in simple linear relation to one another, and the heterogonic or compound-interest formula is no longer required: Fiddler-crab (Uca pugnax): ratio of growth-rates ^ in later stages ^ of claw and body (mgm.) Weight of body less 872 983 1080 1165 1212 1291 1363 1449 claw {x) Weight of large 418 461 537 594 617 670 699 778 claw {y) Do., calculated: 413 480 538 590 617 665 708 759 y=0-6a;-110 * Data from A. Becher, in Archivf. Naturgesch. (A), 1923; see Huxley, Problems of Relative Growth, p. 18, and Biol. Centralbl. loc. cit. Here, and in the previous case of Cydommatus, the equation has been arrived at in a very simple way. Take any two values, x^, x^, and the corresponding values, yi, y^. Then let x-Xj^ ^ y-yi ^-^ Vz-Vx Of -65-3 v-48-3 e.g. or from which a; = 22-7+0-88y. We may with advantage repeat this process with other values of x and y; and take the mean of the results so obtained. 112-6-65-3 1020-48-3^ a; -653 _y-48-3 47-3 "■ 53-7 * TIT] OF HETEROGONY 211 Once again we find close agreement between the observed and calculated values, although the observations are somewhat few and the equation is arrived at in a simple way. We may take it as proven that the relation between the two growth-rates is essentially linear. A compound-interest law of growth occurs, as Malthus knew, in cases, and at times, of rapid and unrestricted growth. But unrestricted growth occurs under special conditions and for brief 50 100 Length of cranial portion (mm.) 150 Fig. 56. Relative growth of the cranial and facial portions of the skull in the sheepdog. Cf. Huxley, p. 18, after Becher, periods; it is the exception rather than the rule, whether in a population or in the single organism. In cases of diiferential growth the compound-interest law manifests itself, for the same reason, when one of the two growth-rates is rapid and "unre- stricted," and when the discrepancy between the two growth-rates is consequently large, for instance in the fiddler-crabs. The compound- interest law is a very natural mode of growth, but its range is 212 THE RATE OF GROWTH [CH. limited. A linear relation, or simple-interest law, seems less likely to occur; but the fact is, it does occur, and occurs commonly. On so-called dimorphism In a well-known paper, Bateson and Brindley 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 DD Fig. 57. Tail-forceps of earwig. From Martin Burr, after Willi Kuhl. 150 ^ 100 mm. 10 Length in mm. Fig. .58. Variability of length of tail-forceps in a sample of earwigs. After Bateson and Brindley, P.Z.S. 1892, p. 588. tail-forceps (Fig. 57), with few instances of intermediate magnitude*. This distribution into two groups, according to magnitude, is illustrated in the accompanying diagram (Fig. 58); and the * W. Bateson and H. H. Brindley, On some cases of variation in secondary sexual characters [Forficula, Xylotrupa], statistically examined, P.Z.S. 1892, pp. 585-594. Cf. D. M. Diakonow, On dimorphic variability of Forficula, Joum. Genet, xv, pp. 201-232, 1925; and Julian Huxley, The bimodal cephalic horn of Xylotrupa, ibid, xviii, pp. 45-53, 1927. Ill OF DIMORPHIC GROWTH 213 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 looks as, though it lay close behind. For the two size-groups into which the tails of the earwigs fall look curiously hke two age-groups such as we have already studied in a fish, where the ages and therefore also the magnitudes of a random sample form a discontinuous series (Fig. 59). 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 for the magnitudes 200 150 100 Ocm. \5 20 25 30 Length in cm. Fig. 59. Length of body in a random sample of plaice. of these organs, just as for the whole body of the fish, and just as for the tails of Bateson's earwigs. The differences, in short, with which Bateson was dealing were a question of magnitude, and it was only natural to refer these diverse magnitudes to diversities of growth; that is to say, it seemed natural to suppose that in this case of "dimorphism," the tails of the one group of earwigs (which Bateson called the "high males") had either grown faster, or had been growing for a longer period of time, than those of the "low males." If the whole random sample of earwigs were of one and the same age, the dimorphism would appear to be due to two alternative values for the mean growth-rate, individual earwigs varying around one mean or the other. If, on the other hand, the 214 THE RATE OF GROWTH [CH. two groups of earwigs were of diiferent ages, or had passed through one moult more or less, the phenomenon would be simple indeed, and there would be no more to be said about it*. Diakonow made the not unimportant observation that in earwigs living in un- favourable conditions only the short-tailed type tended to appear. In apparent close analogy with the case of the earwigs, and in apparent corroboration of their dimorphism being due to age, Fritz Werner measured large numbers of water-fleas, all apparently adult, found his measurements falling into groups and so giving multimodal curves. The several cusps, or modes, he interpreted without difficulty as indicating diiferences of age, or the number of moults which the creatures had passed through f (Fig. BO). 9 /i. 60 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 80 100 120 140 160/1 Length in ^ Fig. 60. Measurements of the dorsal edge in a population of Chydorus sphaericus, a water-flea. From Fritz Werner. An apparently analogous but more difficult case is that of a certain little beetle, Onthofhagus taurus, which bears two "horns" on its head, of variable size or prominence. Linnaeus saw in it a single species, Fabricius saw two; and the question long remained an open one among the eniomologists. We now know that there are two "modes," two predominant sizes in a continuous range of * The number of moults is known to be variable in many species of Orthoptera, and even occasionally in higher insects; and how the number of moults may be influenced by hunger, damp or cold is discussed by P. P. Calvert, Proc. Amer. Pkilos. Soc. Lxviii, p. 246, 1929. On the number of moults in earwigs, see E. B. Worthington, Entomologist, 1926, and W. K. Weyrauch, Biol. Centralbl. 1929, pp. 543-5^8. ■f Fritz Werner, Variationsanalytische Untersuchungen an Chydoren, Ztschr. f. Morphologie u. Oekologie d. Tiere, n, pp. 58-188, 1924. Ill] OF DIMORPHIC GROWTH 215 variation*. In the "complete metamorphosis" of a beetle there is no room for a moult more or less, and the reason for the two modal sizes remains hidden (Fig. 61). But new hght has been thrown on the case of the earwigs, which may help to explain other obscure diversities of shape and size within the class of insects. At metamorphosis, and even in a simple moult, the external organs of an insect may often be seen to unfold, as do, for instance, the wings of a butterfly; they then quickly harden, in a form and of a size with which ordinary gradual growth I 200 Head of 0.'tauru8;two forms of maJe ^N A. 0 0*5 1-0 Length of horn (mm.) Fig. 61. Two forms of the male, in the beetle Ontkophagvs taurus. has had nothing directly to do. This is a very peculiar phenomenon, and marks a singular departure from the usual interdependence of growth and form. When the nymph, or larval earwig, is about to shed its skin for the last time, the tail-forceps, still soft and tender, are folded together and wrapped in a sheath; they need to be distended, or inflated, by a combined pressure of the body-fluid (or haemolymph) and an intake of respiratory air. If all goes well, * Rene Paulian, Bull. Soc. Zool. Fr. 1933; also Le polymorpMsme des mMea de Cddoptires, Paris, 1935, p. 8. 216 THE RATE OF GROWTH [ch. the forceps expand to their full size; if the .reature be weak or underfed, inflation is incomplete and tho tail-forceps remain small. In either case it is an affair of a few critical moments during the final ecdysis; in ten minutes or less, the chitin has hardened, and shape and size change no- more. Willi Kuhl, who has given us this interesting explanation, suggests that the dimorphism observed by Bateson and by Diakonow is not an essential part of the pheno- menon; he has found it in one instance, but in other and much larger samples he has found all gradations, but only a single,- well-marked unimodal peak*. The effect of temperature'f The rates of growth which we have hitherto dealt with are mostly 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 Belgian people. But apart from that "fortuitous" individual variation which we have already considered, it is obvious that the normal rate of growth will be found to vary, in man and in other animals, just as the average stature varies, in different locahties and in different "races." This phenomenon is a very complex one, and is doubtless a resultant of many undefined con- tributory causes; but we at least gain something in regard to it when we discover that rate of growth is directly affected by temperature, and doubtless by other physical 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 temperature, being retarded at temperatures below and somewhat * Willi Kuhl, Die Variabilitat der abdominalen Korperanhange bei Forficula, Ztsch. Morph. u. Oek. d. Tiere, xii, p. 299, 1924. Cf. Malcolm Burr, Discovery, 1939, pp. 340^345. t The temperature limitations of life, and to some extent of growth, are sum- marised for a large number of species by Davenport, Exper. Morphology, cc. viii, xviii, and by Hans Przibram, Exp. Zoologie, iv, c. v.^ X Reaumur, L'art de faire eclorre et elever en toute saison des oiseaux domestiques, soil par le moyen de la chaleur du fumee, soil par le moyen de celle du feu ordinaire, Paris, 1749. He had also studied, a few years before, the effects of heat and cold on growth-rate and duration of life in caterpillars and chrysalids: Memoires, ii, p. 1, de la dnree de la vie des crisalides (1736). See also his Observations du Thermometre, etc., Mem. Acad., Paris, 1735, pp. 345-376. Ill] THE EFFECT' OF TEMPERATURE 217 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 tem- perature is a matter of famiUar 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*. That temperature is only one of the climatic factors determining growth and yield is well known to agriculturists; and a method of "multiple correlation" has been used to analyse the several influences of temperature and of rainfall at different seasons on the future yield of our own crops f. The same joint influence can be recognised in the bamboo; for it is said (by Lock) that the growth-rate of the bamboo in Ceylon is proportional to the humidity of the atmosphere, and again (by Shibata) that it is proportional to the temperature in Japan. But BlackmanJ suggests that in Ceylon 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 low: so that in the one country it is the one factor, and in the other country the other, whose variation is both conspicuous and significant. After all, it is probably rate of evaporation, the joint result of temperature and humidity, which is the crux of the matter§. "Climate" is a subtle thing, and includes a sort of micro-meteorology. A sheltered corner has a climate of its own; one side of the garden -wall has a different climate to the other; and deep in the undergrowth of a wood celandine and anemone enjoy a climate many degrees warmer than what is registered on the screen ||. Among the mould-fungi each several species has its own optimum tempera- ture for germination and growth. At this optimum temperature growth is further accelerated by increase of humidity ; and the further we depart from the optimum temperature, the narrower becomes the range of humidity within which growth can proceed^. Entomologists know, in like manner, how over- abundance of an insect-pest comes, or is apt to come, with a double optimum of temperature and humidity. * Cf. {int. at.) H. de Vries, Materiaux pour la connaissance de I'influence de la temperature sur les plantes, Arch. Neerlandaises, v, pp. 385-401, 1870; C. Linsser, Periodische Erscheinungen des Pflanzenlebens, Mem. Acad, des Sc, 8t Petersbourg (7), XI, XII, 1867-69; Koppen, Warme und Pflanzenwachstum, Bull. Soc. Imp. Nat., Moscou, xliii, pp. 41-110, 1871; H. Hoffmann, Thermisehe Vegetations- constanten, Ztschr. Oesterr. Ges. f. Meteorologie, xvii, pp. 121-131, 1881; Pheno- logische Studien, Meteorolog. Ztschr. iii, pp. H3-120, 1886. t See [int. al.) R. H. Hooker, Journ. Roy. Statist. Soc. 1907, p. 70; Journ. Roy. Meteor. Soc. 1922, p. 46. I F. F. Blackman, Ann. Bot. xix, p. 281, 1905. § Szava-Kovatz, in Petermann's Mitteilungen, 1927, p. 7. II Cf. E. J. Salisbury, On the oecological aspects of Meteorology, Q.J.R. Meteorol. Soc. July 1939. ^ R. G. Tomkins, Proc. R.S. (B), cv, pp. 375^01, 1929. 218 THE RATE OF GROWTH [CH. The annexed diagram (Fig. 62), showing growth in length of the roots of some common plants at various temperatures, is a sufl&cient illustration of the phenomenon. We see that there is always a certain temperature at which the rate is a maximum ; while on either side of the optimum the rate falls off, after the fashion of the normal curve of error. We see further, from the data given by Sachs and others, that the optimum is very much the same for all the common plants of our own chmate. For these it is somewhere about 26° C. 80 70 M^ie 16 20 22 24 26 28 30 32 34 36 38 40° Temp. Fig. 62. Relation of rate of growth to temperature in certain plants. From Sachs's data. (say 77° F.), or about the temperature of a warm summer's day; while it is considerably higher, naturally, in such plants as the melon ot* the maize, which are at home in warmer countries than our own. The bacteria have, in like manner, their various optima, and some- times a high one. The tuberculosis-bacillus, as Koch shewed, only begins to grow at about 28° C, and multipUes most rapidly at 37-38°, the body- temperature of its host. The setting and ripening of fruit is a phase of growth stiU more dependent on temperature; hence it is a "dehcate test of climate," and a proof of its constancy, that the date-palm grows but bears Ill] THE EFFECT OF TEMPERATURE 219 no fruit in Judaea, and the vine bears freely at Eshcol, but not in the hotter country to the south*. Shellfish have their own appro- priate spawning-temperatures; it needs a warm summer for the oyster to shed her spat, and Hippopus and Tridacna, the great clams of the coral-reefs, only do so when the water has reached the high temperature of 30° C. For brown trout, 6° C. is found to be a critical temperature, a minimum short of which they do not grow at all; it follows that in a Highland burn their growth is at a standstill for fully half the yearf. That a rise of temperature accelerates growth is but part of the story, and is not always true. Several insects, experimentally reared, have been found to diminish in size as the temperature increased J; and certain flies have been found to be larger in their winter than their summer broods. The common copepod, Calmius finmarchicus, has spring, summer and autumn broods, which (at Plymouth §) are large, middle-sized and small; but the large spring brood are hatched and reared in the cold "winter" water, and the small autumn-winter brood in the warmest water of the year. In the cold waters of Barents Sea Calanus grows larger still; of an allied genus, a large species lives in the Antarctic, a small one in the tropics, a middle-sized is common in the temperate oceans. The large size of many Arctic animals, coelenterates and crustaceans, is well known; and so is that of many tropical forms, Hke Fungia among the corals, or the great Tritons and Tridacnas among molluscs. Another common phenomenon is the increasing number of males in late summer and autumn, as in the Rotifers and in the above-mentioned Calani. All these things seem somehow related to temperature; but other physical conditions enter iilto the case, for instance the amount of dissolved oxygen in the cold waters, and the physical chemistry of carbonate of Hme in the warm||. The vast profusion of life, both great and small, in Arctic seas, the multitude of individuals and the unusual size to which many species grow, has been often ascribed to a superabundance of dissolved oxygen, but oxygen alone would not go far. The nutrient salts, nitrates and phosphates, are the * Cf. J. W. Gregory, in Geogr. Journ. 1914, and Journ. E. Geogr. Soc. Oct. 1930. t Cf. C. A. Wingfield, op. cit. supra, p. 176. X B. P. Uvarow, Trans. Ent. Soc. Lond. lxxix, p. 38, 1931. § W. H. Golightly and LI. Lloyd, in Nature, July 22, 1939. li Cf. B. G. Bogorow and others, in the Journ. M.B.A. xix, 1933-34. 220 THE RATE OF GROWTH [ch. limiting factor in the growth of that micro-vegetation with which the whole cycle of life begins. The tropical oceans are often very bare of these salts; in our own latitudes there is none too much, and the spring-growth tends to use up the supply. But we have learned from the Discovery Expedition that these salts are so abundant in the Antarctic that plant-growth is never checked for stint of them. Along the Chilean coast and in S.W. Africa, cold Antarctic water wells up from below the warm equatorial current. It is ill-suited for the growth of corals, which build their reefs in the warmer waters of the eastern side ; but it teems with nourishment, breeds a plankton- fauna of the richest kind, which feeds fishes preyed on by innumerable birds, the guano of which is sent all over the world. Now and then persistent winds thrust the cold current aside ; a new warm current, el Nino of the Chileans, upsets the old equilibrium ; the fishes die, the water stinks, the birds starve. The same thing happens also at Walfisch Bay, where on such rare occasions dead fish lie piled up high along the shore. It is curiously characteristic of certain physiological reactions, growth among them, to be affected not merely by the temperature of the moment, but also by that to which the organism has been previously and temporarily exposed. In other words, acclimatisation to a certain temperature may continue for some time afterwards to affect all the temperature relations of the body*. That temporary cld may, under certain circumstances, cause a subsequent accelera- tion of growth is made use of in the remarkable process known as vernalisation. An ingenious man, observing that a winter wheat failed to flower when sown in spring, argued that exposure to the cold of winter was necessary for its subsequent rapid growth ; and this he verified by " chiUing" his seedlings for a month to near freezing-point, after which they grew quickly, and flowered at the same time as the spring wheat. The economic advantages are great of so shortening the growing period of a crop as to protect it from autumn frosts in a cold chmate or summer drought in a hot one ; much has been done, especially by Lysenko in Russia, with this end in viewf. The most diverse physiological processes may be afl'ected by temperature. A great astronomer at Mount, Wilson, in California, used some idle hours to watch the "trail-running" ants, which run all night and all day. Their speed increases so regularly with the temperature that the time taken to run 30 cm. suffices to tell the * Cf. Kenneth Mellanby, On temperature coefficients and acclimatisation, Nature, 3 August 1940. t Of. {int. al.) V. H. Blackman, in Nature, June 13, 1936. Ill] THE TEMPERATURE COEFFICIENT 221 temperature to 1° C. ! Of two allied species, one ran nearly half as fast again as the other, at the same temperature*. While at low temperatures growth is arrested and at temperatures unduly high hfe itself becomes impossible, we have now seen that within the range of more or less congenial temperatures growth proceeds the faster the higher the temperature. The same is true of the ordinary reactions of chemistry, and here Van't Hoff and Arrheniusf have shewn that a definite increase in the velocity of the reaction follows a definite increase of temperature, according to an exponential law: such that, for an interval of n degrees the velocity varies as x", x being called the "temperature coefficient" for the reaction in question J. The law holds good throughout a considerable range, but is departed from when we pass beyond certain normal limits ; moreover, the value of the coefficient is found to keep to a certain order of magnitude — somewhere about 2 for a temperature-interval of 10° C. — which means to say that, the velocity of the reaction is just about doubled, more or less, for a rise of 10° C. This law, which has become a fundamental principle of chemical mechanics, is applicable (with certain qualifications) to the pheno- mena of vital chemistry, as Van't Hoff himself was the first to declare ; and it follows that, on much the same fines, one may speak of a "temperature coefficient" of growth. At the same time we must remember that there is a very important difference (though we need not call it a fundamental one) between the purely physical and the * Harlow Shapley, On the thermokinetics of Dolichoderine ants, Proc. Nat. Acad. Sci. x, pp. 436-439, 1924. t Van't HofF and Cohen, Studien zur chemischen Dynamik, 1896; Sv. Arrhenius, Ztschr. f. phys. Chemie, iv, p. 226. X For various instances of a temperature coefficient in physiological processes, see (e.g.) Cohen, Physical Chemistry f or ... Biologists (Enghsh edition), 1903; Kanitz and Herzog in Zeitschr. f. Elektrochemie, xi, 1905; F. F. Blackman, Ann. Bot. XIX, p. 281, 1905; K. Peter, Arch.f. Entw. Mech. xx, p. 130, 1905; Arrhenius, Ergebn. d. Physiol, vii, p. 480, 1908, and Quantitative Laws in Biological Chemistry, 1915; Krogh in Zeitschr. f.allgem. Physiologie,xyi, -pp. 163,178,1914; James Gray, Proc. E.S. (B), xcv, pp. 6-15, 1923; W. J. Crozier, many papers in Journ. Gen. Physiol. 1924; J. Belehradek, in Biol. Reviews, v, pp. 1-29, 1930. On the general subject, see E. Janisch, Temperaturabhangigkeit biologischer Vorgange und ihrer kurvenmassige Analyse, Pfluger's Archiv, ccix, p. 414, 1925; G. and P. Hertwig, Regulation von Wachstum . . . durch Umweltsfaktoren, in Hdb. d. normal, u. pathol. Physiologie, xvi, 1930. 222 THE RATE OF GROWTH [ch. physiological phenomenon, in that in the former we study (or seek and profess to study) one thing at a time, while in the living body we have constantly to do with factors which interact and interfere; increase in the one case (or change of any kind) tends to be con- tinuous, in the other case it tends to be brought, or to bring itself, to arrest. This is the simple meaning of that Law of Optimum, laid down by Errera and by Sachs as a general principle of physiology ; namely that every physiological process which varies (Hke growth itself) with the amount or intensity of some external influence, does so under such conditions that progressive increase is followed by progressive decrease; in other words, the function has its optimum condition, and its curve shews a definite maximu^n. In the case of temperature, as Jost puts it, it has on the one hand its accelerating effect, which tends to follow Van't Hoff's law. But it has also another and a cumulative effect upon the organism: "Sie schadigt oder sie ermiidet ihn, und je hoher sie steigt desto rascher macht sie die Schadigung geltend und desto schneller scbxeitet sie voran*." It is this double effect of temperature on the organism which gives, or helps to give us our "optim'im" curves, which (like all other curves of frequency or error) are the expression, not of a single solitary phenomenon,' but of a more or less complex resultant. Moreover, as Blackman and others have pointed out, our "optimum" temperature is ill-defined until we take account also of the duration of our experiment; for 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. The mile and the hundred yards are won by different runners; and maximum rate of worldng,^ and maximum amount of work done, are two very different things f. In the case of maize, a certain series of experiments shewed that the growth in length of the roots varied with the temperature as follows J: * On such limiting factors, or counter-reactions, see Putter, Ztschr. f. aUgem. Physiologic, xvi, pp. 574-627, 1914. t Cf. L. Errera, UOptimum, 1896 (Recueil d'oeuvres, Physiologie genirale, pp. 338- 368, 1910) ; Sachs, Physiologie d. Pflanzen, 1882, p. 233; PfeflFer, Pflanzenphysiologie, n, p. 78, 194; and cf, Jost, Ueber die Reactionsgeschwindigkeit ira Organismus, Biol. CentraWl. xxvi, pp. 225-244, 1906. t After Koppen, Bull. Soc. Nat. Moscou, XLin, pp. 41-101, 1871. Ill] THE TEMPERATURE COEFFICIENT 223 Temperature Growth in 48 hours °C. mm. 18-0 11 23-5 10-8 26-6 29-6 23-5 26-5 30-2 64-6 33-5 69-5 36-5 20-7 us write our formula in the form V(>^) = x^, or In laF..._^- \o0V = nA Then choosing two values out of the above experimental series (say the second and the second-last), we have t = 23-5, n = 10, and F, V = 10-8 and 69-5 respectively. . ,. - log 69-5 - log 10-8 , Accordmgly, — ^ — — = log x, 0-8414 - 0-034 or ^^ = 0-0808, and therefore the temperature-coefficient = antilog 0-0808 = 1-204 (for an interval of 1° C). This first approximation might be much improved by taking account of all the experimental values, two only of which we have yet made use of; but even as it is, we see by Fig. 63 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. For an experiment on Lupinus albus, quoted by Asa Gray* I have worked out the corresponding coefficient, but a httle 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. 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 resultsf . * Asa Gray, Botany, p. 387. t I. Leitch, Some experiments on the influence of temperature on the rate of growth in Pisum sativum, Ann. Bot. xxx, pp. 25-46, 1916, especially Table III, p. 45. Cf. Priestley and Pearsall, Growth studies, Ann. Bot. xxxvi, pp. 224-249, 1922. 224 THE RATE OF GROWTH [CH. In Fig. 64 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 shewing a rate of growth of just about 1 mm. per hour at a temperature of 20°. My curve in Fig. 64 is drawn from these determinations; and it will be seen that, while it is by no means exact at the lower temperatures, and will fail us altogether at very high tem- peratures, yet it serves as a satisfactory guide to the relations between rate and temperature within the ordinary limits of healthy growth. Miss Leitch 18 20 22 24 26 28 30 32 34°C Fig. 63. Relation of rate of growth to temperature in maize. Observed values (after Koppen), and calculated curve. 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 combine'd with other factors which we cannot determine or eliminate. 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 as much or more 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 misleading*. We have abundant * F. F. Blackman, Presidential Address in Botany, Brit. Assoc. Dublin, 1908. Ill] OF TEMPERATURE COEFFICIENTS 225 illustrations, however, among animals, in which we may study the temperature-coefficient under circumstances where, though the phenomenon is always complicated, true metabohc growth or chemical combination plays a larger role. Thus Mile. Maltaux and Professor Massart* have studied the rate of division in a certain flagellate, Chilomonas paramoecium, and found the process to take 20 /• — / .7 ■~ 7 / / / _ / V - . / / — / '/. / / 1 — • / / ^ — ^i ^^ • ^ * - • ^ y -^ • * "" per hour 20 1-8 1-6 1-4 1-2 1-0 •8 •6 •4 •2 0 4° 8° 1 2° 1 6° 20° 24° ^ 28° 32° ^ Fig, 64. Relation of rate of growth to temperature in rootlets of pea. From Miss I. Leitch's data. 29 minutes at 15° C, 12 at 25°, and only 5 minutes at 35° C. These velocities are in the ratio of 1 : 24: 5-76, which ratio corresponds precisely to a temperature-coefficient of 24 for each rise of 10°, or about 1-092 for each degree centigrade, precisely the 'Same as we have found for the growth of the pea. By means of this principle we may sometimes throw hght on apparently compUcated experiments. For instance. Fig. 65 is an * Rec. de VInst. Bot. de Brzizelles, vi, 1906. 226 THE RATE OF GROWTH [CH. illustration, which has been often copied, of 0. Hertwig's work on the effect of temperature on the rate of development of the tadpole*. '1 ~fZi Z ^4^ 1 1 1 / / 1 ~/ ~X X- T_ J- L «' / / 1 1 I 1 1 ill 01 «^ .... - t 1 14^ "^ - 4 -T ttU- . I 2^ riii . t^'^ rtlf '^ v^ IJ LE - z^^ Tti-l^t M R 1 J^iItX- »^ VJV ptj ^ ^ t Qj 3 -l f- /- ziy i f~ /^^'^ z it-,/ " JJ- -i^'^-^Z Jl J \ JV . t- z ^^zt'^ \ .,o^j> y^ L/. %^^yjLt £^^^ V ^ / ^^^-""^^^^^^ .-^ >^ ""^^ '^'^^ ^c^^ -^^ ^^^ ^^^^^^^^^ --' -■^ ^ — " ^ — ::h^ "^ "^ j TemneXcLl uie Ceatign.a.d.e «*• «• 2^" «/• /*• //" It,' I-.' IV li" If II' lof 9" a" ^ ef s' Fig. 65. 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. * 0. Hertwig, Einfluss der Temperatur auf die Entwicklung von Rana fusca und R. esculenta, Arch. f. mikrosk. Anat. li, p. 319, 1898. Cf. also K. Bialaszewicz, Beitrage z. Kenntniss d. Wachsthumsvorgange bei Amphibienembryonen, Ball. Acad. Sci. de Cracovie, p. 783, 1908; Abstr. in Arch. f. Entwicklungsmech. xxviii, p. 160, 1909: from which Ernst Cohen determined the value of Q^q (Vortrdge iib. physikal. C hemic f. Arzte, 1901; English edit. 1903). Ill] OF TEMPERATURE COEFFICIENTS 227 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° C. At 10° C. Stage III 2-0 6-5 days „ IV 2-7 8-1 „ „ V 30 10-7 „ ;, VI 4-0 13-5 „ „ VII 50 16-8 ,. Total 16-7 55-6 25'C. 20° 15° 10° 5' Fig. 66. Calculated values, corresponding to preceding figure. That is to say, the time taken to produce a given result at 10° was (on the average) somewhere about 55-6/16-7, or 3-33, times as long as was required at 20° C. We may then put our equation in the simple form, x^^ = 3-33. Or, 10 log X = log 3-33 = 0-52244. Therefore log x = 0-05224, and X = 1-128. 228 THE RATE OF GROWTH [ch. 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. Fig. 66 is calculated throughout from this value; and it will be found 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 on the original curves. In hke manner, the following table shews the "incubation period" for trout-eggs, or interval between fertihsation and hatching, at different temperatures * : Incubation-period of trout-eggs Temperature Days' interval °C. before hatching 2-8 165 3-6 135 3-9 121 4-5 109 50 103 5-7 96 6-3 89 6-6 81 7-3 73 8-0 65 90 56 10-0 47 111 38 12-2 32 Choosing at random a pair of observations, viz. at 3-6° and 10°, and proceeding as before, we have 10° - 3-6° = 64°. Then (64) = '-g, or 6-4 X log X = log 135 — log 47 = 2-1303 - 1-6721 = 0-4582 and log X = 0-4582 - 6-4 = 0-0716, X = 1-179. * Data from James Gray, The growth of fish, Journ. Exper. Biology, vi, p. 126, 1928. Ill] OF TEMPERATURE COEFFICIENTS 229 Using three other pairs of observations, we have the following concordant results : At 12-2° and 2-8^, x - M91 10-0° 3-6° M79 9-0° 5-7°; M78 8-0° 5-0° M65 Mean M8 A very curious point is that (as Gray tells us) the young fish which have hatched slowly at a low temperature are bigger than those whose growth has been hastened by warmth. Again, plaice-eggs were found to hatch and grow to a certain length (4-6 mm.), as follows*: Temperature (° C.) Days 41 230 6-1 181 8-0 13-3 10-1 10-3 120 8-3 From these we obtain, as before, the following constants: At 12° and 8°, x = M3 12° 4-1° M4 10-1° 6-1° M5 8-0° 4-1° M5 Mean M4 The value of x is much the same for the one fish as for the other. Karl Peter t, experimenting on echinoderm eggs, and making use also of Richard Hert wig's experiments on young tadpoles, gives the temperature-coefficients for intervals of 10° C. (commonly written Qio) as follows, to which I have added the corresponding values forg,: Sphaerechinus Qiq =2-15 Qi^ 1-08 Echinus ?'13 1-08 Rana 2-86 Ml * Data from A. C. Johansen and A. Krogh, Influence of temperature, etc., Publ. de Circonstance, No. 68, 1914. The function is here said to be a linear one — which would have been an anomalous and unlikely thing". t Der Grad der Beschleunigung tierischer Entwicklung durch erhohte Tem- peratur, Arch. f. Entw. Mech. xx, p. 130, 1905. More recently Bialaszewicz has determined the coefficient for the rate of segmentation in Rana as being 2-4 per 10° C. 230 THE RATE OF GROWTH fcH. These values are not only concordant, but are of the same order of magnitude as the temperature-coefficient in ordinary chemical reactions. Peter has also discovered the interesting fact that the temperature-coefficient alters with age, usually but not always decreasing as time goes on*: Sphaerechinus Segmentation Q^q =2-29 Q^ = 1-09 Later stages 2-03 1-07 Echinus Segmentation 2-30 1-09 Later stages 2-08 1-08 Rana Segmentation 2-23 1-08 Later stages 3-34 1-13 Furthermore, the temperature-coefficient varies with the tem- perature itself, falhng as the temperature rises — a rule which Van't Hoff shewed to hold in ordinary chemical operations. Thus in Rana the temperature-coefficient (Qiq) 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. As the several stages of development are accelerated by warmth, so is the duration of each and all, and of life itself, proportionately curtailed. The span of life itself may have its temperature- coefficient — in so far as Life is a chemical process, and Death a chemical result. In hot climates puberty comes early, and old age (at least in women) follows soon; fishes grow faster and spawn earlier in the Mediterranean than in the North Sea. Jacques Loeb f found (in complete agreement with the general case) that the larval stages of a fly are abbreviated by rise of temperature; that the mean duration of life at various temperatures can be expressed by a temperature-coefficient of the usual order of magnitude ; that this coefficient tends, as usual, to fall as the temperature rises; and lastly — what is not a little curious — ^that the coefficient is very much the same, in fact all but identical, for the larva, pupa and imago of the fly. * The diflferences are, after all, of small order of magnitude, as is all the better seen when we reduce the ten-degree to one-degree coefficients. t J- Loeb and Northrop, On the influence of food and temperature upon the duration of life, Journ. Biol Chemistry, xxxii, pp. 103-121, 1917. Ill OF TEMPERATURE COEFFICIENTS 231 Temperature-coefficients (Q^q) of Drosophila Larva Pupa Imago 15-20° C. 115 117 118 20-25° C. 106 1-08 107 And Japanese students, studying a little fresh- water crustacean, have carried the experiment much beyond the range of Van't Hoff 's law, and have found length of hfe to rise rapidly to a maximum at about 13-14° C, and to fall slowly, in a skew curve, thereafter* (Fig. 67). 20 30 40 Temperature, °C. Fig. 67. Length of life, at various temperatures, in a water-flea. If we now summarise the various temperature-coefficients (Q^) which we have happened to consider, we are struck by their remarkably close agreement: Yeast Qi = M3 Lupin M6 Maize 1-20 Pea 1-09 Echinoids 1-08 Drosophila (mean)" M2 Frog, segmentation 1-08 „ tadpole M3 Mean M2 * A. Terao and T. Tabaka, Duration of life in a water-flea, Moina sp.; Joum. Imp. Fisheries Inst., Tokyo, xxv, No. 3, March 1930. 232 THE RATE OF GROWTH [ch. The constancy of these results might tempt us to look on the phenomenon as a simple one, though we well know it to be highly complex. But we had better rest content to see, as Arrhenius saw in the beginning, a general resemblance rather than an identity between the temperature-coefficients in physico-chemical and biological processes*. It was seen from the first that to extend Van't Hoff's law from physical chemistry to physiology was a bold assumption, to all appearance largely justified, but always subject to severe and cautious limitations. If it seemed to simplify certain organic phenomena, further study soon shewed how far from simple these phenomena were. Living matter is always heterogeneous, and from one phase to another its reactions change; the temperature- coefficient varies likewise, and indicates at the best a summation, or integration, of phenomena. Nevertheless, attempts have been made to go a little further towards a physical explanation of the physiological coefficient. Van't Hoff suggested a viscosity-correction for the temperature -coefficient even of an ordinary chemical reaction; the viscosity of protoplasm varies in a marked degree, inversely with the temperature, and the viscosity-factor goes, perhaps, a long way to account for the aberrations of the temperature-coefficient. It has even been suggested (by Belehradekf) that the temperature- coefficients of the biologist are merely those of protoplasmic viscosity. For instance, the temperature-coefficients of mitotic cell-division have been shewn to alter from one phase to another of the mitotic process, being much greater at the start than at the end| ; and so, precisely, has it been shewn that protoplasmic viscosity is high at the beginning and low at the end of the mitotic process §. On seasonal growth There is abundant evidence in certain fishes, such as plaice and haddock, that the ascending curve of growth is subject to seasonal fluctuations or interruptions, the rate during the winter months bejng always slower than in the months of summer. Thus the Newfoundland cod have their maximum growth-rate in June, and in January-February they cease to grow; it is as though we super- imposed a periodic annual sine-curve upon the continuous curve of growth. Furthermore, as growth itself grows less and less from year to year, so will the difference between the summer and the ♦ Cf. L. V. Heilbronn, Science, lxii, p. 268, 1925. t J. Belehradek, in Biol. Reviews, v, pp. 30-58, 1930. X Cf. E. Faure-Fremiet, La cinetique du developpement, 1925; also B. Ephrussi, C.R. cLxxxii, p. 810, 1926. § See {int. al.) L. V. Heilbronn, The Colloid Chemistry of Protoplasm, 1928. Ill] OF SEASONAL GROWTH 233 winter rates grow less and less. The fluctuation in rate represents a vibration which is gradually dying out; the amplitude of the sine-curve diminishes till it disappears; in short our phenomenon is simply expressed by what is known as a "damped sine-curve*." Growth in height of German military cadets, in half-yearly periods Increment (em.) Height (cm. ) f A ■> Number observed , ^ Winter ^-year Summer ^-year Age October April October Year 12 11-12 139-4 141-0 143-3 1-6 2-3 3-9 80 12-13 143-0 144-5 1474 1-5 2-9 4-4 146 13-14 147-5 149-5 152-5 2-0 3-0 5-0 162 14-1.-) 152-2 155-0 158-5 2-8 3-5 6-3 162 15-16 158-5 160-8 163-8 2-3 30 5-3 150 16-17 163-5 165-4 167-7 1-9 2-3 4-2 82 17-18 167-7 168-9 170-4 1-2 1-5 2-7 22 18-19 169-8 170-6 171-5 0-8 0-9 1-7 6 19-20 170-7 171-1 171-5 Mean 0-4 1-6 0-4 2-2 0-8 cm. 4 years Fig. 68. Half-yearly increments of growth, in cadets of various ages. From Daffner's data. 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, by help of certain measure- ments of the height of German cadets, measured at half-yearly intervals t- In the accompanying diagram (Fig. 68) the half-yearly increments are set forth from the above table, and it will be seen * The scales, on the other hand, make most of their growth during the int/er- mediate seasons: and with this peculiarity, that a few broad zones are added to the scale in spring, and a larger number of narrow circuli in autumn : see Contrib. to Canadian Biology, iv, pp. 289-305, 1929; Ben Dawes, Growth... in plaice, Journ. M.B.A. xvii, pp. 103-174, 1930. t From Daffner, Da^ Wachstum des Menschen^ p. 329, 1902. 234 THE RATE OF GROWTH CH. that they form two even and entirely separate series. Danish school- boys show just the same periodicity of growth in stature. The seasonal effect on visible growth-rate is much alike in fishes and in man, in spite of the fact that the bodily temperature of the one varies with the milieu externe and that of the other keeps constant to within a fraction of a degree. While temperature is the dominant cause, it is not the only cause of seasonal fluctuations of growth; for alternate scarcity and abundance of food is often, as in herbivorous animals, the ostensible 18 10 a o S / •^A -1 1 1 J 1 1 X / / M / \ / / '^ \ I \ / / V / \ / / / / \\ / \ y / \\ / Steers\ / \ / \ \ / \ v.^ y \ \ / \ 1 1 1 1 1 \ / HeiVfers \ \ 1 1 I.I t 1 .IN JUN AUG OCT DEC FEB APR dUN AUG OCT DEC FEB APR 15 20 25 30 35 Age in months Fig. 69. Seasonal growth of S. African cattle: Sussex half-breeds. After Schiitte. reason. Before turnips came into cultivation in the eighteenth century our own cattle starved for half the year and grew fat the other, and in many countries the same thing happens still. In South Africa the rainy season lasts from November to February; by January the grass is plentiful, by June or July the veldt is parched until rain comes again. Cattle fatten from January to March or April; from July to October they put on Httle weight, or lose weight rather than put it on*. * Cf. D. J. Schiitte, in Onderstepoort Journal, Oct. 1935. J Ill] OF THE GROWTH OF TREES 2^5 The growth of trees Some sixty years ago Sir Robert Christison, a learned and versatile Edinburgh professor, was the first to study the "exact measurement" of the girth of trees*; and his way of putting a girdle round the tree, and fitting a recording device to the girdle, is copied in the " dendrographs " t used in forestry today. The Edinburgh beeches begin to enlarge their trunks in late May or June, when in full leaf, and cease growing some three months later;' the buds sprout and the leaves begin their work before the cambium wakens to activity. The beech-trees in Maryland do likewise, save that the dates are a little earlier in the year; and walnut-trees on high ground in Arizona shew a like short season of growth, differing somewhat in date or "phase," just as it did in Edinburgh, from one year to another. Deciduous trees stop growing after the fall of the leaf, but ever- greens grow all the year round, more or less. This broad fact is illustrated in the following table, which happens to relate to the Mean monthly increase in girth of trees at San Jorge, Uruguay : from C. E. HalVs data. Values given in ^percentages of total annual increment J Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. Evergreens 91 8-8 8-6 8-9 7-7 5-4 4-3 6-0 9-1 Ill 10-8 10-2 Deciduous trees 20-3 14-6 9-0 2-3 0-8 0-3 0-7 13 3-5 9-9 16-7 210 southern hemisphere, and to the climate of Uruguay. The measure- ments taken were those of the girth of the tree, in mm., at three feet from the ground. The evergreens included Pinus, Eucalyptus * Sir R. Christison, On the exact measurement of trees. Trans. Edinb. Botan. Soc. XIV, pp. 164-172, 1882. Cf. also Duhamel du Monceau, Des semis, et plantation des arbres, Paris, 1750. On the general subject see {int. al.) Pfeffer's Physiology of Plants, II, Oxford, 1906; A. Maliock, Growth of trees, Proc. R.S. (B), xc, pp. 186-191, 1919. Maliock used an exceedingly delicate optical method, in which interference-bands, produced by two contiguous glass plates, shew a visible displacement on the slightest angular movement of the plates, even of the order of a millionth of an inch. t W. S. Glock, A. E. Douglass and G. A. Pearson, Principles ... of tree-ring analysis, Carnegie Inst. Washington, No. 486, 1937; D. T. MacDougal, Tree Growth, Leiden, 1938, 240 pp. % Trans. Edinb. Botan. Soc. xviii, p. 456, 1891. 236 THE RATE OF GROWTH [CH. and Acacia; the deciduous trees included Quercus, Populus, Robinia and Melia. The result (Fig. 70) is much as we might expect. The deciduous trees cease to grow in winter-time, and during all the months when the trees are bare; 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 evergreens the amplitude of the Fig. 70. Periodic annual fluctuation in rate of growth of trees in the southern hemisphere. From C. E. Hall's data. annual wave is much less ; there is a notable amount of growth all the year round, and while there is a marked diminution in rate during the coldest months, there is a tendency towards equality over a considerable part of the warmer season. In short, the evergreens, at least in this case, do not grow the faster as the temperature continues to rise; and it seems probable that some of them, especially the pines, are definitely retarded in their growth, either by a temperature above their optimum or by a deficiency of moisture, during the hottest season of the year. Fig. 71 shews how a cypress never ceased to grow, but had alternate Ill OF THE GROWTH OF TREES 237 spells of quicker and slower growth, according to conditions of which we are not informed. Another figure (Fig. 72) illustrates the growth in three successive seasons of the Calif ornian redwood, a near ally of the most gigantic of trees. Evergreen though the redwood is, its growth has periods of abeyance; there is a second minimum about midsummer, and the chief maximum of the year may be that before or after this. Fig. 71. Growth of cypress (C. macrocarpa), shewing seasonal periodicity. From MacDougal's data: smoothed curve. 1931 1932 1933 Fig. 72. Fortnightly increase of girth in Californian redwood (Sequoia sempervirens), shewing seasonal periodicity. After MacDougal. In warm countries tree-growth is apt to shew a double maximum, for the cold of winter and the drought of summer are equally antagonistic tQ it. Trees grow slower — and grow fewer — the farther north we go, till only a few birches and willows remain, -stunted and old; it is nearly a hundred years ago since Auguste Bravais* shewed a steadily decreasing growth-rate in the forests between 50° and 70° N. * Recherches sur la croissance du pin silvestre dans le nord de I'Europe, Mdm. couronnees de VAcad. R. de Belgique, xv, 64 pp., 1840-41. 238 THE RATE OF GROWTH [CH. The delicate measuring apparatus now used shews sundry minor but beautiful phenomena. A daily periodicity of growth is a common thing* (Fig. 73). In the tree-cactuses the trunk expands by day and shrinks again after nightfall ; for the stomata close in sun- light, and transpiration is checked until the sun goes down. But it is more usual for the trunk to shrink from sunrise until evening and to swell from sunset until dawn ; for by dayhght the leaves lose 70° F Black Poplar feet above ground, 59 inches July 24 26 Fig. 73. Growth of black poplar, shewing daily periodicity. After A. Mallock. water faster, and in the dark they lose it slower, than the roots replace it. The rapid midday loss of water even at the top of a tail Sequoia is quickly followed by a measurable constriction of the trunk fifty or even a hundred yards below f. * The diurnal periodicity is beautifully shewn in the case of the hop by Johannes Schmidt, C.R. du Laboratoire Carlsberg, x, pp. 235-248, Copenhagen, 1913. •f This rapid movement is accounted for by Dixon and Joly's "cohesion-theory" of the ascent of sap. The leaves shew innumerable minute menisci, or cup-shaped water-surfaces, in their intercellular air-spaces. As water evaporates from these the little cups deepen, capillarity increases its pull, and suffices to put in motion the strands or columns of water which run continuously through the vessels of wood, and withstand rupture even under a pull of 100-200 atmospheres. See {int. al.) H. H. Dixon and J. Joly, On the ascent of sap, Phil. Trans. (B), clxxxvi, p. 563, 1895; also Dixon's Transpiration and the Ascent of Sap, 8vo, London, 1914. Ill] OF THE SECULAR GROWTH OF TREES 239 In the case of trees, the seasonal periodicity of growth and the direct influence of weather are both so well marked that we are entitled to make use of the phenomenon in a converse way, and to draw deductions (as Leonardo da Vinci did*) as to climate during past years from the varying rates of growth which the tree has recorded for us by the thickness of its annual rings. Mr A. E. Douglass, of the University .of California, has made a careful study of this question, and I received from him (through Professor H. H. Turner) some measurements of the average width of the annual rings in Californian redwood, five hundred yfears old, in which trees the rings are very clearly shewn. For the first hundred years the mean of two trees was used, for the next four hundred years the mean of five; and the means of these (and sometimes of larger numbers) were found to be very concordant. A correction was appHed by drawing a nearly straight fine through the curve for the whole period, which line was assumed to represent the slowly diminishing mean width of annual ring accompanying the increasing size, or age, of the tree; and the actual growth as measured was equated with this diminishing mean. The figures used give, then, the ratio of the actual growth in each year to the mean growth of the tree at that epoch. It was at once manifest that the growth-rate so determined shewed a tendency to fluctuate in a long period of between 100 and 200 years. I then smoothed the yearly values in groups of 100 (by Gauss's method of "moving averages"),, 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, so simply obtained, give us a curve of beautiful and surprising smoothness, from which we draw the direct conclusion that the climate of Arizona, during the last five hundred years, has fluctuated with a regular periodicity of almost precisely 150 years. I have drawn, more recently, and also from Mr Douglass's data, a similar curve for a group of pine trees in Calaveras County |. These trees are about 300 years old, and the * Cf. J. Playfair McMurrich, Leonardo da Vinci, 1930, p. 247. t When this was first written I had not seen Mr Douglass's paper On a method of estimating rainfall by the growth of trees. Bull. Amer. Geograph. Soc. XLVI, 240 THE RATE OF GROWTH [CH. data are reduced, as before, to moving averages of 100 years, but without further correction. The agreement between the growth- rate of these pines and that of the great Sequoias during the same period is very remarkable (Fig. 74). We should be left in doubt, so far as these observations go, whether the essential factor be a fluctuation of temperature or an alternation of drought and humidity; but the character of the Arizona chmate, and the known facts of recent years, encourage the behef that the latter is the more direct and more important factor. In a New England forest many trees of many kinds were studied after a hurricane; they shewed on the whole no correlation^ 1440 CQ Fig. 74. Long-period fluctuation in growth of Arizona redwood (Sequoia), from A.D. 1390 to 1910; and of yellow pine from Calaveras County, from a.d. 1620 to 1920. (Smoothed in 100-year periods.) between growth-rate and temperature, with the remarkable exception (in the conifers) of a clear correlation with the temperature of March and April, a month or two before the season's growth began. In a cold spring the melting snows and early rains ran off into the rivers, in a warm and early one they sank into the soil * ; in other words, humidity was still the controlling factor. An ancient oak' tree in Tunis is said to have recorded fifty years of abundant rain. pp. 321-335, 1914; nor, of course, his great work on Climatic cycles and tree- growth, Carnegie Inst. Publications, 1919, 1928, 1936. Mr Douglass does not fail to notice the long period here described, but he is more interested in the sunspot-cycle and other shorter cycles known to meteorologists. See also (int. al.) E. Huntingdon, The fluctuating climate of North America, Geograph. Journ. Oct. 1912; and Otto Pettersson, Climatic variation in historic and prehistoric time, Svens^M, Hydrografisk-Biolog. Skrifter, v, 1914. ♦ C. J. Lyon, Amer. Assoc. Rep. 1939; Nature, Apr. 13, 1940, p. 595. Ill] OF THE INFLUENCE OF LIGHT 241 with short intervals of drought, during the eighteenth century ; then, after 1790, longer droughts and shorter spells of rainy seasons*. It has been often remarked that our common European trees, such as the elm or the cherry, have larger leaves the farther north we go ; but the phenomenon is due to the longer hours of dayhght throughout the summer, rather than to intensity of illumination or diJBFerence of temperature. On the other hand, long dayhght, by prolonging vegetative growth, retards flowering and fruiting; and late varieties of soya bean may be forced into early ripeness by artificially shortening their dayhght at midsummer f. The effect of ultra-violet hght, or any other portion of the spectrum, is part, and perhaps the chief part, of the same problem. That ultra-violet hght accelerates growth has been shewn both in plants and animals f. In tomatoes, growth is favoured by just such ultra-violet hght as comes very near the end of the solar spectrum §, and as happens, also, to be especially absorbed by ordinary green- house glass II . At the other end of the spectrum, in red or orange light, the leaves become smaller, their petioles longer, the nodes more numerous, the very cells longer and more attenuated. It is a physiological problem, and as such it shews how plant-hfe is adapted, on the whole, to just such rays as the sun sends; but it also shews the morphologist how the secondary effects of chmate may so influence growth as to modify both size and form^. An analogous case is the influence of hght, rather than temperature, in modifying the coloration of organisms, such as certain butterflies. * Le chene Zeem d'Ain Draham, Bull, du Directeur General, Tunisie, 1927. t That the plant grows by turns in darkness and in light, and has its characteristic growth-phases in each, longer >or shorter according to species and variety and normal habitat, is a subject now studied under the name of "photoperiodism," and become of great practical importance for the northerly extension of cereal crops in Canada and Russia. Cf. R. G. Whyte and M. A. Oljhovikov, Nature, Feb. 18, 1939. I Cf. Kuro Suzuki and T. Hatano, in Proc. Imp. Acad, of Japan, in, pp. 94-96, 1927. § Withrow and Benedict, in Bull, of Basic Scient. Research, iii, pp. 161-174, 1931. II Cf. E. C. Teodoresco, Croissance des plantes aux lumieres de diverses longueurs d'onde, ^WTi. Sc. Nat, Bat. (8), pp. 141-336, 1929; N. Pfeiffer, Botan. Gaz. lxxxv, p. 127, 1929; etc. II See D. T. MacDougal, Influence of light and darkness, etc., Mem. N. Y. Botan. Garden, 1903, 392 pp.; Growth in trees, Carnegie Inst. 1921, 1924, etc.; J. Wiesner, Lichtgenuss der Pflunzen, vn, 322 pp., 1907; Earl S. Johnston, Smithson. Misc. Contrib. 18 pp., 1938; etc. On the curious effect of short spells of light and dark- ness, see H. Dickson, Proc. E.S. (B), cxv, pp. 115-123, 1938. 242 THE RATE OF GROWTH [ch. Now if temperature or light affect the rate of growth in strict uniformity, aHke in all parts and in all directions, it will only lead to local races or varieties differing in size, as the Siberian goldfinch or bullfinch differs from our own. But if there be ever so Httle of a discriminating tendency such as to enhance the growth of one tissue or one organ more than another*, then it must soon lead to racial, or even "specific," difference of form. It is hardly to be doubted that chmate has some such dis- criminating influence. The large leaves of our northern trees are an instance of it; and we have a better instance of it still in Alpine plants, whose general habit is dwarfed though their floral organs suffer little or no reduction f. Sunhght of itself would seem to be a hindrance rather than a stimulant to growth; and the familiar fact of a plant turning towards the sun means increased growth on the shady side, or partial inhibition on the other. More curious and still more obscure is the moon's influence on growth, as on the growth and ripening of the eggs of oysters, sea- urchins and crabs. Behef in such lunar influence is as old as Egypt; it is confirmed and justified, in certain cases, nowadays, but the way in which the influence is exerted is quite unknown J. Osmotic factors in growth The curves of growth which we have been studying have a twofold interest, morphological and physiological. To the morpho- logist, who has learned to recognise form as a "function of growth," the most important facts are these: (1) that rate of growth is an orderly phenomenon, with general features common to various organisms, each having its own characteristic rates, or specific constants; (2) that rate of growth varies with temperature, and so with season and with climate, and also with various other physical factors, external and internal to the organism ; (3) that it varies in different parts of the body, and along various directions or axes: * Or as we might say nowadays, have a different "threshold value" in one organ to another. t Cf. for instance, NageH's classical account of the effect of change of habitat on alpine and other plants, Sitzungsber. Baier. Akad. Wiss. 1865, pp. 228-284. X Cf. Munro Fox, Lunar periodicity in reproduction, Proc. R.S. (B), xcv, pp. 523-550, 1935; also Silvio Ranzi, Pubblic. Slaz. Zool. Napoli, xi, 1931. Ill] OF OSMOTIC FACTORS IN GROWTH 243 such variations being harmoniously "graded," or related to one another by a "principle of continuity," so giving rise to the characteristic form and dimensions of the organism and to the changes of form which it exhibits in the course of its development. To the physiologist the phenomenon of growth suggests many other considerations, and especially the relation of growth itself to chemical and physical forces and 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 stimuh, would be but an example of that "lack of particularity" with which we are apt to be all too easily satisfied. But in the case of growth we pass some little way beyond these limitations: to this extent, that an affinity with certain types of chemical and physical reaction has been recognised by a great number of physiologists*. A large part of the phenomenon of growth, 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 rate of growth depends (as we have already seen) as much or more on the amount of water taken up into the living cells f, as on the actual amount of chemical metabohsm performed by them; and sometimes, as in certain insect-larvae, we can even distinguish between tissues which grow by increase of cell-size, the result 'of imbibition, and others which grow by multiplication of their constituent cells |. Of the chemical phenomena which result in the * Cf. F. F. Blackman, Presidential Address in Botany, Brit. Assoc, Dublin, 1908. The idea was first enunciated by Baudrimont and St Ange, Recherches sur le developpement du foetus, Mem. Acad. Sci. xi, p. 469, 1851. t Cf. J. Loeb, Untersuckungen zur physiologischen Morphologie der Tiere, 1892; also Experiments on cleavage, Journ. Morphology, vii, p. 253, 1892; Ueber die Dynamik des tierischen Wachstums, Arch. f. Entw. Mech. xv, p. 669, 1902-3; Davenport, On the role of water in growth, Boston Soc. N.H. 1897; Ida H. Hyde in Amer. Journ. Physiology, xii, p. 241, 1905; Bottazzi, Osmotischer Druck und elektrische Leitungsfahigkeit der Fliissigkeiten der Organismen, in Asher-Spiro's Ergebnisse der Physiologic, vn, pp. 160-402, 1908; H. A. Murray in Journ. Gener. Physiology, ix, p. 1, 1925; J. Gray, The role of water in the evolution of the terrestrial vertebrates, Journ. Exper. Biology, vi, pp. 26-31, 1928; and A. N. J. Heyn, Physiology of cell-elongation, Botan. Review, vi, pp. 515-574, 1940. X Cf. C. A. Berger, Carnegie Inst, of Washington, Contributions to Embryology, xxvu, 1938. 244 THE RATE OF GROWTH [CH. actual increase of protoplasm we shall speak presently, but the role of water in growth deserves a passing word, even in our morpho- logical enquiry. The lower plants only Uve and grow in abundant moisture; lew fungi continue growing when the humidity falls below 85 per cent, of saturation, and the mould-fungi, such as Penicillium, need more moisture still (Fig. 75). Their hmit is reached a little below 90%. Humidity ^ I. 75. Growth of PeniciUium in relation to humidity. Growth of PeniciUium (at 25° C.) * Humidity Growth per hour (% of saturation) (mm.) 1000 7-7 97-0 5-0 94-2 1-0 926 0-5 90-8 0-3 Among th^ coelenterate animals growth and ultimate size depend on Httle more than absorption of water and consequent turgescence, the process, she wing itself in simple ways. A sea-anemone may live to ^n immense agef, but its age and size have Httle to do with one * From R. G. Tomkins, Studies of the growth of moulds, Proc. B.S. (B), cv, pp. 375-^01, 1929. t Like Sir John Graham Dalyell's famous "Granny," and Miss Nelson's family of CereiLS (not Sagartia) of which one still lives at over 80 years old. Cf. J. H. Ashworth and Nelson Annandale, in Trans. R. Physical Soc. Edin. xxv, pp. 1-14 1904. Ill] OF TURGESCENCE 245 another. It has an upper limit of size vaguely characteristic of the species, and if fed well and often it may reach it in a year ; on stinted diet it grows slowly or may dwindle down; it may be kept at wellnigh what size one pleases. Certain full-grown anemones were left untended in war-time, unfed and in water which evaporated down to half its bulk; they shrank down to little beads, and grew up again when fed and cared for. Loeb shewed, in certain zoophytes, that not only must the cells be turgescent in order to grow, but that this turgescence is possible only so long as the salt-water in which the cells lie does not overstep a certain limit of concentration: a limit reached, in the case of Tubularia, when the salinity amounts to about 3-4 per cent. Sea- water contains some 3-0 to 3-5 per cent, of salts in the open sea, but the salinity falls much below this normal, to about 2-2 per cent., before Tubularia exhibits its full turgescence and maximal growth; a further dilution is deleterious to the animal. It is likely enough that osmotic conditions control, after this fashion, the distribution and local abundance of many zoophytes. Loeb has also shewn* that in certain fish-eggs (e.g. of Fundulus) an increasing concentration, leading to a lessening water-content of the egg, retards the rate of segmentation and at last arrests it, though nuclear division goes on for some time longer. The eggs of many insects absorb water in large quantities, even doubling their weight thereby, and fail to develop if drought prevents their doing so ; and sometimes the egg has a thin- walled stalk, or else a "hydropyle," or other structure by which the water is taken inf. In the frog, according to Bialaszewicz J, the growth of the embryo while within the vitelline membrane depends wholly on absorption of water. The rate varies with the temperature, but the amount of water absorbed is constant, whether growth be fast or slow. Moreover, the successive changes of form correspond to definite quantities of water absorbed, much of which water is intracellular. The solid residue, as Davenport Has also shewn, may even diminish * P finger's Archiv, lv, 1893. t Cf. V. B. Wigglesworth, hised Physiology, 1939, p. 2. % Beitrage zur Kenntniss d. Wachstumsvorgange bei Amphibienerabryonen, Bull. Acad. Sci. de Cracovie, 1908, p. 783; also A. Drzwina and C. Bohn, De Taction. . .dea solutions salines sur les larves des batraciens, ibuL 1906. 246 THE RATE OF GROWTH [ch. notably, while all the while the embryo continues to grow in bulk and weight. But later on, and especially in the higher animals, the water-content diminishes as growth proceeds and age advances; and loss of water is followed, or accompanied, by retardation and cessation of growth. A crab loses water as each phase of growth draws to an end and the corresponding moult approaches; but it absorbs water in large quantities as soon as the new period of growth begins*. Moreover, that water is lost as growth goes on has been shewn by Davenport for the frog, by Potts for the chick, and particularly by Fehhng in the case of man. Fehhng's results may be condensed as follows : Age in weeks (man) 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 I The following illustrate Davenport's results for the frog: Age in weeks (frog) 1 2 5 7 9 14 41 84 Percentage of water 56-3 58-5 76-7 89-3 931 950 90-2 87-5 The following table epitomises the drying-off of ripening maize f; it shews how ripening and withering are closely akin, and are but two phases of senescence (Fig. 76): Days (from August 6) 0 22 35 49 56 63 Percentage of water 87 81 77 68 65 58 The bird's egg provides all the food and all the water which the growing embryo needs, and to carry a provision of water is the special purpose of the white of the egg ; the water contained in the albumen at the beginning of incubation is just about what the chick contains at the end. The yolk is not surrounded by water, which would diffuse too quickly into it, nor by a crystalloid solution, whose osmotic value would soon increase; but by a watery albu- minous colloid, whose osmotic pressure changes slowly as its charge of water is gradually withdrawn J. * Cf. A. Krogh, Osmotic regulation in aquatic animals, Cambridge, 1939. t Henry and Morrison, 1917; quoted by Otto Glaser, on Growth, time and form, Biolog. Reviews, xiii, pp. 2-58, 1938. X Cf. James Gray, in Journ. Exper. Biology, iv, pp. 214-225, 1926. Ill OF GAIN OR LOSS OF WATER 247 Distribution of water in a hen's egg Gm. of water contained in Day of incubation ^ Loss by evaporation Gain by combustion Albumen Embryo Yolk 0 29-9 0-0 8-5 00 00 6 27-2 0-4 8-45 2-4 001 12 20-4 4-6 7-8 5-6 0-27 18 9-2 181 2-3 8-8 1-20 20 2-2 27-4 10 9-8 2-00 The actual amount of water, compared with the dry solids in the egg, has been determined as follows: Day of incubation (chick) 5 8 11 14 17 19 Percentage of water 94-7 93-8 92-3 87-7 82-8 82-3 10 20 30 40 Days Fig. 76. Percentage of water in ripening maize. From Otto Glaser. We know very httle of the part which all this water plays: how much is mere "reaction-medium," how much is fixed in hydrated colloids, how much, in short, is bound or unbound. But we see that somehow or other water is lost, and lost in considerable amount, as the embryo draws towards completion and ceases for the time being to grow. All vertebrate animals contain much the same amount of water in their living bodies, say 85 per cent, or thereby, however unequally 248 THE RATE OF GROWTH [ch. distributed in the tissues that water may be*. Land animals have evolved from water animals with Httle change in this respect, though the constant proportion of water is variously achieved. A. newt loses moisture by evaporation with the utmost freedom, and regains it by no less rapid absorption through the skin; while a lizard in his scaly coat is less hable to the one and less capable of the other, and must drink to replace what water it may lose. We are on the verge of a difficult subject when we speak of the role of water in the living tissues, in the growth of the organism, and in the manifold activities of the cell ; and. we soon learn, among other more or less unexpected things, that osmotic equihbrium is neither universal nor yet common in the living organism. The yolk maintains a higher osmotic pressure than the white of the egg — so long as the egg is living; and the watery body of a jellyfish, though aot far off osmotic equilibrium, has a somewhat less salinity than the sea-water. In other words, its surface acts to some extent as a semipermeable membrane, and the fluid which causes turgescence of the tissues is less dense than the sea- water outside!. In most marine invertebrates, however, the body-fluids con- stituting the milieu interne are isotonic with the milieu externe, and vary in these animals pari passu with the large variations to which sea- water itself is subject. On the other hand, the dwellers in fresh-water, whether invertebrates or fishes, have, naturally, a more concentrated medium within than without. As to fishes, diiferent kinds shew remarkable differences. Sharks and dogfish have an osmotic pressure in their blood and their body fluids little diiferent * The vitreous humour is nearly all water, the enamel has next to 'none, the grey matter has some 86 per cent., the bones, say 22 per cent. ; lung and kidney take up mor^ than they can hold, and so become excretory or regulatory organs. Eggs, whether of dogfish, salmon, frogs, ' snakes or birds, are composed, roughly speaking, of half water and half solid matter. t Cf. {int. al.) G. Teissier, Sur la teneur en eau. . .de Chrysaora, Bull. Soc. Biol, de France, 1926, p. 266. And especially A. V. Hill, R. A. Gortner and others. On the state of water in colloidal and living systems. Trans. Faraday Soc. xxvi, pp. 678-704, 1930. For recent literature see (e.g.) Homer Smith, in Q. Rev. Biol. vn, p. 1, 1932; E. K. Marshall, Physiol. Rev. xiv, p. 133, 1934; Lovatt Evans, Recent Advances in Physiology, 4th ed., 1930; M. Duval, Recherches. . .sur le milieu interieur des animaux aquatiques, Thhe, Paris, 1925; Paul Portier, Physiologic des animaux marins. Chap, iii, Paris, 1938; G. P. Wells and I. C. Ledingham, Effects of a hypotonic environment, Journ. Exp. Biol, xvii, pp. 337-352, 1940. Ill] OF OSMOTIC REGULATION 249 from that of the sea-water outside: but with certain chemical diiferences, for instance that the chlorides within are much diminished, and the molecular concentration is eked out by large accumulations of urea in the blood. The marine teleosts, on the other hand, have a much lower osmotic pressure within than that of the sea-water outside, and only a httle higher than that of their fresh-water allies. Some, hke the conger-eel, maintain an all but constant internal concentration, very different from that outside; and this fish, like others, is constantly absorbing water from the sea ; it must be exuding or excreting salt continually*. Other teleosts differ greatly in their powers of regulation and of tolerance, the common stickleback (which we may come across in a pool or in the middle of the North Sea) being exceptionally tolerant or "eury- halinef." Physiology becomes "comparative" when it deals with differences such as these, and Claude Bernard foresaw the existence of just such differences: "Chez tous les etres vivants le milieu interieur, qui est un produit de I'organisme, conserve les rapports necessaires d'echange avec le milieu exterieur; mais a mesure que I'organisme devient plus parfait le milieu organique se specifie, et s'isole en quel que sorte de plus en plus au milieu ambiant J." Claude Bernard was building, if I mistake not, on Bichat's earUer concept, famous in its day, of life as "une alternation habituelle d'action de la part des corps exterieurs,* et de reaction de la part du corps vivant": out of which grew his still more famous aphorism, "La vie est I'ensemble des fonctions qui resistent a la mort§." One crab, like one fish, differs widely from another in its power * Probably by help of Henle's tubules in the kidney, which structures the dogfish does not possess. But the gills have their part to play as water-regulators, as also, for instance, in the crab. t The grey mullets go down to the sea to spawn, but may live and grow in brackish or nearly fresh- water. The several species differ much in their adaptability, and Brunelli sets forth, as follows, the range of salinity which each can tolerate : M. auratus 24-35 per mille saliens 16-40 chelo 10-40 capita 5-40 cephalus 4^0 X Introduction d Vetude de la medecine experimentale, 1855, p. 110. For a dis- cussion of this famous concept see J. Barcroft, "La fixite du milieu interieur est la condition de la vie libre," Biol. Reviews, viii, pp. 24-87, 1932. § Sur la vie et la mart, p. 1. 250 THE RATE OF GROWTH [ch. of self-regulation; and these physiological differences help to explain, in both cases, the limitation of this species or that to more or less brackish, or more or less saline, waters. In deep-sea crabs {Hyas, for instance) the osnjotic pressure of the blood keeps nearly to that of the milieu exteme, and falls quickly and dangerously with any dilution of the latter; but the httle shore-crab (Cardnus moenas) can hve for many days in sea-water diluted down to one-quarter of its usual sahnity. Meanwhile its own fluids dilute slowly, but not near so far; in other words, this crab combines great powers of osmotic regulation with , a large capacity for tolerating osmotic gradients which are beyond its power to regulate. How the unequal balance is maintained is yet but httle understood. But we do know that certain organs or tissues, especially the gills and the antennary gland, absorb, retain or ehminate certain elements, or certain ions, faster than others, and faster than mere diffusion accounts for; in other words, "ionic*' regulation goes hand in hand with "osmotic" regulation, as a distinct and even more fundamental phenomenon*. This at least seems generally true — and only natural — that quickened respiration and increased oxygen-consumption accompany all such one-sided conditions: in other words, the "steady state" is only maintained by the doing of work and the expenditure of energyf. To the dependence of growth on the uptake of water, and to the phenomena of osmotic balance and its regulation, HoberJ and also Loeb were inclined to refer the modifications of form which certain phyllopod Crustacea undergo when the highly sahne waters which they inhabit are further concentrated, or are abnormally diluted. Their growth is retarded by increased concentration, so that individuals from the more saline waters appear stunted and dwarfish ; and they become altered or transformed in other ways, suggestive of "degeneration," or a failure to attain full and perfect develop- * See especially D. A. Webb, Ionic regulation in Cardnus moenas, Proc. R.S. (B), cxxix, pp. 107-136, 1940. t In general the fresh- water Crustacea have a larger oxygen -consumption than the marine. Stenohaline and euryhaline are terms applied nowadays to species which are. confined to a narrow range of salinity, or are tolerant of a wide one. An extreme case of toleration, or adaptability, is that of the Chinese woolly-handed crab, Eriockeir, which has not only acclimatised itself in the North Sea but has ascended the Elbe as far as Dresden. X R. Hober, Bedeutung der Theorie der Losungen fiir Physiologic und Medizin, Biol. Centralbl. xix, p. 272, 1899. Ill] OF OSMOTIC REGULATION 251 merit*. Important physiological changes ensue. The consumption of oxygen increases greatly in the stronger brines, as more and more active " osmo-regulation " is required. The rate of multiplication is increased, and parthenogenetic reproduction is encouraged. In the less sahne waters male individuals, usually rare, become plentiful, and here the females bring forth their young alive ; males disappear altogether in the more concentrated brines, and then the females lay eggs, which, however, only begin to develop when the sahnity 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 nearly two hundred years ago in the salt-pans at Lymington. Among many allied forms, one, A. milhausenii, inhabits the natron- lakes of Egypt and Arabia, where, under the name of "loul," or "Fezzan-worm," it is eaten by the Arabsf. 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 hves. In short Artemia, hke teleostean fishes in the sea, hves constantly in a "hypertonic medium"; the fluids of the body, the milieu interne, are no more salt than are those of any ordinary crustacean or other animal, but contain only some 0-8 per cent, of NaCl J , while the milieu externe may contain from 3 to 30 per cent, of this and other salts; 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 freely. When brought into a lower concentration the animal may grow large and turgescent, until a statical equilibrium, or steady state, is at length attained. Among the structural changes which result from increased con- * Schmankewitsch, Zeitschr. f. wiss. Zool. xxix, p. 429, 1877. Schraankewitsch has made equally interesting observations on change of size and form in other organisms, after some generations in a milieu of altered density ; e.g. in the flagellate infusorian Ascinonema acinus Biitschli. t 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. X See D. J. Kuenen, Notes, systematic and physiological, on Artemia, Arch. N4erland. Zool. iii, pp. 365-449, 1939; cf. also Abonyi, Z.f. w. Z. cxiv, p. 134, 1915. Cf. Mme. Medwedewa, Ueber den osmotischen Druck der Haemolymph v. Artemia; in Ztsch. f. vergl. Physiolog. v, pp. 547-554, 1922. 252 THE RATE OF GROWTH [CH. centration of the brine (partly during the hfe-time of the individual, but more markedly during the short season which suffices for the development of three or four, or perhaps more, successive genera- tions), 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 described as the specific characters of A. milhausenii, and of a still more extreme form, A. koppeniana] 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 u ^wwwWWWH I I Artewia s.str. Callaonella Fig. 77. 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. referred to a separate genus, Callaonella, closely akin to Branchipus (Fig. 77). 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 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 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 Ill] OF THE BRINE-SHRIMPS 253 curve (Fig. 78). In short, the density of the water is so clearly "specific," that we might briefly define Artemia jelskii, for instance, as the Artemia of density 1000-1010 (NaCl), or all but fresh water, and the typical A. salina (or principalis) as the Artemia of density 1018-1025, and so on*. Koppeniana 160 """^ 140 120 100 1000 1020 1080 1100 1040 1060 Density of water Fig. 78, Percentage ratio of length of abdomen to cephalothorax in brine -shrimps, at various salinities. After Abonyi. These Artemiae are capable of living in waters not only of great density, but of very varied chemical composition, and it is hard to say how far they are safeguarded by semi-permeabihty or by specific properties and reactions of the living colloids "j". The natron-lakes, * Different authorities have recognised from one to twenty species of Artemia. Daday de Dees {Ann. sci. nat. 1910) reduces the salt-water forms to one species with four varieties, but keeps A. jelskii in a separate sub-genus. Kuenen suggests two species, A. salina and gracilis, one for the European and one for the American forms. According to Schmankewitsch every systematic character can be shewn to vary with the external medium. Cf. Professor Labbe on change of characters, specific and even generic, of Copepods according to the ^H of saline waters at Le Croisic, Nature, March 10, 1928. t We may compare Wo. Ostwald's old experiments on Daphnia, which died in a pure solution of NaCl isotonic with normal sea-water. Their death was not to be explained on osmotic grounds; but was seemingly due to the fact that the organic gels do not retain their normal water- content save in the presence of such concentrations of MgClj (and other salts) as are present in sea-water. 254 THE RATE OF GROWTH [ch. for instance, contain large quantities of magnesium sulphate; and the Artemiae continue to live equally well in artificial solutions where this salt, or where calcium chloride, has largely replaced the common salt of the more usual habitat. Moreover, such waters as those of the natron-lakes are subject to great changes of chemical composition as evaporation and concentration proceed, owing to the different solubilities of the constituent salts; but it appears that the forms which the Artemiae assume, and the changes which they undergo, are identical, or 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 concen- trations (that is to say, under conditions where the osmotic pressure, and consequently the rate of diffusion, is identical), that the same changes of form and structure are produced and corresponding phases of equihbrium attained. Sea-water has been described as an instance of the "fitness of the environment*" for the maintenance of protoplasm in an appropriate milieu; but our Artemias suffice to shew how nature, when hard put to it, makes shift with an environment which is wholly abnormal and anything but "fit." While Hober and others f 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. We should have to look far in organic nature for what the physicist would call simple osmosis % ; and assuredly there is always at work, besides the passive phenomena of intermolecular * L. H. Henderson, The Fitness of the Environment, 1913. t Cf. Schmankewitsch, Z. f. w. Zool. xxv, ISTi); xxix, 1877,. etc.; transl. in appendix to Packard's Monogr. of N. American Phyllopoda, 1^83, pp. 466-514; Daday de Dees, Ann. Sci. Nat. (Zool), (9), xi, 1910; Samter und Heymons, Abh. d. K. pr. Akad. Wiss. 1902; Bateson, Mat. for the Study of Variation, 1894, pp. 96-101; Anikin, Mitlh. Kais. Univ. Tomsk, xiv: Zool. Centralhl. vi, pp. 756-760, 1908; Abonyi, Z.f. w. Zool. cxiv, pp. 96-168, 1915 (with copious bibliography), etc. % Cf. C. F. A. Pantin, Body fluids in animals, Biol. Reviews, \i, p. 4, 1931; J. Duclaux, Chimie apjMquee a la biologic, 1937, ii, chap. 4. Ill] OF CATALYTIC ACTION 255 diffusion, some other activity to play the part of a regulatory mechanism*. On growth and catalytic action In ordinary chemical reactions we have to deal (1) with a specific velocity proper to the particular reaction, (2) with variations due to temperature and other physical conditions, (3) with variations due to the quantities present of the reacting substances, according to Van't Hoff's "Law of Mass Action," and (4) in certain cases with variations due to the presence of "catalysing agents," as BerzeHus called them a hundred years agof. In the simpler reactions, the law of mass involves a steady slo wing-down of the process as the reaction proceeds and as the initial amount of substance diminishes: a phenomenon, however, which is more or less evaded in the organism, part of whose energies are devoted to the continual bringing-up of supphes. Catalytic action occurs when some substance, often in very minute quantity, is present, and by its presence produces or accelerates a reaction by opening "a way round," without the catalysing agent itself being diminished or used up J. It diminishes the resistance somehow — little as we know what resistance means * According to the empirical canon of physiology, that, as Leon Fredericq expresses it (Arch, rfc Zool. 1885), '*L'etre vivant est agence de telle maniere que chaque influence pertyrbatrice provoque d'elle-meme la mise en activite de Tappareil compensateur qui doit neutraliser et reparer le dommage." Herbert Spencer had conceived a similar principle, and thought he recognised in it the vis medicutrix. Nahirae. It is the physiological analogue of the "principle of Le Chatelier " (1888), with this important difference that the latter is a rigorous and quantitative law, ba8e^ / /Calculauted (autocaialytic) / curve • / •/ y^\ 1 1 1 1 1 1 1 1 1 t 1 Weeks 1 10 li 12 Fig. 80. Growth of sunflower-stem : calculated (autocatalytic) curve. After Reed and Holland. 260 THE RATE OF GROWTH [ch. But if the substance produced exercise a catalytic effect, then the velocity will vary not only as above but will also increase as x increases: the equation becomes V = -T- ^k'x(a — x), Cut which is the elementary equation of autoca.talysis. Integrating, at a — X In our growth-problem it is sometimes found convenient to choose for our epoch, t', the time when growth is half-completed, as the chemist takes the time at which his reaction is half-way through; and we may then write (with a changed constant) This is the physico-chemical formula which Reed and Holland apply to the growing sunflower-stem — a simple case*. For a we take the maximum height attained, viz. 254-5 cm. ; for t\ the epoch when one-half of that height was reached, viz. (by interpolation) about 34-2 days. Taking an observation at random, say that for the 56th day, when the stem was 228-3 cm. high, we have K in this case is found to be 0-043, and the mean of all such determinations t is not far difierent. Applying this formula to successive epochs, we get a calculated curve in close agreement with the observed one; and by well- known statistical methods we confirm, and measure, its "closeness of fit." But jtist as the chemist must vary and develop his funda- mental formula to suit the course of more and more comphcated reactions, so the biologist finds that only the simplest of his curves * H. S. Reed and R. H. Holland, The growth-rate of an annual plant, Helianthus, Proc. Nat. Acad, of Sci. (Washington), v, p. 135, 1919; cf. Lotka, op. cit., p. 74, A sifnilar case is that of a gourd, recorded by A. P. Anderson, Bull. Survey, Minnesota, 1895, and analysed by T. B. Robertson, ibid. pp. 72-75. t Better determined, especially in more complex cases, by the method of least squares. Ill] ITS CHEMICAL ASPECT 261 of growth, or only portions of the rest, can be fitted to this simplest of formulae. In a Hfe-time are many ages; and no all-embracing formula covers the infant in the womb, the suckling child, the growing schoolboy, the old man when his work is done. Besides, we need such a formula as a biologist can understand ! One which gives a mere coincidence of numbers may be of little use or none, unless it go some way to depict and explain the modus operandi of growth. As d'Ancona puts it : "II importe d'apphquer des formules qui correspondent non seulement au point de vue geometrique, mais soient representees par des valeurs de signification biologique." A mere curve-diagram is better than an empirical formula; for it gives us at least a picture of the phenomenon, and a qualitative answer to the problem. Growth of sunflower-stem. (After Reed and Holland) 1st diff. 15-8 24-4 33-3 39-2 38-4 31-6 22-6 14-4 8-5 4-9 2-8 The chemical aspect of growth As soon as we touch 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 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 mainly as a help to our study and comprehension of form. The whole question of diet, of overfeeding and underfeeding*, would present * For example, A. S. Parker has shewn that mice suckled by rats, and conse- quently much overfed, grow so quickly that in three weeks they reach double their normal weight; but their development is not accelerated; Ann. Appl. Biol, xvi, 1929. Height (( cm.) j^ > (days) Observed Calculated 7 17-9 21-9 14 34-4 37-7 21 67-8 62-1 28 981 95-4 35 1310 134-6 42 1690 173-0 49 205-5 204-6 56 228-3 227-2 63 247-1 241-6 70 250-5 250-1 77 253-8 255-0 84 254-5 257-8 262 THE RATE OF GROWTH [ch. itself for discussion*. But without opening up this large subject, we may say one more passing word on the remarkable fact that certain chemical substances, or certain physiological secretions, have the power of accelerating or of retarding or in some way regulating growth, and of so influencing the morphological features of the organism. To begin with there are numerous elements, such as boron, manganese, cobalt, arsenic, which serve to stimulate growth, or whose complete absence impairs or hampers it; just as there are a few others, such as selenium, whose presence in the minutest quantity is injurious or pernicious. The chemistry of the hving body is more complex than we were wont to suppose. Lecithin was shewn long ago to have a remarkable power of stimulating growth in animals t, and accelerators of plant-growth, foretold by Sachs, were demonstrated by Bottomley and others J; the several vitamins are either accelerators of growth, or are indis- pensable in order that it may proceed. In the little duckweed of our ponds and ditches [Lemna minor) the botanists have found a plant in which growth and multiplication are reduced to very simple terms. For it multiplies by budding, grows a rootlet and two or three leaves, and buds again; it is all young tissue, it carries no dead load; while the sun shines it has no lack of nourishment, and may spread to the limits of the pond. In one of Bottomley's early experiments, duckweed was grown (1) in a "culture solution" without stint of space or food, and (2) in the same, with the addition of a little bacterised peat or "auximone." In both cases the little plant spread freely, as in the first, or Malthusian, phase of a population curve; but the peat greatly accelerated the rate, which was not slow before. Without the auximone the population doubled in nine or ten days, and with it in five or six; but in two months the one was seventy-fold the other ! The subject has grown big from small beginnings. We know certain substances, haematin being one, which stimulate the growth of bacteria, and seem to act on them as true catalysts. An obscure but complex body known as "bios" powerfully stimulates the growth of yeast; and the so-called auxins, a name which covers numerous bodies both nitrogenous and non-nitrogenous, serve in * For a brief resume of this subject see Morgan's Experimental Zoology, chap. xvi. t Hatai, Amer. Joum. Physiology, x, p. 57, 1904; Danilewsky, C.R. cxxi, cxxn, 1895-96. X W. B. Bottomley, Proc. R.S. (B), lxxxviii, pp. 237-247, 1914, and other papers. O. Haberlandt, Beitr. z. allgem. Botanik, 1921. Ill] OF HORMONES 263 minute doses to accelerate the growth of the higher plants*. Some of these "growth-substances" have been extracted from moulds or from bacteria, and one remarkable one, to which the name auxin is especially applied, from seedhng oats. This last is no enzyme but a stable non-nitrogfenous substance, which seems to act by softening the cell- wall and so facihtating the expansion of the cell. Lastly the remarkable discovery has been made that certain indol- compounds, comparatively simple bodies, act to all intents and purposes in the same way as the growth-hormones or natural auxins, and one of these "hetero-auxins." an indol-acetic acidf, is already in common and successful use to promote the growth and rooting of cuttings. Growth of duckweed, with and without peat-auodmone Without With ^ ^ eeks Obs. Calc. Obs. Calc. 0 20 20 20 20 1 30 33 38 55 2 52 54 102 153 .1 77 88 326 424 4 135 155 1,100 1,173 5 211 237 3,064 3,250 6 326 390 6,723 8,980 7 550 640 19,763 2,490 8 1052 1048 69,350 68,800 Percentage increase, 164 o/o 2770/0 per week There are kindred matters not less interesting to the morphologist. It has long been known that the pituitary body produces, in its anterior lobe, a substance by which growth is increased and regulated. This is what we now call a "hormone" — a substance produced in one organ or tissue and regulating the functions of another. In this case atrophy of the gland leaves the subject a dwarf, and its hyper- * The older literature is summarised by Stark, Ergebn. d. Biologies n, 1906; the later by N. Nielsen, Jh. wiss. Botan. Lxxm, 1930; by Boyson Jensen, Die Wuchsstojftheorie, 1935; by F. W. Went and K. V. Thimann, Phytohormones, New York, 1937, and by H. L. Pearse, Plant hormones and their practical importance. Imp. Bureau of Horticulture, 1939. Cf. Went, Bee. d. Trav. Botan. Neerl. xxv, p. 1, 1928; A. N. J. Heyn, ibid, xxviii, p. 113, 1931. t Discovered by Kogl and Kostermans, Ztschr. f. physiol. Chem. ccxxxv, p. 201, 1934. Cf. {int. al.) P. W. Zimmermann and F. W. Wilcox in Contrib. Boyce- Thompson Instil. 1935. 264 THE RATE OF GROWTH 264 trophy or over-activity goes to the making of a giant; the Umb- bones of the giant grow longer, their epiphyses get thick and clumsy, and the deformity known as "acromegaly" ensues*. This has become a famihar illustration of functional regulation, by some glandular or "endocrinal" secretion, some enzyme or harmozone as Gley called it, or hormone'f as Bayliss and Starhng called it — in the particular case where the function to be regulated is growth, with its consequent influence on form. But we may be sure that this so-called regulation of growth is no simple and no specific thing, but imphes a far-reaching and complicated influence on the bodily metabohsmj. Some say that in large animals the pituitary is. apt to be dispro- portionately large §; and the giant dinosaur Branchiosaurus, hugest of land animals, is reputed to have the largest hypophyseal recess (or cavity for the pituitary body) ever observed. The thyroid also has its part to play in growth, as Gudernatsch was the first to shew||; perhaps it acts, as Uhlenhorth suggests, by releasing the pituitary hormone. In a curious race of dwarf frogs both thyroid and pituitary were found to be atrophied ^. When tadpoles are fed on thyroid their legs grow out long before the usual time; on the other hand removal of the thyroid delays metamor- phosis, and the tadpoles remain tadpoles to an unusual size**. The great American bull-frog (R. Catesheiana) fives for two or three years in tadpole form; but a diet of thyroid turns the little tadpoles into bull-frogs before they are a month old If- The converse * Cf. E. A. Schafer, The function ofjthe pituitary body, Proc. R.S. (B), lxxxi, p. 442, 1904. t It is not easy to dtaw a line between enzyme and vitamin, or between hormone and enzyme. X The -physiological relations between insulin and the pituitary body might seem to indicate that it is the carbohydrate metabolism which is more especially concerned. Cf. (e.g.) Eric Holmes, Metabolism of the Living Tissue, 1937. § Van der Horst finds this to be the case in Zalophus and in the ostrich, compared with smaller seals or birds; cf. Ariens Kappers, Journ. Anat. lxiv, p. 256, 1930. II Gudernatsch, in Arch. f. Entw. Mech. xxxv, 1912. <;; Eidmann, ibid, xlix, pp. 510-537, 1921. ** Allen, Journ. Exp. Zod. xxiv, p. 499, 1918. Cf. [int. al.) E. Uhlenhuth, Experimental production of gigantism, Journ. Gen. Physiol, ill, p. 347; iv, p. 321, 1921-22. tt W. W. Swingle, Journ. Exp. Zool. xxiv, 1918; xxxvn, 1923; Journ. Gen. Physiol. I, II, 1918-19; etc! Ill] THE THYROID GLAND . 265 experiment has been performed on ordinary tadpoles*; with their thyroids removed they remain normal to all appearance, but the weeks go by and metamorphosis does not take place. Gill-clefts and tail persist, no limbs appear, brain and gut retain their larval features; but months after, or apparently at any time, the belated tadpoles respond to a diet of thyroid, and may be turned into frogs by means of it. The Mexican axolotl is a grown-up tadpole which, when the ponds dry up (as they seldom do), completes its growth and turns into a gill-less, lung-breathing newt or salamander!; but feed it on thyroid, even for a single meal, and its metamorphosis is hastened and ensured {. Much has been done since these pioneering experiments, all going to shew that the thyroid plays its active part in the tissue-changes which accompany and constitute metamorphosis. It looks as though more thyroid meant more respiratory activity, more oxygen- consumption, more oxidative metabohsm, more tissue-change, hence/ earlier bodily development §. Pituitary and thyroid are very different things; the one enhances growth, the other retards it. Thyroid stimulates metabolism and hastens development, but the tissues waste. It is a curious fact, but it has often been observed, that starvation or inanition has, in the long run, a similar effect of hastening metamorphosis II . The meaning of this phenomenon is unknown. An extremely remarkable case is that of the "galls", brought into existence on various plants in response to the prick of a small insect's ovipositor. One tree, an oak for instance, may bear galls * Bennett Allen, Biol. Bull, xxxii, 1917; Journ. Exp. Zool. xxiv, 1918; xxx, 1920; etc. t Colorado axolotls are much more apt to metamorphose than the Mexican variety. J Babak, Ueber die Beziehung der Metamorphose . . . zur inneren Secretion, Centralbl f. Physiol, x, 1913. Cf. Abderhalden, Studien iiber die von einzelrien Organen hervorgebrachten Substanzen mit spezifischer Wirkung, Pfliiger's Archiv, CLXii, 1915. § Certain experiments by M. Morse {Journ. Biol. Chem. xix, 1915) seeihed to shew that the effect of thyroid on metamorphosis depended on iodine; l;^t the case is by no means clear (cf. 0. Shinryo, Sci. Rep. Tohoku Univ. iii, 1928, and others). The axolotl is said to shew little response to experimental iodine, and its ally Necturus none at all (cf. B. M. Allen, in Biol. Reviews, xiii, 1939). !| Cf. Krizensky, Die beschleunigende Ein wirkung des Hungerns auf die Meta- morphose, Biol. Centralbl. xi.iv, 1914. Cf. antea, p. 170. 266 THE RATE OF GROWTH [ch. of many kinds, well-defined and widely different, each caused to grow out of the tissues of the plant by a chemical stimulus contri- buted by the insect, in very minute amount; and the insects are so much alike that the galls are easier to distinguish than the flies. The same insect may produce the same gall on different plants, for instance on several species of willow ; or sometimes on different parts, or tissues, of the same plant. Small pieces of a dead larva have been used to infect a plant, and a gall of the usual kind has resulted. Beyerinck killed the eggs with a hot wire as soon as they were deposited in the tree, yet the galls grew as usual. Here, as Needham has lately pointed out, is a great field for reflection and future experiment. The minute drop of fluid exuded by the insect has marvellous properties. It is not only a stimulant of growth, like any ordinary auxin or hormone; it causes the growth of a peculiar tissue, and shapes it into a new and specific form*. Among other illustrations (which are plentiful) of the subtle influence of some substance upon growth, we have, for instance, the growth of the placental decidua, which Loeb shewed to be due to a substance given off by the corpus luteum, lending to the uterine tissues an enhanced capacity for growth, to be called into action by contact with the ovum or even of a foreign body. Various sexual characters, such as the plumage, comb and spurs of the cock, arise in hke manner in response to an internal secretion or "male hormone " ; and when castration removes the source of the secretion, well-known morphological changes take place. When a converse change takes place the female acquires, in greater or less degree, characters which are proper to the male: as in those extreme cases, known from time immemorial, when an old and barren hen assumes the plumage of the cockf. The mane of the lion, the antlers of the stag, the tail of the peacock, are all examples of intensifled differential growth, or localised and * Joseph Needham, Aspects nouveaux de la chimie et de la biologic de la croia- sance organisee. Folia Morphologica, Warszawa, viii, p. 32, 1938. On galls, see {int. al.) Cobbold, Ross und Hedicke, Die Pflanzengallen, Jena, 1927; etc. And on their "raorphogenic stimulus", cf. Herbst, Biolog. Cblt., 1894-5, passim. t The hen which assumed the voice and plumage of the male was a portent or omen — gallina cecinit. The first scientific account was John Hunter's celebrated Account of an extraordinary pheasant, and Of the appearance of the change of sex in Lady Tynte's peahen, Phil. Trans, lxx, pp. 527, 534, 1780. Ill] THE MALE HORMONES 267 sex-linked hypertrophy; and in the singular and striking plumage of innumerable birds we may easily see how enhanced growth of a tuft of feathers, perhaps exaggeration of a single plume, is at the root of the whole matter. Among extreme instances we may think of the immensely long first primary of the pennant-winged nightjar; of the long feather over the eye in Pteridophora alberti, Fig. 81. A single pair of hypertrophied feathers in a bird-of-paradise, Pteridophora alberti. .-I Fig. 82. Unequal growth in the three pairs of tail-feathers of a humming-bird {Loddigesia). 1, rudimentafy: 2, short and stiff; 3, long and spathulate. or the Fix long plumes over or behind the eye in the six-shafted bird-of-paradise; or among the humming-birds, of the long outer rectrix in Lesbia, the second outer one in Aethusa, or of the extra- ordinary inequalities of the tail-feathers of Loddigesia mirabilis, some rudimentary, some short and straight and stiff, and other two immensely elongated, curved and spathulate. The sexual, hormones have a potent influence on the plumage of a>bird ; they serve, somehow, to orientate and regulate the rate of growth from one feather-tract to another, and from one end to another, even from one side to the other, of a single feather. An extreme case is the occasional pheno- 268 THE RATE OF GROWTH [ch. menon of a " gynandrous " feather, male and female on two sides of the same vane*. While unequal or differential growth is of pecuhar interest to the morphologist, rate of growth pure and simple, with all the agencies which control or accelerate it, remains of deeper importance to the practical man. The live-stock breeder keeps many desirable quahties in view: constitution, fertility, yield and quahty of milk or wool are some of these; but rate of growth, with its corollaries of early maturity and large ultimate size, is generally more important than them all. The inheritance of size is somewhat complicated, and limited from the breeder's point of view by the mother's inability to nourish and bring forth a crossbred offspring of a breed larger than her own. A cart mare, covered by a Shetland sire, produces a good-sized foal; but the Shetland mare, crossed with a carthorse, has a foal a little bigger, but not much bigger, than herself (Fig. 83). In size and rate of growth, as in other qualities, our farm animals differ vastly from their wild progenitors, or from the " un-improved " stock in days before Bake well and the other great breeders began. The improvement has been brought about by "selection"; but what lies behind? Endocrine secretions, especially pituitary, are doubtless at work; and already the stock- raiser and the biochemist may be found hand in hand. If we once admit, as we are now bound to do, the existence of factors 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 saysf that by these chemical influences "Toute une partie de la construction des etres parait s'expliquer d'une fayon toute mecanique. La forteresse, si longtemps inaccessible, du vitahsme est entamee. Car la notion morphogenique etait, suivant le mot de Dastre J , comme ' le dernier reduit de la force vitale'." * See an interesting paper by Frank R, Lillie and Mary Juhn, on The physiology of development of feathers: I, Growth-rate and pattern in the individual feather. Physiological Zoology, v, pp. 124-184, 1932, and many papers quoted therein. f Le Neo-vitalisme, Revue Scientifique, March 1911. X La Vie et la Mart, 1902, p. 43. Ill] OF INHERITANCE OF SIZE 269 The physiological speculations we need not discuss: but, to take a single example from morphology, we begin to understand the possibihty, and to comprehend the probable meaning, of the all but sudden appearance on the earth of such exaggerated and almost monstrous forms as those of the great secondary reptiles and the 500 Pure Shetland 10 20 Age, in months 30 40 Fig. 83. Effect of cross-breeding on rate of growth in Shetland ponies. From Walton and Hammond's data.* great tertiary mammals f. 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 * Walton and Hammond, Proc. R.S. (B), No. 840, p. 317, 1938. t Cf. also Dendy, Evolutionary Biology, 1912, p. 408. 270 THE RATE OF GROWTH [ch. is not to originate* but to remove: donee ad interitum 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 hmited 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 hfe may be changed, and many a refuge found, before the sentence of unfitness is pronounced and the penalty of exter- mination paid. But there comes a time when "variation," in form, dimensions, or other qualities of the organism, goes further 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 Hmits which, as again Lucretius tells us, natural law has set between what may and what may not be, et quid quaeque queant per foedera natural 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 is not to create but to destroy — to weed, to prune, to cut down and to cast into the fire J. * So said Yves Delage {UherediU, 1903, p. 397): "La selection naturelle est un principe admirable et parfaitement juste. Tout le monde est d'accord sur ce point. Mais ou Ton n'est pas d'accord, c'est sur la limite de sa puissance et sur la question de savoir si elle pent engendrer des formes specifiques nouvelles. II semble bien demontre aujourd'hui qu'elle ne le pent pas.'' t Lucret. v, 875, "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. X Even after we have so narrowed its scope and sphere, natural selection is still a hard saying ; for the causes of extinction are wellnigh as hard to understand as are those of the origin of species. If we assert (as has been lightly and too Ill] OF REGENERATIVE GROWTH 271 Of 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. It is a property manifested in a high degree among invertebrates and many cold-blooded vertebrates, diminishing as we ascend the scale, until it lessens down in the warm-blooded animals to that vis smedicatrix which heals a wound. Ever since the days of Aristotle, and still more since the experiments of Trembley, Reaumur and Spallanzani in the eighteenth century, physiologist and psychologist alike have recognised that the pheno- menon is both perplexing and important. "Its discovery," said Spallanzani, " was an immense addition to the riches of organic philo- sophy, and an inexhaustible source of meditation for the philosopher." The general phenomenon is amply treated of elsewhere*, and we need only deal with it in its immediate relation to growth. Regeneration, like growth in other cases, proceeds with a velocity which varies according to a definite law; the rate varies with the time, and we may study it as velocity and as acceleration. Let us take, as an instance, Miss M. L. Durbin's measurements of the rate .of regeneration of tadpoles' tails : the rate being measured in terms of length, or longitudinal increment f. From a number of tadpoles, whose average length was in one experiment 34 mm., and in another 49 mm., about half the tail was cut off, and the average amounts regenerated in successive periods are shewn as follows : Days 3 5 7 10 12 14 17 18 24 28 30 Amount regenerated (mm.): First experiment 1-4 — 3-4 4-3 — 5-2 — 5-5 6-2 — 6o Second „ 0-9 2-2 3-7 5-2 60 6-4 7-1 — 7-6 8-2 8-4 confidently done) that Smilodon perished on account of its gigantic tusks, that Teleosaurus was handicapped by its exaggerated snout, or Stegosaurus weighed down by its intolerable load of armour, we may call to mind kindred forms where similar conditions did not lead to rapid extermination, or where extinction ensued apart from any such apparent and visible disadvantages. Cf. F. A. Lucas, On momentum in variation, Amer. Nat. xli, p. 46, 1907. * See Professor T. H. Morgan's Regeneration (316 pp.), 1901, for a full account and copious bibliography. The early experiments on regeneration, by V^allisneri, Dicquemare, Spallanzani, Reaumur, Trembley, Baster, Bonnet and others, are epitomised by Haller, Elementa Physiologiae, viii, pp. 156 seq. t Journ. Exper. Zool. vii, p. 397, 1909. 272 THE RATE OF GROWTH [CH. Both experiments give us fairly smooth curves of growth within the period of the observations; and, with a shght and easy extra- polation, both curves draw to the base-hne at zero (Fig. 84). More- Fig. 84. Curve of regenerative growth in tadpoles' tails. From M. L. Durbin's data. Fig. 85. Tadpoles' tails : , amount regenerated daily, in mm. (Smoothed curve). over, if from the smoothed curves we deduce the daily increments, we get (Fig. 85) a bell-shaped curve similar to (or to all appearance identical with) a skew curve of error. In point of fact, this instance of regeneration is a very ordinary example of growth, with its Ill] OF REGENERATION 273 S-shaped curve of integration and its bell-shaped differential curve, just as we have seen it in simple cases, or simple phases, of 'the growth of a population or an individual. If we amputate one limb of a pair in some animal with rapid powers of regeneration, we may compare from time to time the dimensions of the regenerating hmb with those of its uninjured fellow, and so deal with a relative rather than an absolute velocity. The legs of insect-larvae are easily restored, but after pupation no further growth or regeneration takes place. An easy experiment, then, is to remove a limb in larvae of various ages, and to compare 100 120 Fig. 86. Regenerative growth in mealworms' legs. at leisure in the pupa the dimensions of the new Hmb with the old. The following much-abbreviated table shews the gradual increase of a regenerating limb in a mealworm, up to final equahty with the normal Hmb, the rate varying according to the usual S-shaped curve* (Fig. 86). Rate of regeneration in the mealworm (Tenebrio moHtor, larva) Days after amputation % ratio of new limb to old 0 16 21 25 34 44 58 70 100 121 0 7 11 20 29 42 71 83 91 100 * From J. Krizenecky, Versuch zur statisch-graphischen Untersuchung . . .der Regenerationsvorgange, Arch. f. Entw. Mech. xxxix, 1914; xlii, 1917. 274 THE RATE OF GROWTH [ch. Some writers have found the curve of regenerative growth to be different from the curve of ordinary growth, and have commented on the apparent difference; but they have been misled (as it seems to me) by the fact that regeneration is seen from the start or very nearly so, while the ordinary curves of growth, as they are usually presented to us, date not from the beginning of growth, but from the comparatively late, and unimportant, and even fallacious epoch of birth. A complete curve of growth, starting from zero, has the same essential characteristics as the regeneration curve. Indeed the more we consider the phenomenon of regeneration, the more plainly does it shew itself to us as but a particular case of the general phenomenon of growth*, 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 underhes this process of heahng and repair. It is a very general rule, though 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 .investi- gators, who, like the Abbe Trembley and hke Voltaire, found a fascination in the study of artificial injury and the regeneration which followed it. Sir John Graham Dalyell, for instance, says, in the course of an admirable paragraph on regeneration!: "The reproductive faculty ... is not confined to one pjortion, 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." * The experiments of Loeb on the growth of Tubularia in various saline solutions, referred to on p. 24.5, might as well or better have been referred to under the heading of regeneration, as they were performed on cut pieces of the zoophyte. (Cf. Morgan, op. cit. p. 35.) t Powers of the Creator, i, p. 7, 1851. See also Rare and Remarkable Animals, u, pp. 17-19, 90, 1847. Ill] OF REGENERATION 275 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 a mere 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 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 /x in diameter, that is to say of a volume equal to about one twenty- seventh of the average entire animal. He arrives at the remarkable conclusion that for this, and for all other species of animals, there is a "minimal organisation mass," that is to say a "minimal mass of definite size consisting of nucleus and cytoplasm within which the organisation of the species can just find its latent expression." And in like manner, 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 Hmit 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 fife itself can be manifested, and which we have discussed in another chapter. The Bermuda "hfe-plant" (Bryophyllum calycinum) has so remarkable a power of regeneration that a single leaf, kept damp, sprouts into fresh leaves and rootlets which only need nourishment to grow into a new plant. If a stem bearing two opposite leaves be split asunder, the two co-equal sister-leaves will produce (as we might indeed expect) equal masses of shoots in equal times, whether these shoots be many or fe^^; and, if one leaf of the pair have part cut off it and the other be left intact, the amount of new growth * F. R. Lillie, The smallest parts of Stentor capable of regeneration, Journ. Morphology, xii, p. 239, 1897. t -Boveri, Entwicklungsfahigkeit kernloser Seeigeleier, etc.. Arch. /. Entw. Mech. u, 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; Pfluger's Archiv, lix, 1894, etc. Bonnet studied the same problem a hundred and seventy years ago, and found that the smallest part of the worm Lumbriculus capable of regenerating was 1^ lines (3-4 mm.) long. For other references and discussion see H. Przibram, Form und Formel, 1922, ch. v. 276 THE RATE OF GROWTH [CH. will be in direct and precise proportion to the mass of the leaf from which it grew. The leaf is all the while a living tissue, manu- facturing material to build its own offshoots ; and we have a simple case of the law of mass action in the relation between the mass of the leaf with its 'included chlorophyll and. that of its regenerated oifshoot*. 16 18 20 days Fig. 87. Relation between the percentage amount of tail removed, the percentage restored, and the time required for its restoration. Constructed from M. M. Ellis's data. A number of phenomena connected with the linear rate of regeneration are illustrated and epitomised in the accompanying diagram (Fig. 87), 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 * Jacques Loeb, The law controlling the quantity and rate of regeneration, Proc. Nat. Acad. Sci. iv, pp. 117-121, 1918; Journ. Gen. Physiol, i, pp. 81-96, 1918; Botan. Oaz. lxv, pp. 150-174, 1918. Ill] OF REGENERATION 277 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 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 of time, when the amount removed approaches 75 per cent, of the whole; and all start from zero, for nothing is regenerated where nothing had been destroyed. The rate of regenerative growth in tadpoles' tails (After M. M. Ellis, Journ. Exp. Zool. vii, ^.421, 1909) Body Tail Amount Per cent. /o amount regenerated in days length length removed of tail r — ^ Series* mm. mm. mm. removed 3 6 9 12 15 18 32 0 39-575 25-895 3-2 12-36 13 31 44 44 44 44 44 P 40-21 26-13 5-28 20-20 10 29 40 44 44 44 44 R 39-86 25-70 10-4 40-50 6 20 31 40 48 48 48 S 40-34 2611 14-8 56-7 0 16 33 39 45 48 48 * Each series gives the mean of 20 experiments. The amount regenerated varies also with the age of the tadpole, and with other factors such as temperature; in short, for any given age or size of tadpole, and for various temperatures, and doubtless for other varying physical conditions, a similar diagram might be constructed!. The power of reproducing, or regenerating, a lost limb is par- t Cf. also C. Zeleny, Factors controlling the rate of regeneration, Illinois Biol- Monographs, iii, p. 1, 1916. 278 THE RATE OF GROWTH [ch. ticularly well developed in arthropod animals, and is sometimes accompanied by remarkable modification of the form of the regenerated limb. A case in point, which has attracted much attention, occurs in connection with the claws of certain Crustacea*. In many of these 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 matter of "chance." It is otherwise in many other Crustacea, where the larger and more powerful claw is always left or right, as the case may be, according to the species: where, in other words, the "probability" of the large or the small claw being left or being right is tantamount to certainty f. As we have already seen, the one claw is the larger because it has grown the faster; it has a higher "coefficient of growth," and accordingly, as age advances, the disproportion between the two claws becomes more and more evident. Moreover, we must assume that the characteristic form of the claw is a "function" of its magnitude ; the knobbiness is a phenomenon coincident with growth, and we never, under any circumstances, find the smaller claw with big crushing teeth and the big claw with Httle serrate ones. There are many other somewhat similar cases where size and form are manifestly correlated, and we have already seen, to some extent, how the phenomenon of growth is often accompanied by such ratios of velocity as lead inevitably to changes of form. Meanwhile, then, we must simply assume that the essential difference between the two claws is one of magnitude, with which a certain differentiation of form is inseparably associated. * Cf. H. Przibram, Scheerenumkehr bei dekapoden Crustaceen, Arch. f. Entw. Mech. XIX, pp. 181-247, 1905; xxv, pp. 266-344, 1907; Emmel, ibid, xxii, p. 542, 1906; Regeneration of lost parts in lobster, Bep. Comm. Inland Fisheries, Rhode Island, XXXV, xxxvi, 1905-6; Science (N.S.), xxvi, pp. 83-87, 1907; Zeleny, Compensatory regulation, Journ. Exp. Zool. ii, pp. 1-102, 347-369, 1905; etc. t Lobsters are occasionally found with two symmetrical claws: which are then usually serrated, sometimes (but very rarely) both blunt-toothed. Cf. W. T. Caiman, P.Z.8. 1906, pp. 633, 634, and reff. Ill] OF REGENERATION 279 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 does so with an accelerated velocity which ceases 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 lesser and weaker one; A and B have apparently changed places. In a third case, as in the hermit-crabs, the A- claw regenerates itself as a small or 5-claw, but the 5-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: that is to say, on the familiar phenomenon of unequal growth, or, as it has been called, heterogony*. It is com- plicated a little, however, by the possibility of the uninjured limb growing all the faster for a time after the animal has been relieved of the other. From the time of amputation, say of A, A begins to grow from zero, with a high "regenerative" velocity; while B, starting from a definite magnitude, continues to increase with its normal or perhaps somewhat accelerated velocity. The ratio between the two velocities of growth will determine whether, by a given time, A has equalled, outstripped, or still fallen short of the magnitude of B. That this is the gist of the whole problem is confirmed (if con- firmation be necessary) by certain experiments of Wilson's. It is * E.' B. Wilson, Reversal of symmetry in Alpheus heterocheles, Biol. Bull, iv, p. 197, 1903. t See p. 205. 280 THE RATE OF GROWTH [ch. 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 little way towards helping us to comprehend the phenomenon of "multiplication by fission," as it is exemplified 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 for its presence here and absence there, in forms which are materially different in a physical sense, though zoologically speaking they are very closely allied. But the phenomena of regeneration, like all the other phenomena of growth, soon carry us far afield, and we must draw this long discussion to a close. * A morphological polarity, or essential difference between one end and the other of a segment, is important even in so simple a case as the internode of a hydroid zoophyte; and an electrical polarity seems always to accompany it. Cf. A. P. Matthews, Amer. Journ. Physiology, viii, j). 294, 1903; E. J. Lund, Journ. Exper. Zool. xxxiy, pp. 477-493; xxxvi, pp. 477^94, 1921-22. Ill] THE RATE OF GROWTH 281 Summary and Conclusion For the main features which appear to be common to all curves of growth we may hope to have, some day, a simple explanation. In particular we should like to know the plain 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 accelera- tion curves, demonstrate the existence of, provided only that they include the initial- stages of the whole phenomenon: just as we should also hke 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 little able to explain, but which presents us with a definite and attractive problem for future solution. It would look as though the abrupt change in velocity must be due, either to a change in that pressure outwards from within by which the "forces of growth" make themselves manifest, or to a change in the resistances against which they act, that is to say the tension of the surface; and this latter force we do not by any means limit to "surface- tension" proper, but may extend to the development of a more or less resistant membrane or "skin," or even to the resistance of fibres or other histological elements binding the boundary layers to the parts within*. I take it that the sudden arrest of velocity is much more likely to be due to a sudden increase of resistance than to a sudden diminution of internal energies: in other words, I suspect that it is coincident with some notable event of histological differentiation, such as the rapid formation of a comparatively firm skin ; and that the dwindling of velocities, or the negative accelera- tion, which follows, is the resultant or composite effect of waning forces of growth on the one hand, and increasing superficial resistance * It is natural to suppose the cell-wall less rigid, or more plastic, in the growing tissue than in the full-grown or resting cell. It has been suggested that this plasticity- is due to, or is increased by, auxins, whether in the course of nature, or in our stimulation of growth by the use of these bodies. Cf. H. Soding, Jahrb. d. wiss. Bot. Lxxiv, p. 127; 1931. 282 THE RATE OF GROWTH [ch. 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 establish- ment of resistances which check its own further increase. Our knowledge of the whole complex phenomenon of growth is so scanty that it may seem rash to advance even this tentative suggestion. But yet there are one or two known facts which seem to bear upon the question, and to indicate at least the manner in which a varying resistance to expansion may affect the velocity of growth. For instance, it has been shewn by Frazee* 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. To another important aspect of regeneration we can do no more than allude. The Planarian worms rival Hydra itself in their powers of regeneration; and in both cases even small bits of the animal are likely to include endoderm cells capable of intracellular digestion, whereby the fragment is enabled to live and to grow. Now if a Planarian worm be cut in separate pieces and these be suffered to grow and regenerate, they do so in a definite and orderly way ; that part of a shce or fragment which had been nearer tq the original head will develop a head, and a tail will be regenerated at the opposite end of the same fragment, the end which Had been tailward in the beginning; the amputated fragments possess sides and ends, a front end and a hind end, like the entire worm; in short, they retain their polarity. This remarkable discovery is due to Child, who has amplified and extended it in various instructive ways. The existence of two poles, positive and negative, implies a "gradient" between them. It means that one part leads and another follows; that one part is dominant, or prepotent over the rest, whether in regenerative growth or embryonic development. We may summarise, as follows, the main results of the foregoing discussion : (1) Except in certain minute organisms, whose form (hke that * Journ. Ezper. ZooL vii, p. 457, 1909. Ill] SUMMARY AND CONCLUSION 283 of a drop of water) is due to the direct action of the molecular forces, we may look upon the form of an organism as a " function of growth," or a direct consequence of growth whose rate varies in its different directions. In a newer language we might call the form of an organism an "event in space-time," and not merely a "configuration in space." (2) Growth varies in rate in an orderly way, or is subject, like other physiological activities, to definite "laws." The rates differ in degree, or form "gradients," from one point of an organism to another; the rates in different parts and in different directions tend to maintain more or less constant ratios to one another in each organism ; , and to the regularity and constancy of these relative rates of growth 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 in course of time, or to fluctuate in an orderly way; and to these progressive changes are due the changes of form which accompany development, and the slower changes which continue perceptibly in after hfe. (4) Rate of growth depends on the age of the organism. It has a maximum somewhat early in hfe, after which epoch of maximum it slowly declines. (5) Rate of growth is directly affected by temperature, and by other physical conditions: the influence of temperature being notably large in the case of cold-blooded or " poecilothermic " animals. Growth tends in these latter to be asymptotic, becoming slower but never ending with old age. (6) It is markedly affected, in the way of acceleration or retarda- tion, at certain physiological epochs of hfe, 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 concomitant of old age. (8) The phenomenon of regeneration is associated with a large transitory increase in the rate of growth (or acceleration of growth) in the region of injury; in other respects regenerative growth is similar to ordinary growth in all its essential phenomena. 284 THE RATE OF GROWTH [ch. In this discussion of growth, we have left out of account a vast number of processes or phenomena in the physiological mechanism of the body, by which 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. That is to say, we rule "heredity" or any such concept out of our present account, however true, however important, however indispen- sable in another setting of the story, such a concept may be. In physics "on admet que I'etat actuel du monde ne depend que du passe le. plus proche, sans etre influence, pour ainsi dire, par le souvenir d'un passe lointain*." This is the concept to which the differential equation gives expression; it is the step which Newton took when he left Kepler behind. In all its aspects, and not least in its relation to form, the growth of organisms has many analogies, some close, some more remote, among inanimate things. As the waves grow when the winds strive with the other forces which govern the movements of the surface of the sea, as the heap grows when we pour corn out of a sack, as the crystal grows when from the surrounding solution the proper molecules fall into their appropriate places: so in all these cases, very much as in the organism itself, is growth accompanied by change of form, and by a development of definite shapes and contours. And in these cases (as in all other mechanical phenomena), we are led to equate our various magnitudes with time, and so to recognise that growth is essentially a question of rate, or of velocity. The diiferences 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 episodes in the life-history of the individual, or manifest themselves as the distinctive charac- teristics of what we call separate species of the race. From one form, or one ratio of magnitude, to another there is but one straight and direct road of transformation, be the journey taken fast or * Cf. H. Poincare, La physique generale et la physique mathematique, Rev. gin. des Sciences, xi, p. 1167, 1900. Ill] SUMMARY AND CONCLUSION 285 slow; and if the transformation take place at all, it will in all likelihood proceed in the self-same way, whether it occur within the Ufetime 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 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. CHAPTER IV ON THE INTERNAL FORM AND STRUCTURE OF THE CELL In the early days of the cell-theory, a hundred 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 morphodynatnique — 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 expression. * Anatomical and Pathological Observations, p. 3, 1845; Anatomical Memoirs, II, p. 392, 1868. This was a notable improvement on the "kleine wirkungsfahige Zentren oder Elementen" of the Cellularpathologie. Goodsir seems to have been seeking an analogy between the living cell and the physical atom, which Faraday, following Boscovich, had been speaking of as a centre of force in the very year before Goodsir published his Observations: see Faraday's Speculations concerning Electrical Conductivity and the Nature of Matter, 1844. For Newton's "molecules" had been turned by his successors into material points; and it was Boscovich (in 1758) who first regarded these material points as mere persistent centres of force. It was the same fertile conception of a centre of force which led Rutherford, later on, to the discovery of the nucleus of the atom. t A. Giard, L'oeuf et les debuts de revolution, Bull. Sci. du Nord de la Fr. vm, pp. 252-258, 1876. CH. IV] THE CELL THEORY . 287 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 which are associated therewith. The overwhelming progress of microscopic observation has multipHed our knowledge of cellular and intra- cellular structure ; and to the multitude of visible structures it has been often easier to attribute virtues than to ascribe intelUgible functions or modes of action. But here and there nevertheless, throughout the whole hteratiire of the subject, we find recognition of the inevitable fact that dynamical problems lie behind the morphological problems of the cell. Biitschli pointed out sixty 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 t, 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 ynotion — "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 X-^' In short it would seem evident that, except in relation to a dynamical investigation, the mere study of cell structure has but * Entwickelungsvorgdnge der Eizelle, 1876; Investigations on Microscopic Foams and Protoplasm, p. 1, 1894. t Journ. Morphology, i, p. 229, 1887. t While it has been very common to look upon the phenomena of mitosis as sufficiently explained by the results towards which they seem to lead, we may find here and there a strong protest against this mode of interpretation. The following is a case in point: ''On a 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 (Dangeard), distribution particuliere et signification dualiste des substances nucleaires (substance kinetique et substance generative ou hereditaire, Ilartmann 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. Alexeieflf, Archiv fiir Protistenkunde, xix, p. 344, 1913. 288 ON THE INTERNAL FORM [ch. 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 centro- somes, 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, chromosomes, 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 Ukely 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, Hke Goodsir himself, first attacked the problem of the cell and originated our conceptions of its nature and functions*. If we speak, as Weismann and others speak, of an "hereditary sitbstance,'' a substance which is spHt 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 capabihty of producing motion, or of doing "work." For, as Newton said, to tell us that a thing "is endowed with an occult specific quahtyf, by which it acts and produces manifest effects, is to tell us nothing ; but to derive two or three general principles of motion { from * See also {int. al.) R. S. Lillie's papers on the physiology of cell-division in the Journ. Exper. Physiology; especially No. vi, Rhythmical changes in the resistance of the dividing sea-urchin egg, ibid, xvi, pp. 369-402, 1916. f Such as the vertu donnitive which accounts for the soporific action of opium. We are now more apt. as Le Dantec says, to substitute for this occult quality the hypothetical substance dormitin. X This is the old philosophic axiom writ large: Ignorato motu, ignoratur -natura; which again is but an adaptation of Aristotle's phrase, 17 dpx^ ^V^ Kivrjcrecxjs, as equivalent to the "Efiicient Cause." FitzGerald holds that "all explanation consists in a description of underlying motions" {Scientific Writings, 1902, p. 385); and Oliver Lodge remarked, "You can move Matter; it is the only thing you can do to it." IV] AND STRUCTURE OF THE CELL 289 phenomena would be a very great step in philosophy, though the causes of those 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, scrutinise, 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. 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 com- phcated mechanism : nor yet does it prove, though this is somewhat less obvious, that the forces at work or latent within it are especially numerous and complex. If we blow into a bowl of soapsuds and raise a great mass of many-hued and variously shaped bubbles, if we explode a rocket and watch the regular and beautiful configura- tion of its falhng streamers, if we consider the wonders of a Hmestone cavern which a filtering stream has filled with stalactites, we soon perceive that in all these cases we have begun with an initial system of very slight complexity, whose structure in no way foreshadowed the result, and whose comparatively simple intrinsic forces only play their part by complex interaction with the equally simple forces of the surrounding medium. In an earlier age, men sought for the visible embryo, even for the homunculus, within the repro- ductive cells; and to this day we scrutinise these cells for visible structure, unable to free ourselves from that old doctrine of ' ' pre-f ormation * . " Moreover, the microscope seemed to substantiate the idea (which * As when Nageli concluded that the organism is, in a certain sense, "vorge- bildet"; Beitr. zur wiss. Botanik, ii, 1860. 290 ON THE INTERNAL FORM [ch. we may trace back to Leibniz* and to Hobbest), that there is no Hmit to the mechanical complexity which we may postulate in an organism, and no limit, therefore, to the hypotheses which we may rest thereon. But no microscopical examination of a stick of sealing- wax, no study of the material of which it is composed, can enlighten 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 potentiaUties 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 formula ted 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 an organisation, within the cell histologists have ever since been trying to comply J. But unless we mean to include thereby invisible, and merely chemicu,l or molecular, structure, we come at once on dangerous ground. For we have seen in a former chapter that organisms are known of magnitudes so nearly approaching the molecalar, that everything which the morphologist is accustomed to conceive as "structure" has become physically impossible; and recent research tends to reduce, rather than to extend, our con- ceptions of the visible structure necessarily inherent in living protoplasm §. The microscopic structure which in the last resort * "La matiere arrangee par une sagesse divine doit etre essentiellement organisee partout. . .il y a machine dans les parties de la machine naturelle a I'infini." Siir le principe de la Vie, p. 431 (Erdmann). This is the very converse of the doctrine of the Atomists, who could not conceive a condition "w6/ dimidiae partis pars semper hahehit Dimidiam partem, nee res praefiniet ulla.'' t 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. % "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 bezeichnen," Briicke, Die Elementarorganismen, Wiener Sitzungsber. xliv, 1861, p. 386; quoted by Wilson, The Cell, etc., p. 289. Cf. also Hardy, Journ. Physiol, xxiv, 1899, p. 159. § The term protoplasm was first used by Purkinje, about 1839 or 1840 (cf. Reichert, Arch. f. Anat. u. Physiol. 1841). But it was better defined and more strictly used by Hugo von Mohl in his paper Ueber die Saftbewegung im Inneren der Zellen, Botan. Zeitung, iv, col. 73-78, 89-94, 1846. IV] AND STRUCTURE OF THE CELL 291 or in the simplest cases it seems to sheWj is that of a more or less viscous colloid, or rather mixtm:e 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 configura- tion and motion which it has at a given instant*." If we cannot assume differences in structure or configuration, we must assume differences in motion, that is to say in energy. And if we cannot do this, then indeed we are thrown back upon modes of reasoning unauthorised in physical science, and shall find ourselves constrained to assume, or to "admit, that the properties of a germ are not those of a purely material system." 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 hfe 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 beheve that it comprises an intricate "mechanism," we mgiy be quite sure, both on direct and indirect evidence, that, hke 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 con- stitute 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 comphcated contrivances of himaan skill. From the physical point of view, we understand by a "mechanism" w^hatsoever checks or controls, and guides into determinate paths, the workings of energy: in other words, what- soever leads in the degradation of energy to its manifestation in some form of work, at a stage short of that ultimate degradation which lapses in uniformly diffused heat. This, as Warburg has well explained, is the general effect or function of the physiological machine, and in particular of that part of it which we call "cell- * Precisely as in the Lueretian concursus, motus, ordo, positura, figurae, whereby bodies mutato ordine mutant naturam. 292 ON THE INTERNAL FORM [ch. structure*." The normal muscle-cell is something which turns energy, derived from oxidation, into work; it is a mechanism which arrests and utihses the chemical energy of oxidation in its downward course; but the same cell when injured or disintegrated loses its "usefulness," and sets free a greatly increased proportion of its energy in the form of heat. It was a saying of Faraday's, that "even a Hfe is but a chemical act prolonged. If death occur, the more rapidly oxygen and the affinities run on to their final state f." Very great and wonderful things are done by means of a mechanism (whether natural or artificial) of extreme simphcity. A pool of water, by virtue of its surface, 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. Music itself is made of simple things — a reed, a pipe, a string. The great cosmic mechanisms are stupendous in their simphcity; and, in point of fact, every great or little aggregate of heterogeneous matter (not identical in "phase") involves, ipso facto, the essentials of a mechanism. Even a non-living colloid, from its intrinsic hetero- geneity, 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 speaks somewhere or other of the colloid state as "the dynamic state of matter"; in the same philosopher's' phrase, it possesses ^' energia%.'^ 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 hght 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 * 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). t See his Life by Bence Jones, ii, p. 299. X Both phrases occur, side by side, in Graham's classical paper on Liquid diffusion applied to analysis, Phil. Trans, cli, p. 184, 1861; Chem. and Phys. Researches (ed. Angus Smith), 1876, p. 554. iv] AND STRUCTURE OF THE CELL 293 direction and magnitude; tli£ nature and specific identity of the force or forces is a very different matter. This latter problem is likely to be difficult of elucidation, for the reason, among others, that very different forces are often 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 "caryo- kinetic" figures of which we are about to speak. One student may, like Rhumbler, choose to account for them by an hj^othesis of mechanical traction, acting on a reticular web of protoplasm*; another, hke Leduc, may shew us how in many of their most striking features they may be admirably simulated by salts diffusing in a colloid medium; others, hke Lamb and Graham Cannon, have compared them to the stream-hnes produced and the field of force set up by bodies vibrating in a fluid; others, like Gallardof and Rhumbler in his earher papers J, insisted on their resemblance to certain phenomena of electricity and magnetism §; while Hartog believed 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 importance, so long as we seek only to account for certain configurations which reveal the direction, rather than the nature, of a force. One and the same system of lines of force may appear in a field of magnetic or of electrical energy, of the osmotic energy of diffusion, of the gravitational energy of a flowing stream. In short, we may expect to learn something of the pure or abstract dynamics long before we can deal with the special physics of the * L. Rhumbler, Mechanische Erklarung der Aehnlichkeit zwischen magne- tischen Kraftliniensystemen und Zelltheilungsfiguren, Arch. f. Entw. Mech. xv, p. 482, 1903. t A. Gallardo, Essai d'interpretation des figures caryocinetiques, Armies del Museo de Buenos- Aires (2), ii, 1896; Arch. f. Entw. Mech. xxvm, 1909, etc. X Arch.f. Entw. Mech. ill, iv, 1896-97. § On various theories of the mechanism of mitosis, see (e.g.) Wilson, The Cell in Development, etc.; Meves, Zelltheilung, in Merkel u. Bonnet's Ergehnisse der Anatomic, etc., vii, viii, 1897-98; Ida H. Hyde, Amer. Journ. Physiol, xii, pp. 241- 275, 1905; and especially A. Prenant, Theories et interpretations physiques de la mitose, Journ. de VAnat. et Physiol, xlvi, pp. 511-578, 1910. See also A. Conard, Sur le mecanisme de la division cellulaire, et sur les bases, morphologiques de la Cytologie, Bruxelles, 1939: a work which I find hard to follow. II M. Hartog, Une force nouvelle: le mitokinetisme, C.R. 11 Juli 1910; Arch. f. Entw. Mech. xxvii, pp. 141-145, 1909; cf. ibid, xl, pp. 33-64, 1914. 294 ON THE INTERNAL FORM [ch. cell. For indeed, just as uniform expansion about a single centre, to whatsoever physical cause it may be due, will lead to the con- figuration of a sphere, so will any two centres or foci of potential (of whatsoever kind) lead to the configurations with which Faraday first made us famihar under the name of "lines of force*"; a^d this is as much as to say that the phenomenon, though physical in the concrete, is in the abstract purely mathematical, and in its very essence is neither more nor less than a property of three-dimensional space. But as a matter of fact, in this instance, that is to say in trying to explain the leading phenomena of the caryokinetic division of the cell, we shall soon perceive that any explanation which is based, like Rhumbler's, on mere mechanical traction, is obviously inade- quate, and we shall find ourselves limited to the hypothesis of some polarised and polarising force, such as we deal with, for instance, in magnetism or electricity, or in certain less familiar phenomena of hydrodynamics. 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 con- tractions or "Ransom's waves "f on the surface of a trout's egg; or again, the surface-tension may be locally unequal and variable, giving rise to an amoeboid figure, as in the egg of HydraX- Within the cell is a nucleus or germinal vesicle, also spherical, * The configurations, as obtained by the usual experimental methods, were of course known long before Faraday's day, and constituted the "convergent and divergent magnetic curves" of eighteenth century mathematicians. As Leslie said, in 1821, they were "regarded with wonder by a certain class of dreaming philosophers, whp did not hesitate to consider them as the actual traces of an invisible fluid, perpetually circulating between the poles of the magnet." Faraday's great advance was to interpret them as indications of stress in a medium — of. tension or attraction along the lines, and of repulsion transverse to the lines, of the diagram. t W. H. Ransom, On the ovum of osseous fishes, Phil. Trans, clvii, pp. 431-502, 1867 (vide p. 463 et. seq.) (Ransom, afterwards a Nottingham physician, was Huxley's friend and class-fellow at University College, and beat him for the medal in Grant's class of zoology.) X Cf, also the curiou.s phenomenon in a dividing egg described as "spinning" by Mrs G. F. Andrews, Journ. Morph. xii, pp. 367-389, 1897. IV] AND STRUCTURE OF THE CELL 295 and consisting 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* remarks that "except during the process of division the nucleus seldom departs from it^ cypical 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 history." On simple dynamical grounds, the contrast is easily explained. So long as the fluid substance of the nucleus is qualitatively different from, and incapable of mixing with, the fluid or semi-fluid protoplasm surrounding it, we shall expect it to be, as it almost always is, of spherical form. For on the one hand, it has a surface of its own whose surface-tension is presumably uniform, and on the other, it is immersed in a medium which transmits on all sides a uniform fluid or "hydrostatic" pressure f; thus the case of the spherical nucleus is closely akin to that of the spherical yolk within the bird's egg. Again, for a similar reason, the contractile vacuole of a protozoon is spherical {. It is just a drop of fluid, bounded by a * Whitman, Journ. Morph. ii, p. 40, 1889. t "Souvent il n'y a qu'une separation physique entre le cytoplasme et le sue hucleaire, comme entre deux liquides immiscibles, etc."; Alexeieff, 8ur la mitose dite primitive, Arch. f. Protistenk. xxix, p. 357, 1913. X The appearance of " vacuolation " is a result of endosmosis, or the diffusion of a less dense fluid into the denser plasma of the cell. But while water is probably taken up at the surface of the cell by purely passive osmotic intake, a definite "vacuole" appears at a place where osmotic work is being actively done. A higher osmotic pressure than that of the external medium is maintained within the cell, but as a "steady state" rather than a condition of equilibrium, in other words by the continual expenditure of energy; and the difference of pressure is at best small. The "contractile vacuole" bursts when it touches the surface of the cell, and bursting may be delayed by manipulating the vacuole towards the interior. It may sometimes burst towards the interior of the cell through inequalities in its own surface-tension, and the collapsing vacuole is then apt to shew a star-shaped figure. The cause of the higher osmotic pressure within the cell is a matter for the colloid chemist, and cannot be discussed here. On the physiology of the contractile vacuole, see {int. al.) H. Z. Gow, Arch. f. Protistenk. lxxxvii, pp. 185- 212, 1936; J. Spek, Einfluss der Salze auf die Plasmkolloide von Actinosphaerium, Acta Zool. 1921; J. A, Kitching, Journ. Exp. Biology, xi, xiii, xv, 1934-38. 296 ON THE INTERNAL FORM [ch. uniform surface-tension, and through whose boundary-film diffusion is taking place; but here, owing to the small difference between the fluid constituting and that surrounding the drop, the surface-tension equihbrium is somewhat unstable; it is apt to vanish, and the rounded outhne of the drop disappears, Uke a burst bubble, in a moment. If, on the other hand, the substance of the cell acquire a greater soUdity, as for instance in a muscle-cell, or by reason of mucous accumulations in an epithehum cell, then the laws of fluid pressure no longer apply, the 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 where "nuclear material" is scattered in small portions throughout the cell as in many Rhizopods, instead of being aggregated in a single nucleus, the simple explanation probably is that the "phase difference" (as the chemists say) between the nuclear and the protoplasmic substance is comparatively shght, and the surface-tension which tends to keep them separate is correspondingly small*. 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 contains, for instance, a rounded nucleolus or "germinal spot," certain conspicuous granules or strands of the peculiar substance called chromatin*)*, and a coarse mesh work of a protoplasmic material known as "linin" or achro- matin; the outer protoplasm, or cytoplasm, is generally believed to consist throughout of a sponge-work, or rather alveolar mesh- work, of more and less fluid substances; it may contain "mito- chondria," appearing in tissue-cultures as small amoeboid bodies; and lastly, there are generally to be detected (in the animal, rarely in the vegetable kingdom) one or more very minute bodies, usually in the cytoplasm sometimes within the nucleus, known as the centrosome or centrosomes. * 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. •j- First so-called by W. Flemming, in his Zellsubstanz, Kern und Zelltheilung, 1882. IV] AND STRUCTURE OF THE CELL 297 The morphologist is accustomed to speak of a "polarity" of 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 hne drawn through the two as the morphological axis of polarity ; an epitheUum cell is morphologically symmetrical about a median axis passing from its free surface to its attached base. Again, by an extension of the term polarity, as is customary in dynamics, we may have a "radial" polarity, between centre and periphery; and lastly, we may have several apparently independent centres of polarity within the single cell. Only in cells of quite irregular or amoeboid form do we fail to recognise a definite and symmetrical polarity. The morphological polarity is accompanied by, and is but the outward expression (or part of it) of a true dynamical polarity, or distribution of forces; and the hues of force are, or may be, rendered visible by concatenation of particles of matter, such as come under the influence of the forces in action. When hnes of force stream inwards from the periphery towards a point in the interior of the cell, 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. In the cells of columnar or cihated epithehum, 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. 149). A simple manifestation of polarity may be well illustrated by the phenomenon of diffusion, where we may conceive, and may automatically reproduce, a field of force, with its poles and its 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 plate of glass, and let fall into the middle of it a drop of indian ink, or of blood, we shall find the coloured particles travelling * Thtorie physico-chimique de la Vie, 1910, p. 73. 298 ON THE INTERNAL FORM [ch 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 fines of diffusion will oppose and repel one another. Or, instead of the uniform layer of salt solution, we may place at a little distance from one another a grain of salt and a drop of blood, representing two opposite poles: and so obtain a picture of a "bipolar field" of diffusion. In either case, we obtain results closely analogous to the morphological, but really dynamical, polarity of the organic cell. But in all probability, the dynamical polarity or asymmetry of the cell is a very complicated phenomenon: for the obvious reason that, in any system, one asymmetry will tend to beget another. A chemical asymmetry will induce an inequafity of surface-tension, which will lead directly to a modification of form ; the chemical asymmetry may in turn be due to a process of electrolysis in a polarised electrical field; and again the chemical heterogeneity may be intensified into a chemical polarity, by the tendency of certain substances to seek a locus of gi^eater or less surface-energy. We need not attempt to grapple with a subject so compHcated, and leading to so many problems which lie beyond the sphere of interest of the morphologist. But yet the morpho- logist, 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 its simpler phenomena. We are now ready, and in some measure prepared, to study the numerous and complex phenomena which accompany the division of the cell, for instance of the fertilised egg. But it is no easy task to epitomise the facts of the case, and none the easier that of late new methods have shewn us new things, and have cast doubt on not a little that we have been accustomed to believe. Division of the cell is of necessity accompanied, or preceded, by a change from a radial or monopolar to a definitely bipolar sym- metry. In the hitherto quiescent or apparently quiescent cell, we perceive certain movements, which correspond precisely to what must accompany and result from a polarisation of forces within: of forces which, whatever be their specific nature, are at least IV] AND STRUCTURE OF THE CELL 299 capable of polarisation, and of producing consequent attraction or repulsion between charged particles. The opposing forces which are distributed in equilibrium throughout the cell become focused in two "centrosomes*," which may or may not be already visible. It generally happens that, in the egg, one of these centrosomes is near to and the other far from the "animal pole," which is both visibly and chemically different from the other, and is where the more conspicuous developmental changes will presently begin. Between the two centrosomes, in stained preparations, a spindle- shaped figure appears (Fig. 88), whose striking resemblance to the Fig. 8». Caryokinetic ligure in a dividing cell (or blastomere) of a trout s egg. After Prenant, from a preparation by Prof. Bouin. lines of force made visible by iron-filings between the poles of a magnet was at once recognised by Hermann Fol, in 1873, when he witnessed the phenomenon for the first timef. On the farther side of the centrosomes are seen star-like figures, or "asters," in which we se^m to recognise the broken lines of force which run externally to those stronger lines which lie nearer to the axis and constitute the "spindle." The lines of force are rendered visible, or materialised, just as in the experiment of the iron-fihngs, by the fact that, in the heterogeneous substance of the cell, certain portions * These centrosomes are the two halves of a single granule, and are said (by Boveri) to come from the middle piece of the original spermatozoon. t He did so in the egg of a medusa {Geryon), Jen. Zeitschr. vii, p. 476, 1873. Similar ideas have been expressed by Strasbiirger, Henneguy, Van Beneden, Errera, Ziegler, Gallardo and others. 300 ON THE INTERNAL FORM [ch. of matter are more "permeable" to the acting force than others, 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, seem to cling to one another as it were in threads*, and are only prevented by the friction of the surrounding medium from approaching and con- gregating around the adjacent poles. As the field of force strengthens, the more will the lines of force be drawn in towards the interpolar axis, and the less evident will be those remoter lines which constitute the terminal, or extrapolar, asters: a clear space, free from materialised fines of force, may thus tend to be set up on either side of the spindle, the so-called ''Biitschfi space" of the histologistsl. On the other hand, the lines of force constituting the spindle will be less concentrated if they find a path of less resistance at the periphery of the ceU : 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 greatly developed, might easily be explained by variations in the potential of the field, the large, conspicuous asters being correlated in turn 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. * Whence the name "mitosis" (Greek /ziros, a thread), applied first by Flemming to the whole phenomenon. Kolimann (Biol. Centralbl. ii, p. 107, 1882) called it divisio per fila, or divisio laqueis implicata. Many of the earlier students, such as Van Beneden (Rech. sur la maturation de I'oeuf, Arch, de Biol, iv, 1883), and Hermann Fol (Zur Lehre v. d. Entstehung d. karyokinetischen Spindel, Arch. f. mikrosk. Anat. x,xxvii, 1891) thought they recognised actual muscular threads, drawing the nuclear material asunder towards the respective foci or poles; and some such view of Zugkrdfte was long maintained by other writers, by Heidenhain especially, by Boveri, Flemming, R. Hertwig, Rhumbler, and many more. In fact, the existence of contractile threads, or the ascription to the spindle rather than to the poles or centrosomes of the active forces concerned in nuclear division, formed the main tenet of all those who declined to go beyond the "contractile properties of protoplasm" for an explanation of the phenomenon (cf. J. W. Jenkinson, Q.J. M.S. XLViii, p. 471, 1904. See also J. Spek's historical account of the theories of cell-division. Arch. f. Entw. Mech. xliv, pp. 5-29, 1918). t Cf. 0, Biitschli, Ueber die kiinstliche Nachahmung der karyokinetischen Figur, Verh. Med. Nat. Ver. Heidelberg, v, pp. 28-41 (1892), 1897. IV AND STRUCTURE OF THE CELL 301 Let us look a little more closely into the structure of this body, and into the changes which it presently undergoes. Within its spherical outHne (Fig. 89 a), it contains an ''alveolar" meshwork (often described, from its appearance in optical section, as a "reticulum"), consisting of more sohd substances with more fluid matter filling up the interalveolar spaces. This phenomenon, familiar to the colloid chemist, is what he calls a "two-phase system," one substance or "phase" forming a continuum through which the other is dispersed; it is closely alhed to what we call in atr racHon - sphere 1 ,C€ntro3omes Fig. 89 A. Fig. 89 B. ordinary language a. froth ot'sl foam*, save that in these latter the disperse phase is represented by air. It is a surface-tension pheno- menon, due to the interaction 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 cellf, * Froth and foam have been much studied of late years for technical reasons, and other factors than surface-tension are foiind to be concerned in their existence and their stability. See (int. al.) Freundlich's Capillarchemie, and various papers by Sasaki, in Bull. Chem. Soc. of Japan, 1936-39. t Biitschli, Untersuchungen iiber mikroskopische Schdume und das Protoplasma, 1892; Untersuchungen uber Strukturen, etc., 1898; L. Rhumbler, Protoplasma als physikalisches System, Ergehn. d. Physiologie, 1914; H. Giersberg, Plasmabau der Amoben, im Hinblick auf die Wabentheorie, Arch. f. Entw. Mech. li, pp. 150-250, 1922; etc. 302 ON THE INTERNAL FORM [ch. and which we now know to be not inherent in the nature of proto- plasm nor of Hving 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 coohng is an example of similar surface-tension phenomena. Such then, in briefest outhne, is the typical structure commonly ascribed to a cell when its latent energies are about to manifest themselves in the phenomenon of cell-division. The account is based on observation not of the hving cell but of the dead : on the assumption, that is to say, that fixed and stained material gives a true picture of reahty. But in Robert Chambers's method of micro- dissection f, the hving cell is manipulated with fine glass needles under a high magnification, and shews us many interesting things. Chambers assures us that the spindle fibres never make their appearance as visible structures until coagulation has set in; and that astral rays are, or appear to be, channels in which the more fluid content of the cell flows towards a centrosome:|:. Within the bounds to which we are at present keeping, these things are of no great moment; for whether the spindle appear early or late, it still bears witness to the fact that matter has arranged itself along bipolar fines of force; and even if the astral rays be only streams or currents, on lines of force they still approximately he. Yet the change from the old story to the new is important, and may make a world of diff'erence when we attempt to define the forces concerned. All our descriptions, all our interpretations, are bound to be influenced by our conception of the mechanism before us; and he * Arrhenius, in describing a typical colloid precipitate, does so in terms that are very closely applicable to the ordinary microscopic appearance of the protoplasm of the cell. The precipitate consists, he says, "en un reseau d'une substance solide contenant peu d'eau, dans les mailles duquel est inclus un fluide contenant un peu de colloide dans beaucoup d'eau. . . . Evidemment cette structure se forme a cause de la petite difference de poids specifique des deux phases, et de la consistance gluante des particules separees, qui s'attachent en forme de reseau " {Rev. Scientifique, Feb. 1911). This, however, is far from being the whole story: cf. (e.g.) S. C. Bradford, On the theory of gels, Biochem. Journ. xvii, p. 230, 1925; W. Seifritz, The alveolar structure of protoplasm, Protoplasma, ix, p. 198, 1930; and A. Frey- Wissling, Submikroskopische Morphologie des Protoplasmas, Berlin, 1938. t See R. Chambers, An apparatus. . .for the dissection and injection of living cells, Anatom. Record, xxiv, 19 pp., 1922. X This centripetal flow of fluid was announced by Biitschli in his early papers, and confirmed by Rhumbler, though attributed to another cause. IV] AND STRUCTURE OF THE CELL 303 who sees threads where another sees channels is hkely to tell a different story about neighbouring and associated things. It has also been suggested that the spindle is somehow due to a re-arrange- ment of protein macromolecules or micelles ; that such changes of orientation of large colloid particles may be a widespread phenomenon; and that coagu- lation itself is but a polymerisation of larger and larger macromolecules*. But here we have touched the brink of a subject so important that we must not pass it by without a word, and yet so contentious that we must not enter into its details. The question involved is simply whether the great mass of recorded observations and accepted beliefs with regard to the visible structure of protoplasm and of the cell constitute a fair picture of the actual living cell, or be based on appearances which are incident to death itself and to the artificial treatment which the microscopist is accustomed to apply. The great bulk of histological work is done by methods which involve the sudden killing of the cell or organism by strong reagents, the assumption being that death is so rapid that the visible phenomena exhibited during life are retained or "fixed" in our preparations. Hermann Fol struck a warning note full sixty years ago: "II importe a I'avenir de I'histologie de combattre la tendance a tirer des conclusions des images obtenues par des moyens artificiels et a leur donner une valeur intrin- seque, sans que ces images aient ete controlees sur le vivantf." Fol was thinking especially of cell-membranes and the delimitation of cells; but still more difficult and precarious is the interpretation of the minute internal net- works, granules, etc., which represent the alleged structure of protoplasm. A colloid body, or colloid solution, is ipso facto heterogeneous; it has after some fashion a structure of its own. And this structure chemical action, under the microscope, may demonstrate, or emphasise, or alter and disguise. As Hardy put it, "It is notorious that the various fixing reagents are co- agulants of organic colloids, and that the figure varies according to the reagent used." A case in point is that of the vitreous humour, to which some histologists have ascribed a fairly complex structure, seeing in it a framework of fibres with the meshes filled with fluid. But it is really a true gel, without any structure in the usual sense of the word. The "fibres" seen in ordinary microscopic preparations are due to the coagulation of micellae by the fixative employed. Under the ultra -microscope the vitreous is optically empty to begin with; then innumerable minute fibrillae appear in the beam of light, criss-crossing one another. Soon these break down into strings of beads, and * Cf. J. D. Bernal, on Molecular architecture of biological systems, Proc. Boy. Inst., 1938; H. Staiidinger, Nature, Aug. I, 1939. t H. Fol, Becherches sur la fecondation et le commencement de VMnogenie chez divers animaux, Geneve, 1879, pp. 241-242. Cf. A. Daleq, in Biol. Beviews, iii, p. 24, 1928: "II serait desirable de nous debarrasser de I'idee que tout ce qu'il y a d'important dans la cellule serait providentiellement colorable par I'hematoxy- line, la safranine ou le violet de gentiane." 304 ON THE INTERNAL FORM [ch. finally only separate dots are seen*. Other sources of error arise from the optical principles concerned in microscopic vision; for the diffraction-pattern which we call the "image" may, under certain circumstances, be very different from the actual object f. Furthermore, the optical properties of living proto- plasm are especially complicated and imperfectly known, as in general those of colloids may be said to be; the minute aggregates of the "disperse phase" of gels produce a scattering action on light,' leading to appearances of turbidity etc., with no other or more real basis J. 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 have cast doubts on the main tenets of recent histology regarding nuclear structure ; and Hardy, discussing the structure of certain gland-cells, declared 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 went on to say "is an artefact. The profound difference in the minute structure of a secretory cell of a mucous gland according to the reagent which is used to fix it would, it seems to me, almost suffice to establish this statement in the absence of other evidence §." Nevertheless, histological study proceeds, especially on the part of the morphologists, with but little change in theory or in method, in spite of these and many other warnings. That certain visible structures, nucleus, vacuoles, "attraction-spheres" or centrosomes, etc., are actually present in the living cell we know for certain; and to this class belong the majority of structures 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. The following is a brief epitome of the visible changes undergone by a t3rpical cell, subsequent to the resting stage, leading up to the act of segmentation, and constituting the phenomenon of mitosis or caryokinetic division. In the fertilised egg of a sea-urchin we see with almost diagrammatic completeness, in fixed and stained specimens, what is set forth here||. * W. S. Duke-Elder, Journ. Physiol, lxviii, pp. 1.54-165, 1930; of. Baurmann, Arch.f. Ophthalm. 1923, 1926; etc. t Abbe, Arch.f. mikrosk. Anat. ix, p. 413, 1874; Gesammelte Ahhandl. i, p. 45, 1904. X Cf. Rayleigh, On the light from the sky, Phil. Mag. (4) xli, p. 107, 1871. § W. B. Hardy, On the structure of cell protoplasm, Journ. Physiol, xxiv, pp. 158-207, 1889; also Hober, Physikalische Chemie der Zelle und der Gewebe, 1902; W. Berg, Beitrage zur Theorie der Fixation, etc., Arch.f. mikr. Anat. LXii,' pp. 367-440, 1903, Cf. {int. al.) Flemming, Zellsubstanz, Kern und Zelltheilung, 1882, p. 51; etc. II My description and diagrams (Figs. 89-93) are mostly based on those of the late Professor E. B. Wilson. IV] AND STRUCTURE OF THE CELL 305 1. The chromatin, which to begin with had been dimly seen as granules on a vague achromatic reticulum (Figs. 89, 90) — perhaps no more than an histological artefact — concentrates to form a skein or spireme, often looked on as a continuous thread, but perhaps discontinuous or fragmented from the first. It, or its several fragments, will presently spht asunder; for it is essentially double, and may even be seen as a double thread, or pair of chromatids, from an early stage. The chromosomss are portions of this double thread, which shorten down to form httle rods,' straight or curved, often chromoaome* Fig. 90 A. Fig. 90 B. bent into a V, sometimes ovoid, round or even annular, and which in the living cell are frequently seen in active, writhing movement, *"hke eels in a box"*; they keep apart from one another, as by some repulsion, and tend to move outward towards the nuclear membrane. Certain deeply staining masses, the nucleoh, may be present in the resting nucleus, but take no part (at least as a rule) in the formation of the chromosomes; they are either cast out of the nucleus and dissolved in the cytoplasm, or else fade away in situ. * T. S. Strangeways, Proc. E.S. (B), xciv, p. 139, 1922. The tendency of the chromatin to form spirals, large or small, while the nucleus is issuing from its resting-stage, is very remarkable. The tensions to which it is due may be overcome, and the chromosomes made to uncoil, by treatment with ammonia or acetic acid vapour. See Y. Kuwada, Botan. Mag. Tokyo, xlvi, p. 307, 1932; and C. D. Darlington, Mechanical aspects of nuclear division, Sci. Journ. B. Coll, of Sci. TV, p. 94, 1934. 306 ON THE INTERNAL FORM [ch. But this rule does not always hold; for they persist in many protozoa, and now and then the nucleolus remains and becomes itself a chromosome, as in the spermogonia of certain insects. 2. Meanwhile a certain deeply staining granule (here extra- nuclear), known as the centrosome*, has divided into two. It is all but universally visible, save in the higher plants; perhaps less stress is laid on it than at one time, but Bovery called i-t the "dynamic centre" of the cellf. The two resulting granules travel to opposite poles Df the nucleus, and there eacii becomes surrounded by a starhke figure, the aster, of which we have sptken already; immediately around the centrosome is a clear space, the centro- sphere. Between the two centrosomes, or the two asters, stretches the spindle. It lies in the long axis, if there be one, of the cell, a rule laid down nearly sixty years ago, and still remembered as "Hertwig's Law" J; but the rule is as much and no more than to say that the spindle sets in the direction of least resistance. Where the egg is laden with food-yolk, as often happens, the latter is heavier than the cytoplasm; and gravity, by orienting the egg itself, thus influences, though only indirectly, the first planes of segmentation §. 3. The definite nuclear outhne is soon lost; for the chemical "phase-difference" between nucleus and cytoplasm has broken down, and where the nucleus was, the chromosomes now he (Figs. 90, 91). The lines of the spindle become visible, the chromosomes arrange themselves midway between its poles, to form the equatorial plate, and are spaced out evenly around the central spindle, again a simple result of mutual repulsion. 4. Each chromosome separates longitudinally into two|| : usually at this stage — but it is to be noted that the spHtting may have taken place as early as the spireme stage (Fig. 92). * The centrosome has a curious history of its own, none too well ascertained. The ovum has a centrosome, and in self- fertilised eggs this is retained; but when a sperm-cell enters the egg the original centrosome degenerates, and its place is taken by the "middle-piece" of the spermatozoon, f The stages 1, 2, 5 and 6 are called by embryologista the prophase, metaphase, anaphase and telophase. X C. Hertwig, Jenaische Ztschr. xviii, 1884. § See James Gray, The effect of gravity on the eggs of Echinus, Jl. Exp. Zool. v, pp. 102-11, 1927. II A fundamental fact, first seen by Flemming in 1880. IV] AND STRUCTURE OF THE CELL 307 5. The halves of the spht chromosomes now separate from and apparently repel one another, travelUng in opposite directions towards the two poles* (Fig. 92 b), for all the world as though they were being pulled asunder by actual threads. Fig. 91 A. Fig. 91 B. central spindle mantle 'fibres split chromosome* Fig. 92 A. Fig. 92 B. 6. Presently the spindle itself changes shape, lengthens and con- tracts, and seems as it were to push the two groups of daughter- * Cf. K. Belar, Beitrage zur Causalanalyse der Mitose, Ztschr. f. Zellforschung, X, pp. 73-124, 1929. 308 ON THE INTERNAL FORM [CH. chromosomes into their new places* (Figs. 92, 93); and its chromo- somes form once more an alveolar reticulum and may occasionally form another spireme at this stage. A boundary-surface, or at least a recognisable phase-difference, now develops round each reconstructed nuclear mass, and the spindle disappears (Fig. 93 b). The centrosome remains, as a rule, outside the nucleus. 7. On the central spindle, in the position of the equatorial plate, a "cell-plate," consisting of deeply staining thickenings, has made its appearance during the migration of the chromosomes. This cell- plate is more conspicuous in plant-cells. otfrottron spncft ,d>iopptQr,ng spmifte Htcon^rucffd dough/trnuei^t Fig. 93 B. 8. Meanwhile a constriction has 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 spHts 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 process takes from half-an-hour to an hour; and this extreme slowness is not. the least remarkable part of the pheno- menon, from a physical point of view. The two halves of the * The spindle has no actual threads or fibres, for Robert Chambers's micro- needles pass freely through it without disturbing the chromosomes: nor is it visible at all in living cells in vitro. It seems to be due to partial gelation of the cytoplasm, under conditions which, whether they be mechanical or chemical, are not easy to understand. IV] AND STRUCTURE OF THE CELL 309 dividing centrosome, while moving apart, take some twenty minutes to travel a distance of 20 /x, or at the rate, say, of two years to a yard. It is a question of inertia, and the inertia of the system must be very large. The beautiful technique of cell-culture in vitro has of late years let this whole succession of phenomena, once only to be deduced from sections, be easily followed as it proceeds within the living tissue or cell. The vivid accounts which have been given of this spectacle add little to the older account as we have related it: save that, when the equatorial constriction begins and the halves of the split chromosomes drift apart, the protoplasm begins to show a curious and even violent activity. The cytoplasm is thrust in and out in bulging pustules or "balloons"; and the granules and fat-globules stream in and out as the pustules rise and fall away. At length the turmoil dies down; and now each half of the cell (not an ovum but a tissue-cell or "fibroplast") pushes out large pseudopodia, flattens into an amoeboid phase, the connecting thread of protoplasm snaps in the divided cell, and the daughter-cells fall apart and crawl away. The two groups of chromosomes, on reaching the poles of the spindle, turn into bunches of short thick rods; these grow diffuse, and form a network of chromatin within a nucleus; and at last the chromosomes, having lost their identity, disappear entirely, and two or more nucleoH are all that is to be seen within the cell. The whole, or very nearly the whole, of these nuclear phenomena may be brought into relation with some such polarisation of forces in the cell as a whole as is indicated 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 He within the range of known physical actions and phenomena, or that known physical phenomena produce effects similar to them. We feel that the whole phenomenon is iiot sui generis, but is some- how or other capable of being referred to dynamical laws, and to 310 ON THE INTERNAL FORM [ch. 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 stop far short of the implication that this or that force is necessarily the very one which is actually at work within the living cell; and certainly we need not attempt the formidable task of trying to reconcile, or to choose between, the various hypotheses which have already been enunciated, or the several assumptions on which they depend. Many other things happen within the cell, especiall)^ in the germ- cell both before and after fertilisation. They also have a physical element, or a mechanical aspect, like the phenomena of cell- division which we are speaking of; but the narrow bounds to which we are keeping hold difficulties enough*. 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 fine joining two points, or poles, and with reference also to the "equatorial" plane equi- distant from both. We have such a field of force in the neigh- bourhood 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 diagram- matic representation by a variation of the direction or the strength of the field (other conditions being the same) from the state manifested in some uniform medium taken as a standard. The medium is said to be permeable to the force, in greater or less degree than the standard medium, according as the variation of the density of the lines of force from the standard case, under otherwise identical conditions, is in excess or defect. A body placed in the medium will tend to move towards regions of greater or less force according as its * Cf. C. D. Darlington, JieretU Advances in Cytology, 1932, and other well-known works. IV] AND STRUCTURE OF THE CELL 311 penneability is greater or less than that of the surrounding medium'^. In the common experiment of placing iron-filings between the two poles of a magnetic field, the filings have a very high permeability; and not only do they themselves become polarised so as to attract one another, but they tend to be attracted from the weaker to the stronger parts of the field, and as we have seen, they would soon gather together around the nearest pole were it not for friction or some other resistance. But if we "repeat the same experiment with such a metal as bismuth, which is very little permeable to the magnetic force, then the conditions are reversed, and the particles, being repelled from the stronger to the weaker parts of the field, tend to take up their position as far from the poles as possible. The particles have become polarised, but in a sense opposite to that of the surrounding, or adjacent, field. Now, in the field of force whose opposite poles are marked by the centrosomes, we may imagine the nucleus to act as a more or less permeable body, as a body more permeable than the surrounding medium, that is to say the " cytoplasm " of the cell. It is accordingly attracted by, and drawn into, the field of force, and tries, as it were, to set itself between the poles and as far as possible from both of them. In other words,' the centrosome-foci will be apparently drawn over its surface, until the nucleus as a whole is involved within the field of force which is visibly marked out by the "spindle" (Fig. 90 b). 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 f: with the double result that the inner alveolar mesh work will be broken up (par. 1), and that the spherical boundary of tKe whole nucleus will disappear (par. 2). The break- up of the alveoli (by thinning and rupture of their partition walls) * If the word penneability be deemed too directly suggestive of the phenomena of magnetism, we may replace it by the more general term of specific iyidiictive 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. t On the effect of electrical influences in altering the surface-tensions of the colloid particles, see Bredig, Anorganische Fermente, pp. 15, 16, 1901. 312 ON THE INTERNAL FORM [ch. 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". 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 eifect (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 Hues 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 into 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 hjrpothesis) 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. 90 a and b represent two so-called "types," of a phase which follows that represented in Fig. 89. According to the usual descriptions we are told that, in such a case as Fig. 90b, 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. 90 a, on the other hand, the primary spindle persists, and subsequently comes to form the main or "central" spindle; while at the same time we see the fading away of the nuclear membrane, the breaking up of the spireme into separate chromosomes, and an ingrowth into the nu- clear area of the "astral rays" — all as in Fig. 91 a, which represents the next succeeding phase of Fig. 90 b. This condition, of Fig. 91 a, occurs in a variety of cases; it is well seen in the epidermal cells of the salamander, and is also on the whole characteristic of the mode of formation of the "polar bodies t." It is clear and obv;ous that the two "types" correspond to mere differences of degree, * The spindle is potentially there, even though (as Chambers assures us) it only becomes visible after post-mortem coagulation. It is also said to become visible under crossed nicols: W. J. Schmidt, Biodynamica, xxii, 1936. t These v/ere first observed in the egg of a pond-snail (Limnaea) by B. Dumortier, Mim. sur Vemhryoginie des mollusques, Bruxelles, 1837. IV] AND STRUCTURE OF THE CELL 313 and are such as would naturally be brought about by differences in the relative permeabilities of the nuclear mass and of the surrounding cytoplasm, or even by differences in the magnitude of the former body. But now an important change takes place, or rather an important difference appears; for, whereas the nucleus as a whole tended to be drawn in to the stronger parts of the field, when it comes to break up we find, on the contrary, that its contained spireme-thread or separate chromosomes tend to be repelled to the weaker parts. Whatever this difference may be due to — whether, for instance, to actual differences of permeability, or possibly to differences in "surface-charge" or to other causes — the fact is that the chromatin substance now behaves after the fashion of a "diamagnetic'' body, and is repelled from the stronger to the weaker parts of the field. In other words, its particles, lying in the inter-polar field, tend to travel towards the equatorial plane thereof (Figs. 91, 92), 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. 91 b). And if this comparatively non-permeable chromatin substance come to consist of separate portions, more or less elongated in form, these portions, or separate "chromosomes," will adjust themselves longitudinally, in a peripheral equatorial circle (Figs. 92 a, b). 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 v^ry 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 t^nd to assume the characteristic convolutions of the "spireme." After the spireme has broken up into separate chromosomes, these bodies come to rest in the equatorial plane, somewhere near its periphery ; and here they tend to set themselves in a symmetrical arrangement (Fig. 94), such as makes for still better equihbrium. 314 ON THE INTERNAL FORM [CH. The particles ir^y be rounded or linear, straight or bent, sometimes annular; they may be all alike, or one or more may differ from the rest. Lying as they do in a semi-fluid medium, and subject (doubtless) to some symmetrical play of forces, it is not to be wondered at that they arrange themselves in a symmetrical con- figuration; and the field of force seems simple enough to let us predict, to some extent, the symmetries open to them. We do not know, we cannot safely surmise, the nature of the forces involved. In discussing Brauer's observations on the sphtting of the chromatic filament, and on the symmetrical arrangement of the separate granules, in Ascaris megalocephala, LiUie* remarks: "This behaviour Fig. 94. Chromosomes, undergoing splitting and separation. After Hatsehek and Flemming, diagrammatised. 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 probable that surface-tensions between the particles and the sur- rounding protoplasm would bring about an identical result, and would sufficiently account for the obvious, and at first sight very curious symmetry. If we float a couple of matches in water, we know that they tend to approach one another till they He close together, side by side; and if we lay upon a smooth wet plate four matches, half broken across, a similar attraction brings the four matches together in the form of a symmetrical cross. Whether one of these, or yet another, be the explanation of the phenomenon, * R. S. Lillie, Conditions determining the disposition of the chromatic filaments, etc., in mitosis; Biol. Bulletin, viii, 1905. IV] AND STRUCTURE OF THE CELL 315 it is at least plain that by some physical cause, some mutual attraotion or common repulsion of the particles, we must seek to account for the symmetry of the so-called "tetrads," and other more or less familiar configurations. The remarkable annular chromosomes, shewn in Fig. 95, can be closely imitated by loops of thread upon a soapy film, when the film within the annulus is broken or its tension reduced ; the balance of forces is here a simple one, between the uniform capillary tension which tends to widen out the ring and the uniform cohesion of its particles which keeps it together. We may find other cases, at once simpler and more varied, where the chromosomes are bodies of rounded form and more or less Fig. 95. Annular chromosomes, formed in the spermatogenesis of the mole-cric'iict. From Wilson, after Vom Rath. uniform size. These also find their way to. an equatorial plate; we gather (and Lamb assures us) that they are repelled from the centrosomes. They may go near the equatorial periphery, but they are not driven there; and we infer that some bond of mutual attraction holds them together. If they be free to move in a fluid medium, subject both to some common repulsion and some mutual attraction, then their circumstances are much like those of Mayer's well-known experiment of the floating magnets. A number of magnetised needles stuck in corks, all with like poles upwards, are set afloat in a basin; they repel one another, and scatter away to the sides. But bring a strong magnet (of unlike pole) overhead, and the little magnets gather in under its common attraction, while still keeping asunder through their own mutual repulsion. The symmetry of forces leads to a symmetrical configuration, which is 316 ON THE INTERNAL FORM [ch. the mathematical expression of a physical equiUbrium — and is the not too remote counterpart of the arrangement of the electrons in an atom. Be that as it may, it is found that a group of three, four or five Httle magnets arrange themselves at the corners of an equilateral triangle, square or pentagon; but a sixth passes within the ring, and comes to rest in the centre of symmetry of the pentagon. If there be seven magnets, six form the ring, and the seventh occupies the centre ; if there be ten, there is a ring of eight and two within it ; and so on, as follows * : Number of magnets 5 6 7 8 9 10 11 12 13 14 15 16 Do. in outer ring Do. in inner ring 5 0 5 1 6 1 7 1 8 1 8 2 8 3 9 3 10 3 10 4 10 5 11 5 When we choose from the published figures cases where the chromosomes are as nearly as possible alike in size and form — the condition necessary for our parallel to hold — then, as LilHe pre- dicted and as Doncaster and Graham Cannon have shewn, their congruent arrangement agrees, even to a surprising degree, with what we are led to expect by theory and analogy (Fig. 96). 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 associated with surface-tension changes, of which the first step is a minute puckering of the surface-skin, a sort of interdigitation with the surrounding medium. For instance, Schewiakoff has observed in Euglyphaf that, just before the break-up of the nucleus, a system of rays appears, concentred about it, but having nothing to do with the polar asters: and during the existence of this striation the nucleus enlarges very considerably, evidently by imbibition of fluid from the surrounding protoplasm. In short, diffusion is at work, hand in hand with, and as it were in opposition to, the surface- tensions which define the nucleus. By diffusion, hand in hand with surface-tension, the alveoli of the nuclear meshwork are formed, enlarged and finally ruptured: diffusion sets up the movements * H. Graham Cannon, On the nature of the centrosomal force, Journ. Genetics, XIII, p. 55, 1923. t Schewiakoff, Ueber die karyokinetische Kerntheilung der Euglypha alveolata, Morph. Jahrb. xiii, pp. 193-258, 1888 (see p. 216). IV] AND STRUCTURE OF THE CELL 317 which give rise to the appearance of rays, or striae, around the nucleus: and through increasing diffusion and weakening surface- tension the rounded outHne of the nucleus finally disappears. 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 ••• •V. ^»N f" % o a o 0 1 ^ ^ (I V J * /•v • V # -^ lOj 0. /•;•;% Fig. 96. Various numbers of chromosomes in the equatorial plate: the ring- diagrams give the arrangements predicted by theory. From Graham Cannon. in particular, the order of succession in which certain of the pheno- mena occur is variable and irregular. The precise order of the phenomena, the tiiiie 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 ''inter- ferences." And it is worthy of particular note that these variations in the order of events and in other subordinate details, while 318 ON THE INTERNAL FORM [ch. doubtless attributable to specific physical conditions, would seem to be without any obvious classificatory meaning or other biological significance. So far as we have now gone, there is no great difficulty in pointing to simple and familiar examples 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 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 chromosomal matter differs in permeability from the medium, that is to say the cytoplasm, in which it hes, let us now make the further assumption that its permeabihty is variable, and depends upon the strength of the field. In Fig. 97, we have a field of force (representing our cell), con- sisting of a homogeneous medium, and including two opposite poles : lines of force are indicated by full lines, and loci of constant magnitude of force are shewn by dotted lines, these latter being what are known as Cayley's equipotential curves*. Let us now consider a body whose permeabihty (/a) depends on the strength of the field F. At two field -strengths, such as F^, F^, let the permeability of the body be equal to that of the medium, and let the curved line in Fig. 98 represent generally its permeabihty at other field-strengths; and let the outer and inner dotted curves in Fig. 97 represent respectively the loci of the field-strengths F^, * Phil. Trans, xiv, p. 142, 1857. Cf. also F. G. Teixeira, TraiU des Courhes, I, p. 372, Coimbra'. 1908. IV] AND STRUCTURE OF THE CELL 319 and Fa. 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 Fb Fig. 97. Fig. 98. in the region intermediate to the two dotted curves, it will tend to move towards regions of weaker field-strength. The locus Fly is therefore a locus of stable position, towards which the body tends to move ; the locus F^ is a locus of unstable position, from which it tends to move. If the body were placed across F^^, 320 ON THE INTERNAL FORM [ch. 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 ¥„, 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 Fa and F^, are meanwhile situated somewhat farther from the axis than in our figure, that (for instance) F^ is situated where we have drawn F^, and that F^ is still farther out. The bodies then tend towards the poles; but the tendency may be very small if, in Fig. 98, the curve and its intersecting straight hne do not diverge very far from one another beyond Fa\ in other words, if, when Fig. 99. situated in this region, the permeability of the bodies is not very much in excess of that of the medium. Let the poles now tend to separate farther and farther from one another, the strength of each pole remaining unaltered; in other words, let the centrosome-foci recede from one another,- as they actually do, drawing out the spindle-threads between them. The loci Fa, 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 Fa is in some median position (Fig. 99). When the locus Fa has passed entirely over the body, the body tends to move towards regions of weaker force; but when, in turn, the locus Fi, has crossed it, then the body again moves towards regions of stronger force, that is to say, towards the nearest pole. IV] AND STRUCTURE OF THE CELL 321 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 histologist*. 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 simpHcity. Doubtless there are other assumptions which would meet the case; for instance, that during the active phase of the chromatin molecule (when it de- composes and sets free nucleic acid) it carries a charge opposite to that which it bears during its resting, or alkahne phase ; and that it would accordingly move towards different poles under the influetice 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 demonstrate, by such illustrations as these, that, whatever be the actual and as yet unknown modus operandi, there are physical conditions and distributions of force which could produce just such phenomena of movement as we see taking place within the living cell. This, and no more, is precisely what Descartes is said to have claimed for his description of the human body as a " mechanism f." While it can scarcely be too often repeated that our enquiry is not directed towards the solution of physiological problems, save 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 * • 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. t Michael Foster, Lectures on the History of Physiology, 1901, p. 62. 322 ON THE INTERNAL FORM [ch. particular forces are at work when the mere visible forms produced are such as to leave this an 6pen 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. LiUie, 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, LilHe comes easily to the conclusion that "electrical theories of mitosis are entitled to more careful consideration than they have hitherto received." * Amer. J. Physiol, viii, pp. 273-283, 1903 {vide supra, p. 314); cf. ibid, xv, pp. 46-84, 1905; xxii, p. 106, 1910; xxvii, p. 289, 1911; Journ. Exp. Zool. xv, p. 23, 1913; etc. t In like manner Hardy shewed 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 wall 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. Physiol, xxrv, pp. 288-304, 1899; also Hardy and H. W. Harvey, Surface electric charges of living cells, Proc. E.S. (B), Lxxxiv, pp. 217-226, 191 1, and papers quoted therein. Cf. also E. N. Harvey's observations on the convection of unicellular organisms in an electric field (Studies on the permeability of cells, Journ. Exp. Zool. x, pp. 508-556, 1911). IV] AND STRUCTURE OF THE CELL 323 Among other investigations all leading towards the same general conclusion, namely that differences of electric potential play their part in the phenomena of cell division, I would mention a note- worthy 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 also that this difference fluctuates, or actually reverses its direction, periodically, at epochs coinciding with successive acts of segmentation or other Fig. 100. Final stage in the first seg- mentation of the egg of Cerebra- tidus. From Prenant, after Coe*. Fig. 101. Diagram of field of force with two similar poles. important phases in the development of the' eggt; just as other physical rhythms, for instance, in the production of COg , had already been shewn to do. Hence we need not be surprised to find that th^ "materialised" hues of force, which in the earlier stages form the * On differences in electrical potential in developing eggs, Amer. Journ. Physiol. XII, pp. 241-275, 1905. This paper contains an excellent summary, for the time being, of physical theories of the segmentation of the cell. t Gray has demonstrated a temporary increase of electrical conductivity in sea-urchin eggs during the process of fertilisation, and ascribes the changes in resistance to polarisation of the surface: Electrical conductivity of echinoderm eggs, etc., Phil. Trans. (B), ccvii, pp. 481-529, 1916. 324 ON THE INTERNAL FORM [CH. 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 in the field by the divided and reconstituted nuclei, are now ahke in their polarity* (Figs. 100, 101). The foregoing account is based on the provisional assumption that the phenomena of caryokinesis are analogous to those of a bipolar electrical field — a comparison which seems to offer a helpful and instructive series of analogies. But there are other forces which lead to similar configurations. For instance, some of Leduc's diffusion-experiments offer very remarkable analogies to the dia- grammatic phenomena of caryokinesis, as shewn in Fig. 102 f. Fig. 102. Artificial caryokinesis (after Leduc), for comparison with Fig. 88, p. 299. Here we have two identical (not opposite) poles of osmotic con- centration, 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 is drawn in opposite directions by equal forces, and so tends to remain undisturbed, in the form of an "equatorial plate." To account for the same mitotic phenomena an elegant hypothesis has been put forward by A. B. Lamb J, and developed by Graham * W. R. Coe, Maturation and fertilisation of the egg of Cerebratulus, Zool. Jahrbiicher {Anat. Abth.), xii, pp, 425^76, 1899. t Op. cit. pp. 110 and 91. t A. B. Lamb, A new explanation of the mechanism of mitosis, Journ. Exp. Zool. V, pp. 27-33. 1908. IV] AND STRUCTURE OF THE CELL 325 Cannon*. It depends on certain investigations of the Bjerknes, father and sonf, which prove that bodies pulsating or oscillating J in a fluid set up a field of force precisely comparable with the lines of force in a magnetic field. Certain old and even familiar observa- tions had pointed towards this phenomenon. Guyot had noticed that bits of paper were attracted towards a vibrating tuning-fork; and Schellbach found that a sounding-board so acts on bodies in its neighbourhood as to attract those which are heavier and repel those which are lighter than the surrounding medium; in air bits of paper are attracted and a gas-flame is repelled. To explain these simple observations, Bjerknes experimented with Httle drums attached to an automatic bellows. He found that two bodies in a fluid field, synchronously pulsating or synchronously oscillating, repel one another when their oscillations are in the same phase, or their pulsations are in opposite phase; and vice versa: while other particles, floating passively in the same fluid, tend (as Schellbach had observed before) to be attracted or repulsed according as they are heavier or lighter than the fluid medium. The two bodies behave towards one another like two electrified bodies, or like two poles of a magnet; we are entitled to speak of them as "hydro- dynamic poles," we might even call them " hydrodynamic magnets" ; and pursuing the analogy, we may call the heavy bodies para- magnetic, and the light ones diamagnetic with regard to them. Lamb's hypothesis then, and Cannon's, is that the centrosomes act as "hydrodynamic magnets." The explanation depends on oscilla- tions which have never been seen, in centrosomes which are not always to be discovered. But it brings together certain curious analogies, and these, where we know so little, may be worth reflecting on. If we assume that each centrosome is endowed with a vibratory motion as it floats in the semi-fluid colloids, or hydrosols (to use Graham's word) of the cell, we ma;^ take it that the visible intra- cellular phenomena will be much the same as those we have * Op. cit. Cf. also Gertrud Woken, Zur Physik der Kernteilung, Z. f. allg, Physiol. XVIII, pp. 39-57, 1918. t V. Bjerknes, Vorlesungen liber hydrodynamische Fernkrdjte, nach C. A. Bjerknes^ Theorie, Leipzig, 1900. J A body is said to pulsate when it undergoes a rhythmic change of volume; it oscillates when it undergoes a rhythmic change of place. 326 ON THE INTERNAL FORM [ch. described under an electrical hypothesis; the lines of force will have the same distribution, and such movements as the chromo- somes undergo, and such symmetrical configurations as they assume, may be accounted for under the one hypothesis pretty much as under the other. There are however other phenomena accompanying mitosis, such as Chambers's astral currents and certain local changes in the viscosity of the egg, which are more easily explained by the hydrodynamic theory. We may assume that the cytoplasm, however complex it may be, is but a sort of microscopically homogeneous emulsion of high dispersion, that is to say one in which the minute particles of one phase are widely scattered throughout, and freely mobile in, the other; and this indeed is what is meant by caUing it a hydrosol. Let us assume also that the particles are a little less dense than the continuous phase in which they are dispersed; and assume lastly (it is not the easiest of our assumptions) that these ultra-minute particles will be affected, just as are the grosser ones, by the forces of the hydrodynamic field. All this being so, the disperse particles will be repelled from the oscillating centrosome, with a force which falls off very rapidly, for Bjerknes tells us that it varies inversely as the seventh power of the distance; a round clear field, hke a drop or a bubble, will be formed round the centrosome ; and the disperse particles, expelled from this region, will tend to accumulate in a crowded spherica- zone immediately beyond it. Outside of this again they will con- tinue to be repulsed, but slowly, and we may expect a second and lesser concentration at the periphery of the cell. A clear central mass, or "centrosphere," will thus come into being; and the surrounding cytoplasm will be rendered denser and more viscous, especially close around the centrosphere and again peripherally, by condensation of the disperse particles. Moreover, all outward movements of these lighter particles entail inward movements of the heavier, which (by hypothesis) are also the more fluid ; stream- lines or visible currents will flow towards the centre, giving rise to the star-shaped "aster," and the best accounts of the sea-urchin's egg* tally well with what is thus deduced from the hydrodynamic * Cf. R. Chambers, in Journ. Exp. Zool. xxni, p. 483, 1917; Trans. R.S. Canada, XII, 1918; Journ. Gen. Physiol, ii, 1919. IV] AND STRUCTURE OF THE CELL 327 hypothesis. The round drop of clear fluid which forms the centre of the aster grows as the aster grows, fluid streaming towards it from all parts of the cell along the channels of the astral rays. The cytoplasni between the rays is in the gel state, but gradually passes into a sol beyond the confines of the aster. Seifritz asserts that the substance of the centrosphere is *'not much more viscous than water," but that the wedges of cytoplasm between the inwardly directed streams are stiff and viscous*. After the centrosome divides we have two oscillating bodies instead of one; they tend to repel one another, and pass easily through the fluid centrosphere to the denser layer around. But now the new centrosomes, on opposite sides of the centrosphere, repel, each on its own side, the disperse particles of the denser zone ; and two new asters are formed, their rays marked by the streams coursing inwards to the centrosome-foci. Thus the amphiaster comes into being; it is not that the old aster divides, as a definite entity; but the old aster ceases to exist when its focus is disturbed, and about the new foci new asters are necessarily and automatically developed. Again this hypothetic account taUies well with Chambers's description. The same attractions and repulsions should be manifested, perhaps better still, in whatsoever bodies he or float within the cell, whether liquid or solid, oil-globules, yolk-particlea, mitochondria, chromo- somes or what not. A zoned, concentric arrangement of yolk- globules is often seen in the egg, with the centrosome as focus; and in certain sea-urchin eggs the mitochondria gather around the centrosome while the amphiaster is forming, collecting together in that very zone to which Chambers ascribes a semi-rigid or viscous consistency!. The Golgi bodies found in various germ-cells are at first black rod-hke bodies embedded in the centrosphere ; they undergo changes and complex movements, now scattering through the cytoplasm and anon crowding again around the centrosome. Some periodic change in the density of these bodies compared with ♦ Cf. W. Seifritz, Some physical properties of protoplasm, Ann. Bot. xxxv, 1921. Wo. Ostwald and M. H. Fischer had thought that the astral rays were due to local changes of the plasma-sol into a gel, Zur physikal. chem. Theorie der Befruchtung, Pfluger's Archiv, cvi, pp. 2^3-266, 1905. t Cf. F. Vejdovsky and A. Mrazek, Umbildung des Cytoplasma wahrend der Befruchtung und Zelltheilung, Arch. f. mikr. Anat. LXii. 431-579, 1903. 328 ON THE INTERNAL FORM . [ch. that of the medium in which they He seems all that is required to account for their excursions; and such changes of density are not only of likely occurrence during the active chemical operations associated with fertilisation and division, but are in all probability inseparable from the changes in viscosity which are known to occur*. The movements and arrangements of the chromosomes, already described, may be easily accounted for if we postulate, in addition to their repulsion from the oscillating centrosomes, induced oscillations in themselves such as to cause them to attract one another. The well-defined length of the spindle and the position of equili- brium in which it comes to rest may be conceived as resultants of the several mutual repulsions of the centrosomes by one another, by the chromosomes or other lighter material of the equatorial plate, and again by such lighter material as may have accumulated at the periphery of the egg ; the first two of these will tend to lengthen the spindle, the last to shorten it; and the last will especially affect its position and direction. When Chambers amputated part of an amphiastral egg, the remains of the amphiaster disappeared, and then came into being again in a new and more symmetrical position ; it or its centrosomal focus had been symmetrically repelled, we may suppose, by the fresh surface. Hertwig's law that the spindle-axis tends to lie in the direction of the largest mass of protoplasm, in other words to point where the cell-surface lies farthest off and its repulsion is least felt, may likewise find its easy explanation. Between these hypotheses we may choose one or other (if we choose at all), according to our judgment. As Henri Poincare tells us, we never know that any one physical hypothesis is true, we take the simplest we can find; and this we call the guiding principle of simphcity ! In this case, the hydrodynamic hypothesis is a simple one; but it all rests on a hypothetic oscillation of the centrosomes, which has never been witnessed. Bayliss has shewn that precisely such reversible states of gelation as we have been speaking of as * Cf. G. Odquist, Viscositatsanderungen des Zellplasmas wahrend der ersten Entwicklungsstufen des Froscheies, Arch. f. Entw. Mech. ia, pp. 610-624, 1922; A. Gurwitsch, Pramissen und anstossgebende Faktoren der Furchung und Zelltheilung, Arch. f. Zellforsch. n, pp. 495-548, 1909; L. V. Heilbrunn, Protoplasmic viscosity-changes during mitosis, Journ. Exp. Zool. xxxiv, pp. 417-447, 1921 ; ibid, xliv, pp. 255-278, 1926; E. Leblond, Passage de I'etat de gel a I'etat de sol dans le protoplasme vivant, C.R. Soc. Biol, lxxxii, p. 1150; of. ibid. p. 1220; etc. rv] AND STRUCTURE OF THE CELL 329 "periodic changes in viscosity" may be induced in living protoplasm by electrical stimulation*. On the other hand, the fact that the hydrodynamic forces fall off as fast as they do with increasing distance Hmits their efficacy ; and the minute disperse particles must, under Stokes's law, be slow to move. Lastly, it may well be (as Lillie has urged) that such work as his own, or Ida Hyde's, or Gray's, on change of potential in developing eggs, taken together with that of many others on the behaviour of colloid particles in an electrical field, has not yet been followed out in all its consequences, either on the physical or the physiological side of the problem. But to return to our general discussion. As regards the actual mechanical division of the cell into two halves, we shall see presently that, in certain cases, such as that of a long cylindrical filament, surface-tension, and what is known as the principle of "minimal areas," 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 none too easy to see what physical cause is at work to disturb its equiUbrium 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 comphcated. Before we can hope to comprehend it, we shall 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 also to take account of the differences of potential which the material arrangements along the lines of force must themselves tend to produce. Only thus can 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 * W. M. Bayliss, Reversible gelation in living protoplasm, Proc. R.S. (B), xci, pp. 196-201, 1920. PO ON THE INTERNAL FORM [ch. 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 localised differences of surface-tension are likely to occur. If not, then changes in the conformation of the cell such as lead immediately to its division must be ascribed not to local changes in its surface-tension, but rather to direct changes in internal pressure, or to mechanical forces due to an induced surface-distribution of electrical potential. We have little reason to be sceptical ; in fact we now know that the cell is so far from being chemically and physically homogeneous that local variations in its surface-tension are more than likely, they are certain to occur. Biitschh suggested more than sixty years ago that cell-division was brought about by an increase of surface-tension in the equatorial region of the cell ; and the suggestion was the more remarkable that it was (I beheve) the very first attempt to invoke surface-tension as a factor in the physical causation of a biological phenomenon*. An increase of equatorial tension would cause the surface-area there to diminish, and the equator to be pinched in, but the total surface- area of the cell would be increased thereby, and the two effects would strike a balance f. But, as Biitschh knew very well, the surface-tension change would not stand alone; it would bring other phenomena in its train, currents would tend to be set up, and tangential strains would be imposed on the cell-membrane or cell- surface as a whole. The secondary if not the direct effects of increased equatorial tension might, after all, suffice for the division of the cell. It was Loeb, in 1895, who first shewed that streaming went on from the equator towards the divided nuclei. To the violence of these streaming movements he attributed the phenomenon of division, and many other physiologists have adopted this hypo- thesis J. The currents of which Loeb spoke call for counter-currents * 0. Butschli, tJber die ersten Entwicklungsvorgange der Eizelle, Abh. Senckenberg. naturf. Gesellsch. x, 1876; Uber Plasmastrqmungen bei der Zell- theilung, Arch. f. Entw. Mech. x, p. 52, 1900. Ryder ascribed the earyokinetic figures to surface-tension in his Dynamics in Evolution, 1894. t A relative, not positive, increase of surface-tension, was part of Giardina's hypothesis: Note sul mecanismo della divisione cellulare, Anat. Anz. xxi. 1902. J J. Loeb, Amer. Journ. Physiol, vi, p. 432, 1902; E. G. Conklin, Protoplasmic movements as a factor in differentiation, Wood's Hole Biol. Lectures, p. 69, etc., 1898-99; J. Spek, Oberflachenspannungsdifferenzen als eine Ursache der Zell- teilung, Arch.f. Entw. Mech. xliv, pp. 54-73, 1918. IV] AND STRUCTURE OF THE CELL 331 towards the equator, in or near the surface of the cell; and theory and observation both indicate that precisely such currents are bound to be set up by the surface-energy involved in the increase of equatorial tension. An opposite view has been held by some, and especially by T. B. Robertson*. Quincke had shewn that the formation of soap at the surface of an oil-droplet lowers the surface-tension of the latter, and that if the saponification be local, that part of the surface tends to enlarge and spread out accordingly. Robertson, in a very curious experiment, found that by laying a thread, moistened with dilute caustic alkali or merely smeared with soap, across a drop of olive oil afloat in water, the drop at once divided into two. A vast amount of controversy has arisen over this experiment, but Spek seems to have shewn conclusively that it is an exceptional case. In a drop of olive-oil, balanced in water f and touched anywhere with an alkaH, there is so copious a formation of lighter soaps that di^erences of density tend to drag the drop in two. But in the case of other oils (and especially the thinner oils, such as oil of bergamot) the saponified portion bulges, as theory directs; and when the alkali is applied to two opposite poles the equatorial region is pinched in, as McClendonJ, in opposition to Robertson, had found it to do. Conversely, if an alkaline thread be looped around the drop, the zone of contact bulges, and instead of dividing at the equator the drop assumes a lens-like form. We may take it then as proven that a relative increase of equatorial surface-tension, whether in oil-drops, mercury-globules or living cells, does lead, or tend to lead, to an equatorial constriction. In all cases a system of surface-currents is set up among the fluid drops towards the zone of increased tension ; and an axial counter-current flows towards the pole or poles of lowered tension. Precisely such currents have been observed to run in various eggs (especially of ♦ T. B. Robertson, Note on the chemical mechanics of cell-division, Arch. f. Entw. Mech. xxvn, p. 29, 1909; xxxii, p. 308, 1911; xxxv, p. 402, 1913. Cf. R. S. Lillie, Joum.-Exp. Zool. xxi, pp. 369^02, 1916; McClendon, loc. cit.; etc. t In these experiments, and in many of Quincke's, a little chloroform is added to the oil, in order to bring its density as near as may be to that of water. X J. F. McClendon, Note on the mechanics of cell -division, Arch. f. Entw. Mech. xxxrv, pp. 263-266, 1912. 332 ON THE INTERNAL FORM [ch. certain Nematodes) during division of the cell; but if the })rocess be slow, more than 7 or 8 minutes long, the slow currents become hard to see. Various contents of the cell are transported by these currents, and clear, yolk-free polar caps and equatorial accumula- tions of yolk and pigment are among the various manifestations of the phenomenon. The extrusion of a polar body, at a small and sharply defined region of lowered tension, is a particular case of the same principle*. But purely chemical changes are not of necessity the fundamental cause of alteration in the surface-tension of the egg, for the action of electrolytes on surface-tension is now well known and easily demonstrated. So, according to other views than those with which we have been dealing, electrical charges are sufficient in themselves to account for alterations of surface-tension, and in turn for that protoplasmic streaming which, as so many investigators agree, initiates the segmentation of the eggf. A great part of our difficulty arises from the fact that in such a case as this the various pheno- mena 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 adsorption has been adduced (as we have already briefly said) in order to account for many of the events and appearances which are associated with the asymmetry, and lead towards the division, of the cell. But our short discussion of this phenomenon may be reserved for another chapter. However, we are not directly concerned here with the phenomena of segmentation or cell-division in themselves, except only in so far as visible changes of form are capable of easy and obvious correla- tion 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 the gist of the matter is that, if you have caryo- kinetic figures developing inside the cell, that of itself indicates that the dynamic system and the localised forces arising from it are in * J. Spek, loc. cit. pp. 108-109. t Cf. D'Arsonval, Relation entre la tension superficielle et certains phenomenes electriques d'origine animale. Arch, de Physiol, i, pp. 460-472, 1889; Ida H. Hyde, op. cit. p. 242. I Cf. Plateau's remarks (Statique des liquides, ii, p. 154) on the tendency towards equilibrium, rather than actual equilibrium, in many of his systems of soap-films. IV] AND STRUCTURE OF THE CELL 333 gradual alteration; and changes in the outward configuration of the system are bound, consequently, to take place. Perhaps we may simplify the case still more. We have learned many things about cell-division, but we do not know much in the end. We have dealt, perhaps, with too many related phenomena, and failed because we tried to combine and account for them all. A physical problem, still more a mathematical one, wants reducing to its simplest terms, and Dr Rashevsky has simplified and general- ised the problem of cell-division (or division of a drop) in a series of papers, which still outrun by far the elementary mathematics of this book. If we cannot follow^ him in all he does, we may find useful lessons in his way of doing it. Cells are of many kinds; they differ in size and shape, in visible structure and chemical com- position. Most have a nucleus, some few have none; most need oxygen, some few do not; some metabolise in one way, some in another. What small residuum of properties remains common to them all? A living cell is a little fluid (or semi-fluid) system, in which work is being done, physical forces are in operation and chemical changes are going on. It is in such intimate relation with the world outside — its own milieu interne with the great ynilieu externe — that substances are continually entering the cell, some to remain there and contribute to its growth, some to pass out again with loss of energy and metabolic change. The picture seems simplicity itself, but it is less simple than it looks. For on either side of the boundary- wall, both in the adjacent medium and in the living protoplasm within, there will be no uniformity, but only degrees of activity, and gradients of concentration. Substances which are being absorbed and consumed will diminish from periphery to centre; those which are diffusing outwards have their greatest concentration near the centre, decrease towards the periphery, and diminish further with increasing distance in the near neighbourhood of the system. Size, shape, diffusibility, permeability, chemical properties of this and that, may affect the gradients, but in the living cell the interchanges are always going on, and the gradients are always there *. * Outward diffusion makes one of the many contrasts between cell-growth and crystal-growth. But the diffusion^gradients round a growing crystal are far more complicated than was once supposed. Cf. W. F. Berg, Crystal growth from solutions, Proc. R.8. (A), clxiv, pp. 79-95, 1938. 334 ON THE INTERNAL FORM [ch. If the cell be homogeneous, taking in and giving out at a constant rate in a uniform way, its shape will be spherical, the concentration- field of force, or concentration-field, will likewise have a spherical symmetry, and the resultant force will be zero. But if the symmetry be ever so little disturbed, and the shape be ever so little deformed, then there will be forces at work tending to increase the deformation, and others tending to equalise the surface-tension and restore the spherical symmetry, and it can be shewn that such agencies are within the range of the chemistry of the- cell. Since surface- tension becomes more and more potent as the size of the drop diminishes, it follows that (under fluid conditions) the smallest solitary cells are least likely to depart from a spherical shape, and that cell-division is only likely to occur in cells above a certain critical order of magnitude; and using such physical constants as are available, Rashevsky finds that this critical magnitude tallies fairly well with the average size of a living cell. The more important lesson to learn, however, is this, that, merely by virtue of its metabolism, every cell contains within itself factors which may lead to its division after it reaches a certain critical size. There are simple corollaries to this simple setting of the case. Since unequal concentration-gradients are the chief cause which renders non-spherical shapes of cell possible, and these last only so long as the cell lives and metabolises, it follows that, as soon as the gradients disappear, whether in death or in a "resting-stage", the cell reverts to a spherical shape and symmetry. Again, not only is there a critical size above which cell-division becomes possible,, and more and more probable, but there must also be a size beyond which the cell is not likely to grow. For the "specific surface" decreases, the metabolic exchanges diminish, the gradients become less steep, and the rate of growth decreases too ; there must come a stage where anabolism just balances katabolism, and growth ceases though life goes on. When streaming currents are visible within the cell, they seem to complicate the problem; but after all, they are part of the result, and proof of the existence, of the gradients •we have described. In any further account of Rashevsky 's theories the mathematical difficulties very soon begin. But it is well to realise that pure theory often carries the mathematical physicist a long way ; and that higher and higher powers of the microscope, and IV] AND STRUCTURE OF THE CELL 335 greater and greater histological skill are not the one and only way to study the physical forces acting within the cell *. 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 forces which are more or less 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, the same attraction being further manifested in a small conical uprising of the surface of the eggt. The nature of the attractive force has been much disputed. Loeb found the spermatozoon to be equally attracted by other substances, even by a bead of glass. It has been held also that the attraction is chemotropic, some substance being secreted by the egg which drew the sperm towards it: just as Pfeifer, having shewn that maUc acid has an attraction for fern-antheridia, supposed this substance to play its attractive part within the mucus of the archegonia. Again, the chemical secretion may be neither attractive nor directive, but yet play a useful part in activating the spermatozoa. However that may be, Gray has shewn reason to believe that an electromotive force is developed in the contact between active spermatozoon and inactive ovum; and that it is the electrical change so set up, and almost instantaneously propagated, which precludes the entry of another spermatozoon J. Whatever the force may be, it is one which acts normally to the surface of the ovum, and after entry the * Cf. N. Rashevsky, Mathematical Biophysics, Chicago, 1938; and many earlier papers. Eg. Physico-matheraatical aspects of cellular multiplication and de- velopment, Cold Spring Harbor Symposia, ii, 1934; The mechanism of division of small liquid systems which are the seat of physico-chemical reactions, Physics, in, pp. 374-379, 1934; papers in Protoplasma, xiv-xx, 1931-33, etc. t With the classical account by H. Fol, C.R. lxxxiii, p. 667, 1876; Mem. Soc. Phys. Geneve, xxvi, p. 89, 1879, cf. Robert Chambers, The mechanism of the entrance of sperm into the star-fish egg, Journ. Gen, Physiol, v, pp. 821-829, 1923. Here a delicate filament is said to run out from the fertilisation -cone and drag the spermatozoon in; but this is disputed and denied by E. Just, Biol. Bull. LVii, pp. 311-325, 1929. I But, under artificial conditions, "polyspermy" may take place, eg. under the action of dilute poisons, or of an abnormally high temperature, these being doubtless also conditions under which the surface-tension is diminished. 336 ON THE INTERNAL FORM [ch. spermatozoon 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 con- sequence of the entry of the spermatozoon is an increase in the surface-tension of the egg: this being but one of the complex reactions exhibited by the surface, or cortex of the cell*. Some- where or other, within the egg, near or far away, lies its own nuclear body, the so-called female pronucleus, and we find that after a while this has fused with the "male pronucleus" or head of the spermatozoon, 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 many years 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 an easy task to calculate the precise path which the two pronuclei would follow, leading to con- jugation and to the central 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 he 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 line 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 little streak of brown pigment, and the fact of the meeting of the pronuclei before reaching the centre has been repeatedly seen by many observers f. * See Mrs Andrews' beautiful observations on "Some spinning activities of protoplasm in starfish and echinoid eggs," J own. Morphol. xii, pp. 307-389, 1897. t W. Pfeffer, Locomotorische Richtungsbewegungen durch chemische Reize, Unters. a. d. Botan. Inst. Tubingen, i, 1884; Physiology of Plants, m, p. 345, Oxford, 1906; W. J. Dakin and M. G. C, Fordham, Journ. Exp. Biol. t. pp. 183-200, 1924. Cf. J. Loeb, Dynamics of Living Matter, 1906, p. 153. IV] AND STRUCTURE OF THE CELL 337 The problem recalls the famous problem of three bodies, which has so occupied the astronomers; and it is obvious that the foregoing brief description is very far from including all possible cases. Many of these are particularly described in the works of Fol, Roux, Whitman and others*. The intracellular phenomena of which we have now spoken have assumed great importance in biological literature and discussion during the last fifty years; but it is open to us to doubt whether they will be found in the end to possess more than a secondary, even a remote, biological significance. Most, if not all of them, would seem to follow immediately and inevitably from certain 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 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 secondary biological importance; and that is, the great variation to which these phenomena are subject in similar or closely related organisms, and the apparent impossibihty of correlating them with the pecuHarities of the organism as a whole. In a broad and general way the phenomena are always the same. Certain structures swell and contract, twine and untwine, split and unite, advance and retire ; certain chemical changes also repeat themselves. But Nature rings the changes on all the details. "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, * H. Fol, Becherches sur la fecondation, 1879; W. Roux, Beitrage zur Erit- wickelungsmechanik des Embryos, Arch. f. Mikr. Anat. xix, 1887; C. 0. Whitman, Ookinesis, Journ. Morph. i, 1887; E. Giglio-Tos, Entwicklungsmechanische Studien, I, Arch. f. Entw. Mech. li, p. 94, 1922, See also Frank R. Lillie, Problems of Fertilisation, Chicago, 1919. t Wilson, The Cell. p. 77; cf. 3rd ed. (1925), p. 120. 338 FORM AND STRUCTURE OF THE CELL [ch. sometimes it is adjacent to, and at other times it lies remote from, the nucleus. The distribution of potential is very often symmetrical and bipolar, as in the case described; but a less symmetrical distribution often occurs, with the result that we have, for a time at least, numerous centres of force, instead of the two main correlated poles : this is the simple explanation of the numerous stellate figures, Haploid number of chr.'raosomes Fig. 103. Summation diagram shewing the % number of instances (among 2,415 phanerogams and 1,070 metazoa), in which the chromosomes do not exceed a given number. Data from M. J. D. White. or "Strahlungen," which have been described in certain eggs, such as those of Chaetopterus. The number of chromosomes may be constant within a group, as in the tailed Amphibia, with 12; or very variable, as in sedges, and in grasshoppers * ; in one and the same species of worm (Ascaris megdlocephala), one group or two groups of chromosomes may be present. And remarkably constant, in general, as the number 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 (Fig. 103); for (as regards * There are varieties of Artemia salina which hardly differ in outward characters, but differ widely in the number of their chromosomes. IV] OF CHROMOSOMES 339 the germ-nuclei) few have less than six chromosomes, and few have more than twenty*. 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 ways alluded to above ; while only here and there, as among the chrysanthemums, do related species or varieties shew their own characteristic chromo- some numbers. In contrast to the narrow range of the chromo- some numbers, we may reflect on the all but infinite possibilities of chemical variabihty. Miescher shewed that a molecule containing 40 C-atoms would admit (arithmetically though not necessarily chemically) of a million possible isomers; and changes in position of the N-atoms of a protein, for instance, might vastly increase that prodigious number. In short, we cannot help perceiving that many nuclear phenomena are not specifically related to the particular organism in which they have been observed, and that some are not even specially and indisputably connected with the organism as such. They include such manifestations of the physical forces, in their various permutations and combinations, as may also be witnessed, under appropriate conditions, in non-living things. When we attempt to separate our purely morphological or "purely * The commonest numbers of (haploid) chromosomes, both in plants and animals, are 8, 12 and 16. The median number is 12 in both, and the lower quartile is 8, likewise in both; but the upper quartile is 24 or thereby in animals, and in the neighbourhood of 16 in plants. If we may judge by the long lists given by E. B. Wilson (The Cell, 3rd ed. pp. 855-865), by M. Ishikawa in Botan. Mag. Tokyo, XXX, 1916, by M. J. D. White in his book on Chromosomes, or by Tischler in Tabulae Biologicae (1927), fully 60 per cent, of the observed cases lie between 6 and 16. As Wilson says (p. 866) "the number of chromosomes is per se a matter of secondary importance"; and (p. 868) "We must admit the present inadequacy of attempts to reduce the chromosome numbers to any single or consistent arithmetical rules." Clifford Dobell had said the same thing: "Nobody nowadays will be prepared to argue that chromosome numbers, as such, have any quantitative or qualitative relation to the characters exhibited by their owners. Complexity of bodily structure is certainly not correlated in any way with multiplicity of chromosomes " ; La Cellule, xxxv, p. 188, 1924. On the other hand, Tischler stoutly maintains that chromosome-numbers give useful evidence of phylogenetic affinity {Biol. Centralbl. XLvni, pp. 321-345, 1928); and there axe a few well-known cases, such as the chrysanthemums, where, undoubtedly, the numbers are constant and specific. Again in certain cases, the number of the chromosomes may- differ in dififerent races (diploid and tetraploid) of the same plant; and the difference is accompanied by differences in cell-size, in rate of growth, and even in the shape of the fruit (of. Sinnott and Blakeslee, Xat. Acad, of Sci. 1938, p. 476). 340 FORM AND STRUCTURE OF THE CELL [ch. embryological " studies from physiological and physical investiga- tions, 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 easily led to the framing of theories as to the ancestral history, the classificatory position, the natural affinities of the several organisms : in fact, to apply our embryological knowledge to the study of phylogeny. When we find, as we are not long of finding, that our phylogenetic hypotheses 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 earher 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 shghtest fink, if any, with the problems of zoological classification. Comparative embryology has its own facts to classify, and its own methods and principles of classification. 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 httle 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 efforts to explain "heredity" by help of "genes" and chromo- somes, which have grown up in the hands of Morgan and others since this book was first written, stand by themselves in a category which is all their own and constitutes a science which is justified of itself. To weigh or criticise these explanations would lie outside my purpose, even were I fitted to attempt the task. When these great discoveries began to be made, Bateson crossed the ocean to see and hear for himself what Morgan and his pupils had to shew and to tell. He came home convinced, and humbly marvelling. And I leave this great subject on one side not because I doubt for a moment the facts nor dispute the hypotheses nor decry the im- IV] OF THE CELL-THEORY 341 portance of one or other; but because we are so much in the dark as to the mysterious field of force in which the chromosomes he, far from the visible horizon of physical science, that the matter lies (for the present) beyond the range of problems which this book professes to discuss, and the trend of reasoning which it endeavours to mamtain. 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 individuahty. The partially segmented differs from the totally segmented egg, the unicellular Infusorian from the minute multi- * The " cell -theory " began early and grew slowly. In a curious passage which Mr Clifford Dobell has shewn me {Nov. Org. ii, 7, ad fin.). Bacon speaks of "cells" in the human body: of a " coUocatio spiritus per corpoream molem, eiusque pori, meatus, venae et cellulae, et rudimenta sive tentamenta corporis organici." It is " surely one of the most strangely prophetic utterances which even Bacon ever made." Apart from this the story begins in the seventeenth century, with Robert Hooke's well-known figure of the "cells" in a piece of cork (1665), with Grew's "bladders" or "bubbles" in the parenchyma of young beans, and Malpighi's "utriculi" or "sacculi" in the parenchyma or "utriculorum substantia" of various plants. Christian Fr. v. Wolff conceived, about the same time, a hypo- thetical "cell-theory," on the analogy of Leibniz's Monads; but the first clear idea of a cellular parenchyma, or contextus cellularis, came from C. Gottlieb Ludwig (1742), and from K. Fr. Wolff, who spoke freely of cells or cellulae. Fontana, author of a curious Traite sur le venin de la vipere (1781), described various histological elements, caught a glimpse of the nucleus, and experi- mented with reagents, using syrup of violets for a stain. Early in the eighteenth century the vessels of the plant played an important role, under Kurt Sprengel and Treviranus; but it was not till 1831 that Hugo v. Mohl recognised that they also arose from "cells." About this time Robert Brown discovered, or re-discovered, the nucleus (1833), which Schleiden called the cytohlast, or "cell- producer." It was Schleiden's idea, and a far-seeing one, that the cell lived a double life, a life of its own and the life of the plant to which it belonged: "jede Zelle fuhrt nun ein zweifaches Leben : ein selbststandiges, nur ihrer eigenen Entwicklung angehorigen, und ein anderes mittelbares, insofern sie integrierender Theil einer Pfianze geworden ist " {Phylogenesis, 1838, p. 1 ). The cell-theory, so long a- building, may be said to have been launched, and christened, with Schwann's Mikroskopische Untersmhungen of 1839. Within the next five years Martin Barry shewed how cell-division starts with the nucleus, Henle described the budding of certain cells, and Goodsir declared that all cells originate in pre-existing cells, a doctrine at once accepted by Remak, and made famous in pathology by Virchow. (Cf. {int. al.) J. G. McKendrick, On the modern cell-theory, etc., Proc Phil. Soc. Glasgow, xix, pp. 1-55, 1887; J. Stephenson, Robert Brown. . .and the cell-theory, Proc. Linn. Soc. 1931-2, pp. 45-54; M. Mobius, Hundert Jahre Zellenlehre, Jen. Ztschr. lxxi, pp. 313-326, 1938.) 342 FORM AND STRUCTURE OF THE CELL [ch. cellular Turbellarian, in the intensity and the range of those surface- tensions which in the one case succeed and in the other fail to form a visible separation between the cells. Adam Sedgwick used to call attention to the fact that very often, even in eggs that appear to be totally segmented, it is yet impossible to discover an actual separation or cleavage, through and through, between the cells which on the surface of the egg are so clearly delimited; so far and no farther have the physical forces effectuated a visible "cleavage." The vacuolation of the protoplasm in Actinophrys or Actinosphaerium is due to localised surface-tensions, quite irrespective of the multi- nuclear nature of the latter organism. In short, the boundary walls due to surface-tension may be present or may be absent, with or without the delimination of the other specific fields of force which are usually correlated with these boundaries and with the inde- pendent 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 bouhdary 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 inteUigible. 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 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 IV] OF THE CELL-THEORY 343 apparently all its later steps can be carried on, independently of nuclear action. Nor, as Verworn pointed out, can the nucleus possibly be regarded as the "sole vehicle of inheritance," since only in the conjunction of cell and nucleus do we find the essentials of cell-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 expressions denoting two differentiated areas in a common structural basis." Endless discussion has taken place regarding the centrosome, some holding that it is a specific and essential structure, a permanent corpuscle derived from a similar pre-existing corpuscle, a "fertilising element" in the spermatozoon, a special "organ of cell-division," a material "dynamic centre" of the cell (as Van Beneden and Boveri call it); while on the other hand, it is pointed out that many cells live and multiply without any visible centrosomes, that a centrosome may disappear and be created anew, and even that under artificial conditions abnormal chemical stimuh 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 con- centration 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 hving cell itself"; or to say that in the animal cell, the cell-wall, because it is "slightly developed," is relatively unimportant compared with the important role which it assumes in plants. On the contrary, it is quite certain that, whether visibly diiferentiated into a semi-permeable membrane or merely con- stituted by a liquid 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 solidified 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 344 FORM AND STRUCTURE OF THE CELL [ch. the entire multicellular organism : as Schwann said of old, in very- precise and adequate words, "the whole organism subsists only by means of the recijpTocal action of the single elementary parts*." As Wilson says again, "the physiological autonomy of the individual cell falls into the background . . . and the apparently composite character which the multicellular organism may exhibit is owing to a secondary distribution of its energies among local centres of action!." I* 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 hteral 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 inde- pendent units of Ufe||. As Goethe said long ago, "Das lebendige ist zwar in Elemente zerlegt, aber man kann es aus diesen nicht wieder zusammenstellen und beleben " ; the dictum of the Cellular- pathologie being just the opposite, "Jedes Thier erscheint als cine Summe vitaler Einheiten, von denen jede den vollen Charakter des Lebens an sich trdgt." Hofmeister and Sachs have taught us that in the plant the growth ♦ Theory of Cells, p. 191. t The Cell in Development, etc., p. 59; cf. 3rd ed. (1925), p. 102.- X E.g. Brticke, 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." § C. 0. Whitman, The inadequacy of the cell-theory, Journ. Morphol. viii, pp. 639-658, 1893; A. Sedgwick, On the inadequacy of the cellular theory of development, Q.J. M.S. xxxvii, pp. 87-101, 1895; xxxviii, pp. 331-337, 1896. Cf. G. C. Bourne, ibid, xxxviii, pp. 137-174, 1896; Clifford Dobell, The principles of Protistology, Arch. f. Protistenk. xxiii, p. 270, 1911. II Cf. 0. Hertwig, 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 ausgedruckt hat, die 'Lebensein- heiten.' Von diesem Gesichtspunkt aus betrachtet, erscheint der Gesammtlebens- prozess eines zusammengesetzten Organismus nichts Anderes zu sein als das hochst yerwickelte Resultat der einzelnen Lebensprozesse seiner zahlreichen, verschieden functionirenden Zellen." But in 1920 Doncaster (Cytology, p. 1) declared that "the old idea of discrete and independent cells is almost abandoned," and that the word cell was coming to be used "rather as a convenient descriptive term than as denoting a fundamental concept of biology"; and James Gray {Experimental Cytology, p. 2) said, in 1931, that "we must be careful to avoid any tacit assumption that the cell is a natural, or even legitimate, unit of life and function." IV] OF THE CELL-THEORY 345 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 liefert die Theile, nicht die Theile das Ganze: letzteres setzt die Theile zusammen, nicht diese jenesf." 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 J. 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 hke 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 biology §." * Journ. Morph. viii, p. 653, 1893. t Neue Grundlegungen zur Kenntniss der Zelle, Morph. Jahrb. viii, pp. 272, 313, 333, 1883. X Cf. e.g. Ch. van Bambeke, A propos de la delimitation cellulaire, Bull. Soc. beige de Microsc. xxiii, pp. 72-87, 1897. § Journ. Morph. ii, p. 49, 1889. CHAPTER V THE FORMS OF CELLS Protoplasm, as we have already said, is a fluid* or a semi-fluid substance, and we need not try to describe the particular properties of the colloid or jelly-like 'substances to which it is alHed, or rather the characteristics of the "colloidal state" in which it and they exist; we should find it no easy matter f. Nor need we appeal to precise theoretical definitions of fluidity, lest we come into a debatable land. It is in the most general sense that protoplasm is "fluid." As Graham said (of colloid matter in general), "its softness partakes of fluidity, and enables the colloid to become a vehicle for liquid diffusion, like water itself J." When we can deal with protoplasm in sufficient quantity we see it /oi^§; particles move freely through it, air-bubbles and liquid droplets shew round or spherical within it; and we shall have much to say about other phenomena manifested by its own surface, which are those especially characteristic of liquids. It may encompass and contain solid bodies, and it may "secrete" solid substances within or around itself; and it often happens in the complex Hving organism that these sohd substances, such as shell or nail or horn or feather, remain when the protoplasm which formed them is dead and gone. But the protoplasm itself is fluid or semi-fluid, and permits of free (though not necessarily rapid) diffusion and easy convection of particles within itself, which simple fact is of elementary importance * Cf. W. Kuhne, Ueber das Protoplasma, 1864. t Sand, or a heap of millet-seed, may in a sense be deemed a "fluid," and such the learned Father Boscovich held them to be {Theoria, p. 427), but at best they are fluids without a surface. Galileo had drawn the same comparison; but went on to contrast the continuity, or infinite subdivision, of a fluid with the finite, dis- continuous subdivision of a fine powder. Cf. Boyer, Concepts of the Calculus, 1939, p. 291. . J Phil Trans, clt, p 183, 1861; Researches, ed. Angus Smith, 1877, p 553. We no longer speak, however, of "colloids' in a specific sense, as Graham did; for any substance can be brought into the "colloidal state" by appropriate means or in an appropriate medmm. § The copious protoplasm of a Myxomycete has been passed unharmed through filter-paper with a pore-size of about 1 /x, or 0001 mm. CH. V] OF THE COLLOID STATE 347 in connection with form, throwing light on what seem to be common characteristics and pecuHarities of the forms of Hving things. Much has been done, and more said, about the nature of protoplasm since this book was written. Calling cytoplasm the cell-protoplasm after deduction of chloroplasts and other gross inclusions, we find it to contain fats, proteins, lecithin and some other substances combined with much water (up to 97 per cent.) to form a sort of watery gel. The microscopic structures attributed to it, alveolar, granular or fibrillar, are inconstant or invalid; but it does appear to possess an invisible or submicroscopic structure, distinguishing it from an ordinary colloid gel, and forming a quasi-solid framework or reticulum. This framework is based on proteid macromolecules, in the form of polypeptide chains, of great length and carrying in side-chains other organic constituents of the cytoplasm*. The polymerised units represent the micellae f which the genius of Nageli predicted or postulated more than sixty years ago; and we may speak of a "micellar framework" as representing in our cytoplasm the dispersed phase of an ordinary colloid. In short, as the cytoplasm is neither true fluid not true solid, neither is it true colloid in the ordinary sense. Its micellar structure gives it a certain rigidity or tendency to retain its shape, a certain plasticity and tensile strength, a certain ductility or capacity to be drawn out in threads; but yet leaves it with a permeability (or semi-permeability), a capacity to swell by imbibition, above all an ability to stream and flow, which justify our calling it "fluid or semi-fluid," and account for its exhibition of surface-tension and other capillary phenomena. The older naturalists, in discussing the differences between organic and inorganic bodies, laid stress upon the circumstance that the latter grow by "agglutination," and the former by what they termed "intussusception." The contrast is true; but it applies rather to solid or crystalline bodies as compared with colloids of all kinds, whether living or dead. But it so happens that the great majority of colloids are of organic origin; and out of them our bodies, and our food, and the very clothes we wear, are almost wholly made. A crystal "grows" by deposition of new molecules, one by one and layer by layer, each one superimposed on the solid substratum * See {int. al.) A. Frey-Wyssling, Subrnikroskopische Morphologie des Protoplasmas, Berlin, 1938; cf. Nature, June 10, 1939, p. 965; also A. R. Moore, in Scientia, LXii, July 1, 1937. On the nature of viscid fluid threads, cf. Larmor, Nature, July 11, 1936, p. 74. f Micella, or micula, diminutive of mica, a crumb, grain or morsel — ynica panis, salis, turis, etc. Nageli used the word to mean an aggregation of molecules, as the molecule is an aggregation of atoms; the one, however, is a physical and the other a chemical concept. Roughly speaking, we may think of micellae as varying from about 1 to 200 /x/x; they play a corresponding part in the "disperse phase" of a colloid to that played by the molecules in an ordinary solution. The macro- molecules of modern chemistry are sometimes distinguished from these as still larger aggregates. See Carl Nageli, Das Mikroskop (2nd ed.), 1877; Theorie der Gahrung, 1879. 348 THE FORMS OF CELLS [ch. already formed. Each particle would seem to be influenced 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. So Lavoisier was the first to say. And as Kelvin and others later on explained the formation and the resulting forms of crystals, so wo beheve that each added particle takes up its position in relation to its immediate neighbours already arranged, in the holes and corners that their arrangement leaves, and in closest contact with the greatest number*; hence we may repeat or imitate this process of arrangement, with great or apparently even with precise accuracy (in the case of the simpler crystalhne systems), by piling up spherical pills or grains of shot. In so doing, we must have regard to the fact that each particle must drop into the place where it can go most easily, or where no easier place offers. In more technical language, each particle is free to take up, and does take up, its position of least potential energy relative to those already there: 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. This has been shewn to lead to the production of plane surfaces f (in all cases where, by the hmitation of material, surfaces must occur); where we have planes, there straight edges and solid angles must obviously occur also, and, if equihbrium is to follow, must occur symmetrically. Our pihng up of shot to make 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. * Cf. Kelvin, On the molecular tactics of a crystal, The Boyle Lecture, Oxford, 1893; Baltimore Lectures, 1904, pp. 612-642. Here Kelvin was mainly following Bravais's (and Frankenheim's) theory of "space-lattices," but he had been largely anticipated by the crystallographers. For an account of the development of the subject in modern crystallography, by Sohncke, von Fedorow, 8chonfliess, Barlow and others, see (e.g.) Tutton's Crystallography, and the many papers by W. E. Bragg and others. t 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. Crystals "seem to be, as it were, the Elemental Figures, or the A B C of Nature's workmg, the reason of whose curious Geometrical Forms (if I may so call them) is very easily explicable" (Robert Hooke, Posthumous Works^ 1745, p. 280). V] OF INTUSSUSCEPTION 349 But the living cell grows in a totally different way, very much as a piece of glue swells up in water, by "imbibition," or by 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 molecules or micellae, and partly, as it would seem, in other ways; so that the entire phenomenon is a complex and even an obscure onef. But, so far as we are concerned, the net result is very simple. For the equilibrium, or tendency to equilibrium, of fluid pressure in all parts of its interior while the process of imbibition is going on, the constant rearrangement of its fluid mass, the contrast in short with the crystalhne method of growth where each particle comes to rest to move (relatively to the whole) no more, lead the mass of jelly to swell up very much as a bladder into which we blow air, and so, by a graded and harmonious distribution of forces, to assume everywhere a rounded and more or less bubble-hke external form J. So, when the same school of older naturahsts called attention to a new distinction or contrast of form between 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 con- sequence of the difference between the processes of agglutination and intussusception §. So far then as growth goes on undisturbed by pressure or other external force, the fluidity of the protoplasm, its mobility internal * This is what Graham called the water of gelatination, on the analogy of water of crystallisation; Chem. and Phys. Researches, p. 597. t On this important phenomenon, see J. R. Katz, Oesetze der Quellung, Dresden, 1916. Swelling is due to "concentrated solution," and is accompanied by increase of volume and liberation of energy, as when the Egyptians split granite by the swelling of wood. X 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. § Intussusception has its shades of meaning; it is excluded from the idea of a crystalline body, but not limited to the ordinary conception of a colloid one. When new micellar strands become interwoven in the micro-structure of a cellulose cell- wall, that is a special kind of "intussusception." 350 THE FORMS OF CELLS [ch. and external*, and the way in which particles move freely hither and thither within, all manifestly tend to the production of swelhng, rounded surfaces, and to their great predominance over plane sur- faces in the contours 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 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 generally in the external form 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. The swelling of any object, organic or inorganic, living or dead, is bound to be influenced by any lack of structural symmetry or homogeneity f. We may take it that all elongated structures, such as hairs, fibres of silk or cotton, fibrillae of tendon and connective tissue, have by virtue of their elongation an invisible as well as a visible polarity. Moreover, the ultimate fibrils are apt to be invested by a protein different from the "collagen" within, and liable to swell more or to swell less. In ordinary tendons there is a "reticular sheath," which swells less, and is apt to burst under pressure from within; it breaks into short lengths, and when the strain is relieved these roll back, and form the familiar annuli. Another instance is the tendency to swell of the "macro- molecules" of many polymerised organic bodies, proteins among them. But the rounded contours which are assumed and exhibited by a piece of hard glue when we throw it into water and see it expand as it sucks the water up, are not near so regular nor so beautiful as are those which appear when we blow a bubble, or form a drop, or even pour water into an elastic bag. For these curving contours depend upon the properties of the bag itself, of the film or membrane which contains the mobile gas, or which 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. * The protoplasm of a sea-urchin's egg has a viscosity only about four times, and that of various plants not more than ten to twenty times, that of water itself. See, for a general discussion, L. V. Heilbrunn, Colloid Symposium Monograph^ 1928. t D. Jordan Lloyd and R. H. Marriott, The swelling of structural proteins, Proc. U.S. (B), No. 810, pp. 439-44.3, 1935. V] OF SURFACE TENSION 351 Among the forces which determine the forms of cells, whether they be sohtary 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. The peculiar beauty of a soap-bubble, solitary or in collocation, depends on the absence (to all intents and purposes) of these ahen forces from the field ; hence Plateau spoke of the films which were the subject of his experiments as "lames fluides sans pesanteur.'' The resulting form is in such a case so pure, and simple that we come to look on it as wellnigh a mathematical abstraction. Surface-tension, then, 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 hke circumstances*. It is a property of Hquids (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 hquid, a solid or a gas. We note here that the term surface is to be interpreted in a wide sense; for wherever we have solid particles embedded in a fluid, wherever we have a non-homogeneous fluid or semi-fluid, or a "two-phase colloid " such as a particle 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-phenomena, not only on the exterior of the mass but also throughout its inter- stices, wherever like and unlike meet. * The idea of a "surface-tension" in liquids was first enunciated by Segner, and ascribed by him to forces of attraction whose range of action was so small "ut nullo adhuc sensu percipi potuerat" {Defiguris super ficier um fiuidarum, in Comment. Soc. Roy. GoUingen, 1751, p. 301). Hooke, in the Micrographia (1665, Obs. viii, etc.), had called attention to the globular or spherical form of the little morsels of steel struck off by a flint, and had shewn how to make a powder of such spherical grains, by heating fine filings to melting point, "This Phaenomenon" he said "proceeds from a propriety which belongs to all kinds of fluid Bodies more or less, and is caused by the Incongruity of the Atnbient and included Fluid, which so acts and modulates each other, that they acquire, as neer as is possible, a spherical or globular form " 352 THE FORMS OF CELLS [ch. A liquid in the mass is devoid of structure ; it is homogeneous, and without direction or polarity. But the very concept of surface- tension forbids this to be true of the surface-layer of a body of liquid, or of the "interphase" between two liquids, or of any film, bubble, drop, or capillary jet or stream. In all these cases, and more empha- tically in the case of a "monolayer," even the liquid has a structure of its own ; and we are reminded once again of how largely the living organism, whether high or low, is composed of colloid matter in precisely such forms and structural conditions. Surface-tension is due to molecular force * : to force, that is to say, arising from the action of one molecule upon another; and since we can only ascribe a small "sphere of action" to each several molecule, this force is manifested only within a narrow range. Within the interior of the liquid mass we imagine that such molecular interactions negative one another ; but at and near the free surface, within a layer or film approximately equal to the range of the molecular force — or to the radius of the aforesaid " sphere of action " — there is a lack of equilibrium and a consequent 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 are being constantly attracted into the interior by such as are just a little more deeply situated; the surface shrinks as molecules keep quitting it for the interior, and this surface-shrinkage exhibits itself as a surface-tension. The process continues till it can go no farther, that is to say until the surface itself becomes a "minimal areaf." This is a sufficient description of the phenomenon in cases where a portion of liquid is subject to no other than its own molecular forces, and (since the sphere has, * While we explain certain phenomena of the organism by reference to atomic or molecular forces, the following words of Du Bois Reymond's seem worth recalling: ' Naturerkennen ist Zuriickfiihren der Veranderungen in der Korperwelt auf Bewegung von Atomen, die durch deren von der Zeit unabhangige Centralkrafte bewirkt werden, oder Auflosung der Naturkrafte in Mechanik der Atome. Es ist eine psychologische Erfahrungstatsache dass, wo solche Auflosung gelangt, unser Causalbediirfniss vorlaiifig sioh befriedigt fiihlt" (Ueber die Grenzen des Natnr- erkennens, Leipzig, 1873). t There must obviously be a certain kinetic energy in the molecules within the drop, to balance the forces which are trying to contract and diminish the surface. V] OF SURFACE ENERGY 353 of all solids, the least 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 innumerable simple organisms f. It accounts also, as we shall presently see, for many much more complicated forms, manifested under less simple conditions. Let us note in passing that surface-tension is a comparatively small force and is easily measurable: for instance that between water and air 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 great curvature, such as that of a minute drop, gives nse to a very great pressure, directed inwards towards the centre of curvature. We may easily calculate this pressure, and so satisfy ourselves that, when the radius of curvature approaches molecular dimensions, the pressure is of the order of thousands of atmospheres — a conclusion which is supported by other physical considerations. The contraction of a liquid surface, and the other phenomena of surface-tension, involve the doing of work, and the power to do work is what we call Energy. The whole energy of the system is diffused throughout its molecules, as is obvious in such a simple case as we have just considered; but of the whole stock of energy only the part residing at or very near the surface normally manifests itself in work, and hence we speak (though the term be open to * Raindrops must be spherical, or they would not produce a rainbow; and the fact that the upper part of the bow is the brightest and sharpest shews that the higher raindrops are more truly spherical, as well as smaller than the lower ones. So also the smallest dewdrops are found to be more iridescent than the large, shewing that they also are the more truly spherical; cf. T. W. Backhouse, in Monthly Meteorol. Mag. March, 1879. Mercury has a high surface-tension, and its globules are very nearly round. t That the offspring of a spherical cell (whether it be raindrop, plant or animal) should be also a spherical cell, would seem to need no other explanation than that both are of identical substance, and each subject to a similar equilibrium of surface-forces; but the biologists have been apt to look for a subtler reason. Giglio-Tos, speaking of a sea-urchin's dividing egg, asks why the daughter-cells are spherical like the mother-cell, and finds the reason in "heredity": "Wenn also die letztere (d. i. die Mutterzelle) eine spharische Form besass, so nehmen auch die Tochterzellen dieselbe ein; ware urspriinglich eine kubische Form vorhanden, so wurden also auch die Tochterzellen dieselbe auch aneignen. Die Ursache warum die Tochterzellen die spharische Form anzunehmen trachten liegt darin, dasa diese die Ur- und Grundform alter Zellen ist, sowohl bet Tieren ivie bei den Pflanzen'' [Arch.f. Entw. Mech. Li, p. 115, 1922). 354 THE FORMS OF CELLS [ch. some objection) of a specific surface-energy. Surface-energy, and the way it is increased and multiplied by the multiphcation of surfaces due to the subdivision of the tissues into cells, is of the highest interest tq the physiologist; and even the morphologist cannot pass it by. For the one finds surface-energy present, often perhaps paramount, in every cell of the body ; and the other may find, if he will only look for it, the form of every solitary cell, hke that of any other drop or bubble, related to if not controlled by capillarity. The theory of "capillarity," or "surface-energy," has been set forth with the utmost possible lucidity by Tait and by Clerk Maxwell, on whom the following paragraphs are based: they having based their teaching on that of Gauss*, who rested on Laplace. Let E be the whole potential energy of a mass M of hquid; let Cq be the energy per unit mass of the interior hquid (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 plus the surface-energy, and that the former is distributed through the whole mass, minus its surface layers. That is to say, in mathematical language, E={M -S. l.tp) e^ + S . I^tpe. But this is equivalent to writing : = MeQ + S .I.tp{e- eo); 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 to the difference between e and Cq, that is to say to the difference between the unit values of the internal and the surface energy. It was Gauss who first shewed how, from the mutual attractions between all the particles, we are led to an expression for what we * See Gauss's Principia generalia Theoriae Figurae Fluidorum in statu equilibriif Gottingen, 1830. The historical student will not overlook the claims to priority of Thomas Young, in his Essay on the cohesion of fluids, Phil. Trans. 1805; see the account given in his Life by Dean Peacock, 18.55, pp. 199-210. V] OF SURFACE ENERGY 355 no\v' call the potential energy* of the system; and we know, as a fundamental theorem of dynamics, as well as of molecular physics, that the potential energy of the system tends to a minimum, and finds in that minimum its stable equihbrium. We see in our last equation that the term Mcq 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 equahty of e and ^q , that is to say by a diminu- tion 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 develop- ment of a celi-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 otherf. Inasmuch as we are concerned with the form of the cell, it is the former which becomes our main postulate: telhng us that the energy-equations of the surface of a cell, or of the free surfaces of cells in partial contact, or of the partition-surfaces of cells in contact with one another, all indicate a minimum of potential energy in the system, by which minimal condition the system is brought, ipso facto, into equihbrium. And we shall not fail to observe, with something more than mere historical interest and curiosity, how * The word Energy was substituted for the old vis viva by Thomas Young early in the nineteenth century, and was used by James Thomson, Lord Kelvin's brother, about 1852, to mean, more generally, "capacity for doing work." The term potential, or latent, in contrast to actual energy, in other words the distinction between " energy of activity and energy of configuration," was proposed by Macquorn Rankine, and suggested to him by Aristotle's use of 8vva/j.L$ and evepyeia; see Rankine's paper On the general law of the transformation of energy, Phil. Soc. Glasgow, Jan. 5, 1853, cf. ibid. Jan. 23, 1867, and Phil. Mag. (4), xxvii, p. 404, 1864. The phrase potential energy was at once adopted, but kinetic was substituted for actical by Thomson and Tait. t The capacity factor, inasmuch as it leads to diminution of surface, is responsible for the concrescence of droplets into drops, of microcrystals into larger units, for the flocculation of colloids, and for many other similar "changes of state." 356 THE FORMS OF CELLS [ch. deeply and intrinsically there enter into this whole class of problems the method of maxima and minima discovered by Fermat, the "loi universelle de repos" of Maupertuis, "dont tous les cas d'equilibre dans la statique ordinaire ne sont que des cas particuhers", and the lineae curvae maximi miniynive proprietatibus gaudentes of Euler, by which principles these old natural philosophers explained correctly a multitude of phenomena, and drew the lines whereon the founda- tions of great part of modern physics are well and truly laid. For that physical laws deal with minima is very generally true, and is highly characteristic of them. The hanging chain so hangs that the height of its centre of gravity is a minimum ; a ray of hght takes the path, however devious, by which the time of its journey is a minimum ; two chemical substances in reaction so behave that their thermodynamic potential tends to a minimum, and so on. The natural philosophers of the eighteenth century were engrossed in minimal problems ; and the differential equations which solve them nowadays are among the most useful and most characteristic equa- tions in mathematical physics. "Voici," said Maupertuis, "dans un assez petit volume a quoi je reduis mes ouvrages mathematiques ! " And when Lagrange, fol- lowing Euler 's lead*, conceived the principle of least action, he regarded it not as a metaphysical principle but as "un resultat simple et general des lois de la mecaniquef." The principle of least action J explains nothing, it tells us nothing of causation, yet it illuminates a host of things. Like Maxwell's equations and other such flashes of genius it clarifies our knowledge, adds weight to our observations, brings order into our stock-in-trade of facts. It embodies and extends that "law of simplicity" which Borelli was the first to lay down: "Lex perpetua Naturae est ut agat minimo labore, mediis et modis simplicissimis, facillimis, certis et * Euler, Traite des Isoperimetres, Lausanne, 1744. t Lagrange, Mecanique Analytique (2), ii, p. 188; ed. in 4to, 1788. { This profound conception, not less metaphysical in the outset than physical, began in the seventeenth century with Fermat, who shewed (in 1629) that a ray of light followed the quickest path available, or, as Leibniz put it, via omnium facillima; it was over this principle that Voltaire quarrelled with Euler and Maupertuis. The mathematician will think also of Hamilton's restatement of the principle, and of its extension to the theory of probabilities by Boltzmann and Willard Gibbs. Cf. (int. al.) A. Mayer, Geschichte des Prinzips der kleinsten Action, 1877. vj THE MEANING OF SYMMETRY 357 tutis: evitando, quam maxime fieri potest, incommoditates et prolixitates." The principle of least action grew up, and grew quickly, out of cruder, narrower notions of "least time" or "least space or distance." Nowadays it is developing into a principle of "least action in space-time," which shall still govern and predict the motions of the universe. The infinite perfection of Nature is expressed and reflected in these concepts, and Aristotle's great aphorism that "Nature does nothing in vain" lies at the bottom of them all. In all cases where the principle of maxima and minima comes into play, as it conspicuously does in films at rest under surface- tension, the configurations so produced are characterised by obvious and remarkable symmetry'^. Such symmetry is highly characteristic of organic forms, and is rarely absent in living things — save in such few cases as Amoeba, where the rest and equilibrium on which symmetry depends are likewise lacking. And if we ask what physical equilibrium has to do with formal symmetry and structural regularity, the reason is not far to seek, nor can it be better put than in these words of Mach'sf: "In every symmetrical system every deformation that tends to destroy the symmetry is com- plemented 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 niaximum or minimum of work corresponds to the form of equiUbrium, is thus supj)ned by symmetry. Regularity is successive symmetry; there is no reason, therefore, to be astonished that the forms of equiUbrium are often symmetrical and regular." A crystal is the perfection of symmetry and of regularity; symmetry is displayed in its external form, and regularity revealed in its internal lattices. Complex and obscure as the attractions, rotations, vibrations and what not within the crystal may be, we rest assured that the configuration, repeated again and again, of * On the mathematical side, cf. Jacob Nteiner, Einfache Beweise der isoperi- metrischen Hauptsatze, Abh. k. Akad. Wisa. Berlin, xxiii, pp. 116-135, 1836 (1838). On the biological side, see {int. al.) F. M. Jaeger, Lectures on the Principle of Symmetry, and its application to the natural sciences, Amsterdam, 1917; also F. T. Lewis, Symmetry. . .in evolution, Amer. Nat. lvii, pp. 5-41, 1923. f Science of Mechanics, 1902, p. 395; see also Mach's article Ueber die physika- lische Bedeutung der Gesetze der Symmetrie, Lotos, xxi, pp. 139-147, 1871. 358 THE FORMS OF CELLS [ch. the component atoms is precisely that for which the energy is a minimum; and we recognise that this minimal distril^ution is of itself tantamount to symmetry and to stability. Moreover, the principle of least action is but a setting of a still more universal law — that the world and all the parts thereof tend ever to pass from less to more probable configurations; in which the physicist recognises the principle of Clausius, or second law of thermodynamics, and with which the biologist must somehow reconcile the whole "theory of evolution." 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 Httle 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 tbwards a surface of minimal area ; and in like manner, by merely melting the tip of a thin rod of glass, Hooke made the little spherical beads which served him for a microscope*. 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 seen hkewise 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 Engelmannf 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 con- tractility, muscular movement, or other special 'physiological func- * 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. See also Hooke's Micrographia, 1665; and Adam's Essay on the Microscope, 1798, p. 8: "No person has carried the use of these globules so far as Father Torre of Naples, etc." Leeuwenhoek. on the other hand, ground his lenses with exquisite skill. t Beitrage zur Physiologie des Protoplasma, Pfluger's Archlv, ii, p. 307, 1869. V] OF SURFACE ACTION 359 tions, but we find ample room to trace the operation of the same cause in producing, under conditions of rest and equiUbrium, certain definite and inevitable forms. It is of great importance to observe that the living cell is one of those cases where the phenomena of surface-tension are by no means limited to the outer surface; for within the heterogeneous emulsion 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 large internal "interfacial contacts" between the protoplasm and its included granules, or its vacuoles filled with the "cell-sap." Here we have a great field for surface-action; and so long ago as 1865, Nageli and Schwendener shewed that the streaming currents of plant cells might be plausibly explained by this phenomenon. Even ten years earlier, Weber had remarked upon the resemblance between the protoplasmic streamings and the currents to be observed in certain inanimate drops for which no cause but capillarity could be assigned*. What sort of chemical changes lead up to, or go hand in hand with, the variations of surface-tension in a hving cell, is a vastly important question. It is hardly one for us to deal with ; but this at least is clear, that the phenomenon is more complicated than its first investigators, such as BUtschli and Quincke, ever took it to be. For the lowered surface-tension which leads, say, to the throwing out of a pseudo- podium, is accompanied first by local acidity, then by local adsorption of proteins, lastly and consequently by gelation; and this last is tantamount to the formation of "ectoplasm" — a step in the direction of encystmentf. The elementary case of Amoeba is none the less a complicated one. The "amoeboid" form is the very negation of rest or of equihbrium; * Poggendorff's Annalen, xciv, pp. 447-459, 1855. Cf. Strethill Wright, Phil. Mag. Feb. 1860; Journ.Anat. and Physiol, i, p. 337, 1867. t Cf. C, J. Pantin, Journ. Mar. Biol. Assoc, xiii, p. 24, 1923; Journ. Exp. Biol. 1923 and 1926; S. 0. Mast, Jo^lrn. Morph. xli, p. 347, 1926; and 0. W. Tiegs, Surface tension and the theory of protoplasmic movement, Protoplasma, iv, pp. 88-139, 1928. See also (int. al.) N. K. Adam, Physics and Chemistry of Surfaces, 1930; also Discussion on colloid science applied to biology (passim), Trans. Faraday Soc. XXVI, pp. 663 seq., 1930. 360 THE F0RM8 OF CELLS [ch. the creature is always moving, from one protean configuration to another; its surface-tension is never constant, but continually varies from here to there. Where the 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 sub- stances*, by the transport of constituent materials from the interior to the surface, or again by actual chemical and fermentative change ; for within the cell, the surface-energies developed about its hetero- geneous contents will continually 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!. 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 multitude of experiments shew how extraordinarily dehcate is the adjustment of the surface-tension forces, and how sensitive they are to the least change of temperature or chemical state. Thus, * Haycraft and Carlier pointed out long ago {Proc. R.S.E. xv, pp. 220-224, 1888) that the amoeboid movements of a white blood-corpuscle are only manifested when the corpuscle is in contact with some solid substance: while floating freely in the plasma or serum of the blood, these corpuscles are spherical, that is to say they are at rest and in equilibrium. The same fact was recorded anew by Ledingham (On phagocytosis from an adsorptive point of view, Journ. Hygiene, XII, p. 324, 1912). On the emission of pseudopodia as brought about by changes in surface tension, see also {int. al.) J. A. Ryder, Dynamics in Evolution, 1894; Jensen, L'eber den Geotropismus niederer Orgahismen, Pfliiger's Archiv, liii, 1893. Jensen remarks that in Orbitolites, the pseudopodia issuing through the pores of the shell first float freely, then as they grow longer bend over till they touch the ground, whereupon they begin to display amoeboid and streaming motions. \'erworn indicates {Ally. Physiol. 189o, p. 429), and Davenport says {Exper. Morphology, ii, p. 376), that "this persistent clinging to the substratum is a ' thigmotropic ' reaction, and one which belongs clearly to the category of ' response '. " Cf. Putter, Thigmotaxis bei Protisten, Arch. 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. V] THE FORM OF AMOEBA 361 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 we bring into its neighbourhood a heated wire, or a glass rod dipped in ether*. The water-colour painter makes good use of the surface- tension effect of the minutest trace of ox-gall. When a plasmodium of Aethalium creeps towards a damp spot or a warm spot, or towards substances which happen to be nutritious, and creeps away from solutions of sugar or of salt, we are dealing with pheno- mena too often ascribed to 'purposeful' action or adaptation, but every one of which can be paralleled by ordinary phenomena of surface-tensiont- The soap-bubble itself is never in equilibrium: for the simple reason that its film, Hke the protoplasm of Amoeba or Aethalium, is exceedingly heterogeneous. Its surface-energies vary from point to point, and chemical changes and changes of temperature increase and magnify the variation. The surface of the bubble is in continual movement, as more 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 restless movement as its parts strive one with another in their interactioAs towards unattainable equilibrium J. On reaching a certain tenuity the bubble bursts: as is bound to happen when the attenuated film has no longer the properties of matter in mass. * 80 Bernstein shewed that a drop of mercury in nitric acid moves towards, or is "attracted by," a crystal of potassium bichromate; Pfliiger's Archiv, lxxx, p. 628, 1900. "^ t The surface-tension theory of protoplasmic movement has been denied by many. Cf. (e.g.) H. S. Jennings, Contributions to the behaviour of the lower organisms, Carnegie Instit. 1904, pp. 130-230; O. P. Dellinger, Locomotion of Amoebae, etc., Journ. Exp. Zool. iii, pp. 337-3o7, 1906; also various papers by Max Heidenhain, in Merkel u. Bonnet's Anatomische Hefte; etc. X These motions 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 epipolique of Dutrochet; until van der Mensbrugghe shewed that differences of surface-tension were enough to account for this whole series of phenomena (Sur la tension super- ficielle des liquides, consideree au point de vue de certains mouvements observes a leur surface, Mem. Cour. Acad, de Belgique, xxxiv, 1869, Phil. Mag. Sept. 1867; cf. Plateau, Statique des Liquides, p. 283). An interesting early paper is by Dr G. Carradini of Pisa, DelF adesione o attrazione di superficie, Mem. di Matem. e di Fisica d. Soc. Ital. d. Sci. (Modena), xi, p. 75, xii, p. 89, 1804-5. 362 THE FORMS OF CELLS [ch. The film becomes a mere bimolecular, or even a monomolecular, layer; and at last we may treat it as a simple "surface of discon- tinuity." So long as the changes due to imperfect equihbrium are taking place very slowly, we speak of the bubble as "at rest"; it is then, as Willard Gibbs remarks, that the characters of a film are most striking and most sharply defined*. So also, and surely not less than the soap-bubble, is every cell- surface a complex affair. Face and interface have a molecular orientation of their own, -depending both on the partition-membrane and on the phases on either side. It is a variable orientation, changing at short intervals of space and time; it coincides with inconstant fields of force, electrical and other; it initiates, and controls or catalyses, chemical reactions of great variety and importance. In short we acknowledge and confess that, in sim- pHfying the surface phenomena of the cell, for the time being and for our purely morphological ends, we may be losing sight, or making abstraction, of some of its most specific physical and physiological characteristics. In the case of the naked protoplasmic cell, as the amoeboid phase is emphatically a phase of freedom and activity, of unstable equi- librium, of chemical and physiological change, so on the other hand does the spherical form indicate a phase of stabihty, of inactivity, of rest. In the one phase we see unequal surface-tensions manifested in the creeping movements of the amoeboid body, in the rounding- off of the ends of its 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, in the alternate phase, when internal homogeneity and equilibrium have been as far as possible attained and the potential energy of the system is at a minimum, the cell assumes a rounded or spherical form, passes into a state of "rest," and (for a reason which we shall presently consider) becomes at the same time encysted f. * On the equilibrium of heterogeneous substances, Collected Works, i, pp. 55-353; Trans. Conn. Acad. 1876-78. f We still speak of the naked protoplasm of Amoeba; but short, and far short, of "encystment," there is always a certain tendency towards adsorptive action, leading to a surface-layer, or "plasma- membrane," still semi-fluid but less fluid than before, and different from the protoplasm within ; it was one of the first and chief things revealed by the new technique of "micro-dissection." Little is known of v] THE FORM OF AMOEBA 363 In their amoeboid phase the various Amoebae are just so many varying distributions of surface-energy, and varying amounts of surface-potential*. An ordinary floating drop is a figure of equi- librium under conditions of which we shall soon have something to say; and if both it and the fluid in which it floats be homogeneous it will be a round drop, a "figure of revolution." But the least chemical heterogeneity will cause the surface-tension to vary here and there, and the drop to change its form accordingly. The httle swarm-spores of many algae lose their flagella as they settle down, and become mere drops of protoplasm for the time being; they "put out pseudopodia" — in other words their outline changes; and presently this amoeboid outHne grows out into characteristic lobes or lappets, a sign of more or less symmetrical heterogeneity in the cell-substance. In a budding yeast-cell (Fig. 103 A), we see a 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 tension has diminished, and the area of that portion expands accordingly ; but in turn the surface-tension of the expanded or ex- truded portion makes itself felt, and the bud rounds itself off into a more or less spherical form. ^^' The yeast-cell with its bud is a simple example of an important principle. Our whole treatment of cell-form in relation to surface- tension depends on the fact (which Errera was the first to give clear expression to) that the incipient cell-wall retains with but little impairment the properties of a liquid filmf, and that the growing cell, in spite of the wall by which it has begun to be surrounded, the physical nature of this so-called membrane. It behaved more or less like a fluid lipoid envelope, immiscible with its surroundings. It is easily injured and easily repaired, and the well-being of the internal protoplasm is said to depend on the maintenance of its integrity. Robert Chambers, Physical Properties of Protoplasm, 1926; The living cell as revealed by microdissection, Harvey Lectures, Ser. xxii, 1926-27; Journ. Gen. Physiol, vm, p. 369, 1926; etc. * See (int. al.) Mary J. Hogue, The effect of media of different densities on the shape of Amoebae, Journ. Exp. Zool. xxii, pp. 565-572, 1917. Scheel had said in 1889 that A. radiosa is only an early stage of ^. proteus {Festschr. z. 70. Geburtstag 0. V. Kuptfer). t Cf. infra, p. 561. 364 THE FORMS OF CELLS [ch. behaves very much Hke a fluid drop. So, to a first approximation, even the yeast-cell shews, by its ovoid and non-spherical form, that it has acquired its shape under some influence other than the uniform and symmetrical surface-tension which makes a soap-bubble into a sphere. This oval or any other asymmetrical form, once acquired, may be retained by virtue of the solidification and consequent rigidity of the membrane-like wall of the cell; and, unless rigidity ensue, it is* plain that such a conformation as that of the yeast-cell with its attached bud could not be long retained 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 its bud. In like manner the developing egg, through all its successive phases of form, is never in complete equilibrium: but is constantly responding to slowly changing conditions, by phases of partial, transitory, unstable and conditional equihbrium. There are innumerable solitary plant-cells, and unicellular organisms in general, which, hke the yeast-cell, do not correspond to any of the simple forms which 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 some com- paratively sudden solidification of the envelope. This is the case, for instance, in the more comphcated 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 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) tc branch, at par- ticular points. We can often trace in each particular stage of growth, or at each particular temporary growing point, the laws of surface tension manifesting themselves in what is for the time being a fluid surface ; nay more, even in the adult and completed structure we have little difficulty in tracing and recognising (for instance in the outline of such a desmid as Euastrum) the rounded lobes which have successively grown or flowed out from the original rounded and V] OF LIQUID FILMS 365 flattened cell. What we see in a many chambered foraminifer, such as Glohigerina or Rotalia, is the same thing, save that the stages are more separate and distinct, and the whole is carried out to greater completeness and perfection. The little organism as a whole is not a figure of equihbrium nor 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 consoHdated one by one into a rigid system. Let us now make some enquiry into the forms which a fluid surface can assume under the mere influence of surface-tension. In doing so we are 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 in general this will be the case when we are dealing with portions of liquid so small that their dimensions come within or near to what we have called the molecular range, or, more generally, in which the "specific surface" is large. In other words it is the small or minute organisms, or small cellular elements of larger organisms, whose forms will be governed by surface-tension; while the forms of the larger organisms are due to other and non-molecular forces. A large surface of water sets itself level because here gravity is predominant; but the surface of water in a narrow tube is curved, for the reason that we are here deahng with particles which he within the range of each other's molecular forces. The like is the case with the cell-surfaces and cell-partitions which we are about to study, and the effect of gravity will be especially counteracted and concealed when the object is immersed in a hquid of nearly its own density. We have already learned, as a fundamental law of "capillarity," that a liquid film in equilibrium assumes a form which gives it a minimal area under the conditions to which it is subject. 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, that the film (if it be a closed one) may have to contain. In the simplest of cases, as when we take up a soap-film on a plane wire ring, the film is exposed to equal atmospheric pressure on both side^, and it ob- 36G THE FORMS OF CELLS [ch. viously 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 comphcated, but is still the smallest possible surface which can be drawn continuously across the uneven boundary. The question of pressure involves not only external pressures acting on the film, but also that which the film itself is capable of exerting. For we have seen that the film is always contracting to the utmost; and when the film is curved, this 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 across 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 Kquid layer, with a free surface within and another without, and each of these two surfaces exercises its own independent and coequal tension and its 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 cy finder 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 down- * Or, more strictly speaking, unless its thickness be less than twice the range of the molecular forces. t It follows that the tension of a bubble, depending only on the surface-conditions, is independent of the thickness of the film. v] OF LIQUID FILMS 367 ward direction. In mathematical language, the pressm-e (p) varies directly as the tension (T), and inversely as the radius of curvature (R) : that is to say, p = T/R, per unit of surface. If instead of a cylinder, whose curvature lies only in one direction, we take a case of curvature in two dimensions (as for. instance a sphere), then the effects of these two curvatures must be added together to give the resulting pressure p: which becomes equal to T/R + TIR\ or 1 1 R'^R' i.= H+-* And if in addition to the pressure p, which is due to surface-tension, we have to take into account other pressures, p\ p", etc., due to gravity or other forces, then we may say that the total pressure p=/+/'+2'(l+l) We may have to take account of the extraneous pressures in some cases, as when we come to speak of the shape of a bird's egg ; but in this first part of our subject we are able for the most part to neglect them. Our equation is an equation of equihbrium. The resistance to compression — the pressure outwards — of our fluid mass is a constant quantity (P); the pressure inwards, T (IjR + l/R'), is also con- stant; and if the surface (unlike that of the mobile amoeba) be homogeneous, so that T is everywhere equal, it follows that 1/i? + l/R' = C (a constant), throughout the whole surface in question. Now equihbrium is reached after the surface-contraction has done its utmost, that is to say when it has reduced the surface to the least possible area. So we arrive at the conclusion, from the physical side, that a surface such that IjR + IjR' = C, in other \<^ords a surface which has the same 7nean curvature at all points, is equivalent to a surface of minimal area for the volume enclosed f ; * This simple but immensely important formula is due to Laplace (Mecanique Celeste, Bk x, suppl. Theorie de Vaction capillaire, 1806). t A surface may be "minimal" in respect of the area occupied, or of the volume enclosed: the former being such as the surface which a soap-film forms when it fills up a ring, whether plane or no. The geometers are apt to restrict the term "minimal surface" to such as these, or, more generally, to all cases where the mean curvature i» nil; the others, being only minimal with respect to the volume con- tained, they call "surfaces of constant mean curvature." 368 THE FORMS OF CELLS [ch. and to the same conclusion we may also come by ways purely mathematical. The plane and the sphere are two obvious examples of such surfaces, for in both the radius of curvature is everywhere constant. From the fact that we may extend a soap-film across any ring of wire, however fantastically the wire be bent, we see that there is no end to the number of surfaces of minimal area which may be constructed or imagined*. While some of these are very com- phcated indeed, others, such as a spiral or helicoid screw, are relatively simple. But if we Umit ourselves to surfaces of revolution (that is to say, to surfaces symmetrical about an axis), w^e find, as Plateau was the first to shew, that those which meet the case are few in number. They are six in all, nam-ely the plane, the sphere, the cyhnder, the catenoid, the unduloid, and a curious surface which Plateau called the nodoid. A B CD Fig. 104. Roulettes of the conic sections. 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 Delaunayf, in 1841) that the plane curves by whose revolution they are generated are themselves generated as "roulettes" of the conic sections. Let us imagine a straight line, or axis, on which a circle, ellipse or other conic section rolls ; the focus of the conic section will describe a line in some relation to the fixed axis, and this fine (or roulette), when we rotate it around the axis, will describe in space one or another of the six surfaces of revolution of which we are speaking. If we imagine an elhpse so to roll on a base-hne, either of its foci will describe a sinuous or wavy fine (Fig. 104, B) at a distance * To fit a minimal surface to the boundary of any given closed curve in space is a problem formulated by Lagrange, and commonly known as the "problem of Plateau," who solved it with his soap-films. f Sur la surface de revolution dont la courbure moyenne est constante, Journ. de M. Liouville, vi, p. 309, 1841. Cf. {int. al.) J. Clerk Maxwell, On the theory of rolUng curves. Trans. R.S.E. xvi, pp. 519-540, 1849; J. K. Wittemore, Minimal surfaces of rotation, Ann. Math. (2), xix, 1917, Amer. Journ. Math, xl, p. 69, 1918; Crino Loria, Courbes planes speciales, theorie ef histoire, Milan, 574 pp., 1930. V] OF MINIMAL SURFACES alternately maximal and minimal from the axis; this wavy Hne, by rotation about the axis, becomes the meridional Hne of the surface which we call the unduloid, and the more unequal the two axes are of our elhpse, the more pronounced will be the undulating sinuosity of the roulette. If the two axes be equal, then our elhpse becomes a circle ; the path described by its roUing centre is a straight line parallel to the axis (A), and the sohd of revolution generated therefrom will be a cylinder: in other words, the cylinder is a "Hmiting case" of the unduloid. If one axis of our ellipse vanish, while the other remains of finite length, then the elhpse is reduced to a straight line with its foci at the two ends, and its roulette will appear as a succession of semicircles touching one another upon the axis (C); the solid of revolution will be a series of equal spheres. If as before one axis of the ellipse vanish, but the other be infinitely long, then the roulette described by the focus of this ellipse will be a circular arc at an infinite distance; i.e. it will be a straight line normal to the axis, and the surface of revolution traced by this straight hne turning about the axis will be 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 elhpse 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 more difficult to imagine, we have the case of the hyperbola. We cannot well imagine the hyperbola rolling upon a fixed straight line so that its focus shall describe a continuous curve. But let us suppose that the fixed hne is, to begin with, asymptotic to one branch of the hyperbola, and that the rolUng proceeds until the hne is now asymptotic to the other branch, that is to say touching it at an infinite distance; there will then be mathematical continuity if we recommence rolling with this second branch, and so in turn with the other, when each has run its course. We shall see, on reflection, that the line traced by one and the same focus will be an "elastic curve," describing a succession of kinks or knots (E), and the solid of revolution described by this meridional line about the axis is the so-called nodoid. I The physical transition of one of these surfaces into another can 370 THE FORMS OF CELLS [ch. be experimentally illustrated by means of soap-bubbles, or better still, after another method of Plateau's, by means of a large globule of oil, supported when necessary by wire rings, and lying in 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. Darhng is a great improvement on Plateau's*. Mr Darhng used the oily hquid 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 a little salt to the lower layers of water, the drop may be made to rest or float upon the denser liquid. Fig. 105. We have seen that, the soap-bubble, spherical to begin with, is transformed into a plane when we release its internal pressure and let the film shrink back upon the orifice of the pipe. If we blow a bubble and then catch it up on a second pipe, so that it stretches between, we may draw the two pipes apart, with the result that the spheroidal surface will be gradually flattened in a longitudinal direction, and the hubble will be transformed into a cyhnder. But if we draw the pipes yet farther apart, the cyhnder narrows 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 cyhnder has, in turn, been converted into an unduloid. When we hold a soft glass tube in the flame and "draw it out," we are in the same identical fashion converting a cylinder into an unduloid (Fig. 105, 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 * 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. v] OF MINIMAL SURFACES 371 essentially the same, save that the two halves of the one change places in the other. That spheres, cylinders and unduloids are of the commonest occurrence among the forms of small unicellular organisms 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 of these phenomena. But before we go further in this enquiry we must consider, to some small extent at least, the curvatures of the six different sur- faces, so far as 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 impinging on the film, and also the very important question of the limiting conditions which, from the nature of the case, set bounds to the extension of certain of the figures. 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 IjR + IjR' = C, a constant; and that this is, in all cases, the condition of our surface being one of minimal area. That is to say, 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, IjR + IjR' = 0. The expression therefore vanishes, and our dy- namical equation of equihbrium 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 sphere the radii are all equal, R= R'; they are also positive, and T (l/R + l/R'), or 2T/R, is a- positive quantity, involving a constant positive pressure P, on the other side of the equation. In the cylinder one radius of curvature has the finite and positive value R; but the other is infinite. Our formula becomes TjR, to * They tend to reappear, no less obviously, in those precipitated structures which simulate organic form in the experiments of Leduc, Herrera and Lillie. 372 THE FORMS OF CELLS [ch. which corresponds a positive pressure P, suppHed by the surface- tension as in the case of the sphere, but evidently of just half the magnitude. In plane, sphere and cyUnder the two principal curvatures are constant, separately and together; but in the unduloid the curva- tures change from one point to another. At the middle of one of the swollen "beads" or bubbles, the curvatures are both positive; the expression (l/R + 1/^') is therefore positive, and it is also finite. The film exercises (like the cyUnder) a positive pressure inwards, to be compensated by an equivalent outward pressure from within. Between two adjacent beads, at the middle of one of the narrow necks, there is obviously a much stronger curvature in the trans- verse direction; but the total pressure is unchanged, and we now see that a negative curvature along the unduloid balances the increased curvature in the transverse direction. The sum of the two must remain positive as well as constant; therefore the convex or positive curvature must always be greater than the concave or negative curvature at the same point, and this is plainly the case in our figure of the unduloid. The catenoid, in this respect a limiting case of the unduloid, has its curvature in one direction equal and opposite to its curvature in the other, this property holding good for all points of the surface ; R = — R'; and the expression becomes {l/R -+- l/R') = (L/R - l/R) = 0. That is to say, the mean curvature is zero, and the catenoid, like the plane itself, has no curvature, and exerts no pressure. None of the other surfaces save these two share this remarkable property; and it follows that we may have at times the plane and the catenoid co-existing as parts of one and the same boundary system, just as the cyHnder or the unduloid may be capped by portions of spheres. It follows also that if we stretch a soap-film between two rings, and so form an annular surface open at both ends, that surface is a catenoid : the simplest case being when the rings are parallel and normal to the axis of the figure*. * A topsail bellied out by the wind is not a catenoid surface, but in vertical section it is everywhere a catenary curve; and Diirer shews beautiful catenary curves in the wrinkles under an Old Man's eyes. A simple experiment is to invert v] OF FIGURES OF EQUILIBRIUM 373 The nodoid is, like the iinduloid, 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 its full mathe- matical treatment is too hard for us. In one of Plateau's experiments, a bubble of oil (protected from gravity by a fluid of equal density to its own) is balanced between annuh; and by adjusting the distance apart of these, it may be brought to assume the form of Fig. 106, that is to say, of a cyhndet with spherical ends; there is then everywhere a pressure inwards on the fluid contents of the bubble a pressure due to the convexity Fig. 106. Fig. 107. of the surface film. This cylinder may be converted into an undu- loid, either by drawing the rings farther apart or by abstracting some of the oil, imtil at length rupture ensues, and the cylinder breaks up into two spherical drops. Or again, if the surrounding liquid be made ever so little heavier or hghter than that which constitutes the drop, then gravity comes into play, the conditions of equihbrium are modified accordingly, and the cylinder becomes part of an unduloid, with its dilated portion above or below as the case may be (Fig. 107). In all cases the unduloid, like the original cyhnder, is capped by spherical ends, the sign and the consequence of a positive pressure produced by the curved walls of the unduloid. But if our initial cyhnder, instead of being tall, be a flat or dumpy one a small funnel in a large one, wet them with soap-solution, and draw them apart; the film which develops between them is a catenoid surface, set perpendicularly to the two funnels. On this and other geometrical illustrations of the fact that a soap-film sets itself at right angles to a solid boundary, see an elegant paper by Mary E. Sinclair, in Annals of Mathematics, viii, 1907. 374 THE FORMS OF CELLS [CH. (with certain definite relations of height to breadth), then new phenomena may occur. For now, if 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 a sufficient 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. 108 B. This figure is a catenoid, which, B Fig. 108. as we have seen, is, like the plane itself, a surface exercising no pressure, and which therefore may coexist with the plane as part of one and the same system. We may continue to withdraw more oil from our bubble, drop by drop, and now the upper and lower surfaces dimple down into concave portions of spheres, as the result of the negative internal pressure ; and thereupon the peri- pheral catenoid surface alters its form (perhaps, on this small scale, imperceptibly), and becomes a portion of a nodoid. It represents, in fact, that portion of the nodoid which in Fig. 109 lies between such points as 0, P. While it is easy to draw the outline, or meridional section, of the nodoid, it is obvious that the solid of revolution to be derived from it can never be reahsed in its entirety : for one part of the solid figure would cut, or entangle with, another. All that we can ever do, accordingly, is to reahse isolated portions of the nodoid*. * This curve resembles the looped Elastic Curve (see Thomson and Tait, ii, p. 148, fig. 7), but has its axis on the other side of the curve. The nodoid was represented upside-down in the first edition of this book, a mistake into which others have fallen, including no less a person than Clerk Maxwell, in his article "Capillarity " in the Encycl. Brit. 9th ed. Fig. 109. V} OF FIGURES OF EQUILIBRIUM 375 In all these cases the ring or annulus is not merely a means of mechanical restraint, controlling the form of the drop or bubble ; it also marks the boundary, or "locus of discontinuity," between one surface and another. If, in a sequel to the preceding experiment of Plateau's, we use solid discs instead of annuli, we may exert pressure on our oil- globule as we exerted traction before. We begin again by adjusting the pressure of these discs so that the oil assumes the form of a cylinder: our discs, that is to say, are adjusted to exercise a mechanical pressure just equal to what in the former case was supplied by the surface-tension of the spherical caps or ends of the bubble. If we now increase the pressure slightly, the peripheral walls become convexly curved, exercising a precisely corresponding pressure; the form assumed by the sides of our figure is now that of a portion of an unduloid. If we increase the pressure, 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 find within certain limits the system remaining 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. 109 lies between 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. 109, from OP, the surface con- cerned in the former case, to MN, that concerned in the present, we shall see that in the two experiments the surface of the Uquid is not the same, but lies on the positive side of the curve in the one case, and on the negative side in the other. These capillary surfaces of Plateau's form a beautiful example of the "materialisation" of mathematical law. Theory leads to certain equations which determine the position of points in a system, and these points we may then plot as curves on a coordinate diagram; but a drop or a bubble may realise in an instant the 376 THE FORMS OF CELLS [ch. whole result of our calculations, and materialise our whole ap- paratus of curves. Such a case is what Bacon calls a "collective instance," bearing witness to the fact that one common law is obeyed by every point or particle of the system. Where the under- lying equations are unknown to us, as happens in so many natural configurations, we may still rest assured that kindred mathematical laws are being automatically followed, and rigorously obeyed, and sometimes half-revealed. Of all the surfaces which we have been describing, the sphere is the only one which can enclose space of itself; the others can only help to do so, in combination with one another or with the sphere. Moreover, the sphere 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, ipso facto, the form which will be assumed by a unicellular organism (just as by a raindrop), if it be practically homogeneous and if, Hke Orhulina floating in the ocean, its surroundings be likewise homo- geneous and its field of force symmetrical f. It is only relatively speaking that the rest of these configurations are surfaces minimae areae', for they are so under conditions which involve various pressures or restraints. Such restraints are imposed by the pipe or annulus which supports and confines our oil-globule or soap-bubble ; and in the case of the organic cell, similar restraints are supplied by solidifications partial or complete, or other modifications local or general, of the cell-surface or cell-wall. One thing we must not fail to bear in mind. In the case of the soap-bubble we look for stabihty or instability, equihbrium or non- equilibrium, in its several configurations. But the living cell is seldom in equihbrium. It is continually using or expending energy; and this ceaseless flow of energy gives rise to a "steady state," taking the place of and simulating equilibrium. In like manner the * On the circle and sphere as giving the smallest boundary for a given content, see (e.g.) Jacob Steiner, Einfache Beweisen der isoperimetrischen Hauptsatze, Berlin. Abhandlungen, 1836, pp. 123-132. t The essential conditions of homogeneity and symmetry are none too common, and a spherical organism is only to be looked for among simple things. The floating (or pelagic) eggs of fishes, the spores of red seaweeds, the oospheres of Fucus or Oedogonium, the plasma-masses escaping from the cells of Vaucheriaf are among the instances which come to mind. OF FIGURES OF EQUILIBRIUM 377 hardly changing outHne of a jet or waterfall is but in pseudo- equiHbrium; it is in a steady state, dynamically speaking. Many puzzling and apparent paradoxes of physiology, such (to take a single instance) as the maintenance of a constant osmotic pressure on either side of a cell-membrane, are accounted for by the fact that energy is being spent and work done, and a steady state or pseudo-equilibrium maintained thereby. Before we pass to biological illustrations of our surface-tension figures we have still another 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 a cyhnder, and then beyond, there comes a certain point at which the bubble breaks in two, and leaves us with two bubbles of w^hich each is a sphere or a portion of a sphere. In short there are certain limits to the dimensions of our figures, within which limits equi- librium is stable, but at which it becomes unstable, and beyond which it breaks down. Moreover, in our composite surfaces, when the cyhnder for instance is capped by two spherical cups or lenticular discs, there are well-defined ratios which regulate their respective curvatures and their respective dimensions. These two matters we may deal with together. Let us imagine a Hquid drop which in appropriate conditions has been made to assume the form of a cylinder; we have already seen that its ends will be capped 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), w^e assume the surface-tension (T) to be everywhere identical; and it follows, since the internal fluid-pressure is also every- where identical, that the expression (IjR + 1/i?') for the cylinder is equal to the corresponding expression, which we may cafl (1/r + 1/r'), in the case of the terminal spheres. But in the 378 THE FORMS OF CELLS [ch. cylinder l/R' =- 0, and in the sphere 1/r = 1/r'. Therefore our relation of equality becomes l/R = 2/r, or r = 2R; which means that 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 (Oa) of the cyhnder, 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 Jtt. In other words, the spherical disc which (under the given conditions) caps our cylinder is not a portion taken at haphazard, but is neither more nor less than that portion of a sphere which is subtended by a cone of 60°. Moreover, it is plain that the height of the spherical cap, de, = Ob — ab = R (2 — ^/3) = 0'27R, 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, more or less, under normal circumstances, in such a case as the cylindrical cell of Spirogyra, when one end is free and capped by a portion of a sphere*. Among the many 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 stabihty; only the plane and the sphere, or any portion of a sphere, are perfectly stable, because they are perfectly symmetrical, figures. Perhaps it were better to say that their symmetry is such that any small disturbance will probably readjust itself, and leave the plane or spherical surface as it was before, while in the other configurations the chances are that a disturbance once set up will travel in one direction or another, increasing as it goes. For equihbrium and probabihty (as Boltzman told us) are nearly alHed: * The conditions of stability of the cyhnder, and also of the catenoid, are explained with the utmost simplicity by Clerk Maxwell, in his article, already quoted, on "Capillarity.'' On the catenoids, see A. Terquem. C.R. xcii, pp. 407-9, 1881. V] OF FIGURES OF EQUILIBRIUM 379 so nearly that that state of a system which is most likely to occur, or most likely to endure, is precisely that which we call the state of equilibrium. For experimental demonstration, the case of the cylinder is the simplest. If we construct a liquid cylinder, either by drawing out a bubble or by supporting a globule of oil between two rings, the experiment proceeds easily until the length of the cylinder becomes just about three times as great as its diameter. But soon afterwards instability begins, and the cylinder alters its form; it narrows at Fig. 111. the waist, so passing into an unduloid, and the deformation pro- gresses quickly until our cylinder breaks in two, and its two halves become portions of spheres. This physical change of one surface into another corresponds to what the mathematicians call a "discon- tinuous solution" of a problem of minima. The theoretical limit of stability, according to Plateau, is when the length of the cylinder is equal to its circumference, that is to say, when L = Inr, or when the ratio of length to diameter is represented by -n. The fact is that any small disturbance takes the form of a wave, and travels along the cylinder. Short waves do not affect the stability of the system ; but waves whose length exceeds that of the circumference tend to grow in amplitude: until, contracting here, expanding there, the cylinder turns into a pronounced unduloid, and soon breaks into two parts or more. Thus the cylinder is a 380 THE FORMS OF CELLS [ch. stable figure until it becomes longer than its own circumference, and then the risk of rupture may be said to begin. But Rayleigh shewed that still longer waves, leading to still greater instability, are needed to break down material resistance*. For, as Plateau knew well, his was a theoretical result, to be departed from under material conditions; it is affected largely by viscosity, and, as in the case of a flowing cylinder or jet, by inertia. When inertia plays a leading part, viscosity being small, the node of maximum in- stability corresponds to nearly half as much again as in the simple or theoretical case: and this result is very near to what Plateau himself had deduced from Savart's experiments on jets of water "j". When the fluid is external >(as when the cyhnder is of air) the wave- length of maximal instability is longer still. Lastly, when viscosity is very large, and becomes paramount, then the wave-length between regions of maximal instability may become very long indeed: so that (as Rayleigh put it) ''long (viscid) threads do not tend to divide themselves into drops at mutual distances comparable with the diameter of the cylinder, but rather to give way by attenuation at few and distant places." It is this that renders possible the making of long glass tubes, or the spinning of threads of "viscose" and like materials; but while these latter preserve their continuity, the principle of Plateau tends to give them something of a wavy, unduloid surface, to the great enhancement of their beauty. We are prepared, then, to find that such cylinders and unduloids as occur in organic nature seldom approach in regularity to those which theory prescribes or a soap-film may be made to shew; but rather exhibit all manner of gradations, from something exquisitely neat and regular to a coarse and distant approximation to the ideal thing {. The unduloid has certain peculiar properties as regards its limita- tions of stabihty, but we need mention two facts only: (1) that when the unduloid, which we produce with our soap-bubble or our * Rayleigh, On the instability of fluid surfaces, Sci. Papers, iii, p. 594. t Cf. E. Tylor, Phil. J^ag. xvi, pp. 504-518, 1933. X Cf. F. Savart, Sur la constitution des veines liquides lancees par des orifices, etc., Ann. de Chimie, liii, pp. 337-386, 1833. Rayleigh, On the instability of a cylinder of viscous liquid, etc., Phil. Mag. (5), xxxiv, 1892, or Sci. Papers, i, p. 361. See also Larmor, On the nature of viscid fluid threads. Nature, July 11, 1936, p. 74. V] OF FIGURES OF EQUILIBRIUM 381 oil-globule, consists of the figure containing a complete constriction, then it has somewhat wide limits of stabiHty; but (2) if it contain the swollen portion, then equilibrium is limited to the case of the figure consisting of one complete unduloid, no less nor more; that is to say when the ends of the figure are constituted by the narrowest portions, and its middle by the widest portion of the entire curve. The theoretical proof of this is difficult ; but if we take the proof for granted, the fact itself will serve to throw light on what we have learned regarding the stabihty of the cyhnder. For, when we remember that the meridional section of our unduloid is generated by the rolhng of an ellipse upon a straight line in its own plane, we 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 outUne it approaches, and in time reaches, the form of the cylinder, as a "Hmiting case"; and jpari passu, the ellipse which generated it passes into a circle, as its foci come closer and closer together. The cyhnder of a length equal to the circum- ference of its generating circle is homologous to an unduloid whose length is equal to the circumference of its generating elhpse; and this is just what we recognise as constituting one complete segment of the unduloid. The cylinder turns so easily into an unduloid, and the unduloid is capable of assuming so many graded differences of form, that we may expect to find it abundantly and variously represented among the simpler living things. For the same reason it is the very stand-by of the glass-blower, whose flasks and bottles are, of necessity, unduloids*. The blown-glass bottle is a true unduloid, and the potter's vase a close approach to an unduloid; but the alabaster bottle, turned on the lathe, is another story. It may be an imitation, or a reminiscence, of the potter's or the glass-blower's work; but it is no unduloid nor any surface of minimal area at all. The catenoid, as we have seen, is a surface of zero pressure, and as such is unlikely to form part (unless momentarily) of the closed boundary of a cell. It forms a limiting case between unduloid and nodoid, and, were it realised, it would seldom be visibly different from the other two. In Trichodina pediculus, a minute infusorian para- * Unless, that is to say, their shape be cramped and their mathematical beauty annihilated, by compression in a mould. 382 THE FORMS OF CELLS [ch. site of the freshwater polype, we have a circular disc bounded (apparently) by two parallel rings of cilia, with a pulley-like groove between. The groove looks very like that catenoid surface which we have produced from two parallel and opposite annuli; and the fact that the lower surface of the little creature is practically plane, where it creeps ^.^^^^^^^^^ over the smooth body of the Hydra, i\ '\\\ vvjN ^ looks like confirming the catenoid Fig. 112. Triclwdiruz pediculus. analogy. But the upper surface of the infusorian, with its ciliated "gullet," gives no assurance of a zero pressure; and we must take it that the equatorial groove of Trichodina resembles, or approaches, but is not mathematically identical with, a catenoid surface. While those figures of equilibrium which are also surfaces of revolution are only six in number, there is an infinite number of other figures of equilibrium, that is to say of surfaces of constant mean curvature, which are jiot 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 bend a wire into a closed curve, plane or not plane, we may always fill the entire area with a con- tinuous film. No more interesting problem has ever been pro- pounded to mathematicians as the outcome of experiment than the general problem so to describe a minimal surface passing through a closed contour; and no complete solution, no general method of approach, has yet been discovered*. Of the regular figures of equihbrium, or 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. * Partial solutions, closely connected with recent developments of mathematical analysis, are due to Riemann, Weierstrass and Schartz. Cf. (int. al.) G. Darboux, Theorie des surfaces, 1914, pp. 490-601; T. Bonneson, Problemes des i^operimetres et des iaipipJianes, Paris, 1929; Hilbert's AnschauUche Geometrie, 1932, p. 237 seq.; a good account also in G. A. Bliss's Calculus of Variations, Chicago, 1925. See also (int. al.) Tibor Rado, Mathem. Ztschr. xxxir, 1930; -Jesse Douglas, Amer. Math. Journ. XXXIII, 1931, Journ. Math. Phys. xv, 1936. V] OF FIGURES OF EQUILIBRIUM 383 This is a helicoid generated by a straight Hne 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," or surfaces capable of being defined by a system of stretched strings*, the plane and the hehcoid are the only two whose mean curvature is null, while the cyhnder is the only one whose curvature is finite and constant. As this simplest of hehcoids 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 hehcoid), 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 cyhnder it is obvious that the resulting figure is indis- tinguishable from the cyhnder itself. In the case of the unduloid we obtain a grooved spiral, and we meet with something very like it in nature (for instance in Spirochaetes, Bodo gracilis, etc.) ; but in point of fact, the screw motion given to an unduloid or catenary curve fails to give a minimal screw surface, as we might have expected it to do. The foregoing considerations deal with a small part only of the theory of surface-tension, or capillarity: with that part, namely, which relates to the surfaces capable of subsisting in equilibrium under the action of that force, either of itself or subject to certain simple constraints. And as yet we have limited ourselves to the case of a single surface, or of a single drop or bubble, leaving to another occasion a discussion of the forms assumed when such drops or vesicles meet and combine together. Ip short, what we have said may help us to understand the form of a cell — considered, as with certain hmitations we may legitimately consider it, as a Hquid drop or liquid vesicle ; the conformation of a tissue or cell-aggregate must be dealt with in the light of another series of theoretical con- siderations. In both cases, we can do no more than touch on the fringe of a large and difficult subject. There are many forms * Or rather, surfaces such that through every point there runs a straight line which lies wholly in the surface. 384 THE FORMS OF CELLS [ch. capable of realisation under surface-tension, and many of them doubtless to be recognised among organisms, which we cannot deal with 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, the simple an4 symmetrical conditions with which the mathematical theory is capable of deahng. And before we pass to illustrate the physical phenomena by biological examples, we must repeat that the simple physical conditions which we presuppose will never be wholly realised in the organic cell. Its substance will never be a perfect fluid, and hence equilibrium will be slowly reached; its surface will seldom be perfectly homo- geneous, and therefore equilibrium will 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 surface-tension effects*. But we shall find that, on the whole, these effects of surface-tension though modified are not obliterated nor even masked; and accordingly the phenomena to which I have devoted the foregoing pages will be found manifestly recurring and repeating themselves among the phenomena of the organic cell. In a spider's web we find exemplified several of the principles of surface-tension which we have now explained. The thread is spun out of a glandular secretion which issues from the spider's body as a semi-fluid cyHnder, the force of expulsion giving it its length and that' of surface-tension giving it its circular section. It is too viscid, and too soon hardened on exposure to the air, to break up into drops or spherules ; but it is otherwise with another sticky secretion which, coming from another gland, is simultaneously poured over the * That "every particular that worketh any effect is a thing compounded more or less of diverse single natures, more manifest and more obscure" is a point made and dwelt on by Bacon. Of the same principle a great astronomer speaks as follows: "It is one of the fundamental characteristics of natural science that we never get beyond an approximation. . .Nature never offers us simple and undivided phenomena to observe, but always infinitely complex compounds of many different phenomena. Each single phenomenon can be described mathematically in terms of the accepted fundamental laws of Nature : . . . but we can never be sure that we have carried the analysis to its full exhaustion, and have isolated one single simple phenomenon." W. de Sitter, in Nature, Jan. 21, 1928, p. 99. V] OF SPIDEKS' WEBS 385 slacker cross-threads as they issue to form the spiral portion of the web. This latter secretion is more fluid than the first, and only dries up after several hours*. By capillarity it "wets" the thread, spreading over it in an even film or liquid cylinder. As such it has its limits of stabihty, and tends to disrupt at points more distant than the theoretical wave-length, owing to the imperfect fluidity of the viscous film and still more to the frictional drag of the inner thread with which it is in contact. Save for this qualifi- cation the cyhnder disrupts in the usual manner, passing first into the wavy outhne of an unduloid, whose swollen internodes swell more and more till the necks between them break asunder, and leave a row of spherical drops or beads strung like dewdrops at regular intervals along the thread. If we try to varnish a thin taut wire we produce automatically the same identical result t; imless our varnish be such as to dry almost instantaneously it gathers into beads, and do what we will we fail to spread it smooth. It follows that, according to the drying quahties of our varnish, the process may stop at any point short of the formation of perfect spherules; and as our final stage we may only obtain half-formed beads or the wavy outlines of an unduloid. The beads may be helped to form by jerking the stretched thread, and so disturbing the unstable equilibrium of the viscid cyhnder. This the spider has been said to do, but Dr G. T. Bennett assures me that she does nothing of the kind. She only draws her thread out a little, and leaves it a trifle slack ; if the gum should break into droplets, well, and good, but it matters httle. The web with its sticky threads is not improved thereby. Another curious phenomenon here presents itself. In Plateau's experimental separation of a cyhnder of oil into two spherical halves, 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 * When we see a web bespangled with dew of a morning, the dewdrops are not drops of pure water, but of water mixed with the sticky, gummy fluid of the cross- threads ; the radii seldom if ever shew dewdrops. See F, Strehlke, Beobachtungen an Spinnengewebe, Poggendorff's Annalen, XL, p. 146, 1937. t Felix Plateau recommends the use of a weighted thread or plumb-line, to be drawn up slowly out of a jar of water or oil; Phil. Mag. xxxiv, p. 246, 1867. 386 THE FORMS OF CELLS [ch. the liquid into a thin thread as they completed their spherical f^rm and receded from one another: the reason being that, after the thread or "neck" has reached a certain tenuity, internal friction prevents or retards a rapid exit of the fluid from the thread to the adjacent spherule. It is for the same reason that we are able to draw a glass rod or tube, which we have heated in the middle, into a long and uniform cyhnder or thread by quickly separating the two ends. But in the case of the glass rod the long thin thread quickly cools and sohdifies, while in the ordinary separation of a liquid cylinder the corresponding intermediate cylinder remains liquid; and therefore, Hke any other liquid cylinder, it is liable to Fig. 113. Dew-drops on a spider's web. break up, provided that its dimensions exceed the limit of stability. And its length is generally such that it breaks at two points, thus leaving two terminal portions continuous and confluent with the spheres, and one median portion w^hich resolves itself into .a tiny spherical drop, midway between the original and larger two. Occasionally, the same process of formation of a connecting thread repeats itself a second time, between the small intermediate spherule and the large spheres; and in this case we 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 now and then with a third order 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 — sometimes interrupted and disturbed by a sKght want of homogeneity in the secreted fluid ; and the same phenomenon is V] OF BEADS OR GLOBULES 387 repeated on a grosser scale when the web is bespangled with dew, and its threads bestrung with pearls innumerable. To the older naturalists, these regularly arranged and beautifully formed globules on the spider's -web were a frequent source of wonderment. Black- wall, counting some twenty globules in a tenth of an inch, calculated that a large garden-spider's web should comprise about 120,000 globules; the net was spun and finished in about forty minutes, and Blackwall was filled with admiration of 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*. Here we see exemphfied what Plateau told us of the law of minimal areas transforming the cylinder into the unduloid and disrupting it Fig. 114. Root-hair of Trianea, in glycerine. After Berthold. into spheres. The httle dehcate beads which stud, the long thin pseudopodia.o.f a foraminifer, such as Gromia, or which appear in like manner on the film of protoplasm coating the long radiating spicules of Globigerina, represent an identical phenomenon. Indeed we may study in a protoplasmic filament the whole process of formation of such beads: if we squeeze out on a shde the viscid contents of a mistletoe-berry, the long sticky threads into which the substance runs shew the whole phenomenon particularly well. True, many long cylindrical cells, such as are common in plants, shew no sign of beading or disruption ; but here .the cell- walls are never fluid but harden as they grow, and the protoplasm within is kept in place and shape by its contact with the cell-wall. It was noticed many years ago by Hofmeisterf, and afterwards explained by Berthold, that if we dip the long root-hairs of certain water-plants, such as Hydrocharis or Trianea, in a denser fluid (a httle sugar-solution or * J. Blackwall, Spiders of Great Britain (Ray Society), 1859, p. 10; Trans. Linn. Soc. XVI, p. 477, 1833. On the strength and elasticity of the spider's web, see J. R. Benton, Amer. Journ. Science, xxiv, pp. 75-78, 1907. t Lehrbuch von der Pflanzenzelley p. 71 ; cf. Nageli, Pflanzenphysiologische Unter- suchungen [Spirogyra), ni, p. 10. 388 THE FORMS OF CELLS [ch. dilute glycerine), the cell-sap tends to diffuse outwards, the proto- plasm parts company with its surrounding and supporting wall, and then hes free as a protoplasmic cylinder in the interior of the cell. Thereupon it soon 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*. We could not wish for a better or a simpler proof of the essential fluidity of the protoplasm f. Similar, but less regular, beads or droplets may be caused to appear, under stimulation by an alternating current, in. the protoplasmic threads within the living cells of the hairs of Tradescantia; the explanation usually given is, that the viscosity of the protoplasm is reduced, or its fluidity increased ; but an increase of the surface-tension would seem a more likely reason}. In one of Robert Chambers's delicate experiments, a filament of protoplasm is drawn off, by a micro-needle, from the fluid surface of a starfish-egg. If drawn too far it breaks, and part returns within the protoplasm while the other rounds itself off on the needle's point. If drawn out less far, it looks hke a row of beads or chain of droplets; if yet more relaxed, the droplets begin to fuse until the whole filament is withdrawn; if drawn out anew the process repeats itself. The whole story is a perfect description of the behaviour of a fltiid jet or cy finder, of varying length and thickness §. 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, * The intermediate spherules appear with great regularity and beauty whenever a liquid jet breaks up into drops. So a bursting soap-bubble scatters a shower of droplets all around, sometimes all alike, but often with a beautiful alternation of great and small. How the breaking up of thread or jet into drops may be helped, regularised, and sometimes complicated, by external vibrations is another and by no means unimportant story. t Though doubtless to speak of the viscid thread as a fluid is but a first approxi- mation; cf. Larmor, in Nature, July 11, 1936. X Kiihne, Untersuchungen ilber das Protoplasma, 1864, p. 75, etc. § Cf. R. Chambers in Colloid Chemistry, theoretical and applied, ii, cap. 24, 1928; also Ann. de Physiol, vi, p. 234, 1930; etc. V] THE SHAPE OF A SPLASH 389 are closely related to those which attend and which bring about the breaking-up of a liquid cylinder or thread. In Mr Worthington's beautiful experiments on splashes*, it was found that the fall of a found pebble into water from a height first formed a dip or hollow in the surface, and then caused a filmy "cup" of water to rise up all round, opening out trumpet-fashion Fig. 115. Phases of a splash. From Worthington. ""''^WiW^JSSM, ''S^^s?*^ Fig. 116. A wave breaking into spray. or closing in like a bubble, according to the height from which the pebble fell. The cup or "crater" tends to be fluted in alternate ridges and grooves, its edges get scolloped into corresponding lobes and notches, and the projecting lobes or prominences tend to break off or break up into drops or beads (Fig. 115). A similar appearance is seen on a great scale in the edge of a breaking wave : for the smooth * A Study of SplasJies, 1908, p. 38, etc.; also various papers in Proc. R.S. 1876-1882, and Phil Trans. (A), 1897 and 1900. 390 THE FORMS OF CELLS [ch. edge becomes notched or sinuous, and the surface near by becomes ribbed or fluted, owing to the internal flow being helped here and hindered there by a viscous shear; and then all of a sudden the uneven edge shoots out an array of tiny jets, which break up into the countless droplets which constitute "spray" (Fig. 116). The naturalist may be reminded also of the beautifully symmetrical notching of the calycles of many hydroid zoophytes, which little cups had begun their existence as Hquid or semi-hquid films before they became stiff and rigid. The next phase of the splash (with which we are less directly concerned) is that the crater subsides, and where it stood a tall column rises up, which also tends, if it be tall enough, to break up into drops. Lastly the column sinks down in its turn, and a ripple runs out from where it stood. The edge of our little cup forms a hquid ring or annulus, com- parable on the one hand to the edge of an advancing wave, and on the other to a hquid thread or cylinder if only we conceive the thread to be bent round into a ring; and accordingly, just as the thread segments first into an unduloid and then into separate spherical drops, so likewise will the edge of cup or annulus tend to do. This phase of notching, or beading, of the edge of the splash is beautifully seen in many of Worthington's experiments*, and still more beautifully in recent work (Frontispiece f). In the second place the fact that the crater rises up means that liquid is flowing in from below; the segmentation of the rim means that channels of easier flow are being created, along which the liquid is led or driven into the protuberances; and these last are thereby exaggerated into the jets or streams which become conspicuous at the edge of the crater. In short any film or film-like fluid or semi-fluid cup will be unstable ; its instability will tend to show itself in a fluting of the surface and a notching of the edge; and just such a fluting and notching are conspicuous features of many minute organic cup-like structures. In the hydroids (Fig. 117), we see that these common features of the * Cf. A Study of Splashes, pp. 17, 77. The same phenomenon is often well seen in the splash of an oar. It is beautifully and continuously evident when a strong jet of water from a tap impinges on a curved surface and then shoots off again. t We owe this picture to the kindness of Mr Harold E. Edgerton, of the Massachusetts Institute of Technology. It shews the splash caused by a drop falling into a thin layer of milk; a second drop of milk is seen above, following the first. The exposure-time was 1/50,000 of a second. The latter phase of a splash: the crater has subsided, a columnar jet has risen up, and the jet is dividing into droplets. From Harold E. Edgerton, Massachusetts Institute of Technology V] THE SHAPE O'F A SPLASH 391 cup and the annulation of the stem are phenomena of the same order. A cord-hke thickening of the edge of the cup is a variant of the same order of phenomena; it is due to the checking at the rim of the flow of hquid from below, and a similar thickening is to be seen, not only in some hydroid calycles but also in many Vorticellae (cf. Fig. 124) and other cup-shaped organisms. And these are by no means the only manifestations of surface-tension in a splash which shew resemblances and analogies to organic form*. The phenomena of an ordinary liquid splash are so swiftly tran- sitory that their study is only rendered possible by photography: 01 [VA~/"'^^^VY Fig. 117. Calycles of Campanularia spp. but this excessive rapidity is not an essential part of the pheno- menon. 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 sandf, 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 proto- plasmic organism, of phenomena which appear and disappear with * The. same phenomena are modified in various ways, and the drops are given off much more freely, when the splash takes place in an electric field — all owing to the general instability of an electrified liquid -surface; and a study of this aspect of the subject might suggest yet more analogies with organic form. Cf. J. Zeleny, Phys. Rev. x, 1917; J. P. Gott, Proc. Cambridge Philos. Soc. xxxi, 1935; etc. t We find now and then in certain brick-clays of glacial origin, hard, quoit- shaped rings, each with an equally indurated, round or flattened ball resting on it. These may be precisely imitated by splashing large drops of water on a smooth surface of fine dry sand. The ring corresponds, apparently, to the crater of the splash, and the ball (or its water content) to the pillar rising in the middle. 392 THE FORMS OF CELLS [ch. evanescent rapidity in a more mobile liquid. Nor is there anything pecuhar in the splash itself; it is simply a convenient method of setting up certain motions or currents, and producing certain surface- forms, in a Hquid medium — or even in such an imperfect fluid as 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 clay or "shp" it works on, 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 hmiting 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 naturahst as also for the physicist: illustrating as it does the development of a host of mathematical configurations and organic conformations which depend essentially on the establishment of a constant and uniform pressure within a closed elastic shell or fluid envelope or bubble. In hke manner the potter's art illustrates the somewhat obscurer and more complex problems (scarcely less frequent in biology) of a figure of equilibrium which is an open surface of revolution. The two series of problems are closely akin; for the glass-blower can make most things which the potter makes, by cutting off portions of his hollow ware; 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 still older craft of the potter. It would seem venturesome to extend our comparison with these liquid surface-tension phenomena from the cup or calycleof the hydrozoon to the little hydroid polyp within: and yet there is something to be learned by such a comparison. The cylindrical body of the tiny polyp, the jet-hke row of tentacles, the beaded V] THE SPLASH AND THE BUBBLE 393 annulations which these tentacles exhibit, the web-like film which sometimes (when they stand a httle 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" con- tracts*, 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 of the splash "f". Some seventy years ago much interest was aroused by Helmholtz's work (and also Kirchhoff's) on "discontinuous motions of a fluidj"; that is to say, on the movements of one body of fluid within another, and the resulting phenomena due to friction at the surfaces between. What Kelvin§ called Helmholtz's "admirable discovery of the law of vortex-motion in 'a perfect fluid" was the chief result of this investigation; and was followed by much experimental work, in order to illustrate and to extend the mathematical conclusions. The drop, the bubble and the splash are parts of a long story; and a "falling drop," or a drop moving through surrounding fluid, is a case deserving to be considered. A drop of water, tinged with fuchsin, is gently released (under a pressure of a couple of milh- metres) at the bottom of a glass of water || . Its momentum enables it to rise through a few centimetres of the surrounding water, and in doing so it communicates motion to the water around. In front the rising drop thrusts its way through, almost like a sohd body; behind it tends to drag the surrounding water after it, by fluid friction^ ; and these two motions together give rise to beautiful vorti- coid configurations, the Strdmungspilze or Tintenpilze of their first discoverers (Fig. 119). Under a higher and more continuous pressure * See a Study of Splashes, p. 54. t There is little or no difference between a splash and a burst bubble. The craters of the moon have been compared with, and explained by, both of these. X Helmholtz, in Berlin. Monatsber. 1868, pp. 215-228; Kirchhoflf, in CreUe'a Journal, lxx, pp. 289-298, lxxi, 237-273, 1869-70. § W. Thomson, in Proc. R.S.E. vi, p. 94, 1867. II See A. Overbeck, Ucber discontinuirliche Fliissigkeitsbewegungen, Wiedemann'' s Annalen, ii, 1877; W. Bezold, Ueber Stromungsfiguren in Fliissigkeiten, ibid. XXIV, pp. 569-593, 1885; P. Czermak, ibid, l, p. 329, 1893; etc. % The frictional drag on the hinder part of the drop is felt alike in the ship, the bird and the aeroplane, and tends to produce retarding vortices in them all. It is always minimised in one way or another, and it is autoraaticaUy minimised in the present instance, as the drop thins off and tapers down. 394 THE FORMS OF CELLS [CH. __Md3:::,,^^ Fig. 118. a, b. More phases of a splash, after Worthington. c. A hydroid polype, after Allman. o ^t Fip. 119. -Liiquid jets. From A. Overbeck. V] OF FALLING DROPS 395 the drop becomes a jet; the, form of the vortex is modified thereby, and may be further modified by sHght differences of temperature (i.e. of density), or by interrupting the rate of flow. To let a drop of ink fall into water is a simple and most beautiful experiment*. The effect is more violent than in the former case. The descending Fig. 120. Falling drops. A, ink in water, after J. J. Thomson and Newall. B, fusel oil in paraffin, after Tomlinson. drop turns into a complete vortex-ring ; it expands and attenuates ; it waves about, and the descending loops again turn into incipient vortices (Fig. 120). Lastly, instead of letting our drop rise or fall freely, we may use a hanging drop, which, while it sinks, remains suspended to the surface. Thus it cannot form a complete annulus, but only a * J. J. Thomson and H. F. Newall, On the formation of vortex-rings by drops, Proc. R.S. XXXIX, pp. 417-436, 1885. Emil Hatschek, On forms assumed by a gelatinising liquid in various coagulating solutions, ibid. (A) xciv, pp. 303-316, 1918. 396 THE FORMS OF CELLS [ch. partial vortex suspended by a thread or column — just as in Over- beck's jet-experiments; and the figure so produced, in either case, is closely analogous to that of a medusa or jellyfish, with its bell or "umbrella," and its clapper or "manubrium" as well. Some years ago Emil Hatschek made such vortex-drops as these of hquid gelatine dropped into a hardening fluid. These " artificial medusae " sometimes show a symmetrical pattern of radial "ribs", due to shrinkage, and this to dehydration by the coagulating fluid. An Fig. 121. Various medusoids: 1, Syncoryne; 2, Cordylophora; 3, Cladonema (after Allmaii). extremely curious result of Hatschek's experiments is to shew how sensitive these vorticoid drops are to physical conditions. For using the same gelatine all the while, and merely varying the density of the fluid in the third decimal place, we obtain a whole range of configurations, from the ordinary hanging drop to the same with a ribbed pattern, and then to medusoid vortices of various graded forms. The living medusa has a geometrical symmetry so marked and regular as to suggest a physical or mechanical element in the little creature's growth and construction. It has, to begin with, its vortex-hke bell or umbrella, with its cylindrical handle or manubrium. The bell is OF VORTICOID OR MEDUSOID DROPS 397 traversed by radial canals, four or in multiples of four; its edge is beset with tentacles, smooth or often beaded, at regular intervals and of graded sizes ; and certain sensory structures, including sohd concretions or "otohths," are also symmetrically interspaced. No sooner made, than it begins to pulsate ; the little bell begins to " ring." m & Fig. 1216. "Medusoid drops", of gelatin. After Hatschek, Buds, miniature replicas of the parent-organism, are very apt to appear on the tentacles, or on the manubrium or sometimes on the edge of the bell; we seem to see one vortex producing others before our eyes. The development of a medusoid deserves to be studied without prejudice, from this point of view. Certain it is that the tiny medusoids of Obelia, for instance, are , budded off with a rapidity and a complete perfection which suggest an automatic and all but instantaneous act of conformation, rather than a gradual process of growth. Moreover, not only do we recognise in a vorti- coid drop a "schema" or analogue of medusoid form, but we seem able to discover -various actual phases of the splash ot drop in the all but in- numerable living types of jellyfish; in Cladoneyna we seem to see an early stage of a breaking drop, and in Cordylophom a beautiful picture of incipient vortices. It is hard indeed to say how much or little all these analogies imply. But they indicate, at the very least, how certain simple organic forms might be naturally assumed by one fluid mass within another, when gravity, surface tension and fluid friction play Fig. 122. Meckbsach- loris, a ciliate infusoiia. 398 THE FORMS OF CELLS [ch. their part, under balanced conditions of temperature, density and chemical composition. A little green infusorian from the Baltic Sea is, as near as may be, a medusa in miniature*. It is curious indeed to find the same medusoid, or as we may now call it vorticoid, configuration occurring in a form so much lower in the scale, and so much less in order of magnitude, than the ordinary medusae. According to Plateau, the viscidity of the liquid, while it retards the breaking up of the cylinder and increases the length of the segments beyond that which theory demands, has never- theless less influence in this direction than we might have expected. On the other hand any external support or adhesion, or mere contact with a sohd body, will be equivalent to a reduction of surface-tension and so will very greatly increase the stability of our cylinder. It is for this reason that the mercury in our thermo- meters seldom separates into drops: though it sometimes does so, much to our inconvenience. And again it is for this reason that the protoplasm in a long tubular or cyhndrical cell need not divide into separate cells and internodes until the length of these far exceeds the theoretical limits. An interesting case is that of a viscous drop immersed in another viscous fluid, and drawn out into a thread by a shearing motion of the latter. The thread seems stable at first, but when left to rest it breaks up into drops of a very definite and uniform, size, the size of the drops, or wave-length of the unduloid of which they are made, depending on the relative viscosities of the two threads f- Plateau's results, though discovered by way of experiment and though (as we have said) they illustrate the " materiahsation " of mathematical law, are nevertheless essentially theoretical results approached rather than realised in material systems. That a hquid cylinder begins to be unstable when its length exceeds ^-nr is all but mathematically true of an all but immaterial soap-bubble ; but very far from true, as Plateau himself was well aware, in a flowing jet, retarded by viscosity and by inertia. The principle is true and universal; but our living cylinders do not follow the abstract laws * Medusachloris phiale, of A. Pascher, Biol. Centralbl. xxxvii, pp. 421-429, 1917. t See especially Rayleigh, Phil. Mag. xxxiv, p. 145, 1892, by whom the subject is carried much further than where Plateau left it. See also {int. al.) G. I. Taylor, Proc. R.S. (A), cxLvi, p. 501, 1934; S. Tomotika, ibid, cl, p. 322, 1935; etc. V] OF VISCOUS THREADS 399 of mathematics, any more than do the drops and jets of ordinary fluids or the quickly drawn and quickly cooling tubes in the glass- worker's hands. Plateau says that in most hquids the influence of viscosity is such as to cause the cylinder to segment when its length is about four times, or even six times, its diameter, instead of a fraction over three times, as theory would demand of a perfect fluid. If we take it at four times, the resulting spherules would have a diameter of about 1-8 times, and their distance apart would be about 2-2 times, the original diameter of the cylinder; and the calculation is not difficult which would shew how these dimensions are altered in the case of a cylinder formed around a solid core, as in the case of a spider's web. Plateau also observed that the time taken in the division of the cyUnder is directly proportional to its diameter, while varying with the nature of the hquid. This question, of the time taken in the division of a cell or filament in relation to its dimensions, has not so far as I know been enquired into by biologists. From the simple fact that the sphere is of aU configurations that whose surface-area for a given volume is an absolute minimum, we have seen it to be the one figure of equilibrium assumed by a drop or vesicle when no disturbing factor is at hand; but such freedom from counter-influences is likely to be rare, and neither does the rain- drop nor the round world itself retain its primal sphericity. For one thing, gravity will always be at hand to drag and distort our drop or bubble, unless its dimensions be so minute that gravity becomes insignificant compared with capillarity. Even the soap-bubble will be flattened or elongated by gravity, according as we support it from below or from above; and the bubble which is thinned out almost to blackness will, from its small mass, be the one which remains most nearly spherical*. Innumerable new conditions will be introduced, in the shape of comphcated tensions and pressures,, when one drop or bubble becomes associated with another, and when a system of inter- mediate 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 * Cf. Dewar, On soap-bubbles of long duration, Proc. Roy. Inst. Jan. 19, 1929. 400 THE FORMS OF CELLS [ch. needed as a preliminary to any such enquiry. Meanwhile let us consider a few cases of the forms of cells, either sohtary, or in such simple aggregates that their individual form is httle disturbed thereby. Let us clearly understand that the cases we are about to consider are those where the perfect sjnnmetry of the sphere is replaced by another symmetry, less complete, such as that of an ellipsoidal or cylindrical cell. The cases of asymmetrical deformation or dis- placement, such as are illustrated in the production of a bud or the development of a lateral branch, are much simpler; for here we need only assume a slight and locahsed variation of surface- tension, such as may be brought about in various ways through the heterogeneous chemistry of the cell. But such diffused and graded asymmetry as brings about for instance the ellipsoidal shape of a yeast-cell is another matter. If the sphere be the one surface of complete symmetry and therefore of independent equilibrium, it follows that in every cell which is otherwise conformed there must be some definite cause of its departure from sphericity; and if this cause be the obvious one of 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 some form of mechanical restraint, as when we submit our bubble to the partial restraint of discs or rings or more compHcated cages of wire ; on the other hand it maybe 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 or its envelope*. * A case which we have not specially considered, but which may be found to deserve consideration in biology, is that of a cell or drop suspended in a liquid of varying density, for instance in the upper layers of a fluid (e.g. sea-water) at whose surface condensation is going on, so as to produce a steady density-gradient. In this case the normally spherical drop will be flattened into an oval form, with its maximum surface-curvature lying at the level where the densities of the drop and the surrounding liquid are just equal. The sectional outline of the drop has been shewn to be not a true oval or ellipse, but a somewhat complicated quartic curve. (Rice, Phil. Mag. Jan. 1915.) A more general case, which also may well deserve consideration by the biologist, is that of a charged bubble in (for instance) a uniform field of force : which will expand or elongate in the direction of the lines of force, and become a spheroidal surface in continuous transformation with the original sphere. V] OF ASYMMETRY AND ANISOTROPY 401 Our formula of equilibrium, or equation to an elastic surface, is P = jDg + (TjR + T' jR), where P is the internal pressure, jo^ 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 seek to account for, R, R' are known quantities; but all the other factors of the equation are 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 sohdification), the cyhndrical cell of Spirogyra^ the elhpsoidal yeast-cell, or (as we shall see in another chapter) even the egg of any bird. In using this formula hitherto we have taken it in a simphfied form, that is to say we have made several limiting assumptions. We have assumed that P was the uniform hydrostatic pressure, equal in all directions, of a body of hquid; we have assumed likewise that the tension T was due to surface-tension in a homogeneous hquid 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, jo„, was non-existent. Now in the case of a bird's egg the external pressure p^, 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 Spirogym, 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 inequahties in the internal pressure P, or else to inequahties 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 respect of the tension, and implicitly the molecular structure, of the cell- wall. Now' the former hypothesis is not impossible. Protoplasm is far from being a perfect fluid; it is the seat of various internal forces, sometimes manifestly polar, and it is quite possible that the forces, osmotic and other, which lead to an increase of the content of the 402 THE FORMS OF CELLS [ch. cell and are manifested in pressure outwardly directed upon its wall niay be unsymmetrical, and such as to deform 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 httle room for the development of unsymmetrical pressures; and in such a case as Spirogyra, where most part of the cavity is filled by watery sap, the conditions are still more obviously, or more nearly, those under which a uniform hydrostatic pressure should be displayed. 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 (TIR + T' jR) = a con- stant; 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, we are no longer dealing with a perfectly hquid surface film; but its departure from 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 manifested in the self-same layer which is still a part of the protoplasm of the cell, or in a layer which is completely differentiated into a distinct and separate membrane. As soon as, by secretion or adsorption, the molecular constitution of the surface-layer is altered, it is clearly conceivable that the alteration, or the secondary chemical changes which follow it, may be such as to produce an anisotropy, and to render the molecular forces less capable in one direction than another of exerting that contractile force by which they are striving to reduce to a minimum the surface area of the cell. A slight inequality in two opposite directions will produce the eUipsoid cell, and a great inequahty will give rise to the cylindrical cell. I take it therefore, that the cylindrical cell of Spirogyra, or any other cyhndrical 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 wall or * Indeed any non-isotropic stiffness, even though T remained uniform, would simulate, and be indistinguishable from, a condition of non-stiffness and non- isotropic T. v] OF ASYMMETRY AND ANISOTROPY 403 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 the form of concentric lamellae, annular and spiral striations, and the like. But there exists yet another heterogeneity, to help us account for the long threads, hairs, fibres, cylinders, which are so often formed. Carl NageH said many years ago that organised bodies, starch-grains, cellulose and protoplasm itself, consisted of invisible particles, each an aggregate of many molecules — he called them micellae; and these were isolated, or "dispersed" as we should say, in a watery medium. This theory was, to begin with, an attempt to account for the colloid state; but at the same time, the particles were supposed to be so ordered and arranged as to render the substance anisotropic, to confer on it vectorial properties as we say nowadays, and so to account for the polarisation of light by a starch- grain or a hair. It was so criticised by Biitschli and von Ebner that it fell into disrepute, if not oblivion; but a great part of it was true. And the micellar structure of wool, cotton, silk and similar substances is now rendered clearly visible by the same X-ray methods as revealed the molecular orientation, or lattice-structure, of a crystal to von Laue. It is now well known that the cell-wall has in many cases a definite structure which depends on molecular assemblages in the material of which it is com- posed, and is made visible by X-rays in the form of "diffraction patterns". The green alga Valonia has very large bubbly cells, 2-3 centimetres long, with cell-walls formed, as usual, of cellulose; this substance is a polysaccharide, with long-chain molecules some 500 Angstrom-units, or say 0-05 /x long, bound together sideways to form a multiple sheet or three-dimensional lattice. In the cell-wall of Valonia one set of chains runs round in a left-handed spiral, another forms meridians from pole to pole, and these two layers are superposed alternately to build the wall. Hemp has two layers, both running in right-handed spirals; flax two layers, crossing and recrossing in spirals of opposite sign. Even the cytoplasm and its contents seem to be influenced by molecular ''lignes directrices,'''' corresponding to the striae of the cell-wall. Analogous but still more complicated results of molecular structure are to be found in wool, cotton and other fibres*. * Cf. R. D. Preston, Phil. Trans. (B), ccxxiv, p. 131, 1934: Preston and Astbury, Proc. R.S. (B), cxxii, pp. 76-97, 1937; and many other important papers by Astbury, van Iterson, Heyn, and others. We are brought by them to a borderland 404 THE FORMS OF CELLS [ch. 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 cy Under 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 Aclilya or in the young zygospore of Spirogyra, we 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 cyhndrical 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 aniso- tropy of its envelope, which when more developed will convert the ovoid into a cyhnder. We may also notice that a truly cyhndrical cell is comparatively rare, for in many cases what we call a cylindrical cell shews a distinct bulging of its sides; it is not truly a cyhnder, but a portion of a spheroid or ellipsoid. Unicellular organisms in general — protozoa, unicellular crypto- gams, various bacteria and the free isolated cells, spores, ova, etc. of higher organisms — are referable for the most part to a small number of typical forms ; but there are many others in which either no symmetry is to be recognised, 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 aberrant Infusoria. We shall return to the consideration of these; but in the meanwhile it will suffice to say (and to repeat) that, inasmuch as their surfaces are not equihbrium-surfaces, so neither are the Uving cells themselves in any stable equihbrium. On the contrary, they are in continual flux and movement, each portion of the between chemical and histological structure, where micellae and long- chain molecules enlarge and alter our conceptions not only of cellulose and keratin, but of pseudopodia and cilia, of bone and muscle, and of the naked surface of the cell. See L. E. R. Picken, The fine structure of biological systems, Biol. Reviews, xv, pp. 133-67, 1940. V] OF STABLE AND UNSTABLE EQUILIBRIUM 405 surface constantly changing its form, passing from one phase to another of an equihbrium which is never stable for more than a moment, and which death restores to the stable equilibrium of a sphere. The former class, which rest in stable equihbrium, must fall (as we have seen) into two classes — those whose equilibrium arises from liquid surface-tension alone, and those in whose con- formation some other pressure or restraint has been superimposed upon ordinary surface-tension. To the fact that all these 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. They vary according to varying physical conditions. Sometimes they do so in so regular and orderly a way that w^e intuitively explain them as "phases of a Hfe-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 call a typical or "characteristic" form or attitude. In the case of the smallest organisms, 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 little "cycle" of hfe; but it will be remarkably different in outward form according to the circum- stances under which we find it, or the histological treatment to which we subject it. Hence the pathological student, commencing the study of bacteriology, is early warned to pay little heed to differences of form, for purposes of recognition or specific identi- fication. Whatever grounds we may have for attributing to these organisms a permanent or stable specific identity (after the fashion of the higher plants and animals), we can seldom safely do so on the ground of definite and always recognisable form: we may often be inclined, in short, to ascribe to them a physiological (sometimes a "pathogenic") rather than a morpho- logical specificity. 406 THE FORMS OF CELLS [CH. Many unicellular forms, and a few other simple organisms, are spherical, and serve to illustrate in the simplest way the point at issue. Unicellular algae, such as Protococcus or Halisphaera, the innumerable floating eggs of fishes, the floating unilocular foraminifer Orbulina, the lavely green multicellular Volvox of our ponds, all these in their several grades of simplicity or complication are so many round drops, spherical because no alien forces have deformed or mis-shapen them. But observe that, with the exception of Volvox, whose spherical body is covered wholly and uniformly with minute ciha, all the above are passive or inactive forms; and in a "resting" or encysted phase the spherical form is common and general in a great range of unicellular organisms. Conversely, we see that those unicellular forms which depart jUBrkedly from sphericity — excluding for the moment the amoeboid forms and those provided with skeletons — are all cihate or flagellate. Ciha and flagella are sui generis ; we know nothing of them from the physical side, we cannot reproduce or imitate them in any non- hving drop or fluid surface. But we can easily see that they have an influence on form, besides serving for locomotion. When our httle Monad or Euglena develops a flagellum, that is in itself an indication of asymmetry or "polarity," of non-homogeneity of the little cell ; and in the various flagellate types the flagellum or its analogues always stand on prominent points, or ends, or edges of the cell — on parts, that is to say, where curvature is high and surface-tension may be expected to be low — for the product of surface-tension by mean curvature tends to be constant. Fig. 123. A flagellate "monad," Distigma proteus Ehr, After Saville Kent. The minute dimensions of a cilium or a flagellum are such that the molecular forces leading to surface-tension must here be under peculiar conditions and restraints; we cannot hope to understand them by comparison with a whip- lash, or through any other analogy drawn from a different order of magnitude. I suspect that a ciliary surface is always electrically charged, and that a point- charge is formed or induced in each cilium or flagellum. Just as we learn the properties of a drop or a jet as phenomena proper to their scale of magnitude, so some day we shall learn the very different physical, but v] OF CILIA AND FLAGELLA 407 microcosmic, properties of these minute, mobile, pointed, fluid or semi-fluid threads.* Cilia, like flagella, tend to occupy positions, or cover surfaces, which would otherwise be unstable; and often indeed (as in a trochosphere larva or even in a Rotifer) a ring of cilia seems to play the very part of one of Plateau's wire rings, supporting and steadying the semi-fluid mass in its otherwise unstable configura- tion. Let us note here (in passing) what seems to be an analogous phenomenon. Chitinous hairs, spines or bristles are common and characteristic structures among the smaller Crustacea, and more or less generally among the Arthropods. We find them at every exposed point or corner; they fringe the sharp edge or border of a limb; as we draw the creature, we seem to know where to put them in ! In short, they tend to occur, as the flagella do, just where the surface-tension would be lowest, if or when the surface was in a fluid condition. Of the other surfaces of Plateau, we find cylinders enough and to spare in Spirogyra and a host of other filamentous algae and fungi. But it is to the vegetable kingdom that we go to find them, where a cellulose envelope enables the cyUnder to develop beyond its ordinary limitations. The unduloid makes its appearance whenever sphere or cyhnder begin to give way. We see the transitory figure of an unduloid in the normal fission of a simple cell, or of the nucleus itself; and we have already seen it to perfection in the incipient headings of a spider's web, or of a pseudopodial thread of protoplasm. A large number of infusoria have unduloid contours, in part at least; and this figure appears and reappears in a great variety of forms. The cups of various Vorticellae (Fig. 124), below the ciliated ring, look like a beautiful series of unduloids, in every gradation of form, from what is all but cylindrical to all but a perfect sphere; moreover successive phases in their hfe-history appear as mere graded changes * It is highly characteristic of a cilium or a flagellum that neither is ever seen motionless, unless the cell to which it belongs is moribund. "I believe the motion to be ceaseless, unconscious and uncontrolled, a direct function of the chemical and physical environment "; tieorge Bidder, in Presidential Address to Section D, British Associqtion, 1927. Cf. also James Gray, Proc. R.S. (B), xcix, p. 398, 1926. 408 THE FORMS OF CELLS [CH. of unduloid form. It has been shewn lately, in one or two instances at least, that species of Vorticella may "metamorphose" into one another : in other words, that contours supposed to charac- terise species are not "specific" These VorticelUd unduloids are >^- Fig. 124. Various species of Vorticella. M Fig. 1 25. Various species of Salpingoeca. Fig. 126. Various species of Tintinnus, Dinohryon and Codonella. After Saville Kent and others. not fully symmetrical ; rather are they such unduloids as develop when we suspend an oil-globule between two unequal rings, or blow a bubble between two unequal pipes. For our Vorticellid bell hangs by two terminal supports, the narrow stallj to which it is attached below, and the thickened ring from which spring its circumoral cilia; and it is most interesting to see how, when the bell leaves OF VARIOUS UNDULOIDS 409 its stalk (as sometimes happens) and swims away, a new ring of cilia comes into being, to encircle and support its narrow end. Similar unduloids may be traced in even greater variety among other famihes or genera of the Infusoria. Sometimes, as in Vorticella itself, the unduloid is seen in the contour of the soft semifluid body of the hving animal. At other times, as in Salpingoeca, Tintinnus, and many other genera, we have a membranous cup containing the animal, but originally secreted by, and moulded upon, its semifluid hving 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, 127. Vaginicola. Fig. 128. FolUculina. as a domicile and shelter for the httle organism within. The mechanical explanation of the physicist (seeking after the "efficient," not the "final" cause) is that it owes its presence, and its actual conformation, to certain chemico-physical conditions: that it was inevitable, under the given conditions, that certain constituent substances present in the protoplasm should be drawn by molecular forces to its surface layer; that under this adsorptive process, the conditions continuing favourable, the particles accumulated and concentrated 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 pellicPe or membrane was inevitably bound, by molecular forces, to contract into 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 410 THE FORMS OF CELLS [CH. being also surfaces minimae areae, were available, the unduloid was manifestly the one permitted, and ipso facto caused, by the dimen- sions of the organism 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 see that the actual outhne of this or that particular unduloid is also a very subordinate matter, such as physico- chemical variants of a minor order would suffice to -bring about; for between the various undu- loids 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 hnes of unequal length. In many cases (of which' Fig. 129 is an example) we have a more or less 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 changes continually from one to another unduloid contour according to the movements of the animal.* We are deahng here with no stable equihbrium, but possibly with a subtle problem of ''stream-lines," as in the difficult but beautiful problems suggested by the form of a fish. But this whole class of cases, and of problems, we merely take note of here; we shall speak of them again, but their treatment is hard. In considering such series of forms as these various unduloids we are brought sharply up (as in the case of our bacteria or micrococci) against the biological concept of organic species. In the intense classificatory activity of the last hundred years it has come about that every form which is apparently characteristic, that is to Say which is capable of being described or portrayed, and of being * Doflein lays stress, in like manner, on the fact that Spirochade, unlike Spirillum, "ist nicht von einer starren Membran umhiillt," and that waves of contraction may be seen passing down its body. Fig. 129. Trachelo- phyllum. After Wreszniowski. V] OF SNOW-CRYSTALS 411 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 deserves "specific rank," or be "only a variety." And this secular labour is pursued in direct obedience to the precept of the Systema Naturae — ''ut sic in summa confusione rerum apparenti, summus conspiciatur Naturae ordo.'' In like manner the physicist records, and is entitled to record, his many hundred "species" of snow-crystals*, or of crystals of calcium carbonate. Indeed the snow-crystal illustrates to perfection how Nature rings the changes on every possible variation and permutation and combination of form: subject only to the condition (in this instance) that a snow-crystal shall be a plane, symmetrical, rectilinear figure, with all its external angles those of a regular hexagon. We may draw what we please on a sheet of "hexagonal paper," keeping to its lines; and when we repeat our drawing, kaleidoscope-fashion, about a centre, the stellate figure so obtained is sure to resemble one or another of the many recorded species of snow-crystals. And this endless beauty of crystalline form is further enhanced when the flakes begin to thaw, and all their feathery outlines soften. But regarding these "species" of his, the physicist makes no assumptions: he records them simpliciter; he notes, as best he can, the circumstances (such as temperature or humidity) under which each occurs, in the hope of elucidating the. conditions which determine their formation f; but above all, he * The case of the snow-crystals is a particularly interesting one; for their "distribution" is 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, and another day with myriads of another while at another time and place we may find species intermingled in all but inexhaustible variety. Cf. e.g. J. Glaisher, Illustrated London News, Feb. 17, 1855 Q.J. M.S. m, pp. 179-185, 1855; Sir Edward Belcher, Last of the Arctic Voyages, II, pp. 288-306 (4 plates), 1855; William Scoresby, An Account of the Arctic Regions, Edinburgh, 1820; G. Hellmann, Schneekrystalle, Berlin, 1893; Bentley and Hum phreys. Snow Crystals, New York, 1931; and the especially beautiful figures of Nakaya and Hasikura in Journ. Fac. Sci. Hokkaido, Dec. 1934. t Every snow-crystal tells, more or less plainly, the story of its own development. The cold upper air is saturated with water- vapour, but this is scanty and rarefied compared with the space in which snow- crystallisation is going on. Hence crystallisation tends to proceed only along the main axes, or cardinal framework, of the crystalline structure of ice ; in so doing it gives a visible picture or actual embodiment of the trigonal-hexagonal space-lattice, in the endless permutations and combinations of its constituent elements. 412 THE FORMS OF CELLS [ch. does not introduce 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 "per- manence of species" is dead and gone, yet a certain quasi-permanency is still connoted by the term. If a tiny foraminiferal shell, a Lagena for instance, be found hving to-day, and a shell indistinguishable from it to the eye be found fossil in the Chalk or some still more 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 milHons of years*. If the ancient forms be Hke to rather than identical with the recent, we still assume an unbroken descent, accompanied by the hereditary transmission of common characters and progressive variations. And if two identical forms be discovered at the ends of the earth, still (with occasional slight reservations on the score of possible "homoplasy") we build hypotheses on this fact of identity, taking it for granted that the two appertain to a common stock, whose dispersal in space must somehow be accounted for, its route traced, its epoch determined, and its causes discussed or discovered. In short, the naturaUst 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 charac- teristics which they present is due to the fact of their being related by descenf\'' But this great and valuable and even fundamental generahsation sometimes carries 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 Silurian 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. V] OF FORM AND SPECIES 413 becomes more manifest from the point of view of physical and mathematical analysis, and whose form is referable, or largely referable, to the direct action of a physical force. When we come to the minute skeletons of the Radiolaria we shall again find our- selves dealing with endless modifications of form, in which it becomes more and more difficult to discern, and at last vain and hopeless to apply, the guiding principle of affihation or '^phylogeny.'' Among the Foraminifera we have an immense variety of forms, which, 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 them are composite structures, formed by the successive imposition of cell upon cell, and these we shall deal with later on ; let us glance here at the simpler conformations exhibited by the single chambered or " monothalamic " genera, and perhaps one or two of the simplest composites. We begin with forms like Astrorhiza (Fig. 320, p. 703), which are large, coarse and highly irregular, and end with others which are minute and delicate, and which manifest a perfect and mathe- matical 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 equihbrium; 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 are especially characteristic of this group of animals, is a sign of unstable surface-equihbrium ; and we must therefore consider, or may at least suspect, that those forms whose shells indicate the most perfect symmetry and equilibrium have secreted these during periods when rest and uniformity of surface-conditions contrasted 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 414 THE FORMS OF CELLS [CH. parts of the ramified mass. When we look more closely at the arenaceous forms, we find the same thing true of them; they represent, in whole or part, approximations to the surfaces of equilibrium, spheres, cylinders and so forth. In Aschemonella we have a precise rephca of the calcareous Ramuliyia; and in Astrorhiza itself, in the forms distinguished by jiaturalists as A. crassatina, what is described as the " subsegmented interior*" seems to shew Fig. 130. Various species of Zagrew«. After Brady. 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 contact with the unyielding arenaceous covering. Passing to the typical calcareous Foraminifera, we have the most symmetrical of all possible types in the perfect sphere of Orbulina ; this is a pelagic organism, whose floating habitat gives it a field of ♦ Brady, Challenger Monograph, pi. xx, p. 233. V] OF HANGING DROPS 415 force of perfect symmetry. 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 that of an unduloid, or, hke that of a flask itself, an unduloid combined with a portion of a sphere. We do not know the circumstances under which the shell oi 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 re- volution is symmetrically formed. 4 5 6 Fig. 131. Roman pottery, for comparison with species oi Lagena. E.g., 1, 2, with L. sulcata; 3, L. orbignyana; 4, L. striata; 5, L. crenata; 6, L. stelligera. 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 Orhulina is a simple spherical drop, Lagena suggests to our minds a hanging drop, drawn out to a longer or shorter neck by * 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 fiacon que Ton remplit de nouvelle eau, on voit au bout d'une heure environ les animaux [Gromia dujardinii] se mettre en mouvement et commencer a grimper. Six heures apres ils tapissent Texterieur du flacon, de sorte que les plus eleves sont a trente-six ou quarante-deux millimetres du fond; le lendemain beaucoup d'entre eux, apres avoir atteint le niveau du liquide, ont continue a rarnper a sa surface, en se laissant pendre au-dessous comme certains mollusques gastero- podes." (F. Dujardin, Observations nouvelles sur les pretendus cephalopodes microscopiques, Ann. des Sci. Nat. (2), iii, p. 312, 1835.) 416 THE FORMS OF CELLS [ch. its own weight, aided by the viscosity of the material. Indeed the various hanging drops, such as Mr C. R. DarUng shews us, are the most beautiful and perfect unduloids, with spherical ends, that it is possible to conceive. A suitable hquid, a httle denser than water and incapable of mixing with it (such as ethyl benzoati), is poured on a surface of water. It spreads over the surface and gradually forms a hanging drop, approximately hemispherical; but as more liquid is added the drop sinks or rather stretches downwards, still adhering to the surface fihn; 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 Sreak it in two. At the moment of rupture, by the way, a tiny Fig. 132. Large "hanging drops" ot oil. After Darling. droplet js formed in the attenuated neck, such as we described in the normal division of a cyhndrical thread. The thin, fusiform, pointed, non-globular Lagenas are less easily explained. Surface-tension, which tends to keep the drop spherical, is overmastered here, and the elongate shape suggests the viscous drag of a shearing fluid*. To pass to a more highly organised class of animals, v^e 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 Pteropod. * Cf. G. I. Taylor, The formation of emulsions in definable fields of flow, Proc. R.S. (A), No. 858, p. 501, 1934. t Cf. Boas, Spolia Atlantica, 1886, pi. 6. V] OF RETICULATE OR WRINKLED CELLS 417 Many species of Lagena are complicated and beautified by a pattern, and some by the superaddition to the shell of plane extensions or "wings." These latter give a secondary, bilateral symmetry to the little shell, and are strongly suggestive of a phase or period of growth in which it lay horizontally on the surface, instead of hanging vertically from the surface-film : in which, that is to say, it was a floating and not a hanging drop. The pattern is of two kinds. Sometimes it consists of a sort of fine reticulation, with rounded or more or less hexagonal interspaces: in other cases it is produced by a symmetrical series of ridges or folds, usually longitudinal, on the body of the flask-shaped cell, but occasionally transversely arranged upon the narrow neck. The reticulated and folded patterns we may consider separately. The netted pattern is very similar to the wrinkled surface of a dried pea, or to the more regular wrinkled patterns on poppy and 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 flexibihty of the skin be also uniform, then the amount of foldings will be uniformly distributed over the surface. Little elevations and depressions will appear, regularly interspaced, and separated by concave or convex folds. These being of equal size (unless the system be otherwise perturbed), each one wiU tend to be surrounded by six others; and when the process has reached its limit, the intermediate boundary-walls, or folds, will be found converted into a more or less regular pattern of hexagons. To these symmetrical wrinkles or shrinkage-patterns we shall return again. But the analogy of the mechanical wrinkHng of the coat of a seed is but a rough and distant one; for we are deahng with molecular rather than with mechanical forces. In one of Darling's experiments, a little heavy tar-oil is dropped on to a saucer of water, over which it spreads in a thin film shewing 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 aU probability at work in a surface-film 418 THE FORMS OF CELLS [ch. of protoplasm covering the shell. As a physical phenomenon the actions involved are by no means fully understood, but surface- tension, diffusion and cohesion play their respective parts 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 conditions of viscous or frictional restraint. A case which (as it seems to me) is closely alhed to that of our foraminiferal shells is described by Quincke t, who let a film of chromatised gelatin 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, as fine as a Fig.^ 133. diffraction grating. If the surface thus folded or wrinkled be a cyhnder, or any other figure with one principal axis * This cellular pattern would seem to be related to the "cohesion figures" described by Tomlinson in various surface-films {Phil. Mag. 1861-70); to the "tesselated structure" on liquid surfaces described by James Thomson in 1882 {Collected Papers, p. 136); and (more remotely) to the tourhillons cellulaires of Benard, Ann. de Chimie (7), xxiii, pp. 62-144, 1901; (8), xxiv, pp. 563-566, 1911, Bev. gener. des Sci. xi, p. 1268, 1900; cf. also E. H. Weber, Mikroskopische Beo- bachtungen sehr gesetzmassiger Bewegungen welche die Bildung von Niederschlagen harziger Korper aus Weingeist begleiten, Poggend. Ann. xciv, pp. 447-459, 1855; etc. Some at least of TomHnson's cohesion-figures arise, according to van Mens- brugghe, from the disengagement of minute bubbles of gas, when a fluid holding gases in solution comes in contact with a fluid of lower surface-tension. The whole phenomenon is of great interest and various appearances have been referred to it, in biology, geology, metallurgy and even astronomy: for the flocculent clouds in the solar photosphere shew an analogous configuration. (See letters by Kerr Grant, Larmor, Wager and others, in Nature, April 16 to June 11, 1914; also Rayleigh, Phil. Mag. xxxii, p. 529, 1916; G. T. Walker, Clouds, natural and artificial,' i?oz/a/ Inst. 8 Feb. 1935; etc.) In many instances, marked by strict symmetry or regularity, it is very possible that the interference of waves or ripples may play its part in the phenomenon. But in the majority of cases, it is fairly certain that localised centres of action, or of diminished tension, are present, such as might be provided by dust-particles in the case of Benard's experiment (cf. infra, p. 503). t Quincke, Ueber physikalische Eigenschaften diinner fester Lamellen, Sitzungsb, Berlin. Akad. 1888, p. 789; Ueber ansichtbare Eigenschaften, etc., Ann. d. Physik, 1920, p. 653. Quincke found that "sehr kleine Menge fremder Substanz haben eine grosse Einfluss auf die Bildung der Schaumwande." V] OF FLUTED OR PLEATED CELLS 419 of symmetry, such as an ellipsoid or unduloid, the folds will tend to be related to the ax's of symmetry, and we may expect accordingly to find regular longitudinal, or regular transverse wrinkling. Now as a matter of fact we almost invariably find in Lagena the former condition: that is to say, in our ellipsoid or unduloid shell, 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 locaHsed at the end of the eUipsoid, or in the region where the unduloid merges into its spherical base*. In the latter region we often meet with a regular series of short longitudinal folds, as in the forms denominated L. semistriata. All these various forms of surface can be imitated, or precisely reproduced, by the art of the glass-blower; and they can be seen in a contracting bubble of saponin, though not in the more fluid soap-bubble. They remind one of the ribs or flutings in the film or sheath which splashes up to envelop a smooth pebble dropped into a hquid, as Mr Worthington has so beautifully shewn. 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 difiicult to see that there must be in general a tendency towards longitudinal puckering or "fluting" in the case of a thin- walled cyhndrical or. other elongated body, rather than a tendency towards transverse puckering, or "pleating." For let us suppose that some change takes place involving an increase of surface-tension in some small area of the curved wall, and leading therefore to an increase of pressure : that is to say let T become T + t, and P become P + p. Our new equation of equilibrium, then, in place of P = T/r -f- T/r', becomes T+t T+t P + P = —— + —J-' * Certain palaeontologists (e.g. Haeusler and Spandel) have asserted that in each family or genus the plain smooth-shelled forms are primitive and ancient, and that the ribbed and otherwise ornamented shells make their appearance at later dates in the course of advancing evolution (cf. Rhumbler, Foraminiferen der Plankton-Expedition, 1911, p. 21). If this were true it would be of fundamental importance: but this book of mine would not deserve to be written. 420 THE FORMS OF CELLS [ch. and by subtraction, p = tlr + tjr'. Now if r< r\ t/r > tjr'. Therefore, in ordei to produce the small increment of pressure p, it is easier to do so by increasing tjr than tjr' ; 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 "ripphng" of the walls takes place owing to small changes of surface-tension, and consequently of pressure, such corrugation is more Hkely 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 wrinkUng of the flask-shaped bodies of our Lagenae, and of the more or less cyhndrical cells of many other Foraminifera (Fig. 134), is in complete accord with the above con- siderations ; but nevertheless, we soon find that .our result is not a general one but is defined by certain hmiting 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 circum- ference, of the neck. We must accordingly take it into account in considering the bending moments at any point in this region of the shell-wall. And it is at once obvious that, in any portion of the narrow neck, flexure of a wall in a transverse direction will be very difficult, while flexure in a longitudinal direction will be compara- tively 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-sheW V] OF FLUTED OR PLEATED CELLS 421 tends to fold or wrinkle, longitudinally in its wider part and trans- versely or annularly in its narrow neck, is thus completely 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 Cam- panularia, we see that in one case the httle 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. In the other Campanularian the vesicles are long, narrow and tubular, and here a transverse folding Fig. 134. Nodosaria scalaria Batsch. 135. Gonangia of Canjpanularians. (a) C. gracilis; [b) C. grandis. After AUman. 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. An everted rim, or short neck, may arise in various ways apart from the phenomenon of the hanging drop. To make a "thistle- head" the glassblower blows a bubble, and from that another one; after blowing the latter up large and thin he crushes it to pieces, and melting down what is left of it he forms the rim. I take it that the neck or rim of the shell in Diffiugia is formed in an analogous way, in connection with the growth of a new individual at the mouth 422 THE FORMS OF CELLS [CH. of the first. There is a very neat expanded orifice in the cyst of Chromulina (Fig. 136); it is doubtless fashioned in just as simple a way, but how I know not. Passing from the sohtary flask-shaped cell of Lagena, but without leaving the Foraminifera, we find in Nodosaria, Rheophax or Sagrina constricted cyUnders, or successive unduloids, such as are repre- sented in Fig. 137. In some of these, as in the arenaceous genus Fig. 136. Flask-shaped shells or cysts, a, b, Chromulina and Deropyxis (Flagellata); c, Difflugia. tig. llil . Various species of Nodosaria, Rheophax. Sagrina. After Brady. Rheophax, we have to do with the ordinary phenomenon of a partially segmenting cylinder. But in others, the structure is not developed out of a continuous protoplasmic cyhnder, 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 which, after it solidified, another drop of protoplasm accu- mulated, 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 similar if not identical phenomenon*. * See Sillijnan's Journal, ii, p. 179, 1820; and cf. Plateau, op. cit. ii, pp. 134, 461. V] OF THE QUIET OF THE DEEP SEA 423 Fishes shew a vast though hmited variety of form; and some of the strangest shapes are found in the great depths of the ocean. Here, in unchanging temperature, in darkness save for a few phosphorescent rays, above all in unruffled stillness and eternal Fig. 138. Deep-sea fishes (Stomiatidae). a,Lamprotoxusflagellibarbis; b, Eustomias dactylobus; c, E. parri; d, E. schmidti; e, E. silvescens. After Tate Regan and Trewavas. calm*, the conditions of hfe are strange indeed. In deep-sea fishes length and attenuation are common characters of the body and of its parts. A barbel below the hp may grow to ten times the whole length of the fish ; it ends, commonly, in a httle bulb or blob ; it may give off threadlike branches, and these last slender filaments * In Overbeck's jet-experiments {supra, p. 394) the water into which the jet is led must first stand for many hours, till all internal movements and temperature- differences are eliminated. 424 THE FORMS OF CELLS [ch. are sometimes finely beaded* ; and slight differences in the beading and branching are said to characterise allied species of fish. Such a barbel looks hke a jet or branching stream of one fluid falhng through another. It may indeed be that in these quiet depths growth easily follows its fines of least. resistance, and that in the shaping of these pecufiar out- growths hydrodynamical and capillary forces are taking the upper hand. We have found it easy to illus- trate the sphere, the cy finder and the unduloid, three of the six "surfaces of Plateau," all with an endless wealth of illustration among the simplest of organisms. The plane we need hardly look for among the finite outlines of a fluid body; and the catenoid, also *a surface of zero mean curvature, can likewise only be a rare and transitory configuration. One last surface still remains, namely the nodoid; and there also remains one very common but most re- markable Protozoan configura- Fig. 139. Stentor, a cUiate infusorian: tion, that of the cifiate Infusoria, from Savile Kent. ii.i, ^.i, j. • ,.- r i. to the most characteristic leature of which we have not so far found a physical analogue. Here the curved contour seems to enter, re-enter, and disappear within the substance of the body, so bounding a deep and twisted space or passage, which merges with the fluid contents and vanishes within the cell, and is called by naturafists the "gullet." This very pecufiar and compficated structure is only kept in equifibrium, and in existence, by the constant activity of cifia over the general surface * See, for instance, C. Tate Regan and E. Trewavas on The Stomiatidae of the Dana Expedition, 1930; W. Beebe, Deep-sea Stomiatoids, Copeia, Dec. 1833. V] OF THE CILIATE INFUSORIA 425 of the body and very especially in the said gullet or re-entrant portion of the surface. Now we have seen the nodoid to be a curved surface, re-entering on itself and endless; no method of support, by wire-rings or otherwise, enables us to construct or reahse more than a small portion of it. But the typical cihate, such as Para- moecium, looks just hke what we might expect a^ nodoid surface to be, if we could only realise it (or a single segment of it) in a drop of fluid, and imagine it to be kept in quasi-equiUbrium by continual cihary activity. I suspect, indeed, that here is nothing more, and* nothing less, than a partial realisation of the nodoid itself; that the so-called gullet is but the characteristic inversion or "kink" in that curve; and that the ciHa, which normally clothe the surface and always line the gullet, are needed to reahse and to maintain the unstable equihbrium of the figure. If this be so — it is a suggestion and no more — we shall have found among our simple organisms the complete realisation, in varying abundance, of each and all of the six surfaces of Plateau. On each and all of them we have a host of beautiful "patterns" of various sorts; all of them so beautiful and so symmetrical that they ought to be capable of geometric representation — and all waiting for their interpreter ! From all these configurations, which the law of minimal area controls and dominates, Aynoeba stands aloof and alone. The rest are all figures of equilibrium, unstable though it may sometimes be. But Amoeba is the characteristic case of a fluid surface without an equihbrium; it is the very negation of stabihty. In composition it is neither constant nor homogeneous ; its chemistry is in constant flux, its surface energies vary from here to there, its fluid substance is drawn hither and thither; within and without it is never still, be its motions swift or be they slow. The heterogeneity of its system points towards a maximal surface-area, .rather than a minimal one ; only here and there, in small portions of its hetero- geneous substance, do we see the rounded contours of a fluid drop, in token of temporary equihbrium. Only when its heterogeneous reactions quieten down and the little living speck enters on its "resting-stage," does the protoplasmic body withdraw itself into a sphere and the law of area minima come into its own. Physically analogous is the case of such comphcated pseudopodia, or " axopodia ", as we find among the Foraminifera and Hehozoa : where the whole 426 THE FORMS OF CELLS [ch. fabric is in a flux, and currents flow and granules are carried hither and thither. Here again there is no statical equilibrium; but surface tension varies, as does the chemistry of the protoplasm, from oiie spot to another. The great oceanic group of the Radiolaria, and the highly com- plicated skeletons which they construct, give us many beautiful illustrations of physical phenomena, among which the efl'ects of surface-tension are as usual prominent. But we shall deal later on with these little skeletons 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 this heterogeneous protoplasm are more fluid portions, and a similar surface-tension causes these also to assume the form of spherical "vacuoles," or of little clear drops within the big one; unless indeed they become numerous and. closely packed, in which case they run together and constitute a "froth," such as we shall study in the next chapter. One or more of such clear spaces may be what is called a "con- tractile vacuole " : that is to say, a droplet whose surface-tension is in unstable equilibrium and is apt to vanish altogether, so that the definite outHne of the vacuole suddenly disappears*. Again, within the protoplasm are one or more nuclei, whose own surface- tension draws them in turn into the shape of spheres. Outwards through the protoplasm, and stretching far beyond the spherical surface of the cell, run stiff linear threads of modified protoplasm, reinforced in some cases by delicate siliceous needles. In either case we know little or nothing about the forces which lead to their production, and we do not hide our ignorance when we ascribe their development to a "radial polarisation" of the cell. In the case of the protoplasmic filament, we may (if we seek for a hypo- thesis) suppose that it is somehow comparable to a viscid stream, or "liquid vein," thrust or spirted out from the body of the cell. But when it is once formed, this long and comparatively rigid filament is separated by a distinct surface from the neighbouring * 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. 295 (footnote), it is probably no more than a physical con- sequence of the different conditions of existence in fresh and in salt water. V] OF THE SUN-ANIMALCULES 427 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. Catting these tensions respectively T/^, T^^,, and T^^^,, equilibrium will be attained when the angle of contact between the fluid T — T protoplasm and the filament is such that cos a = -^~r^ — ~' - 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 all but identical wuth 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 this surface-tension is symmetrical around the filament, the latter will be pulled equally in all directions; in other words the filament 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 (cf. p. 710). In the simpler Radiolaria. the spherical form of the entire organism is equally well marked; and here, as also in the more complicated 428 THE FORMS OF CELLS [ch. Heliozoa (such as Actinosphaerium), the organism is apt to be differentiated into layers, so constituting sphere within sphere, whose inter-surfaces become the seat of adsorption, and the locus of skeletal secretion. One layer at least is close-packed with vacuoles, forming an "alveolar meshwork," with the configura- tions of which we shall attempt in another chapter to correlate certain characteristic types of skeleton. In Actinosphaerium the radial filaments pass through the outer layer, and seem to rest on but do not penetrate the layer below; this must happen if the surface-energy between the one plasma-layer and the other be less than that between the filament and the water around*. 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 dehcate film which shews an undu- latory or ripphng motion. It is a surface of revolution, and as it maintains itself in equihbrium (though a somewhat unstable and fluctuating one) it must be, under the restraining 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 shrink or shrivel 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 ot flagellum, in constant move- ment; and by the action of this flagellum, and doubtless in part also by the intrinsic vibrations of the collar itself, there is set up a constant steady current in the surrounding water, whose direction would seem to be such that it passes up the outside of the collar, down its inner side, and out in the middle in the direction of the flagellum; and there is a distinct eddy, in which foreign particles tend to be caught, around the peripheral margin of the collar f. * Cf. N. K. Koltzoff, AnaL Anzeiger, xli, p. 190, 1912. t The very minute size of Codosiga, whose collar and flagellum measure about 30-40 /i,, and of all such collar-cells, make the apparently complex current-system all the harder to comprehend. Cf. G. Lepage, Notes on C. botrytis, Q.J. M.S. LXix, pp. 471-508, 1925. V] OF COLLARED CELLS 429 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. 390 : this latter condition being the outcome of a definite instability, marking the close of a period of equihbrium. But the vibratile collar of Codosiga is in "a steady state," its equihbrium, such as it is, being constantly renewed and per- petuated, like that of a juggler^s pole, by the motions of the system. Somehow its existence is due to the current motions and to the traction exerted upon it through the friction of the stream which is constantly passing by. In short, I think that it is formed very much in the same way as the cup-hke ring of streaming ribbons, which we see fluttering and vibrating in the air-current of „. ,,^ a ventiktmg fan. If we turn once more to Mr Worthington's Study of Splashes, we may find a curious suggestion of analogy in the beautiful craters encirchng 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. Another exceptional form of cell, and beautiful manifestation of capillarity, occurs in Trypanosomes, those tiny parasites of the blood which are associated with sleeping-sickness and certain other dire maladies of beast and man. These minute organisms consist of elongated sohtary cells down one side of which runs a very delicate frill, or "undulating membrane," the free edge of which is seen to be sHghtly thickened, and the whole of which undergoes rhythmical and beautiful wavy movements. When certain Trypano- somes 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 cihum 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 Hkewise * Cf. Doflein, Lehrbuch der Protozoenkunde, 1911, p. 422. 430 THE FORMS OF CELLS [CH. devoid of an undulating membrane, but are provided with a long free flagellum*. It is a plausible assumption to suppose that, as the flagellum waves about, it comes to lie near and parallel to the body of the cell, and that the frill or undulating membrane is formed by the clear, fluid protoplasm of the surface layer springing up in Fig. 141. A, Trichomonas muris Hartmann; B, Trichomastix serpentis DobeU; C, Trichomonas angusta Alexeieff. After Kofoid. Fig. 142. A "Trypanosome. a film to run up and along the flagellum, just as a soap-film would form under similar circumstances. This mode of formation of the undulating membrane or frill appears to be confirmed by the appearances shewn in Fig, 14L 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: * Cf. Minchin, Introduction to the Study of the Protozoa, 1914, p. 293, Fig. 127. V] OF UNDULATING MEMBRANES 431 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 b. Moreover, this mode of formation has been actually witnessed and described, though in a somewhat exceptional case. The little flagellate monad Herpetomonas is normally desti- tute of an undulating membrane, but possesses a single long terminal flagellum. According to Prof. D. L. Mackinnon, the cytoplasm in a certain stage of growth becomes somewhat "sticky," a phrase which we may in all probabihty 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 „ , ., . ^ ,, . . , , Fig. 143. ^er^e^omonas assuming tail (so to speak), giving a rounded or the undulatory membrane sometimes annular form to the organism, of a Trypanosome. After such as has also been described in certain ^- ^' ^^a^^^^^non. species or stages of Trypanosomes. But again, the long flagellum, if it get bent backwards upon the body, tends to adhere to its surface. "Where the flagellum was pretty long and active, its efforts to continue movement under these abnormal conditions resulted in the gradual lifting up from the cytoplasm of the body of a sort of pseudo-undnlsitmg membrane (Fig. 143). The move- ments of this structure were so exactly those of a true undu- lating membrane that it was difficult to beheve one was not deahn^ with a small, blunt Trypanosome"*. This in short is a precise * Cf. C. A. Kofoid and Olive Swezy, On Trichomonad flagellates, etc., Pr. Amer. Acad, of Arts and Sci. li, pp. 289-378, 1915. Also C. H. Martin and Muriel Robertson, Q.J. M.S. lvii, pp. 53-81, 1912. t D. L. Mackinnon, Herpetomonads from the alimentary tract of certain dungflies, Parasitology, iii, p. 268, 1910. 432 THE FORMS OF CELLS [ch. description of the mode of development which, from theoretical considerations alone, we should conceive to be the natural if not the only possible way in which the undulating membrane could come into existence. There is a genus closely alhed to Trypanosoma, viz. Trypanoplasma, which possesses one free flagellum, together with an undulating membrane; and it resembles the neighbouring genus Bodo, save that the latter has two flagella and no undulating membrane. In lijce manner, Trypanosoma so closely resembles Herpetomonas that, when individuals ascribed to the former genus exhibit a free flagellum only, they are said to be in the "Herpetomonas stage." In short, all through the order, we have pairs of genera which are presumed to be separate and distinct, viz^ Trypanosoma-Herpeto- monas, 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 essen- tially one and the same. The undulating membrane of a Trypanosome, then, according to our interpretation of it, is a hquid film and must obey the law of constant mean curvature. It is under curious limitations of freedom: for by one border it is attached to the comparatively, motionless body, while its free border is constituted by a flagellum which retains its activity and is being constantly thrown, like the lash of a whip, into wavy curves. It follows that the membrane, for every alteration of its longitudinal curvature, must at the same instant become curved in a direction perpendicular thereto; it bends, not as a tape bends, but with the accompaniment of beautiful but tiny waves of double curvature, all tending towards the estabhshment of an " equipotential surface", which indeed, as it is under no pressure on either side, is really a surface of no curvature at all; and its characteristic undulations are not originated by an active mobihty of the melhbrane but are due to the molecular tensions which produce the very same result in a soap-film under similar circumstances. Some of the larger Spirochaetes possess a structure so like to the undulating membrane of the Trypanosomes that it has led some persons to include these peculiar allies of the bacteria among the flagellate protozoa; but it would seem (according to the weight of V] OF UNDULATING MEMBRANES 433 evidence) that the Spirochaete membrane does not undulate, and possesses no thickened border or marginal filament (Randfade)*. 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. 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 winds spirally round the axis, and then dipping the whole into a soapy solution. Fig. 144, Dinenympha gracilis Leidy. A pecuUar type is the flattened spiral of Dinenympha '\ , which reminds us of the cylindrical spiral of a Spirillum among the bacteria. Here we have a symmetrical figure, whose two opposite surfaces each constitute a surface of constant mean curvature; it is evidently * For a discussion of this obscure lamella, and of the crista which seems to correspond with it in other species, see Doflein, Probleme der Protistenkunde, ii, Die Natur der Spirochaeten, Jena, 1911; see also ChfFord Dobell, Arch.f. Protisten- kunde, 1912. t Leidy, Parasites of the termites, Journ. Nat. Sci., Philadelphia, viii, pp. 425- 447, 1874-81; cf. Savile-Kent's Infusoria, ii, p. 551. 434 THE FORMS OF CELLS [ch. a figure of equilibrium under certain special conditions of restraint. The cylindrical coil of the Spirillum, on the other hand, is a surface of constant mean curvature, and therefore of equilibrium, as truly, and in the same sense, as the cylinder itself. A very beautiful " saddle-shaped " surface, of constant mean curvature, is to be found in the little diatom Cmnpylodiscus, and others, a httle more complicated, in the allied genus Surirella*. These undulating and helicoid surfaces are exactly reproduced among certain forms of spermatozoa. The tail of a spermatozoon consists normally of an axis surrounded by clearer and more fluid protoplasm, and the axis sometimes splits up into two or more slender filaments. To surface-tension operating between these and the surface of the fluid protoplasm (just as in the case of the flagellum of the Trypanosome), I ascribe the formation of the undulating membrane which we find, for instance, in the spermatozoa of the newt or salamander; and of the helicoid membrane, wrapped in a far closer and more beautiful spiral than that which we saw in Spirochaeta, which is characteristic of the spermatozoa of many birds. The undulatory membrane which certain ciliate infusoria exhibit is, seemingly, a difl'erent thing. It is not based on a single marginal flagellum, but consists of a row of fine ciha fused together. The membrane can be broken up by certain reagents into fibrillae, and — what is more remarkable — a touch of the micro-dissection needle may split it into a multitude of cilia, all active but beating out of time; a moment more and they unite again, all but dis- appearing from view as they fuse into the optically homogeneous membrane. They unite as quickly and as intimately as though they were so many liquid jets, and they manifestly "partake of fluidity." Neither they, nor cilia in general, have received, nor seem hkely to receive, a simple explanation f. Nevertheless, we may see a httle hght in the darkness after all. It would be overbold to seek for every form of living cell a parallel configuration due to simple capillary forces, as manifested in drop or bubble or jet. And yet, if the simple cases of sphere or cyhnder be the beginning of the story, they assuredly are not the end. The * Van Heurck, Synopsis des Diatomees de Belgique, pis. Ixxiv, 6; Ixxvii, 4. t H. N. Maier, Der feinere Bau der Wimperapparate der Infusorien, Arch. f. Protistenk. ii, p. 73, 1903; R. Chambers and J. A. Dawson, Structure of the undulating membrane in the ciliate Blepharisma, Biol. Bull, xlviii, p. 240, 1925. V] OF FLAGELLATE CELLS 435 pointed and flagellate cell of a Monad, one of the least and com- monest of micro-organisms, is far removed from a simple "drop," and all its characters, to the microscopist's eye, are both generally and Specifically those of a hving thing. But a drop of water falhng through an electric field, as in a thunderstorm, is found to lengthen out to three or four times as long as it is broad; and then, if the strength of the field increase a httle, the prolate drop becomes unstable, it grows spindle-shaped, and suddenly from one of its two pointed spindle-ends (the positive end especially) a long and slender filament shoots out, to the accompaniment of an electrical discharge. We need not assert that the phenomena are identical, nor that th^ forces in action are absolutely the same. Yet it is no small thing to have learned that the pecuhar conformation of the Httle flagellate Monad has its analogue in an electrified drop, and is not unique after all*. 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, of 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 seeins to be incapable of explanation by theories of surface-tension. In all the mammalia (save a very few) the red blood-corpuscles are flattened circular discs, dimpled in upon their two opposite sides. This configuration closely resembles that of an india-rubber ball when we pinch it tightly between finger and thumb •)". The form of the corpuscle is symmetrical ; it is a sohd of revolu- tion, but its surface is not a surface of constant mean curvature. From the surface-tension point of view, the blood-corpuscle is not a surface of equilibrium ; in other words, it is not a fluid drop poised * Cf. W. A. Macky, On the deformation of water-drops in strong electric fields, Proc. B.S. (A), cxxxiii, pp; 565-587, 1931. t On this analogy we might expect the double concavity to pass, with no great difficulty, into the single hollow of a cup or bell, and such a shape the blood- corpuscles are said sometimes to assume. Cf. Weidenreich, Arch. f. mikr. Anat. Lxvii, 1902; and cf. Clerk-Maxwell on "dimples" in Tr. R.S.E. xxvi, p. 11, 1870. 436 THE FORMS OF CELLS [ch. in another liquid. Some other force or forces must be at work to conform it, and the simple effect of mechanical pressure is excluded, because the corpuscle exhibits its characteristic shape while floating freely in the blood. It has been suggested that the corpuscle is perhaps comparable to a sohd of revolution described about one of Cay ley's equipotential curves*, such as we have spoken of briefly on p. 318. Were the corpuscle a sphere, or a thin plate, a gas diffusing inwards would reach all parts equally soon ; but the surface would be small in the one case and -^_C ^ ^ the volume in the other. In so far as the corpuscle resembles or approaches the equipotential form, we might look on it as a com- Fiff 14.^ ^" ^ promise; but however advan- tageous such a shape might be, and however interesting physiologically, we should be as far as ever from understanding how it was produced. In all other vertebrates, from fishes to birds, sluggish or active, warm-blooded or cold, the blood-corpuscles have the simpler form of a flattened oval disc, with somewhat sharp edges and eUipsoidal surfaces, and this again is manifestly not a surface of fluid equilibrium. But there is nothing to choose between the one type and the other in the way of physiological efficiency, nor any apparent need for a refinement of adaptive form in either of them. Two facts are noteworthy in connection with the form of the mammalian blood-corpuscle. In the first place its form is only maintained, that is to say it is only in equilibrium, in specific relation to the medium in which it floats. If we add water to the blood, the corpuscle becomes a spherical drop, a true surface of minimal area and stable equiUbrium; if, conversely, we add a little salt, or a drop of glycerine, the corpuscle shrinks, and its surface becomes puckered and uneven. So far, it merely obeys the laws of diffusion; but the phenomenon is more complex than this|. For the spherical form is assumed just as well in various isotonic solutions, leaving ♦ Cf. H. Hartridge, Journ. Physiol, lii, p. Ixxxi, 1919-20; Eric Ponder, Journ. Gen. Physiol, ix, pp. 197-204, 625-629, 1925-26. I Cf. A. Gough, On the assumption of a spherical form by human blood- corpuscles, Biochem. Journ. xvni, p. 202, 1924. V] OF BLOOD CORPUSCLES 437 the volume unchanged; a httle ammonium oxalate impedes or inhibits the change of form, a little serum brings the spherical corpuscles back to biconcave discs again. We are no longer deaUng with simple diffusion, but with phenomena of a very subtle kind. Secondly, the form of the corpuscle can be imitated artificially by means of other colloid substances. Many years ago Norris made the interesting observation that drops of glue in an emulsion assumed a biconcave form closely resembling that of the mammahan cor- kpuscles*; the glue was impure and doubtless contained lecithin. Waymouth Reid made similar emulsions of cholesterin oleate, in which the same conformation of the drops or particles is beautifully shewn; and Emil Hatschek has made somewhat similar biconcave bodies by dropping gelatine containing potassium ferrocyanide into copper sulphate or a tannin solution. Here Hatschek beheves that his biconcave drops are half -formed vortex-rings, arrested by the formation of a semi-permeable membrane ; but the explanation does not seem to fit the blood-corpuscle. The cholesterin bodies in Waymouth Reid's experiment are such as have a place of their own among Lehmann's "fluid crystals "f; and it becomes at least con- ceivable that obscure forces akin t6 those of crystalHsation may be playing their part along with surface-energy in these strange but famihar conformations. The case is a hard one in every way. From the physiological point of view it is difficult and coinplex^ enough. For the surface of the corpuscle is equivalent to a semi- permeable membrane t, through which certain substances pass freely but not others — for the most part anions and not cations §; and accordingly we have here in life a steady state of osmotic inequih- brium, of negative osmotic tension within, and to this comparatively simple cause the imperfect distension of the corpus(5le may be due. * Proc. M.S. ;xn, pp. 251-257, 1862-63. t Cf. {int. al.) Lehmann, Ueber scheinbar lebende Kristalle und Myelinformen, Arch.f. Entw. Mech. xx\^, p. 483, 1908; Ann. d. Physik, xliv, p. 969, iai4. X That no "true membrane" exists has long been known; cf. [int. al.) Rohring, KoU. Chem. Beihefte, viii, pp. 337-398, 1916. On the other hand the surface of the corpuscle is defined by a monolayer, and very probably by the still more stable condition of two "interpenetrating" monolayers, a proteid and a lipoid. Cf. Eric Ponder, Phys. Rev. xvi, p. 19, 1936; and on "interpenetration," Schulman and Rideal in Proc. M.S. (B), cxxii, pp. 29-57, 1937. § Cf. Hamburger, Z. f. physikal. Chem. lxix, p. 663, 1909; Pfluger's Archiv, 1902, p. 442; etc. 438 THE FORMS OF CELLS [ch. Whatever the forces are which make and keep the man's corpuscle a dimpled disc and the frog's a flattened ellipsoid, they seem to be of a powerful kind. When we submit either to great hydrostatic pressure, it tends to become spherical at last, the natural result of uniform pressure over its whole surface; but the pressure necessary to bring this result about is very great indeed*. Since the form of the blood-corpuscle cannot, then, be rated as a figure of equili- brium, we must be content to regard it as a "steady state"; and this, moreover, is all we can say of its physico-chemical conditional The red blood-corpuscle, especially the non-nucleated one, is in no ordinary sense alive'f. It has no power of movement, of reproduction or of repair; it is a mere haemoglobin-freighted drop of protein; its own metabolism, apart from its alternate give and take of oxygen, is slight indeed or absent altogether. But all the same, chemical change is continually going on; anions (Uke HCO3) pass freely through its walls, simple cations (like Na, K) find it imper- meable; and so, between plasma and corpuscles the conditions are fulfilled for that steady osmotic state known as a "Donnan equi- Kbrium." Somehow, but we know not how, a steady state is maintained alike in the corpuscle's osmotic equihbrium and in its form. In mammalian blood, the running together of the round biconcave corpuscles into "rouleaux" gives a well-known and characteristic picture. When cold, rouleaux are formed slowly, in warmed plasma they form quickly and well, in salt-solution they do not form at all. * The whole phenomenon would become simple and mechanical if we might postulate a stiffer peripheral region to the corpuscle, in the form (for instance) of an elastic ring. Such an annular stiffening, like the " collapse-rings" which an engineer inserts in a boiler or the whalebone ring which a Breton fisherman fits into his beret, has been repeatedly asserted to exist; by Dehler, Arch.f. mikr. Anat. xlvi, 1895; by Meves, ibid. Lxxvii, 1911; and especially by J. Riinnstrom, Was bedingt die Form und die Formveranderungen derSaiigetiererythrocytcn, Arch. f. Entw. Mech. i, pp. 391-409^,1922. It has been denied at least as often; but the remarkable statement has been lately made that in a corpuscle which has been swollen up and then brought back to its biconcave form, the dimples reappear on the same sides as before: apparently in "strong evidence for some sort of fixed cellular structure"; see R. F. Furchgott and Eric Ponder in Jonrn. Exp. Biol. xvii. pp. 30-44, 117-127, 1940. See also, on the whole subject, Eric Ponder, The Mammalian Red Cell, Berlin, 1934. t Cf. A. V. Hill, Trans. Faraday Soc. xxvi, p. 667, 1930; Proc. R.S. (B), 1930; K. R. Dixon, in Current Sci. vii, p. 169, 1938; etc. V] OF CERTAIN OSMOTIC PHENOMENA 439 The phenomenon, though a purely physical one, is none too clear. There is a difference of electrical potential between corpuscles and plasma, and the charged corpuscles tend to repel one another; but they also tend to adhere together, all the, more when they meet broadside on, whether by actual stickiness or through surface- energy. The attractive forces then overcome the repulsive, and the rouleau is formed. But if the potential be reduced, and mutual repulsion reduced with it, then the corpuscles stick together just as they happen to meet; rouleaux are no longer formed, and ordinary "agglutination" takes place. Whatever be the precise nature of the phenomenon, the number of rouleaux and the mean number of Fig. 146. Sperm-cells of Decapod Crustacea (after Koltzoff). a, Inachus scorpio; b, Galathea sqimmijera; c, do. after maceration, to shew spiral fibrillae. corpuscles in each is found, after a given time, to obey a certain law (Smoluchowsky's Law), defining the number of contacts of floating bodies under ordinary physical conditions*. The sperm-cells of the Decapod Crustacea exhibit various singular shapes. In the crayfish they are flattened cells with stiff curved processes radiating outwards Hke St Catherine's wheel; in Inachus there are two such circles of stiff processes ; in Galathea we have a still more comphcated form, with long and slightly twisted processes. * Smoluohowsky, Ztschr. f. physik. Chemie, xcix, p. 129, 1917; Eric Ponder, On Rouleaux-formation, Q. Journ. Exp. Physiol, xvi, pp. 173-194, 1926. 440 THE FORMS OF CELLS [ch. In all these cases, just as in the case of the blood-corpuscle, the structure alters, and finally loses, its characteristic form when the constitution of the surrounding medium is changed*. Here again, as in the blood-cOrpuscle, we have to do with the important force of osmosis, manifested under conditions similar to those of Pfeffer's classical experiments on the plant-cell f. The surface of the cell acts as a semi-permeable membrane, 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." Again, in the hen's egg a delicate yolk-membrane separates the yolk from the white. The morphologist looks on it but as the cell-wall of a vast yolk-laden germ-cell; the physiologist sees in it a semi- permeable membrane, the seat of many complex activities. The end and upshot of these last is that a steady difference of osmotic pressure, the equivalent of some two atmospheres, is maintained between yolk and white ; and yet there is no current flowing through. Somewhere or other in the system there is a constant metabolic flux, a continuous liberation of energy, a continual doing of work, all leading to the maintenance of a steady dynamical state, which is not " equilibrium {." In 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 corresponding to definite concentrations of the salt; and, as a further and final proof that the phenomenon is entirely physical, it is found that other salts produce an identical effect when their concentration is proportionate to their molecular weight, and whatever identical effect is produced * Cf. N. K. KoltzofF, Studien iiber die Gestalt der Zelle, Arch. f. mikrosk. Anat. Lxvii, pp. 365-572, 1905; Biol. Centralhl. xxm, pp. 680-696, 1903; xxvi, pp. 854- 863, 1906; XLvm, pp. 345-369, 1928; Arch. f. Zellforschung, u, pp. 1-65, 1908; vn, pp. 344^23, 1911; Anat. Anzeiger, xli, pp. 183-206, 1912. t W Pfeffer, Osmotische Untersuchungen, Leipzig, 1877. X Cf. J. Straub, Der Unterschied in osmotischer Konzentration zwischen Eigelb und Eiklar, Rer. Trav. Chim. du Pays-Bas, XLvni, p. 49, 1929. V] OF THE SPERM-CELLS OF DECAPODS 441 by various salts in their respective concentrations, a similarly identical effect is produced when these concentrations are doubled or otherwise proportionately changed. KNO Fig. 147. Sperm-cells oi Inachtis, as they appear in saline solutions of varying density. After KoltzoflF. Thus the following table shews the percentage concentrations of certain salts necessary to bring the cell into the forms a and c of Fig. 147; 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 c, compared with that which gives rise to the all but spherical form of a. % concentration of salts in which the sperm- cell of Inachus assumes the form of r ^ a c Sodium chloride 0-6 1-2 Sodium nitrate 0-85 1-7 Potassium nitrate 10 20 Acetic acid 2-2 4-5 Cane sugar 50 100 If we look then upon the spherical form of this cell as its true condition of symmetry and of equiUbrium, 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 inequahty 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 442 THE FORMS OF CELLS [ch. complicated arrangement of more or less rigid or elastic structures in the wall of the cell, which, like the wire framework in Plateau's experiments, restrain and modify the forces acting on the drop. In one form of Plateau's experiments, instead of supporting his drop on rings or frames of wire, he laid upon its surface one or more elastic coils; and then, on withdrawing oil from the centre of his globule, he saw its uniform shrinkage counteracted by the spiral springs, with the result that the centre of each elastic coil seemed to shoot out into a prominence. Just such spiral coils are figured (after "■''• ""-A^rKXcff.' "'""'"■ Koltzoff) in Fig. 148*; and they may be regarded as closely akin to those local thickenings or striations, spiral ard other, which are common in vegetable cells. Physically speaking, the protoplasmic colloids are neither simple nor uniform. We begin by thinking of our cell as a drop of a homo- geneous fluid and on this bold simplifying assumption we account for its form to a first, and often to a near, approximation. For the cell is largely composed of fluid "hydrosols," which are still fluid however viscous they may be, and still tend towards rounded, drop-hke configurations. But it has also its "hydrogels," which shew a certain tenacity, a certain elasticity, a certain reluctance to let their particles move on one another; and of these are formed the scarce distinguishable fibrillae within a host of highly specialised cells, the elastic fibres of a tendon, the incipient cell- walls of a plant, the rudiments of many axial and skeletal structures. The cases which we have just dealt with lead us to another consideration. In a semi-]:^ermeable membrane, through which water passes freely in and out, the conditions of a liquid surface are greatly modified; in the ideal or ultimate case, there is neither surface nor surface-tension at all. And this would lead us some- * As Bethe points out (Zellgestalt, Plateausche Fliissigkeitsfigur und Neuro- fibrille, Armt. 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. V] OF THE PROTOPLASMIC COLLOIDS 443 what to reconsider our position, and to enquire whether the true surface-tension of a Hquid 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 his way, paying little heed to what is no great difficulty. For in surface-tension we have the production of a film with the properties of an elastic membrane, and with the special peculiarity that contraction continues with the same energy however far the process may have already gone ; while the ordinary elastic membrane contracts to a certain extent, and contracts no more. But within wide limits the essential phenomena are the same in both cases. Our fundamental equations apply to both cases alike. And accordingly, so long as our purpose is morpho- logical, so long as what we seek to explain is regularity and definite- ness 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. CHAPTER VI A NOTE ON ADSORPTION An important corollary to, or amplification of, the theory of surface-tension is to be found in the chemico-physical doctrine of Adsorption; which means, in a word, the concentration of a substance at a surface, by reason of that surface-energy of which we have had so much to say*. Charcoal, with its vast internal surface-area of carbonised cell- walls, is the commonest and most famihar of adsorbents, and of it Du Bois Reymond first used the name. In its full statement this subject becomes very complicated, and involves physical conceptions and mathematical treatment which go far beyond our range. But it is necessary for us to take account of the phenomenon, even though it be in the most elemen- tary way. In the brief account of the theory of surface-tension with which our last chapter began, it was shewn that, in a drop of liquid, the potential energy of the system could be diminished, and work mani- fested 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 Uquid and its surrounding medium, be simple substances, and the system be uncomphcated by other distributions of force, then the energy of the system will have done its work when this equihbrium 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 * Some define adsorption as surface-condensation, without reference to the forces which produce it; in other words they recognize chemical, electrical and other forces, including cohesion, as producing analogous or indistinguishable results: of. A. P. Mathews, in Physiological Reviews, i, pp. 553-597, 1921. CH.vi] OF SURFACE ENERGY 445 morphology of the cell, and especially (so far as we have yet gone) of the solitary cell or unicellular organism. But we also saw, according to Gauss's equation, that the potential energy of the system will be diminished (and its diminution will accordingly be manifested in work) if from any cause the specific surface-energy be diminished, that is to say if it be brought more nearly to an equality with the specific energy of the molecules in the interior of the liquid mass. This latter is a phenomenon of great moment in 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. A diminution of the surface-energy may be brought about in various ways. 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 imme- diately 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 the 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. Again, a rfse of temperature diminishes surface-tension, and consequently facilitates the formation of a bubble or a froth. It follows (from the principle of Le Chatelier) that foam is warmer than, the fluid of which it is made, and the difference is all the greater the lower the concentration of the foaming (or capillary-active) substance*. 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 * Cf. Fr. Schiitz, in Nature, April 10, 1937. In the case of 001 per cent, solution of saponin, the temperature-diflFerence is no less than 3-3° C. 446 A NOTE ON ADSORPTION [ch. 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 circum- stances it is found that (quite apart from gravity, which might cause it to float to the surface) the oil has a tendency to be drawn to the surface; and again this phenomenon of molecular attraction or adsorption represents work done, equivalent to the diminished potential energy of the system*. In more general terms, if a Hquid 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 diffusion. In other words, in any "two-phase" system, a change of concentration at the boundary-surface and a diminution of surface-tension there accompany one another of necessity ; positive adsorption means negative surface-tension, and vice versa. Further- more, the lowering of surface-tension (as by saponin) will permit (caeteris paribus) an extension of surface, manifesting itself in "froth." Thus the production of a froth and the concentration of appropriate substances therein are two sides of one and the same phenomenon. In the complex mixtures which constitute the protoplasm of the living cell, this phenomenon of adsorption has abundant play: for many of its constituents, such as fats, soaps, proteins, lecithin, etc., possess the required property of diminishing surface-tension. * The first instance of what we now call an adsorptive phenomenon was observed in soap-bubbles. Leidenfrost was aware that the outer layer of the bubble was covered by an "oily" layer {De aquae communis nonnullis qualitatihus tractatus, Duisburg, 1756). 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 little different from that of a strong one {Theorie mdcanique de la chaleur, 1869, p. 376; cf. Plateau, ii, p. 100). VI] OF SURFACE CONCENTRATION 447 Moreover, the more a substance has the power of lowering the surface-tension of the Hquid 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, not only in visible drops, but also in the finest sus- pension or "colloidal solution"; and 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) in contact •with the surrounding water would have 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*. And all this is as 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, in other words, so as to reduce towards a minimum the potential energy of the system. This tendency to concentration at a 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 lawf, and was arrived at not by experi- mental but by theoretical and hydrodynamical methods. An assemblage of drops or droplets offers a great extension of surface, but so also does an assemblage of equally minute cells or * This identical phenomenon was the basis of Quincke's theory of amoeboid movement (Ueber periodische Ausbreitung von Fliissigkeitsoberflachen, etc., SB. Berlin. Akad. 1888, pp. 791-806; cf. Pfiuger's Archiv, 1879, p. 136). We must bear in mind that to describe an amoeboid cell as "naked" does not imply that its outer layer is identical with its internal substance. t J. Willard 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.) various papers by C. M. Lewis, Phil. Mag. (6), xv, p. 499, 1908; xvii, p. 466, 1909; Zeitschr. f. physik. Chemie, Lxx, p. 129, 1910; Milner, Phil. Mag. (6), xm, p. 96, 1907; A. B. Macallum, The role of surface-tension in determining the distribution of salts in living matter, Trans, loth Int. Congress on Hygiene, etc., Washington, 1912; etc. 448 A NOTE ON ADSORPTION [ch. pores; both alike are "two-phase" systems, and in both ahke the phenomenon of adsorption has free play. The occlusion of gases, including water-vapour, by charcoal is a familiar phenomenon of adsorption, and is due to the minuteness of the pores only in so far as surface-area is increased and multipUed thereby. For surface- energy is surface-strain or surface-tension x surface-area, and is vastly increased by minute subdivision. And surface-energy is such that, whenever a substance is introduced into a two-phase system — which merely means two things in touch (or surface- contact) with one another — it is apt to concentrate itself on the surface where the two phases meet. Absorption implies uniform distribution, as when a gas is absorbed by a liquid; adsorption imphes a heterogeneous field, and a concentration localised on the surfaces therein. 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 sub- stance, that is to say of its direct association with the surface-layer of molecules such as we imagined in the case of an electrical charge ; but rather of a progressive tendency to concentrate more and more, the nearer the surface is 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 repetition 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 con- tinual instabihty: and a constant manifestation of motion, some- times in the mere invisible transfer of molecules, but often in the production of visible currents, or manifest alterations in the form or outline of the system. Cellular activity is of necessity associated with cellular structure, VI] OF THE BOUNDARY STATE 449 even in our simplest interpretation thereof, as a mere increase of surface due to the existence and the multiphcation of cells. In the chemistry of the tissues there may be substances (catalysts and others) which exhibit their proper reactions even though the cells containing them be disintegrated or destroyed ; but other processes, oxidation itself among them, are essentially surface-actions, based on.adsorption at the vast cell-surface of the tissue*. The breaking- down of the cell-walls, the disintegration of cellular structure in a tissue, brings about "a biochemical chaos, a medley of reactions f." Cells are not merely there because the tissue has grown by their multiplication; there are physico-chemical reasons, even of an elementary kind, which render the morphological phenomenon of the cell indispensable to physiological action. The physiologist deals with the surface-phenomena of the cell in ways undreamed of when I began to write this book. To begin with, the concept of a surface (in the old mathematical or quasi- mathematical sense) no longer suffices to describe the boundary conditions of even a "naked" protoplasmic cell. As Rayleigh foretold, and as Irving Langmuir has proved, the "boundary-state" consists of a layer of complex molecules, each one a long array of atoms, all set side by side in an orderly and uniform way. There is not merely a boundary-surface between two phases (as the older colloid chemistry supposed) but a boundary-layer, which itself constitutes a third phase, or interphase, and which part of the surface-energy has gone to the making of. Surface-energy plays a leading part in modern theories of muscular contraction, and has indeed done so ever since FitzGerald and d'Arsonval indicated a connection between them some sixty years or more agof. It plays its part handsomely (we may be sure) in the electric pile of the Torpedo, where two miUion tiny discs present a * Many surface-active substances are known to be among the most active pharmacologically; cf. Michaelis and Rona, Physikal. Chemie, 1930. t A. V. Hill, Proc. R.S. (B), cm, p. 138^; cf. also M. Penrose and J. H. Quastel, on Cell structure and cell activity, ibid, cvii, p. 168. X Cf. G. F. FitzGerald, On the theory of muscular contraction, Brit. Ass. Rep. 1878; also in Scientific Writings, ed. Larmor, 1902, pp. 34, 75. A. d'Arsonval, Relations entre I'electricite animale et la tension superficielle, C.R. cvi, p. 1740, 1888; A. Imbert, Le mecanisme de la contraction musculaire, deduit de la consideration des forces de tension superficielle. Arch, de Phys. (5), ix, pp. 289-301, 1897; A. J. Ewart, Protoplasmic Streami-ng in Plants, Oxford, 1903, pp. 112-119. 450 A NOTE ON ADSORPTION [ch. vast aggregate of interfacial contact. It gives us a new conception, as Wolfgang Ostwald was the first to shew, of the relation of oxygen to the red corpuscles of the blood*. But many more and still more comphcated "film-reactions" are started or intensified by the oriented molecules of the monolayer. The catalytic action of living ferments (a subject vast indeed) is largely .a question of modified adsorption, or of surface-action. The range of bodies so adsorbed is extremply hmited; the specific reactions, which depend on the bacterium engaged, are fewer still ; and sometimes a whole class of substances may be adsorbed, and only one of them thrown specifically into action!. The physiological, and sometimes lethal, actions of various substances are examples of similar effects. The chemistry of the surface-layer in this cell or that may be elucidated by its reactions to various "penetrants," and depends somehow on the molecular orientation of the surfaces, and on the potentials associated with the characteristic electric fields which we may suppose to correspond to the particular molecular arrangements J. It is the dynamic aspect of the case, the ingresses, egresses and metabolic changes associated with the boundary-layer, which interest the physiologist. He finds the monolayer acting in ways not known in a homogeneous hquid — and adsorption is one of these ways. We keep as much as may be to the morphological side of the case rather than to the physiological, to the static side rather than to the dynamic, to the equilibrium attained rather than to the energies to which it is due. We continue to speak of surface, and of surface- * Ueber die Natur der Bindung der Gase im Blut und in seinen Bestandteilen, Kolloid Ztschr. ii, pp. 264-272, 1908; cf. Loewy. Dissociationsspannung des Oxyhaemoglobin im Blut, Arch. f. Anat. u. Physiol. 1904, p. 231. Arrhenius remarked long ago that the forces which produce adsorption are of the same order, and of the same nature, as those which cause the mutual attractions of the molecules of a gas. Hence the order is constant in which various gases are adsorbed by different adsorbents. The question of the inner mechanism of the forces which result in surface-tension, adsorption and allied phenomena, and their relation to electric charge on particles or ions, belongs to the highest parts of physical chemistry. Besides countless recent papers, M. v. Smoluchowski's Versuch einer mathematischen Theorie der Koagulationskritik, Z.f. physik. Chemie, xcii, pp. 129- 168, 1918, is still interesting. t Cf. N. K. Adams, Physics and Chemistry of Surfaces, 1930. Also {int. al.) J. H. Quastel, Mechanism of bacterial action. Trans. Faraday Soc. xxvi, pp. 831- 861, 19.30. + This is Loeb's so-called "membrane-effect," cf. Journ. Biol. Chemistry, xxxii, p. 147, 1917; and J. Gray, Journ. Physiol. Liv, pp. 68-78, 1920. VI] OF THE GIBBS-THOMSON LAW 451 energy and of adsorptive phenomena, in a somewhat old-fashioned way; but even with this simphfying Uniitation we find them helpful, throwing hght upon our subject. In the first place our preHminary account, such as it is, is already tantamount to a description of the process of development of a cell-membrane, or cell- wall. The so-called "secretion" of this cell- wall is nothing more than a sort of exudation, or striving towards the surface, of certain constituent molecules or particles within the cell; and the Gibbs-Thomson law formulates, in part at least, the conditions under which they do so. The adsorbed material may range from an almost unrecognisable pellicle to the distinctly differentiated "ectosarc" of a protozoon, and again to the develop- ment of a fully-formed cell-wall, as in the cellulose partitions of a vegetable tissue. In such cases, the dissolved and adsorbtive material has not only the property of lowering the surface-tension, and hence of itself accumulating at the 'surface, but has also the prope^y 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 exemphfied. Not only do we have the simple case of the building of a shell or '"test" upon the outward surface of a Hving cell, as for instance in a Foraminifer, but in a sub- sequent chapter, when we come to deal with spicules and spicular skeletons such as those of the sponges and of the Radiolaria^ we shall see how highly characteristic it is of the whole process of * We may trace the first steps in the study of this phenomenon to Melsens, who found that thin films of white of egg become firm and insoluble (Sur les modifications apportees a I'albumine. . .par faction purement mecanique, C.R. XXXIII, p. 247; Ann. de chimie et de physique (3), xxxiii, p. 170, 1851); and Harting made similar observations about the same time. Ramsden investigated the same subject, and also the more general phenomenon of the formation of albuminoid and fatty membranes by adsorption, and found {i7it. al.) that on shaking white of egg practically all .the albumin passes gradually into the froth; cf. his Koagulierung der Eiweisskorper auf mechanischer Wege, Arch. f. Anat. u. Phys. {Phys. Ahth.), 1894, p. 517; Abscheidung fester Korper in Oberflachenschichten, Z. f. phys. Chern. XLVii, p. 341, 1902; Proc. R.S. Lxxii, p. 156, 1904. For a general review of the whole subject see H. Zangger, Ueber Membranen und Membranfunktionen, in Asher-8piro's Ergebnisse der Physioloyie, vii, pp. 99-1(50, 1908. 452 A NOTE ON ADSORPTION [ch. spicule-formation for the deposits to be laid down just in the " interfacial " boundaries between cells or vacuoles, and how the form of the spicular structures tends in many cases to be regulated and determined by the arrangement of these boundaries. The so-called collenchyma, in which an excess of cellulose is laid down around the angles of contact of adjacent cells, in a kind of exag- gerated "bourrelet," is another case in point*. No pure liquid ever forms a froth or foam. White of egg is no exception to the rule; for the albumin is somehow changed, or " denatured," and becomes a quasi-solid when we beat it up. But in the frothing liquid there must always be some admixture present to concentrate on, or be adsorbed by, the surfaces and interfaces of the other ; and this dispersion must go on completely and uni- formly, so as to leave the whole system homogeneous. The resulting diminution of surface-tension facihtates the subdivision of the bubbles and dispersion of the air ; and the adsorbed surface-layer gives firmness and stability to the system. The sudden increase of surface diminishes, for the moment, the concentration, or "thick- ness" of the surface-layer; the tension rises accordingly, and the cycle of operations begins anewf . In physical chemistry, a distinction is usually drawn between adsorption and psevdo-adsorption, the former being a reversible, the latter an irreversible or permanent phenomenon. That is to say, adsorption, strictly speaking, implies the surface-concentration of a dissolved substance, under circumstances which, if they be altered or reversed, will cause the concentration to diminish or disappear. But pseudo-adsorption includes cases, doubtless originating in adsorption proper, where subsequent changes leave the concentrated substance incapable of re-entering the liquid system. It is obvious that many (though not all) of our biological illustrations, for instance the formation of spicules or of permanent cell-membranes, belong to the class of so-called pseudo- adsorption phenomena. But the apparent contrast between the two is in the main a secondary one, and however important to the chemist is of little consequence to us. 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 simphfy ♦ Cf. G. Haberlandt, Zelle u. Elementarorgane, Biol. Centralbl. 1925, p. 263. t Cf. F. G. Donnan, Some aspects of the physical chemistry of interfaces, Brit. Ass. Address (Section B), 1923; Nature, Bee. 15, 22, 1923. VI] OF THE FORMATION OF MEMBRANES 453 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. Having reached this conclusion we may wait patiently, and confidently, for more. But when the physico-chemical nature of these phenomena is admitted, and their dependence on adsorption recognised, or at least assumed, we have still to remember that the chemist himself is none too certain of his ground.. He still finds it hardj now and then, to teU how far adsorption and direct chemical action go their way together, what parts they severally play, what shares they take in their. intimate cooperation f. If the process of adsorption, on which the formation of a mem- brane 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 them- selves 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. The holes in a Globigerina-shell are there "to let the pseudopodia through." They may also be described as due to unequal distribution of surface- energy, such as to prevent shell-substance from being adsorbed here and there, and at the same time inducing a pseudopodium to emerge. * Strasbiirger, Ueber Cytoplasmastrukturen, etc., Jahrb. f, wiss. Bot. xxx, 1897; R. A. Harper, Kerntheilung und freie Zellbildung ira Ascus, ibid.; cf. Wilson, The Cell in Development, etc., pp. 53-55. t The "adsorption theory" of dyeing is a case in point, where the precise mode, or modes, of action seem still far from settled. 454 A NOTE ON ADSORPTION [ch. It is found that a rise of temperature greatly reduces the adsorbability of a substance, and this doubtless comes, either in part or whole, from the fact that a rise of temperature is itself a cause of the lowering of surface-tension. We may in all probability ascribe to this fact and to its converse, or at least associate with it, such phenomena as the encystment of unicellular organisms at the approach of winter, or the frequent formation of strong shells or membranous capsules in "winter-eggs." Again, since a film or a froth (which is a system of films) can only be maintained by virtue of a certain viscosity or rigidity of the liquid, it may be quickly caused to disappear by the presence in its neighbourhood of some substance capable of materially reducing the . surface-tension ; for this substance, being adsorbed, may displace from the surface -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 calms the waves but still more quickly dissipates the foam of the breakers. In a very different order of things, the breaking up of an alveolar network, as at a certain stage in the nuclear division of the cell, may be due in part to just such a cause, as well as to the direct lowering of surface-tension by electrical agency. Our last illustration has led us back to the subject of a previous chapter, namely to the visible configuration of the interior of the cell, in so far (at least) as it represents a "dispersed system," coarse enough to be visible; and in connection with this wide subject there are many phenomena on which light is apparently thrown by our knowledge of adsorption, of which we took Httle or no account in our former discussion. One of these phenomena is nothing less than that visible or concrete "polarity," which we have 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 it is manifested, in an epithelial cell, 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 mucus 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 VI] OF MORPHOLOGICAL POLARITY 455 of linear arrangements of particles, which in the elongated or "monopolar" cell run parallel with its axis, but tend to a radial arrangement in the more or less rounded or spherical cell. Of late years great importance has been attached to these various linear or fibrillar arrangements, as they are seen (after staining) in the cell- substance of intestinal epithelium, of spermatocytes, of ganglion cells, and most abundantly and frequently of all in gland cells. Various functions have been assigned, and hard names given to them; for these structures include your mitochondria* and your chondriokonts (both of these being varieties of chondriosomes), your Altmann's granules, your microsomes, pseudo-chromosomes, epi- 'M^m. Ih A B C Fig. 149. A, B, Chondriosomes in kidney- cells, prior to and during secretory activity (after Barratt); C, do. in pancreas of frog (after Mathews). dermal fibrils and basal filaments, your archeoplasm and ergasto- plasm, and probably your idiozomes, plasmosomes, and many other histological minutiae f. The position of these bodies with regard to the other cell- structures is carefully described. Sometimes they he in the neighbourhood of the nucleus itself, that is to say in proximity to the fluid boundary surface which separates the nucleus from the * Mitochondria are threads which move slowly through the protoplasm, some- times break in two, and often tend to radiate from the centrosphere or division-centre of the cell. The nucleoli are two or more Opaque bodies within the nucleus, which keep shifting their position ; within the cytoplasm many small fatty bodies likewise move about, and display the Brownian oscillation, t Cf. A. Gurwitsch, Morphologie und Biologie der Zelle, 1904, pp. 169-185; Meves, Die Chondriosomen als Trager erblicher Anlagen, Arch. 'f. mikrosk. Anat. 1908, p. 72; J. 0. W, Barratt, Changes in chondriosomes, etc., Q.J. M.S. LVin, pp. 553-566, 1913, etc.; A. P. Mathews, Changes in structure of the pancreas cell, etc., Journ. Mc/rph. xv (Suppl.), pp. 171-222, 1899. 456 A NOTE ON ADSORPTION [ch. cytoplasm ; and in this position they often form a somewhat cloudy sphere which constitutes the Nebenkern. In the majority of cases, as in the epithehal cells, they form filamentous structures, and rows of granules, whose main direction is parallel to the axis of the cell ; and which may, in some cases, and in some forms, be conspicuous at the one end, and in some cases at the other end of the cell. But I seldom find the histologists attempting 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 "Fibrillen- konus," which was almost certainly another aspect of the same phenomenon, was held by him and by cytologists hke Breda and Heidenhain to be an apparatus connected in some unexplained way with the mechanism of ciliary movement. Meves looked upon the chondriosomes as the actual carriers or ti*ansmitters of heredity. Altmann invented a new aphorism, Omne granulum e granulo, as a refinement of Virchow's (or Remak's) omnis cellula e ceUula*; and many other histologists, more or less in accord, accepted the chon- driosomes as important entities, sui generis, intermediate in grade between the cell itself and its ultimate molecular components. The extreme cytologists of the Munich school, Popoif, Goldschmidt and others, following Richard Hertwig, declaring these structures to be identical with "chromidia" (under which name Hertwig ranked all extra-nuclear chromatin), w^ould assign them complex functions in maintaining the balance between nuclear and cytoplasmic material; and the "chromidial hypothesis," as every reader of cytological literature knows, has become a very abstruse and comphcated thing f. With the help of the " binuclearity hypothesis" of Schaudinn and his school, it has given us the chromidial net, the ♦ Virchow, Arch. f. pathol. Anat. viii, p. 23, 1855; but used, implicitly, by Remak, in his paper Ueber extracellulare Entstehung thierischer Zellen und iiber die Vermehrung derselben durch Theilung, Muller's Archiv, 1852, pp. 47-57. That cells come, and only come, from pre-existing cells seems to have been clearly understood by John Goodoir, in 1846; see his Anatomical Memoirs, ii, pp. 90, 389. f Cf. Clifford Dobell, Chromidia and the binuclearity hypotheses; a review and a criticism, Q.J. M.S. liii, pp. 279-326, 1909; A. Prenant, Les Mitochondries et I'Ergastoplasme, Journ. de VAnut. et de la Physiol. Xlvi, pp. 217-285, 1910 (both with copious bibliography). VI] OF SOME INTRACELLULAR PHENOMENA 457 chromidial apparatus, the tropliochromidia, idiochromidia, gameto- chromidia, the protogonoplasm, and many other novel and original conceptions. There is apt to be confusion between important and unimportant things ; and the very 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 ; but from the physical point of view we see but little mystery in the whole phenomenon. For, on the one hand, it is likely enough that these various bodies, by vastly extending the intra-cellular surface-area, may serve to increase the physico-chemical activities of the cell; and, on the other hand, we ascribe their very existence, in all probability and in general terms, to the "clumping" together under surface- tension of various constituents .of the heterogeneous cell-contents, and to the drawing out of the little clumps along the axis of the cell towards one extremity or the other, in relation to osmotic currents as these are set up in turn in direct relation to the phenomena of surface-energy and of adsorption*. And all this imphes that the study of these minute structures, even if it taught us nothing else, at least surely and certainly reveals the presence of a definite field of force, and a dynamical polarity within the celll. * 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 Diiferenz der Oberflachenspannungen oder der Oberflachendruck eine Kraft darstellt, welche als treibende Kraft der Osmose, an die Stelle des nicht mit dem Oberflachendruck identischen osmotischen Druckes zu setzen ist, etc." (Oberflachendruck und seine Bedeutung im Organismus, Pfluger's Archiv, cv, p. 559, 1904.) There is, moreover, good reason to believe that physiological "osmosis" is not a general phenomenon common to this or that colloid membrane or dialyser, but depends {int. al.) on a specific affinity between the particular membrane (or the particular material it is moistened with) and the substance dialysed. This statement, made by Kahlenberg in 1906 {Journ. Phys. Cheni^. x, p. 141; also Nature, lxxv, p. 430, 1907), has been confirmed (e.g.) by R. Brinkmann and A. von Szent-Gyorgyi in Biochem. Ztschr'. cxxxix, pp. 261-273, 1923. t C. E. Walker, in an interesting paper on Artefacts as a guide to the chemistry of the cell, Proc. E.S. (B), cm, pp. 397-403, 1928, tells how he took mixtures of albumen, gelatine and lipins, with droplets of methyl myristate (with or without phosphorus) to act as nuclei; and found on treating with osmic acid that the lipins had separated out and arranged themselves very much as do Golgi bodies and other structural elements in ordinary histological preparations. 458 A NOTE ON ADSORPTION [ch. Our next and last illustration of the effects of adsorption, which we owe to the work of the late Professor A. B. Macallum* of Montreal, is of great importance ; for it introduces us to phenomena in regard to which we seem to stand on firmer ground than in some of the foregoing cases, albeit the whole story has not 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 cyhnder or the unduloid ; and we began at once to find ourselves in difficulties when we were confronted by departures from symmetry, even in such a simple case as the ellipsoidal yeast-cell and the production of its bud. We found the cylindrical cell of Spirogyra, with its plane partitions or its spherical ends, a 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 account for and explain. The analogy of the soap-bubble, or of the simple hquid drop, was apt to lead us to suppose that surface-tension w^as, on the -whole, uniform over the surface of the cell; and that its departures from symmetry of form were due to variations in external resistance. But if we have been inchned to make such an assumption we must now 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 more to do with cells whose general heterogeneity of substance leads to quahtative diSerences of surface, and hence to varying distributions of surface-tension. We must accordingly investigate the case of a cell which displays some definite and regular heterogeneity of its liquid surface, just as Amoeba displays a heterogeneity which is complex, irregular and continually fluctuating in amount and distribution. Such heterogeneity as we are speaking of must be essentially chemical, and the preliminary problem is to devise methods of " microchemical " analysis, which shall reveal localised accumulations of particular substances within * See his Methoden u. Ergebnisse der Mikrochemie in der biologischen Forschung; Asher-Spiro's Ergebnisse, vn, 1908. VI] OF MACALLUM'S EXPERIMENTS 459 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 apphcation of it; for the substances which have such action as we are looking for, and which are also actual or possible constituents of the cell, are very numerous, while the means are very seldom at hand to demonstrate their precise distribution and locahsation. But in one or two cases we have such means, and the most notable is in connection with the element potassium. As 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 characteristic black precipitate of cobaltic sulphide f. By this means Macallum demonstrated, years ago, the unexpected presence of potassium (i.e. of chlorides or other potassium salts) accumulated in particular parts of various cells, both sohtary cells and tissue cells J ; and he arrived at the conclusion that the localised accumulations in question were simply evidences of concentration of the dissolved potassium salts, formed and locaUsed in accordance with the Gibbs-Thomson Law. For potassium (as we now know) has a much higher ionic velocity than sodium ; and accordingly the * On the distribution of potassium in animal and vegetable cells, Journ. Physiol. XXXII, p. 95, 1905. (The only substance at all likely "to be confused with potassium in this reaction is creatine.) t The reader will recognise a fundamental difference, and contrast, between such experiments as those of Macallum 's and the ordinary staining processes of the histologist. The latter are (as a general rule) merely 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 cytologic est tout entier dans la microchimie": A. Prenant, Methodes et resultats de la microchimie, Journ. de VAnat. et de la Physiol, xlvi, pp. 343-404, 1910. There is an interesting paper by Brunswick, on the Limitations of microchemical methods in biology, in Die Naturwissenschaften, Nov. 2, 1923. X It is always conspicuously absent, as are chlorides and phosphates in general, from the nuclear substance. 460 A NOTE ON ADSORPTION [CH. K-ioris reach and occupy the adsorbing surfaces of the cell-membranes out of all proportion to their abundance in the external media*. And we may take it also that our potassium salts, hke inorganic substances in general, tend to raise the surface-tension, and will be found concentrated, therefore, at a portion of the surface where the tension is weakf. , iM2^ Fig. 150. Adsorptive concentration of potassium salts in (1) a cell of Pleurocarpus about to conjugate; (2) conjugating cells of Mesocarpus; (3) sprouting spores of Equisetum. After Macallum. In Professor Macallum's figure (Fig. 150, 1) of the little green alga Pleurocarpus , we see that one side of the cell is beginning to bulge out in a wide convexity. This bulge is, in the first place, a sign of weakened surface-tension on one side of the cell, which as a whole had hitherto been a symmetrical cylinder; in the second place, we see that the bulging area corresponds to the position of a great concentration of the potassium salt; while in the third place, * Cf. A. B. Macallum, Address to Section I, Brit. Ass. 1910; Oberflachen- spannung und Lebenserscheinungen, in Asher-Spiro's Ergehnisse der Physiologie, XI, pp. 598-688, 1911; also his important paper on Tonic mobility as a factor in influencing the distribution of potassium in living matter, Proc. R.S. (B), civ, pp. 440-458, 1929; cf. E. F. Burton, Trans. Faraday Soc. xxvi, p. 677, 1930. • t Ii^ accordance with the "principle of Le Chatelier," which is in fact a corollary to the Gibbs-Thomson Law. VI] OF MACALLUM'S EXPERIMENTS 461 from the physiological point of view, we call the phenomenon the first stage in the process of conjugation. In the figure 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 separates the green protoplasm from the cell-sap), there is a very large accumulation precisely at the point where the tension of the originally cylindrical cell is weakening to produce the bulge. But in a still later stage, when the boundary between the two conjugating cells is lost and the cytoplasm of the two cells becomes fused together, then the signs of potassium concentration quickly disappear, the salt becoming generally diffused through the now symmetrical and spherical "zygospore." In a spore of Equisetum, 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. 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. The Acinete protozoa obtain their food through suctorial tentacles extruded from the surface of the cell : their extrusion being doubtless due to a local diminution of surface-tension. A dense concentration of potassium reveals itself, accordingly, in the surface-film of each tiny tentacle. As the tentacles are withdrawn their potassium diffuses into the cytoplasm; when retraction is complete it is again found in surface-concentration, but the surface-films on which it now concentrates are the surfaces of the protein-spherules (or "food- vacuoles") within the body of the cell. In the case of ciliate or flagellate cells, there is to be found a 462 A NOTE ON ADSORPTION [ch. characteristic accumulation of potassium at and near the base of the ciUa. The relation of ciliary movement to surface-tension* lies beyond our range, but the fact which we have just mentioned throws light upon the frequent or general presence of a little protuberance of the cell-surface just where a flagellum is given oif (cf. p. 406), 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 impHed 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 "chromato- phoral," protoplasm, the rest of the cell being wholly destitute of that substance : the inference 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. And again, in certain minute Chytridia-like fungi, parasitic on Spirogyra and the like, the potassium-reaction helps to trace the delicate haustoria of the parasite in their course within the host-cell — a clear indication of low surface-tension at the surface between. Now as Macallum points out, the presence of potassium is known to be a factor, in connection with the chlorophyll-bearing proto- plasm, 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 substances (e.g. formaldehyde) which lower the surface-tension, and so conduce to the concentration of potassium. All we know for certain is that the several phenomena are associated with one another, as ap- parently inseparable parts or inevitable concomitants of a certain complex aqtionf. * Cf. J. Gray, The mechanism of ciliary movement, Proc. U.S. (B), 1922-24. t The distribution of potassium within plant-cells is more complicated than it seemed at first to be; but it is still the general if not the invariable rule to find it associated (by adsorption) with one boundary-surface or another. Cf. E. S. Dowding, Regional and seasonal distribution of potassium in plant tissues, Ann. Bot. xxxix, pp. 459-470, 1925. The whole ciuestion, first adumbrated by Macallum, VI] OF MACALLUM'S EXPERIMENTS 463 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 essentiall> ..^pendent 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 comphance 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 liquid surfaces 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 apphcable and as equi- valent to a universal law. The occasional difficulties or apparent exceptions are such as to call for further enquiry, but fall short of throwing doubt on the hypothesis. Macallum's researches introduce a new element of certainty, a "nail in a sure place," when they demonstrate that in certain movements or changes of form which we should naturally attribute to weakened surface-tension, a chemical concentration which would naturally accompany such weakening actually takes place. They further teach us that in the cell a chemical heterogeneity may exist of a very marked kind, certain substances being accumulated here and absent there, within the narrow bounds of the system. " Such localised accumulations can as yet only be demonstrated in the case of a very few substances, and of a single one in particular ; and these few 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 they are due to the prior accumulation, or local production, of bodies which have this direct effect; though we are by no means limited to this hypothesis. But in spite of some remaining difficulties and un- certainties, we have arrived at the conclusion, as regards unicellular organisms, that not only their general configuration but also their is part of the general subject of ionic regulation, which has since become a matter of great physiological importance; cf. [int. al.) D. A. Webb, Ionic regulation in Carcinus mnenns, Pror. R,S. (B), cxxix, pp. 107-130, 1940, and many works quoted therein. It is curious and interesting that Macallum's first work on unequal ionic distribution in the tissues and Donnan's fundamental conception of the Donnan equilibrium {Journ. Chem,. Sdv. xcix, p. 1554, 1911) came just at the same time. 464 A NOTE ON ADSORPTION [ch. vi departures from symmetry may be correlated with the molecular forces manifested in their fluid or semi-fluid surfaces. Looking at the physiological side, rather than at the morphological which is more properly our own, we see how very important a cellular system is bound to be, even in respect of its surface-area alone. The order of magnitude of the cells which constitute our tissues is such as to give a relation of surface to volume far beyond anything in all the structures or mechanisms devised and fabricated by man. At this extensive surface, capillary energy, a form of energy scarcely utilised by man, plays a large predominant part in the energetics of the organism. Even the warm-blooded animal is not in reality a heat-engine; working as it does at almost constant temperatures its output of energy is bound, by the principle of Carnot, to be small. Nor is it an electrostatic machine, nor yet an electrodynamic one. It is a mechanism in which chemical energy turns into surface-energy, and, working hand in hand, the two are transformed into mechanical energy, by steps which are for the most part unknown*. We are led on by these considerations to reflect on the molecular, rather than the histological, structure of the cell. We have already spoken in passing of " monomolecular layers," such as Henri Devaux imagined some thirty years ago, and afterwards obtained f, and such as Irving Langmuir has lately made his own. The free surface of every liquid (provided the form and symmetry of its molecules permit) presents a single layer of oriented molecules. Such a surface h no mere limit or simple boundary; it becomes a region of great importance and peculiar activity in certain cases, when, for instance, protein molecules of vast complexity are concerned. It is then a morphological field with a molecular structure of its own, and a dynamical field with energetics of its own. It becomes a frontier where this alien molecule may be excluded and that other be passed through: where some must submit to mere adsorption, and others sufl'er chemical change. In a word, we begin to look on a surface-layer or membrane, visible or invisible, as a vastly important thing, a place of delicate operations, and a field of peculiar and potent activity. * Lippmann imagined a moteur electrocapillaire, unique in the history of mechanical invention. Cf. Berthelot, Rev. Sci. Dec. 7, 1913. t CL{int. al.) H. Devaux, La structure moleculaire de la cellule vegetale, Bull. Soc. Bot. de France, lxxv, p. 88, 1928. CHAPTER VII THE FORMS OF TISSUES OR CELL-AGGREGATES We pass from the solitary cell to 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 consider the effect of various forces; but, as in the case of the soUtary 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 manifest itself in surfaces mininiae areae: where, as Plateau was always careful to point out, we must understand by this expression not an absolute but a relative minimum, an area, that is to say, which approximates to an absolute minimum as nearly as the circumstances and material exigencies of the case permit. There are certain fundamental principles, or fundamental equa- tions, besides those we have already considered, which we shall need in our enquiry; for instance, the case which we briefly touched on (on p. 426) of the angle of contact between the protoplasm and the axial filament in a Heliozoan, we shall now find to be but a particular case of a general and elementary theorem. Let us re-state as follows, in terms of Energy, the general principle which underlies the theory of surface-tension or capillarity*. When a fluid is in contact with another fluid, or with a solid or with a gas, a portion of the total 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 is constant for these, save only in so far as it may be modified by changes of temperature or of electrical charge. Equihbrium, which is the condition of minimum potential energy in the system, will accordingly * See Clerk Maxwell's famous article on "Capillarity" in the ninth edition of the Encyclopedia Britannica, revised by Lord Rayleigh in the tenth edition. 466 THE FORMS OF TISSUES [ch. be obtained, caeteris paribus, by the utmost possible reduction of the surfaces in contact. When we have three bodies in contact with one another the same is true, but the case becomes a Httle more complex. Suppose a drop of some fluid, A, to float on another fluid, J5, while both are exposed to air, C Here are three surfaces of contact, that of the drop with the fluid on which it floats, and those of air with the one and other of these two; and the whole surface-energy, E, of the system consists of. three' parts resident in these three surfaces, Fig. 151. or of three specific energies, -S^^, -E'^ic ^bc- "^^^ condition of equihbrium, or minimal potential energy, will be reached by con- tracting those surfaces whose specific energy happens to be large and extending those where it is small — contraction leading to the production of a "drop,'f and extension to a spreading "film." Floating on water, turpentine gathers into a drop, olive-oil spreads out in a film; and these, according to the several specific energies, are the ways by which the total energy of the system is diminished and equihbrium attained. A drop will continue to exist provided its own two surface-energies exceed, per unit area, the specific energy of the water-air surface around: that is to say, provided (Fig. 151) ^AB + ^AC > ^BC But if the one fluid happen to be oil and the other water, then the combined energy per unit-area of the oil-water and the oil-air surfaces together is less than that of the water-air surface : E > E 4- E Hence the oil-air and oil-water surfaces increase, the air-water surface contracts and disappears, the oil spreads over the water, and the "drop" gives place to a "film," In both cases the total surface-area is a minimum under the circumstances of the case, and always provided that no external force, such as gravity, complicates the situation. VII] OF FLOATING DROPS 467 The surface-energy of which we are speaking here is manifested in that contractile force, or tension, of which we have had so much to say*. In any part of the free water-surface, for instance, one surface-particle attracts another surface-particle, and the multi- tudinous attractions result in equilibrium. But a water-particle in the immediate neighbourhood of the drop may be pulled outwards, Pig. 152. so to speak, by another water-particle, but find none on the other side to furnish the counter-pull; the pull required for equiUbrium must therefore be provided by tensions existing in the other two surfaces of contact. In short, if we imagine a single particle placed at the very point of contact, it will be drawn upon by three different forces, whose directions lie in the three surface-planes and whose * It can easily be proved (by equating the increase of energy stored in an increased surface with the work done in increasing that surface), that the tension measured per unit breadth, T^^, is equal to the energy per unit area, EfO,- Surface- tensions are very diverse in magnitude, -but all are positive; Clerk Maxwell conceived the existence of negative surface-tensions, but could not point to any certain instance. When blood-serum meets a solution of common salt, the two fluids hasten to mix, long streamers of the one running into the other; this remarkable phenomenon, first observed by»Almroth Wright {Proc. R.S. (B), xcii, 1921) and called by him " pseudopodial intertraction," was described by Schoneboom (ibid. (A), ci, 1922) as a case of negative surface-tension. But it is a diffusion-phenomenon rather than a capillary one. 468 THE FORMS OF TISSUES [ch. magnitudes are proportional to the specific tensions characteristic of the three " interfacial " surfaces. Now for three forces acting at a point to be in equihbrium they must be capable of representation, in magnitude and direction, by the three sides of a triangle taken in order, in accordance with the theorem of the Triangle of Forces. So, if we know the form of our drop as it floats on the surface (Fig. 152), then by drawing tangents P, R, from 0 (the point of mutual contact), we determine the three angles of our triangle, and know therefore the relative magnitudes of the three surface-tensions proportional to its sides. Conversely, if we know the three tensions acting in the directions P, R, S (viz. Tab, T^c, T^^) we know the three sides of the triangle, and know from its three angles the form of the section of the drop. All points round the edge of the drop being under similar conditions, the drop must be circular and its figure that of a sohd of revolution*. The principle of the triangle of forces is expanded, as follows, in an old seventeenth-century theorem, called Lamy's Theorem : // three forces acting at a point be in equilibrium, each force is proportional to the sine of the angle contained between the directions of the other two. That is to say (in Fig. 152) P : R : S = sin (f) : sin p : sin s, . P R S or ^-7 = -^ — ==-■ — • sm (f) sm p sm s" 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: viz. P^ = R^ + S^+ 2RS cos j^ hes between 45° and 90°, while ^2 ^i^s between 135° VII] OF RIFTS AND CRACKLES 523 and 180°. Hence the rift, dipping down at the angle TT\ will never dip at a greater angle than 45°, and generally much less. Now a spherical bowl whose hp is at 45° to the horizontal has a depth of times its lip diameter, or approximately one-fifth. And one-fifth of the diameter of the column is, approximately, the depth of the deepest bowl. The hexagonal pattern and the three-way corners on which it is based are characteristic (as we have already explained) of a condition of symmetry or uniformity .under which a partition is as hkely to arise in any one direction as in any other, and no series of partitions has precedence over the rest. We have seen, Fig. 2U2. Crackles on a porcelain bowl. From H. Hukusima. in the dragon-fly's wing and in cambium-tissue, how different is the result when primary partitions are first established and con- solidated, to be followed by a secondary and a weaker set. The "crackles" on a porcelain bowl look somewhat like a cellular epithelium; but the porcelain has been under strain in more ways than one. The plastic clay was first shaped upon a wheel, and potential stress-energy so acquired is stored up even in the finished ware ; again, as the ware cools and shrinks after it is drawn from the kiln the glaze is apt to cool quicker and shrink more than the paste below, and tension -energy is stored up till the glaze ruptures and the cracks appear.' Various rates of cooling and of contraction, the nature of paste and glaze, the shape of the ware, and even the way in which the potter worked the clay, may all influence the pattern of the crackle. The primary crack will be perpendicular to the 524 THE FORMS OF TISSUES [ch. main tension; secondaries will tend to be more or less orthogonal to the primary cracks; two secondaries on opposite sides of a primary will tend to be near togetiier, though not opposite; and spiral cracks are often to be seen, the remote effect of plasticity under the potter's wheel. The net result is that certain primary cracks appear, related (more or less clearly) to the circular or spiral shaping of the clay upon the wheel; and each primary crack so reheves the tension in one direction that the secondaries tend to follow in a direction at right angles to the first. While co-equal angles of 120° were likely to occur in certain symmetrical cases, leading to a simultaneous pattern of hexagons, in other cases suc- Fig. 203. Colour-patterns of kidney-beans with diagrammatic contour-lines added; a, b, Japanese "quail- beans"; c, scarlet-runner. After M. Hirata. cessive partitions or cracks tend to be at right angles to one another ; and Sachs's Law becomes truer of the porcelain than of the plant*. In this latter case, and doubtless in many more, we are deahng not 'with a random pattern, but with one based on systematic and predetermined lines. The apparently confused or random pattern of a kidney-bean comes under the same class of configurations, inasmuch as it also is based on an underlying polarity, whose centre of symmetry is in the stalk or "hilus." For simphcity's sake, imagine the bean round hke a pea, and its surface mapped out, orthogonally, by two sets of boundary-hnes, radial and con- centric. Then suppose an asymmetry of growth to be introduced so that the round pea grows into the elhpsoid of a bean; and suppose that the whole system of boundary-lines is subject to the same conformal transformation — which elliptic functions might help * See H. Hukueima, Cracks upon the glazed surface of ceramic wares, Sci. Papers, Inst, of Chem. and Phys. Research, Tokyo, xxvii, pp. 235-243. 1935. VII] OF THE BEE'S CELL 525 us to define. The colour-pattern of the bean will then be found following the direction of the boundary-lines, and occupying areas or patches corresponding to parts of the orthogonal system. The lines are equipotential hnes, or akin thereto. If we varnish an elastic bag, dry it and expand it, the varnish will tend to crack along the same orthogonal boundaries*. The bee^s cell The most famous of all hexagonal conformations, and one of the most beautiful, is the bee's cell. As in the basalt or the coral, we have to deal with an assemblage of co-equal cylinders, of circular section, compressed into regular hexagonal prisms; but in this case we have two layers of such cyhnders or prisms, one facing one way and one the other, and a new problem arises in connection with their inner ends. We may suppose the original cyhnders to have spherical endsf, which is their normal and symmetrical way of terminating; then, for closest packing, it is obvious that the end of any one cylinder in the one layer will touch, and fit in between, the ends of three cylinders in the other. It is just as when we pile round-shot in a heap; we begin with three, a fourth fits into its nest iDetween the three others, and the four form a ''tetrad," or regular tetragonal arrangement. Just as it was obvious, then, that by mutual pressure from the sides of six adjacent cells any one cell would be squeezed into a hexagonal prism, so is it also obvious that, by mutual pressure against the ends of three opposite neighbours, the end of each and every cell will be compressed into a trihedral pyramid. The three sides of this pyramid are set, in. plane projection, at co-equal angles of 120° to one another; but the three apical angles (as in the analogous case already described of a system of soap-bubbles) are, * M. Hirata, Coloured patches in kidney-beans, Set. Papers, Inst. Chem. Researchy Tokyo, XXVI, pp. 122-135, 1936. f In the combs of certain tropical bees the hexagona structure is imperfect and the cells are not far removed from cylinders. They are set in tiers, not contiguous but separated by little pillars of wax, and the base of each cell is a portion of a sphere. They differ from the ordinary honeycomb in the same sort of way as the fasciculate from the massive corals, of which we spoke on p. 512. Cf. Leonard Martin, Sur les Melipones de Bresil, La Nature, 1930, pp. 97-100. 526 THE FORMS OF TISSUES [ch. by the geometry of the case*, co-equal angles of 109° and so many minutes and seconds. If we experiment, not with cyHnders but with spheres, if for instance we pile bread-pills together and then submit the whole to a uniform pressure, as we shall presently find that Buffon did : each ball (hke 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 under compression it will develop twelve plane surfaces. It will 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 is this figure incompletely formed; it represents, so to speak, one-half of that figure, with its apex and the six adjacent corners proper to the rhombic dodecahedron, but six sides continued, as a hexagonal prism, to an open or unfinished endf. The bee's comb is vertical and the cells nearly horizontal, but sloping slightly downwards from mouth to floor; in each prismatic cell two sides stand vertically, and two corners lie above and below. Thus for every honeycomb or "section" of honey, there is one and only one "right way up"; and the work of the hive is so far con- trolled by gravity. Wasps build the other way, with the cells upright and the combs horizontal; in a hornet's nest, or in that of Polistes, the cells stand upright like the wasp's, but their mouths look down- wards in the hornet's nest arid upwards in the wasp's. What Jeremy Taylor called "the discipline of bees and the rare fabric of honeycombs " must have attracted the attention and excited the admiration of mathematicians from time immemorial. "Ma maison est construite," says the bee in the Arabian Nights, "selon * The dihedral angle of 120° is, physically speaking, the essential thing; the Maraldi angle, of 109°, etc., is a geometrical consequence. Cf. G. Cesaro, Sur la forme de Falveole de I'abeille, Bull. Acad. R. Belgique (Sci.), 1920, p. 100. t See especially Haiiy, the crystallographer; Sur le rapport des figures qui existe entre Falveole des abeilles et le grenat dodecaedre, Journ. d'hist. naturelle, n, p. 47, 1792. VII] OF THE BEE'S CELL 527 lesloisd'une severe architecture; et Euclidos lui-meme s'instruirait en admirant la geometrie de ses alveoles*." Ausonius speaks of the geometrica forma favorum,, and Pliny tells of men who gave a lifetime to its study. Pappus the Alexandrine has left us an account of its hexagonal plan, and drew from it the conclusion that the bees were endowed Vith "a certain geometrical forethought "f- "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, sus- pecting indeed that it could hold more honev than either of the •Y^^v;v^<"r Fig. 204. Portion of a honeycomb. After WiJlem. Other two J." Erasmus Bartholin was apparently the first to suggest that the hypothesis of '"economy" was not warranted, and that the hexagonal cell 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 than that of its sides, and came later, in general terms the arrangement was doubtless often studied and described: as for * Ed. Mardrus, xv, p. 173. t (pvaLK7]v ye(i}/j.eTpiKi]u Trpbvoiav. Pappus, Bk. V; cf. Heath, Hist, of Gk. Math, ii, p. 589. St Basil discusses ttiv yewixerplav t^s cro fxeXlaoTjs: Hexaem. vni, p. 172 (Migne); Virgil speaks of the pars divinae mentis of the bee, and Kepler found the bees anxmd praedifas et geomet. riae suo modo capaces. X This was according to the "theorem of Zenodorus." The use by Pappus of "economy" as a guiding principle is remarkable. For it means that, like Hero with his mirrors, he had a pretty clear adumbration of that principle of minima, which culminated in the principle of least action, which guided eighteenth-century physics, was generalised (after Fermat) by Lagrange, inspired Hamilton and Maxwell, and reappears in the latest developments of wave-mechanics. 528 THE FORMS OF TISSUES [ch. instance, in the Garden of Cyrus: "And the Combes themselves so regularly contrived that their mutual intersections make three Lozenges at the bottom of every Cell* which severally regarded make three Rows of neat Rhomboidall Figures, connected at the angles, and so continue three several chains throughout the whole comb " Or as Reaumur put it, a little later on: "trois cellules accolees laissent un vuide pyramidal, precisement semblable a celui de la base d'une autre cellule tourn^e en sens contraire." Kepler had deduced from the space-filling symmetry of the honey- comb that its angles must be those of the rhombic dodecahedron; and Swammerdam also recognised the same geometrical figure in the base of the cell*. But Kepler's discovery passed unnoticed, and Maraldi the astronomer, Cassini's nephew, has the credit of ascertaining for the first time the shape of the rhombs and of the sohd angle which they bound, while watching the bees in "les ruches vitrees dans le jardin de M. Cassini attenant I'Observatoire de Paris -f." The angles of the rhomb, he tells us, are 110° and 70°: "Chaque base d'alveole est formee de trois rombes presque toujours egaux et semblables, qui, suivant les mesures que nous avons 2^rises, ont les deux angles obtus chacun de 110 degres, et par consequent les deux aigus chacun de 70 degres." Further on (p. 312), he observes that on the magnitude of the angles of the three rhombs at the base of the cell depends that of the basal angles of the six trapezia which form its sides; and it occurs to him to ask what must these angles be, if those of the floor and those of the sides be equal one to another. The solution of this problem is that "les angles aigus des rombes etant de 70 degres 32 minutes, et les obtus de 109 degres 28 minutes, ceux des trapezes qui leur sont contigus doivent etre aussi de la meme grandeur." And lastly: "II resulte de cette grandeur d' angle non seulement une plus grande facilite et simpHcite dans la construction, a cause que par cette maniere les abeilles n'employent que deux sortes d'angles, mais il en resulte encore une plus belle simetrie dans la disposition et dans la figure * Kepleri Opera omnia, ed. Fritsch, v, pp. 115, 122, 178, vii, p. 719, 1864; Swammerdam, Tractatus de apibus (observations made in 1673). t Obs. sur les abeilles, Mem. Acad. R. Sciences (1712), 1731, pp. 297-331. Sir C. Wren had used "transparent bee-hives" long before; see his Letter concerning that pleasant and profitable invention, etc., in S. Hartlib's Reformed Common- Wealth of Bees, 1655. VII] OF THE BEE'S CELL 529 de r Alveole." In short, Maraldi takes the two principles of sim- phcity and mathematical beauty as his sure and sufficient guides. The next step was that which had been foreshadowed long before by Pappus. Though Euler had not yet published his famous dissertation on curves maximi minimive proprietate gaudentes, the idea of maxima and minima was in the air as a guiding postulate, an heuristic method, to be used as Maraldi had used his principle of simphcity. So it occurred to Reaumur, as apparently it had not done to Maraldi, that a minimal configuration, and consequent economy of material in the waxen walls of the cell, might be at the root of the matter: and that, just as the close-packed hexagons gave the minimal extent of boundary in a plane, so the figure deter- mined by Maraldi, namely the rhombic dodecahedron, might be that which employs the minimum of surface for a given content: or which, in other words, should hold the most honey for' the least wax. "Convaincu que les abeilles employent le fond pyramidal qui merite d'etre prefere, j'ai soup9onne que la raison, ou une des raisons, qui les avoit decidees etait I'epargne de la cire; qu'entre les cellules de meme capacite et a fond pyramidal, celle qui pouvait etre faite avec moins de matiere ou de cire etoit celle dont chaque rhombe avoit deux angles chacun d'environ 110 degres, et deux chacun d'environ 70°." He set the problem to Samuel Koenig, a young Swiss mathematician: Given an hexagonal cell terminated by three similar and equal rhombs, what is the configuration which requires the least quantity of material for its construction? Koenig confirmed Reaumur's conjecture, and gave 109° 26' and 70° 34' as the angles which should fulfil the condition; and Reaumur then sent him the Memoires de I'Academie for 1712, where Koenig was "agreeably surprised" to find: "que les rombes que sa solution avait determine, avait a deux minutes pres* les angles que M. Maraldi avait trouves par des mesures actuelles a chaque rhombe des cellules d'abeilles. . . . Un tel accord entre la solution et les mesures actuelles a assurement de quoi surprendre." Koenig asserted that the bees had solved a problem beyond the reach of * The discrepancy was due to a mistake of Koenig's, doubtless misled by his tables, in the determination of V2; but Koenig's own paper, sent to Reaumur, remained unpublished and his method of working is unknown. An abridged notice appears in the Mdm. de VAcad. 1739, pp. 30-35. 530 THE FORMS OF TISSUES [ch. the old geometry and requiring the methods of Newton and Leibniz. Whereupon Fontenelle, as Secretaire Perpetuel, summed up the case in a famous judgment, in which he denied intelhgence to the bees but nevertheless found them blindly using the highest mathe- matics by divine guidance and command*. When CoHn Maclaurin studied the honeycomb in Edinburgh, a few years after Maraldi in Paris, he proceeded to solve the problem without using "any higher Geometry than was known to the Antients," and he began his account by saying: "These bases are fornled from Three equal Rhombus's, the obtuse angles of which are found to be the doubles of an Angle that often offers itself to mathematicians in Questions relating to Maxima and Minima. f" It was an angle of 109° 28' 16", with its supplement of 70° 31' 44". And this angle of the bee's cell determined by Maraldi, Koenig and Maclaurin in their several ways, this angle which has for its cosine 1/3 and is double of the angle which has for its tangent V2, is on the one hand an angle of the rhombic dodecahedron, and on the other is that very angle of simple tetrahedral symmetry which the soap- films within the tetrahedral cage spontaneously assume, and whose frequent appearance and wide importance we havo already touched upon J. That "the true theoretical angles were 109° 28' and 70° 32', precisely corresponding ivith the actual measurement of the bee's cell,'' and that the bees had been "proved to be right and the mathema- ticians wrong," was long believed by many. Lord Brougham * La grande merveille est que la determination de ces angles passe de beaucoup les forces de la Geometrie commune, et n'appartient qu'aux nouvelles Methodes fondees sur la Theorie de I'lnfini. Mais a la fin les Abeilles en S9auraient trop, et I'exces de leur gloire en est la ruine. II faut remonter jusqu'a une Intelligence infinie, qui les fait agir aveuglemerit sous ses ordres, sans leur accorder de ces lumieres capables de s'accroitre et de se fortifier par elles-memes, qui font I'honneur de notre Raison." Histolre de VAcademie Royale, 1739, p. 3.j. t Colin Maclaurin, On the bases of the cells wherein the bees deposit their honey, Phil. Trans, xlii, pp. 561-571, 1743; also in the Abridgement, viii, pp. 709- 713, 1809; it was characteristic of Maclaurin to use geometrical methods for wellnigh everything, even in his book on Fluxions, or in his famous essay on the equilibrium of spinning planets. Cf. also Lhuiller, Memoire sur le minimum du cire des alveoles des Abeilles, et en particulier sur un minimum minimorum relatif a cette matiere, Nouv. Mem. de VAcad. de Berlin, 1781 (1783), pp. 277-300. Cf. Castillon, ibid, (commenting on Lhuiller); also Ettore Carruccio, Notizie storiche eulla geometria delle api, Periodica di Mathematiche, (4) xvi, pp. 35-54, 1936. X Supra, p. 497. The faces of a regular octahedron meet at the same angle. VII OF THE BEE'S CELL 531 helped notably to spread these and other errors, and his writings on the bee's cell contain, according to Glaisher, "as striking examples of bad reasoning as are often to be met with in writings relating to mathematical subjects." The fact is that, were the angles and facets of the honevcomb as sharp and smooth, and as constant and uniform, as those of a quartz-crystal, it would still be a dehcate matter to measure the angles within a minute or two of arc, and a technique unknown in Maraldi's day would be required to do it. The minute-hand of a clock (if it move continuously) moves through one degree of arc in ten seconds of time, and through an angle of two minutes m one-third of a second; — and this last is the angle which Maraldi is supposed to have measured. It was eighty years after Maraldi had told Reaumur what the angle was that Boscovich pointed out for the first time that to ascertain the angle to the nearest minute by direct admeasurement of the waxen cell was utterly impossible. Yet Reaumur had certainly beheved, and apparently had persuaded Koenig, that Maraldi's determina- tions, first and last, were the result of measurement ; and Fontenelle, the historian of the Academy, epitomising Koenig's paper, speaks of "les mesures actuelles de M. Maraldi," of the bees being in error to the trifling extent of 2', and of the grande merveille of their so nearly solving a problem belonging to the higher geometry. Boscovich, in a long-forgotten note, rediscovered by Glaisher, puts the case in a nutshell: "Mirum sane si Maraldus ex observatione angulum aestimasset intra minuta, quod in tarn exigua mole fieri utique non poterat. At is (ut satis patet ex ipsa ejus determina- tione) affirmat se invenisse angulos circiter 110° et 70°, nee minuta emit ex observatione sed ex equahtate angulorum pertinentium ad rhombos et ad trapezia ; ad quam habendam Geometria ipsa docuit requiri ilia minuta*." Indeed he goes on to say the wonder is that the angles could be measured even within a few degrees, variable and irregular as they are seen to be, and as even Reaumur f knew * In his note De apium cellulis, appended to the philosophical poem of Benedict Stay, II, pp. 498-504, Romae, 1792. t Op. cit. V, p. 382. Several authors recognised that the cells are far from identical, and do no more than approximate to an average or ideal angle: e.g. Swammerdam in the Biblia Naturae, u, p. 379; G. S. Klugel, Grosstes u. Kleinstes, in Mathem. Worterb. 1803, Castillon, op. cit.; and especially Jeffries Wyman, Notes on the cells of the bee, Proc. Amer. Acad. Sci. and Arts, vn, 1868. 532 THE FORMS OF TISSUES [CH. well they were. The old misunderstanding was at last explained and corrected by Leslie Ellis; and better still by Glaislier, in a Uttle-known but very beautiful paper*. For these two mathe- maticians shewed that, though Maraldi's account of his "measure- ments" led to misunderstanding, yet he had really done well and scientifically when he eked out a rough observation by finer theory, and deemed himself entitled thereby to discuss the cell and its angles in the same precise terms that he would use as a mathematician in speaking of its geometrical prototype f. y Many diverse proofs { have been given of the minimal character of the bee's cell, some few, hke Maclaurin's, purely geometrical, others arrived at by help of the calculus. The following seems as simple as any: ABCDEF, abcdef, is a right prism upon a regular hexagonal base. The corners B, D, F are cut oif by planes through the fines AC, CE, EA, meeting in a point V on the axis VN of the prism, and intersecting Bb, Dd, Ff, in Z, Y, Z. The volume of the figure thus formed is the same as that of the L original prism with its hexagonal ends : for, if the axis cut the hexagon ABCDEF in N, the volumes ACVN, ACBX are equal. pjg 205. It is required to find the inchnation to the axis of the faces forming the / L.eslie Ellis, On the form of bees' cells, in Mathematical and other Writings, 1863, p. 353; J. W. L. Glaisher, do., Phil. Mag. (4), xlvi, pp. 103-122, 1873. t The learned and original Kieser, in his Memoire sur Vorganisation des plantes, 1812, p. iv, gives advice to the same effect: "II est indispensable de se former, avant de dessiner, une idee de Tobjet dans sa plus grande perfection, et de dessiner selon cette idee, et non pas I'objet plus ou moins imparfait, plus ou moins altere par le scalpel.. Voila la methode qu'ont suivi Haller, Albinus et tous les autres grands anatomistes. . . . Mais il faut employer pour cela la plus grande precaution, la circonspection la plus tranquille pour I'observation, etc." I Cf. Koenig, Lhuiller and Boscovich, opp. cit. ; H. Hennessy, Proc. E.S. xxxix, p. 253, 1885; XLi, pp. 4^2, 443, 1886; XLii. pp. 176, 177, 1887. VII] OF THE BEE'S CELL 533 trihedral angle at F, such that the surface of the whole figure may be a minimum. Let the angle NVX, which is the inclination of the plane of the rhonlbus to the axis of the prism, = d\ the side of the hexagon, as AB, = s\ and the height, as Aa, = h. Then AC = 25 cos 30°, = s VS. And, from inspection of the triangle LXB, VX sin 6 Therefore the area of the rhombus VAXC V3 2sin^* And the area of AabX = -{2h— ^VX cos 6), 2i " = ' (2A - \s cot Q). z Therefore the total area of the figure = the hexagonal base ahcdef + 3s (2h — |s cot 6) + 3 —-. — -, (^(area) Ss^ / 1 a/3 cos ^ Therefore — j^ — = -^ ~^~t^ . o n ad 2 Vsm^ d sm^ d. \ ^ , . . . , d (area) ^ , ^1 But this expression vanishes, or = 0, when cos U = -— =, ^ dd V3 that is to say, when 6 = 54° 44' 8" = } (109° 28' 16"). Such then are the conditions under which the total area of the figure has its minimal value. The following is, in substance, Maclaurin's elementary but somewhat lengthy proof of the minimal properties of the bee's cell, using "no higher Geometry than was known to the Antients." Let ABCD, abed, represent one-half of a right prism on a regular hexagonal base; and let AabE, EbcC be the trapezial portions of two adjacent sides, to which one of the three rhombs, AECe, is fitted. Let 0 be the centre of the hexagon, of which AB, EC are adjacent sides; join ^C and OB, intersecting in P. Then, because AOC = ABC, and BE = Oe, 534 THE FORMS OF TISSUES [CH. the solid AECB = AeCO; whence it appears that the solid content of the whole cell will be the same, wherever the point E be taken in Bb, and will in fact be identical with the content of the hexagonal prism. We have -then to enquire where E is to be taken in Bb, in order that the combined surfaces of the rhomb and of the two trapezia may be a minimum. Because Ee is perpendicular to ^C in P, the area of the rhombus = P^.^C'; and the area of the two trapezia = {Aa + Eb) x BC. The total area in question, then, is PE .AC + 2BC .Bb-BC. BE. But BC . Bb is constant ; so the question remains. When is PE.AC — BC.BE a minimum? Let a point L be so taken in Bb that BL : PL :: BC : AC. From the centre P, in the plane PBE, describe the circular arc ER, meeting PL in R; and on PL let fall the perpendiculars ES, BT. Fig. 206. The triangles LES, LBT, LPB are all similar. Therefore LS:LE::LT:LB::LB:LP, and (by hypothesis) : : BC : AC. Hence {LT -LS) : (LB-LE) :: BC : AC, i.e. ST: BE:: BC: AC. Therefore ST.AC = BE.BC and consequently, PE .AC -BE .BC = PE .AC -ST .AC = AC{PE-8T). But PE^PR; therefore AC {PR - ST) ^ AC {PT + RS). VII OF THE BEE'S CELL 535 But AC and FT do not vaiy, while BS varies with the position of E. Accordingly, AC{PT + RS), or PE. AC-BE. BC, is a minimum when RS vanishes: that is to say, when E coincides with L. "Therefore A LCI is the Rhombus of the most advantageous Form in respect of Frugality, when BL is to PL as BC to .46'." Again, since OB = BC, and OP = PB, BC^ = 4PB\ and PC^ = 3PB^, and AC = 2PC = 2V3.PB. Therefore and, by hypothesis. and Therefore BC:AC::2PB:2VS.PB:: 1 : Vs, BLiPL: : BC : AC = 1 Vs PLiPB:: VS : V2, PBiPC : 1 : V3. PL: PC: : 1 : V2. "That is, the angle CLP is that whose Tangent is to the Radius as V2 is to 1, or as 1-4142135 to 1-0000000; and therefore is of 54° 44' 08", and consequently the Angle of the Rhombus of the Best Form is that of 109° 28' 16".". When we have thus ascertained that the characteristic angles of the rhombs are 109° 28' 16" and its supplement 70° 31' 44", the cosine of which latter angle is 1/3, the construction of a model is of the easiest. Fig. 207. Construction of a model of the bee's cell On AD make AB = BC = CD. Let AF = AD meet the perpen- dicular BE in F. Then the angle BAF (whose cosine = 1/3, or whose tangent = 2\/2) = 70° 31' 44". Complete the rhomb ADGF, and repeat three times as indicated. Make a developed hexagonal prism with sides ab, be, = BF . Cut away angles bb'a, bb'c, etc., = BAF. Fold, and attach together. 536 THE FORMS OF TISSUES [ch. A soap-bubble, or soap-film, assumes a minimal configuratiou instantaneously*, however small the saving of surface-area may be. But after learning that the bee's cell has undoubted minimal properties, we should hke to know what saving is actually obtained by substituting a rhomboidal pyramid for a plane base in the hexagonal prism. It turns out, after all, to be a small matter! The calculation was first made by Maclaurin and by Lhuiller, in both cases briefly but correctly. Lhuiller stated that the whole amount used in the bee's cell was to that required for a flat-topped prismatic cell of equal volume as 25 + V6 (or 2745) to 28, the saving being thus a little more than 2 per cent, of the whole quantity of wax required t- Glaisher recalculated the values, taking the cell part by part. Assuming, with Lhuiller, that the radius of the inscribed circle of the hexagon is to the depth of the prismatic cell, when the latter has the same capacity as the real cell, as 1| to 5, then, taking the side of the hexagon as unity, we have for the same depth (viz. the longest side of the trapezium in the real cell) the 25 VS value ; and then (to three places of decimals): Area of the three rhombs, f a/2 „ „ „ six triangles, f V2 „ „ „ six sides of the equivalent prismatic cell, -^^ V 3 „ „ „ hexagonal base, f VS The whole surface of the real cell, accordingly, = (i)-f-(iii)-(ii) = 23-772; * For the most part instantaneously; but sometimes, when there are two positions of nearly equal potential energy, the film "creeps" from the less to the more advantageous of the two. t We must take into account the depth of the cell, or assume a value for it, if we are to estimate the percentage saving of wax on the whole construction. But (as Dr G. T. Bennett says) the whole saving is on the roof, and the height of the house does not matter; the question rather is, what is saved on the rhomboidal sloping roof compared with a flat one? If the short axis of the rhombs be 2 units (the edge of the cube), then 3 rhombs have area 6 V2, the wall-saving is 2 V2, while the flat hexagonal top is 4 Vs. So the actual saving is the diflFerence between 4 V2 and 4 Vs — which looks much less negligible ! But it is only on a small portion of the work. 3-182 (i)- 1-061 (ii). 21-651 (iii). 2-598 (iv). VII] OF THE BEE'S CELL 537 and that of the flat-bottomed hexagonal prism = (iii) + (iv) = 24-249; , 24-249 102 ,.^^. 23772 "" 100 ' ^^ ^ ^^^^ ' so that the saving in the former case amounts to about 2 per cent., as Lhuiller had found it to be. Glaisher sums up the matter as follows: "As the result of a tolerably careful examination of the whole question, I may be permitted to say that I agree with Lhuiller in beheving that the economy of wax has played a very subordinate part in the deter- mination of the form of the cell ; in fact I should not be surprised if it were acknowledged hereafter that the form of the cell had been determined by other considerations, into which the saving of wax did not enter (that is to say did not enter sensibly; of course I do not mean that the amount of wax required was a matter of absolute indifference to the bees). The fact of all the dihedral angles being 120° is, it is not unlikely, the cause that determined the form of the cell." This last fact, that in such a cell every plane cuts every other plane at an angle of 120°, was known both to Kliigel and to Boscovich; it is no mere corollary, but the root of the matter. It is, as Glaisher indicates, the fundamental physical principle of construction from which the apical angles of 109° follow as a geometrical corollary. And it is curious indeed to see how the obtuse angle of the rhomb, and its cosine —J, drew attention all the while; but the dihedral angle of 120° of the rhombohedron, and the inclination of its three short diagonals at 90° to one another, got rare and scanty notice. Darwin had hstened too closely to Brougham and the rest when he spoke of the bee's architecture as "the most wonderful of known instincts"; and when he declared that "beyond this stage of per- fection in architecture natural selection could not lead ; for the comb of the hive-bee, as far as we can see, is absolutely perfect in economising labour and wax." The minimal properties of the cell and all the geometrical reasoning in the case postulate cell -walls of uniform tenuity and edges which are mathematically straight. But the walls, and still more their edges, are always thickened ; the edges are never accurately straight, 538 THE FORMS OF TISSUES [CH. nor the cells strictly horizontal. The base is always thicker than the side- walls; ,its solid angles are by no means sharp, but filled up with curving surfaces of wax, after the fashion, but more coarsely, of Plateau's bourrelet. Hence the Maraldi angle is seldom or never attained; the mean value (according to Vogt) is no more than 106-7° for the workers, and 107-3° for the drones. The hexagonal angles of the prism are fairly constant; about 4° is the limit of departure, and about 1-8° the mean error, on either side. r^-t^ Fig. 208. Brood-comb, with eggs. The bee makes no economies; and whatever economies lie in the theoretical construction, the bee's handiwork is not fine nor accurate enough to take advantage of them*. The cells vary little in size, so little that Thevenot, a friend of Swammerdam's, suggested using their dimensions as a modulus or standard of length; but after all, the constancy is not so great as has been supposed. Swammerdam gives measurements which work out at 5-15 mm. for the mean diameter of the worker-cells, and 7 mm. for those of the drones ; Jeffries Wyman found mean values for the worker-cells from 5-1 to 5-2 mm.; Vogt, after many careful measurements, found a mean of 5-37 mm. for the worker- * All this Heinrich Vogt has abundantly shewn, in part by making casts of the interior of the cells, as Castellan had done a hundred years before. See his admirable paper on the Geometric und Oekonomie der Bienenzelle, in Festschrift d. Universitdt Breslau, 1911, pp. 27-274. VII OF THE BEE'S CELL 539 cells, with an insignificant difference in the various diameters, and ■a mean of 6-9 for the drone-cells, with their horizontal diameter some- what in excess, and averaging 7-1 mm. A curious attempt has been made of late years by Italian bee-keepers to let the bees work on a larger foundation, and so induce them to build larger cells; and some, but by no means all, assert that the young bees reared in the larger cells are themselves of larger stature*. 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 construction. 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 (among some humble- bees) the old cocoons are used to hold honey, the cells built by the "solitary" wasps and bees are of various kinds. They may be formed by partitioning off little chambers in a hollow stem; they may be rounded or oval capsules, often very neatly constructed out of mud or vegetable fibre or httle 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, for 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." * Cf. (int. al.) H. Gontarsi, >Sammelleistungen von Bienen aus vergrosserten Brutzellen, Arch.f. Bienenkunde, xvi, p. 7, 1935; A. Ghetti, Celli ed api piu grandi, IV Congresso nazion. della S.A.I. 1935. 540 THE FORMS OF TISSUES [CH. But the spherical form is only seen on the outside of the mass; for inwardly each cell is flattened into "two, three or m.ore 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 Fig. 209. An early stage of a wasp's nest. Observe the spherical caps, and the irregular shape of the peripheral cells. After R. Bott. 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*." * Origin of Species, ch. viii (6th ed., p. 221). The cells of various bees, humble- bees and social wasps have been described and mathematically investigated by K. MiillenhofF, Pfiuger'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 Buttel-Reepen, Biol. Centralbl. xxxm, pp.. 4, 89, 129, 183, 1903. VII] OF THE BEE'S CELL 541 We had better be content to say that it depends on the same elementary geometry. The question is, To what particular force are we to ascribe the plane surfaces and definite angles which define the sides of the cell in all these cases, and the ends of the cell in cases where one row meets and opposes another? We have seen that Bartholin sug- gested, and it is still commonly beheved, that this result is due to mere 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*. That the bee, if left to itself, "works in segments of circles," or in other words builds a rounded and roughly spherical cell, is ah old contention! which some recent experiments of M. Victor Willem amplify and confirm {. M. Willem describes vividly how each cell begins as a Httle hemispherical basin or "cuvette," how the workers proceed at first with Httle apparent order and method, laying on the wax roughly like the mud when a swallow builds; how presently they concentrate their toil, each burying its head in its own cuvette, and slowly scraping, smoothing and ramming home; how those on the other side gradually adjust themselves * 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- possible, from the extreme thinness of the little plate, that they could have effected this by gnawing away the convex side; and I suspect that the bees in such cases stand on opposite sides and push and bend the ductile and warm wax (which as I have tried is easily done) into its proper intermediate plane, and thus flatten it." Ruber thought the difference in form between the inner and the outer cells a clear proof of intelhgence; it is really a direct proof of the contrary. And while cells differ when their situations and circumstances differ, yet over great stretches of comb extreme uniformity, unbroken by any sign of individual differences, is the strikingly mechanical characteristic of the cells. I It is so stated in the Penny Cyclopedia, 1835, Art. "Bees"; and is expounded by Mr G. H. Waterhouse [Trans. Entom. Soc, London, ii, p. 115, 1864) in an article of which Darwin made good use. Waterhouse shewed that when the bees were given a plate of wax, the separate excavations they made therein remained hemispherical, or were built up into cylindrical tubes; but cells in juxtaposition with one another had their party-walls flattened, and their forms more or less prismatic. X V^ictor Willem, L'architecture des abeilles, Bull. Acad. Roy. de Belgique (5), XIV, pp. 672-705, 1928. 542 THE FORMS OF TISSUES [ch. to their opposite neighbours ; and how the rounded ends of the cells fashion themselves into the rhomboidal* pyramids, "a la suite de I'amincissement progressif des cloisons communes, et des pressions antagonistes exercees sur les deux faces de ces cloisons." Among other curious and instructive observations, M. Willem has watched the bees at work on the waxen "foundations" now com- monly used, on which a rhomboidal pattern is impressed with a view to starting the work and saving the labour of the bees. The bees (he says) disdain these half-laid foundations of their cells; they hollow out the wax, erase the rhombs, and turn the pyramidal hollows into hemispherical "cuvettes" in their usual way; and the vertical walls which they raise, more or less on the lines laid down for them, are not hexagonal but cylindrical to begin with. "La forme plane, en facettes, tant de prismes que des fonds, n'est obtenue que plus tard, progressivement, comme resultat de retouches, d'enlevements et de pressions exercees sur les cloisons qui s'amin- cissent, par des groupes d'ouvrieres operant face a face, de maniere antagoniste." But when all is said and done, it is doubtful whether such retouches, enlevements and pressions antagonistes, such mechanical forces intermittently exercised, could produce the nearly smooth surfaces, the all but constant angles and the close approach to a minimal configuration which characterise the cell, whether it be constructed by the bee of wax or by the wasp of papery pulp. We have the properties of the material to consider; and it seems much more Ukely to me that we have to do with a true tension effect: in other words, that the walls assume their configuration when in a semi-fluid state, while the watery pulp is still liquid or the wax warm under the high temperature of the crowded hive. In the first few cells of a wasp's comb, long before crowding and mutual pressure come into play, we recognise the identical con- figurations which we have seen exhibited by a group of three or four soap-bubbles, the first three or four cells of a segmenting egg. The direct efforts of the wasp or bee may be supposed to be limited, at this stage, to the making of little hemispherical cups, as thin as the nature of the material permits, and packing these little round cups as close as possible together. It is then con- ceiv ble, and indeed probable, that the symmetrical tensions of the VII] OF THE BEE'S CELL 543 semi-fluid films should suffice (however retarded by viscosity) to bring the whole system into equihbrium, that is to say into the configuration which the comb actually assumes. The remarkable passage in which Bufi'on 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 before we leave the subject I wull 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'en- thousiasme et Tadmiration; cette figure, toute geometrique et toute reguliere qu'elle nous parait, et qu'elle est en effet dans la specula- tion, n'est ici qu'un resultat niecanique et assez imparfait qui se trouve souvent dans la nature, 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 fleurs, 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, tons ces cylindres deviendront de colonnes a six pans. On y voit clairement la raison, qui est purement mecanique; chaque graine, dont la figure est cyhndrique, tend par son renflement a occuper le plus d'espace possible dans un espace donne, elles deviennent done toutes neces- sairement 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 * Buffon, Histoire naturelle, iv, p. 99, Paris, 1753. Bonnet criticised BuflFon's explanation, on the ground that his description was incomplete; for Buffon took no account of the Maraldi pyramids. Not a few others discovered impiety in his hypotheses, and some dismissed them with the remark that "philosophical absurdities are the most difficult to refute"; cf. W. Smellie, Philosophy of Natural History, Edinburgh, 1790, p. 424. 544 THE FORMS OF TISSUES [ch. 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 r architecture, mais dont les constructions 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 nuUement 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*." Of parenchymatous cells Just as Bonanni and other early writers sought, as we have seen, to explain hexagonal symmetry on mechanical principles, so other early naturalists, relying more or less on the analogy of the bee's cell, endeavoured to explain the cells of vegetable parenchyma ; and to refer them to the rhombic dodecahedron or garnet-form, which sohd figure, in close-packed association, was believed in their time, and long afterwards, to enclose space with a minimal extent of surface. * Among countless papers on the bee's cell, see John Barclay and others in Ann. of Philosophy, ix, x, 1817; Henry Lord Brougham, in Dissertations... connected with Natural Theology, app. to Paley's Works, i, pp. 218-368, 1839; C.R. Acad. Sci. Paris, xlvi, pp. 1024-1029, 1858; Tracts, Mathematical and Physical, 1860, pp. 103-121, etc.; E. Carruccio, Note storiche sulla geometria delle api, Periodico di Matem. (4), xvi, 20 pp., 1936; G. Cesaro, Sur la forme de I'alveole des abeilles. Bull. Acad. Roy. Belg. (Sci.), Avril 10, 1929; Sam. Haughton, On the form of the cells made by various wasps and by the honey-bee, Proc. Nat. Hist. Soc. Dublin, in, pp. 128-140, 1863; Ann. Mdg. Nat. Hist. (3), xi, pp. 415-429, 1863; A. R. Wallace, Remarks on the foregoing paper, ibid, xii, p. 33; J. 0. Hennum, Arch. f. Math. u. Vidensk., Christiania, ix, p. 301, 1884; F. Huber, Nouv. obs. sur les abeilles, ii, p. 475, 1814; F. W. Hultmann, Tidsskr.'f. Math., Uppsala, I, p. 197, 1868; John Hunter, Observations on bees, Phil. Trans. 1792, pp. 128-195; Jacob, Nouv. Ann. de Math, ii, p. 160, 1843; G. S. Kliigel, Mathem. Betrachtungcn iib. d. kunstreichen Bau d. Bienenzellen, Hannoversches Mag. 1772, pp. 353-368; Leon Lalanne, Note sur I'architecture des abeilles, Ann. Sc. Nat. Zool. (2), XIII, pp. 358-374, 1840; B. PoweU, Proc. Ashmol. Soc. i, p. 10, 1844; K. H Schellbach, Mathem Lehrstunde • Lehre v. Grossten u. Kleinsten, 1860, pp. 35-37; Sam Sharpe, Phil. Mag iv pp 19-21, 1828; J. E. Siegwart, Die Mathematik im Dienste d. Bienenzucht, Schw. Bitnenzeitung, iii, 1880; 0. Terquem, Nouv. Ann. de M h xv, p. 176, 1856, C. M. Willick, On the angle of dock-gates and the bee's cell, Phtl. Mag. (4) xviii, p. 427, 1859; C.R. li, p. 633, 1860; Chauncy Wright, Proc. Amer. Acad. Arts and Sci. iv, p. 432, 1860. VII] OF THE PARTITIONING OF SPACE 545 We have mentioned both Hooke and Grew*, and we have just heard Buffon engaged in such speculations; but the matter was more elaborately treated near the beginning of last century by Dieterich George Kieserf, an ingenious friend and colleague of the celebrated Lorenz Oken. Kieser clearly understood that the cell has not a shape of its own, but merely one impressed on it by physical forces and defined by mathematical laws. In his Memoire sur V organisation des plantes, he gives an admirable historical account of the work of Malpighi, Hooke, Grew, John Hill and other early microscopists ; and then he Fig. 210. A rhombic says "La forme des cellules est variee dans dodecahedron. les plantes diiferentes, mais il y a des formes principales, fondees sur les lois des mathematiques, que la nature suit toujours dans ses formations.. . .La forme la plus commune est celle que prennent necessairement des globules rondes ou allongees, pressees ensemble, celle des corps hexagonaux a parois quadrilaterales, ou d'une colonne tres courte hexagone, coupee horizontalement d'en haut et d'en bas." Here we have, briefly described and sufficiently accounted for, the configuration of what we call a "pavement epithelium," or other simple association of cells in a single layer. But another passage (from the same author's Phytotomie) is worth quoting at length, where he deals with cells in the mass, that is to say with the three-dimensional problem. "Die nach mathematische Gesetzen bestinlmte als nothwendige Grundforra der Zelle der voll- kommenen Zellengewebe ist das langgezogene Rhombododekaheder. . . . Mathematisch liegt das Beweis dass diese Figur die Grundform der vollkommenen Zellengewebe sei darin, dass unter alien mathe- matischen Korpern welche durch Zusammensetzung einen sohden Korper ohne Zwischenraume bilden, das Rhombododekaheder die einzige ist welche mit der wenigsteri Masse des Umkreises den grossten Raum einschliesst. Sollte also aus dem Globus — dem * R. Hooke, Micrographia, 1665, pp. 115-116; Nehemiah Grew, Anatomy of Plants, 1682, pp. 64, 76, 120. t D. G. Kieser, Memoire, etc., Haarlem, 1814, p. 89; Phytotomie, oder Grundziige der Anatomic der Pfianzen, Jena, 1815, p. 4. 546 THE FORMS OF TISSUES [ch. urspriinglichsten Schleimblaschen der Pflanzenzelle — ein eckiger Korper gebildet werden, so musste dieser das Rhombododekaheder sein, well dieser- im Hinsicht des Minimums der Masse zu dem Maximum des eingeschlossenen Raumes dem Globus am nachsten liegt. Als die Urform der Pflanzenzelle ist nicht Globus sondern Ellipsoide, daher muss das Dodekaheder, welche die Grundform der eckigen Pflanzenzelle ist, auch aus dem Ellipsoide entstanden sein. Das Rhombododekaheder wird also vom unten nach oben gestreckt, und die Grundform der eckigen Pflanzenzelle ist das in perpen- dicularer Richtung langsgestreckte Rhombododekaheder." These views and speculations of Kieser's, now all but forgotten, were by no means neglected in their day. Oken accepted them, and taught them*; Schleiden remarks that "the form of cells frequently passes into that of the rhombic dodecahedron, so beauti- fully determined, a priori, by Kieserf"; and De CandoUe thought it necessary to warn his readers that cells are not as geometrically regular as pubhshed figures might lead one to believe J. The same principles apply to various orders of magnitude, and close-packing may be seen even in the inner contents of a cell. In vitally stained "goblet-cells," the mucin gathers into clumps or droplets, of which each appears in optical section to be surrounded by six more. When fixed they draw together, appear in optical section to be hexagonal, and we may take it that they have become, to a first approximation, rhombic dodecahedra§. These then, and such as these, were the not unimportant specu- lations on the forms of cells by men who early grasped the fact that form had a physical cause and a mathematical significance. But their conception of the phenomenon was of necessity hmited to the play of the mechanical forces; for Plateau's Statique des Liquides had not yet shewn what the capillary forces can do, nor opened a way thereby for Berthold and for Errera. A very beautiful hexagonal symmetry as seen in section, or dodecahedral as viewed in the soHd, is presented by the pith of certain rushes (e.g. J uncus effusus), and somewhat less diagram- * Oken, Physiophilosophy (Ray Society), 1847, p. 209. t Muller's Archiv, 1838, p. 146. X Organogenic vegetale, i, p. 13, 1827. § E. S..Duthie, in Proc, B,S. (B), cxiii, pp. 459-463, 1933. VII] OF STELLATE CELLS 547 matically by the pith of the banana. The cells are stellate, and the tissue has the appearance in section of a network of six-rayed stars (Fig. 211), linked together by the tips of the rays, and separated by symmetrical, air-filled intercellular spaces, which give its snow- like whiteness to the pith. In thick sections, the solid twelve-rayed " star-dodecahedra " may be very beautifully seen under the binocular microscope. They are not difficult to understand. Imagine, as before, a system of equal spheres in close contact, each one touching its twelve neighbours, six of them in the equatorial Fig. 211. Stellate cells in pith of Juncus. 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 tend to shrink or shrivel up by the withdrawal of fluid from its interior. The result will be to enlarge the intercellular spaces ; the attachments of each cell to its neighbours will remain fixed, but the walls between these points of attachment will be withdrawn in a symmetrical fashion towards the centre. As the final result w^e have the star- dodecahedron, which appears in plane section as a six-rayed figure. It is necessary not only that the pith-cells should be attached to one another, but also that the outermost should be attached to a boundary wall, to preserve the symmetry of the system. What 548 THE FORMS OF TISSUES [ch. actually occurs in the rush is tantamount to this, but not absolutely identical. It is not so much the pith-cells which tend to shrink within a boundary of constant size, but rather the boundary wall which continues to expand after the pith-cells which it encloses have ceased to grow or to multiply. The points of attachment on the surface of each little pith-cell are drawn asunder, but the content of the cell does not correspondingly increase; and the remaining portions of the surface shrink inwards, accordingly, and gradually constitute the complicated . figure which Kepler called a star- dodecahedron, which is still a symmetrical figure, and is still a surface of minimal area under the new and altered conditions. The tetrakaidekahedron A few years after the pubhcation 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 liquid ''foam," in which the general problem is completely and automatically solved. The general mathematical solution of the problem (as we have already indicated) is, that every interface or partition-wall must have con- stant mean curvature throughout ; that where these partitions meet in an edge, they must intersect at angles such that equal forces, in planes perpendicular to the line of intersection, shall balance ; that no more than three such interfaces may meet in a line or edge, whence it follows (for symmetry) that the angle of intersection of all surfaces or facets must be 120°; and that neither more nor less than four edges meet in a point or corner. An assemblage of rhombic dodecahedra goes far to meet the case. It fills space; its surfaces or interfaces are planes, and therefore surfaces of constant curvature throughout; and they meet together at angles of 120°. Never^ theless, the proof that the rhombic dodecahedron (which we find exemplified in the bee's cell) is a figure of minimal area is not a comprehensive proof; it is Hmited to certain conditions, and * Sir W. Thomson, On the division of space with minimum partitional acea, Phil. Mag. (5), xxiv, pp. 503-514, Dec. 1887; cf. Baltimore Lectures, 1904, p. 615; Molecular tactics of a crystal (Robert Boyle Lecture), 1894, pp. 21-25. VII OF THE PARTITIONING OF SPACE 549 practically amounts to no more than this, that of the ordinary space-filling soHds 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. At each of the latter three, and at each of the former six, dodecahedra meet in a point in close packing; and foiir edges meet in a point in the one case and eight in the other. This is enough to shew that' the conditions for minimal area are not rigorously metr In one of Plateau's most beautiful experiments*, a wire cube is dipped in soap-solution. When hfted out, a film is seen to pass inwards from each of the twelve edges of the cube, and these twelve films meet, three by three, in eight edges, running inwards from the eight corners of the cube; but the twelve films and their eight edges do not meet in a point, but are grouped around a small central quadrilateral fihn (Fig. 212). Two of the eight edges run to each corner of the little square, and, with the two sides of the square itself, make up the four edges meeting in a point which the theory of area minima requires. We may sub- ^^' stitute (by a second dip) a little cube for the little square; now an edge from each corner of the outer cube runs to the corresponding corner of the inner one, and with the three adjacent edges of the little cube itself the number four is still maintained. Twelve films, and eight edges meeting in a point, w^ere essentially unstable; but the introduction of the little square or cube meets most of the conditions of stabihty which Plateau was the first to lay down. One more condition has to be met, namely the equality of angles at which the four edges meet in each conjunction. These co-equal "Maraldi angles" at each corner of the square can only be con- structed by help of a slight curvature of the sides, and the httle square is seen to have its sides curved into circular arcs accordingly; moreover its size and shape, as that of all the other films in the system, are perfectly definite. It is all one, according to the * Also discovered independently by Sir David Brewster, Trans. R.S.E. xxrv, p. 505, 1867; xxv, p. 115, 1869. 550 THE FORMS OF TISSUES [ch. symmetry of the figure, to which side of the skeleton cube the square lies parallel; wherever it may be, if we blow gently on it, then (as M. Van Rees discovered) it alters its place and sets itself parallel now to one and now to another of the paired faces of the cube. The skeleton cube, Hke the tetrahedron which we have already studied, is only one of many interesting cases; for we may vary the shape of our wire cages and obtain other and not less beautiful configurations. An hexagonal prism, if its sides be square or nearly so, gives us six vertical triangular films, whose apices meet the corners of a horizontal hexagon*; also six pairs of truncated triangles, which link the top and bottom edges of the cage to the sides of the median hexagon. But if the height of the hexagonal prism be increased, the six vertical films become curvihnear triangles, with sides concave towards the apex; and the twelve remaining films, which spring from the top and bottom of the hexagon, are curved surfaces, looking like a sort of hexagonal hourglass f. There is a deal of elegant geometry in these various configurations. Lamarle shewed that if, in a figure represented by our wire cage, we suppress (in imagination) one face and all the other faces adjoining it, then the faces which remain are those which appear in the centre of the figure after the cage has been withdrawn from the soap-solution. Thus, in a cube, we suppress one face and the four adjacent to it; only one remains, and it reappears as the central square in the middle of the new configuration; in the tetrahedron, when we have suppressed one face and the three adjacent to it, there is nothing left — save, a median point, corresponding to the opposite corner. In a regular dodecahedron, if we suppress one pentagonal face and its five neighbours, the other half of the whole figure remains; and the dodecahedral cage, after immersion in the soap, shews a central and symmetrical group of six pentagons J. Moreover, while the cage is carrying its configuration of films, we may blow a bubble within it, and so insert a new polyhedron * The angles of a hexagon are too big, as those of a square were too small, to form the Maraldi angles of symmetry; hence the sides of the hexagon are found to be concave, as those of the square bulged out convexly. t Cf. Dewar, op. cit. 1918. X That is to say, if nF„^ be a polyhedron (of n m-faced sides), the corresponding wire cage will exhibit (w - m + 1) i^^ as central fenestrae. VII] OF THE TETRAKAIDEKAHEDRON 551 within the old, and set it in place of the former fenestra. The inner polyhedral bubble so produced may be of any dimensions, but it resembles the outer polyhedral cage precisely, except in the curvature of its sides ; it has all its faces spherical, and all of equal radius of curvature; its edges are either arcs of circles or straight lines. Later on, we shall see that there is no small biological interest attaching to these configurations. Lord Kelvin made the remarkable discovery that the square fenestra with the four quadrilateral films impinging on its sides, in Plateau's experiment, represented the one-sixth part of a symmetrical figure; that this figure when complete was bounded by six squares and eight hexagons; that by means of an assemblage of these Fig. 213. A set of 14-he(lra, to shew close-packing. From F. T. Lewis fourteen-sided figures, or "tetrakaidekahedra," space is filled and homogeneously partitioned — into equal, similar and similarly situated cells — with an economy of surface in relation to volume even greater than in an assemblage of rhombic dodecahedra*. The tetrakaidekahedron, in its most generalised case, is bounded by three pairs of equal and opposite quadrilateral faces, and four pairs of equal and opposite hexagonal faces, neither the quadri- laterals nor the hexagons being necessarily plane. In its simplest case, with all its facets plane and equilateral, it is Kelvin's "ortho- tetrakaidekahedron " ; and also (though Kelvin was unaware of the fact) one of the thirteen semi-regular and isogonal polyhedra, or "Archimedean bodies." In a particular case, the quadrilaterals are plane surfaces with curved edges, but the hexagons are slightly * Kelvin, Boyle Lecture and Baltitnore Lectures. In the first of these Kelvin described the plane-faced tetrakaidekahedron; in the second he shewed how that figure must have its faces warped and edges curved to fulfil all the conditions of minimal area. 552 THE FORMS OF TISSUES [ch. 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. This figure has the remarkable property that, hke the plane rhombic dodecahedron, it so partitions space that three faces meeting in an edge do so everywhere at co-equal angles of 120°; and, unlike the rhombic dodecahedron, four edges meet in each point or corner at co-equal angles of 109° 28'*. We may take it as certain tha^), in a homogeneous system of fluid films Hke the interior of a froth of soap-bubbles, where the films are perfectly free to glide or turn over one another and are of approximately co-equal size, the mass is actually divided into cells of this remarkable conformation: and the possibility of such a configuration being present even in the cells of an ordinary vegetable pareilchyma was suggested in the first edition of this book. It is all a question of restraint, of degrees of mobihty or fluidity. If we squeeze a mass of clay pellets together, hke Buffon's peas, they come out, or all the inner ones do, in neat garnet-shape, or rhombic dodecahedra. But a young student once shewed me (in Yale) that if you wet these clay pellets thoroughly, so that they slide easily on one another and so acquire a sort of pseudo-fluidity in the mass, they no longer come out as regular dodecahedra, but with square and , hexagonal facets recognisable as those of ill-formed or half- formed tetrakaidekahedra. Dr F. T. Lewis has made a long and careful study of various vegetable parenchymas, by simple maceration, wax-plate recon- * Von Fedorov/ had already described (in Russian), unaware that Archimedes had done so, the same figure under the name of cubo-octahedron, or hepta- parallelohedron^ hmited however to the case where all the faces are plane and regular. This cubo-octahedron, together with the cube, the hexagonal prism, the rhombic dodecahedron and the "elongated dodecahedron," constitute the five plane-faced, parallel-sided figures by which space is capable of being completely filled and uniformly partitioned; the series so forming the foundation of Von Fedorow's theory of crystalline structure — though the space-fillers are not all, and cannot all be, crystalline forms. All of these figures, save the hexagonal prism, are related to and derivable from the cube, so we end by recognising two principal types, cubic and hexagonal. We have learned to recognise the dodecahedron, and we may find in still closer packing the cubo-octahedron, in a parenchyma; the elongated dodecahedron is, essentially, the figure of the bee's cell; the cube we have, in essence, in cambium -tissue; the hexagonal prism, dwarf or tall, simple or recognisably deformed, we see in every epithelium. vn] OF THE TETRAKAIDEKAHEDRON 553 struction and otherwise, and has succeeded in shewing that the tetrakaidekahedral form is closely approached, or even attained, in certain simple and homogeneous tissues. After reconstructing a large model of the cells of elder-pith, he finds that the fourteen-sided figure clearly manifests itself as the characteristic or typical form to which the cells approximate, in spite of repeated cell-divisions and consequent inequalities of size. Counting in a hundred cells the number of contacts which each made with its neighbours, that is to say the total number both of actual and potential facets, Lewis found that 74 per cent, of the cells were either 12, 13, 14, 15 or 16-sided, 56 per cent, either 13, 14 or 15-sided, and that the averagje -•-^ Fig. 214. Reconstructed models of cells of elder-pith, shewing a certain approximation to 14-hedral form. From F. T. Lewis. number of facets or contacts was, in this instance, just 13-96. These figures indicate the general symmetry of the cells, their departure from the dodecahedral, and their tendency towards the tetra- kaidekahedral, form*. But after all, the geometry of the 14-hedron, displayed to per- fection by our soap-films in the twinkhng of an eye, is only roughly developed in an organic structure, even one so delicate as elder- pith; the conditions are no longer simple, for friction, viscosity and * F. T. Lewis, The typical shape of polyhedral cells in vegetable parenchyma, and the restoration of that shape following cell-division, Proc. Amer. Acad, of Arts and Sci. lviii, pp. 537-552, 1923, and other papers. See also {int. al.) J. W. Marvin, The aggregation of orthis-tetrakaidekahedra. Science, lxxxiii, p. 188, 1936; E. B. Metzger, An analysis of the orthotetrakaidekahedron, Bull. Torrey Bot. Club, Liv, pp. 341-348, 1927. Professor van Iterson of Delft tells me that Asparagus Sprengeri (a common greenhouse plant) is a good subject for shewing the 14-hedral cells. 554 THE FORMS OF TISSUES [ch. solidification have vastly complicated the case. We get a curious and an unexpected variant of the same phenomenon in the micro- scopic foam-like structure assumed as molten metal cools. If these foam-cells were again 14-hedra, their facets would all be either squares or hexagons; but pentagonal facets are commoner than either, and the cells often approach closely to the form of a regular pentagonal dodecahedron ! The edges of this figure meet at angles of 108°, not far from the characteristic Maraldi angle of 109° 28'; and the faces meet at an angle not far removed from 120°. A slight curvature of the sides is enough to turn our pentagonal dodecahedron into a possible figure of equihbrium for a foam-cell. We cannot close-pack pentagonal dodecahedra, whether equal or unequal, so as to fill space; but still the figure may be, and seems to be, common, interspersed among the polyhedra of various shapes and sizes which are packed together in a metallic foam*. A somewhat similar result, and a curious one, was found by Mr J. W. Marvin, who compressed leaden small-shot in a steel cylinder, as Buffon compressed his peas; but this time the pressure on the plunger ran from 1000 to 35,000 lb. or nearly twenty tons to the square inch. When the shot was introduced carefully, so as to lie in ordinary close packing, the result was an assemblage of regular rhombic dodecahedra, as might be expected and as Buffon had found. But the result was very different when the shot was poured at random into the cyhnder, for the average number of facets on each grain now varied with the pressure, from about 8-5 at 1000 lb. to 12-9 at 10,000 lb., and to no less than 14-16 facets or contacts after all interstices were eliminated, which took the full pressure of 35,000 lb. to do. An average of just over fourteen facets might seem to indicate a tendency to the production of tetrakaidekahedra, just as in the froth of soap-bubbles; but this is not so. The squeezed grains are irregular in shape, and pentagonal facets are much the commonest, just as we found them to be in the microscopic structure of a once-molten metal. At first sight it might seem that, though the experiment has something to teach us about random packing in a limited space, it has no biological significance; but it is curious to find that the pith-cells of * Cf. Cecil H. Desch, The solidification of metals from the liquid state, Joarn. Inst, of Metals, xxii, p. 247, 1919. VII] OF THE PARTITIONING OF SPACE 555 Eupatorium have a similar average configuration, with the same predominance of pentagonal facets*. We learn, in short, from Lewis and from Marvin that the mechanical result of mutual pressure, even in an assemblage of co-equal spheres, is more varied and more complex than we had supposed. The two simple and homogeneous configurations — the rhombo-dodecahedral and tetrakaidekahedral assemblages — are easily and commonly produced, the one by the compression of solid spheres in ordinary close-packing, the other when a liquid system of spheres or bubbles is free to slide and glide into a packing which is closer still. Between these two configurations there is no othei symmetrical or homogeneous arrangement possible; but random packing and degrees of compression leave their random effects, among which are traces here and there of regular shape and symmetry. As a froth has its histological lessons for us, throwing light on the structure of a parenchyma, so may we draw an illustration or two from the analogous characteristics of an emulsion. Both alike are "states of aggregation"; both are "two-phase systems," one phase being dispersed and the other the medium of dispersion. Both phases are liquid in the emulsion, in the froth the dispersed phase is a gas; our living tissue is, so far, more likely to be an emulsion than a froth. The concept widens. A colony of bacteria, the blood corpuscles in their plasma, the filaments of an alga, the heterogeneous texture of any ordinary tissue, may all be brought under the general concept of "phase systems," and share the common character that one phase exposes a large "interface" to the other. If we take milk as a simple emulsion, we see its hquid oil-globules dispersed in a watery medium and rounded by surface tension into spheres. The watery medium, as is usual in such emulsions, contains dissolved substances which tend to lower the interfacial tension; for were that tension high the globules would tend to be larger and their aggregate surface less. Suppose the "phase-ratio" to alter, the globules becoming more numerous and the disperse medium less and less, the globules will be close-packed * J. W. Marvin, The shape of compressed lead-shot, etc.; Amer. Journ. of Botany xxvi, pp. 280-288, 1939; Cell-shape studies in the pith of Eupatorium, ibid. pp. 487-504. 556 THE FORMS OF TISSUES [ch. at last. Then each (provided they be of equal size) will be in touch with twelve neighbours; and if the spheres were solid— were the system not an emulsion but a " suspension "—the matter would end here. But our hquid globules are capable of deformation, and the points of contact are flattened in still closer packing into planes. Ttey become polyhedral, and tend to take the form of rhombic dodecahedra, or it may be even of 14-hedra, and the dispersion- medium is reduced to mere films or pellicles between. At the stage of mere twelve-point contact, the spherules constitute about 74 per cent., and the disperse .medium 26 per cent., of the whole. But in the final stage the phase-ratio has so altered that the disperse- medium is but a small fraction of the whole, the thin film to which it has been reduced has the appearance of a cell-membrane separating the cells, and the microscopic structure of the whole corresponds to the cellular configuration of a parenchymatous tissue*. Of certain groupings of cells 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 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 disbelieve, that the whole configuration, for instance of an egg in the advanced stages of segmentation, is accurately determined by simple physical laws: just as much as in the early stages of two or four cells, during which early stages we are able to recognise and demonstrate the forces and their effects. But when mathematical investigation has become too difficult, physical experiment can often reproduce the pheno- mena which Nature exhibits, and which we are striving to com- prehend. In an admirable research, M. Robert not only shewed some years ago that the early segmentation of the egg of Trochus (a marine univalve mollusc) proceeded in accordance with the laws * Cf. E. Hatschek, Homogeneous partitionings, etc., Phil. Mag. xxxiii, p. 83, 1917. VII] OF POLAR FURROWS 557 of Surface-tension, but he also succeeded in imitating by means of soap-bubbles one stage 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 earHer stages the experiment is easy; we have merely to blow our bubbles in a httle dish, adding one to another, and adjusting their sizes to produce a symmetrical system. One of the simplest and prettiest parts of his investigation concerned the "polar furrow " of which we have spoken on p. 489. On blowing four httle 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 httle above the others and 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. 215, 1-3). The lamina itself is plane when the system is symmetrical, but it responds by a corresponding curvature to the least inequahty of the bubbles on either side. In the experiment, the upper polar furrow is usually a httle shorter than the lower, but parallel to it; that is to say, the lamina is of trapezoidal form : this lack of perfect symmetry being due (in the experimental case) to the lower portion of the bubbles being somewhat drawn asunder by the tension of their attachments to the sides of the dish (Fig. 215, 4). A similar phenomenon is usually found in Trochus, according to -Robert, and many otlier observers have hkewise 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, ^'^hat 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 species. Returning to the experimental case, Robert found that by with- drawing a little air from, and so diminishing the bulk of the two terminal bubbles (i.e. those at the ends of the intermediate lamina), 558 THE FORMS OF TISSUES [CH. the upper polar furrow was caused to elongate, till it became equal in length to the lower; and by continuing the process it became the longer in its turn. These two conditions have again been described by investigators as characteristic of this embryo or that; for instance in Unio, Lillie has described the two furrows as gradually altering their respective lengths*; and Wilson (as LilHe Fig. 215. Aggregations of four soap-bubbles, to shew various arrangements of the intermediate partition and polar furrows. After Robert. remarks) had already pointed out that "the reduction of the apical cross-furrow, as compared with that at the vegetative pole in molluscs and annelids, 'stands in obvious relation to the different size of the cells produced at the two polesf." When the two lateral bubbles are gradually reduced in size, or the two terminal ones enlarged, the upper furrow becomes shorter * F. R. Lillie, Embryology of the Unionidae, Journ. Morph. x, p. 12, 1895. t E. B. Wilson, The cell-lineage of Nereis, Journ. Morph. vi, p. 452, 1892. VII] OF POLAR FURROWS 559 and shorter; and at the moment when it is about to vanish, a new furrow makes its instantaneous appearance in a direction perpen- dicular to the old one; but the inferior furrow, constrained by its attachment to the base, remains unchanged, and it looks as though our two polar furrows, which were formerly parallel, were now at right angles to one another. But in fact, the geometry of the whole system is entirely altered. Before, two furrows left each end of one polar furrow for the same end of the other polar furrow, and the two cells at either end were shaped like "liths" of an orange. Under the new arrangement, two furrows leave each end of one for the two ends of another. The figure is now divided by six similar furrows into four similar curviHnear triangles*; it has become (approximately) a spherical tetrahedron, and the four cells into which it is divided are four similar and symmetrical figures, also tetrahedral, all meeting in a point at the centroid of the figure. Such a four-celled embryo, described as having two polar furrows arranged in a cross, has often been seen and figured 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. So, by slight and delicate modifications, we pass through many, and perhaps through all, of the possible arrangements of external furrows and internal partitions which divide the four cells from one another in a four-celled egg or embryo; and many, or most, or possibly all of these arrangements have been more or less frequently observed in the four-celled stages of various embryos. iVnd all these configurations, which the embryologists have witnessed and described, 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, coelenterates, vertebrates and what not, we meet with now one and now another, in a medley which defies classification. They are not ''vital phenomena," or "functions" of the organism, or special characteristics of this organism or that, but purely physical * That the sphere can be symmetrically divided into four equilateral triangles, after the manner of these embryos (or of many pollen-grains), is an elementary fact of great importance in geometry and trigonometry. 560 THE FORMS OF TISSUES [ch. phenomena. The kindred but more compUcated phenomena analogous to the polar furrow, which arise 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 less capable 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 over- powered and cancelled by agencies of a different order. Here we pass into a region where observation and experiment are not at hand to guide us, and where a man's trend of thought, and way of judging the whole evidence in the case, must shape his philosophy. We must always remember that even in a froth of soap-bubbles we can apply an exact analysis only to the simplest cases and conditions; 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 con- sists; but our knowledge is enough to leave us assured that what we have learned by investigation of the simplest cases includes the principles which determine the most complex. In the case of the growing embryo we know from the beginning that surface-tension is only one of the physical forces at work; and that other forces, including those displayed within the interior of each living cell, play their part in the determination of the system. But we have no evidence whatsoever that at this point, or that point, or at any, the dominion of the physical forces over the material system gives place to a new condition where agencies at present unknown to the physicist impose themselves on the living matter, and become responsible for the conformation of its material fabric. Before we leave for the, present the subject of the segmenting egg, we may 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 VII] OF THE GASTRULA 561 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 estab- lished. 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 diminu- tion 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 several explanations are conflicting, and are far from clear. There is, however, a certain homely phenomenon which goes some way, perhaps a long way, to explain this remarkable configuration. An ordinary gelatine lozenge, or jujube, has (Uke the developing gastrula) a more or less spherical form, depressed or dimpled at one side; this is a very noteworthy conformation, and it arises, auto- matically, by the shrinkage of a sphere. Were the initial sphere of gelatine perfectly homogeneous, and so situated as to shrink with absolute uniformity, it would merely shrink into a smaller sphere; it does nothing of the kind. There is always some part or other which shrinks a little 'hore than the rest J ; and the dimple so formed goes on increasing, until at last a very perfect cup-shaped figure is formed. I imagine that the gastrula is formed in much the * Professor Peddie has given me this interesting result, but the mathematical reasoning is too lengthy to be set forth here. t Cf. Butschli, Arch.f. Entw. Mech. v, p. 592, 1897; Rhumbler, ibid, xiv, p. 401, 1902; Assheton, ibid, xxxi, p. 46, 1910. X Just as there may be some small part which shrinks a little less. But this we should not distinguish from the common case where one small part grows a little more, and so "produces a 6ud," as in the yeast-cell on p. 363. 562 THE FORMS OF TISSUES [ch. same way, save only that the initial dimple, instead of being fortuitous, has its constant place, determined by the physico- chemical heterogeneity of the embryo. We may even go one step further, and see (or imagine we see) in the formation of the gastrula a physico-chemical or physiological turning-point, the segmentation cavity being due (as we have seen) to an inward flow, and a reversal of the current leading to that shrinkage which produces the gastrula. Fig. 216. Effect of shrinkage on a globule of gelatine. After E. Hatschek. A note on shrinkage We have dealt much with growth, but the fact is that negative growth, or shrinkage, is also an important matter; and just as we find a whole series of phenomena to be based on the extension or expansion of bubbles, vesicles, etc., so there is another series, physically ahke and mathematically identical, which depend on the shrinkage of a soHd or semi-fluid mass. After all, growth and its converse go hand in hand, and a special case of shrinkage is that surface-tension to which all the Plateau configurations are due. One clear case, the gastrula, we have touched on, and we have discussed another which led to the stellate dodecahedra of the Rush. As a cube of gelatine, or of paraffin, dries, and shrinks, it alters its shape in a remarkable way*. Its corners become more salient, its sides become concave ; its cross-section has the form of a four- * Emil Hatschek. Kolloid Ztschr. xxxv, pp. 67-76, 1924; Nature, 1st Nov. 1924. VII OF THE EFFECTS OF SHRINKAGE 563 rayed star with rounded angles. The block has dried unequally; its corners and its edges were naturally the first to dry*, and the twelve dried and hardened edges began to play the part of the wire frame in Plateau's soap-bubble experiment. The shrinking cube is tending towards the identical configuration shewn in Fig. 212; it is a minimal configuration, partially reahsed in a coarse material, but reahsable to perfection in a film. A shrinking cylinder (as Plateau knew) she)vs various phenomena, depending on its proportions. A low, squat cyhnder begins to show a pulley-hke groove — a catenoid— around its periphery, precisely like the soap-film between its two wire rings in Fig. 108 ; and as the groove deepens, the plane surfaces of the cyhnder also begin Fig. 217. Shrinkage of cube and cylinder. to dimple in. They become spherical as they grow more concave, and the deepening groove of the pulley passes from a catenoid to a nodoid curve — so at least theory tells usf, for, beautiful as the experimental configurations are, they hardly lend themselves to precise measure- ments of curvature. But this shrunken cyhnder is now wonderfully like the " amphicoelous " vertebra of a cartilaginous fish, the simplest and most ''primitive" vertebra of all. A series of cracks, or sphts, around the circular groove in the vertebra seem to be a final result of irregular shrinkage, not shewn in the more homogeneous gelatine. A long cylinder, or thread, of gelatine tends to become fluted, with three or more ribs or folds, and it is in this way that threads * Just as, conversely, the prominent parts of a crystal tend to grow more rapidly than the rest in a super-saturated solution, and to dissolve more rapidly in one below saturation; cf. 0. Lehmann, Ueber das Wachstum der Krystalle, Ztschr. f. Krystallogr. i, p. 453. t See p 369. 564 THE FORMS OF TISSUES [ch. of viscose, or artificial silk, tend likewise to have a ridged or fluted structure, and gain in lustre thereby. The subject is new, and hardly ripe for full discussion ; but it holds out promise (as it seems to me) of many biological lessons and illustrations. We glanced in passing at such "shrinkage-patterns" as are found, for instance, on the little shells of Lagena, or on those other hanging drops which constitute Emil Hatschek's artificial medusae; it is no small subject. A stretched elastic membrane, circular or spherical, remains spherical or circular when we let its tension relax; but if, to begin with, we coat the rubber with a pliant but non-elastic material such as wax, the waxen layer, failing to con- Fig. 218. Amphicoelous vertebrae of a shark. tract, is thrown into more or less characteristic folds. In a dried pea the seed has shrunken through loss of moisture, and the loose outer coat wrinkles up*. The pretty pattern of a poppy-seed arises in the same way; but so do the wrinkles on an old man's withered skin. When our experimental elastic with its non-contractile coat is suffered to contract, the first sign of the coat's inability to keep pace is the appearance of little domes, or hummocks, or bhsters; and soon from each of these there run out folds, which tend to fork, and the angles between the three branches tend to equalise. They tend, in simple and symmetrical cases, to form a pattern of hexagons, with occasional pentagons or quadrilaterals between ; but where the surface is larger and the coat more flexible the folds form an irregular network, still with the various anticlines mostly meeting in three- * The difference between a smooth and a wrinkled pea, familiar to Mendelians, merely depends, somehow, on amount and rate of shrinkage. VII] OF SHRINKAGE-PATTERNS 565 way nodes*. Nature will ring the changes on the resultant patterns, according as the surface be plane or curved, spherical or cylindrical, coarse or fine, fragile or tough. But on these general fines very many structures, both regular and irregular, spines, bristles, ridges, tubercles and wrinkled patterns, bid fair to find their physical or mechanical interpretation; and it is in the more or less hardened parts of plant or animal that we find them one and all displayed. On the egg of a butterfly, on the grooved and dotted elytron of a beetle, on the notched forehead of a scarab, in thesaw-fike teeth on a grasshopper's leg, in the little fines of dotted tubercles on the shell of a Rissoa, more crudely in the lozenged bark of elm or pine, we see a very few of this innumerable class of " shrinkage-patterns." * The fact that such triplets of divergent ridges or crests are not a feature in the topography of mountain-ranges is a strong argument against the view that general shrinkage accounts for the pattern of the earth's crust. Cf. A. J. Bull, The pattern of a contracting earth, Geolog. Mag. lxix, pp. 73-75, 1932; A. E. B. de Chancourtois, <^.R. lix, p. 348, 1903. CHAPTER VIII THE FORMS OF TISSUES OR CELL-AGGREGATES {continued) The problems which we have been considering, and especially that of the bee's cell, belong to a class of " isoperimetrical " problems, which, deal with figures whose surface is a minimum for a definite content or volume. Such problems soon become difficult*, but we may find many easy examples which lead us towards the explanation of biological phenomena; and the particular subject which we shall find most easy of approach is that of the division, in definite pro- portions, 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 biological phenomena of cell-division. This investigation may be approached in two ways: by con- sidering the partitioning off from some given space or area of one-half (or some other fraction) of its content; or again, by dealing with the partitions necessary for the breaking up of a given space into a definite numbet of compartments. If We begin with the simple case of a cubical cell, it is obvious that, to divide it into two halves, the smallest 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, ahke of one of its sides and of the partition which 'we have interposed parallel 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 twice the size of a cross-partition perpendicular to it; 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 is obviously a Httle scj^uare, with an area of (|a)2 = ^a^. * Minkowski, and others have shewn how hard it is, for instance, to prove the seemingly obvious proposition that the sphere, of all figures, has the greatest volume for a given surface; cf. (e.g.) T. Bonneson, Les problemes des isoperimetres et des iseplphanes, Paris, 1929. For a historical account of this class of problems, see G. Enestrom, in Bibl. Math. 1888. CH.viii] OF SACHS'S RULES 567 From this we may draw the simple rule that, for a rectangular body or parallelepiped to be bisected by means of a partition of minimal area, (1) the partition must cut across the longest axis of the figure; and (2) in successive bisections, each partition must run. at right angles to its immediate predecessor. X /\ t; / / X V / / / ^' / / 7 / Fig. 219. 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 as follows : (1) The cell tends to divide into two co-equal parts. (2) Each new plane of division tends to intersect thCj preceding plane of division at right angles. The first of these rules is a statement of physiological fact, not without its exceptions, but so generally true that it will justify us in limiting our enquiry for the most part to cases of equal sub- division 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 dynamically sym- metrical, as when the parts differ in surface tension or internal pressure, or, speaking generally, in their chemico-physical properties 568 THE FORMS OF TISSUES [ch. and conditions. This latter condition, of asymmetry of field, is frequent in segmenting eggs*, and it then covers or includes the principle upon which Balfour laid stress as leading to "unequal" or to "partial" segmentation of the egg — viz, the unequal or asym- metrical distribution of protoplasm and of food-yolk. The second rule, which also has its exceptions, is true in a large number of cases, and owes its vahdity, as we may judge from the illustration of the repeatedly bisected cube, to the guiding principle of minimal areas. It is in short subordinate to a much more important and fundamental rule, due not to Sachs but to Errera; that (3) the incipient partition- wall of a dividing cell tends to be such that its area is the least possible by which the given space-content can be enclosed. Let us return to the case of our cube, and 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 then easily shifted to a new position in which it constitutes a curved cylindrical wall cutting oif one corner of the cube. It still meets the sides of the cube at right angles (for reasons which we have already con- sidered); and, as we may see from the symmetry of the case, it constitutes one-quarter of a cylinder. Our plane transverse parti- tion had always the same area, wherever it was placed, viz. a^\ and it is obvious that a cylindrical wall, if it cut oif a small corner, may be much less than this. We want, accordingly, to determine what volume might be partitioned oif with equal economy of wall- space in one way as the other, that is to say, what area of cyhndrical * M. Robert {loc. cit. p. 305) has compiled a long list of cages 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 most other writers of his time, he would ascribe this equality or inequality rather to a provision for the future than to a direct effect of immediate physical causation : "II semble assez probable, comme on I'a dit sou vent, 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 segregation. II faut avouer pourtant qu'on est parfois assez embarrasse pour assigner une cause a pareilles differences." VIII] OF AREAE MINIMAE , 569 wall would be neither more nor less than the area a^. The calculation is easy. The surface-area of a cylinder of length a is 27rr . a, and that of our quarter-cylinder is, therefore, a.7Trl2; and this being, by hypothesis, = a^, we have a = 7rr/2, or r = 2a/7r. The volmne of a cyhnder of length a is anr^, and that of our quarter-cyhnder is a . Trr^/i, which (by substituting the value of r) is equal to a^JTi^ 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/n; 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 0-636 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 be 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 oiF one soHd 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 one- quarter of the original cell*. But while both the cyhndrical wall and the Fig. 220. * The principle is well illustrated in an experiment of Sir David Brewster's {Trans. R.S.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 till it becomes attached to the glass covering, and it then immediately changes place, leaving its transverse position to take up that of a spherical segment extending from one side of the wine-glass to its cover, and so enclosing the same volume of air as formerly but with a great economy of surface, precisely as in the case of our spherical partition cutting oflF one corner of a cube. 570 THE FORMS OF TISSUES [CH. spherical wall would be of less area than the plane transverse partition after that Hmit (of one-quarter volume) was passed, the cylindrical would still be the better of the two up to a further hmit. It is only when the volume to be partitioned off is no greater than 1 1 1 1 Fraction of cube partitioned off. 15 \^ - t'4 H' ■ 1-3 1-2 M Plane \^A \ R transverse partition 9 \ X \ '^ - \ .0 "^ \ " u' •2 1 _J \ A'^ B' ^ Fig: 221. C about 0-15, or somewhere about one-seventh of the whole, that the spherical cell-wall in a corner 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. 221) 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 VIII] OF AREAE MINIMAE 571 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 per cent, and a spherical partition would have an area about 50 per cent, greater. The point A in the diagram corresponds to the point where the cylindrical partition would begin to have an advantage over the plane, thati i to say (as we have seen) when the fraction to be cut off is about one-third, or 0-318 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, as in the hairs within the reproductive " conceptacles " of certain Fuci (Sphacelaria, etc.. Fig. 222), or in the "paraphyses" of mosses (Fig. 226). But it is of great theoretical importance : as serving to introduce us to a large class of cases in which, under the guiding principle of minimal areas, the shape and relative dimensions of the original cavity lead to cell-division in very definite and sometimes unexpected ways. It is not easy, nor indeed possible, to give a general account of these cases, for the limiting conditions are somewhat, complex and the mathematical treatment soon becomes hard. But it is easy to comprehend a few simple cases, which carry us a good long way; and which will go far to persuade 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 its subdivision in other definite propor- tions) soon leads us into a geometry which, if not necessarily difficult, is apt to be unfamiliar ; ^ but in such problems we can go some way, and often far enough for our 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 just been considering, and in the case of the plane and cyhndrical partitions by which it has been divided, it is obvious, since these two partitions extend symmetrically from top to bottom of our cube, that we need only have considered 572 THE FORMS OF TISSUES [ch. the manner in which they subdivide the base of the cube; in short the problem of the sohd, up to a certain point, is contained in our plane diagram of Fig. 221. And when our particular soUd is a solid of revolution^ then it is equally 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 Hne which shall be the shortest possible is an easy problem; for it is obvious that, if the triangle be low, a vertical partition will be shortest; if it be high, a horizontal one ; if it be equilateral, the partition may run parallel to any side; and if it be right-angled, the partition may bisect the right angle or run parallel to either side equally well. Let ABC be an isosceles triangle of which A is the apex ; it may be shewn that, for its shortest Hne of bisection, we are limited to three cases: viz. to a vertical line AD, bisecting the angle at ^ and the side BC\ to a transverse line parallel to the base BC\ or to an obUque Hne paraUel to AB or to AC. The lengths of these partition Hues follow at once from the magnitudes of the angles of our triangle. We know, to begin with, since the areas of similar figures vary as the squares of their linear dimensions, that, in order to bisect the area, a line parallel to one side of our triangle must always have a length equal to 1/a/2 of that side. If then, we take our base, BC, in all cases of a length = 2, the transverse partition, EF, drawn parallel to it wiU always have a length equal to 2/V2, or = a/2. The vertical partition, AD, since BD = 1, will always equal tan^; and the oblique partition, GH, being equal to ABjV^, = l/'v/2 cos j8. If then we call our vertical, transverse and obUque partitions F, T, and 0, we have F = tan j8; T == a/2; and 0 - 1/V2 cos j8, or V:T:0 = tan^/V2 : 1 : 1/2 cos^S. And, working out these equations for various values of j8, we soon see that the vertical partition (F) is the least of the three until P = 45°, at which limit F and 0 are each equal to 1/ V2 = 0-707 ; that 0 then becomes the least of the three, and remains so until VIII] OF AREAE MINIMAE 573 ^ = 60"", when cos ^ = 0-5, and 0= T; after which T (whose value always = 1) is the shortest of the three partitions. And, as we have 80° 90° 10° 20° 30° 40" 50° 60" 70' Basal angle of triangle Fig. 224. Comparative length of the partitions, transverse, oblique or vertical, bisecting an isosceles triangle. seen, these results are at once apphcable, not only to the case of the plane triangle, but also to that of the conical or pyramidal cell. 574 THE FORMS OF TISSUES [ch. In like manner, if we have a spheroidal body less than a hemi- sphere, such for instance as a low, watchglass-shaped cell (Fig. 225, A), it is obvious that the smallest partition by which we can divide it into two halves is (as in our flattened disc) a median vertical one; jand 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, 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 Fig. 225. in B), where an oblique partition will be better than either 'the vertical or the transverse ; though here the analogy of our triangle does not suffice to give us the precise hmiting values. We need not examine these hmitations in detail, but we must look at the curvatures which accompany the several conditions. We have seen that a film tends to set itself at equal angles to the surface which it meets, and therefore, when that surface is a sohd, to meet it (or its tangent) at right angles. Our vertical partition is, there- fore, a plane surface, everywhere normal to the original cell-walls. But in the taller, conical cell .with transverse partition, the latter still meets the opposite sides of the cell at right angles, and it follows that it must itself be curved; moreover, since the tension, and therefore the curvature, of the partition is everywhere uniform, it follows that its curved surface must be a portion of a sphere, concave towards the apex of the original cell. In the intermediate case, where we have an obhque partition meeting both the base and the curved sides of the mother-cell, the contact must still be viTi] OF SIGMOID PARTITIONS 575 everywhere at right angles: provided we continue to suppose that the walls of the mother-cell (hke those of our diagrammatic /cube) have become practically rigid before the partition appears, and are therefore not affected and deformed by the tension of the latter. In such a case, and especially when the cell is elliptical in cross- section or still more comphcated in form, the partition may have to assume a complex curvature in order to remain a surface of minimal area. Fig. 226. S-shaped partitions: A, Taonia atomaria (after Rcinke); B, paraphya of Fucus; C, rhizoids of moss; D, paraphyses oi Polytrichmn. While in very many cases the partitions (like the walls of the original cell) will be either plane or spherical, a more complex curvature will sometimes be alssumed. It will be apt to occur when the mother-cell is irregular in shape, and one particular case of such asymmetry will be that in which (as in Fig. 227) the cell has begun to branch before division takes place. 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 576 THE FORMS OF TISSUES [CH. assume an oblique position, then equally anomalous curvatures will tend to make their appearance*. Suppose an oblong cell to divide by means of an obhque 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 of the mother-cell : and it follows that our oblique partition, throughout its whole extent, will assume the form of a complex, saddle-shaped or anticlastic surface. Many such partitions of complex or double curvature exist, but they are not always easy of recognition, nor do they often appear in a terminal cell. We may see them in the roots (or rhizoids) of t " A B C D Fig. 227. Diagrammatic explanation of S-shaped partition. , mosses, especially at the point of development of a new rootlet (Fig. 226, C); and again among mosses, in the "paraphyses" of the male plants (e.g. in Polytrichum), we find more or less similar partitions (D). They are frequent also among Fuci, as in the hairs or paraphyses of Fucus itself (B). In Taonia atofnaria, 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. 226, 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 * Cf Wildeman, Attache des cloisons, etc., pis. 1, 2. f Nova Acta K. Leop. Akad. xi, 1, pi. iv. VIII] OF SIGMOID PARTITIONS 577 tends to appear in a small, rather than a large cross-section of the cell: but we are also ascribing to the cell a longitudinal polarity (Fig. 227, A), and imphcitly asserting that it tends to divide (just as the segmenting egg does), by a partition transverse to its polar axis. Such a polarity may conceivably be due to a chemical asymmetry, or anisotropy, such as we have learned of (from 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. 4607-then oiir "polar axis" would necessaiily be a curved axis, and the partition, being con- strained (again ex hypothesi) to arise transversely to the polar axis, would lie obHquely to the apparent axis of the cell (as in B or C). And if the oblique partition be so situated that it has to meet the opposite walls (as in C), then, in order to do so symmetrically (i.e. either perpendicularly, as when the cell-wall is already solidified, or at least at equal angles on either side), it is evident that the partition, in its course from one side of the cell to the other, must necessarily assume a more or less S-shaped curvature (D). The complex curvature of the partition-walls in such cases as these m^y be illustrated by the following experiment. Set two plates of glass (as in Fig. 228) in a wire . frame, so that they may he parallel or / / at any angle to one another; and dip ^ — ■ — the whole thing in soap-solution, so that a sheet of film is formed between the two plates and is framed by the two / wires which carry them. The film is, ^ of course, a surface of minimal area ; its ^^' mean curva,ture is constant everywhere, and (since the film is an open surface with identical pressure on both sides) the mean curvature is everywhere nil. A related condition is that the film must meet its solid framework, glass or wire, everywhere at right angles or *' orthogonally"; and this last constraint leads to curvatures of extreme complexity, which continually vary as we rotate one plate on the plane of the other. 578 THE FORMS OF TISSUES [ch. 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 a part, are by no means rare. We know that in cambium-cells division often takes place parallel to the long axis of the cell, though 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 excep- tions that Berthold was led to quahfy 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 tension, both external and internal to the cell, play their part in determining its partitions, and that the answer to our problem is not to be given in a word. Hpw 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 consider the simpler and more general phenomena. And let us leave the case of the cubical, quadrangular or cylindrical cell, and examine that 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 place and form of a partition of minimal area, it is plain that we shall be dealing with comparatively simple cases wherever even one dimen- sion of the cell is much less than the other two. Where two dimensions are.small compared with the third, as in a thin cylindrical filament like that of Spirogyra, we have the problem at its simplest ; for it is 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 * Cf. Protoplasmamechanik, p. 229: "Insofern liegen also die Verhaltnisse hier wesentlich anders als bei der Zertheilung hohler Korperformen durclj fliissige Lamellen. Wenn die Membran bei der Zelltheilung die von dem Prinzip der kleinsten Flachen geforderte Lage und Kriimmung annimmt, so werden wit den Grund dafiir in andrer Weise abzuleiten haben." VIII] OF DISCOID CELLS 579 is so far simplified that we see at once that the partition cannot be parallel to the extended plane, but most cut the cell, somehow, at right angles to that plane. In short, the problem of dividing a much flattened sohd 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, and consisting (for a time at any rate) of but a single layer of cells, which, as Berthold shewed, exempUfy this comparatively simple problem;* and we shall find presently that it is admirably illustrated in the cell-divisions which occur in the egg of a frog or a sea-urchin, when it is flattened out under artificial pressure. These same httle algae which serve to exempHfy the partitioning of a disc also illustrate, now and then, a curious feature of its contour. Such a small^ green alga as Castagna (Fig. 229) shews, and many Desmids shew just as well, a sinuous border running out into rounded crenations or lobes. This is a surface-tension phe- nomenon. A little milk poured over an apple-pie gives a homely illustration of Fig. .229. Castagna pdycarpa. the same sinuous outlines; a drop on a ^^^^^^^^ZlT^ ^''''"^ ^^^"*'' greasy plate spreads in the same uneven way, and does so indeed unless the utmost care be taken to ensure absolute cleanhness and surface equihbrium*. Fig. 230 1 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 quarters. We have only to remember that, for the sum total of partitions to * Cf. Quincke's " Ausbreitungserscheinungen," in Poggendorff^s Annalen, cxxxix, p. 37, 1870; also Tomlinson's papers in Phil. Mag. vm-xxxix; and Van der Mensbrugghe, Mem. Cour. de VAcad. R. Belgique, xxxiv, 1870; xxxvii, 1873. t From Berthold's Monograph of the Naples Bangiaceae, 1882. 580 THE FORMS OF TISSUES [CH. be a minimum, three only must meet in a point; and therefore, the four quadrantal walls must shift a Uttle, producing the usual little median partition, or cross-furrow, instead of one common central point of junction. This intermediate partition, however, Fig. 230. Development of Erythrotrichia. After Berthold. will be small, and to all intents and purposes we may deal with the case as though we had now to do with four equal cells, each one of them a perfect quadrant; so our piroblem is, to find the shortest line which shall divide the quadrant of a circle into two halves of V .--^^ \^ / '"•. \ / \ •A v j X \/ V /__ 2 Fig. 231. equal area. A radial partition (Fig. 231, A), starting from the apex of the quadrant, is at once excluded, for the reason just referred to ; our choice must lie between two modes of division such as are illustrated in Fig. 231, where the partition is either (as in B) con- centric with the outer border of the cell, or else (as in C) cuts that VIIl] OF THE SEGMENTATION OF A DISC 581 outer border; in oth^r words, our partition may (B) cut both radial walls, or (C) may cut one radial wall and the periphery. These are the two methods of division which Sachs called, respectively, (B) periclinal, and (C) anticlinal*. We may either treat the wa&s of the dividing quadrant as already sohdified, or at least as having a tension compared with which that of the incipient partition film is inconsiderable; in either case the new partition must meet the old wall, on either side, at right angles, and (its own tension and curvature being everywhere uniform) must take the form of a circular arc. jgX M Fig. 232. We find that a flattened cell which is approximately a quadrant of a circle invariably divides ^fter the manner of Fig. 231, C, that is to say, by an approximately circular, anticlinal wall, and this we now recognise in the eight-celled stage of Eryihrotrichia (Fig. 230) ; let us then consider that Nature has solved our problem, and let us work out the actual geometrio conditions. Let the quadrant OAB (in Fig. 232) be divided into two parts of equal area, by the circular arc MP. It is required to determine * There is, I think, some ambiguity or disagreement among botanists as to the use of this latter term : the sense in which I am using it, viz. for any partition which meets the outer or peripheral wall at right angles (the strictly radial partition being for the present excluded), is, however, clear. 582 THE FORMS OF TISSUES [ch. (i) 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 0P\ 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 equal to OPN , or POB. Then CP = a cot ^; PN = a cos ^; NO = CP cos 6 = a . cos^ ^/sin 6. The area of the portion PMN = iCP^ d - iPN . NO = Ja2 0 cot2 d-iacosd .a cos^ ^/sin 6 = ia^ (d cot2 d - cos3 e-lsin 6). And the area of the portion PNA = ia^ (77/2 -6)- \0N . NP = U^ (77/2 - (9) - Ja sin ^ . a cos ^ = \a^ (7r/2 - ^ - sin ^ . cos d). Therefore the area of the whole portion PMA = a^l2 (77/2 - e + d cot2 d - cos3 ^/sin ^ - sin ^ . cos 9) = a2/2 (77/2 - ^ + ^ cot2 d - cot <9), and also, by hypothesis, = J . area of the quadrant, = Tra'^/S. Hence 6 is defined by the equation 072 (77/2 - e+ d cot2 e - cot d) = 77a2/8, or 77/4:- 6+ d cot2 ^ - cot ^ = 0. We may solve this equation by constructing a table (of which the following is a small portion) for various values of d. ■d 7r/4 -d -cot^ + d cot2 e = x 34° 34' 0-7854 - - 0-6033 - - 1-4514 + 1-2709 = 0-0016 35' 0-7854 0-6036 1-4505 1-2700 0-0013 36' 0-7854 0-6039 1-4496 1-2690 0-0009 37' 0-7854 0-6042 1-4487 1-2680 0-0005 38' 0-7854 0-6045 1-4478 1-2671 00002 39' 0-7854 0-6048 1-4469 1-2661 -00002 40' 0-7854 0-6051 1-4460 1-2652 -0-0005 We see accordingly that the equation is solved (as accurately as need be) when 6 is an angle somewhat over 34° 38', or say VIII] THE BISECTION OF A QUADRANT 583 34° 3^1'. 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 w^ays 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 oi which expression, for the angle which we have just discovered, is 0-3028. That is to say, the radius (or side) of the quadrant will be divided by the new partition into two parts, in the proportions, nearly, of three to seven. (3) The length of the arc MP is equal to ad cot d; and the value of this for the given angle is 0-8751. 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 8| to 10, or seven-eighths, almost exactly. But we must also compare the length of this curved antichnal partition-wall (MP) with that of the concentric, or perichnal, one {RS, Fig. 233) 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 V2; or, in terms of the radius, = 77/2 a/2 = 1-111. So that, not only is the anticlinal partition (such as we actually find in nature) notably the best, but the periclinal one, when it comes to dividing an entire quadrant, is very considerably larger even than a radial partition. The two cells into which our original quadrant is now divided are equal in volume, but of very different shapes; the one is a triangle (MAP) with circular arcs for two of its sides, and the other is a four-sided figure (MOBP), which we may call approximately oblong. How will they continue to divide? We cannot say as 584 THE FORMS OF TISSUES [CH, 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 periclihally. This is precisely what tends actually to take place. In the following diagrams (Fig. 234) of a frog's egg dividing under pressure, that is to say when reduced to the form of a flattened plate, we see, firstly (A), the division into four quadrants (by the partitions L 2); secondly (B), the division of each quadrant by means of an anti- clinal circular arc (3, 3), cutting the peripheral wall of the quaarant approximately in the proportions of three to seven; and thirdly 3 ABO Fig. 234. Segmentation of frog's egg, under artificial compression. After Roux. (C), 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 quadrant will next divide leads us to a well-defined problem which is only a shght extension, making allowance for the circular arcs, of that elementary problem of the triangle we have already con- sidered. We know now that an entire quadrant (in order that its bisecting wall shall have the least possible area) must divide 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. 235, that a periclinal partition is the least by which we can bisect the cell; we want, accordingly, to know the limits below which the periclinal partition is always the VIII] THE BISECTION OF A QUADRANT 585 best, and above which the anticHnal arc has the advantage, as in the case of the whole quadrant. 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 9, according to our equation MP -^ ad cot 6; and the angle 6 has a definite relation to a, the angle of arc. Fig. 235. Moreover, in the case of the periclinal boundary, RS (Fig. 233) (or ab, Fig. 235), we know that, if it bisects the cell, RS = a. a/V2. Accordingly, the arc RS will be just equal to the arc MP when ^ cot ^ = a/v/2. When ^ cot ^ > a/ V2, or MP < RS, then division will take place as in RS, or perichnally. When 6* cot ^ < a/ V2, or MP > RS, then division will take place as in MP, or anticlinally. In the accompanying diagram (Fig. 236), 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 divide), and on the horizontal scale the corresponding values of 6, or the angle which determines the point on the periphery where it is cut by the partition-wall. 586 THE FORMS OF TISSUES [CH. 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 Fig. 23G. 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 that point, can possibly exist is a periclinal one; and this point, VIII] THE BISECTION OF A QUADRANT 587 as our diagram shews us, occurs when the two angles (a and 9) are both rather under 52°. Next I have plotted on the same diagram, and in relation to the same scale 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 terms of the radius of the circle (a), that is to say the side wall, OA, of our cell. The hmiting 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 : J?*S = 0-875 : Mil: (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 D, D'): (3) the point E, where RS and MP intersect, that is to say the point at which the two partitions, periclinal and anticlinal, are of the same magnitude; this is the case, according to our diagram, when the angle of arc is just over 62 J°. We see from this that what we have called an antichnal partition, as MP, is only Ukely to occur in a triangular or prismatic cell whose angle of arc hes 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 periclinal partition has a very slight and almost imperceptible advantage over the anticlinal, the relative proportions being about as MP\ RSwO'lZ.O-n. But if the triangle be a plane equiangular 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 ^ — cot ^ = 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 oif the point on this curve corresponding to a -- 60°; and we find it at 588 THE FORMS OF TISSUES [CH. the point F'" (vertically under F), wbich tells us that the partition now starts 45/100, or nearly halfway, along the radial wall. The foregoing considerations carry us a long way in our investi- gation of the simpler forms of cell-division. Strictly speaking they are hmited to the case of flattened cells, in which we can treat the problem as though we were 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 comphcated solid into two halves, yet, even in such a case they so far enlighten us as to 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 help us greatly in the interpreta- tion 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 Figs. 233 or 237 ; and let us _, now suppose the whole system to grow, in a uniform fashion, as a prelude to further subdivision. The whole quadrant, growing uniformly (or with equal radial increments), will still remain a quadrant, and it is obvious, therefore, that for every new increment of size, more will be added to the margin of its triangular portion than to the narrower margin of the quadrilateral; and the in- . Ml 1 • ^' ^ Fig- 237. crements will be m proportion to the angles of arc, viz. 55° 22' : 34° 38', or as 0-96 : 0-60, i.e. as 8 : 5. 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 mathe- VIII] THE BISECTION OF A QUADRANT 589 matics; but if we simply assume that the increments are propor- tional to the increasing radii of the circle, we have the following equations : Call the triangular cell T, and the quadrilateral Q (Fig. 237); 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 doubhng of Q be called x' . Then we see that the area of the original quadrant = T + Q = Itto" = 0-7854a2, while the area of T =Q = 0-3927a2. The area of the enlarged sector, P'OA', = {a + x)^ X (55° 22') ^ 2 = 0-4831 (a + x)^, and the area OP A = a^x (55° 22') -^ 2 = 0-4831a2. Therefore the area of the added portion, T\ = 0-4831 {{a + x)^ - a^}. And this, by hypothesis, = T = 0-3927a2. We get, accordingly, since a = I, x^ + 2x= 0-3927/0-4831 = 0-810, and, solving, x-{-l = Vrsi = 1-345, or x^ 0-345. Working out x' in the same way, we arrive at the approximate value, x' + 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 Hnear dimensions of the disc have increased by fully a half (from 1 to 1-517). 590 THE FORMS OF TISSUES [CH. The case here illustrated is of no small importance. For it shews us that a uniform and symmetrical growth of the organism (sym- metrical, 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 inequahty among these; and this phenomenon (or to be quite candid, this hypothesis, which is due to Berthold) is independent of any change or variation in surface tensions, and is essentially different from that unequal segmentation (studied by Balfour) to which we. have referred on p. 568. After this fashion we might go on to consider the manner, and the order of succession, in which subsequent cell-divisions should tend to take place, as governed by the principle of minimal areas. I Fig. 238. The calculations would grow more difficult, and the results •got by simple methods would grow less and less exact; at the same time some of the results would be of great interest, and well worth our while to obtain. 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 see that the width of the triangular cell near P (Fig. 238) 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 hke the original quadrant, into a triangular and a quadrilateral portion. But the case will be different next time, because in this new 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 comes a time VIII] THE SEGMENTATION OF A DISC 591 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 parti- tions will arise transversely to their long axes, that is to say, radially (as U, V) ; and we shall consequently have a superficial or peripheral layer of quadrilateral cells, with sides approximately parallel, that is to say what we are accustomed to call an epidermis. And this epidermis or superficial layer will be in clear contrast with the more irregularly shaped cells, the products of triangles and quadrilaterals, which make up the deeper, underlying layers of tissue. In following out these theoretic principles, and others Hke to them, in the actual division of living cells, we must 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 Unks,. and very important links, in a chain of physical causation ; they are always £tt work, but their effects may be overridden and concealed by the operation of other forces. Secondly, we must remember that, in most cases, the cell- system which we have in view is constantly increasing in magnitude by active growth; and by this means the form and also the propor- tions of the cells are continually altering, 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 fines of contact with one another may shift, like the shifting outlines of a system of soap-bubbles. This is the physical cause of the movements frequently seen among segmenting cells, like those to which Rauber called attention in the segmenting ovum of the frog, and hke those more striking movements or accommodations which give rise to a so-called "spiral" type of segmentation. Bearing in mind these considerations, let us see what our fla|itened disc is hkely to look like, after a few successive Cell-divisions. In Fig. 239 a, we have a diagrammatic representation of our disc, after it has divided into four quadrants, and each quadrant into a triangular and a quadrilateral portion; but as yet, this, figure has 592 THE FORMS OF TISSUES [CH. scarcely anything like the normal look of an aggregate of hving cells. But let us go a little further, still hmiting ourselves 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. 485; and, furthermore, the narrower cells, that is to say the quadrilaterals, will have this outer border somewhat more Fig. 239. Diagram of flattened or discoid cell dividing into octants : to shew gradual tendency towards a position of equilibrium. curved than their broader neighbours. We arrive, then, at the condition shewn in Fig. 239 b. Within the cell, also, wherever wall meets wall, the angle of contact must tend, in every case, to be an angle of 120°; 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 involve the curvature of some, if not all, of the partitions (d) which to begin with 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 d; and it will at once be recognised that this figure has vastly more resemblance to an VIII THE SEGMENTATION OF A DISC 593 aggregate of living cells than had the diagram of a, with whi«h we began. Just as we have constructed in this case a series of purely diagrammatic or schematic figures, so will it be possible as a rule to diagrammatise, with but little alteration, the complicated ap- pearances presented by any ordinary aggregate of cells. The accompanying httle figure (Fig. 240), of a germinating spore of a Liverwort (Riccia), after a drawing of D. H. 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. The same is equally true of any one of Hanstein's figures of the hairs on a leaf-bud*, or Berthold's of the small discoid algae. Let us look again at our figures of Erythrotrichia or Chaetopeltis from Berthold's Monograph, and redraw some of the earher stages. Ficr. 240. Fig. 241. Embryo-stages of Chaetopeltis orbicularis. After Berthold. In the following diagrams (Fig. 242) the new partitions, or those just about to form, are in each case outhned ; and in the next succeeding stage they are shewn after settling down into position, and after exercising their respective tractions on the walls previously kid 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 reason that Berthold's figures are taken from different individuals, and so are not strictly and con- secutively continuous. The last of the six drawings in Fig. 230 is already too comphcated for diagrammatisation, that is to say it is too comphcated for us to decipher with certainty the order of appearance of the numerous partitions which it contains. But in Fig. 243 I shew one more diagrammatic figure, of a disc which has * Bot. Zeitung, xxvi, p. 11, xi, xii, 1868. 594 THE FORMS OF TISSUES [CH. divided, according to the" theoretical plan, into about sixty-four cells; and making due allowance for the changes which mutual tensions and tractions bring about, increasing in complexity with each succeeding stage, we can see, even at this advanced and "DC d Fig. 242. Theoretical arrangement of successive partitions in a discoid cell; for comparison with Figs. 230 and 241. complicated stage, a very considerable resemblance between the actual picture and the diagram which we have here constructed in obedience to a few simple rules. Fig. 243. Theoretical division of a discoid cell into sixty-four chambers: no allowance being made for the mutual tractions of the cell- walls. In like manner, in the annexed figures representing sections through a young embryo of a moss, we have little difficulty in discerning the successive stages which must have intervened between the two stages shewn: so as to lead from the just divided or dividing quadrants (a), to the stage (6) in which a well-marked epidermal VIII] THE SEGMENTATION OF A DISC 595 layer surrounds an at first sight irregular agglomeration of "funda- mental tissue". In the last paragraph but one, I have spoken of the difficulty of so arranging the meetmg-places of a number of cells that at each junction only three cell- walls shall meet in a point, and all three shall meet 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 enclosed in a common boundary, say eight as in the case considered, there are various ways in which their walls may 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 some- a — ^-------^ b Fig. 244. Sections of embryo of a moss. After Kienitz-Gerlofif. what complex ; it has been dealt with by Plateau, " and treated mathematically by M. Van Rees*. If within our boundary we have only three cells all meeting 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." In the case of five cells, we require two httle intermediate walls, and two polar furrows; and we soon arrive at the rule that, for n cells, we require ti — 3 Httle longitudinal partitions (and corre- sponding polar furrows), connecting the triple junctions of the cells f ; and these Httle walls, Hke all the rest within the system, tend to * Cit. Plateau, Statique des Liquides, i, p. 358. t There is an obvious analogy between this rule for the number of internal partitions within a polygonal system, and Lamarle's rule for the number of "free films " within a polyhedron. Vide supra, ^. b5Q. 596 THE FORMS OF TISSUES [CH. incline to one another at angles of 120°. Where we have only one such wall (in the case of four cells), or only two (in the case of five cells), there is no room for ambiguity. But where we have three little connecting- walls, in the case of six cells, we can arrange them in three different ways, as Plateau* found his six 3 4 5 "* 6 7 8 Cells' etc. Fig. 245. Various possible arrangements of intermediate positions, in groups of 3, 4, 5, 6, 7 or 8 cells. soap-films to do (Fig. 246). 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 Fig. 246.. twelve different waysf. It does not follow that these various arrangements are all equally good; some are known to be more stable than others, and some are hard to realise in actual experiment. Examples of these various arrangements meet us at every turn, in all sorts of partitioning, whether there be actual walls or mere * Plateau experimented with a wire frame or "cage", in the form of a low hexagonal prism. When this was plunged in soap-solution and withdrawn upright, a vertical film occupied its six quadrangular. sides and nothing more. But when it was drawn out sideways, six films starting from the six vertical edges met some- how in the middle, and divided the hexagon into six cells. Moreover the partition- films automatically solved the problem of meeting one another three-by-three, at co-equal angles of 120°; and did so in more ways than one, which could be controlled more or less, according to the manner and direction of lifting the cage. t Plateau, on Van Rees's authority, says thirteen; but this is wrong — unless he meant to include the case where one cell is wholly surrounded by the seven others. VIII] THE PARTITIONING OF SPACE 597 rifts and cracks in a broken surface. The phenomeiion is in the first instance mathematical, in the second physical ; and. the limited number of possible arrangements appear and reappear in the most diverse fields, and are capable of representation by the same diagrams. We have seen in Fig. 196 how the cracks in drying mud exhibit to perfection the polar furrow joining two three-way nodes, which is the characteristic feature of the four-celled stage of a segmenting egg. The possible arrangements of the intermediate partitions becomes a question of permutations. Let us call the flexure between two consecutive furrows a or 6, according to its direction, right or left; and let a triple conjunction be called c. Then the three possible arrangements in a system of six cells are aa, ah, c; the four in a system of seven cells are aaa, aab, aba, ac; and the twelve possible arrangements in a system of eight cells are as follows* : a aac f aahb b ahc 9 aaba c ad) h aaab d aca i aaaa e bcb J abab k abba I cc i We may classify, and may denote or symbolise, these several arrangements in various ways. In the following table we see: A, the twelve arrangements of the five intermediate partitions which are necessary to enable all the boundary walls of a plane assemblage of eight cells (none being "insular") to meet in three- way junctions; B, the literal permutations which symbolise the same; C, the number of sides (other than the external boundary) which in each case each cell possesses, i.e. the number of contacts each makes with its neighbours.^ The total number of contacts (as we shall see presently) is 26, and the mean number 3-25; if we take the departures from the mean, and sum them irrespective of sign, the sum is shewn under D. * I believe that Kirkman, in a paper of more than 80 years ago, said that the number of S-sided convex, Eulerian polyhedra, with trihedral comers, was thirteen. 598 THE FORMS OF TISSUES [ch. A B C D aac 222 33 44 6 8-5 aca 222 33 44 6 8-5 acb 222 33 4 55 8-5 bcb 22233 455 8-5 abc 222 3 4445 8-0 abab 22 33 4444 60 aabb 22 3333 55 7-0 aaba 22 333 445 6-5 abba 22 333 44 5 6-5 aaab 22 3333 46 7-0 aaaa 2233333 7 7-5 cc 2222 44 55 10-0 I 9 k Nine cells may be arranged in twenty-seven ways. In higher series the numbers increase very rapidly, but the cells will tend to overlap, and so introduce a new complication* We may draw help from the theory of polyhedra (in an elementary, way) if we treat our group of eight cells (none of them "insular") as part of a polyhedron, to be completed by one eight-sided cell, * Max Bruckner states the number of possible arrangements of thirteen cells, with trihedral junctions, as nearly 50,000; of sixteen, nearly 30 miUions; and of eighteen, "bereits iiber einige Billionen" (Proc. Math. Congress, Bologna, 1930, vol. IV, p. 11), It is plain that the study of " cell-lineage," or the mapping out in detail of the cell-arrangements after repeated cell-divisions, is only possible under severe limitations. VIIl] THE PARTITIONING OF SPACE 599 serving (so to speak) as a base under all the rest. Then, calling ^3, ^4, etc., the number of three-sided and four-sided facets, we re- classify our twelve configurations as follows : Polyhedral arrangements of eight cells (none of them ''insular'*)^ considered as part of a nine-faced polyhedron, whose ninth face is octagonal. ^3 ^4 F, F, I ae cd b i h f gk J 2 — — 1 2 — 1 — It is of interest, and of more than mere mathematical interest, to know, not only that these possible arrangements are few, but that they are strictly defined as to the number and form of the respective faces. For we know that we are limited to three-way corners or nodes; and, that being so, the fpllowing simple rule holds for the facets — a rule which we shall use later on in still more curious circumstances, and which may be easily verified in any line of the foregoing table : ZF^ + 2F, + F,± O.F, -F,- 2F, - etc. = 12. We may produce and illustrate all these configurations by blowing bubbles in a dish and here (Fig. 247) is the complete series, up to seven cells. They correspond precisely to the diagrams shewn on p. 596, and their resemblance to embryological diagrams is only cloaked a little by the circular outhne, the artificial boundary of the system. Of the twelve eight-celled arrangements, four seem unstable; these include the one case (i) where one cell of the eight is in contact with all seven others, and the three cases (a, e, h) where one is in contact with six others. The reason of this insta- bility is, I imagine, that the internal angles cannot be angles of 120°, as equilibrium demands, unless the sides be curved, and convex inwards; but this implies a combined pressure from without on the large cell in the middle. While it adjusts its walls, then, to the required angles, the large cell tends to close up, to lose hold 600 THE FORMS OF TISSUES [CH. of the boundary of the system, and to become an island-cell entirely surrounded by the rest. Fig. 247. Group of soap-bubbles, blown in Petri dishes, a, b, c, the normal partitioning^ of groups of three, four or five cells or bubbles, d, the three ways of partitioning a group of six cells or bubbles, e, three of the four ways of partitioning a group of seven cells. Among the published figures of embryonic stages and other cell aggregates, we only discern the httle 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 often recognise, without much difficulty, what the actual arrangement was whereby VIIl] THE SEGMENTATION OF THE EGG 601 the cell- walls met together in equihbrium. I suspect 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 hmitations and narrow range of apphcation we have already considered) is responsible for many inaccurate or incomplete representations of the mutual arrangement of associated cells. In the accompanying series of figures (Figs. 248-255) I have set forth a few aggregates of eight cells, mostly from drawings of Figj 248. Segmenting egg oiTrochus. After Robert. Fig. 249. Two views of segmenting egg of Cynthia partita. After Conklin. Fig. 250. [a) Section of apical cone of Salvinia. After Pringsheim *. [h) Diagram of probable actual arrangement. segmenting eggs. In some cases they shew clearly the manner in which the cell-walls meet one another, always by three-way junctions, at angles of about 120°, more or less, and always with the help of five intermediate boundary walls within the eight-celled system; in other cases I have added a slightly altered drawing, so as to shew, with as Httle change as possible, the arrangement of boundaries * This, like many similar figures, is manifestly drawn under the influence of Sachs's theoretical views, or assumptions, regarding orthogonal trajectories, coaxial circles, confocal ellipses, etc. 602 THE FORMS OF TISSUES [ch. which may have existed, and given rise to the appearance which the observer drew. These drawings may be compared with the diagrams on p. 598, in which the twelve possible arrangements of five inter- mediate partitions for a system of eight cells have been set forth. It will be seen that Robert-Tornow's figure of the segmenting egg of Trochus (Fig. 248) clearly shews the cells grouped after the fashion of l\ while Conkhn's figure of the ascidian egg (Cynthia) shews equally clearly the arrangement e. A sea-urchin egg segmenting Fig. 251. Egg of Pyrosoma. After KorotneflF. Fig. 252. Egg of Echinus, segmenting under pressure. After Driesch. Fig. 253. (a) Part of segmenting egg of Cephalopod (after Watase) ; (6) probable actual arrangement. under pressure, as figured by Driesch, scarcely wants any modifica- tion of the drawing to appear in one case as type /, in another as g. Turning 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 ill drawn, but may be referable, as in my diagram, to type /; after it is figured a very different object, a segmenting egg of the ascidian Pyrosoma, after Korotneif, also, but still more doubtfully, referred to /. In the cuttlefish egg there is again some uncertainty, but it is probably referable to g. Lastly, I have copied from Roux a curious figure of the frog's egg, viewed from the animal pole; it is obviously inaccurate, but may perhaps belong to type e. Of type i, in which the five partitions form VIII] THE SEGMENTATION OF THE EGG 603 four re-entrant angles, that is to say a figure representing the five sides of a hexagon, and one cell is in touch with seven others, I have found no examples among pubhshed figures of segmenting It is obvious enough, without more ado, that these phenomena 2 2 Fig. 254. (a) Egg oi Echinus; (6) do. oi Nereis, under pressure. After Driesch. a b Fig. 255. (a) Eg^of frog, under pressure (after Roux); (6) probable actual arrangement. are in the strictest and completest way common to both plants and animals, in which respect they tally with, and further extend, the fundamental conclusions laid down by Schwann wellnigh a hundred years ago, in his Mikroskopische Untersuthungen iiher die Ueberein- stimmung in der Struktur und dem Wachsthwn der Thiere und Pflanzen*. But now that we have seen how a certain limited number of types of eight-celled segmentation (or of arrangements of eight cell-partitions) appear and reappear here and there throughout the whole world of organisms, there still remains the very important * Berlin, 1839; Sydenham Society, 1847. 604 THE FORMS OF TISSUES [ch. question, whether in each particular organisin 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 published figures of the embryologist are apt to suggest) as a specific character, or at least a constant or normal character, of the particular organism? The answer to this question is a direct negative, but it is only in the work of the most careful and accurate observers that we find it revealed. Rauber (whom, we have more than once had occasion to quote) was one of those embryologists who recorded just what he saw, without prejudice or preconception; as Boerhaave said of Swammerdam, quod vidit id asseruit. Now Rauber has put on record a considerable number of variations in the arrangement of the first eight cells, which form a discoid surface about the dorsal (or "animal") pole of the frog's egg. In a certain number of cases these figures are identical with one another in type, identical (that is to say) save for slight differences in magnitude, relative proportions, or orientation. But I have selected (Fig. 256) 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 might expect in a material system*. Of these six illustrations, two are exceptional. In Fig. 256, 5, we observe that one of the eight cells is insular, and surrounded 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. 256, 6, is * Such preconceptions as Rauber entertained were all in a direction likely to lead him away from such phenomena as he has faithfully depicted. Rauber had no idea whatsoever of the principles by which we are guided in this discussion, nor does he introduce at all the analogy of surface-tension, or any other purely physical concept; but he was deeply under the influence of Sachs's rule of rectangular intersection, and he was accordingly disposed to look upon the configuration represented above in Fig. 256, 6, as the most typical or primitive. His articles on Thier und Pflanze, in Biol. Cbt. iv, 1881, tell us much about this and other biological theories of his time. VIIl] THE SEGMENTATION OF THE EGG 605 again peculiar, and is probably rare, but it is included under the cases considered on p. 491, in which the cells are not in complete fluid contact but are separated by Httle 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. 256, 1 and 2, 4 5 6 Fig. 256. Various modes of grouping of eight cells, at the dorsal or epiblastic pole of the frog's egg. After Rauber. we have the arrangements corresponding to I and g, and in 3 and 4 to c in our table on p. 598. One thing stands out as very certain indeed: that the elementary diagram of the frog's segmenting egg given in textbooks of embryology — in which the cells are depicted as uniformly symmetrical and more or less quadrangular bodies — is entirely inaccurate and grossly misleading*. * Cf. Rauber, Neue Grundlegungen z. K.Mer Zelle, Morphol. Jahrb. viii, p. 273, 1883 : " leh 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 besehrieben, dass es mir nicht einfallen kann, sie liberhaupt nicht anzuerkennen." See also 0. Hertwig, Ueber den Werth d. erste Furchungszelle fiir die Organbildung des Embryo, Arch. f. A7iat. XLin, 1893; here 0. Hertwig maintains that there is no such thing as "cellular homology." 606 THE FORMS OF TISSUES [CH. We begin to realise the remarkable fact, which may even appear a startHng one to the biologist, that all possible groupings or arrangements whatsoever of eight cells in a single layer or surface (none being submerged or wholly enveloped by the rest) are referable to one or another of twelve types or patterns; and that all the thousands and thousands of drawings which dihgent observers have made of such eight-celled embryos or blastoderms, or other eight- celled structures, animal or vegetable, anatomical, histological or Fig. 257. Photographs of lVf)S04) ; the polyhalite is a triple sulphate of lime, magnesia and potash (2CaSO4.MgSO4.K2.SO4 + 2H2O). t Cf. H. WisHcenus, Ztschr. f. Chemie u. Kolloide, vi, 1910; A. Lingelsheim, PHanzenanatomische Strukturbilder in trocknenden Kolloiden, Arch.f. Entw. Mech. XLii, pp. 117-125, 1917. Cf. also Liesegang, Trocknungserscheinungen bei Gelen, Ztschr. f. Ch. u. K. x, p. 229 sq., 1912; Biitschli, Verh. n. h. Ver. Heidelberg, vii, p. 653, 1904. Also {int. al.) Norman Stuart, on Spiral growths in silica gel, Nature, Oct. 2, 1937, p. .589. 668 ON CONCRETIONS, SPICULES, ETC. [ch. bubbles in chrome-gelatine, and which he usecT in one of his early (and none too fortunate) speculations on the nature of the nuclear spindle. We see that the methods by which we attempt to study the chemico-physical characteristics of an inorganic concretion or spicule within the body of an organism soon introduce us to a multitude of phenomena of which our knowledge is extremely scanty, and which we must not attempt to discuss at greater length. As regargls our main point, namely the formation of spicules and other elementary skeletal forms, we have seen that some of them may be safely ascribed to precipitation or crystalhsation of inorganic materials in ways 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 crystalhsable bodies than in others. For instance Harting, and Rainey also, found that calcium oxalate was much less affected by a colloid medium than was calcium carbonate; it shewed in their hands no tendency to form rounded concretions or " calcospherites " in presence of a colloid, but con- tinued to crystallise, either normally or with a tendency to form needles or raphides. It is doubtless for this reason that, as we have seen, crystals of calcium oxalate are so common in the tissues of plants, while those of other calcium salts are rare; but true calco- spherites, or spherocrystals, even 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 crystalhsation in a colloid medium. If the nature of the salt has a marked specific qffect, so also has the gel: silver chromate is thrown down in rings in gelatin but not in agar; replace the silver by lead, and the rings come in agar but not in gelatin; while neither lead nor silver produce them in silicic- acid gel. There link on to such observations as Harting's, and to the statement already quoted that calcareous deposits are associated with the dead residua, or "formed materials," rather than with the living cells of the organism, certain very interesting facts in regard to the solubility of salts in colloid media, which go far to account for the presence (apart from the form) of calcareous pre- * Spharocrystalle von Kalkoxalat bei Kakteen, Ber. d. d. Bot. Gesellsch. p. 178, 1885. IX] OF SOLUBILITY IN COLLOID MEDIA 669 cipitates within the organism*. 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 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 accounted for thereby. 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 phosphate of lime is scarcely more soluble in them 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 on a more soluble salt (as was at one time believed); but to catabolic changes in the proteids of the organism, which throw down salts that had been already formed, but had remained hitherto in albuminous solution. The very shght solubiHty of calcium phosphate under such circumstances accounts for its pre- dominance in mammalian bonef; and, in short, wherever a supply of this salt has been available to the organism. To sum up, we see that, whether from food or from sea-water, calcium sulphate will tend to pass but little into solution in the albuminoid substances of the body: that 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. When carbonate of hme is secreted or precipitated by hving organisms, to form bone, shell, egg-shell, coral and what not, its mineralogical form may vary, but the causes which determine it * W. Pauli u. M. Samec, Ueber Loslichkeitsbeeinfliissung von Elektrolyten durch Eiweisskorper, Biochem. Zeitschr. xvii, p. 235, 1910. Some of these results were known much earher; cf. Fokker in Pflilger's Archiv, vii, p. 274, 1873; also Robert Irvine and Sims Woodhead, op. cit. p. 347. t Which, in 1000 parts of ash. contains about 840 parts of phosphate and 76 parts of calcium carbonate. 670 ON CONCRETIONS, SPICULES, [ch. are all but unknown. It is amorphous in our Ijones. It has the form of calcite in an oyster, a starfish, a Gorgonia, a Globigerina; but of aragonite in most molluscs and in all ordinary corals. It is of calcite in a bird's egg, of aragonite in a tortoise's; of the one in Argonauta, of the other in Nautilus; of the one in an Ammonite, and the other in its Aptychus-\id ; of the one in Ostrea, the other in Unio; of the one in the outer and the other in the inner layers of a limpet or a mussel-shell. Physical chemistry has little to say of the formation of these two, of the parts played by temperature, by the presence of sulphate of hme, or of magnesia or of various impurities; it leaves us in the dark as to what brings the one form or the other into being in the organism*. Organic fibres, animal and vegetable, proteid and non-pro teid, hair and wool, silk, cotton and the rest, may be mentioned here in passing: because, as formed material, they have a certain analogy to the spicular formations with which we are concerned. A hair or a wool-fibre may shew upon its surface the scaly or scurfy remnants of the living cells among which its substance was laid down; but the wool itself is by no means. living, but is so nearly crystalline as to shew, in an X-ray photograph, the Laue interference- figures well known to physicists. Moreover, the same identical figure is obtained from such diverse sources as human hair, merino- wool and porcupine's quill. But if we stretch the thread, whether of hair or wool, the first Laue diagram changes to another; one crystalline arrangement has shifted over into a new form of molecular equihbrium. We are deahng with a crystalline, or crystal-hke, form of keratin, the substance of which hoof and horn, nail, scale and feather are made; and this remarkable substance turns out to be a comparatively simple substance after all, with no very high or protein-like molecule |. From the comparatively small groun of inorganic formations which, arising within living organisms, owe their form to precipita- tion or to crystallisation, that is to say to chemical or other molecular * Cf. Marcel Prenant, Les formes mineralogiques du calcaire chez les etres vivants, Biol. Reviews, ii, pp. 365-393, 1927. •f" The study of wool and other fibres has much technical importance, ancj has gone far during the last few years; cf. W. T. Astbury, in Phil. Trans. (A), ccxxx, pp. 75-100, 1931, and other papers. IX] AND SPICULAR SKELETONS 671 forces, we shall presently pass to that other and larger group which appears to be conformed in direct relation to the forms and the arrangement of cells or other protoplasmic elements*. 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 Hnes 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 Ume or of silica in presence of colloids f. This is the case, for instance, with various small spicules of a globular or spheroidal form, consisting of amorphous silica, con- centrically striated within, and often developing irregular knobs or tiny tubercles over their surfaces. In the aberrant sponge AstroscleraX, we have, to begin with, rounded, striated discs or globules, which in hke manner are nothing more nor less than the calcospherites of Harting's experiments; and as these grow they become closely aggregated together (Fig. 309), 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 the Alcyonaria. These also must be looked upon as the simple result of chemical deposition, the form of the deposit being somewhat modified in conformity with the surrounding tissues: just as in the simple experiment the form of the con- cretionary precipitate is affected by the heterogeneity, visible or invisible, of the matrix. Lastly, the simple needles of amorphous * Cf. Fr. Dreyer, Die Principien der Geriistbildung bei Rhizopoden, Spongien und Echinodermen, Jen. Zeitschr. xxvi, pp. 204-468, 1892. t In a very anomalous Australian sponge, described by Professor Dendy {Nature, May 18, 1916, p. 253) under the name of Collosderophora, the spicules are "gelatinous," consisting of a gel of colloid silica with a high percentage of water. It is not stated whether an organic colloid is present together with the silica. These gelatinous spicules arise as exudations on the outer surface of cells, and come to lie in intercellular spaces or vesicles. J J. J. Lister, in Willey's Zoological Results, pt iv, p. 459, 1900. § The peculiar spicules of Astrosdera are said to consist of spherules, or calco- spherites, of aragonite, spores of a certain red seaweed forming ^ the nuclei or starting-points of the concretions (R. Kirkpatrick, Proc. E.S. (B), lxxxiv, p. 579, 1911). ON CONCRETIONS, SPICULES, [CH. 672 silica which constitute one of the commonest types of spicule call for little in the way of explanation ; they are accretions or deposits about a linear axis, or fine thread of organic material, just as the ordinary rounded calcospherite is deposited about some minute point or centre of crystallisation, and as ordinary crystallisation may be started by a particle of dust; in some cases they also, like the others, are apt to be roughened by more irregular secondary NMNf Fig. .*i09. Close-packed caleospherites, or so-called '"spicules," oi Astrosclera. After Lister. deposits, which probably, as in Harting's experiments, assume this irregular form when material runs short. Our few foregoing examples, diverse as they are in look and kind, 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 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 crystal- lisation or molecular adhesion is required. As we enter on this IX] ANL SPICULAR SKELETONS 673 subject, which is certainly no small nor easy one, it may conduce to simplicity and to brevity if we make a rough classification, by way of forecast, of the conditions we are likely to meet with. Just as we look upon animals as constituted, some of a great number of cells, others of a single cell or of but 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 ll.ing cellular elements are necessarily concerned; but the form and even the arrangement of these bone-forming cells or corpuscles are monotonously simple, and give no physical explana- tion 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 we have other skeletal elements whose whole magnitude is commensurate with that of a 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 dehcate 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 Httle, the massive and the minute, in regard to their controlling forces and resultant forms. And if all this be so, and if the range of action of the molecular forces be now the important and fundamental thing, then we may somewhat extend our statement of the case, and include among our directive or constructive influences 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 674 ON CONCRETIONS, SPICULES, ETC. [ch. characters connote) of the order of magnitude of which we are speaking. Proceeding a Httle farther in our classification, we may conceive each Httle skeletal element to be associated with, and developed by, a single cell or vesicle, or alternatively a cluster or "system" of consociat^d 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 to assume a configuration comparable to the surface of a fluid drop or 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 inter- stices between the cells, in the partition-walls which separate them, or in the still more restricted edges, or junctions between these partition-walls; conditions such as these will go a long way to help us to understand many sponge-spicules and an immense variety of radiolarian skeletons. And lastly (for the present), there is a possible and very interesting case of a skeletal element associated with the surface of a cell, not so as to cover it like a shell, but only so as to pursue a course of its own within it, and subject to the restraints imposed by such confinement to a curved and limited surface. With this curious condition we shall deal immediately. This preliminary and much simplified classification of the lesser skeletal, or micro-skeletal, forms does not pretend (as is evident enough) to completeness. It leaves out of account some conforma- tions and configurations with which we shall attempt to deal, and others which we must perforce omit. But nevertheless it may help to clear or mark our way towards the subjects which this chapter has to consider, and the con4itions by which they are at least partially defined. Among the possible, or conceivable, types of microscopic skeletons let us 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 more complicated form to a restraint imposed by the cell to which it is confined, and within whose bounds it is generated. IX] OF INTRACELLULAR SPICULES 675 The conception of a spicule developed under such conditions came from that very great mathematical physicist, 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 Hnes of the long calcareous spines of Globigerina or Hastigerina, or the similarly radiating but siliceous spicules of many Radiolaria. In physical language, if our crystalline structure has once begun to be laid down in a definite orientation, further additions tend to accrue in a like regular fashion and in an identical direction: corresponding to the phenomenon of so-called "orientirte Adsorption," as described by Lehmann. In Globigerina or in Acanihocystis the long needles grow out freely into the surrounding medium, with nothing to impede their rectihnear growth and approximately radiate symmetry. 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 some simple and constant restraint or compulsion. If we take any two points on a smooth curved surface, such as that of a sphere or spheroid, and imagine a string stretched between them, we obtain what is known in mathematics as a ''geodesic" curve. It is the shortest Une which can be traced between the two points upon the surface itself, and it has always the same direction upon the surface to which it is confined; the most famihar of all cases, from which the name is derived, is that curve, or "rhumb-line," upon the earth's surface which the navi- gator learns to follow in the practice of "great-circle sailing," never altermg his direction nor departing from his nearest road. Where the surface is spherical, the geodesic is Hterally 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 a spheroid, whether prolate or oblate (that is to say a figure of revolution in which an ellipse rotates about its long or its short axis), then the system of geodesies becomes more complicated. For in it the elliptic meridians are all geodesies, and so is the circle of the equator; though the 676 ON CONCRETIONS, SPICULES, ETC. [ch. circles of latitude are not so, any more than in the sphere. But a line which crosses the equator at an oblique angle, if it is to-be geodesic, will go on so far and then turn back again, winding its way in a continual figure-of-eight curve between two extreme latitudes, as when we wind a ball of wool. To say, as we have done, that the geodesic is the shortest Une between two points upon the surface, is as much as to say that it is a trace 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 (save only for its confinement to that surface) in a straight line, the resultant form which it will assume will be that of a geodesic. Let us now imagine a spicule whose natural tendency is to grow into a straight Hnear 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 ; its fine of growth will be a geodesic 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 nearly circular arc; or, more commonly, a plane curve which is a portion of an eUipse; 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 on the angle by which the spicule is inclined to the elhpsoid 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 inspection, to he not in one plane but in a more comphcated twist. This investigation includes a series of forms which are abundantly represented among actual sponge-spicules, as illustrated in Figs. 310 and 311. ^t^, Growth or motion, when confined to some particular curved surface, may appear in various forms and in unexpected places. An amoeba, creeping along the inside or the outside of a glass tube, was found in either case to follow a winding, spiral path : it was really doing its best to go straight — in other words it was following a geodesic or loxodromic path, determined by whatsoever angle of obhquity to the axis of the tube it had chanced to start out upon. :x] OF INTRACELLULAR SPICULES 677 The spiral bands of chlorophyll in Spirogyra, set at varying angles of hehcoid obhquity, are (I take it) very beautiful examples of con- tinuous growth under the restraint of a cylindrical surface. (Q) Fig. 310. Sponge and holothurian spicules. To return to our sponge-spicules. If the spicule be not restricted to linear growth, but have a tendency to expand, or to branch out from a main axis, we shall obtain a series of more complex figures, all related to the geodesic system of curves. A notable 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 Fig. 312. An "amphidisc" of Ilyalonema. 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 by rays or spokes with a reflex curvature, corresponding to the spherical or elhpsoid curvature of the cell. Such spicules as these are exceedingly conmion among various sponges (Fig. 312). 678 ON" CONCRETIONS, SPICULES, ETC. [ch. Furthermore, if these mechanical methods of conformation, and others hke to these, be the true cause of the shapes which the spicules assume, it is plain that the production of these spicular shapes is not a specific function of the sponge, but that we should expect the same or similar spicules to occur in other organisms, wherever the conditions of inorganic secretion within closed cells were very much the same. As a matter of fact, in the sea-cucumbers, 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 indis- tinguishable 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 siUceous: this being just another proof of the fact that in such cases as these the form of the spicule is not due to its chemical nature or molecular structure, but to the external forces to which it is subjected. The broad fact that the skeleton is calcareous in certain large groups of animals and calcareous in others is as remarkable as its causes are obscure. I for one have no idea why some sponge- skeletons are of the one and some the other, with never the least admixture of the two; or why the diatoms and radiolarians are all the one, and the molluscs and corals and foraminifera are all the other!. So much for that small class of sponge-spicules whose forms seem due to the fact that they are developed within, or under the restraint imposed by, the surface of a single cell or vesicle. Such spicules are usually of small size as well as of 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. These are the many and various cases which we explain on the * See for instance the plates in' Theel's Monograph of the Challenger Holo- thuroidea; also Sollas's Telractinellida, p. Ixi. Cf. also E. Merke, Studien am Skelet der Echin^dernien, Zool. Jahrbiicher (Abth. f. allgem. Zoologie), 1916-19. t The particles of lime and silica tend to bear opposite charges; siliceous organisms seem to flourish in the colder waters as the calcareous certainly do in warmer seas. And such facts, or tendencies, as these may help some day to explain the phenomenon. IX] OF'THE SKELETON OF SPONGES 679 assumption that they develop in association (of some sort or another) with the lines of junction, or boundary-edges, of contiguous cells. They include the triradiate spicules of the calcareous sponges, the quadriradiate or " tetractinelHd " spicules which occur sometimes in the same group but more characteristically in certain siHceous sponges known as the TetractineUidae, and perhaps (though these last are somewhat harder to understand) the six-rayed spicules of the HexactinelHds. We shall come later on to more compUcated skeletons of the same type among the Radiolaria. The spicules of the calcareous sponges are commonly triradiate, and the three radii are usually inclined to one another at nearly equal angles; in certain cases, two of the three rays are nearly in a straight line, and at right angles to the third *. They are not always in a plane, but are often inchned to one another in a trihedral angle, not easy of precise measurement under the microscope. The three rays are often supplemented by a fourth, which is set tetra- hedrally, making nearly co-equal 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 sarne writer there is no organic matter in the spicule, either in the form of an axial filament or otherwise, and the appearance of stratifica- tion, often simulating the presence of an axial fibre, is due to "mixed crystallisation" of the various constituents. The spicule is a true crystal, and therefore its existence and its form are primarily due to the molecular forces of crystallisation; moreover it is a single crystal and not a group of crystals, as is seen by its behaviour in polarised fight. But its axes are not crystalline axes, its angles are variable and indefinite, and its form neither agrees with, nor in any way resembles, any one of the countless, polymorphic forms in which calcite is capable of crystallising. It is as though it were carved out of a solid crystal; it is, in fact, a crystal under restraint, a crystal growing, as it were, in an artificial mould, and this mould is constituted by the surrounding cells or structural vesicles of the sponge. * For very numerous illustrations of the triradiate and quadriradiate spicules of the calcareous sponges, see (int. at.), papers by Dendy {Q.J. M.S. xxxv, 1893), Minchin {P.Z.S. 1904), Jenkin {P.Z.S. 1908), etc. 680 ON CONCRETIONS, SPICULES, ETC. [ch. We have already studied in an elementary way, but enough for our purpose, the manner in which three, four or more cells, or bubbles, meet together under the influence of surface-tension, in configurations geometrically similar to what may be brought about by a uniform distribution of mechanical pressure. And we have seen how surface-energy leads to the adsorption of certain chemical substances, first at the corners, then at the edges, lastly in the partition- walls, of such an assemblage of cells. A spicule formed in the interior of such a mass, starting at a corner where four cells meet and extending along the adjacent edges, would then (in theory) have the characteristic form which the geometry of the bee's cell has taught us, of four rays radiating from a point, and set at co-equal angles to one another of 109°, approximately. Precisely such " tetractinellid " spicules are often formed. But when we confine ourselves to a plane assemblage of cells, or to the outer surface of a mass, we need only deal with the simpler geometry of the hexagon. In such a plane assemblage we find the cells meeting one another in threes ; when the cells are uniform in size the partitions are straight lines, and combine to form regular hexagons; but when the cells are unequal, the partitions tend to be curved, and to' combine to form other and less regular polygons. Accordingly, a skeletal secretion originating in a layer or surface of cells will begin at the corners and extend to the edges of the cells, and will thus take the form of triradiate spicules, whose rays (in a typical case) will be set at co-equal angles of 120° (Fig. 313, F). This latter condition of inequahty will be open to modification in various ways. It will be modified by any inequality in the specific tensions of adjacent cells ; as a special case, it will be apt to be greatly modified at the surface of the system, where a spicule happens to be formed in a plane perpendicular to the cell-layer, so that one of its three rays lies between two adjacent cells and the other two are associated with the surface of contact between the cells and the surrounding medium; in such a case (as in the cases considered in connection with the forms of the cells themselves on p. 494), we shall tend to obtain a spicule with two equal angles and one unequal (Fig. 313, A, C) ; in the last case, the two outer, or superficial rays, will tend to be markedly curved. Again, the equiangular condition will be departed from, and more or less curvature will be imparted to the IX OF THE SKELETON OF SPONGES 681 rays, wherever the cells of the system cease to be uniform in size, and when the hexagonal symmetry of the system is lost accordingly. Lastly, although we speak of the rays as meeting at certain definite angles, this statement applies to their axes rather than to the rays themselves. For if the triradiate spicule be developed in the interspace between three juxtaposed cells it is obvious that its sides will tend to be concave, because the space between three contiguous equal circles is an equilateral, curvilinear triangle ; and even if our Fig. 313. Spicules of Granlia and other calcareous sponges. After Haeckel. spicule be deposited, not in the space between our three cells, but in the mere thickness of an intervening wall, then we may recollect that the several partitions never actually meet at sharp angles, but the angle of contact is always bridged over by an 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 Haeckel's, we shall find all these various configurations severally and completely illustrated. 682 ON CONCRETIONS, SPICULES, ETC. [ch. The tetrahedral, or rather tetractineUid, spicule needs no further explanation in detail (Fig. 313, 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 three facets or partitions meet, their edges being inclined to one another at equal angles of about 109° — the "Maraldi" angle. 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 hne of junction between the three cells, and the other three limbs (which will then be curved Hmbs) will corre- spond to the three edges where the three cells meet in pairs 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 for some physical agency to explain the presence of the solid material just at the junctions or interfaces of the cells, and for the forces by which it is confined to, and moulded to the form of, these intercellular or interfacial contacts. We owe to Dreyer the physical or mechanical theory of spicular conforma- tion 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 soHd material, secreted or excreted by the organism, accumu- lates in the interstices between the cells, and is there subjected to mechanical pressure or constraint as the cells get 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 might have to renounce some of our assumptions as to what takes j)lace at the surface of the system. But where a few years ago the concept of secretion seemed precise enough, we turn now-a-days to the phenomenon of adsorption as a further stage towards the elucidation of our facts, and here we have a case in point. In the tissues of our sponge, IX] OF THE SKELETON OF SPON.GES 683 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 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 prob- abihty that certain substances will tend to be concentrated and ultimately deposited .just in these hues of intercellular contact and conjunction. In other words, adsorptive concentration, under osmotic pressure, at and in the surface-film which bounds contiguous cells, and especially in the edges where these films meet and intersect, emerges as an alternative (and, as it seems to me, a highly preferable alternative) to Dreyer's conception of an accumulation under mechanical pressure in the vacant spaces left between one cell and another. But a purely chemical, or purely molecular, adsorption is not the only form of the hypothesis on which we may rely. For from the purely physical point of view, angles and edges of contact between adjacent cells will be loci in the field of distribution of surface- energy, and any material particles whatsoever will tend to undergo a diminution of freedom on entering one of those boundary regions. Let us imagine a couple of soap bubbles in contact with one another ; over the surface of each bubble tiny bubbles and droplets glide in every direction ; but as soon as these find their way into the groove or re-entrant angle between the two bubbles, there their freedom of movement is so far restrained, and out of that groove they have little tendency, or little freedom, to emerge. A cognate phenomenon is to be witnessed in microscopic sections of steel or other metals. Here, together with its crystalline structure, the metal develops a cellular structure by reason of its lack of homogeneity; for in the molten state one constituent tends to separate out into drops, while the other spreads over these and forms a fihny reticulum between — -the disperse phase and the continuous phase of the colloid chemists. In a poHshed section we easily observe 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 684 ON CONCRETIONS, SPICULES, ETC. [ch. freedom which they undergo on entering one of these boundary regions. And the same phenomenon is turned to account in the various "separation-processes" in which metalHc particles are caught up in the interstices of a froth, that is to say in the walls of the foam-cells or Schaumkammern* . 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 considered. It matters httle or not at all for the phenomenon in question, what is the histological nature or "grade" of the vesicular structures on which it depends. In some cases (apart from sponges), they may be no more than httle alveoh of an intracellular protoplasmic network, and this would seem to be the case at least in the protozoan Entosolenia aspera, within the vesicular protoplasm of whose single cell Mobius has described tiny spicules in the shape of little tetra- hedra with sunken or concave sides. It is probably the case also in the small beginnings of Echinoderm spicules, which are likewise intracellular and are of similar shape. Among the sponges we have many varying conditions. In some cases there is reason to believe that the spicule is formed at the boundaries of true cells or histo- logical units ; but in the case of the larger triradiate or tetractineUid spicules they far surpass in size the actual "cells." We find them lying, regularly and symmetrically arranged, between the "pore- canals" or "ciliated chambers," and it is in conformity with the shape and arrangement of these large rounded or spheroidal struc- tures that their shape is assumed. Again, it is not at variance with our hypothesis to find that, in the adult sponge, the larger spicules may greatly outgrow the bounds not only of actual cells but also of the ciliated chambers, and may even appear to project freely from the surface of the sponge. For we have already seen that the spicule is capable of * The crystalline composition of iron was recognised by Hooke in the Micro- graphia (1665); and the cellular or polyhedral structure of the raetal was clearly recognised by Reaumur, in his Art de convertir le fer forge en acier, 1722. IX] OF THE SKELETON OF SPONGES 685 growing, without marked change of form, by further deposition, or crystalhsation, 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 estabhshed its characteristic but non-crystalline 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 little trio of cells; and the little rods meet and fuse together while still very minute, when the whole spicule is only about 200 ^^ ^ milhmetre long. At this stage, it is interesting to learn that the spicule is non- crystaUine; but the new accretions of calcareous matter are soon deposited in crystaUine form. This observation threw 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 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 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 only natural that the spicule should first arise in the form of separate and detached Hmbs or rays. Among the " tetractinelhd " sponges, whose spicules are com- posed of amorphous silica or opal, all or most of the above-described main types of spicule occur, and, as the name of the group implies, the four-rayed, tetrahedral spicules are especially represented. A somewhat frequent type of spicule is one in which one of the four rays is greatly developed, and the other three constitute small prongs diverging at equal angles from the main or axial ray. In 686 ON CONCRETIONS, SPICULES, ETC. [CH. all probability, as Dreyer suggests, we have here had to do with a group of four vesicles, of which three were large and co-equal, while i\ fourth and very much smaller one lay above and between the other three. In certain cases where we have hkewise one large and three much smaller rays, the latter are recurved, as in Fig. 314, a-c. This type, save for the constancy of the number of rays and the limitation of the terminal ones to three, and save also for the more important difference that they occur only at one and not at both ends of the long axis, is similar to the type of V y 1 1 spicule illustrated in Fig. 312, which we \^ 1 ^f fl have explained as being probably de- ^fflr J veloped within an oval cell, by whose ■ ^ 11 walk its branches have been cabined and I I confined. But it is more probable that ' we have here to do with a spicule de- Fig. 314. Spicules of tetracti- i • ^i • i ^ r r j.\. nellid sponges (after SoUas). veloped m the midst of a group of three a-e, anatriaenes; d-f, pro- co-equal and more or less elongated or tnaenes. cylindrical cells or vesicles, the long axial ray corresponding to their common edge or 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 S-shaped spicules formed within the bounds of a single cell, so also in the case of the larger tetrac- tinellid types do we find the same configurations reproduced among the holothuroids as we have dealt with in the sponges. The holo- thurian 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. 315). The very peculiar spicules of the holothurian Synapta, where a tiny anchor is pivoted or hinged on a perforated plate, are a puzzle indeed; but we may at least solve part of the riddle. How the hinge is formed, I do not know; the anchor gets its shape, perhaps, in some such way as we have supposed the " amphidiscs "of -ff^^/^Zo- IX] OF HOLOTHURIAN SPICULES 687 nema to acquire their reflexed spokes, but the perforated plate is more comprehensible. Each plate starts in a httle clump of cells in whose boundary-walls calcareous matter is deposited, doubtless by adsorption, the holes in the finished plate thus corresponding to the cells which formed it. Close-packing leads to an arrangement of six cells round a central one, and the normal pattern of the plate displays this hexagonal configuration. The calcareous plate begins as a little rod whose ends fork, and then fork again: in the same inevitable trinodal pattern which includes the "polar furrow" of the embryologists. The anchor had been first formed, and the Rh r^^O Fig, 315. Various holothurian spicules. After Theel. little plate is added on beneath it. The first spicular rudiment of the plate may lie parallel to the stock of the anchor or it may lie athwart* it. From the physical point of view it would seem to be a mere matter of chance 'which way the cluster of cells happens to he; but this difference of direction will cause a certain difference in the symmetry of the resulting plate. It is this very difference which systematic zoologists at one time seized upon to distinguish S. Buskii from our two commoner "species." The two latter * Cf. S. Becher, Nicht-funktionelle Korrelation in der Bildung selbstandiger Skeletelemente, Zool. Jahrbucher (Physiol.), xxxi, pp. 1-189, 1912; Hedwiga Wilhelmi, Skeletbildung der fiisslosen Holothurien, ibid, xxxvii, pp. 493-547, 1920; Arch. f. Entw. Mech. xlvi, pp. 210-258, 1920. See also W. Woodland, Studies in spicule-formation, Q.J. M.S. xlix, pp. 535-559, 1906; li, pp. 483-509, 1907 and R. Semon, Naturgeschichte der Synaptiden, Mitth. Zool. St. Neapel, vii, pp. 272-299, 1886. On the common species of Synapta, see Koehler, Faune de France, Echinodermes, 1921, pp. 188-9. 688 ON CONCRETIONS, SPICULES, ETC. [CH. {S. inhaerens and S. digitata) are mainly distinguislied from one another by the number of holes in the plate, that is to say, by the average number of cells in the little cluster of which the plate or spicule was formed. In many or perhaps most other holothurians Fig. 316. Development of anchor-plate in Synapta. After Semon. the spicules consist of little perforated plates or baskets, developed in the same way, about cells or vesicles more or less close-packed, and therefore more or less symmetrically arranged (Fig. 316). Fig. 317. Spicules of hexactinellid sponges. After F. E. Schultze. The six-rayed siliceous spicules of the hexactinellid sponges, while they are perhaps the most regular and beautifully formed spicules to be found within the entire group, have beert found very difficult to explain, and Dreyer has confessed his complete inabihty to account for their conformation*. But, though it may only be * Cf. Albr. Schwan, Ueber die Funktion des Hexactinellidenskelets, u. seine Vergleichbarkeit mit dera Radiolarienskelet, Zool. Jb., Ahth. allg. Zool. u. Physiol. XXXIII, pp. 603-616, 1913; cf. V. Hacker, Bericht uber d. Tripyleenausbeute d. d. Tiefsee-Exped, Verh. d. zool. Ges. 1904. IX] OF HEXACTINELLID SPICULES 689 throwing the difficulty a Uttle 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 what is known as '/closest packing," but in hnear 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 dodeca- hedra but of cubes or parallelepipeda. 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 hexactinelHd. And this being so, the spicules formed between the linear, or cubical series of vesicles, will have the same tendency towards a " hexactinelhd " 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 te tract inellid form, when developed in connection with the angles and edges of a system of hexagons, or a system of rhombic dodecahedra. However the hexactinelhd spicules be arranged (and this is none too easy to determine) in relation to the tissues and chambers of the sponge, it is at least clear that, whether they lie separate or be fused together in a composite skeleton, th-ey effect a symmetrical partitioning of space according to the cubical system, in contrast to that closer packing which is represented and effected by the tetrahedral system^*. 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 segmentationf. But in what is described as the "spiral type" of segmentation, it is stated that, as soon as the first hori- zontal 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, * Chall. Hep., Hexactinellida, pis. xvi. liii, Ixxvi, Ixxxviii. t See, for instance, the figures of Ihe segmenting egg of Synapta (after Selenka), in Korschelt and Heider's Vergleichende Entvnckinngsgeschichte. On the spiral type of segmentation as a secondary derivative, due to mechanical causes, of the "radial" type of segmentation, see E. B. Wilson, Cell-lineage of Nereis, Journ, Morph. VI, p. 450, 1892. 690 ON CONCRETIONS, SPICULES, ETC. [ch. by means of a rotation about the vertical axis*." 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 shp into the closest propinquity, or position of minimmn potential, will be found to manifest itself. Lastly, a curious case is presented by the so-called "chessman" spicules of Latrunculia and of a few other sponges, where the spicular shaft is thickened at regular intervals, and the thickenings grow into whorled and flattened lobes. Dendy suggested that the developing spicule is in a state of vibration (due perhaps to the water-currents of the sponge), and that the whorls correspond to nodes, or loci of comparative rest, where the formative cells tend to settle down and do their work undisturbed. The position of the nodes and internodes will depend on many circumstances, on whether the spicule be a fixed rod or a free one, straight or curved, uniform in section or tapering towards either end. In the free bar there should tend, in any case, to be a node in the middle, and two more at definite distances from either end. It so happens that in the forms investigated there are only two whorls, the median and one other; but J. W. Nicholson has calculated the positions of these according to the vibration theory, and the theoretical results are found to agree with those of observation very closely indeed. That one of the whorls should be lacking might seem to imperil the proof; but on the other hand among large numbers of spicules no one was found to have its whorls in a position incon- sistent with the theory, and there was the required agreement between the shape of the spicule and the position of the whorls. The absence of a third whorl is explained as due to a lack of the necessary formative cells at that part of the spiculef. The theory is in a way supported by recent work (by R. W. Wood of Baltimore and others) on "supersonic vibrations," showing excessively rapid * Korschelt and Heider, p. 16. t A. Dendy and J. W. Nicholson, On the influence of vibration upon the form of certain sponge-spicules, Proc. R.S. (B), lxxxix, pp. .'^73-587, 1917; A. Dendy, The chessman spicules of the genus Latrunculia, etc., Journ. Quekett Microsc. Club, xm, pp. 1-16, 1917. IX] OF HEXACTINELLID SPICULES 691 vibrations in quartz rods, more rapid even than Dendy's hypothesis would seem to require. But on the other hand, it is only to few and even exceptional spicules that the theory would seem to apply. This question of the origin and causation of the forms of sponge- spicules, with which ^we have now sought to deal, is all the more important and all the more interesting because it has been discussed time and again, from points of view which are characteristic of very different schools of thought in biology. Haeckel found in the form of the sponge-spicule a typical illustration of his theory of "bio-crystallisation"; he considered that these " biocrystals " re- presented 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-crystalhne 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 differentiated in a vast variety of ways, in the struggle for existence*." What Haeckel precisely meant by these words is not clear to me. F. E. Schultze, perceiving that identical forms of spicule were developed whether the material were crystaUine or non-crystaUine, abandoned all theories based upon crystaUisation ; 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 w^alls of the sponge, and finding clear evidence of "utility" in the specific characters of these skeletal elements, had no difficulty in ascribing them to natural selection. SoUas 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- * "Hierbei nahm der kohlensaure Kalk eine halb-krystallinische Beschaffen- heit an, und gestaltete sich unter Aufnahme von Krystallwasser und in Verbindung mit einer geringen Quantitat von organischer Substanz zu jenen individuellen, festen Korpern, welche durch die naturliche Ziichtung als Spicula zur Skeletbildung benutzt, und spaterhin durch die Wechselwirkung von Anpassung und Vererbung ira Kampfe urns Dasein auf das Vielfaltigste umgebildet und differenziert wurden." Die Kalkschwdmme, i, p. 377, 1872; cf. also pp. 482, 483. 692 ON CONCRETIONS, SPICULES, ETC. [ch. currents; though even here, by the way, if I understand Haeckel aright, he was thinking not of a direct or immediate physical causation, but rather of one manifesting itself through the agency of natural selection*. SoUas laid stress upon the "path of least resistance" as determining the direction of growth; while Dreyer dealt in greater detail with the 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 constituent cells, or by the minute structure of the intracellular protoplasm. But neither of these writers, so far as I can discover, was inchned to doubt for a moment the received canon of biology, which sees in such strucHires as these the characteristics of true organic species, the indications of blood-relationship and family hke- ness, and the evidence by which evolutionary descent throughout geologic time may be deduced or deciphered. Minchin, in a well-known paperf, took sides with F. E. Schultze, and gave his reasons for dissenting from such mechanical theories as those of SoUas and of Dreyer. For example, after pointing out that all protoplasm contains a number of "granules" or micro- somes, contained in an alveolar framework and lodged at the nodes of a 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 similar or identical position. "If vesicular tension cannot in any other instance cause the granules at the nodes to assume a tetraxon form, why should it do so for the sclerites?" The answer is not far to seek. If the force which the "mechanical" hypothesis has in view were simply that of mechanical pressure, as between solid bodies, then indeed we should expect that any substances lying between the impinging spheres would tend to assume the quadriradiate or "tetraxon" form; but this conclusion does not follow at all, in so far as it is to surface-energy that we ascribe the phenomenon. Here the specific nature of the substance makes all the difference. We cannot argue from one ♦ Op. cit. p. 483. "Die geordnete, oft so sehr regelmassige und zierliche Zusam- mensetzung des Skeletsy stems ist zum grossten Theile unmittelbares Product der Wasserstromung; die characteristische Lagerung der Spioula ist von der constanten Richtung des Wasserstroms hervorgebracht; zum kleinsten Theile ist Bie die Folge von Anpassungen an untergeordnete ^ussere Existenzbedingungen." t Materials for a Monograph of the Ascones, Q.J. M.S. XL, pp. 469-587, 1898. IX] OF THE SKELETON OF SPONGES 693 substance to another; adsorptive attraction shews its effect on one and not on another; and we have no reason to be surprised if we find that the Httle granules of protoplasmic material, which as they lie bathed in the more fluid protoplasm have (presumably, and as their shape indicates) a strong surface-tension of their own, behave towards the adjacent vesicles in a very different fashion to the incipient aggregations of calcareous or siliceous matter in a colloid medium. "The ontogeny of the spicules," says Professor Minchin, "points clearly to their regular form being a phylogenetic adaptation, which has becmne fixed and handed on by heredity, appearing in the ontogeny ai a prophetic adaptation.'' And again, "The forms of the spicules are the result of adaptation to the requirements of the sponge as a whole, produced by the action of natural selection upon variation in every direction." It would scarcely be possible to illustrate more briefly and more cogently than by these few words (or the similar words of Haeckel quoted on p. 691), the fundamental difference between the Darwinian conception of the causation and determination of Form, and that which is based on, and characteristic of, the physical sciences. Last of all, Dendy took a middle course. While admitting that the majority of sponge-spicules are "the outcome of conditions which are in large part purely physical," he still saw in them "a very high taxonomic value," as "indications of phylogenetic history" all on the ground that "it seei^is impossible to account in any other way for the"^ fact that we can actually arrange the different forms in such well-graduated series." At the same time he believed that "the vast majority of spicule-characters appear to be non-adaptive," "that no one form of spicule has, as a rule, any greater survival- value than another," and that "the natural selection of favourable varieties can have had very Uttle to do with the matter*." The quest after lines and evidences of descent dominated morpho- logy for many years, and preoccupied the minds of two or three generations of naturahsts. We find it easier to see than they did that a graduated or consecutive series of forms may be based on physical causes, that forms mathematically akin may belong to * Cf. A. Dendy, The Tetraxonid sponge-spicule : a study in evolution, Acta Zoologica, 1921, pp. 136, 146, etc. Cf. also Bye-products of organic evolution, jQurn. Quekett Microscop. Club, xii, pp. 65-82, 1913. 694 ON CONCRETIONS, SPICULES, ETC. [ch. organisms biologically remote, and that, in general, mere formal likeness may be a fallacious guide to evolution in time and to relationship by descent and heredity. If I have dealt comparatively briefly with the inorganic skeletons of sponges, in spite of the interest of the subject from the physical point of view, it has been owing to several reasons. In the first place, though the general trend of the phenomena is clear, it must be admijited that many points are obscure, and could only be discussed at the cost of a long argument. In the second place, the physical theory is too often (as I have shewn) in conflict with the accounts given by embryologists of the development of the spicules, and with the current biological theories which their descriptions embody; it is beyond our scope to deal with such descriptions in detail. Lastly, we find ourselves able to illustrate the same physical principles with greater clearness and greater certitude in another group of animals, namely the Radiolaria. The group of microscopic organisms known as the Radiolaria is extraordinarily rich in diverse forms or "species." I do not know how many of such species have been described and defined by naturaUsts, but some fifty years ago the number was said to be over four thousand, arranged in more than seven hundred genera*; of late years there has been a tendency to reduce the number. But apart from the extraordinary multiplicity of forms among the Radiolaria, there are certain features in this multiphcity which arrest our attention. Their distribution in space is curious and vague ; many species are found all over the world, or at least every here and there, with no evidence of specific limitations of geo- graphical habitat; some occur in the neighbourhood of the two poles, some are confined to warm and others to cold currents of the ocean. In time their distribution is not less vague: so much so that it has been asserted of them that "from the Cambrian age downwards, the families and even genera appear identical with those now Hving." Lastly, except perhaps in the case of a few large "colonial forms," we seldom if ever find, as is usual in most animals, a local predominance of one particular species. On the * Haeckel, in his Challenger Monograph, p. clxxxviii (1887), estimated the number of known forms at 4314 species, included in 739 genera. Of these, 3508 species were described for the first time in that \i?ork. IX] OF SNOW CRYSTALS 695 contrary, in a little pinch of deep-sea mud or of some fossil "radio- larian earth," we shall probably find scores, and it may be even hundreds, of different forms. Moreover, the radiolarian skeletons are of quite extraordinary delicacy and complexity, in spite of their minuteness and the comparative simplicity of the "unicellular" organisms within which they grow; and these complex conforma- tions have a wonderful and unusual appearance of geometric regularity. All these general considerations seem such as to prepare us for some physical hypothesis of causation. The Httle skeletons remind us of such things as snow-crystals (themselves almost endless in their diversity), rather than of a collection of animals, constructed in accordance with functional needs and distributed in accordance with their fitness for particular situations. Nevertheless, great efforts have been made to attach "a biological meaning" to these elaborate structures, and "to justify the hope that in time their utilitarian character will be more completely recognised*." As Ernst Haeckel described and figured many hundred "species" of radiolarian skeletons, so have the physicists depicted snow- crystals in several thousand different forms |. These owe their multitudinous variety to symmetrical repetitions of qne simple crystalline form — a beautiful illustration of Plato's One among the Many, to ev napa ra iroXXd. On the other hand, the radiolarian skeleton rings its endless changes on combinations of certain facets, corners and edges within a filmy and bubbly mass. The broad difference between the two is very plain and instructive. Kepler studied the snowflake with care and insight, though he said that to care for such a trifle was like Socrates measuring the hop of a flea. The first drawings I know are by Dominic Cassini; and if that great astronomer was content with them they shew how the physical sciences lagged behind astronomy. They date from the time when Maraldi, Cassini's nephew, was studying the bee's cell; * Cf. Gamble, Radiolaria (Lankester's Treatise on Zoology), i, p. 131, 1909. Cf. also papers by V. Hacker, in Jen. Zeitschr. xxxix, p. 581, 1905; Z. f. vnss. Zool. Lxxxiii, p. 336, 1905; Arch. f. Protistenkunde, ix, p. 139, 1907; etc. I See above, p. 411; and see (besides the works quoted there) Kepler, De nive sexangula (1611), Opera, ed. Fritsch, vii, pp. 715-730; Erasmus Bartholin, De figura nivis, Diss., Hafniae, 1661; Dom. Cassini, Obs. de la figure de la neige (Abstr.), Mem. Acad. R. des Sciences (1666-1699), x, 1730; J. C. Wilcke, Om de naturliga sno-figurers, K. V. Akad. Handl. xxii, 1761. 696 ON CONCRETIONS, SPICULES, ETC. [CH. and they shew once more how very rough his measurements of the honeycoipb are bound to have been. Crystals He outside the province of this book ; yet snow-crystals, and all the rest besides, have much to teach us about the variety, the beauty and the very nature of form. To begin with, the snow-crystal is a regular hexagonal plate or thin prism; that is to Fig. 318 a. Snow-crystals, or "snow -flowers." From Dominic Cassini (c. 1600). say, it shews hexagonal faces above and below, with edges set at co-equal angles of 120°. Ringing her changes on this fundamental form, Nature superadds to the primary hexagon endless combina- tions of sii^iilar plates or prisms, all with identical angles but varying lengths of side; and she repeats, with an exquisite symmetry. Fig. 318 6. Snow-crystals. From Bentley and Humphreys, 1931. about all three axes of the hexagon, whatsoever she may have done for the adornment and elaboration of one. These snow-crystals seem (as Tutton says) to give visible proof of the space-lattice on which their structure is framed. The beauty of a snow-crystal depends on its mathematical regularity and symmetry; but somehow the association of many IX] OF THE RADIOLARIAN SKELETON 697 variants of a single type, all related but no two the same, vastly increases our pleasure and admiration. Such is the pecuHar beauty which a Japanese artist sees in a bed of rushes or a clump of bamboos, especially when the wind's ablowing; and such (as we saw before) is the phase-beauty of a flowering spray when it shews every gradation from opening bud to fading flower. The snow-crystal is further complicated, and its beauty is notably enhanced, by minute occluded bubbles of air or drops of water, whose symmetrical form and arrangement are very curious and not always easy to explain*. Lastly, we are apt to see our snow^-crystals after a slight thaw has rounded their sharp edges, and has heightened their beauty by softening their contours. In the majority of cases, the skeleton of the Radiolaria is com- posed, like that of so many sponges, of silica; in one large family, the Acantharia, and perhaps in some others, it is made of a very unusual constituent, namely strontium sulphate f. There is no important morphological character in which the shells made of these two constituents differ from one another; and in no case can the chemical properties of these inorganic materials be said to influence the form of the complex skeleton or shell, save only in this general way that, by their hardness, toughness and rigidity, they give rise to a fabric more slender and delicate than we find among calcareous organisms. A shght exception to this rule is found in the presence of true crystals, which occur within the central capsules of certain Radio- laria, for instance the genus Collos'phaera%. Johannes Miiller (whose knowledge and insight never fail to astonish us§) remarked * We may find some suggestive analogies to these occlusions in Emil Hatschek's paper, Gestalt und Orientirung von Gasblasen in Gelen, Kolloid. Ztschr. xx, pp. 226-234, 1914. t Biitschli, Ueber die chemische Natur der Skeletsubstanz der Acantharia, Zool. Anz. XXX, p. 784, 1906. I For figures of these crystals see Brandt, F. n. Fl. d. Golfes von Neapel, xiii, Radiolaria, 1885, pi. v. Cf. Johannes Miiller, Ueber die Thalassicollen, etc., Abh. K. Akad. Wiss. Berlin, 1858. § It is interesting to think of the lesser discoveries or inventions, due to men famous for greater things. Johannes Miiller first used th^e tow-net, and Edward Forbes first borrowed the oyster-man's dredge. When we watch a living polyp under the microscope in its tiny aquarium of a glass-cell, we are doing what John Goodsir was the first to do; and the microtome itself was the invention of that best of laboratory-servants, "old Stirling," Goodsir's right-hand man. 698 ON CONCRETIONS, SPICULES. [ch. that these were identical in form with crystals of celestine, a sulphate of strontium and barium; and Biitschli's discovery of sulphates of strontium and of barium in kindred forms renders it all but certain that they are actually true crystals of celestine*. In its typical form, the radiolarian body consists of a spherical mass of protoplasm, around which, and separated from it by some sort of porous "capsule," lies a frothy protoplasm, bubbled up into a multitude of alveoU or vacuoles, filled with a fluid which can scarcely differ much from sea-waterf. According to their surface- tension conditions, these vacuoles may appear more or less isolated and spherical, or joined together in a "froth" of polyhedral cells; and in the latter, which is the commoner condition, the cells tend to be of equal size, and the resulting polygonal meshwork beautifully regular. In some cases a large number of such simple individual organisms are associated together, forming a floating colony; and it is probable that many others, with whose scattered skeletons we are alone acquainted, had likewise formed part of a colonial organism. In contradistinction to the sponges, in which the skeleton always begins as a loose mass of isolated spicules, which only in a few exceptional cases (such as Euplectella and Farrea) fuse into a continuous network, the characteristic feature of the radiolarians lies in the production of a continuous skeleton, of netted mesh or perforated lacework, sometimes replaced by and oftener associated with minute independent spicules. Before we proceed to treat of the more complex skeletons, we may begin by dealing with those comparatively few simple cases where the skeleton is represented by loose, separate spicules or aciculae, which seem, like the spicules of Alcyonium, to be isolated formations or deposits, precipitated in the colloid matrix, with no relation to cellular or vesicular boun- daries. These simple acicular spicules occupy a definite position in the organism. Sometimes, as for instance among the fresh- water Heliozoa (e.g. Raphidiophrys), they lie on the outer surface of the organism, and not infrequently (when few in number) they tend to * Celestine, or celestite, is SrS04 with some BaO replacing SrO. t With the colloid chemists, we may adopt (as Rhumbler has done) the terms sputnoid or emulsoid to denote an agglomeration of fluid-filled vesicles, restricting the na.me froth to such vesicles when filled with air or some other gas. IX] AND SPICULAR SKELETONS 699 collect rouiul the bases of the [)seiidopo(lia, or around the larger radiating spicules or axial rays in cases where these latter are present. When the spicules are thus localised around some pro- minent centre, they tend to take up a position of symmetry in regard to it ; instead of forming a tangled or felted layer, they come to lie side by side, in a radiating cluster round the focus. In other cases (as for instance in the well-known radiolarian Aulacantha scolymantha) the felted layer of aciculae lies at some depth below the surface, forming a sphere concentric with the entire spherical organism. In either case, whether the layer of spicules be deep or be superficial, it tends to mark a "surface of discontinuity," a meeting place either between two distinct layers of protoplasm or between the protoplasm and the water around; and it is evident that, in either case, there are manifestations of surface-energy at the boundary, which cause the spicules to be retained there and to take up their position in its plane. The case is analogous to that of a cirrus cloud, which marks a surface of discontinuity in a stratified atmosphere. We have, then, to enquire w^hat are the conditions, apart from gravity, which confine an extraneous body to a surface-film ; and we may do this very simply, by con- sidering the surface-energy of the entire system. In Fig. 319 we have two fluids in contact with one another (let us call W them water and protoplasm), and a body (6) which may be immersed in either, or may be restricted to the boundary between. We have here three possible "interfacial contacts," each with its own specific surface- energy per unit of surface area: namely, that between our particle and the water (let us call it a), that between the particle and the protoplasm (^), and that between water and protoplasm (y). When the body lies in the boundary of the two fluids, let us say half in one and half in the other, the surface-energies concerned are equivalent to (>S'/2) a 4- (^/2) ^; but we nmst also remember that, by the presence Fig. 319. 700 ON CONCRETIONS, SPICULES, ETC. [ch. of the particle, a small portion (equal to its sectional area s) of the original contact-surface between water and protoplasm has been obliterated, and with it a proportionate quantity of energy, equi- valent to sy, has been set free. When, on the other hand, the body lies entirely within, one or other fluid, the surface-energies of the system (so far as we are concerned) are equivalent to Sol + sy, or S^ + sy, as the case may be. Accordingly as a be less or greater than ^, the particle will have a tendency to remain immersed in the water or in the protoplasm ; but if (aS/2) (a + ^) - sy be less than either Sol or 8/3, then the condition of minimal potential will be found when the particle lies, as we have said, in the boundary zone, half in one fluid and half in the other; and, if we were to attempt a more general solution of the problem, we should have to deal with possible conditions of equilibrium under which the necessary balance of energies would be attained by the particle rising or sinking in the boundary zone, so as to adjust the relative magnitudes of the surface-areas concerned. This principle may, in certain cases, help us to explain the position even of a radial spicule, which is just a case where the surface of the sohd spicule is dis- tributed between the fluids with a minimal disturbance, or minimal replacement, of the original surface of contact between the one fluid and the other. In like manner we may provide for the case (a common and an important one) where the protoplasm "creeps up" the spicule, covering it with a dehcate film, and forming catenary curves or festoons between one spicule and another; and a less fluid or more tenacious thread of protoplasm may serve, like a solid spicule, to extend the more fluid film, as we see, for instance, in Chlamydomyxa or in Gromia. When the spicules are numerous and close-set, the surface-film of protoplasm stretching between them will tend to look like a layer of concave or inverted bubbles ; and this honeycombed bubbly surface is sometimes beautifully regular, to judge from a well-known figure of the living Globigerina* . In Acanthocystis we have yet another special case, where the radial spicules plunge only a certain distance into the protoplasm of the cell, being arrested at a boundary-surface between an inner and an outer layer of * See H. B. Brady's Challenger Monograph, pi. Ixxvii; and see the figure of Chlamydomyxa in Doflein's Protozoenkunde, p. 374. IX] OF SO-CALLED MYONEMES 701 cytoplasm; here we have only to assume that there is a tension at this surface, between the two layers of protoplasm, sufficient to balance the tensions which act directly on the spicule*. In various Acanthometridae, besides such typical characteristics as the radial symmetry, the concentric layers of protoplasm, and the capillary surfaces in which the outer vacuolated protoplasm is festooned upon the projecting radii, we have another curious feature. On the surface of the protoplasm where it creeps up the sides of the long radial spicules, we find a number of elongated bodies, forming in each case one or several little groups, and lying neatly arranged in parallel bundles. A Russian naturalist, Schewiakoff, whose views have been accepted in the text-books, tells us that these are muscular structures, serving to raise or lower the conical masses of protoplasm about the radial spicules, which latter serve as so many "tent-poles" or masts, on which the protoplasmic membranes are hoisted up; and the little elongated bodies are dignified with various names, such as "myonemes" or "myophriscs," ,in allusion to their supposed muscular nature f. This explanation is by no means convincing. To begin with, we have precisely similar festoons of protoplasm in a multitude of other cases where the "myonemes" are lacking; from their minute size (0-006-0-012 mm.) and the amount of contraction they are said to be capable of, the myonemes can hardly be very efficient instruments of traction; and further, for them to act (as is alleged) for a specific purpose, namely the "hydrostatic regulation" of the organism giving it power to sink or to swim, would seem to imply a mechanism of action and of coordination not easy to conceive in these minute and simple organisms. The fact is that the whole explanation is unnecessary. Just as the supposed "hauling up" of the proto- plasmic festoons may be at once explained by capillary phenomena, so also (in all probability) may the position and arrangement of the little elongated bodies. Whatever the actual nature of these bodies may be, whether they be truly portions of differentiated protoplasm, or whether they be foreign bodies or spicular structures (as bodies occupying a similar position in other cases undoubtedly * Cf. Koltzoff, Zur Frage der Zellgestalt, Anat. Anzeiger, xli, p. 190, 1912. t Mem. de VAcad. des Sci., St Petersbourg, xii, Nr. 10, 1902. 702 ON CONCRETIONS, SPICULES, ETC. [ch. are), we can explain their situation on the surface of the protoplasm, and their arrangement around the radial spicules, all by the prin- ciples of capillarity. This last case is not of the simplest; and I do not forget that my explanation of it, which is wholly theoretical, impHes a doubt of Schewiakoif 's statements, founded on his personal observation. This I am none too wiUing to do; but whether it be justly done in this case or not, I hold that it is in principle justifiable to look with suspicion upon all such statements where the observer has obviously left out of account the physical aspect of the pheno- menon, and all the opportunities of simple explanation which the consideration of that aspect might aiford. Whether it be applicable to this particular and complex case or no, our general theorem of the localisation and arrestment of sohd particles in a surface-film is of great biological significance ; for on it depends the power displayed by many httle naked protoplasmic organisms of covering themselves with an "agglutinated" shell. Sometimes, as in Difflugia, Astrorhiza (Fig. 320) and others, this covering consists of sand-grains picked up from the surrounding medium, and sometimes, on the other hand, as in Quadrula, it consists of sohd particles said to arise as inorganic deposits or concretions within the protoplasm itself, and to find their way outwards to a position of equihbrium in the surface-layer; and in both cases, the mutual capillary attractions between the particles, confined to the boundary-layer but enjoying a certain measure of freedom therein, tends to the orderly arrangement of the particles one with another, and even to the appearance of a regular "pattern" as the result of this arrangement. The "picking up" by the protoplasmic organism of a solid particle with which "to build its house" (for it is hard to avoid this customary use of figures of speech, misleading though it be) is a physical phenomenon akin to that by which an amoeba "swallows" a particle of food. This latter process has been repro- duced or imitated in various pretty experimental ways. For instance, Rhumbler hsts shewn that if a splinter of glass be covered with shellac and brought near a drop of chloroform suspended in water, the drop takes in the spicule, robs it of its shellac covering, IX] OF AGGLUTINATED SKELETONS 703 Fig. 320. Arenaceoi;s Foraminifera; Astrorhiza liinirola and nrc)iari(f. From Brady's Challenger MonrxjrajJt. 704 ON CONCRETIONS, SPICULES, ETC. [ch. and then passes it out again*. In another case a thread of shellac, laid on a drop of chloroform, is drawn in and coiled within it: precisely as we may see a filament of Oscillatoria ingested by an Amoeba, and twisted and coiled within its cell. It is all a question of relative surface-energies, leading to different degrees of " adhesion " between the chloroform and the splinter of glass or its shellac covering. Thus it is that the Amoeba takes in the diatom, dissolves off its proteid covering, and casts out the shell. Furthermore, as the whole phenomenon depends on a distribu- tion of surface-energy, the amount of which is specific to certain particular substances in contact with one another, we have no difficulty in understanding the selective action which is very often a conspicuous feature in the phenomenon")*. Just as some caddis- worms make their houses of twigs, and others of shells and again others of stones, so some Rhizopods construct their agglutinated "test" out of stray sponge-spicules, or frustules of diatoms, or again of tiny mud particles or of larger grains of sand. In all these cases, we have to deal with specific surface-energies, and also doubtless with differences in the total available amount of surface- * Rhumbler, Physikalische Analyse von Lebenserscheinungen der Zelle, Arch, f. Entw. Mech. vii, p. 250, 1898. t The whole phenomenon has been described as a "surprising exhibition of constructive and selective activity," and ascribed, in varying phraseology, to inteUigence, skill, purpose, psychical activity, or "microscopic mentality": that is to say, to Galen's t(x^lk7] (pvais, or "artistic creativeness " (cf. Brock's Galen, 1916, p. xxix); cf. Carpenter, Mental Physiology, 1874, p. 41; Norman, Architectural achievements of Little Masons, etc., Ann. Mag. Nat. Hist. (5), i, p. 284, 1878; Heron- Allen, Contributions ... to the study of the Foraminifera, Phil. Trans. (B), ccvi, pp. 227-279, 1915; Theory and phenomena of purpose and intelligence exhibited by the Protozoa, as illustrated by selection and behaviour in the Foraminifera, Journ. R. Microsc. Soc. 1915, pp. .547-.557; ibid., 1916, pp. 1.37-140. Sir J. A. Thomson {New Statesman, Oct. 23, 1915) describes a certain little foraminifer, whose proto- plasmic body is overlaid by a crust of sponge-spicules, as "a psycho-physical individuality, whose experiments in self-expression include a masterly treatment of sponge-spicules, and illustrate that organic skill which came before the dawn of Art." tSir Kay Lankester finds it "not difficult to conceive of the existence of a mechanism in the protoplasm of the Protozoa which selects and rejects building-material, and determines the shapes of the structures built, comparable to that mechanism which is assumed to exist in the nervous system of insects and other animals which 'automatically' go through wonderfully elaborate series of complicated actions." And he agrees with "Darwin and others [who] have attributed the building up of these inherited mechanisms to the age-long action of Natural Selection, and the survival of those individuals possessing qualities or 'tricks' of life-saving value," ■Iniirn. R. Microsc. Soc. April, 1916, p. 136. IX] OF AGGLUTINATED SKELETONS 705 energy in relation to gravity or other extraneous forces. In my early student days, Wyville Thomson used to tell us that certain deep-sea "Difflugias," after constructing a shell out of particles of the black volcanic sand common in parts of the North Atlantic, finished it off with "a clean white collar" of httle grains of quartz. Even this phenomenon may be accounted for on surface-tension principles, if we may assume that the surface-energy ratios have tended to change, either with the growth of the protoplasm or by reason of external variation of temperature or the like; we are by no means obhged to attribute even this phenomenon to a mani- festation of volition, or taste, or aesthetic skiU, on the part of the microscopic organism. Nor, when certain Radiolaria tend more than others to attract into their own substance diatoms and such- hke foreign bodies, is it scientifically correct to speak, as some text-books do, of species "in which diatom-selection has become a regular habits To do so is an exaggerated misuse of anthropo- morphic phraseology. The formation of an "agglutinated" shell is thus seen to be a purely physical phenomenon, and indeed a special case of a more general physical phenomenon which has important consequences in biology. For the shell to assume the sohd and permanent character which it acquires in Difflugia, we have only to make the further assumption that small quantities of a cementing substance are secreted by the animal, and that this substance flows or creeps by capillary attraction through all the interstices . of the little quartz grains, and ends by binding them together. Rhumbler* has shewn us how these agglutinated tests of spicules or of sand-grains can be precisely imitated, and how they are formed with greater or less ease and greater or less rapidity according to the nature of the materials employed, that is to say according to the specific surface- tensions which are involved. If we mix up a little powdered glass with chloroform, and set a drop of the mixture in water, the glass particles gather neatly round the surface of the drop so quickly that the eye cannot follow the operation. If we do the same with oil and fine sand, dropped into 70 per cent, alcohol, a still more * Rhumbler, Beitrage z. Kenntniss d. Rhizopoden, i-v, Z. f. w. Z. 1891-o; Das Protoplasma als physikalisches System, Jena, p. 591, 1914; also in Arch. J. Ent- wickelungsmech. vii, pp. 279-335, 1898; Biol. Centralbl. xvm, 1898; etc. 706 ON CONCRETIONS, SPICULES, [ch. beautiful artificial Rhizopod-shell is formed, but it takes some three hours to do. Where the action is quick the Uttle test forms as the droplet exudes from the pipette : precisely as in the living Difflugia when new protoplasm, laden with soHd particles, is being extruded from the mouth of the parent-cell. The experiment can be varied, simply and easily. Instead of a spherical drop a pear-shaped one may easily be formed, so exactly like the common Difflugia pyriformis that -Rhumbler himself was unable, sometimes, to tell under the microscope the real from the artefact. Again he found that, when the alcohol dissolved the oily substance of the (^rop and shrinkage took place accordingly, the surface-layer with its solid particles got kinked or folded in — and reproduced in doing so, with startUng accuracy, a httle shell of common occurrence, known by the generic name of Lesqueureusia, or Diffiugia spiralis. The pecuhar shape of this little twisted and bulging shell has been taken to shew that "it had enlarged after its first formation, a very rare occurrence in this group*"; the very opposite is the case. Neither here nor in any. allied form does the agglutinated test, once set in order by capillary forces, yield scope for intercalation and enlargement. At the very tinje when Rhumbler was thus demonstrating the physical nature of the Difflugian shell, Verworn, a very notable person, was studying the same and kindred organisms from the older standpoint of an incipient psychology!. But as Rhumbler himself admits, Verworn (unlike many another) was doing his best not to over-estimate the appearance of vohtion, or selective choice, in the little organism's use of materials to construct its dweUing. This long parenthesis has led us away, for the time being, from the subject of the radiolarian skeleton, and to that subject we must now return. Leaving aside, then, the loose and scattered spicules, which we Have sufficiently discussed, the more perfect radiolarian skeletons consist of a continuous and regular structure; and the sihceous (or other inorganic) material of which this framework is composed tends to be deposited in one or other of two ways or in both combined: (1) in long radial spicules, emanating symmetrically from, and usually conjoined at, the centre of the protoplasmic body; * Cf. Cambridge Natural History, Protozoa, p. 55. t Max Verworn, Psycho -physiologische Protisten-Studien, Jena, 1889 (219 pp.); Biologische Protisten-Studien, Z. f. wiss. Z. L, pp. 445-467, 1890. IX] AND SPIOULAR SKELETONS 707 (2) in the form of a crust, developed either on the outer surface of the organism or in relation to one or more of the internal surfaces which separate its concentric layers or its component vesicles. Not infrequently, this superficial skeleton comes to constitute a spherical shell, or a system of concentric spheres. We have already seen that a great part of the body of the Radiolarian, and especially that outer portion to which Haeckel has given the name of the "calymma," is built up of a mass of "vesicles," forming a sort of stiff froth, and equivalent in the physical though not necessarily in the biological sense to "cells," inasmuch as the little vesicles have their own well-defined boun- daries, and their own surface phenomena. In short, all that we have said of cell-surfaces and cell-conformations in our discussion of cells and of tissues will apply in like manner, and under appro- priate conditions, to these. In certain cases, even in so common and so simple a one as the vacuolated substance of an Actino- sphaerium, we may see a close resemblance, or formal analogy, to a cellular or parenchymatous tissue in the close-packed arrangement and consequent configuration of these vesicles, and even at times in a shght membranous hardening of their walls. Leidy has figured * some curious httle bodies hke small masses of consoUdated froth, which seem to be nothing else than the dead and empty husks, or filmy skeletons, of Actinosphaerium ; and Carnoyf has demon- strated in certain cell-nuclei an all but precisely similar framework, of extreme minuteness and tenuity, formed by adsorption or partial sohdification of interstitial matter in a close-packed system of alveoh (Fig. 321). In short, we are again dealing or about to deal with a network or basketwork, whose meshes correspond to the boundary fines between associated cells or vesicles. It is just in those boimdary walls or films, still more in their edges or at their corners, that surface-energy will be concentrated and adsorption will be hard at work; and the whole arrangement will follow, or tend to follow, the rules of areas minimae — the partition-walls meeting at co-equal angles, three by three in an edge, and their edges meeting four by four in a corner. Let us suppose the outer surface of our Radiolarian to be covered * J. Leidy, Fresh-water Rhizopods of North America, 1879, p. 262, pi. xli, figs. 11,12. t Carnoy, Biologie Cellulaire, p. 244, fig. 108; cf. Dreyer, op. cit. 1892, fig. 185. 708 ON CONCRETIONS, SPICULES, ETC. [ch. by a layer of froth-like vesicles, uniform in size or nearly so. We know that their mutual tensions will tend to conform them into the fashion of a honeycomb, or regular meshwork of hexagons, and that the free end of each hexagonal prism will be a little spherical cap. Suppose now that it be at the outer surface of the protoplasm (in contact with the. surrounding sea- water) that the sihceous particles have a tendency to be secreted or adsorbed; the distribu- tion of surface-energy will lead them to accumulate in the grooves which separate the vesicles, and the result will be the development Fig. 321. '''' Reticulum plasmatique J" After Carnoy. Fig. 322. Aulonia hexagona Hkl. of a dehcate sphere composed of tiny rods arranged, or apparently arranged, in a hexagonal network after the fashion of Carnoy's reticulum jplasmatique, only more solid, and still more neat and regular. Just such a spherical basket, looking like the finest imaginable Chinese ivory ball, is found in the siliceous skeleton of Aulonia, another of Haeckel's Radiolaria from the Challenger. But here a strange thing comes to light. No system of hexagons can enclose space; whether the hexagons be equal or unequal, regular or irregular, it is still under all circumstances mathematically impossible. So we learn from Euler: the array of hexagons may be extended as far as you please, and over a surface either plane or curved, but it never closes in. Neither our reticulum plasmatique IX] OF THE RADIOLARIAN SKELETON 709 nor what seems the very perfection of hexagonal symmetry in Aulonia are as we are wont to conceive them; hexagons indeed predominate in both, but a certain number of facets are and must be other than hexagonal. If we look carefully at Carnoy's careful drawing we see that both pentagons and heptagons are shewn in his reticulum, and Haeckel actually states, in his brief description of his Aulonia hexagona, that a few square or pentagonal facets are to be found among the hexagons. Such skeletal conformations are common: and Nature, as in all her handiwork, is quick to ring the changes on the theme. Among Fig, 323. Actinomma arcadophorum Hkl. its many variants may be found cases (e.g. Actinonima) where the vesicles have been less regular in size; and others in which the meshwork has been developed not on an outer surface only but at successive levels, producing a system of concentric spheres. If the sihceous material be not limited to the linear junctions of the cells but spread over a portion of the outer spherical surfaces or caps, then we shall have the condition represented in Fig. 324 (Eihmo- sphaera), where the shell appears perforated by circular instead of hexagonal apertures and the circular pores are set on shght spheroidal eminences; and, interconnected with such types as this, 710 ON CONCRETIONS, SPICULES, ETC. [CH. we have others in which the accumulating peUicles of skeletal matter have extended from the edges into the substances of the boundary walls and have so produced a system of films, normal to the surface of the sphere, constituting a very perfect honeycomb, as in Ceno- sphaera favosa and vesparia*. In one or two simple forms, such as the fresh-water Clathrulina, just such a spherical perforated shell is produced out of some Fig. 324. Ethmosphaera conoslphonia Fig. 325. Portions of shells Hkl. of two "species" of Cenosphaera : upper figure, C. favosa ; lower, C. vesparia Hkl. organic, acanthin-like substance; and in some examples of Clath- rulina the chitinous lattice-work of the shell is just as regular and delicate, with the meshes for the most part as beautifully hexagonal as in the siHceous shells of the oceanic Radiolaria. This is only another proof (if proof be needed) that the peculiar form and character of these little skeletons are due not to the material of which they are composed, but to the moulding of that material upon an underlying vesicular structure. Let us next suppose that another and outer layer of cells or vesicles develops upon some such lattice- work as has just been * In all these latter cases we recognise a relation to, or extension of, the principle of Plateau's bourrelet, or van der Mensbrugghe's masse annulaire, or Gibbs's ring, of which we have had much to say. IX] OF THE EADIOLARIAN SKELETON 711 described; and that instead of forming a second hexagonal lattice- work, the skeletal matter tends to be developed normally to the surface of the sphere, that is to say along the radial edges where the external vesicles (now compressed into hexagonal prisms) meet one another three by three. The result will be that, if the vesicles be removed, a series of radiating spicules will be left, directed outwards from the angles of the original polyhedron mesh work, all as is seen in Fig. 326. And it may further happen that thes6 radiating skeletal rods branch at their outer ends into divergent rays, forming a triple fork, and corresponding (after the fashion Fig. 326. Aulaatrum triceros Hkl. which we have already described as occurring in certain sponge- spicules) to the superficial furrows between the three adjacent cells; this is, as it were, a halfway stage between simple rods or radial spicules and the full completion of another sphere of latticed hexagons. Another possible case, among many, is when the large, uniform vesicles of the outer protoplasm are replaced by smaller vesicles, piled on one another in concentric layers. In this case the radial rods will no longer be straight, but will be bent zig-zag, with their angles in three vertical planes corresponding to the alternate contacts of the successive layers of cells (Fig. 327). The solid skeleton is confined, in all these cases, to the boundary- lines, or edges, or grooves between adjacent cells or vesicles, but 712 ON CONCRETIONS, SPICULES, ETC. [ch. adsorptive energy may extend throughout the intervening walls. This happens in not a few Radiolaria, and in a certain group called the Nassellaria it produces geometrical forms of peculiar elegance and mathematical beauty. When Plateau made the wire framework of a regular tetrahedron and dipped it in soap-solution, he obtained in an instant (as we well know) a beautifully symmetrical system of six films, meeting /^JvVvi Fig. 327. Fig. 328. . A Nassellarian skeleton, Callimitra agnesae Hkl. (0-15 mm. diameter). three by three in four edges, and these four edges running from the corners of the figure to its centre of symmetry. Here they meet, two by two, at the Maraldi angle; and the films meet three by three, to form the re-entrant soHd angle which we have called a "Maraldi pyramid" in our account of the architecture of the honey- comb. The very same configuration is easily recognised in the minute siliceous skeleton of Callimitra. There are two discrepancies, neither of which need raise any difiiculty. The figure is not a rectilinear but a spherical tetrahedron, such as might be formed IX] OF THE NASSELLARIAN SKELETON 713 by the boundary-edges of a tetrahedral cluster of four co-equal bubbles; and just as Plateau extended his experiment by blowing a small bubble in the centre of his tetrahedral system, so we have a central bubble also here. This bubble may be of any size*; but its situation (if it be present at all) is always the same, and its shape is always such as to give the Maraldi angles at its own four corners. The tensions A B Fig. 329. Diagrammatic construction of Callimitra. A, a bubble suspended within a tetrahedral cage. B, another bubble within a skeleton of the former bubble. of its own walls, and those of the films by which it is supported Or slung, all balance one another. Hence the bubble appears in plane projection as a curvihnear equilateral triangle; and we have only got to convert this plane diagram into the corresponding sohd to obtain the spherical tetrahedron we have been seeking to explain (Fig. 329). We may make a simpHfied model (omitting the central bubble) of the tetrahedral skeleton of Callimitra, after the fashion of that of the bee's cell (p. 535). Take OC =^ CD = DB, and draw a circle with radius OB and diameter AB. Erect a perpendicular Fig. 330. Geometrical construction to AB at C, cutting the circle at E, F. °^ Callimitra-sl.eleton. AOE, ^Oi^ will be (as before) Maraldi angles of 109°; the arcs AE, * Plateau introduced the central bubble into his cube or tetrahedron by dipping the cage a second time, and so adding an extra face-film; under these circum- stances the bubble has a definite magnitude. 714 ON CONCRETIONS, SPICULES, ETC. [ch. AF will be edges of the spherical tetrahedron, and 0 will be the centre of s)niimetry. Make the angle FOG = FOA = EOA. Cut out the circle EAFG, and cut through the radius E0\ fold at AO, FO, GO, and fasten together, using EOG for a flap. Make four such sheets, and fasten together back to back. The model will be much improved if little cusps be left at the corners in the cuttmg out. The geometry of the little inner tetrahedron is not less simple and elegant. Its six edges and four faces are all equal. The films attaching it to the outer skeleton are all planes. Its faces are spherical, and each has its centre in the opposite corner. The edges are circular arcs, with cosine ^ ; each''is in a plane perpendicular to the chord of the arc opposite, and each has its centre in the middle of that chord. Along each edge the two intersecting spheres meet each other at an angle of 120°*. This completes the elelmentary geometry of the figure; but one or two points Temain to be considered. We may notice that the outer edges of the little skeleton are thickened or intensified, and these thickened edges often remain whole or strong while the rest of the surfaces shew signs of imper- fection or of breaking away; moreover, the four corners of the tetrahedron are not re-entrant (as in a group of bubbles) but a surplus of material forms a little point or vcusp at each corner. In all this there is nothing anomalous, and nothing new. For we have already seen that it is at the margins or edges, and a fortiori at the corners, that the surface-energy reaches its maximum — with the double effect of accumulating protoplasmic material in the form of a Gibbs's ring or bourrelet, and of intensifying along the same Hues the adsorptive secretion of skeletal matter. In some other tetrahedral systems analogous to Callimitra, the whole of the skeletal matter is concentrated along the boundary-edges, and none left to spread over the boundary-planes or interfaces: just as among our spherical Radiolaria it was at the boundary-edges of their many cells or vesicles, and often there alone, that skeletal formation occurred, and gave rise to the spherical skeleton and its mesh work * For proof, see Lamarle, op. cit. pp. 6-8. Lamarle shewed that the sphere can be so divided in seven ways, but of these seven figures the tetrahedron alone is stable. The other six are the cube and the regular dodecahedron; prisms, triangular and pentagonal, with equilateral base and a certain ratio of base to height; and two polyhedra constructed of pentagons and quadrilaterals. IX]. OF THE NASSELLARIAN SKELETON 715 of hexagons. In the beautiful form which Haeckel calls Archi- scenium the boundary edges disappear, the, four edges converging on the median point are^ intensified, and only three of the six convergent facets are retained; but, much as the two differ in appearance, the geometry of this and of Callimitra remain essentially the same. We learned also from Plateau that, just as a tetrahedral bubble can be inserted within the tetrahedral skeleton or cage, so may a cubical bubble be introduced within a cubical cage; and the edges of the inn^er cube will be just so curved as to give the Maraldi angles at the corners. We find among Haeckel's Radiolaria one (he calls it Lithocubus geometricus) which precisely corresponds to the skeleton A B Fig. 331. A, bubble suspended within i cubical cage. B, Lithocubus geometricus Hkl. of this inner cubical bubble; and the httle spokes or spikes which project from the corners are parts of the edges which once joined the corners of the enclosing figure to those of the bubble within (Fig. 331). Again, if we construct a cage in the form of an equilateral triangular prism, and proceed as before, we shall probably see a vertical edge in the centre of the prism connecting two nodes near either end, in each of which the Maraldi figure is displayed. But if we gradually shorten our prism there comes a point where the two nodes disappear, a plane curvilinear triangle appears horizon- tally in the middle of the figure, and at each of its three corners four curved edges meet at the famihar angle. Here again we may insert a central bubble, which will now take the form of a curvihnear 716 ON CONCRETIONS, SPICULES, ETC. [ch. equilateral triangular prism; and Haeckel's Prismatium tripodium repeats this configuration (Fig. 332). In a framework of two crossed rectangles, we may insert one bubble after another, producing a chain of superposed vesicles whose shapes vary as we alter the relative positions of the rectangular frames. Various species of Triolampas, Theocyrtis, etc. are more or less akin to these complicated figures of equilibrium. A very beautiful series of forms may be made by introducing successive bubbles within the film-system formed by a tetrahedron or a Fig. 332. Prismatium tripodium Hkl. parallelepipedon. The shape and the curvature of the bubbles and of their suspensory films become extremely beautiful, and we have certain of them reproduced unmistakably in various Nassel- larian genera, such as Podocyrtis and its allies. In Fig. 333 we see a curious little skeletal structure or complex spicule, whose conformation is easily accounted for. Isolated spicules such as this form the skeleton in the genus Dictyocha, and occur scattered over the spherical surface of the organism (Fig. 334). The basket-shaped spicule has evidently been developed about a cluster of four cells or vesicles, lying in or on the surface of the organism, and therefore arranged, not in the three-dimensional, tetrahedral form of Callimitra, but in the manner in which four contiguous cells lying side by side in one plane normally set themselves, like the IX] OF THE NASSELLARIAN SKELETON 717 four cells of a segmenting egg: that is to say with an intervening "polar furrow," whose ends mark the meeting place, at equal angles, of the four cells in groups of three. The Uttle projecting spokes, or spikes, which are set normally to the main basket-work, seem to be uncompleted portions of a larger basket, corresponding to a more numerous aggregation of cells. Similar but more complex forma- Fig. 333. An isolated portion of the skeleton of Dityocha. tions, all explicable as basket-like frameworks developed around a cluster of cells, and adsorbed or secreted in the grooves common to adjacent cells or bubbles, are found in great variety. Kg. 334. Dictyocha stapedia Hkl. The Dicfyocha-si^icule, laid down as a siliceous framework in the grooves between a few clustered cells, is too simple and natural to be confined to one group of animals. We have already seen it, as a calcareous spicule, in the holothurian genus Thyone, and we may find it again, in many various forms, in the protozoan group known as the Silicoflagellata*. Nothing can better illustrate the physico-mathematical character of these configurations than their * See {iyit. al.) G. Deflandre, Les Silicoflagelles, etc., Bull. Soc. Fr. de Microscopic, I, p. 1, 1932: the figures in which article are mostly drawn from Ehrenberg's Mikrogeologie, 1854. 718 ON CONCRETIONS, SPICULES, ETC. [ch. common occurrence in diverse groups of organisms. And the simple fact is, that we seem to know less and less of these things on the biological side, the more we come to understand their physical and Fig. 335. Various species of Distephanus (SilicojEIagellata). From Deflandre, after Ehrenberg. mathematical characters. I have lost faith in Haeckel's four thousand "species" of Radiolaria. In Callimitra itself, and elsewhere where the boundary- walls (and not merely their edges) are sihcified, the skeletal matter is not Fig. 336. Holothurian spicules. A, of Thyone (Mortensen) ; B, Holothuria ladea (Perrier); C, D, Holothuria and Phyllophorus (Deichman). deposited in an even layer, like the waxen walls of a bee's cell, but in a close mesh work of fine curvilinear threads; and the curves seem to form three main series more or less closely related to the three edges of the partition. Sometimes (as may also be seen in our figure) the system is further comphcated by a radial series running from the centre towards the free edge of each partition. IX] OF THE NASSELLARIAN SKELETON 719 As to the former, their arrangement is such as would result if deposition or soKdification had proceeded in waves, starting inde- pendently from each of the three boundary-edges of the Httle partition- wall, and something of this kind is doubtless what actually happened. We are reminded of the wave-hke periodicity of the Liesegang phenomenon, and especially, perhaps, of the criss-cross rings which Liesegang observed in frozen gelatine {supra, p. 663). But there may be other explanations. For instance the film, hquid or other, which originally constituted the partition, might conceivably be thrown into vibrations, and then (hke the dust upon a Chladni plate) minute particles in or on the film would tend to take up position in an orderly way, in relation to the nodal points or fines of the vibrating surface*. Some such hypothetical vibration may (to my thinking) account for the minute and varied and very beautiful patterns upon many diatoms, the resemblance of which patterns to the Chladni figures (in certain of their simpler cases) seems here and there striking and obvious. But I have not attempted to investigate the many special problems suggested by the diatom- skeleton. The cusps at the four corners of the tetrahedral skeleton are a marked peculiarity of our Nassellarian shell, and we should by no means expect to see them in a skeleton formed at the boundary- edges of ai simple tetrahedral pyramid of four bubbles or cells. But when we introduce another bubble into the centre of a system of four, then, as Plateau shewed, the tensions of its walls and of the surrounding partitions so balance one another that it becomes a regular curvifinear tetrahedron, or, as seen in plane projection (Fig. 337), a curvilinear, equilateral triangle, with prominent, not re-entrant angles. A drop of fluid tends to accumulate at each corner where four edges meet, and forms a bourreletf ; it is drawn out in the directions of the four films which impinge upon it, and * Cf. Faraday's beautiful experiments, On the moving groups of particles found on vibrating elastic surfaces, etc., Phil. Trans. 1831, p. 299; Researches, 1859, pp. 314-358. t The bourrelet is not on4y, as Plateau expresses it, a "surface of continuity," but we also recognise that it tends (so far as material is available for its production) to further lessen the free surface-area. On its relation to vapour-pressure and to the stability of foam, see FitzGerald's interesting note in Nature, Feb. 1, 1894 {Works, p. 309); and on its effect in thinning the soap-bubble to bursting-point, see Willard Gibbs, Coll. Papers, i, p. 307 seq. 720 ON CONCRETIONS, SPICULES, ETC. [ch. so tends to assume in miniature the very same shape as the tetra- hedron to whose corner it is attached. Out of these bourrelets, then, the cusps at the four corners of our httle skeleton are formed. A large and curiously beautiful family of radiolarian (or "poly- cystine") skeletons look, in a general way, Uke tiny hehnets or Pickelhauben, with spike above, and three (or sometimes six) curved lobes, Hke helmet-straps, below. We recognise a family likeness, even a mathematical identity, between this figure and the last, for both alike are based on a tetrahedral symmetry: the body of the helmet corresponding to the inner vesicle of Callimitra, and the spike and the three straps to the four edges which ran out from the inner to the outer tetrahedron. In the one case an inner vesicle is surrounded by a tetrahedral figure whose outer walls, indeed, are absent, but its edges remain, and so do the walls connecting the outer and inner vesicles. In the other case the outer edges are gone, and so are the filmy partition- walls, save parts which corre- spond to the four internal edges between them*. There are apt to be two slight discrepancies. The helmet is often of somewhat complicated form, easily explained as due to the presence of two superposed bubbles instead of one. The other apparent anomaly is that the three helmet-straps are curved, while the corresponding edges are straight in Plateau's figure of the regular tetrahedron. But it is a paramount necessity (as we well know) for each set of four edges in a system of fluid films to meet in a point two and two at the Maraldi angle \ just as it is necessary for the faces to meet * Looking through Haeckel's very numerous figures, we see that now and then something more is left than the mere edges of the partition-walls. IX] OF THE RADIOLARIAN SKELETON 721 CO 4^ •2 a 00 CO 722 ON CONCRETIONS, SPICULES, ETC. [ch. at 120°; and faces and edges become curved, whenever necessary, in order that these conditions may be fulfilled. The need may arise in various ways. Suppose (as in Fig. 339) that our httle central bubble be no longer in the centre of symmetry, but near one corner of the enclosing tetrahedron; the short edge running out from that corner will tend to remain straight (and so form the spike of the helmet); while the other three will each form an S-shaped curve, as a condition of making co-equal angles at their two extremities. An analogous case is figured in one of Sir David Brewster's papers where he repeats and amplifies some experiments of Plateau's*. Fig. 339. Diagram of one of the He made a tetrahedral cage, and fitted helmet-shaped radiolaria, e.g. it with three more wires, leading from Podocyrtis: to shew its tetra- ,^ , ,-, • t t, • , n i i i hedral symmetry. ^^^ ^P®^ ^^ the middle pomt of each basal edge. On dipping this into soap-solution, various comphcations were seen. At the apex, six films must not meet together, for no more than three surfaces may meet in an edge ; intermediate or interstitial films make their appearance, with which we are not greatly concerned. But six films now ascend towards the apex from the base instead of three, and the three which come from the corners have a longer path than the other three which come from the mid-points of the basal edges ; they must be curved in different degrees in order that all t&ree may make at either end their co-equal angles. And if we now introduce a bubble (or two bubbles) into the interior of the system, we obtain the characteristic form of our helmet- shaped Radiolarian — the spike above, the single or double vesicle of the body, and the straps or lappets with their peculiar and characteristic curvatures. The Httle shell is perforated with many rounded holes, and it remains to account for these. They are required, so we are told, for the passage of pseudopodia; well and good. We have referred the Dictyocha-si^icule and the hexagonal meshwork of Aulonia to froth-like associations of vesicles and to adsorption taking place * On the figures of equihbrium in liquid films, Trans. R.S.E. xxiv, 1866. IX] OF THE RADIOLARIAN SKELETON 723 between; so far so good, again. But the irregular lacework of the Uttle helmets does not suit this explanation very well, and there may be yet other possibilities. We have already mentioned Tomhnson's "cohesion figures*." Experimenting with a great variety of substances, Tomhnson studied the innumerable ways in which drops, jets or floating films "cohere," disrupt or otherwise behave, under the resultant influences of surface-tension, cohesion, viscosity and friction; in one case a film runs out into a wavy or broken edge, in another it gives way here and there, and makes rents or holes in its surface. I take the Httle holes in our poly cystine skeleton to be cohesion-figures, in Tomhnson's sense of the word — spots where the delicate film has given way and run into holes, and where surface-tension has rounded off the broken edges, and made the rents into rounded apertures. In the foregoing examples of Radiolaria, the symmetry which the organism displays seems identical with that symmetry of forces which results from the play and interplay of surface-tensions in the whole system : this symmetry being displayed, in one class of cases, in a more or less spherical mass of froth, and in another class in a simpler aggregation of a few, otherwise isolated, vesicles. In either case skeletons are formed, in great variety, by one and the same kind of surface-action, namely by the adsorptive deposition of sihca in walls and edges, corresponding to the manifold surfaces and interfaces of the system. But among the vast number of known Radiolaria, there are certain forms (especially among the Phaeodaria and Acantharia) which display a no less remarkable symmetry the origin of which is by no means clear, though surface- tension may play a part in its causation. Even this is doubtful; for the fact that three-way nodes are no longer to be seen at the junctions of the cells suggests that another law than that of minimal areas had been in action here. They are cases in which (as in some of those already described) the skeleton consists (1) of radiating spicular rods, definite in number and position, and (2) of inter- connecting rods or plates, tangential to the more or less spherical body of the organism, whose form becomes, accordingly, that of a geometric, polyhedral sohd. The great regularity, the numerical ♦ Cf. supra, p. 418 724 ON CONCRETIONS, SPICULES, ETC. [ch. symmetries and the apparent simplicity of these latter forms makes of them a class apart, and suggests problems which have not been solved or even investigated. The matter is partially illustrated by the accompanying figures (Fig. 340) from Haeckel's Monograph of the Challenger Radiolaria*. In one of these we see a regular octahedron, in another a regular, or pentagonal, dodecahedron, in a third a regular icosahedron. In all cases the figure appears to be perfectly symmetrical, though neither the triangular facets of the octahedron and icosahedron, nor the pentagonal facets of the dodecahedron, are necessarily plane surfaces. In all of these cases, the radial spicules correspond to the comers of the figure; and they are, accordingly, six in number in the octahedron, twenty in the dodecahedron, and twelve in the icosahedron. If we add to these three figures the regular tetra- hedron which we have just been studying, and the cube (which is represented, at least in outline, in the skeleton of the hexactinelhd sponges), we have completed the series of the five regular poly hedra known to geometers, the Platonic bodies "f of the older mathema- ticians. It is at first sight all the more remarkable that we should here meet with the whole five regular polyhedra, when we remember that, among the vast variety of crystalline forms known among minerals, the regular dodecahedron and icosahedron, simple as they are from the mathematical point of view never occur. Not only do these latter never occur in crystallography but (as is explained in textbooks of that science) it has been shewn that they cannot occur, owing to the fact that their indices (or numbers expressing the relation of the faces to the three primary axes) involve an irrational quantity: whereas it is a fundamental law of crystallo- graphy, involved in the whole theory of space-partitioning, that "the indices of any and every face of a crystal are small whole numbers J." At the same time, an imperfect pentagonal dodeca- * Of the many thousand figures in the hundred and forty plates of this beautifully illustrated book, there is scarcely one which does not depict some subtle and elegant yeornetrical configuration. t They were known long before Plato- llXdnov 5^ kuI iv tovtols irvdayopl^ei. X If the equation of any plane face of a crystal be written in the form hx + ky -\-lz-\, then h, k, I are the indices of which we are speaking. They are the reciprocals of the parameters, or reciprocals of the distances from the origin at which the plane meets the several axes. In the case of the regular or pentagonal do«lecahedron these indices are 2, 1 -r Vo, 0. Kepler described as follows, briefly IX] OF JOHANNES MtJLLER'S LAW 725 hedron, whose pentagonal sides are non-equilateral, is common among crystals. If we may safely judge from Haeckel's figures, the pentagonal dodecahedron of the Radiolarian (Circorhegma) is perfectly regular, and we may rest assured, accordingly, that it is not brought about by principles of space-partitioning similar to those which manifest themselves in the phenomenon of crystallisa- tion. It will be observed that in all these radiolarian polyhedral shells, the surface of each external facet is formed of a minute hexagonal network, whose probable origin, in relation to a vesicular structure, is such as we have already discussed. In certain allied Radiolaria of the family Acanthometridae (Fig. 341), which have twenty radial spines, the arrangement of these spines is commonly described in a somewhat singular way. The twenty spines are referred to five whorls of four spines each, arranged as parallel circles on the sphere, and corresponding to the equator, the tropics and the polar circles. This rule was laid down by the celebrated Johannes Miiller, and has ever since been used and quoted as Miiller's law*. But when we come to examine the figure, we find that Miiller's law hardly does justice to the facts, and seems to overlook a simpler symmetry. We see in the first place that here, unlike our former cases, the twenty radial spines issue through the facets (and all the facets) of the polyhedron, instead of coming from its corners; and that our twenty .spines correspond, therefore, not to the corners of a dodecahedron, but to the facets of some sort of an icosahedron. We see, in the next place, that this icosahedron is composed of faces of two kinds, hexagonal and pentagonal ; and that the whole figure may be described as a hexagonal prism, whose twelve corners are truncated, and replaced by pentagonal facets. Both hexagons and pentagons appear to be equilateral, but if we but adequately, the common characteristics of the dodecahedron and icosahedron : "Duo sunt corpora regularia, dodecaedron et icosaedron, quorum illud quin- quangulis figuratur expresse, hoc triangulis quidem sed in quinquanguli formam coaptati^. Utriusque horum corporum Jpsiusque adeo quinquanguli structura perfici non potest sine proportione ilia, quam hodierni geometrae divinam appellant'' (De nive sexangula (1611), Opera, ed. Fritsch, vn, p. 723). Here Kepler was dealing, somewhat after the manner of Sir Thomas Browne, with the mysteries of the quincunx, and also of the hexagon; and was seeking for an explanation of the mysterious or even mystical beauty of the 5-petalled or 3-petalled flower — pulchri- tudinis aut proprietatis fgurae, quae animam harum plantarum characterisavit. * See Johannes Miiller, Ueber die Thalassicollen, Polycistinen und Acantho- metren des Mittelmeeres, Abh. d. Akad. Wiss. Berlin, 1858, pp. 1-62, 11 pi. 726 ON CONCRETIONS, SPICULES, ETC. [ch. try to construct a plane-sided polyhedron of this kind, we find it to be impossible; for into the angles between the six equatorial Fig. 340. Skeletons of various Radiolarians, after Haeckel. 1, Circoporus sexfurcus; 2, C. odahe.drus; 3, Circogonia icosahcdra; 4, Circospathis novena; 5, Circorrhegma dodecahedra. regular hexagons six regular pentagons will not fit. The figure, however, can be easily constructed if we replace the straight edges IX] OF JOHANNES MtJLLER'S LAW 727 (or some of them) by curves, the plane facets by slightly curved surfaces, or the regular by non-equilateral polygons*. In some cases, such as Haeckel's Phatnaspis cristata (Fig. 342), we have an elHpsoidal body from which the spines emerge in the order described, but which is not obviously divided into facets. In Fig. 234 I have indicated the facets corresponding to the rays, and dividing the surface in the usual symmetrical way. 6F, eF. -6F, Fig. 341. Dorataspis cristata Hkl. A, viewed according to Muller's law: a, four polar plates; h, four intermediate or "tropical" plates; c, four equatorial plates. B, an alternative description: Fg, two polar and six equatorial hexagonal plates; F5, two rows of six intermediate pentagonal plates. About any polyhedron (within or without) we may describe another whose corners correspond to the sides, and whose sides to the corners, of the original figure; or the one configuration may be developed from the other by bevelling off, to a certain definite extent, the corners of the original polyhedron. The two figures, thus reciprocal to one another, form a "conjugate pair," and the principle is known as the "principle of duahty" in polyhedral . Of the regular sohds, cube and octahedron, dodecahedron and * Muller's interpretation was emended by Brandt, and what is known as Brandt's Law, viz. that the symmetry consists of two polar rays and three whorls of six each, coincides so far with the above description : save only that Brandt says plainly that the intermediate whorls stand equidistant between the equator and the poles, i.e. in latitude 45", which, though not very far wrong, is geometrically inaecurate. But Brandt, if I understand him rightly, did not propose his "law" as a substitute for Muller's, but rather as a second law, applicable to a few special cases. t First proved by Legendre, EUm. de Geometrie, vii. Prop. 25, 1794. 728 ON CONCRETIONS, SPICULES, ETC. [ch. Kg. 342 A. Phatnaspis cristata Hkl. Fig. 342 B. The same, diagrammatic. IX] OF JOHANNES MULLER'S LAW 729 icosahedron, are conjugate pairs; but the first and simplest of all solid figures, the tetrahedron, has no conjugate but itself. In our httle shell of Dorataspis, the twenty spicules (as we have seen) spring from and correspond to the twenty facets of the poly- hedron, twelve pentagonal and eight hexagonal, meeting at thirty-six corners, in all cases three by three; we may write the formula of the polyhedron, accordingly, as 12i^5 + 8i?'6 + 36C3. If we now connect up the twenty spicules, three by three, we shall obtain thirty-six triangles, completely covering the figure; but we shall find that of the twenty corners twelve are surrounded by five, and eight by six triangles. The formula is now 362^3 +I2C5+8C6, and the two figures are fully reciprocal or conjugate. I do not know of any radiolarian in which this configuration is to be found; nor does it seem a likely one, owing to the large and variable number of edges which meet in its corners. But we may have polyhedra related to, or derived from, one another in a less full and perfect degree. For instance, letting the twenty spicules of Dorataspis again serve as corners for the new figure, let four facets meet in each corner; or (which comes to the same thing) let each spicule give off four branches or offshoots, which shall meet their corresponding neighbours, and form the boundary-edges of a new network of facets. The result (Fig. 34'3) is a symmetrical figure, not geometrically perfect but elegant in its own way, which we recognise in a number of described forms*. It shews eight triangular and fourteen rhomboidal facets; and its formula is 8^3+142^4 + 2004. Many subsidiary varieties may arise in turn: when, for instance, certain of the little branches fail to meet, or others grow large and widely confluent, always in symmetrical fashion. We now see how in all such cases as these there is a double symmetry involved, that of two superimposed, and conjugate or semi-conjugate, figures. And the ambiguity which attends such descriptions as that which Johannes Miiller embodied in his "law" * Cf. W. Mielck, Acanthometren aus Neu-Pommern, Diss., Kiel, 1907. 730 ON CONCRETIONS, SPICULES, ETC. [CH. seems due to a failure to recognise this twofold or alternative symmetry. In all these latter cases it is the arrangement of the axial rods — the "polar symmetry" of the entire organism — which hes at "the root of the matter; and which, if only we could account for it, would make it comparatively easy to explain the superficial con- figuration. But there are no obvious mechanical forces by which we can so explain this peculiar polarity. This at least is evident, that it arises in the central mass of protoplasm, which is the essential living portion of the organism as distinguished from that frothy Fig. 343. Acanthometra sp. A derivative of the Dorataspis figure. After Mielck. Fig, 344. Phractaapis prototypus Hkl. peripheral mass whose structure has helped us to explain so many phenomena of the superficial or external skeleton. To say that the arrangement depends upon a specific polarisation of the cell is merely to refer the problem to other terms, and to set it aside for future solution. But it is possible that we may learn something about the fines in which to seek for such a solution by considering the cage of Lehmann's "fluid crystals," and the light which they throw upon the phenomena of molecular aggregation. The phenomenon of "fluid crystallisation" is found in a number of chemical bodies; it is exhibited at a specific temperature for each substance; and it would seem to be limited to bodies in which there is an elongated, or "long-chain" arrangement of the atoms in the molecule. Such bodies, at the appropriate temperature, tend to aggregate themselves into masses, which are sometimes spherical IX] OF LIQUID OR FLUID CRYSTALS 731 drops or globules (the so-called "spherulites"), and sometimes have the definite form of needle-like or prismatic crystals. In either case they remain Hquid, and are also doubly refractive, polarising Hght in brilhant colours. Together with them are formed ordinary sohd crystals, also with characteristic polarisation, and into such soHd crystals all the fluid material ultimately turns. It seems that in these hquid crystals, though the molecules are freely mobile, just as are those of water, they are yet subject to, or endowed w4th, a ''directive force," a force which confers upon them a definite configuration or "polarity," the " Gestaltungskraft " of Lehmann. Such an hypothesis as this has been gradually extruded from the theories of mathematical crystallography*; and it has come to be understood that the s^onmetrical conformation of a homogeneous crystalhne structure is sufficiently explained by the mere mechanical fitting together of appropriate structural units along the easiest and simplest lines of "close packing": just as a pile of oranges becomes definite, both in outward form and inward structural arrangement, without the play of any specific directive force. But w^hile our conceptions of the tactical arrangement of crystalline molecules remain the same as before, and our hypotheses of "modes of packing" or of "space-lattices" remain as useful and as adequate as ever for the definition and explanation of the miolecular arrange- ments, a new conception is introduced when we find something hke such space-lattices maintained in what has hitherto been considered the molecular freedom of a liquid field; and Lehmann would per- suade us, accordingly, to postulate a specific molecular force, or " Gestaltungskraft " (not unlike Kepler's "facultas formatrix"), to account for the phenomenon^. Now just as some sort of specific " Gestaltungskraft " had been of old the deus ex machina accounting for all crystalline phenomena (gnara totius geometriae, et in ea exercita, as Kepler said), and as * Cf. Tutton, Crystallography, 1911, p. 932. t Kepler, if I understand him aright, saw his way to account for the shape of the bee's cell or the pomegranate-seed; and it was for want of any such mechanical explanation, and as little more than a confession of ignorance, that he fell back on a facultas formatrix to account for the six rays of the snow-crystal or the five petals of the flower. He was equally ready, unfortunately, to explain, by the same facultas formatrix in acre, the appearance of a plague of locusts or a swarm of flies. 732 A NOTE ON POLYHEDRA [ch. such an hypothesis, after being dethroned and repudiated, has now fought its way back and claims a right to be heard, so it may be also in biology. We begin by an easy and general assumption of specific properties, by which each organism assumes its own specific form ; we learn later (as it is the purpose of this book to shew) that throughout the whole range of organic morphology there are innu- merable phenomena of form which are not pecuhar to living things, but which are more or less simple manifestations of ordinary physical law. But every now and then we come to deep-seated signs of protoplasmic symmetry or polarisation, which seem to lie beyond the reach of the ordinary physical forces. It by no means follows that the forces in question are not essentially physical forces, more obscure and less familiar to us than the rest; and this would seem to be a great part of the lesson for us to draw from Lehmann's beautiful discovery. For Lehmann claims to have demonstrated, in non-living, chemical bodies, the existence of just such a deter- minant, just such a "Gestaltungskraft," as would be of infinite help to us if we might postulate it for the explanation (for instance) of our Radiolarian's axial symmetry. Further than this we cannot go; such analogy as we seem to see in the Lehmann phenomenon soon evades us, and refuses to be pressed home. The symmetry of crystallisation, which Haeckel tried hard to discover and to reveal in these and other organisms, resolves itself into remote analogies from which no conclusions can be drawn. Many a beautiful protozoan form has lent itself to easy physico-mathematical ex- planation ; others, no less simple and no more beautiful prove harder to explain. That Nature keeps some of her secrets longer than others — that she tells the secret of the rainbow and hides that of the northern hghts — is a lesson taught me when I was a boy. A note on Polyhedra. The theory of Polyhedra, Euler's doctrina solidorum, is a branch of geometry which deals with the more or less regular solids; and the rudiments of the theory may help us to study certain more or less symmetrical organic forms. Euler, a contemporary of Lin- naeus, is the most celebrated of the many mathematicians who have carried this subject beyond where Pythagoras, Plato, Euclid and Archimedes had left it. He drew up a classification of poly- IX] OF EULER'S LAW 733 hedral solids, using a binomial nomenclature based on the number of their corners or vertices, and sides or faces. Thus, for example, he called a figure with eight corners and seven faces Octogonum hsptaedrum; and the analogy between this and Linnaeus's botanical classification and nomenclature — e.g. Hexandria trigynia and the rest — is very close and curious. A simple theorem, of which Euler was vastly proud and which we still speak of as Euler 's Law*, is fundamental to the theory of polyhedra. It tells us that in every polyhedron whatsoever, the faces and corners together outnumber the edges by two t : C-E+F=2 (1). Another fundamental theorem follows. We know from Euchd that the three angles of a triangle are equal to two right angles; consequently, that in a polygon of C angles, the sum of the angles -= 2 (C — 2) right angles. And there follows from this — but by no means expectedly — the analogous and extremely simple relation * Euler, Elementa doctrinae solidorum, Novi Comment. Acad. Sci. Imp. Petropol. IV, p. 109 seq. (ad annos 17.52 et 1753), 1758: "In omni solido hedris planis inclusum, aggregatura ex numero angulorum solidorum et ex numero hedraruiu binario excedit numerum acierum." For a proof, see {int. al.) De Morgan, article Polyhedron in the Penny Cyclopaedia. There is reason to believe that Descartes was acquainted with this theorem between 1672 and 1676; cf. Foucher de Careil, (Euvres inedites de Descartes, Paris, ii, p. 214. Cf. Baltzer, Monatsber. Berlin. Akad. 1861, p. 1043; and de Jonquieres, C.R. 1890, p. 261. (The student will be struck by the resemblance between this formula and the phase rule of Willard Gibbs.) t If we include, besides the corners, edges and faces (i.e. points, lines and surfaces) the solid figure itself, Euler's Law becomes C-E + F-S = \. And in this form the theorem extends to n dimensions, as follows: With equal beauty and simplicity, the simplest figure in each ^-dimensional space is given as follows: Kq rC% A^2 "'S "^4 ^^^» w = 0 1 " =1 (point) 12-1 =1 (line) 2 3-3 +1 =1 (triangle) 3 4 -6 +4 -1 —1 (tetrahedron) 4 5 -10 +10 -5 +1 =1 (pentahedroid) etc. And, in a figure of /i- dimensions, the sum of the plane angles =2*"-^' (C-2) right angles. 734 A NOTE ON POLYHEDRA [ch. that in a polyhedron of C corners, the sum of the plane angles = 4 (C - 2) right angles (2)*. Hence, if the polyhedron be isogonal, the sum of the plane angles at each corner 4 (C - 2) C 90°, or U - ^i right angles (3). The five regular sohds, or Platonic bodies — there can be no more — ^have been known from remote antiquity ; they have their corners all ahke and their face's all alike, they are isogonal and isohedral. Three of them, the tetrahedron, octahedron and icosahedron, have triangular faces; three of them, the tetrahedron, cube and dodeca- hedron, have trihedral or three-way corners. One or other of these, triangles or three-way corners, must (as we shall soon see) be present in every polyhedron whatsoever. The semi-regular solids are -regular in one respect or other, but not in both; they are either isogonal or isohedral — isohedral, when every face is an identical polygon and isogonal when at every corner the same set of faces is combined. The semi-regular isogonal gohds, with all their corners ahke but with two or more kinds of regular polygons for their faces, are thirteen in number — there can be no more; they were all described by Archimedes, and we call them by his name. One of them, with six square and eight hexagonal facets, derived by truncating the octahedron or the cube, we have found to be of pecuhar interest, and it has become famihar to us as, of all homogeneous space-fillers, the one which encloses a given volume within ,a minimal area of surface. It is the cuho-octahedron of Kepler or of Fedorow, the tetrakaidekahedron of Kelvin, which latter name we commonly use. Of semi-regular isohedral bodies, with all their sides alike (though no longer regular polygons) and their corners of two kinds or more, only one was known to antiquity; it is the rhombic dodecahedron, which is the crystalline form of the garnet, and appears in part * On this remarkable parallel see Jacob Steiner, Gesammelte Werke, i, p. 97. It follows that the sum of the plane angles in a polyhedron, as in a plane polygon, is at once determined by the number of its comers: a result which dehghted Euler, and led him to base his primary or generic classification of polyhedra on their comers rather than their sides. IX] OF THE SEMIREGULAR SOLIDS 735 again as a "space-filler" at the base of the bee's cell. A closely related rhombic icosahedron was known to Kepler; but it was left to Catalan* to discover, only some seventy-five years ago, that the isohedral bodies were thirteen in number, and were precisely comparable with and reciprocal to the Archimedean solids. The semi-regular solids, both of Archimedes and of Catalan, are all, like the Platonic bodies, related to the sphere |, for a circum- scribing sphere meets all the corners of an Archimedean solid, and an inscribed sphere touches all the faces of a solid of Catalan; and while the isogonal bodies can be constructed by various simple geometrical means, the general method of constructing the thirteen isohedral bodies is by dividing the sphere into so many similar and equal areas J. It is a matter of spherical trigonometry rather than of simple geometry, and the problem, for that very reason, remained long unsolved. The thirteen Archimedean bodies are derivable from the five Platonic bodies, in most cases easily, by so truncating their corners and their edges as to produce new and rpgularly polygonal faces in place of the old faces, corners and edges, and the possible number of faces in the new figure will be easily derived from the edges, corners and faces of the old. Part of the old faces will remain; each truncated corner will yield one new face; but each edge may be truncated, or bevelled, more than once, so as to yield one, two, or possibly three new faces. In short, if the faces, corners and edges of a regular solid be F, C, E, those of the Archimedean solids derivable from it {Fj) will be F^ = F + mC + nE, where m = 0 or 1, and n = 0, 1, or 2. From the cube six Archimedean bodies may be derived, from the dodecahedron six, and from the tetrahedron one. * Journal de Vecole imper. polytechnique, xli, pp. 1-71, 1865. t It follows that the Chinese carved and perforated ivory balls, which are based on regular and symmetrical division of the sphere, can all be referred to one or another of the Platonic or Archimedean bodies. J As a matter of fact, the Catalan bodies can be formed by adding to the Platonic bodies, just as (but not so easily as) the Archimedean bodies can be formed by truncating: them. 736 A NOTE ON POLYHEDRA [ch. The derivatives of the cube (with its six sides and eight corners) have the following numbers of sides : F^ = F+C =6 + 8 =14 F+C + E =6 + 8+12 = 26 i^+C+2j6;=6 + 8 + 24 = 38. The derivatives of the dodecahedron have, in hke manner, 32, 62 or 92 sides; while the tetrahedron yields, by truncation of its four corners, a solid with eight sides. The growth and form of crystals is a subject alien to our own, yet near enough to attract and tempt us. It is a curious thing (probably traceable to the Index Law of the crystallographer) that the Archimedean or isogonal bodies seldom occur and certainly play no conspicuous part in crystallography, while several of Catalan's isohedral figures are the characteristic forms of well-known minerals *. Just as we pass from the Platonic to the Archimedean bodies by truncating the corners or edges of the former, so conversely, by producing their faces to a limit we obtain another family of figures — in all cases save the tetrahedron, which admits of no such extension; and the figures of this family are remarkable for the "twinned," or duphcate or multiple appearance which they present. If we extend the faces of an octahedron we get what looks hke two tetrahedra, "twinned with" or interpenetrating one another; but there has been no interpenetration in the construction of this twin- like figure, only further accretion upon, and extension of, the facets of the octahedron. Among the higher polyhedra there are many figures which look, in a far more complicated way, like the twinning of simpler but still comphcated forms; and these also have been constructed, not by interpenetration, but by the mere superposition of new parts on old. An elementary, even a very elementary, knowledge of the theory of polyhedra becomes useful to the naturahst in various ways. Among organic structures we often find many-sided boxes (or what may be regarded as such), like the capsular seed-vessels of plants, the skeletons of certain Radiolaria, the shells of the Peridinia, the carapace of a tortoise, and a great many more. Or we may go * E.g. the triakis, tetrakis and hexakis octahedra of fluor-spar. However the Archimedean tetrakaidekahedron [QF^ SFq) occurs in ahmi. IX] OF EULER'S LAW 737 further and treat any cluster of cells, such as a segmenting ovum, as a species of polyhedron and study it from the point of view of Euler's Law and its associated theorems. We should have to include, as the geometer seldom does, the case of two-sided facets — facets with two corners and two curved sides or edges — hke the " hths " * of a peeled orange ; but the general formula would include these as a matter of course. On the other hand, we need very seldom consider any other than trihedral or three-way corners. When we limit ourselves to polyhedra with trihedral corners the following formula applies : 4A+ 3/3+ 2/4+/6± 0./,-/, - 2/3- ... = 12 (4). That this formula apphes to the tetrahedron with its four triangles, the cube with its six squares and the dodecahedron with its twelve pentagons, is at once obvious. The now famihar case of our four- celled egg with its polar furrows (Fig. 486, B, etc.) appears in two forms, according as the polar furrows run criss-cross or parallel. In the one case we have a curvilinear tetrahedron, in the other a figure with two two-sided and two four-sided facets; in either case the formula is obviously satisfied. But the main lesson for us to learn is the broad, general principle that we cannot group as we please any number and sort of polygons into a polyhedron, but that the number and kind of facets in the latter is strictly Umited to a narrow range of possibiHties. For example, the case of Aulonia has already taught us that a poly- hedron composed entirely of hexagons is a mathematical impossi- bihtyt; and the zero-coefficient which defines the number of Jiexagons in the above formula (4) is the mathematical statement of the fact J. We can state it still more simply by the following corollary, hkewise hmited to the case of three-way corners : * Lith, a useful Scottish word for a joint X)t segment. Cromwell, according to Carlyle, "gar'd kings ken they had a lith in their necks." t That hexagons cannot enclose space, or form a "three-way graph," has been recognised as a significant fact in organic chemistry : where, for instance, it limits, somewhat unexpectedly, the ways in which a closed cyclol, or space-enclosing protein molecule, can be imagined to be built up. Cf. Dorothy Wrinch, in Proc. R. S. (A), No. 907, p. 510, 1937. J Euler shewed at the same time the singular fact that no polyhedron can exist with seven edges. 738 A NOTE ON POLYHEDRA [ch. This applies at once to the tetrahedron, the cube and the regular dodecahedron, and at once excludes the possibihty of the closed hexagonal network. We found in Dorataspis a closed shell consisting only of hexagons and pentagons; without counting these latter we know, by our formula, that they must be twelve in number, neither more nor less. Lord Kelvin's tetrakaidekahedron consists only of squares and hex:agons; the squares are, and must be, six in number. In a typical Peridinian, such as Goniodoma, there are twelve plates, all meeting by three-way nodes or corners; we know^ and we have no difficulty in verifying the fact, that the twelve plates are all pentagonal. T. 345. Goniodoma, from above and below. Without going beyond the elements of our subject we may want to extend our last formula, and remove the restriction to three-way corners under which it lay. We know that a tetrahedron has four triangles and four trihedral corners ; that a cube has six squares and eight trihedral corners ; an octahedron eight triangles and six four- way corners ; an icosahedron twenty triangles and twelve five-way corners. By inspection of these numbers we are led to the following rule, and may estabhsh it as a deduction from Euler's Law : ifs + c,) = 8 + 0if, + c,) + {f, + c,) + 2if, + c,)+etc (5). This important formula further illustrates the limitations to which all polyhedra are subject; for it shews us, among other things, that (if we neglect the exceptional case of dihedral facets or 'Uiths") every polyhedron must possess either triangular faces or trihedral corners, and that these taken together are never less than eight in number. IX] OF EULER'S LAW 739 We may add yet two more formulae, both related to the last, and all derivable, ultimately, from Euler's Law * : 3/3+2/4+/5-^12 + 0.C3+2c4+4c5+... + 0./e+/7 + 2/8 (6) and 3C3+ 2c4 + C5 = 12 + 0,/3+ 2/4+ 4/5 + ... + O.Ce + C7 + 2c, ...(7). These imply that in every polyhedron the triangular, quadrangular and pentagonal faces (or corners) must, taken together and multi- plied as above, be at least twelve in number. Therefore no poly- hedron can exist which has not a certain number of triangles, squares or pentagons in its composition; and the impossibihty of a polyhedron consisting only of hexagons is demonstrated once again. Formulae (5), (6) and (7) further shew us that not only is a three- way polyhedron of hexagons impossible, but also a four- way polyhedron of quadrangles, or one of six- way corners and triangular facets ; all of which become the more obvious when we reflect that the plane angles meeting in each point or node must be, on the average, in the first case 3 x 120°, in the second 4 x 90°, and in the third 6 X 60°. Lastly, having now considered the case of other than trihedral corners, we may learn a simple but very curious relation between the number of faces and corners, arising (like so much else) out of Euler's Law. In a polyhedron whose corners are all n-hedral, nC=2E; therefore (by Euler) 7iC/2 + 2 = F + C; therefore 2F ={n-2)C + 4. Therefore, if n=3, 2F=4:+ C] = 4, =4 + 2cl (8). = 5, = 4 + 3Cj Let us look again at the microscopic skeleton of Dorataspis (Fig. 341). We have seen that some of its facets are hexagonal, the rest pentagonal; there happen to be eight of the former, and therefore (as we now know) there must be twelve of the latter. * Derivable from Euler together with the formulae for the "edge-counts," viz. llnF^ = 2E, and SnC„ = 2^; which merely mean that each edge separates two faces, and joins two corners. 740 A NOTE ON POLYHEDRA [ch. ix We know also that, having no triangular facets, the polyhedral skeleton must possess trihedral corners; and these, moreover, must (by equation 5) be eight in number, plus the number of pentagons, plus twice the number of hexagons. The total, 8 +/5+ 2/6= 8 + 12 + (2 X 8) = 36, is precisely the number of corners in the figure, all of them trihedral. We also know (from equation 2) that the sum of its plane angles = 4 (36 - 2) X 90° =12,240°, which agrees with the sum of the angles of twelve pentagons and eight hexagons. The configuration, then, is a possible one. So here and elsewhere an apparently infinite variety of form is defined by mathematical laws and theorems, and hmited by the properties of space and number. And the whole matter is a running commentary on the cardinal fact that, under such foedera Natural "as Lucretius recognised of old, there are things which are possible, and things which are impossible, even to Nature herself. CHAPTER X A PARENTHETIC NOTE ON GEODESICS We have made use in the last chapter of the mathematical principle of Geodesies (or Geodetics) in order to explain the conformation of a certain class of sponge-spicules ; but the principle is of much wider appHcation in morphology, and would seem to deserve atten- tion which it has not yet received. The subject is not an easy one, and if we are to avoid mathematical difficulties we must keep within narrow bounds. Fig. 346. Annular and spiral thickenings in the walls of plant-cells. Defining, meanwhile, our geodesic hne (as we have already done) as the shortest distance between two points on the surface of a soUd of revolution, we find that the cyhnder gives us some of the simplest of cases. Here it is plain that the geodesies are of three kinds: (1) a series of annuh around the cyhnder, that is to say, a system of circles, in planes parallel to one another and at right angles to the axis of the cylinder (Fig. 346, A); (2) a series of straight lines parallel to the axis; and (3) a series of spiral curves winding round the wall of the cyhnder (B, C). These three systems 742 A PARENTHETIC NOTE [ch. are all of frequent occurrence, and are illustrated in the local thickenings of the wall of the cyHndrical cells or vessels of plants. The spiral, or rather helical, geodesic is particularly common in cylindrical structures, and is beautifully shewn for instance in the spiral coil which stiffens the tracheal tubes of an insect, or the so-called tracheides of a woody stem. A Hke phenomenon is often witnessed in the splitting of a glass tube. If a crack appear in a test-tube it has a tendency to be prolonged in its own direction, and the more isotropic be the glass the more evenly will the split tend to follow the straight course in which it began. As a result, the crack often continues till our test-tube is split into a continuous spiral ribbon. One may stretch a tape along a cylinder, but it no sooner swerves to one side than it begins to wind itself around; it is tracing its geodesic*. In a circular cone, the spiral geodesic falls into closer and closer coils as the cone narrows, till it comes to the end, and then it winds back the same way; and a beautiful geodesic of this kind is exemplified in the sutural line of a spiral shell, such as Turritella, or in the striations which run parallel with the spiral suture. On a prolate spheroid, the coils of a spiral geodesic come closer together as they approach the ends of the long axis of the ellipse, and wind back and forward from one pole to the other "j". We have a case of this kind in an Equisetum-sipoTe, when the integument splits into the spiral "elaters," though the spire is not long enough to shew all its geodesic features in detail. We begin to see that our first definition of a geodesic requires to be modified; for it is only subject to conditions that it is ''the shortest distance between two points on the surface of the solid," and one of the commonest of these restricting conditions is that our geodesic may be constrained to go twice, or many times, round the surface on its way. In short, we may re-define a geodesic, as a curve drawn upon a surface such that, if we take any two adjacent * It is not that the geodesic is rectified into a straight line; but that a straight line (the midline of the tape or ribbon)^ is converted into the geodesic on the given surface. t In all these cases, r cos a = a constant, where r is the radius of the circular section, and a the angle at which it is crossed by the geodesic. And the constant is measured by the smallest circle which the geodesic can reach. X] ON GEODESICS 743 points on the curve, the curve gives the shortest distance between them. It often happens, in the geodesic systems which we meet with in morphology, that two opposite spirals or rather helices run separate and distinct from one another, as in Fig. 346, C; and it is also common to find the two interfering with one another, and forming a criss-cross or reticulated arrangement. This indeed is a common source of reticulated patterns. The microscopic and even ultramicroscopic structure of the cell- wall shews analogous configurations: as in the large cells of the alga Valonia, where the wall consists of many lamellae, each com- posed of parallel fibrillae running in spiral geodesies, and alternating in direction from one lamella to another. Here, and not less clearly in the young parenchyma of seedHng oats, it is the long-chain cellulose molecules which follow a spiral course around the cell- wall, right-handed or left-handed as the case may be, and inchned more or less steeply according to the elongation of the cell. But these highly interesting questions of molecular, or micellar, structure lie beyond our scope*. Among the ciliated infusoria, we have a variety of beautiful geodesic curves in the spiral patterns in which their ciha are arranged; though it is probable enough that in some comphcated cases these are not simple geodesies, but developments of curves other than a straight hne upon the surface of the organism. In other words, they seem to be instances of "geodesic curvature." Lastly, an instructive case is furnished by the arrangement of the muscular fibres on the surface of a hollow organ, such as the heart or the stomach. Here we may consider the phenomenon from the point of view of mechanical efficiency, as well as from that of descriptive anatomy. In fact we have a right to expect that the muscular fibres covering such hollow organs will coincide with geodesic lines, in the sense in which we are using the term. For if we imagine a contractile fibre, .or an elastic band, to be fixed by its two ends upon a curved surface, it is obvious that its first effort of contraction will tend to expend itself in accommodating * Cf. (e.g.) C. Correns, Innere Struktur einiger Algenmembranen, Beitr. zur Morphol. u. Physiol, d. Pflanzenzelle, 1893, p. 260; W. T. Astbury and others, Proc, E.S. (B), cix, p. 443, 1932, and other papers; G. van Iterson, jr.. Nature, cxxxviii, p. 364, 1936; R. D. Preston, Proc. U.S. (B), cxxv, p. 772, 1938; etc. 744 A PARENTHETIC NOTE [ch. the band to the form of the surface, in "stretching it tight," or in other words in causing it to assume a. direction which is the shortest possible hne upon the surface between the two extremes: and it is only then that further contraction will have the effect of constricting the tube and so exercising pressure on its contents. Thus the muscular fibres, as they wind over the curved surface of an organ, arrange themselves automatically in geodesic curves: in precisely the same manner as we also automatically construct complex systems of geodesies whenever we wind a ball of wool or a spindle of tow, or when the skilful surgeon bandages a Hmb ; indeed the surgeon must fold and crease his bandage if it is not to keep on geodesic lines. It is as a simple, necessary result of geodesic principles that we see those " figures-of -eight " produced, to which, in the case for instance of the heart-muscles, Pettigrew and other anatomists have ascribed pecuhar importance. In the case of both heart and stomach we must look upon these organs as de- veloped from a simple cyhndrical tube, after the fashion of the glass-blower, as is further discussed on p. 1049 of this book, the modification of the simple cyhnder consisting of various degrees of dilatation and of twisting. In the primitive undistorted cyhnder, as in an artery or in the intestine, the muscles run in simple geodesic fines, and constitute the circular and longitudinal coats which form (or are said to form) the normal musculature of all tubular organs, or the cylindrical body of a worm. However, we can often recog- nise, in a small artery for instance, that the so-called circular fibres tend to take a shghtly obhque or spiral course; and that the so-called annular muscle-fibres are really spirals is an old statement which may very hkely be true*. If we consider each muscular fibre as an elastic strand embedded in the elastic membrane which constitutes the wall of the organ, it is evident that, whatever be the distortion suffered by the entire organ, the individual fibre will follow its own course, which will still, in a sense, be geodesic. But if the distortion be considerable, as for instance if the tube * See A Discourse concerning the Spiral, instead of the supposed Annular, structure of the Fibres of the Intestins; discover'd and shewn by the Learn'd and Inquisitive Dr. William Cole to the Royal Society, Phil. Trans, xi, pp. 603-609, 1676. Cf. Eben J. Carey, Studies on the. . .small intestine, Anat. Record, xxi, pp. 189-215, 1921 ; F. T. Lewis, The spiral trend of intestinal muscle fibres. Science, LV, June 30, 1922. X] ON GEODESICS 745 become bent upon itself, or if at some point its walls bulge outwards in a diverticulum or pouch, then the old system of geodesies will only mark the shortest distance between two points more or less approximate to one another, and new systems of geodesies, peculiar to the new surface, will tend to appear, and link up points more remote from one another. This is evidently the case in the human stomach. We still have the systems, or their unobhterated remains, of circular and longitudinal muscles; but we also see two new systems of fibres, both obviously geodesic (or rather, when we look more closely, both parts of one and the same geodesic system), in the form of annuh encircling the pouch or diverticulum at the cardiac end of the stomach, and of obhque fibres taking a spiral course from the neighbourhood of the oesophagus over the sides of the organ. In the heart we have a similar, but more complicated pheno- menon. Its musculature consists, in great part, of the original simple system of circular and longitudinal muscles which enveloped the original arterial tubes, which tubes, after a process of local thickening, expansion, and especially tuisting, came together to constitute the composite, or double, mammalian heart; and these systems of muscular fibres, geodesic to begin with, remain geodesic (in the sense in which we are using the word) after all the twisting which the primitive cylindrical tube or tubes have undergone. That is to say, these fibres still run their shortest possible course, from start to finish, over the complicated curved surface of the organ; and, as Borelli well understood, it is only because they do so that their contraction, or longitudinal shortening, is able to produce its direct eifect in the contraction or systole of the heart*. As a parenthetic corollary to the case o-f the spiral pattern upon the wall of a cyhndrical cell, we may consider for a moment the spiral fine which many small organisms tend to follow in their path * The spiral fibres, or a large portion of them, constitute what Searle called "the rope of the heart" (Todd's Cyclopaedia, ii, p. 621, 1836). The "twisted sinews of the heart" were known to early anatomists, and have been frequently and elaborately studied: for instance, by Gerdy {Bull. Fac. Med. Paris, 1820, pp. 40-148), and by Pettigrew (Phil. Trans. 1864), and again by J. B. Macallum {Johns Hopkins Hospital Report, ix, 1900) and by Franklin P. Mall {Amer. Journ. Anat. XI, 1911). 746 A PARENTHETIC NOTE [ch. of locomotion*. A certain physiologist observed that an Amoeba, crawling within a narrow tube, wound its slow way in a spiral course instead of going straight along the tube. The creature was going nowhere in particular, but merely following the direction in which it had begun : in curious illustration of a familiar statement in the "dynamics of a particle," that a particle moving on a surface without constraint will describe geodesic lines. But it is after a different fashion, and without any constraint to a surface, that the smaller ciliated organisms, such as the ciliate and flagellate infusoria, the rotifers, the swarm-spores of various Protista, and so forth, shew a tendency to pursue a spiral path in their ordinary locomotion. The means of locomotion which they possess in their cilia are at best somewhat primitive and inefficient; they have no apparent means of steering, or modifying their direction; and, if their course tended to swerve ever so little to one side, the result would be to bring them round and round again in an approximately circular path (such as a man astray on the prairie is said to follow), with little or no progress in a definite longitudinal direction. But as a matter of fact, by reason of a more or less unsymmetrical form of the body, all these creatures tend more or less to rotate about their long axis while they swim. And this axial rotation, just as in the case of a rifle-bullet, causes their natural swerve, which is always in the same direction as regards their own bodies, to be in a continually changing direction as regards space: in short, to make a spiral course around, and more or less near to, a straight axial line|. In this short chapter we have touched on phenomena where form repeats itself, and mathematical analogies recur, in very different things and very different orders of magnitude. The spiral muscles of heart or stomach are the mechanical outcome of twists which these tubular organs have undergone in the course of their develop- ment, and come, accordingly, under the general category of organic * Cf. Butschli, "Protozoa," in Bronn's Thierreich, ii, p. 848, in, p. 1785, etc., 1883-87; Jennings, Amer. Nat. xxxv, p. 369, 1901; Putter, Thigmotaxie bei Protisten, Arch. f. Anat. u. Phys. {Phys. Abth. Swppl.), pp. 243-302, 1900. t Cf. W. Ludwig, Ueber die Schraubenbahnen niederer Organismen, Arch. f. vergl. Physiologic, ix, 1919. x] ON GEODESICS 747 or embryological growth. But the spiral thickenings in the woody fibres of a plant are of another order of things, and lie in the region of molecular phenomena. The delicate spirals of the cell-wall of a cotton-hair are based on a complicated cellulose space-lattice, recaUing Nageh's micellar hypothesis in a new setting; and giving us a glimpse of organic growth after the very fashion of crystaUine growth, that is to say from the starting-point of molecular structure and configuration*. * W. Lawrence Balls, Determiners of cellulose structure as seen in the cell-wall of cotton-hairs, Proc. R.S. (B), xcv, pp. 72-89, 1923, and other papers. Cf. also Wilfred Robinson, Microscopical features of mechanical strains in timber, and the bearing of these on the structure of the cell-wall in plants, Phil. Trans. (B), ccx, pp. 49-82, 1920. CHAPTER XI THE EQUIANGULAR SPIRAL The very numerous examples of spiral conformation which we meet with in our studies of organic form are pecuUarly adapted to mathematical methods of investigation. But ere we begin to study them we must take care to define our terms, and we had better also attempt some rough preliminary classification of the objects with which we shall have to deal. In general terms, a Spiral is a curve which, starting from a point of origin, continually diminishes in curvature as it recedes from that point; or, in other words, whose radius of curvature continually increases. This definition is wide enough to include a number of different curves, but on the other hand it excludes at least one which in popular speech we are apt to confuse with a true spiral. This latter curve is the simple screw, or cyhndrical helix, which curve neither starts from a definite origin nor changes its curvature as it proceeds. The "spiral" thickening of a woody plant-cell, the "spiral" thread within an insect's tracheal tube, or the "spiral" twist and twine of a climbing stem are not, mathe- matically speaking, spirals at all, but screws or helices. They belong to a distinct, though not very remote, family of curves. Of true organic spirals we have no lack*. We think at once of horns of ruminants, and of still more exquisitely beautiful molluscan shells — in which (as Pliny says) tnagna ludentis Naturae varietas. Closely related spirals may be traced in the florets of a sunflower; a true spiral, though not, by the way, so easy of investigation, is seen in the outhne of a cordiform leaf; and yet again, we can recognise typical though transitory spirals in a lock of hair, in a staple of woolt, in the coil of an elephant's trunk, in the "circling spires" * A great number of spiral forms, both organic and artificial, are described and beautifully illustrated in Sir T. A. Cook's Spirals in Nature and Art, 1903, and Curves of Life, 1914. t On this interesting case see, e.g. J. E. Duerden, in Science, May 25, 1934. CH. XI] THE EQUIANGULAR SPIRAL 749 of a snake, in the coils of a cuttle-fish's arm, or of a monkey's or a chameleon's tail. Fig. 347. The shell of Nautilus pompilius, from a radiograph: to shew the equiangular spiral of the shell, together with the arrangement of the internal septa. From Green and Gardiner, in Proc. Malacol. Soc. ii, 1897. Among such forms as these, and the many others which we might easily add to them, it is obvious that we have to do with things which, though mathematically similar, are biologically 750 THE EQUIANGULAR SPIRAL [ch. speaking fundamentally different ; and not only are they biologically remote, but they are also physically different, in regard to the causes to which they are severally due. For in the first place, the spiral coil of the elephant's trunk or of the chameleon's tail is, as we have said, but a transitory configuration, and is plainly the result of certain muscular forces acting upon a structure of a definite, and normally an essentially different, form. It is rather a position, or an attitude, than a form., in the sense in which we have been using this latter term; and, unhke most of the forms which we have been studying, it has little or no direct relation to the phenomenon of growth. Fig. 348. A foraminiferal shell {Pulvinnlina). Again, there is a difference between such' a spiral conformation as is built up by the separate and successive florets in. the sunflower, and that which, in the snail or Nautilus shell, is apparently a single and indivisible unit. And a similar if not identical difference is apparent between the Nautilus shell and the minute shells of the Foraminifera which so closely simulate it: inasmuch as the spiral shells of these latter are composite structures, combined out of successive and separate chambers, while the molluscan shell, though it may (as in Nautilus) become secondarily subdivided, has grown as one continuous tube. It follows from all this that there cannot be a physical or dynamical, though there may well be a mathematical law of growth, which is common to, and which defines, the spiral form in Nautilus, in Globigerina, in the ram's horn, and in the inflorescence of the sunflower. Nature at least exhibits in them all ^'un reflet des formes rigoureuses qu'etudie la geometrie"^ .'' * Haton de la Goupilliere, in the introduction to his important study of the Surfaces Nautiloides, Annaes sci. da Acad. Polytechnica do Porto, Coimbra, ni, 1908. XI] OF SUNDRY SPIRALS 751 Of the spiral forms which we have now mentioned, every one (with the single exception of the cordate outline of the leaf) is an example of the remarkable curve known as the equiangular or logarithmic spiral. But before we enter upon the mathematics of the equiangular spiral, let us carefully observe that the whole of the organic forms in which it is clearly and permanently exhibited, however different they may be from one another in outward appear- ance, in nature and in origin, nevertheless all belong, in a certain sense, to one particular class of conformations. In the great majority of cases, when we consider an organism in part or whole, when we look (for instance) at our own hand or foot, or contemplate an insect or a worm, we have no reason (or very Httle) to consider one part of the existing structure as older than another; through and through, the newer particles have been merged and commingled among the old ; the outhne, such as it is, is due to forces which for the most part are still at work to shape it, and which in shaping it have shaped it as a whole. But the horn, or the snail-shell, is curiously different; for in these the presently existing structure is, so to speak, partly old and partly new. It has been conformed by successive and continuous increments; and each successive stage of growth, starting from the origin, remains as an integral and un- changing portion of the growing structure. We may go further, and see that horn and shell, though they belong to the living, are in no sense alive*. They are by-products of the animal; they consist of "formed material," as it is sometimes called ; their growth is not of their own doing, hut comes of Hving cells beneath them or around. The many structures which display the logarithmic spiral increase, or accumulate, rather than grow. The shell of nautilus or snail, the chambered shell of a foraminifer, the elephant's tusk, the beaver's tooth, the cat's claws or the canary-bird's— all these shew the same simple and very beautiful spiral curve. And all alike consist of stuff secreted or deposited by living cells ; all grow, as an edifice grows, by accretion of accumulated * For Oken and Goodsir the logarithmic spiral had a profound significance, for they saw in it a manifestation of life itself. For a like reason Sir Theodore Cook spoke of the Curves of Life; and Alfred Lartigues says (in his Biodynamique generale, 1930, p. 60): "Nous verrons la Conchy liologie apporter une magnifique contribution a la Stereodynamique du tourbillon vital." The fact that the spiral is always formed of non-living matter helps to contradict these mystical conceptions. 752 THE EQUIANGULAR SPIRAL [ch. material; and in all alike the parts once formed remain in being, and are thenceforward incapable of change. In a shghtly different, but closely cognate way, the same is true of the spirally arranged florets of the sunflower. For here again we are regarding serially arranged portions of a composite structure, which portions, similar to one another in form, differ in age; and differ also in magnitude in the strict ratio of their age. Somehow or other, in the equiangular spiral the time-element always enters in; and to this important fact, full of curious biological as well as mathematical significance, we shall afterwards return. In the elementary mathematics of a spiral, we speak of the point of origin as the pole (0) ; a straight hne having its extremity in the pole, and revolving about it, is called the radius vector; and a point (P), travelling along the radius vector under definite conditions of velocity, will then describe our spiral curve. Of several mathematical curves whose form and development may be so conceived, the two most important (and the only two with which we need deal) are those which are known as (1) the equable spiral, or spiral of Archimedes, and (2) the equiangular or logarithmic spiral. The former may be roughly illustrated by the way a sailor coils a rope upon the deck; as the rope is of uniform thickness, so in the whole spiral coil is each whorl of the same breadth as that which precedes and as that which follows it. Using its ancient definition, we may define it by saying, that ''If a straight fine revolve uniformly about its extremity, a point which Hkewise travels uniformly along it will describe the equable spiral*." Or, putting the same thing into our more modern words, "If, while the radius vector revolve uniformly about the pole, a point (P) travel with uniform velocity along it, the curve described will be that called the equable spiral, or spiral of Archimedes." It is plain that the spiral of Archimedes may be compared, but again roughly, to a cylinder coiled up. It is plain also that a radius {r = OP), made up of the successive and equal whorls, will increase in arithmetical progression : and will equal a certain constant quantity (a) multiplied * Leslie's Geometry of Curved Lines, 1821, p. 417. This is practically identical with Archimedes' own definition (ed. Torelli, p. 219); cf. Cantor, Geschichte der Mathematik, i, p. 262, 1880. XI] AND THE SPIRAL OF ARCHIMEDES 753 by the whole number of whorls, (or more strictly speaking) multiplied by the whole angle (6) through which it has revolved: so that r = ad. And it is also plain that the radius meets the curve (or its tangent) at an angle which changes slowly but continuously, and which tends towards a right angle as the whorls increase in number and become more and more nearly circular. But, in contrast to this, in the equiangular spiral of the Nautilus or the snail-shell or Globigerina, the whorls continually increase in breadth, and do so in a steady and unchanging ratio. Our definition is as follows: "If, instead of travelhng with a uniform velocity, our point move along the radius vector with a velocity increasing as its distance from the pole, then the path described is Fig. 349. The spiral of Archimedes. called an equiangular spiral." Each whorl which the radius vector intersects will be broader than its predecessor in a definite ratio; the radius vector will increase in length in geometrical progression, as it sweeps through successive equal angles ; and the equation to the spiral will be r = a^. As the spiral of Archimedes, in our example of the coiled rope, might be looked upon as a coiled cyfinder, so (but equally roughly) may the equiangular spiral, in the case of the shell, be pictured as a cone coiled upon itself; and it is the conical shape of the elephant's trunk or the chameleon's tail which makes them coil into a rough simulacrum of an equiangular spiral. While the one spiral was known in ancient times, and was investigated if not discovered by Archimedes, the other was first 754 THE EQUIANGULAR SPIRAL [ch. recognised by Descartes, and discussed in the year 1638 in his letters to Mersenne*. Starting with the conception of a growing curve which should cut each radius vector at a constant angle — just as a circle does — Descartes shewed how tt would necessarily follow that radii at equal angles to one another at the pole would be in con- tinued proportion ; that the same is therefore true of the parts cut off from a common radius vector by successive whorls or convolutions of the spire; and furthermore, that distances measured along the curve from its origin, and intercepted by any radii, as at B, C, are proportional to the lengths of these radii, OB, OC. It follows that Fig. 350. The equiangular spiral. the sectors cut off by successive radii, at equal vectorial angles, are similar to one another in every respect; and it further follows that the figure may be conceived as growing continuously without ever changing its shape the while. If the whorls increase very slowly, the equiangular spiral will come to look like a spiral of Archimedes. The Nummulite is a case in point. Here we have a large number of whorls, very narrow, very close together, and apparently of equal breadth, which give rise to an appearance similar to that of our coiled rope. And, in a case of this kind, we might actually find that the whorls were of equal breadth, being produced (as is apparently the case in the Nummulite) not by any very slow and gradual growth in thickness of a con- tinuous tube, but by a succession of similar cells or chambers laid on, round and round, determined as to their size by constant surface-tension conditions and therefore of unvarying dimensions. The Nummulite must always have a central core, or initial cell, around which the coil is not only wrapped, but out of which it springs ; and this initial chamber corresponds to our a' in the expression r = a' + ad cot a. * (Euvres, ed. Adam et Tannery, Paris, 1898, p. 360. XI] OR LOGARITHMIC SPIRAL 755 The many specific properties of the equiangular spiral are so interrelated to one another that we may choose pretty well any one of them as the basis of our definition, and deduce the others from it either by analytical methods or by elementary geometry. In algebra, when m^ = n, x is called the logarithm of n to the base m. Hence, in this instance, the equation r = a^ may be written in the form log r = 6 log a, or 6 = log r/log a, or (since a is a constant) 6 = A; log /•*. Which is as much as to say that (as Descartes dis- covered) the vector angles about the pole are proportional to the logarithms of the successive radii; from which circumstance the alternative name of the "logarithmic spiral" is derivedf. Moreover, for as many properties as the curve exhibits, so many names may it more or less appropriately receive. ^James Bernoulli called it the logarithmic spiral, as we still often do ; P. Nicolas called it the geometrical spiral, because radii at equal polar angles are in geometrical progression; Halley, the proportional spiral, because the parts of a radius cut off by successive whorls are in continued proportion; and lastly, Roger Cotes, going back to Descartes' first description or first definition of all, called it the equiangular spiral J. We may also recall Newton's remarkable demonstration that, had the force of gravity varied inversely as the cube instead of the square of the distance, the planets, instead of being bound to their * Instead of r=a^, we might write r = rQa^; in which case r^ is the value of r for zero value of 6. f Of the two names for this spiral, equiangular and logarithmic, I used the latter in my first edition, but equiangular spiral seems to be the better name; for the constant angle is its most distinguishing characteristic, and that which leads to its remarkable property of continuous self-similarity. Equiangular spiral is its name in geometry; it is the analyst who derives from its geometrical pro- perties its relation to the logarithm. The mechanical as well^as the mathematical properties of this curve are very numerous. A Swedish admiral, in the eighteenth century, shewed an equiangular spiral (of a certain angle) to be the best form for an anchor-fluke {Sv. Vet. Akad. Hdl. xv, pp. 1-24, 1796), and in a parrot's beak it has the same efl&ciency. Macquorn Rankine shewed its advantages in the pitch of a cam or non-circular wheel {^Manual of Mechanics, 1859, pp. 99-102; cf. R. C. Archibald, Scripta Mathem. m (4), p. 366, 1935). X James Bernoulli, in Acta Eruditorum, 1691, p. 282; P. Nicolas, De novis spiralibus, Tolosae, 1693, p. 27; E. Halley, Phil. Trans, xix, p. 58, 1696; Roger Cotes, ibid. 1714, and Harmonia Mensurarum, 1722, p. 19. For the further history of the curve see (e.g.) Gomes de Teixeira, Traite des courbes remarqicables, Coimbre, 1909, pp. 76-86; Gino Loria, Spezielle algebrdische Kurven, ii, p. 60 scq., 1911; R. C. Archibald (to whom I am much indebted) in Amer. Mathem. Monthly, xxv, pp. 189-193, 1918, and in Jay Hambidge's Dynamic Symmetry, 1920, pp. 146-157. 756 THE EQUIANGULAR SPIRAL [CH. ellipses, would have been shot off in spiral orbits from the sun, the equiangular spiral being one case thereof.* A singular instance of the same spiral is given by the route which certain insects follow towards a candle. Owing to the structure of their compound eyes, these insects do not look straight ahead but make for a light which they see abeam, at a certain angle. As. they continually adjust their path to this constant angle, a spiral pathway brings them to their destination at last|. In mechanical structures, curvature is essentially a mechanical phenomenon. It is found in flexible structures as the result of Fig. 351. Spiral path of an insect, as it draws towards a light. From Wigglesworth (after van Buddenbroek). Fig. 352. Dynamical aspect of the equiangular spiral. bending, or it may be introduced into the construction for the purpose of resisting such a bending-moment. But neither shell nor tooth nor claw are flexible structures ; they have not been bent into their pecuhar curvature, they have grown into it. We may for a moment, however, regard the equiangular or logarithmic spiral of our shell from the dynamical point of view, by looking on growth itself as the force concerned. In the growing structure, let growth at any point P be resolved into a force F acting along the line joining P to a pole 0, and a force T acting in a direction perpendicular to OP; and let the magnitude of these forces (or of these rates of growth) remain constant. It follows that * Principia, I, 9; ii, 15, On these "Cotes's spirals" see Tait and Steele, p. 147. t Cf. W. Buddenbroek, Sitzungsber. Heidelb. Akad., 1917; V. H. Wigglesworth, Insect Physiology, 1839, p. 167. XI] IN ITS DYNAMICAL ASPECT 757 the resultant of the forces F and T (as PQ) makes a constant angle with the radius vector. But a constant angle between tangent and radius vector is a fundamental property of the "equiangular" spiral: the very property with which Descartes started his investigation, and that which gives its alternative name to the curve. In such a spiral, radial growth and growth in the direction of the curve bear a constant ratio to one another. For, if we consider a consecutive radius vector, 0P\ whose increment as compared with OP is dr, while ds is the small arc PP', then dr/ds = cos a = constant. In the growth of a shell, we can conceive no simpler law than this, namely, that it shall widen and lengthen in the same unvarying proportions: and this simplest of laws is that which Nature tends to follow. The shell, hke the creature within it, grows in size but does not change its shape; and the existence of this constant relativity of growth, or constant similarity of form, is of the essence, and may be made the basis of a definition, of the equiangular spiral*. Such a. definition, though not commonly used by mathematicians, has been occasionally employed; and it is one from which the other properties of the curve can be deduced with great ease and sim- plicity. In mathematical language it would run as follows: "Any [plane] curve proceeding from a fixed point (which is called the pole), and such that the arc intercepted between any two radii at a given angle to one another is always similar to itself, is called an equiangular, or logarithmic, spiral." In this definition, we have the most fundamental and "intrinsic" property of the curve, namely the property of continual similarity, and the very property by reason of which it is associated with organic growth in such structures as the horn or the shell. For it is peculiarly characteristic of the spiral shell, for instance, that it does not alter as it grows ; each increment is similar to its predecessor, and the whole, after every spurt of growth, is just Uke what it was before. We feel no surprise when the animal which secretes the shell, or any other animal whatsoever, grows by such sym- metrical expansion as to preserve- its form unchanged; though even there, as we have already seen, the unchanging form denotes a nice balance between the rates of growth in various directions, which is * See an interesting paper by W. A. Whitworth, The equiangular spiral, its chief properties proved geometrically, Messenger of Mathematics (1), i, p. 5, 1862. The celebrated Christian Wiener gave an explanation on these lines of the logarithmic spiral of the shell, in his highly original Grundzuge der Weltordnung, 1863. 758 THE EQUIANGULAR SPIRAL [ch. but seldom accurately maintained for long. But the shell retains its unchanging form in spite of its asymmetrical growth; it grows at one end only, and so does the horn. And this remarkable property of increasing by terminal growth, but nevertheless retaining .unchanged the form of the entire figure, is cnaracteristic of the equiangular spiral, and of no other mathematical curve. It well deserves the name, by which James Bernoulli was wont to call it, of spira mirahilis. We may at once illustrate this curious phenomenon by drawing the outline of a Uttle Nautilus shell within a big one. We know, or we may see at once, that they are of precisely the same shape ; so that, if we look at the little shell through a magnifying glass, it becomes identical with the big one. But we know, on 'the other hand, that the little Nautilus shell grows into the big one, not by growth or magnification in all parts and directions, as when the boy grows into the man, but by growing at one end only. If we should want further proof or illustration of the fact that the spiral shell remains of the same shape while increasing in magnitude by its terminal growth, we may find it by help of our ratio W : L^, which remains constant so long as the shape remains unchanged. Here are weights and measurements of a series of small land-shells {Clausilia):* W (mgm.) L (mm.) €tv) by the dots whose addition keeps the annexed figures perfect squares*: • • • • • • • • • • • There are other gnomonic figures more curious still. For example, if we make a rectangle (Fig. 355) such that the two sides are in the * Cf. Treutlein, Ztschr. /. Math. u. Phya. {Hist. litt. Abth.), xxviii, p. 209, 1883. XI] CONCERNING GNOMONS 761 ratio of 1 : a/2, it is obvious that, on doubling it, we obtain a similar figure; for 1 : V2 : : a/2 : 2; and each half of the figure, accordingly, is now a gnomon to the other. Were we to make our paper of such a shape (say, roughly, 10 in. x 7 in.), we might fold and fold it, and the shape of folio, quarto and octavo pages would be all the same. For another elegant example, let us start with a rectangle (A) whose sides are in the proportion of the "divine" or "golden section*" that is to say as 1 : J {Vb — 1), or, approximately, as 1:0-618.... The gnomon to this rectangle is the square (B) erected on its longer side, and so on successively (Fig. 356). 1 V2 1 V2 V2 1 Fig. 355. 0 Fig. 356. In any triangle, as Hero of Alexandria tells us, one part is always a gnomon to the other part. For instance, in the triangle ABC (Fig. 357), let us draw BD, so as to make the angle CBD equal to the angle A. Then the part BCD is a triangle similar to the whole triangle ABC, and ABD is a gnomon to BCD. A very elegant case is when the original triangle ABC is an isosceles triangle having one angle of 36°, and the other two angles, therefore, each equal to 72° (Fig. 358). Then, by bisecting one of the angles of the base, we subdivide the large isosceles triangle into two isosceles triangles, of which one is similar to the whole figure and the other is its gnomon f. There is good reason to believe that this triangle was especially studied by the Pythagoreans; for it lies at the root of * Euclid, Ti, 11. t This is the so-called Dreifachgleichschenkelige Dreieck; cf. Naber, op. infra cit. The ratio 1 : 0-618 is again not hard to find in this construction. 762 THE EQUIANGULAR SPIRAL [ch. many interesting geometrical constructions, such as the regular pentagon, and its mystical "pentalpha," and a whole range of other curious figures beloved of the ancient mathematicians * : culminating A C B C B Fig. 357. Fig. 358. in the regular, or pentagonal, dodecahedron, which symbolised the universe itself, and with which Euclidean geometry ends. If we take any one of these figures, for instance the isosceles triangle which we have just described, and add to it (or subtract Fig. 359. from it) in succession a series of gnomons, so converting it into larger and larger (or smaller and smaller) triangles all similar to the first, we find that the apices (or other corresponding points) of all these * See, on the mathematical history of the gnomon, Heath's Euclid, i, passim, 1908; Zeuthen, Theoreme de Pythagore, Geneve, 1904; also a curious and interesting book, Das Theorem des Pythagoras, by Dr H. A. Naber, Haarlem, 1908. XI] CONCERNING GNOMONS 763 triangles have their locus upon a equiangular spiral: a result which follows directly froni that alternative definition of the equiangular spiral which I have quoted from Whit worth (p. 757). If in this, or any other isosceles triangle, we take corresponding median lines of the successive triangles, by joining C to the mid-' point (M) of AB, and D to the mid-point (N) of BC, then the pole of the spiral, or centre of similitude of ABC and BCD, is the point of intersection of CM and DN*. Again, ,we may build up a series of right-angled triangles, each of which is a gnomon to the preceding figure; and here again, an equiangular spiral is the locus of corresponding points in these suc- cessive triangles. And lastly, whensoever we fill up space with a collection of equal and similar figures, as in Figs. 360, 361, there we can always discover a series of equiangular spirals in their successive multiples*)*. Once more, then, we may modify our definition, and say that: "Any plane curve proceeding from a fixed point (or pole), and such that the vectorial area of any sector is always a gnomon to the whole preceding figure, is called an equiangular, or logarithmic, spiral." And we may now introduce this new concept and nomen- clature into our description of the Nautilus shell and other related organic forms, by saying that: (1) if a growing structure be built up of successive parts, similar in form, magnified in geometrical progression, and similarly situated with respect' to a centre of similitude, we can always trace through corresponding points a series of equiangular spirals; and (2) it is characteristic of the * I owe this simple but novel construction, like so much else, to Dr G. T. Bennett. t In each and all of these gnomonic figures we may now recognise a never- ending polygon, with equal angles at its corners, and with its successive sides in geometrical progression; and such a polygon we may look upon as the natural precursor of the equiangular spiral. If we call the exterior or "bending" angle of th'^ polygon j3, and the ratio of its sides A, then the vertices lie on an equiangular spiral of angle a, given by logg A = j3 cot a. In the spiral of Fig. 359 the constant angle is thus found to be about 75° 40', in "that of Fig. 355, 77° 40', and in that of Fig. 356, 72° 50'. The calculation is as follows. Taking, for example, the successive triangles of Fig. 359, the ratio (A) of the sides, as BC : AC, is that of the golden section, 1 : 1-618. The external angle (^), as ADB, is 108°, or in radians 1-885. Then log 1-618=0-209, from which log, 1-618=0-481 and cota = !^ = ^^=0-255 = cot75°4y. 764 THE EQUIANGULAR SPIRAL [ch. growth of the horn, of the shell, and of all other organic forms in which an equiangular spiral can be recognised, that each successive increment of growth is similar, and similarly magnified, and similarly 1 \\ \ \ . \ \ ^ \\ \ V \ u ^ \ \ / \ \ / 1 \ \, \ / / ^^ \ z ^ y Fig. 360*. Logarithmic spiral derived from corresponding points in a system of squares. Fig. 361. The same in a system of hexagons. From Naber. situated to its predecessor, and is in consequence a gnomon to the entire pre-existing structure. Conversely (3) it follows that in the spiral * This diagram was at fault in my first edition (p. 512), as Dr G. T. Bennett shews me. The curve met its chords at equal angles at either end: whereas it ought to meet the further end at a lesser angle than the other, and ought in consequence to intersect the hues of the coordinate framework. The constant angle of this spiral is about 66° 11' (tan a=7r/2 logg2). XI] CONCERNING GNOMONS 765 outline of the shell or of the horn we can always inscribe an endless variety of other gnomonic figures, having no necessary relation, save as a mathematical accident, to the nature or mode of develop- ment of the actual structure*. But observe that the gnomons to a square may form increments of any size, and the same is true of the gnomons to a Haliotis-sheW ; but" in the higher symmetry of a chambered Nautilus, or of the successive triangles in Fig. 359, growth goes on by a progressive series of gnomons, each one of which is the gnomon to another. Fig. 362. A shell of Haliotis, with two of the many hues of growtii, or generating curves, marked out in black: the areas bounded by these hnes of growth being in all cases gnomons to the pre-existing shell. Of these three propositions, the second is of great use and advantage for our easy understanding and simple description of the molluscan shell, and of a great variety of other structures whose mode of growth is analogous, and whose mathematical properties are therefore identical. We see that the successive chambers of a spiral Nautilus or of a straight Orthocems, each whorl or part of a whorl of a periwinkle or other gastropod, each new increment of the operculum of a gastropod, each additional increment of an elephant's tusk, or each new chamber of a spiral foraminifer, has its leading characteristic at once described and its form so far explained by the * For many beautiful geometrical constructions based on the molluscan shell, see S. Colman and C. A. Coan, Nature a Harmonic Unity (ch. ix, Conchology), New York, 1912. 766 THE EQUIANGULAR SPIRAL [ch. simple statement that it constitutes a gnomon to the whole previously existing structure. And herein lies the explanation of that "time- element" in the development of organic spirals of which we have spoken already; for it follows as a simple corollary to this theory M^iZ Fig. 363. A spiral foraminifer {Pnlvinulina), to shew how each successive chanibei continues the symmetry of, or constitutes a gnomon to, the rest of the structure. of gnomons that we nmst never expect to find the logarithmic spiral manifested in a structure whose parts are simultaneously produced, as for instance in the maririn of a leaf, or among the many curves that make the contour of a fish. But we most look for it wherever , - ' the organism retains, and still presents at a single view, the successive phases of preceding growth : the successive magni- tudes attained, the successive outhnes occui)ied, as growth pursued the even tenor of its way. And it follows from this that it is in the hard parts of organisms, and not the soft, fleshy, actively growing parts, that this spiral is commonly and characteristically found: not in the fresh mobile tisssue whose form is constrained merely by the active forces of the moment; but in things like shell and tusk, and horn and claw, visibly composed Fig. 364. Another spiral fora-. minifer, Cristellaria. XI THE CYMOSE INFLORESCENCE 767 of parts successively and permanently laid down. The shell-less molluscs are never spiral; the snail is spiral but not the slug*. In short, it is the shell which curves the sr^ail, and not the snail which curves the shell. The logarithmic spiral is characteristic, not of the hving tissues, but of the dead. And for the same reason if will always or nearly always be accompanied, and adorned, by a pattern formed of "hues of growth," the lasting record of successive stages of form and magnitude!. The cymose inflorescences of the botanists are analogous in a curious and instructive way to the equiangular spiral. In Fig. 365 B (which represents the Cicinnus of Schimper, or cyme unipare scorpioide of Bravais, as seen in the Borage), we begin with a primary shoot from which is given off, at a certain definite angle, a secondary shoot: and from that in turn, on the same side and at the same angle, another shoot, and so on. The deflection, or curvature, is continuous and progressive, for it is caused by no external force but only by causes intrinsic in the system. And the whole system is symmetrical: the angles at which the successive shoots are given off being all equal, and the lengths of the shoots diminishing in constant ratio. The result is that the successive shoots, or successive increments of growth, are tangents to a curve, and this curve is a true logarithmic spiral. Or in other words, we may regard each successive shoot as forming, or defining, a gnomon to the preceding structure. While in this simple case the successive shoots are depicted as lying in a plane, it may also happen that, in addition to their successive angular divergence from one another wildiin that plane, they also tend to Fig. 365. A, a helicoid; B, jbl scorpioid cyme. * Note also that Chiton, where the pieces of the shell are disconnected, shews no sign of spirality. t That the invert to an equiangular spiral is identical with the original curve does not concern us in our study of organic form, but it is one of the most beautiful and most singular properties of the curve. It was this which led James Bernoulli, in imitation of Archimedfes, to have the logarithmic spiral inscribed upon his tomb; and on John Goodsir's grave near Edinburgh the same symbol is reinscribed. Bernoulli's account of the matter is interesting and remarkable: "Cum autem ob proprietatem tam singularem tamque admirabilem mire mihi placeat spira haec mirabilis, sic ut ejus contemplatione satiari vix nequeam: cogitavi iUam ad varias res symbolice repraesentandas non inconcinne adhiberi posse. Quoniam enim semper sibi et eandera spiram gignit, utcunque volvatur. evjlvatur, radiet, hinc poterit esse vei sobolis parentibus per omnia similis Emblema: Simillima Filia Matri; vel (si rem aeternae veritatis Fidel mysteriis accommodare non eat 768 THE EQUIANGULAR SPIRAL [ch. diverge by successive equal angles from that plane of reference; and by this means, there will be superposed upon the equiangular spiral a twist or screw. And, in the particular case where this latter angle of divergence is just equal to 180°, or two right angles, the successive shoots will once more come to lie in a plane, but they will appear to come off from one another on alternate sides, as in Fig. 365 A. This is the Schrauhel or Bostryx of Schimper, the cyme unipare helicoide of Bravais. The equiangular spiral is still latent in it, as in the other; but is concealed from view by the deformation resulting from the helicoid. Many botanists did not recognise (as the brothers Bravais did) the mathematical significance of the latter case, but were led by the snail-like spiral of the scorpoid cyme to transfer the name "helicoid" to it*. The spiral curve of the shell is, in a sense, a vector diagram of its own growth; for it shews at each instant of time the direction, radial and tangential, of growth, and the unchanging ratio of velocities in these directions. Regarding the actual velocity of growth in the shell, we know very little by way of experimental measurement; but if we make a certain simple assumption, then we may go a good deal further in our description of the equiangular spiral as it appears in this concrete case. Let us make the assumption that similar increments are added to the shell in equal times; that is to say, that the amount of growth in unit time is measured by the areas subtended by equal angles. Thus, in the outer whorl of a spiral shell a definite area marked out by ridges, tubercles, etc., has very different linear dimensions to the corresponding areas of an inner whorl, but the symmetry of the figure imphes that it subtends an equal angle with these; and it is reasonable to suppose that the successive regions, marked out in this way by successive natural boundaries or patterns, are produced in equal intervals of time. prohibitum) ipsius aeternae generationis Filii, qui Patris veluti Imago, et ab illo ut Lumen a Lumine emanans, eidem bixoLovaLo% existit, qualiscunque adumbratio. Aut, si mavis, quia Curva nostra mirabilis in ipsa mutatione semper sibi constantissime manet similis at numero eadem, poterit esse vel fortitudinis et constantiae in adversitatibus, vel etiam Carnis nostrae post varias alterationes et tandem ipsam quoque mortem, ejusdem numero resurrecturae symbolum: adeo quidem, ut si Archimedem imitandi hodiernum consuetudo obtineret, libenter Spiram hanc tumulo meo juberem incidi, cum Epigraphe, Eadem numero mutata resurgeV; Acta Erudi- torum, M. Mali, 1692, p. 213. Cf. L. Isely, Epigraphes tumulaires de mathe- maticiens, Bull. Soc. Set. nat. Neuchdtel. xxvii, p. 171, 1899. * The names of these structures have been often confused and misunderstood ; cf. S. H. Vines, The history of the scorpioid cyme, Journ. Bot. (n.s.), x, pp. 3-9, 1881. XI] ITS MATHEMATICAL PROPERTIES 769 If this be so, the radii measured from the pole to the boundary of the shell will in each case be proportional to the velocity of growth at this point upon the circumference, and at the time when it corresponded with the outer hp, or region of active growth ; and while the direction of the radius vector corresponds with the direction of growth in thickness of the animal, so does the tangent to bhe curve correspond with the direction, for the time being, of the animal's growth in length. The successive radii are a measure of the acceleration of growth, and the spiral curve of the shell itself, if the radius rotate uniformly, is no other than the hodograph of the growth of the contained organism*. So far as we have now gone, we have studied the elementary properties of the equiangular spiral, including its fundamental property of continued si?nilarity; and we have accordingly learned that the shell or the horn tends necessarily to assume the form of this mathematical figure, because in these structures growth proceeds by successive increments which are ailways similar in form, similarly situated, and of constant relative magnitude one to another. Our chief objects in enquiring further into the mathe- matical properties of the equiangular spiral will be: (1) to find means of confirming and verifying the fact that the shell (or other organic curve) is actually an equiangular spiral; (2) to learn how, by the properties of the curve, we may further extend our knowledge or simphfy our descriptions of the shell; and (3) to understand the factors by which the characteristic form of any particular equiangular spiral is determined, and so to comprehend the nature of the specific or generic differences between one spiral shell and another. Of the elementary properties of the equiangular spiral the following are those which we may most easily investigate in the concrete case of the molluscan shell: (1) that the polar radii whose vectorial angles are in arithmetical progression are themselves in geometrical progression; hence (2) that the vectorial angles are pro- portional to the logarithms of the corresponding radii ; and (3) that the tangent at any point of an equiangular spiral makes a constant angle (called the angle of the spiral) with the polar radius vector. * The hodograph of a logarithmic spiral (i.e. of a point which lies on a uniformly revolving radius and describes a logarithmic spiral) is likewise a logarithmic spiral : W. Walton, Collection of Problerha in Theoretical Mechanics (3rd ed.), 1876, p. 296. 770 THE EQUIANGULAR SPIRAL [ch. The first of these propositions may be written in a simpler form, as follows: radii which form equal angles about the pole of the equiangular spira] are themselves continued proportionals. That is to say, in Fig. 366, when the angle ROQ is equal to the angle QOP, then OP:OQ::OQ: OR. A particular case of this proposition is when the equal angles are each angles of 360° : that is to say when in each case the radius vector makes a complete revolution, and when, therefore, P, Q and R all he upon tjie same radius. It was by observing with the help of very careftil measurement this continued proportionality, that Moseley was enabled to verify his first assumption, based on the general appearance of the shell, that the shell of Nautilus was actually an equiangular spiral, and this demonstration he was soon after- Via ^ftfi wards in a position to generalise by extending it to all spiral Ammonitoid and Gastropod mollusca*. For, taking a median transverse section of a Nautilus pompilius, and carefully measuring the successive breadths of the whorls (from the dark line which marks what was originally the outer surface, before it was covered up Tjy fresh deposits on the part of the growing and advancing shell), Moseley found that "the distance of any two of its whorls measured upon a radius vector is one-third that of the two next whorls measured upon the same radius vectorf. Thus (in * The Rev. H. Moseley, On the geometrical forms of turbinated and discoid shells, Phil. Trans. 1838, Pt. i, pp. 351-370. Reaumur, in describing the snail-shell (Mem. Acad, des Sci. 1709, p. 378), had a glimpse of the same geometrical law: " Le diametre de chaque tour des spirale, ou sa plus grande longueur, est a peu pres double de celui qui la precede et la moitie de celui qui la suit." Leslie (in his Geometry of Curved Lines, 1822, p. 438) compared the "general form and the elegant septa of the Nautilus'" to an equiangular spiral and a series of its involutes. f It will be observed that here Moseley, speaking as a mathematician and considering the linear spiral, speaks of whorls when he means the linear boundaries, or lines traced by the revolving radius vector; while the conchologist usually applies the term whorl to the whole space between the two boundaries. As con- chologists, therefore, we call the breadth of a whorl what Moseley looked updn as the distance between two consecutive whorls. But this latter nomenclature Moseley himself often uses. Observe also that Moseley gets a very good approximate result by his measurements "upon a radius vector," although he has to be content with a very rough determination of the pole. XI] OF THE NAUTILUS SHELL 771 Fig. 367), ab is one-third of 6c, de of e/, gh of hi, and kl of Im. The curve is therefore an equiangular spiral." The numerical ratio in the case of the Nautilus happens to be one of unusual simplicity. Let us take, with ^oseley, a somewhat more complicated example. Fig. 367. Spiral of the Naviilua. Fig. 368. Turritella dupli- cata (L.), Moseley's . Turbo duplicatus. From Chenu. x ^. From the apex of a large Turritella {Turbo) duplicata* a line was drawn across its whorls, and their widths were measured upon it in succession, beginning with the last but one. The measure- * In the case of ''Turbo'", and all other turbinate shells, we are dealing not with a plane logarithmic spiral, as in Nautilus, but with a "gauche" spiral, such that the radius vector no longer revolves in a plane perpendicular to the axis of the system, but is inclined to that axis at some constant angle (j3). The figure still preserves its continued similarity, and may be called a logarithmic spiral in space; indeed it is commonly spoken of as a logarithmic spiral wrapped upon a cone, its pole coinciding with the apex of the cone. It follows that the distances of successive whorls of the spiral measured on the same straight Line passing through the apex of the cone are in geometrical progression, and conversely; just as in the former case. But the ratio between any two consecutive interspaces (i.e. ^3 - R2IR2 - Ri) is now equal to e^'^^i^^cota^^ being the semi-angle of the enveloping cone. (Cf. Moseley, Phil. Mag. xxi, p. 300, 1842.) 772 THE EQUIANGULAR SPIRAL [cH. ments were, as before, made with a fine pair of compasses and a diagonal scale. The sight was assisted by a magnifying glass. In a parallel colimin to the following admeasurements are the terms of a geometric progression, whose first term is the width of the widest whorl measured, and whose common ratio is 1-1804. Turritella duplicata Widths of successive Terms of a geometrical progression, whorls, measured in whose first term is the width of inches and parts the widest whorl, and whose of an inch common ratio is 1-1804 1-31 1-310 1-12 1110 0-94 0-940 0-80 0-797 0-67 0-675 0-57 0-572 0-48 0-484 0-41 0-410 The close coincidence between the observed and the calculated figures is very remarkable, and is amply sufiicient to justify the conclusion that we are here deaUng with a true logarithmic spiral*. Nevertheless, in order to verify his conclusion still further, and to get partially rid of the inaccuracies due to successive small measurements, Moseley proceeded to investigate the same shell, measuring not single whorls but groups of whorls taken several at a time: making use of the following property of a geometrical progression, that ''if /x represent the ratio of the sum of every even number (m) of its terms to the sum of half that number of terms, then the common ratio (r) of the series is represented by the formula 2 r={fjL- 1)^." * Moseley, writing a hundred years ago, uses an obsolete nomenclature which is apt to be very misleading. His Turbo duplicatus, of Linnaeus, is now Turritella duplicata, the common large Indian Turritella, a slender, tapering shell with a very beautiful spiral, about six or seven inches long. But the operculum which he describes as that of Turbo does indeed belong to that genus, sensu stricto; it is the well-known calcareous operculum or "eyestone" of some such common species as Turbo petholatus. Turritella has a very different kind of operculum, a thin chitinous disc in the form of a close spiral coil, not nearly fiUing up the aperture of the shell. Moseley's Turbo phasianus is again no true Turbo, but is (to judge from his figure) Phasianella bulimoides Lam. =P. australis (Gmelin); and his Buccinum aubulatum is Terebra subulata (L.). XI] OF CERTAIN OPERCULA 773 Accordingly, Moseley made the following measurements, begin- ning from the second and third whorls respectively: Width of Ratio fi Six whorls Three whorls 5-37 4-55 203 1-72 2-645 2-645 Four whorls Two whorls 415 3-52 1-74 1-47 2-385 2-394 "By the ratios of the two first admeasurements, the formula gives /•= (1-645)*= M804. By the mean of the ratios deduced from the second two admeasure- ments, it gives r = (l-389)i = M806. "It is scarcely possible to imagine a more accurate verification than is deduced from these larger admeasurements, and we may with safety annex to the species Turbo duplicatus the characteristic number M8." By similar and equally concordant observations, Moseley found for Turbo phasianus the characteristic ratio, 1-75; and for Bucci- num subulatum that of 1*13. From the measurements of Turritella dupUcata (on p. 772), it is perhaps worth while to illustrate the logarithmic statement of the same thing: that is to say, the elementary fact, or corollary, that if the successive radii be in geometric progression, their logarithms will differ from one another by a constant amount. Turritella duplicata Widths of suc- Logarithms Differences Ratios of suc- cessive whorls of do. of logarithms cessive widths 131 211727 — — 112 2-04922 0-06805 1-170 94 1-97313 , 0-07609 1-191 80 1-90309 0-07004 1-175 67 1-82607 0-07702 M94 57 1-75587 0-07020 1-175 48 1-68124 0-07463 1-188 41 1-61278 0-06846 1-171 Mean 0-07207 M806 And 0-07207 is the logarithm of 1-1805. 774 THE EQUIANGULAR SPIRAL [CH. Lastly, we may if we please, in this simple case, reduce the whole matter to arithmetic, and, dividing the width of each whorl by that of the next, see that these quotients are nearly identical, and that their mean value, or common ratio, is precisely that which we have already found. We may shew, in the same simple fashion, by measurements of Terebra (Fig. 397), how the relative widths of successive whorls fall into a geometric progression, the criterion of a logarithmic spiral. ^ Measurements of a large specimen (15-5 cm.) of Terebra maculata, along three several tangents (a, 6, c) to the whorls. {After Chr. Peterson, 1921.) Width (mm.) Ratio 25 1-25 20 1-33 15 1-25 12 Width Ratio Width Ratio 24-5 23 •25 18^5 1-32 17-5 1-31 •33 14 132 13-3 1-31 •25 10-75 130 9-75 136 •33 8 1-34 7-25 1-34 Mean 1-29 1-32 1-33 Mean ratio, b31 The logarithmic spiral is not only very beautifully manifested in the molluscan shell*, but also, in certain cases, in the little Hd or "operculum" by which the entrance to the tubular shell is closed after the animal has withdrawn itself withinf. In the spiral shell of Turbo, for instance, the operculum is a thick calcareous structure, with a beautifully curved outhne, which grows by successive incre- ments applied to one portion of its edge, and shews, accordingly, a spiral line of growth upon its surface. The successive increments leave their traces on the surface of the operculum (Fig. 370), which traces have the form of curved lines in Turbo, and of straight hnes * It has even been proposed to use a logarithmic spiral in place of a table of logarithms. Cf. Ant. Favaro, Statique graphique, Paris, 1885; Hele-Shaw, in Brit. Ass. Rep. 1892, p. 403. t Cf. Fred. Haussay, Hecherches sur Vopercule, Diss., Paris, 1884. XI OF CERTAIN OPERCULA 775 in (e.g.) Nerita (Fig. 371); that is to say, apart from the side con- stituting the outer edge of the operculum (which side is always and of necessity curved) the successive increments constitute curvihnear Fig. 369. Operculum of Turbo. triangles in the one case, and rectilinear triangles in the other. The sides of these triangles are tangents to the spiral Une of the Fig. 370. Fig. 371. Figs. 370, 371. Opercula of Turbo and Nerita. After Moseley. operculum, and may be supposed to generate it by their consecutive intersections. In a number of such opercula, Moseley measured the breadths 776 THE EQUIANGULAR SPIRAL [ch. of tlie successive whorls along a radius vector*, just in the same way as he did with the entire shell in the foregoing cases; here is one example of his results. Operculum of Turbo sp. ; breadth {in inches) of successive whorls, mea^redfrom the pole and Distance Ratio Distance Ratio Distance Ratio Distance Ratio 0-24 016 0-2 0-18 2-28 2-31 2-30 2-30 0-55 0-37 0-6 0-42 2-32 2-30 2-30 2-24 1-28 0-85 1-38 0-94 The ratio is approximately constant, and this spiral also is, therefore, a logarithmic spiral. But here comes in a very beautiful illustration of that property of the logarithmic spiral which causes its whole shape to remain unchanged, in spite of its apparently unsymmetrical, or unilateral, mode of growth. For the mouth of the tubular shell, into which the operculum has to fit, is growing or widening on all sides: while the operculum is increasing, not by additions made at the same time all round its margin, but by additions made only on one side of it at each successive stage. One edge of the operculum thus remains unaltered as it advances into its new position, and comes to occupy a new-formed section of the tube, similar to but greater than the last. Nevertheless, the two apposed structures, the chamber and its plug, at all times fit one another to perfection. The mechanical problem (by no means an easy one) is thus solved: "How to shape a tube of a variable section, so that a piston driven along it shall, by one side of its margin, coincide continually with its surface as it advances, provided only that the piston be made at the same time continually to revolve in its own plane." As Moseley puts it: "That the same edge which fitted a portion of the first less section should be capable of adjustment, so as to fit a portion of the next similar but greater section, supposes a geometrical provision in the curved form of the chamber of great * As the successive increments evidently constitute similar figures, similarly related to the pole (P), it follows that their linear dimensions are to one another as the radii vectores drawn to similar points in them: for instance as PPj, PP2> which (in Fig. 370) are radii vectores drawn to the points where they meet the common boundary. XI] OF CERTAIN OPERCULA 777 apparent complication and difficulty. But God hath bestowed upon this humble architect the practical skill of a learned geo- metrician, and he makes this provision with admirable precision in that curvature of the logarithmic spiral which he gives to the section of the shell. This curvature obtaining, he has only to turn his operculum shghtly round in its own plane as he advances it into each newly formed portion of his chamber, to adapt one margin of it to a new and larger surface and a different curvature, leaving the space to be filled up by increasing the operculum wholly on the other margin." The fact is that self-similar or gnomonic growth is taking place both in the shell and its operculum; in both of them growth is in reference to a fixed centre, and to a fixed axis through that centre; and in both of them growth proceeds in geometric progression from the centre while rotation takes place in arithmetic progression about the axis. The same architecture which builds the house constructs the door. Moreover, not only are house and door governed by the same law of growth, but, growing together, door and doorway adapt themselves to one another. The operculum of the gastropods varies from a more or less close -wound spiral, as in Turritella, Trochus or Pleurotomaria, to cases in which accretion takes place, by concentric (or more. or less excentric) rings, all round. But these latter cases, so Mr Winckworth tells me, are not very common. Paludina and Ampullaria come near to having a concentric operculum, and so do some of the Murices, such as M. tribulus, and a few Turrids, and the genus Helicina; but even these opercula probably begin as spirals, adding on their gnomonic increments at one end or side, and only growing on all sides later on. There would seem to be a truly concentric operculum in the Siphonium group of Vermetus, where the spiral of the shell itself is lost, or nearly so; but it is usually overgrown with Melobesia, and hard to see. Fig. 372. One more proposition, an all but self-evident one, we may make passing mention of here: If upon any polar radius vector OP, 778 THE EQUIANGULAR SPIRAL ' [ch. a triangle OPQ be drawn similar to a given triangle, the locus of the vertex Q will* be a spiral similar to the original spiral. We may extend this proposition (as given by Whitworth) from the simple case of the triangle to any similar figures whatsoever ; and see from it how every spot or ridge or tubercle repeated symmetrically from one radius vector (or one generating curve) to another becomes part of a spiral pattern on the shell. Viewed in regard to its own fundamental properties and to those of its hmiting cases, the equiangular spiral is one of the simplest of all known curves ; aiid the rigid uniformity of the simple laws by which it is developed sufficiently account for its frequent manifesta- tion in the structures built up by the slow and steady growth of organisms. In order to translate into precise terms the whole form and growth of a spiral shell, we should have to employ a mathematical notation considerably more complicated than any that I have attempted to make use of in this book. But we may at least try to describe in elementary language the general method, and some of the variations, of the mathematical development of the shell. But here it is high time to observe that, while we have been speaking of the shell (which is a surface) as a logarithmic spiral (which is a line), we have been simphfying the case, in a provisional or preparatory way. The logarithmic spiral is but one factor in the case, albeit the chief or dominating one. The problem is one not of plane but of sohd geometry, and the solid in question is described by the movement in space of a certain area, or closed curve*. Let us imagine a closed curve in space, whether circular or eUiptical or of some other and more complex specific form, not necessarily in a plane: such a curve as we see before us when we consider the mouth, or terminal orifice, of our tubular shell. Let * For a more advanced study of the family of surfaces of which the Nautilus is a simple case, see M. Haton de la Goupilliere {op. cit.). The turbinate shells represent a sub-family, which may be called that of the "surfaces cerithioides " ; and "surfaces a front generateur" is a short title of the whole family. The form of the generating curve, its rate of expansion, the direction of its advance, and the angle which the generating front makes with the directrix, define, and give a wide extension to, the family. These parameters are all severally to be recognised in the growth of the living object; and they make of a collection of shells an unusually beautiful materialisation of the rigorous definitions of geometry. XI] OF THE MOLLUSCAN SHELL 779 us call this closed curve the "generating curve"; the surface which it bounds we may call (if need arise) the "generating front," and let us imagine some one characteristic point within this closed curve, such as its centre of gravity. Then, starting from a fixed origin, let this characteristic point describe an equiangular spiral Fig. |373. Melo ethiopicus L. in space about a fixed axis or "conductrix" (namely the axis of the shell), while at the same time the generating curve grows with each increment of rotation in such a way as to preserve the symmetry of the entire figure, with or without a simultaneous movement of translation along the axis. The resulting shell may now be looked upon in either of two ways. It is, on the one hand, an ensemble gf similar closed curves, spirally arranged in space, and gradually increasing in dimensions in pro- portion to the increase of their vector-angle from the pole*. In * The plumber, the copper-smith and the glass-blower are at pains to conserve in every part of their tubular constructions, however these branch or bend, the constant form which their cross-sections ought to have. Throughout the spiral twisting of the shell, throughout the windings and branchings of the blood-vessels, the same uniformity is maintained. 780 THE EQUIANGULAR SPIRAL [ch. other words, we can imagine our shell cut up into a system of rings, following one another in continuous spiral succession, from that terminal and largest one which constitutes the hp of the orifice of the shell. Or on the other hand, we may figure to ourselves the whole shell as made up of an ensemble of spiral lines in space, each spiral having been traced out by the gradual growth and revolution of a radius vector from the pole to a given point on the boundary of the generating curve. -r" ■^^^ -j|KjSp^^ M. -J,,... K2 m \'^^ - ' ' ' ■'■■.' / • ■ i WSrWWW^^II^B f I 2 Fig. 374. 1, Harpa; 2, Dolium. The ridges on the shell correspond in (1) to generating curves, in (2) to generating spirals. Both systems of fines, the generating spirals (as these latter may be called)! and the closed generating curves corresponding to suc- cessive margins or hps of the shell, may be easily traced in a great variety of cases. Thus, for example, in Dolium, Eburna, and a host of others, the generating spirals are beautifully marked out by ridges, tubercles or bands of colour. In Trophon, Scalaria, and (among countless others) in the Ammonites, it is the successive generating curves which more conspicuously leave their impress on XI] OF THE MOLLUSCAN SHELL 781 the shell. And in not a few cases, as in Harpa, Dolium perdix, etc., both ahke are conspicuous, ridges and colour-bands^ intersecting one another in a beautiful isogonal system. In ordinary gastropods the shell is formed at or near the mantle- edge. Here, near the mantle-border, is a groove Hned with a secretory epithehum which produces the horny cuticle or perio stracum of the shell*. A narrow zone of the mantle just behind this secretes Hme abundantly, depositing it in a layer below the periostracum ; and for some httle way back more hme may be secreted, and pigment superadded from appropriate glands. Growth and secretion are periodic rather than continuous. Even in a snail- shell it is easy to see how the shell is built up of narrow annular increments; and many other shells record, in conspicuous colour- patterns, the alternate periods of rest and of activity which their pigment-glands have undergone. The periodic accelerations and retardations in the growth of a shell are marked in various ways. Often we have nothing more than an increased activity from time to time at or near the mantle- edge — enough to give rise to shght successive ridges, each corre- sponding to a "generating curve" in the conformation of the shell. But in many other cases, as in Murex, Ranella and the like, the mantle-edge has its alternate phases of rest and of turgescence, its outline being plain and even in the one and folded and contorted in the other; and these recurring folds or pleatings of the edge leave their impress in the form of various ridges, ruffles or comb-like rows of spines upon the shell f. In not a few cases the colour-pattern shews, or seems to shew, how some play of forces has fashioned and transformed the first elementary pattern of pigmentary drops or jets. As the book- binder drops or dusts a little colour on a viscous fluid, and then produces the beautiful streamlines of his marbled papers by stirring * That the shell grows by accretion at the mantle-edge was one of Reaumur's countless discoveries {Mem. Acad. Roy. des Sc. 1709, p. 364 seq.). It follows that the mathematical "generating curves," as Moseley chose them, correspond to the material increments of the shell. t The periodic appearance of a ridge, or row of tubercles, or other ornament on the growing shell is illustrated or even exaggerated in the delicate "combs" of Murex aculeatus. Here normal growth is. interrupted for the time being, the mantle-edge is temporarily folded and reflexed, and shell-substance is poured out into the folds. 782 THE EQUIANGULAR SPIRAL [ch. and combing the colloid mass, so we may see, in the harp-shells or the volutes, how a few simple spots or hnes have been drawn out into analogous wavy patterns by streaming movements during the formation of the shell. ' In the complete mathematical formula for any given turbinate shell, we may include, with Moseley, factors for the following elements: (1) for the specific form of a section of the tube, or (as we have called it) the generating curve; (2) for the specific rate of growth of this generating curve; (3) for its incHnation to the directrix, or to the axis ; (4) for its specific rate of angular rotation about the pole, in a projection perpendicular to the axis; and (5) in turbinate (as opposed to nautiloid) shells, for its rate of screw- translation, parallel to the axis, as measured by the angle between a tangent to the whorls and the axis of the shell*. It seems a compHcated affair; but it is only a pathway winding at a steady slope up a conical hill. This uniform gradient is traced* by any given point on the generating curve while the vector angle increases in arithmetical progression, and the scale changes in geometrical progression; and a certain ensemble, or bunch, of these spiral curves in space constitutes the self-similar surface of the shell. But after all this is not the only way, neither is it the easiest way, to approach our problem of the turbinate shell. The conchologist turned mathematician is apt to think of the generating curve by which the spiral surface is described as necessarily identical, or coincident, with the mouth or Hp of the shell; for this is where growth actually goes on, and where the successive increments of shell-growth are visibly accumulated. But it does oiot follow that this particular generating curve is chosen for the best from the mathematical point of view; and the mathematician, unconcerned with the physiological side of the case and regardless of the suc- cession of the parts in time, is free to choose any other generating curve which the geometry of the figure may suggest to him. We are following Moseley's example (as is usually done) when we think of no other generating curve but that which takes the form of a * Note that this tangent touches the curve at a series of points, whorl by whorl, instead of at one only. Observe also that we may have various tangent-cones, all centred on the apex of the shell. In an open spiral, like a ram's horn, or a half-open spiral like the shell Solarium, w^ have two cones, one touching the outside, the other the inside of the shell. XI] OF THE MOLLUSCAN SHELL 783 frontal plane, outlined by the lip, and sliding along the axis while revolving round it; but the geometer takes a better and a simpler way. For, when of two similar figures in space one is derived from the other by a screw-displacement accompanied by change of scale — as in the case of a big whelk and a little whelk — there is a unique (apical) point which suffers no displacement; and if we choose for our generating curve a sectional figure centred on the apical point and passing through the axis of rotation, the whole development of the surface may be simply described as due to a rotation of this generating figure about the axis (z), together with a change of scale with the point 0 as centre of simihtude. We need not, and now must not, think of a slide or shear as part of the operation; the translation along the axis is merely part and parcel of the magnifica- tion of the new generating curve. It follows that angular rotation in arithmetical progression, combined with change of scale (from 0) in geometrical progression, causes any arbitrary point on the generating curve to trace a path of uniform gradient round a circular cone, or in other words to describe a helico-spiral or gauche equiangular spiral in space. The spiral curve cuts all the straight- fine generators of the cone at the same angle ; and it further follows that the successive increments are, and the whole figure constantly remains, "self -similar"*. Apart from the specific form of the generating curve, it is the ratios which happen to exist between the various factors, the ratio for instance between the growth-factor and the rate of angular revolution, which give the endless possibihties - of permutation of form. For example, a certain rate of growth in the generating curve, together with a certain rate of vectorial rotation, will give us a spiral shell of which each successive whorl will just touch its predecessor and no more; with a slower growth-factor the whorls will stand asunder, as in a ram's horn; with a quicker growth-factor * The equation to the surface of a turbinate shell is discussed by Moseley both in terms of polar and of rectangular coordinates, and the method of polar co- ordmates is used also by Haton de la Goupilliere; but both accounts are subject to mathematical objection. Dr G. T. Bennett, choosing his generating curve (as described above) in the axial plane from which the vertical angles are measured (the plane ^ =0), would state his equation in cylindrical coordinates, / {za^, ra^) =0: that is to say in terms of z, conjointly with ordinary plane cylindrical coordi- nates. 784 THE EQUIANGULAR SPIRAL [ch. each will cut or intersect its predecessor, as in an Ammonite or the majority of gastropods, and so on. A similar relation of velocities suffices to determine the apical angle of the resulting cone, and give us the diflFerence, for example, between the sharp, pointed cone of Turritella, the less acute one of Fusus or Buccinum, and the obtuse one of Harpa or^of Dolium. In short it is obvious that all the differences of form which we observe between one shell and another are referable to inatters of degree, depending, one and all, upon the relative magnitudes of the various factors in the complex equation to the curve. This is an immensely important thing. To learn that all the multitudinous shapes of shells, in their all but infinite variety, may be reduced to the variant properties of a single simple curve, is a great achieve- ment. It exemphfies very beautifully what Bacoi^ meant in saying that the forms or differences of things are simple and few, and the degrees and coordinations of these make all their variety*. And after such a fashion as this John Goodsir imagined that the naturahst of the future would determine and classify his shells, so that conchology should presently become, hke mineralogy, a mathe- matical science I . The paper in which, more than a hundred years ago, Canon Moseleyf gave a simple mathematical account, on fines fike tfies^, of the spiral forms of univalve shells, is one of the classics of Natural History. But other students before, and sometimes long before, him had begun to recognise the same simpficity of form and structure. About the year 1818 Reinecke had declared Nautilus to be a well-defined geometrical figure, whose chambers followed * For a discussion of this idea, and of the views of Bacon and of J. S. Mill, see J. M. Keynes, op. cit. p. 271. t On the employment of mathematical modes of investigation in the determina- tion of organic forms^ in Anatomical Memoirs, ii, p. 205, 1868 (posthumous pubhcation). X The Rev. Henry Moseley (1801-1872), of St John's College, Cambridge, Canon of Bristol, Professor of Natural Philosophy in King's College, London, was a man of great and versatile ability. He was father of H. N. Moseley, naturalist on board the Challenger and Professor of Zoology in Oxford; and he was grand- father of H. G. J. Moseley (1887-1915) — Moseley of the Moseley numbers — whose death at Gallipoli, long ere his prime, was one of the major tragedies of the Four Years War. XI] OF THE MOLLUSCAN SHELL 785 one another in a constant ratio or continued proportion*; and Leopold von Buch and others accepted and even developed the idea. Long before, Swammerdam had grasped with a deeper insight the root of the whole matter; for, taking a few diverse examples, such as Helix and Spirula, he shewed that they and all other spiral shells whatsoever were referable to one common type, namely to that of a simple tube, variously curved according to definite mathe- matical laws; that all manner of ornamentation, in the way of spines, tuberosities, colour-bands and so forth, might be superposed upon them, but the type was one throughout and specific differences were of a geometrical kind. "Omnis enim quae inter eas anim- advertitur differentia ex sola nascitur diversitate gyrationum: quibus si insuper externa quaedam adjunguntur ornamenta pin- narum, sinuum, anfractuum, planitierum, eminentiarum, profundi- tatum, extensionum, impressionum, circumvolutionum, colorumque : . . . tunc deinceps facile est, quarumcumque Cochlearum figuras geometricas, curvosque, obliquos atque rectos angulos, ad unicam omnes speciem redigere : ad oblongum videhcet tubulum, qui vario modo curvatus, crispatus, extrorsum et introrsum flexus, ita concrevitf." Nay more, we may go back yet another hundred years and find Sir Christopher Wren contemplating the architecture of a snail-shell, and finding in it the logarithmic spiral. For AValhsf, after defining and describing this curve with great care and simplicity, tells us that Wren not only conceived the spiral shell to be a sort of cone or pyramid coiled round a vertical axis, but also saw that on the magnitude of the angle of the spire depended the specific form of the shell: "Hanc ipsam curvam . . . fcontemplatus est Wrennius noster. Nee tantum curvae longitudinem, partiumque ipsius, et * J. C. M. Reinecke, Maris protogaei Nautilos, etc., Coburg, 1818, p. 17: "In eius forma, quae canalis spiram convoluti formam et proportiones simul sub- ministrat, totius testae forma quoddammodo data est. Restaret solum scire, quota cuj usque anfractus pars sequent! inclusa^it, ut testam geometrice construere possimus." Cf. Leopold von Buch, Ueber die Ammoniten in den alteren Gebirgs- schichten, Ahh. Berlin. Akad., Phys. Kl. 1830, pp. 135-158; Ann. Sc. Nat. xxvin, pp. 5-43, 1833; cf. Elie de Beaumont, Sur I'enroulement des Ammonites, Soc. Philom., Pr. verb. 1841, pp. 45-4^. t Biblia Naturae sive Historia Insectorum, Leydae,.1737, p. 152. X Job. Wallis, Tractatus duo, de Cydoide, etc., Oxon., 1659, pp. 107, 108. 786 THE EQUIANGULAR SPIRAL [ch. magnitudinem adjacentis plani; sed et, ipsius ope, Limacum et Conchiliorum domunculos metitur. Existimat utique, magna veri- similitudine, domunculos hosce non alios esse quam Pjrramides convolutas: quarum Axis sit, istiusmodo Spiralis: non quidem in piano jacens, sed sensim in convolutione (circa erectum axim) assurgens: pro variis autem curvae, sive ad rectam circumductam sive ad subjacens planum, " angulis, variae Conchiliorum formae enascantur. Atque hac hjrpothesi, mensurata Pyramide, metitur etiam ea conchiliorum spatia." For some years after the appearance of Moseley's paper, a number of writers followed in his footsteps, and attempted in various ways to put his conclusions to practical use. For instance, d'Orbigny ^ — ^—^^-^^^ '''■• \ \. ^ ^^,^ j^^^^^^ ■ ''|i'iTi|irir|iiii|i'iii(iiiiji iijiii()iiiimH Mntimi! IITTTImitii iiiiiliii iiiiiliii 1 — ' — 1 — ^ — 1 — ^ ° o « L L J — 1 — - — \ — ^—-\ — ■ — 1 ? % Fig. 375. d'Orbigny's helicometer. devised a very simple protractor, which he called a Helicometer*, and which is represented in Fig. 375. By means of this little instrument the apical angle of the turbinate shell was immediately read off, and could then be used as a specific and diagnostic character. By keeping one limb of the protractor parallel to the side of the cone wjiile the other was brought into hne with the suture between two adjacent whorls, another specific angle, the '^sutural angle," could in like manner be recorded. And, by the hnear scale upon the instrument, the relative breadths of the consecutive whorls, and that of the terminal chamber to the rest of the shell, might * Alcide d'Orbigny, Bull, de la*soc. geol. Fr. xiii, p. 200, 1842; Cours elem. de PaUontologie, ii, p. 5, 1851. A somewhat similar instrument was described by Boubee, in Bull. soc. geol. i, p. 232, 1831. Naumann's conchyliometer {Poggend, Ann. Liv, p. 544, 1845) was an application of the screw-micrometer; it was provided also with a rotating stage for angular measurement. It was adapted for the study of a discoid or ammonitoid shell, while d'Orbigny 's instrument was meant for the study of a turbinate shell. XI] OF THE MOLLUSCAN SHELL 787 ako, though somewhat roughly, be determined. For instance, in Terebra dimidiata the apical angle was found to be 13°, the sutural angle 109°, and so forth. It was at once obvious that, in such a shell as is represented in Figs. 369 and 375 the entire outUne (always excepting that of the immediate neighbourhood of the mouth) could be restored from a broken fragment. For if we draw our tangents to the cone, it follows from the symmetry of the figure that we can continue the projection of the sutural hne, and so mark off the successive whorls, by simply drawing a' series of consecutive parallels, and by then filling into the quadrilaterals so marked off a series of curves similar to one another, and to the whorls which are still intact in the broken shell. But the use of the hehcometer soon shewed that it was by no means universally the case that one and the same cone was tangent to all the turbinate whorls; in other words, there was not always one specific apical angle which held good for the entire system. In the great majority of cases, it is true, the same tangent touches all the whorls, and is a straight Hne. But in others, as in the large Cerithium nodosum, such a hne is sKghtly concave to the axis of the shell; and in the short spire of Doliumj for instance, the concavity is marked, ^nd the apex of the spire is a distinct cusp. On the other hand, in Pupa and Clausilia the conmion tangent is convex to the axis of the shell. So also is it, as we shall presently see, among the Ammonites: where there are some species in which the ratio of whorl to whorl remains, to all appearance, perfectly constant; others in which it gi^dually though only shghtly increases; and others again in which it slightly and gradually falls away. It is obvious that, among the manifold possibihties of growth, such conditions as these are very easily conceivable. It is much more remarkable that, among these shells, the relative velocities of growth in various dimensions should be as constant as they are than that there should be an occasional departure from perfect regularity. In these latter cases the logarithmic law of growth is only approximately true. The shell is no longer to be represented simply as a cone which has been rolled up, but as a cone which (while rolling up) had grown trumpet-shaped, or conversely whose mouth had narrowed in, and which in longitudinal section is a curvilinear instead of a rectihnear 788 THE EQUIANGULAR SPIEAL [CH. triangle. But all that has happened is that a new factor, usually of small or all but imperceptible magnitude^ has been introduced into the case ; so that the ratio, log r = 6 log a, is no longer con- stant but varies shghtly, and in accordance with some simple law. Some ' writers, such as Naumann* and Grabau, maintained that the moUuscan spiral was no true logarithmic spiral, but differed from it specifically, and they gave it the name of Conchospiral. They said that the logarithmic spiral originates in a mathematical point, while the molluscan shell starts with a Httle embryonic, shell, or central chamber (the "protoconch" of the conchologists), around which the spiral is subsequently wrapped. But this need not affect the logarithmic law of the shell as a whole ; indeed we have already allowed for it by writing our equation in the form r = ma^. And Grabauf, while he clung to Naumann's conchospiral against Moseley's logarithmic spiral, confessed that they were so much ahke that ordinary measurements would seldom shew a difference between them. There would seem, by the way, to be considerable confusion in the books with regard to the so-called "protoconch." In many cases it is a definite structure, of simple form, representing the more or less globular embyyonic shell before it began to elongate into its conical or spiral form. But in many cases what is described as the "protoconch" is merely an empty space in the middle of the spiral coil, resulting from the fact that the actual spiral .shell must have some magnitude to begin with, and that we cannot follow it down to its vanishing point in infinity. For instance, in the accompanying figure, ■ the large space a is styled the protoconch, but it is the little bulbous or hemispherical chamber within it, at the end of the spire, which is the real beginning of the tubular shell. The form and mag- nitude of the spa^ce a are determined by the "angle of retardation," or ratio of rate of growth between the inner and outer curves of the spiral shell. They Fig. 376. * C. F. Naumann, Beitrag zur Konchyliometrie, Poggend. Ann. l, p. 223, 1840; Ueber die Spiralen der Ammoniten, ibid, li, p. 245, 1840; ibid, liv, p. 541, 1845; etc. (See also p. 755.) Cf. also Lehmann, Die von Seyfriedsche Konchyliensammlung und das Windungsgesetz von einigen Planorben, Constanz, 1855. f A. H. Grabau, Ueber die Naumannsche Conchospirale, und ihre Bedeutung fiir die Conchyliometrie, Inauguraldiss., Leipzig, 1872; Ueber die Spiralen der Conchylien, etc., Leipzig Progr. No. 502, 1880; cf. Sb. naturf. Gesellsch. Leipzig, 1881, pp. 23-32. XI] ITS MATHEMATICAL PROPERTIES 789 are independent of the shape and size of the embryo, and depend only (as we shall see better presently) on the direction and relative rate of growth of the double contour of the shell*. rd0 Now that we have dealt, m a. general way, with some of the more obvious properties of the equiangular or logarithmic spiral, let us consider certain of them a httle more particularly, keeping in view as our chief object of study the range of variation of the molluscan shell. There is yet another equation to the logarithmic spiral, very commonly employed, and without the help of which we cannot get far. It is as follows : This follows directly from the fact that the angle a (the angle between the radius vector and the tangent to the curve) is constant. For then, tan a (= tan (f>) = rdd/dr; therefore dr/r = dO cot a, and, integrating, log r = ^ cot a. or Fig. 377. It is easy to see (we might indeed have noted it before) that the logarithmic spiral is but a plotting in polar coordinates of increase by compound interest. For if A be the "amount" of £1 in one year (A = 1 + a, where a is the rate of interest), and PA the amount of P in one year, then the whole amount, M, in t years is M = PA^\ this, provided that interest is payable once a year. But, as we are taught by algebra, and as we have seen in our study of growth, this formula becomes Pe"* when the intervals of time between the payments of interest decrease without limit, that is to say, wh^n we may consider growth to be continuous. And this formula PeP-^ is precisely that of our logarithmic spiral, when we represent the time * J. F. Blake (cf. infra, p. 793) says of Naumann's formula: "By such a modification he hoped to bring the measurements of actual shells more into harmony with calculation. The errors of observation, however, are always greater than this change would correct — if founded on fact, which is doubtful; and all practical advantage is lost by the compUcation of the equations." 790 THE EQUIANGULAR SPIRAL [CH. by a vector angle d, and when for a, the particular rate of interest in the cas.e, we write cot a, the constant measure of growth of the particular spiral. As we have seen throughout our preliminary discussion, the two most important constants (or "specific Characters," as the naturalist would say) in an equiangular or logarithmic spiral are (1) the magni- tude of the angle of the spiral, or "constant angle" a, and (2) the rate of increase of the radius vector for any given angle of revolution, 6. But our two magnitudes, that of the constant angle and that of the ratio of the radii or breadths of whorl, are directly related to one another, so that we may determine either of them by measurement and calculate the other. Fig. 378. In any complete spiral, such as that, of Nautilus, it is (as we have seen) easy to measure any two radii (r), or the breadths in a radial direction of any two whorls* (If). We have then merely to apply the formula ^"+1 ^ g^cota W. or «+l ^ g^cota which we may simply write r = e^^°*", etc., when one radius or whorl is regarded, for the purpose of comparison, as equal to unity. Thus, in Fig. 378, OC/OE, or EF/BD, or DC/'EF, being in each case radii, or diameters, at right angles to one another, are - * all equal to e^"^" ". While in like manner, EO/OF, EG/FH, or GO/HO, all equal e^^o^a. ^nd BC/BA, or CO/OB = e2^cota^ XI] ITS MATHEMATICAL PROPERTIES 791 As soon, then, as we have prepared tables for these values, the determination of the constant angle a in a particular shell becomes a very simple matter. A complete table would be cumbrous, and it will be sufficient to deal with the simple case of the ratio between the breadths of adjacent, or immediately succeeding, whorls. Here we have r = e27rcota^ qp log/ = log e x 27r x cot a, from which we obtain the following figures*: The shape of a nautiloid spiral Ratio of breadth of each whorl to the next preceding Constant angl< r/1 a 11 89° 8' 1-25 87 58 1-5 86 18 2-0 83 42 2-5 81 42 30 80 5 3-5 78 43 40 77 34 4-5 76 32 50 75 38 100 69 53 200 64 31 500 58 5 1000 53 46 1000-0 42 17 10,000 34 19 100,000 28 37 1,000,000 24 28 10,000,000 21 18 100,000,000 18 50 1,000,000,000 16 52 We learn several interesting things from this short table. We see, in the first place, that where each whorl is about three times the breadth of its neighbour and predecessor, as is the case in Nautilus, the constant angle is in the neighbourhood of 80°; and hence also that, in all the ordinlary ammonitoid shells, and in all the t3rpically spiral shells of the gastropods f, the constant angle is also a large one, being very seldom less than 80°, and usually between 80° and 85°. In the next place, we see that with smaller * It is obvious that the ratios of opposite whorls, or of radii 180° apart, are represented by the square roots of these values ; and the ratios of whorls or radii 90° apart, by the square roots of these again. ■j" For the correction to be applied in the case of the helicoid, or "turbinate" shells, see p. 816. 792 THE EQUIANGULAR SPIRAL [ch. angles the apparent form of the spiral is greatly altered, and the very fact of its being a spiral soon ceases to be apparent (Figs. 379, 380). Suppose one whorl to be an inch in breadth, then, if the angle of the spiral were 80°, the next whorl would (as we have just seen) be about three inches broad; if it were 70°, the next whprl would be nearly ten inches, and if it were 60°, the next whorl would be nearly four feet broad. If the angle were 28°, the next whorl would be a mile and a half in breadth; and if it were 17°, the next would be spme 15,000 miles broad. In other words, the spiral shells of gentle curvature, or of small constailt angle, such as Dentalium or Cristellaria, are true equi- angular spirals, just as are those of Nautilus or Rotalia: from Fig. 379. Fig. 380. which they differ only in degree, in the magnitude of an angular constant. But this diminished magnitude of the angle causes the spiral to dilate with such immense rapidity that, so to speak, it never comes 'round; and so, in such a shell as Dentalium, we never see but a small portion of a single whorl. We might perhaps be inclined to suppose that, in such a shell as Dentalium^ the lack of a visible spiral convolution was only due to our seeing but a small portion of the curve, at a distance from the pole, and when, therefore, its curvature had already greatly diminished. That is to say we might suppose that, however small the angle a, and however rapidly the whorls accordingly increased, there would nevertheless be a manifest spiral convolution in the immediate neighbourhood of the pole, as the starting point of the curve. But it is easy to see that it is not so. It is not that there cease to be con- volutions of the, spiral round the pole when a is a small angle ; on the contrary, there are infinitely many, mathematically speaking. But as a diminishes, and cot a increases towards infinity, the ratio between the breadth of one whorl and the next increases very rapidly. Our taTble shews us that even when a is no less than 40°, and our shell stiU looks strongly curved, one whorl is a thousandt'i part of the breadth of the next, and a thousandfold that XI] ITS MATHEMATICAL PROPERTIES 793 of the one before ; we cannot expect to see either of them under the materialised conditions of the actual shell. Our shells of small constant angle and gentle curvature, such as Dentalium, are accordingly as much as we can ever expect to see of their respective spirals. The spiral whose constant angle is 45° is both a simple case and a mathematical curiosity; for, since the tangent of 45° is unity, we need merely write r = e^\ which is as much as to say that the natural logarithms of the radii give us, without more ado, the vector angles. In this spiral the ratio between the breadths of two consecutive whorls becomes r = e^^ ■= e2x3-i4i6_ Reducing this from Naperian to common logs, we have log r = 2-729; whicH tells us (by our tables) that the radius vector is multiphed about 535J times after a whole polar revolution; it is doubled after turning through a polar angle of less than 40°. Spirals of so low an angle as 45° are common enough in tooth and claw, but rare among molluscan shells; but one or two of the more strongly curved Dentaliums, like D. elejphantinum, come near the mark. It is not easy to determine the pole, nor to measure the constant angle, in forms like these. Let us return to the problem of how to ascertain, by direct measurement, the spiral angle of any particular shell. The method aL-eady employed is only applicable to complete spirals, that is to say to those in which the angle of the spiral is large, and further- more it is inapplicable to portions, or broken fragments, of a shell. In the case of the broken fragment, it is plain that the determination of the angle is not merely of theoretic interest; but may be of great practical use to the conchologist as the one and only way by which he may restore the outline of the missing portions. We have a con- siderable choice of methods, which have been summarised by, and are partly due to, a very careful student of the Cephalopoda, the late Rev. J. F. Blake*. (1) When an equiangular spiral rolls on a straight line, the pole traces another straight line at an ^ngle to the first equal to the complement of the constant angle of the spiral; for the contact point is the instantaneous centre of the rotational movement, and the line joining, it to the pole of the spiral is normal to the roulette path of that point. But the difficult^ of determining the pole * On the measurement of the curves formed by Cephalopods and other MoUusks, Phil. Mag. (5), vi, pp. 241-263, 1878. 794 THE EQUIANGULAR SPIRAL [ch. (which is indeed asymptotic) makfes this of Httle use as a method of determining the constant angle. It is, however, a beautiful property of the curve, and all the more interesting that Clerk Maxwell dis- covered it when he was a boy*. (2) The following method is useful and easy when we have a portion of a single whorl, such as to shew both its inner and its outer edge. A broken whorl of an Ammonite, a curved shell such as Dentalium, or a horn of similar form to the latter, will fall under this head. We have merely to draw a tangent, GEH, to the outer whorl at any point E; then draw to the inner whorl a tangent parallel to GEH, touching the curve in some point F. The straight hne joining the points of contact, EF, must evidently pass through the pole : and^ accordingly, the angle GEF is the angle required. In shells which bear longitudinal striae or other ornaments, any pair of these will suffice for our purpose, instead of thfe actual boundaries of the whorl. But it is obvious that this method will be apt to fail us when the angle a is very small; and when, consequently, the points E and F are very remote. (3) In shells (or horns) shewing rings or other transverse ornamentation, we may take it that these ornaments are set at Fig. 381. Fig. 382. An Ammonite, to shew corrugated surface- pattern. * Clerk Maxwell, On the theory of rolling curves. Trans. B.S.E. xvi, pp. 519- 540, 1849; Sci. Papers, i, pp. 4-29. XI] ITS MATHEMATICAL PROPERTIES 795 a constant angle to the spire, and therefore to the radii. The angle (6) between two of them, as AC, BD, is therefore equal to the angle 9 between the polar radii from A and B, or from C and D; and therefore BD/AC = e^cota^ which gives us the angle a in terms of known quantities. (4) If only the outer edge be available, we have the ordinary- geometrical problem — given an arc of an equiangular spiral, to find its pole and spiral angle. The methods we may employ depend (i) on determining directly the position of the pole, and (ii) on determining the radius of curvature. The first method is theoretically simple, but difficult in practice; for it requires great accuracy in determining th'fe points. Let AD, DB be two tangents drawn to the curve. Then a circle drawn through the points A, B, D will pass through the pole 0, since the angles OAT>, QBE (the supplement of OBD) are equal. The point 0 may be deter- mined by the intersection of two such circles; and the angle DBO is then the angle, a, required. Or we may determine graphically, ' p- ^^ at two points, the radii of curvature P1P2. Then, if s be the length of the arc between them (which may be determined with fair accuracy by rolling the margin of the shell along a ruler), cot a = (pi — P2)is. The following method*, given by Blake, will save actual determination of the radii of curvature. Measure along a tangent to the curve the distance, AC, at which a certain small offset, CD, is made by the curve; and from another point B, measure the distance at which the curve makes an equal offset. Then, calling the offset jLt; the arc AB, s; and AC, BE, respectively x^, x^, we have a; 2^_ 2 Pj_ = ^ , approximately, /^ and cota = 2ixs Of all these methods by which the mathematical constants, or specific characters, of a given spiral shell ipay be determined, the * For an example of this method, see Blake, loc. cit. p. 251. 796 THE EQUIANGULAR SPIRAL [ch. only one of which much use has been made is that which Moseley first employed, namely, the simple method of determining the relative breadths of the whorl at distances separated by some convenient vectorial angle such as 90°, 180°, or 360°. Very elaborate measurements of a number of Ammonites have been made by Naumann*, by Grabau, by Sandberger f, and by Miiller, among which we may choose a couple of cases for considera- tion f. In the following table I have taken a portion of Grabau's Ammonites intuslabiatus Ratio of breadth of Ith of whorls successive whorls The angle (a) J0° apart) (360° apart) as calculated 0-30 mm. — — — 0-30 1-333 87° 23' 0-40 1-500 86 19 0-45 1-500 86 19 0-60 1-444 86 39 0-65 1-417 86 49 0-85 1-692 85 13 MO 1-588 85 47 1-35 1-545 86 2 1-70 1-630 85 33 2-20 1-441 86 40 2-45 1-432 86 43 315 1-735 85 0 4-25 1-683 85 16 5-30 1-482 86 25 6-30 1-519 86 12 8-05 1-635 85 32 10-30 1-416 86 50 11-40 1-252 87 57 12-90 Mean — — 86° 15' * C. F. Naumann, Ueber die Spiralen von Conchylien, Ahh. k. sacks. Ges. 1846, pp. 153-196; Ueber die cyclocentrische Conchospirale u. iiber das Windungsgesetz von Planorhis corneus, ibid, i, pp. 171-195, 1849; Spirale von Nautilus u. Ammonites galeatus, Ber. k. sacks. Ges. ir, p. 26, 1848; Spirale von Amm. Ramsaueri, ibid, xvi, p. 21, 1864. Oken, reviewing Naumann's work (in Isis, 1847, p. 867) foretold how some day the naturalist and the mathematician would each learn of the other: "Um die Sache zu Vollendung zu bringen wird der Mathematiker Zoolog und Physiolog, und dieae Mathematiker werden miissen." I G. Sandberger, Clymenia subnautilina, Jakresber. d. Ver. f. Naturk. im Herzogtk. Nassau, 1855, p. 127; Spiralen des Ammonites Amaltkeus, A. Gaytani und Goniatites intumescens, Ztschr. d. d. Geolog. Gesellsch. x, pp. 446-^49, 1858. Also Miiller, Beitrag zur Konchyliometrie, Poggend. Ann. lxxxvi, p. 533, 1850; ibid, xc, p. 323, 1853. These two authors upheld the logarithmic law against Naumann and Grabau. J See also Chr. Petersen, Das Quotientengesetz, eine biologisch-statistische Unter- suckung, 119 pp., Copenhagen, 1921; E. Sporn, Ueber die Gesetzmassigkeit im Baue der Muschelgehaiiser, Arch. f. Entw. Meek, cviii, pp. 228-242, 1926. XI] OF SHELLS GENERALLY 797 determinations of the breadth of the whorls in Ammonites (Arcestes) intuslabiatus; these measurements Grabau gives for every 45° of arc, but I have only set forth successive whorls measured along one diameter on both sides of the pole. The ratio between alternate measurements is therefore the same ratio as Moseley adopted, namely the ratio of breadth between contiguous whorls along a radius vector. I have then added to these observed values the corresponding calculated values of the angle a, as obtained from our usual formula. There is considerable irregularity in the ratios derived from these measurements, but it will be seen that this irregularity only implies a variation of the angle of the spiral between about 85° and 87°; and the values fluctuate pretty regularly about the mean, which is 86° 15'. Considering the difficulty of measuring the whorls, especially towards the centre, and in particular the difficulty of determining with precise accuracy the position of the pole, it is clear that in such a case as this we are not justified in asserting that the law of the equiangular spiral is departed from. Ammonites tornatus Ratio of breadth of The spiral eadth of whorls successive whorls angle (a) as (180' apart) (360" apart) calculated 0-25 mm. — -^ — . 0-30 1-400 86° 56' 0-35 1-667 85 21 0-50 2-000 83 42 0-70 2-000 83 42 1-00 2-000 83 42 1-40 2-100 83 16 2-10 2-lW 82 56 305 2-238 82 42 4-70 2-492 81 44 7-60 2-574 81 27 1210 2-546 81 33 19-35 — Mean 2-11 83° 22' In some cases, however, it is undoubtedly departed from. Here for instance is another table from Grabau, shewing the corre- sponding ratios in an Ammonite of the group of Arcestes tornatus. In this case we see a distinct tendency of the ratios to increase as we pass from the centre of the coil outwards, and consequently for the values of the angle a to diminish. The case is comparable to 7.98 THE EQUIANGULAR SPIRAL [ch. that of a cone with shghtly curving sides : in which, that is to say, there is a sHght acceleration of growth in a transverse as compared with the longitudinal direction. In a tubular spiral, whether plane or hehcoid, the consecutive whorls may either be (1) isolated and remote from one another; or (2) they may precisely meet, so that the outer border of one and the inner border of the next just coincide; or (3) they may overlap, the vector plane of each outer whorl cutting that of its immediate predecessor or predecessors. Looking, as we have done, upon the spiral shell as being essentially a cone roUed up*, it is plain that, for a given spiral angle, intersection or non-intersection of the successive whorls will depend upon the apical angle of the original cone. For the wider the cone, the more will its inner border tend to encroach on the preceding whorl. But it is also plain that the greater the apical angle of the cone, and the broader, consequently, the cone itself, the greater difference will there be between the total lengths of its inner and outer borders. And, since the inner and outer borders are describing precisely the same spiral about the pole, we may consider the inner border as being retarded in growth as compared with the outer, and as being always identical with a smaller and earUer part of the latter. If A be the ratio of growth between the outer and the inner curve, then, the outer curve being represented by r = ae^ootcc^ the equation to the inner one will >^e / = aAe^cota^ or r' = ae^^-y^^^^"", * To speak of a cone "rolling up," and becoming a nautiloid spiral by doing so, is a rough and non-mathematical description; nor is it easy to see how a cone of wide angle could roll up, and yet remain a cone. But if (i) the centre of a sphere move along a straight line and its radius keep proportional to the distance the centre has moved, the sphere generates as its envelope a circular cone of which the straight line is the axis; and so, similarly, if (ii) the centre of a sphere move along an equiangular spiral and its radius keep proportional to the arc-distance along the spiral back to the pole, the sphere generates as its envelope a self-similar shell-surface, or nautiloid spiral. XI] OF SHELLS GENERALLY 799 and y may then be called the angle of retardation, to which the inner curve is subject by virtue of its slower rate of growth. Dispensing with mathematical formulae, the several conditions may be illustrated as follows : In the diagrams (Fig. 385), OF-^F^F^, etc. represents a radius, on which P^, F^, Pg are the points attained by the outer border of the tubular shell after as many entire consecutive revolutions. And Pi', P2', P3' are the points similarly intersected by the inner border ; OFjOF' being always = A, which is the ratio of growth, or ' ' cutting-down factor. ' ' Then, obviously, ( 1 ) when OP^ is less than Fig. 385. OP2' the whorls will be separated by an interspace (a); (2) when OF^ = OP2' they will be in contact (6), and (3) when OF^ is greater than OF^ there will be a greater or less extent of overlapping, that is to say of concealment of the surfaces of the earher by the later whorls (c). And as a further case (4), it is plain that if A be very large, that is to say if OF-^ be greater, not only than OF^ but also than OP3', OP4', etc., we shall have complete, or all but complete, concealment by the last formed whorl of the whole of its predecessors. This latter condition is completely attained in Nautilus pompilius, and approached, though not quite attained, in N. umbilicatus] and the difference between these two forms, or "species," is constituted accordingly by a difference in the value of A. (5) There is also a final case, not easily distinguishable 800 THE EQUIANGULAR SPIRAL [ch. externally from (4), where P' lies on the opposite side of the radius vector to P, and is therefore imaginary. This final condition is exhibited in Argonauta. The Hmiting values of A are easily ascertained. In Fig. 386 we have portions of two successive whorls, whose corre- sponding points on the same radius vector (as R and R) are, therefore, ^^* ' at a distance apart corresponding to 277-. Let r and r' refer to the inner, and R, R' to the outer sides of the two whorls. Then, if we consider it follows that R' = ae<^+2;r) cot a^ and r' = Aae^^+^^^^^o*^ = j^g(^f27r-7)cota_ Now in the three cases (a, b, c) represented in Fig. 385, it is plain that r' = R, respectively. That is to say, Xae^0+2n)COta j ae^cota^ and Ae^^^ota | i^ The case in which Ae^'^^^*'^ = 1, or — log A = 277- cot a log e, is the case represented in Fig. 385, b: that is to say, the particular case, for each value of a, where the consecutive whorls just touch, without interspace or overlap. For such cases, then, we may tabulate the values of A as follows: Cdiistaiit angle a Uatio (A) of rate of growth of inner border of tube. of siiiral as coiiiji ared with that of tlie outer l)order 89° 0-896 88 0-803 87 0-720 86 0-645 85 0-577 80 0-330 75 0-234 70 0-1016 65 0-0534 We see, accordingly, that in plane spirals whose constant angle lies, say, between 65° and 70°, we can only obtain contact between XI] OF SHELLS GENERALLY 801 consecutive whorls if the rate of growth of the inner border of the tube be a small fraction — a tenth or a twentieth — of that of the outer border. In spirals whose con- stant angle is 80°, contact is attained when the respective rates of growth are, approximately, as 3 to 1; while in spirals of constant angle from about 85° to 89°,contact is attained when the rates of growth are in the ratio of from about f to j%. If on the other hand we have, for any given value of a, a value of A greater or less than the value given in the above table, then we have, respectively, the conditions of separa- tion or of overlap which are exempUfied in Fig. 385, a and c. And, just as we have constructed this table for the particular case of simple contact, so we could construct similar tables for various degrees of separation or of overlap. For instance, a case which admits of simple solution is that in which the interspace between the whorls is everywhere a mean pro- portional between the breadths of the whorls themselves (Fig. 387). In this case, let us call OA = R, OC = R^, and OB = r. We then have Fig. 387. i^2=OC = ae(^+2'^>^^*^ And r2= (1/A)2 . €2^«0t«, whence, equating. 1/A = e TTCota * It has been pointed out to me that it does not follow at once and obviously that, because the interspace AB i& & mean proportional between the breadths of the adjacent whorls, therefore the whole distance OB is a mean proportional between OA and OC. This is a corollary which requires to be proved; but the proof is easy.. 802 THE EQUIANGULAR SPIRAL [ch. The corresponding values of A are as follows : Ratio (\) of rates of growth of outer and inner border, such as to produce a spiral with interspaces between the wliorls, the breadth of which intei-spaces is a mean proportional between the Constant angle (a) breadths of the whorls themselves 90° 1-00 (imaginary) 89 0-95 88 0-89 87 0-85 86 0-81 85 0-76 80 0-57 75 0-43 70 0-32 65 0-23 60 018 55 013 50 0090 45 0-063 40 0042 35 0026 30 0016 . As regards the angle of retardation, y, in the formula / = Ae^^o*«, or / = e(^-^)cota^ and in the case r' = e(2^-y)cota^ ^j. _ log A - (277 - y) cot a, it is evident that when y = 27t, that will mean that A = 1. In other words, the outer and inner borders of the tube are identical, and the tube is constituted by one continuous hne. When A is a very small fraction, that is to say when the rates of growth of the two borders of the tube are very diverse, then y wfll tend towards infinity — tend that is to say towards a condition in which the inner border of the tube never grows at all. This condition is not infrequently approached in nature. I take it that Cyfraea is such a case. But the nearly parallel-sided cone of Dentalium, or the widely separated whorls of Lituites, are cases where A nearly approaches unity in the one case, and is still large in the other, y being correspondingly small ; while we can easily find cases where y is very large, and A is a small fraction, for instance in Haliotis, in Calyptraea, or in Gryphaea. For the purposes of the morphologist, thoji, the main result of this last general investigation is to shew that all the various types of ''open" and "closed" spirals, all the various degrees of separation XI] OF SHELLS GENERALLY 803 or overlap of the successive whorls, are simply the outward ex- pression of a varying ratio in the rate of growth of the outer as compared with the inner border of the "tubular shell. The foregoing problem of contact, or intersection, of successive whorls is a very simple one in the case of the discoid shell but a more complex one in the turbinate. For in the discoid shell contact will evidently take place when the retardation of the inner as compared with the outer whorl is just 360°, and the shape of the whorls need not be considered. As the angle of retardation diminishes from 360°, the whorls stand further and further apart' in an open coil; as it increases beyond 360°, they overlap more and more ; and when the angle of retardation is infinite, that is to say when the true inner edge of the whorl does not grow at all, then the shell is said to be completely involute. Of this latter condition we have a striking example in Argonauta, and one a Httle more obscure in Nautilus pompilius. In the turbinate shell the problem of contact is twofold, for v.\, have to deal with the possibihties of contact on the same side of the axis (which is what we have dealt with in the discoid) and also with the new possibiHty of contact or mtersection on the opposite side; it is this latter case which will .determine the presence or absence of an open umbilicus. It is further obvious that, in the case of the turbinate, the question of contact or no contact will depend on the shape of the generating curve ; and if we take the simple case where this generating curve may be considered as an eUipse, then contact will be found to depend on the angle which the major axis of this elhpse makes with the axis of the shell. The question becomes a compHcated one, and the student will find it treated in Blake's paper already referred to. When one whorl overlaps another, so that the generating curve cuts its predecessor (at a distance of 27r) on the same radius vector, the locus of intersection will follow^ a spiral fine upon the shell, whjch is called the "suture" by conchologists. It is one of that ensemble of spiral lines in space of which, as we have seen, the whole shell may be conceived to be constituted ; and we might call it a "contact-spiral," or "spiral of intersection." In discoid shells, such as an Ammonite or a Planorbis, or in Nautilus umbilicatus, there are obviously two such contact-spirals, one on each side of 804 THE EQUIANGULAR SPIRAL [ch. the shell, that is to say one on each side of a plane perpendicular to the axis. In turbinate shells such a condition is also possible, but is somewhat rare. We have it for instance in Solarium per- spectivum, where the one contact-spiral is visible on the exterior of Fig. 388. Solarium perspectivum. the shell, and the other lies internally, winding round the open cone of the umbihcus*; but this second contact-spiral is usually imaginary, or concealed within the whorls of the turbinated shell. Fig. 389. Haliotis tuberculata L. ; the ormer, or ear shell. Fig. 390. Scalaria pretiosa L. ; the wentletrap. From Cooke's Spirals. Again, in Haliotis, one of the contact-spirals is non-existent, because of the extreme obUquity of the plane of the generating curve. In * A beautiful construction: stupendum Naturae artijicium, Linnaeus. XI] OF SHELLS GENERALLY 805 Scalaria pretiosa and in Spirula* there is no contact-spiral, because the growth of the generating curve has been too slow in comparison with the vector rotation of its plane. In Argonauta and in Cypraea there is no contact-spiral, because the growth of the generating curve has been too quick. Nor, of course, is there any contact- spiral in Patella or in Dentalium, because the angle a is too small Fig. 392. Turhinella napus Lam.; an Indian chank- shell. From Chenu. Fig. 391. Thatcheria mirabilis Angas; from a radiograph by Dr A. Miiller. ever to give us a complete revolution of the spire. Thatcheria mirabilis is a peculiar and beautiful shell, in which the outhne of the Up is sharply triangular, instead of being a smooth curve : with the result that the apex of the triangle forms a conspicuous "gene- rating spiral", which winds round the shell and is more conspicuous than the suture itself. In the great majority of hehcoid or turbinate shells the innermost * "It [Spirula] is curved so as its roundness is kept, and the Parts do not touch one another": R. Hooke, Posthumoics Works, 1745, p. 284. 806 THE EQUIANGULAR SPIRAL [ch. or axial portions of the whorls tend to form a soHd axis or "columella"; and to this is attached the columellar muscle which on the one hand withdraws the animal within its shell, and on the other hand provides the controlhng force or trammel, by which (in the gastropod) the growing shell is kept in its spiral course. This muscle is apt to leave a winding groove upon the columella (Fig. 373) ; now and then the muscle is spht into strands or bundles, and then it leaves parallel grooves with ridges or pleats between, and the number of these folds or pleats may vary with the species, as in the Volutes, or even with race or locahty. Thus, among the curiosities of conchology, the chank-shells on the Trincomah coast have four columellar folds or ridges; but all those from Tranquebar, just north of Adam's Bridge, have only three (Fig. 392)*. The various forms of straight or spiral shells among the Cephalo- pods, which we have seen to be capable of complete definition by the help of elementary mathematics, have received a very com- phcated descriptive nomenclature from the palaeontologists. For instance, the straight cones are spoken of as orthoceracones or bactriticories, the loosely coiled forms as gyroceracones or mimo- ceracones, the more closely coiled shells, in which one whorl overlaps the other, as nautilicones or ammoniticones, and so forth. In such a series of forms the palaeontologist sees undoubted and unquestioned evidence of ancestral descent. For instance We read in Zittel's Palaeontologyf : " The bactriticone obviously represents the primitive or primary radical of the Ammonoidea, and the mimoceracone the next or secondary radical of this order " ; while precisely the opposite conclusion was drawn by Owen, who supposed that the straight chambered shells of such fossil Cephalopods as Orthoceras had been produced by the gradual unwinding of a coiled nautiloid shell {. The mathematical study of the forms of shells lends no support to these * Cf. R. Winckworth, Proc. Malacol Soc. xxiii, p. 345, 1939. t English edition, 1900, p. 537. The chapter is revised by Professor Alpheus Hyatt, to whom the nomenclature is largely due. For a more copious terminology, see Hyatt, Phytogeny of an Acquired Characteristic, 1894, p. 422 seq. Cf. also L. F. Spath, The evolution of the Cephalopoda, Biol. Reviews, viii, pp. 418-462, 1933 . X Th is latter conclusion is adopted by Willey, Zoological Results, 1902, p. 747. Cf. also Graham Kerr, on Spirula: Dana Reports, No. 8, Copenhagen, 1931. XI] OF VARIOUS CEPHALOPODS 807 or any stichlihe phylogenetic hypotheses*. If we have two shells in which the constant angle of the spire be respectively 80° and 60°, that fact in itself does not at all justify an assertion that the one is more- primitive, more ancient, or more "ancestral" than the other. Nor, if we find a third in which the angle happens to be 70°, does that fact entitle us to say that this shell is intermediate between the other two, in time, or in blood relationship, or in any other sense whatsoever save only the strictly formal and mathematical one. For it is evident that, though these particular arithmetical constants manifest themselves in visible and recog- nisable differences of form, yet they are not necessarily more deep-seated or significant than are those which manifest themselves only in difference of magnitude; and the student of phylogeny scarcely ventures to draw conclusions as to the relative antiquity of two allied organisms on the ground that one happens to be bigger or less, or longer or shorter, than the other. At the same time, while.it is obviously unsafe to rest conclusions upon such features as these, unless they be strongly supported and corroborated in other ways — for the simple reason that there is unlimited room for coincidence, or separate and independent attainment of this or that magnitude or numerical ratio — yet on the other hand it is certain that, in particular cases, the evolution of a race has actually involved gradual increase or decrease in some one or more numerical factors, magnitude itself included — that is to say increase or decrease in some one or more of the actual and relative velocities of growth. When we do meet with a clear and unmistakable series of such progressive magnitudes or ratios, manifesting themselves in a progressive series of ''aUied" forms, then we have the phenomenon of "orthogenesis."' For orthogenesis is simply that phenomenon of continuous Unes or series of form (and also of functional or physiological capacity), * Phylogenetic speculation, fifty years ago the chief preoccupation of the biologist, has had its caustic critics. Cf. {int. al.) Rhumbler, in Arch. f. Entw. Mech. VII, p. 104, 1898: " Phylogenetische Speculationen . . , werden immer auf Anklang bei den Fachgenossen rechnen dlirfen, sofern nicht ein anderer Fachgenosse auf demselben Gebiet rait gleicher Kenntniss der Dinge und mit gleicher Scharfainn zufallig zu einer anderen Theorie gekommen ist. . . .Die Richtigkeit 'guter' phylo- genetischer Schliisse lasst sich im schlimmsten Falle anzweifeln, aber direkt widerlegen lasst sich in der Regel«iicht." 808 THE EQUIANGULAR SPIRAL [ch. which was the foundation of the Theory of Evolution, aUke to Lamarck and to Darwin and Wallace; and which we see to exist whatever be our ideas of the "origin of species," or of the nature and origin of "functional adaptations." And to my mind, the mathematical (as distinguished from the purely physical) study of morphology bids fair to help us to recognise this phenomenon of orthogenesis in many cases where it is not at once patent to the eye; and, on the other hand, to warn us in many other cases that even strong and apparently complex resemblances in form may be capable of arising independently, and may sometimes signify no more than the equally accidental numerical coincidences which are manifested in identity of length or weight or any other simple magnitudes. I have already referred to the fact that, while in general a very great and remarkable regularity of form is characteristic of the molluscan shell, yet that complete regularity is apt to be departed from. We have clear cases of such a departure in Pujpa, Clausilia and various Bulimi, where the spire is not conical, but its -sides are curved and narrow in. The following measurements of three specimens of Clausilia shew a gradual change in the ratio to one another of successive whorls, or in other words a marked departure from the logarithmic law: Clausilia lamellosa. (From Chr. Petersen*.) Width of successive Ratios, or ' 'quotients' 'of whorls (mm. ) successive A whorls T~ II III I II III Mean a 2-42 2-51 2-49 ajb 1-43 1-45 1-42 1-44 b 1-69 1-72 1-75 b/c 1-36 1-33 1-31 1-33 c 1-24 1-30 1-33 cjd 1-21 1-29 1-23 1-24 d e 102 0-83 100 0-83 1-08 0-86 die 1-22 1-20 1-26 1-23 In many ammonites, where the helicoid factor does not enter into the case, we have a clear illustration of how gradual and marked * From Chr. Petersen, Das Quotientengesetz, p. 36. After making a careful statistical study of 1000 Clausilias, Peterson found the following mean ratios of the successive whorls, aJb, b/c, etc.: 1-37, 1-33, 1-27, 1-24, 1-22, 119. XI] OF VARIOUS CEPHALOPODS 809 changes in the spiral angle may be detected even in ammonites which present nothing abnormal to the eye. But let us suppose that the spiral angle increases somewhat rapidly; we shall then get a spiral with gradually narrowing whorls, which condition is characteristic of Oekotraustes, a subgenus of Ammonites. If on the other hand, the angle a gradually diminishes, and even falls away to zero, we shall have the spiral curve opening out, as it does in Scaphites, Ancyloceras Fig. 393. An ammonitoid shell (Macroscaphites) to shew change of curvature. and Lituites, until the spiral coil is replaced by a spiral curye so gentle as to seem all but straight. Lastly, there are a few cases, such as BelleropJion expansus and some Goniatites, where the outer spiral does not perceptibly change, but the whorls become more "embracing" or the whole shell more involute. Here it is the angle of retardation, the ratio of growth between the outer and inner parts of the whorl, which undergoes a gradual change. In order to understand the relation of a close-coiled shell to its straighter congeners, to compare (for example) an Ammonite with an Orthoceras, it is necessary to estimate the length of the right cone which has, so to speak, been coiled up into the spiral shell. Our problem is, to find the length of a plane equiangular spiral, in terms of the radius and the constant angle a. Then, if OP be a radius vector, OQ a line of reference perpendicular to OP, and PQ a tangent to the curve, PQ, or sec a, is equal in length to the spiral arc OP. In other words, the arc measured from the pole is equal to the polar tangent*. And this is practically obvious: for * Descartes made this discovery, and records it in a letter to Mersenne, 1638. The equiangular spiral was thus the first transcendental curve to be "rectified." 810 THE EQUIANGULAR SPIRAL [ch. PP'/PR' = dsjdr = sec a, and therefore sec a = sjr, or the ratio of arc to radius vector. Fig. 394. Accordingly, the ratio of I, the total length, to r, the radius vector up to which the total length is to be measured, is expressed by a simple table of secants ; as follows : a Hr 5° 1004 10 1015 20 1064 30 1165 40 1-305 50 1-56 60 20 70 2-9 75 3-9 80 5-8 85 11-5 86 14-3 a llT 87° 191 88 28-7 89 57-3 89° 10' 68-8 20 85-9 30 114-6 40 171-9 50 343-8 55 687-5 59 3437-7 90 Infinite Putting the same table inversely, so as to shew the total length in terms of the radius, we have as follows : Total length (in terms of the radius) Constant angl< 2 60° 3 70 3r 4 75 32 5 78 28 10 84 16 20 87 8 30 88 6 40 88 34 50 88 51 100 89 26 1000 89 56' 36" 10,000 89 59 30 XI] OF VARIOUS CEPHALOPODS 811 Accordingly, we see that (1), when the constant angle of the spiral is small, the shell (or for that matter the tooth, or horn or claw) is scarcely to be distinguished from a straight cone or cylinder; and this remains pretty much the case for a considerable increase of angle, say from 0° to 20° or more; (2) for a considerably greater increase of the constant angle, say to 50° or rnore, the shell would still only have the appearance of a gentle curve; (3) the charac- teristic close coils of the Nautilus or Ammonite would be typically represented only when the constant angle lies within a few degrees on either side of about 80°. The coiled up spiral of a Nautilus, with a constant angle of about 80°, is about six times the length of its radius vector, or rather more than three times its own diameter ; while that of an Ammonite, with a constant angle of, say, from 85° to 88°, is from about six to fifteen times as long as its own diameter. And (4) as we approach an angle of 90° (at which point the spiral vanishes in a circle), the length of the coil increases with enormous rapidity. Our spiral would soon assume the appearance of the close coils of a Nummuhte, and the successive increments of breadth in the successive whorls would become inappreciable to the eye. The geometrical form of the shell involves many other beautiful properties, of great interest to the mathematician but which it is not possible to reduce to such simple expressions as we have been content to use. For instance, we may obtain an equation which shall express completely the surface of any shell, in terms of polar or of rectangular coordinates (as has been done by Moseley and by Blake), or in Hamiltonian vector notation*. It is likewise pos- sible (though of little interest to the naturalist) to determine the area of a conchoid al surface or the volume of a conchoidal solid, and to find the centre of gravity of either surface or solid f. And Blake has further shewn, with considerable elaboration, how we may deal with the symmetrical distortion due to pressure which fossil shells are often found to have undergone, and how we may re- constitute by calculation their original undistorted form — a problem which, were the available methods only a Kttle easier, would be * Cf. H. W. L. Hime's Outlines of Quaternions, 1894, pp. 171-173. t See Moseley, op. cit. p. 361 seq. Also, for more complete and elaborate treat- ment, Haton de la Goupilliere, op. cit. 1908, pp. 5-46, 69-204. 812 THE EQUIANGULAR SPIRAL [ch very helpful to the palaeontologist; for, as Blake himself has shewn, it is easy to mistake a symmetrically distorted specimen of (for instance) an Ammonite for a new and distinct species of the same genus. But it is evident that to deal fully with the mathematical problems contained in, or suggested by, the spiral shell, would require a whole treatise, rather than a single chapter of this elementary book. Let us then, leaving mathematics aside, attempt to summarise, and perhaps to extend, what has been said about the general possibilities of form in this class of organisms. The univalve shell: a summary The surface of any shell, whether discoid or turbinate, may be imagined to be generated by the revolution about a fixed axis of a closed curve, which, remaining always geometrically similar to itself, increases its dimensions continually: and, since the scale of the figure increases in geometrical progression while the angle of rotation increases in arithmetical, and the centre of similitude remains fixed, the curve traced in space by corresponding points in the generating curve is, in all such cases, an equiangular spiral. In discoid shells, the generating figure revolves in a plane perpendicular to the axis, as in the Nautilus, the Argonaut and the Ammonite. In turbinate shells, it follows a skew path with respect to the axis of revolution, and the curve in space generated by any given point makes a constant angle to the axis of the enveloping cone, and partakes, therefore, of the character of a helix, as well as of a logarithmic spiral.; it may be strictly entitled a helico-spiral. Such turbinate or helico- spiral shells include the snail, the periwinkle and all the common typical Gastropods. When the envelope of the shell is a right cone — and it is seldom far from being so — then our helico-spiral is a loxodromic curve, and is obviously identical with a projection, parallel with the axis, of the logarithmic spiral of the base. As this spiral cuts all radii at a constant angle, so its orthogonal projection on the surface intersects all generatrices, and consequently all parallel circles, under a constant angle* this being the definition of a loxodromic curve on a surface of revolution. Guido Grandi describes this curve for the first time in a letter to Ceva, printed at the end of his Demonstratio theorematum Hugenianorum circa. . .logarithmicam lineam, 1701 *. * See R. C. Archibald, op. cit. 1918. Ohvier discussed it again {Rev. de geom. descriptive, 1843) calling it a "conical equiangular" or "conical logarithmic" XI] OF VARIOUS UNIVALVES 813 The generating figure may be taken as any section of' the shell, whether parallel, normal, or otherwise incHned to the axis. It is very commonly assumed to be identical with the mouth of the shell; in which case it is sometimes a plane curve of simple form ; in other and more numerous cases, it becomes compHcated in form and its boun- daries do not he in one plane: but in such cases as these we may replace it by its "trace," on a plane at some definite angle to the direction of growth, for instance by its form as it appears in a section through the axis of the helicoid shell. The generating curve is of very various shapes. It is circular in Scalaria or Cyclostoma, and in Spirula ; it may be considered as a segment of a circle in Natica or in Planorbis. It is triangular in Conus or Thatcheria, and rhomboidal in Solarium or Potamides. It is very commonly more or less elliptical : the long axis of the ellipse being parallel to the axis of the shell in Oliva and Cypraea; all but perpendicular to it in many Trochi ; and oblique to it in many well-marked cases,^such as Stomatella, Lamellaria, Sigaretus halio- toides (Fig. 396) and Haliotis. In Nautilus pomjpilius it is approximately a semi-ellipse, Fig. 395. Section of a spiral and in A^. umbilicatus rather more than a ^^^^^^> Triton corrugatus Lam. jrom Woodward. semi-ellipse, the lo'ng axis lying in both cases perpendicular to the axis of the shell*. Its form is seldom open to easy mathematical expression, save when it is an actual circle or spiral. Paul Serret (Th. nouv. . .des lignes a double courbure, 1860, p. 101) called it ""helice cylindroconique'" ; Haton de la Goupilliere calls it a '' conhelice.'" It has also been studied by {int. at.) Tissot, Nouv. ann. de mathem. 1852; G. Pirondini, Mathesis, xix, pp. 153-8, 1899; etc. * In Nautilus, the "hood" has somewhat dififerent dimensions in the two sexes, and these differences are impressed upon the shell, that is to say upon its "generating curve." The latter constitutes a somewhat broader ellipse in the male than in the female. But this difference is not to be detected in the young; in other words, the form of the generating cu'rve perceptibly alters with advancing age. Somewhat similar differences in the shells of Ammonites were long ago suspected, by d'Orbigny, to be due to sexual differences. (Cf. Willey, Natural Science, vi, p. 411, 1895; Zoological Results, 1902, p. 742.) 814 THE EQUIANGULAR SPIRAL [ch. ellipse; but an exception to this rule may be found in certain Ammonites, forming the group " Cordati," where (as Blake points out) the curve is very nearly represented by a cardioid, whose equation is r = a (1 + cos 6). When the generating curves of successive whorls cut one another, the hne of intersection forms the conspicuous heHco-spiral or loxodromic curve called the suture by conchologists. The generating curve may grow slowly or quickly; its growth- factor is very slow in Dentalium or Turritella, very rapid in Nerita, or Pileopsis, or Haliotis or the Limpet. It may contain the axis in its plane, as in Nautilus ; it may be parallel to the axis, as in the majority of Gastropods; or it may be inclined to the axis, as it is in a very marked degree in Haliotis, In fact, in Haliotis the generating B A Fig. 396. A, Lanydlaria perspicua; B, Sigaretus haliotoides. After Woodward. curve is so oblique to the axis of the shell that the latter appears to grow by additions to one margin only (cf. Fig. 362), as in the case of the opercula of Turbo and Nerita referred to on p. 775; and this is what Moseley supposed it to do. The general appearance of the entire shell is determined (apart from the form of its generating curve) by the magnitude of three angles; and these in turn are determined, as has been sufficiently explained, by the ratios of certain velocities of growth. These angles are (1) the constant angle of the equiangular spiral (a); (2) in turbinate shells, the enveloping angle of the cone, or (taking half that angle) the angle (p) which a tangent to the whorls makes with the axis of the shell; and (3) an angle called the "angle of retarda- tion" (y), which expresses the retardation in growth of the inner as compared with the outer part of each whorl, and therefore measures the extent to which one whorl overlaps, or the extent to which it is separated from, another. XI] OF VARIOUS UNIVALVES 815 The spiral angle (a) is very small in a limpet, where it is usually- taken as =0°; but it is evidently of a significant amount, though obscured by the shortness of the tubular shell. In Dentalium it is still small, but sufficient to give the appearance of a regular curve; it amounts here probably to about 30° to 40°. In Haliotis it is from about 70° to 75°; in Nautilus about 80°; and it lies between 80° and 85° or even more, in the majority of Gastropods*. The case of Fissurella is curious. Here we have, apparently, a conical shell with no trace of spiral curvature, or (in other words) with a spiral angle which approximates to 0°; but in the minute embryonic shell (as in that of the limpet) a spiral convolution is distinctly to be seen. It would seem, then, that what we have ^to do with here is an unusually large growth-factor in the generating curve, causing the shell to dilate into a cone of very wide angle, the apical portion of which has become lost or absorbed, and the remaining part of which is too short to show clearly its intrinsic curvature. In the closely allied Emarginula, there is hkewise a well-marked spiral in the embryo, which however is still manifested in the curvature of the adult, nearly conical, shell. In both cases we have to do with a very wide-angled cone, and with a high retardation-factor for its inner, or posterior, border. The series is continued, from the apparently simple cone to the complete spiral, through such forms as Calyptraea. The angle a, as we have seen, is not always, nor rigorously, a constant angle. In some Ammonites it may increase with age, the whorls becoming closer and closer; in others it may decrease rapidly and even fall to zero, the coiled shell then straightening out, as in Lituites and similar forms. It diminishes somewhat, also, in many Orthocerata, which are slightly curved in youth but straight in age. It tends to increase notably in some common land-shells, the Pupae and Bulimi ; and it decreases in Succinea. Directly related to the angle a is the ratio which subsists between the breadths of successive whorls. The following table gives a few * What is sometimes called, as by Leslie, the angle of deflection is the complement of what we have called the spiral angle (a), or obliquity of the spiral. When the angle of deflection is 6° 17' 41", or the spiral angle 83'' 42' 19", the radiants, or breadths of successive whorls, are doubled at each entire circuit. 816 THE EQUIANGULAR SPIRAL [ch. illustrations of this ratio in particular cases, in addition to those which we have already studied. Ratio of breadth of consecutive whorls Pointed Turbinates Telescopium fuscum ... 1-14 Terebra suhulata ... ... 1-16 * Turritella terebellata ... 1-18 *Turritella imbricata... ... 1-20 Cerithium palustre ... ... 1 "22 Turritella duplicata ... ... 1-23 Melanopsis terebralis ... 1-23 Cerithium nodulosum ... 1-24 * Turritella car inata ... ... 1-25 Terebra crenulata ... ... 1-25 Terebra maculata (Fig. 397) 1-25 *Cerithium lignitarum ' ... 1-26 Terebra dimidiata 1-28 Cerithium sulcatum ... ... 1-32 Fusus longissimus ... ... 1*34 *Pleurotomaria conoidea ... 1-34 Trochus niloticus (Fig. 398) 1-41 Mitra episcopalis ... ... 1-43 Fusus antiquus ... ... 1 -50 Scalaria pretiosa ... ... 1-56 Fusus colosseus ... ... 1-71 PhasiaTvella australis ... 1-80 Helicostyla polychroa ... 2-00 Obtuse Turbinates and Discoids Conus virgo ... ... ... 1-25 XClymenia laevigata ... ... 1-33 Conus litteratus ... ... 1*40 Conus betulinus ... ... 1-43 XClymenia arietina ... ... 1-50 %Goniatites bifer ... ... 1-50 * Helix nemoralis ... ... 1-50 * Solarium perspectivum ... 1*50 Solarium trochleare ... 1-62 Solarium magnificum ... 1-75 *Natica aperta ... ... 2-00 Euomphalus pentangulatus 2-00 Planorbis corneus ... ... 2-00 Solaropsis pellis-serpentis ... 2-00 Dolium zondtum ... ... 2-10 XGoniatites carinatus ... 2-50 *Natica glaucina ... ... 3-00 Nautilus pompilius ... 3-00 Haliotis excavatus ... ... 4-20 Hal^otis parvus ... ... 6*00 Delphinula atrata ... ... 6-00 Haliotis rugoso-plicata ... 9-30 Haliotis viridis ... ... 10-00 Those marked * from Naumann; J from Miiller; the rest from Macalisterf. In the case of turbinate shells, we muSt take into account the angle y^, in order to determine the spiral angle a from the ratio of the breadths of consecutive whorls; for the short table given on p. 791 is only applicable to discoid shells, in which the angle fi is an angle of 90°. Our formula, as mentioned on p. 771, now becomes J^ _ ^27rsin/?cota For this formula, I have worked out the following table. f Alex. Macahster, Observations on the mode of growth of discoid and turbinated shells, Proc. R.S. xviii, pp. 529-532, 1870; Ann. Mag. N.H. (6), iv, p 160, 1870. Cf. also his Law of Symmetry as exemplified in animal form, Journ. B. Dublin Soc. 1869, p. 327. XI] OF VARIOUS UNIVALVES 817 r t^OiOt^iOCOeCMOiCiOOiMO O5l>?DC0^O5Q0l>COiOO5TtCO § 00»0'-^^000»0»0»0»0>-0 0i0 Ot^r-iO©0»00©000»00 o (M CO w ici r-i ^ c^ eo O -^ -^ 00 CO Tj* lO O »0 CO ooi>cou:)kO-<*'^coeoco(N(M'-*i-t o Xi-HOOCO«iOc0C0-H 818 THE EQUIANGULAR SPIRAL [ch. From this table, by interpolation, we may easily fill in the approximate values of a, as soon as we have determined the apical angle ^ and measured the ratio R\ as follows: Turritella sp. 112 7° 81' Cerithium nodvloaum 1-24 15 82 Conus virgo 1-25 70 88 Mitra episcopalis . . . 143 16 78 Scalaria pretiosa ... 1-56 26 81 Phasianella australis 1-80 26 80 Solarium perspectivum 1-50 53 85 Natica aperta 2-00 70 83 Planorbis corneus 2-00 90 84 Euomphalus pentangviat us 2-00 90 84 We see from this that shells so different in appearance as Cerithium, Solarium, Natica and Planorbis differ very httle indeed in the • magnitude of the spiral angle a, that is to say in the relative velocities of radial and tangential growth. It is upon the angle /? that the difference in their form mainly depends. The angle, or rather semi-angle (/?), of the tangent cone may be taken as 90° in the discoid shells, such as Nautilus and Planorbis. It is still a large angle, of 70° or 75°, in Conus or in Cymba, somewhat less in Cassis, Harpa, Dolium or Natica ; it is about 50° to 55° in the various species of Solarium, about 35° in the typical Trochi, such as T. niloticus or T. zizyphinus, and about 25° or 26° in Scalaria pretiosa and Phasianella bul- - hides', it becomes a very acute angle. Fig. 397. TerebramacuJutal.. ^^ ^^o ^ ^^o ^ ^^ ^^^^ j^^^^ -^ ^^^^.^^^ Turritella or Cerithium. The British species of ' Fusus ' form a series in which the apical angle ranges from about 28° in F. antiquus, through F. Norvegicus, F. berniciensis, F. Turtoni, F. Islandicus, to about 17° in F. gracilis. It varies much among the Cones; and the costly Conus gloria-maris, one of the great treasures of the conchologist, differs from its congeners in no important particular XI] OF VARIOUS UNIVALVES 819 save in the somewhat "produced" spire, that is to say in the com- paratively low value of the angle p. A variation with advancing age of ^ is common, but (as Blake points out) it is often not to be distinguished or disentangled from an alteration of a. Whether alone, or combined with a cKange in a, we find it in all those many gastropods whose whorls cannot all be touched by the same enveloping cone, and whose spire is accordingly described as concave or convex. The former condition, as we have Fig. 398. Trochus niloticus L. it in Cerithium, and in the cusp-Hke spire of Cassis, Dolium and some Cones, is much the commoner of the two*. In the vast majority of spiral univalves the shell winds to the right, or turns clockwise, as we look along it in the direction in which the animal crawls and puts out its head. The thread of a carpenter's screw (except in China) runs the same way, and we call it a "right- handed screw." Save that it takes a right-handed movement to * Many measurements of the linear dimensions of univalve shells have bgen made of late years, and studied by statistical methods in order to detect local races and other instances of variation and variability. But conchological statisticians seem to be content with some arbitrary linear ratio as a measure of "squatness" or the reverse; and the measurements chosen give Uttle or no help towards the determination either of the apical or of the spiral angle. Cf, (e.g.) A. E. Boycott, Conchometry, Proc. Malacol. Soc. xvn, p. 8, 1928; C. Price- Jones, ibid, xix, p. 146, 1930; etc. See also G. Duncker, Methode der Variations-Statistik, Arch, f. Entw. Mech. vm, pp. 112-183, 1899. 820 THE EQUIANGULAR SPIRAL [ch. drive in a "right-handed" screw, the terms right-handed and left- handed are purely conventional; and the mathematicians and the naturalists, unfortunately, use them in opposite ways. Thus the mathematicians call the snail-shell or the joiner's screw kiotropic; and Listing for one has much to say about lack of precision or even confusion on the part of the conchologists and the botanists, from Linnaeus downwards, in their attempts to deal with right-handed and left-handed spirals or screws*. The convolvulus twines to the right, the hop to the left; vine- tendrils are said to be mostly right- handed. At any rate, Clerk Maxwell spoke of hop-spirals and vine- ^irals, trying to avoid the confusion or ambiguity of left and right. Some climbing plants are one and some the other; and, the architect shews httle preference, but builds his spiral staircases or twisted columns either way. But in all these, shells and all, the spiral runs one way; it is isotropic, while the fir-cone shews spirals running both ways at once, and we call them heterotropic, or diadroinic. When we find a "reversed shell," a whelk or a snail winding the wrong way, we describe it mathematically by the simple statement that the apical angle (^) has changed sign. Such left-handed shells occur as a well-known but rare abnormality; and the men who handle snails in the Paris market or whelks in Billingsgate keep a sharp look-out for them. In rare instances they become common. While left-handed whelks (Buccinum or Neptunea) are very rare nowadays, it was otherwise in the epoch of the Red Crag; for Neptunea was then extremely common, but right-handed specimens were as rare as left-handed are today. In the beautiful genus Ampullaria, or apple-snails, which inhabit tropical and sub-tropical rivers, there is unusual diversity; for the spire turns to the right in some species, and to the left in others, and again some are flat or "discoid," with no spire at all; and there are plenty of half-way stages, with right and left-handed spires of varying steepness or acutenessf; in short, within the limits of this singular genus the apical angle (y^) may vary from about ± 35° to ± 125°. But we need not imagine that the direction of gK)wth actually changes over from right-handed to left-handed; it is enough to suppose * See Listing's Topologie, p. 36; and cf. Clerk Maxwell's Electricity and Magnetism, i, p. 24. t See figures in Arnold Lang's Comparative Anatomy (English translation), II, p. ICl, 1902. XI] OF VARIOUS UNIVALVES 821 that the ^ew movement along the axis has changed its direction. For if I take a roll of tape and push the core out to one side or to the other, or if I keep the centre of the roll fixed and push the rim to the one side or to the other, I thereby convert the flat roll into a hollow cone, or (in other words) a plane into a gauche spiral. Whether we push one way or other, whether the spiral coil be plane or gauche, positively or negatively deformed, it remains right-handed or left- handed as the case may be ; but it does change its direction as soon as we turn it upside down, or as soon as the animal does so in assuming its natural attitude. The linear spirals within and without the cone may change places but must remain congruent with one another; for they are merely the two edges of the ribbon, and as such are inseparable and identical twins. But of the shell itself we may reasonably say that a^right-handed has given place to a left-handed spiral. Of these, the one is a mirror-image of the other; and the passing from one to the other through the plane of symmetry (which has no "handedness*') is an operation which Listing called perversion. The flat or discoid apple-snails are Uke our roll of tape, which can be converted into a conical spire and perverted in one direction or the other; and in this genus, by a rare exception, it seems wellnigh as easy to depart one way as the other from the plane of sjmametry. But why, in the general run of shells, all the world over, in the past and in the present, one direction of twist is so overwhelmingly commoner than the other, no man knows. The phenomenon of reversal, or "sinistrality," has an interest of its own from j:he side of development and heredity. For careful study of certain pond-snails has shewn that dextral and sinistral varieties appear, not one by one, but by whole broods of the one sort or^the other; a discovery which goes some way to account for the predominant left-handedness of Fusus amhiguus in the Red Crag. The right-handed, or ordinary form, is found to be "dominant" to the other; but the Mendelian heredity is of a curious and complicated kind. For the direction of the twist appears to be predetermined in the germ even prior to its fertilisation; and a left-handed pond-snail will produce a brood of left-handed young even when fertilised by a normal, or right-handed, individual*. * See A, E. Boycott and others, Abnormal forms of Limnaea peregra . . .and their inheritance, Phil. Trans. (B), ccxxix, p. 51, 1930; and other papers. 822 THE EQUIANGULAR SPIRAL [ch. The angle of retardation (y ) is very small in . Dentalium and Patella ; it is very large in Haliotis ; it becomes infinite in Argonauta and in Cypraea. Connected with the angle of retardation are the various possibilities of contact or separation, in various degrees, between adjacent whorls in the discoid shell, and between both adjacent and opposite whorls in the turbinate. But with these phenomena we have already dealt sufficiently. The beautiful shell of the paper-nautilus {Argonauta argo L.) differs in sundry ways both from the Nautilus and from ordinary univalves. Only the female Argonaut possesses it; it is not attached to its owner, but is (so to speak) worn loose; it is rather a temporary cradle for the young than a true shell or bodily covering; and it is not secreted in the usual way, but is plastered on from the outside by two of the eight arms of the Uttle Octopus to which it belongs. The shell shews a single whorl, or but Httle more; and the spiral is hard to measure, for this reason. It ha« been supposed by some to obey a law other than the logarithmi-c spiral. For my part I have made no special study of it, nor has any one else, to my knowledge, of recent years; but the simple fact that it conserves its shape as it grows, or that each increment is a gnomon to the rest, is enough to shew that this dehcate and beautiful shell is mathematically, though not morphologically, homologous with all the others. Of bivalve shells Hitherto we have dealt only with univalve shells, and it is in these that all the mathematical problems connected .with the spiral, or hehco-spiral, configuration are best illustrated. But the case of the bivalve shell, whether of the lamelHbranch or the brachiopod, presents no essential difference, save only that we have here to do with two conjugate spirals, whose two axes have a definite relation to one another, and some independent freedom of rotatory movement relatively to one another. The bivalve or lamelHbranch moUusca are very different creatures from the rest. The univalves or gastropods, hke their cousins the cephalopods, go about their business and get their living in an ordinary way; but the bivalves are linintelHgent, "acephalous" animals, and imbibe the invisible plankton-food which ciliary currents bring automatically to their mouths. There is something XI] OF THE ARGONAUT 823 <§■ OS CO 824 THE EQUIANGULAR SPIRAL [ch. to be said for withdrawing them, as brachiopods and others have been withdrawn, from Cuvier's great class of the Mollusca. But whether bivalves and univalves be near relations or no is not the question. Both of them secrete a shell, and in both the shell grows by the successive addition of similar parts, gnomon after gnomon; so that in both the equiangular spiral makes, and is bound to make, its appearance. There is a mathematical analogy between the two; but it has no more bearing on zoological classi- fication than has the still closer Hkeness between Nautilus and the nautiloid Foraminifera. The generating curve is particularly well seen in the bivalve, where it simply constitutes what we call "the outline of the shell." It is for the most part a plane curve, but not always ; for there are forms such as Hippopus, Tridacna and many Cockles, or Rhynchonella and Spirifer among the Brachiopods, in which the edges of the two valves interlock, and others, such as Pholas, Mya, etc., where they gape asunder. In such cases as these the generating curves, though not plafie, are still conjugate, having a similar relation, but of opposite sign, to a median plane of reference or of projection. There are a few exceptional cases, e.g. Area (Parallelepipedon) tortuosa, where there is no median plane of symmetry, but the generating curve, and therefore the outline of the shell itself, is a tortuous curve in three dimensions. A great variety of form is exhibited among the bivalves by these generating curves. In many cases the curve or outline is all but circular, as in Anomia, Sphaerium, Artemis, Isocardia] it is nearly semicircular in Argiope; it is approximately elliptical in Anodon, Lutraria, Orthis; it may be called semi-elliptical in Spirifer; it is a nearly rectilinear triangle in Lithocardium, and a curvilinear triangle in Mactra. Many apparently diverse but more or less related forms may be shewn to be deformations of a common type, by a simple application of the mathematical theory of "trans- formations," which we shall have to study in a later chapter. In such a series as is furnished, for instance, by Gervillea, Perna, Avicula, Modiola, Mytilus, etc., a "simple shear" accounts for most, if not all, of the apparent differences. Upon the surface of the bivalve shell we usually see with great clearness the "lines of growth" which represent the successive XI] OF BIVALVE SHELLS 825 margins of the shell, or in other words the successive positions assumed during growth by the growing generating curve; and we have a good illustration, accordingly, of how it is characteristic of the generating curve that it should constantly increase, while never altering its geometric similarity. Underlying these lines of growth, which are so characteristic of a moUuscan shell (and of not a few other organic formations), there is, then, a law of growth which we may attempt to enquire into and which may be illustrated in various ways. The simplest cases are those in which we can study the lines of growth on a more or less flattened shell, such as the one valve of an oyster, a Pecten or a Tellina, or some such bivalve mollusc. Here around an origin, the so-called "'umbo" of the shell, we have a series of curves, some- times nearly circular, sometimes elliptical, often asymmetrical; and such curves are obviously not "concentric," though we are often apt to call them so, but have a common centre of similitude. This arrangement may be illustrated by various analogies. We might for instance compare it to a series of waves, radiating outwards from a point, through a medium which offered a resistance increasing, with the angle of divergence, according to some simple law. We may find another and perhaps a simpler illustration as follows: In a simple and beautiful theorem, Galileo shewed that, if we imagine a number of inclined planes, or gutters, sloping downwards (in a vertical plane) at various angles from a common starting-point, and if we imagine a number of balls rolling each down its own gutter under the influence of gravity (and without hindrance from friction), then, at any given instant, the locus of all these moving bodies is a circle passing through the point of origin. For the acceleration along any one of the sloping paths, for instance AB (Fig. 400), is such that AB = ig cos d.t^ Therefore ^ig.AB/AC.t^. t^=2lg.AC. 826 THE EQUIANGULAR SPIRAL . [ch. That is to say, all the balls reach the circumference of the circle at the same moment as the ball which drops vertically from ^ toC. Where, then, as often happens, the generating curve of the shell is approximately a circle passing through the point of origin, we may consider the acceleration of growth along various radiants to be governed by a simple mathematical law, closely akin to that simple law of acceleration which governs the movements of a faUing body. And, mutatis mutandis, a similar definite law underlies the cases where the generating curve is continually elliptical, or where it assumes some more complex, but still regular and constant form. It is easy to extend the proposition to the particular case where the lines of growth may be considered elliptical. In such a case we have x^/a^ + y^jb^ = 1, where a and b are the major and minor axes of the ellipse. Or, changing the origin to the vertex of the figure, x^ 2aj 1/2 ^ . . (x— a)^ y^ Then, transferring to polar coordinates, where r . cos 6 = x, r . sin 6 = y, we have r . cos^ 6 2 cos 6 r . sin^ 0 which is equivalent to 2a62 cos d r = b^cos^d + a^sin^d' or, simpHfying, by eliminating the sine-function, 2ab^ cos d r = (62_a2)cos2<9 + a2* Obviously, in the case when a = b, this gives us the circular system which we have already considered. For other values, or ratios, of a and 6, and for all values of 6, we can easily construct a table, of which the following is a sample : XI OF BIVALVE SHELLS 827 Chords of an ellipse, whose major and minor axes {a, b) are in certain given ratios 0° 10 20 30 40 50 60 70 80 90 a/6 = 1/3 10 101 105 1115 1-21 1-34 1-50 1-59 1-235 00 1/2 10 101 103 1065 Ml 1145 1142 101^ 0-635 0-0 2/3 10 1-002 1005 1-005 0-995 0-952 0-857 0-670 0-375 0-0 1/1 1-0 0-985 0-940 0-866 0-766 0-643 0-500 0-342 0-174 0-0 3/2 10 0-948 0-820 0-666 0-505 0-372 0-258 0-163 0-078 0-0 2/1 1-0 0-902 0-695 0-495 0-342 0-232 0-152 0-092 0-045 0-0 3/1 1-0 0-793 0-485 0-289 0-178 0-113 0-071 0-042 0-020 0-0 The ellipses whicli we then draw, from the values given in the table, are such as are shewn in Fig. 401 for the ratio a/6 = f, and in Fig. 402 for the ratio a/6 = J ; these are fair approximations to the actual outlines, and to the actual arrangement of the lines of growth, in such forms as Solecurtus or Cultellus, and in Tellina or Psammobia. It is not difficult to in- troduce a constant into our equation to meet the case of a shell which is somewhat unsymmetrical on either side of the median axis. It is a some- what more troublesome matter, however, to bring these configurations into relation with a "law of growth," as was so easily done in the case of the circular figure: in other words, to °° formulate a law of acceleration according to which ^^^' ^^^ • points starting from the origin 0, and moving along radial lines, would all lie, at any future epoch, on an ellipse passing through 0 ; and this calculation we n^ed not enter into. All that we are immediately concerned with is the simple fact that where a velocity, such as our rate of growth, varies with its direction — varies that is to say as a function of the angular divergfence from a certain axis — then, in a certain simple case, we get lines of growth laid down as a system of coaxial circles, and, in some- what less simple cases, we obtain a system of ellipses or of other more complicated coaxial figures, which may or may not be symmetrical on either side of the axis. Among our bivalve mollusca we shall find the Unes of growth to be approximately circular in, for instance, Anomia] in Lima (e.g. L. subauriculata) we have 828 THE EQUIANGULAR SPIRAL [ch. a system of nearly symmetrical ellipses with the vertical axis about twice the transverse; in Solen pellucidus, we have again a system of lines of growth which are not far from being symmetrical eUipses, Ct 10 Fig. 402. in which however the transverse is between three and four times as great as the vertical axis. In the great majority of cases, we have a similar phenomenon with the further complication of slight, but occasionally very considerable, lateral asymmetry. In the above account of the mathematical form of the bivalve shell, we have supposed, for simplicity's sake, that the pole or origin of the system is at a point where all the successive curves touch one another. But such an arrangement is neither theoretically probable, nor is it actually the case; for it would mean that in a certain direction growth fell, not merely to a minimum, but to zero. As a matter of fact, the centre of the system (the "umbo" of the conchologists) lies not at the edge of the system, but very near to it; in other words, there is a certain amount of growth all round. But to take account of this condition would involve more troublesome mathe- matics, and it is obvious that the foregoing illustrations are a sufficiently near approximation to the actual case. In certain little Crustacea (of the genus Estheria) the carapace takes the form of a bivalve shell, closely simulating that of a lamellibranchiate mollusc, and bearing lines of growth in all respects analogous to or even identical with those of the latter. The explana- tion is very curious and interesting. In ordinary Crustacea the carapace, like the rest of the chitinised and calcified integument, is shed off in successive moults, and is restored again as a whole. But in Estheria (and one or two other small Crustacea) the moult is OF BIVALVE SHELLS 829 XI] incomplete: the old carapace is retained, and the new, growing up underneath it, adheres to it like a lining, and projects beyond its edge: so that in course of time the margins of successive old carapaces appear as "lines of growth" upon the surface of the shell. In this mode of formation, then (but not in the usual one), we obtain a structure which "is partly old and partly new," and whose suc- cessive increments are all similar, similarly situated, and enlarged Fig. 403. Hemicardium inver- sum Lam, From Chenu. Fig. 405. Section of Productus '{Strophonema) sp. From Woods. Fig. 404. Caprinella adversa. After Woodward. in a continued progression. We have, in short, all the conditions appropriate and necessary for the development of a logarithmic spiral; and this logarithmic spiral (though it is one of small angle) gives its own character to the structure, and causes the little carapace to partake of the characteristic conformation of the molluscan shell. Among the bivalves the spiral angle (a) is very small in the flattened shells, such as Orthis, Lingula or Anomia. It is larger, as a rule, in the LameUibranchs than in the Brachiopods, but in the latter it is of considerable magnitude among the Pentameri. 830 THE EQUIANGULAR SPIRAL [ch. Among the Lamellibranclis it is largest in such forms as Isocardia and Diceras, and in the very curious genus Caprinella; in all of these last-named genera its magnitude leads to the production of a spiral shell of several whorls, precisely as in the univalves. The angle is usually equal, but of opposite sign, in the two valves of the LamelHbranch, and usually of opposite sign but unequal in the two valves of the Brachiopod. It is very unequal in many Ostreidae, and especially in such forms as Gryphaea, or in Caprinella, which is a kind of exaggerated Gryphaea; in the cretaceous genus Requienia, the two valves of the shell closely resemble a turbinate gastropod with its flat calcified operculum. Occasionally it is of the same sign in both valves (that is to say, both valves curve the same way) as we see sometimes in Anomia, and better in Productus or Strophonema. It will be observed, and it may not be difficult to explain, that the more the bivalve shell curves in the one direction the more it curves in the other; each valve tends, to be spheroidal, or ellipsoidal, rather than cylindroidal. The cyhndroidal form occurs, excep- tionally, in Solen. But Pecten, Gryphaea, Terebratula are all cases of bivalve shells where one valve is flat and the other curved from side to side ; and the flat valve tends to remain flat in the longitudinal direction also, while the curved valve grows into its logarithmic spiral. In the genus Gryphaea, an oyster-hke bivalve from the Jurassic, the creature lay on its side with' its left valve downward, as oysters and scallops also do; and this valve adhered to the ground while the animal was young. The upper valve stays flat, and looks Uke a mere operculum; but the lower or deep valve grows into a more or less pronounced spiral. So is it also in the neighbouring genus Pecten, where P. Jacobaeus has its under-valve much deeper and more curved than, say, P. opercularis ', but Gryphaea incurva is more spirally curved than any of these, and G. arcuata has a spiral angle very near to that of Nautilus itself. In both the spiral is a typical equiangular one, built up of a succession of gnomonic incre- ments, which in turn depend on a constant ratio between the expansion of a generating figure and its rotation about a centre of simihtude. Rate of growth is at the root of the whole matter. Now Gryphaea, Hke some Ammonites of which we spoke before, is XI] OF BIVALVE SHELLS 831 one of those cases in which not only does the form of the shell vary, but geologists recognise, now and then, a trend, or progressive sequence of variation, from one stratum or one "horizon" to anotlier. In short, as time goes on, we seem to see the shell growing thicker or widey, or more and more spirally curved, before our eyes. What meaning shall we give, what importance should we assign, to these changes, and what sort or grade of evolution do they imply? Some hold that these palaeontological features are "strictly comparable with those on which the geneticist bases his factorial studies"; and that as such they may shew "linkage of characters," as when "in the evolution of Gryjphaea the area of attachment retrogresses as the arching progresses"*. These are debatable matters. But in so far as the changes depend on mere gradations of magnitude, they lead indeed to variety but fall short of the full concept of evolution. For to quote Aristotle once again (though we need not go to Aristotle to learn it) : "some things shew increase but suffer no alteration ; because increase is one thing and alteration is another." The so-called "spiral arms" of Spirifer and many other Brachio- pods are not difficult to explain. They begin as a single structure, in the form of a loop of shelly substance, attached to the dorsal valve of the shell, in the neighbourhood of the hinge, and forming a skeletal support for two cihate and tentaculate arms. These grow to a considerable length, coiling up within the shell that they may do so. In Terehratula the loop reilaains short and simple, and is merely flattened and distorted somewhat by the restraining pressure of the ventral valve ; but in Spirifer, Atrypa, Athyris and •x / ^^^'^ Fig. 406. Skeletal loop of Terg- many more it forms a watchsprmg coil on j,^^^^ p^^^ ^^^^^^ either side, corresponding to the close- coiled arms of which it was the support and skeleton. In these curious and characteristic structures we see no sign of progressive * H. H. Swinnerton, Unit characters in fossils, Biol. Reviews, vii, pp. 321-335, 1932; of. A. E. Truman, Oeol. Mag. lix, p. 258, lxi, p. 358, 1922-24. 832 THE EQUIANGULAR SPIRAL [ch. growth, no successional increments, no "gnomons," no self- similarity in the figm:e. In short it has nothing to do with a logarithmic or equiangular spiral, but is a mere twist, or tapering helix, and it points now one way, now another. The cases in which the heUcoid spires point towards, or point away from, the middle line are ascribed, in zoological classification, to particular "families" of Brachiopods, the former condition defining (or helping to define) the Atrypidae and the latter the Spiriferidae Fig. 407. Spiral arms of Spirifer. Fig. 408. Inwardly directed spiral arms of Atrypa. and Athyridae. It is obvious that the incipient curvature of the arms, and consequently the form and direction of the spirals, will be influenced by the surrounding pressures, and these in turn by the general shape of the shell. We shall expect, accordingly, to find the long outwardly directed spirals associated with shells which are transversely elongated, as Spirifer is; while the more rounded Atrypa will tend to the opposite condition. In a few cases, as in Cyrtina or Reticularia* where the shell is comparatively narrow but long, and where the uncoiled basal support of the arms is long also, the coils into which the latter grow are turned backwards, in the direction where there is most room for them. And in the few cases where the shell is very considerably flattened, the spirals (if they find room to grow at all) will be constrained to do so in a discoid or nearly discoid fashion, and this is actually the case in such flattened forms as Koninckina or Thecidium. The shells of Pteropods While mathematically speaking we are entitled to look upon the bivalve shell of the Lamellibranch as consisting of two distinct elements, each comparable to the entire shell of the univalve, we XI THE SHELLS OF PTEROPODS 833 have no biological grounds for such a statement ; for the shell arises from a single embryonic origin, and afterwards becomes spHt into portions which constitute the two separate valves. \¥e can perhaps throw some indirect light upon this phenomenon, and upon several other phenomena connected with shell-growth, by a consideration of the simple conical or tubular shells of the Pteropods. The shells of the latter are in few cases suitable for simple mathematical investigation, but nevertheless they are of very considerable interest in connection with our general problem. The morphology of the Pteropods is bv no means well understood, and in speaking of them B Fig. 410. Diagrammatic transverse sections, or outlines of the mouth, in certain Pteropod shells: Fig. 409. Pteropod shells: ^' ^' ^^^^^''^ australis; C, C. pyramidalis; (1) Cuvierimi columnella; D, CMUintium; E, C. cuspidata. After Boas. (2) Cleodora chierchiae; (3) C. pygmaea. After Boas. I will assume that there are still grounds for beheving (in spite of Boas' and Pelseneer's arguments) that they are directly related to, or may at least be directly compared with, the Cephalopoda*. The simplest shells among the Pteropods have the form of a tube, more or less cylindrical {Cuvierinay, more often conical (Creseis, Clio) ; and this tubular shell (as we have already had occasion to remark, on p. 416), frequently tends, when it is very small and delicate, to assume the character of an unduloid. (In such a case it is more than likely that the tiny shell, or that portion of it which * We need not assume a close relationship, nor indeed any more than such a one as permits us to compare the shell of a Nautilus with that of a Gastropod. 834 THE EQUIANGULAR SPIRAL '[ch. constitutes the unduloid, lias not grown by successive increments or ''rings of growth," but has developed as a whole.) A thickened "rib" is often, perhaps generally, present on thef dorsal side of the little conical shell. In a few cases (Limacina, Peraclis) the tube becomes spirally coiled, in a normal equiangular spiral or helico- spiral. In certain cases (e.g. Cleodora, Hyalaea) the tube or cone is curiously modified. In the first place, its cross-section, originally circular or nearly so, becomes flattened or compressed dorsoventrally ; and 3 -' 4 Fig. 411. Shells of thecosome Pteropods (after Boas). (1) Cleodora cuspidata; (2) Hyalaea trispinosa; (3) H. globulosa; (4) H. uncinata; (5) H. inflexa. the angle, or rather edge, where dorsal and ventral walls meet, becomes more and more drawn out into a ridge or keel. Along the free margin, both of the dorsal and the ventral portion of the shell, growth proceeds with a regularly varying velocity, so that these margins, or lips, of the shell become regularly curved or markedly sinuous. At the same time, growth in a transverse direction pro- ceeds with an acceleration which manifests itself in a curvature of the sides, replacing the straight borders of the original cone. In other words, the cross-section of the cone, or what we have been calling the generating curve, increases its dimensions more rapidly than its di^ance from the pole. XI] THE SHELLS OF PTEROPODS 835 In the above figures, for instance in that of Cleodora cicspidata, the markings of the shell which represent the successive edges of the lip at former stages of growth furnish us at once with a ''graph" 'Fig. 412. Cleodora ctispidata. of the varying velocities of growth as measured, radially, from the apex. We can reveal more clearly the nature of these variations in the following way, which is simply tantamount to converting our radial into rectangular coordinates. Neglecting curvature (if any) b c d e f g Y X O X Fig. 413. Curves obtained by transforming radial ordinates, as in Fig. 412, into vertical equidistant ordinates. 1, Hyalaea trispinosa; 2, Cleodora cicspidata. of the sides and treating the shell (for simpUcity's sake) as a right cone, we lay off equal angles from the apex 0, along the radii Oa, Ob, etc. If we then plot, as vertical equidistant ordinates, the 836 THE EQUIANGULAR SPIRAL [CH. magnitudes Oa, Oh ... OY, and again on to Oa' , we obtain a diagram such as follows in Fig. 413: by help of which we not only see more clearly the way in which the growth-rate varies from point to point, but we also recognise better than before the nature of the law which governs this variation in the different species. Furthermore, the young shell having become differentiated into a dorsal and a ventral part, marked off from one another by a lateral edge or keel, and the inequality of growth being such as to cause each portion to increase most rapidly in the median line, it follows that the entire shell will appear to have been split into a dorsal and a ventral plate, both connected with, and projecting from, what remains of the original undivided cone. Putting the same thing in other words, we may say that the generating figure^ which Fig. 414. Development of the shell of Hyalaea (Cavolinia) tridentata Forskal: the earlier stages being the ''' Pleuropits longifiUs'' of Troschel. After Tesch. lay at first in a plane perpendicular to the axis of the cone, has now, by unequal growth, been sharply bent or folded, so as to lie approximately in two planes, parallel to .the anterior and posterior faces of the cone. We have only to imagine the apical connecting portion to be further reduced, and finally to disappear or rupture, and we should have a bivalve shell developed out- of the original simple cone. In its outer and growing portion, the shell of our Pteropod now consists of two parts which, though still connected together at the apex, may be treated as growing practically independently. The shell is no longer a simple tube, or simple cone, in which regular inequalitiies of growth will lead to the development of a spiral ; and this for the simple reason that we have now two opposite maxima XI] THE SHELLS OF PTEROPODS 837 of growth, instead of a maximum on the one side and a minimum on the other side of our tubular shell. As a matter of fact, the dorsal and the ventral plate tend to curve in opposite directions, towards the middle line, the dorsal curving ventrally and the ventral curving towards the dorsal side. In the case of the Lamelhbranch or the Brachiopod, it is quite possible for both valves to grow into more or less pronounced spirals, for the simple reason that they are hinged upon one another; and each growing, edge, instead of being brought to a standstill by the growth of its opposite neighbour, is free to move out of the way, by the rotation about the hinge of the plane in which it Hes. But where there is no such hinge, as in the Pteropod, the dorsal and ventral halves of the shell (or dorsal and ventral valves, if we Fig. 415. Pteropod shells, from the side: (1) Cleodora cnspidafa; (2) Hyalaea longirostris; (3) H. trispinosa. After Boas. may call them so) would soon interfere with one another's progress if they curved towards one another (as they do in a cockle), and the development of a pair of conjugate spirals would become impossible. Nevertheless, there is obviously, in both dorsal and ventral valve, a tendency to the development of a spiral curve, that of the ventral valve being paore marked than that of the larger and overlapping dorsal one, exactly as in the two unequal valves of Terebratula, In many cases (e.g. Cleodora cuspidata), the dorsal valve or plate, strengthened and stiffened by its midrib, is nearly straight, while the curvature of the other is well displayed. But the case will be materially altered and simplified if growth be arrested or retarded in either half of the shell. Suppose for instance that the dorsal valve grew so slowly that after a while, in comparison with the other, we might speak of it as being absent altogether: 838 THE EQUIANGULAR SPIRAL [ch. or suppose that it merely became so reduced in relative size as to form no impediment to the continued growth of the ventral one; the latter would continue to grow in the direction of its natural curvature, and would end by forming a complete and coiled logarithmic spiral. It would be precisely analogous to the spiral shell of Nautilus, and, in regard to its ventral position, concave towards the dorsal side, it would even deserve to be called directly homologous with it. Suppose, on the other hand, that the ventral valve were to be greatly reduced, and even to disappear, the dorsal valve would then pursue its unopposed growth; and, were it to be markedly curved, it would come to form a logarithmic spiral, concave towards the ventral side, as is the case in the shell of Spirula*. Were the dorsal valve to be destitute of any marked curvature (or in other words, to have but a low spiral angle), it would form a simple plate, as in the shells of Sepia or Loligo. Indeed, in the shells of these latter, and especially in that of Sepia, we seem to recognise a manifest resemblance to the dorsal plate of the Pteropod shell, as we have it (e.g.) in Cleodora or Hyalaea ; the Httle " rostrum " of Sepia is but the apex of the primitive cone, and the rounded anterior extremity has grown according to a law precisely such as that which has produced the curved margin of the dorsal valve in the Pteropod. The ventral portion of the original cone is nearly, but not wholly, wanting ; it is represented by the so-called posterior wall of the "siphuncular space." In many decapod cuttle-fishes also (e.g. Todarodes, Illex, etc.) we still see at the posterior end of the "pen" a vestige of the primitive cone, whose dorsal ^nargin only has continued to grow; and the same phenomenon, on an exaggerated scale, is represented in the Belemnites. It is not at all impossible that we may explain on the same lines the development of the curious "operculum" of the Ammonites. This consists of a single horny plate (Anaptychus), or of a thicker, more calcified plate divided into two symmetrical halves (Aptychi), often found inside the terminal chamber of the Ammonite, and occasionally to be seen lying in situ, as an operculum which partially closes the mouth of the shell ; this structure is known to exist even * Cf. Owen, "These shells [Nautilus and Ammonites] are revolutely spiral or coiled over the back of the animal, not involute like Spirula^': Palaeontology, 1861, p. 97; cf. Memoir cm the Pearly Nautilus, 4832; also P.Z.S. 1878, p. 95.5. XI] THE SHELLS OF CEPHALOPODS 839 in connection with the early embryonic shell. In form the Anap- tychus, or the pair of conjoined Aptychi, shew an upper and a lower border, the latter strongly convex, the former sometimes shghtly concave, sometimes shghtly convex, and usually shewing a median projection or slightly developed rostrum. From this rostral border the curves of growth start, and course round parallel to, finally constituting, the convex border. It is this convex border which fits into the free margin of the mouth of the Ammonite's shell, while the other is appHed to and overlaps the preceding whorl of the spire. Now this relationship is precisely what we should expect, were we to imagine as our starting-point a shell similar to that of Hyalaea : in which however the dorsal part of the split cone had become separate from the ventral half, had remained fiat, and had grown comparatively slowly, while at the same time it kept slipping forward over the growing and coiUng spire into which the ventral half of the original shell develops*. In short, I think there is reason to believe, or at least to suspect, that we have in the shell and Aptychus of the Ammonites, two portions of a once united structure; of which other Cephalopods retain not both parts but only one or other, one as the ventrally situated shell of Nautilus, the other as the dorsally plated shell for example of Sepia or of Spirula. In the case of the bivalv-e shells of the Lamellibranchs or of the Brachiopods, we have to deal w^ith a phenomenon precisely analogous to the split and flattened cone of our Pteropods, save only that the primitive cone has been split into two portions, not incompletely, as in the Pteropod {Hyalaea), but completely, so as to forin two separate valves. Though somewhat greater freedom is given to growth now that the two valves are separate and hinged, yet still the two valves oppose and hamper one another, so that in the longitudinal direction each is capable of only a moderate curvature. This curvature, as we have seen, is recognisable as an equiangular spiral, but only now and then does the growth of the spiral continue so far as to develop successive coils : as it does in a few symmetrical forms such as Isocardia cor ; and as it does still more conspicuously in a few others, such as Gryphaea and Caprinella, where one of the * The case of Terebratula or of Gryphaea would be closely analogous, if the smaller valve were less closely connected and co- articulated with the larger. 840 THE EQUIANGULAR SPIRAL [ch. two valves is stunted, and the growth of the other is (relatively speaking) unopposed. Of septa Before we leave the subject of the molluscan shell, we have still another problem to deal with, in regard to the form and arrangement of the septa which divide up the tubular shell into chambers, in the Nautilus, the Ammonite and their allies. The existence of septa in a nautiloid shell may probably be accounted for as follows. We have seen that it is a property of a cone that, while growing by increments at one end only, it con- serves its original shape : therefore the animal within, which (though growing by a different law) also conserves its shape, will continue to fill the shell if it actually fills it to begin with: as does a snail or other Gastropod. But suppose that our mollusc fills a part only of a conical shell (as it does in the case of Nautilus) ; then, unless it alter its shape, it must move upward as it grows in the growing con^, until it comes to occupy a space similar in form to that which it occupied before: just, indeed, as a little ball drops far down into the cone, but a big one must stay farther up. Then, when the animal after a period of growth has moved farther up in the shell, the mantle- surface continues or resumes its secretory activity, and that portion which had been in contact with' the former septum secretes a septum anew. In short, at any given epoch, the creature is not secreting a tube and a septum by separate operations, but is secreting a shelly case about its rounded body, of which case one part appears as the continuation of the tube, and the other part, merging with it by indistinguishable boundaries, appears as the septum*. The various- forms assumed by the septa in spiral shells | present us with a number of problems of great beauty, simple in their essence, but whose full investigation would soon lead us into difficult mathematics. * "It has been suggested, and I think in some quarters adopted as a dogma, that the formation of successive septa [in Nautilus] is correlated with the recurrence of reproductive periods. This is not the case, since, according to my observations, propagation only takes place after the last septum is formed"; Willey, Zoological Results, 1902, p. 746. f Cf. Henry Woodward, On the structure of camerated shells, Pop. Sci. Rev. XI, pp. 113-120, 1872. XI] OF SEPTA 841 We do not know how these septa are laid down in an Ammonite, but in the Nautilus the essential facts are clear *. The septum begins as a very thin cuticular membrane (composed of a substance called conchjolin), which is secreted by the skin, or mantle-surface, of the animal ; and upon this membrane nacreous matter is gradually laid down on the mantle-side (that is to say between the animal's' body and the cuticular membrane which has been thrown off from it), so that the membrane remains as a thin pellicle over the hinder surface of the septum, and so that, to begin with, th.e membranous septum is moulded on the flexible and elastic surface of the animal, within which the fluids of the body must exercise a uniform, or nearly uniform pressure. Let us think, then, of the septa as they would appear in their uncalcified condition, formed of, or at least superposed upon, an elastic membrane. They must follow the general law, applicable to all elastic membranes under uniform pressure, that the tension varies inversely as the radius of curvature ; and we come back once more to our old equation of Laplace and Plateau, that p = r ^r^v] Moreover, since the cavity below the septum is practically closed, and is filled either with air or with water, P will be constant over the whole area of the septum. And further, we must assume, at least to begin with, that the membrane constituting the incipient septum is homogeneous or isotropic. Let us take first the case of a straight cone, of circular section, more or less like an Orthoceras ; and let us suppose that the septum is attached to the shell in a plane perpendicular to its axis. The septum itself must then obviously be spherical. Moreover the extent of the spherical surface is constant, and easily determined. For obviously, in Fig. 417, the angle LCL' equals the supplement of the angle (LOU) of the cone; that is to say, the circle of contact subtends an angle at the centre of the spherical surface, which is constant, and which is equal to tt — 2/?. The case is not excluded where, owing to an asymmetry of tensions, the septum meets the * See Willey, op. cit., p. 749. Cf. also Bather, Shell-growth in Cephalopoda, Ann. Mag. N.H. (6), i, pp. 298-310, 1888; ibid. pp. 421-427, and other papers by Blake, Riefstahl, etc. quoted therein. 842 THE l^QUIANGULAR SPIRAL [ch. side walls of the cone at other than a right angle, as in Fig. 416; and here, while the septa still remain portions of spheres, the geometrical construction for the position of their centres is equally easy. If, on the other hand, the attachment of the septum to the' inner walls of the cone be in a plane oblique to the axis, then the outhne of the septum will be an elhpse, but its surface will still be spheroidal. If Fig. 41fi. the attachment of the septum be not in one plane, but forms a sinuous line of contact with the cone, then the septum will be a saddle-shaped surface, of great complexity and beauty. In all cases, provided only that the membrane be isotropic, the form assumed will be precisely that of a soap-bubble under similar conditions of attachment : that is to say, it will be (with the usual limitations or conditions) a surface of minimal area, and of constant mean curvature. If our cone be no longer straight, but curved, then the septa will by symmetrically deformed in consequence. A beautiful and in- teresting case is afforded us by Nautilus itself. Here the outline of the septum, referred to a plane, is approximately bounded by two elliptic curves, similar and similarly situated, whose areas are to one another in a definite ratio, namely as A_^l^'l_ 4^C0ta ^a ^2^ 2 and a similar ratio exists in Ammonites and all other close-whorled spirals, in whi^h however we cannot always make the simple XI] TRE SETTA 0¥ NAUTILUS 843 assumption of elliptical form. In a median section of Nautilus, we see each septum forming a tangent to the inner and to the outer wall, just as it did in a section of the strsiight Orthoceras; but the Fig. 418. Section of Xaufilus, shewing the contour of the septa in the median plane. curvatures in the neighbourhood of these two points of contact are not identical, for they now vary inversely as the radii, drawn from the pole of the spiral shell. The contour of the septum in this median plane is a spiral curve — the conformal spiral transformation of the spherical septum of the rectihnear Orthoceratite. 844 THE EQUIANGULAR SPIRAL [ch. But while the outline of the septum in median section is simple and easy to determine, the curved surface of the septum in its entirety is a very complicated matter, even in Nautilus which is one of the simplest of actual cases. For, in the first place, since the form of the septum, as seen in median section, is that of a logarithmic spiral, and as therefore its curvature is constantly Fig. 419. C&,Bi oi the mteviov oi Nautilus-, to shew the contours of the septa at their junction with the shell-waU. altering, it follows that, in successive transverse sections, the curva- ture is also constantly altering. But in the case of Nautilus, there are other aspects of the phenomenon, which we can illustrate, but only in part, in the following simple manner. Let us imagine a ^ack of cards, in which we have cut out of each card a similar concave arc of a logarithmic spiral, such as we actually see in the median section of the septum of a Nautilus. Then, while we hold the cards together, foursquare, in the ordinary position of the pack, we have a simple ''ruled" surface, which in any longitudinal section has the XI] THE SEVTA 0¥ NAUTILUS 845 form of a logarithmic spiral but in any transverse section is a straight horizontal line. If we shear or slide the cards upon one. another, thrusting the middle cards of the pack forward in advance of the others, till the one end of the pack is a convex, and the other a concave, elUpse, the cut edges which combine to represent our septum will now form a curved surface of much greater complexity ; and this is part, but not by any means all, of the deformation produced as a direct consequence of the form in Nautilus of the section of the tube within which the septum has to He. The complex curvature of the surface will be manifested in a sinuous outline of the edge, or line of attachment of the septum to the tube, and will vary according to the configuration of the latter. In the case of Nautilus, it is easy to shew empirically (though not perhaps easy to demonstrate mathematically), that the sinuous or saddle- shaped contour of the " suture" (or line of attachment of the septum to the tube) is such as can be precisely accounted for in this manner ; and we may find other forms, such as Ceratites, where the septal outline is only a httle more sinuous, and still precisely analogous to that of Nautilus. It is also easy to see that, when the section of the tube (or "generating curve") is more comphcated in form, when it is flattened, grooved, or otherwise ornamented, the curvature of the septum and the outline of its sutural attachment will become very complicated indeed * ; but it will be comparatively simple in the "case of the first few sutures of the young shell, laid down before any over- lapping of whorls has taken place, and this comparative simplicity of the first-formed sutures is a marked feature among Ammonites f. * The "lobes" and "saddles" which arise in this manner, and on whose arrange- ment the modern classification of the nautiloid and ammonitoid shells largely depends, were first recognised and named by Leopold von Buch, Ann. Sci. Nat. XXVII, xxvni, 1829. t Blake has remarked upon the fact (op. cit. p. 248) that in some Cyrtocerata we may have a curved shell in which the ornaments approximately run at a constant angular distance from the pole, while the septa approximate to a radial direction; and that "thus one law of growth is illustrated by the inside, and another by the outside." In this there is nothing at which we need wonder. It is merely a case where the generating curve is set very obliquely to the axis of the shell ; but where the septa, which have no necessary relation to the mouth of the shell, take their places, as usual, at a certain definite' angle to the walls of the tube. This relation of the septa to the walls of the tube arises after the tube itself is fully formed, and the obliquity of growth of the open end of the tube has no relation to the matter. 846 THE EQUIANGULAR SPIRAL [ch. We have other sources of complication, besides those which are at once .introduced by the sectional form of the tube. For instance, th'e siphuncle, or little inner tube which perforates the septa, exercises a certain amount of tension, sometimes evidently considerable, upon the latter: which tension is made manifest in Spirula (and slightly so even in Nautilus) by a dip in the septal floor where it meets the siphuncle. We can no longer, then, consider each septum as an isotropic surface under uniform pressure; and there may be other structural modifications, or inequalities, in that portion of the Fig. 420. Ammonites Sowerbyi. From Zittel. animal's body with which the septum is in contact, and by which it is conformed. It is hardly likely, for all these reasons, tha»t we shall ever attain to a full and particular explanation of the septal surfaces and their sutural outlines throughout the whole range of Cephalopod shells; but in general terms, the problem is probably not beyond the reach of mathematical analysis. The problem might be approached experimentally, after the manner of Plateau's experi- ments, by bending a wire into the complicated^ form of the suture-line, and studying the form of the liquid film which constitutes the corresponding surface 'minimae areae. In certain Ammonites the septal outline is further complicatec" in another way. Superposed upon the usual sinuous outline, witl XI] THE SEPTA OF AMMONITES 847 its "lobes" and "saddles," we have here- a minutely ramified, or arborescent outline, in which all the branches terminate in wavy, more or less circular arcs — looking just like the "landscape marble" from the Bristol Rhaetic. We have no difficulty in recognising in this a surface-tension phenomenon. The figures are precisely such as we can imitate (for instance) by pouring a few drops of milk upon a greasy plate, or of oil upon an alkaUne solution*; they are what Charles Tomlinson called "cohesion figures." Fig. 421. Suture-line of a Triassic Ammonite {Pinacoceras). From Zittel. We must not forget that while the nautilus and the ammonite resemble one another, and are mathematically identical in their spiral curves, they are really very different things. The one is an external, the other an internal shell. The nautilus occupies the large terminal chamber of the many-chambered shell, and "Still as the spiral grew. He left the past year's dwelling for the new." But even the largest ammonites never contained the body of the animal, but lay hidden, as Spirula does, deep within the substance of the mantle. How the comphcated septa and septal outHnes of the ammonites are produced I do not knowt. We have very far from exhausted, we have perhaps little more than begun, the study of the logarithmic spiral and the associated curves which find exemplification in the multitudinous diversities of molluscan shells. But, with a closing word or two, we must now bring this chapter to an end. * "The Fimbriae, or Edges, appeared on the Surface like the Outlines of some curious Foliage. This, upon Examination of them, I found to proceed from the Fulness of the Edges of the Diaphragms, whereby the Edges were waved or plaited somewhat in the manner of a Ruff" (R. Hooke, op. cit.). t In certain rare cases the complicated sutural pattern of an ammonite is found upside down, but unchanged otherwise. Cf. Otto Haas, A case of inversion of suture hues in Hysteroceras, Amer. Jl. of Sci. ccxxxix, p. 661, 1941. 848 THE EQUIANGULAK SPIRAL [ch. In the spiral shell we have a problem, or a phenomenon, of growth, immensely simplified by the fact that each successive increment is no sooner formed than it is fixed irrevocably, instead of remaining in a state of flux and sharing in the further changes which the organism undergoes. In such a structure, then, we have certain primary phenomena of growth manifested in their original simplicity, undisturbed by secondary and conflicting phenomena. What actually grows is merely the lip of an orifice, where there is produced a ring of solid material, whose form we have discussed under the name of the generating curve ; and this generating curve grows in magnitude without alteration of its form. Besides its increase in areal magnitude, the growing curve has certain strictly limited degrees of freedom, which define its motions in space. And, though we may know nothing whatsoever about the actual velocities of any of these motions, we do know that they are so correlated together that their relative velocities remain constant, and accordingly the form and symmetry of the whole system remain in general unchanged. But there is a vast range of possibilities in regard to every one of these factors : the generating curve may be of various forms, and even when of simple form, such as an elhpse, its axes may be s4t at various angles to the system; the plane also in which it lies may vary, almost indefinitely, in its angle relatively to that of any plane of reference in the system; and in the several velocities of growth, of rotation and of translation, and therefore in the ratios between all these, we have again a vast range of possibihties. We have then a certain definite type, or group of forms, mathematically isomorphous, but presenting infinite diversities of outward appear- ance: which diversities, as Swammerdam said, ex sola nascuntur diver sitate gyrationum ; and which accordingly are seen to have their origin in differences of rate, or of magnitude, and so to be, essentially, neither more nor less than differences, of degree. In nature, we find these forms presenting themselves with but little relation to the character of the creature by which they are produced. Spiral forms of certain particular kinds are common to Gastropods and to Cephalopods, and to diverse famihes of each ; while outside the class of molluscs altogether, among the Foraminifera and among the worms (as in Spirorbis, Spirographis, and in the Dentalium -like shell of Ditrupa), we again meet with similar and corresponding spirals. XI] CONCLUSION 849 Again, we find the same forms, or forms which (save for external ornament) are mathematically identical, repeating themselves in all periods of the world's geological history; and we see them mixed up, one with another, irrespective of cHmate or local conditions, in the depths and on' the shores of every sea. It is hard indeed (to my mind) to see in such a case as this where Natural Selection necessarily enters in, or to admit that it has had any share what- soever in the production of these varied conformations. Unless indeed we use. .the term Natural Selection in a sense so wide as to deprive it of any purely biological significance; and so recognise as a sort of natural selection whatsoever nexus of causes suffices to differentiate between the likely and the unlikely, the scarce and the frequent, the easy and the hard: and leads accordingly, under the pecuhar conditions, limitations and restraints which we call "ordinary circumstances," one type of crystal, one form of cloud, one chemical compound, to be of frequent occurrence and another to be rare*. * Cf. Bacon, Advancement of Learning, Bk. ii (p. 254) : " Doth any give the reason, why some things in nature are so common and in so great mass, and others so rare and in so small quantity?" CHAPTER XII THE SPIRAL SHELLS OF THE FORAMINIFERA We have already dealt in a few simple cases with the shells x)f the Foraminifera * ; and we have seen that wherever the shell is but a single unit or single chamber, its form may be explained in general by the laws of surface-tension : the argument (or assumption) being that the little mass of protoplasm which makes the simple shell behaves as a fluid drop, the form of which is perpetuated when the protoplasm acquires its solid covering. Thus the spherical Orbulinae and the flask-shaped Lagenae represent drops in equilibrium, under various conditions of freedom or constraint; while the irregular, amoeboid body of Astrorhiza is a manifestation not of equilibrium, but of a varying and fluctuating distribution of surface energy. When the foraminiferal shell becomes multilocular, the same general principles continue to hold; the growing protoplasm increases drop by drop, and each successive drop has its particular phenomena of surface energy, manifested at its fluid surface, and tending to confer upon it a certain place in the system and a certain shape of its own. It is characteristic and even diagnostic of this particular group of Protozoa (1) that development proceeds by a well-marked alterna- tion of rest and of activity — of activity during which the protoplasm increases, and of rest during which the shell is formed; (2) that the shell is formed at the outer surface of the protoplasmic organism, and tends to constitute a continuous or all but continuous covermg ; and it follows (3) from these two factors taken together that each successive increment is added on outside of and distinct from its predecessors, that the successive parts or chambers of the shell are of different and successive ages, so that one part of the shell is always relatively new, and the rest old in various grades of seniority. The forms which we set together in the sister-group of Radiolaria are very differently characterised. Here the cells or vesicles of which each little composite organism is made up are but httle * Cf. pp. 420, 702, etc. CH. XII] THE FORAMINIFERA 851 separated, and in no way walled off, from one another; the hard skeletal matter tends to be deposited in the form of isolated spicules or of little connected rods or plates, at the angles, the edges or the interfaces of the vesicles; the cells or vesicles form a coordinated and cotemporaneous rather than a successive series. In a word, the whole quasi-fluid protoplasmic body may be likened to a little mass of froth or foam : that is to say, to an aggregation of simul- taneously formed drops or bubbles, whose-physical properties and geometrical relations are very different from those of a system of Fig. 422. Hastigerina sp.; to shew the "mouth." drops or bubbles which are formed one after another, each solidifying before the next is formed. With the actual origin or mode of development of the foraminiferal shell we are now but little concerned. The main factor is the adsorption, and subsequent precipitation at the surface of the organism, of calcium carbonate — the shell so formed being interrupted by pores or by some larger interspace or "mouth" (Fig. 422), which interruptions we may doubtless interpret as being due to unequal distributions of surface energy. In many cases the fluid protoplasm "picks up" sand-grains and other foreign particles, after a fashion which we have already described (p. 702); and it cements these together with more or less of calcareous material. The calcareous shell is a crystalline structure, and the micro-crystals of calcium carbonate are so set that their little prisms radiate outwards in each chamber through the thickness of the wall — which symmetry is 852 THE SPIRAL SHELLS [ch. subject to corresponding modification when the spherical chambers are more or less symmetrically deformed*. In various ways the rounded, drop-Uke shells of the Foraminifera, both simple and compound, have been artificially imitated. Thus, if small globules of mercury be immersed in water in which a little chromic acid is allowed to dissolve, as the little beads of quicksilver become slowly covered with a crystalline coat of mercuric chromate they assume various forms reminiscent of the monothalamic Fora- minifera. "the mercuric chromate has a higher atomic volume than the mercury which it replaces, and therefore the fluid contents of the drop are under pressure, which increases with /the thickness of the pellicle ; hence at some weak spot in the latter the contents will presently burst forth, so forming a mouth to the little shell. Sometimes a long thread is formed, just as in Rhabdammina linearis; and sometimes unduloid swelhngs make their appearance on such, a thread, just as in R. discreta. And again, by appropriate modi- fications of the experimental conditions, it is possible (as Rhumbler has shewn) to build up a chambered shell f. In a few forms, such as Globigerina and its close allies, the shell is beset during life with excessively long and dehcate calcareous spines or needles. It is only in oceanic forms that these are present, because only when poised in water can such delicate structures endure; in dead shells, such as we are much more familiar with, every trace of them is broken and rubbed away. The growth of these long needles may be partly explained (as we have already said on p. 675) by the phenomenon which Lehmann calls orientirte Adsorption — the tendency for a crystalline structure to grow by accretion, not necessarily in the outward form of a "crystal," but continuing in any direction or orientation which has once been impressed upon it: in this case the spicular growth is in direct continuation of the 'radial symmetry of the micro-crystalhne * In a few cases, according to Awerinzew and Rhumbler, where the chambers are added on in concentric series, as in Orbitolites, we have the crystalline structure arranged radially in the radial walls but tangentially in the concentric ones: whereby we tend to obtain, on a minute scale, a system of orthogonal trajectories, comparable to that which we shall presently study in connection with the structure of bone. Cf. S. Awerinzew, Kalkschale der Rhizopoden, Z. f. w. Z. Lxxiv, pp. 478-490, 1903. t L. Rhumbler, Die Doppelschalen von Orbitolites und anderer Foraminiferen, etc., Arch. f. Protistenkunde, i, pp. 193-296, 1902; and other papers. Also Die Foraminiferen der Planktonexpedition, i, pp. 50-56, 1911. XII] OF THE FORAMINIFERA 853 elements of the shell-wall. But the calcareous needles are secreted in, or by, no less long and dehcate pseudopodia or "filopodia," and much has been learned since this book was written of the molecular, or micellar, orientation of the protoplasm in such filamentous structures; it is known that the long pseudopodia of the Foraminifera are doubly refractive, and it follows that their molecules are anisotropically arranged*. Whether the slender form and asymmetrical structure of calcareous rod and protoplasmic thread be independent phenomena, or merely two aspects of one and the same phenomenon, is a hard question, and not one for us to discuss. Nor can we profitably discuss (much as we should like to know) how far these patterns of molecular structure in threads, films and surface-pelHcles aifect the "fluidity" of the substance, and conflict with the capillary ' forces which influence its outward form. But we may safely say that the effects of surface-tension on cell-form have been so plainly seen all through this book that any counter-effects due to protoplasmic asymmetry must be phenomena of a second order, and inconspicuous on the whole. Over the whole surface of the shell of Globigerina the radiating spicules tend to occur in a hexagonal pattern, symmetrically grouped around the pores which perforate the shell. Rhumbler has suggested that this arrangement is due to diffusion-currents, forming little eddies about the base of the pseudopodia issuing from the pores: the idea being borrowed from Benard, to whom is Hue the discovery of this type or order of vortices f. In one of Benard's experiments a thin layer of parafiin is strewn with particles of graphite, then warmed to melting, whereupon each little sohd granule becomes the centre of a vortex ; by the interaction of these vortices the particles tend to be repelled to equal distances from one another, and in the end they are found to be arranged in a hexagonal pattern J. * Cf. W. J. Schmidt, Die Bausteine des Tierkorpers in polarisiertem Lichte, Bonn, 1924; Ueber den Feinbau der Filopodien; insb. ihre Doppelbrechung bei Miliola, Protoplasma, xxvii, p. 587, 1937; also D. L. Mackinnon, Optical pro- perties of contractile organs in Heliozoa, Jl. Physiol, xxxviii, p. 254, 1909; R. 0. Herzog, Lineare u. laminHre Feinstrukturen, Kolloidzschr. lxi, p. 280, 1932. See, for discussion and bibliography, L. E. Picken, op. cit. t H. Benard, Les tourbillons cellulaires, Ann. de Chimie (8), xxiv, 1901. Cf. also the pattern of- cilia on an Infusorian, as figured by Biitschli in Bronn's Protozoa, iii, p. 1281, 1887. X A similar hexagonal pattern is obtained by the mutual repulsion of floating magnets in Mr R. W. Wood's experiments, Phil. Mag. xlvi, pp. 162-164, 1898. 854 THE SPIRAL SHELLS [ch. The analogy is plain between this experiment and those diffusion experiments by which Leduc produces his beautiful hexagonal systems of artificial cells, with which we have dealt in a previous chapter. But let us come back to the shell itself, and consider particularly its spiral form. That the shell in the Foraminifera should tend towards a spiral form need not surprise us ; for we have learned that one of the fundamental conditions of the production of a concrete spiral is just precisely what we have here, namely the develop- ment of a structure by means of successive graded increments superadded to its exterior, which then form part, successively, of a permanent and rigid structure. This condition is obviously forth- coming in the foraminiferal, but not at all in the radiolarian, shell. Our second fundamental condition of the production of a logarithmic spiral is that each successive increment shall be so posited and so conformed that its addition to the system leaves the form of the whole system unchanged. We have now to enquire into this latter condition; and to determine whether the successive increments, or successive chambers, of the foraminiferal shell actually constitute gnomons to the entire structure. It is obvious enough that the spiral shells of the Foraminifera closely resemble true logarithmic spirals. Indeed so precisely do the minute shells of many Foraminifera repeat or simulate the spiral shells of Nautilus and its allies that to the naturalists of the early nineteenth century they were known as the Cephalopodes micro- scopiques*, until Dujardin shewed that their little bodies comprised no complex anatomy of organs, but consisted merely of that sUme-like organic matter which he taught us to call "sarcode," and which we learned afterwards from Schwann to speak of as "protoplasm." One striking -difference, however, is apparent between the shell of Nautilus and the little nautiloid or rotaline shells of the Fora- minifera: namely that the septa in these latter, and in all other chambered Foraminifera, are convex outwards (Fig. 423), whereas they are concave outwards in Nautilus (Fig. 347) and in the rest of the chambered molluscan shells. The reason is perfectly simple. * Cf. Ale. d'Orbigny, Tableau methodique de la classe des Cephalopodes, Ann. des Sci. Nat. (1), vii, pp. 245-315, 1826; Felix Dujardin, Observations nouvelles sur les pretendus Cephalopodes microscopiques, ibid. (2), iii, pp. 108, 109, 312-315, 1835, Recherches sur les organismes inferieurs, ibid, iv, pp. 343-377, 1835; etc. XII] OF THE FORAMINIFERA 855 In both cases the curvature of the septum was determined before it became rigid, and at a time when it had the properties of a fluid film or an elastic membrane. In both cases the actual curvature is determined by the tensions of the membrane and the pressures to which it was exposed. Now it is obvious that the extrinsic pressure which the tension of the membrane has to with- stand is on opposite sides in the two cases. In Nautilus, the pressure to" be resisted is that produced by the growing body of the animal, lying to the outer side of the septum, in the outer, wider portion of the tubular shell. In the Foraminifer the septum at the time of its formation was no septum at all ; it was but a portion of the convex Fig. 423. Nummulina antiquior R. and V. After V. von Moller. surface of a drop — that portion namely which afterwards became overlapped and enclosed by the succeeding drop ; and the curvature of the septum is concave towards the pressure to be resisted, which latter is inside the septum, being simply the hydrostatic pressure of the fluid contents of the drop. The one septum is, speaking generally, the reverse of the other; the organism, so to speak, is outside the one and inside the other; and in both cases alike, the septum tends to assume the form of a surface of minimal area, as permitted, or as defined, by all the circumstances of the case. The logarithmic spiral is easily recognisable in typical cases* (and * It is obvious that the actual outline of a foraminiferal, just as of a molluscan shell, may depart widely from a logarithmic spiral. When we say here, for short, that the shell is a logarithmic spiral, we merely mean that it is essentially related to one: that it can be inscribed in such a spiral, or that corresponding points (such, for instance, as the centres of gravity of successive chambers, or the extremities of successive septa) will be found to lie upon such a sj)iral. 856 THE SPIRAL SHELLS [ch. especially where the spire makes more than one visible revolution about the pole), by its fundamental property of continued similarity: that is to say, by reason of the fact that the big many-chambered shell is of just the same shape as the smaller and younger shell — which phenomenon is as apparent and even obvious in the nautiloid Foraminifera, as in Nautilus itself: but nevertheless the* nature of the curve must be verified by careful measurement, just as Moseley determined or verified it in his original study of Nautilus (cf. p. 770). This has accordingly been done, by various writers : and in the first instance by Valerian von Moller, in an elaborate study of Fusulina — a palaeozoic genus whose little shells have built up vast tracts of carboniferous limestone in European Russia*. In this genus a growing surface of protoplasm may be conceived as wrapping round and round a small initial chamber, in such a way as to produce a fusiform or ellipsoidal shell — a transverse section of which reveals the close- wound spiral coil. The following are measurements of the successive whorls in a couple of species of this genus: — F. cylindrica Fischer F. Bocki v. Moller Breadth (in miUimetres) Vhofl Observed Calculated Observed Calculated I 0-132 — 0-079 — II 0-195 0198 0120 0119 III 0-300 0-297 0180 0179 IV 0-449 0-445 0-264 0-267 V — — 0-396 0-401 In both cases the successive whorls are very nearly in the ratio of 1 : 1-5; and on this ratio the calculated values are based. Here is another of von MoUer's series of measurements of F. cylindrica, the measurements being those of opposite whorls — that is to say of whorls 180° apart: Breadth (mm.) 0096 0117 0-144 0-176 0-216 ; 0-264 ' 0-422 0-323 0-395 Log. of do. 0-982 0-068 0-158 0-246 0-334 0-509 0-597 Diff. of logs. — 0-086 0090 0-088 0-088 0-088 0-087 0-088 The mean logarithmic difference is here 0-088, = log 1-225; or the mean difference of alternate logs (corresponding to a vector angle of 27T, i.e. to consecutive measurements along the same radius) is 0-176, = log 1-5, the same value as before. And this ratio of 1-5 between the breadths of successive whorls corresponds (as we see * V. von Moller, Die spiral-gewundenen Foraminifera des riissischen Kohlen- kalks, Mim. de VAcad. Imp. Sci.f St Petersbourg (7), xxv, 1878. XIl] OF THE FORAMINIFERA 857 by our table on p. 791) to a constant angle of about 86°, or just such a spiral as we commonly meet with in the Ammonites (cf. p. 796). In Fusulina, and in some few other Foraminifera (cf. Fig. 424, A), the spire seems to wind evenly on, with little or no external sign of the successive periods of growth, or successive chambers of the shell. The septa which mark off the chambers, and correspond to retardations or cessations in the periodicity of growth, are still to be found in sections of the shell of Fusulina, but they are somewhat irregular and comjiaratively inconspicuous; the measurements we have just spoken of are taken without reference to the segments or chambers, but only with reference to the whorls, or in other words with direct reference to the vectorial angle. A B Fig. 424. A, Cornuspira foliacea Phil.; B, Operculinu complatiata Defr. The linear dimensions of successive chambers have been measured in a number of cases. Van Iterson* has done so in various Miliolinidae, with such results as the following: Triloculina rotunda d'Orb. No. of chamber 12 3 4 5 6 Breadth of chamber in /x — 34 45 61 7 142 8 182 9 246 10 319 __ __ . 84 114 Breadth of chamber in /x, calculated — 34 45 60 79 105 * G. van Iterson, Mathem. u. mikrosk.-anat. Studien iiber Blattstellungen, nebst Betrachtungen liber den Schalenbau der MilioUnen, 331 pp., Jena, 1907. 140 187 243 319 858 THE SPIRAL SHELLS [ch. Here the mean ratio of breadth of consecutive chambers may be taken as 1-323 (that is to say, the eighth root of 319/34); and the calculated values, as given above, are based on this determination. Again, Rhumbler has measured the linear dimensions of a number of rotaline forms, for instance Pulvinulina menardi (Fig. 363) : in which common species he finds the mean linear ratio of consecutive chambers to be about 1-187. In both cases, and especially in the latter, the ratio is not strictly constant from chamber to chamber, but is subject to a small secondary fluctuation*. When the linear dimensions of successive chambers are in con- tinued proportion, then, in order that the whole shell may constitute a logarithmic spiral, it is necessary that the several chambers should subtend equal angles of revolution at the pole. In the case of the Miliolidae this is obviously the case (Fig. 425); for in this family the chambers lie in two rows (Biloculina), or three rows (Triloculina), or in some other small number of series : so that the angles subtended by them are large, simple fractions of the circular arc, such as 180° or 120°. In many of the nautiloid forms, such as Cyclammina (Fig. 426), the angles subtended, though of less magnitude, are still remarkably constant, as we may see by Fig. 427; where the angle subtended by each chamber is made equal to 20°, and this diagram- matic figure is not perceptibly different from the other. In some cases the subtended angle is less constant; and in these it would be necessary to equate the several linear dimensions with the corresponding vector angles, according to our equation r = e^cota It is probable that, by so taking account of variations of 6, such variations of r as (according to Rhumbler's measurements) Pul- vinulina and other genera appear to shew, would be found to diminish or even to disappear. The law of increase by which each chamber bears a constant ratio of magnitude to the next may be looked upon as a simple * Hans Przibram asserts that the linear ratio of successive chambers tends in many Foraminifera to approximate to 1-26, which = V2; in other words, that the volumes of successive chambers tend to double. This Przibram would bring into relation with another law, viz. that insects and other arthropods tend to moult, or to metamorphose, just when they double their weights, or increase their linear dimensions in the ratio of 1 : V 2. (Die Kammerprogression der Foraminiferen als Parallele zur Hautungsprogression der Mantiden, Arch. f. Entw. Meek, xxxiv, p. 680, 1813.) Neither rule seems to me to be well grounded (see above, p. 165). XII] OF THE FORAMINIFERA 859 Fig. 425. 1, 2, Miliolina pulchella d'Orb.; 3-5, M. liyinaeana d'Orb, After Brady. Fig. 426. Cyclammina cancellata Brady, 860 THE SPIRAL SHELLS [ch. consequence of the structural uniformity or homogeneity of the orgai^ism ; we have merely to suppose (as this uniformity would naturally lead us to do) that the rate of increase is at each instant proportional to the whole existing mass. For if Vq, Vi, etc. be Fig. 427. Cydammitia sp. (J>iagramniatic.) the volumes of the successive chambers, let Fj bear a constant proportion to Fq, so that V^ = qV^, and let Fg bear the same proportion to the whole pre-existing volume: then n = ? (Fo + Fi) =^ q (F„ + ?F„) = ?F„ (1 + q) and VJV^ = 1 + ?. This ratio of 1/(1 + q) is easily shewn to be the constant ratio running through the whole series, from chamber to chamber; and if this ratio of volumes be constant, so also are the ratios of corre- sponding surfaces, and of corresponding linear dimensions, provided always that the successive increments, or successive chambers, are similar in form. We have still to discuss the similarity of form and the symmetry of position which characterise the successive chambers, and which, together with the law of continued proportionahty of size, are the distinctive characters and the indispensable conditions of a series of "gnomons." The minute size of the foraminiferal shell or at least of each XII] OF THE FORAMINIFERA 861 successive increment thereof, taken in connection with the fluid or semi-fluid nature of the protoplasmic substance, is enough to suggest that the molecular forces, and especially the force of surface-tension, must exercise a controlling influence over the form of the whole structure; and this suggestion, or behef, is already implied in our statement that each successive increment of growing protoplasm constitutes a separate drop. These "drops," partially concealed by their successors, but still shewing in part their rounded outhnes, are easily recognisable in the various foraminiferal shells which are illustrated in this chapter. Fig. 428. Orbulina imiversa d'Orb. The accompanying figure represents, to begin with, the spherical shell characteristic of the common, floating, oceanic Orbulina. In the specimen illustrated, a second chamber, superadded to the first, has arisen as a drop of protoplasm which exuded through the pores of the first chamber, accumulated on its surface, and spread over the latter till it came to rest in a position of equihbrium. We may take it that this position of equilibrium is determined, at least in the first instance, by the "law of the constant angle," which holds, or tends to hold, in all cases where the free surface of a given Uquid is in contact with a given sohd, in presence of another liquid or a gas. The corresponding equations *are precisely the same as those which we have used in discussing the form of a drop (on p. 466); though some slight modification must be made in our definitions, inasmuch as the consideration of snriace-tension is no longer appropriate at the solid surfaces, and the concept of snviace-energy must take its 862 THE SPIRAL SHELLS [ch. place. Be that as it may, it is enough for us to observe that, in such a case a^ ours, when a given fluid, (namely protoplasm) is in surface contact with a solid (viz. a calcareous shell), in presence of another fluid (sea- water), then the angle of contact, or angle by which the common surface (or interface) of the two liquids abuts against the solid wall, tends to be constant: and that being so, the drop will have a certain definite form, depending {inter alia) on the form of the surface with which it is in contact. After a period of rest, during which the surface of our second drop becomes rigid by calcification, a new period of growth will recur and a new drop of protoplasm be accumulated. Circumstances remaining the same, this new drop will meet the solid surface of the shell at the same angle as did the former one; and, the other forces at work on the system remaining the same, the form of the whole drop, or chamber, will be the same as before. According to Rhumbler, this "law of the constant angle" is the fundamental principle in the mechanical conformation of the foraminiferal shell, and proV-ides for the symmetry of form as well as of position in each succeeding drop of protoplasm: which form and position, once acquired, become rigid and fixed with the onset of calcification. But Rhumbler's explanation brings with it its own difficulties. It is by no means easy of verification, for on the very complicated curved surfaces of the shell it seems to me extraordinarily difficult to measure, or even to recognise, the actual angle of contact : of which angle of contact, by the way, but little is known, save only in the particular case where one of the three bodies is air, as when a surface of water is exposed to air and in contact with glass. It is easy moreover to see that in many of our Foraminifera the angle of contact, though it may be constant in homologous positions from chamber to chamber, is by no means constant at all points along the boundary of each chamber. In Cristellaria, for instance (Fig. 429), it would seem to be (and Rhumbler asserts that it actually is) about 90° on the outer side and only about 50° on the inner side of each septal partition; in Pulvinulina (Fig. 363), according to Rhumbler, the angles adjacent to the mouth are of 90°, and the opposite angles are of 60°, in each chamber. For these and other similar discrepancies Rhumbler would account by simply invoking the heterogeneity of the protoplasmic drop: that is to say, by XII] OF THE FORAMINIFERA 863 assuming that the protoplasm has a different composition and different properties (including a very different distribution of surface-energy), at points near to and remote from the mouth of the shell. Whether the. differences in angle of contact be as great as Rhumbler takes them to be, whether marked heterogeneities of the protoplasm occur, and whether these be enough to account for the differences of angle, I cannot tell. But it seems to me that we had better rest content with a general statement, and that Rhumbler has taken too precise and narrow a view. Fig. 429. Cristellaria reniformis d'Orb In the molecular growth of a crystal, although we must of necessity assume that each molecule settles down in a position of minimum potential energy, we find it very hard indeed to explain precisely, even in simple cases and after all the labours of modern crystallographers, why this or that position is actually a place of minimum potential. In the case of our httle Foraminifer (just as in the case of the crystal), let us then be content to assert that each drop or bead of protoplasm takes up a position of minimum potential energy, in relation to all the circumstances of the case ; and let us not attempt, in the present state of our knowledge, to define that position of minimum potential by reference to angle of contact or any other particular condition of equilibrium. In most cases the whole exposed surface, on some portion of which the drop must come to rest, is an extremely complicated one, and the forces in- volved constitute a system which, in its entirety, is more complicated 864 THE SPIEAL SHELLS [ch. still; but from the symmetry of the case and the continuity of the whole phenomenon, we are entitled to believe that the conditions are just the same, or .very nearly the same, time after time, from one chamber to another : as the one chamber is conformed so will the next tend to be, and as the one is situated relatively to the system so will its successor tend to be situated in turn. The physical law of minimum potential (including also the law of minimal area) is all that we need in order to explain, in general terms, the continued similarity of one chamber to another; and the physiological law of growth, by which a continued proportionality of size tends to run through the series of successive chambers, impresses the form of a logarithmic spiral upon this series of similar increments. In each particular case the nature of the logarithmic spiral, as defined by its constant angle, will be chiefly determined by the rate of growth ; that is to say by the particular ratio in which each new chamber exceeds its predecessor in magnitude. But shells having the same constant angle (a) may still differ from one another in many ways — in the general form and relative position of the chambers, in their extent of overlap, and hence in the actual contour and appearance of the shell; 'and these variations must correspond to par- ticular distributions of energy within the system, which is governed as a whole by the law of minimum potential. Our problem, then, becomes reduced to that of investigating the possible configurations which may be derived from the successive symmetrical apposition of similar bodies whose magnitudes are in continued proportion; and it is obvious, mathematically speaking, that the various possible arrangements all come under the head of the logarithmic spiral, together with the limiting cases which it includes. Since the difference between one such form and another depends upon the numerical value of certain coefficients of mag- nitude, it is plain that any One must tend to pass into any other by small and continuous gradations; in other words, that a classi- fication of these forms must (like any classification whatsoever of logarithmic spirals or of any other mathematical curves) be theoretic or "artificial." But we may easily make such an artificial classi- fication, and shall probably find it to agree, more or less, with the usual methods of classification recognised by biological students of the Foraminifera. XIl] OF THE FORAMINIFERA 865 Firstly we have the typically spiral shells, which occur in great variety, and which (for our present purpose) we need hardly describe further. We may merely notice how in certain cases, for instance Globigerina, the individual chambers are little removed from spheres; in other words, the area of contact between the adjacent chambers is small. In such forms as Cyclamynwa and Pulvinulina, on the other hand , each chamber is greatly overlapped by its successor, and the spherical form of each is lost in a marked asymmetry. Furthermore, in Glo- higerina and some others we have a tendency to the development of a gauche spiral in space, as in so many of our univalve molluscan shells. The mathematical problem of how a shell should grow, under the assumptions which we have made, would probably find its most general statement in such a case as that of Globigerina, where the whole organism lives and grows freely poised in a medium whose density is little different from its ow^n. The majority of spiral forms, on the other hand, are plane or discoid spirals, and we may take it that in these cases some force has exercised a controlling influence, so as to keep all the chambers in a plane. This is especially the case in forms like Rotalia or Biscorhiiui (Fig. 430), where the organism lives attached to a rock or a frond of sea- weed ; for here (just as in the case of the coiled tubes which little worms such as Serpula and Spirorbis make, under similar conditions) the spiral disc is itself asymmetrical its whorls being markedly flattened on their attached surfaces. m:Wt^- "Fig. 430. Discorbina bertheloti d'Orb. 866 THE SPIRAL SHELLS [ch. We may also conceive, among other conditions, the very curious case in which the protoplasm may entirely overspread the surface of the shell without reaching a position of equilibrium; in which case a new shell will be formed enclosing the old one, whether the old one be in the form of a single, solitary chamber, or have already attained to the form of a chambered or spiral shell. This is precisely what often happens in the case of Orbulina, when within the spherical shell we find a small, but perfectly formed, spiral ''Globigerina*.'^ The various Miliolidae (Fig. 425) only differ from the typical spiral, or rotaline forms, in the large angle subtended by each chamber, and the consequent abruptness of their inclination to each other. In these cases the outward appearance of a spiral tends to be lost; and it behoves us to recollect, all the more, that our spiral curve is not necessarily identical with the outline of the shell, but is always a line drawn through corresponding points in the successive chambers of the latter. We reach a limiting case of the logarithmic spiral when the chambers are arranged in a straight Line ; and the eye will tend to associate with this limiting case the much more numerous forms in which the spiral angle is small, and the shell only exhibits a gentle curve with no succession of enveloping whorls. This constitutes the Nodosarian type (Fig. 134, p. 421); and here again, we must postulate some force which has tended to keep the chambers in a rectilinear series : such for instance as gravity, acting on a system of "hanging drops." In Textularia and its allies (Fig. 431) we have a precise parallel to the helicoid cyme of the botanists (cf. p. 767): that is to say we have a screw translation, perpendicular to the plane of the underlying logarithmic spiral. In other words, in tracing a genetic spiral through the whole succession of chambers, we do so by a continuous vector rotation through successive angles of 180° (or 120° in some cases), while the pole moves along an axis perpendicular to the original plane of the spiral. Another type is furnished by the "cyclic" shells of the Orbi- tolitidae, where small and numerous chambers tend to be added * Cf. G. Schacko, Ueber Glohigerina-^in^chluBS bei Orbulina, Wiegmann's Archiv, XLix, p. 428, 1883; Brady, Chall. Rep. 1884, p. 607. XII]' OF THE FORAMINIFERA 867 on round and round the system, so building up a circular flattened disc. This again we perceive to be, mathematically, a hmiting case of the logarithmic spiral; the spiral has become wellnigh a circle and the constant angle is wellnigh 90°. Lastly there are a certain number of Forarninifera in which, without more ado, we may simply say that the arrangement of the chambers is irregular, neither the law of constant ratio of magnitude nor that of constant form being obeyed. The chambers are heaped pell-mell upon one another, and such forms are known to naturahsts as the Acervularidae. A B Fig. 431. A, Textularia trochus d'Orb. B, T. concava Karrer. While in these last we have an extreme lack of regularity, we must not exaggerate the regularity or constancy which the more ordinary forms display. We may think it hard to believe that the simple causes, or simple laws, which we have described should operate, and operate again and again, in millions of individuals to produce the same delicate and complex conformations. But we are taking a good deal for granted if we assert that they do so, and in particular we are assuming, with very little proof, the "constancy of species" in this group of animals. Just as Verworn has shewn that the typical Amoeba proteus, when a trace of alkali is added to the water in which it lives, tends, by alteration of surface tensions, to protrude the more delicate pseudopodia characteristic of A. radiosa — and again when the water is rendered a little more alkaline, to turn apparently into the so-called A. liTnax — so it is evident that a very slight modification in the surface-energies concerned might tend 868 THE SPIRAL SHELLS [ch. to turn one so-called species into another among the Foraminifera. To what extent this process actually occurs, we do not know. But that this, or something of the kind, does actually occur we can scarcely doubt. For example in the genus Peneroplis, the first portion of the shell consists of a series of chambers arranged in a spiral or nautiloid series; but as age advances the spiral is apt to be modified in various ways*. Sometimes the successive chambers grow rapidly broader, the whole shell becoming fan-shaped. Some- times the chambers become narrower, till they no longer enfold the earlier chambers but only come- in contact each with its immediate predecessor: the result being that the shell straightens out, and (taking into account the earlier spiral portion) may be described as crozier-shaped. Between these extremes of shape, and in regard to other variations of thickness or thinness, roughness or smoothness, and so on, there are innumerable gradations passing one into another and intermixed without regard to geographical distribution: — " wherever Peneroplides abound this wide variation exists, and nothing can be more easy than to pick out a number of striking specimens and give to each a distinctive name, but in no other way can they be divided into ' species.' "f Some writers have wondered at the peculiar variability of this particular shellj; but for all we know of the life-history of the Foraminifera, it may well be that a great number of the other forms which we distinguish as separate species and even genera are no more than temporary manifestations of the same variabiUty§. * Cf. H. B. Brady, Challenger Rep., Foraminifera, 1884, p. 203, pi. xin. t Brady, op. cit. p. 206; Batsch, one of the earliest writers on Foraminifera, had already noticed that this whole series of ear-shaped and crozier-shaped shells was filled in by gradational forms; Conchylien des Seesandes, 1791, p. 4, pi. vi, fig. 15 a-/. See also, in particular, Dreyer, Peneroplis ; eine Studie zur biologischen Morphologie und zur Speciesfrage, Leipzig, 1898; also Eimer und Fickert, Artbildung und Verwandschaft bei den Foraminiferen, Tuhinger zool. Arbeiten, ni, p. 35, 1899. X Doflein, Protozoenlcunde, 1911, p. 263: "Was diese Art veranlasst in dieser Weise gelegentlich zu variiren, ist vorlaufig noch ganz rathselhaft." § In the case of Globigerina, some fourteen species (out of a very much larger number of described forms) were allowed by Brady (in 1884) to be distinct; and this list has been, I believe, rather added to than diminished. But these so-called species depend for the most part on slight differences of degree, differences in the angle of the spiral, in the ratio of magnitude of the segments, or in their area of contact one with another. Moreover with the exception of one or two "dwarf" XII] OF THE FORAMINIFERA 869 ConclvMon If we can comprehend and interpret on some such lines as these the form and mode of growth of the foraminiferal shell we may also begin to understand two striking features of the group, on the one hand the large number of diverse types or famihes which exist and the large number of species and varieties within each, and on the other the persistence of forms which in many cases seem to have undergone little change or none at all from the Cretaceous or even frorri earlier periods to the present day. In few other groups, perhaps only among the Radiolaria, do we seem to possess so nearly complete a picture of all possible transitions between form and form, and of the whole branching system of the evolutionary tree : as though little or nothing of it had ever perished, and the whole web of life, past and present, were as complete as ever. It leads one to imagine that these shells have grown according to laws so simple, so much in harmony with their material, with their environ- ment, and with all the forces internal and external to which they are exposed, that none is better than another and none fitter or less fit to survive. It invites one also to contemplate. the possibility of the lines of possible variation being here so narrow and determinate that identical forms may have come independently into being again and again. While we can trace in the most complete and beautiful manner the passage of one form into another among these httle shells, and ascribe them all at last (if we please) to a series which starts with the simple sphere of Orhulina or with the amoeboid body of Astrorhiza, the question stares us in the face whether this be an "evolution" which we have any right to correlate with historic time. The mathematician can trace one conic section into another, and " evolve " for example, through innumerable graded ellipses, the circle from the straight line: which tracing of continuous steps is a true "evolution," though time has no part therein. It was after this fashion that Hegel, and for that matter Aristotle himself, was an forms, said to be limited to Arctic and Antarctic waters, there is no principle of geographical distribution to be discerned amongst them. A species found fossil in New Britain turns up in the North Atlantic; a species described from the West Indies is rediscovered at the ice- barrier of the Antarctic. 870 THE SPIRAL SHELLS [ch. evolutionist — to whom evolution was a mental concept, involving order and continuity in thought but not an actual sequence of events in time. Such a conception of evolution is not easy for the modern biologist to grasp, and is harder still to appreciate. And so it is that even those who, like Dreyer* and like Rhumbler, study the foraminiferal shell as a physical system, who recognise that its whole plan and mode of growth is closely akin to the phenomena exhibited by fluid drops under particular conditions, and who explain the conformation of the shell by help of the same physical principles and mathematical laws^yet all the while abate no jot or tittle of the ordinary postulates of modern biology, nor doubt the validity and universal applicability.of the concepts of Darwinian evolution. For these writers the biogenetisches Grundgesetz remains impregnable. The Foraminifera remain for them a great family tree, whose actual pedigree is traceable to the remotest ages; in which historical evolution has coincided with progressive change; and in which structural fitness for a particular function (or functions) has exercised its selective action and ensured "the survival of the fittest." By successive stages of historic evolution we are supposed to pass from the irregular Astrorhiza to a Rhahdammina with its more concentrated disc; to the forms of the same genus which consist of but a single tube with central chamber; to those where this chamber is more and more distinctly segmented; so to the typical many-chambered Nodosariae; and from these, by another definite advance and later evolution to the spiral Trochamminae. After this fashion, throughout the whole varied series of the Fora- minifera, Dreyer and Rhumbler (following Neumayr) recognise so many successions of related forms, one passing into another and standing towards it in a definite relationship of ancestry or descent. Each evolution of form, from simpler to more complex, is deemed to have been attended by an advantage to the organism, an enhancement of its chances of survival or perpetuation ; hence the historically older forms are on the whole structurally the simpler; or conversely, the simpler forms, such as the simple sphere, were the first to come into being in primeval seas; and finally, the gradual development and increasing comphcation of the individual * F. Dreyer, Prinzipien der Gerustbildung bei Rhizopoden, etc., Jen. Zeitschr. XXVI, pp. 204-468, 1892. XII] or THE FORAMINIFERA 871 within its own lifetime is hfeld to be at least a partial recapitulation of the unknown history of its race and dynasty* We encounter many difficulties when we try to extend such concepts as these to the Foraminifera. We are led for instance to assert, as Rhumbler does, that the increasing complexity of the shell, and of the manner in which one chamber is fitted on another, makes for advantage; and the particular advantage on which Rhumbler rests his argument is strength. Increase of strength, die Festigkeitssteigerung, is according to him the guiding principle in foraminiferal evolution, and marks the historic stages of their development in geologic time. But in days gone by I used to see the beach of a little Connemara bay bestrewn with millions upon millions of foraminiferal shells, simple Lagenae, less simple Nodosariae, more complex Rotaliae : all drifted by wave and gentle current from their sea-cradle to their sandy grave: all lying bleached and dead: one more delicate than another, but all (or vast multitudes of them) perfect and unbroken. And so I am not incUned to beheve that niceties of form affect the case very much : nor in general that foraminiferal life involves a struggle for existence wherein breakage is a danger to be averted, and strength an advantage to be ensured f. In the course of the same argument Rhumbler remarks that Foraminifera are absent from the coarse sands and gravels J, as Williamson indeed had observed many years ago: so averting, or at least escaping, the dangers of concussion. But this is after all * A difficulty arises in the case of forms (like Peneroplis) where the young shell appears to be more complex than the old, the jfirst-formed portion being closely coiled while the later additions become straight and simple: "die biformen Arten verhalten sich, kurz gesagt, gerade umgekehrt als man nach dem biogenetischen Grundgesetz erwarten sollte," Rhumbler, op. cit. p. 33, etc. f "Das Festigkeitsprinzip als Movens der Weiterentwicklung ist zu interessant und fiir die Aufstellung meiries Systems zu wichtig um die Frage unerortert zu lassen, warum diese Bevorziigung der Festigkeit stattgefunden hat. Meiner Ansicht nach lautet die Antwort auf diese Frage einfach, well die Foraminiferen meistens unter Verhaltnissen leben, die ihre Schalen in hohem Grade der Gefahr des Zerbrechens aussetzen; es muss also eine fortwahrende Auslese des Festeren stattfinden," Rhumbler, op. cit. p. 22. X " Die Foraminiferen kiesige oder grobsandige Gebiete des Meeresbodens nicht lieben, u.s.w.": where the last two words have no particular meaning, save only that (as M. Aurelius says) "of things that use to be, we say commonly that they love to be." 872 THE SPIRAL SHELLS [ch. a very simple matter of mechanical analysis. The coarseness or fineness of the sediment on the sea-bottom is a measure of the current: where the current is strong the larger stones are washed clean, where there is perfect stillness the finest mud settles down; and the light, fragile shells of the Foraminifera find their appropriate place, like every other graded sediment in this spontaneous order of levigation. The theorem of Organic Evolution is one thing; the problem of deciphering the lines of evolution, the order of phylogeny, the degrees of relationship and consanguinity, is quite another. Among the higher organisms we arrive at conclusions regarding these things by weighing much circumstantial evidence, by dealing with the resultant of many variations, and by considering the probability or improbability of many coincidences of cause and effect; but even then our conclusions are at best uncertain, our judgments are continually open to revision and subject to appeal, and all the proof and confirmation we can ever have is that which comes from the direct, but fragmentary evidence of palaeontology*. But in so far as forms can be shewn to depend on the play of physical forces, and the variations of form to be directly due to simple quantitative variations in these, just so far are we thrown back on our guard before the biological conception of consanguinity, and compelled to revise the vague canons which connect classification with phylogeny. The physicist explains in terms of the properties of matter, and classifies according to a mathematical analysis, all the drops and forms of drops and associations of drops, all the kinds of froth and foam, which he may discover among inanimate things; and his task ends there. But when such forms, such conformations and configurations, occur among living things, then at once the biologist introduces his concepts of heredity, of historical evolution, of suc- cession in time, of recapitulation of remote ancestry in individual growlh, of common origin (unless contradicted by direct evidence) of similar forms remotely separated by geographic space or geologic time, of fitness for a function, of adaptation to an environment, of higher and lower, of " better " and " worse." This is the fundamental * In regard to the Foraminifera, "die Palaeontologie lasst uns leider an Anfang der Stammesgeschichte fast ganzlich im Stiche," Rhumbler, op. cit. p. 14. XII] OF THE FORAMINIFERA 873 difference between the "explanations" of the physicist and those of the biologist. In the order of physical and mathematical complexity there is no question of the sequence of historic time. The forces that bring about the sphere, the cylinder or the ellipsoid are the same yesterday and to-morrow. A snow-crystal is the same to-day as when the first snows fell. The physical forces which mould the forms of Orbulina, of Astrorhiza, of Lagena or of Nodosaria to-day were still the same, and for aught we have reason to believe the physical conditions under which they worked were not appreciably different, in that yesterday which we call the Cretaceous epoch; or, for aught we know, throughout all that duration of time which is marked, but not measured, by the geological record. In a word, the minuteness of our organism brings its conformation as a whole within the range of the molecular forces; the laws of its growth and form appear to lie on simple lines; what Bergson calls* the "ideal kinship" is plain and certain, but the "material afiihation" is problematic and obscure; and, in the end and upshot, it seems to me by no means certain that the biologist's usual mode of reasoning is appropriate to the case, or that the concept of con- tinuous historical evolution must necessarily, or may safely and legitimately, be employed. That things not only alter but improve is an article of faith, and the boldest of evolutionary conceptions. How far it be true -were very hard to say; but I for one imagine that a pterodactyl flew no less w^ell than does an albatross, and that Old Red Sandstone fishes swam as well and easily as the fishes of our own seas. * The evolutionist theory, as Bergson puts it, "consists above all in establishing relations of ideal kinship, and in maintaining that wherever there is this relation of, so to speak, logical affiliation between forms, there is also a relation of chronological succession between the species in which these forms are materialised''^ {Creative Evolution, 1911, p. 26). Cf. supra, p. 412. CHAPTEPv XIII THE SHAPES OF HORNS, AND OF TEETH OR TUSKS: WITH A NOTE ON TORSION We have had so much to say on' the subject of shell-spirals that we must deal briefly with the analogous problems which are presented by the horns of sheep, goats, antelopes and other horned quadrupeds; and all the more, because these horn-spirals are on the whole less symmetrical, less easy of measurement than those of the shell, and in other ways also are less easy of investigation. Let us dispense altogether in this case with mathematics; and be content with a very simple account of the configuration of a horn. There are three types of horn which deserve separate consideration : firstly, the horn of the rhinoceros; secondly, the horns of the sheep, the goat, the ox or the antelope, that is to say, of the so-called hollow-horned ruminants; and thirdly, the solid bony horns, or "antlers," which are characteristic of the deer. The horn of the rhinoceros presents no difficulty. It is physio- logically equivalent to a mass of consolidated hairs, and, like ordinary hair, it consists of non-living or "formed" material, continually added to by the living tissues at its base. In section the horn is elliptical, with the long axis fore-and-aft, or in some species nearly circular. Its longitudinal growth proceeds with a maximum velocity anteriorly, and a minimum posteriorly; and the ratio of these velocities being constant, the horn curves into the form of a loga- rithmic spiral in the manner that we have already studied. The spiral is of small angle, but in the longer-horned species, such as the great white rhinoceros (Ceratorhinus), the spiral curvature is distinctly recognised. As the horn occupies a median position on the head — a position, that is to say, of symmetry in respect to the field of force on either side — there is no tendency towards a lateral twist, and the horn accordingly develops as a plane logarithmic spiral. When two median horns coexist, the hinder one is much the smaller of the two: which is as much as to say that the force, CH. XIII] THE HORNS OF SHEEP AND GOATS 875 or rate, of growth diminishes as we pass backwards, just as it does within the limits of the single horn. And accordingly, while both horns have essentially the same shape, the spiral curvature is less manifest in the second one, by the mere reason of its shortness. The paired horns of the ordinary hollow-horned ruminants, such as the -sheep or the goat, grow under conditions which are in some respects similar, but which differ in other and important respects from the conditions under which the horn grows in the rhinoceros. As regards its structure, the entire horn now consists of a bony core with a covering of skin ; the inner, or dermal, layer of the latter is richly supplied with nutrient blood-vessels, while the outer layer, or epidermis, develops the fibrous or chitinous material, chemically and morphologically akin to a mass of cemented or consolidated hairs, which constitutes the " sheath" of the horn. A zone of active growth at the base of the horn keeps adding to this sheath, ring by ring, and the specific form of this annular zone may be taken as the "generating curve" of the horn*. Each horn no longer lies, as it does in the rhinoceros, in the plane of symmetry of the animal of which it forms a part; and the limited field of force con- cerned in the genesis and growth of the horn is bound, accordingly, to be more or less laterally asymmetrical. But the two horns are in symmetry one with another; they form "conjugate" spirals, one being the "mirror-image" of the other. Just as in the hairy coat of the animal each hair, on either side of the median "parting," tends to have a certain definite direction of its owji, inclined away from the median axial plane of the whole system, so is it both with the bony core of the horn and with the consolidated mass of hairs or hair-like substance which constitutes its sheath; the primary axis of the horn is more or less inclined to, and may even be nearly perpendicular to, the axial plane of the animal. The growth of the horny sheath is not continuous, but more or less definitely periodic : sometimes, as in the sheep, this periodicity is particularly well-marked, and causes the horny sheath to be com- * In this chapter we keep to Moseley's way of regarding the equiangular spiral in space, of shell or horn, as generated by a certain figure which (a) grows, (6) revolves about an axis, and (c) is translated along or parallel to the said axis, all at certain appropriate and specific velocities. This method is simple, and even adequate, from the naturalist's point of view; but not so, or much less so, from the mathe- matician's, as we have found in the last chapter (p. 782). 876 THE SHAPES OF HORNS [ch. posed of a series of all but separate rings, which are supposed to be formed year by year, and so to record the age of the animal*. Just as Moseley sought for the true generating curve in the orifice, or "lip," of the moUuscan shell, so we begin by assuming that in the spiral horn the generating curve corresponds to the lip or margin of one of the horny rings or annuli. This annular margin, or boundary of the ring, is usually a sinuous curve, not lying in a plane, but such as would form the boundary of an anticlastic surface of great complexity: to the meaning and origin of which phenomenon we shall return presently. But, as we have already seen in the case of the molluscan shell, the complexities of the Up itself, or of the corresponding lines of growth upon the shell, need Fig. 432. The Argali sheep; Ovis Ammon. From Cook's Spirals in Nature and Art. not concern us in our study of the development of the spiral: inasmuch as we may substitute for these actual boundary lines, their "trace," or projection on a plane perpendicular to the axis — in other words the simple outline of a transverse section of the whorl. In the horn, this transverse section is often circular or nearly so, as in the oxen and many antelopes: it now and then becomes of * Cf. R. S. Hindekoper, On the Age of the Domestic Animals, Philadelphia and London, 1891, p. 173. In the case of the ram's horn, the assumption that the rings are annual is probably justified. In cattle they are much less conspicuous, but are sometimes well-marked in the cow; and in Sweden they are then called "calf-rings," from a belief that they record the number of offspring. That is to say, the growth of the horn is supposed to be retarded during gestation, and to be accelerated after parturition, when superfluous nourishment seeks a new outlet. (Cf. Lonnberg, P.Z.S. 1900, p. 689.) XIII] OF SHEEP AND GOATS 877 somewhat complicated polygonal outline, as in a highland ram ; but in many antelopes, and in most of the sheep, the outline is that of an isosceles or sometimes nearly equilateral triangle, a form which is typically displayed, for instance, in Ovis Ammon. The horn in this latter case is a trihedral prism, whose three faces are (1) an upper, or frontal face, in continuation of the plane of the frontal bone; (2) an outer, or orbital, starting from the upper margin of the orbit; and (3) an inner, or nuchal, abutting on the parietal bone*. Along these three faces, and their corresponding angles or edges, we can trace in the fibrous substance of the horn a series of homologous spirals, such as w^e have called in a preceding chapter the ''ensemble of generating spirals" which define or constitute the surface. The case of the horn differs in ways of its own from that of the molluscan shell. For one thing, the horn is always tubular — its generating curve is actually, as well as theoretically, a closed curve; there is no such thing as "involution," or the wrapping of one whorl within another, or successive intersection of the generating curve. Again, while the calcareous substance of the shell is laid down once for all, fixed and immovable, there is reason to believe that the young horn has, to begin with, a certain measure of flexi- bility, a certain freedom, even though it be slight, to bend or fold or wrinkle. And this being so, while it is no harder in the horn than in the shell to recognise the general field of force or general direction of growth, the actual conditions are somewhat more complex. In some few cases, of which the male musk ox is- one of the most notable, the horn is not developed in a continuous spiral curve. It changes its shape as growth proceeds; and this, as we have seen, is enough to show that it does not constitute a logarithmic spiral. The reason is that the bony exostoses, or horn-cores, about which the horny sheath is shaped and moulded, neither grow continuously nor even remain of constant size after attaining their full growth. But as the horns grow heavy the bony core is bent downwards by their weight, and so guides the growth of the horn in a new direction. Moreover as age advances, the core is further weakened and to a great extent absorbed: and the horny sheath or horn proper, * Cf. Sir V. Brooke, On the large sheep of the Thian Shan, P.Z.S. 1875, p. 511. 878 THE SHAPES OF HORNS [CH. deprived of its support, continues -to grow, but in a flattened curve very different from its original spiral*. The chamois is a somewhat analogous case. Here the terminal, or oldest, part of the horn is curved; it tends to assume a spiral form, though from its com- parative shortness it seems merely to be bent into a hook. But later on the bony core within, as it grows and strengthens, stiffens the horn and guides it into a straighter course or form. The same phenomenon of change of curvature, manifesting itself at the time when, or the place where, the horn is freed from the support of the internal core, is seen in a good many other antelopes (such- as the Fig. 4.33. Diagram of ram's horns, a, frontal; h, orbital; c, nuchal surface. After Sir Vincent Brooke, from P.Z.S. hartebeest) and in many buffaloes; and the cases where it is most manifest appear to be those where the bony core is relatively short, or relatively weak. All these illustrate the cardinal difference between the growth of the horn and that of the bone below: the one dead, the other alive; the one adding and retaining its successive increments, the other mobile, plastic, and in eontinual flux through- out. But in the great majority of horns we have no difficulty in recognising a continuous logarithmic spiral, nor in correlating it with an unequal rate of growth (parallel to the axis) on two opposite sides of the horn, the inequality maintaining a constant ratio as long as growth proceeds. In certain antelopes, such as the gemsbok, the spiral angle is very smafl, or in other words the horn * Cf. E. Lonnberg, On the structure of the musk ox, P.Z.S. 1900, pp. 686-718. XIII] OF SHEEP AND GOATS 879 is very nearly straight; in other species of the same genus OryXy such as the Beisa antelope and the Leucoryx, a gentle curve (not unlike though generally less than that of a Dentalium shell) is evident; and the spiral angle, according to the few measurements I have made, is found to measure from about >20° to nearly 40°. In some of the large wild goats, such as the Scinde wild goat, we have Fig. 484. Head of Arabian wild goat, Capra sinaitica. After Sclater, from P.Z.S. a beautiful logarithmic spiral, with a constant angle of rather less than 70°; and we may easily arrange a series of forms, such for example as the Siberian ibex, the moufflon, Ovis Ammon, etc., and ending with the long-horned Highland ram: in which, as we pass from one to anothei', we recognise precisely homologous spirals with an increasing angular constant^ the spiral angle being, for instance, about 75° or rather less in Ovis Ammon, and in the Highland ram a very little more. We have already seen that in the neighbourhood of 70° or 80° a small change of angle makes a marked difference in 880 THE SHAPES OF HORNS [ch. the appearance of the spire ; and we know also that the actual length of the horn makes a very striking difference, for the spiral becomes especially conspicuous to the eye when horn or shell is long enough to shew several whorls, or at least a considerable part of one entire convolution. Even in the simplest cases, such as the wild goats, the spiral is never a plane but always a gauche spiral : in greater or less degree there is always superposed upon the plane logarithmic spiral a helical spiral in space. Sometimes the latter is scarcely apparent, for the horn (though long, as in the said wild goats) is not nearly long enough to shew a complete convolution : at other times, as in the ram, and still better in many antelopes such as the koodoo, the corkscrew curve of the horn becomes its most characteristic feature. So we may study, as in the molluscan shell, the helicoid component of the spire — in other words the variation in what we have called (on p. 816) the angle j^. This factor it is which, more than the constant angle of the logarithmic spiral, imparts a characteristic appearance to the various species of sheep, for instance to the various closely allied species of Asiatic wild sheep, or Argali. In all of'these the constant angle of the logarithmic spiral is very much the same, but the enveloping angle of the cone differs greatly. Thus the long drawn out horns of Ovis Poli, four feet or more from tip to tip, diifer conspicuously from those of Ovis Ammon or 0. hodgsoni, in which a very similar logarithmic spiral is wound (as it were) round a much blunter cone. Let us continue to dispense with mathematics, for the mathe- matical treatment of a gauche spiral is never very simple, and let us deal with the matter by experiment. We have seen that the generating curve, or transverse section, of a typical ram's horn is triangular in form. Measuring (along the curve of the horn) the length of the three edges of the trihedral structure in a specimen of Ovis Ammon, and calling them respectively the outer, inner, and hinder edges (from their position at the base of the horn, relatively to the skull), I find the outer edge to measure 80 cm., the inner 74 cm., and the posterior 45 cm. ; let us say that, roughly, they are in the ratio of 9:8:5. Then, if we make a number of little cardboard triangles, equip each with three httle legs (I make them of cork), whose relative lengths are as 9:8:5, and pile them up XIII] OF SHEEP AND GOATS 881 and stick them all together, we straightway build up a curve ol double curvature precisely analogous to the ram's horn: except only that, in this first approximation, we have not allowed for the gradual increment (or decrement) of the triangular surfaces, that is to say, for the tapering of the horn due to the magnification of the generating curve. In this case then, and in most other trihedral or three-sided horns, one of the three components, or three unequal velocities of growth, is of relatively small magnitude, but the other two are nearly equal one to the other; it would involve but little change for these latter to become precisely equal; and again but little to turn the balance of inequality the other way. But the immediate consequence of this altered ratio of growth would be that the horn would appear to wind the other way, as it does in the antelopes, and also in certain goats, e.g. the markhor, Capra falconeri. For these two' opposite directions of twist Dr Wherry has suggested a convenient nomenclature. When the horn winds so that we follow it from base to apex in the direction of the hands of a watch, it is customary to call it a "left-handed" spiral. Such a spiral we have in the horn on the left-hand side of a ram's head. Accordingly, Dr Wherry calls the condition homonymous, where, as in the sheep, a right-handed spiral is on the right side of the head, and a left-handed spiral on the left sid,e ; while he calls the opposite condition heteronymmis, as we have it in the antelopes, where the right-handed twist is on the left side of the head, and the left-handed twist on the right-hand side. Among the goats, we may have either condition. Thus the domestic and most of the wild goats agree with the sheep; but in the markhor the twisted horns are heteronymous, as in the antelopes. The difference, as we have seen, is easily explained; and (very much as in the case of our opposite spirals in the apple-snail, referred to on p. 820) it has no very deep importance Summarised then in a very few words, the argument by which we account for the spiral conformation of the horn is as follows : The horn elongates by dint of continual growth within a narrow zone, or annulus, at its base. If the rate of growth be identical on all sides of this zone, the horn will "grow straight; if it be greater on one side than on the other, the horn will become curved; and it probably will be greater on one side than on the other, because each single horn occupies an unsymmetrical field with reference to the plane of symmetry of the animal. If the maximal and minimal velocities of growth be precisely at opposite sides of the zone of 882 THE SHAPES OF HORNS [ch. growth, the resultant spiral will be a plane spiral ; but if they be not precisely or diametrically opposite, then the spiral will be a gauche spiral in space ; and it is by no means likely that the maximum and minimum will occur at precisely opposite ends of a diameter, for no such plane of symmetry is manifested in the field of force to which the growing annulus corresponds or appertains. Now we must carefully remember that the rates of growth of which we are here speaking are the net rates of longitudinal increment, in which increment the activity of the living cells in the zone of growth at the base of the horn is only one (though it is the fundamental) factor. In other words, if the horny sheath were continually being added to with equal rapidity all round its zone of active growth. Fig. 4,*i5. Marco Polo's sheep: Ovis FoU. From Cook. but at the same time had its elongation more retarded on one side than the other (prior to its complete solidification) by varying degrees of adhesion or membranous attachment to bhe bony core within, then the net result would be a spiral curve precisely such as would have arisen from initial inequalities in the rate of growth itself. It seems probable that this is an important factor, and sometimes even the chief factor in the case. The same phenomenon of attachment to the bony core, and the consequent friction or retardation with which the sheath slides over its surface, will lead to various subsidiary phenomena: among others to the presence of transverse folds or corrugations upon the horn, and to their unequal distribution upon its several faces or edges. And while it is perfectly true that nearly all the characters of the hotn ca n be accounted for by unequal velocities XIII] OF SHEEP AND GOATS 883 of longitudinal growth upon its different sides, it is also plain that the actual field of force is a very complicated one indeed. For example, we can easily see (at least in the great majority of cases) that the direction of growth of the horny fibres of the sheath is by no means parallel to the axis of the core within; accordingly these fibres will tend to wind in a system of helicoid curves around the core, and not only this hehcoid twist but any other tendency to spiral curvature on the part of the sheath will tend to be opposed or modified by the resistance of the core within. On the other hand living bone is a very plastic structure, and yields easily though slowly to any forces tending to its deformation; and so, to a considerable extent, the bony core itself will tend to be modelled by the curvature which the Fig. 43(5. Head of Ovis Ammon, shewing St Venant's curves. growing sheath assumes, and the final result will be determined by an equilibrium between these two systems. While it is not very safe, perhaps, to lay down any general rule as to what horns are more and what are less spirally curved, I think it may be said that, on the whole, the thicker the horn the greater is its spiral curvature. It is the slender horns, of such forms as the Beisa antelope, which are gently curved, and it is the robust horns of goats or of sheep in which the curvature is more pronounced. Other things being the same, this is what we should expect to find ; for it is where the transverse section of the horn is large that we may expect to find the more marked differences in the intensity of the field of force, whether of active growth or of retardation, on opposite sides or in different sectors thereof. 884 THE SHAPES OF HORNS [ch. But there is yet another and a very remarkable phenomenon which we may discern in the growth of a horn when it takes the form of a curve of double curvature^ namely, an effect of torsional strain; and this it is which gives rise to the sinuous " lines of growth,"' or sinuous boundaries of the separate horny rings, of which we have already spoken. It is not at first sight obvious that a mechanical strain of torsion is necessarily involved in the growth of the horn. In our experimental illustration (p. 880), we built up a twisted coil of separate elements, and no torsional strain attended the development of the system. So would it be if the horny sheath grew by successive annular increments, free save for their relation to one another and having no attachment to the 'Solid core within. But as a matter of fact there is such an attachment, by subcutaneous connective tissue, to the bony core; and accordingly a torsional strain will be set up in the growing horny sheath, again provided that the forces of growth therein be directed more or less obliquely to the axis of the core; for a -'couple" is thus introduced, giving rise to a strain which the sheath would not experience were it free (so to speak) to shp along, impelled only by the pressure of its own growth from below. And furthermore, the successive small increments of the growing horn (that is to say, of the horny sheath) are not instan- taneously converted from living to solid and rigid substance; Init there is an intermediate stage, probably long-continued, during which the new-formed horny substance in the neighbourhood of the zone of active growth is still plastic and capable of deformation. Now we know, from the celebrated experiments of St Venant*, that in the torsion of an elastic body, other than a cylinder of circular section, a very remarkable state of strain is introduced. If the body be thus cylindrical (whether solid or hollow), then a twist leaves each circular section unchanged, in dimensions and in figure. But in all other cases, such as an elliptic rod or a prism of any particular sectional form, forces are introduced which act parallel to the axis of the structure, and which warp each section into a complex " anticlastic " surface. Thus^ in the case of a triangular and ♦ St Venant, De la torsion ties prismes, avec des considerations sur leur flexion, etc., Mem. des Savants J'jtrangers, Paris, xiv, pp. 233-560, 1856. Karl Pearson dedicated part of his History of the Theory of Elasticity to the memory of this ingenious and original man. For a modern account of the subject see Love's Elasticity (2nd ed.), chaj), xiv. XIII ST VENANT'S EXPERIMENTS 885 e(|uilateral prism, such as is shewn in section in Fig. 437 A, if the part of the rod represented in the section be twisted by a force acting in the direction of the arrow, then the originally plane section will be warped as indicated in the diagram — where the full contour-lines represent elevation above, and the dotted lines represent depression below, the original level. On the external surface of the prism, then," contour-lines which were originally parallel and horizontal will be found warped into sinuous curves, such that, on each of the three faces, the curve will be convex upwards on one half, and concave upwards on the other half of the face. The ram's horn, and still better that of Ovis Ammon, is comparable to such a prism, A Fig. 437. save that in section it is not quite equilateral, and that its three faces are not plane. The warping is therefore not precisely identical on the three faces of the horn; but, in the general distribution of the curves, it is in complete accordance with theory*. Similar anticlastic curves ?ire well seen in many, antelopes; but they are conspicuous by their absence in the cylindrical horns of oxen. The better to illustrate this phenomenon, the nature of which is indeed obvious enough from a superficial examination of the horn, I made a plaster cast of one of the horny rings in a horn of Ovis Arnmon, so as to get an accurate pattern of its sinuous edge : and then, filling the mould up -with wet clay, I modelled an anticlastic * The case of a thin conical shell under torsion is more complicated than either that of the cylinder or of a prismatic rod ; and the more tapering, horns doubtless deserve further study from this point of view. Cf. R. V. Southwell, On the torsion of conical shells, Proc. R.S. (A), clxiii, pp. 337-355, 1937. 886 THE SHAPES OF HORNS [ch. surface, such as to correspond as nearly as possible with the sinuous outhne*. Finally, after making a plaster oast of this sectional surface, I drew its contour-lines (as shewn in Fig. 437 B) with the help of a simple form of spherometer. It will be seen that in great part this diagram is precisely similar to St Venant's diagram of the •cross-section of a twisted triangular prism; and this is especially the case in the neighbourhood of the sharp angle of our prismatic section. That in parts the diagram is somewhat asymmetrical is not to be wondered at: and (apart from inaccuracies due to the somewhat rough means by which it was made) this asymmetry can be sufficiently accounted for by anisotropy of the material, by inequalities in thickness of different parts of the horny sheath, and especially (I think) by unequal distributions of rigidity due to the presence of the smaller corrugations of the horn. It is on account of these minor corrugations that in such horns as the Highland ram's, where they are strongly marked, the main St Venant effect is not nearly so well shewn as in smoother horns, such as those of Ovis Ammon and its congeners f. The distribution of forces which manifest themselves in the growth and configuration of a horn is no simple nor merely super- ficial matter. One thing is coordinated with another; the direction of the axis of the horn, the form of its sectional boundary, the specific rates of growth in the mean spiral and at various parts of its periphery — all' these play their parts, controlled in turn by the supply of nutriment which the character of the adjacent tissues and the distribution of the blood-vessels combine to determine. To suppose. that this or that size or shape of horn has been pro- duced or altered, acquired or lost, by Natural Selection, whensoever one type rather than another proved serviceable for defence or attack or any other purpose, is an hypothesis harder to define and to substantiate than some imagine it to be. There are still one or two small matters to speak of before we leave these spiral horns. It is the way of sportsmen to keep record of big game by measuring the length along the curve of the horn and the span from tip to tip. Now if we study such measurements (as * This is not difficult to do, with considerable accuracy, if the clay be kept well wetted or semi-fluid, and the smoothing be done with a large wet brush. t The curves are well shewn in most of Sir V. Brooke's figures of the various species of Argali, in the paper quoted above, on p. 877. xm] OF SHEEP AND GOATS 887 they may be found in Mr Rowland Ward's book*), we shall soon see that the two measurements do not tally one with the other: but that a pair of horns, the longer when measured along the curve, may be the shorter from tip to tip, and vice versa. We might set this down to mere variability of form, but the true reason is simpler still. If the axes of the two horns stood straight out, at right angles to the median plane, then growth in length and in width of span would go on together. But if the two horns diverge at any lesser angle, then as the horns grow their spiral curvature will tend to bring their tips nearer and farther apart alternately. There is one last, but not least curious property to be seen in a ram's horns. However large and heavy the horns may be — and in Ovis Poll 50 or 60 lb. is no unusual weight for the pair to grow to — the ram carries them with grace and ease, and they neither endanger his poise nor encumber his movements. The reason is that head and horns are very perfectly balanced, in such a way that no bending moment tends to turn the head up or down about its fulcrum in the atlas vertebra ; if one puts two fingers into the foramen magnum one may lift up the heavy skull, and find it hang in perfect equili- brium. Moreover, the horns go on growing, but this equipoise is never lost nor changed ; for the centre of gravity of the logarithmic spiral remains constant. There are other cases where heavy horns, well balanced as they doubtless are, yet visibly affect the set and balance of the head. The stag carries his head higher than a horse, and an Indian buffalo tilts his muzzle higher than a cow. A further note upon torsion The phenomenon of torsion, to which we have been thus intro- duced, opens up many wide questions in connection with form. Some of the associated phenomena are admirably illustrated in the case of climbing plants ; but we can only deal with these still more briefly and parenthetically. The subject has been elaborately dealt with not only in Darwin's books f, but also by a great number of earlier and later writers. In "twining" plants, which constitute the greater number of "climbers," the essential phenomenon is a tendency of the growing shoot to revolve about a vertical axis — * Records of Big Game, 9th edition, 1928. t Climbing Plants, 1865 (2nd ed. 1875); Power of Movement in Plants, 1880. 888 THE SHAPES OF HORNS [ch. a tendency long ago discussed by de Candolle and investigated by Palm, H. von Mohl and Dutrochet*. This tendency to revolution — circumvolution, as Darwin calls it, revolving nutation, as Sachs puts it— is very closely comparable to the process by which an antelope's horn (such as the koodoo's) acquires its spiral twist, and is due, in like manner, to inequahties in the rate of growth of the growing stem : with this difference between the two, that in the antelope's horn the zone of active growth is confined to the base of the horn, while in the climbing stem the same phenomenon is at work throughout the whole length of the growing structure. This growth is in the main due to "turgescence," that is to the extension, or elongation, of ready-formed cells through the imbibition of water ; it is a phe- nomenon due to osmotic pressure. The particular stimulus to which these movements (that is to say, these inequalities of growth) have been ascribed can hardly be discussed here; but it was hotly debated fifty years ago and for many years thereafter, the point at issue being no other than whether direct physical causation, or the Darwinian concept of fitness or adaptation, should be invoked as an "explanation" of biological phenomena. The old Natur- philosophie had been inclined to look for spirals everywhere, and to attribute them to very simple causes: "Man wird nicht gross irren" (said Okenf) "wenn man sagt, alle Pflanzen entstehen als Spirale, und zwar weil sie feststehen und ein End gegen die Sonne kehren, die taglich einen Spiralgang um sie ihacht, u.s.w." When de Candolle saw a shoot curve under the influence of light (by helio- tropism, as we are told to call it), he was content to regard the curvature as the result of difi'erent rates of growth on one side or other of the shoot, and these in turn as the direct result of differences of illumination. But by the Darwins, father and son, and by Sachs and by the Wiirzburg school, the curvature was ascribed to "irritability," a "stimulus" on one side of the shoot being followed by a "motor-reaction" on the other. The curvature was thus taken to be a "response" to external stimuli (such as light and gravity); and stimulus and response were supposed to have * Palm, Ueber das Winden der Pflanzen, 1827; H. von Mohl, Bau und Windert der Ranken, etc., 1827; R. H. J. Dutrochet, Sur la volubilite des tiges de certains vegetaux, et sur la cause de ce phenomene, Ann. Sc. Nat. {Bot.), ii, pp. 156-167, 1844, and other papers, t I sis, I, p. 222, 1817. XIII] A NOTE UPON TORSION 889 evolved together in the course of ages, to bring about something more and more fitted for survival in the struggle for existence. They were, in short, of the nature of acquired habits, rather than physical phenomena. But there was no gainsaying the fact that the immediate cause of curvature was inequahty of growth on opposite sides*. A simple stem growing upright in the dark, or in uniformly diffused light, would be in a position of equilibrium to a field of force radially symmetrical about its vertical axis. But this complete radial sym- metry will not often occur; and the radial anomalies may be such as arise intrinsically from structural pecuHarities in "the stem itself, or externally to it by reason of unequal illumination or through various other locahsed forces. The essential fact, so far as we are concerned, is that in twining plants we have a very marked tendency to inequahties in longitudinal growth on different aspects of the stem^a tendency which is but an exaggerated manifestation of one which is more or less present, under certain conditions, in all plants whatsoever. Just as in the case of the ruminants' horns so we find here that this inequality may be, so to speak, positive or negative, the maximum lying to the one side or the other of the twining stem ; and so it coiftes to pass that some climbers twine to the one side and some to the other : the hop and the honeysuckle following the sun, and the field-convolvulus twining in the reverse direction ; there are also some, like the woody nightshade (/So/ai, in the horizontal plane, but also by an angle of slope {6), or merely by linear distance from its basal plane; for the height of B above the level of ^, in comparison with the diameter of the cylinder, will obviously make a great difference in the appearance of the whole system. But this matter botanical students have not concerned themselves with ; in other words, their studies have been limited (or mainly limited) to the relation of the leaves to one ailother in azimuth — in other words, to the angle ^ and its multiples. Whatever relation we have found between A and B, let precisely the same relation subsist between B and C: and so on. Let the growth of the system, that is to say, be continuous and uniform; it is then evident that we have the elementary conditions for the development of a simple cyhndrical helix; and this "primary helix" or "genetic spiral" we can now trace, winding round and round the stem, through A, B,C, etc. But if we can trace such a helix through A, B, C, it follows from the symmetry of the system, that we have only to join A to some other leaf to trace another spiral helix, such, for instance, as ^, C, E, etc. ; parallel to which will run another and similar one, namely in this case B, D, F, etc. And these spirals will run in the opposite direction to the spiral ABC*. In short, the existence of one helical arrangement of points im- plies and involves the existence of another and then another helical pattern, just as, in the pattern of a wall-paper, our eye travels from one linear series to another. A modification of the helical system will be introduced when, instead of the leaves appearing, or standing, in singular succession, w^ get two or more appearing simultaneously upon the same level. If there be two such, then we shall have two generating spirals * For the spiral ACE to be different from ABC, the angle of divergence, or angle of azimuth for one step, must exceed 90°, so that the nearer way from ^ to C is backwards; otherwise the spiral ACE is ABCDE, or ABC over again. 918 ON LEAF-ARRANGEMENT [ch. precisely equivalent to one another; and we may call them A, B, C, etc., and A', B', C\ and so on. These are the cases which we call "whorled" leaves; or in the simplest case, where the whorl consists of two opposite leaves only, we call them *' decussate." Among the phenomena of phyllotaxis, two points in particular have been found difficult of explanation, and have aroused dis- cussion. These are (1), the presence of the logarithmic spirals such as we have already spoken of in the sunflower; and (2) the fact that, as regards the number of the helical or spiral rows, certain numerical coincidences are apt to recur again and again, to the exclusion of others, and so to become characteristic features of the phenomenon. As to the first of these, we have seen that the curves resemble, and sometimes closely resemble, the logarithmic spiral; but that they are, strictly speaking, logarithmic is neither proved nor capable of proof. That they appear as spiral curves (whether equable or logarithmic) is then a mere matter of mathematical "deformation." The stem which we have begun to speak of as a cyhnder is not strictly so, inasmuch as it tapers off towards its summit. The curve which winds evenly around this stem is, accordingly, not a true helix, for that term is confined to the curve which winds evenly around the cylinder : it is a curve in space which (like the spiral curve we have studied in our turbinate shells) partakes of the characters of a helix and of a spiral, and which is in fact a spiral with its pole drawn out of its original plane by a force acting in the direction of the axis. If we imagine a tapering cylinder, or cone, projected by vertical projection on a plane, it becomes a circular disc; and a helix described about the cone becomes in the disc a spiral described about a pole which corresponds to the apex of our cone. In like manner we may project an identical spiral in space upon such surfaces as (for instance) a portion of a sphere or of an ellipsoid ; and in all these cases we preserve the spiral configuration, which is the more clearly brought into view the more we reduce the vertical component by which it was accompanied. The converse is equally true, and equally obvious, namely that any spiral traced upon a circular disc or spheroidal surface will be transformed into a corresponding spiral helix when the plane or spheroidal disc is extended into an elongated XIV] OR PHYLLOTAXIS 919 cone approximating to a cylinder. This mathematical conception is translated, in botany, into actual fact. The fir-cone may be looked upon as a cylindrical axis coiltracted at both ends, until it becomes approximately an ellipsoidal solid of revolution, generated about the long axis of the ellipse ; and the semi-ellipsoidal capitulum of the teasel, the more or less hemispherical one of the thistle, and the flattened but still convex one of the sunflower, are all beautiful and successive deformations of what is typically a long, conical, and all but cylindrical stem. On the other hand, every stem as it grows out into its long cylindrical shape is but a deformation of the little spheroidal or ellipsoidal or conical surface which was its forerunner in the bud. This identity of the helical spirals around the stem with spirals projected on a plane was clearly recognised by Hofmeister, who was accustomed to represent his diagrams of leaf-arrangement either in one way or the other, though not in a strictly geometrical projection*. According to Mr A. H. Church f, who has dealt carefully and elaborately with the whole question of phyllotaxis, the spirals such as we see in the disc of the sunflower have a far greater importance and a far deeper meaning than this brief treatment of mine would accord to them : and Sir Theodore Cook, in his book on the Curves of Life, adopted and helped to expound and popularise Mr Church's investigations. Mr Church, regarding the problem as one of "uniform growth," easily arrives at the conclusion that, if this growth can be conceived as taking place sjnnmetrically about a central point or "pole," the uniform growth would then manifest itself in logarithmic spirals, including df course the limiting cases of the circle and straight line. With this statement I have Httle fault to find; it is in essence identical with much that I have said in a previous chapter. But other statements of Mr Church's, and many theories woven about them by Sir T. Cook and himself, I am less able to follow. Mr Church tells us that the essential phenomenon in the sunflower disc is a series of orthogonally intersecting logarithmic spirals. Unless I wholly misapprehend Mr Church's meaning, I should say that this * Allgemeine Morphologie der Gewdchse, 1868, p. 442, etc. t Relation of Phyllotaxis to Mechanical Laws, Oxford, 1901-1903; cf. Ann. Bot, XV, p. 481, 1901. 920 ON LEAF-ARRANGEMENT [ch. is very far from essential. The spirals intersect isogonally, but orthogonal intersection would be only one particular case, and in all probability a very infrequent one, in the intersection of logarithmic spirals developed about a common pole. Again on the analogy of the hydrodynamic lines of force in certain vortex movements, and of similar lines of force in certain magnetic phenomena, Mr Church proceeds to argue that the energies of life follow lines comparable to those of electric energy, and that the logarithmic spirals of the sunflower are, so to speak, lines of equipotential*. And Sir T. Cook remarks that thife "theory, if correct, would be fundamental for all forms of growth, though it would be more easily observed in plant construction than in animals." But the physical analogies are remote, and the deductions I am not able to follow. Mr Church sees in phyllotaxis an organic mystery, a something for which we are unable to suggest any precise cause : a phenomenon which is to be referred, somehow, to waves of growth emanating from a centre, but on the other hand not to be explained by the division of an apical cell, or any other histological factor. As Sir T. Cook puts it, "at the growing point of a plant where the new members are being formed, there is simply nothing to see.'' But it is impossible to deal satisfactorily, in brief space, either with Mr Church's theories, or my own objections to themj*. Let it suffice to say that I, for my part, see no subtle mystery in the matter, other than what lies in the steady production of similar growing parts, similarly situated, at similar successive intervals of time. If such be the case, then we are bound to have in consequence * "The proposition is that the genetic spiral is a logarithmic spiral, homologous with the line of current-flow in a spiral vortex ; and that in such a system the action of orthogonal forces will be mapped out by other orthogonally intersecting logarithmic spirals— the ' parastichies ' " ; Church, op. cit. i, p. 42. t Mr Church's whole theory, if it be not based upon, is interwoven with, Sachs's theory of the orthogonal intersection of cell- walls, and the elaborate theories of the symmetry of a growing point or apical cell which are connected therewith. According to Mr Church, "the law of the orthogonal intersection of cell- walls at a growing apex may be taken as generally accepted" (p. 32); but I have taken a very different view of Sachs's law, in the eighth chapter of the present book. With regard to his own and Sachs's hypotheses, Mr Church .makes the following curious remark (p. 42) : "Nor are the hypotheses here put forward more imaginative than that of the paraboloid apex of Sachs which remains incapable of proof, or his construction for the apical cell of Pteris which does not satisfy the evidence of his own drawings." XIV] OR PHYLLOTAXIS \ 921 a series of symmetrical patterns, whose nature will depend upon the form of the entire surface. If the surface be that of a cylinder, we shall have a system, or systems, of spiral helices: if it be a plane with an infinitely distant focus, such as we obtain by "unwrapping" our cylindrical surface, we shall have straight lines; if it be a plane containing the focus within itself, or if it be any other symmetrical surface containing the focus, then we shall have a system of loga- rithmic spirals. The appearance of these spirals is sometimes spoken of as a "subjective" phenomenon, but the description is inaccurate: it is a purely mathematical phenomenon, an inseparable secondary result of other arrangements which we, for the time being, regard as primary. When the bricklayer builds a factory chimney, he lays his bricks in a certain steady, orderly way, with no thought of the spiral patterns to which this orderly sequence inevitably leads, and which spiral patterns are by no means "subjective." The designer of a wall-paper not only has no intention of producing a pattern of criss-cross lines, but on the contrary he does his best to avoid them ; nevertheless, so long as his design is a symmetrical one, the criss-cross intersections inevitably come. And as the train carries us past an orchard we see not one single symmetrical configuration, but a multiplicity of collineations among the trees. Let us, however, lea'C^e this discussion, and return to the facts of the case. Our second question, which relates to the numerical coincidences so familiar to all students of phyllotaxis, is not to be set and answered in a word. Let us, for simplicity's sake, avoid consideration of simultaneous or whorled leaf origins, and consider only the more frequent cases where a single "genetic spiral" can be traced throughout the 'entire system. It is seldom that this primary, genetic spiral catches the eye, for the leaves which immediately succeed one another in this genetic order are usually far apart on the circumference of the stem, and it is only in close-packed arrangements that the eye readily apprehends the continuous series. Accordingly in such a case as a fir-cone, for instance, it is certain of the secondary spirals or " parastichies " which catch the eye ; and among fir-cones, we can easily count these, 922 ON LEAF-ARRANGEMENT [ch. and we find them to be on the whole very constant in number, according to the species. Thus in many cones, such as those of the Norway spruce, we can trace five rows of scales winding steeply up the cone in one direction, and three rows winding less steeply the other way ; in certain other species, such as the common larch, the normal number is eight rows in the one direction and five in the other; while in the American larch we have again three in the one direction and five in the other. It not seldom happens that two arrangements grade into one another on different parts of one and the same cone. Among other cases in which such spiral series are readily visible we have, for instance, the crowded leaves of the stone-crops and mesembryanthemums, and (as we have said) the crowded florets of the composites. Among these we may find plenty of examples in which the numbers of the serial rows are similar to those of the fir-cones ; but in some cases, as in the daisy and others of the smaller composites, we shall be able to trace thirteen rows in one direction and twenty-one in the other, or perhaps twenty-one and thirty-four; while in a great big sunflower we may find (in one and the same species) thirty-four and fifty-five, fifty-five and eighty-nine, or even as many as eighty-nine and one hundred and forty-four. On the other hand, in an ordinary " pentamerous " flower, such as a ranunculus, we may be able to trace, in the arrangement of its sepals, petals and stamens, shorter spiral series, three in one direction and two in the other; and the scales on the little cone of a Cypress shew the same numerical simplicity. It will be at once observed that these arrangements manifest themselves in connection with very different things, in the orderly interspacing of single leaves and of entire florets, and among all kinds of leaf-like structures, foliage-leaves, bracts, cone-scales, and the various parts or members of the flower. Again we must be careful to note that, while the above numerical characters are by much the most common, so much so as to be deemed "normal," many other combinations are known to occur. The arrangement, as we have seen, is apt to vary when the entire structure varies greatly in size, as in the disc of the sunflower. It is also subject to less regular variation within one and the same species, as can always be discovered when we examine a sufficiently large sample of fir-cones. For instance, out of 505 cones of the XIV] OR PHYLLOTAXIS 923 Norway spruce, Beal* found 92 per cent, in which the spirals were in five and eight rows ; in 6 per cent, the rows were four and seven, and in 4 per cent, they were four and six. In each case they were nearly equally divided as regards direction ; for instance, of the 467 cones shewing the five-eight arrangement, the five-series ran in right-handed spirals in 224 cases, and in left-handed spirals in 243. Omitting the "abnormal" cases, such as we have seen to occur in a small percentage of our cones of the spruce, the arrangements which we have just mentioned may be set forth as follows (the fractional number used being simply an abbreviated symbol for the number of associated helices or parastichies which we can count running in the opposite directions): 2/3, 3/5, 5/8, 8/13, 13/21, 21/34, 34/55, 55/89, 89/144. Now these numbers form a very interesting series, which happens to have a number of curious mathematical properties "f. We see, for instance, that the denominator of each * Amer. Naturalist, vn, p. 449, 1873. t This celebrated series corresponds to the continued fraction 1+1 etc., 1 + and converges to 1-618..., the numerical equivalent of the sectio divina, or ''Golden Mean." The series of numbers, 1, 1, 2, 3, 5, 8, ..., of which each is the sum of the preceding two, was used by Leonardo of Pisa (c. 1170-1250), nicknamed Fi Bonacci, or films bonassi, in his Liber Abbaci, a work dedicated to magister mens, summus philosophus, Michael Scot {Scritti, i, pp. 283-284, 1857). This learned man was educated in Morocco, where his father was clerk or dragoman to Pisan merchants; and he is said to have been the first to bring the Arabic numerals, or " novem figurae Indorum,'' into Europe. The Fibonacci numbers were first so-caUed by Eduard Lucas, Bollettino di Bibliogr. e Storia dei Sci. Matem. e Fis. x, p. 129, 1877. The general expression for the series V^Y (\-V5\ ^^{(^T-C~)"l' was known to Euler and to Daniel Bernoulli {Coram. Acad. Sci. Imp. Petropol. 1732, p. 90), and was rediscovered by Binet, C.E. xviii, p. 563, 1843; xix, p. 939, 1844) and by Lame, ibid, xix, p. 867, after whom it is sometimes called Lame's series. But the Greeks were familiar with the series 2, 3: 5, 7 : 12, 17, etc.; which converges to V2, as the other does to the Golden Mean ; and so closely related are the two series, that it seems impossible that the Greeks could have known the one and remained ignorant of the other. (See a paper of mine, on "Excess and Defect, etc.," in Mind, xxxviii. No. 149, 1928.) The Fibonacci (or Lame) series was well known to Kepler, who, in his paper De nive sexangula (1611, cf. supra, p. 695), discussed it in connection with the form of the dodecahedron and icosahedron, and with the ternary or quinary symmetry of the flower. (Cf. F. Ludwig, Kepler iiber das Vorkommen der Fibonaccireihe im Pflanzenreich, Bot. Centralbl. lxviii, p. 7, 1896.) Professor William Allman, Professor of Botany in Dublin (father of the historian of Greek geometry). 924 ON LEAF-ARRANGEMENT [ch. fraction is the numerator of the next; and further, that each suc- cessive numerator, or denominator, is the sum of the preceding two. Our immediate problem, then, is to determine, if possible, how these numerical coincidences come about, and why these particular numbers should be so commonly met with as to be considered "normal" and characteristic features of the general phenomenon of phyllotaxis. The following account is based on a short paper by Professor P. Gr. Tait*. Of the two following diagrams, Fig. 450 represents the general case, and Fig. 451 a particular one, for the sake of possibly greater simplicity. Both diagrams represent a portion of a branch, or fir- cone, regarded as cylindrical, and unwrapped to form a plane surface. A, a, at the two ends of the base-line, represent the same initial leaf or scale; 0 is a leaf which can be reached from ^ by m steps O Fig. 450. in a right-hand spiral (developed into the straight line AO), and by n steps from a in a left-handed spiral aO. Now it is obvious in our fir-cone, that we can include all the scales upon the cone by taking so many spirals in the one direction, and again include them all by so speculating on the same facts, put forward the curious suggestion that the cellular tissue of the dicotyledons, or exogens, would be found to consist of dodecahedra, and that of the monocotyledons or endogens of icosahedra {On the mathematical connection between the parts of vegetables: abstract of a Memoir read before the Royal Society in the year 1811 (privately printed, n.d.). Cf. De Candolle, Organo- genie vegetale, i, p. 534. See also C. E. Wasteels, Over de Fibonaccigetalen, Me Natuur. Congres, Antwerpen, 1899, pp. 25-37; R. C. Archibald, in Jay Hambidge's Dynawfc Fiymmetry, 1920, pp. 146-157; and, on the many mathematical properties of the series, L. E. Dickson, Theory of Numbers, i, pp. 393-411. 1919. Of these famous and fascinating numbers a mathematical, friend writes to me: "All the romance of continued fractions, linear recurrence relations, surd approxi- mations to integers and the rest, lies in them, and they are a source of endless curiosity. How interesting it is to see them striving to attain the unattainable, the golden ratio, for instance; and this is only one of hundreds of such relations." * Proc. R.S.E. vn, p. 391, 1872. xiv] OR PHYLLOTAXIS 925 many in the other. Accordingly, in our diagrammatic construction, the spirals AO and aO must, and always can, be so taken that m spirals parallel to aO, and n spirals parallel to AO, shall separately include all the leaves upon the stem or cone. If m and n have a common factor, I, it can easily be shewn that the arrangement is composite, and that there are I fundamental, or genetic spirals, and I leaves (including A) which are situated Fig. 451. exactly on the line. Aa. That is to say, we have here a whorled arrangement, which we have agreed to leave unconsidered in favour of the simpler case. We restrict ourselves, accordingly, to the cases where there is but one genetic spiral, and when therefore m and n are prime to one another. Our fundamental, or genetic, spiral, as we have seen, is that which passes from A (or a) to the leaf which is situated nearest to the base-line Aa. The fundamental spiral will thus be right-handed (A, P, etc.) if F, which is nearer to A than to a, be this leaf — left- handed if it be jo. That is to say, we make it a convention that we 926 ON LEAF-ARRANGEMENT [ch. shall always, for our fundamental spiral, run round the system, from one leaf to the next, hy the shortest way. Now it is obvious, from the symmetry of the figure (as further shewn in Fig. 451),- that, besides the spirals running along AO and aO, we have a series running /rom the steps on aO to the steps on ^0. In other words we can find a leaf (S) upon AO, which, like the leaf 0, is reached directly by a spiral series from A and from a, such that aS includes n steps, and AS (being part of the old spiral fine AO) now includes m — n steps. And, since m and n are prime to one another (for otherwise the system would have been a composite or whorled one), it is evident that we can continue this process of convergence until we come down to a 1, 1 arrangement, that is to say to a leaf which is reached by a single step, in opposite directions from A and from a,, which leaf is therefore the first leaf, next to A, of the fundamental or generating spiral. If our original lines along AO and aO contain, for instance, 13 and 8 steps respectively (i.e. m = 13, n= 8), then our next series, observable in the same cone, will be 8 and (13 — 8) or 5; the next 5 and (8 — 5) or 3; the next 3, 2; and the next 2, 1 ; leading to the ultimate condition of 1, 1. These are the very series which we have found to be common, or normal; and so far as our investigation has yet gone, it has proved to us that, if one of these exists, it entails, ipso facto, the presence of the rest. In following down our series, according to the above construction, we have seen that at every step we have changed direction, the longer and the shorter sides of our triangle changing places every time. Let us stop for a moment, when we come to the 1, 2 series, or AT, aT of Fig. 451. It is obvious that there is nothing to prevent us making a new 1, 3 series if we please, by continuing the generating spiral through three leaves, and connecting the leaf so reached directly with our initial one. But in the case represented in Fig. 451, it is obvious that these two series {A, 1, 2, 3, etc., and a, 3, 6, etc.) will be running in the same direction; i.e. they will both be right- handed, or both left-handed spirals. The simple meaning of this is that the third leaf of the generating spiral was distant from our initial leaf by more than the circumference of the cyhndrical stem; in other words, that there were more than two, but less than three leaves in a single- turn of the fundamental spiral. XIV] OR PHYLLOTAXIS 927 Less tnan two there can obviously never be. When there are exactly two, we have the simplest of all possible arrangements, namely that in which the leaves are placed alternately on opposite sides of the stem. When there are more than two, but less than three, we have the elementary condition for the production of the series which we have been considering, namely 1, 2; 2, 3; 3, 5, etc. To put the latter part of this argument in more precise language, let us say that: If, in our descending series, we come to steps 1 and t, where t is determined by the condition that 1 and t + 1 would give spirals both right-handed, or both left-handed; it follows that there are less than t + 1 leaves in a single turn of the fundamental spiral. And, determined in this manner, it is found in the great majority of cases, in fir-cones, and a host of other examples of phyllotaxis, that t= 2. In other words, in the great majority of cases, we have what corresponds to an arrangement next in order of simphcity to the simplest case of all : next, that is to say, to the arrangement which consists of opposite and alternate leaves. "These simple considerations," as Tait says, "explain completely the so-called mysterious appearance of terms of the recurring series 1, 2, 3, 5, 8, 13, etc.* The other natural series, usually but mis- leadingly represented by convergents to an infinitely extended continuous fraction, are easily explained, as above, by taking ^ = 3, 4, 5, etc., etc." Many examples of these latter series have been recorded, as more or less rare abnormalities, by Dickson f and other writers. We have now learned, among other elementary facts, that wherever any one system of spiral steps is present, certain others invariably and of necessity accompany it, and are definitely related to it. In any diagram, such as Fig. 451, in which we represent our leaf- arrangement by means of uniform and regularly interspaced dots, we can draw one series of spirals after another, and one as easily * The necessary existence of these recurring spirals is also proved, in a somewhat different way, by Leslie Ellis, On the theory of vegetable spirals, in Mathematical and other Writings, 1863, pp. 358-372. Leslie Ellis, Whewell's brother-in-law, was a man of great originahty. He is best remembered, perhaps, for his views on the Theory of Probabilities (cf. J. M. Keynes, Treatise an Probabilities, 1921, p. 92), and for his association with Stebbing as editor of Bacon. t Proc. R.S.E. vn, p. 397, 1872; Trans. E.S.E. xxvi, pp. 505-520, 1872. 928 ON LEAF-ARRANGEMENT [ch. as another. In a fire-cone one particular series, or rather two conjugate series, are always conspicuous, but the related series may be sought and found with httle difficulty. The spruce-fir is com- monly said to have a phyllotaxis of 8/13; but we may count still Fig. 452. steeper and nearly vertical rows of scales to the number of 13/21; and if we take pains to number all the scales consecutively, we may find the lower series, 5/8, 3/5, and even 1/2, with ease and certainty. The phenomenon is illustrated by Fig. 452, a-d. The ground-plan of all these diagrams is identically the same. The generating spiral XIV] PR PHYLLOTAXIS 929 in each case represents a divergence of 3/8, or 135° of azimuth; and the points succeed one another at the same successional distances parallel to the axis. The rectangular outhnes, which correspond to the exposed surface of the leaves or cone-scales, are of equal area, and of equal number. Nevertheless the appearances presented by these diagrams are very different; for in one the eye catches a 5/8 arrangement, in another a 3/5 ; and so on, down to an arrangement of 1/1. The mathematical side of this very curious phenomenon I have not attempted to investigate. But it is quite obvious that, in a system within which various spirals are imphcitly contained, the conspicuousness of one set or another does not depend upon angular divergence. It depends on the relative proportions in length and breadth of the leaves themselves; or, more strictly speaking, on the ratio of the diagonals of the rhomboidal figure by which each leaf-area is circumscribed. When, as in the fir-cone, the scales by mutual compression conform to these rhomboidal outlines, their inclined edges at once guide the eye in the direction of some one particular spiral; and we shall not fail to notice that in such cases the usual result is to give us arrangements corresponding to the middle diagrams in Fig. 452, which are the configurations in which the quadrilateral outhnes approach most nearly to a rectangular form, and give us accordingly the least possible ratio (under the given conditions) of sectional boundary-wall to surface area. The manner in which one system of spirals may be caused to slide, so to speak, into another, has been ingeniously demonstrated by Schwendener on a mechanical model, consisting essentially of a framework which can be opened or closed to correspond with one another of the above series of diagrams*. The same curious fact, that one Fibonacci series leads to, or involves the rest, is further shewn, in a very simple way, in the following diagrammatic Table (p. 930). It shews, in the first instance, the numerical order of the scales on a fir-cone, in so-called 5/8 phyllotaxis; that is to say, it represents the cone unwrapped, with the two principal spirals lying along the axes of a rectangular system. Starting from 0, the abscissae increase by 5, the ordinates by 8; or, in other words, any given number m = bx + d>y\ it is * A common form of pail-shaped waste-paper basket, with wide rhomboidal meshes of cane, is well-nigh as good a model as is required. 2 3 5 8 13 21 34 55 2 -1 1 0 1 1 2 3 -1 1 0 1 i 2 3 5 930 ON LEAF-ARRANGEMENT [ch. easy, then, to number the entire system. The generating spiral, 0, 1, 2, 3, ... , and the various secondary Fibonacci spirals, are then easily recognised. A Fibonacci series, unwrapped from a cone or cylinder. m = 6x-\- Sy. 1 6 11 16 21 26 31 36 T 2 3' 8 13 18 23 28 15 To 5" 0 5 10 15 20 ^r8l3'83 2 7 12 ■5T'2B2Tl6rT 6 T4 The place of the first scale in each series is then found to be as follows : Series 1 y~ 2 And this is the Fibonacci series over again. We also see how the several spirals, of which these are the beginnings, alternate to the right and left of an asymptotic line, where x/y = 0-618. . . . The Fibonacci numbers, so conspicuous in the fir-cone, make their appearance also in the flower. The commonest of floral numbers are 3 and 5; among the Composites we find 8 ray-florets in the single dahlia, 13 in the ragwort, 21 in the ox-eye daisy or the mari- gold. In the last two, heads with 34 ray-florets are apt to be produced at certain times or in certain places*; and in C. segetum these florets are said to vary in a bimo(fe,l curve of frequency, with a high maximum at 13 and a lower at 21t. The simplest explanation (though perhaps it does not go far) is to suppose that a ligulate floret terminates, or tends to terminate, each of the principal spiral series. But among the higher numbers these numerical relations are only approximate, and the whole matter rests, so far, on some- what scanty evidence. The determination of the precise angle of divergence of two con- secutive leaves of the generating spiral does not enter into the above general investigation (though Tait gives, in the same paper, a method * Cf. G. Henslow, On the origin of dimerous and trimerous whorls among the flowers of Dicotyledons, Trans. Linn. Soc. (Bot.) (2), vii, p. 161, 1908. f Cf. A. Gravis, Elements de Physiologic vegetale, 1921, p. 122. XIV] OR PHYLLOTAXIS 931 by which it may be easily determined); and the very fact that it does not so enter shews it to be essentially unimportant. The determination of so-called ''orthostichies," or precisely vertical suc- cessions of leaves, is also unimportant. We have no means, other than observation, of determining that one leaf is vertically above another, and spiral series such as we have been dealing with will appear, whether such orthostichies exist, whether they be near or remote, or whether the angle of divergence be such that no precise vertical superposition ever occurs. And lastly, the fact that the successional numbers, expressed as fractions, 1/2, 2/3, 3/5, represent a convergent series, whose final term is equal to 0-61803..., the sectio aurea or "golden mean" of unity, is seen to be a mathematical coincidence, devoid of biological significance; it is but a particular case of Lagrange's theorem that the roots of every numerical equation of the second degree can be expressed by a periodic continued frac- tion. The same number has a multitude of curious arithmetical properties. It is the final term of all similar series to that with which we have been dealing, such for instance as 1/3, 3/4, 4/7, etc., or 1/4, 4/5, 5/9, etc. It is a number beloved of the circle-squarer, and of all those who seek to find, and then to penetrate, the secrets of the Great Pyramid. It is deep-set in the regular pentagon and dodecahedron, the triumphs of Pythagorean or Euclidean geometry. It enters (as the chord of an angle of 36°) into the thrice-isosceles triangle of which we have spoken on p. 762 ; it is a number which becomes (by the addition of unity) its own reciprocal — its properties never end. To Kepler (as Naber tells us) it was a symbol of Creation, or Generation. Its recent application to biology and art-criticism by Sir Theodore Cook and others is not new. Naber' s book, already quoted, is full of it. Zeising*, in 1854, found in it the key to all * A. Zeising, Neue Lehre von der Proportion des menschlichen Korpers aus einem bisher unerkannt gebliebenen die game Natur und Kunst durchdringenden morpho- logischen Grundgesetze entwickelt, Leipzig, 1854, 457 pp.; ibid. Deutsche Viertel- jahrsschrift, 1868, p. 261; also, posthumously, Der Goldene Schnitt, Leipzig, 1884, 24 pp. Cf. S. Gunther, Adolph Zeising als Mathematiker, Ztschr. f. Math. u. Physik. {Hist. Lit. Abth.), xxi, pp. 157-165, 1876; also F. X. PfeiflFer, Die Propor- tionen des goldenen Schnittes an den Blattern u. Stengelen der Pflanzen, Ztschr. f. math. u. naturw. Unterricht, xv, pp. 325-338, 1885. For other references, see R. C. Archibald, op. cit. Among modern books on similar lines, the following are curious, interesting and beautiful (whether we agree with them or not): Jay Hambidge, Dynamic Symmetry, Yale, 1920; C. Arthur Coan, Nature's Harmonic Unity, New York, 1912. 932 ON LEAF-ARRANGEMENT [ch. morphology, and the same writer, later on, declared it to dominate both architectm^e and music. But indeed, to use Sir Thomas Browne's words (though it was of another number that he spoke): "To enlarge this contemplation into all the mysteries and secrets accommodable unto this number, were inexcusable Pythagorisme." That this number has any serious claim at all to enter into the biological question of phyllotaxis seems to depend on the assertion, .first made by Chauncey Wright*, that, if the successive leaves of the fundamental spiral be placed at the particular azimuth which divides the circle in this "sectio aurea," then no two leaves will ever be superposed f; and thus we are said to have "the most thorough and rapid distribution of the leaves round the stem, each new or higher leaf falling over the angular space between the two older ones which are nearest in direction, so as to divide it in the same ratio (K), in which the first two or any two successive ones divide the circumference. Now 5/8 and all successive fractions differ inappreciably from Z." To this view there are many simple objections. In the first place, even 5/8, or 0-625, is but a moderately close approximation to the "golden mean"; and furthermore, the arrangements by which a better approximation is got, such as 8/13, 13/21, and the very close approximations such as 34/55, 55/89, 89/144, etc., are comparatively rare, while the much less close approximations of 3/5 or 2/3, or even 1/2, are extremely common. Again, the general type of argument such as that which asserts that the plant is "aiming at" something which we may call an "ideal angle" is one which cannot commend itself to a plain student of physical science : nor is the hypothesis rendered more acceptable when Sir T. Cook qualifies it by telHng us that "all that a plant can do is to vary, to make blind shots at constructions, or to ' mutate ' as it is now termed : and the most suitable of these con- structions will in the long run be isolated by the action of Natural Selection." Thirdly, we must not suppose the Fibonacci numbers ♦ On the uses and origin of the arrangement of leaves in plants, Mem. Amer. Acadfix, p. 380, 1871, Cambridge, Mass. Cf. J. Wiesner, Ueberdie Beziehungen der Stellungsverhdltnisse der Laubbldtter zur Beleuchtung, Wien, 1902. t This is what Ruskin spoke of as " the vacant space " ; Mod. Painters, v, chap, vi, p. 44, 1860. Leonardo had in like manner explained the leaf-arrangement as serving to let air pass between the leaves, keep one from overshadowing another, and let rain-drops fall from the one leaf to the one below. XIV] OR PHYLLOTAXIS 933 to liave any exclusive relation to the Golden Mean; for arithmetic teaches us that, beginning with any two numbers whatsoever, we are led by successive summations toward one out. of innumerable series of numbers whose ratios one to another converge to the Golden Mean*. Fourthly, the supposed isolation of the leaves, or their most complete "distribution to the action of the surrounding atmosphere" is manifestly very little affected by any conditions which are confined to the angle of azimuth. For if it be (so to speak) Nature's object to set them farther apart than they actually are, to give them freer exposure to the air or to the sunlight than they actually have, then it is surely manifest that the simple way to do so is to elongate the axis, and to set the leaves farther apart, lengthways on the stem. This has at once a far more potent effect than any nice manipulation of the "angle of divergence." Lastly, and this seems the simplest, the most cogent and most unanswerable objection of them all, if it be indeed desirable that no leaf should be superimposed above another, the one condition necessary is that the common angle of azimuth should not be a rational multiple of a right angle — should not be equivalent to — I - j . One irrational angle is as good as another : there is no special merit in any one of them, not even in the ratio divina. We come then without more ado to the conclusion that while the Fibonacci series stares us in the face in the fir-cone, it does so for mathematical reasons; and its supposed usefulness, and the hypo- thesis of its introduction into plant-structure through natural selection, are matters which deserve no place in the plain study of botanical phenomena. As Sachs shrewdly recognised years ago, all such speculations as these hark back to a school of mystical idealism. * Thus, instead of beginning with 1, 1, let us begin 1, 7. The summation-series is then 1, 7, 8, 15, 23, 38, 61, 99, 160, 259, ..., etc.; and 99/160=0-618... and 259/160= 1-619...; and so on. But after all, the old Fibonacci numbers are not far away. For we may write the new series in the form : 7 (0, 1, 1, 2, 3, 5, 8, ...) + 1 (1,0, 1, 1,2,3,5,'...). CHAPTER XV ON THE SHAPES OF EGGS, AND OF CERTAIN OTHER HOLLOW STRUCTURES The eggs of birds and all other hard-shelled eggs, such as those of the tortoise and the crocodile, are simple solids of revolution ; but they differ greatly in form, according to the configuration of the plane curve by the revolution of which the egg is, in a mathematical sense, generated. Some few eggs, such as those of the owl, the penguin, or the tortoise, are spherical or very nearly so; a few more, such as the grebe's, the cormorant's or the pelican's, are approxi- mately ellipsoidal, with symmetrical or nearly symmetrical ends, and somewhat similar are the so-called "cylindrical" eggs of the megapodes and the sand-grouse; the great majority, like the hen's egg, are " ovoid," a little blunter at one end than the other ; and some, by an exaggeration of this lack of antero-posterior symmetry, are blunt at one end but characteristically pointed at the other, as is the case with the eggs of the guillemot and puffin, the sandpiper, plover and curlew. It is an obvious but by no means negligible fact that the egg, while often pointed, is never flattened or discoidal; it is a prolate, but never an oblate, spheroid. Its oval outhne has one maximal and two minimal radii of curvature, one minimum being less than the other. The evolute to a curve often emphasises, even exaggerates, its features ; and the evolutes to a series of eggs (i.e. to their generating curves) are more conspicuously different than the eggs themselves (Fig. 453)*. The careful study and collection of birds' eggs would seem to have begun with the Coujit de Marsiglil, the same celebrated naturalist * Cf. A. Mallock, On the shapes of birds' eggs. Nature, cxvi, p. 311, 1925. The evolute may be easily if somewhat roughly drawn by erecting perpendiculars on a sufficient nunrber of tangents to the curve. The .evolute then appears as an envelope, the perpendiculars all being tangents to it. t De avibus circa aquas Danubii vagantihus et de ipsarum nidis (Vol. v of the Danubius Panonico-Mysicus), Hagae Com. 1726. Count Giuseppi Ginanni, or Zinanni, came soon afterwards with his book Delle uove e dei nidi degli uccelli, Vcnezia, 1737. CH. XV] ON THE SHAPES OF EGGS 935 who first studied the "flowers" of the coral, and who wrote the Histoire physique de la mer; and the specific form as well as the colour and other attributes of the egg have been again and again discussed, and not least by the many dilettanti naturalists of the • eighteenth century who soon followed in Marsigli's footsteps*. We need do no more than mention Aristotle's belief, doubtless old in his time, that the more pointed egg produces the male chicken, and the blunter egg the hen ; though this theory survived into modern times t and still lingers on (cf. p. 943). Several naturahsts, such as Giinther (1772) and Biihle (1818), have taken the trouble to disprove it by experiment. A more modern and more generally accepted ^ A .A- a b c d e Fig. 453. Typical forms of birds' eggs: from A. Mallock. The figures below are pinhole photographs of the eggs. The upper figures (drawn to a uniform scale) shew the generating curves and their evolutes. a Green plover. c Crow. e Kingfisher. b Humming-bird. d Pheasant. / Owl. explanation has been that the form of the egg is in direct relation to that of the bird which has to be hatched within — a view that would seem to have been first set forth by Naumann and Biihle, in their great treatise on eggs J, and adopted by Des Murs§ and many other well-known writers. In a treatise by de Lafresnaye||, an elaborate comparison is made * But Sir Thomas Browne had a collection of eggs at Norwich in 1671, according to Evelyn. f Cf. Lapierre, in Buffon's Histoire Naturelh, ed. Sonnini, 1800. % Eier der Vogel Deutschlands, 1818-28 (cit. Des Murs, p. 36). § Traite dVologie, 1860. II F. de Lafre.snaye, Comparaison des ceufs des oiseaux avec leurs squelettes, comme seul moyen de reconnaitre la cause de leurs differentes formes. Rev. Zool. 1845, pp. 180-187, 239-244. 936 ON THE SHAPES OF EGGS [ch. between the skeleton and the egg of various birds, to shew, for instance, how those birds with a deep-keeled sternum laid rounded eggs, which alone could accommodate the form of the young. According to this yiew, that "Nature had foreseen*" the form adapted to and necessary for the growing embryo, it was easy to. correlate the owl with its spherical egg, the diver with its elhptical one, and in like manner the round egg of the tortoise and the elongated one of the crocodile, with the shape of the creatures which had afterwards to be hatched therein. A few writers, such as Thienemannf, looked at the same facts the other way, and asserted that the form of the egg was determined by that of the bird by which it was laid and in whose body it had been conformed. In more recent times, other theories, based upon the principles of Natural Selection, have been current and very generally accepted to account for these diversities of form. The pointed, conical egg of the guillemot is generally supposed to be an adaptation, advan- tageous to the species in the circumstances under which the egg is laid; the pointed egg is less apt than a spherical one to roll off the narrow ledge of rock on which this bird is said to lay its solitary egg, and the more pointed the egg, so niuch the fitter and hkeher is it to survive. The fact that the plover or the sandpiper, breeding in very different situations, lay eggs that are also conical, ehcits another explanation, to the effect that here the conical form permits the many large eggs to be packed closely under the mother bird J. Whatever truth there be in these apparent adaptations to existing circumstances, it is oiily by a very hasty logic that we can accept them as a vera causa, or adequate explanation of the facts ; and it is obvious that in the bird's egg we have an admirable case for direct investigation of the mechanical or physical significance of its form§. * Cf. Des Murs, p. 67: "Elle devait encore penser au moment ou ce germe aurait besoin de I'espace necessaire a son accroissement, a ce moment ou . . . il devra remplir exactement I'intervalle circonscrit par sa fragile prison, etc." f F. A. L. Thienemann, Syst. Darstellung der Fortpflanzung der Vogel Europas, Leipzig, 1825-38. X Cf. Newton's Dictionary of Birds, 1893, p. 191 ; Szielasko, Gestalt der Vogeleier, Journ. f. Ornith. Lm, pp. 273-297, 1905. § Jacob Steiner suggested a Cartesian oval, r + mr' — c, as a general formula for all eggs (cf. Fechner, Ber. sacks. Ges. 1849, p. 57); but this formula (which fails in such a case as the guillemot) is purely empirical, and has no mechanical foundation. XV] AND OTHER HOLLOW STRUCTURES 937 Of all the many naturalists of the eighteenth and nineteenth centuries who wrote on the subject of eggs, only two (so far as I am aware) ascribed the form of the egg to direct mechanical causes. Giinther*, in 1772, declared that the more or less rounded or pointed form of the egg is a mechanical consequence of the pressure of the oviduct at a time when the shell is yet unformed or unsolidified ; and that accordingly, to explain the round egg of the owl or the kingfisher, we have only to admit that the oviduct of these birds is somewhat larger than that of most others, or less subject to violent contractions. This statement contains, in essence, the whole story of the mechanical conformation of the egg. A hundred and twenty years after, Dr J. Ryder of Philadelphia gave, as near as may be, the same explanation f. Let us consider, very briefly, the conditions to which the egg is subject in its passage down the oviduct. (1) The "egg," as it enters the oviduct, consists of the yolk only, enclosed in its vitelHne membrane. As it passes down the first portion of the oviduct the white is gradually superadded, and becomes in turn surrounded by the "shell-membrane." About this latter the shell is secreted, rapidly and at a late period: the egg having meanwhile passed on into a wider portion of the oviducal tube; called (by loose analogy, as Owen says) the "uterus." Here 'the egg assumes its permanent form, here it ultimately becomes rigid, and it is to this portion of the oviduct that our argument principally refers. (2) Both the yolk and the entire egg tend to fill completely their respective membranes, and, whether this be due to growth or imbibition on the part of the contents or to contraction on the part of the surrounding membranes, the resulting tendency is for both yolk and egg to be, in the first instance, spherical, unless or until distorted by external pressure. (3) The egg is subject to pressure within the oviduct, which is an elastic, muscular tube, along the walls of which pass peristaltic * F, C. Giinther, Sammlung von Nestern und Eyern verschiedener Vogel, Niirnb. 1772. Cf. also Raymond Pearl, Morphogenetic activity of the oviduct, J. Exp. Zool. VI, pp. 339-359, 1909. - t J. Ryder, The mechanical genesis of the form of the fowl's egg, Proc\ Amer. Philosoph. Soc. Philadelphia, xxxi, pp. 203-209, 1893; cf. A. S. Packard, Inheritance of acquired characters, 'Proc. Amer. Acad. 1894, p. 360. 938 ON THE SHAPES OF EGGS [ch. waves of contraction. These muscular contractions hiay be de- scribed as the contraction of successive annuli of muscle, giving annular (or radial) pressure to successive portions of the egg ; they drive the egg forward against the frictional resistance of the tube, while tending at the same time to distort its form. While nothing is known, so far as I am aware, of the muscular physiology of the oviduct, it is well known in the case of the intestine that the presence of an obstruction leads to the development of violent contractions in its rear, which waves of contraction die away, and are scarcely if at all propagated in advance of the obstruction ; indeed in normal intestinal peristalsis a wave of relaxation travels close ahead of the wave of constriction. (4) The egg is, to all intents and purposes, a solid of revolution ; in other words, its transverse sections are all but perfect circles, sd nearly perfect that, chucked in the lathe, an egg "runs true." This may be taken to shew that the direct pressure of the oviduct, whether elastic or muscular,* is large compared with the weight of the egg. Even in ostrich eggs, where if anywhere gravitational deformation should be found, the greatest and least equatorial diameters do not differ by 1 per cent., and sometimes by less than one part in a thousand*. (5) It is known by observation that a hen's egg is always laid blunt end foremost f. (6) It can be shewn, at least as a very common rule, that those eggs which are most unsjnnmetrical, or most tapered off posteriorly, are also eggs of a large size relatively to the parent bird. The guillemot is a notable case in point, and so also are the curlews, sandpipers, phaleropes and terns. We may accordingly presume that the more pointed eggs are those that are large relatively to the tube or oviduct through which they have to pass, or, in other words, are those which are subject to the greatest pressure while * Cf. Mailock, op. cit. t This was known to Albertus Magnus, though his explanation was wrong, "Ova autem habentia duos colores non sunt oranino penitus rotunda, sed ex una parte sunt acuta habentia angulum sphericum acutum, sicut sunt composita ex duobus seraispheris, in una parte extensis ad angulum acutum et in alia parte sphericis noo extensis in loco ubi est polus ovi. . . . Et in exitu ovi acutus angulus exit uliimo, eo quod ipse porrectus est ad interiora matricis versus parietem ubi ovum cum matrice continuatur in sui generatione" {De animalibas, lib. xvii, tract. 1, c. 3). XV] AND OTHER HOLLOW STRUCTURES 939 being forced along. So general is this relation that we may go still further, and presume with great plausibihty in the few exceptional cases (of which the apteryx is the most conspicuous) where the egg is relatively large though not markedly unsymmetrical, that in these cases the oviduct itself is in all probability large (as Giinther had suggested) in proportion to the size of the bird. In the case of the common fowl we can trace a direct relation between the size and shape of the egg, for the first eggs laid by a young pullet are usually smaller, and at the same time are much more nearly spherical than the later ones ; and, moreover, some breeds of fowls lay proportionately smaller eggs than others, and on the whole the former eggs tend to be rounder than the latter*. We may now proceed to enquire more particularly how the form of the egg is controlled by the pressures to which it is subjected. The egg, just prior to the formation of the shell, is, as we have seen, a fluid body, tending to a spherical shape and enclosed within a membrane. Our problem, then, is: Given an incompressible fluid, contained in a deformable capsule, which is either (a) entirely inextensible, or (6) slightly extensible, and which is placed in a long elastic tube the walls of which are radially contractile, to determine the shape under some given distribution of pressure. We may assume, at least to begin with, that the shell-membrane is homogeneous and isotropic — uniform in all parts and in all directions. If the capsule be spherical, inextensible, and completely filled with the fluid, absolutely no deformation can take place. The few eggs that are actually or approximately spherical, such as those of the tortoise or the owl, may thus be alternatively explained as cases where little or no deforming pressure has been applied prior to the solidification of the shell, or else as cases where the capsule was so * In so far as our explanation involves a shaping or moulding of the egg by the uterus or oviduct (an agency supplemented by the proper tensions of the egg), it is curious to note that this is very much the same as that old view of Telesius regarding the formation of the embryo {De rerum natura, vi, cc. 4 and 10), which he had inherited from Galen, and of which Bacon speaks {Nov. Org. cap. .50; cf. Ellis's note). Bacon expressly remarks that "Telesius should have been able to shew the like formation in the shells of eggs." This old theory of embryonic modelling survives in our usage of the term "matrix" for a "mould." 940 ON THE SHAPES OF EGGS [ch. little capable of extension and so completely filled as to preclude the possibility of deformation. If the capsule be not spherical, but be inextensible, then only such deformation can take place as tends to make the shape more nearly spherical; and as the surface area is thereby decreased, the envelope must either shrink or pucker. In other words, an incompressible fluid contained in an inextensible envelope cannot be deformed without puckering of the envelope. But let us next assume, as the condition by which this result may be avoided, that the envelope is to some extent extensible and that deformation is so far permitted. It is obvious that, on the presumption that the envelope is only moderately extensible, the whole structure can only be distorted to a moderate degree away from the spherical or spheroidal form. At all points the shape is determined by the law of the distribution of radial pressure withih the given region of the tube, surface friction helping to maintain the egg in position. If the egg be under pressure from the oviduct, but without any marked component either in a forward or backward direction, the egg will be compressed in the middle, and will tend more or less to the form of a cylinder with spherical ends. The eggs of the grebe, cormorant, or crocodile may be supposed to receive their shape in such circumstances. When the egg is subject to the peristaltic contraction of the oviduct during its formation, then from the nature and direction of motion of the peristaltic wave the pressure will be greatest some- where behind the middle of the egg; in other words, the tube is converted for the time being into a more conical form, and the simple result follows that the anterior end of the -egg becomes the broader and the posterior end the narrower. The peristalsis of the oviduct thus plays a double part, in pro- pelling the' egg down the oviduct and in impressing on it its ovoid form; but the whole process is a very slow one, for the hen's oviduct is only a few inches long, and the egg is some ten or twelve hours upon its way. We shall consider presently certain shells which may be regarded as so many drops or vesicles deformed by gravity; that is a statical problem. Compared with it the problem of the egg is a dynamical one; and yet it becomes a quasi-statical one, because the action is so very slow. It is an action without lag XV] AND OTHER HOLLOW STRUCTURES 941 and without momentum; and the question, common in dynamical problems, of the relation between the period of the appHcation of the force and the free period of response or adjustment to it need not concern us at all. Again, the case of the egg is somewhat akin to a hydrodynamical problem ; for as it lies in the oviduct we may look on it as a stationary body round which waves are flowing, with the same result as when a body moves through a fluid at rest. Thus we may treat it as a hydrodynamical problem, but a very simple one — simphfied by the absence of all eddies and every form of turbulence; and we come to look on the egg as a streamlined structure, though its streamlines are of a very simple kind. The mathematical statement of the case begins as follows: In our egg, consisting of an extensible membrane filled with an incompressible fluid and under external pressure, the equation of the envelope is ;p„ + T (1/r + \jr') = P, where p^ is the normal component of external pressure at a point where r and r' are the radii of curvature, T is the tension of the envelope, and P the internal fluid pressure. This is simply the equation of an elastic surface where T represents the coefficient of elasticity; in other words, a flexible elastic shell has the same mathematical properties as our fluid, membrane-covered egg. And this is the identical equation which we have already had so frequent occasion to employ in our discussion of the forms of cells; save only that in these latter we had chiefly to study the tension T (i.e. the surface-tension of the senii-fluid cell) and had little or nothing to do with the factor of external pressure (^„), which in the case of the egg becomes of chief importance. To enquire how an elastic sphere or spheroid will be deformed in passing down a peristaltic tube is an ill-defined and indeterminate problem ; but we can study the effect produced in the shape of any particular egg, and so far infer the forces which have been in action. We need only study a single meridian of the egg, inasmuch as we have found it to be a solid of revolution. At successive points along this meridian, let us determine the amount of curvature, that is to say the principal radii of curvature, in latitude and longitude, in the Gaussian formula P = pn + T (1/r 4- l/r') : or, as we may write it if we have any reason to doubt the uniformity or isotropy of the 942 ON THE SHAPES OF EGGS [ch. membrane, Tjr + T'jr'. The sum of these curvatures varies from point to point; the internal or hydrodynamical pressure, P, is constant; and therefore the external pressure, jo„, varies from point to point with the curvature, and is a direct function of the shape of the egg. Some few eggs, such as the owl's and the kingfisher's, are so nearly spherical that we are apt to speak of them as spheres ; but they are all prolate more or less, and no egg is so nearly circular in meridional section as all eggs are in their circles of latitude. When the egg is all but spherical that shape may be due (as we have seen) to various causes : to a relatively, small size of the egg, allowing it to descend the tube under a minimum of peristaltic pressure; perhaps to an unusually strong shell-membrane, resistant of deformation; in general terms, to a possible diminution of j?„ , or a possible increase of T. But all eggs have approximately spherical ends, and the big anterior end of the large conical eggs of plover or curlew or guillemot is conspicuously so. Here the egg projects into the wide cavity of the uncontracted oviduct, external or peristaltic pressure does not exist, the shell-membrane has to resist internal pressure without further external support, and the resultant spherical cur- vature is an indication of the uniformity, or isotropy, of the mem- brane. The lesser of the two spherical ends, that is to say the posterior end, has by much the greater curvature, and the tension there is correspondingly great. It would seem that the membrane ought to be thicker or stronger at this pointed end than elsewhere, but it is not known to be so. In any case, it is just here, in this presumably weakest part, that we are most apt to find the irregulari- ties and deformities of misshapen eggs. Within the egg lies the yolk, and the yolk is invariably spherical or very nearly so, whatever be the form of the entire egg. The reason is simple, and lies in the fact that the fluid yolk is itself enclosed within another membrane, between which and the shell- membrane Hes the fluid albumin, which transmits a uniform hydro- static pressure to the yolk*. The lack of friction between the yolk- membrane and the white of the egg is indicated by the well-known fact that the "germinal spot" on the surface of. the yolk is always * In like manner, the cell-nucleus is "usually globular, except in certain specialised tissues, or when it degenerates" (Darlington). Whether it possesses a membrane is matter in dispute, but it at all events possesses a surface, with a phase- difference between it and the surrounding cytoplasm. Cf. above, p. 295. XV] AND OTHER HOLLOW STRUCTURES 943 found uppermost, however we may place and wherever we may open the egg ; that is to say, the yolk easily rotates within the egg, bringing its lighter pole uppermost. In its passage down the oviduct the egg is not merely thrust but also screwed along; and its spiral course leaves traces on wellnigh all its structure save the shell. When we have broken the shell of a hard-boiled egg the shell-membrane below peels off in spiral strips, and even the white tends to flake off in layers, spirally. In the fresh unboiled egg two knotted cords — the treadles or chalazae — are connected with the yolk, and lie fore-and-aft of it, loose in the albimien. These represent the free ends of a yolk-membrane, which got caught in the constricted oviduct while the yolk between them was being screwed along: very much as we may wrap an apple in a handkerchief, hold the two ends fast, and twirl the apple round. These, then, are the general principles involved in, and illustrated by, the configuration of an egg ; and they take us as far as we can safely go without actual quantitative determination, in each par- ticular case, of the forces concerned*. In certain cases among the invertebrates, we again find instances of hard-shelled eggs which have obviously been moulded by the oviduct, or so-called "ootype," in which they have lain: and not merely in such a way as to shew the effects of peristaltic pressure upon a uniform elastic envelope, but so as to impress upon the egg the more or less irregular form of the cavity within which it had been for a time contained and compressed. , After this fashion is explained the curious form of the egg in Bilharzia (Schistosoma) haematobium, a formidable parasitic worm to which is due a disease wide-spread in Africa and Arabia, and an especial scourge of the Mecca pilgrims. The egg in this worm is provided at one end with a little spine, which is explained as having been moulded within a little funnel-shaped expansion of the uterus, just where it communi- cates with the common duct leading from the ovary and yolk-gland. Owing to some anatomical difference in the uterus, the httle * It is a common but unfounded belief among poultry-men that shape and size are related to the sex of the egg, the longer eggs producing mostly male chicks. That there is no such correlation between sex on the one hand and weight, length or shape on the other, has been clearly demonstrated. Cf. M. A. Jull and J. P. Quinn, Journ. Agr. Research, xxix, pp. 195-201, 1924. 944 ON THE SHAPES OF EGGS [ch. spine may be at the end or towards the side of the egg : and this -visible difference has led to the recognition of a new species, S. mansoni*. In a third species, S. japonicum, the egg is described as bulging into a so-called "calotte," or bubble-like convexity at the end opposite to the spine. This, I think, may, with very httle doubt, be ascribed to hardening of the egg-shell having taken place just at the time when partial relief from pressure was being experienced by the egg in the neighbourhood of the dilated orifice of the oviduct. This case of Bilharzia is not, from our present point of view, a very important one, but nevertheless it is interesting. It ascribes to a mechanical cause a curious pecuUarity of form ; and it shews, by reference to this mechanical principle, how two simple mechanical modifications of the same thing may not only seem very different to the systematic naturaUst's eye, but may actually lead to the recognition of a new species, with its own geographical distribution, and its own pathogenic characteristics. On the form of sea-urchins As a corollary to the problem of the bird's egg, we may consider for a moment the forms assumed by the shells of the sea-urchins. These latter are commonly divided into two classes — the Regujar and the Irregular Echinids. The regular sea-urchins, save in sHglit details which do not affect our problem, have a complete axial symmetry. The axis of the animal's body is vertical, with mouth below and the intestinal outlet above; and around this axis the shell is built as a symmetrical system. It follows that in horizontal section the shell is everywhere circular, and we need only consider its form as seen in vertical section or projection. The irregular urchins (very inaccurately so-called) have the anal extremity of the body removed from its central, dorsal situation; and it follows that they have now a single plane of symmetry, about which the organism, shell and all, is bilaterally symmetrical. We need not concern ourselves in detail with the shapes of their shells, which may be very simply interpreted, by the help of radial coordinates, as deformations of the circular or "tegular" type. * L. W. Sambon, Proc. Zool. Soc. 1907 (i), p. 283; also in Journ. Trop. Med. and Hygiene, Sept. 15, 1926. XV] AND OF SEA-URCHINS 945 The sea-urchin shell consists of a membrane, stiffened into rigidity by calcareous deposits, which constitute a beautiful skeleton of separate, neatly fitting "ossicles." The rigidity of the shell is more apparent than real, for the entire structure is, in a sluggish way, plastic ; inasmuch as each Httle ossicle is capable of growth, and the entire shell grows by increments to each and all of these multi- tudinous elements, whose individual growth involves a certain amount of freedom to move relatively to one another ; in a few cases the ossicles are so little developed that the whole shell appears soft and flexible. The viscera of the animal occupy but a small part of the space within the shell, the cavity being mainly filled by a large quantity of watery fluid, whose density must be very near to that of the external sea-water. Apart from the fact that the sea-urchin continues to grow, it is plain that we have here the same general conditions as in the egg-shell, and that the form of the sea-urchin is subject to a similar equihbrium of forces. But there is this important difference, that an external muscular pressure (such as the .oviduct administers during the consolidation of the egg-shell) is now lacking. In its place we have the steady continuous influence of gravity, and there is yet another force which in all probability we require to take into consideration. While the sea-urchin is alive, an immense number of delicate "tube-feet," with suckers at their tips, pass through minute pores in the shell, and, like so many long cables, moor the animal to the ground. They constitute a symmetrical system of forces, with one resultant downwards, in the direction of gravity, and another out- wards in a radial direction ; and if we look upon the shell as originally spherical, both will tend to depress the sphere into a flattened cake. We need not consider the radial component, but may treat the case as that of a spherical shell symmetrically depressed under the influence of gravity. This is precisely the condition which we have to deal with in a drop of liquid lying on a plate ; the form of which is determined by its own uniform surface-tension, plus gravity, acting against the internal hydrostatic pressure. Simple as this system is, the full mathematical investigation of the form of a drop is not easy, and we can scarcely hope that the systematic study of the Echinodermata will ever be conducted by methods based 946 ON THE SHAPES OF EGGS [ch. on Laplace's diiferential equation*; but we have little difficulty in seeing that the various forms represented in a series of sea-urchin shells are no other than those which we may easily and perfectly imitate in drops. In the case of the drop of water (or of any other particular liquid) the specific surface-tension is always constant, and the pressure varies inversely as the radius of curvature; therefore the smaller the drop the more nearly is it able to conserve the spherical form, and the larger the drop the more does it become flattened under gravityf. We can imitate this phenomenon by using india-rubber balls filled with water, of different sizes ; the httle ones will remain very nearly spherical, but the larger will fall down "of their own weight," into the form of more and more flattened cakes; and we see the same thing when we let drops of heavy oil (such as the orthotoluidene spoken of on p. 370) fall through a tall column of water, the httle ones remaining round, and the big ones getting more and more flattened as they sink. In the case of the sea-urchin, the same series of forms may be assumed to occur, irrespective of size, through variations in T, the specific tension, or "strength" of the enveloping shell. Accordingly we may study, entirely from this point of view, such a series as the following (Fig. 454). In a very few cases, such as the fossil Paheechinus, we have an approximately spherical shell, that is to say a shell so strong that the influence of gravity becomes negligible as a cause of deformation, just as (to compare small things with great) the surface tension of mercury is so high that small drops of it seem perfectly spherical {. The ordinary species of Echinus begin to display a pronounced depression, and this reaches its maximum in such soft-shelled flexible forms as Phormosoma. On the general question I took the oppor- ♦ Cf. Bashforth and Adams, Theoretical Forms of Drops, etc., Cambridge, 1883. t The drops must be spherical, or very nearly so, to produce a rainbow. But the bow is said to be always better defined near the top than down below; which seems to shew that the lower and larger raindrops are the less perfect spheres. (Cf. T. W. Backhouse, Symons's M. Met. Mag. 1879, p. 25.) For the small round droplets in the cloud tend to cannon off one another, and remain small and spherical. But when there comes a diflference of potential between cloud and cloud, or be- tween earth and sky, then the spherules become distorted, one droplet coalesces with another, and the big drops begin to fall. X Cf. A. Ferguson, On the theoretical shape of large bubbles and drops, Phil. Mag. (6), XXV, pp. 507-520, 1913. XV] AND OF SEA-UKCHINS 947 tunity of consulting Mr C. R. Darling, who is an acknowledged expert in drops, and he at once agreed" with me that such forms as are represented in Fig. 454 are no other than diagrammatic illustrations of various kinds of drops, "most of which can easily be reproduced in outline by the aid of liquids of approximately equal density to water, although some of them are fugitive." He found a difficulty in the case of the outline which represents Asthenosoma, but the reason for the anomaly is obvious; the flexible shell has flattened Fig. 454. Diagrammatic vertical outlines of various sea-urchins: A, Palaeechinus; B, Echinus acutits; C, Cidaris; D, D', Coelopleurus; E, E', Genicopatagus ; F, Phormosoma liiculenter ; G, P. tenuis; H, Asthenosoma; I, Urechinus. down until it has come in contact with the hard skeleton of the jaws, or " Aristotlci's lantern," within, and the curvature of the outline is accordingly disturbed. The elevated, conical shells such as those of Urechinus and Coelopleurus evidently call for some further ex- planation; for there is here some cause at work to elevate, rather than to depress the shell. Mr Darling tells me that these forms "are nearly identical in shape with globules I have frequently obtained, in which, on standing, bubbles of gas rose to the summit and pressed the skin upwards, without being able to escape." The same condition may be at work in the sea-urchin; but a similar 948 ON THE FORM AND BRANCHING [ch. tendency would also be manifested by the presence in the upper part of the shell of any accumulation of substance Hghter than water, such as is actually present in the masses of fatty, oily eggs. On the form and branching of blood-vessels Passing to what may seem a very different subject, we may investigate a number of interesting points in connection with the form and structure of the blood-vessels, and we shall find ourselves helped, at least in the outset, by the same equations as those we have used in studying the egg-shell. We know that the fluid pressure (P) within the vessel is balanced by (1) the tension (T) of the wall, divided by the radius of curvature, and (2) the external pressure (pn), normal to the wall: according to our formula P = y„+r(l/r+l//). If. we neglect the external pressure, that is to say any support which may be given to the vessel by the surrounding tissues, and if we deal only with a cylindrical vein or artery, this formula becomes simpUfied to the form P = T/R. That is to say, under constant pressure, the tension varies as the radius. But the tension, per unit area of the vessel, depends upon the thickness of the wall, that is to say on the amount of membranous and especially of muscular tissue of which it is Qomposed. Therefore, so long as the pressure is constant, the thickness of the wall should vary as the radius, or as the diameter, of the blood-vessel. But it is not the case that the pressure is constant, for it gradually falls off, by loss through friction, as we pass from the large arteries to the small; and accordingly we find that while, for a time, the cross-sections of the larger and smaller vessels are symmetrical figures, with the wall-thickness proportional to the size of the tube, this proportion is gradually lost, and the walls of the small arteries, and still more of the capillaries, become exceedingly thin, and more so than in strict proportion to the narrowing of the tube. In the case of the heart we have, within each of its cavities, a pressure which, at any given moment, is constant over the whole wall- I XV] OF BLOOD-VESSELS 949 area, but the thickness of the wall varies very considerably. For instance, in the left ventricle the apex is by much the thinnest por- tion, as it is also that with the greatest curvature. We may assume, therefore (or at least suspect), that the formula, ^ (1/r + 1/r') = C, holds good ; that is to say, that the thickness {tj of the wall varies inversely as the mean curvature. This may be tested experimentally, by dilating a heart with alcohol under a known pressure, and then measuring the thickness of the walls in various parts after the whole organ has become hardened. By this means it is found that, for each of the cavities, the law holds good with great accuracy*. Moreover, if we begin by dilating the right ventricle and then dilate the left in like manner, until the whole heart is equally and sym- metrically dilated, we find (1) that we have had to use a pressure in the left ventricle from six to seven times as great as in the right ventricle, and (2) that the thickness of the walls i» just in the same proportion f. Many problems of a hydrodynamical kind arise in connection with the flow of blood through the blood-vessels; and while these are of primary importance to the physiologist they interest the morphologist in so far as they bear on questions of structure and form. As an example of such mechanical problems we may take the con- ditions which go to determine the manner of branching of an artery, or the angle at which its branches are given off; for, as John Hunter said J, "To keep up a circulation sufficient for the part, and no more, Nature has varied the angle of the origin of the arteries accordingly." This is a vastly important theme, and leads us a deal farther than does the problem, petty in comparison, of the shape of an egg. For the theorem which John Hunter has set forth in these simple words is no other than thaf " principle of minimal work" which is funda- mental in physiology, and which some have deemed the very criterion * R. H. Woods, On a physical theorem applied to tense membranes, Journ. of Anat. and Phys. xxvi, pp. 362-371, 1892. A similar investigation of the tensions in the uterine wall, and of the varying thickness of its muscles, was attempted by Haughton in his Animal Mechanics, 1873, pp. 151-158. t This corresponds with a determination of the normal pressures (in systole) by Knohl, as being in the ratio of 1 : 6-8. % Essays, edited by Owen, i, p. 134, 1861. The subject greatly interested Keats. See his Notebook, edited by M. B. Forman, 1932, p, 7; and cf. Keats as a Medical Student, by Sir Wm Hale- White, in Guy's Hospital Reports, lxxiii, pp. 249-262, 1925. 950 ON THE FORM AND BRANCHING [ch. of "organisation*." For the principle of Lagrange, the "principle of virtual work," is the key to physiological equilibrium, and physiology itself has been called a problem in maxima and minima f. This principle, overflowing into morphology, helps to bring the morphological and the physiological concepts together. We have dealt with problems of maxima and minima in many simple con- figurations, where form alone seemed to be in question; and we meet with the same principle again wherever work has to be done and mechanism is at hand to do it. That this mechanism is the best possible under all the circumstaaces of the case, that its work is done with a maximum of efiiciency and at a minimum of cost, may not always lie within oui; range of quantitative demonstration, but to believe it to be so is part of our common faith in the perfection of Nature's harndiwork. All the experience and the very instinct of the physiologist tells him it is true ; he comes to use it as a postulate, or meihodus inveniendi, and it does not lead him astray. The dis- covery of the circulation of the blood was implicit in, or followed quickly after, the recognition of the fact that the valves of heart and veins are adapted to a one-way circulation ; anA we may begin likewise by assuming a perfect fitness or adaptation in all the minor details of the circulation. As part of our concept of organisation we assume that the cost of operating a physiological system is a minimum, what we mean by cost being measurable in calories and ergs, units whose dimensions are equivalent to those of work. The circulation teems with illustra- tions of thi^ great and cardinal principle. " To keep up a circulation sufiicient for the part and no more" Nature has not only varied the angle of branching of the blood-vessels to suit her purpose, she has regulated the dimensions of every branch and stem and twig and capillary; the normal operation of the heart is perfection itself, even the amount of oxygen which enters and leaves the capillaries is such that the work involved in its exchange and transport is a minimum. In short, oxygen transport is the main object of the circulation, and it seems that through all the trials and errors of * Cf. Cecil D. Murray, The physiological principle of minimal work, in the vascular system, and the cost of blood-volume, Proc. Acad. Nat. Sci. xii, pp. 207-214, 1926; The angle of branching of the arteJries, Journ. Gen. Physiol, ix, pp. 835-841, 1926; On the branching-angles of trees, ibid, x, p. 72;'), 1927. t By Dr F. H. Pike, quoted by C. D. Murray. XV] OF BLOOD-VESSELS 951 growth and evolution an efficient mode of transport has been attained. To prove that it is the very best of all possible modes of transport may be beyond our powers and beyond our needs; but to assume that it is perfectly economical is a sound working hypothesis*. And by this working hypothesis we seek to understand the form and dimensions of this structure or that, in terms of the work which it has to do. The general principle, then, is that the form and arrangement of the blood-vessels is such that the circulation proceeds with a minimum of effort, and with a minimum of wall-surface, the latter condition leading to a minimum of friction and being therefore included in the first. What, then, should be the angle of branching, such that there shall be the least possible loss of energy in the course of the circulation? In order to solve this problem in any particular case we should obviously require to know (1) how the loss of energy depends upon the distance travelled, and (2) how the loss of energy varies with the diameter of the vessel. The loss of energy is evidently greater in a narrow tube than in a wide one, and greater, obviously, in a long journey than a short. If the large artery, AB, gives off a com- paratively narrow branch leading to P (such as CP, or DP), the route ACP is evidently shorter than ADP, but on the other hand, by the latter path, the blood has tarried longer in the wide vessel ^B, and has '^ ^. ^g had a shorter course in the narrow branch. The relative advantage of the two paths will depend on the loss of energy in the portion CD, as compared with that in the alternative portion CD', the one being short and narrow, the other long and wide. If we ask, then, which factor is the more important, length * Cf. A. W. Volkmann, Die Haemodynamik nach Verauchen, Leipzig, 1850 (a work of great originality); G. Schwalbe, Ueber. . .die Gestaltung des Arterien- systems, Jen. Zeitschr. xii, p. 267, 1878; W. Hess, Eine mechanischbedingte Gesetz- massigkeit im Bau des Blutgefasssystems, A.f. Entw. Me'ch. xvi, p. 632, 1903 ; Ueber die peripherische Reguiierung der Blutzirkulation, Pfluger's Archiv, CLXvm, pp. 439-490, 1917; R. Thoma, Die mittlere Durchflussmengen der Arteriert des Menschen ais Funktion des Gefassradius, ibid, clxxxix, pp. 282-310, cxcra, pp. 385-406, 1921-22; E. Blum, Querschnittsbeziehungen zwischen Stamm u. Asten im Arteriensystem, ibid, clxxv, pp. 1-19, 1919. 952 ON THE FORM AND BRANCHING [CH. or width, we may safely take it that the question is one of degree ; and that the factor of width will become the more important of the two wherever artery and branch are markedly unequal in size. In other words, it would seem that for small branches a large angle of bifurcation, and for large branches a small one, is always the better. Roux has laid down certain rules in regard to the branching of arteries, which cojrespond with the general conclusions which we have just arrived at. The most important of these are as follows: (1) If an artery bifurcates into two equal branches, these branches come off at equal angles to the main stem. (2) If one of the two branches be smaller than the other, then the main branch, or continuation of the original artery, makes with the latter a smaller angle than does the smaller or "lateral" branch. And (3) all branches which are so small that they scarcely seem to weaken or diminish the main stem come off from it at a large angle, from about 70° to 90°. We may follow Hess in a further investigation of the phenomenon. Let AB be an artery, from which a branch has to be given off so as to reach P, and let ACP, ADP, etc., be alternative courses which the branch may follow: CD, DE, etc., in the diagram, being equal distances (= I) along AB. Let us call the angles PCD, PDE, Xi, X2, etc.: and the distances CD', DE', by which each branch exceeds the next in length, we shall call li,l2, etc. Now it is evident that, of the courses shewn, ACP is the shortest which the blood can take, but it is also that by which its transit through the narrow branch is the longest. We may reduce its transit through the narrow branch more and more, till we come to CGP, or rather to a point where the branch comes off at right angles to the main stem ; but in so doing we very considerably increase the whole distance travelled. We may take it that there will be some intermediate point which will strike the balance of advantage. Fig. 456. XV] OF BLOOD-VESSELS 953 Now it is easy to shew that if, in Fig. 456, the route ADP and AEP (two contiguous routes) be equally favourable, then any other route on either side of these, such as ACP or AFP, must be less favourable than either. Let ADP and AEP, then, be equally favourable; that is to say, let the loss of energy which the blood suffers in its passage along these two routes be equal. Then, if we make the distance DE very small, the angles Xg and x^ are nearly equal, and may be so treated. And again, if DE be very small, then DE'E becomes a right angle, and I2 (or DE') = I cos Xg. But if L be the loss of energy per unit distance in the wide tube AB, and U be the corresponding loss of energy in the narrow tube DP, etc., then IL = I2L' , because, as we have assumed, the loss of energy on the route DP is equal to that on the whole route DEP. Therefore IL = IL cos Xg, and cos x^ = LjU . That is to say, the most favour- able angle of branching will be such that the cosine of the angle is equal to the ratio of the loss of energy which the blood undergoes, per unit of length, in the main vessel, as compared with that which it undergoes in the branch. The path of a ray of light from one refractive medium to another is an analogous but much more famous problem; and the analogy becomes a close one when we look upon the branching artery as the special case of "grazing incidence." After thus dealing with the most suitable angle of branching, we have still to consider the appropriate cross-section of the branches compared with the main trunk, for instance in the special case where a main artery bifurcates into two. That the sectional area of the two branches may together equal the area of the parent trunk, it is (of course) only necessary that the diameters of trunk and branch should be as\/2 : 1, or (say) as 14 : 10, or (still more roughly) as 10 : 7; and in the great vessels, this simple ratio comes very nearly true. We have, for instance, the following measurements of the common ihac arteries, into which the abdominal aorta subdivides: Internal diameter of abdominal arteries* Aorta abdom. (mm.) 15-2 120 141 13-9 Iliaca comm. d. Iliaca comm. s. 10-8 8-8 10-7 8-6 10-4 9-5 8-6 10-0 Mean of do. 10-8 8-7 100 9-3 Ratio 71 72 69 Av. 70 p.c 67 p.c. * From R. Thoma, op. cit. P- 388. 954 ON THE FORM AND BRANCHING [ch. But the increasing surface of the branches soon means increased friction, and a slower pace of the blood travelling through; and therefore the branches must be more capacious than at first appears. It becomes a question not of capacity but of resistance; and in general terms the answer is that the ratio of resistance to cross- section shall be equal in every part of the system, before and after bifurcation, as a condition of least possible resistance in the whole system; the total cross-section of the branches, therefore, must be greater than that of the trunk in proportion to the increased resistance. An approximate result, familiar to students of hydrodynamics, is that the resistance is a minimum, and the condition an optimum, when the cross-section of the main stem is to the sum of the cross-sections of the branches as 1 : ^2, or 1 : 1-26. Accordingly, in the case of a blood-vessel bifurcating into two equal branches, the diameter of each should be to that of the main stem (approximately) as J 1-26 -^-il, or (say) 8: 10. While these statements are so far true, and while they undoubtedly cover a great number of observed facts, yet it is plain that, as in all such cases, we must regard them not as a complete explanation, but as factors in a complicated phenomenon : not forgetting that (as one of the most learned of all students of the heart and arteries, Dr Thomas Young, said in his Croonian lecture*) all such questions as these, and all matters connected with the muscular and elastic powers of the blood-vessels, "belong to the most refined departments of hydrauhcs"; and Euler himself had commented on the "in- * On the functions of the heart and arteries, Phil. Trans. 1809, pp. 1-31, of. 1808, pp. 164-186; Collected Works, i, pp. 511-534, 1855. The same lesson is conveyed by all such work as that of Volkmann, E. H. Weber and Poiseuille. Cf. Stephen Hales's Statical Essays, ii, Introduction: "Especially considering that they [i.e. animal Bodies] are in a manner framed of one continued Maze of innumerable Canals, in which Fluids are incessantly circulating, some with great Force and Rapidity, others with very different Degrees of rebated Velocity: Hence, etc." Even Leonardo had brought his knowledge of hydrodynamics to bear on the valves of the heart and the vortex-like eddies of the blood. Cf. J. Playfair McMurrich, L. da Vinci, the Anatomist, 1930, p. 165; etc. How com- plicated the physiological aspect of the case becomes may be judged by Thoma's papers quoted above. XV] OF BLOOD-VESSELS 955 superable difficulties" of this sort of problem*. Some other explanation must be sought in order to account for a phenomenon which particularly impressed John Hunter's mind, namely the gradually altering angle at which the successive intercostal arteries are given off from the thoracic aorta: the special interest of this case arising from the regularity and symmetry of the series, for "there is not another set of arteries in the body whose origins are so much the same, whose offices are so much the same, whose dis- tances from their origin.to the place of use, and whose uses [? sizes] f are so much the same." The mechanical and hydrodynamical aspect of the circulation was as plain to John Hunter's mind as it had been to William Harvey or to Stephen Hales, or as it was" afterwards to Thomas Young; but it was not always plain to other men. When a turtle's heart has been removed from its body, the blood may still be seen moving in the capillaries for some short while thereafter; and Haller, seeing this, "attributed it to some unknown power which he conceived to be exerted by the solid tissues on the blood and also by the globules of the blood on each other; to which power, until further investigation should elucidate its nature, he gave the name of attraction'' So said WilHam Sharpey, the father of modern Enghsh physiology; and Sharpey went on to say that "many physiologists accordingly maintain the existence of a peculiar pro- pulsive power in the coats of the capillary vessels different from contractility, or that the globules of the blood are possessed of the power of spontaneous motion." Alison, great physician and famous vitalist, "extended this view, in so far as he regards the motion of the blood in the capillaries as one of the effects produced by what he calls vital attraction and repulsion, powers which he conceives to be general attributes of hving matter." But Sharpey 's own clear insight so far overcame his faith in Alison that he found it "not impossible that a certain degree of agitation might be occa- sioned in the blood by the elastic resilience of the vessels reacting on it, after the distending force of the heart has been withdrawn"; and, in short, that the evidence in the case did not "warrant the * In a tract entitled Principia pro motu sanguinis per arterias determinando Op. posth. XI, pp. 814-823, 1862. f "Sizes" is Owen's editorial emendation, which seems amply justified. 956 ON THE FORM AND BRANCHING [ch. assumption of a peculiar power acting on the blood, of whose existence in the animal economy we have as yet no other evidence *," Sir Charles Bell, whose anatomical skill was great but his mathe- matical insight small, drew the conclusion, of no small historic interest, that "the laws of hydraulics, though illustrative, are not strictly applicable to the explanation of the circulation of the blood, nor to the actions of the living frame." He goes on to say: *' Although \^e perceive admirable mechanism in the heart, and in the adjustment of the tubes on hydraulic principles: and although the arteries and veins have form, calibre and curves suited^ to the conveyance of fluid, according to our knowledge of hydraulic engines: yet the laws of life, or of physiology, are. essential to the explanation of the circulation of the blood. And this conclusion we draw, not only from the extent and minuteness of the vessels, but also from the peculiar nature of the blood itself. Life is in both, and a mutual influence prevails f." This peculiar form of vitalism savours more of Bichat and the French school than of the teaching of John Hunter or Thomas Young. It is precisely that idea of "organic control" or "organic coordination," which the physiologists are always reluctant to accept, always unwilling to abandon: which is said to be inherent in every process or operation of the body, and to differentiate biology from all the physical sciences : and of which in our own day Haldane has been the chief and great protagonist. But it is a subject with which this book is not concerned. , To conclude, we may now approach the question of economical size of the blood-vessels in a broader way. They must not be too small, or the work of driving blood through them will be too great; * See Sharpey's article on Cilia, in Todd's Cyclopaedia, i, p. 637, 1836; also Allen Thomson's admirable article on the Circulation, ibid. p. 672. Alison's views were based not only on Haller, but largely on Dr James Black's Essay on the Capillary Circulation, London, 1825. t Practical Essays, 1842, p. 88. When Sir Charles Bell declared that hydraulic principles were not enough, but that "the laws of life" were needed to explain the circulation of the blood, he was right from his point of view. He was slow to see, and unwilling to admit, that hydrodynamical principles suffice to explain a large, essential part of the problem ; but as a physiologist he had every reason to know that that part was not the whole. He may have had many things in mind: the arrest of the circulation in inflammation, as we see it in a frog's web ; that a cut artery bleeds to death while a torn one does not bleed at all; that blood does not coagulate when stagnant within its own vessels — a fact which, as John Hunter said, "has ever appeared to me the most interesting fact in physiology." XV] OF BLOOD-VESSELS 957 they must not be too large, or they will hold more blood than is needed — and blood is a costly thing. We rely once more on Poiseuille's Law*, which tells us the amount of work done in causing so much fluid to flow through a tube against resistance, the said resistance being measured by the viscosity of the fluid, the coefficient of friction and the dimensions of the tube; but we have also to account for the blood itself, whose maintenance requires a share of the bodily fuel, and whose cost per c.c. may (in theory at least) be expressed in calories, or in ergs per day. The total cost, then, of operating a given section of artery will be measured by (1) the work done in overcoming its resistance, and (2) the work, done in providing blood to fill it; we have come again to a differential equation, leading to an equation of maximal -efficiency. The general result! is as follows: it can be made a quantitative one by intro- ducing known experimental values. Were blood a cheaper thing than it is we might expect all arteries to be uniformly larger than they are,, for thereby the burden on the heart (the flow remaining equal) would be greatly reduced — thus if the blood-vessels were doubled in diameter, and their volume thereby quadrupled, the work of the heart would be reduced to one-sixteenth. On the other hand, were blood a scarcer and still costlier fluid, narrower blood- vessels would hold the available supply; but a larger and stronger heart would be needed to overcome the increased resistance. * Owing to faulty determination of the fall of pressure in the capillaries, Poiseuille's equation used to be deemed inapplicable to them ; but Krogh's recent work removes, or tends to remove, the inconsistency {Anatomy and Physiology of the Capillaries, Yale University Press, 1922). t Cf. C. D. Murray, op. cit. p. 211. CHAPTER XVI ON FORM AND MECHANICAL EFFICIENCY There is a certain large class of morphological problems of which we have not yet spoken, and of which we shall be able to say but Uttle. Nevertheless they are so important, so full of deep theoretical significance, and so bound up with the general question of form and its determination as a result of growth, that an essay on growth and form is bound to take account of them, however im- perfectly and briefly. The phenomena which I have in mind are just those many cases where adaptation in the strictest sense is obviously present, in the clearly demonstrable form of mechanical fitness for the exercise of some particular function or action which has become inseparable from the life and well-being of the organism. When we discuss certain so-called "adaptations" to outward circumstance, in the way of form, colour and so forth, we are often apt to use illustrations convincing enough to certain minds but unsatisfying to others — in other words, incapable of demonstration. With regard to coloration, for instance, it is by colours "cryptic," "\Yarning," "signalling," "mimetic," and so on*, that we prosaically expound, and slavishly profess to justify, the vast AristoteUan synthesis that Nature makes all things with a purpose and "does nothing in vain." Only for a moment let us glance at some few instances by which the modern teleologist accounts for this or that manifestation of colour, and is led on and on to beUefs and doctrines to which it becomes more and more difficult to subscribe. Some dangerous and mahgnant animals are said (in sober earnest) to wear a perpetual war-paint, in order to "remind their enemies that they had better leave them alonef." The wasp and the hornet, * For a more elaborate classification, into colours cryptic, procryptic, anti- cryptic, apatetic, epigamic, sematic, episematic, aposematic, etc., see Poulton's Colours of Animals (Int. Scientific Series, Lxvin, 1890; cf. also R. Meldola, Variable protective colouring in insects, P.Z.S. 1873, pp. 153-162; etc. The subject is well and fully set forth by H. B. Cott, Adaption coloration in Animals, 1940. t Dendy, Evolutionary Biology, 1912, p. 336. CH. XVI] THE INTERPRETATION OF COLOUR 959 in gallant black and gold, are terrible as an army with banners; and the Gila Monster (the poison-lizard of the Arizona desert) is splashed with scarlet — its dread and black complexion stained with heraldry more dismal. But the wasp-Hke livery of the noisy, idle hover-flies and drone-flies is but stage armour, and in their tinsel suits the little counterfeit cowardly knaves mimic the fighting crew. The jewelled splendour of the peacock and the humming-bird, and the less effulgent glory of the lyre-bird and the Argus pheasant, are ascribed to the unquestioned prevalence of vanity in the one sex and wantonness in the other*. The zebra is striped that it may graze unnoticed on the plain, the tiger that it may lurk undiscovered in the jungle; the banded Chaetodont and Pomacentrid fishes are further bedizened to the hues of the coral-reefs in which they dwell f- The tawny lion is yellow as the desert sand; but the leopard wears its dappled hide to blend, as it crouches on the branch, with the sun-flecks peeping through the leaves. The ptarmigan and the snowy owl, the arctic fox and the polar bear, are white among the snows; but go he north or go he south, the raven (Uke the jackdaw) is boldly and impudently black. The rabbit has his white scut, and sundry antelopes their piebald flanks, that one timorous fugitive may hie after another, spying the warning signal. The primeval terrier or colHe-dog had brown spots over his eyes that he might seem awake when he was sleeping J : so that an enemy might let the sleeping dog lie, for the singular reason that he imagined him to be awake. And a flock of flamingos, * Delight in beauty is one of the pleasures of the imagination; there is no limit to its indulgence, and no end to the results which we may ascribe to its exercise. But as for the particular "standard of beauty" which the oird (for instance) admires and selects (as Darwin says in the Origin, p. 70, edit. 1884), we are very much in the dark, and we run the risk of arguing in a circle ; for wellnigh all we can safely say is what Addison says (in the 412th Spectator) — that each different species "is most aflFected with the beauties of its own kind, . . ,Hinc merula in nigro 86 oblectat nigra marito; . . .hinc noctua tetram Canitiem alarum et glaucos miratur ocellos." t Cf. T. W. Bridge, Cambridge Natural History (Fishes), vn, p. 173, 1904; also K. V. Frisch, Ueber farbige Anpassung bei Fische, Zool. Jahrh. {Abt. Allg. ZooL), xxxn, pp. 171-230, 1914. But Reighard, in what Raymond Pearl calls "one of the most beautiful experimental studies of natural selection which has ever been made," found no relation between the colours of coral-reef fishes and their elimination by natural enemies {Carnegie Inst. Publication 103, pp. 257-325, 1908). } Nature, L, p. 672; u, pp. 33, 57, 533, 1894-95. 960 ON FORM AND MECHANICAL EFFICIENCY [ch. wearing on rosy breast and crimson wings a garment of in- visibility, fades away into the sky at dawn or sunset like a cloud incarnadine*. To buttress the theory of natural selection the sa'me instances of "adaptation" (and many more) are used, as in an earher but not distant age testified to the wisdom of the Creator and revealed to simple piety the immediate finger of God. In the words of a certain learned theologian f, ''The free use of final causes to explain what seems obscure was temptingly easy-. . . . Hence the finalist was often the man who made a liberal use of the ignava ratio, or lazy argument: when you failed to explain a thing by the ordinary process of causahty, you could 'explain' it by reference to some purpose of nature or of its Creator. This method lent itself with dangerous facility to the well-meant endeavours of the older theologians to expound and emphasise the beneficence of the divine purpose." Mutatis mutandis, the passage carries its plain message to the naturahst. The fate of such arguments or illustrations is always the same. They attract and captivate for awhile; they go to the building of a creed, which contemporary orthodoxy % defends under its severest penalties: but the time comes when they lose their fascination, they somehow cease to satisfy and to convince, their foundations * They are "wonderfully fitted for 'vanishment' against the flushed, rich- coloured skies of early morning and evening. . .their chief feeding - times " ; and "look like a real sunset or dawn, repeated on the opposite side of the heavens — either east or west as the case may be" (Thayer, Concealing -coloration in the Animal Kingdom, New York, 1909, pp. 154-155). This hypothesis, like the rest, is not free from difficulty. Twilight is apt to be short in the homes of the flamingo; moreover, Mr Abel Chapman watched them on the Guadalquivir feeding by day, as I also have seen them at Walfisch Bay. f Principal Galloway, Philosophy of Religion, 1914, p. 344. J Professor D. M. S. Watson, addressing the British Association in 1929 on Adaptation, parted company with what I had called contemporary orthodoxy in 1917. Speaking of such morphological differences as "have commonly been assumed to be of an adaptive nature," he said: "That these structural diflferences are adaptive is for the most part pure assumption. . . .There is no branch of zoology in which assumption has played a greater part, or evidence a less part, than in the study of such presumed adaptations. ' ' Hume, in his Dialogue concerning Natural Religion, shewed similar caut-ion: "Steps of a stair are plainly constructed that human legs may use them in mounting; and this inference is certain and infallible. Human legs are also contrived for walking and mounting ; and this inference, I allow, is not altogether so certain." XVI] THE PROBLEM OF ADAPTATION %1 are discovered to be insecure, and in the end no man troubles to controvert them*. But of a very different order from all such "adaptations" as these are those very perfect adaptations of form which, for instance, fit a fish for swimming or a bird for flight. Here we are far above the region of mere hypothesis, for we have to deal with questions of mechanical efficiency where statical and dynamical considerations can be appHed and established in detail. The naval architect learns a great part of his lesson from the stream-lining of a fish|; the yachtsman learns that his sails are nothing more than a great bird's wing, causing the slender hull to fly along % ; and the mathematical study of the stream-lines of a bird, and of the principles underlying the areas and curvatures of its wings and tail, has helped to lay the very foundations of 'the modern science of aeronautics. We know, for example, how in strict accord with theorv i\t was George Cayley who explained it first) the wing, whether of bird or insect, stands stiff along its "leading edge," like the mast before the sail; and how, conversely, it thins out exquisitely fine along its rear or "trailing edge," where sharp discontinuity favours the formation of uplifting eddies. And pQ see how, alike in the flying wing, in the penguin's swimming wing and in the whale's flipper, the same design of stiff fore-edge and thin fine trailing edge, both curving away evenly to meet at the tip, is continually exemphfied. We learn how lifting power not only depends on area but has a linear factor besides, such that a long narrow wing is more stable and effective both for speedy and for soaring flight than a short and broad one of equal area ; and how in this respect the hawkmoth differs from the butterfly, the swallow from the thrush. We are taught how every wing, and every kite or sail, must have a certain * The influence of environment on coloration is one thing, and the hypothesis of protective colouring is quite another. That arctic animals are often white, and desert animals sandy-hued or isabelline, are simple and undisputed facts; but such field-naturalists as Theodore Roosevelt, Selous (in his African Nature Notes), Buxton {Animal Life in Deserts), and Abel Chapman {Savage Sudan, and Retrospect) reject with one accord the theory of colour-protection. t No creature shews more perfect stream-lining than a fur-seal swimming. Every curve is a continuous curve, the very ears and eye-slits and whiskers falling into the scheme, and the flippers folding close against the body. J Cf. Manfred Curry, Yacht Racing, and the Aerodynamics of Sails, London, 1928. 962 ON FORM AND MECHANICAL EFFICIENCY [ch. amount of arch* or "belly," slight in the rapid fliers, deeper in the slow, flattened in the strong wind, bulging in the gentler breeze; and how advantageous is all possible stiffening of sail or wing, and why accordingly the yachtsman inserts "battens" and the Chinaman bamboos in his sail. We are shewn by Lilienthal himself how a powerful eddy, the so-called "ram," forms under the fore-edge, and is sometimes caught in a pocket of the bird's under wing-coverts and made use of as a forward drive. We have lately learned how the gaps or slots between the primary wing-feathers of a crow, and a slight power of the wing-feathers to twist, like the slats of a Venetian blind, play their necessary 1 Fig. 457. Ligaments in a swan's wing. 1, 2, 3, remiges; A, B, longitudinal ligaments, with their oblique branches; C, small subcutaneous ligaments. From Marey, after Pettigrew. part under certain conditions in the perfect working of the machine. Nothing can be simpler than the mechanism by whi(!h all this -is done. Delicate Ugaments run along the base of the wing from feather to feather, and send a branch to every quill (Fig. 457); by these, as the wing extends, the quills are raised into their places, and kept at their due and even distances apart. Not only that, but every separate ligamentous strand curls a httle way round its feather where it is inserted into it; and thereby the feather is not only elevated into its place, but is given the little twist which brings it to its proper and precise obliquity. * On the curvatura veli, cf. J. Bernoulli, Acta Ertidit. Lips. 1692, p. 202. Studied also by Eiffel, Resistance de Vair et V aviation, Paris, 1910. XVI] THE MECHANISM OF FLIGHT 963 All this is part of the automatic mechanism of the wing, than which there is nothing prettier in all anatomy. ' The triceps muscle, massed on the shoulder, extends the elbow- j oint in the usual way. But another muscle (a long flexor of the wrist) has its origin above the elbow and its insertion below the wrist. It passes over two j oints ; it is what German anatomists call a zweigelenkiger Muskel. It transmits to the one joint the movements of the other; and in birds of powerful flight it becomes less and less muscular, more and more tendinous, and so more and more completely automatic. The wing itself is kept hght; its chief muscle is far back on the shoulder ; a contraction of that re- mote muscle throws the whole wing into gear. The little ligaments we have been speaking of are so linked up with the rest of the mechanism that we have only to hold a bird's wing by the arm-boue and extend its elbow- joint, to see the whole wing spring into action, with every joint extended, and every feather tense and in its place*. Again on the fore-edge of the wing there lies a tiny mobile " thumb," whose little tuft of stiff, strong feathers forms the so-called "bastard wing." We used to look on it as a "vestigial organ," a functionless rudiment, a something which from ancient times had "lagged superfluous on the stage"; until a man of genius saw that it was just the very thing required to break the leading. vortices, keep the flow stream-lined at a larger angle of incidence than before, and thereby help the plane to land. So he invented, to his great profit and advantage, the "slotted wing." We learn many and many another interesting thing. How a stiff "comb" along the leading edge, a broad soft fringe along the trailing edge (the fringe acting as a damper and preventing "fluttering"), and a soft, downy upper surface of the wing, all help as silencers, and give the owl her noiseless flight. How the wing- loading of the owls is lower than in any other birds, lower even than in the eagles; and how owl and eagle have power to spare to carry easily their prey of mouse or mountain-hare. How the deep terminal wing-slots aid the heavy rook or heron in their slow * The mole's forelimb has a somewhat similar action, by which, as the arm and hand are pulled violently backward, the claws are powerfully and automatically flexed for digging. The "suspensory ligament " of the horse, which is, or was, a short flexor of the digit, is an analogous mechanism. See a paper of mine On the nature and action of certain ligaments, Journ. of Anat. and Physiol, xviii, pp. 406-410, 1884. 964 ON FOKM AND MECHANICAL EFFICIENCY [ch. flapping flight ; how the sea-gull does not need them, for his load is lighter and his wings move slowly. There is never a discovery made in the theory of aerodynamics but we find it adopted already by Nature, and exemplified in the construction of the wing*. We may illustrate some few of the principles ijivolved in the con- struction of the bird's wing with a half-sheet of paper, whose laws, as it planes or glides downwards, Clerk Maxwell explained many years ago|. To improve this first and roughest of models, we see that its leading edge had better be as long as possible, and that sharp 1 / / 1 \ / \ / \ L i Fig. 458. A diagrammatic bird. corners are bound to cause disturbance ; let us get rid of the corners and turn the leading edge into a continuous curve (Fig. 458). The leading edge is now doing most of the ^^ork, and the area within and behind is doing little good. Vorticoid air-currents are beating down on either side on this inner area; moreover, air is ''sliding out" below, and tending to curl round the tip and edges of the * Note that the aeroplane copies the beetle rather than the bird, as Lilienthal himself points out, in Vom Gleitflug zu Segelflug, Berlin, 1923. f On a particular case of the descent of a heavy body in a resisting medium, Camb. and Dublin Math. Jonrn. ix, pp. 145-148, 1854; Sci. Papers, i, pp. 115-118. This elegant and celebrated little paper was written by Clerk Maxwell while an undergraduate at Trinity College, Cambridge. XVI] OF STREAM-LINES 965 wing, all with so much waste of energy*. In short the broad wing is less efficient than the narrow; and on either side of our sheet of paper we may cut out a portion whiph is useless and in the way : for the same reasons we may. cut out the middle part of the tail, which also is doing more harm than goodf. Hard as the problem is, and harder as it becomes, we may venture on. In aeronautics, as in hydrodynamics, we try to determine the resistances encountered by bodies of various shapes, moving through various fluids at various speeds; and in so doing we learn the enormous, the paramount importance of "stream-lining." There would be no need for stream-lining in a "perfect fluid," but in air or water it makes all the difference in the world. Stream-lining impHes a shape round which the medium streams so smoothly that resistance is at last practically nil; there only remains the slight "skin- friction," which can be reduced or minimised in various ways. But the least imperfection of the stream-lining leads to whirls and "pockets" of dead water or dead air, which mean large resistance and waste of energy. The converse and more general problem soon emerges, of how in natural objects stream-lining comes to be; and whether or no the more or l^ss stream-lined shape tends to be impressed on a deformable or plastic body by its own steady motion through a fluid J. The principle of least action, the "loi de repos," is enough to suggest that the stream will tend to impress its stream- lines on the plastic body, causing it to yield or "give," until it ends by ofl'ering a minimum of resistance; and experiment goes some way to support the hypothesis. A bubble of mercury, poised in a tube through which air is blown, assumes a stream-lined shape, in so far as the forces due to the moving current avail against the * This is why "slotting" so improves the broad wing of the crow. See on this and other matters, R. R. Graham's papers on Safety devices in the wings of birds, in British Birds, xxiv, 1930. t Pettigrew shewed long ago {Tr. R.S.E. xxvi, p. 361, 1872) that the wing-area (in insects) "is usually greatly in excess of what is absolutely required for flight," and that the posterior or traihng edge could be largely trimmed away without the power of flight being at all diminished. We see how in the swallow this trailing edge is "trimmed away" till a bare minimum is left, and how (at least for a certain kind of flight) the wing is thereby greatly improved. X Cf. Enoch Farrer, The shape assumed by a deformable body immersed in a moving fluid, Journ. Franklin Inst. 1921, pp. 737-756; also Vaughan Cornish, on Waves of Sand and Snow. 966 ON FORM 'AND MECHANICAL EFFICIENCY [cii. other forces of restraint. We have seen how the egg is automatically stream-lined, after a simple fashion, by the muscular pressure which drives it on its way. The contours of a snowdrift, of a wind- swept sand-dune, even of the flame of a lamp, shew endless illustrations of stream-lines or eddy-curves which the stream itself imposes, and which are oftentimes of great elegance and complexity. Always the stream tends to mould the bodies it streams over, facilitating its own flow; and the same principle must somehow come into play, at least as a contributory factor, in the making of a fish or of a bird. But it is obvious in both of these that even though the stream-lining be perfected in the individual it is also an inheritance of the race ; and the twofold problem of accumulated inheritance, and of perfect structural adaptation, confronts us once again and passes all our understanding*. When, after attempting to comprehend the exquisite adaptation of the swallow or the albatross to the navigation of the air, we try to pass beyond the empirical study and contemplation of such per- fection of mechanical fitness, and to ask how such fitness came to be, then indeed we may be excused if we stand wrapt in wonderment, and if our minds be occupied and even satisfied with the conception of a final cause. And yet all the while, with no loss of wonderment nor lack of reverence, do we find ourselves constrained to believe that somehow or other, in dynamical principles and natural law,' there lie hidden the steps and stages of physical causation by which the material structure was so shapen to its endsf. The problems associated with these phenomena are difficult at every stage, even long before we approach to the unsolved secrets of causation ; and for my part I confess I lack the requisite know- ledge for even an elementary discussion of the form of a fish, or of an insect, or of a bird. But in the form of a bone we have a problem * Mechanical perfection has often little to do with immunity from accident or with capacity to survive. Legs and wings of locust or mayfly are indescribably perfect for their brief spell of life and narrow sphere of toil; but they may be torn asunder in a moment, and whole populations perish in an hour. Careful of the type, but careless of the single life, Nature seems ruthless and indiscriminate in the sacrifice of these little lives. t Cf. Professor Flint, in his Preface to Affleck's translation of Janet's Catises finales: "We are, no doubt, still a long way from a mechanical theory of organic growth, but it may be said to be the quaesitum of modern science, and no one can say that it is a chimaera." XVI] THE STRENGTH OF MATERIALS 967 of the same kind and order, so far simplified and particularised that we may to some extent deal with it, and may possibly even find, in our partial comprehension of it, a partial clue to the principles of causation underlying this whole class of phenomena. Before we speak of the form of a bone, let us say a word about the mechanical properties of the material of which it is built*, in relation to the strength it has to manifest or the forces it has to resist: understanding always that we mean thereby the properties of fresh or living bone, with all its organic as well as inorganic constituents, for dead, dry bone is a very different thing. In all the structures raised by the engineer, in beams, pillars and girders of every kind, provision has to be made, somehow or other, for strength of two kinds, strength to resist com- pression or crushing, and strength to resist tension or pulling asunder. The evenly loaded column is designed with a view to supporting a downward pressure^ the wire-rope, like the tendon ^^' of a muscle, is adapted only to resist a tensile stress ; but in many or most cases the two functions are very closely inter-related and com- bined. The case of a loaded beam is a familiar one; though, by the way, we are now told that it is by no means so simple as it looks, and indeed that "the stresses and strains in this. log of timber are so complex that the problem has not yet been solved in a manner that .reasonably accords with the known strength of the beam as found by actual experiment!." However, be that as it may. we know, roughly, that when the beam is loaded in the middle and supported at both ends, it tends to be bent into an arc. in which condition its lower fibres are being stretched, or are undergoing a tensile * Cf. Sir Donald MacAlister, How a bone is built, Engl. III. Mag. 1884, t Professor Claxton Fidler, On Bridge Construction, p. 22 (4th ed.), 1909; cf. {int. al.) Love's Elasticity, p. 20 {Historical Introduction), 2nd ed., 1906, where the bending of the beam, and the distortion or warping of its cross-section, are studied after the manner of St Venant, in his Memoir on Torsion (1855). How complex the question has become may be judged from such papers as Price, On the structure of wood in relation to .its elastic properties, Phil. Trans. (A), ccvni, 1928; or D. B. Smith and R. V. Southwell, On the stresses induced by flexure in a deep rectangular beam, Proc. U.S. (A), cXLiii, pp. 271-285, 1934. 968 ON FORM AND MECHANICAL EFFICIENCY [ch. stress, while its upper fibres are undergoing compression. It follows that in some intermediate layer there is a "neutral zone," where the fibres of the wood are subject to no stress of either kind. The phenomenon of a compression-member side by side with a tension- member may be illustrated in many simple ways. Ruskin (in Deucalion) describes it in a glacier. He then bids us warm a stick of sealing-wax and bend it in a horseshoe: "you will then see, through a lens of moderate power, the most exquisite facsimile of glacier fissures produced by extension on its convex surface, and as faithful an image of glacier surge produced by com- pression on its concave side." A still more beautiful way of exhibiting the distribution of strain is to use gelatin, into which bubbles of gas have been .introduced with the help of sodium bicarbonate. A bar of such gelatin, when bent into a hoop, shews on the one side the bubbles elongated by tension and on the other those shortened by compression*. In like manner a vertical pillar, if unevenly loaded (as for in- stance the shaft of our thigh-bone normally is), will tend to bend, and so to endure compression on its concave, and tensile stress upon its convex side. In many cases it is the business of the engineer to separate out, as far as possible, the pressure-lines from the tension-lines, in order to use separate modes of construction, or even different materials for each. In a suspension-bridge, for instance, a great part of the fabric is subject to tensile strain only, and is built throughout of ropes or wires; but the massive piers at either end of the bridge carry the weight of the whole structure and of its load, and endure all the "compression-strains" which are inherent in the system. Very much the same is the case in that wonderful arrangement of struts and ties which constitute, or complete, the skeleton of an animal. The "skeleton," as we see it in a Museum, is a poor and even, a misleading picture of mechanical efficiency f. From the engineer's point of view, it is a diagram shewing all the compression-lines, but by no means all the tension-lines of the construction; it shews all the struts, but few of the ties, and perhaps we might even say none of the principal * Cf. Emil Hatschek, Gestait und Orientirung von Gasblasen in Gelen, KoUoid- Ztschr. XV, pp. 226-234, 1914. t In preparing or "macerating" a skeleton, the naturalist nowadays carries on the process till nothing is left but the whitened bones. But the old anatomists, whose object was not the study of "comparative morphology" but the wider theme of comparative physiology, were wont to macerate by easy stages; and in many of their most instructive preparations the ligaments were intentionally left in connection with the bones, and as part of the "skeleton." XVI] THE STRENGTH OF M]A T|E R I A L S 969 ones; it falls all to pieces unless we clamp it together, as ])est we can, in a more or less clumsy and immobilised way. But in life, that fabric of struts is surrounded and interwoven with a compli- cated system of ties — "its living mantles jointed strong, With ghstering band and silvery thong*": ligament and membrane, muscle and tendon, run between bone and bone; and the beauty and strength of the mechanical construction he not in one part or in another, but in the harmonious concatenation which all the parts, soft and hard, rigid and flexible, tension-bearing and pressure-bearing, make up together j*. However much we may find a tendency, whether in Nature or art, to separate these two constituent factors of tension and com- pression, we cannot do so completely ; and accordingly the engineer seeks for a material which shall, as nearly as possible, offer equal resistance to both kinds of strain J. From the engineer's point of view, bone may seem weak indeed ; but it has the great advantage that it is very nearly as good for a tie as for a strut, nearly as strong to withstand rupture, or tearing asunder, as to resist crushing. The strength of timber varies with the kind, but it always stands up better to tension than to compression, and wrought iron, with its greater strength, does much the same; but in cast-iron there is a still greater discrepancy the other way, for it makes a good strut but a very bad tie indeed. Mild steel, which has displaced the old-fashioned wrought iron in all engineering construc- tions, is not only a much stronger material, but it also possesses, like- bone, the two kinds of strength in no very great relative dis- proportion §. * See Oliver Wendell Holmes' Anatomist's Hymn. t In a few anatomical diagrams, for instance in some of the drawings in Schmaltz's Atlas der Anatomie des Pferdes, we may see the system of "ties" diagrammatically inserted in the figure of the skeleton. Cf. W. K. Gregory, On the principles of quadrupedal locomotion, Ann. N. Y. Acad, of Sciences, xxii, p. 289, 1912. X The strength of materials is not easy to discuss, and is still harder to tabulate. The wide range of qualities in each material, in timber the wide differences according to the direction in which the block is cut, and in all cases the wide difference between yield-point and fracture-point, are some of the difficulties in the way of a succinct statement. § In the modern device of "reinforced concrete," blocks of cement and rods of steel are so combined together as to resist both compression and tension in due or equal measure. 970 ON FORM AND MECHANICAL EFFICIENCY [ch. When the engineer constructs an iron or steel girder, to take the place of the primitive wooden beam, we know that he takes advantage of the elementary principle we have spoken of, and saves weight and economises material by leaving out as far as possible all the middle portion, all the parts in the neighbourhood of the "neutral zone"; and in so doing he reduces his girder to an upper and lower "flange," connected together by a "web," the whole resembling, in cross-section, an I or an I . But it is obvious that, if the strains in the two flanges are to be equal as well as opposite, and if the material be such as cast-iron or wrought-iron, one or other flange must be made much thicker than the other in order that they may be equally strong* ; and if at times the two flanges have, as it were, to change places, or play each other's parts, then there must be introduced a margin of safety by making both flanges thick enough to meet that kind of stress in regard to which the material happens to be weakest. There is great economy, then, in any material which is, as nearly as possible, equally strong in both ways; and so we see that, from the engineer's or contractor's point of view, bone is a good and suitable material for purposes of construction. The I or the H-girder or rail is designed to resist bending in one particular direction, but if, as in a tall pillar, it be necessary to resist bending in all directions alike, it is obvious that the tubular or cylindrical construction best meets the case; for it is plain that this hollow tubular pillar is but the I-girder turned round every way, in a "solid of revolution," so that on any two opposite sides compression and tension are equally met and resisted, and there is now no need for any substance at all in the way of web or "filHng" within the hollow core of the tube. And it is not only in the supporting pillar that such a construction is useful ; it is appropriate in every case where stiffness is required, where bending has to be resisted. A sheet of paper becomes a stiff rod when you roll it up, and hollow tubes of thin bent wood withstand powerful thrusts in aeroplane construction. The long bone of a bird's wing has Httle or no weight to carry, but it has to withstand powerful bending * This principle was recognized as soon as iron came into common use as a structural material. The great suspension bridges only became possible, in Telford's hands, when wrought iron became available. XVI] OF TUBULAR STRUCTURES 971 moments; and in the arm-bone of a long-winged bird, such as an albatross, we see the tubular construction manifested in its perfection, the bony substance being reduced to a thin, perfectly cylindrical, and almost empty shell*. The quill of the bird's feather, the hollow shaft of a reed, the thin tube of the wheat- straw bearing its heavy burden in the ear, are all illustrations which Gahleo used in his account of this mechanical principle f; and the working of his practical mind is exemplified by this catalogue of varied instances w^hich one demonstration suffices to explain. The same principle is beautifully shewn in the hollow body and tubular limbs of an insect or a crustacean ; and these complicated and elaborately jointed structures have doubtless many constructional lessons to teach us. We know, for instance, that a thin cylindrical tube, under bending stress, tends to flatten before it buckles, and also to become "lobed" on the compression side of the bend; and we often recognise both of these phenomena in the joints of a crab's legj. Two points, both of considerable importance, present themselves here, and w^e may deal with them before we go further on. In the first place, it is not difficult to see that in our bending beam the stress is greatest at its middle ; if we press our walking-stick hard against the ground, it will tend to snap midway. Hence, if our cylindrical column be exposed to strong bending stresses, it will be prudent and economical to make its walls thickest in the middle and thinning off gradually towards the ends; and if we look at a longitudinal section of a thigh-bone, we shall see that this is just what Nature has done. The presence of a "danger-point" has been avoided, and the thickness of the walls becomes nothing less than * Marsigli (op. cit.) was acquainted with the hollow wfng-bones of the pelican; and Bulfon deals with the whole subject in his Discours sur la nature des oiseaux. f Galileo, Dialogues concerning Two New Sciences (1638). Crew and Salvio's translation. New York, 1914, p. 150; Opere, ed. Favaro, viii, p. 186. (According to R, A. Millikan, "we owe crur present day civilisation to Galileo.") Cf. Borelli, De Motu Animaliiim, i, prop, clxxx, 168.5. Cf. also P. Camper. La structure des OS dans les oiseaux, 0pp. in, p. 459, ed. 1803; A. Rauber, Gahleo iiber Knochen- formen, Morphol. Jahrb. vii, pp. 327, 328, 1881; Paolo Enriques, Delia economia di sostanza nelle osse cave, Arch. f. Entw. Mech. xx, pp. 427-465, 1906. Galileo's views on the mechanism of the human body are also discussed by 0. Fischer, in his article on Physiologische Mechanik, in the Encycl. d. mathem. Wissenschaften, 1904. X Cf. L: G. Brazier, On the flexure of thin cylindrical shells, etc., Proc. R.S. (A), cx^^, p. 104, 1927. 972 ON FORM AND MECHANICAL EFFICIENCY [ch. a diagram, or "graph," of the bending-moments from one point to another along the length of the bone. The second point requires a Uttle more explanation. If we imagine our loaded beam to be supported at one end only (for instance, by being built into a wall), so as to form what is called H i j a "bracket" or "cantilever," then we can see, without much difficulty, that the lines of stress in the beam run somewhat as in the accompanying diagram. Immediately under the load, the "compression-lines " tend to run ■ ^ vertically downward, but where the bracket is ^^' ' fastened to the wall there is pressure directed horizontally against the wall in the lower part of the surface of attachment; and the vertical beginning and the horizontal end of these pressure-lines must be continued into one another in the form of some even mathematical curve — which, as it happens, is part of a parabola. The tension-lines are identical in form with the compression-lines, of which they constitute the "mirror-image"; and where the two systems , intercross they do so at right angles, or "orthogonally" to one another. Such systems of stress-lines as these we shall deal with again; but let us take note here of the important though well-nigh obvious fact, that while in the beam they both unite to carry the load, yet it is often possible to weaken one set of lines at the expense of the other, and in some cases to do altogether away with one set or the other. For example, when we replace our end-supported beam by a curved bracket, bent upwards or downwards as the case may be, we have evidently cut away in the one case the greater part of the tension-hnes, and in the other the greater part of the compression-hnes. And if instead of bridging a stream with our beam of wood we bridge it with a rope, it is evident that this new construction contains all the tension-hnes, but none of the compression-hnes of the old. The biological interest connected with this principle Hes chiefly in the mechanical construc- tion of the rush or the straw, or any other typically cylindrical stem. The material of which the stalk is constructed is very weak ,to withstand compression, but parts of it have a very great tensile strength. Schwendener, who was both botanist and engineer, has elaborately investigated the factor of strength in the cylindrical XVI] OF STRESS AND STRAIN 973 stem, which Gahleo was the first to call attention to. Schwendener* shewed that its strength was concentrated in the little bundles of "bast-tissue," but that these bast-fibres had a tensile strength per square mm. of section not less, up to the limit of elasticity, than that of steel-wire of such quality as was in use in his day. For instance, we see in the following table the load which various fibres, and various wires, were found capable of. sustaining, not up to the breaking-point but up to the "elastic Hmit," or point beyond which complete recovery to the original length took place no longer after release of the load. Stress, or load in Strain, gms. per sq. nun., at Limit of Elasticity Do., in tons per sq. inch or amount of stretching, per mille Secale cereale 15-20 9-4-12-5 4-4 Lilium auratum 19 11-8 7-6 Phormium tenax 20 12-5 130 Papyrus antiquorum Molinia coerulea 20 22 12-5 13-8 15-2 110 Pincenectia recurvata 25 15-6 14-5 Copper wire Brass „ 121 13-3 7-6 8-5 10 1-35 Iron „ 21-9 13-7 10 Steel 24-&* 15-4 1-2 * This hgure should be considerably higher for the best modern steel. In other respects, it is true, the plant-fibres were inferior to the wires; for the former broke asunder very soon after the limit of elasticity was passed, while the iron- wire could stand, before snapping, about twice the load which was measured by its limit of elasticity : in the language of a modern engineer, the bast-fibres had a low "yield-point," little above the elastic limit. Nature seems content, as Schwendener puts it, if the strength of the fibre be ensured up to the elastic limit; for the equihbrium of the structure is lost as soon as that limit is passed, and it then matters httle how far off the actual breaking-point may bef. But nevertheless, within cer- tain limits, plant-fibre and wire were just as good and strong one * S. Schwendener, Das mechanische Princip im anatomischen Bau der Monocotyleen, Leipzig, 1874; Zur Lehre von der Festigkeit der Gewachse, Sb. Berlin. Akad. 1884, pp. 1045-1070. t The great extensibility of the plant-fibre is due to the spiral arrangement of the ultramicroscopic micellae of which the bast-fibre is built up : the spiral untwisting as the fibre stretches, in a right or left-hand spiral according to the species. Cf. C. Stein bruck, Die Micellartheorie auf botanischem Gebiete, Biol. Centralbl. 1925, p. 1. 974 ON FORM AND MECHANICAL EFFICIENCY [ch. as the other. And then Schwendener proceeds to shew, in many beautiful diagrams, the various ways in which these strands of strong tensile tissue are arranged in various stems : sometimes, in the simpler cases, forming numerous small bundles arranged in a peripheral ring, not quite at the periphery, for a certain amount of space has to be left for living and active tissue; sometimes in a sparser ring of larger and stronger bundles; sometimes with these bundles further strengthened by radial balks or ridges; sometimes with all the fibres set close together in a continuous hollow cylinder. In the case figured in Fig. 461, Schwendener calculated that the resistance to bending was at least twenty-five times as great as it would have been had the six main bundles been brought close together in a solid core. In many cases the centre of the stem is altogether empty ; in all other cases it is filled with soft tissue, suitable for various functions, but never such as to confer mechanical rigidity. In a tall Fig. 461. conical stem, such as that of a palm-tree, we can see not only these principles in the construction of the cylindrical trunk, but we can observe, towards the apex, the bundles of fibre curving over and intercrossing orthogonally with one another, exactly after the fashion of our stress-lines in Fig. 460; but of course, in this case, we are still dealing with tensile members, the opposite bundles taking on in turn, as the tree sways, the alternate function of resisting tensile strain*. * For further botanical illustrations, see {int. al.) R. Hegler, Einfiuss der Zug- kraften auf die Festigkeit und die Ausbildung meehanischer Gewebe in Pflanzen, SB. sacks. Ges. d. Wiss. 1891, p. 638; Einfluss des mechanischen Zuges auf das Wachstum der Pflanze, Cohn's Beitrdge, vi, pp. 383-432, 1893; 0. M. Ball, Einfluss von Zug auf die Ausbildung der Festigkeitsgewebe, Jakrb. d. wiss. Bot. xxxix, pp. 305-341, 1903; L. Kny, Einfluss von Zug und Druck auf die Richtung der Scheidewande in sich teilenden Pflanzenzellen, Ber. d. hot. Gesellsch. xiv, pp. 378- 391, 1896; Sachs, Mechanomorphose und Phylogenie, Flora, Lxxvin, 1894; of. also Pfliiger, Einwirkung der Schwerkraft, etc., uber die Richtung der ZeUtheilung, Archiv, xxxiv, 1884; G. Haberlandt's Physiological Plant Anatomy, tr. by Montagu Drummond, 1914, pp. 1.50-213. On the engineering side of the case, see Angus R. Fulton, Experiments to show how failure under stress occurs in timber, etc., Trans. R.S.E. XLViii, pp. 417-440, 1912; Fulton shews {int. a/.) that "the initial cause of fracture in timbers lies in the medullary rays." XVI] THE STRUCTURE OF BONE 975 The Forth Bridge, from which the anatomist may learn many a lesson, is built of tubes, which correspond even in detail to the structure of a cylindrical branch or stem. The main diagonal struts are tubes twelve feet in diameter, and withm the wall of each of these lie six T-shaped "stiffeners," corresponding precisely to the fibro- vascular bundles of Fig. 461 ; in the same great tubular struts the tendency to "buckle" is resisted, just as m the jointed stem of a bamboo, by "stiffening rings," or perforated diaphragms set twenty feet apart within the tube. We may draw one more curious, albeit parenthetic, comparison. An engineering construction, no less than the skeleton of plant or animal, has to grow; but the living thing is in a sense complete during every phase of its existence while the engineer is often hard put to it to ensure sufficient strength in his unfinished and imperfect structure. The young twig stands more upright than the old, and between winter and summer the weight of leafage affects all the curving outlines of the tree. A slight upward curvature, a matter of a few inches, was deliberately given to the great diagonal tubes of the bridge during their piecemeal construction; and it was a triumph of engineering foresight to see how, like the twig, as length and weight increased, they at last came straight and true. Let us now come, at last, to the mechanical structure of bone, of which we find a well-known and classical illustration in the various bones of the human leg. In the case of the tibia, the bone is somewhat widened out aboVe, and its hollow shaft is capped by an almost flattened roof, on which the weight of the body directly rests. It is obvious that, under these circumstances, the engineer would find it necessary to devise means for supporting this flat roof, and for distributing the vertical pressures which impinge upon it to the cylindrical walls of the shaft. In the long wing-bones of a bird the hollow of the bone is empty, save for a thin layer of living tissue lining the cylinder of bone; but in our own bones, and all weight-carrying bones in general, the hollow space is filled with marrow, blood-vessels and other tissues; and amidst these living tissues lies a fine lattice-work of little interlaced "trabeculae" of bone, forming the so-called "cancellous tissue." The older anatomists were content to describe this can- 976 ON FORM AND MECHANICAL EFFICIENCY [ch. cellous tissue as a sort of spongy network or irregular honeycomb * ; but at length its orderly construction began to be perceived, and attempts were made to find a meaning or "purpose" in the arrange- ment. Sir Charles Bell had a glimpse of the truth when he asserted f that "this minute lattice- work, or the cancelli which constitute the interior structure of bone, have still reference to the forces acting on the bone"; but he did not succeed in shewing what these forces are, nor how the arrangement of the cancelli is related to them. Jeffries Wyman, of Boston, came much nearer to the truth in a paper long neglected and forgotten {. He gives the gist of the whole matter in two short paragraphs: "1. The cancelli of such bones as assist in supporting the weight of the body are arranged either in the direction of that weight, or in such a manner as to support and brace those cancelli which are in that direction. In a mechanical point of view they may be regarded in nearly all these bones as a series of 'studs' and 'braces.' 2. The direction of these fibres in some of the bones of the human skeleton is characteristic and, it is believed, has a definite relation to the erect position which is naturally assumed by man alone." A few years afterwards the story was told again, and this time with convincing accuracy. It was shewn by Hermann Meyer (and afterwards in greater detail by Julius Wolff and others) that the trabeculae, as seen in a longitudinal section of the femur, spread in beautiful curving lines from the head to the hollow shaft of the bone ; and that these linear bundles are crossed by others, with so nice a regularity of arrangement that each intercrossing is as nearly as possible an orthogonal one: that is to say, the one set of fibres or cancelli cross the other everywhere at right angles. A great engineer. Professor Culmann of Zurich, to whom by the way we owe the whole modern method of "graphic statics," happened (in the year 1866) to come into his colleague Meyer's dissecting-room, where the anatomist was contemplating * Sir John Herschel described a bone as a "framework of the most curious carpentry: in which occurs not a single straight line nor any known geometrical curve, yet all evidently systematic, and constructed by rules which defy our research" (On the Study of Natural Philosophy, 1830, p. 203). f In Animal Mechanics, or Proofs of Design in the Animal Frame, 1827. X Animal mechanics : on the cancellated structure of spme of the bones of the human body, Boston Soc. of Nat. Hist. 1849. Reprinted, together with Sir C. Bell's work, by^ Morrill Wyman, Cambridge, Mass., 1902. i XVI] THE STRUCTURE OF BONE 977 the section of a bone*. The engineer, who had been busy designing a new and powerful crane, saw in a moment that the arrangement of the bony trabeculae was nothing more nor less than a diagram of the lines of stress, or directions of tension and compression, in the loaded structure: in short, that Nature was strengthening the bone in precisely the manner and directioi^ in which strength was required; and he is said to have cried out, "That's my crane!" Fig. 462. Head of the human femur in section. After Schafer, from a photo by Professor A. Robinson. In the accompanying diagram of Culmann's crane-head, we recognise a simple modification, due entirely to the curved shape of the structure, of the still simpler Hues of tension and compression which we have already seen in our end-supported beam, as represented in Fig. 460. In the shaft of the crane the concave or inner side, * The first metatarsal, rather than the femur, is said to have been the bone which Meyer was demonstrating when Culmann first recognised the orthogonal intercrossing of the cancelli in tension and compression; cf. A. Kirchner, Architektur der Metatarsalien des Menschen, Arch. f. Entw. Mech. xxiv, pp. 539-^16, 1907. 978 ON FORM AND MECHANICAL EFFICIENCY [ch. overhung by the loaded head, is the "compression-member"; the outer side is the "tension-member"; the pressure-Unes, starting from the loaded surface, gather themselves together, always in the direction of the resultant pressure, till they form a close bundle running down the compressed side of the shaft : while the tension- lines, running upwards along the opposite side of the shaft, spread out through the head, orthogonally to, and linking together, the system of compression-lines. The head of the femur (Fig. 462) is Fig. 463. Crane-head and femur. After Culmann and J. Wolff. a little more complicated in form and a little less symmetrical than Culmann's diagrammatic crane, from which it chiefly differs in the fact that its load is divided into two parts, that namely which is borne by the head of the bone, and that smaller portion which rests upon the great trochanter ; but this merely amounts to saying that a notch has been cut out of the curved upper surface of the structure, and we have no difficulty in seeing that the anatomical arrangement of the trabeculae follows precisely the mechanical distribution of compressive and tensile stress or, in other words, accords perfectly with the theoretical stress-diagram of the crane. XVI] THE STRUCTURE OF BONE 979 The lines of stress are bundled close together along the sides of the shaft, and lost or concealed there- in the substance of the solid wall of bone; but in and near the head of the bone, a peripheral shell of bone does not suffice to contain them, and they spread out through the central mass in the actual concrete form of bony trabeculae*. Mutatis mutandis, the same phenomenon may be traced in any other bone which carries weight and is liable to flexure; and in the OS calcis and the tibia, and more or less in all the bones of the lower limb, the arrangement is found to be very simple and clear. Thus, in the os calcis, the weight resting on the head of the bone has to be transmitted partly through the backward-projecting heel to the ground, and partly forwards through its articulation with * Among other works on the mechanical construction of bone see: Bourgery, Traite de Vanatomie {I. Osteologie), 1832 (with admirable illustrations of trabecular structure); L. Fick, Die Ursachen der Knochenformen, Gottingen, 1857; H. Meyer, Die Architektur der Spongiosa, Arch. f. Anat. und Physiol, xlvii, pp. 615-628, 1867; Statik u. Mechanik des menschlichen Knochengeriistes, Leipzig, 1873; H. Wolfer- mann, Beitrag zur K. der Architektur der Knochen, Arch. f. Anat. und Physiol. 1872, p. 312; J. Wolff, Die innere Architektur der Knochen, Arch. f. Anat. und Phya. L, 1870; Das Gesetz der Transformation bei Knochen, 1892; Y. Dwight, The significance of bone-architecture, Mem. Boston Soc. N.H. iv, p. 1, 1886; V. von Ebner, Der feinere Bau der Knochensubstan^, Wiener Bericht, Lxxn, 1875; Anton Rauber, Elastizitdt und Festigkeit der Knochen, Leipzig, 1876; 0. Meserer, EUlsI. u. Festigk. d. menschlichen Knochen, Stuttgart, 1880; Sir Donald MacAlister, How a bone is buil^, English Illustr. Mag. 1884, pp. 640-649; Rasumowsky, Architektonik des Fussskelets, Int. Monatsschr. f. ^na<. .1889, p. 197; Zschokke, Weitere Unters. uher das Verhdltnis der Knochenhildung zur Statik und Mechanik des Vertehratenskelets, Ziirich, 1892; W. Roux, Ges. Ahhandlungen uher Entuncklungs- mechanik der Organismen, Bd. I, Funktionelle Anpassung, Leipzig, 1895; J. Wolff, Die Lehre von der funktionellen Knochengestalt, Virchow's Archiv, clv, 1899; R. Schmidt, Vergl. anat. Studien liber den mechanischen Bau der Knochen und seine Vererbung, Z. f. w. Z. lxv, p. 65, 1899; B. Solger, Der gegenwartige Stand der Lehre von der Knochenarchitektur, in Moleschott's Unters. z. Naturlehre des Menschen. xvi, p. 187, 1899; H. Triepel, Die Stossfestigkeit der Knochen, Arch, f. Anat. und Phys. 1900; Gebhardt, Funktionellwichtige Anordnungsweisen der feineren und" groberen Bauelemente des Wirbelthierknochens, etc. Arch. f. ErUw. Mech. 1900-10; Revenstorf, Ueber die Transformation der Calcaneus -architektur. Arch. f. Entw. Mech. xxiii, p. 379, 1907; H. Bernhardt, Vererbung der inneren Knochenarchitektur beim Menschen, und die Teleologie bei J. Wolff, Inaug. Diss., Miinchen, 1907; Herm. Triepel, Die trajectoriellen Structuren (in Einf. in die Physikalische Anatomie, 1908); A. F. Dixon, Architecture of the cancellous tissue forming the upper end of the femur, Journ. Anai. and Phys. (3), xliv, pp. 223-230, 1910; A. Benninghoff, Ueber Leitsystem der Knochencompacta ; Studien zur Architektur der Knochen, Beitr. z. Anat. funktioneller Systeme, i, 1930. 980 ON FORM AND MECHANICAL EFFICIENCY [ch. the cuboid bone, to the arch of the foot. We thus have, very much as in a triangular roof- tree,, two compression-members sloping apart from one another; and these have to be bound together by a "tie" or tension-member, corresponding to the third, horizontal member of the truss. It is a simple corollary, confirmed by observation, that the trabeculae have a very different distribution in animals whose actions and attitudes are materially different, as in the aquatic mammals, such as the beaver and the seal*. And in much less extreme cases there are lessons to be learned from a study of the Fig, 464. Diagram of stress-lines in the human foot. From Sir D. MacAlister, after H. Meyer. same bone in different animals, as the loads alter in direction and magnitude. The gorilla's heelbone resembles man's, but the load on the heel is much less, for the erect posture is imperfectly achieved : in a common monkey the heel is carried high, and consequently the direction of the trabeculae is still more changed. The bear walks on the sole of his foot, though less perfectly than does man, and the lie of the trabeculae is plainly analogous in the two; but in the bear more powerful strands than in the os calcis of man transmit the load forward to the toes, and less of it through the heel to the ground. In the leopard we see the full effect of tip-toe, or digitigrade, progression. The long hind part (or tuberosity) of * Cf. G. de M. Rudolf, Habit and the architecture of the mammalian femur, Jovrn. Anatomy, lvi, pp. 139-146, 1922. xvi] OF WARREN'S TRUSS 981 the heel is now more a mere lever than a pillar of support; it is little more than a stiffened rod, with compression-members and tension-members in opposite bundles, inosculating orthogonally at the two ends*. In the bird the small bones of the hand, dwarfed as they are in size, have still a deal to do in carrying the long primary flight- feathers, and in forming a rigid axis for the terminal part of the wing. The simple tubular construction, which answers well for the long, slender arm-bones, does not suffice where a still more efficient stiffening is required. In all the mechanical ^ide of anatomy Fig. 465. Metacarpal bone from a vulture's wing; stiffened after the manner of a Warren's truss. From 0. Prochnow, Formenkunst der Natur, nothing can be more beautiful than the construction of a vulture's metacarpal bone, as figured here (Fig. 465). The engineer sees in it a perfect Warren's truss, just such a one as is often used for a main rib in an aeroplane. Not only so, but the bone is better than the truss; for the engineer has to be contefit to set his V-shaped struts all in one plane, while in the bone they are put, with obvious but inimitable advantage, in a three-dimensional configuration. So far, dealing wholly with the stresses and strains due to tension and compression, we have omitted to speak of a third very im- portant factor in the engineer's calculations, namely what is known as "shearing stress." A shearing force is one which produces * Cf. Fr. Weidenreich, Ueber formbestimmende Ursachen am Skelett, und die Erblichkeit der Knochenform, Arch. f. Entw. Mech. li, pp. 438-481, 1922. 982 ON FORM AND MECHANICAL EFFICIENCY [ch. "angular distortion" in a figure, or (what comes to the same thing) which tends to cause its particles to slide over one another, A shearing stress is a somewhat complicated thing, and we must try- to illustrate it (however imperfectly) in the simplest possible way. If we build up a pillar, for instance, of flat horizontal slates, or of a pack of cards, a vertical load placed upon it will produce com- pression, but will have no tendency to cause one card to slide, or shear, upon another; and in like manner, if we make up a cable of parallel wires and, letting it hang vertically, load it evenly with Fig. 466. Trabecular structure of the os calcis. From MacAIister. a weight, again the tensile stress produced has no tendency to cause one wire to slip or shear upon another. But the case would haye been very different if we had built up our pillar of cards or slates lying obliquely to the lines of pressure, for then at once there would have been a tendency for the elements of the pile to slip and shde asunder, and to produce what the geologists call "a fault" in the structure. Somewhat more generally, if AB be a bar, or pillar, of cross-section a under a direct load P, giving a direct and uniformly distributed stress per unit area =p, then the whole pressure P=pa. Let CD be an oblique section, inclined at an angle 6 to the cross-section ; the pressure on CD will evidently be =pa cos 6. But at any point 0 in CD, the pressure P may be resolved into the shearing force Q acting along CD, and the direct force N perpendicular to i t : where N = P cos 6 = pa cos 6, and ^ = P sin ^ = ^a sin 6. The shearing force XVI OF SHEARING STRESS Q upon CD = q . area of CD, which is = g . a/cos 6. Therefore qa/ cos 6= pa sin By therefore q=p sin 6 cos 6 = ^p sin 2^. Therefore when sin 20= 1, that is, when 6 = 45°, g is a maximum, and =p/2; and when sin 20 = 0, that is when 0 = 0° or 90°, then q vanishes altogether. This is as much as to say, that under this form of loading there is no shearing stress along or perpendicular to the lines of principal stress, or along the Unes of maximum compression or tension; but shear has a definite value on all other planes, and a maximum value when it is inclined at 45° to the cross-section. This may be further illustrated in various simple ways. When we submit a cubical block of iron to compression in the testing machine, it does not tend to give way by crumbling all to pieces, but always disrupts by shearing, and along some plane approxi- mately at 45° to the axis of compression ; this is known as Coulomb's 'Theory of Fracture, and, while subject to many qualifications, it is still an important first approximation to the truth. Again, in the beam which we have already con- sidered under a bending moment, we know that if we substitute for it a pack of cards, they will be strongly sheared on one another ; and the shearing stress is greatest in the "neutral zone," where neither tension nor compression is manifested: that is to say in the Hne which cuts at equal angles of 45° the orthogonally intersecting lines of pressure and tension. In short we see that, while shearing stresses can by no means be got rid of, the danger of rupture or breaking-down under shearing stress is lessened the more we arrange the materials of our con- struction along the pressure-lines and tension-lines of the system; for along these lines there is no shear*. To apply these principles to the growth and development of our bone, we have only to imagine a little trabecula (or group of trabeculae) being secreted and laid down fortuitously in any direction within the substance of the bone. If it lie in the direction of one of the pressure-lines, for instance, it will be in a position of comparative Fig. 467. It is also obvious that a free surface is always a region of zero-shear. 984 ON FORM AND MECHANICAL EFFICIENCY [ch. equilibrimn, or minimal disturbance ; but if it be inclined obliquely to the pressure-lines, the shearing force will at once tend to act upon it and move it away. This is neither more nor less than what happens when we comb our hair, or card a lock of wool : filaments lying in the direction of the comb's path remain where they were ; but the others, under the influence of an obUque component of pressure, are sheared out of their places till they too come into coincidence with the Unes of force. So straws show how the wind blows — or rather how it has been blowing. For every straw that lies askew to the wind's path tends to be sheared into it; but as soon as it has come to lie the way of the wind it tends to be disturbed no more, save (of course) by a violence such as to hurl it bodily away. In the biological aspect of the case, we must always remember that our bone is not only a living, but a highly plastic structure; the httle trabeculae are constantly being formed and deformed, demohshed and formed anew. Here, for once, it is safe to say that "heredity" need not and cannot be invoked to account for the configuration and arrangement of the trabeculae: for we can see them at any time of hfe in the making, under the direct action and control of the forces to which the system is exposed. If a bone be broken and so repaired that its parts lie somewhat out of their former place, so that the pressure- and tension-lines have now a new distribution, before many weeks are over the trabecular system will be found to have been entirely remodelled, so as to fall into line with the new system of forces. And as Wolff pointed out, this process of reconstruction extends a long way off from the seat of injury, and, so cannot be looked upon as a mere accident of the physiological process of healing and repair; for instance, it may happen that, after a fracture of the shaft of a long bone, the trabecular meshwork is wholly altered and reconstructed within the distant extremities of the bone. Moreover, in cases of transplantation of bone, for example when a diseased metacarpal is repaired by means of a portion taken from the lower end of the ulna, with astonishing quickness the plastic capabilities of the bony tissue are so manifested that neither in outward form nor inward structure can the old portion be distinguished from the new. Herein then lies, so far as we can discern it, a great part at least XVI] ON STRESS AND STRAIN 985 of the physical causation of what at first sight strikes us as a purely functional adaptation: as a phenomenon, in other words, whose physical cause is as obscure as its final cause or end is apparently manifest. Partly associated with the same phenomenon, and partly to be looked upon (meanwhile at least) as a fact apart, is the very im- portant physiological truth that a condition of strain, the result of a stress, is a direct stimulus to growth itself. This indeed is no less than one of the cardinal facts of theoretical biology. The soles of our boots wear thin, but the soles of our feet grow thick, the more we walk upon them: for it would seem that the living cells are "stimulated" by pressure, or by what we call "exercise," -to increase and multiply. The surgeon knows, when he bandages a broken limb, that his bandage is doing something more ^an merely keeping the parts together: and that the even, constant • pressure which he skilfully applies is a direct encouragement of growth and an active agent in the process of repair. In the classical experiments of Sedillot*, the greater part ef the shaft of the tibia was excised in some young puppies, leaving the whole weight of the body to rest upon the fibula. The latter bone is normally about one-fifth or sixth of the diameter of the tibia ; but under the new conditions, and under the "stimulus" of the increased load, it grew till it was as thick or even thicker than the normal bulk of the larger bone. Among plant tissues this phenomenon is very apparent, and in a somewhat remarkable way; for a strain caused by a constant or increasing weight (such as that in the stalk of a pear while the pear is growing and ripening) produces a very marked increase of strength without any necessary increase of bulk, but rather by some histo- logical, or molecular, alteration of the tissues. Hegler, Pfeifer, and others have investigated this subject, by loading the young shoot of a plant nearly to its breaking point, and then redetermining the breaking-strength after a few days. Some young shoots of the sunflower were found to break with a strain of 160 gm.; but when loaded with 150 gm., and retested after two days, they were able to support 250 gm. ; and being again loaded with something short * Sedillot, De I'influence des fonctions sur la structure et la forme des organes, C.R. Lix, p. 539, 1864; ef. lx, p. 97, 1865; Lxvm, p. 1444, 1869. 986 ON FORM AND MECHANICAL EFFICIENCY [ch. of this, by next day they sustained 300 gm., and a few days later even 400 gm.* The kneading of dough is an analogous phenomenon. The viscosity and perhaps other properties of the stuff are affected by the strains to which we have submitted it, and may thus be said to depend not only on the nature of the substance but on its history f . It is a long way from this simple instance, but we stretch across it easily in imagination, to the experimental growth of a nerve-fibre within a mass of clotted lymph : where, when we draw out the clot in one direction or another we lay down traction-lines, or tension-lines, and make of them a path for growth to follow J. Such experiments have been amply confirmed, but so far as I am aware we do not know much more about the matter: we do not know, for instance, how far the change is accompanied by increase in number of the bast-fibres, through transformation of other tissues ; or how far it is due to increase in size of these fibres; or whether it be not simply due to strengthening of the original fibres by some molecular change. But I should be much inclined to suspect that this last had a good deal to do with the phenomenon. We know nowadays that a railway axle, or any other piece of steel, is weakened by a constant succession of frequently interrupted strains; it is said to be "fatigued," and its strength is restored by a period of rest. The converse effect of continued strain in a uniform direc- tion may be illustrated by a homely example. The confectioner takes a mass of boiled sugar or treacle (in a particular molecular condition determined by the temperature to which it has been raised), and draws the soft sticky mass out into a rope; and then, folding it up lengthways, he repeats the process again and again. At first the rope is pulled out of the ductile mass without difficulty ; but as the work goes on it gets harder to do, until all the man's force is used to stretch the rope. Here we have the phenomenon * Op. (it. Hegler's results are criticised by 0. M. Ball, Einfluss von Zug aiif die Ausbildung der Festigungsgewebe, Jb. d. wiss. Botanik, xxxix, pp. 305-341, 1903, and by H. Keller, Einfluss von Belastung und Lage auf-die Ausbildung des Gewebes in Fruchtstielen, Inaug. Diss. Kiel, 1904. t Cf. R. K. Schofield and G. W. S. Blair, On dough, Proc. R.S. (A), cxxxviii, p. 707; cxxxix, p. 557, 1932-33; also Nadai and Wahl's Plasticity, 1931. For analogous properties of hairs and fibres, see Shorter, Journ. Textile Inst, xv, 1824; etc. X Cf. Ross Harrison's Croonian Lecture, 1933. XVI] ON STRESS AND STRAIN 987 of increasing strength, following mechanically on a rearrangement of molecules, as the original isotropic condition is transmuted more and more into molecular asymmetry or anisotropy; and the rope apparently "adapts itself" to the increased strain which it is called on to bear, all after a fashion which at least suggests a parallel to the increasing strength of the stretched and weighted fibre in the plant. For increase of strength by rearrangement of the particles we have already a rough illustration in our lock of wool or hank of tow. The tow will carry but little weight while its fibres are tangled and awry: but as soon as we have carded or "hatchelled" it out, and brought all its long fibres parallel and side by side, we make of it a strong and useful cord*. But the lessons which we learn from dough and treacle are nowadays plain enough in steel and iron, and become immensely more important in these. For here again plasticity is associated with a certain capacity for structural rearrangement, and increased strength again results therefrom. Elaborate processess of rolling, drawing, bending, hammering, and so on, are regularly employed to toughen and strengthen the material. The "mechanical structure" of solids has become an important subject. And when the engineer talks of repeated loading, of elastic fatigue, of hysteresis, and other phenomena associated with plasticity and strain, the physiological analogues of these physical phenomena are perhaps not far away. In some such ways as these, then, it would seem that we may coordinate, or hope to coordinate, the phenomenon of growth with certain of the beautiful structural phenomena which present them- selves to our eyes as "provisions," or mechanical adaptations f, for the display of strength where strength is most required. That is to say the origin, or causation, of the phenomenon would seem to lie partly in the tendency of growth to be accelerated under strain : and partly in the automatic effect of shearing strain, by which it tends to displace parts which grow obliquely to the direct lines of tension and of pressure, while leaving those in place which happen to he parallel or perpendicular to those lines : an automatic effect * Cf. Sir Charles Bell's Animal Mechanics, chap, v, "Of the tendons compared with cordage." f So P. Enriques (op. cit. supra, p. 5), writing on the economy of material in the construction of a bone, admits that "una certa impronta di teleologismo qua e la e rimasta, mio malgrado, in questo scritto." 988 ON FORM AND MECHANICAL EFFICIENCY [ch. which we can probably trace as working on all scales of magnitude, and as accounting therefore for the rearrangement of minute particles in the metal or the fibre, as well as for the bringing into Hne of the fibres within the plant, or of the trabeculae within the bone. But we may now attempt to pass from the study of the individual bone to the much wider and not less beautiful problems of mechanical construction which are presented to us by the skeleton as a whole. Certain problems of this class are by no means neglected by writers on anatomy, and many have been/ handed down from Borelli, and even from older writers. For instance, it is an old tradition of anatomical teaching to point out in the human body examples of the three orders of levers * ; again, the principle that the limb-bones tend to be shortened in order to support the weight of a very heavy animal is well understood by comparative anatomists, in accordance with Euler's law, that the weight which a column liable to flexure is capable of supporting varies inversely as the square of its length ; and again, the statical equilibrium of the body, in relation foi instance to the erect posture of man, has long been a favourite theme of the philosophical anatomist. But the general method, based upon that of graphic statics, to which we have been introduced in our study of a bone, has not, so far as J know, been apphed to the general fabric of the skeleton. Yet it is plain that each bone plays a part in relation to the whole body, analogous to that which a little trabecula, or a little group of trabeculae, plays within the bone itself: that is to say, in the normal distribution of forces in the body the bones tend to follow the lines of stress, and especially the pressure-Hnes. To demonstrate this in a comprehensive way would doubtless be difficult ; for we should be dealing with a frame- work of very great complexity, and should have to take account of * E.g. (1) the head, nodding backwards and forwards on a fulcrum, represented by the atlas vertebra, lying between the weight and the power; (2) the foot, raising on tip-toe the weight of the body against the fulcrum of the ground, where the weight is between the fulcrum and the power, the latter being" represented by the tendo Achillis; (3) the arm, lifting a weight in the hand, with the power (i.e. the biceps muscle) between the fulcrum and the weight. (The second case, by the way, has been much disputed; cf. Hay craft in Scha,{eT'B Textbook of Physiology, 1900, p. 251.) Cf. (int. al.) G. H. Meyer, Statik u. Mechanik der menschlicken Knochen- geriiste, 1873, pp. 13-25. XVI] COMPARATIVE ANATOMY OF BRIDGES 989 a great variety of conditions*. This framework is complicated as we see it in the skeleton, where (as we have said) it is only, or chiefly, the struts of the whole fabric which are represented ; but to under- stand the mechanical structure in detail, we should have to follow out the still more complex arrangement of the ties, as represented by the muscles and ligaments, and we should also require much detailed information as to the weights of the various parts and as to the other forces concerned. Without these latter data we can only treat the question in a preUminary and imperfect way. But, to take once again a small and simplified part of a big problem, let us think of a quadruped (for instance, a horse) in a standing posture, and see whether the methods and terminology of the engineer may not help us, as they did in regard to the minute structure of the single bone. And let us note in passing that the "standing posture," whether on two legs or on four, is no very common thing; but is (so to speak), with all its correlated anatomy, a privilege of the few. Standing four-square upon its fore-legs and hind-legs, with the weight of the body suspended between, the quadruped at once suggests to us the analogy of a bridge, carried by its two piers. And if it occurs to us, as naturalists, that we never look at a standing quadruped without contemplating a bridge, so, conversely, a similar idea has occurred to the engineer ; for Professor Fidler, . in this Treatise on Bridge-Construction, deals with the chief descrip- tive part of his subject under the heading of "The Comparative Anatomy of Bridges"]"." The designation is most just, for in studying the various types of bridge we are studying a series of well-planned skeletons %] and (at the cost of a little pedantry) * Our problem is analogous to Thomas Young's problem of the best disposition of the timbers in a wooden ship {Phil. Trans. 1814, p. 303). He was not long of finding that the forces which act upon the fabric are very numerous and very variable, and that the best mode of resisting them, or best structural arrangement for ultimate strength, becomes an immensely complicated problem. f By a bolder metaphor Fontenelle said of Newton that he had "fait I'anatomie de la lumiere." X In like manner. Clerk Maxwell could not help employing the term "skeleton" in defining the mathematical conception of a "frame," constituted by points and their interconnecting lines : in studying the equilibrium of which, we consider its different points as rnutually acting on each other with forces whose directions are those of the lines joining each pair of points. Hence (says Maxwell), "in order to exhibit the mechanical action of the frame in the most elementary manner, we may 990 ON FOKM AND MECHANICAL EFFICIENCY [ch. we might go even .further, and study (after the fashion of the anatomist) the "osteology" and "desmology" of the structure, that is to say the bones which are represented by "struts," and the Hgaments, etc., which are represented by "ties." Furthermore after the methods of the comparative anatomist, we may classify the families, genera and species of bridges according to their dis- tinctive mechanical features, which correspond to certain definite conditions and functions. Fig. 468. Skeleton of an American bison. (An unusually weU-mounted skeleton, of American workmanship, now in the Anatomical Museum of Edinburgh University.) In more ways than one, the quadrupedal bridge is a remarkable one; and perhaps its most remarkable peculiarity is that it is a jointed and flexible bridge, remaining in equilibrium under con- siderable and sometimes great modifications of its curvature, such as we see, for instance, when a cat humps or flattens her back. The fact thsit flexibility is an essential feature in the quadrupedal draw it as a skeleton, in which the different points are joined by straight lines, and we may indicate by numbers attached to these lines the tensions or com- pressions in the corresponding pieces of the frame" {Trans. R.S.E. xxvi, p. 1, 1870). It follows that the diagram so constructed represents a "diagram of forces," in this limited sense that it is geometrical as regards the position and direction of the forces, but arithmetical as regards their magnitude. It is to just such a diagram that the animal's skeleton tends to approximate. XVI] COMPARATIVE ANATOMY OF BRIDGES 991 bridge, while it is the last thing which an engineer desires and the first which he seeks to provide against, will impose certain important limiting conditions upon the design of the skeletal fabric. But let us begin by considering the quadruped at rest, when he stands upright and motionless upon his feet, and when his legs exercise no function save only to carry the weight of the whole body. So far as that function is concerned, we might now perhaps compare the horse's legs with the tall and slender piers of some railway bridge; but it is obvious that these jointed legs are ill- adapted to receive the horizontal thrust of any arch that may be placed atop of them. Hence it follows that the curved backbone of the horse, which appears to cross like an arch the span between his shoulders and his flanks, cannot be regarded as an arch, in the Fig. 469. a, tied arch; b, bowstring girder. engineer's sense of the word. It resembles an arch in form, but not in function, for it cannot act as an arch unless it be held back at each end (as every arch is held back) by ahutinents capable of resisting the horizontal thrust; and these necessary abutments are not present in the structure. But in various ways the engineer can modify his superstructure so as to supply the place of these external reactions, which in the simple arch are obviously indis- pensable. Thus, for example, we may begin by inserting a straight steel tie, AB (Fig. 469), uniting the ends of the curved rib AaB\ and this tie will supply the place of the external reactions, converting the structure into a "tied arch," such as we may see in the roofs of many railway stations. Or we may go on to fill in the space between arch and tie by a "web-system," converting it into what the engineer describes as a "parabolic bowstring girder" (Fig. 4696). In either case, the structure becomes an . independent "detached 992 ON FORM AND MECHANICAL EFFICIENCY [ch. girder," supported at each end but not otherwise fixed, and con- sisting essentially of an upper compression-member, AaB, and a lower tension-member, AB. But again, in the skeleton of the quadruped, the necessary tie, AB, of the simple bow-girder is not to be found ; and it follows that these comparatively simple types of bridge do not correspond to, nor do they help us to understand, the type of bridge which Nature has designed in the skeleton of the quadruped. Nevertheless if we try to look, as an engineer would look, at the actual design of the animal skeleton and the actual distribution of its load, we find that the one is most admirably adapted to the other, according to the strict principles of engineering construction. The structure is not an arch, nor a tied arch, nor a bowstring girder: but it is strictly and beautifully comparable to the main girder of a double-armed cantilever bridge. Fig. 470. A two-armed cantilever of the Forth Bridge. Thick lines, com- pression-members (bones); thin lines, tension -members (ligaments). Obviously, in our quadrupedal bridge, the superstructure does not terminate (as it did in our former diagram) at the two points of support, but it extends beyond them, carrying the head at one end and sometimes a heavy tail at the other, upon projecting arms or "cantilevers." In a typical cantilever bridge, such as the Forth Bridge (Fig. 470), a certain simplification is introduced. For each pier carries, in this case, its own double-armed cantilever, linked by a short connecting girder to the next, but so jointed to it that no weight is transmitted from one cantilever to another. The bridge in short is cut into separate sections, practically independent of one another; at the joints a certain amount of bending is not precluded, but shearing strain is evaded; and each pier carries only its own load. By this arrangement the engineer finds that design and construction are alike simplified and faciUtated. In the horse or the ox, it is XVI] COMPARATIVE ANATOMY OF BRIDGES 993 obvious that the two piers of the bridge, that is to say the fore-legs and the hind-legs, do not bear (as they do in the Forth Bridge) separate and independent loads, but the whole system forms a continuous structure. In this case, the calculation of the loads will be a little more difficult and the corresponding design of the structure a Httle more complicated. We shall accordingly simplify our problem very considerably if, to begin with, we look upon the quadrupedal skeleton as constituted of two separate systems, that is to say of two balanced cantilevers, one supported on the fore-legs and the other on the hind; and we may deal afterwards with the fact that these two cantilevers are not independent, but are bound up in one common field of force and plan of construction. In both horse and ox it is plain that the two cantilever systems into which we may thus analyse the quadrupedal bridge are unequal in magnitude and importance. The fore-part of the animal is much bulkier than its hind-quarters, and the fact that the fore-legs carry, as they so evidently do, a greater weight than the hind-legs has long been known and is easily proved ; we have only to walk a horse on to a weigh-bridge, weigh fijst his fore-legs and then his hind-legs, to discover that what we may call his front half weighs a good deal more than what is carried on his hind feet, say about three-fifths of the whole weight of the animal. The great (or anterior) cantilever then, in the horse, is constituted by the heavy head and still heavier neck on one- side of that pier which is represented by the fore-legs, and by the dorsal vertebrae carrying a large part of the weight of the trunk upon the other side; and this weight is so balanced over the fore-legs that the cantilever, while "anchored" to the other parts of the structure, transmits but little of its weight to the hind-legs, and the amount so transmitted will vary with the attitude of tne head and with the position of any artificial load*. Under certain conditions, as when the head is thrust well forward, it is evident that the hind-legs will be actually relieved of a portion of the comparatively small load which is their normal share. * When the jockey, crouches over the neck of his race-horse, and when Tod Sloan introduced the "American seat," the avowed object in both cases is to relieve the hind-legs of weight, and so leave them free for the work of propulsion. On the share taken by the hind-limbs in this latter duty, and other matters, cf. Stillman, The. Tlnrne, in Motion, 1882, p. 69. 994 ON FORM AND MECHANICAL EFFICIENCY [ch. But here we pass from the, statical problem to the dynamical, from the horse at rest to the horse in motion, from the observed fact that weight lies mainly over the fore-legs to the question of what advantage is gained by such a distribution of the load. Taking the hind-legs as the main propulsive agency, as we may now safely do, the moment of propulsion is about the hind-hooves; then (as we see in Fig. 471) we may take the weight, W = A sin a, and the W H propulsive force, f= A cos a, and -y = y, WL = fH being the balanced condition. From the statical point of view the load must balance over the fore-legs; from the dynamical point of view it might well lie even farther forward. And when the jockey crouches Fig. 471. over the horse's neck, and when Tod Sloan introduced the " American seat," both shew a remarkable, though perhaps unconscious, insight into the dynamical proposition. Our next problem is to discover, in a rough and approximate way, some of the structural details which the balanced load upon the double cantilever will impress upon the fabric. Working by the methods of graphic statics, the engineer's task is, in theory, one of great simplicity. He begins by drawing in outline the structure which he desires to erect; he calculates the stresses and bending-moments necessitated by the dimensions and load on the structure; he draws a new diagram representing these forces, and he designs and builds his fabric on the lines of this statical diagram. He does, in short, precisely what we have seen nature doing in the case of the bone. For if we had begun, as it were. xvi] THE SKELETON AS A BRIDGE 995 by blocking out the femur roughly, and considering its position and dimensions, its means of support and the load which it has to bear, we could have proceeded at once to draw the system of stress-lines which must occupy that field of force : and to precisely B Fig. 472. A, Span of proposed bridge. B, Stress diagram, or diagram of bending-moments*. those stress-lines has Nature kept in the building of the bone, down to the minute arrangement of its trabecular The essential function of a bridge is to stretch across a certain span, and carry a certain definite load; and this being so, the chief XfJwTX Fig. 473. The bridge construnted, as a parabolic girder. problem in the designing of a bridge is to provide due resistance to the "bending-moments" which result from the load. These bending-moments will vary from point to point along the girder, * This and the following diagrams are borrowed and adapted from Professor Fidler's Bridge Construction. We may reflect with advantage on Clerk Maxwell's saying that "the use of diagrams is a particular instance of that method of symbols which is so powerful an aid in the advancement of science" ; and on his explanation that "a diagram differs from a picture in this respect that in a diagram no attempt is made to represent those factors of the actual material system which, are not the special objects of our study." 996 ON FORM AND MECHANICAL EFFICIENCY [ch. and taking the simplest case of a uniform load, whether supported at one or both ends, they will be represented by points on a parabola. If the girder be of uniform depth and section, that is to say if its two flanges, respectively under tension and compression, be equal and parallel to one another, then the stress upon these flanges will vary as the bending-moments, and will accordingly be very severe in the middle and will dwindle towards the ends. But if we make the dejpih of the girder everjrwhere proportional to the bending- moments, that is to say if we copy in the girder the outlines of the bending-moment diagram, then our design will automatically meet the circumstances of the case, for the horizontal stress in each flange will now be uniform throughout the length of the girder. In short. / w Fig. 474. in Professor Fidler's words, "Every diagram of moments represents the outline of a framed structure which will carry the given load with a uniform horizontal stress in the principal members." In the above diagrams (Fig. 474, a, b) (which are taken from the original ones of Culmann), we see at once that the loaded beam or bracket {a) has a "danger-point" close to its fixed base, that is to say at the .point remotest from its load. But in the parabolic bracket (6) there is no danger-point at all, for the dimensions of the structure are made to increase jpari passu with the bending- moments: stress and resistance vary together. Again in Fig. 475, we have a simple span (A), with its stress diagram (B); and in (C) we have the corresponding parabolic girder, whose stresses are now uniform throughout. In fact we see that, by a process of conversion, the stress diagram in each case becomes the structural XVI OF RECIPROCAL DIAGRAMS 997 diagram in the other*. Now all this is but the modern rendering of one of Galileo's most famous propositions. In the Dialogue which we have already quoted more tlfan oncef, Sagredo says "It would be a fine thing if one could discover the proper shape to give a solid in order to make it equally resistant at every point, in which case a load placed at the middle would not produce fracture more easily I Fig. 475. than if placed at any other point J." And Gahleo (in the person of Salviati) first puts the problem into its more gen'eral form; and then shews us how, by giving a parabolic outline to our beam, we have its simple and comprehensive solution. It was such teaching as this that led R. A. Millikan to say that "we owe our present-day civilisation to Galileo." * The method of constructying reciprocal diagrams, of which one should represent the outlines of a frame and the other the system of forces necessary to keep it in equilibrium, was first indicated in Culmann's Graphische Statik; it was greatly developed soon afterwards by Macquorn Rankine (Phil. Mag. Feb, 1864, and Applied Mechanics, passim), to whom the application of the principle to engineering practice is mainly due. See also Fleeming Jenkin, On the practical application of reciprocal figures to the calculation of strains in framework, Trans. R.S.E. XXV, pp. 441^48, 1869; and Clerk Maxwell, ibid, xxvi, p. 9, 1870, and Phil. Mag. April 1864. t Dialogues concerning Two New Sciences (1638); Crew and Salvio's translation p. 140 seq. X As in the great case of the Eiffel Tower, supra, p. 29. 998 ON FOKM AND MECHANICAL EFFICIENCY [ch. In the case of our cantilever bridge, we shew the primitive girder in Fig. 475, A, with its bending-moment diagram (B); and it is evident that, if we turn this diagram upside down, it will still be illustrative, just as before,, of the bending-moments from point to point: for as yet it is merely a diagram, or graph, of relative magnitudes. To either of these two stress diagrams, direct or inverted, we may fit the design of the construction, as in Figs. 475, C and 476. Fig. 476. Now in different animals the amount and distribution of the load differ so greatly that we can expect no single diagram, drawn from the comparative anatomy of bridges, to apply equally well to all the cases met with in the comparative anatomy of quadrupeds; but nevertheless we have already gained an insight into the general principles of "structural design" in the quadrupedal bridge. . In our last diagram the upper member of the cantilever is under tension; it is represented in the quadruped by the ligamentum nuchae on the one side of the cantilever, and by the supraspinous ligaments of the dorsal vertebrae on the other. The compression-member is similarly represented, on both sides of the cantilever, by the vertebral column, or rather by the bodies of the vertebrae; while the web, or "filling," of the girders, that is to say the upright or sloping members which extend from one flange to the other, is represented on the one hand by the spines of the vertebrae, and on the other hand by the obli(|ue interspinous ligaments and muscles — that is to say, by compression -members and tension-members inclined in opposite directions to one another. The high spines over the quadruped's withers are no other than the high sisruts which rise over the supporting piers in the parabolic girder, and correspond to the position of the maximal bending-moments. The fact that these tall vertebrae of the withers usually slope backwards, some- XVI] OF RECIPROCAL DIAGRAMS 999 times steeply, in a quadruped, is easily and obviously explained*. For each vertebra tends to act as a "hinged lever," and its spine, acted on by the tensions transmitted by the ligaments on either side, takes up its position as the diagonal of the parallelogram of forces to which it is exposed. It happens that in these comparatively simple types of cantilever bridge the whole of the parabolic curvature is transferred to one or other of the principal members, either the tension-member or the compression-member as the case may be. But it is of course equally permissible to have both members curved, in opposite directions. This, though not exactly the case in the Forth Bridge, is approximately so; for here the main compression-member is curved or arched, and the main tension-member slopes downwards on either side from its maximal height above the piers. In short, the Forth Bridge (Fig. 470) is a nearer approach than either of the other bridges which we have illustrated to the plan of the quadrupedal skeleton; for the main compression-member almost exactly recalls the form of the backbone, while the main tension- meraber, though not so closely similar to the supraspinous and nuchal ligaments, corresponds to the plan of these in a somewhat simplified form. We may now pass without difficulty from the two-armed canti- lever supported on a single pier, as it is in each separate section of the Forth Bridge, or as we have imagined it to be in the fore-quarters of a horse, to the condition which actually exists in a quadruped, when a two-armed cantilever has its load distributed over two separate piers. This is not precisely what an engineer calls a "continuous" girder, for that term is applied to a girder which, as a continuous structure, has three supports and crosses two or more spans, while here there is only one. But nevertheless, this girder * The form and direction of the vertebral spines have been frequently and elaborately described; cf. (e.g.) H. Gottlieb, Die Anticlinie der Wirbelsaule der Saugethiere, Morphol. Jahrb. lxix, pp. 179-220, 1915, and many works quoted therein. According to Morita, Ueber die Ursachen der Richtung und Gestalt der thoracalen Dornfortsatze der 8augethierwirbelsaule {ibi cit. p. 201), various changes take place in the direction or inclination of these processes in rabbits, after section of the interspinous ligaments and muscles. These changes seem to be very much what we should expect, on simple mechanical grounds. See also 0. Fischer, Theoretische Grundlagen fur eine Mechanik der lebenden Korper, Leipzig, 1906, pp. X, 372. 1000 ON FOR]\J AND MECHANICAL EFFICIENCY [ch. is effectively continuous from the head to the tip of the tail; and at each point of support (A and 5) it is subjected to the negative bending-moment due to the overhanging load on each of the projecting cantilever arms AH and BT. The diagram of bending- moments will (according to the ordinary conventions) lie below the base line (because the moments are negative), and must take some such forfn as that shewn in the diagram : for the girder must suffer its greatest bending stress not at the centre, but at the two points of support A and 5, where the moments are measured by the vertical ordinates. It is plain that this figure only differs, from a representation of two independent two-armed cantilevers in the fact that there is no point midway in the span where the bending- moment vanishes, but only a region between the two piers in which it tends' to diminish. H A B T Fig. 477. Two-armed cantilever and its stress diagram. The diagram effects a graphic summation of the positive and negative moments, but its form may assume various modific^ttions according to the method of graphic summation which we choose to adopt; and it is obvious also that the form of the diagram rriay assume many modifications of detail according to the actual distribution of the load. In all cases the essential points to be observed are these: firstly that the girder which is to resist the bending-moments induced by the load must possess its two principal members — an upper tension-member or tie, represented by ligament (whose tension doubtless varies along its length), and a lower compression-member represented by bone: these members being united by a web represented by the vertebral spines with their interspinous ligaments, and being placed one above the other in the order named because the moments are negative; secondly we observe that the depth of the web, or distance apart of the principal xvij OF RECIPROCAL DIAGRAMS 1001 members — that is to say the height of the vertebral spines — must be proportional to the bending-moment at each point along the length of the girder. In the case of an animal carrying most oi his weight upon his fore-legs, as the horse or the ox do, the bending-moment diagram will be unsymmetrical, after the fashion of Fig. 478, the precise form depending on the distribution of weights and distances. 7a/7 Head Fig. 478. Stress -diagram of horse's backbone. On the other hand the Dinosaur, with his light head and enormous tail would give us a moment-diagram with the opposite kind of asymmetry, the greatest bending stress being now found over the haunclies, at B (Fig. 479). A glance at the skeleton of Diplodocus will shew us the high vertebral spines over the loins, in precise correspondence with the requirements of this diagram: just as in the horse, under the opposite conditions of load, the highest vertebral spines are those of the withers, that is to say those of the posterior cervical and anterior dorsal vertebrae. We have now not only dealt with the general resemblance, both in structure and in function, of the quadrupedal backbone with its associated ligaments to a double-armed cantilever girder, but we have begun to see how the characters of the vertebral system must differ in different quadrupeds, according to the conditions imposed by the varying distribution of the load : and in particular how the height of the vertebral spines which constitute the web will be in a definite relation, as regards magnitude and position, to the bending-moments induced thereby. We should require much de- tailed information as to the actual weights of the several parts of the body before we could follow out quantitatively the mechanical efficiency of each type of skeleton; but in an approximate way what we have already learnt will enable us to trace many interesting correspondences between structure and function in this particular part of comparative anatomy. We must, however, be careful to note that the great cantilever system is not of necessity constituted 1002 ON FORM AND MECHANICAL EFFICIENCY [ch. by the vertebral column and its ligaments alone, but that the pelvis, firmly united as it is to the sacral vertebrae, and stretching back- wards far beyond the acetabulum, becomes an intrinsic part of the system ; and heljnng (as it does) to carry the load of the abdominal viscera, it constitutes a great portion of the posterior cantilever arm, or even its chief portion in cases where the size and weight of the tail are insignificant, as is the case in the majority of terrestrial mammals. Tail Head Fig. 479. Stress- diagram of backbone of Dinosaur. We may also note here, that just as a bridge is often a " combined" or composite structure, exhibiting a combination of principles in its construction, so in the quadruped we have, as it were, another girder supported by the same piers to carry the viscera; and con- sisting of an inverted parabolic girder, whose compression-member is again consti'tuted by the backbone, its tension-member by the line of the sternum and the abdominal muscles, while the ribs and intercostal muscles play the part of the web or filling. A very few instances must suffice to illustrate the chief variations in the load, and therefore in the bending-moment diagram, and therefore also in the plan of construction, of various quadrupeds. But let us begin by setting forth, in a few cases, the actual weights which are borne by the fore-limbs and the hind-limbs, in our quadrupedal bridge*. On On %on %on Gross weight fore-feet hind-feet fore-feet hind -feet ton cwt. cwt. cwt. Camel (Bactrian) - 14-25 9-25 4-5 67-3 32-7 Llama - 2-75 1-75 0-875 66-7 33-3 Elephant (Indian) 1 15-75 20-5 14-75 58-2 41-8 Horse - 8-25 4-75 3-5 57-6 42-4 Horse (large Clydes- - 15-5 8-5 7-0 54-8 45-2 dale) * I owe the first four of these determinations to the kindness of Sir P. Chalmers Mitchell, who had them made for me at the Zoological Society's Gardens; while the great Clydesdale carthorse was weighed for me by a friend in Dundee. XVI] IN THE SKELETON OF QUADRUPEDS 1003 It will be observed that in all these animals the load upon the fore-feet preponderates considerably over that upon the hind, the preponderance being rather greater in the elephant than in the horse, and markedly greater in the camel and the llama than in the other two. But while these weights are helpful and suggestive, it is obvious that they do not go nearly far enough to give us a full insight into the constructional diagram to which the animals are conformed. For such a purpose we should require to weigh the total load, not in two portions but in many; and we should also have to take close account of the general form of the animal, of the relation between that form and the distribution of the load, and of the actual directions of each bone and ligament by which the forces of compression and tension >were transmitted. All this lies beyond us for the present; but nevertheless we may consider. Tail Head Fig. 480. Stress- diagram of Titanotheriuni. very briefly, the principal cases involved in our enquiry, of which the above animals form a partial and preliminary illustration. (1) Wherever we have a heavily loaded anterior cantilever arm, that is to say whenever the head and neck represent a considerable fraction of the whole weight of the body, we tend to have large bending-moments over the fore-legs, and correspondingly high spines over the vertebrae of the withers. This is the case in the great majority of four-footed terrestrial animals, the chief exceptions being found in animals with comparatively small heads but large and heavy tails, such as the anteaters or the Dinosaurian reptiles, and also (very naturally) in animals such as the crocodile, where the " bridge " can scarcely be said to be developed, for the long heavy body sags down to rest upon the ground. The case is sufficiently exemplified by the horse, and still more notably by the stag, the ox, or the pig. It is illustrated in the skeleton of a bison (Fig. 468), or 1004 ON FORM AND MECHANICAL EFFICIENCY [ch. in the accompanying diagram of the conditions in the great extinct Titanotherium. (2) In the elephant and the camel we have similar conditions, but slightly modified. In both cases, and especially in the latter, the weight on the fore-quarters is relatively krge ; and in both cases the bending-moments are all the larger, by reason of the length and forward extension of the camel's neck and the forward position of the heavy tusks of the elephant. In both cases the dorsal spines are large, but they do not strike us as exceptionally so; but in both cases, and especially in the elephant, they slope backwards in a marked degree. Each spine, as already explained, must in all cases assume the position of the diagonal in the parallelogram of forces defined by the tensions acting on it at its extremity; for it constitutes a "hinged lever," by which the bending-moments on either side are automatically balanced ; and it is plain that the more the spine slopes backwards the more it indicates a relatively large strain thrown upon the great ligament of the neck, and a relief of strain upon the more directly acting, but weaker, Hgaments of the back and loins. In both cases, the bending-moments . would seem to be more evenly distributed over the region of the back than, for instance, in the stag, with its light hind-quarters and heavy load of antlers: and in bo^h cases the high "girder" is considerably prolonged, by an extension of the tall spines backwards in the direction of the loins. When we come to such a case as the mam- moth, with its immensely heavy and immensely elongated tusks, we perceive at once that the bending-moments over the fore-legs are now very severe; and we see also that the dorsal spines in this region are. much more conspicuously elevated than in the ordinary elephant. (3) In the case of the giraffe we have, without doubt, a very heavy load upon the fore-legs, though no weighings are at hand to define the ratio; but as far as possible this disproportionate load would seem to be relieved by help of a downward as well as backward thrust, through the sloping back to the unusually low hind-quarters. The dorsal spines of the vertebrae are very high and strong, and the whole girder-system very perfectly formed. The elevated rather than protruding position of the head lessens the anterior bending-moment as far as possible, but it leads to a strong XVI] IN THE SKELETON OF QUADRUPEDS 1005 compressional stress transmitted almost directly downwards through the neck: in correlation with which we observe that the bodies of the cervical vertebrae are exceptionally large and strong, and steadily increase in size and strength from the head downwards. (4) In the kangaroo, the fore-limbs are entirely relieved of theii* load, and accordingly the tall spines over the withers, which were so conspicuous in all heavy-headed quadrupeds, have now completely vanished. The creature has become bipedal, and body and tail form the extremities of a single balanced cantilever, whose maximal bending-moments are marked by strong, high lumbar and sacral vertebrae, and by iliac bones of peculiar form, of exceptional strength and nearly upright position. Precisely the same condition is illustrated in the Iguanodon, and better still by reason of the great bulk of the creature and of the heavy load which falls to be supported by the great cantilever and by the hind-legs which form its piers. The long heavy body and neck require a balance-weight (as in the kangaroo) in the form* of a long heavy tail; and the double-armed cantilever, so constituted, shews a beautiful parabolic curvature in the graded heights of the whole series of vertebral spines, which rise to a maximum over the haunches and die away slowly towards the neck and towards the tip of the tail. (5) In the case of some of the great American fossil reptiles such as Diplodocus, it has always been a more or less disputed question whether or not they assumed, like Iguanodon, an erect, bipedal attitude. In all of them we see an elongated pelvis, and, in still more marked degree, we see elevated spinous processes of the vertebrae over the hind-limbs ; in all of them we have a long heavy tail, and in most of them we have a marked reduction in size and weight both of the fore-limb and of the head itself. The great size of these animals is not of itself a proof against the erect attitude; because it might well have been accompanied by an aquatic or partially submerged habitat, and the crushing stress of the creature's huge bulk proportionately relieved. But we must consider each such case in the whole light of its own evidence ; and it is easy to see that, just as the quadrupedal mammal may carry the greater part but not all of its w^eight upon its fore-limbs, so a heavy-tailed reptile may carry the greater part upon its hind-limbs, without 1006 ON FORM AND MECHANICAL EFFICIENCY [ch. this process going so far as to relieve its fore-limbs of all weight whatsoever. This would seem to be the case in such a form as Diplodocus, ■ and also, in Stegosaurus, whose restoration by Marsh is doubtless substantially correct*. The fore-limbs, though com- paratively small, are obviously fashioned for support, but the weight which they have to tarry is far less than that which the hind-limbs bear. The head is small and the neck short, while on the other hand the hind-quarters and the tail are big and massive. The backbone bends into a great double-armed cantilever, culminating over the pelvis and the hind-limbs, and here furnished with its highest and strongest spines to separate the tension-member from the com- Fig. 481. Diagram of Stegosaurus. * pression-member of the girder. The fore-legs form a secondary supporting pier to this great continuous cantilever, the greater part of whose weight is poised upon the hind-limbs alone. (6) In the slender body of a weasel, neither head nor tail is such as to form an efficient cantilever; and though the lithe body is arched in active exercise, our parallel of the bridge no longer works well. What else to compare it with is far from clear; but the mechanism has some resemblance (perhaps) to an elastic spring. Animals of this habit of body are all small; their bodily weight is a light burden, and gravity becomes an ineffectual force, * This pose of Diplodocus, and of other Sauropodous reptiles, has been much discussed. Cf. {int. al.) 0. Abel, Ahh. k. k. zool. hot. Ges. Wien, v, 1909-10 (60 pp.); Tornier, SB. Ges. Naturf. Fr. Berlin, 1909, pp. 193-209; 0. P. Hay, Amer. Nat. Oct. 1908; Tr. Wash. Acad. Sci. xlii, pp. 1-25, 1910; Holland, Amer. Nat. May 1910, pp. 259-283; Matthew, ibid. pp. 547-560; C. W. Gilmore [Restoration of Stegosaurus), Pr. U.S. Nat. Miiseum, 1915. XVI] IN AQUATIC ANIMALS 1007 (7) An abnormal and very curious case is that of the sloth, which hangs by hooked hands and feet, head downwards, from high branches in the Brazilian forest. The vertebrae are unusually numerous, they are all much alike one to another, and (as we might well suppose) the whole pensile chain of vertebrae hangs in what closely approximates to a catenary curve*. (8) We find a highly important corollary in the case of aquatic animals. For here the eifect of gravity is neutralised; we have neither piers nor cantilevers; and we find accordingly in all aquatic mammals of whatsoever group — whales, seals or sea-cows — that the high arched vertebral spines over the withers, or corresponding structures over the hind-limbs, have both entirely disappeared. But in the whale or dolphin (and not less so in the aquatic bird), stiffness must be ensured in order to enable the muscles to act against the resistance of the water in the act of swimming; and accordingly Nature must provide against bending-moments irrespective of gravity. In the dolphin, at any rate as regards its tail-end, the conditions will be not very different from those of a column or beam with fixed ends, in which, under deflection, there will be two points of contrary flexure, as at C, D, in Fig. 482. Here, between C and D we have a varying bending-moment, represented by a continuous curve with its maximal elevation mid- way between the points of inflection. And correspondingly, in our dolphin, we have a continuous series of high dorsal spines, rising to a maximum about the middle of the animal's body, and falling to nil at some distance from the end of the tail. It is their business (as usual) to keep the tension-member, represented by the strong supraspinous hgaments, wide apart from the compression- member, which is as usual represented by the backbone itself. But in our diagram we see that on the farther side of C and D we have a negative curve of bending-moments, or bending-moments in a contrary direction. Without enquiring how these stresses are precisely met ♦ A hmvy cord, or a cord carrying equal weights for equal distances along its line, hangs in a catenary: imagine it frozen and inverted, and we have an arch, carrying Ihe same sort of load, and under compression only. On the other hand, a flexible cable (itself of neghgible weight), carrying a uniform load along the line of its horizontal projection, hangs in the form of a parabola. 1008 ON FORM AND MECHANICAL EFFICIENCY [ch. towards the dolphin's head (where the coalesced cervical vertebrae suggest themselves as a partial explanation), we see at once that towards the tail they are met by the strong series of chevron-bones, which in the caudal region, where tall dorsal spines are no longer needed, take their place below the vertebrae, in precise correspondence with the bending-moment diagram. In many cases other than these aquatic ones, when we have to deal with animals with long and heavy tails (like the Iguanodon and the kangaroo of which we have already spoken), w^e are apt to meet with similar, though usually shorter chevron-bones; and in all these cases we may see without difficulty that a negative bending-moment in the vertical direction has to be resisted or controlled. In the dolphin we may find an illustration of the fact that not only is it necessary to provide for rigidity in the vertical direction but often also in the horizontal, where a tendency to bending must be resisted on either side. This function is effected in part by the ribs with their associated muscles, but they extend but a little way and their efficacy for this purpose can be but small. We have, however, behind the region of the ribs and on either side of the backbone a strong series of elongated and flattened transverse processes, forming a web for the support of a tension-member in the usual form of Hgament, and so playing a part precisely analogous to that performed by the dorsal spines in the same animal. In an ordinary fish, such as a cod or a haddock, we see precisely the same thing: the backbone is stiffened by the indispensable help of its three series of ligament-connected processes, the dorsal and the two transverse series ; but there are no such stiffeners in the eel. When we come to the region of the tail, where rigidity gives place to lateral flexibility, the three stiffeners give place to two — the dorsal and haemal s|)ines of the caudal vertebrae. And here we see that the three series of processes, or struts, tend (when all three are present) to be arranged Fig. 483. a, dorsal and b, caudal vertebrae of haddock. XVI] ON STRENGTH AND FLEXIBILITY 1009 well-nigh at equal angles, of 120°, with one another, giving the greatest and most uniform strength of which such a system is capable. On the other hand, in a flat fish, such as a plaice, where from the natural mode of progression it is necessary that the back- bone should be flexible in one direction while stiffened in another, we find the whole outline of the fish comparable to that of a double bowstring girder, the compression-member being (as usual) the backbone itself, the tension-member on either side being constituted by the interspinous ligaments and muscles, while the web or filUng is very beautifully represented by the long and evenly graded neural and haema) spines, which spring symmetrically up and down from each individual vertebra. In the skeleton of the flat fishes, the web of the otherwise perfect parabolic girder has to be cut away and encroached on to make room' for the viscera. When the body is long and the vertebrae many, as in the sole, the space required is small compared with the length of the girder, and the strength of the latter is not much impaired. In the shorter, rounder kinds with fewer vertebrae, like the turbot, the visceral cavity is large compared with the length of the fish, and its presence would seem to weaken the girder very seriously. But Nature repairs the breach by framing in the hinder part of the space with a strong curved bracket or angle-iron, which takes the place very efl5.ciently of the bony struts which have been cut away. The main result at which we have now arrived, in regard to the construction of the vertebral column and its associated parts, is that we may look upon it as a certain type of girder, whose depth is everywhere very nearly proportional to the height of the corre- sponding ordinate in the diagram of moments: just as it is in a girder designed by a modern engineer. In short, after the nineteenth or twentieth century engineer has done his best in framing the design of a big cantilever, he may find that some of his best ideas had, so to speak, been anticipated ages again the fabric of the great saurians and the larger mammals. But it is possible that the modern engineer might 4)e disposed to criticise the skeleton girder at two or three points ; and in particular he might think the girder, as we see it for instance in Diplodocus or Stegosaurus, not deep enough for carrying the animal's enormous 1010 ON FORM AND MECHANICAL EFFICIENCY [ch. weight of some twenty tons. If we adopt a much greater depth (or ratio of depth to length) as in the modern cantilever, we shall greatly increase the strength of the structure ; but at the same time we should greatly increase its rigidity, and this is precisely what, in the circumstances of the case, it would seem that Nature is bound to avoid. We need not suppose that the great saurian was by any means active and limber; but a certain amount of activity and flexibility he was bound to have, and in a thousand ways he would find the need of a backbone that should hQ flexible as well as strong. Now this opens up a new aspect of the matter and is the beginning of a long, long story, for in every direction this double requirement of strength and flexibility imposes new conditions upon the design. To represent all the porrelated quantities we should have to construct not only a diagram of moments but also a diagram of elastic deflection and its so-called ''curvature"; and the engineer would want to know something more about the material of the ligamentous tension-member — its flexibility, its modulus of elasticity in direct tension, its elastic limit, and its safe working stress. In various ways our structural problem is beset by "limiting conditions." Not only must rigidity be associated with flexibility, but also stability must be ensured in various positions and attitudes ; and the primary function of support or weight-carrying must be combined with the provision of points d'appui iot the muscles con- cerned in locomotion. We canr^ot hope to arrive at a numerical or quantitative solution of this complicate problem, but we have found it possible to trace it out in part towards a qualitative solution. And speaking broadly we may certainly say that in each case the problem has been solved by Nature herself, very much as she solves the difficult problems of minimal areas in a system of soap-bubbles ; so that each animal is fitted with a backbone adapted to his own individual needs, or (in other words) corresponding to the mean resultant of the many stresses to which as a mechanical system it is exposed. The mechanical construction of a bird is a more elaborate aff'air than a quadruped's, inasmuch as it has a double part to play, the bird's whole weight being borne now by its legs and now by its wings. As it stands on the ground our bird is a balanced cantilever, carried XVI ON STRENGTH AND FLEXIBILITY 1011 on two legs as on a pier, the cantilever being constituted by the pelvic bones, drawn out fore and aft and firmly welded to a long stretch of vertebral column. The centre of gravity is kept in a line passing through the acetabulum^, and the long toes help to preserve an unstable but well-ad j usted equilibrium . One arm of the cantilever carries head, neck and wings, the other, the shorter arm, carries the abdomen; but the whole weight of the viscera hangs in the abdomen as in a bag, and on the other hand head and neck are kept small and light, and their purchase on the fulcrum is under constant modifica- tion and control. A stork or a heron is continually balancing itself; Fig. 484. Pelvis oi Apteryx. The line AB is vertical, or nearly so, in the standing posture of the bird. as the beak is thrust forward a leg stretches back, as the bird walks along its whole body sways in keeping. No less elegant is the perfect balance of the same birds at rest — the heron standing on one leg, even on a tree top, the flamingo also on one long leg, with its neck close coiled and its head tucked amongst the feathers. The approximately parabolic form of the great pelvic cantilever is best seen in the ostrich and other running birds, but more commonly the strength of the cantilever is got in other ways. Usually, as in the fowl, it consists of a thin shell of bone curved over like the bonnet of a motor car and stiffened, or "cambered," by ridges converging on the acetabulum. A doubled sheet of paper, cut roughly to the shape of the pelvis and then pinched up into folds 1012 ON FORM AND MECHANICAL EFFICIENCY [ch on either side, as in Fig. 485, will serve as a model of the skeletal cantilever and shew how its limp surface is stiffened by the folds. Save in the ostrich, and a few other flightless birds, the breast-bone or sternum is a broad, flat bone, produced into a deep, descending ridge or "keel," Very .firmly fixed to the sternum on either side is a short strong bone, the coracoid; attached to it again, and bending backwards over the ribs, is the scapula; and at the junction of scapula and coracoid is the socket, or glenoid cavity, for the wing. The clavicles, fused into a "merry-thought," run from near the glenoid cavity to the front end of the keel; in strong-flying birds they are stout and curved, and a continuous curve sweeps Fig. 485. Rough paper model of a fowl's pelvis. round from scapula to sternal keel. The keel is commonly explained as necessary to give space enough for the attachment of the muscles of flight, but this explanation is inadequate, even untrue; for one thing, the great pectoral muscle springs from the edge, not from the broad surface of the keel. The keel is essentially a flange, and, as in a piece of T iron, adds immensely to the strength and stiffness of the construction*; that it tends to give the fibres of the muscle more stretch and play, and a straighter pull on the arm-bone to which they run, is a secondary advantage. Strong as they are, these bones are exquisitely light and thin. A great frigate-bird, with a 7-foot span of wings, weighs a little over a couple of pounds, and all its bones weigh about four ounces. The bones weigh less than the feathers |. ♦ T irons, if I am not mistaken, were among the many inventions of Robert Stephenson, in his construction of the Menai tubular bridge a hundred years ago. t Cf. R. C. Murphy, Natural History, Oct. 1939. XVI] ON STRENGTH AND FLEXIBILITY 1013 While the bird stands/on the ground its backbone is, as in ourselves, its skeletal axis, and it, including the great cantilever associated with Fig. 485. a, Sternum and shoulder-girdle of a skua gull, b, section of do., through the hne AB. St, sternum; Ca, its carina or keel; Co, coracoid bone; Sc, scapula; F, merry-thought or furcula. it, carries, and transmit^ to the legs, the whole weight of the body. But as soon as a bird spreads its wings and rests upon the air, legs, Fig. 487. A flying bird. Sternum, shoulder -girdle and wings combine to support the body; and all the rest lies as a dead weight thereon. backbone, cantilever and all become merely so much weight to be carried; and the whole rests, as on a floor, on the strong, stiff platform made of sternum and shoulder-girdle, which the wings (so 1014 ON FORM AND MECHANICAL EFFICIENCY [ch. to speak) take hold of and support, and which is now, mechanically speaking, the axis of the body. The bird has two points of suspension, which it uses alternately: the one through the two acetabula, the other through the glenoid cavities and the outstretched wings. Glenoid cavity and acetabulum are but a little way apart, and the bird swings its weight over from one to the other easily and smoothly. At first sight it seems a curious feature of the bird's skeleton that breast-bone, shoulder-girdle, wings and all are but very slightly attached to the rest of the body, and to what we look on, usually, as its main axis of support ; the only skeletal attachment is by the framework of the ribs, and these are sUght and slender. The fact is that the two skeletal axes, the backbone and the breast-bone, have their separate and independent roles, and each is but loosely connected with the other. Fig. 488. Diagram of a continuous girder. The curvature of the bird's neck is very beautiful: one curve leads on to another; and indeed the bird's whole axial skeleton, from head to tail, is one even and continuous curve. Where a bridge crosses the gap between two piers, it sags as the load passes over; where successive girders cross successive gaps, each sags in its turn under the travelHng load. But suppose one continuous girder to cross two gaps; it bends in a more complicated way, and one half tends to bend up while the other is sagging down. We cannot analyse the whole field of force to which the bird is subject, but we reahse that it is a continuous field, in which what the engineer calls a "continuous girder" has its great part to play. The continuous girder is apt to sag and bend and sway in an erratic fashion unless its ends be firm and secure, and the bird's head must, of necessity, be under some analogous control; the semicircular canals are the potent factors in equilibrmm, and the bird "keeps XVI] ON STRENGTH AND FLEXIBILITY 1015 a level head " by their help and guidance. Then, between the level of the skull and the level of the great pelvic cantilever a continuous field of force governs and defines the S-shaped curvature. Man's vertebral column shews, mutatis mutandis, the same phenomenon of continuous but alternating curvature. The dorsal region is, of necessity, concave towards the cavity of the chest, and as a simple consequence the cervical and lumbar regions curve the other way. The typically aquatic birds, such as swim under water as penguins and divers do, have characteristic features and adaptations of their own. Just as the cantilever girder becomes obsolete in the aquatic mammal so does it tend to weaken and disappear in the aquatic bird. There is a marked contrast between the high-arched strongly built pelvis in the ostrich or the hen, and the long, thin, comparatively straight and apparently weakly bone which represents it in a diver, a grebe or a penguin. Wings large enough foi; air would be an obstruction under water, and small wings are enough ; for they have to produce thrust only, not Uft, and the former is but a small fraction of the latter load. The feet also are now mainly concerned with the same forward thrust, and we begin to see how the long narrow pelvis gives just the point d^appui which that thrust requires. The woodcock, as ornithologists are aware, shews us an osteological paradox, which is commonly described by saying that this bird's ear is in front of its eye ! If we hold a woodcock's skull level, beak and all, this indeed seems to be the case, but no woodcock does so. Standing or flying, the woodcock holds its beak pointing downwards, and its skull is then level, like that of other birds ; in other words, its beak is not in a line with the basi-cranial axis, as a guillemot's is, but bends sharply downwards. When the axis of the skull is horizontal, the beak points downwards at an angle of nearly 60°, and the auditory aperture is then as much behind the eye as in other birds. There is a certain other principle much to the fore in the con- struction of the skeleton, well known to the designer of a hydroplane or "flying boat," and not wholly neglected by the bridge-building engineer; it is the principle of non-rigid, flexible or elastic stability*. A homely comparison between a basket and a tin-can tells us in a moment what it means, and shews us some at least of its peculiar advantages. This method of construction helps to distribute the load, * India-rubber has great elastic stability. It is not compressible, but is almost as incbmpressible as water itself, as J. D. Forbes discovered a century ago. 1016 ON FORM AND MECHANICAL EFFICIENCY [en. bridges over points or areas where pressure might be unduly con- centrated or confined, adapts itself to a sudden impact or concentrated stress, helps to lessen or to guard against shock, and imparts to the whole structure a quality which we may call, for short, resiliency. The engineer finds it easiest of attainment when his principal members are in tension; hence elastic movement and resilience are apt to be conspicuous in a suspension-bridge. One way and another, resihence shews to perfection in a bird. The S-shaped curve of the neck carrying the Hght weight of the head, the zig-zag flexures of the legs bearing the balanced burden of the body, the supple basket Fig. 489.. A woodcock's skull, in (or nearly in) the natural attitude. A, B, the baMS cranii; E, auditory meatus; 0, orbit; Q, quadrate bone. of the ribs, each rib in two halves one flexed on the other, all these are such as to make the whole framework act like an elastic spring, absorbing every shock as the bird lights on or rises from the ground. Bird, beast and man exhibit this resilience, each in its degree; a springy step is part of the joy of youth, and its loss is one of the first infirmities of age. Nature's engineering is marvellous in our eyes, and our finest work is narrow in scope and clumsy in execution compared to her construction and design. But following her example, wittingly or unwittingly, our own problems evolve and our ambitions enlarge towards the conception of an "organised structure." In such XVI] ON THE SKELETON AS A WHOLE 1017 triumphs of modern mechanism as a torpedo, a racing aeroplane, a high-speed rail way- train, the whole construction is knit together in a new way. It finds its streamlined outline in what seems to be a simple and natural way; it is solid and robust, it is graceful as well as strong; it is no longer a bundle of parts, it has become an organic whole : its likeness, even its outward likeness, to a living organism has becorne patent and clear. Throughout this short discussion of the principles of construction, we see the same general principles at work in the skeleton as a whole as we recognised in the plan and construction of an individual bone. That is to say, we see a tendency for material to be laid down just in the lines of stress, and so to evade thereby the distortions and disruptions due to shear. In these phenomena there lies a definite law of growth, whatever its ultimate expression or explanation may come to be. Let us not press either argument or hypothesis too far: but be content to see that skeletal form, as brought about by growth, is to a very large extent determined by mechanical considerations, and tends to manifest itself as a diagram, or reflected image, of mechanical stress. If we fail, owing to the immense complexity of the case, to unravel all the mathematical principles involved in the construction of the skeleton, we yet gain something, and not a little, by applying this method to the famihar objects of anatomical study: ohvia conspicimus, nubem pellente tnathesi*. Before we leave this subject of mechanical adaptation, let us dwell once more for a moment upon the considerations which arise from our conception of a field of force, or field of stress, in which tension and compression (for instance) are inevitably combined, and are met by the materials naturally fitted to resist them. It has been remarked over and over again how harmoniously the whole organism hangs together, and how throughout its fabric one part is related and fitted to another in strictly functional correlation. But this conception, though never denied, is sometimes apt to be forgotten in the course of that process of more and more minute * The motto was Macquorn Rankine's, \vl 1857; cf. Trans. R.S.K. xxvi, p. 715, 1872. 1018 ON FORM AND MECHANICAL EFFICIENCY [ch. dnalysis by which, for simpUcity's sake, we seek to unravel the intricacies of a complex organism. As we analyse a thing into its parts or into its properties, we tend to magnify these, to exaggerate their apparent independence, and to hide from ourselves (at least for a time) the essential integrity and individuality of the composite wh^le. We divide the body into its organs, the skeleton into its bones, as in very much the same fashion we make a subjective analysis of the mind, according to the teachings of psychology, into component factors : but we know very well that judgment and knowledge, courage or gentleness, love or fear, have no separate existence, but are somehow mere manifestations, or imaginary coefficients, of a most complex integral. And likewise, as biologists, we may go so far as to say that even the bones themselves are only in a limited and even a deceptive sense, separate and individual things. The skeleton begins as a continuum, and a continuum it remains all life long. The things that link bone with bone, cartilage, ligaments, membranes, are fashioned out of the same primordial tissue, and come into being pari passu with the bones themselves. The entire fabric has its soft parts and its hard, its rigid and its flexible parts; but until we disrupt and dismember its bony, gristly and fibrous parts one from another, it exists simply as a "skeleton," as one integral and individual whole. A bridge was once upon a time a loose heap of pillars and rods and rivets of steel. But the identity of these is lost, just as if they were fused into a solid mass, when once the bridge is built; their separate functions are only to be recognised and analysed in so far as we can analyse the stresses, the tensions and the pressures, which affect this part of the structure or that; and these forces are not themselves separate entities, but are the resultants of an analysis of the whole field of force. Moreover wJien the bridge is broken it is no longer a bridge, and all its strength is gone. So is it precisely with the skeleton. In it is reflected a field of force: and keeping pace, as it were, in action and interaction with this field of force, the whole skeleton and every part thereof, down to the minute intrinsic structure of the bones themselves, is related in form and in position to the lines of force, to the resistances it has to encounter; for by one of the mysteries of biology, resistance begets resistance, and XVI] THE PROBLEM OF PHYLOGENY 1019 where pressure falls there growth springs up in strength to meet it. And, pursuing the same train of thought, we see that all this is true not of the skeleton alone but of the whole fabric of the body. Muscle and bone, for instance, are inseparably associated and connected; they are moulded one with another ; they come into being together, and act and react together*. We may study them apart, but it is as a concession to our weakness and to the narrow outlook of our minds. We see, dimly perhaps but yet with all the assurance of conviction, that between muscle and bone there can be no change in the one but it is correlated with changes in the other; that through and through they are linked in indissoluble association; that they are only separate entities m this limited and suboi'dinate sense, that they are parts of a whole which, when it loses its com- posite integrity, ceases to exist. The biologist, as well as the philosopher, learns to recognise that the whole is not merely the sum of its parts. It is this, and much more than this. For it is not a bundle of parts but an organisation of parts, of parts in their mutual arrangement, fitting one with another, in what Aristotle calls "a single and indivisible principle of unity " ; and this is no merely metaphysical conception, but is in biology t])e fundamental truth which lies at the basis of Geoffroy's (or Goethe's) law of "compensation," or "balancement of growth," Nevertheless Darwin found no difficulty in believing that "natural selection will tend in the long run to reduce any part of the organisa- tion, as soon as, through changed habits, it becomes superfluous: without by any means causing some oth^r part to be largely developed in a corresponding degree. And conversely, that natural selection may perfectly well succeed in largely developing an organ without requiring as a necessary compensation the reduction of some ad- joining partf " This view has been developed into a doctrine of the "independence of single characters" (not to be confused with the germinal "unit characters" of Mendelism), especially by the palaeontologists. Thus Osborn asserts a "principle of hereditary correlation," combined with a "principle of hereditary separability ^ * John Hunter was seldom wrong; but I cannot believe that he was right when he said {Scientific Works, ed. Owen, i, p. 371), "The bones, in a mechanical view, appear to be the first that are to be considered. We can study their shape, connections, number, uses, etc., without considering any other part of the body.'' t Origin of Species, 6th ed. p. 118. 1020 ON FORM AND MECHANICAL EFFICIENCY [ch. whereby the body is a colony, a mosaic, of single individual and separable characters*." I cannot think that there is more than a very small element of truth in this doctrine. As Kant said, "die Ursache der Art der Existenz bei jedem Theile eines lebenden Korpers ist im Ganzen enthalten.'" And, according to the 'trend or aspect of our thought, we may look upon the coordinated parts,, now as related and fitted to the end or function o/the whole, and now as related to or resulting from the physical causes inherent in the entire system of forces to which the whole has been exposed, and under whose influence it has come into being f. In John Hunter's day the anatomist studied every bone of the skeleton in its own place, in order to discover its useful purpose and understand its mechanical perfection. The morphologist of a hun- dred years later preferred to study an isolated bone from many animals, collar-bones or shoulder-blades by themselves, apart from the field of force in which their work was done, in the search for signs of blood-relationship and common ancestry. Truth lies both ways; immediate use and old inheritance are blended in Nature's handiwork as in our own. In the marble columns and architraves of a Greek temple we still trace the timbers of its wooden prototype, and see beyond these the tree-trunks of a primeval sacred grove; roof and eaves of a pagoda recall the sagging mats which roofed an * Amer. Naturalist, April, 1915, p. 198, etc. Cf. infra, p. 1036. t Drie.sch saw in "Entelechy" that something which differentiates the whole from the sum of its parts in the case of the organism: "The organism, we know, is a system the single constituents of which are inorganic in themselves; only the whole constituted by them in their typical order or arrangement owes its specificity to 'Entelechy'" {Gijford Lectures, 1908, p. 229): and I think it could be shewn that many other philosophers have said precisely the same thing. So far as the argument goes, I fail to see how this Entelechy is shewn to be peculiarly or specifically related to the living organism. The conception (at the bottom of General Smuts's 'Holism'') that the whole is always something very different from its parts is a very ancient doctrine. The reader will perhaps remember how, in another vein, the theme is treated by Martinus Scriblerus (Huxley quoted it once, for his own ends): "In every Jack there is a meat-roasting Quality, which neither resides in the fly, nor in the weight, nor in any particular wheel of the Jack, but is the result of the whole composition; etc., etc." Indeed it was at that very time, in the early eighteenth century, that the terms organism and organisation were coming into use, to connote that harmonious combination of parts "qui conspirent toutes ensembles a produire cet efFet general que nous nommons la vie" (Buffon). Cf. Ch. Robin, Recherches sur Torigine et le sens des termes organisme et organisation, Jl. de VAnat. lx, pp. l-.'i.^, 1880. XVI THE PROBLEM OF PHYLOGENY 1021 earlier edifice; Anglo-Saxon land-tenure influences the planning of our streets, and the cliff-dwelling and the cave-dwelling linger on in the construction of our homes! So we see enduring traces of the past in the living organism — landmarks which have lasted on through altered functions and altered needs ; and yet at every stage new needs are met and new functions effectively performed. When we consider (for instance) the several bones in a fish's shoulder-girdle — clavicle, supra-clavicle, post-clavicle, post-temporal and so on — and recognise these in this fish or that under countless minor transformations, we have something which is not only wide- Fig. 490. Skeleton of moonfish, Vomer sp. From L. Agassiz. spread but is rooted in antiquity, and whose full significance seems beyond our reach. But take the skeleton of some particular fish, a moonfish or a John Dory will do very well, and look at its shoulder- girdle from the mechanical point of view. It is a deal more than is needed for the support of the small, weak pectoral fin; but another function, and its perfect adaptation for that function, are not hard to see. The flattened body of the fish is built (as we have seen also in the plaice) on the plan of a parabolic girder; but out of this girder a great gap has had to be cut, to hold the viscera. The great shoulder- girdle serves to strengthen and complete the girder, to bind its upper and lower members together, and to compensate for the part 1022 ON FORM AND MECHANICAL EFFICIENCY [ch. which has been taken away^^ It fulfils this function by various means; by the way in which the two sides of the girdle are conjoined into a single arch ; by its strong attachment to the head, and again to the pelvis, and through the latter to the chain of ossicles which bound or constitute the abdominal border of the fish ; and a large part of the stress upon the shoulder-girdle proper is taken up, or relieved, by the strong post-clavicular bones, which form a supple- mentary arch running downwards from the clavicle (just where it begins to incline forward), straight to the ventral border, to be firmly attached there to the ventral ossicles. Similarly we notice at the hinder border of the abdominal cavity, a strong curved bone running from the anterior part of the ventral fin to a solid attach- ment with the vertebral column, stiffening the ventral part, and helping the shoulder-girdle to restore full strength to the girder after it had been reduced, so to speak, to the brink of inevitable collapse. The skull itself is not only streamlined with the rest of the body, but is an intrinsic part of the whole engineering construction. The lines of stress run simply and clearly through the skeleton, and a bone can no longer teach us its full and proper lesson after we have taken it apart. To look on the hereditary or evolutionary factor as tlie guiding principle in morphology is to give to that science a one-sided and fallacious simplicity*. It would seem to me that the mechanical principles and phenomena which we have dealt with in this chapter are of no small importance to the morphologist, all the more when he is inclined to direct his study of the skeleton exclusively to the problem of phylogeny ; and especially when, according to the methods of modern comparative morphology, he is apt to take the skeleton to pieces, and to draw from the comparison of a series of scapulae, humeri, or individual vertebrae, conclusions as to the descent and relationship of the animals to which they belong. It would, I dare say, be an exaggeration to see in every bone nothing more than a resultant of immediate and direct physical or mechanical conditions ; for to do so would be to deny the existence, * The extreme evolutionary, or phylogenetic, aspect of morphology was being questioned even forty years ago. "Where we once thought we detected relation- ships we now know we were often being misled, and the old-time supposition that mere community of structure is necessarily an index of community of origin has gone to the wall" (G. B. Howes, in Nature, Jan. 10, 1901). XVI] THE PROBLEM OF PHYLOGENY 1023 in this connection, of a principle of heredity. And though I have tried throughout this book to lay emphasis on the direct action of causes other than heredity, in short to circumscribe the employment of the latter as a working hypothesis in morphology, there can still be no question whatsoever but that heredity is a vastly important as well as a mysterious thing ; it is one of the great factors in biology, however we may attempt to figure to ourselves, or howsoever we may fail even to imagine, its underlying physical explanation. But I maintain that it is no less an exaggeration if we tend to neglect these direct physical and mechanical modes of causation altogether, and to see in the characters of a bone merely the results of variation and of heredity, and to trust, in consequence, to those characters as a sure and certain and unquestioned guide to affinity and phylo- geny. Comparative anatomy has its physiological side, which filled men's minds in John Hunter's day, and in Owen's day; it has its classificatory and phylogenetic aspect, which all but filled men's minds in the early days of Darwinism; and we can lose sight of neither aspect without risk of error and misconception. It is certain that the question of phylogeny, always difficult, becomes especially so in cases where a great change of physical or mechanical conditions has come about, and where accordingly the former physical and physiological constraints are altered or removed. The great depths of the sea differ from other habitations of the living, not least in their eternal quietude. The fishes which dwell therein are quaint and strange; their huge heads, prodigious jaws, and long tails and tentacles are, as it were, gross exaggerations of the common and conventional forms. We look in vain for any purposeful cause or physiological explanation of these enormities; and are left under a vague impression that life has been going on in the security of all but perfect equilibrium, and that the resulting forms, liberated from many ordinary constraints, have grown with unusual freedom*. To discuss these questions at length would be to enter on a discussion of Lamarck's philosophy of biology, and of many other things besides. But let us take one single illustration. The affinities of the whales constitute, as will be readily admitted, a very hard problem in phylogenetic classification. We know now that the * Cf. supra, p. 423. 1024 ON FORM AND MECHANICAL EFFICIENCY [ch. extinct Zeuglodons are related to the old Creodont carnivores, and thereby (though distantly) to the seals*; and it is supposed, but it is by no means so certain, that in turn they are to be considered as representing, or as allied to, the ancestors of the modern toothed whales t. The proof of any such a contention becomes, to my mind, extraordinarily difficult and complicated ; and the arguments commonly used in such cases may be said (in Bacon's phrase) to allure, rather than to extort assent. Though the Zeuglodons were aquatic animals, we do not know, and we have no right to suppose or to assume, that they swam after the fashion of a whale (any more than the seal does), that they dived like a whale, or leaped like a whale. But the fact that the whale does these things, and the way in which he does them, is reflected in many parts of his skeleton — perhaps more or less in all: so much so that the lines of stress which these actions impose are the very plan and working- diagram of great part of his structure. That the Zeuglodon has a scapula Hke that of a whale is to my mind no necessary argument that he is akin by blood-relationship to a whale: that his dorsal vertebrae are very different from a whale's is no conclusive argument that such blood-relationship is lacking. The former fact goes a long way to prove that he used his flippers very much as a whale does; the latter goes still farther to prove that his general movements and equilibrium in the water were totally different. The whale may be descended from the Carnivora, or might for that matter, as an older school of naturalists believed, be descended fropa the Ungulates; but whether or no, we need not expect to find in him the scapula, the pelvis or the vertebral column of the lion or of the cow, for it would be physically impossible that he could live the life he does with any one of them. In short, when we hope to find the missing links between a whale and his terrestrial ancestors, it must be not by means of conclusions drawn from a scapula, an axis, or * See {int. al.) my paper On the affinities oi Zeuglodon in Studies from the Museum of University College, Dundee, 1889. t "There can be no doubt that Fraas is correct in regarding this type (Procetus) as an annectant form between the Zeuglodonts and the Creodonta, but. although the origin of the Zeuglodonts is thus made clear, it still seems to be by no means so certain as that author beheves, that they may not themselves be the ancestral forms of the Odontoceti" (Andrews, Tertiary Vertebrata of the Fayum, 1906, p. 235). i XVI] THE PROBLEM OF PHYLOGENY 1025 even from a tooth, but by the discovery of forms so intermediate in their general structure as to indicate an organisation and, ipso facto, a mode of life, intermediate between the terrestrial and the Cetacean form. There is no vahd syllogism to the effect that A has a flat curved scapula like a seal's, and B has a fl.a,t curved scapula like a seal's : and therefore A and B are related to the seals and to each other; it is merely a flagrant case of an "undistributed middle." But there is validity in an argument that B shews in its general structure, extending over this bone and that bone, resemblances both to A and to the seals : and that therefore he may be presumed to be related to both, in his hereditary habits of life and in actual kinship by blood. It is cognate to this argument that (as every palaeontologist knows) we find clues to affinity more easily, that is to say with less confusion and perplexity, in certain structures than in others. The deep-seated rhythms of growth which, as I venture to think, are the chief basis of morphological heredity, bring about similarities of form which endure in the absence of conflicting forces ; but a new system of forces, introduced by altered environment and habits, impinging on those particular parts of the fabric which he within this particular field of force, will assuredly not be long of manifesting itself in notable and inevitable modifications of form. And if this be really so, it will further imply that modifications of form will tend to manifest themselves, not so much in small and isolated phenomena, in this part of the fabric or in that, in a scapula for instance or a humerus : but rather in some slow, general, and more or less uniform or graded modifica- tion, spread over a number of correlated parts, and at times extending over the whole, or over great portions, of the body. Whether any such general tendency to widespread and correlated transformation exists, we shall attempt to discuss in the following chapter. CHAPTER XVII ON THE THEORY OF TRANSFORMATIONS, OR THE COMPARISON OF RELATED FORMS In the foregoing chapters of this book we have attempted to study the inter-relations of growth and form, and the part which the physical forces play in this complex interaction; and, as part of the same enquiry, we have tried in comparatively simple cases to use mathematical methods and mathematical terminology to describe and define the forms of organisms. We have learned in so doing that our own study of organic form, which we call by Goethe's name of Morphology, is but a portion of that wider Science of Form wtjch deals with the forms assumed by matter under all aspects and conditions, and, in a still wider sense, with forms which are theoretically im^aginable. The study of form may be descriptive merely, or it may become analytical. We begin by describing the shape of an object in the simple words of common speech : we end by defining it in the precise language of mathematics; aild the one method tends to follow the other in strict scientific order and historical continuity. Thus, for instance, the form of the earth, of a raindrop or a rainbow, the shape of the hanging chain, or the path of a stone thrown up into the air, may all be described, however inadequately, in common words ; but when we have learned to comprehend and to define the sphere, the catenary, or the parabola, we have made a wonderful ai^d perhaps a manifold advance. The mathematical definition of a "form" has a quahty of precision which was^ quite lacking in our earlier stage of mere description; it is expressed in few words or in still briefer symbols, and these words or symbols are so pregnant with meaning that thought itself is economised; we are brought by means of it in touch with Galileo's aphorism (as old as Plato, as old as Pythagoras, as old perhaps as the wisdom of the Egyptians), that "the Book of Nature is written in characters of Geometry*." ♦ Cf. Plutarch, Symp. viii, 2, on the meaning of Plato's aphorism ("if it actually was Plato's"): ttwj HXdrufv iXeye t6v d^bv ad yeuifxeTpe'tv. CH. XVII] COMPARISON OF RELATED FORMS 1027 We are apt to think of mathematical definitions as too strict and rigid for common use, but their rigour is combined with all but endless freedom. The precise definition of an ellipse introduces us to all the elHpses in the world; the definition of a "conic section" enlarges our concept, and a "curve of higher order" all the toore extends our range of freedom*. By means of these large limitations, by this controlled and regulated freedom, we reach through mathe- matical analysis to mathematical synthesis. We discover homologies or identities which were not obvious before, and which our descrip- tions obscured rather than revealed : as for instance, when we learn that, however we hold our chain, or however we fixe our bullet, the contour of the one or the path of the other is always mathematically homologous. Once more, and this is the greatest gain of all, we pass quickly and easily from the mathematical concept of form in its statical aspect to form in its dynanncal relations: we rise from the conception of form to an understanding of the forces which gave rise to it; and in the representation of form and in the comparison of kindred forms, we see in the one case a diagram of forces in equihbrium, and in the other case we discern the magnitude and the direction of the forces which have sufficed to convert the one form into the other. Here, since a change of material form is only effected by the movement of matter |, we have once again the support of the schoolman's and the philosopher's axiom, Ignorato motu, ignoratur Natural' * So said Gustav Theodor Fechner, the author of Fechner's Law, a hundred years ago. (Ueber die mathematische Behandlung organischer Gestalten und Processe, Berichte d. k. sacks. Gesellsch., Maih.-phys. CI., Leipzig, 1849, pp. 50-64.) Fechner's treatment is more purely mathematical and less physical in its scope and bearing than ours, and his paper is but a short one, but the conclusions to which he is led differ Uttle from our own. Let me quote a single sentence which, together with its context, runs precisely on Aie lines which we have followed in this book : "So ist also die mathematische Bestimmbarkeit im Gebiete des Organischen ganz eben so gut vorhanden als in dem des Unorganischen, und in letzterem eben solchen oder aquivalenten Beschrankungen unterworfen als in ersterem ; und nur sofern die unorganischen Formen und das unorganische Geschehen sich einer einfacheren Gesetzlichkeit mehr nahern als die organischen, kann die Approximation im unorganischen Gebiet leichter und weiter getrieben werden als im organischen. Dies ware der ganze, sonach rein relative, Unterschied." Here, in a nutshell, is the gist of the whole matter. t "We can move matter, that is all we can do to it" (Oliver Lodge). 1028 THE THEORY OF TRANSFORMATIONS [ch. There is yet another way — we learn it of Henri Poincare — to regard the function of mathematics, and to reahse why its laws and its methods are hound to underlie all parts of physical science. Every natural phenomenon, however simple, is really composite, and every visible action and effect is a summation of countless subordinate actions. Here mathematics shews her pecuUar power, to combine and to generahse. The concept of an average, the equation to a curve, the description of a froth or cellular tissue, all come within the scope of mathematics for no other reason than that they are summations of more elementary principles or phenomena. Growth and Form are throughout of this composite nature; therefore the laws of mathematics are bound to underlie them, and her methods to be peculiarly fitted to interpret them. In the morphology of living things the use of mathematical methods and symbols has made slow progress ; and there are various reasons for this failure to employ a method whose advantages are so obvious in the investigation of other physical forms. To begin with, there would seem to be a psychological reason, lying in the fact that the student of living things is by nature and training an observer of concrete objects and phenomena and the habit of mind which he possesses and cultivates is alien to that of the theoretical mathematician. But this is by no means the only reason; for in the kindred subject of mineralogy, for instance, crystals were still treated in the days of Linnaeus as wholly within the province of the naturalist, and were described by him after the simple methods in use for animals and plants: but as soon as Haiiy shewed the application of mathematics to the description and classification of crystals, his methods were immediately adopted and a new science came into being. A large part of the neglect and suspicion of mathematical methods in organic morphology is due (as we have partly seen in our opening chapter) to an ingrained and deep-seated belief that even when we seem to discern a regular mathematical figure in an organism, the sphere, the hexagon, or the spiral which we so recognise merely resembles, but is never entirely explained by, its mathematical analogue ; in short, that the details in which the figure differs from its mathematical prototype are more important and more interesting XVII] THE COMPARISON OF RELATED FORMS 1029 than the features in which it agrees; and even that the pecuUar aesthetic pleasure with which we regard a living thing is spmehow bound up with the departure from mathematical regularity which it manifests as a peculiar attribute of life. This view seems to me to involve a misapprehension. There is no such essential difference between these phenomena of organic form and those which are manifested in portions of inanimate matter*. The mathematician knows better than we do the value of an approximate resultf. The child's skipping-rope is but an approximation to Huygens's catenary curve — but in the catenary curve lies the whole gist of the matter. We may be dismayed too easily by contingencies which are nothing short of irrelevant compared to the main issue ; there is a jyrinciple ' of negligibility. Someone has said that if Tycho Brahe's instruments had been ten times as exact there would have been no Kepler, no Newton, and no astronomy. If no chain hangs in a perfect catenary and no raindrop is a perfect sphere, this is for the reason that forces and resistances other than the main one are inevitably at work. The same is true of organic form, but it is for the mathematician to unravel the conflicting forces which are at work together. And this process of investigation may lead us on step by step to new phenomena, as it has done in physics, where sometimes a knowledge of form leads us to the interpretation of forces, and at other times a knowledge of the forces at work guides us towards a better insight into form. After the fundamental advance had been made which taught us that the world * M, Bergson repudiates, with peculiar confidence, the appHcation of mathe- matics to biology; cf. Creative Evolution, p. 21, "Calculation touches, at most, certain phenomena of organic destruction. Organic creation, on the contrary, the evolutionary phenomena which properly constitute life, we cannot in any way subject to a mathematical treatment." Bergson thus follows Bichat: "C'est peu connaitre les fonctions animales que de vouloir les soumettre au moindre calcul, parceque leur instabUite est extreme. Les phenomenes restent toujours les memes, et c'est ce qui nous importe; mais leurs variations, en plus ou en moins, sont sans nombre" (La Vie et la Mort, p. 257). t When we make a 'first approximation' to the solution of a physical problem, we usually mean that we are solving one part while neglecting others. Geometry deals with pure forms (such as a straight line), defined by a single law; but these are few compared with the mixed forms, like the surface of a polyhedron, or a segment of a sphere, or any ordinary mechanical construction or any ordinary physical phenomenon. It is only in a purely mathematical treatment of physics that the "single law" can be dealt with alone, and the approximate solution dispensed with accordingly. 1030 THE THEORY OF TRANSFORMATIONS [ch. was round, Newton shewed that the forces at work upon it must lead to its being imperfectly spherical, and in the course of time its oblate spheroidal shape was actually verified. But now, in turn, it has been shewn that its form is still more complicated, and the next step is to seek for the forces that have deformed the oblate spheroid. As Newton somewhere says, "Nature dehghts in trans- formations." The organic forms which we can define more or less precisely in mathematical terms, and afterwards proceed to explain and to account for in terms of force, are of many kinds, as we have seen; but nevertheless they are few in number compared with Nature's all but infinite variety. The reason for this is not far to seek. The living organism represents, or occupies, a field of force which is never simple, and which as a rule is of immense complexity. And just as in the very simplest of actual cases we meet with a departure from such symmetry as could only exist under conditions of ideal simplicity, so do we pass quickly to cases where the interference of numerous, though still perhaps very simple, causes leads to a recultant complexity far beyond our powers of analysis. Nor must we foi*get that the biologist is much more exacting in his requirement^, as regards form, than the physicist; for the latter is usually content with either an ideal or a general description of form, while the student of living things must needs be specific. Material things, be they living or dead, shew us but a shadow of mathematical perfection*. The physicist or mathematician can give us perfectly satisfying expressions for the form of a wave, or even of a heap of sand ; but we never ask him to define the form of any particular wave of the sea, nor the actual form of any mountain-peak or hill. In this there lies a certain justification for a saying of Minot's, of the greater part of which, nevertheless, I am heartily inclined to disapprove. "We biologists," he says, "cannot deplore too frequently or too emphatically the great mathematical delusion by which men often of great if limited ability have been misled into becoming advocates of an erroneous conception of accuracy. The delusion is that no science is accurate imtil its results can be expressed mathematically. The error comes from the assumption that mathematics can express complex relations. Unfortunately mathematics have a very limited scope, and are based upon a few extremely rudimentary * Cf. Haton de la GoupiUiere, op. cit.: "On a souvent I'occasion de saisir dans la nature un reflet des formes rigoureuses qu'etudie la geometrie," XVII] THE COMPARISON OF RELATED FORMS 1031 experiences, which we make as very little children and of which no adult has any recollection. The fact that from this basis men of genius have evolved wonderful methods of dealing with numerical relations should not blind us to another fact, namely, that the observational basis of mathematics is, psychologically speaking, very minute compared with the observational basis of even a single minor branch of biology. . . . While therefore here and there the mathematical methods may aid us, we need a kind and degree of accuracy of which mathematics is absolutely incapable. . . . With human minds constituted as they actually are, we cannot anticipate that there will ever be a mathe- matical expression for any organ or even a single cell, although formulae will continue to be useful for dealing now and then with isolated details..." {op. cit. p. 19, 1911). It were easy to discuss and criticise these sweeping assertions, which perhaps had their origin and parentage in an obiter dictum of Huxley's, to the effect that "Mathematics is that study which knows nothing of observation, nothing of experiment, nothing of induction, nothing of causation" {cit. Cajori, Hist, of Elem. Mathematics, p. 283). But Gauss, "rex mathematicorum," called mathematics "a science of the eye"; and Sylvester assures us that "most, if not all, of the great ideas of modem mathematics have had their origin in observation" {Brit. Ass. Address, 1869, and Laws of Verse, p. 120, 1870. Reaumur said the same thing two hundred years ago {Mem. i, p. 49, 1734). Maupertuis, he said, was both naturalist and mathematician; and all his mathematics "n'ont en rien affaibli son gout pour les insectes, personne peut-etre n'a plus d' amour pour eux." He goes on to say: "L' esprit d'observation qu'on regarde comme le ' caractere d' esprit essentiel aux naturalistes, est egalement necessaire pour faire des progres en quelque science que ce soit. C'est I'esprit d'observation qui fait appercevoir ce qui a echappe aux autres, qui fait saisir des rapports qui sont entre des choses qui semblent differentes, ou qui fait trouver les differences qui sont entre celles qui paroissent semblables. On ne resoud les problemes les plus epineux de Geometric qu'apres avoir S9u observer des rapports, qui ne se decouvrent qu'a un esprit penetrant, et extremement attentif. Ce sont des observations qui mettent en etat de resoudre les problemes de physique comme ceux d'histoire naturelle, car I'histoire natureUe a ses problemes a resoudre, et elle n'en a meme que trop qui ne sont pas resolus." It is in a deeper sense than this, however, that the modem physicist looks on mathematics as an "empirical" science, and no longer a matter of pure intuition, or "reine Anschauung." Cf. Max Bom, on Some philosophical aspects of modem physics, Proc. R.S.E. Lvn, pp. 1-18, 1936.. For one reason or another there are very many organic forms which we cannot describe, still less define, in mathematical terms: just as there are problems even in physical science beyond the mathematics of our age. We never even seek for a formula to define this fish or that, or this or that vertebrate skull. But we may already use mathematical language to describe, even to define in general terms, 1032 THE THEORY OF TRANSFORMATIONS [ch. the shape of a snail-shell, the twist of a horn, the outline of a leaf, the texture of a bone, the fabric of a skeleton, the stream-lines of fish or bird, the fairy lace-work of an insect's wing. Even to do this we must learn from the mathematician to ehminate and to discard; to keep the 'type in mind and leave the single case, with all its accidents, alone ; and to find in this sacrifice of what matters little and conservation of what matters much one of the pecuUar excellences of the method of mathematics*. In a very large part of morphology, our essential task lies in the comparison of related forms rather than in the precise definition of each; and the deformation of a complicated figure may be a phenomenon easy of comprehension, though the figure itself have to be left unanalysed and undefined. This process of comparison, of recognising in one form a definite permutation or deformation of another, apart altogether from a precise and adequate understanding of the original "type" or standard of comparison, Hes within the immediate province of mathematics, and finds its solution in the elementary use of a certain method of the mathematician. This method is the Method of Coordinates, on which is based the Theory of Transformations f. I imagine that when Descartes conceived the method of co- ordinates, as a generalisation from the proportional diagrams of the artist and the architect, and long before the immense possibilities of this analysis could be foreseen, he had in mind a very simple purpose ; it was perhaps no more than to find a way of translating the form of a curve (as well as the position of a point) into numbers and into words. This is precisely what we do, by the method of coordinates, every time we study a statistical curve ; and conversely, we translate numbers into form whenever we "plot a curve," to illustrate a table of mortality, a rate of growth, or the daily variation of temperature or barometric pressure. In precisely the same way * Cf. W. H. Young, The mathematical method and its limitations, Congresso dei Matematici, Bologna, 1928. f The mathematical Theory of Transformations is part of the Theory of Groups, of great importance in modern mathematics. A distinction is drawn between Substitution-groups and Transformation-groups, the former being discontinuous, the latter continuous — in such a way that within one and the same group each transformation is infinitely little different from another. The distinction among biologists between a mutation and a variation is curiously analogous. XVII] THE COMPARISON OF RELATED FORMS 1033 it is possible to inscribe in a net of rectangular coordinates the outline, for instance, of a fish, and so to translate it into a table of numbers, from which again we may at pleasure reconstruct the curve. But it is the next step in the employment of coordinates which is of special interest and use to the morpnologist ; and this step consists in the alteration, or deformation, of our system of coordinates, and in the study of the corresponding transformation of the curve or figure inscribed in the coordinate network. Let us inscribe in a system of Cartesian coordinates the outUne of an organism, however complicated, or a part thereof: such as a fish, a crab, or a mammalian skull. We may now treat this complicated figure, in general terms, as a function of x, y. If we submit our rectangular system to deformation on simple and recognised lines, altering, for instance, the direction of the axes, the ratio of xjy, or substituting for x and y some more complicated expressions, then we obtain a new system of coordinates, whose deformation from the original type the inscribed figure will precisely follow/ In other words, we obtain a new figure which represents the old figure under a more or less homogeneous strain, and is a function of the new coordinates in precisely the same way as the old figure was of the original coordinates x and y. The problem is closely akin to that of the cartographer who transfers identical data to one projection or another*; and whose object is to secure (if it be possible) a complete Correspondence, in each small unit of area, between the one representation and the other. The morphologist will not seek to draw his organic forms in a new and artificial projection; but, in the converse aspect of the problem, he will enquire whether two different but more or less obviously related forms can be so analysed and interpreted that each may be shewn to be a transformed representation of the other. This once demonstrated, it will be a comparatively easy task (in all probability) to postulate the direction and magnitude of the force capable of effecting the required transformation. Again, if such a simple alteration of the system of forces can be proved adequate to meet the case, we may find ourselves able to dispense with many * Cf. (e.g.) Tissot, Memoire sur la representation des surfaces, et les projections des cartes geographiques, Paris, 1881. 1034 THE THEORY OF TRANSFORMATIONS [ch. widely current and more complicated hypotheses of biological causation. For it is a maxim in physics that an effect ought not to be ascribed to the joint operation of many causes if few are adequate to the production of it : Frustra fit per plura, quod fieri potest per pauciora. We might suppose that by the combined action of appropriate forces any material form could be transformed into any other: just as out of a "shapeless" mass of clay the potter or the sculptor models his artistic product; or just as we attribute to Nature herself the power to effect the gradual and successive transformation of the simple germ into the complex organism. But we need not let these considerations deter us from our method of comparison of related forms. We shall strictly limit ourselves to cases where the transformation necessary to effect a comparison shall be of a simple kind, and where the transformed, as well as the original, coordinates shall constitute an harm<5ni6us and more or less sym- metrical system. We should fall into deserved and inevitable confusion if, whether by the mathematical or any other method, we attempted to compare organisms separated far apart in Nature and in zoological classification. We are limited, both by our method , and by the whole nature of the case, to the comparison of organisms such as are manifestly related to one another and belong to the same zoological class. For it is a grave sophism, in natural history as in logic, to make a transition into another kind*. Our enquiry lies, in short, just within the limits which Aristotle himself laid down when, in defining a "genus," he shewed that (apart from those superficial characters, such as colour, which he called "accidents") the essential differences between one "species" and another are merely differences of proportion, of relative mag- nitude, or (as he phrased it) of "excess and defect." "Save only for a difference in the way of excess or defect, the parts are identical in the case of such animals as are of one and the saiiie genus ; and by ' genus ' I mean, for instance, Bird or Fish." And again : " Within the limits of the same genus, as a general rule, most of the parts exhibit differences ... in the way of multitude or fewness, magnitude * The saying heterogenea comparari non possunt is discussed by Coleridge in his Aids to Reflexion. XVII] THE COMPARISON OF RELATED FORMS 1035 or parvitude, in short, in the way of excess or defect. For 'the more' and 'the less' may be represented as 'excess' and * defect'*." It is precisely this difference of relative magnitudes, this Aristotelian "excess and defect" in the case of form, which our coordinate method is especially adapted to analyse, and to reveal and demon- strate as the main cause of what (again in the Aristotelian sense) we term "specific" differences. The appUcability of our method to particular cases will depend upon, or be further Hmited by, certain practical considerations or qualifications. Of these the chief, and indeed the essential, con- dition is, that the form of the entire structure under investigation should be found to vary in a more or less uniform manner, after the fashion of an approximately homogeneous and isotropic body. But an imperfect isotropy, provided always that some "principle of continuity" run through its variations, will not seriously interfere with our method ; it will only cause our transformed coordinates to be somewhat less regular and harmonious than are those, for instance, by which the physicist depicts the motions of a perfect fluid, or a theoretic field of force in a uniform medium. Again, it is essential that our structure vary in its entirety, or at least that " independent variants " should be relatively few. That independent variations occur, that localised centres of diminished or exaggerated growth will now and then be found, is not only probable but manifest; and they may even be so pronounced as to appear to constitute new formations altogether. Such independent variants as these Aristotle himself clearly recognised: "It happens further that some have parts which others have not; for instance, some [birds] have spurs and others not, some have crests, or combs, and others not; but, as a general rule, most parts and those that go to make up the bulk of the body are either identical with one another, or differ from one another in the way of contrast and of excess and defect. For ' the more ' and ' the less ' may be represented as 'excess' or ' defect '|." If, in the evolution of a fish, for instance, it be the case that its * Historia Animalium i, 1. •j; Aristotle's argument is even more subtle and far-reaching; for the diflferences of which he speaks are not merely those between one bird and another, but between them all and the very type itself, or Platonic "idea" of a bird. 1036 THE THEORY OF TRANSFORMATIONS [ch. several and constituent parts — head, body and tail, or this fin and that fin — represent so many independent variants, then our coordinate system will at once become too complex to be intelligible ; we shall be making not one comparison but several separate com- parisons, and our general method will be found inapplicable. Now precisely this independent variability of parts and organs — here, there, and everywhere within the organism — would appear to be implicit in our ordinary accepted notions regarding variation; and, unless I am greatly mistaken, it is precisely on such a conception of the easy, frequent, and normally independent variability of parts that our conception of the process of natural selection is fundamentally based. For the morphologist, when comparing one organism with another, describes the differences between them point by point, and "character" by "character*." If he is from time to time con- strained to admit the existence of "correlation" between characters (as a hundred years ago Cuvier first shewed the way), yet all the while he recognises this fact of correlation somewhat vaguely, as a phenomenon due to causes which, except in rare instances, he can hardly hope to trace ; and he falls readily into the habit of thinking and talking of evolution as though it had proceeded on the lines of his own descriptions, point by point, and character by character f. With the "characters" of Mendelian genetics there is no fault to be found; tall and short, rough and smooth, plain or coloured are opposite tendencies or contrasting qualities, in plain logical contradistinction. But when the morphologist compares one animal with another, point by point or character by character, these are too often the mere outcome of artificial dissection and analysis. Rather is the living body one integral and indivisible whole, in * Cf. supra, p. 1020. t Cf. H. F. Osbom, On the origin of single characters, as observed in fossil and living animals and plants, Amer. Nat. xlix, pp. 193-239, 1915 (and other papers); ibid. p. 194, "Each individual is composed of a vast number of somewhat similar new or old characters, each character has its independent and separate history, each character is in a certain stage of evolution, each character is correlated with the other characters of the individual The real problem has always been that of the origin and development of characters. Since the Origin of Species appeared, the terms variation and variability have always referred to single characters; if a species is said to be variable, we mean that a considerable number of the single characters or groups of characters of which it is composed are variable," etc. XVII] THE COMPARISON OF RELATED FORMS 1037 wEich we cannot find, when we come to look for it, any strict dividing line even between the head and the body, the muscle and the tendon, the sinew and the bone. Characters which we have differentiated insist on integrating themselves again; and aspects of the organism are seen to be conjoined which only our mental analysis had put asunder. The coordinate diagram throws into rehef the integral solidarity of the organism, and enables us to see how simple a certain kind of correlation is which had been apt to seem a subtle and a complex thing. But if, on the other hand, diverse and dissimilar fishes can be referred as a whole to identical functions of very different coordinate systems, this fact will of itself constitute a proof that variation has proceeded on definite and orderly line?, that a comprehensive "law of growth" has pervaded the whole structure in its integrity, and that some more or less simple and recognisable system of forces has been in control. It will not only shew how real and deep-seated is the phenomenon of "correlation," in regard to form, but it will also demonstrate the fact that a correlation which had seemed too complex for analysis or comprehension is, in many cases, capable of very simple graphic expression. This, after many trials, I believe to be in general the case, bearing always in mind that the occurrence of independent or localised variations must sometimes be considered. We are deahng in this chapter with the forms of related organisms, in order to shew that the differences between them are as a general rule simple and symmetrical, and just such as might have been brought about by a slight and simple change in the system of forces to which the living and growing organism was exposed. Mathematically speaking, the phenomenon is identical with one met with by the geologist, when he finds a bed of fossils squeezed flat or other- wise symmetrically deformed by the pressures to which they, and the strata which contain them, have been subjected. In the first step towards fossilisation, when the body of a fish or shellfish is silted over and buried, we may take it that the wet sand or mud exercises, approximately, a hydrostatic pressure — that is to say a pressure which is uniform in all directions, and by which the form of the buried object will not be appreciably changed. As the strata consolidate and accumulate, the fosail Organisms which they contain will tend to be flattened by the vast superincumbent load, just as the stratum which contains them will also be compressed and will have its molecular arrangement more or less modified*. But the deformation due to direct vertical pressure in a horizontal stratum is not nearly so striking as are the deformations produced by the oblique or shearing stresses to which inclined * Cf. Sorby, Quart. Journ. Geol. Soc. (Proc), 1879, p. 88. 1038 THE THEORY OF TRANSFORMATIONS [ch. and folded strata have been exposed, and by which their various "dislocations " have been brought about. And especially in mountain regions, where "these dislocations are especially numerous and complicated, the contained fossils are apt to be so curiously and yet so symmetrically deformed (usually by a simple shear) that they may easily be interpreted as so many distinct and separate "species*." A great number of described species, and here and there a new genus (as the genus Ellipsolithes for an obliquely deformed Goniatite or Nautilus), are said to rest on no other foundation f. If we begin by drawing a net of rectangular equidistant coordinates (about tlie axes x and y), we may alter or deform; this network in various ways, several of which are very simple indeed. Thus (1) we may alter the dimensions of our system, extending it along one or / /^ N \ / / \ \ 1 \ V y / Fig. 491. Fig. 492. other axis, and so converting each little square into a corresponding and proportionate oblong (Figs. 491, 492). It follows that any figure which we may have inscribed in the original net, and which we transfer to the new, will thereby be deformed in strict proportion to the deformation of the entire configuration, being still defined by corresponding points in the network and being throughout in conformity with the original figure. For instance, a circle inscribed * Cf. Ale. D'Orbigny, Cmirs dim. de PaUontologie, etc., i, pp. 144-148, 1849; see also Daniel Sharpe, On slaty cleavage, Q.J.G.S. m, p. 74, 1847. t Thus Ammonites erugatus, when compressed, has been described &a A . planorbis : cf. J. F. Blake, Phil. Mag. (5), vi, p. 260, 1878. Wettstein has shewn that^several species of the fish-genus Lepidopus have been based on specimens artificially deformed in various ways: Ueber die Fischfauna des Tertiaren Glarnerschiefers, Abh. Schw. Palaeont. Gesellsch. xiii, 1886 (see especially pp. 23-38, pi. i). The whole subject, interesting as it is, has been httle studied ; both Blake and Wettstein deal with it mathematically. XVII] THE COMPARISON OF RELATED FORMS 1039 in the original "Cartesian" net will now, after extension in the ^/-direction, be found elongated into an ellipse. In elementary mathematical language, fo;- the original x and y we have substituted iCj and c?/i , and the equation to our original circle, x^ ^ ip' = a^, becomes that of the ellipse, x^ + c^yx ^ ^^• If I draw the cannon-bone of an ox (Fig. 493, A), for instance, within a system of rectangular coordinates, and then transfer the same drawing, point for point, to a system in which for the x of the original diagram we substitute x' = 2x/3, we obtain a drawing (B) which is a very close approximation to the cannon-bone of the sheep. In other words, the main (and perhaps the only) difference r < 7 X / 1 — . \ \ U si ft Fi ?.4 94. A B C Fig. 493. between the two bones is simply that that of the sheep is elongated along the vertical axis as compared with that of the ox, in the pro- portion of 3/2. And similarly, the long slender cannon-bone of the giraife (C) is referable to the same identical t3rpe, subject to a reduction of breadth, or increase of length, corresponding to x" = x/3. (2) The second type is that where extension is not equal or uniform at all distances from the origin: but grows greater or less, as, for instance, when we stretch sl tapering elastic band. In such cases, as I have represented it in Fig. 494, the ordinate increases logarithmically, and for y we substitute e^. It is obvious that this logarithmic extension may involve both abscissae and ordinates, X becoming e^ while y becomes e^. The circle in our original figure is now deformed into some such shape as that of Fig. 495. This 1040 THE THEORY OF TRAN^SFORMATIONS [ch. method of deformation is a common one, and will often be of use to us in our comparison of organic forms. Fig. 495. Fig. 496. (3) Our third type is the "simple shear," where the rectangular coordinates become "obHque," their axes being inclined to one another at a certain angle oj. Our original rectangle now becomes such a figure as that of Fig. 496. The system may now be described in terms of the oblique axes X, Y; or may be directly referred to new rectangular coordinates f, rj by the simple transposition X = i — 7) cot oj, y = r] cosec co. Fig. 497. (4) Yet another important class of deformations may be repre- sented by the use of radial coordinates, in which one set of lines are XVII] THE COMPARISON OF RELATED FORMS 1041 represented as radiating from a point or "focus," while the other set are transformed into circular arcs cutting the radii orthogonally. These radial coordinates are especially apphcable to cases where there exists (either within or without the figure) some part which is supposed to suffer no deformation ; a simple illustration is afforded by the diagrams which illustrate the flexure of a beam (Fig. 497). In biology these coordinaites will be especially applicable in cases where the growing structure includes a "node," or point where growth is absent or at a minimum; and about which node the rate of growth may be assumed to increase symmetrically. Precisely Fig. 498. such a case is furnished us in a leaf of an ordinary dicotyledon. The leaf of a typical monocotyledon — such as a grass or a hyacinth, for instance — grows continuously from its base, and exhibits no node or "point of arrest." Its sides taper off gradually from its broad base to its slender tip, according to some law of decrement specific to the plant; and any alteration in the relative velocities of longitudinal and transverse growth will merely make the leaf a little broader or narrower, and will effect no other conspicuous alteration in its contour. But if there once come into existence a node, or "locus of no growth," about which we may assume growth — which in the hyacinth leaf was longitudinal and trans- verse— to take place radially and transversely to the radii, then we 1042 THE THEORY OF TRANSFORMATIONS [ch. shall soon see the sloping sides of the hyacinth leaf give place to a more typical and "leaf -like" shape. If we alter the ratio between the radial and tangential velocities of growth — in other words, if we increase the angles between corresponding^radii — we pass successively through the various configurations which the botanist describes as the lanceolate, the ovate, and the cordiform leaf. These successive changes may to some extent, and in appropriate cases, be traced as the individua,l leaf grows to maturity ; but as a much more general rule, the balance of forces, the ratio between radial and tangential velocities of growth, remains so nicely and constantly balanced that the leaf increases in size without conspicuous modification of form. It is rather what we may call a long-period variation, a tendency for the relative velocities to alter from one generation to another, whose result is brought into view by this method of illustration. There are various corollaries to this method of describing the form of a leaf which may be here alluded to. For instance, the so-called A unsymmetrical leaf* of a begonia, in which one side of the leaf may be merely ovate while the other has a cordate outhne, is seen to be really a case of unequal, and not truly asymmetrical, growth on either side of the midrib. There is nothing more mysterious in its conformation than, for instance, in that of a forked twig in which one limb of the fork has grown longer than the other. The case of the begonia leaf is of sufficient interest to deserve illus- tration, and in Fig. 499 I have outlined a leaf of the large Begonia daedalea. On the smaller left-hand Fig. 499 Begonia daedalea. ^^^^ ^f ^he leaf I have taken at random three points a, 6, c, and have measured the angles, AOa, etc., * Cf. Sir Thomas Browne, in The Garden of Cyrus: "But why ofttimes one side of the leaf is unequal! unto the other, |is in Hazell and Oaks, why on either side the master vein the lesser and derivative channels stand not directly opposite, not at equall angles, respectively unto the adverse side, but those of one side do often exceed the other, as the Wallnut and many more, deserves another enquiry." xvii] THE COMPARISON OF RELATED FORMS 1043 which the radii from the hilus of the leaf to these points make with the median axis. On the other side of the leaf I have marked the points a\ h\ c', such that the radii drawn to this margin of the leaf are equal to the former, Oa' to Oa, etc. Now if the two sides of the leaf are mathematically similar to one another, it is obvious that the respective angles should be in continued proportion, i.e. as AOa is to AOa', so should AOh be to AOh'. This proves to be very nearly the case. For I have measured the three angles on one side, and one on the other, and have then compared, as follows, the calculated with the observed values of the other two : AOa AOb AOc AOa' AOh' AOc' Observed values 12° 28-5° 88° — — 157° Calculated „ — — — 21-5° 5M° — Observed „ — — — 20 52 — The agreement is very close, and what discrepancy there is may be amply accounted for, firstly, by the slight irregularity of the sinuous margin of the leaf; and secondly, by the fact that the true axis or midrib of the leaf is not straight but shghtly curved, and therefore that it is curvilinear and not rectihnear triangles which we ought to have measured. When we understand these few points regarding the peripheral curvature of the leaf, it is easy to see that its principal veins approximate closely to a beautiful system of isogonal coordinates. It is also obvious that we can easily pass, by a process of shearing, from those cases where the principal veins start from the base of the leaf to those where they arise successively from the midrib, as they do in most dicotyledons. It may sometimes happen that the node*, or "point of arrest," is at the upper instead of the lower end of the leaf-blade; and occasionally there is a node at both ends. In the former case, as we have it in the daisy, the form of the leaf will be, as it were, inverted, the broad, more or less heart-shaped, outhne appearing at the upper end, while below the leaf tapers gradually downwards to an ill-defined base. In the latter case, as in Dionaea, we obtain a leaf equally expanded, and similarly ovate or cordate, at both ends. We may notice, lastly, that the shape of a solid fruit, such as an apple or a cherry, is a solid of revolution, developed from similar curves and to be explained on the same principle. In the * "Node," in the botanical, not the mathematical, sense. 1044 THE THEORY OF TRANSFORMATIONS [ch. cherry we have a "point of arrest" at the base of the berry, where it joins its peduncle, and about this point the fruit (in imaginary section) swells out into a cordate outhne ; while in the apple we have two such well-marked points of arrest, above and below, and about both of them the same conformation tends to arise. The bean and the human kidney owe their "reniform" shape to precisely the same phenomenon, namely, to the existence of a node or "hilus," about which the forces of growth are radially and symmetrically arranged. When the seed is small and the pod roomy, the seed may grow round, or nearly so, like a pea ; but it is flattened and bean-shaped, or elliptical like a kidney-bean, when compressed within a narrow and elongated pod. If the original seed have any simple pattern, of the nature for instance of meridians or parallels of latitude, it is easy to see how these will suffer a conformal trans- formation, corresponding to the deformation of the sphere*. We might go farther, and farther than we have room for here, to illustrate the shapes of leaves by means of radial coordinates, and even to attempt to define them by polar equations. In* a former chapter we learned to look upon the curve of sines as an easy, gradual and natural transition — perhaps the simplest and most natural of all — from minimum to corresponding maximum, and so on alternately and continuously; and we found the same curve going round like the hands of a clock, when plotted on radial co- ordinates and (so to speak) prevented from leaving its place. Either way it represents a "simple harmonic motion." Now we have just seen an ordinary dicotyledonous leaf to have a "point of arrest," or zero-growth in a certain direction, while in the opposite direction towards the tip it has grown with a maximum velocity. This progress from zero to maximum suggests one-half of the sine-curve ; in other words, if we look on the outline of the leaf as a vector- diagram of its own growth, at rates varying from zero to zero in a complete circuit of 360°, this suggests, as a possible and very simple case, the plotting of r = sin ^/2. Doing so, we obtain a curve (Fig. 500) closely resembling what the botanists call a reniform (or kidney-shaped) leaf, that is to say, with a cordate outline at the base formed of two "auricles," one on either side, and then rounded * Vide supra, p. 524. XVII] THE COMPARISON OF RELATED FORMS 1045 off with no projecting apex*. The ground-ivy and the dog-violet (Fig. 501) illustrate such a leaf; and sometimes, as in the violet, the veins -of the leaf show similar curves congruent with the outer edge. Moreover the violet is a good example of how the reniform leaf may be drawn out more and more into an acute and ovate form. From sin^/2 we may proceed to any other given fraction of ^, and plot, for instance, r = sin 5^/3, as in Fig. 502 ; which now no longer represents a single leaf but has become a diagram of the Fig. 501. Violet leaf. Fig. 500. Curve resembling the out- line of a reniform leaf: r=:8in^/2. five petals of a pentamerous flower. Abbot Guido Grandi, a Pisan mathematician of the early eighteenth century, drew such a curve and pointed out its botanical analogies ; and we still call the curves of this family "Grandi's curvesj*." The gamopetalous corolla is easily transferred to polar coordinates, in which the radius vector riow consists of two parts, the one a constant, the other expressing the amplitude (or half-amplitude) of the sine-curve ; we may write the formula r = a + b cos nO. In Fig. 503 w ^ 5 ; in this figure, if the radius of the outermost circle be taken as unity, the outer of the two sinuous curves has a:b as * Fig. 500 illustrates the whole leaf, but only shows one-half of the sine-curve. The rest is got by reflecting the moiety already drawn in the horizontal axis (^=7r/2). t Dom. Guido Grandus, Flores geometrici ex rhodonearum et cloeliarum curvarum descriptione resultantes . . . , Florentiae, 1728. Cf. Alfred Lartigue, Biodynamique generale, Paris, 1930 — a curious but eccentric book. 1046 THE THEORY OF TRANSFORMATIONS [ch. 9:1, and the inner curve as 3 : 1 ; while the five petals become separate when a = b, and the formula reduces to r = cos^ — . z In Fig. 504 we have what looks like a first approximation to a horse-chestnut leaf. It consists of so many separate leaflets, akin to the five petals in Fig. 503; but these are now inscribed in (or have a locus m) the cordate or reniform outline of Fig. 500. The new curve is, in short, a composite one; and its general formula is r = sin 6 12. sin nd. The small size of the two leaflets adjacent to the petiole is characteristic of the curve, and helps to explain the development of "stipules." Fig. 502. Grandi's curves based on r = sin5^, and illustrating the five petals of a simple flower. Fig. 503. Diagram illustrating a corolla of five petals, or of five lobes, are based on the equation r = a + b cos 6. In this last case we have combined one curve with another, and the doing so opens out a new range of possibilities. On the outhne of the simple leaf, whether ovate, lanceolate or cordate, we may superpose secondary sine-curves of lesser period and varying ampli- tude, after the fashion of a Fourier series ; and the results will vary from a mere crenate outline to the digitate lobes of an ivy-leaf, or to separate leaflets such as we have just studied in the horse-chestnut. Or again, we may inscribe the separate petals of Fig. 505 within a spiral curve, equable or equiangular as the case may be ; and then, continuing the series on and on, we shall obtain a figure resembling the clustered leaves of a stonecrop, or the petals of a water-lily or other polypetalous flower. XVII] THE COMPARISON OF RELATED FORMS 1047 Most of the transformations which we have hitherto considered (other than that of the simple shear) are particular cases of a general transformation, obtainable by the method of conjugate functions and equivalent to the projection of the original figure on a new plane. Appropriate transformations, on these general lines, provide for the cases of a coaxial system where the Cartesian coordinates are replaced by coaxial circles, or a confocal system in which they are replaced by confocal ellipses and hyperbolas. Fig. 504. Outline of a compound leaf, like a horse-chestnut, based on a composite sine-curve, of the form r = sin 012 . sin nd. Fig. 505. Yet another curious and important transformation, belonging to the same class, is that by which a system of straight lines becomes transformed into a conformal system of logarithmic spirals: the straight line Y — AX = c corresponding to the logarithmic spiral ^ — ^ log r = c (Fig. 505). This beautiful and simple transforma- tion lets us at once convert, for instance, the straight conical shell of the Pteropod or the Orthoceras into the logarithmic spiral of the Nautiloid; it involves a mathematical symbolism which is but a slight extension of that which we have employed in our elementary treatment of the logarithmic spiral. These various systems of coordinates, which we have now briefly considered, are sometimes called "isothermal coordinates," from the fact that, when employed in this particular branch of physics, they perfectly represent the phenomena of the conduction of heat, the 1048 THE THEORY OF TRANSFORMATIONS [ch. contour lines of equal temperature appearing, under appropriate conditions, as the orthogonal lines of the coordinate system. And it follows that the "law of growth" which our biological analysis by means of orthogonal coordinate systems presupposes, or at least foreshadows, is one according to which the organism grows or develops along stream-lines, which may be defined by a suitable mathematical transformation. When the system becomes no longer orthogonal, as in many of the following illustrations — for instance, that of Orthagoriscus (Fig. 526) — then the transformation is no longer within the reach of comparatively simple mathematical analysis. Such departure from the typical symmetry of a "stream-line" system is, in the first instance, sufficiently accounted for by the simple fact that the developing organism is very far from being homogeneous and isotropic, or, in other words, does not behave like a perfect fluid. But though under such circumstances our coordinate systems may be no longer capable of strict mathematical analy'sis, they will still indicate graphically the relation of the new coordinate system to the old, and conversely will furnish us with some guidance as to the "law of growth," or play of forces, by which the transformation has been effected. Before we pass from this brief discussion of transformations in general, let us glance at one or two cases in which the forces applied are more or less intelligible, but the resulting transformations are, from the mathematical point of view, exceedingly compHcated. The "marbled papers" of the bookbinder are a beautiful illustra- tion of visible "stream-lines." On a dishful of a sort of semi-liquid gum the workman dusts a few simple lines or patches of colouring matter; and then, by passing a comb through the Hquid,.he draws the colour-bands into the streaks, waves, and spirals which con- stitute the marbled pattern, and which he then transfers to sheets of paper laid down upon the gum. By some such system of shears, by the effect of unequal traction or unequal growth in various directions and superposed on an originally simple pattern, we may account for the not dissimilar marbled patterns which we recognise, for instance, on a large serpent's skin. But it must be remarked, in the case of the marbled paper, that though the method of application XVII] THE COMPAKISON OF RELATED FORMS 1049 of the forces is simple, yet in the aggregate the system of forces set up by the many teeth of the comb is exceedingly complex, and its complexity is revealed in the complicated "diagram of forces" which constitutes the pattern. ' To take another and still more instructive illustration. To turn one circle (or sphere) into two circles (or spheres) would be, from the point of view of the mathematician, an extraordinarily difficult transformation; but, physically speaking, its achievement may be extremely simple. The little round gourd grows naturally, by its symmetrical forces of expansive growth, into a big, round, or some- what oval pumpkin or melon*. But the Moorish husbandman ties a rag round its middle, and the same forces of growth, unaltered save for the presence of this trammel, now expand the globular structure into two superposed and connected globes. And again, by varying the position of the encircling band, or by applying several such ligatures instead of one, a great variety of artificial forms of "gourd" may be, and actually are, produced. It is clear, I think, that we may account for many ordinary biological processes of development or transformation of form by the existence of trammels or lines of constraint, which limit and determine the action of the expansive forces of growth that would otherwise be uniform and symmetrical. This case has a close parallel in the operations of the glass-blower, to which we have already, more than once, referred in passing f. The glass-blower starts his operations with a tube, which he first closes at one end so as to form a hollow vesicle, within which his blast of air exercises a uniform pressure on all sides ; but the spherical conformation which this uniform expansive force would naturally tend to produce is modified into all kinds of forms by the trammels or resistances set up as the workman lets one part or another of his bubble be unequally heated or cooled. It was Oliver Wendell * Analogous structural differences, especially in the fibrovascular bundles, help to explain the differences between (e.g.) a smooth melon and a cantelupe, or between various elongate, flattened and globular varieties. These breed true to type, and obey, when crossed, the laws of Mendelian inheritance. Cf. E. W. Sinnett, Inherit- ance of fruit-shape in Cucurbita, Botan. Gazette, lxxiv, pp. 95-103, 1922, and other papers. t Where gourds are common, the glass-blower is still apt to take them for a prototype, as the prehistoric potter also did. For instance, a tall, annulated Florence oil-flask is an exact but no longer a conscious imitation of a gourd which has been converted into a bottle in the manner described. 1050 THE THEORY OF TRANSFORMATIONS [ch. Holmes who first shewed this curious parallel between the operations of the glass-blower and those of Nature, when she starts, as she so often does, with a simple tube*. The alimentary canal, the arterial system including the heart, the central nervous system of the vertebrate, including the brain itself, all begin as simple tubular structures. And with them Nature does just what the glass-blower does, and, we might even say, no more than he. For she can expand the tube here and narrow it there; thicken its walls or thin them; blow off a lateral offshoot or caecal diverticulum; bend the tube, or twist and coil it; and infold or crimp its walls as, so to speak, she pleases. Such a form as that of the human stomach is easily explained when it is regarded from this point of view ; it is simply an ill-blown bubble, a bubble that has been rendered lopsided by a trammel or restraint along one side, such as to prevent its symmetrical expansion — such a trammel as is produced if the glass- blower lets one side of his bubble get cold, and such as is actually present in the stomach itself in the form of a muscular band. The Florence flask, or any other handiwork of the glass-blower, is always beautiful, because its graded contours are, as in its living analogues, a picture of the graded forces by which it was conformed. It is an example of mathematical beauty, of which the machine-made, moulded bottle has no trace at all. An alabaster bottle is different again. It is no longer an unduloid figure of equilibrium. Turned on a lathe, it is a solid of revolution, and not without beauty ; but it is not near so beautiful as the blown flask or bubble. The gravitational field is part of the complex field of force by which the form of the organism is influenced and determined. Its share is seldom easy to define, but there is a resultant due to gravity in hanging breasts and tired eyelids and all the sagging wrinkles of the old. Now and then we see gravity at work in the normal construction of the body, and can describe its effect on form in a general, or qualitative, way. Each pair of ribs in man forms a hoop which droops of its own weight in front, so flattening the chest, and at the same time twisting the rib on either hand near its point of suspension f. But in the dog each costal hoop is dragged * Cf. Elsie Venner, chap. ii. t See T. P. Anderson Stuart, How the form of the thorax is partly determined by gravitation, Proc. R.S. xlix, p.l43, 1891. XVII] THE COMPARISON OF RELATED FORMS 1051 straight downwards, into a vertical instead of a transverse ellipse, and is even narrowed to a point at the sternal border. We may now proceed to consider and illustrate a few permutations or transformations of organic form, out of the vast multitude which are equally open to this method of enquiry. We have already compared in a preliminary fashion the metacarpal or cannon-bone of the ox, the sheep, and the giraffe (Fig, 493); and we have seen that the essential difference in form between these three bones is a matter of relative length and breadth, such that, if we reduce the figures to an identical standard of length (or identical values of y), the breadth (or value of x) will be approximately two-thirds that of the ox in the case of the sheep and one-third that of the ox in the case of the giraffe. We may easily, for the sake of closer comparison, determine these ratios more accurately, for instance, if it be our purpose to compare the different racial varieties within the limits of a single species. And in such cases, by the way, as when we compare with one another various breeds or races of cattle or of horses, the ratios of length and breadth in this particular bone are extremely significant*. If, instead of limiting ourselves to the cannon-bone, we inscribe the entire foot of our several Ungulates in a coordinate system, the same ratios of x that served us for the cannon-bones still give us a first approximation to the required comparison; but even in the case of such closely allied forms as the ox and the sheep there is evidently something wanting in the comparison. The reason is that the relative elongation of the several parts, or individual bones, has not proceeded equally or proportionately in all cases ; in other words, that the equations for x will not suffice without some simultaneous modification of the values of y (Fig. 506). In such a case it may be found possible to satisfy the varying values of y by some logarithmic * This significance is particularly remarkable in connection with the develop- ment of speed, for the metacarpal region is the seat of very important leverage in the propulsion of the body. In a certain Scottish Museum there stand side by side the skeleton of an immense carthorse (celebrated for having drawn all the stones of the Bell Rock Lighthouse to the shore), and a beautiful skeleton of a racehorse, long supposed to be the actual skeleton of Eclipse. When I was a boy my grandfather used to point out to me that the cannon-bone of the little racer is not only relatively, but actually, longer than that of the great Clydesdale. 1052 THE THEORY OF TRANSFORMATIONS [ch. or other formula ; but, even if that be possible, it will probably be somewhat difficult of discovery or verification in such a case as the present, owing to the fact that we have too few well-marked points of correspondence between the one object and the other, and that especially along the shaft of such long bones as the cannon-bone of the ox, the deer, the llama, or the giraffe there is a complete lack of easily recognisable corresponding points. In such a case a brief tabular statement of apparently corresponding values of y, or of those obviously corresponding values which coincide with the boundaries of the several* bones of the foot, will, as in the following example, enable us to dispense with a fresh equation. y (Ox) y' (Sheep) y" (Giraffe) b 27 19 10 42 36 24 100 100 100 This summary of values of y' , coupled with the equations for the value of X, will enable us, from any drawing of the ox's foot, to con- struct a figure of that of the sheep or of the giraffe with remarkable accuracy. ^n i 100 Sheep Giraffe 100 Fig. 506. Fig. 507. That underlying the varying amounts of extension to which the parts or segments of the limb have been subject there is a law, or principle of continuity, may be discerned from such a diagram as the above (Fig. 507), where the values of y in the case of the ox are plotted as a straight line, and the corresponding values for the XVII] THE COMPARISON OF RELATED FORMS 1053 sheep (extracted from the above table) are seen to form a more or less regular and even curve. This simple graphic result implies the existence of a comparatively simple equatiqn between y and y'. An elementary application of the principle of coordinates to the study of proportion, as we have here used it to illustrate the varying proportions of a bone, was in common use in the sixteenth and seventeenth centuries by artists in their study of the human form. The method is probably much more ancient, and may even be classical*; it is fully described and put in practice by Albert Diirer in his Geometry, and especially in his Treatise on Proportio7f\ . In this latter work, the manner in which the human figure, features, and facial expression are all transformed and modified by slight variations in the relative magnitude of the parts is admirably and copiously illustrated (Fig. 508). :^Z3t Fig. 508. (After Albert Durer.) In a tapir's foot there is a striking difference, and yet at the same time there is an obvious underlying resemblance, between the middle toe and either of its unsymmetrical lateral neighbours. Let us take the median terminal phalanx and inscribe its outline in a net of rectangular equidistant coordinates (Fig. 509, a). Let us then make a similar network about axes which are no longer at right angles, but inclined to one another at an angle of about 50° (h). * Cf. Vitruvius, ni, 1. t Les quatres livres d' Albert Diirer de la proportion des parties et pourtraicts des corps humains, Arnheim, 1613, folio (and earlier editions) . Cf. also Lavater, Essays on Physiognomy, iii, p. 271, 1799; also H. Meige, La geometrie des visages d'apres Albert Diirer, La Nature, Dec. 1927. On Diirer as mathematician, cf. Cantor, ii, p. 459; S. Giinther, Die geometrische Ndherungsconstructione Albrecht DUrers, Ansbach, 1866; H. Staigmuller, Diirer als Mathematiker, Stuttgart, 1891. 1054 THE THEORY OF TRANSFORMATIONS [ch. If into this new network we fill in, point for point, an outline precisely corresponding to our original drawing of the middle toe, we shall find that we have already represented the main features of the adjacent lateral one. We shall, however, perceive that our new diagram looks a httle too bulky on one side, the inner side, of the lateral toe. If now we substitute for our equidistant ordinates, ' 2 ( \ ^ , ^ a b c d <' K •-^ N, ^ / \ / (^ J ^ ^^ — ' Fig. 509. ordinates which get gradually closer and closer together as we pass towards the median side of the toe, then we shall obtain a diagram which differs in no essential respect from an actual outline copy of the lateral toe (c). In short, the difference between the outline of the middle toe of the tapir and the next lateral toe may be almost completely expressed by saying that if the one be represented by rectangular equidistant coordinates, the other will be represented by oblique coordinates, whose axes make an angle of 50°, and in i (After Albert Diirer.) which the abscissal interspaces decrease in a certain logarithmic ratio. We treated our original complex curve or projection of the tapir's toe as a function of the form F (x, y) = 0. The figure of the tapir's lateral toe is a precisely identical function of the form f (e*, ^i) = 0, where x^, 2/1 ^^® oblique coordinate axes inclined to one another at an angle of 50°. XVII] THE COMPARISON OF RELATED FORMS 1055 Diirer was acquainted with these oblique coordinates also, and I have copied two illustrative figures from his book*. In Fig. 511 I have sketched the common Copepod Oiihona nana, and have inscribed it in a rectangular net, with abscissae three-fifths the length of the ordinates. Side by side (Fig. 512) is drawn a very different Copepod, of the genus Sapphirina ; and about it is drawn a network such that each coordinate passes (as nearly as possible) through points corresponding to those of the former figure. It will be seen that two differences are apparent. (1) The values of y in Fig. 512 are large in the upper part of the figure, and diminish Fig. 511. Oithona nana. Fig. 512. Sapphirina. rapidly towards its base. (2) The values of x are very large in the neighbourhood of the origin, but diminish rapidly as we pass towards either side, away from the median vertical axis; and it is probable that they do so according to a definite, but somewhat complicated, * It was these very drawings of Diirer's that gave to Peter Camper his notion of the "facial angle." Camper's method of comparison was the very same as ours, save that he only drew the axes, without filling in the network, of his coordinate system ; he saw clearly the essential fact, that the skull varies as a whole, and that the "facial angle" is the index to a general deformation. "The great object was to shew that natural differences might be reduced to rules, of which the direction of the facial line forms the norma or canon ; and that these directions and inclinations are always accompanied by correspondent form, size and position of the other parts of the cranium," etc.; from Dr T. Cogan's preface to Camper's work On the, Connexion between the Science of Anatomy and the Arts of Drawing, Painting and Sculpture (1768?), quoted in Dr R. Hamilton's Memoir of Camper, in lAves of Eminent Naturalists (Nat. Libr.), Edinburgh, 1840. See also P. Camper, Dissertation sur les differences reelles que presentent les Traits du Visage chez les hommes de differents pays et de differents ages, Paris, 1791 {op. posth.); cf. P. Topinard, ]6tudes 8ur Pierre Camper, et sur Tangle facial dit de Camper, Bev. d'Anthropol. n. 1874. 1056 THE THEORY OF TRANSFORMATIONS [ch. ratio. If, instead of seeking for an actual equation, we simply tabulate our values of x and y in the second figure as compared with the first (just as we did in comparing the feet of the Ungulates), we get the dimensions of a net in which, by simply projecting the figure of Oithona, we obtain that of Sapphirina without further trouble, e.g.: X (Oithona) x' {Sapphirina) 0 0 3 8 6 10 9 12 12 13. 15 14 — y {Oithona) y' {Sapphirina) 0 0 5- 2 10 7 15 3 20 23 25 32 30 40 In this manner, with a single model or type to copy from, we may record in very brief space the data requisite for the production of approximate outlines of a great number of forms. For instance, the difference, at first sight immense, between the attenuated body of a Caprella and the thick-set body of a Cyamus is obviously httle, and is probably nothing more than a difference of relative mag- nitudes, capable of tabulation by numbers and of complete expression by means of rectihnear coordinates. The Crustacea afford innumerable instances of more complex deformations. Thus we may compare various higher Crustacea with one another, even in the case of such dissimilar forms as a lobster and a crab. It is obvious that the whole body of the former is elongated as compared with the latter, and that the crab is relatively broad in the region of the carapace, while it tapers off rapidly towards its attenuated and abbreviated tail. In a general way, the elongated rectangular system of coordinates in which we may inscribe the outhne of the lobster becomes a shortened triangle in the case of the crab. In a little more detail we may compare the outline of the carapace in various crabs one with another : and the comparison will be found easy and significant, even, .in many cases, down to minute details, such as the number and situation of the marginal spines, though these are in other cases subject to independent variabihty. If we choose, to begin with, such a crab as Geryon (Fig. 513, 1) and inscribe it in our equidistant rectangular coordinates, we shall see that we pass easily to forms more elongated in a transverse direction, such as Matuta or Lupa (5), and conversely, by transverse compression, to such a form as Corystes (2). In certain other cases I XVII] THE COMPARISON OF RELATED FORMS 1057 the carapace conforms to a triangular diagram, more or less curvi- linear, as in Fig. 513, 4, which represents the genus Paralomis. Here we can easily see that the posterior border is transversely elongated as compared with that of Geryon, while at the same time the anterior , ^ Fig. 513. Carapaces of various crabs. I, Geryon; 2, Corystes; 3, Scyramathia; 4, Paralomis; 5, Lupa; 6, Chorinus. part is longitudinally extended as compared with the posterior. A system of slightly curved and converging ordinates, with ortho- gonal and logarithmically interspaced abscissal hues, as shewn in the figure, appears to satisfy the conditions. In an interesting series of cases, such as the genus Chorinus, or Scyramathia, and in the spider-crabs generally, we appear to have 1058 THE THEORY OF TRANSFORMATIONS [ch. just the Qonverse of this. While the carapace of these crabs presents a somewhat triangular form, which seems at first sight more or less similar to those just described, we soon see that the actual posterior border is now narrow instead of broad, the broadest part of the carapace corresponding precisely, not to that which is broadest in Paralomis, but to that which was broadest in Geryon; while the most striking difference from the latter Hes in an antero-posterior lengthening of the forepart of the carapace, culminating in a great elongation of Ihe frontal region, with its two spines or "horns." The curved ordinates here converge posteriorly and diverge widely in front (Fig. 513, 3 and 6), while the decremental interspacing of the abscissae is very marked indeed. We put our method to a severer test when we attempt to sketch an entire and compHcated animal than when we simply compare corresponding parts such as the carapaces of various Malacostraca, or related bones as in the case of the tapir's toes. Nevertheless, up to a certain point, the method stands the test very well. In other words, one particular mode and direction of variation is often (or even usually) so prominent and so paramount throughout the entire organism, that one comprehensive system of coordinates suffices to give a fair picture of the actual phenomenon. To take another illustration from the Crustacea, I have drawn roughly in Fig. 514, 1 a little amphipod of the family Phoxocephalidae (Harpinia sp.). Deforming the coordinates of the figure into the curved orthogonal * system in Fig. 514, 2, we at once obtain a very fair representation of an alhed genus, belonging to a different family of amphipods, namely Stegocephalus. As we proceed further from our type our coordinates will require greater deformation, and the resultant figure will usually be somewhat less accurate. In Fig. 514, 3 I shew a network, to which, if we transfer our diagram of Harpinia or of Stegocephalus, we shall obtain a tolerable representation of the aberrant genus Hyperia'f, with its narrow abdomen, its reduced pleural lappets, its great eyes, and its inflated head. * Similar coordinates are treated of by Lame, Lemons sur les coordonnees curvilignes, Paris, 1859. t For an analogous, but more detailed comparison, see H. Mogk, Versuch einer Formanalyse bei Hyperiden, Int. Rev. d. ges. Hydrobiol., etc., xiv, pp. 276-311, 1923; xvn, pp. 1-98, 1926. XVII] THE COMPARISON OF RELATED FORMS 1059 The hydroid zoophytes constitute a "polymorphic" group, within which a vast number of species have already been distinguished; and the labours of the systematic naturahst are constantly adding to the number. The specific distinctions are for the most part based, not upon characters directly presented by the living animal, but upon the .form, size and arrangement of the little cups, or "calycles," secreted and inhabited by the little individual polyps Fig. 514. 1, Harpinia plumosa Kr.; 2, Stegocephalus infiatus Kr.; 3, Hyperia galha. which compose the compound organism. The variations, which are apparently infinite, of these conformations are easily seen to be a question of relative magnitudes, and are capable of complete expression, sometimes by very simple, 'sometimes by somewhat more complex, coordinate networks. For instance, the varying shapes of the simple wineglass-shaped cups of the Campanularidae are at once sufficiently represented and compared by means of simple Cartesian coordinates (Fig. 515). In the two allied famihes of Plumulariidae and Aglaopheniidae the 1060 THE THEORY OF TRANSFOEMATIONS [ch. calycles are set unilaterally upon a jointed stem, and small cup-like structures (holding rudimentary polyps) are associated with the large calycles in definite number and position. These small calycuh IVWIl/W \ ^ I yw\mj \- -4 \ 1 Ai- V\/VTl/V/V/| \ \ \ / \ \ 1 / \ L • ah c Fig. 515. a, Campanularia macroscyphus Allm.; b, Gonothyraea hyalina Hincks; c, Clytia Johnstoni Alder. are variable in number, but in the great majority of cases they accompany the large calycle in groups of three — two standing by its upper border, and one, which is especially variable in form and magnitude, lying at its base. The stem is liable to flexure and. Fig. 516. a, Cladocarpus crenatus F.; &, Aglaophenia plurna X,.; c, A. rhynchocarpa A.; d, A. cornuta K.; e, A. ramulosa K. in a high degree, to extension or compression; and these variations extend, often on an exaggerated scale, to the related calycles. As a result we find that we can draw various systems of curved or sinuous coordinates, which express, all but completely, the con- figuration of the various hydroids which we inscribe therein (Fig. 516). XVII] THE COMPARISON OF RELATED FORMS 1061 The comparative smoothness of denticulation of the margin of the calycle, and the number of its denticles, constitutes an independent variation, and requires separate description; we have already seen (p. 391) that this denticulation is in all probability due to a par- ticular physical cause. Among countless other invertebrate animals which we might illustrate, did space and time permit, we should find the bivalve molluscs shewing certain things extremely well. If we start with a more or less oblong shell, such as Anodon or Mya or Psammobia, we can see how easily it may be transformed into a more circular or orbicular, but still closely related form; while on the other hand a simple shear is well-nigh all that is needed to transform the oblong Anodon into the triangular, pointed Mytilus, Avicula or Pinna. Now suppose we draw the shell of Anodon in the usual rectangular coordinates, and deform this network into the corre- sponding oblique coordinates of Mytilus, we may then proceed to draw within the same two nets tEe anatomy of the same two molluscs. Then of the two adductor muscles, coequal in Anodon, one becomes small, the other large, when transferred to the oblique network of Mytilus ; at the same time the foot becomes stunted and the siphonal aperture enlarged. In short, having "transformed" one shell into the other we may perform an identical transformation on their contained anatomy : and so (provided the two are not too distantly related) deduce the bodily structure of the one from our knowledge of the other, to a first but by no means negligible approximation. Among the fishes we discover a great variety of deformations, some of them of a very simple kind, while others are more striking and more unexpected. A comparatively simple case, involving a simple shear, is illustrated by Figs. 517 and 518. The one represents; within Cartesian coordinates, a certain little oceanic fish known as Argyropelecus Olfersi. The other represents precisely the same out- line, transferred to a system of oblique coordinates whose axes are inclined at an angle of 70° ; but this is now (as far as can be seen on the scale of the drawing) a very good figure of an allied fish, assigned to a different genus, under the name of Sternoptyx diaphana. The deformation illustrated by this case of Argyropelecus is precisely analogous to the simplest and commonest kind of deformation to 1062 THE THEORY OF TRANSFORMATIONS [ch. which fossils are subject (as we have "seen on p. 811) as the result of shearing-stresses in the solid rock. Fig. 519 is an outUne diagram of a typical Scaroid fish. Let us deform its rectihnear coordinates into a system of (approximately) coaxial circles, as in Fig. 520, and then filling into the new system. Fig. 517. Argyropelecua Olfersi. Fig. 518. Stemoptyx diaphana. space by space and point by point, our former diagram of Scarus, we obtain a very good outline of an allied fish, belonging to a neigh- bouring family, of the genus Pomacanihus. This case is all the more interesting, because upon the body of our Pomacanihus there are striking colour bands, which correspond in direction very closely Fig. 519. Scarus sp. Fig. 520. Pomacanihus. to the lines of our new curved ordinates. In like manner, the still more bizarre outUnes of other fi.shes of the same family of Chaetodonts will be found to correspond to very slight modifications of similar coordinates; in other words, to small variations in the values of the constants of the coaxial curves. In Figs. 521-524 I have represented another series of Acantho- pterygian fishes, not very distantly related to the foregoing. If we XVII] THE COMPARISON OF RELATED FORMS 1063 start this series with the figure of Polyprion, in Fig. 521, we see that the outlines of Pseudopriacanthus (Fig. 522) and of Sebastes or Scorpaena (Fig. 523) are easily derived by substituting a system Fig. 521. Polyprion. Fig. 522. Pseudopriacanthus alius. of triangular, or radial, coordinates for the rectangular ones in which we had inscribed Polyprion. The very curious fish Antigonia capros, an oceanic relative of our own boar-fish,